The Electronic Band Structure of Bismuth.

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LSU Historical Dissertations and Theses Graduate School

1963

The Electronic Band Structure of Bismuth. The Electronic Band Structure of Bismuth.

Henry James Mackey Louisiana State University and Agricultural & Mechanical College

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T h is d i s s e r t a t io n h a s b e e n 6 4 —152 m ic r o f i lm e d e x a c t ly a s r e c e i v e d

M ACK EY, H en ry J a m e s , 193 5 - THE E L E C T R O N IC B A N D ST R U C T U R E O P BISMUTH.

L o u is ia n a S tate U n iv e r s i t y , P h .D . , 1963 P h y s i c s , s o l id sta te

U n ivers ity Microfilms, Inc., Ann Arbor, M ichigan

THE ELECTRONIC BAND STRUCTURE OF BISMUTH

A D is s e r ta t io n

Submitted to the Graduate Faculty of the Louisiana S ta te U n iv e rs i ty and

A g r ic u l tu ra l and Mechanical College In p a r t i a l f u l f i l l m e n t o f the requirements for the degree of

Doctor of Philosophy

I n

The Department of Physics and Astronomy

byHenry James Mackey

M.S., Louisiana State U n iv e rs i ty , 1959 June, 1963

ACKNOWLEDGMENT

The author wishes to express his deep g ra t i tu d e to Prof.

Claude G. Grenier and Prof . J. M. Reynolds for t h e i r constant

guidance, advice and assistance throughout the course of th is

study. The author is indebted to Dr. R. J. Gil l ingham, S. J.

who provided the data on which th is study is based. He wishes

to thank a l l the other members of the low temperature group

whose various c ontr ib u t ion s aided in th is work.

TABLE OF CONTENTS

I . THE DYNAMICAL EQUATIONS FOR THE GALVANOMAGNETIC AND THERMOMAGNET 1C EFFECTS IN METALS.................................................................................. 1

I I . RESULTS OF SONDHEIMER-WILSON THEORY AND PROPERTIESOF THE SOLUTIONS FOR THE TENSOR ELEMENTS............................................................... 5

I I I . DETERMINATION OF THE BAND PARAMETERS BY LEAST SQUARESCURVE F I T T IN G ........................................................................................................................... 12

IV. COMPARISON OF RESULTS TO THEORY................................................................. 19

APPENDIX A - THE TANGENT METHOD..................................................................................29

APPENOIX B - A FORTRAN I I LEAST SQUARES CURVE FITTING PROGRAMFOR BOTH LINEAR AND NON-LINEAR PARAMETERS............................... 33

APPENDIX C - DATA TABLES AND CURVES........................................................................ k2

SELECTED BIBLIOGRAPHY ................................................................................................. 1M

V I T A ..............................................................................................................................................1^2

I 1 I

44

45

46

47

48

49

50

52

55

58

61

64

LI ST OF TABLES

Parameters From Curve F i t t i n g for Ha^

Using 4 Bands with W = Y^2 ...........................

Parameters From Curve F i t t i n g for Her

Using 3 Bands with = Y^2 ............................

Parameters From Curve F i t t i n g for a- 2Using 3 Bands with W = Y ...........................

Parameters From Curve F i t t i n g for


Parameters From Curve F i t t i n g for e1


Parameters From Curve F i t t i n g for


Parameters From Curve F i t t i n g for Hc'jj


Parameters From Curve F i t t i n g for He’j'j

Using 3 Bands with = 1 ...............................

Plot Back for HCjj a t 4 ,2°K , 4 Band Model

With W. = yT2 ...........................................................k k

Plot Back for Her . a t 4. 2°K, 3 Band Model

..............................................................

Plot Back for H a^ a t 3*5°K, 4 Band Model

With W. = y‘ 2 ...........................................................k k

Plot Back for Ho^ a t 3-5°K, 3 Band Model

With W. = Y~2 ...........................................................k k

Plot Back for H a^ a t 2.66°K, 4 Band Model

With W, = y72 ...........................................................k k

LIST OF TABLES (CONTINUED)

TABLE PAGE

XIV. Plot

Wi th

Back

W =k

for

\ 2

Hon at 2 . 66°K , 3 Band Model

...................67

XV. Plot

w; thBack

W =k

for

\ 2

H° 1 1 at 2. 1°K, k Band Model

...................70

XVI . Plot

Wi th

Back

W =k

for

\ e

Hol l at 2 . 1°K, 3 Band Model

.................. 73

XVI 1 . Plot

Wi th

Back

W = wk

for

Ca i 2 at 2°K, 3 Band Model

.................. 76

X V I11. Plot

wi thBack

W =k

for

vk2a l 2 at k .2 °K , 2 Band Model

.................. 79

XIX. Plot

Wi th

Back

Wk “

for- 2V

a 12 at 3-5°K, 3 Band Model

XX. Plot

Wi th

Back

W =k

for

vk2

°12 at 3-5°K, 2 Band Model

.................. 85

XX I . Plot

With

Back

wk -

for

\ e

° \ 2 at 2 .66°K, 3 Band Model

.................. 88

XX II . Plot

Wi th

Back

W =k

for °12 a t 2 . 66°k , 2 Band Model

.................. 91

X X I11. Plot

Wi th

Back

W =k

for

vk2a l 2 a t 2 . 1°K, 3 Band Model

.................. 9^

XXIV. Plot

Wi th

Back

W =k

for

\ 2° 12 at 2 . 1°K, 2 Band Model

...................97

XXV. Plot

With

Back

w -k

fo r

\ e

p* 1 12 a t k .2 °K , k Band Model

...................100

XXVI . P lo t Wi th

BackW =k

for

^k2e"12 a t k .2 °K , 3 Band Model .........103

XXVI 1. Plot

Wi th

Back

Wk =

for

v2c"12 a t 3*5°K, *+ Band Model

...................106

112

115

118

121

12*+

127

130

133

136

139

LI ST OF TABLES (CONTINUED)

Plot Back for a t 3*5°K> 3 Band Hodel

With W = V“ 2 ............................................................k k

P lot Back for c'jg a t 2 .66°K, 3 Band Model

With W. = Y~2 ............................................................k k

Plo t Back for e'jg a t 2. 1°K, 3 Band Model

With W. = Y72 ............................................................k k

Plot Back for He'^ at U.2°K, 3 Band Model

Plot Back fo r He'jj a t k. 2°K, 3 Band Model

With W = 1 ................................................................k

P lot Back fo r Hc'jj a t 3*5°K, 3 Band Model

With W. = Y" 2 ............................................................k k

P lo t Back for He'^ a t 3*5°K, 3 Band Model

With W. = 1 ................................................................k

Plot Back fo r He'j'j a t 2 .66°K, 3 Band Model

P lot Back for He'^ a t 2 .66°K, 3 Band Model

Wi th W. = 1 ................................................................k

Plot Back fo r He'jj a t 2. 1°K, 3 Band Model

With W. = Y" 2 ............................................................k k

Plot Back for He'jj a t 2. 1°K, 3 Band Model

Wi th W. = 1 ................................................................k

v i

LI ST OF FIGURES

FIGURE

1. Construction fo r Tangent Method . . . .

2. Construction for Proof of Tangent Method

3 . HOjj Versus H at 4 .2 °K — b Band Model

With W. = Y~2 .......................................................k k

b. Han Versus H at 4 .2 °K — 3 Band Model

With w. = y “ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .k k

5 . Hau Versus H a t 3-5°K — Band Model

w l th wk - \ s .......................................................

6 . H a ^ Versus H a t 3 -5°K — 3 Band Model

With W. = Y' 2 .......................................................k k

7 . Ha Versus H a t 2 . 66°K — b Band Model- 2With W, = Y, .......................................................k k

8 . Han Versus H a t 2 . 66°K — 3 Band Model

With W. = Y~2 ............................................................k k

9- Versus H a t 2. 1°K — b Band Model

With W, = Y“2 ........................................................k k

10. Versus H at 2 . 1°K — 3 Band Model

wi th wk - Yk8 .........................................................

11. a Versus H a t i+.2°K — 3 Band Model

wi th wk = \ s .......................................................

12 . a Versus H at 4 .2 °K — 2 Band Model

With W, = Y~2 .......................................................k k

13. Versus H at 3*5°K “ 3 Band Model

Hi th wk = \ 8 .......................................................

14. 0^2 Versus H a t 3 -5°K ~ 2 Band Model

With W. = Y72 .......................................................k k

90

93

96

99

102

105

108

111

I l k

117

120

123

126

LI ST OF FIGURES (CONTINUED)

0^2 Versus H a t 2 . 66°K — 3 Band Model

With W. = Y" 2 ....................................................k k

0 j 2 Versus H a t 2 .66°K — 2 Band Model

“ , t h wk = Yk2 ....................................................

0^2 Versus H a t 2. 1°K — 3 Band Model

With W. = Y" 2 ....................................................k kVersus H a t 2. 1°K — 2 Band Model

Wi th W. = Y“ 2 ....................................................k k

€12 Versus H a t ^ •2 °K — Band ModelWith W, = y72 ....................................................k k

€ j 2 Versus H a t 4 .2 ° K — 3 Band Model

With W. = Y' 2 ....................................................k k

e12 ^ersus H a t 3*5°K — ** Band ModelWi th W. = Y~2 ....................................................k k

€ 12 Versus H a t 3*5°K ~ 3 Band ModelWi th W, = y" 2 ....................................................k k

e12 Versus H a t 2 . 66°K — 3 Band ModelWith W. = Y~2 ....................................................k k

e12 Versus H a t 2 * lOK ~ 3 Band Model

wi th wk = Y"k2 ....................................................

He1* Versus H a t 4 -2 °K - 3 Band Model

With W = y7 2 ....................................................k kHg'jj Versus H a t *+. 2°K — 3 Band Model

With W. - 1 .........................................................k

Hg'j’ j Versus H a t 3*5°K — 3 Band Model

wi th \ - \ s ....................................................

He'i'i Versus H a t 3 *5°K — 3 Band Model

Wi th W, = 1 .........................................................k

v i i t

LIST OF FIGURES (CONTINUED)

FIGURE PAGE

29. He'j'j Versus H at 2.66°K — 3 Band Model

With W. = Y"2 ......................................................................................................129k k

30. Versus H at 2.66°K — 3 Band Model

With Wk = 1 ...........................................................................................................132

31. Hc'j' j Versus H at 2. 1°K - 3 Band Model

Wi th Wk = Y^2 ......................................................................................................135

32. He'jj Versus H at 2 . 1°K - 3 Band Model

With Wk = 1 ...........................................................................................................138

ix

ABSTRACT

Data for the elements of the isothermal e l e c t r i c conduct iv i ty tensorA Aa and the thermoelectr ic c o e f f ic ie n t tensor e" for bismuth over a magnetic

f i e l d range of 0-18,000 gauss and a t temperatures of 4 .2°K , 3-5°K,

2.66°K and 2 . 1°K are analyzed on the basis of Sondheimer-WiIson theory.

The analysis rests on the use of an IBM 1620 d i g i t a l computer to f i t the

data curves in the least squares sense by varying c e r ta in parameters which

occur in the functions proposed by the theory. The approach to least

squares problems involv ing nonlinear parameters is discussed and a

Fortran I I computer program is given which is designed for curve f i t t i n g

with an a r b i t r a r y function. Up to e ight parameters have been varied

s imu 1taneously.

The parameters determined by th is method are: n. - the density

of c a r r ie rs in the i c o n t r i b u t i n g band, a j n j - where a, is a weight

factor re la ted to the degree of anisotropy in the basal plane, Z. - the

density of states of the i t 1 band, c .Z . - where c. = a . , and H. - the 1 i i i i i

so -ca l led sa turat ion f ie ld s of the various bands.

The ind iv idua l curve f i t t i n g s are qu ite good, but the various e f fe c ts

show d i f f e r e n t numbers of bands with d i f f e r e n t sa turat ion f i e ld s . A l l

the e f fe c ts e x h ib i t an e lec tron band with low sa turat ion f i e l d and varying

numbers of hole bands a t higher sa tu ra t ion f i e ld s . The value of a.

found for the e lec tron band is much smaller than values deduced from the

data of other invest igators based on d i f f e r e n t e f fe c ts such as cyclotron

resonance, e tc . I t is shown that the assumption of an an isotrop ic

re laxa t ion time can exp la in the e f f e c t i v e l y higher degree of isotropy in

the galvanomagnetic and thermomagnetic e f f e c ts . The assumption of a

x

common mean f ree path for holes and e lectrons is shown to p red ic t a

higher s a tu ra t io n f i e l d for the e lectrons than for the holes contrary

to the present re s u l ts . I t is suggested that the hole band found with

a sa tu ra t ion f i e l d *= 20,000 gauss is due to the presence of an acceptor

impuri ty which may account for the fact tha t the present data for the

thermomagnetic e f f e c t s are some ten times la rger than would be expected

on the basis of the Sondhetmer-WiIson theory.

CHAPTER I

THE DYNAMICAL EQUATIONS FOR THE GALVANOMAGNETIC

AND THERMOMAGNET 1C EFFECTS IN METALS

The l in e a r i z e d dynamical equations which describe the physical

s i tu a t io n in a metal in which there e x is ts an e l e c t r i c current and a

heat current in the presence of an e l e c t r i c f i e l d , a magnetic f i e l d ,

and a gradient of temperature, may be w r i t t e n in several ways.* Three

such sets are given here using tensor no ta t ion . The f i r s t se t ,

J = oE* - e"G (1—a )

W* = -n"E* + *"G (1-b)

expresses the f luxes and V£*, the e l e c t r i c and heat current d e n s i t ie s

re s p e c t iv e ly , as functions of the a f f i n i t i e s or d r iv in g forces £ * and

G. E/v is the negative gradient of the e lectrochemica l p o t e n t i a l , and<*.

G is the negative of the temperature g ra d ie n t . a is the isothermalA

e l e c t r i c a l c o n d u c t iv i ty tensor, e" is the therm oe lec tr ic c o e f f i c i e n tA A

tensor, it" is the P e l t i e r tensor, and A" is the is o p o te n t ia l thermal

co n d u c t iv i ty tensor. The components of these tensors are known as the

k in e t ic c o e f f i c ie n t s and are functions of the in tensive parameters of

the system.

* J. R. Sybert , "Transport Phenomena in a Single Crys ta l of Bismuth a t Helium Temperatures," D is s e r ta t io n , Louisiana State U n iv e rs i ty ,1961.

1

The second set of dynamical equations is:

E* = p_J + cG (2- a)

W* - -nJ + % (2-b)

Here .J and £ are taken as independent v a r ia b le s , p is the isothermal

e l e c t r i c a l r e s i s t i v i t y tensor, e Is the absolute thermoelectr ic tensor ,

n is the isothermal P e l t i e r tensor, and % is the thermal co n d u c t iv i ty

tensor. The components of these tensors are c a l le d the isothermal

coeff ic i e n ts .

The t h i r d set of equations is:

E* = p' J + e'W* (3 -a)

G = rt'J + yW* (3“b)

c' is the therm oelectr ic tensor, S' is the P e l t i e r tensor, p' Is the

a d ia b a t ic e l e c t r i c a l r e s i s t i v i t y tensor, and y is the thermal r e s i s t i v i t y

tensor. The components of these tensors are c a l le d the a d ia b a t ic c o e f f i

c ie n ts .2

I t may be shown fo r the case of bismuth, that i f the magnetic

f i e l d is d i r e c t l y along the t r ig o n a l a x is , wh i le the f luxes are confined

to the basal plane, a l l of these tensors reduce to the form

0 1

where a ^ and may be the corresponding elements of any of the above

tensors .

^1 b id . p. 5 -

The data for the elements of p, e ' , and X were reported by Gi l l ingham.A

X is e f f e c t i v e l y not a funct ion of the magnetic f i e l d , H, due to very

l i t t l e e le c t r o n ic c o n t r ib u t io n to the heat curent in bismuth.

is e f f e c t i v e l y zero. was measured over a temperature range of 2. 1°K

to k.k^°K. p is a slowly vary ing funct ion of temperature, and e' is

s trong ly temperature dependent. Both p and e1 elements were measured

over a range of H from 0 -18 ,00 0 gauss. This was done a t temperatures o f

2 . 1°K, 2 .66°K, 3 . 5°K and k . 2°K in the case of e ' , but low f i e l d values

of p were not taken a t 2 . 6 6 ° K and 3-5°K-

The elements of the e l e c t r i c a l cond uc t iv i ty tensor 8 and the thermo

e l e c t r i c tensor e" have been computed t h e o r e t i c a l l y by Sondheimer and

Wilson. A l i t t l e tensor a lgebra gives the r e la t io n between the e x p e r i

m enta l ly measured q u a n t i t ie s and the t h e o r e t ic a l q u a n t i t i e s as:

3 = p' 1 ( V a )

e" - Aae' (4-b)

Low f i e l d values fo r the elements of 3 a t 2 .66°K and 3*5°K were i n t e r

polated from the 4 .2 °K and 2 . 1°K data. This paper is concerned w i th

an analys is of the experimental data in the modified form of the 9 and

e" tensor elements in the l i g h t of the Sondheimer-WiIson theory. This

s e m i -c la s s ic a 1 treatment is not concerned w i th the quantum o s c i l l a t io n s

■3R. J. G i l l ingham, "Galvanomagnetic and Thermomagnetic E f fe c ts in

Bismuth and Pressure Dependence of de Hass-van Alphen-Type O s c i l la t io n sin Zinc and Bismuth," D is s e r ta t io n , Louisiana State U n iv e r s i ty , 1963*

4A. H. Wilson, The Theory of Metals (Cambridge: The Univers i ty

Press, 1958), pp. 193 f f *

k

which appear superimposed on the gross e f f e c t a t high magnetic f i e l d .

In order to apply th is theory the fo l low ing procedure was adopted.

The envelope was c a r e f u l l y drawn to the o s c i l l a t o r y port ion of the data.

This envelope was then s p l i t a t se lected f i e l d values and the mean

values were taken as data for the gross e f f e c t .

CHAPTER I I

RESULTS OF THE SONDHEIMER-WILSON THEORY AND PROPERTIES

OF THE SOLUTIONS FOR THE TENSOR ELEMENTS

Sondheimer and Wilson have given a s e m i -c la s s ic a l treatment o f the

thermomagnet Ic and ga1vanomagnet 1c e f f e c t s in metals.^ This treatment is

based on a s o lu t io n to the Boltzmann transport equation assuming an

e f f e c t i v e re la x a t io n time r which is taken to be a constant and id e n t ic a l

for both the galvanomagnetic and thermomagnet Ic e f f e c t s . I t is assumed

that the metal is i s o t r o p ic , and tha t the constant energy surfaces in

momentum space are s p h e r ic a l . I t is a lso assumed that the temperature

is small compared to the Fermi temperature, and tha t the q u a n t iza t io n

of the e le c t ro n o r b i ts in the magnetic f i e l d may be neglected. The

resu lts of these considerat ions are given below expressed in gaussian

units assuming a s ing le band.

H

° n ° ecn a S a ( 5 ‘ a)11 H + Hs

Ha = ± ecn — ------- - (5~b)

s

2 2 Hc'l'l - + ~ T ~ cTZ ■ 5 ( 5 - c )

H + Hs

■ • 2 ¥ c t z t 4 (5- d>■> H + Hs

^WiIson, I b i d . , p. 198 f f .

5

6

In Equations (5) the upper sign re fe rs to holes, and the lower sign re fe rs

to e lec tro ns , e is the magnitude o f the e le c t r o n ic charge, c is the

v e lo c i ty of l i g h t , n is the density of c a r r i e r s , k is Boltzmann's constant,

T is the absolute temperature, Z is the density of s ta te s , and H is the

magnetic f i e l d s trength . The q u a n t i ty is defined as

u _s eT

where m* is the e f f e c t i v e c a r r i e r mass, and t is the e f f e c t i v e re la xa t io n

time. H is c a l le d the s a tu ra t io n f i e l d of the band. A f ree c a r r i e r ofs

mass m* and charge e would made a p a r t i a l o r b i t of one radian in time t

under the in f luence of a magnetic f i e l d o f magnitude H^.

Although bismuth e x h ib i ts non-spherical energy surfaces in momentum

space, the preceeding equations may be used i f one includes weighting

parameters. The weight fa c to r fo r Equation (5"b) may be shown to be equal

to un i ty . * * For the case of more than one band we introduce a sum

over the bands and obtain:

H.

°11 ocV l - 2 / ^ - <6' a >i

a iP = E * CCni p H p (6 -b >12 i H + H21

2Sybert , ojd. ci t . , p. 57-

■5

B. Abeles and 5. Meiboom, "Galvanomagnetic E f fe c ts in Bismuth,"The Phy s i ca 1 Rev i ew, Cl, (1956) , 544-550*

iiJ. W. G a l t , W. A. Yager, F.R. M e r r i t t and B. B- C e l t in , "Cyclotron

Absorpt i on in Meta 11i s Bi smuth and i ts A1loys, " The Physical Revi ew, CXI V, (1959), 1396-1413.

7

2 ,2 H.€'• - L + c TZ ' u (6-c )

11 i ' 3 ' H2 + H2i

€12 = ? - b i TZ! " T ^ (6" °1 J H + Hj

In Equations (6 ) the subscript i re fe rs to the i b a n d of c a r r i e r s .

I t is of in te r e s t to examine the geometric p roper t ies of the above

funct ions. F i r s t one notes that and e'j'j are formed from sums of

s im i la r funct ions. E i th er funct ion may be w r i t t e n as:

Y ■ f i n h ; (7)

2 2where B = H , B. = H.; and for O . . .B . = e c a .n .H . ; w h i le fo r eV,» B. =H ’ K i i * 1 1 ’ i i i i ' 1 1 ' i

2, 2 .1 c. — ------ Z . H . . A p lo t of the type o^j vs. p or e'j'j vs. 0 wi 11 be

c a l le d a beta p lo t . We note that Ojg/H an<* e12^H bave *be form of

Equation (7 ) with B. given as below:

for a ,_ /H B. = ± ecn.12 1 1

2 . 2and for eV^/H = "b. n ■ C TZ.12 1 1 3 '

The funct ion ex h ib i te d in Equation (7 ) is seen to be a sum of hyperbolas

with v e r t i c a l asymptotes a t p - ~^i* t an9ent l in e a t p = 0 is given

by

I n

B. 0 B.Y » ? ( - - 5- . ) (8)‘ ' pf pi

the case where the bands c o n t r ib u t in g terms to O j p g'j jj a l 2^H ° r

g12^H bave w idely separated values of H., an examination of the low

f i e l d data p lo t te d against 0 may y e i Id f a i r approximations to the parameters

associated w i th the band having the lowest s a tu ra t io n f i e l d . The c o n t r ib u t in g

8

bands w i l l be numbered in the order of th e i r sa turat ion f i e l d magnitudes

s ta r t in g with the lowest value. Examination of Equation (8 ) shows that

under the above conditions band I w i l l contr ibute most strongly to the low

f i e l d data since p is the smallest of the p.. from Equation (8 ) we have:

B BY‘ = ? T T * i t (9‘ a)

p^o ' pi pit

dYdp

B. B.

" a <9_b)£*=0 • Pj pj

a t Y = 0 L B . /p .t i iP = 5 * P. ( 9 - c )

l B . /p rI I

Thus by constructing the tangent l ine to the data curve at p - 0 and using

Equations (9) one may estimate band 1 parameters. This procedure w i l l be

ca l led the s lop e - in te rc e p t method.

2 2 2 H O j j , H e'jj* ^°12 anC* Hg12 maY P^otted a9a i nst 7 = 1/H at high

f i e l d in order to obtain estimates of the parameters of the las t band.

These functions may be w r i t t e n as:

C.Y = E — - 1 (10)

i 7 + 7j

2 2where y = 1/H , y. = 1/H. and C, has the fo l lowing values:i l l

eca.n.,2_ i ifor H o C. =I

c . n2k2cT Z . fo r H e'1, C. = ’ '11 i + 3H.

ecn.for Ha,„ C. = + '

12 1 - H2i

b.it2k2cTZ.

for Hg12 Ci “ - ' H2 'i

Data p lo t te d in th is manner w i l l be c a l le d a gamma p l o t . Comparison of

Equations (7 ) and (10) show that the band which has i t s asymptote nearest

the o r ig in in a beta p lo t w i l l appear w i th i t s asymptote fu r th e s t from

the o r ig in in a gamma p l o t , and v ice versa. Therefore one may use the

s lo p e - in te rc e p t method in conjunction w i th a gamma p lo t to est imate the

parameters o f the band w i th the highest s a tu ra t io n f i e l d in analogy to

Equation (9 ) , although these estimates are q u i te rough since one must

e x tra p o la te the data to the 7 = 0 (H =00 ) ax is .

A b e t te r method fo r these estimates has been developed which does

not require an e x t ra p o la t io n of the data. I t is shown in Appendix A

that for the la s t band one has:

CLY ~ -----7 + 7 l ( 1 1- a )

7 - a y 1r L - - f n - a t u - b )

L (a - l ) 3

where, r e fe r r in g to Figure 1, DB is the tangent l ine to Y at 7 = 7 ,

AC is the tangent l in e to Y a t 7 = 7 ^ , T = AD, T ‘ = BC and a - T ' / T .

This construct ion is c a l le d the tangent method.

Again r e fe r r in g to Equations (6 ) i t is seen th a t the functions hcr^,

a 12’ ^e l l an( €12 *"iave f ° rrri a sum terms l ik e

A. HY = - r 1 5 (1 2 -a )

H + H7i

This funct ion passes through the o r ig in , reaches an extremum a t H - H,

and goes to zero l i k e A . /H as H becomes i n f i n i t e . The peak value is

CONSTRUCTION FOR TANGENT METHOD

11

Y . - A./2H. ( 1 2 -b )peak i i

This funct ion w i l l be re fe r red to as a peaking funct ion. Equation (12-a )

may be put in to another form which is the most useful of a l l the various

representat ions because of the symmetry i t introduces. i t amounts to ju s t

p l o t t in g Equation (12-a ) against H on semi- log paper. A n a l y t i c a l l y one

may proceed as fol lows:

Let a ^ In H a ( = l n H .

then ( 12-a ) becomes

A.Y = sech (a - a . ) (13)

i

Equation (13) is seen to be symmetric about a = a . w i th a peak magnitude

as given by (1 2 -b ) . Because of th is g reat a id to v is u a l i z a t i o n a l l data

were p lo t te d in peaking funct ion form on semi- log paper. The values of

A. are given below:

for Ha., A. = eca .n .H . (14 -a )11 i i i i

for a ]2 A. = ± ecn. (14-b)

2 .2 T c . n k cTfor He',', A. = + --------- Z.H. ( l ^ - c )

11 i 3 1 1

b .n 2k2cTfor e'>2 A. Z. ( l k - d )

CHAPTER I I I

DETERMINATION OF BAND PARAMETERS BY LEAST

SQUARES CURVE FITTING

One may obta in f i r s t estimates of the parameters of the f i r s t and

la s t bands by the methods discussed in the previous chapter . More crude

estimates of c e n t r a l l y located bands may be had by the same means i f one

subtracts out the f i r s t and la s t bands from the experimental data.

Improved values for a l l parameters may then be had by p l o t t in g the data

in the peaking funct ion representat ion and s h i f t i n g the values of the

parameters u n t i l the sum of the various bands reproduces the data curve

approximately. This procedure is q u i te laborious , and the po int a t which

to stop is somewhat a r b i t r a r y . Some c r i t e r i a must be adopted as a measure

of best f i t . The c r i t e r i a used in th is ana lys is is to minimize the root

mean square per cent e r ro r (R. M. S. per cent e r r o r ) . I t w i l l be shown

th a t the s o lu t ion of the governing equations y ie lds the same set of

parameters in the case of the hyperbol ic form as in the case of the

peaking funct ion form of the data , so long as the R. M. S. per cent e r r o r

is minimized. The two forms of the data w i l l lead to two d i f f e r e n t sets

of parameters i f ord inary leas t squares c r i t e r i a is used, that is , m in imizat ion

of the R. M. S. e r r o r .

The least squares approach is as fo l low s. Suppose there is given a

funct ion y of an independent v a r ia b le (or v a r ia b le s ) x, and of c e r t a in

constant parameters p . , i = 1, 2, NP (NP is mnemonic fo r Number of

Parameters). Assume there is given a set of data points (x^, Y^)*

12

13

k = 1, 2, NO (ND is mnemonic for Number of Data p o in ts ) . The

parameters p. are to be chosen so as to minimize the fo l low ing function:

ND _ p• = E [ y (xk , p . ) - yk ] Wk (15)

k=l

in th is expression p. stands for a l l of the parameters, and W, is aI K

weight attached to the k**1 data po int . I f = 1 for a l l k then the-2

best f i t c r i t e r i a is the minimum R. M. S. e r r o r . I f V#k = yk for

a l l k then the c r i t e r i a is minimum R. M. S. per cent e r r o r . In p r in c ip le

any a r b i t r a r y set of numbers may be assigned to the Wk so that th e i r

r e l a t i v e magnitudes express r e l a t i v e confidences in the various data

points . Also one may c lu s te r many more points in one region of the

data curve than in another to force a b e t te r f i t of the f i r s t region.

We note here that i f the fo l low ing s u b s t i tu t io n s are made in Equation

(1 5 ) ,

yk = x k yk

n — x Yl

and i f

WLr = (xny. ) 2 = Y l 2 = X 2 n w

then Equation (15) remains the same except for the primes on the yk , Wk

and yk - This shows that the same parameters are determined when

the peaking funct ion representat ion is used as when the hyperbol ic

representat ion is used as long as R. M. S. per cent e r ro r is minimized.

li+

Whatever the choice of the one may proceed by forming the

p a r t i a l d e r iv a t iv e s of $ w i th respect to the p. and s e t t in g these

expressions equal to zero:

ND P j )5 T = * 2 [ y k < V p i ) ' yk ] 5 ^ wk ( l 6 ‘ a)J k=l Kj

or

ND _ ^yk * V p i *kf , yk < V •*«> S p J -^ w> (16-b)

ND dyk (xk , p . )

yk s r “ i.k= 1 i

This set of NP equations ( l 6- b ) , obtained by s e t t in g j = 1, 2, NP,

are c a l le d the normal equations of the system.

Two s i tu a t io n s now a r is e . The f i r s t is tha t the parameters p.

a l l e n te r the funct ion y l i n e a r l y . In th is case

* 7

is independent of the p. and the normal equations are a set of NP

l in e a r equations in the NP unknowns, p . . These may be solved p re c is e ly

with no d i f f i c u 1t y .

I f we are dea l ing w i th only one band we may l i n e a r i z e the

problem as fol lows:

2Form \|f = L h k=l k

and se t 5 a * SB * 0 (17)

A and B may be determined exac t ly from ( 17) although i t is not c lear

what the best f i t c r i t e r i a means in th is s i tu a t io n .

When more than one band is involved the above l i n e a r i z a t i o n scheme

is not p r a c t ic a l . We may then turn to an i t e r a t i v e procedure in order

to f ind a solu t ion to ( l 6- b ) . The procedure used here has been c a l led

the Gauss method, the Gauss-Newton method, the Gauss~Seidel method,

the Seidel method, and the Newton-Raphson method. Credit fo r the

basic idea and development of the approach is usual ly given to Gauss.*

Let p. be a representat ive point of the NP dimensional parameter

space in the neighborhood of the point p . Q which minimizes in Equation

(15) . Expanding y^tx^, P j ) in a Tay lo r 's series about p. and keeping

only f i rs t order terms:

*H. F. T r o t te r , "Gauss's Work (1802-1826) on the Theory of Least Squares," an English T rans la t io n , AEC-TR-30^9 (1957).

16

np dyk ( * . , p )

V v "io* ' yk(v pi> + .f, sf; api

whe re

A p . = p. - p.r i r iO i

Since y (x , p. ) is a s o lu t io n to Equation ( l 6 -b )K K i O

ND NP dy. (x ,p ) ^y. (x , P; )

kc, lV v pi> * V ' ^ I 1 ap u>k=l (=1 r i r j

ND t x£ y 5yk<V pj) u

k-1 op. kJ

or

ND dy. (x , p . )

l y k (xk- p ^ ■ V § 5 ]------------------ wi,

ND NP dy (x , p ) dyL L----------— ------- ^ W,AP. (18). . . . dp. o p . k ik~ 1 1=1 i r |

Equations (18) are a set of NP l i n e a r equations in the A p . . S t a r t i n g

w i th i n i t i a l est imates of the p . , Equation (18) may be used to f in d thei

A p , , Theni

<Pi>L ♦ 1 = ‘ P ^ L + ^ Pi>L

gives an improved set of parameters fo r the (L + l ) *^1 i t e r a t i o n in

IT

terms of the parameters used in the previous i t e r a t i o n . This procedure

is continued u n t i l A p , /p . « 1. Convergence of th is method depends upon

the s e n s i t i v i t y of the procedure to the coarseness of the f i r s t est imates.

This is in turn dependent upon the complexity of 7 and on how w e l l the

data may be represented by y. Experience indicates that the Ap. ge ne ra l ly

have the cor rec t r a t i o so that Ap. is d i re c te d from the point p. towards

the v i c i n i t y o f po int P jc * But sometimes the magnitude of Ap. is so

large as to overstep the minimum a t p . Q and cause o s c i l l a t i o n or divergence.

In such a s i t u a t io n Ap. may be m u l t ip l i e d by a numerical fa c to r less

than one to reduce the step s ize . Genera l ly when p. is s u f f i c i e n t l y

close to p .^ no d i f f i c u l t y is found in ob ta in ing convergence. in some

cases a fa c to r g re a te r than one may be used successfu l ly to speed up

convergence. In the curve f i t t i n g reported here no d i f f i c u l t y was

experienced in obta in ing convergence except when too many terms were

included in y.

I t was found h igh ly convenient to use the s lo p e - in te rc e p t or tangent

method to est imate only the H. and then to solve fo r the l in e a r parameters

by least squares tak ing the H, as constants. Then th is e n t i r e group

of estimates were used as f i r s t approximations in the above described

least squares c a lc u la t io n .

Appendix B gives a Fortran I I program for the s o lu t ion of Equation

(18 ) . This program was run on an IBM 1620 d i g i t a l computer. I t was

found that c a lc u la t io n s w i th standard e igh t d i g i t p rec is ion gave absurd

re s u l ts , while those made w i th f i f t e e n and twenty d i g i t p rec is ion agreed

to about e igh t d i g i t s . F i f te e n s ig n i f i c a n t d i g i t s were used in a l l

computation to avoid th is t runcat ion or rounding e r r o r , although

physical s ig n i f ic a n c e in so fa r as number of meaningful s ig n i f i c a n t

18

f igures in the answers for the parameters must be judged by the accuracy

of the data. Running time is dependent upon the complexity of y, the

number of data po in ts , and upon the number of d i g i t s used in the a r i th m e t ic .

When y is the sum of four peaking functions (e ig h t parameters) , and when

f i f t y data points are used wi th f i f t e e n d i g i t p re c is ion , each i t e r a t i o n

requires about f i v e minutes. On the average about ten i t e r a t io n s were

s u f f i c i e n t to obtain a so lu t ion .

19

CHAPTER IV

COMPARISON OF RESULTS TO THEORY

The data tables and the corresponding f igures showing the f i t t e d

curves are found in Appendix C. Note that in the tab le t i t l e s E is

w r i t t e n fo r e" and S is w r i t t e n fo r a. These tables were p r in ted

d i r e c t l y from I.B.M. punched cards, and no spec ia l symbols were a v a i la b le .

An explanat ion of the tables is found in Appendix B where the computer

program is discussed.

Tables I through V I I I summarize the resu l ts obtained fo r the various

parameters by curve f i t t i n g . Each data curve was f i t t e d using as many

terms as possib le without divergence. Then each curve was f i t t e d with

one less term fo r comparison. With the exception of the H€jj data a l l- 2

curve f i t t i n g s were done w i th = y^ . The He1 data were f i t t e d using

-pboth = y^ and = 1. Convergence was not obtained for four terms,

while a t least three are obviously needed as seen by the shape of the data

curves. Therefore th is data was f i t t e d w i th three terms only.

Examination of the tables and curves ind ica te that the functions

proposed by Sondheimer-WiIson theory may be used to represent the

in d iv id u a l data q u i te w e l l , but d i f f i c u l t i e s a r is e upon comparison of

the various e f f e c t s . I t is seen that the d i f f e r e n t e f f e c t s e x h ib i t

d i f f e r e n t numbers of terms or bands, and that these bands have qu i te

d i f f e r e n t sa tu ra t ion f i e l d s . and are seen to e x h ib i t a band

with a s a tu ra t io n f i e l d in the neighborhood of 20,000 gauss, which is

e n t i r e l y absent in a^ . I t appears as a very small c o n t r ib u t io n to

at it .2 °K and 3-5°K but is n e g l ig ib le a t 2 .66°K and 2 - 1°K.

20

In bismuth the Fermi surface in momentum space consists of pieces

of both p o s i t iv e and negative curvature due to overlap or underlap at

B r i l l o u i n zone boundaries. Pieces w i th the same sign of curvature may

be t ran s la ted to a common o r ig in to produce approximately e l l i p s o i d a l

pockets of c a r r i e r s . The geometry of these pockets may be mapped by

f i e l d o r ie n ta t io n studies of the quantum o s c i l l a t io n s in various

e f f e c t s . The most commonly accepted model is the fo l low ing . The

Fermi surface in momentum space for e lec trons consists of three e l l i p s o i d s

re la ted by ro ta t ion s of 120° about the t r ig o n a l a x is . One of the

p r in c ip a l axes of each e l l i p s o i d is p a r a l l e l to a b inary a x is . The

p a r t i c u la r e l l i p s o i d w i th p r in c ip a l ax is along the x -a x is is given by

Ql l px + a s ^ y + a 33pz + a j 23pvpz = 2r;

where the momenta are measured from the center o f the e l l i p s o i d .

is the chemical p o te n t ia l fo r the e le c t ro n s , and the a . , are elements' J

of the rec iproca l e f f e c t i v e mass tensor a . The surface for holes consists

of an e l l i p s o i d of revo lu t io n w i th the symmetry ax is in the t r ig o n a l

d i r e c t io n . I t s equation is

fJl l (px + py> + P33Pz "

where is the chemical p o te n t ia l of the holes, and the p.^ are elements

of the rec iproca l mass tensor fo r holes. The weight fa c to rs a , b and

c, introduced in Equations (6 ) , are a measure of the departure of the

basal plane cross sections of the pockets from c i r c u l a r symmetry. These

q u a n t i t ie s are given as

21

r m2n 1 /2a = I — J = cm,

(19-a)b = 1

for a s ingle e l l i p s o i d with p r in c ip a l axes or iented p a r a l l e l to the co

ord inate axes, so that the e f f e c t i v e mass tensor is d iagonal ized.

Note that

a = l ^ ] ' / 2 = c ( ,9 -b )2

and i f R is the r a t i o of the major and minor axes of the basal cross

section then

a = R = c (19~c)

I f a second e l l i p s o i d is added equ iva len t to the f i r s t but ro ta ted 90°

in the basal plane then

2 R 2 m, ms (£0)

b = 1

In the case of several e l l i p s o i d s re la ted by r o ta t io n about the t r ig o n a l

axis s im i la r expressions a l low them to be t rea ted as one pocket. Here

again, a is a measure of the anisotropy.

In the case of two crossed e l l i p s o i d s t i l t e d out of the basal

plane, m is not diagonal but is of the form

Then Equation (20) becomes

a - _1_2

(mj + m. 2 -- )m. 1/2

(m2m 2 \ 1/2

V m\1/2 (21)

In terms of the tensor elements of a Equation (21) becomes simply

a = c = — (22 )

Note that Equations (21) and (22) reduce to Equation (20) i f = 0.

Even for m / 0 one s t i l l has

a = c = | ( R + I ) (23)

A p a r t i a l survey of the l i t e r a t u r e has been made, and a has been

computed for the e lectron band on the basis of Equations (21) , (22) or

(23) from the data of several invest igato rs . The re s u l t in g values of

23

a are: 11 .3, 1 *+-5, 2 4 .6 , 3 5-35,** 6- 35 ,5 5 - 5 5 ,6 and 4. 75. 7

None of the above data was from di rect measurements upon the

galvanomagnetic e f f e c ts , and represent values averaged over many f i e l d

direc t ion s . However, data from the galvanomagnetic studies of Abeles

and Meiboom gives a = 3-22 which is seen to be somewhat small by comparisong

ind ica t ing a more isotrop ic s i tu a t io n . Our data a lso leads to a

r e l a t i v e l y small value for a in the case of the e lec tron band. Even

thouqh 0 . , and o . „ e x h ib i t d i f f e r e n t numbers of bands with d i f fe r e n t 3 11 12

saturat ion f i e ld s , the value of a for the e lec tro n band may be approxi

mated by taking the extreme values for a^n^ from the cr^ f i t t i n g s

with three and four bands, and n from the a f i t t i n g s with two and

-17three bands. Then a n x 10 ranges from 4 .09 to 4 .98 while nj x- 1710 ranges from 2 .04 to 2 .10. This puts a^ in the range 1.95 to

2.44. Sybert mapped the hole e l l i p s o i d by a f i e l d o r ie n ta t io n study

17and found the density of holes to be 3-4 x 10 . I f a two band model

is assumed then the same value is expected for the e lec tron density.

“ 17Taking n x 10 - 3-*+ with ' n the range 4 .09 to 4-98 one

obtains in the range 1 .20 -1 .46 . Thus the present data indicates an

increase in the e f f e c t i v e isotropy when the f i e l d is along the t r ig o n a l axis .

*0. Schoenberg, P h i l . Trans. A245, 1 (1952).

^J. K. Galt , e_t a_ . , oj). ci t . 1396-1413--3

J. E. Aubrey and R. G. Chambers, J. Phys. Chem. Solids 3,128 (1957).

**J. E. Aubrey, J. Phys. Chem. Solids 19* 321 ( 1961) .

\ . S. Lerner, Phys. Rev. T2J, 1480 ( 1962) ./J

D. Weiner, Phys. Rev. 125, 1226 ( 1962).

^A. L. Jain and S. H. Doenig, Phys. Rev. 127, 442 (1962)-

^Abeles and Meiboom, og>. ci t . .p. 544-550.

Some ins ight in to th is e f f e c t may be obtained by the fo l low ing

considerat ions. The Sondheimer-WiIson theory is based on the assumption

of a constant time of re la x a t io n t , which is the same for both the

galvanomagnetic and thermomagnetic e f f e c t s due to a s ingle pocket of

c a r r ie r s . For the case of a s ing le e l l i p s o i d or iented w i th i t s p r in c ip a l

axes p a r a l l e l to the co-ord inate axes, Sondheimer-WiIson gives

2 2 nc m _ m.m c .

" n = < H + a i V > < * >e t

This expression may be re w r i t te n as

Ha = neca — — ( 25~a)

11 H + Hs

where a = ( ^ ) * / 2 Hs = (mi m2 ) 1 /2 ” (25“b)

Now i f the r e la x a t io n time i s introduced as a diagonal tensor

T 1 00 ^AT ~ 0 T 2 0

1 °0

T3 j

Equations (25_b) become

25

The re la t io n s a = c, b = 1 s t i l l hold. Note that i f T j = = t ,

Equations (26) reduce to Equations (25)- Now TjT^ * may be adjusted

independently of T jT2 50 that i f t j t 2 ^ a ' s reduced and an

apparent decrease in anisotropy resu l ts without necessar i ly changing

H .5

To see the e f f e c t upon the r e la t iv e saturat ion f i e ld s for holes

and e lectrons, assume a two band model consist ing of a spherical pocket

of holes and an e l l i p s o i d of e lectrons or iented with i ts p r in c ip a l

axes p a r a l l e l to the co-ordinate axes. Associate nij, t , and

with the extrema of the e lec tron pocket, and associate m1 and t ' with

the hole pocket. Now w r i t in g and as the sa turat ion f ie ld s for the

holes and e lectrons Equation (2 6 -b) gives

Hu i •— = (27)H m*t '

t / V i / 2where t -.' = (t j t 2 ' ,Tr'r = (mjm2 ^

The fact that we f ind greater than may be accounted for i f t * »

t ’ when the magnetic f i e l d is along the t r ig o n a l axis .

I t is in te re s t in g to note the e f f e c t of assuming a mean free path

A which is the same for e lectrons and holes. Assume the two band

model used above, and le t v and v^ be the e lec trons ' extremal v e lo c i t i e s

in the basal o rb i t as defined in Equation (29) where is the e le c t ro n ic

chemical p o te n t ia l . In the same manner associate v' and £ with the

holes. l e t

v ' t ' = v ^ j = = A (28-a)

26

thenT p v ' 2

f1?) = ”------ (28-b)T ,y VjV2

On the Fermi surfaces

V i = 2^e = m2v2

m'v ' 2 = 2^h

(29)

then

, 2 t . m*( ? ) - (30)

and Equation (27) becomes

(31)

I f the values of m', m*, and as found in the l i t e r a t u r e from

o r ie n ta t io n studies are subst i tu ted in Equation (31) i t predic ts

H > H, contrary to our data. Using Equations (28) and (29) one e h

f i nds

t v m 1/2

r = v = t r * (32)2 V1 2

Then Equation (26-a) becomes

27

( 3 3 )

p re d ic t in g a much more is o t ro p ic s i tu a t io n than when t is considered

a sea la r quant i ty.

Another model may be used which is in te r e s t in g because i t is a

geom etr ica l ly na tura l choice. I t may be supposed that the e lec trons and

the f ie ld s necessary to cause the corresponding o r b i t circumferences

to be equal to 2nA. O r ie n ta t io n studies ind ica te that the momentum

space cross sections of the e le c t ro n and hole pockets are about equal.

This implies near e q u a l i t y of the areas in real space. Assuming an

e l l i p t i c a l o r b i t for the e lectrons with axes of length a and b, and a

c i r c u l a r hole o r b i t of radius r one has

holes have a common mean f ree path A, and Hg and may be def ined as

nab = nr 2

Ch 2nr

where C and C. are the circumferences o f the o r b i t s . Then e h

(3*0

where R - ab-1

This implies H > H. which is contrary to our data, e h

28

The above discussion of various models which might account fo r the

e f f e c t i v e isotropy in the galvanomagnetic e f f e c ts when the magnetic

f i e l d is p a r a l l e l to the t r ig o n a l ax is may be appl ied to the thermo-

magnetic e f fe c ts since we have a = c. Sondheimer-WiIson theory predic ts

thermomagnetic e f f e c t s which are only about one-tenth as large as those

found. This suggests the presence of a mechanism fo r heat f low not

considered in Sondheimer-WiIson theory, which takes in to account only

the v a r i a t i o n of the d i s t r i b u t i o n funct ion due to a temperature

grad ient . The band of very high s a tu ra t io n f i e l d which appears in a l l

the e f f e c t s except a ^ is p e c u l ia r in that i t e x h ib i ts an extremely

small density of c a r r i e r s and a very large density of s ta te s . I t is

suggested that th is band is due to the presence of an acceptor impurity

which would account for the excess of holes over e le c t ro ns . I t might

a lso account for the low r e s i s t i v i t y r a t i o

Prt<293°K)

P0 (2. l°K j " 1,0

Bismuth is a semi-metal and the presence of the impurity might bring

about a near semi-conductor s i t u a t io n In which the production and re

combination of e le c t ro n -h o le pa irs is an important mechanism in heat

t ran spor t . This could account fo r the large thermomagnetic e f f e c t s .

However, i f a s ing le impurity leve l is considered s l i g h t l y above the

Fermi energy i t should y i e l d an apparent decrease in the dens ity of

sta tes as a funct ion of temperature. This is not the case in the present

data , but the e f f e c t of a d i s t r i b u t i o n of impuri ty levels could be studied

i f speculat ions were made as to the d i s t r i b u t i o n of impuri ty le v e ls , e tc .

APPENOIX A

THE TANGENT METHOD

Consider data composed of several bands p lo t te d in the hyperbol ic

representat ion such as curve Y in Figure 2. Assume tha t band I

contr ibutes e s s e n t i a l l y a l l the curvature to Y in the low f i e l d region.

R eferr ing to Figure 2, the tangent l ines to Y are constructed at

P = p^ and £3 = Pq . V e r t ic a l l ines are drawn a t these same po ints .

We have:

where the f i r s t term is due to band I , and the la s t terms are due to

the other bands. From the f ig u re

D = Y + A

D - D (p ' ) D - D(B ) ________o_ _ ob - p; = p - pq

then

A(B) - 0 (po> ' m e ' b

Now 0(B ) - Y (P )o o

d(b;) = y(b;)29

CONSTRUCTION FOR PROOF OF TANGENT METHOD

\ Y\

\ \

31

Then

■ l po + Pj ' + 3 , + m(eia . ) g , , Bi i^ (p ♦ p, ) 2

(p0 - ° i

eo (e l + f5o) ( B l + ^

t ' =£>3

(po - pi> Bi

Bi (B1 + Bo) ( P l + pi>'

From Figure 2

t 1 = T*3 - 3''o o

t = -T3 - p * o o

a = v_ B1 + Bo

t ' p i +

orPi =

3 - a 3 ‘o Ko1 a - 1

This expression allows an easy estimate from the construct ion. Now;

T = - t ( 3 „ - 3 ' ) = V p0 - pi>, * 2

° <p, - P i > O j + e0 ) 2

ThusTtt (po - p.)

1 " (a - I )3

32

This expression may be used to est imate Bj- Note that t and t ' are

slopes wi th a lgebra ic signs, but T and T 1 are to be considered un-

di rected d istances•

APPENDIX B

A FORTRAN I t LEAST SQUARES CURVE FITTING PROGRAM FOR

BOTH LINEAR AND NON-LINEAR PARAMETERS

This program performs the i t e r a t i v e c a lc u la t io n s for the s o lu t io n

of Equation (18 )- Both in the case of l in e a r and n o n - l in e ar parameters,

i n i t i a l estimates fo r the parameters are required. I f a l l the parameters

are l in e a r these estimates are a r b i t r a r y , and one may merely read in

blank cards since Fortran I I reads blanks as zeros. The f i r s t i t e r a t i o n

w i l l give the so lu t io n . No set ru le can be given as to how accu ra te ly

no n- l in ear estimates must be estimated in order to insure convergence.

During the curve f i t t i n g reported in th is paper, no d i f f i c u l t y was

experienced, except when superf luous terms were included in the funct ion,

even though some estimates were o f f by 100 per cent.

A numerical fa c to r is m u l t i p l i e d times eachAp. before i t is

added to the corresponding parameter p . . This fac to r is c a l le d FAC

in the program. FAC is i n i t i a l l y set equal to one by the program.

Sense Switch 3 >s in ter roga ted once each i t e r a t i o n , and i f i t is found

on, a message is typed in d ic a t in g that FAC is to be entered v ia the

ty p e w r i te r . FAC may be set less than one i f the Ap, are of the same

order as the p . , in order to attempt to avoid divergence. This method

has been used very successfu l ly . Conversely, i f convergence is slow,

one may set FAC gre a te r than one to speed up the convergence.

The program is w r i t t e n in a general way so that in p r in c i p le one

may do curve f i t t i n g w i th any funct ion whatsoever. The funct ion and

i t s f i r s t p a r t i a l d e r iv a t iv e s w i th respect to the parameters must be

33

3*+

defined in a separate ly compiled subroutine named SUB. The usual

rules must be followed for l ink ing the subroutine to the main program.

In p a r t ic u la r a COMMON card must appear in the subroutine l i s t i n g the

same q u an t i t ie s in the same order as the COMMON card in the main program.

Examples w i l l be given. The data points are read in by the names

X(K) and Y(K). The subprogram must use K as th is p a r t ic u la r sub

s c r ip t a lso. The function must be named F, and the parameters must

be subscripted var iab les named P ( I )- Although the subscript need not

be I , i t must not be K. The same is true for the PF( I ) which represent

the p a r t i a l der iva t ives of F with respect to the indicated parameters.

Provision is made fo r three choices of weight ing. A code number IWT

indicates the choice. These are:

I WT = +1 W(K) = Y(K) " 2

I WT = 0 W(K) = 1

I WT = -1 W(K) = Y(K) * 1

Data is read in the fo l lowing form:

Card 1 col . 1-3 NP = number of parameters

c o l . k -6 ND = number of data points

col . 7-8 IWT = weighting code

Format (2I3> 12)

Cards 2 - (NP + 1)

co l . 1-20 P ( I ) in order

Format (E20.0)

35

Cards (NP + 2) - (NP + 1) + ND

col . 1-15 X (K)

c o l . 16-30 Y(K)

Format (2E15-8)

Provision has been made fo r reading in data by the subroutine

the f i r s t time i t is c a l le d by the main program. A number INSUB is

set equal to one by the main program previous to the f i r s t c a l l i n g of

SUB. Immediately a f t e r SUB returns to the main program INSUB is set

equal to two. Thus SUB may te s t INSUB to determine whether or not

to read data. I f data is to be read the cards should fo l low those

l i s te d above u l t i l i z i n g the Format s p e c i f ic a t io n s of the subroutine.

Use was made of th is prov is ion in the subroutine which t rea ts the H.

as constants and finds the best values for the l in e a r parameters. In

th is example the H. were read in by the subroutine.

During each i t e r a t i o n the variance or R.M.S- e r ro r is computed

I f Sense Switch 1 is on, the program produces a status punch a t the end

of each i t e r a t i o n g iv ing the present values of p . , A p . , VAR and the

i t e r a t i o n count. I f Sense Switch 2 is on, a complete p lo t back is

punched g iv ing the fo l low ing q u a n t i t ie s in tab le form:

as

ND£ t Y(K) - F(K)J2

X(K) - each value of X read in as data

Y(K) - each value o f Y read in as data

YBAR(K) - the values of F a t each X(K) using the present values of the parameters

36

DEL(K) - each value of YBAR(K) - Y(K)

PD{K) - the per cent d i f fe r e n c e = 100*DEL( K)/Y(K)

PDMEAN - the average of the PD( K) without regard to sign

PDRMS - the R. M- $■ per cent e r ro r

The value of VAR and the i t e r a t i o n count is a lso punched. The program

continues to run a f t e r status punch and p lo t back. Sense Switches

1 and 2 may be turned on and o f f wh i le the program is running. No

e x i t or term inat ion te s t is b u i l t in the program, and the operator

must judge when convergence is s a t i s f a c t o r y . A l l numerical tables in

th is paper were punched by the program with Sense Switches 1 and 2 on.

The t i t l e cards were inserted before p r in t in g on an IBM 407 Accounting

Machine e s p e c ia l ly wired to compress the data for standard s ize paper.

Another useful fe a tu re is incorporated using Sense Switch 4. With

i t on, the program reads the data (prepared as indicated above) and

branches d i r e c t l y to the p lo t back rout ine w i th no i t e r a t i v e computation.

This in s tan t p lo t back is q u i te useful in the adjustment of parameters

before attempting to i t e r a t e for a so lu t io n . At the completion of the

p lo t back the values of the parameters which were read in are punched,

and a branch is executed back to the i n i t i a l read statement in the

program. Switches may be set as desired, and the program is ready to

run. Sense Switch 4 should not be turned on during i t e r a t i v e compu

ta t io n and should be l e f t on during use of instant p lo t back.

A l i s t i n g of the main program fol lowed by several subroutines

is given next. The funct ion of each subroutine is indipated by i t s

t i t l e cards. The f i r s t card In each program is a contro l card which

t e l l s the compiler to generate f i f t e e n d i g i t f l o a t i n g po int a r i th m e t ic

and four d i g i t f ix e d point a r i th m e t ic .

37

* 1 5 0 4C C u R V E F I T T E S F O R B O T H L I N E A R AMD N O N - L I N E A WC B Y L E A S T S Q U A R E S M E T H O D D U E TO G A u b SC H J MA D

T Y P F 9 RMB C Q W M A T ( 1 ON F O H S T A T U S P l N C H )

T Y P F 9 79 7 F O R M A T ( ? O H S * ? ON F O R P L O T B A C K )

T Y P K M 69 6 F O R M A T | j a r i 5 W 3 ON T O A C C l P T F A C , OT M t R' * 1 F"

T Y P E 9 59 5 F O R M A T ( ? 8 H S W A ON F O R J N S T A n T P L O T H A G ' V / / )

P A U S E0 I M F N S I O N C F ( 1 0 , 1 1 ) » X ( I S O ) i Y ( I S O ) • a1 ( 1 0 0 ) , P (

H I M p N S I O N O P M 0 )r D M M O N X * Y , P i P F i F i N P B ’ i I M S j P

1 V I RK A P 1 Or*' , N P , N D , I ' * T ♦ ( P ( I ) , I = 1 , N P )1 '■> F Q R M A T < 7 I 7 . I 2 2 ( r 2 0 • D ) )

R ^ A O l C l , ( X < K ) , Y ( < ) , K = 1 , N O )1 0 1 F O R M A T < P E 1 5 . 8 5

R = N 0 1N S u b ■ i| F L G F N S E . S W I T C H 4 ) 7 , 2 9 9

2 9 9 I F ( I W T ) 7 S C t 3 0 1 , 3 0 2 3 0 ^ S 0 3N. SK = 1 , N O5 - - W ( < ) = 1 • / Y ( K )

S C T 0 A 0 0 n 1 0 0 3 5 1 K = 1 , NO15 1 W ( K ) = 1 .

G O T 0 4 0 0 3 0 2 H Q 3 * ^ 2 * = 1 , NO3 5 2 a, ( K ) = 1 . / < Y ( < ) * Y ( < > )A DC’ F A C = 1 .

1 C * 0N P 1 = N P + 1

I S O S V A R = C ,0 0 1 7 1 = 1 , N P C O 1 3 J = I , N P 1

1 3 C F t I , J ) = 0 ,n o 1 K = 1 | N D C A L L .5 U R I N S U B = 7

F P = Y ( K ) - F V A R * V A R + E R * E R S O 1 1 = 1 * N P T E M P = P F ( I ) * W ( K )C F ( I , N P 1 ) = C F ( I , N P 1 ) + E R * T f c M P

S O I J * I * N P 1 C F ( I * J ) = C F ( I i J l + P F ( J ) * T E M P

0 0 3 I = > 2 , N P L M N * I - 1 0 0 3 J = 1 » l m n

3 C F ( I • J ) = C F { J , I 1V A R a S Q R T F ( V A R / R )

PAWAMF.Tt RS

< F Y L S •'

AC = 1 • )

1 C ) , P F ( 1 0 )

38

C ENTFR G f l j S S AT 48

4 8 ' ' O f . C l i P . N P: ' 0 5 1 J = I . N P 1

- 1 CF ( 1 — 1 i J ) = f F ( [ —1 i J ) / CF ( 1 — 1 * 1“ 1 )D 0 6 Q L * I * N P 0 0 5 0 J * I * N P |

5 0 C F ( L » J ) s C F ( L « v / ) - C F ( L t i - l > * C < j - I . J )O P ( N P ) = C F ( N P * N P 1 ) / C F ( N P . M P |D052 I « NP

L = N P i - I

I J * L + IS O M s 0 .0 0 5 9 J * I J i N P

f q 5 J M * S U M + C F ( L « J ) * D P ( J >5 ? i ) P < L ) » C F ( L « N P 1 ) - S U F

C E X I T G A U b h A T o pI F t S ^ N F P S W I T C H 3 ) 1 0 * 1 1

I r. j V P p 9 4■)« F O R M A T < 2 8 H T Y P t . I N F A l t o o l N S A O K C I V A L )

A C C E P T 2 C 4 . F A CI I 0 0 1 2 1 * 1 * N P1 2 p ( I ) = P ( I ) + F A C * D P ( r )

1 C = 1 C + 1S W I T C H 1 ) 5 * 6

^ P U N C H 2 E 1 * ,X1 H I , H X A H P ( ! ) i 3 1 X 5 H D P I I ) / ( 1 4 * 5 X F 2 1 * 1 4 ,

T 5 X F 3 1 • 14 ) )P U N C H P r . P * V A P , I c

2 0 2 F O R M A T ( 3 X 4 H V A R * * E 1 0 * 4 * 5 X 1 r i H N O * O F ] T h . R A T I O N S = « I 3 / / / >6 I F { F N S F S W I T L H 2 ) 7 , 1 C 0 C

C P L O T B A C K7 P U N O H 2 0 33 0 3 F O R M A T ( 5 X 1 H X * 1 4 X | H Y * 1 2 X 4 H Y H A P * 1 ? X 3 H D t . L ♦ 1 3X?HPl) )

A b P O = 0 .R M S P D = O •0 0 8 2 = 1 ♦ N O C A L L M J 3 O F L = F - Y < 2 )P D * 1 0 0 . * D F L / Y < « )A B P D = A R P D + A B 5 F ( P C )p m s p o = r m s p o + p d * P D

8 P U N C . H 2 0 4 * X ( K ) , Y ( < ) * F • D E L . P D2 0 4 F O R M A T ( 5 ( E 1 1 . 4 * 4 X ) )

POMEa n =a b p d / r

W M S P D * S Q R T F ( R M S P D / P )I F ( S E N S E S W I T C H 4 ) 2 0 6 . 2 0 7

3 0 6 P U N C H 2 0 B i P O M E A N • R M S P DP O P F O R M A T ( 7 H D D M E A N = F 8 . 3 * 6 X 6 H P 0 H M S - F 8 . 3 / 3 2 )

P U N C H 2 0 9 3 0 9 F O R M A T ( 3 X 1 H I « 1 7 X 4 H P ( I ) )

P U N C H 3 0 0 * < I * P ( 1 ) . 1 = 1 * N P )c 0 r ' F O R M A T ( I 4 , 5 X E 2 1 • 1 4 )

P U N C H 2 1 0 2 1 C F O R M A T ( / / / )

39

G O T O 1 0 0 1P. 3 7 P U N C H 2 0 F , 1C . V 4 H . P D M t A N , * v y S P D2 0 F F 0 R M A T ( 1 9 H N O . OF-' I T E R A T I O N S * * I 3 * 4 X 4 H V A W = * K 1 0 . 4 *

* 1 x 7 h P D W F A N » i F H . ~ h I X 6 H P O W M S = * F 8 . 7 / / / )G O T O l 0 0 0 F NO

40

C S U B P R O G R A M F O P G U M O F P E A K F U N C T I O N SS I IB R O U T I N F SUr?D I M E N S I O N X ( l C 0 | * Y ( t 0 C ) i P ( 1 0 ) . P F ( | 0 ) COMMON X • Y«P ' P F <F . NP. K * INbUB F * 0 ,D O ? I = 2 . N P , ?T s P ( I ) + X I K ) * X ( < )P F < 1 —1 | = X I K ) / T T ? = P F ( I — 1 ) / T F = F + P ( I - 1 ) # P F ( I - I )

? P F ( I ) ■ —P ( 1 - 1 ) * T 2r f t u r nF N O

* 1C S U B P R O G R A M f o r E S T I M A T I O N O F L I N E A R0 P A R A M E T E R S I N S u m wF P E A K F U N C T I O N S

S U H R O U T I n E S u p

■> 1 M E N S I O N X ( 1 0 0 ) t Y ( 1 OC ) t P ( 1 0 ) * P F ( 1 0 ) i H 2 ( 4 ) C O M M O N X * Y i P « P F t F f N P « K * I N S U F X ? - X f < ) * X { < )rr — r~~ •I F ( I N S U P - 1 ) 1 « 1 »?

1 R E A D 1 C O ♦ < H ? ( I ) . I = 1 » N P )1 TO F O R M A T ( F ? 0 . 0 )2 0 0 3 1 = 1 . N P

P F ( I ) = X ( < ) / ( H ? • Y < 1 CO > * P < 1 0 > • P F ( l 0 ) C O M M O N X • Y . P * P F , F * N P * K • I N S u P F = C •D O ? I ~ 2 * N P . 2P F ( I * 1 ) = 1 • / ( P ( I ) + X ( < ) )T s P ( I - 1 ) # P F ( I - 1 )P F ( I ) * —T * P F ( I - 1 )F = F + T R E T U R N F NO

k l

s u b P w o r.P flM r o w a p o l y n o m i a ls u b p o u t i n f S U PD I M F N S I O N X ( 1 0 0 ) t v ( l 0 0 1 t P ( 1 0 ) i P c ( I 0 1 C O M M O N X * Y * P * P F * f * N P * K » I N S U B F eP ( 1 )P F ( 1 > = 1 •0 0 1 1 = Z * N P J = I - 1 / e x (< ) * * J p- = F + P ( I ) * 7

1 P F ( 1 ) * 7pFTuPN F NO

APPENDIX C

DATA TABLES AND CURVES

In ad d it io n to tables g iv ing the numerical resu lts fo r the band

parameters at various temperatures (Tables I - V I I I ) th is appendix

contains data tables in the form of the output from the curve f i t t i n g

computer program. These p lo t backs are explained In d e ta i l in

Appendix B where the computer program is discussed. These tables were

p rin ted on an IBM *+07 Accounting Machine d i r e c t ly from punched cards.

Because of the lack of specia l symbols enis w r i t te n as E, and a is

w r it te n as S on the t i t l e cards. The curve p lo ts are arranged adjacent

to the corresponding tables fo r easy comparison.

During the curve f i t t i n g i t was sometimes necessary to stop

computation on the computer and then s ta r t over a t a la te r time

using the improved values a lready obtained fo r the parameters as f i r s t

estim ates. R estart ing the program caused the i te r a t io n count to be

reset to one so tha t the f in a l i t e r a t io n count was not in d ic a t iv e of

the true number of i te ra t io n s used. For th is reason a l l i t e r a t io n

counts were set equal to one in the p lo t backs given in the tab les .

The actua l number of i te r a t io n s required depends upon the f i r s t

estimates of the parameters and averaged about ten In th is work.

*+2

TABLE NO. I

PARAMETERS FROM CURVE FITTING FOR

Ho.. USING A BANOS WITH W. = Y~2 11 k k

T°K

a n j x 10

cm

“ 17

gauss

a2n2 x 10

-3cm

17 H2 x 10 17

gauss cm-3H3 ai A * 10

17

gauss cm

RMS ° /o

gauss Error

4 .2 4.093

3-5 4.219

2.66 4.334

2.1 4 .372

23-42 2-739

23-21 3-136

23-30 3-346

23-77 3-477

127-2

152.4

173-3

I 86.9

1-715

1.124

7504

,5414

464.0

582.3

721.4

913-4

.2205

.2232

.2258

.2509

17902 1.92

18898 1.99

19290 1.96

23814 2.02

£~00

TABLE NO. I I


Ho.. USING 3 BANOS WITH W. = Y~2 11 k k

x 10 ^ Hj a2°2 X 10 ^ H2 a3n3 X 10 ^ H3 RMS °^ ° -R -7 -R°K cm J gauss cm gauss cm J gauss Error

4 .2 4.975 26*65 3.505 280.5 .2284 12396 4.43

3 .5 4.896 27.31 3-581 272.8 . 2288 12752 4 .50

2.66 4 .839 26.47 3.583 264.5 .2342 12208 4 .49

2.1 4 .786 26.51 3 .567 260.7 . 2436 12257 4.71

TABLE NO. I l l


a 10 USING 3 BANOS WITH W. = y" 2 12 k k

T°K

nj x 10

-3cm J

-1 7Hi

gauss

n2 x 10

cm ^

-17H2

gauss

x 10-17

cm-3H3

gauss

RMS ° /o

Error

k .2

3-5

2.66

2.1

■ 2.089

■ 2.096

■2.103

■2.102

2^.94

2k. 08

23.31

23.03

2.031

2.120

2.186

2 . 203

32^.0

323.0

320.2

320.8

.2080

. 1226

.0631

. 0M+5

625.6

719.9

963.7

1193

3-89

3-1*7

3-1*7

3-71

-p~V I

TABLE NO. IV


a. _ USING 2 BANOS WITH W. = Y~2 12 k k

n x 10"17T

°K cm

*4.2 -2.0*42

3-5 -2.0*45

2.66 - 2.050

2.1 - 2.056

Ht n2 x 10‘ 17

"3gauss cm

2*4.52 2.190

23.6*4 2.190

22.86 2-192

22.62 2.197

H2 RMS ° /o

gauss Error

358.7

352

3*4*4. 7

3*41.*4

*4. 17

3.81

*4.00

*4.32

•CON

TABLE NO. V


e1' USING 4 BANDS WITH W. = Y~212 k k

b jZ j x 10"31 ^ b2Z2 x 10 '31 H2 b ^ x 10"31 H3 b ^ x lO*31 RMS ° /o

0I -1 -3 -1 -3 -1 -3 *1 -3UK erg - cm gauss erg - cm gauss erg - cm gauss erg - cm gauss Error

4 .2 9.811 15.43 21.85 41.35 14.89 299.3 24-29 21090 1.04

3-5 19.39 17-90 23.56 56.95 12.65 573.7 33.60 23420 I .87

-e-- j

TABLE NO. VI


c'j'g USING 3 BANDS WITH Wk = y ” 2

T°K

b jZ j x 10"31

"1 ” 3erg - cm gauss

b Z x 10"312 2

-1 -3erg - cm

H2

gauss

b3Z3 x 10-31

“ 1 -3erg - cm

H3

gauss

4 .2

3 -5

2.66

2.1

22.98

26.20

23*53

26.17

21.77

20.36

19.67

21.51

21.88

22.29

21.52

22.59

148.1

100.8

97.66

113.9

13.23

13.27

21-27

18.50

9416

2306

2917

2632

RMS ° /o

Error

3.04

3-06

2.27

2.76

■p-00

TABLE NO. V I I


He'j'j USING 3 BANOS WITH « k = Y* 2

T°K

C j Z j x 10 3 ^

-1 - 3e rg - c m

H1

gauss

c Z x 10" 312 2

-1 - 3e rg - c m

H2

gauss

c Z x 10 33 3

-1 -3e rg - cm

H3

gauss

PORMS RMS

E rro r

4 .2 14-33 3 9 -0 5 - 11.11 4 2 5 .2 -2 1 .3 0 16212 6 .8 5 5 -9 5

3 -5 16 .3 4 3 5 .8 7 - I O .8 7 4 5 9 .0 - 2 6 .6 7 17209 5 .0 2 4 .5 3

2 .6 6 15.40 3 3 *8 4 -1 1 *5 9 3 9 2 .0 -2 8 .6 4 14693 9 -0 0 6 .0 2

2 .1 I 6 .7 7 35*99 -1 7 *2 4 3 5 3 -9 -3 1 *7 3 16191 4 .4 7 3*51

-p-VD

TABLE NO. VII I


He'j'j USING 3 BANDS WITH WR = 1

c.Z. x 10’ 31 c Z x 10"31 H* c Z x 10 31 PDRMS RMST 1 1 1 2 2 2 3 3 3

°K - 1 - 3erg -c m gauss - 1 - 3erg -c m gauss - 1 "3erg - cm gauss Error

1+. 2

3 - 5

2.66

2 . 1

13-33

15-82

1M 2

\6 .k 2

3 6 . 7 1*

3 5 .6 1

3 3 .5 2

3 5 -7 2

10.52

IO .6 5

■11.1+7

■17.09

W l . l

1+8 1 .2

1+6 0 .6

363.2

■2 1 .6 7

-26.76

■29.1+5

■3 2 .2 0

17271 10 . 1+ 4 .1 9

17V +2 6 .5 5 1+.09

16421 18 .8 1+ .61+

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8 A N D M O D F L T a 2 • 6 6 D F G W E F S K E L V I N W ( < ) = l . / ( Y ( K ) * Y ( K ) )

I P ( 1 ) D P ( I >1 1 . 6 1 7 6 0 9 0 9 8 2 0 9 1 7E + 08 3 . 3 8 9 7 9 9 0 2 9 6 9 9 9 4 E +042 5 . 4 4 2 1 4 7 3 4 0 5 9 0 2 2 E + O 2 9 . 1 3 7 7 3 7 1 4 3 7 8 4 8 1 E - 0 23 9 . 2 7 6 1 5 9 9 3 8 2 8 9 2 4 E + O 0 4 * 4 1 6 4 6 5 2 0 2 9 2 P 3 2 E + 054 3 . 0 0 2 J 5 8 9 9 0 0 9 2 6 5 E +04 2 • 0 3 3 9 3 5 7 6 1 1 7 7 2 0 E + 0 13 8 . 6 6 1 4 0 6 1 4 2 7 0 3 5 B E + 0 8 - 3 . 3 6 6 3 8 9 6 3 2 4 1 6 3 7 E +086 5 . 2 0 3 5 7 9 1 6 8 1 8 18 3 E + 0 5 6 • 6 0 2 8 4 3 8 8 0 1 5 5 3 2 E +027 6 . 9 6 9 3 1 1 2 9 1 3 5 3 6 0 E + 0 9 1 • 0 6 6 2 7 0 3 9 7 7 9 6 9 8 E + 0 60 3 . 7 2 0 9 8 8 1 O 8 5 3 9 9 9 E + O 0 1 * 7 8 3 8 8 0 0 2 1 1 1 0 8 1 E + 0 5V A 9 = . 6 1 1 4 E + 0 5 N O . O F 1 7 E 9 A T I O N S o 1

X Y y b a n DEL PD4 I 0 O E - 0 0 1 15 6 3 E + 0 6 1 10 3 4 E+06 — 6 2862E + o *■ - 4 5 71 7E —0081 OOF-O 0 2 13 8 3 E + 0 6 2 0 8 6 7 E + 0 6 - 5 1 508E + 04 - 2 4OB0 E- 000 2 2 0 E + C 1 2 8 7 1 0E+O6 2 8 8 0 6 E + 0 6 8 706 OE + 03 3 0 3 1 5 E - 013 6 2 0 E + 0 1 3 3 8 2 9 E + 0 6 3 4 6 0 3 E + 0 6 7 7469E + 04 2 2 9 0 6 E - 0070 30E + C 1 3 7 6 0 2 E + C 6 3 8 5 1 9E+ 06 9 1 7835 + 04 2 4 4 0 9 E - 000 4 4 0 E + 0 1 3 9 5 9 2 E + 0 6 4 0 9 4 2 E + 0 6 1 3504E + 05 3 4 1 0 7 E - 003 8 5 0 E + 0 1 4 10 6 9 E + C 6 4 2 2 96E+C6 1 2 2 6 I E + 08 2 9 8 5 5 E - 007 2 5 0 E + 0 I 4 18 2 H E + 0 6 4 2 9 3 0 E + 0 6 1 1 025E + 05 2 6 3 6 0 E - 004 0 7 0 E + 0 1 4 2 9 9 6 E + 0 6 4 3 0 3 1 E+06 3 5502E + 03 0 2 5 7 2 E - 020 8 8 0 E + 0 1 4 3 2 1 0 E + 0 6 4 2 5 0 0 E + 0 6 - 7 0 9 5 7 5 + 04 - 1 6 4 2 1 E - 007 6 9 0 E + 0 1 4 2 Q 7 3 E + 0 6 4 18 6 7 E + 0 6 - 1 0 0 5 9 5 + 05 - 2 3 4 6 4 E - 004 3 1OE+C1 4 2 5 7 2 E + 0 6 4 1 3 1 9E +06 -1 252 1 E + 05 - 2 9 4 1 2 E —0013 2 C E + 0 1 4 2 1 2 6 E + 0 6 4 0 9 0 7 E + 0 6 -1 2 1 B7E + 05 - 2 8 9 3 0 E - 008 1 3 0 E + 0 1 4 17 6 3 E + 0 6 4 Q62 0E+0 6 -1 I 427E + 05 - 2 7 3 6 3 E - 0017 6 0 E + 0 1 4 1207E+O6 4 0 3 1 6 F + 0 6 -R 908 35 + 04 - 2 1 6 1 8 E - 005 3 9 0 F + 0 1 4 0 4 4 5 E + 0 6 4 0 1 9 7 F + 0 6 - 2 4 7 3 3 E + 04 - 6 115 3 E - 010 9 0 1 E+02 3 9 8 9 7 E + 0 6 4 0 1 1 4 5 + 0 6 2 1 74 IE + 04 5 4 4 9 4 E —012 2 6 4 E + 0 2 3 9 3 6 7 E+06 3 9 9 8 2 E + 0 6 6 1 504E + 04 1 5 6 2 3 E - 003 6 2 7 E + 0 2 3 0 9 7 3 E + O 6 3 9 7 6 2 5 + 0 6 7 8950E + 04 2 0 2 5 7 E - 004 9 8 9 E + 0 2 3 8 5 2 1 E+06 3 9 4 4 6 E + 0 6 9 251 IE + 04 2 40 15 E - 006 3 5 2 E + 0 2 3 8 ] 0 0 E + 0 6 3 9 0 3 8 5 + 0 6 9 3 0 0 3 5 + 04 2 4 6 2 0E —0077 15 E + 0 2 3 7 7 3 2 E + 0 6 3 8 5 5 1 E+06 B 1 9 5 4 5 + 04 2 172 OE —000 0 5 0 E + 02 3 7 1 5 2 E + 0 6 3 7 5 6 1 E +0 6 4 29 30 E + O 1 15 5 5 E - 004 5 0 0 E + 0 2 3 5 6 2 3 E + 0 6 3 54 0 4 E + O6 -1 3896E + 04 - 3 9011 E - 019 7 0 0 E + 0 2 3 3 4 7 1 E+ 0 6 3 2 9 9 7 E + 0 6 - 4 73 15E + 04 - 1 4 1 3 6 E - 008 3 5 0 E + 0 2 3 0 2 4 2 E + C 6 2 9 3 3 8 E + 0 6 - 9 0326E + 04 - 2 9 8 6 7 E - 007 1 0 0 E + 0 2 2 7 3 3 6 E + 0 6 2 6 3 5 7 E + 0 6 - 9 7 8 2 4 5 + 04 - 3 5 7 8 6 E - 005 B 0 0 E + 0 2 2 4 4 4 5 E + 0 6 2 3 9 7 1 E+ 0 6 - 4 73 75E + 04 - 1 9 3 8 0 E - 004 6 0 0 E + 0 2 2 2 2 2 8 E + 0 6 2 1 9 8 4 5 + 0 6 - 2 4 34 8E + 04 -1 0 9 5 4 E —003 2 6 0 E + 0 2 2 0 2 8 5 E + 0 6 2 0 3 3 6 E + 0 6 5 1 856E + 03 2 5 5 6 4 E —012 1 0 0 E + 0 2 1 8 7 1 8 E + 0 6 1 8 8 9 2 E + 0 6 1 7466E + 04 9 3 3 1 2 E - 010 8 0 0 E + 0 2 1 7 3 9 7 E + 0 6 1 7 6 5 8 E + 0 6 2 50 1 5E + 04 1 4 8 3 9 E - 009 5 0 0 E + 0 2 1 6 1 7 8 E + 0 6 1 6 5 6 5 E + 0 6 3 87 0 0 E + 04 2 3921 E - 00

61+

65

1 2 5 8 0 E + 0 3 1 36 36 E+0 6 1 • 3 9 3 7E + 06 3 • 0 1B 8 E + 0 4 2 2 1 3 8 5 —001 4 3 1OE + 0 3 1 2 3 8 6 E + 0 6 1 • 2 6 1 1 E+ 0 6 2 • 2 3 5 7 E + 0 4 1 B04 7 E - 0 01 6 0 3 0 E + 0 3 1 1 3 7 1 E+ 0 6 1 • 15 2 0 E + 06 1 • 4 9 32 E+04 1 3 1 3 2 E - 0 01 7 6 4 0 E + 0 3 1 0 6 2 4 E + 0 6 1 • 0 6 5 9 F +06 3 • 8 4 8 1 E +03 3 3 3 9 7 5 - 0 11 9 3 2 3 E + 0 3 9 9 4 2 0 E + 0 5 9 • U9 3 0 E + 0"- - 4 • A98 7E +03 - 4 9 2 4 2 E - 0 12 0 9 7 0 E + 0 3 9 3 6 3 1 E + 0 5 9 • 2 4 6 3E + 0 - 1 • 1 6 7 4 E + 0 4 - 1 2 4 6 9 E - 0 02 2 5 9 0 E + 0 3 3 8 9 3 6 E + 0 5 a • 6 9 5 0 E + 0 8 - 1 • 9 8 5 4 E + 04 - 2 2 3 2 3 5 - 0 02 4 1 8 0 E + C 1 8 4 7 7 E E + 0 5 8 . 2 2 0 8 5 +0' ' - 2 • 5 6 6 9 E + 0 4 - 3 0 2 7 9 5 - 0 02 8 2 7 0 E + C 3 7 18 9 0 E + 08 7 • 2 3 6 0 F +08 4 • 70 345 +0 7 6 6 4 2 5 5 - 0 13 5 350E + C3 0 1C 4 0 E + 0 5 6 • 0 6 8 1 E + 0 5 - . 3 • 580 1 E+0 3 _ 8 6 r'2E - 014 2 4 3 0 E + 0 3 8 3 5 0 0 E + 0 8 5 • 3 0 5 8 F + 0 8 - 4 • 4 1 6 0 E + 0 3 - 8 2 5 4 2 E - 0 16 0 4 0 0 E + 0 3 4 7 5 9 0 E + 0 5 4 • 72 69E + 0 I- - 3 • 2 0 2 7 E + 0 3 - 6 7 2 9 8 5 - 0 16 01 30 E + 0 3 4 2 9 3 0 E + 0 5 4 . 2 6 6 9 E + O b - 3 • 6 0 9 3 E + 0 3 - 8 4 0 7 4 E - 0 17 0 7 5 0 E + 0 3 3 8 4 8 0 E + 0 5 3 • 9 1 8 6 E + 0 5 7 • 0 6 0 4 E + 0 3 1 8 3 4 8 E - 0 08 0 OOOE + 03 3 7 2 8 0 E + 0 5 3 • 71 36F + 0 - - 1 • 4 3 9 7 E + 0 3 - 3 8 6 1 9 E - 0 11 0 0 00 E + 04 3 4 2 0 0 E + 0 5 3 • 4 2 6 9 F + 0 9 6 • 9 9 4 4 E + 0 2 2 0 4 6 1 E - 0 11 2C0OE+04 3 2 2 8 0 E + 0 5 3 . 2 4 7 2 5 + 0^ I * 9 2 9 75 + 0 9 5 9 7 8 0 E - 0 11 4 0 0GE+C4 3 C 9 4 0 E + 0 5 3 • 1 125F + 05 1 • 8^4 75 +0 7 6 994 6 E— 0 11 70 00 E+C4 ■p> 9 6 7 0 E+ 0 5 2 . 9 4 1 4 E + 06 - 2 • ‘ - 8 2 6 E + 0 3 - 8 60 3 3 E - 0 1NC • O F I T E R A T I O N S * 1 V A R = • 6 1 1 4 E + 0 5 P O M i - AN" 1 . 6 ^ 7 DORMS- 1 . 9 8 8

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3 BAND MOOEL

EXPERIMENTAL POtNTS

LEAST SQUARES CURVE

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SEPARATE BANDS

BAND I

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x n X a n .1. B B D <t B fy r i B <0 CO r i X vC ~1 CD B X y> r i B y> B O' O' ■a 11*.s» • 0 c in B Oo o O O o o+ 1 + 1 4 4 CD :0 r i n ip x <3 — o r i n ■» in IT <*■ D 1 V. r i n X — i-fB r yj ir yj in n a X X CO mX X X 111 id . 1 1 1 t 1 1 t i 1 1 1 1 i i jO (T x B C\J X—« ® to n B CMin CO cvj a O I X X a X X Li t li Ui tv ■y LI 1 n L" U, id Ul IT ■a V m tn IT IT. n n m a a a a XB O' o r> (V x C o o o o o o C o o o o o O o o O o O o o o O 0 0 0 O 0 0 0 0 0 0 0 0

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LD r i O' Cj o _j o O' B r i o CO o X 0 B If y) B B o B B X yj X B m X 4 O X n O O' yj — 0 0•c D a m a X id "1 !P (V n X B X X yj «■ X <v <t O O <T ■S) «o O » o> o 0 0 O X r i in 0 X — ® XX B n r i x n r i X r i a o C C ri X vf) r i X yj X B — cc o B o 41 -V- c l>4D 30 X to a a r i 0 O' X Xn O' O B a a o CVJ O' r i r, X ■ rv B B B rr< o n B CD » 0 X O' X- X n\ O yj m X 0 X X v 4 X 0

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t— + 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4d Id X Id Id Id id Iri U; L; Ul Id U1 id Ul U‘ UJ U1 Id d Id hj UJ U u u* u id Ul Id UJ X X X X Xo rr O' B X 0 B 0 ■0 X yj X o X U3 o r i O IV X X iT n 0 r i O' U) O' 0 O' B B n nid < a ® X O X n •0 01 B B o o X B (D CO B O' o B 0 tvi — X in X U) X B X O' X nB EC X r i X O X B O' I B X o X X O' X I CD o D B O' X O' 0 a X a O' in a IT

30 t\l 0 a 0 X — > o o CD 03 r i <0 b X r i o o X CD B B B B IS. D n X D X X O' B X n Xo o O o o o+ 4 4 + + + U r i X X x m ■y <1 <s- 4 <3 X n X X X X X X X X X X X X — — —I — rix X ±1 X x X oO O' —i c- d r ix a a b B x •x r i r i a in B o <0 <0 vO <0 VJ yj iC yj yj x d y) D yi yj V3 <0 yj U) yj y) yj yj >0 U) yj U) yj <0 y> yj U) 0 X Xa X o r i 03 B z c o o o c c o c c o o o o o o o o o c c c 0 0 0 0 c 0 0 0 0 c 0 0 0 c

a ® o X CVI m ® + 4 4 + 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4CO n O' o tn a UJ id Id ■jj id UJ id UJ Ul Ul kJ id U' .d id id Ul UJ U id Ul u u id u Id UJ UJ UJ UJ X X X X X_ a B 'O 5 "1 " i D O' X 'V 0 IE vD 0 X ■ij y) n B •r B B n >- O X X X X X in <D tn CD B 33 X yjo in O' in o r i > <! CO —• ty o o O tv O' r i B B X x O <t O' sO B X o n in X B 4 X a X <D r i O' B 0 Xo in in B » X in x B X yj X o CO O X X r r i B X < ® FT', 0 IT r i B — y) 4 X X a X X B X r i X yjo X in >0 n <t >0 * 4 CO n B o X n tv X — r i o O' b X 03 X B B m X 0 B a X O ® B 0 in na IT in o B r . o • • • • • • • • * * • • « • • • « • • • • • • • • 1 • • • • • • • • •0 B « ® X o 4 — tv X ■n rr' ID <3 <* <r y If j f <r ■ri X XI |V\ X rXi X X X X X X r i — — —a o O' B O' Idc o ir O' IT a X• • • # • • <0V B •r* lO ~ r i o O r i r i r i r i r i r i r i r i r i r i X X X X X X X X X X X X X X X X X X X

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O o o o O o O o o o o o O o o O O Oru ry — o o o o o o o o o o o o o o o

T * 2.1* K

4 BAND MOOEL

4 - EXPERiMENTAL POINTS

LEAST SQUARES CURVE

W(K) * YOO**

SEPARATE BANDS^BAND I

BAND H

BAND 1ZBAND HI

IOOOOO1 0 0 0 01000100H (GAUSS)

Figure 9

4

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H S l 1 V S H

B A N D M O D E L T « 2 . 1 D E G R E E S K E L V I N W ( < > * 1 . / < Y ( K ) # V ( K ) )

I P ( I ) OP t I )1 1 • 6 6 3 I 9 3 3 6 3 6 9 72 0E + 08 1 • 1 5 79 3 0 2 6 6 7 6 6 5 0 E + 0 32 S . 6 5 2 1 B1 9 7 8 1 3 5 6 4 E + 0 2 4 . 0 7 8 5 5 7 8 4 2 4 7 5 7 1 E - 0 33 1 • 0 3 9 6 1 9 0 0 B 8 6 0 4 3 E + 0 9 1 . 8 3 1 9 2 5 4 4 B 3 6 9 8 H E + 0 44 3 * 4 9 3 0 1 7 5 0 9 4 6 9 4 1E+04 9 , 3 8 4 6 3 8 2 9 6 8 3 5 1 O E - O 15 7 . 9 1 3 1 5 2 3 B 1 3 3 0 2 5 E + 0 B - 1 * 0 3 0 7 6 0 2 2 3 1 0 2 5 7 E + 0 46 8* 3 4 3 0 5 4 8 6 7 6 2 4 03E + 05 6 . 3 3 4 64 0 9 9 8 6 6 9 58E +017 9 . 5 6 1 6 6 3 2 6 7 1 0 0 7 I E + 0 9 2 . 6 4 3 7 0 3 8 5 4 3 5 2 0 9 E + 059 5 . 6 7 0 9 9 8 4 3 3 9 1 7 0 9 E + 0 8 2 . 4 2 8 5 5 2 3 6 9 1 4050E +04VAR = • 6 4 7 9 E + 0 5 NO. OE ITERATIONS* 1

X Y Y R A R DEL PD41 00 E- 0 0 1 15 2 9 E + 0 6 1 0 8 7 9 E + 0 6 - 6 4 9 6 2 E + 0 4 6 3 4 6 E -e i o o e - oo 2 1 1 3 8 E + 0 6 2 0 6 0 9 E + 0 6 - 8 2 8 7 3E +04 - 2 5 0 1 3 E -0 2 2 0 E + 0 1 2 8 4 5 0 E + 0 6 2 8 6 1 3E + 0 6 6 3 8 5 8 6 + 0 3 3 2 4 4 6 E -3 6 3 0 E + 0 1 3 3 6 3 0 E + 0 6 3 4 352E + 06 r 2 2 4 5 6 + 0 4 2 14 8 2 E -7 0 3 0 E + 0 1 3 7 1 8 0 E + 0 6 8 3 0 9 E + 0 6 1 1 2 9 6 6 + 0 5 3 0 3 8 2 6 -0 4 4 0 E + 0 1 3 9 4 1 OE + 06 4 0 7 9 1 E+0 6 1 3 8 1 9 E + 0 5 3 5 0 6 6 6 -3 8 5 0 E + 0 I 4 0 9 0 0 E + 0 6 4 2 1 9 3 E + 0 6 1 2 9 3 3 E + 0 5 3 16 2 2 E -7 2 5 0 E + 0 I 4 16 4 0 E + 0 6 4 2 8 6 0 E + 0 6 1 2 2 0 0 E + 0 5 2 9 2 9 6 6 -0 6 6 0 E + 0 1 4 2 4 6 0 E + 0 6 4 3 0 5 8 E + 0 6 ri, 9 8 7 5 E + 0 4 1 4 1 0 1 E -4 0 7 0 E + 0 1 4 2 8 3 0 E + 0 6 4 2 9 7 3 6 + 0 6 1 4 3 0 1 6 + 0 4 3 3 3 9 2 6 -0 8 8 0 E + 0 1 4 3 0 1 0E +06 4 2 4 0 7 E + 0 6 - 6 C245E +04 - 1 4 0 0 7 E -7 6 9 0 E + 0 1 4 26 30 E+ 06 4 1 7 1 8 6 + 0 6 - 9 1 1 5 6 6 + 0 4 - 2 138 36 —4 5 1 0 E + 0 1 4 2 3 0 0 E + 0 6 4 1 1 1 1 6 + 0 6 - 1 1 8 8 5 6 + 0 5 - 2 8 0 9 8 E -1 3 2 0 E + 0 1 4 18 8 0 E + 0 6 4 0 6 4 4 E + 0 6 - 1 2 3 5 0 E + 0 5 - 2 9 4 6 9 E -8 1 30E+01 4 14 9 0 E + 0 6 4 0 3 1 5 E + 0 6 - 1 17 4 9 E + 0 5 - 2 8 3 1 9 E -4 9 4 0 E + 0 1 4 14 4 0 E + 0 6 4 0 0 9 7 6 + 0 6 - 1 3422F. + 05 - 3 2 3 9 0 E -1 7 6 C E + 0 1 4 10 4 0 E + 0 6 3 9 9 6 4 E + 0 S - 1 075 3E+05 - 2 62 0 1 E -5 3 9 0 E + 0 1 4 0 2 5 0 E + 0 6 3 9 8 5 0 E + 0 6 9 9 6 OE + 04 - 9 9 2 B 0 E -0 9 0 1 E+02 3 9 6 8 0 E + 0 6 3 9 8 1 0 6 + 0 6 1 3085E +04 3 2 9 7 7 E -2 2 6 4 E + 0 2 3 92 4CE+06 3 9 7 4 8 E + 0 6 5 0 8 2 9 E + 0 4 1 2 9 5 3 E -3 627E+C2 3 8 8 4 0 E + 0 6 3 9 6 1 2E+C6 7 7 2 1 2 6 + 0 4 1 9 8 7 9 E -4 9 8 9 E + 0 2 3 8 3 7 0 5 + 0 6 3 9 3 8 2 E + 0 6 1 0 1 2 4 E + 0 5 2 6 3 8 7 E —6 3 52 E + 03 3 794 0 E + 06 3 9 0 5 7 6 + 0 6 1 1 17 j e + 05 2 9 4 4 5 6 -7715E+C2 -i 7 5 60 E + 06 3 8644 E+G6 1 0 8 4 4 E + 0 5 2 0 8 7 2 E -1 '"'606 + 02 3 6 8 0 0 E + 0 6 3 73 49 E+C6 5 4 9 9 9 E + 0 4 1 4 9 4 5 E -450CE+C2 3 5 4 8 0 6 + 0 6 3 5 7 5 8 E + 0 5 2 7 8 0 0 E + 0 4 7 B 3 5 4 E -9 7 0C E+02 3 3 2 6 0 E + 0 6 3 3 2 2 6 E + 0 6 * t! 3 5 9 6 E + 0 3 - 7 0 9 4 7 E -8 3 5 0 E + 0 2 3 0 0 5 0 E + 0 6 2 9 3 8 2 6 + 0 6 - 6 6 7 0 8 E + 0 4 - 2 2 1 9 9 E -7 1 0 0 E + 02 2 7 0 8 0 E + 0 6 2 6 2 0 0 E + 0 6 -a 7 9 7 8 E + 0 4 - 3 2 4 8 8 E —5760E+C2 2 4 2 2 0 E + 0 6 2 3 6 6 5 E + 0 6 - 5 3 4 6 3 E + 0 4 - 2 2 0 7 4 E -9 2 6 0 E + 0 2 2 3 2 6 0 E + 0 6 2 2 8 1 3E +06 - 4 4 6 4 4 E + 0 4 -1 9 1 9 3 E -4 6 0 0 E + C 2 2 196CE+06 2 1614 E + 0 6 - 3 4 5 8 9 E + 0 4 -1 5 7 5 1 E-3 2 6 0 E + 0 2 2 00 7CE+06 1 9 9 4 3 6 + 0 6 -1 2 6 8 9 E + 0 4 - 6 3 2 2 6 E -

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i • 2 4 8 0 F + 0 3 - 2 . ‘SAACE + O 1 - 2 4 4 6 7 E + 0 1 9 a 7 2 4 8 F — 0 1 - 3 0 2 2 B F -l • 4 2 1 CE+03 1 . 8 C 1CE+ C1 1 0 6 8 0 E + 01 6 • 7 0 9 O F - 0 1 3 7 2 3 2 E -l • 5 9 3 0 E + 0 3 4 a 6270E+F! 1 4 2 3 0 3 F +01 - 3 a9 6 6 7 E - 0 0 - B ^ 7 3 0 6 -i • 7 6 4 0 E + 0 3 3 a B4 50 E + 01 3 3 3 0 1 E + 0 1 - 1a 1 4 0 0 E - 0 0 - b 3 8 3 0 E -i • 9 3 2 0 F + 0 3 6 . 5 7 8 0 E + 0 l 6 2 2 1 1 E + 0 l - 3 • 3 6 0 BE — 00 “ 5 4 2 5 4 E -2 a 0 9 7 0 E + C 3 7 . 0 0 5 0 E + 0 1 6 3 6 P 3 E + 0 1 - 4 • 4 2 6 0 E - O O - 6 3 1 9 b E -2 a 2 8 9 0 E + 0 3 7 . C 6 7 0 E + 0 1 6 6 9 7 7 E + 0 1 - 3 a 6 9 2 78 — 0 0 - 3 22 8 3 E —£ a 4 1 8 0 F + 0 3 7 . 1 3 6 0 E + 01 6 71 0 3E + 0 1 - 4 a 4 7 6 4 h - 0 0 - 6 28 3 7E-2 a 3 7 3 0 E + 0 3 7 . 0 9 6 0 E + 0 1 6 6 8 0 1 E + 0 1 - 4 a 4 3 0 3 E - 0 D - 6 2 0 2 HE-j a 1 8 1 0E + 03 6 a C 82 0 E +01 6 1 1 3 4 E + 0 1 i a 34 76 F — 01 3 ^ 0 4 F t “4 • 2 4 3 0 E + 0 3 3 a 0 33 0 E +0 1 3 0 3 0 1 E +01 - 2 a 8 6 9 1E — 02 6 7 U 0 E —4 • 9 8 1 OE + C 3 4 a 3 9 0 0 E + 01 4 4 6 5 2 p-‘ + 0 1 7 . 3 2 B 9 F - 0 I 1 7 1 9 0 E -6 a 0 1 3 0 E + 0 3 3 . 6 9 9 0 E + P 1 3 7 778E + C 1 7a 8 6 6 9 E — C 1 2 132 1 E -7 . 0 7 5 O E + 0 3 3 a 161 OF + O1 3 ? 6 12F +0 I 1 a 0 0P7F - 0 0 3 1 7 P ] E -B a4 9 5 0 E + C 3 2 . 6 6 3 0 E + 0 1 2 7 4 9 7 8 + 0 1 8 a 1 7? OF - 0 1 ' 6 2 9 E -1 • C 0 6 5 E + 0 4 2 . 2 4 4 0 E + 0 1 2 3 193E + 0 1 9 a 3 10 O F —91 4 2 4 6 BE -1 • 19 3 0 E + 0 4 ] . H H 9 0 E + 0 1 1 9 8 4 7E +0 1 V . 3 7 3 4 8 - C 1 0 6 9 C E -1 a 4 85 5 E+ C4 1 a 3 8 7 0 E + 0 1 1 6 14 2 r + 0 t r . 7 2 1 9 F - 0 1 4 9 3 9 " E -1 a 7 7 6 0 E + 0 4 1 a 30 4 0 E + 0 1 1 34 3 18 + 0 1 .'7 3 9 4 8 - 0 1 3 0 2 1 E -NC . OP- i r p . H A r i O N S * 1 VA 9 = • 1C40E +04 POP*!; A Nl = 8 . 3 2 3 Pi; w.v S = 2 . 0

oo0000000000oc0000C 10200000000oo0000oc9 4

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-CM

f1T = 4 2° K 2 BAND MOOEL

"’C'BAM ) H• EXPERIMENTAL POINTS LEAST SQUARES CURVE

W(K)»Y(K) '2

SEPARATE BANDS

BAND E

,Vj~BAM) I

BAND IH4

125 X lO' GAUSS CM'3) 1.25 X lO13H6

i l lI 10 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0

H (GAUSS) —J00Figure 12

z

36I1122

.1J445667091111J12J345677a9lll

T AHLP NO. XVI I 1

SI 2 VS H

BAND MODEL T = 4 . 2 DEGREES K L L V l N * ( K ) * 1 , / ( V ( K | » Y | K ) )

I P ( I ) D P I I )1 - 3 . 2 6 7 9 8 9 6 2 2 1 1 102E + 06 - 6 • 2 2 I I 1C 2 6 4 3 9 0 C 9 E —012 6 , 0 1 4 0 9 0 9 4 0 7 4 4 4 1 E + 0 2 1 • 7 ^ 744 4 1 2 8 3 7 3 1 7 E - 0 >3 3 . 6 0 4 0 7 6 1 7 6 3 2 9 R ^ E + 0 6 3 . 5 2 2 9 8 5 0 9 0 2 4 3 4 2 5 - 0 24 1 • 2 0 7 0 6 1 4.374 1 3 8 65 + 07 - 1 • 2 5 8 6 1 4 2 4 7 2 8 4 7 2 5 - 0 3V A 9 = . 9 6 9 3 E + 03 N O . OF I T E R A T I O N S * 1

X Y YB AR DLL PD4 I 0 0 E - C 0 - 2 0 0 0 0 E + O 4 - 1 8 0 8 6 E + 0 4 1 9 1 4 7E + 03 - 9 57 3 70 - 008 1 OOE-OO - 3 5 2 B 0 E + 0 4 - 3 4 1 7 0 E + 0 4 1 1Q9BE +03 - 3 1 4 C6E - 000 2 2 0 E + 0 1 - 4 7 1 7 0 t + 0 4 - 4 f 0 30E + 04 1 3 1 3 1 E + 0 2 - 2 70 3 7E- 0 13 6 3 O E + 0 1 - 6 6 2 9 0 E + 0 4 — 6 6 2 1 4 E + 0 4 - 9 2 4 1 6 E + 0 2 1 6 7 14 E- 0070 3 0 6 + 0 1 - 6 C 13OE+04 - 6 19 6 9 E + 0 4 -1 03961: + 0 3 7 7 89 25 - 000 4 4 O E + C 1 - 6 2 6 6 0 E + 0 4 — 6 49B4E + 04 _ -j 324 41"+ 0 3 3 7 0 9 6 E- 003 8 5 0 E + C 1 - 6 8 6 1 OE+94 - 6 B 9 5 7 E + 0 4 - 2 3 4 7 C E + 0 3 1 68 9 7E - 007 2 5 C E + C 1 - 6 3 1 QOE+04 - 6 n 6 2 3 E + 0 4 - 2 4 2 3 2 E +0 3 “3 0 4 0 2 E- 000 6 6 9 E + 0 1 - 6 1470E +04 - 6 4 1 7 7E + 04 - 2 7 0 3 0 5 + 0 3 4 39 7 3E - 004 O 7 C E + 0 1 - 6 0 8 4 0 E + 9 4 — 6 2 2 6 4 E + 0 4 — 2 02 4 2E + 03 i 3 6 0 3L - 000 8 6 0 E + C 1 - 6 6 1 5 0 E + 0 4 -0 7 6 0 6 E +04 - 1 B 3 6 9 E + 0 3 3 73 72E - 007 6 9 0 E + 0 1 - 6 2 4 5 0 E + 0 4 - 6 2 9 1 0 E + 04 - 4 6 8 9 5 E + 02 8 9 4 0 9 E - 014 6 1 0 E + 0 1 - 4 0 4 5 0 5 + 0 4 - 4 04O9E+O4 4 0 9 4 4 E +01 - 8 32 7 05 - 021 3 2 0 E + 0 I - 4 4 8 6 OE + 04 - 4 4 32 7E + 04 T67QC +02 - 1 1 9 6 4 E- 00H13OF+01 - 4 19 0 C E + 0 4 - 4 0 6 7 4 5 + 0 4 1 2 2 5 3 F +03 - 2 9 2 4 8 E - 004 94 0 E + 0 1 - 3 8 7 1 0E +04 — 3 M .1«F + 04 1 2 7 5 1 E + 0 3 - 3 2 9 39E - CO1 B 6 0 E + 0 1 - 3 5 9 6 0 E + 0 4 - 3 4 6 1 5T+ 04 1 4 4 4 e'E + 03 - 4 0 1 70E - CO5 3 9 C F + 0 1 _ -> 0 9 6 0 E + 4 — 2 9 709E+ 04 1 2 50 2E+0 3 - 4 0 3 8 3 E—000 9 0 1 E+02 - 2 6 B 3 0 E + 0 4 - 2 6 8 1 7E +04 1 0 1 2 35 + 0 3 - 3 77 3 1 E—002 2 6 4 5 + 0 2 - 2 3 4 5 0 E + 0 4 - 2 2 6 32E + 04 8 1 7 1 OE+02 - 3 4 84 4 5 - 003 6 2 7 E + 0 2 - 2 0 6 6 C E + 0 4 -1 79871- + 0 4 6 72H4E +02 - 3 2 6 6 7 1 - 004 9 8 9 E + 0 2 - 1 8 2 2 C E + 0 4 - 1 7 759E+ 04 4 6 0 1 7E +02 - 2 6 2 6 6 E - 006 352 E + 0 2 - 1 6 1 3 0 E + 0 4 -1 ■j 0 5 9 5 + 04 2 7Q49E +02 - 1 6 7 6 9 5 - 00771 BE + 0 2 - 1 4 3 7 0 L + 04 - 1 4 2 2 3E + 04 L 4 6 8 6 E +02 - 1 02 1 9 5 - 003 6 0 0 E + 0 2 - 9 4 4 2 0 E + 0 3 - 9 2 7 9 6 C + 0 8 1 6,? 4 HE. + 02 - 1 72 0 0 E- 0004 10 E + 0 2 - 5 4 1 4 0 E + C 3 -'•7 869.7 F - + 0 7 - 4 4 52 8? +02 H 22 4 75 - 007 3 5 0 E + C ? - 3 6 75 0 E + 0 3 - 7 8 32 3'. + 0 3 - I 0 7 3 7 E +02 4 2B23E - 007 7 9 0 E + 0 2 - 2 0 7 4 0 E + C 3 - 2 1 3 0 7 f +0 7 6 74 7E + 01 2 7 3 6 I E - 004 6 6 0 E + 0 2 - 1 4 2 7 0 E + 0 3 - 1 4 0 62 E + 0 7 - 6 9 2 0 4 L +01 4 1 4 8BE —005 2 5 0 F + 0 2 - 8 3 2 2 0 E + 0 2 - 0 7 7 6 8 E“+ 02 - 4 6 4 8 0 5 + 0 1 6 4 6 6 1 E- 002 2 6 0 E + 0 2 - 5 8 8 1 0 E + 0 2 - 6 2 7 0 0 E + 0 2 - 3 09 0 BE" + 01 6 6 1 5 9 5 - 009 2 7 0 E + 0 2 - 4 3 3 5 0 E + 0 2 - 4 4 9 7 1 E + 0 2 - 1 6 2 1 6 E + 0 1 3 7 4 0 7 5 - 009 7 8 0 E + 0 2 - 2 6 0 2 0 E + 0 2 - 2 7 1 7 2 E + 0 2 - 1 15 2 0 E + 0 1 4 42 7 75 - 006 7 9 0 E + 0 2 - 1 B 4 0 0 E + 0 2 - I 9 1 2 0 5 + 0 2 - 7 2 0 9 0 E - 0 0 3 9 1 7 9 5 —0010 8 0 E + 0 3 - 8 6 6 2 0 E + 0 1 -B 5 6 7 1 E + 0 1 1 J4 85E —00 - 1 2.1 04 5 - 002 4 8 0 E + 0 3 - 2 5 4 4 0 E + 0 1 - 2 4 1 2 7 E + 0 I 1 3 1 2 1 E - 0 0 - 5 1 5 7 9 E - 004 2 1 0 E + 0 3 1 BO 1GE + 0 1 1 9 0 6 7 E + C 1 1 0 5 7 5 E —00 5 8 7 2 1 5 - 00

79

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1 1 i i 1 1 1 I 1rn — J 0 0 ‘u. ft IV — V 'jJ 0 j' ft ft CJ Cm u> • • • » • • • • • • • • ■ • • • •Z O u 'J X £ — o V -J o o — - jti U) ft Ip 0 o ft ft f t ft 0 0 — ft -J 0 J 0

o ■n! — u o ft 0 o £ 0 o £ O U J £» u 0 o 0 X 0 'J £1 £ ft V f t £ Jl r V• m 3 m rn m m m TT 3 m m m rn m m in m0 ( 1 i I I I t 1 1 i l i 1 i i I i£ 3 O o o 3 o o o O o a o <3 o o o o

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— UJ U) ro — “ 0 V — 0 -J 0 0 0 0 ffi2 • • • • • • • • • • * • • • • • •U*! U rv U) S f t n X OJ ft — •- o m X -4 UJn rv ft 0 0 rv 0 o 'J Ui £ U o -4 rv £>

0 — ro ffi f t ft CJ — ft — — £ rv o ffi 0£ £ —. ft 0 V 0 V ^1 *0 o 0 3 •“ f t —

• rn m m m m m m rn m m m m m m rni m rn£ 1 l I t I t i I I i i i I i 1 i i'T'c>- o o o o o o o o o o o o o o o o o0 o o o o o o — — o o o o o o o o

3

2

I

0

- I

2

3.4

-5

6

•7

8

9

T -3 .5 *K

3 BAND MODEL EXPERIMENTAL POINTSLEAST SQUARES CURVE

-2W( K) • Y (K )

SEPARATE BANDS\ BAND 3E

BAND I BAND M

1 -1 .25 X 10BAND I

/ s ? n d i

I

1 ' I . Z i X I O 1 (GAUSS-CM )

100 1000

H(G AU SS)

1 0 0 0 0 1 0 0

Figure 13

3

J6111222344b66a911111122234567a9911

TABLE NO. XIX

SI 2 VS H

ri A NO MODEL T = 3 . 5 DEGUfclS K L L V I N 'M ( K. ) * 1 . / ( Y ( K ) * Y < K >)

I1?3456

0 ( 1 )- 3 . ’ 5 3 5 8 1 1 7 5 8 R 9 4 2 E + 0 6

S . 7 9 8 3 97 3 3 7 5 0 l 6 3E+0? 3 • 3 9 2 * 9 9 5 2 3 9 1 0 62E+O6 I . 0 4 3 0 5 6 2 3 1 1 7 6 ? R L + 0 ‘-' I . 9 6 1 73 381 3 9 ' ’ ?9HE + 0B 5 . 1 8 2 0 6 7 2 6 0 0 0 6 ? 3 E+OS

DO ( I )7 . 4 0 0 6 1 0 6 8 5 5 7 0 9 5 E +01

- 8 . 1 0 0 1 1 8 3 7 8 5 7 1 4QE- 0 3 3 . 0 4 9 1 9 1 8 5 2 3 16 14 E +0? 1 . 661 9 3 7 5 2 8 8 * 2 6 4 5 + 0 1

- 3 . 3 7 8 3 6 0 6 7 0 2 9 9 2 35 +0 2 7 . 2 3 2 6 0 0 0 6 2 3 98 325. + 02

VAR= .11 1SE+04 NO. OF I TE9A T IONS = 1

X Y YBAW DEL PD4 1 0 0 E - C 0 -2 16 5 0 E + 0 4 - 1 9 2 2 2 E + 0 4 2 • 4 2 7 7 E + 0 3 - 1 1 2 1 3E+018 1 OOE-09 - 3 7 9 0 0 E + 0 4 - 3 6 2 4 5 F + 0 6 1• 6 5 4 3 E + 0 3 - 4 3 6 4 9 E - 0 00 2 2 C E + 0 1 - 6 O7O0E+O4 - 4 9 7 5 0 E + 0 4 9 • 4 9 4 4 E + 02 - 1 H 7 2 6 E - 0 036 30 E + 01 — 6 8 5 3 0 E + 0 4 _ 9 2 5 4 E + 0 4 - 7 • 2 4 9 1 E+0? 1 2 3 8 6 E - 0 07 0 3 C E + 0 1 -6 2 7 0 0 E + 0 4 -6 6 0 9 7E + 0 4 -2 • 3 9 7 0 E + O 3 3 8 2 3 1 E- 0 00 44 0 5 + 3 1 -6 4 6 3 3 E + 0 4 -6 8 0 3 9 E + C 4 - 3 • 4C99E + 03 5 2 7 6 0 E - 0 03 8 5 0 E + 0 1 -6 5 2 0 0 E + 0 4 -6 8 8 5 0 F +04 — 3 • 6 5 0 8 E + 0 3 5 4 9 9 4 E - 0 07 2 5 0 E + 0 1 -6 4 4 6 0 E + 0 4 -6 8 2 1 6 E + 0 4 - 3 • 7 5 5 0 E + 0 3 £=. 8 2 * 3 E - 0 04 07 0 E + C 1 -6 1 2 4 0 E + 04 -6 4 5 3 3 E + 0 4 — j • 2 9 3 0 E +0 3 5 3 7 7 2 E - 0 00 880 E + 0 1 7 1 9 0 E + 0 4 — 5 9 5 7 9 E + 0 4 -2 • 3 8 9 2 E + 0 3 4 1 7 7 W E - 0 0769C E + 0 1 — 6 3 2 1 CE+34 - 5 4 4 9 B E + 0 4 -1 . 2 8 8 4 E + 0 3 2 4 2 14 E - 0 04 61 9E + C 1 - 4 9 0 9 0 E + 0 4 - 4 9 7 325 + 04 -6 * 4 2 5 1 E +0? 1 3 0 8 8 E - 0 01 J 2 0 E + 0 1 - 4 5 2 7 0 E + 04 - 4 5 4 3 4 E + 04 - 1. 64 9 3 E +02 3 64 3 3 t - 0 18 1 3 C E + 0 1 - 4 2 2 6 0 E + 0 4 - 4 I 61 OF +04 6 • 4 9 9 8 E +C2 -1 3 3 8 0 5 - 0 017 6 0 E + 0 1 - 3 62OOE+04 - 3 >2 1 3E + 04 9 • 8 6 2 2 E +0? -2 7 2 4 3 E - 0 05 3 9 0 E + 0 1 - 3 1 1 2 0 E + 0 4 - 3 0 1 6 1 E+C4 9 • 4 8 7 3 E + 0 2 - 3 0 8 0 7E — 000 9 0 1 E+02 -2 6 9 9 0 E + 0 4 -2 6 10 9 E + 0 4 8 • 8 0 6 3 E + 0 ? — 3 2 6 2 8 E - 0 02 2 6 4 E + 0 ? -2 3 5 7 0 E + O 4 -2 2 7 9 8 E + 0 4 7 • 7 1 40E + 0? _ 3 2 7 3 1 £ - 0 03 6 2 7 E + 0 2 -2 0 7 C 0 E + 0 4 -2 0 0 5 2 E + 0 4 6 . 4 759E +0 2 - 3 12 8 4 E - 0 04 9 8 9 E + C 2 - 1 8 2 6 0 E + 0 4 -1 7 7 4 5 5 + 0 4 ">. 1 4 9 2 E + 0 ? -2 8 1 9 9 E - 0 06 3 5 2 E + 0 2 - 1 6 17 0 E + 0 4 -1 6 7 8 1 E + 04 3 • 8 8 8 2 6 + 0 2 -2 4 0 4 6 E — OC77 15 E + 0 2 -1 4 3 7 0 E + 04 -1 4 C 9 4 E + 0 4 -j • 752 OE +0 2 - 1 9 1 6 1E — 00OC5 0 E + 0 2 - 1 2 2 1 0 E + 0 4 -1 1 7 1 0E+C4 4 • 9 9 3 5 E+ 0 2 - 4 0 e 9 7 E - 0 04 7 C 0 E + 0 2 -8 6 9 5 0 E + 0 3 -6 4 169E + 03 1* 7 9 0 1E +02 -2 0 8 2 7E— 009 7 0 C E + 0 2 — 5 7 2 8 0 E + 0 3 - 4 B8 78E + 0 3 - 1 . 5 o 8 0 E + 0 ? 2 7 8 9 8 E - 0 08 3 5 0 E + C 2 - 3 4 0 7 0 E + 0 3 - 3 4 2 15 E + 0 3 -1 • 46? 3E +01 4 2 6 2 9 E - 0 17 1 0 0 E + 0 2 -2 0 7 5 0 E + 0 3 -2 0 7 7 3 E + 0 3 -2 • 34 C 1E —00 1 12 7 7 E - 0 15 8 0 0 E + 0 2 -1 2 6 9 0 E + 0 3 -1 3 1 2 6 E + 0 7 - 4 . 3 6 1 0 E + 0 1 3 4 3 6 5 E - 0 04 6 0 0 E + C 2 -8 2 5 8 0 E + 0 2 -8 4 7 0 8 E + 0 2 _2 • 12 8 6 E + 0 1 T> 6 7 7 7 E - 0 03 2 6 0 E + C 2 — 5 2 6 7 0 E + 0 2 - 5 5 9 1 0 E + 0 2 - 3 • 2 4 0 9 E + 0 1 6 15 3 3 E - 0 02 1 OOE+02 - 3 5 1 9 0 E + 0 2 - 3 6 7 6 3 E + 0 2 - 1• B 7 3 8 E + 0 1 4 4 7 2 5 E - 0 0C 8 0 0 E + C 2 -2 4 6 5 0 E + 0 2 -2 4 1 3 7 E + 0 2 5 • 1 2 1 8 E - 0 0 -2 0 7 7 8 E - 009 5 0 0 E + 0 2 - 1 5 2 6 0 E + 0 2 -1 5 4 3 8 E + 0 2 -1 * 7 8 3 6 E - 0 0 1 16 8 8 E - 0 0CB30E+D3 - 9 3 4 0 0 t + 0 1 - 9 2 6 4 1E + 0 1 7 . 5 8 1 6 E - 0 1 -8 1 1 7 5 E - 0 12 5 8 0 E + C 3 - 1 7 8 8 0 E + 0 1 -1 7 ? 88E + 0 1 5 . 9 1 9 0 E - 0 1 - 3 3 1 0 4 E - 0 0

82

83

1 4 31 CE+03 2 . 2 2 2 0 E + C 11 6 0 3 0 E + 0 3 4 . 4 7 9 0 E + 0 11 7 6 4 0 E + C 3 5 * 762 OE + 0 I1 9 3 2 C E + 0 3 6 * 4 5 5 OE + 0 I2 0 97 0E+ C3 6 . 8 4 7 0 E + 0 12 2 59QE + 0 3 6 . 9 3 5 0 E + 0 I2 4 1 5 0 E + 0 3 6 . 9 9 6 C E + 0 I2 4 7 3 0 E + 0 3 6 . 9 0 0 0 E + 0 I2 S27CE+0 3 6 * 6 1 50E + 0 13 1 8 1 0E + 0 3 6 . 2 1 90E+010 5 3 5 0 E + 0 3 5 • 7 7 7 3 E + C l3 8 8 9 0 E + 0 3 8 . 4 0 6 0 E + O14 2 4 3 0 E + 0 3 4 • 9 7 7 0 E + 0 14 6 8 5 0 E + C 3 4 . 6 2 8 0 E + 0 13 180CE+C 3 4 • ? 793 E +015 4B20E+ 33 4 . C 12 0 E +016 0 1 3 0 E + 0 J 3 . 7 0 0 0 E + 0 17 00C0E+C3 3 . 1 6 5 C E + 0 1a 0CC0E+0 3 2 . 8 0 0 0 E+019 0 0 0 0 E + 0 J ? . 4 9 5 C E + 0 1l D0 90 E+04 2 . 2 5 0 0 E + 0 1l 10C0E+C4 2 • 0 4 7 Q E + 0 1i 2 0 0 0 E + C 4 1 . 8 8 0 0 E + 0 1i 39 0 0 E + C 4 1 • 7 3 5 0 E + 1"11! 40 0 0 E + C 4 1 . 6 1 0 0 E + 0 11 50C0E+C4 1 . 5 0 5 C E + 0 11 6 0 0 0 E + 0 4 1 . 4 10 0 E + 3 11 7 300F + C4 1 • 3 2 5 C E + 0 1NO* OF ITERATIONS*: I VAR

2 2 4 2 6 E + 0 1 2 06 1 I E - C l 9 2 75 9E - 0 14 4 0 9 5 E + C 1 - 6 94 3 1 E — 0 1 - 1 550 1 E- 0 08 54 2 9 E + 0 1 — 2 19 0 6 E - 0 0 - 3 80 1 8F - 0 06 1B 8 9 E + 0 1 - ? 6 6 0 2 E - C 0 - 4 1 2 1 2 E- 0 06 8 0 1 9 E + 0 1 '3 4 5 0 4 E - 0 0 - 5 0 39 38 - 0 06 6 1 9 4 E + 0 1 * 3 15 5 5 F - 0 0 - 4 55 0 1 E- 0 06 6 2 1 4 E + 0 1 - 3 7 4 5 2 E - 0 0 - 5 3534E - 0 06 6 C 3 0 E + 0 1 _ /j 9 6 9 2 E - H 0 - 4 30 3 1 E- o c6 .3*3 94 E + 0 1 - 2 8 - J 5 4 E - 0 0 - 3 B6 30E - 0 06 Cl 1 or- +0 I - 2 0 79 3E-CC - 3 34 34E - 0 0l;i 6422E +0 1 - 1 3 4 7 7 E - C 0 - 2 3329E - 0 0C-T 28 63 E + C 1 —! 19 6 9 E - 0 0 _ p 21 4 r E-CO4 9 K6 6 F + C 1 — 3 1 34 9 F - 0 1 - 4 2 8 9 5 E - 0 14 5 8 2 9 F + 0 1 - 4 5 0 8 0 F - C 1 - 9 74 0 7 E- 0 14 2 525E + 01 6 4 4 9 E - C 1 - 6 1 81 1 E- 0 14 0 1 67E +C i 4 7 2 2 9 E - 0 2 1 1 7 72 E- 0 13 7 0 3 8 E + C 1 3 8 8 4 2 E — 0 2 1 r A 9 F - 0 13 2 2 8 1 E + 0 1 6 3 17 1t - 0 1 1 99 5 9E - 0 02 8 8 1 b t + C 1 8 I 3 7 8 8 - 0 1 1 84 2 OE - 0 02 o 8 11 t + C 1 8 6 1 7 2 E - 0 1 2 251 4E - 0 02 3 0 6 6 E + 0 1 e. 6 6 4 BE —01 P. 5 1 77E - 0 02 1 0 4 0 E + C 1 6 7 0 8 3 E - 0 1 '-J 7886E - 0 01 9 3 3 7 E + 0 1 5 3 7 1 8 E - 0 1 2 85 7 3 E - 0 01 78 SEE + 01 s 3 S 4 F E - C 1 3 C862E - 0 01 6 6 34 E + CI 1 \ 3 4 2 7 E - C 1 31 84E - 0 01 5 5 4 5 E + 0 1 4 9 5 1 7 F - 0 1 -> 2 9 0 1 E- 0 01 4 5 B 0 E + C 1 4 8 8 8 3 E — 0 I 4 6 6 8 E - 0 01 3 742E + 0 1 4 9254 E — 01 3 71 73E - 0 0* 1 1 1 6E +04 POMEAN= 2 . 9 4 1 PDRMS= 3 . 4 7 3

54

3

2

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2

3

4

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6

7

8

9

T * 3 . 5 * K

2 BAND MODEL EXPERIMENTAL POINTS LEAST SQUARES CURVE

W(K)'Y(K)"26 -

SEPARATE BANDS - p p » I . 2 5 X 10

2 -BAND I

1.25 X IO , 5 (G A U S S ~ '-C m '3 )

- 6 -

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1 0 -

12 -

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1111

10 100 1000 H (G A U S S)

Figure 14

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SELECTED BIBLIOGRAPHY

Abeles, B ., and Melboan, S ., Phys. Rev. 101, 544 ( 1956) .

Aubrey, J. E . , J. Phys. Chem. Solids 19, 321, (1961).

Aubrey, J. E-, and Chambers, R. G., J. Phys. Chem. Solids 128 (1957)*

Cal len , H. B-, Phys. Rev. 8^ , 16 (1952).

Cal len , H. B . , Thermodynamics, John Wiley and Sons, In c . , New York,I960.

G alt , J. K . , Yager, W. A . , M e r r i t t , F. R . , C e l t in , B. B . , and B r a i ls fo r d , A. D., Phys. Rev. 114, I 396 (1959).

Jain , A. L. and Koenig, S. H. , Phys. Rev. 127, 44-2 (1962).

K i t t e l , C., Sol id State Physics, John Wiley and Sons, I n c . , New Yor,2nd e d . , 1956"^

Lerner, L. S . , Phys. Rev. 127, 1480 ( 1962) .

Schoenberg, D . , P h i l . Trans. A245, 1 (1952).

Weiner, D . , Phys. Rev. 125, 1226 ( 1962) .

141

VITA

Henry James Mackey was b o r n in V i c k s b u r y , M i s s i s s i p p i on

November 25? 1935- He was g r a d u a t e d f rom h i g h s c h o o l in C l a r k s d a l e ,

M i s s i s s i p p i in 1953? wh e r e u p o n he e n t e r e d H i nds J u n i o r C o l l e g e in

Raymond, M i s s i s s i p p i - In 1955 He r e c e i v e d h i s J u n i o r C o l l e g e Diploma

and e n t e r e d L o u i s i a n a S t a t e U n i v e r s i t y as a j u n i o r . He r e c e i v e d h i s

B a c h e l o r o f S c i e n c e d e g r e e i n P h y s i c s in l y 5 7 - Hi s M a s t e r o f Sc i ence

d e g r e e was o b t a i n e d i n 1959 f r om t he same i n s t i t u t i o n . He is now a

c a n d i d a t e f o r t h e d e g r e e o f D o c t o r o f P h i l o s o p h y in t h e Depar tment

o f P h y s i c s and As t r o n o my o f L o u i s i a n a S t a t e U n i v e r s i t y .

142

EXAMINATION AND THESIS REPORT

Candidate: Henry Jaines Mackey

Major Field: Physics

Title of Thesis: The E l e c t r o n i c Band S t r u c t u r e o f Bismuth

Approved:

M ajor Profe Chairman

Dean of the Graduate School

EXAMINING COMMITTEE:

-JLjfer

Q z 7 / J

■ / ’XW '

Date of Examination:

/& z J £ I P * / f £ . V ___

The Electronic Band Structure of Bismuth.

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Transcript of The Electronic Band Structure of Bismuth.