THE HIGH-FIELD GALVANOMAGNETIC PROPERTIES
OF AuA12,AuGa2. and AuIn2

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY ‘
JOSEPH T. LONGO
1968

This is to certify that the

thesis entitled

THE HIGH-FIELD GALVANOMAGNETIC PROPERTIES

OF AuAl AuGa and AuIn

2’ 2’ 2

presented by

Joseph T. Longo

has been accepted towards fulfillment
of the requirements for

Ph-D- degree in-_ 211323108

////'/ /// /
/ ;/ _-- I
v‘/

____.—-

“/Illajor professor

Date Sept- 9: 1968

0-169

 

 

u;

D-

«4‘

- \
r.h

ABSTRACT

THE HIGH-FIELD GALVANOMAGNETIC PROPERTIES
OF AuAl2, AuGag, and AuIn2

By

Joseph T. Longo

The Fermi surface topologies of AuX2 (X = A1, Ga, In)
are investigated using high—field galvanomagnetic measure-
ments. The high-field galvanomagnetic prOperties of the
nearly-free-electron (NFE) model of the Fermi surface of
Aux2 are also determined with the aid of the Harrison con-
struction. The most important result is that the "open“
fourth zone electron sheet has hole orbits for_§ ll <lll>
in AuAl

and AuGa in disagreement with the NFE model. New

2 2
models are proposed for AuAl2 and AuGa2 which are in good
agreement with experiment. Incomplete results for AuIn2

indicate that its Fermi surface may be similar to that of

AuGag.

THE HIGH-FIELD GALVANOMAGNETIC PROPERTIES
OF AuA12, AuGa2, and AuIn2

By
.,“I=

Joseph T? Longo

3.

A THESIS

Submitted to

Michigan State University
in partial ful illment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Physics

1968

ACKNOWLEDGEMENTS

I wish to thank Professor Peter A. Schroeder for his
advice, assistance, and encouragement during the course of
this research. I am indebted to Professors P. A. Schroeder
and F. J. Blatt for the support necessary for the completion
of this work.

I am grateful to Professor D. J. Sellmyer of the
Massachusetts Institute of Technology for the provision of
his apparatus during runs at the Francis Bitter National
Magnet Laboratory and for many informative discussions about
interpretation of results. I sincerely appreciate the
assistance I received from Andrew Davidson during these
runs.

I am indebted to my wife, Marge, and Dr. Gordon J.
Edwards for their constructive criticism of the rough draft
of this thesis.

Financial support by the National Science Foundation

is gratefully acknowledged.

ii

TABLE OF CONTENTS

Page

1. Introduction 1
2. Theory of High-Field Galvanomagnetism in AuX2 9
Conductivity in High Magnetic Fields 10

Free Electrons-Closed Orbits 13

A Cylindrical Surface - Open Orbits 15

Open and Closed Orbits 16

A11 Closed Orbits 18

Real Fermi Surfaces 20
Description of the NFE Model of the Fermi 25

Surface of AuX

2 ‘I

Calculation of Ap/p for AuX2 in the {100} plane 41

3. Growth and Preparation of Samples 52
4. Apparatus and Experimental Techniques 65
Analysis of Tipping Arrangement 69
Experimental Difficulties 72
Summary of High-Field Galvanomagnetic Properties 75

5. Experimental Results and Discussion 77
Hall Effect in Aux2 78
General Field Directions 78
Symmetry Directions 84

The High-Field Magnetoresistance of AuX2 9)4

The {100} Plane 9%
Two-Dimensional Regions in AuAl2 and AuGa2 96
"Whiskers" in AuGa2 127

Models 136
Summary 140

6- Conclusions 142

iii

List of References 144

Appendix A
The Harrison Construction Program 147

Appendix B
A Program to Calculate Ap/p in {100} for AuX2 158

iv

U...

I“

.~.c

I...

 

LIST OF TABLES

Page
I List of compounds 4
II List of semiconductors and metals (after 6
Sellmyer, ref. 3)
III Dependence of Ap/p on the number of electrons 19
on open orbits
IV Properties of the AuX2 compounds (after Jan 53
_e_§_a_l_., ref. 13)
V RRR of AuX2 crystals 60
VI Impurities in Aux2 crystals 62
VII Samples selected for experiments 64

VIII Summary of High-Field Galvanomagnetic Properties 76
IX Hall data, general field directions 83
X Hall data, <1oo> and <111>; n = ne - nh ; An 88

XI Calculations based on a model with a Fermi sphere 91
fit to give experimental nlll values. NFE radius
= 1.495, fit AuGa2 radius = 1.532, fit AuAl2
radius = 1.552. As usual 2wh/a = 1. There are
hole orbits until 2 = .013 in the AuGa2 model and
z = .062 in AuA12; see figure 9. Data in paren-
theses is from early results of a study soon to
be published by P. A. Schroeder, M. Springford,

and J. T. Longo.

XII Higher order open orbits for Fermi surfaces with 135
<IOO> or <111>-directed primary open orbits.

.0

I?»

l.
2.

\J'I

10.

11.

LIST OF FIGURES

A spherical Fermi surface
A cylindrical Fermi surface

Crystal structure of the AuX2 compounds. X is
symbolized by the darkened Spheres. (After Jan
.§§.al., ref. 13)

Holes in the second zone of the NFE model of AuX2.
(After Jan gt a1., ref. 13)

Holes in the third zone of the NFE model of AuX2
in the reduced and repeated zone schemes. (After
Jan gt a}., ref. 13)

Section of the NFE surface in the flmuth zone.
(After Jan g§.al., ref. 13)

NFE surfaces in the fifth zone (left) and sixth
zone (right). (After Jan_g§-al., ref. 13)

Cross sections of the NFE surfaces in the third
(clear) and fourth (shaded) zones at the pz values
given for p || (100). Height of the unit cell =
200 0

Cross sections of the NFE surfaces in the third
(Clear) and fourth (shaded) zones at the pz values
given for E ||<lll>. Height of the unit cell =
1/(3)1/2.= .577 .

Cross section of the fourth zone surface for §
290 from [100] in a (010) plane with orbits Open
in [010].

Cross section of the fourth zone surface for_§ in

vi

Page
14
14
26

26

3O

33

33

33

12.

13.

14.

15.

l6.

17.

18a.

Page
{110} with orbits open in <llO>.

Cross section of third and fourth zone surfaces 33
for E in {111} with orbits open in <lll>.

Cross section of fourth zone surface for_§ in 33
{210} with orbits open in (210).

Cross section of the fourth zone with p at point 36
a in figure 18. Open orbits are separated by

closed electron and hole orbits; net direction

is 10° from <loo> in the {100} plane.

Cross section of the fourth zone with g at point 36
b in figure 18. Open orbits are separated only by

an occasional closed electron orbit; note how

closed hole orbits have unfolded to form sections

of open orbits or, in one case, pinched off the

open orbit altogether; net direction is 180 from
<loo> in the {100} plane.

Cross section of the fourth zone with p at point 36
c in figure 18. Extended orbits on the fourth

zone surface several degrees from the edge of the
two—dimensional region.

Cross sections of the NFE surfaces in the third 36
(clear) and fourth (shaded) zones at the pz

values given for_§ parallel to (110).
Magnetoresistance stereogram for the NFE model in 38

the fourth zone.

b. Standard 100 stereogram

l9.

. Magnetoresistance stereogram for the NFE model in

the third zone.

Open and closed orbits on a "log-pile" surface with 43
the same topology as that of the fourth zone of the
NFE model

vii

*1

-\

'\D

20.

21.
22.
23.
24.
25.

26a.

27.

28.

29.
30.

31.

32.

33.

34.

variation of'a with angle for the "log-pile”
model.

Ap/p in {100} plane for log-pile model.

RRR of AuGa2 crystals vs. excess Ga concentration.

Phase diagram of Au-Ga.
Schematic diagram of apparatus and circuitry.
Experimental apparatus (after Sellmyer, ref. 28)

Rotation and tipping geometry in the sample coor-
dinate system. Magnetoresistance leads are con-
nected at 1, 2; Hall leads at 3, 4, 5, 6.

. Stereogram showing the effect in the sample coor-

dinate system of rotating and tipping the sample
in the magnetic field.

Rotation and tipping geometry in the laboratory
coordinate system.

VH vs. B in AuAl2
supporting no open orbits.

for general field directions

VH vs. B in AuIn2 for singular field directions

Simple model for the magnetic breakdown of orbits
C4.

Ap/p vs. V for all three compounds in {100}.

B = 140 kG.

Ap/p vs. V with m changed by .60 from figure 31.
B = 140 kG.

Ap/p vs. v in {ill} for AuAl2 at 83.1, 99.8, and
129.9 kG. The exponents computed from these
graphs for B || (110) are m = 1.7 at 91.5 kG and
m = 1.5 at 115 kG.

Ap/p vs. v for A1<100> and G3<lOO> along paths
indicated in the stereograms of figures 35 and 36.

viii

Page
48

7O

80

87
93

95

95

98

101

Page
B = 145 kG.

Magnetoresistance stereogram for Al<lOO>. Bars 103
and dots indicate m>1.5; open circles mean m<.7 .

Magnetoresistance stereogram for G3<lOO>. 104
Magnetoresistance stereogram for the NFE model 105
in the fourth zone.

Ap/p vs. v for G3{100} along path indicated in 108
figure 39 at fields of 130 and 145 kG.
Magnetoresistance stereogram for G3{100}. 109
Ap/p vs. v for G3<110> along path indicated in 111
figure 42.

Ap/p vs. B for_§ oriented along three crystallo- 113
graphic directions in G3<lOO>. m = 0.0, 0.2, 1.8.

Magnetoresistance stereogram for G3<110>. 115

Ao/p vs. W for A1<100> along paths indicated in 118
figure 44. B = 130 kG.

Magnetoresistance stereogram for Al<lOO>. 120
Ap/p vs. v for A1<111>. B = 130 kG. 122
Ap/p vs. I for A2(random). B = 130, 145 kG. 125
Ap/p vs. v for G3<110>. 126
Ap/p vs. v for G3<110>. 128
Magnetoresistance stereogram for G3<110>. 130
"Whiskers" and two-dimensional regions in AuGae. 132
Ap/p vs. v for G3<110> for six values of p. 133

Magnetoresistance stereograms comparing eXperimen- 138
tal open orbit regions (shaded) with the type of
orbits (c = closed, 0 a open) determined on the
models.

ix

1. Introduction

The determination of a metal's high-field galvano-
magnetic properties has played an important and well docu-
mented role in the understanding of its Fermi surface tOp-
ology(1). Until recently(2’3’u’5’6) measurements have been
performed only on very pure metallic elements at liquid
helium temperatures so that a11_carriers perform many cyclo-
tron orbits before being scattered. This high-field con-
dition, mbg >> 1 for all carriers, is so rigorous that a
metal crystal in a field of 20 k0 must typically have an
impurity content of less than 10 parts per million to sat-
isfy it. With the advent of zone refining techniquele),
single crystals of non-transition elements meeting this re-
quirement became available in the late 50's and were the
object of extensive galvanomagnetic measurements. More
recently, electron beam zone refining methods applied to the
high melting point transition metals have been successful in
increasing their relaxation time sufficiently to attain the

high-field region in the laboratory(8’9)

, though the rare
earth and transuranic elements are still only available
with 39's purity and escape investigation. ,
From the above discussion it follows immediately that
the high-field condition cannot be satisfied in disordered

alloys. Consider a .1‘5 concentration of element X in host

element Y; this is a 39's element Y and requires the use of
a megagauss magnetic field. More concentrated alloys than
this have been studied by the de Haas-van Alphen (deA)
effect and by the use of magnetothermal oscillations.

These methods have the less restrictive requirement that
ab? >> 1 for only a subset of all the carriers. This sub-
set may be, e.g., the electron needles in zinc for which

mc = .01 me(lo), thus increasing wk accordingly over its
value for free electrons. Dilute alloys of up to 1%
impurity concentration are, in fact, now being extensively

investigated(11)

because one eXpects large relative changes
in small low effective mass pieces of the Fermi surface
upon adding an impurity of valence different from that of
the host.

There is one class of metals, viz., metallic inter-
metallic compounds, which could in principle satisfy the
high field condition. Consider a compound AxBy in which x
and y are integers and the A and B types of atoms each have
a unique set of basis vectors in the unit cell. In such a
compound, the potential would be perfectly periodic and the
relaxation time, 7, would approach a as the temperature
approached zero .

Thorsen and Berlincourt were the first to observe the

deA effect in a metallic compound, InBi, in 1961(12).

Since then, Pearson and co-workers at the National Research

3

Council (NRC) in Ottawa, Canada, have observed deA oscil-
lations in several binary metallic compounds and completed
a study of AuAle, AuGae, and Auln2(13). The significant
part of this research to someone envisioning a high field
galvanomagnetic study of a metallic compound was that the
residual resistance ratios, RRR = p(295°K) / o(4.2°K), of
some of the samples approached 160. This is roughly equiv-
alent to an impurity content of 60 parts per million (cf.
page 59). In a field of 50 k0, one could expect that
enough carriers would be in the high field region to give
useful tOpological information. On this basis Sellmyer and
Schroeder undertook a successful study of AuSn(2’3) in 1965.
Later galvanomagnetic studies of metallic compounds
included ZrB2(4), ordered Cu-Zn(5), and AuX2(6)

(X = A1, Ga, In). Work is underway on AuSn(14) and
Aqu2(15). Table I indicates the experimental progress to
date.

The face centered cubic fluorite compounds, Aux2 are
of considerable interest, because changes in the electronic
structure from one compound to the other should be explain-
able in terms of the differing electronic cores at the X
sites. An energy band calculation has not been carried out
for these compounds, but one can Speculate on relative
changes with the aid of the "Phillips cancellation
"(16)

theory If ltk> is the state vector of a conduction

band electron, then

Table I .

Compound
AuSn

ZrB2

Cu-Zn

AuAl2

AuGa2

AuIn2

Aqu2

List of Compounds.

RRR(highest)

160
110
418
550
904

75

B(highest)

150 kG
12.7 RG
150 kG
150 kG
150 kG
150 kG
150 kG

_publications
2, 3, 14
4

5
6
6
6

l5

5

p2

[an +‘VJ l¢k> = Eklek>
where V is the periodic potential of the lattice. Orthog-
onalize Itk> to the core states Ink) by letting

ltk> = IXk> - EE_lnk> <nk|k> .

Here le> is some smoothly varying function which is a

solution of

L§S+ v + VR] lxk> = EK lxk>

where Vh is a non-local repulsive potential which can be
shown to better cancel V as the core states become a more
complete set of basis functions in which to expand lik>.
If V + VR ~ 0, then ka> as eilc-T— unless E is near a
Brillouin zone boundary at which the periodic V + VR mixes
plane wave states to produce an energy gap. This is the
basis of the nearly free electron (NFE) or one
orthoganalized plane wave (1-OPW) Fermi surface model.

Since the heavy elements have the largest number of
core states, one expects that V +‘VR and therefore the
energy gap should be a decreasing function of the row
number of the periodic table. Table II, reproduced from
Sellmyer's thesis(3), gives actual examples of this effect;
the energy gaps separating the valence and conduction bands
in semiconductors are listed. In a metal energy gaps at
the Brillouin zone boundaries separate the conduction bands.

These gaps should also decrease with increasing Z in a

given column so that large Z elements should be more

Table II. List of semiconductors and metals. (After
Sellmyer, ref. 3)

 

 

 

 

Position* Material Energy Gap_(eV)** Crystal Structure
(2:4) C(diamond) 6 diamond
(3.4) Si 1.12 diamond
(4.4) Ge 0.75 diamond
(5,4) Sn(grey) 0.08 diamond
(5,4) Sn(white) metallic tetragonal
(6,4) Pb metallic f.c.c.
(3,5) InP 1.30 zincblende
(4:5) InAS 0.33 zincblende
(5,5) InSb 0.17 zincblende
(6,5) InBi metallic tetragonal
(3,”) M2231 0.77 fluorite
(4.4) Mnge 0.55 fluorite
(5,4) Megsn 0.25 fluorite
(6,4) Mgng metallic fluorite

* Position, (i,J), means 1th row, 3th column in periodic
' table. For compounds, (i,J) refers only to the position
of the second listed element in the compound.

** Most of the energy gaps are taken from W. D. Lawson and
‘ S. Vielson,_Preparation of Single Crystals, (Butterworths
Scientific Publications, London, 1958), pp. 241, 242.

7

NFE-like than the smaller Z elements. In the limit V +

Vh = 0, a metal would become free—electron like (i.e.
magnetic breakdown would occur with unit probability at
every zone boundary). Several effects distort this simple
picture. Relativistic corrections are important for
elements with Z>55(17). Tin can be a metal or a semi-
conductor depending on its crystal structure; thus
structure changes in the columns of the periodic table
present complications. Also, energy gaps due to Spin-orbit
coupling increase with Z. Finally, if there is mixing of
the high-energy core and conduction band states, this
formalism falls by assumption; the noble and transition
metals are in this category. Understandably then, excepI
tions to the rule occur: Na is the most free electron like
of the bee alkalis, but Be is less NFE-like than Mg. It is
difficult to compare the elements Ca and mg because of
differing crystal structures; the same is true of Al, Ga,
and In. A comparison of the fluorite compounds, AuXépwould
avoid this difficulty. Since the troublesome Au atom is
common to all three, it may be that distortion from NFE
behavior is primarily due to Au and secondly to the core
states at the X sites. We conclude that AuAl should be

2

the least and AuIn2 the most NFE-like.

The first experimental evidence bearing on this

(13). The extremal

hypothesis came from deA measurements
cross section of necks in the third zone had the behavior

predicted above; however, the "waist" areas suggested that

8

2 was the most NFE-like and AuIn2 the least. Results

on the octahedron in the second zone showed the same

AuAl

deviation from prediction; the AuAl2 extremal areas were
closer to the NFE values than those of AuIn2, while the
existence of this surface in AuGa2 had not been decisively
determined. The only comparison possible for the multiply
connected surface in the fourth zone was the extremal area
of the (100) directed necks. AuGa2 and AuAl2 both had
values in close agreement with the NFE model. These
results, published in the early stages of a magnetoresis-
tance study of AuGa2, provided the incentive for this
comparison of the Fermi surface topologies of all three

compounds and the 1-OPW model through a determination of

their galvanomagnetic properties.

5 I

V

2. Theory of High-Field Galvanomagnetism in AuX2

Kohler realized in 1949 that high-field magnetoresis-
tance and Hall effect data contain important information
concerning the shape of the Fermi surfaces of metals (18);
but the remarkable anisotrOpy to be found in magnetoresis-
tance as a function of crystal orientation, discovered in
1938 (19), remained a mystery for 18 years. Lifshitz,
Azbel, and Kaganov demonstrated in 1956 that, if all car-
riers completed many cyclotron orbits before being scat—
tered, the variation of the field dependence was indepen-
dent of collision processes and determined solely by

(20)

geometric features of the Fermi surface. In 1964

Coleman, Funes, Plaskett, and Tapp (CFPT) performed the first
calculation of the absolute value of the magnetoresistance in
several symmetry planes for a simple open Fermi surface

(21) Their

using a single—relaxation—time approximation.
work on the noble metals contained the assumption that the
Fermi surface consisted of a Sphere pierced by narrow
cylinders along (111) directions. They were successful
because they applied a simplified geometrical theory to this
geometrically simple model.

The l-OPW or NFE model of a Fermi surface is geomet-
rically simple to construct when done in the manner of

(22), and prompted an attempt on our part to extend

Harrison
the single-relaxation-time treatment to cover the more com-

plicated NFE surfaces. In the section, we develop the

lO

theory and calculate the magnetoresistance from a NFE-like
model of AuX2.

For readers suspicious of a constant-relaxation time
treatment, we have included a table listing those galvano—
magnetic properties which do not depend on this assumption

in section four.

Conductivity in High Magnetic Fields
The Boltzmann transport equation describing the motion

of a system of particles in phase Space is:

V- T"
at

0f +.:°vrf + p~v f = of
ct p coll.

(1)

f is the statistical distribution function which Specifies
the probability of finding a particle of the system with

its position and momentum in the interval between_g and

3 + d3 in real space and between_p and p + dp in momentum
Space. In an isothermal metal vrf may be safely set equal
to zero; we wish to consider dc effects only so that of/at =
0. Finally we note that the scattering term must vanish in
equilibrium when f = fo, the Fermi-Dirac distribution, and
return the system to equilibrium when a deviation is intro-

duced. The simplest possible form which satisfies these

requirements is

(Si-Lon. "' ’ 311:0”)

5*.

NJ.

 

11

and the Boltzmann equation then reduces to

T1 5 O (3)

The equations of motion of an electron in a magnetic

field are

.2 = "let! X.§ : .X = vpc (4)
In cartesian coordinates
15X = -IeIva I 15y = -IeIBvx

pz = O , é = vpcgp = O

if,§ (l 2. The electron moves on a curve of constant energy
and constant pz which suggests a change to variables 6, pz,
and a third variable u describing the motion tangent to the

trajectory. We define

du =_.___E- d t = \eIBdt. (5)

I‘

VXZ

——

Clearly, Q divided by a mass is a cyclotron frequency.

In the presence of a small electric field in addition

to the large magnetic field, we have

'=. -=—ev-*'
e vpe p l L_.2

l2

bz = -IeI§z (6)
e =1912t = IeIB(1 “-5112
det Bv

In terms of these variables, the Boltzmann equation

is

éof + p of + def = — f - f
'5; zdpz '5; T(e)o ' (7)

We seek solutions linear in the electric field (Ohm's law

region) and thus set

where_y is to be independent of.3, y =_y(e, pz, u)- Keeping

only terms linear in_§, we have

- e v-? of + e B e T:- SW of = —IeL;gy of
l l—JEEO l l l l3 Sag-go 60

Note that this is equivalent to neglecting 3 in the equa—

tions for pz and d. Since the electric field is arbitrary,
+ dV = d! 3 am = m =__l_ (9)

The solution of this equation is

L1
Jim) = Ute-mil ea”! 1(u‘)du' (10)

-CD

13

Because
- ‘ f A
a8 a} ' e5“ du' = l ,

1 is a weighted velocity average of_y along the orbit for.a
distance of about l/a in the direction from which the elec—
tron has come.

The electric current density is
I‘ '-
=-2 1 Vde =0.”
.9 .Jfifl. __ p .2

integrated over the Fermi surface inside each partially

filled Brillouin zone. The approximation OfO/de = -6(e - u)
dedpdpz

will be excellent for low temperatures. Now de

allows us to write

5 = 2e21 r _y.£ dpdpz. (11)

(gm); ddB.Z.

This eXpression can be readily evaluated for free electrons
and for field directions perpendicular and parallel to the

axis of a cylindrical Fermi surface.

Free Electrons - Closed Orbits

From figure 1,

14

P2

 

 

Px

Figure l A Spherical Fermi surface

l)z

 

 

l3x

 

Figure 2 A cylindrical Fermi surface

15

<
II

x vLcos(u'/m) , vy = vLSin (p'/m).

Then

€-
II

vLcosB.COS(u/m - B)

from equation 10. Here 0 s 3 s v/2 and cot B = am. 8 is

the Hall angle. Integrating,

E V V du = mwvi-cosg.sins
13.2. y X

2 .
ny — (nee T/Qm)COSB-Sln8

ne is the number of electrons in a primitive cell and Q is

the cell's volume. The final result is

cosea —cosg sing O

' 2

0 = (nee2T/Qm) cosg sins cos a 0 (12)
0 0 1

Note that a metal with equal numbers of free electrons and
free holes has vanishing off diagonal elements since vy =
'YLSin(u'/m) for the holes. We will later prove this result

for any carriers in the high field limit am < 1.

A Cylindrical Surface - Open Orbits

Consider the case of B 11 2 in figure 2.

p
o “A?

on v5

.1
0:11

5.,

..v.

from equation 10.

u
5°
\
<
\o
'o
l\)
u
in
m
+
'o
m

'40

1 is the momentum length separating Bragg reflection planes.

Then,
u
o 2 2 2 1/2
vidu=uV =:§(p -p)

noeer

0’ =

W e
2mm

In a similar manner we obtain the other elements of the

conductivity tensor.

 

O O O
o 2
.6 = nee T O 1 O (13)
29’” o o 1

Open and Closed Orbits
A zero'th order model of an "open" Fermi surface
might consist of allotting nfi "free holes” to a first zone
closed surface, n: electrons to a second zone cylinder, and

n: free electrons to a third zone closed sheet. The con-

ductivity tensor for such a model is

17

c c 2 c c
(ne+nh)cos B (nh-ne)cosBsins 0
3 = e21 (nc—nc)cosssin8 nO/2 + (nc+nc)cos23 O (14)
e h e e h
m0 0 c c o
O ne+nh+ne/2

For experimental simplicity we measure the resistivity
tensor in the high-field limit, a = v72. The transverse and

longitudinal parts are:

 

 

nO/2a2m2 (nfi-nc)/dm
at m ec c c g
r
2 o c c c c 2 (n -n )/dm n +n
e T[ne/2(ne+nh)+(ne-nh) ] e h e h
0 = m
22 2 f c o 2
e TLne+nhrne/ ]

and pyz vanish for this model. At 331 = 0 (3:0),

C C O
l/(ne+nh) O
E = m 0 l/(no/2 +nc+nc) O
...._._. e e h
e2 o c c
T O O l/(ne/2 +ne+nh)

The longitudinal magnetoresistance, (pzz(B)-pzz(0))/Ozz(0)a

vanishes, but the transverse magnetoresistance does not:

-A3 =[pxx(B)'pxx(O)l / pXX(O)
P

=(ng/2xngmgwwcn2 ,1 > O
(n2/2Xn2+nfi) + (ng-nfil2

g...

.»r

. —

au-u

van

.Pl

nu

$\3

ha

.1,

18

Three possibilities are listed in Table III. The first

case is realized in copper, silver, and gold with nC = 0.

h

CFPT's careful analysis of the noble metal topology showed
that Ap/p = A§(wcr)2- l for B in a symmetry plane. A is a
constant which includes the number of conduction electrons,
approximately ng, and a measure of the cylinder area normal
to B. d measures the effective width of the cylinder area
parallel to_§ -- the other three (111) cylinders cause some
of the orbits to close back upon themselves. Nevertheless
our simple model should estimate the largest Ag. From CFPT
one can easily calculate that the maximum n: = 1/5. Thus the
maximum n: / 2n: = .125 which is in rough agreement with their
largest A =.23.

Cases II and III may be common occurences in metals
with a large number of valence electrons per primitive cell,

but the burden of calculation is now truly monumental since

one cannot assume, in the manner of CFPT, that

 

 

0 (Open)
0 w IVY;
xx
_ 0 closed 0 closed)
oxx(closed)0yy(open) xy( ) yx(
All Closed Orbits
c c
DEM = pzz = nh-ne . wcT (16)
c c
and
c c 2
EH =3” 2 nh+n€ (l7)

 

er

I...

 

l9

 

 

 

 

 

Table III. Dependence of Ap/p on the number of electrons on
open orbits.
Case I Case II Case III
no 0 c c 2 o
ߤ< <(ne' h)2 29. It (ne'nh) is > (n; nh)2
2 c c 2 c c 2 c c
ne+nh ne+nh ne+nh
2 2
n°(nf_.j+ n°)(w I)? (wCI) (wot) - 1
.AA - ._§_ — l I
p 2(n: — h)2

:-
,-
v0-
pry“
.-
-4.-
I
‘ C
l .
a
I
i“
-.
“‘A

rn

2O

allow us to make estimates of n +nfi and ch from the experi-

(D+O

C

c
mentally measurable pxx, p , and nh-ne.

xy’ pzz
Real Fermi Surfaces
For a more complicated Fermi surface we can simplify

y for closed and open periodic orbits:

 

- H I
1(11) = one ‘1qu loge “3(u')du' + I ea“ X(u')dI-I' } . (18)
-cc 0

But

0 '14 t
l ew'flu'mu' + 120 ea“ _Y(u')du‘ + ------ =

'“o ' uo

O a ' -d1 P O du'
j e “_y(p0du' + e ‘03 e _g(p')du' t ........ ,

-uo ’“0
so that

No Q ,
v( ) = ae’wf l I eau'v(p')dp,' +3 ea“ _v(u')du'(.(l9)
_ H L d _. O
eauo - l O

“o is the period of the orbit. In the high-field region,

we can expand_y in powers of a:

1(u) “,1(0) + av(l) + a 2v(2) + .......... (20)
1(0) =l-__ IHO X(Hl)dlll ,, (21)
no 0

u
111(1) = 1 I140 u.'.Y(u')du' - 11.120 x(u‘)du' + "lo X(u')du' (22)
u

‘7‘

21
2 Ho , , Ho {“0 2
,%__l .x(u )du -.u_) utx(u')du‘ +._1_I uky(u0du'
“O O “O O ENOUO
-U H
- It) £(u')du' + l u'1(u')du' - (23)
no do
Only-139) is trivial;

=.l_fuoz X (X X 2) du' = (g X AE)/UO (24)

A2 is the momentum change from the beginning to the end of

the period.
Consider the case of all closed orbits. We use the

.uo u .u

notation < '> = I dpi and < '> = J du' . i = x or y.
”0 0

(o)- .
W2 - (Vz >/uo
Vél) = (u'vz'>/uo - “(vz'>/uo + <vz'>u

ll

(0) _
W1 (Vi'>/uo - O

Til)

I I U
(u'vi >/“o + <vi >

2
W: ) ‘ uW§l) + <u'2vi‘>/2uo + (U'Vi'>u

To determine 3 we must evaluate several integrals.

Tue (0) _ 2 25
JO Vsz du — (Vz'> /uo ( )
(“0 ¢(1) I>< >/ + (V (V 1)“)

v1 2 du = -<vz “Vi “o i z (26)

22

L1

v v dp = (v <v I>u> = I 0 dp _de
l0 x y x y .0 y lo x
= -Area(e1ectrons) + Area(holes) (27)
Ho (2) 2 ‘
£0 ViWi du = -<pvi> /uo -<uvi<vi'>“> + <vi<p'vi'>“> (28,29)
. (“o (l) .
The demonstration that J Vivi du = O is unnecessary: the
O
Onsager relations ok1(B) a olk(-B) predict that diagonal

elements can only be even in a and reduce the number of in-
dependent off-diagonal elements to three.

For orbits open in Q, the only changes are

W§O) = (Vy'>/“o = ‘Apx/“o

ng) = (u'vy'>/uo - uApX/uo + <vy'>u

Then,

5:0 VZW§O)du = Apx<vz>/uo (30)
(:0 vxv§0)du - 0

E20 vxv§1)du = -Apx<uvx>/uo + <vx<vyt>u> (31)
I'“° vinO’m = (nape/II, (32>

0
In the most general case, we cannot expect integrations
over dpz to cause the vanishing of any of the functions of pz

represented by these integrals. Thus the conductivity tensor

has the form

(I.

23

2
a
coat aocCL aoc0L
' 2
0= 3. a. +8, + o 7:
oca o oc0L ao aoc‘3L ’ (/3)
a a +
oca o aoc0L aoc

a0 is a coefficient to be evaluated for open orbits only,

aoc for both open and closed orbits. Inversion of this
tensor gives us the experimentally measured resistivity
tensor. Normally each piJ depends on all nine of the Okl’

but for the case of no open orbits a simplification occurs:

= (-l)i+J cofactor(oij)/determinant(5) ;

p13
Since determinant(5) = O( 2) = -c o o
a 22 xy yx,
~B° 1 o ~B°
/ xy
= ~ ~ 9
p l/Gyx B B , ( )
.qBO ’QBO .QBO

Of the nine elements, only pxy and pyx do not saturate in

the high-field region. These terms have a simple form since

—-

o = - 282Ig V [Ap(e) — Ap(h)! dpz
(Eva); 3.2?
_ n -n (35)
BO
0 is the volume of the primitive cell, ne and nh are the

number of occupied electron and hole states, respectively,

v .
nrv
v so

3‘.-

.V‘

24

per primitive cell of the crystal. The prediction of ne-nh

for any metal is given by setting the ”known” number of
electrons in the conduction band equal to the number of

states occupied in the various zones of momentum space,

nV = 2F + ne+ (2J - nh) . (36)

nV is the number of valence electrons per primitive cell
in the crystal, F is the number of zones completely filled

with electrons, 2J — n is the number of electrons in par—

h
tially filled zones with hole surfaces. Notice that, if nV
is even, ne-nh may vanish. This actually occurs for all
even-valence non-magnetic metals whose Fermi surfaces have
been investigated and leads to a completely different re-
sistivity tensor because the determinant(3) = 0(a4). These
"compensated" metals are primarily characterized by elements
Dxx’pyy = 0(B2) in contrast to the odd-valence'hncompensated”
metals.

For“singular” field directions(l) to be discussed in
detail later, we must amend OXy:

0xy = -\el(ne-nh*An) (37)
B0

 

An measures the number of carriers which have changed charac—
ter on an open sheet. Thus a compensated metal can undergo

ngeometric discompensation” along certain high symmetry axes

 

 

25

(<0001> in Mg and Zn), or an uncompensated metal could be-
come compensated (this is almost the case for Cu, E ||
<111>).

If there are open orbits on the Fermi surface, the
determinant(5) does not simplify although it is still of

order 32. The form of 3 is

32 B so
a.» B Bo Bo (38)
0 B0 B0

Thus pxx goes as B2 with a coefficient dependent in a com—

plicated way on the shape of the orbits since

0 G -O' 0'
pxx = lyy zz zy_yz (39)
U 0' -U U :10 +0 0 U
L yy 22 zy yz xx yz xy zx
+0

0 o a o o o o
zy xz yx yy zx xz zz xy yx

For certain symmetry directions, e.g. <211> and (110),
it may be possible for a surface to support two bands of non-

intersecting open orbits with different average directions.

In this case all the elements of p saturate.

Description of the NFE Model of the Fermi Surface of Aux2

Aux2 has the fluorite structure with the gold atoms

lying on a face-centered cubic lattice and the X atoms

occupying all the tetrahedral sites between the gold atoms

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(3
r\/’ /C)
4,

 

 

 

Figure 3 Crystal structure of the AuX2 compounds.

X is symbolized by the darkened spheres.
(After Jan §§.al., ref. 13)

 

Figure 4 Holes in the second zone of the NFE model

of AuXe. (After Jan 2; al., ref. 13)

 

 

27

(figure 3). This structure belongs to the Space group ijm,
so the Fermi surface will have full cubic symmetry. The
NFE Fermi surface was first constructed by the NRC group(13)
using the method of Harrison(22). A free electron Sphere
whose volume equals the number of valence electrons times
one-half the volume of each Brillouin zone is positioned
about each body—centered cubic lattice point in momentum
space. The occupied electronic states in the n'th zone are
made up of all points located within n or more Spheres. We
can thus construct the Fermi surface for each zone in the
repeated zone scheme without considering the placement of
the zone boundaries. For our model seven nearly free
electrons are assumed and one-half the volume of each
Brillouin zone is (1/2)(4)(2wh/a)3. The factor 1/2 arises
because each zone can accommodate two electrons per primi-
tive cell of the real lattice; we choose 2wh/a as a unit in
momentum space to render the model independent of the
lattice parameters which vary among the three compounds.

To facilitate a study of this surface a computer
program was written which performs the Harrison construc-
tion calculations and plots the results with the aid of a
30" x-y plotter. A description of this program, which also
plots Brillouin zone boundaries, is contained in Appendix A
along with a program listing. The cross sections shown in
figures 8-17 are from the computer plots.

The first zone is full. The surface in the second

zone has the shape of an octahedron holding about .05 holes

 

 

Figure 5 Holes in the third zone of the NFE model
of Aux2 in the reduced and repeated zone
schemes. (After Jan gt al., ref. 13)

29

(figure 4). There is good experimental evidence concerning
this surface in AuAl2 and AuIn2 from the deA experiment.
The open surface in the third zone, containing .34 holes,
makes contact with the hexagonal faces of the zone as do
the noble metals (figure 5). deA data indicate that this
contact area is reduced to about one-third of the NFE value

in AuGa2 and AuIn and 1/15 of that value in AuAl this

2 2’
will considerably reduce the width of the open orbit layers,
particularly in AuA12. The open electron sheet in the
fourth zone (figure 6) has "arms" along the directions
(100). deA evidence indicates that some of the AuAl2
extremal areas on this surface have values in good agree-
ment with NFE predictions. It holds 1.14 electrons. The
surfaces in the fifth and sixth zones contain .20 and .05
electrons, reSpectively (figure 7). Recently experimental
evidence from deA confirms the existence of a surface in
the fifth zone. (See reference in Table XI)

The galvanomagnetic prOperties give no direct informa-
tion on closed surfaces, but simply determine the number of
full plus hole zones. A measurement of the Hall effect in
AuX2 for general field directions supporting no Open orbits
will give ne - nh; since closed and open surfaces contribute
to ne and nh one suSpects that all the surfaces must be
considered in computing this quantity for any model. For
the NFE model of AuX2, ne-nh =
= 1.0. A close examination of equation 36 reveals, however,

.05 + .20 + 1.14 - .34 - .05

3O

 

a———"———-‘-‘ ,
_- w...

 

 

 

 

 

.——-—-——'——.— a- --
#—

Figure 6 Section of the NFE surface in the fourth
zone. (After Jan 23 al., ref. 13)

 

Figure 7 NFE surfaces in the fifth zone (left) and
sixth zone (right). (After Jan 5232 54.,
ref. 13)

31

that any model having seven valence electrons and a combina-

tion of three full and hole zones also has ne - = 1.0.

nh
In addition to the general field directions, there are
two singular field directions in the NFE model, <100) and
(111). They are defined as axes Of higher than two-fold
symmetry which are at the center Of a region Of aperiodic
open orbits. At the singular direction the Open orbits
intersect tO form closed orbits Of character Opposite to
that Of the Open surface. Figures 8 and 9 are cross sec-
tions Of the Fermi surface for g 11 (100) and (111) in the
third and fourth zones. Clearly, there are closed hole
orbits on the fourth zone electron sheet for g 11 (100) and
there are closed electron orbits on the third zone hole
surface for g 11 (111). To calculate n - n

e h
(100), we must subtract from ne those electrons which have

for E 11

changed character, né , and add to nh the new holes, nfi.

This is equivalent to subtracting the total volume occupied
by né and nfi multiplied by 20/(2rh)3; this volume is A°d
where A is the cross-sectional area Of a cell with A i; A(pz)
and d is the pz width over which the orbits have changed
character. An = né + nfi is to be added tO ne - nh for
electrons on a hole sheet and subtracted from it for holes
on an electron surface. Then

B0 -
a. u

The calculated values give n111 = 1.035 electrons per prim-

itive cell and n = 0.372 holes per primitive cell.

100

Figure 8

Figure 9

Figure

Figure

Figure

Figure

10

ll

12

13

Cross sections of the NFE surfaces in the
third (clear) and fourth (shaded) zones at

the pz values given for_§ || (100). Height
Of the unit cell = 2.0

Cross sections of the NFE surfaces in the
third (clear) and fourth (shaded) zones at
the pz values given for B (( (111). Height
of the unit cell = (3)1/E/3 = .577

Cross section of fourth zone surface for E

O
29 from [100] in a (010) plane with orbits
open in [OlO].

Cross section of fourth zone surface for E
in {110; with orbits open in <llo>.

Cross section of third and fourth zone sur-

faces for E in {111} with orbits open in <111>-

Cross section of fourth zone surface for E
in {210; with orbits open in <zlo>

35

I gs I;

 

(110) -)

 

Fig. 9

34

These values differ significantly from the experimental
values we will present later.

We now turn to the investigation Of those field
directions supporting Open orbits. From figure 6 it can
be seen that as the field is tilted away from [100] in a
(010) plane, orbits which are open in the direction [010]
will occur for some values of pz on the fourth zone
surface. A cross section for g 29° from [100] demonstrates
this effect in figure 10. These periodically repeating
orbits are called "primary" Open because they make repeated
use of the same [010] arm of the Fermi surface when they
cross the zone boundary. Secondary periodic Open orbits
occur for some field directions in the (110) plane by
repeated use Of [100] and [010] arms to give a [110]-
directed Open orbit (figure 11). Fourth zone tertiary
orbits Open in the directions (111) and (210) have also
been investigated (figures 12 and 13).

If.§ is applied in a direction close to the [100] axis
in a non-symmetry plane (point a in figure 18), a plane
perpendicular toug will intersect the Fermi surface to form
alternating bands Of closed electron and hole orbits which
are separated by two-dimensional aperiodic Open orbits as
seen in figure 14. They are called two~dimensional since
they are generated for a solid angle of field directions
which is represented by an area on a stereogram, and they
are called aperiodic since the direction cosines of1§ are

incommensurable. As the angle between E and [100]

Figure 14

Figure l5

Figure 16

Figure 17

35

Cross section of fourth zone with E at point

a in figure 18. Open orbits separated by closed
electron and hole orbits, net direction is 100
from <lOO> in the {100} plane.

Cross section of fourth zone with E at point b
in figure 18. Open orbits separated only by
an occasional closed electron orbit; note how
closed hole orbits have unfolded to form sec-
tions Of open orbits or in one case pinched
off the Open orbit altogether; net direction

is 18° from <lOO> in the {100} plane.

Cross section of fourth zone with g at point c
in figure 18. Extended orbits on the fourth
zone surface several degrees from the edge of
the two-dimensional region.

Cross sections Of the NFE surfaces in the
third (clear) and fourth (shaded) zones at the
pz values given for 3 parallel to <llO>.

 

37

increases, the number Of hole orbits gets progressively
narrower and disappears; Observe figure 15 and point b in
figure 18. If the field now angles towards (111), the
number Of Open orbits begins tO decrease, and finally all
the Open orbits coalesce tO form closed orbits "extended"
over several zones; refer to figure 16 and point c in
figure 18. If the field moves toward (110) from point b,
the Open orbits persist even for a parallel to that axis.
The cross sections for g 11 (110) shown in figure 1? reveal
only closed electron orbits for small values Of pz, closed
hole orbits for intermediate values, and Open orbits for
the largest values Of pz. The presence of closed hole
orbits for E 11 (110) requires that (110) be surrounded by
a two-dimensional region Of Open orbits in order that the
closed hole orbits may be "unfolded" by an Open orbit and
then "refolded" into the electron orbits we observe for a
general field direction. But the width, d, of the hole
orbit layer cannot be determined on this model through the

Hall voltage since the Open orbits change ox into a very

y
different form than it has in equation 40. For the field
parallel to (211), there are non-intersecting orbits Open
in directions (111) and (110), but there are no hole orbits
on the electron sheet. All of these results are summarized
in figure 18a which is a stereogram Of field directions.
Shaded areas represent field directions giving rise tO
orbits Open in a single direction and, therefore, to a B2

dependence of the magnetoresistance. 115 field directions

 

 

 

 

E]

Figure 18a
Magnetoresistance
stereogram for
the NFE model in
the fourth zone .

”<lOO>

Figure 18b
Standard 103

stereogram.

Figure 18c
FagnetoresiSEance
stereogram for
the NFE model in

the third zone.

p
V
U

I.

PA

in

VV

Cl. 0

a»

‘U.

or

AU

39

were sampled for open orbits by the computer.

A similar analysis has been carried out on the third
zone hole surface. (111) is singular but the layer of
electron orbits on the hole sheet is so narrow that the
extent Of the two-dimensional region is estimated at less
than One degree. (100) is not singular. There is a two-
dimensional region of Open orbits surrounding (110) due to the
presence of a layer Of electron orbits when p is parallel
to that direction. The complete results are summarized in
figure 180. If the necks are diminished in size, we can
expect a decrease in the size Of the two—dimensional
regions and possibly the angular extent of the secondary
periodic Open orbits, (100) and (110). The (111) primary
open orbits will probably only be restricted in width
parallel to B.

Lifshitz and Peschanskii have analyzed several types
of Fermi surfaces which were derived from an analytic
expression for e(p). (23) For one surface which consisted
of a three-dimensional grid of undulating cylinders whose
axes are parallel to the directions <lOO>, <llO>, and (111),
the (111) and (110) two-dimensional regions overlapped.
Inside this overlap, layers Of Open trajectories with
different average directions are formed; thus the magneto-
resistance must saturate destroying the connectivity Of the
two—dimensional regions. These aperiodic open orbits must
be intersecting rather than non—intersecting since all

values Of pz will be sampled if repetition is not possible.

.
v F. s“
.u v ‘4

5-.

:1.

40

Both our NFE results (figure 18) and our experimental data
(figure 35) show that their analysis is not Of general
validity. The reason for this is the artificiality Of a
model with three sets of open "arms". Such a model cannot
give non-intersecting orbits in two directions for

B 11 (211) or (110), a known feature Of at least four Fermi

(24,8)

surfaces and our NFE model. Their model also cannot

produce (100) directed orbits when H 11 (110), which occurs

(25)

for the copper Fermi surface and for our NFE model of

Aux2 also. Thus we see that their model is of limited
value. Two interesting variations Of their model dO occur
on the NFE model Of Auxe. The most Obvious is the fact
that the third zone has (111) directed arms while the
fourth zone has (100) directed arms. The region Of over-
lapping aperiodic Open orbits is centered entirely about
(110). Our computer plots, however, indicate that the
average direction Of Open orbits from both zones is the
same. Secondly, the rather abrupt termination Of the two-
dimensional region about (110) as §_moves away from (110)
within the (110) - (111) - (211) Spherical triangle is
caused by the intersection Of orbits Open in different
average directions. One set Of orbits is derived from the
(100) - directed orbits seen for g 11 (110); the other set
arises from the unfolding Of the hole orbits for g 11 (110).
For 5 not far from this axis these orbits "constructively

interfere". But with g deviating by more than 10° from
(110) they "close" each other Off.

(I)

(I?

(3-

 

Calculation Of Ap/p for Aux in the {100} Plane

2

We have developed a theory capable Of calculating the
absolute value of the magnetoresistance for a complicated
Fermi surface model with a single value of wa. We now wish
to apply this theory to the NFE model Of Aux2 in order to
predict the value Of the magnetoresistance. Unfortunately,
this project is, in fact, a major undertaking and we resort
approximating the Fermi surfaces Of zones 2,3,5, and 6 by
Spheres and assuming the fourth zone necks have square cross-
sections. The drastic nature of these approximations sug—
gests that calculations on this model should be considered
primarily as a guide to a more exact later calculation.
However, a comparison between Oyy determined for the NFE
model and this model was made and it indicates that the ap-
proximation may be fairly good for a less than 300 from (100)
(See figure 20).

Consider figure 19. This "log-pile" surface has the

same topology as the NFE Fermi surface Of AuX in the fourth

2
zone. For simplicity Of calculation, we have chosen the
necks tO have a width Of 1/2 in units 2wh/a = 1. We con-
sider orbits in the {100} plane only and note that there are
six types, labelled A,B,C,D,E, and H. We must calculate ,
weighted velocity averages of these orbits and integrate over
dpz to Obtain the various 013's.

For our model each Open orbit is composed Of five

1
types of sections which we label in figure 19 as K, K. 1,

3

41

42

Figure 19 Open and closed orbits on a "log-pile” surface
with the same topology as that Of the fourth
zone Of the NFE model.

43

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

v
i
E Figure 19
B. p
z ¢ p/z
T= '/4o . 41
/ dpl (A) = 8/2
p,
L:””””§; I

'/z

1+4

v, and v'. The closed electron and hole orbits are each com-
posed Of only two types. From the figure, K = -dpt/vL =
-(-3/4)/stinv = 3/4vs = m/2s. The Fermi velocity, VF = v,

is given by nkF/m = 1.495/m m 3/2m. Similarly K' is m/2cosv =
m/2c. Also v = m/3c and v' = m/3s. Since x changes with
angle and pz, we leave it as l = I/v =(Xo2m1/3; X is the
momentum length of x in the usual units 2wh/a = 1. The slow
variation of.! with pz is rather troublesome; a considerable
Simplification is Obtained if we average.! over the appro-
priate range of pz for each type Of orbit. These values are
I(A) = l/4c, 7(3) = l/4c, 7(0) = l/c, I(D) = 5/40 for the
Open orbits. The ranges Of dpz are all s/2 if tanv<l/3. If
1/3 (tanv< 1/2, the range of dpz for the D orbits is c/2 - s
and RD) = 1/4s + l/2c. For 1/2 <tanv< 1, the D orbits
vanish while the range Of dpz for the C orbits changes to
l/(2-(c-s)). The value of:E(C) remains the same however.

.IKE) and i(H) follow in a similar manner.

The periodic lengths Of the orbits can now be eval-

uated:

2K + 2x(A) + v

(10(1))

m/s + m/3c + m/3c

and so on.

uO(B)

45
m(l/s + 2/3c)

2K' + 2x(B) + v'

(m/3)(4/c + l/s),

A check of equations 25 through 32 shows that we must

still compute the following quantities for each type Of

. I H t H | H
orbit. (v2), <uvx>, (vx<vy > >, (VX<VZ > >, (uvx<vX > >,

and <vx<u‘vx'>“> all integrated over the appropriate range

Of dpz.

multiplication.

the A orbits.

By using I, this integration amounts to a simple

We will now compute these quantities for

<vz> =
= 3c/2s — S/2c
rK+)\
(uvx> = j deu + J
K K+l+v
= -v(l+v)l
= —m/(802)
u
. u = F 0 d
(vx<vZ > > JO vX 0 IO
= —V(VZV)vl
= s/8c2
H = - =
<vx<vy'> > v(vyv)vl

VCK + 0'1 + (-vsv) + 0') + VCK

PK+k+v+X

u(-V)du

f n

.v 1‘

(U

‘i

46

Similarly,

(uvx<vx'>u> = -v212(x/3 + 0/2) = -l/72c3

= -(vx<uvx'>“>-
The evalution of all the other quantities follows in a
Similar manner.
In order to take account Of the NFE surfaces in the

second, third, fifth, and sixth zones, we have employed the
following approximations:

Oxx z 2Loxx(fourth zone) +(.39 + .25huJ

022:3 2[Ozz(fourth zone) +(.39 + .25)/mJ

O z 2[Oxz(fourth zone)J

xz
Oxy = 2L0Xy(fourth zone) + .39 _ '25j (41)
Oyz = 2Loyz(fourth zone)]
Oyy = 2[ny(fourth zone)J

The first three equations contain free electron approxim-
tions for the .39 holes in the second and third zones and
the -25 electrons in the fifth and sixth zones; we assume
that the third zone necks are pinched Off. The factor 2
arises because there are two sets of each tYPe Of orbit in
the fourth zone and because the momentum volume occupied by
.39 holes, for example, is .78(2wh/a)3. (In the high-field

region, the Hall angle = w/2 in equation 12). We drop the

Th

‘A
iv

47

factor 2e27/(27rh)3 for simplicity. The last three equations
are exact in the high-field region.

The results Of a computer program.written to carry out
such calculations (Appendix B) are presented in the graphs
Of figures 20 and 21. In figure 20 we have plotted the var-
iation of 5 with angle in the {100} plane. We are using the
convention 2wn/a = l, 2e2T/(27rh)5 = l, and a = 1. As a

result terms of order a0, O O , appear in units

yy’ yZ’
l/m; terms of order a, OXy and oxz’ are dimensionless, while

d O
an zz
w o 2 o o T
Uxx’ which is Of order a , 18 measured in units of m. o
determi e
n pxx’

0 0 -0 0
pXX = __1yy zz zy yz , (39)

_ + 0 0 0
(OnyZZ Uzycyz)oxx yx zy XZ

_ O - O O O J
+ szoyzoxy Oszyy zx zz xy yx
we first note that both factors in the numerator are given
in units of (l/m)2 while all six factors in the denominator

are given in units of l/m. It is obvious, then, from figure

20 that all terms containing O or OZX will be small so

 

xz
that
- O
pxx argyyozz Ozy yz . (42)
- -O — O O O
(nygzz Uzyoyz) .xx _zz xy yx

The magnetoresistance can be calculated from pXX by noting

that

(Tu

 

 

 

 

48

O'yy

\ .
.. NFE fourth zone

_. NFE third zone
_ log — pile

open
Itotol

 

 

 

 

 

open
1" e'o' I ’ \p
(:00) (no) (me) (no)
O'yz 0'"
v y
I/In'
I/m:
: aw
~I/m- 090“ -I- open
'9otol
a . n I |
('00) ". '°. (:10) (:00) W ("0)
Figure 20 Variation of'E with angle for the log-

 

 

 

'p’u

'-. total

 

I/m‘

cIoeed

 

 

(mm.

    

a." 310' r: '4’
(no)
O’xz

closed

 

(100)

'o ‘ o I
In so (no)

pile model.

49
2 2 -
331;: ' 2Wfi 3ma 1 =.m§3_° (wcT)2
(27W: )3 a3 2821’
We estimate 1 from the resistivity at B = 0 using a free
electron approximation, p = maB/negw - n equals the number
Of conduction electrons per cell Of volume a}. This number

is 28, so that

Ap/p = lAm-pxmeT)2 - 1 . (43)

For some Of the samples measured in this work, wcvze 5,

which gives

Ap/p m 350m°pxx - l . (44)

The results Of these calculations are given in figure 21.
There are five distinct regions Of pxx depending on the

number Of Open orbits. They are:

00-5O Case I behavior: a small layer of

open orbits.

50-8O Case II behavior: a moderate number
Of Open orbits whose importance is
enhanced by the rapid decrease
Of Oxy due to the thinning layer of
hole orbits, H, and the thinning
layer Of electron orbits, E; all
terms in (42) are important.

80-13o Case III behavior: a thick layer

Figure 21
Ap/p in
{100} plane
for log-pile
model.

 

 

 

 

51

of Open orbits aided by the van—

ishing Of OXy at 120 results in

pxx.“ l/axx'

130-20O Case II and III behavior: a thicker
layer Of Open orbits is moderated
by a decreasing Oxx so that the
first term in the denominator of
(44) isgs constant. The second term
is increasing, however, causing pxx
to decrease.

o o . ~ -

2O -45 Case III behavior. pxx ~ l/Oxx ,
Oxx is in turn dependent on fourth
zone Open orbits for its variation.

In addition to the complete calculation Of 0 for this
model, we have determined Oyy from the third and fourth zone
Open orbits of the NFE model. The results of that calcula—
tion, also shown in figure 20, suggest that the log-pile
model is a good approximation to the NFE fourth zone and
that the Open orbits from these surfaces do dominate those
of the third zone even without the known reduction in the
size of its copper-like necks. The dip in the NFE fourth
zone curve occuring over the range 300-45O warns us to look
at pxx in this region with some suspicion since Uxx may also

look somewhat differently in this interval.

52

3. Growth and Preparation Of Samples

In this section we discuss the techniques used to grow

, AuGa and AuIn . We also con-

2 2’ 2
sider the X-raying, sparkcutting, and mounting of the

single crystals Of AuAl

crystals. The general procedure for making the alloys was
as follows. A small high-purity graphite crucible, usually
with an alumina insert, was outgassed in a Lepel induction
furnace, the temperature Of which was increased in several
steps up tO l200°C in such a manner that the pressure
remained at 10'“ mm Hg. The pure metals were etched if
necessary, washed with distilled water, and rinsed with
ethyl alcohol. The desired amount Of A1, Ga, or In
(typically 2.5, 6, and 8 grams) was then placed in the
crucible. Its weight was determined to .l milligram on a
Mettler balance and the amount of gold necessary was
computed and deposited in the crucible. The crucible was
then placed in the induction furnace and heated to 100°C
above the melting point of the compound (of. Table IV)
either at a pressure of less than 10-4 mm Hg or in an argon
atmosphere. Mixing was accomplished by agitating the melt
mechanically and by the action Of the rf field.

Several methods Of growth were attempted before high
purity single crystals were produced. The first success

was Obtained by vertical zone refining of AuGa with an rf

2
coil. The samples, prepared with the exact stoichiometry,

had residual resistance ratios up tO 250 at the bottom of

a;
5
.n‘

53

Table IV. Properties Of the AuX2 compounds. (After Jan_§§
_gl., ref. 13)

AuAl2 AuGa2 AuIn2
Resistivity at 2950K 8 13 8
(on-cm)
Lattice parameter at 5.988 6.055 6.487

4.2°K (angstroms)

Melting Point (00) 1060 492 544

54

the crystal. The graphite boat used was grooved so that
the dimensions Of the crystals were 1/16" x 1/16" x 4".
Three experimental runs on these crystals were disappoint-
ing. At 55 kilogauss, the largest magnetoresistance
Observed was 5 and no decision could be made on the state
of compensation of AuGa2 since the field dependences were

Bl'O-i .4

for all measured directions. The third crystal
investigated, which had i approximately parallel to (111),
did have a deep minimum with B ~v() (112) in its transverse
magnetoresistance. Hall effect measurements were inconclu-

sive: 1.2(ne-n (1.5 for several different general field

h
directions.

For three reasons we decided to switch our concentra-
tion to AuA12. Straumanis and ChOpra had determined that
the extent Of the AuAl2 phase is 78.18 - 78.94 % weight
Au<26). At the stoichiometric ratio there are .152 empty
lattice Sites per unit cell in the Al sublattice and .076
empty lattice sites in the Au sublattice. But at the
Al-rich border there was strong evidence that all of the
A1 vacancies were filled. A crystal grown at the Al-rich
border of the phase should be appreciably better than one
Prepared at stoichiometry. The standard Bridgeman tech-
nique was apparently not very successful when AuGa2 was
prepared up to two atomic percent Off stoichiometry(27).

Finally, AuGa has a rather high room temperature resis-

2
tivity Of 13 uO-cm, compared to AuAl2‘s 8 ufl-cm; this means

 

 

55

that oh? (free electron) for AuAl will be about 50% larger

2

than that for a AuGa crystal with the same residual resis-

2
tance ratio.

Our first attempt to grow Al-rich AuAl2 in a graphite
crucible was a failure. The crystal, which wet the
crucible, had a residual resistance ratio of 33. We dis-
covered that near 1000°c Al forms a carbide with the
graphite; so we decided to place an alumina insert inside
the graphite. The dimensions Of this insert were 1.5" long
and .42" inside diameter. The bottom of this alumina
crucible was bowl-shaped so there was some premature
concern that it would be difficult to grow single crystals.
The crucible and its contents were, as usual, sealed in a
vycor tube filled with argon and lowered thru a three turn
rf coil at a Speed of 1/2" per hour. The temperature of
the graphite was measured with an Optical pyrometer and the
rf current adjusted so that the hottest portion of the

crucible was 60°C above the melting point Of AuAl Upon

2.
breaking the vycor, we discovered that AuAl2 had wet the
alumina; however, the alumina insert had cracked due to
differential contraction upon cooling, and it was possible
to pry Off the pieces of alumina clinging to the AuA12
slug. Back reflection X-ray photographs indicated that the
crystal was single. The first two samples spark-cut from
this slug had residual resistance ratios of 400 and 550
with J's parallel to (111) and (100) respectively. We

designate these samples as A1 (111) and A1 (100); here A

56

refers to the compound, 1 to the slug, and the numbers in
brackets to the current direction.
Since studies similar to that of Straumanis and ChOpra

had not been published for AuGa and AuInz, it was neces-

2
sary to determine experimentally the dependence Of their
residual resistance ratios on the excess concentration Of
one Of their constituents. Slugs of 0.0, 0.1, 0.2, 0.3,
0.4, and 0.5% weight excess Ga were prepared in the same
manner as A1. The average residual resistance ratios Of
crystals cut from these slugs were 190, 540, 680, 710, 660,
and 200 respectively. The points in figure 22 representing
individual samples show a considerable Spread about the
average. It should be noted here that traces of Ga were
found on the surface Of the slugs with 0.2 - 0.5% excess
Ga. This may indicate that the phase exists to about 2%
weight excess Ga; as a larger amount Of Ga is added it is
energetically more favorable for the charge to reject this
Ga, but at 0.5% the Ga phase begins to coexist with the
AuGa2 phase inside the slug. See figure 23.

A AuIn2 slug Of exact stoichiometry had a residual
resistance ratio of 60. By varying the composition to both
sides, the highest value achieved was 75 in a .1% In excess
AuIn2 Slug. Several growth methods were tried in an
attempt to improve on this value; these included vertical

zone refining and horizontal zone refining and leveling.

Crystals prepared in such a manner had residual resistance

ratios less than 60.

.soaooapsoocoo mu mmooxo .m> manomzao

so:
omvrw.‘3
passages

57

moos< no mam mm osswam

 

so. so :2 so. so. :3
o. e. n. w. t
i i i i i
Ll
r
. ..000
IT
. hr
.rooo_
wooa< :
m
xqas
4‘

 

 

 

 

WEIGHT PER CENT GALLIUM

IO
L4

20
1.11

30
1

50
I

70 90

ill

 

Kmo—J

lo

800 "'

600 ""

400 —

TEMPERATURE,
l

200 '-

 

é—- AuGa

AuGa

AuGa2

 

 

 

 

r

 

T
60

 

ATOMIC PER CENT GALLIUM

Figure 23 Phase diagram of Au-Ga.

59

Table V is puzzling for at least two reasons: AuIn2
cannot be prepared to within even a factor of ten as highly
ordered as AuGa2, and AuA12 of exact stoichiometry can be
grown with residual resistance ratios greater than 100
despite the fact that 1.9% of all the sites are empty
according to Straumanis and Chopra. This latter effect is
extremely peculiar because Sellmyer(3) has shown that AuSn
qualitatively obeys the same empirical law as many of the

elements,

RRR ~r10u / I , l 5.1 $.10” (45)

Here I is impurity or vacancy content in parts per million
(p.p.m.). Thus 49's, 59's, and 69's metals typically have

1
3, and 10 . This is

residual resistance ratios of 102, 10
reasonable because metals have room temperature resistivi-
ties of l - lO uO-cm, while the resistivity due to impuri-
ties or vacancies in dilute alloys is from 1 — 10 uO-cm.

per atomic %. Assuming that AuAl obeys this law, a resin-

2
ual resistance ratio of 140 is equivalent to I = 71 p.p.m.
or .007l%. This compares very unfavorably with the 1.9%
from the Straumanis and Chopra study. The difference
cannot be explained by the fact that compounds prepared
with stoichiometric proportions may grow off stoichiometry;
consider the limiting case of the Al-rich border: we
expect I = .634% but, experimentally, I o'.0018%. A possi-

ble explanation is that vacancies are segregated and not

diSpersed throughout the material; adding extra Al or Ga

60

Table V. RRR of AuX2 crystals

AuAl2 AuGa2 AuIn2
prepared prepared prepared

exact Al-rich exact Ga—rich exact In—rich

largest 140(13) 550 250 904 60 75
RRR

 

61

simply cuts down on the size and number of aggregates. The
variation of residual resistance ratio over the slugs tends
to support this belief.

In this discussion we have neglected the residual
resistivity due to the impurities in the Au, Al, Ga, and
In. The ASARCO gold and indium had I ~'9 p.p.m. For the
MRC aluminum, I ~'2 p.p.m., and for the ALCOA gallium,

I «'1 p.p.m. Table VI shows that the residual resistance
ratios of both AuAl2 and AuGa2 are appreciably affected by
the impurity content of the starting material if the law
RRR ~104 / I holds. For some combinations of impurity and

host, a law in which 104 is replaced by 105

better fits the
resistivity data; this is the case for mg in Cd and Sn in
In. The numbers in parentheses are the appropriate changes
which, in this case, clearly indicate order limiting of the
residual resistance ratio. The use of 69's gold in these
compounds could determine a suitable form of the law and
possibly provide a most desirable increase in the average
relaxation time.

Samples were obtained by placing each slug in a small
brass cup and securing with one metal and five nylon
retaining screws. This cup was screwed into the face of a
goniometer and the slug oriented to within 10 of the
desired current axis with the standard Laue back reflection
technique. The entire goniometer assembly was now mounted

on a platform which was the high voltage side of a Servo

Met Spark cutter. A stainless steel tube attached to the

.‘..L

“V:

0".

—C

Li:

/-—- 5 a.

vrl

Table VI. Impurities in AuX2

AuAl2
prepared RRR 550
I from eqn. 45 l
(p-p-m-) (180)
I from starting 4.3
material (p.p.m.)
I from ordering 13-7
(p p.m ) (175.7)
RRR (if limited 2300

only by impurities) (23000)

RRR (if limited 770
only by ordering) (570)

62

crystals.
AuGa2
904

11
(110)

3.7

7.3

(106.3)
3720

(37200)

1360
(940)

AuIn2

75

133
(1330)

124
(1321)

1110
(11100)

80
(75)

 

 

63

working arm at ground potential then cut out a cylindrical
sample. Non-metallic debris was removed through a side arm
Of the tube connected to a water pump. Since it is impossi-
ble to obtain X-ray pictures from a Spark-cut surface of
these compounds and etching is also of no benefit, the
crystals were spark-planed on four sides to give a
rectangular cross section. This is an extremely tedious
process but well worth the effort when attempting to mount
four orthogonal Hall probes. The final shape was roughly
10mm x 1.5mm x 1.5mm. Usually, major symmetry axes were
perpendicular to each face.

Mounting six potential and two current leads to a
sample this size requires that it be firmly mounted to a
large heat Sink. The Sites selected for probe placement
were lightly abraded with a pointed object or a pencil sand
blaster. This area was then tinned with solder until a
very small bead was formed (less than .5 mm in diameter).
The leads could then be quickly soldered to such sites.
Wood's metal solder and Sta-Clean flux were used for
AuGaé and AuIn . Rose's alloy was found superior for

2

AuA12. Table VII lists those samples selected for

experiments.

.
.ukl

‘v‘

(3‘;

64

Table VII. Samples selected for experiments.

Date Crystal RRR B(kG) Type of Data
8-65 G(randoml) 230 55 MR and Hall
3-66 G(random2) 205 55 HR and Hall
6-66 G<lll> 160 55 MR
9-66 Gl<1lO> 190 85 MR
Gl<100> 200 140 MR
G<lll> 160 140 MR
C(randoml) 230 140 MR
* Il<100> 60 140 MR
* Al<lOO> 550 140 MR
* Al<lll> 400 140 MR
8-67 04(110 620 50 MR and Hall
04(111 775 50 MR and Hall
11-67 * 05(100} 475 150 MR and Hall
* 03<100> 725 150 MR and Hall
* 03<110> 904 150 MR and Hall
* Al<100> 500 150 MR and Hall
* A2(random) 550 150 MR and Hall
* I2<110> 75 150 MR and Hall

* Interpretation of data in section 5 is based on evidence

from these samples.

65

4. Apparatus and Experimental Techniques

The experiment consists of measurements of the magne-
toresistance and Hall coefficient as a function of the mag-
nitude and direction of the magnetic field. The apparatus
used at Michigan State has already been described.(3) Since
we will just present data taken at the 150 kG fields avail-
able at the Francis Bitter National Magnet Laboratory, we
will only consider the apparatus kindly provided for our
use there by Dr. D. J. Sellmyer.(°°)

Figure 24 is a schematic diagram of the apparatus and
circuitry. The apparatus, shown in figure 25, permits the
field to be oriented along any crystallographic direction
for an arbitrary sample axis. This is accomplished by use
of the worm gear which changes the tip angle m, and the
spiral gear which changes v. The sample is mounted on an
insert, I, which is removeable so that the crystal can be
positioned by Laue back reflection techniques until a certain
axis is parallel to BB'. The advantage of this positioning
is that all rotations of t for any n will be straight lines
on a stereogram centered at <lmn> if_§ is in the plane {imp}.
This is seen in figures 26 a and b. The drive rod D is con-
nected at the top of the cryostat to a motor whose speed was
normally adjusted to achieve a 180° rotation of v in 10
minutes. Rotation plots were recorded continuously on an
X4Y recorder with the X axis signal coming from a linear

ten-turn potentiometer coupled to drive rod D. We estimate

66

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_i _lHrI

1! II,

 

 

 

 

 

 

 

 

 

 

 

 

 

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oaocoaom J amppam

.ox. oma
. . 1;
seesaw HHHA
mama zoa, sopoauao>onmc —>
, _
was assesses
magnum
unoaasO
nmpmsoo

 

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Omaaqad COpoonaam

67

I
I711.» DRIVE R00 0
HI
anNG "

  
  

STAINLESS .
STEEL TUBE

RING RETAINING SCREW

WORM
DRNE l
GEAR I

    

SPIRAL

SCREW- IN
PIVOTS

  
 
  
 
 
 

 

 

 

TWSEND an, \\a
MACHWED \T‘
OFF TO

CLEAR -, *
wunwc ’/

TEFLON SPACER

SPIRAL DRIVE GEAR

Figure 25 Experimental apparatus (after Sellmyer, ref. 23)

68

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laws on» Ca OHQEmm

on» wcaaaap ocm

wsHoSpoa Mo Seaman

massaoaooo oaaEmm

one CH poommo one
waasonn EmawOOSOSm new oazwam

 

5,-9-

 

: x

..o .m .s..n so needs ease

mmea no oouoocgoo ops
momma oonmpmamoaOpocwsz
.EOszm Opocaoaooo
oaaemm one CH zapoaomw
mcaoaflp oco Coapduom

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69

that the maximum total error in our knowledge oflg with

respect to the crystallographic axes is f2°.

This estimate
is based on accumulating the errors due to x-raying, Spark-
cutting and gear backlash. If there are Sharp extrema in

a rotation plot,the position is usually known to f.5°.

Analysis of Tipping Arrangement
Consider figure 27. In the xyz coordinate system, the
current density is
l O 0 008m 0 -Sinm J
-l
J = nyhJ" 0 cost —sinv O l O 0

O sinv cosy sinm O cosm O

cosm O —sinm J

o

= -sinvsinm cosv -sinvcosm O
costsinm sinw cochosm 0
COSm

= -SinWSinm 0 J

costsinm

The double primed coordinate system is the sample‘s. We

wish to measure voltages in the sample system. Thus,

_E_H= R ER

Let us begin by considering the case of all closed orbits.

70

 

IO
~<

 

 

 

 

Figure 27 Rotation and tipping geometry in the laboratory
coordinate System.

 

   

aB° p aB°
xy
.. O O
p = pyx aB aB
aB° aB° aB°

By straightforward matrix multiplication,

aB° coseocosIIpXy sin‘IIpyx
5" = cosmcosirpyX aB° —SinmCOSIpyX
sinvpXy —sianOSVpr aB°
Therefore,
E; aB°
E; = cosmcoswpyx . JO .
E; SinIpr

By measuring E; and E; we can determine pyx’

 

1 2
p = l EIVI2 + :EI‘IZI2 \/ (46)
yx '—— .__,
Jo cos2rpco:2 I + sin QII
Careful analysis of figure 27 Shows that
2 2
cos°mcos°v + sin W = cos B .
B is the angle of departure of the crystal from the x-y
plane. Thus

 

Now consider g to be || to x and allow orbits to be

72

open in a direction which makes an angle 7 with x and, of

course,v/2 with B. One can easily show that

2 2

B COS 7 —B2cosysin7 B°

- 2

p d -B cos7sin7 B°sin27 BO . (48)
BO BO Bo

If the crystal is now rotated by arbitrary angles m and I,

we find that
2
E; a Ap/p a B (cosycosw + SiHYSinmSin¢)2. (49)
Inspection of figure 27 gives us
2 2
Ap/o s B cos d . (50)

d is the angle between the current and open orbit directions.
2

The B dependence due to open orbits is washed out if the

open direction is about 900 from g. Note that this must

always occur when B is near.§-

Experimental Difficulties
The ideal magnetoresistance behavior of B° or B2 is
not usually achieved in practice because <w0T> is not much
greater than 1. Estimates of wcv give values of 3—10 at
150 k0 for three of the samples (Table IX). The wet of
G3<llO> is possibly greater than 10,while that of both

AuIn2 crystals is less than 3. Copper samples with wchz 10

have a "quadratic” behavior of Bl'°’2’°. (‘9) The reason

for the exponent not achieving 2.0 is simply explained.

73

Assume orbits Open in the x-direction; for simplicity, take

0 O - O 0
xx O O - O -
( yy 22 zy°yz) OXX Uzzoxyoyx

If a = l/eBT is not much less than 1,

_ 2 4

°xx T a axx + a bxx’
2
0 = a + a b
yy yy YY’
O = a + d°b
zz 22 22’
O = d a + 02b
xy Xy xy’
Oyz = ayz + d byz'
Then,

_ 2

pxx — al(wb7) + a2 (wa) + a3 + .....

The deviation from quadratic behavior depends on the value
of wbv and factors containedhlthe aij and bij'

Chambers(3°) has formulated an explanation of poor
saturation if there are extended orbits for a certain
direction of the field. For an electron which only
traverses a section along one side of a closed orbit before

colliding, the orbit appears to be Open. Thus in the field

2
region for which wclosed.7 >> 1 )> mext T, pxx‘¢ B , at

fields such that wext'T )) l, pxx will saturate. Since

we are only in the region wext'T ) 1, poor saturation is to

be expected. (ext = extended)
Values of the exponent, m, are calculated by sweeping
the field to 150 kG at fixed angle or by performing two

rotations at different fields, usually 130 and 145 kG.
Unless otherwise noted m is the high-field exponent.

74

Because of experimental limitations, the procedure of
reversing both current and field directions should be used

in making measurements. We define VMR as that component of

the voltage measured on contacts 1-2 in figure 26 which

reverses sign with current but not with field. VH is

defined as that part of the voltage on probes 3-5 or 4-6

which is odd in both current and field. It can be shown(3)

that

v [v(+1, +13) + V(+I, -B) -V(-I, +13) -v(-1, 43)]

1
MR 'E
1

V

H - 17 [V(+I, +13) — V(+I, -B) -v(-1, +13) +V(—I, 43)]

These current and field reversals eliminate thermal volt-
ages from both'VMR and VH' They also eliminate magneto-
resistive voltages from VH and Hall voltages from VMR

caused by probe misalignment. Since the unwanted voltages
appearing on the transverse probes can be as large as VH’

it is imperative that VH be measured in this manner. The
unwanted voltages appearing on contacts 1-2 are usually
small for field directions supporting Open orbits so that
VMR = V12(+I, +H) to a very close approximation. For
general field directions, however, V12(+I, +H) can be quite
small («'5 uv) and errors can be appreciable. Unfortunately,
the amount of magnetoresistance data required for a complete
study of a metal far outweighs the necessary amount of Hall
data. A compromise solution is in order: selected measure-

ments on 1-2 are made in the rigorously correct manner to

75

estimate the magnitude of the discrepancies to be expected
when only V12(+I, +H) is measured.

From Eq. (48) one can see that the dominant voltage on
the transverse probes will be prOportional to Becos a sin a
when there are Open orbits. We have not studied these
transverse - even voltages in any systematic way because
the time-consuming field and current reversals must be
employed in this case also.

Voltage measurements were not appreciably affected by
the noise level of .05 ~ .5 microvolts at the highest
fields. With a two ampere sample current, Hall voltages
were typically 10 u volts and resistive voltages from 5 to
500 u volts at 150 kG.

Summary of High-Field Galvanomagnetic Properties
The constant relaxation time treatment given in

section 2 enabled us to calculate the magnitude of all the
01J as well as their field dependences. A summary of those
high-field galvanomagnetic properties which do not depend
on any assumptions about the relaxation time is given in
Table VIII. These results of the Lifshitz theory are
dependent only on the requirements that a semiclassical
treatment is valid and that a certain field, Bo’ is
exceeded. B0 is that field at which all carriers complete

many cyclotron orbits before being scattered.

76

Table VIII Summary of High-Field Galvanomagnetic Properties

Type of orbit and Magnetoresistance Hall Field *
state of compensation

 

. All closed and uncom- I~B° -QB
pensated (neh nh) (ne-nh)le(coss
. All closed and com- ~52 ~13
pensated (ne= nh)
2 2
*-)(- ~ ~
. Open in one direction B cos a B
O -l
. Open in two directions ~B ~B
. Singular field direction '9B0 —03

 

(ne-nhIAn)|e\cosB

* i.e. electric field per unit current density; 8 is the
complement of the angle between g and B.

**.Q makes an angle a with the Open orbit direction.

5. Experimental Results and Discussion

Since measurements of the effective number Of carriers
per primitive cell most clearly indicate deviations from
NFE behavior in AuX2, we begin with a presentation of Hall
data and a discussion of possible changes in the model. We
follow this section with evidence from the magnetoresis-
tance behavior of these compounds which corroborates our
interpretation of the Hall data.

High-field magnetoresistance is, potentially, a more
useful phenomenon for investigating a Fermi surface than
the Hall effect, but it is also more difficult for two
reasons. The first is experimental: a large amount Of
data is required to make a quantitative comparison with a
model and, more importantly, data determining the angular
extent of the two~dimensional regions must be taken at
field directions which have carriers of unusually large
cyclotron masses. Schoenberg has said that "the poor man's
dHVA effect involves looking only at low mass pieces of the
Fermi surface". In a Similar vein it might be said that
the poor man's magnetoresistance experiment is concerned
with measurements in high symmetry planes only. Secondly,
if a good model of the surface is not available in an
analytic form so that a computer can look for field direc—
tions supporting open orbits, one must be both clever and
diligent to make quantitative comparisons between theory

and experiment.

77

Hall Effect in Aux2

General Field Directions
From Table VIII and the discussion on page'71, we have

E = ~ 0 B
H ( II a
ne - nh) e cos

 

in the high field region at a general field direction
supporting no open orbits. For the NFE model, Aux2 have
seven conduction elections per primitive cell, one full
zone, and two hole zones giving ne - nh = 1. Measurements
of VH vs. B enable us to determine ne ~ nh and hence to
check the above assumptions about the NFE model.

The results for a AuA12 crystal with a residual resis-
tance ratio of 550, (A2(random)), are shown in figure 28.
Each curve represents a general field direction for which
the magnetoresistance approaches B° dependence. For
typical sample cross section (1.5 mm x 1.5 mm), current
(2 amps), and at 150 kG, vH = - 8 u volts/(he - nh). we
have shifted these curves vertically so that a line tangent
to them at 150 kG passes through the origin; we can then
simply use the value of VH(150 kG) to determine he - nh.

Table IX lists the values of he - nh for AuAl2 and
AuGa . The last three AuGa

2 2
a residual resistance ratio of 475, G3 {100}. The first

values are for a crystal with

value is for a crystal with a residual resistance ratio of
725, G3 (100). The exponent, m, of B in the magnetoresis-
tance is also given. This eXponent is a better measure Of

the attainment of the high-field region than the linearity

78

79

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m

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81

Of the Hall curves. Generally, the higher m's correspond

to larger deviations from he - hh = 1. Note that AuAl2
tends to have he - nh ( 1, while in AuGa2 he - nh is
usually greater than 1. Any deviation of ne - nh from

integral values indicates that some carriers are still not
in the high-field region because of their low mobility

(high cyclotron mass). The sign of the deviation indicates

(1)

whether such carriers are electrons or holes . Thus in
AuAl2 electrons appear to be the lower mobility carriers

while in AuGa2 the situation is reversed. The AuAl2
behavior is fairly easy to understand on the model. The

Hall

vast majority of the fourth zone closed electron orbits

are extended over several zones. From plots Similar to
figure 16, one can easily Show that the cyclotron masses

of these orbits are several times the free electron mass.
From the same plots, the cyclotron masses of the third zone
hole orbits are calculated to be usually less than the free
electron mass. Thus we predict that the number of electrons
not in the high field region will be greater than the cor-
rSSpOhding number of holes, ne - n ( 1. On the NFE model,

h

the AuGa Hall values are hard to understand. Since VH/IB

2
is known within 1%, the only other source of important
experimental error is in the measurement of the sample
dimensions. These were made with two micrometers; values
were averaged for several attempts. We estimate the

possible overall experimental error at less than 5%. There

is only one group of field directions on the NFE model for

82

which the third zone extended orbits have cyclotron masses
larger than those of the fourth zone. It is just outside
the two-dimensional region of figure 19 and within the
(110), (111), (211) Spherical triangle. The AuGa2 values
were not determined here.

Hall measurements can be combined with magnetoresis—
tance measurements to give an estimate of ne + nh and (we?)
(equns. 16 and 17). For G3 (100),

2 2

n + n n + n
_.._ .4..... == (_IL...E) = h e
922 3.0 uv nh - he .98

nh + h6 = 1.22
x nh —
_1 =7.8= + eCDT
pxx h
me? = 9-5

From the NFE model, nh + he .39 + 1-39 = 1-78- The

experimental approximation is probably too small; from deA
data(13), we estimate that he + nh ~'1.6. A free-electron
(F.E.) calculation of mb(f), using the resistivity at 4.2°K,
predicts a value of 4.5. The experimental wb's are cer~
tainly as large as mb(F.E.), and thus the (7) Obtained from
D (4.2°K, 0 k0) seems to be an underestimate. Similar
analyses of two other crystals have been carried out with
the results also listed in Table IX. We conclude that for

general field directions our data for AuAl and AuGa and

2 2
calculations based on it are in substantial agreement with

the NFE model which predicts an effective carrier

83

Table IX. Hall data, general field directions.

m ne-nh ne+nh wcT(calc.) wCT(exp.)
F.E. 0.00 7.00 7.00
N.F.E. 0.00 1.00 1.78
G3<lOO> .20 .98 1.22 4.5 9.5
I
G37100} .51 1.13 1.26 2.8 5.2
.42 1.25 1.49 2.8 4.4
.74 1.01 1.88 2.8 3.5
A2(random) .38 .81 1.32 4.6 6.5
89 1.15 2 90 4.6 3.0
.50 .99 1.70 4.6 5 l
.25 .95 1.38 4.6 6.0

84

concentration of l electron/primitive cell. Due to time
limitations, we did not attempt measurements on the rather

impure AuIn for general field directions.

2
Symmetry Directions

Four symmetry directions are Of interest to us; they
are (211), (110), (111), and (100).(2°) With the field
parallel to (211), Hall voltages in all three compounds
were buried within .5 uv of peak-to-peak noise at 150 kG.
A slight monotonic decrease with increasing field was noted.
When this behavior is coupled with a saturating magnetore-
sistance, as it is here, we can state with some certainty
that there are non-intersecting orbits open in the two
directions <110> and <111> (Case 4., Table VIII). This is
in agreement with the NFE model: on the fourth zone
surface there are orbits open in (111) and (110); in the
third zone, there are orbits open in (111) only.

The behavior for B 11 (110) is somewhat clearer now
than it was in an earlier report(°), but by no means trans~
parent. The Hall voltage is linear in B with a slope which
depends critically on alignment with minima associated with
(110) in Ap/p vs. t curves. The magnetoresistance itself
is a rapidly varying function near (110) with exponents
ranging from .25 to 1.5 at the minima. We will later argue
for Open orbits when B 11 (110) in agreement with the NFE
model. The Hall behavior in all three compounds is con-
sistent with the model also (Case 3, Table VIII).

./\

85

Major discrepancies with the model occur when B is
parallel to (111) and (100). Figure 29 displays the Hall
voltages in AuIn2 for the NFE model. The eXperimental
curves have been shifted vertically as in figure 28. The
value of n100 = :68 means that there are .68 holes/per
primitive cell compared to .372 on the NFE model.

Electrons Oh the fourth zone sheet being replaced by holes
must entirely account for this value if deA measurements
Of the third zone necks are correct in predicting a smaller
area so that no An arises from this hole surface. For

B 11 (111) we see that V first swings positive and then

H
crosses back at 60 kG. Further, the curve has been dis-
placed more than its total voltage drOp. The AD/p vs. B
sweep here gives m = 1.08 at 150 kG. Thus the value

n111 = +1.34 cannot be relied upon. The difficulty is
caused by the low residual resistance ratio (75) of this
sample. Table X contains all of the results including

111 for AuAl2 and AuGae.

values for AuGa2 and AuAl2 are less than 1.0

reliable values Of n
The h111
even with a 5% experimental error. We must conclude, then,
that there are hole orbits on an electron sheet for this
field direction in these compounds. Inspection of figure 9
reveals the sensible way for this to occur. If the NRC
group<13> is correct in postulating that the fifth and

sixth zone electron pockets have been emptied by the

lattice potential, the remaining three zones must contain

them. From figure 9 it is clear that the cut at p2: 0.0

86

Fiaure 2 ' °
C 9 VH VS. B in AuIn2 for Singular field directions.

 

‘
.......

87.

 

 

 

 

A
201“. VH (’LLV) ‘ I . ///’ "‘ .372
"l0“
e~\‘ I I 7
[’7 -\\‘\\ +l.34
9‘ “‘~\\\ AUInZ
“~+|.035
_.-.10-- I-OPW
-_ V := .158.- ,LLV CII I50
_20___ n = no."nh¥An

88

Table X. Hall data, <lOO> and (111); n = ne-nhztm.

m n100 m n111
F.E. 0.0 7.00 0.0 7.00
N.F.E. 0.0 -.372 0.0 1.035
I2<110> .57 —.68 1.08 (1.34)
03{100} .32 -.63
G3<llO> 0.0 -.62 .48 .89
03<100> .38 —.68 .50 .92

Al<lOO> .43 —.79 .36 .57

89

produces electron orbits which are closer to contact than
the orbits Of any other section. Thus an excess of
electrons in this zone could produce the required hole
orbits. There is a catch, however; the NRC group has
tentatively assigned a dHyA frequency to these very
electron orbits in AuAla, called 04 in figure 6. They
point out that this frequency Should continuously join on
to the frequency they have postulated for B4 as the field
is swept towards <110> in the {110} plane. In fact, both
frequencies are restricted to 5° intervals from these
major symmetry axes. The reason for the restricted angular
range near (111) appears obvious from our Hall data; their
frequency corresponds to the area of the hole orbit, which
will clearly vanish for some angle of deviation from (111).
unfortunately, the sign of the effective mass of m*(Cu) is
not known. Note that there are two types of hole orbits;
the one centered at V is extremal while the other, centered
at the corners of the hexagonal unit cell, is not.

The easiest way to make quantitative checks of this
postulate is to increase the radius of the Fermi Sphere
until the desired n111 is reached and then calculate the
area and angular extent of the orbit by running the
Harrison construction program in the usual manner. We can
only hope for an estimate with this method since the NFE
fifth and sixth zone electrons are almost certainly prefer-
entially located in the ravines hear the sharp tips of 04.

This "corner rounding" occurs because in general the

90

periodic potential of the crystal lattice acts to reduce
the exposed area of the Fermi surface.(1°) Thus the (100)
arms probably are more cylinder-like than appears to be
the case in figure 6. The results of the calculation are
given in Table XI. The models which were fit to give the
experimental n111 are in better agreement with deA and
n100 results than the NFE model. The A4 results are
noticeably bad but we expect this as a result of "corner-
rounding".

There is one possibility of salvaging the NFE model
and that is to postulate magnetic breakdown of electron
orbits like C4 to form the hole orbits Observed. Several
experimental facts discredit this postulate. We would

expect AuGa to be most easily broken down from the argu~

2

ments in section 1, but our nloo and n111 values indicate

the Opposite occurrence. A much simpler eXplanation is

that AuGa is more nearly free-electron-like than AuAl

2 2
and that both breakdown fields are greater than 150 kG.

If breakdown is occurring, we can set an upper limit
on the field at which breakdown will be complete. In AuAl2
our Hall curves give values of 10111 = .57 and h100 = -.79
within 5% for B > 50 kG. For our best AuGa2 crystal, a 5%
tolerance is maintained down to 25 k0, while for the
poorest crystal, VH vs. B is linear to one part in 20 above
40 kG. We can demonstrate that breakdown is not complete
at 25 - 50 k0 if we assume that a simple breakdown model

has validity here.

91

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92

The probability of transition between two orbits
coupled by magnetic breakdown is given by(°1)

-B /B
P = e °

where
B = KA2 mc/e eh K m 1
o F ’

A is the gap separating two energy bands. Our data indi-
cates that Bo <( 50 k0 or B0 >> 150 kG when the field is
parallel to <111> or (100). Assume B0 = 10 k0 (250 kG)
then with 6F = 9.4 ev, A ~'.l ev (.5 ev). Now we make a
rough estimate of A and Bo from the dHVA results by noting
from figure 9 that C4 must breakdown via C3 to give the
experimentally observed area for the hole orbit. We con-
struct a simplified two-dimensional model (figure 30) in
which C4 is represented by an electron overlap into the

second zone, and C by first zone holes. Then we have

 

3
1 5A
In AuA12 m*(cu) = 1.5 me, the area of the NFE C4 =
1.7(2wh/a)2, and the measured area = l.4(2vh/a)°. Therefore
~ 2 107 - 10 L a. __
A2 21~(1.5 me} ’ '6 9V
BO o: 360 kG

Similarly, with m*(03) = .58 me, a measured C3 area of
.6 (21rh/a)2 and a NFE area of .65 (2wh/a)2, we obtain

A = .26 ev , BO «'68 kG

1

 

93

 

 

 

 

 

.\\\\\\\\

 

 

94

Evidence that this estimated breakdown field of 68kG
is too low is given by the normal behavior of the magneto-
resistance vs. B curve which does not have the predicted
anomaly at a field:~ Bo/2 .(32)

We conclude that our Hall measurements along (100)
and (111) give strong support to a fourth zone electron
sheet which has hole orbits for B (1 <lll> caused by the

contact of orbits 04'

The High-Field Magnetoresistance of AuX2

The {100} Plane
Figure 31 displays the magnetoresistance of all three

compounds in the {100} plane. Calculations of wcT for
Al<lOO> and G3{lOO} give values near 5 (Table IX) so that
the curves for these compounds can be compared directly to
the calculated curve of figure 21. The agreement is good
considering the assumptions about the relaxation time and
the shapes of the surfaces that went into the calculation.
Actually, the AuGa2 curve should be multiplied by the factor
l/cosga = 1.16 because.§ is 22° from <lOO> in G3{1oo}. This

improves the fit slightly. The fact that there is no dip in

o
the experimental curves at v a218 suggests that hole orbits

persist on the real fourth zone surface for several degrees

beIYond the 18° range on the log-pile model. Figure 20 shows

11 then be
why: oxy(closed), axx(closed), and O’xx(t0’cal) W1

larger at 18°, while (oxy(total)| will be smaller. Hence

 

A N.”

I

IZO‘

 

   
 

 

 

 

I l Goo)" 6
-.;_.;r
-- 4
-- 2
r; .-
000) "r. "°° (no)

Figure 31

Ao/p vs. v for

all three com-‘

r no;
pounds in klcc)

B = 140 kG

95

 

AP/P

I

IZO'

 

 

 

 

 

 

~<uo)

Ap/p vs. W with

m changed by .
from fig. 31.
B = 140 kG

g0

V

 

96

pxx.~ l/Oxx’ unlike the model, resulting in a monotonic in-
crease in the magnetoresistance. This suggestion seems rea-
sonable because the arms of the log-pile model do have a
smaller cross sectional area than the measured value Ag
through the symmetry point W. (13) No pertinent deA data
is available to aid in affirming this suggestion.

We estimate that g does not deviate from {100} by more
than .50 in the AuAl2 and AuGa2 curves of figure 31. We
arrive at this estimate from the fact that a change of T
in figure 25 by i .60 produced curves similar to those of
figure 32. (Values of m are given). Unfortunately, meas—
urements of m were not made for the rotations of figure 31.
Nevertheless, we know from the field dependence of the mag—
netoresistance at peaks observed in Ap/p vs. V curves for
other crystals that the entire {100} plane supports open
orbits except for_§ \| <lOO> and <llO>. The <lOO> axis is
singular. The magnetoresistance at (110) was determined to
be "quadratic”, m = 1.5 at 115 kG, by rotational measure-
ments on Al<lll> at three different fields. See figure 33.
The very large magnetoresistance observed in figure 31 in-
dicates that Ap/p is probably quadratic at (110) in AuGa2
also. We attribute the sharp drop in magnetoresistance near

0
<llO> in figure 32 to the cosga term of Table VIII, a = 90 .

Figure 35 provides an illustration of this effect.

Two-Dimensional Regions in AuAl2 and AuGa2
The region between 15° and 30° in figure 32 is begin-

97

Figure 33 Ap/p vs. 4; in {111}- for AuAl2 at 83.1, 99.8.
and 129.9 kG. The eXponents computed from
these graphs for B (l <llO> are m = 1.7 at
91.5 kG and m = 1.5 at 115 kG.

 

 

A P/P

‘40--

20--

 

 

 

 

 

98
l I . l
AuAl2
Befinbu} "
13° I o

 

<2”)

(“0)

' Figure 33

(an)

I
-HV..--—.hofll i--‘.. -_, .-. ..

_\.’.“~-_~..._~w . —---

a” s. .— .~...-.'.--m _-..

~—~—v~n.-AI-‘ "-2 .,

99

ning to saturate because the orbits here are most likely not
open but extended. By tipping n to several angles and per-
forming rotations at two different fields, a systematic ex-
ploration of field directions supporting open orbits can be
carried out. We begin with a presentation of the results of
such an analysis on Al<lOO> and G3<lOO>. We will also fol-
low this with a discussion of our results on several other
crystals.

Figure 34 displays the magnetoresistance of Al<lOO>
and G3<lOO> along the paths indicated in the stereograms of
figures 35 and 36. In the latter figures we use the symbol
of an open circle to represent "saturation", i.e., Ap/p =Bm,
m (.7; and we use a solid line or dot to represent the
extent of "quadratic" field dependence, m >l.5. Intermed-
iate or unknown values of m are not marked. Shaded areas of
all stereograms represent probable regions of open orbits
primarily as determined by data on one crystal. If infor-
mation on a certain section of a stereogram is not available
from this data, we have supplied this information from re-
sults on other crystals. The similarity of the two curves
of figure 34 is as remarkable as their disagreement with the
NFE model which predicts a B2 dependence along the entire
rotation path. A summary of the results of several similar
rotations is given in figures 35 and 36. The behavior pre—
dicted by the NFE model in the fourth zone (figure37) is
not seen experimentally. Open orbits from the third zone

of the NFE model could give a qualitative explanation of the

100

Figure 34 Ap/p vs. W for Al<lOO> and G3<lOO> along

paths indicated in the stereograms of figures
35 and 36. B = 145 kG.

101

 

 

 

 

 

 

1 l L L l
A

WP

200— (,6 = 63° . AuGa, -2oo
leo— ' -l60
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106

(110) two-dimensional region in AuGa2. However the re-

striction of the neck size observed in deA almost certainly

eliminates this possibility: calculations of the extent

of this region with a Fermi radius increased.§.5 per cent

to give the deA area of the third zone neck show that open

orbits exist only within 10° of <llo> in AuGa2 and 5° in

AuA12.
We have a considerable amount of supplementary data

on AuGa2. Figure 38 is the result of a rotation of G3{100}

along an arc of the great circle indicated in figure 39.

Seven other rotations yielded the remaining data on this cry-

stal with the results symbolized in figure 39. Note that m

assumes more intermediate values here than it did for G3<lOO>,

a better crystal. The saturation observed inside the sug-

gested <llO> two—dimensional region results from the cosga =

O are being shifted from the {110} plane since_§ is 220

from (100).

Our best data relevant to the directions of.§ suppor-
ting open orbits in AuGa2 comes from G3<110>. We observed
extremely sharp structure in rotations, e.g., figure 40 and
only a minority of the measurements of m gave equivocal
results. (See figure 41) The analysis of nine pairs of
Ap/p vs. V curves at fields of 125.0 and 14”.? kG resulted
in figure 42. Most of the dots represent higher-order open
orbits which are "excited” as the field crosses low symmetry
planes. A complete analysis of the extent of these "whiskers"

from the major symmetry directions has been carried out

107

\
I

Figure 38 Ap/p vs. v for G3£lOOJ along path indicated
in figure 39 at fields of 130 and 140 kG.

 

 

 

 

 

 

 

 

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116

based on information from another set of runs on this same
crystal with m at the more favorable values near 90°. We
will conclude this section with those results.

In 1966 several rotations at one field were performed
on Al<100> and Al<lll> with the (110) axis parallel to BB‘
of figure 25. We cannot make any definitive statements
about 32 regions of the stereogram from this data but
experience has shown that, usually, saturation is associated

with low, slowly varying parts of the curves and quadratic

behavior with sharp peaks and the highest parts of the curves.

Thus we can interpret these curves in a qualitative fashion.
Consider figure 43. In the upper trace, we speculate that
Ap/p is quadratic from 35° to 650 except at the minima where
m takes on an intermediate value. From the lower curve, we
guess that the two-dimensional region about (111) extends

out approximately 10°, while the region about <110> measures
150 when the field is in the plane {211}. Figure 44 shows
that these predictions, which were made before the 1967 runs,
generally conform with the established results taken from
figure 35.

Figure 45 is a rotation of A1<111> along the same path
as the upper curve of figure 43 to within 2°. The difference
in the shape of the two curves is explained by the term
cos2a. The peak at 180 in figure 45 is missing in figure 43
because the angle between the open orbit direction [010]
and the current direction [100] is 90°. A similar analysis

explains the difference in the relative height of other B2

“”3.

 

 

 

117

Figure 4
3 Ap/p vs. v for A1<100> along paths indicated
in figure 44. B = 130 kG

W

 

 

 

118
' 30° '

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A114,

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Figure 43

 

(no)

 

119

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120

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121

Figure 45 Ap/p vs. 1 for A1<111>.

B

‘
—

130 kG.

L443.

 

w

122

 

 

 

 

 

 

 

 

 

 

 

 

. 1 J '1
'AP/p AuAIZ- Ai (Ill)
.. ' 4
4) = 18.9°
40-
ac—
1 ‘ l
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123

regions.
In all of the experimental stereograms, we have shown

unexplained B2

regions near the axis (211). Some of these
are close to, or in, the planes I210} and{2ll} and could be
attributed to one-dimensional regions of open orbits. Others
are not in high symmetry planes and evade explanation unless
there is a two-dimensional region of Open orbits surrounding
(211). (211) is not surrounded by aperiodic open orbits on
the NFE model.

Our data indicates that the (100) and <110> two-
dimensional regions are not connected along {100} unless
this region is very narrow, i.e., less than .60 (see figure
32 and the discussion concerning it on page 96). Our data
indicates that the (110) and (111) Open orbit regions are
probably connected in AuAl2 but not in AuGaz. However,
there are some minor discrepancies. Figure 46 is a rotation
of A2 (random) in which 3 is known to be 2 2 degrees out-
side of 11101 at the heads of the arrows depicting the
region of quadratic behavior. The upper limit of 40 on
the connecting area is in disagreement with figures 43 and
45. Here we must appeal to the questionability of our inter-
pretation of those figures and to the fact that there could
be a 2° error in the knowledge of the field in figures 43
and 45 also. Figure 47 is a rotation of G3<110> very close
to the {110} plane. At 12° and 63° we know that §.is 2° 1°
from {110}. The rather large magnetoresistance seen here

(values of m are unknown) suggests that the orbits are open

.10. "I

 

Figure 46

124

Ap/p vs. v for A2(random).

B

130, 145 kG.

 

 

 

 

 

 

125
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160- f
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Figure 46

 

90°

126

 

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Figure 4? Ap/o vs. v for G3<110>.

127

in disagreement with our other results. The fact that these
points are minima, however, leaves open the possibility

that these orbits are merely extended.

“Whiskers” in AuGa2

The sharpness of the {100} peak in figure 47 indicates
that carriers on <lOO> directed open orbits in G3<110> are
rather far into the highefield region. (33) Using this cry—
stal, which has the highest residual resistance ratio of
any intermetallic compound on which data has been published,
we easily resolved sixth and higher order open orbits in the
magnetoresistance. The peaks on the left side of figure 48
are due to orbits open in directions <llO>, <210>, <310>,
(511), (100), (611), (511), (311), (211), (533), (322), (111),
<553>, <774>, (221), <331>. The two-dimensional regions of
open orbits which produce the broad peaks are centered on
(111) and (110). Note again the sharpness of the peak ob—
served when p crosses {100} at about v = 130°. Several
similar rotations with m varying between 60° and 90° pro-
duced figures 49 and 50. The lengths of the whiskers in
these figures are determined by noting the disappearance of
a peak as m is changed. We demonstrate this in figure 51.
At m = 72.10 we are inside the (100) two—dimensional region
until v = 43°; the magnetoresistance is slowly varying be-
cause only the coefficient of B2 is changing. When m is
changed to 75.3°, we skirt the edge of the two—dimensional

region. For g in certain symmetry planes only, open orbits

128

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Figure 50 lWhiskers" and two-dimensional regions in

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134

still exist; the magnetoresistance acquires structure be—
cause of the transition between saturating and quadratic
behavior. At T = 81.7° the "fuzz" due to very high order
open orbits has disappeared. Finally, at m = 90°, the
210 whisker has ended and the peak due to the crossing of the
{110} plane has merged into a B2 background near <211> on
the stereogram.
The assignment of the 774 and 553 whiskers in figure
50 must be considered tentative since the planes {332} and
{443} are within 1° of the position of.§ on the stereogram
where the magnetoresistance peaks occur. However, the data
is sharp enough that we estimate a maximum error of i .50
in v, which gives considerable weight to our assignment.
Table XII gives the order of the open orbits arising
from <lll> and (100) - directed necks according to the
simple scheme used in section 2. (34) In copper the
"whiskers" have a strength and extent that decreases with
increasing order of the orbit. (25) From Table XII and
figure 50, we see that if this property held for a complica-
ted surface like that of the fourth zone of Aux2, the
whiskers extending out of the (110) axis must be partially
derived from the third zone surface with its <lll>—directed
necks. We have already shown, however, that in the NFE
model of the fourth zone the (111) open orbits exist for
_§ anywhere in {111} while the lower order (110) open orbits
vanish for p near (111) in (110}. Thus the lengths of the

whiskers in AuGa2 may be explicable without appealing to

135

Table XII Higher order open orbits for Fermi surfaces with
(100) or <lll>-directed primary open orbits.

 

Order Possible open orbit directions

1. <lOO> <lll>

2 <llO> <lOO>, (110)

3 <lll>, <210> <3ll>, (331)

4 <211>, <310> <210>, <211>, <221>

5 <311>, <221>, (511), (531), <551>,
<4lO>, <320> (533), <553>

6 <4ll>, <321>, <310>, <320>, <32l>,

<510> <322>, <332>

136

third zone open orbits.

Models

A comparison of any of the experimental stereograms
in this section with the NFE stereograms of figures 18 and
37 shows that the NFE model does not explain our experi—
mental results on the extent of two-dimensional regions
about (100) and (110). With a smaller third zone neck, it
does not even predict the existence of one about <lll>.
Another point of disagreement concerns the extent of Open
orbits in the {110} plane. In all three compounds the
magnetoresistance in {110} is quadratic unless the field is
close to (211). On the NFE model {110} is not a one—
dimensional region of open orbits near <lll>.

As we pointed out at the beginning of this section, we
must have an analytic model to obtain results we can compare
in a quantitative way with experiment. The obvious choice
is NFE models with radii swollen to give the n111 values
since these models were successful in interpreting the Hall
data at <lOO> and various deA areas. If the Hall effect
and magnetoresistance data are consistent, these models should
also yield two—dimensional regions of approximately correct
dimensions. Because of the cost involved C~ twenty-five dol-
lars per angle), we limited ourselves to four symmetry and
ten non-symmetry directions of.§ for each model. On the NFE
model only one of the non-symmetry directions of'g gives the

experimental result.

137

In figures 52a,b we summarize the analysis of general
field directions and compare it with experimental results.
Agreement is excellent for the AuAl2 model: all ten field
directions produce the type of orbit eXperiment found. In
the AuGa2 model four of the field directions give erroneous
results and indicate that the <lOO> two—dimensional region
is too large while that about <lll> is too small. The AuAl2
model actually gives a better fit to the AuGa2 experimental
results than the AuGa2 model for the ten non-symmetry direc-
tions. The fact that the experimental two—dimensional re-
gion near <lOO> in AuGa2 does not extend a few degrees be-
yond the range observed in AuAl2 is somewhat surprising
since the experimental Hall coefficient at (100), as well
as that obtained from the models, indicates that there are
fewer holes on the electron sheet in AuGa2 than there are in
AuA12( Table XI). This is a minor but interesting point for
which we have no explanation. The behavior of the AuGa2
model near <lll> and (110) is in better agreement with ex-
perimental data. Although the two dimensional region about
(111) is too small, we must remember that the hole orbit
layer is very thin on the model (less than five per cent of
the height of the unit cell) and the tendency for electrons
in excess of the NFE number is to deviate from rigid-band
placement, resulting in cornerbrounding. Thus, it would
seem that a small change in the model near the hole orbit

layer could produce the experimental angular extent of the

aperiodic cpen orbits near <lll>. Of great importance is the

“—.. ‘-- A..~u .—

igure 52

 

 

 

Magnetoresistance

stereograms
Acomparing experimental open
orbit regions (shaded) with

the type of orbits (c=closed,
o=open) determined on the
models.

 

 

 

 

 

139

fact that the two-dimensional region about <lll> on the AuAl2

model is clearly larger than the region about this axis on the

AuGa2 model and furthermore that there is evidence that the

(111) and (110) two-dimensional regions on the AuAl2 model are

connected, while on the AuGa model they are not. These fea-

2
tures are in agreement with experiment. Thus, with the field

near <lll>, our magnetoresistance data corroborates our inter—
pretation of the Hall data, i.e., that the Open surfaces of

AuGa may be more NFE-like than those Of AuAl

20
We find that there are extended closed hole orbits for

2

general field directions on these models. This is in disa-
greement with our Hall data since ne-nh = 0 then. We take
this as evidence that excess electrons do deviate from rigid~
band placement.

The behavior Of the models at (100) and <lll> has al—
ready been described. Both models were fit to give the ex-

perimental n and then gave better values than the NFE model

111

for the eXperimental n . With g 1| to (110), the fourth zone

100
Open orbit layer has diminished in width, while the hole orbit
layer has increased in width. This should result in an in-
crease in the size of the two-dimensional region about <llO>
over the NFE value in agreement with experiment. At <2ll>
there are still orbits Open in two directions. Furthermore,
on the AuAl2 model, there is a small band Of hole orbits on
the electron sheet, which Offers a possible explanation for

the suspected aperiodic open orbits near this axis. NO hole

orbits were observed on the AuGa2 model.

 

140

We were unable to measure the two—dimensional regions
of open orbits in AuIn2 due to the small relaxation times Of
our samples. An approximate B2 dependence of the magneto-
resistance was measured in {100} and {110}. The magnetore-
sistance at (211) saturates, while at (110), m.~ 1.5. Hall
effect measurements at (100) indicate that the Fermi surface
of AuIn2 may be similar to that of AuGa2.

We have presented eXperimental results on whiskers in
AuGa2 primarily for later researchers who might want to
compare two analytic models of the Fermi surface which give
good fits to the experimental two-dimensional regions. In
such a case, the extent Of the whiskers can aid in deciding

(25) For the present they merely

which model is superior.
serve as an indicator of the large value of wcT achieved in

G3<llO>.

Summary

In summary, the NFE model does not agree with the re-
sults of the magnetoresistance experiments because it does
not yield Open orbits near <lll> and because it does yield
them in the region between (100) and <lll>. An NFE model
with a radius swollen from 1.495(2Wh/a) to 1.552(2Wn/a) to
give the experimental n111 in AuAl2 provides an excellent fit
to the magnetoresistance data on AuAl2 for a sampling Of
fourteen field directions. An NFE model with a radius swol—
len from 1.495(2wh/a) to 1.532(2Wh/a) to give the experimental

nlll in AuGa2 produces the experimental magnetoresistance Of

 

141

AuGa2 for most Of the fourteen directions of the field which
were sampled. Therefore our magnetoresistance results give
strong support to our interpretation of the Hall data, i.e.,
there are hole orbits on an electron sheet for g \\ <lll>

and the Open surfaces Of AuGa2 may be more NFE—like than

those Of AuA12.

6. Conclusions

Extensive galvanomagnetic measurements on the AuX2
compounds have demonstrated the feasibility Of Obtaining
rather detailed topological information on the Fermi sur-
faces Of metallic compounds. In AuGa2 we were able, for
the first time, to consistently Observe magnetoresistance
ridges due to higher order open orbits, while in both AuAl2

and AuGa we were able to determine the angular extent of

2
aperiodic open orbits.

Hall effect and magnetoresistance data in AuAl2 and

AuGa cannot be explained by the NFE model. A modified

2
NFE model which increases the number Of fourth zone elec-

trons better explains both galvanomagnetic and deA data.

High-field galvanomagnetic properties primarily con
sist Of weighted velocity averages over the open surfaces.
Thus we have presented evidence that the open surfaces of

AuGa may be more NFE-like than those of AuAle. This re-

2
sult is consistent with the OPW predictions Of section 1.
The evidence available indicates that the Fermi surface of
AuIn2 is similar to that of AuGa2. Relativistic effects
may play an important role in this case.

We have shown that our extension Of a single-relax-
ation time theory applied to a NFE-like model Of AuX2 gives
a fair approximation to the experimental magnetoresistance

in the {100} plane. Perhaps, more importantly, we have

demonstrated that for one complicated Fermi surface, the

142

143

magnetoresistance falls into the same three categories we
described for a combination Of cylindrical and Spherical

Fermi surfaces.

LIST OF REFERENCES

l.

2.

7.
8.

9.
10.

ll.

12.

13.

l4.
15.

LIST OF REFERENCES

E. Fawcett, Advan. Physics_l§, 39 (1964)-

D. J. Sellmyer and P. A. Schroeder, Phys. Letters 16,
100 (1965).

D. J. Sellmyer, Ph.D. thesis, Michigan State University
(1965), unpublished.

J. Piper, J. Phys. Chem. Solids_g7, 1907 (1966).

D. J. Sellmyer, J. Ahn, and J.—P. Jan, Phys. Rev. 161,
618 (1967).

J. T. LongO, P. A. Schroeder, and D. J. Sellmyer, Phys.
Letters, 2§A, 747 (1967).

w. G. Pfann, Zone Melting (John Wiley, Inc., 1958).

E. Fawcett, W. A. Reed, and R. R. Soden, Phys. Rev. 159,
533 (1967).

E. Fawcett and w. A. Reed, Phys. Rev. 123, A723 (1964)-

A. S. Joseph and W. L. Gordon, Phys. Rev. 126, 489
(1962).

J. P. G. Shepherd and W. L. Gordon, Phys. Rev. (to
be published).

A. C. Thorsen and T. G. Berlincourt, Nature, London,

.122, 959 (1961).

J.—P. Jan, W. B. Pearson, Y. Saito, M. Springford, and
I. M. Templeton, Phil. Mag., 12, 1271 (1965).

D. J. Sellmyer (to be published).

J. Ahn and D. J. Sellmyer, Bull. Amer. Phys. Soc.,.13,
485 (1968).

144

16.

17.

18.
19.

20.

21.

22.

23.

24.

25.

26.

27.
28.

29.

30.

31.

145

c. Kittel, Quantum Theory_2£ Solids (John Wiley and
Sons, Inc., New York, 1963) pp. 249-266.

T. L. Loucks, Augmented Plane Wave Method (W. A.
Benjamin, Inc., New York, 1967) p. 15.

M. Kohler, Ann. Physik.§, 99 (1949)-

E. Justi and H. Scheffers, Metallwirtschaft_11, 1359
(1938).

I. M. Lifshitz, M. Ya Azbel', and M. I. Kaganov, Zh.
skeperim. i Teor. Fiz..31, 63 (1956). [English transl:
Soviet Phys. — JETP 4, 41 (1957)].

R. V. Coleman, A. J. Funes, J. S. Plaskett, and C. M.
Tapp, Phys. Rev. 192, A521 (1964).

w. A. Harrison, Phys. Rev. 118, 1190 (1960).

I. M. Lifshitz and V. G. Peschanskii, Zh. Eksperim. i
Teor. Fiz..3§, 188 (1960). [English transl:Soviet
Phys. - JETP_11, 137 (1960)].

J. E. Kunzler and J. R. Klauder, Phil. Mag. 6, 1045
(1961).

J. R. Klauder, W. A. Reed, G. F. Brennert, and J. E.
Kunzler, Phys. Rev. 141, 592 (1966).

M. E. Straumanis and K. S. Chopra, Zeitschrift fur
Physikalische Chemie Neue Folge 42, p. 344 (1964).

D. J. Sellmyer, private communication.
D. J. Sellmyer, Rev. Sci. Instr. 28, 434 (1967).

A. J. Funes and R. V. Coleman, Phys. Rev. 121, 2084
(1963).

R. G. Chambers, The Fermi Surface (John Wiley, Inc.,
1960) p.114.

E. I. Blount, Phys. Rev._lg§, 1636 (1962).

146

32. L. M. Falicov and Paul R. Sievert, Phys. Rev., 128,
A88 (1965).

33. I. M. Lifshitz and V. G. Peschanskii, Zh. Eksperim. i.
Teor. Fiz..3§, 1251 (1958). [English transl.: Soviet

Phys. - JETP Q, 875 (1959)]-

34. J. R. Klauder and J. E. Kunzler, The Fermi Surface
(John Wiley, Inc., 1960) p. 125.

APPENDIX A

APPENDIX A

The Harrison Construction Program

This program is conveniently divided into the following

sections:
I.
II.

III.

Read in

Calculations

A. Rotation Of lattice and zone boundaries
B. Plotting Of zone boundaries

C. Plotting Of Fermi circles

Subroutines

A. Matrix multiplication

B. Coordinate elimination

Definitions of important variables are:

1.
2.

HOT is the Euler angle rotation matrix.
THETA, PHI, and PSI are the Euler angles in radians
as given by Goldstein.

X are the coordinates Of the lattice points in
momentum space. EX are the rotated latice points.

VARRAD are the radii Of all the Fermi spheres at
their intersection with the plane Z.

AB and 0RD are the x, y values Of points on the
Fermi circles.

C and S are the values Of the cosine and sine.

V gives the directions of the vectors perpendicu-
lar to the Bragg reflection planes and the distance
Of the plane from the origin. CE is the rotated

version of V.

147

10.

11.

12.

13.

14.

148

KT and YT are trial values for coordinates
describing the line formed by the intersection Of
Z and the Bragg reflection plane. XX and YY are
the acceptable values of XT and YT (those which
are not outside plotter bounds)

The two intersecting planes may be written
ax + By + 7z = 5, z = 6; DE = 6-76.

VOLBRZ is the volume of the Brillouin zone in
units 2vn/a = 1. VALENCE is the number of valence
electrons per primitive cell.

SK and SY are scaling factors for the plotter. SX
= .67 means that a line in the x direction of
length 1.5 will be plotted with a length of 1".

NC counts the number of cross-sections plotted in
the x direction. RNC is the distance the plotter
pen moves back to x = 0 after a row of cross-
sections has been completed.

NPLNS is the number of cuts parallel to B, i.e.
the number of Z planes.

In CALL PLOT(Y,X,n,SY,SX), Y and x are the values
to which the pen moves; if n=1, the pen is down;
if n = 2, the pen is up.

Because the Harrison construction only requires a

knowledge Of the crystal structure and the number of elec-

trons in the conduction band, it is easy to adjust this pro—

grammed version Of it to other metals by appropriately

changing X, V, VOLBRZ, and VALENCE. This has been done for

a body centered tetragonal metal (white tin) and for the

hexagonal metals (cadmium, magnesium, and zinc). An adap—

149

tion of this program by Professor Sellmyer‘s students at
MIT has calculated the NFE model of Aqu2.(15)

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5 LEIJIH‘T‘; legal—"(c.375/ _<.I_4_i_$f{_):__‘i_0'(_.ffe7rVAL__E’\ICEI .1-.. -1
““ “““Jq”e"¢o.<i + :3 ‘ ‘
__~' CALL PLQT (94, x. II-_ ~_ , "l - --
‘I‘I‘“' “”30'9‘nr'1“e'1,‘~A'"’
_lilf3.« 1"_T5-Ts(_) ; CANVMfMIdATAIEIMl_-l_ _ 5v ,1 l __
PHI(I) : “Iv, . PHY(L)
\“ __ _"PSI‘(I) : I M I.95Y( # _. “1...... 1-....-__,____, _-_f__-__“ ___ J
AA = COSIDHIIIII
.M.“. 88 : SIVIPaIU )I__”pmr , 1-“- __m"wlumll-li.EMMilm-wm-ms
"II”"””tC‘E clszsst<iII'
rm—“i- ;_____ 90 =_ SIVIDSHLI) _ * ~_,_ 1-1.1.1."... ____________-_-_________._-_,___-_ «___
EE 3 CIDITI-TAILII
«1.- _ FF ': S I \‘I T -IETA(L)) _ _,,____M..._.____ ”1---“- _ _--—...-r—«——«—-» —~
“I“ " RuT(1.1I : CchA- E:t84«30
M.;~_ NOT (1 : C", 0 33": :OA\A*“1).D_ -.-._ .- -1. ..- --....._--..._..-..-_.-—~ ,-l..._.__.._.--, A-
Mfl~~*-m03T(I,~ 2..-; “Icrrfl'.

 

/ .3/ z / / I? f T. A 7 E 4 .4 FE. I wL \ \ K A I. A a K x ALI.

 

 

 

 

 

 

 

 

 

 

 

' _ 151

.flb41A . A _". .QQZIZHIEZQQQLWLLN
R01£2L§l.3 ”DafAA-EEfiBRtCC
ROTT2N2) ="-na*as+ eiAAtcc ' .“""
'ROT(2. 3L“3.Q§3F£*Mmu-
RoT(3,1) : FF.88 T —”“
ROTI3LZI_E.7FF15€“_
ROT(3. 3) = EE ’"“""'

c ROTATE WAG‘tTIL [17: L0 To THETA(L). PHItL). PSIIL)

 

n..- a-
u...-

 

CALL MT4x~:L(Q3T, x. Ex.NLP 3)
CALL "IBIJLQQ”QIL:LLQ§EL-NPT314)
C PRINT THE THREE MATRICES AND THETA(L). PHItL). PSIIL)
PRINT_1fljflJ__V_QLBBZ, VALENpg, RADIUS
1010 FORMATI1 1. 25x *T4E HARRISON CONSTRUCTION. //// tVOLUME 0F BR id
1N6 = 9, r1“. .6 ,iX‘ “VALEX'CF =*'{_F_710. 61/
2 .HADI 'S “or :ER‘4! SU'TFACE = *3 F1006 ll)
PRINT 4060.9THATA( >, DHY(L). PSY<L) E
4000 F0HMAT(1H ,tIflE ROTATION MATRIX CORRESDONDING TO THETA = ..
1F8. §1“:,¢ 2X_IEH1”:QAF§:3' in* 2X *PSI 3*. F8. 3: IS*/)

401" FOTEEILQEXAW3519L31“

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PRINT 4011
4011_£98m£I_L113§1L3_IHS.LAIII§E-EQLNTS ARE~ E
1//10y * X(1) X(2) X(3)*10X* EX(1) EX(2)
-——’ 2_QEXJBI:L)WQL __ “m.“ _!
PRINT 4030. ( J, (VII J): I: 1 3). (EX(IDJ)5 I=1s3). J = 1. NLP)
——- 4Q11_EQ$%£ILLEMA,ILO. 3’ 10 4g LlXa 3510 4)_ .wgmmm-m-m_m_-_~flmu-
4040 PRINT 4050
—m 405a FOENA T<//5nx1";v:4_34xl tCEEZI _wa_--_n
pRINT 40‘3 , (Jo 1:114), (CE(IDJ)J I31:3)o J31: NPTS)
—~MN_QADQQ_EQEQAL£1¥Afljlo. 4EJQ491QJOXL,§ELOI4) -E A m"
NPLNS : 1n.0 . EE 3 1.0
—-——._~_- EDD fiiflfiLZME-IAQN?L33_WE- ~mw~~

 

 

G = ~ - 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

“~445le 7 L2,) , LGMfi" 0‘ , 1.”- _"________ , L.-. __.______.__..,m _____
CALL pLOT ( 0.00 0.0: 0) SY: 5X)
‘*~~——---- -- MCQLL_QL 0,711 1.31 “'19. 5 .A) I- 2 I “53.1 S X ) ..—— r——-———-~-— --
CALL PLJT( 9.0: 0.1: 0. SY. 5X)
'---*—WE.N£:EVK.= 1.. E1“_ 0---,Qmmnm-_. .~-__m_-m-,fl_h~
b0'5‘nF j g 1' NPLNS
-~~—__.QQ__ DQIJT_,4'n.-?<N>U_m ”.JMLEWQL Q____Q__Q,Hw___uhmr.‘
5410 FOTM£T(Ln 2, .2 :g' 76.3)
~~133m£.,-w“QR§erQNE_gq¢anARIES __W_M___ . _ ..LL-QLH-
p INT 1,; 9
_-_ 108 F34MAT(31nx «I ( “waum_ DEII,_K) RIQ SflALme_§83~M“MU__
"“"37""””””””;*' 1x7~dy ”'vs xx<Ns> XT(I) YT(I)*)
‘- .~_-“DD_2 CAI 5 I. IPTSH" w HJ_”Q E -w-w.r
1 ' : "4 \IN
“‘“-—5-. .22-:P38;N_1AT;_PE=N3(KI 'QC_!.CONSIJ HHERE.0(KLNE.GARLK)Q!.DLH-
UE(I.K) = GAJ(<) «1(4.I) - CE<3 I)*Z(M)
~-—-L;__N _ME ACE " I_v1w3 THE 9AIR 0F SIV_ULTANEO_QS EQUATIONS “,5

c ‘ Ax . a? . (z = a. z = CONST. THUS 1E CONSIDER THE VARIouS '

4““-QA~M_QRQSSIBILITI55,A=Q:J. 5:0:. 9: 0; A. Mi. QQANDLR N-.-0.ELQWWM
TF (CE(1.I) .=0. 0.0 .AND. CE<2.I) .EO. 0.0) so T0 2nd

 

 

*--»_QQ__MLF (15(1 1) 50, 0.01 GS To 150- w.-_--_“.MEE mnflvnwnum_33wumu
IF (CF(2.I) .Efl 0.0) “0 TO 140

'“‘-~—~_____QQW13N_J 5.1 L11LL”-_EHHE-mmAW- ,. ,r- —u~—u~ ”mm m» w—~
AJ = J

“““ ,.E-m,-_._.HML-__

‘5”.
'-—-‘—...______ ..__
——_...__.. .—._..A ---.. - ., -A.- E 7“ ,E .. ..... .

 

-——*

IwmsA *_,_

---——.~.* -. a...

152

 

__.-0_2-[2 0. [6]...“- - _
-3.0 + AJ/2.o

 

 

 

 

—— 13n vrsz IIIE‘I"I"."KYT":E(1":HTKTTUTI‘V‘T'ETz.II .-.-.-_ -
C NE ELIWNATE UNACCE’TABLE VALUES 0F Y AVD DIMENSION A NEW
-—— c vAHTABLE”?Y‘AHICH”5AEY“C0NTAINS“NN"KCE€PTKBIE“VALUES“ " ““““‘""‘
CALL ELI”NAT€(YT.XT.YY,XX,NN)

 

 

IF "FF AFC—FEET”? 'I"'G‘e)*'°FT'3W1F‘DFO

DO 131 J = 11231

 

 

AJ J

 

 

YTIJI = -3.g :WAJ/?.0
131 MM) = muI.AT213571“???r233???ecI,I)

CALL ELI”NATE(XT.YT.XX,YY,NN)

_.

 

C')

N 0 ST FIFST-IWI ‘5: "C ‘4 AA I A] TA“?! ATE—_I F‘— T?" RFE’ I S—EFITIV "O‘N'EWKCQCEP'TFA’B'CE ’V'MA LU E' " ‘

IF I NM .LT. 2) GO TO 61

 

'5')

NOV FE"FTND THE ”LA? EST VYFANU'TWE.§FACE:ST YY AND—DRAW AMIIAE
BETWFEN THFI

 

 

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NS 1

 

 

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IF (YY((XI.G SHALL) c9 To 67

 

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67

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IF (YY(KL)

 

---— “—.-*.. _-_. fiar _,

 

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0 816’ GO TO 69

cw —-~ .» o..—

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ma KK
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4
‘E04T.T(6 215. 5x.“1F11T3T"Ia. Fin. 3. IS. F10.3I
2N3:;H2+=L5 ”__. ,W_mmw -
9.A[L 5'5IKLih4"575
XX(“_) ; (‘(V5Awiw3:i,__.__n--
XXIV“) : ‘Y‘f-3’ + 5.5
A CALL PIJT ALI. YXINS)1__?. SY. 5X) C.--

CALL PIJ' (2': G, (XIVU), 1: SY! 5X)

F0 Th PU“
"RT? 13‘“;
IF ‘XT.F1’

 

Lg7t)&)

IT. .GT. 3.n) ogfllp 200

WT(I)W
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PPI» 141. I. X.
N..." 141 LQQMAT(or, 233, Sr, f n. I an}, F}n: _) _ _”“M
X T-(VIFIM': 3' TI I I “4 ‘2‘ . '3

--_ ......... CA'L P'O’I? “ XTIIIo-3¢ §YJ,§I’CCC~__ 7* 1~
““FCALI pI;.r‘A.v, .TgI), 1, 8%, SK)

CC._ ______ Ago Ti 20 .-__mw- _-- _-__ -__ ~
15r TITI”E”IEIT,XI / TE<2,I) .

--~_ IF (YT’ I) LT '3-‘ -0L:-XT£I?-2§L;.§-“’ “0 To 2”” 2"

FF““"7m:-h jQ ,"IT AffliTfik),vT<H

‘“~—-~ 471_f04' I’°’ _2P5! 5X!-513;§;M293;.F10'3l ---“-w-

. YT(I) §_$:(1) + 5.3 ‘

-~CC~“- CALL 9L0 <II.2 5.U2,_§YFM§XLY«__,”____~_H.“*--_"w_uw___n_-___
F" CACL pLj TIYT;' 6.5. I, SV. 5*)

‘“““~-n~.,m,-,‘;0 T' éch _._2__ -_ A __ "_4___, _*_.____."”__,~-_,m~«_.___”____,u _fl,.-._ -_.J -
6’ 9RT: T A2, I , K, DE<I K): “N -

KM... 67 F 0% 1») T ( b' " . 2 y ‘3. g 5 X.- F11”. ‘6’ 7.nx_'-__.1_2‘_)_~,_, _g—o.u-~~‘.. --» -~—- "‘fl‘ MM"“‘ " "V'- .-
ZOZY Ch ITTAU’EF"' "'

———~ _ .fl.».._.-.—_ ---.- . .

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g

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‘ 153
IEEJA — -._. 4212.0 1.6.2---.” - - 2.2..

 

 

 

 

 

c
'13, seen no 5409 J = 1. NL5”"
ARG = PA:IHS*4AWIU3 ;,__gZTM)- -Ex(3. J))¢«9

 

"noa- ,-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

”’* IF (ARC - .0001) =000, 5090.5100 "
509n VARR$D(J) 3mg.o ___ _
so TO 5400 “”
510n VARRAD(J) : SJEIFLESC)-.*
c wE wILL \or PLOT CIRCLES IF THEY wOULn EXCEED THE 5C6??EE"§BENE§”"W
CHECKI = EBSLEXLAEJIJHE_YABBEQSJ> __
IF (CHEC/l .LT. 5.%) GO To 5389
538a PRINT §;L34.J_ __2_d_"_ ‘ -WWL“
5381 FORMAT(1>. *CHECKl FO< J =*.13) '
638° CHE-‘02 = we -.J._z._:_._.v.EEE_A.2.LJ.2 2-2.-
IF (CHE 3r2 r. ~5.5) 30 TO 5110
530" PRINT *3“1_2l__ - _fl_
5591 FOQMATtl), .CHECk2 FOR J =*.13) '
__ fiflmliwfifiifl -- —~»~~

 

 

511n IF (VAHQ£D(JT .GF. 1.00) STEP = 1

IEMJNAfi9fQiJl.EBEJEJEDWJANDEMMARRADLJ) ELJAHJJEQAH_SIEB"E_ZW__w"-m-

IF (VA3?IU(J) .LT. .50) STEP = 3 _

-.__.__- .230 52E? 1.;_51EE 22.51EP . ._____.M_HE ”m_.w_n-
A8<I> = Fx<1 J) + VARRAD(J) * C(I) + 5.5

_.WHE_EEEELQPJLL1.;-EYKZLJIWEWMABBADAJIW3ESJI) +_§ES 2%
CALL 9237 (3“0(72). 58(72). 2. SYISX)

.___n_“*wm. n3 :21.__T ; qTEE, 22, QTEEH___, _ _“"“_,,Mwn

5209 CA; L p'jT (J‘U(I), AR‘I): 1; 5Y1 5X)

“n ‘ --~n--o---.-

 

 

 

 

 

 

 

 

 

—~WE_QZLLJXTW;M'EPM_ __2__m*wmw‘ u"_”“mmw__-*npi
5433 co=71une -
_..__..._._____________ P_RI.‘."7_541_2 ___- ____ --_._._....._.. __ -- W2 ,

 

5419 F3 niT<x tx~WTAD(J) AREt) ,
aqznr a41« (JARRAjjj),_J";_lJ”flLEJ __, -wm.wmu-nmm.

“—..—u.-- -.....-,“_, -4.'. .-- M. _ .- . I
5411 FOEHATT1>, 1EF10.33
—-——__~-~~HNCEE K ;_HffiECKfljNJWEEW_-_.-g —~-—~~—-' "‘
IF (" CH:- .-F". .'3. NC) 60 TO 5499
-_ww__*_~§AL}HPL3TL11.E, -45.D.-21.SYLUSX) -«- ~-~~—~w~w~wu-~~~W'~W

CALL PLJT (0.0. 0.0. 0. SY. SX)

 

 

 

--

 

“III-h.- w--—.—wk!CjE: :K _: ml ~ - _._.......-.. "' ‘.- —‘ _ ”‘--__.____‘“~,- -_-.~‘—."Oil~-m-.'..‘" —---—*¢i '-
GO T“ 53;, r}

 

 

-—-—__.5£9°.Fu"2~'37 (i 9 L 0. 2, SY._SX1_
CA L 9')? (n ,0. o. n ,9V.SX)
-iififl-£J,_J1f:=;
900“ C0 T!‘
‘~»~~__«_""_QALL_ 3!;

CALL PLi

“_~*-* W W ~——————~——-.-—. ..._._,,7
- -.. o

 

 

5‘ ' -"-_-_._-.‘__

 

fin“

 

_l“(1 U ”’2 O. 2: SYISXJHh_", -r__mflmr“fln-w~wfl,-ww.w
T (0.:.0.0.-1: SY,SX)

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SUGPGUTI‘E ”TQXML(?JT;XoEX.NLP.L)
‘ ”"' DI”EHSIO‘ INITISTSII'R(LBIHJYYF"EYTKTNC53 “‘”“"”'“'
no 3010 9 7w“,,;
# DO 3010 I. = 3
Ex (I. J) = q Q,wi . J. _ ._ III- I
I
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Ff, ' DO 3d10 ' = I
3010 EX‘I: J) : EX(
RETUVEW.__"*-

END

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APPENDIX B

APPENDIX B

A Program to Calculate Ap/p in {100} for AuX2

The terminology in the program is obvious, e.go: u

MU, pxx = ROXX, <vz> = VZA, dpz(A) = DPZA, (uvx<vx‘>u>

A B

MUXXB, etc. Note that the factor 2 in equations H1 is

hidden in the statements following statement 32.

158

.. - - _—.--. .— -..-..—.. ---.- »-- . _ -- _,.___ 7
--- -..—J- -... - o- .. _

fIN503ANlliiu-u.

V159

"-»a-:.£:~-a. ‘---‘.--—¢ . - .‘ v.-V.. ~ 7‘”-

PROGRAV FOURBZ

_ ._...—. --.-....--

“”“""“"””‘1”MUXXE. MUVXH, MUCH. MuxxH
C
' m"" ""°”""PRINT 15
' 15 FORMATtlHi)
m "”‘"m””'*”00”40'v 2 1,45'
‘ I N

"_““"m*"“” "PH! E 5&3.14159/180,n

s ufsxvcth)
”“““‘C”-JCos<PHI>

Y I 8/6

“_——~— .N c .—..—»..-> ..«uuw. - _ - .. . _ H. 7“ » . - ,i

C OPEN OQBITS

"fiwmwuw‘_hM~V2“= .50(3,0*C/5 OSIC’
DP 3 02.0

'V‘mawwvflUVXA's’v1.0/( 8.0*C**2)
~ VXVZA : S/(8.0*C**?)
-mufmvjm"*”VXVYA I 91,0/(8.0*C)
WUOA 3 1.0/S * 2.0/(3.0*C)
DPZA = S/2.0
V29 : .5t(¢-3.0tS)/C + C/S)

-- -

. V—_‘-*

“09/29/68

“-m“vhn.i.l. MUVXB s .(1.0/(12.n,c)).(1,o/(2,0*C) + 1.0/S)

VXVZB 3 '1.0/(8a0*5)
”WVXVYB 8 VXVYA
nuns =12.0/3.0)r(2.0/C + .5/5)
DPZB : DFZA A
VZC : (92,0tS)/C
”UVXC : e.gtMUVXA
VXVZC 3 4,Q*VXVZA
“'vxvvc : 4,0«vxva
“uac : 8.0/(3.ovan
'DPZC12 = DF’ZA ‘
DPZCS 3 ,5¢(C-S)
“NV201 = VZB

Nuvxfii = -c5.0/<12.0*C)>*<5.0/<2.0'9’ ‘ 1.0/S)

’ VXVZDl 2 S,O*VXVZB
VXVYDl = 5.0thVYA -
' WMUODi : (2.0/3.0)tt4.0/C + .5/5>
DPZDi z DPZA
"‘“VZDZ = VZB

‘_W1.c,)
vxvzoz : (..25/S)t(C/(2.G*S) + 1.0)
““VXVYDZ =1..25w(.5/S . 1.0/C)
“U302 ¢ 1.0/3.0)a(5,o/C # 2.0/S)
““ DPZD2 C/2.o - S
«urxA -1.0/<72.0~C*o3)
’““““"“‘“ wuxxs
wuxxc

-7.g/(18.0.Co*3)

OZL‘: 1.0/(4.0.5) « 1.0/(2.0*C)

‘””‘ “' ‘“‘ “flU!XUZ = _n2L*,2*(2.o/3,0)0(32L/3c0 + 1

XMUXA a -MuxiA
* “xnuxs : -Muxxa

O.

'1.0/(°6.0*C**2)*(1.0/(3.0‘9) * 1.0/S)

'“**“wuxx01 - =n5.0/(96.0*C«*23*(5.0/(3.o*C) * 1.0/5’

.0/(4.0*S))

"YYPE REAL HUVXA. MUOA, MUVXBL MuOB. MUVXC; Huoc, MUVXDI, MUDDl.
1 Muvxoz. Muooz, MuxxA. muxxB. MuxxC,flMuxx01. Huxxnz.

HUVXE' HUGE,

nuvxpz =—<1.0/2.0)c(1.0/s . 2.0/<3,0~C))w(1.0/¢4.ots) + 1.0/(2.0

 

”__-.._7______~_,-__.,___,-.._we -.,._.._ l- .. -.,-_,-..,. .. - w 160. a. _ _. -.... -- l. -
[quJA‘MWEMW-~_M__."WlWMmHHHHso“,..H-. .. m a Mn . . -l 09/29/¢8

”...—......—

””-"“*iflux01 g .nuxxni“‘
c“,

”"”CL03ED"onssz

 

XMch : ymuxxc -
XMUXDZ 8 BHUXXDZ
VZE : 0.0

”HUVXE x 91:0/(3.0tCti2,
vxvze a S/(9.0~C**2)

 

-. cw. ..-~-—-- ~

... _q...‘-l_......._,

‘-.~~

-,. v- ' um-.. .—V_-.

‘00
I

”—... .....-

" DPZE“5'0.5.<C . S)

‘M‘DPZH =“o.s;<c—3,o.3)

"‘ “xnuxH : e‘nuxxH

"” IF‘YT .LT; .333) so T0 20
"'"IF ( T gar. .5)' so T0 22 ' “' *~ ‘ ;

“"223‘

21 YY a 03t*2*( DPZA/VUO

VXVVE‘im-1,o/(4.0:C)
NUDE = 4,0/(3,9«C)i_

flUXXEHQWf“5.0/(72.0t0tt3)
XMUXE : - MUXXE ' '
VZH = 0,0

“’Huva'i‘9.6.Muvxs ”'

VXVZB c ~9.o-vxvze
VXVYH a q 9.0fiVXVYE ” "‘ ‘ *"‘ - ~ ~ - w-1—. w "m . w
HUGH = 3,0‘Muoe . {

wuxxu a 27.0.Muxx5

IF c 7 .LT. .5 .AND. T .67. .333) 60 T0 21

 

'OPEN Oaslrs

__J?0 YY = D°**2*( DPZA/“UOA * DPZB/MUOB ¢ DPZCIZIMUOC + UPlflllMUODl)

#22 = va..2*anA/MUOA + vza*.2*opza/Muos ¢ vzc..2.DeZClZ/MUOC ¢

iVZDit*2*DP201/MU001
Y2 = 03*(vatDPZA/VUOA + vZBwDPZB/Muoa + VZC~DP2612IMUOC o

1V291~092011Mu001)
XY =~(DP*MUVXA/MUOA - VXVYA)*DPZA - (DPtMuva/MUOB ~VXVY8)tDP£8 -

“p.1‘0°*MUVXC/Muoc . vxvvcwaPZC12 = (DrrMUVXD1/MU001 - vxvvo1).upzu1
*2 =‘(VzfitNUVXA/HUOA-VXVZA)tDPZA -

CVZBtHUVXB/MUOB-VXVZB)cDPZB -
1(VZC'HJVXC/MUDC=VXVZC)*DPZCIZ - (VZDlfiNUVxnl/HUODl-VXVZDI)*03291

’ "XX'- (~MUVYAtt2/MU0A a MUXXA + XMUXA )oDPZA

1* ( -MvaB.«2/Muos - MUXXB + XMUX81¢DPZB

“il‘tsvuvxc..2/Huoc . Muxxc + XMUXC)t9PZC12
v 10 c -MJVXD1.¢2(MU001 = MUxxD1 + XMUx91 )*DPZD1

CLasED onstrs

YYC : 0.0 E

= (VZEfit2/HUOEitDPZE ¢ (VzHrr2/MU0H)*DPZH + .39 ‘ '25
YZC : 0.0 ~
XYC = VXVYEtflFZE + vxva*onH . .39 . .25
XZC = -(sz*MuvxE«nP7E)/Nu0E + vxszrDPZE

"1“thZHwMUVXH*DPzH)/MUOH + VXVZH.DPZH

xxc : (=NUVXEtv2/MUOE - HUXXE + XMUxE)oDPZE + .39 * ~25..

‘ 1 *(-HUVXva2/MU0H - Huxxw + XMUXHI'UPZH

39 I9 39.”

OPEN o:axrs
A + DPZBIMuca + DPZC1?/Nuoc ¢ uonz/Muooa)

Z! s VZAtt’tDPZA/MUOA + V23**2*DPZQIHUDB o VZC«92¢DgZC12/MUUC +

1vzn2..2«0p702/Mu002
VZ a DDtCVZA€DPZA/MUOA + VZB*
1v202*03202/Mu002)

DPZBIMUOB t V70fiDPZC12/Mdoc ¢

_ I _'-_——»- —-.-..-.—-—.-

ZIN5.3A;l“i-.

-.-.—-

.-... ---

...—...

......

me. ~-— v-

nan-'—

w

...a-un.

. any a—‘u‘o—«uu

”C

_us—w — ...-q.-

.. »--_--_.. . ...
a

-. 5.-“ -. s.‘ -~'

....“ -.—- -..... - -..» - -..-— ..- . . -- . . ...l V- l- .-

161 it
a-.“-¢_mg,-v.h.-h___ - _,u “,hv , ,, , _,“M_09/29/68

_, ..- ...-
-...‘

XV ='(DP*MUVXA/HUOA - VXVYA)*DPZA 9 (DPtMUVXB/HUOB -VXVY9)*DPZB -

‘”“”‘“'1lDPtMuvxc/Muoc - VYVYC)*DP2012 . (Dpwnuvxnzxmuooz - vayuz).Dqu2

X2 =9(VZ‘tHUVXA/MUoA-VXVZA)‘DPZA - IVZBtHUVXB/MUOB-VXVZB)tDPZB 9
11VZC¢MJVXCIMUOC=VXVZC)tDP2512 - cv2u2.Muvx02/MuoDz-vxvzoz>*DP102

XX : {-HUVXArt2/MUOA c HUXXA + XMUXA )tDPZA -
15”<”9Muvxs..2/~uoa - HUXXB ¢ XHUXB)tDPZB ‘ '“ “
1. C-MUVXC«v2/MUOC - Huxxc + XHUXC)¢UPZC12

16“¢‘enva02..2/MU002 a Huxxnz o XMUXDZ )tDPzDZ

CLOSED oasrrs“

. -.Wmivyc g 0.0 le-_la ”i A‘.H.H _n--ufl , il.lim
220 x (VZE..2/MUOE)cDPzE ¢ .39 ¢ .2:
VIC i'0§0 ' " '
XYC : VXVYErDPZE ¢ .39 c .25
-'XZC’= -(VZE9MUVXEtHPZE)/MUOE * VXVZEtDPZE
XXC 8 (GMUVXEtt2/MUOE - MUXXE + XMUxE)~DPZE + .39 ¢ .25_
'GO TO 29 _ “ “ ‘ ‘ ‘“ ’ ‘ ‘

no
i

22

YY : _
”‘22 = VZA‘tZtDPZA/MUOA ¢ vzathtDPZB/MUOB + VZCtthDPZC3/"UOC i
Y1 = DDt(v7A¢DPZA/HUOA o v28*DPzB/Muoa ¢ VZC~DPZCSIMUJC) '

“W*”xv aecopoMUVXA/MUOA - VXVYA)*DPZA -'(DPtMUVXB/MUOB -VXVYB)*DPZ8 ~

1(Da-PUVXC/Huoc =.VXVYC)*DPZC3
""xz =-(VZAcHUVxA/MUOA-VXVZA)tDPZA - tVZBtHuva/MUOB-vXVZB)tDPZS -

' —..

'“OPEN ORBITS

D°**20( DPZA/HUOA + DPZB/NUOB 0 DPZCSIMUOC)

I“

 

1(VZCoHJVXC/HUOC:VXVZC)rDPZC3

' "xx = <-MUVxA..2/MUOA «MUNA * XMUXA "DPZ‘

1* ¢ ~Mdvx9..2/uuos - nuxxa ¢ XMuxB)&DPZB
1‘ (~PUVXCto2/MUOC . Muxxc + XMUXC)'UPZC3
3L?SED ORBITS

‘YYC : “'0 . , -. .
21: = (VZEot2/NUOE3tDPZE ¢ .39 ¢ .2:
YZC : 9.0 ‘
X7: 8 VXVYEoDPZE + .39 u .25

" XZS’: 6(VZEtWUVXEtDPZE)/MUOE * VXVZEwDPZE

xx: : c-MUVXE..2/Muoe - MUxXE f XMUXE)0DPZE . .39 . .25

' "GO TO 29

C
C
W ”'29
'60
51

PRINT 30. H. YV. ZZ._YZ: XV: X2: XX
Foamnrc/x. 110; 6F10-4)

"PRINT 3;. vvc, zzc; vzc. xvc} xzc. xxc

704MA1¢/11x. 6F10.4>
‘YY = YY + YYC

22 = 22 . 22:
"Y2 = YZ o YzC

XY = xv t XYC
"X2 = X2 0 YZC

XX = xx . XXC

32

'3RINT 32. vv. 22. V2? XYi X2: XX
FO?MAT(/11Y. 6710.4)

'YYZZ = 4.0.YYaZZ

ZYYZ = UYZtYZi4,0

' vvzzxx - vvzz.XX*2.0

ZYYZ'XX’ZGO

-‘fz..x‘(.XZ~8,0
YZXYZX _
9YY3X7.*XZ'800

Zszxx
YZYYZX
Zszvx
‘YYZXXZ

162

_ ___ ... - , A 7 -,‘ A . l - 7 -- . ._ _ ,... ...
, - ‘._ .7- .. P......- .. ... , -»
-u-—~.—_—-..__.-. ‘ -.-~ - . .

ETN5.3A---.MmH www~_- _06/29/68

"~zlxin;?i-22txvrxytsio

c PRINT 38. vvzz; zvvzi vvzzxx; zvvzxx. Yvazx} zvxzvx. vyzxng
miwmw-,mmm122xvyx -.7-. V -. L .
as :oaMAT</11x. sr1o,§)
u on. ‘.'7. .‘I-w'tt- c ' ... .~ ~--+ u v . o .. -. a- .- . n '
902x = cvvzz - zvvz) / gvyzzxx - zvvzxx . vzxvzx + ZYXZYX_?
"‘"“““”"‘”“1vvzxxz a’zzxvvx) .

°RlNT 390 POXX .
-~"~“"“*"‘39 ruaMA1¢I11X. F10:4’
c

"*“ ‘ 40 SOvTINuE '

WE“. . _ -
END

”rt I-FI'W!