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Studies of nonconventional superfluids: Ultrasound propagation in helium-3-boron and the microwave surface impedance of the heavy -fermion superconductor uranium platinum(3)

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S tu d ies o f nonconventional superfluids: U ltrasou n d p ropagation
in 3H e-B and th e m icrow ave surface im p ed an ce o f th e
heavy-ferm ion superconductor U P t3
Zhao, Zuyu, Ph.D.
Northwestern University, 1990
UMI
300 N. Zeeb Rd.
Ann Arbor, MI 48106
NORTHWESTERN UNIVERSITY
STUDIES OF NON-CONVENTIONAL SUPERFLUIDS:
ULTRASOUND PROPAGATION IN 3HE-B AND THE MICROWAVE
SURFACE IMPEDANCE OF THE HEAVY-FERMION
SUPERCONDUCTOR UPt3
A DISSERTATION
SUBMITTED TO THE GRADUATE SCHOOL
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
for the degree
DOCTOR OF PHILOSOPHY
Field of Physics
By
ZUYU ZHAO
Evanston, Illinois
June 1990
ABSTRACT
STUDIES OF NON-CONVENTIONAL SUPERFLUIDS:
ULTRASOUND PROPAGATION IN 3HE-B AND THE MICROWAVE
SURFACE IMPEDANCE OF THE HEAVY-FERMION
SUPERCONDUCTOR UPt3
ZUYU ZHAO
Two nonconventional superfluids, superfluid 3He-B and the heavy
fermion
superconductor
techniques:
cell
with
UPt3
have
been
studied
using
different
(1) A study of 3He-B was performed in an acoustic sound
a path
length
of
acoustic impedance technique.
381/jm using
the
single-ended,
c.w.,
The fundamental frequency of the x-cut
quartz transducer employed in the experiments was 12.80 MHz.
following studies were performed:
The
(a) A systematic measurement was
made on the pair-breaking edge in zero magnetic field with ultrasonic
frequencies of 64.3 MHz, 90.1 MHz, 141.6 MHz and 167.4 MHz, in the
pressure range from 3 bar to 28 bar.
indirectly support the
The results of our measurements
temperature scale of Greywall and
the weak
coupling plus (WCP) model of Rainer and Serene for the gap function.
The pair-breaking edge was also measured in magnetic fields up to
1.36 kG perpendicular
to
the
sound propagation direction and
the
predicted shift of the effective pair-breaking threshold (from 2A(T)
in zero field) by 0 - — r-1------ (the renormalized Larmor frequency)
l4~Fa (2+Y)
has been observed.
(b) ifie (Imaginary) squashing mode was excited
with
sound
frequencies
of
141.6
MHz
and
115.8 MHz.
splitting (of about 0.3 MHz) of this mode was observed.
ii
A
This
doublet
doublet splitting was
found to be strongly pressure
and frequency
dependent, but independent of the magnetic field (at the low fields
studied).
Possible causes of this splitting include superfluid flow
induced texture effects and finite wavevector (dispersion) effects,
(c) Structure was observed with a sound frequency of 64.3 MHz in the
vicinity of 2A(T) in a magnetic field of about 580 Gauss which is
thought to be Jz - -1 component of the J - 1* collective mode.
surface
impedance
performed with
study
of heavy
an X-band
Fermion
microwave
(2) A
superconductor UPt3
spectrometer
(f “
11.42
was
GHz)
integrated with an Oxford 400 TLE dilution refrigerator so as to have
top-loading
measurements
capability.
of
observed along
the
the c
H ^
axis
(3)
Ui ..ig a
on UPt^
were
top
loading
performed
and in the basal
magnetometer,
and
plane.
kinks
The
were
results
support a model of unconventional superconductivity by Hess, Tokuyasu
and Sauls.
iii
i
ACKNOWLEDGEMENTS
I would like to begin this traditional part of a Ph.D. thesis by
acknowledging my thesis advisor, John B. Ketterson;
it is only with
his wide experience, scientific perception and persistence that this
work was accomplished.
As a friend 1 also express my sincere grat­
itude for all his help during my stay in Evanston.
Along with the
rest of his students, I enjoyed his BBQ ribs and American folk songs.
I
am
obliged
to
William
P.
Halperin
conversations about ^He physics and cryogenics.
also appreciated.
for
many
useful
His encouragement is
It was a pleasure to share the laboratory space
with his students, Harvey Hensley and Peter Hamot.
Many thanks go to James A.
Sauls,
for his theoretical guidance
and who kindly gave us the computer program which was indispensable
in the analysis of our experimental results.
It was a great pleasure to work with my colleague and friend,
Shireen
Adenwalla,
whose
uniform
dedication
contribution to the success of this work.
made
an
invaluable
Also appreciated are many
helpful and interesting discussions with her,
both about academics
and daily life.
Bimal K. Sarma deserves sincere thanks for his constant concern
about the research of this group.
All his help, particularly during
the difficult period when this work was getting started and during
the fruitful cooperation thereafter, is greatly appreciated.
Whenever I encountered tough machining or mechanical problems,
iv
my last resort was always my friend Ron Mcllick, who could always be
counted on to come up with a useful answer.
A lot of machine was done by Joe Hahn, and later Steve Jacobson,
and their fellow workers in the Physics Shop, whose skill, patience
and effort are appreciated.
I am indebted to my dear friend Jia-qi Zheng for his help in so
many things.
of on-going
electronics
Mian-Zhong Lin rescued us more than once in the middle
^He experiments
as
they
would
by promptly
break
down.
fixing various pieces
Both
Young-jia
Qian
of
and
Shengnian Song deserve thanks for their contributions in the latter
part of this work.
The contribution from Hai-qin Yang and Yuan-hua
Shen are appreciated.
I have greatly enjoyed my friendship with Gerard Garino and I
thank him for his help in making the drawings in this thesis.
Many thanks to Arlene Jackson, who typed this thesis, as well as
the preprints of my papers, with remarkable speed and accuracy.
I would
like
to express
my deepest thanks
to my
sisters
and
brothers for their tremendous concern and support.
My wife deserves a special and wholehearted acknowledgement, for
her constant understanding and encouragement during all these years.
Last, but not least, boundless gratitude goes to my dear mother.
It is only with her caring support,
deep concern and constant love
from her heart of gold, that I have been able to survive this long
separation.
v
Memory of my Father
TABLE OF CONTENTS
Chapter 1.
Introduction.............................................. 1
3
Chapter 2. The Fermi Liquid He .................................. 5
2.1
Introduction.................................... 5
2.2
Normal Liquid ^He...... .................... .
7
2.2.1 Landau fermi liquid theory..................7
2.2.2 Acoustic properties of the
normal liquid ^He.......................... 13
2.3
Superfluid ^He.................................. 16
2.3.1 Phase diagram and order parameter.......... 16
2.3.2 Zero sound propagation in ^He-B............ 21
2.4
Review of the Recent B-Phase Experimental Studies.25
Chapter 3, Experimental Techniques.............................. 34
3.1
Introduction.................................... 34
3.2
Acoustic Cell................................... 34
3.3
The rmome try..................................... 39
3.4
^He High Pressure System........................ 43
3.5
Gas Handling System and Modification
to the Cryostat.................................. 46
3.6
The c.w. Acoustic Impedance Technique........... 49
3.6.1
Comment on the time-of-flight pulse
technique................................. 49
3.6.2
Acoustic impedance........................ 50
vii
3.6.3
3.7
The r.f. spectrometer..................... 53
Experimental Performance........................ 56
Chapter 4. Results and Discussion............................... 59
4.1
Introduction.................................... 59
4.2
Pair-Breaking Edge in Superfluid ^He-B...........60
4.2.1 Theory and early experiments............... 60
4.2.2 Pair-breaking edge in zero magneticfield...62
4.2.3 Pair-braking edge in finite magneticfield..76
4.2.3.1
The location of the pair-breaking edge
in a magnetic field.................... .76
4.2.3.2
Results and Discussion.................. 78
4.3
Doublet splitting of the Squashing Mode..........79
4.4
Observation of a New Structure near 2A in Finite
Magnetic Fields..................................98
4.4.1 Summary of previous work................... 98
4.4.2 Observation of a new structure near2A...... 99
Chapter 5. The Heavy Fermion Superconductor UPt3 .......
105
5.1
Basic properties.......................... ,...105
5.2
Proposed experiments on UPt3 .....................114
Chapter 6. The Northwestern Low Temperature
Microwave Spectrometer................................ 118
6.1
Introduction.................................... 118
6.2
The Microwave Spectrometer...................... 119
6.2.1
An approach to top loading................ 119
viii
6.2.2
Cryogenic aspects........................ 121
6.2.3
Microwave aspects........................ 128
6.2.4
Adjusting the coupling to the cavity..... 135
Chapter 7. Microwave Results and
Discussion..................... 137
7.1
Introduction.................................... 137
7.2
EPR Studies of Ca(tatbp)I....................... 137
7.2.1 Introduction.............................. 137
7.2.2 Results and discussion.................... 141
7.3
S-wave superconductors surface impedance
studies: Sn, Zn, and In.......................... 146
7.3.1 Introduction...............................146
7.3.2 Measurements of the surfaceimpedance
response of s-wave superconductors:
Sn. Zn, and In............................ 147
7.4.
Progress with the Heavy Fermion
Superconductor, UPt3
...................... 157
7.4.1 Samples................................ ...157
7.4.2 The upper critical field measurements
(resistively)............................. 163
7.4.3
Microwave surface impedance study......... 171
7.4.4 d.c. Susceptibility measurements
of UPt3 ...................................171
7.4.4.1 Experimental details.....................176
7.4.4. 2 Results and discussions.................. 181
ix
Chapter 8. Summary.............................................. 186
REFERENCES...................................................... 189
VITA............................................................ 198
x
LIST OF TABLES
Tables
CHAPTER 2
2-1.
Temperature dependence of the normalized BCS gap
function, Ab c s (T)/Ar c s (0) (Ref. 19)................... 31
2-2(a)Data table of the properties of liquid ^He (Ref. 33)...32
2-2(b)Polynomial fit of the properties of liquid
He,
Q(p) - 2a pn , where the coefficients a are taken
n n
n
from the table and the units of Q(p) are given in
table 2-2(a) (Ref. 33)................................ 33
CHAPTER 4
4-1.
Measurements of the pair-breaking edge (Ref. 3)........ 103
4-2.
The calculated values of the superflow induced
splitting of the SQ mode.............................. 104
CHAPTER 5
5-1.
Basic properties of the normal state of UPt^
compared to those of Na.
(Ref. 1).................... 117
CHAPTER 7
7-1.
Measurements of the upper critical field of
superconducting UPt^ (Ref. 22)........................ 185
xi
LIST OF FIGURES
CHAPTER 2
2-1
Phase diagram of ^He. (Ref. 1).........................6
2-2
Attenuation and velocity of sound in Normal ^He
at 0.32 atmosphere.
2-3
(Ref. 7).......................... 17
Excited states of the ^He-B Order Parameter.
(Ref. 34)..............................................26
2-4
Doublet splitting of
- 0 state of the real-
squashing mode, P — 5.3 bar. (Ref. 36)................. 27
2-5
Acoustic attenuation due to the coupling to the J — 1*
collective mode P - 3.9 bar.
2-6
(Ref. 33)................ 29
Zeeman splitting of the (imaginary) squashing mode.
(Ref. 42)....:........................................ 30
CHAPTER 3
3-1
Experimental tower.................................... 35
3-2
Assembly of the acoustic cell......................... 36
3-3
Measurement of the spacing between the two
transducers.....................................
3-4
40
Calibration curve of the upper LCMN thermometer
(DEM. ; H - 0)........................................ 42
3-5
The melting curve of
xii
3
He at low temperature........... 44
3
He...........
3-6
Gas handling system for
3-7
Calibration curve of the pressure gauge........
3-8
Gas handling system of the dilution refrigerator.......48
3-9
The equivalent circuit for a piezoelectric
transducer........
AS
47
52
3-10 Block diagram of the c.w. acoustic
impedance spectrometer................................54
CHAPTER 4
4-1 Typical temperature sweeps
(a) f - 141.6 MHz P - 28.5 bar......................... 65
(b) f - 64.3 MHz P - 5,4 bar........................... 67
4-2 Pair breaking edge in the pressure-temperature plane....68
4-3 Frequency of the pair-breaking edge (normalized to
A+ (0)) vs. T/Tc ........................................70
4-4 The coefficient of the edge calculated using
(a)
theHelsinki temperature scale with the BCS gap......71
(b)
theHelsinki temperature scale with the weakcoupling-plus gap...................................72
(c)
theGreywall temperature scale with the BCS gap..... 73
(d)
theGreywall temperature scale with the weakcoupling-plus gap...................................74
4-5
A sketch of the pair-breaking edge in a magnetic field..77
xiii
4-6
The measured shift of the pair-breaking edge in a
magnetic field (P - 27.7 bar, f - 141.6 MHz)........... 80
4-7
Typical demagnetization traces of the acoustic
impedance signal, showing the doublet splitting in
.............................. 83
the squashing mode
4-8
A Lorenzian fit to the doublet splitting of the
imaginary squashing mode (P - 19.2 bar, f - 115.8 MHz
and H - 0)...............
4-9
(a)
85
Pressure dependence of the doublet splitting
(f - 141.0 MHz, H - 0)............................ 88
(b) The frequency splitting vs. T/Tc ....................89
4-10 (a) Comparison of a ^ between two
sets of measurements............................... 90
(b) Pressure dependence of ag^ in zerofield............ 91
4-11 (a) Field dependence of the doubletsplitting ofthe
squashing mode in ^He-B for f - 141.6
MHz and
P - 27.7 bar...................................... 92
(c) Field dependence of agq>
(P - 27.7 bar and f - 141.6
MHz).................. 93
4-12 (a) The calculated values of the normalized
splitting between J
z
- ± 2 and J
components as a function of
xiv
z
- 0
T/Tcand Vg ............ 96
(b) The calculated values of the normalized
splitting between J
z
± 1 and J
z
- 0
components as a function of T/Tc and
4-13
............ 97
A "bump-like" structure in the vicinity
of 2A (P - 7.2 bar, f - 64.3 MHz)
and H - 0.563 KG......................................100
4-14
Separation between the center of the "bump" and the
effective pair-breaking edge vs. T/Tc and a comparison
with the positions of both
components of
the J — 1"
collective modes (f - 64.3 MHz, H - 0.58 kG).......... 101
CHAPTER 5
5-1
A typical temperature sweep showing the longitudinal
ultrasonic attenuation in UPt^ . (Ref. 14)..............109
5-2
H-T plot of HpL(T).................................... 110
5-3
Specific heat data in UPt^ indicating two
discontinuities in the vicinity of the
transition temperature (Ref. 18)...................... 112
5-4
A proposed phase diagram for superconducting
UPt^ derived from the heat capacity
measurements for Hi c-axis......
xv
113
5-5
(a)
The upper critical field for the orientations,
H||c and H||a (Ref. 21)................................. 115
(b)
The lower critical field for the
same orientations (Ref. 22)........................... 116
CHAPTER 6
6-1.
Schematic drawing of the microwave cavity
showing the Gordon coupler...............
6-2.
120
Coupling of the Gordon coupler as a
function of the insertion depth of the
dielectric rod.......................................122
6-3.
Symbolic overall view of the microwave line
as it extends through the cryostat................... 124
6-4.
The rectangular to circular waveguide
adapter, load lock and positioner mechanism
for the dielectric rod.........................
6-5.
127
Block diagram of the homodyne microwavebridge spectrometer with AFC circuitry............... 129
6-6.
Anode drive (Op-Amp) circuit diagram.................. 131
6-7,
Diagram of the sweep circuit for the
bipolar d.c. magnet power supply..................... 134
xvi
CHAPTER 7
7-1.
The chemical structure of the Cu(tatbp)Imolecule......138
7-2.
The chain-like structure Cu(tatbp)I................... 140
7-3.
(a)
Idealized absorption line shape for a
randomly oriented system having uniaxial
symmetry and a S function line shape (g^ > gj)
(Ref. 6)............................................. 143
(b)
Computed line shapes for randomly oriented systems
having uniaxial symmetry (Ref. 7)....... ............ 143
7-4.
EPR measurements on a Cu(tatbp)I powderedsample ....... 144
7-5.
A family of g-factor measurements on
Cu(tatbp)I powder ranging in temperature
from 1.98 K to 2.77 K .................................145
7-6.
Q vs. temperature of a 10 GHz microwave
cavity with the inner wall coated with Sn
(Ref. 20)............................................ 148
7-7.
Measurements of the microwave absorption
from Sn during a temperature sweep in zero field.......150
7-8.
Measurement of the microwave absorption
from Sn during a field sweep at 2.13 K ................ 151
7-9.
Measurement of the microwave absorption
from Zn during a temperature sweep in zero field.......154
xvii
7-10. Measurement of the microwave absorption
from Zn during a field sweep at 0.53 K,
using a bipolar magnet power supply.................. .155
7-11. Temperature dependence of the critical
field of Zn...............................
156
7-12. Measurement of the microwave absorption
arising from the In 0-rings...............
158
7-13. Calibration curve of the 100-0 Matsushita
resistance thermometer................................ 161
7-14. Resistivity of UPt3 vs. temperature in
zero magnetic field................................... 162
7-15. (a)
vs- temperatures with H||c-axis................ 164
(b)
vs- temperatures with H||a-axis................ 165
(c)
H£2 vs. temperatures with H)|b-axis............... 166
(d)
The upper critical field vs.
temperatures when the applied field is
in the basal plane.................................... 167
7-16. The measured transition width of UPt3 vs.
the applied field..........................
170
7-17. Typical trace of the microwave surface resistance
measurements during a temperature sweep............... 172
7.18. Power law fit on the microwave surface
resistance of UPt3 ....................
xviii
173
7.19. Power law fit on the bulk resistance of UPt^......... .174
7.20.
2
$R surface vs. T.. ................................... 175
7.22.
Schematic diagram of themagnetometer.................. 178
3
Schematic diagram of the HeGas-Handling System........ 179
7.23.
A typical magnetometer trace
7.21.
taken
at T - 230 mK ........................................ 180
7.24. The lower critical fields vs. temperature
for H|| c-axis ....................................... 183
7.25. The lower critical fields vs. temperature
for Hi c-axis........................................ 184
xix
CHAPTER 1
INTRODUCTION
More Chan a decade separated Che discoveries of superfluid
(1972) , with Tc ~ 2mK,
(1984).
and
the heavy Fermion
(as evidenced by a large many-body enhancement of the
density of states)
and providing a fertile ground for study,
theoretical and experimental.
a
non-conventional
pairing.
It
is
a
superfluid
believed
non-conventional
both
It has been confirmed that the former
that
responsible for the pairing.
also
superconductor UPt^
Both of these Fermion systems are highly correlated at low
temperatures
is
3
He
with
spin
triplet
spin
polarization
and
p-wave
effects
are
Evidence is accumulating that UPt^ in
superconductor;
however
the
mechanism
leading to superconductivity is still not clear.
Three
stable phases (A, B, and Al) have been found for
3
superfluid He; however the phase diagram of superconducting UPt^ is
the
subject
specific
of
heat,
much
current
propagation
controversy.
of
ultrasound,
Measurements
torsional
of
the
oscillation
response, a.c. and r.f. susceptibility and the upper critical field
suggest that multiple phases may co-exist in superconducting UPt^.
of
One of the most Interesting properties of the order parameters
3
a non-conventional superfluid (e.g.
He) is the existence of
oscillatory excited states, referred to as order parameter collective
modes;
these modes result from mean fields created when the normal
and anomalous components of the distribution function are distorted
from their equilibrium values.
Substantial progress has been
achieved in studying the collective modes in normal and superfluid
3
He and ultrasound has played an important role.
However, because of
technical difficulties (which will be discussed in Chapter 3) various
features in high attenuation regimes,
e.g.,
in the vicinity of the
pair-breaking edge (including the pair-breaking edge itself), or the
texture induced behavior in the imaginary squashing mode,
have not
been fully explored.
Information concerning possible order parameter excitations
UPtg, either theoretical
or experimental,
is lacking
Intuitively
at
of the
we
expect
frequencies to be of the
least
some
in
at this time.
collective
order of the gapfrequency,
mode
A(T), which
scale as T . Since the transition temperature of UPt^ is ~ 500 mK or
3
order 200 times that of He (where a frequency — 50 MHz provides
useful data) collective modes in UPt^ might profitably be studied at
frequencies of over 10 GHz, i.e., in the microwave region.
This thesis was motivated by the following two goals:
a) explore the high frequency high-attenuation region in
3
superfluid He-B by using the previously developed acoustic
impedance technique (which is particularly suited for this
regime) but in an acoustic cell with a very short path length
(in our design, the path length was reduced by a factor of
20 compared to that used in previous Northwestern work);
b)
develop an X-band microwave spectrometer for use on a
dilution refrigerator system and perform microwave surface
impedance measurements on
.
3
3
The following phenomena were studied in He-B:
i)
field
A systematic measurement of the pair-breaking edge in zero
has
been
performed.
The
results
support
the
temperature scale of Greywall and the weak-coupling-plus model
(of
Rainer and Serene) for the gap function.
indirectly
ii) In agreement with the
theory, a shift of the pair-breaking edge with magnetic field (from
its zero
field value) was observed.
pair-breaking edge,
iii)
a "bump-like11 feature
In the vicinity of the
was
observed only
magnetic field; its position Is consistent with the J
collective mode.
“
in a
J - 1'
iv) The imaginary squashing mode was studied in the
high pressure regime
(from 19 bar to 28 bar), with
high frequencies of 115.8 MHz and 141.6 MHz.
the relatively
A doublet splitting has
been resolved which shows a strong pressure, but no field dependence.
In connection
400 TLE
dilution
with the second thrust of our work
refrigerator
has been
permit measurements of microwave response
an
Oxford Model
installedand modified
at X-band.
to
A microwave
surface impedance study on several s-wave superconductor (Zn, Sn and
In) and the heavy fermion superconductor UPt^ has been performed.
The remaining chapters of this thesis are organized as follows:
3
Chapter 2 reviews the theory of the Fermi liquid He as well
as
3
recent ultrasound experimental
results on
He-B. Details of our
ultrasound experimental techniques are described in Chapter 3, and
the results
are discussed in Chapter 4.
Chapter 5 gives a short
review of some important physical properties of UPt^. Chapters 6 and
7 present,
respectively,
the details of the microwave spectrometer
4
and a status
surface
report on
impedance
the experimental
measurements.
In
results
Chapter
8,
phesis and propose some possible future experiments.
)
of
we
the microwave
summarize
the
CHAPTER 2
2.1
3
He
nature.
subject
THE FERMI LIQUID 3He
Introduction
is surely one of the most fascinating systems occurring
The study of its properties has been and continues
of world wide
interest.
We
now briefly
summarize
in
to be a
some
of
these properties:
1) Fig. 2-1 shows the phase diagram of ^Het^-l; note that it
does not solidify, even at absolute zero, unless a pressure
of at least 29 bar is applied. This property results from a
combination of its large zero point energy and the very
weak van der Waals interaction between the
3
He atoms (the
latter also accounts for the very low boiling point, 3.2 K,
of 3He).
2) Because of its closed electronic Is
2
shell,
3
He atoms have
a total spin of 1/2 arising from the nucleus;
are subject to Fermi-Dirac statistics.
below the Fermi temperature,
thus the atoms
At temperatures well
- 1 K, many of the properties
3
of liquid He are similar to those of a degenerate Fermi
gas.
The effect of the interaction between the atoms can be
included by using Landau's phenomenological theory of a Fermi
liquid.t2]
3) When the temperature is lowered to a temperature of the order
of a millikelvin (depending on the pressure) the liquid
5
6
WELTING
PLANE
'8* LIQUID
90
S 20
S 10
- ZERO FIELD PLANE
TEMPERATURE (mK)
Fig, 2-1
Phase diagram of ^He (Ref. 1)
7
undergoes a second order phase transition from the normal
state to various superfluid s t a t e s . A l t h o u g h such a
transition was predicted on the basis of various
generalizations of the BCS theory,
(4] more than a
decade separated the prediction and the observation.
The short-range repulsion (due to the hard core of the
3
3
He- He interactions potential) rules out an L - 0 pairing
state.
Actually it has been demonstrated that the superfluid
phase of
3
He is an S - 1 (spin triplet) L - 1 (p wave) state,
which implies that superfluid
properties.
3
He displays anisotropic
Indeed, one of the observed phases (the A
phase) shows markedly anisotropic behavior.
The
above
discussed)
mentioned
unusual
properties
(and many
we
have
3
He merits extensive studies,
suggest that liquid
not
both in
the normal and superfluid states.
2.2
Normal Liquid ^He
2.2.1
As
Landau Fermi Liquid Theory
the
momentum
temperature
of
(corresponding
atoms
to
of
liquid
3
He
decreases
and
thermal
motion)
the
the
is
lowered,
de
Broglie
becomes
larger.
the
average
wavelength
When
the
wavelength becomes comparable with the distance between the atoms,
the macroscopic
effects.
properties
of
Almost all substances
the
liquid are
determined
by
quantum
solidify well before quantum effects
3
become
and
important;
He) ,
and
some
nuclear
exceptions
matter
(a
are
both
neutron
isotopes
of Helium
star).
The
( He
previously
mentioned weak van der Waals attraction and large zero point energy
combine
to keep
3
He
and
4
He
in
(quantum)
liquid
states
at
zero
pressure.
In his
phenomenological
theory of
a
Fermi
liquid,
Landau
[21
introduced a one-to-one correspondence between the states of a Fermi
3
gas and those of an interacting system, like liquid He.
The basic
idea is that if we take a non-interacting Fermi gas in the ground
state, and adiabatically turn on the interacting between particles,
we will go from the gas to the liquid and the classification of the
energy
levels
will
remain unchanged.
In
this
classification
the
original role of the gas particles is replaced by new entities called
quasiparticles (or elementary excitations).^
This
Idea suggests that each quasiparticle possess a definite
momentum p - hk and spin S - 1/2 with projections Sz - ± 1/2.
The
3
quasiparticle may loosely be thought of as a He atom moving through
background of all the other atoms, which must flow around the atom In
question; this latter phenomena is referred to as "back-flow".
The
quasiparticles are assumed to obey the Pauli exclusion principle.
Landau
showed
quasiparticles has
that
the same
the
distribution
form as
for
" p , ? “ expf[e-* J ^ l l / k T + 1
p ia
a
function
for
the
the non-interacting
gas.
(2'1}
where fi and k^ are respectively the chemical potential (or equivalently
the Fermi energy) and Boltzmann's constant.
Eq. 2-1 is deceptively simple because (as we will discuss later)
the
quasipartlcle
quasiparticle
energy,
£-*
itself
P,a
distribution function, i.e.
depends
on
the
entire
e-» -* is a functional of
p.CT
2-1 is a rather complicated implicit equation
n-»
therefore, Eq.
p,o
for n-*
That the energy of a quasiparticle should depend on the
p,cr
distribution function is clear from our earlier heuristic picture; if
we alter the occupation of some fraction of the other quasiparticles,
3
it will alter the back flow around a given moving
He atom.
At T - 0, Eq. 2-1 yields
1,
n-»
P.tf
^
/i
£->.»>
p.
P' a
0;
(2-2)
P.^
A major assumption of Landau's
theory (which he later proved)
was that the Fermi momentum of the interacting liquid is the same as
that for a free gas
1/3
PF -
h(3,r2 v]
<2' 3>
where N and V are the total number of
3
He atoms and the volume of the
system respectively
At T - 0 the Fermi sea consists of occupied states up to the
limiting
momentum.
Landau
def \sd
the
Fermi
velocity
of
a
quasiparticle as
ra£S -*>
rde-»
VF ‘ — r ~ \
1 a
3p
Jp-j
P JP-pF
HH
(2-4)
which is the velocity of a quasiparticle at the Fermi surface.
He
10
also defined the following quantities:
a)
b)
Effective mass of the quasiparticle
*
Pp
m — ~ ;
(2-5)
Density of quasiparticle states at the fermi surface
i —
r
K (0 ) - i
i
f ^
p
,a
— 3n° ■*
„ - #
--$)> _
(2-6)
■»
P,C7
p ,er
-+
p io
where
-> and n$ -> are, respectively, the quasiparticle
p,cr
p,<7
energy and the distribution function at absolute zero.
It should be emphasized that the quasiparticle states are well
defined only in the immediate vicinity of /*, and therefore it is not
possible to write the total energy of the liquid as a sum of the
quasiparticle
E ^
)
i*~
energies
(as
one
does
for
an
ideal
J n-> -> c.> -> dr
p,cr
p ,a
gas),
i.e.
(2-7)
a
2
3
where dr - ------ d p. However
itis possible
to express the change
(2xh)J
in the
total energy of the system as a function of a change
occupation
of
states
near
n, which
we
write
in
5 tvl ~ ) / c-> -*Sn-* dr
1-VJ
hr
P ,a p ,a
a
the
of
form
(2 -8 )
Thus
.» is the functional derivative of the total energy per unit
P-<7
volume with respect to the distribution function.
Up
to
now
we
have
notdiscussed
theinteraction
between
quasiparticles
in a Fermi liquid.
Landau defined the interaction
energy of two quasiparticles, f-> -> _>
, as the change of the energy
p.ff-p',a'
of the first quasiparticle with momentum p and spin a produced by a
change
in
the
occupation
momentum p' and spin a '.
from
a
change
6n
of
of
the
second
quasiparticle
state
with
The total change in the energy resulting
some
specified
fraction
of
all
the
other
quasiparticles would then be given by
1
(2-9)
a
We may then define £+-»-».»
as the functional derivative of the
p,o.p',a’
quasiparticle energy with respect to the distribution function and
hence the second functional derivative of the total energy density;
f-*
-* -* being a second functional derivative of some quantity, must
p.cr-p' ,a'
therefore be symmetric in p,p' and a,o'.
For an isotropic liquid, the interaction function f-» ->
can
p.o.p' ,cr'
only depend on the relative orientation between p and p', so the two
spin operators can appear in f-» .» .» only as a scalar product.
p.ff.p',o'
Thus the quasiparticle interaction function has, on the Fermisurface, the form
- f(s)(0) + fa (9) o-o'
( 2 - 10 )
where 9 is the angle between p and p', and a denotes the vector Pauli
spin matrix.
12
It
is
convenient
to
expand
S
f {$)
and
Sl
f (8)
in
Legendre
polynomials,
fS(p.p') - ^ f / P ^ c o s * )
(2 -lla)
£
fa (p.p’> - J j / p ^ c o s * ) .
£
We define
(2 -llb)
- N(0)f/
f/
(2-12a)
and
f / - NCOf/,
where
N(0)
isthe
quantities F^
S
(2 -1 2 b)
density
and F^
of
states
at
the
Fermi
surface;
the
are so-called Landau parameters.
Landau derived the following expressions for various equilibrium
properties of normal
a)
effective mass
*
m
,
1
_ s
S - 1 + 3F1 •
where m is the mass of the bare
b)
<2-13>
3
He atom;
specific heat
m Pp
C
3h
c)
2
k T;
B
spin susceptibility
*N “
where
(2-14)
1 yh is the magnetic moment of the 3He atom.
- —
<2’15)
13
3
Acoustic properties of the normal liquid He
2.2.2
As we have seen, any given quasiparticle is subject to the self
consistent
field
of
the
remaining particles
in
the
medium.
In
equilibrium, under homogeneous conditions, the quasiparticles are not
subject
to
a
quasiparticle
net
force.
If
distribution
takes
however,
place
a
as
distortion
a
result
of
the
of
some
perturbation, the average interaction force no longer vanishes.
This
restoring force can lead to a new phenomena, not present in the ideal
Fermi gas:
collective oscillations of the system.
In this case, the
distribution function, n_* -» will depend on position, r and time, t,
P.<7
—¥
i.e. n — n-> -»(r,t).
In order to discuss collective modes we assume
P ,cr
that a wave is propagating in the liquid with angular frequency w and
wave vector k.
value
of
ur
collisions.
The nature of the wave depends essentially on the
where
mean time between quasiparticle
.2
At low temperature r a T . Thus, for a given frequency,
r
is
the
there will be a temperature above which wr < | and below which wr >
|.
When one is well into these two regimes the collective modes are
referred
to
as
hydrodynamic
and
collisionless
respectively.
A
collisionless collective mode involves a coherent motion of all the
particles under the influence of their (collective) mean field.
For
a hydrodynamic mode rapid collisions between particles act Indirectly
as a restoring force.
can
be
obtained
by
Both regimes and the transition between them
solving
d
-»
Tr n_> .* + V-»£n-> ->*v • V-»n-» -> •
oc p,o
r p ,o
p p,ff
the
(Landau)
^ - I(n)
r p.tr
kinetic
equation
(2-16)
14
where I(n) is the collision integral.
When
ur
«
1,
the
solution
of
equation
influenced by the collision integral,
organized
density
oscillation
I.
(2-16)
Under
is possible,
is
strongly
this condition an
i.e.
we
have
ordinary
hydrodynamic sound waves which are also called first sound waves.
Landau showed that the velocity of first sound could be derived
from
his
Fermi
liquid
theory;
C1 " 3 t1 + F0S) t1 + 3F12K
the
resulting
expression
2-
is
(2'17a)
The attenuation may be derived from classical hydrodynamics if
one
introduced
hydrodynamics
the
viscosity,
transport
as
r/,
coefficients
a
parameter
cannot
be
(the
various
accurately
derived
from first principles but may be parameterized by introducing the socalled scattering amplitudes).
The well known hydrodynamic result
for the attenuation, a, is
2 w2
“ — j- 1
(2- 17b)
where p is the mass density.
When ur »
contribution
1 the collision are sufficiently infrequent that the
from
the
collision
integral
may be
neglected.
The
resulting wave is called zero sound, which is a striking property of
normal
3
He.
.
3
It is also a very useful probe to study superfluid He.
Neglecting
the
collision
integral
Eq.
a
->
7T n-» .* + V-»5n-» .» • v - V_>n-> .» • V.+ c-> .» - 0.
p.cr
r p,o
p p ,o
r p ,a
2-17
becomes
(2-18)
15
At T — 0, n-> .* is a step function at p - p , We therefore assume
p,a
*
that Eq. 2-17 has the solution
5n-> ->(r,t) - £<£-£„)R(p)ei^k *r'tl>t^
P.*
F
where R(p) has the units of energy and depends on
(2-19)
the direction
A
p (9,4>).
Eq.
(2-18)
can
be
linearized
c-» -»(r,t) and
P ’a
(Note R(p) is independent of spin in zero
n_* -»(r,t) to first order.
P.<7
sound).
A solution of Eq.
(2-18)
by
r2 1
expanding
can be obtained as follows:
>
2 m*p
- cost R(f^) - — r-5- cos8jfS (,e,<t>]$‘4>'')
F
J
h
fc
—
1
where
■ ^
cq
is
the
zero
sound velocity.
dft'
If we
are
(2 -2 0 )
to
avoid
encountering a singularity when solving the integral equation (2 -2 0 )
it will require c > v_.
o
F
The velocity, cq , and the attenuation, a^, of the zero sound with
frequency w can be expressed ast®]
Co
2
VF2m*
- C1 - 15 “c^nT
<2'21a>
and
at
o
v 2
*
2
F 1 m*
— 15
tt’ —
q
—
—
c,3 t m
1
a
Thetransition
experiment
from first
v
(2 - 2 lb)'
'
to
zero
sound
in apropagation
was first observed by Abel et al.E2^ They found
that as
the temperature was lowered the sound attenuation increased, reached
16
a maximum, and then decreased.
was proportional to T
high
temperatures
2
At low temperatures the attenuation
and independent of sound frequency, while at
the
attenuation was
proportional
2
2
to w /T . The
sound velocity was essentially temperature independent at both high
and low temperature,
but near the attenuation maximum the velocity
changed by a few percent
(Fig.
2-2). I?] All of these experimental
results are in agreement with the prediction of Landau's theory.
The
observations
the
of
zero
sound
removed
any
remaining
doubts
in
theoretical community regarding the validity of Landau's Fermi liquid
theory.
2.3
Superfluid ^He
2.3.1
Phase diagram and Order parameter
Unlike Fermi liquid theory, which did not initially make a large
impact (in the west),
the BCS theory of superconductivity was much
more rapidly accepted,
partly because of a large pre-existing data
base.
In this model the ground state is made up of Cooper pairs -
highly correlated pairs of electrons with opposite momenta and total
spin
S
-
0
"bound"
together
by
an
attractive
electron-electron
interaction mediated by phonons.
study another Fermi system,
This model was soon generalized to
3
liquid He, and a superfluid state was
predicted.[®] >[1 0 ]
However,
not
until
1972,
Osheroff, Richardson and Lee.
3
superfluid
He discovered by
r3 i
The first
evidence came from two
was
anomalous points, denoted as A and B, in the cooling characteristics
of a Pomeranchuk cell. These were later shown from NMR
17
400 —
a
«U
MHt
t
A4
IMl
(- too —
30
40
<00
T -MiDEGREES KELVW
94
••
92
l» 8
MH
Fig. 2 -2
Attenuation and velocity of sound in Normal ^He
at 0.32 atmosphere. (Ref. 7).
18
experimentstto
correspond to transitions
involving two separate
superfluid phases.
According to the phase diagram shown in Fig, 2-1, the superfluid
transition,
which marks
the onset of superfluity,[12],[13]
extends
from a temperature of 2.49 mKt^l on the melting curve (at a pressure
of 34 bar) to approximately 0.89 mK at zero pressure.
distinct superfluid phases:
the Al-phase,
There are three
the A-phase
and the B-
phase.
It
is
now
universally
accepted
that
the
ground
state
of
superfluid -^He is composed of BCS-like spin triplet (S — 1), p-wave
(L - 0) Cooper pairs.
Note that the additional degrees of freedom
suggest that states with total angular momentum, J -
0
,
1
,
2
may be
important.
The
complex
order
scalar
parameter
of
as
s-wave
instead
to
a
notation
superconductor,
of
being
a
a 3x3 complex
2_
matrix, which measures the (complex) amplitude of the ^ (2J+1) - 9
J-0
possible combinations of the spin and orbital angular momentum.
According
in an
3
He,
superfluid
introduced
is
originally
by
Balian
and
Werthamer,t1 0 ] the order parameter for a spin triplet system can be
expressed as
± - i [a • d(k)]a2
(2 -2 2 )
-»
-> ^
where a is the vector Pauli spin matrix and d(k) is a vector in spin
3
space.
The ground states which occur in He (at zero field) are so
called unitary states having the property
tx
1
; for such states
19
->JL
the vector d has the property d x d
- 0.
Near Tc one may expand the components of
the vector d(k)
in
terms of a representation of the three L - 1 orbital eigenfunctions;
by convention the representation chosen is the set
kv - sinflsinA and k
j
z
- cos$.
- sinflcos^,
Thus we write:
“ i < k > - L Ai „ k ,
<2 - 2 3 >
p
where i corresponds to any three orthogonal directions in spin space
and
to
p
momentum
three
space.
(generally
The
matrix
different)
A^
orthogonal
provides
(near
directions
Tc)
a
in
complete
3
specification of the order parameter of superfluid
He.
In the A phase, also referred to as the Anderson-Brinkman-Morel
(ABM) phase, [9]
A
A
A
A
- j 3 / 2 Ad (n +in„)
lp
1
1
i. p
A
A
(2-24)
A
A
where n- and n„ are any two orthogonal (n*n - 0) unit vectors in kl
Z
12
A
A
A
space; the orbital angular momentum has the orientation £ - n^ x n^.
The gap function, A - A^in#,
is anisotropic (where 19 is the angle
A
measured from £); note there is a node on the Fermi surface along the
A
direction £.
In the B, or Balian-Werthamer (BW), phase,[10]
- A(T) ei^Ri^(n,8)
(2-25)
A
where
(n,0) is a rotation matrix connecting the spin and orbital
A
coordinates by a rotation through an angle $ about an axis n; <f> is a
20
3
The dipole-dipole interaction between the He nuclei
phase factor.
A
results in a value of 8 of 104®; the axis n is arbitrary.
external
magnetic
or
electric
fields,
boundaries
and
a
However,
superflow
A
velocity
field
will
affect
the
orientation of n (and also the
3
structure of the order parameter) in He-B.
All three spin triplet
1
3
components |tt>,
and
[11
+ 14-] , are present in He-B.
4.
The
modulus
square
of
the
gap,
TrA (T)A(T),
is
isotropic
and
A(T) can be approximated by the weak coupling BCS
theory:[15]
'
*kBTc
A(0)
.m
|c
h
.
. j2 AC
J3 s
1
/2 '
(2-26)
where A(0) — 1.764 kgTc and AC/C^ are respectively the weak-coupling
BCS gap function at T - 0 and the experimentally determined heat
capacity jump at T .
'[^1 •
1 Values of A
C
(T) are tabulated in
bCu
Table 2-1 (Ref. 19).
In the weak coupling theory
stable phase at zero field.
high pressure must arise
the B phase
should be
the
only
Thus the existence of the A phase at
from strong coupling effects.
A strong
coupling model involving spin-fluctuations has been
suggested. ^
Strong
coupling
effects
present in the B phase at high pressures,
measured heat capacity jump, AC/C^,
larger than the weak-coupling
are
also
known
to
be
since the experimentally
at the superfluid transition is
(BCS) value of 1.43.
A theoretical
model for including some of the strong coupling effects is the socalled weak-coupling plus model of Rainer and Serene
r211
(with the
21
corresponding gap function denoted as A+ (T)).
the work performed
for
this
thesis was
A substantial part of
carried
out
in
the high
pressure regime (P > 21 bar) and A+ (T) model has been used to analyze
some of the experimental results.
2.3.2
3
Zero sound propagation in
Clearly
it
is
of
considerable
He-B*
interest
to
study
the
order
parameter
of the ground state and various excited states of
3
superfluid He. Zero sound has proven to be one of the most powerful
probes for this purpose.
Sound
W ol f l e , ^ ^
attenuation
Serene^"^
in
3
He-B
and Maki
was
studied
, et al.
theoretically
by
They showed that two
mechanisms have great contributes to the sound attenuation.
One is
pair breaking by zero sound quanta when the hw > 2A(T); this topic
will be discussed
involves
the
in detail
excitation
exicton modes)
in Chapter 4.
of various
The
collective
second mechanism
modes
of the order parameter which couple
(also
called
to zero sound.
Such modes may be thought of as excited states of the Cooper pairs,
The nine complex parameters specifying possible states of
suggest
the
existence
of eighteen
collective modes
of which
3
He
nine
correspond to independent combinations the real parts of the order
parameter and the other nine correspond to the imaginary parts.
former tend to involve spin density fluctuations while
The
the latter
All the work on superfluid JHe to be reported in this thesis was
performed on the B phase so the remaining discussion will be confined
to this phase only.
22
involve
density
fluctuations.
Intuitively
we
expect
a
stronger
coupling to the latter class since zero sound Is itself a density
fluctuation.
BCS
gap
Maki
r241
total
Some of the excited states are located well Inside the
while
others
lie
near
or
within
the
continuum
region.
was the first to classify the collective modes based on the
angular
between zero
momentum.
f231
Serene1
sound and a J -
hw - 7x2/5 M T ) .
predicted
a
strong
coupling
2 collective mode with a frequency
Since the mode involves imaginary components of the
order parameter, it is denoted as J - 2' mode and is referred to as
the
"imaginary"
squashing mode.
Nagai et al[^5]> an^ Tewordt et.
alt26] predicted the existence of another collective mode with J - 2,
at a frequency hw - 7g/5 M T ) .
the
order parameter,
and
is
This mode involves the real part of
denoted as
the
J
-
2+ mode;
it
is
referred to as the real squashing mode.
This latter mode was
sound.
Cornell
However,
initially not
experiments
performed
thought to couple to zero
at Northwestern[^ ] an<j
[27] indicated the presence of an additional collective mode.
It was quickly realized that the presence of an asymmetry between the
energy of particle and hole states located symmetrically about the
Fermi surface would provide the needed coupling mechanism to this
newly observed collective mode.
Tewordt and Schopohl made a detailed study of the effect of a
magnetic field on the collective modes.
2J
+
1
fold
Zeeman-like
squashing modes.[28]
splitting
A dramatic prediction was a
of
both
real
and
imaginary
23
Later Sauls and Serene[29] calculated the frequency of the Bphase collective modes in the presence of an i - 3 contribution to
the pairing interaction and all Landau parameters through & - 4.
A
new J - 4' collective mode was predicted, the frequency of which is
the solution of the following equation:
l+ipSA
w
4 *3*5
+ x A "2 - r
- 4A‘•)a] * v
[J
- 4A2 )*]
2
P
20
9
+ 4«2(“ 2
-1
where they defined xj> “ v-^ _v^
coefficients
of
the
BCS
a2 + tiF:
- 2
A
- 4A2
5
- 5
+ 81 F4 ‘
(2-27)
(A) - 0
and v^ are the Legendre expansion
pairing
interaction
v(p.p');
where
the
function A(w,T)(-A2A) is given by
X - 4
Based
i
f dc t-anME/2T)________
(4e2.»2) . / - 2 - 2
J E -A
on
the
experimental
(2-28)
•
results
of
the
London
group[^0] ,
Shopohl and Tewordt [-^1 recalculated the frequency,
line width and
the coupling constant of the J - 1" collective mode.
They concluded
that, although no coupling to zero sound occurs to the J - 1” mode at
zero field, the J
- ± 1 components do couple at finite fields with
frequencies given by
hw - 2A(T) + 0.39
(2-29)
24
where the function ft is defined by
ft - —
---l+4 Fa (2 +Y)
3 o
(2-30)
and Y(T) is the Yosida function.
Tewordt and Schopohl also predicted that, although the coupling
between
zero
sound
and
the
J
—
-1
component
ordinary resonance in the sound attenuation,
would
give
relative
to
rise
the
to
an
antiresonance, a
background
(continuum)
would
the J
dip
-
in
involve
+ 1
the
attenuation.
an
component
attenuation
This
latter
phenomena is called a Fano resonance.132]
The frequencies of all the order parameter collective modes may
be parameterized as follows[33J
2 2
WJ±^T,P,mJ ,q,H^ “ WJ±(T ’P^ + aj± raJ7H + ^J± mJ ^a (O)
7
•V2 H 2 .
. <qvF)2. ,
. 2 (qVF ) 2
J+ A(0) + SJ± ’q A(0) + bJ±,q “j A(0)
(2-31)
where the subscript J± indicates the mode designation according to
its angular momentum classification and whether it involves real (+)
or imaginary (-) components of the order parameter; H, y, q and v„
are respectively the magnetic field, the gyromagnetic ratio, the zerosound wavevector and the Fermi velocity; wJ+(T,P) is the temperature
and pressure
dependent mode
coefficients
aj+ >
functions of
temperature and pressure.
Pj+ ^
7
frequencyfor H - 0 and q
j+'
aj+ q ’ and
We
^j+ q
are
- 0.
The
general
parameterize the linear
Zeeman effect by the coefficient a^+ and thequadratic Zeeman effect
25
(or
Paschen-Back
effect),
coefficients B
and
^J±
arising
from
gap
distortion,
by
the
... The coefficients a., -» and b T. -> are used
J±
J±,q
J±,q
7
to parameterize the finite wave vector dispersion of the collective
modes.
A schematic energy level
d iagram
[53] of the excited states of
the superfluid ^He-B Cooper pair is shown in Fig. 2-3.
2.4
M
a
Review of the Recent B-phase Experimental Studies
s
t
and
Shivaram[35]
have reviewed the experimental studies
performed on ^He-B prior to their work.
will
be
available
in
the
near
A rather complete review
future. [33]
Here
we
will
briefly
summarize the ultrasonic studies performed on ^He-B since 1982.
In 1982,
(Fig.
2-4)
Shivaram
of about
squashing mode,
threshold
et
0.5 MHz
in the
value)
al.[35]
of
presence
applied
observed a
the
- 0
"doublet"
state
of a magnetic
perpendicular
to
the
of
splitting
the
real-
field
(above
some
sound
propagation
direction.
The observed doublet splitting is pressure dependent but
"saturates"
for fields
field applied).
ranging from 50 mT to
160 mT
This phenomena has been attributed
effect on the dispersion of the real-squashing
A continuous depressurization technique
m o d e .[56
(the highest
to a texture
],t 37],[38]
for ultrasonic studies
on ^He-B was developed simultaneously by the Cornell and Northwestern
groups.
the
Using this technique, the group velocity of zero sound near
squashing
mode
was
measured.[59]
Qne
may
pass
through
a
collective mode by sweeping either the pressure or the temperature.
The group velocity was observed to decrease by more than a factor of
15 in these experiments.
The pressure sweeping technique was also
26
continuum
J = lJ = 4 “ 1 = 3, S
&
c
w
-
J = 2~
SQ Modes
J = 2+
RSQ Modes
J =0
Zero Sound
<D
tc
CO
0
H =0
Fig. 2-3
H * 0
Excited states of the ^He-B Order Parameter
(Ref. 34)
27
a* -
0.12
^IT)
0.06
*
2.0
0
1 0
AUMli)
Fig. 2-4
Doublet splitting of J - 0 state of the real
squashing mode, P - 5.5 bar. (Ref. 37).
28
used by Adenwalla et al.
to study the J - 2" collective mode over
a wide range of pressures.
In 1987, Ling et al.t^2] measured the acoustic attenuation of
^He-B in a magnetic field.
short path
They used a sonic cell involving a very
length which allowed
them to study regimes where very
large attenuations are encountered.
The difference between the zero
field attenuation and that in the presence of a field displayed a
resonance-
"anti-resonance" structure (Fig.
attributed to the J - 1", J
z
2-5) which the authors,
- ±1 modes.
The Zeeman splitting of the squashing mode in -^He-B in magnetic
field was deduced by Movshovich et al.t^^-l (Fig. 2-6) in 1988.
Lande g-factor was found to be 0.042 for T «
By
using
transducers,
frequencies
the
aluminized
Fraenkel
at
et
temperature
al.
PVDF
approaching
frequency sweep at fixed pressures. ]
Tc at P - 19 bar.
(polyvinylidene
measured
the
T
The
-
real
0
fluoride)
film
squashing
mode
with
a
continuous
29
60
tO
C 30
AO mT
«0
20 mT
•o
0 mT
0.05
0.10
0.15
0 2 0
1 - T/T,
Fig. 2-5
Acoustic attenuation due to the coupling to the J - 1"
collective mode P - 3.9 bar. (Ref. 33).
30
1.72
W»q/Aecs
1.69
1.66
1.63
1.60
0.0
0.1
0.2
0.3
0.4
0.5
H, tesla
Fig. 2-6
Zeeman splitting of the (imaginary) squashing mode.
(Ref. 42).
Table 2-1
Temperature dependence of the normalized BCS gap function,
ABCs (t )/Ab CS<0> (Ref. 19).
T/ Tc
W
T> / W 0)
1 .0 0
0 .0 0 0 0
0.98
0.96
0.94
0.92
0.90
0.2436
0.3416
0.4148
0.4749
0.5263
0.5715
0.6117
0.6480
0.6810
0.7110
0.7386
0.7640
0.7874
0.8089
0.8288
0.8471
0.8640
0.8796
0.8939
0.9070
0.9190
0 .8 8
0 . 8 6
0.84
0.82
0.80
0. 78
0.76
0.74
0.72
0.70
0 . 6 8
0 . 6 6
0.64
0.62
0.60
0.58
T/ Tc
0.56
0.54
0.52
0.50
0.48
0.46
0.44
0.42
0.40
0.38
0.36
0.34
0.32
0.30
0.28
0.26
0.24
0 .2 2
0 . 2 0
0.18
0.16
0.14
W
T> ' W 0)
0.9299
0.9299
0.9488
0.9569
0.9641
0.9704
0.9760
0.9809
0.9850
0.9885
0.9915
0.9938
0.9957
0.9971
0.9982
0.9989
1.9994
1.9997
0.9999
1 .0 0 0 0
1 .0 0 0 0
1 .0 0 0 0
32
Table 2-2(a)
Data Table of the properites of liquid ^He (Ref.
*
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Table 2-2(b)
Polynomlnal fit of the properties of liquid ^He, Q(p) -
w***re
the coefficients a are taken from the table and the units of Q(p)
n
are given In table 2>2(a).
(Ref. 33).
■o
•i
3 6 .S 2
-1 .1 8 5
8 .6 0 9 x 1 0 -*
-4 .1 8 7 x 1 0 " *
1 .0 7 1 X 1 O 4
5 9 .0 3
- 1 .9 6 7
8 .3 4 9 x 1 0 " *
- 2 .3 0 4 x 1 0 *
2 .5 5 3 x 1 0 *
s
18 3 .1
1 7 .0 0
• 8 .9 4 0 x 1 0 " '
3 .8 2 4 x 1 0"2
-8 .9 4 3 X 1 0 "4
8 .4 8 4 x 1 0 -*
(eo-<H ) * ,
3 .2 6 8 x 1 0 -*
-4 .1 5 0 X 1 0 -*
3 .4 2 6 X 1 0 " 4
■1.597x10"®
3 .7 6 8 X 1 0 "7
-3 .4 8 7 x 1 0 - 9
C Jn R T
2 .7 8 0
7 .1 5 1 x 1 0 -*
-1 .6 7 8 x 1 0 "3
5 .3 4 2 x 1 0 *
-6 .1 6 0 X 1 0 "7
m '/m
2 .8 0 0
1 -2 9 2 X 1 0 '1
- 3 .1 8 8 x 1 0"3
9 .3 7 2 x 1 0 " 5
-1 ,0 3 0 x 1 0 "®
T*
3 .5 8 9 x 1 0 - '
•Z 471X 10*
2 .9 7 1 x 1 0 " *
- 2 .5 3 2 x 1 0 *
1 .1 9 9 x 1 0 *
- 2 .8 2 9 x 1 0 *
• 7 .9 1 8 x 1 0 " '
2 .6 3 5 x 1 0 " '
-2 8 1 1 X 1 0 " 2
1 .5 6 0 X 1 0 "3
•4 .1 1 4 x 1 0 -*
4 .1 4 7 x 1 0 *
- 7 .o o 7 x io " '
•6 .2 3 2 x 1 O '3
2 .0 5 7 x 1 O'4
-1 .8 2 3 x 1 0 * ®
• 5 .6 7 8 x 1 0 " '
- 4 .7 5 3 X 1 0 2
1 .791X 10"3
- 2 .2 7 3 x 1 0 " *
• 8 .7 7 4 x 1 0 " 1
4 .7 5 3 x 1 0 -2
-6 .3 9 9 x 1 0 “ *
9 .2 9 4 x 1 0 '
1 .3 8 7 x 1 0 " ’
-6 .9 3 0 X 1 0 "3
2 .5 6 9 x 1 0"4
- 5 .7 2 5 x 1 0 *
5 .3 0 1 x 1 0 -*
a +( o y v e
1 .7 7 4
7 .9 3 0 x 1 0 * 2
-4 .1 3 7 x 1 0 - *
1 .2 4 4 x 1 0*5
-1 .4 0 6 x 1 0 "7
&ac
1 .4 8 4
3 .4 6 9 x 1 0 * 2
-U 1 9 x 1 0 "3
4 .3 5 1 x 1 0 " ®
- 4 ,7 1 0 x 1 O 7
2 .9 6 4 x 1 0 " '
• 2 .9 1 1 x 1 0 - 2
2 .4 8 8 X 1 0 "3
- 1 .2 2 8 x 1 O '4
3 .0 4 2 x 1 0 "®
-2 .9 3 1 x 1 0 "®
- 2 .3 4 2 x 1 0 -2
2 . 028X 10*3
-1 .0 0 4 X 1 0 * 4
2 .4 9 4 x 1 0 "®
•2.411X 10"®
V
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•*
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CHAPTER 3.
3.1
EXPERIMENTAL TECHNIQUES
Introduction
The experiments on superfluid
reported in this thesis were
performed on a cryostat constructed previously at Northwestern; it is
equipped with a SHE Dilution Refrigerator (Model DRP-36) and a copper
nuclear demagnetization stage,
all of which have been described in
detail e l s e w h e r e . ^ ^ W e
will emphasize only those new features
which are relevant to our experiments.
A sketch of the experimental
silver tower, which extends above the nuclear stage heat exchanger, is
shown in Fig. 3-1.
3.2
Acoustic Cell
An acoustic cell,
residing within the tower, was assembled as
shown in Fig. 3-2.
Between the two silver halves making up the body
of
two
the
cell
were
x-cut
quartz
transducers
having
the
same
fundamental resonance frequency of 12.8 MHz; they were separated by a
pair of arcs of about 150° made of gold plated tungsten wire.
Two
wires were cut to the required length and then bent to form the two
arcs.
The radius of the arcs was equal to the mean value of the
outer
gold-
transducers.
plated-grounding-ring
The ends of
about half a minute
cutting.
to
of
the
plated
the wires were dipped into "Aqua Regia" for
etch away
the sharp edges resulting
The two pieceswere then varnished
transducers; two openings,
coaxially
near the rim of
positioned atthe topand the bottom,
34
from
the
MIXING CHAMBER
HEAT SWITCH
UPPER LCMN
SIVER HEAT LINK
SILVER FLANGES
VESPEL SPACER
ACOUSTIC CELL
SILVER TUBE
LOWER LCMN
PRESSURE GAUGE
HEAT EXCHANGER
COPPER BUNDLE
Fig. 3-1
Experimental Tower
SPLIT SILVER
HOUSING
190.5/im
CENTER PIN
OF CO-AX
/
Be-Cu SPRING
X-CUT QUARTZ
TRANSDUCERS
*
TEFLON WASHER
P t SPRINGS
GOLD PLATED
W WIRE
CO-AXIAL SPACER
Fig. 3-2 Assembly of Acoustic Cell
37
permitted heat transport through liquid -*He to the space between the
two transducers.
Behind one of the transducers was a 3.81 mm co-axial spacer with
a cross section identical to that of the transducer.
parts were made of
Sterling silver
1266 Stycast epoxy.
Two slots were cut onthe spacer rim to further
facilitate heat transport.
and the
The conducting
insulating
Two springs (made of .0039" Pt wire) were
used to make
electrical contact
ducers.
two halves of the cell body were
The
annulusfrom
to the active
area of
the trans­
held together
by a
pair of Be-Cu springs and four silver pins, which were silver painted
after being pushed into position.
coax lines.
Within the tower were two homemade
The central conductor was made from teflon-coated-silver
wire, and made contact at one end to a Pt spring.
ductor
was
a
ID — 0.031")
piece
of annealed
copper
tubing
The outer con­
(OD -
0.063"
and it was silver painted to the silver housing.
and
The
silver housing was then screwed on to a vespel tube which, in turn,
was epoxied to the cell flange of the tower
(see Fig.
3-1).
The
co-axes passed through openings in the cell flange and the copper
outer conductor was soft soldered to a silver bushing which, in turn,
was soft soldered to the cell flange;
this procedure assumed a leak
tight connection for the outer conductor of the co-ax.
The center
pin of a microdot connector was soft soldered to the inner conductor
of the co-ax.
An epoxy seal was then made at the annulus between the
inner and outer conductors of the
raicrodot
connector
was
co-ax.The outer
soft soldered
to
a
silver
shell of
sleeve;
the
this
combination was
then positioned around the central pin and silver
painted on to a silver ring (which had previously been soldered to
the copper sleeve of the co-ax prior to making the seal).
The
spacing between the
two
transducers was
measured by
two
different methods:
a)
the thickness of the transducer pair was directly
measured, with a micrometer, before and after the
tungsten wires were mounted, the difference being the
effective room temperature distance between two transducers.
b)
The single-ended, c.w., acoustic-impedance technique was
used to measure the phase change, 5<f>, via the transducer
driving point impedance of the acoustic standing wave
pattern In the cell at 1.2°K as a function of pressure
change,
6
P.
Since the sound velocity, c(P), Is a known
function of the pressure, the change, 5c(P), can be used to
determine the distance between the two transducers, d, by the
relation
d -
6±
(3-1)
c
where 8<t>, fq , P^, P^, c are, respectively,
the total phase change,
the r.f. frequency, the initial and final pressures and the pressuredependent
sound
measurement was
velocity.
from 7.9 bar
The
actual
to 9.7 bar
pressure
(5P -
corresponding phase change is 8<j> - 10»r (Fig. 3-3).
change
1.8 bar)
in
our
and the
For this small a
pressure range we may assume the sound velocity is a linear function
39
of
the
pressure,
9c
—
i.e.
is
a
constant,
which
simplifies
the
calculation.
The results of both approaches agree well with each other and we
obtained a spacing d — 190.5 pin.
Note that, since a single ended
technique was used in all our experiments, the actual path length is
that of a round trip; i.e., 381 pm.
3.3
Thermometry
Two LCMN thermometers
f41
were available for thermometry.
One
was mounted on a second silver flange (above the cell flange) and the
other on the nuclear stage heat-exchanger (Fig. 3-1).
We found that
3
the upper LCMN thermometer tracked the temperature of the He in the
sound path much better than the lower one.^^
All the temperatures
reported in this thesis were determined from the upper thermometer.
The upper LCMN thermometer consisted of a thin wall (0.0035")
stycast 1266 epoxy former on which one primary (ac driving) coil and
two
opposing
secondary
(pick
susceptibility measurements.
positioned
69.9
(NO^Jj^'^HgO] .
mg
of
up)
coils
were
wound
for
the
At the center of the upper secondary we
LCMN
powder
K
La0
95
Ceo 05^2
Mg3
The epoxy former was then epoxied to a silver tube
which had been previously soft soldered to the upper flange.
The
epoxy former was shielded with an annular Nb tube, which was further
shielded with a concentric NbTi tube.
Finally the entire thermometer
was wrapped with a 0.066" thick sheet of lead.
A Curie-Weiss behavior is commonly assumed for the temperature
dependence of the LCMN susceptibility, i.e.
7.9bar
= 9.7bar
I------------ <5* = 10rr-------------1
Fig. 3-3
Measurement of the Spacing
Between the Two Transducers
41
*<T > - f f i * ' .
<3-2)
Here x(T) is the measured LCMN susceptibility at "low" temperatures
(in range where the experiments are actually performed), xQ is the
bridge
reading
at a
sufficiently
high
temperature,
IK,
that
the
bridge reading assumes a constant behavior C (the Curie constant) and
8 (loosely, the Neel temperature) are constants to be determined.
The superfluid transition signatures detected from the acoustic
impedance
change
are
used
as
"fixed
points"
to
calibrate
thermometer (i.e. to fit the constants C, 8 and X Q*)•
the
Calibrations
are performed over a wide pressure range, between 3 bar and 28 bar,
r 6 I
using the temperature scale of Greywall.
6
is
sufficiently
(within
the
experiments
small
accuracy
are
that a
of
performed
our
magnetization,
magnetic
these
fields;
good calibration
measurements)
under
demagnetization,
In practice we find that
various
with
8
experimental
depressurization
different
can be
conditions
-
0.
Our
conditions:
and
result
obtained
in
in
finite
different
temperature gradients between the sample and the LCMN thermometer.
Therefore calibrations of the upper LCMN thermometer were established
for each of these cases.
A typical calibration curve is shown in
Fig. 3-4.
Because
temperature
the
is
LCMN
below
thermometer
several
is
sensitive only when the
3
millikelvin, a
He melting curve
*In practice Xo *-s also used as a parameter to improve the fit in the
region where data are actually taken.
42
0.8
0.8
1/T
(m K
0.7
0.6
0.4
0 .3 0
0.36
0.40
LCMN THERMOMETER READING a A
Fig. 3-4 Calibration curve of the upper LCMN thermometer
(DEM.; H - 0)
0 .5 0
43
thermometer (MCT) is mounted on the nuclear stage heat exchanger to
monitor the temperature during the precooling.
It
is well known that
slope,
i.e.
dP/dT < 0,
3
the
when the
He melting curve has
temperature
is below
a negative
318 mK.^^
f61
3
Greywall1
has used a fifth order polynomial to fit the He melting
curve in that temperature region.
P
'
Pmi„
-
L
V
"
:
< 3 '3 >
n- 0
here Pm^n “ 29.13 bar is the pressure at the minimum of the melting
curve (Fig. 3-5).
The melting curve thermometer is basically a capacitance gauge
3
containing a fixed amount of He. The pressure is calibrated against
the
capacitance
at
1.2°K with
a capacitance bridge,
versus P curve is fitted to a straight line.
melting curve minimum,
and
the
1/C
The capacitance at the
carefully measured as the gauge
(nuclear stage) is cooled.
Using Eq.
millikelvin
3.3 the temperature can be determined down to a few
from
satisfactory
the
accuracy.
corresponding
The
capacitance
sensitivity
of
reading
our
with
capacitance
measurement is about 0.005 pF corresponding to a pressure resolution
.3
of 1.2 x 10
bar at 34.26 bar and a temperature of 0.3 mK at 5
mK.t3]
3
He High Pressure System
3
A sketch of the He high pressure system is shown in Fig. 3.6.
3.4
The pressure in the acoustic cell is measured with a Paroscientific
44
souo
LIQUID
3.1,
1000
>00
fin*)
Fig.
3-5
The
melting
curve
of
^He
at
low
temperature
45
MCT
IcoHo
TRAP
TO CELL
<X> <$)
PARO
SCIENTIFIC
PRESSURE
GAUGE
coto
REFERENCE
GAS INLET
TRAP
EDLE VALVE
1S00
ps1’
SMALL
DUMP
LARGE
DUMP
(1-71)
Fig. 3-6
Gas handling system for -*He.
46
pressure gauge located at room temperature;
conditions
(e.g.
depressurization
when
measurements
are
technique), the pressure
in
however,
under dynamic
performed
the cell
with an in-situ Straty-Adams type capacitance gauge
f 81
with
the
is measured
(see Fig. 3-1)
which is calibrated at 1.2°K against the Paroscientific gauge.
Fig,
3-7 shows the pressure, P, versus the reciprocal of the capacitance,
1/C, and the polynomial fit to the form
r
, 79.41 1108.54
1532.41
P - 31.32 + — —
- -- —
+ — 3 --- ,
c
(3.4)
c
where P is the pressure (in psi) and C is the capacitance (in pF).
3.5
Gas Handling System and Modifications to the Cryostat
The
gas
handling
system
(GHS)
associated
with
our
dilution
refrigerator is shown in Fig. 3-8.
The following two modifications on the cryostat were made since
the last description of the system. F31J
a)
In order to reduce the radiation heat leak,
the original
brass mixing-chamber shield was replaced by a custom machined pure
copper tube.
Vertical slots were cut along the tube axis in order to
minimize eddy current heating.
aluminized mylar (0 .0
0 1
In addition, it is wrapped with thin
") to further reduce the radiation heat leak.
b) Two of the three nylon screws which supported the dilution
refrigerator were found to be broken and were replaced with vespel
47
o
n
in
CM
I
a,
o '— 8
U
in
o
o
o
o
o
o
o
CM
(isd )
Fig.
3-7
o
o
o
a n n ssa n d
Calibration
curve
of
the
pressure
gauge
48
G3
G4
#3
G2
GAS TANK
G1
o—
TC
He ROTARY
PUMP
Fig.
3-8
Gas
handling
He BOOSTER
PUMP
system
of
a
dilution
refrigerator
49
studs
with
brass
nuts
(note
nylon
is
very
brittle
at
low
where P is the pressure (in psi) and C is the capacitance (in pF).
temperatures).
mechanical
return
This
procedure
touch between the
line
to
the
removed
a
hitherto
undetected
first heat exchanger
flange and
chamber;
the
mixing
thereafter
the
precooling
temperature was lowered from 25 mK to 17mK.
3. 6
The c.w. Acoustic Impedance Technique
3.6.1
Comment on the time-of-flight pulse technique
The time-of-flight pulse technique with X-cut quartz transducers,
is frequently used to study the order parameter collective modes (OPCM)
3
is superfluid He. The advantages of this approach are that both the
sound attenuation and velocity can be measured directly.
technique inherently involves a finite bandwidth, Af -
7
However this
— , where Ar is
ZiT
the width of the pulse, which can lead to an artificial broadening of
the mode width.
overcome
One might initially think that this effect could be
simply by using a longer pulse.
However
the
very high
acoustic attenuations that occur near the collective modes (e.g.
cm ^ for
the squashing mode)
require
short
acoustic
path
lengths
which restrict the width of the pulse due to r.f. cross talk.
workers
have
partially
quartz
delay
line
to
circumvented
this
resolve
transmitted
the
problem
by
100
Some
inserting
signal
from
a
the
electrical cross talk.
However the impedance mismatch between the
quartz
greatly
and
approach.
the
liquid
reduces
the
sensitivity
of
this
50
3,6.2
Acoustic impedance
r91
For a classical liquid the acoustic impedance, Z, is defined1
as the ratio of the generalized pressure, P, to the liquid velocity,
v, at the transducer interface (at z —
0
); i.e.
Z - P/v,
(3.5)
Avenel et al. [91 have proposed an expression for Z for the case
3
of superfluid He.
It is semi-phenomenological and based on two
boundary conditions at the transducer interface.
c+v+ + c v , where p
The first is P/VQ “
is the liquid density and c+ , v+, c , v
are
the quantities referring to the two solutions for the mode velocities
and fluid velocities of the coupled equation of motion for the order
parameter distortion and the hydrodynamic flow (see Eq. 2-12 in Ref.
35
The second boundary condition is — — - &/$, where
10).
6
is a gap
distortion parameter and f is some accommodation length (of the order
of a coherence length,
- hvF/7rkgTc , of superfluid
3
He-B).
With
these assumptions the acoustic impedance of the liquid was calculated
as
p0 c+c_+iwfZ„
Zo P0 c+c_+iw?Z0 ’
Z
(3.6)
where
_
7
o
°
-2
P°
c+ + cc 9 "f* c+c_
and
c? + c .c
«>
o
+ c2 - c2c2/c2
c
T
+ c
are the impedance for w —
0
and <*> respectively.
The equivalent c i r c u i t o f
Fig. 3-9.
the loaded transducer is shown in
The total electrical impedance (including the contribution
of the shunt capacitance C^) is given by
V
ZT
Z
-f
—
<3 -7>
—
iwCQ
with
Zm - %
where
- 1ZQ “ = ( ¥ )
- f>0c0
h2co
1
Xq “ ^
wavelength,
+ H
Z i + iZQ Ca” f^)]
acoustic impedance of the quartz transducer;
; and A, Z, A and h are,
the
thickness,
the
respectively,
cross-sectional
the acoustic
area
and
the
piezoelectrical constant of the quartz transducer.
Changes
impedance
of
in
the
the
real
liquid
and
result
frequency of the transducer and
resonance.
In the
limit
power
frequency,
imaginary,
points),
AfR , are
Af^pp
parts
of
in
shift
in
Z
f 111
) a
the
acoustic
resonance
ti
(which
is appropriate
in
our
that the width of the resonance
and
related to
1
the
) a broadening of the width of the
Z «
L.
experiments), it can be shown
(half
2
imaginary
a
the
change inthe
change
in the
AXT ,parts of the acoustic impedance of
“
The relations are given by
resonante
real,
AR^ and
the liquid, ZT .
L
-o
V
■o
Fig. 3-9
The equivalent circuit for a piezoelectric transducer.
For our case of a symmetrically loaded transducer,
Z1 -Z2 “Z^j. In the text, Zq“Zc and H-h.
53
where f
is the fundamental resonance frequency.
The
above
expressions
are
best
exploited
in
a
"ring
down"
experiment where the decay characteristics of transducer, previously
excited to a steady state, are studied.
This is simply because a
relatively clean separation between the transducer response and the
frequency characteristics
of
interconnecting
r.f.
components
(e.g.
the transmission line) can then be obtained (due to the relatively
fast
response
separation
is
of
the
more
latter).
In
problematic;
a
c.w.
however
experiment
the
c.w.
a
clean
approach
is
generally more sensitive which tends to off-set some of the ambiguity
associated with separating the in-phase and quarature response.
3.6.3
The r.f. spectrometer
All the data reported in this thesis were taken with the singleended,
frequency modulated (f.m.), continuous wave (c.w.),
impedance technique,
which has
acoustic
two distinct advantages compared to
the r.f. pulse technique.
First, the frequency resolution is high,
which
are
is
features.
important
Secondly,
if we
to
resolve
spectroscopically
the high sensitivity of this
technique
sharp
is an
advantage in detecting signals in the high attenuation region (e.g.
near the squashing mode or in the vicinity of the pair breaking edge
at 2A).
A block diagram of our spectrometer is shown in Fig.
frequency
modulated
signal,
tuned
to
an
odd
harmonic
3-10.
of
A
the
fundamental resonance frequency of the transducer, is applied to arm
1 of a quarature hybrid.
Arms 2, 3, and 4 of the quadrature hybrid
I
Fig. 3-10
ELECTRONICS
Block
1
Diagram
SIGNAL
GENERATOR
HP608F
"7
of the CW Acoustic
FREQUENCY
COUNTER
snr
SYNCHRONIZER
Impedance
TUNING AMP
i
2
QUAD
HYBRID
tvhd
£
ACOUSTIC m
3
LOCK-IN AMP *
(f-mode)
4
DETECTOR
WBA600E
x-t CHART
Spectrometer
f
LOCK-IN AMP
{2f-mode)
x-t CHART
4>
55
are connected to a stub tuner, the co-ax line from the transducer,
and a low noise figure
(2dB)
r.f. preamplifier
(with 20 dB gain)
respectively.
The signal generator
(HP608F) is locked to a synchronizer
(HP
8708) to stabilize the frequency of the output signal frequency to
about
1
part
in
10®.
The
synchronizer
also
allows
frequency
modulation by a reference signal derived from a lock-in amplifier
(PAR HR-8 );
Because
the
of
modulation
thesharp
frequency,
f , was
(resonant) frequency
typically
400
Hz.
of
the
characteristics
transducer, an f.m. signal applied to the transducer will acquire an
amplitude-modulated (a.m.)
component.
The output of arm 4 of the
quadrature hybrid will then contain an f.m. part (the applied signal)
as
well
as
transducer)
The
an
[1 2 ]
a.m.
part
(the signal
reflected
back
from
the
both of which are modulated with the same frequency.
(wide band)
detector
(Arenberg WBA-600-E), however,
will
only
respond to the a.m. signal and the output will be an a.c. signal at
the modulation frequency,
which
is,
in turn,
converted to a d.c.
signal by the lock-in amplifier.
The r.f.
(using the
Passage
background signal
stubtuner
ofa collective
at
arm 4
is approximately
nulled
in arm 2) at a temperature just above
mode
or
the occurrence
of various
T .
other
phenomena in the liquid (e.g. the superfluid transition) will result
in a change of the acoustic impedance of the liquid, which, in turn,
results in a characteristic signature at the output of the lock-in
amplifier.
A standing wave pattern can be set up in the acoustic
56
cell in the low attenuation regime.
Changes in the phase velocity of
the sound wave will result in a change of the driving point acoustic
impedance
of
the
transducer;
changes
in
the
attenuation
can
be
detected by observing the envelope of the oscillations caused by a
changes in the phase velocity (which is,
change in temperature or pressure).
in turn,
caused by the a
details of these phenomena will
be discussed in the next chapter.
In ending this section, we should again point out that the real
and imaginary parts, ReZ and ImZ, of the total impedance, Z, are not
simply
related
to
sound
attenuation
and
velocity.
Numerical
simulations could be performed to fit the experimental data.
In
a
region
where
the
attenuation
is
high
enough
so
that
reflected waves can be neglected, but still low enough to not affect
Z, a change
in the real part of the acoustic
induced simply by changing the pressure
impedance could be
(and hence
the velocity);
this promising method was not exploited in the present work.
3.7 Experimental Performance
The data reported in this thesis were taken according to the
following procedure:
1)
The acoustic cell was usually loaded up to a pressure of
one or two bars above the desired value, the copper bundle
(and the attached acoustic cell) were then precooled (for 48
hours) down to about 17 mk with the demagnetization magnet
maintained at 50A (8 T).
If the experiment was to be
performed In a finite magnetic field, the cell magnet
57
(henceforth referred to as the Kampwirth magnet) was
ramped up to the desired value (0.866 KG/A) at the outset.
2) The heat switch was then opened and a demagnetization
performed.
The starting demagnetization rate was usually
about 1.5 gauss/sec., and, when approaching the superfluid
transition, was reduced to about 0.28 gauss/sec.
If the experiments were to be performed with a high pressure
(P £ 21 bar) in the cell, the starting demagnetization rate
was decreased to 0.25 T/hour in order to keep the bundle
and the cell in reasonable thermal equilibrium.
3)
At a temperature near Tfi (typically 5 mK) the depressurizing
t e c h n i q u e w a s used to generate an unambiguous acoustic
signal (the changing phase velocity) to permit a final
optimization of the tuning of the spectrometer.
The
experimental magnet (referred to as Kampwirth magnetf^J)
was also checked to see that its field had not shifted
*
significantly (due to induction from the main magnet).
4)
When the liquid was further cooled down, the normal to
superfluid transition signal was recorded, for
calibration of the LCMN thermometer.
Most of the data reported in this thesis were taken during a
demagnetization and remagnetization cycle.
with the depressurizing technique.
The remainder were taken
The input level from the signal
generator was limited to a few millivolts (peak to peak)
heating;
to reduce
it was usually necessary to make a compromise between the
58
signal to noise and the (power-level dependent) time available in the
superfluid.
CHAPTER 4
4.1
Introduction
The
major
results
RESULTS AND DISCUSSION
obtained
from
our
ultrasonic
studies
of
superfluid ^He-B involve the following phenomena:
(1) Systematic measurements of the pair breaking edge have been
performed at frequencies of 64.3 MHz, 90.1 MHz, 141.6 MHz and 167.4
MHz (which are respectively the 5th, 7th, 11th and 13th harmonics of
the
fundamental
transducer)
12.8
MHz
with pressures
magnetic field.
resonant
ranging
frequencies
from
3 bar
of
the
to 28 bar
quartz
in zero
Our results are consistent with predictions made for
the strong-coupling corrections to the energy gap.
In addition, the
data indirectly support the temperature scale of Greywall.
Also,
the pair breaking edge was measured in magnetic fields
ranging from zero to 1.36 kG with a sound frequency of 141.6 MHz at
28 bar.
A downward shift of the pair-breaking edge proportional to
the applied field has been observed.
(2) The imaginary squashing mode was studied with frequencies of
115.8 MHz and 141.6 MHz at pressures between 19 bar and 28 bar.
doublet splitting of the squashing mode has been observed.
A
The width
of the splitting is dependent on the pressure but has no substantial
dependence on a magnetic field applied perpendicular
to
the sound
propagation direction.
(3) An anomalous "bump-like" structure in the acoustic impedance
has been observed in the vicinity of 2A(T) using a frequency of 64.3
59
MHz in a magnetic field of 0.58 KG with pressures ranging from 4 bar
to 7 bar.
The frequency shift of the position of this structure is
about 3 MHz above the pair-breaking signature in the same field (when
the g-factor is taken as
consistent with the J
4.2
z
the theoretical value
of 0.392.);
it is
- -1, J — 1" mode.
Pair Breaking Edge in Superfluid ^He-B
4.2.1
Theory and early experiments
As discussed in Chapter 2, the gap function of superfluid -^He-B
in zero magnetic field is isotropic with an amplitude A(T,P)
2-26).
If
the
zero-sound
frequency
is higher
than
2A(T,P),
(Eq.
the
Cooper pairs can be broken into two ^He quasiparticles (excited into
the
continuum
mechanism
region).
results
in
The
a
large
absorption
of
attenuation.
referred to as (acoustic) pair breaking.
zero
sound
This
by
phenomenon
this
is
Wolfle has calculated the
pair breaking contribution to the attenuation ast^-1
tanh
8 (hw-2A(T)) (4.1)
where q, c q and c^ are the wavevector and the zero and the first sound
velocities respectively,
and the remaining terms have been defined
previously (or have their conventional definitions).
The step function:
- 2A(T)], ensures that the contribution
to the sound attenuation arises only from phonons with frequencies
above the pair breaking threshold (which is 2A(T)
in zero magnetic
field); as the temperature is lowered a sharp drop of the attenuation
61
occurs when hw — 2A(T) (as plotted in Ref, 2), which will be referred
to as the pair-breaking "edge".
Serene predicted a phase velocity change at the pair breaking
edge as given by Halperin and
Ac _ CQ - C 1 6
£
c^
5
Varoquauxt^]
-1.05 [ A(T)]
2 {l-4[A(T)/hc]2}2
[ ho?
1
1
-
2 . 2
t a n h ^ <?{2A(T) - hw)
B
(4.2)
The phase velocity, c,, almost remains constant when hw > 2A(T)
9
(because of the step function) and an abrupt change of c , is expected
9
at the pair breaking change (as plotted in Ref. 4).
Summarized briefly, both a sharp drop of the sound attenuation,
a, and an abrupt change of the phase velocity, c,,
are expected as
9
the temperature is lowered through the pair breaking edge at constant
sound frequency.
Measurements of the acoustic pair breaking edge are
of considerable interest since they provide a direct measurement of
the energy gap,
A(T) .
The results may
then be used to evaluate
various theoretical models for the gap function.
models are the weak-coupling BCS m o d e l a n d
Two commonly used
the weak-coupling-plus
model of Rainer and Serenely ,[7], i^e corresponding gap functions
will be denoted as Ag^,g(T) and A+ (T) in the following sections.
In 1980 Giannetta et al.t®] first observed a separation between
the pair-breaking and the squashing mode contributions to the sound
attenuation at 5.3 bar and 60.0 MHz.
In 1983 Daniels et al,[2] were
62
able to more completely resolve the pair-breaking attenuation peak
from the squashing mode at 1.36 and 2.44 bar with frequencies of 44
MHz and 54 MHz by using an acoustic cell with the quartz delay line
and a short path length of 250 pm.
Meisel et al.f^J also resolved
these two attenuation features and,
in addition, detected an abrupt
change of the phase velocity at the pair-breaking edge.
results
are
listed
in Table
4-1.
Quite
recently,
The above
Movshovich
et
al.l^l succeeded in using higher frequencies (107 MHz, 137.6 MHz and
168.3 MHz) to probe the pair- breaking edge in the low T/Tc (~0.35)
region in a magnetic field of 0.46 Tesla.
4.2.2
Pair-breaking edge in zero magnetic field
Before presenting our experimental results, we note that certain
properties
of
superfluid
^He
show
predictions of weak-coupling theory.
marked
deviations
from
the
According to the weak-coupling
theory, the specific heat jump at the transition should be AC/C^ =
1.43; the measured specific heat jump in -*He varies from nearly the
BCS value at the vapor pressure to 1.9 at the melting pressure.f10)
Moreover, the B-phase is the only stable phase predicted by the weakcoupling theory, but in -*He, at pressures above 21 bar,
is also stable.
the A-phase
These deviations from the weak-coupling theory are
the result of strong-coupling corrections.
Theoretical models
for
incorporating the effect of strong-coupling on the size of the gap,
A,
include
the renormalized
gap, [H]
A -
(AC/CH)'*'^Anri , and
»
uub
the
weak-coupling-plus model of Rainier and S e r e n e . ,[7] jn the weak„
coupling-plus
model,the
contributions
to
the
free
energy
63
difference between the superfluid and normal states are arranged in
powers of
The BCS free energy is the leading term in this
expansion
and
is
2
(Tc/Tp) . The
of order
retains terms to order (Tc/Tp)
quasi particle
3
weak-coupling-plus
model
and these corrections depend upon the
scattering amplitudes
for
the normal
For most superconductors, in which T^/T^, - 10
-4
liquid.
-5
, the superfluid
3
state is well described by the weak-coupling BCS theory.
In He,
however,
Tc/TF -
.3
and there
10
-10
Fermi
is a strong residual
interaction
between quasiparticles, which is an order of magnitude stronger than
in most superconducting metals.
the (Tc/Tp)
In
3
term important in
attempting
coupling
effects
to
one
These two effects combine to make
3
He.
experimentally
must
keep
in
determine
mind
that
certain
there
disagreement concerning the correct temperature scale.t
strong
is
^
still
] ,[13]
Thus measurements of the gap function involve two problem areas: what
is the best model for A(T) and which is the most reliable temperature
scale.
In an attempt to separate these two areas of uncertainty we
performed
the
strategies
pair
breaking
in mind:
(a)
A
measurements
measurement
with
of
the
the
following
two
at very
low
gap
pressures (where the weak-coupling theory is generally agreed to be
applicable) and at low T/Tc (where we essentially measure the zero
temperature
gap)
would
unambiguously
determine
the
transition
temperature at that (low) pressure through the BCS relation A
1.764kgTc . ^c (P)
different
is
temperature
then
scales
compared
(two
with
that
predicted
of which 1^2],[13]
BCS
(0) -
by
the
currently
are
64
related to each other by a multiplicative constant).
(b) Since (on
model-independent
of
breaking
edge
kinematic
must
be
2
grounds)
, then
the
we
can,
coefficient
based
on
the
the
pair-
"correct"
temperature scale obtained from step (a) , select the gap model that
will maximize the cluster of our data points around the condition
hw —
2
A.
Step (a) was first attempted with a sound frequency f - 90.05
MHz at a pressure of 2.620 bar (where Tc - 1.25 mK according to the
Greywall temperature scale).
Then the pair-breaking edge is expected
to be located at T - 0.53 mK (T/T
coupling model.
Unfortunately,
- 0.434) according to the weak-
the lowest temperature we attained
was 0.822 mK and the edge was not observed.
systematic
measurement
of
the
frequencies ranging from 64.3 MHz
We then performed a
pair-breaking
to 167.4 MHz,
edge
with
pressures
sound
ranging
from 2 bar to 28 bar, and over a temperature range from 1 mK to 2.1
mK; this corresponds to T/Tc ranging from 0.62 to 0.95.
The experimental technique has been reviewed in Chapter 3.
To
analyze the data, we used the temperature scale, and the values of
A C / C a s a function of pressure, reported by Greywall.t*-0]
A typical temperature trace is shown in Fig. 4-l(a).
As we cool
into the superfluid, there is a step in the impedance at T . Below
Tc>
the attenuation
is high
(due
to damping by
process) and continues to increase as we cool.
the pair-breaking
At a temperature T^n,
P d
where hi/ -
2
A(TpB>, the sound attenuation decreases abruptly and we
observe the onset of oscillations due to the presence of standing
65
m
f\i
<N
CD
•“ ^
T
O
-O 2
(O (O
w
00
CM
*-
II
O
05
D
II
Q.
H
X
C
05
in W
a,
S
w
H
u
Q
0
as1 —
Or
CO
_
ro
1V N 0 I S
Fig. 4-1(a)
Typical temperature sweeps. These traces are during a
demagnetization and are taken as a function of time,
with the approximate temperatures as shown,
f - 141.6 MHz P - 28.5 bar
66
waves in the cell.
(The oscillations are caused by the changes in
the sound phase velocity with temperature or pressure and can only
appear
when
the
(reflected)
response).
wave
attenuation
causes
a
is
low
enough
measurable
that
shift
in
the
the
returning
transducer
The decrease in the period of the oscillations is due to
the approach of the squashing mode (the velocity changes very rapidly
near this strongly coupled collective mode).
mode peak is split;
Note that the squashing
this phenomena will be discussed in Sec. 4.3.
The point at which the oscillations appear (implying the presence of
a standing wave pattern in the cell and hence a sudden decrease in
attenuation) is taken as the pair-breaking edge, 2A.
The pair-breaking edge was measured by varying both the pressure
and the
temperature.
alternating between
We have
temperature
described
the
and pressure
technique,
sweeps
involving
elsewhere.[1 ^ 1
The technique was also used recently to study the Zeeman splitting of
the squashing
m o d e . [15]
The
pressure sweeps give a clearer signature
of the edge (especially at low T/Tc), since the velocity changes more
rapidly with pressure than with temperature, causing the oscillations
to have a shorter period.
Another trace with 64.3 MHz and 5.4 bar is
shown in Fig. 4-l(b).
In Fig. 4-2 we have plotted our data for the pair-breaking edge
in the pressure-temperature plane.
The
curves
calculated values for 2A obtained using both A
correspond to
the
and A+ (the weak-
coupling-plus model of Rainer and S e r e n e *[7]) for the frequencies
64.33 MHz, 90.05 MHz, 141.6 MHz and 167.4 MHz.
From this figure, it
67
eo
to
n
- a
* 0s5
w
Q
O
s
I
or
o
CM
cn
H
<3
05
w
O
h
s
2
2
w
H
1VN 0 I S
Fig. 4-l(b)
Typical temperature sweeps. These traces are during a
demagnetization and are taken as a function of time,
with the approximate temperatures as shown,
f - 64.3 MHz P - 5.4 bar
68
30
167.4MHz
PRESSURE
(bars)
25
*
20
15
'm
10
64.33MHz
5
FREQ. DEP.
64.3
♦
90.0
•
141
■
167
m
DEM.
o
o
□
x
0
1.0
2.0
2.5
TEMPERATURE ( m K )
Fig. 4-2
Pair breaking edge in the pressure-temperature plane. The
solid curve corresponds to 2 AfiCS and the dashed curve to 24
for each frequency. The data were taken during
depressurizations (DP) and demagnetizations (DM).
69
can be seen that our data fits the weak-coupling-plus model better
than BCS.
The scatter in our data is chiefly due to our thermometry.
As mentioned earlier the LCMN thermometer is located in the low
field region, resulting in some temperature uncertainty.
To minimize
any temperature gradients between the region of the transducers and
the
thermometer,
conditions,
sweep.
all
data
reported
here
are
taken
under
similar
that is while doing a temperature sweep or a pressure
Note that the data was taken over a long period of
6
months,
covering a wide range of frequency, pressure and temperature, and it
was necessary to warm the cryostat to room temperature on several
occasions.
This
recycling
introduces
an
uncertainty
in
the
thermometry between different cool downs.
The
frequency
of
the
pair-breaking
edge
(normalized
to
the
calculated value of A+ (0)) as a function of the reduced temperature
is shown in Fig. 4-3.
2A+
(at 29 bar,
The two solid curves are 2A+ (at 0 bar) and
which is the highest pressure at which data were
taken) and join together at T/Tc - 0.6 and T/Tc - 1.
The scatter in
our data precludes our seeing any tendency for the higher pressure
data to cluster near the upper curve.
However most data points lie
between the two curves.
As stressed earlier, the coefficient of the gap, a^, (defined ht/
- a^A) for the pair-breaking edge, must be 2.
coefficient
for
all
our
data
points
We have calculated the
using
both
the
Helsinki
temperature scale[12] ancj the Greywall temperature scale[10] as shown
in Figs. 4-4(a)-4-4(d).
Note that the points near Tc inevitably have
2.0
2A+(P ,T)/A +(P ,0)
2A+(2 9 BAR)
2A+(0 BAR)
0
10
20
25
B A R - 10
BAR-20
BAR-25
BAR-29
BAR
BAR
BAR
BAR
0 .5
0.0
0.6
0 .7
0.8
0.9
TA C
Fig. 4-3
Frequency of the pair-breaking edge (normalized to
A+ (0)) vs. T/Tc. The two solid curves are 2A+ (at
zero bar) and 2A+ (at 29 bar).
1.0
2 .4
O
P<21 BAR
♦
P>21 BAR
2.2
2.0
a =2
1 .8
1.6
0 .6
Fig. 4-4(a)
0 .7
0 .8
0 .9
T /T c ( H e ls in k i te m p , s c a le )
The coefficient of the edge calculated using the
Helsinki temperature scale with the BCS gap.
1.0
72
2 .4
P<21 BAR
P>21 BAR
hi//A
2.2
2.0
a =2
«o
1.8
1.6
0 .7 0
Fig. 4-4(b)
0 .7 5
0 .8 0
0 .8 5
0 .9 0
0 .9 5
1.00
T /T ^ (H e ls in k i t e m p e r a t u r e s c a le )
The coefficient of the edge calculated using the Helsinki
temperature scale with the weak-coupling-plus gap.
2 .4
2.2
co
o
OQ
2.0
= 2
1.8
P<2I BAR
1.6
0 .6
Fig. 4-4(c)
P>21 BAR
0 .7
0 .8
0 .9
T/Tc (Grejrwrall)
The coefficient of the edge calculated using the
Greywall temperature scale with the BCS gap.
1.0
2.2
O
h y /A
+
♦ .*
2.0
a =2
—
“
*
°
*
♦*
o°
1.8
1.6
-I
0 .6
O
P<21 BAR
♦
P>21 BAR
-- 1--1__1__I__I__I__I
__I ' ‘ ■ ■__ 1__I__ 1__I
__u
1
0 .7
T /T
Fig. 4-4(d)
C
0 .0
0 .9
(G re y w a ll)
The coefficient of the edge calculated using
the Greywall temperature scale with the weak
coupling-plus gap.
1.0
75
a greater scatter; however,
points are
it is still apparent that,
to cluster around
2
, it
is necessary
to use both
Greywall temperature scale and the weak-coupling-plus
4(d)).
These
results
provide
Greywall temperature scale.
an
indirect
if our data
gap
the
(Fig.
4-
of
the
confirmation
Independent measurements of the pair-
breaking edge by Movshovich et al. at low temperatures also reach the
same conclusions.^]
We have
broadening.
not
taken
into
account
the
effect
of
quasiparticle
The effect of broadening would be to shift our signature
of the pair-breaking edge to temperatures lower than 2A; however the
nearly vertical character of the edge.t^]
short
path
length
tends
to
minimize
combined with our very
the
effect.
There
suggestion in the data that the coefficients at T/Tc greater than 2.
tends
to
capacity
pressures
a
will be
This would imply that the weak-coupling-plus model
underestimate
measurements
( >
0
is
the
(by
25 bar)
strong-coupling
Alvesalotl^]
also
indicate
and
that
corrections.
G r e y w a l l [1®]
other
strong
Heat
at
high
coupling
effects may be important.
To summarize this section, we have made systematic measurements
3
of A(P,T) in the B-phase of He in zero magnetic field. Our results
support the weak-coupling-plus model and, indirectly, the temperature
scale of Greywall.
76
4.2.3
Measurements of the pair-breaking edge in the finite
magnetic field.
4.2.3.1
The location of the pair-breaking edge in a magnetic
field
The presence of magnetic field has the following two-fold effect
on the pair-breaking edge.
(1)
Quasiparticle Zeeman Splitting Effect
When a magnetic field is applied to normal liquid
3
He,
it will
split the "up-spin" and "down-spin" Fermi surfaces by an energy ± pH
where
m3
--------------------------------------------------- (4-3)
1 + Fa
o
4
here p^ is the magnetic moment of a
Fermi liquid parameter.
3
3
He nucleus and Fq is a Landau
When the liquid enters the superfluid state,
an energy gap appears between the ground state and the continuum.
The gap associated with the "up-spin" surface (referred to as A(H) in
Fig.
4-5) is smaller than it is in zero field (referred to as A(0)in
Fig.
4-5), as discussed in the previous sections.
Obviously
h^pg
-
Schopohl
2A(T,H),
et
the
will
effective
shift
to
al.calculated
pair-breaking
lower
the
edge,
frequency
shift,
(see
and writing
defined
Fig.
it
4-5).
in
terms of a renormalized Larmor frequency we have
fi - 2A(0) - 2A(H)---- -— -----1 + i Fa(2+y)
as
(4-4)
the
77
2.0
O
o
u
1.0
•ound
CVJ
0.2
0.4
Q.6
T /T
'
Fig. 4-5
0.8
1.0
c
A sketch of the pair-breaking edge in a magnetic field.
78
where
- 3.243
7
MHz/kG,
H, FQa and Y -f (dfl/4x]J ^ d c ^
are respectively the gyromagnetic ratio of liquid
magnetic field,
the
Yosida
Ek “ A
(2)
3
E
He, the applied
the zerothorder antisymmetric Landauparameter
function;
:2
sech2 ~
the
excitation
energies
are
+ a2 (0 )
and
given by
(4'5)
Gap distortion effect.[19]
The magnetic field at low temperatures will cause a distortion of
the energy gap of
2
He-B resulting in a decrease in the longitudinal
energy gap, A||, and an increase in the transverse energy gap, Aj_,
corresponding respectively to directions parallel and perpendicular
to the applied magnetic
(An) enters
field.
Apparently the smaller energy gap
the expression of the pair-breaking edge and causes a
shift of the pair-breaking edge to a lower frequency by an amount of
n
1
c0*/Ao (P.T) , where the coefficient c goes to ^ as T -+ 0.
So
the
final
expression
for
the
effective
pair-breaking
frequency to the second order in the magnetic field is
w p B
-
2 A 0 ( P .T )
-
O
-
c n 2/ A 0 ( P , T )
4.2.3.2
Results and discussion
Clearly
it
would
be
of
considerable
(4-6)
interest
to
perform
a
systematic study of the pair-breaking edge in a magnetic field.£9] On
completing our study of zero
largely
field pair-breaking edge our interest
shifted to the newly observed splitting
squashing mode,
of the
imaginary
which will be discussed in the following section.
79
However,
some
limited
measurements
were
performed
on
the
pair-
breaking edge for magnetic fields up to 1.36 kG (which is quite low
in order to study the quadratic effect of the magnetic field), at a
pressure of 28 bar with a sound frequency of 141.6 MHz.
frequency shift of the pair- breaking edge from
which
was
proportional
experimental
calculated
results
are
renormalized
to
the
external
plotted
in Fig.
Larmor
frequency,
2
A(0 ) was observed
magnetic
4-6
A downward
field.
compared with
nT,Ui-nnv
(when
The
the
only
1 H L U K Y
containing the linear part of
the field effect) .
Note
observed field dependent shift of the pair-breaking edge
small:
that
the
is rather
the maximum shift for our field of 1.36 kG is approximately
12.5 MHz.
A 2% shift in the temperature scale (which represents the
uncertainty for our measured temperatures) corresponds to a 3.5 MHz
shift of the edge; this uncertainty is depicted as the error bars in
Fig. 4-6.
We conclude that the measured shift of the effective pair-
breaking edge in a magnetic field is in satisfactory agreement with
theory.
4.3 Doublet Splitting of the Squashing Mode
As discussed in Chapter 2 the equilibrium order parameter of the
superfluid ^He-B in zero magnetic field may be expressed by a 3 x 3
matrix
AiP “
V T> RiP
<*-7>
where the amplitude 4q (T) is the energy gap and
matrix connecting
the spin and orbital
coordinates
is a rotation
by
a rotation
80
RENORMALIZED
LARMOR
15
N
W
10
>o
THEORY
w
p
O'
w
5
Cl*
0
0.0
0.5
1.0
1.5
MAGNETIC FIELD (KG)
Fig. 4-6
The measured shift of the pair-breaking edge in a
magnetic field; the straight line is the theoretical
value; P - 27.7 bar, f - 141.6 MHz.
2.0
81
A
through an angle
"intensity"
of
6 about
the
interaction between
104° I-*]; n
gap
an axis n;
TrA+A
the
^He
is arbitrary.
is
ip is
a phase
isotropic.
nuclei
The
results
However,
in
external
factor.
magnetic
a value
magnetic
of
or
The
dipole
0
of
electric
fields,
boundaries and a superflow velocity field affect the
A
orientation of n (and also the structure of the order parameter).
The study of such effects,
generally termed texture effects,
is an
active (and rather complicated) field.
In 1982, an interesting feature observed by Shivaram et al.t^O]
was a "doublet" splitting of a
mode
-
0
state of the real -squashing
when a magnetic field perpendicular to the sound propagation
direction was applied.
The observed doublet splitting depended on
the pressure, but was field independent in the range 50mT to 160mT
(thelatter was
hysteresis
the highest
behavior
with
field
field.
applied);
Various
it also displayed
explanations
have
been
proposed by Volovik,[21] Brusov et al.,[22] Fishman et al.,[23]
explain this novel feature of the real-squashing mode;
a
to
they relate
the splitting to details of the texture.
The study of texture or dispersion effects on the (imaginary)
squashing mode
is also
clearly of
interest.
However,
phenomenon
similar to those observed for the real-squashing mode have not been
reported to date.
This
is primarily due
attenuations that occur near the J — 2
to the
collective mode. Two ways to
approach this experimental problem are to use
cell
(for
propagation
experiments)
enormously high
a short path length
or theacoustic
impedance
82
technique
which
is
especially
sensitive
to
the
high
attenuation
region.
In this thesis we report
a study of the squashing mode with
techniques similar to those of Ref. 20 but with the path length in
the sound
cell reduced by a factor
of
twenty.
technique was fully described In Chapter 3.
the previous section,
This
short
path
From the discussion in
the Greywall temperature scale and the weak-
coupling-plus model for the gap function, A+ (T) , are used to analyze
the data.
The data to be reported here were taken at frequencies of 115.8
MHz and 141.6 MHz, corresponding to the 9th and 11th harmonics of the
transducer.
A typical temperature trace Is shown in Fig. 4-7.
The
new phenomenon which we are reporting here is the doublet splitting
observed at the squashing mode peak.
The behavior of this splitting
has been studied for pressures ranging from 19.2 bar to 27.7 bar in
zero
magnetic
field.
Measurements with
the magnetic
field
-»
perpendicular to q (the sound propagation direction) were performed
up to 1.36 kG at a single pressure of 27.7 bar.
The main features
observed may be summarized as follows:
(i) The doublet splitting of the squashing mode is observed in both
zero
(Fig.
4-7);
i.e.
4-l(a)
there
and
is no
Fig.
4-8)
and
finite
threshold value
for
magnetic
field
(Fig.
the field required to
produce the splitting (as was the case for the real-squashing mode).
83
in
o
JQ
hs
r^.
CM
2
tO
‘
i-
a
w
£>
EPh
h
a,
S
w
E-i
in
o
eo
1VN0IS
Fig. 4-7
Typical demagnetization traces of the acoustic impedance
signal showing the doublet splitting in the squashing mode.
The traces are taken as a function of time with the
approximate temperatures as shown.
P - 27.7 bar, f - 141.6 MHz, and H - 1.07 kG.
84
(ii) A Lorentzian fit of the form
(^/w)(T-To)sin^
dcosd
+
(4-8)
(T-To)2 +(w)
2
yields a good representation of the data;
respectively,
the
peak
position,
the
here T , w and <f> are,
half-width
(HWHM) and the phase of the detected signal
at
half-maximum
(the appearance of an
unknown phase angle results from peculiarities of the c.w. acoustic
impedance method which we will not discuss here).
Lorentzian
is
conventional
given
by
non-linear
Fig.
least
4-8
square
fit
The area under a
shows
an
to
temperature
a
example
of
a
sweep
performed at a pressure of 19.2 bar with a sound frequency of 115.8
MHz
in
zero
field.
In
order
to
best
represent
feature,
the fit was performed in the range
although
the
range.
resulting curve
splitting
1.51 mK to 1.55 mK,
Is plotted over
a wider
temperature
Two Lorentzian lines were assumed which are depicted by the
dot-dash line (A) and dashed line (B) .
as
the
the
solid
line,
is
in
reasonably
The resultant trace,
good
agreement
with
experimental data, which are depicted by the filled circles.
shown
the
The
fitting yields the following parameters: <f> — 25.5°
0A - 1.3x10
O
-4
- 13.7x10
Tq a - 1.533mK,
-1
2-1
wA - 1.73x10 K
A
Tq b - 1.517mK,
w_
D
-1
2-1
- 0.91x10 K.
The coupling strength, X, of the collective mode components will
be assumed to be proportional to the area under the corresponding
Lorentzian; by this criteria the ratio of the coupling strength of
(arbitrary
unit)
85
12
10
19.2bar
115.8MHz
a
INTENSITY
a
4
SIGNAL
2
0
—
1.45
1.50
1.55
1.60
TEMPERATURE
1.70
(mK)
Fig. 4-8 A Lorentzian fit to the doublet splitting of the imaginary
squashing mode: P - 19.2 bar, f - 115.8 MHz and H - 0.
The dots are the experimental data; the fits to the
two components referred to as A, and B, of the collective
mode are shown as the dot-dash and dashed line
respectively. The solid line is the resultant curve.
86
these
two
components
generated a model
Is
~
to interpret
0.18.
Avenel
the acoustic
et
al.t^]
impedance
have
signal when
only one of the SQ mode components couples to zero sound; this model
could
easily
be
generalized.
We
adopted
a
Lorentzian
form
for
simplicity.
(iii)
A
pressure dependence
of the zero
field
splitting
is
Clearly,
the
observed and the results are plotted in Fig. 4-9(a),
splitting
increases
as
the
pressure
is
increased.
The
T/T
dependence of the splitting (at the frequencies studied) is plotted
in Fig. 4-9(b).
Note that the absolute value of the splitting around
27 bar
for f - 141.6
between
twosets
months.
For
do
believe
not
procedures.
of
MHz
could not be quantitatively
measurements
which
were
reproduced
separated
by
three
the small scatter of as(j> as shown in Fig. 4-10(a), we
this
arises
from
artifacts
of
our
measurement
Most likely it involves different heat flows within the
cell as discussed later.
Varoquaux et al. report two "horns" at SQ
mode in ^He-Bl^] and they found that the behavior of this splitting
(using our term) was related to the power level of the input signal
which may also
induce different heat flows.
However,
we
did not
carry out a systematic study of our doublet splitting as a function
of the input power level.
^-f^) > *-s calculated
for
against
Fig.
obtained,
pressure
which
in
agrees
The squashing mode coefficient,
the
center
4-10(b).
reasonably
of
the
A
value
well
peak
with
of
a
-
and
is plotted
a
-
that
1.62
is
obtained
by
Movshovich et al.t^-^1 Also a slight pressure dependence of a
can be
87
observed,
(iv)
As seen from Fig.
4-11 (a) no magnetic field dependence of
the splitting (at the fixed pressure of 27.7 bar with f -> 141.6 MHz)
is observed.
plotted
As a byproduct,
against
the
field
in
a
for the center of the peak is
Fig.
4-ll(b),
It
should
not
be
surprising that no visible field dependence of ag^ is observed here.
According to Meisel et al.t^l, the quadratic coefficient,
in magnetic field is about 21 MHz
were taken.
sq
w
of a
for T/Tc - 0.75 where our data
The highest field employed in our experiments was 0.136
Tesla corresponding to a mode frequency shift of
8u>
r,
- 0.39 MHz, or
Sa
sq
a
sq
sq
- 2.8 x 10'3 .
(4-9)
which is far smaller than the scatter in the data.
(v) This new feature of the SQ mode has not been resolved for
sound frequencies 90.1 MHz and 64.3 MHz.
In short the observed doublet splitting of SQ mode
is strongly
pressure dependent but magnetic field Independent phenomena (in the
range studied).
It was unexpected that a doublet splitting rather than a three­
fold splitting (as in the case of the
dispersion induced splitting
for RSQ mode), was observed since for
a total angular
J
be
-
2
a
three-fold
splitting
superflow or an electric field.
would
induced
by
momentum of
dispersion,
88
10
O
55
M
-5
-10
20
22
24
PRESSURE
Fig. 4-9(a)
26
28
30
(bar)
Pressure dependence of the width (in temperature) of
the doublet splitting of SQ mode, where f — 141.6 MHz
and H — 0. 0(«) are data taken during demagnetization
(magnetization). The dashed lines are guides to eyes.
(MHz)
0.6
FREQUENCY
SPLITTING
0.4
0.2
0.0
-
0.2
-
0.4
0.40
0.45
0.50
0.55
T/T
Fig. 4-9(b)
0.60
0.65
c
The measured temperature splitting (shown in
Fig. 4-9(a)) is converted to frequency splitting
and plotting against T/T . The squares represent
data taken from different runs. The dashed lines
are guide to eyes.
90
1.8
1.6
1.4
1.2
20
22
24
26
28
P RE SSU R E (BAR)
Fig. 4-10(a)
Comparison of a
between two sets of measurements.
The solid linesSire guides to the eye.
91
= E cj/ A +(T)
1.8
1.6
1.4
1.2
20
22
24
26
PRE SSURE (BAR)
Fig. 4 -10(b)
Pressure dependence of a
in zero field.
28
SPLITTING
(MHz)
92
0.8
0.6
FREQUENCY
0.4
0.2
0.0 —1
0.00
0.25
0.50
0.75
1.00
1,25
1.50
MAGNETIC FIELD (KG)
Fig. 4-ll(a)
Field dependence of the doublet splitting of the
squashing mode in ^He-B where f — 141,6 MHz and
P - 27.7 bar. The solid line is a guide to the eye.
93
=K cj/ 4 +(T)
l.B
o'
a
1 .2
0.00
0.25
0.50
0.75
1.00
1.25
MAGNETIC FIELD (KG)
Fig. 4-11(b)
Field dependence of a , P - 27.7 bar and f - 141.6
MHz.
Sq
1.50
94
There arise two possibilities:
either a two-fold splitting has been
observed or only two components of a three-fold splitting have been
resolved.
Two arguments that would support the existence of a two­
fold splitting are:
a) a texture effect induced by the restricted
geometry; or b) the possible existence of some other phase near the
transducer interface.
Fujita et al.[25] have shown that the B-phase
may
2D-phase,
evolve
A(T)
into
a
^+5^2^j2 ’
cl°se
to
with
a
spectrum
order
boundary.
spectrum of the collective modes
part of the
an
parameter
Calculations
in such a 2D phase [25]
in the 2D-phase
is
the
same
on
the
show that
as the A-phase
(e.g. the clapping mode, pair-breaking or the super-flapping mode).
It was estimated that the difference between SQ mode in -^He-B and the
super-flapping-mode of the 2D-phase is of the order of a few tens of
(iK at T/Tc " 0.7.
This is quite close to our experimental results.
A three-fold splitting could arise from either dispersion induced
splitting
(DIS)
or
superflow
induced
splitting
(SIS).£26][27]
However, note the following two features of the DIS:
the splitting
decrease with increase T/Tc and the mode spectrum has the ordering of
W0 ”W1
[with the ratio of splitting r - ---- ~
> to.. >
U
1
2
subscripts
o>Q-a>2
is
equal
to
l^2 l*
Apparently
these
two
1
—1
4 J
where the
features
are
contradictory to our observed results.
In the case of SIS,
order
parameter:
it
the superflow has a two-fold effect on the
aligns
the
direction
of
vector
n
along
the
superflow velocity, V^; and also leads to a gap distortion transverse
A
2
— A
2
2
+ Cl
and parallel A
2
— A
2
2
+ aCl
to the
direction
of
superflow V , where a and ft are functions of the superflow velocity
and
reduced
B r u s o v [ 2 7 ]
temperature
(T/Tc) ,
atuj Nasten'ka et
a l . [ 2 8 ]
The
g a v e
calculation
performed
by
the following frequencies for
the superflow induced splitting of SQ mode
J
6a+ll
z
-
(4-10(a))
0
2
(4.10(b))
10
2 _ 12
2
3*0 + 3 ) n 2
(4-10(c))
where the branches of SQ-mode with | |
- 1,2 couple to the zero sound
via the texture which in this case is created by simultaneous effect
of superflow and restricted geometry.
->
From Eq.
4-9
the value of
2
°(T,VS) and fl (T,Vs) in Ref. 29, the SIS values of the SQ mode have
been estimated for several different values of
within a temperature
range 0.3 < T/Tc ^0.6, with the results listed in Table 4-2 (Also see
Fig. 4-12(a) and (b)).
Here 4g^,g(0) — 1.76 kgTc •
the ordering of
; in addition the splitting increases with
increasing T/Tc for the same
to
superflow.
Note that the frequency spectrum has
The
agreement
(Fig. 4-12), which is probably unique
of
these
two
features
with
the
experimental results strongly suggests the superflow interpretation.
The nonreproducibility of data obtained from different runs (Fig. 49(b)) provides further evidence favoring the SIS interpretation.
The
absolute temperature values of SIS (estimated by taking Tc - 2.4 mK,
96
(u8 -«o)/A(,cS(t=0)
0.06
0.6
0.03
0.4
T /T
0.2
0
4
6
12
16
Vs(m m /s )
Fig. 4-12(a)
The calculated values of the normalized splitting
between J — ± 2 and J — 0 components as a function
of T/T anct V .
Z
c
s
97
(“ r wo > / \ c s < T=0>
0.04
0.6
16
Vs(mm/a)
Fig. 4-12(b)
The calculated values of the normalized splitting
between J — ± 1 and J — 0 components as a function
of T/T and V .
2
c
s
98
which is very close to that in our experiments) are also listed in
Table 4-2.
In conclusion we have observed a doublet splitting of the SQ mode
which
is ascribed
to superflow.
From the measured value
of
the
splitting, it is possible to estimate the superflow velocity, Vs ; since
this is a difficult quantity to measure directly one might use the
doublet splitting as a probe of the superfluid velocity in future
experiments
4.4
designed
to
study
superflow
induced
phenomena.
Observation of a New Structure near 2A(T) in a Finite
Magnetic Field.
4.4.1
The
Summary of previous work
3
studying of He-B in the vicinity of 2A(T)
is
of great
interest because of the theoretically predicted collective modes in
that region and their interaction with the continuum of quasiparticle
states.
Schopohl et al.t^O] calculated the coupling strength between the J
-
1
" collective mode (noted earlier by
Wolflet^l])
and zero sound.
They also predicted a Zeeman splitting of this mode in finite
magnetic fields 1 ^1 ] according to
" j-l'-J
- + 1 ~ 2 h ‘ l A ( T> + 6 J Z n
(4 -1 1 )
’ z
where
g - 0.39
is a g factor,
frequency, as defined in Eq. 4-4.
and Cl is
the
Note that the
mode should not couple to zero sound.
renormalized Larmor
- 0 branch of this
An estimate of the coupling
strength results in an "antipeak" - peak structure of the attenuation;
99
i.e., a reduction in the attenuation at the mode crossing.
Observations of J - 1" collective mode phenomena near 2A are
complicated by the possibility of other collective modes, in particular
the one predicted by Sauls and Serene 132] which arise from 2 —
3
contributions to the pairing potential.
Experimentally, Daniels et al.£2]
acoustic
attenuation lying
temperature.
higher
found a new structure in the
than
the
pair-breaking
edge
in
It was suggested this new structure might arise from the
J - 4" collective mode.
mode (Ling et al.
1 9 8 7
(Schopohl et al. 1 9 8 4 or from the J - 1‘
[^3])
The latter group observed the "antipeak"-
peak structure of the acoustic attenuation and the measured g-factor,
0.40, was in good agreement with that expected for the J — 1‘ mode (for
pure p-wave pairing).
4.4.2
Observation of a New Structure Near 2A(T)
The single ended c.w. acoustic impedance technique was applied to
probe
superfluid
^He-B in
the vicinity of
hw -
2A with
a
sound
frequency of 64.3 MHz and at pressures ranging from 4 bar to 7 bar with
a magnetic
field
of
about
0.58
direction of sound propagation.
shown in Fig. 4-13.
kG
applied
perpendicular
to
the
A typical demagnetization trace is
Note the "bump-like" structure preceding the pair-
breaking edge by a temperature of about 20 pK (about 3.5 MHz); this
structure could not be observed in zero field.
The anomalous feature
in the detected acoustic impedance may result from the coupling of a
collective mode with zero sound.
The
One candidate is the J" - 1 mode.
— ± 1 components are expected to couple with zero sound and to
100
SQ-MODE
SIGNAL
p=7.2bar
f=64.3MHz
H=0.56KG
C
M
1.35
1.45
TEMPERATURE
Fig. 4-13.
1.55
1.65
(m K )
A demagnetization trace of the acoustic impedance signal.
The trace is taken as a function of time with the
approximate temperature as shown. Note the "bump-like"
structure in the vicinity of 24: F — 7.2 bar, f - 64.3
MHz, and H - 0.563 KG.
2A+(H) (MHz)
J =+1,
J=1
FROM
POSITON
OF
THE
"BUMP
10
0.82
0.84
0.88
0.88
T/T
Fig. 4-14.
0.90
0.92
0.94
c
Separation between the center of the "bump" and the
effective pair-breaking edge vs. T/T and a comparison
with the positions of both components of the J — 1‘
modes; f - 64.3 MHz, H - 0.58 kG.
102
be shifted from the zero-field pair-breaking edge,
where g - 0,392.
2
A(0 ), by ± gO™..,.
IHfc
However in a magnetic field the pair-breaking edge is
reduced relative to its zero field value according to 2A(H) - 2A(0) h^THE• Thus >
t^ie magnetic field, the separation between the pair-
breaking edge and the two components of the J — 1" collective mode
should be (1 ± S)^THg ■ or. 1.932
A-13
showed
a
clear
signature
and 0.608 BTHE- The trace in Fig.
of
the
pair-breaking
edge.
The
separation between the sharp finite field pair-breaking edge and the
center of the "bump" in the acoustic impedance, in frequency units, is
plotted against T/T^ in Fig. 4-14 with the error bar AT/T -2%; observe
that the experimental data are clustering around the J
If the data were to cluster around the
- -1 component.
- +1 component (requiring an
upward frequency shift of about 5.0 MHz) it would imply a temperature
uncertainty of about
60/ik for our thermometer;
although we cannot
entirely rule this possibility out, we regard it as highly unlikely.
Therefore our present
interpretation of
"bump-like" feature observed is the
the data is
that the
‘ 1 component of the J - 1"
collective mode.
We
note
in
passing
that
if
higher
components
of
the
pair
potential (starting with v^) are sufficiently large other collective
modes may occur near 2A(0) as calculated by Sauls and Serene.[32]
Table 4-1
Measurements of the pair-breaking edge.
“/2*
Giannetta et al. (1980)
5.3 bar
T/Tc
60.0 MHz
0.889
^ pb/Abcs (T)
1.97 ± 0.03
Meisel
(1983)
1 .0
bar
12.7 MHz
0.990
1 .8 8
Meisel
(1983)
1 .0
bar
12.7 MHz
0.991
1.98 ± 0.10
Ling et al.
(1987)
3.9 bar
42.2 MHz
0.935
2.03 ± 0.04
This work
(1989)
3-28 bar
64.3 MHz
±
0 .1 0
0.62 - 0.95 2.00 ± 0.08
90.1 MHz
141.6 MHz
167.4 MHz
103
104
Table 4-2
The Calculated Values of the
Superflow Induced Splitting of the SQ Mode
n2
VS
"2-"0
a b c s
^
U1
' W0
ab c s
01
<°>
“2
,2
Bcs<
>
’ w0
wl ' w 0
(M K)
(/*K>
(mm/s)
0.3
14.8
0.0067
0.0043(5)
-4.19
0.010(5)
28.5
18,4
0.3
2 1 . 0
0 . 0 2 2 0
0.0078
-2
0.059(5)
91.1
33.1
0.4
13.7
0.0096
0.0039(9)
-2.25
0.024
40.4
16.9
0.4
19.4
0.0310
0 .0 1 0 1
-1 . 8 6
0.087
130.9
42.9
0.5
1 2 .6
0.0150
0.0055
-2
0.040
63. 6
23.2
0.5
17.7
0.0420
0.0146
-1.94
0 .1 1 0
176, 5
61.7
0 . 6
1 1 .2
0.0188
0.0050
-1.63
0.054
79.3
0 . 6
15.9
0.0460
0.0144
-1.84
0.114
188.1
. 0 0
. 0 0
2 1 .0
61.0
THE HEAVY FERMION SUPERCONDUCTOR UPt3
CHAPTER 5
5.1 Basic Properties
Heavy ferraion systems (HFS)t^-l, first discovered by Steglich et
al.,[2 ] offer an alternative to liquid ^He (discussed in the previous
chapters),
states.
for
studying
highly
correlated
quasiparticle
ground
UPt^ was the third HFS that was found to be
superconducting,t3J with a transition temperature, T , about 0.5K in
zero magnetic field.
We
present
characteristics
here
of
a
brief
UPt^, which
description
taken
of
together
some
suggest
important
it
is
an
interesting material to study.
1.
Crystal structure^]
The crystal structure of UPt^ is hexagonal closed-packed, with
point group symmetry Dgh- The unit cell contains two uranium atoms
and
six
platinum
atoms.
The
a — b — 5.764A and c — 4.899A.
dy
, is4.1
A,
much
larger
lengths
of
basis
vectors
are
The spacing between uranium atoms,
than
the
so-called
Hill
limitE^]
of
3.25 - 3.50A, beyond which he noted that direct f-f overlap ceases
and, unless sufficient f-hybridization with s, p, and d electrons is
present,
magnetism
should
occur.
At
room
temperature
thermally
induced disorder destroys any tendency toward a "correlated"
and the f electrons of UPt^ may be regarded as localized.
105
state
2.
Normal state
1)
Specific heat
The specific heat of the normal state of UPt^ at low temperatures
(and in zero field) has the formt-*]
C =
7
T + £T3 + £T3 inT.
(5-1)
We note the following
2
(a)
7
(0 ) s 450 mJ mol'^ K
, which is about 300 times
as large as that of the alkali metal, Na (see table
5-1).
This anomalously large heat capacity implies
a large average effective mass of the conduction
electrons, m*.
If compared to the effective mass
obtained from a band structure calculation, referred
to as the band-mass and denoted as
(b)
one would
predict a many body enhancement factor of 17.
3
A large contribution From the iT J?nT term in the
specific heat suggests that spin fluctuations may play
an important role in heavy fermion systems as is the
case for liquid
2)
He.[7]
Magnetic susceptibilityt®1[9]
The susceptibility of UPt^ obeys
temperatures; i.e.,
the Curie-Weiss
law at high
107
where 6*
is negative.
At low temperature, x becomes large and also a large anisotropy
of
the
susceptibility
susceptibility is
the basal plane.
addition,
a
peak
in
the
Asymptotically,
2
*^^H||c-axis
Resistivity in zero field
UPtj has an
room
In
observed around 19K when the magnetic field is in
* ^^Hflbasal plane
3)
develops.
f31
anomalously large resistivity
temperature; the ratio p(300K)/p(4k) ~ 10.
(~
200
p£l - cm) at
The resistivity is
well represented by the form
p
-
pQ
+ PlT 2
(5-3)
in the temperature range between Tc and - 2K, where the first term is
typically due to impurities and the second one is usually interpreted
as indicative of strong electron-electron scattering.
3.
Superconducting State
Superconductivity in UPt^ was first found by Stewart et al.t^]
from
measurements
specific heat.
of
the
resistivity,
Following this work
a.c.susceptibility
there have been a
of
the
and
number
of
efforts
directed at determining the nature
superconducting
state.
Based on several experiments, which will be discussed below,
there is now a growing evidence that the superconducting state of
UPt^ is unconventional
108
(a)
Ultrasonic Measurements
Ultrasonic measurements on single crystal U P t ^ f 1't ^ ] revealed
a power-law like temperature dependence,
a
-
a
o
+
a i T n
L
of the ultrasound attenuation, a, with n ranging From 1 - 3, when
T «
Tc ; here
contrasted
aD
with
«X(T)
is
a
the
background
attenuation.
exponential
BCS
This
is
temperature
to
be
dependence
2
“ l<T c> - 1+eA ( T ) A B T
expected for a conventional BCS superconductor.
An anisotropic behavior of the ultrasonic attenuation has also
been
observed.tIt
explained in terms of
has
been
suggested
can
be
an anisotropic energy gap which vanishes
at
either points or lineson the Fermi surface.
these
results
A typical trace of
the
ultrasonic attenuation vs. 'temperature is shown in Fig. 5-1.
Ultrasonic
attenuation
in superconducting
studied in a magnetic field. £^ ][ 16]
UPt^
The
position
of this
orientation of the field
peak
relative
also
been
attenuation peak at a field
(designated as Hp^ in Ref. 15), was observed below H
sound.
has
is
to
for longitudinal
strongly dependent
the c-axis
(Fig.
on
the
5-2),
this
ultrasound anomaly may indicate a transition between two different
superconducting phases of UPt^ in a magnetic field.
A model involving
a vortex core transition associated with a specific unconventional
109
TEMPERRTURE SWEEP FOR q
O R MRL
CRT 2.5
TESLfl)
RTTENURTION
(dB)
8
b , H ✓✓ b
3
SC (RT 0 F£ELD)
0
0.5
TEMPERATURE
Fig. 5- 1
I
(KELVIN)
Typical temperature sweep for longitudinal ultrasonic
attenuation in UPt^. (Ref. 16)
110
1.5 '■ ’ »
r
\
\
h
1.5
V
0.5
pi i » i i i i
0
30
60
90
0 .5
o
Jxr\.
0 .2
0.4
TEMPERATURE (K)
Fig. 5-2
0.6
H-T plot of H (T) (filled symbols) and the A peak
(open symbols), for different orientations of the
magnetic field with respect to the c axis. (T: 0°,
♦, 0 : 45“, #,0: 85°, ▲: 90°). Inset: the anisotropy
of Hfl(*). (Ref. 15)
Ill
superconducting (gauge) symmetry group has been proposed.f^ ]>[^ ] The
Identification of multiple phases in superconducting UPt^ is of great
importance since it provides perhaps the strongest-evidence that UPt^
is indeed an unconventional superconductor.
b)
Specific heat measurements
Specific heat measurements on single crystal specimens of UPt^
have been performed by Hasselbach et al.[*®]
The results
may be
summarized as follows:
(1)
A T
2
dependence of the specific heat of superconducting UPt^
is observed down to
(2)
100
mK.
Two sharp discontinuities separated by 60 mK, have been
reported in the specific heat in zero magnetic field in the
vicinity of T , strongly suggesting two phase transitions
(Fig. 5-3).
The separation between these peaks becomes
smaller when a magnetic field is applied in the basal plane,
ultimately vanishing for H = 5.0 kG (Fig. 5-4).
It was
suggested that there is some kind of a critical point at
this field.
It was further noted that the high temperature
extrapolation of Hp^(T) appears to pass very close to this
critical field.
c)
Critical field measurements
Strong anisotropies have been observed in both the upper and lower
critical fields (H , and H « ) . "I^l] The large value of dH 0/dT has
cl
c2
e
c2
been interpreted as evidence for strong spin-orbit scattering.E1 1
112
•00
600
0 43 K
0 49K
200
Somplt 2
u
,430
400
0 42 K
200
UPl
0.2
Fig. 5-3
0.4
06
T CK)
08
Double heat capacity jump in UPt» at the transition
temperature. (Ref. 46).
113
.0
UPt
H1 c
0.5
0.0
0
Fig. 5-4
T(K)
Phase diagram for superconducting UPt3 From heat capacity
measurements for H_Lc. The closed circles are the heat
capacity data, the open squares are H c 2 data on a different
sample and the open triangles are the HpL data for H in the
basal plane. (Ref. 18).
114
Another
interesting observation is an abrupt
(referred to as a "kink"
change
of slope
in both the H , and H _ curves,[22 ],[ 23 ]
cl
c2
(Fig. 5-5), when the magnetic field is perpendicular to the c-axis;
this phenomenon may also suggest the existence of multiple phases in
superconducting UPt^ ■ Note that the "kink"
in the
curve occurs
near H - 5.0kG; i.e., it is likely a different signature of the same
critical point seen in the heat capacity.
5.2
Proposed Experiments on UPt^
Eased on the experiments discussed in the previous section one
may
tentatively
superconductor.
conclude
that
By analogy with
UPt^
^He
is
this
an
unconventional
suggests
the
possible
existence of order parameter collective modes.
3
We recall that He has a transition temperature of order 2 mK.
Qualitatively we expect the collective mode frequencies to be of the
order of the gap, A, which scales as T . Scaling by the ratio of the
transition
occur
at
temperatures,
frequencies
collective modes,
if
250
any,
3
He.
in UPt^
should
times higher than
Most of the
3
collective mode phenomena in He are observable at a frequency of
order 40 MHz which scales to 10 GHz for UPt^.
Based on this simple
minded argument, an X-band (—11 GHz) microwave spectrometer has been
developed
which
refrigerator.
is
compatible
with
our
Oxford
400
TLE
dilution
The details of this microwave spectrometer, as well as
a preliminary study of the microwave surface impedance of UPt^, will
be discussed in the following two chapters.
115
• H || c-axis
<o
to
o>
I—
OS
0
3
OS
0
7
0 1
t I
T/Tc
Fig. 5-5 (a) The upper critical field for both the orientations,
H||c-axis and H||a-axis. The data for H|)c axis were
obtained by the inductive technique measurements.
For H||a axis, the data were obtained resistively. The
lines used to determined the temperatures Tc+, Tc -, and
Tjj given in Table I are also shown. (Ref. 21)
116
ro
iq
in
OJ
ro
tn
QJ
H || c-axas
9
0
3
0.4
0
3
0 6
0 7
0 6
0
9
I0
T/Tc
Fig. 5-5(b)
The lower critical fields for the same orientations,
the "kink" also occurs in the H||a-axis curve. (Ref. 22)
Table 5-1 Basic properties of the normal state
of UPt3 compared to those of Na. (Ref. 1)
7(0)
(mJmol'^-K'2 )
X(0)
Po
(1 0 *3 emucm*3) (^fl-cm)
UPt3
450
0.19/0.10
Na
1.5
0.008
Pee
(^O-cmK*^)
0.5
0.5
0.9xl0' 3
l.OxlO’ 6
"^cw
Peff 'p (T-300K)
(fitl cm)
(k)
(p b >
200
4.6
130
--
5
--
CHAPTER 6 . THE NORTHWESTERN LOW TEMPERATURE MICROWAVE SPECTROMETER
6
.1
Introduction
The
temperature
magnitude,
to the
range
0.1
to
frequency range
1°
K
corresponds,
2.3 GHz
generally in the microwave regime.
to
in order
23 GHz;
i.e.
of
it is
This range of temperatures
is
easily produced by a dilution refrigerator.
There are many interesting microwave experiments to be performed
at
temperatures below IK.
conventional
various
It is clearly of
EPR studies,
kinds
of
especially on
magnetic
ordering.
interest
to perform
systems which may undergo
A
wide
range
of
microwave
surface Impedance measurements on metals deserve further examination.
Included
are
studies
superconductors
collective
(possibly
of
(particularly
modes
in
involving
very
conventional
with
pure
collective
and
respect
to
the
metals), magnetic
modes
of
a
unconventional
mixed
existence
superconductors
magnetic
superconducting character), and electron lifetime studies
(in both the very pure and very dirty limits).
of
and
in metals
Certain ENDOR studies
may also be of interest.
To be maximally useful, a low temperature microwave spectrometer
should have a top loading capability;
i.e.,
one should be able to
directly load samples into the microwave cavity from room temperature
without raising the low temperature region above, say, 4 K.
The
main
difficulty
associated with
including
a
top-loading
feature arises from the small access available through the coupling
118
119
hole (of about
2
millimeters in diameter) between the waveguide and a
cavity resonator.
In
this
thesis
we
describe
modifications
of
Oxford 400
TLE
cryostat equipped with a 12 Tesla magnet permitting X-band microwave
measurements
at
dilution
refrigerator
temperatures
involving
a
top-loading capability.
.2
The Microwave Spectrometer
6.2.1
An approach to top loading
6
One
strategy
for
top
loading
is
to use
a Gordon
between the waveguide and the cavity (see Fig.
6-1).
coupler^]
This device
functions by installing, in front of the cavity, a short section of
waveguide
that
(with a tapered lead-in),
the cut-off frequency
the diameter of which is such
is above
the
operating frequency.
By
judiciously choosing the diameter we may create a situation whereby
inserting a polyethylene dielectric rod (with the dielectric constant
€ - 2.3) into this restricted-diameter-region,
of
this
section of
frequency.
the
line
may be
the cut-off frequency
lowered below
the
operating
Clearly by varying the depth of insertion of the tuning
rod into this region we may vary the coupling to the cavity until it
is optimized (critical).
In our design the rod does not fill the
entire diameter of the restricted region;
with a coaxial,
tapered,
the remainder
is filled
dense-polyethylene section (which is fixed
to the copper section with two 3-56 G-10 set screws).
The tuning rod
is a piece of 1/4" polyethylene dielectric rod with the tip machined
into a cone (with an apex angle of -90°).
A 1/4" knob is screwed on
120
POLYETHYLENE
I/* POLYETHYLENE
ROD { £ . 2.3 )
LOADINC TUBE
COPPER
2 X 3-56
G-10 SCREWS
SAMPLE
LEFT-HAND
HOLDER
THREAD
In O-RINCS
RIGHT-HAND
THREAD
CAVITY
SAMPLE
SAMPLE HOLDER
CAVITY
BOTTOM
HOLES FOR CRT 6 CRT
THERMOMETERS
Fig. 6 - L.
Schematic drawing of the microwave cavity showing
the Gordon coupler.
121
the polyethylene rod which is used for coarse positioning.
Since the reduced diameter section of transmission line extends
directly
into
the cavity,
removing
the dielectric
access hole through which we may load the sample(s) .
rod creates
an
The sample is
loaded into the cavity with a stainless steel sample-loading rod into
which the sample holder
is screwed with a left hand thread.
The
lower part of the sample holder has a tapered section followed by a
right-hand thread, which ensures good thermal contact to the cavity
bottom.
The sample holder is first screwed snuggly into the end of
the sample loading rod.
cryostat
On inserting the sample loading rod into the
the sample holder may be screwed
into
the cavity bottom
where, on seating at the bottom the torque rises to the point where
the holder unscrews from the loading rod.
to remove the sample holder.
This process is reversed
The sample itself is attached to the
top of the holder with either silver paint or GE 7031 varnish.
The performance of the Gordon coupler was measured by butting up
a self-contained waveguide-crystal detector equipped with a double
stub tuner at the bottom side of the coupler.
Although we are then
attaching a
guide,
rectangular guide
to
a circular
the
optimized
detector output is a useful check on the performance of the coupler.
Fig.
6.2 shows the detector output (optimized at each point)
as a
function of the position of the dielectric rod.
6.2.2 Cryogenic aspects
The
tubing
largest
sizes)
diameter
that
circular
could be
waveguide
inserted
in our
(involving
standard
cryostat was
0.687"
122
OUTPUT OF THE DETECTOR ( ARB. UNITS )
10
DIELECTRIC ROD
8-
POLYETHYLENE SLEEVE
REDUCED DIAMETER
TRANSMISSION LINE { R }
d< 0
WAVE GUIDE DETECTOR
( SYMBOLIC )
2
-4
-3
-
-2
INSERTION DISTANCE
Fig. 6-2,
d
cb
)
Coupling of the Gordon coupler as a function of the
insertion depth of the dielectric rod; the microwave
detector is shown schematically as a diode.
123
(11/16"); the I.D.
was
.667".
This results in a cut-off frequency
of 10.15 GHz, which is toward the upper end of the X-band.
shows
a schematic,
involving
the
over-all view of the elements
microwave
line
and
cavity.
Fig. 6-3
of the cryostat
The
stainless
steel
waveguide starts at a rectangular-to-circular adapter (just above the
cryostat) and extends through a "quick connector" vacuum seal, along
the central line of the cryostat, to the dilution refrigerator; here
it makes
threads
a
transition
into
the
to
mixing
a copper
chamber
section the
and
extends
outside
down
to
of which
the
Gordon
coupler and cavity (the copper section thermally grounds the cavity,
and with it the sample, to the mixing chamber).
The
stainless
steel
guide
is
thermally
corresponding in temperature to -150K,
grounded
7OK and 4K;
at
points
the grounds are
made by soldering double sided r.f. finger stock into an annulus; the
three
grounds
are
then
tied
at
the
proper
positions
with
nylon
(parachute) cord and pulled into the annulus formed by the waveguide
and
the
central
guide
tube
of
the
cryostat
(certain
geometric
restrictions in the Oxford design for our unit forbad the use of the
central access
tube itself as the waveguide;
minor changes
in the
original design would allow this approach).
The
low temperature portion of
the
line contains four more thermal grounds.
stainless
steel
microwave
They involve a copper-clamp-
ring surrounding the line, to which copper braid is silver soldered;
the far side of the braid is silver soldered to a small copper block
which is screwed to the desired heat sink point.
Such grounds were
124
TOP OF CRYOSTAT
jffj,
CIRCULAR WAVE GUIDE
Cu FINGER STOCK
THERMAL GROUNDS
1*K *H« POT
POLYETHYLENE FOAM
BAFFLES
VACUUM CAN
STILL
ANNULUS
CONTINUOUS HEAT EXCHANGER
STEP HEAT EXCHANGER
Cu BRAID
THERMAL GROUNDS
GORDEN COUPLER
SAMPLE
MIXING CHAMBER
DIELECTRIC ROD
FIELD MODULATION
COIL
RESONATOR
( CAVITY )
12T S.C. MAGNET
Fig. 6-3.
THERMOMETERS
Symbolic overall view of the microwave line as it extends
through the cryostat.
125
positioned at the 4He pot, still,
the flange between the continuous
and step heat exchangers and the mixing chamber.
These clamps have
the property that they allow limited rotation of the microwave line
around the central axis, about which we shall comment later.
In order
to minimize
the
thermal
temperature region of the cryostat,
radiation
entering
the
low
the inside of the waveguide is
filled with a stack of polyethylene foam annuli, with a 1/4" I.D.,
from the point just below the rectangular/circular waveguide adaptor
down to the top of the Gordon coupler.
A rectangular/circular waveguide adapter is shown in Fig. 6.4.
It consists of a short section of rectangular x-band waveguide
on
both ends of which standard choke flanges (each of which contains an
O-ring groove) are soldered.
adapter.
One flange serves as the input to the
The other (which is positioned as close as possible to the
quick connector sealing the exiting circular waveguide)
to a
short
shorting
section
block.
of
This
rectangular waveguide
block,
and
the
is attached
containing
accompanying
a
moving
threaded
positioner, were obtained by disassembling one arm of a commercial E-H
tuner.
The vacuum seal was accomplished with two sheets
which were sealed by O-rings
in the choke flanges.
of mylar
Since we are
varying only one parameter (and two are required, in general, to make
an arbitrary match), we included a stub turner at the driving point
of the adapter.
126
An arrangement involving three 1/4" ball valves and a pop off
valve is soldered to the top side of the waveguide adapter (See Fig.
6-4).
The in-line valve is opened to pass either the dielectric rod
(which adjusts the Gordon coupler) or the sample loading rod (when
changing samples).
When removing either rod this value is closed
when the end of the rod clears it.
Two other valves are used to pump
the space above the in-line valve and admit helium gas
condenses
into
the
cavity)
respectively;
the
(which then
liquid
helium
sometimes useful for enhancing thermal contact to the sample.
is
Leak
tight junctions between the Gordon coupler, cavity and cavity bottom
are
sealed
with
indium
0-rings,
which
also
assures
reliable
electrical contact between these components and eliminate microwave
leakage
(which
thermometers).
can
adversely
affect
The pop-off valve
avoids
the
performance
an over pressure
of
the
if the
helium is inadvertently left in the cavity on warming up.
At the top of the load lock is an arrangement formed from a
bellows and the previously mentioned 1/4" "quick connector"
through
which the tuning rod passes.
The bellows are incorporated to allow a
fine,
of
continuous
adjustment
the
position
of
the
end
of
the
dielectric rod in the Gordon coupler; the position is controlled with
a threaded section (which itself is made from a 7/8" diameter "quick
connector") and a retainer ring (to keep the bellows from collapsing
under pressure).
127
1/*" QUICK
CONNECTOR
RETAINING RING
B«-Cu BELLOW
7/8" QUICK
PUMP OUT PORT !
DIELECTRIC ROD
3/8" UCRL 6 VALVE
RELIEF VALVE (3p»i)
:
CAS ENTRY POINT i
3/8" UCRL & VALVE
LOAD LOCK VALVE
MYLAR SHEET
RECTANGULAR/CIRCULAR
WAVE GUIDE ADAPTER
RECTANCULAR
WAVE CUIDE
CIRCULAR WAVE GUIDE
Fig.
>-4.
The rectangular to circular waveguide adapter, load
lock and positioner mechanism for the dielectric rod.
128
The
temperature
is
measured
with
either
a
Matsushita-carbon
resistor 12] or a germanium thermometer, both of which are located in
holes drilled in the cavity bottom.
Temperatures in a magnetic field
were measured with the Matsushita carbon resistor thermometer.
The lowest temperature we could reach was just below 50 mK.
was presumably limited by residual
thermal radiation which
scattered or attenuated by the polyethylene foam and rod.
It
is not
With liquid
4
He condensed into the cavity the base temperature was 210 mK (there is
no apparent reason why the system would not be cooled down to 50 mK if
liquid
3
4
He replaced the He).
A partially filled cavity should be
avoided since vibrations of the liquid level then modulate the cavity
resonance frequency.
6.2.3 Microwave aspects
Fig. 6-5 shows a block diagram of the microwave spectrometer.
Although it is somewhat conventional in design we include it for the
sake of completeness.
One arm
provides
supplies
The klystron output is split into two arms.
the power
the homodyne
which also
receives
to the sample
reference
the
signal
reflected
cavity.
for
signal
The other arm
the balanced
from
the sample
detector
cavity.
This latter arm also contains a wavemeter and crystal detector;
the
klystron may be locked to the wavemeter using an automatic frequency
control (AFC) circuit
(Fig. 6-5).
Alternatively the klystron may be
locked to the sample cavity.
The AFC,
which is quite conventional,
reference output
of the
first
works
lock-in detector
as follows:
is applied
the
to the
129
POWEB FORHEATER,CATHODE * REFLECTOR
WATER COOLEDPORT
TOR
MM
WAVE
M0C6 A,SYS.LOCXEDTO WAVE METER
MODE B.SV& LOCKEDTO CAVITY
OCTCCTOR
CHI
cncuuwwwBour
SW2
Fig. 6-5.
LOCftM
EXT REF M
Block diagram of the homodyne microwave-bridge spectrometer
with AFC circuitry.
130
repeller sweep input feature of the HP 716A klystron power supply
which,
for the proper gain setting,
modulation
typically
of
a
the
small
klystron
causes a narrow band frequency
microwave
fraction
output;
the
band
t*ie bandwidth
°f
cavity (which was generally narrower than the wavemeter).
of the crystal
detector will
modulation frequency,
cavity resonance.
then have
an a.c.
width
the
is
sample
The output
component
at
the
the phase of which reverses on crossing the
The detected lock-in output can then be used as a
frequency control signal.
In some klystron power supplies there is a feature where this
signal can be used to alter the repeller voltage and hence correct
the frequency.
However, the HP 716A does not have this feature and
therefore we adopted the following approach.
Since we could correct
neither the cathode nor the repeller voltage we choose to correct the
anode voltage.
In most
klystrons
the
anode
is
grounded
to
the
waveguide; however by inserting an insulating (mylar) sheet between
the klystron and the rest of the spectrometer (which is grounded) the
klystron can be floated.
(A)
or
(A')
can be
The outputs of the AFC lock-in amplifiers
applied
to
circuit, which has two inputs;
an
operational
amplifier
one input directly feeds a
(op-amp)
(power)
operational amplifier, with a gain of two and a maximum output of
±13V
and
100 mA,
while
before the op-amp (Fig.
the
6).
other
contains
an
integrator
circuit
The output of the power op-amp was
applied to the klystron anode, with the common side attached to the
klystron power supply ground.
In this way the lock-in output could
Fig. 6-6.
INTEGRATE
+15V
>* DIRECT
Anode
150K
100K
•Wr
“ W r
0.01 MF
10K
1 MF
drive
IN #1
(Op-Amp)
1S0K 3
0.1 MF.
20K
LF366
LF366
MPS
AOS
470
OUT
100K
IN #2
circuit
Wr
10K
MPS
A55
diagram.
-15V
131
132
swing the klystron anode ± 13V relative to the klystron power-supply
ground.
This gives sufficient control of the klystron frequency to
permit stable AFC operation.
long-term drift.
Note
The integrator is used to control the
that
the
effective
output
impedance
of
an
operational amplifier is essentially zero and hence manually altering
the cathode or repeller voltages has no effect on the klystron anode
voltage
relative
circuit
allowed
carried.
full
induced
anode
100 mA output capability of
current
(typically
40
ma)
this
to
be
modulation),
cooled
addition,
water-cooled
coupling
positioned on the klystron microwave output port,
provided
extends
flange,
the
the
A fan, mounted independently of the klystron (to avoid any
mechanically
which
to common;
its
frequency
life.
In
a
the
klystron,
further thermal stabilization.
The homodyne
detector
involves
phase shifter and an attenuator.
cryostat
through
reference
outputs
are
are
a
applied
in
to
a
reference
arm
containing
a
The other arm is applied to the
circulator.
combined
a
The
a balanced
lock-in
circulator
detector;
amplifier
output
the
(B)
two
and
the
detector
operating
in
a
differential input mode and the lock-in output is applied to a chart
recorder.
Since
changes
in
the
temperature
of
the
room
can cause
the
resonant frequency of the wavemeter to drift we usually locked the
klystron to the cavity.
SW1 in Fig. 5 allows operation in either a
sample cavity or wavemeter locked mode.
133
When performing
EPR studies
we
must
field (the diagram of the sweep circuit
power
supply
is
shown
in
the
d.c.
magnetic
for the bipolar d.c. magnet
Fig. 6-7).
modulate the d.c. magnetic field.
sweep
It is
also
convenient
to
This is accomplished by winding
two layers of #32 (d-0.202 mm) enameled Cu wire for a length of 5" on
the section of the vacuum can surrounding the cavity (this was the
maximum number of layers that could be safely accommodated between
the
vacuum
can
tail
and
the
superconducting
corresponds to about 100 Gauss/A.
a.c.
magnetic
field
operating frequency of 13Hz.
bore);
this
The use of superconducting wire
would allow still larger modulation fields.
the
magnet
into
the
Adequate penetration of
cavity
was
obtained
at
the
The necessary modulation current was
obtained from a commercial audio amplifier.
The dimensions of the experimental microwave cavity were chosen
so that the TM (010) and TE
(111) modeswere nearby in frequency, but
not so close as
confusion.
to cause
The TE
(111)
mode has
property that the microwave field on the bottom of the
strongest
in
the
center
(where
the
sample
is
the
cavity is
located)
and
perpendicular to the cavity axis.
This is the desired arrangement
for
mode
EPR
studies.
degenerate
modes.
perturbation,
cavity.
such
The
The
as
TE
(111)
actually
degeneracy may be
a
radial
protrusion
consists
lifted by
on
the
of
two
installing
bottom
of
a
the
For some experiments it is useful to purposely include such
a mode splitting perturbation (it could be the sample itself).
We
may then alter the polarization in the cavity simply by rotating the
Fig. 6-7.
Diagram of the sweep
power supply.
«
circuit
R1 ~ R7
the bipolar
1N
5231B *2
for
d.c. magnet
1. RESISTORVALUESINOHM
S.
+ 15V
2. CAPACITORVALUESINM
ICROFARADS.
3. INTEGRATOROUTPUTUPTO«V.IFHIGHEROUTPUTEXPECTEDREPLACE1N82SAND
ACCESSORYRESISTOR.
CO
(>
135
cavity about its axis: it is for this reason we designed the entire
microwave line so it could be rotated.
It is easy to null out one of
the modes by rotating the cavity and observing the pattern on the
scope when the klystron is operated in a wide-band frequency-swept
mode.
6.2.A Adjusting the coupling to the cavity
In a system employing a long microwave line (75" in our system),
finding the true microwave resonance can be difficult due to closely
spaced
standing
locating
the
wave
nodes.
precise
In
resonance
tuning
the
frequency,
system
and
we
the
begin
by
approximate
position of the dielectric rod for optimal coupling of the cavity,
with a test set up employing a very short line.
Basically it is the
same as the full set up, only with a short section of stainless steel
tubing
replacing
the
long
tube
in
the cryostat.
A
satisfactory
position for the sliding short in the circular-to-rectangular adapter
is first
withdrawn.
spurious
adjusted with
the moving probe
of the
stub
tuner
fully
This helps to avoid confusing the cavity resonance with a
resonance
involving
the
stub
tuner
and
other
elements.
After the cavity resonance frequency and Gordon coupler positions are
reasonably well determined, the probe of the stub tuner can then be
inserted and positioned to optimize coupling. With the data obtained
with the short line, it is then relatively easy to find the resonance
and adjust the coupling with the cavity attached to the long line. If
the system is monitored as it is cooled down, the frequency shift and
the necessary reduction of the coupling (due to the increased Q of
136
the cavity) may be tracked; the frequency shift occuring when liquid
4
He is condensed may also be traced.
Once the appropriate settings
in the low temperature configuration are known,
different
samples
may
be
readily
followed.
changes
Finally,
induced by
we
always
include a small sample of DPPH (2,2-phenyl-1-picrylhydrazyl), the EPR
signal of which is relatively easily found.
This gives us a valid
signal originating in the cavity and not from some spurious line (or
other) resonance.
The above adjustments are made in the absence of the homodyne
reference signal; i.e., with the crystal self-biased by the reflected
microwave power.
When the correct sample cavity resonant frequency
and coupling power are determined,
decreased
(by about
reference
amplitude
attenuator
included
15 dB)
is
in
and
the applied microwave
the homodyne
conveniently
this
arm).
altered
The
signal
by
reference
power
is
applied
(the
adjusting
the
phase
is
then
adjusted to obtain a dispersive response (vanishing first harmonic).
(When converting from self-biased detection to homodyne detection it
is sometimes convenient to lock the klystron to the wavemeter,
the
latter having been previously tuned to the sample cavity resonance
frequency;
this
procedure
avoids
homodyne phase is being optimized.)
drift
to the
klystron while
the
CHAPTER 7.
7.1
As
EXPERIMENTAL PERFORMANCE AND RESULTS
Introduction
discussed
in
Chapter
5,
one
of
our
primary
to develop a low temperature microwave spectrometer.
goals
was
A challenging
problem in low temperature physics is to detect collective modes in
superconductors,
UPt^,
which
in particular in the heavy fermion superconductor
may
be
a
candidate
for
unconventional
metallic
superconductivity.
In
order
conventional
to
accumulate
EPR
superconductors
experience,
experiments
(involving
and
shifts
the following two sections.
trial
in both
components of the surface impedance).
we
began
with
studies
the
real
of
and
more
s-wave
imaginary
These topics are discussed in
In Section 4, surface impedance studies
of the heavy fermion superconductor UPt^ will be discussed.
7.2
7.2.1
EPR Studies of Ca(tatbp)!.
Introduction
Porphyrinic
"molecular
m
e
molecular
t
a
l
s
"
(a
metalsare
term
contradictory only a decade ago).
which
important
would
examples
have
been
of
self­
These highly conducting molecular
crystals are prepared from porphyrin and phthalocyanine
complexes.
The fundamental structural molecular element of interest is shown in
Fig. 7-1.
When this porphyrinic structure is reacted with iodine, needle
like crystals form which exhibit a relatively high conductivity along
137
138
I
Fig, 7-1.
Cu
The chemical structure of the Cu(tatbp)! molecule.
139
the needle axis
the parent
features
son>® 12 orders of magnitude greater than that of
[3] of course
m o l e c u l e . ,
of
molecular
crystals
could,
the metallic-like
potentially,
conducting
open
up
new
technologies.
Also, the chemical flexibility of porphyrin-like
metallomacrocycles provides an opportunity to prepare a large class
of
new materials that are
particularly valuable for the exploration
of
the relations between molecular stereoelectronic characteristics
and solid-state transport properties.
The sample discussed
molecular
metal,
in this thesis is a quasi one-dimensional
copper(triazatetrabenzporphyrinato)(II)
iodide,
Cu(tatbp)I; its chemical structure is shown in Fig. 7-1.
Cu(tatbp)I is a ring-oxidized conductor with metallic behavior
that contains a dense array of localized Cu+2 moments embedded in the
conduction
band
(Fig.
7-2).
The
Cu-Cu
spacing
along
the
one­
dimensional conducting stack is 3. 2A whereas the closest interstack
distance between Cu ions is 14A.
can be
found in Ref.
4.
More detailed data from Cu(tatbp)I
A detailed description
(e.g.
synthesis;
X-ray diffraction Raman spectra; NMR spectra; magnetic susceptibility
measurement; EPR measurement, etc.) can be found elsewhere.t^1
What is of interest to us here is the g| value from previous
X-band EPR measurements
at low temperatures.15]
Below 20K,
g|| was
found to increase dramatically and reach a maximum by T - 4.OK of ~
2.255, far greater than 2.171, the value extrapolated from the hightemperature regime.
As the temperature is lowered further to T -
140
Cu(tatbp)I
7-2.
The chain-like structure Cu(tatbp)!.
141
2.4K
(about
the
lowest
temperature
reached
in
the
previous
measurements), g|| decreased slightly.
Clearly it is of interest to repeat the previous measurements
and perform further measurements of g|| at lower temperatures.
7.2.2
Results and discussion
Conventional
temperature
EPR
experiments,
microwave
spectrometer
using
the
discussed
Northwestern
in
performed to measure the g-factor of Cu(tatbp)I.
ground
into
a
fine
powder
and
mixed
with
1266
Chapter
low-
6,
were
The crystal was
stycast
resulting in a ball shape sample with a diameter - 3.0 mm.
epoxy,
On top of
this sample a speck of DPPH was varnished (which has an accurately
determined g value of 2.0036, and was used to calibrate the external
d.c. magnetic field).
Note that in a powder the paramagnetic system may assume all
possible orientations relative to the direction of the d.c. magnetic
field, and hence one expects to have the EPR spectra spread over the
entire field range, AH, governed by the extremes of the principal g
components of the system.
and
possessing
tetragonal
assumes
uniaxial
symmetry
that,
of
A model for a system with S =
symmetry
our
(which
compound),
is
has
consistent
been
L = 0,
with
the
proposed.^]
it
upon grinding such a crystal to a fine
powder,
orientations along the field axis will be equally probable.
some crystallites are in resonance for all fields,
all
Hence
H , between H
(the field corresponding to g^) and Hj| (the field corresponding to
gii).
For our tetragonal system (or any uniaxial system) H^ is given
Where
and 6 are,
respectively,
the Bohr magneton and the angle
between the tetragonal axis and the magnetic field direction.
The probability P(H) (which will be proportional to the EPR line
intensity) versus the angle, 6, is given by
P(H) " ()r)2 3f 2 1 2,—
0
Hr (g|j - gjcostf
Fig.
7
(7‘2)
-3 17] shows: (a) an idealized absorption line shape for a
randomly oriented system having uniaxial symmetry and a S function
line shape (in this case g^ > gj|)i ant^ (b) the computed line shapes
for the case of finite line widths of (1) 1. G, (2) 10 G, (3) 50 G,
and
(4)
100
independent
of
G
respectively
angle).
qualitatively produces
Note
a mirror
(here
that
the
in
line
our
width
case
g^
image of the curves
is
<
g|
shown
assumed
which,
in Fig.
7-3.
A typical EPR trace at T - 3.45K is shown in Fig. 7-4.
apparent that it fits the above model quite well.
It is
Point A indicates
the DPPH absorption peak (which fixed our field calibration at 1.224
KG/A) ; point B indicates the position of g^ and point C that of gp .
At this temperature of 3.45K we obtain g^ = 2.045
respectively.
and g| = 2.158
143
(<7 )
3000
3200
3400
3600
3800
Magntlic f<•Id,G
(3)
Fig. 7-3.
(a) Idealized absorption line shape for a randomly
oriented system having uniaxial symmetry and a 5 function
line shape (gn >
. (Ref. 6).
(b) Computed"line shapes for randomly oriented systems
having uniaxial symmetry. The Lorenzian line width from
a single crystallite are given as (a) 1 G, (b) 10 G,
(c) 50 G and (d) 100 G. (Ref. 7)
SIGNAL
(arbitrary
units)
144
EPR
0
-1
34
3.0
3.8
4.0
4.2
MAGNETIC FILED (k G )
Fig. 7-4.
EPR measurements on a Cu(tatbp)! powdered sample with
a speck of DPPH on top of it. "A" indicates the absorption
peak of DPPH; "B" and "C" indicate respectively the
positions of
and gn.
145
A family of EPR curves were
traced at different
temperatures
ranging from 0.92 K to 3.5 K (some of which are shown in Fig. 7-5);
contrary to the result presented in Ref. 5, no visible change of g|j
is observed.
7.3
S-wave superconductor surface impedance studies: Sn,
r
Zn and In
7.3.1
Introduction
There
was
a
considerable
superconductivity in mercury by
of surface
(e.g.
delay
between
Sn,
and
Zn).
Such
development of microwave techniques,
London in 1940t^J,
discovery
of
in 1911, and the initiation
0nnest®3
impedance studies on pure bulk
Al,
the
(s-wave)
experiments
superconductors
had
to
await
the
and were first performed by H.
and further developed by Pippard t
3 and other
workers.tH~l^]
Based
Bardeenf
on
3
independently
the
BCS
(using
model [15]
Df a
Boltzmann-like
Abrikosov,
Gorkov,
superconductor,
transport
and
Mattis
equations)
Khalatnikovt^-18]
and
and
(using
thermal Green's function and analytic continuation techniques) have
derived a general theory of the anomalous skin effect for both the
normal and superconducting states.
A commonly used technique to study the surface impedance is the
so-called resonance
method, 3^-^3 where
one
drives
a
cavity at
its
resonant frequency and measures the change of the quality factor, AQ,
when the sample undergoes the superconducting transition.
The change
of the surface resistance, AR, is inversely proportional to AQ,
It
146
10
5
T=1.98K
EPR SIGNAL (arbitrary
unit)
0
0
3.50
3.75
4.00
4.25
MAGNETIC FIELD (KG)
Fig. 7-5.
A family of g-factor measurements on Cu(tatbp)I powder
ranging in temperature from 1.98 K to 2.77 K.
147
should
also
reactance,
be
AX,
mentioned
here
that
the
is directly proportional
change
to
the
of
the
resonant
surface
frequency
shift of the cavity, Af.
Fig. 7-6 shows a Q-value measurement versus temperature of a 10
GHz microwave resonator with the inner wall coated with Sn performed
by
[20]
when
Sridha.
the
cavity
is
cooled
down
to
3.72K,
the
superconducting transition temperature of Sn in zero magnetic field,
the
Q-value
to
increase
absorption dramatically decreases).
As the
below
suddenly
T , more
starts
electrons
condense
into
(i.e.
the
temperature
pairs
which
microwave
is lowered
can
carry
a
current, but do not dissipate.
7.3.2
Measurements of the surface impedance response of s-wave
superconductors: Sn, Zn and In
We
s-wave
have
performed
microwave
superconductors,
Sn,
Zn
absorption
and
In,
measurements
using
spectrometer with the following two motivations:
our
on
the
microwave
(a) to verify that
our microwave spectrometer (which does not employ a superconducting
resonator
and
hence
will
function
in
a
magnetic
required sensitivity to detect the transitions;
and
field)
(b)
had
the
to improve
the sensitivity and stability with a well understood signal.
A major
(but well known) improvement resulted from using homodyne detection
(as
opposed
to
simply
rectifying
the
signal
reflected
from
the
MICROWAVE
CAVITY
Q VALUE
148
»•*
>SUPERCONDUCTING STATE
• NORMAL STATE
I©*
T
Cl
«*
TEMPERATURE
Fig. 7-6.
(K )
Q vs. temperature of a 10 GHz microwave cavity with the
inner wall coated with Sn; the superconducting transition
temperature, Tc is 3.72 K (Ref. 20).
149
cavity).*
The experimental results for the three metals studied are
discussed in the following sections.
A)
Sn
Sample
#1 was
a cylindrical
piece
of
diameter of 3.5 mm and a height of 3.5 mm.
6'9's
pure
Sn
with
a
Sn has a superconducting
transition temperature of 3.72 K at zero magnetic field; the critical
field for Sn at zero temperature is 305 Gauss.
The resonant frequency of the cavity was determined by employing
the internal sweeping feature of the klystron power supply to sweep
the klystron back and forth through the cavity resonance (in a non­
homodyne detection mode; i.e. using straight diode detection).
wave meter was
sweep
then tuned to the cavity resonance.
signal was
applied.
then removed and a small
The klystron was
external
The
The
internal
sweep voltage
then locked to the wavemeter using the
output signal from a lock in amplifier (driven by a diode detector in
the wave meter arm) as the AFC signal.
was
then
applied
harmonic.
The
and
its
change
phase
in
the
The homodyne mixing signal
adjusted
second
to
maximize
harmonic
the
second
amplitude
is
proportional to the dissipative response of the cavity.
A typical temperature sweep in zero magnetic field as well as a
field
sweep
at
2.13
K
are
shown
in
Fig.
7-7
and
Fig.
7-8
*A
still
larger
increase
would
have
resulted
from
using
superhetrodyne detection.
This requires a second microwave source.
With the present availability of low noise, wide band amplifiers the
stability of the local oscillator is not critical.
Some workers
employ a Gunn oscillator as the second source.
150
CAVITY
Q VALUE
160
125
•**
c
3.
&
i i i—r
• •
100
i ”i i i— — ,
—
—
|—
i
|
r -
T , , -| r , , 1 | i- i i i S a m p l e : Sn
f = 11.420GHz
•*
•
V5
c
•
-
cC
In
•
3u.
<d
—
•
25
•
—
111 J 11111
— t
4
6
----- ------ ------ ------ 1—
------ ------ ------ ------ ------ ------1------L _ J
1
1
1
1
TEMPERATURE
Fig. 7-7.
—
H = 0
T = 3.72K
•
50
MICROWAVE
|™
I
]
1
8
1
1
i
T
1
1
1
10
(K)
Measurements of the microwave absorption from Sn during
a temperature sweep in zero field.
.
12
MICROWAVE
CAVITY
Q VALUE
151
125
Sam ple: Sn
T = 2.1 3 K
G
3
100
>>
u
a
u
+->
75
f = l 1 . 4 2 0 GHz
50
25
0
50
100
150
200
250
MAGNETIC FIELD (GAUSS)
Fig. 7-8.
Measurement of the microwave absorption from Sn during a
field sweep at 2.13 K.
152
respectively.
We can make the following brief comments concerning
the observed response (a) it is evident from the discussion in the
previous section that our microwave spectrometer is sensitive
(conventional)
superconducting transition;
(b)
the
field sweep
to a
the
second harmonic amplitude is proportional to the dissipative which is
quite sharp (as expected for an s-wave superconductor with a rapidly
opening gap).
B)
Zn
Sample #2 was a ball-like piece of 5'9's Zn with a diameter of
about 4.1 mm.
Zn has
a superconducting transition temperature of
0.875K in zero magnetic field;
and a critical field of 53 Gauss at
zero temperature.
The homodyne
detector
was
again
dissipative component of the cavity;
was
11.420 GHz.
However,
it was
adjusted
was
directions.
employed
The
to
width
ramp
of
determined
the
the
respond
to
the
the cavity resonance frequency
technique could be improved as follows:
supply
to
that
the measurement
(a) A bipolar magnet power
d.c.
pedestal
magnetic
in
field
Fig.
7-10
in
both
actually
represents twice the critical field.* (b) When measuring Q changes,
more stable operation is obtained if the Klystron is locked to the
experimental
cavity
(rather
than
the
wave
meter).
With
*Note the use of a bipolar power supply allows one to circumvent
the problem of trapped flux that is generally encountered with a
superconducting magnet which has been previously raised to high
fields. One may either use the midpoint of the bipolar sweep as the
zero field value or demagnetize (degauss) the solenoid by applying a
sine
wave
and
gradually
lowering
the
amplitude
to
zero.
153
the homodyne detector adjusted to maximize the second harmonic at the
cavity resonant frequency (as before) the first harmonic may be used
as the input to the
AFC. This compensates
wave
at
meter
temperature
(located
changes)
room
and the
for a drift between the
temperature
experimental
and
subject
cavity.
to
room
One may
still
measure a frequency shift of the cavity, which is proportional to the
imaginary
part
of
the
surface
impedance,
by
detecting
the
first
harmonic signal generated by the wave meter (although it implicitly
contains the aforementioned source of drift).
A typical temperature
sweep at 0.53 K are
sweep in zero field
shownin Fig.
7-9 andFig.
as well as a field
7-10 respectively.
The results are discussed as follows:
a)
The
interpretation
of
the
results
obtained
during
temperature sweep is as straightforward as that shown in Fig.
When
the
sample
is
cooled
down
to
0.875
K,
the
7-6.
superconducting
transition temperature of Zn in zero field, the rapid decrease of the
surface resistance results in a dramatic increase of the cavity Qvalue.
(b) In Fig. 7-10 we show the signal obtained from a field sweep
at 0.53 K.
This
superconducting
magnetic
field
approach again yields
transition.
were
now
Note
that
generated by
a sharp
both
the
signature
of
directions of
bipolar
power
the
the
supply;
therefore the width of the pedestal Is again 2Hc . The critical field,
Hc , has been plotted for different temperatures
extrapolation results
in a transition temperature
in Fig.
7-11.
of about
An
830 mK,
the
154
Q- Value ( Arbitrary Unit)
1001
i|
i
|
i
i
|
|
i
|
r
|
Sample: Zn
H -0
*>eo -
40 -
20
-
0C
JI
II
I
L
0.7
0.8
0.9
1.0
I
I I .I.,.
1.1
1.2
Ii
1.3
T/TC
Fig. 7-9.
Measurement of microwave absorption from Zn during a
temperature sweep in zero field. The solid line is a
guide to the eye.
1.4
Q-Value
(ArbitraryUnits)
155
Sample: Zn
T-053K
H c*19J4Q uw
f « 11.4200Hz
2.0
2HC-30.67 Guaae
1.0
-8 0
80
- 40
- 20
0
20
40
80
80
Magnetic Field H (Guass)
Fig. 7-10. Measurement of the microwave absorption from Zn during
a field sweep at 0.53 K, using a bipolar magnet power
supply.
156
30
He (Gauss)
Sample: Zn
f * 11.420 GHz
20
0.40
0.60
0.70
Temperature (K )
Fig. 7-11. Temperature dependence of the critical field of Zn.
solid is a guide to the eye.
The
157
about 5% lower than the literature value of 875 mK.
c)
As
spectrometer
a
result
(arising
of
the
increase
from homodyne
in
detection
the
and
sensitivity
locking
to
of
the
experimental cavity), an unexpected superconducting transition signal
was then observed below 3.0 K.
Fig.
7-12.
signal
(Note the sharp dip at high fields is the EPR absorption
from DPPH.
critical
field
transition
A typical field sweep is shown in
Carrying
at
signal
different
was
later
out a series
temperatures,
identified
as
of measurements
this
of
the
superconducting
arising
from
the
pure
(99.999%) In O-rings used to provide leak tight joints in the cavity.
7.4 Progress with the Heavy Fermion Superconductor UPt3
7.4.1
Samples
All samples were obtained from the same ingot out of which the
so-called
"sample
#2"
of
references
5-16,
was
made.
We
made
measurements on the following
a)
One sample was a piece of a hemisphere (which will be
referred to as sample A) having a diameter of about 2.8 mm,
and the c-axis normal to the flat surface.
The hemisphere
resulted from the accidental cleaving of a sphere on the
final stages of spark planing (the sphere was being
machined for susceptibility measurements; a new sphere
was subsequently machined). After annealing this sample
was used to perform the preliminary microwave surface
impedance measurements because it was thought that a
cleaved
surface
might
be
especially
favorable
for
our
unit)
158
(arbitrary
dpph
Q VALUE
2H
MICROWAVE
CAVITY
SAMPLE: In
T=l.72K
f= 11.420GHz
x
-6.0
x
-4.0
-2.0
0
2.0
MAGNETIC FIELD (KG)
Fig. 7*12. Measurement of the microwave absorption arising from the In
O-rings (which seal our microwave cavity) during a field
sweep at 1.72 K.
The sharp dip is the DPPH EPR absorption.
159
microwave response.
The measurements will be discussed
in Section 7.4.3.
b) From this same hemispherical piece, a needle-shaped specimen
(referred to as sample B) was later spark-cut from the region
near the flat surface, the dimensions were
0.077 x 0.077 x 2.76 mm^.
This sample (B) was used to
perform conventional four-wire resistivity and upper
critical field measurements.
c)
A disk-like sample (4.2 mm diameter 1.5 and mm thick)
referred to as sample C, was also spark-cut from the original
ingot with the c-axis perpendicular to the flat surfaces of
the disk.
This sample was used for further microwave
impedance studies.
d)
Also a spherical sample, (4.03 ± 0.01) mm in diameter, was
produced by spark cutting technique from the original
ingot.
It was referred to as sample D and used for the
lower critical measurements of UPt3 .
Conventional four-wire
a.c.
resistance measurements,
0.22 A/cm^ at 9.3 Hz, have been performed on sample B.
with J =
The sample
was varnished on the bottom section (which is made of OFHC) of the
top loading transport probe which was tightly screwed into the bottom
of the cavity such that the sample was positioned at the center of
the field.
Temperatures were measured with a 100 fl - Matsushita
resistance
thermometer
bottom
the
of
cavity
(pushed
(Fig.
into
6-1),
a
which
tight-fitting
was
hole
calibrated
on
the
against
a
160
(precalibrated)
curve
is
Germanium
shown
in
Fig.
resistance
thermometer.
The
7-13.
discernible
field
No
observed on the Matsushita thermometer.
between the sample
and the
calibration
effect
was
Reliable thermal equilibrium
thermometer was achieved by condensing
about 3 cm^ of liquid ^He into the cavity.
Some properties
measurements
are
our
listed
sample
as
from
follows
the
four-wire
(also
see
resistance
Fig.
7-14):
p(300K) — 259.4 pfl-cm;
(7-3a)
P.{ 30OK) _ gg
p (IK)
(7-3b)
3
.
p(T) = P q + Pl T 1
with p
o
(7-3c)
, 5
-0.96 ufl-cm
p1 - 2.0 pii-cm K '
The
of
superconducting
2
(T
< T < 1.8 K)
transition
temperature,
T , is
553mK with
width about 22 mK in average, which was determined in the following
way (see the insert of Fig. 7-14):
on
the
traces
superconducting
just
before,
transition.
at
The
three straight lines were drawn
the
two
mid
point,
and
intersections
respectively, as the"onset temperature" T^ and the "zero
temperature Tq . The
transition
width is defined as
after
are
the
defined,
resistance"
AT=* T^
-T^.
a
161
(K)
4
TEMPERATURE
3
2
1
0.00
0.25
0.50
0.75
RESISTANCE
1.00
1.25
(OHM)
Fig. 7-13. Calibration of lOOfl-Matsushita resistance thermometer.
RESISTIVITY
(jio h m -c m )
162
Fig. 7-14. Resistivity of UPt3 vs. temperature in zero magnetic field.
The inset shows how the transition temperature, Tc , as well
as the width, AT - Tq , are defined.
163
7.A.2
The upper critical field measurements (resistively)
The upper critical field has been measured
(resistively) using
the following orientations and procedures
a)
Measurements were performed with the field alongc-axis
(referred to as the longitudinal
b)
critical field
Basal plane measurements were carried out with the field
along a or b - axis (referred to as H ^ ) • Notethe
b-axis
is here defined as 30° away from the a-axis.
c)
The sample was cooled to IK in a field of 1 Tesla, applied
at ~ 20 K.
The field was then ramped to zero, prior to
further cooling and initiating the critical field
measurements.
The
reason
antiferromagnetic
for
the
latter
ordering
has
procedure
been
is
observed
discussed
with
T^
next.
=
5K
An
from
neutron diffraction experiments.t^1] with a 1 Tesla external magnetic
field applied as the sample was cooled down from ~ 20 K, the q vector
associated with the antiferromagnetic phase ordering was presumably
"locked" at low temperatures to a particular direction in the basal
plane perpendicular to the filed.
We were curious
to see
if any
differences in the upper critical fields could be observed with and
without the ordering being "locked" along a certain orientation.
The measured upper
critical fields
are plotted in Figs.
7-15
(a) - (d) with th^ following results:
1)
with
No difference has been detected in the upper critical field
or without the
application of the
"q vector
locking"
field,
164
---1---1---1---1---1---1---1---1---1---1---1---1---1---1---1---1---1---r
T
H ||c -ax is
15
10
UPPER
CRITICAL
FIELD
(kG)
20
w
—
>
i
i
i
200
1
300
i
i
i
i
I
400
i
i
i
I
■
*^
i
ig
i
500
TEMPERATURE (m K )
Fig. 7-15. (a)
vs. temperatures with H||c-axis.
600
165
15
10
UPPER
CRITICAL
FIELD
(kG)
20
200
300
400
TEMPERATURE
Fig. 7-15.
500
600
(mK)
(b ) Hx9 v s . temperatures with H||a-axis.
»(0): data taken
with ^without) 1 Tesla magnetic field applied when the
sample was cooled down.
166
(kG)
20
10
UPPER
CRITICAL
FIELD
H||b— axis
ts
200
aoo
400
500
TEMPERATURE (m K )
Fig. 7-15. (c) H-*-_ vs. temperatures with H||b-axis • (O) : data
taken with (without) 1 Tesla magnetic field applied
when the sample was cooled down.
600
167
HJLc—a x i s
is
to
UPPER
CRITICAL
FIELD
(kG)
20
200
300
400
500
TEMPERATURE (m K )
Fig. 7-15. (d) The upper critical field vs. temperatures when
the applied field is in the basal plane. Note the
change of slope at T - 418 mK.
600
168
2)
An anisotropy of the upper critical field was observed.
A least square fit of the data in the vicinity to Tc (H~0) gave
the
following
dHL
slopes
of
the
upper
critical
fields
at
T
- 6.1 (T/K)
dT
=
:
(7.4a)
and
dH^2
- - 4.3 (T/K),
(7.4b)
which are in good agreement with the previous measurements (see Table
7
-it22]j
From
superconductor
anisotropic
with
upper
the
anisotropic
uniaxial
critical
symmetry
field
Ginzburg-Landau
(like
governed
UPt^)
by
theory,
should
the
forms
have
a
an
oft 23]
II
and
H
1
0
c2
0o
2wfx2
- -------r
(7 - 5b)
where <j>Q is the flux quantum; f
and f
are,
respectively,
the L-G
coherence lengths parallel and perpendicular to the anisotropy axis,
and are given by
r
h2
i 1/f2
fx “ L4mxa ( T c -T)J
and
(7-6a)
169
f
. r
i1/2
(7-6b)
Umz a ( T c - T ) J
2
where
sf
m , m
are
K
z
the
"effective masses"
along
the
corresponding
directions (not to be confused with the effective masses derived from
the
electronic
band
structure
and
a
is
the
coefficient
of
the
temperature dependent factor in the L-G free energy.
From the measured slopes of the upper critical field,(Eqs.
7-4)
the L-G coherence length is estimated as
fx - 87.5 (Tc - T ) - V 2 (AK1/2)
(7 -7a )
fz - 61.7 (Tc - T ) ' V 2
(7 -7b)
3)
(AK1/2)
Observation of a kink on the H1„ curve.
A change of the slope of the upper critical field curve
dl4
.
which we will refer to as a kink, was observed for H parallel to both
the a and b axes at T„ - 420 mK and H„ - 5.85 kG (see Fig. 7-15d).
n
H
The slopes above and below
K
are,
respectively,
T/K (with a ratio of 1.53); T^+ and T^
identified as 553 mK and 508 mK.
6.62 T/K and 4.4 3
(Fig. 7-16d) are respectively
These results are in good agreement
with those of Ref. 24.
4)
Transition width
The transition width, AT, is plotted as a function of field in
Fig. 7-16.
Although the scatter of the data is large (particularly
at low field), no discernible width broadening is observed when the
magnetic field is increased.
TRANSITION
WIDTH
(mK)
170
UPt
60 I—
3
40 (—
<51
O
I-cA j
DsW > oODtg°
30
°
°
a S ^ c C-1
p
------------ < t ~ * -
D
> » * „
©%
o
o
i—
0
i
i—
i—
1i
55
1
a
«
□
i
i
o
o
°
i
i
10
i
■
i
i
I
■
i
i
i
15
MAGNETIC FIELD (KG)
Fig. 7-16, The measured transition width of UPt3 vs. the applied
field.
20
171
7.4.3
Microwave surface impedance study
A preliminary microwave (f - 11.420 GHz) surface impedance study
was performed on sample A with the spherical bottom varnished onto
the bottom of the microwave resonator.
trace is shown in Fig. 7-17.
A typical tempereature sweep
There is no superconducting transition
signal observed neal Tc . A possible reason could be the bad thermal
contact
between
the
sample
and
the
cavity
at
low
temperatures.
Recall the small temperature discrepancy of the measrued transition
temperature
of
Zn
(Tc
-
0.875
mK)
we
discuss
the
UPt 3
surface
impedance results in the temperature range from 1.0 K to 2.0 K.
Assume
the
surface
impedance
of normal UPt3
obeys
power
law,
i.e.
5R - ATn
Fig.
(7-8)
7-18
shows a plotting of £R with
logarithmic scale.
The
linear
temperature
square fit gives
n -
in
a full
1.43
in
the
temperature range, 1.0 K < T < 2.OK which is quite close to that of
the bulk resistance, n - 1.49 (see Fig.
7-19).
surface impedance is plotted vs T^ in Fig.
The change of the
7-20 and apaprently,
n
appraoches to two when the temperature is decreasing.
7.4.4
d.c.
Hess,
Susceptibility Measurements on UPt3
Tokuyasu,
superconducting
parameter
gives
and Sauls
states
rise
symmetry which exist
(HTS)
of UPt-j
to
two
p r o p o s e d t ^ 5 ]
in which
a
superconducting
in adjacent
two
a model
for
dimensional
phases
temperature domains.
of
the
order
different
Hence
this
model accounts for the existence of two jumps in the specific heat.
u n its )
(arb itrary
RESISTANCE
SURFACE
130
f = 1 1 .4 2 0 G H z
00
50
10
0 .4 0
0 .5 5
0 .6 5
1 .1 0
2 .2 5 4 .0 0
TEMPERATURE (K)
Fig. 7-17. Typical trace of the microwave surface resistance
measurements during a temperature sweep.
173
1.8
units)
Sample:
<5R -
(arbitrary
i.e
UPt
AT
n = 1.43
log
<5R
1.4
1.2
o.o
o.i
0.2
0.3
0.4
log T
Fig. 7-18.
Power law fit on the microwave surface resistance of
UPt3 ( 1 , O K S T < 2 . 0 K )
174
(arbitrary
units)
1.2
1.0
Sample:
UPt
<5R
n = 1.49
0.8
log
<5R
o.o
0.0
0.1
0.2
0.3
log T
Fig. 7-19.
Power law fit on the bulk resistance of UPt
(1.0 K 5 T <; 2.0 K)
0.4
S U R F A C E RESISTANCE (arbitrary units)
o
■
“
©
to
©
U
©
*■
©
Ol
©
©
©
o
to
O
surface
Ol
73
<
W
fO
175
176
Furthermore, the model predicts an abrupt change in the slope of the
upper critical field phase line when the field is in the basal plane,
signaling the transition between the two superconducting phases at a
finite field.
The lower critical field phase line is also expected
to display a kink for all field orientation at a temperature very
close to the zero-field transition.
With an rf resonance technique, which only probes
the surface
(either the skin or London depths) of the sample, Shivaram et al.[24]
recently reported a kink in
with the field in the basal plane;
however the reported ratio of the slopes is much larger
predicted by the theory.
than that
No kink was confirmed with the field along
the c-axis.
Here we present the lower critical field measurements on sample
D with a d.c. technique.£^6]
The results support the predictions of
the HTS model for all field orientations.
We emphasize that in order to obtain the lower critical fields
reliably,
on
a
it is essential to perform the magnetization measurements
single
demagnetizing
crystal
factor,
ellipsoidal
sample
with
A
sample
is
spherical
a
well
perhaps
the
known
most
convenient with a demagnetizing factor of 1/3 for all
principal axes orientations.
7.4.4.1
Experimental details
As shown in Fig. 7-21, the magnetometer contains a primary coil
(for a.c. susceptibility measurements) with 6200 turns #36 Cu wire;
the
two
secondary
coils
(for
both
d.c.
and
a.c.
susceptibility
177
measurements
are
#40 Cu wire).
r.f.
connected
in
opposition
each
having
2750
turns
Between the two secondary coils is a fourth coil, the
coil, with 200 turns #36 Cu wire, which forms part of an LC
circuit with a resonant frequency of 218 kHz when the UPt^ sample
is positioned at the center of the r.f. coil, the resonant frequency
shifts
to
223 kHz.
This
coil
is used
to
accurately
locate
the
position of the sample.
All coils .are wound concentrically on a leak tight epoxy coil
form which is cast on an OFHC copper flange so as to be compatible
with that on the Cu tube thermally connected to the mixing chamber
(Chapter
6) .
installed
A
inside
calibrated
the coil
Mastushita
form.
The
resistance
thermometer
top loading Oxford
was
Dilution
Refrigerator (Chapter 6) with the 12T magnet was used for the d.c,
susceptibility measurement.
The sample was top-loaded to the center
of one of the secondary coils with an epoxy sample holder which was
attached to a 1/4" G-10 rod.
A -^He Gas-handling- system, as shown in
Fig. 7-22, was connected to the magnetometer.
3
liquid
He condensed into the magnetometer,
With about 2 c.c. of
the
sample
could
be
cooled below 68 mK.
The residual
magent
field
has
been
magnetic
field,
usually encountered when a superconducting
previously
measruements
energizied,
(as
when
can
seriously
determining
Hci
affect
of
low
UPt3 ).
Hence all desired low field data were collected before proceeding to
high field measurements.
TO MIXING CHAM BER
and R O O M T E M P E R A T U R E
S A M PLE
LOADINC ROD
POLY ETHYLENE FOAM
t LIQUID J H»~
VACUUM CAN
I CAST EPOXY
C u TUD IN C
TEFLO N O -R IN G
PRIM ARY COIL
SECONDARY COIL
rf COIL
FIELD
P IC K U P COIL
THERMOMETERS
12T S C
7-21.
MAGNET
Schematic diagram of the magnetometer
179
G2
tu
e»
CO
l-J
o
He
GAS TANK
TO THE CAVITY
3
Fig. 22.
Schematic diagram of the
He Gas-handling system
e
3
T=230mK
30
>»
L.
ClSPHERB
at
Xi
z
o
p
<
S
10
fr-*
w
z
o
<
se
40
120
200
M A G N E T I C FIELD (Gauss)
Fig, 23.
A typical magnetometer trace taken at T — 230 mK.
181
While sweeping the magnet,
superconducting switch.
some currents will pass
across
the
In order to obtain an accurate reading of
the field at the magnet center, the modulation coil (Chapter 6) was
connected to
a second
integrator circuit, the output
then proportional to the flux
in the magnet.
of which
is
The
integrator was
net
einf
periodically zeroed in a zero field.
When
the
external
"secondary" coils
s a m p l e . [26]
sample
is
is
ramped, the
is proportional to
Direct
magnetization.
field
Since
accurately
integrationof
the
the dc susceptibility
this
demagnetizaton
know,
the
from
signal
factor
Meissner slope
of
of the
gives
our
can
the
the
spherical
be used
to
calibrate the magnetization scale precisely.
7.4.4.2 Results and discussion
Typically the low field magnetization data were taken by warming
the sample above Tc in zero field to drive off any trapped flux, then
cooling to a fixed temperature.
Gauss/sec.
The field was then ramped at about 1
to obtain the low field magnetization data.
A typical
trace taken at T = 230 mK is shown in Fig. 7-22.
The
lower
critical
field
of
the
sphere,
referred
to
as
(H 1) ,
, was taken as the field at which the initial slope of the
cl sphere
r
magnetization curve just began to deviate from the Meissner value,
then
Hcl
2 ^cl^sphere
(7-9)
182
Figure
7-23
shows
the
lower critical
fields
as a funciton of
temperature for the field parallel to the c axis, while Figure 7-24
shows the data for the field parallel to the above a and b axes.
In
all cases a kink near 0.4 K is apparent.
The
straight
lines
for
the
combined a-
fitted by the following procedure.
and b-axes data
were
The data were partitioned into
low and intermediate field groups and separately fitted to straight
lines.
The total r.m.s. error was then minimized as a function of
the partitioning point.
The fitting gives the parameters of the
HTS
model T^, Tc ^+^ and T ^
^ respectively as 395.0 mK, 541 mK and
517
mK.
Points lying in the dashed region of the lines were not included
in the fit.
The slope ratios of the H
cl
vs, T curve above and below
T*
is estimated to be 1.19
c for the external field in the basal plane
r
and is in good agreement with the HTS model.
The c-axis data appears
to exhibit some curvature; however for comparison with the existing
theory (which predicts straight line behavior) the same two-straight
line
procedure
was
also adopted for the c axis data.
The
•k
intersection occurs at Tc - 395.5 mK. Also the ratio of the slope of
the two fitted lines is estiamted to be 1.36.
of the kink
at Tc
is
Although the existence
in general agreement with
details need further exploration.
HTS
theory,
the
183
ISO
H ^ c — axis
100
o
so
100
200
300
400
soo
600
TEM PERATURE ( m K )
Fig. 7-24.
The lower critical fields vs. temperature for H|| c axis. The filled circles and the solid lines are
respectively the experimental data and the results of
a linear least squares fit (see text).
184
(G auss)
150
100
u
0
100
200
300
400
500
TEM PERATURE ( m K )
Fig. 7-25.
The lower critical fields vs. temperature for
Hi c-axis. The squares (circles) are the
experimental data taken with H|| a- (b-) axis.
The solid lines are a linear least square fit
(see text).
600
Table 7-1 Measurements of the upper critical field
of superconducting UPt^. (Ref. 22.)
Pee
Current
density
J(A/cnr)
Sample
Chen et al.
J||c
Tc
p(300K)
(K) (pfl-cm)
0.52
165
p(3Q0K) P0
P(1K)
150
(pfl-cm)
-0.4
pfl-cm
K1.6
-0.7
0.49
Rauchswalbe
et al.
|dHC2/d T |.
(kOe/K)
63
60
40
40
Hc(T-0)
(kOe)
-18
-19
-18
-19
Shivaram et
L.
Jfic
- 0.1
0.53
125
155
0.20
0.54
77.6
45.1
---
Shivaram et
L.
J||b
0.53
230
110
0.59
1.44
77.2
45.9
21.1 25.9
0.55
259
86
0.96
28.1
- 0.2
This work
J a
61.5
43.3
.......
CHAPTER 8 . SUMMARY
The single-ended c.w, acoustic impedance technique has been used
to study
3
He-B in a sound cell employing a short path length (- 381/im) ;
the chief results are as follows:
a)
the unambiguous signature at the pair-breaking edge in
3
He-B has allowed us to perform a systematic measurement
of A(P,T,H).
The results indirectly support the temperature
scale of Greywall and the weak-coupling-plus model of
Rainer and Serene for the gap function.
A few measurements
of the pair-breaking edge in low magnetic fields (< 1.36 kG)
showed a linear shift with field from the zero field value, by
an amount consistent with the renormalized Larmor frequency,
Cl.
b)
An anomalous doublet splitting was observed at the imaginary
squashing mode which has a strong pressure dependence.
This
observation will hopefully stimulate both theorists and
experimentalists to study the effects of superflow on the
squashing mode.
c)
An anomaly was observed in a magnetic field in the vicinity of
the pair-breaking edge in the form of "bump-like" feature,
which deserves further study.
We
have
also
performed modifications
to
our
Oxford
dilution
refrigerator cryostat which permit X-band microwave experiments.
It
is envisioned that this will open up a new chapter in low-temperature
186
187
physics research at Northwestern.
Based on
the above mentioned experimental
remaining unanswered
questions
that
have
been
observations
generated
we
and
the
suggest
some possible future experiments:
1)
Measurements using an acoustic cell with an even shorter
path length (of order 1/10 of that used here) should be
undertaken.
Since the single ended c.w. acoustic impedance
technique can continue to be employed, two different path
lengths (differing from each other by a factor of, say, 5)
could be used with each of the two quartz transducers.
Such measurements should permit standing wave oscillations
in the acoustic signal (induced by changes in the phase
velocity) to be followed through the squashing mode, which
would be useful, not only for studying the phase velocity
change near this collective mode, but also for obtaining
more information about the doublet splitting.
2)
More careful studies of the magnetic field dependence of the
"bump-like" feature near the pair-breaking edge should be
examined with the shorter path acoustic cell.
3)
Incorporating a relatively minor change on the acoustic cell
(which is already in progress) would allow us to apply a
large electrical field across the liquid
column.
3
He sound
The theoretically predicted three-fold
electric-field-induced splitting of the collective mode
in
3
He-B could be searched for.
188
4)
Our earlier construction of a novel compressional cell,I
which allows us to continuously sweep the
in either direction,
3
He pressure
should be applied to the study of sound
propagation in the spin polarized liquid or solid ^He.
5)
For the microwave spectrometer system,
should be replaced by a sealed
with a dipstick.
With liquid
the
He gas tank
3
He gas system outfitted
3
He condensed in the cavity,
transport, or magnetization measurements can be performed
down to lower temperatures (- 50 m K ) . Upper and lower
critical field measurements on a UPt^ sphere, with
diameter - 4.0 mm, will be initiated shortly.
6)
The possibility of making UPt^ thin films (now being
attempted by the thin film group) will provide a
new type of sample for surface impedance and other
measurements (particularly flux structures and
tunneling on this important heavy Fermion
superconductor).
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2.
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S. Adenwalla, Ph.D. Thesis, Northwestern University
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36
CHAPTER 4
1.
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2.
M.E. Daniels, E.R. Dobbs, J. Sauders, and P.L. Ward,
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W.P. Halperin and J.B. Ketterson Phys. Rev.
(1982).
D.B. Mast,
Lett. 49, 1646
21. G.E. Volovik, JETP Lett. 39, 365 (1984);
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58, 265 (1985).
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R.S. Fishman, J.A. Sauls, Phys. Rev. Lett. 61, 287 (1988).
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P.N. Brusov, J. Low Temp. Phys. 58, 268 (1985).
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194
31.
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33.
Renzhi Ling, J. Saunders, and E.R. Dobbs, Phys. Rev. Lett.
59, 461 (1987).
CHAPTER 5
1.
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Solid Helium. K.U. Bennemann and J.B. Ketterson, Eds.
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8.
P.H. Frings, J.J.M. Franse, F.R. de Boer and A. Menovsky
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B.S. Shivaram, Y.H. Jeong, T.F. Rosenbaum and D. Hinks,
Rev. Lett. 56, 1078 (1986); ibid 57, 1259 (1986).
13.
Y.J. Qian, M.-F. Xu, A. Schenstrom, H.P. Baum, J.B. Ketterson,
D. Hinks, M. Levy and B.K. Sarma, Solid State Commun. 63, 599
(1987).
Phys.
14. S. Adenwalla, Ph.D. Thesis, Northwestern University, 1989.
15. V. Muller, Ch. Roth, D. Maurer, E.W. Scheldt, K. Luders,
E. Bucher and H.E. Bommel, Phys. Rev. Lett. 58, 1224 (1987).
16. A. Schenstrom, M.-F. Xu, Y. Hong, D. Beln, M. Levy, B.K. Sarma
S. Adenwalla, Z. Zhao, T. Tokuyasu, D.W. Hess, J.B. Ketterson
and J .A . Sauls Phys. Rev. Lett. 62, 332 (1989).
17. T. Tokuyasu, D. Hess and J, Sauls, 1989 Phys. Rev,
B at press.
18. K. Hasselbach, L. Talllefer, and J. Flouquet, Phys. Rev. Lett,
63, 93 (1989).
19. J.W. Chen, S.E. Lambert, M.B. Maple, Z. Fisk, J.L. Smith,
G.R. Stewart, and J.O. Wills, Phys. Rev. B30, 1583 (1984).
20.
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J.J. Franse, Z. Phys. B-Condensed Matter 60, 379-386 (1985).
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57, 1259 (1986).
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Lett. 63, 1723 (1989).
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1, 8135-8145 (1989).
CHAPTER 6
1.
J.P. Gordon, Rev. Scl. Instr. 32, 658 (1961).
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Matsushita Corp., Kadoma, Osaka, Japan.
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B.M. Hoffman and J.A. Ibers, Acc. Chem. Res., 16, 15-21 (1983)
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M.Y. Ogawa, J. Martinsen, S.M. Palmer, J.L. Stanton, J. Tanaka
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109. 1115 (1987).
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G. Quirion, M. Poirier, K.K. Liou, M.Y. Ogawa and B.M.
Hoffman, Phys. Rev. 37, 4272 (1988).
196
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K.K.
5.
K.K. Liou, M.Y. Ogawa, T.P. Newcomb, G. Quirion, M. Lee, M.
Poirier, W.P. Halperin, B.M. Hoffman, and J.A. Ibers, Inorganic
Chemistry, 28, 3889 (1989).
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J.E. Wertz and J.R. Bolton, Electron Spin Resonance Elementary
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pp 154-157.
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12. P.L. Richards, Phys. Rev. 126, 912 (1962).
13. R.T. Lewis, Phys. Rev. 134, Al (1964).
14. S.A. Zemon and H.A. Boorse, Phys.
15.
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16. D.C. Mattis and J. Bardeen, Phys.
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18.
I.M. Khalatnikov and A.A. Abrikosov, Advances in Physics,
edited by N.F, Mott (Taylor and Francis, Ltd., London, 1959),
8, p. 45.
19.
M. Spiewak, Phys. Rev. B113, 1479 (1959).
20.
S. Sridha, J. Appl. Phys. 63, 159 (1988).
21.
G. Aeppli et al., J. Magn. Mag. Mater., 76-77, 385 (1988),
22.
B.S. Shivaram, T.F. Rosenbaum, and D.G. Hinks, Phys. Rev.
Lett. 57, 1259 (1986).
23.
B.Y. Jin and J.B. Ketterson, Advances in Physics, 38, 189
(1989).
24.
B.S. Shivaram and J.J. Gannon, Jr., Phys. Rev. Lett. 63.
1723 (1989)
25.
D.W. Hess, T.A. Tokuyasu and J.A. Sauls, J. Phys. Condens.
Matt. 1, 8135 (1989).
26.
F. Behroozi, Am. J. Phys. JH., 28 (1983).
CHAPTER 8
8-1
Zuyu Zhao, H.Q. Yang, S. Adenwalla and J.B. Ketterson;
Bimal K. Sarma, Can. J. Phys. 65, 1534 (1987).
VITA
I.
PERSONAL DATA
Name:
Date of Birth:
Place of Birth:
Sex:
II.
III.
IV.
Zuyu Zhao
August 23, 1947
Shanghai, P.R. China
Male
EDUCATIONAL BACKGROUND
Bachelor of Science (B.Sc)
Fudan University, 1982
Shanghai, P.R. China
Master of Science (M.S.)
Northwestern University, 1985
Evanston, Illinois
Doctor of Philosophy (Ph.D)
Northwestern University, 1990
EMPLOYMENT (U.S. only)
Teaching Assistant:
Physics Department
Northwestern University, 1984-1985
Evanston, Illinois
Research Assistant:
Physics Department
Northwestern University, 1985-1989
Evanston, Illinois
Post Doctor:
Physics Departments
Northwestern University, 1990Evanston, Illinios
RESEARCH INTERESTS
Quantum fluids
Materials at low temperatures
Heavy fermion superconductor
Microwave suface impedance
198
199
V.
RESEARCH EXPERIENCE
Cryogenics
Ultrasonics
Microwave techniques
VI .
SCHOLARSHIPS
Fudan University Scholarship, 1980—1981
World Bank Scholarship, 1983-1984
VIII. PUBLICATIONS
Novel Compressional Cell for Liquid and Solid 3He Experiments
Z . Zhao. H.Q. Yang, S. Adenwalla, J.B. Ketterson and B.K.Sarma
Can. J. Phys. 65, 1534 (1987).
Ultrasonic Properties of Oriented Ceramic High Tc Superconductors
M. Levy, M.F. Xu, B.K. Sarma, Z . Zhao. S. Adenwalla,
Q. Robinson and J.B. Ketterson
IEEE 1988 Ultrasonics Symposium Proceedings pp. 1097-1103.
Ultrasonic Attenuation Measurements of the Flux Lattice Phase
Transition in the Heavy Fermion Superconductor UPt3
S. Adenwalla, Z . Zhao. J.B. Ketterson, D.G. Hinks.
A. Schenstrom, Y. Hong, M.F. Xu, M. Levy and B.K, Sarma
IEEE 1988 Ultrasonics Symposium Proceedings pp. 1085-1088.
Ultrasonic Velocity Anomalies in Superconducting Sinter-Forged
YBa2Cu307_s
Z . Zhao. S. Adenwalla, A. Moreau, J.B. Ketterson,
Q. Robinson, D.L. Johnson, S.J. Hwu, K.R. Poeppelmeier,
M.F. Xu,Y. Hong, R.F. Wiegert, M. Levy and B.K. Sarma
Phys. Rev. B39 721 (1989).
Ultrasonic Attenuation Measurements in Sinter—Forged YBa2Cu307_tf
M.F. Xu, D. Bein, R.F. Wiegert, B.K. Sarma, M. Levy,
Z . Zhao. S. Adenwalla, A. Moreau, Q. Robinson, D.L. Johnson,
S.J. Hwu, K.R. Poeppelmeier, and J.B. Ketterson
Phys. Rev. B39 843 (1989).
Elastic Constant Anomalies in the Sinter Forged High-Tc
Superconductor YBa2Cu307_tf
Z . Zhao. S. Adenwalla, A. Moreau, J.B. Ketterson, Q. Robinson,
D.L. Johnson, S. J. Hwu, K.R. Poeppelmeier, M.-F. Xu, Y. Hong,
M. Levy and Bimal K, Sarma
J. Less Comm. Met., 149 (1989) 451-454.
200
Ultrasonic Attenuation in Sinter Forged High Tc YBa2 Cu3 0 7 _tf
M.F. Xu, D. Bein. Y. Hong, B.K. Sarma, M. Levy, Z . Zhao.
S. Adenwalla, A. Moreau, Q. Robinson, D.J. Johnson, S.J. Hwu,
K.R. Poeppelmeier and J.B. Ketterson
J. Less Comm. Met., 149 (1989) 447-450.
Hysteresis in Ultrasonic Attentuation in UPt3 in Low Magnetic
Fields
A. Schenstrom, M.F. Xu, Y. Hong, M. Levy, B.K. Sarma,
S. Adenwalla, Z . Zhao. J.B. Ketterson, D.G. Hinks
J. Less Comm. Metals, 149 (1989) 353-356.
Magnetic Field Dependent Sound Attenuation in UPt3
A. Schenstrom, M.F. Xu, Y. Hong, M. Levy, B.K. Sarma,
S. Adenwalla, Z . Zhao. J.B. Ketterson, and D.G. Hinks
J. Less Comm. Metals, 149 (1989) 349—351.
Anisotropy of the Magnetic-Field-Induced Phase Transition in
Superconducting UPt3
A. Schenstrom, M.F. Xu, D. Bein, M. Levy, B.K. Sarma,
S. Adenwalla, Z . Zhao. T. Tokoyasu, D. Hess,
J.B. Ketterson, J.A. Sauls, and D.G. Hinks
Phys. Rev. Lett. 6£ 332 (1989).
*“
A Novel Technique to Measure the Group Velocity of Sound in
Dispersive Media
Z . Zhao. S. Adenwalla, J.B. Ketterson and B.K. Sarma
IEEE Trans, on Ultrasonics, Ferroelec. and Freq. Ctrl. ,36, 481
(1989).
A Study of the J-2 Collective Mode of Superfluid 3He Using a
Dynamic Pressure Sweeping Technique
S. Adenwalla, Z . Zhao. J.B. Ketterson, and B.K. Sarma
J. Low Temp. Phys. JJi, 1 (1989)
Observation of a Doublet Splitting of Squashing Mode in 3He-B
Z . Zhao. S. Adenwalla, J.B. Ketterson and B.K. Sarma
International Symposium of Quantum Fluids and Solids, 1989, Edited
by G.G. Ihas and Y. Takano, AIP Conference Proceedings, 194 109
(Amer. Inst. Phys., N.Y. 1989)
Pair-Breaking Edge in Superfluid 3He-B
S. Adenwalla, Z . Zhao. J.B. Ketterson and B.K. Sarma
International Symposium of Quantum Fluids and Solids, 1989, Edited
by G.G. Ihas and Y. Takano, AIP Conference Proceeding, 194. 107
(Amer. Inst. Phys., N.Y. 1989)
201
Shubnikov-de Haas Effect in Thin Epitaxial Films of Gray Tin
L.W. Tu, G.K. Wong, S.N. Song, Z . Zhao. and J.B. Ketterson
Appl. Phys. Lett. 55, 2643 (1989)
Measurements of the Pair-Breaking Edge in Superfluid 3He-B
S. Adenwalla, Z . Zhao. J.B. Ketterson, and B.K. Sarma
Phys, Rev. Lett. 6£, 1811 (1989).
Growth and Characterization of Substrate-Stabilized
Heteroepitaxial Grey Tin Films
L.W. Tu, G.K. Wong, S.N. Song, Z. Zhao and J.B. Ketterson,
Semicond. Sci. Technol. 5 XXX (1990)
A Top-Loading Very-Low-Temperature, X-Band, Microwave Spectrometer
M.Z. Lin, Z . Zhao. Y.H. Shen, J.B. Ketterson, Y.J. Qian and
Bimal K. Sarma
(to be published)
Resolution of a Doublet Splitting of the Squashing Modein 3He-B
Zuvu Zhao. S. Adenwalla, J.B. Ketterson,and B.K. Sarma
(submitted to Phys. Rev. Lett.)
Observation of New Structures in the Collective Mode Spectrum of
3He-B
Z . Zhao. S. Adenwalla, J.B. Ketterson, and B.K. Sarma
(to be published)
Superconducting Lower Critical Fields in UPt3
Zuvu Zhao. F. Behroozi, J.B. Ketterson, Yongmin Guan,
Bimal K. Sarma and D.G. Hinks
(to be published)
Phase Diagram of Superconducting UPtg
S. Adenwalla, Z . Zhao. J.B. Ketterson, M.
(to be published)
Levy and Bimal K. Sarma
Dimensional Crossover of Shubnikov-de Hass Oscillations in Thin
Films Gray Tin
S.N. Song, X.J. Yi, J.Q. Zheng, Z . Zhao. L.W. Tu, G.K. Wong, and
J.B. Ketterson
(submitted to Phys. Rev. Lett.)
IX.
ABSTRACTS AND POSTERS
Measurements of Group Velocity of Sound Using an F.M. Modulating
C.W. Method.
Z . Zhao. S. Adenwalla and J.B. Ketterson; B.K. Sarma,
Bull. Amer. Phys. Soc., 33. No. 3. 408 (1988).
202
Probing Collective Mode in 3He via continuous Pressure Sweeps.
S. Adenwalla, Z . Zhao and J.B. Ketterson; B.K. Sarma,
Bull. Amer. Phys. Soc., 33, No. 3, 408 (1988).
Elastic Constants of Sinter—forged YBagC^O^j.
S. Adenwalla, Z . Zhao. A. Moreau, Q. Robinson, D.L. Johnson,
S.J. Hwu, K.R. Poeppelmeier and J.B. Ketterson; M.F. Xu, M. Levy
and B.K. Sarma,
Bull. Amer..Phys. Soc., .33, No. 3, 513 (1988).
Ultrasonic Attenuation Measurements in Sinter-Forged YBa2Cu307_<J.
M.F. Xu, M. Levy and B.K. Sarma; Z . Zhao. S. Adenwalla,
A. Moreau, Q. Robinson, D.L. Johnson, S.J. Hwu,
K.R. Poeppelmeier, and J.B. Ketterson.
Bull. Amer. Phys. Soc., 3 3 , No. 3 513 (1988).
Ultrasonic Measurements in Sinter Forged High Tc Superconductor
YBa2Cu307_,j
M.F. Xu, A. Schenstrom, Y. Hong, D. Bein, B.K. Sarma, M. Levy,
Z . Zhao. S. Adenwalla, A. Moreau, Q. Robinson, D.L. Johnson.
S.J. Hwu, K.R. Poeppelmeier, and J.B. Ketterson
Proc. of the 1988 Appld. Supercon. Conf., San Francisco.
EPR Measurements of Cu (tatpb) I
Z. Zhao. M.Z. Lin, S. Adenwalla, Y.H. Shen, J.A. Thompson,
B.M. Hoffman and J.B. Ketterson.
Bull. Amer. Soc., 34, No. 3, 736 (1989).
Ultrasonic Attenuation Measurements in UPt3.
S. Adenwalla, Z . Zhao. and J.B. Ketterson; D.G. Hinks.
A. Schenstrom, Y. Hong, M.F. Xu, M. Levy and B.K. Sarma.
Bull. Amer. Soc., 34, No. 3, 849 (1989).
Dispersion Induced Splitting of Squashing Mode in 3He-B.
Z . Zhao. S. Adenwalla and J.B. Ketterson; B.K. Sarma.
APS March Meeting 1989 (St. Louis).
Observation of the Pair-Breaking Edge in Superfluid 3He-B.
S. Adenwalla, Z . Zhao and J.B. Ketterson; B.K. Sarma.
APS March Meeting 1989 (St. Louis).
Angular Dependence of Shubnifov-de Haas Oscillation in MBE Grown
Grey Tin Films.
S.N. Song, Z . Zhao. L.W. Tu, G.K. Wong and J.B. Ketterson.
APS March Meeting, 1989 (St. Louis).
Observed Anomalies in Ultrasound Studies on Superfluid 3He-B.
Z . Zhao. S. Adenwalla, and J.B. Ketterson; Bimal K. Sarma.
APS March Meeting 1990 (Anaheim).
Microwave Study of the Surface Impedance of the Heavy Fermion
Superconducting UPt .
Z . Zhao. S.N. Song, and J.B. Ketterson; D.G. Hinks.
APS March Meeting 1990 (Anaheim)
Effects of 2D-3D Crossover on Localization and Superconductivi
in Ultra-Thin Niobium Films.
S.N. Song, Z . Zhao. and J.B. Ketterson.
APS March Meeting 1990 (Anaheim).
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