FERMI SURFACE STUDIES OF ALUMINUM
AIND DILUTE ALUMINUM—MAGNESIUM ALLOYS
USING THE MAGNETOTIIERMAL OSCILLATION. TECHNIQUE

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
JOHN C. ABELE
196.8

1-H“ I‘:

This is to certify that the

thesis entitled

FERMI SURFACE STUDIES OF ALUMINUM
AND DILUTE ALUMINUM - MAGNESIUM ALLOYS

USING THE MAGNETOTHERMAL OSCILLATION TECHNIQUE
presented by

John C. Abele

has been accepted towards fulfillment
of the requirements for

PhoDo degree in PhYSiCS

6/ / fl/AH/

0-169

ABSTRACT

FERMI SURFACE STUDIES OF ALUMINUM
AND DILUTE ALUMINUM-MAGNESIUM ALLOYS
USING THE MAGNETOTHERMAL OSCILLATION TECHNIQUE

by John C. Abele

The third zone Fermi surface of aluminum and dilute
aluminum - magnesium alloys has been measured using
the magnetothermal oscillation (MTO) technique. Measure-
ments for pure aluminum.are found to be in good agree-
ment with the results of Gunnerson, Gordon, Bohm and
Vol'skii. These results can all be interpreted on the
basis of a pseudopotential model due to Ashcroft.

For aluminum alloys with up to 0.4 at.% magnesium
the extremal cross-sectional areas of the Fermi surface
were found to decrease by as much as 10%. This effect
can be accounted for by a simple rigid band valence
model.

The so-called Dingle temperatures, which represent
the collision broadening of the Landau levels, have also
been measured for the alloys. A comparison is made of
the lifetimes calculated from the Dingle temperatures
(via A.D. Brailsford's theory) with those obtained from

the electrical resistivity.

FERMI SURFACE STUDIES OF ALUMINUM
AND DILUTE ALUMINUM-MAGNESIUM ALLOYS
USING THE MAGNETOTHERMAL OSCILLATION TECHNIQUE

BY

John C. Abele

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics

1968

ACKNOWLEDGMENTS

I am indebted to Professor Frank J. Blatt who
provided both guidance and encouragement during the
course of this research.

Support from the National Science Foundation,
the National Aeronautics and Space Administration,
and the Michigan Institute of Science and Technology

is also gratefully acknowledged.

ii

TABLE OF CONTENTS

ACKNOWLEDGMENTS. . . . . . . . . . . . . . . . .

LIST OF TABLES . . . . . . . . . . . . . . . . . . .

LIST OF FIGURES . . . . . . . . . . . . . . . . . .

LIST OF APPENDICES . . . . . . . . . . . . . .

CHAPTER I

CHAPTER II

CHAPTER III

INTRODUCTION TO FERMI SURFACE STUDIES.
1.1 Introduction . . . . . . . . .

1.2 The Worth of Fermi Surface
Measurements . . . . . . . .

1.3 Experimental Fermi Surface Studies

1.4 Theoretical Methods for
Determining the Fermi Surface. .

1.5 Comparison of Theory and
EXperiment . . . . . . . .

THE THEORY OF OSCILLATORY QUANTUM
EFFECTS . . . . . . . . . . . . . . .

2.1 Introduction . . . . . . . .

2.2 History of Landau Diamagnetism .
2.3 Lifshitz - Onsager Theory.

2.4 The General Result . . . . . .

THERMDDYNAMIC ANALYSIS OF THE MAGNETO-
THERMAL EXPERIMENT . . . . . . . . . .

3.1 Introduction and Working Hypothesis.
3.2 Thermodynamic Calculation. . .

iii

Page

10
10
10
12

15

19
19

20

CHAPTER IV

CHAPTER V

CHAPTER VI

CHAPTER VII

CHAPTER VIII

BIBLIOGRAPHY

iv

THE EXPERIMENTAL ARRANGEMENT USED
TO STUDY MTO. . . . . .

4.1 Introduction. . . . . . . .
4.2 The Experimental Cryostat . .

4.3 The Measurements of Small Diff-
erential Temperatures . . . .

4.4 Magnetic Field Measurement. .
4.5 Field Sweep Techniques. . . . .
4.6 Data Acquisition and Reduction.
SAMPLE PREPARATION. . . . . . . .
5.1 Introduction. . . . . . . . .
5.2 Necessary Precautions ... . . .
5.3 Crystal Growth. . . . . . . .
5.4 Resultant Crystals. . . . . .
THE FERMI SURFACE OF PURE ALUMINUM.
6.1 Introduction. . . . . . .

6.2 Existing Fermi Surface Models
and Data . . . . . . . . .

EXPERIMENTAL RESULTS FOR AL-MG
ALLOY S O O O O O O O O O O O O O O

7.1 Introduction. . . . . . .

7.2 Changes in the Aluminum Fermi
Surface Induced by Alloying . .

7.3 Amplitude Dependence. .
7.4 Amplitude Comparisons . . . . .

GENERAL CONCLUSIONS . . . .

Page
28
28

28

33
4O
41
47
58
58
58
59
65
69

69

69

75

75

76
81
94

98

102

LIST OF TABLES

Table Page
5.1 . . . Sample Information . . . . . . . . . . 68
7.1 . . . Dingle Temperature Results . . . . . . 87
7.2 . . . Scattering Results . . . . . . . . . . 92
7.3 . . . Amplitude Comparison . . . . . . . . . 98

LIST OF FIGURES

Figure

2.1 Landau Levels and the Fermi Surface.

3.
4.
4.
4.

4.
4.
4.
4.
4.
4.

4.10

4.11
4.12
5.
5.

5.
6.

l
1
2

3

4
5
6
7
8
9

l

2

3
1

Thermodynamic System . . . . . . . .
Vacuum System
Rotatable Sample Holder.

Temperature Dependence of Resistance of
Carbon Resistors . . . . . . . . .

Thermal Bridge and Associated Electronics.

D.C. Equivalent Circuit.

Single Amplifier Sweep Control . .
Manual Integrator Sweep Control. . . . .
Automated Sweep Control.

Dependence of Ca upon Magnetic Field .

Hyperbolic Dependence of MTO and deA
Amplitudes upon T/H. . . . . . . . .

Steady Field Rotational Techniques .
Field Mbdulation Electronics .
AluminumrMagnesium Phase Diagram .

Magnesium Concentration after Single Pass
Zone Refining . . . . . .

Sample Mounting. . . .

Nearly Free Electron Construction for
Aluminum 0 O O O O O O O O O O O O 0

vi

Page
14
19
29

32

35
37
39
42
42
44

50

52
55
57

61

64
66

71

vii

Figure

6.2

6.3

7.1

7.2

7.3

7.4A

7.4B

7.5
7.6
7.7

7.8

7.9
A.l
B.1
C.1
D.l

E.1

F.1

Ashcroft's Model for the Third Zone Surface
of Aluminum. . . . . . . . . . . . . . . . .

Comparison of Data with Ashcroft' 3 Model in
{110} Plane. . . . . . . . . . . . .

Angular Variation of [S Orbit near (110)
in {100} Plane . . . . . . . . . . . .

Variation of [3 Orbit Area along (110) with
Mg Concentration . . . . . . .

Variation in [3 Orbit Area along<100) with
'Mg Concentration . . . . . . . . . . . .

Angular Variation of 3‘ Orbits near (110)
in the {110} Plane . . . . . . . . . . . .

Variation in ‘6‘ Orbit Area along (110) with
Mg Concentration . . . . . . . . .

Angular Variation of X‘ Orbits in {100}P1ane .

Amplitude Dependence in the {100} Plane.
Dingle Plots along (110) . . . . . . . . .

Results of Brailsford's Scattering Calcula-
tion 0 O O O O O O O O C 0

Temperature Dependence of the MTO Amplitude.
Extraneous Heating Effects . . . . . . . .
Crush-Type Lead O-Ring Seal.

Differential Thermal Bridge Schematic.
Magneto-resistance Probe . .

Harvey-Wells Electromagnet Field
Characteristics. . . .

Squaring Circuit for-% Sweep .

Page

73

74

77

79

80

82

83
84
85

87

90
93
107
110
112

114

115

117

viii

Figure

G.1 Least Squares Program for Finding
Frequencies. . . . . . . . . . .

G.2 Least Squares Program for Finding
Frequencies, cont. . . . . . .

G.3 Least Squares Program for Finding
Frequencies: Typical Output. .

H.1 Filter Circuits.

H.2 Example of Application of Narrow Band
Filter using 1/H Sweep . . . . . . .

H.3 Low Frequency Data . . . . . . .

H.4 High Frequency Data .

Page

119

120

121

122

123
124

125

LIST OF APPENDICES

Appendix

A Heat Input via the Rotating Gear Mechanism .

B

C

Crush-Type Lead O-Ring Seals .
Differential Thermal Bridge Schematic.
‘Magneto-resistance Probe . . .

Harvey-Wells Electromagnet Field Character-
istics . . . . . . . . . . . . . . . . . .

1
Squaring Circuit for‘fi Sweep .

Least Squares Program for Finding Fre-
quencies . . . . . . . . . . .

Filter Circuits.

ix

Page
107
110
112

113

115

116

119

122

lllll‘wllll.

CHAPTER I

INTRODUCTION TO FERMI SURFACE STUDIES

1.1 Introduction

 

Over the last ten years an enormous amount of effort
by both experimental and theoretical solid state physicists
has been directed toward the determination of the Fermi
surface of a large number of metal and alloy systems. The
purpose of this first chapter is to relate some of the
reasons for such an extensive study of Fermi surfaces and
to summarize the progress, both eXperimental and theo-
retical, that has been made. As the nature of Fermi sur-
face measurements is unraveled it will become apparent
that the techniques have become diversified as well as
sophisticated. This is to say that many various types of
measurements utilizing different physical phenomena can
be used to probe the Fermi surface. On the theoretical
side there have been some new ideas but much of the work
has been the adaptation of the earlier methods to high
Speed computer calculations.

1;2_ The Worth of Fermi Surface Measurements

The reSponse of metallic elements as well as many other

solids to their environment depends in many interesting

1

2

cases on the nature of the electron distribution. One way
to characterize the distribution is by plotting surfaces
of constant energy in a wave number or‘k Space. Many of
the important prOperties of the carriers can be related

to such surfaces.

at 1
(e.g. mc* =i—1rg-é , v =5(—3—%).... )

AS the temperature of a Fermi Dirac system is reduced,
the low-lying energy states become filled up to some
fixed energy Ef and higher energy states are unfilled.

A well known butvery important preperty of such a system
is that the usual excitations which result in the trans-
port of energy are largely limited to those states whose
energy is very near Ef. Thus the contours of energy in

'3 Space having energy near Ef are of extreme importance
to the understanding of electronic processes in solids.
The physically interesting parameters such as mg, v, etc.
previously mentioned are to be evaluated at Ef. This then
is briefly the motivation for Studying the Fermi surface
of metals.

1.3 Experimental Fermi Surface Studies

There are many methods for determining Fermi surface
parameters. Some of the methods currently in use will

be briefly surveyed here and references to much more

complete descriptions given.

3

First, there are the oscillatory effects due to the
Landau diamagnetism. These effects fall roughly into
two categories: those related to equilibrium phenomena
and those which require some kind of additional excita-
tion mechanism (e.g. electric field, thermal gradient,
etc.) and are thus non-equilibrium phenomena. Examples
of the first type are the deHaas-van Alphenlu7 and
magnetothermal oscillationsa'lo. The deHaas-van Alphen
measurement, which is perhaps the most common of all
the Fermi surface measurements, has undergone quite a
series of improvements from the old "point by point"
susceptibility balance to the field modulation (i.e.
derivative) technique of the present. Those included in
the second category are oscillatory electrical resis-
tivityll’12 (Shubnikov-deHaas effect), oscillatory
thermal resistance, and oscillatory thermopower13’14.
The frequency of the oscillations observed in all of
these eXperiments is proportional to the extremal cross-
sectional areas of Fermi surface in a plane perpendicular
to the applied field H. From the temperature and field
dependence of the amplitude of these oscillations one
should be able to obtain information about the scattering

of the carriers as well as their effective masses. The

amplitude measurements are probably more difficult to

4
interpret in the nonequilibrium phenomena because the
amplitude of the one set of oscillations is not indep-
endent of other parts of the Fermi surface.

Another important class of eXperimentS is the caliper
type measurements using geometric resonances. Among
them are the Gantmakher r.f. size effect16 and ultra-
sonic attenuation17-20. Both of the experiments measure
the caliper or extremal diameters of the Fermi surface
in a plane perpendicular to the applied field H. Since
r.f. size effect resonances are produced by the relation
between the diameter of the carriers‘orbits and the
width of the sample, the preparation of smooth plane
samples is critical. It Should in principle be possible
to deduce some information about the boundary Scattering
using this eXperiment.

Yet another important type of measurement is the cyclo-
tron resonance eXperimentZI. This is a temporal resonance
effect in.which conduction electrons pursuing helical
paths under the influence of a magnetic field H applied
parallel to the surface are intermittently brought into
the skin depth of a metal where they are acted upon by
the microwave electric field. A resonance can occur when
the component of the microwave electric field in the

plane of the surface undergoes an integral number of

5
cycles during the time it takes a carrier to make one
orbit. This experiment gives the average value of the
effective mass over the electron orbit.

The last class of Fermi surface measurements to be
mentioned is magnetoresistance22’23. At low fields it
is difficult to separate Fermi surface anisotrOpies
from relaxation time anisotrOpies. In the high field
region the behavior of the magnetoresistance is largely
dependent upon existence or non-existence of open orbits.
Thus the method can be used as a t0pologica1 check of
prOposed Fermi surfaces or to choose between alternative
models.

There are other techniques which can be used to shed
light on the Fermi surface of a particular material
but these are the ones most commonly being used.

In addition to the relatively straight-forward eXper-
iments on pure materials, one can try to alter either
the band structure or the Fermi energy. The alterations
are made in the nature of perturbations of the usual
Fermi surface. One possible alteration is to introduce
an impurity which can affect the scattering, the electron

24,25

concentration, or the lattice structure . Another

way to manipulate the surface would be to apply hydro-

26,27

static pressure . One would hope that changes due

6

to such effects could be accounted for within the frame-
work of the current theories, although sometimes this
process is muddled because of the limited amount of
knowledge available about the physical changes that
accompany alloying.

Much effort (both experimental and theoretical) has
been focused on the problem of magnetic breakdown in

28-30. Magnetic breakdown can give

the last few years

information about the nature of the band gaps in various

directions and is an interesting effect on its own but

it has not been used as a general tool to probe the

Fermi surface.

1.4 Theoretical Methods for Determining the Fermi Surface
At present there exist several methods by which one

may calculate the band structure and hence infer the

Fermi surface for a given material. The applicability

of a given method depends largely upon the type of

electronic states the valence electrons occupy. The

methods most commonly used are the tight binding method,

the cellular method, the augmented plane wave (APW)

method, the orthogonalized plane wave (OPW) method, the

pseudopotential method and the Green's function method.

The applications of these methods are described very

briefly below and references to much more complete

expositions are given.

The tight binding method is best applied to core
States and localized outer electrons such as one finds
in the transition metals. The cellular method dates
back to the work of E. P. Wigner and F. Seitz. The APW
method is based upon the cellular method, which assumes
a Spherical potential inside non-overlapping Spheres
centered on each lattice point and a constant one outside
the Sphere331’32. If there is directional bonding the
simple APW method will provide unsatisfactory results.
There are many materials whose energy bands can be
classified as nearly free electron (NFE). Among these
materials are the alkali and alkali-earth metals; the
Simple polyvalent metals A1, Cd, Zn, Mg, etc., as well
as semi-metals and semiconductors. For many of these
materials a tolerably good approximation to the actual
Fermi surface can be obtained by the one OPW or NFE

33 .
. The accuracy 13 increased

construction of Harrison
when more OPW's are included, eSpecially near the zone
boundaries. The pseudOpotential method is based on the
OPW method, although it could be based on an APW-type

calculation. The OPW method requires that electrons be

classified as either core or itinerant electrons. Conse-

quently for materials containing partially-filled d

8
bands the OPW or OPW-based pseudOpotantial cannot be
expected to yield a good description. The Green's
function method is, like the APW, based on the muffin-
tin crystal potential model34. In the Green's function
method the coefficients in the reciprocal lattice
vector expansion are Specified and coefficients in the
atomic orbital expansion are determined variationally,
This is just the reverse of the APW method and is use-
ful in that the secular equation is usually much
smaller than in the APW method. The Similarity between
these two methods has been eXplored by Ziman35. Relati-
vistic corrections become important as the atomic number
becomes greater than about 55 to 70.

1.5. Comparison of Theory and Experiment

 

The Fermi surfaces for the simple metals have by
this time been fairly well determined, both experi-
mentally and theoretically. These are the metals whose
outer shells contain only S and p type electrons and
are thus amenable to OPW or OPW-based pseudopotential
methods. Although these methods have a few variable
parameters which are adjusted to fit the experimental
data the agreement of the angular variation of the
extremal cross-sectional areas is remarkable.

The experimental results on the noble metals are

9
well accounted for by the APW calculations.

The transition metals are of current interest. The
problem of obtaining good samples is rapidly being
overcome. The computer APW calculations (some relativ-
istic) which can be easily adapted for different mater-
ials are producing models which agree well with eXper-
iment for both these elements and associated intermetallic
compounds.

The rare earths are still relatively unexplored
because of their complicated metallurgy?6

In addition to completing measurements of Fermi sur-
faces the next few years will witness the measurement of
the parameters ’C and mi and their anisotrOpy over the
Fermi surface. As will be shown later these quantities
can be determined from Studies of the amplitude varia-
tion with temperature and field using the techniques
presently available. Such values are not averaged over
pieces of Fermi surface but are averaged over only the
extremal orbit causing the oscillations. The effect of
impurities upon these quantities and upon the Fermi

surface itself will also involve much study.

CHAPTER II

THE THEORY OF OSCILLATORY QUANTUM EFFECTS

2.1 Introduction

 

In this chapter a short historical Sketch of Landau
diamagnetism is given. This is followed by a brief
derivation of the Onsager - Lifshitz relation and a
presentation of the formal result of Lifshitz and
Kosevitch for the oscillatory free energy. The exten-
sions of this theory as well as some interesting side
effects are presented.

2.2 History of Landau Diamagnetism

 

The oscillatory effects which are periodic in recip-
rocal applied field have a common historical origin. In
Leiden in 1930 W. J. deHaas and P. M; van Alphen37’1
first observed the oscillatory behavior of the magnetic
susceptibility of a Single crystal of bismuth. This
effect, which bears their names and will hereafter be
referred to as the deA effect, was at first thought to
be a particular prOperty of bismuth. It had been shown
by Bohr38 in 1911 that the diamagnetic susceptibility
39 in

of a classical free electron gas is zero. Landau

1930 showed that Bohr's result is not valid when one

10

11
applies quantum mechanics to the motion of the electrons.
By 1939 the early work of D. Shoenberg4O had shown that
the deA effect could provide detailed information
about the electron energy Spectrum near maxima or minima
in the energy bands as well as the number in each part.
In a supplement to this article Landau, using the
effective mass approximation, enumerated the eigenvalues
and eigenfunctions for electrons in a magnetic field
which obey a quadratic diSpersion relation. Apart from
the region around a band edge there is little justifica-
tion for the use of a quadratic diSpersion relation.

By the time that Onsager41 had related the 1/H per-
iodicity of the deA oscillations to the extremal cross-
sections of Fermi surface normal to the field direction,
the effect had been seen in many materials: Bi, Sb, Mg,
Hg, Cd, Zn, Be, C, Ga, In, Sn, T1, and Al. The central
result of the Onsager relation is that even though the
energy may be quantized in a complicated manner the
orbital areas of the charge carriers in momentum Space
can be quantized in a simple way. Shortly thereafter,

IgM. Lifshitz and A.M. Kosevitch42

used the quasi-
classical Bohr-Sommerfeld quantization condition to
quantize the areas and obtained the grand canonical

potential for charged quasi-particles obeying a general

12
diSpersion law 8 = 8 (kx, ky, kz). This result forms
the basis of the presently accepted result.

2.3 Lifshitz - Onsager Theopy

 

Consider the motion of a charged quasi-particle in
the state k under the arbitrary diSpersion relation
5 = g (kx, ky, kz) (2.1)

From the Lorentz equation,
k ‘2 v x H 2 2
-— _ tic — -— ( ‘ )

it is easily seen that since changes in‘k are normal
to §,the quasi-particle's k-Space trajectory remains
in a plane normal to H. Since 3 is normal to surfaces

1
of constant energy ()5 = " VkE ) we see that the

'fi
quasi-particle trajectory in k-Space is along the
surface of constant energy in a plane normal to E. It
follows from.Eqn. (2.2) that the projection of the
real Space orbit has the same Shape as the k-Space
orbit though it is scaled by the factor -EE and

rotated by' Tf/Z.

In a magnetic field the momentum is generalized to

s. =fis+ A

film

It is convenient to direct the magnetic field along the

z axis and use the vector potential ‘A = (0, Hx, 0).

13

In this case the Bohr-Sommerfeld quantization relation

{2'3
becomes

12% feds

so that the k-Space orbit along a line of constant

(n +'r )h, n = 0,1,... ,

(n +II')h .

energy encloses an area A normal to the field H and

quantized by

ZZTeH

fie (n +79) , n=0,1,2,... (2.3)

A(E, kz) =
This is the Onsager result.

Eqn. (2.3) can be inverted to yield E = gn(kz,H).
As a simple example consider an ellipsoidal Fermi sur-
face. Cylinders with the allowed cross-sectional area
are Shown in Fig. (2.1)A. Fig. (2.1)B Shows that E is
not a constant for a given Landau tube but varies with
kz.

At T=0 the occupied States are those with energy
equal to or below the Fermi energy, but the quasiparticles
are restricted to orbits with enclosed area given by
Eqn. (2.3), the Onsager relation. Thus as H is increased

the number of quasi-particles on the outermost cylinder

with k < kf must decrease as its area

14

mommunm Heuom one pom mam>mq downed Aa.mv .wwm

 

 

 

Lit

 

 

 

 

 

15

surpasses that of the Fermi Sphere. The degeneracy of
the Landau tubes is very high and the density is pro-
portional to the field H. The Fermi energy does depend
upon the field but this is a second order effect and of
little importance unless one is in a region where only
a few quantum levels are left inside the Fermi surface
(i.e. the quantum limit). In addition to substantiating
these assertions the theory of Lifshitz and Kosevitch
shows that by far the major contribution to the oscill-
atory free energy comes from an extremum in the cross-
sectional area of the Fermi surface. One would expect
this Since in this region a large number of particles
must simultaneously make the transition to lower levels.

From Eqn. (2.3) we can infer the frequency of the
oscillations. We will bring the j th level up to the

Fermi level when H is such that

h.
AF c = j +\“
21-reH
The oscillations are thus periodic in l/H ‘with

frequency

Aphc
2fre

2.4 The General Result

The result of Lifshitz and Kosevitch for the oscill-

l6

atory part of the free energy is:

lhc

 

 

 

 

 

 

___. - fl’
eH 3/ E - ZHTF '—
ZWTCEE) C°S(6H Aext( F) + 4)
F " 2 X
osc IOZACEF’R ) [QB/2833111217 lkT
hmbz
l a kz2 ext
- 1r]:
Mr smc* “was 1:
cos e (2.4)
2m
where V = volume
k = Boltzmann's constant
T = absolute temperature in 0K
L32A1EF,kz) % curvature of the Fermi surface
I 3 k 2 = along the field direction
2 ext evaluated at the extremum
m*= cyclotron effective mass = E 2.2..
217' 36 Ef’ext
eH
wc*= —-
m*C

r'= phase constant, = 1/2 for quadratic diSper-
sion relation

g = Spectrosc0pic Spin Splitting factor
[Ts lifetime of a state on the extremal cross-

section of Fermi surface following Brails-

17
For Eqn. (2.4) to be valid it is sufficient that

we*"“ EF: k'I‘(( EF, and f/H )> 1.

The e-” 1 /u.) c*’E'

term in Eqn. (2.4) accounts for
collision broadening and was derived for lifetime
independent of energy and kz, for particles having
quadratic diSpersion relation by Dingle43. This cal-
culation has Since been generalized by Williamson44 for
an arbitrary diSpersion relation and by Brailsford45
for arbitrary Landau level line shape.

The cos-Agifgfif term46, modified for arbitrary g
47

value , is due to the Spin of the particles. From
this term we can see that when gmc*/mo = 1,3,5.... ,
the amplitude of the fundamental frequency will be zero.
Since mc*/mo can be determined from the temperature
dependence of the amplitude, anisotropic values of g
can be determined. Several workers have noted departures
from g = 2 of as much as 30% 48.
The effective mass mc* used in the Lifshitz and
Kosevitch calculation has been shown to be enhanced by
the electron-phonon and electron-electron inter-
actions49'51. A calculation of such an enhancement has

been discussed by Harrison in the pseudopotential

formalism”

l8
Eqn. (2.4) does not take account of magnetic break-
down effects since they are greatly dependent upon the
band Structure of a given material. The result also
does not include the additional harmonic dependence seen
in the so-called B-H effect. This effect has been
discussed at length by Pippardsz, Shoenberg53, and

Condon54.

CHAPTER III
THERMODYNAMIC ANALYSIS OF THE MAGNETOTHERMAL EXPERIMENT

3.1 Introduction and Working Hyppthesis

In this chapter the formal result of Lifshitz and
Kosevitch for the oscillatory free energy is used to
gain an expression for the oscillatory temperature of
a Single crystal. The calculation is made for the

simplified but practical system seen in Fig. (3.1).

Thermodynamic System
/

”7v 0
Bath ’ do 50m”'"°'b°" ‘7’” JM 0""
4_+ rumor. (Md ~—>— eddy curnnt
(thermal , beat. leak support hcatm
puma”); hmP. =7; ‘9

W7. .

 

 

 

 

 

 

‘Ha'I

Fig. (3.1)

In this calculation the Specific heat of the carbon
resistor mounted on the sample, together with that of
the associated wire and graphite support, is added to
the Specific heat of the sample. It is also assumed that

19

20
changes in the sample temperature are Small relative to
the average sample temperature.

Due to the power dissipation in the sample resistor
and eddy current heating arising from changing the mag-
netic field strength, the temperature of an adiabatically
isolated sample will gradually increase without bound.

A thermal contact or heat leak.with conductivity K and

CHL
K11

to maintain the average sample temperature constant. The

exponential time constant ’tq = is thus necessary
effect of this heat leak has also been included in the
analysis.
3.2 Thermodynamic Calculation

For a magnetic system the first law of thermo-

dynamics

dQ dU + aw

becomes

dQ dU + dw' - HdI (3.1)

By analogy with the chemical system one replaces P by
~H, and V by I.
Considering U = U(H,T) and I = I(U,T) we have

_ 19. 112
(ill - OH)T dH '1" 3T)H (1T

21

and

d1 = '3‘ng d'I‘ +%__)T dH (3.2)

For this system the ”energy equation”

bu _ 31’
537M” ‘ T 3T)V "
becomes
_ 3H
)T - -T '55)]: + H
Multiplying both Sides by'4§%)T we have
93 3;; 3H 3: H31
31".1‘ 3H)T T T)I BH>T+ HaH)T (3’3)
Applying
Bx L bz = _
TY>z n'xa x'r 1

to the first term on the right Side of (3.3) we have

311 T 113—156°“

Upon substitution of Eqns. (3.2) into (3.1) we find,
- .= LU - a1 22 - aI
dQ aw [3T)HH 5T>H:IdT + [3H)T H SENT] dH (3.5)

From the modified energy equation, Eqn. (3.4), the

second term on the right side of Eqn. (3.5) is obviously

22

...?Jl 3.!
CP - 3T)P + P 3T>P

so that in this system

ECU b1

CH = T)H ' H'ST)H

With these definitions Eqn. (3.5) becomes
.. . _ 21.

where
dQ is positive for heat flow into the System

dW' is positive for work done by the System
dW' has been reserved for joule heating of the carbon re-
Sistors, eddy current heating due to sweeping the field
or mechanical vibration of the sample relative to the
field as indicated in Fig. (3.1). The sources of the
temperature difference between the sample and the bath

T - TB are the dissipative effects dW' and the Landau

S
diamagnetism.

Now the rate at which heat flows between the bath

and the sample is

99.. ° _ - =- = 41's 21 d3 £W__'.(3.6)

where ,K = % , and AT = TS'TB

In order to continue this calculation one needs an

 

 

 

 

 

expression for €%%)H . Using I = % jLEM , Eqn. (2.4) is
rewritten in the form,
-b X
l D
2m’f
———-1 H
FOSC . VA,m3/2 “3‘ H + I! ) e (3.7)
sinh b1 T/H
where ‘ °' 1
F...= 2 Fosc
=|
3/2 gm"
2K(h-ea) coal mo
A‘ = 32A 35 3/2
Z ext
8‘ = 1.47x105 gauss/OK
fl = -21r£r ; Tr/4
XD =n/zrr K1:
fic
f1 = l—é—Aexth) = If
Thus .1
_. _1 )F _ - .1. 3ngc _ 1
Iosc — V‘Tfigfic )T " V 3H )T — Iosc
1. 1
2m: “2331
1" = _A Tlfl' [H3/2COSC-_Lj1 + Y1) e
08C ' 3 sinh bflT/H

This result may be simplified since for all cases discussed
21rf

here ‘71—'37) bIT/H and EXJH' Thus in taking deriva~

tives the slowly changing exponential can be ignored

relative to the trigonometric terms.

24

 

 

 

Thus
277’f ..b X /H
I. ATzwf SinLH£+VQ)e1D
03° — - l H sinh bLT/H
so that
I
BI S -A221Tf£ . 21Tf _b X
3T0 C)H=_T—H2 S1n('—fi”¢'+fil ) e .9 D/H x
( l-b’pT/H coth byT/H ) (3.8)
sinh be/H
Thus Eqn. (3.6) becomes
1
_ dAT 3108c 5111 dW'

where it is assumed that TB = constant, so

dt dt dt

To solve Eqn. (3.9) in the general case is a diff-
icult problem, but fortunately one can make some well-
justified simplifying assumptions. The magnitude of the
oscillations is a few millidegrees at most, and a heat
leak is used to insure the equilibrium value. Thus
AT/TS << 1 so that TS IV To, a constant. The separation
in field between adjacent peaks at the fundamental
( fl = 1) frequency is at most 5% of the field, so since

the dominant terms in Eqn. (3.8) change very Slightly

25
over a few cycles of oscillations, Eqn. (3.9) is
solved with the parameter evaluated at some arbitrary
field HO. One can make the following explicit assump-
tions for the oscillatory temperature evaluated at

mean temperature T and mean field H0.

0
HO-+¢K t

:13
ll

CH = constant

K 3 LA = constant

L
awl
'5;- = -P, a constant and not necessarily zero
since we will use this later to calibrate
the heat leak
Zng
H + 71 is eXpanded as
21rf
(——1— + ——J~ (1-%—+ 3.10
“6

(with error ((‘é—E) 2)
Ho

26

where
YI=Y-_2_1T._f.l and w gglr—f-L—
l 1 H ’ 1 H2
0
Thus Eqn. (3.9) can be written as
<1A1?_dw' co
0 I
CH—dt+1°AT=dt - E B181n(btit- fl) (3.11)
13/

where

2
B = ("”511 2c )3/ <2osflzqrrmo * 431E,
1 __9__z_ 2A 3/2 H035 1
3K2 '54

 

l-bgTQ/HO coth bLTO/HO )

and the other symbols have their previous definitions.

The general solution of Eqn. (3.11) is

‘ 00
AT 2+c-1€Ht+1EV-1r§——B
= e — x
10 CHI, (Cfi)+%2

. l . C 00
8111((031: -Y’ - tan 1 "Hi—A)

Thus the final expression for the oscillatory temper-
ature dependence (including addenda due to thermal
leaks) of a single crystal upon the magnetic field

expanded about T = TO and H = H0 is,

TS'TB = AT =AT

27
3/2 E 3/2
+ATp= -_2_KBTQH0 ( he )

OSC 2
I b A(gF,KZ) %

 

 

 

 

CH
I F K22 ext
00
21113 - 21ft '
sin[——L “2771» -T- W -tan 1( ...—.9.
H I Tosc ) cos(11rgm3)
2 3 2 2mo
13' V1 + (Tosc/ZWIQ ) fl /
-b X
44H 1-b,*1~0/H0 coth bgTO/HO P "E:-
e - -- ) +"+Ce 9‘ (3.12)
where
KB = Boltzmann's constant
’hc
f2 = I? Aext(EF)
Z‘N'ZK *c 5 1%"
b2 = BmQ 1 = 1.469x10 fig" gauss/OK
eh
tosc = period of oscillations
’UQ = CHL/KA = thermal time constant
XD = Dingle temperature = ’h/Z TrKB’l'
Z = lifetime of State on extremal cross-section
of Fermi surface
ATosc = oscillatory temperature change due to the
Landau diamagnetism
ATP = temperature change of the sample due to

 

dissipative heating effects

X

CHAPTER IV

THE EXPERIMENTAL ARRANGEMENT USED TO STUDY MTO

4.1 Introduction

 

 

In this chapter the design and operation of the low
temperature cryostat used in the present MTO Studies is
presented and discussed. The techniques necessary for
the measurement of small differential temperatures
are given. Various methods of data acquisition and the
subsequent data reduction techniques for each method
are also discussed. The chapter closes with a discuss-
ion of some new methods that were tried and suggests
further improvements.

4.2 The Experimental Cryostat

There were actually two cryostats designed and
built for this Study. The second design includes all
of the features found in the original cryostat; hence
only it will be discussed in any detail.

A schematic of this cryostat and pumping facilities
is shown in Fig. (4.1).

4 bath type. A large

The cryostat is a single He
mechanical pump, pumping on this bath, provides temp-

eratures down to 1.1°K. The pressure is regulated by

28

 

29

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

      

 

 

 

to mov to back
line try cartcsmn mamstat Pump ":9
-, i a I
3.?" rc gauge ' - I
ll .
90m ? e
S: F %
Walker mmostac
29"“,
deg/or
meshangz ,
a”? as}? #— *
fin _
00! 00’ and ’9
WP manqmws
. wad.
14006
PUMP
Pg“ raga bunsan,
«macho» or v23.
9° ,‘
f I
A
Sample
chombcr

 

 

 

 

 

Fig. (4.1) Vacuum System

30
either a Walker-type condom manostat or a cartesian
manostat. The bath pressure is measured by conventional
methods using either mercury or oil manometers except
for the lowest temperatures where it is determined
using a Universal Todd type MCLeod gauge. The pressure
in this pumping line can also be conveniently read on
a U.S. f 30 inch vacuum gauge and at low pressures
monitored by a thermocouple gauge.

In order to observe the MTO effect it is necessary
to have the sample thermally isolated. The degree of
isolation necessary depends on the time period of the
oscillations. In most cases this requires that the
pressure as read on the ionization gauge in the sample
vacuum line be < 5x10'4mm Hg. Since fluctuations in
this pressure would greatly influence amplitude measure-
ments and are difficult to control, the sample chamber
was evacuated to pressures of ( 10'5 mm Hg. A water-
cooled oil diffusion pump provided this pressure, which
was monitored by a Veeco ionization gauge. The water-
cooled pump was chosen Since an air-cooled model was
found to introduce vibrations in the cryostat via their
common supports.

The second apparatus has a rotatable sample holder.

This holder is rotatable about two mutually perpen-

31

dicular axes and is pictured in Fig. (4.2).

Because it is difficult, if not impossible, to make
a rotary vacuum seal which works reliably at He temper-
atures, the control rods pass up through the sample
chamber vacuum line to a pair of rotary Veeco seals at
room temperature. The control rods are made of 3/16
inch diameter Stainless steel tubes having a wall thick-
ness of 0.016 inches. Even so, enough heat passed down
them to make the sample holder heat up to about 100K
in about a two hour period while the bath remained at
4.2°K. The problem was really that these rods passed
directly from room temperature to the sample holder
which.was fairly well isolated from the bath. This was
overcome by wrapping a bare #22 Cu wire several times
around each rod and bolting it to the brass O-ring
flange. In addition, an approximately 8 inch section of
Glastic rod is used in place of the stainless tubing
near the room temperature flange because it has a much
lower thermal conductivity near room temperature.

Because turning the gears in the rotating device
produces frictional heating effects (calculation in
Appendix A), it was also necessary to solder a pillbox

3

of about 5 cm capacity onto the bottom of the sample

holder. This is filled through a curled lflb inch dia-

thermal
Sinking

lead

0- ring fl'

seal

Samp/Cb

vac uum —-’

 

 

Chamber

term nal
board

d

 

 

spiral 9¢or
set

helium
bath

 

 

 

1‘s

  

‘ tailor)
bush/n s
and kupqcrs

Siam/ass

 

Cont rol rods

m (a chamber
/— sa p
vacuum can

tqflon bushings

 

 

 

 

 

 

 

\\

and keep 2r:

/— detachable
lo war umt

,—— worm gear
-— pinion gear

b¢v¢l g¢ar

 

liamd
/— [ml/um [bi/lb“

lead o-rmg
scal

 

Fig. (4.2) Rotatable Sample Holder

pi.

33
meter stainless steel tube from the helium bath. Near
the lower end of this tube a brass flange and lead 0-
ring arrangement was used to maintain the sample chamber
vacuum.

The general type of O-ring seal used throughout
these apparati was suggested by M. Garber and is briefly
discussed in Appendix B.

The electrical leads for the carbon resistors and
other additional probes were twisted in pairs, encased
in teflon Spaghetti, and routed down the sample vacuum
line. The leads used are #36 Manganin wire with a
nearly temperature independent resistance of about
31 {I/foot. These leads, eight in number, are connected
to a terminal board a few inches above the sample,
facilitating easy changing of carbon resistors, etc.

4.3 The Measurements of Small Differential Temperatures

 

 

To study the MTO effect one needs a technique which
is able to measure differential temperatures of about
lO'SOK relative to a background temperature of 1°K to
40K. In order not to damp the oscillations, such a
sensor must also have a heat capacity which is small
relative to that of the sample. A differential thermal
bridge utilizing ordinary carbon resistors which

satisfies both these criterion is described below.

34
The use of carbon resistors as thermal sensors has
become quite common Since their introduction around
1950. The temperature dependence of their resistance is

usually fairly well represented by the equations

p

dR B' A/T
logeR ='T + B and '5; = :2

 

(4.1)

Thus as the temperature is reduced the resistance as
well as dR/dT increases quite markedly. Plots obtained
for the various resistors used in this study are shown
in Fig. (4.3). The choosing of resistors for use in a
given temperature range will be discussed after the
bridge circuit and its sensitivity considerations have
been presented.

The resistors to be used are modified in the follow-
ing fashion. A large number of resistors are glued to
a flat metal block and lapped on fine emery paper.
After this is done to both sides of the resistors the
metal leads are cut back as far as possible and twisted
pairs of #31 Manganin wire are soldered to each resis-
tor. Since two resistors are used in the differential
technique it is worthwhile to attempt some matching
procedure. In principle one should match the resistors
for thermal and magneto-resistance characteristics;

however, a simple matching of room temperature resis-

35

30,0000

10m

§

4000'

Re SIS tance (cums)

 

300.

 

l I l l
.2 .4- .6' .8 [.0

Reciprocal Tcmpcraturc ( °K” )

Fig. (4.3) Temperature Dependence of Resistance of
Carbon Resistors

36

tances after grinding has proved satisfactory. Electri-
cal insulation between the resistor and sample is
provided by first gluing a small piece of cigarette
paper to the sample and then gluing the resistor to this
paper. The glue used is G.E. 7031 varnish. The resistors
are placed so that they both have the same orientation '
relative to the magnetic field. After several (5-10)
thermal cyclings between 4.2°K and room temperature the
resistors tend to become noisy and are replaced. This
is probably due to mechanical strains in the resistors
as well as degradation of the glue bonds. In addition
to having greater thermal conductivity than most glues
the G.E. 7031 withstands thermal cycling far better
than the more quick-drying varieties such as Duco Cement.

A block diagram of the electronics associated with
the differential thermal bridge is Shown in Fig. (4.4).
The differential bridge with lock-in detection is
essentially the same as the method developed by Garber
and LePageSS. The a.c. method is used to reduce the
effect of thermal emf's in the leads as well as enabling
lock-in detection to be used. The low operating
frequency (20 to 60 Hz) of the bridge is chosen so that

the capacitive off-balance remains small and can either

be balanced out in the bridge or neglected. The lock-

37

 

 

{attenuator [-—

     
   

calibration
rtsnwons J:_2,

I nfc rmcz

l‘

j‘

cryostat

 

 

 

 

mfcmcc samp It

 

 

low mm.
IVI'afifla

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

output lock-In “PM" I
J 41
(NU/(JOIN! and
L— post -d¢t¢c£I0fl . 5 “VP -chart Anglo
- {Hans recorder

 

 

 

 

 

 

mum

field

Fig. (4.4) Thermal Bridge and Associated Electronics

38
in detection scheme reduces both the electrical pickup
and the effects of the sample vibrating in the magnetic
field. Lock-in detection at the fundamental bridge
frequency 00 eliminates the IZR self-heating effects
of the carbon resistors since these occur at 2a).

A complete schematic of the bridge circuit itself
is given in Appendix C. The simple d.c. equivalent
circuit Shown in Fig. (4.5) can be analyzed to yield
the voltage across the detector input due to a small
resistive change A.. The result is

Vdetector = A 85/)
[(2+} /R) r+}>][r+R+A] + AR

 

However in practice A((r,f) ,R, so that

Ydetector 1; A8 P (4'2)

[(2+ P/R)r+fl [1+9]

 

From Eqn. (4.1)
A T

Thus Eqn. (4.2) yields

==£Llsplrctr
iv— detector T2 E2+P /R) r+})][r+lg

EELJ) Rr dT
T2 Er+R)}) +2rlgE+lg

Hence the voltage output of the detector is propor-

(4.3)

39

 

detector

 

 

 

 

Fig. (4.5) D.C. Equivalent Circuit

40
tional to the temperature change of the sample. To
determine the absolute temperature changes one must
calibrate both the sample resistor to determine A and
the electronics using the Simple Stepping calibration
resistor which takes the place of A in Eqn. (4.2).
The power dissipation in the sample resistor is
typically around 10"11 watts. For the 33I1, 1/8 watt
resistors at 1.2°K a temperature change of l/HOK
correSponds to a voltage input to the detector (with
50K input impedance) of .005/1V. This figure represents
the practical lower limit of observation.

4.4 ‘Mggnetic Field Measurement

 

 

The accuracy with which one can determine the
frequency of the MTO oscillations and hence the cross-
sectional areas of the Fermi surface ultimately depends
upon the measurement of the magnetic field. The mag-
netic field was measured by three methods in these
Studies. Initially a rotating coil gaussmeter (Rawson
model 720) calibrated with an NMR probe was used. A
magneto-resistance probe (American AeroSpace Controls,
Inc. model MRA-ll) was also calibrated with the idea of
using it in a high field solenoid. The resistance of
the probe was found to be quite dependent on its

orientation relative to the field. While this is not

‘ I I II

41
desirable for field calibration work it could be used
for accurate orientation in a magnetic field. The
effects of thermal cycling and temperature dependence
in the l0 to 4°K range were less than .3%. A direct
reading calibration circuit was designed for use with
this probe and is presented together with the trans-
verse calibration curve in Appendix D.

A digital hall probe gaussmeter (F.W. Bell Inc.
model 660) was also used. The error of this device
when used with the NMR factory calibration curve was
reputed to be less than .22 .

The output of either of these devices was recorded
concurrently with the thermal oscillations on either a
2 pen Varian or a 3 pen Texas Instruments (Servo/riterII)
strip chart recorder.

4.5 Field Sweep Techniques

 

During the course of these measurements a very
useful adaptation of the sweep unit originally designed
by J. LePage was made56. The LePage circuit shown in
Fig. (4.6) is based on the fact that the Harvey-Wells
electromagnet control will produce a current through
the magnet preportional to an applied reference voltage.
Since the H(I) curve for the iron electromagnet is not

linear (Appendix E) one cannot simply make I=constant

42

./ F’fi

f .1”

magnet
to magnd} :rcfcrma
power sum/y

 

 

 

 

Fig. (4.6) Single Amplifier Sweep Control

 

 

 

 

 

 

magnet
1'" "”9”“ “'1 reference

power supply

 

 

Fig. (4.7) Manual Integrator Sweep Control

43
to get a linear field sweep. In the LePage circuit
dfi /dt was compared with a constant voltage and the
error signal used as a reference for the magnet control.
In order to make a very linear device with one
amplifier stage it was necessary to use very high
gain () 104) for which the circuit would oscillate.
In order to eliminate this problem the output of an
operational integrator was also added to the magnet
reference as shown in Fig. (4.7). The voltage which
was integrated could be varied manually in such a
manner as to keep the output of the error amplifier
minimized.

This sweep unit was automated by Simply using a
part of the error Signal to drive the integrator. One
then has a quick reSponding error amplifier and a
longer time constant electronic servo-integrator
operating together in a closed loop. Since the loop
is closed the accuracy of the control does not depend
upon the linearity of the integrator and its rate of
integration can be adjusted to minimize the output of
the error amplifier. A modified sweep unit utilizing
these principles has recently been constructed. A
simplified schematic of this revised circuit is shown

in Fig. (4.8).

44

 

 

 

 

 

Volta} 3171711)“

Val! Samwtr’ ' L/’ l
for : for 7' 6
1fi“f§ : l/‘IS

V «P5997 6 coast

 

Va% '0’”\ 1r
/ a I

“I L.\, '5

/ \ 6‘7

 

 

 

I /

 

 

 

 

 

 

 

__ may/m t

( 02339:: V “fiance and

is magnet) _ Power supply

 

 

 

 

Fig. (4.8) Automated Sweep Control

45

The versatility of this sweep control becomes
apparent when one considers the possibility of non-
linear sweep modes. The voltage reference with which
dyi/dt (or H) is compared by the error amplifier clearly
determines the sweep mode.

For example, if Vref=CH, then (CH-bH) is minimized
so dH/H =N3/b)dt, which results in a sweep which is
exponential in time.

In studying oscillatory effects which are periodic
in l/H (i.e. sin 21rf/H) it would be very useful to
have l/H proportional to time.

Since the time separation of two adjacent oscill-
ations is

(n+1) - (n) = f/H1 - f/H2 =(f/H1H2)(H2-H1)
thus
H2 - H1 = AH = Hle/f or AH”H2/f

Thus for a sweep rate H the time between successive

peaks will be
_ '= 2 ’
’L'OSC - AH/H H /fH . (4.4)
So for time periodic oscillations we set
HZ/fH = A (a constant)
Thus for the above sweep control the reference with

- 2
which H must be compared is simply proportional to H .

Time-based l/H sweeps have been described in the

46

literaturezo’57

. The conditions necessary for their
operation can be obtained from Eqn. (4.4).

dH/H2 = dt/f tosc
so that

l/H = -t/f EOSC + C
or

H = f tosc/ [(f tosc/H0)'€I °

But since
tosc = HOZ/HOf 9

HQZHO

(Ho'fiot)2 where H0 = H)t=0'

_ 2 ' ° _
H — HO / (HO-Hot) or H —

Thus direct or differential comparison of the field
with a time-based reference requires adjustment of the
initial sweep rate HO relative to H0. For this reason
the time-based sweep, whether mechanically or electron-
ically driven, is inherently more difficult to adjust
initially for the l/H characteristic. In the differen-
tial H method described above the time between oscill-
ations can be changed in the middle of the sweep by
merely adjusting the amount of H2 reference voltage
which the error amplifier compares with H.

It should be noted that this technique is not

limited to electromagnets but is in fact more directly

47
applied to superconducting solenoids. The field in a
solenoid is very nearly prOportional to the current
I. The voltage drop across a superconducting solenoid,
being almost entirely inductive, is LI which is propor-
tional to H - the correct pickup for the differential
l/H sweep. The reference with which to compare this
can be obtained by squaring the current I.

In practice the magnetic field is sampled by the
rotating coil gaussmeter and this signal is then
squared. The squaring circuit first takes the log of
H, then multiplies by 2, and then takes the antilog.
As the accuracy of the log and antilog functions are
crucial, a series of about thirty transitors were
tested to determine which ones had the best logarithmic
prOperties. Appendix F contains the relevant circuits
and notes on operation. More accurate squaring and
logarithmic devices with greater voltage range can be
obtained from the George A. Philbrick Co.58. Their
units are heartily recommended for the construction
of future sweep units.

4.6 Data Acqpisition and Reduction

In this section the various conditions under which

data is recorded are considered. Data analysis techni-

ques are discussed concurrently with each method.

48
Additional methods that were tried as well as suggest-
ions for modifications are also presented.

To obtain the frequency of a set of oscillations
periodic in 11H (i.e.a(sin 2fl'f/H) is in principle a
simple task. One merely plots peak numbers versus 1/H
and determines the frequency from the SlOpe. A linear
least squares program to do this has been.written and
is listed in Appendix G. A typical output complete
with statistical parameters, errors for each peak and
a 95% confidence interval for the period is also given
there. The error intervals are to a large degree in-
dependent of linear errors in the magnetic field
calibration and really reflect only the statistical
uncertainty in locating oscillation maxima (or minima).

To seperate out simultaneously occurring frequencies,

use can be made of the high pass filter characteristic

 

which the thermal leak introduces. (1//\/1+(tosc/21rtQ)2)
Examples of data taken using this technique are Shown in
Appendix H in Fig. (H.3) and Fig. (H.4). Two filters
designed to be used with the l/H sweep unit are also
shown in Appendix H with an example of data output.

For the more dilute alloys the amplitude of the

<ascillations is large enough relative to the noise

49
and drift to permit analysis. Excluding geometric
factors, Spin Splitting and Simple power law dependence
upon temperature and field, the amplitude is affected

by four factors. From.Eqn. (3.12) these four terms are

 

 

1 l-blT/H coth Ia T/H
I]. = ’ 12 =
fi]1+(tosc/2Wtq)§ sinh 111/11
I = ‘bIXD/H I = 1
3 e ’ 4 32A 1/2
3KZ ext

 

 

where bfl = 1.469xlO5 mc*/mo Q gauSS/OK
11 is the thermal damping due to the heat leak. For

field sweeps linear in time, I} increases with

osc
increasing field and thus in order to quantitatively
interpret amplitude effects one must measure UQ= CHL/KA.
'1; osc is very simple to obtain from the strip chart
records. A plot of 'UQ(H) which was measured at several
temperatures for the .08 at.% sample is shown in Fig.(4.9).
These measurements were made by noting the temperature
change of the sample caused by a calibrated heater via
the PL/KA term in Eqn. (3.12). With the l/H sweep
rose is constant, so that 11 is solely a function of
ta. From Fig. (4.9) ATE?- g; 7% over the field range

*where the Dingle temperatures were determined. Since

50

2.5-”

244

2.2." / . / 2 .03 .K

2.0d /'/././

 

 

I I

6 5 1315131316 To 2'2 24
H(ka) ~

Fig. (4.9) Dependence of Ta upon Magnetic Field

51
for most sweeps Tosc a: 21)"?2 (the cutoff region). From
Fig. (4.9) it follows that since the relative change

in tQ is only+ that of H

 

g: I 2(TOSC)29_E3=ESJ=_L§_I£_ o

By recombining the Taylor series expansion of 11(H)

H2-H H
IZHO / H

about HO we have

11 = I1)HO e

Thus from the ln(amplitude/H3/2) versus l/H plot we do

not measure XD but

2
- will 1:: XD - .040K

xD

where mp/mjalZ and H changes from 12 to 20 KG.

The term 12 arises from thermal broadening of the
Landau levels. In the deA effect the correSponding
term issdmply l/sinh(b£T/H). A plot of both of these
functions is given in Fig. (4.10) The cyclotron
effective mass mg ( = nee, /1.469x1o5 c/OK) can be
obtained by measuring the amplitude of the oscillations
at a given field as a function of temperature. From
Fig. (4.10) it is apparent that the deA amplitude is

more sensitive to changes in.blT/H than is the MTO

mE. com: momauwag <38 mam 0H2 mo moampamamn owaonummtmm 2.3.3 .wwm

 

52

 

 

 

 

I the
we 4... c... a. e... um .... eh. ... .... .... c... m. m. .1. e. o
I . -fifi
U‘o
1N.
.n
the ash \
.t
ewe 58 $3.;
the
to.
rho
.9
I? seem \
~ .9.
e3

 

53
effect, eSpecially in the region where bRT/H‘vl.6
Thus m* can be determined more accurately with the deA
effect than with MTO.

The term I3 = efbhxD/H represents the effects of
collisions. The so-called Dingle factor XD can be
determined by measuring the field dependence of the
amplitude at fixed temperature. Because the factor 12
for MTO goes through maximum at bnT/Havl.6 , for a
given m* (and hence bl) one can fix T so that
12 = .31 t 2% while H has been increased by a factor
of v 1.6. For the part of the aluminum Fermi surface
studied here (3rd zone), HM z..12 mo so that for fields
between 13.5 and 22 KG, I2 will be in this constant
region if T13 1.50K. Although this makes the measure-

ment of m* difficult it is very fortuitous for measure-

ments of XD. Thus for l/H sweeps where 11 = constant

log-SEE __._Bl(.D—+C (4.5)
H3/2 H

over a fairly wide field range and a Simple log plot
easily gives bLXD.

In addition to field sweeps another method of
taking data was used. This consists of holding the
magnitude of the field constant and rotating the sample

relative to the field or vice versa. In practice since

54
the magnet was motor-driven and had a simple angular
index it was most convenient to rotate the field. For
a given field strength H the Landau level whose cross-
sectional area is about equal to extremal area has
quantum'number
n = f/H
where f is proportional to the extremal cross-sectional
area of the Fermi surface. At an angle 9 away from the
first,
n(9) = f(9 )/H

Thus if one does some field sweeps to find f(e) in
major symmetry directions and to get a qualitative
idea of the angular variation of f, the steady field
rotation method can be used as a very accurate inter-
polation scheme. Data taken using this method is shown
in Fig. (4.11).

To deduce f(6) we count peaks through a monotonic
region of 9 from a known f. Then

f(6) = T H(n(6)-n) + f .

Since the quantum numbers n = f/H are usually from
30 to 500 in most of these measurements the method can
be quite accurate.

An interesting alternative technique which has not

been tried would be to parallel field modulate over a

55

89v

 

possessooa Hmcowumuom caowm utmoum AHH.qV .wflm

com

ohm

ON

Kauznbatf

 

56
given oscillation and ”lock” onto this oscillation as
the field rotates, adjusting the d.c. field to
accomplish this locking in on a gigep oscillation.
Thus with the phase of the oscillations constant

f/H = constant or f(e) a( H(e) . (4.6)
Hence a record of the H(e) necessary to satisfy
Eqn. (4.6) immediately gives f(9) to within a simple
multiplicative constant.

In addition to the a.c. bridge and lock-in amp-
lifier technique a field modulation method was also
tried. A block diagram of the electronics is shown in
Fig. (4.12). In this method the modulation signal via
PSD 2 could be compared with the standard a.c. bridge
technique via PSD 1. Although the signal to noise ratio
of the modulation technique could be increased by
increasing reference power to the bridge it was still
lower by a factor of about 5 than the Standard technique.

This method was therefore not pursued.

57

 

 

 

23 He 23 H:
h’._ am ’0’!!!
osCII/ator an:
IIItCV

 

 

 

 

 

 

 

 

P130513 ..‘f +7 Mlxcr ,

5'0”! 00! .
smsitlvc 73 1‘23 ”I _l0”d {Wail

3'52““; IN #-
. p ‘ K \[ K
thermal I 4F}
r

 

 

 

 

 

 

 

 

 

 

 

 

bridge

 

 

 

 

 

 

 

II I / r \
* pmllgiénoduhb'w

 

 

law [7018‘
p". - amp

I
Ii.—

rccordzr 4.1."; 50”! PM“
unset-Iva

Z dctcctor

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. (4.12) Field Modulation Electronics

CHAPTER V

SAMPLE PREPARATION

5.1 Introduction

 

 

In this chapter the method used to grow the dilute
alloy single crystals is discussed. The effects of non-
uniformity in solute concentration and crystal structure
are also considered. The techniques used to determine
solute concentration and crystal imperfections are de-
scribed. The chapter is concluded with a presentation of
the results of such measurements on the present samples.

5.2- Necessary Precautions

 

 

It seems obvious from the start that if one wants to
study the Fermi surface of a metal for different alloy
concentrations at least two requirements must be met.

Since the solute concentration is expected to have
some effect upon the Size and shape of the Fermi surface,
we require that the solute concentration for any given
sample be as constant as possible. Spatial regions of
the crystal where the Mg concentration differs greatly
from the mean will lead to prOperties typical of that
concentration. Thus an assembly of Such regions Should be

expected to broaden the range of measured values for any
58

59
parameter whose value depends upon solute concentration.
This effect will reduce the confidence in the values
which one measures for carrier lifetimes at the Fermi
energy. The effect will also produce a damping of the
amplitude of the oscillations.

The second requirement is related to the Single cry-
stal nature of the sample. Since the Fermi surface for
any polyvalent metal differs greatly from a Sphere the
extremal cross-sectional areas will depend upon the orien-
tation of the magnetic field relative to the crystal-
graphic axes. If the crystal orientation varies with pos-
ition, the extremal cross-sections of Fermi surface and
hence the frequency of the oscillations, will be a func-

tion of position. Thus the oscillations observed from

 

such a sample will vary as EE:};H'Zfl(f§+4§fi) . For low
field or high frequency oscilldtions where fo/H is very
large Afi will have a much greater effect than in the
high field region. Thus microstructure can lead to a field
dependent damping which could be confused with a Dingle
temperature (although it should have greater effect

upon high than low frequency oscillations).

5.3 Crystal Growth

The theory and associated techniques which made

possible the preparation of samples necessary for this

60
study are largely a product of the last fifteen yearssg.
Much of the stimulus for this develOpment has been
provided by the semiconductor industry. Since we are
chiefly concerned in this study with the growth of
alloy single crystals the following discussion will
center on this problem.

It is convenient to begin the discussion with some
assumptions and definitions. Suppose one begins with
a slug of Mg - Al alloy of roughly uniform concen-
tration. In our Study such a slug was obtained by
chill-casting a well mixed solution of the alloy in
a reduced nitrogen atmOSphere. The distribution
coefficient, K0, is defined as the ratio of the solute
concentration in the solid (C3) to its concentration in
the liquid (CL). This ratio depends upon the temperature
at which the zone refining is done, but as seen in
Fig. (5.1) the value for low concentration alloys of
Mg - Al typically is about 0.3 .

The zone refining technique is to melt a Small
region of metal while the remaining material remains
solid. The material is then slowly moved relative to
the heat source so that a molten zone is gradually
swept through the starting ingot. In the samples pre-

pared for this study the zone was produced by a radiant

6l

WEIGHT PERCENT MRSNESIUM
1: .. a: “P 5.0 an

 

 

 

 

 

 

 

 

 

no 1
“0'
450 \
boo \
sso \ '3
06
\ c
U :r
“50' v 2
«so \

m . b
at /(17.4) lag-:0
HOOP “3‘ ',
//

 

 

 

Trimaran-runs :c
S
O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

zso
zoo
ISO
24 37-5
MIMI ea!
0 ID ID 30 40 5'0 60 70
m nTomc pen CENT MAGNESIUM

Fig. (5.1) Aluminum-Magnesium Phase Diagram

‘ I lull - I

62
energy projection lamp obtained from the Materials
Research Corp. In addition to the lamp and associated
elliptical reflector an additional heater was used to
raise the temperature of the remainder of the alloy
to within about 300°C of its melting temperature.
With this technique it was possible to maintain a
molten zone of about 1/2 inch in length. The ingot is
placed in a graphite boat which is then sealed off in
a Vycor tube in a reduced atmOSphere of nitrogen. At
the working temperature the pressure of the nitrogen
gas is very near atmoSpheric. This assembly is then
placed under the lamp and a molten zone initiated at
one end of the Slug. The concentration of solute CL
(in this case Mg) is in this first zone just the
initial equilibrium concentration Co. As the zone moves
on, some solidification begins at the end of the rod.
The solute concentration of the solid initially frozen

18

c8 = KOCO .

Since for Mg - Al this CS is less than Co, the amount
of solute in the molten zone increases. The solute
concentration keeps increasing until C = Co/K. At this
point the amount of solute being melted is equal to

the amount freezing out. This Situation continues until

63
the leading edge of the zone reaches the end of the
Slug. At the instant this happens the solute concen-
tration C in the molten zone is CO/K, which is substan-
tially larger than Co' Thus as the zone passes out
through the last zone length of the crystal the concen-
tration of solute in both the liquid and solid states
increases in an exponential fashion. Fig. (5.2) shows
what one expects for the solute concentration as a
function of position after a single pass. By repeated
unidirectional passes a very uniform gradient of
solute concentration can be established. For low K it
is possible to obtain impurity concentrations as
low as one part in ten billion at the initial end. An
important variation of this technique was used to grow
the crystals used in this study.

The first pass was performed in essentially the
above manner. The second pass was made in the opposite
direction. Reversing the direction makes the first
molten zone of the second pass have initial concen-
tration, C = CO/K, the concentration necessary to
establish the region of constant solute. Thus after
the second pass the crystal should have at least a
middle region of very constant solute concentration.

An additional result is brought about by the zone

64

 

   

 

50/060. Concentration C

‘9 length of rod 1-

 

 

 

C. = average salute conccntratlon
x = 6...

Cu,“
K. = aymlcbnum distribution cat/flcnnt

1 = length of zone

= distribution coefficmnt

 

Fig. (5.2) Magnesium Concentration after Single Pass
Zone Refining

65
refining method. The resulting crystal is found to be
single over much of its length.

5.4 Resultant Crystals

 

Four samples with magnesium concentrations ranging
from 0.1 to 0. 52 at.% were prepared in the above
fashion. The resultant rods were examined using Laue
back-reflection X-ray techniques. The photographs
Showed that most samples were single over at least
the central three inches. (One of the intended samples
showed that 5 crystals had been formed and that they
persisted over this central length.) The samples to
be used were cut from the central regions of their res-
pective crystals using a Servo-met Spark machine. These
samples were then X-rayed, oriented'with(@00>axis along
the X-ray beam axis using a convenient goniometer
arrangement and a{100}face Sparkplaned with the crystal
still mounted in the goniometer. The samples were then
glued by this{100}plane to an accurately machined
graphite holder. (Fig. (5.3)) Using this method it was
possible to estimate the initial sample orientation to
within about 2 degrees about one axis and to within
4 degrees about a perpendicular one.

Samples prepared in the above fashion were weighed

and their residual resistivity ratios (RRR) measured

66

{act of Samp/c
spark - planed
J, (100)

 

('9')

   

rotation

plane ,2) oral/cl
{ace 9

Fig. (5.3) Sample Mounting

67

between 4.20K and room temperature. These data as well
as the physical dimensions of the samples are given in
Table (5.1).

Using the data of B. Serin6O for the contribution
to the resistivity due to impurity scattering as a
function of magnesium concentration, one arrives at
the calculated values of magnesium concentration as
listed in the final column of Table (5.1). This cal-
culation assumes that the low temperature resistivity
is due to impurity scattering. The correlation
between the impurity concentration before mixing and

that calculated from the resistivity ratio is reassuring.

Table (5.1) Sample Information

 

 

 

 

 

 

sample at.% Mg before Residual Resis-* at.% Mg
number zone refining tance Ratio(RRR) from RRR

l .13 108 .08

2 .40 22 .39

7 .22 66 .13
pure Al 6-9's oriented

 

single crystal

 

 

 

These samples including the pure Al sample were
roughly in the form of cubes; the masses ranged from
.3 to .5 grams.

 

*
RRR = J03oooK 4,0 4.20K

 

CHAPTER VI

THE FERMI SURFACE OF PURE ALUMINUM

6.1 Introduction

 

 

One of the reasons for undertaking a dilute alloy
study in aluminum is that the Fermi surface has been
relatively well established by a wide variety of tech-
niques. In this chapter the agreement between the
experimental data and various theoretical calculations
is discussed and a presentation of the relevant data
obtained in this study is also made.

6.2 Existing Fermi Surface Models and Data

The early work of Heine61 (1957) showed that the
valence wave function in aluminum is well represented
by a single orthogonalized plane wave (OPW) except
near the Brillouin zone boundary. A nearly free
electron (NFE) construction in which the contours of
constant energy are Spherical was first used by
A. V. Gold62 to interpret his deA data in lead. In a
magnetic field the effect of the lattice is to reflect
the orbiting charge carriers at the Bragg reflection

planes and to thus separate the carriers of one zone

from those of another. By translating the correSponding
69

70
pieces of each zone back into the central zone Gold
was able to form the Fermi surfaces corresponding to
carriers of a given zone. The remarkable thing is that
this naive model was able to explain the lead deA
results. This reduced zone remapping procedure is
eSpecially tedious in three dimensions. In 1959

63 greatly Simplified the method when

W. A. Harrison
he realized that instead of considering only one Sphere
and the Bragg reflection planes, the same reduced
surfaces could be obtained directly from the inter-
section of Fermi Spheres centered on each lattice
point. Thus since Heine's work had shown that aluminum
is a good NFE metal it was selected by Harrison for

the first trial of the method. The Harrison construct-
ion, Fig. (6.l), was quite successful in interpreting
Gunnerson's preliminary results64 for the third zone
surface. More recently (1963) Ashcroft65 has used an
OPW pseudo-potential with coefficients adjusted to fit
the third zone results of Gunnerson and also those of
Gordon66. Although Ashcroft's ”final model" confirms
the Harrison construction in general, it predicts a
different connectivity near the W point of the zone.

This is, of course, where one would expect the one OPW

calculation to fail. The Ashcroft model showing the

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3td ZONE-REGIONS 0F EL'NS

Fig.

71

  
    

 

--- .. --- —‘

’
f
s. ”

20d ZONE-POCKET OF HOLES

1"“~~ 0“."
I T s
I u \‘
I, ' ‘
,‘ w I s
I <I> ' ‘
I ' '
l D l
' I*\ '
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~‘+”’

4th ZONE-POCKET?) OF EL'NS

X

(6.1) Nearly Free Electron Construction for Aluminum

72
dismemberment of the third zone ”monster” into “rings
of four” is shown in Fig. (6.2).

The Ashcroft model which agrees quantitatively
with Gordon's data is shown in Fig. (6.3). The pure
aluminum data taken in the present study is also
shown. These measurements are not intended to repre-
sent a Fermi surface determination but instead, a
check of the experimental method being used. The an-

67 as well as

omalous skin effect results of Vol'skii
the ultrasonic attenuation measurements of Kamm and
Bohm68 are in good agreement with this model. The
existence of the /3 orbits is not consistent with

Harrison's model but is predicted by Ashcroft's, thus

confirming the latter model.

73

0( ORBIT

\

IRBFT

2! (man

Fig. (6.2) Ashcroft's M del for the Third Zone Surface

of Aluminum

74

90‘ H «I {no}

 

 

 

gal
—— Asbcrofb model
. Present study
‘6‘

of."

2

{$21

35

u

C

o

H.

 

 

I I I l I I 1 I
on) m. 20‘ ad 40‘ séznlbw 70° arm!»

Fig. (6.3) Comparison of Data with Ashcroft's Model
in {110} Plane

CHAPTER VII

EXPERIMENTAL RESULTS FOR AL-MG ALLOYS

7.1 Introduction

 

 

Experimental data has been taken with the intention
of analyzing it to determine the shape and size changes
of the Fermi surface caused by alloying. One might
expect the effects to be small for dilute alloys with
solute concentration less than l/2 at.% . It is found,
however, that changes in extremal cross-sections in the
small third zone electron surface of Al can be as large
as 10%.

The amplitude of the oscillations has also been
studied as a function of temperature, field, and angle.
From these measurements one can determine Dingle
scattering factors.

When the magnetic field is rotated away from a
symmetry direction the analysis becomes more difficult
due to the larger number of frequencies present. This
difficulty is eSpecially apparent for the more concen-
trated alloys where the amplitude of the signal_has been
greatly reduced via the Dingle factor.

Throughout the following discussion the oscillations
7S

76
are divided into high (8‘) and low (M andfi) frequency
groups as indicated in Fig. (6.2). Rotation angles from
(110) in a {100} or a {110} plane are denoted by 6 and
¢ reSpectively.

7.2 Changes in the Aluminum Fermi Surface Induced by

Alloying
The angular variation of the /3 orbit near4<lld>

in a {106} plane has been studied for pure Al and three
alloys. The results are shown in Fig. (7.1). A system-
atic decrease in extremal cross-sectional area with
increasing Mg concentration is apparent. The theoretical
prediction for the angular variation of frequency in

a {100} plane shows four frequencies converging toward

a common value at <lld>. Because these branches could
not be resolved near <llQ> in the most concentrated
sample it is difficult to put much emphasis on changes
in the structure of the angular variation curves.

Since the cross-sections of the third zone electron
surface are decreasing as di-valent Mg replaces tri-
valent Al, one is tempted to try interpreting these
changes in terms of a rigid valence band model. By
analogy with the NFE Harrison construction, changing the
electron concentration scales the radius of the Fermi

Sphere. Since the third zone surface is formed by an

78
overlap of such Spheres it should be quite sensitive to
changes in kF' Consider a simple model which assumes
that the Fermi surface cross-sections scale quadratically
with the amount that the Fermi ”radius" overlaps the

zone boundary. Thus along (llO)

A 2 A 2
'Xo = (kF‘kw> and '50 = (kF-kK)
l/3
where kF o( Z .

This model has been used successfully by Gordon
et al66 to interpret their Al-Zn alloy data. In Fig. (7.2)
the <110>I30rbit data for Al-Mg is presented. The solid
line correSponds to the above-mentioned calculation.
This simple model agrees with the present data to within
the eXperimental error although it is not applicable to
all alloy systems.* A similiar plot is shown in Fig. (7.3)
for the .08 at.% alloy along (led).

The amplitude of the high frequency oscillations
is only about 20% of that of the low frequency, and
consequently their analysis presented difficulty. The
only simplifying fact is that since the nearly cylin-

drical ”arms” lie along (11d) there is very little

 

*

Higgins and Marcus30 found that in Zn alloys changes
in the axial c/a ratio were at least equally as important
as the valence effect described above.

79

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81
beating of these X oscillations when the field is
within 200 of this axis. The frequency of these oscill-
ations near (110) in a {110} plane for pure Al and the
two most dilute alloys is shown in Fig. (7.4)A. In
steady field rotations in a {100} plane the oscillations
come principally from the low frequency 3‘ oscillations.

The variation of the ‘8 frequency with magnesium
concentration along (110) is shown in Fig. (7.4)B.

This frequency correSponds to the extremal cross-
sectional area when the magnetic field is aligned with
the axis of arms.

Steady field rotation plots, described in Section
(4.6), were obtained for the two lowest concentration
alloys. For the .08 at.% alloy it was possible to obtain
good data throughout the {106} plane. In the .13 at.%
alloy the signal was smaller and its interpretation
was difficult for 69 greater than about 15°. Plots of
these frequencies versus 9 are compared with the pure
Al data in Fig. (7.5).

7.3 Amplitude Dependence

 

As the field is rotated away from <lld> the amplitude
of both the high andlow frequencies decreases. Rough
measurements of this decrease were made for all of the

samples studied and are shown in Fig. (7.6). As the

82

 

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83

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69
one

+

% ofpure 4/ arm [/~2.9x/a‘)
9 >3 :2 2 a? 3
._§—.

 

 

o .64 .68 .I'2 .I's .25 .2;- .258 352
M5 concmtratzon (at.%)

Fig. (7.4)B Variation in 3‘ Orbit Area along (110)
with Mg Concentration

84

434

42‘

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Fig. (7.5) Angular Variation of 3‘ Orbits in {100}

Plane

8S

 

 

 

Amplitudo (Orbifrary units )
o

 

 

O —'
Fig. (7.6) Amplitude Dependence in the {100} Plane
upper: High Frequency 3‘ Orbits

lower: Low frequency o( and f3 Orbits

86

angular dependence of the amplitude is due to aniso-
tropies in g, m*, t and I4, it could not be simply
accounted for. In principle the effect of each of these
parameters could be sorted out, though this would be a
very difficult series of experiments to perform.

Field plots to obtain the Dingle temperatures as
described in Section (4.6) were made for both the f3
and '3‘ oscillations near (110). For some of the samples
the Dingle temperature was determined at a series of
temperatures between 1 and 20K. Examples of the Dingle
plots for high and low frequencies are shown in Fig. (7.7)
The beating of the 3‘ oscillations by the higher
amplitude /3 ones is manifest in Fig. (7.7)A. The effect
of the 2:1 beating of the /3 by the o( oscillations
is apparent in Fig. (7.7)B. Since in both cases the
oscillations were followed over several cycles of the
beat pattern this should have little effect upon the
value of XD, which was measured. The results of the
Dingle temperature measurements are shown in Table (7.1).

The measured Dingle temperature can be related to
the lifetime ’6 via

XI, = h/anBt. (7.1)

The €7as determined from Eqn. (7.1) is really very

Specific. It is the lifetime of a state correSponding

87

 
    

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5 is 9 I 8 5
4! 06‘6")
Fig. (7.7) Dingle Plots along (110)

upper: High Frequency 3‘ Oscillations
lower: Low Frequency Oscillations

88

Table (7.1) Dingle Temperature Results

 

 

 

 

 

 

 

 

 

 

 

 

 

approximate
frequency of temperature
alloy oscillations and * for measure-
sample orientation XD(°K) ment(°K)
#1
.08 at.% Mg sxlo5c 2.88 2.03
(llQ)
2.62 1.73
2.91 1.49
2.80 1.41
2.22 1.24
3x106G 2.69 1.46
(110)
#2 5
.39 at.% Mg 5x10 C 4.65 1.21
(110)
#7 5
.13 at.% Mg 5x10 C 2.30 1.89
(119)
2.37 1.72
2.6 1.60
2.5 1.45
3x106 3.2 1.89
(110)
3.1 1.72
3.8 1.61
4.2 1.42
*

using m*/m = .118 for the 5x105G (110) osc.
and .130 for the 3x106G’l(110> osc.

89

to an extremal orbit at the Fermi energy. Thus the 3’
as measured gives very Specific information about the
scattering from one part of the Fermi surface. We can
also calculate TI}; from the electrical resistivity/D .

Z); = m/nezf (7.2)

This is of course not the same C as has been

measured. '9) of Eqn. (7.2) is the "average” lifetime
of all the conduction electrons. It is composed not only
of third zone states but also those states of the large
second zone. The masses of these zones are reSpectively
~0.12 andAIl.3 free electron masses. Thus in consid-
ering an "average” m to use in Eqn. (7.2) it is probably

not a bad approximation to use m = m the free electron

0,
mass. Having measured the .PBOOOK /f4.2°K resistance
ratio it is then a simple matter to calculate 8'} .

4
In the above-mentioned paper Brailsford calculated

7"
Z‘ £p(9)(1-cose)sin9 d9
_._._.__. 71’ ' de
3}) f0 (9)81n9

 

P(e) is the probability of scattering through an angle

9 . Considering a simple shielded coulomb potential

-qor
e
W—
I'

he obtained the result of Fig. (7.8).

The results of these calculations are compared with

90

 

 

 

j

o I' 2
?o/2 kF"—"

.m-

Fig. (7.8) Results of Brailsford's Scattering Calculation

91

the experimental values in Table (7.2). The increase
of ”CI/1?}, for high alloy concentrations shows that a
more SOphisticated scattering theory is probably
necessary.*

By comparing the amplitude of the oscillations in
a given field range at several temperatures we can
study the temperature dependence of the amplitude. After
correcting for the simple temperature dependence of the

amplitude (i.e. T/ CHfiJ1+E;%g%§)2 ) we would expect

 

from Section (4.6) that

AT

-bXD
osc a< 1-bT/H coth bT/H ( 8 —1T—-) H3/2 (7.3)

amp sinh bT7H

 

 

Thus for fixed field we would expect the temper-

ature to vary as

l-bT/H coth bT/H
sinh bT/H

 

A family of constant field curves showing this
temperature dependence of the amplitude is shown in
Fig. (7.9) for the .08 at.% Mg sample. As we go suc-
cessively to curves of higher field we would expect
(from Eqn. (7.1)) the temperature of the maximum to

increase proportionally. This behavior is seen in

i If 57$}: 4 .7 (the Thomas-Fermi value) then the
1% Al-Zn deA results of Gordon must have their amplitude
at 17 KG reduced by a factor of «ae'lz relative to pure A1.

92

 

 

 

 

 

 

Table (7.2) Scattering Results (110)
‘approximate
fre uency XD 333 1%; ‘gy
at.%Mg (10 G) (0K) (10 sec) RR (10 sec) ’L}
.08 5 2.8 4.3 106 7.4 .58
30 2.7 4.5 7.4 .61
.13 S 2.5 4.8 66 4.6 1.07
30 3.7 3.3 4.6 .72
.39 5 4.7 2.6 22 1.53 1.44

 

 

 

 

 

 

 

 

 

93

— theoretical

..... expmmcntol

_____....--o 20.9 as

’
"O'

’

6’

 

(e-bn )f/a/Z

Smh bk,

 

 

.( /- bhicothbp”

 

 

\3JkG

o-- - "' - - -o “.3“. ~ ‘ was“.

amp/I t udc
k
\
I
I
I
5
I

 

 

O T I I I I j r U j
u n I: 1.4 I": I. 6 n l.8 L9 2.0 7'1
Tcmpcrot on C 0K )

Fig. (7.9) Temperature Dependence of the MTO Amplitude

94

Fig. (7.9) although the maxima appear sharper than
one would expect from the theory. Since the temperature
of the amplitude maximum for a given field curve
depends only upon the effective mass and fundamental
constants we can obtain a value of mfi/m from each
curve. m*/m determined in this fashion varies from
.104 to .121 with a mean of .113 . This value compares
very favorably with m*/m :: .118 calculated by Gordon66
from the temperature dependence of the amplitudes (from
deA data).
7.4 Amplitude Comparisons

From the theoretical expression for the MTO effect
we can calculate the absolute amplitude of the fund-
amental oscillation, -bXD

3/2 3/2 +_
ATosc = ZKBTOHo (e/hc) e (1-1r—coth'fi— (7.4)

32A 35 4’ 1+(Z' /2wZ’ )Zsin H
ext CH osc Q ha—

akzz
We make the initial assumptions:

 

 

 

 

CH is mainly electronic (i.e.“XT, 987. isin A1

 

at 10K),
1711-?- coth 131-1:
bT is in the constant region and
Sinh-EF therefore g .31,

 

tosc e: ZIrCQ so 1/’\/1+(toso/27TZ’Q)2 2,71

95

 

Thus
-bx
3/2 D
H "H -14
AT = ‘3 2.21x10 0K
osc TP/A 32A 32
bkzz ext

 

where H is in gauss

Xis in ergs/mole deg2

fis the mass density in gm/cm3
A is the atomic weight (e.g.l” 27 for A1)

For Al using m*/m = .118 and H = 17KG,

 

 

 

3 62xlo-40K e"XD
ATOSC_ . a 2A 32' (7.5)
2
3k2 ext

To evaluate the denominator in Eqn. (7.5) we can
use the angular variation of frequency data. Consider
a piece of Fermi surface having cylindrical symmetry
about some axis determined by the magnetic field. The
radius of an orbit tipped at an angle 9 with reSpect
to this axis is expanded as

r = r0(1 +049 2) .
The area A(e )ext= 7Tr02(1 + «92) (7.6)
At a distance rsina along kz from the extremal orbit
at €3= 0
A = 7T(rcosg)2¢.' 1-92 + 2 0(62

It then follows that

96

a 1 a
-/\]41rla<- 7| (7.7)

 

IH

Since for a cylinder X = 2 we have

A<9>7A<i§iinder _ l 2
A — o<- 2 )9
(9 =0)
from Eqn. (7.6).
For the 6‘1 orbits alongl<110> we can calculate

a(- i g .097. Thus from Eqn. (7.7)

2
b A %’\/---417'(. 097)= l. 10

at;

 

 

 

For these orbits there are two equivalent pieces of

Fermi surface per Brillioun zone and thus
2(3.62x10'40K)e’XD
" 1.10
xfilm

The corresponding measured amplitude for pure Al was

..XD

ATosc = 6.6x10'40K e

 

 

6.5 if 1 x10'40K.

The [3 and 0( orbits arise from regions of the
Fermi surface near the junction of the arms. Since
these pieces do not lie alongl<110> it is difficult

to evaluate the Shape factor
32A 36
Bkzz

We would expect

97
1
92M; kzz [35 to be greater for these sections

 

than for a Sphere where

1 1
IDZA/Ibkzz [a = afihr' = '4

We thus find for the 5x105 G frequency along<<110)

 

a» 1.75x10“3°x .
04,3

The correSponding measured amplitude is 4.35x10’30K.

ATOSC

 

We can also compare the relative amplitudes of
the alloys as determined from their measured Dingle
temperatures. This comparison, made at fixed temper-

ature and field, is shown in Table (7.3).

98

Table (7.3) Amplitude Comparison <110>

 

 

 

 

 

 

 

 

 

 

Dingle ratio of
approximate factor amplitude
frequency measured XD amplitude to pure A1
at.%Mg (105G) A TOSCPK) (OK) reduction amplitude
pure 5 4.35x10'3 0 l l
30 .65x10'3 0 1 1
-4 2 .1 .1.
.08 5 3.65x10 .8 17 13
30 .4x10”4 2.7 l- -l
2 1 l 6
.13 5 4.6x10'4 2.5 .l .1
1 2 9 .5
- 1 1
30 1.5 10 5 3.7 —- ‘—
x 64 (I5
-4 l .1.
.39 5 1.2x10 4.65 105 36

 

 

 

CHAPTER VIII
GENERAL CONCLUSIONS

These measurements of the third zone Fermi sur-
face of pure aluminum and dilute aluminum-magnesium
alloys show that the extremal cross-sectional areas
along (110) and (100) decrease as the concentration
of magnesium is increased. The electron nature of this
surface is thus verified. A simple free electron
perturbation of the model proposed by Ashcroft can be
used to explain these changes. In particular the
cross-sectional areas correSponding to a frequency
of about 5x105G where a fairly substantial decrease
in area is observed (~10%) fit the model quite well.
(Fig. (7.2)) The higher frequency ‘3 oscillations appear
to be more dependent upon the alloying than would be
expected from the model. Uncertainties in the high
frequency pure aluminum frequencies of about 2%
could account for most of this discrepancy.

The MTO technique has been used to study the
variation in both the amplitude and frequency as the
magnetic field is rotated relative to the crystal.

Although the amplitude variation is difficult to
99

lOO

interpret it was possible to measure the angular
variation of the ‘8’orbit very thoroughly.

The variation of the amplitude of the oscillations
with both temperature and magnetic field was also
studied. The different hyperbolic dependence of the
MTO and deA oscillations enables one to determine
the effective mass more directly with the deA measure-
ment. A technique which enables the MTO effect to be
used to measure Dingle temperatures has been deve10ped.
This technique takes advantage of the relatively small
dependence of the MTO amplitude upon the hyperbolic
T/H term. Dingle temperatures determined using this
method agree qualitatively with those obtained from
the electrical resistivity. The effective mass
measured along <110> for the low frequency (5x105G)
was found to be m*/m = .113 f .008 , in agreement
with the value measured by Gordon of .118.

The absolute amplitude,which could be calculated
quite accurately for the X oscillations, was found
to agree with the measured value to within the
experimental error of about 15%.

A crude amplitude calculation for the low fre-

quency oscillations also agreed with experiment.

101
The latter comparison cannot be considered a rigorous
test of the theory because of the difficulty involved
in evaluating the shape factor for this part of the
Fermi surface.

The absolute amplitude of the alloys was found
to Scale approximately as expected from the Dingle
factor. The actual reduction in amplitude was in all
cases less than would be expected from the Dingle
theory. Effective Dingle temperatures determined from
the scaling of the amplitudes of the alloys relative
to pure A1 are 10 to 20% less that the actual Dingle

temperatures measured for each alloy.

10.

11.

12.

13.

14.

BIBLIOGRAPHY
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103
Y. Shapira and B. Lax, Phys. Rev. 138, 1191 (1965).
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A.B. Pippard, Phil. Mag. 2, 1147 (1957).
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M.H. Cohen, MgJ. Harrison, and W.A. Harrison, Phys.
Rev. 117, 937 (1960).

J.B. Ketterson and Y. Eckstein, Rev. Sci. Instr.
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M. Ya. Azbel' and E.A. Kaner, Soviet Phys.-JETP 3,
772 (1956); Soviet Phys.-JETP 5, 730 (1957).

I.M. Lifshitz and V.G. Peschanskii, Soviet Phys.-
JETP 8, 875 (1959); Soviet Phys.-JETP 11, 137 (1960).

E. Fawcett, Adv. Phys. 13, 139 (1964).

V. Heine, Proc. Phys. Soc. (London) 624, 505 (1956).
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J.B. Schirber, Physics of Solids at High Pressure,
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Inc., New York, 1965).

J.R. Anderson, W.J. O'Sullivan, J.E. Schirber, and
D.E. Soule, Phys. Rev. 164, 1038 (1967).

A.B. Pippard, Proc. Roy. Soc. (London) 270A, 1(1962).
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R.J. Higgins and J.A. Marcus, Phys. Rev. 141, 553
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J.C. Slater, Advances in Quantum Chemistry I (Acad-
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T.L. Loucks, Augmented Plane Wave Method (W.A.

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42.

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104
Benjamin, Inc., New York, 1967).

W.A. Harrison, Pseudo-potentials in the Theory of
Metals (W.A. Benjamin, Inc., New York, 1966).

J.C. Slater, Phys. Rev. 145, 599 (1966).
JgM. Ziman, Proc. Phys. Soc. (London) 86, 337(1965).

Tanuma, Tshizawa, Nagasawa, Sugawara, Phys. Letters
25A, 669 (1967).

W.J. deHaas and P.M. Van Alphen, Proc. Amsterdam
Acad. 33, 1106 (1930).

N. Bohr, Dissertation, Copenhagen (1911).

L. Landau, Zets. fur Phys. 64, 629 (1930).

D. Shoenberg, Proc. Roy. Soc. 119A, 341 (1939).
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I.M. Lifshitz and A. Kosevitch, Soviet Phys.-JETP
22, 730 (1955).

R.B. Dingle, Proc. Roy. Soc. (London) A211, 517
(1952).

S.J. Williamson, S. Forner and R.A. Smith, Phys.
Rev. 136A, 1065 (1964).

A.D. Brailsford, Phys. Rev. 149, 456 (1966).

R.B. Dingle, Proc. Roy. Soc. (London), A211, 500
(1952).

M.H. Cohen and E.I. Blount, Phil. Mag. 5, 115 (1960).
M.H. Halloran, F.S.L. Hsu, and J.E. Kunzler, Bull.
Am. Phys. Soc. 13, 59 (1968); and comment by

L.R. Windmiller.

R.B. Prange and L.P. Kadanoff, Phys. Rev. 134, 566
(1964).

D. Pines, Elementary Excitations in Solids (W.A.
Benjamin, Inc., New York, 1964).

Il'ul III-.1! It'll-lull III1. 1| .
\

51.

52.

53.

54.

55.

56.

57.

58.

59.

60.

61.
62.

63.

64.

65.
66.

67.

105

J.W. Wilkins and J.W.F. Woo, Phys. Letters 11, 89
(1965).

A.B. Pippard, Proc. Roy. Soc. (London) 272A, 192
(1963).

D. Shoenberg, Phil. Trans. Roy. Soc. (London)
255A, 85 (1962).

J.B. Condon, Phys. Rev. 145, 526 (1966).

J. LePage, Dissertation, Michigan State University
(1965).

J.C. Abele and H.J. Trodahl (to be published).

W.L. Gordon, Phys. Rev. 126, 489 (1962).

G.A. Philbrick Researches, Inc., Applications
Manual for Computing Amplifiers (Nimrod Press Inc.,
Boston, 1966).

W.G. Pfann, Zone Melting (John Wiley and Sons, Inc.,
New York, 1958).

W.A. Harrison, Pseudo-potentials in the Theory of
Metals (W.A. Benjamin, Inc., New York, 1966)p. 150.

V. Heine, Proc. Roy. Soc. (London) 240A, 340 (1957).

A.V. Gold, Phil. Trans. Roy. Soc. (London) A251,
85 (1958).

W.A. Harrison, Phys. Rev. 116, 555 (1959).

E.M. Gunnerson, Phil. Trans. Roy. Soc. (London)

' 249A, 299 (1957).

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private communication mentioned in Ashcroft (1963)
and later published; C.O. Larson and W.L. Gordon,
Phys. Rev. 126, 703 (1967); J.P.G. Sheperd, C.O.
Larson, D. Roberts, and W.L. Gordon, Low Temper-
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E.P. Vol'skii, Soviet Phys.-JETP _12, 89 (1964).

106

68. G.N. Kamm and H.V. Bohm, Phys. Rev. 131, 111
(1963).

llli
I'u'l‘ III II I! I'll-

APPENDIX A

Heat Input via the Rotating Gear Mechanism

The heat input to the sample holder via the rotating

gear mechanism is the result of two different effects:

heat conducted down the control rods and frictional

heating during the turning of the gears.

Because the

sample holder is located in a vacuum chamber and ther-

mally isolated from.the helium bath these heating effects

are particularly troublesome. The situation is shown in

Fig. (A.l).

mmdml
Gountcr

 

 

' tampcraturt

 

control rod —\

.oté” wall, 15¢.”dm.
5.5. tub/n]

W

 

[:_

 

.2241
L

 

 

  

 

 

 

 

flange.

W
. «b
l.:cn;::f}76wym:

rt in“
F‘s-010'?“ "View.

,-— ~5on mm
VOCUUW7 49

Ms: sample Mr
64:»? 15 344ch

 

 

JK— samplc. can

 

Rig. (A.l) Extraneous Heating Effects
‘ 107

108
For each of the control rods Operating between
3000K and 40K the heat flow per second is about
2.5x10"2 watts.
If we consider a c0pper wire thermal leak of
length L, cross-sectional area A and conductivity K the

temperature rise is about

P L L
AT 2 w— 3 5x10'3 X cm°1<

RA
To turn the control rods requires a torque of about
1 inch-ounce or 7x10"3 newton-meters. Thus for one
complete revolution of these shafts, correSponding to

2 joules

a few degree change in the gears, about 4x10-
of work must be done. If this energy all goes into
frictional heating of the gears during a period of

15 seconds the temperature rise will be,

PL 2 L
A. _ ~ " — 0
where K ~ .5 joules/cm sec OK . The time constant

for returning to the equilibrium temperature is

C L L
=‘EX‘ 2.2x10‘3‘K sec-cm

Since the length L is about 30 cm and a reasonable
A is about 1x10"2 cmz (i.e. #17 wire), L/A is about

3x103 cm'l, giving AT ~'15°K. It became obvious in

109
early experiments that another type of heat leak was
necessary. A helium pillbox was soldered to the bottom
of the sample holder thus reducing L to about 2 cm.
Since glastic rod has a much lower thermal conductivity

1 ) a
100
30 cm section of this was incorporated in the higher

than stainless steel at room temperature (

 

temperature regions of each control rod. The control
rods were brought in thermal contact with the bath at
the helium flange by wrapping copper wire around each
and attaching this wire to the flange. With these mod-
ifications the conduction heating effect is less than
10'30K as measured using the carbon resistors with
either M100 microns of helium exchange gas in the
sample can or a good vacuum. The thermal time constant

from rotating the gears is about 1 sec.

. APPENDIX B

Crush-Type Lead O-Ring Seals

This particular type of lead seal was introduced
to our laboratory by M. Garber. A possible seal is

shown in Fig. (B.l).

 

[and (.0014

 

 

 

Fig. (B.l)

A lead wire of diameter ~ .02 inches is greased
lightly with Apiezon M 1ubricant,wrapped around the
flange and twisted tightly. When the flanges are clamped
together a simple but very reliable vacuum seal is

formed. When these seals were used in the present

110

111

apparatus no failures were encountered even upon re-

peated thermal cycling between 1 and 3000K.

APPEND IX C

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112

APPENDIX D
Magneto-resistance Probe

The magneto-resistance probe which was referred
to in Sec. (4.4) was examined as a means of monitoring
the magnetic field. Since it is useful to have contin-
uous measure of the field available a simple power
supply to accomplish this was built. This circuit is
shown in Fig. (D.1)A.

The current through the probe varies by up to
2% when the resistance of the probe changes from 100:1
to SKSl. This is unimportant however, since the
magneto-resistance is not linear anyway. Once a cal-
ibration has been made the circuit can be re-zeroed by
maintaining the voltage across the 4.7K resistor
constant. The drift in this voltage is only a few parts
in 104 during an 8 hour period. The bias circuit is
used to buck out thermal voltages which also amount
to a few parts in 104 for the magnetic fields of interest.

A calibration of this probe is shown in Fig. (D.l}B.

113

114

 

 

 

 

 

 

 

 

 

 

?

N
9

Magnetic Field (*6)
3:

 

 

I2-

3.

4,

0 u T v u u
0' I 2 £3 4‘ 5'

Resistance. (RA)

Fig. (D.l) Magneto-resistance Probe
upper: Power Supply Circuit
lower: Magnetic Field Calibration

APPENDIX E

Harvey-Wells Electromagnet Field Characteristics

“1
w

'8‘

cut Flt/d
E- ? 3

may:
‘3 3

 

ob ioddoa'ooiommmmaio

Current (amps)

 

‘2’".

2t6k6—\

 
 
  

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K—I£u:6

30‘

 

 

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o

 

0 -3 .s :75 Lb L's i0 5.2:
Out on¢¢ mm C Onttr {IO/J (In/ms)
Fig. (E.l) Harvey-Wells Electromagnet Field Characteristics
upper: Field versus Current Relationship
lower: Field Profile Perpendicular to Field
Axis and Equidistant from.Pole Faces
115

APPENDIX F

l
Squaring Circuit for-E Sweep

1
For the magnetic sweep control to maintain g o( t

2 is necessary. A circuit which

an external reference 0( H
provides arbitrary powers between about 1 and 5 of an
input voltage over three decades of output is described.
This circuit is shown in Fig. (F.1). Approximately
thirty transistors were checked to determine their
logarithmic properties.

For the present studies the field reference was
obtained from a rotating coil gaussmeter*. Because the
reference voltage for the sweep must be floating it is
necessary to keep from grounding the tip of the gauss-
meter on magnet pole faces or connecting the power
common of the amplifiers and their power supply to any
fixed grounds (e.g. racks or chassis).

For the transistors currently employed in the
1 l

.( t sweep circuit an error of less than 1/2% in‘fi

H

was measured for sweeps between 9 and 22 KG. The cir-
cuit parameters were initially adjusted as follows:

i) using the meter mode switch all amplifiers

 

9:
Rawson-Lush model 720

116

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118
are zeroed and returned to their "Operate”
positions (a zeroing error of .lv correSponds
to 100 nanoamps )
ii) with the adder off, the multiplier output
is set to 2.370 v output for 1.000 v input
using the calibration source and the mult-
iplier potentiometer(s)
iii) with the calibration source off the adder is
adjusted so that the multiplier output
reads -.190v
The amount of H2 signal being supplied to the error
amplifier of the sweep control determines the-% sweep
rate. Thus for a given set of deA oscillations with
frequency f their time period can be set by adjusting
the input attenuator potentiometer of the sweep control.
The experimentally determined relation connecting these
quantities is:

tosdsec) = 3°86X107 gauss Sec
f (gauss) N

where N is the potentiometer setting (0 +10).

APPENDIX G

Least Squares Program for Finding Frequencies

PR3GRAY LIMFIT .
‘TTIST snag LIN QFGRESS ' PERIOD’HNCERT'IS‘§§“fi7E”CT
DIMPW3IO\ A(80)o8(80\oH(3) ‘8

PEA) 1300.KMAK,(H(J) J= 1 3)

1000 FOQ4AT(110 3A8) H

PRIJT 1001. (H(J).J=1a3)
1001 F0914T(1H1 20X, 5A8) ,- _,,1,Lweiiii, 1111,
_ —_WU_1170 J=1.RMA{_ -"_ “_‘pA

READ 1002. K, N
1002 FORWAT (215) 7

READ 1003.(A(I).I=1.N)
1003 FORAAT (8F10.3)

D0 1130 1311-” ___,
'113UmRTTTW?UTTU7ATI1Mflflfiu'”*17W”77m HAW” CPR—"7 "
AN 2 ’
XSJ1 : .
YSJW = .
XSQR : .
YSQR : .
“‘”“““13€T"T[1[1rtr 'T””Ef‘3I":BQ”" “"“‘ ""‘ "”““’ ' " I’ " ”““'“"Pf'*”"_‘“_"_'P"'

l

L

 

[ocacno

RI 2
XSJM
1140 YSJM YSUM + 9(1) .
XMEAM : (XSU81/AN WNW”
YMEAM = (YSUM)/A~ "

XSUM + RI f7“'

 

XVAHN : XVARN 4 (B! 7 XMEAN)t*2
___.... '—'YVWP_E_'VV7§RT\1“¥ ‘ ’1 ‘HT‘TT"'-—YME‘A‘N"" in?” 7 ‘ F"'"—‘"M“ —
1150 XYPRODN = XYPROUN + 81*B(I)
xvaw = XVAQH/AM
YVAQ = YVAnN/AN
XYPRflD = xvppnnN/AM
SL3PF = (XYPRHD - YMFAthMEAN)/XVAR
-— flmnrrrmsn 0.78tUP1-‘"'
PRINT 1004. K
1004 FORMAT (ll/l/lOX.*FRFQ. IDENT. Nn.v. 110)
PRINT 1007
1007 FORMAT (llflX.wPEAK Nn.*.15X.¢YERR-IJ
VARYN = 0.
—‘_————1N77113TU—1—'E”IjJV__ ""““*“*—m"“"*—W'——'- i- —
RI 2 I .
Ytau = SLopE~(HI—XMEAN> + YMFAN . 8(1)
VAQYN = VARYN + YERRtt?
”Effiv = VARVN/AN*

. a
Fig. (G.1) Program Listing
119

 

 

 

11160 PRINT 1008.1. YERR .

11008 FORWAT_(13¥,12;18x,E11;4)” 1 — W *“*“"“ “'“- ”-‘-' W*“_r-~
PRINT 1035 . AREAS _

'1005 FORMAT (/1nx.~FRFQ. n? 080. «.E11.4.v GAUSS*)

‘ TCOEFF = SQRTF((AN - 2.)*XVAR/VAQY)
Pt?) = SLOPE/100000.
FRR = 2.1/( TCOEFFtIHOOOO.)

““———’PRTNT'1150;"DERD;“:Rw “ ‘ ‘ ’ 1"“ “hr * ‘“*‘“‘“"‘—*"'

1100 FDQMAT (I10X.*PERIOD OF USE. «LE11.4.9 + - *.E9,2,* G-lt)
_fl—"TPRINT"1006{mVARY, V H‘“w__-""_m_“fl""vm_"“mm_kmm"”_”w‘_w_

1006 FO?MAT(/10Y.*VARIAMCF = *. E11.4 . //.
‘1 10X,tNUMB:R 0F pEAKS = *IIS)
1110 CONTINJE
ENn

Fig. (G.2) Program Listing, Continued

120

121

 

 

 

 

 

 

 

 

 

FREQ. IUFHT. No. 10
PEAK N3. YFRR
1 4.1554-003
2 7,8527-004
3 9,8808-003
--*-——4'_“"-*"fi4~~r~WLe-'ww.5473-003 -"***~~**t * _1 *v*
5 9.5532-005
6 1.2968-003
7 “1.1189‘003
a -7.0539-004
9 -7.2902-0n4
1U"*—“"‘—*“*"“—“*“““21;1813-003 '"*~ '““*‘*—‘ ““““““ ’
11 -9.0558-003
12 -x.3434-0n3
13 -9.0330-003
14 ~1.1964-on3
15 9.0990-003
. 16 ~0.0811-on4 .
_i___17_"__1__0 '*‘*‘ “'-1;4386-003 " ‘i ‘“ -1111111111
18 -9.4075-003
19 -3.8063'003
an -9,8845-on3
21 7.5505-004
22 -?.4864-003
‘“ "23““ “M” "*“"’*"“"“-‘x.5958-004 ' ‘*‘**"’***"“‘” "* M‘“
24 1.2385-003
25 9.3191'003
26 9,8932-003
27 9,9713-0n3
28 2.5637-003
”""_”29*“‘“”““"”m"“_—“‘”“1:6806-003 "'*"""“r“”‘""fi'r*“
so 9,7146-003
61 3.2255-003
32 0.0982'004
33 4.4114-004
34 -9.7698-003
*———"35 —"*"“+4:1781:un3“-*"'“-*“"”*”'““”*Wrr‘
FREQ. or 090. ~2.8606+006 Gauss

PERIOD 0F OSC. -3.4957-007 + - 8.08-010 G-l

 

-VARiANCE“:~—r—-4;9229.onn-m-"~v~--w~—~m""—-w~w&w~~J“J—r
NUMBER OF DEAKQ = x5

Fig. (G.3) Typical Output

APPENDIX H

Filter Circuits

5),:de coo VnP TANTALEX

 

 

 

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3744
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m

 

5.6

 

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([— R’s zzthcr
5'60 K ~ I.” ~16 sec.

or 390K ~t~IZsta

 

 

 

 

30m
éfll
944 " 5M!
—1% :1—
50K
"n hflh
O—_w" \
In

 

 

 

'C :- 113/7V— 5“-
N : lbw/got scum, 0‘10

13
l:—:L/////,/' OUt
Fig. (H.l) Filter Circuits

upper: Wide Band Integrator-Differentiator

Filter
lower: Narrow Band Bridged-tee Filter

122

123

{i/tcnd
l H 1‘ n /

 

 

 

 

 

 

 

 

 

 

 

\— unfnltzmd

o( magnetic {m /a’

 

.6 .'7 :8
|/H ( Ié‘d')

Fig. (H.2) Example of Application of Narrow Band Filter
using l/H Sweep

I .s'

124

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