close

Вход

Забыли?

вход по аккаунту

?

EFFECTS OF MICROWAVES ON THE ANNIHILATION RATES OF ORTHOPOSITRONIUM LOCALIZED IN FLUID ETHANE

код для вставкиСкачать
INFORMATION TO USERS
This reproduction was made from a copy o f a document sent to us for microfilming.
While the most advanced technology has been used to photograph and reproduce
this document, the quality of the reproduction is heavily dependent upon the
quality o f the material submitted.
The following explanation o f techniques is provided to help clarify markings or
notations which may appear on this reproduction.
1.The sign or “ target” for pages apparently lacking from the document
photographed is “Missing Page(s)” . I f it was possible to obtain the missing
page(s) or section, they are spliced into the film along with adjacent pages. This
may have necessitated cutting through an image and duplicating adjacent pages
to assure complete continuity.
2. When an image on the film is obliterated with a round black mark, it is an
indication o f either blurred copy because o f movement during exposure,
duplicate copy, or copyrighted materials that should not have been filmed. For
blurred pages, a good image o f the page can be found in the adjacent frame. I f
copyrighted materials were deleted, a target note will appear listing the pages in
the adjacent frame.
3. When a map, drawing or chart, etc., is part o f the material being photographed,
a definite method o f “sectioning” the material has been followed. It is
customary to begin filming at the upper left hand comer of a large sheet and to
continue from left to right in equal sections with small overlaps. I f necessary,
sectioning is continued again—beginning below the first row and continuing on
until complete.
4. For illustrations that cannot be satisfactorily reproduced by xerographic
means, photographic prints can be purchased at additional cost and inserted
into your xerographic copy. These prints are available upon request from the
Dissertations Customer Services Department.
5. Some pages in any document may have indistinct print. In all cases the best
available copy has been filmed.
University
Microfilms
International
300 N. Zeeb Road
Ann Arbor, Ml 48106
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Order N um ber 1331891
Effects o f microwaves on the annihilation rates o f orthopositronium
localized in fluid ethane
Arganbright, Robert Harvey, M.S.
The University of Texas at Arlington, 1987
U
M
I
300 N. ZeebRd.
Ann Arbor, M I 48106
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
EFFECTS OF MICROWAVES ON THE ANNIHILATION RATES OF ORTHOPOSITRONIUM
LOCALIZED IN FLUID ETHANE
by
ROBERT HARVEY ARGANBRIGHT
Presented to the Faculty of the Graduate School of
The University of Texas at Arlington in Partial Fulfillment
of the Requirements
for the Degree of
MASTER OF SCIENCE IN PHYSICS
THE UNIVERSITY OF TEXAS AT ARLINGTON
August 1987
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
EFFECTS OF MICROWAVES ON THE ANNIHILATION RATES OF ORTHOPOSITRONIUM
LOCALIZED IN FLUID ETHANE
APPROVED:
C.
(Supervising Professor)
A
IffllloWkljhy '
ijy
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i
DEDICATION
To the loving memory of Shirley Jean and David, who instilled a love within me
for travelling never ending paths to knowledge.
iii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ACKNOWLEDGEMENTS
Foremost acknowledgement for this work is due Suresh C. Sharma for providing the
original concept of the experiment. Also, I am deeply grateful for his time, guidance, patience,
and understanding, without which this thesis could not have been completed.
I would also like to express my appreciation to the many other people who provided
assistance to this work. Mike Ward provided aid in collecting and analyzing data as well as
computer programming. Charley Dark also lent his skills as a programmer. Wallace Lutes and
Doug Coyne provided neccessary technical advice and assistance. Dr. J. M. Kowalski provided
invaluable theoretical discussions. Dr. John Owens of the UTA Department of Electrical
Engineering and Elizabeth Hardin of Hewlett Packard Corp. provided for the loan of various
components of the microwave system.
Finally I am grateful to the Robert A. Welch Foundation in its support for this research
and to A. O. Pipkin Jr. of ARL Inc. for student financial support.
July 23.1987
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ABSTRACT
EFFECTS OF MICROWAVES ON THE ANNIHILATION RATES OF ORTHOPOSrTRONIUM
LOCALIZED IN FLUID ETHANE
Robert Harvey Arganbright, M.S.
The University of Texas at Arlington, 1987
Supervising Professor: Suresh C. Sharma
Orthopositronium atoms localized in density fluctuations in fluid ethane have been
subjected to microwaves in the 90-100 GHz range. The resulting pick-off annihilation rates
of orthopositronium are significantly reduced at certain density dependent values of the
incident microwave frequency. A mechanism for the absorption of microwaves by non-polar
ethane molecules is described in terms of pressure induced dipoles arising through molecular
quadrupole interactions. Two possible mechanisms for the resulting decrease in the
annihilation rates of orthopositronium are discussed. The first model assumes a reduction in
the density of electrons around a localized positronium atom as a result of the response of the
fluid-compressibility to the microwave field at some resonant frequency. The second model
describes the reduction of annihilation rates as due to positronium annihilation with molecules
in rotationally excited states.
v
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS...................................................................................................
iv
ABSTRACT ......................................................................................................................
v
Chapter
I.
IN TR O D U C TIO N ..................................................................................................
II. MODELS FOR LOCALIZATION OF ORTHOPOSITRONIUM IN DENSE
F L U ID S ................................................................................................................
Positronium Self-Trapping......................................................................
1
3
3
The Simple Cavity M o d e l.......................................................................
3
The Self-consistent Field M o d e l..........................................................
12
The Density Fluctuation M o d e l...................................................................
I I I . EXPERIMENTAL D E TA ILS ................................................................................
Apparatus......................................................................................................
14
17
17
Gas Handling S ystem ................................................................................
17
Heating System ......................................................................................
20
R F Generator.........................................................................................
20 .
Positron S ource......................................................................................
21
vi
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Lifetime Spectrometer................................................................................
Set U p ......................................................................................................
23
Time Calibration......................................................................................
24
Data Collection.........................................................................................
24
Analysis D etails......................................................................................
27
Density Calculations................................................................................
28
IV. RESULTS AND DISCUSSIO N...........................................................................
V.
23
31
Orthopositronium Annihilation Rate Behavior at 98 a m ag a t................
31
Absorption Mechanism.............................................................................
33
Mechanisms for Microwave Induced Reductions in x0.P s ....................
38
Rotionally Excited Molecules.............................................................
38
Density Fluctuation Excitations..........................................................
39
Orthopositronium Annihilation Rate Behavior at 60,37, and
99 a m ag a t.............................................................................................
46
x0.Ps at 60 amagat and 307.15 K ...................................................
46
x0.Ps at 37 amagat and 307.15 K ...................................................
50
x0.Ps at 99 amagat and 307.15 K ...................................................
50
CONCLUSIONS...............................................................................
59
REFERENCES..........................................................................................................
61
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UST OF ILLUSTRATIONS
2-1
V
2-2
Potential well depth versus density for ethane at 306.4 K ........................
8
2-3
Equilibrium cavity radius versus density of ethane at 306.4 K .................
9
2-4
Ground state binding energy versus density of ethane at 306.4 K . . . .
10
2-5
Change in the free energy versus the characteristic size
ps
versus density at 306.4 K ....................................................................
of the wave function He at 10 K and ethane at 306.4 K .............................
3-1
5
15
Schematic of gas handling and heating systems
and the microwave system .............................................................................
19
3-2
Decay scheme of Sodium-22........................................................................
22
3-3
Schematic of the lifetime spectrometer...........................................................
25
3-4
Typical Lifetime Spectrum..............................................................................
26
4-1
>.0_ps versus frequency in ethane at 98 amagat and 307.15 K .................
32
4-2
versus frequency in ethane at 98 amagat and 307.15K ..................
42
4-3
A>20^>s versus frequency in ethane at 98 amagat and 307.15 K ..................
43
4-4
A?T20_ps versus q2 in ethane at 98 amagat and 307.15 K ..............................
44
4-5
A-x*20_ps versus q2 in ethane at 98 amagat and 307.15 K ..............................
45
4-6
* 0_ps versus frequency in ethane at 60 amagat and 307.15 K ........................
47
viii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4-7
x^p, versus frequency in ethane at 60 amagat and 307.15 K
.......................
48
4-8
A * 2*-*., versus frequency in ethane at 60 amagat and 307.15 K .................
49
4-9
A x '2,,-?, versus q2 in ethane at 60 amagat and 307.15 K .............................
51
4-10
versus frequency in ethane at 37 amagat and 307.15K ...........................
53
4-11
A x2^ , versus frequency in ethane at 37 amagat and 307.15 K ....................
54
4-12
A x2e_p, versus frequency in ethane at 37 amagat and 307.15 K ....................
55
4-13
Xo.P, versus frequency in ethane at 99 amagat and 307.15 K ...........................
56
4-14
Ax2^ , versus frequency in ethane at 99 amagat and 307.15 K ....................
57
4-15
A x '2e_p, versus q2 in ethane at 99 amagat and 307.15 K ..............................
58
ix
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UST OF TABLES
1.
Comparison between observed values of N* and R*
and those calculated with the self-consistent field m o d e l.................................
2.
14
Impurities found in a representative sample of the
ethane used in the experiment..............................................................................
x
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
18
CHAPTER I
INTRODUCTION
The localization of orthopositronium (o-Ps) in regions of lower than average density
within fluids under certain PVT conditions has in recent years received considerable
theoretical and experimental interest. Two models, the cavity model and the density
fluctuation model have been separately successful in different fluids at different
temperatures. In the cavity mode! the positronium, through exclusion forces, creates a cavity
in which it resides. The density fluctuation model assumes that the positronium
preferentially samples pre-existing regions of lower than average density. Both models are
discussed in Chapter 2.
This investigation began for the purpose of testing the simple cavity model. It was
assumed that, if this model is correct, irradiation of the sample with electromagnetic quanta
having energies equal to the binding energy of o-Ps in the cavity would eject the o-Ps from the
cavity into the denser medium surrounding it. The delocalization of the o-Ps atom would
result in an increase in the annihilation rate ( x ^ ) over that when no radiation was
present. The results from this simple experiment revealed a suprising effect of the radiation
upon V p s. Instead of increasing at a certain single incident energy, as expected from the
simple cavity model,
actually decreased below that predicted for an already localized
o-Ps atom. This effect was observed for several densities for frequencies in the 90-100 GHz
1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2
range.
This thesis is primarily concerned with the absorption of microwaves by ethane at
307.15K and the subsequent reduction of \ , _ p s . Two questions raised by the experimental
results are: 1) how does the non-polar ethane molecule absorb radiation? and 2) how does
the absorption of the radiation cause the o-Ps atoms to annihilate at a lower rate? Ample
literature is available concerning pressure induced dipoles in non-polar molecules. The first
question will be approached in terms of this model.
The reduction of
will be discussed in terms of two models. The first assumes that
the density of electrons around a localized o-Ps atom is reduced as a result of the response of
the fluid compressibilty to the microwave field at certain resonant frequencies. The second
model describes the annihilation rate reduction as due to positronium annihilating with
rotationally excited molecules.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTERII
MODELS FOR LOCAUZATION OF ORTHOPOSITRONIUM IN DENSE FLUIDS
Positronium Self-Trapping
At suitable temperatures and pressures a cavity may form around Ps in a non-polar
fluid. Its formation is due to short-range Pauli repulsive forces which dominate over the
attractive, long-range polarization force. The size of the cavity is governed by a balance
between the outward pressure of the o-Ps's zero point energy, due to its localization, and the
inward pressure due to the external pressure and the surface tension of the fluid. As a result
it is expected that the density of electrons within the cavity is less than the average density of
the surrounding medium. This model has been used to understand the non-linear behavior of
\>-ps in C2H6 and other fluids including He, Ne, and H2.
The Simple Cavity Model
At low densities x0_ps exhibits a linear dependence on the gas density according to the
equation
V p * = >v*e+ 4Trr02 cnQ% u D
(2.1)
where xvac is the annihilation rate of orthopositronium in vacuum, rQis the classical radius of
the electron, c is the speed of light, n0 is the standard number density, % f f is the effective
number of atomic electrons per molecule "seen" by o-Ps, and D is the density of the gas in
3
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4
amagat. Figure 2-1 shows \ ^ , s versus density for ethane at 307.K. The dashed line
represents Equation (2.1). The non-linear xe_pj versus density behavior for a cavity devoid of
gas molecules is given by
\-P s ~
0 " P © )\/ie + P o ( W Q D ) ,
( 2 .2 )
where P0 is the fraction of the positronium’s probability density extending beyond the cavity
and q= 4-rr r02 cn01Z*ff. Assuming a spherical square well potential of depth V and radius R,
one obtains1,
P0 = sin^ / (1 -n cot n )
(2.3)
and
n /sin n = R (2mV/h2)1/2,
(2.4)
where m is the mass of the positronium atom. The simplifying assumption that the potential
well of the cavity is shallow enough to allow only the ground state limits the yalue of n to the
range 7t/2 to n.
The total energy of the system is the sum of three terms; the zero point energy of the
positronium, the work done against the external pressure in creating the cavity, and: if the
fluid is a liquid, the work done against surface tension. Hence
E = Ti2k2/2m +4ir/3 R3 P +4 tiR24 \
(2.5)
where P is the external pressure, <f is the surface tension and k is the Ps center of mass
wavenumber. The equilibrium radius of the cavity is obtained by minimizing E with respect
to R,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5
o
o
CO
co
csT^
co
co
©
CO
co
3
O'
LU
>s
X)
c
.2:
o>
o
o
o
C\J
w
CD
C
CO
O)
CO
E
•
2
*
as
CO
o
CO
CD
£
H
c
o
Q
CO
o
CO
LO
o
o
05
6
CO
c
CD
“O
CO
3
£2
CD
>
If)
CVJ
©
3
CO
O)
u.
o
CJ
o
o
(t . su)S d ' 0 ^
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
SE/5R « 0
(2.6)
For a gaseous medium, <f<=0, and .the depth of the spherical square well potential is given by
V «{2nP(ft2/2m)3/2ln2(ncotn-l )/(sin4n cosn)]}(2/5).
(2.7)
The ground state or zero point energy is given by
E2 = V sin2n
(2.8)
The binding energy is therefore given by
Eb = Ez-V
(2.9)
To apply this model to the data, we measure V p s and use Equations (2.2-2.4) to
calculate n. Equations (2.5-2.9) are then used to obtain vaiues of the equilibrium radius R
and the potential well depth V. Thus, Equations (2.2 and 2.3) give a value of n through the
measured >>0_ps then from Equations (2.4,2.6, and 2.7) "measured" values of R and V are
obtained. Assumptions implicit in this simplified model are, that the ground state is the only
occupied state, and both the molecular penetration of the cavity and the diffuse nature of its
surface are negligible.
This model has recently been applied2,2 to \,_ps data published4 for ethane at 306.4 K as
shown in Figure 2-1. At low densities up to about 11 amagat >,e^ s varies linearly with the
density in accordance with Equation (2.1). For densities between 11 and 180 amagat the
departure of v * , from Equation (2.1) increases with the density. At higher densities V -ps
again increases linearly with the density but at a steeper rate. Up to about 100 amagat these
results are qualitatively similar to those seen in He. The cavity model has been particularly
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
successful, in the case of helium, in predicting a scattering length, a=0.77 ± 0.04
A, in good
agreement with theory5. For ethane at 306.4 K the results for the well depth, V, the cavity
radius, R, and the binding energy are shown in Figures 2-2,2-3, and 2-4 respectively. The
potential barrier experienced by the oPs atom in ethane at 306.4 K varies between 0.14 and
0.56 eV over the range of densities shown. In the case of helium the density dependent
potential agrees well with an optical potential given by
V0 = (h2/m)27t n0aD
(2.9)
where a is the scattering length for o-Ps-He collisions mentioned above. Since the Ps-C2H6
scattering length was not known, an estimate was made using the known scattering length of
0.55
Afor the case of
Ps-CH46 . The calculated density dependence of the potential, using
Equation (2.10), is represented by the solid line in Figure 2-2. The "measured" potential
does not agree with the optical potential. A value of 0.14 eV at densities approximately equal
to 10 amagat is too large to describe a cavity devoid of gas molecules7. The sharp increase in
the potential at densities higher than about 250 amagat is completely unaccounted for by the
optical potential. A multiple scattering potential was also used2 to see if it would affect a
signifigant correction. The mutiple scattering potential is given by
V ms = v o I1 + a(4.5n n 0D)1/3].
This is the optical potential with a correction term for higher densities in terms of a fluid
structure factor8 . This equation is represented by the dashed line in Figure 2-2. This
correction hardly improves the fit of the model to the data.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.10)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.6
0.4
0.2
ms
• «* ®*
0.0
0
100
200
Density (amagat)
Figure 2-2. Potential well depth versus density for a spherical square-well potential in ethane
at 306.4 K. The solid and the dotted lines represent the optical and multiple scattering
potentials, respectively, for a=0.55
A2’3.
300
9
o
o
CO
CO
CM
*
<6
O
CO
CO
CO
CO
O)
o
o
©
c
CM
CO
©
o
CO
u>
E
CO
CO
tn
c
o
X)
CO
CO
c
0)
D
o
o
3
w
w
©
>
«
3
TJ
«C
>
CO
o
E
3
3
cr
tu
co
l
CM
©
t.
3
O)
U.
CO
CO
(y )
sm peu
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.00 « ®•
e® • •
•%
Hm
-0.05
k
UJ
-
0.10
3 0 6 .4
-0.15
K
X
X
100
200
300
Density (amagat)
Figure 2-4. The ground state binding energy of positronium atom to the potential well of the
cavity versus the density of ethane at 306.4 K 2>3.
O
The measured density dependence of the equilibrium radius is shown in Figure 2-3. It
differs considerably from the results obtained in He at 1.7 and 4.2K9’10. In He the cavity
radius is seen to decrease exponentially with the pressure; it seems reasonable to expect the
radius should decrease with increasing density. This is not at all the case for ethane up to 180
amagat. In fact the radius increases lineary with D between 50 and 180 amagat. At densities
greater than 180 amagat the equilibrium radius falls rapidly with the density, approaching a
value of 3.5
Afor the highest densities measured.
This is approximately the size of the ethane
molecule. If the cavity ever existed, it has by this point collapsed so that o-Ps samples the
average density of the gas according to Equation (2.1). This fits with the fact t h a t v ^ seems
to approach Equation (2.1) at high densities. In Figure 2-4 the measured binding energies
are plotted against density. Even at the relative maximum occurring at 180 amagat, the
binding energy of 0.15 kT is considered much too small to support a cavity should one form2.
Detailed theoretical calculations of self-trapping in He have further shown that the cavity
model has considerable difficulty in describing the non-linear behavior of V p s when the
binding energy is less than about 6kT19.
The Self-consistent Field Model;
A Further Test of the Cavity Model
The self-consistent field model for investigating self-trapped states of o-Ps has been
summarized by lakubov and Krahpak11. A simple outline of this model is used here to
calculate the transition density, D*, which is characterized as being that density at which
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
12
o-Ps makes a transition from an extended state to a localized state. The localization of the
o-Ps creates a cavity within the fluid having a radius R* and binding energy, EB\
In this
model interactions between the o-Ps and fluid molecules give rise to a molecular correlation
in the neighborhood of o-Ps given by
N(r) ■ N exp[-B V(r)],
(2.11)
where N is the average number density of the gas, V(r) is the interaction potential between
the o-Ps and the molecules, and 13=1/kT. The change in the free energy of the system due to
the trapping of a single Ps atom is given by
BAF = fi2/2m/ (V'f'(r)|2dr3 - N jfd r^ e xp I-B V ^ j+ B V ^ -l}
(2.12)
where
^(r) = / d r ,3V(r-r,)|v * (r ) |2
(2.13)
is the quantum mechanical average of the interaction potential. Assuming a psuedopotential,
V(r) = 2n h 2 a S(r)/m,
(2.14)
where a is the scattering length for the Ps-molecule interaction, and approximating a trial
wave function for the ground state,
'P(r)=(3/2 7tx3)1/2exp(-r3/x3),
(2.15)
the change in the free energy is obtained:
BAF=[3r(1/3)/27/3n2](>>3/>.)2-N/ dr3{exp[(-3Bk2a/mx3)exp(-2r3/x3)]
+(3Bk2a/mx3) expf^r3/* 3)-!} >
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
^ 16)
13
where >p= h/(3mkT)1/2 is the be Broglie wavelength of the o-Ps atom and \ characteristic
range of its wavefunction. By changing the variable of integration to z=exp(-r3/v 3), and
keeping the first few terms of the expansion of exp[3Bh2az/mx3], an approximate integral for
Equation (2.16) is obtained giving for the change in free energy2,
8AF=[3 T(1/3)/27/3 ir2] ^ ) 2 [27N>.b4 a2/(32 A
3)](1-xB2 a/2A 3).
(2.17)
From Equation (2.17) and the conditions for self-trappining, namely, Ba F=0 anc
6(I3a F)/5>.=0, the transition density and characteristic wavelength can be obtained frc
N*=32/27[T(1/3)/a5x B4]
(2.18)
and
x *=(2/ti2)
(a * q2)1/3-
(2.19)
The last three expressions, resulting from our calculations, do not agree with those given by
lakubov et al. It is believed that the authors are in error due to either the actual calculation
or the transcription of the results.
Figure 2-5 shows the dependence of Ba F on * ' 1 for He at 10 K and ethane at 306.4 K.
In the case of helium the change in the free energy is positive for all densities less than
N*=4.55x1021 cm'3 so that cavity formation is unlikely. For densities greater than N*,
over a certain range of x's, Ba F is less than zero and self-trapping is energetically possible.
For ethane at 306.4K, Ba F is greater than zero over the entire range of densities for which
annihilation rates have been measured. Only at densities greater than about 8.22x1022 cm'3
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
14
(3000 amagat) is the change in free energy less than zero and in favor of cavity formation. That
the localization of the positronium can occur only at such extremely large and unphysical
densities above 3000 amagat demonstrates the failure of the cavity model in describing the
non-linear behavior of V p s These calulations are summarized below in Table 1.
Table 1.
Temperature
Sample
Transition density, N*
(K)
(cm‘3)
Calculated2
He
10
77
c 2h 6
Cavity radius, R*
4.55 x1021
1 .77x10
22
306.4
8.22 X1022
305.45
8.20 x1022
(A)
Observed4,9
1.4 x1021
5 x 1 0 21
3 x 1 0 2°
Calculated2
21.0
Observed4,9
12.7
10.7
5.97
10
Density Fluctuation Model
In the density fluctuation model the reduction of the local density of electrons
experienced by o-Ps is shown to result from its localization in density fluctuations2,12,13.
That is, orthopositronium preferentially samples pre-existing and spontaneously forming
regions of lower than average density in the fluid. According to the density fluctuation model
the difference between the measured orthopositronium decay rate and the decay rate predicted
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
15
10t
A/*=4.55 x1021 cm"3(~169 am agat)
o .m
y
1.25A/
CO.
0.02
0.06,
0.04
/V*=8.22x1022 cm“ 3(~3x103am agat)
0.75/V*
1.25/V
0.05
0.15
0.25
(A -1)
Figure 2-5. The change in the free energy, as calculated from the density with the
self-consistent field model, versus the inverse of the characteristic size of the wave function for
(a) He at 10 K and (b) ethane at 306.4 K 2’3.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
16
by Equation (2.1), a
v
*>„ is related to the isothermal compressibility according to
A V P s = 4jtro2cno1zeff D(kTK0/2nV0)1/2,
(2.20)
where K0 = V 1(3V/3P)t is the isothermal compressibility of the fluid. This equation has
worked well for ethane12,12, methane14, nitrogen1® and carbon dioxide1® as long as the
density fluctuations are small.
A subtle but important distinction exists between these two models for localization of
o-Ps. In the cavity model, the localization of the o-Ps atom is seen to take an active role in the
molecular dynamics of the fluid by the creation of its own low density cavity. While on the
other hand, the density fluctuation model places o-Ps in a passive role; that is, it simply falls
into or migrates to pre-existing regions of lower than average electron density. In the latter
view o-Ps is viewed as an excellent probe of density fluctuations in fluids.
The original intent of this research was to experimentally test the simple cavity model.
If o-Ps is localized in such a cavity, then it is reasonable to expect to be able to dissociate the
"cavitron" by striking it with photons having energies close to its binding energy. This would
result in an increase in v * ,* to a value predicted by Equation (2.1).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER I I I
EXPERIMENTAL DETAILS
Apparatus
The Gas Handling System
A schematic representation of the gas handling system is shown in Figure 3-1. The
sample holder consists of a cylindrical copper block three inches long and two inches in
diameter into which a cylindrical cavity 7/8 inches in diameter and 2.4 inches deep has been
bored. The holder was tapped at the top in order to mount a Swage.lok plug with an entry port
for the gas. The plug mounting leaves a volume 1.7 inches deep and 1 inch in diameter to be
occupied by the gas sample. In order to admit microwaves into the sample chamber two optical
sapphire windows were installed on opposite sides of the holder. Each window is 3/4 inch in
diameter and 1/8 inch thick. When mounted on the vessel, each window has an exposed surface
1/2 inch in diameter through which the microwaves can pass. The positron source assembly
was suspended by a nichrome wire from the inside of the Swag.lok plug so that it was held near
the center of the holder and off to the side of the window axis. The gas used was research grade
(99.99% pure) ethane. The manufacturer’s analysis of the impurities found in a
representative sample of the gas is shown in Table 2.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE 2
IMPURITIES FOUND IN A
REPRESENTATIVE SAMPLE
OF THE ETHANE USED IN THE EXPERIMENT
Impurity
Oxygen
Nitrogen
Carbon Dioxide
Hydrogen
Methane
Acetylene
Ethylene
Propane
Propylene
Isobutane
n-Butane
Water
parts/million (ppm)
ft
5
8it
2
5*
1*
40
h
2it
2*
2*
1*
‘ undetected, less than quantity indicated
The sample holder was connected to the gas bottle and pressure gauge through stainless
steel tubing. The pressure was measured with a Heise Model 7200 Bourdon type pressure
gauge having a precision of ±1 psi. The pressure was manually read and recorded at the
beginning and end of each run. Preparatory to introduction of the sample gas, the entire
system was subjected to standard baking and flushing techniques, including evacuation of the
system with a diffusion pump while heating the sample holder and tubing. After doing this
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
DC Power Supply
Pressure Gauge
Heise Model 7200
Apple HE PC
Research
Grade Ethane
Hewlett Packard 8755B
Sweep Amplitude Analyzer
Thermistor
Positro
Source
Nichrome
Heating Wire
m um
Waveguide^
(Receiver)
Waveguide
(Transmitter)
Microwave Generation System:
Hewlett Packard 8350B Mainframe &
Hughes 4772X Sweep Generator
Figure 3-1. Schematic of the experimental set-up including the gas handling and heating
systems and the microwave system.
<0
20
several times over several hours, ethane gas was introduced to the system and evacuated three
times to further purify the system. The sample gas was loaded and, after allowing sufficient
time to reach equilibrium, the pressure was read and recorded and the valve connecting the
sample chamber to the pressure guage was closed. The density of the gas was determined from
the temperature and pressure of the gas just prior to isolating it from the gauge.
The Heating System
The heating system is shown schematically in Figure 3-1. The temperature was
moniiered and controlled by an Instec Temperature Controller; a hardware/software package
run by an Apple HE computer. A thermistor embedded in the copper sample holder provides
the temperature, in terms of its resistance, to the computer which computes the heater power
neccessary to maintain the programmed temperature setting and regulates the current from a
DC power supply to the nichrome heater wire wrapped around the sample vessel. For
electrical insulation Teflon tape was wrapped around the vessel before the heater wire was
installed. This system provides for the control of the temperature within ± 0.05 K.
RF Generator
The microwave source was a Hughes 4772XH Millimeter-wave Plug-in Sweep
Generator, powered and controlled by a Hewlett Packard 8350A Mainframe with a Levelling
Loop. Figure 3-1 shows the arrangement for introducing microwaves into the sample holder.
The radiation was transmitted through the millimeter wave guide to the transmitting horn
which was pressed flush against one of the sapphire windows. There are two detectors along
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the guide; one is used to detect the original or reference signal from the generator and the
other is used to detect any reflected radiation. Pressed against the opposite window and
parallel to the transmitter was a receiving antenna or horn. The signal transmitted through
the gas was detected here. The signals from the reference, reflection, and transmission were
displayed graphically on a Hewlett Packard 8755B Sweep Amplitude Analyzer. Most of the
runs were made in the center frequency mode with a frequency width of 0.10 GHz having a
sweep time of 0.01 seconds. The maximum output power of the generator was about 1 mW.
PQSitron.5.CMC.e
The positron source was 22NaCI. Figure 3-2 shows the decay scheme for ^ N a . By the
emission of a positron, ^ N a decays to 22Ne*; where the asterik indicates that 22Ne is in an
excited state. In a very short time (about 10'11 seconds) 22Ne* decays to the ground state of
22Ne by the emission of a 1.275 MeV Y-ray. Because of the short time between the decay and
the de-excitation, the 1.275 MeV y-ray was used to provide the START signal to the lifetime
spectrometer to begin the lifetime measurement. The annihilation of a positron-electron pair
produces a 0.511 MeV photon. Detection of this photon sends a STOP signal to the
spectrometer. An approximately 50 jxCi source was deposited on a piece of 0.8 mg/cm2 Mylar
film. The source was dryed and then covered by another piece of Mylar. The edges of the foil
envelope were secured between two thin concentric rings leaving the source exposed in the
center of the assembly. The diameter of the source assembly was about 15 mm. The assembly
was suspended in the gas by a fine nichrome wire such that its face was parallel to and level
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
22
22Na(2.58 yrs.)
EC.
9.4%
(0.544
22.
Ne*
(1.275
Mev
9 0 .6 % /
sej
Mev)
(1.820
Mev)
0.004 %
~100%
22.
on
Figure 3-2. The decay scheme of Sodium-22.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
23
with the microwave beam.
Lifetime Spectrometer
Setup
Figure 3-3 shows a block diagram of the lifetime spectrometer. The spectrometer uses
the 1.275 MeV emission y-ray and the 0.511 MeV y-ray resulting from the positron's
annihilation to collect the lifetime spectrum.
The two detectors for the spectrometer were RCA 8575 photomultiplier tubes(PMT)
with Ortec Model 265 bases. The PMT’s were powered by an Ortec high voltage power supply.
The gamma rays were detected by plastic scintillators optically coupled to the photocathodes of
the PMT's. The output pulses from the PMT's were sent to an Ortec Model 583 Constant
Fraction Differential Discriminator where pulse height (energy) selection was made to choose
about fifty percent upper Compton distribution for the 1.275 MeV (start) and 0.511 MeV
(stop) gamma rays. The output from the discriminators served as timing signals'. After
energy selection, the puises were sent to an Ortec Model 457 biased time to pulse height
converter (TPHC). The TPHC converts the difference in the arrival times of the two inputs to
an electric pulse whose height is directly proportional to the difference in arrival times
between the start and stop pulses. This pulse height spectrum is stored in a mutichannel
analyzer(MCA). The lifetime spectrum analysis program POSITRONFIT-EXTENDED17 must
have an accurate resolution function for measuring short lifetimes. A 60Co source was
measured with the spectrometer to produce a "prompt" peak. Using the peak's full width at
half maximum (FWHM), the time resolution was determined by fitting a sum of three
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
24
gaussians to the data. A typical lifetime spectrum is shown in Figure 3-4.
Time Calibration
Each channel of the MCA corresponds to a certain time interval. The interval or, the
time-per-channel(A) was determined with an Ortec Model 462 Time Calibrator module. The
time calibrator produces start and stop pulses at fixed intervals over an adjustable time range,
generating multiple peaks simultaneously. The multiple peak spectrum was collected and
transferred to an IBM computer where a least-squares fitting program was used to determine a
and its error.
Data Collection
After thoroughly purging the system, testing for pressure leaks, and adjusting the
temperature control, a large amount of ethane gas was transferred from the gas bottle to the
experimental vessel and the valve between them closed. The system was allowed at least four
hours to reach equilibrium; following which, the pressure was read and recorded and the valve
connecting the test vessel to the pressure guage was closed. The microwave frequency, band
width, and sweep time were adjusted and the antennae aligned to the vessel's windows. The
positron lifetime spectrum was collected on the MCA and, after a sufficient number of total
counts, was transferred from the MCA to the IBM computer for analysis by POSITRONFITEXTENDED. To begin the next run a different microwave frequency was selected and the
temperature control verified. Usually, five minutes were allowed between changing the RF
frequency and beginning a new run. A series of runs consisted of measuring the lifetime
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
25
High Voltage
Power supply
PM Tube
PM Tube
Source
CFDD
CFDD
S ta rt
Delay Box
TPHC
MCA
Macintosh
512 K
Computer
P lo tte r
IBM 4341
Computer
Figure 3-3. Schematic of the lifetime spectrometer.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
26
o
o
°O
CO
0)
*£
E
3
z
a>
c
c
ca
:js.
:.i
x:
O
o
o
CM
Figure 3-4. A typical lifetime spectrum.
in
O
O
ujii.Lt
in
co
CM
o
o
o
jduueqo jad siunoQ
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
27
spectrum for a roughly even spread of frequencies between 90 and 100 GHz. To begin a new
series of runs the gas density was changed by either admitting gas to, or releasing it from the
vessel. In this fashion V p s was measured as a function of incident microwave frequency in the
range of 90 to 100 GHz for gas densities of 37,6 0,9 8 , and 99 amagat. The resolution
function and timing calibration were checked periodically throughout each series of
experimental runs.
Analysis Details
The lifetime spectra were analyzed with the standard computer program POSITRONFITEXTENDED. The program employs a standard non-linear least squares analysis method which
assumes that the spectrum is a sum of decaying exponentials convoluted with a gaussian
resolution function with a constant background added. The analysis applied in this thesis used
three exponential terms given by,
N{ (t) = Nf(0) exp(-Xf t)
V sW =
N q-Ps W = No-Ps(°) e x P f-V p s 1*
where Nj{0) is the number of species j and Xj annihilation rate of the species j. The
resolution function is given by
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
{3-1)
28
^
R(,,'
v
f
r
i
t
-
ath
2 -»
6xp{ - [ —
}
«»«
where the </oj are the relative intensities of the gaussians such that Zwj = 1, T0 is the center of
the primary gaussian, and AT; is the displacement of the centroid of the ith gaussian from T0.
The <fj's are related to the the full width at half maximum by
FWHM = 2(ln2)1/2<T.
(3.3)
The spectrometer set-up and the data analysis are components of the standard technique
employed for the determination of Ps lifetimes. More complete details of this method can be
found in numerous works (see, for example, References 1,3,7, and 12).
Density Calculations
The density of the gas was calculated with an iterative computer program which uses
both the vir'al and the Benedict, Webb, Rubin (BWR) equations of state18.
The vir'al equation of state is given by 19
PV/RT = 1 + B(T)<f + C(T)<f2
(3.4)
<r=D/Dc
(3.5)
where
is the reduced density. B(T) and C(T) are the dimensionless second and third virial
coeffecients. B(T) is given by
B(T)=B1+ B^x + Bg/X2 + B4/ x4-5,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.6)
29
where
x= T/T*
is the reduced temperature, and
Tc=305.32
B1=0.522671
B2=-1.106244
B3=-0.592947
B4=-0.041944
Dc= 6.74 mole/liter (150.1 amagat)20
C(T) is given by
C(T)=(C1/x +C2/X3 +C3/x5)(1-To/T)
(3.7)
where
T q =217.8K
C 1=0.24423
C2=0.83253
C3=0.53488
The virial equation of state works well in the realm of low to moderate densities.
The BWR equation of state, which was designed to fit P-V-T data in hydrocarbons up to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30
twice the critical density, is given by18
P-RTD+(B0 RT-A0 -Cc/ T 2)D2+(bRT-a)D3+
aaD+(cD3/T2)(1+Y D 2 )exp(- y D2 ).
(3.8)
where P is the absolute pressure in atmospheres, D is the density in moles/liter, T is the
absolute temperature in Kelvin, and
Ao=4.15556 liter2 atm/mole
B0= 0.0627724 liter/mole
C0=1.79592 x 105 liter2 atm/mole2 K2
a=0.345160 liter3 atm/mole3
b=0.0111220 lite^/mole2
c=3.27676 x 104 liter3 atm K2/mole3
a =2.43389 x 10'4 lite^/mole3
y
= 1
.18006 x 1
0 '2
litei^/mole2
R= 0.08207 liter atm ./mole K .
The computer program that calculates the gas density uses the measured values of the
temperature and pressure to calculate a density with the viral equation of state. This initial
estimate of the density is then used in the BWR equation of state to calculate a pressure. This
pressure is compared with the measured pressure. If the calculated pressure does not fall
within a tolerance of 0.005 PSI, the density is iterated and a new BWR pressure is calculated.
The iterative process continues until the measured and the BWR pressures are within the
prescribed tolerance.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER IV
RESULTS AND DISCUSSION
Orthopositronium Annihilation Rate Behavior
Figure 4-1 shows the orthopositronium decay rate, %0_ps, in fluid ethane at 307.15 K
and at a constant average density of 98 amagat as a function of microwave frequency. In
this series of measurements the RF generator was operated in the center frequency (CF) mode
having a band width of ±0.200 GHz and a sweep time of 0.01 second. The dashed horizontal line
represents a decay rate of 0.04234 ± 0.0012 ns'1 measured with no microwaves incident on
the fluid. This value is in close agreement with the previously published experimental value
of 0.03963±0.0005 ns'1 for ethane at 98 amagat and 307.15 K with no microwaves
present20. Comparison of the "off' value of
to that predicted by Equation (3.1) shows
the o-Ps is localized in a region of lower than average density. The decay rate is thus seen as a
measure of the time-averaged local density of the fluid experienced by orthopositronium
during its lifetime.
The data in Figure 4-1 shows two sharp, resonance-like "dips" in \^ > s centered at
92.0 and 96.2 GHz. The indication is that microwaves are absorbed by ethane at certain
frequencies and this absorption induces local density fluctuations and thereby reduces the
density of electrons in the neighborhood of o-Ps during its lifetime. Two questions are
31
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.044
0.043-
0.042
0.041
0.040
98 amagat
0.039
0.038
89
91
93
95
97
99
101
Frequency (GHz)
Figure 4-1. >,0_r>f versus frequency in ethane at 98 amagat and 307.15 K. The solid curve
hand drawn to guide the eye. The dotted horizontal line represents the decay rate with no
microwaves.
“
33
immediately raised by these data: 1) by what mechanism is radiation absorbed by ethane
molecules having no permanent dipole moment? and 2) how does the absorbed radiation induce
local density fluctuations?
The first question is addressed in terms of pressure induced molecular dipoles. Then,
once we have allowed for the fact that microwaves are indeed absorbed by the ethane molecule,
two possible reasons for the resulting increase in the lifetime of o-Ps are discussed. The first
involves an analysis of theoverlap integrals between the wave functions of o-Ps and the ethane
molecule in the ground state and then, in
the first few rotationally excitedstates.The second
possibility explored is that of density fluctuation excitations wherein the bulk
compressibility of the fluid is related to the frequency of the incident radiation.
Absorption Mechanism
Consider a molecule whose unperturbed quantum mechanical Hamiltonian operator is
H(°). The corresponding unperturbed time dependent wave function,
satisfies the
Schroedinger equation:
- E f f ^ - t i / i (d*|<°)/dt)
(4.1)
where E, refers to the energy of the Ith levei. The wave function can be factored into time
dependendent and time independent parts;
<°> -= )*, (°)exp(-i E, t/h),
where
(4.2)
is the time independent wave function of the unperturbed molecule.
Let us assume that the molecule is subjected to electromagnetic radiation of frequency
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
34
v0 = c/x0 propogating in the z-direction, and polarized in the x-z plane. Further assuming
the wavelength of the electromagnetic radiation to be large compared to the size of the
molecule, the electric field component of the radiation can be considered constant over the
molecule. Also, since the speeds of the molecular electrons are small compared to the speed of
light, the magnetic field intensity of the radiation field can be ignored. Under these
assumptions the only component of the electromagnetic field remaining for consideration is
EX=E0 sin2nv0t, so that the total Hamiltonian for the molecule subjected to the radiation is
H= H<0>-(E0 sin2jtvot ) I iqixj,
(4.3)
where the qj and Xj are the charge and x-coordinate of the ith particle in the molecule. The
Schrodinger equation for the perturbed molecule is
{H<°) -(E0 sin2nvot ) I jqjxj}'i'| = -fi/i (9 */a t).
The wave function of the perturbed system,
(4.4)
can be expanded in terms of the unperturbed
wavefunction,
(4.5)
Substituting this equation into Equation (4.4), multiplying by
and integrating the
resulting expression over all space, the following differential equations for the expansion
coeffecients
(fi/i)
are obtained:
“ (E0 Sin27tv0 ) . | ^ ai j ( f l x |nj exp[i(Em -Ej -hv0)t/h]}
where
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.6)
35
I qi Xi
(4-7)
are the matrix elements of the x-component of the molecule's dipole moment.
If at a time t, the molecule is in a state I, a^ =1 and all other a,m=0. During a short time
interval the probabability of a transition taking place is so small that the values aM, a^, a^,.
..
are negligibly small compared to a« so that Equation (4.6) becomes
(h/i) da|m/dt = E0 sin 2ji v0( ^ ) m| exp[i(Em-E,)t/h]
■cEo ( ^ m |( 'i/2)(exPli(Em-El+hvo)t/hI‘ expri(Em*Er hv,o)t/hl}
{4>8)
Direct integration of this equation gives
aim= E0(px)m|(-i/2){[exp[i(Em-El+hvo)t^]-1V(Em-El+hvo)
[exp[i(Em-E,-hv0)tm]-1]/ (Em-E,-hv0).
(4.9)
The quantity |atm|2= a ^ * a ^ is the transition probability for the molecule to go to the state m
from the state I during a short time under the influence of polarized light of frequency v0.
If |Em-E|U v0, the problem is one of resonance absorption. Equation (4.9) was
derived on the basis that the radiation field was monochromatic. In practice this is never the
case; so we assume the frequency covers a narrow band from v, to v2 including |Em-E1|/h.
i
Then by taking the absolute value of the square of Equation (4.9) and integrating from v, to v2,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
is obtained, where the integration variable is ?= (lE^-Ed-hvJt/'h. If the frequency width is
large compared to the natural width of the absorption line, not too much error is introduced by
letting $(v,) and s(v2) go to plus and minus infinity. Integration then gives
(4.11)
where I is the intensity of the radiation. The transition probability is thus proportional to the
intensity of the incident radiation, the time and to the square of the matrix component of the
dipole moment.
This relation has been derived for polarized radiation. For an isotropic electromagnetic
field the relation above becomes
(4.12)
where
M 2 - l<M*U2+ l<M,>n/+ IW n /'
(4.13)
For an isolated symmetric top molecule such as ethane the matrix components of the
dipole moment vanish and thus absorption of electromagnetic radiation is forbidden according
to Equation (4.12). However a colliding pair of molecules may produce a transient dipole
during the collision. If the gas is dense enough these collisions can occur with sufficient
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
37
frequency to produce an induced dipole moment time-correlation function owing, primarily,
to quadrapole interactions between pairs of molecules. At higher densities one must consider
shorter ranging octapole and hexadecapole interactions as well. The dipole moment time
correlation function gives rise to an absorption coeffecient per unit path length in a fluid of
volume V in thermal equilibrium at temperature T given by21
oc[ uj )
= (4 rr2 w/3hcV)(1-exp(-Bfi<Aj))G(t/o),
(4.14)
where w is the angular frequency and B'1=kT. The spectral density, G(w), is given by
G (w ) = (2n)'1 / exp(-i wt)4>(t)dt,
(4.15)
where $(t) is the quantum-mechanical correlation function defined by
$(t)= (N2/2)<fL(0)-p(t)>,
(4.16)
where N is the number of molecules contained in V and p. is the pair moment due to
bimolecular interactions. For the past few decades the study of molecular absorption in the
far-infrared and microwave bands has been a valuable tool in determining molecular
intereactions and structure. As an obvious example, the quadrapole and higher order
multipoles of several molecules, including CH41( C 0 22, and H23 , have been measured from
such data.
To our knowledge only two groups have investigated microwave absorption in
compressed ethane. Dagg et al, using a microwave frequency of 137 GHz, measured slight
absorption that decreased with density in the 20-50 amagat range 23. Birnbaum and Maryott
had earlier reported negligible absorption at 24.35 GHz in ethane at 298.15 K in the range of
0- 38 amagat24 To our knowlwdge no one has investigated the absorption of microwaves by
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
38
ethane in the 90-100 GHz range. However, the collision induced spectrum of methane
exhibits absorption at about 95 GHz.25. We do not have a measure of the relative absorption
of microwaves at 92 and 96.2 GHz as we were not equipped to measure this directly.
Nonetheless, the observed changes in the decay rates over the range 90-100 GHz indicate an
absorption of microwaves at these two frequencies.
Mechanisms for Microwave-Induced Reductions in > , Pr
Rotation ally Excited Molecules
It is possible that the absorbed microwave radiation contributes to the rotational
degrees of freedom of the ethane molecules. Quantum mechanically, the probability of
positronium undergoing pick off annihilation within a molecule's electron cloud is
proportional to the value of the overlap integral defined as
/ l *
where vPnim and
n
J
2 l * n
J
2
<
* -
<4 - 1 7 >
are the wave functions of positronium and the molecule respectively. A
simple model for investigating the effect of rotationally excited molecules on the annihilation
rate of o-Ps assumes the molecule to be hydrogen-like. Evaluation of the overlap integrals of
the ground state of orthopositronium with the hydrogen atom in different I states shows that as
the hydrogen is excited to higher rotational states the value of the overlap integral and hence,
the annihilation rate, decreases. Thus o-Ps is expected to have a longer lifetime in the fields
of rotationally excited molecules.
Assuming that the microwave energy absorbed at 92.0 and 96.2 GHz., as seen in Figure
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
39
4-1, raises the ethane molecule to a rotationally excited state, we obtain an estimate for the
quantum number J and the moment of inertia, I, by using,
v ■ 2B(J+1) -4D(J+1)3,
(4.18)
where v is the frequency of the absorbed radiation, B=h/(8n2 I) is the rotational constant for
the ethane molecule with a moment of inertia I, and D is the centrifugal stretching constant
whose estimated value for ethane is „10® Hz26. Using Equation (4.17) and the two
absorption frequencies from Figure 4-1 we obtain J = 15, and l= 2.4 x10'45 kg m2. The
moment of inertia is in close agreement with that of 1= 0.422 x 10*43 kg m2 reported by
Dagg, Smith, and Read23. There are no data available for microwave absorption in ethane in
the 90-100 GHz range. However, transitions between such high quantum numbers have been
reported for methane 25.
Density Fluctuation Excitations
Measurements of
in ethane, methane, nitrogen, and carbon dioxide as functions of
density and temperature have shown positronium to be localized in density fluctuations. The
density fluctuation model relates the difference between the measured orthopositronium decay
rate and the decay rate predicted by Equation (3-1) to the isothermal compressibility of the
fluid according to Equation (3.20),
Ax0.Ps“4"ro2crV)1zeffD{kTKo/2,tVo)1/2'
where K0=\T^ (dN/dP)x is the isothermal compressibility of the fluid. Positronium decay
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4‘19)
40
rates can therefore provide information about the equilibrium properties of a fluid.
At temperatures close to the liquid-vapor equilibrium point, the isothermal compressibility
of the fluid becomes infinitely large and, therefore, highly succeptable to external
pertubations. In this analaysis an analogy is made between the highly compressable fluid
subjected to microwaves and the classic harmonic oscillator set into motion by an external
driving force. In the latter case the vibration of a system of particles is described by a linear
combination of normal modes with each normal oscillation occurring at the frequency of the
driving force. In the following we consider the possibility that the changes seen in the decay
rates of orthopositronium in ethane subjected to microwaves are a result of the response of
the fluid's compressibility to the incident radiation.
In a simple fluid, the Ornstein-Zernike form of the susceptibility (isothermal
compressibility) is given by
K (q )« C p /(U $ 2 q2 ),
(4.20)
where Cp is the specific heat at constant pressure, $ is the correlation length and q is the
wavenumber of the scattered radiation. The wavenumber is given by
q(0) =(47in/>.) sin(0/2),
(4.21)
where n is the index of refraction, > is the wavelength of the incident radiation, and 0 is the
scattering angle. Relating the positronium decay rates to the wave number dependence of the
compressibility by combining Equations (4.19) and (4.20), we obtain
K (q)« A * 20_Psa Cp/(1+$2 q2 ).
Therefore,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.22)
41
AVPs'2 - ( 0/Cp)(1+*2q2)’
where a is a constant. The measured values of
a v
(4'23)
* , 2 versus frequency at 98 amagat are
plotted in Figures 4-2 and 4-3. The curves are obtained by a computer program that fits the
data to a Lorentzian peak. These data refelect sharp increases in the fluid’s compressibility at
92.0 and 96.2 GHz.
Following Equation (4.23), A x ^ '2 is plotted against q2 at 98 amagat in Figures 4-4
and 4-5. An average value of q =4nA , obtained by integrating q(6) over all possible
scattering angles, was assumed for these calculations. Also the index of refraction, n, was
assumed to be 1. These data are clearly in agreement with Equation (4.23). The straight line
is obtained from a linear least squares fit of the data. From the absolute values of the slopes
and intercepts of these data a correlation length $=(m/b)1/2 is obtained; m and b are the
slope and intercepts of the straight lines given by Equation (4.23). From the four straight
line segments in Figures 4-4 and 4-5 we obtain an average value of ?=(2.5 ± 0.1) x106 A .
This correlation length is considerably larger than the expected value of a few Angstroms for
correlation lengths measured in fluids such as CO2 in light scattering experiments27. The
reason for this discrepency is not immediately clear.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
98 amagat
3.5
or
■
o
©
V)
2.5
«
a.
©I o
/C
<
90.9
91.9
92.9
Frequency (GHz)
Figure 4-2. & \ 20+ s versus frequency in ethane at 98 amagat and 307.15 K for the peak
centered at 92.0 GHz. The solid curve is hand drawn to guide the eye.
■rw
ro
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3.5
-
2.5
-
98 amagat
cT•
o
s
n
■o
~
M
a
■
CM O
/<
<
1.5
95.95
95.55
96.35
Frequency (G Hz)
Figure 4-3.
versus frequency in ethane at 98 amagat and 307.15 K for the peak
centered at 96.2 GHz. The solid curve is hand drawn to guide the eye.
CO
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
98 amagat
«
o
®
n
e
co
o
T
«
a
CM
1.47
1.45
1.49
q2 (10 3cm * 2
Figure 4-4.
a*
1.51
)
-2^ , versus q2 in ethane at 98 amagat and 307.15 K for the peak centered at
92.0 GHz. The solid lines represent the results of a weighted least-squares fit.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CM
O
©
co
c
M
O
CO
a
Vc
<
98 amagat
1.61
1.62
1.63
1.64
q 2 (10 3cm * 2)
Figure 4-5.
versus q2 in ethane at 98 amagat and 307.15 K for the peak centered at
96.2 GHz. The solid lines represent the results of a weighted least-squares fit.
cn
46
Orthopositronium Annihilation Rate Behavior at 99.60. and 37 Amaaat
In the following, data similar to that shown for 98 amagat are presented for ethane
densities of 99,60, and 37 amagat at 307.15 K. For each of these densities, although not as
clearly, the annihilation rates seemed to exhibit "dips" at certain density-dependent values of
the microwave frequency.
60 Amaaat
Figure 4-6 shows V p s as a function of frequency at 60 amagat and 307.15 K. The data
were taken in order of increasing microwave frequency over a period of two days. A dip in the
decay rates is suggested at 94.0 GHz. Figure 4-7 represents data taken over the following
week in an attempt to duplicate the data in Fig 4-6. Once again a characteristic dip in the
annihilation rate is suggested, but the dip at 94.0 GHz in Figure 4-6 has shifted to 93.5 GHz
in Figure 4-7. These data were collected with the microwave generator in the continuous
wave (CW) mode where the RF power is essentially monochromatic having a small band width
of only ±0.00038% of the selected frequency. The results using microwaves in the CF mode,
at 98 amagat, seem to give better resolution of the dips than when operated in the CW mode at
60 amagat. This might be attributable to the fact that the transition probability is
proportional to the integral in Equation (4.10) which decreases with decreasing band width.
Figure 4-8 shows
versus frequency at 60 amagat for the five frequencies
centered around and including 94.0 GHz. The curve is a computer fit of the data to a
Lorentzian.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.034
60 amagat
0.032
.o
©
0)
c
m
a
0.030
0.028
i
89
91
93
95
97
99
Frequency (GHz)
Figure 4-6.
versus frequency in ethane at 60 amagat and 307.15 K. The solid curve is
hand drawn to guide the eye. The dotted horizontal line represents the decay rate with no
microwaves.
101
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.034 -
60 amagat
o
®
»
c
n
a
0.030
_L
0.026
88
90
92
94
96
98
100
Frequency
Figure 4-7.
versus frequency in ethane at 60 amagat and 307.15 K. The solid curve is
hand drawn to guide the eye. These data were collected in the week following the data in Figure.
4-6. The dotted horizontal line represents the decay rate with no microwaves.
03
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.0
60 amagat
0.8
-
0.6
_
0.4
_
CM
i
O
©
CO
in c
■o
n
Q.
CM O
0.2 _
0.0
93
92
94
95
96
Frequency (GHz)
Figure 4-8.
a * 20-p*
versus frequency in ethane at 60 amagat and 307.15 K for the peak
centered at 94.0 GHz shown in Figure 4-6. The solid curve is hand drawn to guide the eye.
-U
CO
In Figure 4-9 A V p / 2 is plotted against q2 for the data shown in Figure 4-8. The straight
lines are linear least square fits to the data. From the slopes and intercepts of the two lines
an average value of the correlation length $=(2.5 ± 0.05) x106 A is obtained. Once again
this is much larger than expected, but it is in excellent agreement with the correlation length
obtained from the 98 amagat data.
37 Amaoat
The frequency dependance of V p s for 37 amagat at 307.15 K exhibited dips at three
frequencies; 93.0,95.5, and 99.0 GHz as shown in Figure 4-10. The horizontal dashed line
in the figure represents the measured decay rate of 25.70 ±0.86 ps'1 with no microwaves.
This is close to the value of 26.1 ps*1 predicted by the linear Equation (3-1) which
indicates the density to be near the point, below which localization of o-Ps will not occur.
Figures 4-11 and 4-12 show A \0_ps2 versus frequency for the dips centered at 93.0
and 95.5 GHz. The solid curves are Lorentzians fitted to the data. The dip at 99.0 GHz did not
provide a good fit to a Lorentzian. When subjected to the same analysis as for the 98,60, and
99 amagat the data for 37 amagat gave poor results with regard to showing the dips to be
Lorentzian and in determining correlation lengths.
99 Amaaat
Figure 4-13 shows V p s versus frequency at 99 amagat and 307.15 K. We consider the
data at this density as distinct from those at 98 amagat because of the differences in pressure
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.0
60 amagat
CM
O
3.0
©
«
e
o
0)
CL
CM o
’/<
2.0
<
A
1.0
1.50
1.52
A
1.54
A
1.56
1.58
1.60
q 2 (10 3cm ‘ 2 )
Figure 4-9.
a * " 2o-p *
versus q2 in ethane at 60 amagat and 307.15 K for the peak centered at
94.0 GHz shown in Figure 4-6. The solid lines represent the results of a weighted leastsquares fit.
cn
52
readings at the times each experiment was run. The pressure reading taken at the
time the vessel was sealed prior to collecting the data for 98 amagat was 695.0 PSIG. After
collecting the data ethane was removed from the vessel for a series of runs made at a lower
density. After this, ethane was readmitted to the vessel in an attempt to get as close as possible
to 98 amagat again. The resulting pressure turned out to be 696.5 PSIG. Within the precision
of the Heisse Gauge, there is signifigant difference between the two pressures measured and
therefore, the densities calculated are distinct.
The data in Figure 4-13 shows a "closeup" of a dip in the decay rate centered at 96.4
GHz. Since 99 amagat is very close to 98 amagat one would expect that this is the same dip as
the one exhibited at 96.2 GHz in the 98 amagat data. Assuming this to be the case, the shift to
the higher frequency reflects a decrease with the density either in the molecule’s moment of
inertia or in its distortion constant according to Equation (4.4).
In Figure 4-14 A>0_ps2 is plotted against the frequency. The curve is a Lorentzian fit
to the data.
Figure 4-15 shows A * ^ ' 2 for this density. From the slopes and intercepts of the
straight line segments a correlation length, £=(2.48 ± 0.02) x106 A, is obtained. Which is,
once again, in excellent agreement with those obtained at 98 and 60 amagat, but is much
larger than expected.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
37 amagat
0.028
.o
©
n
c
«
CL
0.024
0.020
89
91
93
95
97
99
101
Frequency (GHz)
Figure 4-10.
versus frequency in ethane at 37 amagat and 307.15 K. The solid curve is
hand drawn to guide the eye. The dotted horizontal line represents the decay rate with no
microwaves.
cn
CO
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
37 amagat
CM O
I
94.5
Frequency (GHz)
Figure 4-11. A>>20^ , versus frequency in ethane at 37 amagat and 307.15 K for the peak
centered at 93.0 GHz shown in Figure 4-10. These data provide a poor lorentzian fit.
C7I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
25 -
37 amagat
CM
•o
ffl
0)
c
15 -
o>
a
■
CM O
5
-
4
-5
93
94
95
96
Frequency (GHz)
Figure 4-12.
a x 20_j>,
versus frequency in ethane at 37 amagat and 307.15 K for the peak
centered at 95.5 GHz shown in Figure 4-10. These data provide a poor lorentzian fit.
Ol
cn
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.044
■u
®
V)
c
«
Q.
0.042
99 amagat
0.040
95.9
96.1
96.3
96.5
96.7
96.9
Frequency (GHz)
Figure 4-13. >>0 ^ , versus frequency in ethane at 99 amagat and 307.15 K. The solid curve is
hand drawn to guide the eye. The dotted horizontal line represents the decay rate with no
microwaves.
Ol
o>
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3.3
99 amagat
e*
'o
2.8
©
0)
C
i
o
<i
0)
Q.
•
<M O
2.3
X
1.8
95.8
9G.0
X
X
96.2
96.4
X
96.6
96.8
97.0
Frequency (GHz.)
Figure 4-14.
a >,20_j>s
versus frequency in ethane at 99 amagat and 307.15 K for the data shown
in Figure 4-10. The solid cun/e is hand drawn to guide the eye.
tn
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.0
do
©
CO
c
o
o
CO
CL
4.0
V
99 amagat
3.0
i
1.61
1 . 62
1.63
1.64
1.65
q 2 (10 3cm ' 2 )
Figure 4-15.
a >t 2o_p,
versus q2 in ethane at 99 amagat and 307.15 K for the data presented in
Figure 4-14. The solid lines represent the results of a weighted least-squares fit.
cn
CO
CHAPTER V
CONCLUSIONS
The experiment performed for this thesis provided for observations of the changes in
the decay rates of o-Ps in fluid ethane subjected to microwave radiation. The sharp dips seen
in the decay rates at certain frequencies were totally new and unexpected. These resonant-like
dips indicate some absorption of the microwaves incident upon the ethane molecules. This
absorption is described in terms of transient dipoles induced through quadrapole-quadrapole
interactions during collisions between molecules.
Two possible mechanisms for the decrease in
resulting from the absorption of
microwave energy are proposed. The first describes the decrease in V p s as being due to the
interaction between the localized positronium and rotationally excited molecules, i. e. The
overlap integral between the o-Ps atom and a molecule in the state J=1 is smaller than when
the molecule is in the state J+1. Applying this idea to the data for ethane at 98 amagat and
307.4 K show that the transitions in the 90 to 100 GHz range occur in the neighborhood of
J=14. Also the moment of inertia for the ethane molecule was calculated to be 1= 2.4 x10‘45
kg m2, in close agreement with published values.
The second mechanism proposed describes the frequency dependent dips in >0_ps in terms
of density fluctuation excitations which involve the response of the fluid's compressibility to
the microwave field at some resonant frequency. The dips in the annihilation rates thus
59
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
60
reflect sharp increases in the compressibility at certain microwave frequencies. The peaks
observed in the 98,60, and 99 amagat were shown to be Lorentzians centered about the
resonant frequency. The fit for the peaks at 37 amagat was not as good. In this analysis the
density fluctuation model for localization of o-Ps in ethane relates the incident radiation to the
Ornstein-Zernike form of the isothermal compressibility of the fluid. This relation predicts
that, near the resonant frequency, a v p ,'2 is proportional to the the square of the frequency.
The data for all densities investigated, except at 37 amagat, did yield this result. Furthermore
the slopes of the straight lines from the data gave correlation lengths which were all in the
range of 2.45 to 2.55 x 106 A. The consistency of the calculated correlation lengths is
satisfying but we have yet to reconcile the discrepency between the value of around a million
Angstroms measured in this experiment and published values of a few Angstroms in other
non-polar gases.
The data obtained at densities other than 98 amagat suggest problems in controlling the
density of the gas. It is hoped that increased accuracy in the the control of the pressure,
temperature and hence, the density, will provide future researchers with more convincing
evidence for microwave induced density fluctuations. If such is the case Ps may prove to to be
an effective probe of the molecular dynamics of fluids.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
REFERENCES
1. K. F. Canter, J. D. McNutt, and L. 0 . Roellig, Phys. Rev. A 12,375 (1975)
2. E. M. Juengerman, R. H. Arganbright, M. H. Ward, and S. C. Sharma,
J. Phys. B: At. Mol. Phys. 2Q, 867 (1987)
3. E. M. Juengerman, Masters Thesis, Univ. of Texas at Arlington, 1982 (Unpublished)
4. S. C. Sharma, A. Eftekhari, and J. D. McNutt, Phys. Rev. Lett. 45, 953 (1982)
5. R. J. Drachman and S. K. Houston, J. Phys. B: At. Mol. Phys. 2.1657 (1970)
6. E. Bruche, Ann. Phys. 4 , 387 (1930)
7. J. D. McNutt, S. C. Sharma, M. H. Franklin, and M. A. Woodall II, Phys. Rev. A 2<2.357
(1979)
8. L. L. Tankersley, J. Low Temp. Phys. H , 451 (1973)
9. P. Hautojarvi, K. Rytsola, P. Tuovinen, and P. Jauho, Phys. Lett. 57A. 175 (1976)
10. J.P. Hernandez, Phys. Rev. A 14,1579 (1976)
11. I. T. lakubov and A. G. Khrapak, Rep. Prog. Phys. 4 5 , 697 (1982)
12. S. C. Sharma, A. Eftekhari, J. D. McNutt, and R. A. Hejl, J. Chem. Phys. Z 5 , 1226
(1981)
14. J. D. McNutt and S. C. Sharma, J. Chem. Phys. 68.130 (1978)
15. T. Kawatarani, Y. Nakayama, andT. Mizogawa, Phys. Lett. A 122,75 (1985)
16. G. L. Wright, M. Charlton, G. Clark, T. C. Griffith, and G. R. Heyland, J. Phys. B: At.
Mol. Phys. 12,4065 (1983)
17. P. Kirkegaard and M. Eldrup, Comput. Phys. Commun.2,401 (1974)
18. M. Benedict, G. B. Webb, and L. C. Rubin, Chem. Eng. Prog. U47U, 419 (1951)
19. D. A. Gyrog and E. F. Ohert, AlchE J. IQ, 621 (1964)
20. A. Eftekhari, Ph.D Dissertation, Univ. of Texas at Arlington, 1982 (Unpublished)
61
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
62
21. G. Birnbaum. Intermolecular Spectroscopy and Dynamical Proprties of Dense Systems.
Proceddinas of the International School of Phvsics. Enrico Fermi. (1978)
22. G. Birnbaum, J. Chem. Phys. £2,59 (1975)
23. I. R. Dagg, W. Smith, and L. A. A. Read, Can. J. Phys. £Q, 16 (1982)
24. A. A. Maryott and G. Birnbaum, J. Chem. Phys. 3£, 2026 (1962)
25. I. Ozler, M. C. L. Gerry, and A. G. Roblette, J. Phys. Chem. Ref. Data IQ, 1085 (1981)
26. C. H. Townes and A. L. Schawlow, Microwave Spectroscopy. McGraw-Hill. New York,
(1959), p. 78
27. H. L. Swinney and D. L Henry, Phys. Rev. A £ , 2586 (1973)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Документ
Категория
Без категории
Просмотров
0
Размер файла
1 947 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа