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THERMOELECTRIC POWER AND RESISTIVITY OF

COPPER—NICKEL ALLOYS

by

Robert Allen Wolf

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Physics and Astronomy
College of Natural Science

1963

ABSTRACT

The purpose of this program was (1) to obtain in-
formation about the Fermi surfaces of copper—nickel alloys
through a study of the phonon drag contribution to the
thermopower and (2) to study the dependence of the resis-
tivity of copper-nickel alloys on temperature and
concentration.

Two wires about one meter long by 0.02 centimeter
in diameter were prepared for each of four alloys (0%,

0.85 wt. % Ni, 3.45 wt. % Ni, and 17.10 wt. % Ni). One

of the wires, the thermoelectric voltage sample, was used
as part of a Lead-alloy thermocouple with the cold junction
in liquid nitrogen or helium and the hot junction in a cop-
per block of known temperature. The other wire, the resis-
tivity sample, was wound on a quartz rod, annealed, and
placed in a cavity in the same copper block.

The thermoelectric voltage and resistivity of each
of the four alloys were measured from 4.2 to 3000K. From
these data, plots were made of (l) thermopower versus
temperature; (2) resistivity versus temperature; (3) charac-
teristic temperature, 0R, versus temperature; (4) log(p-po)
versus 103 temperature; and (5) thermopower at 2800K versus

resistivity-l at 2800K.

It was found that all of the alloy samples had laxf‘

negative diffusion thermopowers and only that of the lowes
concentration sample (0.85% nickel) had a positive phonon
drag component. The absence of the expected phonon drag
thermopower peaks in tne other alloys, along with the
relatively high resistivities of the alloys, are in agree-
ment with other experimental evidence suggesting that
Bd-band vacancies exist in copper—nickel alloys containi.g

as little as 3% nickel.

1..

U

A C KNOW LEDG MENTS

The author wishes to thank Dr. Peter A. Schroeder
for suggesting this problem and for his assistance and
encouragement throughout the course of the investigation.

The author is grateful to the National Science

Foundation for their support of this program.

iii.

TABLE OF CONTENTS

Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . 1
II. THEORY . . . . . . . . . . . . . . . . . . . 2

Thermoelectric Power

Diffusion thermopower

Phonon drag contribution to thermopower
Electrical Resistivity

Characteristic temperature, 9R

Temperature dependence of resistivity
III. EXPERIMENTAL PROCEDURE . . . . . . . . . . . 12

Production of Alloy Samples
Production of alloys
Production of wires
Construction of Cryostat
Design of sample holder
Wiring of sample holder
Measurement of Thermoelectric voltage
and resistivity
Design of measuring circuit
Procedure used in making measurements

IV. RESULTS . . . . . . . . . . . . . . . . . . 24

Thermopower
Electrical Resistivity

Characteristic Temperature, 9R

Temperature dependence of resistivity

V. ANALYSIS OF RESULTS . . . . . . . . . . . . 35
Thermopower
Discussion of ”absolute” and ”measured”
thermopower

Summary of the Schroeder and Henry copper-
zinc results
Expected relation between copper-nickel
and copper-zinc results
Discussion of copper-nickel thermopower
results
Resistivity

VI. CONCLUSIONS . . . . . . . . . . . . . . . . 52

iv.

LIST OF FIGURES

Figure No. Page

1. Normal phonon—electron interaction 8
(schematic).

2. Umklapp phonon-electron interaction 8
(schematic).

3. Cryostat. l5

4. Copper Block. 16

5. Electrical circuit. 21

6. Thermopower of copper—nickel alloys. 25

7. Characteristic temperature, OR, of copper-
nickel alloys. 27

8. Ideal resistivity of copper-nickel alloys. 28

9. log (p-po) versus log T for copper—nickel
alloys. 29

10. Thermopower (2800K) versus resistivity-l
(2800K) for copper-nickel alloys. 54

ll. Schematic drawing of thermopower experiment.36

l2. Thermopower of copper-zinc alloys. 58

15. Fermi surface which has pulled away from
the Brillouin Zone. (Schematic) 41

14. Saturation magnetic moment, 0, and Curie
temperature, 00, for copper-nickel alloys. 44

15. Atomic susceptibility of the copper-nickel

and silver—palladium systems. 47

LIST OF TABLES

Table No. Page
1. 9R (2800K) values for copper-nickel and
copper-zinc alloys. 30
2. Slope of the log(p-po) versus log T plots
for copper-nickel alloys. 30
3. Electrical resistivity of copper-nickel
and copper-zinc alloys. 31
4. Specific heat constants for copper-nickel
alloys. 48

vi.

LIST OF APPENDICES

Appendix No. Page
I. Sources and chemical analyses of
materials. 54
II. Details of the melting and annealing
processes. 55

vii.

I. INTRODUCTION

Although there are several methods of obtaining
precise information concerning the Fermi surface in
extremely pure metals, there is no direct experiment
which will give this information for concentrated alloys.
It is necessary to deduce information about the Fermi
surface of alloys from measurements of bulk properties
of the sample. Two such properties are thermoelectric
power and electrical resistivity.

After reviewing the results of Schroeder and
Henry (1963) concerning the thermopower of copper—zinc
alloys, it was decided that a study of copper-nickel
alloys might give similarly interesting results.

This thesis (1) describes the construction of an
apparatus for the measurement of both thermopower and
electrical resistivity of copper-nickel alloys from 4.20
to 3000K, (2) describes the procedure used in making the

alloy samples, and (3) presents an analysis of the data.

II. THEORY

A. Thermoelectric Power

 

l. Diffusion thermopower. If, in the derivation

 

of thermopower, we take account of the scattering
processes but neglect the effect of the net phonon drift
velocity, then we arrive at an expression for the dif-
fusion thermopower of a material.

Mott and Jones (1936) state that the diffusion

component of the thermopower of a material is

2 2
w K T (B [ w
5 e .EE *E=n

S (diffusion) =

where C(E) is the electrical conductivity of the material
and n is the Fermi energy. The above expression holds at
low temperatures if impurity scattering predominates and
at high temperatures (T > 0).

If the phonons have a net drift velocity, which is
always the case when there is a temperature gradient in
the material, then they will interact with the electrons
and thereby affect the thermopower. The change in thermo-
power due to this interaction is called the phonon drag
contribution to the thermopower. The total thermopower of
a material is the sum of the diffusion and phonon drag

components.

Gold 22.21- (1960) suggest that, under certain
conditions, it is possible to separate the impurity scat-
tering and thermal scattering components of the diffusion
thermopower. In the case of dilute alloys containing two
or more scattering mechanisms which are independent of
each other, the total diffusion thermopower can be ex-
pressed by the Kohler relation

; wisjL
S(diff.) = Atr—__.
% wi

where Wi(T) and Si(T) are the thermal resistivity and
characteristic thermoelectric power (at temperature T)

of the ith

scattering mechanism.
Assuming that the thermal resistivities wi and
electrical resistivities, pi, are related by the Wiedemann-

Franz Law

 

 

p.
wi = LlT
o
W2K2
where L0 = é , then the above equation can be written
Be
in the form of the Nordheim-Gorter relationship
Zo.S
. _ l i
S(d1ff) — 281

For the case of copper—nickel alloys, this relationship

becomes

4

pNiSNi + pThSTh

S(diff) = p p

p
Th
SNi +' s (STh ‘ SNi)

where p = + pTh’ SNi(T) and STh(T) are the characteristic

pNi

thermopowers at temperature T due to the nickel and thermal
scattering mechanisms, and pNi and pTh are the electrical
resistivities due to nickel and thermal scattering.

Since SNi(T)’ S T), and pTh are constants (at a

Th(

given temperature), and since 9 = + could be ex-

pTh pNi
pected to vary linearly with concentration, a plot of

S(diff) versus %-would be expected to give a straight line
with the intercept on the S(diff.) axis equal to sNi.

2. Phonon drag contribution £2 thermopower. For

 

 

the sake of simplicity, let us assume that the energy con-
tained in the vibrating lattice is in the form of quantized
”packets” of energy which we will call phonons. Our prob-
lem now is to determine the manner in which the phonon-
electron interactions will affect the thermopower.

At very low temperatures (< 50K) we can assume,
according to MacDonald (1962), that the number of phonon-
electron interactions will be much greater than the number
of phonon-phonon interactions and that only one type of

phonon-electron collision (normal) takes place.

MacDonald suggests that, under these conditions,
the phonons may be compared to the molecules of an ideal
gas; that is, a phonon energy density U(T) will exert a
pressure

_ .1

on the conduction electrons. If a temperature gradient

g; is present then there will be a pressure gradient %%-,

giving rise to a net force per unit volume

 

F = _ s2 = _.l GU = -.1.2!.§T = _.; c .92
x dx 3 dx 9 0T dx 9 g dx
on the conduction electrons. Here, Cm =-%¥ is defined
ck

as the lattice Specific heat.

This force on the conduction electrons will give
rise to an electric current which is proportional to, but
in the opposite direction from, the temperature gradient.
This contribution to the total current is called the phonon
drag current.

If an electric field Ex is placed across the con-
ductor so that no phonon drag current is allowed to flow,

then we can say

d
d}

I£1

 

NeEX = -FX =

\JJI I—’

CC

l"

where N is the number of conduction electrons per unit

volume.

Therefore, the phonon drag contribution to the

thermopower can be written

EV Cg

SS: 1. 2......—
dT;dX 3N8

The preceding derivation assumes that only one
type (normal) of phonon-electron collision occurs. How-
ever, a different type (Umklapp) of phonon—electron col-
lision must also be taken into account. The two types of
collisions are discussed below.

In the normal type interaction, the change in the
electron wave vector 5 is just equal to the wave vector 3
of the phonon which is emitted or absorbed (see Fig. 1).

That is, the normal process conserves not only energy

I _ - 1
Ex ' Ex ‘ i “ Cs9-

but also momentum

§'-i=is

h2K2

2m

Here, E

 

K = and Cs is the speed of sound in the

material.

Dekker (1957) points out that since the energy of
the phonon is nu0.01 e.v. and the energy of the electrons
on the Fermi surface is a» several e.v., we can conclude
that the energy of the electron will remain (nearly) con-
stant even though it may be scattered through a large
angle; that is, scattering occurs between points on

(approximately) the same Fermi surface.

The situation shown in Figure l is one in which a
phonon having wave vector q is traveling (roughly) from
left to right. his corresponds to the physical situation
in which the left end of the lattice is hotter than the
right end, thus giving the phonons a net drift velocity to
the right. A collision of the type shown in Figure 1 re-
sults in the electron gaining momentum* in the +x direction
(to the right), thereby making the cold (right) end more
negative.

This small increase in the momentum of the electron
results in a small negative contribution to the thermo-
power.

The second type of phonon—electron interaction is
known as the Umklapp process. In this interaction, the
electron wave vector, E, is not conserved. The momentum
equation for the Umklapp.process is (see Figure 2)

K - K' = i.g +.g

where g is a reciprocal lattice vector.

In the Umklapp collision shown in Figure 2, both
the electron and phonon are going (roughly) to the right
(again, toward the cold end of the lattice) before the

collision, but the interaction changes the direction of

the electron, sending it to the left (toward the warm end).

 

*
This is not as obvious as it seems at first
0 * O O 0
Since m may change during the interaction.

CO

 

ILKV /{M5v
f
‘l’f '3 ””75!“
£1
_,. XX

 

 

 

 

 

Figure 1: Normal phonon-electron interaction (Schematic)

/\’r

A

 

 

 

 

 

 

 

 

 

Figure 2: Umklapp phonon-electron interaction (Schematic)

KO

This large change in the momentum of the electron
and the reversal of its sign results in a large positive
contribution to the thermopower.

We have found that Sg (normal) and Sg (Umklapp)
give contributions of differing magnitudes and signs to
S g(tota1). The problem is further complicated by the
fact that, except at extremely low temperatures, phonon-
impurity scattering and phonon-phonon scattering is also
important.

MacDonald suggests that all of these effects can be
accounted for reasonably well if we modify our simple low
temperature expression for Sg to take account of the re-
laxation times of the various processes.

T
Cg o
jNe T + T
o pe

Sg =

 

where Tpe is the relaxation time for phonon—electron inter-
actions and To is the relaxation time for all other inter-
actions.

At relatively high temperatures, T > 0 the ex-

D)
pression for Sg can be analyzed as follows: Since Tpe is
approximately constant and is much larger than To, and

. I
Since TO m-T, we can say

To 1
Sg a-?—— a-T for T > 0

D
pe

10

This suggests that Sg decreases as 7%- at higher
temperatures, which is roughly correct. However, it gives
a numerical result of several uv/OK at room temperature,
whereas experiments show that Sg'z O at room temperature.
IacDonald resolves this discrepancy by assuming that the

normal and Umklapp processes cancel each other out at

higher temperatures.

B. Electrical Resistivity

 

1. Characteristic temperature, 9 The variation

R.

of electrical resistance with temperature is given by the

 

Bloch-Grflneisen formula as

.22.
e

R)

R = K G(-,—f—

2

R

where G is a universal function of T having the properties

that G goes to l as T goes to m, and G goes to 497.6 (2%)
9

R

as T goes to O. 9R is the associated characteristic

temperature and K is the phonon-electron interaction con—
stant. The Bloch—Grflneisen formula is valid only for pure
metals since it assumes that the Debye Model of Specific
heat is valid.

Since K can be neither calculated nor measured very
accurately, it is necessary to eliminate it before 9R can

.7

be calculated. If we follow method 9 given by Kelly and

ll

MacDonald (1953) and assume that 9R is constant, then we
can differentiate the Bloch-Grflneisen formula to obtain

dR/dT _ 1
‘77—“ “

+ d log G

 

Q C
'R
d 10s (—T——)

Kelly and MacDonald calculate the value of the right hand
9

side of this equation and plot it against-fig. From this

graph, one can easily find the value of GR corresponding

to any experimentally found value of-9%é%$

2. Temperature dependence of the electrical resis-

 

 

tivity. In the limits of very high and very low temperatures,

the Bloch-Grflneisen formula becomes

R-——+ (3%)T as T-——+ w

9R

R ——> («E-{6w5 as T ——+ OOK
9R

Assuming that K and 9R are constants, we would

expect R to be directly proportional to T at high tempera-
5

tures and proportional to T at low temperatures.

Note that the low temperature formula is extremely

. . , 6
R Since it contains GR.
Also, the assumption that K is constant at low temperatures

sensitive to any variation in Q

is doubtful.

12

III. EXPERIMENTAL PROCEDURE

The work accomplished on this program can be
divided into four general areas:
A. Production of alloy samples.
B. Construction of cryostat.
C. Measurement of thermoelectric voltage and
resistivity.

D. Analysis of the data.

The first three of these areas will be discussed

below:

A. Production of Alloy Samples.

1. Production of alloys. The alloys were produced

 

by melting weighed amounts of copper1 and nickel1 in an

3

induction furnace under a vacuum of 10‘ mm of mercury or
better.

Two of the alloys (0.85% and 17.10% nickel) were
melted in a vycor crucible while the third (5.45% nickel)
was made in an alumina1 crucible. The pure copper sample

could be made directly from the copper rod without melting.

Graphite was not a suitable crucible material because

 

1The sources and chemical analyses of materials

are given in Appendix I.

|_J
\fi

nickel forms a carbide. In general, the two types of
crucibles worked equally well although the vycor crucibles
sometimes broke while heating.

After the copper and nickel were outgassed and melted
under vacuum, the alloy was chill cast by pouring it into a
heavy copper mold having a cavity about one inch deep by
three-eighths inch in diameter. These castings were re-
melted in the induction furnace and re-poured (this time
into a mold about one—quarter inch in diameter) in order to

produce more homogeneous alloys.

2. Production of wires. Wires on the order of one

 

meter long by 0.02 centimeter in diameter were produced by
rolling the above mentioned castings into rod-like forms
about one-eighth inch in diameter and pulling these rods
through a series of about thirty steel and diamond dies.
The wires were etched lightly in nitric acid after every
third die to remove surface impurities.

The resistivity sample wire was wound on its vycor
rod (about two inches long by one—quarter inch in diameter)
and both sample wires were placed in a one—half inch
diameter vycor tube which was then evacuated and sealed.
This tube was placed in a platinum wound furnace and the

alloys were annealed2 and cooled slowly to room temperature.

2Details of the melting and annealing procedures
are given in Appendix II.

14

B. Construction of Cryostat.

The cryostat was designed so that the thermoelectric
voltage and resistivity of an alloy could be measured
simultaneously from 4.2OK to 3000K. The high cost of liquid
helium and the relatively long time required for the helium
run (about 8 hours) required that heat leaks into the system
by kept to a minimum.

Basically, the cryostat (see Fig. 3) consists of a
double-dewar in which is suspended an insulated copper
block containing most of the important elements (thermo-
meter, sample junction, heater) of the system. The copper
block, which can be electrically heated, has a thin-walled
stainless steel tube attached to it and extending down into
the liquid helium or nitrogen. This copper block and its

stainless steel tail is called the sample holder.

 

1. Design of the sample holder. The cryostat was
designed around a copper block (see Fig. 4) containing
inserts for the resistivity sample, platinum resistance
thermometer, carbon resistance thermometer, and the hot
junction of the thermoelectric voltage sample. Other im-
portant features of the block are (1) a projection around
which a heater is wrapped, (2) horizontal and vertical
grooves to simplify the wiring, (3) three small holes in
the base through which leads may be run, and (4) a threaded

base.

 

 

 

\\\"\W m L\

 

 

 

 

/////

 

 

 

 

 

 

 

 

 

Figure 3:

 

 

 

 

 

 

 

 

Cryostat

‘F/////00pper collar

1/

/1L///'stainless steel
tubing

’Z/’//’ styrofoam

co er can
1 / pp
co er block
1' / pp

Bakelite disk

 

///,/-stainless steel tubing

V/,/’ 21 pin plug

16

 

 

 

 

I \Ntflaflgdw. a»)... c 6 «ms 2t:

 

 

.w\

\IL

 

 

 

 

 

 

gill

HI I. Ill-III Hh.ku\ km‘l\“.~>o.w £MMQIQNN. L

 

 

A

 

 

\WN
M

ti
‘1

JL

’-’

JG

Cover for Jimp/e fig"

53. mp/c Aé/c/cr

Copper Block

Figure 4:

17

A copper ”can” or cover fits over the block and
screws onto its threaded base. This cover, with a one-inch
thickness of styrofoam over it, provides an approximately
isothermal enclosure.

Each of the electrical leads entering or leaving the
enclosure is wrapped around its assigned horizontal Slot
six times in order to insure that the leads, copper block,
and thermometers are all at the same temperature. The
vertical grooves, which are deeper than the horizontal
ones, are used to carry the wires from the horizontal
grooves to the active elements. With this arrangement,
any lead may be removed or replaced without disturbing any

of the other leads.

2. Wiring of the sample holder. The stainless

 

steel tube, previously called the tail of the sample holder,
serves two purposes: (1) it provides a good place to attach
a 2l-prong plug, a resistance heater, and the cold-junction
of the thermoelectric voltage sample; (2) it provides a
heat leak from the copper block from and to the liquid
helium or nitrogen. These two purposes will be discussed
below.

It is very helpful to be able to disconnect the
sample holder from the measuring circuit so that it may be
taken to a workbench for the purpose of changing samples or

making repairs. In order to avoid producing unwanted

thermoelectric voltages at these connections, the plug must
be put into the circuit at a place where there will be no
temperature gradient across it. It was convenient for us
to put the plug at the bottom of the stainless steel tail
where it would be covered with either liquid helium or
liquid nitrogen during the entire experiment.

In the original design, all leads from the measuring
circuit went into the dewar, down to the 2l-prong plug, and
then up (taped to the tail) to the copper block. Later, in
the interest of conserving liquid helium, the leads to the
top heater were brought directly into the block without
going through the plug and the liquid helium.

The cold—junction of the Lead—alloy thermocouple is
made at one pin of the 2l-prong plug. It is interesting to
note that having this arrangement eliminates the need for
one of the two Lead wires in the Lead-alloy-Lead system.
That is: instead of a copper-Lead-alloy—Lead-copper
system, we used a copper-Lead-alloy-copper system, where
the Lead wire at the cold-junction could be omitted since
the entire length of it would be at 4.20 or 770K. This
arrangement is especially useful since Lead wire is soft
and easily broken.

A five hundred ohm resistor attached near the bottom
of the tail is used as a heater to evaporate liquid nitrogen

left in the bottom of the dewar after an experiment or to

lower the level of liquid nitrogen during the experiment
when necessary.

In agreement with accepted practice, the resistivity
sample, platinum resistance thermometer, and carbon resis-
tance thermometer each have two current and two voltage
leads. The only other leads going into the dewar are the E
two copper leads to the Lead-alloy thermocouple and the two ‘

pairs of leads which supply current to the two heaters.

 

If the tail did not provide a heat leak from the

copper block to the liquid helium or nitrogen, the block
would lose heat so slowly that it would be extremely dif-
ficult to reach thermal equilibrium at temperatures near
that of the cold-junction. In fact, it was found that
additional heat leaks were necessary when the block was
less than BOOK above the temperature of the cold-junction.
The extra heat leaks were provided by hanging short lengths
of No. 50 gauge copper wire from the copper block down into
the tail of the dewar. The lengths of these wires were
chosen so that they were long enough to reach into the
liquid nitrogen or helium at the start of the experiment
and short enough to be out of the liquid by the time the
temperature of the block had been raised 300K.

When the temperature difference between the copper
block and cold-junction was greater than 300, enough heat

leaked down the tail so that good temperature stability was

20

relatively easy to maintain. We found that the temperature
stability improved as the temperature of the copper block

increased.

C. Measurement of Thermoelectric Voltage and Resistivity

1. Design of the measuring circuit. Both the

 

thermopower and resistivity measurements required that the
temperature of the copper block be known within O.lOK
throughout the range from 4.20 to 3000K. This temperature
measurement was accomplished by using a C.R.T.5 in the
range from 4.2OK to 160K and a P.R.T.3 in the range from
look to 3000K. Since we wanted to limit the power put into
the cryostat to microwatts, it was decided that the C.R.T.
(R4.2 A/lO,OOO ohms) current should be 10 microamps and

the P.R.T. (R r~'30 ohms) current should be 2 milliamps.

275
Using a current of 2 milliamps for the resistivity

sample (R ,V 1 ohm) and putting the appropriate shunt

275
across the C.R.T. enabled us to use the same current source,
a six volt storage battery, for all elements (see Fig. 5).
The 2 milliamp current is regulated manually with a O—2OO
ohm heliopot and is monitored throughout each run by a L

and N type K3 potentiometer which reads the voltage produced

across a one ohm standard resistor.

 

3C.R.T.
P.R.T.

Carbon resistance thermometer.
Platinum resistance thermometer.

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Bringing the potential leads of the C.R.T. and P.R.T.
to one input (No. l) of the three—input Cambridge Microstep
potentiometer used in conjunction with a photocell amplifier
makes it possible to determine the temperature to within O.lOK.

The potential leads of the resistivity sample are
connected to input No. 2 of the Cambridge Microstep potentio-
meter. Current reversing switches are used to eliminate
unwanted thermoelectric voltage from the P.R.T. and resis-
tivity readings.

The voltage readings of the Lead—alloy thermocouple
are measured on input No. 3 of the Cambridge Microstep
potentiometer. A reversing switch allows for the change
in Sign of the thermoelectric voltage (needed only for
”pure” copper), and a shorting switch is used at intervals
to insure that the voltage being read is really coming
from the Lead—alloy thermocouple, rather than from other

sources such as the potentiometer junctions.

2. Procedure used in making measurements. The

 

 

thermoelectric voltage and resistivity of each of the four

sets Of samples was measured from 4.20 to;3‘lOOOK with the
cold-junction in liquid helium and from 770 to zaooox with
the cold junction in liquid nitrogen.

Initially, the liquid helium or nitrogen partly
covered the copper block, thus insuring that both the copper

block and the cold—junction were at either 4.20 or 770K.

This provided a good preliminary check on all measurements.
The temperature of the copper blo R was raised in
steps of 30 by putting power (controlled by a variac) into
the top heater. After bringing the system to thermal equili-
brium at each temperature we would make the following
measurements: P.R.T.--thermoelectric voltage-~P.R.T.--
resistivity sample voltage--P.R.T.--C.R.T.u——P.R.T. The
current was reversed after each P.R.T. and resistivity

reading, and each set of readings was averaged.

 

4

C.R.T. readings were taken only from 4.20 to l60K.

IV. RESULTS

A. Thermopower

 

The thermopower of pure copper was found to be
positive above 270K and have a local maximum of 1.25 pv/OK
at about 600K. This result has been well established by
other investigators [Blatt and Kropschot (1960); Gold,
MacDonald, Pearson, and Templeton (1960)], and our data
agrees closely with their published findings. The peak
at 600K is attributed to the Umklapp phonon drag contribution.

We found' that the thermopowers of the alloys were
negative at all temperatures and increasingly negative with
increasing concentration (see Figure 6). There appears to
be a positive phonon drag contribution of about 1/2 pv/OK
at 600K in the 0.85% nickel sample as would be expected
from the copper—zinc results. In contrast to the copper-
zinc results, however, there did not appear to be any phonon
drag contribution to the thermopower of either the 3.45% or

17.10% nickel sample.

:8. Electrical Resistivity

 

1. Characteristic temperature, 0R. With the ex—

 

ception of the 17.10% nickel sample, which did not give

25

 

 

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26

meaningful 0 results* it was found that (l) 0 for a given

R R
alloy is about constant or increases slightly with tempera-
ture and (2) 9R at any given temperature decreases with

concentration (see Figure 7).

Values of QR at 2800K are given in Table l for both

 

 

the c0pper—nickel alloys and the corresponding copper-zinc E
alloys.
t
2. Temperature dependence of the resistivity. The
resistivity versus temperature curves for the four samples E

are shown in Figure 8. The resistivity was found to be
approximately linear with temperature at high temperatures
and (with the exception of the 17.10% nickel sample)
roughly proportional to T5 at low temperatures. The log
(D - 00) versus log T curves from which the power of the
temperature dependence was calculated, are shown in
Figure 9. The slopes of these curves over the region of
interest, 150K to 300K, are listed in Table 2.

Values of the residual resistivities of both the
copper-nickel and copper-zinc alloys and also the resis-
tivities of the copper-nickel alloys at 2800K are listed in

Table 5.

 

*-
GB for the 17.10% sample was 2300 at 600K, rose

to a peak of 3000 at 1000K, and then fell to 1550 at 2800K.

, . , a . . dR
Since 0H is extremely oGHSltlve to ET”

error in these results is probably due to a relatively small

the large (assumed)

error in the resistivity versus temperature data.

27

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(25372 /

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ll JLILJllllll
4O 80 120 160 200 240 280
Temperature (OK)
Figure 8: Ideal Resistivity versus Temperature

for Copper-Nickel Alloys.

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50

Table 1

0R (2800K) values for copper-nickel

and copper-zinc alloys.

 

 

0R (2800K) in OK

Sample (% Ni or Zn) Cu-Ni Cu-Zn
Pure copper 580 “[570
0.8595 321 N 557

3-45% 273 rv332

17.10% -- r¢274

 

 

Table 2

Slope of log(p-po) versus log T plots
for copper-nickel alloys.

 

 

Sample Slope (Ll Ohm-Cm)
1
Pure copper 4.8 i 0.5
0.85% Ni 4.6 i 0.5
5.45% Ni 4.1 i 0.3
17.10% Ni 2.9 i 0.3

 

 

Table 3

Electrical resistivity of copper-nickel

and copper-zinc alloys.

Resistivity (a Ohm-Cm)

 

 

 

Cu-Zn Cu-Ni
Sample (% Ni or Zn) p4.2 p4.2 0280
Pure Copper 0.002 0.0058 1.59
0.85% a40.225 1.06 2.70
5.455% 250.750 4.29 6.12
17.10% Q52.7O 19.46 21.59

 

 

 

 

32

From these values we find that, for ”pure” copper

9 so
DQLO 159/ = 418

'EETE "070038
In order to determine the effect of work hardening
due to wrapping the resistivity sample on the quartz rod,
two identical copper samples were prepared exactly as the
sample had been. One was annealed before being wrapped on
its quartz rod, and the other was annealed afterwards. The
residual resistivities of these test samples were found

to be

 

p4.2 (annealed after winding) :(2 60)_1

p4.2 (annealed before winding)

Therefore, the ”true” resistivity ratio of our ”pure”

copper was
g—E—Eg—g = 418 x 2.60 = 1087.

This value compares favorably with the value given by

O7
Schroeder and Henry for A. S. and R. copper (ggig = 1316

after annealing in argon) and by Blatt and Kropschot

R . " R) R .___I _ R2,-

( 27% ('2 = 185 unannealed and 27% ‘°2
“4.2 4.2

 

 

= 540 annealed).

Another interesting finding was that both the 0.85%

and 3.45% samples had resistivity minimums of about
-6 1 707 ' '
0.01 x 10 ohm - cm at about 19 K. Neither the pure copper

nor the 17.10% sample had a resistivity minimum.

A plot of S(28OOK) versus 1 showed that the
(‘7 O
o(2o0 K)

 

points for the pure copper, 0.855 nickel, and 5.45% nickel
samples lay on a straight line as predicted by the Nordheim-
Gorter relationship (see Figure 10). The point correspond—
ing to the 17.10% nickel alloy was not expected to fall on
the straight line since it is a concentrated alloy.
Schroeder and Henry (1965) found that a similar plot for
copper-zinc was linear up to 12% zinc.

The extrapolated straight line in the S versus-%

graph intersected the S-axis at

LIV
OT;-

IX.

 

WOT _
sNi(280 x) _ -l9.0

    
   
 

0.85% Ni

5.45% Ni

-22 h

-24 _

 

-26 — 0 17.10% Ni
L

re Copper

 

t a _ l 1 I l
0 0.1 0.2 0.5 0.4 0.5 0.6 0.7
1 (Ohm—cm x 10'6)‘1

 

p280

Figure 10: Thermopower (2800) versus restivity-l (2800)
for Copper—Nickel Alloys

 

01
UI

V. ANALYSIS OF RESULTS

A. Thermopower Results

 

1. Relation between the measured thermopower and

 

the absolute thermopower of the alloys. Consider a simple

 

xperimental setup consisting of the sample wire A and lead
wires B (see Figure 11). If we attempt to measure the
thermoelectric voltage V34 across A due to the temperature
difference T2 - Tl’ we will actually measure the thermo-
electric voltage produced by the entire system (A + B).
If A and B are the same material then, by simple

conservation arguments, we must have V34 = 0.

If A and B are different materials, we will measure

 

2 r dVB dVA a
‘74:] "T'T IdT
) T L_Cl d J
l
2T2
=k/T [SB - SA] dT
1
so that v32+ > 0 if (SB - SA) > 0
If T1 is kept constant then
dV54
Smeasured : dT = SBJT2 - SA]T2 : SB - SA

It is obvious that either SB or SA must be known

before the other can be determined. Since the absolute

thermopower of Lead has been determined by Christian,_gt a1.

Figure 11:

72
5
3
A
f
5
7f
T dV dV
2 B A
V54 = fl. [dri— ‘ 'aT—JdT
l
dvja
S(measured) = —dT_ = SB — SA

Schematic Drawing of Thermopower Experiment

\H
N

(1958), it was decided to use Lead lead wires to our
thermopower samples. Therefore,

S(A) = S(B) - S(measured)
becomes

S(Alloy) = S(Lead) - S(measured)

It is interesting to note that S(Lead) is negative

at all temperatures while S(copper) is positive except
below 200K. We found that S(alloy) was always negative for

our copper—nickel alloys.

2. Summary of the copper—zinc thermopower results

 

of Schroeder and Henry_(1963). Since nickel and zinc are

 

the two immediate neighbors of copper in the periodic table,
it is reasonable to expect that the thermopowers of copper-
nickel alloys may resemble those of copper-zinc alloys. For
the purposes of comparison, a short summary of the Schroeder
and Henry copper-zinc results follows (see Figure 12).
Schroeder and Henry found that a positive phonon
drag component of thermopower existed for all concentrations
of zinc. The main features of their data were:
(a) There exists a phonon drag peak between 100 and
800K followed by a curve of positive curvature.
(b) .The magnitude of he peak decreases with con—
centration up to about 10 atomic % zinc, and then increases.
(0) The position of the peak shifts to lower tem-

peratures as the concentration increases.

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The initial decrease in the phonon drag peak was
attributed to increased scattering which gives a lower TO.
The resurgence of the peak for concentrations higher than
10 atomic % was attributed to the appr ach of the Fermi

surface to the €200J face of the Brillouin Zone.

.7

5. Expected relationship between the copper-nickel

 

and the copper-zinc results. The degree to which the

 

copper-nickel results should resemble the copper—zinc

 

results depends on how closely (l) the value of To and (2) a
the shape of the Fermi surface in copper-nickel resembles
the same quantities in copper-zinc.

The three main factors which affect the impurity
scattering, and therefore, T0, are:

(a) Mass of the impurity atom: Since nickel, copper,
and zinc have atomic numbers of 28, 29, and 30, reSpectively,
we can say that M(Ni) 2:M(Zn) and, therefore, nickel atoms
should give about the same amount of scattering as those of
zinc.

(b) Distortion of the lattice: It has been found
that nickel atoms distort the lattice less than zinc atoms.
This would lead to less scattering and, therefore, to higher
To and higher Sg for nickel than for zinc.

(c) Inter—atomic forces: The inter-atomic forces

of copper-nickel are approximately equal to those of copper—

zinc since 9D (Cu — Ni) 239D (Cu - Zn) as shown by Guthrie,
EE.§ln (1965) and Rayne (1957).

Each of the factors listed above indicate that To,
and therefore Sg, should be at least as large in copper—
nickel alloys as it was in copper-zinc. Therefore, we would
expect phonon drag peaks in copper-nickel alloys to be very
prominent for all of our samples.

The other factor having an important effect on 83
is the shape of the Fermi surface. If we assume that the
holes in the unfilled 5d band of nickel take free electrons
away from copper atoms (when nickel is added to copper)
then the energy of the copper atoms is reduced-- that is,
the Fermi energy of a copper-nickel alloy decreases as
nickel is added. Therefore, we can say that the Fermi
surface should shrink and pull away from the Brillouin Zone
as the concentration of nickel is increased. This process
would produce a situation highly favorable to Umklapp inter—
actions giving large changes in electron momentum (see
Figure 13) which, in turn, would produce a large positive
phonon drag contribution.

It would seem, then, that any changes in either T0
or the shape of the Fermi surface would be such that the
phonon drag peak in copper-nickel would be greater than

that in the corresponding copper-zinc alloy.

 

 

f
,, J
7i
i:
£9:
5d.
7

Figure l}: Fermi Surface which has pulled away from the
Brillouin Zone boundary (Schematic).

 

 

 

 

 

 

42

4. Discussion of copper-nickel thermopower results.

 

A comparison of the copper-nickel and copper—zinc thermo-
power results shows two striking differences: (1) Copper-
nickel alloys have large negative diffusion thermopower,
while those of copper—zinc are small for all concentrations,
and positive for concentration less than 8% zinc. (2) Only
the 0.85p nickel alloy showed a phonon drag component of
thermopower, while all of the copper-zinc alloys showed
phonon drag peaks.

What are the possible causes of these large dif-
ferences? As discussed in the preceding section, they cannot
be attributed to changes in T0 or the shape of the Fermi
surface, since a change in either would be expected to
increase the positive phonon drag contribution. Hopefully,
any explanation of the thermopower differences would also
explain the fact that nickel atoms in copper increase the
resistivity much more than an equal number of zinc atoms.

One place to look for a possible explanation is the
electronic structure of the nickel and zinc atoms. The

8

outer electron configuration of atomic nickel is 3d 432
while that of zinc is Bolouse. The most striking dif—
ference in the configurations is that nickel has vacancies
or ”holes” in the Bd-band while zinc does not.

It is reaSonable to assume that if the holes in the

Bd-band of nickel are not filled when it is alloyed with

copper, then many of the As electrons from copper atoms
would undergo collisions with phonons and be scattered into
these empty energy states (s—d scattering). This would
(1) ”use up” many of the phonons so that they could not
contribute to the phonon drag peak, and (2) lower the mean
free path of the conduction electrons, thus increasing the
resistivity.

At first glance, it would seem highly improbable
that there would be any Ed—band holes unfilled in alloys
having as little as 4% nickel. As pointed out by Coles
(1952), the most important experimental evidence to support
the belief that the Ed-band holes of copper-nickel alloys
are filled for nickel concentrations of less than #0 atomic
per cent is provided by a study of the ferromagnetic pro-
perties of these alloys. Both the saturation magnetic
moments, o, and Curie temperatures, QC, of copper-nickel
alloys decrease linearly with copper concentration and
extrapolate to zero at 60% copper (see Figure 14). It
should be remembered, however, that there are twp_conditions
which must be satisfied in order to have a ferromagnetic
material: (1) buried vacancies in the d-band and (2) correct
lattice spacing. It is possible that the disappearance of
ferromagnetism above 60% copper is due to the second, rather
than the first, of these reasons.

Coles goes on to say that there is a great deal of

experimental evidence which can only be explained by

44

 

 

 

 

 

22
O
O
‘3 ooo -
400 —
r.
/
200 -' / /, - 0.1
’/
._ //
//
0 II 1 W1 11 I l
O 20 40 60 80 108

Atomic % Ni

Figure 14: Saturation Magnetic Moment, 0, and Curie
Temperature, QC, for Copper-Nickel Alloys.

From Coles (1952)

“KM4nu-u- n:- q."

0(Bohr Magnetons per Atom)

assuming that Bd—band holes exist in copper—nickel alloys
having concentrations as low as 5% nickel.

The most striking evidence that Bd—band holes exist
down to the very low concentration of nickel comes from an
examination of the paramagnetic properties of copper-nickel
alloys. We know that the presence of d-band holes in non—
ferromagnetic materials is indicated by high paramagnetic
susceptibilities, roughly inversely proportional to tempera-
ture at high temperatures. When all the d-band states are
occupied, diamagnetism is expected; the weak temperature
independent paramagnetism of the conduction electrons being
insufficient to overcome the diamagnetism of the full inner
shells.

In order to get a clear example of how well some ex~
perimental results agree with this theory, it is instructive
to look at the silver-palladium alloy system. Silver-
palladium resembles the copper-nickel system in that (l)
palladium precedes silver in the periodic table just as
nickel precedes copper; (2) silver and copper have ”similar”

104s1 and AleBSl, respectively;

electronic structures--jd
(5) palladium and nickel both have closed ”outer” shells,
namely Adlo and 5d84s2, respectively; (A) all four of the
elements are face centered cubic; and (5) both alloy

systems show complete solid solubility.

f

40

The room temperature magnetic susceptibility of
silver-palladium alloys shows exactly the behavior to be
expected if all the d-band holes have disappeared when a
silver concentration of about 60% silver has been reached
(see Figure 15). The atomic susceptibility of copper-
nickel alloys, however, is quite different. The suscepti-
bility is negative only up to about 5% nickel and is
increasingly positive for higher concentrations of nickel
(see Figure 15). This strongly suggests that Bd-band
holes exist for concentration as low as 3% nickel, and
possibly lower.

Another indication that the 5d-band holes are vacant
at relatively low concentrations of nickel is given by the
electronic specific heat. This specific heat can be expressed
as C8 = 7T where 7 is proportional to the density of energy
states at the Fermi surface. The high density of states
in the unfilled Bd-band gives transition metals a high
specific heat (compared to that of copper). The values
found for the specific heat of various copper—nickel alloys
shows that Ce goes from a low of A20.2 for pure copper to
n«0.5 for 22% nickel and levels off at A.1.7 (units of
103 cal/degB/mol) for concentrations above 40% nickel.

This suggests that the Fermi surface is in the Ed-band for
22% nickel, and, therefore, vacancies may exist at this
concentration. There is no data between pure copper and

22% nickel (see Table A).

47

 

 

 

 

 

 

500 -'

400 _

‘9: 300 —-
r—-{

K —
(D
4.)

'5 200 -
5

U; h—
s1

O 100 -
.CU

>< ._

o
)—
-1oo -
I ll J l l J l l l
o 20 40 60 80 100

Atomic % Nickel or Palladium

Figure 15: Atomic Susceptibility of the Copper-Nickel
and Silver-Palladium Systems.
From Coles (1952)

Table 4
Specific Heat Constants for

Copper-Nickel Alloys
From Coles (1952)

 

% Nickel 7(103 califiecg/mol)
100.0 1-7“
81.61 1-58
61.97 1'52
42.07 1.66
21.58 0-457
0 0.178

 

119

In agreement with the susceptibility and Specific
heat data, a comparison of the resistivity results for
copper—nickel and silver-palladium also suggests the
presence of unfilled Bd-band vacancies at low nickel con—
centrations. The resistivity of the silver-palladium system
has been shown by Mott and Jones (1956) to be due to s-s
scattering over the entire range of concentration and an
added component of s—d scattering in the alloys having
more than 40% palladium. That is, s-d scattering was found
only in the range where 4d-band vacancies were thought to
exist, as expected. In contrast, the resistivity of
copper-nickel alloys suggests that, if a s—d scattering
component does exist, it exists over the entire range of
concentration (not just above 40% nickel where the alloy
is ferromagnetic).

To quote Coles (1952):

Although a simple band model and collective electron
treatment are appropriate in the treatment of all
available data on silver-palladium alloys, they fail
entirely to account for the effects found in copper-
nickel alloys containing less than 40% nickel. In
fact, all physical properties of copper-nickel alloys
are in agreement with each other in indicating the
presence of jd—band holes in alloys having as little
as rv5% nickel.

Although it is obvious from the experimental evidenCe
given above that the conduction electrons ”think” that there
are vacancies in the Bd-band, it is extremely difficult

to find a model which satisfies all of the known conditions.

For instance, any conduction model which results in the

correct optical energy levels will not lead to the correct
resistivity and susceptibility results. It has been sug-
gested that even at low concentrations, nickel atoms might
form a d—band of their own-_ that is, the energy levels
of the Bd-band nickel atoms would not be modified by the
copper lattice. As yet, no single model can successfully
explain all of the physical properties of copper-nickel

alloys.

B. Discussion of the electrical resistivity results

 

The resistivity of each of the four samples was
found to be approximately proportional to T at high tempera—
tures as predicted by the Bloch-Grflneisen theory. At low
temperatures, however, the temperature dependence differs
from the predicted T5. Pure copper is closest with a TM'8

4'6, 3.45% with Tu'l,

dependence, followed by 0.85% with T

. y , 2.9 . . . . 5
and 17.10% With T . The increaS1ng variation from T
with concentration is not surprising since the Bloch-
Grflneisen theory is valid only for pure metals and makes
several approximations. The AJTB dependence of the 17.10%
sample suggests that another temperature-dependent
mechanism, other than thermal scattering, is present.

The fact that the residual resistivities of the

copper-nickel alloys are significantly higher than those
of the corresponding copper-zinc alloys is thought to be

due to s~d scattering to the jd—band of the nickel.

51

The fact that the data points on the thermopower
versus resistivity"l plot fell on a straight line inter—

. a 0 [V o o o
secting the S-aX1S at -19.0 %—-18 interesting for two
K

reasons: (1) The Kohler relation, which is the basis of
the derivation of the Nordheim-Gorter relation, is valid
for the diffusion thermopower of an alloy. Since our data

at 2800K fall on a straight line when we plot the total

(rather than the diffusion) thermopower against resistivity-1,

we can conclude that the phonon-drag component of the
thermopower is ¢a0 at 2800K. This is in agreement with
other published results (MacDonald, 1962). (2) The inter-
section of the straight line at -19.o gX-means, by defini-
tion, that

SNi(2800K) = -19.o-5—

Gold 22.21: (1960) have found that SNi(1SOK) = —1.1 gX-.
K

If S T) were directly proportional to T, then we would

Iii(
expect that

, o ,
280 K)(-i.i IN) = —2o.5.%X

280) = ( O O
15 K K K

 

 

A comparison of our result and this predicted result shows

that SNi(T) is indeed nearly proportional to temperature.

VI. CONCLUSIONS

The thermoelectric powers and resistivities of a
series of copper—nickel alloys have been measured and
compared to those of copper—zinc and silver-palladium
alloys.

It was noted that the large negative thermopower,
lack of phonon drag peaks in all but the 0.85% thermopower
results, and relatively high electrical resistivity of the
sample alloys were consistent with other experimental
evidence suggesting that Bd-band vacancies exist in alloys
having as little as 3% nickel.

A study of the thermopower and resistivity of silver-
palladium alloys would give additional insight into the

results found for copper-nickel alloys and provide further

information about the Fermi surface of concentrated alloys.

U1
\2.‘

REFERENCES

Blatt, F. J. and Kropschot, R. H. (1959). Phys. Rev._ir§,
617.

Coles, B. R. (1952). Proc. Phys. Soc. London 65, 227.

Christian, J. W., Jan, J. P., Pearson, w. B., and
Templeton, I. M. (1958). Proc. Roy. Soc._2£5, 213.

Dekker, A. J. (1957). Solid State Physics (Prentice Hall,

 

Inc.) p. 291.

Gold, A. V., MacDonald, D. K. C., Pearson, W. B., and
Templeton, I. M. (1960). Phil. Mag. 5, 765.

Guthrie, Friedberg, and Goldman (1959). Phys. Rev. 112, 45.

Kelly, F. M. and MacDonald, D. K. C. (1953). Can. J. Phys.
31, 147.

MacDonald, D. K. c. (1962). Thermoelectrieity (John Wiley
and Sons) p. 91.

Mott, N. F. and Jones, H. (1936). The Theory of the

 

Properties of Metals and Alloys. (The Clarendon

 

 

Press, Oxford). p. 310.

Rayne, J. A. (1957). Phys. Rev. 19§, 22.

Schroeder, P. A. and Henry, W. G. (1963). The Low Tempera-
ture Resistivities and Thermopowers 23.372EEEE

Copper-zinc Alloys. (unpublished)

 

Sources and Chemical Analyses

Material

copper
7'
(§” diam. rod)

nickel
(powder)

alumina
(crucibles)

nitric acid
(used to etch
wire)

Lead
(10 mil wire)

Appendix I

Source

American Smelting and

Refining Co.

_Grade ASARCO A-58

Johnson and Matthey
Cat. No. J.M. 891

McDaniel Refractory
Porcelain Co.
Cat. No. AP35

Fisher Scientific Co.

Cat. No. A-EOO

Comico Electronic
Materials—-Spokane,
Washington

lAdvertised purity.

of Materials

Purity

99.999% Purel
0.01% Silver2
”Trace” Iron2

99.999% Purel
0.04% Iron2
1
99% A1203

o.ooooe% Ironl

99.9999% Purel

2Spectrographic Testing Laboratory, Detroit 12, Michigan
”Trace” = < 0.01%.

Appendix II

Details of the melting and annealing processes

 

Sample Annealing time Original wts. of Cu and Ni
Pure Cu 5 hours at 58000 10 gm Cu
*
0.85% ii 20 hours at 74000 9.29 gm Cu
0.08 gm Ni
* /'
5.45% Ni 20 hours at 7400C 10.05 gm Cu
0.36 gm Ni
*-
17.10% Ni 40 hours at 78000 8.89 gm Cu
1.55 gm Ni

Comments:

The pure copper sample was wound on a 1/4” diameter
copper rod (insulated with mylar) after annealing. All
other samples were wound on a 1/4” diameter vycor rod before
annealing.

All samples were cooled slowly (od24 hours) in the
furnace.

All samples were annealed in an evacuated vycor tube.

*
Spectrographic Testing Laboratory; Detroit 12, Michigan.