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ABSTRACT

TRANSPORT PROPERTIES OF DILUTE
COPPER ALLOYS

by Burton w. Scott

Seventeen dilute alloys of Cu-Ag, Cu-Au, Cu-Sn, Cu-Zn-In,
Cu-In-Cd, and Cu-Zn-Ga were prepared by induction melting
and chill casting. The thermoelectric power of these
alloys and of pure copper was measured over the range 80K
to 3000K. The residual resistivity of the Cu-Ag and Cu-Au
alloys was determined from the resistance ratio. On the re-
maining alloys measurements of thermal conductivity and
electrical resistivity were made simultaneously with, and on
the same sample, as the thermoelectric measurements. The
resistivity measurements extend from 80K to 3000K while the
thermal conductivity measurements extend from 80K to 1000K.
Wherever possible, the thermoelectric power of dilute binary
alloys was used to predict the thermoelectric power of the
ternaries. Substantial agreement between predicted and
measured thermopower provides excellent confirmation that
(1) the thermopower of pure c0pper has been separated
correctly into its diffusion and phonon drag components and
(2) that the theory of Kohler correctly predicts the dif-
fusion term while the theory of Blatt and KrOpschot correctly
predicts the phonon drag thermopower of dilute alloys.
Attempts to correlate the lattice thermal resistance with
the phonon drag thermoelectric power was not successful.

The low temperature lattice thermal resistance agrees with
the Pippard theory. Above 500K our results are relatively
inaccurate, however, the lattice thermal resistivity is in
quantitative agreement with the Leibfried and Schloemann
theory of anharmonic thermal scattering, and shows evidence
of anisobaric scattering in the less dilute samples.

TRANSPORT PROPERTIES OF DILUTE COPPER ALLOYS

BY.
“N‘

Burton W) cott
A THESIS

Submitted to the School for Advanced Graduate Studies of
Michigan State University
in partial fulfillment of
the requirements for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1962

Acknowledgements

I thank Dr. F. J. Blatt for arousing my interest in
the study of transport phenomena, for prOposing the re-
search problem, and for his patience and guidance through-
out the project.

To Dr. Meyer Garber for many helpful suggestions
about the design of the apparatus, and to Dr. Peter
Schroeder for suggestions concerning interpretation of the
data I wish to express my gratitude.

I appreciate the cooperation of the Department of
Metallurgical Engineering and the Hoskins Corporation both
of whom extended the use of equipment for shaping the
samples.

For financial aid I wish to thank the U.S. Army Office

of Ordinance Researdh and the National Science Foundation.

11

TABLE OF CONTENTS

Chapter
1. Introduction
2. Theory

Thermoelectric Power
Effect of Alloying on Thermoelectric Power
Thermal and Electrical Conductivities of the
Electrons
Lattice Conductivity
Preparation of Samples
Apparatus
Description of Results
Thermoelectric Power of Pure Copper
Alloys of Monovalent Metals
Alloys of Polyvalent Metals
Analysis of Results
Electrical Resistivity
Thermoelectric Power of Pure COpper
Thermoelectric Power of Alloys
Thermal Conductivity
References
Appendix I

Appendix II

111

Page

10

l4
17
22
29
45
45
45
45
73
73
74
79
104
125

128

133

II.
III.
IV.
V.

VI.

Tables

Weight and composition of samples
Annealing and homogenizing schedule
Residual resistivity of binary alloys
Residual resistivity of ternary alloys
300

Values of Sx

Lattice thermal resistivity constants

iv

Page

26
28
46
75

85

110

Figures

Page
1. The thermoelectric circuit 4
2. Induction melting unit 23
3. Graphite crucible 24
4. and 4a. Thermoelectric power, thermal conduct- 31

ivity, and electrical resistivity apparatus
5. Details of heater and 16 lead platinum seal 33
6. Details of temperature measuring plate and 34

platinum seals

7. Heater circuits 36
8. Temperature measuring circuits 37
9. Resistance measuring circuit 33
10. Thermocouple calibration apparatus 40
11. Resistance measuring circuit 42
12. Thermoelectric force apparatus 43

13. Absolute thermoelectric power of pure copper 47

14. Absolute thermoelectric power of pure copper 48

15. Thermoelectric power of Cu-Ag vs. Cu 49
16. Thermoelectric power of Cu-Au vs. Cu 50
17. Thermoelectric power of Cu-Sn vs. Pb 53
18. Thermoelectric power of Cu-In-Cd vs. Pb 54
19. Thermoelectric power of Cu-In-Zn vs. Pb 55
20. Thermoelectric power of Cu—Zn-Ga vs. Pb 56
21. Absolute thermoelectric power of Cu-Sn 57
22. Absolute thermoelectric power of Cu-Cd-In 58

23.

39.

40.

41.

42.

43.

Absolute thermoelectric power

Absolute thermoelectric power

Change of thermOpower due to alloying - Cu-Sn

of Cu-Zn-In

of Cu—Zn-Ga

Change of thermOpower due to alloying - Cu-Cd-In

Change of thermOpower due to alloying - Cu-Zn-In

Change of thermOpower due to alloying - Cu-Zn-Ga

Total thermal conductivity Cu-Sn

Total thermal conductivity Cu-Cd-In

Total thermal conductivity Cu-Zn-In

Total thermal conductivity Cu-Zn-Ga

Resistivity of Cu-Sn alloys
Resistivity of Cu-In-Cd
Resistivity of Cu-Zn-In

Resistivity of Cu-ansa

Phonon drag thermOpower of pure copper

Predicted and observed change
to alloying Cu-Ag No. 31
Predicted and observed change
to alloying Cu-Ag No. 33
Predicted and observed change
to alloyimg Cu-Ag No. 32
Predicted and observed change
to alloying Cu-Au No. 35
Predicted and observed change
to alloying Cu-Au No. 34
Predicted and observed change

to alloying Cu-Sn #2

vi

of thermOpower

of thermopower

of thermOpower

of thermOpower

of thermOpower

of thermopower

due

due

due

due

due

due

Page

59
6O
61
62
63
64
65
66
67
68
69
7O
71
72

86

87

88

89

9O

91

92

44.

45.

46.

47.

48.

49.

so.

51.

52.

53.

54.

55.
56.

57.

Predicted and observed change
to alloying Cu-Sn #6
Predicted and observed change
to alloying Cu-Sn #8
Predicted and observed Change
to alloying Cu-In-Cd #14
Predicted and observed change
to alloying Cu-In-Cd #15
Predicted and observed Change
to alloying Cu-In-Cd #17
Predicted and observed change
to alloying Cu-Zn-In #101
Predicted and observed change
to alloying Cu-Sn-In #102
Predicted and observed change
to alloying Cu-Zn-In #103
Predicted and observed change
to alloying Cu-Zn-Ga #104
Predicted and observed change
to alloying Cu-Zn-Ga #105
Predicted and observed change

to alloying Cu-Zn-Ga #106

of

of

of

of

of

of

of

of

of

of

of

thermOpower

thermopower

thermopower

thermOpower

thermOpower

thermOpower

thermOpower

thermOpower

thermopower

thermopower

thermopower

Ideal thermal resistivity of pure copper

due

due

due

due

due

due

due

due

due

due

due

Total, lattice and electronic thermal conduct-

ivities of Cu-Sn #2

Total, lattice and electronic thermal conduct-

ivities of Cu-Sn #6

vii

Page

93

94

95

97

98

99

100

101

102

103

111

112

113

58.

59.

60.

61.

62.

63.

64.

65.

66.

67.

Total, lattice and electronic
ivities of Cu-Sn #8

Total, lattice and electronic
ivities of Cu-In-Cd #14
Total, lattice and electronic
ivities of Cu-In-Cd #15
Total, lattice and electronic
ivities of Cu-In-Cd #17
Total, lattice and electronic
ivities of Cu-Zn-In #101
Total, lattice and electronic
ivities of Cu-Zn-In #102
Total, lattice and electronic
ivities of Cu-Zn-In #103
Total, lattice and electronic
ivities of Cu-Ga-Zn #104
Total, lattice and electronic
ivities of Cu-Ga-Zn #105
Total, lattice and electronic
ivities of Cu-Ga-Zn #106

Correlation of l/B with

viii

thermal

thermal

thermal

thermal

thermal

thermal

thermal

thermal

thermal

thermal

conduct-

conduct-

conduct-

conduct-

conduct-

conduct-

conduct-

conduct-

conduct-

conduct-

115

116

117

118

119

120

121

122

123

124

1. Introduction

To describe the properties of conduction electrons
in a metal one needs detailed knowledge of the relaxation
mechanisms and of the band structure,e3pecially near the
Fermi surface. With extremely pure metals at very low

temperatures it has been possible to 'map' the Fermi sur-

(l)'

face of many metals by means of anomalous skin effect

(2) (3)

deHaas-van Alphen effect
(4)

cyclotron resonance , and ultrasonic attenuation(5). The

, high field magneto resistance

noble metals crystallize in the face centered cubic struc-
ture and have one electron per atom. In this scheme, if
the Fermi surface were spherical, they would not touch the
zone boundaries. The above measurements have shown however
that the Fermi surfaces are distorted spheres having thick
necks passing through the zone boundaries in the [113
direction‘l-S).

An essential requirement for the success of these
topological techniques is that “*7'>l ’where a)! is the
cyclotron resonance frequency of the charge carriers of
effective mass,m*,and ?' is their relaxation time. Con-
sequently these techniques fail at high temperatures and in
alloys. Ordinary transport phenomena are not so restricted
however, and thus have provided much of the existing
information on the band structure of alloys.

It has been established‘6) that the transport proper-

ties of some metals are unusually sensitive to the presence

of transition metal impurities even in extremely small

1

I

2

amounts. Consequently measurements on 'pure' copper
performed in different laboratories may be.and frequently
areIin conflict. We shall show shortly that a complete
analysis of observed thermoelectric power data requires
knowledge of thermal and electrical conductivities. One
of our aims was to account for the effects of these im-
purities by measuring simultaneously on the same sample,
all three transport prOperties; thermoelectric power,
thermal conductivity, and electrical resistivity.

The Object of this study is the investigation of the
effects on the thermoelectric power of impurities in vary-
ing concentrations and combinations. As will be seen in
the detailed analysis, this concentration dependence pro-
vides one test of the validity of the rigid band model.
Also we wished to see if the thermoelectric power of ternary
alloys could be accurately predicted from the known thermo-

electric power of binary alloys.

2. Theory

Since our principal interest in these investigations
is to study the effects of impurities on the thermo-
electric power, the results of the free electron theory
of the thermoelectric power is presented first, followed
by the theory of thermal and electrical conductivities.

The calculations involved in deriving the various results
are somewhat tedious and will not be reproduced here.
Complete treatments can be found in the various references
cited.

Thermoelectric power

If a temperature gradient is maintained across a metal
sample in which no current is allowed to flow, the
electrons will assume a steady state distribution in which
there is an unbalance of charge and hence an emf across the
sample. The absolute thermoelectric power, S, of the

metal is given by the derivative of this emf:

AE__" .
fi-s-[gdr 1

where /u is the Thompson coefficient. If two dissimilar
metals are connected in a circuit as in figure 1, their
absolute thermoelectric power and the observed emf are re-

E = jTYSrIHT 2.

T

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\/

NEPAL B

lig. 1. The thermoclcctriv circuit

5

This is the Seebeck effect. It is the effect underlying
the Operation of simple thermocouples.

The equilibrium distribution is brought about through
interaction between the electron system and the lattice.

In the first approximation the exact nature of the inter-
action is not important; one need only assume that a
meChanism exists through which a disturbed electron distri-
bution function can relax toward equilibrium. The thermo-
electric power calculated with these assumptions is called
the diffusion thermOpower, 5d'

The thermoelectric effect depends on the establishment
of a temperature gradient across the metal, and this implies
a heat flow, of whidh a portion is carried by the lattice.
When these phonons interact with the electrons, crystal
momentum and energy must be conserved. In the presence of
a temperature gradient an electron will be more likely to
absorb a phonon whose momentum is directed toward the cold
than the hot end of the sample since there is a net flow of
phonons toward the cold region. Consequently, the phonon-
electron interaction provides an additional contribution
to the thermoelectric power, called "phonon drag", and
denoted by Sg.

The free electron theory gives for the "diffusion

thermopower":(7)

36

84: 31.111 [ML’y 3,

where K0 is Boltzmann's constant, T the absolute tempera-

ture, e the electron charge, 0' the electrical conductivity,,

6
C; the electron energy, and.<; the Fermi energy. If a

relaxation time (wican be defined and Spherical energy

surfaces are assumed, the conductivity is given by:
" 4 6" N 31’

where m* is an appropriate effective mass, N(c) the density
of states, and fo the distribution function. This integral
can be readily evaluated at low temperatures where point
imperfections provide the dominant scattering mechanism
and at high temperatures where lattice vibrations dominate.

For these two ranges we have:

W) = 1'. e"

for point imperfections 5.

T(€) =7 e for lattice scattering
’ range 6.

and the thermoelectric power for these two situations is

then:
1
SJ ""J&—IL’ for point imperfections 7.
3e;
S ,_11:££LI:_ for lattice scattering
at " i 3'; range 8.

Thus for these two simple situations, S is a linear
function of T with the prOportionality constant differing
by a factor of three. In the region in which no single re-
laxation time can be defined, the following formula due to

Sondheimer(8) and Wilson‘7) may be used:

n J.
8,: n k.‘ 7‘ Ap+39.{'+m-o 1") TF‘J'F}

say he +9,{|+i§—m3.(fl) -2" f,
I

9.

7

where A? is the residual resistivity due to impurity
scattering, e. the ideal resistivity due to thermal lattice
waves, 9 the Debye temperature, 5,

3
D h,
where n is the number of free electrons per unit volume and

nS is the number of atoms per unit volume,

 

e _ zndz __
J“ (3:) ‘ £(|_ e.z)(e¢_l) 10.

where z = 5;.qu (q being the wave vector of an electron
with a frequency V ).

At high temperatures the thermoelectric power of
c0pper indeed exhibits a linear T dependence(6), but the
preportionality constant is considerably less than pre-
dicted, and worse, the sign is wrong. Ziman‘g) has dis-
cussed the possible ways in which a more complete theory
might account for the positive thermOpower. The hOpe lies
in a more general expression for the electrical conduct-
ivity, particularly one which correctly takes account of
the variation of relaxation time with energy and orienta-
tion near the Fermi surface*. Since Eq. (3) is quite
general, the linear T dependence will remain in a more
complete theory.

Calculation of the phonon drag thermoelectric power
requires detailed knowledge of phonon scattering processes,

particularly the phonon-electron interactions. Bailyn<1o)

(11)

and Ziman have shown that the two types of scattering

""(12)

*Taylor has recently calculated S following Ziman's
suggestions and obtained a negative result.

8

processes, Normal and Umklapp, give rise to phonon drag
thermOpowers of Opposite sign. Thus a complete theory
would require knowledge of the relative importance of
these events. At present the theory of the phonon drag
is either not very reliable or in such a form as to be
difficult to apply to a metal with a complicated Fermi
surface. However, some valuable qualitative information
can be gleaned from the calculations of Hanna and Sondheimer(13)
which are based on some simplifying assumptions:
sphericity of the Fermi surfaces, and neglect of Umklapp
processes.

Hanna and Sondheimer show that at high temperatures,

(T>'9)’where anharmonic scattering of phonons is the

dominant phonon scattering mechanism’Sg can be described

by:
it 7- ‘J—J—N 3“ 11
SJ Jko'T‘lC' .

where Na is the number of electrons per atom, Kg the
lattice conductivity, and Ki the ideal electronic thermal
conductivity. Since Sd goes as T in this region (cf Eq. 8),
and we will show Shortly that K1 is constant and Kg goes as
l/T for these temperatures, it follows that Sg must
decrease as l/T.

In a pure metal at low temperatures, CP<<6) where
electron phonon scattering dominates over other phonon

scattering events, Hanna and Sondheimer give:

 

 

\ __ .
3 3 ”.6 - N.e 12.

Thus we see that the phonon drag thermopower goes

3

as T at low temperatures, passes a maximum, then falls

off as l/T. Extensions of the theory by Ziman<9>

predict

that the maximum will fall near T/9 = 1/4 for pure copper.
In Eq. (12) we consider the phonons as being scattered

only by electrons. If more than one scattering mechanism

is Operative, we can write:

8:

.L.§.a.[.._L ]- .CLQP
33N.e

I

7;+'I;e - 3- N. 13'
where T” is a suitably averaged relaxation time for
electron phonon scattering and 1; is the relaxation time
for all other scattering events. When phonon electron
scattering is completely dominant, 79,4 <<I , and Eq. (13)
approaches Eq. 12. P

Blatt and KrOpschot(l4) have identified a contribu-
tion to the thermoelectric power of COpper as due to
phonon drag. They did this by showing that impurities of
greatly different mass in the COpper lattice quenched the
phonon drag by presenting a large scattering cross section

(15)

to the phonons. DeVroomen et al, have identified a

phonon drag contribution to the thermOpower of aluminum at
very low temperatures by virtue of its T3 temperature
dependence. The alkali metals, because of their simplicity

would be eXpected to compare best with the theory.

10

(16: 17' 18) have investigated

MacDonald and collaborators
the alkali metals and some alloys and have identified the
phonon drag thermoelectric power there. The effect of
impurity scattering of phonons and the relative importance
of Umklapp electron-phonon processes received particular
attention in their work. Their results were compared with
the theory of Bailyn(lo).
Below 200K the thermoelectric power of "pure" copper
often attains large negative values which vary widely for
samples of similar purity. Gold et alfi6) have cited
instances ranging between -0.05‘/tV/OK and -6.0,uV/OK for
the maximum negative value of TEP. However, a sample of
very pure natural COpper showed a small positive thermo-
electric power below 13OK which is more in keeping with the
predictions of the free electron theory, and Ziman's
extensions thereof, that the thermoelectric power at low
temperature should be of the same sign as at high tempera-
tures. The anomalous negative results are now thought to
be due to scattering of electrons by small traces of iron.

Bailyn‘lg), Kasuya‘zo)

, and de Vroomen and Potters<21) have
shown theoretically that traces of transition metal im-
purities can be expected to induce this behavior.
Effect of Alloyinggon.Thermoelectric Power

As was shown first byMatthiessen<Cf 22), the increase
in the resistance of a metal due to a small concentration
of another metal in solid solution is in general independent

of temperature. Thus:

11

Q:AQ+€. 14.

where 6Q is the residual resistivity due to the solvent
metal atoms (and other impurities), and 96 is the
intrinsic resistivity of the ideally pure metal (ideal
resistivity).

-1

Since 9:0- Eq. (3) may be rewritten as follows:

84': M lilieL] 15a.

3GB “3

.. -fl‘k."T' .- ' 3A0 3 0
SA‘ 39 ((4\Q*r9.)(a?+ as) 15b“

 

 

I
54" wufigflfifl +(ng3Q Q. 96 L 15C.

Thus
0
QSdzAQAS4+Q°Sa 16.
O s.
where S 3 J! k. _Ie:[% 95%;]64 is the thermoelectric

power of the pure metal.

‘ I
and A54: #[53 1945—] 2 is the thermoelectric

power associated with the alloying atoms which produce a

concomitant increase 69 in the residual resistivity.

23)

Kohler(, using a more general approach based on a

variational principle'derived the relation:
I

12

VSd ”\n/asz‘IAM/AS‘ 17.

where W0 is the electronic thermal resistance of the pure
metal, AW the electronic thermal resistance due to the
solvent atoms, and W the total electronic thermal re-
sistance. Clearly $6) and (i?) are equivalent provided
the Wiedemann-Franz law may be applied individually to

each of the resistivities appearing in (15), i.e. provided

. - Ag ..
if'wr'h 18'

where L0 is the Lorentz number.

In that temperature region in which (18) fails, (17)
should be used instead of (16) since (17) is based on a
more general approach to the solution of the transport

equation. For equation (17) to be valid, So and A Sd

d
must be functions of temperature only, being independent of
concentration (i.e. the presence of impurity atoms must
not alter the band structure of the solvent metal). This
condition implies the ”rigid band model".

Kohler's equation has been used successfully by

(24) to describe the various

Gold(6), et a1. and Pearson
contributions to the thermoelectric power of the noble
metals.

The solute atoms in alloys affect the phonon drag

thermOpower by providing additional scattering centers for

phonons, thus reducing Kg in (11) or - what is equivalent-

13

reducing P in Eq. (13). Scattering of phonons by im-
purities depends on the relative mass difference of
solute and solvent atoms and on the elastic distortion
of the lattice in the vicinity of the solute. Assuming
that mass difference is the most important factor, one
finds<25):

AM 1
oc(—--

I
f M

19.
To indicate how important anisobaric scattering may be,
we shall use an example from the present investigation.

For three typical impurities in c0pper we have:

Zn in copper (a!) 2= 8.41 x 10'4
u _

Ag in COpper G?) 2= 4.85 x 10 1

Au in COpper (9%.) 2: 4.41 x 1

As will be shown later, we found that 1/2 atomic percent

of Ag in Cu eliminated approximately 90 percent of the
phonon drag thermOpower, i.e. P in Eq. (13) is diminished
by a factor of 0.1. It follows that a similar concentration
of Au in copper would reduce P by a factor of approximately

10'2

, while Zn in COpper would not alter P noticeably.
These effects of different impurities on the phonon drag
term in copper were first shown and explained by Blatt and
KrOpschot(14).

To summarize, the total thermoelectric power of a
dilute copper alloy is the sum of two terms:

S = 8d + 59 20.

where the diffusion term, Sd' is given by Eq. (17), and

14

the phonon drag term, S is given by (11) or (13).

gl
It is useful to define a “change due to alloying";

Cu d
where Sd' is the Change in the diffusion term and 89' is

salloy-s = 5': s '+ sg' 21.

the change in the phonon drag term.

From Eq. (17) it follows that:

all. c

u_ o

W

22.

 

and from the discussion of phonon drag we expect that:

all. cu

S '= 8 -S9 2.! 0 23.

g g
for dilute alloys of Cu with its neighbors in the periodic
table.

However, we anticipate that

S ,____ S all.“s Cue -S Cu 24.

g g 9 9
if solutes of greatly different mass than COpper are
present in concentrations exceeding about 1/2 atomic per-
cent.
Thermal and Electrical Conductivities of the Electrons

There are two mechanisms of heat conduction in a metal:

transfer of heat by the electrons and by the lattice waves.
Since these two mechanisms Operate in parallel, the total
thermal conductivity K is given by:

K = K +K 25.

e 9

where Ke is the electronic conductivity and Kg is the
lattice conductivity.

The presence of free electrons in a metal has an

effect on the lattice conductivity that is absent in

15
insulators; that of scattering of phonons by electrons.
Partly because of this, the electronic thermal conductivity
predominates in pure metals, and the lattice conductivity
can be neglected. Since the electronic thermal conduct-
ivity is of course closely related to the electrical con-
ductivity we will consider these together.

In a pure metal the resistance to the tranSport of
heat and electricity by conduction electrons is due mainly
to scattering by the lattice vibrations: in fact, in an
ideally pure metal these conductivities should increase
without limit as the temperature approaches absolute zero.
This situation is of course never realized since the
presence of lattice defects and impurity atoms provide
scattering centers at extremely low temperatures.

Calculations of the behavior of the conduction elect-
rons under the combined influence of an external thermal
or electrical field and the various scattering mechanisms
result in the following expressions for the 'ideal'

electronic thermal and electrical resistivities‘ze):

Ae(-§-)‘(%)' 4%)

6’.

V4 2

H

7.? A*(-§§(_gj_rg ”01%) - 1%), (gafffilg 27.

where Ae and At are known constants. Eq. (26) is the

Bloch-Grfineisen formula.

16

In the high temperature limit, (T:-6) these equa-

tions reduce to:
.A c '7‘
9° twee“)

- J3:—
M LgT' 29.
Eq. (29) is of course the Wiedemann-Franz law.

At moderately low temperatures, below about 9/10

Eq. (26) becomes:

90 = '2” “(81%):

As the temperature is further reduced the resistiv-
ity asymptotically approaches a constant value, A? , the
residual resistivity, whose magnitude depends on the number
and kind of stationary lattice imperfections that may be
present. In this low temperature region, the electronic

thermal resistivity is given by:

:2 + an: N. xALC T 31.
4L. 93‘ 9

where Na is the number of free electrons per atom.

In summary, the BlochuGruneisen theory predicts that

5 at low temperatures and to T at

high temperatures. Similarly, W0 is proportional to T2 at

p is prOportional to T

0

low temperatures and is constant at high temperatures.

I .—
According to Matthiessen 3 law p 'Qo +AQ where AQis a

17

constant: correspondingly W = wo + AW’where AW is
prOportional to T—l.

The comparison of experimental results with this
theory is generally good, at least for dilute alloys.
For reviews of experimental and theoretical work see

Gerritson<27), MacDonald(28)

(29)

and Olsen and Rosenberg .
Extensions of the theory with applications to COpper have
been made by Ziman<26).
Lattice Conductivity

The electronic thermal conductivity is reduced con-
siderably when a small amount of impurity is present. If
impurity scattering of electrons becomes a dominant cause
of thermal resistance over a wide temperature range, in-
stead of being effective at only the lowest temperatures,
the electronic thermal conductivity may be reduced to a
point where it is comparable with the lattice conductivity,
whiCh is not affected so strongly. Therefore, a complete
study of thermal conductivity of alloys must include con-
sideration of the lattice component.

The thermal conductivity of a crystal lattice can be

described by the relation(25):

K = 1/3 {C(w)v(w)[(w)o(w 32.
where C0») is the contribution to the specific heat per
frequency interval from lattice waves of frequencyoo, v00)
is their velocity, and [(0) ) their mean free path.

If the frequency and temperature dependence of the

mean free path associated with a given interaction mechanism

18

is known, the one can, in principle, deduce the temperature
dependence of the corresponding contribution to the

lattice resistivity by a simple application of Eq. (32).
For example at low temperatures where

3-n
K~T if [(w)~w’“. Klemens (25)

has obtained estimates
for the frequency variation and magnitude of various
scattering mechanisms. The results show that for a dilute
alloy, the lattice conductivity should fall off as 1/T at
high temperatures where anharmonic interaction of the
phonons is the dominant scattering mechanism. Point
defects also give a l/T contribution. At low temperatures
electron-phonon interactions and dislocations each give a

T2 contribution. If more than one process is Operative at

the same time '4“), = ; “2:75) , where 4(a)) is
the mean free path for each scattering process. Thus the
various contributions to the lattice thermal resistivity
are not additive but may be approximately so in some cases.
The vibrations in a lattice can be resolved into a
set of travelling waves. For each wave vector, k, there
correspond three waves with mutually perpendicular direc-
tions of polarization and three frequencies,a>. In an
elastic continuum one polarization is parallel to 4K
(longitudinal model) and two are perpendicular (transverse
modes). The longitudinal wave has the highest frequency.
In a crystal the longitudinal and transverse modes are not

necessarily so simply related, but in any case the one with

the highest frequency is called the longitudinal wave.

l9
Phonon—electron scattering depends strongly on how
the different modes of lattice vibrations interact with
the electrons. In the simplest situation, that of com—
plete spherical symmetry of the electron wave functions

(30) that there will

and energy surfaces, it can be shown
be no interaction of electrons with phonons having truly
transverse polarization. The Bloch theory is based on
this assumption and on the assumption of tight coupling
between the various branches of polarization of phonons.
Thus the lattice modes will all have approximately the
same relaxation time with only the longitudinal modes
interacting with the electrons.

Since the Fermi surface of copper is known to depart
significantly from a spherical shape, the Bloch approx—
imations are invalid in this case. A more apprOpriate
scheme to use would be that described by Iv’iakinson‘cf 25),

in which all modes of lattice vibrations interact independ -

ently with the electrons. The Makinson theory predicts:

9

Kg = 313 K; (7") (g N:% 33.
in the limit of low temperatures, where Ki is the ideal
electronic thermal conductivity, and N“ is the electron
concentration.

Measurements of the lattice component, Kg, involve
a measurement of the total thermal conductivity, K, and a
calculation of the electronic thermal conductivity, Ke from

resistance data. KG is then deduced by subtraction. In a

)(31)

J

pure monovalent metal, Ke/Kg'fi'lo2 (cf Dekker . It is

20

not possible to measure the total conductivity of such a
metal with sufficient precision to deduce the lattice
term reliably. The lattice conductivity of the pure

metal has been deduced, however, by extrapolating measured
lattice conductivities of dilute alloys of various concen-
trations to zero solute content. The lattice conductivity
of COpper derived in this manner agrees quite well with
that predicted theoretically by Eq. (33). Much experi-
mental data on the low temperature thermal conductivity of
alloys have been obtained in the last few years. The

data have generally shown that the total conductivity
below 9/25 can be expressed by Eq. (25) which at these

temperatures can be rewritten:

K=£QI+BTx 34.
A?

The second term in Eq. (34) represents the lattice thermal
conductivity whose magnitude is limited by phonon-electron
scattering. These measurements have exhibited a variation

of B with solute content whiCh has been shown by Lindenfeld

(32)

and Pennebaker to be principally due to a variation in

the mean free path of the electrons. Their data and that

(33)

of Tainsh and White show that there is a close corre-

lation between the value of B and the residual electrical

resistivity. A few years ago Pippard<35)

prOposed a theory
of thermal conductivity valid when the electron mean free
path becomes comparable to or smaller than the wave length
of those phonons which make the dominant contribution to

the lattice thermal conductivity. The results of Tainsh

21
and White, and of Lindenfeld and Pennebaker support
Pippard's theory, and, moreover, add further support to
Makinson's contention that both longitudinal and trans-

verse lattice waves interact with the electrons.

3. Preparation of Samples

The alloys and the pure metal samples were prepared
from 99.999 percent pure metals supplied by the American
Smelting and Refining Company. After thorough cleaning
in an acid etch followed by distilled water rinse, the
constituents for each alloy were weighed on a Mettler
balance to a precision of 10 micrograms.

The constituents were next placed in a graphite
crucible and melted by induction heating. The induction
melting unit is shown in Fig. 2. The graphite crucibles
(Fig. 3) were designed for Optimum heating with the 450
kilocycle, 2-1/2 kilowatt Lepel induction heater according
to the formula of Brewer(35). The crucibles were fabri-
cated by the United Carbon Company and were purified by
them after fabrication. In order to drive off adsorbed
gases, each crucible was held at an elevated temperature
in vacuum before the alloy constituents were added. In
SOme cases the solute elements were added to the already
molten COpper by means of the hOpper shown at the tOp of
the figure. The molten alloys were held at a high tempera-
tlire for approximately thirty seconds to utilize the effect
of induction stirring, and then poured into a cool COpper
mold. The melting, casting, and annealing were done in
(high vacuum, or, in the case of high vapor pressure con-

Stituents’a helium atmosphere.

22

23

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

  

 

2.
I
N.
N
E
K
N
N
N
N
N
N
N
3 N
h
\
4 5 t
/— ‘ N
; .. \
3 a \
~ :2 : r t
5 i N
3: I Q
\ ‘ :- :
s N ’ g “ w
\ ,
\ -\—.
\ IO
\
\ D
:- .
N
R\ ‘ \\\\\\\\\\\Y]

 

 

 

 

F1<3. 2. Induction Melting Unit. 1. Window 2. Rotating
Orring seal 3. HOpper assembly 4. Leads to induction
furnace 5. Graphite crucible 6. Vycor insulation 7. In-
duction coil 8. Rubber stOpper 9. Rotating O-ring seal
10. Copper mold ll. Vacuum line

24

 

.‘

 

 

 

 

'v

 

{N

[I—

4

 

ll

 

I'-"--J

 

5.6

 

 

'-"""--

 

 

 

 

Graphite crucible

Fig. 3.

25

After casting, the alloys were reweighed on the
Mettler balance. The results of the weighing are shown
in Table I. In some cases part of the melt missed the
mold and was not recovered. No final weight is shown for
these cases. The cast slugs were approximately 3/16" by
2". These slugs were next homogenized in sealed vycor
capsules at 6000C. for six days.

All but the copper-gold, COpper-silver, and pure
COpper samples were swaged to 1/4 h. by 5-10 cm.* After
a heavy etch in nitric acid, the alloys were given a
final anneal in sealed vycor capsules before mounting in
the measuring apparatus. The copper-gold, COpper-silver,
and pure COpper samples were rolled to 0.01" thickness in
a hardened steel rolling mill and trimmed into strips 10"
to 15" long by approximately 0.015" wide before the final
etch and anneal. The annealing schedule is shown in
Table II. Finally, after the experimental work was com-
pleted on each alloy, the 1/4 cm. x 5-10 cm. rods were
chemically analysed**. The results of the analysis are

shown in Table I.

 

*The swaging was done in the metallurgy laboratory of
Hoskins Manufacturing Company, Detroit, Michigan.

**The chemical analysis was performed by Spectrographic
Testing Laboratory, Detroit, Michigan.

26

 

TABLE I
Sample Weight before Weight after At.% At.% from
melting(grams) melting(grams) from chemical
weight analysis*
No. 2
Copper 7.52806
Tin 0.25715 1.837 1.86
Total 7.58521 7.56735
No. 6
COpper 7.92436
Tin 0.18415 1.217 1.60
Total 8.10851 8.09119
No. 8
COpper 9.62986
Tin 0.11808 0.653 0.61
Total 9.74794 9.72173
No. 14
COpper 7.83163
Cadmium 0.07792 .553 0.22
Indium 0.15403 .966 0.76
Total 8.06358 8.00328
No.'15
COpper 8.28976
Cadmium 0.08054 .543 .10
Indium 0.08049 .532 .38
Total 8.45679
No. 17
Copper 8.51667
Cadmium 0.18117 1.186 .48
Indium 0.08480 .541 .24
Total 8.78264
NO. 101
COpper 12.93543
Zinc .07420 .554 .47
Indium .10306 .481 .29
TOtal 13.11269
NO. 102
COpper 20,30441
Zinc .10610 .505 .49
Indium .36228 .960 .77
Total 20.87279 20,75987
No. 103
COpper 17.81873
Zinc .17534 .942 .99
Indium .15343 .467 .41
Total 18.14750

27

TABLE 1 (Continued)....

Weight before Weight after

 

Sample At.% At.% from

melting(grams) melting(grams) from chemical
weight analysis*

No. 104

Copper 17.88554

Zinc .65655 .288 .33

Gallium .06994 .335 .30

Total 19.01203 18.12258

No. 105

Copper 17.79133

Zinc 0.08737 .467 .34

Gallium .18688 .942 .80

Total 18.06558

No. 106

Copper 14.74170

Zinc .14766 .955 .86

Gallium .08458 .516 .38

Total 14.97394 14.97

31

COpper 10.03

Silver .16507 0.96 -

Total 10.19307 10.11297

32 10.300

Silver 0.07969 .454

Total 10.37969

33

Copper 12.500

Silver 0.42822 1.977

Total 12.92822 12.88807

34

COpper 8.5442

Gold 0.21444 0.81

Total 8.75864 8.74812

35

COpper 8.92604

Gold 0.11106 0.40

Total 9.03710 9.01876

*The chemical analysis is estimated to be accurate to 0.01

at.%

 

Annealing and Homogenizing Schedule

28

TABLE II

 

 

Sample Homogenizing and Annealing
annealing atmosphere temperature
and time
2 high vacuum 60030. 5 hrs.
750 C. 1 hr.
6 high vacuum 60030. 5 hrs.
750 C. 1 hr.
8 high vacuum 6002C. 5 hrs.
750 C. 1 hr.
14 2/3 atm. helium 600°C 3 hrs.
15 2/3 atm. helium 600°C. 3 hrs.
17 2/3 atm. helium 6000C. 3 hrs.
101 2/3 atm. helium 750°C. 2 hrs.
102 2/3 atm. helium 750°C. 2 hrs.
103 2/3 atm. helium 750°C. 2 hrs.
104 2/3 atm. helium 750°C. 2 hrs.
105 2/3 atm. helium 750°C. 2 hrs.
106 2/3 atm. helium 750°C. 2 hrs.
31 high vacuum 700°C. 3 hrs.
32 high vacuum 7000C. 3 hrs.
33 high vacuum 700°C. 3 hrs.
34 high vacuum 700°C. 3 hrs.
35 high vacuum 700°C. 3 hrs.
pure Cu high vacuum 8000C. 5 hrs.

4. Apparatus

It is impossible to choose a form factor (cross
sectional area/length) that lends itself ideally to the
measurement of thermoelectric power, thermal conductivity,
and electrical resistivity of the same specimen. Each
measurement is plagued by its own Special difficulties.

The electrical resistivity of dilute COpper alloys is

quite low, so in order to get a measurable emf one must

go to high currents or long, thin samples. But when
measuring thermal conductivity, it is necessary to minimize
the effects of radiation and conduction down the leads.
This calls for either a shorter, thicker sample, whose
conductivity is high compared to the extraneous effects,

or a more complicated apparatus, such as that of Powell, et
al.(36). In the measurement of thermoelectric power one
must use reasonably large thermal gradients to obtain
measurable voltages.

In this project it was decided to use a form factor
of approximately 1/100 cm. The sample dimensions are
5-10 cm. long by 1/4 cm. diameter. These dimensions allow
for most of our alloys a temperature difference across the
sample ranging from 10K at the lower temperatures to 50K
at the higher temperatures with a power input of 0.001 watt
to 0.25 watt. These temperature differences gave thermo-
electric emf's of 0.15 ,uv to 15 ,uv. The dewar flask that

holds the cryogenic liquid is fitted to another vacuum

system to enable pumping on the liquid helium to lower its

29

30

temperature. Details of the sample and mount are shown in
Figs. 4 and 4a. Some of the separate parts are shown in
greater detail in Figs. 5 and 6. All but the Pb leads

are brought into the vacuum can through the 16 lead plati-
num seal (Fig. 5)*. No. 40 COpper wire was used for all
leads from seal to sample. The Pb leads are connected
directly to the COpper leads coming from the potentiometer
by means of the seals shown in Fig. 6. The sample is
supported by a "glasstic" rod**. 'Glasstic" is a material
made with full length glass fibres bound together by plas-
tic and has the desirable prOperty of matching closely the
thermal expansion coefficient of COpper, thus minimizing
strains when the assembly is cooled. The rod is hung

from a copper plate which is in good contact with the cryo-
genic bath. This plate is used as the heat sink in thermal
conductivity measurements and as a reference for tempera-
ture measurements. To facilitate rapid changing of
Samples, all connections to the sample are physical. Nylon
Ixolts and 0.001" sheets of mylar plastic are used through-
CNit the apparatus where thermal contact and electrical
iJnsulation are desired between parts.

At each end of the sample is a heater (Fig. 5): the
JJDwer one maintains a thermal gradient across the sample
while the upper one raises the temperature of the sample
t6) some desired value above that of the liquid bath. The

x

* 'The platinum to glass connectors were made by Mr. W.H.
Haak of Lafayette, Ind.

'**Glasstic was obtained from the Glasstic Corp., Cleveland.

31

 

 

PLATINUM-

PUMPING LINE LASS SEAL
2,, 16 LEADS
CONTACT

WITH BATH

 

 

 

F 8. 9|

W

HEAT SINK AND TEMP.REFERENCE
l6 LEAD PLATINUM-GLASS SEAL
HEATER '

TEMPERATURE NEASURING PLATE_
CARBON RESISTOR '
THERMOCOUPLE JUNCTION

NYLON SCREW ‘

GLASSTIC SUPPORT

SAMPLE

 

 

NYLON SCREW
THERMOCOUPLE JUNCTION

CARBON RESISTOR

TEMPERATURE MEASURING PLATE
HEATER '
VACUUM CAN

 

 

 

\_ ,
F19. 4. Apparatus for measuring thermoelectric power,
- theJfiual conductivity, and electrical resistivity

>OH>HumHmmu Hmofiuuuwam Ccm >DH>HOUSCCCU Hmruoru

.HOBOQ

U «HuUmVHOOfu OAR

msuhhoouo
:3 Jam

 

33

 

 

u---d

 

 

 

o“

 

-----.l b-“c-d
--—-

 

 

..-----q‘.------

r----- L—

 

 

 

 

EATER WIRE

 

 

 

 

 

16 PLATINUM LEADS
GLASS INSULATION

PLATINUM TUB E

\V

GLASS BEAD

 

 

 

 

 

 

Fig. 5. Details of heater and 16 lead platinum seal.

 

 

34

 

 

     

 

 
    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r . '11 ‘ ,
g" d i
L ‘ i (DR-J
. a ’ i ~ 5
.oz"/§ E 4’ ii . .02"
0207' ' i :l: I: star
I
* ta\\\\Q\\\ * I '-
_,_.
64
COPPER TO POTENTIOMETEIR
GLASS BEAD avg PLATINUM TUBING
?
N V.
Pb TO SAMPLE
_.-\

 

 

E19. 6. Details Of temperature measuring plate and plati-
num seals . v

35

heaters are each wound with thirty feet of 0.003" manganin
wire, giving a resistance of about 1000 ohms. The high
resistance value was used to insure that power losses
along the leads are negligible compared with the power
dissipated by the heater. The circuit used to measure the
power input to the sample is shown in Fig. 7. It consists
simply of a series precision resistor to measure the cur-
rent and a parallel precision voltage divider to measure
the voltage across the heater. A heat leak, a No. 14
copper wire from the upper heater to the cold sink, com-
pletes the thermal path. Since the upper heater merely
raises the temperature of the whole sample, its power dis-
Sipation is not important in thermal conductivity measure-
ments, and was therefore not measured.

Near each heater is a temperature measuring assembly
CFig; 6), consisting of a carbon resistor and a thermo-
CKNJple junction. The circuitry associated with these
tfiknperature measuring devices is shown in Fig. 8. One
thermocouple is used to measure the difference between hot
arui cold ends, while the other measures the temperature of
tile cold end relative to the bath.

The 0.003" Pb wires, used as a reference for the
15hermoelectric measurements, and as the potential leads
for the resistivity measurements, are soldered to the
temperature measuring assembly. Pb was chosen because its
thermoelectric power has been widely studied and is

generally accepted as the best thermoelectric standard

because of its insensitivity to small amounts of impurity.

 

36

 

 

 

 

 

 

POTENTIOMETER

 

 

 

 

 

 

 

 

1 "“““ S
W 1 OHM [:POTENTIOMETER

 

——-+|I|I

21 VOLTS

 

 

UPPER
HEATER

 

 

 

 

 

VARIAC

 

 

 

 

 

Fig. 7. ‘ Heater circuits

 

 

37

 

 

[POTENTIOMETER]

 

 

 

 

 

 

 

(r

 

   

 

 

CARBON STANDARD

 

 

 

 

(SWITCH

 

 

 

 

‘ THERMOCOUPLES
2 V.

 

 

 

Fig. 8. Temperature measuring circuits;

38

 

' 7 VOLTS

REVERSING SWITCH

H

 

 

 

SAMPLE

 

 

 

 

 

 

 

 

 

 

 

 

 

POTENTIOMETER

 

 

 

 

 

Pig. 9. Resistance measuring circuit

V__'-__ _..__A_‘_~V_~,fi_ll

39
In addition to this advantage, is the fact that below
the superconducting transition temperature, 7.20K, its
thermoelectric power is zero, thus allowing a direct
measure of the thermoelectric power of the alloy in this
range.

The current leads for the resistance measurements
are connected to the heater blocks at each end. The
associated circuitry for the resistivity measurement is
shown in Fig. 9. Temperatures above 50K were measured by
means of thermocouples made of gold ~2.l at.% cobalt vs.
silver —37 at.% gold wires supplied by Sigmund Cohn
Corporation. They were calibrated against a Leeds and
Northrup platinum resistance thermometer (number 1215326)
which had been certified by the National Bureau of
Standards. The calibration apparatus is shown in Fig. 10.
It consists of two copper blocks in an insulating vacuum,
one in contact with the cold bath, the other wound with
Inanganin heater wire. The temperature of the heated one
‘vas measured with the platinum resistance thermometer, and
tflne emf generated in the thermocouple by the temperature
<iifference between the two blocks was recorded. The
tfihermoelectric force of the couple is tabulated in
ékppendix II.

An attempt was made to measure the thermopower and
tihermal conductivity below 4.2OK, using carbon resistors
tn: determine the temperatures. Accordingly, the resistors

were calibrated by recording resistance as a function of

4O

 

COLD
Juucrxofi_\\\\\\\
K

 

 

 

 

 

 

 

 

 

 

\~
222:.“ I 5 mama
HOT §§ w/E THERMOMETER
JUNCTION §§ JH ;

\ 5

§

\

\

 

 

 

 

 

\
F19 . 10 . The rmocoupl e calibration apparatus

 

41

helium pressure while the resistors were in thermal con-
tact with the helium. Temperature of the liquid helium

was determined from the standard tables<37). It was

found that the heat leak, while making the higher tempera-
ture measurements convenient, was not large enough to make
good measurements below 4.20K, i.e. a power input whidh
maintained an average sample temperature of 20K with the
heat sink at 10K produced only 0.010K temperature differ-
ence across the sample. This temperature difference was
entirely too small for reliable measurements.

0n the pure COpper, COpper-gold, and COpper-silver
Samples, no thermal conductivity measurements were made,
and resistivity was measured at only 4.20K and 2730K. The
long thin samples were wound around 0.16" x 4" micarta rods
for the resistance measurements. Six rods were mounted in
a holder whiCh was lowered into a liquid helium bath or an
ice water bath. The samples were wired as in Fig. 11 so
that six could be measured simultaneously.

The thermoelectric measurements of the pure Cu, Cu-Ag
and Cu—Au samples were made with the apparatus shown in
Fig. 12. One inch lengths of COpper pipe were placed
arOund each end of 8" x 1/4" wooden dowels. One pipe was
SL1bmerged in the cold bath while the other was wrapped with
‘heater wire and insulated with glass wool. The samples
‘WQIe assembled in thermocouples of alloy vs. pure COpper
and Pb vs. pure COpper and then mounted on the wood dowels
W1ththeir junctions glued to the COpper pipes. Immediately

Ehfljacent to each sample thermocouple a measuring thermocouple

42

 

 

 

 

 

POTENTIOMETER

 

 

SAMPLES

 

 

 

Fig. 11. Resistance measuring circuit

 

 

 

 

h: ‘—

POTENTIOMETER (1'1) POTENTIOMETER

 

 

 

 

 

 

 

 

 

 

 

JJJ'|”W
V
A

 

 

 

t o
I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I ‘DETAIL 0F THERMOCOUPLE

’ fl l ' JUNCTION

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. Ii. Thermoelectric force apparatus

44
of Au -2.1 at.% Co vs. Cu was mounted. The reference
thermocouples used with this apparatus were all made
with wire calibrated by M.D. Bunch at National Bureau of
Standards. Two runs were made with each sample, using
liquid helium and liquid air, reSpectively as cold baths.
The heated copper pipe was slowly raised in temperature
and the emf generated in the sample thermocouple was re-
corded while the Au Co vs. Cu thermocouple measured the

temperature difference.

45

5. Description of Results

Thermoelectric_power of pure COpper

In the measurement of the thermoelectric power of
pure copper, Pb was used as "metal B" in Fig. l. The
apparatus of Fig. 12 was used for the measurements with
T, the temperature of the cold bath (either liquid helium
or liquid nitrogen) and T2 the variable temperature of the

heater. emf vs. T was plotted and the derivative of this

2

curve was taken to obtain plots of S -S The plot of

Cu Pb'
S was obtained by adding to S -S the thermoelectric

Cu Cu Pb
power of pure Pb as given by Christian, et al.<38). The
thermoelectric power of pure COpper found in this way is
shown in Figs. 13 and 14.
The resistance of the pure COpper was measured at

4.20K and at 2730K in the apparatus of Fig. (14). The re-

sistance ratio, r¢L is shown in Table III.

r17: ‘ 73.2..
galloys of monovalent metals

The thermoelectric powers of the Cu-Au and Cu-Ag
Eilloys were measured in the same way as that of pure
Chapper except that a length of pure copper was used as
nletal B. Thus the measured thermoelectric power of these
tihermocouples was the "change due to alloying". The
tihermoelectric power of the Cu-Au vs. Cu and Cu-Ag vs. Cu
t”-hermocouples is shown in Figs. 15 and 16. The resistance
iratios are shown in Table III.
ifiLloys with polyvalent metals

All of the remaining samples were measured in the

aDparatus of Fig. (4). In this device Pb is used as metal

46

Table III

Residual Resistivity of Binary Alloys

 

 

Residual At. % At. %

Sample r4.2 resistivity predicted by
r273-r4.2 microhm-cm by Linde analysis

No. 31 Cu-Ag 0.0649 0.1006 0.72 0.96
No. 32 Cu-Ag 0.0336 0.0480 0.345 0.454
No. 33 Cu-Ag 0.1105 0.166 1.19 1.977
No. 34 Cu-Au 0.262 0.401 0.61 0.81
No. 35 Cu-Au 0.141 0.218 0.32 0.40
No. 2 Cu-Sn - 4.86 1.72 *1.86
No. 6 Cu-Sn - 3.38 1.24 *1.60
No. 8 Cu-Sn - 1.75 0.614 *0.754
pure Cu 2.7ix10'3 4.62x10‘3 - -

‘MAt. % of Cu-Sn is by chemical analysis, others by weight
a.nalysis.

47

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§

MO mm384mmm2m9
Omv 00v 0mm 00m 0mm CON OmH OOH 0m
7 A a A u

 

 

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ON CA
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sameness . a L

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48

 

 

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. x0 mmoedmmmzma A

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_ _ q _ _ _ _ _ _ _
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UmHSmmm: I a

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x0 mmbfidmmmyfin.

 

 

 

 

 

OQN O¢H CON 00H ONH 0m 0?
d — 1 — — u —
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Hm
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NM

 

 

 

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50

x0 mmDBEmmmsz

no .m> mdssu uo umzom vauuomamosumne .oa .mam

 

 

 

 

 

 

 

ONm omN OfiN CON OQH ONH om 0v 0.
. _ . _ _ . a u n o
Ivao
ll mm
fim
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Xo/yqn/

51

B. Both T1 and T2 were higher than the reference cold
bath (liquid helium or liquid nitrogen). Thermoelectric

power was computed directly from the data by the relation:

5 = §§.«~I QE. . 35-
dT'AT

This relation is valid if dS/dT is small and/or the
temperature interval AT is small. Below ZOOK dS/dT of
some of the samples was as high as .05 v/(OK)2. In this
region temperature intervals of 1°K or smaller were used
so the maximum error that might be introduced by this
treatment would be .05 gar/OK. The actual error is prob-
ably much less. At higher temperatures dS/dT was much
lower (0.0031%XE77——) and the temperature difference used
was correspondingly greater (5°K -10°K). Figs. 17-20 show
the thermoelectric powers of these alloys vs. Pb. The
thermoelectric power of pure Pb was added to the thermo-
electric power of the Pb-alloy couple to obtain the absolute
thermoelectric power of the alloys, shown in Figs. 21-24.
The difference between the absolute thermoelectric power
of the alloy and the absolute thermoelectric power of pure
copper (Fig. 13) is the “change due to alloying“ and is
shown in Figs. 25-28.

The thermal conductivity was found by measuring the
heat input required to maintain the thermal gradients in
the thermoelectric measurements. The thermal conductivity
of the alloys are shown in Figs. 28-32.

At various temperatures a current was passed through
the sample and the potential difference between the
thermoelectric (Pb) leads was measured. The resistivity

was calculated and is shown in Figs. 33-36.

52

The experimental data is tabulated in Appendix II.
The curves in this section represent smoothed values of
these data. As we gained experience using the apparatus,
we discovered that it was to our advantage to take fewer
data points, allowing the sample more time to come to
equilibrium at each point. Thus an inspection of the
data will show more points, with a larger Spread of points
on the earlier samples, and fewer data points with very
little Spread on the most recently measured ones. As an
example of an earlier sample we have shown all the data
Exoints on sample No. 2 in the figures, and as an example

of a recently measured one, the data for No. 106 is shown.

53

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TEMPERATURE °K

Fig. 29. Total thermal conductivity of Cu-Sn alloys

 

 

 

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TEMPERATURE OK

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67

 

 

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M IC ROIIM—CM

72

 

 

 

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73

6. Analysis of EXperimental Results

Analysis of the resistance measurements

Since the residual resistivity is a function of the
type and concentration of impurities present, it provides
an additional method of determining the composition of
binary alloys. Conversely, knowledge of the composition
of a ternary alloy will enable one to predict the residual
resistivity, and moreover, that part of the residual re-
sistivity due to each solute can be deduced. Since know-
ledge of the residual resistance is essential to the
analysis of the thermoelectric power it seems the most
logical place to begin our analysis.

The residual resistivity of the Cu-Ag and Cu-Ag alloys
was calculated from the resistance ratio assuming
Matthiessen's rule to be valid. The value of 1.55 microhm—
cm. was used for the resistivity of pure COpper at 273°K(39).
Impurity concentrations were calculated from the residual

(40). The residual re-

resistance using the data of Linde
sistivity of the remaining samples was calculated from the
measured resistance at 4.20K and the geometry of the
sample. The residual resistance and impurity concentration
indicated by Linde's results is shown in Table III for the
binary alloys.

The difference between the impurity concentration
predicted by Linde's data and that indicated by the weight
of the constituents is probably due either to loss of some

of the solutes in preparation or to innaccuracies in

Linde's results. As shown in Chapter 3, the weight before

74
and after casting precludes the possibility of solutes
being lost in these samples. Since similar discrep-

ancies were reported by KrOpschot(4l), and by MacDonal

d(42)
and Pearson we prefer to think that innaccuracies of
Linde's results are the true cause of the difference.

In the case of the ternary alloys, we need to know
what portion of the residual resistance should be attri-
buted to each constituent. Since most of these alloys
contain a high vapor pressure constituent, and since the
analysis of weight of the samples before and after melting
indicates that a significant amount was lost in casting,
the composition obtained by chemical analysis was con-
sidered the most reliable. Linde's data was used to
calculate the M2 that each constituent should cause. The
sum of the predicted 40's is slightly less than the measured
Ag of the sample. However, since the differences are small,
we deduced the residual resistivity attributed to each
constituent by assuming that the ratio of the residual
resistivities per atomic gmucent is given correctly by
Linde's results. Residual resistivities for the ternary
samples are shown in Table IV.

Thermoelectric Power of Pure Cooper

The principal aim of this program is the study of the
effects of alloying on the various contributions to the
thermoelectric power of COpper, Specifically, to see if
the free electron theory satisfactorily eXplains the
changes in the diffusion term, and to investigate the

effects of various impurities on the phonon drag term.

75

TABLE IV

Residual Resistivity of Ternary Alloys

 

Sample At.% by 4Q predicted I‘deasured A 4? deduced
chemical by Linde microhm-cm. by scaling
analysis microhm-cm. Linde's results

14 0.225% Cd 0.0472 0.0567

1.061
0.762% In 0.838 1.005
15 0.104% Cd 0.0218 0.0266
0.549
0.389% In 0.428 0.522
17 0.477% Cd 0.100 0.121
0.446
0.245% In 0.269 0.325
101 0.47% Zn 0.1575 0.157
0.481
0.299% In 0.324 0.324
102 0.49% Zn 0.1642 0.164
1.03
0.77% In 0.848 0.868
103 0.995% Zn 0.332 0.307
0.730
0.416% In 0.458 0.423
104 0.331 2.. 0.111 0.121
0.585
0.305 5% 0.427 0.464
105 0.34% Zn .114 0.1305
1.418
0.805%13a 1.125 1.285
106 0.856% Zn 0.287 0.326
0.718
0.37 % 3a 0.528 0.600

76
To do this we must first separate the thermoelectric
power of pure COpper into its diffusion and phonon drag
components.

Focusing our attention first on "pure" COpper, we
recall that presumably Eq. (9), the Sondheimer-Wilson
formula, gives the correct temperature dependence for the
diffusion thermoelectric power of iron-free copper. From
Eq. (11) we see that 3g Should be negligible compared to
Sd at sufficiently high temperatures. We also need to
consider the effect of extraneous impurities in "pure
COpper". According to Eq. (22) the "change“ due to ex-
traneous impurities will depend on the factor - '.

This factor will be nearly unity at low temperatures where

Av , ) W 2 > Esq.)
Aub+wg ' U

 

impurity scattering of electrons is dominant (
From Eq. (29) and the discussion following it we can con-
clude that this factor will decrease as the temperature
increases, and will have a l/T dependence at sufficiently
high temperatures where lattice scattering of electrons is
dominant (AV >>U°). Since So and AS go as T in this range

(from Eqs. (7), (8)) , the change due to extraneous im-
purities will approach a constant, Which presumably will be
negligible compared with the total thermoelectric power
(which is still increasing as T) at sufficiently high
temperatures.

Since the effects of extraneous impurities and of

phonon drag are negligible at high temperatures, the

Sondheimer-Wilson formula Should provide a reasonably close

77
fit for the total thermoelectric power at sufficiently

(41) has matched the

high temperatures. Kropschot
Sondheimer-Wilson formula to the highest temperature data
available, that of Nystrom. Since our own data for pure
COpper matches that of KrOpSChOt within a few percent,

we used his evaluation of the Sondheimer-Wilson formula.
Eq. (9) evaluated in this way is shown with the pure
COpper thermoelectric power in Figs. 13 and 14.

Iron impurities presumably cause the large negative
hump below 200K. According to Blatt and KrOpschot(14),
sg near 100K (the minimum of the hump) is at least an order
of magnitude smaller than S, the total thermoelectric
power of copper. Consequently, we may here apply Kohlers
formula, Eq. (17) to the total thermoelectric power:

ws = (WO+AW)S§ + ersFe 36.
where W is the total electronic thermal resistivity, WFe
is that part due to iron impurities, W is that due to all
other impurities and W0 is that due to lattice vibrations.
S is the total observed thermoelectric power of pure
COpper, SFe the characteristic thermoelectric power of
iron in copper, and $3 is the thermoelectric power that
one would measure if iron were absent.

At 100K, the Wiedemann-Franz law, Eq. (29), is valid
(T<<9)7 therefore, the total electronic thermal resist-
ivity can be calculated from the residual electrical re-

0

sistivity. Assuming Sd is correctly given by the Sondheimer-

Wilson formula and using the data of Gold, et al.(6) for

78

sFe' we can now evaluate WFe and AW’at 100K. According
to Eq. (31) these thermal resistivities, arising from
'residual' impurities, have a l/T dependence. The ideal
thermal resistivity of pure COpper, W0, is shown in Fig.
55. The data for this figure comes from G.K. White(43)
for the region 100K to 500K, Berman and MacDonald<44)
between 500K and 900K, and from the Wiedemann-Franz law
above 900K.
We now have sufficient information to evaluate

Kohler's formula for the diffusion thermoelectric power

ingludipg iron effects:

Cu
d Feer
Cu

Sd calculated in this manner is shown as curve c”in Fig.

S w = 53(340-0- (WM 8 37.
14.

We have now accounted for the effects of iron on the
thermOpower of COpper and therefore assume that the depart-
ure of the observed thermoelectric power from curve c in
Fig. 14 is due to phonon drag. Curve A in Fig. 37 is the
difference between curves A and C of Fig. 14 and is pre-
sumably the phonon drag thermoelectric power of pure Cu.
Except at temperatures below 200K it is in substantial
agreement with the corresponding curve by Kr0pschot and
Blatt. The discrepancy between our postulated phonon drag
thermOpower and that of KrOpschot and Blatt below 200K

arises because the latter did not make prOper allowance for

effects of iron in "pure" COpper.

79
Silver Alloys

When the solutes are silver atoms, the "change due to
alloying" of the diffusion thermoelectric power follows

from Eq. 22:

Ag d. ”Is: a 38.

where wAg is the electronic thermal resistivity due to the
silver atoms and SAg is the characteristic thermoelectric
power of silver in COpper.

From curve A Fig. 37 and also from the work of
KrOpschot and Blatt it is evident that phonon drag effects
are indeed small at sufficiently high temperatures, in
agreement with theory. Consequently Eq. (38) which applies
strictly only to the diffusion term, is valid for the total
thermoelectric power at and above 3000K. Thus, using the
experimental data from 1% Ag for Sd', and ”Ag (deduced from
the residual resistivities and the Wiedemann-Franz law)

300

Ag , the characteristic thermo-

Eq. (38) can be solved for S
electric power of silver in COpper at 3000K.

Since SAg is a thermoelectric power arising from
"impurity" atoms, its temperature dependence should be ACT.
We now have sufficient information to evaluate Sd' for all
T. 8' calculated at selected temperatures is shown in

(14)

According to Kropschot and Blatt the total change

I
due to alloying is given by Eq. (21) with S a given by

J

Eq. (24) for this concentration of Silver atoms. The total

80

change due to alloying calculated at selected tempera-
tures is shown in Fig. 38. The influence of heavy atoms
on the phonon drag effect is clearly essential to account
for the total change in thermoelectric power.

Alternatively we can invert the procedure by assum-
ing that 5d. accurately shows the change in the diffusion
thermoelectric power produced by the silver atoms, and
that the difference between Sd' and the observed change
is due to the absence of the phonon drag term in the alloy.
The phonon drag term deduced in this manner from the
Cu-1% Ag alloy is compared with that deduced from the data
on pure copper in Fig. 37.

The fact that the two curves are in qualitative agree-
ment supports the validity of our separation of the phonon
drag thermOpower of pure COpper. A possible eXplanation
of the discrepancy that does exist might be found in a
better interpretation of the iron effects in pure copper.
Clearly the values we have calculated for 3d are very

sensitive to the value taken for 8 Possibly the sample

Fe'
of pure COpper used in the CuAg measurements and the

sample used in the pure copper measurement differed slight-
ly in iron content even though they were prepared from the
same bar. Since we do not have any reason to prefer curve
8 over curve A in Fig. 37 we shall continue to rely on our
interpretation of the pure COpper thermopower and use

curve A for the phonon drag thermOpower in the interpreta-

tion of the remaining alloys.

81

Once SAg is known it should be applicable to dilute
alloys of slightly different concentration. Thus one
should be able to predict the thermoelectric power of a
Cu-2 at.% Ag alloy at 300°. This is of course predicted
on the validity of the rigid band model which predicts
that SAg be independent of concentration for small con-
centrations.

Using 5 from the 1 percent silver analysis, the

A9
predicted and experimentally observed change in the thermo-
electric power due to an addition of 2 at.% Ag are compared
in Fig. 39. The change in the diffusion term is shown
separately.

The 1/2 percent Silver sample was analyzed in the same
way, again using the same value for the parameter SAg' In
this case the best agreement between experiment and theory
could be obtained if

s; 2 -0.9 5;”
as per curve A of Fig. 37.

The experimental results of the thermoelectric power
of the dilute COpper gold alloys were analyzed in the same
way as those of the COpper silver alloys. The character-

istic thermoelectric power of gold in COpper, S was

Au'
deduced from the 1/2 % data at 3000K and was used to pre-
dict the change in the diffusion thermoelectric power for
both the l/ZZand l % samples. Since even as little as 1/2 %
gold will completely eliminate the phonon drag term

according to our theory, (-S§u) was added to the change

82
in the diffusion term to predict the total change. The
measured change, theoretically predicted change, and the
change in the diffusion term are Shown separately in
Figs. 41 and 42.

The experimentally observed change due to alloying
with tin was compared with the theory also in the same
way, with SSn being determined from the 3000K data on
sample No. 2. The phonon drag term was presumed complete-
ly quenched in each case. The theoretically predicted
total change and the change in the diffusion term are
compared with the experimentally observed change in Figs.
43-45.

When two solutes are present, as in the case of the

Cu-Cd-In alloys, Eq. (22) becomes:

 

Cu
n. _ r -
Dd ‘ ‘WCdSCd+NInsIn) (WCd+WIn)sd 39'
W
where ch and WIn are the electronic thermal resistivities
due to the Cd and In atoms, respectively, and SCd and sIn

are the characteristic diffusion thermopowers of indium
and cadmium in copper.

The values of SCd and SIn have been previously deter-

(41). We Should be able to use these

mined by Kropschot
values, which were deduced from measurements on dilute
binary alloys, to predict the change in thermoelectric
power produced by these same solutes in ternary alloys.
The method of separating the portion of the residual

resistivity due to each type of impurity is described

earlier in this chapter. W and w

In Cd were computed from

83

the appropriate residual resistivities by the Wiedemann-
Franz law. Since a sufficient concentration of heavy
atoms was present in the lattice to eliminate the phonon
drag thermOpower in these alloys, Eq. (24) was used to
predict the change in Sg due to alloying. The predicted
total change is compared with the eXperimentally observed
change in Figs. 46-48. The change in the diffusion term
is shown separately in the figures.

Obviously, the agreement between experiment and
theory is not as good for these ternary alloys as it was
for the binaries. This is not surprising Since in the
analysis of the binaries, Sx’ the characteristic thermo-
electric power of impurity x in COpper, was evaluated
with the experimental data while the Sx's in the Cu-Cd-In
analysis were taken from a separate source‘41).

The Cu—In-Zn alloys were analyzed in the same way as
the Cu-Cd-In, again using Kropschot's values for Sx‘

These alloys contain both light and heavy atoms as im-
purities. Again the concentration of heavy impurities
seemed great enough to make Eq. (24) appropriate for all
but the most dilute sample. In Sample 101, the assumption,
(S? = -0.9 Sgu) provided the best comparison with the
measured change due to alloying. The predicted and
measured changes are Shown in Figs. 49-51.

The Cu-Ga-Zn alloys were also analyzed in the same
way as the other ternaries. Unfortunately a value for

8,.a was not available from other sources so this parameter

Q

84
had to be deduced from the data. The change due to alloy-
ing of sample 106 at 3000K was used to evaluate SGa' This
value, together with KrOpschot's value for SZn were used to
predict the change in the diffusion thermoelectric power
of all three samples. Since the impurities in these
samples are all of approximately the same mass as copper,
we anticipate no change in Sg. The predicted and observed
values of S' are shown in Figs. 52-54. The agreement be-
tween theory and experiment appears quite good for samples
104 and 106, but not so good for sample 105. The diffi-
culty appears to be in the treatment of the diffusion term
and not the phonon drag term since the qualitative features
of the experimental and theoretical curves are in agreement.
Since this sample contains the greatest concentration of
Ga among the three, one might be inclined to think that the
value of 33a used was incorrect. A different value of SGa
would in fact affect the theoretical curve of this sample
more than it would the other two.

Another possible source of error in the analysis of
all the ternary alloys is the evaluation of the 49': as
described in earlier in this chapter. It is perhaps sig-
nificant that the greatest discrepancy between our measured
residual resistivities and those predicted by Linde occurs
in the alloy systems Cu-In-Cd and Cu-Zn-Ga, the same systems
which demonstrate the greatest discrepancy in the analysis

of the thermoelectric power.

3
Sx

00

Cu-Sn
Cu-In
Cu-Cd
Cu-Zn
Cu-Ga
Cu-Ag

Cu-Au

-1021
+3.88

-10018

85

Table V

Values of $300

 

from KrOpschott

S300

x from present measurements

-1.32

+ .624

+ .084

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104

In conclusion we believe that the results give
strong support to the theory set forth in Chapter 2 on
the effects of impurities on the phonon drag and dif-
fusion thermoelectric power of copper. The discrepan-
cies that do exist can be attributed to lack of precise
knowledge of the parameters involved, and do not suggest
a basic flaw in the theory. The results Show, as theory
predicts, that the 5&3 are indeed independent of concen-
tration for small concentrations. Further, again in
agreement with theory, the presence of more than 1/2 at.%
of solutes of greatly different mass effectively quenches
the phonon drag thermoelectric power of Cu regardless of
the presence of other impurities.

Thermal Conductivity

Our original objective in combining thermal conduct-

.ivity measurements with thermoelectric power measurements

Vvas to correlate the effect of anisobaric scattering of
Eahonons on the lattice conductivity with quenching of the
Ephonon drag thermopower. Unfortunately, we were unsuccess-
2Eu1. It is now apparent that the low temperature lattice
<:onductivity of dilute alloys is explained by Pippard's
t:heory in which it is correlated with the residual electri-
<:al resistivity. At higher temperatures the anisobaric
effect is apparently masked by scattering due to differ-
ences in binding forces between the solute atoms and their
neighbors, distortion of the lattice in the vicinity of

solute atoms, and anharmonic thermal scattering. In this

105

section we will present our data on the lattice conduct-
ivity, compare the low temperature results with those of
Pennebaker and Lindenfeld, and attempt to cite some
theoretical and experimental estimates of contributions
to the lattice thermal resistivity that mask the aniso-
baric effect.

The intrinsic electronic resistivity is shown in
Fig. 55. To this was added the residual electronic thermal
resistivity for each alloy calculated from the electrical
residual resistivity by the Wiedemann-Franz law. The
reciprocal of this sum, the total electronic thermal con-
ductivity, was subtracted from the measured total conduct-
ivity to obtain the lattice term.

Unfortunately, the measurements above 500K were not
as accurate as the measurements at lower temperature
because of difficulty in attaining a true steady state
condition (see Appendix I). Furthermore the lattice con-
ductivity is only about 10% of the total at 1000K in many
of our alloys. Consequently our estimated error of 5% in
the total conductivity is as great as 50% of the lattice
term. However, near 300K, at the lattice conductivity
Inaximum, the situation is not so bad and we can determine
‘the lattice term with some confidence there. Above 500K
the principal contributions to the lattice thermal re-
sistivity are eXpected to be anharmonic scattering and
anisobaric scattering, both of which cause the lattice

-l(25)

conductivity to vary as CT Assuming that this was

106

the case with our alloys, we chose a curve having this
temperature dependence above 500K. We feel that this
treatment of the data enables us to determine the prOp-
ortionality constant, C, with greater reliability than
our error analysis implies.

In Figs. 56-67 are plotted the total, electronic,
and lattice thermal conductivities. Estimated errors are
shown by brackets at three points. Below 200K the

lattice conductivity is given by:

Kg = 3T2 40.
While above 500 it goes as:
K a CT"1 41.
9

Values of l/C and 1/8 are listed in Table VI.

Fig. 68 shows a comparison between our values of 1/8 and
those of Lindenfeld and Pennebaker(32). The Makinson
value (Eq. 33) for pure COpper is also shown.

Turning our attention to the region above 500K, we
will first estimate the amount of lattice resistivity that
might be attributed to thermal scattering. White and
Woods<48) have deduced this term by measuring the conduct-
ivity of Cu-0.056 % Fe alloy. They reasoned that such a
small concentration of iron atoms in the COpper lattice
would not scatter phonons appreciably. Further, iron
presents such a large scattering cross section to electrons
that the electronic thermal conductivity is reduced

sufficiently to permit a reasonably good determination of

lattice conductivity. White and Woods give:

107

WTT-1= 1/C = 0.028 cm/w due to anharmonic scatter-

ing, 42.

The best theoretical estimate is given by Leibfried

and Schloemann<49):
WTT-l= l/C = 0.016 cm/w due to anharmonic scatter-
in(3 43.

The value given by White and Woods is quite close to our
estimated value for all the samples with resistivities

of l microhm-cm or less (the more dilute samples). It is
also quite close to that of sample 105 whiCh contains no
heavy atoms even though its residual resistivity is 1.418
microhm-cm.

In addition to anharmonic thermal scattering, aniso-
baric scattering should give an additional contribution to
the thermal resistance above 500K. This contribution
should have the same temperature dependence as the an-
harmonic thermal scattering and the two should be approx-
imately additive(25).

Anisobaric scattering is given by:

w = 11:23. (331ch

A OAIIv‘ n 44.

where a3 is the atomic volume in the crystal lattice, Ar1
is the difference between the mass of the solute and sol-
vent atoms, M is the mass of the solvent atoms, v is the

phonon velocity, and C is the solute concentration.

For copper one obtains WAT-l: 0.034 ( Aim/3732 cap/wills),

Eq. (44) has been approximately verified for Cu-Au alloys<46),

but agreement has not been so good in the case of

108
Cu-Zn(47). The Cu-Zn alloys exhibit an abnormally large
thermal resistance apparently because of additional
scattering from changes in binding forces between the
solute and its neighbors, and because of distortion of the
lattice in the vicinity of the solute.
The total thermal resistivity above 500K is given
approximately by the sum:
-1 —l

l/k- z {iA‘ + WTL 45.

Values of l/C theo. for each alloy are shown in Table V
using the experimental value of White and Woods for WTT-l
and the value given by Eq. (44) for WAT-l. The agreement
appears quite good, eSpecially in view of our large ex-
perimental error, which only permitted us to make a reason-
able estimate of l/C.

We see that while the thermoelectric studies of
copper alloys indicate that solutes whose mass differs
greatly from that of COpper have a pronounced effect on the
phonon drag thermoelectric power, the correSponding
change in the lattice thermal conductivity appears to be
relatively small.

Looking for a possible explanation of this phenomenon,
we turn again to the work of Hanna and Sondheimer(13).

For their most general expressions for lattice conduct-

ivity and phonon drag thermopower they give:

"3 “(31%; (2%.!) “4’- 46°

109

3 %
391°C (g) ;[ {(SJz)Zs—Jz 47.

where JAG is related to the high temperature mean free
path of electrons scattered by phonons of polarization 5
and z = .23;;) Where (q) is the frequency of a phonon

of wave vegtor q. From Eqns. (46) and (47) we see that
the phonon drag thermopower is more sensitive than the
lattice conductivity to non equilibrium of phonons of
high frequency. Since the frequency dependence of point
defect scattering is ar-T4, it is exactly these high fre-
quency phonons Which are most effectively scattered by the
addition of impurities.

Only careful calculations can determine if it is
reasonable that certain impurities cause order of magni—
tude effects in the phonon drag thermopower and relatively
small changes in the lattice conductivity. More reliable
thermal conductivity measurements need also to be made

since our results in this region are only suggestive.

110

Table VI

Lattice Thermal Resistivity Constants

 

 

Sample Microhm-cm l/Bobs. l/Cobs. l/Ctheo.
2 4.86 2500iloo 0.06451.01 .076
6 3.33 1750i70 0.061i.01 .070
s 1.75 1000150 0.05:.01 .043
14 1.06 1000:75 .05 .050
15 0.549 550:50 .028 .039
17 0.612 590:50 .025 .044
101 0.491 700:50 .033 .035
102 1.03 910150 .05 .045
103 0.73 590:50 .032 .037
104 0.595 740150 .025 .023
105 1.418 800150 .029 .029

106 0.718 700150 .018 .028

WATT - cm‘1

.28

.24

.20

.16

.12 —

.08

.04

111

 

 

 

l

 

 

Fig.

55.

4O

80

TEMPERATURE
Ideal thermal resistivity of pure COpper

I
120

WY

1
160

l
200

 

112

 

.5.—

 

   

 

a ECTRONIC
LATTICE
7 ”1"
5
l
S
g .
.cxs _
.01. u
l I L
10 so 100
TEMPERATURE 9K
Fig. 56.

ductivity of Cu-Sn No. 2

Total, lattice, and electronic thermal con-

 

WATT -CM ‘1

113

 

   

 

 

.5 _
.1 _
a ECTRONIC
LATTICE
.435 _
~01 ..
l . I I
10 50 100

TEMPERATURE °K

Fig. 57. Total, lattice, and electronic thermal con-
ductivity of Cu-Sn No. 6

 

114

 

    

 

 

 

 

l.-
u ECTRONIC
7 s
5 -.
E:
g
LATTICE
\\
.1, b
I l j
10 50 100

TEMPERATURE °K

Fig. 58. Total, lattice, and electronic thermal con-
ductivity of Cu-Sn No. 8

WATT CM'1

.1

115

 

LATTICE

 

 

 

J ' L l

 

10 50 100
TEMPERATURE °K

Fig. 59. Total, lattice and electronic thermal con-
ductivity of Cu-In-Cd No. 14

 

WATT CM-1

116

 

   

ECTRONIC

 

o1.

 

l . , . .

 

 

’_ 10 so 100

TEMPERATURE °K

Fig. 60. Total, lattice, and electronic thermal con-
ductivity of Cu-In-Cd No. 15

117

 

 

LATTICE

 

 

 

 

 

.JL—
1 l l
10 A 50 100
TEMPERATURE °K
Fig. 61. Total, electronic, and lattice conductivity

of Cu-Cd-In NO. 17

wATT cm.‘1

118

 

    

ELECTRONIC

LATTICE

 

l I 1

 

 

 

10 50 100
TEMPERATURE OK

Fig. 62. Total, electronic, and lattice conductivities

of Cu-Zn-In No. 101

WATT CM. ’1

0
U

119

 

 

 

LATTICE

 

 

 

Fig. 63. Total,
of ell-211-1!) NO.

I
50
TEMPERATURE 0K

100

electronic and lattice conductivities

102

WATT CM’1

120

 

    

.5
F—
ELECTRONIC
1. __
.5 LATTICE

 

 

 

 

 

10 ' 50 100
TEMPERATURE 9K

Fig. 64. Total, lattice, and electronic thermal con-
ductivities Cu-Zn-In No. 103

WATT CM '1

121

 

 

 
    
 

ELECTRONIC

 

LATTICE

 

 

1 n 1
50 100

 

TEMPERATURE °K

Fig. 65. Total, electronic,
of Cn—Ga-Zn No. - 104 ~

and lattice conductivities

WATT CM”1
p;

122

 

TOTAL

   

LECTRONIC

LATTICE

 

l I J

 

 

10 50 100
TEMPERATURE °K

E’g. 66. Total, lattice, and electronic thermal con-
ductivities of Cu-Ga-Zn No. 105

 

WATT - cM“1

123

 

 

 

 

 

5>,__
TOTAL £
LECTRONIC
1 _—-
LATTIC
.,5 __
- l I I
- 10 50 100

TEMPERATURE °K

Fig. 67. Total, lattice, and electronic thermal con-
ductivitiesof Cu-Ga-Zn No. 106

124

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125

References

Pippard, A.B., Phil. Trans. £229, 325 (1957).
Shoenberg, D., Phil. Mag. 2, 105 (1960).

Alekseevskii and Gaidukov. J. exp. theo. Phys.
Moscow, 31, 672 (1959).

Langenberg, D.N. and Moore, T.w., Phys. Rev. Letters,
3, 328 (1959).

Gavenda and Morse, (1959), Bull. Amer. Phy. Soc. 3,
463.

Gold, A.V., MacDonald, D.K.C., Pearson, W.B., and
Templeton, I.M., Phil. Mag. 2, 765 (1960).

Wilson, A.w., "The Theory of Metals" 2nd Ed. Cambridge
Univ. Press (1954).

Sondheimer, 3.3. Proc. Camb. Phil. Soc. 4;, 571 (1947).
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Bailyn, M., Phys. Rev. 112, 1597 (1958).

Ziman, J.m., "Electrons and Phonons", Oxford Univer-
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Taylor, P., private communication.

Hanna, I.I., and Sondheimer, E.H., Proc. Roy. Soc.
Aggg, 247 (1957).

Blatt, F.J., and KrOpschot, R.H., Phys. Rev. 113, 480
(1960).

DeVroomen, A.R., Van Baarle, C., and Cuelenaere, A.J.,
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MacDonald, D.K.C., Pearson, W.B., and Templeton, I.M.,

Proc. Roy. Soc. A248, 107 (1953).

17.

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34.

126
MacDonald, D.K.C., Pearson, w.B. and Templeton, I.M.,
Proc. Roy. Soc. 2229, 334 (1960).
Guenault, A.M., and MacDonald, D.K.C., Proc. Roy.
Soc. aggg, 41 (1961).
Bailyn, M., Scientific paper 029-BOOO-P1, Westinghouse
Research Laboratories (1961).
Kasuya, T., Progr. Theoret. Phys. 22, 227 (1959).
DeVroomen, A.R., and Patten, Physica. 21, 657 (1961).
Mott, N.F.. and Jones H., “Theory of the PrOperties of
Metals and Alloys”, Clarendon Press (1936), Dover,
reprint (1958).
Kohler, M., Z. Phys. 22g, 481 (1949).
Pearson, w.B., Phys. Rev. 222 (1960).
Klemens, P.G.. “Solid State Physics", vol. 7 Academic
Press (1958).
Ziman, J.M., Proc. Roy. Soc. 222g, 436 (1954).
Gerritson, ”Handbuch der Physik“, vol. 19, Springer-
Verlag/Berlin (1959).
MacDonald, D.K.C., "Handbuch der Physik", vol. 14 (1956).
Olsen, J.L., and Rosenberg, W.M., Phil. Mag. Supple-
ment vol. 2, No. 5 (1953).
Bloch, F., Z. Physik 22, 555 (1928).
Dekker, A.J., "Solid State Physics“, Prentice-Hall (1957).
Lindenfeld, P., and Pennebaker, w.B., Phys. Rev. 221,
1881 (1962).
Tainsh, R.J., and White, G.K., Phys. Chem. Solids 22,
1329 (1962).

Pippard, A.B.. Phil. Mag. fig, 1104 (1955).

127

35. Seybolt, A.U., and Burke, J.E., "Procedures in
Experimental metallurg‘", Wiley, (1953).

36. Powell, R.L., Rogers, w.n., and Coffin, D.O., J.
Research Natl. Bur. Standards g2, 349 (1957).

37. Brickwedde, F.G., van Dijk, 8., Durieux, J.R., and
Logan, J.K., J. Research Natl. Bur. Standards @42,
l (1960).

38. Christian, J.W., Jan. J.P., Pearson, w.B., and
Templeton, I.W., Proc. Roy. Soc. @232, 213 (1958).

39. "Metals Handbook", American Society for Metals (1948).

40. Linde, J.O., Ann. Physik 2;, 219 (1932).

41. KrOpschot, R.H., Thesis, Mich. State U. (1959).

42. MacDonald, D.K.C., and Pearson, w.B., Phys. Rev. _2,
149 (1952).

43. White, G.K., Aust. J. Phys. g, 397 (1953).

44. Berman, R., and MacDonald, D.K.C., Proc. Roy. Soc.
222;, 122 (1952).

45. White, G.K., Aust. J. Phys. 22, 256 (1960).

46. Klemens, P.G., Prog. Phys. Soc. (London) 262, 1113
(1955).

47. Kemp, W.R.G., Klemens, P.G., and Tainsh, R.J., Aust. J.
Phys. lg, 458 (1957).

48. White, G.K. and Woods, 8.8., Phil. Mag. 32, 1343 (1954).

49. Leibfried, G., and Schloemann, Nachr. Akad. Wiss.
Gessell. Gottingen IIa, No. 4, 71 (1954).

50. Scott, R.8., "Cryogenic Engineering”, D. Van Nostrand
(1959).

51. Chemical Rubber handbook 37th edition (1956).

A

'U
L

128

129

Analysis of Possible Errors

Caiibration of thermocouples

In the calibration of the Au-2.l at. % Co vs Ag-31
at. % Au thermocouples, the thermocouple emf was deter-
mined to an accuracy of 0.01‘pv. This corresponds to a
temperature uncertainty of approximately 0.001 OK at 100K
and 0.00050K at 1000K. The platinum resistance thermometer
was measured to an accuracy of 0.1 ohm, correSponding to a
temperature uncertainty of 0.010K at 100K and 0.0050K at
1000K. The thermocouple that was calibrated was not the
one used in the experiments but was made with wire taken
from the same spool. In addition, another sample was taken
from the same spools and checked for the emf produced be-
tween 4.20K and 2730K. This couple exhibited a 1% higher
emf than the one on which the calibration was done. Other
workers‘so) have noted differences of 1/2% in the thermo-
electric force of various sections of a single Au-Co wire
and have attributed these differences to inhomogeneities.
Presumably this is the source of our discrepancies and is
the limiting factor in our temperature measurements.
Thermoelectric power measurements

The thermoelectric emf's of the samples were measured
to a precision of 0.005‘#v. Under the most unfavorable con-
dition, (below 100K) the emf‘s were as low as 0.15’pv so
that the error in this region may be as much as 3%. For
the most part however, the emf's ranged between 1 and lSluv

making the error in voltage measurements negligible.

130

Resistance measurements

 

The current which passed through the sample was
measured to 0.1% by monitoring the voltage across a pre-
cision resistor in series with the sample. The voltage
change across the sample was measured with the same leads
used for the thermoelectric power measurements. In this
case however, the voltage was Sluv or more so errors in
its measurement were negligible. The shape factor,
(length/érea) was measured to a precision of 2%. An
error in this quantity would effect results by a constant
factor on a given sample.

Thermal Conductivity

 

In the measurement of thermal conductivity the power
input to the heater was determined to a precision of 0.1%.
The resistance of the leads was only 0.1% of the resistance
of the heater coil so loss along the leads was negligible.
Eighteen leads go from the cold bath to the sample and its
attachments. Each lead is 3 mil diameter or less. Since
the sample has a diameter of 0.1", its cross section is 100
times the area of all of the leads. In addition, the
leads are longer than the sample so the possible error due
to transfer of heat down the leads is presumed less than 1%.

The possibility of heat transfer by the residual gas
near the sample was investigated by the following test:

A gradient was maintained across the sample as in a normal
measurement, great care being made to assure that it was in
a true steady state condition. Lelium was then slowly

_C:
introduced until the pressure in the system rose from 10 “mm

131
(normal operating vacuum) to 5xlO-3mm Hg. No change was
observed in the thermal conductivity.
The rate at which a surface radiates is given by the
Stefan-Boltzman equation,
W = O'AT4e
where e is the emissivity at temperature T, A is the area,

12 2

andma'a constant having the value 5.67x10' watt cm-

(OK)-4. Assuming that the wall at 4.20K will adsorb all

of the radiation incident it, we can calculate the total

heat transfer by radiation from the above equation. Using

a value e = 0.017(50) for the etched copper rod, we find
that for 100K our samples will radiate approximately 7.51410"9
watts. The heat being conducted by the sample at these
temperatures is approximately 10.2 watts so clearly radi-
ation is no problem here. At 1000K the loss will be

7.51110"5

whiCh constitutes approximately 0.3% of the heat
conducted by the sample at this temperature.

A muCh greater problem, and one whiCh most certainly
limits the accuracy of thermal conductivity measurements
above 500K, is that of achieving a true steady state con-
dition. The smallest temperature drift detectable was
about 0.10K/min. For our sample of about 15 grams, the
power required to cause the temperature of the whole sample

3

to drift at this rate is about 5x10‘ watts at 840K where

copper has a specific heat of 0.05(51) cal. This is 2% or
gmUK
more of the heat required to maintain the temperature

gradient at these temperatures. Achieving this minimum

132
drift was exceedingly tedious, requiring 15 to 30 minutes
for each reading. At lower temperatures the Specific heat
of COpper drops sharply thus at 200K the specific heat is
only 0.003(51)g§l; and achieving a steady state is no
9' mob;
longer a problem.

To summarize, we should say that the thermoelectric
power measurements were limited principally by the 1%
error in calibration of the thermocouples. In a region of
rapidly varying thermopower, such as we find below 200K in
some of our measurements, this error in temperature measure-
ment could cause as much as 5% error in thermoelectric
power. However, for the most part the error would be 1%
or less. The limiting error in the resistance measure-
ments would be the 2% uncertainty in the shape factor. The
thermal conductivity measurements were limited principally
by the problems in attaining a true steady state condition.
We estimate the total error in these measurements at 1% at
200K, 2% at 300K, and 5% at 500K. The error in the shape
factor need not be considered in comparing electrical con-
ductivity with thermal conductivity (i.e. in calculating
lattice conductivity) as it appears to the same extent in

both measurements.

APPENDIX I I

133

134

Pure Copper vs. Pb

 

 

 

 

TOK Thermoelectric TOK Thermoelectric
force (Ev) force (pv)
coldhgppaiip2.SgKliquid coldhgppfigipfioggKgiquid
9.5 5.2 36.6 3.07
11.16 6.87 37.59 1.80
13.5 8.6 58.6 0.90
15.6 10.1 39.55 -O.45
17.33 10.9 40.5 -1.50
19.1 11.45 42.4 4.02
21.0 11.75 46.1 9.25
22.2 11.80 49.7 12.40
23.65 11.75 53.08 20.7
25.0 11.60 56.4 26.5
26.35 11.00 59.7 32.3
27.6 10.50 62.79 38.0
28.8 9.85 65.9 43.6
29.95 9.10 68.9 49.5
31.33 8.00 71.9 54.9
32.40 7.23 74.9 60.0
33.36 6.32 77.8 65.8
54.4 5.62 80x5 70.8
35.5 4.25 83.5 76.0
36.6 3.07 86.3 81.3

135

Pure Copper vs. Pb (c0nt.)

 

 

TOK Thermoelectric TOK Thermoelectric
force (uv) force (UV)
°°ldh31133171331<31qum ”whi‘ifirfii‘fifiégfiiqum
89.1 86.5 56.55 -6.9
91.9 91.8 32.25 -7.8
90.53 89.9 27.59 -ll.2
84.92 79.9 22.2 --
82.1 74.7 15.6 11.8
83.5 77.1 11.17 9.08
80.0 69.8 13.5 10.2
77.8 64.5 15.6 11.3
75.57 58.1 17.55 12.1
70.41 53.2 19.1 12.6
67.4 48.0 21.0 12.7
64.32 41.8 22.2 13.0
61.19 36 23.65 12.9
58.05 50.8 25.0 12.6
59.75 25.1 26.35 12.5
51.40 19.1 27.6 11.5
37.90 26.3 28.8 10.95
44.35 13.6 29.95 10.00
40.50 2.2 31.33 .00

R)
.1:-

40.50 2.25 32.40

136

Pure Copper vs. Pb (c0nt.)

 

 

TOK Thermoelectric TOK Thermoelectric
force (pv) force (uv)
cold Junction at liquid air cold Junction at liquid air
(79.20K) (79.20K)
80.62 3.07 88.19 18.50
81.15 5.67 85.69 9.45
82.55 7.70 82.55 7.08
85.11 9.20 82.55 8.58
85.69 10.70 85.11 9.92
84.25 11.90 85.67 11.50
84.79 15.20 84.25 12.50
85.55 14.17 84.79 15.58
85.90 14.85 ' 85.55 14.85
86.49 15.86 85.90 14.85
87.05 16.97 86.49 16.25
87.62 17.88 87.05 17.56
88.19 19.05 87.62 18.70
88.75 20.08 90.94 25.9
89.51 21.05 88.19 18.87
89.88 22.2 85.90 12.88
90.40 23.15 95.59 29.25
90.94 24.08 96.21 55.80
88.19 19.02 98.86 42.8

85.90 14.12 101.41 46.9

137

Pure Copper vs. Pb (c0nt.)

 

 

TOK Thermoelectric TOK Thermoelectric
force (0v) force (pv)_
cold Junction at liquid air cold Junction at liquid air
(79 20K) (79.20K

103.88 53.2 277.95 325.0
108.82 63.8 233.60 339.0
113-56 75-7 239~5O 351.5
118.25 88.6 245.00 365.2
122.74 96.3 250.05 381.0
127.16 112.0 255.08 395.0
131.49 117.5 261.01 408.5
139.91 136.5 266.30 421.0
147.85 154.0 271.7 ' 435.0
155.0 168.2 276.8 449.5
163.0 184.0 281.9 464.0
170.0 198.6 286.9 473.0
177.3 212.0 291.8 491.0
184.1 226.1 296.7 505.0
190.75 241.0

197-25 255.5

203.3 270.0

210.0 284.5

216.10 295.2

222.0 314.5

138

Sample No. 2, Cu-1.86% Sn

T°K

 

4

.2

10.6

15.
.95
.42
20.
.25
28.
.95
.50
40.
47.
58.
65.
69.
88.
92.

15
17

24

31
34

98.
101.

94.

F1 (p \o t— a) 4: .e o

[00
U1

S-S

u

Pb'E—

.0595
.210
.377
.373
.513
.544
.553
649
.553
.514

.500

.451
.441
.425
.591
.397
.594
.400
.426

.43

v
K

Thermal Conduc
watt cm"

.095
.135
.180

.210
.270
.340
.400
.455
.480
.510
.510
-530
.580
.581
.637
.651
.662
.660

iivity

Electrical
Resistance
microhm-cm

 

4.86

139

Sample No. 2, Cu-1.86% Sn (cont.)

 

 

 

TOK S'SPb'gx' Thermal Conduciivity 8881:8888:

K watt cm“ ' microhm-cm
106.05 .45 -- --
108.8 .45 -- 5.58
108.6 .400 -- __
80.0 -- -- 5.18
120.5 .404 -- --
131.7 .406 -- --
140.0 .420 -- --
154.1 .430 -- --
172.5 .455 -- --
177.6 .462 -- 5.93
184.4 .472 -— --
195.5 .490 -- --
178.2 .465 -- -_
220.5 .540 -- -_
251.6 .560 -- --
199.0 .500 -- ~-
204.6 .510 -- --
188.9 .485 —- --
305.0

.740 -- 6.90

140

Sample No. 6, Cu-l.60% Sn

 

2
6
11.5
14.2
17.7
20.8
24.95
26.90
28.65
52.40
58.45
45.0
53.75
55.99
61.25
77.4

95.0
102.2
102.6
105.0

s-s “V

Pb 0K

.402
.825
.986
.92

.867
.837
.825
.761
.735
.690
.653
.542
.555
.525
.500
.518
.510
.514
.505
.510

Thermal Conduc
watt cm'

iivity

 

.062
.150
.210
.300
.415
.525
.540
.590
.625
.660
.690
.675
.690
.700
.770
.760

Electrical
Resistance
microhm-cm

 

5.58
5.58

3-37

141

Sample No. 6, Cu-l.60% Sn (cont.)

 

 

TOK S-SPb 4:1 Thermal Conduciiivity 3831:8882:
K watt cm microhm-cm
98 .4 .514 —— ..
116.0 .526 -- --
126 .5 .548 -- 4.05
155.4 553 -- --
172 .5 .564 _- ..
172.9 .578 -- --
126 .8 .600 .. --
128.0 .597 .. ..
138.3 .572 -- __
142 .6 .541 —— __
87.4 .45 -- -_
90.11 .497 -- 3,68
102 .6 .526 —— __
224 .4 .782." -- --
247 .8 .81 -_ __
278 .4 .91 -- -_
295.0 .96 —- 5.29

Sample 8, Cu-O.6l% Sn

1-3
O
W

 

CID-P:

o p. C) <9 03 \N -¢ a) C) (h a) 04 n) (n ox C) \x -q -< k) R)

19.
23.
32.
43.
65.
82.
102.
94.
105.
106.
118.
120.
127.
141.
161.
165.
186.
2205.
2354.

s-s “V

Pb'ag
.650
.688
.668
.685
.639
.599
.622
.657
.640
.689
.685
.681
.679
.681
.672
.685
.688
-697

1.04

142

Thermal Conduc
watt cm“

14 F4 14 F4 +4

.20
.66
.77
.94
.01
.13
.25
.55
.28

Eivity

Electrical
Resistance
microhm-cm

 

1-75

143

Sample 8, Cu-0.6l% Sn (cont.)

 

TOK _ s-sPb g3!- Thermal Conductivity
K watt cm"l
505.0 1.14 --
255.0 1.04 --

167.0 0.98 --

Electrical
Resistance
microhm-cm

 

5.40
5.28

144

Sample 14, Cu-0.225% Cd 0.762% In

TOK

16

21
28
35
45

I100.
198.
.124.
124.
125.

.25
12.
.55
17.
.25
.15
35
.75
52.
65.
79-
85.
91.
99-
101.

CI)
U1

«F'NKDKOm-P-‘U'I-PKOI-J

S-S

Pb?)—

.254
.206
.514

.321

.400

.412
.441
.510
.544
535
.560
.589
.570
.565

- 573
- 573
747

uv
K

Thermal Conduc
watt cm"

Eivity

 

#4 r4 F4 +4 F1 +4 F3 +4 P4

.225

Electrical
Resistance
microhm—cm

 

1.057

145

Sample 14. 0.1-0.225% Cd 0.762% In (cont.)

TOK

148.8

169.5
185.6
184.2
206.5
201.1

S-S

uv
Pb o

.850
.84
.88.
.879
.951
.948

Thermal Conduc
watt cm‘

Eivity

Electrical
Resistance
microhm-cm

 

 

146

Sample 15, Cu-0.104% Cd 0.389% In

 

 

 

TOK s-st-gi Thermal Conductivity g:§i§§1§i%y
K watt cm‘“l microhm-cm
4.2 -- -- -5“9
8.3 .181 .480 -_
14.2 .26 1.05 --
15-0 .339 1.10 _-
17.35 -363 1.34 -552
19-7 .455 1.53 --
25.85 .517 1.92 -_
27.6 .499 2.15 .562
36.9 .556 2.28 -_
49.0 .688 2.45 --
66.4 .758 2.45 --
82.8 .898 2.55 --
89.8 .88 2.75 .898
89.8 .891 2.78 --
99.0 .902 2.70 .941
92.6 .907 2.69 __
87.0 .87 -- --
86 15 .86 -- 851
78.0 .78 -- ~-
70.65 .72 -- --

65.50 .70 -- --

Sample 15, Cu-O.104% Cd 0.389% In (cont.)

TOK

 

54.25
84.60
89.60
92.0
98.
106.
121.
122.
127.
1154.
173.
.184.
199.
2254.
1233.
223.

\ONUIUTU'ICNUIKOCDWOW

S-S

H H H H H H H H H H H

u

Pb'E-

.68

.89

.88

.911
.908
.05

.097
.101
.146
.333
.301
.504
.372
.501
.469
.415

147

V

Thermal Conduc
K watt cm‘

 

Electrical
Resistivity
microhm-cm

 

148

Sample 17, Cu-0.477% Cd 0.294% In

TOK

 

15.
20.
30.
36.
49.
58.
75.
79-
90.
98.
101.
107.
107.
323.
318.
312.
314.
315.
83 .

ONKOKOQKOKOWID
U‘IOU'IOOKOU‘I

\O-P-‘KIHUWCD-fi'Kfi<F—‘4=‘\O

S-S

pv

Pb'cT'

K

 

+4 r0 [D m m m F4 +4 +4 +4 H H H H F’

.551
.499
.581
.788
.879
.053
.198
.469
.503
.504
.506
.60

.61

.61

.21
.19
.18
.18

.50

Thermal Conductivity
watt cm"l

 

-‘

0.61

1
l

2

\NI'DIUNIDIDI'U

.15
.65
.60
-75
.80
.78
.90
.98
-95
.01

Electrical
Resistivity
microhm-cm

 

.446
.445
.445
.450
.958

149

Sample 17, Cu-O.477% Cd 0.294% In (cont.)

 

 

 

 

TOK S'SPb'8X Thermal Conductivity gigiggigiiy
K watt cm"l microhm—cm
124.5 1.70 -- --
247.7 2.10 -- --
267.5 2.2 -- --
211.5 2.01 -- 1.62
196.6 1.95 -- 1.61
185.3 1.88 -- --
172.8 1.80 -- --
165.5 1.81 -- --
142.8 1.78 -- 1.13
155.4 1.70 -- . --
131.4 1.71 -- 1.10

150

Sample No. 32, Cu-O.454% Ag

 

 

TOK 03h388088884454 :gréeAg ‘TOK c$h3§4°§ieéfiéfi ::r%eAg
_14V) (uVJ
cold Junction afi.%gggid helium cold junction ?4.2g%%id helium
12.2 29.0 47.90 31.5
13.05 32.5 49.69 25.0
13.91 34.7 51.40 17.8
14.78 38.2 53.08 10.8
15.54 41.7 54.75 3.0
17.78 48.7 56.39 -lO.6
19.20 55.2 58.00 -17.4
20.73 60.5 59.65 -23.1
22.2 64.7 61.19 -30.8
25.0 70.1 63.08 -37.1
28.08 74.0 64.32 -43.2
29.95 75.2 65.89 -49.1
32.25 74.8 67.39 -54.8
54.45 68.9 68.90 -6o.o
36.54 66.6 70.40 -65.7
38.55 62.1 71.91 -70.6
40.52 57.0 73.38 -76.9
42.45 51.0 74.85 -82.5
44.35 42.5 76.31 -88.1
46.10 58.5 77.76 -95.9

151

Sample No. 32, Cu-0.454% Ag (cont.)

 

 

 

TOK cghsg8088884454 :gréeAg TOK cghngogiegffiéfi §§r%eig
(4V) (4V)
cold Junction a4.%g%%id helium cold junction aEBlgguid air
79.22 -996 106.56 86. 2
82.09 -107.8 108.82 92.3
83.50 -113.2 111.2 98.6
84.92 -ll8.0 113.56 103.6
86.33 -123.0 118.25 113.8
87.72 -128.2 122.74 122.0
89.13 —123.0 127.16 130.0
90.52 -229.8 131.49 138.6
89.13 -135.0 135.70 144.7
87.72 -131.16 139.91 152.0
86.30 -137.0 143.85 158.4
56.39 ~13.2 147.85 164.2
cold JunCti?8882°K3qUid air 151.86 169.6
85.35 24.5 155.53 174.4
88.17 33.1 159-33 179-5
90.94 41.5 165.05 189.0
95.59 49.8 166.7 188.0
96.21 58.0 123.7 195.0
98.86 66.0 177.3 200.0
101.40 73.1 180.69 203.0
103.88 60.0 184.1 205.0

152

Sample No. 32, Cu-O.454% Ag (cont.)

 

 

th$E°Sieé‘fiéfi 32:28... .Ehsmligiigz :22...
(11V) (11V)
cold Junctiogaagoiiquid air
187.4 208.4 250.05 252.0
190.7 212.0 255.2 255.0
195.9 215.1 255.08 254.0
197.2 216.0 258.4 255.0
200.5 220.0 261.01 256.0
205.5 224.0 266.5 258.0
206.8 228 269.0 260.0
210.0 229.0 271.0 261.0
215.0 252.0 274.0 265.0
216.0 252.0 276.0 264.0
219.0 255.0 279.0 265.0
222.0 255.0 281.0 266.0
224.0 257.0 284.0 +268.0
227.95 259.0 286.0 +268.0
250.8 241.5 289.0 269.0
236.6 244.0 291.0 270.0
258.5 245.0 294.0 271.0
242.2 247.0 296.0 272.0
245.0 248.0 299.0 275.0

153

Sample No. 33, Cu-1.977 at % Ag

 

 

 

 

TOK Thermoelectric force .TOK Thermoelectric force
Cu vs. Cu 1.977 at% Ag Cu vs. Cu 1.977 at_%Ag
cold Junction at liquid helium cold Junction at liquid helium
(4.2 K) (4.20K)

13.45 17.1 90.66 31.1

24.55 40.6 85.19 35.8

33.68 53.2 79.34 40.5

40.64 58.5 71.16 47.0

48.51 59.4 64.23 52.96

55.41 57.3 58-35 57.2

61.75 53.1 53.46 60.5

68.68 48.3 47.0 61.8

75.54 42.5 39.72 62.0

82.41 36.4 32.15 59.0

89.28 51.2 28.05 54.0

95.2 26.8 15.55 25.0
104.0 21.0 10.78 12.5
109.9 16.9 6.30 10.4
118.0 12.5 8.85 10.0
129 O 7.4 cold junctign8%§)liquid air
122.7 10.8 80.43 .2
115.1 14.5 80.53 6.1
107.0 19.5 92.18 8.2

98.75 25.0 104.15 19.1

154

Sample No. 55, Cu-l.977 at % Ag (cont.)

 

 

 

TOK Thermoelectric force TOK Thermoelectric force
Cu vs. Cu 1.977 at % Ag Cu vs. Cu 1.977 at;% Ag
cold Junction at liquid air cold Junction gt liquid air
(79.80K) (79.8 K)

111.20 17.7 277.0 68.0

118.40 27.0 285.5 68.8

128.22 32.4 294.5 69.0

135.94 35.8 303.3 70.0

145.09 59.6 511.2 70.6

154.03 43.0 317.3 71.0

162.29 45.8 322.0 70.8

169.94 48.0 331.3 71.8

177.84 50.1 339.1 72.0

185.37 53.0 351.4 72.4

192.81 55.0 565.8 75.0

200.7 56.8 390.7 74.0

208.1 58.0 567.0 75.6

215.4 59.8 346.4 72.4

222.7 61.0 322.3 71.6

230.0 62.0 298.7 70.0

237.7 64.0 292.2 69.0

248.5 65.0 267.2 67.0

259.5 66.0 239.2 63.6

267.8 67.0 212.8 58.6

155

Sample No. 33, Cu-l.977 at % Ag (cont.)

TOK Thermoelectric force
Cu vs. Cu 1.977 at% Ag

 

 

cold Junction at liquid air

(79.80K)
218.6 60.4
185.52 52.0
148.60 41.8
126.50 32.0

113.18 24.5

156

Sample No. 34, Cu-0.81% Au

0 Thermoelectric force 0 Thermoelectric force
T K Cu vs. Cu-0.81% Au uv T K Cu vs. Cu-O.8l% Au uv

 

 

cold Junction at liquid helium cold Junction at liquid helium

(4.20K) (4.2OK)

5.1 1.0 101.23 14.1
17.7 27.1 91.74 22.2
23.77 40.0 83.38 29.0
27.58 46.7 74.82 36.6
33.12 53.2 65.65 44.8
37.24 56.3 56.68 52.4
41.05 58.1 47.04 57.9
45.12 58.1 43.41 59.4
51.78 55.5 40.40 59.4
55.95 52.0 36-95 58.2
59.00 49.2 33.95 56.6
62.77 46.6 30.87 54.6
69.71 40.4 29.00 52.3
78.40 32.0 25.39 47.5
84.38 27.0 23.05 43.9
92.75 19.9 21.93 41.6

102.7 13.0 19.97 37.4
111.7 6.4 cold JunctggnnggKfiiquid air
123.6 -0.3 80.43 0

115.0 3.5 86.53 2.1

157

Sample No. 34, Cu—0.8l% Au (cont.)

 

 

TOK Thermoelectric force TOK Thermoelectric force
Cu vs. Cu-0.8l% Au pv Cu vs. Cu-O.81% Au uv

cold JunctggnuggKliquid air cold JunctggnuggKliquid air

92.18 3.9 259.5 42.1
104.15 7.3 267.8 43.7
111.20 11.0 277.0 44.7
118.40 13.6 285.5 46.0
128.22 16.6 294.5 47.2
135.94 19.2 303.3 48.2
145.09 21.6 311.2 49.4
154.03 23.7 317.2 50.3
162.29 25.5 322.0 51.6
169.94 27.2 331.3 52.2
177.84 28.4 339.1 53.7
185-33 30.7 351.4 55.5
192.81 51.7 565.8 58.0
200.70 33.1 390.7 61.5
208.10 34.5
215.40 35-5
222.70 36.7
230.00 38.1
237-7 39.0
248.5 40.5

158

Samples 55, Cu-O.40% Au; and 51, Cu-0.96% Ag

TOK Thermoelectric Force Thermoelectric Force
Cu vs. Cu-O.4Q% Au(uv) Cu vs. Cu—0.96% Ag

 

 

cold Junction at liquid air (78.4OK)

81.17 2.0 2.2
82.56 3.7 4.0
85.36 7.1 8.2
88.20 10.2 12.0
93.60 14.4 17.0
96.20 16.5 19.5
101.4 20.0 24.0
106.56 25.5 28.0
111.20 27.0 52.0
115.90 30.2 35.5
120.50 33.5 39.0
125.0 55.5 42.0
129.34 37.7 44.5
155.6 39.7 47.0
157.8 41.7 49.7
141.82 43.2 51.0
145.90 45.1 55.1
155.67 48.5 57.0
157.42 50.0 58.5
161.20 51.5 60.5

159

Samples 35, Cu-0.40% Au; and 31, Cu-0.96% Ag (cont.)

TOK Thermoelectric Force Thermoelectric Force
Cu vs. Cu-0.40% Au(uv) Cu vs. Cu-0.96% Ag

 

 

164.88 55.0 62.0
168.48 54.5 63.3
172.05 56.0 65.5
175.5 57.0 66.5
179.0 58.1 67.5
182.4 59.1 68.9
189.1 61.5 70.8
195.6 65.5 75.1
200.4 65.5 75.4
208.5 66.8 77.1
214.5 68.7 79.0
226.5 71.5 82.0
252.25 75.0 85.2
258.0 75.0 84.9
243.6 76.4 86.0
249.1 77.8 87.1
254.5 78.7 88.3
259.7 80.0 89.5
265.0 81.1 90.5
270.5 82.1 91.5

160

Samples 35, Cu-O.40% Au; and 31, Cu-0.96% Ag (cont.)

 

 

TOK Thermoelectric Force Thermoelectric Force
Cu vs. Cu-0.40% Au(uv) Cu vs. Cu-0.96% Ag
275.0 83.0 92.5
280.2 84.6 93.3
285.6 86.0 94.3
290.5 87.0 96.3
295.5 88.2 97.0
200.5 89.5 97.0
cold Junction at liquid helium (4.2OK)
12.4 -12.6 -5.5
13.5 -13.9 -6.2
15.6 ~16.1 ~7.5
19.1 ~19.5 ~9.0
22.2 -2l.5 -9.2
25.0 -22.0 -9.3
27.6 -22.0 -7.5
29.9 ~21.4 -7.0
32.4 -20.5 -5.2
34.4 -19.2 -3.6
36.6 -18.0 -1.2
38.6 -16.1 +1.5
40.5 -14.0 +3.9
42.4 -11.6 +6.9

161

Samples 35, Cu-0.40% Au; and 31, Cu-0.96% Ag (cont.)

 

 

 

T K c$h$§4°§$538$$ 1316351 ”38382883333683?
44 .5 -9.4 +8 -9
46 .1 -7.8 +11 .0
47.9 -5.2 +12.5
51 .4 -3.2 +15.6
55.9 -1.4 +17.2
56 .4 +2 .0 +21 .2
59.7 +5.2 +24.3
63.6 +9.2 +29.1
65.9 +16.0 +33.2
68.9 +19.5 +36.2
71.9 +22.8 +40.1
74.9 +26.0 +47.8
77.8 +29.2 +51.9
80.6 +32.0 +55.0
83.5 +35.0 +58.2
86 .3 +37 .9 +62 .0
89.1 +40.5 +64.9
91-9 +45-9 +68.2

162

Sample 101, Cu .47 at. % Zn, .294 at. % In

 

u,

23
29
39
48
57
71
93
82
77
77
107

151.
180.
210.
234.
-75
-55
~75
.10

19
28
39
47

.2
15.
.45
.85
.00
.25
.90
.05
.20
.80
.90
.40
.10

uv
OK

Thermal Conduc
watt cm“

iivity

 

1
1
2
2
2
2
2
2
2
2

.141
.851
.253
.496
.601
.632
.651
.650
.806

Electrical
Resistivity
microhm cm.

 

.485
.492
.486
.503
.505
.550
.596
.638
.802
.735
.707
.692
.900
1.21
1.42
1.62
1.78

Sample 101, Cu

TOK

 

62.
70.
79.
85.
92.
98.
102.
111.
129.
93-
121.
143.
168.
182.
212.
236.
280.

HWWOWONNONF—‘mm

S-S

2.1:
Pb o

163

Thermal Conduc
watt cm‘

fivity

 

 

10 m H1 F4 +4 14 H

.857
.889
.918
.927
.907
.950
.975
.01
.11
.969
.100
.250
.425
.51
.76
.14
.68

.47 at. % Zn, .294 at. % In (cont.)

Electrical
Resistivity
microhm cm.

 

164

Sample 102, Cu vs. 0.49 at. % Zn, 0.77 at. % In

 

 

 

TOK S—SPb-gz Thermal Conductivity giziggisiiy
K watt cm' microhm-cm
4,2 _- -- 1.050
12.38 .497 .44 __
18.90 .725 .83 --
25.60 .754 t 1.04 _-
55.40 .660 1.25 -_
50.45 .635 1.49 -_
66.20 .649 1.55 --
79 6O .654 1.68 -_
97.80 .729 1.74 —-
91.8 .662 1.75 _-
99.2 .748 1.70 1.420
77.4 -- -- 1.290
110.0 .707 -- 1.459
124.3 .842 -- 1.614
141.1 .982 -- 1.747
166.1 1.38 -- 1.902
185.8 1.49 -- 2.045
230.0 1.56 -- --
260.2 1.91 -- --
293.1 2.25 -- __

165

Sample 103, Cu-0.4l6 at. % In 0.995 at. % Zn

 

 

 

TOK s-st-gX Thermal Conductivity gigigiigiiy
K watt cm' microhm—cm.
4.2 -- -— .752
8.2 .144 .421 --
11.85 .505 .610 —-
14.00 .667 .799 .728
17-55 -735 1.06 .728
24.70 .762 1.45 .755
55.10 .750 1.84 .757
49.90 .715 2.05 .812
62.40 .750 2.07 .875
70.5 .815 2.25 .967
95.9 .870 2.25 1.092
96.0 .850 2.28 1.115
118.2 1.02 -- __
95.5 .866 —- 1.095
155.6 1.22 -- 1.528
154.1 1.22 -- _-
174.6 1.54 -- _-
196.0 1.50 -- 1.850
217.5 1.66 —- --
242.7 2.05 -- --
280.0 1 12 -- 2.424

166

Sample 104, Cu-0.331 at. % Zn, 0.35 at. % Ga

 

 

 

 

TOK s-st-gz Thermal Conductivity Sigiggigiiy
K watt cm" microhm-cm.
2 .. .. .586
8.1 0 .415 __
11.1 .0814 .58 -—
13.6 .364 .76 --
16.22 .658 .99 .584
19.25 .807 1.28 .581
22.6 .962 1.70 .581
51.1 1.555 2.20 .599
41.0 1.668 2.48 .626
51.4 1.805 2.58 .664
66.2 1.825 2.55 .756
76.9 1.77 2.70 .824
87.3 1.699 2.65 .884
95.4 1.67 2.68 .958
90.9 1.68 2.60 .892
107.8 1.66 -- 1.026
148.5 1.706 -- 1.552
152.0 1.66 -- 1.200
168.0 1.725 -- 1.478
210.7 1.96 -- 1.855
221.0 2.15 —- 1.882

167

Sample 104, Cu-0.33l at. % Zn, 0.35 at. % Ga (cont.)

TOK'
240.3
240.8
269.4

uv
S'SPb '5—
2.36
2.36

2.43

Thermal conductivity
watt cm”l

Electrical
Resistivity
microhm-cm.

2.060

2.251

168

Sample 105, Cu-OL34 at. 5 Zn, 0.805 at. % Ga

 

 

 

TOK S-st-gi Thermal Conductivity gigigiiiiiy
K watt cm" microhm-cm .
99.35 1.46 1.65 1.86
88.75 1.51 1.55 1.762
77.00 1.51 1.65 1.668
60.57 1.59 1.55 1.566
4.2 -- -- 1.417
7.41 -— -- --
9.72 .108 .285 __
15.5 .569 .45 1.420
16.6 .532 .601 1.407
21.75 .764 .900 1.415
51.40 1.079 1.250 1.450
40.75 1.433 1.42 1.46
48.85 1.545 1.52 1.49
208.7 1.61 —- 2.50
222.0 1.67 -- 2.71
238.7 1.75 -- 2.81
77.4 -- -- 1.675
96.4 1.45 1.60 1.745
115.8 1.44 -- 1.895
142.0 1.42 -- 2.111
168.2 1.46 -- 2.515

Sample

192.0

266.0
285.3

169

105, Cu-0.34 at. % Zn, 0.805 at. 5 Ga (cont.)

S-SPb-gi Thermal Conductivity
K watt cm‘

1-53 ~-

1.98 --

2.08 --

Electrical
Resistivity
microhm-cm.

2-53
3-09
3-23

Sample 106, Cu-0.856 at. % Zn, 0.37 at. % Ga

TOK

 

ll
14
22
52
43

73
97

.25
.57
.25
.25
.25
-75
59-
.35
.70

95.60

114.
128.
147.
169.
208.
213.
221.
248.
272.
291.

87.

81.

I'D-\lm
OOO

#N'IOHU‘IIDIDm-P'

S-S

u

Pb'6"

.13
.452
.686

-995
.258

v
K

170

Thermal Conductivity
t cm"l

wat

 

[UNIDNIUNH

.334
.475
.62
.22
.15
.25
.35
.40
.38
.35

Electrical
Resistivity
microhm-ch

 

.910
.905
.905
.905
.904
.924
-956
1.036
1.127
.248
.406
.572
.650
.805
.005
.16
.22
.44
.58
.70
.272
.196

FJ :4 n) [0 n) :0 n) 10 p. +4 F4 +4 F’

Calibration of Au-2.1 at % Co vs. Ag-37 at % Au Thermocouples

TOK Thermoelectric TOK Thermoelectric

force (uv) forceLuv)

 

 

 

 

cold JunctggnufigKliquid air cold JunctginuigKliquid air
81.44 -0.76 172.58 5446.9
85.61 85.08 180.88 5767.4
88.18 247.46 190.89 4169.9
90.22 402.1 200.08 4542.5
97.45 580.9 210.57 4966.5
102.17 752.6 219.51 5519.6
107.11 955.6 229.51 5726.6
111.51 1098.5 259.18 6122.6
115.79 1258.6 248.15 6487.5
119.99 1414.5 257.55 6865.5
124.71 1597.4 267.97 7299.8
129.22 1768.18 276.80 7646.2
155.17 1962.47 295.07 8581.2
80.68 1.14 297.55 8481.0
134'15 1953'0 cold Junction at liquid
158.58 2120.0 helium (4'20K)
145.55 2506.0 14.51 86.99
148.80 2512.4 15.68 151.8 '
155.60 2700.2 22.21 201.1
161.87 5025.1 25.28 255.6

Calibration of Au-2.1 at % C0 vs. Ag-37 at % Au Thermocouples

 

 

 

 

(cont.)
TOK Thermoelectric TOK Thermoelectric
force (uv) force (uvl

coldhgunfifiiofi.ggK1iquid 001dh812331?4.28K31quid
30-35 355.7 15.401 100.54
35.10 458.3 17.50 127.64
39.23 558.6 54.25 962.7
43.42 664.8 58.98 1102.5
47.42 770.9 62.47 1209.0
50.80 864.6

67.447 1365.1

76.715 1669.3

83.047 1881.7

86.509 1996-9

8.1 27.43

8.7 32.26

9.2 35-93

9.65 40.29

10.4 48.75

11.55 56.75

12.43 66.86

13.22 75-17

14.00 83.74

14.606 91.

0‘1
\1

ca" P87816342}; 1:8.

MICHIGnN STRTE UNIV. LIBRnRIEs
1|WNWWI”ll”WIHll”HIlIHmIWWW“HI
31293017640024