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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Recommended Citation Recommended Citation Mackey, Henry James, "The Electronic Band Structure of Bismuth." (1963). LSU Historical Dissertations and Theses. 849. https://digitalcommons.lsu.edu/gradschool_disstheses/849
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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
PAGE
43
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
Using 2 Bands with = Y^2 ............................
Parameters From Curve F i t t i n g for e1
Using 4 Bands with = Y^2 ............................
Parameters From Curve F i t t i n g for
Using 3 Bands with = Y^2 ............................
Parameters From Curve F i t t i n g for Hc'jj
Using 3 Bands with = Y^2 ............................
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
PAGE
109
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
PAGE
87
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 , <B ~ B°> , 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 * < I « P ( I ) « P P ( I ) * I = 1 * N P )0 0 ! F O R M A T C>,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 ? < I ) + X ? I 'I f = F + p f ( I ) # P ( I )
R E T U R N t. NO
■ S U B P R O G R A M f q w SUM OF h y p e r b o l a s
S U B R O U T I N E ' S U BD I M E N S I O N X ( 1 0 0 > • 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
PARAMETERS FROM CURVE FITTING FOR
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
PARAMETERS FROM CURVE FITTING FOR
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
PARAMETERS FROM CURVE FITTING FOR
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
PARAMETERS FROM CURVE FITTING FOR
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
PARAMETERS FROM CURVE FITTING FOR
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
PARAMETERS FROM CURVE FITTING FOR
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
PARAMETERS FROM CURVE FITTING FOR
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*
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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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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
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T * 2.1* K
4 BAND MOOEL
4 - EXPERiMENTAL POINTS
LEAST SQUARES CURVE
W(K) * YOO**
SEPARATE BANDS^BAND I
BAND H
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4
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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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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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• 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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0
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2
3
4
5
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 -
BAND X- 8 -
1 0 -
12 -
• •14-
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 ___
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