ATreWJATION CF ULTRASONIC SHEAR WAVES TRUE7IT …

ATreWJATION CF ULTRASONIC SHEAR WAVES

IN SUPERCCNDOCTING MOLYBDEMW

by

TRUE7IT IWCMAS AUSTIN, B.S,

A TTiESIS

IN

PHYSICS

Suhnu-tted to the Graduate Faculty of Texas Tech University in Partial Fulfilhiient of the Requirements for

the degree of

MASTER OF SCIENCE

ApprcDved

Accepted

August 1976

* • ^ '

ACKNOWLE DGMENT S

I am very grateful to Dr. B. J. Marshall for his

direction of this thesis and to the Robert A. Welch Founda

tion for financial support. I also wish to thank Dr. C. R.

Cleavelin for his invaluable assistance and advice and Dr.

Virgil E. Bottom for arousing my interest in studying

physics. I wish to thank Wayne Bailey and Eddie Breashears

for hours of valuable discussion which they have contributed

I am also grateful to Joyce Rowe and Marcia Yarbrough for

typing the manuscript.

11

TABLE OF CONTENTS

ACKNOWLEDGMENTS ii

LIST OF TABLES V

LIST OF FIGURES vi

I. INTRODUCTION 1

1.1 Superconductivity 1

1.2 Background and Purpose of the Research . 11

1.3 Ultrasonic Attenuation of Shear Waves in Normal Metals 15

1.4 Ultrasonic Attenuation of Shear Waves

in Superconductors 21

1.5 Molybdenum 2 3

II. EXPERIMENTAL 27

II. 1 Introduction 27

II.2 The Pulse-Echo Method of Attenuation

Measurement 2 8

II. 3 The Attenuation Coefficient a 30

11.4 Electronic Equipment 34 3

11.5 He Refrigerator-Cryostat System . . . . 40

II. 6 The Thermometry System 44

II.7 Transducers and Transducer Bonding . . . 47

II. 8 The Sample 49

III. RESULTS AND INTERPRETATIONS 5 3

111.1 Ultrasonic Attenuation Data 5 3

111.2 Analysis of Data 57 iii

III. 3 Results 61

A. Energy Gap Determination 61

Transition Temperature 61

Results for q||[100], ^||[110J . . . 61

Results for ct||[110], s'||[ir0J . . . 69

Results for q||[110], s"||[001] . . . 75

Results for q | | [111] , s Arbitrary . 75

Summary of Energy Gap Results . . . 79

B, Electronic Attenuation as a Function of Frequency for Longitudinal Waves. 80

Transverse Wave to Longitudinal Wave Attenuation Ratios 82

Results for q||[100J 86

Results for q||[110] 89

Results for q||[lll] 92

C. Electronic Attenuation as a Function of Frequency for Transverse Waves. . 95

Results for q||[100], s| | [110] . . . 98

Results for q| I [110], s||[ir0] . . . 101

Results for qj I [110], s||I001] . . . 101

Results for q||[lll], s Arbitrary. . 101

III. 4 Interpretation of Results 108

III. 5 Summary Ill

LIST OF REFERENCES 112

APPENDIX . . . . . 118

iv

LIST OF TABLES

Table

1. Comparison of Energy Gaps in Molybdenum as Determined in Various Investigations. .

2, Energy Gaps From Shear Wave Attenuation Data

3, Electronic Attenuation vs. Frequency Data ,

4, Ratios of Transverse to Longitudinal Electronic Attenuation

Page

63

64

83

84

5. Estimated Electron Mean Free Paths 97

6,

7.

8.

9.

10.

11.

12.

Data for q Echo No.

Data for q Echo No.

Data for q Echo No,

Data for q Echo No,

Data for q Echo No,

Data for q Echo No.

Data for q Echo No.

13. Data for q Echo No.

14.

15.

16.

Data for q Echo No.

Data for q Echo No.

Data for q Echo No.

[100], 3| 10

I [100], s|

[100] , s

I [110], t\

[110], s|

I[110], s|

I [110], t

I[110], t

[110], 9.0 MHz,

[110], 29.92 MHz,

[110], 51.85 MHz,

[110], 9.3 MHz,

[110], 30.23 MHz,

[110], 50.41 MHz,

[001], 9.22 MHz,

[001] , 32.9 MHz,

I [111], s "arbitrary, 10.64 MHz,

[111], s arbitrary, 31.55 MHz,

[111], s arbitrary, 52.04 MHz,

119

119

120

120

121

121

122

122

123

123

124

LIST OF FIGURES

Figure Page

1. Schematic Representation of Shear-Wave Attenuation 14

2. Schematic Representation of the Residual Attenuation 19

3. Fermi Surface and Energy Bands for

Molybdenum 26

4. Specimen, Bond and Transducer 29

5. Ultrasonic Echo Train Display 29

6. Block Diagram of Electronic System 35

7. Selected Waveforms from Electronic System . . . 37 3

8. He Refrigerator-Cryostat System 41

9. Composite Recorder Trace of Attenuation Data . 54

10. Plot of Ln (—2. _ 1) Versus G(t)/t at 9.0 MHz for ql I [100] ^ 62 11. Energy Gap Function A^t) for q||[100] at 9.0 MHz 66

12. Energy Gap Function A^t) for q||[100] at 29.92 MHz 67

13. Energy Gap Function A^t) for q|| [100] at 51.85 MHz 68

14. Energy Gap Function drit) for q||[110], $11 [110] at 9 .3 MHz" 70

15. Energy Gap Function A^t) for q | | [ 1 1 0 ] , ^11 [ 1 1 0 ] , a t 30.23 MHz 71

16. Energy Gap Function A^t) for q | | [ 1 1 0 J , ^ | | [ l l 0 ] , a t 50 .41 MHz 72

17. Energy Gap Function A^t) for q | | [ 1 1 0 J , ^ | | I001J a t 9.22 MHz 73

vi

18. Energy Gap Function Ut) for c||[110], sI I[001] at 32.9 MHz. . . . . . . . 74

19. Energy Gap Function A^t) for q||[lll] at 1 0 . 6 4 MHz 76

20. Energy Gap Funct ion A^t) for q | | [ l l l ] a t 31.55 MHz 77

2 1 . Energy Gap Funct ion /\4t) for q | | [ l l l ] a t 52.04 MHz 78

22. Longitudinal Wave Electronic Attenuation vs. Frequency for q||[100], 0-100 MHz . . . . 87



25. Longitudinal Wave Electronic Attenuation vs. Frequency for q||[110], 0-1000 MHz. . . . 91

26. Longitudinal Wave Electronic Attenuation vs. Frequency for q||[lll], 0-100 MHz . . . . 93

27. Longitudinal Wave Electronic Attenuation vs. Frequency for q||[lll], 0-1000 MHz. . . . 94

28. Shear Wave Electronic Atter^uation vs. Frequency for q| I [100] , s||[110] 99

29. Shear Wave Electronic Atterjnation vs. Frequency for q||[110], s||[110] 102

30. Shear Wave Electronic Attenuation vs. Frequency forq||[110], s||[001] 103

31. Shear Wave Electronic Attenuation vs. Frequency for q||[111], Arbitrary Polarization 104

32. Shear Wave and Longitudinal Wave Attenuation for q| I [111] 107

vii

CHAPTER I

INTRODUCTION

I.l Superconductivity

Superconductivity is a remarkable phenomenon exhibited

by certain metals which display two unusual properties at

very low temperatures: (1) the abrupt disappearance of all

electrical resistance; and (2) the expulsion from the metal

of all lines of magnetic flux. The discovery of supercon

ductivity occurred when the first of these two properties,

zero resistance, was observed in the metal mercury by the

Dutch physicist Heike Kamerlingh Onnes in 1911.^ Shortly

thereafter, he observed the same behavior in lead, tin, and

indium. In each case, the change occurred below a certain

temperature characteristic of the given metal, viz., the

critical or transition temperature (T^). Today there are

twenty-four known elemental superconductors, with transi

tion temperatures ranging from 0.01 K (tungsten) to 9.3 K

(niobium). A large and growing number of superconducting

compounds and alloys have also been found. The search for

alloys with higher transition temperatures is a research

area of great interest today; the highest temperature 3

recorded to date is for Nb^Ge at 23.2 K.

The striking things about the disappearance of resis

tance, as Onnes pointed out in his 1913 Nobel speech,^ were

that it was abrupt rather than gradual, and it was complete

rather than leaving a small residual resistance which would

be expected from crystal imperfections. Onnes also found

that in addition to being destroyed above the critical

temperature, the property could also be destroyed by exceed

ing a certain critical current density or critical magnetic

field within the sample.

The second property was not discovered until 19 33

when Meissner and Ochsenfeld^ discovered that in addition

to being perfect conductors, superconductors were also per

fect diamagnets, that is, they expelled all lines of magnetic

flux. This second property, the "Meissner Effect," although

discovered much later is today considered to be the more

fundamental property and has served as the basis for several

of the modern theories of superconductivity.°'^

The properties of infinite conductivity and perfect

diamagnetism are closely related, however, in that external

magnetic fields can be thought of as being completely can

celled within the body of a superconductor by the magnetic

field produced by a resistanceless "supercurrent" flowing on

the surface of the superconductor. The supercurrent was

predicted by the phenomonological theory of F. and H. London

in 1935^ to have a penetration depth, X^, in which the mag

netic flux density fell to 1/e of its surface value. This

3

penetration depth was later confirmed experimentally^ and

led to the concept of coherence length which will be dis

cussed later.

Below their transition temperatures, superconductors

exhibit Ixjth normal state and superconducting characteris

tics, the former being most evident just below T^ and fall

ing off rapidly with decreasing temperature until, at abso

lute zero, the metal exhibits only the superconducting

characteristics. This behavior is evidenced by measurements

of the temperature dependence of the specific heat and

thermal conductivity at low temperatures. The fact that

there is a jump in the specific heat but no latent heat at

the transition temperature makes the transition a second

order phase transition, and it also indicates that there is

an increase in the rate at which the entropy of the system

decreases as the temperature is reduced.

In 1934 a model was proposed by C. J. Garter and

H. B. G. Casimir-^^ which could explain qualitatively the

behavior described above. The "two-fluid" model, as it was

called, assumed the existence of two interpenetrating "seas

of electrons'", one sea being the normal electrons, the other

being highly-ordered superelectrons. The fraction of super-

electrons would increase from zero at the transition tempera

ture to 100% at 0 K. Although the two-fluid model was a

significant advancement, it was not exactly correct quantita

tively and did not give a detailed picture of what might be

taking place on the microscopic level.

4

Several important advances, both experimental and

theoretical, began to occur in the early 1950's which laid

the foundation for the development of a successful micro

scopic theory. The first of these was the independent dis

covery in 1950 by E. Maxwelll2 and by Reynolds,^3 ^t al., of

the isotope effect, that is, that the transition temperature

Tc is proportional to M"-^/^, where M is the isotopic mass of

the superconductor. The isotope effect indicated that the

mechanism which produces superconductivity involves some

type of interaction between electrons and vibrations of the

crystal lattice, or phonons. H. Frohlich"^^ showed inde

pendently in the same year, on theoretical grounds, that

superconductivity could be produced by an electron-phonon

interaction.

Another important idea was first hinted at by Fritz

London in 1935, viz., that an energy gap existed between

the normal electrons and the superelectrons. This received

some strong experimental support in 1954 from specific heat

measurements^^ on superconductors. These revealed that the

functional dependence at very low temperatures, especially

below T^/3, was exponential in T rather than following the

T^ relation which it seemed to fit at higher temperatures.

An exponential dependence of specific heat would be the

natural result of an exponential probability of normal

electrons being raised above an energy gap to become super-

electrons.^

5 1 7 John Bardeen-^' subsequently showed that an important

concept advanced by A. B. Pippard^^ in 1953, that the highly

ordered state of the superelectrons was nonlocal, would

likely follow from an energy-gap model. Pippard's predic

tion was that the orderliness would extend over a distance

^ which he called the coherence length (M0~^ cm in pure

metals). This theory of Pippard's was really a crystalliza

tion of a conjecture of F. London in his book published in

19 1950 that a superconductor is a "quantum structure on a

macroscopic scale (which is a) kind of solidification or

condensation of the average momentum distribution" of the

electrons.

The concept of an energy gap was confirmed experimen

tally in 1957 when Glover and Tinkham^O published the re

sults of infrared and microwave absorption spectroscopy

measurements of thin film superconductors. A gap in the

electron energy spectrum of about 3KgT^, where Kg is Boltz-

mann's constant, was found.

The above ideas and experimental results each played

an important role in the development of the successful

microscopic theory brought forth by Bardeen, Cooper, and

Schreiffer^l in 1957. The BCS theory, as it is named after

its authors, forms most of the basis for the analysis of

the experimental results obtained in the present paper.

Since the mathematical details are presented in a number of

other places, 21,22 only a qualitative discussion of the

salient features will be given here.

6

The BCS theory shows that below the superconducting

transition temperature it becomes energetically favorable for

the conduction electrons of a superconductor to become part of

a new, highly ordered state by combining to form "Cooper-

pairs." These Cooper-pairs all have the same energy and are

no longer independent of one another, as are the normal elec

trons, which are fermions. Hence the superconducting ground

state, which is the common energy state occupied by the Cooper-

pairs, acts in many respects like a form of Bose condensation

which is a completely correlated set of bosons.

The attractive interaction which produces this pairing

of electrons is the one first shown to be possible by Frohlich-*- ;

the electrons are coupled by the exchange of a virtual phonon,

or excitation of the crystal lattice. The excitation is pro

duced by the interaction between the passing electron and the

positive ion site. The first electron, with wave vector k-, ,

emits a phonon with wave vector q = hv^/Vg, where hv^ is the

energy of the phonon and Vg is the velocity of sound. The

wave vector of the first electron becomes k . The resulting

interaction can be expressed by

(1) k; = k^ + q.

Then the second electron absorbs the phonon, resulting in a

change in its wave vector also, as given by

(2) k2 + q = k2.

The complete interaction conserves the total wave vector:

->- -•- -• / -*• /

(3) k^ + k2 = ki + k2.

The total energy is also conserved by the complete inter

action (3), but the two individual steps (2) and (3) do not

conserve energy; this is possible due to the energy-time

uncertainty relation, AG«AT==)f(. The fact that the energy is

not conserved during the two intermediate steps is what

makes the phonon a virtual phonon. Such virtual excitations

are possible if the interaction time is very short.

The natural frequency of the virtual phonon would be

given by hv^ = e^ - e^, where e^ and e^ are the energies of

the first electron before and after the emission of the

virtual phonon. Its actual frequency, v^, however, would

correspond to the average lattice frequency, which could be

higher or lower due to the uncertainty principle. If higher,

the resulting force between the two electrons can be shown

by detailed quantum mechanics to be attractive rather than

23 repulsive. Under these circumstances a Cooper-pair would

be formed providing that two further conditions are satis

fied: (1) The attractive force resulting from the above

electron-phonon interaction dominates the normal Coulomb

repulsion between electrons; and (2) thermal phonons do not

interrupt the process.

The first condition is satisfied in metals which natu

rally have strong electron-phonon interactions, a property

which in the normal state produces the reverse characteristic,

poor conductivity. This is due to scattering of the conduc

tion electrons in the normal state by thermal phonons. The 24

good conductors like the alkali metals eind noble metals are

8

not superconductors for the same reason.

The second condition begins to exist when the tempera

ture falls to a sufficiently low value that the dominant

thermal phonon mode no longer has enough energy to interrupt

the pairing process, i.e., at the transition temperature and

below.

Conservation of momentum requires that the total momen

tum of two interacting electrons remain unchanged. The pair

ing interaction, or scattering via a virtual phonon, may

occur many times for a given pair, always with the same re

sulting total momentum. The maximum decrease in energy for

a given pair occurs when the maximum number of states is

available to which they can scatter, and this can be shown

to be the case when the two electrons have equal and oppo

site momenta. Also, due to the quantum mechanical exchange

"force," the greatest probability of two electrons being

close to one another, and hence the maximum lowering of

energy, will occur when they have opposite spins.

The net result of two electrons associating together

under an attractive force is a decrease in potential energy.

However, both electrons are changing their kinetic energies

to the new states represented by Jc, and k2; these states

must first be vacant for this to occur. Since Fermi-Dirac

statistics predicts that all states would be filled up to

the Fermi level at cU solute zero for a normal metal, two

electrons trying to form a Cooper-pair at that temperature

would have to move to new states slightly above the Fermi

9

level, resulting in an increase in the total kinetic energy

of 2(e^ - e^), where GT is the average energy (kinetic) of

the new states and e^ is the Fermi energy. Hence, in order

for the pair formation to be energetically favorable, that

being the case when the decrease in potential energy exceeds

the increase in kinetic energy, both electrons must be

initially fairly close to the Fermi level so that only a

small change in the kinetic energy would be necessary.

The BCS theory predicts that only electrons lying

within an interval of the order of magnitude K_T (where o C

Kg is Boltzmann's constant) above or below the Fermi surface

will enter into the pairing processes. It also predicts

that there will be vacant states below, and occupied states 25 above the Fermi level, even at absolute zero. The number

of electrons in states just below the Fermi level must be

large, however, to provide a sufficient density to allow

superconductivity.

The BCS theory takes as its fundamental assumption

the idea that the above-described pairing interaction between

two electrons is the only interaction of importcince between

26 electrons in the superconducting state. It then proceeds

to calculate the probability function h. for the occupation

of pair states by determining the conditions under which the

free energy of the system will be a minimum when the correla

tion potential of the pairing interaction is considered.

At T = OK, the results are

10 1 ^1 " ^f

(4) h « ^ [1 i^ ^ wo i ^f

where e^ is the Fermi energy, e. is the average energy

(negative interaction) of an electron pair relative to the

Fermi energy, and A is the energy gap, a quantity of funda

mental importance. The quantity A is given by

(5) A = 2hv^ e x p l - j ^ ^ j

where v^ is the average lattice frequency (about one-half

the maximum Debye frequency), N(e^) is the density of states

(ignoring spin) of the normal metal for electrons at the

Fermi surface and -V is the interaction potential associated

with the virtual phonon process.

The total energy required to break a pair into two

quasiparticle states k. and k. would be given by

(6) E = {(e^-e^)^ + A^}^/^ + {{e^-e^)^ + A^}^/^

since in reality not one but two possible pair states would

be destroyed. Thus even under conditions of minimum exci

tation (i.e., e- = £. = e ^ ) , it would be impossible to break

up the pair without applying a minimum energy of 2A. The

BCS theory predicts that the energy gap at OK is related to

the transition temperature by

(7) 2 A (0) = 3.5 KgT^

Above absolute zero, the energy gap is a function of temp

erature, A = A C T ) , falling in value with increasing temp

erature to zero at T^. c

11

1,2 Background and Purpose of the Research

The experimental research described in this thesis

pertains to the measurement of the attenuation of trans

verse waves of ultrasound propagating in molybdenum at very

low temperatures, viz,, in the neighborhood of and below

the transition temperature T^ C^ 0,92 K) at which molybdenum

becomes a superconductor. Transverse (or shear) waves are

those with atoms vibrating perpendicular to the direction

of travel or propagation of the sound wave.

The ultrasonic shear waves were in the frequency range

from ten to seventy megahertz, and the attenuation measure

ments were made for the waves propagating in each of the

three basic crystallographic directions of molybdenum, viz.,

the [100], [110], and [111] directions, as denoted by the

Miller indices. In addition, for the [110] direction,

measurements were made for each of two possible planes of

polarization of the sound wave, the [001] and [ifo] planes,

for which results could be expected to be distinctly dif

ferent.

The results of the above measurements have been 21

analyzed in terms of the BCS theory of superconductivity

27

and the Pippard theory of ultrasonic attenuation in metals,

both of which are discussed elsewhere in this thesis. The

main objective of this work was the determination of the

superconducting energy gap in molybdenum for each of the

above directions and polarizations. Also the frequency

12

dependence of the ultrasonic attenuation has been measured

for comparison with predictions from Pippard's theory.

This experiment is a supplement to a previous study

made on the same molybdenum crystals by O'Hara and Marshall28

in which longitudinal sound waves were used to determine

the superconducting energy gaps. The electron-phonon inter

action for shear waves takes place by a completely different

mechanism than that for a longitudinal wave. Consequently,

the investigation of the superconducting properties of

molybdenum by shear ultrasound waves gives results which

are independent of the results found for longitudinal

ultrasound waves. Furthermore by properly orienting the

polarization of the shear wave additional properties of the

Fermi surface can be studied.

In general, transverse waves travel slower than longi

tudinal waves; they also are more selective, interacting

with a smaller part of the Fermi surface of a metal. The

Fermi surface is the surface of constant energy EJ in k-space,

where e, represents the energy of the outermost filled energy

level at 0 K, the Fermi energy, and k is the wavevector of

the electron on the Fermi surface. In particular, shear

waves interact primarily with electrons moving in the so-

called effective zones of the Fermi surface parallel to the

direction of polarization of the sound wave.^^ Effective

zones are regions of the Fermi surface which are parallel

to the sound propagation direction so that the outwardly-

13

directed Fermi velocities of the electrons move essentially

in phase with the sound wave.

The actual nature of the interaction is also different.

Attenuation vs. temperature plots for longitudinal waves

experimentally satisfy the functional form predicted by the

BCS theory quite well; that is

a (8) rr- ^ 2 F(A) = P7-± a ' A/K T

n e^'^B + 1

where a^ and a^ are the longitudinal-wave attenuation

coefficients in the superconducting and normal states,

respectively, and F is the Fermi function of the temperature-

dependent superconducting energy gap.

The attenuation of transverse waves has been found

experimentally to behave quite differently; just below the

transition temperature the attenuation undergoes a sharp drop

within a very small, but finite temperature region. This is

known as the "rapid-fall" region. The discontinuity has

been observed to be dependent upon the value of the product

qZ, where q is the sound wave vector 27r/X and t is the

electron mean free path. After this initial sharp drop,

the residual attenuation a seems to obey the BCS relation.

Equation 8, about as well as does the longitudinal attenua-31 tion. Figure 1 illustrates the two regions.

This behavior has been qualitatively well explained

by the Pippard theory of attenuation of ultrasonic shear

waves in normal metals, with a modification to include the

14

4J C

w-

I

1

c 0 -H Cn (U «

£

rH r-i <d b

73 -H

a (d a;

c: 0

•H D 0) «

c 0

•H •P fO J3 C 0) 4J -M <

iH (d :3

-a •H (Q 0) o:

3

C3

O

Oi

N

u •H

•H ^ 0

c ja o 4J

o <D

f-i

0)

(1) >

•H

s a 0 u *u *—'

<0 H ^ 1

u (0 Q)

j : : (0

«M 0

c o

•H 4J «d 4J

c (U m 0) M

Q) M P 4J <0 V4 (1) a g 0) •p

(0 p CO

u <u >

(0

c: 04 O Q) }^

o-•H 4J OJ g (1)

c 0 H 4J ^ trS r-i 3 ro C Q •

X 4J «M O W

4-> 0) (0 U

cn CM

CN

15 32

superconducting region. In some more recent experiments,

however, there have been cases in which no clear-cut "rapid-

fall" region could be distinguished. This has occurred

primarily in the superconducting transition elements, such

as vanadium and niobium, in which resistivity ratio measure-

ments have indicated that qil<l. This result is not too

surprising, since theory predicts that the rapid-fall

phenomenon, if detectable, would be quite small for qJl<l,

More recently Leibowitz has failed to observe a

rapid-fall region in niobium under conditions where q£ well

34 35 exceeded one, ' Under these conditions a large rapid-fall

36 region would be expected. Similar results have been

asserted for zinc, cadmium, and molybden\m\ in unpublished

37 work by Almond and Rayne.

Magnetoacoustic measurements of electron mean free

paths in the molybdenum crystals used in this experiment

indicate that in the frequency range investigated, the qi 28 product is always greater than unity. Molybdenum is also

one of the transition metals. One of the objectives of this

experiment therefore was to try to observe whether rapid-

fall occurred in the various sound propagation directions

and, if so, its extent euid dependence on frequency.

1*3 Ultrasonic Attenuation of Shear Waves in Normal Me tali" "

Ultrasonic attenuation measurements provided an early

38 39 test of the detailed structure of the BCS theory ' and

16

provide one of the more effective methods of measuring the

superconducting energy gap. In order to understand ultra

sonic attenuation in superconductors, however, some under

standing of ultrasonic attenuation in normal metals is needed,

Ultrasonic sound waves are primarily attenuated at

low temperatures both in normal metals and in superconductors

by interacting with (1) impurities and crystal lattice dis

locations, and (2) conduction electrons. Attenuation due

to impurities and dislocations has been found in general

to be temperature-independent. The remainder of the attenua

tion, the electronic attenuation, will vary with temperature

and is the part which is of interest in this experiment.

These electron-phonon interactions are most frequent when

the wavelength of the sound is comparable to the electron

mean free path. At temperat\ires well above absolute zero,

the electron mean free path is limited to relatively small

values due to collisions with thermal phonons; it becomes

longer with decreasing temperature due to the decreasing

density and energy of the thermal phonons. Hence the con

duction electrons begin to be more available to interact

with ultrasound phonons and this electron-phonon interaction

becomes the dominant attenuation mechanism at very low

temperatures.

Morse,^^ Mason, and Kittel each showed, using

semi-classical treatments, that the electron-phonon inter

action is the cause of the large increase in ultrasonic

17

attenuation in normal metals at low temperatures. The mean

free paths of the electrons become limited by impurities at

a given low temperature, causing electronic attenuation to

lev^l off to a constant value below that temperature.

The first successful semi-quantitative theory for

ultrasonic attenuation of both longitudinal and shear waves

in normal metals was published by A. Pippard in 1955.^^

The theory which he derived for shear wave attenuation was

based upon a purely electromagnetic interaction between the

stress wave and the conduction electrons. This theory was

based upon an "ideal" metal (one obeying the free-electron

theory). The theory involved a well-defined relaxation time,

T, for restoration of an equilibrium electron configuration.

Pippard showed that in the region where electron mean free

paths are limited by impurities, the normal state attenua

tion a of the transverse wave is given by

Nm , (9) a = ^ (i^)

n p^VgT g where

(10) g = _ 3 (iai)i_t^ tan-^ (q^)-!} 2(q£)'^ "^^

In the above, N is the number of electrons per unit volume,

m is the free electron mass, p is the density of the metal, e o

v is the transverse wave velocity, q is the wave number of s

the sound wave (27r/X) , and I is the electron mean free path.

Two additional types of interaction were later pro

posed. Holstein^^ showed that a "collision-drag" effect could

18

contribute to the electronic attenuation, and Morse sug

gested that a contribution to the attenuation would be made

by the "shear-deformation" of a nonuniform Fermi surface by

a passing stress wave. The latter would occur in a "real"

metal, i.e., a metal which did not obey the free-electron

theory. Both of these processes would produce fictitious

forces on the electrons which would give rise to dissipative

effects when the lattice strain is time-varying.

27 Pippard subsequently treated the real metal case,

including both the shear-deformation and collision-drag, or

"relative velocity" effect, as he called it. The resulting

general expression he derived for normal state electronic

attenuation was

(11) a = hcL TJHialJI , eL p -Ji I ) ^ 477" pVg Jl + (q^) cos (t> 47r' hq ^ ^ "

where

(12) I, = rq£ ((IDq^COS ^ ) dS ^ ^ ^ 1 + {qJO cos"^ (j)

The angle (|) is the angle between v^and q, v^being the Fermi

velocity; I. is the vector mean free path; andJD = V *(u/u),

the component of in the particle velocity direction, where

"p represents both the collision-drag and shear-deformation

interactions.

(13) "p = K + ic cos (|)

where K is a deformation peurameter, and k is the electron

wave vector. Finally the parameter P^.^ is the resistivity

tensor of the periodic electric field associated with the

19

c o

•H 4J Its 3 C 0) 4J 4J

3 73 •H (0 0)

0) X u

o c o

•H +J . (d UH 4J C 0) (0 (1)

0)

u N 4J

O i - H 0)

u o

-H (d B Q)

X u 1 I .

CM

•H (X4

•H

E O V4

('jP^'o/H t)

20

acoustic wave, characterized by wave number q. Equation

(11) reduces to Equation (9) when the integrals are eval

uated for the free electron case where K = 0.

Equation (9) can be evaluated in three regions of

interest by considering the approximations resulting from

different values of ql in Equation (10). These regimes are

as follows: ^ Nm V. T 2 ft 9

(14) a = ^ 3 0) a - iqir), ql«l o s

4Nm v^ (15) a = ^ \ (A), ql»l and v<v

3lTp V

^o s Nm

(16) a = — - — q£>>l and v>>v ^ p v^T °

•o s

where v^ is the Fermi velocity, w is the angular velocity

2 3 1/2 of the wave, with v = (37r v^ o/l) ' being the critical o s

frequency at which attenuation begins to fall off to the

constant value given in Equation (15), and o being the con-9

ductivity. This would occur at about 10 Hz in most metals.

It can be seen from (14) and (15) that, as for longitudinal

waves, the total shear wave attenuation for qt«\ should be

proportional to q£ and to the square of the frequency. For

q£>>l, it should be independent of q£ and be proportional

to frequency. However, below the temperature at which the

electron mean free path becomes limited by impurities, the

normal state attenuation a assumes a constant value which n

it maintains down to zero Kelvin.

21

1.4 Ultrasonic Attenuation of Shear Waves in Superconductors

Below the critical temperature in a superconductor,

normal electrons begin to form Cooper-pairs. The energy

required to break up a Cooper-pair into two quasiparticles

is at least 2A = 3K T^ ^ 10"^ eV, which is much larger than

the energy of each phonon of a sound wave of less than 10

Hz (less than 10'^ eV). Hence the sound wave can interact

only with the normal electrons near the Fermi surface. When

the transition temperature is passed the ultrasonic attenua

tion will fall off rapidly, reflecting the decreasing normal

electron population. The normal electron population level

is predicted by the BCS theory to follow the Fermi distribu

tion function, resulting in the temperature dependence for

ultrasonic attenuation which was given in Equation (8).

(8) !£= 2F(A) - ^ A B T , , ' n e ' 5 + 1

where A is the energy gap and K^ is Boltzmann's constant.

Longitudinal ultrasound wave attenuation has been found to

fit this type of temperature dependence quite well.

Transverse wave ultrasonic attenuation drops almost

discontinuously in the rapid-fall region at T^ to a residual

value a^ which then appears to follow the same functional

dependence given in Equation (8). Tsuneto"* in fact has

shown that the temperature dependence of the residual atten

uation should satisfy the BCS relation when the electron

mean free path is impurity limited.

22

A qualitative explanation for the rapid-fall of

shear wave attenuation was proposed by Morse^^ based upon

44 a suggestion by Holstein. Morse proposed that the electromagnetic part of the attenuation was shorted out below T

c

by the Meissner effect, leaving a residual part which could

be caused by shear-deformation. Holstein^^ also had sug

gested that a residual attenuation could be caused by the

collision-drag effect. Pippard^^ generalized his theory to

include all three contributipns, as discussed previously,

31

and Leibowitz then showed how turning off the electromag

netic part within the Pippard formalism (second term of

equation [111] would give the rapid-fall region plus a

residual attenuation which behaved like the longitudinal

attenuation. The collision-drag part of the residual

attenuation decreases monotonically with frequency, so that

at high qt values the residual attenuation becomes just the

shear-deformation attenuation, which is constant with fre

quency .

In the present experiment the frequency range was well

below the value at which the residual attenuation should be

come constant; therefore it would be expected to follow the

functional form predicted by Equation (8). As mentioned

previously, the ql values in the molybdenum crystals in

this study were greater than unity, so that a rapid-fall

region would also be expected to occur.

23

1.5 Molybdenum

Molybdenum is a transition element (Gp. VI) of body-

centered-cubic (BCC) crystal structure. It is a weak-

coupling, type I superconductor, having the reversible transi

tion which is characteristic of type I superconductors.^^

Superconductivity in molybdenum was not discovered

until 1962, when Geballe, Matthias, Corenzwit, and Hull "

obtained an extremely pure single crystal grown by E. Buehler.

The delay in discovery resulted from the fact that the transi

tion temperature of molybdenum is extremely sensitive to

traces of impurities, particularly those which are magnetic.

Most measurements to present indicate a transition tempera

ture of about 0.91 - 0.92 ± .01 K.^®'^^'^^ The most accurate

measurement to date, based on sample purity and width of the

transition ('1.2 mdeg) , is probably one made by Matthias,

et al.,^^ who found values of 0.918 K and 0.917 K for two

small samples of natural isotopic composition.

The average width of the energy gap at 0 K, as found

by measurements of specific heat, critical magnetic

field, - ' ' ^ thermal conductivity,^^ and ultrasonic

attenuation,^^'^^'^^ is reasonably consistent with the BCS

value of 3.52 K^T^. Anisotropy in the energy gap, or pos

sible existence of two gaps, was suggested by the specific

heat investigation of Rorer, et al.^^ Anisotropy was

observed in the ultrasonic investigation of Jones and

Rayne.^^ O'Hara and Marshall^® also found the energy gap

24

to be anisotropic, as well as frequency dependent; in addi

tion their data also indicated the possibility of two or

more energy gaps in the [100] direction. The anisotropy

found in both cases is not large and hence is consistent

with the calculation of Garland,^^ which predicted that the

anisotropy of the energy gap for a transition metal will be

smaller than that for a simple (i.e., non-transition) metal.

The Fermi surface of molybdenum has been investigated

experimentally by several te.chniques, including magneto-

resistance,^^ magnetoacoustic,^^ and de Haas-van Alphen

effect^' measurements, and detailed theoretical calculations^®'^^

have been made of the surface structure and the electron

density of states. Magneto-resistance measurements indicate

that the prediction of a free electron model, which would

give a spherical Fermi surface, is not a good approximation

due to the lack of any evidence for open-orbits. A model

has been proposed for the Fermi surface by Lomer^^ which

seems to be in good agreement with all of the above experi

mental results. The features of molybdenum's electronic

energy band structure which make it an interesting subject

for study are the multiplicity of Fermi surface sheets and

a low density of states at the Fermi energy. A descrip-

58 tion of the four sheets is given by Koelling, et al., as

follows: (i) The largest surface, which arises from the

fourth band, is the electron jack centered at the point T

of the Brillouin zone; (ii) the next largest surface is

25

the third-band hole octahedron at H; (iii) the six nearly

ellipsoidal third-band hole surfaces are at the N points;

(iv) the smallest surfaces are the six electron fifth-band

lenses located on the A line. These features are illustrated

in Figure 3 below. Note that the majority of the density of

58

states is contributed by the first two sheets. The princi

pal features of Lomer's model, in addition to the four sheets

described above, are that all of the energy surfaces are

closed, and the number of holes is equal to the number of

electrons.^^'^°

26

3a} r, X are the electron zones; H are the hole zones.

3b) N are the small hole zones.

3c) Nonrelativistic energy bands Mo -(from Iverson and Hodges, 58)

for ref.

Figure 3.—Fermi Surface and Energy Bands of Molybdenum

CHAPTER II

EXPERIMENTAL

II.1 Introduction

The attenuation of ultrasonic shear waves in both

normal and superconducting molybdenum was measured in this

experiment as a function of .temperature. Several systems

of equipment were required for this study.

First, an electronic system was needed which would be

capable of generating, receiving and measuring electronic

pulses of the required frequency range to be used for pro

ducing the ultrasonic waves. Secondly, a refrigerator sys

tem was needed which would lower the sample temperature to

well below the transition temperature of molybdenum, which

is approximately 0.92 K. The refrigerator system had to be

capable of continuously maintaining this temperature range

for several hours and yet permit cycling of the temperature

up and down slowly. Third was the need for an accurate

thermometry system which would continuously monitor the

sample temperature.

In this chapter a description of the method of ultra

sonic attenuation measurement and a definition of the atten

uation coefficient a will be given first. Then each of the

27

28

cUDOve three systems and their associated equipment will be

described. Finally, transducers, transducer bonding, and

the sample itself will be discussed.

II.2 The Pulse-Echo Method of Ultrasonic Attenuation Measurement

The method used for measuring the attenuation of

ultrasonic shear waves in the molybdenum sample was the

standard single transducer pulse-echo technique. The

sample is cut so that it has two plane parallel faces nor

mal to the crystallographic direction to be studied. The

pulse is introduced into the crystal at one of the faces

by means of a piezoelectric transducer (in this case, quartz)

which is bonded to that face (See Figure 4). A radio fre

quency (RF) electromagnetic pulse of the fundamental fre

quency or of one of the odd harmonics of the transducer is

generated by a pulsed oscillator and is impressed across

the transducer faces, which are plated. The transducer con

verts the electrical pulse into a mechanical stress wave

pulse of the same frequency which propagates through the

sample. Upon reaching the opposite face, the ultrasonic

pulse is reflected and travels back to the initial face,

where a small fraction of it is reconverted by the trans

ducer back into an electrical pulse. This is the first

echo. The major portion of the ultrasonic sound wave, how

ever, is reflected again at the sample-transducer interface

and travels back through the crystal, making another round

29

Plated Electrodes (Gold) Y-Cut Quartz

Transducer

Dow Corning DC-11 Bond"

Mo Sample

Figure 4.—Specimen, Bond and Transducer

i- ft

(jiilibyjyMi

!! ( I I

I I

Figure 5.—Ultrasonic Echo Train Display

30

trip before returning as the next echo. With each succes

sive round trip, part of the energy of the stress wave is

absorbed, so that each successive echo is smaller than the

previous one (or attenuated). Before each pulse has returned

to the transducer, the pulsed oscillator has been turned

off, so that the initial pulses do not interfere with the

returning echoes. The echoes are conveyed as rf pulses to

an amplifier system and displayed on an oscilloscope. The

display will show the initiaj. pulse followed by a train of

echoes which are decreasing in amplitude approximately

exponentially. A typical example of this pattern as seen

on an oscilloscope is shown in Figure 5. The attenuation is

obtained by monitoring the height of one of these echoes as

the sample temperature is cycled up or down.

II.3 The Attenuation Coefficient a

In this section a quantitative expression will be ob

tained to describe the loss of amplitude of the ultrasonic

pulse as it travels through the sample. This will be ex

pressed as an attenuation coefficient a, which will repre

sent all energy loss processes in the sample. It will then

be shown how the attenuation of an ultrasonic pulse within

the sample can be expressed in terms of the change in

voltage height of one echo displayed on the oscilloscope.

Finally an expression for the ratio OL^/OL^ as a function of

temperature will be obtained which can be compared with the

BCS expression in Equation 8.

31

Assume; that the ultrasonic wave produced by the trans

ducer is a monochomatic plane wave which is uniformly atten

uated with distance traveled through the sample. The ampli

tude of this plane wave can then be expressed as a function

of distance and time by

(14) a-(x,t) = a e'^'V^"'^"^^^ o

where a^ is the amplitude of the initial stress pulse at the

transducer, w is the angular frequency of the pulse, and k

is the propagation vector.

Since we are only interested in the amplitude of the

envelope, we can eliminate the complex exponential term

which expresses the sinusoidal variation. The envelope

amplitude is then

(15) a(x) = o^e'^^

The total distance traveled through the sample by the

j ultrasonic echo is given by j(2L), where L is the length

of the crystal. Hence the amplitude of this echo will be (16) a. =a^e-'23I')

This equation can be put in linear form by taking

the natural log of both sides:

(17) In a. = In a^ - a(2jL)

Solving for , we have

(18) a = ^ in ^

where a is expressed in nepers per unit length. Attenua

tion is more commonly expressed in decibels (dB) per centi

meter,^^ in terms of which it l>ecomes

20 , ^o 32

(19) a = 2 ^ log^, ^

D The amplitude of the stress wave at the transducer is

proportional to the voltage on the transducer. Hence, if

the voltage at the transducer is amplified linearly, as

was the case in this experiment, then the stress amplitude

is proportional to the voltage height of the corresponding

echo displayed on the oscilloscope;

(20) a. = KV. .

Attenuation can then be expressed in terms of the echo

heights as

^2^^ ^ = 23r ^°^io v^ •

We are interested in measuring the change in a as the

temperature is varied, so we write equation (21) as a func

tion of the temperature:

on V^Cr) (22) a(T) = 23r 1°^10 V ^ •

The above expression for a(T) represents the total

attenuation and includes not only electronic attenuation,

but also attenuation due to many non-electronic sources.

Fortunately the latter are found to be temperature inde

pendent for the most part at low temperatures and can be

subtracted out as background attenuation a„; V (T)

(23) a(T) = ^ log^Q V T T T T " ^B

where a(T) now represents only the electronic attenuation.

33

In this experiment the voltage of the initial pulse,

V (T), is held constant during a given temperature cycling,

so that it cam be taken as an arbitrary reference voltage:

(24) V^(T) = V Q

The superconducting attenuation, a (T), and the nor-

mal state attenuation, a (T), can now be expressed by Equa

tion (23) as V

(25) ag(T) = j f r l o g i o V T - T T T " «B 3S

and V

(26) a^(T) = 23^ ^10 V . J T ) • " B •

V. (T) and V. (T) represent the voltages of the j echo at js jn

a given temperature in the superconducting and normal states,

respectively.

From Equation (8) it can be seen that the BCS theory

assumes that at zero Kelvin no unpaired conduction electrons

are left to attenuate the ultrasonic wave. Therefore at

zero Kelvin,

(27) ag(0) = 0

which implies that V

20 T o (28) «B = 2jL °^10 vTTTor

3 s Substituting Equation (28) into Equations (25) and

(26) and then obtaining the ratio ^ s/ 'n, we have

V.g(O)

(29) f- v! (0)

34

Now since the normal state attenuation of molybdenum

is constant with temperatiure below T (since this is within c

the impurity-limited region), V. (T) = V. (T ) , and equa-jn jn c ^

tion (29) becomes

"s l°5l0 vTTifr (30) -i := ii! 21

°n l°9io Vjs(O)

We can now determine a /a in terms of the above s n

experimentally measureable quantities and compare its temp-

erature dependence with that given in Equation (8).

II.4 Electronic Equipment

The generation, reception, amplification and display

of the RF pulse and the returning ultrasonic echo train was

accomplished by a system similar to conventional ultrasonic

6 3

attenuation systems. One improvement patterned after cir

cuits described by Claiborne and Einspruch and Hemphill

was incorporated into our system. This was the inclusion

of phase sensitive detection to enhance the signal-to-noise

ratio in the amplification of the echoes, making possible

the measurement of very small attenuation changes (<1 db).

Since the electronic system already has been described in

detail elsewhere,^^'^^'^^'^^ only the basic features will

be presented here.

Figure 6 shows a block diagram of the electronic sys

tem. The first Hewlett-Packard model 214A pulse generator,

operating on internal trigger, produced simultaneously an

800 Hz trigger signal which was sent to the Hewlett-Packard

• ^

A P E N 6 E RQ PULSED OSC/LLATOft

PUi_SE G E N E R A T O R - 2

A

AND

GATP -<-

PULSE

GEN ERATOR- »

" T E c "

* ^ ^ ^ * ^ " ^ ^ w^*^^«n*

PRE.->;N-P

>^

AR£(N8ERe W10E6ANO AMPLlFi IQ

CftYOSTAT

T V

VERT. AMP,

I DISPLAY ISCAWHER

BINARY COUNTER

REF. S I 9.

Y

P. A . R. i L O C K - I N I

A M P H F I E P

Y Y - A ) ^ i S

M O S E L E Y

x - Y RECO<?OEC

T O S C O P E TRiSGCR f X - A X I S

- ^

1 ^

sCRT s s

CONSTftNT

CURRENT

SOUPvCE

^ AfV\PLIF lER

SR H P. Olo»TRL

I

V O L T M E T E R i

35

Figure 6 . — B l o c k Diagram of E l e c t r o n i c System

36

model 175A oscilloscope and an 800 Hz pulse of about five

microseconds duration which was sent to a binary counter

and an AND gate. This 800 Hz pulse is illustrated in

Figure 7a. The four-stage binary counter divided the 800

Hz signal by sixteen to produce a 50 Hz square wave which

is shown in Figure 7b. The 50 Hz square wave was sent

simultaneously to a PAR model 121 lock-in amplifier as a

reference frequency for the phase detection and to the AND

gate. The AND gate used the 50 Hz signal to gate the 800

Hz signal, producing the output shown in Figure 7c. The

800 Hz signal gated at 50 Hz was used to trigger the second

Hewlett-Packard model 214A pulse generator, which was used

primarily to invert the signal into negative pulses. This

800 Hz signal gated with 50 Hz negative pulses was used to

trigger the Arenberg model PG-650C pulsed oscillator, which

produced a more stable output when triggered by negative

pulses.

The output of the Arenberg pulsed oscillator was a

series of RF pulses at a pulse repetition rate of 800 Hz

gated at 50 Hz, each pulse having a width of about two micro

seconds (adjustable) and an amplitude of from about one volt

to over 300 volts peak to peak (VPP) into 93 ohms. The

frequency of the RF pulses could be tuned within the re

quired range of ten to two hundred megahertz by use of the

appropriate coil in the oscillator circuit. Below 100 MHz,

the frequency was measured by a Hewlett-Packard model 5245L

VOLTAGE

«

j l i i

TIME 7a) Pulse Generator One Output

VOLTAGE

TIME

7b) Reference Signal

VOLTAGE

TIME

7<i And Gate Output

VOLTAGE y' ^

TIME

7d) Display Scanner Output

37

Figure 7.—Selected Waveforms from Electronic System

38

electronic counter. These RF pulses from the pulsed

oscillator were sent through an electrical "Tee" into the

cryostat and to the crystal as shown in Figure 6, where

each pulse produced an echo train as described in section

II.2.

The RF signal corresponding to the echo train, upon

returning to the Tee from the cryostat, was then sent to

either one of two receiver systems. If the pulse frequency

was 65 MHz or less, the signal was sent first to an Aren

berg model PA-620SN preamplifier which could be tuned so

that its input impedance matched that of the cryostat. The

output impedance of the preamplifier was 93 ohms. The out

put signal from the' preamplifier was next sent to an Arenberg

model WA-600-D wide-band amplifier with a 93 ohm input

impedance.

High frequency pulses in the range from 65 to 200 MHz

were sent from the cryostat to an Arenberg model VR-720 VHF

receiver which performed the Scime functions as the above

preamplifier-amplifier system. The output of either receiver

system was an amplified video signal which was sent to the

vertical input of the Hewlett-Packard model 175A oscillo

scope where it was displayed on the CRT as shown in Figure

5. The maximum video signal strength for either receiver

system was maintained at less than ten volts to insure that

they each were operating within their linear amplification

regions.

39

The Hewlett-Packard oscilloscope was equipped with a

Hewlett-Packard model 1782A display scanner plug-in unit

which made it possible to sample the amplitude of any

point on the CRT trace. The point sampled was manually set

on the peak of one given echo, and the display scanner out

put was then a 50 Hz square wave with an amplitude propor

tional to the echo height. This 50 Hz signal was sent to

the PAR model 121 lock-in amplifier where it was detected

by comparing it with the 50 Hz reference frequency while

noise of all other frequencies was discriminated against.

The output of the lock-in amplifier was a DC signal propor

tional to the voltage of the echo being sampled. This out

put was used to drive the vertical input of the model 2X-2A

Moseley X-Y recorder.

The display scanner was calibrated at 200 mV/cm prior

to each experiment by adjusting its output to 1200 mV when

the CRT baseline trace was set on the top centimeter scale

line on the scope face, and then adjusting its output to

200 mV with the CRT trace set on the next-to-lowest scale

line on the scope face. There were ten one-centimeter

divisions between these two lines. This procedure resulted

in a very linear response.

The lock-in amplifier sensitivity was normally set

on 50, 100, or 200 mv, depending on the size of the atten

uation change being measured and the amount of noise present

The time consteuit was set on either 300 milliseconds or 1

40

second response, also depending on the noise level. The

lock-in amplifier also contained a calibrated suppression

voltage ranging from 0 to ± 100 volts which allowed the

zero level of a given echo voltage to be shifted down (or

up) . This made it possible to measure large voltage signals

which had relatively very small voltage changes. The chart

recorder vertical sensitivity was set on 1 volt /inch.

3

II.5 He Refrigerator-Cryostat System

Temperatures from 1 K to well below the transition

temperature of molybdenum ^ 0.92 K) were maintained by 3

means of a He refrigerator-cryostat designed by Robbins

and Marshall and modified for ultrasonic attenuation

studies. Details of the design criteria have been thoroughly

covered elsewhere,^^'^ ,67,68 ^ ^^^^ ^^^ basic features

will be described here.

The refrigerator-cryostat system consisted of three

stages, as shown in Figure 8. The outer two stages were

two concentric glass dewars, the outer dewar being filled

with liquid nitrogen (LN2) at 77.4 K, and the inner dewar, 4

which was sealable, being filled with liquid helium (LHe ). The inner dewar had a stopcock to permit evacuation of its

3 vacuum jacket. The third stage was the He cryostat, which

extended down into a high vacuum can situated within the

LHe^ bath. The high vacuum can insulated the He can at

the bottom of the cryostat from the LHe bath temperature.

41

BNC Connector To He Piamp

He^ Pump Line

To He Pump

Tp_iU-Vac Pump

r\

Hi-Vac Pump Line

R.F. Transmission Line

Radiation Trap

Stycast Thermal Short Hi-Vacuum Can

Plunger Contact Specimen

Superconducting CU Table Magnet CU Stand and Heater Coil Liquid He3

Nj Dewar

He Dewar

Figure S.^^e^ Refrigerator-Cryostat System

42 4

Pumping on the LHe lowered its pressure, causing its

temperature to be reduced to about 1.4 K, which is below

the condensation point for He" . When He" gas was admitted

from its storage tank into the He" pumping line leading down 3.

to the He can, it condensed on the wall of that portion of 3

the He line which was above the high vacuum can and began

to drip down into the He can. Upon reaching the can, the 3

liquid He initially evaporated again immediately, carrying

heat away from the relatively warm can. This cyclic

process continued until the condensation temperature of

He was reached by the can and liquid He began to collect 3

in the can. Then by pumping on the liquid He , the temper-3

ature of the He can and the sample could be lowered to

about 0.38 K. The temperature could be increased by allow-

3 3 3

ing He gas to leak back into the He line and He can

through one (or both) of two valves—a Hoke coarse valve

and a Whitey type 316 micrometer needle valve. This caused

an increase in pressure and hence in temperature. By slowly

increasing or decreasing the leak rate by opening or closing

one of the valves, the temperature could be cycled up or

down.

The refrigerator-cryostat system involved three

vacuum systems. One was the vacuum system within the inner

LHe^ dewar, which was produced by a high volume (105 cubic

feet per minute) vacuum pump. The high vacuum system which

insulated the He can was produced by a mechanical forepump

43

The third vacuum system was the sealed He^ system, consist

ing of a specially sealed five cubic feet per minute Welch

Duo-Seal mechanical vacuum-pump, a He" pump cut-off valve, .• 3

ri ..t; gas storage tank, an Ashcroft vacuum gauge, and the

aforementioned coarse- arid fine-leak valves.

The sample was prepared for an experiment by bonding

a transducer onto it and mounting it on the copper table

with a spring-loaded button from the RF transmission line

making contact with the transducer. Electrical function-

ability of the thermometers, solenoid and heater coil, and

the quality of the echoes were all checked at liquid nitro

gen temperature. This was accomplished by submerging the 3

He can and sample in LN2 with the high vacuum can removed.

After warming the system to room temperature, the high

vacuum can was soldered to the cryostat system using Wood's

metal solder. The cryostat was then installed into the He

refrigerator system with all vacuum connections sealed.

Then the mechanical forepump was turned on to pump out both 3

the He system and the high vacuum can. At about 1.00 Torr pressure the liquid nitrogen cold trap was filled and the

3 He system was closed off from the high vacuum system.

The vacuum attainable in the high vacuum system by

pumping with both the forepump and the diffusion pump was

about 10~ Torr within about three days. The addition of

LN^ to the outer dewar would drop the pressure further to

—6 about 6 X 10 Torr as the temperature within the high

44

vacuum can approached 78 K. After transferring LHe"* into

the inner dewar and pumping on it to reduce the pressure to —5

approximately 10 Torr and the temperature to 1.4 K, the

pressure in the high vacuum system was approximately 2 x — 6

10 Torr. At this point the high vacuum isolation valve

was closed and cryopumping reduced the pressure in the high

vacuum can to about 10~ to 10~^° Torr. The coarse and fine

leak valves were then opened to allow the He^ gas to con-3 dense in the He can. This required about an hour and a

3 3

half. The He pump was then turned on and the He pump

cut-off valve was opened. The system was now ready for

data-taking.

II.6 The Thermometry System

The temperature of the molybdenum sample was moni

tored by using carbon resistance thermometers (CRT), the

resistance of which varied considerably with temperature

at low temperatures. The resistances of the thermometers

were measured by the four-terminal method as shown in

Figure 6. A constant current source supplied either a

1 yamp or a 10 vamp current to either a IKQ or a lOKiT pre

cision 0.05% standard resistor (SR) in series with the CRT.

The voltage drop across the SR was continuously monitored

by a Hewlett-Packard model 3440A digital voltmeter. Thus

the current through the CRT was constant and was accurately

known. Using this value of the current the resistance of

the CRT could be found by measuring the voltage drop across

45

it with a second Hewlett-Packard digital voltmeter. During

data-taking the CRT output voltage was amplified by a Leeds

and Northrup Model 9835-B D.C. microvolt amplifier and used

as the input for the x-axis of the x-y recorder.

At the beginning of each experiment, the carbon resis

tance thermometers were calibrated against the vapor pres-3

sure of He as measured by a precision Wallace and Tiernan

absolute pressure gauge. The temperatures of the carbon

resistance thermometers were determined from the corres-3 3

ponding He pressures by referring to the T62 He Temper-69

ature Tables prepared by Sherman, Sydoriak, and Roberts.

The Wallace and Tiernan gauge measured the absolute

pressure in the range of 20.00 Torr to 0.10 Torr with an

absolute accuracy of 0.33% of full scale anywhere within

this range. The inaccuracy of the measurement became rela

tively large below 1.00 Torr. Also, the correction needed 3

between the He pressure measured at room temperature and 3

the actual He pressure above the liquid was negligible for 70 pressure measurements above 1.00 Torr. Hence the cali-

3 bration of the carbon resistors against He pressure was

3 carried out between 20.0 Torr and 1.00 Torr He pressure

3 as measured on the Wallace and Tiernan. These He pressures

corresponded to temperatures of 1.3 K and 0.65 K, respec

tively. Hence in order to interpolate between the calibra

tion points and also to extend the temperature range down

to 0.37 K, the calibration data were fitted to the Clement-

71 Quinnel equation:

46

(31) Log R + K/Log R = B(i) + A

In this equation R is the resistance of the CRT, T is

the absolute temperature corresponding to that resistance,

and A, B, and K are experimental constants. In order to

fit the calibration data to equation (31), a computer pro

gram for the IBM VS/370 computer was used to determine the

values of A, B, and K using the method of least squares.

When values for A, B, and K were determined, a complete

table of resistance versus temperature values could be gen

erated using equation (31).

The carbon resistance thermometers were two one-half

watt Allen Bradley carbon resistors labeled as CRT A and

CRT B. CRT A had a nominal room temperature resistance of

2.7 ohms, and CRT B had a nominal room temperature resis

tance of 5.1 ohms. Both thermometers were inserted into

holes drilled into the pure copper table, after first being

coated with DC-11 silicone grease to enhance the thermal

contact with the table. One lead of each CRT was also

soldered directly to the table to provide a thermal path.

The other lead was thermally connected to the table by

means of a 750 pf silver mica capacitor which was soldered

to the lead and to the table. The capacitor not only pro

vided a good thermal path while blocking the flow of direct

current, but also provided a short circuit for any stray

RF signals which might be picked up by the leads outside

the cryostat. This prevented RF heating in the resistor.

47

It was determined experimentally that a thermal

gradient between the copper crystal table and the CRT

developed when the CRT output voltage exceeded 10 milli

volts. This gradient was caused by Joule heating in the

CRT. In the present study, this "self-heating" was mini

mized by using a 1.0 microampere current instead of a 10

microampere current through either CRT during data-taking.

With a 1.0 microampere current CRT B could be used above

0.5 K without detectable self-heating, and CRT A would not

self-heat even at the lowest temperature obtainable. How

ever, CRT B was an order of magnitude more sensitive than

CRT A in the transition region of molybdenum ('vo.92 K) .

Hence CRT B was used in the approximate temperature range

of 1.2 K to 0.5 K, and CRT A was used in the approximate

range of 0.6 K to 0.37 K. This overlap provided a check

on thermometer accuracy.

II. 7 Transducers and Transducer Bonding

Thin quartz wafers obtained from Valpey Crystal

Corporation were used as the transducers in this study. The

quartz transducers were of the "Y-cut" type i.e., due to the

orientation of their cut, they vibrated in the transverse mode

Their shape was a cylindrical wafer except for a slot on one

edge which indicated the direction of polarization of the

transverse sound wave. The wafers were about 3/16 inch in

48

diameter and had a thickness which corresponded to a quarter

wavelength of their fundamental frequency, 10 MHz. Since they

v/ere bonded rigidly on one face, they would also only vibrate

at odd harmonics of 10 MHz. The RF electric field was im

pressed across the transducers by means of a thin layer of

gold plated over a thin layer of chrome on each transducer

face. Gold was used as the outer plating layer due to its

high conductivity and resistance to tarnish. Chrome was

plated onto the quartz as the first layer since gold will not

adhere directly to quartz.

The transducer was bonded to the molybdenum sample

with a thin layer of Dow Corning DC-11 silicone grease. This

material could be applied easily to the sample at room temp

eratures, yet produced a rigid bond upon freezing at 210 K.

DC-11 silicone grease also satisfied several important re

quirements: (1) Its thermal expansion coefficient was com

patible with the expansion coefficients of both the trans

ducer and the sample, enabling the bond to be cycled between

300 K and 0.37 K repeatedly without breaking; (2) Its vapor

pressure was low (less than 10"^ Torr at room temperature);

(3) It could be applied in a very thin, uniform layer, re

sulting in minimum sound amplitude losses due to binder atten

uation and non-parallelism. Prior to beginning each experi

ment, the transducer bond was tested by submerging the lower

portion of the cryostat in liquid nitrogen (77.4 K) and check

ing whether an exponential echo pattern could be obtained.

49

II.8 The Sample

The two molybdenum crystals used in this study were the

same ones used by O'Hara and Marshall in their longitudinal

ultrasonic attenuation study. Both were cut from a 99.95%

pure single crystal grown by Metals Research Limited. The

crystals cut from the original crystal were oriented in such

a manner that it was possible to study all three basic crys

tallographic directions with only two crystals.

Two requirements had to be met by the dimensions of

the crystal. First, the length in the crystallographic

direction being studied could not be too great or excessive

ultrasonic attenuation would result. Secondly, a minimum

sample length in the direction of study was required to pre

vent echo overlap. In other words, the time required for a

given echo to traverse the crystal and return to the trans

ducer had to be greater than the width of the pulse. This

can be expressed as

(32) ^ > P.W. Vs

where L is the sample length, Vg is the sound velocity in

the given direction, and P.W. is the pulse width, which was

about 2.0 microseconds. From the measured values of the

72

elastic constants of molybdenum, O'Hara and Marshall calcu

lated the velocities of longitudinal sound in the three basic

crystallographic directions to be: V[100] = 6.62 x 10^ cm/sec

V[110] = 6.52 X 10^ cm/sec

T U A S TECH LIBRARY

50

V[lll] = 6.49 X 10^ cm/sec

which would give a minimum sample length of about 0.5 cm for

any of the three directions. Hence the first sample was cut

as a right circular cylinder with the cylinder axis oriented

in the [100] direction and a distance between end faces of

0.5118 cm. Since the [110] direction is perpendicular to

the [100] direction for a body-centered cubic structure such

as molybdenum, this direction was obtained by polishing two

parallel faces perpendicular to the [110] direction into the

sides of the cylindrical surface of the first sample. These

faces were 0.872 3 cm apart. The second sample was cut to

give two parallel faces, aligned perpendicular to the [111]

direction, which were 0.5758 cm apart.

The above sample dimensions were considered to be

acceptable for the present study since the transverse sound

73

velocities as calculated from the measured elastic con

stants are lower in all three directions than the longitudi

nal sound velocities. The transverse sound velocities are

V^[100] = 3.485 X 10^ cm/sec

Vg[110], S||[001] = 3.485 X 10^ cm/sec

VgEllO], S||[110] = 3.675 X 10^ cm/sec

Vg[lll] = 3.615 X 10^ cm/sec

where S is the sound polarization vector. The corresponding

round trip transit times would be

2.935 X 10"^ sec. [100]

4.993 X 10"^ s e c . [ 1 1 0 ] , [001] p o l a r i z a t i o n

51

4.734 X 10" sec. [110], [110] polarization

3.185 X 10"^ sec. [Ill]

Cutting of the two samples by O'Hara and Marshall had

beer, performed on an Elox model TQH-31 electric discharge

machine to minimize dislocations and strains in each crystal.

Orientation of the samples was determined to be within 1**

of perpendicular to the desired crystallographic direction

in each case by use of the conventional have-back-reflection

technique. ' Each pair of sample end faces had then been

further polished to within 0.0005 cm of parallel on the Elox

electric discharge machine.

A magnetoacoustic study of the crystal using the tech-

76 77 nique of Deaton and Gavenda ' had also been performed by

O'Hara and Marshall to obtain an estimate of the electron

mean free paths. This experiment was performed only for the

78 [100] orientation. However, Fawcett's measurements of

magnetoresistance in molybdenum indicate that the electron

relaxation time is isotropic to the nearest order of magni

tude. Hence the electron mean free path I for all three

crystallographic directions in molybdenum should be reason

ably close to those obtained for the [100] direction. The

mean free path values for the [100] direction of molybdenum

ranged from 0.022 cm to 0.025 cm depending on the orientation

of the magnetic field used for the magnetoacoustic measure

ment. This would give, for frequencies ranging from 10 MHz

to 70 MHz, ql values ranging from about 4.0 to about 30.0

52

respectively. Hence the ql values for all orientations

in our sample could be expected to be larger than unity

for frequencies of 10 MHz or greater.

CHAPTER III

RESULTS AND INTERPRETATIONS

III.l Ultrasonic Attenuation Data

The attenuation data were taken as described in chapter

II by plotting the voltage height of a given echo as a func

tion of temperature (measured as CRT resistance). These

plots were recorded on the Moseley X-Y recorder while cycl

ing the temperature either up or down in the approximate

temperature range of 1.2K to 0.37K. Each plot was taken in

three or more segments to allow for changing of the ther

mometers or resistance scales for maximum sensitivity in

each part of the temperature range.

A typical composite recorder tracing is shown in

Figure 9. The voltages shown in the figure are the ones

used in Equation (30) to calculate the ratio of supercon

ducting to normal state attenuation for comparison with the

BCS theory. ^js^^^ T V (T)

's ^°^10 ^ (30) -^ = ^^ a lr.rr V. (O)

V. T^) jn c

In addition, the total electronic attenuation a can be

obtained in terms of these voltages from Equations (26) and

(28) :

53

v_/

A

CVi

54

\

O

o<

'v.

U

6

^.

•~i •—

I \

\

O

r ^ v"^'/

• >

o - ^ O o 6

>

r3

ra o c o

a c 4-) 4->

<

o u 03 U

U TJ U O O (U

(1) 4J •H (0

o a e o

ON

o u

•H

55 on V. (0)

(33) a^ = a„(T) = ^0 ]s^ e n 2nL ^10 v. vT )

jr. c

The frequency dependence of the total electronic attenuation

can then be compared with predictions of the Pippard theory

using Equation (33).

In order to study the variation with temperature of

the electronic attenuation, i.e., that caused by electron-

phonon interactions, it was assumed that all other mechan

isms contributing to ultrasonic attenuation in metals at

low temperatures are temperature independent. As discussed

in Section 1.3, this assumption has been found to be true in

general. However, one other type of temperature-dependent

attenuation has been found to exist in metals at low tempera

tures. This attenuation, called amplitude dependence, is

thought to be due to interactions between large amplitude

ultrasound vibrations and dislocations in the crystal lat-38

tice. It was first observed by Bommel in lead and later 79 80

by Love, Shaw, and Fate and others in other supercon

ductors. Theoretical explanations for this effect have been 81 82

presented by Granato and Lucke and by Titmann and Bommel.

Amplitude dependence causes an increase in ultrasonic attenua

tion as the temperature decreases, as shown by the dashed line

in Figure 9. This increase opposes the decrease in attenua

tion caused by decreasing electron-phonon interactions at

lower temperatures and hence its presence would be readily

discernable. 73 the name implies, amplitude dependence can

be very large at high ultrasound amplitudes or negligibly

56

small at low amplitudes. In the present study, data was taken

in each crystallographic direction at several different trans

ducer voltages ranging up to 50 VPP with no discernable

air.plitude dependence. As a precautionary measure, however,

transducer voltages during experiments were normally kept

well below ten volts. This also minimized the possibility

of a thermal gradient acjross the sample caused by RF heating

in the transducer. Such a thermal gradient would cause the

sample temperature to remain above that of the thermometers

at very low temperatures, preventing the attenuation from

decreasing to the value it would otherwise attain at the

temperature indicated by the CRT.

Another assumption made in obtaining Equations (30) and

(33) was that the normal state attenuation a below T was XI w

a constant and equal to the value of a at T_. From the ^ n c

purity of the molybdenum samples used in this study, the mean

free path should become impurity limited and a should become

constant for all temperatures below about 2.0 K. The normal

state attenuation was measured below T by applying a mag-

netic field of about 150 gauss across the sample. This

field somewhat exceeded the zero Kelvin critical field value 46

of about 100 gauss estimated by Rorer, Onn, and Meyer.

Measurements of a taken in this manner indicated that the

normal state attenuation below T was constant with respect

to temperature and was equal to the constant value of a^

observed between T and 1.2 Kelvin.

57

In order to determine whether temperature hysteresis was

present, two sets of data were taken at each transducer volt

age, one with the temperature rising and one with the temp

erature falling. A small amount of hysteresis appeared to

be present in some cases, as evidenced by a slight shift in

the transition temperature. In order to minimize this hyster

esis and maintain thermal equilibrium as closely as possible

while taking data, the temperature was cycled very slowly

between about 1.0 K and 0.37 K, with from twenty to thirty

minutes spent in traversing this range.

III.2 Analysis of Data

In order to fit the attenuation data to the BCS theory.

Equation (8) was written in a linearized form

A - 2a^(t) (34) A (t) ^ ^ , n . ^ ^ K^T^ l^e ^a3(t) ^^

where the temperature dependence of A(T) has been normalized

by introducing the reduced temperature t = T/T . The tempera-

ture dependence of A(T) can be expressed in terms of the

limiting effective energy gap A(0) by introducing the func

tion G(t), where

(35) A(t) = A(0) •G(t) .

The function G(t) has been tabulated for an ideal supercon-

83

ductor from the BCS theory by Miihlschlegel. For calculat

ing A CO) from the attenuation data, however, it has been

found to be more convenient to use an analytic form of G (t)

58 84 given by Clem:

(36) GCt) = 1.7367(l-t)^/^' [l-0.4095(l-t)-0.0626(l-t)^]

This expression has been shown to agree with Muhlschlegel's

values with an error of less than 0.1% for t>0.40. The

minimum reduced temperature reached in this investigation

was 0.404 Kelvin.

Substituting Equation (35) back into Equation (34)

gives p/.x 2a (t)

(37) A (0) ^ ^ = In (-^Vrr " D t e a (t) '

s From Equation (37) it can be seen that the BCS theory pre-

2a (t) diets that if In^ ( ^ ^ 1) is plotted as a function of

calculated values of G(t)/t, the result will be a straight

line with a slope equal to the zero Kelvin energy gap,

expressed in units of K_T . The values for c, /a at each a c n s

reduced temperature t in the above equation are obtained

from the voltage height ratios as given by equation (30).

Since no data could be obtained below about t = 0.41 K,

however, the value used for the echo voltage height in the

superconducting state at zero Kelvin, V. (0), had to be 3 s

estimated. The estimate was obtained by treating V. (0) as 3 s

an adjustable parameter while fitting each set of attenua

tion data to Equation (37) by the method of least squares

using a Fortran computer program. Using this method of

estimating V._CQ), the data was found to fit the BCS equa-j s

tion very well.

59

As discussed in Section 1.5, the possibility of ir.ulti-

ple energy gaps in molybdenum has been suggested by specific

heat data.^^ O'Hara and Marshall^^ also found some indication

of multiple gaps in the [lOOJ direction. They discussed

their results in terms of a multiband model employed pre

viously by Perz and Dobbs,^^'^^ in which a variation of

equation (8) is used: a^j A.

(38) -r. = E . . 1 a i exp[A. (T)/K T] + 1

In this equation each energy gap Aj, (T) is associated with a

given band and it is assumed that no interband interaction

occurs. Each Aj|_ (T) is assumed to have the temperature

dependence given by Equation (8), and the constants A must

satisfy the condition

(39) lA. = 2 i ^

It can be seen from Equation (38) that in this model each

energy gap would contribute to a /a throughout the entire

temperature range, but the term containing the smallest gap

would become dominant as the reduced temperature approaches

zero Kelvin. At higher temperatures, the contributions from

all terms would become more equal, and the apparent energy

gap would be an average of all the contributing terms.

In order to investigate the possibility of multiple

energy gaps, O'Hara and Marshall defined a temperature-

dependent average energy gap function M^-\ 4- 2a^(t)

^'^^ 2 ) =|(|[=^ln^( n_^^,j

60

which they evaluated at temperatures above zero Kelvin,

constant value for A^(t) in the range between zero Kelvin

and T^, from the above discussion, might indicate either a

single energy gap or multiple energy gaps of approximately

the same size. However, a A2(t) function which decreased

linearly to the value of A(o) at zero Kelvin, as was the

case for the [100] direction in O'Hara and Marshall's inves

tigation, would indicate the possibility of multiple energy

gaps of an appreciable size range.

In this investigation both the zero Kelvin limiting

energy gap A (o) and the temperature dependent energy gap

A^ Ct) were calculated for each crystallographic direction

and ultrasonic frequency. Results for A(o) and A^ (t) eval

uated at t = 0.46 Kelvin and 0.90 Kelvin are listed in Table

2. Plots of A^(t) as a function of reduced temperature t are

also given for each direction in Figures 11 through 21. The

total electronic attenuation a given by Equation (33) has

also been plotted as a function of frequency for each direc

tion in Figures 22 through 32, and is discussed in terms of

the Pippard theory in the results section.

61

III.3 RESULTS

A. Energy Gap Determination

Transition Temperature

The average temperature at which the echo voltage

height changed abruptly as the sample passed from the normal

to the superconducting state and vice versa was deterir.ined

for each crystallographic direction in order to analyze the

data in each direction. The average value for all of the

data sets for all three directions is T = 0.912 ± 0.075, c '

where the estimated error is taken as twice the standard

deviation for all data sets. This is the same value for T c

as that obtained by O'Hara and Marshall.

Results for q||[100], s||[110]

A typical fit of attenuation data to Equation (37)

for determining the limiting energy gap, i.e. , the energy

gap at zero Kelvin 2A(0), is shown in Figure 10 for the [100]

direction for 10 MHz. The data used to plot Figure 10

is given in Appendix A, along with representative data sets

for each of the other crystallographic directions and fre

quencies. For each direction, the average zero Kelvin

energy gaps at 10, 30 and 50 MHz were determined from all

of the data sets taken at each frequency and are shown in

Table 2. The stated error given with each value of 2A(0)

and 2A2Ct) was estimated in each case from the scatter

between data sets.

o in

4i

O O o

62

o o

u o

N

O in

o o

O n 3 (0 u >

o o

o o <N

O O

O m

I

elm a m

CM I

I

£5 (0 CM | 8

O

o m O

O

- - o

0)

3

•H

63

TABLE 1

COMPARISON OF EFFECTIVE ENERGY GAPS IN MOLYBDENUM AS DETERMINED IN VARIOUS INVESTIGATIO:;S

Authors Method 2A(0)/KBTC [orientation)

Horwitz and Bohm^^

Jones and Rayne^®

Garfunkel et al. 21

Rorer et al. 22

Waleh and Zebouni 23

O'Hara and Marshall

Present work

ultrasonic (239 MHz)

ultrasonic (200 MHz to 1000 MHz)

hypersonic (lOGHz)

specific heat

thermal conductivity critical field

ultrasonic (10 MHz to 50 MHz) longitudinal waves

ultrasonic (10 MHz to 50 MHz) shear waves

0.921.01

0.92±.01

0.914

0.913±.002

0.903 0.903

0.912±.010

0.912

3.51.21100]

3.3i.2[100j 3.5±.2[110) 3.1±.2(111J

2.5±.4(111)

3.5±.2(avg.)

3.2±.l[avg.) 3.4±.1[avg.j

3.2±.2(100) 3.4±.2{110) 3.0±.2[111)

3.39±0.08(100] 3.3110.13(110)

^lldTO) 3.42±0.13(110]

s| 1(001] 3.28+0.13(111]

9 "y

64

TABLE 2

ENERGY GAPS FI^OM SHEAR WAVE ATTENUATION DATA

Approx. Dir. Freq.(MHz) ql ae(dB/an) 2A(0) 2AJ-60) 2A2<-S0)

3.33 ± 0.06 3.48 ± 0.07 3.56 ± 0.10

3.58 ± 0.06 3.72 ± 0.03 3.93 ± 0.06

3.33 ± 0.04 3.45 ± 0.14 3.49 ± 0.25

1100)

l U O )

s 11 [lIO]

[110]

S1 1 1001)

[111]

9.00

29.92

51.85

9.30

30.23

50.41

9.22

32.90

50.00

10.64

31.55

52.04

1 0 , 0

3 0 . 0

50.'0

6 . 0

1 8 . 0

3 0 . 0

6 . 3 3

1 9 . 0

—

2 . 6

7 . 8

1 3 . 0

1.38

4.45

8.69

1.38

4.79

8.78

1.14

4.56

—"—

1.33

5.13

5.75

3.39 ± 0.08

3.25 ± 0.18 3.35 ± 0.10 3.40 ± 0.11

3.45 ± 0.30 3.63 ± 0.22 3.70 ± 0.22

3.36 ± 0.14 3.54 ± 0.30 3.61 ± 0.33

5.31 ± 6.13

3.48 ± 0.06 3.54 ± 0.10 3.61 ± 0.17

3.35 ± 0.25 3.24 ± 0.22 3.10 ± 0.34

3.42 1 0.13

3.37 ± 0.25 3.49 ± 0.10 3.62 ± 0.15

3,09 ± 0,05 3,18 ± 0.06 3.25 ± 0.08

3.18 ± 0.01 3.26 ± 0.02 3.30 ± 0.04

3.28 ± 0.13

65

It can be seen from Table 2 that for the [100] direc

tion the average zero Kelvin energy gap of 30 MHz is larger

than the value obtained at 10 MHz. A similar increase was

obtained by O'Hara and Marshall for this frequency and

direction as shown in Table 1. In the present study, how

ever, a decrease was observed for 50 MHz, a frequency for

which O'Hara and Marshall obtained no data in the [100]

direction. The average zero Kelvin energy gap for all

three frequencies in the present study is 3.39 ±0.08 K T ,

which is appreciably larger than the longitudinal wave

value.

The function A2(t) is evaluated at reduced temperatures

of 0.60 and 0.80 for each frequency and direction. Above

about t = 0.80, rapid changes and fluctuations typically

occur in the data, making analysis difficult. As can be

seen from Table 2, the values of A2(t) at t = 0.60 and t =

0.80 for all three frequencies in the [100] direction show

a pronounced decrease with decreasing temperature. Figures

11, 12, and 13 show that the decrease is linear below about

t = 0.80 and extrapolates to the limiting (zero Kelvin)

87 energy gap value at t = 0.0. O'Hara speculated that a

limiting type of behavior might be displayed by the A2(t)

function with increasing frequency based upon an observed

decrease in slope of the function from 10 to 30 MHz.

In the present study a decrease in slope does appear

between 30 and 50 MHz; however the slope observed at 30 MHz

00

o

66

!• VO

o

N

o ON

rj

o o

o

CM

U ^ EH

< ^

o

C o

u c D

a o

* - «

o c I

u

o in

-f-o

— I -

in

o

67

%

ON

o

• o

0.7

0

.6

t

N X

CM

29

.

4J

10

0]

o

o

o

o •P CQ

< ^

CM

O

•

o .

in

CM

—r o CM

nr in

in

o

0

c 0 •H

u c

a u >i

0)

c

I CM

0 u D

68

o

00 •

o

r-•

o

VO •

o

• N

m*~»

X m 00

• .-» i n

4->

m

o o i-i

in

tn

o

o

o

^ EH

< ^

in

CM

—T O

CM

in r

o

—f in

ten u o

c o •H 4J

u c

a o >^ cn u

•c u I I •

O U D

•H

69

is larger than that at 10 MHz. Hence it is difficult to

draw any conclusion about the frequency dependence of the

behavior of the ^2^^^ function from this data.

Results foi- q||[110], s||[ll0]

The limiting energy gap averaged over 10, 30, and 50

MHz for q||[110] with s1 | [lIO] is 3.31 ± 0.13 K^T^, slightly

less than the value of 3.4 ± 0.2 obtained using longitudinal

waves. The energy gap at 30 MHz for shear waves is some

what larger than the 10 MHz value, as was the case for

the [100] direction, and it again decreases at 50 MHz.

This behavior is different from that shown in Table 1

for longitudinal waves, where no significant change in

the energy gap appears at the different frequencies.

A (t) vs. reduced temperature t is plotted for 10,

30, and 50 MHz in Figures 14, 15, and 16. From these

graphs, A2(t) can be seen to assume a constant slope

below about t = 0.80 and to extrapolate to the limiting

energy gap at zero Kelvin, displaying a temperature depend

ence similar to that for the [100] direction. No signifi

cant change in slope occurs between the three frequencies;

hence there is no evidence of frequency dependence in the

behavior of the A2(t) function for this direction.

70

• o

00 •

o

r-•

o

•

MH

z

m •

ON

4J <0

o i H

+ w

O O r-l

in

o

o

CO .

o

CM

o

o .p m < ^

o —r in .

CM

O

CM

in o in

o

u o

c o •H o c D

a «a o >i

C

rH T

U 3

71

ON

o

00

o

o

in

<n 9

O

CM

O

o

o

< ^

—r in

CM

—r-

o CM

in

-T"

O in a

o

o

N

X

CM

•P (0

o H

-f w

o O -H

f tr

o

c o •H +J

o c o a (0 o

u

r. u I I •

in

0)

•H (14

72

ON

o

N X

00

o in

P (0

o

so o

o

ten

in u o

o

< ^

o

ro •

o

CM •

o

c 0

•H 4J O c 3 ^

a 10

o > i

CT Wi 0) c u 1 1

o in .

CM

o .

CM

—r in

O in *

o

VO

0) u d

•H

73

ON •

o

00 •

o

r* •

o

• N

i CM CM

• ON

4J <0

^ O o •—'

.

o

in

o

o

, 0 ^

4i %_ <

1

i n •

CM

o E-<

A Ui

^m^

O •

CM

o

CM

O

- t -

o m o

ten

u O

(M

c o

• H 4-> O c D

C L .

a fd o > i DN u 0) c

0)

3 IP

74

ON

o

o

o

CM *

O

N

ON

CM

00 m

O 4J •0

r- o o

o .—

VO o

O iH

O m CM

r o CM

; r in

—r o

T -

in .

o

o

tcr u o

c o

•H p o c 3

a (d o > i

i-( 0) C u I I

^ 00

U 3

75

Results for q||[110], s I | fOOll

Data were obtained for 10 and 30 MHz for the [110]

direction with [001] sound polarization. The limiting

energy gap at 30 MHz is smaller than the value obtained

at 10 MHz. This behavior is different from that of the

preceding two directions. The change is small in this

case, lying within the experimental error. The limiting

energy gap averaged for both frequencies is 3.42 ± 0.13

KgT^ which is very close to O'Hara and Marshall's value.

No clear evidence for either temperature or frequency

dependence of the A2(t) function can be seen in the data.

Table 2 shows a slight decrease in AjCt) as the reduced

temperature is decreased from t = 0.80 to t = 0.60, but at

30 MHz the tendency is in the opposite direction. On the

whole it seems lilcely that there is no temperature depend

ence for this direction and polarization. Representative

graphs of ^2^^^ ^^' ^ ^^^ both frequencies are shown in

Figures 17 and 18.

Results for q||[111] with Arbitrary Polarization

The behavior of the limiting energy gap for the [ill]

direction vs. frequency is similar to its behavior for

q|I[110] with s| I [001] in that a decrease occurs between

10 and 30 MHz followed by an increase in the gap at 50

MHz. This behavior is opposite to the results for

q||[100) and q| | [110] with s[110]; it is also different

from the results for longitudinal waves for which mar)ced

76

ON

o

00

o

in

o

m

O

< ^

(N

o

in

CM

- I —

o CM

in o in

o

N

r» •

o

VO

o

^ VO

• o t H

4J «t3

.-^

ft

o

o •H +J O c 3

[14

a o CP

<u c u I I

•

0)

3 DN

• H

77

ON

o

00

o

o

N X

in in

m P

VO

o

in

o

t cr u o

o

CM

o

c o

•H 4J o c 3

a rd

o > 1 DN u 0)

c

ii ^

o «

i n

CM

-r-o CM

in o *

in

o

o CM

u 3

• H l>4

78

o

00 •

o

r* •

o

VO •

o

• N X X

i n o

• CM i n

P ITJ

.-» rH i - t

in

o

O

it ^

+ 0

o

• o

ro •

o

CM •

o

c 0 •H p u c 3 CM

a «d o > 1 D Wt Q) c f

0)

3 D

in

CM

o CM

in o *

in

o

79

increases in the limiting values of the energy gap occur as

the frequency is increased to 30 and 50 MHz. The average

limiting energy gap for all three frequencies for shear

waves is 3.28 ± 0.13 K^T^, which is somewhat smaller than

for any of the other directions. However, the value is

still much higher than that for the [111] longitudinal

value.

The A2(t) function for all three frequencies exhibits

a linear decrease with decreasing reduced temperature and

extrapolates to a value very close to the limiting energy

gap at each frequency (see Figures 19, 20, and 21). The

slopes of the A2(t) vs. t plots also decrease linearly with

increasing frequency, as can be seen in Table 2.

Summary of Energy Gap Results

The average zero Kelvin limiting energy gaps from

shear wave data show considerably less anisotropy for the

different directions than do the longitudinal wave values.

The limiting energy gaps for q| | [110] with s|| [1101 or

s||[001] are rather close to the longitudinal wave value

obtained for the [110] direction. The shear wave limiting

energy gap for the [100] and [111] directions are much higher

than the longitudinal wave values. No definite trends toward

larger energy gaps (or smaller ones) with increasing frequency

can be supported by the shear wave data for any direction.

The function A2(t) exhibits a consistent temperature de

pendence for every direction except for the [110] direction with

80

s||[001]. For this case, the behavior of A2(t) with respect

to reduced temperature is slightly erratic but overall roughly

constant. The possible limiting type of behavior suggested

by O'Hara and Marshall for the [100] direction is not indi

cated by the present study. Nor is it exhibited for the

[110] direction with either polarization within the limits

of experimental error. However this type of limiting behavior

does appear to be possibly occurring in the [111) direction,

where the slope of the A2(t) function decreases approximately

linearly with increasing frequency.

B. Electronic Attenuation as a Function of Frequency for Longitudinal Waves

In this section and in the following one will be pre

sented the results of analyzing, in terms of Pippard's theory,

the attenuation vs. frequency data from both O'Hara and

Marshall's investigation and the present study. It will be

shown that these results indicate that q^ >1 for at least two

of the crystallographic directions of the samples studies

in these investigations throughout the range of frequencies

used. This is an important conclusion because of the fact

that no rapid-fall region was observed in the data for any

direction or frequency studied. As was discussed in

Section 1.2, a lack of observation of a rapid fall region

for transverse wave attenuation when the ql product is

greater than unity is contrary to the predictions of current

theoretical literature. Current theories predict on general

81

grounds the onset at ql>l of the electromagnetic coupling

of transverse phonons and electrons. This electromagnetic

coupling, as was discussed earlier, should be suddenly

quenched at the transition temperature, producing an initial

almost discontinuous drop in attenuation. Only one previous

transverse wave ultrasonic attenuation investigation has

been published which reports a lac)c of observation of a

rapid fall region for q^>l. This was the recent investi

gation of the transition metal niobium by Leibowitz, et al.,

' ' in which ql values as high as 20 were obtained.

In the present investigation ql values as high as 50 are

thought to have been obtained with no trace of the rapid

fall phenomenon.

The first part of this section will present a discus

sion of the transverse wave to longitudinal wave attenuation

ratios, which indicate that the ql values are greater than

unity throughout the frequency range of this investiga

tion for the [100] and [110] directions. Secondly results

of fitting O'Hara and Marshall's data to the Pippard theory

of longitudinal wave attenuation will be discussed. An

estimate of the mean free path and ql values for each direc

tion and frequency for longitudinal waves has been obtained.

From the ql values estimated from the longitudinal wave data,

an estimate can be made of the transverse wave ql values, as

will be discussed in the next section. The Pippard theory

a vs. V curves used to fit O'Hara and Marshall's data will n

82

also be compared to the higher frequency a vs. v curves

obtained experimentally by Jones and Rayne.

Transverse Wave to Longitudinal Wave Attenuation Ratios

Table 3 presents the electronic attenuation vs. fre

quency data from both O'Hara and Marshall's longitudinal

wave study and the present investigation. Since the same

samples and frequencies were used in both investigations,

a good indication of whether q^>i or q^<l can be obtained

for each crystallographic direction by comparing the trans

verse wave to longitudinal wave attenuation ratios to the

theoretically predicted values. Morse (32) has given the

following simple equation for each case:

a . -5 V^- 3 (41) = I (y^) ' ' <1)

nl St

nl St

where a and a , are the transverse and longitudinal wave nt nl

electronic attenuations, respectively, and V^^ and V^^ are

the transverse and longitudinal sound velocities, respectively.

The calculated velocities given in Section II.8 have been

used to calculate the predicted ratios given in part A of

Table 4. For ql<l the values can be seen to lie somewhere

between 4.0 and 5.0, whereas for q^>l, the values are

between 2.0 and 3.0. By comparison, the ratios of the

measured values in Table 4, part B, indicate that for our

samples and frequencies, q^>l for all but the [111] direction.

83

TABLE 3

ELECTRONIC ATTENUATION VS . FREQUEIICY DATA

T r a n s v e r s e Waves

• D i r

[ 1 0 0 ]

[ i i o l Sll[i loi

( 1 1 0 ] i l | [ 0 0 1 ]

1111]

FreqCMH:-)

9 . 0 0

2 9 . 9 2

5 1 . 8 5

9 . 3 0

3 0 . 2 3

5 0 . 4 1

9 . 2 2

3 2 . 9 0

5 0 . 0 0

1 0 . 6 4

3 1 . 5 5

5 2 . 0 4

\pprox q ^ 1 0 . 0

3 0 . 0

5 0 . 0

6 . 0

1 8 . 0

3 0 . 0

6 3 3

1 9 . 0

— —

2 . 6

7 . 8

1 3 . 0

On(dB/cm)

1 . 3 8 ± 0 . 0 2 t

4 . 4 5 ± 0 . 3 3

8 . 6 9 ± 1 . 4 2

1 . 3 8 ± 0 . 0 8

4 . 7 9 ± 0 . 3 3

8 . 7 8 ± 2 . 0 6 5

1 . 1 4 ± 0 . 0 5

4 . 5 6 ± 0 . 6 5

^ mm

1 . 3 3 ± 0 . 1 6

5 . 1 3 ± 1 . 2 0

5 . 7 5 + 0 . 2 8 5

^ — — ^ —

L o n g i t u d i n a l Waves

Dir !

(lOOl

[110 ]

[ 1 1 1 ]

Froq^Vwiz)

1 1 . 0

3 2 . 6

9 . 6

3 2 . 3

51 .7

8 .7

3 0 . 7

5 1 . 0

\pprox1 q ^ 2 . 3

6 .9

2 . 0

6 . 0

1 0 . 0

1 .0

3 . 0

5 . 0

a ^ (dB/cm)

0 . 6 4 0 1 0 . 0 2 8

2 .917 1 0 . 1 3 1

0 . 5 3 2 ± 0 . 0 2 5

1 .70 ± 0 . 1 5

4 . 1 7 ± 0 . 1 6

0 . 1 8 6 1 0 . 1 4

1 . 1 2 8 1 0 . 6 0

1 . 0 6 6 1 0 . 8 8

84

TABLE 4

RATIOS OF TRANS /ERSE TO LONGITUDINAL ELECTRONIC ATTENUATION

Direction

[100]

[110], S\ |[irO]

[110], ^1|[001]

[111]

A. Predicted ^nt/<^nl Ratios

q^ < 1

5.14

4.19

4.91

4.34

q^ > 1

2.92

2.55

2.83

2.61

B. Measured ^nt/^nl Ratios

Direction Frequency MHz Ratio

[100]

[110], S||[110]

[110], 21|[001]

[111]

10 30

10 30 50

10 30

10 30 50

2.16 1.53 } ql > 1

2.59 2.82) q£ > 1 2.11

2.14 2.69 } q > 1

7.15 4.55) q < 1 5.39

85

In this section, O'Hara and Marshall's attenuation vs.

frequency data is fitted to the Pippard theory^^ of longi

tudinal ultrasonic attenuation. Pippard predicted from the

free electon model that the total longitudinal wave electronic

attenuation would depend upon ql according to the relation

N m / /»V 2 - 1 „

(43) a = ? _ { (q^) tan UQD -1 . n 0 V T -1 *

^o si 3 [ql - tan " (q )] where N is the number of electrons per unit volume, m is

e

the mass of a free electron, p^ is the density of the metal,

^Q1 is the longitudinal wave velocity and T is the isotropic

electron relaxation time. This expression is similar to

Pippard's expression for transverse wave attenuation given

in Equation (9). Equation (4 3) likewise reduces to two

simpler expressions for the q^>l and q^<l regimes, as

follows: Nm V^

^ ^ 'n = 6TV-72 • <. iqKl) o si ^

2Nm V 03

^^5) «n = 15p V 2 • q^ (q^<i) o s

In the above equations V^ is the Fermi velocity and ^ is the

angular velocity of the ultrasonic wave. It can be seen

from Equations (44) and (4 5) that for q'^>l, a varies

linearly with frequency and is independent of q^, while for

ql<l, CL varies as the square of the frequency (since q=-'y--) 3

and has a temperature dependence from q^. Equation (4 3) can be expressed in terms of the limiting attenuation a for

86

q^>l given in Equation (44) as follows:

^ ^ ' ^ 3 [q£-tan-l (ql) ] - 1 }

To obtain a curve from Equation (46) which best fit O'Hara

and Marshall's data, a " and I were treated as adjustable

parameters. A first estimate for a ' was obtained by plot

ting n/v vs. V to obtain the limiting value. The magneto

acoustic value for I was used as a first estimate for it.

Varying a " and I in Equation (46) to obtain the best fit

to the data yielded an estimate for the electron mean free

path and approximate ql values for each crystallographic

direction. The fitting of the Pippard curve to the data was

facilitated by the indication of whether ql<l or q^>l given

by the ' / nl ^^^^^ and by comparing each curve to the

higher frequency longitudinal wave data of Jones and Rayne.

The samples used by Jones and Rayne were of approximately

the same purity as the ones used in this experiment, lx>th

being electron beam zone refined. Hence the electron mean

free paths were throught lilcely to be similar in magnitude

for both investigations. It was found that the Pippard theory

(a vs. v) curves used to fit O'Hara and Marshall's data n

corresponded very well to the curves which fit Jones and

Rayne's a vs. v data except for the [100] direction, n

Results for q||(100)

In Figure 22 are plotted the total electronic attenua

tion values a as a function of frequency for q||[100]. Data

87

0)

o 4J

c >1 <0

•P VH •H m •0 •0 u ta

c R)

(0

a (u 10 a C 4J

•H o fH P< »0 D

o

o

o

N X s:

o o o .H O

r-i I

o

o

o CD

o o

• cr u o

«M

o c

cr a> u

0) > c o

«

C <D +J •J <

U •H

o

o 0)

M

> <0

5

c

c

I I •

CM CM

Q)

Cr>

1 o

•

... ,

o so

1

o i n

o

^

1

o «n

O •

CM

• • • • • r

o • •H

88

B u

=8

— — t m n

' 1'

o CM

1

m i-i

r o

o o ^

o in CO

o o ro

o in CM

o o CM

o in .-»

o o •-I

o in

o r>

o •H

• O

N X £ o o 'T 1 o ^

. — 1

o o fH •_<

• cr u o «M

> 1

u c 0) 3 tr 0) V4 KM

.

n >

C o •H *> «J 3 C 0) 4J 4J <

U • H

c o V4 +J u o tH

u 9) > « ^

r-t « c •H •o 3 4J •H O c s 1 1 •

m CM

0) h 3 C7 •H t.

in

89

points were obtained for this direction for 11.0 and 32.6

MHz. It was found that a Pippard theory curve could be fit

to these two data points fairly well for an estimated mean

free path value of 0.022 cm, which corresponds to the magneto

acoustic estimate of 0.022 to 0.025 cm for mean free path

obtained by O'Hara and Marshall. This curve did not align

with the curve determined by Jones and Rayne's data, although

the 11.0 MHz data point was found to lie almost exactly on

their curve. With no higher frequency data available from

O'Hara and Marshall than 30 MHz, the 0.022 cm value was taken

as the best estimate for the mean free path. This gave

estimated ql values of 2.3 and 6.9 for the 11.0 and 32.6 MHz

data points. This agrees well with the ' / nl ^^^° which

indicates that the ql product is greater than unity for both

frequencies. The frequency for which ql = 1 can be estimated

from the equation

where I is taken to be 0.022 cm. The curve in Figure 22 can

be seen to approach linearty above about 5 MHz, in accordance

with the Pippard theory for q^>l.

Results for q|I [110 3

The data for the [110] direction is plotted in Figure

24r where the dashed line represents the fit of the data to

the Pippard theory. The estimated ql values are approximately

2.0, 6.Or and 10.0 for 9.6, 32.3, and 51.7 MHz respectively.

90

N X s: o o «H I

o

f cr u o

u c 0) D cr 0)

0) > c o

-H •P O 3 C «) 4J

<

c o u o 0)

u o > 2

c •H •o 3

•H

c

CM

ki 3

91

n o o o

m ^ «) c > i Id C6

TJ c m « 0) Id C -P 0 <d •o Q

« £ V) M «

s •o c <d

Id M «Q Id K +J - (d O Q

o o

o o 00

o o

N X

o o o r-l I

O

icr

u o

> i o c

3

M

n > c o

•H

« 3 C O 4J

<

c o 4J O o

O >

Id c

•H •O 3 4J •H cn c

in

CM

0)

3

92

This is in agreement with the prediction of q^>l from the

°*nt/%l "tios. The curve is consequently approximately

linear above 10 MHz as expected for q^>l. The estimated

mean free path value obtained from this fit is 0.0203 cm,

which is close to the estimate of 0.022 to 0.025 cm obtained

for the [100] direction by the magnetoacoustic measurement

of O'Hara and Marshall. The estimated frequency for which

q^l is 5.10 MHz. The data also can be seen to fit very well

the curve obtained from Jones and Payne's data, as shown in

Figure 25.

Results for q||[111]

The [111] direction is indicated by the a /o . ratios

to have ql<l for all three frequencies. In Figure 26 the

data can be seen to display an unexpected behavior with

either the data point at 30.7 MHz too high, or the data point

at 51.0 MHz too low. A fit of Equation (46) to the 51.0 MHz

data point results in q-d values approaching infinity, whereas

a fit to the 30.7 MHz point gives approximate ql values of

1.0, 3.0, and 5.0 for 8.7, 30.7, and 51.0 MHz respectively.

Furthermore, both the 8.7 MHz and 30.7 MHz data points fit

the curve obtained from Jones and Rayne's data, as seen in

Figure 27, with the 8.7 MHz point on the curve and the 30.7

MHz point slightly below it. The behavior of the data point

at 51.0 MHz is highly interesting because the 50 MHz data

point obtained with shear waves for the [111] direction

displays almost exactly the same behavior, as will be discussed

93

0) 0) c o ^

o

Id 4J

« V4 Id Id X -P - Id O Q

Id x; 0 w 4J O -P 4 J -

+J -H u*

»« V4

n ^ 0) c

•ri t-l > 4J rH

-ri Id V) (l4 X 0) (0

> i ' a TJ v Id

Id 0^ CU a t j

•H & 4

c

•H )-( Id u Id 2 c a •H cuna O H c

Id u P4 Id

o o

. o

o 00

o VO

o in

o

• o cn

N X T, o o rH I

o

• cr u o

u c

U4

>

c o

•H

« 3 C 0)

c o u p o 0)

•H

u >

2e

Id c

•H

•o 3 P •H D> C

I I •

so IN

0)

3

94 N

O O O f H I

O

tcr u o

> 1 O c 0) 3 tr 0) ki

>

c o

•H .P Id 3 C O 4J •P <

o u p u Q)

rH U

« > Id 5

c •H •o 3 •P •H cn c I I

CM

u 3 cn

o in o o o

CM

95

later. The estimated mean free path value obtained from the

fit represented by the dashed line in Figure 26 is 0.011 cm,

a value less than but within an order of magnitude of the

magnetoacoustic estimate for the [100] direction. The

estimated ql values are greater than unity, but are in the

approximate region of q ^ l, where the attenuation behavior

is changing. The estimated frequency for which q^=l is

10.33 MHz.

C. Electronic Attenuation as a Function of Frequency for Transverse Waves

In the preceding section O'Hara and Marshall's longi

tudinal wave attenuation data has been analyzed in terms of

the Pippard theory of ultrasonic attenuation to obtain an

estimate of the electron mean free path for each crystallo

graphic direction. The Pippard theory curves have also been

compared to the a vs. v curves obtained by Jones and Rayne

for higher frequencies and can be seen to correspond very

well except for the [100] direction. The ql values estimated

from fitting O'Hara and Marshall's data to the Pippard theory

are greater than unity for the [100] and [110] directions

and range from approximately unity to a slightly larger value

for the [111] direction in the 10 to 50 MHz range. The

a /a , ratios also indicate that the q values for the [lOOl nt' nl

and [110] directions are in the q^>l regime, but that the q

values for the [111] direction are in the q^<l regime.

96

The Fermi surface model proposed by Loroer^^ and

illustrated in Figure 1 is fairly isotropic. Hence the

electron mean free path estimates obtained from transverse

wave and longitudinal wave studies should be approximately

the same, despite the fact that transverse and longitudinal

waves interact with different parts of the Fermi surface.

If such is the case, then the transverse wave ql values can

be estimated from the longitudinal wave ql values as follows:

(48) (q£) . iql) (Z lj

^ ^st

where V^^^ euid V^^ are the longitudinal and transverse wave

sound velocities, respectively, for each direction. This relation follows from the definition

-2jr_f,

s

From the sound velocities given in Section II.8, the q^ values

for the present study therefore would be approximately twice

as large as those for O'Hara and Marshall's investigation.

This section will present the results of fitting the

transverse wave attenuation data from the present study to

the Pippard theory for transverse waves. Estimates of elec

tron mean free paths which were obtained in this manner for

each direction are given in Table 6 along with the longitudi

nal wave estimates. It can be seen from Table 6 that, con

trary to our above assumption, the transverse and longitudinal

wave electron mean free path estimates are not equal. In

stead the tremsverse wave electron mean free path estimates

(49) ql E i^) I

97

(0

>

5

tl

U O

D-0) V4

M-l

N N

i n

in

cn

•4

H i J CD < EH

Z O P EH CJ W 1-5 M P W H g H EH cn H

IT5 C

•H

•p •H

O

01 (U > Id

in u Q) > cn C (d

EH

o

u

o

cr u

o

•H Q

CM

O *

CO

o CN O

VO

o o o

o o

N a: X in

Cs so in o

o o

N w 2 m V£>

t^ K S 00 in

N

00 in CO o

o in m o

CN

r-f cn

o o

>^w

98

are from 1.5 to 2.5 times as large as those obtained using

longitudinal waves, and the estimated ql values for the

transverse wave study are correspondingly larger than pre

dicted.

The transverse wave data were fit to Equation (9): Nm

(9) a = ^, ^ — (k:£) "" Po s^T g

where

(10) g = - ^ { ^^4-^-1 tan-^ ql - 1} 2 (q^) q

and all other quantities have been defined previously. For

high ql values, a approaches the limiting value a " given by n n -'

E q u a t i o n ( 1 5 ) : 4Nm V^ o . , ^ 9

(15) a " = e f ^ 27TV , q > > l v<10 Hz n ~ 37T p V o s

a can be expressed in terms of in a similar manner as n n

for longitudinal waves as follows:

(50) -^ = Al r_JLlir-A (iz2:) i

where a ^ and ql are again treated as adjustable parameters

to obtain the best fit.

Results for q| | [lOOJ , s 1110]

The data for the IIOOJ direction is plotted in Figure

28, where the dashed line represents the fit of the data to

the Pippard theory. It can be seen that the curve given in

Figure 28 lies somewhat above the 30 MHz data point, and

slightly below the 50 MHz point. This curve corresponds

to a q^ value of approximately 50 at 50 MHz and to the

99

t «

tcr u o

u c 0)

u

>

c o

•H •P « 3 C «) P

<

c o u o rH

u >

Id

x: CO I I •

00 CM

0) w 3 tJ»

100

electron mean free path estimate of 0.057 cm given in Table

5. This mean free path estimate is somewhat larger than the

0.022 cm estimate obtained in O'Hara and Marshall's magneto

acoustic measurement.

The 30 MHz data point appears to be anomalously low.

A curve with a smaller slope lying closer to this data point

would require much larger ql values and a corresponding

increase in the mean free path estimate. For example, a

curve equidistant to the 30 and 50 MHz data points would

give a q^ value of 125 at 50 MHz. On the other hand, the

ql estimates would have to be decreased to \inrealistically

small values (<<1) to obtain a curve which could be fit to

both the 30 and 50 MHz points. Such a curve could not then

be fit to the 10 MHz data point, which is the most reliable

point from the stcindpoint of scatter. Hence the curve given

in Figure 28 appears to be the best fit to the data and also

gives teasonable values for the ql product and mean free path

estimate.

It is interesting that the shear wave data for the [110]

direction appears to display a similar tendency for the 30 MHz

data point to be anomalously low, as can be seen in Figure

29. This was the case also for the longitudinal wave data

for the [110] direction, as shown in Figure 24. Unfortunately

insufficient longitudinal wave data for the [100] direction

are available for comparison. As mentioned previously, the

[111] direction appears to exhibit a different type of anomaly.

101

Results for q| | [110] , sH flloi

In Figure 29 is plotted the data for the fllOJ direc

tion with sound polarization in the (110) plane. The Pip

pard fit to the data is linear for the frequency range studied,

indicating that ql>l for these frequencies. The q^ esti

mates are given in Table 3. The estimated mean free path

is 0.0358 cm, a value wliich corresponds well to the longi

tudinal wave estimate.

Results for q| | [110], s| | [001]

Data were obtained for two frequencies for the [110]

direction with sound polarization in the (001) plane. These

points are plotted in Figure 30. The Pippard fit to these

two points, indicated by the dashed line, corresponds quite

well to the curve used to fit the data for q| | [110], s| | [llo]

in Figure 29. The ql values and mean free path estimate for

ql I [110] and s| | [001] can be seen in Tables 3 and 5 to be

almost the same as those obtained for q| | [110] and s| |[110].

Results for q||[111] with arbitrary polarization

The shear wave data for q| | [111] exhibit the same

anomalous behavior which was described previously for the

longitudinal wave data in the same direction. From the

shear wave data shown in Figure 31 it can be seen that either

the 30 MHz data point is anomalously high or the 50 MHz

point is anomalously low. The latter seems much more likely

since a Pippard curve passing through the 50 MHz data point

could not include either the 10 or 30 MHz points. Such a

102

o 1^

+ «

• cr u o

u c V 3 cr

n > c o

• H •P

«s 3 C o •p

c o •p o 4) •H

u >

Id

(o I I •

CM

0) kl 3 t7»

103

o o

o

o o

4W

o 00

o r-

o \o

'

o in

o

.

o «n

•

o CM

1 1 .

o f H

•H

——

f tr u 0

«M

> i

u c o 3 cr o V4 k i

. to >

C o -H • P « 3 C 0) P

<

u •H c o IH •p o Q)

f H

u V > Id :c V4 « 0) £ (/) 1 1 •

o ro

«) V4 3 cr>

104

U «

- H

tcr u o

u c 3 tr 0) u

> c o

• H •P « 3 C o

<

c o • p o 0)

f H • u c

o tt) -H > "p fi Id

-H u u « Id 0) f H x: o CO CU I I

0)

3

105

curve would also give unrealistically high estimates for g^

values and the electron mean free path. The curve shown in

Figure 31 gives q^ values of 2.6, 7.8, and 13.0 for 10, 30

and 50 MHz, respectively, and a mean free path estimate of

0.014 cm. This mean free path estimate is close to the

longitudinal wave mean free path estimate of 0.011 cm. It

is also reasonably close to the estimate of 0.022 - 0,025 cm

obtained magnetoacoustically by O'Hara and Marshall for the

[100] direction. In fact, since the a ^/a , ratios in Table nt nl

4 indicate that q^<l for the Jill] direction, it is possible

that the 30 MHz data point is also slightly below the value

which might be expected for both the longitudinal veve and

shear wave data (Figures 26 and 31) . A slight upward shift

of the slope of the Pippard curve would result in q^<l for

both cases. Such behavior would correspond to the tendency

of the 30 MHz data point to lie below the Pippard curve for

the [100] and [110] directions for both longitudinal and

shear wave data. This tendency was mentioned previously.

The anomalous behavior of the shear wave data initially

prompted the question of whether some other cause, such as

eimplifier saturation might have given rise to the attenuation

decrease in the 50 MHz data point. However, care was talcen

to prevent amplifier saturation during the experiment, and

it is considered unliJcely that saturation occurred. It is

considered even more unlikely that amplifier saturation would

also have caused the almost Identical behavior to be observed

106

in the longitudinal wave data, particularly since lower

power levels were used at 50 MHz than at 30 MHz in the

longitudinal wave investigation. Shear and longitudinal

wave data are plotted together for comparison in Figure 32,

using different scales for each.

9 -

107

8 . 1.60

7 , 1.40

1.20

5 .1.00

3 .

0.80

0.60

1 .

0.40

n

(db/cm)

0.20 ^

10 20

6

v(MHz)

1—

30 40

i

A O'Hara and Marshall's Data (Right-hand Scale)

e Present Study (Left-hand scale)

50 60 70

Figure 32. —Shear Wave and Longitudinal Wave Attenuation for i||[llll.

108

III.4 Interpretation of Results

The shear wave attenuation data obtained in this study

have been found to fit the BCS equation very well, giving the

energy gap values presented in Table 2. The zero Kelvin limiting

energy gap values are fairly isotropic, with a difference of

about 4% between the minimum and maximum average values for the

various crystallographic directions. This isotropic behavior

of the limiting energy gap agrees well with the fairly isotropic

Fermi surface model proposed by Lomer^^ and illustrated in

Figure 1.

The energy gap function A (t)defined in Section III.2,

Equation (40) , has been found to decrease linearly with re

duced temperature and to extrapolate to the limiting zero

Kelvin energy gap for every direction except for q||[1101

with s||[001]. As was discussed in Section III.2, this type

of temperature dependence indicates the possibility of

multiple energy gaps of an appreciable size range for these

directions. Similarly, the fairly constant behavior of the

Ap(t) function with reduced temperature for q||[110] with

s||[001] might indicate either a single energy gap or multiple

energy gaps of approximately the same size for this direction

and polarization. This is one possible interpretation of

86 the data, based upon the multi-band model of Perz and Dobbs

given by Equation (38). One assumption inherent in this

model, as mentioned previously, is that no interband inter

action occurs. No information appears to be available in

109

the literature on molybdenum to either support or contradict

this assumption.

O'Hara and Marshall observed a frequency dependence of

the zero Kelvin energy gap for the [100] and [110] directions

for longitidunal waves. Since the electron mean free path is

impurity-limited in the temperature range of both their

study and the present one, such a frequency dependence would p Q

correspond to a dependence upon the ql value. Leibowitz

has shown that for q- 1 the "effective zone," or region of

the Fermi surface to which the electron-phonon interaction is

primarily confined, does depend on the ql value for both

longitudinal and transverse waves. This dependence is given

by

(51) cos 9 = ^ + i Vf ql

where 6 is the angle between the Fermi velocity vector,

V^, and the sound wavevector q. Since the Fermi velocity is

typically much larger than the sound velocity, the interaction

favors electrons close to the equator of the Fermi surface

perpendicular to the phonon wavevector. For q^>>l, the

effective zone narrows to a thin band along the equator of the

Fermi surface. This narrowing of the effective zone to a

limiting value as q^ -> » suggests that both the electronic

ultrasonic attenuation and the energy gap should be dependent

upon ql for low values of ql. Furthermore the q dependence

should disappear as q£ -• -.

110

In the present investigation the zero Kelvin enerqy

gap shows no trend of change within the limits of experimental

error for frequency changes within the range from 10 to 50 MHz

for any direction of sound propagation or polarization measured.

This result is thought to be due to the q/ values being in

the q^>>l range for the [100] and [110] directions. For the

[111] direction, although no frequency dependence of the zero

Kelvin gap is indicated, a change in the slope of the A(t)

function was observed with increasing frequency, suggesting

that the slope is approaching a limiting value. These results

agree well with ql value estimates obtained from the Pippard

theory^ fit of the electronic ultrasonic attenuation vs.

frequency data.

For transverse waves, the electrons within the equatorial

band which interact most strongly with the ultrasonic phonons

-+•

are those which have velocity V^ parallel to the sound polari

zation vector s. Hence the effective zone for transverse waves

is further restricted to that portion of the equatorial band

which is approximately perpendicular to s. For an anisotropic

Fermi surface, this suggests that for a given sound propagation

direction, different sound polarizations would result in

different values for the electronic attenuation and for the

energy gap. The fact that the ql value estimates for the [110]

direction are approximately equal for both s||[001] and

s||[lIO] is a further indication that the Fermi surface is

fairly isotropic with respect to the [110] direction.

Ill

III.5 Summary

Presented in this worJc has been a determination of the

superconducting energy gap 2A(0) in molybdenum for q||[100],

q|I[110] with both s| UlIO) and s||[001], and q\|[111] direc

tions using ultrasonic shear waves. These measurements have

been made for comparison to previous measurements on the same

crystal by O'Hara and Marshall using longitudinal waves. The

energy gap values obtained in the present investigation show

less anisotropy than do those obtained with longitudinal waves

and agree well with the isotropic Fermi surface model proposed

by Lomer. Some evidence has also been found to suggest

the possibility of multiple energy gaps for each of the above

directions in molybdenum.

In addition, the total electronic attenuation vs. fre

quency from both the longitudinal and transverse wave studies

18 has been compared to Pippard's theory. Results of this

analysis indicate that the ql values in the transverse wave

investigation were greater than unity for all three direc

tions. For the [100] direction, ql values as high as 50 are

thought to have been obtained. The well-known "rapid-fall"

region which should be expected to appear under these condi

tions for transverse waves was not observed.

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116

63. Ibid., pp. 134-144,

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APPENDIX

This appendix includes representative data sets for

each crystallographic direction, sound polarization and

frequency for which data were taken. The data consist of

echo voltage height values (in volts) as a function of

reduced temperature in increments of 0.02 reduced tempera

ture. These data were taken from X-Y recorder plots and

were read into the computer program which calculated the

energy gap and total electronic attenuation values pre

sented in Chapter III.

118

TA&JiE 6

119

Data for q|

t

a.,00 ,98 ,96 ,94 ,92 .90 ,88 ,86 ,84 ,82 ,80 ,78 ,76 ,74 ,72

[100], s!|[iio]

V

3.21 4,86 5.60 6.25 6.84 7.42 7.83 8,25 8.65 9.09 9.50 9.90 10,31 iO.75 11.13

, 9.0 MHz,

t

.70

.68

.66

.64

.62

.60

.58

.56

.54

.52

.50

.48

.46

.44

.42

Echo No. 10

V

11.52 11.91 12.28 12.64 12.99 13.33 13.65 13.96 14.24 14.53 14.76 15.03 15.26 15.46

^ •*

TABLE 7

Pata for q||[100], s||[110], 29.92 MHz, Echo No. 2

i,00 ,98 ,96

4 6 6

80 00 58

,94 ,92 ,90 ,88 ,86 ,84 ,82 .80 .78 ,76 ,74 .72 .70

7.09 7.50 7.84 8.17 8.48 8.83 9.27 9.58 9.82

i0.04 10.25 10.45 10.67

.62

.60

.58

.56

.54

.52

.50

.48

.46

.44

.42

.40

.38

11.39 11.55 11.70 11.85 11,98 12.10 12.22 12.33 12.43 12.53 12.62 12.69

TABLE 8

120

D a t a f o r q

t

1 . 0 0 . 9 8 . 9 6 . 9 4 , 9 2 . 9 0 . 8 8 . 8 6 . 8 4 . 8 2 . 8 0 . 7 8 . 7 6 . 7 4 . 7 2 . 7 0

l | [ i o o ] , s i U u o i

V

2 . 1 3 3 . 8 2 4 . 4 8 5 . 0 7 5 . 6 4 6 . 2 0 6 . 7 2 7 . 2 2 7 . 7 0 8 . 4 3 8 . 9 3 9 . 3 8 9 . 8 0

1 0 . 2 7 1 0 . 7 0 1 1 . 1 0

, 5 1 . 8 5

t

. 6 8

. 6 6

. 6 4

. 6 2

. 6 0

. 5 8

. 5 6

. 5 4

. 5 2

. 5 0

. 4 8

. 4 6

. 4 4

. 4 2

. 4 0

. 3 8

."iHz, 7,cho No. 2

V

1 1 . 5 1 1 1 . 9 2 12 .3*; 1 2 . 7 1 1 3 . 0 9 1 3 . 4 5 1 3 . 7 8 1 1 4 . 0 8 1 4 . 4 1 1 4 . 6 9 1 4 . 9 7 1 5 . 2 3 1 5 . 4 9 1 5 . 6 7 1 5 . 9 1

-

D a t a f o r q |

t

1 . 0 0 . 9 8 . 9 6 . 9 4 . 9 2 . 9 0 . 8 8 . 8 6 . 8 4 . 8 2 . 8 0 . 7 8 . 7 6

TABLE 9

| [ i i o ] , t\ | [ iTo]

V

3 . 0 5 4 . 4 0 4 . 9 5 5 . 3 7 5 . 7 0 5 . 9 6 6 . 2 1 6 . 4 5 6 . 6 9 6 . 9 2 7 . 1 5 7 . 3 7 7 . 6 0

, 9 . 3 MHz,

t

. 7 4

. 7 2

. 7 0

. 6 8

. 6 6

. 6 4

. 6 2

. 6 0

. 5 8

. 5 6

. 5 4

. 5 2

. 5 0

Echo No. 4

V

7 . 8 1 8 . 0 3 8 . 2 3 8 , 4 3 8 . 6 2 8 . 8 2 8 . 9 9 9 . 1 7 9 . 3 3 9 . 4 8 9 . 6 2 9 . 7 6

•

TABLE 10

121

Patafor q||[110], sl|[110], 30.23 MHz. Echo No. 1

t

1.00 .98 .96 .94 .92 .90 .88 .86 .84 .82 .80 .78

V

6.55 8.68 9.57

10.19 10.74 11.20 11.62 12.01 12.40 12.78 13.17 13.53

t

.76-

.74

.72

.70

.68

. 66

.64

.62

.60

.58

.56

.54

V

13.87 14.22 14.57 14.91 15.23 15.52 15.82 16.10 16.35 16.59 16.84 17.06

TABLE 11

Data for q

t

1.00 .98 .96 .94 .92 .90 .88 .86

1 |[110], s| |[1T01

V

4.60 6.03 6.72 7.30 7.76 8.22 8.66 9.11

, 50.41

t

.84

.82

.80

.78

.76

.74

.72

.70

MHz, Echo No. 1

V

9.53 9.96

10.37 10.77 11.15 11.51 11.89 12.24

-J

122

Data for q||[110]

t

. 1.00 .98 .96 .94 .92 .90 .88 .86 .84 .82 .80 .78 .76 .74 .72

TABLE

/ S| |[oo:

V

3.37 4.78 5.42 6.00 6.50 6.94 7.36 7.77 8.11 8.45 8.76 9.08 9.38 9.70

10.00

TABLE

12

L].

13

9.22 MHz, Echo No. 6

t

.70

.68

.66

.64

.62

.60

.58

.56

.54

.52

.50

.48

.46

.44 ,42

V

10.28 10.58 10.85 11.11 11.36 11.61 11.83 12.05 12.27 12.47 12.64 12.82 12.97 13.13 - -

Data for q[|[110], s||[001], 32.9 MHz, Echo No. 1

t

1.00 .98 .96 .94 .92 .90 .88 .86 .84 .82 .80 .78 .76 .74 .72

V

5.35 6.00

. 6.37 6.67 6.90 7.14 7.37 7.60 7.80 7.99 8.18 8.36 8.54 8,71 8.88

t

.70

.68

.66

.64

.62

.60

.58

.56

.54

.52

.50

.48

.46

.44

.42

V

9.03 9.20 9.35 9.50 9.63 9.76 9.87 9.98

10.09 10.20 10.29 10.38 10.45 10.54 10.60

TABLE 14

123

Data f o r q ! | [ 1 1 1 ] ,

t

1 . 0 0 . 98 . 9 6 .94 . 9 2 .90 .88 . 8 6 .84 .82 . 80 .78 . 7 6 .74

. • ' .

s a r b i t r a r y

V

2 1 . 8 5 2 3 . 5 0 24 .30 25 .10 25 .90 2 6 . 5 0 2 6 . 9 5 27 .40 2 7 . 8 2 2 8 . 2 0 2 8 . 5 3 28 .87 29 .20 2 9 . 4 9

, 10.6^'.

t

. 72

.70

.68

.66

.64

.62

.60

.58

.56

.54

.52

.50

.48

.46

MHz, Echo No. 3 ,

V

29 .79 3 0 . 0 5 30 .33 30 .57 3 0 . 8 1 31 .03 31 .25 31 .47 31 .66 31 .82 3 2 . 0 1 32 .14 32 .20

TABLE 15

Data for q||[lll], s arbitrary, 31.55 MHz, Echo No. 1

V

. 8 6

.84

. 8 2

. 8 0 • 78

• 1 8 . 6 3 1 8 . 9 5 1 9 . 2 6 1 9 . 5 5 . 1 9 . 8 4

. 62

. 6 0

.58

. 5 6

.54

21 .76 21 .96 2 2 . 1 3 2 2 . 2 9 2 2 . 4 7

TABLE 16

124

Data for ql |[111] , s a r b itrary, 52.04 MHz, Echo No

1.00 .98 .96 .94 .92 .90 .88 .86 .84 .82 .80

4.45 5.1^ 5.51 5.77 5.99 6.19 6.37 6.52 6.68 6.83 6.97

.78

.76

.74

.72

.70

.68

.66

.64

.62

.60

.58

7 7 7 7 7 7 7 7

11 26 39 52 64 76 ,87 ,98

8.08 8.18

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