PART I '
AN ELECTRON SPlN RESONANCE STUDY
0F RADECALS EN {RRADIATED SlNGLE. ,
CRYSTALS 0F MANDEUC ACID "

PART I:
A ammummu as METHGDS ma
nmammme g Tamas

Thesis for the Degree of Ph. D.
MECHIGAN STATE UNWERSHY
WiLLlAM GEORGE WALLER
1973

 

    
  

  

LIBRAR Y
Michigan State

 

 

 

ABSTRACT

PART I
AN ELECTRON SPIN RESONANCE STUDY OF RADICALS
IN IRRADIATED SINGLE CRYSTALS
OF MANDELIC ACID
PART II

A GENERALIZATION OF METHODS FOR
DETERMINING g TENSORS

BY

William George Waller

In Part I, the electron spin resonance spectra of
irradiated single crystals of mandelic acid, C6H6CHOHCOOH,
have been studied. Two radicals, a-hydroxybenzyl (CGHSCHOH)
and cyclohexadienyl-glycolic acid (C6H6CHOHCOOH), were
identified and the ESR parameters determined for each.

The a-hydrogen hyperfine splitting tensor of the
a-hydroxybenzyl radical is nearly isotropic with principal
values (-15.0,-15.3,-18.3) gauss and the g tensor is nearly
isotropic with principal values (2.0022,2.0033,2.0039).

The ESR data are those expected for a planar n-electron
radical and molecular orbital calculations (INDO method)
confirm this and provide detailed geometry. The benzene

rings appear to have reoriented upon irradiation.

William George Waller

The cyclohexadienyl-glycolic acid radical shows a
large hyperfine splitting by the methylene protons; the
tensor is nearly axially symmetric and quite anisotropic,
with principal values (-28.S,-52.S,-S7.3) gauss.

In Part II, a generalization of the usual methods
for obtaining the principal values, and directions of the
principal axes, of the g tensor from single—crystal ESR
data has been derived. The formalism of Part II converts
the inherent overspecification of tensor elements into a
determination of three rotational misalignments, and so
improves the accuracy of the g-tensor parameters. The
procedures developed have been applied to the determi-
nation of g tensors from rotations about orthogonal axes,
monoclinic axes, coplanar axes, or general axes. The
coplanar and general cases should prove useful in
determining g tensors for needle-shaped crystals and for
crystals with inconvenient face development. The equations
have been cast in a convenient form for computer programming

and a program has been written.

PART I
AN ELECTRON SPIN RESONANCE STUDY OF RADICALS
IN IRRADIATED SINGLE CRYSTALS
OF MANDELIC ACID
Part II

A GENERALIZATION OF METHODS FOR
DETERMINING g TENSORS

BY

William George Waller

A'THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics
Program in Chemical Physics

1973

a“
5

To

My Parents

ACKNOWLEDGMENTS

The author wishes to express his sincere appreci-
ation to Professor M. T. Rogers for his expert guidance
and interest, and for the freedom allowed during this
investigation. The author also wishes to thank Dr. S.
Subramanian of the Indian Institute of Technology, Madras,
India for helpful discussions and for obtaining the ENDOR
spectra.

The aid of the following people is gratefully
acknowledged: Mr. George Giddings for performing the
y-irradiations, Dr. A. Tulinsky for use of the x-ray
equipment, Dr. M. Neuman for guidance in the crystallo-
graphic interpretation, and Dr. R. Lloyd of the University
of Connecticut for useful discussions.

The author would like to thank the Atomic Energy
Commission for financial support during the course of this

study.

iii

TABLE OF CONTENTS

Page
LIST OF TABLES o I O O 0 O 0 0 O 0 O O 0 Vi i
LIST OF FIGURES. O O O I O 0 O O 0 O O 0 ix

PART I. AN ELECTRON SPIN RESONANCE STUDY OF RADICALS
IN IRRADIATED SINGLE CRYSTALS
OF MANDELIC ACID

INTRODUCTION. . . . . . . . . . . . . . 1
HISTORICAL BACKGROUND. . . . . . . . . . . 3
I. History. . . . . . . . . . . . . 3
II. ESR Literature . . . . . . . . . . 4
III. 9 and A Tensors . . . . . . . . . . 5
IV. Pi-Electron Radicals . . . . . . . . 6
THEORETICAL . . . . . . . . . . . . . . 8
I. Introduction . . . . . . . . . . . 8
II. Time Reversal and Kramers Degeneracy . . . 9
III. 9 and A Tensors . . . . . . . . . . 13
IV. Isotropic Interaction . . . . . . . . 18
V. Spin-Orbit Coupling. . . . . . . . . 22
VI. Spin-Spin Coupling . . . . . . . . . 25
VII. Orbital Angular Momentum . . . . . . . 26
VIII. Nuclear g Tensor. . . . . . . . 27
IX. Electric Quadrupole Coupling. . . . . . 27
X. Spin Hamiltonians . . . . . . . . . 29
A. Electron Zeeman Hamiltonian . . . . 30
B. Hamiltonian A Plus Hyperfine Inter-
action . . . . . . . . . . 30
C. Hamiltonian B Plus Zero-Field
Splitting . . . . . . . . . 33
D. Hamiltonian C Plus Electric Quadrupole
and Nuclear Zeeman Terms . . . . 34

iv

XI. Determination of ESR Parameters . .
EXPERIMENTAL . . . . . . . . . . .

I. Spectrometers . . . . . . .
II. Crystal Irradiation and Mounting . .
III. ENDOR Measurements. . . . . . .
IV. X-ray Crystallography. . . . . .
V. Parameters from Tensor Variations. .

RESULTS 0 O O O O O O O O O O O O

I. X-ray Crystallography. . . . . .
II. Irradiation and Spectra . . . . .
III. Alpha-Hydroxybenzyl Radical. . . .

A. Determination of Radical. . .
B g-Tensor Evaluation . . . .
C. A-Tensor Evaluation . . . .
D. Variable Temperature Study . .

IV. Cyclohexadienyl-Glycolic Acid Radical

A. Determination of Radical. . .
B. g-Value and A-Tensor Evaluation
C. Variable-Temperature Study . .

V. Electron Irradiation at 77° K . . .
VI. d- Mandelic Acid and l- Mandelic Acid
Irradiation . . . . . . . .

DISCUSSION 0 O O O O O O O O O O O
I. Alpha-Hydroxybenzyl Radical. .

II. Cyclohexadienyl- Glycolic Acid Radical

III. Summary . . . . . . . . . .

REFERENCES 0 O O O O O O O O I O .

APPENDIX. Computer Listing of SubrOutines Used to
Determine the g and A Parameters from Tensor

Variations . . . . . . . .

Page
37
43
43
45
46
46
49
S4
54
54
60
60
61
63
71
71
71
79
80
80
81
83
83
89
91

92

99

PART II. A GENERALIZATION OF METHODS FOR

DETERMINING g TENSORS
INTRODUCTION . . . . . . . . . . . . .
DATA FOR g-TENSOR DETERMINATION. . . . . . .

Experimental . . . . . . . . . . .
Parameterization of the Data . . . . . . .

GENERAL THEORY . . . . . . . . . . . .
Nomenclature . . . . . . . . . . . .
General Formulation . . . . . . . .
Determination of Starting- Angle Shifts. . . .

THEORY--DERIVATION OF FORMULAS . . . . . . .

Preliminary Equations . . . . . . . . .
Case of Three Coplanar Axes . . . . . . .
Case of Three Monoclinic Axes. . . . . . .
Case of Three Orthorhombic Axes . . . . . .
General Case . . . . . . . . . . . .

DISCUSSION. 0 O I O O O O O O O O O O

Iterative Solutions . . . . . . . . . .
Exact Solutions . . . . . . . . . . .

COLLECTED FORMULAS . . . . . . . . . . .
Coplanar Case . . . . . . . . . . . .
Monoclinic Case . . . . . . . . . . .
Orthorhombic Case. . . . . . . . . . .
General Case . . . . . . . . . . . .

REFERENCES. . . . . . . . . . . . . .

APENDICES

Appendix

A. Solutions to a Trigonometric Equation . .
B. Properties of Second-Order Tensors . . .
C, Computer Listing of Subroutines Used for the

Calculation of g Tensors for the Coplanar,
Monoclinic, Orthorhombic, and General Case

vi

Page

107
109

109
109

112
112
116
117
120
120
123
124
125
127
131

131
133

135
135
137
138
139

141

142

143

145

LIST OF TABLES

Table Page
PART I

1. Conditions limiting the possible reflections
for dlfmandelic acid . . . . . . . . 55

2. Values of the hyperfine splitting constants
of the alpha-hydroxybenzyl radical in
various solutions and in a single-crystal
matrix. . . . . . . . . . . . . 55

3. The principal values and direction cosines
for the g tensor and the alpha-hydrogen
hyperfine splitting tensor of the alpha-
hydroxybenzyl radical in glfmandelic acid . 64

4. Correlation between the individual sites
and the resulting overlapped lines for the
alpha-hydroxybenzyl radical. . . . . . 65

5. Values of the hyperfine constants of the
cyclohexadienyl-glycolic acid radical in
various solutions and in a single-crystal
matrix. . . . . . . . . . . . . 72

6. Correlation between the individual sites and
the resulting overlapped lines for the
cyclohexadienyl-glycolic acid radical . . 72

7. The principal values and direction cosines
for the CH2 hyperfine splitting tensor of
the cyclohexadienyl-glycolic acid radical
in dlfmandelic acid . . . . . . . . 73

8. Unpaired electron spin and excess charge
densities calculated for the alpha-
hydroxybenzyl radical by the McLachlan
method. . . . . . . . . . . . . 73

vii

Table Page
PART I I
l. w coefficients and null functions with the

necessary "theoretical" initial
orientations . . . . . . . . . . . 136

viii

Figure

10.

LIST OF FIGURES

PART I

Spin-polarization of the C-H bond in a
fl-electron radical . . . . . . . . .

Orthographic projection of a crystal of die
mandelic acid. . . . . . . . . . .

A typical ESR spectrum at room temperature of
glfmandelic acid irradiated at 77°K . . .

Plot of line positions for the alpha-
hydroxybenzyl radical as a function of field
orientation in the be plane . . . . . .

Plot of line positions for the alpha-
hydroxybenzyl radical as a function of field
orientation in the ca plane . . . . . .

Plot of line positions for the alpha-
hydroxybenzyl radical as a function of field
orientation in the ab plane . . . . . .

Hyperfine splitting of the alpha proton of the
alpha-hydroxybenzyl radical KE' angle of
rotation in the ho plane for lines A and D .

Hyperfine splitting of the alpha proton of the
alpha-hydroxybenzyl radical X5. angle of
rotation in the be plane for lines B and C .

Hyperfine splitting of the alpha proton of the
alpha-hydroxybenzyl radical X§° angle of
rotation in the ca plane . . . . . . .

Hyperfine splitting of the alpha proton of the

alpha-hydroxybenzyl radical 22° angle of
rotation in the ab plane . . . . . . .

ix

Page

20

48

56

S7

58

59

66

67

68

69

Figure

11.

12.

13.

14.

15.

16.

17.

18.

The wing lines of an ESR spectrum at room
temperature of dlfmandelic acid irradiated
at 77°K O O O O O I I O O O O I 0

Plot of line positions for the cyclohexadienyl-
glycolic acid radical as a function of field
orientation in the ca plane . . . . . .

The CH2 hyperfine splittings of the
cyclohexadienyl-glycolic acid radical 25°
angle of rotation in the be plane . . . .

The CH2 hyperfine splittings of the
cyclohexadienyl-glycolic acid radical 25°
angle of rotation in the ca plane . . . .

The CH2 hyperfine splittings of the
cyclohexadienyl-glycolic acid radical Kg,
angle of rotation in the ab plane . . . .

The atom positions of the planar alpha-
hydroxybenzyl radical as calculated by INDO .

The relationship between the eigenvectors of
the g and A tensors and the geometry of the
planar alpha-hydroxybenzyl radical . . . .

A possible scheme for the formation of the two

radicals in the irradiation of dlfmandelic
acid at 77°K . . . . . . . . . . .

PART II

Systems of axes . . . . . . . . . . .

Page

74

75

76

77

78

84

85

86

115

PART I

AN ELECTRON SPIN RESONANCE STUDY OF RADICALS

IN IRRADIATED SINGLE CRYSTALS

OF MANDELIC ACID

INTRODUCTION

One distinctive difference between quantum mechani-
cal and classical properties of systems is that there are
sharply defined energy levels in the former as compared‘to
a continuous range in the latter. In situations where the
quantum mechanical energy levels are not too close together,
a resonance technique can sometimes allow an accurate
determination of energy differences between levels.

There are at least four "simple" types of resonance
phenomena--nuclear, paramagnetic, ferromagnetic, and
antiferromagnetic. The first is concerned with inter-
actions of the nuclear dipoles, while the last three deal
with electron dipoles. The last two phenomena deal with
magnetic systems where the electron dipoles are strongly
coupled by exchange forces. Paramagnetic resonance is
confined to loosely-coupled systems where the paramagnetic
units may be regarded as individuals.

Such loosely-coupled paramagnetic species some-
times can be formed in irradiated single crystals. The
damaged molecules can become trapped and oriented in the

crystal. The interaction of the odd electron with nuclei

in the damaged molecule would cause its energy levels to
change and divide. These effects can be determined by
electron spin resonance (ESR) spectroscopy and can be used
to identify the radical and its orientation. Molecular
information concerning the radical also can be deduced.

While there has been a lot of work done on ESR
studies of aliphatic radicals in organic single crystals,
the literature on ring compounds is small. There has been
an interest in discovering the benzyl radical or a simple
modification of it in single crystals. The nature of the
delocalization of the unpaired electron could provide
information concerning the structure of the radical. d1:
Mandelic acid was chosen because it is a relatively simple
ring system. On irradiation, two ring structures, the
alpha-hydroxybenzyl radical and the cyclohexadienyl-
glycolic acid radical, appear to be formed. The analysis
of the ESR spectra of these radicals has been carried out
and their geometrical and electronic structures are
discussed in Part I of this thesis.

Part II of this thesis consists of a theoretical
treatment of methods for determining g tensors. A gener-
alized procedure is derived that can be applied to any
symmetric second rank tensors such as the zero-field
splitting and hyperfine interaction tensors B and A,
respectively. The computer programs used for Part I and

Part II are also listed in the Appendices.

HISTORICAL BACKGROUND

I. History

Some important dates in the history of magnetic

reasonance are as follows:

l936--First prediction of magnetic resonance absorption by

Gorter.l

l938--First observation of magnetic resonance absorption
in molecular beams by Rabi, Zacharias, Millman and

Kusch.2

l945--First observation of the electron spin resonance

phenomenon in liquids by Zavoisky.3

l946--First report of nuclear magnetic resonance by
Purcell, Torrey and Pound4 and by Bloch, Hansen and
Packard.S

l947--First ESR free radical spectrum was observed by

Kozyrev and Salikhov.6

l949--First report of ESR hyperfine structure by Penrose.7
l949--First evidence of quadrupole interaction by Ingram.8

l949--First ESR study of naturally occurring organic free

radicals by Holden, Kittel, Merritt and Yager.9

l951--First analysis of the hyperfine structure in the ESR

spectra of paramagnetic salts by Abragam and

Pryce.10

l951——First ESR study of free radicals formed by radiation

damage by Schneider, Day and Stein.11

l956--First ESR study of oriented organic radicals in a

single crystal by Uebersfeld and Erb.12

l956--First ENDOR experiment by Feher.l3

l958--First ESR study of a triplet state by Hutchison and

Mangum.l4

l959--First complete analysis of the ESR spectrum of an

oriented organic radical in a single crystal matrix

by Cole, Heller and McConnell,15 by Ghosh and

16 17

Whiffen, and by Miyagawa and Gordy.

II. ESR Literature

There are many surveys of the ESR literature”-31

26

including in particular an early review by Morton of

radicals in single crystals. Recent work on ESR of organic

radicals is discussed in a review by Kochi and Krusic.31

There are also frequent review articles in the Annual

Reviews of Physical Chemistry32 and in the Annual Reports

33 Other sources of information

34-38

of the Chemical Society.

include the proceedings of ESR symposia and col-

lections of ESR data.39’40

There are also a large number of books on ESR.
Textbooks giving a complete introduction to magnetic

41
resonance are those by Carrington and McLachlan and by

Wertz and Bolton.42 ESR theory is presented in a new book

by Poole and Farach.43 A more mathematical book with

emphasis on interaction mechanisms was written by Slichter.44
There is a comprehensive reference book by Abragam and
Bleaney45 that does not contain many formula derivations.

A textbook approach in deriving the necessary mathematics
for ESR theory has been used in a work by Griffith.46
Comprehensive books covering the experimental techniques

include those by Poole47 and by Alger.48

III. g and A Tensors
In ESR, the most commonly measured quantities are
the g and A tensors in crystals or their isotropic values
in liquids, powders, or glasses. These quantities are
defined by their contribution to the following spin
Hamiltonian.

+==+ + =+
7i = BH-g-s + glp AP 5

where g is the magnetic field intensity vector, and Ip and
S are the spin angular momentum vectors of the pth nucleus
and the unpaired electron, respectively. Given a specific
magnetic field direction, the value of the g tensor
describes the Zeeman contribution to the energy of the

system as a linear function of magnetic field strength.

The "shape" of the tensor describes the variation of this

Zeeman energy as a function of magnetic field direction
in the crystal. In a similar manner, the Ap tensor
describes the hyperfine energy contribution due to the
interaction of the unpaired electron with the magnetic
moment of the pth nucleus.

The g and Ap tensors each have three principal
values associated with three orthogonal directions. These
principal values can be equivalently specified by their
average isotropic part and their anisotropy.

From ESR studies of a variety of oriented radicals,
investigators have found patterns in the average values
and anisotropies of the g and A tensors. These are useful
in identifying the radical. The principal g values are
related to the orbitals occupied by the free electron. The
principal values of :P are related to the number of bonds
and the geometry between the pth nucleus and the site

where the electron is located.

IV. Pi—Electron Radicals

 

One of the common types of organic radicals,
concerning which there is a great deal of theoretical and
experimental literature, is the pi-electron radical. This
is a paramagnetic molecule containing a number of coplanar
atoms, usually carbon and hydrogen, such that the spin
paramagnetism is largely distributed in atomic orbitals
having a node in the molecular plane. The experimental

Ap tensors have been related to the interaction between

the unpaired electron and a protons (C—Ha), 8 protons

l3

(C-C-HB) and C of the central carbon atom.

For the a protons, the theoretical relationship for

the isotropic part of 3 i528’ 49-54

H n

where Q is approximately a constant having a value of —22.5
gauss and 01T is the fl-electron spin density on the carbon
atom. The anisotropic part has been found in general to
consist of the three principal values (+10,0,-10) gauss.
The first value corresponds to the H-C bond direction, the
second value to a direction perpendicular to the radical
plane, and the third value to a direction perpendicular to
the two previous directions.26' 55’ 56
The K tensor for B protons is generally quite

isotropic and has been described by the equation57' 58

a = B + B cos2 8

where B0 is a constant with a value between 0 and 4 gauss,
B2 is about 50 gauss, and 6 is the angle between the
projections of the axis of the unpaired electron n orbital

and the C-HB bond onto a plane perpendicular to the C-C

bond. Tables of a- and B-proton splittings have been

given in the Ph.D. theses of Kispert59 and Watson.60

THEORET ICAL

I. Introduction

 

The theoretical development and the inherent
limitations of ESR are dependent upon the approximations

used to solve the nonrelativistic Schroedinger equation
‘Hw = BY. (1)

The problem is that W is a function of the
positions, momenta, and spin states of all the particles
in the system. The Hamiltonian?‘ is a quantum mechanical
operator analogous to the classical mechanical formula for
the total energy of the system. This is complex in that
it includes all interactions between the particles. The
main approximation needed to solve Equation (1) is taken

from the following equation:
W = E ciwi. (2)

where W1 are the functions which (as is postulated) span
the space. We assume that we can pick some set of
functions ¢i so that the infinity in the summation is

replaced by a finite reasonably small number.

If Equation (2) is applied in a perturbation

formalism where

W=7Llo +71. '%0¢o(j) = 130%(3) I (3)

and the effect of ‘H' is small compared to that of‘H, then

the coefficients ci of Equation (2) are of the order of

th (k)

(E0(k)-EO(J))—r1 for the n order perturbation where E0

and Eo(j) are the exact energies in Equation (3). So if
we order the functions ¢0(j) according to their energies
E0(j), then we terminate the summation in Equation (2) when
Eo(j) becomes "large" since that corresponding coefficient
is "small." The functions ¢i that we choose form a manifold
in the infinite dimensional W1 vector space.

Electron spin resonance is concerned with the
interaction of an electron with other electrons, nuclei,
and external magnetic fields. The number of functions
needed in the manifold to describe reasonably this inter-

action is determined by considering the time-reversal

operator and Kramers theorem.

II. Time Reversal and Kramers Degeneracy
The time-reversal operator 8 can be defined by its

effect on wave functions W1 in

3(ewi(t)) awi(t)
at ‘ " _§E_"' '

 

10

Thus 8 would have the effect of reversing every momentum

while not affecting the positions of the particles. We,

therefore, can define 8 by its effect on operators:61

 

. (4)

We can apply this similarity transformation to the quantum

mechanical operators that correspond to the following

classical quantities:62

 

+ ++ + 1+++
L = fmrxv dT , u = EEfrxJ(r') d1 ,
H(r) = %fo|i(:') dT ,
r-r

+ + +

where L is the angular momentum, and p and H are the
magnetic moment and magnetic field, respectively, of the
current distribution 3(f'). We can thus write in terms of

operators

where the last relationship follows from the exact mathe-
matical analogue of the postulated intrinsic spin to the
angular momentum in quantum mechanics.

It can be seen then that the spin-spin and spin-
orbital interactions of the electrons are invariant with
respect to the time-reversal similarity transformation,
as is the kinetic energy and any time-independent potential

energy. If these are the only terms we allow in the

11

Hamiltonian of the isolated system of the electron, then we
see a - = , an 15 a symme ry e emen .
thte‘HGIW (19' t 1 t

Consider the following equations:

(mgr) =‘Hw (6)
amiss-$1) = (e’He‘hew
canes-3w =‘Hew . (7)

They can be taken to mean that the isolated system evolves
forwardly in time (Equation (6)) in the same way as it
would appear if viewed by someone going backwards in time
(Equation (7)). Continuing the physical reasoning, we
would deduce that the transition probabilities between
states is the same in the time-forward or time-backwards

view of the system. This gives us the mathematical

 

assumption that

|(<I>,‘i’)l2 = l(e¢,e‘¥)lz°
Using this equation, it is shown63 that
62a w = Za*ew - 92 = +1 (8)
i i i i i i’ —

and thus we call 9 an antilinear operator. From Equation
(4), 6 has been determined for a one-electron system in

the coordinate representation

12
where we use the standard Pauli matrices
_ 01 _ 0-1) _ 1 o
0x ’ (1 o) 0y ‘ (i 0 Oz ‘ (o —1) '
and we define
_ 1 0 _ 0 l : *
E“(o 1) C‘(—1 0) Kow'w '
which can be used to derive
-l 2 2

9 = “KOC = -CKO ; C = 9 = ’1

and the formulas for the many-electron system

n _1 n
O I
9:1 p 09:1 p (9)
2 _ . . even
6 — (1)1 if n is (odd ).

If an isolated system has an odd number of
electrons, then 62=-1 from Equation (9» and it has been

shown64

that if N is an eigenfunction, then 4 = 6w is
another orthogonal eigenfunction of the same energy. The
ground state of the isolated system of the electrons is
then at least two-fold degenerate and this two-dimensional
space is spanned by what is called the Kramers doublet.

If we now add to the Hamiltonian the interaction

-> —>
due to an external magnetic field )J'He , then 74 is not

xt

invariant under the 9 similarity transformation because

—> + —]_ _ 4 —]_ . + —l
9(U Hext)e - (BUB ) (eHexte )
1 + -+
= ('U)°(H 99 ) = -1J°

13

and fiext is not part of the isolated system under study.

Likewise, if the nucleus has a magnetic moment, than

+ + I C I
uI-Hel 18 not 1nvar1ant because

(aile'l)-(e§ele ) = (K ) , (10)

+
where “I is the magnetic moment of the nucleus (which is

not part of the system) and fie is the magnetic field

1
produced by the electrons at the nucleus.

In ESR, these interactions are considered pertur-
bations which lift the degeneracy of the Kramers doublet
and thus we assume that the excited states of the electron

system are much higher in energy than either of the external

interactions.

III. 9 and A Tensors
The 9 tensor is developed by considering the

matrix elements of 5. It has been shown64 that,since
936- =-; from Equation (5), the expectation values of
uq(q=x,y,z) in the two Kramers-conjugate states add to
zero. The matrix representing ”q in the Kramers manifold
must be Hermitian since “q is a physically measurable
quantity. So, in general, we can write

|w> I61)?
- IQ> zq Xq-lyq
q I8Q> xq+iyq -zq .

t“

14

This can be written in terms of Pauli matrices as

_ .—
.— —

= xqox+y

t“

z 0 = §r(q)j0' (11)

+
q q Y q 2 3

and we can put this in terms of a 3X3 9 "matrix"

_ _ 2
gqj Eg'r<q>j
1 Be =
" uq-ѤҤgqjj

The basis functions w and 6w are arbitrary in that we can

choose two other states as in

WI

aw + b(9W)

(GW')

a*(9¢) - b* T)

where we have used the relations in Equation (8). We can

write that
D = (43* a*) ' pq .__ qu'D

represent the similarity-transformation matrix and uq' in

65

the new basis manifold. It can be shown that the change

to the new matrix jg. can also be represented by a proper

rotation of the vector ;q(j) in Equation (11). This is to

say

8

= e
Z R..r .o. = — Z -1 = .
j i 31 ‘q’1 3 T3' i‘R ’iquin

15

After changing the basis set and obtaining the new matrices

jg , we can also rotate the coordinate system by the

rotation matrix E to give new jg":

+1! _ =S=.+l u _ is l
u — w . ur — rquq
q
,.. 3 II ____ is i I
r
q rq q
and
B B
= e l = e =
u " = - ——| X S g .( ).. = - —— Zg ."O ,
r 2 .. r 1 1 2
q31 q q 3 J 3 r3 3
where
§n=;;?1=?;?.

For the "matrix" g to be considered a second-rank (co-
variant) tensor, we must arbitrarily decide that for every
rotation E, we will change the Kramers basis functions so

that the equivalent mathematical rotation R is set equal

 

to E.

Now, if we consider the g2 matrix formed by setting

= fl=
G = ° G = . .
g 9 . pg Esplgql
(12)
== =T= = ===.. ===_
G" = g" 'g" = R'3T°§T'S°9°R 1 = R°G'R 1.
Using the relationships

(EE3T=EE1E,E-=§T,el=?,

16

we see that we need not put any restrictions on E, and that

it is a "true" symmetric tensor.

This tensor is useful when we consider the

eigenvalues of the Zeeman HamiltonianflJ' = —E°H:
B H fz fx-ify
Be = e = =
= _ = _ Q. = =

2 Eng (§% Ci 2 3g ngioi gfioi f +ify -f ,

where
+ BeH
Hq = Q’qIH' I fi = T ggngi (q=XIYIz) I

and the fig are the direction cosines of the magnetic field
with respect to the original coordinate system. The

eigenvalues offij'E=EE are obtained by setting

f -E fX-ify

 

z = 0
fx+1fy -fz-E
E = + #f 2+f 2+f 2 = +(Zfi 2) 1/2 =
_ x y 2
B H
8H11/2 2 1/2
9 2 . (229 =:—-(11 G
:7 i1“; pgpl) ngi 2 pqpqpq) °
Now, when G is diagonal, we have
Be H B H
1/2 e
= +——— 2 = +
E 2 (12 p Gpp) _9-§-

so that a transition occurs when an incoming photon has

the energy

l7

 

AB = hv = gBeH with g = /§L:g:+2;g3+£:g:j
The diagonalization of E by the Jacobi method66 gives the
squares of the principal components (92,93, 9:) and the
eigenvectors give the direction cosines of the new principal
axes with respect to the original coordinate system.

The other mentioned interaction, fiI'Hel (Equation
(10)), also lifts the degeneracy of the ground state.

Following the same procedure as used for the E tensor, we

can form a pseudotensor a and a "real" tensor A from

(Eel =-§-1—B—Zal
9n n 1
=-_—_T.= _
A — a a , pq - gapiaqi . (13)

Similarly, we may determine the energy levels of

+ +
. , = _ . .
the perturbat1on‘H’ “I He1 to obtain

with

 

=/22 XX1). 2+2 )2’AY‘HLZAZ ,

>"

where Ax'A , and A2 are the principal values of after it

Y
is diagonalized.

>fl

The simultaneous diagonalization of the G and
tensors would require that the similarity transform R in

Equation (12) diagonalize K also or, in other words, that

18

one choice of coordinate axes and basis states will lead
to a diagonal representation of E and 3. It has been

determined67

that,if environmental interactions are small
with respect to the energy difference between the J
multiplets of the species, 3 and 3 can be diagonalized
simultaneously. This is also true for the species with
rhombic symmetry, but is not true for those with trigonal
symmetry.

Now, by using the relation

0 i + 10
x 2

+
S:

nflH
:m

j +

NH4

Y
we can write two terms of a spin Hamiltonian

=+

6H = «at-3.9.3 +

H+

3557. (14)

IV. Isotropic Interaction
The second operator in Equation (14) involves the
interaction of the electrons and the nuclei. The Hamil-
tonian for an electron in a magnetic field can be written

as

)2

I0
0 5+

~15 +v++ avg
‘H-Tm‘+ (r) geeH .

68 69

+ 0 O o I
where A is the vector potential. Fermi and Milford have

shown that the Hamiltonian has a singular part at r = 0

and a nonsingular part elsewhere. A careful evaluation of

74' gives

 

 

9 9 B 8 PL ..
74' - e N 63” geQNBeBN{I-§ 3(s-3?) (1%)} '
hr r3 r5
+ 817 B B j? +6 + _ —> + + + = + + + + 1
i—gegN e N s (r) — 51 L I D s aisI s ( 5)

+
where 6(r) is the Dirac delta function and the prime on the
second term indicates that evaluation takes place when

r f 0. The second-order term e2A2/2mc2 has been neglected

since it is small under any normal experimental conditions.
The third term is the Fermi contact interaction giving rise

to a.

is' the isotropic part of :. The 5(?) operator

specifies the electron density at point r = 0 because the

evaluation of <6(;)> gives
(ME) = fw*(‘£>6(?>w(‘£)dr = Iw<o>l2.

The derivation of the Fermi interaction part of
Equation (15) involves electromagnetic theory. It describes
the energy between the nuclear magnetic moment and the
magnetic field at the nucleus due to the magnetic moment of
the electron. There are other interactions though, and
these can give rise to non-classical results such as a
negative value for ais'

Consider the two structures of Figure l,70 where
one pi electron occupies a 2pz carbon orbital (which is

perpendicular to the plane of three sp2 trigonal bonds).

Since the spacial interaction between the sigma and pi

21

bonds is zero, there would be no energy preference between
(a) and (b). The spin interaction, however, would favor
(a) and this slight polarization of the sigma bond would
induce a reverse polarization of the hydrogen electron
spin. Now it is this electron spin on the hydrogen atom

+
which is interacting with the hydrogen I term. Since

+

SH = -§C for configuration (b), we have a negative inter-
+===+ "*

action I'a°S (where S is the electron spin operator on the

carbon atom), which is imputed to a negative value of ais'
To be more exact, we can write a more complete contact

Hamiltonian, which is a function of all the electrons, as

_81[ +--+ +.+
7(c ” 3—9e8egNBNE5(rk rN)Sk I ' (16)
where we sum over the different electrons k, and we can

define the unpaired electron density at the nucleus rN as

+ _ f *22 5 + +
p(rN) _' q) k (Sz)k (rk-rN) WdT I
where 2(Sz)k = +1 or -1, depending on whether the spin is

a or B, respectively; w is now the wave function of all the

electrons k over which the summation is taken. Then we can

write
all _ 4n +
ais - 3 geBegNBNp(rN)
(H = “f.:.‘s*all '
c
- +all . .
where S is the complete spin operator for all the

electrons summed in Equation (16).

22

V. Spin-Orbit Coupling

 

There is also an interaction between the intrinsic
spin of an electron and its own angular momentum. This
interaction and the angular momentum of the electron are
both "caused" by the rotation of the electron about the
nucleus. The general formalism that can be used to
describe the interaction is that of an effective magnetic
field.

The Hamiltonian in Equation (14) can be written as

M. :3 .+ = B g'.-§
“e

(17)

H+

Ca '

Ev - -§-: + 1
E;

where fi' is a "real" vector, because the right~hand side of
Equation (17) has the transformation properties of a
vector. The revolving electron also experiences a magnetic
field caused by the electric field of the nucleus which, in
the electron's reference frame, is revolving in the

opposite direction. The field produced is thus E" =

+
+
- %»x E, so we have an effective magnetic field

+

s =§'+§"=§'- XE. (18)

eff

Ol<+

This will produce a torque on the intrinsic angular

momentum, or the "spin" of the electron,

(mg) + +
—_—dt = uXBeff I (19)

23

where

It can be shown71

that if there is an equation of the form
of Equation (19), and geff is a constant with respect to
time, then the particle will rotate with an angular velocity

vector

 

+_ _—_-e_+' ++.
w — - ch + 2 VXE (20)

8+
_(mE)Beff
mc

We now can show that Eeff is constant. E' is
constant because in Equation (17) H is the constant external
magnetic field, I is constant if there are no NMR tran-
sitions, and 3 and 2 are parameters of the species. In
considering the second part of Equation (18), we know that
the electric force can be written as eE = -§V(r,6,¢). We

now assume that V is spherically symmetric so that
+ + dV
eE - $V(r) ‘; —; I (21)

then

 

+
and we know that the quantum number associated with L is
constant if the electron does not change orbits. Thus,

+
Beff is constant and we obtain the result in Equation (20).

There is yet another correction to Equation (19)

and this comes from special relativity theory which states

24

that a set of space axes that is both moving and being

accelerated has, as observed from an inertial reference

 

 

system, an angular velocity of precession72’ 73
_a _
wT - 2 [(1-3—) 1/2 - 1],
V 2
C
or
.. §x+ V
“0:- a,when—(<1,
T 2c2 C

The coordinates of the electron have the added motion

+ +

+ _ -V X (F/m) _ i—e
“’T‘ 2 ' 2
2C 2mc

+ +
VXE,

 

 

so that a corresponding ET can be written with the same

ratio as in Equation (20):

g; .. i
g T v X E 1+

T = -(e7mc) = 2c = "28

 

This factor of one-half is called the Thomas factor.
Finally, we obtain a "corrected" effective magnetic field

and angular velocity,

 

+

+ _-> _+'_L +
Bcorr — Beff + gT _ B 2c x E ’
+ 1 f dV

= __E ' -_S ___ ___.__
wcorr mcg + ( mc)( 2c)(mer)dr

__ 9"! l+li¥

_ ch + 2 2L r dr

mc

by using Equations (20) and (21). The magnetic interaction

is

' = ? 0+ — i + .+ = + O
4" Bcorr U mc Bcorr S fl wCOI‘I‘
=45... . are: is
2mc r r
_ efi + = —+ -+ = + zfi -+ + l dV
_fn_5HgS+IaS+_—§LSFE?
2mc
'0. 77" -_- Be§.:.§ + fez-cg + AE’E ,
where A = L. l. g;
2mc2 r dr

A is the spin-orbit coupling constant.

VI. Spin-Spin Coupling
There is a tensor form of coupling between any

. + +
nuclear spins Ii and Ij’
_ 2 2 + .= .
(HD — gNBN I. DI I

which is derived in the same way as the second term in

Equation (15). This is caused by the interaction of the

dipole moments of the nuclei. The effect of this term in

liquids is proportional to the trace of 31' A simple

evaluation in any coordinate system shows the trace to be
74

zero.

There can also be an isotropic term

where Jij is an interaction not completely understood. One

important interaction mechanism is the correlation of

26

nuclear spins through the polarization of the intervening
electron Spins,7S which is analogous to the mechanism

described above concerning Figure 1.

VII. Orbital Angular Momentum

 

t“+

There is also a classical energy term E = -H° ,
where i is the orbital angular momentum of the electron.
This is a first-order approximation, in contrast to the
exact energy of —H°I, because the former case does not
consist of an infinitesimally localized current distribution

as is postulated for the "spin."76

The higher-order terms
are neglected though, and we have the added quantum

mechanical analogue

++

04‘ = H'L

operator'

This term sometimes does not contribute much to the
expectation value of the Hamiltonian, or in other words,
to the energy of the system.77 This is because we can

evaluate (L2) as
(L2) = (wl-ifi §$Iw> = -ififw*§-$wdT
and, if w = w*, then
(L2) = —ihf%— .375 (W*lb)d'r = iK .

Now K=0 because the expectation value must be real. So,
whenever the wavefunction for the system can be chosen to

be real, we have (L2) = 0. This can always be done in

27

non-degenerate systems by multiplying by an appropriate
phase factor, which is always allowed.

Species which do not have orbitally degenerate
ground states would then have a "quenched" orbital angular
momentum interaction. This would also be true for the
radicals studied in this work since the Kramers degeneracy

is lifted by the magnetic field.

VIII. Nuclear g Tensor

 

The Hamiltonian can include a nuclear term78

similar to that of the electron in the first term of
Equation (14). The tensor quality would come about by
changes in the magnetic field felt by the nucleus that were
caused by changes in the electron wave function. The
distortion of the electronic cloud can be caused by, and be
approximately proportional to, the external magnetic field

H. This is not important in our studies.

IX. Electric Quadrupole Coupling

 

There is also a nuclear electric quadrupole
coupling. This comes about when the interaction energy of
a charge distribution and an electric potential V due to

external sources is expanded about the origin:79

QIH

++
Energy = Q¢(0) - P'E(0) -

E Q .v.. + ...,
13 13 l]

28

where
Q - = f(3x x -6. r2)p dT
i) i j ij
3E.(O)
Vi' = x
3 1

By using the Wigner-Eckart theorem,80 it can be
shown that the quadrupole term can be changed from an
expression involving integration over coordinates to an
equivalent form in Ix'Iy’ and I2 by multiplying the former
by a constant. This changes the form to

_ 80 3 _ _ +.=.*
fiHQ - 61 21_1 3% Vij[2(Iin+Iin) éijI ] I P I.
From the properties of second-order partial

derivatives, we have Vi'= Then

V..
J 31
the symmetric real matrix E

which implies Pij=Pji'
can be diagonalized by choosing

a new set of axes to give the form81

2 2 2
CHQ PXIX+Pny+PzIz

_ 2 1 1 2 2
_ K + p” [{Iz-§I(I+1)} + g n (I+ ' 1-)]:
where
1 2
K = 5(Ix+ly+lz)[I(I+I)-Iz]
2

 

_ 3 _ 3eQ _ 8 V
P” ’ 212 “ ZI(21-I) q ‘ a 2

= — = + '
n (Ix Iy)/Iz I:- I _ II

29

and we have K=0 because the Laplace equation VZV = Zvii=0
i
implies P +P +P = 0.
x y 2
We see that the n term is a second-order effect
compared to the other term in brackets, and when axial

symmetry is present, n = O.

X. Spin Hamiltonians

 

The energy transitions observed in any type of
magnetic resonance experiment can generally be described in

terms of an appropriate spin Hamiltonian. The Schroedinger
equation to be solved is generally very complicated. The
various terms of a general spin Hamiltonian can be re-
written in terms of raising and lowering operators to give
expressions which are in a form suitable for a computer.82
Approximate formulas for the energy have been obtained from
some Hamiltonians by specifying simplifying conditions and
using first- and second-order perturbation theory.

We list below some of the spin Hamiltonians and
various formulas that have been derived. In the pertur-
bation cases, it is assumed that the first term has a much
greater contribution to the energy than the rest.

Let M be the quantum state of 5,
Let m be the quantum state of I,

Let W be the energy of‘a quantum state,

Let hv be the energy needed for an (M-l)++(M) transition.

30

A. Electron Zeeman Hamiltonian

 

 

 

41.! : Bfi°§'§
Condition: H has direction cosines (gx’zy’fiz) with
reSpect to the principal axes of 3: (3X,3y,§z).
The exact solution i582' 83
W = gBHM
hv = gBH ,
where
(1) if 3 has different principal values,
_ ‘72 '2 2 2 2‘
g — /gxflx + gyly + gzfiz (22)
(2) if 3 is axially symmetric, i.e., gx=gy=g”, gy=€LJ
+
and H makes an angle 9 with 3” ,
i
g = /§fi c0526 + gfsinze . (23)

B. Hamiltonian A Plus Hyperfine Interaction

.+

+ = + +
”H = BH‘g°S + I- -s

w"

Conditions: H has direction cosines (ix’fiy’lz) with
respect to the principal axes of 3. These
axes define the coordinate system.

(1) The First-Order Approximation:
W = gBHM + AmM

(24)
hv

gBH + Am ,

31

where
(a) if E and A have different principal axes,84
A = {(2 g A + 2 g A + R g A )2
x x xx y y yx z 2 2x
2
+(2XgXAXy + RygyAyy + fizngzy) (25)
2 l/2
1 1
+(2ngAXZ + ygyAyz + 2ng22) } /g .
and g is given by Equation (22);
(b) if 3 and A have the same principal axes,
2 2 2 2 2 2 2 2 2 1/2
= 2
A {2"ngX + lygyAy + zngz} /g, (26)

with 9 given by Equation (22);

(c) if 3 and A have the same principal axes, and
are axially symmetric with E making an angle 6

with 3",

2 2 2 2 }1/2 /9

A = {gfiAficos 6 + giAlsin 9 (27)

with 9 given by Equation (23).
(2) The Second-Order Approximation85

(a) If 3 and A have the same principal axes, and
the direction of H is described in this
principal-axis coordinate system by the

standard polar angles 6 and ¢,

32

A A A
X Z

ZgBHA

2
ll

gBHM + AmM - [S(S+l)-M2]m + ZM

hv

A A A
gBH + Am + 333%33 [2M+l]m + z,

2 -A2)2
X 4y coszesin2¢cosz¢} [I(I
A

2
P

(A
2

+1)-m2]
4gBH

A = (g 2Agsinze + g:A:COSZG)1/2/gr

A = (gx 2Azcos2 ¢ + g sin2¢)l/2/g

2A
P YA

N‘<3,N

21/2

2 . 2
(gPSin 9 + gzcos

6)l/2 , g = (g: cos 2¢ + gisin2 ¢)

P

Q
II

(b) if 3 and A have the same principal axes, and
are axially symmetric with H making an angle 6

. ->
WIth g",

'2‘.
ll

gBHM + T

hv

gBH + UA ,

where

33

 

 

T = ZM - fijA” [S(S+l)-M2]m 28
A ZgEHA ( )
AiAII
UA = Z + m [2M+l]m, (29)
9% Qi‘Afi ’Ai)2 2 2 2 Ai(A%Afi ) 2
Z = 5 2 m sin 6cos 9 + 2 [I(I+l)-m ]
29 BHA 4gBHA

and g and A are given by Equations (23) and (27),

respectively.
C. Hamiltonian nglus Zero-Field Splittigg
+==+ +=+ +==+
’H= BH'g'S + I-A-s + s-D-s

= +
,D have different principal axes. H has

V“

Conditions: 3,

direction cosines (Rx'fi .12) with respect to

Y

any arbitrary coordinate system (x,y,z).

3,:, and B are expressed in this coordinate

The second-order approximation i586

_ 1 _
hv — gBH + AM + 2§Bfi'(TxxTyy TxyTyx)(2M+1)m

T2 +T2 +T2 +T2 T2 +T2

xx gyy, xy yx _ 2 xz z 2
+ 49“ [I(I+l) m 1 + “7381};— m

'2 + '2
xz Ayz
ZgBH

 

D

 

[4S(S+1)-24M(M+l)-9]

I I 2
(Dxx - Dyy) +4D

BgBH

'2
X1 [28(S+l)-6M(M+l)-3] ,

 

34

where

U .V .A

i'
3 mn mi n3 mn
I
0.. = )[U .U .D
1] mi n] mn
mn

l/2
A = ( X K.K.A. A. ) /g
ijk i 3 ik 3k
_ 1/2 _
g _ (.2 ILifijgikgjk) ’ Kn _ E:’?“mgmn
ljk m
KZ/Y KlKB/gY Kl/g
U - -Kl/Y K2K3/gY Kz/g
0 -Y/g K3/9
Y=VKi+K§a

v is obtained from 3 by changing Ki to Ki=ZKjAji and g to
j

Ag.

D. Hamiltonian C plus Electric Quadrupole and Nuclear

Zeeman Terms

+

H+
l
ID
33
“5,".
L:
é+

W“

+ +
08+ I.

U"

+ "F
OS + S.

m"

2” = B§'§°S + i-

,E,F, and 3(1)

W"

Conditions: 3, have the same principal

+
axes. H has direction cosines (RX,£ ,lz) with

Y
respect to the principal axes of 3, which

define the coordinate system.87

35
(l) The First-Order Approximation:

W = gBHM + AmM + D"(M-%) + P{m2-%I(I+1)} - G m

I

hV gBH+AIn+D I

where g and A are given by Equations (22) and (26),

respectively, and

" _ 2 2 2 2 2 2 2
D _ 3(2ngox + zygyDy + QZgZDz)/g
_ 3 2 2 2 2 2 2 2 2 2 2 2
p _ 2(2ngAxPx + 2ygyAyPy + zzngsz)/g A
= 2 (I) 2 (I) 2 (I)
GI BHULngAXgX + iygyAygy Rzngzgz )/gA ,

(2) If we also have axial symmetry for all the
tensors, we have the Second-Order Approxi-

mation:

2‘.
II

+
gBHM + TA TD + TP + T9

hv = gBH + UA + UD + UP ,
+=+
where TA'UA correspond only to the I°A°S term,
TD’UD correspond only to the §°B°§ term,
+==+
TP'UP correspond only to the I°P°I term,
+=(I)

Tg corresponds only to the -BH-g °I term,
so that any Hamiltonian made up of only some of the tensors
has only those corresponding terms in the equations for W

and hv.

36

and U are given by Equations (28) and (29), respectively.

+

A

%D{(3gfi /g2)cosze-l}{M2-%S(S+l)}

(g2 gi/g4)(choszesinze/ZX)M{8M2+1-4S(8+1)}

(gfi/g4)(Dzsin46/8X)M{ZS(S+l)-2M2-l}

D{Zgfi cosze—gisin26}/g2

(gfi gE/g4)(D2c05263in26/2X){24M(M-1)+9-4S(S+1)}

(gfi/g4)(Dzsin46/8X){28(S+l)-6M(M-l)-3}

‘ 2
9 9 A A
P{m2--;'-I(I+l)} + (H '2 g 'L) {Pfi sin226/8AM}m
g A
{8m2+1-4I(I+1)}
(gLAL/gA)4{Pfi sin46/8AM}m{21(I+l)-2m2-l}

-(gll gLA“ Ag/92A2)2{Pfi sin228/8AM(M-l)}m{8m2+1—4I(I+1)}

(gLAg/gA)4{Pfi sin4e/8AM(M-1)}m{2I(I+I)-2m2-1}

‘(BHm/gA)(g(I)g A c0526 + g(I)

. 2
H II N .L giAiSID 6)

3
7 DH

98H

3 2 2 2 2 2 . 2
Eig” A P cos 6 + gLALFLSin 6}/g

2 2
H II A

37
g and A are given by Equations (23) and (27), respectively.

XI. Determination of ESR Parameters
In general, the parameters of the 3 and A tensors
are determined from isofrequency plots, where the frequency
v is constant and H is swept through a range for each of
several orientations of a crystal rotated about a specific
axis. To determine the values of the magnetic field at
which resonance occurs, we can use Equation (24) for a

first-order approximation:

where Ap is the hyperfine interaction with the pth nucleus

and has appropriate energy related units (usually MHz), Ap

corresponds to Ap but has units of gauss, and mp is the

quantum number of the pth

nucleus with values from -Ip to
+Ip. Halfway between the outer lines of any hyperfine
multiplet would be the place where (perhaps only mathe-
matically) mp = 0. Choosing the midpoint between the
outermost lines of the whole spectrum will give H0 =
hv/gB. By considering only these midpoints, we have a
"reciprocal g-value" plot for the crystal. The values of
Ap are determined by measuring the distance between the

outer lines of the corresponding multiplet and dividing

by 21p. By plotting these values versus the angle of

38

rotation of the crystal, we obtain A-value plots. We thus
have two symmetric covariant second-rank tensors to
determine.

Since g is given by Equation (22), it can be

rewritten as

If we form the symmetric diagonalized g tensor as

ij = giéij = Gji r ‘30)

where 6ij is the Kronecker delta function, then we can

write
2 2 2 '
g = 2 ~29 5 -g- = X {G -G
1 ij i 1] i l] l i ij )1
=l =I =_ = =0 =_ = =l __ =
= )2?(G °G ).. = {2?(R 1-R c °R 1-R-G °R °R)..
. 1 ll . 1 11
l l
- 2 -1 =,=',=-1‘,=.=',=-1
_ .2 iRimHR G R )(R G R )lmn ni
imn
= .2 (Run. 1) (Rnifii) (G.G)mn
imn
. 2 _ ' '
.. g — Z£m£nwmn. (31)
run
where
W = 32 , fi’ — ET - (32)

39

Here 3 is an orthogonal matrix which represents a proper
rotation to a new coordinate system, and 2A and E. are the
direction cosines of H and the new representation of E in
this new coordinate system, respectively. Equation (31) is
of the same form as Equation (4) in Part II of this thesis,
so the values of the W matrix can be determined by any of
the methods presented there.

To solve for 3, we can first diagonalize W by a

similarity transformation
===T ='
S°W°S = W (diagonalized) ° (33)

Then,by forming a diagonal matrix whose diagonal elements

0
are the square roots of the corresponding ones in W , one

obtains
6:67, (34)
but, from Equation (33), we have

(73:54???

E) = 5-3 ,

2“
ll

=§T

0"

so -E'°§ is the solution to Equation (32).

We now want to determine A for the most general
case in which 3 and A have different principal axes. If
we multiply the terms out and rearrange them in Equation

(25), and use primes to signify the fact that the 3 and A

terms are in the 3 principal-value coordinate system, we get

40

2 2 _ ' ' ' . . __
izjxijg’ig'j I (1!) - XIYIZ)

where a typical term is

ij = 9. 1g. 3(Aixij+Aiijy+AizAjz)

91 9- [AikA jk .

Using Equation (30) and the fact that K is symmetric from

Equation (13), we can write

.Z.(giAik' )(g jAjkm 2).
13

ngz

= kEjdgiéilA lk)(zgm6mjAmk)£12j

= )(Zcil A'lkMZAka mjmiz

kij 3
Z ( ) kF" ‘3 2'2'
= G. .A' A .G 0 0 0
kij k3 1 3
=' =' =' =' ' I
= §:(G -A -A ’G )i.£i£
ij 3 J
. 2 2 I l I
'° 9 A - fig 13 123 I

where

41

Hence, in any coordinate system, we can write

xn
H

on
éu
$n
on

which again is of the same form as Equation (4) of Part II.

We can solve for K by determining X from angular variation
of the quantity ngz by any procedure in Part II, then

solving the equation

-1 1

0;.03-

y“
0“

Z.

by the method mentioned for Equation (34). This result has

88

been stated elsewhere. If 3 and K have the same principal

>H

axes, then the principal values of are found by diago-

a: =l
nalizing X to give X and using the equation
0 1 I
A.. = VX../G.. .
ii 11 11

If the relative anisotropy of g is much smaller than that

of A, we can treat A exactly as we did 9, that is

 

 

' Ag
g. A.
180 180
then
I
A JQZAz + £2A2 + £2A2
x x y y z
.0 A2 = {2; 2W 1
m n mn

42

the largest anisotropy
the largest anisotropy
is the isotropic value

is the isotropic value

of

of

of

of

«MI >H 0H

>H

EXPERIMENTAL

I. §pectrometers

 

Three ESR spectrometers were used, two of them X-
band systems and one a Q-band system. The X-band systems
were used with a magnetic field of about 3300 gauss and a
resonant frequency of about 9.5 GHz. while the Q-band
system was used with the corresponding values 12000 gauss
and 35 GHz., respectively.

One spectrometer was the Varian V-4502 X-band
system with a lZ-inch magnet and 100 KHz. modulation.
First- or second-derivative spectra were taken on various
XY-recorders. The appropriate derivative of the absorption
mode was plotted as a function of the magnetic field
intensity. The magnetic field was measured by accurately
determining the proton NMR resonance frequency of a water

sample and using the equation 89

H(gauss) = 0.2348682 vwater(MHz),

A homemade marginal oscillator90 was used to detect the
proton resonance and the frequency was measured with a
Hewlett-Packard Model 524C electronic counter. After each

spectrum was taken on the XY—recorder, two accurate

43

44

magnetic field determinations were made on top of the
spectrum. From this, it was found that the linearity of

the magnetic field sweep and the stability of the absolute
magnetic field on the spectra were within the experimental
error of the NMR probe measurements. The klystron frequency
was measured using a TS-148/UP U.S. Navy spectrum analyzer
with a calibration chart; the accuracy was about 1 part in
105.

For some preliminary spectra, a Varian E—4 X-band
spectrometer was used. This also had 100 KHz. modulation,
but the magnet had 4-inch diameter pole pieces. The
absolute magnetic field, the magnetic field sweep, and the
klystron frequency were read from the dials on the machine.

The Q-band spectrometer was a Varian V-4503 system
with a 12-inch magnet and 100 KHz. modulation. There was no
external probe to measure the magnetic field accurately.
The klystron frequency was determined by the wavemeter of
the Varian V-4561 35 GHz. Microwave Bridge. Repeated
measurements indicated that the stability of the klystron
frequency was only 1 part in 104.

The X-band spectrometers had provisions for keeping
samples immersed in liquid nitrogen and all three spectro-
meters could be used with a Varian V-4540 variable temper-
ature controller. This instrument regulates the sample

temperature by passing over it a stream of gaseous nitrogen

whose temperature is controlled by either being passed

45

through a helical tube immersed in liquid nitrogen and/or
warmed by heating filaments.

\ A common sequence of measurements involves mounting
an irradiated crystal with a chosen axis vertical. When the
X-band system was used, magnetic field sweeps were made
with the crystal progressively rotated about that axis.
With the Q—band system, the magnetic field was instead
rotated about the sample. This is the more desirable
arrangement since moving the crystal in the cavity changes
the Q of the system which tends to change the klystron
frequency and the detector current leakage. The magnet on

the V-4502 was not rotated because the connecting hoses

were too short.

II. Crystal Irradiation and Mounting
\ There are two methods for mounting crystals that
must be kept at liquid nitrogen temperature. One method is
to irradiate the crystal, then mount it between flexible
brass strips. This method, which is described in the thesis

60 has problems associated with it. It is

by Watson,
difficult to align and ascertain the alignment of the
crystal. The lack of space caused by the presence of the
brass strips in the liquid nitrogen Dewar makes liquid
nitrogen bubbling more likely because of "hot Spots" on
sharp edges of the brass. Also, any bubbling that occurs

tends to shake the crystal causing electronic stability

problems and creates a possibility of accidental realignment

46

of the crystal. The metal in the cavity also lowers the
sensitivity of the instrument.

Another method is first to glue the crystal
(Pliobond cement, Goodyear Rubber Co., Akron, OhiO) onto
a copper wire which in turn is imbedded into a thin glass
rod. This allows one to adjust accurately the axis of
rotation. The whole assembly is then irradiated and used
in the same way as the crystal holder in the first method.
An extra signal at g = 2.0026 is introduced from the
irradiated glue.

There were two sources of irradiation used. One

60

was a Co y-ray source that had an intensity of 1.0 X 106

rad/hr. The other was a 1 Mev electron source (G.E. XRD-l
Resonant Transformer) that had a dose rate of 1.8 X 107

rad/hr.

III. ENDOR Measurements

 

Some electron-nuclear double-resonance (ENDOR)
measurements were made on crystals of dlfmandelic acid.
A Varian E—700 ENDOR system was used with the V-4502 X-band
spectrometer. A Monsanto Model 1100A counter-timer was

used to determine the rf pulse frequencies.

IV. X-ray Crystallography

 

dlfMandelic acid, C6H6CHOHCOOH, in powdered form

was obtained from Matheson Coleman and Bell Co. Crystals

were grown from saturated aqueous solutions by slow

47

evaporation. The morphology of the crystals was the same
as determined by Rose.91 He reported that the crystal
system was orthorhombic with cell dimensions a = 9.66:
0.05 i, b = 16.20 i 0.08 3., and c = 9.94 i 0.05 i, and that
the number of molecules per unit cell was eight. An
orthographic projection of dlfmandelic acid is shown in
Figure 2. Small crystals of about 0.1 mm. length were
grown in a shallow dish by slow evaporation from water.
The crystal chosen had a clear-cut morphology that allowed
us to mount it specifically about the a axis. This was
confirmed by taking oscillation photographs92 using a
Weissenberg Camera (Supper Co., Watertown, Mass.) with
filtered Cu (AKa = 1.5418 A) radiation. The cell
dimensions were determined from these photographs and
compared with those reported by Rose.91 There was excellent
agreement within experimental error. The oscillation
photographs were used also to align the rotation axis of
the crystal accurately along the a axis. The zero-,
first-, and second—layer photographs were taken about the
a axis. By considering the symmetry of the missing
reflections, we were able to determine the limiting
conditions for reflection in terms of the hkl indices.
This uniquely determined the space group out of the 74

possible ones for the orthorhombic case.

48

.cflom oaamoamEme mo Hmummuo m mo cofluommoum UHnmmumonuuo

AI‘.NNo:IV

 

0.0

/3\
\~r

.. Tr:

 

 

 

.m musmflh

49

V. Parameters From Tensor Variations
The purpose of this section is to describe a way in
which the 3 and A tensor parameters can be determined when
there is restricted information from the isofrequency
plots. This was the case for mandelic acid when the normal

93 did not work. The

analysis utilizing Schonland's method
reason that Schonland's method, and the method outlined in
Part II for the orthogonal case, both tend to have diffi-
culty is because they depend heavily on the high accuracy
of the g-values for all three rotations. This is difficult
to achieve since three separate experiments are involved
with the spectrometer and each time the tuning character-
istics differ. Also, if there are broad, overlapping
lines, as was the case for mandelic acid, further un-
certainty is added.

It appears that some of the most accurate data that
can be gathered from an isofrequency plot are the g
variation (Ag) between the maximum and minimum values of g
for that plot, and the angle at which gmax occurs (emax).
This gives six pieces of data from the three isofrequency
plots. We need formulas relating these quantities to the
principal values and eigenvectors of 3.

At first we need to calculate the parameters 8 and
Y as defined by consideration of the value of g at a
specific magnetic orientation given by Equations (1) and

(4) in Part II:

50

2 _ _ .
g — Zwijzizj — a + Bcos20 + YSin28 ,

ij
where W = 3': and the li are direction cosines of the
magnetic field vector with respect to the coordinate

system. From Equation (29) of Part II, we have

_ 1 _
8 — 2 ijwijmisj MiMj)

(36)
Y'== {w .S.M. ,

ij 1) l j

where Si and Mi are the direction cosines of the vectors
g and M, which represent the magnetic field direction at
the start of the isofrequency plot (0 = 0°) and at the
middle of the isofrequency plot (0 = 90°).

There are two coordinate systems involved, one
associated with the crystal axes, which is left unprimed,
and one associated with the eigenvectors of the g tensor,
which is primed. Using Figure lb in Part II, we can write,
in the crystal-axis coordinate system, for the three

standard rotations about the orthogonal axes,
S. = 6. , M. = 6. , Q = 0 , £8 = cose , 1m = sinfl ,(37)

where the subscripts n,s,m refer to the rotation, starting;
and middle-axis numbers. Putting Equation (37) into

Equation (36) gives

SS mm

In the principal-axis coordinate system of the g tensor we

have

and to change into the unprimed system, we set

= = =' =
W = RT'W °R
w = NW) = 2R. g?6..R.
Pq ij Piw ij qu ij 1P 1 1) Jq
2
W = .R. R.
pq £91 lp 1q
. _ l 2 2 _ 2
o o B - .5 ggi(RiS Rim)
Y = ZgiR is Rim
8 = 1 arctan(Y/B)
ex 2 '
where eex 15 an angle assoc1ated w1th gmax or 9min and the

principal g-value eigenvectors in terms of the unprimed co-
ordinate system are along the rows of R; or, likewise, the
crystal-axis eigenvectors in terms of the primed coordinate
system are along the columns of 3. Since the arctangent

function has two solutions 180° apart, we see that emax and

52

6min must be 90° apart. The two values for the extrema of

g are calculated from Equation (22):

2 _ 2 '2
9ex — £9111 (eex)

' _ T
2i(eex) — E<R )ika(eex)

_ 2 1/2
9 " {1211911391 (eex)}

I

where £n(eex) = 0 , 23(eex) = cosBeX , £m(eex) = s1n8ex .

).

The g-value difference Ag is then (gmaX-gmin

It is impractical to determine the values of gi and
the eigenvector matrix R from the values of Ag and emax'
Instead, the formulas calculate Ag and emax for the three
orthogonal rotations, given a specified 9 tensor. There
are three independent parameters describing the eigenvector
matrix, but there are nine elements in 3. So instead of
varying the elements in R, it is best to vary some set of
three independent parameters and calculate R from them.

One such set is the Euler angles (0,8,Y) as defined
by Rose.94 We can construct an orthogonal matrix by using

a formula by Rose.95

53

ca°cb-cc-sa°sc sa-cb-cc+ca-sc -sb°cc

R = -ca°cb°sc-sa°cc -sa-cb°sc+ca°cc sb-sc
ca°sb sa°sb Cb
(38)
ca 5 cos(a) , cb E cos(B) , cc 3 cos(Y)

Ill
HI

sin(Y) .

sa sin(a) , sb sin(8) , so

A computer subroutine was written such that given
three Euler angles, three principal g values, the experi-
mental values for Ag and emax, and the corresponding
weighting factors, it calculated the theoretical values for

Ag and 6m and determined a squared-d1fference-weighted

ax
error. This subroutine was used with a general minimization
routine that varied the parameters to determine a minimum
error. These subroutines are listed in Appendix I.

The A tensor can likewise be calculated by using
this procedure for the quantity gA, or A, as was discussed

under Determination of ESR Parameters in the Theoretical

Section.

RESULTS

1. X-ray Crystallography

 

The limiting conditions for reflection in terms of
the hkl indices were determined as described in the
Experimental section. These are listed in Table 1. There
is a completely symmetric relationship among the hkl indices
in these rules. By comparing these conditions with those

96 it was determined

listed in the International Tables,
that the space group is Pbca. This space group has four
equivalent right-handed positions related by three screw
axes about the a,b, and c axes. It also has four left-
handed positions which are similarly related. These eight
positions could be considered as corresponding with the

four gfmandelic acid and the four 1—mandelic acid molecules

in the unit cell.

II. Irradiation and Spectra

The crystals generally were irradiated with a 60Co

y-ray source at a dose of 3 X 106 rads. This treatment
caused the crystals to change from a colorless to a slightly

yellow appearance. When single crystals of dlfmandelic

acid were irradiated at 77°K, and the ESR spectrum was

54

55

 

moozm

 

XHO3 mHSB mm.m
x003 mHna Hom.m . ao.mc HN.@H mmm ommum>¢
mm mH.H om.H mm.H mH.m mm.m om.q mm.aH cHom Hoamnum
mm 06.0 mo.H am.H sH.m mm.m am.v mm.aH :Hom causmouesnmuume
mm mo.o on.H sm.H 5H.m am.m Hm.a mm.VH cHom Honoon d888,».H
mm am.o am.H om.H «H.m mm.m om.a mm.qH cHom Hocmgum
mm mm.o so.H om.H 6H.m mm.m mm.a km.aH cHom HonoOHm Hsucmm
I/O\ws/: 8 Siov SH SH 26 2%.... $4. 2.3 58 H918? 2.20m
mocmummmm 1mmva Amuse Amuse lomva Aamca immca lkmca ucmecouH>cm

 

.xwuuma Hmummuo onch m ca new mcoHpsHOm msoHum> cH HMUHpmu
H>Ncmn>xou©>£|mnmam gnu mo mucmumcoo msfluumem mchummmn on» mo mmsHm>|u.m mHnme

 

 

lam u Ho . H o 0

law u xv u o x 0

law u no u o o a

an n s u o x s

cm a H a H o s

cm a x u H x o

mcoHqucoo on u H x n
mcoHqucou mwowocH

 

.oHom OHHmeamaumm mom mcoHuoonmu «HnHmmom may mcHuHeHH maOHuHocoouu.H mHame

56

II .xohb um
omumHomuuH oHom UHHmpcmanaw mo ousumummfimu Eoou um Esuuommm mmm Hmowmwu d

 

 

 

 

 

 

 

 

0 n6

0 c.0—

.m musmflm

 

i?
1., $1.3...

0.: 1 .\ k
l”, K
I .
a
..II

.3 . .

‘ a ,. -
\» kegs ..
sees

 

 

 

 

58

.mcmHm no 0:» ca coHumucwHHo onHm mo
coHuoc5m m mm HMUHcmu axucmnmxouomslmzmHm map How mcoHuHmom mcHH mo uon .m muomwm

on: 00

 

o=r.lll

 

 

 

1:1 .2:

 

59

.OCMHQ no man CH coHumucmauo 0H 0
coHuoc5m m mm HMUHUmH Hwncmnmxoupwnlmnmam map How mcoHuHmom ucaapwo.W0mm

 

o_’L
_ _ _ . _ .

.1. e. I

. 0o. _
o=z — _ _ _ _ _

 

.w muswflm

o

oo

 

 

om—

60

taken before the crystals were warmed up, it consisted of
one very broad line at the free-spin value for electrons.
Upon warming up, a "normal" spectrum resulted. When the
crystals were recooled to 77°K, the same type of broad line
resulted, but it was smaller in amplitude. A typical first-
derivative normal spectrum is shown in Figure 3.

This spectrum consists of a central portion with
large peaks, and two other portions on either side with
considerably smaller peaks. The central portion consists
of between seven and about twenty-two resonances over a
range of about 33 gauss, centered approximately about the
g free-spin value. The two groups of side peaks are
symmetrically placed on either side of the central portion
with a distance of about 92 gauss between the centers of
the two groups. Each side portion consists of three to

twelve resonances over a range of about 29 gauss.

III. Alpha-Hydroxybenzyl Radical

A. Determination of Radical

The central spectra can be analyzed as consisting
of two sets of lines for the rotations about the b and c
axes, and four sets of lines for the rotation about the a
axis (Figures 4-6). Each set of lines consists of two
groups of quartets which have one line overlapping to form

12C and 16O have no magnetic

seven lines (Figure 3). Since
moments, only the hydrogen nuclei produce hyperfine

splittings. The observed set of lines could be produced

61

if one hydrogen caused a splitting of 16.0 gauss and three
equivalent hydrogens produced a splitting of 5.3 gauss each.
We postulate that the species is the alpha-
hydroxybenzyl radical, the alpha hydrogen causing the large
splitting, the nearly equivalent ortho and para hydrogens
the smaller splittings; the meta and hydroxyhydrogen
splittings are too small to be observed in the broad lines.
A comparison of the splittings for the alpha-hydroxybenzyl
radical as reported in solution and the average ESR values
of this investigation is shown in Table 2. The agreement
appears good. Also, in the Table is listed the value of
5.83 gauss obtained from an ENDOR experiment on d1fmandelic

acid consistent with the value for the para hydrogen.

B. g—Tensor Evaluation

93 of deter—

We tried to use Schonland's method
mining the tensor parameters for E and K, but it failed.
We then used the method described above in Parameters from
Tensor Variations in the Experimental section. There was
an ambiguity about which set of lines in each isofrequency
plot belonged to a specific site of the radical. Each
combination was tried in the six—parameter error-
minimization routine also described above. We forced one
of the principal values to remain the same during each
computer run of this routine. We would then change this

value independently to give the correct g-value average

and give the other five final parameters as starting values

62

for another run of the program. This procedure stabilized
the search for an absolute error minimum in the six-
dimensional parameter space. At most four runs of the
program were needed to reach a stable minimum. Although
there was no guarantee that the minimum reached was not
just a relative minimum, the searching method of the
program was written to reduce this possibility.

One combination produced a considerably lower
fitting-error than any other one. According to the space
group of the undamaged crystal, there are four sites related
by three twofold screw axes. To determine the eigen-
vectors of a radical related by a twofold screw about any
axis, we need only change the signs of the direction
cosines associated with that axis. When we formed these
four theoretical isofrequency plots and put them together,
they overlapped quite perfectly to form half of the sets
of lines observed experimentally. Since the other half of
the sets of lines were caused by the same radical, we tried
to fit them by varying only the three Euler angles while
keeping the same principal g values determined previously.
Each remaining combination of lines was tried in a three-
parameter minimization routine and one combination fit much
better than the others. This second fit was not as good
as the one obtained in the previous procedure. We now
determined the remaining four sites required by the

symmetry of the undamaged crystal. When all eight

63

theoretical plots were put together, they accounted for all
the lines. The principal values, and direction cosines of
the radical, for all eight sites are shown in Table 3 and
the various site overlappings are shown in Table 4.

There are three causes for the large number of
overlapping isofrequency curves. First, all the iso-
frequency plots have a near-mirror symmetry about the 8 =
0° and 0 = 90° positions (which point along crystal axes).
This requires mathematically that sites must overlap in
pairs. Secondly, the isofrequency plot for the ab plane
shows that the extrema occur along the axes, which also
requires sites to overlap in pairs. Thirdly, there is an
accidental, experimental near-overlapping of lines for the

rotation in the ac plane.

C. A-Tensor Evaluation.

The alpha hydrogen splitting was the only one whose
anisotropy was large enough to be measured. Even in this
case, there was a large scatter of experimental A-value
points. This can be seen in Figures 7-10 where the A-value
plots are labelled according to the g—value lines with which
they are associated in Table 4. A rough calculation was
made of the ratio between the fractional changes of the g
and A tensors as defined in Equation (35). Since the ratio
was small (0.008), we treated the A plots as representing
a diagonalizable second—order tensor rather than the plots

of the quantity gA.

64

 

mm

 

+ u + + + u u u u u + + aaOH.o hmmm.o mama.o am.aH- a
- I - - + + + n + + + . ommm.o ooma.o moam.o mm.mH- «Na
+ u u + + + - u + u + . mHom.o HmH~.o Hamm.o mm.mH- HHa
o n m o b m o n m o n m
m muHm a muHm m muHm m muHm
. + u a u + + + + + u . mmam.o oamm.o mmmm.o am.aH- mma
. + + u u u + + - + u + mmms.o amHm.o smma.o mm.mH- mma
+ + + + u u u + u u u + MHmm.o smHm.o 8000.0 mm.mH- HH<
o n m o b m 0 Q m o b m
a muHm m muHm N muHm H muHm
. u + u + I + u u + + + osmm.o amma.o masm.o mmoo.m mmm
+ - u + + + - - + u + . mHan.o mamm.o mmHN.o mmoo.~ Nam
. n u u + + + u + + + . mHoa.o momH.o oHom.o mmoo.~ HH@
0 b w o n m o b m o n m
m muflm a muHm m muHm m muHm
+ n + + + - u u u - + + moam.o maom.o oamm.o mmoo.~ mmm
+ u u + + + u u + u + . mmsm.o nHmm.o ammv.o mmoo.m Nam
- u u - + + + u + + + . Namm.o mmmm.o ammH.o mmoo.m HHm
o b m 0 Q m o b m o n m 0 Q m
a muHm m muHm N muHm H muHm

mmuHm msoHum> map How mcofim mwchoU aofluomuflo mmsHm>

. Hmmflocflum

 

2H HmoHcmu Hmucmnwxonpxnumanm

:msmHm on» cam Homcmu m mnu now

.aHom oHHmocmsme

may mo HOmcmu mcfluuHHmm mchummwn cmmouoms
mmchoo coHuomqu cam.mmsam> HmmHocHum mseul.m memB

65

Table 4.--Correlation between the individual sites and the
resulting overlapped lines for the alpha-
hydroxybenzyl radical.

 

Axis of Correlation of Center-of-Spectrum
Rotation Sites to Lines Lines

 

 

 

 

 

 

 

 

 

c 180'
a Site: 1 2 3 4 5 6 7 8
Line: B C C B A D D A
b ‘70,
c 0°
‘1 - ”'0'
b Site: 1 2 3 4 S 6 7 8 A
Line: B A B A B A B A
6 40°
0.
00
‘ 170°
0 Site: 1 2 3 4 5 6 7 8 1 fl 5 Q0:
L1ne: A A A A B B B B
a 0'

 

 

 

 

66

.Q cam a mmcwH “Om mcmHm on mnu CH cowumuou mo mamas .MN HMUHUMH
Hmucmbmxoucmnlmnan 03» mo cououm mcmHm on» no meauuflamm mchuwmwm

8, 02 o: 8— 02 o 8 8 9. on 0

II mfipl
o.mH|
mop...
PK TI
NNHI

@le

 

0.9...

(ssneb) v

.5 musmwm

67

O cam m mmcHH How mamam on on» ca coflumuou mo mamas .MM Hmoacmu

HANcmnmxoupmnlmnmam may mo cOpoum manm mnu mo mcfiuuaamm mcamummhm

09

\s

00— 03 cup 09

ooHl

‘0 HI

IO

 

|1

mdpl

U ocfla D

m mafia O NNTI

..I ONHI

.m musmflm

(ssneb) v

68

09

02

0

m 05H o
a 0:3 o

03

O

.mcmHQ mo opp cH coHumMOH mo m was .MN
HmucmbmxoucmnlmnaHm 0:» mo cououm mamas on“ mo mckuHHmm mammmwmwm

Cup

'1

'4

n

00..

5
S

cm

9

\s

'1

.m musmflm

(ssneb) V

69

.mcmHm no mnu ca coHumuou mo mamas .MN Hmofipmu

Hawcmbmxoupmnumnmam mzu mo cououm mamas may mo mcHuUHHmm mcflmummwm .OH musmfim
o
312.2111... 2. a s e a o
____________J
Ilofll
O O O O OH
,. 0.2:
Q all
. («.670
o ll. .)
6
.l. m.
.. m «NT
U

    

m 05H 0

a S: 0 ST

-
‘

U
.l. ONHI

70

Using the experimental values for AA and emax for
the A-value plots, we again used the six-parameter mini-
mization routine on each combination of lines. One
combination of lines had a considerably smaller fitting-
error than the other ones. This set of A-value lines
corresponded to the same g-value lines (site 1) which had
given the best fit for the g tensor. Again using Equation
(38), we produced a matrix iv representing the eigenvectors
of A for site 1.

Since the A and g tensors are for the same radical,
it was expected that the tensors were similarly related to
each other for all eight sites. Because of the similar
symmetry of the A and 9 plots, we could also produce half
of the A-value line sets by applying the same three screw-
axis-transformations to :v' This trivially guaranteed that
the g and A tensors were similarly related for sites 1,2,3,
and 4. For the g tensor, sites 5,6,7, and 8 were symme-
trically related to each other, and site 5 was related to
site 1 by some nonsymmetric similarity transformation
matrix which was calculated. By applying this transfor-
mation to :v' we produced A-tensor eigenvectors for site 5
that had the proper relationship to the g tensor for site
5. We again derived the A-tensor eigenvectors for sites
6,7, and 8 from the one for site 5 by applying the screw-

axis operations. When all eight A tensors were considered

together, they accounted for all of the lines in the A

71

plots. The principal A values and direction cosines of the
alpha-hydrogen splitting for all eight sites are shown in

Table 3.

D. Variable Temperature Study

A variable temperature ESR study of the alpha-
hydroxybenzyl radical was done. The Q-band instrument was
used and the temperature was varied from -150°C to +120°C.
There was no discernible difference in the spectra.

IV. Cyclohexadienyl-Glycolic
Acid Radical
A. Determination of Radical

From the spectrum shown in Figure 11, we see that
each set of peaks can be analyzed as consisting of a
triplet of triplets with splittings of 8.99 gauss and 2.65
gauss. The large 95.5 gauss separation would not normally
be caused by a hydrogen atom, so we can assume that there
is a triplet splitting of 47.7 gauss. The middle portion
of the spectrum would be lost in the larger peaks of the
alpha-hydroxybenzyl radical.

We postulate that the side peaks are caused by the
cyclohexadienyl-glycolic acid radical. A comparison of the
splittings for this radical as reported in solution with
the average ESR values of this work is shown in Table 5.
The agreement appears good. We attribute the largest

triplet splitting to the two para protons, the second

72

 

 

 

 

 

 

 

 

 

 

TE
.6 d w m N N "mch u
2. %m .. vNNH "muHm
.qe. s c
.6 a N s N s ”mch n
e« m u a m N H "muHm
Q
L
.Qu_ a
.o N o N w m N "mch m
w a v m N H "muHm
.o
x
.6». u
muoam msam>|¢ mmcHA ou mmuflm coHumuom
mo coHumamuuou mo maXd

 

.HMOHomu pflom oaaoowamuncmemxmnoHomo

on» you mmcHH cmmmmHum>o mcHuHSmmu on» cam mwuflm HMDUH>H©cH on» cmm3umb coHumHmuuoolu.m anme

E: E
El 2 I

2

I
\
:r
2:
I

 

AHmumwuuanmchv

xuoz mHne o.m om.m v.0v HHHV mo mmm mmmum>m
NOH.HOH mo.N mm.m HN.Nv cHom mcmHomxmnoHuso-a.H cH AHV
00H m.N v.0H m.Na cHom wamucmn cH 1H5
mocmuwmmm m4 Nd H4 ucmacouw>cm

 

.xwuume HmumuuonHmCHm m 2H can mCOHusHom msoHum> :H HMUHUMH pflom
owHoo>HmlH>cmemxonoHomo mnu mo mucmumcoo wchummws on» no mosam>nu.m manme

73

Table 7.--The principal values and direction cosines for the CH2
hyperfine splitting tensor of the cyclohexadienyl-glycolic
acid radical in1g17mandelic acid.

 

Signs for the Various Sites

 

 

Principal Direction Cosines
Values site 1 site 2 site 3 site 4
a b c a b c a b c a b c a b c
All = -28.50 0.6323 0.5228 0.5717 - + - - + + - - - + + -
A22 = -52.48 0.7236 0.1350 0.6769 + + - + + + + - - - + -
A33 = -57.26 0.2767 0.8417 0.4637 - — - - - + - + - + - -

 

Table 8.--Unpaired electron spin and excess charge densities calculated
for the alpha-hydroxybenzyl radical by the McLachlan

 

method.107
Atom 0 E
c 0.001 -0.053
1 o
c 0.159 -0.036 H\é/ \H
2 7
c3 -0.053 0.034
c4 0.214 -0.145
05 -0.065 0.054
c6 0.187 -0.108
07 0.489 0.028

O 0.067 -O.774

 

74

 

 

 

.xosp um emumHomuuH aHom oHHmocmeumm
mo musumquEmu Eoou um Esnuommm mmm cm mo mwcwa mcw3 one

.HH musmHm

 

 

 

 

 

 

 

 

75

5::

0::

u—r.

 

 

 

 

.mCMHm so on» CH COHumquHHo onHm mo COHuoCCM 8 mm HMOHpmu
cHom UHHoomeuHmmepmwaoHomo on» How mCoHuHmom wCHH mo p

on .NH musmHm

 

\2\

.1.

 

o

00

cm—

 

76

cHom UHHoo>HmuHHCchmxmono>o mCu mo mmCHuuHHmm wCHHHmm>C

omH

00H

.mCMHm on on“ CH CoHpmuou mo mHma .MM MUHCMH

 

CD

on

no 0&9

_ u.

.IIOMI.

.IImmn.

. IIIIOVI

VI.

VVI.

.Ilovr.

.llmvr.

.Ilmnl.

.NH musmHm

(ssnefi) v

77

.mCMHm mo on» CH CoHumuou mo mHmCm .MN HMUHUMH
oHom OHHoomHmleCchmchoHomo on» no mmCHuuHHmm wCHmquMC mmu one

o
09 02 0: cup 00— ow 00 CV ON C

__________________I.

V;

 

'1

   

N GCHH D
H. mcHHo

n
O
o
l
l\
T

0

.HH musmHm

(ssn96)v

78

.mCMHm no on» CH CoHumuou mo mHmCm .MM HmoHpmu

oHom oHHoomenHmCmHUmxmnoHowo mnu mo mmCHuuHHmm mCHHHmmwn «no one .mH musmHm

(ssneb) V

 

N OCHH D
O

 

H mcHH

79

largest triplet splitting to the two meta protons, and the
smallest triplet splitting to the two ortho protons.

This second radical is found under more stringent
conditions than the main radical. Irradiation of 91:
mandelic acid at room temperature gives the main radical
but not the second radical. Irradiation of dfmandelic or
lfmandelic acid at either 77°K or room temperature appear
to give the main radical. The lines of the side radical,
if formed, are considerably weaker than in the case of its

formation when dlfmandelic acid is irradiated at 77°K.

B. g:Value and A-Tensor Evaluation

 

In the ab plane, the isofrequency plot for the side
radical showed a clear pattern of twelve lines paired into
two sets of six lines. This is shown in Figure 12. The
ortho proton splitting was discernable only on some outer
lines for some orientations. The other planes had similar
isofrequency plots, but only the outer line positions could
be clearly followed throughout the rotation. The A-value
plots for the CH2 splitting are shown in Figures 13-15 where
the plots are labelled according to the scheme in Table 6.

The data from the isofrequency plots were not
precise enough to determine any g-tensor anisotropy. The
isotropic value for g was calculated to be 2.0026.

The anisotropy of the CH2 splitting was con-
siderable. We used the same procedure as described for

the alpha-hydroxybenzyl radical to determine the tensor

80

parameters. There were two A-plot curves for each

rotation. This gave eight combinations of lines. One
combination gave a drastically lower fitting-error than

the others. The eigenvectors of this site were transformed
into the eigenvectors of the other three sites by performing
the three twofold screw axis operations on them. When

these theoretical plots were considered together, they

accounted for all of the A—value curves. The principal
values and direction cosines of the radical for all four
sites are shown in Table 7 and the various site overlappings

are shown in Table 6.

C. Variable-Temperature Study

A variable temperature ESR study of the
cyclohexadienyl-glycolic acid radical was made. The
Q—band instrument was used and the temperature was varied
from -150°C to +120°C. As in the case of the alpha-
hydroxybenzyl radical, no significant differences were

observed in the spectra with change in temperature.

V. Electron Irradiation at 77°K
A single crystal of d1fmandelic acid was irradi-
ated at 77°K with a dose of 6 X 107 rads using the 1 Mev
electron source described in the Experimental section. This
treatment caused the crystal to turn slightly yellow as was
the case on y-irradiation. The spectra were identical to

the ones obtained from y-irradiation at 77°K.

81

VI. d-Mandelic Acid and l-Mandelic
Acid Irradiation

 

dfMandelic acid and lfmandelic acid in powdered
form were obtained from Aldrich Co. All crystals were
grown from saturated aqueous solutions by slow evaporation.
The crystal system of dfmandelic acid was reported to be

monoclinic103

104

while that of 1fmandelic acid was orthor—
hombic. The morphology of the gfmandelic acid crystals
agreed with that reported in the literature and this was
similarly true for the l-mandelic acid crystals.

dfMandelic acid crystals were y-irradiated at room
temperature and their spectra were taken on the Q-band
spectrometer at various orientations. These spectra had
the same type of seven-line pattern with no side peaks as
was found for the case of dlfmandelic acid irradiated at
room temperature. When dfmandelic acid was irradiated at
77°K, it still showed the same spectra with no side peaks
while the spectra of glfmandelic acid irradiated at 77°K
would have the extra side peaks. This showed that the
alpha-hydroxybenzyl radical is formed in both the d: and
dlfmandelic acid crystals at either 77°K or room temperature,
but that the cyclohexadienyl-glycolic acid radical is
formed only in dlfmandelic acid at 77°K an never in g:
mandelic acid.

The same procedure was applied to lfmandelic acid

with the same results. Only the alpha-hydroxybenzyl

82

radical is formed when 1fmandelic acid is irradiated at

77°K or room temperature.

DISCUSSION

I. Alpha-Hydroxybenzyl Radical

 

The alpha-hydroxybenzyl radical appears to be formed
by the removal of a carboxyl group from a 97 or lfmandelic
acid molecule. It is assumed that the hydrogen and hydroxy
groups would move from their original tetrahedral con-
figuration about the central carbon atom to give a planar
species.

To check the geometry of this radical, energy
calculations were made for various configurations using
INDO.105 At first, we varied the positions of the CHOH
atoms in the plane of the benzene ring using previously

106 until an

determined parameters for the benzyl radical
energy minimum was reached. It was then found that any
rotation of the CHOH group produced a higher energy.
Therefore, INDO calculations would predict a planar
radical. The final parameters which minimized the energy
are shown in Figure 16.

As was discussed in the section on Results, the
central isofrequency lines were accounted for by assuming

that there were two sets of four sites. In each group,

the four sites were related by the three twofold

83

84

 

T
u:
|\
no
C
\(08 "°\ [.40 mo /I.OI
C/’ ’10 C [10‘
I no \\1’//
I.” [.30
\
/|\ I114 II,
IN C In C
V ) \m
L00 \\\\\\fl \”‘/ ”3 ‘1‘
\PC/
(11
Bond lengths (angstroms)
Angles (degrees)
I”!
It!!!

  

Figure 16. The atom positions of the planar alpha—
hydroxybenzyl radical as calculated by INDO.

85

.AmH ousmHm CH mm on on oofismmo mH HooHpou on» mo

wuuoEoom onuv oCon onv CH ohm muouoo>CowHo mCHCHoEoH onu oCm HMUHpmu
onu mo oCmHm ona ou HMHDUHUComHom oum mCOHuoouHc em.VHI u Acmvd oCm
mmoo.m u m one .HooHoou Henconmxouoenloanm HoCmHQ onp mo MHuoEoom on»
0C8 muomCou 4 oCo m on» no muouoo>ComHo onu Coozuon mHnmCoHuoHoH one

 

 

.HH mnsmHm

86

Mandelic acid

 

H
+
H H
H
alpha-Hydroxybenzyl
radical
H H
H H
HH

Cyclohexadienyl-glycolic acid radical

Figure 18. A possible scheme for the formation of the two
radicals in the irradiation of glfmandelic
acid at 77°K.

87

screw axes. The undamaged crystal had two sets of four
molecules per unit cell, one set being dfmandelic acid
molecules and the other being 1—mandelic acid molecules.
We could deduce then that sites 1,2,3, and 4 of Table 3
correspond to the dfmandelic acid species and sites

5,6,7, and 8 correspond to the 1fmande1ic acid species, or
vice versa.

In the undamaged crystal, the £7 and lfmandelic acid
molecules are related by a center of inversion. This would
require that the benzene ring plane of each of the four
sites of the dfmandelic acid molecules must be parallel to
the benzene ring plane of a corresponding 1fmandelic acid
molecule. Because the alpha-hydroxybenzyl species ia a
pi-electron radical, we would expect that one of the
principal axes of the g tensor would be perpendicular to
the radical plane and thus to the benzene ring plane. Then,
if the benzene ring planes did not reorient when the
crystal was damaged, we would expect that one eigenvector
in each of the sites 1,2,3, and 4 would be parallel to the
eigenvectors in each of the sites 5,6,7, and 8. It is
evident from Table 3, that this is not the case. We would
conclude that the benzene ring planes change their orien-
tation significantly as a result of the radiation.

Previous calculations of the spin densities for

107

the alpha-hydroxybenzyl radical have been made by using

McLachlan's approximate self-consistent field method.108

88

An "excess charge density" was also calculated by using the

relation proposed by Colpa and Bolton,109: 110

a = (Q + K€)pl

where a is the isotropic proton hyperfine coupling constant,
e the excess charge density, p the unpaired electron spin
density on the adjacent carbon atom, and Q and K are
constants that were chosen to be -30 gauss and -15 gauss,
respectively, to give the best fit to the values of a. This
relationship was used instead of McConnell's equation
because it produces better results for radicals in which
there is an appreciable excess charge, or considerably
different excess charges at various sites of the radical,
both conditions of which apply to the alpha-hydroxybenzyl

radical. The results are listed in Table 8.

It was found that the relationship between the
eigenvectors of the g and A tensors could be related to the
geometry of the molecule. This is shown in Figure 17,
where the principal g and A values correspond to those in
Table 3. In this figure, the eigenvectors of the g tensor
point so that 933 is perpendicular to the CHOH plane,
gll=gmax points along the exocyclic carbon-carbon bond,
and 922:9min is orthogonal to 911 and 933. Assuming this,
it was then found that the eigenvectors of the A tensor of

the alpha hydrogen point so that A is perpendicular

33=Amin

to the CHOH plane within 2°, A 2 points along the C-H bond

2

o . o = . '
w1th1n 15 , and A11 Amax lS orthogonal to All and A33.

89

If we subtract the isotropic value of —16.21 gauss
from the principal values of A(H7) in Table 3, we obtain
(-2.12,+0.88,+l.24) gauss for the anisotropy of the
alpha-proton. The typical values for alpha-proton
anisotropy are (-10.0,+10) gauss.26 One theoretical

111 to calculating the anisotropy approximates it

approach
as a magnetic dipole interaction between the hydrogen and
the electron spin magnetization that is distributed in the
23 and 2p atomic orbitals on the neighboring carbon atom.
This treatment gives values of (-14,-2,+15) gauss for a
case of one pi electron on the adjacent carbon atom. If
we consider the dipole effect to be proportional to the
unpaired spin density on the carbon atom and use the value
of 0.489 from Table 8, we would still expect theoretically
an anisotropy of (-5,0,+5) gauss, which is considerably
greater than measured experimentally. Considering the
intransigence of the g and A tensors to normal analysis,
and the uncertainties in determining the parameters from
the experimental information used, we might expect that
the calculated anisotropy values have compounded un-
certainties associated with them.
II. Cyclohexadienyl-Glycolic
Acid Radical

The cyclohexadienyl-glycolic acid radical appears

to be formed by the addition of a hydrogen atom at the para

position of a d- or l-mandelic acid molecule. This extra

90

hydrogen atom could come from the dissociation of a free
carboxyl group into CO2 and H, where the carboxyl group
was previously removed from a d: or 1fmandelic acid mole-
cule in forming the alpha-hydroxybenzyl radical discussed
above.

There are eight possible sites in the undamaged
crystal, but only four sites are distinct in ESR spectra
for the cyclohexadienyl-glycolic acid radical as compared
to eight sites for the alpha-hydroxbenzyl radical. This
would imply that the A tensors for the former radical are
paired together while they are not for the latter radical.
The A tensor for the CH2 group would obviously be fixed
with respect to the benzene-ring plane. As discussed
previously, the g7 and 1fmandelic acid molecules are
related by a center of symmetry so that the A tensors
associated with these molecules would have the same
eigenvectors if the benzene rings did not reorient. So we
can conclude that the benzene rings did not significantly
reorient themselves upon irradiation to form the
cyclohexadienyl-glycolic acid radical. A possible scheme
for the formation of the alpha-hydroxybenzyl and
cyclohexadienyl-glycolic acid radicals is shown in

Figure 18.

91

III. Summary

The ESR spectra of the radicals produced in single
crystals of dlfmandelic acid by irradiation with y-rays
have been obtained and analyzed. The alpha-hydroxybenzyl
radical and the cyclohexadienyl-glycolic acid radical are
formed. The alpha-hydroxybenzyl radical appears to be
planar, its benzene ring having reoriented during the
radical formation process. The cyclohexadienyl-glycolic
acid radical appears to be formed only at 77°K and the
hyperfine splitting tensors of the d and 1 radicals are

parallel.

PART I

REFERENCES

10.

11.

12.
13.
14.

15.

REFERENCES

Gorter, Physica 3, 995 (1936).

Rabi, J. Zacharias, S. Millman, and P. Kusch,
Phys. Rev. 23, 318 (1938).

Zavoisky, J. Phys. USSR 2, 211 (1945).

Purcell, H. Torrey, and R. Pound, Phys. Rev. 62, 37
(1946).

Bloch, W. Hansen, and M. Packard, Phys. Rev. 69,
127 (1946).

Kozyrev and S. Salikhov, Doklady Akad. Nau, SSSR
28, 1023 (1947).

Penrose, Nature (London) 163, 992 (1949).
Ingram, Proc. Phys. Soc. (London) A62, 664 (1949).

Holden, C. Kittel, R. Merritt, and W. Yager, Phys.
Rev. 12, 1614 (1949).

Abragam and M. Pryce, Proc. Roy. Soc. (London) A206,
164 (1951).

Schneider, M. Day, and G. Stein, Nature (London)
168, 644 (1951).

Uebersfeld and E. Erb, Compt. Rend. 242, 478 (1956).

Feher, Phys. Rev. 103, 834 (1956).

Hutchison Jr. and B. Mangum, J. Chem. Phys. 22,
952 (1958).

Cole, C. Heller, and H. McConnell, Proc. Natl.
Acad. Sci. U.S. 45, 525 (1959).

92

16.

17.

18.

19.
20.
21.

22.

23.
24.
25.
26.
27.
28.

29.

30.

31.

32.

33.

34.

35.

93

D. Ghosh and D. Whiffen, Mol. Phys. 1, 285 (1959).

I. Miyagawa and W. Gordy, J. Chem. Phys. 10, 1590
(1959).

B. Bleaney and K. Stevens, Rep. Prog. Phys. 19, 108
(1953).

K. Bowers and J. Owen, Rep. Prog. Phys. 18, 304 (1955).
B. Bleaney, Phil. Mag. 21, 441 (1951).
D. Whiffen, Quart. Rev. 11, 250 (1958).

A. Carrington and H. Longuet-Higgins, Quart. Rev. 15,
427 (1960).

A. Carrington, Quart. Rev. 11, 67 (1963).

(.4

Wertz, Chem. Rev. 12, 829 (1955).

G. Russell, Science 161, 423 (1968).

J. Morton, Chem. Rev. 61, 453 (1964).

D. Eargle, Analyt. Chem. 22, 303R (1968).

K. Sales, Adv. in Free Rad. Chem. 1, 139 (1969).

M. C. R. Symons, Adv. Phys. Organic Chem. 1, 284
(1963).

R. Norman and B. Gilbert, Adv. Phys. Organic Chem. 2,
53 (1967).

J. Kochi and P. Krusic, "Electron Spin Resonance of
Free Radicals in Non-aqueous Solutions," in
Chemical Society, Special Publication #24, London
(1970).

Annuals Reviews of Physical Chemistry, 1955-1964,

Annual Reports of the Chemical Society, 1957, 1960,
1962, 1964, 1966-1971.

Symposium on Electron Spin Resonance, East Lansing,
Michigan, 1-3 August, 1966. (Also contained in the
first issue of J. Phys. Chem. 11 (1967)).

Fifth Annual George H. Hudson Symposium, Plattsburg,
New York, 20-22 October, 1969.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

94

Second ESR Conference, Athens, Georgia, 7-9 December,

1970. (Also contained in the 22nd issue of J. Phys.
Chem. 12 (1971)).

Yen, "Electron Spin Resonance of Metal Complexes,"
(Symposium on ESR of Metal Chelates at the Pitts-
burgh Conference on Analytical Chemistry and Applied
Spectroscopy, Cleveland, Ohio, 4-8 March, 1968),
Plenum Press, New York (1969).

Coogan, N. Ham, S. Stuart, J. Pilbrow, and G.
Wilson, "Magnetic Resonance," (Proceedings of the
International Symposium on Electron and Nuclear
Magnetic Resonance, Melbourne, Australia, 11-15
August, 1969), Plenum Press, New York (1970).

Bielski and J. Gebicki, "Atlas of Electron Spin
Resonance Spectra," Academic Press, New York (1967).

Fisher, "Landolt-Bornstein, New Series Gp. II,"
Vol. 1, "Magnetic Properties of Free Radicals,"
Springer-Verlag, Berlin (1965).

Carrington and A. McLachlan, "Introduction to
Magnetic Resonance," Harper, New York (1967).

Wertz and J. Bolton, "Electron Spin Resonance,"
McGraw-Hill, New York (1972).

Poole Jr. and H. Farach, "Theory of Magnetic
Resonance," John Wiley, New York (1972).

Slichter, "Principles of Magnetic Resonance,"
Harper, New York (1963).

Abragam and B. Bleaney, "Electron Paramagnetic
Resonance of Transition Ions," Oxford University
Press, London (1970).

Griffith, "The Theory of Transition-Metal Ions,"
Cambridge University Press, London (1961).

Poole Jr., "Electron Spin Resonance,‘ Interscience,

New York (1967).

Alger, "Electron Paramagnetic Resonance: Tech-
niques and Applications," Wiley, New York (1968).

McConnell and D. Chestnut, J. Chem. Phys. 18, 107
(1958).

McConnell, J. Chem. Phys. 38, 1188 (1958).

51.
52.

53.

54.

55.

56.
57.

58.

59.

60.

61.

62.

63.

64.

65.

66.

67.

68.

69.

95

Colpa and J. Bolton, Mol. Phys. 6, 273 (1963).
Bolton, J. Chem. Phys. 11, 309 (1965).

Giacometti, P. Nordio, and M. Pavan, Theoret. Chim.
Acta 1, 404 (1963).

Moss and G. Fraenkel, J. Chem. Phys. 52! 252 (1969).

McConnell and J. Strathdee, Mol. Phys. 1, 129
(1959).

Ghosh and D. Whiffen, J. Chem. Soc., 1869 (1960).
Stone and A. Maki, J. Chem. Phys. 11, 1326 (1962).

Horsfield, J. Morton, and D. Whiffen, Mol. Phys. 4,
425 (1961).

Kispert, Ph. D. Thesis, Michigan State University,
East Lansing, Michigan (1966).

Watson, Ph. D. Thesis, Michigan State University,
East Lansing, Michigan (1970).

Hammermesh, "Group Theory and Its Application to
Physical Problems," Addison-Wesley, Reading, Mass.
(1962). PP. 86-87.

Jackson, "Classical Electrodynamics," John Wiley,
New York (1967), pp. 137, 146, 148.

Wigner, "Group Theory and Its Application to Quantum
Mechanics of Atomic Spectra," Academic Press, New
York (1959). PP. 326-327.

Abragam and B. Bleaney, 92. cit., pp. 646-648.

Wigner, 22, cit., pp. 158-161 (Note that the Sy and
S2 matrices are defined differently).

Greenstadt, "Mathematical Methods for Digital
Computers," (A. Ralston and H. Wilf, Eds.), John
Wiley, New York (1960), Vol. I, p. 84.

Abragam and B. Bleaney, 92, cit., pp. 653-656.
Fermi, Z. Phys. 60, 320 (1930).

Milford, Am. J. Phys. 18, 521 (1960), (Note that
ge is assumed to be exactly 2).

70.
71.

72.

73.
74.
75.
76.
77.
78.
79.

80.

81.

82.

83.
84.

85.

86.

87.

88.

89.

96

Carrington and A. McLachlan, 92, cit., pp. 81—82.
Slichter, 92, cit., pp. 10-12.

M¢ller, "The Theory of Relativity," Oxford Uni-
versity Press, New York (1952), pp. 53-56.

Furry, Am. J. Phys. 29, 517 (1955).

Carrington and A. McLachlan, 92, 919., pp. 29-30.
Carrington and A. McLachlan, 92, 919., pp. 64-66.
Jackson, 92. 919., pp. 148-150.

Slichter, 92, 919., pp. 65-68.

Abragam and B. Bleaney, 92, 919., p. 167.
Slichter, 92, 919., pp. 161-171.

Rose, "Elementary Theory of Angular Momentum," John
Wiley, New York (1957), pp. 85-88.

Abragam and B. Bleaney, 92, cit., p. 166 (Note we
have used I1 and I_ in place of Ix and I in
Equation (3.40c)). y

Swalen and H. Gladney, IBM J., 515 (1964), (Note
that there is a mistake in Equations (10) and (11)
where o and 8 should be interchanged).

Abragam and B. Bleaney, 92, cit., p. 135.

. Abragam and B. Bleaney, 92, cit., pp. 167-171.

McClung, Can. J. Phys. 99, 2271 (1968), (Note that
the relationships between our A and 0 and his are
A(our) = MA(his) and ¢(our) = 180°-¢(his)).

Lin, Mol. Phys. 29, 247 (1973).

Abragam and B. Bleaney, 92, cit., pp. 157, 171,
181, 182.

Lund and T. Vanngérd, J. Chem. Phys. 92, 2979
(1965).

Taylor, W. Parker, and D. Langenberg, Rev. Mod.
Phys. 41, 375 (1969).

90.
91.

92.

93.
94.
95.

96.

97.

98.

99.
100.

101.

102.

103.

104.

105.

106.
107.

108.

M.

97

Buss and L. Bogart, Rev. Sci. Instr. 91, 204 (1960).
Rose, Analytical Chem. 29, 1680 (1952).

Woolfson, "X-ray Crystallography," Cambridge Uni-
versity Press, Cambridge (1970).

Schonland, Proc. Phys. Soc. (London) 19, 788 (1959).
Rose, 92, cit., pp. 50-51.

Rose, 92, cit., p. 65 (Equation (4.43)).

"International Tables for X—ray Crystallography,"

Kynoch Press, Birmingham, England (1969), Vol. I,
p. 150.

Fischer, Z. Naturforsch. 29, 488 (1965).

Livingston and H. Zeldes, J. Chem. Phys. 99, 1245
(1966).

Wilson, J. Chem. Soc. (B), 528 (1968).

Ohnishi, T. Tanei, and I. Nitta, J. Chem. Phys. 91,
2402 (1962).

Fessenden and R. Schuler, J. Chem. Phys. 99, 773
(1963).

Fessenden and R. Schuler, J. Chem. Phys. 99, 2147
(1963).

Groth, "Chemische Krystallographie," Wilhelm
Engelmann, Leipzig (1917), Vol. IV, p. 559.

Groth, "Chemische Krystallographie," Wilhelm
Engelmann, Leipzig (1917), Vol. IV, p. 560.

Pople and D. Beveridge, "Approximate Molecular
Orbital Theory," McGraw-Hill, New York (1970);
QCPE #142 from Quantum Chemistry Program Exchange,
Department of Chemistry, Indiana University,
Bloomington, Ind. 47401.

Benson and A. Hudson, Mol. Phys. 29, 185 (1971).
Wilson, J. Chem. Soc.(B), 84 (1968).

McLachlan, Mol. Phys. 9, 233 (1960).

98

109. J. Colpa and J. Bolton, Mol. Phys. 9, 273 (1963).
110. J. Bolton, J. Chem. Phys, 99, 309 (1965).

111. H. McConnell and J. Strathdee, Mol. Phys. 2, 129
(1959).

PART I

APPENDIX

COMPUTER LISTING OF SUBROUTINES USED TO
DETERMINE THE 9 AND A PARAMETERS FROM

TENSOR VARIATIONS

000000063065000OOOOOOOOOOOOG

APPENDIX

COMPUTER LISTING OF SUBROUTINES USED TO
DETERMINE THE 9 AND A PARAMETERS FROM
TENSOR VARIATIONS

SUHRDUIINE HINN
DIMFNSIDN XXIIOI-LPOSIIO).SOCOSIIO)~IVECT(IO)
CONNON/GRAD/IGRAD.GRAD.DCOSIII
COHMON/XREG/XOFDII)lsrALI/SCALF(II/LOCATE/V.XIII
COMMON/PARAOEN/NDIM.KTN.KOUT.DIST.DTSTMx.VM
CDHMON/MINN/KDFRV.NVOTM.NGRAD .
couNON/Tlug/TIMELIM.TRFG.TEND.ANOH.IMP.IPP.IVP.IXP
COMMON/TITLT/IIITLE(S).NDATE(3)
THIS SUPROUTINF IS A GENERAL ROUTINE FOR LOCATING THE MININUM OF A
FUNCTION OF ANY NUMBER OF PARAMETERS. ITS BASIC METHOD IS TO USE THE
SURROUTINE PARAGEN wHICH HILL SEARCH FOR A MINIMUM IN ANY GIVEN
DIRECTION BY FITTING THE CURVE TO A PARABDLA. THE MAIN PROGRAM
CONSISTS OF METHODS or DETERMINING IN wHIcH DIRECTION PARAGEN SHOULD
SEARCH. :
THIS PROGRAM REQUIRES: RARAGEN.VALUE.TIh5.NORMGRD.CALNDER.
LAREIIFD CONHONS USED: IGRAD/./XREG/./LOCATF/.ISCALE/./RARAGEN/./MINN/
o/TIMC/./TITLE/.
PUT THE ROUTINE-DETERMINING INSTRUCTIONS IN CENTRAL EVALUATION CENTER.
DIST IS THE LENGTH OF A NORMAL STEP.
KDERV = (0.1) WHFN THF SURROUTINE VALUE (DOES NOT.DDES) CALCULATE
THF GRADIENT COMPONINTS.
LPOS(HVDIM) TELLS wHICH OF THE DIMENSION NUMBERS 1.20....oNDIM ARE TO
BE VARIED.
NDIM IS THE NUMnER OF PARAMETERS IN THE SUBROUTINE VALUE.
NGRAD IS THE NUHRER OF TIMES THF GRADIENT METHOD IS USED BEFORE THE
RROGRAu USES THF UNTFORM-DASIC—DIRECTIONS ROUTINE.
NVDIH IS THE NUMRER OF THE NDIH PARAMETERS HHICH ARE TO RE VARIED.
SCALFINOIHI APF THF SCALING FACTOR: TO BE USED IN THE FOUATION IN
SURPDUTINE VALUE: IBIII=XBEGIIIosCALEIII-XAIII. UHERE XA IS BROUGHT
IN MY MINN AND VALUE USES THE VARIABLES x8.
TIMELIH IS THE NUMBER OF CENTRAL-MEMORY SECONDS FOR THE JOB.
TBEG.TENO ARE THE NUHRERS OF SECONDS OF HIGH-OUTPUT FROM THE
BEGINNING DR FND or THE PROGRAM.
XREGINDINI ARE THE BEGINNING VALUES OF THE NDIM COMPONENTS OF x.
STOP NUVRFRS USED: IO].102o101o104.105.105.I07.
DIMENSIONIZE XXoLPDSoSDCOSoIVFCTo A5 (NZ) HHERE NZ.GE.NDIM.
1000 FORNATIIHI.27x.8AIo.TI?A.T?.Ix.IA1.o 19¢.I2.//)
1001 FORMAT(T50oR.... MINIMIZATION OF A FUNCTION ....°.//
A TAP.°NDIM.THE NUHRrR OF VARIABLES:o.IIA.//
8 TIR.-NVDIM.THE NUMRER OF VARIARLES HHICH ADE TO RE VARIED:O
C 'Ihil/O 132.9LPOSoTHE VARIARIE NUMRERS TO BE VARIFD:°oIOIb)
1002 FORHAI( /. T5.¢KDERV=I0.II.(NO.YEST.DOFS SUHROUTINE VALUE CALCULAT
AE ThE GRADIENT:o.IAA // TlO.oHGRAD.NUMnER OF TIMES THAT I
BHE GRADIENT ROUTINE HILL HE USED:-.IIA.///
C 144.RXREG.IHE BEGINNING VALUES OF THE PARAMETERS ARE:*//(T?3o6615
0090/),
1003 FORMATII. T33.°SCAIE.THE SCALING FACTOR IN THE FOUATION-XHII)=XREG
I(IIOAAIIIRclH’oRSCALEII) ARE:R. II. (T21o6615.9./))
1004 FORMATI I. I43o°DIST.THF DISTANCE OF A NORMAL STEP 1S30o1615.9o/
AI. T40.-DISTMX.THE MAXIMUM DISTANCF OF A SILP TS:¢.IGIS.9.//
D T19.°TIMFLIM.THE TIMF LIMIT FOR THE PQDGPAW 1S29o1615.90//
c T6.oTBEG.THF NUWRFQ oF SECONDS 0F HIGH-OUTPUT FROM BEGINNING OF
DTHE PROGRAM IS:--IGIS.9 II. TOoOTENO.THE NUMHER 0F SECONDS 0
EF HIGH-OUTPUT FROH THF END OF THE PROGRAM IS:°.IGIS.9.//.
'le.2745H . )0/I

99

‘100

IIRS FORMAT(lozxo'IIRS-ANOH.VH 8'.IDIS.9.IIISoIGIS.9)
INOFX=0 S I"ED=0 S IYN=4H NO
IFIKDERV.EQ.I) ITN=AH YES
OISIMX=?0.0°DISI
CAlI (AINDER(NOATE)

... INITIAIIZF LPOS
ITT=0
DO 10 I=I.NDIM
IFISCALEII).LE.0.0) GO TO 10
ITT=ITT¢I s LPOSIITTI=I
IO CONTINuE
PRILI 1000.ITITLE.NDATE
PRINT 1001.NDIM.NVOIM.(IPDS(L)oL=IoNVDIHI
RRIAI 1002~IYN.NGRAO.(XBEGILIoL=I.NOIM)
PRINT 1003.(SCALEIL)9L=I-NDIM)
PRINT IOOA.DIST.DISTMI.TIMFLIMoTREGoTEND
C ..... CHECK FOR 8A0 PARAMETERS ....
IF(NUIM.LT.1.0R.NDIM.GT.IO) STOP 101
IFINVDIM.LT.I.DR.HVOIM.GT.I0.0R.NVOIM.NE.ITT) STOP 102
IFIOIST.LT.I.OF-1.OR.DIST.GT.100.0) STOP 103
IFINGRAD.LI.0I SIDP 104
IF(TIMELIH.LT.0.0.DR.TIMELIM.GT.500.0) STOP IDS
IF(THFO.LI.0.0.0R.TDEG.GT.500.0) STOP 106
IF(TENo.LT.0.0.DR.TENO.GT.500.0I STOP 107
CALL TMINIT
c ... INITIALIZE THE x VECTOR ...
00 2 I=IQNDIH
2 XII)=O.D
[GRADro
10001 CAIL VAIUEIX.V)
60 I0 300
C ...... GRADIENT METHOD .....
50 CONTINUE
60001 CAIL GRADNT(LPOS)
8000! CALL PARAGFN(XX)
IRED=IRED.I ‘
IFIIKDUT.ANO.IBI.NE.IB) IREO=0
IFIIRFD.LI.2) GO In S?
IREC=0 S DIST=DIST/2.0 T OISTMX=20.OODIST
52 DD 54 I=1.NDIM
SA X(I)=XX(I)
V=VN
GO TO 300

.... UNIFORM BASIC-DIRECTIONS ROUTINE ...

160 KIN=0
DD 19S NONFS=1.HVDIN
RFN=1.0/SORT(FLOAT(NONESII
DO 162 IZ=1.NONES

162 IVFCTII7I=NONESol-IZ
GO TO 172

164 IOHJ=I $ IFND=NVDIM

166 IFIIVFCT(IOHJ).LT.IEND) GO TO 168
IF(IOHJ.GE.NONES) GO TO 195
IORJ=IOBIOI % IFND=IEMn-I I GO TO 1H6

168 IVECTIIORJI=IVECTIIOHJ)OI
IFIIOBJ.EO.I) GO TO 172
IORJIzIOnJ-l % ISTAND=IVECTIIORJIoIOBJ
00 170 JN=I.IOHJ1

I70 IVECIIJHI=ISTAND-JN

172 Do 171 IN=I.NDIM

173 OCOSIINI=0.0

ITS NONFSI=NONFS-I T I7=IVECTINONESI S DCDSILROSIIZI)=REN
JINO=SHIFTIIR.NONFSI)
DO 190 I=I.JINO
II=I-I S IFINONESI.E0.0I GO TO R0002
Do 180 IK=IoNONESI
AI=REN T IFIIII.A.IDI.EQ.IB) AL=-RFM
IZ=IVFCT(IK) S DCoSILRoSI17))=AL

180 II=II/2

80002 CALL PARAGEN(XX)

C

C

03135000000

10].

pRINT IIRS.ANOH.VM
DO 186 JY=1oNDIM
185 XIJY)=XX(JY)
V=VM 5 CALL TIME
190 CONTINUE
GO TO 164
195 CONTIMUF
00.... CFNTPAL EVALUATION CENIEQ O...
300 INDEX=INDEX°I
IFIINDEX-NGQAO) 80.50.160
FND

qnqnouTlNE TIME
COMHON/TIME/TIuELTu.TREGoTEND.A~ow.TMP.IPP.IVR.TXP
GO IO (10020910) IARC
Io ANOH=§ECOMDIA1 % OIE1=ANOH-RE6
IFIDIFI.LT.THEG) RETURN
IAHC=? $ IMR=0 ‘ IPD=-l S IVP=0 S RETURN
20 ANOH=RECON0(A) $ OIF2=TIMELIM-ANOH
IFtOIF2.GT.TENO) RETURN
IARC=1 $ IM°=I $ 199:! $ IVP=1 s RETURN
ENTRY TMINIT
IARC=1 T IMP=I s Tpp=l s TVP=I s REG=SECONOIAT
30 PFTURM
Fun

SUBROUTINE GRAONT(IPO§)
COMMON/TOCATE/V.X(I)/GRAD/IGRAO.GRAOoDCOS(l)
COMMON/PARAGEN/NOIMoKTNoKOUT~01§T¢OISTMXoVM
COMMON/MINNIKDEPV.MVDIMoNGRAD
OTMFNQION LPO§(10).XX(10)oSDCOSIIOT
... THIS RURDOUTINE CAICUIATES THE NORMALIZEO GRADIENT OF A FUNCTION 0
OTHER HOROS. THE UNIT VECTOR DOINTING IN THE DIRECTION OF STEEPEST ASC
DEOUIPE§ IARFLLED COMMON: /LnCAIF/0IGQAn/o/"IHN/o/PARAGFN/.
DCOS IR A VECTOR CONTAINING THE PESHLTANT COMPONENTS OF THE NORMALI7E0
GRADIENT ORoIN OTHFR VORDRo THE DIRECTION COSINCS OF THE GRADIENT.
GRAO Tc THF MAGNITHDF OF THE fiQADIFMT
NOT" IS THL MHvHER OF OIMFN§IONH IN THE SPACE OF THE FUNCTION
X Ifi A VFCIOQ CONIAINIMG THE INITIAL COOPnINAIFfi HF IHE FUNCTION
V IS THE VALUE OF THE EUNCTIOH AT THE INITIOI COOROTNATFQ
..... OIMENQIOMIZE: LDQR(N7I.XX(N7).RDCOSIH7) WHERE N7.GE.NDIH
IE(NDIH.LE.0.0P.NVOIM.LE.0) GO TO 0]
IFIKDERV.EO.1) GO TO 81
IGRAD=0 $ GPAD=O.0 % OERSTEP=0.01°DIST
OO 10 13190101”
DCOS(I)=0.0
10 XK(I)=X(I)
On 20 I=I.NVDIM
II=LPOS(I)
XXIIII=XIIIIODERSTFD

70001 COLL VALUEIXXoVV)

XXIII)=X(II1
01 = (VV-VIIDERSTEP
DCOS(III=h7

20 ownn=nnnn.nz'n7
GpnnzfiopTIGPAD)
IEIGQAH.FO.0.0) GO TO 91
0O 30 I=I 9'10!“

30 DCOSII)=DCO$(II/GQAD
QFIURN

51 IGDOD=1

70002 CAIL VAIHFIX.V)

00 S2 I=1~UnIW

SDCOSII1=DCOS(1)
52 DCOSII1=0.0

DO SO I=1~NVDIM

C

nnnfinfinnnnnnnnnnnnnnfififin Wfi.fififififinfiflfifififin

102

55 DCOSILPOSIIII’SDCOSIII
...... DENHQHAIIZE THE “COS VECIOR ...
IGNOD=0 S 7730.0
00 56 I=I~NOIM
S6 Zz=ZZOOCOSII)°‘2
ZZ=SQQIIZZI
IFIZZ.CQ.0.0) GO TO 9!
DO 53 I=IQNDIM
58 DCOSIII=DCOSIIIIZZ
RETURN
91 PQINI I91ONDIMONVDIHOKOERVO{Gu&006;ADO7ZOnISICDERSTEpOI.‘IOVOVVOO7
I. (XILI CI-=IONI)I'410 (LPOSILI.L=1.NVDI'H
IQI FQRHAII//o. CQROQ I” GRADNT-NOIM;NVDIM.KDERV9IGRADsGRADo72 3 .0
A “I502X026I5.90//o TI69°DISTODERSTEDOIOIIQVOVVODZ 3 '9 7615.992Xo
82I90?X03015.90//0T160.(XIL)oL=loNDIH10(LPOSILIQL310NVDIM) = .9
C (15h04615o99/II
QIOP 246
FND

SUBROUTINE PARAGENIUUI
COMMON/GRAO/IGRAO.GRA0.0COS(l)ILOCATE/V.XIII
COMMON/PARAGFN/NDIM.KINoKOUTonIST-DISTMX9VH
COMMON/TIME/TIMELIMoTREG.TEND.AN0w.IMP.IRP.IVP.IXP
DIMENSION STEPIIOI.XX(10).YY(IOIoZZ(10IoUUIIOI
H.G.HAIIFR - CHEN OER. MICH STATE H: 0 JUL 1972 - 17S CARDS
REOUTRFS LARELLFD COMMON: IT[NE/c[LOCATE/oIGRAD/vlpARAGEN/o
PEOUIPES SURROUTINF VAIUE.
INPUT: x.V.OCOS.DIST.UOIM.DISTNX.KIN. OUTPUT: KOUT.UU.VM
PARAGFN IS TO BE USED IN A PROGRAM THAT SEARCHES FOR THE MINIMUM OF A
FUNCTION OF NDIM VARIARLES. THE FUNCTION ITSFLF IS HRITTEN AS THE
SURROUTINF VALUFIXA.ANST. wHEPF XA(NOIM) ARE THE VARIARLES AND ANS IS
THE FUNCTIONAL VALUE. DARAGFN IS GIVEN A STARTING SET OF PARAMETERS IN
THE NDIM DIMENSIONAL SPACF ANO A NORMALIZEU VFCTOR wHICH MAY POINT
ALONG THE GRADIENT. PAPAGFN BEGINS RV TAKING A STEP IN THE OPPOSITE
DIRECTION OF THE vrCTOR. IT THEN TAKES A 7ND STEP AND. IDEALLY. FITS
THF 3 POINTS TO A PARAROLA. IT STEPS AT THE UOTTOM OF THE PARAHOLA AND
RETURNS THE VALUE OF THE FUNCTION. DARAGEN IS SET UP TO TAKE CARE OF
AINOST ANY CONCEIVIRLF SHAPE OF CURVE ALONO HHICM IT MALKS. IT VILL
ONLY RFTURN A NEV POSITION IN THE NDIM DIMENSIONAL SPACE AND A NEW
FUNCTIONAI VALUE IF IT IS LESS THAN THE FUNCTIONAL VALUF GIVFN TO IT.

IF IT FINDS ITSFLF ON A ”FRFFCILY STRAIGHT LINE. II HILL PRINT EVERY-

THING AND STOP.

XINDIMI ARE THE VAIUFS GIVING THE IUITIAI POSITION IN THE NDIH
DIMENSIONAL SPACE.

V IS THE INITIAL VALUE OF THE FUNCTION AT THE INITIAL POSITION.

DCOSINDIMI ARE THE VALUES OF THE NORMALI7EO VECTOR ALONG HHICH THE
SURROUTINE PARACEN NILI SEARCH FOR A MINIMUM.

DIST IS THE LENGTH OF THr STEP THAT PARAGEN SHOULD TAKE.

NDIM IS THF NUMRFR OF VARIARIES THAT DEFINE THE FUNCTION.

DISTMx IS THE MAXIMUM REASONABLE DISTANCE TO BE TRAVELLEO IN I STEP.

KKIN = (1.0) IF THE VECTOR OCOSINDINI (DOCSQDOES NOT) POINT ALONG THE
POSITIVE GRADIENT OF THE FUNCTION.

KOUT IS A FLAG INTEGER WHICH IS 0R UNLESS PARAGEN USES SPECIAL
ROUTINFS. RIT POSITIONS-1.7.3.4.S.6 APE SET EOUAL TO I IF PARAGEN-
(II.TURNS AROUND.(2I.FAILS AT FINAL CHECK-OR GOES TO THE (3I.HIDOLE
9IAI.DOUBLE-VALLEYoIS).HILL-WOlKo OR (6).CUT-OEF ROUTINES.

UUINOIM) RFPRFSENTS THE FINAL MINIMUM POSITION ALONG THE VECTOR DCOS.

VM IS INF FINAL VALUE OF THF FUNCTION AT POSITION UU.

PARAGFN USES ONLY THE VARARLF IPP IN THF LANELLED COMMON ITIMF/.

IPP=(O.II NMFN (NOTHING.FVEDTHING IS TO HF PRINTED. IF IPP=ANOTHER
NUMBER. THEN ONLY FORMATS 2041.2IIO.?I?0.2502.?500 ARE PRINTED.

DIMENSIONIZE: STEP.xx.YY.77.UU AS (NZ) MHFRE NZ IS GREATER THAN OR
EQUAL TO THE LARGEST VALUE OF NDIM TO RF USED.

7001 rnuuA1(/.T]o.o...,, ENTEOING OARAOFN: DIST = °.IGIA.9./.(T3A.OOCOS

I = °96hlb.Qo/)I

POI" FUDNAII" PAQOGI‘H“VIQ‘IP'Q‘XOYY -'- 0.2G15.9./.(T'Io‘BGISJ’o/H
?0‘00 an”AII/OI200.000 NO LOWE-p VALUE FOUND! IIRYQVIOVZOV30VM = ..llz.

12X04616.9o/)

103

20A] FORMAT(/.T10.OSSS..... AFTER A TRIFS ANO USING A STEP SI7F 0F 0.
11616.0. 0. NO LONE” VOIUES HER? InHND on... LfiflvaG pAQAGEN uncoo‘
P“'o/I

20h2 FOO“AI(/of309'..o LEAVING PAPAGEN: V1.v?.v3.VM = 0.6616.o./)

7100 FORNATII-O MIDDLE.PARAGFN?IOO-IOIV-DSTT.VIoVEoXXoYYoSTEP = ’oISo
M2x.3615.°./.(TIA.SOIS.9./11 '

2110 FORMAT(//.T28o°... POUULE VALLEY-PARAGEUZI10-V1.V2.V3.XX.YY.ZZ = 0
001515.90/0ITQOQGIqoqc/II

ZIPO FORNATIIvPOX.'........ FAILURE AT HIDOIE ROUTINE ......“o/)

2200 FORMAT(/° HILLWALV—PAOAGFNP200-ISIFPovl.V2.V3cxxoYYo77 = 'oISoPXo
HWGIS.“o/o(Tl°99615.9~/)1

PSO? FORI‘AIIISo"... CUTOFF..DARAI‘.F‘I?‘—.O;'-x«iUM = 0.101O.91

P000 FnuvATI //.T2So¢...DARAGENPGOO-DIVIDER IS ZERO. SO PARAGEN STOPS H
PEPE: v1.v2.v3.VM.OSTT.xx.YY.ZZ.UU-DCOS = o./I(TRoSGIS.9o/1)

ISTFon $ ITRY=0 S KOUT=OB
IFIIPP.EQ.11 PQIUT ?OOl-DIST.(DCOSIL).L=19NDIMI
DSTT=DIST
2 V1=V
DO 3 I=1oNDIM
1 xx111=X(II
5 Do 10 I=1.NDIM
STEP(II=DSTT°OCOS(I)
10 YY(I)=XX(I)-STEP(II
70001 CALL VALUEIYYoV?)
IF(v2.LT.V1) 16.115
11S IFIKIN.EO.11 GO TO 10h
C.. TURN-AROUND..
12 Do 13 1:1.NDIM
VT=XX(I) S XXII1=YYIII S STEP111=-STEP111
IT YYIII=VT
VT=V1 S v1=v2 S V2=VT s DSTT=~DSTT s KOUT=KOUT.0R.18
16 Do 30 I=1.NDIM
10 ZZIII=YY(I)-STEP(11
70002 CALL VALUF(7Z.V3)
DIVIOFR = Vl-V2-V20V3
IF(OIVIDER) 200.600.)?
37 DELTA = O.S"DSTT°(3.06V1-¢.0°V20V1)IDIVIDER
IFIAUSIDFLTA).GT.DISTMX1 GO TO 800
38 Do 40 I=I.NDIM
#0 UUII):XX(II-DFLTAPDCO<(II
7000} CALL VALUFIUU.VH1
c.0000 FINAL CHECKOQ
VMIN=AMINIIV2.VR.VM1
IE(VHIN.LT.VI GO TO A?
TTRYzIIRYoI 3 KOUTzKDUT.OR.ZR
I‘IIppoEQol) I’RINT '201009ITRY9V10V29V39V’4
IFIIIRY.OI.A) GO TO 41
DSTT = -O.2°DSTT
GO To 2
41 IFITPP.NE.01 PRINT 2041.0STT
6? IFIIPPoEQ.II ppINT 204?.V19V29V30VM
IF(VM1N.EQ.VMI RETURN
IEIVMIN.EO.V21 GO TO 55
DO ‘30 131.5“)1”.
SO UU(II=ZZ(II
VM=V3 s RFTURN
SS DO #0 I=1.NOIM
60 UUIII=YY(II
VH=V2 S RETURN
C so. MIDDLE "HUIINE .00
100 101v=n S KOUT=KOUT.OR.AR
IrIIPP.Fo,1) PRINT 2100oIOIV.nsTT.v1.V?.(lX(L1-L=1oNDIMIo(YY(L1.L=
11.NDIVIo(SIEPIL).L=1.NDIM1
10$ VT=V2 S OSTT=DSTT/2.0 S IDIV=IDIV~I
On 110 I=1.NDIM
STEP(I)=STEP(II/2.0 S ZZII)=YY(II
110 YY(I)=YX(I)-STFP(IT
70004 CALL vALUEIYY.V21
IFIV2.LT.VRI GO TO IPO
C ....... DOUBLE-VALLEY HARNING .....

120

2104

xnnlzxoUT.OP.10H
IFIIPP.NL.01 PRINT 2110-V19V2ovIoIXXILI.LzloNDIM).(YYILIoLtloNDIMI

10(77I1’0L2I0NDIHI

IFIV?.IT.VII GO TO 17
IEIIDIV.LT.31 GO TO 103
IFIIP“.NE.0) PRINT 2190
GO IO 12

C 0.00.. HILL-HALK QnUT1NEoo

200

20S

210
220

C 00.
500

$01
50?

503
504

$06

$08
509

S?0
S?5

530
70008
533
540

Sh?

C a...

600

ISTFP=ISTFH°1
IFIIPP.EQ.11 ppxnv 2200.1STEp.v1.v2.v1.(xx1L1.L:1.N01u).(YY1L1.L=1

1.NDIM10(ZZ(IIoL=1-NDIM)

IF<~OO11STFP.31.EQ.01 GO TO 210
DO 20S I=I-NOIM
XXIII=YY¢II
YY(I)=ZZ(II
V1=V2 S V2=V3 s KOUT=KOUT.OR.20O S 60 TO 16
DO 220 I=1.NDIM
XXII1=ZZ(I1 _
v1=v3 I OSTT=DSTToOSTT S 60 TO S
CUTOFF ROUTINE ...
FOUT=KOUT.OR.60R
VMIN=AMIN1(V1oV?vV3)
IFIVMIN.EO.V1) GO TO 902
IFIVMIN.EO.VII GO TO 504
DO 501 I=IQNDIM
xx111=YY1I1
GO TO SOA
Do 501 1:1.NOIM
xx(11=ZZ(11
160:] s XNUM=ABSIOFLTAIDISTMXI s V1=V3
IFIIPP.NF.01 PRINT 2502.?NUM
IFIXNUM.GT.3.OI GO TO 506
DST=DFLTA/?.0 S NTIME=P A 60 TO 520
IFIXNUM.GT.91 GO TO SOB
DST=DFLTA/3.0 S NTIME=3 5 GO TO 520
DST=SIGNIDISTNxoDELTA1 S NTIME=2 S IGO=2 S 60 T0 520
NTINE = IFIXISORTIXNUM-2.01)
DST=DFLTAIIFLOAT(NTIMEII S IGO=I S GO TO S20
DO 52$ I=IONDIM
STFPIII=DSTPDCOS(II
DO SAD J=1oNTIMF
00 $30 I=I~NOIM
YYII1=XXIII-STEP(II
FAIL vAlHFlYYoVPI
IFIV2.GT.V1) GO TO SO?
00 533 IAzloNDIM
XXIIA1=YY(IAI
V1=V2
CONTINUE
GO TO (S42.50°) IGO
DSTT=SIGN1DISToDSTT1 c 60 TO S
IF DIVIOF” IS 7FPO. EVERYTHING IS PRINTED AND PARAOEN STOPS.
PRIAT 2600oV1oV2oVT~VMoOSTTo(XK(L).L=I.NDIMIoIYYILI~L=IoNDIM1oI77

I(L).L=19NDIM10(UU(L)0L=19NDIVIQ(DCOSILIoL=loNPIM)

STOP 600
END

SHRPOHTINE VALHEIXAQANSI

COMMON/VALuE/QA.OO.YMPNT(TI.ONFNT111.611)
COMVUN/XUEG/XRFGIRI/§CALE/SCALF(6)

COMMON/GRAD/IGPAO.GRAD.OCOS(O)
COMVOM/TIHFIIIMFLIHOTWFGQTENOQANOWOINpoIpp-IVDoIXP

DIMFNSIO” DFAMGIJIonFGVOLI31-GXI710RI301)oXAIfi)0X9I6IoDDFANGI69310

DDDFGVOLI6-3IQDGXIZIoDRI‘oWIoUGI?)oDXHIfiIoN(310$AVE(2~3I

DATA RO/57.2957795131/

C REQUIQFS LAHILLED CDVNON: IVALUE/o/XHFG/qISCALH/o/GRhD/o/TIMEI.

C ALL

VALUES IN THE LORFLLFD COMMONS MUST BE DRFVIOUSLY INITIALIZED

C EXCEPT FOR GRAD AND DCOSIGI IN /GRAD/.

RIO’OCTOTOrsfirSrTO‘erO Oran wrwnrwrsnsw‘Sn

105

NO OTHER guanouTlNfig Rg DEUHIREH.

INDU’: XAIOI. OUTDUT: AmS.

STOP NUNDFRS USFO: 121A.

IHIS SONROUTINF 15 TO OF USED WITH A A PARAMETER MINIHI7ATION PPOGPAN
TO DETERMINE THE PRINCIPLE G-VAIUFS AND Iu-IR O'RECTION COSINES FROM 1
ISOFRFOUCNCY PLOTS. IT DETFRHIUFS HON HiLL TNF THEORETICAI CURVES
PRODUCED FROM THE IRPHT PAPAMFTFRS FIT THF F‘PEPIMFNIAL ISOFRFOUENCY
PLOTS. IT ALSO DETERMINFS TMI CPADIFNT OF THAT VALUE. THF DIRECTION
COSINFS OF THE GPAOIFNT ARE IN DCOSIG) ANO GRAD CONTAINS THE MAGNITUOF
OF INF (RADIFNT.

Inpnn:(0.|) WIIL IUI OOH'IFNT HF LAICUIAIFD: (NOoYFSI.

TRIO: VALUES ARE OUITFN FROM SCALING AND PFPOSITIONINO THE INPUT
XAIO) VALUES. A.n.C=xR(Io?.11 ARE 1 FUIFP ANGLES THAT ARE USED TO
PRODUCE A ROTATION MATRIX 011.1).

0131 IS A VECTOR CONTAINTNO THF 3 PRINCIRIF G«VALUES.

xnta.u.61 ARE THE 3 VARIARIF G-VALUE PARAMFTERS. THE 3 ELEMENTS 0E
GIT) ARE EOUATED TO XRIA.S.61.

R(3.1) Iq THE DnITARy MATRIX THAT IS PRODUCED FROM A.R.C.

ANS IS THE DISHLIAHI [ODOR ASSOCIATED WITH IHF INPUT DARANETFRS.

QA IS THE STANDARD DFVIATION IN THE ANGLE NHFRE G-MAXIMUN OCCURS.

0G IS THE STANDARD DEVIATTON OF THE G-VALUE RANGE FROM G-MAXIMUN TO
G-MINIMUM. ‘

XMEUT IS A VECTOR CONTAINING THE EXPERIMENTAL G-MAX ANGLES FOR THE 3
PLANES OF ROTATION.

GMFNT IS A VECTOR CONTAINING THE EXPERIMENTAL G-VALUE RANGES FOR THE
1 PIANFS OF ROTATION.

90R rnouATIA VALUE-RRA:CPAO.OCOS = 6.7G1S.°1
990 FORMAT(0 VAIUF-qu.A.P.C.C.DFANO.DFOVA1.FRANG.ERGVAI.ANS =0.//.
AZAOI‘IQISogozf)A//03’QE~IGIS.')Q?XI9//o70*03I(-1150902X)I

54321 FORNAII° EPHOP IN VALUF:°o/o(T109§GIS.90/)I

C

00 3 1:195

3 XBIII=AREGIII°XAII1’§CALFII1
AZXUIII S R=XHIZI S C=XBI31 $ GIII=XQIQI E GIZI=XRI§1 S GI3I=X8I6I
CA=COSIAI $ SA=SINIAI $ CB=COSIBI $SB=SINIRI SCC=COSIC> $SC=SINICI

le'l, 3 CA“CH°CC-§A’§C
"(I031 3 SA°CR°CCOCA°§C
”(1031 = “QR“CC
”(7'11 =-CA°Cq”9C-§O”FC
”(202) =‘SA°C”°SCOCA”CC
9(7931 = §R°§C
9(3011 = CA“SR
9‘19?) = Sfiuse
”(393) 3 CB

0.... UNITARY CHECK ...
[HNOR‘ 0.0

DO 10 I=lo3 S 00 10 J=101
T = RIIQI)SPIJol)¢Q(Io2I°RIJ92)0RII~3I°RIJQ3I
IFII-FOQJ, T=T-l.0
10 FRPCR=FRROR°ARSIII
IFIERRDR.GT.1.0F-7) GO TO 12365
N(1)=1 S NI21=2 S NI31=3
DO 22? Ip2101
Nl=h(l) $ N(1)=N(2) $ N(2)=N(3IS NI3I=N7 S 7130.0 S F2=0.0
DO 12 J=103
F1=F1°GINIJII““?”°IN(JIQN(II1°RINIJ10NIZII
I? F2=F20GINIJI1'”2°(°(N(J)oNIII)°’2-R(N(J)9N(2))S¢?)
F2=F2/?.O
X = (AIANIFIIFZIIIZ.O S CX=COSIX1 S $X=SIN(X)
DO 20 11:19?
GXIIII=0.0
00 ‘8 J=I01
QAVEIIIOJI = GINIJ11°I9INIJIOHII)1°CXORINIJ10NIZII”§X)
18 OKIIII=OXIIIIASAVF(II.JI°°2
GXIIII=SDRI(GXIII)) $ 22=CX S cx=-SX 5 SX=ZZ
20 CONTINUE
CX=-C¥ S SX=~SX,
X” = (“Rn S IRFV=O
IFIGXII).GF.GXIP)I GO TO 2]
ID = XD-SIGN(Q0.0.XD) S IREV=I
77 3 GXII) S GXIII = 6X12) S GXIZI = Z?

3106

21 DEANGIIPI=ARSI!0I-1MENTIIRI S DFGVALIIPI=GXIII-GXIZI-GHENT(IPI
IEIIGOAD.FQ.OI DO TO 222
C ... CAICULATE GRADIENT
00 )0 I=lo6
30 DXHIII=0.0
DO 100 IJK=196
Dxn(IIKI=<CALE(IJKI S DAerRII) S DR=UXR(?) S OC=DXRI3I
00(II=DYBIOI S DGI2I=0XBISI S DGIBIwaRI6)
DCA=~£A°DA S HSA=CA°UA S DCR=-SD”DB S HSH=CB°DB $ DCC:-SC°DC
DSC=CCROC
ORII.II=UCA’CR°CC0CA°0CP°FCOCAOCRODCC-DSA05C-SA005C
DRII.2)=D3AoCRocc4sAoOCR«CCoSAocnvDrCoDCAoSCoCAoDQC
no(|.3.=—DRR»(c-sguocc
un(2.II=~DcAocRo<c-CAoncuesc-cAocReDSC-DSARCC~SA¢DCC
09(2.7)=-D<AoCRo§C-<Aoncnosc-qucnoDsc.OCAoccoCAGDcc
DRIPo1I=DSHRSF0SR”0SC
DRI‘~II=HCA0SRoCAoDRH
DR(3.2I=0§A°SRo§AeDSH
DRI3.1)=DCR
DFI=0.0 S DE2=0.0
DO 40 J=I.3
GI=GINIJII S DDJ=OOINIJII S RJI=RINIJ).N(I)I S DRJI=DRIN¢JIoNIIII
PJ2=RINIJI.N(2II S DQJ2=DQ(N(J)9N(?))
CVI=DFITGJ¢(?.O°DGI°PJ109J2°GJ°(DQIl'PJZ‘RJ|“DRJ2)) ’
40 DFP=DF2°GJ*(?.0°UGJ“(DJI““2-°J2“°?I06J“(2.0°RJ1”nPJl-?.O°RJ?°DRJ?I
D) S DI2=DE2/2.0
Dx=((COSI2.0°xIIaa2/?.0I«(IDEI/Ezi-(FI/EZPRZIGOFZI
DCX=~SXRDX S DSX=CX°DX
n") 60 11:10?
DY=0.0
Do SO J=l.3
GJ=G(N(JII S OGJ=DDINIJII S RJI=RINIJI-N(I)I S DRJI=DR(N(J)-N(II)
RJ2=RINIJI.N(2)I S nPJ?=DR(N(J).N(?II
0Y=DY°(OGJ“(RJI“CX00JP°SXIOGJ°(DPJI°CXORJI°DCXODRJZ’SXORJ2°OSXII.
G?.0°SAVF(IIoJ)
SO CONleuf
DGXIIIIan/(2.0°GXIIII)
77=0Cx S DCX=~DSX S DSX=ZZ S ZZ=CX S cx=-sx S SX=ZZ
60 CONTINUE
Cx=-Cr S Rx=—SX .
ODEANG(IJK.IPI=OX°DD F‘
IEIXD.LT.0.0I DDEANGIIJK.IPI=-ODFANGIIJKoIP)
DOEGVAL(IJK.I°I=DG¥(lI-DGXI?I
IFIIREV.EO.II DDFGVAL(IJK~IPI=-DDFGVAL(IJqup)
DxHIIJKI=0.0
IOO CONTINUE
??? CONTINUE
(DANG:(DFAHGII)°°?¢DFAHG(?I°°P°DFANG(1)°‘?)/0A¢.2
ERGVAI=(0EGVALI1)9020nEGVAL(2)““20DFGVAL(3)“"PI/OG'“?
AH§=IQAMGOFPGVAL 2
IEIIVP.EO.I) PRINT ROQ.A.B.CoD.DEAND.DFGVALoEHANG-EPGVALoANS 5
IF(IGUAH.EO.OI PFTHPN
DRAC=D.O
Do TOO Irl.6
UEOANn = P.0’I0FAMGIII°0DFANGII~1)ODFAMG(2)”DDFANG(19?)O
EDEANHI3I°DDFAUGIIo33)l05°°?
DERCVAL = 7.0“(UFGVALI1)°ODFGVAI(Io))ODFGVAL(?)°DDFGVAL(IOZ)’
FDFGVAI(3)°DDFGVAL(101I)/QG“°?
DCOS‘I)=DCR5NG‘DEPGVAL
300 GQADanADoHCO§(I)ac7
DRAD=SQRTIDRAOI
DO JIO I=I.6
310 DCOSIII=DFOSIIIIGRAO
IEIIVP.[Q.I) PRINT 098.60A0.0COS
RETURN
12345 pulNI 510.52]QAOHQCOGQCAOCROCCOSAQSR.SC.RQERROR
STOP I?14
END

 

PART II

A GENERALIZATION OF METHODS FOR

DETERMINING g TENSORS

 

INTRODUCTION

Several authors have derived procedures for
determining the principal values, and the directions of the
principal axes, of the g tensor from magnetic resonance
data.1-4 These involve measurements of g=hv/BH at orien-
tations provided by rotating the crystal about each of
three axes. Any symmetric second-rank tensor may be
evaluated by the same procedures, and examples include the
zero-field splitting and hyperfine interaction tensors E
and K, respectively.5 The necessary equations have been
reported only for the cases of rotation about three
orthogonal axes]‘_5 and three monoclinic axes3 and contain
an overdetermination of some of the tensor elements. It
was suggested that additional information in the form of
rotational misalignments can also be obtained from the data
and a technique for so doing in the orthorhombic case was
presented.4

In this article a general formalism is developed
which converts the inherent over-specification of tensor

elements into a determination of the three rotational mis—

alignments. It is applied to the orthorhombic and

107

 

108

monoclinic cases, the potentially useful case of three
coplanar axes, and the general case of rotation about any
three axes. The latter two cases have not been discussed
previously. Since it is frequently possible to mount a
crystal most precisely by choosing general or coplanar
rotation axes, these latter methods should prove useful in
experimental investigations. Also, the determination of
the three azimuthal misalignments in each case will serve
to make fullest use of data and to improve the accuracy of

the derived tensors.

DATA FOR g-TENSOR DETERMINATION

Experimental

 

Experimentally the crystal is mounted so it may be
rotated about a specific axis and ESR spectra are recorded
for various orientations of the magnetic field in the plane
perpendicular to that axis. The 9 values are plotted as a
function of rotation angle 6, which is taken as positive
when the crystal is rotated clockwise (or magnetic field
rotated counterclockwise) as viewed from above. Three such
plots for the three chosen rotation axes constitute the

experimental data.

Parameterization of the Data

Depending on the quality of the plots, three levels
of analysis may be employed. The first method (called the
aBY method below) utilizes either all the data in a curve-
fitting procedure or only the g extrema with the corre-
sponding angles 6. In the second method (ax method), only
the g extrema are used.

As will be shown below (Equation (28)), the g
values for rotation about any axis must obey the equation

(in the notation of reference 3)

109

 

u-u-xnzmr '

110

92 = a + BcosZO + ysin26. (l)

The most accurate values of the parameters a, B, Y would be
obtained by a least-squares fit of the data to Equation (1).
Alternatively, OBY may be evaluated from the equations for

the extrema:

2 .
a = (g: + gE)/2, B = (g: - g_)cosZG+/2, Y = (g: - g3)51n26+/2,

(2)

obtained by differentiation of Equation (1); 9+, g_ are the
maximum and minimum 9 values in the given plot (or vice
versa) and 6+ is the angle corresponding to g+; 6+ should
be taken from the best extremum.

For a rotation about any axis, each experimental
spectrum is associated with an angle 6 read on a protractor.
The numerical values of 6 then depend on the arbitrarily
chosen initial placement of the crystal. The position of
the crystal associated with 9 = 0° is called the "experi—
mental" initial orientation. Since 6' = e + 180° also
satisfied Equation (1), we see that the initial conditions
specified have a twofold ambiguity. In developing the
theory, it is necessary to specify the orientation of the
crystal in the cavity at the beginning of each rotation,
i.e., the orientation of the crystal axes a, b, 0 when
6 = 0° for each rotation is required. These are called

the three "theoretical" initial orientations. The main

error in orienting crystals is the azimuthal angle error,

111

which is the rotation error 66; we neglect in these
calculations errors in the orientation of the vertical
axis. The aximuthal angle 66 from the actual "experimental"
to the "theoretical" initial orientation is called the
starting-angle shift. The three starting—angle shifts can
always be evaluated.

The data may also be analyzed using only the g

extrema by employing the parameters
2 2 2 2
a = (9+ + 9_)/2. A = lg+ - 9_|/2. (3)

The a, B, Y parameters are used below to solve the general,
coplanar, orthorhombic, and monoclinic cases while the
later two are also solved explicitly using the a, A
parameters. In addition, all cases may be treated using
only the a, A parameters by setting 8 = A and Y = 0 (which
is equivalent to introducing an unknown starting-angle

shift) and employing the aBY method.

GENERAL THEORY

The value of g at a specific orientation has been

shown to be given by1

92: Z w 2!; (w..=w..), (4)

where 21, £2, £3 are the direction cosines of the uniform
magnetic field with respect to a set of orthogonal axes l,
2, 3 fixed with respect to the crystal. The relationship
between the W and g tensors is N = 32. When the symmetric
tensor W is diagonalized, the squares of the principal g
values, and their direction cosines, are determined. The
Jacobi method6 is a very good way to diagonalize the
matrix. The general problem is thus reduced to determining

the six coefficients (matrix elements) W W

11' ”22' W33' 12'

w23' w13'

Nomenclature
The three experimental rotation axes, fixed with
respect to the crystal, are labeled a, b, c. Along these
.+

_ + +
axes are placed three rotation vectors a, b, c chosen so

that rotation of the magnetic field is in a right-handed

112

113

sense about the vectors. Where appropriate, the letters

a, b, c are used as subscripts to the parameters a, B, Y,
A, 0+, 9+, g_, g, M obtained from the respective rotations.
A coordinate system consisting of a specific arbitrary set
of three orthogonal axes labeled 1, 2, 3, also fixed with
respect to the crystal, is chosen for each of the cases
treated in this investigation. It must be remembered that
the direction cosines obtained by diagonalizing the W
matrix will be relative to the l 2 3 coordinate system
chosen.

Considerable simplification of the equations was
found to result from choosing the l 2 3 coordinate system
such that rotation vector 3 points along the positive 3 axis
and b lies in the I 3 plane at an angle ¢ from 3 as shown
in Figure la. Here I 3 3 are three orthonormal vectors in
a right-handed relationship and a positive or negative sign
is given to ¢ according to whether it represents a right-
or left-handed screw sense rotation about 5 (Figure l).

The four experimental cases considered--coplanar, mono-
clinic, orthorhombic and general--then correspond,
respectively, to the conditions (a) vector 3 in the I 3
plane, (b) vector 3 along 3, (c) vector 3 along 5 and ¢ =

+90°, and (d) vector 3 in none of the above relationships

0 + I I C O O
(i.e., c pOints in an arbitrary direction).

114

Figure l.+ Systems ef axee: (a) foretheeprelimi-
nery+equations: a, b, 83, Sb in 3 plane, Ma, Mb along
+ , a along + 3 (in this figure ¢ > 0:); (b) for an arbi-
trary rotation: N axis of rotation, § starting vector
(9+: 0°), M middle vector (6 = 90°), L magnetic field in +
g M plane at aggle 9 to S; (c) for coplanar case: a, b, c,
Sa, Sb, Sc in l 3 plane, Ma, Mb, MC along +3 (in this +
giguge, e > 9°, w < 0°); (Q)+for monoelinic case: e, b,
ea, §b, SC, Mp are in the l 3 plane, Sa, Mc along +1, c,
Ma, Mb along + +(in this figure >+O°%; (e) for ortho-

. +
ghombic case: a, Sc, Mb along + , b, a; M; along++ , c,
Ibfi Ma along +5; (f) for general case: a, b, Sa, Sb in

plane, Ma, Mb along +2, c in arbitrery direetion+(in
none ef the abeve special directions), Sc i.c, Se l.c and
in 1 plane, M5 222 in l 2 plane.

 

115

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) (b) “
a
b ¢ (I
AA
Ma; 2 :M
b
so
¢ Sb L
0
1 S
3
(c) 3 (d) I
0111!] C b (I) 1 a
b ¢ Sc
M M
M: M:
r 2 Av 2
NA C
C QC
0 Sb
1
3
(e) (f) 3
“use ¢ “(1
Iwb b c
: 2 Sc / Mar 2
NE, 5
b S O at
Ma Sb 5c

Figure 1.

116

General Formulation

 

There are nine parameters to determine the six W
coefficients and the three starting-angle errors in the

aBY method. The nine functions wll(aaBaYaabBbeachYc)'

W22(...), W33(...), W12(...), W23(...), W13(...). €1(...),
e2(...), and €3(...) are set up, where the values of the
six W functions for any set of a, B, Y values represent the
approximate values of the W coefficients and the values of
the three 6 functions represent three measures of the
magnitude of the starting-angle shifts. The parameters a,
B, Y from Equation (1) are functions of the starting-angle
shifts 66a, 69b, 68C since a, B, Y depend on the choice of
the "theoretical" initial orientations. When the "true"
starting-angle shifts 68:, 66; and 68: are used, the values
of a, B, Y become the "true" parameters at E a(66:,66t,56:),
at : 3(ae:,59t,ae:), yt z y(66:,66t,66:), while the e

t t tBt t t t t

. t _ _
functions become zero, 81(a B beacBCYC) - €2(...) —

a aYaab

€3(...) = 0, and

t t t t
YbaCB Y ), W22(...), etc.,

become the "true" values of the components of the W tensor.
The 61, £2, 63 functions are called null functions because
the condition that they all be simultaneously zero is used

to determine the true starting-angle shifts and thus the

true a, B, Y parameters and the W tensor.

117

In the OA method there are six parameters, the six
w coefficients, to determine. After N:is diagonalized, the
values of 0+ with respect to any rotation and any arbitrary
initial orientation may be calculated by using the formulas
of Equation (29) with the relation 6+ = (1/2)tan-1(Y/B)
obtained from Equation (2). The difference between the
calculated and the experimental 6+ (each with respect to
the same initial orientation chosen) is equal to the

starting-angle shift for that rotation.

Determination of Starting-Angle Shifts

In this section, formulas are derived giving the
dependence of a, B, Y on the starting-angle shifts 66a,
56b, 66C and the original a, B, Y values. The formulas are
used, along with a condition on the null functions developed
here, to obtain the "true" starting-angle shifts. From the
latter, the "true" a, B, Y values are then obtained and the
"true" components of N computed.

If, in a certain rotation, the angle assigned to
each orientation is changed by an amount 68 so that 6 =

9' + 66, then Equation (1) becomes

a + Bcos(26' + 256) + Ysin(26' + 266)

AG
II

a + (8cos266 + Ysin256)cos26' + (YcosZOB - Bsin266)sin28'

a' + B'cos26' + Y'sin26', (5)

118

from which, using Equations (2) and (3), we see that

a' = a, 8' = Acosg, Y' = Asinc, (6)
where C = 29+ — 266. When the "true" 66i are used, we
obtain the "true" values for at 'BI’YI’ and CE. The values

of the three null functions then become

tt ttt _ _
W(G YHabBbY tGCBCYC) — €i(aaBaYa ab 8 bbY a CB cY C) + 6&1 _ 0,
(7a)
58. = -E. (i = 1,2,3), (7b)

where 6ei is defined by Equation (7a). Also, from Equations

(6) we can rewrite this in terms of the variables Ci as

e. “(A Ab A C,ct;;ct) = o (i =1,2,3). (8)

In some cases the exact solution of the three
simultaneous equations (7) or (8) is prohibitively compli-
cated. It can then be determined by an iterative procedure
using simpler equations. It is, in general, more con-
venient to define the quantities Ai = 266i (i = a,b,c);

then, to first order in A,

5a].- = 0 ai = ail
681 = YiAi, Bi = Bi + 68
6Y = -B A Y. = y. + 6y. (i = a,b,c). (9)

119

These are substituted into Equations (7), which are then

solved for Aa' Ab, AC.

THEORY--DERIVATION OF FORMULAS

PreliminarygEquations

A general rotation may be expressed in the following
way. Let N, g, M be right-handed orthonormal vectors as
shown in Figure lb with N representing the axis of
rotation, § the magnetic field direction at the start of
the rotation and M the magnetic field direction at the
"middle" of the rotation (6 = 90°). Let f be a unit vector
representing the direction of the magnetic field as it
sweeps the g N plane in a right-handed sense about N
(Figure 1b). Let 6 of Equation (1) be the angle between E
and § (rotating f by +9 is the same as rotating the crystal

by -9), then
+ + + -> + + ,
M = N x S and L = Scose + MSinG. (10)

With respect to any axes l, 2, 3 one has, from Equations

(4) and (10),

ii = Sicosfi + Misine (i = 1,2,3). (11)

Choosing the l 2 3 coordinate system in Figure la,

one sees that for the rotation about the a axis, the

120

121

arbitrary initial orientation (where 0 = 0°) is specified
to be along the 1 axis, so that Na = 3, ga = I, and Na = 2.
From Equation (11) one has 2i = Slcose + Mlsine = cose,

22 = Szcosa + M231n9 = Sine and £3 = S3cose + M381n6 = 0,

which, when substituted into Equation (4), give

92 = W1100526 + W225in26 + 2W12cosesin6
= 1(W + W ) + 1(W - W )cosZB + W sin26 (12)
2' 11 22 2’ 11 22 12

and, by comparison with Equation (1), it may be seen that

_ 1 _ 1 _ _
cla " "2’(W11 + ”22’ ' Ba ‘ Emil W22) ' Ya ‘ W12 (13)
and, solving for the W coefficients, one obtains
W =a+8, w =a-B, W =Y. (14)

11 a a 22 a a 12 a

For the rotation about axis b we specify that the
initial orientation occurs when the magnetic field is in

the l 3 plane. From Figure la we see that Nb = 3cos¢ +

Isin¢ g = 1cos¢ - 38in¢ and + = 2 Again- A =
' b ' Mb ' ° 1
cos¢cose,£2 = sine and £3 = -sin¢cos6, and substituting

these into Equation (4) one obtains

2

_ 2 2
g - (Wllcos ¢ + W

. 2 . 2 .
3381n ¢ - 2W13cos¢Sin¢)cos 9 + (W22)s1n 6

+ 2(W12cos¢ - W23Sin¢)COSBSin6. (15)

Since Equation (15) is an exact analogue of Equation (12),

the solutions analogous to Equation (14) are written

122

2 . 2 . _
Wllcos ¢ + W33Sln ¢ - 2Wl3cos¢Sin¢ — ab + Bb, (16)
W22 = “b ’ Bb’ (17)
lecos¢ - W2351n¢ = Yb. (18)

Using Equations (14) in (18), the W23 coefficient may be

defined as

W23 = (Yacos¢ - Yb)/sin¢. (19)

Comparing Equations (14) and (17) leads to the relation

ab - 8b - aa + B = 0: (20)

a

which may be used as a null function. Substituting from
Equation (14) into (16) and using Equation (20) to cancel

the ab term gives

sin2¢(W - sin2¢(W33 - aa) = Ba(1 + cosz¢) - 28b.

13)
(21)

From the six parameters a , B

a a' Ya, ab: Bb’ Yb used

so far, four relationships inVOIVing W11, W22, W12, W23, one
relationship involving W13 and W33, and one null relation—
ship have been defined. The third rotation, which differs
in each of the four cases, will be treated separately for

each individual case below.

123

Case of Three Coplanar Axes

 

If the third rotation axis c lies in the l 3 plane,
along with the a and b axes, we have the coplanar case.
Let ¢, w be the angles between 3 and b, c, respectively;
these have a positive or negative sign according to whether
they represent a right- or left-handed screw sense about 2,
and let the initial orientation vectors gb' EC lie in the
1 3 plane (Figure 1c). From Equations (l9)-(21), with w

and c replacing ¢ and b, we obtain

W23 = (Tacosw - YCI/sinw, (22)
ac - 8C - aa + 8a = 0, (23)
sin2w(wl3) - sin2¢(w33 - aa) = ea(1 + coszw) - 28¢. (24)

Equations (21) and (24) can now be solved to give W13 and
W33. Equations (20) and (23) provide two null functions
and the third is derived from a comparison of Equation (19)

with Equation (22). For the iterative solution, we apply

Equations (7) and (9) to the null function 61 leading to
61 = 6gb - 68b - 6aa + 68a = 0 - YbAb - 0 + YaAa = ”El’
Ab = (El/Yb) + (Ya/Yb)Aa = B + B Aa. (25)

Similarly, applying Equations (7) and (9) to the analogous

52 null function gives

AC = (Ez/YC) + (Ya/YC)Aa = c + c Aa.

124

Again, with e3 and the above relationships,

ll

66

3 -3aAasin(w - ¢) + BbAbsinw - BcAcsin¢

(BstinW - BCCsin¢) - [Basin(w - ¢) -BbB'sinw + BCC'sin¢]Aa
= -€3,
which is solved for Aa’ These results for the approximate
solution are listed in the Collected Formulas, as are some

properties of the exact solution of Equation (8).

Case of Three Monoclinic Axes

If the c axis, about which the magnetic field is
rotated in a right-handed sense, is perpendicular to the l
3 plane, we have the monoclinic case. Let the c axis point
along the +2axis (Figure 1d). Note, however, that if the
experimental c axis actually pointed along the -2 axis,
then one must change He + -9C (or equivalently Yc + -YC) to
use the formulas. The initial orientation for this rotation
is selected so that the magnetic field is along the 3 axis
for 6 = 0° hence N = 2, § = 3 and M = 1. Therefore, from
Equation (11), £1 = sine, £2 = 0, £3 = c036, and from

Equation (4),

2 _ 2 . 2 .
g — W33cos 6 + WIISID 6 + 2W1351n9C086,

which is the same form as Equation (12). So, according to

Equation (14),

W33 = ac + 60, W = a — B , w = Y . (26)

125

Equations (14) and (26) are used to define W W and

13' 33
the null function

Then, equating Equations (14) and (17) gives an expression
for Ba which, when substituted in 81, gives 82 = ac - BC +

8b - 2aa. Substituting 62 and all the relations of
Equation (26) in Equation (16) and dividing by two gives
63 = (ab — 66) + Sin¢(chos¢ - BCSID¢).

Applying Equation (7) to the three null functions
gives 661 = ‘81 = -YcAc - YaAa’ 682 = -62 = -YcAc - YbAb
and 683 = -63 = Sin¢(-BCACcos¢ - YCAC81D¢). From these the
Ai's are obtained directly for the formulas of the iterative

method. The exact solution has also been obtained from

Equation (8) and the formulas for the OA method worked out.

Case of Three Orthorhombic Axes

The initial orientations are best chosen in a cyclic
manner (Figure 1e), because of the high symmetry of the a,
b, c axes, i.e., for rotation about a, b or c the 6 = 0°
orientation is chosen in the ab, bc or ca plane,
respectively. The problem is then set up so that there
is a cyclic relationship betweeen the three rotations.
The formulas in Equation (14) for a rotation about the a
axis can then be extended in a cyclic manner to the other

rotations by changing the subscripts of a, B, Y, and W as

126

as follows: a + b + c, 11 + 22 + 33, 12 + 23 + 13. The

result of this is

Rotation a Rotation b Rotation c
w11 = “a + 8a W22 = “b + 8b W33 = “c + 8c
w22 = “a ’ Ba w33 = “b ‘ 8b w11 = “c ’ Bc
w12 = Ya w23 = Yb w13 = Yc (27)

The first and third rows of Equation (27) define the W
coefficients. Subtracting the first row of relationships

from the second row, dividing by two, and using the relation

-Bb -Bc = ac - ab obtained from the two formulas for W33
gives 81; 62, 83 are then obtained by cyclic permutations
of 61.

Using Equation (9) we obtain for the iterative
solution 661 = -YaAa = -81 so that Aa = el/Ya.

For the exact solution, we have from Equations (8),
ac - ab - Aacosl;a = 0 which is straightforwardly solved

for Ca. The solutions for Ab, A Cb’ Cc are obtained from

C!

those of Aa and Ca by a cyclic permutation.
For the OA method, it is seen from 61 that the

"true" parameters obey the relation Ba = a - ab, which

211/2 =

c
gives W11 = aa - ab + ac and Ya = i.[A: - (ac - ab)
W12, and cyclically the remaining W coefficients. All the

results are listed in the Collected Formulas.

127

General Case

 

If the third axis of rotation does not satisfy the

conditions for any of the previous cases, we have the

general case (Figure 1f).

The direction of the rotation

axis c (about which the magnetic field rotates in a right-

handed sense) must be known, while the initial orientation

need only be determined within a twofold ambiguity.

Let

—+ +
1 + N 2 + N

+ —>
1 2 33' S

+ + + + +
S 1 + S 2 + S and M = N X S

+
1 2 33'

be unit vectors representing the rotation axis, the initial

orientation (6 = 0°) and
The three starting-angle
data, are independent of

solve for them; hence we

convenient system to simplify the solution.

the 6 = 90° direction, respectively.
errors, being a property of the

the coordinate system chosen to

are allowed to choose a more

One such

system is obtained by choosing a different initial unit

*
vector E in the g M plane with the property that 83 =

+ +* +*
Let E be the angle between S and 8 so that S =

+. +*
MSinE and M =

+ + .
McosE - SSinE.

*
0.

§cos€ +

Then S must satisfy the

relation tan€ = -S3/M3. One such transformation that will
+ + . +* +* .
change S and M into S and M is
* A * A
81 (81M3 - 83M1)/ , Ml - ($183 + M1M3)/ ,
* A * A
32 — (82M3 — S3M2)/ , M2 — (3233 + M2M3)/ ,
'k 'k A
S3 — 0' M3 — ’

128

 

.j
where A 2 “(Si + Mg). Similarly the corresponding

transformation between the old a B , Yc' A and the new

c' c c
t * t * .
ac, BC, Yc' Ac parameters would be from Equations (3) and

(5)

* *
a = a , A = A ,
c c c
8* = B cos(2€) + y sin(2€) = [8 (M2 - 52) - 2y M s J/A2
c c c c 3 3 c 3 3 ’
Y* = Y cos(2€) - B sin(2€) = [Y (M2 - 82) + 28 M S ]/A2
c c c c 3 3 c 3 3 '

Substituting Equation (11) into Equation (4), remembering

that wij = wji' and noting the analogy with Equation (12),
we arrive at the result
92 = 23, w. M.

i,j=l 13 1 3

( Z w..S.S.)cosze + ( Z W..M.M.)sin29 + 2( Z W..S.M.)cosesin8
i’j 1] 1 J i'j 13 1 J i'j 13 1 3

“C + BCCOSZG + chinZG (28)

analogous to Equation (13), where

_ 1
0‘c ‘ 7.2.Wij(sisj + MiMj) '
13
B = IZW (SS-41M)
(2 '§.. ij :1;j i j '
1]
YC = ZijwijSiMj' (29)

Since Equations (28) and (29) hold for rotations about any

axis, they may be used as a check on the correctness of the

129

W tensor by reproducing the original 0, B, Y parameters.

*
For the case with S = 0, Equation (29) gives

3
* + 8* — *2 *2 2 * 'k
“c c ‘ W1151 + w2252 + w125152' (3°)
* 8* - *2 *2 + w *2 2 * * 2w * *
a ’ ‘ w11M1 + W22M2 33M3 + w12M1M2 + 23M2M3
* *
* * * * * * * * *
Yc ‘ w1131M1 + szszmz + w12(51M2 + 52M1)
* * * * 2
+ w2332M3 + ”1351M3° (3 )
. .d . . *2 *2 _ 1 d * * * * _
with the 1 entities S1 + 82 — an $1M1 + SZMZ _

O, we use Equations (l4), (19), (32), and (16) to define
W11, W22, w12' w23, W13, and W33, respectively. By
comparing Equations (14) and (17), the first null function
El = ab - 8b - ca + Ba 13 derived. U51ng the formulas for

the W coefficients and Equation (30), the second null

function

E - B 2 *2 1 2 * * * 8*
2 _ aa + a( Sl - ) + YaslSZ - ac - c

is obtained. The third null function 8 listed in Table l,

3'
follows similarly from Equation (31). Considering the
approximate solution, we see that since 61 is the same as
in the coplanar case, Ab is given by Equation (25). Using

82 in Equations (7) and (9) leads to the relationship

Y (25"2 1) 28 s*s*
__ * a 1 - - a 1 2 _ 0
A — (CZ/Yo) +‘{ ‘}Aa — C + C Aa.

 

c Y}
c

130

In order to use the third null function C3 it is necessary

6w and GW ~ these are

to obtain 6W23, 13’ 33.

-BaAacos¢ + BbAb -BaAacos¢ + BbB + BbB'Aa

 

 

 

 

 

6w23 = sin¢ = siné
' —
= (BbB + BbB Bacos¢ A = P + P'A
sin$ sin¢ a a’
8*c Ps*M§
5W13 = c i * 2
S1M3
—B*c' + 2 3*M* + B (s*M* + S*M*) - P'S*M*
+ c Ya 2 2 a 1 2 2 1 2 3 A
*fif a
S1M3
= Q + Q'Aar

 

E + Qsin2¢
_ 1 ZQ'cos¢ _ .
6w33 — ( Sin2¢ ) + (Ya + Tan )Aa — R + R Aa.

Finally, putting 83 into Equation (7) we obtain

*

66 e 2 * * 2 M * *2 28*2 1 28 8* *
= + + + - -
3 ( 2 + PM2M3 Q 1M3 RM3 ) [Ya( 1 ) a 152
*2 *2 28 t t ' * * 2 ' * * ' *2 A
+ Ya(M1 - M2 ) - aM1M2 + 2P M2M3 + Q M1M3 + R M3 ] a
= —€3'

from which the formula for Aa given below (Collected

Formulas) is obtained.

DISCUSSION

A computer program has been written in Fortran IV
for the calculation of g tensors in any of the four cases
described. By employing this program with experimental g
tensors, the equations of this article were checked. a,

B, Y parameters were first obtained from Equations (29) and
these were used as input for the computer program. Agree-
ment of the g tensors derived by the program with the
original tensors provided a check on the internal con-

sistency of the equations for each of the four cases.

Iterative Solutions

 

The iterative procedures for the four cases
described above may not converge if a starting-angle error
is too large (as may happen in using the al method of
Equation (3)), in which case the problem may be solved by

setting

D
II

Ai(ca1culated), iflAi(Calcd)l<ch

RC x SIGNAi(calcd), ifIAi(calcd)I>Rc (i = a,b,c),

131

132

where SIGNx = x/le. RC should be chosen to be within the
range of stable convergence which is different for each of
the four cases considered below. A safe value for Rc is
~0.2.

From the approximate changes in angles Ai one may
proceed in two ways:

(1) Calculate exactly the new values of the parameters

from
I
ai - ai,
' I
Bi = BicosAi + Yi51nAi,
U
Yi = YicosAi - Bi51nAi (1 = a,b,c) (33)

obtained from Equation (5) and use these again in the
equations for the iterative solution to obtain a new set
of Ai's. This process is continued until either the
absolute fractional changes of the a, 8, Y parameters are
below a certain low value (for the CDC 6500 computer this

11

was set at 1.0 x 10- ), or until the values of the null

functions are negligible. If the approximations for Aa'
Ab, AC, are continually summed, and the final values
divided by two, three starting-angle shifts 66a, 69b, 69C
(i.e., the angles from the experimental alignments to the

"true" theoretical initial orientations) are obtained.

133

(ii) Calculate approximately the new values of the

parameters from

a. = a., B. = B. + y.A., Yi = Yi - BiAi (i = a,b,c)

and continue the process as in the first method. This
procedure will converge because the method is self-
correcting. The starting angle shifts can then be
determined by solving Equations (33), using Appendix A with

the initial parameters Bi' 7.

1, Ai and the final parameters

Bi' Yi' to obtain

691 = [cos-1(Bi/Ai)-SIGNYi i cos-1(BE/Ai)] (i = a,b,c).

Mud

From Equations (8) we can calculate the number of
solutions for the starting-angle shifts in each of the
four cases. The approximate methods will only give the

solution with the smallest starting-angle shifts.

Exact Solutions

The principal values of the g tensor are the square
roots of the equation fiii = Eiii (i = 1,2,3). The principal
values Ei solely determine the "shape" of the tensor. The
eigenvectors ii determine the orientation of the principal
axes and thus the orientation of the tensor. The number
of eigenvalue solutions is the number of diagonalized W
tensors which are "shaped" differently. The number of
eigenvector solutions is the number of diagonalized W

tensors which are "shaped" or oriented differently. If we

134

consider the degeneracy of the solutions to be the number
of eigenvector solutions per eigenvalue solution, we find
from Equations (8) that the degeneracies of the coplanar,
monoclinic, orthorhombic, and general cases to be 2, 2, 4,
and 1, respectively. If two of the possible W tensors have
the same principal values, and the eigenvectors of one
tensor become the eigenvectors of the second tensor when
rotated 180° about the axis i, then we call the axis 1 an
axis of degeneracy. The axes of degeneracy account for the

degeneracy of the solutions and are listed below.

COLLECTED FORMULAS

The a, B, y parameters depend on the choice of
initial orientations and, to use the formulas of this
article, they should be determined with the orientation
conventions of Table l. The W coefficients and null
functions for each case are also given in Table 1. The

solutions are listed below.

Coplanar Case

Possible number of eigenvalue solutions: 0, 1,

Possible number of eigenvector solutions: 0, 2,

The one axis of degeneracy is the 3 axis (the 2
axis).
The iterative solution is
+ . _ .
e3 8b351nw BCC31n¢

Aa = Basin(¢ - ¢) - BbB‘sinw + BCC‘sin$ ’

D
II

I _. I
b B + B Aa. B - Sl/Yb. B Ya/Yb.

_ I = I
A — C + C Aa' C ez/YC, C Ya/Yc.

135

1.315

 

o o n mm
«m + 06 N.

n a ma

3 3 + l I 8N +
I a

 

 

 

 

a a
nxwzn~zn + «saturu + o caa\.o~camnax I ounce. m + a. I an + as. I mm: .oaaam
A an I Has m + A n: I a. a I mzsmx.nz~mm~x I .Hzam + a: we r I a: m mm + rs I mas o I «was: gm acoau = o
N. N. N. c c c o c I c c c c c c
1
mm I we I mmmmu>~ + A” I «Haws-m + as I ecu-\Anr I once >. I nu: ocuam a a as x n
on + an I Am I no I a» I «A: no I «a I «N: do + do I Has undan A a :w a a
noun Huuocoo
um I no I no I o» I mg! n» I ma: ocaam a 0 sq m u
QQIUUIGUI flflfldx UQ+OUIMM3 OSHQUQCdS D
an I Ad I on I 3m + no I «N: on + «a I Has ocuan A 1 ca m a
omou caneonuonuuo
“summon I enouu>vocwu + .5 I A6 I ec«u\An> I swoon». I max oadam u a :4 m 0
can I nm I as I on I on I um + on I mm: or I as: I» I as: sedan n a as a 2
cm I so I on I on I am I on I «a: so + no I AA: ocdam n a :a z 4
undo aucaauoco:
A9 I avcwmacaaocwnu 4 no I an
o a n a 3
Omaha” mm I Aammou + a. m_ I a~c«m_ an I “om-00 + a. m.
scamo» + sedan» I as I sycamor I u .0 m synamacaaeMwmn a I max acuam o n a ad a o
owns». um I .awaou + H. m. I amcam_ um I .oNIoo I as m.
u 0
am + as I m I u I ecauxin» I swoon». I n~3 ocasm o n I as a a
an + as I am I as I a» I «as am I «a I «N: am + «a I Has «scan 0 n a as a a
oumu Hacaaaoo
acowuocbh Hafiz mucoauauuooo 3 co I m can: a Man 00x4
accuuavaoo aawuacu noduauoa
.ucOHuoucoauo Haauaca Ideoauouoonu: >ununouon and nu“) «noduucdu dad: can auscuuauuoou 3 .H candy

137

The exact solution to Equations (7) is complex and
thus it is more feasible to use an iterative procedure than
the closed form. Some relationships among the various
exact solutions of Equations (7) are: If (ca, Cb' cc) is
a solution, then (-Ca, -§b, -CC) is another solution with
the properties that (a) its eigenvalues are the same as
those of (Ca, Cb' cc), (b) the signs of W12, W23 are
changed, and (c) the eigenvectors are rotated 180° about

the 2 axis.

Monoclinic Case

 

Possible number of eigenvalue solutions: 0, 2, 4.

Possible number of eigenvector solutions: 0, 4, 8.

The one axis of degeneracy is the 3 axis (the 2
axis).

The iterative solution is: AC = 63/[sin¢(8ccos¢

+ Yc51n¢)]I
Ab = (62 - YCAC)/Yb. A = (61 - YCAc)/Ya.

a

The exact solution is

where

138
_ . -1 _ . _ -1 _ _
EC — Sin [(01a ab)/Ac51n¢], Ea — cos [(aC aa AccosCC)/ka],

-l
Eb _ cos [(ab + ac - 2aa - AccosCC)/Xb].

For each of the two choices for CC there are four solutions
for Ca and:b corresponding to the four choices of sign for
these: (+£a,+£b), (--), (+-), (-+), where the first two
solutions represent one eigenvalue solution and the last
two represent another. Also a rotation of the eigenvectors
about the c axis will transform the solutions as follows:
(++)+(--)I (--)+(++)I (+-)+(-+). (-+)+(+-).

The a1 solution may be obtained by calculating Bi
and Yi from Bi = Xicosci, Yi = X.sinCi(i = a,b,c), using

1

the solutions given above for Ca' C Cc (which depend only

bl
on mi and 11). These are then employed in the formulas for

the W coefficients.

Orthorhombic Case
Possible number of eigenvalue solutions: 0, l, 2.
Possible number of eigenvector solutions: 0, 4, 8.
The three axes of degeneracy are the 3, b, E axes
+

+ +
(the 3, 1, 2 axes).

The iterative solution 18 Aa = El/Ya, Ab = Ez/YbI

D
II

EB/Yc'

The exact solution is
c = + cos-1[(a - )/l ] C = + cos—1[(a - a )/A 1
_. c ab a ' b — a c b '

Cc = : cos—1[(0Lb — aa)/AC].

139

The solutions may be ordered according to the eight
possible combinations of signs for the Ci: (+Ca. +CbI +CC)I
(+--), (-+-), (--+), (---), (-++), (+-+), (++-). The first
four solutions represent one eigenvalue and the last four
solutions represent the other. A rotation of 180° about
axis a will transform the solutions as follows: (+++)+
(+--), (+--)+(+++), (---)+(-++), (-++)+(---). Similarly,
for a rotation of 180° about axis b: (+++)+(-+-), (—+-)+
(+++), (---)+(+-+), (+-+)+(---). Finally, for a rotation
of 180° about axis c: (+++)+(--+), (--+)+(+++), (---)+
(++-)I (++-)+(---).

The aA solution is

wll=aa—%+ac' w22:0‘b-mc+0la' W33=ac-aa+0b'

 

 

_ 2_ _ 2 = 2_ _ 2
“Hz-:63 (ac ab)]. w23 :v/[ij (0Lal 010)].

 

2 2
wl3-ivfnc- (Ob-Ga) 1°

General Case

Possible number of eigenvalue solutions: 0, l, 2,
Possible number of eigenvector solutions: 0, 1,

There are no axes of degeneracy.

The iterative solution is

140

* * * *
A * 3 2 3 - 2PM2M3 — 2QM1M3
- i * T *

+ 2P'M*M* + 2 'M*M*

2 3 Q :1 3

A == (2 + (:‘A ,

C a
where

2

I I It *
B = el/Yb. B = Ya/YbI c = €2/YCI c = [Ya(281 - 1) - 28aslszl/YC.

P = BbB/sin¢, P' = (B B' - B cos¢)/sin¢,
b a

I I
M

_ BI I I
Q - (- CC ‘ PSZM3)/Sl 3,

'k

2

B I I I I ' I I
+ a(SlM + S Ml) - P SzM3l/S

I I
2 2 M

I 8*l *
Q ‘ [' cC + 2yaszM 1 3'

R = (61 + Qsin2¢)/sin2¢, R' = Ya + (2Q'cos¢/sin¢).

The exact solution is obtained by substituting
Equations (6) into Equations (7) which gives the expressions
to be solved for Ca, Cb, CC. Most of the sixteen possible
solutions will generally not be acceptable since they will

not be real.

PART II

REFERENCES

REFERENCES

Weil and J. Anderson, J. Chem Phys. 23, 864 (1958).
Geusic and L. Brown, Phys. Rev. 112, 64 (1958).
Schonland, Proc, Phys. Soc. (London) 12, 788 (1959).

Billing and B. Hathaway, J. Chem. Phys. 29, 2258
(1969).

‘ O
Lund and T. Vanngard, J. Chem. Phys. ggJ 2979
(1965).

Greenstadt, "Mathematical Methods for Digital

Computers" (A. Ralson and H. Wilf, Eds.), John
Wiley, New York (1960), Vol. I, p. 84.

141

PART II

APPENDICES

APPENDIX A

SOLUTIONS TO A TRIGONOMETRIC EQUATION

APPENDIX A

SOLUTIONS TO A TRIGONOMETRIC EQUATION

There are at least four different ways in which the

solution to
Acose + Bsine = C (Al)

may be written. The most convenient for the present purpose
is derived below.

Equation (Al) may be written as Mcos(6 - A) = C,
where McosA = A and MsinA = B. The relative signs of A and
B determine the quadrant of the angle A. By setting M =

+ (A2 + B2) it is seen that A is uniquely determined by

A = cos'1[A/ (A2 + 32)] x SIGN(B), (A2)

where SIGN(B) B/|B|. Equation (A1) now becomes cosE =
C//(A2 + BZ), where E = 6 - A. Since 6 is double valued,

the two solutions to Equation (A1) are

e = cos'ltA/ (A2 + Bz)]°SIGN(B) i cos‘lm/ (A2 + 32)].

(A3)

142

APPENDIX B

PROPERTIES OF SECOND-ORDER TENSORS

APPENDIX B

PROPERTIES OF SECOND-ORDER TENSORS

If a second-order symmetric tensor W is diagonalized

1

by the similarity transformation R. WE, then the columns

of the unitary matrix R represent the eigenvectors of the
tensor. Changing the signs of any column of R does not
change the eigenvalues (the "shape" of the tensor), or the
orientation, of the corresponding diagonalized tensor W.
Changing all the signs of the i-th row of E does not change
the eigenvalues but rotates the eigenvectors 180° about
axis 1, which is the same as reflecting the eigenvectors
through the jk plane (j,k # i). These results are
similarly accomplished by changing the signs of four of

the off-diagonal elements of the undiagonalized W tensor:

W.. = W.

1] ji' wik = Wki (3 # k). This can be shown by con-

=* *
sidering another matrix R such that qu = :_qu[(p,q =
1,3), (1) sign if p(:) i]. Applying this new similarity

transformation to W gives

143

144

w*' — R* ..l * — * *
lm ‘ 3L< )ljwijkm ‘ gLlekawjk

Z R. R. w.. _ _
j#i 31 3m 33 + ( Ril)( Rim)wii + j#k lekaij

+ j;k(le)(-Rim)wji + j;k(-Ril)(ka)wik'
k=i j=i

from which we see that if the similarity transform matrix
R diagonalizes W, then R diagonalizes the matrix W with
the signs changed on the four off-diagonal elements Wji =

wij' wik = wki'

The eight tensors resulting from the eight possible
sign permutations of the off-diagonal elements W12, W23,
wl3 are (+W120 +W23, +W13)I ('+-)r ('-+)r (+--)I (---)!
(+-+), (++-), (-++). The first four of these have one set
of eigenvalues and the last four a second set. A rotation
of 180° about axis f changes (+++)+(—+-) and (---)+(+-+),

about axis 3 changes (+++)+(—-+) and (--—)+(++—), and about

axis 3 changes (+++)+(+—-) and (--—)+(-++).

APPENDIX C

COMPUTER LISTING OF SUBROUTINES USED FOR THE
CALCULATION OF 9 TENSORS FOR THE COPLANAR,
MONOCLINIC, ORTHORHOMBIC, AND

GENERAL CASE

flfififlfiflfiflfififlfiflflflfi

APPENDIX C

COMPUTER LISIING or SUBROUTINES USED FOR THE
CALCULAVION 0F 6 TENSORS FOR THE COPLANAR.
MONOCLINICo ORIHORHONIIC. AND
GENERAL CASE

PROGRAM GTENSORIINPUToOUTPUT) GTNSR
REAL NNIvNN20NN3 OTNSR
CDHHON/PARAH/ALA.BEAOGAAQALDOBEBoGABOALCo.ECOOAC GTNSR
COMMON/U/HII.H229U330U129UZJCH13 GTNSR
99 FORMATIII) GTNSR
98 PORMATCJEID.8) GTNSR
READ 999INDE! GTNSR
READ 989ALA98EAOGAA.ALBOBEBOGABQALCOBECOOAC GTNSR
GO TO (10920930060) INDEX GTNSR
THIS IS A SAMPLE PROGRAH SHOWING HOH ONE MAY HRITE A PROCEDURE THAT GTNSR
UOULD CALL ANY OF THE FOUR SUBROUTINES. ONLY THE SUIROUTINE OF GTMSR
INTEREST NEED BE LOADED INTO THE COMPUTER. GTNSR

ALAoALBoALC ARE THE ALPHA PARAMETERS FOR RDTATIONS ABOUT AXES AQRoC. GTNSR
BEAoBEBQBEC ARE THE BETA PARAMETERS FOR ROTATIONS ABOUT AXES A.a.c. GTNSR
GAAvGAaoGAC ARE THE GAMMA PARAMETERS FOR ROTATIONS ABOUT AXES A989C. GTNSP
THESE PARAMETERS ARE TRANSMITTED THROUGH THE LAIELLED COHMON [PARAM/. GTNSR
THE H-COEEFICIENTS UlloUZZoUJJoHIZoUZJoUIJ ARE CALCULATED BY EACH OE GTNSR
THE SUBROUTINES AND ARE LOCATED IN THE LAIELLED COMMON lU/o EACH GTNSR

SUBROUTINE CALLS DISPLAY AND PASSES THE H-CDEEEICIENTS THROUOH IHI. 'GTNSR
SUBROUTINE DISPLAY CALLS DIAGI WHICH IS A STANDARD DIAGONALIZATION GTNSR
SUBRDUT INE. OTHER SUBROUTINES ARE INCLUDED. CUT ARE NEVER CALLED. GTNSR
THESE SUBROUTINES CAN BE USED TO TEST OR DOUBLE-CHECK THE RESULTS OE GTNSR
THIS PROGRAH. THE CDHHENT CARDS IN EACH SUHROUTINE DESCRI'E THE GTNSR
QUANTITIES CALCULATED. GTNSR
0.0.. COPLANAR CASE .0... IINDEA . I) GTNSR
IO PEAD QBOPHIOPSI GTNSR
CALL COPL ANR (PHI OPSII GTNSR
STOP IIII GTNSR
00000 HONOCL INIC CASE on... (INDEX . 2’ GTNSR
20 READ 980PHI GTNSR
CALL HONOCLNIPHII GTNSR
STOP 2222 GTNSR
0.000 ORTHORHOHBIC CASE .. coo IINDEX . 3) GTNSR
30 CALL RHOHBIC GTNSR
STOP 3333 GTNSR
00000 GENERAL CASE 0.... (INDEX ' “I GTNSR
AD READ 980PHI GTNSR
READ 989NNIONN20NN3OSSI95529553 GTNSR
CALL GENERAL IPHI ONNI ONNZONN3OSSI 05520 $53, GTNSR
STOP “‘0’“. GTNSR

END GTNSR
SUBROUTINE COPLANRIPHIQPSI) COPLN
...... PHI AND PSI ARE IN DEGREES. COPLN
DIMENSION ANGDEGT3) COPLN
COHHDN/PARAH/ALA.BEAQGAAOALBOBEBQGADOALCODECOOAC COPLN
COHHON/H/HII9H229U330N129H239UI3 COPLN
DATA PI/3.I4I592653S9/9TODEGZ/28.60788976/9TORAD/O.OI745329252/9 COPLN
DNSTEPS/IBO/ONTRIESIZ3/ COPLN

ALAoALBQALC ARE THE ALPHA PARAMETERS FOR RDTATIONS ABOUT AXES AvfloC. COPLN
BEAvflEBoBEC ARE THE BETA PARAMETERS FOR ROTATIONS ABOUT AXES AofloC. COPLN

GAAQGABQGAC ARE THE GAMMA PARAHETERS FDR ROTATIONS AHOUT AXES AQHQC. COPLN
TODEGZ 3 57.2957795I/2.0 I (ISO/PIIIZ. TORAD - PI/IBO. COPLN
60° FORMRTIIHI'“IXO 23".... COPL‘NRR CASE .0000I0x06HpHI = oIF9.4.3X. COPLN

AOHPSI ' QIF9.'RO//039XO?BHTHE ORIGINAL PARAMETERS AREIQ I) COPLN
60I EORHATI37XQSHALPHA9IIXQAHBETAQI?X.SHGAHHAo//.24X07HAXIS AIOIE16.HOCOPLN
82E16.9//24X07HAXIS 989IEI6.892EI6.9//?hXo7HAXIS CIOE16.8.2E16.9//)COPLN
602 EORHATIZZXO 38HTHE STARTING-ANGLE ERRORS (IN DEGREES). I9 22X. COPLN
CZQHABOUT THE THREE ROTATION AXESo/923X96HAXIS Av6X96HAXIS 806x. COPLN
DGHAXIS CI COPLN
603 E0RHATI3X0I2HSOLUTIDN No.9 I236H... 0E9.403X9E9.403X0E9.43 COPLN

145

OOQOWPUNI‘

006666060066

146

GOA FORMAT(/.38X. ABHTHIS SOLUTION HAS THE FOLLOWING ALPHAclETAcGAMMAoCOPLN

El. 38X961HPARANETERS FOR THE THREE axes or ROTATION)
605 FORMATI/o27(SH o

).l/O‘qu927H... END OF C‘LCULATIONS ...,

COPLN

COPLN

TOO FORHATI/l. 65H THERE IS AN ERROR. AN ITERATIVE PROCESS DID NOT CONCOPLN
FVERGE HITHIN . IAQIIH ITERATIONSo/o36H THE VALUE OF THE NULL FUNCTCOPLN

OIONS AREo/O OH El ' OEIS.906H F? I OEIS.996H E3 I oElS.9o/o
HID" ANGAQANGPoANGC ' o 3EIS.Qo/)

TOI FORMATIIRH ERROR IN COPLANRI. lo (39X.3EIS.9./))

10

PE 3 PHI'TORAD

5T ' PSIPTDRAD

SNEE'SINIEE)

SNZEE'SIN(2.OPFE)

SNEEZ'SNFE"?

SNSI'SIN(SII

SNZSI'SINIZ.D.SII

SNSIZ'SNSIPPZ

CSSI'COSISI)

CSSIZ'CSSI..Z

CSEE'COSIEEI

CSEEZ'CSFE"?

SNSIFE'SINISI-EE)
DEN-2.0.SNFEPSNSI'SNSIFE
ADSTEP I PI/FLOATINSTEPSI

ACCUR I ABSTEP/IOO0.0

PRINT GOOOPHI.PSI

PRINT GOICALAQHEAQGAAOALBOBEBOGABOALCOBECOOAC
zLHDAAISORT(HEAPPZOGAAPPZ)
ILHDABPSORT(HEBPPZOGABPPZI
zLHDAC'SORTIPECPPE°GACPPZI
DPLUSA ' SIGNIACOSIBEA/ZLHDAA).GAA)
DPLUSB ' SIGNIACDSIBEH/ZLHDAHI06A.)
DPLUSC ' SIGNIACOS‘BEC/ZLHDAC)OGAC)
NSDL'O

STEP 3 SIGNIABSTEPODPLUSA)

XI ' I-lLHDAH-ALBOALAIYZLHDAA
AZ ' I'ZLHDAC-ALCOALAI/ZLMDAA
ARI. I‘ZLHDAH-ALH‘ALAI/ZLHDAA
AXE. IOZLHDAC-ALCOALAIIZLHDAA
ALDHER'AHAXIIXIOX?9OI.O)
XUPPER'AMINI(XXI.XXZO°I.O)
IEIXUPPER-XLDUER) 29Q02999IO
ITAHAXIACDSIXLOHER’
ITAHIN'ACDSIXUPPER)

IBEG 3 IFIXIZTAMIN/ABSISTEPIIOI
IEND 3 IFIXIITAMAX/ABSISTEPIIOI
PNB . 01.0

PNC . -I.O

DO ZOI LHN'IOA

T'PNB

PNBS'PNC

PNC'T

KLL'O

KSCH'O

INDEX=D

CROSS'.I.O

DO 200 I=I8EGoIEND

..... INDEX 3 NUMBER OF STEPS IN ALLOUED REGION.

KLL ' O FORBIDDEN REGION

KLL ' I AILONED REGION

KLLSV ' VALUE OF KLL FOR PREVIOUS STEP. ENTERING THE 00 LOOP
RSCH 8 (001) IF SEARCH FOR CROSSOVER (DOES NOIQDOFSI OCCUR.
KSCHSV ' VALUE OF KSCH FOR PREVIOUS STEP.

FOR THE FIRST TIME CAUSES KLLSV TO BE SET TO 0.

CROSS IS LESS THAN OR EQUAL T0 0.0 ONLY IF A CROSSOVER IS TO

BE SEARCHED FOR.

FUNC 8 TEST FUNCTION 3 E3. NE SEARCH FOR {HE ZERO VALUES.
SFUNC ' LAST VALUE OF FUNC
SSFUNC ' SECOND TO THE LAST VALUE OF FUNC

ZTA'FLOATIIT'STEP

COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN
COPLN

3” ' ‘

 

00

60000 0 0

00 66

147'

cszra-cos¢ZTA) COPLN .9
KLLSVPKLL COPLN 90
NSCHSVPKSCH COPLN 9I
KSCHPO COPLN 92
SSEUNCPSEUNC COPLN 93
SEUNCPEUNC COPLN 9A
CSZTD P IALD-ALAOELHOAAPCSZTA)IZLHOAO COPLN 9S
I'IADSICSETHI'I.OI 38.38.I90 COPLN 96

3. CSETC P IALC-ALAOELHOAAPCSZTAIIzLHDAC COPLN 9T
IPIADSICSETCI'I.0I 39.390I90 COPLN 9D
....... ALLOHED ZONE ..... COPLN 99
39 KLLPI COPLNIOO
INDEAPINDEX’I COPLNIOI
..... CALCULATE EUNC ... COPLNI02
SNZYA P SINIETAI COPLNI03
SNITO P SIGNIISORTII.0-CSZTDPP2II.PNOI COPLNIOO
SNZTC P SIGNIISORTII.0-CSZTCPP2)IQPNCI COPLNIOS
PUNC P ELHDAAPSNZTAPSNSIEE'ZLHDABPSNZTBPSNSIOZLHOACPSNETCPSNFE COPLNIOO
..... IE LAST STEP HAS IN THE FORBIDDEN REGION. CHECN EOR CROSSOVER. COPLNIOT
IEINLLSVI 500I00 COPLNIOD
..... Ir INDEAPI. THIS IS THE FIRST STEP AND ANOTHER STEP IS TAKEN. COPLNIO9
IE PUNCP0.0. THEN THIS HILL DE DETERMINED DY THE PRODUCT SEUNCPEUNC COPLNIIO
SO IEIINDEX‘II 2000200052 COPLNIII
SZ CROSS P SEUNCPFUNC COPLNIIZ
..... IE CROSS IS NEGATIVE OR ZERO, A CROSSOVER HAS OCCURED. COPLNIIS
IEICROSSI I009I00.SA COPLNII“
..... INDEX HUST BE AT LEAST 3 FOR THE 'OLLOHING CHECK TO DE VALID COPLNIIS
SO IEIINDEX-3I 200056.56 COPLNIIO
..... NE TEST TO SEE IE THE FUNCTION HAS APPROACHEO TOUARO AND THEN COPLNIIT
RETREATED EROH THE FUNCP0.0 AXIS. THERE ARE THO CONDITIONS THAT . COPLNIIH
HUST DE HET FOR THIS TO HAVE OCCURED. COPLNII9
III. IEUNC‘SEUNCIPISEUNC'SSEUNCI HUST DE LESS THAN OR EOUALITO ZERO COPLNIZO
IZI. PUNCPIEUNC-SEUNCI HUST BE GREATER THAN OR EDUAL TO ZERO. COPLNIZI
S6 CHECK P IFUNC‘SEUNCIPISEUNC'SSEUNCI COPLNIZZ
IEICHECKI 58.580200 COPLNIZJ

SD CHECK P EUNCPIEUNC'SFUNCI COPLNIZG
IEICHECKI 2009I000I00 COPLNIZS
...... ROUTINE EOR SEARCHING FOR CROSSOVER POINTS ........ COPLNIZO
CALCULATE THE THREE ANGLES ABOUT HHICH THE SEARCH IS TO BEGIN COPLNIZT
I00 KSCHPI COPLNIZH
..... ANGA.ANGO.ANGC ARE THE (DOUBLE)'STARTIND-ANGLE-SHIETS EOR COPLNIZ9
ROTATIONS ABOUT AXES AOBOC RESPECTIVELT. COPLNI30
ANGA I DPLUSA-ZTA COPLNI3I
ANGD P OPLUSO'SIGNIACOSICSZTBI.PN.) COPLNI32
ANGC P DPLUSC'SIGNIACOSICSZTCI.PNCI COPLNI33
JELAGPD COPLNI36

GO TO IIZ COPLNI35
IIO ANGAPANGAODELA COPLNI36
ANGHPANGBODELB COPLNI37
ANGCPANGC0DELC COPLNI3B
IIZ CSAPCOSIANGAI COPLNI39
CSBPCOSIANGBI COPLNIAO
CSCPCOSIANGCI COPLNIAI
SNAPSINIANGAI COPLNlhz
SHOPSINIANGD) COPLNIA3
SNCPSINIANGCI COPLNIAA
OEAKPBEAPCSAOGAAPSNA COPLNIAS
OEBKPBEBPCSBOGAOPSNB COPLNIAO
DECKPHECPCSCOGACPSNC COPLNIAT
GAAKPGAAPCSA-REAPSNA COPLNIAH
GAOKPGABPCSB-HEHPSNR COPLNIA9
GACKPGACPCSC-HECPSNC COPLNISD

EI P ALD-BEBK-ALAOBEAK COPLNISI

E? P ALC-PECK-ALAOBEAK COPLNISZ

E3 P GAAKPSNSIEE-GABKPSNSI’GACKPSNEE COPLNIS3

O P EI/GABK COPLNISA

OP P 0AAK/GARK COPLNISS

C P EZIGACK COPLNISG

CP P GAAK/GACK COPLNIST

 

C CHANGE THE STARYING-ANGLE SHIFT SO THAT ITS NAONITUDE 15 LESS THAN 90

C DEGREES. TO ADD OR SUBTRACT 180 DEGREES DOES NOT AFFECT THE SOLUYION.

606

148

'1 P EJ’BEBKPB'SNSI'8ECKPC‘SNEE
'2 P BEAKPSNS1FE-BE8KPBPPSNSI.BECKPCP'SNFE
OELAPrl/FZ
OELBPBOBPPDELA
OELCPCOCPPDELA
AAA P ABS‘DELA)
1FCJ'LA01 299911¢9119
11‘ 1FIXXA-ABSYEP) 11991199115
115 1'1CROSS1 119.1199200
119 JFLAG P JFLAG°1
1'1JFLAG-NYR1ES1 12091209195
120 XXB P ABS(DELB1
AAC P ABS¢DELC1
XA P ANAXI‘XXA9XXB9XXC)
‘1' 1AA¢ACCUR1 13091309123
123 1F1XXA¢AHSTEP1 1259125912“
12“ UCLA P SIGN(ABSTEP9DELA1
125 1F1XX8¢ABSTEP1 1279127o126
126 DELB P SIGN(ABSTEP9DELB1
127 1F(XXC-ABSTEP) 11091109128
120 DEL: P 516N1ABSTEP9DELC)
130 OD 180 KABCP192

COPLNISB
COPLN159
COPLN160
COPLN161
COPLN162
COPLN163
COPLN164
COPLNIGS
CORLN166
COPLN167
COPLN168
COPLN169
COPLN170
COPLN171
CDPLN172
COPLN173
COPLN176
COPLN17S
COPLN176
COPLN177
CDPLN178
COPLN179

9.... FIRSY TIME THROUGH THE 00 LOOP ANGAIANOAoETC. AND OAAKIGAAK9ETCCOPLN180

0.... NEA' VINE. ANGA P ZPDPLUSA-ANGA9E1C. AND CAAKP-GAAK9ETC.
ANGA P FLOA112'KA8C1PANGAOFLOAT(KAHCP11.12.0.DPLUSA-ANDA1
ANGD P FLOA1(?'KABC1.ANGBOFLOAT(KAHC‘l1.12.0.OPLUSH‘AND'1

ANGC P FL0A1(2'KA9C1PANGCOFLOA1(KABC'11.12.0.OPLUSC'ANOC1

GAAN P FLOATCJ‘Z”KAHC1'GAAK

GAB“ P FLOAT13‘29KA8C1PGABN

GACN P FLOAT‘J'Z'KABC1PGACK

ANGOEGC11 P ANGAPTODEGZ

ANGDEG‘Z1 P ANGBPVODEGZ

ANGOEG‘J’ P ANGCPTODEGZ

00 1A0 K8193
136 X! P ABSCANGDEGtK11-90oo
1'1X21 14091409137
137 ANGOEG‘K’ P ANGDEG(K)'SIGNC180.09ANO0EG‘K11
60 '0 136
1‘0 CONTINUE
NSOLPNSOL°1
PRIN' 602
PR1N7 6039N50L9ANGDEG
PRINT 604
PR1N' 601.ALA9BEA9GAA9AL89HE89GAB9ALC9HEC96AC
U11 P ALAOBEAK
'22 P ALA-BEAN
'12 P GAAK
H23 P (GAAK'CSFE-GA8K1/SNFE
01 P BEAK.(1.0‘CSFEZ1-2.0'HEBK
02 P HEAR.(1.0’CSS121-?.0'HECK
H33 P ALAO(01'SNZS1-02.SN2FE1/DEN
'13 P 101’5NS12-02’5NFE21/DEN
CALL DISPLAYC11
180 CON71NUE
00 to 200

0.... FORBIDDEN ZONE oo. 1? LAST STEP HAS 1N ALLOHED ZONE AND A
CROSSDVER SEARCH HAS NOT PERFORMED9 THEN CHECK FOR CROSSOVER POINT

AFYER RESETYING THE PREVIOUS VALUES OF ZTA9CSZTA9CSZTB9CSZYC.
190 1NDEX80
KLL=0
1F(KLLSV1 1929200
192 17(KSCHSV) 200.193
193 ZTAleA-SYEP
CSZYAICOSCZYA)
CSZYB I (ALB-ALAOZLHDAAOCSZTA)IZLMDAB
CSZYC P (ALC-ALA‘ZLHDAA'CSZTA)IILHDAC
GO TO 100

COPLNIBI
COPLNIBZ
COPLN183
COPLN186
CDPLNlBS
COPLN186
COPLNIRT
COPLNIBO
COPLN189
COPLN19O
COPLN191
COPLN192
COPLN193
COPLN194
CDPLN195
COPLN196
COPLN197
COPLN198
COPLN199
COPLNZOO
COPLN201
COPLN202
COPLN203
COPLN206
COPLNZOS
COPLNPOG
COPLN207
COPLN208
COPLN209
COPLNZIO
COPLNZII
COPLNZIZ
COPLN213
COPLNZIA
COPLNZIS
COPLNZIG
COPLNZIT
COPLNZIB
COPLNZI9
COPLNZZO
COPLN221
COPLN???
COPLN273
COPLNZP6
COPLNBPS
COPLN226

 

149

601 FORMATI37X9$HALPHA91IchHHETAoIZXoSHGAHHAo/loZAXQTHAXIS AloIFlégaoHONOC

C coo... NO CROSSOVER HAS BEEN FOUND: IF CROSS IS LESS THAN 0.09 ERROR COPLNZZT
19S IFICROSS) 19701970200 COPLNZZB
197 CROSSI°1.0 COPLN229

PRINT TDOONTRIESOEIoEZcE39ANGA9ANGBOANGC COPLN230

200 CONTINUE COPLN231
201 CONTINUE COPLNZJZ
PRINT 605 COPLN233
RETURN COPLNZJA

299 PRINT 701oZLHDAAcZLHDABoZLHDACvALAoALIvALCoX1oXZoXchXXZoXLOHERo COPLNZJS
XXUPPER COPLNZ36
RETURN COPLN237
END COPLNZJB
SUBROUTINE HONOCLN(PHI) NDNOC 1

C .0000. PHI IS IN DEGREES. HONOC Z
DIMENSION ANGOEG(3I NONOC 3
CONNON/PARAH/ALAcBEAoGAAoALBOBEfloGABoALCoIECoOAC HONOC k
CONNON/H/UII9H220U339U129H230H13 NONOC 5
DATA PI/3.16159265359/9TODEG2/Zao64788976/9TORAD/0.01745329252/o HONOC 6
DNTRIES/ZJ/ HONOC 7

C ALAQALBoALC ARE THE ALPHA PARAMETERS FOR ROTATIONS ABOUT AXES AoOoC. NONOC B

C BEAOBEBOBEC ARE THE BETA PARAMETERS FOR ROTATIONS ABOUT AXES AoBoC. HONOC 9

C GAAoGABvGAC ARE THE GAHHA PARAMETERS FOR ROTATIONS ABOUT AAES AoHoC. HONOC 10

C TODEGZ I 57.29577951/2o0 I (180/PI)/2. TORAD I PI/IBO. HONOC 11
600 FORHATCIHI. 37X925H..o. MONOCLINIC CASE o...9101¢6HPHI I 01F9.69 HONOC 12

A/IOJOXQZRHTHE ORIGINAL PARAMETERS ARE'. I) HONOC 13
16
15

C

603
60‘

82E16o9/IZAX07HAXIS 8!o1F16.802E16.9//ZQX0THAXIS C89F16.892E16.9I/)H0NOC

602 FORMAT(22X9 38HTHE STARTING-ANGLE ERRORS (IN DEGREES). lo 2219 HONOC
CZ9HABOUT THE THREE ROTATION AXESo/023X06HAXIS AoonbHAXIS 806K, HONOC
D6HAXIS C) HONOC

FORHATCJXoIZHSOLUTION No.9 IvaH... 9F9.49310F9.49310F9.6) HONOC
FORHATI/oJBXv ASHTHIS SOLUTION HAS THE FOLLOWING ALPHAoIETAoGAHHAoHONOC
E/o 38X061HPARAHETERS FOR THE THREE AXES OF ROTATION) HONOC
FORMAT(1027(SH I )9/I969XoZTH... END OF CALCULATIONS ...I HONOC

605
700

FVERGE UITHIN o

FORHATI/lo 65H THERE IS AN ERROR. AN

GIONS AREo/o 6H E1 I 9E15o9obH E2 3 9E150906H E3 I 9E15o9o/0

H18H

1999

10

12

ANGAoANGRoANGC I 9 3E15o99/1

FORMATCITH ERROR IN HONDCLNo/o (9X96E15o99/I)
FE I PHI'TORAD

PRINT 6000PHI

PRINT 601oALAOBEAOGAACALBOBEBOGABQALCOBECvOAC
NSOL I 0

SNEE I SINCEEI

CSFE I COSIFE)

ACCUR I PI/IBDODoD

NTRIESI30

ZLHOAAISORT(PEAIIZ0GAAIIZI
ILHDABISQRT(FER’IZ‘GABIIZI
ILHDACISORT(BECI'2OGAC'IZI

OPLUSA I SIGNIRCOSIBEA/ZLNUAAIOGAA)

DPLUSB I SIGNIACOSIBEB/ZLHDAHI95A.)

OPLUSC I SIGNLACOS(BEC/ZLMDACIQGACI

SNCHIC I (ALA-ALBI/IZLHDACISNFEI
IFIADS(SNCHICI-1.0) 8930999

CHIC I ASINISNCHICI

DO 80 KK'IOZ

ZETAC I FLOAT(2-KK)’(CHICOFEIOFLOAT(KR-1I'IPI'CHICOFE)
CSZTAC I COSCZETAC)

CSCHIB I (ALHOALC’ZoD’ALA-ZLNDAC'CSZIACI/ZLNDA.
IFIABSICSCHIB)-1o0) 109109999

CHIS I ACOSICSCHIB)

CSCHIA I (ALC-ALA-ZLHDAC'CSZTAC)IZLHDAA
IF(ABS(CSCHIA)-1.0) 120129999

CHIA I ACOSICSCHIAI

DO 80 LLIIOZ

ZETAA I CHIA

ZETAB I FLOATT3-2’LLIICHIB

ooo ANGAoANGBoANGC ARE THE (DOUBLE)-STARTINS-ANGLE-SHIFTS FOR

ITERATIVE PROCESS DID NOT CONHONOC
IkollH ITERATIONSo/o36H THE VALUE OF THE NULL FUNCTHONOC

HONOC
HONOC
HONOC
NONOC
HONOC
NONOC
HONOC
MONOC
HONOC
NONOC
NONOC
MONOC
HONOC
HONOC
HONOC
HONOC
HONOC
HONOC
HONOC
HONOC
HONOC
HONOC
HONOC
HONOC
HONOC
HONOC
HONOC
HONOC
HONOC
HONOC
HONOC
HONOC
HONOC

 

150

 

 

C RDTATIONS ABOUT AXES AoBoC RESPECTIVELY. NONOC SO
ANGA I DPLUSA-ZETAA HONOC 59
ANGB I DPLUSH-ZETAB NONOC 60
ANGC I DPLUSC-ZETAC HONOC 61
JFLAGI0 NDNOC 62
GO TO 17 NONOC 63

15 ANGA I ANGAODELA HDNOC 66
ANGa-I ANGBODELB ' NDNOC 65
ANGC I ANGCODELC NONOC 66

17 CSR I COSIANGAI NDNOC 67
C58 I COS(ANGB) HONOC 68
CSC I COSTANGC) NONOC 69
SN‘ I SINIANGAI HONOC 7D
5N8 I SIN(ANGB) NONOC 71
SNC I SIN(ANGC) NDNOC 72
IEAKIBEAICSAOGAAISNA NONOC 73
IEBKIBEBICSBOGABISNB HDNOC 76
.ECKIBECICSCOGACISNC NONOC 75
GAAKIGAAICSA-BEAISNA NONDC 76
GABKIGAB'CSB-BEBISNB NONOC 77
GACKIGAC'CSC-BEC'SNC HONOC 78
E1 I ALC-BECK-ALA-BEAK HDNDC 79
E2 I ALC-BECKOALH-BEBK-Z.OIALA NDNDC 60
E3 I ALB-ALAOSNFE.(GACK'CSFE-BECKISNFEI NONOC 81
DELC I E3/(SNFE'(BECK'CSFEOGACK'SNFEII NDNOC 82
DELB I (E2°GACK'DELC)/GA8K NONOC B3
DELA I (EIIGACK'DELCIIGAAK HONOC 86
JFLAGIJFLAG01 HONOC 85
IFIJFLAG-NTRIESI 20019919 HDNOC 86

19 PRINT 7009NTRIESOE1oEZoE3oANGAoANGDcANGC NONOC 67
GO TO 25 NONOC 88

20 XXA I AHS‘DELA) NDNOC 89
XXB I ABS(DELBI NONOC 90
XXC I AHSIDELC) HONOC 91
XX I AHAXIIXXAQXXBDXXC) HONOC 92
IFIXX-ACCUR) 25025015 NDNOC 93

25 DO 80 HNIIOZ HONOC 96

C 0.... FIRST TIME THROUGH THE 00 LOOP ANGIIANGCOETCo AND GARKIGABKOETCNDNOC 95

C 0.... NEXT TIME. ANGB I 2°DPLUSBOANGIOETC. AND GABKI-GARKOETC. HONOC 96
ANGB I FLOATTZ-MHIIANGBOFLOAT(MN-11.12.0'DPLUSD-ANGDI HONOC 97
ANGA I FLOAT(2'HHIIANGA0FLOAT(NH-11.12.0'DPLUSA-ANGA1 NDNOC 98
GABK I FLOAT(3'2'NN)IGA8K HONOC 99
GAAK I FLOAT(3-2.HH1IGAAK HONOCIDO
ANGDEG(11 I ANGAOTODEGZ NONOCIOI
RNGDEGIZI I ANGBITODEGZ HONOCIOZ
ANGDEGI31 I ANGC'TODEGZ HONOC103

C CHANGE THE STARTING-ANGLE SHIFT SO THAT ITS NAONITUDE IS LESS THAN 90 HONOC104

C DEGREES. TO ADD OR SUBTRACT 180 DEGREES DOES NOT AFFECT THE SOLUTION. HONOCIOS
DO 50 KI193 HONOC106

‘6 X2 I ABS(ANGOEG(K))-90.0 HONOC107
IF(XZI 50950947 NONOCIOB

67 ANGDEGIK) I ANGDEGIKI-SIGN1180.09ANGDEG(KII NONOC109

. GO TO 66 HONOCIIO

50 CONTINUE HONOCIII ‘
NSOL=NSOL01 HONOCIIZ
PRINT 602 HONOC113
PRINT 6039NSOL0ANGDEG HONOCIIQ
PRINT 604 HDNOCIIS
PRINT 601oALAvBEAKoGAAKoALBOBEBKOGABKoALCoIECKoGACK HONOC116
U11 I ALAOBEAK NONOC117
H22 I ALA-BEAK HONOCIIB
N33 I ALCOBECK MONOC119
H12 I GAAK NONOC120
N13 I GACK HONOC121
N23 I (GAAK’CSFE-GABKI/SNFE HONOCIZZ
CALL DISPLAYIZ) HONOC123

80 CONTINUE HONOCIZA
PRINT 605 HONOC125

RETURN HONOC126

151

999 PRINT I9999ALA9BEA9GAA9ALB9BE89OAD9ALC9OEC9GAC9BEAK9GAAK9BE'K9GABKHONOC127
K98ECK9GACK9E19E29E39DELA9DELB9DELC9ANGA9ANGI9ANGC9DPLUSA9DPLUSB9 MONOCIZB

BDPLUSC9ZETAA92ETAB9ZETAC9ACCUR MONOC129
RETURN MONOC130

END MONOC131
SUBROUTINE RHOMBIC RHOMB 1
DIMENSION ANGLESI39B)9LL(3)9LSIGN(39B)9LSI(3939B)9NSOLIB) RHOMB 2
COMMON/PARAM/ALA9BEA9GAA9ALBvBEBoGAB9ALC9IEC9BAC RHOMB 3
COMMON/N/UI19H229H339H1299239H13 RHOMB 6

DATA PI/3.16159265359/9TOOEGZ/28.64788976/9NTRIES/30/9KHA/1HA/9 RHOMB 5
DKHB/IHB/oKHC/IHC/9IY1/1/91Y2/2/9IY3/3/9IY6/4/9IYS/S/9IY6/6/9 RHOMB 6
DIY7/7/9IYB/B/ RHOMB 7

C ALA9ALB9ALC ARE THE ALPHA PARAMETERS FOR ROTATIONS ABOUT AXES A9B9C. RHOMB B
C BEA9BEB9BEC ARE THE BETA PARAMETERS FOR ROTATIONS ABOUT AXES A9B9C. RHOMB 9
C GAA9GAB9OAC ARE THE GAMMA PARAMETERS FOR ROTATIONS ABOUT AXES A9B9C. RHOMB 10
C TOOEG2 I S7.29S779SI/2.0 I (180/PI)/2. RHOMB 11
600 FORMAT11H19 37X927H.... ORTHORHOMIIC CASE ....9 ll939X9 ZBHTHE ORIRHOMB 12

AGINAL PARAMETERS ARE89/) RHOMB 13
601 FORMATI37X9SHALPHA911X9AHBETA912XoSHGAMMA9/I9ZAX97HAXIS AI91F16.B9RHOMB 16
B2E16o9l/26X97HAXIS B!91F16.892E16.9/l26X97HAXIS C'9F16.B92E16.9//)RHOMB 15

602 'ORHRTIZZXO 38HTHE STARTINGIANGLE ERRORS (IN DEGREESIO ’9 22X. RHONB 16
C29HABOUT THE THREE ROTATION AXES9/923X96HAIIS A96X96HAXIS B96X9 RHOMB 17
D6HAXIS C) RHOMB 16

603 FORMAT(3X912HSOLUTION No.9 IZ9AH... 9F9.493X9F9.493X9F9.6) RHONB I9

60‘ FORMAT(/9 27X962HTHESE B SOLUTIONS HAVE THE SAME ALPHA ANO IETA PARHOMR 20
ERAMETERS. THE9 l9 27X962HSIGNS OF THE GAMMA PARAMETERS ARE DIFFERRHOMB 21

EENT FOR EACH SOLUTION99/9 35X9 8(6HSOLTN )9/938X95HALPHA910X9 - RHOHB 22
ORHBETA915X95HGAMMA97X9B(9H NO.912)9/9 121X9SHAXIS 91A191HI9F16.89 RHOHH 23
H6X9E159995H I‘-)9E19.993X9 8(2H (91A193H) 19/11 RHOHB 2“
605 FORMATI/9 17X916HTHE H TENSOR I539 2X931E15.992X19//35X93(E15.99 RHOHB 25
I2X19/l35X93IE15.992X)9/1 RHONB 26

606 FORMAT! 7X9129HTHE SIGNS TO BE ASSOCIATED UITH THE ELEMENTS OF THERHOMB 27
J HITENSOR GIVEN ABOVE ARE ARRANGED IN A CORRESPONDING MATRIX FOR ERHOMB 28
KACH SOLUTION9 //9 2X9 8(SX99HSOLTN NO.912)9/9 3(9X9 8(3X91A191X9 RHOMB 29

L1A191X91A199BX19 l1) RHOMB 30
607 FORMATI 2X926HTHE RESULTS FOR SOLUTIONS 91191H91191H9I191H9II9 RHOHB 31
NIOH ORE GIVEN HELOH9/9 27ISH I )1 RHONB 32

608 FORMATI l9 19X941HABOVE ARE THE RESULTS FOR SOLUTION NUMBER9IZ9 RHOMB 33
NIH.9/9 19X982HTO OBTAIN THE OTHER SOLUTIONS9 CHANGE THE SIGNS OF ORHOMB 36
ONE COLUMN OF DIRECTION COSINES9/9 19X973HAND CHANGE THE CORRESPONDRHOHB 3S
PING COLUMN OF ANGLES BY SUBTRACTING 180 DEGREES.9 //9 (I9X919HFOR RHOMB 36

ISOLUTION NUMBER9 I29 29H CHANGE THE COLUHN UNDER AXIS9 129/)) RHOMB 37
609 FORMATIZTISH I )9/) RHOMB 38
610 FORMATIA9X927H... END OF CALCULATIONS ...) RHOMB 39
611 FORMAT(//936X924H... ERROR IN RHOMBIC! XZ91A1911H I COS(ZETA91A19 RHOMB 60

RAH) I 9IF10.6) RHOMB 41
612 FORMATISSX92OHHAS BEEN CHANGED TO 91F10.69/) RHOMB 42

700 FORMATI/l9 65H THERE IS AN ERROR. AN ITERATIVE PROCESS DID NOT CONRHOMB A3
SVERGE WITHIN 9 IA911H ITERATIONS9/936H THE VALUE OF THE NULL FUNCTRHOMR AA

TIONS ARE9/9 6H E1 I 9E15.996H E2 I 9E15.996H E3 I 9E15.99/9 RHOHB AS

U1BH ANGA9ANGB9ANGC I 9 3E15.99/) RHOMB A6
PRINT 600 RHOHH 67

PRINT 6019ALA9HEAQGAA9ALB9HEBOGA89ALC9BEC9OAC RHOHR 68

ACCUR I PI/IBOO0.0 RHOHB 69
ZLMDAAISORT(BEA'IZ’GAAIIZI RHOHB 50
ZLMDABISQRT(BEB'IZ‘GABIIZ) RHOMB SI
ZLHOACISQRT(HECIIZ‘GACIIZI RHOMR 52
DPLUSA I SIGNIACOSIBEA/ZLHOAA)9GAA) RHOHH S3
DPLUSB I SIGNIACOSIBEB/ZLHDAB)9GAB) RHOMH 54
DPLUSC I SIGNIACOSIBECIZLMOAC)9GAC) RHOHR 55
XZAIIALC-ALBIIZLMDAA RHONB 56
XZBIIALA-ALCIIZLMDAB RHONB S7
szIIRLBIALAIIZLHDAC RHOMR 58
KERROR I O RHOHH 59

C KERROR I (0911 IF EITHER XZA9XZB9OR XZC (HAS NOToHAS) BEEN CHANGED TO RHOHH 60
C TO ALLOU A SOLUTION TO EXIST. RHOMB 61
TV I AHS(XZA1 RHOHB 62
IFIYYI1901 49492 RHOMH 63

2 KERROR I 1 RHOMB 6A

 

152

PRINT 6119KHA9KHA9XZA
XIA I SIGN(1.09XZA)
PRINT 6129XZA
6 YY I ABSIXIB)
IFIVY'1901 89896
6 KERROR I 1
PRINT 6119KHB9KHB9XZB
A29 I SIGN(1.09XZB)
PRINT 6129XZB
O 77 I ABSIXZC)
IFIYY’I-O) 12912910
10 KERROR I I
PRINT 6119KHC9KHC9XZC
XIC I SIGN(1.09XZC)
PRINT 6129XZC
I2 IETAA I ACOSIXZA)
IETAB I ACOSIXZB)
IEIAC I ACOSIXZC)
P" I ‘IOO
DO 25 III192
PMIIPM
C 9999. ANGA9ANGB9ANGC ARE THE (DOUBLE)ISTARTINO-ANOLE‘SHIFTS FOR
C ROTATIONS ABOUT AXES A9B9C RESPECTIVELY.
RNGA I DPLUSA'SIGNIZETAA9PH1
ANGB I DPLUSBISIGNIZETAB9PM)
ANGC I DPLUSC-SIGNIZETAC9PM)
JFLAGIO
I5 CSAICOSIANGA)
CSBICOSIANGB)
CSCICOSIANGC)
SNRISINIANGA)
SNBISINIANGB)
SNCISINIANGC)
'ERNIBEAICSAOGAAISNA
'EBKIBEBICSBOGABISNB
DECKIBECICSCOGACISNC
GAAKIGAAICSA-HEAISNA
OABKIGABICSB-BEBISNB
GACKIGACICSC-BECISNC
E1 I ALC-ALB-BEAK
E2 I ALA-ALC-BEBK
E3 I ALB-ALA-BECK
DELA I EIIGAAK
DELB I E2/GABK
DELC I E3/GACK
C IF KERROR I 19 THEN HE CAN NOT FIND A SELF-CONSISTENT SOLUTION9 SO HE
C ALLOW THIS APPROXIMATION TO BYPASS THE CHECKING OF DELA9DELB9DELC.
IFIKERROR) 21917921
17 ANGAIANGAODELA
ANGBIANGRODELB
ANGCIANGCODELC
JFLAGIJFLAGOI
IFIJFLAG-NTRIES) 19919918
16 PRINT 7009NTR1E59E19E29E39ANGA9ANGB9ANGC
GO TO 21
19 XXA I ABSIOELA)
XXB I ABS(OELB)
XXC I ABS(DELC)
XX I AHAXIIXXA9XXB9XXC)
IFIXX-ACCUR) 21915915
21 IJIAIII'J
ANGLESI19IJ12ANGA
ANGLESI291JIIANGH
ANGLESI39IJ1=ANGC
25 CONTINUE
XAIIANGLESI1951-ANGLES(191I1/290
X8I1ANGLESTZ95)-ANGLES(291))/2.0
XCIIANGLES(3951-ANGLES(391I1/2.0
BEAKIZLMDAA'COSIXA)

RHOHB 6S
RHOMB 66
RHOMB 67
RHOMB 6a
RHOMB 69
RHOMR 7o
RHOMB 7|
RHOMB 72
RHOHB 73
RHOMB 76
RHOHB 7S
RHOHB 76
RHOMB 77
RHOHB 78
RHOHB 79
RHOHB so
RHOHB 8|
RHOHB 82
PHONE 83
RHOMB ah
RHOMB 85
RHOMB 86
RHOMB 87
RHOMB 88
RHOMB B9
RHOMB 9o
RHOMB 91
RHOMB 92
RHOMB 93
RHOHB 9a
RHOMB 9S
RHOHR 96
RHOHR 97
RHOMB 98
RHOMR 99
RHOMBIOO
RHpMBlOl
RHOHBIOZ
RHOMBIO3
RHOMRIOA
RHOMBIOS
RHOMRIO6
RHOMBIOT
RHOMBIOB
RHOMRIO9
RHOMBIIO
RHOMRIII
RHOMRIIZ
RHOMBIIJ
RHOMBIIA
RHOMBIIS
RHOMBII6
RHOMRIIT
RHOMBIIB
RHOMRII9
anonalzo
RHOMBIZI
RHOMBIZZ
RHOM8123
RHOMBIZA
RHOMBIZS
RHOMBI26
RHOMBIZ?
RHOMBIZB
RHOM9129
RHOHBIBO
RHOMBIBI
RHOMRIBZ
RHOMBI33

 

' J

C CHANGE THE STARTING-ANGLE SHIFT SO THAT ITS MAGNITUOE IS LESS THAN 90
C DEGREES. TO ADD OR SUBTRACT 180 DEGREES DOES NOT AFFECT THE SOLUTION.

27
28
30

62

AS

50

60

153

GAAKIZLMOAA'SINIXA)
BEBKIZLMDABICOS(XB)
GABKIZLMOAB'SIN(XB)
DECKIZLMOAC'COS(XC)
GACKIZLMDAC‘SINTXC)
GAAB I ABSIGAAK)
GA88 I ABS(GABK)
GACB I ABS(GACK)

DO 30 KI19596

DO 30 JI193

XX I ANGLES(J9K)ITODEG2
X2 I A85(XXI-909O
IFIXZ) 30930928

XX I XX'SIGN(180.09XX)
GO TO 27

ANGLESIJ9K) I XX

DO 32 II193

LSIGNII91) I 1H0
LSIGNII95) I 1H0
IFIGAAK) 33934934
LSIGNIIOII I IN-
IFIGABK) 35936936
LSIGNI2911 I 1H-
IFIGACK) 37938938
LSIGNI391) I 1H-

DO 50 II193

MM I LSIGNTI911-1HO
IFIHM) 40939940
LSIGNII95) I 1”“
CONTINUE

LLIIIIOZ

LLI21I-2

LLI31I02

DO 65 I8I296

DO 62 IAI193
ANGLESTIA918) I ANGLESIIA93°LLIIA11
ANGLESIIA91894) I ANGLESIIA93ILLIIAII

LSIGNIIAoIB) I LSIGNIIA93°LLIIAII
LSIGNIIA9IBOA) I LSIGNIIA93-LLIIA))
ITILLI3)

LLI3IILL(2)

LLIZIILLIII

LLIIIIIT

CONTINUE

DO 50 II198
LSOII919III1HO
LSOI2929I)‘1H°
LSOI3939I)I1HO
LSOII929I) I LSIGNI19I)
LSOI2919I) I LSIGNII9I)
LSOII939I) I LSIGNI39I)
LSOI3919I) I LSIGN(39I)
LSOIZ939I) I LSIGN129I)
LSOI3929II I LSIGNIZvII
CONTINUE

H11 I ALAOBEAK

H22 I ALROBEHK

H33 I ALCOBECK

H12 I GAAK

H23 I GABK

H13 I GACK

DO 60 II198

NSOLII) I I

PRINT 602

PRINT 6O39III9IANGLESIJ9I)9J=193))9II198)

PRINT 6OA9NSOL9KHA9ALA98EAK96AAH9(LSIGNI19KI9NI19819

1KH89AL898E8K9GABR9ILSIGNIZ9KI9K=1981

9KHC9ALC9HECK96AC89

RHOMBI36
RHOMBIJS

RHOMBI36
RHOM8137
RHOMRIJB
RHOM8139
RHOHBIAO
RHonnlhl

RHOMBIAZ
nHonele
RHOMBIAA
RHOMRIAS
RHOMBlbb
RHOMBIA7
RHOMBIAB
RHOM8169
RHOMBlSO
RHOMBISI

RHOMBISZ
RHOMBISJ
RHOHBISA
RHOHBISS
RHOM8156
RHONBIS?
RHOHBISB
RHOM8159
RHON816O
RHOMBI61

RHOMBI62
RHOMRI63
RHOM816A
RHOM8165
RHOMBI66
RHOM8167
RHOMfilbfl
RHOM8169
RHOMBI70
RHOHBI71
RHOMBI72
RHOMBIT3
RHOMBITA
RHOMRI75
RHOMBI76
RHOMBI77
RHOMBI78
RHOMBIT9
RHOMBIBO
RHOMBIRI

RHOMBIRZ
RHOM9183
RHonalea
RHOMBIBS
RHOH8186
RHOM8187
RHOMBIBB
RHOM8189
RHOM8190
RHOMRI9I

RHOM8192
RHOMBI93
RHOMRIQA
RHOMBI9S
RHOMnl96
RHOMRI97
RHOMRI9H
RHOMRI99
RHOMRPOO
RHOMBEOI

RHOMRZOZ

 

154

006066

non

2(L5IGNI39K)9K=198) RHOMB203
PRINT 6059H119GAAB9GACB9GAA89H229GABB9GACI9GAID9H33 RHOHBZO“
PRINT 6069NSOL9(((LSO(I9J9K)9J=193)9KI198)9II193) RHOHBZOS
PRINT 6079IY19IY291Y39IY0 RHOH8206
CALL DISPLAY(3) RHOMBZO7
PRINT 6089IY19IY29IY19IY391729IY49IY3 RHOMBZOB
PRINT 609 RHOMB209
H12I-H12 RHOMB210
H23IIH23 RHOMBZII
H13IIH13 RHOMB212
PRINT 6079IYS9IY69IY79IY8 RHOM8213
CALL DISPLAY(3) RHOM8216
PRINT 6089IYS9IY69IYI9IY79IY29IY89IY3 RHOM8215
PRINT 609 RHOM8216
PRINT 610 RHOM8217
RETURN RHOM8218

END RHOM8219
SUBROUTINE GENERAL(PHI9NN19NN29NN3955195529553) GENER 1
999999 PHI IS IN DEGREES. GENER 2
NN19NN29NN3 ARE THE DIRECTION COSINES HITH RESPECT TO AXES 19293 GENER 3
RESPECTIVELY OF THE ARBITRARY ROTATION AXIS C. GENER I
55195529553 ARE THE DIRECTION COSINES HITH RESPECT TO AXES 19293 GENER 5
RESPECTIVELY OF THE STARTING DIRECTION FOR THE ROTATION ABOUT AXIS C GENER 6
NEITHER (NN19NN29NN3) NOR (55195529553) NEED 8E NORHALIZED. GENER 7
REAL H19H29H39MH19MM29HH39N19N29N39NN19NN29NN39ANGDEG(3) GENER 8
COHHON/PARAM/ALA98EA9GAA9ALB9BE89GA89ALC9'EC9GAC GENER 9
COHHON/H/HI19H229H339H129H239H13 GENER 10
DATA PI/3.16159265359/9TODEG/57.29577951/9TODEG2/28.6A788976/9 GENER 11
DTDRAD/OoOI765329252/9NSTEP/360/9NTRIES/15/ GENER 12
ALA9AL89ALC ARE THE ALPHA PARAMETERS FOR ROTATIONS ABOUT AXES A9B9C. GENER 13
8EA98E898EC ARE THE BETA PARAMETERS FOR ROTATIONS ABOUT AXES A9B9C. GENER I“
GAA9GAB9GAC ARE THE GAMMA PARAMETERS FOR RDTATIONS ABOUT AXES A9B9C. GENER 15
600 FORHAT(1H19 AOX922H.... GENERAL CASE ....910X96HPHI I 91F9.69//9 GENER I6
A39X928HTHE ORIGINAL PARAMETERS AREI9I) GENER 17
601 FORHAT(37X95HALPHA911X9AHBETA912X95HGAMMA9l/924X97HAXIS A391F16.B9GENER 18
82E16o9/I2AX97HAXIS 8191F16.892E16.9//2AX97HAXIS C89F16.892E16.9//)GENER 19
602 FORNAT(28X9 72HTHE VECTOR (N19N29N3) REPRESENTING THE DIRECTION OFGENER 20
C AXIS C 15 GIVEN AS (91F9.691H91F9.691H91F9.691H)9/9 69X931HTHIS IGENER 21

D5 NORMALIZED TO PRODUCE (91F9.691H91F9.691H91F9.691H)) GENER 22
603 FORHAT(100H THE VECTOR (51952953) REPRESENTING THE STARTING OIRECTGENER 23
EION FOR A ROTATION ABOUT AXIS C 15 GIVEN AS (91F9.691H91F9.691H9 GENER 26
F1F9.691H)) GENER 25
60A FORHAT(84H THIS VECTOR IS PROJECTED ONTO THE PLANE PERPENDICULAR TGENER 26
GO (N19N29N3) AND NORMALIZED.9/976X9 ZAHTHESE CORRECTIONS GIVE (9 GENER 27
H1F9o691H91F9.691H91F9.691H)9/9 AOX960HTHE STARTING-DIRECTION VECTOGENER 28

IR (51952953) 15 NOH CHANGED TO (91F9.691H91F9.691H91F9.691H)) GENER 29
605 FORHAT(60X917HBY A ROTATION OF F9.A925H DEGREES ABOUT (N19N29N3)/)GENER 30
606 FORHAT(43H THIS CHANGES THE FOLLOWING THO PARAMETERSI9/9 35H BETAGENER 31
J FOR ROTATION ABOUT AXIS C I 91E15.99/935H GAMMA FOR ROTATION AROUGENER 32

KT AXIS C I 91E15.99//) GENER 33
607 FORHAT(22X9 38HTHE STARTING-ANGLE ERRORS (IN DEGREE519 I9 22X9 GENER 39
L29HABOUT THE THREE ROTATION AXES9/923X96HAXIS A96X96HAXIS 896X9 GENER 35
H6HAXIS C) GENER 36
608 FORMATI3X912HSOLUTION N099 I204"... 0F90493XOF9.“93XOF994I GENER 37
609 EORHAT(/936X9 “BHTHIS SOLUTION HAS THE FOLLOHING ALPHA98ETA9GAHHA9GENER 38
N/9 36X941HPARAMETERS FOR THE THREE AXES OF ROTATION) GENER 39
610 FORHAT(/927(SH I )9//949X927H9.. END OF CALCULATIONS ...) GENER 60
700 FORHATI/I9 65H THERE IS AN ERROR. AN ITERATIVE PROCESS DID NOT CONGENER AI
OVERGE HITHIN 9 IA911H ITERATIONS9/936H THE VALUE OF THE NULL FUNCTGENER 62
PIONS ARE9/9 6H E1 I 9E15.996H E2 I 9E15.996H E3 I 9E15.99/9 GENER 63
018H ANGA9ANGB9ANGC I 9 3E15.99/) GENER 66
1999 FORHATI19H ERROR IN GENERAL! 92169/9(39X93E15.99/) ) GENER 65
FE I PHI'TORAD GENER 66
NSOL I 0 GENER A7
ABSTEPIPI/FLOAT(NSTEP) GENER 68
STEPIABSTEP GENER A9
ACCUR=STEPIIOOO0.0 GENER 50
CONV I 3.09ABSTEP GENER 51

PRINT 6009PHI GENER 52

0600

155

PRINT 6019ALA98EA9GAA9ALB9BEB9GA89ALC98EC9OAC GENER 53
999.9 NORMALIZE (NN19NN29NN3) TO FORM (N19N29N3) 9.. GENER 54
RENORN . SORT (NN 1 .NNI ONNZ'NNZ’NNB'NNJ) GENER 55
I'(RENORN) “9999 GENER 56

6 NI'NNIIRENORN GENER 57
N2'NNZ/RENORH GENER 58
N33NN3/RENORM GENER 59
PRINT 6029NN19NN29NN39N19N29N3 GENER 60
PRINT 603955195529553 GENER 61
99999 PROJECT (55195529553) ONTO THE PLANE PERPENDICULAR TO THE GENER 62
VECTOR (N19N29N3) ..9 GENER 63

DCSN ' 551'N1’552'NZ’553’N3 GENER 66
551‘551-0CSN'N1 GENER 65
SSZ'SSZ'DCSN'NZ GENER 66
553'553-DC5N0N3 GENER 67
99999 NDRHALIZE THE NEH (55195529553) GENER 68
RENORH ' SORT(551.551’552'552’553'553) GENER 69
IE(RENORM) 59999 GENER 70

5 SSI'SSIIRENORH GENER 71
SSZ’SSZ/RENORN GENER 72
5533553/RENORM GENER 73
99999 FORM (HM19MH29HH3) FROM (NI9N29N3) CROSS (55195529553) 99. GENER 76
HHI'NBPSS3’N3'SSZ GENER 75
HHZ'N3'SSI'NI'SS3 GENER 76
HH33NIPSSZ'N2'SSI GENER 77

..9.. CHANGE TO A NEH STARTING DIRECTION (51932953) UHERE S3l0.0 AND GENER 78

A CORRESPONDING NEH MIDDLE DIRECTION (N1.N2.N31 ... GENER 79
Trtuna) 6.999.6 GENER so

6 1r15531 8.7 GENER a1
7 MllMHI GENER 82
NZINMZ GENER 83
N3-MN3 GENER 86
518551 GENER 85
52.552 GENER 86
538553 GENER a?
BECSaeEC GENER 88
GACSIGAC GENER 89
x1 I 0.0 GENER 90
GO TO 18 GENER 91

8 ALNOA - SORTTSS3'SS39HM3'MM3) GENER 92
51 . (SSloMNT-SSJPMN11/ALNDA GENER 93
52 . (SSZOMM3-5S3'MH21/ALHDA GENER 99
53 h 0.0 GENER 95
N1 - (SSI'SSJOMMI'HM311ALMDA GENER 96
N2 I (552-5530NN20NN31/ALNDA GENER 97
N3 9 ALNOA GENER 9a
TANK! . ~553/NN3 GENER 99
x1 x ATAN(TANXI1OTODEG GENERloo
IFJNMJ) 10.10.15 GENERIOI
10 x1 = x1-516N11eo.o.5531 GENERloz
x1 15 THE AuGLE OF ROTATION ABOUT (N19N2.N31 To PRODUCE (51.52.53). GENERIOJ
..... NEH BETA AND GAMMA PARAMETERS FOR ROTATION AIDUT Asz c. GENERloa
OLD VALUE: BEC.GAC ARE ASSOCIATED HITH (551.552.5531... GENER105

NEH VALUES BECS.GACS ARE A55OCIATED HITH (51.82.53). GENER106

15 DECS s (RECO(NN3-02—5530-21-2.O'GAc-NN3-593)/(ALNDA~ALNDA1 GENERlOT
GACS : (GACGTNN30-2-5530-21o2.0°85c-NN30551)/(ALNDA-ALNDA1 GENERloe
18 PRINT 609.55195529553.Sl.52.53 GENER109
PRINT 605.x1 GENER110
PRINT 606.85C5.GAC5 GENER111
ILNOAA-SORT(REA-ozoGAAo-21 GENER112
ILMDABISORT(BEB'GZOGA89'21 GENER113
ZLMDACISORT(HECS'°ZOGACS"2) GENERlla
DPLUSA . SIGN(ACOS(REAIZLMDAA1.GAA) GENERIIS
OPLUSB - SIGN(ACOS(BE8/2LHDABD.GAB) GENER116
DPLUSC - SIGN(ACOS(BECSIZLM0AC196ACS) GENER117
SNFE . SIN(FE1 GENER118
csre x COS(FE1 GENER119
SNZFE - SIN(2.0'FE1 GENER120
SNFEZISNFE"? GENERIZI

35“."!

0000000560060

156

CSFEZ‘CS'E"?
.ElK'BEA
'EBK'BEB
BECK'BECS
GA‘K3G‘A
GABK‘GAB
G‘CK3GACS
CH!" 3 (ALA'ILB’ZLHDAB)/ZLMDAA
CN‘X 3 (AL‘-‘L8’ZLHDAB)/2LNOAA
CH1N 3 AHAXICCMINO'loO)
CH‘X 3 ANINI(CHAX.01.O)
XNAX 3 ACOS(CH!N’
X"!N 3 .COS(CNAX,
IBEG 3 ‘FtX‘XNIN/STEP,
[END 3 I'IX(‘H‘X/SYEP).‘
IF(IEND'lBEG) 9990999020
20 DO 20‘ Itl3‘02
STEP 3 “STEP
PNB 3 .‘00
PNC ' -100
00 201 LNN3‘0‘
‘3PNB
PNB3-PNC
PNC3'
KLL'O
KSCH30
INDEX'O
CROSS3°loo
DO 200 I'lHEGOIEND
00000 ‘NDEX 3 NUMBER OF SVEPS ‘N ‘LLO'EO “[010".
00000 KLL ' 0 FORBIDDEN REGION
00000 KLL 3 l ALLOWED REGION
00000 KLLSV 3 VALUE OF KLL FOR PREVIOUS-STEP. ENTERlNG INC 00 L009
00000 KSCH 3 (001’ IF SEARCH FOR CROSSOVER (DOES NOIODOES. OCCUR0
00000 KSCHSV 3 V‘LUE 0F KSCH FOR PREVIOUS STEP0
FOR 'HE FIPSY T!HE CAUSES KLLSV 10 SE SE! to 00
00000 CROSS [5 LESS THAN 0R EQUAL 10 000 ONLY IF ‘ CROSSOVER IS TO
BE SE‘RCHED '09.
00000 FUNC 3 VEST FUNCTION 3 E30 HE SEARCH FOR THE ZERO V‘LUESO
00000 SFUNC 3 L557 VALUE OF FUNC
00000 SSFUNC 3 SECOND To YHE LAST V‘LUE 0' FUNC
ZYA3FLOAV(I).STEP
CSZYA3C05(ZTA)
KLLSV3KLL
KSCHSV'KSCH
KSCH30
SSFUNc3S'UNC
SFUNC’FUNC
CSZTBB(ZLMDAA'CSZTAOALB-ALA)IZLHDAB
1F‘IBS(CSZTB)-100) 380380190
38 SNZTA 3 SIN‘ZY"

GENERIZZ
GENERIZJ
GENERIZQ
GENERIZS
GENERIZO
GENERIZ?
GENERIZO
GENERIZ9
GENERIJO
GENERIJI
GENERIJZ
GENEPIBJ
GENERIJ“
GENEPIJS
GENER136
GENER137
GENERIJB
GENERI39
GENERIbO
GENERlb‘
GENERIbZ
GENERle
GENERIbb
GENERIhS
GENERIhb
GENERIb?
GENERIbB
GEN£R|69
GENERISO
GENERISI
GENERISZ
GENERIS3
GENERISA
GENERISS
GENERIS6
GENERIST
GENER158
GENERIS9
GENERIbO
GENERI6I
GENER162
GENER163
GENERlbh
GENERI6S
GENERI66
GENERl67
bGENERle
GENERI69
GENERI70
GENERl7l
GENERI72

CSZTC8(ZLMDAA'((2.'Sl'Sl-l.)'CSZTA°2.'SIOSZ'SNZTA)OALA-lLC’IZLHDACGENEPl73

IFCABS(CSZTC)-l.0) 399390190

C 0000000 ‘LLOHEO ZONE 00000

C

39 KLL'I
lNOEX3iNDEX’1

00000 C‘LCULAVE FUNC 000
SNZTB 3 SIGN((SQRT(100‘CSZTB..2’)IPNB,
SNZ'C 3 SIGN((SORT(10°'CSZTC..2))OPNC’

ach . ZLHDAAOCSZTA
aeex . ZLHDAR'CSZTB
atcx - ZLHDAC'CSZYC
GAAK 3 ZLMOAAOSNZTA
GABK - zLqunosnzre

GACK 8 ZLMDAC'SNZYC

9233(GAAK'CSFE-GABK)ISNFE
H133(GACK02.0'BEAK'SZ'M2-GAAK0($19M20520Hl)-U23'SZ'H3)/(SI'H3)
U33=CALBOBEBK-(ALAOBEAK)'CSFE20H13“SN2FE)/SNFEZ

GENERI?“
GENEPI7S
GENERI76
GENERI77
GENERITB
GENERI79
GENERIBO
GENERIRI
GENERIBZ
GENERIBJ
GENERIBQ
GENERIBS
GENFRlfib
GENERIBT
GENERIBB
GENERIB9

FUNC'RLAO(1.0-M39'Z)OBEAK*(H10¢2-M2“’2)02.0‘6AAK9H1.H20?.0’H23'M2'GENER190

 

00

00000 0 0

0000

157

'H39200'H13."!.HJOHJJ'HJO'Z-ALCOBECK
0000. 1' LAST STEP HAS IN THE FORBIDDEN RE01ON0 CHECK 'OR CROSSOVERo
1F1KLLSV1 500100
00000 1' TNDEX310 THIS 15 THE VIRST STEP AND ANOTHER STEP TS TAKEN.
1F 'UNC30.00 THEN THIS HILL DE DETERMINED IV THE PRODUCT S'UNC'FUNC‘
50 1F(1NDEX-1) 2000200052
52 CROSS 3 SFUNC'FUNC
00... 1' CROSS TS NEGATIVE OR zERO0 A CROSSOVER HAS OCCUREDo
1F1CROSS) 100010005“
00000 1NDEX MUST BE AT LEAST 3 FOR THE FOLLOHING CHECK T0 IE VALID
5. 1F11NDEX-3) 200056056
00000 HE TEST TO SEE IF THE FUNCTION HAS A'PROACHED TOHARD AND THEN
RETREATED FROH THE VUNC30o0 AAISo THERE ARE THO CONDITIONS THAT
MUST BE NET FOR THIS TO HAVE OCCURED.
(110 (FUNC‘SFUNC1‘(SFUNC-SSFUNC) MUST BE LESS THAN OR ECUAU T0 ZERO
(Z10 FUNC.(FUNC-SFUNC) MUST BE GREATER THAN 0R EDUAL T0 ZERO.
56 CHECK 3 (FUNC-SFUNC).(SFUNC-SSFUNC)
IFTCHECK) 580580200
50 CHECK 3 'UNC'(FUNC-SFUNC1
1FCCHECK1 20001000100
000000 ROUTTNE FOR SEARCHING FOR CROSSOVER ROTNTS 00000000
CALCULATE THE THREE ANGLES ABOUT HHICH THE SEARCH 15 TO DE01N
00000 ANGA0ANGB0ANGC ARE THE (DOUBLE)-START1NC-ANGLE'SH1FTS FOR
ROTATIONS ABOUT AXES AOBOC RESPECT1VELY.
100 KSCH31
ANGA 3 DPLUSA¢ZTA
ANGB 3 DRLUSB'SIGNCACOS(CSITD)0PM.)
ANGC 3 DRLUSC-S1GNCACOS(CSZTC10PNC)
JFLAG'O
GO TO 112
110 ANGAIANGAODELA
ANGBSANGBODELB
ANGCSANGC.DELC
112 CSA=COS(ANGA)
CSB=COS(ANGB)
CSC3COS(ANGC1
SNA351NTANGA1
5N8351N1ANGB1
SNC3S1N(ANGC)
DEAK38EA9CSAOGAA.SNA
BEBK3BEB'CSBOGAB.SNB
DECK3BECS’CSCOGACS'SNC
GAAK'GAA'CSA-BEA'SNA
GADK‘GAD'CSD-BEB'SNB
GACK3GACS‘CSC-BECS’SNC
H113ALA°BEAK
HZZ3ALA-REAK
H123GAAK
H23315AAK'CSFE'GABK)/SNFE
H133(GACK02.098EAK°SZ°HZ-GAAK.(Sl'MZ‘SZ'Hl1-H23'529H31/1513H3)
HJJ'TALHOHEBK-(ALA08EAK1.CSFE2°H13°SN2FE1ISNFEE
E13AL8-BEHK-ALAOHEAK
E23ALA°BEAK’(2o0’Sl.'2-1.0).2.0'GAAK“SI.SZ'ALC-IECK

GENER191
GENERI92
GENER1§3
GENERIQ“
GENERI¢5
GENERIQO
GEN£9197
GENER19O
GENERI99
GENERZOO
GENERZOI
GENERZOZ
GENER203
GENERZOb
GENERZOS
GENERZOO
GENERZOT
GENERZOB
GENERZO9
GENERZIO
GENERZII
GENERZIZ
GENERZIJ
GENERZIb
GEMERZIS
GENERZIG
GENERZIT
GENERZIO
GENER219
GENERZZO
GENERZZI
GENERZZZ
GENERZZ3
GENER220
GENERZZS
GENERZZb
GENERZZ?
GENERZZB
GENERZZ9
GENERZ30
GENERZJI
GENERZ3Z
GENERZ33
GENERZJh
GENERZJS
GENER236
GENER237
GENERZJB
GEN69239
GENE926O
GENE9261
GENE926Z
GENERZAJ

E3’ALA'11o0‘M39'21OBEAK'(H1"2-H2"2)02oO'BAAK'N1“M202.O'HZJ'HZ'HJGENERZAA

E02o0'H13°M1'M39H33‘H3"2-ALCOBECK
BB'El/GARK
DBPSGAAKIGABK
CC'EZ/GACK
CCP'IGAAK9(2.0'SI"Z-1.01-2o0'BEAKOSl'521/OACK
PPIBEBK'HBISNFE
PPPI(BEBK'BBD-BEAK’CSFE)ISNFE
003-(BECK’CC0PP'SZ'H3)[(SI‘HJ)

GENERZAS
GENERZhb
GENER267
GENERZAB
GENER249
GENERZSO
GENERZSI
GENERZSZ

OOP=C-BECK'CCPOZ.'GAAK‘SZ'HZOBEAK9(Sl'HZ'SZ'Hl1-PPP'52'H3)[($1'H3166NER253

Rna(61ooo-SN2FE)/snr£2

RRP=GAAK0(2.0'OOPOCSFE/SNFE)

r1 2 -EJ-£2-RR~M3-°2-2.o0pp»H2-M3-2.o-000uloua

r2 2 GAAK-(n1-uzosn-«2-Mzo-2-szooz)-2.o-Ieaxo(Ml-M2o51052)o
r Rap-uaouzoz.o-ppp-nzonaoz.oooopoun-ua

DELAIFl/FZ

GENE9296
GENERZSS
GENE9256
GENERZS7
GENE9258
GENERZS9

 

000

158

DELB3BBODBP'DELA
DELC3CC0CCP°DELA
XXA I ABS(DELA)
IFIJFLAG) 11401140119
11‘ IFIXXA-CONV’ 11901190115
115 IFICROSS) 11901190200
[19 JrLAG-JFLAGol
IFIJFLAG-NTRIESI 12001200195
120 SKI 3 ABSIDELB)
XAC 3 ABS(DELC)
XX 3 AMAXIIXXA0XXB0XXC)
IFIXX-ACCHR) 13001300123
123 IFIXXA-ABSTEPI 1250125012“
123 DELA 3 SIGNIABSTEP0DELA)
125 IFIXXB-ABSTEPI 12701270126
126 DELD 3 SIGNIABSTEP0DELB)
127 IFIXXC-ABSTEP) 11001100128
120 DELC 3 SIGNIABSTEP0DELCI
GO TO 110
130 ANGDEGIl) 3 ANGAOTODEG2
ANGDEGIZ) 3 ANGB'TODEGZ
ANGDEGI3) 3 ANGC'TODEG2

CHANGE THE STARTING-ANGLE SHIFT SO THAT ITS HAINITUDE
DEGREES. TO ADD 0R SUBTRACT 180 DEGREES DOES NOT AFFECT THE SOLUTION.

DO 140 K3103
136 X2 3 ARS(ANGDEG(K))-90.0
IF(XZT 14001400137

13? ANGDEG(K) 3 ANGDEG(K)-SIGN(180.00ANGDEGIK))

GO TO 136
160 CONTINUE
NSOL'NSOLOI
PRINT 607
PRINT 6080NSOL0ANGDEG
PRINT 609

PRINT 6010ALA0BEAK0GAAK0AL80BEBK0GA8K0ALC0DECK0GACK

CALL DISPLAY(A)
GO TO 200

0.00. FORBIDDEN ZONE ... IF LAST STEP HAS IN ALLOHED ZONE AND A
THEN CHECK FOR CROSSOVER POINT
AFTER RESETTING THE PREVIOUS VALUES 0' lTA0CSZTA0CSZTB0CSZTC.

CROSSOVER SEARCH HAS NOT PERFORMED0

190 INDEX30
KLL30
IFIKLLSV) 1920200
192 IFCKSCHSV) 2000193
193 ITAIZTA-STEP
CSZTA!COS(ZTA)
SNZTA I SIN(ZTA)

CSZTB'(ZLHDAA¢CSZTAOAL8-ALA1IZLHDAI

IS LESS THAN 90

GENERZ60
GENERZ61
GENER262
GENER263
GENERZOA
GENERZbS
GENERZOO
GENERZ67
GENERZOO
GENER269
GENERZTO
GENERZTI
GENERZTZ
GENERZTJ
GENERZTb
GENERZTS
GENER276
GENERZTT
GENER278
GENER279
GENERZBO
GENERZRI
GENERZBZ
GENER283
GENERZBA
GENERZBS
GENER286
GENERZBT
GENERZBB
GENERZR9
GENER290
GENER29I
GENER29Z
GENER293
GENER29A
GENER29S
GENER296
GENER297
GENER298
GENER299
GENERJOO
GENERJOI
GENERJOZ
GENER303
GENERJOA
GENER305
GENER306
GENER307

CSZTC'IILHDAA.((2.“51'51-1.1'CSZTA92.'SIPSZ'SNZTA)OALA-ALCTIZLMDACGENER308

GO TO 100
...... NO CROSSOVER HAS BEEN FOUND.
195 IVCCROSS) 19701970200
197 CRO$S=°1.0

PRINT 7000NTRIES0E10E20E30ANGA0ANGD0ANGC

200 CONTINUE

201 CONTINUE
PRINT 610
RETURN

999 PRINT 19990IREG0IEND0RENORM0NN10NN20NN30SSI0SSZ0SS30HM10MH20HH30

IF CROSS 15 LESS THAN 0.00

GENER309
GENER310
GENER311
GENER312
GENER313
GENERJIA
GENER315
GENER316
GENERJIT
GENER318

PXHIN0XHAX0CHIN0CHAX0STEP0ZLHDAA0ZLHDAR0lLHDAC0DPLUSA0DPLUSB0DPLUSCGENER3I9

RETURN
END
SUBROUTINE DISPLAY(INDEX1

DIMENSION H(30310R(303)0RA(303)0EIGI31

COHHON/H/HI10H220H330H120H230H13
DATA TORAD/57.Z9577951/

100 FORHAT(20X016HTHF H TENSOR ISC0ZX03IE16.901X)0//038X03(516.901X10

A/I038X03IE16.901X10/1

101 FORHAT113X0 IITHEACH OF THE PRINCIPAL VALUES

(HHICH ARE THE SQUAREDISPL

GENERJZO
GENER32I
DISPL
DISPL
DISPL
OISPL
OISPL
DISPL

NOW’UNH

 

I

. .

' V

Viv T-.—-A_

C

C

flfiflflflflfiflfififlfifififlflflflfififlflfi

159

B ROOTS OF THE EIGENVALUES) HAS A PRINCIPAL AXIS ASSOCIATED UITH ITDISPL
C.0 l0 13X0118HTHE DIRECTIONS OF EACH PRINCIPAL AXIS ARE GIVEN BY TDISPL
ONE PROJECTIONS OF THIS AXIS (I.E. DIRECTION COSINEST ON THE 3 AXESDISPL
E0 /0 13X0110HOF THE ORIGINAL COORDINATE SYSTEM0 AND THE ANGLE BETHDISPL

FEEN THIS PRINCIPAL AXIS AND THE 3 COORDINATE SYSTEM AXES.0 IT

DISPL

102 FORMATI 69X025HDIRECTION COSINES IETUEEN0 8X027HANGLES (IN DEGREESDISPL

61 BETHEENO ’0 56X09HPRINCIPAL04X022HTHE PRINCIPAL AXIS ANDOIIXO
HZZHTHE PRINCIPAL AXIS AND0/056X06HVALUES08X06HAXIS 105X06HAXIS 20
I§X06HAXIS 307X06HAXIS 105X06HAXIS 20SX06HAXIS 3 T

DISPL
DISPL
DISPL

103 FORMAT(3(1028X022HPRINCIPAL AXIS NUMBER 01102H I0F13.90ZX0FB.60ZX0DISPL

JF8060210F80606H 9 0F80403X0F8.403X0F8.“0/10/1

.1101) 3 III
HI1021 3 H12
NI1031 3 H13
.12011 3 H12
HI2021 3 H22
H(2031 3 U23
HI301) 3 U13
UI3021 3 N23

UI303T 3 H33
IFIINDEX-3T 10201
I PRINT 1000IIUII0JT0J310310I3103)
0.... OIAGI RETURNS THE EIGENVECTORS IN THE COLUMNS OF R ...
2 CALL DIAG1(30101.0E-80H0R0EIGT
DO 3 13103
3 516(1) 3 SORTIEIGIITI
DO 10 13103
00 10 J3103
XX 3 RIIOJT
YY 3 ABSIXXT
IFIYY-1001 1001008
8 XX 3 SIGN11000XX)
10 RAII0JT 3 ACOSIXXI'TORAD
PRINT 101
PRINT 102
0.00.THE EIGENVECTORS ARE PRINTED OUT IN ROHS ...
PRINT 1030III0EIGIIT0IRIJoIT0J=103T0IRAIJ0IT0J3103TT0I3103)
RETURN
END
SUBROUTINE DIAGI(NSIZE0IFLAG0ERROR0A0R0EIDT
DIMENSION AI303T0RI303T0TTI303T0EIGIJT
INPUT! NSIZE0IFLAG0ERROR0AINZ0NZT. OUTPUT! RINZoNZT0EIGINZT.
NEED ONLY THE TOP HALF AND DIAGONAL OF MATRIX AINSIZEONSIZE1.
OIAGI DIAGONALIZES REAL SYMMETRIC MATRICES HHOSE DIMENSIONS ARE LESS

DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DISPL
DIAG

DIAG

DIAG

DIAG

DIAG

THAN OR EOUAL TO THOSE IN THE DIMENSION STATEMENT. IT DOES NOT DESTROYDIAG

INPUT. IT USES THE JACOB! METHOD HITH A A-SOUARE ROOT METHOD FOR
EVALUATING COS AND SIN. A SINGLE-PRECISION SQUARE ROOT IS USED. IT
SEARCHES FOR ABSOLUTE VALUES LESS THAN RHO HHICH IS PROGRESSIVELY
REDUCED. IF IT IS TOLD TO DOUBLE-CHECK THE ANSHER0 IT DOES SO BY
SEEING IF THE ABSOLUTE VALUE OF ANY OFF-DIAGONAL ELEMENT IS GREATER
THAN ERROR. IF 500 DIAGI CONTINUES THE JACOB] PROCESS.

NSIZE0 THE SIZE OF THE MATRIX UHICH MUST HE .LE. N20 UHERE NZ IS THE

DIMENSION USED IN A(NZ0NZT0R(NZ0NZ)0TT(NZ0NZ)0EIGINZT.
IFLAG3IO0IT0 (NO0YES) THE DIAGONALIZATION SHOULD BE DOUBLE-CHECKED.

DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG

ERROR0 THE LARGEST ALLOHED ARSOULUTE VALUE OF AN OFF-DIAGONAL ELEMENTDIAG

IN THE MATRIX FORMED BY MULTIPLYING RAR(-IT. .
AINI0NZT0 THE INPUT MATRIX CONTAINING A(NSIZE0NSIZE) HHICH IS THE
REAL SYMMETRIC MATRIX TO BE DIAGONALIZED.
RINZ0NZT CONTAINS RINSIZE0NSIZE) UHICH IS THE OUTPUT UNITARY MATRIX
UITH THE EIGENVECTORS IN THE ROHS.
EIGINZT CONTAINS EIG(NSIZET WHICH IS THE OUTPUT EIGENVALUES. THE NTH
EIGENVALUE IN ETG HAS ITS EIGENVECTOR IN THE NTH ROH OF R.
MATRIX TT ONLY NEEDED IF THE DIAGONALIZATION IS TO IE DOUBLE-CHECKED.
.... DIMENSIONIZE A0R0TT AS (NZ0NZT AND EIG AS (NZ) UHERE NZ.6E.NSIZE
BIGIABS(A(102)T S NSZI=NSIZE-1 S ERR130.8'ERROR
DO 10 J310NSIZE
IBEG=J°1 S RTJ0J131.0 S EIGIJTSAIJ0J)
IFIJ-NSIZET 5012012
5 DO 10 I3IBEG0NSIZE

DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG

 

{far-.0: n- -_~..

160

C TAKE THE UPPER-HALF ELEMENTS AND PUT THEM IN THE LOHER HALF. THE LOUERDIAG

C HALE IS DESTROTEO BY BEING OIAGONALIZED. SET UP R AS A UNIT NATRIX
ATN3AIJ011 S A(I0JT3ATH S RIIOJ13000 S RIJ01130.0
.163ANAAIIBIG0ABSIAIJOI)11

10 CONTINUE

12 IFIBIO~ERROR1 13014014
13 RETURN

14 RHO 3 DIG

15 RHO 3 001'RHO

20 1ND 3 0

C SEARCH FOR THE ELEMENTS GREATER THAN RHO
DD 300 IP310NSZI
IDEG31P°1
D0 300 IO3IBEG0NSIZE
XX 3 AIIO0IPT
VT 3 ADSIXXT
IFIYV-RHOT 300040040

C CALCULATE CX AND SX BY THE ALGEBRAIC 4-SOUARE ROOT METHOD

40 APO 3 XX
U 3 0.5.(EIGIIPT-EIGIIOTT
U 3 -APO/SDRTIAPO'.2‘U'.21
IFIU1 41042042
41 H 3 3U
42 5X 3 U/ISORT12.0.I1.0°SORT(1.0-H.921TIT
CX 3 SORT1100'SX9921
L30
90 L3L’1
1FIL3NSIZET 920920200
92 1FIL'IDI 94090094
94 IFIL'IPT 1000900110
100 T13AIIP0L1 S GO TO 120
110 T13AIL0IPT
120 IFIL'IDT 13001400140
130 T23AIIO0LT S A(IO0LT3TIRSX0T2'CX S GO TO 150
140 T23AIL0IQT S A(L0I013T1’SXOT2'CX
150 IFIL‘IPI 16001800180
160 A(IP0L13T1'CX-T2'SX S GO TO 90
130 AILOIP13T1.CX-T2.SX‘S GO TO 90
200 APP3EIGIIPT S AQQ3EIGIIOT S A(IO0IPT30.0
EIGI IPT 'APP'CX'CXOAQQRSXRSX32.OPAPQ.SX.CX
EIGIIQ)3APP'SXRSX0AQQ'CX'CX02.O'APQ'SXPCX
DO 250 I310NSIZE
T13RII0IP) S T2=R(IOIQ)
RII0IPT3T1'CX-T2'SX
250 RII0IOT3T1'SX0T2'CX
IND3IND°1
300 CONTINUE
IFIINDT 3050305020
305 IEIRHO’ERR1T 3070307015
C IF FLAG31 DOUBLE-CHECK BY MULTIPLYING RARI-IT
307 IFIIFLAG-IT 30803090308
308 RETURN
309 00 330 I310NSIZE S 00 330 J310NSIZE
TOT=00°
DO 320 L310NSIZE
IFIL‘IT 31003150315
310 EL3AIL0IT S 60 T0 320
315 EL3AII0LT
320 TOTSTOT‘EL'RIL0JT
330 TTIIOJI'TOT
DO 350 I310NSIZE S 00 350 J310NSIZE
VALUE3000
DO 340 L310NSIZE
340 VALUE‘VALUEORIL0IT'TTIL0J)
IFII.ED.JT 3450349
345 EIGIIT3VALUE S GO TO 350
349 IEIABSIVALUE106T0ERROR1 GO TO 355
350 CONTINUE
RETURN

DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG

 

‘ I

161.

c 1r OFF-DIAGONAL ELEMENTS ARE TOO LARGE. CALCULATE A AND CONTINUE

360

362
364
370

355 00 370 J310NSIZE S 00 370 I3J0NSIZE

TOT30.0

DO 360 L310NSIZE

TOT3TOT°RIL0ITPTTIL0JT

IFII’J) 36403620364

EIGIJ)3TOT S GO TO 370

AII0J)3TOT

CONTINUE

ERRI 3 0.5.ERR1

GO TO 20

END

SUBROUTINE PARAMIINDEX0G0R0FE0SI0N10N20N30KSTART0SI0520530APG)
REAL OI3T0RI303)0GGIZ)0THETA(2)0NI3)0S(3)0N10N20N30A96I9)

C THIS SUBROUTINE CALCULATES THE ALPHA0IETA0GAMMA PARAMETERS FOR THREE

flflfifififlfififlfiOfiflflflnOfiflfiflflflfiflfiflflfifinflflflfififl

C

.... SUBROUTINE REQUIRES THE SUBROUTINE
999 FORMATT/l017H ERROR IN PARAM802160(3X06E15.B0/))

ROTATIONS OF A G-TENSOR.
IT DOES THIS FOR THE COPLANAR0MONOCLINIC0ORTHORH0MIIC0AND GENERAL CASEPARAM

INPUTI INDEX0GI3)0RI303)

OPTIONAL INPUT! FE0SI0N10N20N30KSTART051052053

OUTPUT! ABGI9) UHICH CONTAINS 9 VALUES IN THE FOLLOWING ORDERI ALA0
DEA0GAA0AL808EB0GAB0ALC0BEC0GAC

ALA0ALB0ALC ARE THE ALPHA PARAMETERS FOR ROTATIONS ABOUT AXES A0R0C.

DEAODE808EC ARE THE BETA PARAMETERS FOR ROTATIONS ABOUT AXES AODOC.

GAAoGA80GAC ARE THE GAMMA PARAMETERS FOR ROTATIONS ABOUT AXES AOROC.

INDEX3II020304) FOR THE (COPLANAR0MONOCLINIC0ORTHORHOMBIC06ENERAL)

CASES RESPECTIVELTo

6(3) CONTAINS THE 3 PRINCIPLE G-VALUES.

RI303) CONTAINS THE 3 EIGENVECTORS 1N ROHS.
IN RI'ITPGIDIAGONAL).R 3 H0

FE IS THE ANGLE PHIIIN RADIANS) ABOUT AXIS 3 HHICH DETERMINES THE
DIRECTION OF THE 2ND ROTATIONIAXIS B) IN THE COPLANAR0MONOCLINIC0
AND GENERAL CASES.

51 IS THE ANGLE PSIIIN RADIANS) ABOUT AXIS 3 WHICH DETERMINES THE
DIRECTION OF THE 3RD ROTATION (AXIS C) IN THE COPLANAR CASE.

IT IS THE UNITARV MATRIX

N10N20N3 ARE THE 3 COMPONENTS OF A VECTOR SPECIFYING THE DIRECTION OF

DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
DIAG
PARAM
PARAM
PARAM
PARAM

PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM

THE 3RD AXIS OF ROTATIONIAXIS C) FOR THE GENERAL CASE. THIS VECTORPARAM

NEED NOT BE OF UNIT LENGTH.

51052053 ARE THE 3 COMPONENTS OF A VECTOR SPECIFYING THE STARTING

DIRECTION (WHEN THETA3ZERO DEGREES) FOR THE 3RD AXIS OF ROTATION
(AXIS C) FOR THE GENERAL CASE.
THESE VALUES NEED NOT RE SPECIFIED.

KSTART 3 I100) HHENOFOR THE GENERAL CASE0 51052053 IARE0ARE NOT)
SPECIFIED.

NI3) IS A UNIT VECTOR ABOUT HHICH THE MAGNETIC FIELD IS ROTATED.

5(3) IS A UNIT VECTOR THAT INDICATES THE STARTING DIRECTION (WHEN
THETA3ZERO DEGREES).

M13) IS A UNIT VECTOR INDICATING THE HIDOLE DIRECTION (WHEN THETA 3
90 DEGREES).

GGIZ) AND THETAI2) ARE NOT USED BUT ARE NEEDED AS F.P.S IN ALHEGA.
ABGI9) CONTAINS THE 9 ALPHA0HETA0GAMMA PARAMETERS IN THE ORDER.
ALAOBEAOGAAOALHOBEBOGABOALCOHECOGAC

ALHEGA 000000000000

ICHECK 3 (INDEX-IT'I4-INDEX)

IFIICHECK) 1990505
5 IADD 3 1

KEY 3 1

JZGO 3 2'IINDEX-1)

N11) 3 000

N12) 3 000

N13) 3 1.0

SCI) 3 1.0

$12) = 0.0

5(3) 3 0.0

GO TO 100
00000 COPLANAR CASE 000.0
11 SNFE'3 SINIFE)

CSFE 3 COSIFF)

SNSI 3 SINISI)

PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM
PARAM

100
101
102
103
104
105
106
107
108
109
110
111

0CD~IO£ROWdhhfl

 

CSSI 3 COS(5I)
N(1) 3 SNFE
"(2) . 0.0
N(3) 3 CSFE
5(1) 3 CSFE
5(2) 3 000
5(3) 3 3SNFE
GO TO 100

12 N(1) 3 SNSI
N(2) 3 000
N(3) 3 C551
5(1) 3 C551
5(2) 3 000
5(3) 3 .SN51
60 TO 100

CO... "ONOCL1~IC C‘SE .0...
21 SNFE 3 SIN(FE)

CSFE 3 COS(FE)
N(1) 3 SNFE
"(2) 3 00°
N(3) 3 CSFE
5(1) 3 CSFE
5(2) 3 000
5(3) 3 - NFE
GO TO 100

22 N(1) 3 0.0
N(2) 3 1.0
N(3) 3 000
5(1) 3 000
5(2) 3 000
5(3) 3 1.0
60 TO 100

00000 ORTHORHOHDIC CASE 00000

31 N(1) 3 1.0
N(2) 3 000
N(3) 3 0.0
5(1) 3 0.0
5(2) 3 100
5(3) 3 000
GO TO 100

32 N(1) 3 0.0
"(2) 3 1.0
"(31 3 000
5(1) 3 000
5(2) 3 0.0
5(3) 3 10
GO TO 100

00000 GENERAL CASE 00000
41 SNFE 3 SIN(FE)

CSFE 3 COS(FE)
N(1) 3 SNFE
N(Z) 3 000
N(3) 3 CSFE
5(1) 3 CSFE
5(2) 3 0.0
5(3) 3 -SNFE
GO TO 100

42 RN 3 SORTIN10'20N29920N3'92)
IF(RN) 1990199043
43 N(1) 3 N1/RN
N(Z) 3 NZIRN
N(3) 3 N3/RN
IF(KSTART-I) 44050044
44 DN 3 SORT(N(1)"2‘N(2)"2)
IF(DN) 1990199045
45 5(1) 3 N(2)/DN
5(2) 3 -N(1)/DN
5(3) 3 0.0
GO TO 100

162

PARAM SB
PARAM S9
PARAM 60
PARAM 61
PARAM 62
PARAM 63
PARAM 64
PARAM 6S
PARAM 66
PARAM 67
PARAM 6B
PARAM 69
PARAM 70
PARAM 71
PARAM 72
PARAM 73
PARAM 74
PARAM 75
PARAM 76
PARAM 77
PARAM 78
PARAM 79
PARAM 80
PARAM 81
PARAM 82
PARAM 83
PARAM B4
PARAM 65
PARAM B6
PARAM 87
PARAM 88
PARAM B9
PARAM 90
PARAM 91
PARAM 92
PARAM 93
PARAM 94
PARAM 9S
PARAM 96
PARAM 97
PARAM 9B
PARAM 99
PARAMIOO
PARAMIOI
PARAM102
PARAM103
PARAM104
PARAMIOS
PARAM106
PARAM107
PARAMIOB
PARAM109
PARAMIIO
PARAMIII
PARAMIIZ
PARAM113
PARAM114
PARAMIIS
PARAM116
PARAM1I7
PARAMIIB
PARAMII9
PARAMIZO
PARAM121
PARAMIZZ
PARAM123
PARAM1?4
PARAM1?5
PARAM126

 

163

50 RN3SORT(51'51‘52'52‘53'53)
IF(RN) 1990199055
55 5(1) 3 SI/RN
5(2) 3 52/RN
5(3) 3 53/RN
10° CALL ALBEGA(G0R0N0S0AL08E0GA0GG0THETA)
ABG(KEY) 3 AL
ABG(KEY’1) 3 BE
ABO(KEY°2) 3 GA
IF(IAOD‘3) 10201040104
102 KEY I KEYOJ
IADD 3 IADD.)
JIGO 3 JZGO’I
GO TO (11012021022031032041042) JZGO
104 RETURN
199 PRINT 9990INDEX0K5TART0N10N20N30$10520S30EE051oRNOONONOSOOOROGGO
PTHETAoABG
END
SUBROUTINE ALBEGA(G0R0N0S0AL08E0GA0GG0THETA)
REAL G(3)0R(303)0N(3)05(3)0N(3)0GG(2)0THETA(2)0U(303)
DATA TORAD/00174532925199/
THIS SUBROUTINE CALCULATES THE ALPHA0OETA0GANNA PARAMETERS 'OR A
SPECIFIC ROTATION DE A G-TENSOR ABOUT AN ARBITRARY A3150
6(3) CONTAINS THE 3 PRINCIPLE G-VALUES0
INPUT) GORON05 OUTPUT! AL0BE0GA0GG0THETA
R(303) CONTAINS THE 3 EIGENVECTORS IN ROH50 IT IS THE UNITARY HATRIX
1N R(-1).G(DIAGONAL)'R 3 U0
5(3) 15 A UNIT VECTOR THAT INDICATES THE STARTING DIRECTION (UHEN
THETA32ER0 DEGREES)0
"(3) 15 A UNIT VECTOR INDICATING THE MIDDLE DIRECTION (WHEN THETA 3
9O DEGREES)0
56(2) CONTAINS RESPECTIVELY THE HAXIHUN AND MINIMUM VALUES OF 60
THETA(2) CONTAINS THE ANGLES (IN DEGREES) HHICH CORRESPOND TO THE
VALUES IN 56(2) RESPECTIVELY0

THEIR VALUES OF THE IST CALCULATION0 IF THE FRACTIONAL CHANGE OF ANY
OF THE 3 PARAMETERS IS GREATER THAN 100E-90 IT HILL PRINT EVERYTHING.
199 EORHAT(17H ERROR IN ALREGAIO I0 (2X06E15090/))
"(1) 3 N(Z).5(3)-N(3)35(2)
"(2) 3 N(3).S(1)-N(1)'S(3)
"(3) 3 N(1).5(2)‘N(2)35(1)
DO 10 13103
00 10 J3103
N(IOJ) 3 000
00 10 K3103
10 N(IoJ) 3 H(I0J)9(G(K).'2).R(K0I).R(K0J)
AL 3 000
IE 3 000
GA 3 0.0
00 20 13103
00 2° J3103
AL 3 AL°0.5’(U(I0J).(S(I)*5(J)0H(I)°N(J)))
.E 3 HE’O0S.(H(IOJ).(S(1).S(J)-H(I).M(J)))
GA 3 GA°N(IOJ).S(1).M(J)
20 CONTINUE
IF(BE) 30025
25 IF(GA) 99028

nnnnnnnnnnnnnnnnnn

C If BE3000 AND GA=0000 THEN GGTI)=GG(2)3AL AND THETAIANYTHING
28 66(1) 3 AL
66(2) 3 AL
THE 3 000
GO TO 40

30 THE30053ATANTGA/BE)
35 CS 3 COST200'THE)
SN 3 SIN(2000THE)
ADD 3 8E3C5‘GA'SN
66(1) 3 SORTTALOADD)

PARAN127
PARAM128
PARAM129
PARAM130
PARAH131
PARAN132
PARAM133
PARAN134
PARAMIJS
PARAN136
PARAMIJT
PARAN138
PARAM139
PARAH140
PARAN141
PARAM142
PARAM143
PARAN144
ALOEG
ALBEG
ALOEG
ALBEG
ALBEG
ALBEG
ALOEG
ALBEG
ALBEG
ALBEG 10
ALBEG 11
ALBEG 12
ALBEG 13
ALBEG l4
ALBEG 15
ALBEG 16

03‘10015UN3'

AL09E0GA ARE THE ALPHA0BETA0GAHMA VALUES CALCULATED FOR THIS ROTATIONALREG 17
IHEY ARE THEN USED TO CALCULATE GHAX0GMIN AND THE CORRESPONDING ANGLESALBEG 13
HITH THESE VALUE90 ALPHA08ETA06AHHA ARE RECALCULATED AND COMPARED HITHALBEG l9

ALBEG 20
ALBEG 21
ALBEG 22
ALBEG 23
ALOEG 24
ALBEG 25
ALBEG 26
ALBEG 27
ALBEG 28
ALBEG 29
ALBEG 30
ALBEG 31
ALOEG 32
ALBEG 33
ALBEG 34
ALBEG 35
ALREG 36
ALBEG 37
ALBEG 38
ALBEG 39
ALBEG 40
ALREG 41
ALBEG 42
ALBEG 43
ALBEG 44
ALBEG 45
ALBEG 46
ALBEG 47
ALBEG 48
ALBEG 49
ALBEG 50
ALBEG 51

 

37
38

40

02
03

99

164

66(2) 3 SORT(AL-ADD)

ALI 0053(66(1)..2¢66(2)992)
UIK GO(1)332-66(2)332
3E1 OoS'UIK‘CS

6A1 O053U1K’5N

EAL ABS((AL-AL1)/AL)
ESE 3 ABS((BE'8E1)/BE)
E6A300O

IF(GA) 37035
E6A3((6A-6A1/6A))
EE3ANAK1(EAL0EBE0E6A)
IF(EE3100E-9) 40040099
THETA(1) 3 THE'TORAO
THETA(2) 3 THETA(1).9000
XX 3 66(1)'GG(2)

IF(X‘) (03092042

RETURN

T 3 60(1)

66(1) 3 66(2)

66(2) 3 T

T 3 THETA(1)

THETA(1) 3 THETA(2)
THETA(2) 3 T

RETURN

PRINT 199060R0N0S0AL08E06A0660THETAOALI0'E10OA1
RETURN

END

ALOEG
ALIEG
ALOEG
ALOEG
ALIEG
ALBEG
ALIEG
ALBEG
ALBEG
ALBEG
ALIEG
ALOEG
ALBEG
ALBEG
ALBEG
ALBEG
ALBEG
ALBEG
ALBEG
ALBEG
ALBEG
ALBEG
ALBEG
ALBEG
ALBEG
ALBEG
ALBEG