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.TMEQIS
LIBRARY
Michigan State
”ﬂiv."ity‘-l
This is to certify that the
dissertation entitled
I E} EZTLICIAJJ TE? ULVJL. 1'le 33:31?” 1‘ JJELOW 14:
OK COLD ”CPS TED POTACSSIJC
presented by
Shi Yin
has been accepted towards fulﬁllment
of the requirements for
Ph .D . degree in Phys 33.0 3
0%wa
(x/ /V~ ’Mrajor professor
7 / U / 92
/ /
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ELECTRICAL TRANSPORT MEASUREMENTS BELOW 1K
ON COLD WORKED POTASSIUM
BY
Shi Yin
A DISSERTATION
Submitted to
Michigpn State University
in partial fullfillnent of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy
1987
ABSTRACT
ELECTRICAL TRAl‘BPORT MEASURBIENTS am 1K
on COLD BURIED POTASSIUM
Shi Yin
The electron-dislocation interaction has been studied with an
emphasis on resistivity for temperatures below 1K. Dislocations are
introduced into the ample by busting using a device driven by
pressurized helium gas, and meamments of the temperature derivative
of the resistivity. dp/d‘l‘. are made down to ZOmK. Zm-diameter samples
are used to avoid the complication due to a size effect when the sample
diameter is of the order of the electron mean free path. For deformed
K(Rb) samples or deformed K samples with vacancies. a vibrating
dislocation mechanism is observed: for deformed K samples in which the
vacancies are annealed out at 60K, a new betavior in dp/d'r is observed
which can be fit by a localized electronic-energy-level model together
with a residual contribution from the vibrating dislocations. A
possible Charge Density wave contribution is also discussed. A
comparison with previous experimental work is made. Thermoelectric
ratio measurements are also reported and discussed.
TO MY WIFE
iv
ACKNOWLEDGMENTS
I wish to thank Professor William P. Pratt Jr. for his supervision
and invaluable aid in all aspects of this dissertation. I am
particularly grateful for the may late nights worked while taking
data.
I extend a thanks to Professor Peter A. Schroeder for many late
nights worked together and new helpful advices.
I also extend a thanks to Mr. Fred Freiheit for his help in
designing the sample can.
I acknowledge the finacial support of the National Science
Foundation and Michigan State University.
TABLE OF CONTENTS
‘ Page
LIST OF TABLES - - i ------------ » ————————————— viii
LIST OF FIGURES ---------------------------- 1x
INTRODUCTION - - - - - ------------------------- 1
Chapter I
I Basic Electrical Transport Theory ------------------- 3
1.1 Resistivity - - - - w - n n u u n — ~ — — ~ « - n - ~ « « ~3
1.1.1 The residual resistivity due to dislocations - - - - 5
1.1.2 Electron-phonon scattering -------------- 11
1.1.3 Electroneelectron scattering --------------- 16
1.1.4 Electron-dislocation scattering ------------ 19
1.1.5 Electron-phason scattering and Charge Density
waves (CDU) --------------------------- 23
1.2 Thermoelectric power ------------- - ----------- 26
1.2.1 Diffusion thermopower ------------- . ------ 26
1.2.2 Phonon drag thermopower ---------------- 28
1.2.3 Thermoelectric ratio ------------------ 29
II Experimental Techniques ------------------ ~ ~32
2.1 main equipment for measurements ----------------- 32
2.1.1 The dilution refrigerator ---------- - ~~~~~~~ 32
2.1.2 The high precision resistance bridge ---------- 33
2.1.3 The screened room and floating pad - - - — ~ — - ~ - ~33
v1
2.1.4 A schematic diagram of the ultralow temperature
part of the cryostat ---------------------- 33
2.2 The thermometry ------------------------ 35
2.3 Sample can ----------- - ---------------- 40
.2.4 Plastic deformation of the sample -------------- 45
2.5 Sample preparation --------- - ------------ 46
2.6 measurements ------------------------------ 53
2.7 Heat loss ----------------- - ------ ~ ------- 60
2.8 Heat generated while deforming the sample - - - - u f — ~ ~63
2.9 Uncertainties ------------------ - ---------- 65
III Experimental Results ------------ - ----------- 67
3.1 Resistivity - - - - w ~ ~ ~ ~ ~ ~ ~ ~ - ~ ~ - ~ ~ - ~ - - ~67
3.1.1 Residual resistivity -------- ~ ~ — - ————— ~ 67
3.1.2 K(Rb) data ----------- - ---------------- 70
3.1.3 Pure K data without annealing - - ~ ~ - — ~ ~ ~ w w -75
3.1.4- Pure K data after annealing ----------- ~ - - ~64
3.1.5 Annealing at 60K after deforming at 9.3K vs
deforming directly at 60! for pure K.samples - — - - 92
3.1.6 Comparison of 60K annealed pure
K sample with those of Harele 2§'_;. -------- 92
3.2 Thermoelectric Ratio ----------- ‘ ----------- 95
IV Discussion and Conclusions --------------- . ------- 103
LISTOFREFERENCES----~-----~------------------'-------109
vii
LIST OF TABLES
Table . Page
2.1 A list of parts in sample can - -. ------------- 42
2.2 Chemical amlysis of potassium -------------- 51
3.1 K (0.087 atx Rb) alloy sample --------------- 72
3.2 Pure K sample without annealing ------- '- ------ 79
3.3 Pure K sample with A as a variable parameter ------- 88
3.4 Pure K samples after annealing or deformation at 60K - -- - 90
3.5 Parameters in G for the K(Rb) and K samples -------- 98
viii
LIST OF FIGURES
Figures Page
1.1 An edge dislocation (a) and a screw dislocation (b) ----- 6
1.2 Screw dislocations
Two neighbouring planes in a simple cubic lattice rotated
slightly with respect to each other. A regular pattern
of screw dislocations is visible. The (001) direction
is perpendicular to the page --------------- 7
1.3 The phonon processes -------------------- 12
2.1 The low temperature part of the cryostat ---------- 34
2.2 The CMN thermometer - - - - - - - - - - - e -------- 37
2.3 Circuits of the CNN mutual inductance bridges ------- 39
2.4 Sample can ------------------------- 41
2.5 A perspective view of the sample holder ---------- 44
2.6 A perspective drawing of the twister mechanism ------ 46
2.7 The He pressurization system for the bellows -------- 49
2.8 The press and die for extruding the sample- ------- -52
2.9 The low temperature circuit ---------------- 55
2.10 The Ag sample mount
The Ag sample mount on which the thermometers etc.
are mounted ------------------------ 51
3.1 ‘Ap. vs 8
hp. is plotted as a function of the angle of twist for
sample K+5(open circle). The full circles are the data
from van Vucht 33 EA. ------------------- 69
3.2 qp/dT vs T for K(Rb) samples
dp/dT is plotted as a function of T for the K(Rb)
samples in a series of twists. The details are given
3.3
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
in Table 3.1 ------------------------ 71
dp/dT - (2AT-C') vs T for K(Rb) samples
dp/dT - (ZAT-C') is plotted as a function of T for the
K(Rb) samples. A step-like function is seen for samples
KRb-4 and KRb-S ---------------------- 74
(pu/p)dp/dT vs T for pure K samples
The samples are twisted at 9.3K without annealing. The
details of twist are given in Table 3.2. --------- 77
(Mp )dp/dT - 2A'T vs T for the unannealed K samples
A step-like function is seen. The solid and dashed
curves are the best computer fits. Details are given in
the text ------------------------- 78
A two-frequency fit vs the one-frequencey for sample K95
An improved fit is obtained ---------------- 81
D vs ap. the KCRb) and K samples
The coefficient D of Eq.(3.3) or Eq.(3.6) is plotted
as a function of 5p, for the K( Rb) samples or pure K
samples without annealing. The data from Haerle 25 21.
are also shown. The dashed line is the fit to the K
samples after the vacancy contribution is corrected
for, see text for details ----------------- 82
E vs ap. for K(Rb) and K samples
The characteristic energy E is plotted as a function of
.p. for the K(Rb) and K samples. The data from Haerle
35 31. are in agreement with ours. ------------- 85
Ap)dp/dT - 2A'T vs T for the annealed pure K samples
A ak is seen at 0.2K. The dashed curve is the best fit
for the unannealed sample K-5. The solid curves are
explained in the text ------------------- 87
The improved fit to K97 with A as a variable parameter
The dashed curve is without the CT’ term. A CT’ term helps
to fit the rising tail of the data. ------------- 89
A comparison between annealing at 60K after twisting
at 9.3K (K97) and twisting directly at 60K (K2-2)
A similar behavior is seen. The data of Haerle gt_§l. are
also presented ----------------------- 93
G vs T for the KCRb) samples
Details of the fit are given in the text ---------- 96
G vs T for the pure K samples --------------- 97
3.14 The Gorter-Nordheim plot for both K(Rb) and K samples
34 seems to be the same in both cases ----------- 130
3.15 b vs so. for both K(Rb) and K samples
A systematic change is visible- -------------- 102
xi
INTRODUCTION
Potassium has long been a material of considerable theoretical and
experimental interest. Most people believe that potassium has the
simplest electronic structure in comparison with non-alkali metals. It
has a nearly spherical Fermi surface which is entirely contained with
in the first Brillouin zone, and it has no unfilled d- or fb shells.
Thus the nearly free electron model is a good approximation for many
calculations, making transport theory comparatively easy to carry out.
Furthermore, in contrast to potassium, lithium, sodium. and possibly
even rubidium(l) undergo Martensitic phase transformations at low
temperatures which complicate the. interpretation of measurements.
Potassium is also a good testing ground for the existence of Charge
Density Waves (CDW), which are a broken translational symmetry of the
ground state. Recently, two new experiments have addressed this CD!
hypothesis: the observation of CDU satellites in potassium by
Giebultowicz gt 21.(2) in their neutron diffraction experiment and the
failure in detecting such satellites by H. Yen gt, 2;. (3) in their
synchrotron X-ray diffraction experiment. If found to exist, CDW’would
radically change our basic understanding of transport in such a simple
metal.
In this study we are going to report the effect of
electron-dislocation interaction on the transport properties of
potassium, where the dislocations are produced through deformation of
polycrystalline potassium samples. Since deformation of K might also
change the Q-domain textures of CDw's in potassium (4), the effect of
CDVI's will also be discussed.
This study was begun by Mark I... Haerle _e_t 1149(6). and the
results reported here are a continuation of their work with three
significant improvements: Firstly, we introduce the dislocations into
our sample by twisting with a device driven by pressurized helium gas,
while Pherle g; 91. did it by squashing the sample between two plates.
In our method of deformation the sample geometry change MAIL), where
A is the sample cross sectional area and I. is the sample length, is
much better controlled «1%) than that or rserle‘s (20 mK) than that of Haerle 95 31.
(T>80 mK). This new region of temperature is crucial for determining
the low-energy vibration spectrum associated with the dislocations.
Finally, we used 2mm diameter samples to avoid a possible size effect
which made the amlysis of Haerle's 0.9mm diameter samples more
complicated.
Chapter I Basic Electrical Transport Theory
The fundamental basis for electrical transport study is the
following set of equations
J - LuE + an'r (1.1)
Q " Lug + Lair (1.2)
where J is the electric current density, Q is the heat flow current
density, E is the electric field, and vT is the temperature gradient.
The Ir.)- coefficients are tensors in general, but they can be reduced to
scalars here because potassium ins cubic symetry.
In this work we mainly measure the electrical resistivity p and
the thermal power S. They are defined as ,follows
4
p a - (JIE)LT;L.. (1.3)
s - (ENNIS: -I.,,/l.,, (1.4)
1.1 Resistivity
Theory predicts that the resistivity for Bloch electrons in a
perfect lattice is zero, and the finite resistance in a real metal
comes from the interactions of the conduction electrons with the
imperfections of the lattice such as impurities, dislocations, lattice
vibrations, etc. If Matthiessen's rule is obeyed, one can write down
the resistivity of potassium as a sum of terms
p-p.+ap+p“+p“ (1.5)
where p. is the residual resistivity, which is independent of
temperature and is due to scattering by various static imperfections in
the crystal. The second term 3? is usually a function of temperature: a,
- p.P(T). It is due to electrons scattering off phonons. The third term
p“ is the electron-electron scattering term, and it is also
temperature dependent: pu- p.e(T). This term is usually 10* times
smaller ttmn the residml resistivity p. in pure potassium at 1K.
The last term can be expressed as a sum of terms for other
possible contributions to p. The resistivity due to electrons
scattering off phasons, p 01* which are the elementary excitation of
CDWs, is one of these possible terms. Because of the recent observation
of CD“ satellites in potassium by Giebultowicz .e_t_ g1.(2), a
phason-scattering term in p can not be ruled out.
Another possible term is p“ , which is the resistivity due to
interactions between electrons and dislocations. Normally one would
expect pd to be temperature independent and thus only contribute to F3.“
However, the work of Haerle has shown that there is also a temperature
dependent term which is of interest in this thesis. A dislocation is an
extended line-defect in a crystal. There are basically two kinds of
dislocations: the edge dislocation and screw dislocation. For an edge
dislocation the Burgers vector b is perpendicular to the dislocation
line, and for a screw dislocation b is parallel to the dislocation
line(Fig. 1.1).
In our experiment we introduce the dislocations into our sample by
twisting. Ideally, torsional deformation along (001) direction in a
cubic symmetry crystal will produce networks of equally spaced screw
dislocations in two perpendicular directions (Fig. 1.2). However, fer
our polycrystalline samples the sample axis is not parallel to any
particular symmetry direction, and there may be imperfections like
vacancies or other dislocations already present in the sample before we.
deform it. we therefore expect to produce a very irregular dislocation
structure during deformation. As we will see later, the temperature at
which we deform the sample and the possible subsequent annealing will
play an important role in affecting the detailed structure of these
imperfections.
1.1.1 The residual resistivity due to dislocations
There are several theories that explain the origin of the
residual resistivity. Of particular interest here are those which
address electron scattering by dislocations.
Using a geometrical model of obstacles at which the conduction
electron energy dissipates, Tsivinskii (7) calculated classically the
residual resistivity due to impurities, vacancies and dislocations.
- «ﬁshnet»,
(M
(a)
An edge dislocation (a) and a screw dislocation (b).
Figure 1.1
Figure 1.2
Screw dislocations
Two neighbouring planes in a simple cubic lattice rotated slightly with
respect to each other. A regular pattern of screw dislocations is
visible. The ( 001) direction is perpendicular to the page.
From this classical theory the resistivity is given by
p - (ZMVIe‘n.)(1/l) (1.6)
where e and m are electron charge and mass, respectively, n. is the
free electron concentration per unit volume, v is the Fermi velocity,
and a is the mean free nth.
For a dislocation contribution to p, Tsivinskii armed that the
mean free path 1 can be replaced by the following formula:
A ' 1/bN (1.7)
where N is the dislocation density(#lcm2) and b is approximately equal
to the magnitude of the Burgers vector. In this way, we obtain
p - 2mvbN/e2n,- UN . (1.8)
For potassium, Eq. (1.8) predicts tint
-‘9 3
W - 1.6x10 (11cm )e (1.9)
This value of w is- almost a factor of 2 smaller than the experimental
value obtained by Basinski 333 51. (a).
For the contribution of impurities or vacancies Tsivinskii uses
A - 1/[1rn(xz-x2;)] (1.10)
in which n is the atomic concentration per unit volume of the metal,
and x and x, are ionic and impurity radii, respectively. For vacancies
x,-O.
Another theory was proposed by Basinski 'et al.(8) in which they
calculated the mean atomic displacement from equilibrium due to the
strain field around a dislocation. They asked what temperature is
needed to give the same mean displacement via random thermal motion.
Then they assumed that the increase in p due to the dislocation is the
same as the resistivity due to the electron-phonon interaction at this
temperature. They derived VI - «lo-”ncﬁ’ in formula (1.8) for
potassium, which is in good agreement with the experiment they
performed.
In,a third theory, Brown(9) calculated the resistivity due to
resonant s-wave scattering of the electrons. The resonance is created
by virtual bound states which are associated with the dislocation
cores. He predicted that
~10
where for potassium w - 8x10 11cm3 which is a factor of 2 larger than
the experimental value from Basinski(8) ggﬂg1. Brown argued against a
correction factor used by Basinski in obtaining their experiment result
which, if omitted, would bring Brown's theory in closer agreement with
lO
experiment.
Gurney and Gugan(10) performed an early study of the effects on
the residual resistivityof annealing a wire of deformed potassium. In
their experiment the residual resistivity was measured first, and then
the sample was plastically deformed at 4.2K. Again p. was measured.
Next the sample was amealed at a hiduer temperature and then slowly
cooled down to 4.2K where po was measured. They kept annealing and
measuring oo in stages until the initial value of pa was recovered. The
Gurney and Gugan interpretations, which are significant to this thesis,
are as follows:
1. Between 3 and 7K, the relatively few interstitials formed '
during deformation ameal out, causing about 5% resistivity decrease.
2. Between 10 and 20K, about 405 of the extra resistivity caused
by deformation disappears. This is ascribed to the long-range mimtion
of monovacancies. There is evidence that the dominant processes at the
end of this stage involve the annihilation of defects at dislocation
sinks.
3. Between 20 and 80K, there is a region with no significant
peaks in the recovery rate, but over this temperature range about 30%
of the deformation-prochced resistivity disappears. Part of this is
attributed to the detrapping of point defects, probably vacancies, from
impurities.
4. Between 80 and 150K, the recovery rate shows a strong peak at
about 110K and accounts for 25% of the total recovery. This is
associated with the annealing of dislocations during recrystallization.
11
5. Between 5 and 20K, a major length recovery occurs considerably
before the resistivity recovery begins. Gugan(11) ascribes this to a
rearrangement of dislocations rather than a decline in the number of
dislocations.
These temperature ranges are rough guesses based on the breaking
points in the complicated annealing curves of residual resistivity vs
the annealing temperature, and some of the proposed mechanisms which
explain the various recovery stages are unsubstantiated. Nonetheless,
this study does provide a basis on which the effects of dislocations on
the resistivity may be examined.
1.1.2 Electron-phonon scattering
In calculating the electron-phonon interaction, the following
condition must be satisfied in order to obtain a non—vanishing matrix
element:
71“ -E-:H+E (1.12)
where K and K' are the wave vectors for the incoming and scattered
electrons, respectively,'3' is the phonon wave vector, and ‘5 is the
reciprocal lattice vector. The plus or minus sign before q represents
the phonon absorption or emission processes. If Etc, the process is
called Umklapp: and if'ELo, it is the normal process.
The above phonon processes are illustrated in Fig. 1.3, where
12
2' ) Unululopp procggg
2 ) normal process
3) phonon absorption phonon omission
Figure 1.3
The phonon processes
13
potassium is assumed to have a nearly spherical Fermi surface which
does not touch the first Brillouin zone. This fimre illustrates
1) The Umklapp process. When the magnitude of the emitted or
absorbed phonon wave vector q exceeds the minimum value — q“, this
process can occur. The characteristic feature of this process is that
even when the value of q is small there can be a large difference
between the directions of E and K'.
2) The normal process. The crystal momentum is conserved here.
3) The phonon absorption or emission.
By assuming an equilibrium phonon distribution, a relazntion time
approximation for the Boltzmann equation, no Umklapp scattering, and a
Debye spectrum for phonons, Bloch(12) predicted that the electrical
resistivity die to electron-phonon scattering has the following
temperature dependences for the indicated limits:
p (T) as T , (1.13)
for r > 0.59,, and
p (1') - 1'5 . (1.14)
for T < 0.1 8. ,where 80 is the Debye temperature.
Since 8, for potassium is about 100K, we therefore expect to have
in our experiment below 1K
14
Pep- 6T5! - (1.15)
where C is a constant. However, measurements done by Gugan(13)
(T>1.2K), min and Maxfield(14) (T>1.5K) and van Kempen g; a1.(15)
(T>1.1K) did not show this T5 term. Instead their results could be fit
to an equation of the form
QPU') - ﬁrm-671') (1.16)
with n~1 and 8'~ 20K. When T is less than or equal to 1K, this
eaqaonential term is negligible: and the data from fherle 31: a1.(6)
(T<1K) also failed to show the 1'5 behavior in their unstrainod
potassium samples.
The main reason for the failure of this model is attributed to the
presence of phonon drag (13). Basically phonon drag arises when the
phonon distribution is disturbed from its equilibrium state at low
temperatures. According to Danino, Kaveh and Wiser(16), the electrical
resistivity is caused by the electron system transfering its excess
momentum, gained from the electric field, to the phonon system via
electron-phonon scattering. However, at low temperatures, much of this
excess momentum is not dissipated by the phonons (not lost to the
lattice), but is returned to the electron system via phonon-electron
scattering. The electrons are thereby "dragged" along by the phonons
(phonon drag), so that there are less effective electron-phonon
scatterings, and therefore the electrons experience a much reduced
15
resistivity. In potassium at low temperatures, such a phonon drag
process can largely eliminate the normal scattering contribution (CTS)
to the resistivity(17): and at high temperatures this effect is
quenched because phononephonon Umklapp scattering equilibrates the
phonons with the lattice.
The exponential beraviour seen experimentally in p¢P(T) can be
understood in terms of electron-phonon umklapp scattering, which is the
scattering between two Brillouin zones. For potassium, this scattering
requires phonons with at least the minimum momentum (q,) to Jump the
gap between the two cells. Phonons obey Bose-Einstein statistics, and
at low temperatures the density of phonons which can.participate in an '
Umklapp process goes like
exp(-th‘lkT) (1.17)
where v is the velocity of the phonons. This factor dominates the
electron-phonon resistivity seen experimentally: and at temperatures
below 1K; which is the temperature range of present interest, this term
is negligible.
There are mechanisms other than phonon-phonon Umklapp scattering
that can pull the phonons into equilibrium. Fer example, a high
concentration of impurities or dislocations(18) could interact with the
phonons to provide a way for the phonons to lose the momentum given to
them by the electrons, resulting in an equilibrium distribution.
Danino, Kaveh and Wiser(19) pointed out that the quenching of
16
phonon drag by phonon-dislocation scattering is likely to be negligible
for potassium above 2K unless Nd >10” cur).2 ,where N; is the dislocation
density. For dislocation density Nd ~ 109 on“, they and independently
Engquist(20) proposed a new electron-dislocation interaction mechanism,
which is based on the anisotropy of electron—dislocation scattering, to
explain the suppression of phonon drag in the electrical resistivity of
potassium.
1 . l . 3 Electron-electron scattering
Electron-electron scattering can contribute to the resistivity.
Let f, and if; be the initial wave vectors of the two scattering
electrons, and let 7?, and K, be the final wave vectors. Then we have
the following momentum conservation rule
K, +K2-K3+K4+G (1.18)
If 3-0, we have a normal process, and otherwise we have an Umklapp
process.
In a calculation of the resistivity, we know from Ziman(21) that
an approximate solution to the Boltzman equation for a normal e—e
scattering process contains a factor of the following form:
(mi, )‘ic‘, + «Rpm-1:653 )Ea-t (if, )‘f:‘,)- in? “if, iris-71E.) (1 . 19)
17
where t(K) is the electron relaxation time, and U is a unit vector in
the direction of the electric field. When the scattering is isotropic,
u: is a constant, and this term vanishes. Thus the normal process will
not contribute to the resistivity. In the second order approximation
the Umklapp process can take place, and the delta function given above
is replaced by
“Kirk‘s-Era) . (1.20)
So we see that Umklapp scattering gives a finite contribution to
. resistivity even with the isotropic relaxation time approximation. The
calculation predicts a T1 dependence of the H scattering contribution
to the resistivity for 1! << lama. ’
This szehaviour has been observed experimentally by van Kempen 35
31. (22) and Levy gt a_l_. (23). However, some controversy arose when
Rowlands _e_t_ 51424), who were the first to carry out high precision
measurements below 1.2K, found that the resistivity of their potassium
samplo(0.79 mm in diameter) behaved more like 1"" than like 1'?
Overhauser(25) tried to explain this 'r"5behaviour on a cow basis. Later
Lee .e_t_ 31426) measured p(T) for a number of thicker samples and
confirmed the existence of a T2 term dom to 0.4K. Then Yu 35 11427)
measured a series of samples with diameters ranging from 0.09 to 1.5 mm
and found important deviations from 'r2 behaviour in samples thinner
than 1mm. Their interpretation invokes an effect proposed by Gurzhi(28)
involving interference between normal electron-electron scattering and
18
surface scattering. Since Haerle gt; a_l.(5) used 0.9mm diameter samples
in their deformation experiment, this size effect was present. Our
experiment is better because we used 2mm diameter samples where the
size effect is negligible.
As for the coefficient A of this T2 term, a theory was worked out
for potassium by Lawrence and Wilkins(29) for a screened Coulomb
interaction. This theory was later refined by MacDonald 91' 11430), who
included both screened Coulomb scattering and phonon exchange
scattering. MacDonald 93 91. found that the screened Coulomb
interaction gave a much smaller contribution than the earlier work of
Lawrence and Wilkins, and instead they found that the dominant term
was due to phonon excrange scattering. Both theories predicted an AT2
term with A - 1.7 fan/K2.
However, this coefficient A no been found by various groups
mentioned above to be sample dependent, which conflicts with the
fundamental ideas underlying the calculations of A. A way had to be
found to introduce a non-intrinsic property, varying from sample to
suple, which affected the magnitude of A. As we mentioned earlier, the
normal electron-electron scattering does not contribute to the
resistiviw if the relaxation time t is isotropic, as shown in Eq.
(1.19). However, if t is not isotropic, this term will not vanish and
therefore can make ‘a siguificant contrihition to p. Kaveh and Wiser
(31) argued that dislocations are the best candidate for such an
anisotropic scatterer. At very low temperatures the dominant mechanisms
are impuritygo“) and dislocation(p“) scatterings, and impurity
19
scattering is believed to be almost isotropic. Thus the ratio 34/13”. is
a measure of the relative amount of anisotropic scattering, where
po . Per. 4. Pod. (1'21)
Kaveh and Wiser obtained
A - A. + A, ej/(g, + p0, )2 (1.22)
where A, is the Umklapp contribution in the isotropic limit, and A.
corresponds to the maximwn contribution from normal scattering. An
estimate of A, is difficult because the anisotropy of the
electron-dislocation scattering time is not known. Kaveh and Wiser
estimated this anisotropy and were able to fit Eq.(l.22) to the data,
although somewhat arbitrary estimates of ed lo,3 were made for each set
of data. They found that A,- 3.5 film/K2 and A,- 0.5 film/K2 . The
squashing experiments of Haerlo _e_t_ a_l. (6) were designed to test this
theory by introducing a k_no_wn_ value for ad. Haerle 3 91. observed that
p(T) for deformed potassium did not exhibit the predicted T2 behavior,
which is in ag‘oement with the results to be presented here. Thus the
theory of Kaveh and Wiser, although uportant for motivating these
deformation experiments, cannot explain these results, and different
electron—dislocation scattering mechanisms are needed.
1 . 1 .4 Electron-dislocation scattering
20
In his attempt to explain the residual resistivity due to
dislocations, Brown(9) proposed a theory where the electrons are
scattered primarily by the dislocation cores rather than by the
surrounding strain fields. As a result, the large angle scattering by a
segment of a dislocation line is independent of the proximity of other
dislocations. He also stated that there is no reason to distinguish
between the cores of edge, screw, or mixed character dislocations in
this regard. He suggested that dislocations could have virtual bound
states for electrons, with an energy slightly above the Fermi surface.
He estimated that these relative energy levels fer potassium are about
10‘4ev. These energy levels could be localized near the cores of the
dislocations. Recently, Fockel(32) pointed out that the potential of
the dislocation core contains resonance states below the Fermi energy.
Fockel used the pseudo-potential concept in which he treated the core
as discrete and the surrounding matrix as a non-linear elastic
continuum.
Let us suppose that there are some localized electronic levels
near the dislocation cores at some height E above (or below) the Fermi
level and that they become occupied (or unoccupied) as temperature
rises. Therefore the effective core charge changes with, temperature.
Gantmakher and Kulesco(33) derived the following equation.assuming that
the electrons were scattered elastically off the localized levels:
pa “(1) - a(l+bexp(E/k8T))-1 (1.23)
21
where b is the spin degeneracy of the level and a is a preportionality
constant.
There are other calculations which apply to interactions with
localized energy levels in crystals. Fulde and Pesohel(34) calculated
the resistivity due to electrons scattering inelastically off localized
energy levels produced by a -crystalline electric field splitting of
rare earth ions dissolved in metals. By using Matthiessen's rule, they
derived the following expression '
<12
] (1.24)
1 + (2/a)sinh $59.?)
p (T) -mp°[ 1 +
where G and g, are constants, and D is the energy level splitting for
an assumed doublet. Even though this expression is not directly
associated with dislocations, we think this theory might be applicable
to scattering from local electron states caused by dislocations such as
the virtual bound states predicted by Brown.
O'Hara and Anderson(35)(36), in their-stuch' of the lattice thermal
conductivity on some superconducting metals, found the existence of a
resonant phonon-dislocation interaction at certain frequencies. This
work supports the existence of dynamic dislocations at low
temperatures. They pr0posed two models for the possible vibrating
dislocations. One was originated by Granato(37) in which the
dislocation is treated as an elastic band stretched betwoen two pinning
points a distance L apart and the natural resonant frequency v is given,
22
by
v - V/3L (1.25)
where V is the transverse phonon velocity and L is the dislocation
length. Hence_the frequency is inversely proportional to L. The pinning
source might7 be the intersection. of’ a dislocation with other
dislocation lines, vacancies or impurities. The other model is
associated with the Peierls potential, which is important for bcc
metals such as potassium for a reason.we. are going to discuss below.
Here the dislocation might oscillate in its potential well with a
frequency essentially independent of the length of the segment
inNolved(35): ' ‘
v - [tP/4nzpmbzli (1.26)
where u:P is the Peia‘ls stress, p" is the mass density and b is the
magnitude of the Burgers vector.
Potassium shows a rapid increase of critical flow stress with
decreasing temperature.(38) For example, from 20K to 4K its critical
shear stress increases by almost a factor of 2. This and other
significant differences from fcc metals are ascribed to a limited
mobility of the (a/2)<111> screw dislocation due to a high Peierls
potential(38). In contrast to screw dislocations, edge dislocations in
potassium do not feel the effect of a large Peierls potential and are
23
comparatively mobile(39). For this reason we expect deformations at 9K
to produce more screw dislocations than edge dislocations: and for
defOrmations at 60K, where the Peierls stress is less important, we
expect to have more edge dislocations. Since screw dislocations are
thought not to be pinned by impurities or vacancies due to their lack
of a dilatational strain, we expect Eq.(l.26) is more viable for screw
dislocations, and because edge dislocations are thought to be pinned
by impurities or vacancies and 'not interact strongly with.the Peierls
potential, we expect Eq.(l.25) to be more viable for edge
dislocations(6).
The inelastic interaction of these local modes in the phonon
spectrum with the electrons was calculated by Gantmakher and
Kulesco(33) who approximated the local modes by a single frequency-the
Einstein oscillator. They found an additional electrical resistance
due to such an interaction of the form:
-2
p“(T) - (DI4T)sinh (ﬁﬂ'IZkBT) (1.27)
where a) is the ground state frequency of the oscillator and D is a
proportionality constant. Note that Eq.(l.27) predicts 34“.)“ T for T
>> fwd/k3.
1.1.5 Electron-phason scattering and Charge Density Waves (CDW)
According to the CDW theory prOposed by Overhausor(40), the
24
electron density in an alkali metal is modulated due to the
electron-electron interaction:
p(r) =- p" (1 + pcos(6-? + 4’) (1.28)
where p is the modulation amplitude, '6 is the characteristic vector of
the CDV ( Q‘I‘Qk F), and 4‘ is an arbitrary prase term. The lattice then
deforms simsoidally in order to maintain overall charge neutrality.
(The positive—ion lattice is approximated by a deformable-jellium
model.) The wave vector 6 is incommensurate with the reciprocal lattice
vector 6. Since the presence of CDVls in potassium would orange its
nearly spherical Fermi surface into a complex interconnected one, this
would have a profound effect on the transport properties of potassium.
For example, the interconnected Fermi surfaces would reduce the
magnitude of the minimum phonon wave vector qﬁhin the electron-phonon
Umklapp scattering which would enhance such scattering. In addition the
electron-electron Umklapp scattering would also be enhanced by another
channel Edit-133+?“ +75 in addition to the origiml one E-e-Ez-ifa {£45.
Therefore its contribution to resistivity is enl'nnced. This enrancement
would also depend on the relative orientation of the electric field
and the "0" domains, which are defined as regions over which long range.
correlations of the CW exist. The orientation of the "Q" domain is
anticipated to be sample dependent, e.g. it can be changed by rapid
cooling and deformation, and therefore the electron-electron
scattering should. also be sample dependent. According to Overhauser,
25
CDW's can have elementary excitations in which the phase ¢ varies
periodically in time and space. These excitations are called phasons,
and they behave differently from phonons as far as their dispersion
relation is concerned. Using a phason-scattering mechanism Bishop and
Overrnuserul) tried to explain the Tubehavior in the resistivity
measurement done by Rowlands gt al.(24); but from the work done by Lee
2§_gl,(42). Black,(43) and Yu ‘3; g;.,(27) we now know that the sample
size can be responsible for the deviation from T2 behavior in potassium
samples thinner than 1mm.
Bishop and Lawrence(4) have combined the above-mentioned CDW
electron-Umklapp scattering and phason scattering to explain the.
variability in A that Kaveh and Wiser tried to explain by
electron-dislocation scattering. Bishop and Lawrence argued that
different Q-domain textures could cause different amounts of phason and
Umklapp contributions to p(T). Since defamation could modify the
Q-domain textures, our defbrmation studies have relevance to the CDW
hypothesis. We were particularly ‘interested in observing what small
defbrmations might do tolp(T), which twisting the sample allows.
To identify the existence of CDWs in potassium has long been a
subject of interest. Giebultowicz g£_§l.(2), as we mentioned earlier,
reported the observation of CDW satellites in their single crystal
potassium sample by neutron diffraction. On the other hand, H. You 25‘
al.(3) reported recently that they failed to observe such satellites in
their synchrotron Xéray diffraction experiments for their mosaic single
crystal potassium. Indeed we need more experiments to positively
26
identify the existence of CDWs in potassium.
1.2 Thermoelectric Power S
From the basic transport equations given earlier
J - Lula: + 1.,sz (1.1)
Q - 1.2,]: + 1.,sz , (1.2)
we obtain the thermopower S:
s - (E/v'mm- -L,,/L.. (1.29)
which is measured experimentally as follows: If we induce an
infinitesimal temperature drop across the sample, there should be a
thermoelectric voltage: and the thermopower is obtained if we divided
this voltage by the temperature drOp.
Theory predicts that the thermOpower usually consists of two
parts: the diffusion thermopower and the phonon drag thermopower.
1.2.1 Diffusion thermopower
The diffusion thermoelectric power contribution is usually
associated with a system of electrons that interacts with a random
distribution of scattering centers which are assumed to be in thermal
27
equilibrium at the local temperature ’1'. As we will see in the next
section, this assumption is a very poor approximation in the real
situation: an additional contribution will appear when the assumption
of local thermal equilibrium is lifted.
Assuming the conduction electrons constitute a degenerate Fermi
gas, if one uses the relaxation.time approximation, and if the higher
order terms in the expansion of the Fermi-Dirac function are neglected,
then one obtains the Mott(21) expression for the diffusion thermopower:
_ «Zea-r eln ans)
3 e e E “'30)
‘E sip
where o is the conductiviiar and s:F is the Fermi energy.
When impurity scattering dominates, one expects for potassium that
aln o(E)/0E will be independent of temperature. Thus the thermopower in
potassium should vary linearly with temperature T.
If there are two kinds of scattering processes involved in a
system, for example, impurity scattering and dislocation scattering,
and if Matthiessen's rule applies, then
pulpi +pd , (1.31)
where p: and p“ are considered as independent impurity and dislocation
contributions to the resistivity, respectively.
Using Eqs.(l.30) and (1.31), we have the Gorter-Nordheim
relation(44) for the diffusion thermopower
28
S - (llp)[p..S.. + pd Sd] - S, + (pl. /p)[Si - Sd] (1.32)
where S; and Sd represent the diffusion thermOpowers due to impurity
scattering and dislocation scattering, respectively. Therefore, if we
assume‘p; is not changing during deformation and plot S as a function
of 149, we should get a straight line which intercepts the S axis at Sd
1.2.2 Phonon drag thermopower
As we mentioned earlier, the assumption of thermal equilibrium in
the calculation of the thermopower can be a poor one because the
application of a temperature gradient across the sample causes the
phonon distribution to go out of equilibrium. This phonon flow will
"drag" the electrons to the end of the sample until the electric field
formed by the piled-up electrons is large enough to stop further
electron dragging. This will cause an additional thermopower term, the
phonon drag thermopower.(45)
. The phonon drag thermOpower is usually divided into two parts: one
due to the normal electron-phonon process and the other one due to the
umkiapp process. The general theory in both cases tends to be rather
complicated, but fortunately we will be working in the low temperature
limit, T < 1K. I
The normal process inNolves scattering of electrons within a
29
single Brillouin zone. In the low temperature limit this contribution
to thermOpower goes roughly as the lattice specific heat, which has a T3
temperature dependence:
3; .. T3. (1.33)
The Umklapp process is more complicated. As we have seen earlier,
a minimum phonon wave vector q...“ is needed to scatter from one Brillouin
zone to another if the Fermi surface does not touch the zone boundary,
as in the case of potassium. We can estimate the number of such phonons
to be proportional to exp(-fqu_/kT).
Guenault and MacDonald(46) fit their data to. a simple equation of
the form
3 - 3,? + 131'3 + Cexp(-e’/'r) (1.34)
where S,T is the diffusion term, 8T3 is the normal electron-phonon
drag term and Cexp(—67T) is the Umklapp electron-phonon drag term. S,
and B were found to be negative, and C was found to be positive, with
659 21!. For temperatures lower than 1K the Umklapp term is usually
negligible.
1.2.3 Thermoelectric ratio G
If we want to measure the thermopower, we must produce and measure
30
a temperature difference across the sample. EXperimentally this turns
out to be very difficult for potassium because it has a very high
thermal conductivity, and it requires a huge heat flow across the
sample to produce a large enough temperature difference for
measurement. This heat flow usually exceeds the cooling power of the
dilution refrigerator. Instead, the thermoelectric ratio G was actually
measured in this work:
a - (J/QHE: LHL/I.22 (1.35)
From (1.1), (1.2), and (1.29), we know that the resistivity p -'
1/L,,, the thermopower S - ilk/L” , and the thermal resistivity W =-
’1/1'22' So G can be written as
c - Ln/Lzé- (Ln/LHKLHILZZ) - am: (1.36)
Since the Lorenz ratio L is defined as
L - p/WT , (1.37)
we have
G ‘ SILT e (1.38)
Now the thermopower can be related directly to the thermoelectric ratio
31
G. Ideally L should be a constant: and for T << 1K where elastic
impurity scattering dominates, this is indeed the case where
‘3
1. - Lo- wr’kalaez- 2.445x10 vz/K2 . (1.39)
From the results of Haerle 93 al_. (47), we know that L/L,- 0.97 at
1K and that the ratio becomes even closer to 1 at lower temperatures.
Since our interests are below 1K where the exponential Umklapp term is
frozen out, G is expected to have the following simple form
(1 - c, + 131'2 (1.40)
where G, is the diffusion term and sz is the normal phonon drag term.
Any departure from this expression below 1K would indicate the presence
of some other scattering mechanism which has not been taken into
account in‘ the above theory.
Chapter II Experimental Techniques
In this chapter, the major equipment used in the experiment will
be described. Details will be given.about the thermometry, the sample
container, its functioning in deforming the sample, sample preparation,
and the .measurement procedures. In addition, possible heat flow
problems and the measurement uncertainties will be discussed.
2.1 Main Equipment Used For Measurements
2.1.1 The dilution refrigerator ~
Since our emphasis on the properties of potassium is below 1K, a
locally-built dilution refrigerator, which is capable of reaching 10
mK, has been used in the experiment. This dilution refrigerator was
originally built by J. Imes and W. Pratt Jr.,(48)(49) and then modified
by V. Heinen.(50) It can.cool from room temperature to liquid nitrogen
temperature in about 12 hours. It takes about 15 liters of liquid
helium to reach 4.2!; The lowest temperature (about 10 mK) can be
reached after 5-6 hours of 3He-ihe mixture circulation. For a general
description. of a dilution refrigerator and its operation see
Lounasmaa(51).
32
33
2.1.2 The high precision resistance bridge
As the discussion in section 2.6 will make clear, we measure dp/dT
in our experiments. For a typical unstrained potassium sample at 1K,
the relative change of p, solo, is about 10.5 with a temperature change
of AT - 0.1 (K. Since dp/dT is usually smaller at lower temperatures,
rather precise measurements of‘Ap/p are required. we have used a
high-precision current-comparator system with a SQUID' null detector.
The system was built by D. Edmunds gt, al.(52) and can resolve the
qmntity Ap/p to a precision of better trnn 0.1 ppm.
2.1.3 The screened room and floating pad
A commercial, double-layered screened room (from Erik A. Lingren
and Associates, Inc.) surrounds the cryostat and screens out any
radio-frequency noise which might affect the Operation of the
rf-biased SQUID. Since mechanical vibrations in the presence of the
earth's magnetic field can.induoe currents which exceed the dynamic
range of the extremely sensitive SQUID. the refrigerator body is
magnetically shielded with high-p. metal and mounted on a vibration
isolation table. All the pumps are kept outside the screened room and
are connected to the refrigerator through flexible bellows.
2.1.4 A schematic diagram of the ultralow temperature part of the
cryostat
34
li'He
WWO"! can
\\
mixing chamber
stainless steel
weak thermal link
50”.!!! Ag piece
\0‘ ”.0 m.
\ News...
9.. presswismien line ' \ amen. c...
\
f I
Fiaire 2.1
The low temperature part of the cryostat
35
Figure 2.1 shows the low temperature part inside the cryostat. The
details will be discussed in later sections.
2.2 The thermometry
The temperature was measured in this experiment with two germanium
resistance thermometers and one Cerrous magnesium Nitrate(CMN)
susceptibility thermometer. The calibration of these thermometers was
done by C.W. Lee and V. Heinen gtﬂgl.(50) The two germanium resistors
are mounted in the holes in the bottom piece of the sample mount using
Apiezon N grease for thermal contact. The susceptometers of the CNN
thermometer are mounted.against and thermally isolated from the plastic
wall of the. mixing chamber. Silver wires of l-mm-diameter provide
thermal contact between the bottom piece of the sample mount and both
the CNN sensor and its susceptibility coils.
For temperatures above approximately 1.5K, "R6" is used, which is
a Lakeshore Cryotronics germanium resistor. The calibration was done as
follows: First, the susceptibility of a CMN sample was measured against
SRM767 and SRM768, which are the superconducting fixed point devices
from the National Bureau of Standards. Since the susceptibility of CNN
is proportional to T”1 over the temperature range of interest here,
temperatures other than the fixed points could be easily determined by
interpolation. Then by using a least-squares fit, the resistance of R6
was fit to the temperatures given by the CMN for temperatures between
36
1K and 4.2K using the following equations
AI
Leg T a Z an(Log 1’1)ﬂ (2.1)
680
LOg a - ﬁbnaog T)" (2.2)
"80
with N - 7. With these fits, the temperature given by 86 is estimated
to be within 0.3% of the absolute temperature.
For temperatures between approximately 40 mi! and 1.5K, "R7" is
used, which is a Cryocal CRSO germanium resistor. The method of
calibration was similar to that used for R6, except that the
susceptibility of an irregular single crystal of 10% CMN and 90x'
LMN(Lanthanam Magnesium Nitrate) was used for interpolating between the
fixed points. The resistance of R7 is fit to the temperatures given by
the susceptibility of the CMN-LMN using the equations given above with
13-9. The error is estimated to be within 0.7% of the absolute
temperature.
Below approximately 50 ml! the CNN thermometer is used which
consists of a susceptometer (Fig. 2.2) and a CNN pill. The CMN pill is
a 50:50 volume mixture of CNN and 700—3. Ag powder pressed onto a 0.012"
diameter Ag(0.4 atx Au) wire. The pill is a right circular cylinder
(height-diameter-lla"). containing approximately 18 mg of CNN and 95 mg
of Ag. The Ag(Au) alloy wire is used to reduce any possible eddy
currents caused by the 17 Hz magnetic field used to measure the
susceptibility. The eddy currents can produce fields which would affect
the measured susceptibility and could also cause heating of the CNN
37
liebit-
_ v m C"
—ltobit- m.
3
Secondary "-— -— Pm" Coil
Coils
IT
': H CMN en
*1 e
Coil-foil Tubs
=1 4 Niobiu-
J ‘ End Cap
‘g(‘u) wit.
Figure 2.2
The CMN thermometer
38
pill. This wire is approximately 1" long and is spotwelded to a pure Ag
wire which is attached to the bottom Ag sample mount. For a more
detailed description of the CMN thermometer see V. Heinen's Ph.D.
thesis.(50)
The CMN thermometer was calibrated against the SRM 768 which is
the low temperature standard. It has the following fixed points: W
(15.5 mK), Be (22.92 mK), Ir (99.13 mK), AuAl (160.43 mK), and AuIn
(204.36 mK). The 22.92 mK point has not been used because this
superconducting transition point was in complete disagreement with the
other fixed points. A linear least-squares fit has been done to the 15,
99, 160, and 204 mK points assuming a Curie law behavior for the
susceptibility of the CNN. In the overlapping temperature range of the
CNN thermometer and R7, the temperature difference was found to be less
than 1%.
We have used two thermometers for temperature regulation of the
mixing chamber. A carbon resistor is used as the temperature sensor
for T 3 50 mK. The other sensor is a second identical CMN thermometer,
used for T < 50 mK, since CMN has a much better temperature response in
this range. This system has ,the unique feature that only one SQUID is
used as the null detector for both CMN mutual inductance bridges.
Figure 2.3 shows how this is done. Two independent AC oscillators, V1
and‘Vz , drive the two bridges. Each Intersil #ICL 8038 sine wave
generator is powered by a separate 6V battery, one operates at 17 Hz
and the other at 40 Hz. Their maximum output is 1 V p-p. There is also
an optical coupler output for the reference channel of each lock-in
39
V1
V3 .
Figure 2.3
Circuits of the CNN mutual inductance bridges
4O
amplifier. Two HR-B lock-in amplifiers from Princeton Applied Research
Corp. are used to independently extract the two AC signals at the SQUID
output. No interference between these two CMN bridges has been observed
in this experiment.
Also a silicon diode thermometer from Lake Shore Cryotronics, Inc.
is used for determining the temperatures above 4.2K during twisting and
annealing of the sample. This thermometer is thermally connected to the
top end of the sample. The voltage across the diode is measured with 10
HA reversed biased current.
2.3 The sample can
Since potassium reacts with oxygen and water vapor, a
self-contained sample can has been used which permits both our
measuring the electrical properties of the sample and deforming the
sample at low temperatures.
Figure 2.4 shows a drawing of the sample can which is capable of
twisting the sample while mounted on the dilution refrigerator. The
sample can and dilution refrigerator are both mounted inside a main
vacuum can, which is surrounded by liquid helium, as shown in Fig. 2.1.
The sample can can be separated into three assemblies: The top flange D
on which the feed—throughs B and the sample holder H are mounted, the
central body cylinder F, and the bottom flange P on which the twister
is mounted.
The three assemblies are sealed together by two replaceable indium
41
.32.:
The sample can
Fimre 2.4
42
Table 2.1 A list of parts in sample can (shown in Figs. 2.4 - 2.6, not
all parts are shown in each figure)
A central rod
S bellows
B feed-throuzh T pulling gear bar
0 central rod housing . U ratchet cg”
D top flange V pawl
B tOp indium '0' ring W ngm bar
3' bottom indium '0' ring a flippgr pin hole
F central body cylinder b ratchet cam pin
G top sample mount
H nylon sample holder
I potassium sample
J bottom sample mount
K twisting shaft
L torque coupler and rotation detector
L twisting key
L" rotation detector
M flipper
M flipper arm
M" flipper axis
inner track
outer track
N
0
P bottom flange
0 track housing
R
pressurizing line
43
"0" rings located at E and s', respectively. After the sample is
mounted, the tap flange E is sealed last while the can is still
inside the argon glove—box. Then the whole can is taken out of the
glove-box.and mounted underneath the mixing chamber of the dilution
refrigerator.
The top flange is made of brass. On it there are the central rod A
and two feed-throughs B. A is made of OFHC capper and is epoxied onto a
housing 0, which is made of "Vespel" (SP—22 Polymide, from Dupont,
Inc.), a very good heat insUlator at low temperatures. The rod serves
as both the heat path and one of the current leads of the sample. One
of the feed-throughs B is used for various electrical leads, and the
other one is used as a safety valve, made by soldering a thin brass
foil onto it. The central body cylinder F is made of capper’ to which
the two upper and lower flange housings are hard soldered. There is a
heater (not shown in the figure) mounted on the outside of the cylinder
to control both the deforming and annealing temperatures. The lower
flange assembly P consists of the bellows and the twister systems, and
their functions will be described in the following section.
Figure 2.5 is a perspective drawing of the sample holder H, which
is made of nylon. The two sample mounts. G and J. are made of OFHC
capper, and G is part of the central rod A. The lower sample mount J
can be turned by the torque coupler L, which is also made of "Vespel".
The coupler is driven by key L', which is on the output shaft of the
twister. The current is run through A and J, and the voltage is picked
Up through the Veprobes in the figure. A heater is mounted on the back
44
I".
Figure 2.5 p
A perspective view of the sample holder
45
side of J for the thermoelectric ratio G measurements (not shown in the
figure). Nets a potentiometer L" is mounted on the torque coupler so
that the angle through which the sample is twisted can be read.
2.4 Plastic deformation of the sample
The main feature of the twister is as follows: At any temperature
above 4.2 K, if we apply about 30 psi pressure of helium gas to the
bellows S, we can obtain a maximum torque output through L' of about 5
pound-inch, which will twist potassium samples of 3mm diameter throngh
about :80 degrees. The direction of twist can be chosen at will and
twisting,only occurs during the upward power stroke of bellows S. To
obtain a larger amount of deformation, one has to twist the sample back
and forth many times. Since the SQUID circuit is very sensitive to
stray magnetic fields, a purely mechanical method of setting the
direction of twist has been used, rather than the more obvious
electro-mechanical method.
The detailed description of this twister can be understood either
from Figure 2.4 or the simplified perspective drawings Figures 2.5 and
2.6.
Inside the twister there is a flipper M at the bottom of which is
a wheel. This wheel can sit at a position above either the inner track
N or the outer track 0. When the tracks, together with their housing Q,
are driven up by the pressurized bellows, one of the tracks will hit
the wheel and force it to move along the slope. Since the attached
46
Fiaire 2.6
A perspective drawing of the twister mechanism
47
flipper is connected via M" and M' to the central shaft, this motion
will force the central shaft (with key L' at its tOp) to turn through
an angle. If the flipper sits above the other track whiCh has the
opposite slape. L' will turn in the opposite direction.
There is a gear arm T attached to the housing of the tracks Q.
When the pressure inside the bellows is lowered, the housing will
return to its lower position. On its way back the gear arm T will turn
the ratchet cam U. Nets that there is a curved groove in the ratchet
cam U, and pin b is engaged in it. When U turns, the curved groove
Will force the pin b and hence bar W to move horizontallY, which.will
bring the flipper to the other position. When T rises, a pawl V keeps U
from turning. Usually several small up and down motions of T are
required to switch flipper M between tracks. During these oscillations
of T, the flipper M does not contact the track. Thus the sample is not
twisted.
When.the bellows is driven up again by pressurized helium gas, the
central shaft this time will turn in the opposite direction. Thus a
back and forth rotation is accomplished. In addition to the rotation
detector, two similar position detectors have. been put inside the
twister to detect the actual position of the bellows and the actual
position of the flipperCnot shown in the Figures).
The He gas in the bellows mUst be isolated from sample area inside
‘the can because any helium gas there would provide a heat path between
‘uhe sample and the wall of the sample can. The residual argon gas
brought from the argon glove-box is solidified at helium temperature.
48
The plumbing for the pressurization system connected to the
bellows is shown in Figure 2.7, which is similar to that of
Haerle's.(47) Nermally when operating the bellows valve #1 is closed
and helium is slowly applied through valves #2 and #3. The two-line
system has been built for the following reasons: If a solid air plug
should form when pressurizing at around 10K, the bellows could still be
evacuated by using the second line because any plug Would take place in
the first line above the liquid helium level, which would be above the
place where the two lines Joined to form one thin line going into the
vaCuum can. The ballast tanks are about 10 Cubic inches in volume and
are used to damp out any Taconis oscillations.
2.5 Sample preparation .
The samples were prepared inside a commercial argon glove-box
(Vacuum Atmosphere Company) with the gas purifying system built
locally. In comparison with another VAC helium glove-box having a
nominal oxygen contamination of less then 0.4 ppm, the time that
potassium remained shiny inside the argon glove-box was longer or equal
to that of the potassium inside the helium one. So we know that the
oxygen concentration inside the argon glove-box is less than or equal
to 0.4 ppm. The water vapor content has not been directly measured but
exposed potassium remains shiny for hours. In order to allow the sample
can to outgas and the purification system time to remove any residual
contamination brought in with the can, all needed materials were placed
49
Liquid He
V
_L— "L'T
3; E
s 3 .
L1H H— (:3 I 5
a --—-- i
a \5
Qi/
Pump He gas ® Valve
Figure 2.7
The He pressurization system for the bellows
50
in the air-lock and pumped on for at least 24 hours. Then they were
placed in the glove-box for another 24 hours before the sample was
made. To further reduce the contaminants inside the can, an oxygen
getter was used. Before the can was closed, a thin Copper sheet about
2"x7" in size was smeared on one side with a thin laYer of K or Rb and
then plaCed inside the can with the unsmeared side against the inner
surface of the can body. In this way the sample surface remains
reasonably shiny even after a run of several weeks. It is very
important that the sample surface remain clean since thick deposits of
K compounds on the sample surface have been observed to significantly
alter the mechanical properties of the sample.
The pure potassium sample is made of 99.95% potassium obtained
from Callery Chemical Company, a division of Mine Safety Appliances
Company. Table 2.2 shows the chemical composition of a similar batch of
the potassium. The potassium came in glass ampoules sealed under argon
gas. The ampoules were Opened inside the glove-box, and the potassium
was melted and transfered to the stainless steel press (Figure 2.8).
If the sample was K(Rb) allOY, the potassium was first melted and
poured into a hot glass beaker; and then some Rb was melted into the
beaker before the alloy was poured into the press.
The samples were extruded through a 2mm diameter die. The samples
were about 40mm long between the voltage probes (Figure 2.4) and were
cold welded onto the copper mounts J and G which had been smeared with
some potassium first. Then the sample ends were clamped onto the mounts
by two previously potassium-smeared capper clamps. This assures that
51
Table 2.2 Chemical analysis of potassium
Element PPM Element PPM Element PPM
to <5 Cr <5 Sr <1
3 <10 Si 25 Ba <3
Co <5 Ti <5 Ca 8
Mn 1 Ni <5 Ma 15
A1 <2 Mo <3 Pb <5
Mg 2 v <1 2: <10
Sn <5 Be <1
Cu (1/p)dp/dT a (1/p.)dp/dT . (2.12b)
Below about 1K, the right hand part of (2.12b) is obeyed since the
total variation of p below 1K is about ap/p =- 10"? Therefore dp/dT can
be obtained by multiplying (2.12b) by p. in the plots of the data.
One advantage of this temperature—modulation method is that we can
double-check our thermometry here. As mentioned above, we know the
amount of heat Q being put into the heaters H01 and HLl in order to
produce a temperature difference AT across RL' the Ag(Au) resistor in
Figure 2.9. The Wiedemann—Franz law states that
RL/W - L,T ' (2.13)
where RL is the resistance, W - aT/Q is the heat resistance and L. is
the Lorenz number. We obtain that
RL - L,TAT/Q . (2.14)
If our thermometry were perfect, the value of RL we calculate at
different temperatures using (2.14) should be a constant. Thus we check
our thermometry by calculating RL occasionally during our runs. We
usually measure RI. at T - 0.03, 0.05, 0.15 and 0.6K during each run.
For example, in our run with sample K-l the average of RL was 46.65 M
and the standard deviation was 1.1%; In our run with K-5 the average of
60
RL was 46.5 leand the standard deviation was 2.6%. The overall
standard deviation of RL in each run was less than 3%.
The current put through the sample is usually SOmA, but
occasionally checks are made to be sure that there is no current
dependence by using 25mA.
Another quantity measured in the experiment is G, the
thermoelectric ratio. G is defined as the ratio of the electrical
current to the heat flow at zero voltage drOp across the sample:
. . a
G - I - I I R .
where IQ is the current passing through the sample to counteract the
thermal voltage resulting from the heat flow through the sample: Q - Is'
Rh . Ih is the G heater current, and Rh is its resistance.
The measurement is done as follows. First we pass a known heat
flow Q through the sample, and this causes the SQUID to go off the null
value because a thermal voltage is generated. Then we put a balance
current I9 through the sample by adjusting the dials on the current
comparator until the SQUID indicates the null condition again. No
currents passed through Rr during these measurements. Then from (2.15)
the G value is obtained.
2.7 Heat loss
It is important that all the thermometers which are mounted on the
61
stainless steel to mixing chamber
o A“ o
a. é
\ ., In
L. a
A, sample mean:
A ’
to CMNI ‘Q to sample
8-32 brass screw
Figure 2.10
The Ag sample mount
The Ag sample mount on which the thermometers etc. are mounted
62
silver sample mount (Fig. 2.10) be at the same temperature as the
sample. Thus any heat flow between the sample and this sample mount
must be kept to a minimum level. The heat losses due to the
superconducting leads are negligible since they are very poor heat
conductors at these low temperatures.
Since the body of the sample can is directly attached to the
mixing chamber by a Ag wire, any heat paths to the can from the sample
will be considered to be heat leaks. From Fig. 2.4 we see two such
paths. One path is via the central rod housing C, and the other is via
the torque coupler L. They are both made of "Vespel". The empirical
formula for its thermal conductivity at low temperature is
K - 17 r2 pw/(cmx). (2.16)
We want all the heat generated by heater HLl to flow only through the
weak thermal link RL (Figs.2.l, 2.9, 2.10), which is a Ag(Au) wire
with a resistance oflv 50 p11 . By using the Wiedemann-Franz law, we
know that the thermal resistance of this wire is about 2x103 (K/Watt).
The thermal resistance of the housing C is therefore designed to be
about 105 (K/Watt), roughly 50 times that of RI. at 1K. The bottom
sample mount J is made of OFHC copper and is hard soldered to a
stainless steel shaft K. The heat loss due to the torque coupler L is
estimated to be much less than that of the housing C.
The electrical resistance from the top sample mount G to the
silver tab (Fig. 2.10) was measured to be less than 2.4 pn at 4.2K, the
63
estimated temperature difference between the sample and the
thermometers is less than 0.2 mK for the worst case (at 1K).
2.8 Heat generated while deforming the sample
As we mentioned in Chap. 1 (Section 1.1.1 and 1.1.4), for
temperatures near 10K, the actual temperature at which the sample was
deformed must be known since screw dislocations appear to rearrange
themselves near 10K and vacancies begin to anneal out above 10K. It is
therefore necessary to calculate the temperature increase while the
sample is being twisted. Unfortunately, we do not have a thermometer
mounted directly on the sample. However, the diode thermometer is
mounted nearby on the top sample mount (Fig. 2.9) to which the sample
is cold welded. First let us calculate the heat relaxation time t
of the sample. We know that the temperature distribution function
U(x,t) for a bar of length L is of the form
nix
- =32.
U(X.t) - X C“ e SIMT (2.17)
“‘9
where the relaxation time tn is defined as
t n - (Llnﬂzllaz, and ‘ (2.16)
where L is the length of the bar and a2 - k/cp in which k is the
thermal conductivity, c is the specific heat andqp is the mass density.
64
For potassium at 10K, we have k-4.0 watt/ch, c=0.07 Joule/gK and
p'0.66 g/cm3 . Therefore we have
t,8 0.034 sec , (2.19)
which is the longest relaxation time in tn. Our twist is typically
done in about 1 second, which means that the whole sample is at a
uniform temperature during most of the deformation if, as a worst-case
example, we assume no heat is flowing out of the ends of the sample.
Now let us calculate the work needed to twist the potassium
sample. The yield stress a'for potassium at 10K is about 1 Kg/mm2'(57)
and according to Cottrell(56), the torque T' needed to twist a bar of
radius r and shear stress 0'13
1" = (1r/2)r30’. (2.20)
For our sample we have r-lmm, therefore we get T'81.57 (kg-mm). If we
twist the_sample by'n, the work done is av 0.05 Joule. The specific
heat of potassium near 10K is 0.07J/gK. and the sample mass is about
0.15 g. Thus the temperature increase due to this heating is
AT 3' 5 K . ~ ' (2.21)
The actual temperature increase is much less than this because the
sample is in contact with the sample mounts which have a much bigger
65
heat capacity. For example, let the total heat of deformation flow
through the whole sample and out of the upper end which is in contact
with the central rod. Since t << 1 sec, we can assume that a uniform
temperature gradient is rapidly established across the sample and
obtain for the temperature drop across the sample
AT - W0 . (2.22)
where W is the thermal resistance of the sample and Q is the heating
rate due to twisting the sample: 0 - 0.05 Watt. We assume that the
sample end which is in touch with the central rod is at a constant
temperature (10K). The thermal resistance of our K sample is about 36
KlWatt at 10K, and therefore the temperature drop AT 3 21:. Since we
assumed that all the heat flowed the whole length of the sample, AT is
an overestimate. In our experiment we have seen no noticeable
temperature increase on our diode thermometer while smoothly deforming
the sample.
2.9 Uncertainties
In our (lhO)dP/dT measurements the biggest uncertainty below 0.15K
comes from the determination of AC, whose uncertainty may exceed 10% at
the lowest temperatures. Thus possible systematic uncertainties in IAT
are not significant below about 0.15K. For temperatures above 0.15K the
uncertainty in AC is usually less than 2%. For 0.031: < 'r < 0.6K, we
66
estimated in section 2.6 that the systematic uncertainty in T was
less than 3%. If we restrict this estimate to T > 0.15K, we obtain 2%.
Thus for T > 0.15K we estimate the uncertainty in (le)dp/dT to be 3%.
In converting (100)dp/dT'to do/dT, we must include the uncertainty inoD
which we estimate to be t3%.
In our G measurements the major source of uncertainty is from the
thermal drifts of the system which cause the effective zero of the
SQUID null detector to drift during the measurement of IQ . The
uncertainty in IQ is less than 2% for temperatures above 0.1K, and it
could be as large as 5 to 10% at the lowest temperatures (T< 0.05K),
since thermoelectric voltage is usually very small in this temperature
region. Other possible uncertainties are also negligible compared to
those given above.
Chapter III Experimental Results
3.1 Resistivity
In our experiment two kinds of samples have been used: One is a
pure potassium sample with Residual Resistance Ratio (RRR), which is
defined as
RRR - R(293K)/R(4.2K). (3.1)
and is of the order of 4100: and the other sample is a dilute K(Rb)
alloy (£90.087 at % Rb).
We used the K(Rb) sample so that the electron scattering rate
would be dominated by a known impurity, Rb. For the most heavily
deformed K(Rb) sample, we will see that this condition is still met
since pd /p; < 0.09, where p" - pa + pi . In this limit the theory of Kaveh
and Wiser would predict a small increase in A with deformation. The
presence of Rb impurities might also modify the dynamical properties of
the dislocations.
3.1.1 Residual resistivity
From Eq.(l.11) we know the increase in.p due to dislocations goes
0
67
68
linearly with the change in the dislocation density Nd:
The coefficient w is predicted by Basinski (a) to be 4x10." ncm3.
If we use the simple dislocation model shown in Fig.’ 1.2, we can
obtain a linear relation betwun the dislocation density Nd and the
twist angle 6 (53)
N4 - 26/bL _ (3.2)
where L is the sample length and b is the Burgers Vector.
From Eqs.(3.2) and (1.11), we see that the change in residual
resistivity is predicted to vary linearly with the angle of twist. Fig.
3.1 is a plot of the change in ,0“ vs the twist angle 6 for our K sample
in a series of twists at 9.3K (open circles). It can be seen from the
plot that a nice linear relation is obtained. As we will see later,
about 70% of so. is due to vacancies, therefore one must assume that
the vacancy concentration is also proportional to 6. The full circles
are the data from van Vucht g; _a_l.(54). The data shown are corrected
because their sample was 10 cm long and ours was 4 cm long. The higher
slope of their data might be related to the higher yield stress of K at
4.2K. For our K—S sample which was twisted 1329° at 9.3K, we have a
change in o of Ag 3' 0.7' nncm. The corresponding change in AA due to
the dislocations is estimated to be 3, 3'- 0.21 nﬂcm, which is the value
69
AP. (90m)
10
(mecMeOel 4.2K)
(Yin eral 9K)
9( 100')
Figure 3.1
A vs 9
AB is plotted as a function of the angle of twist forKsamples
(open circle). The full circles are the data from van Vucht '95 21.
70
of AP. after the sample was annealed at 60K. With L~4cm, b~5A for K,
and e - 1329‘ , Eqs.(3.2) and (1.11) predict that pd~0.l nacm. Thus
the theoretical and experimental values of'g, agree within a factor of
2.
We can also use (1.11) to estimate the dislocation density for our
deformed sample. For our 1329‘ twisted sample K-S with p d 3 0.2 nﬂcm,
Eq.(1.ll) gives N~5xlo' cm'z.
3.1.2 K(Rb) data
In Fig. 3.2 we present a set of data obtained in a series of
deformations. The theory of Kaveh.and Wiser (19) would predict that the
introduction of dislocations would only slightly increase the
coefficient A of the e-e scattering Tz' term, since ,ed 00; < 0.09.
Instead we see a rather different behavior for dp/dT as we introduce
dislocations. See Table 3.1 for details about the deformation
procedures. Note that annealing at 2003 very effectively removes the
dislocations and essentially restores the behavior of qo/dT to that
seen before deformation.
We shall try to fit the data with
'2
p(T) - ATZ - C'T + (D/4T)sinh (BIZT). (3.3)
where AT2 is the e-e effective scattering term and -C'T is an
anomalous term observed in our laboratory for unstrained alloy samples
71
dp .
dT ( me/K) r ‘
12 ' 2 22:3 ‘
2 53222 e
8 3::
1o - -
8 _ "' -I
e - . ‘
4 - I ‘
f
2 _ .
O
o -
o .2 e4 .6 .8 1.0
1' (K)
Figure 3.2
dp/dT vs T for K(Rb) samples
dp/dT is plotted as a function of T for the K(Rb) samples in a series
of twists. The details are given in Table 3.1.
Table 3.1
K(0.087 at% Rb) Alloy Sample
Sample pm(nncm) 5: (Mom) E.(K)
KRb-l
KRb-Z
KRb-S
KRb-4
KRb-S
KRb-6
KRb-4
A' - 4.0110.01
0' - 0.22:0.01
11.94
12.17
12.38
13.03
12.62
12.06
13.03
11.61
11.85
12.04
12.68
12.29
11.73
12.68
(ﬁlm/K2)
(mm/x )
72
u, (fan)
0.354io.oee 0.174t0.094
0.40810.028 0.336t0.046
053930.044 1.20:0.13
0.45210.032 0.6220.09
0.745t0.051 1.21:0.09
22m)
0.296t0.042 0.180t0.077
02(fan)
The temperature is limited to T < 0.7K in the least-squares fit.
KRb-l
KRb-2
KRb-S
KRb-d
KRb-S
KRb-G
Untwisted
480" twisted at 60K
4800' twisted (total) at 60K
additional 4300' at 9.3K then annealed at 36K for 30 min.
annealed at 60K for 30 min.
annealed at 200K for 2 hrs.
I11
th?
73
with high concentration of impurities(55), which is not of interest to
this work. The third term is the vibrating dislocation term modeled by
Gantmakher and Kulesco (Eq. 1.27), which was also used by M. Haerle gt
21.1n their work. The coefficients A and C' were obtained from the fit
of the undeformed sample KRb-l and were then kept constant in the fits
of the deformed samples KRb-2 to KRb-S where the vibrating dislocation
term (1.27) was introduced.
Table 3.1 also shows how the various parameters change as the
deformation is increased.
In Fig. 3.2 we obtain reasonably good fits with the vibrating
dislocation model. For sample KRb-4 we see a significant deviation
below 0.23. This fit can be improved by assuming that there is ‘more
than one frequency in the spectrum of the vibrating dislocations. Fig.
3.3 shows such an improved fit. In this figure we plotted dp/dT -
(2AT-C') vs the temperature. In this way we see more clearly the
step-like function in qo/dT caused by deforming the samples. The dotted
lines are the fits plotted in Fig. 3.2, and the solid curve is the fit
using a two-frquency model
pm - In"- our + (D,/4T)sinh.2(F../2T) + (oz/ansixnzcszlz'r) . (3.4)
The parameters of this two frequency fit for KRb-4 are also given in
Table 3.1.
Our success in making dp/dT measurements well below 80 mK was
necessary in order to establish the need for this two-frequency fit. We
74
_I
‘5
O
f'bv. ‘0. 9- 9
/
-0-‘--°------- o
I 0 0°
.9
E
I
I
I
i;
O J-Q9-QSA-¢9.¢°--99. ‘ﬂ' '6' .u-.°-Da.---uQ- .—
0 .2 .4 .6 .8 1.0
T (K)
Figure 3.3
do/dT - (2AT-C') vs T for K(Rb) samples
dp/dT - (2AT-C') is plotted as a function of T for the K(Rb) samples.
A step-like function is seen for samples KRb-4 and KRb-S.
75
recall that making measurements down to about 20 mK was one of the
goals of this research.
It is worth mentioning here that the one-frequency model seems to
be adequate for samples KRb—Z, KRb-3 and KRb-S, which are deformed at
60K or annealed at 60K after deformation at 9.3K; From Chapter I we
know that the all vacancies in potassium will not be annealed out until
60K. Since KRb-4 was annealed only at 36K, we might ascribe this
multi-frequency behavior to the presence of vacancies in the sample.
From Table 3.1 we see a saturation in the ,residual resistivity
when twisting at 60K. For KRb-2 we twisted 480°, and we got an increase
in g of about 0.23 nncm. In KRb-3 we twisted about 10 times more, but
the change in p. is only 0.44 nn cm. We may ascribe this to the
following fact: After -the sample has been twisted back and forth
several times, there will exist dislocations with opposite-sign Burgers
vectors which can annihilate if they are close enough. The dislocation
density will then saturate when the annihilation and generation rates)
are equal.
3.1.3 Pure K data without annealing
To see the effect of small angle twists, we did a series of
deformations in a pure K sample with RRR ~14100. we saw behaviors which
were similar to the K(Rb) samples. The magnitude of the step-like
function went up systematically as the amount of deformation was
increased. Recall that Fig. 3.1 shows thatjq increases linearly with
76
the total angle of twist for this sample. In Figs. 3.4 and 3.5 and in
Table 3.2 we show the data and parameters obtained in a least-squares
computer fit.
we first fit K-l by using
p<'r) = ATz (3.5)
to obtain the value of A. We then kept A constant in the fits of the
strained samples. For K92, K93, K94 and K95 we used
2 ’2
p(T) 8 AT + (D,/4T)sinh (E,/2T) (3.6)
with T < 0.6K so that the electron-phonon scattering terms were
negligible. In Fig. 3.4 we plot (ad/p )dp/dT vs T and see a reasonably
good fit. Fig. 3.5 shows the plot of (pig/P)dP/dT-2A'T where the
step-like behavior is clearer. Since (3.2/Io )dp/dT is what we actually
measure during the experiment, we fit our equations to this form of the
data: and A' which appears in Figs. 3.5 and 3.6 is a parameter related
to A by: A - (I)o lpu)A'. Typically p42 is 15% larger than p0 , which we
define to be p at about 30mK. All the parameters in the tables have
been properly corrected. Since pa.2 is very close to o, for the K(Rb)
sample, we chose to convert that data directly to dp/dT.
We see a rapid increase in Figs. 3.4 and 3.5 for T > 0.6K, and
this has been ascribed to the quenching of phonon drag, which results
in the reappearance of the normal electron-phonon scattering term 0T5.
77
(Kim/K)
t 5°
his“;
I
l
12
x4
K-2
K-J
be
10 - -
4o>°0
I I a
I I
I I
' I
,’ o I ’
I ” ’
,' a ’ a
a I
l ' I " d
— ' I .
v
0 I
4 ' X ‘
" /°
.//°/ a °
2 - a." -
O ..
O .2 .4 .6 .8 1.0
'l' (K)
Figure 3.4
(on/p)dp/dT vs T for the pure K samples.
The samples are twisted at 9.3K without annealing. The details of
twist are given in Table 3.2.
78
ﬁ I I If I
" d
o K-l
O K-2
A K-3
_ 6 [-4 1
V s-s V
I- -I
' d
d
0
O
-o- -- - - - - - --
O
A
- '1
Q
aﬁ ‘
a a a
l J l L L
0
Figure 3 .5
.2 .4 .6 .8 1.0
1' (K)
(pulp)dp/d'1' - 2A'T vs T for the unannealed K samples
A step-like function is seen.
best computer fits. Details are given in the text.
The solid and dashed curves are the
Table 3.2
Pure K Sample Without Annealing
Sample e‘énncm) p. (nﬁcm) E,(K)
K-l
K-2
K-3
K94
K95
A - 2.6610.02 (fem/K?)
1.776
1.805
1.908
2.056
2.489
2.489
1.497
1.531
1.632
1.773
2.193
2.193
(mm/K )
c - 0.300
K-l untwisted
sea 77‘ at 9.3K
K-3 267' at 9.3K
540‘ at 9.3K
1329' at 9.3K
79
D,(fan)
0.219i0.060 0.0l2i0.009
0.152i0.046 0.01930.011
0.179:0.023 0.041t0.010
0.242:0.036 0.12410.036
0.16210.014 0.04110.009
E,(K) D,(£nmx)
0.58030.050 0.210t0.023
80
The solid curve for K95 shows the improved fit with a CTsterm added to
Eq.(3.6). From the Table 3.2 we see that the value of C is less than
0.35 film/K5 , the theoretical value predicted by Frobose (56) for
totally quenched phonon drag.
As we did with KRb-4, we can also improve the low temperature fit
for T < 0.6K by using a two frequency model
2 . ‘2 '2
p(T) 8 AT + (D,/4T)sinh(E,/2T) + (Dz/4T)sinh (El/2T) (3.7)
Fig. 3.6 shows such a plot. We fit Eq.(3.7) only to K-5 because the
improvement there was significant. The parameters are also given in
Table 3.2.
In Fig. 3.7 we plot the parameter D in Eq.(3.6) as a function of
A9, for both K(Rb) and K samples. The circles are the data from the K
sample which was twisted at 9.3K without annealing, and the triangles
are the data from the K(Rb) samples. We see in the plot that
deformation systematically increased 0 for both K(Rb) and K samples.
Indeed D varies almost linearly with 4% which is in agreement with our
expectation that D~JNa. For a comparison, we also plot Haerle's data in
the figure: the diamond is from his K(Rb) sample KRbhb, which was
deformed at 60K: and the square is from his K sample Kh9b, which was
deformed at 4.2K. We see that they are in agreement with our data
points even though a rather different method of deformation was used.
It should be pointed out here that the K(Rb) samples were deformed or
annealed at 60K, where the vacancies are believed to anneal out.
81
I9. 18, __ ,
7;.“ 2A? (mm/K) . .
6.. an
5_ .
i l L i l
0 .1 .2 .3 .4 .5
'I' (K)
Figure 3.6
A two-frequency fit vs the one frequency-fit for sample K-5.
An improved fit is obtained.
82
D (meK)
1.4- _ -
“.b (Yin are”
1.2 '- :15 (He-risers!) - d
00 0'
L L l L
O .2 .4 .6 .6 1.0 1.2
Figure 3.7 A Po ("0"“)
D vs 53 for K(Rb) and K samples
The coefficient D of Eq.(3.3) or Eq.(3.6) is plotted as a function of
#8 for the K(Rb) samples or pure K samples with out annealing. The
data from Haerle 35,31. are also shown. The dashed line is the fit to
the K samples after the vacancy contribution is corrected for. see
text for details.
83
However, the K samples were deformed at 9.3K, where vacancies, as well
as screw dislocations, were produced. From Tables 3.2 and 3.3 below we
see that for our sample K-5, which was deformed at 9.3K, AP, = 0.696 nn
cm, and for K—7, which was annealed at 60K,‘qq - 0.207 nncm. We see
that 70% of the increase in hp. with deformation at 9.3K is due to
vacancies. If we correct for this and plot D vs 9d (due to dislocation
contribution only), we get the dotted line in the figure.
We see from the plot that the.K(Rb) data have a slope higher than
that of the K sample. We know for a bcc metal that at 9K, deformation
produces more screw dislocations than edge dislocations, and at 60K,
deforming K is thought to produce more edge dislocations(6). Perhaps
the difference in the slopes is due to the different dislocations in
these K(Rb) and K samples or to the presence of Rb in one of them.
For the vibrating dislocation mechanism, we have discussed two
possible models in Chapter I. One is the Granato model (37) in which
the dislocation is considered to be a vibrating elastic band stretched
between two pinning points. The resonant frequency is given by
v= V/3L (1.25)
where V is the transverse phonon velocity and L is the pinned
dislocation length. From Eq.(l.11) we know thatp‘j is proportional to
the dislocation density N3: and if we simply assume N3; lle, then
v Apart (3.6)
84
Thus the characteristic frequency (or energy) is proportional to the
square root of 9d . The other model is associated with the Peierls
potential in which the dislocation oscillates. The characterestic
frequency is given in Chapter I as
v - (t,,/¢m’,:>mb)‘k (1.26)
where t:P is the Peierls stress, on is the mass density and b is the
Burgers vector. An important feature of this model is that the _
vibration frequency is independent of the dislocation segment length L
and consequently, is independent of pd .
In Fig. 3.8 we plot the characteristic energy 8, obtained by
fitting Eq. (3.3) or (3.6) to the K(Rb) data and the unannealed K data
as a function of Ape . We also plot the data from Haerle e_t 51. in the
figure, and we see that they are also in agreement with our data. For
our K(Rb) data, there seems to be a systematic increase in E with 43.
which might imply that the Granato model is applicable. Unfortunately,
the error bars are sufficiently large that we cannot discriminate
between E~ap° and E~(Ag )‘k. For our K samples: we do not see a really
significant M3, dependence in 8.
3.1.4 Pure K data after annealing
When we anneal the K(Rb) samples at 60K, we see a drop in the
85
E (K)
axes
as ...>(VIn em)
<>Klb
("eerie-cal)
as)
A P. (nﬂcm)
Figure 3.6
B vs do for K( Rb) and K samples
The ctnracteristic energy E is plotted as a fuhction of so for the
K( Rb) and K samples. The data from Raerle 95 31. are in agreement
with ours.
86
height of the step-like function in dp/dT (Fig. 3.3). However, when we
anneal the K samples at 60K, we see a peak rising even above the
height of the step of the unannealed samples. This is shown in Fig 3.9
in which we again plotted (eh 1/p)dp/dT—21I‘T as a function of T. The
dashed curve is the best fit to K95, which was twisted at 9.3K without
annealing. The solid curves are the fits to the data which we will
discuss below.
To fit this peak we tried 'several models. The vibrating
dislocation model Eq.(3.6) failed to fit this peak, even with two
frequencies. Then we tried to let A be a variable parameter in
Eq.(3.6). In Table 3.3 we present the fitting parameters obtained by '
using this method. Figure 3.10 is such a plot in which we show the fit
to K-7(solid curve). The parameter A' in Fig. 3.10 is from Table 3.2
and is derived from A!2.66 fan/K2; Since A in Table 3.3 is smaller than
2.66 film/K2, the fit in Fig. 3.10 has a negative slope at higher
temperatures. The fit was done for T < 0.55K where the phonon
contribution is negligible. To fit the data for T > 0.6K, a CT5 term
has to be added where C was found to be 0.30 film/K5, which is less
than the maximum value 0.35 film/K5 predicted by Frobose(56). We had an
improved fit. However, the resulting drop in A turned out to be hard to
explain since the theory of Kaveh and Wiser (19) predicted that the
introduction of dislocation would only increase A.
To avoid letting A become smaller, we then tried the following
method: we thought that the peak might be associated with some other
mechanism and used the following:
87
E5 _ -
° :-:
£5 - a x-s r
‘1 __ q
(3 e12 e“ e55 e53 1JC’
'l’ (K)
Figure 3.9
he)do/dT - 2A'T vs T for the annealed pure K samples
A peak is seen at 0.2K. The dashed curve is the best fit for the
unannealed sample K-S. The solid curves are explained in the text.
Table 3.3
Pure K sample with A as a variable parameter
Sample p¢.2(nncm) p. (nﬂcm) A (mm/K2) E (K)
K-l
K92
K93
K94
K95
K96
K-7
K98
1.776
1.805
1.908
2.056
2.489
2.095
1.985
1.853
1.497
1.531
1.632
1.773
2.193
1.806
1.704
1.574
88
2.655i0.018
2.422t0.073
2.31920.066
2.441t0.062
3.16710.129
2.289t0.137
1.638t0.088
1.959t0.074
D (ian)
0.242t0.046 0.022t0.009
0.156t0.023 0.02310.008
0.18320.016 0.04610.008
0.218t0.025 0.08910.020
0.254t0.020 0.174:0.028
0.22210.018 0.13210.021
0.14310.020 0.027i0.006
The temperature range is limited to T < 0.551!
89
. d . '
313-? -2AT (fﬂm/K)
p
6 - -
55 - v s-r q
4 - ..
3 - .
2 -[ / -
§'~'~.
1 - \ .
0 J _
0 .2 .4 .6 .8 1.0
I (K)
Figure 3.10
The improved fit to K97 with A as a variable parameter
The dashed curve is without the CT5term. A CT5 term helps to fit the
rising tail of the data.
90
Table 3.4
Pure K samples after annealing or deformation at 60K
(The parameters are for Eq. (3.10).)
Sample an1(nncm)p.(nncm) E,(K) D (M) 82(K) 8 (film)
K96 2.095 1.806 0.18410.014 0.07210.012 0.530t0.020 0.22210.025
K-8 1.853 1.574 0.51:0.36 0.18i0.23 0.182t0.052 0.145t0.079
K2-2 1.879 1.618 0.17210.021 0.05310.013 0.43010.016 0.240t0.019
A - 2.66io.02 (fem/x2)
K-6 annealed at 35K for 30 min.
K-7 annealed at 60K for 30 min.
K-6 annealed at 100K for 30 min.
A - 24320.02 (mm/x2)
K2-l untwisted
K2-2 2510° at 60K
91
-2 -
p(T) I AT1+ (D/4T)sinh (E,/2T) + a(l+bexp(Ez/T)) ‘ (3.9)
where the last term is the model of the localized electronic levels
associated with dislocations, also proposed by Gantmakher and Kulesco
[Eq.(l.23)]. With A still fixed, we had a very good fit: unfortunately,
the parameters a and b were so strongly correlated with each other that
we could not obtain any sensible values for them.
We finally used the following formula which gave us a curve that
was almost identical to that for Eq.(3.9):
-2 _l
p(T) - AT2+ (D/4T)sinh (E,/2T) + B[1 + (2/3)sinh2(Ez/2T)] (3.10)
where the third term, proposed by Fulde and Peschel(35), is due to
inelastic scattering off localized energy levels produced by a
crystalline electric field. The advantage of using Eq.(3.10) instead
Eq.(3.9) is that the parameter 8 was well-behaved in the computer fit,
which made it easier to analyse the data.
In Fig. 3.9 the solid curves show the fits using Eq.(3.10). We
limit the temperature range to T < 0.6K where the phonon terms are
negligible. The parameters from the least-squares fit using Eq.(3.10)
are given in Table 3.4. We see that the new term in Eq.(3.10) has been
used to fit primarily the prominent peak in the annealed K data. No
such term is needed for the annealed K(Rb) data.
92
3.1.5 Annealing at 60K after deformation at 9.3K vs
deforming directly at 60K for K samples.
If the peak arising in dp/dT-ZA'T for K after annealing (Fig. 3.9)
is due to some complicated process which occurred during annealing,
then the question arises as to whether the peak will still be there if
we directly deform our K sample at 60K. In Fig. 3.11 we make such a
comparison. K97 is the sample which was twisted by l329'at 9.3K and
then annealed at 60K. K2-2 is another K sample which was twisted by
25100 at 60K. The fitting parameters for these samples are given in
Table 3.4. In Fig. 3.11 we see that these two samples have remarkably
similar behaviors which correlate well with their similar values of 4p.
where ao’- 0.21 nncm and 0.24 n£1cm for K97 and K2-2, respectively.
Note that K2-2 required a much larger angle of twist at 60K. Thus the
electron scattering characteristics of the dislocations seem to depend
only on the' fact that the sample was heated to 60K and not on the
process of deformation and annealing.
3.1.6 Comparison of 60K annealed pure K sample with those
of Haerle 95 21.
In Figs. 3.7 and 3.8 our values of D and B were in good areement
with those of Haerle gt ‘31. We wish now to compare our results
presented in Fig. 3.9 - 3.11 with theirs. Since Haerle gt 31. were able
to fit their data for samples deformed at 60K with a single-energy
93
Ez_d£_ ’
P d'l' 2AT (film/K)
I I I I I
6 - est (Heads oral) ~
0....
e I7I-7 ,
. 0 “2-2 (7:0 e: a”
5 - ' . ~
e
e
e
e
4 - . , -
. e e
J L 1 ° 1 L
0 .2 .4 .6 .8 1.0
'l’ (K )
Figure 3.11
A comparison between annealing at 60K after twisting at 9.3K (K—7)
and twisting directly at GCK (K2-2)
A similar behavior is seen. The data of Haerle gt a_1_. are also
presented.
94
vibrating dislocation model [Eq.(3.6)], there would seem to be a
contradiction between our respective results. We now believe that this
disagreement is not real and that the "size-effect" contribution to
their undeformed 0.9-mm-diameter samples was probably not corrected for
properly in their deformed samples. In undeformed samples this size
effect shows up as an apparent e-e term inyo of the form: T" where n<2.
If their very lowest temperature data were constrained to fit a T2
behavior, then they obtained A - 1.5 film/Kz'which is much smaller than
typical values of A ( 2.5 mmle ) for our 2-Imn-diameter samples. It is
not known what happens to the size effect contribution when the sample
is deformed. Haerle g§,§l. fit Eq.(3.6) to their data and obtained 1.0
2 A z 1.5 film/1K2 for their deformed samples. Since the A's before and
after deformation were comparable, it was implicitly assumed by Haerle
_t '31. that the size effect was not significantly changed by
deformation. If, on the contrary, we assume that severe deformation
eliminated the size effect in the data of Haerle g§_§1. and raised A to
about 2.5 ﬁlm/K2, then their results look very much like ours. In Fig.
3.11 we present their data for sample K6hf which was severely deformed
at 60K with Ap.- 0.73 nﬂcm where A = 2.5 ﬁlm/Karather than their value
of 1.34:0.07 ism/K2. ‘ Plotted in this way, their data behave in a very
similar manner to ours. To uncover this unusual behavior in our 60K
annealed sample, 2-mm-diameter samples were necessary so that this size
effect was eliminated.
95
3.2 Thermoelectric ratio
From Chapter I we know that the thermoelectric ratio G for
potassium below 1K is expected to obey
c = c,+ 8T2, ; (1.40)
where c, is the diffusion term and sz'is the normal phonon drag term.
In Figs. 3.12 and 3.13 we present the 0 data for the K(Rb) and K
samples. We can fit K91 and ‘KRb-l reasonably well(solid curves) by
using Eq.(l.40), but we see the fit is not as good for the strained .
samples (the dashed curves). For most of the strained samples, there is
a maximum in G at about 0.5K. We then tried the following empirical
formula 1
0 - c,+ aT + bT2 , (3.11)
and we obtained much improved fits. The solid curves except K91 and
KRb-l show these fits. The reason for the down-turn at the lowest
temperatures, below 0.1K, is not known at this time.
The parameters obtained from the least-squares fit are given in
Table 3.5. It is clear that dislocations make a negative contribution
to Go in a systematic way.
In Chapter I we have shown the Garter-Nerdheim relation for
96
O Klb-l
+ Kit-2
A Kit-3
V KIb-4
O Klb-S
D KIb-O
5+
Ob
+5
Figure 3.12
G vs T for the K(Rb) samples
Details of the fit are given in the text.
1.2
97
J l
(DK-I
+ K-J
ax-e‘
VK-s
a K-e
oK-T
0
.2 .4 .6
'I' (K)
Figure 3.13
0 vs T for the pure K samples.
1.2
Table 3.5
Parameters in G for the K(Rb) and K samples
Suple P. (nﬂcm) G°(l/V)
KRb-l
KRb-2
KRb-3
KRb-4
KRb-S
KRb-6
11.61
11.85
12.04
12.68
12.29
11.73
1.497
1.531
1.632
1.773
2.193
1.806
1.704
1.574
0.415i0.001
0.36210.002
0.337t0.002
0.23010.001
0.30310.002
0.42010.002
-0.06510.003
-0.11910.004
-0.23010.006
-0.36910.003
-0.677t0.003
-0.408t0.004
-0.312i0.004
04(KRb) - -l.7ei0.19 (l/v)
04(K) - -i.967i0.014 (l/V)
98
a (l/VK)
0.09910.008
0.09310.010
0.07210.004
0.092t0.005
0.12410.014
0.11810.023
0.06310.013
0.10510.009
0.19310.015
0.204t0.015
0.17710.012
b (l/VK?)
-0.260i0.003
-0.185t0.006
-0.153i0.008
-0.115t0.003
-0.lsei0.003
-0.256t0s003
-0.30310.004
-0.358t0.010
-0.29610.018
-0.178t0.010
-0.18710.006
-0.276i0.011
-0.29210.011
99
diffusion thermopower:
S = Sd + (p:Ap)[Si- Sd] (1.32)
where Si and 6,, are the diffusion thermopowers due to impurity
scattering and dislocation scattering, respectively. We know that the
thermopower S can be related to the thermoelectric ratio 0 by
G = S/LT (1.38)
where L is the Lorenz ratio which is approximately a constant below 1K.
Therefore we have
G,3 644- (P; lp)[G.-- Gd] . (3.12)
where Gd and 0} correspond to the diffusion thermoelectric ratio due to
dislocation scattering and impurity scattering, respectively. If we
plot 0, as a function of l/p we should get a straight line intercepting
the 0, axis at Gd . Fig. 3.14 is such a plot for both K(Rb) and K
samples. Note that q,for the K(Rb) and K samples has the same value
within the experimental error: Gd(K(Rb)) a -l.76 t 0.19 (l/V) and Gd(K)
= -1.97 i 0.02 (l/V). This means that Gd is independent of the type of
impurity which is present in the sample. For K(Rb) the dominant
impurity is Rb, and for our pure K sample we have unknown impurities
with vacancies present in samples K2-K5. Nets samples K-6 and K-7 have
100
c. (v")
l l
0 .2 .4 .6
( R (nﬂcm) >"
Figure 3.14
A Gorter-Nordheim plot for both K(Rb) and K samples
04 seems to be the same in both cases.
101
vacancies being annealed out, and yet their G;s also follow the same
straight line as K92 to K-S. The simplest interpretation of this
unusual result is that Go for vacancies is very similar to Gd .
Fig. 3.15 is the plot of the coefficient b of the phonon drag term
vs,Ap5 for both K(Rb) and K samples. As is expected, we see that
dislocations also suppress the phonon drag term, which is consistent
with our dp/dT measurements. Note that the presence of Rb impurities
also tends to suppress phonon drag.
No systematic changes have been observed for a, the coefficient of
the linear term we used in G. Its value is about 0.10 (l/VK) for all
samples except for K96, K97 and K96 where it jumps to 0.20 (l/VK). NO'
theory has been found to explain this term so far.
102
b (VK )
_.1_ OK .1
e
i
d i
...:z.. s
é
7’ l
I i 1
"3?. I -
i
i
"40 :2 .14 .16 .8 1.0 1.2
AP. (nﬂcm)
Figure 3.15
b vs no for both K(Rb) and K samples
A systematic change is visible.
Chapter IV Discussion and Conclusions
Deformation has a profound influence on the electrical properties
of potassium. The residual resistivity of pure K has been found to
increase linearly with the twist-angle of deformation for small angles
(6 < 1400') at 9.3K. However, a tendency toward saturation in no. is
also observed when the twist angle 9 > 4600° at 60K, and this is
probably due to the high mobility of dislocations at this temperature
so that close dislocations with opposite Burgers vectors are more
likely to annihilate each other. For the K(Rb) samples which are
deformed at 60K or the K samples which are deformed at 9.3K} without
annealing (where impurities or vacancies are present), the
electron-dislocation interaction can be described by a vibrating
dislocation model proposed by Gantmakher and Kulesco (Eq. 1.27).
However, a two-frequency model has been used to give a better fit for
sample KRb-4, which is twisted 4600' at 9.3K and then annealed at 35K,
and sample K-5, which is twisted l329°at 9.3K without annealing. The
temperature is limited to T < 0.6K where the phonon contribution is
negligible. Deformation also suppresses the phonon drag which exists in
the unstrained samples. The coefficient C of this Nbrmal
electron-phonon term CT5 has been found to be ~ 0.30 fﬂ m/Ks , which is
less than its maximum value 0.35 frlm/K5 predicted by Frobdse (56).
The coefficient of the vibrating dislocation term 0 increases
103
104
close to linearly with the change in 9. due to deformation. This is
expected because D should be proportional to the total dislocation
length in the sample. D vs Apa for K(Rb) suples apparently has a much
higher slope (Fig. 3.7) than the pure K samples. However, we know the
increase in p. for K when it is deformed at 9.3K contains also the
contribution of vacancies which could be as large as 70% of the total
increase in ap. . If the vacancy contribution is corrected for, this
slope for the K samples increases and becomes about 67% of the slope
for K(Rb). The remaining discrepancy in the slopes might be ascribed to
the higher yield stress in potassium when the Rb impurities are added,
since it is observed that K(Rb) has a higher yield stress than pure K
at room temperature by us and at liquid nitrogen temperature by Hands
and Rosenberg.(59)
The characteristic energy E of this vibrating dislocation term
seems to have a go, dependence for the K(Rb) sample (Fig.3.6). For the
K sample this dependence appears to be smaller. However, if we correct
“q. for the vacancy contribution to the pure K sample in the same manner
as in Fig.3.7, then the slopes in Fig. 3.6 could be the same for both
samples, with K(Rb) having on average a larger E. If we use the Granato
elastic band model of the vibrating dislocations, we might be able to
explain the 5p. dependence in E, since the frequency is proportional to
the square root of the dislocation density. However, both slopes do not
extrapolate to £90 as apo-rO, and this might suggest some other
mechanisms. 7
The average Rb-Rb atom distance in our alloy is about 50A. If we
105
use the Tsivinskii model (Eq.l.10) with an ion radius for K of x =
1.33A and x. 8 O for the case of vacancy, we can obtain the average
vacancybvacancy distance for the most severly deformed sample K-S (;%==
0.5 nlicm) to be about 240A. If we use Eq.(1.6), the correSponding
dislocation separation is about 4000A ( pd- 0.2 nﬂcm for K—S). If, we
think that the dislocation is a pinned elastic band of length 40001,
its resonant vibrating frequency is about 1.5x109 Hz or the
characteristic energy E is about 0.1K for sample K-S, which is a
reasonable value if we compare it with the experimental value for K-5:
5: 2: 0.2K . The K(Rb) apparently has a higher value for 1: than that of
the K sample, even if the vacancy contribution is corrected for in the
pure K samples. This might be ascribed to the, presence of the Rb
impurity which modifies partially the pinning distances of the
oscillators so that the average pinning distance becomes smaller, and
thus the characteristic energy E is higher. If we use the Rb or
vacancy separation length as the pinning length, then we would obtain
an E with a value much higher than our experimental ones. If we use
the model (Eq.1.26) in which the dislocations oscillate within the
Peierls potential, we might. not expect any App dependence in E.
However, this is not always true: if the yield stress d} depends on the
concentration of either impurities or dislocations, we might have a AR,
dependence in B. We are planning to explore further the impurity
dependence of E and D by deforming K(Rb) alloys with different Rb
concentrations. We also want to see if there exists a vacancy
contribution to E or D for K(Rb) samples, since our K(Rb) samples were
106
all annealed above 35K where most vacancies were annealed out. Thus we
plan to deform K(Rb) alloys at 9.3K and measure dp/dT below 1K.
For our 60K.annealed pure K samples where the vacancies are
annealed out, we see a peak in qp/dT which cannot be fit by the
vibrating dislocation model, even with a two-frequency one. It is found
that this peak does not depend on the process of annealing, since our
sample K2-2, which was directly deformed at 60K, also shows the peak in
dp/dT. This peak is apparently suppressed by the impurities because we
do not see any of this in the K(Rb) samples which were also annealed at
60K. By letting A vary we can get an improved fit even with the
one-frequency vibrating dislocation model. However, the drop in A is '
hard to explain considering that the theory of Kaveh and Wiser(19)
predicts an increase in A after deformation. The drop in A might be
associated with the rearrangement of the Q—domain structure predicted
by Bishop and Lawrence(4), but a more detailed theory is needed before
this idea can be explored experimentally. This unusual peak can be fit
instead by keeping A fixed and by adding a new term to the vibrating
dislocation model, and this new term is the localized-energy-level
model associated with dislocations which was proposed also by
Gantmakher and Kulesco in the same paper.(33) An alternative term from
Fulde and Peschel (Eq.1.24) was actually used to obtain the same fit
because it gave more sensible values for the fitting parameters.
The appearance of this localized-energy—level term raises the
possibility that annealing the K sample at 60K porduces a rearrangement
of this energy-level distribution and that the energy-level
107
distribution before annealing produces a behavior in dp/dT which looks
like the vibrating dislocation model. This idea also needs to be
further explored.
For our G data we obtain reasonably good fits by using Eq. (1.40)
for our unstrained samples. However, a new peak is visible for the
strained K(Rb) and K: samples. An empirical term, aT, which has no
theoretical explanation, has been added to obtain the best fit. The
Gorter-Nordheim plot (Fig.3.14) exhibits good straight-line behaviors
for both the K(Rb) and K samples. The characteristic diffusion term Gk
due to dislocation scattering is found to be the same for both samples.
Analysis of this Gorter—Nordheim plot for pure K suggests that the
dislocation scattering and vacancy scattering produce quite similar
contribution to G.. If this is not the case, then the Gugan and Gurney
assertion that vacancies anneal out above about 10K must be
re-examined. For example, the above behavior for the Ggof pure K could
be explained as being due to lowering of the dislocation density rather
than a reduction of vacancy density as the sample is warmed from 9K to
60K.
This study has been a continuation of M. Haerle's work, with
emphasis on the electron-dislocation interaction in K below 1K. By
using a completely different mechanism of deformation with much
better control of sample geometry, we have observed similar behaviors
in dp/dT. By using a second dilution refrigerator, we extended the
lowest temperature down to 20 mK, which is much lower than that of
Haerle's (BOmK). This extended region has helped us in determining the
108
multi—frequency spectrum of the vibrating dislocations. We used 2mm
diameter samples to avoid the complication of the size effect observed
in Haerle's work.
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