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Search for parity nonconservation in metastable hydrogen using microwave and VHF spectroscopy

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O rd er N u m b e r 9 0 3 5 S 3 6
Search for parity nonconservation in m etastab le hydrogen u sin g
m icrow ave and V H F spectroscopy
Edwards, John W illiam, Ph.D .
Yale University, 1989
UMI
300 N. Zeeb Rd.
Ann Arbor, MI 48106
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SEARCH FOR PARITY NONCONSERVATION
IN METASTABLE HYDROGEN USING
MICROWAVE AND VHP SPECTROSCOPY
A Dissertation
Presented to the Faculty of the Graduate School
of
Yale University
in Candidacy for the Degree of
Doctor of Philosophy
by
John William Edwards
May 1990
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ABSTRACT
SEARCH FOR PARITY NONCONSERVATION
IN METASTABLE HYDROGEN USING
MICROWAVE AND VHF SPECTROSCOPY
John William Edwards
Yale University
1990
This thesis describes an experiment in which the allowed magnetic
dipole transition between the hyperfine levels of the metastable 2s^/2
state is interfered with an electric dipole transition.
This electric
dipole transition is forbidden if parity is conserved but allowed to a
small extent by the neutral-current weak interaction between the electron
and the proton.
The electric dipole transition can be mimicked by an
admixture of the nearby p-states owing to the presence of stray electric
fields.
In the work reported the necessary sensitivity to measure
neutral-current effects was not achieved owing to the presence of such
stray fields.
This stray amplitude, if naively interpreted as a weak
interaction, corresponds to a value of the phenomenological coupling
constant C 2p of 1.3 x 10^.
By analyzing the frequency dependence of
all available signals it is shown that this stray amplitude is inconsistent
with weak interactions.
Many hypotheses are tested in an attempt to
Identify this systematic effect and some further approaches are suggested.
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TABLE OF CONTENTS
Page
Chapter I.
Introduction.................................
Chapter II.
1
Overview of the Experiment.......................... 14
Chapter III.
Source............................................. 36
Chapter IV.
Detection....................................... .... .53
Chapter V.
State Selection...................................... 74
Chapter VI.
Beam and Background................................. 84
Chapter VII.
Transition Regions................................. 98
Chapter VIII.
Precession....................................... 137
Chapter IX.
Data Collection and
Control........................154
Chapter X.
Results............................................. 174
References
.................................................... 201
Appendix A: Two-State Time-Dependent PerturbationTheory......... 203
Appendix B: Lineshapes.......................................... 208
Appendix C: Precessor Field..................................... 220
Appendix D: State Selection of 22Si/2 HyperfineLevels of
Hydrogen in Zero Magnetic Field...................... 228
(i)
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TABLE OF FIGURES
Page
Figure 1.1
Figure 2.1
Feynman Diagrams for electromagnetic electron-proton
scattering, electron capture, and a second-order
electron-proton diagram involving the exchange of
two charged vector bosons.
Breit-Rabi diagram for the n=2, J=l/2 levels of hydrogen.
5
17
Figure 2.2
Lineshape for transition amplitude due to uniform
oscillating field.
19
Figure 2.3
Lineshape for transition amplitude due to antisymmetric
oscillating field.
22
Figure 2.4
Exterior view of the atomic beam machine.
24
Figure 2.5
(a) Schematic cross section of the source.
cross section of the beam machine.
Figure 2.6
Cross section of El-region transmission line showing
electric and magnetic field lines of the TEM mode.
Coordinate axes and expected transition amplitudes.
33
Figure 3.1
Schematic cross section of the the source.
37
Figure 3.2
Duoplasmatron ion source and electrostatic lenses.
38
Figure 3.3
Proton beam extraction.
43
Figure 3.4
Cesium charge transfer canal.
46
Figure 3.5
The cross section for formation of H(2s) in collisionsof
protons with cesium atoms (adapted from Pradell (PR74)).
49
Figure 3.6
Fractional yield of protons (F+), ground state hydrogen
(Fg), metastable (2s) hydrogen (Fm), and negative
hydrogen ions (F-) as a function of cesium target
thickness.
50
Figure 4.1
Schematic cross section of detector.
above the atomic beam.
The view is from
54
Figure 4.2
Electric field on axis of quench tube as a function of
distance from gap.
58
Figure 4.3
Intensity of quench light on axis of quench tube for
various field strengths.
61
Figure 4.4
Geometrical parameters for window transmission.
63
Figure 4.5
Transmission of disk window for an unpolarized Lyman-a
source.
(ii)
65
(b) Schematic
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25
Page
Figure 4.6
Variation in transmission for two disk windows for
unpolarized source on window axis, for various window
spacings.
67
Figure 4.7
Transmision of cylindrical window as a function of aspect
ratio showing contributions, of reflection/refraction and
attenuation.
69
Figure 4.8
Ion chamberelectrodes:
(a) axial wire; (b) coathanger
anode with collecting wire parallel to window.
72
Figure 5.1
Allowed quenching transitions between states of 2s\/2
and 2pi/2 for an oscillating electric field in the
z-direction.
Figure 5.2
State selector transmissions as a function of microwave
power detected in a pick-up loop.
81
Figure 5.3
Contours ofconstant leakage and transparency in the L qL i
plane.
83
Figure 6.1
Apertures and obstructions in the beam line.
86
Figure 6.2
Moveable knife edges.
87
Figure 6.3
Beam intensity and background for a 50 mil (0.050 inch)
slit.
89
Figure 6.4
Metastable background versus pressure for an 80 cm length
of beam.
92
Figure 6.5
Figure 6.6
Measured leakage versus 1-selector transmission.
Measured noise versus beam intensity.
93
Figure 7.1
Magnetic field lines in Ml-region parallel plate
transmission line and field strength on axis.
103
Figure 7.2
Ml-region transmission line.
108
Figure 7.3
Ml-region transition probability and theoretical
lineshape for atoms moving on axis.
109
Figure 7.4
El-region transmission line.
Ill
Figure 7.5
Magnetic field lines for travelling wave in El-region.
Field strength on beam axis.
112
Figure 7.6
Coordinate axes and field directions in El-region.
119
Figure 7.7
Magnetic dipole transition amplitude in the El-region.
122
(ill)
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76
Page
Figure 7.8
Block diagram of VRF electronics illustrating the phases
<(> and 6, and the excess field strength a.
126
Figure 7.9
Expanded block diagram of VHF electronics.
128
Figure 7.10 Counterpropagated waves in El-region. A node of magnetic
field and an antinode of electric field are placed at the
center of the atomic beam.
129
Figure 7.11 ir-shifter schematic.
131
Figure 7.12 Method used to measure phase shift of ir/2-shifter.
134
Figure 7.13 Varactor tuned (analog) phase shifter.
135
Figure 8.1
Precessor coils (rectangular loops wrapped on cylinder)
and field strength on beam axis.
143
Figure 8.2
Cone of precession.
146
Figure 8.3
Any two points on a sphere can be connected by precession
on a cone whose axis is in the xy-plane because the cones
connecting the two points define a family of circles whose
line of centers cuts the equator.
147
Figure 8.4
Precessor arrangement.
149
Figure 9.1
Electronic noise measured in a 1 Hz bandwidth.
156
Figure 9.2
(a) The resonance at 60 Hertz, (b) The anomalous
resonance near 195 Hertz, a mechanical resonance of the
detector anode.
157
Block diagram of CAMAC interface.
160
Figure 9.4
Complicated square waves from the product of elementary
square waves.
163
Figure 9.5
Noise amplification factor resulting from feedback to a
controlled signal.
169
.Figure 9.3
Figure 10.1 Argand diagram showing how the sum of magnetic dipole
amplitudes for the two counterpropagated travelling waves
varies with the magnitude and phase of one of the waves
177
< 42))Figure 10.2 Interference signal between Ml-region and El-region
181
Figure 10.3 Argand diagram showing the relation between the mode-axes
and the phase-axes.
189
(iv)
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Figure 10.4 The induced Stark amplitude (A£Xe ) and the 6-wobble
signal in the vertical direction (AAX ).
Figure 10.5 Argand diagrams of amplitudes from Table 10.3.
Figure B.l
Lineshapes for El-region amplitudes.
Figure C.l
Precessor coils and field strength on beam axis.
Figure C.2
A split circular current loop centered at the origin.
(v)
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I.
A.
INTRODUCTION
Parity Non-Conservation
In quantifying observations of physical phenomena, certain
choices are made which are arbitrary or conventional, as when Benjamin
Franklin elected to describe the charge acquired by a glass rod when
rubbed with silk as "positive".
Had he chosen to regard this charge
as negative, beginning students of physics might be able to delay
distinguishing between current density and mass flux, but the tangible
phenomena of interacting charges and currents would be unaffected.
Nowadays, we would say that electromagnetism is invariant under the
operation of reversing the signs of all charges (and therefore of all
electric and magnetic fields), or that electromagnetism possesses
charge-reversal symmetry.
The view that Nature and Nature’s Laws
cannot depend on the arbitrary choices of Man has been fruitful in
suggesting and unveiling the symmetries underlying physical phenomena
and in limiting the class of theories eligible to describe new
discoveries.
It is therefore wonderfully humbling that Nature
disregards this view in what at first sight is a perfectly arbitrary
mathematical choice, the handedness of the coordinate system used to
quantify positions.
If we choose three mutually perpendicular axes against which to
measure the positions of points in space it makes a difference which
ends of the axes are assigned positive numerical values, since
reversing the signs produces a coordinate system which cannot be
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rotated back to the original one.
But the points in space are not
themselves affected by our choice of positive and negative labels, and
neither are the positions and velocities of physical particles, and so
it was long held that physical laws should be invariant under the
operation of reversing the signs of all coordinates, the so-called
parity operation.
Newton's law of gravitation and Maxwell's equations for
electromagnetism, the two most complete and successful physical
theories at the turn of the century, are invariant under parity, and
Einstein's general theory of relativity is founded on general
covariance— the invariance of physical laws with respect to general
coordinate transformations.
Moreover, the concepts of quantum
mechanics when combined with the symmetries of electromagnetism, were
able to explain the observation of Otto Laporte that the bright lines
in atomic spectra could be accounted for if two sets of energy levels
were supposed and if transitions could take place between the two
sets, but not within a single set.
The quantum mechanical states of
definite energy can be classified as having even or odd "parity"
depending on whether the wave function for the state changes sign (odd
parity) or does not change sign (even parity) when the signs of all
coordinates are reversed.
Laporte's rule in this language says that
the strongly allowed atomic transitions change the parity of the
atomic state.
This is what one expects if the atom is described as
interacting with a classical external electromagnetic wave through the
charges of its constituent particles.
The strong transitions are the
electric dipole transitions, the action of the field on the atom being
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given by a term -
where 5 is the dipole moment of the atom which
changes sign when the atomic coordinates are reversed.
Note, though,
that the interaction is even under parity when all coordinates,
including those determining the electromagnetic field, are reversed.
Invariance under parity implies that doing two experiments which
differ only in the reversal of all coordinates will yield the same
values for measured quantities.
The surprise came in 1957 when C.S.
Wu and her collaborators demonstrated that parity invariance is not
correct in the beta-decay of 60Co, as predicted by Lee and Yang.
Suppose an electron is emitted from the cobalt nucleus with its
intrinsic angular momentum (spin) aligned with its velocity.
When the
parity operation is performed on this process, the direction of motion
is reversed but the spin direction is not reversed, so that the new
process has the spin of the emitted electron aligned opposite to its
direction of motion.
If the law governing beta-decay were invariant
under parity, the number of electrons with spins parallel to their
velocities should equal the number emitted with spins antiparallel to
their velocities.
It was found, however, that more electrons are
emitted with spins antiparallel to their velocities than parallel,
indicating a violation of parity invariance.
In quantum mechanics the possible states of a system are assigned
the number +1 if they have even parity, and -1 if they have odd
parity.
If the laws governing a physical process are invariant under
the parity operation, then the parity of the final state will be the
same as the parity of the initial state.
One says that the quantity
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"parity" is conserved.
In the decay of S0Co the final states included
both parities and so parity was not conserved.
Laporte's rule for
atomic transitions is that for electric dipole transitions the parity
of the atomic state must change, but this does not represent parity
non-conservation because the state of the system includes the emitted
or absorbed photon and the parity of the combined system remains
constant.
If, however, one could drive an electric dipole transition
between atomic states of the same parity, this would represent parity
non-conservation.
This dissertation describes an attempt to induce
and detect an electric dipole transition between the hyperfine states
in the 2s level of atomic hydrogen, in violation of the Laporte rule.
Except for improvements in inexpensive high vacuum technology there is
no reason why this experiment could not have been attempted thirty
years ago when parity non-conservation was first demonstrated, but at
that time there was no reason to expect measurable parity
non-conservation In atoms.
B.
Weak Interactions
The canonical weak-interaction process is the decay of a free
neutron into a proton, an electron, and an antineutrino, that is,
beta-decay, but for purposes of comparing with electromagnetic
processes we can consider the inverse process in which a proton (in a
nucleus) absorbs an electron and emits a neutrino.
This can be
compared (see Figure 1.1) to the scattering of a proton and an
electron.
The (electromagnetic) scattering of charged particles is
described in quantum electrodynamics (QED) by the exchange of a photon
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w
Figure 1.1.
Feynman Diagrams for electromagnetic electron-proton
scattering, electron capture, and a second-order
electron-proton diagram involving the exchange of
two charged vector bosons.
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whose momentum is equal to the momentum transfer in the process.
The
photon can also carry angular momentum in being emitted by one
particle and absorbed by the other, but as it is neutral it does not
carry charge and the particles have the same charge going out as they
did coming in.
By contrast, in the weak interaction of inverse beta
decay the proton changes to a neutron and the electron changes to a
neutrino and so the mediating particle must carry charge from one
particle to the other.
It also carries momentum and angular momentum
(and possibly other properties).
Until 1973 such charge-changing
processes were the only observed form of weak interaction.
Such
processes are referred to as charged-current interactions, while those
without a transfer of charge are called neutral-current interactions.
The absence of neutral currents in strangeness-changing decays of
kaons was well established, so it was believed that weak interactions
always involved charged currents.
With this assumption weak
interactions could manifest themselves in atomic spectra through a
second order interaction involving the exchange of two charged quanta
(see Figure 1.1).
Estimates based on this mechanism made the effects
hopelessly small, so no atomic experiments were undertaken.
The interpretation of weak interaction events as involving the
exchange of a charged particle, the Intermediate vector boson, was
motivated entirely by analogy with QED.
The phenomena of weak
interactions were adequately accounted for by taking first order
matrix elements of a phenomenological Hamiltonian Introduced by Fermi
in 1936 in which the four fermion fields of the entering and exiting
particles appeared with all possible combinations of Dirac matrices,
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each term with its own coupling constant, to be determine
experimentally.
These "contact” interactions were adequate because of
the extremely short range of the weak interactions, but they did not
constitute a fundamental theory because quantum field theory allows
the Hamiltonian to act in all orders and the contact interactions led
to irreconcilable infinities in higher order.
The short range of the weak interactions meant that the
intermediate vector boson must be massive, which proved to be an
obstacle to constructing a suitable theory.
A promising approach by
Yang and Mills was to construct an interaction by requiring invariance
under an SU(2) gauge transformation.
The gauge invariance generated
two charged and one neutral vector boson corresponding to the three
generators of SU(2), but adding a mass term destroyed the gauge
invariance and made the theory non-renormalizable.
The mass problem
was eventually solved by Steven Weinberg who introduced scalar fields
coupled to Che vector bosons.
The scalar fields preserved the gauge
invariance of the theory, but by giving them a quartic
self-interaction whose minima occurred at non-zero values of the
scalar fields the vector bosons acquired an effective mass.
The
underlying gauge symmetry was still there, but it was broken by
choosing a vacuum state.
It was suggested by Weinberg that this might
allow the theory to be renormalized, and that turned out to be
correct, as shown four years later (in 1971) by Gerard t'Hooft.
Weinberg's theory was a gauge theory based on the group
SU(2)xU(l).
It produced interactions mediated by four particles, two
charged vector bosons with equal masses, one neutral vector boson
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whose mass could differ from that of the charged bosons, and a
massless neutral vector boson.
The massless boson coupled with equal
strength to both the right and left handed components of the charged
fermion field, and so corresponds to photons and quantum
electrodynamics.
The charged vector bosons accounted for all the data
of weak interactions.
The "red-headed stepchild" of the theory was
the neutral vector boson, dubbed the Z°.
The problem with the Z° was that severe limits had been set for
neutral current events in strangeness-changing decays and there was no
compelling reason to suppose a difference between strangeness-changing
and strangeness-conserving weak decays.
This situation changed when
in 1970 Glashow, Iliopoulos, and Maiani (GIM) postulated a fourth
quark (the "charmed" quark) to suppress strangeness-changing decays.
It was not immediately realized that this rescued the Z°.
Since the
three quarks (up, down, and strange) of Gell-Mann's model accounted
for all the known elementary particles, the charm hypothesis was to
some extent ad hoc, invoked for the purpose of explaining the K^-Kj
mass difference.
As the intimate connection between charm and weak interactions
was elaborated, and as t'Hooft's proof of renormalizability diffused
into the theoretical community (even Weinberg did not believe it at
first), the SU(2)xU(l) theory began to influence the experimental
program of high energy physics.
In 1973 two groups (Gargamelle at
CERN and a Harvard-Wisconsin-Pennsylvania-Fermilab collaboration at
FNAL) produced evidence of strangeness-conserving neutral current
events (the elastic scattering of muon-neutrinos).
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gauge theories sprang up in the wake of these discoveries.
C.
Parity Non-Conservation (PNC) in Atoms
Weinberg's SU(2)xU(l) theory, now called the GWS model after
Glashow, Weinberg, and Salam, is the simplest of an arbitrarily large
class of theories.
It has two coupling constants which are
constrained by the need to reproduce electromagnetism and the results
of charged-current experiments.
The constraints leave a single degree
of freedom usually written as sin2 0W where 8W is the weak mixing
angle or the Weinberg angle.
The masses of the charged (W*) vector
bosons and the neutral vector boson (Z°) are related by Mjj = Mz
cos 0W , which is to say that the mass of the Z° is greater than the
mass of the W's, and so the range of the interaction is shorter.
We
can thus follow the pattern of Fermi's phenomenological treatment of
beta decay, writing the effective interaction as a four-fermion
contact interaction, also called a current-current interaction.
Accordingly, we may write two Hamiltonian densities for parity
non-conserving semileptonic interactions between atomic electrons and
nucleons:
GF
■ Jy
^2N =
%
7
=
C 1N O e V s ' U
C2N (+ . V J
O
nV
n)
("VuVn)
in which ”N" stands for p or n —
proton or neutron.
Here Gp is the
Fermi constant (89.6 eV fm3 ; 1.027 x 10-5 mpc2 (h/mpC)3) which is
related to parameters in the GWS model by
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The Hamiltonian density*)^iu couples an axial electron current to a
vector nucleon current, w h i l e c o u p l e s a vector electron current
to an axial nucleon current.
Treating the nucleons as having wave functions which give
6-function probability densities at the origin and treating the atomic
electrons non-relativistically, one obtains the effective perturbation
F
H
pnc
r 5(r) r_
„
,+
— I “V tCM V pe ' 2H V pe
mc/2 N
“e*
si
+ Hermitian conjugate
where the sum is over the nucleons.
In 1974 it was pointed out by
M.A. Bouchiat and C.C. Bouchiat that matrix elements of Hpnc between
s- and p-states will depend cubically on the nuclear charge in heavy
atoms.
One factor of Z comes from the number of nucleons, one from
the probability density of the s-electron at the origin, and one from
the derivative of the p-state wave function.
Owing to this Z
O
enhancement many atomic experiments have been done in the heavy atoms
bismuth, thallium, lead, and cesium.
primarily with the
These experiments are concerned
term which is additive over the nucleons.
The C2N term depends on the nucleon spins, which are mostly paired
to give zero.
The effect of Hpnc is to induce an off-diagonal electric dipole
moment between states of the same parity.
This can be written as a
second order perturbation of energy eigenstates (in the absence of
Vc>
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<f|Hpnc|k><k|£l|i>
** = k f
in which
*£ - \
<f|tl|k><k|H n c |i>
+
Ei -
S'
1
= -Z e rj is the electric dipole moment operator.
If
the states f and i are connected by an allowed magnetic dipole
transition then the amplitudes for magnetic and electric dipole
transitions interfere to produce an optically active medium; that is,
the indices of refraction for right and left handed circularly
polarized light are slightly different by an amount proportional to
Dpnc/M where M is the off-diagonal magnetic dipole coupling the
states.
As a result, the plane of polarization of linearly polarized
light will be rotated as it passes through a vapor of these atoms.
This is the basis of the experiments in bismuth and in lead.
The
thallium and cesium experiments each use a highly forbidden transition
which becomes allowed in the presence of a static electric field.
In
this case the fluorescence signal when the sample is illuminated with
circularly polarized light changes when the electric field is
reversed.
These experiments have succeeded in determining the ratio
Dpnc/M to an accuracy of 20% and in full agreement with the GWS
model.
The difficulty of these experiments is formidable, inasmuch as
the measured ratio is of order 10-7.
From the first, the heavy atom experiments were criticized on the
grounds that the atomic theory was not good enough to provide a
meaningful measurement of the Weinberg angle, which is more or less
true but very presumptuous inasmuch as a general phenomenology of
semileptonic neutral current interactions admits ten free parameters.
These parameters are all specified in the GWS model once 0W is
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known, of course, but there are innumerable other models.
From this
point of view the heavy atom experiments have placed quite strict
limits on two of the phenomenological parameters.
It is also true
that the atomic calculations have improved considerably in response to
this criticism so there is no doubt that the measured results are
consistent with the GWS model.
The hydrogen atom does not enjoy a Z3 enhancement of the
sp-matrix elements of Hpnc, but it does have s- and p-states which
are very close in energy, namely 2s\/2 and 2pi/2» and the atomic
theory for hydrogen is essentially exact.
Because the difference in
energies between the s- and p-states appears in the denominator in the
expression for Dpnc this compensates for the smallness of the
interaction to make experiments in hydrogen competitive (in principle)
with the heavy atom experiments.
In the Dirac theory of hydrogen the
2si/2 an<* ^Pl/2 states are degenerate so that the energy
denominator in this case would be reduced to half the p-state decay
rate (times fi)— the imaginary part of the p-state energy.
In fact
these states are not degenerate but are split by radiative corrections
to the Dirac theory (i.e., the Lamb shift).
This splitting is still
very small (1058 MHz) but is twenty times the decay width of the
p-state.
A very clever idea to minimize the energy denominator is to apply
a static magnetic field so that the Zeeman effect causes the s- and
p-states to cross making the real part of the energy difference
vanish.
The denominator is then reduced to 50 MHz.
This was the
basis for three parity non-conservation experiments initiated in 1975
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and 1976 at the University of Michigan, the University of Washington,
and at Yale University.
The Yale level-crossing experiment was later
scrapped when it was realized that while the dipole moment Dpnc
increases near the level crossing, so does the quenching of the
2 si/2 atomic beam by the oscillating electric field used to drive
the (forbidden) electric dipole transition (HI 80).
Because of this
the experimental sensitivity that can be achieved is nearly
independent of magnetic field.
The new Yale experiment— that is, the
present work— was done in zero static magnetic field to preclude
mixing of s- and p-states by motional (v x 2) electric fields.
None of the hydrogen experiments has fulfilled its promise
although an upper limit on C2p of 620 has been published by the
Michigan group.
This has since been bettered by heavy atom
experiments, even though they are not particularly sensitive to this
coupling constant.
For a more comprehensive review of atomic pnc
experiments, see "Atomic Parity Nonconservation Experiments" by E.N.
Fortson and L.L. Lewis (Physics Reports 113, 289 (1984)).
For a
proper history of the early neutral current experiments see "How the
First Neutral Current Experiments Ended” by Peter Galison (Reviews of
Modern Physics 55, 477 (1983)).
A readable discussion of the SU(2) x
U(l) theory (with many typographical errors) can be found in
"Flavordynamics of Quarks and Leptons” by H. Fritzsch and P. Minkowski
(Physics Reports 73, 67 (1981)).
These reviews contain extensive
bibliographies.
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. II.
A.
OVERVIEW OF THE EXPERIMENT
Principles
The experiment described in this thesis is essentially a zero-
field atomic beam magnetic resonance experiment of the so-called
"flop-in" variety.
A beam of metastable hydrogen is passed through a
state selector which transmits one of the hyperfine components of the
^sl/2 state, and, further downstream, through a selector which
transmits the other component.
Between the selectors VHF fields drive
the hyperfine transition so that metastables detected at the end of
the line have made the transition (they have "flopped in").
The
transition is an allowed magnetic dipole (Ml) transition which is
interfered with a (forbidden) electric dipole (E2) transition in two
regions of separated oscillatory fields.
A true Ml-El interaction,
free of systematic effects, would constitute a violation of parity.
The atomic states needed to discuss the experiment are the
2s 1/2
and 2pj/2 states of hydrogen.
The 2p3/2 states can be
neglected because they lie above the 2s i /2 level at ten times the
splitting between the 2s!/2 and 2p1/2 levels.
Each of the levels
2s i /2 and 2p1/2 contains four states corresponding to Mj = ± 1/2
Mj = ± 1/2 (the proton has spin 1/2).
Each of these levels is split
into two eigen-energies by the hyperfine interaction between the
electron and the proton, corresponding to total angular momenta F=0
and F=l.
The centers of gravity of the 2 8 ^ 2 energies and the
2Pi /2 energies are separated by the Lamb shift (~ 1058 MHz) which is
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due to the interaction of the atom with the electromagnetic vacuum
(so-called radiative corrections).
All of these interactions, the
Coulomb attraction between the proton and the electron, the hyperfine
coupling, and the radiative corrections, are electromagnetic, and as
such, invariant under coordinate inversion— the parity operation.
As
the parity operator commutes with the Hamiltonian, each state has not
only an eigenenergy and quantum numbers associated with total angular
momentum and one component of total angular momentum (F and M), but
also a definite signature under parity.
The s-states have even (+1)
parity and the p-states have odd (-1) parity.
A very important
consequence of invariance under parity is that the 2 s±f2 hyperfine
transition cannot be driven by an oscillating electric field, that is,
it is not an allowed electric dipole (El) transition.
Electric dipole
transitions must occur between states of opposite parity.
This situation is changed if, in addition to the electromagnetic
interaction between the proton and the electron, there is a parity-,
non-conserving interaction such as one expects from the GlashowWeinberg-Salam model of weak interactions.
In that case the energy
eigenstates are no longer states of definite parity:
Each of the
”s”-states contains a small admixture of p-state, and each of the
”p”-states contains some small admixture of s-state.
It is then
possible to couple the s-part of one ”s”-state to the p-part of
another with an electric field, and so drive the hyperfine
transition.
The "s”-states can also be coupled, as in the absence of
the parity-non-conserving interaction, by a magnetic field which
couples the s-parts.
The experiment described here interferes this
allowed magnetic dipole (Ml) transition amplitude with the forbidden
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electric dipole (El) transition amplitude; the interference term is
then linear in the parity-non-conserving interaction.
The transition used in this experiment is that between the
hyperfine levels of 2s^f2‘ Restricted to this manifold of states
A
the Hamiltonian in a magnetic field B = z B can be written
H = EQ + * Ehfs(2^2"3> + ^BFz + y'B(|00X10| + |l0><00|)
(2.1)
in which the states are labelled by eigenvalues of f2 and Fz in the
conventional way, viz., ?2 |f M> = F(F+1)|FM> and Fz |FM> =
m |f m >.
The magnetic coefficients are given by
w = (8Jy0 + S l V /2
v'm
■ S i V /2
where gj is the gyromagnetic ratio of the electronic state, and
gI (= 2) the gyromagnetic ratio of the proton.
Since yN« y 0 the
moments y and y' are practically equal to the Bohr magneton
(2tt x 1.4 MHz/gauss).
We can choose as the zero of energy the center of gravity of the
hyperfine levels, Eq » and separate the Hamiltonian into field-free
and field-dependent parts:
0
4 hfs
(2?2-3)
(2 .2 )
Hj = yBFz + y 'B( 10 0 X 1 0 1 + |l0X00|)
Diagonalizlng H=Ho+Hl(B) gives the eigenenergies plotted in the
Breit-Rabi diagram of Figure 2.1.
The same analysis applies to the
spectrum of the 2pj /2 levels, with Ehfs and gj appropriately
modified.
These energy levels are also shown in the figure.
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1000
750
500
N
250
X
m
F=I
>©
q:
uj
UJ
F=0
|h >
loo>
-2 5 0
-500
-750
-1000
-1250
0
Figure 2.1.
IOO
200
300
400
500
MAGNETIC FIELD (GAUSS)
600
Breit-Rabi diagram for the n=2, J=l/2 levels of hydrogen.
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The only states in the 2sj/2 manifold nixed by the field z B
are those corresponding to the energies labelled 2 and 4 in Figure
2.1, corresponding to the off-diagonal term of Hj in the parentheses
of Eq. 2.2.
This mixing causes the curvature, or repulsion, of the
eigen-energies as the magnetic field is increased from zero.
If the
field B is an oscillating one, thie mixing causes transitions to be
driven between the states |00> and |l0>.
parity, magnetic dipole transition.
This is the allowed, even
The transition amplitude caused
a
by a field zfJ coscot which interacts with an atom initially in the
00-state for the period t=-Ti to t’H-Tj is given by
(2.3)
t
A 2 ± (u’S)2
in which v = co - Ejjfs measures the detuning from resonance.
The
operator U(t’,t) is the time evolution operator for the system.
This
very standard two-state problem is presented in Appendix A.
The lineshape factor of the form (sin x)/x in equation 2.3 is
shown in Figure 2.2.
For x small the function is nearly one,
corresponding to short transit time near resonant frequency in weak
field.
These are the conditions in our apparatus for the reference
magnetic dipole transition, so for this preliminary discussion we take
(2.4)
We would like to interfere this allowed Ml amplitude with an El
amplitude for the same transition.
This would be permitted if the
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1.0
sin x
0.5-
x / tt
Figure 2.2.
Lineshape for transition amplitude due to uniform
oscillating field.
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internal Hamiltonian for the hydrogen atom includes a term odd under
the parity operation, but rotationally scalar.
It is this parity
non-conserving term in the Hamiltonian we are trying to measure; call
it Hpnc*
Taking Hpnc to be very small we can define an effective
electric-dipole matrix element as follows:
(2.5)
This and other second-order amplitudes are derived in Chapter VII.
The dipole D written above contains the detuning v, which is very much
smaller in our experiment than the energy difference (Eio “ ®k)>
since the nearest intermediate k-states are in the 2pj/2 level of
Figure 2.1.
The dipole D can be regarded as independent of frequency
for practical purposes.
If it had been possible to establish a region of oscillating
electric field similar to that of the oscillating magnetic field we
would have a lineshape given by equation 2.3 with y'g replaced by De.
In order to obtain an oscillating electric field in the
correctdirection it was necessary to use a transmission line with an
electric field antisymmetric with respect to the center.
The
oscillating electric field is given approximately by
e cos(u>t + <|>) ;
-T2 < t < 0
-e cos(ut + <f>) ;
0 < t <T2
(at t=0 the atom is at the center of the region).
be set experimentally using phase shifters.
(2 .6 )
The phase <J> is to
The lineshape for this
field distribution is calculated in Appendix B.
The resulting
electric dipole amplitude is
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sin2(|vT2)
V
- T2*
e' 1+
(i VT2 )
(2.7)
•
The function (sin2 x)/x is illustrated in Figure 2.3.
It is an odd
function of x, with a large maximum at x/ir = 0.3710, where it achieves
the value 0.7246.
It is at this maximum (and at the one on the
opposite side of line-center) that we operate our experiment.
The
transit time of the second region (where the El amplitude is
generated) is much longer than that of the first (where the Ml
amplitude is generated).
This allows us to have the first
maximum of
Agi at a frequency near the peak of Ajji .
By changing the phase cj> to $ +
it
we are able to reverse the sign
of Ae i * We thus have two signals available:
with the difference
S+ - S. - 4
Agj).
This is, of course, the signal of interest in the experiment.
Before
outlining the various complications it seems useful to describe the
beamline.
B . Apparatus
Figures 2.4 and 2.5 represent the atomic beam machine.
The first
figure, traced from a photograph taken with a wide-angle lens, shows
the sequence of vacuum chambers through which the beam passes.
There
are, of course, support structures, gaslines, cooling lines, microwave
cables, and pumps which are not shown.
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21
To promote high vacuum the
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1.0 -
0 .5 -
x /7 r
-2
Q5
1.0
Figure 2.3.
Lineshape for transition amplitude due to anti-symmetric
oscillating field.
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various microwave cavities and transmission lines were kept free, as
nearly as was possible, of internal structures.
It is more common to
build a vacuum system into which many different devices can be
placed.
Here we have made the vacuum system itself into a sequence of
devices.
The first step in the experiment is the production of a clean
beam of hydrogen atoms in the 2s level.
The portion of the apparatus
which accomplishes this is called the source; it comprises the
duoplasmatron (ion source), the cesium (charge transfer) canal, and
the deflector.
The vacuum in the source
downstream beamline.
is not as good as that in the
The deflector contains two series of apertures
to permit differential pumping.
gas at about 1 Torr.
The duoplasmatron contains hydrogen
The charge transfer region, when the beam is
running, has a pressure of a few 10~6 Torr.
divided by a wall.
The upstream pressure is
downstream pressure is about 1 x 10
Torr.
The deflector region is
several 10~8 Torr, the
The lowest pressures are
in the El-region, where the parity-forbidden transition is to be
driven, and in the detector.
These are typically a few 10-10 Torr.
The low pressures maintained in the apparatus reduce excitation from
the ground state due to collisions with residual gas which is a source
of background signal.
Our duoplasmatron is taken from drawings of an ion source
developed at Los Alamos National Laboratory for the triton
accelerator.
It makes a plasma by a high current dc arc which is
constrained in a magnetic field.
downstream through a pinhole.
The plasma bleeds into the vacuum
Since the whole duoplasmatron is raised
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
DUOPLASMATRON
CHARGE TRANSFER CANAL
CHARGED PARTICLE DEFLECTION
AND DIFFERENTIAL PUMPING
F-0 SELECTOR
El REGION AND
MAGNETIC SHIELDS
LARGE SOLID ANGLE
LYMANCl DETECTOR
r ■I
SELECTOR
BEAM DUMP
PUMP
PUMP
PUMP
Figure 2.4.
Exterior view of the atomic beam machine.
I
Figure
2.5(a).
Schematic
cross section
of the source.
T
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PRECESSORS
Ml-REGION
El-REGION
O-SELECTOR
ro
on
f
m
SELECTOR
PUMPING
STATION
#
MAGNETIC
SHIELDS
Figure 2.5(b).
Schematic cross section of the beam machine.
DETECTOR
BEAM
PUMP
4
to a potential of 500 volts, the protons in the.plasma are accelerated
to an energy of 500 eV.
Between the duoplasmatron and the charge
transfer region there are electrostatic lenses to extract the protons
and to focus the resulting proton beam.
The proton beam is fired into a tube containing cesium vapor.
At
the energy of the beam there is a peak in the charge transfer
cross-section for the formation of hydrogen atoms in the 2sjY2
level.
In addition to the large quantity of 2s atoms emerging from
the cesium canal tube there are many atoms in the Is ground state, a
few un-neutralized protons, H“ ions, and some hydrogen atoms in
excited states above 2s (PR74).
The ions are swept away in the
deflector region and what emerges is a beam roughly half in the Is
level, half in the 2s level, with a small portion distributed
throughout the excited state spectrum.
The
atoms are produced
with equal numbers in each of the four states with total angular
momentum F=0, 1 (ED79, included as Appendix D of this thesis).
In
order to see the transition from F=0 to F=l, we must destroy this
equality of numbers.
The beam passes through a cylindrical microwave
cavity tuned to mix the F=1 states with the 2pj/2 states, which
promptly decay to the ground state.
The net effect is that the
emergent beam is almost entirely in the F=0 state.
This "0-selector"
is followed by two transmission lines in which the F=0 to F=1
transition is driven (at a frequency near 177MHz, the 2S i /2
hyperfine splitting).
The first transmission line drives the Ml transition and has for
its conductors two parallel plates, mounted vertically and passing on
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either side of the beam.
This arrangement produces an oscillating
magnetic field in the direction that the beam is moving.
As mentioned
before the transit time of this Ml-region is kept short so that its
linewidth is large compared to that of the E1-region.
The El-region is where the parity violating amplitude to be
measured is generated.
Its transit time is made long since the
transition amplitude is proportional to it.
The El-region is a large,
transversely mounted "coaxial" transmission line whose center
conductor is a vertical flat plate.
The electric field in the
transmission line is antisymmetric with respect to this center
conductor.
This antisymmetry gives rise to an amplitude lineshape
qualitatively similar to that shown in Figure 2.3.
It should be noted that the oscillating electric field (177MHz)
of the El-region can mix s- and p-states, effectively driving the
Lamb-shift transitions (1058 MHz) non-resonantly.
This causes
quenching of the metastable beam apart from driving the forbidden El
hyperfine amplitude.
The optimal field strength is a compromise
between increasing the parity-forbidden transition amplitude and
quenching the metastable beam. This optimum is such that the
metastable beam is reduced to about 1/e of its full intensity (HI
79).
This effect is neglected in this thesis because, owing to
technical difficulties, the optimal field strength could not be
reached; the quenching was, in fact, negligible (< 2%).
Between the Ml-region and the El-region is an evacuated tube
surrounded by small coils.
These coils, called precessors, permit us
in effect to change the direction of the oscillating magnetic field in
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the Ml-region.
These precessors have turned out to be far more
important to the experiment than was at first evident.
For the
present discussion it is enough to know that the result of the
Ml-transition and subsequent precession is the same as being able to
choose the direction of the oscillating magnetic field.
In normal operation the Ml-region contains a travelling wave
which is deposited in a 50 0 load when it leaves the transmission
line.
The El-region*s operation is much more complicated in order to
permit the introduction of travelling waves propagating in opposite
directions so that the Ml-amplitude in the El-region can be made to
vanish.
The 177 MHz radiation for both the Ml- and El-regions is
produced by doubling the frequency of the signal from a synthesizer.
Active and passive elements split this signal into three travelling
waves:
one for the Ml-region, two to counterpropagate in the
El-region.
The relative phase <f> (Equation 2.6) between the Ml leg of
the circuit and the two counterpropagated El legs is variable, as is
the relative phase between these two El legs.
The magnitudes of all
three fields can be varied.
The Ml-region, El-region, and the precessors will be collected
under the heading "transition region.”
This transition region is
followed by a second state selector, also a cylindrical microwave
cavity operated in its lowest mode.
The resonant frequency is chosen
to mix the F=0 state with the 2p1/2 state (the other Lamb-shift
transition) and so quench it to the ground state.
The sequence is
thus (1) select F=0, (2) drive transitions to F=l, and (3) select the
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F=1 states.
This second state selector is called the ”l-selector.”
Except for residual background and the small percentage of highly
excited states, what passes the 1-selector is a beam of atoms in the
Is ground state together with those atoms in the 2s state which have
made the F=0 to F=1 transition.
Since the 1-selector is not pumped directly, it is followed by a
pumping station.
Inside this pumping station we have placed a set of
moveable apertures which allow examination of a small portion of the
beam at a time.
The beam itself is approximately three centimeters
high and one centimeter wide at this point.
The pumping station is
followed by the detector.
Inside the detector the beam passes through a rectangular conduit
(labelled QUENCHER TUBE in Figure 4.1).
This conduit is divided into
halves, the upstream half is grounded, the downstream half held at
about a kilovolt.
At the junction of the two halves there is a static
electric field which quenches the remaining 2s beam to the ground
state.
The upstream portion of the conduit is grounded to prevent
this happening prematurely.
When the 2s atoms are quenched they
radiate ultraviolet light, the well-known Lyman-ot line, which emerges
through open mesh.
It is this ultraviolet light which is detected, so
that the detector is blind to atoms in the ground state.
The Lyman-a
detector comprises two Ion chambers filled with nitric oxide (NO)
which has an especially high quantum efficienty at this wavelength,
around 80% (WA59).
The two "turrets” of the detector have thin
windows of magnesium fluoride which are transparent to the ultraviolet
light.
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To prevent the beam, which is nearly all ground state atoms at
this point, from causing an increase in pressure, the detector is
followed by a low—conductance tube and an ion pump.
This is the "beam
dump•”
Each portion of the apparatus is described in detail in
subsequent chapters.
The preceeding discussion should suffice to aid
the imagination in the following overview of techniques used and
problems encountered in the experiment.
C.
Technique
Because we are able in effect to rotate the direction of the
oscillating magnetic field produced in the Ml-region, the entire
discussion can be reduced to a contemplation of the El-region. In a
way to be made precise later
inthe thesis, the relevant amplitudes
can be regarded as spatial vectors of complex numbers.
The Ml-
amplitude of equations 2.3 and 2.4 is merely the z-component of the
vector amplitude for the Ml-transitions.
These vector amplitudes have
a spatial character determined by the fields which produce them.
amplitude from the Ml-region
The
isto be interfered with whatever
amplitudes are generated in the El-region.
It will be rotated
in
space to pick out one or another combination of the El-region
amplitudes, and its phase will be varied, by varying the relative
phase of the VHF (177 MHz) signal delivered to the Ml- and El-region.
In this way we can explore directions, magnitudes, and phases of the
various amplitues generated in the El-region.
This rotable reference amplitude will be called ^Ml*
There are
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three amplitudes to consider which are generated in the El-region.
Figure 2.5(b) showed the atomic beam machine in a schematic
cross-section down its center.
The cross-section of the El-region is
repeated in Figure 2.6 with the electric and magnetic field lines of
the TEM travelling wave drawn in.
Since the amplitudes to be
considered have a spatial character determined by the fields producing
them, it is well to hold this picture in mind throughout the
discussion.
Included in Figure 2.6 is a set of coordinate axes which
are also useful.
beam.
The z-axis is taken in the direction of the atomic
The x-axis points up, and the y-axis is out of the page,
transverse to the beam horizontally.
Note that in the region of the atomic beam the oscillating
electric field, e, points in the z-direction, while the associated
oscillating magnetic field, £, points along the x-axis.
(This field,
5, is not to be confused with that of the Ml-region which produces
Ajji*)
The vector transition amplitude associated with the allowed
Ml-transition taking place in the El-region has the direction of the
x-axis, namely, up.
To distinguish it from the magnetic dipole
amplitude in the reference region, and as a reminder that its
direction is associated with the field
called Ag.
be called
this amplitude willbe
Similarly, the (forbidden) electric dipole amplitude will
Its direction is along e (the z-direction).
There is one more amplitude to mention which is not generated on
purpose, although it is always there when we turn on the machine.
This amplitude amounts to an allowed electric dipole transition
between the hyperfine levels of 2s i /2 •
It is induced by stray
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Figure 2.6.
Cross section of El-region transmission line showing
electric and magnetic field lines of the TEM mode.
Coordinate axes and expected transition amplitudes.
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electric fields in the El-region, which mix the s-states and the
p-states.
This p-state admixture allows the oscillating electric
field to drive the F=0 to F=1 transition.
The off-diagonal transition
dipole is given by a second-order expression analagous to Equation
2.5, with H^nc replaced by the Stark interaction, er»S.
The
direction of the corrsponding amplitude is along e x S where £ is
the stray electric field.
We do not know where 1 points, but we do
know the direction of e, and so we can say that this Stark-induced El
amplitude lies in the xy-plane.
It has become customary in our
labortory to call this amplitude
By adding two planes of fine wires parallel to the yz-plane above
and below the atomic beam we are able to apply an electric field in
the x-direction, creating a Stark amplitude in the y-direction.
With these definitions and conventions it is easy to summarize
the work described in this thesis.
By counter-propagating waves in
the El-region we are able to position a node of magnetic field so that
the resulting magnetic dipole amplitude
p
precisely counterbalances
the x-component of the stray Xg^ark which shifts the node from the
center by a tiny bit.
Associated with this node of magnetic field is
an anti-node of VHF electric field pointing in the z-direction.
By
applying a voltage to the wire planes we are able to create a Stark
amplitude in the y-direction which precisely counterbalances the
y-component of the stray Ag^ark*
What remains ought to be an
amplitude in the z-direction which is just the parity-forbidden
amplitude X g .
It should lie in the z—direction, it should be
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extremely small, and it should have the lineshape corresponding to
Figure 2.3.
What we find in the z-direction, though, is an amplitude
about half as large as that of the stray ^stark’
a lineshape
unrelated to Figure 2.3.
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III.
SOURCE
The source comprises Che duoplasmatron ion source, the cesium
charge-transfer canal, and the deflector, which are illustrated in
Figure 3.1.
The duoplasmatron has two active plasma regions, the
discharge and extraction regions.
A high power (one kilowatt)
discharge ionizes the gas introduced into the duoplasmatron volume, in
this case hydrogen.
This direct current discharge takes place along
the axis of cylindrical symmetry, the arc being confined by a magnetic
field (see Figure 3.2).
The anode supporting the discharge has in it
a pinhole through which gas and plasma bleed into the evacuated volume
downstream.
Electrostatic lenses extract a proton beam from the
effusing plasma.
This beam is accelerated to 500 eV, the potential of
the plasma, and collides with a cesium vapor target.
At this
collision energy there is a broad resonant peak in the cross section
for forming hydrogen in the metastable 2s state, as shown in Figure
3.5.
Other products from the collisions are hydrogen atoms in the
ground state, un-neutralized protons, and negative hydrogen ions (see
Figure 3.6), as well as some excited hydrogen atoms.
3.6).
(See Figure
The positive and negative ions are swept away by the
deflector.
What remains is a neutral beam of atomic hydrogen,
approximately half in the ground state, half in the metastable (2s)
state, with a small fraction of highly excited atoms.
Figure 3.2 is a schematic cross-section of the duoplasmatron.
Our version was assembled from blueprints contributed by Los Alamos
- 36 -
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
COOLED
APERTURES
CESI UM CANAL
DUO P L A S MAT R ON
DEFLECTOR
£3
TO P U M P
(DIFFUSION)
TO ION
PUMP# I
■<--------------------------------------
Figure 3.1.
I m
Schematic cross section of the source.
TO ION
PUM P#2
►
MAGNET WINDINGS
72Z Z 2EZ Z Z Z Z ZZ 22Z
EXPANSION CUP
ACCEL
FILAMENT
HOLDER
,<r
n^DECEL
*
is
PROBE.
ANODE
DISCHARGE
FILAMENT
nrrr,
tu
K
2
Figure 3.2.
5
4
3
2
I
0
-I
-2
-3
-4
-5
-6
-7
500 VOLTS
1____
T
Duoplasmatron ion source and electrostatic lenses.
- 38 -
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
National Laboratory where it was developed (MC68).
The duoplasmatron
housing, probe, and expansion cup are made from soft (i.e. permeable)
iron which makes the field in the region of the discharge high and
creates a cusp near the probe tip for trapping of ions and electrons.
The magnet windings and the probe are cooled by freon.
There is no
direct cooling of the anode, so it is made of copper to better
transport its thermal load to the magnet housing.
Failure of the
cooling system results in melting of the anode, which is expendable,
and of the magnet windings, which are not expendable and are costly to
replace.
All parts shown in Figure 3.2, with the exception of the filament
supports and filament, are azimuthally symmetric about the beam axis,
normally, although a great deal of mobility is built in to permit
tuning.
The filament holder is rigidly bolted to the probe, but the
probe assembly can be translated in the two dimensions perpendicular
to the axis.
The magnet assembly can similarly be translated.
accel(eration)-lens, held at a negative potential of about
The
six
kilovolts, can be moved in all three directions, but cannot be
rotated.
The decel-lens at ground potential, is stationary.
The discharge which ionizes the hydrogen gas takes place between
the filament and the anode.
The filament is made from nickel mesh,
coated with a paste made from a mysterious powder called "emission
carbonate" and amyl acetate, a volatile liquid smelling like bananas.
The emission carbonate, also called tri-carbonate or triple carbonate,
is a standard of the vacuum tube industry.
It was obtained from
General Electric at no cost in lots of about a kilogram.
Evidently it
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is difficult to purchase this substance in small quantities, as it is
normally sold by the ton.
It is a mixture of salts, such as barium
carbonate, which have a low thermoelectric work function.
The high current density, of order 250 amps/cm2 (TR 73), is
needed to produce atomic ions rather than molecular ions.
At this
high current density the electrons are essentially space-charge
limited so that the ionizing volume of the arc is at a potential close
to that of the anode, which is five hundred volts.
net energy of the emerging proton beam.
This
provides the
The discharge usually
transports a current of about twelve amperes at a voltage of something
near a hundred volts.
These parameters vary with the cleanliness of
the hydrogen, deformation of the filament, gas pressure, magnet
current, and so on.
In order to strike the arc, a heating current of order fifty
amperes at line frequency is supplied to the filament.
The heat
causes electron emission to build up, which will eventually initiate
the discharge.
This heating procedure is done slowly, gradually
increasing the heating current and adjusting the gas
something near 0.2 Torr.
pressure to
Even after the temperatures have stabilized,
the emission of the electrons continues to increase.
This is
attributed to a cleansing of the filament by ion bombardment.
During
the heating procedure the potential difference between filament and
anode has been about fifty volts.
the arc current monitored.
It is now gradually increased, and
At a voltage somewhere near a hundred
volts the arc softly strikes and a current of a fraction of an ampere
is seen to bridge the gap.
The arc is now a ten ohm (or so) short
- 40 -
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
circuit in series with a high-power resistor.
Increasing the voltage
across this chain no longer increases the potential difference between
filament and anode.
The current through the discharge does
increase, and can, in our machine, be made as large as twenty
amperes.
Beyond that the dissipated heat cannot be carried away by
the cooling system.
This is dangerous for the magnet windings and
harmful to the anode.
Occasionally the anode melts, sealing the
pinhole through which the plasma escapes.
The "soft strike" described here is to be contrasted with what
happens if the potential difference is turned up too rapidly, as
sometimes happens when the filaments are old and their emission is
feeble.
The arc can still be made to strike, but explosively.
In
these circumstances the filament is the most abused, its tip
frequently melting and cracking so that it can no longer carry current
to be heated.
Sometimes the anode suffers, although it seems more
forgiving of short term abuse.
The pressure in the discharge region is influenced by the heat
produced by the arc and by the extraction of ions in the extraction
region.
The two effects tend in opposite directions, but both can
cause the arc to wander into hazardous territory.
A rise in pressure
can increase the thermal load, melting filament and anode or
overburdening the freon system, while a decrease in pressure may
extinguish the arc.
Once the arc is struck it is usual for us to run
with about a Torr of hydrogen inside the duoplasmatron.
Increasing
the pressure from where the arc will strike to where we can get high
proton currents is somewhat tricky, because the current in the
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discharge drops, and this can also cause extinction.
It is best to
procede slowly, eyes on all meters.
The magnet current is not irrelevant here.
high, the arc will not strike.
If the field is too
If it is too low, the arc will strike
to the probe, and from the probe to the inside wall of the magnet.
Once this happens there is no way to make the arc run along the axis
as it should.
All things being otherwise equal, about three and a
half amperes of magnet current works to get things going.
The whole
process of heating, striking, and adjusting the pressure takes about
an hour or so.
The pinhole in the anode (0.020 inches in diameter) allows the
plasma created by the arc to leak out of the discharge region.
Figure
3.3 depicts the expansion cup and extraction electrodes in this
evacuated region.
The ions and electrons effusing through the pinhole
enter a conical well called the expansion cup and are more or less
bound to it by their image charges when the entire region is at a
uniform potential.
When a negative potential is applied to the
extractor the positive Ions are drawn away from the expansion cup and
through a hole in the extractor.
The behavior of the extracted plasma
can be understood qualitatively through Child's Law which relates the
current density (j), the potential (V), and the distance from the
emitting surface (x) in a space-charge limited non-neutral plasma (CO
41). That is,
1 /2
i -
(I 1 )
tt 3
-^7
9irx
f 3 - 1)
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R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
PROTON BEAM
Q-1CO
DECEL
LENS
EXPANSION
CUP
ACCEL LENS
Figure 3.3.
Proton beam extraction.
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where the emitting surface is taken at ground potential, and the
number of emitted ions is assumed so large that the electric field
vanishes on some surface within the plasma, neutralized by the field
of the ions in the beam.
This surface, the so-called emitting surface
in the expansion cup, is concave, as shown in Figure 3.3, and free to
move so as to satisfy equation 3.1 (TR 73).
The result is that the
ion beam which is initially accelerated perpendicular to this surface
is focused through the hole in the extractor.
The cylindrical beam of positive ions, with a mean energy of 500
eV and a spread of 1% (TR 73), passes out of the extraction region
into the cesium box (see Figure 3.1).
Since the beam is positively
charged one expects a radial field which will cause the beam to expand
rapidly.
This is what happens in very high vacuum, but at higher
pressures the protons in the beam ionize background gas and attract
the liberated electrons while repelling the postive ions.
A steady
state results when the electrons have neutralized the charge of the
positively charged beam.
In our machine (and in others like it) this
neutralization of the space-charge of the beam comes from the
ionization of cesium escaping from the cesium canal
back toward the
duoplasmatron and not from the background gas pressure (a few times
10~6 xorr).
One can watch this spectacular effect through a window
in our cesium box.
As the cesium canal is heated, sending an
increasing spray of cesim atoms toward the duoplasmatron, there is a
sudden transition from an expanding ion beam to a narrow pencil when
the cesium density reaches some critical value.
(What one sees, of
course, is the fluorescence of the background gas as the beam collides
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with it.)
The cesium canal, depicted in Figure 3.4, is a stainless steel
tube, concentric with the axis of the duoplasmatron, but fixed in
space.
At the center of the tube is a hole connecting it with a
reservoir of liquid cesium.
are heated electrically.
The reservoir and the center of the canal
Inside the canal, and pressed firmly against
the wall, are seven layers of fine stainless steel mesh.
are copper blocks which are water cooled.
called the wick.
At the ends
The mesh lining the tube is
Two platinum resistance thermometers monitor the
surface temperature of the reservoir and the wick housing (the
canal).
The result is a tube filled with cesium vapor, hot in the
middle, cool on the ends, and open to vacuum at each end.
Some cesium
vapor inevitably escapes through the open ends, but any cesium hitting
the wick near the water-cooled ends will liquify and be drawn back to
the center by capillary forces, assuming the ends are not so cold that
the cesium solidifies there.
In winter, when the city water supply
gets quite cool, this solidification can be expected and must be
prevented.
The purpose of the wick and end-cooling is to minimize cesium
losses, and so keep the machine running for as many months as possible
before refilling the reservoir.
Cesium reacts with nearly everything,
and so this refilling process is difficult and time consuming.
The
entire cesium canal assembly is removed, cleaned with water and nitric
acid, rinsed in acetone and ethanol, and dried before being placed in
a polyethylene glovebag.
The air in the glovebag is removed by
pumping most of it out, refilling with argon, repumping and
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WATER COOLED ENDS
WICK
Cs RESERVOIR
HEATED
MOUNTING
BLOCK
HEATER
WIRES
— VALVE
STEM
LUCITE .
INSULATOR
Figure 3.4.
Cesium charge transfer canal.
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refilling.
It takes three or four cycles of pumping and filling to
achieve the purity necessary to accomplish the reloading of the
reservoir without the cesium reacting significantly with the gas.
The
whole thing can be done in one working day, provided no wrench has
been left out of the bag by mistake, and provided the cesium does not
spill and eat through the plastic while it is being loaded into the
reservoir.
Checklists and patience have both proved their value, as
also have fire extinguishers.
The wick minimizes cesium losses, but some cesium escapes.
The
cesium escaping in the direction of the duoplasmatron is essential to
the neutralization of space charge, but the cesium escaping downstream
is undesirable.
This obnoxious beam of cesium is collimated in the
deflector region when it hits the columns of apertures indicated in
Figure 3.1.
The apertures are cooled by a freon refrigerator so that
cesium will stick to them, eliminating nearly all the cesium which
would otherwise travel into the apparatus.
We have never found
deposits of cesium beyond the source, although it shows up everywhere
within the source.
If the machine is mistakenly let up to atmospheric
pressure with room air, cesium hydroxide forms on the apertures making
it difficult to pump down again.
By adjusting the temperature of the middle of the cesium canal
the density of cesium vapor can be varied.
At very low density the
protons entering the canal will collide only once with the cesium.
The initial state for the collision is a bare proton at 500 eV and a
cesium atom in its 6S ground state.
The final states of interest are
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those with an outgoing hydrogen atom in any state with a Cs+ ion
left behind.
At incident energies near 500 eV the outgoing hydrogen
atom is primarily in the n=l and n=2 levels.
The initial state, with
a proton at large distance from the neutral Cs target atom in its
ground state, is close in energy to the final state, with a hydrogen
atom in the n=2 level departing from a Cs+ ion.
The binding energy
for n=2 is approximately 3.4 eV while the first ionization potential
of cesium is 3.9 volts.
The final state is thus 0.5 eV higher in
energy than the initial state.
The nearest-energy final state after
this is that for capture to the n=3 level, 1.9 eV higher in energy (a
defect of 2.4 eV).
In this sense the charge capture to n=2 can be
regarded as occuring at small energy defect, and is correspondingly
large.
(For treatments of this as a two-level problem, see DE 64 and
ST 70.)
As can be seen from Figure 3.5, there is a broad peak in the
H(2s) formation cross section near 500 eV (PR 74).
This is the reason
for operating at this energy.
At higher cesium densities multiple collisions in the vapor
target limit the amount of H(2s) that can be formed.
Figure 3.6 shows
the fractions of un-neutralized protons, ground state hydgrogen atoms,
metastables, and negative hydrogen ions.
As the target thickness
increases, the number of surviving protons vanishes while the
fractions of H(ls) and H(2s) increase.
The H“ fraction increases
quadratically at low density which is, not surprisingly, indicative of
a two-step process.
As double scattering becomes probable the
metastable atoms formed in one collision are deexcited by the second
collision.
This deexcitation contributes to the ground state
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CM
E
o
O
|
fc
LU
CO
CO
CO
O
CC
O
2
0
3
ENERGY (keV)
Figure 3.5.
The cross section for formation of H(2s) in
collisions of protons with cesium atoms
(adapted from Pradell (PR74)).
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CM
CM
(O
E
CM
2
a
ID
L
i
.
CO
CO
Ll I
z
o
X
H
H
UJ
CO
<r
<
CM
hx
CO
u
ID
CM
SQ13IA IVNOIlOVyj
Figure 3.6.
Fractional yield of protons (F+ ), ground state hydrogen
(F ), metastable (2s) hydrogen (F ), and negative hydrogen
ions (F_) as a function of cesium target thickness. (PR74).
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population which continues to increase.
Eventually deexcitation
overwhelms the formation of metastables and the metastable fraction
drops off from its peak of 31%.
The aim in adjusting the wick
temperature is to sit on the peak of the metastable yield.
74).
(See PR
The fraction of hydrogen in states with n>2 is not well
characterized at this collision energy (LE 82).
It has been noted earlier that cesium sprayed from the canal
toward the duoplasmatron is ionized by the passing proton beam and
that the liberated electrons are trapped and neutralize the space
charge while the ions diffuse away.
At the high density of cesium
inside the canal the same ionization occurs, but the mean free path of
the cesium ions is much shorter.
The result is that inside the canal
there is a radial electric field from the build up of cesium ions
which causes the proton beam to expand and collide with the walls of
the canal. Consequently, while three to five milliamperes of protons
are injected into the cesium canal, no more than half a microampere of
metastable atoms can be got out of it.
The primary evidence for this
mechanism limiting the metastable current is that in pulsed sources
the peak metastable current is roughly two orders of magnitude higher
than can be obtained from continuously running sources.
The belief is
that after a pulse of protons has passed there is time for the cesium
atoms to diffuse to the walls and be neutralized so that there is no
build up of ionic charge in the target.
If a way could be found to
neutralize the residual cesium ions, say by injecting electrons into
the canal or by liberating them photoelectrically within the canal, it
might be possible to Increase the metastable yield.
Cursory attempts
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to photoionize the liquid cesium in the canal with a helium-cadmium
laser were undertaken, but no effect on the metastable yield was
observed.
This does not rule out the technique, as the laser was weak
and very little of the canal was illuminated.
The duoplasmatron, when operating, contains hydrogen at a
pressure of about a Torr.
The hydrogen leaking through the anode
consititutes the main gas load on the diffusion pump serving the
cesium box.
The pressure when operating is typically several
10~6 Torr in the cesium box.
times
The apertures and two ion pumps on the
deflector box allow the pressure to be stepped down so that at the
output end of the source the pressure is a few times 10-9 Torr.
The deflectors themselves are
a set of slanted vanes (like
Venetian blinds) set at an angle of 7.2° and held at a potential
difference of ten volts.
This design sweeps away the protons and
negative hydrogen ions from the beam.
The angle was chosen to
minimize the number of ions hitting the deflectors.
The small field (10 V/cm) of the deflectors causes some loss of
metastable atoms (10%) through Stark quenching, but this is
acceptable.
Emerging from the source is a beam of hydrogen atoms,
mostly in the states Is and 2 s, with some atoms in highly excited
states.
Comparison of the metastable beam measured in the detector
with a measurement of the total beam current in a Faraday cup at the
end of the apparatus suggests that the Is and 2s populations are
roughly equal, but this result is only good to a factor of two owing
to uncertainty in the detector efficiency.
The total current detected
at unit gain corresponding to the full (unselected) metastable beam is
typically 300 nA to 350 nA, with a record performance of 450 nA.
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IV.
A.
DETECTION
Description of Detector
When an electric field is applied to them the metastable atoms
emit ultraviolet (Lyman-a) light.
Ion chambers detect the light.
Figure 4.1 shows the detector viewed from above in a schematic cross
section.
The atoms enter a long rectangular quench tube which is
split into two sections, one preceding the ion chambers and one
following.
The upstream tube is grounded to the beam pipe to prevent
the atoms seeing an electric field before they are in view of the
detector.
The downstream tube is biased at one kilovolt to provide a
strong electric field in the tubes in the volume between the two
ionization chambers.
The electric field mixes the s- and p-states,
and since the p-states decay rapidly to the groundstate so do the
states mixed by the electric field, with the associated Lyman—a
emission (LA 50).
The ionization chambers contain nitric oxide (NO) which ionizes
with high efficiency at this wavelength (81% at 121.6 nm) (WA 54 and
ST 63).
The body of the detector is maintained at a negative voltage
so the electrons will be collected on the anode wires which are
nominally held at ground potential.
The detected current is amplified
by a low noise, high gain Ithaco preamplifier which produces a voltage
proportional to the current.
This voltage is used to set the
frequency of a voltage controlled oscillator.
The train of TTL pulses
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Reproduced with permission of the copyright owner
FLOATING SHIELD
“ FEED-THRU
ANODE
2" DIAM. MgF2
WINDOW
Further reproduo,ion prohibited without permission
beam
IN
in
QUENCHER TUBE
mesh
LARGE SOLID ANGLE BAKEABLE UV DETECTOR
Figure A.I.
Schematic cross section of detector.
above the atomic beam.
The view is from
from the oscillator is then free of additional noise sources and can
be transmitted to scalers on the other side of the laboratory.
The two-millimeter thick windows which separate the volume of
nitric oxide from the high vacuum where the metastable beam passes are
made from magnesium fluoride crystals.
These disk-shaped windows are
unusually large, with a two-inch diameter, so that together the
two windows subtend half a sphere, or 2tt in solid angle.
There turns
out to be little advantage to making flat windows larger than this or
to placing them closer to the atomic beam because attenuation and
reflection of the light at large angle of incidence preclude the
detection of more of the emitted light.
The sides of the quench tubes which face the magnesium fluoride
windows are made from a coarse flat mesh, not woven but eroded from a
thin sheet of stainless steel ("electromesh”).
The squares of the
mesh are a tenth of an inch on a side with ninety percent of the area
open.
This allows the quench light to pass while still providing the
quenching field.
The same mesh is used on the nitric oxide side of
the magnesium fluoride windows to provide an accelerating field for
the electrons and a neutralizing electrode for the N0+ ions.
Originally we had the mesh on the vacuum side of the windows, but the
build-up of ionic charge on the window tended to reduce the efficiency
of the detector.
Because the mesh is so coarse there continues to be
a slight problem with charging of the windows owing to the slow
migration of ions on the dielectric surface.
This is manifest by a
slow settling of the detected current when the gain in the ion chamber
is increased.
The gain is turned up under conditions when the
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detected metastable current is small.
We overcome this long settling
time by exposing the detector to the full metastable beam intensity
for a couple of seconds.
Upon reaching this steady-state operation
the detector behaves linearly with a fast response to fluctuations in
metastable beam intensity.
The full metastable beam, comprising populations in all four of
the 2si/2 states, produces an electron current of more than three
hundred nanoamps when the ion chamber is operated at unity gain.
Typical full-beam currents day-to-day are between three hundred and
three hundred fifty nanoamps maintained for six to ten hours.
The
record current for our machine is 450 nA, but this evidently cannot be
maintained stably.
Under the conditions of the parity experiment the
metastable current is reduced to 3 x 10~5 of the full beam and the
gain of the ion chambers is increased to ten.
The remainder of this chapter is devoted to detailed calculations
which aided the design of the detector, included to augment the
qualitative descriptions already given.
The electrostatic potential in the quench tube can
by the usual boundary-value methods (JA 75 Chapters 2 &
becalculated
3) in the
limit of infinitessimal gap between the upstream and downstream
tubes.
For a tube of height
a
and width
b
the potential is
given by
tt/ n
1 tt
z f i 16
V( r) = 2 V0 Izl t 1 “ ~
I I
it
V 1
niT'x
I SnslnT
n,m
odd
' *■
. miry
Sin—
Iz I
mn| | i
6
f
where the coordinates are as illustrated in Figure 4.4 and
k2mn =
(n/a)2 + (m/b)2 ; kmn> 0 .
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Along the z-axis the electric field has only a z-component given by
the partial derivative with respect to z of the above potential (which
just brings down (-k^) in each term of the sum).
Direct
computation of the field on-axis from this equation requires nearly
two hundred terms to achieve a one-percent accuracy.
This computation
was carried out for the case a/b = 3 used in our design of the quench
tube.
(This ratio is the ratio of height to width of the metastable
beam).
What one wants to know is where the ultraviolet light will come
from, and approximately how much.
The field along the axis will die
away more slowly than that along any parallel line as one moves away
from the gap, so that it provides a worst case estimate of the volume
in which quenching occurs (one would like this volume to be small,
centered between the two magnesium fluoride windows).
To make these
estimates it would suffice to have tabulated values for the field, but
it turned out to be simpler and satisfactorily accurate to fit the
computed values and use the fitting function to do the necessary
integrals.
The fitting function gives the following for the field
along the axis of the tube:
where
Figure 4.2 shows the computed and fitted fields.
A = — /lO
a
The discrepancy is
no worse than the uncertainty from the truncated series.
The
relatively simple form of the approximating function was discovered
by plotting the computed values semi-logarithmlcally and noticing that
57
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.8
0.6
o
UJ
s
UJ
0.4
0.2
0.0
8z/b
Figure 4.2.
Electric field on axis of quench tube as a function of distance
from gap. The dashed curve is computed from 200 terms of the
orthogonal function series, the solid curve is a simple approximate
form.
the result was an hyperbola.
The asymptotic form is clear from the
mode in the series with the slowest decay, namely kmn = h n *
I
have found no good reason why the constant term under the radical
should be unity, but it is.
The rest is normalization.
The decay rate for the metastable atoms depends on the electric
field, and therefore upon z.
For atoms moving along the axis the
amount of light emitted is proportional to the local decay rate and to
the number of atoms surviving up to that point.
This intensity is
therefore given by
I(z) = T(z) exp {- /z dz' T(z')/v}
— 00
To a good approximation (SE 64) the decay rate in an electric field is
given by
where
% = 2/“ J
and AESp is the Lamb shift.
ea 0 |£|/AEgp= |£|/(475 V/cm)
This decay rate can be obtained as the
zero frequency case of the quenching theory presented in Appendix A of
this thesis (remembering to restore a factor of two in the matrix
element because it is no longer appropriate to separate rotating and
counter—rotating waves), but it is also presented by Bethe and
Salpeter in Quantum Mechanics of One- and Two- Electron Atoms (section
67).
The calculated luminosity of the quenched beam along the axis of
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the quench tube is shown in Figure 4.3.
As the electric field is
increased (go) the atoms are quenched further upstream so that at
very
high fields (go^lOO) all of the atoms are quenched before the
beam reaches the gap in the quencher tube.
The area under this
advanced curve is proportional to the total incoming beam.
At the
more modest field realized in our quencher (corresponding to £o=2 ,
or a potential difference of 1 kV) not all of the metastable atoms are
quenched and the area under the curve is smaller.
serious loss.
This is not a
The important point to take from Figure 4.3 is that
nearly all the atoms can be made to decay in a very short segment of
the tube, in our case half an inch (the tube dimensions are 0.5" x
1.5").
C. Windows
Another interesting design problem is to understand the role of
the magnesium fluoride windows.
Some folklore is involved in the
assignment of window parameters.
The AIP Handbook (second edition)
gives an attenuation coefficient for lithium fluoride, which used to
be the standard vacuum ultraviolet window material, but not for
magnesium fluoride.
Now lithium fluoride absorbs water to an extent
that always made it difficult to work with, but prior to 1966 it was
the preferred VUV window material.
This was evidently because the
industrial standard of purity for magnesium fluoride crystals was
sufficiently poor that magnesium fluoride, which does not have the
hydrophillic problem, was far inferior as a VUV window. Improvements
in crystal purity reduced the attentuation to "as good as or better
than lithium fluoride,” according to the manufacturer (Harshaw
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INTENSITY
—
Figure 4.3.
ELECTRIC
FIELD
Intensity of quench light on axis of quench tube for
various field strengths. The strength of the electric
field versus position on the tube axis is also indicated
£or 50 - 5-
_61 .
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p erm ission.
Chemicals).
They were unable to supply me with a number, but they did
refer me to a group which had reported a measurement of the index of
refraction at the relevant wavelength (WI 79).
Magnesium fluoride is
birefringent, so they ought to have reported two indices of
refraction, to which they responded, "Well— it's not very
birefringent.”
In consequence I have taken the attenuation
coefficient for lithium fluoride and the reported single index of
refraction for magnesium fluoride in these calculations.
It ought to
be about right.
The fraction (T) of light transmitted by a window can be written
T -
h Sia i
( Ti + T? ) e _ a t
Where
a = absorption coefficient of window, assumed to be the
same for both polarizations
t = apparent thickness of window for refracted ray
TJ_,Ta = amplitude transmission coefficients for light
polarized (£) perpendicular and parallel to the
plane of incidence (i.e., the plane containing
the incident ray and the normal to the window).
For a point source on the axis of a disk-shaped window this reduces to
t = j
sin 9 (Ti + t J) e"at-
The geometry is illustrated in Figure 4.4, defining the incident angle
0 , the angle of refraction r, and the window thickness to*
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Figure 4.4.
Geometrical parameters for window transmission.
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The thickness of window traversed by the refracted ray is clearly
t = tg/cos r.
Ignoring multiple reflections between the interfaces
we have the (window) transmissions
(JA 75 Section 7.3)
2 cos 9________ .
_
(“2.
2~
cos 0 + /n -sin 0
2n cos r__________
I
2
2
n cos r + vl-n sin r
and
T
_
8
2n cos 0___________
n
2
r 2
2~
cos 0 + /n -sin 0
. 2 n cos r__________
r
2
2~
cos r + n / 1-n sin r
in which the sine and cosine of the angle of refraction are derivable
from Snell's law:
sin r = (sin 0 )/n
;
I
2
2
cos r = /I - sin 0 /n.
For the case of interest we take o = 1.0/cm, n = 1.6, and to = 0*2
cm.
The window transmission T is plotted in Figure 4.5 along with
the fractional solid angle 0/4ir.
Evidently, little is gained (10%) by
making the window infintely large rather than stopping with 9may =
0.4ir.
The integral for the transmission T was done numerically.
The atomic beam in our apparatus is very large, roughly one
centimeter wide and three tall.
homogeneity of the detector.
This large size raises the issue of
How does the detection efficiency vary
with the position of the decaying atom?
The curve in Figure 4.5 can
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T R A N S M IS S IO N OF L y - a
T H R U DISK W IN D O W
0 .4
SO LIDANGLE
O
0 .3
co
CO
0.2
£
o
o
z
0.0
0.0
0.2
0 .3
0 .4
0 .5
7T
Figure 4.5.
Transmission of disk window for an unpolarized Lyman-a
source. The lower curve represents the actual design
of our detector.
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be used to determine the combined transmission of two disk shaped
windows when the source is on the window axis.
This calculation was
done for the windows of our detector, which are
for several window separations.
one inch in radius,
Figure 4.6 shows the fractional
variation of the combined transmission as the source of light is moved
away from the center of the detector along the window axis.
The
reference transmission is that corresponding to the center of the
detector (hence all the curves pass through zero there).
It might
have been unfortunate that this optimization was not considered before
the detector was in the machine shop being assembled were it not for
other considerations which make the calculation fairly meaningless in
retrospect.
The window separation in the detector we built
corresponds to s=0.375 inches which gives about three percent
variation in the transmission over the beam width.
Although this is
quite satisfactory it is clear from Figure 4.7 that far less is in
principle achievable (s=0.5”).
In any case when the windows were
brazed to their mounts one of them came back highly metalized and had
to be replaced.
This throws considerable doubt on the transmission of
the windows used and demotes model calculations to the rank of
heuristic.
Neither of Figures 4.5 and 4.6 were available at the time the
detector drawings were put in the shop.
The favored design (except
for its prohibitive cost) prior to this was a cylindrical window and I
had done calculations for this configuration which were used to set
rough limits on the scale of the actual detector built.
The
calculations were done for lithium fluoride which as noted has
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66
-
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I—2 S -H
MgR
0.03- FRACTIONAL VARIATION
INTRANSMISSION
RELATIVE TO CENTER
0.02-
0 .01-
S-0.5
-0.3
-
0.2
-
0.1
'01
0.2
- 0.02-
S*0.3
-0.03-
-0.04-
-0.05H
Figure 4.6.
Variation in transmission for two disk windows for
unpolarized source on window axis, for various
window spacings.
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properties similar to magnesium fluoride.
Figure 4.7 shows the fraction of light transmitted through a two
millimeter thick LiF window as a function of the ratio L/D of length
to diameter.
cylinder.
The source is taken to be at the center of the
Evidently you gain little by making L/D greater than two,
which is, of course, similar to the disk window case.
The more
interesting information to be had from Figure 4.7 is the relative
contribution made by reflections and by absorption of the light.
Specifically an absorbing layer of unit index of refraction transmits
less than a non-absorbing layer with an index of refraction when the
numerical values are those of LiF, but both contribute significantly
to the reduction.
If the window were made thinner, the absorptive '
effect would be reduced, but the reflections would still cause
considerable loss.
D.
Ion Chambers
I have discussed the quenching of the metastable beam and the
transmission of the resulting light through the detector windows.
Once the light is inside the detector (that is, in the ion chambers)
it ionizes the detector gas which is nitric oxide.
On its own the
ionized gas would recombine, but an electric field is present to sweep
the electrons to the anode and to drive the ions to the cathode where
they neutralize.
If the field is strong then the electrons gain
sufficient energy between collisions with the gas to ionize additional
molecules.
This is the mechanism of gas gain familiar from
proportional counters, wire chambers, and geiger counters.
-
68
A lot is
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Lv- g
TRANSMISSION OF LiF WINDOW
CYLINDER
SOLID
ANGLE
ONLY
£
o
o
REFLECTION
- AND
a =1.17/C
REFRACTION
ONLY
ABSORPTION ONLY
IvERYTWNG
a=l . l 7/ c m
n=l.6
X
©
3
o
cr
x
©
o
<
a:
L
i
.
0.2 cm
ISOTROPIC
UNPOLARIZED
SOURCE
Figure 4.7.
Transmission of cylindrical window as a function of
aspect ratio showing contributions of reflection/
refraction and attenuation.
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known (SA 77) about the diffusion of electrons and amplification of
the current in such chambers but no detailed modelling of our detector
has been done*
Nitric oxide is a fairly standard gas for vacuum ultraviolet
detection.
It has a low ionization potential and a high quantum
efficiency, which is the probability that some molecule will ionize as
a photon passes through a thick medium (which is not the same, for
instance, as the probability that a single scattering will cause
ionization).
This is said to be eighty or eighty-one percent for
nitric oxide
(WA 54).
For comparison the quantum efficiency of
benzene— another gas used for vacuum ultraviolet detection— is about
twenty-five percent (BR
79).
To achieve this high quantum efficiency it is necessary to purify
commercial nitric oxide by distillation.
The nitric oxide Is
condensed in a bottle bathed in ethanol which has been cooled to a
waxy state by mixing it with liquid nitrogen.
The liquified gas is
then exposed briefly to vacuum to pump away volatile components.
The
bath of cold ethanol is replaced by one of acetone mixed with dry ice
and the ethanol bath is attached to a second bottle.
When the two
bottles are connected the (relatively) warm bottle bathed in acetone
releases nitric oxide which is condensed in the second bottle, now
cold.
What remains behind are components of the mixture which are
less volatile than nitric oxide, presumably polymers (NO)2>3...
and the other four oxides of nitrogen (N2 0, N02 , N2 C>3 , and N 2 05).
This process of cooling, pumping over the top, warming and
transferring is done three times before the purified gas is admitted
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to the ion chamber.
The ion chamber itself has been baked under
vacuum at a modest temperature (110° C).
It was designed to go to a
much higher temperature, but the cost of replacing a window, both in
money and in lost time, has made us cautious.
We have never attempted
to discover how high a temperature is required to break the window or
to
open a leakin the brazed join.
Evidently
time.
operating the detector causes the gas to degrade over
A gain curve taken after months of running will reveal the
deterioration.
Distilling new gas and refilling the detector suffices
to rejuvenate it.
A small valved bottle is attached to the detector
for short-term in situ distillation to remove the polymers, but a full
replacement is eventually needed because the detection efficiency
depends (for given voltage) upon the gas pressure.
In situ
distillation gradually reduces the gas pressure.
A single wire along the axis of the ion chamber gave a gain curve
with no clear unity-gain plateau from which to calibrate the gain and
efficacy of detection.
This anode was replaced by a bent one shaped
like a wire coat-hanger so that the collecting part of the anode ran
parallel to thewindow (Figure 4.9).
This design
gave a flat plateau.
At voltages less than the plateau region the electrons tend to
recombine with the nitric oxide ions and so are not collected.
As the
voltage is increased, the electrons gain enough energy between
collisions to avoid recombination.
Above the plateau the electrons
gain enough energy en route to the anode to ionize additional nitric
oxide molecules.
The electrons liberated by these ionizing collisions
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(a )
/ 7 M \ \
(b)
Figure 4.8.
Ion chamber electrodes: (a) axial wire;
(b) coathanger anode with collecting wire
parallel to window.
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are then collected also, so that the ion chamber exhibits gain.
Since
the electric field is strongest near the wire most of the gain comes
from collisions occuring near the wire.
The efficiency of the detector was not measured, but an estimate
can be made based on the analysis given here.
The detector was built
with two 2 inch diameter MgF2 windows 2 mm thick, separated by 0.75
inches.
Two meshes, each 90% open, lay between the quenched beam and
the nitric oxide.
The nitric oxide has a reported (WA54, ST63)
quantum efficiency of 0.81.
The transmission factor for two 2 inch
windows 3/8 inch from the beam is 0.53 (using Figure 4.5).
Multiplying these factors, one estimates an overall efficiency of
35%.
Under the conditions of the parity experiment, namely very small
metastable current, it was necessary to run the detector with a gain
of 10 so that the noise on the beam signal was equal to or greater
than the electronic noise at the input to the current-sensitive
pre-amplifier used to measure the beam current.
In addition to the
large solid angle subtended by the detector and the correspondingly
high overall efficiency, the detector was found to operate well for
extremely long periods (of order a year) without the need to replace
the nitric oxide.
This is presumably a consequence of the large
volume of gas in the ionization chambers.
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V. STATE SELECTION
To observe the hyperfine transition from F=0 to F=1 in the 2s
level, it is necessary to create a population difference.
State
selectors are placed upstream and downstream of the regions driving
the transition.
The upstream state selector is designed to pass the
F=0 state and the downstream one transmits the F=1 states.
called the 0-selector and the 1-selector.
These are
When the hyperfine
transition is not driven the 2s atoms should be absent from the
detector.
This ideal is not realized in practice of course, but can
be well approximated.
Specifically, the selectors taken together can
be made 31% transparent to hyperfine transitions with a fractional
leakage of 2s atoms of 3 x 10“5 when no transition is driven.
State selection is accomplished by quenching the hyperfine
components to the ground state (invisible to the detector) by way of
the nearby Ivx/i level which decays rapidly.
There are three
resonant frequencies in these Lamb-shift transitions corresponding to
the FM=00, 10, and 1±1 states of 2si/2*
The short p-state lifetime
makes these quenching resonances very broad (FWHM = 100 MHz) compared
to the hyperfine splittings (Af(2s) = 177 MHz, Af(2p)= 59 MHz).
The
resonances overlap so that it is impossible to quench one component
without also quenching the others to some extent.
One can make the
ratio of one component to another arbitrarily high but only at the
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expense of transmitted beam intensity.
The quenching of a long lived 2s state (natural lifetime (1/7)
sec) has been treated by Lamb and Retherford using the Bethe-Lamb
prescription in which decay rates are added to the Hamiltonian as
imaginary energies (LA 50a).
Appendix A presents the calculation
for two states in the rotating wave approximation.
When states |s>
and jp> are coupled by an electric field e = eQ cos ait, an atom
initially in state |s> decays.
This decay is approximately
exponential with rate r given by
r ■ »
where X = ui - (E “ E^» ^ = \
2ir x 50 MHz.
.[4 2
|e
an<^ ^ =
This result is a good approximation for |v |2 «
Y2»
i.e., at low power in the oscillating field.
Figure 5.1 shows the n = 2, J =
magnetic field.
states of hydrogen in zero
Here the appropriate quantum numbers are the orbital
angular momentum, L, the total angular momentum, F, and M its
projection along the field
sq.
The three quenching resonances are
illustrated by Lorentzian line shapes in the same figure.
decay rates
r_
r i?
s p
The induced
for a uniform oscillating field at frequency
id
be written as
01
,
1 12 . 2
(io-<D01) + Y
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can
F
TRANSITION
I
2
3
4x)0|
1/2
Figure 5.1.
(O.O) — (I. O)
(I,± 1)— (l,± I)
( 1,0 )— (0 .0 )
1
1/2
<*>|| U »,0
Allowed quenching transitions between states of 2s
and 2pj^ 2 ^or an oscillating electric field in the
z-direction.
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CO
where
idoi/27T
= 909.6 MHz, u)n/2ir = 1087.1 MHz, and aJio/2ir = 1146.3
MHz.
These results from the two-state atom are valid because the
electric field couples each s-state to exactly one p-state.
reference, the explicit decoupling of the states
For
|LFM> into product
states |LML> x jmsmi> is presented in Table 5.1, and the
electric dipole matrix elements are presented in Table 5.2.
The
Clebsch-Gordan coefficients were taken from the Particle Properties
Data Booklet (PP 84)which are in the phase convention of A..R.
Edmonds.
Because the quenching fields are in the z-direction, only
the eg elements are involved, the z-
being identically zero.
Since the four s-states are quenched independently, and since
they are formed incoherently the beam may be represented by the
diagonal elements of the density operator for the 2 s level, formed
into a vector R.
Initially R is given by
ordering is FM =* (li,10,1-1,00).
(1,1,1,1) where the
After passing through the
0-selector, for instance, R is given by
R = -i- (e“rilt, e~ri0t, e~rllt, e“r01C).
In this expression the T's
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TABLE 5.1:
EXPLICITLY DECOUPLED STATES OF 2ZS
AND 22Pjy2 LEVELS
|LFM> = l± A ± (Ln^ > |m gmI>
|sOO> = | s O > ^ —
sl-l> -
sO>
[ |++> - |++>]
++>
|slO> = j s O > ^ —
[ |++> + |++> ]
|sll> = |sO> |++>
jpOO> =
jp+> |++> /3
--—
S T
pl-l> =
^
|pio> = - —
I
/3
+ -—
/5
|pll> =
^
/6
|pO> j++>
|pO> U + > + ---- |p-> |++>
1
1
/3
1
|pO>|++>-/273 jp->|++>
|p+> U+> - - —
1
1
/6
|pO> U + >
1
1
Ip0> U+>
1
1
- - -- |p-> |++>
/3 ---- 1
|p+> |++> - :^= z
1
|pO> |++>
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TABLE 5.2:
DIPOLE MATRIX ELEMENTS
r • e
✓ 3 ear
sOO>
sl-l>
slO>
sll>
+e+
<Poo|
+s_
+eo
<pl-l|
-e
-e+
<P10|
+e_
-e+
+€_
<pll|
TABLE 5.3:
,( 0)
+eo
EXTINCTION EXPONENTS
(</0) - ("oi^2 + y2
= 13.2
a =
a)n )2 + Y 2
, (0) -
,( 0)
(u)
10
b =
,( 0)
*01
_
= 14.3
.2
,CD
a
■ r(D
10
u)(0 )/2tt = 1119 MHz
j. 2
+ Y
u>l0 )
ii
r( D
- -5L-
2 + y2
^
U)10)
r (1)
10
a>m
.
2
+ Y
.2 . 2
(u>(1) - «o11) + Y
( J 1)
-
.2 . 2
“ 10) + Y
, (1 )
.2
2
(u)
- a>01) + Y
=
1.66
= 23.95
to^1 ^/2ir * 892 MHz
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are chose corresponding to the operating frequency and power of the
O-selector.
The same expression gives the beam condition after
passing through the 1-selector (with the O-selector off) if the r's
are modified to its frequency and power.
Figure 5.2 shows the
trasmitted beam fraction for the 0—selector and 1—selector operated
singly as a function of cavity power.
It is useful to note that for a given state selector the ratio of
any two T's is independent of power, since each r is proportional to
the square of the electric field.
the transparency T q = e
*“*r
01
power then we can write R =
exponents.
t
If for the O-selector
we define
where Tqi is proportional to the
1
3
b
3
(TQ, TQ, TQ, TQ ) with power-independent
For the 1-selector we choose the FM=10 state to define
•F
the transparency Tj= e
represented by
10
r
so that in single operation the beam is
(T^, T^, T^, T^).
The leakage of metastable atoms
in combined operation, without hyperfine transitions, is then given by
L=i
(2T0 T1 + T0 T1 + V l
)
and the overall transparency to |00>-»-|lO> transitions is defined to be
T
T0T1.
(5.2)
In choosing design parameters one wants a large transparency and a
small leakage.
The definitions of the extinction exponents a, b, c,
and d and their values for our system are given in Table 5.3.
In our
standard operation T0 = 0.44 and Tj = 0.7. which give T = 0.31 and
L = 3 x 10~5 .
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i.o<
z\ (b)
(a) 0 - S e l e c t o r Transmission
I - Selector Transmission
0 .5
P, = 0 .7 0 m W
PQ = 2.0mW
Transmission
I/I
o
-i
,0 5
-t
0.0
2.0
4.0
6.0
8.0
10X>
12.0
0 - S e l e c t o r Loop Power (mW)
Figure 5.2.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
I-Selector Loop Power (mW)
State selector transmissions as a function of
microwave power detected in a pick-up loop. The
intercepts of the lines extrapolated from high power
are 0.25 and 0.75 showing that the four states in
the levels with F=0 and F=1 are equally populated.
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This parametrization has proved useful because it refers everything
to the measured beam and does not rely on power calibrations, which
changed as microwave components were swapped around.
Since L =L(Tg,Tx),
we may define the two operational parameters Lg = L(Tg, 1) and
=
LCljTj) which are the fractions of beam transmitted by the O-selector
and the 1-selector when each is operated alone.
Figure 5.3 is a contour
plot showing lines of constant leakage and transparency in the Lg-Lx
plane (Figure 5.3).
The figure shows that a leakage as small as 10“6
can be had with a transparency of 0.2, at least in principle.
Unfortunately, this is not realized.
As will be shown in the next chapter, the 2s beam is repopulated
in flight between state selectors so that the best operating conditions
are those in our standard operation, namely T = 0.31, L = 3 x 10“5 .
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- 83
h b ite d vJ''-’noU^ Pett0 'SS' ° n
vepto^uC^on P
Fu^ eTfep
VI.
BEAM AND BACKGROUND
The hydrogen beam contains roughly equal amounts of ground state
and metastable atoms, along with a small number of atoms in the higher
lying excited states.
These atoms travel from the cesium canal, where
they are formed, to the dectector, which detects ultraviolet light.
On the way from one end of the beam machine to the other the atoms
pass through holes in various collimators, microwave cavities, and
transmission lines, which provide surfaces from which the atoms might
scatter.
Since the kinetic energy of the atoms is 500 eV and the
first excited state of hydrogen lies 10.2 electron volts above the
ground state, there Is plenty of energy for such collisions to result
in excitation.
This is one source of background.
In addition to the
scattering of atoms, the various apertures can also scatter light
produced by the discharge in the duoplasmatron or radiated from the
state selectors when they quench the unwanted atomic states.
These
sources of background can be controlled by careful choice of
collimating apertures along the beamline and do not, therefore,
constitute the main limitation on our aparatus.
A third source of
background, and the most difficult to treat, is the repopulation of
the metastable beam by cascade decays from highly excited states.
This appears (by a process of elimination) to be the dominant source
of background signal.
The relevant obstructions near the beam path are depicted in
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Figure 6.1, which is an asymmetrical dilatation of the beamline:
the
transverse scale is expanded relative to the scale in the direction of
travel.
The trajectories of atoms in the beam are still straight
lines in such a distended representation.
The cooled apertures are
circular knife edges one centimeter in diameter.
All of the apertures
which follow are elongated slots made by a three eighths inch end mill
moved three quarters of an inch (this has been our "standard aperture”
with an area of 2.5 cm ).
While the circular, cooled apertures are
bevelled to produce knife edge collimators, the apertures in the
microwave cavities and transmission lines are not themselves bevelled,
but are screened by additional knife edges so that the flat walls of
the holes lie in the shadows of the sharp apertures.
To choose the
sizes of these screening apertures the hydrogen beam was taken to
originate at the downstream end of the cesium canal, which provides a
worst case for an extended source.
These screening apertures are marked in Figure 6.1 by the letters
A through D.
The aperture at C had to be made small in order to
screen the three apertures of the El region all at once (the aim was
to avoid having atoms travel close to walls and surfaces in the region
where the parity non-conserving amplitude is generated).
The last knife edge the beam passes is moveable.
Actually it is
made from two bevelled plates, each mounted on a linear motion
feedthrough.
Figure 6.2 shows these moveable knife edges.
The
diagonal apertures cut in the plates can be used to sample a small
portion of the beam when the two plates overlap, as-shown in the
figure.
Only one set of knife edges— the vertical ones or the
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
STANDARD
APERTURE
3/4".
COOLED
APERTURES
CESIUM
CANAL NJ
AB
MOVABLE
SLIT
T
Icm
Figure 6.1.
1
—
Im --- ^
Apertures and obstructions in the beam line.
1
><\VVvV^
Figure 6.2.
XVAWWATO3
Moveable knife edges. A vertical slit can be formed
by placing the vertical edges close together, or the
diagonal apertures can be overlapped to produce a
small square aperture.
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diagonal ones— intersects the atomic beam at a given time.
Since
these moveable apertures are on micrometer drives it is possible to
map the beam intensity (and all other signals derivable from the
detected beam) as a function of the position of the small square
formed by intersecting the diagonal apertures, or as a function of the
position of a slit formed by placing the vertical edges close to one
another.
The latter method is the less tedious.
The background is
here defined as the detected current when the maximum available
microwave power was admitted to the state selectors— in each case
several times that power used for measuring transition signals.
Figure 6.3, shows histograms of beam intensity and background as
a fifty mil slit was moved across the beam.
The background from the
edges of the beam is much larger than that from the middle.
This
large background at the extremities is attributed to the atoms
colliding with the walls of the rectangular quench tube in the
detector.
Whenever the beamline was realigned it was necessary to
find the edge of the beam which produced no such background.
If
there were no sources of background other than metastable atoms that
survive the state selectors these would produce a current less than
10
of the full beam when the state selectors are run at maximum
power.
What we actually find is a much greater
background signal (1
x 10-** of the full beam, or about 30 pA).
Part of the background is due to ultraviolet light from the
duoplasmatron.
It is conceivable that light from the state selectors
can also cause a background, but modulating the power in the
1-selector, which is closer to the detector, did not reveal any
contribution to the background from this source.
-
88
We did find that
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INTENSITY, nA
BEAM
BACKGROUND,pA
2.0
20
1.5
1.6
1.7
1.9
2.0
S L IT P O S IT IO N , IN C H E S
Figure 6.3.
Beam intensity and background for a 50 mil (0.050 inch) slit.
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blackening the last flange with soot from a candle greatly reduced the
background from source light— that is, the current detected when the
acceleration voltage is reduced to zero so that no atomic beam travels
down the pipe.
In the end the contribution to the background from
■ scattered source light was reduced to about three picoamps.
This is
not really very small, but It is less than the dominant contribution,
so it was deemed unprofitable to put more soot in the vacuum system.
The largest contribution to the background comes from metastable
atoms formed as the beam travels.
for this metastable production:
Three mechanisms suggest themselves
scattering from walls and apertures
(discussed above), excitation by collisions with residual gas in the
vacuum system, and cascading from the highly excited states produced
in the source.
The residual gas pressure in the apparatus downstream
O
from the source Is everywhere less than 10
Torr, and close to the
pumps it is less than 10”9 Torr, even when the atomic beam is
contributing to the load on the pumps.
By partly closing the
gatevalve over the ion pump servicing the detector and the moveable
slit we increased the background associated with residual gas (Figure
6.4).
The length of the beam path affected by this increased pressure
is approximately 80 centimeters (between the 1-selector and the
detector).
These data can be used to estimate the background due to
excitation by collisions with residual gas:
If the pressure In the
300 cm of beamline between the 0-selector and the detector Is taken to
Q
C
be 10” Torr then one expects an excitation background of 2.7 x 10“
of the full metastable current; this assumes that the numbers of
metastable atoms and of ground state atoms occur with a fixed ratio
every time the machine is run.
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The intercept in Figure 6.4 is the amount of metastable beam
produced between the 1-selector and the detector in complete vacuum,
assuming that the 1-selector extinguishes the 2s population.
When
Figure 6.4 was produced the full metastable current was 110 nA, so
this residual repopulation of 16.5 pA represents 5.9 x 10 “5 of the
full beam.
If we take this to mean that the metastable beam is being
repopulated at a constant rate (5.9 x 10~5 per 80 cm of flight) we
would expect that between the state selectors, a distance of 115 cm,
the beam will regain a metastable fraction of 8.1 x 10"5 from the same
mechanism.
Adding to this a contribution from scattering with
p
residual gas (taken at 10
Torr) we expect a repopulation of
9.4 x 10”5 of the full beam.
This estimate is based entirely on
mesurements of what happens after the state selectors when they are
run at such high power that we are persuaded the metastable beam is
eliminated at the 1-selector.
It is to be compared with the following
analysis of how the detected current depends on state selector
parameters.
By setting the power in the 0-selector (upstream) and varying the
power in the 1-selector, a slice through the contour plot of Figure
5.3 (theoretical Leakage and Transparency) is produced.
Figure 6.5 is
such a slice, with the 0-selector set to transmit 0.11 of the full
beam (Lq = 0.11).
The open circles are the measured points with the
background subtracted.
Figure 5.3.
The' lower curve is the theory corresponding to
The background is here defined to be the detected current
when full power is in the state selectors, and is attributed to
scattered light.
Assuming that between the state selectors a fraction
f
of the
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METASTABLE (QUENCHABLE)BACKGROUND,pA
30
25
20
50
100
150
200
P R E S S U R E , n T o rr
Figure 6.4.
Metastable background versus pressure for an 80 cm
length of beam. The full metastable beam intensity
was 150 nA when these data were taken.
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_
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250
Ld
O
<
'X.
<
H
- i
-4
10
_i
L . I-SELEC TO R TRANSM ISSION
Figure 6. i.
Measured leakage
f ^ r a S ^ l
“ “ e r ^ u ^ e ' l s ^ h a ^ »Ith
srare aeleerors.
1:
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full beam is reconstituted, the leakage formula (Equation 5.1) must be
modified to
L = J- ( 2(Tq + f)Tj + (t £ + f)Tx + (Tq + f)Tj).
(6.1)
The upper curve in Figure 6.5 (which more nearly fits the data) is
drawn from Eq. 6.1 with f =* 1 x 10”1*.
This is consistent with the
intercept of Figure 6.4.
One hypothesis for how this repopulation occurs is that even
though the beam has been collimated by sharpened apertures it might
still be scattering from the knife edges.
If this were the cause of
the repopulation one would expect that as the moveable knife edges are
brought together the fraction of background produced by these edges
should increase as the area is reduced.
This is not the case.
Once
the portion of the beam which scatters in the quench tube has been cut
off, the ratio of background to full beam remains constant.
It is
therefore unlikely that the collimating apertures are the source of
this repopulation.
Why is the metastable beam repopulated during its flight?
We
have accounted for the scattering by the residual gas which can at
most produce 3.4 x 10“5 of the full beam.
We have shown that the
moveable aperture, at least, does not produce the repopulation;
thus seems unlikely that the other apertures have caused the
repopulation.
What remains in the list of possibilities is
repopulation from the decay of highly excited atoms.
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It
In this experiment it is desirable to keep the transition
probability small for the following reasons.
The detected current is
proportional to B + |AM1±Ag j2 where B is the background, A ^ is, as
before, the transition amplitude generated in the Ml-region, and A£
is the parity-forbidden amplitude generated in the El-region.
The
relative sign between the two amplitudes is modulated by reversing the
phase of the VHP field in the El-region.
The interference signal
detected by demodulating the detected current is then proportional to
2AM ]Ae . As the parity-forbidden amplitude A£ is expected to be
very small, the fractional change in the signal is very nearly
AS
2 ^Sll^
s
B + V *
'
This fraction is maximized when A ^ = B, in which case the optimal
"asymmetry" is
T
= Ae/fAMl*
The smaller the background, the larger this asymmetry can be made by
reducing the magnetic dipole amplitude, A^i*
Systematic
imperfections must be made smaller than this if the parity-forbidden
amplitude is to be seen (or, if they are not smaller, they limit the.
sensitivity of the experiment).
The smaller this optimal A^i can be
made, the less the demand on the technolgy becomes.
A similar
conclusion follows from the examination of the signal-to-noise ratio
(see HI 80).
The noise measured on the metastable beam depends in a strange
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way on the intensity*
A reasonable expectation is that the beam
should carry the usual Gaussian noise associated with Poisson
statistics, and, if the plasma in the source is unstable or
oscillates, there should be a part which is linear in the intensity if
the intensity is changed by methods not affecting the source.
fractional noise should then vary as (constant + 1/I1^2 ).
The
Figure
6.6 shows the fractional noise as a function of intensity in a
semi-logarithmic plot.
The noise was measured using the
computer-controlled data collection apparatus to be described in
Chapter IX, with no VHP power supplied to the transition regions (that
is, by performing non-experiments).
The beam intensity was reduced by
running the state selectors and by blocking the beam with moveable
beam stops.
It is seen in Figure 6.6 that at low intensity the noise
approaches the Gaussian limit, but that at higher intensity it rises
faster than one expects (in the figure, the fractional noise ought to
approach a constant value at high intensity).
Inasmuch as one cannot improve upon the Gaussian limit, the
experiments were done with the beam state selected to a background
current less than 10 pA.
It was then necessary to increase the
detector gain from unity to 10 in order to overcome the electronic
noise of the current preamplifier which collects the signal from the
ionization chambers.
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CM
FRACTIONAL
NOISE, Hz
I
5
10
20
50
100
B E A M C U R R E N T, pA
Figure 6.6.
Measured noise (dots) versus beam intensity. The
Gaussian noise line is the noise expected from Poisson
statistics.
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VII.
A.
TransiCion Regions
Cartesian States and Vector Amplitudes
This chapter describes the two transmission lines, the Ml-region
and the El-region, which carry the oscillating fields that drive the
F=0 to F=1 transition, along with the electronics which manipulate the
VHF fields applied to the atoms.
The atomic response to these fields
is described more conveniently if the F=1 level is resolved into a
Cartesian basis rather than the more customary basis of angular
momentum eigenstates, |FM>.
These Cartesian states are defined to be
| x> = / r
( | n > - l 1”1^
|y> = /-p
(|n > + |1-1>)
z> =
o> =
F=1
(7.1)
|l0>
|00>
F=0
where the last line gives a shortened notation for the F=0 state.
This resolution can be done for both the 2S±/2 level and the 2P±/2
level; when necessary they will be distinguished as jsx> and jpx>, and
so on.
What makes this basis useful is that it transforms as you
would expect under rotations.
The x, y, and z states rotate into one
another, the state |o> transforms into itself.
The transformation under rotations is particularly easy to see
for a rotation through an angle <J> about the axis of quantization
(z-axis):
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Each of the states x, y, and z is an eigenstate of the corresponding
component of total angular momentum f, with eigenvalue zero.
That is,
Fx|x> = °*
Fy|y> - o,
and
( 7.3 )
Fz|Z> = O'
Because of this the state |x> is not affected by a rotation around the
x-axis, and similarly for y and z.
After traversing a region of oscillating field the atom,
initially in the state |so> will emerge in a linear combination of the
four states so, sx, sy, and sz.
The portion still in the initial
state can be Ignored because it will ultimately be quenched away, by
the 1-selector.
We can therefore write the final state (F=l) as
|f> = Ax |x> + Ay |y> + Az |z>.
(7.4)
This is summarized by writing the coefficients as a single vector
transition amplitude
X=
(A^, Ay , Az).
as a vector under rotations.
The amplitude
X
transforms
In general the components A^, Ay,
and Az are complex numbers, not necessarily having the same phase.
An arbitrary vector of complex numbers can be decomposed into the form
X=
e
(v + iw)
with v.w = 0, where v and w are real 3-vectors.
(7.5)
Thus, except for an
overall phase, a complex vector can be regarded as comprising a main
vector (v) and a "tail” (w) which is out of phase (by ir/2) and at
right angles.
One could equally well decompose this as
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where the first term in square brackets gives a linear polarization in
the direction of v and the second term in square brackets represents a
circular polarization in the vw-plane.
The decomposition of Equation
7.5 is more natural for this experiment.
To put an arbitrary complex
vector in the form of Equation 7.5, we write
■jf
A =
ia .
(A^e ,
£ =
16
. iyx
,
, A^e ) and construct
Re(e10 X),
then maximize
(7.6)
with respect to 0.
2
Setting -Is- ( ^ ) = 0
gives
2
Aj cos (0 + a) sin (0 + a) + A 2 cos (0 + 0) sin (0 + 8)
+ A^ cos (0 + y) sin (0+ y) = 0
(7.7)
This can be manipulated to give
A? sin 2a + A? sin 26 + A? sin 2y
tan 20 = ------------- *-=--------- ---------2
2
(7.8).
2
Ajcos 2a + A 2 cos 26 + A^ cos 2y
Equation
(7.8)has two solutions,
giving a minimum.
one giving a
maximum for
and one
Choosing the maximal solution, we set
* = ^ (9max) = Re (ei9 X)
(7*9)
and we let
w = Im (e10 t)
(7.10)
1 = e"i9(v + iw).
(7.11)
so that
The orthogonality condition v • w
= 0 is justequation(7.7) which
the defining equation for 0.
This decomposition into a maximal vector and a (minimal) tail
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100
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is
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and unavailable from author or university.
Filmed as received.
101-102
University Microfilms International
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I
c
LOJ
0 .5 -
1.0
-0 .5
0 .0
0 .5
z/a
Figure 7.
Magnetic field lines in Ml-region parallel plate
transmission line and field strength on axis.
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In the spherical tensor basis the dot product y • £ is given by
y • § = yQ bQ - u+ b_ - y_ b+ ;
(7.16)
or in our case, with the field given by Equation 7.12,
♦
bx
-y • d ** ---
on the interval ("T^, Tj).
cos ait (y+ - y-)
(7.17)
The operators y+ act like raising and
lowering operators on Mp (this is the substance of the Wigner-Eckart
Theorem for rank-1 operators).
That means y+ connects |FM> = |00>
with j11> and y_ connects |00> with |l-l>.
aij^joo
= a1Ellt
cos
ost
We have, then
e'^OO*
(7.18)
and
<l-l|HI |00> = e^l-l*
cos ait e ^ O O *
(7.19)
In evaluating the integral for U(t) we need only retain the term
1/2 e *alt: from the cosine (this is the rotating wave approximation).
This is because this term gives an exponent -i^-CE^-Ego)) which is
small if ai-E11-E0 0
that is, near resonance.
With this approximation we find
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V T 1 sin v+T,
|final F=1 state> = U(t)|00> - ^ --- (— ^
) |ll>
b uTi sin v_Tx
x * f
) h-i>
v_Ti
J I
rr
(7.20)
where
v+ =uj- (Ej^-Egg)
and
V_
=0)-
( E j ^ - E g g )
(7 .2 1 )
Now in zero (static) magnetic field the states
and 11—1> are de­
generate so that v+ = v_ and
I
V
T 1
,s i n v ± T l
,
"<‘>!°°>
M
,
[ | H > - |1-1>]
sinv^Tj
v±Ti ) |x>-
= -bxpTi (
(7.22)
(See Equation 7.1). In zero magnetic field the oscillating field
the x-direction produces a pure |x> final state.
in
(An oscillating
field in the y- or z-directions produce pure |y> or |z> states, too.)
If, on the other hand, there is a non-zero static magnetic field
5 = zB in the Ml-region, then the final state is given by equation
7.20.
The frequencies v_j. and v_ are no longer equal because the
magnetic
fieldlifts the degeneracy.
b uT,
In the Cartesian basis
sin V4.T,
u ( c ) |o o >
sin v_T,
*
ib pTj
+H h
sin v+Tj
105
) |*>
sin v_Tj
-----------^
(—
-
- -vlT)
7—
) ly>
<7-23>
-
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The final state thus has the form
Ax |x> + iAy |y>
(7.24)
which cannot, in general, be rotated into a pure vector.
That is, the
transition amplitude has a main vector part and an out of phase tail.
There are two points to take from all this.
First, that the
Cartesian basis is really the natural basis for this experiment, since
the static magnetic field is everywhere very close to zero.
As a
result, transitions from the F=0 state to the F=1 states can be
characterized by vector transition amplitudes.
The vector amplitude
for the magnetic dipole transition has the same spatial direction as
the oscillating magnetic field driving it, if the static magnetic field
is actually zero.
The second point is that the purity of the state
produced by the Ml-region is affected by imposing a small magnetic
field.
The magnetic field lines in Figure 7.1 curve around the plates
of the transmission line so that atoms travelling off-axis, but
parallel to it, experience in effect a field driving the hyperfine
transition whose direction changes; this produces a transition
amplitude with a tail which can be removed by applying a small magnetic
field to the Ml-region.
The technique for purifying the Ml-region*s
final state will be covered in the next chapter.
It should be clear from Figure 7.1 that the oscillating magnetic
field is not, in fact, uniform.
The field distribution in Figure 7.1
was obtained by numerically integrating Laplace's equation in two
dimensions for the given boundaries.
The cylindrical wall is held at
ground potential for this calculation while the two plates are given
equal and opposite voltages.
line.
This is the antisymmetric TEM mode of the
The topology of the transmission line also admits a symmetric
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mode corresponding to equal voltages on the plates with the wall again
at ground.
These two modes form the basis for all possible TEM modes
of the line.
To constrain the transmission line to operate in the
antisymmetric mode a balun impedance transformer was assembled at each
end.
These baluns ensure that the VHF voltages supplied to the plates
have oppsite signs. The baluns also match the device into fifty ohm
cables at the input and output ends.
Figure 7.2 gives the dimensions
of the Ml-region.
Since the oscillating field is not really uniform the transition
amplitude is not quite of the form (sin x)/x used in Chapter II to
facilitate discussion.
The field is still weak so that first-order
perturbation theory gives an accurate representation
lineshape.
of the
In this approximation, as we have seen, the transition
amplitude is the Fourier transform of the field envelope of Figure
7.1.
Figure 7.3 shows both the computed transition probability for
atoms travelling along the axis and the measured transition
probability for the entire beam.
One feature to note is the full
width at half-maximum, which is 6.9 megahertz.
(For details of the
numerical integration consult Appendix B.)
C. El-Region Transmission Line
The heart of this experiment is the El-region, where an
oscillating electric field is applied to the metastable hydrogen atoms
to drive the 2s hyperfine transition.
Ideally one would like to
create a region to be traversed by the atomic beam in which there is a
uniform oscillating electric field with no other electric or magnetic
fields present.
In that case the only possible interaction the atoms
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C I"
I11
*«
3#
Figure 7.2.
If
Ml-region transmission line.
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m 120
-J 100
80
FWHM
6.9 MHz
60
40-
20
FREQUENCY, MHz
Figure 7.3.
Ml-region transition probability (circles) and theoretical linesliape for
atoms moving on axis.
can have is with the oscillating electric field, so that a transition
from the F=0 state to an F=1 state would have to come about through a
coupling of the states by the electric field.
look for an electric dipole coupling.
In our experiment we
This creates a vector transition
amplitude whose spatial direction is along the electric field.
The
difficulty comes in trying to realize a region where there is only an
oscillating electric field.
An oscillating electric field is always
accompanied by an oscillating magnetic field (possibly in some other
location).
The earth produces a magnetic field, as do the magnets in
ion pumps, while stray electric fields due to contact potentials,
charging of dielectric surfaces, ions in the background gas and so on
are invariably present.
These additional fields complicate the problem
of isolating the electric dipole transition amplitude— that is, the
parity violating signal.
The El-region is a cylindrical "coaxial" transmission line whose
center conductor is a flat vertical plate (see Figure 7.4).
enveloped in two concentric magnetic shields.
outside of the shields, is an ion pump.
It is
On each end of the line
Although an Infinite line of
this cross-section would have a 50 J2 characteristic impedance it is
still necessary to match into it.
Disks at each end of the center
conductor add the necessary capacitance to effect matching.
The atomic
beam travels perpendicularly through the transmission line.
In the
region where the atomic beam passes there are fine wires above and
below the beam path to provide a vertical static electric field (Figure
7.5).
The "coaxial" cross section of the transmission line supports a
single TEM mode.
(Figure 7.5) The electric field lies in
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El REGION
Outer Magnetic
Shield
Impedance
Matching Disk
N-Type
Feedthrough
Center
Conductor
Atomic
Beam
To Ion Pump
End
Range
To Ion Pump
Figure 7.4.
El-region transmission line.
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V/\RE.S
2.0 -
1.0 -
0 .5
1.0
z/a
Figure 7.5.
Magnetic field lines for travelling wave in El-region.
Field strength on beam axis. The wires shown in the
drawing provide a vertical static electric field.
-
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the direction of travel, while the magnetic field is transverse to the
atomic velocity.
In designing this region one of the chief aims was
to obtain an oscillating electric field in the beam direction.
The
Ml-region produces an oscillating magnetic field in the beam direction
and since this is to be interfered with the forbidden electric dipole
transition the fields must, in effect, be made to lie in the same
direction.
It has turned out that this can be accomplished by
precessing the Ml transition amplitude in magnetic fields, as
discussed in Chapter VIII, but even so we had originally hoped to do
the minimum amount of manipulation in picking out the Ml-El
interference.
A consequence of having an electric field in the
direction of the atomic velocity is that the field distribution is
anti-symmetric with respect to the center conductor.
The experiment
must be done at a frequency slightly away from resonance, as
on-resonance any amplitude gained on one side of the center conductor
is lost on the other.
Since it is in the El-region that the parity violating amplitude
is to be generated, and since the direction of the amplitude is one of
the discriminants to be used in picking it out, it is important to
know the VHP field distribution.
That is, we want the
electromagnetic field to be in the TEM mode as nearly as possible, at
least where the atoms are.
In changing from an N-type coaxial cable
to a line with a 12 centimeter radius there is necessarily a certain
amount of confusion at the junction, but since the hyperfine frequency
(177 MHz) is far below the cut-off frequency (~ 625 MHz) for the TM
and TE modes of the line these will eventually decay away (JA 75
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Chapter 8).
inches).
For this reason we made the El-region quite long (85
Approximating the left half, say, of the El-region by a
rectangular hollow waveguide with the same cross sectional area and
one side equal to the radius gives an attenuation length for the
lowest TE mode of 3 centimeters.
The El-region is eighty-five inches
long so that from the center to either end is about thirty attenuation
lengths for the fields (e“30=lO“13).
As mentioned in Chapter II the experiment is done by counterpropagating waves in the El-region.
Because of this we were careful to
make the line as symmetrical as possible.
In particular the matching
at each end was effected simultaneously by maintaining the dimensions
of the conductor and of the matching disks the same.
By plotting the
reflected power versus disk radius and choosing the minimum the
reflected power was reduced to -30 dB of the input power.
The
bandwidth of the matching was found to be much wider (« 40 MHz) than
the range of frequencies of interest (that is, the response was flat
over that range), so that when matched we had a transmission line with
little standing wave.
This matching is 10 dB better than we were able
to obtain on any other component in the circuit.
The El-region is encased in two cylindrical magnetic shields.
The outer one is 60 inches long, 17.8 inches in diameter and a tenth
of an inch thick, while the inner shield is 30 inches long, 12 inches
in diameter and a tenth of an inch thick.
Using the design formulae
from Sumner (SU79) with a dynamic permeability of 25,000 one finds
that the outer shield has a dynamic shielding factor for axial fields
of 40, for transverse fields of about 100 (taking into account that
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the cylinder is made from two halves clamped together along the seam,
which causes a thirty percent reduction in the shielding factor).
The
inner shield has dynamic shielding factors of 65 and 140 for axial and
transverse magnetic fields.
The shielding factors for the two layers
taken together are 1400 (axial) and 8000 (transverse).
Since the
static permeability is twenty times as great as the dynamic
permeability the static shielding factors for the two shields taken
together are 400 times as large as the dynamic shielding factors.
The
magnetic noise measured in our laboratory corresponds to field changes
of order ten milligauss.
If this is an axial field, seven microgauss
is the expected variation of the field inside the two shields.
When
the shields have been demagnetized, we estimate that the magnetic
field inside the El-region is of order seven microgauss.
The demagnetization is carried out by saturating the shields,
then reversing the magnetic field (the current in the demagnetizing
coils) and reducing it by a fixed factor.
This reduced reversal is
repeated some forty times with a reduction factor of 0.9.
The
saturating field corresponds to a current in our coils of twenty
amperes or more.
With axial and transverse Hall probes we were unable
after demagnetization to detect any magnetic field between the shields
at the level of 100 microgauss.
The field inside the inner shield is
presumably less than 1/130th of this, i.e., less than the noise level
of seven microgauss expected day-to-day.
On the outside of the E1-transmission line (and inside the inner
magnetic shield) we have fashioned a pair of Helmholz coils to provide
an axial magnetic field (that is, along the axis of the transmission
line).
As will be seen in the next chapter, this field is useful for
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determining the direction of the El-region axis.
inside the transmission line.
We added wire planes
The wires were gold-plated tungsten,
thirty microns in diameter, supported on glass rods set six inches
away from the beam path.
The wire planes were placed three
centimeters above and three centimeters below the centerline of the
atomic beam.
The voltages biasing the wire planes were fed in through
tiny semi-rigid coaxial cables (1 mm in diameter) whose axes were
parallel to the axis of the transmission line.
This design minimized
the perturbation of the TEM mode.
D.
Transition Amplitudes in the El-Region
As the atoms pass through the El-region, transitions are driven
from the initial state, |so>, to the final states |sx>,
The transitions are characterized by some complex vector amplitude
X.
As mentioned earlier, it is possible to purify the transition
amplitude generated by the Ml-region so that it has no tail.
It is
possible to rotate this Ml-region amplitude so that it points in any
direction, by means of the precessors described in the next chapter.
By varying the phase of the VHF which drives the Ml-region one can
vary the phase of the complex transition amplitude.
With these
capabilities, the Ml-amplitude can be used to study the transition
amplitude,
generated in the El-region.
The transition probability for all transitions to the F=1 level
can be written
(7.25)
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in which 5
is the Ml-region transition amplitude, with 3 a real
three-vector.
(The meaning of the absolute square is that the vectors
are dotted into their complex conjugate vectors.)
By changing the
phase of the Ml-region amplitude from <J> to $ + ir and subtracting the
transition probabilities one obtains
P(<f>) - P(<j> + ir) = S = 4 Re (e"i4> 3 • X).
(7.26)
Clearly, by choosing the direction of 3 we can pick out any given com­
ponent of
X, and
by varying <|>, we can explore both the real and
imaginary parts of that component.
The essential features of the
experiment can thereby be reduced to a discussion of the amplitude
X generated in the El-region.
The static magnetic field in the El-region is very nearly zero,
so that the states of the F=1 level are degenerate.
This degeneracy
has important implications for the directions of the transition
amplitudes caused by the oscillating magnetic and electric fields of
the transmission line.
In addition to the applied oscillating
electric and magnetic fields, and the static electric field from the
wire planes, there is also an uncontrolled electric field ^stray or
which is to be measured and, hopefully, counteracted.
These static
electric fields, stray and from the wire planes, can mix s- and
p-states and so allow the oscillating electric field to drive the
2s i /2 hyperfine transition.
This Stark-induced electric dipole
transition Is not the parity forbidden electric dipole transition, but
can in principle be distinguished from the parity-non-conserving
amplitude.
There are three amplitudes to consider.
The first is just
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the normal magnetic dipole transition amplitude A
p
from the
oscillating magnetic field in the TEM mode of the line.
The second is
the Stark-induced electric dipole amplitude Agtar^ and the third is
the parity forbidden electric dipole amplitude A^.
Because the
transition amplitudes are very small compared to one, they simply add
to make the total El-region amplitude, A.
We therefore decompose the
El-region amplitude in the form
1 = h
+ ^Stark + K
(7*27)
where the subscripts 8 and e refer to the oscillating magnetic and
electric fields.
Figure 7.6 illustrates the coordinate system to be used in
A
A
discussing these amplitudes (z normal to the center conductor, x is
A
up, and y is such as to make the coordinate system right-handed).
The
oscillating magnetic field on the beam axis lies in the x-direction,
the oscillating electric field lies in the z—direction.
The interaction of the atoms with the eletromagnetic fields can
be treated semi-classically, as has been done elsewhere in the
thesis.
The terms in the Hamiltonian added to take account of the
external fields are the following:
Hg = -uB |.
H
e
(£ + 2 §) - gj pN $. t
= e r • e
(7.28)
(7.29)
and
H. « e r • I.
Ci
(7.30)
We also need to include the parity non-conserving interaction (DU 78,
equations 2.35 and 2.36) with matrix elements
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Figure 7.6.
Coordinate axes and field directions in El-region.
M represents the Ml-region amplitude used to probe
El-region amplitudes. The direction and phase of
M are adjustable.
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It is the quantity
measure.
in these matrix elements that we intend to
That is, we are looking for a term in the Hamiltonian which
couples s- and p-states.
It is part of the internal Hamiltonian and
is therefore a scalar under rotations of the atom.
The magnetic term H^ can be simplified by ignoring the nuclear
moment compared to the Bohr magneton (it is smaller by a factor
mg/mp = 1/1836) and by remembering that L=0 in the 2si/2 manifold.
We thus take
Hg = -2yB£.3.
(7.32)
The matrix elements of H„ and H In the basis of Cartesian states for
P
£
the 2S}/2 snd 2pj/2 levels are presented in Tables 7.1 and 7.2.
The matrix elements for H£ were presented in Tables 5.2 in connection
with state selection.
(Note that the matrix for H^, is the same as for
H£ with s replaced by E.)
As mentioned in the discussion of the
Ml-region, the magnetic field couples the initial state, |so>, to a
final state whose Cartesian components are the same as those of the
magnetic field.
Since the oscillating magnetic field in the El-region
is in the x-direction, it drives transitions to the final state
|sx>.
That is, the transition amplitude called^Ag has only an
x-component.
(These statements are not affected by the simplification
of Hg.)
When a travelling wave is injected into the El-region the
magnetic dipole transitions dominate overwhelmingly.
Figure 7.7 shows
the transition amplitude as a function of frequency.
The calculated
amplitude is taken from the first-order time-dependent perturbation
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TABLE 7.1
2B»S
|so>
<so
0
<sx|
<sy|
<sz|
jsx>
*x
0
Bx
8y
if5z
*z
-i8y
|sy>
|sz>
ey
*z
’ifJz
IB
0
-iB
iB
0
y
X
TABLE 7.2
hi
V 1
Jso>
jsx>
|sy>
|sz>
<po
0
c
c
e„
z
<px
e
0
-ie
<pyj
e
iez
<pz
e
y
X
X
y
-ie
z
-
0
ie X
y
121
z
ie
y
-ie X
0
-
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0.4
■S 0.2
Ql
0.0
175
176
177
178
179
180
Frequency, MHz
CO
P -0.2
-
Figure 1.1.
0.4
Magnetic dipole transition amplitude in the El-region. The
dots are the square-roots of transition currents, the solid
line is first-order time-dependent perturbation theory using
the computed fields.
-
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theory, using the field envelope of Figure 7.5 (see Appendix B).
The
measured points are the square root of the transition signal
(metastable current) when a travelling wave was in the El-region.
Only
the height of the computed curve has been adjusted to match the data.
None of the operators Hg, Hg and
other s-states.
Hpnc couples s-states to
In order for the oscillating field e to drive
transitions from one s-state to another it must act in conjunction with
one of the other two operators.
That is, we compute the transition
amplitude using second-order perturbation theory.
The second-order
term for the (interaction picture) time evolution operator is given by
(ME58)
t,
U (2) = (-i)2 Jt*
t,
dTl ft* dx2 H i(t1)Hi(t2)
(7.33)
where
HT(t) = elH0t (H + IL. + H
) e ^ V 1
I
e E
pnc
(7.34)
In evaluating this expression for our experiment most of the terms in
the product of Hj(xi) with Hi(x2) can be dropped because they are
not resonant.
retained.
Only the terms that involve Hg once need to be
The hyperfine transition amplitude <sj|u(2 )|s0> (where
j = x, y, or z) is found by inserting IkjkXkj between Hj^xx) and
Hi(x2)«
We can restrict the intermediate states |k> to the 2p ^/2
level because these states are close in energy to the 2sx/2 states—
that is, the energy denominators that show up upon effecting the
integration make the contributions from other states negligible.
compute the amplitude A
, we set H
= 0 and take the electric
btiflrK
pnc
fields to be given by
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To
and
e = E q f(t) cosiut
(7.35)
t = tQ g(t)
(7.36)
where f(t) and g(t) are slowly varying envelopes.
<s3|DStark|s0> ’ ~
We find
V sJ|e? • ®o|k><t|e? ' So|s0> V “>
- £k <sj|er • E0 |k><kjer • eQ |sO> Jk (w)
(7.37)
Where the integrals 1^ and Jk are given by
i ^ " 1
V«)
= fdzifdT2 e
Jk (oj) = /dt1/dT2e
—
(*^1""^2)
f(xx) cosojti e
iE TX
-IE
J
g(xx)e
— -fp _
g(x2)e
(t X-t 2-)
/-]
oo\
0 2
"iEoT2
f(x2) coscox2e
(7.39)
The important point is that for those states |k> having non-zero
matrix elements, that is, for k=px, py, and pz, the energies E^ are
all the same.
field.
These states are degenerate in zero static magnetic
The integrals 1^ and
do not depend on k and can be
factored out of the sums.
Effecting the sum over states (using Table 7.2 and its conjugate
table) we find for j=z,
<sz|UStark|s0> = 1 3eV
(ey V ex V
(I(<1)) " J(ai))‘
This is the z-component of the amplitude ^gtarXc*
(7,40)
Combining the
results for j=x,y, and z we can write
^ a r k = -3i eV
(z0
(I(a)) " J(uj))-
(7*41)
In computing the parity non-conserving transition amplitude
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X£> the only intermediate state contributing to the first sum (I(ai))
is |po> and the only state contributing to the second sum is |pj>,
when Hpnc is substituted for er • E in equation 7.37.
As a result
we find
1
.-ie/J
eao (V0
~ V1
(7 .4 2 )
Here it is understood that g(t) is identically one in evaluating the
the integrals I(u>) and J(u>) for
E.
VHF Circuitry
The 177 MHz VHF signal'is divided into three legs. One of these
legs is sent through the Ml-region transmission line into a
terminating load.
This leg drives the Ml transition amplitude with
which El-region amplitudes are interfered.
the El-region.
The other two legs serve
These two travelling waves are injected into opposite
ends of the El-region transmission line as counterpropagating waves.
By adjusting the magnitude of the field in one leg and the phase of
the other, we create a node of oscillating magnetic field at the
center of the atomic beam.
Figure
legs.
7.8 is a simplified schematic representation of the three
The power in the Ml-leg and in one of the El-legs is left
constant during an experimental run.
Three parameters are adjustable,
namely the phase <{> between the Ml-region and the El-region, the phase
5 of one of the El-region legs, and the fractional excess field
amplitude a of the remaining El-leg.
These parameters, a, 6 , and <(>,
are adjusted by manual and electronic devices.
Some of the devices
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El
REGION
Ml
REGION
Figure 7.8.
”
Block diagram of VHF electronics illustrating the
phases <j> and 6 , and the excess field strength a.
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are controlled by the computer (a PDP 11/03) which runs the
experiment.
A more detailed schematic of the VHF hardware is shown in Figure
7.9.
The source of oscillating field is a synthesizer whose output
frequency is doubled and filtered to give the (roughly) 177 MHz of the
2s^/2 hyperfine splitting.
The filtered signal is split into an
Ml-leg and an El-leg, and the El-leg is then split again.
A line
stretcher, adjusted manually, and two TTL driven two-state phase
shifters provide control of <|>. One of the phase shifters switches
between shifts of 0 and ir; the other switches between shifts of 0 and
tt /
2 .
These are the ir-shifter and ir/2-shifter.
The low-power signal
is then amplified and injected into the Ml-transmission line and
deposited in a load.
A directional coupler samples the forward power,
providing a signal which when rectified is used to level the high
power VHF to the Ml-region.
The control voltage is applied to a
p.i.n. attenuator at the low power input to the amplifier.
The 5-variable leg of the El-region circuit contains a
varactor-tuned phase shifter marked by the letter 6 in Figure 7.9.
This phase shifter is controlled by a voltage from a digital-to-analog
converter under computer control.
The signal from the metastable beam
is used to set the DAC so that there is a node of magnetic field at
the beam (Fig. 7.10).
This same El-leg also has a line stretcher for
manually setting 6 so that the phase shifter can operate in the middle
of its 30° range.
The low power VHF is amplified and levelled in the
same way as the Ml-leg.
The remaining El-leg of the circuit is amplified and levelled,
but the reference voltage supplied to the levelling circuit is
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SYNTHESIZER
DOUBLER
e x
FILTER
5 PLITTEI
LINE
STRETCHER
r"
7T
SPLITTEI
I----
lin e
STRETCHER
N
AMP
AMP
[_EXT.
REF.
EXT.
REF.
Ml
REGION
CIRCU­
LATOR/
LOAD
LOAD
|
I__
REGION
DAC
L -fDAC
- 4 ttl
- A t t l CAMAC
COMPUTER
Figure
7.9.
Expanded block diagram of VHF electronics.
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COMPUTER
177 MHz OSCILLATOR
VHF SPLITTER
PHASE SHIFTER
$
MAGNETIC
NODE
ELECTRIC
ANTI- NODE
Figure 7.10.
Counterpropagated waves in El-region. A node of magnetic
field and an antinode of electric field are placed at
the center of the atomic beam.
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provided by another computer controlled DAC»
parameter o.
This regulates the
The two El-legs each go through a circulator before
entering the El-region transmission line.
The circulator is a
three-port ferrite device which cycles power from one port to the
next, but not backward (in 1 out 2, in 2 out 3, in 3 out 1).
permits the power
This
from each of thecounterpropagated waves to be
dumped into separate loads, rather than heading toward the other
wave's amplifier.
they overheat and
The circulatorswork up to about 10W at which point
become unstable.Below this limiting power in the
counterpropagated travelling waves the loops are able to lock stably.
The gain in each of the levelling loops (including the Ml-region
loop) is about two hundred, meaning that a five percent modulation of
the input power shows up as a modulation of 2.5 x 10-lf at the high
power end.
The rise time of a locked square modulation is 0.2
millisecond, well within the 1 msec gate used to block transients (see
Chapter IX).
The levelling loops on the high power amplifier serve to
reduce (by a factor of 200) the power modulations which attend phase
shifting in the low power circuit.
The central component in the ir-shifter is a double balanced mixer
(see Figure 7.11).
When current is injected into the IF port one or
another pair of diodes is forward biased, allowing the VHF signal to
pass from one RF port to the other.
The sign of the VHF is reversed
if the direction of the current is reversed (turning off one pair of
diodes and turning the other pair on).
Two regulated current sources
are used to provide the diode biasing.
A TTL driven
transistor switch determines which of the two current sources is doing
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IF
TTL
Figure 7.11.
iT-shifter schematic.
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the biasing.
The output of the current sources is determined by an
external feedback resistor.
One of the current sources is given a
variable feedback resistor so that the VHF power transmitted in the
two states (corresponding to phase shift of 0 and
tt)
can be made the
same.
The ir/2-shifter was made from a solid state switch originally
purchased for use in the state selector electronics.
designed to operate near 1 GHz, not 177 MHz.
That is, it was
As a result it was very
lossy (20 dB of attenuation) and the transmitted power for the two
states differed by 30%.
The switch either transmitted power directly
between the input and output ports, or added a length of cable to the
path so as to introduce a phase shift near ir/2.
After levelling at
the high power end, the 30% modulation becomes 0.0015, so that
measurements of interference signals in the 0-phase and the ir/2-phase
can only be compared to that accuracy.
This is actually more than
adequate for the measurements in this thesis.
To measure the actual phase shift introduced by the ir/2-shifter
we used the horizontal and vertical inputs of a fast oscilloscope to
make a Lissajous pattern.
The trace on the oscilloscope becomes a
line whenever the phase difference between the two signals is a
multiple of
tt.
If $ is the phase shift introduced by the shifter
(either zero or approximately ir/2) then these lines occur at
frequencies
fn= ($ + mr)/2nt
where t is the time delay between the two paths (L/c where L is the
difference in the lengths of cable and c is the speed of light).
If
we measure the frequencies at which the signals are in phase (i.e.,
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PLEASE NOTE:
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andunavailablefromauthororuniversity.
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SPLITTER
OSCILLOSCOPE
L=ct
1.5
IM
-To. 0.5-" f'P H A S E
A f =0.0693 MHz
'O-PHASE
0.0
0
2
6
4
8
10
n
Figure 7.12.
Method used to measure phase shift of
it/2-shifter.
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IN
OUT
180
D.C.
BIAS
Figure 7.13.
Varactor tuned (analog) phase shifter.
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mixer, hybrid coupler) can function in either direction so that
changes in the impedance of one leg, which affect the reflected power
in that leg, can go back through a splitter to another leg.
The
second mechanism was direct coupling of power radiated from cables and
amplifiers into other cables.
This was mostly due to power radiated
from the high power amplifiers into the low power cables across the
room, but it was also found that near-by 8NC cables radiate into one
another.
The impedance problem was solved by inserting an attenuator
and a low-power amplifier after each splitter and between successive
devices whose state could change.
The radiation problem was solved by
encasing each device (especially the high power devices) in completely
grounded enclosures: boxes with boxes within boxes.
In addition, the
BNC cables connecting low power devices were sealed in copper tape
with a conducting adhesive, and the connections were also sealed in
copper tape.
These precautions— balancing the ir-shifter currents, buffering
circuit elements with attenuators and amplifiers, and sealing elements
in Faraday cages to reduce cross-talk, along with the levelling at the
high power end— reduced the power asymmetries associated with each VHF
leg of the circuit to just below 10”6 .
These asymmetries were
measured directly by modulating the iT-shifter with only one leg
driving magnetic dipole transitions from the F=0 level to F=l.
other two legs were fed into dummy loads.
The corresponding amplitude
(or field) asymmetries associated with modulating the
thus less than 5 x 10”7 .
The
tt-shifter
are
This is the instrumental limit on the size
of asymmetry detectable by the experiment.
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V III.
A.
PRECESSION
Theory of Precession
Between the Ml-region transmission line and the El-region
transmission line is a sequence of precessors which rotate the vector
amplitude generated in the Ml-region so that it will interfere with
any desired spatial component of the amplitude in the El-region.
The
precessors are coils of wire producing a modest magnetic field (about
two gauss) in a length of the beam tube five centimeters long.
As the
hydrogen atoms pass through a precessor the amplitude in the F=1
states evolves so as to rotate on a cone whose axis is in the
direction of the magnetic field.
The magnetic field also mixes the
F=0 state with the state parallel to the magnetic field.
This mixing
is adiabatic, inducing no transitions, but causing a small phase shift
of the component of the F=1 amplitude which is parallel to the field
(the component which doesn't rotate).
The precessor field is
lnhomogeneous, but its direction is approximately constant along the
beam path.
This has the useful consequence that the effects of
rotation and of mixing with the F=0 state can be separated, as shown
below.
Restricted to the 2s j /2 manifold the Hamiltonian in a magnetic
field § can be written (BE57)
H = ■5Ehfs(2^2"3)
“ V'o ^
■ Sl^N ^
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C8-1*
in which E ^ s is the hyperfine splitting,
is the nuclear magneton, g
S
electron and proton.
and g
J>
is the Bohr magneton, yN
are the
gyromagnetic ratios of the
The operators ? and f are the spin angular
momenta of the electron and the proton, while § = Ij + ? is the total
angular momentum (for s-states, L «* 0, so £ does not appear in the
Hamiltonian).
Defining y - SsUq/2 and n = SjWjj/CggUo^
Hamiltonian
becomes
H = | E h f g (2f2 -3)-y[(l+n) (5 + t).£ +(l-n) (§ - t ) 4 ] ( 8 . 2 )
in which (S + I) can be replaced by F.
The matrices for £•£ and £•(£-£) are presented as Tables 8.1 and
8.2. For brevity I will make the following definitions:
H0 - £ w
2f2- 3>
R' = -y(1+n) S.?
(8.3)
M» = -y(l-n) S*(S-f)
(the primes on M and R are temporary).
Since
the operator ?2 commutes
with any component of f, such as the one parallel to £, it is clear
that Ho and R' commute. Because they commute, the interaction
picture representative of R* is just itself:
R = e1H0C (R’) e""iH0C = R'
(8.4)
The same cannot be said of the interaction picture representative of
M':
M - eiH0£ (M') e’^ O 1
so its matrix is presented as Table 8.3.
(8.5)
The (interaction picture)
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TABLE 8.1
sz>
sy>
sx>
so>
<so
iB.
-IB,
<sx
-iB.
IB
iB.
-iB.
<sz
TABLE 8.2
B- ( S - l 5
sz>
sy>
sx>
so>
<so
<sx
<sz
TABLE 8.3
1H t
-iH t
so>
sx>
-iEt
<so
<sx
<sy
<sz
sy>
-iEt
sz>
-iEt
iEt
iEt
iEt
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time-evolution operator, U(t), is the solution of the equation
i U
=
(R+M) U
(8.6)
which at t=0 is the identity.
partitioned into
n
If the time interval from 0 to t is
equal segments of duration
t
, then the
time-evolution operator can be approximated as
U(t) = (l-i(Rn-tt!n ) T )
(l-i(R14M1)t)
(8.7)
in which Rj and Mj are the values of R and M at the beginning of
the
jth time interval.
This is just Euler’s
integrating a differential equation.
method ofnumerically
Theapproximation
in the limit that n goes to infinity.
becomes exact
Now if the direction of the
magnetic field is constant, even though the magnitude changes with
time, then the product of any Rj with any Mg is zero in either
order; such a product is proportional to (S* - I^)(for § ■ zB) which
vanishes since S=0>l/2.
This implies that
l-i(Rj-Htj)T - (1—iRjt) (1-iMjt)
(8.8)
and it implies that each factor with an Mj commutes with any factor
with an Rfc.
Thus,
U(t) - (l-iRnx) ... (1-iRiX) (l-iMnx) ...
(1-iMiX)
(8.9)
U(t) = (l-iMnx) ... (1-iMiX) (l-IRnx) ...
(1-iRjx)
(8.10)
and
That is, we can write
"(t) ' V
where
■ um “r
(8al)
Ug satisfies
i \
and
m
- R «R
(8.12)
satisfies
1
- M 0M
(8.13)
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We can solve for Ur, which accounts for rotations, and for % ,
which accounts for the mixing of the F=0 state with the F<*1 state
parallel to the magnetic field, then multiply them to obcain the
time-evolution operator for the 2 SJ/2 states in a magnetic field of
constant direction.
To solve for Ur we write
R(t) - Rof(t)
(8.14)
where Rq is a constant matrix, and the magnetic field is given by
.
A
A
B = n Bq f(t) where n is a unit vector.
given by Ro * -u(l+n) Bo n*?.
Then the matrix Rq is
Because Ro is a constant matrix
the solution of the equation for Ur is easy:
UR - exp ( -iRQ / q dt' f(t’) ).
(8.15)
This can be seen by changing to a basis in which Ro is diagonal— a
time-independent transformation— and then changing back to the
original basis.
Returning to a more transparent notation,
Ur “ e
$
=» ny(l+n) /dt B(t)
it is clear that Ur is a rotation operator.
(8.16)
The axis of rocatlon is
along the direction of the magnetic field, and the angle is a magnetic
moment times the integrated field.
To solve for %
we can use first order time-dependent
perturbation theory, if the matrix element y(l-ri)B is small compared
to the splitting, Ehfs*
Taking the magnetic field
to be
in the
y-directlon, passage through a precessor results in a transition
amplitude from the F=0 state to the state |sy> given by
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<sy|ll|so> = -i /dt <sy| eiHOty(l-n )By (t)(Sy-Iy ) e"iHOt |so>
= -iy(l-n)
/dt
e1EhfsC By (t)
(8.17)
The transition amplitude Is proportional to the Fourier component of
the magnetic field at the hyperfine frequency Ehfs*
B.
Precessor Field and Phase Shift
The precessors are made from two rectangular coils wrapped around
a cylinder, shown idealized by single loops in Figure 8.1.
In the
lower part of the figure is graphed the magnetic field on the cylinder
axis for loops 5 cm wide wrapped on a cylinder of 3 cm radius (that
is, a = 3 cm, b = 2.5 cm).
In Appendix C it is shown that the Fourier
transform of this on-axis field is given by
B(oj) = 2 B t sinmT [^(wt) -u)tK0 (u>t)]
in which
t
(8.18)
= a/v = 0 . 1 ysec, T = b/v = 0.0833 ysec, and the average
field is given by
B = 8 I/ac » 0.267 I gauss/ampere.
(8.19)
This is the integrated magnetic field divided by the transit time (2T
= 1/6 ysec— See Appendix C).
Equations 8.18 for <u =
Evaluating the Bessel functions in
gives an estimate of the transition
amplitude (Eq. 8.17):
<sy|u|so> = -i(2yBT) x 7.95 x 10"1*8 .
(8.20)
Since the precession angle 2yBT (ignoring the small correction from
the factors (l±n)) need never be greater than ir, this gives a very
small transition amplitude.
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2a
2
b
0.8-
0.6-
0.4-
0.2-
-6
-4
Z
Figure 8.1.
Precessor coils (rectangular loops wrapped on cylinder)
and field strength on beam axis.
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That no transitions are driven as the atom passes through the
precessor is equivalent to saying the evolution is adiabatic; the atom
evolves so as to remain in an energy eigenstate at all times.
A
non-rigorous but generally useful criterion can be derived from
Equation 8.17.
Integrating by parts and noting that B^(t) vanishes at
±“ , one has
<sy|u|so> = -y(l-n) ^ —
I 1
hfs
/dt
eiEhfsC (3B /3t)
7
(8.21)
which if the derivative is presumed constant becomes
3B /3t
<sylu|so> -p(l-n) —
— (-cos E
t)
II
'
\fl
hts
(8.22)
2
so that the evolution is considered adiabatic if ji(3B / S O / E ^ g is
small.
(Compare Schiff (SC 68 ), Equation 35.27, p. 291.)
Evaluating
the derivative from Figure 8.1 at z=2.5 cm, one finds 3By/3t = 0.72
B/T so that
_U
^ 2 .= UBT (0.72) = it ,
(0.72) = 2>g x 10_it>
Ehfs
3t
<EhfsT >
(8.23)
(92‘7>
The shift in the energy of the F=1 state parallel to the
precessor's magnetic field appears as a phase shift when the atom
emerges into a region of zero field.
The states perpendicular to the
precessor field are rotated with no additional phase shift.
In
Appendix C the phase shift of the parallel state is estimated to be
less than 0.053 or 3°.
C.
Geometry of Precession
The precessors drive no transitions from the F=0 state to the F=1
states.
They cause the F=1 amplitude generated upstream in the
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Ml-region to rotate around the direction of the magnetic field in the
precessors, and they cause an unimportant phase shift of the component
of the Ml-region amplitude which is parallel to the precessor's field.
The Ml-region amplitude,
can be visualized as moving on a
cone whose axis is the direction of the magnetic field of the
precessor (Figure 8.2).
The tail of the amplitude and the main vector
part move on separate cones.
Since a cone cuts the sphere with a circle, the orbits of 3 ^
also be visualized as circles on the sphere.
can
A useful geometrical
fact is that two points on the sphere define a family of circles whose
line of centers is a great circle.
Every great circle Intersects
every other great circle, so in particular the line of centers of the
circles defined by two points on the sphere will intersect the equator
(the great circle in the xy-plane).
8.3.
This is illustrated in Figure
The practical consequence of this is that no matter what
amplitude the Ml-region delivers, its main vector part can be made to
point in the z-direction with a single precessor whose axis is in the
xy-plane, but is otherwise variable (i.e., it can be rotated on the
cylindrical tube on which it is wound).
D.
Arrangement of Precessors
Four precessors are wound on the tube connecting the Ml-region to
the El-region.
The first one, immediately following the Ml-region, is
a preliminary precessor whose purpose is to align the Ml-region
amplitude so that it points in the z-direction (down beam).
This
”pre-precessor" has its axis in the xy-plane— that is, in the
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Precession
✓ A xis
y
Figure 8.2*
Cone of precession.
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Figure 8 .3.
Any two points on a sphere can be connected by precession
on a cone whose axis is in the xy-plane because the cones
connecting the two points define a family of circles whose
line of centers cuts the equator.
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equator— and can be varied by rotating the sleeve to which the coils
are glued.
By adjusting the current in the coils and the orientation
of the axis the Ml-region amplitude is "initialized" to a
z-orientation.
The three subsequent precessors are labelled by their
axes as the y-precessor, x-precessor, and z-precessor.
The
z-precessor is not of the type described in section B of this chapter,
but is instead a solenoid, five centimeters long, wound on the tube.
This arrangement of pre-precessor, y-, x-, and z-precessors is shown
in Figure 8.4.
Since rotations do not commute, the order of operations is
significant.
If the y-precessor is used to bring the Ml-region
amplitude to the vertical (x-) direction, then subsequent x-precession
will cause a small phase shift, but won't change the direction of the
amplitude.
Similarly, when the pre-precessor has been adjusted so the
Ml-region amplitude is unchanged by the z-precessor, the
initialization is done.
Surrounding the Ml-region and the precessor tube are large coils
to adjust the ambient field.
Before the precessors are adjusted
themselves this ambient field is reduced to an inhomogeneous field
which is everywhere less than fifty milligauss.
could do with the correction coils.
This is as well as we
The effect of the residual field
is to shift slightly the effective axes of the various precessors.
That is, a y-precession followed by motion in the residual field is
not quite a precession about the y-axis inside the El-region.
To identify the'directions, we use the interference of the
Ml-region amplitude with the allowed Ml transition amplitude driven by
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arrangement.
Precessor
8.4.
Figure
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a travelling wave in the El-region.
Because of the shielding, the
many attenuation lengths for non-TEM modes, and the smallness of the
other transition amplitudes, this El-region Ml-amplitude is well
defined to be in the x-direction.
Ml-region amplitude,
The interference between the
and the Ml-amplitude in the El-region, X ,
is detected by modulating the phase of the VHF supplying the two
regions between <J> and <j> + ir, and demodulating the detected current
with a lock-in amplifier.
The overall phase <fi can be adjusted by a
line stretcher in the cable to the Ml-region, and can be switched
to
<f> + ir/2 with the electronic phase shifter discussed in the last
chapter.
Only the component of
parallel to
is detected. To adjust
the pre-precessor, its current is varied until no interference is
seen.
All the other precessors are off.
amplitude
by the time it reaches the El-region, lies somewhere in
the yz-plane— the plane perpendicular to
null-plane.
This means that the
Xap .
I will call this the
Now, the z-precessor is turned on.
The z-precessor is
close to the magnetic shields surrounding the El-region, and its
ambient field is zeroed with particular care
to the level
of a
milligauss.
axis is very
closeto the
We believe, therefore, that its
z-axis of the El-region.
If
has been put in the null-plane by the
pre-precessor, then the z-precessor will bring it out of the
null-plane on a cone whose axis is the z-axis, and the interference
signal will reappear.
The interference is maximized when the rotation
angle of the z-precession is
tt/2.
With the z-precessor set at this
rotation, the axis and current of the pre-precessor are adjusted to
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return
to the null-plane.
Then the z-precessor is turned off and
the pre-precessor is again adjusted to give no interference.
The
process converges rapidly, so that when the pre-precessor is set, the
z-precessor leaves the interference signal zeroed.
The Ml-region
amplitude is now initialized to the z-direction.
The y-precessor is first oriented so that its axis is parallel to
the physical axis of the El-region.
With the pre-precessor
initialized, the current in the y-precessor is adjusted to give the
maximum interference signal.
Its axis is then rotated to make this
maximum (as a function of current) as large as possible.
small correction to the physical orientation.
This is a
The phase (J> is now
adjusted with the line-stretcher until it, too, gives a maximum.
(This is just the physical version of the decomposition which
identifies the main vector part of
The x-precessor axis, which
is oriented nominally vertical, is then adjusted to give no change in
this maximal interference signal.
If the y-precessor is now turned
off, the x-precessor should leave the Ml-region amplitude in the
null-plane, and if it doesn’t, it can be further adjusted so that it
does.
E.
The precessors are now ready to be used as intended.
Removal of Tails
When the pre-precessor is aligned, the y-precessor adjusted to
give the maximum interference, and the phase $ between the Ml-region
VHF and the El-region VHF adjusted to give the maximum interference,
then we are looking at the main vector part of
p
.
interfering with
This defines what I shall call the "real" phase.
If <j> is now
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increased by
tt/2,
the interference signal vanishes.
This is, in fact,
the better way to define the imaginary phase, since a zero is more
sensitive than a maximum.
The tail of X ^ is not visible in this
configuration because, though we are looking in the imaginary phase,
the tail is at right angles to
restores the real part of X ^
Xp .
Turning the y-precessor off
to the z-direction, so that the
imaginary part is in the xy-plane somewhere.
interference in the imaginary phase.
There is then an
By using the z-precessor to
rotate the tail by ir/2 , the x- and y- components of the tail can be
measured.
Since the tail can be measured, it can be removed.
is very tedious.
The process
The Ml-region is equipped with coils to produce a
magnetic field in the same place that the transitions are driven.
Changing this magnetic field changes the tail of X ^ .
Unfortunately,
it also changes the direction and phase of the main vector part of
X ^ , so in order to measure the tail after making a change in the
Ml-region's magnetic field the pre-precessor must be re-aligned, and
the phase <j> between the Ml- and El-regions must be adjusted to
identify the "real” phase and "imaginary" phase all over again.
The
changes are so drastic that no quick, convergent, minimization
procedure exists for eliminating the tail.
Nevertheless, by
persistently adjusting and measuring, the tail is reduced until it is
one ten-thousandth the size of the main vector part.
Rather, the
interference of X ^ with a travelling wave in the El-region, in the
imaginary phase, is adjusted to this level.
It is no longer
meaningful to separate A 0 from the other El-region transition
p
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amplitudes at this level.
F.
The t y Directions
The El-region is equipped with a pair of Helmholz coils under the
shield to define the direction of the y-axis.
This Helmholz pair acts
like a large y-precessor inside the El-region itself.
pair therefore has no effect on
y-direction.
The Helmholz
when it is oriented in the
To reach the ±y-directions the (tailless) amplitude
is put in the null-plane with the pre-precessor (this is the usual
alignment).
Then the x-precessor is turned on to rotate from the
z-direction to the y-direction.
2^
When the Helmholz pair is turned on,
will rotate out of the null-plane unless it is oriented along the
y-axis to begin with.
A certain current in the x-precessor reaches the +y-direction,
and a different current (almost its negative) reaches the
-y-direction.
To define the z-direction more precisely than with the
z-precessor, we split the difference between the +y-direction and
-y-directions.
That is, we orient
half-way between + and -y.
to be in the null-plane,
This is very nearly the same direction
obtained by the usual alignment of the pre-precessor.
By use of these procedures the direction of
can be determined
to ± 2 °.
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IX. -DATA COLLECTION AND CONTROL
A.
Detector Electronics
The detector collects a current proportional to the flux of
metastable hydrogen atoms reaching it.
This current is converted to a
voltage by a preamplifier (Ithaco model 164) with a transconductance
of 10
volts/amp.
Under typical running conditions, with the beam
state-selected and with the Ml-region driving transitions at a rate •
equal to the background signal, the detected current at unity gain is
between ten and fifteen picoamps.
1.5 x 10 15 A Hz
The statistical noise at 15 pA is
(Noise = /Ie , where e is the elementary
charge). The equivalent input noise current of the Ithaco preamplifier
is 1.3 x 10 ^
A Hz
which is ten times as large as the beam noise
and accounts for all the electronic noise observed when the detector
is not connected to the rest of the system.
That is, there is no
significant contribution to the noise from subsequent stages of
detector electronics.
In order to achieve the noise limit imposed by
counting statistics, we must increase the gain of the detector to
ten.
2
-9
In general, we must have G I>10
ampere, where G is the
detector gain and I is the unity-gain current.
The two ionization chambers of the detector are connected in
parallel by coaxial cables and the combined current is run into the
preamplifier by a thin, flexible, coaxial cable.
The preamplifier is
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PLEASE NOTE:
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andunavailablefromauthororuniversity.
Filmedasreceived.
UMI
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1400
1200
co
h
180
Z 1000
_
60
o
120
a.
|
°
o
in
800
O
600
b.
195
240
300
400
50
100
150
200
LOCK-IN FREQUENCY, Hz
Figure 9.1.
250
Electronic noise measured in a 1 Hz bandwidth.
One unit on the ordinate corresponds to 6.54 x 10- *7 A
Hz- 1/2 equivalent noise current at the input of the
Ithaco preamplifier.
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300
20
FWHM—0 .8 Hz
54
Figure 9.2(a)
56
58
60
f t Hz
62
64
The resonance at 60 Hertz.
The width of this scan is approximately equal to
the instrumental resolution.
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1200
1000
800
600
400
200
0
193
Figure 9.2(b)
194
195
196
197
198
The anomalous resonance near 195 Hertz, a mechanical
resonance of the detector anode.
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B.
Computer and Interface
The modulation of experimental parameters and the assimilation of
the signal from the voltage controlled oscillator are managed by a
PDP 11/03 computer through devices in a CAMAC interface bus.
Besides
the controller, linked to the computer, the crate contains a preset
scaler, two scalers, two digital-to-analog converters (DACs), and
twelve logic output registers.
The preset scaler counts a
predetermined number of TTL pulses from a quartz oscillator clock (1
kHz).
It then sends an interrupt to the computer, initiating the
reading of the scalers and the adjustment of DAC voltages and logic
levels for the output registers.
A new number is set on the preset
scaler determining the time period of the next cycle of operation.
Figure 9.3 shows a block diagram of the interface.
The two scalers are connected to the output of the voltage
controlled oscillator so that the number of pulses received in acollection interval represents the metastable signal during that
interval.
Only one scaler collects the signal in a given interval,
the other one being gated off.
This produces two "channels" of data
corresponding to having the Ml—region amplitude precessed to the
vertical (x-) direction, or having it precessed to some direction in
the null (yz-) plane.
The modulation of the precessor (the
y-precessor) and the gating of the scalers is handled by one of the
output registers.
Only four of the output registers are actually used.
controls the y-precessor.
One
The other three control phases of the VHF.
One drives the ir-shifter, one the
it/2-shifter,
and the third
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anooe
COMPUTER
(through
crate
controller)
1 kHz
CLOCK
QUENCHER TUBE
DETECTOR
PRESET
SCALER
PREAMP
VOLTAGE
CONTROLLED
OSCILLATOR
gate
read
clear
OUTPUT
REGISTER
1
2
Ch 0
Ch 1
DUAL SCALER
y-precessor
if /2-shifter
*r-shifter
DUAL
12 BIT
DAC
Figure 9.3.
CH
Block diagram of CAMAC interface.
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introduces a modulation of the voltage controlling the variable phase
shifter.
The base level of this controlling voltage is supplied by
one of the two DAC's in the CAMAC crate. The remaining DAC allows
the computer to control the power in the adjustable leg of the
El-region VHF circuit.
The range of each DAC is 0-10 volts set by an
integer from 0 to 4095 (12 bits).
Manually adjusted offsets added to
the DAC voltages permit operation near the middle of the DAC's range.
Each time the output registers are set and the preset scaler
counts one pulse from the clock.
This "output stabilization time"
ensures that the output register has a definite logic level.
The
preset scaler then counts off another few milliseconds (typically 2
msec, determined at the software level) to stabilize transients in the
devices controlled by the output registers.
Then the scalers are
zeroed and the preset scaler is set to count off the data collection
time.
At the end of the collection interval the preset scaler gates
the scalers and interrupts the computer.
The scalers are read, the
output registers are set, and the cycle begins again.
We call this a
"basic period."
C.
Orthogonal Square Waveforms
The number of VCO pulses counted by the scalers in the collection
interval of a basic period represents the metastable beam intensity
when the apparatus is in the state specified by the logic values of
the output registers. The datum from each basic period is binned in
one of sixteen locations in a linear array which we call a "cycle."
specified number of cycles is averaged together to produce a
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A
"datapoint."
A datapoint is thus a vector of sixteen numbers each
corresponding to the metastable beam intensity for a different state
of the apparatus.
To recover particular signals from the.datapoint -
to demodulate the signal - we use a many-channel phase-sensitive
detection method based on orthogonal square waveforms (HA 71).
The orthogonal square waveforms, or "modes,” are products of
elementary modes defined by
M
(t) = sign ( i
-
frac(t/2n+1))
= sign c|
-
frac((k-|)/2n+1))
(9.1)
2
where "frac" denotes the fractional part of a number, and "sign" is +1
if the argument is non-negative and -1 if it is negative.
Here 2n+is
the period of the square wave, and k, representing time, is the bin
number.
It is best to label the elementary waveforms by binary
numbers, as
M^ q instead of M ^
and
M100 instead of M22.
Products of elementary modes are labelled by the bitwise exclusive-or
(beor) of the binary labels of the elementary ones.
This rule is
correct, in fact, for the product of any two waveforms, not just
elementary ones.
M101 ■ M100*M1 de£lnes Mioi<Mi<n<W - « i o o «
The waveforms made of products of elementary waveforms are more
complicated square waves, as indicated in Figure 9.4.
of the bitwise exclusive-or (beor) rule, the product of
As an example
and
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M 1011 *s C^e wave^orm Mn o = ^ 100*^10 because (1101 )beor(1011 )=(0110 ).
In addition to the elementary waveforms M^,
M^qq and so on, we
need Mg which is identically one for all time (or for all bin
numbers).
Because the cycle length is sixteen basic periods we are limited
to a four bit system of waveforms.
The modes M q q q q to
sixteen dimensional space of the datapoint.
span the
The software can
accomodate a six bit system (64 bins) by an easy change of parameters
in the set-up file controlling the run program, but the added
complexity was not needed.
mode.
Thus, to each four bit number is assigned a
The modes are orthogonal in that
“ a" M b - £ V O M b " 5’ - 16Jab.
(,.2)
That is, the "scalar product” of two modes is zero unless the modes are
identical, or equivalently the integral of the product of waveforms
over time is zero.
The signal in a particular mode (S ) is given by the dot product
SL
of that mode with a datapoint (X):
S = M .X = £ M (k)X(k)
a
a
, a
k
(9.3)
For example, consider the dependence of the metastable beam intensity
when the ir-shifter is switched from phase 0 to phase
tt.
The detected
current is proportional to the background signal plus the transition
rate.
This includes the interference term between the Ml- and
El-regions which changes sign when the phase is switched between 0 and
it.
The dot product of M
&
with X is proportional to the inteference term
in the transition probability.
Similarly, Mq .X
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measures the average current during a datapoint.
This is analagous to
the method of lock-in detection when the signal is linear in the
modulated parameter.
The system also accomodates a more subtle kind of modulation than
the usual linear case.
Suppose that the ir/2-shifter is modulated in
mode b, so that when M^(k) is +1 the shifter is in the 0-phase and
when M^(k) is -1 it is in the ir/2-phase.
The function (HM^)/2 is
then unity for the O-phase and zero for the ir/2-phase.
In that case
X.(14M^)/2 picks out the contribution from the O-phase alone.
If we
wish to know the interference term (ir-shifter) when the ir/2-shifter is
in the zero phase, while modulating both devices then we want
(Ma*(l+M^)/2).X where
modulates the ir-shifter.
The star product
and dot product are linear and distributive (but you have to perform
stars before dots).
We then have
(Ma*(l+Mb)/2).X = (Ma*(M04ttb )/2).X
= (M .X + (M *M. ).X/2
3
3 D
(9.4)
- (Sa + S« W b ) t t
Similarly, the interference term in the v/2-phase is given by
(M - M *M. ) .X/2.
a
a b'
A last and important algebraic property is the following:
M . ( M ^ X ) = (Ma*Mh).X
(9.5)
Its importance comes from the method of channelling.the data.
The
computer actually calculates separate datapoints for the two scalers.
The scalers are gated so that only one of them accepts the VCO signal
at a time, according to whether
is precessed to interfere with the
vertical amplitude in the El-region, or whether it is precessed to
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interfere with an amplitude in the null plane whose direction is set
by the x-precessor.
channels,
This means that half the bins of the two data
and TL^y always contain zeroes.
The system was originally
intended to process separate streams of data from the two ion chambers
simultaneously, but it proved more useful to use the separate channels
for the two precessor states instead.
If
is the mode used to
modulate the y-precessor and the gating of the scalers, and
is the
mode used to switch the ir-shifter then the beam current (as a function
of time) is given by
I = constant + 4(1+M )*M A.+ 4(1-M )*M A.
2
y
ir 1 2
y
(9.6)
ir 2
Where Aj represents the interference term when the precessor is set to
vertical and the VCO signal goes to Channel 1.
A 2 represents the
interference term with the precessor is set so that
null plane with the VCO signal going to Channel 2.
lies in the
The switching of
star and dot products in Equation 9.11 allows one to show that M^'X^
is proportional to A.^ while
*s ProPort*onal to A2 *
That is,
there is no problem with analyzing the separate channels as though
they held uninterrupted streams of data for the separate
configurations.
D.
Feedback
The two digital-to-analog converters (DACs) provide computer-set
control voltages which can be used to eliminate chosen signals.
of these signals is the interference between
dipole transition in the El-region.
and
One
the magnetic
This signal ismeasured
by
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modulating the ir-shifter when the y-precessor has tilted
vertical.
to
A measurement of this signal is used to increment the DAC
controlling the voltage-variable phase shifter (the phase 6 ).
The
generic ideal control problem of this sort can be phrased as follows.
A signal S is, on average, equal to a (scaled) control parameter x.
Each measurement of S is used to set the value of x for the next
measurement:
x . = x + Ax = x n+l
n
n
n
where Sn = x n + Noise.
gS .
n
(9.7)
Ignoring
the noise for the moment we have
w o
Xn+1 =
(9,8)
This is a difference equation with the evident solution
xn = (l-S)nxQ.
(9.9)
This sequence converges to x=0 provided 0<g<2.
The convergence is
most rapid when 3 =1 .
If to each measurement we add a normally distributed independent
random variable,’ so that S„
n = xn +Nn , then subsequent measurements are
affected by memory of past noise induced by the feedback.
For the
n-th measurement we have
Sn = x 0 (l-B)n + N 03(l-3)n_1 + N 18(l-3)n"2
+
... + Nn_13(l-8)° + Nn .
(9.10)
If the Nj are Identically distributed normal variables we can find the
effective noise (standard deviation) on such n-th measurements by
adding the standard deviations in quadrature.
One finds
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As n becomes large this approaches a limit:
a(S) = a(SQ )
f
(9.12)
The radical is plotted in Figure 9.5 showing that feedback causes
"noise amplification.” This noise amplification is to be contrasted
with the standard deviation of the mean (SDM) of the locked signal.
Our software permitted restarting data acquisition once the loops were
locked, that is, once the feedback parameter had stabilized, so that
we can drop the X q terms in the signals Sn » The mean of
measurements Sq ,
the
, ... Sji is then given by
§M = —
[NM + (1 _ 6) NM-1 + •*• + d-B)MN0]
M+l
(9-13)
which as M becomes large has the limiting standard deviation
= M+f
✓ 8(2-8)
°(V *
(9.14)
in the locked channel, which diminishes as 1/(M+1 )— much faster than
the usual 1//M for the SDM of independent random measurements.
Although the analytic forms for the standard deviations in this
section apply, strictly speaking, only to an ensemble of runs under
identical conditions, and not to statistics computed in a given run,
they should be approximately valid.
In any case, these results
account for the puzzling
effect of a large standard deviationcomputed
for the few measurements
taken in a type-out period while
atthe same
time the SDM in this locked channel was far below that in any noise
channel.
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0 .5
0.0
~ 0 .0
0 .5
1.0
1 .5
2 .0
£
Figure 9.5.
Noise amplification factor resulting from feedback to a
controlled signal.
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E.
Drifts
Drifts are changes in the experimental parameters which occur
slowly compared to the modulated parameters.
The beam intensity
gradually decreases until it gets low enough to require stopping the
experiment and re-optimizing the source.
The value of control voltage
which gives a zero signal may drift as the components reach
equilibrium temperature.
operation.
The detector gas is slowly polluted by its
The effect of drifts can be reduced by using sufficiently
complicated waveforms (HA 71).
Over the time scale of a cycle (16 basic periods) a drifting
signal can be expanded in a Taylor series.
The sensitivity of a given
mode to linear, quadratic, and higher terms of the drift can be
computed.
The mode
is sensitive to linear (and higher) drifts, as
are the other elementary modes, while
M j^ q , an<* ^ioi are not
sensitive to linear drifts but will pick up a spurious signal if the
drift is quadratic.
Mostly one is concerned with identifying modes
which give zero for drifts of a given degree (and lower), but a more
general measure can be defined.
Defining the vector of integers to a
given power by Zn = (ln, 2n,...,16n) we can take
M ..Zn
D° - - j
J
V
(9 .1 5 )
2"
as a measure of the sensitivity of mode
normalized to the signal in mode 0.
to a drift of n-th degree,
Table 9.1 shows the 16 modes of
our four bit system, and Table 9.2 displays the values of
for these
modes, for the first few values of n.
There is at least one use for the nonzero D*j values.
If the beam
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TABLE 9 . 1
BIN NUMBER
MODE
1
0000
1
0001
1
0010
1
0011
1
0100
1
0101
1
0110
1
0111
1
1000
1
1001
1
1010
1
1011
1
1100
1
1101
1
1110
1
1111
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
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16
TABLE 9 . 2
DEGREE (n)
1
2
3
4
0000
1.000
1.000
1.000
1.000
0001
-0.059
-0.091
-0.121
-0.151
0010
-0.118
-0.182
-0.241
-0.299
0.011
0.022
0.036
-0.364
-0.471
-0.571
MODE (j)
0011
0100
0
-0.235
0101
0
0.021
0.044
0.071
0110
0
0.043
0.088
0.142
0111
0
0
-0.005
-0.013
-0.727
-0.860
-0.928
1000
-0.471
1001
0
0.043
0.088
0.130
1010 .
0
0.086
0.176
0.259
1011
0
0
-0.010
-0.027
1100
0
0.171
0.353
0.505
1101
0
0
-0.021
-0.054
1110
0
0
-0.042
-0.107
1111
0
0
0
0.006
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intensity is constant except for statistical fluctuations then the
standard deviation of signals in various modes will be the same (more
or less).
But if the beam intensity slowly oscillates (a "breathing"
mode) so that in any one cycle it appears to drift linearly, then some
modes will detect a drift signal and others will not.
Statistics
computed on the mode-analyzed datapoints will reflect this.
The
standard deviation computed from a sample of datapoints should be
twice as large in the mode M 1Q0 as in M 1Q, with that for
much smaller (i.e., closer to the statistical noise).
very
This effect is
very clear in "runs" taken with no experimental parameters
modulated— looking at the beam alone.
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X.
A.
RESULTS
Procedure
The experimental results of this chapter reveal an unanticipated
large transition amplitude in the El-region in the direction of the
oscillating electric field.
The data did not uncover the cause of
this transition amplitude but many hypotheses have been disqualified.
The signals under direct control behave precisely as expected; in
particular, the suppresion of El-reeion amplitudes in the directions
perpendicular to the oscillating electric field, where systematic
signals were expected, is effective within the noise of the
measurements made.
A fairly uniform procedure emerged for taking data
the main features of which have already been discussed in connection
with the relevant apparatus.
A beam of metastable hydrogen atoms was first established by
igniting the discharge in the duoplasmatron and optimizing source
parameters to yield the largest possible detected current.
The
detected current with the detector at unity gain lay between 250 and
350 nanoamperes.
Next the power in the 0-selector was set to reduce
the full beam to 0.11 of itself and the 1-selector power was set to
reduce the beam to 0.46 of the full intensity.
At these settings the
background current is twice what it would be if there were no
repopulation of the metastable beam in flight— that is, near the break
in Figure 6.5.
The background is thus about 6 x 10“5 of the full
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metastable beam, or 15 pA in the case of a 250 nA beam.
At this point
the gain of the detector is raised to ten and the state selectors are
turned off for a few seconds to expose the detector to high current.
This circumvents a long settling time.
When the state selection is restored the system is 31%
transparent to hyperfine transitions.
The power in the Ml-region VHF
circuit is set to double the metastable current so that the transition
amplitude generated in the Ml-region is about 0.03.
(That is,
(transition probability A ^ ) x (fraction of full beam in F=0 state (=
1/4)) x (transparency) x (full beam) = transition current.)
Since the
amplitudes to be measured in the El-region are small, the background
plus the transition current is the nominal average beam intensity.
With this current the gain of the amplifier preceding the voltagecontrolled oscillator is set to give an output of 5.0 volts so that
the VCO is in the center of its operating range.
Power (10 watts) is now admitted to one leg of the El-region VHF
circuit and the ir-shifter is modulated with a TTL squarewave from a
signal generator.
This squarewave is also the external reference for
the lock-in amplifier.
The signal from the Ithaco preamplifier drives
a follower whose output provides the signal to the lock-in.
The
lock-in amplifier thus detects the interference of amplitudes in the
Ml- and El-regions.
With the ir/2-shifter set in the 0-phase the
y-precessor is adjusted to produce a maximal interference (the
pre-precessor having already been set to produce a zero unaffected by
current in the z-precessor).
The ir/2-shifter is switched to the
ir/2-phase and the line stretcher in the Ml-leg is varied to produce a
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zero on the lock-in.
This defines the 0- and
To set the parameters
it/2-phases.
of the remaining leg of the El-region
circuit, the computer and interface are turned on, and a program is
run which sets the DACs to their midpoints (2048 = 5.0 volts).
power in the remaining leg is turned on.
0-phase, the power of this
the lock-in.
The
With the ir/2-shifter
inthe
second leg is adjusted to produce a
zeroon
In the ir/2-phase the phase 5 between the two
counterpropagated waves is adjusted with a line-stretcher to give a
zero.
The Argand diagram of Figure 10.1 illustrates these
adjustments.
A change of a (power) moves the sum of the amplitudes
from the separate waves to a different circle, while a change of 6
moves this sum around a circle of given radius.
The two amplitudes
are equal and opposite when the sum is at the origin— on the circle
tangent to the imaginary axis.
It should be clear from the diagram
that when the amplitudes of the counterpropagated waves nearly cancel
the 0-phase is sensitive to changes of magnitude (power) and the
it/2-phase
is sensitive to changes of phase.
Figure 10.1 can be used to show that a modulation of the phase 6
between small positive and negative values of equal size can be used
to detect a magnetic interference in the ir/2-phase when the average
signal is kept zero.
We have used this 5-wobble signal to calibrate
the field in the El-region when counterpropagating and to define the
null-plane, perpendicular to Ag.
With the waves in the El-region counterpropagated to give a zero
signal in both phases it is now possible to take data.
and
it/2-shifter
The ir-shifter
are switched to control by the output registers.
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In
Figure 10.1.
Argand diagram showing how the sum of magnetic
dipole amplitudes for the two counterpropagated
travelling waves varies with the magnitude and
phase of one of the waves (a (2)).
p
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addition the output registers control a small voltage added to the
variable phase shifter controlling 6 (to give the 6-wobble), and they
control the y-precessor which is either on, so that
is precessed
to vertical, or off, so that 1^1 lies in the null-plane.
The
controlling waveforms (modes) are as follows:
ir-shifter
1111
ir/2-shifter
1
y-precessor
10
6-wobble
100.
The y-precessor mode also controls the channelling of data into the
two scalers.
Channel 1 corresponds to Am i vertical and Channel 2 to
Am i *n fche null-plane.
The data are analyzed in (star) product
modes of the modulation modes.
These are 1111 (M^), 1110
(Mir*Mir/2), 1 0 U (M1T*M5), and 1010 (Mir*M6*M1T/2).
Table
10.1 shows the array of signals available.
The difference ( S i m “ s1110> in Channel 1 can be used to
control a, the power difference in the two El-legs.
The sum ( S i m
+ S m Q ) can he used to control 5, the relative phase between the
waves.
Similarly, the sum in Channel 2 of Sio n anc* ^1010 can he
fed back to the pre-precessor to keep the amplitude ^ m i lying in the
null-plane.
Only two of these controls can be implemented at once
because there are only two DACs.
The signals of greatest interest are the interferences in the
null-plane (1111 and 1110 in Channel 2).
The position in the
null-plane is determined by the x-precessor current.
The currents
corresponding to the ±y-directions are determined as described at the
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TABLE 1 0 . 1
M
Modulation:
IT
1111
Mode:
A*/2 + A°
1110
1011
1010
A*/2 - A°
X
X
AA*/2 + AA0
X
X
AA*/2 - AA°
X
X
AA"/2 + AA°
AA*/2 - AA°
o
Channel 2:
M M M /0
n a tt/2
<1
I
A"/2 + A°
X
X
M M
ir a
N
V
%
Channel 1:
MM..,
IT tt/2
where
is the interference with A ^ vertical, averaged
over the 6-wobble, with the
0-phase.
«ir/2
x
is the interference with
it/2-shifter
in the
vertical, averaged
over the 6-wobble, with the ir/2-shifter in the
ir/2-phase.
is the inteference with
in the null-plane,
averaged over the 5-wobble, in the 0-phase.
•ir/2
is the interference in the null-plane in the
7r/2-phase, averaged over 6-wobble.
AA
x
is the vertical interference in the 0-phase,
demodulated by the 6-wobble.
it/2
AA
is the null-plane interference in the
ir/2-phase, demodulated by the 6-wobble.
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end of Chapter VIII, and the difference is split to specify the
z-direction.
This procedure for measuring interference signals can, in
principle, be done at any VHF frequency, but it can be done with the
least trouble only at a few frequencies*
(These special frequencies
require no changes to extend the range of phase shifters (no extra
segments of cable), and do not require the inversion of gains in the
feedback loops which control the zeroed signals.)
the interference signal on the lock-in when
Figure 10.2 shows
is interfered with
Xg for a single travelling wave (i.e., not counterpropagated).
Except for small shifts of phase from the lengths of cable carrying
the VHF field, this figure illustrates the Ramsey fringes in the
interference of the separated oscillating fields of the Ml- and
El-regions.
The positive peaks of this Ml-Ml interference represent
equivalent places to do experiments.
B.
Calibration
Two methods have been used to convert measurements of the
El-region interference signals into "measurements" of C2p» the
proton-spin-dependent weak coupling constant.
In each method the
amplitude identified as Ag. is compared to a known amplitude, either
Ag or Ag.^.
Appendix B presents the following numerical
expressions for the three amplitudes (here I have removed an overall
minus sign):
Xg « 1.76 |Q Lp
X
_ = 99i (e x
exE
v o
(10.1)
I )L
(10.2)
_
o' exE
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SIGNAL, VOLTS
LOCK-IN
10
-
4-
f/2
Figure 10.2.
(SYNTHESIZER FREQUENCY), MHz
Interference signal between Ml-region and El-region magnetic
dipole amplitudes. The synthesizer frequency is written as
f/2 because this is doubled to produce the hyperfine
frequency, f. The line center, fh^s/2 is at 88.77843 MHz.
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1e
(10.3)
= 2.46 x 10-8 C
where the fields ate in e.g.s. Gaussian units and the factors Lg,
Ls , and Ls x e are dimensionless lineshape factors graphed in Figure
B.l(Appendix B).
The first method compares Ag to Ag.
The magnetic dipole
interference is measured with a single travelling wave in the
El-region, the (forbidden) electric dipole amplitude must be measured
with the two waves counterpropagated.
Under ideal conditions, the
electric field would then be twice what the magnetic field was for one
wave.
This gives
A
C„
= 3.6 x 107
(2 waves)
e
A„ (1 wave)
(10.4)
(the lineshape factors are equal, and cancel).
This method has the advantage that the lineshapes for the two
amplitudes are equal, but it is unreliable for two reasons.
First,
the experiments must be done at different times— so one must be aware
of drift problems.
Second, and more important, the assumption that
two waves counterpropagated doubles the field over one wave is not
good.
The gyrators which permit the counterpropagation are lossy
devices and change their transmissions from one configuration to the
other.
This can be taken into account, in principle, using the signal
labelled A A ^ ^ in Table 10.1.
This gives a measure of the magnetic
dipole amplitude taken simultaneously with Ag— that is, with waves
counterpropagated.
In practice it was more sensible to compare to the
induced Stark amplitude.
The Stark amplitude generated with the electric field from the
-
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planes of wires above-and below the beam (see Figure 7.5) has a
different lineshape from A£ , but comparing these amplitudes has the
advantage that they can be measured simultaneously.
field strength cancels in the ratio.
C2 - 4 X 109 E0 ^
r
e
Also, the VHF
We find that
P —
exE
(10.5)
As will be seen this discussion is somewhat fatuous inasmuch as the
signal indentified as Ag is much too large to be due to weak
interactions and has a frequency-dependent phase inconsistent with
that expected.
C.
Measurements of El-Region Amplitudes
The data of Tables 10.2 and 10.3 are typical in size and
precision of measurements taken to explore the El-region amplitudes.
They have been selected because they characterize these amplitudes in
and of themselves; the bulk of the experiments were aimed at finding
some experimental variable which would alter the signal or test a
hypothetical cause.
section.
These will be described qualitatively in the next
Here we concentrate on the amplitudes "as found".
The voltage listed in Table 10.2 is that applied to the upper
plane of wires (the lower supplied with its negative) to create a
static electric field in the x-direction (Eo = -V/3cm).
The signal
in Channel 2 in the modes 1111 and 1110 is then the induced Stark
amplitude calculated in Appendix B and given in Equation 10.2.
The
synthesizer frequency (f/2) was set to 89.470 MHz for these runs (see
Table 10.4 for numerical values of the lineshape factors).
From these
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TABLE 1 0 . 2 :
STARK AMPLITUDE (A g x E )
INDUCED
USING WIRE PLANES IN El-REGION
VOLTAGE
DIR
0.00
x
y
1.00
2.00
3.00
4.00
5.00
CH
1
2
0000/103
1111
1110
1011
1010
48
-13(4)
24(4)
41(1)
67(1)
44
29(2)
- 6(1)
- 2(3)
3(2)
x
1
47
-51(3)
29(4)
-43(2)
65(1)
y
2
44
-240(2)
-584(2)
2(2)
0(2)
x
1
46
-60(4)
43(6)
44(1)
62(3)
y
2
43
-507(2)
-1146(2)
4(2)
- 1(2)
x
1
46
-72(2)
65(7)
46(2)
66(1)
y
2
44
-775(2)
-1706(3)
4(1)
- 4(3)
1
46
-102(5)
75(7)
45(2)
62(2)
2
44
-1058(2)
-2214(4)
6(1)
- 5(2)
41(3)
61(2)
6(2)
- 7(4)
1
46
-94(5)
2
43
-1323(3)
118(11)
-2723(4)
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TABLE 1 0 . 3 :
FREQUENCY AND DIRECTION DEPENDENCE
OF E1-REGION TRANSITION AMPLITUDES
f/2
89.47843
DIR
X
y
X
z
89.20043
X
y
X
z
88.92843
X
y
X
z
88.51743
X
y
X
z
88.24443
X
y
X
z
CH
0000/103
1111
1110
4(10)
75(5)
1011
1010
60(7)
1(9)
73(2)
2(4)
53(5)
0(8)
76(4)
9(5)
73(8)
- 7(6)
84(6)
- 4(3)
1
2
55
53
-60(3)
70(3)
1
2
54
54
-22(7)
-69(4)
4(6)
136(3)
1
2
55
53
72(5)
159(5)
-52(13)
66(8)
1
2
54
54
-20(18)
-18(8)
29(17)
167(8)
1
2
58
54
-59(12)
164(5)
3(5)
-53(4)
40(4)
- 4(6)
30(3)
5(5)
1
2
56
56
24(13)
188(5)
22(9)
261(3)
56(5)
2(4)
37(3)
- 1(2)
1
2
58
56
45(7)
-99(7)
-69(2)
228(5)
41(8)
3(6)
48(3)
7(2)
1
2
56
56
- 1(5)
-185(5)
-20(5)
-37(1)
46(3)
0(6)
47(4)
3(5)
1
2
57
54
-33(4)
29(2)
”16(6)
221(2)
58(3)
- 2(4)
60(3)
- K2 )
1
2
55
55
-67(5)
-132(3)
61(5)
58(2)
60(2)
0(2)
60(2)
- 2(2)
64(8)
2(11)
83(10)
7(9)
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TABLE 1 0 . 4 :
f/2
LINESHAPE FACTORS FOR TABLES 1 0 . 2 AND 1 0 . 3
v T2/ tt
exE
V
LS
89.470
1.107
0.681
0.596
89.47843
1.120
0.678
0.589
89.20043
0.765
0.640
0.671
88.92843
0.240
0.255
0.309
88.51743
-0.418
-0.424
-0.499
88.24443
-0.854
-
0.666
-0.672
-
186
-
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data one finds
Aexg = 5.5 x 105 E q
cu/
(statvolt/cm)
where "cu” stands for "computer units", meaning thenumbers in the
tables.
Using Equation 10.5 one has, then
C„
= 8.3 x 103 A (with A in cu's).
2p
e
e
The observed signal of 152 computer units corresponds (at this
frequency, etc.) to a C2p of 1.3 x 10^.
The induced Stark amplitude Agxg is supposed to lie in the
y-direction, which is identified as the direction in which the probe
amplitude is unaffected by a magnetic field parallel to the El-region
axis.
An idea of how well the x-direction, or vertical, is set by
maximizing the interference with the magnetic dipole amplitude, Ag,
can be had from Table 10.2 by noticing that some of this Stark
amplitude leaks into the x-direction signal (Channel 1, modes 1111 and
1110).
This leakage corresponds to a misalignment of 3°.
In
principle the induced
Stark amplitude could be used to define the
vertical (rather, the
xz-plane) more accurately,
done.
butthis wasnot
A
similar leakage
(corresponding to a2°misalignment) was
found in
the z-direction
signal,where "z"
wasidentifiedby splitting
the x-precessor current between that for the +y-direction and the
-y-direction in the null-plane defined by zeroing out the magnetic
dipole interference with one travelling wave.
Because the
z-direction is that of the parity signal, this method was used to
define it more precisely.
One can also use the 6-modulated signal in the x-direction to
calibrate the 6-wobble.
The phase is found to be modulated by 0.0174,
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or 1° (this is the total modulation).
This is consistent with the
calibration with one travelling wave done in a separate experiment.
A small error has been made in writing Equation 10.2 (also 10.3)
by ignoring the effect of the damping term (-iy) on the phase of the
amplitude.
When this is taken into account one finds that the
amplitude should not be purely imaginary but should have an additional
phase factor
or ei5.8 _
p^ase information can be
got from the data, as can be seen from Table 10.1.
If we pair the
signals as (1111, 1110) and (1011, 1010) we have what is essentially
an Argand diagram rotated by 45°, as is illustrated in Figure 10.3.
(We could easily deconvolve the data into real and imaginary parts,
but that compounds the uncertainties, so it is better just to plot the
amplitudes in the 11-10-plane.
Because of the small phase shifts
associated with changing to two counterpropagated travelling waves
from a single wave, it is best to refer all phases to the 5-wobble
signal in Channel 1, AAX (Channel 1, modes 1011 and 1010).
This
signal is very nearly purely imaginary, which is to say that the
0-phase and n/2-phase established for the experiment get shifted
slightly (but together by the same amount).
When this is done one
finds an angle of 8°±2° between A£Xg and AAjj, which is consistent
with the 5.8° expected.
(See Figure 10.4).
The data of Table 10.2, and in fact all the data on the induced
Stark amplitude, are very gratifying.
In every way this amplitude
behaved as we expected it should based on our ab^ initio modelling.
This is to be contrasted with the stray amplitude which prevented the
experiment from reaching its goal.
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Figure 10.3.
Argand diagram showing the relation between
the mode-axes and the phase-axes.
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AA.
£xE
Figure 10.4.
The induced Stark amplitude
and the 6-wobble
signal in the vertical direction (AA ). The
amplitudes are taken from the 5 volt (fata of
Table 10.2, and are plotted on different but
conformal scales. The box around AA^ represents
the uncertainty in the data, and accounts for a 2°
uncertainty in the relative phase of the amplitudes.
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The stray amplitude is what remains when the large amplitude in
the x-direction (principally a magnetic dipole amplitude) has been
zeroed by counterpropagation.
Table 10.3 contains data for all the
experimentally equivalent frequencies with the x-precessor set so that
Am i points either along the y-axis or the z-axis (in Channel
2— Channel 1 always points in the x-direction).
We thus have
magnitude, phase, and direction information for the stray amplitude at
five frequencies.
These data are represented in Figure 10.5 by their
Argand diagrams in the 11-10-plane.
As the frequency changes the
components Ay and Ag change magnitude and phase, but it is clear
that they "go around together” in the Argand diagram and thus are not
separate systematic effects, but just different components of the same
effect (whatever it is).
The magnitude of these signals is in each
case of order 100 to 200 computer units, which is a million times too
large to discover weak interaction effects at the level expected.
The graph in the lower right corner of Figure 10.3 shows the phase
of Ay relative to the reference amplitude AAX as a function of
frequency.
If the two points taken at frequencies below line center
are shifted by 180°, as indicated, then all the points lie on a line
whose slope is about 120°/MHz.
This shift across line center is
motivated by analogy with the anti-symmetric lineshapes of the other
amplitudes, Ag and Agxg.
As noted in Appendix B, a linear
dependence of the phase with the frequency is suggestive of a localized
source of field, but it is also consistent with any anti-symmetric
field distribution whose center of symmetry is shifted from the center
conductor.
The shift can be deduced from the slope to be 4.2 cm,
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39.1*7843
AA
X
—
29*1°
8 8 .5 1 7 4 3
8 8 .9 2 8 4 3
/
X
AA
g200 _
322°
8 8 .2 4 4 4 3
100
f/2, MHz
88.0
Figure 10.5.
8 8 .5
89 .O
8 9 -5
Argand diagrams of amplitudes from Table 10.3. Each
semi-axis has been drawn 100 computer units long for
scale. The phase of Ay relative to AA^ is plotted in
the graph. The two points to the left of line center,
when shifted by 180° as indicated, lie on the same line
as the others.
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which is not associated with any obstruction in the beam path, as one
might have hoped.
Many other guesses as to the cause of this stray
amplitude have been tested and eliminated as will be described in the
next section.
D.
Search for a Cause
Various subtleties have been avoided in the discussion because
although it was long known that these might complicate the experiment,
it was hoped that they wouldn’t, and attempts to analyze them proved
difficult.
In discussing the state selectors, the precessors, and the
transition regions the "typical atom” has been taken as travelling on
the centerline of an ideally aligned apparatus.
Yet it is clear from
diagrams such as Figure 6.1 that the beam, which is 1 cm wide and 3 cm
high, comprises atoms moving off axis and diagonally through the
machine which will experience slightly modified fields— even if the
machine is perfectly aligned.
The beamline was equipped with retractable centered pointers
which allowed the machine to be aligned to a milliradian or better
using a transit, and all the imperfections in the direction of the
oscillating field in the El-region were made symmetrical by design, so
that if the atomic beam had a uniform density these imperfections
would cancel when averaged over the beam.
The stray fields, however,
have no particular symmetry, and the uniformity of the beam density is
only approximate.
The key point is that we had a nonzero amplitude in
the z-direction where we expected none.
The inhomogeneity in the
direction of the oscillating electric field— to which the stray Stark
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amplitude must be locally perpendicular— is too small to account for
the z-amplitude if the y-amplitude is used to estimate the stray static
electric field.
The z-amplitude should be less than 10~3 of the
y-amplitude, yet they are of the same magnitude.
We checked that the signal (z-amplitude) was a real interference
by looking for transient effects and for directionality (a
non-directional signal might be due to grounding currents, for
instance).
All of the lock loops and precessor currents had been made
to have settling times less than a tenth of a millisecond, whereas the
shortest transient gate was one millisecond; but the slow settling
observed in the detector when its gain is changed seemed like a
possible problem (though unlikely— once "flashed" with a high current,
the detector produced sharp-edged pulses when the Ml-region power was
switched on and off).
No change in the z-amplitude was noticed as we
lengthened the gate from 1 msec to 10 msec.• Another test that the
signal was real was to precess the Ml-amplitude to the negative
z-direction and see whether the interference changed sign or had some
"scalar" component.
It was found to change sign.
The argument that a stray Stark amplitude must be perpendicular to
the oscillating electric field depends on having zero static magnetic
field in the region.
To double check that we hadn't inadvertently left
a static magnetic field we did a careful demagnetization while taking
data— leaving all the shim currents and so on constant— and found all
signals unchanged.
The counterpropagated waves were adjusted by a feedback loop to
zero out the x-amplitude in the 0-phase (<S-locking) while the magnitude
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of the waves was not usually locked (cx-locking) in order that the
other feedback loop could be used to lock in the null-plane by
controlling the pre-precessor.
There was always a residual signal in
the x-amplitude, even though the loop was stable (it would track and
lock out a mechanically induced phase shift).
To see if the
z-amplitude might somehow be associated with this imperfection we ran
without 5-locking, varied 6, and again found no change within the
noise.
The large y-precession needed to modulate between the x- and
z—directions was also suspect.
It is in the x-sensitive position that
we lock the counterpropagating waves, define the vertical (and hence
the null-plane), and it might be that the inhomogeneity of the
precessor field over the beam introduced a mixing of the amplitudes in
some unspecified way.
We looked for a change in the z-amplitude as
the y-precessor current was reduced so that we were switching between
the z-direction and a direction in the zx-plane requiring less
precession.
At very small angles there isn't enough x-signal to lock
to, but we got down to 20° with no change in the z-amplitude and
concluded that the problem was not in switching directions.
To look
for an alignment effect we scanned the z-amplitude while sampling a
vertical slice of the atomic beam with the moveable slits and did runs
with various slit-widths.
We altered the roll, pitch and yaw as well
as the altitude of the El-region, all with no change except to reduce
the beam and all signals proportionately as we got too far off center.
It seemed certain that the stray amplitude must be a Stark
amplitude.
The El-region is made of aluminum which is known for
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charging on its surfaces.
If this charging were due to the atomic
beam hitting the center conductor or the entrance and exit apertures,
then the resulting electric field should reduce with beam intensity
and the Stark amplitude should go down faster than linearly with beam
intensity.
But varying the beam intensity changed the z-amplitude
proportionately.
If the z-amplitude were associated with an oscillating magnetic
field and not an oscillating electric field, then placing an antinode
of magnetic field should increase the signal by a factor of a thousand
or more.
(An oscillating magnetic field would be present if the
counterpropagation were cancelling a large Stark amplitude in the
x-direction, for example.)
We created an antinode of magnetic field
and found only a small signal consistent with our ability to define
the z-direction, and certainly not a thousand times as large (it was
of the same order of magnitude).
We also checked to see if the
z-amplitude might be some effect associated with the tail of the
Ml-amplitude, and so made the tail 50 times as large, with no change.
The aim in all these tests was to find some "knob” we could turn
which would change the z-amplitude in any way.
If we could change it,
we stood a chance of identifying the cause and eliminating it, but
none of these things made a discernible difference.
What became clear
as we did these experiments was that the aspects of the apparatus
under our direct control behaved as we thought they should based on
our modelling.
The precessors precessed the atoms with very little
dephasing; their motions were independent so long as the precession
angle was less than 180°; different ways of establishing directions
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were consistent with each other.
The Stark amplitude generated by the
wire planes lay in the right direction and was of the right size and
phase.
Biasing the center conductor with a static potential produced
no amplitude— as expected, since in that case the static and
oscillating fields are everywhere parallel and so the cross product is
everywhere zero.
One can even take a positive attitude towards the
null results of the various tests inasmuch as the various
imperfections which we thought should be unimportant were in fact
unimportant.
The cause of the stray amplitude, in particular the z-amplitude
although that seems to be just a part of the same thing as the
y-amplitude, is still a mystery.
possibility.
There remains at least one
The aspect of the El-region which is least well
understood is the effect of the beam holes on the oscillating fields.
In calculating the far-field perturbation due to an aperture one
replaces the aperture by an effective electric or magnetic dipole, but
the far-fields on opposite sides of the conducing wall are derived
from effective dipoles of opposite sign!
The fields in the aperture,
which by Huygens's principle propagate out to the far-field, must vary
dramatically from one side to the other.
Moreover, since the fields
in the aperture do not satisfy TEM boundary conditions, one cannot use
a Laplace integrator to compute the fields.
One cannot assume that an
aperture excited by an antinode of electric field will have only
electric fields in the hole, which opens the possibility that the
stray amplitude is not necessarily a Stark amplitude only.
What this
suggests is that the "coaxial" El-region be replaced by a structure
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similar to the Ml-region (a parallel plate line) so that the electric
and magnetic fields vanish near the beam holes for the antisymmetric
TEM mode.
If there is no field near the holes, it cannot matter
whether they are there or not.
The original intent of the "coaxial" geometry was to make the
parity amplitude, Ag, parallel to the Ml-amplitude from the
Ml-region.
It wasn't clear at first how the alignment would be
accomplished (that is, the precessors had not been introduced) and
some thought was given to mechanical modulation of the transition
regions.
Even when the precessors were introduced it wasn't at all
clear that they would work well, especially at large precession angles
where the magnetic field inhomogeneities might be 'damaging for such a
large beam cross section.
But the precessors work quite well, and so
the criterion of parallelism is obviated.
Switching to a
parallel-plate transmission line, however, is a major undertaking and
has been left for future research.
The search for parity non-conservation in hydrogen is proceeding
with the above modification with good prospect for reducing the limit
on C2 p by five orders of magnitude over the Michigan limit of 620.
A precise measurement at the level of the GWS model predictions is
evidently not in the offing, although various experimental
improvements, such as the development of an intense cold source of
metastable hydrogen for the level-crossing experiments, could improve
this situation, as would techniques to reduce stray electric fields.
Despite the formidable difficulty of this experiment, or perhaps
because of it, several technologic advances have come out of the work,
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notably our long-lived high-efficiency Lyman-a detector and the VHF
electronics to permit precise phase changes without power shifts,
which our microwave engineer assured us could not be done at the level
required.
Some minor contributions to the theoretical understanding
and formulation of sensitive interference experiments can also be
claimed, as Ed Hind’s observation that going to a level crossing is
not necessarily a good thing when the experimental parameters are
optimized, the method of parametrizing the state selection system in
terms of transparencies and power-independent exponents rather than
exponentials, the elucidation of vector amplitudes in a J=1 system,
and the statistical properties of discreet lock loops, as well as the
use of Schmidt waveforms (orthogonal square waves) to deconvolve
other-than-linear signals.
The effort has been long and arduous and the results are somewhat
disappointing, though hopefully edifying.
For my part, I am dismayed
that the enthusiasm for these parity non-conservation experiments in
hydrogen has waned in the wake of the successful UA1 and UA2
experiments at CERN, which produced candidate events for the decay of
the Z° predicted by the GWS model and crude estimates of the mass.
The feeling seems to be that these high energy experiments have
wrapped up the weak interaction questions.
The main features of the
GWS model have been verified, but open questions remain about the
correct way to do higher order calculations within the model, effects
which play a more significant role in low energy experiments than at
high energy.
Moreover, atomic (i.e., low energy) experiments can
establish limits on the masses of additional neutral bosons.
It is
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true that the present work has not produced a measurement of these
effects, but it has pressed the technology beyond what was formerly
available— an ongoing process which will in the end bear fruit if it
is not abandoned.
-
200
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BE 57
H. A. Bethe and E. E. Salpeter. Springer-Verlag, Berlin (1957).
Quantum Mechanics of One-and Two-Electron Atoms.
BL 40
F. Bloch and A. Siegert. Phys. Rev. 5£, 522 (1940).
"Magnetic Resonance for Nonrotating Fields."
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A. Breskin, G. Charpak, S. Majewski, G. Melchart, G. Petersen, and
F. Sauli. N.I.M. 161. 19 (1979). "The Multistep Avalanche Chamber.
A New Family of Fast, High-Rate Particle Detectors."
CO 41
J. D. Cobine. Dover Publications, Inc. (1958) (orig. McGraw-Hill
(1941)). Gaseous Conductors.
DE 64
Yu. N. Demkov. Sov. Phys. JETP 1j5, 138 (1964).
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DO 78
R. W. Dunford. Doctoral Dissertation, University of Michigan,
1978. Parity Nonconservation in the Hydrogen Atom.
ED 82
John Vm. Edwards, G. L. Green, and E. A^Hinds. Nucl. Instru. Meth.
197. 581 (1982). "State Selection of 2 S./2 Hydrogen in Zero
Magnetic Field."
FO 84
E. N. Fortson and L. L. Lewis. Phys. Rept. 113. 289 (1984).
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FR 81
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Role of Level-Crossing Experiments of Parity Nonconservation using
Metastable Atoms."
HMF
A. Abramowitz and I. A. Stegun. National Bureau of Standards
(1964), Dover (1965). Handbook of Mathematical Functions.
JA 75
J. D. Jackson.
(John Wiley and Sons, 1975).
Electrodynamics. Second Edition.
LA 50
Willis E. Lamb and Robert C. Retherford.
"Fine Structure of the H Atom, Part 1."
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MC 68
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ME 58
Albert Messiah. Nortb-Holland Publishing Co. and John Wiley & Sons
(New York, 1958). Quantum Mechanics.
PP 84
Particle Data Group, Technical Information Division,
Lawrence-Berkely Laboratory (1984). Particle Properties Data
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F. Sauli. (CERN Publication 77-09» 3 May 1977). "Principles of
Operation of Multiwire Proportional and Drift Chambers."
SC 68
L. I. Schiff.
Mechanics.
SE 64
I. A. Sellin. Phys. Rev., A1245 (1964). "Experiments on the
Production and Extinction of the 2s State of the Hydrogen Atom."
ST 63
A. K. Stober, R. Scolnik, and J. P. Hermes, Appl. Opt. 2, 735
(1963).
ST 70
David Storm and Donald Ropp. J. Chem. Phys. 53» 1333 (1970). "On
the Relation between Symmetric and Asymmetric Charge Exchange."
SU 79
T. J. Sumner. D. Phil. Thesis. D. of Sussex. (ILL Special Rept. 80
SU 09S). Progress towards a New Experiment to Search for the
Electric Dipole Moment of the Neutron Using Ultra-Cold Neutrons.
TISP
I. S. Gradshteyn and I. M. Ryzhik. Academic Press, New York
(1965). Table of Integrals. Series, and Products.
TR 73
T. A. Trainor (thesis). University of North Carolina at Chapel Hill
(1973). golarized Ion Source Development and the Lowest T = 3/2
Level in — Sc.
WI 79
M. W. Williams and E. T. Arakawa.
n(lz-d, MgFg)
Phys. Rev. A 10, 797 (1974).
McGraw-Hill Book Co., New York (1968).
-
202
Quantum
Appl. Opt. J8, 1477 (1979).
-
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APPENDIX A:
TWO-STATE TIME-DEPENDENT PERTURBATION THEORY
For two states coupled by an oscillating field the Hamiltonian in
the Schroedinger picture can be approximated as
(A.l)
with E2>E1 and Yi,Y2 >
The inclusion of imaginary terms in the
energy eigenvalues is known as the Bethe-Lamb prescription.
If we
:e V=0 and |i|i(0)> = J1> then
2 = e
.jhe Y «s
decay of each state.
are thusidentified with half the rate of
This prescription is not always justified, but
in the system of 2s j /2 and 2p^/2 states of hydrogen the method has
a rigorous foundation (HI79).
Another approximation made in writing equation (A.l) lies in
ignoring the counter-rotating wave.
An oscillating field (~ cosrnt)
will generally contribute both terms e+^u,t and e~ialt to each
matrix element.
One of these terms has been dropped in each
off-diagonal element of A.I.
approximation.
coupling.
This is known as the rotating wave
It is justified when
uj=E2-E1;
that is, near resonant
In that case the effect of the counter-rotating wave, of
frequency (-ui), is dwarfed by the resonance.
In many cases the
missing matrix elements account for a small shift in the location of
the resonance (BL40).
This is negligible for all cases in this
thesis.
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The merit of the approximations is that the problem posed by
(A.l) is exactly solvable in closed form.
Throughout the thesis the
interaction picture is preferred to the Schroedinger picture.
The
Schroedinger equation is
itfs " V
but if Hg = H q + %
s
(A. 2)
where Hq is time-independent, then we may
let ijjg = e“*®oti|> define an interaction picture state i|». With this
substitution, Equation A.2 becomes
iif =
iH t
-iHt
(e
Hge
)ij>.
(A. 3)
The situation is least confusing if Hq is chosen Hermitian.
E,
Ho = ( q
Taking
0
e 2^
(A. 4)
we find
iH0t
Hint S 6
-iH t
HS e
where v = uj-(E2 -Ei).
.
-iy.1
„ -xvt
Ve
:
TI. ivt i
V*e
.
iy.
^2
(A. 5)
The solution to (A.3) can be expressed as
i|i(t) = U(t)ij»0
(A.6)
where U(t) is the (interaction picture) time evolution operator,
satisifying the boundary condition U(0) = 1.
With
V± = - \ ( i v + Y j + y ^ ± j /(iv+(Y1-Y2 ))2 “4 |v |2
we have for
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(A-7 )
the solution
IT
ivt
U11 = e
ivt
U. . = — ttt—
12
IV
-iV
21 -
U
=
22
\ W++Y2
jivnC
u+c
v- ^ 2
w-c (
6
’ »+-n_ 6
(
(M.+Y,)(n_-hr9)
— ±— *------—
y+-p_
y+t
u_t
(e + - e
)
(A>9)
, H*
y- \
(e
- e
>
-1
i
... . *■*+'
|(y_+Y2
)e
- (u++ Y 2) e
y+- u_
Two special cases are applicable to this thesis.
j
For the hyperfine
transition in the 2s\j2 manifold we may regard the states as
infinitely long-lived.
Then y* = Y 2 =
Thetl c^e exponents y± are
imaginary.
Letting
= /
v 2+4|v
|'
(A.10)
we find
U,, =
H
ivt/2
^--- (n cos -=flt - iv sin — fit)
^
2
2
° U ■ slVC/2
U21 = e'1'":/2
U 22
=
sin \
fit
CA-U)
,ln I a,;
-ivt/2
.
.
jj-- (0 cos y Ot + iv sin
ftt)
Note that if we use first-order time-dependent perturbation theory
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t
U(t) = 1-i / dt Hint(t)
(A.12)
o
to compute U 2 1 , we find
•ivt/2
(A.13)
12/ 2
1 V /v « 1
First order perturbation theory is thus warrantable away from
resonance.
In the event that
| v | t « l (short transit times) this can
be extended close to the line center since for v = 0
sin -j vt
1
(A.14)
This situation is realized in both the Ml-region and El-region of the
present experiment so that first-order time-dependent perturbation
theory can be used to obtain the lineshape (Appendix B) when V (i.e.,
the field strength) is not constant but varies slowly in time.
The second special case of A.9 is that for which Yi=0» Y2=Y*0.
This is the circumstance relevant to the state selectors where state
|l> is an s-state and state |2> is a rapidly decaying p-state.
In
this case we have (for jV|2<<y2)
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U+ = -
= ~
J
(iv+Y) ± J
\ (iv+y) ± j (iv-y)(l -
/ ( i v - y )2 -
4
|v|
I 12
2)
(A. 15)
(iv-y)
-Y
, ,2
r
t,
2y
V
-iv ; r =
2 * 1
2 ' 1 2
v + y
After a short time the terms of A.9 with e^+c die away leaving only
the e^-c terms.
Un
In particular
- eivt e"-* - e-rt/2.
This is the amplitude to remain in the s-state.
(A. 16)
The survival
probability is then given by
Uu |2 = e"rt
with
T =
2y IV
T~T "»
2. 2
v +y
(A. 17)
This provides the basis for the analysis of
Chapter V.
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APPENDIX B :
A.
LINESHAPES
Ml-Region
The Ml-region is a parallel-plate transmission line in a grounded
cylinder (Chapter VII, Figure 7.2).
It is operated in its
anti-symmetric TEM mode (i.e., the plates have opposite voltages).
Table B.l tabulates the magnetic field on the beam axis, midway
between the plates.
The field was derived from electrostatic
potentials obtained by relaxing a grid with the boundaries held at
constant potentials.
(At each step in the relaxation method a grid
point is assigned the average of the potentials at the surrounding
four grid points.)
equation.
This provides a numerical solution to Laplace's
The unit field for Table B.l is that obtained by dividing
the potential difference of the plates by their separation.
(In
e.g.s. units electric and magnetic fields have the same dimensions.)
From the field distribution one obtains the transition amplitude
lineshape by Fourier transforming.
time-dependent perturbation theory.
That is, we use first-order
Each triplet of points in Table
B.l was interpolated by a parabola as in Simpson's Rule.
field distribution is even,
Since the
onlythe cosinetransform wasneeded.
The
contribution from each parabolic segment was evaluated using
dt cos
t = — sin
dt t cos
and
dt t
cos
t = —
t = —
t
cos
cos
t + — sin
t
t + (—--- — ) sin
(B.l)
t.
The conversion to time from position is effected through z/R = t/T
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TABLE B . l :
32z/R
M l-R E G IO N FIELD
Bz/BQ
32z/R
Bz/BQ
0
1
2
3
4
0.963
0.962
0.958
0.954
0.947
15
16
17
18
19
0.645
0.597
0.548
0.497
0.447
5
6
7
8
9
0.938
0.925
0.909
0.890
0.867
20
21
22
23
24
0.399
0.352
0.308
0.265
0.228
10
11
12
13
14
0.839
0.808
0.773
0.734
0.691
25
26
27
28
29
0.190
0.155
0.120
0.101
0.066
30
31
32
0.042
0.021
0.000
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with Tj = 0.058 ysec.
v = 30 cm/ysec.)
B.
(That is,
= a/v with a = 1.75 cm,
This gives the solid curve of Figure 7.3.
Idealized El-Region
The simplified anti-symmetric lineshape of Chapter II (Figure
2.2) is obtained by Fourier transforming the anti-symmetric
distribution
-1: -T2 < t < 0
f(t) -
1;
0 < t < T2
0;
elsewhere
(B.2)
We have then
T
A - / dt e *Vt f(t)
-T
r
= 2iT2
C.
sin2 i vT2
j------ ’
(B,3)
El-Region Magnetic Dipole Amplitude
The El-region is a cylindrical "coaxial” transmission line.
Its
center conductor is a thin vertical plate whose height is 0.80 of the
diameter of the cylinder.
impedance.
This gives a fifty ohm characteristic
The cylinder radius is 12 cm.
Table B.2 tabulates the
electric field on the beam axis (perpendicular to the center
conductor).
The unit field is the potential difference between the
center conductor and the cylinder, divided by the cylinder radius, R.
The field is anti-symmetric in z (see Figure 7.5 which is drawn from
Table B.2).
As noted before, the oscillating magnetic field in a TEM
mode has the same magnitude as the oscillating electric field.
The field values of Table B.2 are reasonably well fit by the form
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TABLE B.2:
E1-REGION FIELDS
z«e/£o
x.E/E
0.000
0.031
0.063
0.094
0.125
1.315
1.315
1.311
1.305
1.296
0.000
0.093
0.183
0.295
0.386
0.000
0.122
0.240
0.385
0.500
0.156
0.188
0.219
0.250
0.281
1.283
1.264
1.245
1.228
1.212
0.466
0.531
0.580
0.618
0.646
0.598
0.671
0.722
0.759
0.783
0.313
0.344
0.375
0.406
0.438
1.190
1.165
1.139
1.114
1.088
0.666
0.678
0.685
0.687
0.684
0.793
0.790
0.780
0.765
0.744
0.469
0.500
0.531
0.563
0.594
1.059
1.027
0.995
0.963
0.931
0.676
0.661
0.639
0.610
0.572
0.716
0.679
0.636
0.587
0.533
0.625
0.656
0.688
0.719
0.750
0.899
0.867
0.835
0.803
0.771
0.524
0.469
0.409
0.349
0.293
0.471
0.407
0.342
0.280
0.226
0.781
0.813
0.844
0.875
0.906
0.746
0.723
0.698
0.672
0.647
0.241
0.195
0.155
0.118
0.086
0.180
0.141
0.108
0.079
0.056
0.938
0.969
0.628
0.615
0.056
0.027
0.035
0.017
z/R
-
211
o
le * E K
E <,
-
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8 = 8q [A + B cos(Cz/R)]
(B.4)
with A = 0.941, B = 0.386, and C = 2.717.
To obtain 8 as a function
of time we replace z/R by t/T2 with T2 = 0.40 psec.
With
A
0
= -if 2dt
-T,
e-iVC p8
(B.5)
we find
A8 = 7 U0o T2
" cos
v T2>
+
[1 "
c °s (vT2-W>]
+
[1 " coe(vT2-C)]} = - j p8oT2Le(vT2)
(B.6)
This gives the solid curve of Figure 7.7.
D.
El-Region Second-Order Amplitudes
In Chapter VII a non-trivial point was omitted in the derivation
of second order amplitudes.
There the emphasis was on the directional
dependence of the Stark and parity amplitudes on the static and
oscillating electric fields.
The zero-field Hamiltonian was taken to
be Hermitian in writing equation 7.34:
Hi -
iH t
e
<He + He + Hpnc)
-iH t
e
.
(B.7)
In deriving the interaction picture Hamiltonian H^ it is the inverse
of the zero-field evolution operator which enters on the left and not
the Hermitian conjugate.
This means that Equation B.7 (or 7.34) is
correct as written, even when H
o
is not Hermitian.
The H
o
in the
first exponential factor should not be conjugated.
The energies W^ appearing in, say, Equations 7.38 and 7.39 are to
be replaced by (E^ - iy) where y is half the p-state decay rate.
-
212
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In
the integrals of Equations 7.38 and 7.39 we then have terms of the
form
K =
/dTj /dr2
el0tTl
e1^ 2 F(ti ) G(t 2)
e~Y(Tl'"r2)
(B.8)
In writing this I have emphasized that the upper limit of the x2
integration is xj^
damping factor.
That is, t2<t1 so that the last factor is a
Because of this damping the slowly-varying envelope
G(t 2 ) can be expanded in a Taylor series around Xj.
Retaining only
the zeroth order term G(Xj) gives
K -
/dx,
FtT.) OCT,)
(B.9)
18 + Y
In computing lineshapes only the terms near resonance (i.e., a+8=o)
need to be retained so that the constant in the bracket can be
dropped.
In this approximation the integrals I(oj) and J(u>) of
equations 7.38 and 7.39 are given by
I(m) =
JdTj
e
-ivx,
f(Tl)g ^
i(Ek-WQ )T 7
(B.10)
j(co) =
Here
fdTi
v =oj-(W^-Wo )
e
-ivt,
f(Tl)g
_w
...
i(Ek-Wo-m)+y
is the difference of the field frequency from
resonance with the 2s,/2 hyperfine splitting.
The energy
denominators can be separated out to give the lineshape function
L
£X1j
(vT2) = -i/dr
m
e"ivT f(x) g(x).
(B.ll)
t2
The parity non-conserving amplitude X
the same lineshape as
Xa
(Figure
P
has g(x)sl so that X
has
7.7). For the Stark amplitude due to
stray fields the envelope g(r) is unknown.
Note, however, that if the
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stray electric field is localized (£ = S
L(v) =
o
5(t-T)) that we obtain
e“ivT f(T) g(T)
(B.12)
so that the phase is linear in the frequency.
E.
Stark Amplitude from Applied Field
Four wire planes provide a vertical electric field in the
El-region.
Each plane contains four wires run parallel to the axis of
the transmission line.
They are located at distances of 2, 4, 6, and
8 centimeters from the center conductor.
Two planes are up beam of
the center conductor, two are down beam, located at distances of 3 cm
above and below the beam axis (see Figure 7.5).
When the lower planes
are biased positively and the upper planes negatively by the same
amount, there is a vertical electric field on the beam axis.
The
center conductor and the cylindrical shield are held at d.c. ground.
Table B.2 contains the VHF electric field and this static
electric field from the wires as well as their cross-product as
functions of position on the z-axis (z=0 at center conductor).
usual
eq
As
and EQare the naive fields obtained by dividing the
potential differences by the radius of the cylinder (eQ) and the plane
separation (Eq).
The numbers listed under |e x SJ/eQEo represent the product
f(t)g(T) in Equation B.ll.
To produce a lineshape conveniently I
first fit the e x I distribution with a polynomial.
can be had by taking |e x
= eQEo h(z/R) where
h(5) = a ? (l-5)(l+b§); £ > 0
with a ** 5.43, b =* 1.02.
A reasonable fit
(B.13)
For negative values we take h(£) = -h(-£).
The lineshape is then given by
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=
2f1 dC sin (vT2C) h(S)
0
(B.14)
(T2 is the transit time for half the El-region.)Since h(£) = a£ + a(b-l)s2“bg3 we need integrals of the form
1
/
d? sin a? • ?n.
(B.15)
0
These integrals can be had by taking derivatives of the following two
integrals:
X1 (®) = /
I2 (a)
d5 sin a? =
---^°S a
= / d£ cos a? =
0
(B.16)
(B.17)
°
In particular we need
»
*2 («)
II (a)
*
“ “J d? sin a? • ?
0
_ cos a _ sin a
a
2
a
1
= "/ d? sin a? •
0
cos a
a
(B.18)
o
(B.19)
2 sin a , 2
(1 - cos a)
a2
a3
and
I2(3) (a) = /r1 d? sin aS • 3
0
_
cos a , 3 sin a . 6 cos a
(B.20)
6 sin a
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TABLE B . 3 :
v T2/it
E1-REGION AMPLITUDES
Le x E
L. ,L
8’ e
0.1
0.2
0.3
0.4
0.5
0.110
0.214
0.316
0.410
0.490
0.134
0.262
0.379
0.481
0.564
0.6
0.7
0.8
0.9
1.0
0.560
0.614
0.654
0.676
0.688
0.642
0.662
0.676
0.669
0.642
1.1
1.2
1.3
1.4
1.5
0.682
0.662
0.630
0.592
0.544
0.600
0.545
0.482
0.416
0.352
1.6
1.7
1.8
1.9
2.0
0.492
0.436
0.380
0.322
0.268
0.291
0.239
0.196
0.165
0.144
2.1
2.2
2.3
2.4
2.5
0.218
0.176
0.136
0.102
0.076
0.135
0.134
0.141
0.153
0.168
2.6
2.7
2.8
2.9
3.0
0.056
0.042
0.034
0.028
0.026
0.182
0.194
0.203
0.206
0.204
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1.0
-TTRANg/r/O*/
AMPLfTUDB
0.0
C.S
/•O
/. 5
2.0
vT
i r
Figure B.l.
Lineshapes for El-region amplitudes.
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Twice this integral (LexE) is tabulated in Table B.3, along with the
corresponding integral for
Lg (or he).
They are both plotted in
Figure B.l.
F.
Explicit Values for El-region Amplitudes
The lineshape factors
dimensionless functions of
vT, = (to - E, , )T~.
^
nrs *
%
m T
Lg, L£, and 1*^^ have been defined
the dimensionless variable
The El-region amplitudes are then given by
I2 T2 LS
CB-22>
- 3(eao>2 i(T - * eo>t 2 U
t = ea i ~
s
o
2
to be
{---- 2 _ _
-a>oo —ly
-
0'---T y -
-1Y ) L CKE
- ---l _ 4 l
<B -23>
(B.24)
1
in which the VHF fields appear halved because only one of the
exponentials in the expansion of coseot gives a resonant contribution
(■j e""^u)t) . Evaluated numerically these expressions become
la
P
XS X E
Xe
= -1.76 g
O
La gauss-1
(B.25)
p
= -99 i (e xS )L
„
= - 2.46 x 10-8 C„
2p
i
O
O
(stat volt/cm)-2
o
e
e
L
(stat volt/cm)-1
(B.26)
(B.27)
The following formulae are given for reference:
T2 = 12cm/(30cm/usec) = 0 . 4 psec
ojqQ
= WS0 Wp0
— 2x(968.8MHz)
oill
= WS1 — ^Pl = 2ir(1087.1 MHz)
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oiq2.
= ^so — ^pi — 2ir(909.6 MHz)
(ij|Q
3
^S1 *
Ehfs= WS1
Y
^PO
MHz)
= 2 tt ( 1 1 4 6 » 3
“ WS0 = 2ir(177.55686 MHz)
= 2ir(50MHz)
p = 2ir(1.40 MHz/gauss)
eaQ = 2ir(384 MHz/(stat volt/cm))
v =
GpCt
__
(£)<♦
2ir/2a
_ _ _
n
= 8.o3 x io-2 sec-l
/n
t = $o f(t) cos
(Jit
e = Co f(t) cos
oit
t = t0 g(t)
_
_
Lfi " L p
e
=
./To dt-ivr o,
.
j
T
e
f(T)
-T2 2
1
. /To dT -Ivt
LexE = 1 / !2 f j e
f(T)g(t)
“T2
v 0 = V (C
+ 3C
P
)
P
vx = V (C - C )
1P
p
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APPENDIX C:
A.
PRECESSOR FIELD
Precessor Field and Fourier Transform
The precessors are made from two rectangular current loops
wrapped around a cylinder.
Figure C.l shows the flow of current in
the two straight segments, parallel to the cylinder axis (z-axis), and
in the two split circular loops.
The current in each of the straight
segments is 21 and the y-component (see Figure C.l)
the z-axis is the same for each.
of the field on
The other components cancel.
Together the straight segments contribute a field
B (1) = 2 (2I/c)
7
/ bdz'
“b
-----------r,
,v2 . 2i3/2
[(z-z1) + a ]
(C.l)
in which c is the speed of light and a is the radius of the cylinder
(that is, the straight segments are at x=±a).
For a split circular *loop in the xy-plane with the currents in
the two semi-circular arcs running generally in the x-direction
(Figure C.2) the field on the z-axis is also in the
y-direction. The
field from such a loop is
*
A
B = y
la
—
c
-4z
— s--- o o/T
[z2 + a2]372
(C,2)
For the precessors there are split loops centered at z=±b.
The loop
at +b has its currents running in the +x direction and the one at -b
has then running in the -x-direction.
The combined field from the two
loops is then
w (2)
*
41a
'
,
b—z_______
[ < W > 2 + a2]3/2
b+z
[ O H - ^ + a 2]3'2
and from the straight segments
-
220
-
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2a
2b
0 .8-
0 .6-
0.4-
0 . 2-
-4
z
Figure C.l.
Precessor coils (rectangular loops wrapped on cylinder)
and field strength on beam axis.
-
221
-
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I
Figure C.2.
A split circular current loop centered at the origin.
-
222
-
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B(1)= —
7
>'Z
aC
+
b+Z
/ (b-z)2 + a2
/
(C.4)
(b+25)2 + a2
Setting z=vt, b=vT,- and a=vx, where v is the beam velocity, we obtain
as a function of time the field seen by an atom travelling along the
z-axis. We have
By(t)
-
B<1) (t)
+
B<2) (c )
I=£----y
J(T—t , 2
“
.(2). 41
By
2
“
+
2E
+ T2
, W
T-t
CC.5)
2T
.
[(I-t,2 + T2]372
t2
T+t
[(T+t)2 + + 2]3/2
To compute the transition amplitude on passing through the precessor
we need the Fourier transforms of these fields.
Each of
and
B ^ ) is of the form
f(t)
=
g(T-t)
+
g(T+t)
(C.6)
so that
f(ui) =
/ dt
e iu)t f(t)
=
/ 3t
gd-.).'1*'
+ ! 3c g(T+t)e—2ojC
-00
(C‘7)
*09
Letting £ = t-T in the first integral and £ = t+T in the second we
have
- e 1 “ '1
; 3 s g C - E ) . ' 1" 5 + e ‘ 1“ T / 3 ? g C E ) . ' 1" 5
— CO
For each of
and
b (2)
(C.8)
— CO
we have g(-£) = ~g(?)* Thus
f(u) = -(elu)T - e"iuT)
/ %
g(C)e"iujC
(C.9)
— 00
Again, because g is odd, only the sin tog in e iu)^ gives a non-zero
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term and the integration can be done on (0,»). We have then,
f(aj) = -4 sin u>T /Q”d§ g(?) sinco?
Evidently
(C.10)
f(-w)=f(co) so we can take co to be positive and replace it
by 10)| at the end.
We must evaluate the two integrals:
ii =
°°
g sin cog
J o d5 .
/ 2 . 2
✓g + T
(c-n )
lo
* =
-
/L«"“d«s ■ i z T zz% 2
( c - 12>
and
[ 5 Z + T ]■
To do this we use
.
(n<*>
dx
0
cos ax
ax
•
/ “2
H
+ x
- Krt(af3) ; a>0, Reg>0
(TISP 3.754.2)
2
and
/ "dx V
0
[g
Note that
X2 =
-ln f W
+ x ]
—dn- - =
L
=
V
=
-r—
a K_(ag); a>0, Reg>0
°
cos- H^—
(TISP 3.754.3)
So that
“T) = TK0(u,T)
-t K i (u)t )
(HMF 9.6.27)
And with the second integral (TISP 3.754.3) we have
12
=
(OK q ( cO T )
The functions Kg and
are modified Bessel functions. For the
Fourier transform of the magnetic field we have then
B(co) =
t
sincoT [Ki (cot)
-cot K o (cot)]
(C.13)
For large arguments the Bessel functions are given by
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K0(x) = Ki(x) a 1.253 e-X/Vx
(C.14)
(HMF 9.8.6 and 9.8.8).
For small arguments we have the ascending series
K0(z) = - (In | + Y)Iq (z ) + |z2 + ...
(C. 15)
I0(z) = 1 + ^ z 2 +
(C. 16)
with
(^ z 2 ) / ( 2 ! ) 2
+
...
(HMF 9.6.13 and 9.6.12). That Is,
K 0(z) = -(ln-| + Y)
and since
0(z2,z2ln z); y = 0.57721
(C.17)
= -Kg, we have
Kj^(z) - 1/z
Thus as
+
+
0(z, z In z).
(C.18)
uh -0,
sinwT
K i (ojt)
T/t
sinuiT
K o (w t ) -*■ 0.
(C.19)
«•
81
Then B(0) = — (2T).
field.
»
00
But B(0) = /
dt B(t), the integrated magnetic
This means that for rotationswe can model the precessor as a
region of uniform field B = 8I/ac applied to theatoms
in the time
interval from -T to T.
B.
Transition Amplitude and Phase Shift
To estimate the transition amplitude on passing through a
precessor we let w = 2ir x 177 MHz = Ehfs, and we evaluate the
times
t
and T from the dimensions a=3cm, b=2.5cm.
at 30 cm/psec this gives T=l/12 psec and
uT = 92.7 and
iot
= 111.
t
With the velocity
= 1/10 psec.
We then have
The Bessel functions evaluated at the large
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argument of 111 give
-50
(8.37)
K0 (lll) = K^(lll) = 5.97 x 10
so that
(8.38)
B(Ehfs) = (2BT) (t /T) sin 92.7 (5.97 x 10~50
-111 x 5.98 x 1(T50)
or,
(8.39)
B(Ehfs) = (2BT) (7.95 x 10"48)
The transition amplitude (Equation 8.17) is thus
<syju| so> = iii(l-n) B(Ehfs)
(8.40)
= i(2W (l-n)BT) (7.95 x 10~48)
The quantity 2u(l-n)BT is the angle through which the F=1 states are
precessed by the field, which need never be larger than
transition amplitude is therefore completely negligible.
it.
The
This should
be contrasted with the transition amplitude one predicts by taking the
field to be uniform in the interval from -T to T.
results of Appendix A are applicable.
In that case the
We find
(8.41)
<sy|u|so> = -i(2y(l-n)BT) —
in which q = y (l-n)B/E<Tr/185 = 0.017.
This gives a transition
amplitude of 0.017 for a rotation through ir.
wrong.
This is enormous, and
We were unable to detect any transitions owing to passage
through the precessors— which is consistent with the more careful
calculation.
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That no transitions are driven means that the evolution through
the precessors is adiabatic.
The atom evolves so as to remain in an
energy eigenstate at all times.
Since the operator M' * u(l-n)S» (is-f)
mixes the state |so> with the state parallel to the magnetic field,
these states repell.
That is, the eigenenergies are shifted from
their zero field values.
This shift in energy means that the F=1
state parallel to the magnetic field undergoes a phase shift relative
to its zero field phase.
This phase shift is given by
A<f> = - {dt AE(t)
in which AE is the shift in energy in the field
B(t).
Diagonalizing
H q + M* gives the shift as
AE =
^Ehfs (A + 4 q 2
-1);
q«p(l-n)B/Ehf g
or, approximately,
“
-
In the uniform-field model this gives a phase shift of at most
(0.017)2 (2ir x 177 MHz x
1/6 ysec) = 0.053, or 3°.
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A P P E N D IX
D
581
N uclear Instrum ents and M ethods 197 (1982) 581-584
N orth-H olland Publishing Company
STATE SELECTION O F 22S 1/2 HYPERFINE LEVELS O F HYDROGEN IN ZERO
M AGNETIC FIELD
John Wm. EDW ARDS, G.L. G R EEN E and E.A. H IN D S
Physics Department, J. W. Gibbs Laboratory, Yale University, New Haven C T 06520, U.S.A.
Received 12 June 1981 and in revised form 6 Novem ber 1981
We describe a m ethod of hyperfine level selection using rf quenching of a metastable (2S) hydrogen beam in zero m agnetic field.
T he method provides excellent suppression o f unwanted states with acceptable transmission of desired states. The technique has been
used to study a vety weakly induced hyperfine transition.
nique which utilizes certain unique features of the B = 0
level crossing. Donnally [7] has developed an rf tech­
nique in a weak magnetic field to select the lower of the
two hyperfine levels for subsequent conversion into
proton polarization by increasing the magnetic field.
Each of the methods mentioned above requires the
use of a static magnetic field. Lundeen and Pipkin [8],
1. Introduction
The n — 2 manifold of hydrogen has traditionally
proved to be an exciting laboratory for quantum physics
[1], Experimentally this is largely because the 2S states
are metastable, their long natural lifetime being very
convenient for resonance or other techniques. In an
electric field these metastable atoms are quenched and
emit ultraviolet light (Lyman a) which can be detected
with low noise. Hydrogen is of considerable importance
due to its theoretical simplicity.
Lamb and Retherford [2] pioneered microwave spec­
troscopy on » = 2 hydrogen with the very significant
measurement of the 22S1/2-2 2P,/2 interval (the Lamb
shift) which provided impetus for resolving divergence
problems in quantum electrodynamics. Since then, a
variety of precise and important experiments have been
carried out on metastable 2S systems. Lamb anticipated
the state selection schemes used in these experiments,
most of which exploit the short lifetime of the 22P,/2
states. The 2S states can be mixed with 2P states by
J an
electric field so that subsequently they decay. Fig. 1
shows the Breit-Rabi diagram for these states. The
Greek and Latin labels are those of Lamb; the sub­
scripts are the z-projections of total angular momentum.
In many previous experiments, use has been made of
the fie level crossings near B = 575 G to remove the fi
states. Where two states cross, the fie mixing due to a
static electric field far exceeds any mixing of a states so
that the fi's may be selectively removed from the 2S
population. This method has been used by Heberle et al.
[3] in a precise measurement of the 2S hyperfine inter­
val, and by Robiscoe [4] in a precise determination of
the Lamb shift An ingenious method of selection using
simultaneous rf and dc electric fields at the 575 G level
crossings has been developed by Ohlsen and McKibben
[5]. Sona [6] has described a nuclear polarization tech-
tooo
730
500
25 0
1
>
re
zLlI
-5 0 0
-7 5 0
-1000
-1230
O
100
200
300
400
MAGNETIC FIELD
300
600
(GAUSS)
Fig. 1. The B reit-R abi diagram for the n = 2, J = 1 /2 states of
hydrogen. States fi0 and f0 have total angular m omentum F = 0 ,
all other states have F = I.
0167-5087/82/0000-0000/502.75 © 1982 North-Holland
-
228
-
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J . Wm. Edwards el al. / 2~St2 hyperfine levels of hydrogen
582
in their presice Lamb shift determination, have selec­
tively detected metastable sublevels at zero magnetic
field by a purely rf quenching technique. Our scheme
applies rf quenching to select each of the hyperfine
levels separately so that weakly induced hyperfine tran­
sitions may be studied.
The quenching of a long lived 2S state (natural
lifetime £s) has been treated by Lamb and Retherford
[2]. When states |S) and |P) are coupled by an electric
field t = «0 cos ut, an atom initially in state |S) decays.
This decay is approximately exponential with rate T
given by
T ..2y\V\2
(0
where X = u — (E P —E s)/h, V = ^ < P \ e r - e 0\S>/h
and y = (2rP)-1 = let X 50 MHz. This result is a good
approximation for | V\2 « y 2,i.e., at low power in the rf
field.
Fig. 2 shows the n = 2, J = 1/2 states of hydrogen in
zero magnetic Held. -Here the appropriate quantum
numbers are the orbital angular momentum L, the total
angular momentum F, and M its z-projection. Three
quenching resonances exist between the S and P levels
as illustrated by Lorentzian lineshapes in the same
figure. The induced decay rates Tf m for a uniform rf
field at frequency a can be written as
r
A0I
r„ =
r,n =
2rm2
/
\2
("-"oi) + r2’
2y| V \2
(«-«,, )*+ Y2’
2 t | V \2
(2)
(n-co.o^ + Y2’
where w0,/2w = 909.6 MHz, ajm/2it = 1087.1 MHz,
and u l0/2 tr = 1146.3 MHz.
n
(O.O)— (I.O)
(I,±11— (I.*I)
(1.01— (O.O)
II2<
P
11/20-
Fig. 2. The allowed quenching transitions between various sublevels of 22S |/2 and 22Pl/f2-The z-direction is taken to be that
of the quenching field. The frequencies of these transitions are
given in the te x t
2. Experimental technique
The metastable H(2S) beam is produced by passing
500 eV protons through cesium vapor. The cross section
for charge exchange into the 2S level has a broad peak
at this energy [9]. The proton beam is extracted from a
duoplasmatron ion source of the same type widely used
for polarized proton work [10]. The beam emerging
from the charge exchange canal contains approximately
equal amounts of H(2S), H(1S) and ions [11]. The
charged beam is swept away with a 10 V/cm transverse
electric field, leaving a neutral beam whose population
of metastable atoms is about 50%. The duoplasmatron,
cesium canal, and charge deflector are collectively called
the source.
Following the source is the first of the two state
selective rf quench regions. This is a cylindrical micro­
wave cavity, coaxial with the beam velocity, operated in
the TM010 mode at frequency « 0/2 tr = 1119 MHz which
preferentially quenches the F = 1 states. We shall refer
to this cavity as the 0-selector. The frequency was
chosen to maximize the ratio of selected F = 0 atoms to
unwanted F — 1 atoms. The second state selecting
quench region consists of a similar cylindrical micro­
wave cavity operated at a frequency u t/2v = 892 MHz
which is optimum for the selection of F = 1 atoms. This
second cavity is the 1-selector [12]. The metastable
beam surviving these regions is detected by dc quench­
ing in an electric field and the Lyman a light emitted is
detected in a commercial ion chamber.
The cavities are cylinders of stainless steel with their
internal dimensions machined to the appropriate dimen­
sions for the TM01C mode. The stainless steel cylinders
serve an additional function as vacuum vessels. There
are two coupling loops placed diametrically opposite
one another on the cylindrical surface. One of these is
used to couple power into the cavity and the other as a
pick-up loop to monitor the power level in the cavity.
The transmission T of the state selectors may be
expressed as T = T/I0 where I and I0 are the detector
currents with and without microwave power applied to
the cavities. Experimental transmission data are shown
in fig. 3 as a function of microwave power monitored by
the pickup loops. When either state selector acts alone
(no power in the other selector), the theoretical trans­
missions T0 for the 0-selector and T, for the 1-selector
are given by T0 = 2 /ie x p ( - /f 0,F0) and T, =
2 f-4«tp(—Af/Pf). The summations are taken over the
four hyperfine states, each with a weight of 1/ 4 since
one expects the vector and tensor polarizations of the
beam to be small for an S1/2 state, P0 and P, are the
power levels in the state selectors. The coefficients A 0i
and A u are obtained using eq. (2) and are constant for a
fixed cavity frequency. The predicted transmission
curves for the two selectors are shown by the solid
curves in figs. 3a and 3b. These curves are one parame-
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583
J . Wm. Edwards el al. / 2 ~St i hyperfine levels of hydrogen
(a) O -Selector Transmission
I - Selector Transmission
0 .5
P0 = Q
Pq = 2.0mW
— P, = 0 .70m W
o
H
H
c
o
U)
w
E
in
c
o
k_
I-
.0 5 - 1
-I
0.0
2.0
4.0
6.0
8.0
10.0
12.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
I-S elector Loop Power (mW)
0 - Selector Loop Power.(mW)
Fig. 3. Figures a and b represent the transmission as a function of pickup loop power for the 0- and 1-selectors, respectively. T he solid
lines are the theoretical transmissions for each selector with no power in the other selector. The dashed lines are the theoretical
transm issions for each selector with power held constant in the other one.
ter fits to experimental data, this one parameter can be
viewed as a calibration of the pick-up loop.
If both selectors operate simultaneously, the trans­
mission through the pair will be a more complicated
sum of the form T — 2,iexp(—A 0IP0 — A uP t). Fig.3
also shows the measured transmission as a function of
power in one selector with the power in other held
constant The dashed curve is a theoretical prediction
based only on the assumption of no polarization in the
initial beam.
This state selection scheme, in addition to being
simple both in principle and in practice, provides a
highly selected beam with very low background leakage.
The combination of this high selectivity with operation
at zero magnetic field is a most attractive feature of this
method. It is, for example, ideal for sensitive experi­
ments in which residual magnetic fields may cause
unwanted effects. A demonstration of the very high
selectivity is seen in the resonance displayed in fig. 4.
This was obtained by inducing F = 0 to F = 1 transi-
5.0
b
HO
m
3.0
Z
o
S 2.0
sto
z
g
i-
i .o
0.0
165
170
175
180
185
190
FREOUENCY ( MHz)
Fig. 4. Beam transmission as a function o f frequency in a third
cavity placed between the state selectors.
- 230 -
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584
J. Wrn. Edwards el al. / 2~S/, hyperfine levels o f hydrogen
tions between the state selectors through the use of a
third cavity tuned to this Ml transition. It should be
emphasized that the transition probability was ap­
proximately 10-3 yet no sophisticated signal integration
was required. The data in fig. 4 were read directly from
an electrometer dial.
The authors would like to acknowledge R. Fong-Tom
and W. Howe for their expert technical assistance and J.
Prestage for his work on the hydrogen source. This work
was supported in part by the National Science Founda­
tion under Grant No. PH779-25353.
References
[1] D. K leppner, Atomic physics and astrophysics, eds., M.
C hretien and E. Lipworth (G ordon and Breach, New
York, 1971).
[2] W.E. Lam b and R.C. Retherford, Phys. Rev. 79 (1950)
549.
[3] J.W. Heberle, H.A. Reich and P. Kusch, Phys. Rev. 101
(1956) 612. See alo Phys. Rev. 104 (1956) 1585.
[4] R.T. Robiscoe, Phys. Rev. 138 (1964) A22.
[5] G.G. Ohlsen and J.L. McKibben, Los A lam os Publication
LA-3725 (1967).
[6] P.G. Sona, Energia Nucleare 14 (1967) 295.
[7] B.L. Donnally, Bull. Am . Phys. Soc. 12 (1967) 509;
B.L. Donnally, Proc. 3rd Int. Symp. on Polarization phe­
nomena in nuclear reactions, eds., H.H. Barschall and N.
Haeberli (University o f Wise. Press, M adison, 1971) pp.
295ff, esp. p. 309.
[8] S.R. Lundeen, Ph.D. Thesis, H arvard University (1975);
S.R. Lundeen and F.M . Pipkin, Phys. Rev. Lett. 46 (1981)
232.
[9] B.L. Donnally et al., Phys. Rev. Lett. 12 (1964) 502.
[10] J.L. M cKibben, in H igh energy physics with polarized
beams and targets, ed., M.L. M arshak (A IP, N ew York,
1976) p. 375.
[11] P. Pradel, F. Roussel. A .S. Schlachter, G. Spi ess and A.
Valance, Phys. Rev. A 10 (1974) 797.
[12] At the given frequency there is a slightly higher transmis­
sion for ( F , 3 f ) = ( 1,0) than for (1, = 1).
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