iail t ,

ABSTRACT

ELECTRICAL RESISTIVITY OF ALUMINUM ALLOYS
AT LOW TEMPERATURES

BY

Ronald Leon Carter

The electrical resistivities of aluminum and alloys
of aluminum with magnesium, copper, zinc, (n: gallium have
been measured from 2.5°K to 300°K. The importance of cor-
rections for change in volume upon alloying and thermal
expansion in determining the magnitude and temperature
derivative of deviations from Matthiessen's Rule at high
temperatures is stressed. Two hypotheses were found to be
consistent with the deviations from Matthiessen's Rule
measured at low temperatures. The conclusion that the
electron states on the third zone sheet of the Fermi sur-
face have an average relaxation time which is considerably
different from that for the hole states on the second

sheet was drawn from the application of the two—band model.

h
ph'

were shown to be consistent with thermoelectric power data.

The relative resistivities of the two bands, 03h > p

The data were also found to be consistent with a theory
due to Mills, which considers the additional resistivity
for a process which involves simultaneous scattering from

an impurity ion and the emission or absorption of a phonon.

ELECTRICAL RESISTIVITY OF ALUMINUM ALLOYS

AT LOW TEMPERATURES

BY

Ronald Leon Carter

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics

1971

PLEASE NOTE:

Some Pages have indistinct
print. Filmed as received.

UNIVERSITY MICROFILMS

ACKNOWLEDGMENTS

I would like to express my appreciation to Professor
Frank J. Blatt, who first suggested this study, and whose
advice and assistance on theoretical and experimental
matters was invaluable and cheerfully offered. And to
Professor Peter A. Schroeder, who heard many tales of woe
and experimental failure, thank you for your freely offered
help. I also acknowledge Professors Blatt and Schroeder
and the National Science Foundation for financial support
received during this study.

I would like to acknowledge Professor Carl L. Foiles
for his work in doing the resistivity at high pressure
measurements which were useful in the analysis of this
study, and for many helpful discussions on transport meas-
urements. Thanks are due also to Professor W.M. Hartmann
for providing the results of his phonon dispersion calcula-
tions for aluminum before publication.

The efforts of Professors B.H. Wildenthal and
W.H. Kelly in providing computer time on the XDS Sigma
Seven are greatly appreciated for the time their efforts
saved for me.

I would like to acknowledge the help of Mr. J.
Thomas and Miss G. Pucilowski in setting up the computer

programs, and to Miss Pucilowski for doing the drawings.

ii

I am pleased to also acknowledge the help of Mr. B.
Shumaker in the manufacture of some of the alloy samples.
My deepest appreciation is for my wife, Carole, who
accepted many things that were less than they might be,
offered encouragement when needed, typed this manuscript,

and who worked very hard to make this study possible.

iii

TABLE OF CONTENTS

Chapter

INTRODUCTION . . . . . . . .

Matthiessen's Rule . . . . .

Deviations from Matthiessen's Rule

Deviations from Matthiessen's Rule
in Aluminum . . . . . .

EXPERIMENTAL APPARATUS . . . .

Cryostat and Sample Mounting .
Current Control . . . . . .
Temperature Control . . . .
Potentiometric Measurements . .

SAMPLE PREPARATION . . . . . .

Manufacture of the Alloys . .
Forming of Wires . . . . .
Resistance Ratios . . . . .
Annealing . . . . . . . .
Sample Characterization . . .

RESISTIVITY DATA . . . . . .

Introduction . . . . . . .
Room Temperature Resistivity
Measurements . . . . . .
Temperature Dependence of the
Resistance . . . . . . .
Calculating the Resistivity from
the Data . . . . . . .

RESULTS AND CONCLUSIONS . . . .

The Phonon Resistivity of Aluminum
The Deviations from Matthiessen's
in Aluminum Alloys . . . .
The Two-Band Model . . . . .
The Ehrlich Theory . . . . .
The Mills Theory . . . . .

iv

Page

11
13
15
l9
19
20
21
22
23
25
25
25
28
29
33
33
38
45

66
67

Chapter Page

SUMMARY AND IMPLICATIONS FOR FURTHER

STUDY . . . . . . . . . . . . . . 73
REFERENCES . . . . . . . . . . . . . . 75
APPENDICES . . . . . . . . . . . . . . 78

APPENDIX
A. FORTRAN Listing of Computer Program
for Calculating Resistivities . . . 78
B. Typical Data Output . . . . . . . 90
C. Analysis of Errors . . . . . . . . 97

LIST OF TABLES

Table Page
1. Lot Assay for HPM 2404 . . . . . . . . 20
2. Coefficients of the T2 and T5 Components
of the Resistivity of Aluminum . . . . 35
3. Volume Corrections . . . . . . . . . 44

. Residual Resistivity of Aluminum Alloys . . 45

4
C-1. Standard Deviations for Measurements
Using Equation C.l . . . . . . . . 97

C—2. Phonon Resistivity of Aluminum at
273.160K O O O O O O 0 O O O O 100

vi

LIST OF FIGURES

Figure Page

1. Schematic Representation of the

Cryostat . . . . . . . . . . . . 10
2. Constant Current Source . . . . . . . 12
3. Temperature Controller . . . . . . . . 14

4. Schematic Representation of Experimental
Apparatus . . . . . . . . . . . l6

5. Spring Loaded Electrical Contacts . . . . l7

6. Temperature Dependent Resistivity of
Pure Aluminum . . . . . . . . . . 34

7. Deviations from Matthiessen's Rule in
Al—Mg Alloys 0 c o o o o o o o o 39

8. Deviations from Matthiessen's Rule in
Al-Cu Alloys 0 O O O O O O O O O 40

9. Deviations from Matthiessen's Rule in
Al-Zn Alloys 0 o o o o o o o o o 41

10. Deviations from Matthiessen's Rule in
Al-Ga Alloys 0 O O O O O O O O O 42

11. Low Temperature Deviations from
Matthiessen's Rule in Aluminum Alloys . . 43

12. Residual Resistivities of Aluminum

Alloys . . . . . . . . . . . . 46
13. Fermi Surface of Aluminum . . . . . . . 47
14. Kohler-Sondheimer-Wilson Plot at 30°K . . . 53
15. Kohler-Sondheimer-Wilson Plot at 70°K . . . 54
16. Kohler-Sondheimer-Wilson Plot at 295°K . . 55

vii

Figure

17.

18.
19.

20.

21.

22.

23.

Page

KSW Parameters a and b as a Function

of Temperature . . . . . . . . . . 57
Phonon Resistivity of the Hole Band . . . 59
Phonon Resistivity of the Electron Band . . 6O
Backscattering Event on Third Zone

Extremal Cross-Section . . . . . . . 62
Phonon Dispersion Spectrum for Aluminum . . 63
Fit of the Coefficients a and B to the

Ehrlich Theory . . . . . . . . . . 68
Coefficients of the T3 and T2 Terms

as a Function of pO . . . . . . . . 71

viii

INTRODUCTION

Matthiessen's Rule

 

More than a century ago, Matthiessen (1,2) observed
that near room temperature the temperature derivative of
the resistivity of a dilute metal alloy is the same as for

the pure base metal. Thus, the Matthiessen Rule (MR) is,

dpp _ dpA 1.1

dT ‘dT
where T is the absolute temperature, and DP and CA are the
resistivities of the pure metal and alloy respectively.
When temperatures near the absolute zero became
experimentally attainable, it was observed that the limit-

ing behavior of a "perfectly pure" metal was

oP(T=0) o, 1.2

and for an alloy

Where p0, the residual resistivity, appears in the inte-

grated form of the MR:

When Bloch (3,4) and later Grfineisen (5), formu-
lated early theories of the effect of the thermal motion
of the ions (phonons) on the electrical resistivity of a

pure metal, was identified as pph’ the resistivity due

“P
to scattering of the electrons by the phonons. The
residual resistivity, po, was then identified as due to
scattering of the electrons by imperfections in the
lattice. Thus, the MR was given the more general inter-
pretation: the partial resistivities arising from the

scattering of the conduction electrons by different types

of scatterers are additive.

Deviations from Matthiessen's Rule

 

In practice, deviations from the MR are usually
observed in dilute metal alloys. The temperature dependence
of the alloy resistivity is not identical to the temperature

dependence of the pure resistivity. Thus,

(T)

_(T)
0A -0

ph + pO + A(T), 1.5

where A(T), the deviation from the MR, depends on the
nature and concentration of defects and impurities in a
complex manner.

Departures from the MR can arise for a number of
reasons. Some of the most important (for nonmagnetic im-

purities) are:

A1: The addition of impurities may change the
phonon spectrum of the metal.

2: The electronic band structure may be altered
by the impurity.

A3: Scattering from the thermally oscillating
impurities may give a temperature dependent
impurity resistivity (phonon-assisted impurity
scattering).

A4: Interference terms between scattering by the
vibrating impurities and the host ions.

A5: Two (or more) groups of electrons may con-
tribute to the conductivity, giving (as will
be shown later) an apparent deviation from
the MR.

A6: The simultaneous presence of thermally induced
Umklapp processes and impurity scattering.

In this study, dilute alloys were considered; the

maximum impurity content being less than two atomic per
cent (2 a/o). In this case, de Haas-van Alphen measure-
ments (6) support the assumption that changes in the
electronic band structure are negligible. Furthermore,
the results of Massbauer experiments (7,8) indicate that
the phonon spectrum is little changed.

Extensive reviews of the theoretical and experi-

mental work on departures from the MR have been given by

Lengler, Schilling and Wenzl (9), Stewart and Huebener (10).

and Seth and Woods (11). A brief discussion of the theo~
retical points important to this study follows.

Kagan and Zhernov (12) have developed a theory of
the electrical resistivity of a metal with impurities which
takes into account deformation of the phonon spectrum due
to the presence of the impurity ions and the electron
scattering by the thermally oscillating impurity ions. In
the low temperature range, scattering by the oscillating
impurity ions gives a term proportional to poTz, inter-
ference between scattering by the perturbed phonon spectrum
and the oscillating impurity ion gives a term proportional
to poT4, and scattering by the deformed phonon spectrum
gives a term prOportional to poTS. The temperature vari-
ation was shown to exhibit an anomaly in the case of heavy
impurity atoms when a quasilocal level appears in the
phonon spectrum. At high temperatures, the impurity part
of the resistivity was shown to vary linearly with temper-
ature, the sign of the derivative depending on the relative
positions of the impurity (solute) and matrix (solvent)
atoms in the periodic table.

A theory for the phonon-assisted impurity scatter-
ing due to the average strain induced in the lattice by
the thermally oscillating impurity atoms has been formu-
lated by Klemens (13). Assuming the change in the impurity

potential to be prOportional to strain, Klemens showed the

additional resistivity to be prOportional to po<ez>. The

mean-square thermal strain of the lattice, <€2>, is pro-
portional to the thermal vibrational energy of the lattice,
so A3apoT4(T<<6D), and A3¢poT(T26D).

Ehrlich (14) has calculated the low temperature
resistivity for the case for which the Fermi surface touches
or nearly touches the Brillouin zone boundaries, extending
the Klemens-Jackson (15) theory to include the simultaneous
presence of impurity and thermally induced umklapp scatter-
ing. Assuming an isotropic, temperature independent, im—
purity relaxation time, Ehrlich solved the Boltzmann
equation in terms of a Legendre series in the scattering
angle for current-carrying electrons. For the case in
which the phonon resistivity, pph’ in the absence of

5

umklapp processes is proportional to T , the numerical

solution for the total resistivity is of the form
_ 2 S
p(T) - p0 + a(P,€)T + b(P,e)T 1.6

where P = Zoo/pph and e is the angle of contact of the
Fermi surface on the Brillouin zone. The combined effect
of impurity and umklapp scattering processes introduces a
T2 term as well as enhancing the T5 term.

Mills (16) has calculated the resistivity for a
process which involves simultaneous scattering from an
impurity ion and the emission or absorption of a phonon.

3

Breakdown of momentum conservation leads to a T depend-

ence for this resistive contribution. Further, as a result

of the small energy difference between initial and inter-
mediate states for the process, this contribution to A4 is
weakly dependent on the impurity concentration.

Dugdale and Basinski (l7) interpreted the results
of measurements on copper and gold based alloys using a
simple two-band model. Originally proposed by Sondheimer
and Wilson (18) and Kohler (19) the two-band model approxi-
mates anistropies of the relaxation times associated with
phonon and impurity scattering with two non-interacting
conduction bands with isotropic relaxation times.

Deviations from Matthiessen's Rule
in Aluminum

 

 

Aluminum was chosen for this study of deviations
from the MR for three reasons. First, there is reason to
suspect a priori that the sources for deviations A3, A4,
A5, and A6 should be manifest in aluminum alloys. Second,
the cubic symmetry of the face-centered cubic aluminum
lattice permitted the use of wire resistance samples.
Last, deviations from the MR had been previously reported
in aluminum systems (11, 20-24).

The light aluminum ions should experience rather
large amplitude thermal vibrations, so the scattering
effects of thermal strain at impurity sites might be ex-
pected to be pronounced. The relatively light mass of the

aluminum ion should favor creation of in-band local modes

or quasilocal levels in the phonon spectrum upon alloying

with heavy solutes. This might be expected to give the
anomalous impurity resistivity anticipated by Kagan and
Zhernov (12). The reversal of the Hall field in pure
aluminum reported by Luck (25) and Forsvoll and Holwech
(26) from the low field electron-like limit to the high
field hole-like limit supports the hypothesis that aluminum
may be treated as a two-band metal. With this hypothesis,
the Kohler-Sondheimer-Wilson equation (18,19) may be used

to fit the deviations, A from the MR, and thus infer the

5:
relative relaxation times for scattering by impurities and
phonons for electrons in the two bands.

Alley and Serin (20) have reported departures from
the MR in alloys of aluminum with zinc, magnesium, ger—
manium, or silver from 4.2 to 300°K. This work did not
include dimensional measurements, so deviations could only
be inferred and not quantitatively compared with theory.
Van Zytveld and Bass (21) have carefully documented size
dependent deviations from the MR in thin foils and fine
wires of aluminum from 1.3 to above 40°K. Panova, Zhernov
and Kutaisev (22) have reported deviations from the MR in
alloys of aluminum with gold or silver solutes from 4.2 to
300°K, interpreting their results on the basis of the Kagan
and Zhernov (12) theory. Panova §t_gl, did not, however,
study the effects of change in valence or of light solutes.

Recently, Seth and Woods (11) have reported deviations

from the MR in alloys of aluminum with magnesium or silver

from 4.2 to 300°K. Caplin and Rizzuto (23) report devia-
tions from the MR in alloys of aluminum with magnesium,
iron, silicon, cobalt, manganese, copper, chromium, or
vanadium. Campbell, Caplin and Rizzuto (24) later inter-
preted these deviations in light of the Mills theory (16).
In the following chapters, the deviations from
Matthiessen's rule are reported for aluminum with magnesium,

zinc, copper, or gallium solutes.

EXPERIMENTAL APPARATU S

Cryostat and Sample Mounting

 

The cryostat used was of the sample holder-within a
cannister-within a vacuum jacket-within an outer cannister-
design. A schematic representation of the cryostat is
shown in Figure l.

The sample holder was made of aluminum to minimize
the effects of differential thermal expansion between the
samples and the sample holder. The inner diameter of the
hollow cylindrical sample holder was designed for a loose
fit on the oxygen free high conductivity (OFHC) copper
block (to minimize strains due to differential thermal ex—
pansion), with firm thermal contact insured by mechanically
bolting the sample holder to the OFHC block at a single
point. The outside of the sample holder was insulated by
cigarette paper attached with GE 7031 Varnish. Samples
were annealed in a coil slightly larger than the sample
holder diameter, placed on the sample holder, and held in
place by spring-loaded clamps. Thus, strains induced in
the samples due to mounting or thermal cycling were mini-
mized. Platinum and germanium resistance thermometers were
mounted in the thermometer well in the interior of the OFHC

c0pper block for measurement of the sample temperature.

10

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Potential and current leads for the samples and
thermometers were run through the stainless steel inner can
pumping line. These B&S 38 copper wires were thermally
attached to the OFHC copper block and the binding post in
the pumping line (which was provided with a heat leak to
the cryogenic liquid). A second variable heat leak could
be provided by introducing exchange gas into the vacuum
jacket between the inner and outer cannisters. Exchange
gas could also be introduced into the vacuum space in the

inner cannister.

Current Control

 

The sample and thermometer currents were held con-
stant to better than .001% during the period of a measure-
ment (about 30 minutes) using the constant current source
shown in Figure 2. In this configuration, the Operational
amplifier output causes current to flow in the feed-back
loop (the load, R1 and SR) and then through CR, with such a
magnitude that the potential drop across CR matches the
offset potential of the mercury cell, RM-42R. Variation of
CR may then be used to adjust the load current, which may
be measured by the potential drOp across the standard re-
sistor, SR.

The Analog Devices 118A is capable of supplying
currents up to 5 mA. For the low temperature resistivity
measurements, up to 100 mA was required. The current was

boosted using transistor Ql by breaking the feed-back loop

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at X and making the connections shown in dashed lines. In
this way, the available current was limited only by the

power dissipation of 01 and the +15 volt source.

Temperature Control

 

The temperature of the OFHC block was held constant
to within .01°K during a 30 minute period by use of the
temperature controller shown in Figure 3.

A Micro-Measurements Cryogenics Linear Temperature
Sensor (CLTS) was mounted on the outer surface of the inner
cannister with GE 7031 Varnish. The nominal resistance of
the CLTS is 2909 at 300°K and the sensitivity is nearly
constant at .2390/°K down to 4°K. The CLTS is used as one
arm of a bridge with the off-balance voltage amplified by
an Analog Devices (AD) 118K. The AD 118A is connected in
a standard differentiator-integrator control configuration.
The control voltage of the AD 118A is converted to a phase-
controlled pulse by the 2N167lB unijunction transistor
circuit. Large control voltages generate a pulse early in
the 60 cps line voltage half-cycle, while small control
voltages cause a pulse later in the 60 cps line voltage
half-cycle. These pulses are then applied to the gate of
the RCA 40485 triac which then conducts for the remainder
of the half-cycle. Since the triac is in series with a
bifilar resistance heater wound on the outside of the

outer cannister, the control voltage causes more or less

14

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15

average power to be put into the cryostat, causing heating

or permitting cooling of the samples.

Potentiometric Measurements

 

The sample potentials, sample current, thermometer
potentials, and thermometer current were measured in a
circuit schematically represented in Figure 4. In practice,
up to ten samples were connected in series for a given run.
Copper potential and current leads were attached to the
samples using the spring loaded system shown in Figure 5.
The insulated washers were made by attaching cigarette
paper to one side of a brass washer with GE 7031 Varnish.
The insulating coating was removed from the end of the
current or potential lead and the bared c0pper wire placed
between the insulated washer and the sample. The nylon
screw was then tightened compressing the spring, which then
held the c0pper lead rigidly against the sample wire.
Creeping of the lead along the sample was not observed,
as inferred by the fact that the resistance of the sample
at a specific temperature did not change upon thermal
cycling.

The distribution switch for the potentiometers was
made of three Chicago Dynamics Industries two pole-sixteen
position printed circuit switches. This enabled connecting.
the digital voltmeter, K-3 potentiometer, or 6-Dia1 poten-
tiometer independently to any of the potential measuring

_circuits. These switches are of noble-metal construction;

16

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18

and low thermal solder was used for all connections.
Spurious thermal emfs observed upon reversal of the current
were less than .02 uV.

The Leeds and Northrup K—3 potentiometer was used
to measure the thermometer potentials and the potential
drops across the standard resistors in the current supply
circuits. The precision for this instrument is .015% or
2uV whichever is greater in the range used. The Rubicon
2768 6-Dial potentiometer was used to measure the potential
drops across the samples. The precision for this instrument

is .01% or .OluV whichever is larger in the range used.

SAMPLE PREPARATION

Manufacture of the Alloys

 

All of the alloys (except the Mg series) in this
investigation were made from 6-9 grade aluminum, lot HPM
2404, purchased from Cominco American. The lot assay is
shown in Table l. The total impurity level was less than
one part per million (ppm). A piece of lot HPM 2404 was
also used for the pure aluminum sample, as well as some
Gallard Schlesinger B 3015 zone refined aluminum. The
doping level for the alloys ranged from .05 a/o (500 ppm)
to 2 a/o. The doping material used was 5-9 or 6-9 grade
material.

A master alloy (maximum concentration) for each
series was made by R.F. induction melting the solute and
solvent (pure aluminum) in a vitreous graphite crucible.
After about two minutes of mixing by the stirring action of
the induction heater, the molten alloy was poured into an
aluminum chill-cast mold. A portion of the master alloy
was melted and poured into the chill-cast mold a second
time, for better mixing, and then used for the maximum con-
centration alloy. The remainder of the master alloy was
further diluted with pure aluminum and chill-cast for the

more dilute alloy samples. In some cases (as described

19

20

TABLE l.--Lot assay for HPM 2404.

 

 

Impurity Concentration (in ppm)
Ca .1
Cn .1
Fe .3
Mg .1
Mn .3
Total .9

 

below) chill-casting did not yield a homogeneous dopant
concentration throughout the casting. These samples were
given a homogenization period of one week at a temperature
of about 550°C in an argon atmosphere.

The magnesium series of alloys was kindly provided
in wire form by Dr. R. P. Huebener of the Argonne National
Laboratory. Spectrographic analysis on the least concen-
trated alloy of the magnesium series revealed <10 ppm
copper as the only detectable impurity. Electron micro-
probe analysis found copper and carbon present in amounts

barely sufficient for detection (much less than 100 ppm).

Forming of Wires

 

The castings were removed from the chill-cast mold,
a piece of the casting cut out and etched. This piece was

then passed through a rolling mill until it formed a wire

21

of 1 mm square cross section. This wire was then drawn
through tungsten carbide dies to 0.75 mm diameter. The
final drawing operations were performed using diamond dies
to a final diameter of 0.5 mm or 0.25 mm. After each pass
through the rolling mill or dies the wires were carefully
wiped clean using a lint-free tissue and ethyl alcohol.
Sufficient time was allowed for the ethyl alcohol to evapo-
rate before subsequent milling or drawing operations. The
dies and rolling mill were also carefully cleaned with
ethyl alcohol prior to use. Triple-distilled water was

used if lubrication was needed during the drawing operations.

Resistance Ratios

 

For characterization of the samples, the resis-
tivity ratio RR = R(295°K)/R(4.2°K), where R is the sample
resistance, was found to be quite useful. The procedure
was as follows: A piece of wire sample about 7 cm long was
fixed with current and potential leads. This sample was
placed in the room temperature resistivity apparatus and
given time to come into thermal equilibrium with the appa-
ratus. The resistance and temperature were then measured.
Matthiessen's rule was assumed to calculate the resistance
at 295°K. The sample was then placed in liquid helium and
the resistance at 4.2°K was measured. The ratio RR, thus
calculated, was then used in the characterization pro-
cedures. It should be noted that the resistance ratio thus

found is not wholly accurate (since this investigation is

22

indeed founded on the experimental fact that Matthiessen's
rule is not obeyed for aluminum alloys). This procedure
can be prOperly used to infer the effect of annealing on
residual resistivity and the homogeneity of alloy concen-
tration along a wire, since a decrease or increase in RR,
so defined, can be unequivocally used to infer a commen-

surate increase or decrease in the residual resistivity.

Annealing

 

A study of the effect of annealing on both pure and
alloy samples revealed the following: No advantage could
be found in annealing in an inert atmosphere as compared to
air. All samples were therefore annealed in an aluminum
foil "envelope" in air. After one hour of annealing at
approximately 500°C, the residual resistivity decreased (as
inferred by an increase in the resistance ratio) to within
1% of the minimum observed for a given sample. In the next
one to two hours the minimum residual resistivity was
observed to be reached. For further annealing no consistent
variation was observed, until ten to twenty hours elapsed
after which the residual resistivity began to increase.

This was probably due to contamination from the air or
aluminum foil. All samples were annealed in 10 cm diameter
coils for two hours, then the room temperature resistivity
was measured. The wire was then coiled to the diameter of
the sample holder and given one more hour of annealing time.

The resistivity ratio RR was observed to be constant to

23

within experimental error before and after the second

anneal for all samples tested.

Sample Characterization

 

After formation into wires, and the initial two-
hour anneal, the samples were tested by resistance-ratio,
electron microprobe and chemical analysis to characterize
the usefulness for this study.

A wire 90 cm in length was required for the
resistivity measurements. Additional 10 cm lengths from
each end were used for resistance ratio analysis. The
sample was considered satisfactory for this study only if
the resistance ratio of the two end pieces matched to within
5%. Furthermore, when installed in the cryostat, the sample
from the central piece was required to have a resistance
ratio within 3% of the average of the end pieces. This
criterion was established to infer that fluctuations in
alloy concentration of the samples used were about 3% of
the average concentration. It was particularly difficult
to achieve a sample with zinc, cadium, or silver as the
solute which could pass this test. Even after one week of
homogenization, only two zinc and one silver alloy samples
were homogeneous enough to pass this test.

An Applied Research Laboratories electron micro-
probe was used to qualitatively analyze the least concen-
trated gallium and most concentrated copper samples for

impurities. The electron microprobe is not suitable for

24

quantitative analysis of impurity levels below 300 ppm.
However, it is possible to detect the presence of trace
impurity levels to below 10 ppm. In addition to aluminum
and the desired dopant, carbon, silicon, manganese, and
copper were each observed at the threshold detection level.
The lot analysis for the pure aluminum used included copper
and manganese at below the 1 ppm level. It was assumed
that carbon and silicon were picked up from the vitreous
carbon crucible and vycor glass sleeve used in the melting
operation.

Chemical analysis for the quantity of solute mate-
rial was performed by Schwarszpf Microanalytical Labora-

tories.

RES ISTIVITY DATA

Introduction

 

The resistivities of pure aluminum and fourteen
aluminum alloys were measured from 4.2°K (as low as 2°K in
some cases) to 300°K. The specific resistivity of each
sample was determined at room temperature. Resistances
were measured for each sample as mounted in the cryostat
as a function of temperature. These data were then con-
verted to resistivities by normalization to the measured

specific resistivity at room temperature.

Room Temperature Resistivity Measurements
The resistivity of a cylindrical wire sample of
resistance R, length 2, and cross-sectional area A is

given by

The resistivity of each wire sample used in this study was
carefully determined at room temperature by measurement of
the resistance per unit length and the mass, m, of a known
length, 2, of wire. The area was calculated from the hand-
book value (27) for the density, d, according to the rela-

tionship

25

26

_ m
A — 35—. 4.2
0

Each wire sample was placed on an Invar scale
mounted in a channel milled out of a massive, thermally
insulated, aluminum block. Razor blades were used for
potential probes. Simultaneous position, potential, and
current measurements were used to determine the resistance
per unit length of the wire sample. The temperature was
measured using a platinum resistance thermometer which was
mounted in the aluminum block. Potentiometric measure-
ments were made with the circuitry described under Experi-
mental Apparatus, and shown in Figure 5. At the greatest
length, £0, the razor blade potential probes were pressed
down, cutting off £0 of the wire. This amount of wire was
weighed for its mass, m, using a Cahn microbalance (if
microgram precision was required) or a Mettler model B-6
semi-micro balance. Corrections for the bouyancy of air
were made when the Mettler Balance was used, but were not
needed with the Cahn microbalance since the mass standards
used were aluminum alloys within .3% of the density of the
wire samples.

This procedure is applicable if the cross-sectional
area is uniform along the entire length of the wire. To
insure this, R/£ was measured as a function of the position
along the wire. The variation in R/£ was assumed to be due

to variation in A rather than alloy concentration. This

27

was assumed by the restrictions on the residual resistivity
described previously.

If any part of the sample had a variation of R/l of
more than .03% from the mean, an alternate procedure was
used. If the wire is supposed to be composed of segments
1i in length with each segment Ai in cross-sectional area,

the mass of the length, 20 = nzi, of wire is

Now, since the resistance of each segment, R1 is

R. = 0 %E. 4.4

_ ii
Ai—pfi-O— 4.5
1
giving
n
m=d2..zp z i. 4.6
l . R.
1=l 1

 

28

In practice, then, the wire was divided into n
sections such that the values of Ri from adjacent sections
varied by less than .03%, then Ri was measured for each 2i

and the resistivity calculated using the above expression.

Temperature Dependence of the Resistance

 

In practice, up to ten wire samples were connected
in series in the cryostat for a given run. One of these
samples was always pure aluminum. The data were taken in
the sequence: potential for the pure aluminum sample (for-
ward current), potential for alloy sample (forward current).
thermometer potential, potential for alloy sample (reverse
current), potential for pure aluminum sample (reverse
current). In this way, the resistances of the alloy and
pure aluminum were obtained for the same average tempera-
ture, minimizing the effect of uncertainty in temperature
measurement.

The cryostat was immersed in or above liquid
helium when taking data from 2°K to 65°K. The helium was
pumped for data points below 4.2°K and temperature control
achieved by regulation of the pumping speed. Good thermal
contact with the bath was effected by introducing a small
amount of helium gas into the inner and outer cannisters.
For data points above 4.2°K, the outer cannister was pumped
out and the temperature controller used to adjust the sample

temperature and hold it constant.

29

The cryostat was immersed in liquid nitrogen when
taking data above 65°K. In the range from 65°K to 77°K,
the nitrogen was pumped and temperature control was
achieved by regulation of the pumping speed. Helium gas
was introduced into the inner and outer cannisters. For
temperatures above 77°K, the outer cannister was evacuated
and the temperature controller used to adjust the sample

temperature and hold it constant.

Calculating the Resistivity from the Data

 

It is not sufficient over the entire range of tem-
perature studied to simply normalize the resistance to the
room temperature resistivity by using a temperature inde-
pendent shape factor. Corrections must be made for change
in volume due to thermal expansion and alloying.

Dugdale (17) has pointed out that deviations from
Matthiessen's rule can be properly inferred only if the
resistivities are all measured at the same atomic volume.
The resistivity at a volume V was corrected for volume

change to V0 by the relation

V -V
_ V 8p 0
0(VO) — 0(V)[l+-5—3—\—,(-—V—)]. 4.3

The volume derivative may be calculated from the pressure

derivative

30

Y.§E = _ 1.334- l.§X)—l 4.9a
pBV paP VBP ’
= - 1.22.1 4.9b
0 BP x'

where x is the compressibility of the sample material. The
pressure derivative for the resistivity of an alloy may be
separated into phonon and impurity resistivity terms. This

is suggested by taking the pressure derivative of

Matthiessen's rule.

= 4.10
0t pph + po
80 30 89 4 11
t _ ph o '
a? ” 3P + 5?
Defining
1 439 h 4 12
3"‘3—"5’g" '
ph
and
30
_ _ IL._JE 4.13
Y— D 8P!
0
we have
1 apt _ _ 8 92h _ fig 4.14
"‘ 3P ’ Y

31

Bridgman (28) has measured the pressure derivative
of aluminum between 90°K and 273°K. This data was extended
to 4°K using the relation (29)

114

_ 1
3I3 8P]high T _ [E aPllow T'

Bridgman (30) has also reported the pressure derivatives of
the resistivity of Al-Mg and Al-Zn alloys at 273°K. Foiles
(31) has measured the pressure derivative at 77°K of the
Al-Mg alloys used in this investigation. For Al-Mg alloys,
Y is 1.5(10_6)cm2/kg, and is independent of temperature.
For Al-Zn alloys, 7 is -3.8(10-6)cm2/kg. For pure aluminum
at 273°K, e is 4.03(1o'6)cm2/kg.

Corrections were made to the resistivities follow-
ing the two-step program suggested by Dugdale (17). First,
ideal phonon resistivity, pph' has been corrected for
volume dilatation on alloying. Thus

corr 38(a-ao

pph = Ophil + X a )1: 4.16

 

0

where a0 and a are the lattice parameters of the pure mate-
rial and the alloy respectively as reported by Pearson (27).
Second, the residual resistivity of the alloy, po, has been
corrected for the volume change due to thermal expansion.

Thus

_ 3Y _
00(T) — po[1 + 57(00 a)]. 4.17

32

where a and do are the fractional linear expansion at
temperature T and 0°K relative to 293°K as tabulated
by Corruccini and Gniewek (32). The deviation from

Matthiessen's rule at the temperature T is then

COI'I'

A(T) = ot(T) - 00(T) - oph

O 4.18

Since 7 was known for the Al—Mg and Al-Zn systems only,
complete corrections have not been made for alloys in the
Al-Cu, Al-Ga, and Al-Ag systems. The maximum error thus
incurred is at high temperature. If we assume the y value

observed for Al-Zn is large, an approximate maximum for the

correction is

 

-6 2
2‘90"“)90 = 3(3.8)(10 12m /§g(.00415)po 4.19a
X 1.34(1o )cm /kg
= .045 p . 4.19b
0

Consequently, for the Al-Cu, Al-Ga, and Al-Ag systems, it
has been assumed uncertainties, Spvol’ due to volume cor-

rections are
dpvol S .05 00. 4.20

A FORTRAN listing of a computer program used to
calculate resistivities from the voltage data is shown in
Appendix A. A typical data run computer output is in

Appendix B. An error analysis is in Appendix C.

RESULTS AND CONCLUS IONS

The Phonon Resistivity of Aluminum

The temperature dependent resistivity, 6(T) =
p(T) - po, for three pure aluminum wire samples is shown
as 6/T2 vs T3 in Figure 6. Reich (33) measured the resis-
tivity of three samples from l4°K to 20°K. The power
series fit to 6 = aT2 + bT5 obtained by Reich for p0 =
.068 nQ-cm sample is represented by a solid line in the
region of data, and extended with a dashed line. On the
basis of the data of Aleksandrov and D'yakov (34),
Aleksandrov (35) proposed the fit shown as the dash-dot
line in Figure 6. The coefficients a and b reported by
Garland and Bowers (36), Reich (33), and Aleksandrov (35)
and fit to the data of this investigation between 25°K
and 40°K are summarized in Table 2. Below 25°K, the data
deviates considerably from the straight line fit. This
feature will be discussed later in terms of deviation
from the MR. Above 45°K, the temperature dependence is
weaker than T5; eventually falling to a nearly linear
functional behavior for temperatures above 100°K.

The Bloch—Grfineisen (3-5) theory for the resis-

tivity due to phonon scattering predicts

p a T5, T << 9D. 5.1

ph
33

34

 

 

 

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Anxe me
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8 9. or F on on “Miami 0.0. 3

35

TABLE 2.--Coefficients of the T2

resistivity of aluminum.

5

components of the

 

 

Wire p 10
. . o axlO bx10
Investigator Diagfiter (nQ-cm) (uQ—cm/°K2) (uflcm/°K
Present In—
vestigation .50 1.42 1.25 1.56
Present In-
vestigation .25 2.03 1.50 1.46
Present In-
vestigation .25 .97 1.55 1.61
Garland & Bowers 1.05 1.04 0.44
Reich 1.25 .22 0.64 1.18
Reich 5.22 .068 0.375 1.38
Reich 5.22 .055 0.511 1.22
Aleksandrov m .092 0.535 1.17

 

Such behavior is not always observed. The simplifying
assumptions of the Bloch-Grfineisen model are:
1. Elastic scattering of the electrons by the
phonons is assumed.
2. A Debye spectrum is assumed for the phonons.
3. The microscopic scattering cross-section for
the electron-phonon interaction is assumed to
be independent of the scattering angle. (The
relaxation time is assumed isotropic.)
4. Umklapp (U) processes in which a reciprocal
lattice vector is included in momentum con-

servation are not included.

36

With these approximations, it is surprising that the re-
sistivity of a real metal would even approximate the
T5 law.

Pytte (37), Kaveh and Wiser (38), and Klemens and
Jackson (15) have shown that consideration of U processes
alters the phonon resistivity considerably. In calcu-
1ating the resistivity of aluminum, Pytte (37) has shown
that when U process are included, a T4 term appears, and
for temperatures below the cutoff for U processes, the
resistivity falls off more rapidly than T4. Kaveh and
Wiser (38) calculated the resistivity for sodium, and
having taken account of U processes and the momentum de-
pendence of the electron-phonon scattering amplitude they
accounted for the observed change from T5(T 2> 9°K) to
T6(T < 9°K) in the temperature dependence of the resis-
tivity. They assert further that the T5 behavior of the
resistivity of a metal has nothing to do with the Bloch
result, but is merely a fortuitous numerical accident.
The theory of Klemens and Jackson (15) predicts the U
processes will not affect the temperature dependence, but
will enhance the coefficient of the T5 term. In consider-
ing the effect of impurity scattering in the Klemens-
Jackson model, Ehrlich (14) derived an added term prOpor-
tional to T2 as well as an additional enhancement to the

coefficient of the TS term. Klemens and Jackson and

5

Ehrlich assumed pph « T in the absence of U processes.

37

Dugdale and Basinski (17) have pointed out that
presence of deviations from the MR due to the two-band
effect can also enhance the coefficient of the pph power
law term in the limit, p0 >> pph'

Although there is little agreement among the
above authors as to what the power, n, should be for
pph « Tn, there is agreement on the premise that n is as
large as or larger than the highest power observed in
6(T) = p(T) - p0, and that it is difficult to underesti-
mate pph from the experimental data.

The resistivity due to phonon scattering in pure

aluminum has been assumed to be the smaller of

pph = 1.56 x 10-10(u9cm/°K5)T5, 5.2

the low temperature limit, or at high temperatures,

pph _ — Q o 5.3

ppure o

The cross-over from use of Equation 5.2 to Equation 5.3
usually occurred at ~40°K. The T2 term has been associated
with electron-electron scattering (35, 36) but, as will be
discussed later is most likely a deviation from the MR.

The values for b reported by Reich (33) were rejected

since his data were taken in the region pph 3 po, which

as will be seen later, is the region where additional de-
viations from MR are large, thus affecting the apparent

coefficient of the T5 term. Aleksandrov (35) relied on

38

data (34) from the region T > 50°K, the data taken in this

study indicates n < 5, so the T5

coefficient reported by
Aleksandrov was considered unreliable. The b value for
the sample measured in this study which had the lowest
residual resistance was not used, since it was relatively
small (.25 mm diameter) and size effects have been shown
to exhibit temperature dependent deviations from the MR
(21). The largest diameter sample (.5 mm) with the lowest
residual resistivity (p0 = .0014 uQ—cm) was chosen as the
most reliable, thus giving the value for b quoted in
Equation 5.2

The Deviations from Matthiessen's Rule
in Aluminum Alloys

 

 

The deviations, A(T), from the MR for thirteen
aluminum alloy wire samples are shown as A/pO vs tempera-
ture in Figures 7 through 10. The AlfMg system is repre-
sented in Figure 7, AlfCu in Figure 8, Alen in Figure 9,
and AlfGa in Figure 10.

The qualitative behavior of the deviations ob-
served was the same in all cases. The temperature depend-
ence is characterized by a sharp rise from zero for low
temperatures (T < 40°K), a peak or saturation plateau in
the region 40°K < T < 100°K followed by a region with
dp/dT 1’0 as 300°K is approached. Figure 11 shows the

deviations from the MR for the same alloys as log A/pO vs

39

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CE mmzk<mmaEMF

 

 

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41

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43

 

  

 

 

 

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log(T)

Figure ll.——Low temperature deviations from Matthiessen's
Rule in aluminum alloys.

44

log temperature. The dominant feature of Figure 11 is

that for T < 20°K
A(T) = cT 5.4

for all alloys.

Table 3 shows the relative corrections at 295°K
due to volume change on alloying (Goph/pph) and thermal
expansion (500/00) for the most concentrated alloys of
each series. The measured deviation from MR at 295°K for
the 2 at/O Zn alloy before the volume corrections was much
smaller and displayed a negative temperature derivative.
Thus, without volume corrections, information derived from
the magnitude or temperature derivative of deviations from
the MR is unreliable, particularly for the more concen—
trated alloys.

The residual resistivities as a function of alloy

concentration for the AlfMg, Alen, AlfGa, and AlfCu system

TABLE 3.--Volume corrections.

 

 

p
0
Alloy (uQ-cm) Goo/p0 épph/pph dA/A
Al + 1.03 Cu .604 -- -.013 -.73
Al + .66 Ga .142 -- .001 +.06
A1 + 1.5 Mg .759 .013 .015 +.84

A1 + 2.1 Zn .401 -.034 -.003 -.34

 

45

are shown in Figure 12. Table 4 lists the residual resis-

tivities per atomic per cent for each of the systems.

TABLE 4.--Residual resistivity of aluminum alloys.

 

 

Alloy System p0(uQ-cm/atomic per cent)
Al-Mg .52
Al-Cu .58
Al-Zn .19
Al-Ga .21

 

From Figures 7, 8, 9, and 10, it is apparent that
the deviations from the MR in the Al-Mg, Al-Cu, Al—Zn,
and Al-Ga alloys are not linearly dependent on the resid-
ual resistivity, p0, or in other words, the concentration.
It will be shown that the data are consistent with a much
weaker functional dependence. The data will be discussed
in terms of the theories appropriate to such a weak func-
tional dependence--the two-band model, the Ehrlich Theory,

and the Mills Theory.

The Two-Band Model

 

Figure 13a shows the single OPW construction (39)
of the Fermi surface of a trivalent metal such as aluminum.
The pseudo-potential model of the Fermi surface, consistent
with de Haas-van Alphen data, generated by the calculations

of Ashcroft (40) is in topological agreement with the

.m>oHHm ESCHEDHM mo mmHuH>HpmHmmu Hospflmmmil.ma musmflm

Czwommm oiohds o
N . o

 

46

 

A d o6

 

No
ad
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mo
nu
w
d lmoo (
26...: «o -._< >
964.: zN-._<u
96...: no .._< < o no
93.2 22...: o 1

 

 

47

 

 

Figure 13.-—Fermi surface of aluminum. (a) Single OPW
construction. (b) Ashcroft model of the
third Brillouin zone Fermi surface.

48

single OPW model. The essential features of the Ashcroft
model are:

l. The first zone is full.

2. The fourth zone is empty.

3. The second zone hole surface is constructed
from remapped segments of free electron
spheres. The density of states is about one
hole per atom and the effective mass m* 3 me.

4. The third zone surface is multiply connected
and significantly distorted from the single
OPW construction. The third zone "monster"
surface is reconnected to form "rings of four"
lying in the {100} planes as shown in Figure
12b.. The density of states is = .02 electron
per atom and m* = 0.1 me.

This resembles the heavy hole-light electron con—
figuration of the semiconductor. One distinctive differ-
ence remains between the trivalent metal and the semicon-
ductor. In contrast to the energy gap of the semiconduc-
tor, the Ashcroft model predicts 24 points of contact onto
the third zone from the second zone at points identified
as R by Ashcroft, lying near the symmetry point W.

The reversal of the Hall field for aluminum and
aluminum alloys has been reported by Luck (25) and
Forsvoll and Holwech (26). The Hall coefficient is con-

sistent with an electron density of three electrons per

49

atom in the low field limit, and one hole per atom for the
high field limit. The model proposed by Forsvoll and
Holwech follows: The electrons on the second zone Fermi
surface will behave approximately like free electrons as
long as they are not undergoing Bragg reflections. The
probability for Bragg reflections is small in low fields
and for short mean free path, because then the electrons
only travel a small part of a closed orbit on the Fermi
surface between collisions. Under these conditions one
would therefore expect the Hall coefficient to be close to
the free electron value. In high fields most of the elec-
trons will undergo reflections in their normal lifetime,
and because the second zone Fermi surface encloses a hole
region, the reflections will tend to change the sign of
the overall curvature of the electron paths. The elec-
trons will then tend to drift towards the "wrong" side of
the specimen, and give rise to a positive Hall coefficient.
In the high field limit, when all the electrons travel
several times around the orbit, the Hall coefficient is

given by the simple formula R = l/ec(nH-ne), where n is

H H
the number of states enclosed by the second zone surface
and ne the number of states enclosed by the third zone
surface.

.In light of the evidence provided by the Hall

measurements it may be apprOpriate to treat all transport

phenomena in aluminum as two-band in character.

50

Sondheimer and Wilson (18) have pointed out that
the MR does not hold if the electrons are considered to be
independently distributed in two bands in order to approxi-
mate the anisotropy of the relaxation time. The conduc-
tivities of the two bands are additive, while the phonon
and impurity resistivities are separately additive in each
band. Thus, denoting band number by superscripts and con-

ductivities by 0:

otOt — 01 + 03, 5.5
where l—-= i = i + i 5 6
oi D pph DO: -
and l—-= p1 = pj + p]. 5.7
03 ph 0

At the absolute zero of temperature,

 

1 Dj
_ _ o o
o o
and for the ideally pure metal (00 = 0)
i j
_ pph pph
p '- I u 509
ph 01 + p]
ph ph

For an alloy, the resistivity as a function of temperature

is then given by

51

(3-8)2000 h
= 0(T) = pph + 00 + 2 p 2 ' 5'10
a(1+B) 00 + 8(1+a) pph

 

where the last term is the apparent deviation from the MR.

The parameters a and B are defined as:

i i
p p

a = —%27 and B = —%. 5.11
pph po

Kohler (19) obtained a similar result based on a varia-
tional solution of the collision operators for the im-
purity and phonon scattering. Kohler pointed out that
the deviation represented in Equation 5.10 could be zero
only if relaxation times could be unambiguously defined
for both the phonon and impurity scattering and if their
ratio is independent of electron momentum.

The expression,

2
(a~B) p p
A(T) = 2 °Ph 2 , 5.12
a(1+8) 90 + 8(1+a) oph

 

or the equivalent expression,

0 _
_Z._ p0 + b, 5.13

q(l+B)2
‘(a-B)2

where m =

52

=Ellfifli

and ‘ b
(a-B)2

are referred to as the Kohler-Sondheimer-Wilson (KSW)
equations. If the deviations from the MR can be shown to
fit the KSW equations, then this constitutes an element
of evidence that two-band effects are responsible for the
observed deviations from the MR.

The deviations from the MR in alloys of aluminum
with magnesium, copper, zinc, or gallium fit the KSW
equation. Figures 14, 15, and 16 show the A(T) data for
T = 30°K, 70°K, and 295°K respectively, plotted in the
form of Equation 5.13. The fit in Figure 16 is not par-
ticularly good, as the accuracy of determination of A may
be as poor as 50 per cent. The effect of the volume cor-
rections to the phonon and residual resistivities on the
calculations of A is shown in Figure 16. The data reported
by Seth and Woods (11) is also shown. Using the slopes
and intercepts of the straight line fit for each alloy
series on the KSW plot at a given temperature, the param—
eters a and 8 required for a fit to the KSW equation
were calculated from 8°K to 100°K. Above 100°K, uncer-
tainties in measurements and volume corrections did not
permit unequivocal determination of a and 8.

Equation 5.12 shows that A > 0 for all no # 0.
Consequently, a non-zero deviation from the MR is implicit

in the phonon resistivity calculation of Equation 5.3.

53

 

I l I I 1 T l l
aoI- AL-MG ALLOYS 0..

 

AL‘CU ALLOYS
7o- AL-ZN ALLOYS
AL-GA ALLOYS
60 "
P
.2 5 d
A

 

 

 

 

2 30 °K .1
Io- '/ -
0 .I . , .

B (pH-cm)

Figure 14.—-Koh1er-Sondheimer-Wilson plot at 30°K.

54

 

2() I l l

GAL-MG alloys
AAL-CU “

VAL -GA
'5r xAL-ZN "

 

 

 

 

O. I 1 1
O .2 £1 .6
p,(/.LQ-cm.I

Figure 15.--Kohler-Sondheimer-Wilson plot at 70°K.

55

 

 

.xommm um “Cam COmHABIHmEflmspcomlumanomii.oH mnsmwm

28-0.3 ml

 

 

     
 

 

IMII
r 3.44 1 m Q
4 30.44
w§3a§$
E I 62...; EC-
0 G O @234
m’gbowmmoo “Moo Waco
2‘39, oz o. 365% 6.34 Am.
_

b b _ P

 

56

In the temperature regions for which Equation 5.3 was the
appropriate definition for pph’ corrections were made to
pph to insure the inferred deviation from the MR in the
pure sample was consistent with that observed in the alloy
samples. A KSW plot was generated using the uncorrected

p . The average intercept

ph

lim Do

""""'I
pd+0 A

b:

was used to calculate a pph from the pure data consistent
with the alloy data, pph(T) = pP(T) - po(l + 1/b), which
was then used to generate another KSW plot. Usually one
more iteration yielded self—consistency to within the ac-
curacy of the dimensional measurements.

The parameters a and B are shown in Figure 17.
The parameter a is strongly temperature dependent varying
from ~104 at 8°K to ~2 at 100°K, while B is relatively in-
sensitive to change in temperature in this range. Two in-
consistencies with the two-band model are apparent in
Figure 17. The variation in B with temperature is not
anticipated since p: and pg are assumed to be temperature
independent. The two band approximation may not fully repre-
sent the relaxation time anisotropies, or additional mechan-
isms may give a temperature dependent 90 for one or both of
the bands. The KSW scheme for extracting a and 8, however,
rely heavily on a constant impurity resistivity. Below

20°K the value for B is quite sensitive to the choice of

57

 

20000 I I IInIII I ITIIIHI T I IIII
IOOOO:- ! -—__1
500g: 5 a .8 ALLOY j
_ O 0 AL MG ..
2000- A A AL CU _
I D AL ZN
V V
IOOO__— ' AL GA _—
E 2 :
500'- q

I
m

 

 

 

200“ _.
ICK):- ‘x _—
.. :- . :
c3 50- _.
. v —1
20" g : —-I
A
9 V
_. I .—_
IO: 6 5A 1
I- . :
5: g o “I
: g g§°o‘: ..
2+ 6AA! -
38",
I 1 1111111L 1 1111111I 1 1111111
| 2 5 I0 20 50 I00 200 500
T (°K)

Figure l7.--KSW parameters a and B as a function of tempera-
ture. '

58

intercept in the KSW plot, which in turn is quite sensi-
tive to deviations in the MR implicit in Equation 5.2.

If the T5

coefficient were about 20% smaller, 8 = 4 :_2
would fit to within the precision of the experiment. A
flaw of an even more serious nature is evident at about
90°K, where Figure 17 shows a = B. This would be consis-
tent with the two-band model only if deviations from
Matthiessen's rule were zero at the temperature for which
a = B. This is the region where thermal expansion begins
to be important, and the error bars implicit make a choice
of a > 8 possible up to 300°K.

Figure 18 shows p; = (a+l)pph/a as a function of

5

T, showing that for T : 60°K, is of the form bT . The

3'
Dph
total phonon resistivity pph follows the same power law
for T < 40°K, but follows a weaker temperature dependence

at higher temperatures. The error bars shown indicate the

magnitude of the uncertainties for the points representing

3' . j . _
pph and pph. Below 30 K, a >> 1 and pph cannot be dlS

. . j i =
tinguished from pph' In contrast to pph’ pph(T) (a+l)pph

is linear in T as shown in Figure 19. Below 20K the in-
determinant nature of a makes the calculated values for
pgh quite uncertain. Using the evidence summarized in
Figures 18 and 19, it is proposed that the bands i and j
represent the states on the third zone electron and second
zone hole surfaces of the aluminum Fermi surface, respec-

tively.

59

 

 

 

 

 

'04:; j I IIIIIII I II III' I I TIITT-‘IOZ
I '0 AL-MG Alloys :
r blaAtcu : 1
-13 lPh DIALTIZDI '
'0 F LVAL-GA " q l0
C T5 «T :
”' I. Z 1
b 1
'01? . —. IO°
E - a
h : 4+. 1 h
Pph ,Pph _ ‘ " Pph 0P”
(II-$7.1m)r :1 #97ch
'01? 1 I0”.
IO-Sp ' P"= 9%! @55le
E + Pph E
-6 L 1 L1 0'3
'0 |.O IO .0 l. . .

T (’K)

Figure 18.--Phonon resistivity of the hole band.

60

.pcmn couuomam man no wuw>flumflmmn coconmll.mH munmflm

35 .r

 

 

OON OO— Om ON 0. m N _

I—. 114 q q _ d —« 51 0% - a A _O.
.m>01_l_4 4011.4 4 . NO.
um>0.._l_4 ZNll_4 o 1
rm>0134 3011.4 4 - .810
@634 3.-.: o w\ H no.1.
n m .n
n. l. _0 U

4..
. .3 W
T 1
n “no
1 .r. . p —P__ . p . . .lL 4 Ava

 

 

 

61

Phonon resistivity linear in temperature is usu-
ally associated with segments of the Fermi surface with
dimensions such that the phonon wave vector, Eb, necessary
for large electron wave vector scattering angles has a

population density, n(qb) 3 1. Where

-1

éhw/kt-l) . 5.17

+
n(qb) = (
The characteristic temperature for which n(qb) = l is de-

fined by

+ hm
T(qb) = k , 5.18

where mg is the frequency of the phonon ab. Figure 20
shows such an event scattering an electron from the state
ki to kf on the external cross-section of an arm of the
third zone of the Ashcroft model of the aluminum Fermi
surface. Hartmann (41) has fit neutron diffraction data
to a dynamical model of the aluminum lattice giving the
phonon dispersion shown in Figure 21. The phonons 3b

7 cm-l), which are appropriate for the scattering

event shown in Figure 20 have mg 2 40 x 1011 sec-1. Thus

(2 10

11 —1
sec I 5.19a

-1 I

(10-27)erg-sec(40x10

~16

T(qb) =
1.4 x 10 erg °K

giving

T(qb) 2 3o °K. 5.19b

62

 

 

 

BRAGG PLANEx
(' 'I\

l J
1

' 01 ALU.

Figure 20.-—Backscattering event on third zone extremal
cross-section.

63

 

 

f Janu‘w
600 ”x'0

500

 

 

400

 

 

 

 
 

  

 

 

 

r it ___- -e __t
1‘”, (3/4 3/4 O)(ll0)(|00)(l'/4 V4) 1‘”,

 

 

 

Figure 21.~-Phonon dispersion spectrum for aluminum.

64

Since the phonon wave vector ab crosses a Bragg plane, a
reciprocal lattice vector also enters into the momentum
conservation for this event. Phonons such that q :_qb can
also participate in similar backscattering events. If
T(qb) = 30°K then T(q < qb) < 30°K and the population of
phonons capable of resistive backscattering is large
enough to expect a linear temperature dependent resistivity
even below 30°K. Thus, it is reasonable to interpret the
linear temperature dependence of the resistivity, pgh, in-
ferred from the KSW equations as associated with the third
zone electron states. We set

i _ e
pph " pph 0

This leaves the interpretation of the resistivity

p; as being associated with the second zone hole states.
Ehrlich (14) has calculated the low temperature resistivity
of monovalent metals whose Fermi surface touches or nearly
touches the Brillouin zone boundaries, extending the
Klemens-Jackson (15) theory to include the simultaneous
presence of impurity and thermally-induced Umklapp scatter-
ing. The second zone hole sheet of the trivalent aluminum
Fermi surface when considered alone and in the extended
zone scheme, is analogous to the case considered by
Ehrlich. The application of the Ehrlich theory to devia-
tions from the MR in pure aluminum will be treated

later. In this case, however, the resistivity, does

3'
pph.

65

not exhibit the T2 component which Ehrlich predicts for
the case of U processes in the presence of impurity scat-
tering. The T5 component is compatible with the Ehrlich
theory, assuming, as does the Ehrlich-Klemens-Jackson
approach, that pph s T5 for no U scattering processes.

In presenting the Bloch-Grfineisen pph a T5 formal-
ism, Ziman (42) asserts that the fifth power of T, or as
has been pointed out (14, 15,37, and 38) some other high
power of T, is a characteristic quantum effect in the
limit of phonon momentum too small for effective large-
angle resistive backscattering. Phonons capable of direct
resistive backscatter on the hole surface must extend to

the region of the {100} and {110} planes in reciprocal

space (see Figure 21), so 350 x 1011 < mg < 600 x 1011
rad./sec. giving 250 < T: < 430 °K. This band then, is

considerably removed, in terms of available phonon momen-
tum, from the region where single event, large-angle
scattering is anticipated. We thus set

pph = pph. 5.21
Since a > 1 at low temperatures, the phonon resis-

tivities of the hole and electron bands are in the order

pgh > pgh. Huebener (43) has applied a two-band model to
the low temperature phonon drag thermopower he has ob-

served in aluminum alloys, and concluded that 02h > 02h

is consistent with the data.

66

Although the data fit the KSW equations, applica-
tion of the two-band model to the deviations from the MR
in these aluminum alloy systems cannot be carried out
without the ambiguities previously discussed. The phonon
resistivities inferred for the two bands are consistent
in temperature dependence with the dimensions of the
third zone sheet of electron states and second zone sheet

of hole states in momentum space.

The Ehrlich Theory
The parameters a and b, of the Ehrlich (14)

Theory, in Equation 1.6 are related by

for 8 (angle of contact of the Fermi surface with the
Brillouin zone boundary) held constant. This Can be shown
by dimensional analysis. The quantity a/b must have di-

mensions (temperature)3. Now the only variable parameter

0 5.23

 

5

must be fixed in terms of po and b0, where boT is the

phonon resistivity in the absence of U processes. Thus
3 2"

_ 0 3/5
T _ (B—O'fi’)’ 5024

67
so

a a 3/5
B (00) . 5.25
Figure 22 shows log(a/b) vs log p0 for the data in Table 2

and one Al-Ga alloy. SlOpes corresponding to po3/5, the

2/5 have been drawn. Al—

theoretical relationship, and po
though the Ehrlich theory predicts the correct temperature
behavior for deviations from the MR for relatively pure
aluminum or very dilute aluminum alloys, the coefficients

3/5 2

do not follow the predicted po law. Further, the T

term could not be detected in the low temperature resis-

tivity of the more concentrated alloys even though the T2

3 term which was

term should be about as strong as the T
observed for T < lO°K.

Although the processes which the Ehrlich Theory
considers are certainly appropriate to aluminum and

aluminum alloys, the data do not support the predictions

of the theory.

The Mills Theory

 

Mills (16) has calculated the additional resis-
tivity A(T) for the second order process involving scat-
tering from an impurity and the .Subsequent, emission or

absorption of a phonon. The result is the form

 

A(T) o: 0 5.26
O

68

.suomaa noaausm map on a cam m mpcmfiofimmooo man no unmnu.mm museum
.80..ch m. i
O. Om ON 3 m. N.
. _

Jj-_ _E-u fidJl

_. 00. N0 _0.
1... .. . MO—

 

l
J

IIIIITI T

111111 1

o o
D\NQ O. o

..o.

amhm. 12mm n. .
>omoz<mxmu< O
zo_._.<o_._.mm>z_ .rzmmmmm 0
>041? <05... 9

__.k... _ ___. __L I. __....r. r 00—

l

l

 

 

TlllllT

llLl l l

 

 

 

 

69

where y(T) is proportional to the electron-phonon inter-

action relaxation rate, and y(T) ~ 0 In the limit

ph'
90 >> Y(T) I

A(T) = cT3, 5.27

which is consistent with the data below 20°K as shown in
Figure 11. The form of Equation 5.26 is also consistent
with the situation that T3 behavior is followed to higher

temperatures for the more concentrated alloys. For larger

p higher temperatures must be reached before the y(T)

0!
term becomes important, thus altering the T3 temperature
dependence.

In order to determine the dependence of A(T) on

the residual resistivity, y(T) must be included to first

order. In the limit pO >> pph ~ y(T):

3
T
A(T) “ 5.28a
1 + ——YzT) '
O
a T3 (1 - 1131), 5.28b
Do
a - T3 1n(11319. 5.28c
p0

Thus, for the T3 region of the data,

A(T) = cT3 « T3 ln pO-T3 ln T 5.29

70
so,

c a 1n p0. 5.30

Figure 23 shows the value obtained for c from a fit of
A(T) to Equation 5.27 as a function of log 00. The dashed
straight line was taken from data reported by Caplin and
Rizzuto (23) for 14°K.

The points for Ekin and Maxfield (44) were taken
from their published data from 1.2°K to 7°K for a mono-
crystalline wire (p0 = .85 nQ-cm), a polycrystalline
ribbon (p0 = .42 nQ—cm) and a polycrystalline wire
(00 = .40 nQ-cm).

At temperatures above 20°K the pure samples have a
T2 deviation from the MR. Yet, as shown in Figure 6, the
T2 deviation is replaced by some higher power of T below

20°K. Figure 11 suggests that the deviations follow a T3

law below 15°K. The T3 at low temperatures--T2 at high

temperatures behavior is not consistent with a simple
power law composition but may be understood as the effect

of the coefficient of the T3 term modulating the tempera-

ture dependence from the T3 low temperature limit to T2
at higher temperatures. This is consistent with the re-

sult in Equation 5.26 if pph ~ p0 and

00 + Y(T) or T. 5.31

71

C (pfl- cm/°K3)

.oQ mo :OHvOCSM m mm mEHmu B can

a man no muqmnoammmouuu.mm musmHE

 

m m
.Eo-d1. m.
00_ _.0_ NF? m..O_ .70. 0.9
0’114.. « . ~..q.m. . J —qdq+.u . . :...«1. q 1... 11W. O
.2252 .. 5.5 o . . .. .
9.3. 5.80--- . xx
.. 4012. D xxx
”032.: .. ZN-_< D xxx libs: D
2 DUI—q 4 C \\\m 1m,
. :22 22.... o m .. xx 6
_< 8525 x .xxx .. r .
.4 2.5 + p xxx 0 m
D \
O_xN.| I \\ /
s- kzmoiumoo ._._..o x 1 o
n u xxx 65:09.24 9 w.o_xN M2
. o D xxgxx 5:... u (
p a xxvx mtosom w 23:00 4 .
xxx 6:4 40:3 D
#9me a xx 5:03.32: 2.39... o
4 D O\\ 10'0—Xm
xxx HszEum—oo uh u D
V\O\
KNO—xv :P.-h . » —:.P-rr _ rP-er. r » P:p..b - r _pppp.~ . p

 

 

 

 

 
 

72

If this were the situation, one would expect the
coefficients a, of the T2 term, and b, of the T3 term, to

be proportional, and thus

a m lnpo. 5.32

The coefficientsxaslisted in Table 2 and for the least
dilute Al-Ga alloy are also plotted as a function of
lnpo in Figure 22. The change in scale was calculated

from the average experimental ratio, c/a = 0.105°K-l, for

3 and T2 terms observed in the three pure samples

the T
measured in this study. The agreement in functional de-
pendence upon po indicates that it is consistent to con-
sider the T2 and T3 terms to be the intermediate and low
temperature manifestations of the same effect. As pointed
out in Equation 5.31, the relaxation rate must have a
nearly linear temperature dependence for such a modulation
effect to take place. In the pure matrix, one expects (45)
that. the relaxation rate y(T) « T3 for T << 6D. Conse-
quently the assumption of Equation 5.31 is not consistent
with theoretical anticipation.

The Mills Theory is successful in predicting the
concentration and temperature dependence of the deviations
from the MR in aluminum alloys at low temperatures. The
theory is not successful at higher temperatures, since the

observed temperature dependence is not consistent with

that theoretically anticipated for y(T).

SUMMARY AND IMPLICATIONS FOR FURTHER STUDY

At low temperatures, the two-band model and the
Mills Theory are both consistent with the deviations from
the MR observed in the resistivity of aluminum and dilute
aluminum alloys. Both theories (Equations 5.12 and 5.26)

predict deviations from the MR of the form

oof('1‘)
+ g (T)

 

A(T) 6.1

Do

so it is necessary to weigh the relative merits of the two
theories on the basis of the functions f(T) and g(T). The
a and 8 parameters of the two-band model and the y(T) and
T3 terms of the Mills Theory have all shown inconsistencies
between the experimentally-determined and theoretically-
anticipated values.

Two-band effects have been used in considering
Hall field reversals (46) in aluminum and phonon drag
thermopowers of aluminum alloys (43). A coordinated effort
of resistivity (and associated pressure measurements),
thermopower, Hall effect, magneto resistance and thermal

conductivity measurements on the same high purity aluminum

and well-characterized dilute aluminum alloy samples could

73

74

provide evidence as to the relative importance of the
Mills or two-band theories.

There is a further, more specific test. The Mills
Theory may be construed as implying (see Figure 22) that
for p0 ~ 10-5uQ-cm, the T2 and T3 terms in the resistivity
of aluminum should be negligible. The two-band model
(Equation 5.12) predicts that for all pure samples in the
region, po >> pph’ the deviation from the MR is proportional
to the phonon reSistivity. An experimental measurement of
the resistivity of pure aluminum (RR > 105) would be par-

ticularly useful in weighing the relative merits of the

two theories.

REFERENCES

10.

11.

12.

l3.
14.
15.
l6.

17.

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76
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24. I.A. Campbell, A.D. Caplin and C. Rizzuto, Phys. Rev.
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25. R. Luck, Phys. Stat. Sol. 18, 49(1966).
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29. G.T. Meaden, Electrical Resistance of Metals (Plenum
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Ch'iang and V.V. Eremenko, Zh. Eksperim. i Teor.
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N.W. Ashcroft, Phys. kondens. Materie 3, 45(1969).

APPENDICES

APPENDIX A

FORTRAN LISTING OF COMPUTER PROGRAM FOR

CALCULATING RESISTIVITIES

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ll“! N) N)
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000000 ~ooo.moom .ooo.mmmm .ooo.mmm: .ooo.mowm ~00). .ome .ooo.m¢mmHH m
«accoch \mcn <~<i ma
\oHooo.o .ooooo. .omooo. ~m¢ooo. .mnooo. ‘mHHoo. .ocHoa. H mm
~Homoo. .o:moo. ~nnmoo. .mHmoo. .MJmoo. xonm oo. memoo. Ho moo. H :m
.oowoc. .m03cc. .c.¢oo. .«.¢oo. .:.:oo. .m.¢on. .m.soo1..mH¢oo a flu
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APPENDIX B

TYPICAL DATA OUTPUT

 

 

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APPENDIX C

ANALYSIS OF ERRORS

ANALYS IS OF ERRORS l

Resistivity Measurements

 

For most samples, the room temperature resistivity
was calculated by substituting Equation 4.2 into Equation

4.1 to get

where m is the mass of a wire sample of length 20, resis—
tance R and density d. Table C.l lists the variable, the
assumed standard deviations, s, approximate magnitudes, and
the fractional standard deviations,<Sof the variables.

TABLE C-l.--Standard deviations for measurements using
Equation C.l.

 

 

Variable (units) Approximate s 6
Magnitude

m (grams) .10000a .00001 10-4

m (grams) .050000b .000001 2x10-5

R (0 ohms)C .50000 .00007 1.4x10‘4

d (gm/cm3) 2.6984d .0003 10-4

20 (cm) 80.000 .008 10"4

 

aWhen using Mettler B-6 semi-micro balance.
bWhen using Cahn microbalance.

cMeasured by comparing voltage drop with that
across a 10 ohm, .005% standard resistor.

dThe value for pure aluminum.

97

98
The density of an alloy was calculated using

d (alloy) = 2.6984 + 3% c. c.2

where c is the alloy concentration, and Bd/Bc was calcu-
lated from data tabulated by Pearson (27). The data for
Bd/ac values is reliable to about 5%. So, assuming the
reliability of c is 1% and since for copper, zinc and
gallium as solutes, Bd/Bc = .01 gm/cm3—at%, a one atomic
per cent alloy has a fractional standard deviation for
the density of about 3 x 10’4. Using these data, the
anticipated fractional standard deviation for the room

temperature resistivity of a dilute alloy is

/2 -4

l
2 + (1.4 + 2)2) x 10 C.3a

5 = (12 + 1

ll

.04%. C.3b

And for a concentrated alloy,

1/2 _4

2 + (1.4 + 2)2) x 10 C.4a

6 = (12 + 3

ll

.05%. C.4b

The errors in measurement of £0 and R were assumed to be
completely correlated, and as such were added linearly.

The former estimate is more appropriate for the concentra-
ted magnesium alloys, since Bd/Bc is << .01 for magnesium

alloys.

99

The analysis of errors for the cases in which
Equation 4.7 was used is more easily done if the denomi-

nator of Equation 4.7 is rewritten such that

 

Now since nli = 20 as defined for Equation C.l, the frac-
tional standard deviations for all terms except the summa-
tion are the same as in the above case. The fractional
standard deviation of nii was kept the same as for 20 by
"leap—frogging" the potential probes in the formation of
the separate fli. The standard deviation of £1 is the same
as for 20, so the fractional standard deviation for the
summation is

8 = n'1/2 (n6

2 + SR), C.6

2

noting again that Ri and ii are correlated. Assuming the
correlation between £0 and the summation to be weak, for

n = 6 (the usual case), we have

1/2
60 = (12 + 12 + 12 + (2.5 + %L§)2) x10‘4 C.7a

= .04%. C.7b

If n had to be chosen much greater than 6 or 7, another

wire sample was used.

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19:33-15 d . -,<‘ -‘ . ."L .-<-v. o 3:" Cd bad.

100

Temperature-Dependent Resistivity

 

The resistance of the wire samples as mounted in
the cryostat was determined by voltage comparison with a
standard resistor. Since only the resistance and the room
temperature resistivity enter into calculations of the
temperature-dependent resistivity, the fractional standard
deviation for errors in the temperature-dependent resis-
tivity were assumed to be = .05%. Table C-2 compares the
room temperature resistivity reported by other investi-

gators with that measured in this study.

TABLE C-2.--Phonon resistivity of aluminum at 273.16°K.

 

 

Investigator p(uQ-cm)
This study 2.429
This study (two-band correction) 2.427
Seth and Woods (11) 2.429
a 2.44

 

aL.A. Hall, Survey of Electrical Resistivity
Measurements on 16 Pure Metals in the Temperature Range
0 to 273°K (U.S. Natl. Bur. Std. Technical Note 365, 1968).

Errors in measurement of temperature were .Ol°K
in the range of the platinum resistance thermometer and
decreasing from .01°K to .001°K at 4°K for the germanium

resistance thermometer range.

101

Errors in Calculations of Deviations
from Matthiessen's Rule

 

Deviations from the MR were calculated using
Equation 1.5. We examine the error at 295°K, 70°K, and in
the very low temperature region.

At 295°K, i .05% of pph is i .001 uQ—cm and the

error in the term, — po, ranges from :_.001 uQ—cm for

DA
dilute alloys to i .002 uQ-cm for the concentrated alloys.
Thus errors in A(295) are about i .002 uQ-cm. Now from

Figure 16 it may be seen that

 

D
o _ _ -1 .
T — l + 14 (HQ CHI) 00. C.8
The fractional error in A/pO is given by
+ .OOZuQcm 0 ~ _
— p T0 = ———°°°§“90m [1 + 14(00cm) 100]. c.9

O O

For the most dilute alloy, 00 = .008 uQ—cm, so the frac-
tional error is 25%. For the most concentrated alloy,
00 = .76 ufl-cm, so the fractional error is 3% which is
the order of magnitude of the agreement with Seth and Woods
(11) indicated in Figure 16. As derived in Equation 4.20,
the errors due the pressure corrections may be as large
as 5%.

At 70°K, A is as large or larger than it was at

295°K. However pph has fallen to about one-sixth of its

room temperature value. The net result is that A has

102

error bars from 3% for the concentrated alloys to about
8% for the dilute alloys.

At low temperatures, Equation 5.4 shows that

A = cT3. c.1o

Since this is in the region where pph is increasingly neg-
ligible with decreasing temperature, one can calculate the
temperature at which A has an anticipated error of 100% as

the temperature where

since this means 0A - po equals the standard deviation of

both. Solving for T from C.ll and C.10, we have

5 x 10'4 p0 1/3
T(100%) = 0.12

 

C

-7 3

For 00 = .10 09cm, c = 3 x 10 uflcm/°K , so

 

[E x 10"4 x 10’1 1/3
T(100%) = _7 W °K C.13a
L. 3 x 10
= 5°K. C.l3b

The error decreases to the values discussed for 70°K as

T-3.

 

lY. Beers, Introduction to the Theory of Errors
(Addison-Wesley, Reading, Massachusetts, 1958).

 

 

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