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Field dependent tensors in solid state physics
Pourghazi, Azam
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Pourghazi, Azam (1977) Field dependent tensors in solid state physics, Durham theses, Durham University.Available at Durham E-Theses Online: http://etheses.dur.ac.uk/8306/
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FIELD DEPENDENT TENSORS IN
SOLID STATE PHYSICS
by
AZAM POURGHAZI, H.Sc. Graduate Society
A thesis submitted to the University of Durham
i n candidature for the degree of
Doctor of Philosophy
August 1977
The copyright of this thesis rests with the author.
No quotation from it should be published without
his prior written consent and information derived
from it should be acknowledged.
To my parents
ABSTRACT
The concept of field-dependent tensors (tensors the components
of which depend on the direction of an applied f i e l d ) i s generalized to
define quantities which have dif f e r e n t t e n s o r i a l character i n di f f e r e n t
subspaces ( f i e l d subspace and geometrical or physical subspace) of the
whole space. The transformation laws for these field-dependent tensors
are worked out and the weighted and r e l a t i v e f i e l d dependent tensors
are defined.
The calculus of f i e l d dependent tensors i s established through
which the operations of addition, subtraction, inner and outer m u l t i p l i
cation, contraction and d i f f e r e n t i a t i o n of f i e l d dependent tensors are
defined and the conditions under which each operation can be performed
are discussed, and the quotient law for f i e l d dependent tensors i s also
worked out.
The e f f e c t of magnetic c r y s t a l symmetry on the forms of f i e l d -
dependent tensors i s considered and a generalized Neumann's pr i n c i p l e
i s defined. Furthermore, i t i s proved that i n working out the magnetic
point group of symmetry operations, i d e n t i f i c a t i o n of the magnetic moment
inversion operator with the time-inversion operator i s not correct.
The e f f e c t of the symmetry operations of the magnetic point
groups on the f i e l d dependent tensors representing the transport proper
t i e s of a magnetic c r y s t a l i s considered. Different prescriptions
(A, B and C) given by different workers for finding the magnetic symmetry -»•
r e s t r i c t e d forms of the magnetoconductivity tensor 0 ^ j ( 3 ) are discussed,
the objections to each prescription are pointed out and then a new pre
s c r i p t i o n (D) i s given. Following prescription D, the r e s t r i c t i o n on
the forms of the magnetoconductlvity tensor ( 8 ) , the magnetoresis--»• •> t i v i t y tensor ( 8 ), the oagnetothermoelectric power ( 8 ), the
•+ magnetothermal conductivity (B) and the magneto-Peltier e f f e c t
* i j ^ ' iMPOBe^i b v t n e symmetry of c r y s t a l s belonging to each magnetic
point group are found.
F i n a l l y the way i n which the permittivity of a c r y s t a l belonging
to a magnetic point group can be represented by a f i e l d dependent
tensor i s discussed.
ACKNOWLEDGEMENTS
I n the p r e p a r a t i o n o f t h i s t h e s i s under the su p e r v i s i o n
of Professor G.A. Saunders, I have b e n e f i t t e d g r e a t l y from h i s
encouragement, guidance and advice, during a l l phases o f the work,
f o r which I am s i n c e r e l y g r a t e f u l .
My thanks also go t o Dr. J.S. Thorp f o r h i s help and
encouragement.
I would l i k e t o thank Professors >D.A. Wright and
G.G. Roberts, s e q u e n t i a l heads o f the department, f o r making
a v a i l a b l e the f a c i l i t i e s i n the Department o f Applied Physics
and E l e c t r o n i c s during my work.
I am g r a t e f u l t o Isfahan U n i v e r s i t y (Isfahan, I r a n ) f o r
t h e i r f i n a n c i a l support and encouragement, i n p a r t i c u l a r
Dr. G. Motamedi ex-chancellor of Isfahan U n i v e r s i t y (now M i n i s t e r
of Science and Higher Education) f o r h i s sustained enthusiasm and
help.
I would also l i k e t o thank the s t a f f o f Durham U n i v e r s i t y
Science L i b r a r y f o r the help and assistance i n using the f a c i l i t i e s
i n p a r t i c u l a r Mrs. CTean M. Chishclin ( i n t e r - l i b r a r y l o a n s ) .
L a s t l y , b ut by no means l e a s t , I would l i k e t o express my
deep and sincere g r a t i t u d e t o my husband, Tdghi, f o r h i s constant
encouragement and i n t e r e s t i n the work.
CONTENTS
Page No.
A b s t r a c t Acknowledgements
No.
A b s t r a c t Acknowledgements
CHAPTER 1 GENERAL INTRODUCTION 1.1 Tensors 1 1.2 Generalized symmetry and f i e l d 2
dependent tensors 1.3 Some remarks on the nomenclature 4
CHAPTER 2 FIELD INDEPENDENT TENSORS 5 2.1 Tensors 5 2.2 Linear orthogonal t r a n s f o r m a t i o n
o f coordinate axes and the summation convention 6
2.3 Linear t r a n s f o r m a t i o n o f axes 9 2.4 General t r a n s f o r m a t i o n of
coordinate axes 10 2.5 Transformation by invariance 11 2.6 Transformation by contravariance
and covariance 12 2.7 D e f i n i t i o n o f tensors by t h e i r
t r a n s f o r m a t i o n laws 14 2.8 Symmetrical and antisymmetrical
tensors 16 2.9 Constant tensors and tensor f i e l d 17 2.10 R e l a t i v e or weighted tensors (polar
and a x i a l tensors) 18 2.11 Algebra of tensors 19 2.12 Quotient Law 22 2.13 D i f f e r e n t i a t i o n o f tensors 24
CHAPTER 3 FIELD DEPENDENT TENSORS ( I ) : DEFINITION AND TRANSFORMATION LAWS 3.1 I n t r o d u c t i o n 27 3.2 Expansion o f the f i e l d dependent
tensor components i n terms of f i e l d components 27
3.3 Grabner and Swanson t r a n s f o r m a t i o n law f o r f i e l d dependent tensors 29
3.4 Inadequacy o f Grabner and Swanson tr a n s f o r m a t i o n law 31
3.5 N-dimensional space 32 3.6 Subspace 34 3.7 Formal D e f i n i t i o n o f a f i e l d
dependent tensor 35 3.8 The nature o f coordinates i n the
f i e l d subspace 41
CHAPTER 4 FIELD DEPENDENT TENSORS ( I I ) : CALCULUS 4.1 I n t r o d u c t i o n 46 4.2 A d d i t i o n and s u b t r a c t i o n 46 4.3 Outer m u l t i p l i c a t i o n o f two f i e l d 49
dependent tensors 4.4 Contr a c t i o n of f i e l d dependent 51
tensors 4.5 Inner m u l t i p l i c a t i o n o f f i e l d 53
dependent tensors 4.6 The q u o t i e n t law o f f i e l d dependent
tensors 54 4.7 D i f f e r e n t i a t i o n o f f i e l d dependent
tensors 57
CHAPTER 5 NEUMANN'S PRINCIPLE .'EFFECT OB" SYMMETRY ON TENSOR COMPONENTS 5.1 I n t r o d u c t i o n 60 5.2 Macroscopic symmetry operations 61
and p o i n t groups 5.3 The symmetry o f c r y s t a l l i n e l a t t i c e 62
and c r y s t a l classes 5.4 Neumann's p r i n c i p l e and the e f f e c t 62
of symmetry on p h y s i c a l p r o p e r t i e s
5.5 The symmetry o f magnetic c r y s t a l s ( I ) 76 5.6 The symmetry o f magnetic c r y s t a l s ( I I ) 7g 5.7 Time i n v e r s i o n o p e r a t i o n and magnetic 77
groups 5.8 Generalized Neumann's p r i n c i p l e ^ 5.9 Neumann's p r i n c i p l e f o r symmetry 82
operations which act on the coordinates of both subspaces 5.9.1 Example 82 5.9.2 The general case 85
CHAPTER 6 FIELD DEPENDENT TENSOR PROPERTIES OF CRYSTALS 6.1 I n t r o d u c t i o n 87 6.2 The c o n d u c t i v i t y o f a c r y s t a l 87 6.3 The magnetoconductivity p r o p e r t y o f 89
a c r y s t a l 6.4 Magnetic symmetry r e s t r i c t i o n s on the 90
forms of the m a g n e t o r e s i s t i v i t y and the magnetoconductivity tensors 6.4.1 P r e s c r i p t i o n A - by B i r s s 91 6.4.2 P r e s c r i p t i o n B - by K l e i n e r 94 6.4.3 P r e s c r i p t i o n C - by Cracknell 97 6.4.4 P r e s c r i p t i o n D - present work 101
6.5 Symmetry r e s t r i c t i o n s on the forms o f 104 other t r a n s p o r t p r o p e r t i e s
6.6 The p e r m i t t i v i t y o f a c r y s t a l 105
APPENDIX I SPATIAL SYMMETRY RESTRICTED FORMS OF FIELD 107 DEPENDENT TENSORS OF RANK ZERO IN THE FIELD SUBSPACE, WHERE FIELD COMPONENTS ARE THEMSELVES TENSORS IN GEOMETRICAL SUBSPACE A.l I n t r o d u c t i o n 107 A.2 A p p l i c a t i o n o f Neumann's P r i n c i p l e 108 A.3 Tabulation of the forms of the f i e l d
dependent tensors 111 A.4 Use of the t a b u l a t i o n t o f i n d T
2 ^ F i ^ f o r
the p o i n t group 3m l i 3 A.5 Conclusion 114
REFERENCES 156
- 1 -
CHAPTER 1
GENERAL INTRODUCTION
1.1 Tensors
Most physical properties of c r y s t a l s , which are defined by relations
between two or more measurable quantities associated with the c r y s t a l s ,
can be represented by mathematical e n t i t i e s c a l l e d tensors. Tensors are
defined by t h e i r transformation laws from one set of coordinate axis to
another, and are c l a s s i f i e d by t h e i r rank. Scalars and vectors are tensors
of zero and f i r s t rank respectively. A physical property which i n t e r
r e l a t e s two vectors i s a second rank tensor. As an example of t h i s , the
conductivity, °f a c r y s t a l may be considered where
J l = °11 E1 + °12 E2 + °13 E3
J 2 = a 2 1 E 2 + °22 E2 + a 2 3 E 3
J = a E + a E + a E 3 31 1 32 2 33 3
There are tensors of higher rank also. A tensor of rank h has ( n ) h
components i n an n dimensional space. C r y s t a l symmetry dictates some
rela t i o n s between the components of a tensor representing a property of
the c r y s t a l and therefore reduces the number of i t s independent components.
The detailed s i m p l i f i c a t i o n which can be made i n the forms of tensor proper
t i e s are considered, for example, i n trie book by Nye (1972) , for tensors
of various ranks and for a l l the 32 crystallographic point groups.
There are a number of physical properties of c r y s t a l s which are
best described by tensors, the components of which are themselves functions
of an applied f i e l d . The magnetoconductivity tensor o _. (H) , the magneto--»-
thermoelectric power (H) and the magnetothermal conductivity (H)
2N0Vf977 I l l t t *
- 2 -
of a magnetic c r y s t a l are examples of t h i s kind of tensor. The components
of a l l these tensors depend on the magnetic f i e l d . Grabner and Swanson
(1962) have ca l l e d these tensors "field-dependent tensors" and have worked
out t h e i r transformation law under a s p a t i a l transformation operation.
They have stated that a symmetry operation acts on both the ( f i e l d -
dependent) tensor and i t s arguments. Fai l u r e to recognise t h i s has led
some e a r l i e r workers (Birss 1963, 1966 and Cracknell 1973) to incorrect
s i m p l i f i c a t i o n of certain tensors. Akgoz and Saunders (1975 a,b) have
shown how to use the transformation law of f i e l d dependent tensors i n con
junction with Onsager's relations ( i n t r i n s i c symmetry) to es t a b l i s h the
form of the magnetoconductivity tensor c .(H) and the magnetothermoelectric
power a^_. (H) for each of the 32 c l a s s i c a l point groups G. Although AkgSz
and Saunders have recognised the need to use the transformation law of
f i e l d dependent tensors, a detailed mathematical substructure to t h e i r
work has up u n t i l now been missing. One major objective of t h i s thesis
has been to develop the mathematics of f i e l d dependent tensors on a formal
basis. Thus i n part the work i s devoted to giving a general d e f i n i t i o n
of f i e l d dependent tensors and to establish t h e i r transformation laws
under any. transformation operation either in the geometrical subspace or
i n the f i e l d subspace (Chapter 3). Furthermore, the calculus of f i e l d
dependent tensors i s worked out (Chapter 4). This investigation has put
the studies of Akgoz and Saunders on a firmer foundation.
1.2 Generalized symmetry and f i e l d dependent tensors
By introducing the concept of antisymmetry, Shubnikov (see for
example Shubnikov and Belov 1964) added a new dimension to the study of
symmetry of the geometrical objects. He employed an antisymmetry operation
i n addition to the s p a t i a l symmetry operations to develop what are now
- 3 -
known as the type I , I I and I I I Heesch-Shubnikov groups. By i d e n t i f i c a
t i o n o f the magnetic moment i n v e r s i o n o p e r a t i o n w i t h t h i s antisymmetry
o p e r a t i o n i t has proved p o s s i b l e t o put the s t r u c t u r e and p r o p e r t i e s o f
magnetic m a t e r i a l s on a group t h e o r e t i c a l basis (see Mackay 1957,
Donnay e t a l 1958 and Cracknell 1969). Furthermore, since the opera t i o n
of time i n v e r s i o n has the e f f e c t of re v e r s i n g the d i r e c t i o n o f the magnetic
moment, i t has become a common p r a c t i c e t o i d e n t i f y the time i n v e r s i o n
operator w i t h the magnetic moment i n v e r s i o n operator (see, f o r example,
Opechowski and Guccione 1965, Tavger and Z a i t s e v 1956, Bi r s s 1966, K l e i n e r
1966, 1967 and 1969, and Cracknell 1973). I n t h i s t h e s i s i t i s shown t h a t
t h i s i d e n t i f i c a t i o n i s net always c o r r e c t . Furthermore there are a number
of magnetic s t r u c t u r e s t h a t cannot adequately be described by using the
magnetic moment i n v e r s i o n operation (or time i n v e r s i o n o p e r a t i o n ) . To
describe these s t r u c t u r e s i t i s necessary t o adopt other operations
( r o t a t i o n s ) on the magnetic moment (or spin) o f the atoms as w e l l (see
Brinkman and E l l i o t t 1966 a,b and L i t v i n 1973). This leads t o what i s
c a l l e d the generalized symmetry (see Cr a c k n e l l 1975).
Some workers (Birss 1966, K l e i n e r 1966, 1967, 1969 and Cra c k n e l l
1973) have t r i e d t o f i n d the r e s t r i c t i o n s imposed on the form o f magnetic
field-dependent t r a n s p o r t tensors by the simple magnetic symmetry where
only the opera t i o n o f i n v e r s i o n on the magnetic moment i s i n v o l v e d . But
due t o lack of a tra n s f o r m a t i o n law f o r field-dependent tensors under the
transformation operations o f the f i e l d subspacs and also because c f the
i n c o r r e c t i d e n t i f i c a t i o n of the magnetic moment i n v e r s i o n o p e r a t i o n w i t h
time i n v e r s i o n o p e r a t i o n , the r e s u l t s are n e i t h e r c o r r e c t nor c o n s i s t e n t
w i t h each other. Akgoz and Saunders (1975 a,b) have found the r e s t r i c
t i o n s on the forms o f magnetoconductivity and other t r a n s p o r t p r o p e r t i e s
imposed by s p a t i a l symmetry operations. Now t h i s has been extended here
- 4 -
t o f i n d the r e s t r i c t i o n s imposed by the magnetic symmetry operations
(without i d e n t i f y i n g the magnetic moment i n v e r s i o n operator w i t h the time-
i n v e r s i o n operator) using the tr a n s f o r m a t i o n law f o r field-dependent tensors
(see Section 6.3.4).
1.3 Some remarks on the nomenclature
The terms field-dpendent tensor, field-independent tensor, constant
tensor and tensor f i e l d are used f r e q u e n t l y i n the t e x t . Thus a thorough
d e f i n i t i o n o f them may be found u s e f u l . A constant tensor i s a tensor,
the components of which have the same value a t every p o i n t i n the space.
That i s the components o f a constant tensor, measured a t a p o i n t i n the
space, do not depend on the coordinates o f the p o i n t . While a tensor f i e l d
i s a tensor the components o f which have values which vary from one p o i n t
i n space t o another. A field-dependent tensor i s a tensor the components
o f which not only may depend on the coordinates of a p o i n t , b ut also on
the d i r e c t i o n o f an ap p l i e d f i e l d . This d e f i n i t i o n i s generalized i n
Chapter 3 t o describe a field-dependent tensor as a tensor the components
of which have one type of t e n s o r i a l dependence on the coordinates i n the
geometrical subspace and another type o f t e n s o r i a l dependence on the
coordinates i n the f i e l d subspace. A field-independent tensor i s a tensor
the components o f which do not depend on the coordinates i n the f i e l d
subspace. A f u r t h e r extension o f the nomenclature would correspond t o
a pro p e r t y which i s a f i e l d dependent tensor which has values which can
vary from one p o i n t i n space t o another - such a system would comprise a
f i e l d dependent tensor f i e l d .
- 5 -
CHAPTER 2
FIELD INDEPENDENT TENSORS
This chapter c o n s t i t u t e s the necessary background f o r the study o f
f i e l d dependent tensors and t h e i r a n a l y s i s , and i t i s not intended t o cover
the whole sub j e c t o f f i e l d independent tensors (tensors which have com
ponents independent o f any f i e l d ) and t h e i r a p p l i c a t i o n s . For the sake
o f b r e v i t y , throughout t h i s chapter the word tensor has been used t o
denote f i e l d independent tensors.
2.1 Tensors
Tensors are a b s t r a c t o b j e c t s (such as p h y s i c a l q u a n t i t i e s ) whose
p r o p e r t i e s are independent o f the reference frame used t o describe them.
As an example the p o s i t i o n of a p o i n t i n geometrical space can be con
sidered. A p o i n t i n t h i s three dimensional space can be denoted by an
ordered set o f three numbers i n a s p e c i a l c a r t e s i a n coordinate system.
I n another c a r t e s i a n coordinate system the same p o i n t w i l l be represented
by a d i f f e r e n t s e t o f numbers. The p o s i t i o n v e c t o r of the p o i n t i s a
tensor, which i s independent o f the choice o f the reference frame, and
each set of three numbers forms i t s components w i t h respect t o t h e cor
responding coordinate system.
Scalars are a s p e c i a l k i n d o f tensor (tensors o f rank zero) . I n o
an n dimensional space a scal a r has one component m = 1 ) which i s the
same i n a l l the d i f f e r e n t coordinate systems. Vectors are another k i n d
o f tensor (being tensors o f the f i r s t rank) which i n an n dimensional
space have n components ( n 1 = n) w i t h respect t o each coordinate system
(three i n a three dimensional space). These n components transform i n t o
a new se t o f n components i n a new coordinate system. Then there are
- 6 -
tensors of higher rank (h) which have n components w i t h respect t o each
coordinate system (n i s the dimension o f the space and h i s the rank o f
the t e n s o r ) . Tensors can be more p r e c i s e l y defined by the tra n s f o r m a t i o n
laws o f t h e i r components from one reference frame t o another.
2.2 Linear orthogonal t r a n s f o r m a t i o n o f coordinate axes and the
summation convention
F i r s t we consider the simple case of t r a n s f o r m a t i o n o f one rec
t a n g u l a r c a r t e s i a n system o f coordinates x(X^ , , X j ] i n t o another
rectangular c a r t e s i a n system of coordinates X' (Xj , X^ , X^) which describe
the three dimensional geometrical space. I n a d d i t i o n we assume t h a t
t h e re i s no t r a n s l a t i o n of axes, t h a t i s , the o r i g i n 0 remains the same
a f t e r the t r a n s f o r m a t i o n . The r e l a t i o n between the axes i n the o l d
system X and the new system X1 may be s p e c i f i e d by the f o l l o w i n g t a b l e :
new
X i X 2
X1
11
*21
l31
o l d
X,
12
*22
'32
X.
13
'23
'33
(2.1)
where a.. i s the d i r e c t i o n cosine o f X' w i t h respect t o x. and i s a ID i 3 constant f o r a given c a r t e s i a n coordinate t r a n s f o r m a t i o n . A p o i n t B
1 2 3 i n the space can be described e i t h e r by i t s coordinates (x , x , x )
1 2 3
w i t h respect t o the X-system-or eq u a l l y w e l l by (x* , x' , x' ) w i t h
respect t o the X' - system. The f o l l o w i n g r e l a t i o n s hold between these
two sets o f coordinates:
- 7 -
,1 1 2 ^ 3 x' = an x + a!2 x a i 3 x
.2 1 ^ 2 ^ 3 X = a21 X + a22 X + a23 X
. 3 1 2 3 X = a31 X + a32 X + a33 X
Equations (2.2) may also be w r i t t e n i n m a t r i x n o t a t i o n s ,
(2.2)
_ x .1
..2
L v , 3 j
a l l a i 2 d13
a21 a22 a23
331 a32 a33
(2.3)
where the array a ^ i s c a l l e d the tr a n s f o r m a t i o n m a t r i x and i s the
operator which transforms one set o f axes X i n t o another set X 1. The
equations (2.2) may also be concisely w r i t t e n as
x ' 1 - a i j x j ( i ' 3 = 1 ' 2 ' 3=1
3) (2.4)
We w i l l now introduce the E i n s t e i n summation convention; namely, i f a
s u f f i x i s repeated i n a term, then i t i s understood t h a t a summation
w i t h respect t o t h a t s u f f i x over the range 1,2 , N i s i m p l i e d .
An unrepeated s u f f i x i s c a l l e d a f r e e s u f f i x , w h i l e , the repeated one
i s c a l l e d a dummy or umbral s u f f i x , as i t can be replaced by any other
symbol except the ones already used t o express f r e e s u f f i x e s . The sum
mation convention w i l l be employed throughout the r e s t o f t h i s t h e s i s .
Therefore equation (2.4) may be w r i t t e n as
i i 3 X = a i j X ( i , j = 1 , 2 , 3 ) (2.5)
- 8 -
I n t h i s chapter the range of values o f a l l s u f f i x e s i s 1 , 2 , 3 unless
otherwise s t a t e d .
I n the s p e c i a l case of rectangular c a r t e s i a n coordinates the 9- similar
reverse t r a n s f o r m a t i o n w i l l have \too& aaae t r a n s f o r m a t i o n operator,
t h a t i s
x 1 = ^ j j . * ' 3 ( 2 , 6 )
The nine a ^ are not independent of one another, and i n f a c t (Nye 1972)
a., a . = 6. (2.7) i k pk i p
where 6. i s the Kronecker d e l t a , IP
= 1 when i = p (2.8)
= 0 when i p.
The r e l a t i o n s (2.7) are c a l l e d the o r t h o g o n a l i t y r e l a t i o n s . Transfor
mations i n which the c o e f f i c i e n t s s a t i s f y the o r t h o g o n a l i t y r e l a t i o n s
are c a l l e d l i n e a r orthogonal t r a n s f o r m a t i o n s . The t r a n s f o r m a t i o n m a t r i x
f o r such a t r a n s f o r m a t i o n has an inverse equal t o i t s transpose and i s
c a l l e d an orthogonal m a t r i x .
When two l i n e a r orthogonal transformations of coordinate axes are
ap p l i e d successively, the r e s u l t i n g t r a n s f o r m a t i o n from the i n i t i a l
coordinate system t o the f i n a l one, which i s c a l l e d the product t r a n s
f o r m a t i o n , i s also l i n e a r orthogonal. To show t h i s , l e t the t r a n s
formations be
and
x ' k = a k x j (2.9)
x " 1 = b i k x , k. (2.10)
- 9 -
And l e t the product transformation be
it i = C x i j (2.11)
Substituting for x | J from (2.9) into (2.10), we get
i i i = b i k *3- (2.12) x
Comparing (2.12) and (2.11), we obtain
i j = b ik \ i . (2.13)
Equation (2.13) shows that i s an orthogonal matrix, as the product
of two orthogonal matrices i s orthogonal. Therefore the product trans
formation i s also a l i n e a r orthogonal transformation. This r e s u l t may
be extended to the product of any number of transformations.
Another c h a r a c t e r i s t i c property of l i n e a r orthogonal transforma
tions i s that the determinant of the transformation matrix ( a . . ) ,
which we write as | [ , i s equal to - 1 or + 1 according to whether the
hand of the axes i s changed or unchanged by the transformation
(Jeffreys 1931).
2.3 Linear transformation of axes
The l i n e a r orthogonal transformation of coordinate axes i s a
spec i a l case of l i n e a r transformation of axes. When a l i n e a r transfor
mation of axes i s applied i n an n dimensional space, there e x i s t s a
set of n l i n e a r r e l a t i o n s between the new coordinates and the old ones,
that i s
x'^ = a ^ x^ , ( a ^ i s constant i , j = 1, ....n). (2.14)
We s h a l l suppose that the transformation i s non-singular, \ a^\ ^ 0 /
so that the s e t of n l i n e a r equations (2.14) can be solved for the x 1
- 1 0 -
i n terms of x'*. The solution of equations (2.14) for the x's y i e l d s
x = - J i x , : ] (2.15) a
where A. . i s the co-factor of the element a. . i n l a . .1 = a. (A..,/;,)
forms the transformation matrix for the reverse transformation.
I f we have two successive l i n e a r transformations
x , D - a ^ x 3 (2.16)
and
x" k = b ^ x' 1, ( i , j , k = l n) (2.17)
then the d i r e c t transformation from x 1 coordinate system to x" 1
coordinate i s
x" k = b k i a x j , ( i , j ,k m l,...n), (2.18)
t h i s i s c a l l e d the product transformation. Writing t h i s transformation
i n matrix notation y i e l d s
X" = B A X (2.19)
Since the product BA , i n general, i s not equal to AB , we see that
the order i n which the transformations are performed i s not immaterial.
2.4 General transformation of coordinate axes
Let us consider the coordinate system x 1 ( i = 1, ...n) i n an
n-dimensional space. A general transformation of coordinates to a new
reference frame can be defined by n independent equations
x' 1 - / (x 1 , x 2 , , x n ) , ( i = 1, n) (2.20)
where the ^ are single-valued continuous d i f f e r e n t i a t e functions of
- l i
the coordinates. A necessary and s u f f i c i e n t condition for n equations
(2.20) to be independent i s that the Jacobian determinant formed from
the p a r t i a l derivatives 3x , i/3x :' does not vanish (Sokolnikoff 1956).
Under t h i s condition we can solve equations (2.20) for the x 1 as
functions of the x' 1 and obtain
x 1 - I / I 1 (x' 1 , x' 2 , , x , n ) ( i = 1 , 2 , n). (2.21)
Now d i f f e r e n t i a t i o n of (2.20) y i e l d s
t J u l 1 _
dx' = 52£-_ d x r . (2.22) 3x r
i r Here matrix formed from the p a r t i a l derivatives 3x' /3x = a. i s the i r transformation matrix. Generally, the elements of t h i s matrix are not
constant with respect to the corresponding coordinate axes. But i n the i r 2
case of l i n e a r transformation the = 3x' /3x are a set of n constants and the transformation, as i t has been shown before (2.14), can be rewritten as
• * i i Bx' 1 r _ x' = a x or x 1 x . (2.23) 3x r
And, i n the case of orthogonal l i n e a r transformation we have the
condition ,
2 2 i l i = 2 « L . (2.24) r I 3 x r 3x«
2.5 Transformation by invariance
Let f(P) = f ( x * , ...fxn) be a function of the n components x* of
the point P with respect to X - coordinate system i n an n-dimensional
space. This function i s c a l l e d an "invariant" or a " s c a l a r function"
or, simply, a " s c a l a r " with respect to a transformation of coordinates
- 12 -
from X-system. into X'-system , i f f(P) = f ' ( P ) , where f(P) and f 1 ( P ) are
the values of t h i s function i n X and X'-coordinate systems. The func
t i o n a l form of f(P) i n the X"-coordinate system w i l l be f ( x ' 1 , , x ' n ) .
Therefore
fp ( x ' 1 , , x , n ) , . x ^x' 1, ,x,n)J = f' ( x ' 1 , , x , n ) , (2.25)
since the value of f ( x 1 , . . . . ,x n) at PCx 1,...^") i s the same (invariant)
as that of f ' ( x 1 1 , , x , n ) at P l x ' 1 , , x , n ) . We can regard equation
(2.25) as being the d e f i n i t i o n of a scalar function; that i s , the
functions having the transformation laws t y p i f i e d by formulates,, 25) are
scalar functions. A transformation of the kind (2.25) i s c a l l e d x
"transformation by invariance".
2.6 Transformation by contravariance and covariance
A set of n functions A^(x) of the n coordinates x* are said to be
the components of a contravariant vector, associated with the X-coordinate
system, i f they transform according to the equation
A' (x 1) = ^ — AU (x) (2.26)
on the change of the coordinates x 1 to x' 1. Equations (2.26) define the
n components of the vector i n the new coordinate system X 1. When we
examine equation (2.22)
dx' 1 = $*L- dx r (2.22) 3x r
we see that the d i f f e r e n t i a l s , dx1, which determine the displacement I n 1 1 vector from a point P^(x ,.... ,x ) to another point P 2 (x + dx , ....,
x n + dx 1 1), form the components of a contravariant vector, whose com
ponents i n any other system of coordinates are the set of d i f f e r e n t i a l s ,
- 13 -
dx' 1 , i n that system. The coordinates x w i l l only behave l i k e the
components of a contravariant vector with respect to l i n e a r transfor
mations. A transformation of the kind (2.26) i s c a l l e d a "transformation
by contravariance".
There e x i s t s another law of transformation of vectors, which i s
quite d i f f e r e n t from the law t y p i f i e d by formula (2.26), and i s c a l l e d
the "transformation by covariance". I t can be defined i n the following
way:
A set of n functions A^(x) of the n coordinates x 1, associated with
the X-coordinate system, are the n components of a covariant vector i n
an n-dimensional space, i f they transform according to the equation
ax° A! (x 1) - H—r A (x) (2.27)
i a x ' 1 a
on the change of the coordinates x 1 to x' 1. Equations (2.27) define
the n components of the vector i n the new coordinate system X 1. As an
example of a covariant vector consider the set of n p a r t i a l derivatives
of a scal a r function f. Since af/ax' 1 = (af/3x a) O x ^ a x ' i ) , i t
follows immediately from (2.27) that the quantities df/dx 1 are the com
ponents of a covariant vector, whose components are the corresponding
p a r t i a l derivatives df/dx' 1. Such a covariant vector i s c a l l e d "the
gradient of f".
A covariant vector w i l l always be denoted by a single subscript,
while a single superscript w i l l denote a contravariant vector, unless
otherwise stated. The coordinates x* are an exception to t h i s , since
with respect to general transformations of coordinates, the x 1 do not
form the components of a contravariant vector.
There i s no d i s t i n c t i o n between contravariant and covariant vectors
when we r e s t r i c t ourselves to l i n e a r orthogonal transformations, since
- 14 -
according to (2.24)
3x a/3x' i = 3x , Jy3x a .
2.7 Definition of tensors by t h e i r transformation laws
We can define a tensor as "a system of numbers or functions
(components of the tensor) representing a quantity i n an n-dimensional
space, which obeys a cer t a i n law of transformation when the coordinates
undergo a transformation." We have already defined the transformation
law for s c a l a r s or tensors of rank zero (equation (2.25)), and also the
two different transformation laws for contravariant and covariant vectors
or tensors of rank one (equations (2.26) and (2.27)).
The next requirement i s to state the transformation law for tensors
of ranks higher than one. To do t h i s , we s t a r t with second rank tensors. 2 i l
Consider the s e t of h functions A (x) whose transformation law i s
A' i 3(x') = A** ( X ) , ( 2 . 2 8 )
3x* ax*
then we c a l l A ^ ( x ) the components of a contravariant tensor of second 2
rank. S i m i l a r l y , i f we have a s e t of n functions whose transformation law i s
k I A' Ax') = -22—- A * (x) , (2.29)
1 3 3x' 3x , : i
we c a l l A^j(x) the components of a covariant tensor of second rank. 2 i
Further, i f we have a set of n functions A^(x) whose transforma
tion law i s
A'^x') = ^ ?JL. A* ( X ) F (2.30) 3 3x* 3x' 3
- 15 -
we c a l l A*(x) the components of a mixed tensor of second rank. Super
s c r i p t s and subscripts always correspond to respectively contravariance
and covariance properties.
From what has been said above, we can generalize the de f i n i t i o n
of a tensor i n the following way: s+ t l t 2 " " t s i A set of n * functions A (x) of the n coordinate system x ,
q i V " p
associated with X-coordinate system are said to be the components of a
mixed tensor of rank (s+p), contravariant of the s-th rank and covariant
of the p-th rank, i f they transform according to the equation
V 2 . . . u s ,^1 ^ vy-s
(2.31)
on the change of coordinates x^ to x'1". Equations (2.31) define the
n S + ^ components i n the new coordinate system X 1.
Now we e s t a b l i s h a useful property of the law of the transformation
of tensors. I f we are given the components P? (x) of a contravariant
vector i n X-coordinate system, by using the transformation law of. a
contravariant vector, (equation (2.26)), A | i ( x ' ) = Sx'V^x"' 2? ( x ) , the
components of the same vector i n X'-system, A , x ( x ' ) can be found. On k i
multiplying equation (2.26) by 3x /3x* and summing over the index
i from 1 to n, we obtain
3x . , i , 3x 3x' j 3x » j , » j.k _ j , . _k, j- A* (x 1) = t — j — A J (x) = — T * AJ (x) =6. A J (x) = A (x). 3x' 3x' 3x 3 Sx 3 3
That i s (2.32)
A*(x) «= 22—r A' (x«). (2.33) 3 x , a
- 16 -
Generalising, we can state that:
"Let the components of a mixed tensor in the X-coordinate system
s 1 . . . s s
be denoted by (x) and i t s components i n the X"-coordinate °r--°r
system by A' (x')« Then, from the law of transformation of mixed l j . . . l r
tensors we can write
» , 31 3 s , ,> 3* 1 3x r 9x' 1 3x' 5 _ s , , A . (X 1) = • J ' • *• • T~" ' R " ' " E A (x) • V ' ^ r 3X' 1! 3 x ^ a 7 * a r - - a r (2.34)
3 i * " 3 s On the other hand, i f we are given the components A'. ( x 1 ) , the 1 l " " 1 r
components A. (x) of the same tensor i n the X-coordinate system a,.. .a 1 r
are determined by
A (x) » — - — ••• — " — • r - ••• 3 — A" ( x 1 ) . " 2.35) V " ° r a l a r j l 3 s i i 1 r 3x 1 3x r 3x' 1 3x' S X r " i r
From the two equations (2.34) and (2.35) we can deduce an important
theorem, namely
" I f . a l l components of a tensor vanish i n a p a r t i c u l a r coordinate
system, then they necessarily vanish i n every other coordinate system."
The significance of t h i s theorem i n finding the n u l l properties of
c r y s t a l s has been discussed by B i r s s (1966, p.71).
2.8 Symmetrical and antisymmetrical tensors
When two covariant (or contravariant) indices i n the components i . . . . i X g
A. (x) of a tensor can be interchanged without a l t e r i n g the value 3 r , , 3 r
of those components, the tensor i s said to be symmetric with respect to
- 17 -
those indices. I t can be shown that the symmetry of the components of
a symmetric tensor i s preserved under the coordinate transformations
(Sokolnikoff, 1956).
A tensor i s said to be antisymmetric with respect to two c e r t a i n
indices (both covariant or both contravariant), i f an interchange of those
two indices i n the components changes the sign and not the magnitude of
the components. The antisymmetry i s also an invariant property
(Sokolnikoff, 1956). A second order symmetric tensor has got ^ n (n + 1)
d i f f e r e n t non-zero components.
2.9 Constant tensors and tensor f i e l d
The words "constant tensor" and "tensor f i e l d " w i l l be used
frequently throughout t h i s t h e s i s , therefore i t i s b e n e f i c i a l to give
a short d e f i n i t i o n of them here.
(a) Constant tensors
Constant tensors are those tensors whose components have the same
value at every point i n the space (or the subspace) over which they have
been defined. These are the type of tensors which are usually encountered
i n s o l i d state physics and have been described i n d e t a i l i n standard
texts such as that by Nye (1972). The transformation law for these
tensors i s
s . / - - 3 s . 3 x j _ *jLL • <!*L1 A " > - ' 8 B i l x. i is. a 'a a
1 r ax- 1 ax« r ax 1 ax 6 1
(b) Tensor-field
A tensor f i e l d i s a tensor which has d i f f e r e n t values for i t s
components at different points in the space over which the tensor has
been defined, i n other words the components of a t e n s o r - f i e l d depend on
the position of the point, at which the tensor has been defined, i n the
- 18 -
space. Equation (2.34) i s the transformation law for these tensors.
Constant tensors are a sp e c i a l kind of tensor f i e l d . In t h i s t h e s i s ,
the word tensor has been used to mean tensor-field unless otherwise
stated.
2.10 Relative or weighted tensors (polar and a x i a l tensors)
We s h a l l now extend the d e f i n i t i o n of a tensor to include r e l a t i v e
or weighted tensors. By a r e l a t i v e t e n s o r - f i e l d of weight u> ( or,is a
constant), we s h a l l mean an object with components whose transformation X
law d i f f e r s from the transformation law of a tensor-field by the appear-fch
ance of the determinant of the transformation matrix to the u power
i n the transformation law. That i s , 31"' *^s A. , s (x 1) = V r
3x 6
3 x , j 3 x«i 3 x « s a x . i i 7 x ^ W
(2.37)
where 3x , J
denotes the Jacobian of the transformation; i n other
words, the determinant of the transformation matrix.
In the s p e c i a l case of l i n e a r orthogonal transformation,
|3x p/dx' | = ± 1 (+ sign for transformation which keep the hands of
axes unchanged and - sign for those which change the hands of axes).
Therefore, considering the fa c t that for l i n e a r orthogonal transforma
tion of coordinate axes, there i s no difference between the covariance
and contravariance properties (see Section 2.6), the transformation law
of tensors, for such transformation of axes, w i l l be
B{ , (x') » (±l) u a ...a B (x) (2.38) V - ' S V l V a V ' ° s
where a. . constants are the elements of the corresponding transformation mm
- 19 -
matrix. Therefore, we w i l l have two kinds of tensors:
(i) tensors which transform according to the following law
B! , (x 1) = a. V " s
B. (x) (2.39) *1^1 x s 3 s • ' l " * " ^ s
these are tensors of even weight and are referred to as "polar tensors"•
( i i ) tensors which transform according to
B' (x') = ±a ...a B (x) V " s ^ l 3 ! V s V " 3 s
or.
(2.40)
B' . (x') V ' s
a. . a-3 1-1 s Js J l J s (2.41)
these are tensors of odd weight and are referred to as " a x i a l tensors"
2.11 Algebra of tensors
(i) Addition and subtraction
Consider two unweighted tensors A(x) and B(x) of the same type
and rank defined a t the same point P, and the corresponding laws of
transformation
^ l * * * ^ s 3x 1
B1 . s (x') = • 2 X - — L1'"L* 8K' 1
3x
3x'
3x' 3 l
B_ (x) (2.42) 3x
B s V " - a r
a ? r , , j s . 3x A (X 1) =» —
i r . . i r
3x Ll l r 3x* 1 3x' r
3x , 31
3x
ax' 3 s i - " e s
3x s 1 r
Now we prove that the sum of (or the difference between) these two
- 20 -
tensors i s again a tensor of the sampe type and rank as the two tensors.
Performing the summation we get,
B' . (x'> ± A' (x') = i r * - i
r i r ' * i r
/ a x " 1 **x \ /a X' j l 3x' j s >\ [ j V ^ s , * r " 6 s , / I — H '[— 3 7 j * V ' « r V " ° r ( X )
> 3x' 1 3x' V \3x 1 3x s / L 1 1 1 r
(2.44)
and i t immediately follows that A(x) + B(x) i s a tensor which has the
same number of covariant and the same number of contravariant indices as
the o r i g i n a l tensors, and each of i t s components i s the sum of the
corresponding components of the summand tensors. The summation operation
can be extended to three or more tensors. I t can also be e a s i l y shown
that, i f each component of a tensor i s multiplied by a constant, the
resu l t i n g set of functions i s a tensor of the same type and rank. There
fore, any lin e a r combination of tensors of the same rank and the same
type i s a tensor of the same type and the same rank.
We can generalize the addition of tensors to include r e l a t i v e
tensors as well. I t can be e a s i l y shown that r e l a t i v e tensors of the
same weight and rank and the same type (having equal number of covariance
and equal number of contravariance indices) can be added together, and
the sum w i l l be a r e l a t i v e tensor of the same weight, rank and type.
For the case of a l i n e a r orthogonal transformation, the addition of a
polar and an a x i a l tensor i s not defined. That i s , i f two tensors are
to be added together, they should be of the same rank and the same
pol a r i t y .
- 21 -
( i i ) Multiplication; Outer product
Consider two tensors (not necessarily of the same rank or type
•••x or weight) A q (x) of weight u^, and B s (x) of weight u^-
" • l p 1 r The set of quantities consisting of the product of each component of
• • d.. • • • J. the tensor B. s (x) by each component of the tensor A. , q ( x )
V r 3r"° P
forms a tensor C ( x ) , c a l l e d the "outer product" of tensors B(x) and
A(x). By using the transformation law for r e l a t i v e tensors A(x) and
B(x) (equation(2.37) i t can be e a s i l y shown that t h i s tensor, C ( x ) ,
i s contravariant of rank q + s and covariant of rank p + r , and i t s
weight i s + u^. I n X-coordinate system the components of tensor
C(x) are given by the following formula:
• • i ^ • • * i q £^ • • *£g i • • • C (x) = B (x) A (2.45)
( i i i ) . Tensor contraction i 1 . . . i s
Consider a mixed tensor, A. . (x), and equate a covariant J l r
and a contravariant index and sum with respect to that index. The r+s~2
resulting s e t of n sums i s a mixed tensor, covariant of rank r-1 and contravariant of rank s-1. To show t h i s , consider a mixed tensor
i j k
l i k e Ag^ ( x ) , from the transformation law of tensors (equation(2.31))
we have A ' " t ( x ' ) - SL- 2L- -22S- i 2 _ A. (x) (2.46)
M 3X 1 3x 3 3x k 3x' P 3x' q
Now i f we equate t and q we w i l l get
- 22 -
. . r a t , 1 N 3x' r 3x' s 3x , t : dxl 3x m
Kl j k , . A' (x 1) = r T r- - A„ (x)
p t Sx 1 3x 3 3x k 3x'^ ^
3x' r 3x' s 3 X* » i j x Sx 1 3x j 3x'P 3
. r s t . 3 x , r 3 x , s 3 X^ i 3 k
A'" (x.1) = - ^ - - r -S-^r - ~ A., (x) . (2.47)
P t Sx 1 3x j 3x'P
i j k
Therefore A ^ i s a mixed tensor, contravariant of the second rank and
covariant of the f i r s t rank. This process of reducing the rank of a
tensor i n contravariant and covariant indices i s c a l l e d the operation
of contraction and i t can be applied to a l l the indices of a tensor,
one a f t e r another, as far as there are indices of both types present.
The operation of contraction on a r e l a t i v e tensor y i e l d s a r e l a t i v e
tensor of the same weight as the o r i g i n a l tensor. (iv) Inner product
Inner multiplication i s performed by combining the two opera
tions of outer multiplication and contraction, and the r e s u l t i n g tensor
i s c a l l e d the inner product of the o r i g i n a l tensors upon which the
operations are applied. To find the inner product of two tensors,
f i r s t we find the outer product of them, and then we apply the
operation of contraction ( i f i t i s p o s s i b l e ) .
2.12 Quotient law
The d i r e c t method of establishing the tensor character of sets
of functions i s to find out how they transform from one coordinate
system to another. The quotient law provides a means of doing t h i s
- 23 -
without having to study transformation properties i n d e t a i l , which i n k i
practice i s troublesome. I t states that n functions of x coordinates
form the components of a tensor of rank h, (with c e r t a i n contravariant
and qovariant character), provided that an inner product of these
functions with an arbitrary tensor i s i t s e l f a tensor, (the term"inner
product"has been used here for sums of the form A(a, i _ , i . ) A , l h a
a or A(a, i _ , i . ) A , whether the set of functions A ( i , , i . i . ) i n l 2 n
represents a tensor or not). I t w i l l s u f f i c e to est a b l i s h the proof
for the p a r t i c u l a r case of the set of n^ functions A ( i , j , k ) , which has
a l l features of the more involved cases. Now l e t us suppose that the iP ip inner product A ( i , j , k ) B^ (x) for an arbitrary tensor B. (x) y i e l d s
pk a tensor of the type C (x):
«(i.j.k) B^ P (x) C p k (x) (2.48)
The transformed quantities, referred to a new system of coordinates
x' 1, s a t i s f y the equations
A«(i,j,k) B j i p ( x ' ) = C , p l c (x') (2.49)
Substituting for B^ i p(x') and C , p k ( x ' ) from t h e i r transformation laws
(equation(2.31)), we have
»• <i.i,w &4 *4 K £ w - ^ c q r <*> <2-5°> 3x* 3x" 3x'3 n Sx"1 3x r
qr
Substituting for C (x) from equation (2.48) into equation (2.50) and
changing the dummy indices, we get (x) = 0 (2.51) dx
- 24 -
S P On inner multiplication by 3x /3x we obtain
A ' ( i , j , k ) — 7 r - A(£,u,r) B [ 3x* 3x' j 3x rJ u
£s (x) - 0. (2.52)
Is
Since B^ (x) i s an arbitrary tensor, the expression i n brafcgets i s
i d e n t i c a l l y zero, that i s
A'(i»jfW • S 2 L T - a s r - M£,u,r) ^ L - . ( 2 . 5 3 ) 3x Sx' 3 3x r
3x 3x . Inner multiplication of t h i s equation by — — — yi e l d s 3x' 3x
A'(s,t,k) = - — ^ - T T ^ - r A(m,u,r) , (2.54) 3 x , s 3x U 3x r
which shows that A(m,u,r) i s a tensor of t h i r d rank and contravariant i n mu
m and n, and covariant i n r , A f .
2.13 D i f f e r e n t i a t i o n of tensors
We have seen i n Section(2.6)that the set of p a r t i a l derivatives
(gradient) of a sc a l a r function, f ( x ) , forms a covariant tensor of rank
one. I f we form the set of p a r t i a l derivatives of t h i s covariant vector 3f
— j - we get 3x X
3 2 f » / 3f 3 x a \ _ 3 2 f 3x 3 3x a + 3f 3 2 x a i _ /_3f_ _3x^\
1 1 \ 3 x a ax'V " a x ^ S x * 3 X 1 \ 3 xa a x ' 1 / 3 x a a x 6 S x ' 1 dx' 1 3x° a x ' ^ x ' 1
(2.55)
Therefore for general transformation of coordinate axes, where
3 x /(3x' 3x' ) £ o, t h i s s e t of p a r t i a l d erivatives of the covariant 3f
tensor — — does not represent a tensor. This can be generalized to any 3x
kind of vector, to show that the set of p a r t i a l derivatives of a
- 25 -
contravariant or a covariant vector do not form a tensor for general
transformation of coordinates.
Consider a covariant vector A (x), then ot
A!(x'> = - ~ T A« M (2.56)
Differentiating t h i s with respect to coordinates x' 1, we get
3 A ' i ( x , ) 3x° 3x* 3V X> + 32x* A „ + ; r A ~ <x> (2.57) 3X' 1 dx' 1 3 x , j 3x S S x ' ^ x ' 3 a
which i s not the transformation rule for a tensor, that is , 3Aa (x)
i s not a tensor. ax* S i m i l a r l y , for a contravariant vector A°(x), we get
H i f L = _*4 i d + i . w i k L . J * ! ( 2. 5 8 )
3x B 3x' 3 Sx" 3 x 0 3 x B ax' 3
3A (x) which shows that 1
a — i s not a tensor. 3x B
This can be extended to mixed tensors of any rank to show that
the s e t of p a r t i a l derivatives of a tensor does not form a tensor. But ci d.
for l i n e a r transformation where 3x /3'x i s a constant the p a r t i a l
derivative operator acts l i k e a covariant vector, and adds one covariant
index to the indices of the o r i g i n a l tensor. That i s , the tensor formed i < . • • j-g
by the set of p a r t i a l derivatives of the tensor A. . (x) w i l l have s contravariant and r + 1 covariant indices. The transformation law for such a tensor w i l l be ,
i . . . . i a.••«o 3A' . s(X') o i . i 0. 0„ A * _ (x) *1 - " 3 g 3x* 3x' 1 3xj_2 3x1. 3 x r B r " B r
3 x , k 3 x , l c 3 x a l 3x s 3x , : ) l 3x' r 3x
(2.59)
- 26 -
In the preparation of t h i s chapter, text books by the
following authors have been consulted: Sokolnikoff (1956),
Brand (1948), Spain (1956) , B r i l l o u i n (1964),
McConnell (1957), Nye (1972) and B i r s s (1966).
- 27 -
CHAPTER 3
FIELD DEPENDENT TENSORS ( I ) ;
DEFINITION AND TRANSFORMATION LAWS
3.1 Introduction
In the study of tensor properties of materials i n the presence of
an external f i e l d , such as a magnetic f i e l d or s t r a i n , one encounters
properties, for instance the conductivity, with components which depend
on the applied f i e l d . These tensors have been named " f i e l d dependent"
tensors (Grabner and Swanson 1962).
In Sections 3.2, 3.3 and 3.4, the way f i e l d dependent tensor pro
p e r t i e s have been dealt with i n the past has been discussed and the need
for a more general approach has been recognized. The r e s t of t h i s chapter
i s devoted to giving a general de f i n i t i o n of f i e l d dependent tensors and
to finding t h e i r transformation laws from one system of coordinates to
another.
3.2 Expansion of the f i e l d dependent tensor components i n terms of
f i e l d components
In dealing with f i e l d dependent tensor properties, previous practice
has been to expand the tensor components as a power se r i e s i n the applied
f i e l d and to r e s t r i c t the calculations and measurements to the low f i e l d
c o e f f i c i e n t s (which are constant with respect to the f i e l d components).
The formulation of t h i s procedure for some sp e c i a l properties, i n orthog
onal coordinate system, can be found i n many a r t i c l e s (for example see
Akgoz and Saunders 1974, B i r s s 1966, Juretschke 1955, Mason et a l 1953).
We generalize t h i s to state that:
- 28 -
In a general coordinate system a f i e l d dependent tensor,
i . . . . i 1 s
A. . (F) can be expanded according to the following formulas: ¥ , , ] r
(a) i n the presence of a f i e l d F, which behaves l i k e a covariant vector,
i , . . . i ^ i . - . . i (0) i . . . . i k . ( l ) i . . . . i k.k.(2) A / / ( | ) = A . 1 . S a / , S 1 F„ + A.1 . S 1 2 F F +
J l J r J l J r J l J r 1 J l J r 1 2
(3.1)
(b) i n the presence of a f i e l d F, which behaves l i k e a contravariant vector,
i . . . i g + i 1 . . . i g ( 0 ) i . . . i (1) k t i 1 . . . i g ( 2 ) k k 2
(3.2)
i r . . i s (N) The c o e f f i c i e n t s A or A
1" r 1"' il l " " " ^ r
i , . . . i k,...k M(N) 1 S l N are now f i e l d
independent tensors and are given by
i , . . . i , 1 s
(N) N i . . . . i 3 A,1 . S (F)
•'r k
3 F N F = 0 (3.3)
and
i . . . . i k....k (N) , 1 s 1 N
rN i
3, •••3 J l J r (3.4)
The new set of components of each of the c o e f f i c i e n t tensors, i n a new
reference frame, can be found by employing the transformation law for
- 29 -
f i e l d independent tensors (equation 2.31). By substituting these trans
formed c o e f f i c i e n t s into equation (3.1) and (3.2), the transformed s e t i l * • m i- B +
of components of the tensor A (F) i s then obtained. Experimental
measurements and theoretical interpretation of the low f i e l d ( f i e l d
independent) c o e f f i c i e n t s for some f i e l d dependent properties have been
done i n the past. As an example the magnetoconductivity can be considered:
For the magnetoconductivity tensor (B) ( i n an orthogonal
reference frame) t h i s procedure gives (see Drabble et a l 1956, Akgoz and
Saunders 1974):
V* • °il ( 0 ) + • i j k " \ + tfljk1i|) \ • ... (3.5) 2
where ' N
(N) , 4 . / 9o..(B) i j k 1 k 2 . . . k N
VN! ' V 3 B k ... 3B. V B - 0 (3.6)
Experimental measurements of these low f i e l d c o e f f i c i e n t s have then
usually been interpreted i n terms of c a r r i e r densities and m o b i l i t i e s ,
and Fermi surface parameters. (For more d e t a i l see Z i t t e r 1962, OktU and
Saunders 1967, Jeavons and Saunders 1969, Michenatfd and I s s i 1972, AkgBz
and Saunders 1974, Turetken e t a l 1974).
3.3 Grabner and Swanson transformation law for f i e l d dependent tensors
A transformation law for a f i e l d dependent tensor d . ( F ) , 1jK np«••
with respect to a l i n e a r orthogonal transformation of coordinates was
f i r s t suggested by Grabner and Swanson (1962) which can be written as
d' . (F* ) - a 4 a L t . . . d (F ) , (3.7) i j k . . . np... i r j s k t r s t . . . mo...
- 30 -
where i s t h e transformation m a t r i x . That i s , the transformation law
f o r a f i e l d dependent tensor r e q u i r e s t h a t both the matter components and
t h e i r arguments be transformed. Using t r a n s f o r m a t i o n law (3.7) enables
one t o work .with f i e l d dependent tensor components r a t h e r than the low
f i e l d c o e f f i c i e n t s , and important p r a c t i c a l advamtages accrue from t h i s .
( i ) they are easier t o measure because the applied f i e l d can take any
value - there i s no longer any necessity t o ensure t h a t the low f i e l d
l i m i t i s achieved - and so i t can be assured t h a t the induced e f f e c t s are
largerA,
( i i ) since the measurements can be made over a wide range o f f i e l d ,
much more comprehensive data can be obtained,
( i i i ) i n many cases the number of samples needed t o o b t a i n the r e q u i r e d
i n f o r m a t i o n i s l e s s than t h a t demanded by the low f i e l d method,
( i v ) d i r e c t s tudies o f non-linear e f f e c t s i n the p h y s i c a l p r o p e r t i e s
of s o l i d s - an area of much i n t e r e s t a t present - can be made.
S o l u t i o n of the Boltzmann t r a n s p o r t equation has enabled expression of
f i e l d dependent t r a n s p o r t tensors i n terms o f band model parameters f o r
m a t e r i a l s w i t h simple many-valleyed Fermi surfaces. (For more d e t a i l see
Fuchser e t a l 1970, Aubrey 1971, and Sumengen e t a l 1972). As a r e s u l t
the microscopic f o r m u l a t i o n i s no longer r e s t r i c t e d t o the low f i e l d
c o n d i t i o n ( UB << 1) but now includes the intermediate f i e l d r e g i o n f o r
which galvanomagnetic data could not p r e v i o u s l y be i n t e r p r e t e d q u a n t i t a
t i v e l y . Analysis o f experimental data of the f i e l d and o r i e n t a t i o n
dependence o f p ^ j ( B ) (the m a g n e t o r e s i s t i v i t y tensor) arid ^ (E) (the
magnetothermoelectric power tensor) f o r bismuth (see Sumengen e t a l 1972,
- 31 -
Saunders et a l 1972, and Tiiretken et a l 1974) and arsenic-antimony alloy
(see Akgoz et a l 1974) has established that the band model parameters
found by the f i e l d dependent tensor components are the same within experi
mental error as those obtained from the low f i e l d c o e f f i c i e n t s . Thus
using the transformation law (3.7) has great advantages.
3.4 Inadequacy of Grabner and Swanson transformation law
As long as the s p a t i a l transformation operators are being con
sidered, there i s no d i f f i c u l t y i n obtaining the transformed tensor com
ponents of a f i e l d dependent tensor. But there are cases where more
complicated transformation operators are considered, that i s transformation
operators composed of s p a t i a l transformation operators and operators which
operate only on the f i e l d components. As an example of t h i s kind of trans
formation, a transformation operator in "spin space" (see L i t v i n 1973,
Brinkman et a l 1966 a,b, and Cracknell 1975) can be considered. This
operator can be denoted by { (R^j R | v^p^' where ( } i s a t r a n s l a t i o n ,
{ R^} a rotation or rotation-inversion operator acting on the s p a t i a l
coordinates of the spin arrangement, and { R^} a rotation or rotation-
inversion operator acting on the spin directions of that spin arrangement
(through the r e s t of t h i s t h e s i s operators have been no t i f i e d with barred
l e t t e r s l i k e A or R to avoid confusion). Now f o r a tensor defined in the
geometrical space with components depending on the spin directions, the
e f f e c t of the { R, i v, } part of the transformation ooere.tor can be J L 1 lp
obtained using the transformation law (3.7), but the { R } part needs J?
further consideration. Before t h i s can be done i t i s necessary to
define the space of the physical problem under consideration.
- 32 -
3.5 N-dimensional space
An N-dimensional space, denoted by V N, can be defined as any
set of objects that can be placed i n a one-to-one correspondence with the
t o t a l i t y of ordered sets of N numbers x, , x„, , x . These numbers 1 2 N
are c a l l e d coordinates of the objects (points) (see Sokolnikoff 1956,
page 9) . As an example consider physical space - a three dimensional
space with the objects being the geometrical points and with the coordinates
of a point being x^, x^, x 3 with respect to three coordinate axes. In
general by "objects" a broader meaning than j u s t geometrical points i s
understood: i n different cases "objects" may imply di f f e r e n t sets of
physical properties such as e l e c t r i c f i e l d , magnetic f i e l d , spin of a
p a r t i c l e , position vector, temperature gradient, pressure, time, etc.
These physical properties can be the coordinate axes of a space. I t must
be remembered that i n a de f i n i t i o n of an N-dimensional space, there i s no
suggestion of the concept of distance between p a i r s of points (objects).
I f a rule for the measurement of the distance i n a space i s defined, that
space i s c a l l e d a "metric space."
In addition to the points i n an N-dimensional space, we need a
second kind of mathematical entity, that i s the vector, which i s related
to the points through the following postulates:
( i ) every pair of points i n the N-dimensional space determines an
entit y which i s called a vector,
( i i ) any two vectors A and B have a sum A + B which obeys the
following laws: - > - - » • - > - - >
(a) commutative law A + B = B + A , - > - * • - * • • * • -> ->
(b) associative law (A + B)+ C = A + (B + C) ,
- 33 -
(c) i f A and B are v e c t o r s , there e x i s t s a unique vector X such - » -->->
t h a t A - B + X, thus i t f o l l o w s t h a t the operation o f s u b t r a c t i o n of -> -+-»•->•
vectors i s unique and there e x i s t s a vector 0 (zero) such t h a t A + 0 = A ,
( i i i ) f o r every r e a l number a and a vect o r A there e x i s t s a vector ->• -> ah - Aoc such t h a t
( a i + a 2 ) A = ct^A + c^A ,
and
and
-+ -y -y
a (A + B) = aA + aB ,
a i ( a 2 A ) = ( a1
a 2 ) A
The t o t a l i t y of vectors defined i n t h i s way i n the N-dimensional space
c o n s t i t u t e s a l i n e a r vector space o f K dimensions- A l i n e a r vector space ->- ->
i s c a l l e d a me t r i c vector space when t o every p a i r of vectors A and B of -> ->-
the space a number A.Bis assigned, c a l l e d the "inner" or " s c a l a r " product
o f these v e c t o r s , such t h a t
A . A > 0 , unless
-+ -> -y -y A « B = B » A |
-> -y -> -> -v A . ( B + C ) = A . B + A . C ,
-> -* -y ->
a (A . B) = aA . B .
I n an N-dimensional space there e x i s t N independent v e c t o r s ,
c a l l e d basis v e c t o r s , and every set o f N + 1 vectors i s l i n e a r l y depen
dent. When a basis X^, k = 1, ... , N, i s defined i n an N-dimensional ~r
vector space, then each vector A of the space i s uniquely determined by
- 34 -
a set o f N numbers a , k = 1, ... , N, such t h a t
A = \ 3
k
k The numbers a are c a l l e d the components o f the v e c t o r A w i t h respect t o
the given basis. These basis vectors form the coordinate axes and the
t r a n s f o r m a t i o n m a t r i x introduced i n equation (2.22) represents a change
of basis v e c t o r s.
A subspace can be defined as any subset o f a vector space which
s a t i s f i e s the f o l l o w i n g c o n d i t i o n (see Gerretsen 1962) :
I f A and B are vectors belonging t o the subset, then a l l the ->• -*•
vectors aA + £B , where a and 3 are a r b i t r a r y r e a l numbers, also belong
t o the subset.
The whole space i s a subspace of i t s e l f , c a l l e d the "improper
subspace". A "proper subspace" i s a subspace which does not c o n t a i n a l l
the elements o f the whole space. The set of basis vectors o f a proper
subspace i s a subset o f the set of basis vectors o f the whole space, and
t h e r e f o r e the dimension o f the subspace, t h a t i s the number of vectors
i n i t s b a s i s , i s l e s s than the dimension o f the whole space.
A t r a n s f o r m a t i o n operator corresponding t o a change o f basis
vectors i n an N-dimensional space can then be decomposed i n t o an operator
a c t i n g on the basis vectors o f a subspace and an operator which acts on
the other basis vectors of the whole space.
This approach w i l l be r e l e v a n t when we consider l a t e r (Section
3.7), magnetic m a t e r i a l s (which can have complicated magnetic o r d e r i n g ) ,
i n which case we w i l l be d e a l i n g w i t h a six-dimensional space, V (si x
3.6 Subspace
- 35 -
degrees of freedom) composed o f the three-dimensional geometrical sub-
space V, and the three-dimensional magnetic subspace M. I n t h i s s i x -
dimensional space the tr a n s f o r m a t i o n operator corresponding t o a change
of basic vectors can be w r i t t e n as { R IR.IV. } (see Section 3.4), p i ' xp
(Brinkman e t a l 1966 a,b and L i t v i n 1973) where {R.} acts on the co-x
o r d i n a t e s i n the geometrical subspace and { ffO acts on the coordinates
i n the magnetic subspace, and } i s a t r a n s l a t i o n operator.
3.7 Formal D e f i n i t i o n o f a f i e l d dependent tensor
I n a v a r i e t y o f circumstances one encounters s i t u a t i o n s where a
p h y s i c a l p r o p e r t y , which can be i d e n t i f i e d as a tensor i n one subspace of
the whole space, has components depending on the v a r i a b l e s defined i n the
oth e r subspaces. We generalize the idea one step f u r t h e r t o say t h a t the
dependence of the tenser components defined i n one subspace, on the basis
vectors of another subspace might be more complicated than being j u s t a
s c a l a r f u n c t i o n .
For example i n the six-dimensional space of the l a s t s e c t i o n , a
p h y s i c a l property can be considered which i n the absence of magnetic f i e l d
behaves l i k e a tensor of rank n, t h a t i s , i t i s a tensor o f rank n i n the
geometrical subspace V. A f t e r i n t r o d u c i n g the magnetic f i e l d the t r a n s
formation operator w i l l become more complicated and w i l l i n v o l v e {IR^} •
Therefore the t r a n s f o r m a t i o n law f o r t h i s p h y s i c a l p r o p e r t y (tensor o f rank
n i n V) has t o be generalised t o take operations i J i n t o account. To
do t h i s we have t o seek the t r a n s f o r m a t i o n law c f the p r o p e r t y corresponding
t o a change of basis vectors i n magnetic subspace M, and f i n d out what s o r t
o f t e n s o r i a l dependence t h i s p r o p e r t y has on the magnetic subspace co
o r d i n a t e s . That i s , what i s i t s rank, £,in t h i s space; t h i s rank L does
not have t o be the same as n, the rank of the p r o p e r t y i n V. Z depends
- 36 -
e n t i r e l y on each s p e c i a l p r o p e r t y and the p h y s i c a l laws concerning t h a t
p r o p e r t y . The process of f i n d i n g t i s s i m i l a r t o the o r d i n a r y case of
f i e l d independent tensors (namely q u o t i e n t law, see Section 4.6).
The transformation law f o r a tensor property A w i t h rank n i n V
and rank L i n M, which t o maintain the g e n e r a l i t y i s considered t o be a
tensor f i e l d i n both subspaces can be w r i t t e n
{ R J R . } A ,± i , (B, r ) =
m in r r
I n
= A' a a U*J m / ( R . } r ) , (3.8) m m r .... r
i j A £ a l a £ where m , , m , m , , m in d i c e s r e f e r t o the coordinate axes
31 j n Bl &n i n M subspace and r r , r r i n d i c e s r e f e r t o the coo r d i n a t e axes i n V subspace, and they a l l range from one t o three (number o f coordinate axes i n a three-dimensional space). The t r a n s f o r m a t i o n
operator {R } has the ma t r i x r e p r e s e n t a t i o n w i t h components (R ) , , and P . P i e a m m
the components o f the r e p r e s e n t a t i o n m a t r i x f o r operation { R } are
(R.) . Therefore the operator { R jR.} on the l e f t hand side of 1 i ft p 1
3 e 3 f r r equation (3.8) can be represented by
{ R IR. } = (R ) . (R ) (R ) P i P • P • P ^
m m m m m m
(R.) . (R.) (R.) (3.9 ^ 1 S l 1 h *2 1 j n \
X X X X X X
- 37 -
The combined operation { R^| lO acts i n the following way. f i r s t operation
transforms the 3 components of the tensor A, i n the subspace V,
into t h e i r new values i n the new reference frame i n V, then each of these
components which has 3 components i n M i s operated upon by { R } and i s I P
transformed into a s e t of 3 new values i n the new set of m (which for
instance could be magnetic coordinates). The number 3 i s the dimension
of the subspace i n each case. Therefore our tensor property has indeed n i £+n
(3) x (3) = 3 components. In general, i n a space composed of two subspaces V(h^ - dimensional) and M(h^ - dimensional), a tensor of rank n
n &
in V and rank Z i n M w i l l have (h^) + (h^) components. Supposing that
t h i s tensor has t contravariant and s covariant indices i n V (T + s = n)
and v contravariant and u covariant indices i n M (V + u = 4.), the trans
formation law for i t s components w i l l be
v t g l g v j l * t m ....m r . . . . r
A' (ra\r') = 3m i , i k, k - ° 1 u 1 s am m ....m r . . . . r
s j 6 6
3r'' 1 1 3r 3r s
g l 3m' 9 v 8 0 1 am 3m 3 u
5i 3r' 1
1 a « v dm
3m' 1 1 3m- l u
a l m .
a V .. • m Y l . r ...
Y t • r A (m,r) (3.10)
m . 3 u
.. .m
6 1 r ...
6 s
• r 3r z 3 1 3r' s
For a r e l a t i v e , f i e l d dependent tensor, B, of weight i n V, and of
weight i> 2 i n M, and of the same rank and covariance and contravariance
property as the tensor A i n equations (3.10), the transformation law
w i l l be
- 38 -
g i g v j l j t m ... .m r .... r On' ,r')
i . i k. k 1 u 1 s m ,...m r . . . . r
3n> 3m' 1
3 m .g 3m1
Sm"1 3m a v
3m 3m
3m1
3r
3m« X u
3r'
3r' :
a
fir' 1 3r' 3r
s m B
.12
3r
'1 ' t r .... r
3r u 3r'
(m,r) ,
3r k
i s m
u 1 s .m r — . r
(3.11)
where 3m 3m ,g
and 3r' 3r i 3
are the determinants of the transformation
matrices i n M and \? subspace. Equation (3.11) i s the general transforma
tion law for f i e l d dependent tensors. For a transformation operator,
which i s composed of orthogonal transformation of coordinate axes i n a l l
the subspaces of the whole space, there i s no difference between covariant
and contravariant indices and the transformation law (3.11) can be written
as
A' m. •m. i , r . ,r. (m',r') -3m
3m o
3m' . 3 3r' "
3 % , 3m~'~
3m O-i
3m'
3r„
3r'
3r„ n
3r' 'n
on .m. r g r g (m,r> , (3.12)
where 0 for even (polar i n M) ,
and 1 for odd UJ^ a x i a l i n M) ,
'0 for even (polar i n V) ,
1 for odd to ( a x i a l i n v ) .
- 39 -
As an example the c o n d u c t i v i t y tensor? i n the space composed of the
three-dimensional geometrical subspace and the three-dimensional momentum
subspace, can be considered. I n t h i s space a tr a n s f o r m a t i o n o f coordinates
may be represented by the operator { E IsR.lv. } w i t h iR a r o t a t i o n p i ' i p p
operation i n the momentum subspace, a r o t a t i o n o p e r a t i o n i n the geomet
r i c a l subspace, and a t r a n s l a t i o n o p e r a t i o n . The form o f the Ohm's
law i n the three-dimensional geometrical subspace i s
J i * °ij E j ' ( 3 - 1 3 )
w i t h J and E a c t i n g as v e c t o r s , or f i r s t rank tensors and a as a second
rank tensor. Therefore the e f f e c t o f IR. on a, . can be e a s i l y found. But 1 ID
f o r R^, the behaviour of J, E and o i n the momentum subspace has t o be
found f i r s t . Since e l e c t r i c f i e l d E a t a p o i n t depends only on the distance
between the p o i n t and the charged body, which i s c r e a t i n g the f i e l d , every
r o t a t i o n o f coordinate axes i n the momentum subspace leaves E unchanged.
Therefore e l e c t r i c f i e l d E behaves l i k e a scal a r i n the momentum subspace.
The c u r r e n t d e n s i t y J can be w r i t t e n as the product of e l e c t r i c charge
d e n s i t y times e l e c t r i c charge v e l o c i t y ; J = P.v. E l e c t r i c charge de n s i t y
P i s a scalar i n momentum subspace and v the charge v e l o c i t y i s a ve c t o r .
Therefore using the q u o t i e n t law (see Section 4.6), J, the c u r r e n t d e n s i t y ,
i s a tensor o f f i r s t rank, or a vector, i n the momentum subspace, and can be w r i t t e n as (J.) , w i t h nine components (m- r e f e r s t o coordinate axes i n
m k
the momentum subspace). Now by using the q u o t i e n t law i t can be shown
t h a t c o n d u c t i v i t y i s a l s o a vector i n the momentum subspace, and can be
w r i t t e n as (a. ) w i t h twenty seven components. Thus we can conclude
the generalized Ohm's law i n the space composed of the geometrical sub-
space and the momentum subspace : (J.) (o..) E. (3.14) ^ k 1 3 -k 3
- 40 -
Spin-tensor q u a n t i t i e s can be considered as another example o f
q u a n t i t i e s w i t h such m u l t i - t e n s o r i a l character. Spinors have been defined
i n the f o l l o w i n g way ^Corson 1953):
Spinors are geometrical objects which are defined over a two
dimensional space (the spin-space) and obey the f o l l o w i n g t r a n s f o r m a t i o n
law ,
*A " SA (A, B = 1, 2) (3.15)
w i t h S the t r a n s f o r m a t i o n m a t r i x i n s-space (the spin-space).
defined i n t h i s way i s a c o v a r i a n t spinor of f i r s t rank i n the spin-space.
I n our terminology t h i s w i l l be a c o v a r i a n t tensor o f f i r s t rank i n the
s p i n subspace. There are also higher rank spinors and weighted spinors.
I n a d d i t i o n spin-tensors are defined as those q u a n t i t i e s which have both
tensor and spinor i n d i c e s and obey the t r a n s f o r m a t i o n laws
<b' = & V 9 {3.16) vpAB (J. VAB
v under coordinate t r a n s f o r m a t i o n a i n the geometrical subspace (a i n
u |J.V n o t a t i o n s used i n t h i s t h e s i s ) , and
= S? S° * „„ , (3.17) VUAB A B UCD
under s p i n t r a n s f o r m a t i o n i n the spin-subspace. A l l these q u a n t i t i e s
might be defined i n terms of f i e l d dependent tensor terminology. I t i s
apparent t h a t there w i l l be many other q u a n t i t i e s which a r i s e i n d i f f e r e n t
f i e l d o f physics t h a t can be brought under a b l a n k e t d e f i n i t i o n o f f i e l d
dependent tensors.
- 41 -
3.8 The n a t u r e o f c o o r d i n a t e s i n t h e f i e l d subspace
So f a r i n t h e d e f i n i t i o n o f f i e l d dependent t e n s o r s and t h e i r
t r a n s f o r m a t i o n laws we have j u s t c o n s i d e r e d t h e dependence o f t h e f i e l d
dependent t e n s o r components on e x t r a c o o r d i n a t e s ( c o o r d i n a t e s i n t h e
f i e l d s u b s p a c e ) , w i t h o u t any s p e c i a l r e f e r e n c e t o t h e n a t u r e o f t h e f i e l d .
By t h e n a t u r e o f f i e l d components, we mean t h e r e l a t i o n between t h e co
o r d i n a t e s i n t h e f i e l d subspace and t h e g e o m e t r i c a l subspace. T h i s r e l a t i o n
w i l l be i n v e s t i g a t e d i n t h i s s e c t i o n .
F i r s t we c o n s i d e r t h e s i m p l e case where t h e f i e l d c o o r d i n a t e s a r e
s c a l a r q u a n t i t i e s i n t h e g e o m e t r i c a l subspace ( q u a n t i t i e s w h i c h a r e i n
dependent o f t h e c h o i c e o f c o o r d i n a t e s i n t h e g e o m e t r i c a l s u b s p a c e ) , such
as t e m p e r a t u r e , c o l o u r o r shape o f t h e i n d i v i d u a l o b j e c t ( d i f f e r e n t c o l o u r s
o r d i f f e r e n t shapes can be c h a r a c t e r i z e d by d i f f e r e n t numbers). I n t h i s
case t r a n s f o r m a t i o n o p e r a t o r s o f t h e g e o m e t r i c a l subspace do n o t have any
e f f e c t on t h e c q o r d i n a t e s i n t h e f i e l d subspace, and e q u a t i o n s ( 3 . 3 ) ,
( 3 . 9 ) , ( 3 . 1 0 ) , (3.11) and (3.12) can be used t o f i n d t h e t r a n s f o r m e d s e t
o f f i e l d dependent t e n s o r components.
There a r e o t h e r cases where t h e c o o r d i n a t e s i n t h e f i e l d subspace
depend on t h e d i r e c t i o n o f c o o r d i n a t e axes i n t h e g e o m e t r i c a l subspace.
Examples o f t h i s a r e t h e space composed o f momentum and g e o m e t r i c a l sub-
spaces, d e s c r i b e d i n t h e l a s t s e c t i o n , where momentum i s i t s e l f a v e c t o r
i n t h e g e o m e t r i c a l subspace, o r t h e space o f a m a g n e t i c c r y s t a l where t h e
m a g n e t i c moments ( c o o r d i n a t e s o f t h e f i e l d subspacs) are v e c t o r s i n t h e
g e o m e t r i c a l subspace. I n c o n s t r u c t i n g t h e s e t o f symmetry o p e r a t i o n s o f
such spaces t h e e f f e c t o f each symmetry o p e r a t i o n on e v e r y c o o r d i n a t e
a x i s s h o u l d be s p e c i f i e d . I n t h i s sense t h e r e a r e d i f f e r e n t p o s s i b i l i t i e s
f o r symmetry o p e r a t i o n s ;
- 42 -
( i ) t h e symmetry o p e r a t i o n s o f t h e g e o m e t r i c a l subspace a r e s u f f i c
i e n t t o s a t i s f y t h e symmetry r e q u i r e m e n t s o f t h e f i e l d subspace,
( i i ) e x t r a o p e r a t i o n s i n t h e f i e l d subspace must be added t o each
t r a n s f o r m a t i o n o p e r a t i o n o f t h e g e o m e t r i c a l subspace t o s a t i s f y t h e
symmetry r e q u i r e m e n t s o f t h e f i e l d subspace,
( i i i ) t h e o p e r a t i o n s w h i c h s a t i s f y t h e symmetry r e q u i r e m e n t s o f t h e
f i e l d subspace a r e q u i t e d i f f e r e n t , f r o m t h o s e i n t h e g e o m e t r i c a l sub-
space, w h i c h a r e supposed t o l e a v e t h e c o o r d i n a t e s i n t h e f i e l d subspace
unchanged. Here, t o f i n d t h e s e t o f t r a n s f o r m e d components o f a f i e l d
dependent t e n s o r , t h e t r a n s f o r m a t i o n law (one o f t h e e q u a t i o n s ( 3 . 8 ) ,
( 3 . 9 ) , ( 3 . 1 0 ) , (3.11) o r (3.12) a c c o r d i n g t o t h e k i n d o f t e n s o r under
c o n s i d e r a t i o n ) o f t h e l a s t s e c t i o n can be employed.
For spaces o f t h e k i n d ( i ) and ( i i ) where one t r a n s f o r m a t i o n
o p e r a t o r may a c t on b o t h o f t h e subspaces t h e s i t u a t i o n i s s l i g h t l y
d i f f e r e n t . To e x p l a i n t h i s l e t us c o n s i d e r t e n s o r p r o p e r t i e s w h i c h
a c t l i k e s c a l a r f u n c t i o n s i n t h e f i e l d subspace. Under a t r a n s f o r m a t i o n
o f c o o r d i n a t e s i n t h e g e o m e t r i c a l subspace, denoted by o p e r a t o r \fR },
t h e t r a n s f o r m a t i o n law f o r such a t e n s o r p r o p e r t y w i l l have one o f t h e
f o l l o w i n g f o r m s :
(a) when t h e c o o r d i n a t e s i n t h e f i e l d subspace a r e v e c t o r s i n t h e
g e o m e t r i c a l subspace;
- 43 -
8r r r , 6 k B * * On , r ) , (3.18)
a r . s «i 6 s a
3 r r ... . r
where i s t h e |R! . . element o f t h e t r a n s f o r m a t i o n m a t r i x . (R) , 3 r ' 3 ^
(b) when t h e c o o r d i n a t e s i n t h e f i e l d subspace a r e second ran k t e n s o r s
i n t h e g e o m e t r i c a l subspace-.
a7 n \ 3. J f ar 3 r ' 9 8 8 r ! 1 3r' fc
: m , -x— r' / = .... 3r' 2 1 2 d r ^ 3r * 3r u
6 6 Y t
3r 1 3 r r •••• r & —k7 — ~ B « 6 (Vc* < r >' ( 3' 1 9 )
3r« 1 3r> 3 r ^ . . . r S 1 2
where m. . a r e t h e components o f t h e f i e l d t e n s o r a l o n g t h e c o o r d i n a t e 1 1 1 2
axes i n t h e g e o m e t r i c a l subspace, ( t h e f i e l d t e n s o r c o u l d have c o n t r a -
v a r i a n t i n d i c e s as w e l l ) f
(c) when t h e c o o r d i n a t e s i n t h e f i e l d subspace a r e vor.sors o f r a n k s
h i g h e r t h a n two:
- 44 -
B 1' 1 s r . r
3r or m_
3r • 1 3r' a. .. .a 1 a
3 r ' g 3 TT" r ' 3r /
3r'
3r
3r ,3 t
3r
3r
3r'
3 r
3 r '
. r
. r
(3.20)
t h e f i e l d t e n s o r c o u l d a l s o have c o n t r a v a r i a n t i n d i c e s .
We may e x t e n d what has been s a i d a l s o t o t h e case o f f i e l d depen
d e n t t e n s o r p r o p e r t i e s w h i c h have r a n k s h i g h e r t h a n z e r o i n t h e f i e l d sub-
space/ where t h e f i e l d components a r e t h e m s e l v e s t e n s o r s i n t h e g e o m e t r i c a l
subspace. Then f o r spaces o f k i n d ( i ) and ( i i ) , a t r a n s f o r m a t i o n o p e r a t o r
{ IR} i n t h e g e o m e t r i c a l subspace w h i c h can s a t i s f y a p a r t o f o r a l l t h e
symmetry r e q u i r e m e n t s o f t h e f i e l d subspace w i l l t r a n s f o r m t h e components
o f t h e f i e l d t e n s o r i n t h e g e o m e t r i c a l subspace i n t h e f o l l o w i n g way:
i l 1
m , _ 3r 3r U
m - — j r — m. „,, 1 d 1 . . u i u <jr' d r 1
where m' a r e t h e f i e l d components a l o n g t h e c o o r d i n a t e axes i n t h e 1 u
g e o m e t r i c a l subspace. Now we can c o n s t r u c t t h e t r a n s f o r m a t i o n o p e r a t o r
{ Rp } i n t h e f i e l d subspace, c o r r e s p o n d i n g t o t h e - c r a n j i f o m i u t i o n o p e r a t o r
{ R } i n t h e g e o m e t r i c a l subspace= We can w r i t e
= { R p } mfc (3.22) ab
where m' = m' „ , and m. = m. , and 3 ^ _ - ** O T _ - _ _ 1 a a,...- a b i . . . . . i
1 u 1 u
- 45 -
~ 1 - u
< KP> - ^ V ( 3- 2 3 1
ar 3r
Then e q u a t i o n (3.8) can be used as t h e t r a n s f o r m a t i o n law f o r f i e l d
dependent t e n s o r s w i t h o p e r a t i o n i R } as d e f i n e d i n e q u a t i o n ( 3 . 2 3 ) .
- 46 -
CHAPTER 4
FIELD DEPENDENT TENSORS ( I I ) ;
CALCULUS
4.1 Introduction
The basic concepts and theorems of the algebra of f i e l d independent
tensors (Sections 2.11, 2.12 and 2.13) such as addition and subtraction
operations can be generalized to establish the algebra o f f i e l d dependent
tensors. This i s done here for the f i r s t time. In th i s chapter t h e way
the operations of addition, subtraction, inner and outer multiplication,
contraction and dif f e r e n t i a t i o n o f ' f i e l d dependent tensors act are worked
out, and the conditions under which these operations can be performed are
discussed. Furthermore, the quotient law i s generalized t o include f i e l d
dependent tensors.
4.2 Addition and subtraction
Theorem 1:
In the space composed of two subspaces R and M, i t w i l l be shown
that the sum (or difference) of two f i e l d dependent tensors A(r,m) and
B(r,m) (r and m refer to the coordinate axes i n R and M subspaces) which
have the same rank h (with the same number of covariant f .aiid the same
number of contravariant e indices (f + e = h)j in R and t h e same rank n
(with the same number of covariant p and t h e same number o f cd.travariant
q indices (p + q = n)j i n M, defined a t t h e same point i s again a f i e l d
dependent tensor C(r,m) of the same rank and type i n R and t h e sama rank
and type i n M.
Proof: The corresponding transformation iaws (equation 3.10) for A(r,m)
and B(r,m) are
- 47 -
(r\m') = 9 i
3m' *
dip
m
,3m' H
a 3m <J
a .m
.m
3m
cm
dm
3m'lp
(r,m) ,
(4.1)
q e q p
p 3 m 3m 3 m m m om (r' ,m') B 1 3 in 1 1 dm c>in om m m
1 1 1 m m (r,m) B
1 3 3 m
(4.2)
Then adding each component of A' from equation (4.1) cc the corresponding
component of B* from equation (4.2), we arr i v e at a set with the same
number of components which obeys the following transformation lav;
- 48 -
91 g q in . . . . m j l r ... .
je ....m q j l j e r . ... r
A' (r 1,m') B" 1 P ni ... .m
k l r ... K
. r s m
i m P r .... r
3m'91 3m'9q
6. 3m • 3 m ^ 3r' 3 r ! e 3 r
a i 3m
a 3m q 3m' 1 3m' l p
Y, 3 r 1
Y 3r 8 3f'
a, a Y, Y a, a y , Y v 1 q 1 e 1 q ' 1 e \ m ....m r . . . . r m ....m r . . . . r ' A (r,m) + B (r,rc) j
" 6 , 8 fi. 6^ / *1 ep 61 6 f " 61 6p 61 " f / m ....m r r m . ...m r , . . . r
Therefore the set of components ,
q fll Q q Y l Y e m ....m r . . . . r
C (r,n) = S l &
P 61 5 f m ....m r . . . . r
P
a. a y , y a. a y, y 1 q 1 e i q 11 e m .... m r .... r m ... . m r . .. . r A (r,m) ± B ir.m) ,
B. B 6. 6. 0. 6 6, 6., l p l r l p i j . m ....m r . . . . r m ....m r . . . . r
forms a f i e l d dependent tensor of the same rank and type ES the two
or i g i n a l tensors A and B, in each subspace-
- 49 -
G e n e r a l e x t e n s i o n o f t h e theorem: t h e o p e r a t i o n o f a d d i t i o n can
be i m m e d i a t e l y e x t e n d e d t o f i n d t h e ( a l g e b r a i c ) sum o f any number o f
f i e l d dependent t e n s o r s , p r o v i d e d t h e y a l l have t h e same r a n k and a r e
a l l o f t h e same t y p e ( e q u a l number o f c o v a r i a n t i n d i c e s and e q u a l number
o f c o n t r a v a r i a n t i n d i c e s ) i n every subspace o f t h e whole space. But t h e
o p e r a t i o n o f a d d i t i o n does n o t a p p l y t o f i e l d dependent t e n s o r s w h i c h have
d i f f e r e n t r a n k s o r a r e o f d i f f e r e n t t y p e s i n a subspace.
Theorem I I :
The s e t o f f u n c t i o n s formed by m u l t i p l y i n g each component o f a
f i e l d dependent t e n s o r by a c o n s t a n t ^ i s i t s e l f a f i e l d dependent t e n s o r
w i t h t h e same t e n s o r i a l c h a r a c t e r ( t h e same r a n k and t y p e i n e v e r y sub-
space) . T h i s i s r e a d i l y p r o v e d by m u l t i p l y i n g b o t h s i d e s o f e q u a t i o n
(4.1) by a c o n s t a n t . From t h i s f a c t t o g e t h e r w i t h t h e d e f i n i t i o n o f
t h e a d d i t i o n o p e r a t i o n , i t can be s t a t e d t h a t :
Any l i n e a r c o m b i n a t i o n o f f i e l d dependent t e n s o r s o f t h e same t e n s o r i a l
c h a r a c t e r i s a f i e l d dependent t e n s o r o f t h e same c h a r a c t e r .
4.3 O u t e r m u l t i p l i c a t i o n o f two f i e l d dependent t e n s o r s
D e f i n i t i o n :
I n a space composed o f two subspaces M ( w i t h m^ as c o o r d i n a t e axes)
and R ( w i t h r ^ as c o o r d i n a t e a x e s ) , c o n s i d e r two f i e l d dependent t e n s o r s :
A ( r , i u ) o f r a n k k ( w i t h s c o n t r a v a r i a n t and p c o v a r i a n t i n d i c u o a + p = k)
i n M subspace and o f r a n k Z ( w i t h t c o n t r a v a r i a n t and q c o v a r i a r . t i n d i c e s
t + q = Z) i n R subspace, and B ( r , n ) o f r a n k n ( w i t h i c o n t r a v a r i a n t find
u c o v a r i a n t i n d i c e s i + u = n) i n M subspace and o f o rank ( w i d i j con
t r a v a r i a n t and v c o v a r i a n t j + v = a ) i n R subspace. The o u t e r p r o d u c t
o f t h e s e two f i e l d dependent t e n s o r s can be d e f i n e d as t h e s e t o f
50 -
elements C(r,m) formed by multiplying each component of A(r,m) by each
component of the tensor B(r,m). That i s
t i j
a a b b e e f f 1 s " l t e l i x i E j m ....m r .... r m ..••m r .... r A c, c d d ( r ' m ) ' B g g h h ( r ' m ) S
1 p 1 q y l 9 u 1 v m ....in r ..«.r m ....m r , . . . r
p q u
s + i t + j a a e e b b f " l s 1 i 1 t 1 3 m ... .m ID ... .m r . . . . r r ..«.r
S C (r,m) . (4.5) c, e g , g d, d h, h
m...««m m ... .m x «... r r .....r
p + u q + v
After the transformation of coordinate axes (r,m) into new axes (r',m')
the outer product can be written as
V a s E i e i B i K V " j m ....m m ....m r . . . . r r . . . . r
M 'p 1 u 1 q *1 *v m ... .m m ....m r . . . • r r . . . . r
a l a s *1 * t e l £ i n l n j m ....m r • • • • r m .... m r .... r A' (r' ,m'') - B ! ( r ! ,ms) .
Y l Y p 6j « Xj X u P l y v
m • • • • I D r •. • • r m .. • .m r . . • • r
(4.6)
Changing the dummy suffix e s i n equations (4.1) and (4.2), and substituting
for A 1(r,m) and B'(r,m) from these equations into equation (4.6) shows that
the s e t of elements of the outer product C(r,m) forms a f i e l d dependent
- 51 -
tensor of rank k + n (with s + i contravariant and p + u covariant indices
s + i + p + u = k + n ) i n H and of rank t + 0 (with t + j contravariant
and q + v covariant indices t + j + q + v = £ + 0 ) i n R subspace.
4.4 Contraction of f i e l d dependent tensors
In a space composed of two subspaces R (n-dimensional) and M (£-
dimensional) consider a f i e l d dependent tensor
a, a b. b 1 s i t m ....m r . . . . r A _ - (r,m) of rank s + p i n M and rank t + q i n R (s c 4 c a. a m 1 m P r 1 r q
m ....m i •... r
and t contravariant, p and q covariant). Equating a covariant index m
and a contravariant index m i n the same subspace M and summing with
respect to this.index w i l l r e s u l t i n a s e t of (£) S +^~ 2(m) t + (* functions. This
i s the operation of contraction. I t w i l l be shown that t h i s s e t of
functions forms a f i e l d dependent tensor of rank s + p - 2 i n M subspace
and t + q i n R subspace. I n order to avoid writing out long formulae with
many covariant and contravariant indices i n each subspace we i l l u s t r a t e
the proof by considering a f i e l d dependent tensor
a, a„ b. b_ m 1 2 1 2 A (r,m). The transformation law for t h i s tensor i s (see
c c c d 1 2 3 " l m m m r
equation 4.1)
a. a„ b. b A ,
m 1 m 2 r 1 r 2 , , 1 % ( 3 m ' 3 m ' * 2 3m Y 2 3 j ^ _ \ ' (r' ,m') =1 — - — • — - — « — • — • — ) c, c„ c„ d, \ « 1 . 2 , , 1 * , 2 *,2' 1 2 3 1 3m 3m 3m' 3m' 3m' m m m
0 3 a 1 m m (r.m) (4.7) 6 6 3
3r m m m
- 52 -
a c 2 3 Now i f we equate m and m i n equation (4.7) we obtain
b b a« a». 1 2 a. a_ Y. Y„ Y m 1 m 2 r r a . 1 , , 2 . T l a_ T 2 Y 3 i „ „ A I . I . I I . *5»! M* 3m 3m 3m A- c c d (r\m') . c. 2 3 l a. a
3r' 1 3r' 2 3r 1 " m r r
*d~ Y * Y Y Y 6 ( r ' m )
a ,1 T 3 Y l Y 2 Y 3 °1 or' m m m m r
That i s :
m l m m r . 1 . 2 «. . 1 . , 2 « , 2 3m 3m 3m' 3m* 3m'
b. b„ 6. Q l °2 B l B2
ft A * A~ A v v v A ( r ' m )
9 B i a
P2 . , d l Y l Y 2 Y 3 61 3r 3r 3r' m m m r
Therefore ,
m 1 m 2 r 1 r 2 a i Y l Y 2 b l b2
°1 C2 C 3 d l " l C l C2 B l B2 m m m J r 3m 3m' 3m' 3r 3r
6, °2 *1 °2 B l B2 a r 1 m m m r r
a. a„ b. b„ . 1 2 1 2 a Y. Y, b, b. " 1 * . ___ . __L . isJ_ . «£!_ . l£li
m m m r 3m 3m" 3m' 3r 3r
3r " l °2 B l B2 m m r r
d~ A Y Y a 6 (r'm) ' a , 1 T l T2 a2 °1 or" m m m r
(4.8)
- 53 -
a a b b 1 2 1 2 m m r r Equation (4.8) shows that A a d ( r» m) i s a f i e l d dependent
1 2 2 1 m m m r
a l a2 b l b 2 m m r r tensor of the same rank as A (r,m) i n R subspace but of
C l C2 C3 d l m m m r
a a b b 1 2 1 2 m m r r rank l e s s than A (r,m) by two (one contravariant, one
C l C2 C 3 d l m m m r
covariant), i n N subspace. I t should be borne i n mind that the operation
of contraction (lowering the number of covariant and contravariant indices
by one) can only be applied to two indices which are of d i f f e r e n t types
and refer to the axes defined i n one subspace. Moreover the operation of
contraction can be c a r r i e d out for a l l the indices of a f i e l d dependent
tensor one a f t e r another, as far as there are indices of both types present
i n the same subspace.
4.5 Inner multiplication of f i e l d dependent tensors
The operation of inner multiplication of two f i e l d dependent
tensors i s a combination of the operations of outer multiplication and
tensor contraction. That i s , to find the inner product of two f i e l d
dependent tensors, f i r s t we find t h e i r outer product and then we apply
the operation of contraction. As an example, consider two f i e l d depen
dent tensors
1 1 e. e„ m r m 1 2 A , (r,m) and B . (r,m); then forming t h e i r outer product,
1 2 1 g l 1 m m r m m
we obtain
- 54 -
a e e b a b e e 1 1 2 1 1 1 1 2 m m m r m r m m c«, «, «, a, h, (r'") " A =2 -t
(r"' B«, », (r"" • m m m r r m m r m m
a l 6 1 6 2 b l m m m r The f i e l d dependent tensor C , . (r,m) can be contracted
C l C 2 g l d l h l m m m r r
three times i n M subspace and once i n R subspace (these contractions can
be performed i n several ways: two dif f e r e n t ways i n R and s i x d i f f e r e n t
ways i n M).
4.6 The quotient law of f i e l d dependent tensors
The quotient law, defined i n Section 2.12 for f i e l d independent
tensors, can be generalized to include f i e l d dependent tensors i n the
following way. That i s , i t can be stated that i n the space composed of h k
two subspaces R (n-dimensional) and M (^-dimensional), the set of (n ) { t )
functions of coordinates r and m forms a f i e l d dependent tensor of rank
h i n R subspace and of rank k in M subspace (with a certain number of
contravariant and covariant i n d i c e s ) , provided that an inner product of
t h i s s et of functions with an arb i t r a r y f i e l d dependent tensor i s i t s e l f
a f i e l d dependent tensor, (the inner product of a tensor and a set of
functions means the sum of the form > f ° ± 2 ± k j l \ , 1 ( a H K j l v y , A(m ,m ...,m ,r , . . . , r )A , or A(m ,m ,...,m ,r , . . . , r )A ) .
m
Writing out the general proof for t h i s statement involves long formulae
with many contravariant and covariant indices. To avoit t h i s the proof
i s given here for the following s p e c i a l case which has a l l the features 3 2
of the general case. Consider the set of (m ) x (t ) functions . a b c d e
A(m ,m ,m ,r ,r ) . Now l e t us suppose that the inner product ,
- 55 -
a f , , m r a b c d e A(m ,m ,m ,r ,r ) B (r,m). c m
a f ra ,r for an arbitrary f i e l d dependent tensor B (r,m) y i e l d s a f i e l d b f mc
m r dependent tensor C d (r,m) of f i r s t rank (contravariant) i n M and of
r r t h i r d rank (two covariant and one contravariant) i n R. That i s
v a f b f . . a b c d e . „ m r , . „m r . A(m fm ,m ,r ,r ) B (r,m) = C (r,m) (4.9) c e a m r r
The transformed quantities, referred to a new system of coordinates
(r',m')» s a t i s f y the following equation
a f b f h d m r m r A'(ma,m ,mC,r , r e ) B' (r',m') = C . (r^m') (4.10) c e a m r r
a f b f m r m r Substituting for B' (r',m') and C . (r^m 1) from t h e i r trans-c e a m r r
formation laws (equation 4.1) into equation (4.10), we obtain
a b c d e. 3m'a 3mY 3 r , £ Dm a r M . . A' (m ,m ,m ,r f r ) — — • — — B (r,m)
3m 3m' 3r mY
- B u . ,b . e . ,f . 6 m r C . (r,m) (4.11)
3m 0 3 r ' e 3 r P 3 r ' d r £ r*
B u m r Substituting for C £ g(r,m) from equation (4.9) into equation (4.11)
r r and changing the dummy indices, we get
- 56 -
3 r , £ l ~ ,, a b c d e, 3m" A* (m ,m ,m ,r ,r ) — -3 r " L 3m
3m1
3m «# a B Y 6 e. - A(m , m ,m ,r ,r ) .
, • . a n 3m' 3 '—g 3m 3m
e . o - i m r R — "Z-T B (r fm) = 0 (4.12) m , e 3r« J raY
a ra r Equation (4.12) should hold for every B (r,m), the arbitrary tensor.
m
Therefore the expression i n the bracket i s i d e n t i c a l l y zero, that i s
... a b c d e, 3ma 3m , b 3m , C 3 r E 3 r 6
A' (m ,m ,m ,r ,r ) = • > — — • 3m'3 3mB 3mY 3 r , e 3 r , d
A(m ,m ,m ,r ,r ) , (4.13)
which shows that A(ma,m^,mY, r ^ , r E ) i s a f i e l d dependent tensor of B Y
rank three i n H subspace (contravariant i n m and m , and covariant i n
ma ) and of rank two i n R subspace (covariant i n both r^ and r e ) , that i s
B Y a t mm
A(ma,m ,mY,r , r E ) >= A (r,m) - (4.14) Ot 0 £ m r r
The ten s o r i a l character of a set of functions can also be investigated
i n each subspace separately by means of the quotient law. The quotient
law can then be stated in the following way:
A set of functions forms a f i e l d dependent tensor of certain rank(with
certain covariant and contravariant indices) i n a subspace i f an inner
product of t h i s s e t of functions with an orbitrary f i e l d dependent tensor
i n that subspace i s again a f i e l d dependent tensor i n the same subspace.
- 57 -
4.7 Differentiation of f i e l d dependent tensors
The way i n which the d i f f e r e n t i a t i o n operator has been dealt
with for f i e l d independent tensors i n Section 2.13 can be followed for
the components of a f i e l d dependent tensor i n one subspace. That i s ,
i t can be stated that, i n a space composed of two subspaces R(n-n 3
dimensional) and M (^-dimensional), the set of p a r t i a l derivatives 3ma
(with respect to the coordinates i n M) of a f i e l d dependent tensor which
behaves l i k e a s c a l a r function i n M, forms a f i e l d dependent tensor
which acts l i k e a covariant tensor of rank one i n M. The ten s o r i a l
character of t h i s f i e l d dependent tensor i n the subspace R does not change
a f t e r d i f f e r e n t i a t i o n . This f i e l d dependent tensor (scalar i n the sub-
space M) can be denoted as A ^ ( r f m ) , where the s u f f i x (r) shows that the tensor A (r,m) has an undefined character i n the subspace R. The set of
r 3A (r,m) p a r t i a l derivatives can then be denoted as — — . Now i f we form
3m the set of p a r t i a l derivatives with respect to the coordinates i n the
3A (r,m) subspace M of the f i e l d dependent covariant vector (to avoid
3m
complication we drop the s u f f i x (r) bearing i n mind that A(r,m) i s not a
sc a l a r function i n R) we get
2 3 A(r.m) 3m,a3m'a 3m
3 / 3A(r,m) 3m° \ 3m-a \ 3ma 3m'a /
2 R « _ , . 2 a 3"A(rym) 3m 3m' 3A(rfm) 3 m 3m°3mB 3m'a 3m'a 3m° 3m , a3ra , a
(4.15)
Therefore for a general transformation of coordinate axes i n the sub-
space M, where 3 2m a/(3m , a3m , a) ^ 0, th i s s e t of p a r t i a l derivatives of
the covariant tensor i n the subspace M does not represent a tensor. This
can be generalized to any kind of f i e l d dependent tensor to show that the
- 58 -
p a r t i a l derivatives of a f i e l d dependent tensor, with respect to the
coordinate axes i n one subspace, where a general transformation of co
ordinate axes i n that subspace i s considered, does not form the com
ponents of a tensor i n that subspace. But for a l i n e a r transformation
of coordinate axes i n the subspace under consideration, where dm°/dm'a
i s a constant the p a r t i a l derivative operator i n every subspace acts l i k e
a covariant vector i n that subspace and adds one covariant index to the
indices of. the o r i g i n a l f i e l d dependent tensor i n that subspace. That i s ,
the f i e l d dependent tensor formed by the set of p a r t i a l derivatives,
with respect to coordinate axes i n the subspace M, of the f i e l d dependent a, a b. b. 1 g 1 h m •..•m r . . . . r
tensor A (r,m) with g contravariant and p covariant c. c d. d 1 p i q m . •. .m r . . . . r
indices in M, and h contravariant and q covariant indices i n R, w i l l
have g contravariant and p + 1 covariant indices in M and h contravariant
and q covariant indices i n R. The transformation law for such a f i e l d
dependent tensor w i l l be
a. a b, b. 1 g 1 h y, y . m m * r r e ' 1 . p o . . . . 9m 3m ... 3m A* , (r ' ,m') = *>' „ 1 „ p , 1 ,, q 3m' . , 1 - p m ....m r r . . . . r 9m 3m *
p + 1.
a a b b 5. 6 3m' 1 3m" 9 3r' 1 3r' h 3r 3r q
a, a 0. 0. d, d dm 1 3m 9 3r 1 3r h 3r' 1 3r« q
9 h
a l <*g 0i B h JJ m ....m r . . . . r • — - j - A (r,m)» (4.16) * * i Y 61 6 q
m ,...m r .... r
- 59 -
In the sane way p a r t i a l d i f f e r e n t i a t i o n i n the subspace R w i l l also
add one covariant index to the indices of a f i e l d dependent tensor in
that subspace, only where a li n e a r transformation of coordinate axes
i n that subspace i s concerned.
- 60 -
CHAPTER 5
NEUMANN'S PRINCIPLE
EFFECT OF SYMMETRY ON TENSOR COMPONENTS
5.1 Introduction
The symmetry properties of the space, upon which a tensor i s de
fined can r e s t r i c t the form of that tensor i n that space. These r e s t r i c
tions are dictated by a pr i n c i p l e due to Neumann (see, for example, Nye
1957, p.20, or B i r s s , 1966, p.44). Through Neumann's p r i n c i p l e every
symmetry operation of the space under consideration defines r e l a t i o n s
between the components of the tensor defined on that space, and therefore
reduces the number of i t s independent components.
Some authors have cast doubt on the v a l i d i t y of Neumann's p r i n c i p l e
when applied to some special kind of tensor (viz transport property
tensors) defined over some spec i a l spaces (space composed of magnetic and
geometric subspace, or i n other words,space of a magnetic c r y s t a l ) . The
aim of t h i s chapter i s to ret a i n the v a l i d i t y of Neumann's p r i n c i p l e and
to show that the origin of the p a r t i c u l a r doubts expressed by these workers
i s u n j u s t i f i e d and stems from use of symmetry operations, which are not
exhibited by the space under consideration ( i . e . the space of magnetic
c r y s t a l s ) . To f a c i l i t a t e the argument a short review of the nature of
symmetry operations and the symmetry groups of ordinary c r y s t a l s (ordinary
geometrical space, where each point i s i d e n t i f i e d only by i t s position
vector) i s given f i r s t and Neumann's pr i n c i p l e i s defined. Then the gen
e r a l i z e d symmetry, that i s the symmetry of the space composed of coloured
objects or the space composed of magnetic objects i s discussed. Moreover
i t i s shown that the i d e n t i f i c a t i o n of time inversion operator with the
magnetic moment (or spin) inversion operator leads to incorrect r e s u l t s
(the reason for doubt about the v a l i d i t y of Neumann's p r i n c i p l e by
previous workers).
- 61 -
5.2 Macroscopic symmetry operations and point groups
Symmetry operations are the displacements of an i n f i n i t e r i g i d body
which leave the body i n a position undistinguishable from i t s o r i g i n a l
one. These operations can involve pure translation, pure rotation, pure
r e f l e c t i o n (or inversion), and, i n general, combination of rotation,
inversion and translation. A symmetry operation can be shown by } >
where v denotes a translation and denote rotation or rotation-inversion
operations. Moreover the symmetry operations of a r i g i d body can be
i d e n t i f i e d with the transformation of coordinate axes (section 2.4) and
represented by transformation matrices.
The complete set of symmetry operations which restore a r i g i d body
to i t s e l f forms a group and i s referred to as a space group. That i s :
( i ) the product of every two elements A and B of the set i s i t s e l f
a member of the s e t ,
( i i ) combination (called multiplication) of the elements of the set i s
associative, that i s , for any three elements A, B and C
( A B) C = A( B C) ,
( i i i ) one of the elements of the set i s the identity operator IE, the
symmetry operator which leaves the r i g i d body unmoved and for every
element A of the set /ft E = E A = A,
(iv) for every element A of the set there ex i s t s an element B i n the
se t such that A 8 = B A = IE, the element B i s c a l l e d the
inverse of A and i s denoted by B = A
In dealing with macroscopic properties, however, we are concerned
with the point group, which i s a s p e c i a l sub-group of the space group,
i n which the t r a n s l a t i o n a l part of symmetry operations are completely
suppressed and only rotation and rotation-inversion symmetry operations
are considered.
- 62 -
For i d e n t i f i c a t i o n of the group of symmetry elements of a r i g i d body
only a few of the symmetry elements are necessary. These are ca l l e d the
generating elements of the group and the whole set of elements i n the group
can be generated by forming different combinations of these generating
elements.
5.3 The symmetry of c r y s t a l l i n e l a t t i c e and c r y s t a l classes
The symmetry of a c r y s t a l l i n e l a t t i c e i s described by considering
i t to be a three-dimensional array formed by repetition of physical objects,
atoms or molecules or e l e c t r i c charges and current distributions. The
c r y s t a l structure i s then defined by spe c i f i c a t i o n of the pattern of
repetition plus a complete description of the contents of the repetition
unit (the unit c e l l ) . The existence of the s p a t i a l l a t t i c e (the three-
dimensional array) r e s t r i c t s the number of possible rotation axes to f i v e :
one-, two-, three-, four- and s i x - f o l d axes of rotations. The space group
of symmetry operations of an extended c r y s t a l l i n e l a t t i c e i s then described
i n terms of these rotation and inversion operations, together with the
f i n i t e displacements of the l a t t i c e p a r a l l e l to p a r t i c u l a r directions.
There are 230 di f f e r e n t space groups. Suppressing the tr a n s l a t i o n a l com
ponents of the symmetry operations of the space group r e s u l t s i n 32 possible
point groups for a c r y s t a l l i n e l a t t i c e . Detailed information about these
point groups may be found i n many books (see for example volume 1 of the
'"international tables for X-ray crystallography", Henry and Lonsdale 1965).
5.4 Neumann's p r i n c i p l e and the e f f e c t of symmetry on physical
properties
Neumann's pr i n c i p l e states that (see for example, Nye, 1957, p. 20
or B i r s s , 1966, p.44, or Bhagavantam, 1966, p.72) "the symmetry operations
of any physical property of a c r y s t a l must include the symmetry operations
- 63 -
of the point group of the c r y s t a l " . In other words when by symmetry
considerations two different directions i n a c r y s t a l are exactly the
same, any physical property must have the same c h a r a c t e r i s t i c s along
these two directions. A physical property can have symmetry operations
which do not e x i s t i n the point group of the c r y s t a l . These symmetry
operations are ca l l e d the i n t r i n s i c symmetry of a property. Many workers
have studied the influence of macroscopic s p a t i a l symmetry on the tensor
properties of c r y s t a l s (see, for example, Voigt 1928, Love 1927, Wooster
1938, Mason 1947, Fumi 1952 a,b, F i e s c h i 1957, Nye 1972, B i r s s 1962,
1963, 1966).
The way a symmetry operation manifests i t s e l f i n a physical tensor
property can be shown i n the following way: any symmetry operation, A,
can be represented by a transformation of coordinate axes (with trans
formation matrix ( a ^ ) ) , which dictates the following relations between
the components B. (x) of a tensor i n the old and new coordinate X - • • • • X 1 s
system (equations 2.39 and 2.41).
B i 1 . . . . i ( x k ) = V j . - ' - i j v . . . j ( a k * V { S- l )
1 S 1 1 S S l s
for a polar tensor, and
K * = U J 4 | a, 4 a , B. . . . /no* i r . . . i s k i j l i 1 j 1 i g j s D 1 - . . O g ( a k £ x £ ) (5.2)
for an a x i a l tensor. Then through Neumann's p r i n c i p l e ,
B v . . i ( xk> • Bv . . i ( V - ( 5 - 3 )
I s I s
Substituting for B' (x') from equations(5.1) and (5.2). for polar X . • • • X JC 1 s
and a x i a l tensors, into equation (5.3) we get i r . . i g ~k i j j j i s D s 3 l . . . . j B I
for a polar tensor and
- 64 -
B i . . . i (V = ' a i j l a i , V a i j B
3 l . . . j ( a k * x * } ( 5 ' 5 )
1 s 1 1 s s J l s
for an a x i a l tensor. Equations (5.4) or (5.5) are the a n a l y t i c a l form
of Neumann's p r i n c i p l e , and define relations between the components of
a polar or an a x i a l tensor and thus reduce the number of independent com
ponents. The form of the Neumann's pri n c i p l e for a general (mixed and J . - . . J
weighted) f i e l d independent tensor B -'(x. ) can be e a s i l y deduced X - » • • 1 ic 1 s as
1 1 1 115
B ( X ) = V' r 3x • 3
3x
• j l ' j s h B r a a / * \ 9x 3x 3x 1 3x „V s | 3 x * I .... . .... B I TT X J
ox 3x 3x 3x 1 r \ '
(5.6)
and for a constant tensor (tensor with constant components with respect
to the coordinate axes) i n a cartesian frame of coordinates Neumann's
pri n c i p l e can be written as
B i . . . . i = a i , j / - - a i j B j , . . . j ( 5 ' 7 )
1 s 1J1 s J s J l J s
for a polar tensor and
B J 3 = la. . I a, . ...a. . B. (5.8) i r . . i s iD l l D l V s D l..O s
for an a x i a l tensor. To find the form of a physical tensor property
for a c r y s t a l belonging to a special c r y s t a l system, the symmetry opera
tions i n the corresponding point group w i l l be inserted one afte r another
into the appropriate equation for Neumann's p r i n c i p l e and a l l the re
s t r i c t i o n s found i n t h i s way are put together. To secure the maximum
si m p l i f i c a t i o n i n the form of a tensor, i t i s not always necessary to
employ a l l the symmetry operators of a point group. The set of generating
elements of the corresponding point group s u f f i c e s to serve t h i s purpose
(see, for example, B i r s s , 1966, p.48). Different sets of generating
- 65 -
elements might be chosen for one c r y s t a l c l a s s . The following assemblage
of generating elements used by B i r s s ( B i r s s , 1966, p.48) has been employed
in t h i s t h e s i s :
(0) -[,] 1
0
0
0
1
0
0
0
1
(1) -J] -
-1
0
0
0
-1
0
0
0
-1
(2) • eg -
0
< 4 ) - r s i -y
(6)
(8)
-1
0
0
0
0
- ft] -0
1
0
1
0
-1
0
-1
0
0
-1
0
0
1 J
-1/2 /3/2 0
0
0
-1
(3) • M -
(5)
(7)
(9)
-rg-
- l
o
o
I
0
0
0
0
0
-1
0
0 1
0
1
0
0
0
1
0
-1
0
1
Choosing t h i s assemblage has the advantage that the number of generating
matrices i n any c r y s t a l point group i s small and the re l a t i o n s between
tensor components (obtained from one of the equations (5.4), (5.5),
- 66 -
(5.7) or (5.8) are simple. Sets of generating matrices (chosen from
t h i s assemblage) for the 32 c r y s t a l c l a s s are l i s t e d i n Table 1.
Using equations (5.7) and (5.8), the forms of constant tensor
up to rank four i n every c r y s t a l c l a s s have been worked out and t h e i r
non-zero components have been l i s t e d by B i r s s ( B i r s s , 1966, tables 4a,
4b, 4c, 4d, 4e and 4 f ) .
However, these tables cannot be used to investigate the r e s t r i c
tions on tensor f i e l d and f i e l d dependent tensors such as transport pro
pe r t i e s of sol i d s i n the presence of an external f i e l d , and tensors i n
c r y s t a l s with more complicated symmetry operations, such as magnetic
moment inversion operation. The reason for th i s i s that, to find the
r e s t r i c t i o n s on the form of tensor f i e l d s , equations (5.4) and (5.5)
should be employed instead of equations (5.7) and (5.8). That i s the
symmetry operation acts on the argument of the tensor components as we l l
as on tensor components themselves. The non-zero components of such
tensors have been worked out i n Appendix I and l i s t e d i n Tables A.3-A. 14 i n
that appendix. As for f i e l d dependent tensors and tensors i n the so-
ca l l e d generalized symmetric spaces, the a n a l y t i c a l form of Neumann's
pr i n c i p l e w i l l be derived i n the subsequent sections of t h i s chapter
and t h e i r form w i l l be given i n the Tables A.1-A.14.
5.5 The symmetry of magnetic c r y s t a l s ( I ) :
Heesch-Shubnikov groups
In describing the s p a t i a l symmetry of a c r y s t a l the c r y s t a l l i n e
matter i s conveniently regarded as a composition of si m i l a r groups of
atoms or molecules located on de f i n i t e points, without any reference
to the movements of atoms around thei r equilibrium position or th e i r
magnetic moment direction or the spin direction of the electrons.
- 67 -
TABLE 1
! Symbol of symmetry c l a s s Number Generating matrices
System International of symmetry Generating matrices
Abbreviated F u l l Schbnflies Shubniko\ operations
T r i c l i n i c 1 1 C l 1 1 (0) o
1 -1 c l ( s 2 ) 2 2 o ( 1 )
Monoclinic 2 2 C 2 2 2 o ( 3 )
m m m 2 o ( 5 )
2/m 2 m C2h 2:m 4 a ( 1 ) . o ( 3 )
Orthorhombic 222 222 D 2(V) 2:2 4 (2) (3) a , a
mm2 mm2 C 2 v 2.m 4 (3) (4) a , 0
mmm 222 mmm D 2 h ( V m. 2:m 8 (1) (2) (3) a , a # a
Tetragonal 4 4 C4 4 4 (7) a
4 4 S4 4 4 a ( 8 )
4/m 4 m C4h 4:m 8 a ( 1 ) , a ( 7 >
422 422 D4 4:2 8 (2) (7) a ,o
4mm 4mm C4v 4.m 8 „ ( 4 ) « ( 7 ) o i a
42m 42m 4.m 8 a ,o
4/nunni 422 mnwn °4h m.4:m 16 (1) (2) (7) o ,a ,a
Table 1 continued overleaf
- 68 -
TABLE 1 (Continued)
System Symbol of symmetry c l a s s Number
of symmetry operation
Generating matrices System International
Number of symmetry operation
Generating matrices
Abbreviated F u l l Schbnflies Shubniko\
Number of symmetry operation
Trigonal 3 3 C 3 3 3 a < 6 )
3 3 C 3 i ( S 6 ) 6 6 a ( 1 > , a ( 6 )
32 32 D 3 3:2 6 a< 2>,o ( 6 )
3m 3m C3v 3.m 6 o ( 4 ) , a ( 6 )
3m 32-m °3d 6.m 12 o ( 1 )( o ( 2 ' ( a ( 6 )
Hexagonal 6 6 C 6 6 6 r,<3> „<6> 0 , 0
6 6 C3h 3:m 6 a , a
6/m 6 m C6h 6:m 12 (1) (3) (6) a , a ,o
622 622 °6 6:2 12 (2) (3) (6) o , a , a
6mm 6mm C6v 6.m 12 (3) (4) (6) o , a ,o
6m2 6m2 °3h m. 3 :m 12 o ( 2 \ o ( 5 ) , o ( 6 )
6/nnnin 6 2 2 m m m D6h m. 6:m 24 a ( D a ( 2 ) a(3) J6
Cubic 23 23 T 3/2 12 a ,o
m3 2-3 m M 24 „(1) „(3) (9)
432 432 0 3/4 24 a , a
43m 43m T d 3/4 24 a ,a
m3m 4-3 2 m m °h 6/4 48 „(D J7) (9)
0 , 0 , 0
- 69 -
Such a description i s s t a t i s t i c a l . Therefore i n dealing with proper
t i e s , such as transport properties of c r y s t a l s , which depend on the
dynamical c h a r a c t e r i s t i c s of c r y s t a l s , t h i s aspect of c r y s t a l symmetry
( s p a t i a l symmetry) f a i l s to give correct predictions of the symmetry
r e s t r i c t i o n s on these properties. In determination of the appropriate
symmetry operations for these cases, the symmetry of the individual
properties of groups of atoms (unit c e l l s ) as well as s p a t i a l symmetry
must be taken into account. To do t h i s new coordinate axesr-representing
the appropriate c h a r a c t e r i s t i c s of the unit c e l l , such as the direction
of i t s movements or the direction of i t s magnetic moment etc.r-may be
introduced then the symmetry operators corresponding to the transforma
tion of these coordinate axes can be combined together with the s p a t i a l
transformation operators to give the appropriate symmetry operators.
This has been done by Shubnikov i n 1951 (see Shubnikov and Belov, 1964)
for a simple case where one extra coordinate with two possible values
has been introduced (atoms can be considered as black or white objects
with the new coordinate being t h e i r colour, or i n the case of simple
magnetic structures, the new coordinate can be the direction of the mag
net i c moment of the atoms - p a r a l l e l or a n t i p a r a l l e l to a given a x i s ) .
Then he defined the operation of antisymmetry as the operation which
changes the value of t h i s coordinate (black to white and white to black,
for example). This concept of antisymmetry had been f i r s t introduced
by Heesch (1929 a,b, 1930 a-b) but i t was ignored because i t did not
have any immediate significance i n physics at that time. The existence
of black and white atoms (or atoms with two different directions of
magnetic moment) w i l l destroy the s p a t i a l symmetry of the c r y s t a l ,
unless they are distributed i n a regular fashion;- i n that case, part
of c l a s s i c a l symmetry (symmetry of c r y s t a l with s i m i l a r atoms) w i l l
survive. That i s , introducing the antisymmetry operation w i l l r e s u l t
- 70 -
in new sets of possible point groups and space groups, which are ca l l e d
Heesch-Shubnikov groups. There are three types of Heesch-Shubnikov
point groups, G*, which are usually described as follows:
Type I the c l a s s i c a l point groups (there are 32 of t h i s type),
Type I I : the grey point groups (there are 32 of t h i s type),
Type I I I : the black and white point groups which are also c a l l e d the
magnetic point groups (there are 58 of t h i s type),
the t o t a l number of these Heesch-Shubnikov point groups i s therefore 122.
Type I point groups describe the symmetry of c r y s t a l s composed
of similar groups of atoms ( a l l black or a l l white, or a l l with the mag
netic moments i n one d i r e c t i o n ) , and they do not contain the operation of
antisymmetry, A, that i s they are the c l a s s i c a l or ordinary point groups,
G, i . e . G* = G.
Type I I point groups, describe the symmetry of c r y s t a l s with atoms
of both kinds (black and white or with magnetic moments p a r a l l e l and a n t i -
p a r a l l e l to a given direction) present at each s i t e simultaneously, so that
any operation of a c l a s s i c a l point group, G, leaves the fourth coordinate
(the colour or the direction of the magnetic moment) unchanged, and the
operation of antisymmetry, A, times any operation of G changes two d i f
ferent values of the fourth coordinate into each other, thereby leaving
i t unchanged. Thus G' «= G + A G . That i s , a grey point group contains
twice the number of symmetry elements as the number of symmetry elements
i n the correspond point group, with the antisymmetry operation, A, as an
element on i t s own.
In the type I I I Heesch-shubnikov, the antisymmetry operation, A,
appears only i n combination, A (R, with the elements, R, of the cor
responding c l a s s i c a l point group, G, i . e . G' = H + A(G-H), where H i s a
halving subgroup of the c l a s s i c a l point group G.
- 71 -
The generating elements of the type I Heesch-Shubnikov point
groups are exactly those given in Table 1. For the type I I Heesch-
Shubnikov point groups, the antisymmetry operation, A, must be added
to the generating elements of the corresponding c l a s s i c a l point group
to form the s e t of t h e i r generating elements.
The i d e n t i f i c a t i o n of the elements i n each of the 58 type I I I
Heesch-Shubnikov point groups has been done by Tavger and Zaitsev (see
Tavger and Zaitsev, 1956) and by other workers and the generating elements
for these point groups are given in Table 2 (see B i r s s , 1966). Here the
underlining of a symmetry operator indicates multiplication by the a n t i
symmetry operator, /A, i . e . 6^ = A 6^
Heesch-Shubnikov (or coloured) space groups can be defined i n
the following way:
Type I the c l a s s i c a l space groups (there are 230).
Type I I the grey space groups (there are 230).
These two kinds of space groups can be formed by combining the
corresponding point group together with the t r a n s l a t i o n a l sub-group.
Type I I I black and white space groups, which are derived from
the c l a s s i c a l space groups (there are 674).
Type IV Heesch-Shubnikov space group, G', which i s another
black and white space group and i s based on a black and white Bravais
l a t t i c e and consists of a l l the operations of a c l a s s i c a l space group,
G, together with an equal number of operations that involve the a n t i
symmetry operation, A, i . e . G* = G + A | IE | t^JG (there are 517 of
these kind of space groups). The black and white Bravais l a t t i c e s were
derived by Belov et a l (1955), and the black and white space groups were
f i r s t derived by Zamorzaev i n 1953 (see Belov et a l . , 1957; Shubnikov
and Belov, 1964; Opechowski and Guccione, 1965; Bradley and Cracknell,1972).
- 72 -
TABLE 2
System
• • • 1 Symbol of Symmetry c l a s s and magnetic
Symbol of c l a s s i c a l
i Generating matrices of
group G' System point group G' sub-grou 9 H Generating Additional International Shubnikoy International
i Shubnikov matrices of
sub-group H generating
matrix
r r i c l i n i c I \ 1 1 a ( 0 ) <L(1)
Monoclinic 2 2 1 1 a ( 0 ) 1 ( 3 )
m m 1 1 a ( 0 ) 1 ( 5 )
2/m 2_:m T 2 o ( 1 ) £ ( 3 )
2/m 2:m 2 2 a ( 3 ) o ( U
2/m 2 m m m a ( 5 ) 2.(1)
Orthorhombic 222 2:2 2 2 a ( 3 ) o ( 2 )
mm 2 2i.m 2 2 a ( 3 ) o ( 4 )
2mm 2.m m m a ( 5 ) £ ( 4 >
.mmm m. 2 :m 2/m 2:m o ( 1 ) , o ( 3 ) 1 ( 2 )
mmm m. 2:m 222 2:2 (2) (3) a , a
mmm m. 2 :m mm 2 2.m a ( 3 ) , o ( 4 ) 2.(1)
Tetragonal A 4 2 2 a ( 3 ) 1 ( 7 )
4 4 2 2 a ( 3 ) S L ( 8 >
4/m 4:m 2/m 2:m a ( 1 ) , o ( 3 )
4/m 4:m 4 4 a ( 7 ) £ ( 1 )
4/m 4:m 4 4 a ( 8 ) 2.(1>
422 4:2 222 2:2 (2) (3) jo , a i
/Table 2 continuec overleaf
- 73 -
Table 2 (Continued)
System Symbol of Symmetry c l a s s and magnetic
Symbol of c l a s s i c a l
Generating group
matrices of G' System point group G' sub-grou P H Generating Additional
International Shubniko\ International Shubniko\ matrices of sub-group H
generating matrix
Tetragonal 422 4:2 4 4 (7) 0 £ ( 2 )
4mm 4.m mm2 2.m (3) (4) a r a (7) g.
4mm 4.m 4 4 (7) a
(4) g_
42m 4_.m 222 2:2 (2) (3) a , a (8)
4m2_ 4.m mm2 2.m (3) (4) a r o (8)
42m 4.m 4 4 (8) a
(2) a_
4/mmra m. 4 :m mmm m. 2:m (1) (2) (3) a > a a (7) a.
4/mmm m.4:m 4/m 4:m (1) (7) o f a (2) a.
4/mmm m. 4 :m 422 4:2 (2) (7) a » a (1) a.
4/mmm m.4:m 4mm 4.m (4) (7) 0 / 0
(1) SL
4/mmm m.4:m 42m 4.m (2) (8) a * a (1) _o
Trigonal 3_ § 3 3 (6) a
(1) a;
32 3:2 3 3 (6) a
(2) a;
3m 3.m 3 3 (6) a
(4)
3m 6.m 3 6 o » a (2) a.
3m 32 3:2 (2) (6) 0 / 0
(1) i l
3m 6.m 3m 3.m (4) (6) a # o (1) a.
/Table 2 continued overleaf
- 74 -Table 2 (Continued)
Symbol of Symmetry cl a s s and magnetic
Symbol of c l a s s i c a l
Generating matrices of group G*
System point group G' sub-group H Generating j 1 additional International Shubnikob International Shubnikov matrices of g sub-group H
enerating matrix
Hexagonal 6 6 3 3 (6) 0 £ ( 3 )
6 3 :m 3 3 (6) a
(5)
6/m 6 :m 3 6 ! 1 1
o , 0 (3)
6/m 6:m 6 6 (3) (6) 0 , 0
(1) _a
6/m 6 :m 6 3 :m (5) (6) o i a (1)
a.
6 2 2 6:2 32 3:2 (2) (6) o f a (3)
a.
622 6:2 6 6 (3) (6) O f 0
(2) a.
6mm 6.m 3m 3.m (4) (6) a i a
(3)
6mm 6.m 6 6 (3) (6) a t a (4)
SL
62m m. 3:m 32 3:2 (2) (6) a , a (5) o;
6m2 m. 3:m 3m 3.m (4) (6) a , a £ ( 5 >
6m2 m. 3 :m 6 3 :m o ( 5 ) , o ( 6 )
6/mmm m.6:m 3m 6.m o ( i ) , J 2 y 6 ) £ l 2 >
6/mmm m. 6 :m 6/m 6 :m (1) (3) (6) a , a p (2)
6/mmm m.6:m 622 6:2 (2) (3) (6) a ,o f o (1) 0
6/mmm m. 6 :m 6mm 6.m (3) (4) (6) a , a ,a o ( 1 )
6/mmm m.6:m 6m 2 m. 3 :m (2) (5) (6) o / a ,o «i) £
Cubic m3 6/2 23 3/2 (3) (9) a , o (1) 0,
432 3/4 23 3/2 (3) (9) o , a (7) £
43m 3/4 23 3/2 (3) (9) a , a (8) £
m3m 6/4 m3 6/2 (1) (3) (9 a , a ,a (7) £
m3m 6/4 432 3/4 (7) (9) 0 f 0
(1) 0.
m3m 6/4 432 3/4 (0) (9) a ,o (1) £
}
- 75 -
The symmetry of most of the magnetic materials (diamagnetic, para
magnetic, ferromagnetic, ferrimagnetic) can be described by the three
types of Heesch-Shubnikov point groups, when the antisymmetry operator
i s i d e n t i f i e d with the magnetic moment (or spin) inversion operator (see,
for example, Mackay, 1957). In that case a grey point group w i l l contain
the operation of magnetic moment inversion, A, on i t s own; therefore
i f a spontaneous magnetic moment were to develop at any point within the
c r y s t a l , described by that point group, the presence of /A would require
that an equal magnetic moment should appear at the same point in the
opposite direction. Therefore the symmetry of c r y s t a l s with magnetic
ordering cannot be described by a grey point group, but the symmetry of
diamagnetic materials (in which the atoms or ions which constitute the
c r y s t a l have zero magnetic moments) and paramagnetic materials i n the
absence of an external magnetic f i e l d (where the orientations of the mag
netic moments of atoms or ions are random and the c r y s t a l can be con-*
sidered as being invariant under the operation /A) can be described by
grey groups.
The symmetry of ferromagnetic c r y s t a l s (which consist of atoms or
ions with magnetic moments aligned along certain directions i n the
absence of an external magnetic f i e l d ) and ferrimagnetic c r y s t a l s (with
some of the magnetic moments p a r a l l e l to a p a r t i c u l a r direction and the
r e s t a n t i - p a r a l l e l to that direction in the absence of an external mag
net i c f i e l d ) and some of the antiferromagnetic materials which possess
a spontaneous net magnetic moment can be described by type I and type I I
Heesch-Shubnikov point group, where the magnetic moment inversion as a
symmetry element on i t s own i s absent. Application of these groups to
magnetic structure determinations has been done by several groups of
workers i n t h i s f i e l d (see, for example, Donnay et a l , 1958 and
l e Corre, 1958).
- 76 -
5.6 The symmetry of magnetic c r y s t a l s ( I I ) ;
Generalized symmetry
The symmetry of a large number of magnetically ordered materials
can be described by using the Heesch-Shubnikov groups. However, there
are some other magnetic materials whose symmetry cannot adequately be
described by these groups. Examples of these are materials with h e l i c a l
or conical antiferromagnetic structures. The ordering of magnetic moments
i n these structures i s such that there are more than two possible direc
tions of magnetic moment present i n the c r y s t a l . Therefore further
generalization of c l a s s i c a l point groups i s necessary. One possible
generalization i s to consider the polychromatic point groups and space
groups. But th i s cannot be used as a general approach, since the poly
chromatic groups can only describe the symmetry of magnetic materials where o o
the angle between two successive spins i s a discrete value (120 , 90 or
60°) which i s not always the case (see, for example, Cracknell 1975).
More comprehensive generalization can be achieved by using the spin
groups developed by L i t v i n (1973). The idea of spin groups i s closely
related to the generalized magnetic groups defined by Naish (1963) and
Kitz (1965), and i s c a l l e d 'spin space' groups by Brinknlan and E l l i o t
(1966). A symmetry operation i n the generalized magnetic group, as i t i s
c a l l e d i n t h i s t h e s i s , (or spin group) can be denoted by { R |
where the rotation on the far l e f t side i n the bracket, IR , acts only i n P
spin space, that i s , on the components of the spins, and the rotation, R., and translation, \v. , act only i n the geometrical space on the co-i ip ordinates of the atoms. I n spec i a l cases i t i s possible to have some
symmetry operations that involve one rotational operation IR^ for both
the physical space coordinates and the spin space coordinates. Such a
symmetry operation can be denoted by { R.| R | v }. Further s p e c i f i c a t i o n
- 77 -
of the generalized magnetic groups can be found i n a r t i c l e s by
Opechnowski (1971) and Opechowski and Dreyfus (1971).
5.7 Time inversion operation and magnetic groups
I t has been shown (Section 5.5) that by i d e n t i f i c a t i o n of the mag
netic moment inversion operator with the antisymmetry operator i n Heesch-
Shubnikov groups the structure of most magnetic materials can be put on
a group th e o r e t i c a l basis. However, since the operation of time inversion,
according to standard assumptions of the electromagnetic theory, has the
eff e c t of reversing the direction of magnetic moment (or s p i n ) , i t has
become a common practice to identify the time inversion operator, 9 ,
with the magnetic moment inversion operator (see, for example, Opechowski
and Guccione 1965, Tavger and Zaitsev 1956, B i r s s 1966, Zocher and Torok
1953, Dimmock 1963 a,b, Dimmock and Wheeler 1962 a,b, and Dimmock and
Wheeler 1964). In t h i s section i t w i l l be pointed out that t h i s i d e n t i f i
cation i s not always correct and sometimes leads to incorrect r e s u l t s .
Time inversion operation has actually the ef f e c t of reversing the
direction (or changing the sign) of every dynamic property which depends
on time to an odd degree, i . e . time-antisymmetric properties including
magnetic moment. Thus i n adopting t h i s operation as a symmetry operation
of a magnetic c r y s t a l s p e c i a l care must be taken to make sure that there
i s no other time-antisymmetric property present i n the c r y s t a l . I n other
words when a time-antisymmetric property, such as one of the transport
properties ( l i k e current density), of a c r y s t a l i s under consideration
adopting time inversion operator imposes some r e s t r i c t i o n on the form
of property which i n actual f a c t should not be there. The way i n which
i d e n t i f i c a t i o n of the magnetic moment inversion operator with time
inversion operator leads to incorrect r e s u l t s i n the study of current
- 78 -
density i n a magnetic c r y s t a l i s disccussed i n Section 6.3.1.
The problems that a r i s e as a r e s u l t of i d e n t i f i c a t i o n of the
magnetic moment inversion operator with the time inversion operator have
been noticed by several workers: B i r s s (1966) suggests that the problem
may a r i s e because Neumann's pr i n c i p l e does not hold for transport proper
t i e s under the symmetry operations i n space-time. This cannot be true
because Neumann's pri n c i p l e i s axiomatic i n i t s nature. Pantula and
Sudarshan (1970 a,b) have reached the conclusion that i n the non-
equilibrium steady state the c r y s t a l does not possess time-inversion sym
metry. But then to remove the problem caused by time inversion they have
only used the sub-group of time invariant operations of the appropriate
group (Bhagacantam and Pantula 1967).
In addition to leading to wrong r e s u l t s , i n some cases, the time
inversion operator, 0, proves to be inadequate to be i d e n t i f i e d with mag
netic moment inversion operator i n the sense that 6 can only provide two
senses, while many magnetic c r y s t a l s with complex magnetic ordering have
more than two possible directions of magnetic moment (Section 5.6, thus
more complex rotation-inversion operations on the spin (or magnetic moment)
rather than j u s t an inversion are needed to s a t i s f y symmetry requirements
(see, for example, Cracknell 1975, L i t v i n 1973 and Brinkman and E l l i o t
1966 a,b). In works on the symmetry of these magnetic materials some
authors have considered time-inversion operator i n combination with a
f u l l rotation group (including an inversion operation) on the spin arrange
ment (see, for example, L i t v i n 1973). Thus when using the generalized
magnetic groups (or spin groups) worked out in t h e i r a r t i c l e s , i t must
be made quite c l e a r that there i s no time-antisymmetry property other
than magnetic moment present i n the calculations.
- 79 -
The symmetry and antisymmetry r e s t r i c t i o n s on the form of the
transport tensors for materials with simple magnetic ordering have
recently been established (Pourghazi, Saunders and Akgoz 1976) using the
transformation law for f i e l d dependent tensors (equation 3.10). For
reasons mentioned above, the use of the time inversion operator 6 , i n
that work has been avoided and the magnetic moment inversion operator
has been employed.
5.8 Generalized Neumann's pr i n c i p l e
The e f f e c t of symmetry operations, corresponding to the transforma
tion i n physical space, on tensor properties of materials has been formu
lated through Neumann's p r i n c i p l e (see Section 5.4). Equations (5.7)
and (5.8) have been used by several authors to determine the symmetry
r e s t r i c t i o n s on the form of f i e l d independent tensor properties of
materials (see, for example, B i r s s 1966, Nye 1972, Bhagavantam 1966, and
Wooster 1973). However in dealing with f i e l d dependent tensor properties
of materials, when a symmetry operation corresponding to the transforma
tion of f i e l d components i s under consideration, there i s some ambiguity
i n the v a l i d i t y and form of the Neumann's pr i n c i p l e (see B i r s s 1966).
We believe that t h i s d i f f i c u l t y i s due to the incorrect i d e n t i f i c a t i o n
of the magnetic moment inversion operator with the time inversion operator,
(see Section 5.7). Furthermore, since the transformation laws (3.10 and
3.11) should be used i n constructing the a n a l y t i c a l form of Neumann's
pr i n c i p l e for f i e l d dependent tensors, rather than the transformation
laws (2.39), (2.41), Neumann's principle w i l l have a s l i g h t l y d i f f e r e n t
a n a l y t i c a l form for f i e l d dependent tensors. The transformation law for
a f i e l d dependent tensor with t contravariant and s covariant indices
(t + s = n) and of weight to i n subspace v (the physical space, for
- 80 -
example) and v contravariant and u covariant Indices (v + u = Z) and of
weight in subspace M (the magnetic subspace or spin space for example)
(see Section 3.7) can be written as (equation 3.11): v t
B g i g v j l - t m ....m r • • • • r
i i k. k 1 u 1 s m ....m r . . . . r
(m'a , r - b ) =
u
g dm
U l j 3r
3m'i 3m'k
«1 3r • • • • 3r
W2 g l 3m* 3m' dm
Bu j l 3m 3r'
3m a v M 3m 3m1 3m' 3r
«1 *s a a Y j Y t / a b \ 3r 3r nm m r r I 3m' i|> . 3r' u \
3* 3r' m ....m r , . . . r \ /
3r'
3r
(5.8)
Now through Neumann's p r i n c i p l e we can write
91 g v j l j t 91 9 v j l j t „,m ,...m r . . . . r , ,a ,b. m ....m r . . . . r ,_a b B i i k k ( m ' r > = B i , i k, k ( m ' r > l u l s _ l u l s m ....m r . . . . r m ....m r . . . . r
(5.9)
m 9 1 ^ j l j t a h Substituting for B m r r (m' , r' ) from equatifcn (5.9)
m i k, k _ u 1 s .m r .... r
into equation (5.8) we get:
- 81 -
B 1 j 1 ... .m r • • • • IT
i k 1 u .... m u r • • • •
3mg
9m'1 3 r'
3r'
3r
6.
(ro , r ) =-
3m ,g l
3r k l
ar' 1
dm
3r
3r*
_m
m
% Bl 3m' 3m
a. 3m
a n
3m . 1
v '1 .m r r a.
3m
3m' u
3r'
3r
u 1 .m r
.r t / 3 m ' a * 3 1 1 m * ~ °s\3m* 3
• r
(5.10)
This i s the most general, a n a l y t i c a l form of Neumann's p r i n c i p l e for f i e l d
dependent tensors. Equation (5.10) can be equally w e l l expressed i n terms
of operator form of the transformation matrices, and 9 (~r) a n d ("^r) ' v 3mJ J N 3r J ' i . e . { R I R } , where I R acts on the coordinates i n M subspace and (R m r m r r
acts on the coordinates i n V subspace. That i s for an unweighted tensor
we can write:
g l g v j l j t m ....m r . . . . r , a D , B i , i k, k ( m ' r } =
1 u 1 s m ....m r . . . . r
a l a v Y l Y t = { R | R } B™ — B r » " r ({ <R } t R } r ) . (!
m r B i Bu 61 6 s m r
m ....m r . . . . r
In the spec i a l case where there i s orthogonal l i n e a r transformation of
coordinates i n both subspaces to be considered, equation (5.10) can be
written as:
- 82 -
p q ) = | R I | R J ( R )
1 m' ' r m B ( R ( R ) ( R ) r. (m ,r m m m n 1 1 n n
( R ) 5.12) R B m ru m • m m a* B B au 1 1
where ( R ) and ( R ) are the corresponding elements i n the transforma-m . r i a DB
tion matrices, and p and q are constants and have values zero or one
according to the weight of tensor i n each subspace ( i . e . p = 0 i f the tensor
has even weight, that i s i f i t i s polar, i n M and p = 1 i f the tensor has
odd weight, that i s i f i t i s a x i a l i n M).
Equations (5.10), (5.11) and (5.12) are di f f e r e n t a n a l y t i c a l forms
of Neumann's p r i n c i p l e for f i e l d dependent tensors defined upon spaces
where the transformation operators can be divided into two parts, each
acting on the coordinates i n one subspace only. But there are some cases
where the transformation operator cannot be divided i n t h i s fashion (see
Section 3.8) because of the dependence of the coordinates of the f i e l d
subspace on the coordinates of geometrical subspace. This w i l l be
investigated i n the next section.
5.9 Neumann's pr i n c i p l e for symmetry operations which act on the
coordinates of both subspaces
To construct the a n a l y t i c a l form of Neumann's p r i n c i p l e for s p a t i a l
symmetry operations which act on the coordinates i n the f i e l d subspace,
the appropriate transformation law should be chosen from one of the
equations (3.18), (3.19) and (3.20).
5.9.1 Example
As an example consider a space composed of two subspaces - f i e l d
- 83 -
subspace and geometrical subspace. In that space a f i e l d dependent tensor
property B can now be considered which acts l i k e a scalar function (zero
rank tensor f i e l d ) i n the f i e l d subspace. Under a transformation of
coordinates | A } i n geometrical subspace, Neumann's p r i n c i p l e can have one
of the following forms depending on the nature of coordinates i n the f i e l d
subspace:
(a) when the coordinates i n the f i e l d subspace behave l i k e (covariant)
vectors i n geometrical subspace ,
B , . 1 - m f • \ r |= • • • • — k l M a r ' 1 a 3r* / * I *1 r . . . r \ / 3r 3r 3r
6 s Y l Y t • B* ...r. (m , r B ) , (5.13) k o4 6 a „ , s 1 s 3r' r r
where r i s the a.. element of the transformation matrix (A) 3 r . j ID
(equation 5.13) can equally well be written for f i e l d s which are contra-
variant vectors i n geometrical subspace, and also for weighted f i e l d
vectors). For li n e a r orthogonal transformation of coordinates (A)
equation (5.13) can be written as:
B (± a. m » a. r.) = a, ...a, B (ni » r„) , V'-rk 1 0 1 " ]B 6 Vl Vs V " r Y a 6
I s I s
(5.14)
where ± signs i n the bracket on the £.h.s. of equation correspond to
whether the f i e l d i s a polar or an a x i a l vector.
(b) When the coordinates i n the f i e l d subspace are second rank tensors
i n geometrical subspace ,
- 84 -
3\ j a 1 9r' 3r B m a. a 1 1 2 3r
s p B t r ) a a 1 2 s s 9r 3r (5.15)
where m . are the components of the f i e l d tensor along the coordinate 1 2
axes i n the geometrical subspace (the f i e l d tensor can have contravariant
indices as w e l l ) .
When a lin e a r orthogonal transformation of coordinates, denoted
by A - where there i s no difference between covariant and contravariant
indices - i s considered equation (5.15) can be written as
B . (± a a m , a r ) = V " r k Vl 1 2 a 2 a i a 2 j B B
1 s
a . .a. ^ B (m , r Q ) , (5.16) 1 1 s s Y, Y 1 2 1 s
where ± signs i n the bracket on the left-hand side of equation (5.16)
refer to the f i e l d tensor being either a polar or an a x i a l tensor.
(c) When the coordinate in the f i e l d subspace act l i k e tensors of
ranks (u) higher than two ,
j j j a u p B m p a 1 • u u •
j 1 s 6 i r " ) m B 1 s
(5.17)
- 85 -
The f i e l d tensor can also have contravariant indices. When a l i n e a r
transformation of coordinates axes i n geometrical subspace i s considered
equation (5.17) can be written as
B (± a. a. m > a _ r Q ) = r. . . . r . i , a i a a,...a j£ B k, k 1 1 u u l u 6 1 s
a, ...a, B (m , r Q ) , (5.i8) k,Y, k Y r . . . r a,.. .a ' B 1 1 s s y. y 1 u
1 s
where again the ± signs, in the bracket on the l e f t hand side of equation
(5.18), r e f e r to f i e l d tensor being a polar or an a x i a l tensor.
Using equation (5.14), (5.16) and (5.18), the r e s t r i c t i o n s imposed
by the ordinary c r y s t a l groups ( i . e . type I Heesch-Shubnikov groups, where
only s p a t i a l symmetry operations are present) on the f i e l d dependent
tensor properties, which have been defined through transformation laws
(3.18), (3.19) and (3.20), have been investigated i n Appendix I . The
r e s u l t s for f i e l d dependent tensors up to rank four in geometrical sub-
space, with zero and f i r s t and second rank f i e l d tensors are l i s t e d i n
Tables A3
5.9.2 The general case
We may generalize the concepts of Section 5.9.1 to write Neumann's
pr i n c i p l e for f i e l d dependent tensor properties which have ranks higher
than zero i n the f i e l d subspace, where f i e l d components are themselves
tensors i n the geometrical subspace. To do t h i s the symmetry operations
{ B } which act on the f i e l d components can be derived from equation m (3.23), as:
OD
1 i 3r u
{ R } = m 1 (5.19)
3 r' u
where u i s the rank of the f i e l d tensor i n the geometrical subspace, and i l a l
3r /3r' i s an element of the transformation operation { jO which acts
on the coordinates i n the geometrical subspace. Now equation (5.10) can
be used as Neumann's p r i n c i p l e .
There i s also the p o s s i b i l i t y that the symmetry operations i n the
f i e l d subspace can be a combination of the operations of the form derived
from equation (5.19), and other operations which are defined only i n the
f i e l d subspace, where again Neumann's pr i n c i p l e w i l l have the form of
equation (5.10) and the geometrical part of symmetry operation should be
worked out from equation (5.19).
In addition to the r e s t r i c t i o n s imposed by the symmetry operations
of the space under consideration, a f i e l d dependent tensor may exhibit
i n t r i n s i c symmetry. I n other words the physical nature of the property
may dictate certain r e s t r i c t i o n s , on the f i e l d dependent tensor com
ponents. An example of i n t r i n s i c symmetry i s provided by Onsager's
theorem (Onsager 1931 a,b) which i s based on thermodynamical arguments
(for further d e t a i l s see any textbook on i r r e v e r s i b l e thermodynamics such
as, for example, de Groot 1951). This theorem has been used to find the
r e s t r i c t i o n s on the form of transport properties i n the next chapter.
- 87 -
CHAPTER 6
FIELD DEPENDENT TENSOR PROPERTIES OF CRYSTALS
6.1 Introduction
The concepts of the l a s t chapter reveal that when a f i e l d -
dependent physical property, such as one of the transport properties, of
a c r y s t a l i s under consideration the usual form of Neumann's pri n c i p l e
(equations 5.4 to 5.8) f a i l s to give correct r e s u l t s as far as the symmetry
operations of the f i e l d coordinates are concerned. In t h i s chapter the
magnetoresistivity tensor for magnetic materials i s considered. F i r s t
the ways other workers have tackled the problem of finding the e f f e c t that
symmetry operations acting on the magnetic f i e l d coordinates can have on
the magnetoresistivity tensor and the discrepancy i n the r e s u l t s obtained
are discussed. Then making use of the transformation law of f i e l d depen
dent tensors (equation 3.10) and the generalized form of Neumann's
prin c i p l e (equation 5.10) the r e s t r i c t i o n s on the form of the magneto
r e s i s t i v i t y tensor have been found.
Furthermore, the forms of the magnetothermoelectric power tensor Q i j ^ f ° r e a < * °^ t* i e m a 9 n e t i c point groups have been obtained using the
same method as for the magnetoresistivity tensor. I n addition the way
that the permittivity tensor behaves as a f i e l d dependent tensor i n a
magnetic material i s described.
6.2 The conductivity of a c r y s t a l ->
When an e l e c t r i c f i e l d , described by a polar vector E, acts i n -*•
a conductor, an e l e c t r i c current of density J flows. In an iso t r o p i c •+ -*• medium J i s p a r a l l e l to E and the relationship between t h e i r components
- 88 -
i s given by (Ohm's law):
J OE , (6.1)
where a i s ca l l e d the conductivity of the medium. That i s
Jl = o E 1 , J 2 = 0 E 2 , J 3 = a E 3 (6.2)
where E^, E^, E^ and J ^ , J ^ , are the components of the vectors E and -y
J i n a rectangular cartesian system of coordinate axes, Ox^, Ox^, Ox^.
Now when the medium i s a c r y s t a l the conductivity i s not necessarily
i s o t r o p i c , the vectors E and J need no longer be p a r a l l e l ; the conduc
t i v i t y cannot be represented by a s c a l a r . However usually each ompoonent -> -v
of J i s s t i l l l i n e a r l y related to a l l three components of E, and Ohm's law (equation 6.1) may now be represented by
J l = °11 E1 + °12 E2 + °13 E3 '
J„ = o_, E, + CJ E + a E_ , (6.3) 2 21 1 22 2 23 3
J = o„. E. + CJ E„ + cu_ E„ 3 31 1 32 2 33 3
Or
J . = o i j E j , (6.4)
where each a^ i s .constant. Thus according to the quotient law (see
Section 2.12), i s a second rank polar tensor ( f i e l d independent).
The symmetry operations of the c r y s t a l impose r e s t r i c t i o n s on the form
of (see Section 5.4). Moreover there e x i s t s a r e s t r i c t i o n imposed
by the i n t r i n s i c symmetry dictated by Onsager rec i p r o c i t y r e l a t i o n
which can be shown by
- 8 9 -
oj . = a. . 6 . 5 )
The s p a t i a l symmetry r e s t r i c t e d form the c o n d u c t i v i t y tensor o i s w e l l
known, see f o r example, Nye ( 1 9 7 2 ) , B i r s s ( 1 9 6 6 ) and Bhagavantam ( 1 9 6 6 ) .
6 . 3 The magnetoconductivlty p r o p e r t y of a c r y s t a l
I f one now considers e i t h e r a c r y s t a l t h a t e x h i b i t s spontaneous
magnetic o r d e r i n g or a nonmagnetic c r y s t a l t h a t i s subjected t o an
ex t e r n a l magnetic f i e l d , the r e l a t i o n s h i p between J and E i s s t i l l l i n e a r
but the c o n d u c t i v i t y o _. must be replaced i n equation ( 6 . 4 ) by a tensor
a ( B ) which depends on the d i r e c t i o n o f the magnetic i n d u c t i o n v e c t o r , -> B , so t h a t
J. = o. . ( B ) E ( 6 . 6 ) i 13 j
-y
where (B) i s c a l l e d the magnetoconductivity tensor, and i s a second
rank p o l a r (magnetic) f i e l d dependent tensor. Ohm's law (equation 6 . 6 )
can equally w e l l be expressed i n the f o l l o w i n g form:
E = p . (?) J. , ( 6 . 7 ) i iD 3
where ( B ) i s the m a g n e t o r e s i s t i v i t y tensor and i s the inverse o f the
magnetoconductivity tensor. I n t h i s case the d e t a i l s o f the a p p l i c a t i o n
of Onsager's theorem (equation 6 . 5 ) and Neumann's p r i n c i p l e (see
Section 5 . 8 ) have t o be re-examined. I f a c r y s t a l i s subjected t o a
magnetic f i e l d H, one t h a t may be an e x t e r n a l f i e l d or may a r i s e as an
i n t e r n a l f i e l d i n the c r y s t a l , i t i s then p o s s i b l e t o show (see Onsager
1 9 3 1 a,b and DeGroot 1 9 5 1 ) t h a t equation ( 6 . 5 ) has t o be replaced by
o .(B) = a..(-B) , ( 6 . 8 ) i ] Di
- 90 -
and as f o r the m a g n e t o r e s i s t i v i t y tensor, the Onsager r e c i p r o c i t y
r e l a t i o n w i l l d i c t a t e
P. . (B) = p . . (-B ) . (6.9) ID 33-
I n o b t a i n i n g the r e s t r i c t i o n s imposed by the symmetry
p r o p e r t i e s of c r y s t a l s on the forms of the magnetoconductivity and the
m a g n e t o r e s i s t i v i t y tensors through Neumann's p r i n c i p l e there has been
some controversy among d i f f e r e n t workers (see B i r s s 1963, 1966, K l e i n e r
1966, 1967, 1969, Crackne l l 1969, 1973, 1975 and AkgSz and Saunders 1975a).
I n the next s e c t i o n we s h a l l give a short review of each o f the methods
adopted,by d i f f e r e n t workers and comment on them and a t the end we s h a l l
g i v e our own method.
6.4 Magnetic symmetry r e s t r i c t i o n s on the forms o f the magneto
r e s i s t i v i t y and the magnetoconductivity tensors •
The symmetry o f a magnetically ordered c r y s t a l or o f a nonmagnetic
c r y s t a l i n an e x t e r n a l magnetic f i e l d H can be described by some magnetic
p o i n t group G' which can be w r i t t e n as (see Sections 5.5, 5.6, 5.7)
G' = H + /A (G - H)
where G i s the corresponding c l a s s i c a l ( s p a t i a l ) p o i n t group and H i s one
of i t s subgroups and /A i s the anti-symmetry operator, which can be
i d e n t i f i e d w i t h the magnetic moment (spin) i n v e r s i o n operator. Previous
workers have i d e n t i f i e d the magnetic moment i n v e r s i o n operator w i t h the
ti m e - i n v e r s i o n operator. This together w i t h the f a i l u r e t o t r e a t the
magnetoconductivity and the m a g n e t o r e s i s t i v i t y tensors as field-dependent
tensors (see Akgdz and Saunders 1975a) has l e d the procedures adopted
- 91 -
by these workers ( p r e s c r i p t i o n s A, B and C) t o wrong r e s u l t .
6.4.1 P r e s c r i p t i o n A - By Bi r s s
I n t h i s p r e s c r i p t i o n B i r s s f o l l o w s previous workers i n the f i e l d
(Juretschke 1955, Mason e t a l 1953, and others) t o express the components -*•
o f the m a g n e t o r e s i s t i v i t y tensor p ^(B) as ser i e s expansions i n ascending
powers of H, the magnetic f i e l d . That i s (Birss 1966, p.112, equation 3.26)
E = p J. + R. ., J. H + (6.10) i i ] ] i ] k j k
Now p , R . . I ... are second, t h i r d , ... rank f i e l d independent tensors, i j i j k To correspond w i t h these expansion c o e f f i c i e n t s , t h e usual experimental
p r a c t i c e i s t o r e s t r i c t measurement t o the low magnetic f i e l d r e g i o n , so
t h a t terms higher than second order can then be neglected.
I n d e a l i n g w i t h the anti-symmetry o p e r a t i o n , B i r s s f i r s t i d e n t i f i e s
t h i s operation w i t h the t i m e - i n v e r s i o n o p e r a t i o n on the ground t h a t since
e l e c t r o n s p i n i s associated w i t h angular momentum which i s a time-
antisymmetric p r o p e r t y , t i m e - i n v e r s i o n o p e r a t i o n reverses the d i r e c t i o n o f
sp i n and t h e r e f o r e the d i r e c t i o n o f magnetic moment ( f o r more d e t a i l see
B i r s s 1966, pp.74-78). Then he s t a t e s t h a t f o r c r y s t a l s where t r a n s p o r t
p r o p e r t i e s are present there e x i s t s an i n t r i n s i c a l l y p r e f e r r e d d i r e c t i o n
o f time, t h a t i s , t r a n s p o r t p r o p e r t i e s are time-antisymmetric) the a p p l i c a
t i o n o f Neumann's p r i n c i p l e y i e l d s the erroneous r e s u l t t h a t a l l t r a n s p o r t
e f f e c t s are p r o h i b i t e d f o r non-magnetic ( i . e . time-symmetric) c r y s t a l s .
Therefore he concludes t h a t "Neumann's p r i n c i p l e cannot be a p p l i e d i n
space-time." Then B i r s s gives the f o l l o w i n g p r e s c r i p t i o n ( r e f e r r e d t o as
p r e s c r i p t i o n A by K l e i n e r 1966) f o r f i n d i n g the r e s t r i c t i o n s on the forms
o f t r a n s p o r t p r o p e r t y tensors, based on what has been s a i d above.
- 92 -
( i ) Use Onsager's theorem, i . e . ~ P j j / ~ H * '
( i i ) use Neumann's p r i n c i p l e only f o r the u n i t a r y elements of the
magnetic p o i n t group (elements which do not co n t a i n t i m e - i n v e r s i o n
operators i n any form),
( i i i ) ignore the a n t i u n i t a r y elements o f the magnetic p o i n t group
(elements which con t a i n t i m e - i n v e r s i o n o p e r a t o r ) .
The o b j e c t i o n s on t h i s procedure are as f o l l o w s :
(a) Working w i t h the low f i e l d c o e f f i c i e n t s (which are f i e l d -
independent tensors) has some disadvantages, which are described i n
Section 3.3.
(b) K l e i n e r (1966,1967,1969) o b j e c t s t o B i r s s ' procedure on the
grounds t h a t by i g n o r i n g the anti-symmetry elements, i t does not e x p l o i t
the f u l l y symmetry o f the s i t u a t i o n .
(c) I d e n t i f i c a t i o n of the s p i n - i n v e r s i o n o p e r a t i o n w i t h the time-
i n v e r s i o n o p e r a t i o n i s not always c o r r e c t (see Section 5.7), e s p e c i a l l y
when there i s some t r a n s p o r t phenomenon present i n the c r y s t a l . The way
i n which t h i s i d e n t i f i c a t i o n leads t o i n c o r r e c t r e s u l t s f o r the conduc
t i v i t y tensor o f a paramagnetic m a t e r i a l w i l l now be discussed here.
Consider the c u r r e n t d e n s i t y J i n a paramagnetic c r y s t a l , subjected t o
an e l e c t r i c f i e l d E i n the absence of any e x t e r n a l magnetic f i e l d ( i . e .
H = 0 ) . Symmetry o f the c r y s t a l w i l l be described by one o f the type I I
Heesch-Shubnikov p o i n t groups, i n which /A, the magnetic moment (spin)
i n v e r s i o n operator i s a symmetry operator on i t s own as w e l l as i n com
b i n a t i o n /AiS w i t h the s p a t i a l symmetry operators iS o f the c r y s t a l p o i n t
group. Now i f 8 the opera t i o n o f time i n v e r s i o n i s i d e n t i f i e d w i t h /A,
- 93 -
i n consequence of the f a c t t h a t the c u r r e n t d e n s i t y ? i s a time a n t i
symmetric p r o p e r t y , then
0 J = - J . (6.11)
-*• -> Taken t h i s together w i t h Neumann's p r i n c i p l e (J = 8J) leads t o
J 8 J = - J . (6.12)
That i s the c u r r e n t d e n s i t y i s n u l l . This cannot be t r u e . From the
p o i n t o f view o f the c o n d u c t i v i t y tensor, from Ohm's law ,
J. = o £ . , (6.13) i i j 3
a f t e r the opera t i o n o f 8 ,
6 J , = 8 O . . 8 E J (6.14) i i j J
Ej i s a time-symmetric p r o p e r t y ,
8 E. = E. . (6.15) 3 3
S u b s t i t u t i n g from equations (6.11) and (6.15) i n t o (6.14) we get
6 ° i j = ~°ij * (6.16)
But Neumann's p r i n c i p l e d i c t a t e s .-
0 O i j = +°ij ' ( 6 ' 1 7 )
t h e r e f o r e
o = - o ^ = 0 . (6.18)
- 94 -
This i n d i c a t e s t h a t a l l the paramagnetic c r y s t a l s should be i n s u l a t o r s
i n the absence o f an e x t e r n a l magnetic f i e l d . This i s not t r u e . Thus
use of 8, the op e r a t i o n of t i m e - i n v e r s i o n does lead t o the i n c o r r e c t
r e s u l t t h a t the c o n d u c t i v i t y tensor i s n u l l . We o b j e c t t o B i r s s ' s t a t e
ment t h a t Neumann's p r i n c i p l e does not h o l d f o r t r a n s p o r t phenomena i n
magnetic c r y s t a l ; f u r t h e r we consider t h a t i f a p p l i c a t i o n o f Neumann's
p r i n c i p l e does lead t o an i n c o r r e c t r e s u l t , then such a f i n d i n g casts
doubt not on Neumann's p r i n c i p l e , but on the i d e n t i f i c a t i o n o f the time-
i n v e r s i o n operator, 8 w i t h /k. Furthermore as the j u s t i f i c a t i o n f o r
i d e n t i f i c a t i o n o f 8 w i t h /A, B i r s s proves t h a t the t i m e - i n v e r s i o n operator
has the e f f e c t o f reve r s i n g the d i r e c t i o n o f the s p i n ; however he does
not prove t h a t the two operations have e x a c t l y s i m i l a r e f f e c t s which i s
a necessary c o n d i t i o n f o r such an i d e n t i f i c a t i o n . I n f a c t the time-
i n v e r s i o n o p e r a t i o n has a much broader e f f e c t than the simpler o p e r a t i o n
o f magnetic moment i n v e r s i o n : 8 a c t u a l l y changes the d i r e c t i o n (or sign)
o f a l l the dynamic phenomena which depend on time t o an odd degree.
6.4.2 P r e s c r i p t i o n B; By K l e i n e r
K l e i n e r (1966,1967,1969) o b j e c t s t o the procedure adopted i n
p r e s c r i p t i o n A on the grounds t h a t no use i s being made of the antisymmetry
operations o f the corresponding magnetic p o i n t group. He proposes a
procedure which he c a l l s " p r e s c r i p t i o n B", i n which he t r e a t s the problem
on microscopic grounds and obtains the f o l l o w i n g equations.
T B j A v < U ' l ) - E E V x ( U , S ) ^ ^ ^ ^ ° ( A ) ( U ) A V ' ( 6 - 1 9 )
f o r the s p a t i a l symmetry operators of the corresponding p o i n t group ,
and
- 95 -
V A. (b'» = E £ V ai 'V d ( B ) ( AV* ° ( A ) ( a ) x v ( 6 - 2 0 )
u v K A A K
f o r the symmetry operators which i n v o l v e 8 , the operation o f time-
i n v e r s i o n . I n these equations T (U>,B) are the components o f a t r a n s p o r t B A
tensor, u i s the angular frequency (which f o r Ohm's law i s z e r o ) , 8 i s (A) (B)
the magnetic f i e l d , D ( u^Xv a n d D ^ K u a r e t* i e t r a n s f o r m a t i ° n
matrices corresponding t o the u n i t a r y ( s p a t i a l ) symmetry opera t i o n s , and (A) (B)
D ( a ) , and D (a) are the ones corresponding t o a n t i u n i t a r y symmetry A V K|X
operations (those which c o n t a i n 6 ) . Equation (6.19) i s the same as the
commonly accepted equation based on the d e f i n i t i o n o f a field-independent
second rank tensor and the use o f Neumann's p r i n c i p l e . Equation (6.20)
d i f f e r s from equation (6.19) i n t h a t the t r a n s f o r m a t i o n matrices on the
rig h t - h a n d side o f equation (6.20) are the complex conjugates of those
i n equation (6.19), and also the t r a n s p o s i t i o n o f the s u f f i x e s on the
righ t - h a n d side o f equation (6.20) does not occur i n equation (6.19).
I n K l e i n e r ' s procedure, r e l a t i o n s between t r a n s p o r t c o e f f i c i e n t s are
found by using i n equations (6.19) and (6.20) the u n i t a r y and a n t i u n i t a r y
elements, r e s p e c t i v e l y , t h a t are contained i n the group G' (or J or J(B)
i n K l e i n e r ' s n o t a t i o n ) : G' i s the group o f the operations which leave
the Hamiltonian #(8) i n v a r i a n t (G1 i s c a l l e d the Schrodinger group o f
the system and i s u s u a l l y the same as the magnetic p o i n t group or the
magnetic space group o f the c r y s t a l ) = I t i s p o s s i b l e t o choose, on
p h y s i c a l grounds, a Hamiltonian w i t h more (not less) symmetry than G' ,
the magnetic p o i n t group. Considering t h i s f a c t , i n K l e i n e r ' s p r o
cedure, Onsager's theorem has been d e a l t w i t h i n a d i f f e r e n t way from
t h a t which B i r s s employs. Instead o f using the group G' t h a t includes -»•
only the operations which leave the Hamiltonian #(H) o f the system
- 96 -
i n v a r i a n t , K l e i n e r constructs another group K ( 8 ) . I n a d d i t i o n t o the •+
operations o f the magnetic group G', A(B ) also includes a l l the operations -*• -*• -* t h a t send H(B) i n t o # ( - 8 ) . By using the l a r g e r group X(B) i n s t e a d o f
G', Kl e i n e r i s able t o d e r i v e the generalized Onsager r e c i p r o c a l r e l a t i o n s
as a consequence o f r e q u i r i n g the tensor t o be i n v a r i a n t under the opera-
t i o n s o f the group K(B). Therefore i n p r e s c r i p t i o n B, Onsager's r e c i p r o c i t y
r e l a t i o n s are not used,because they are assumed t o be covered by the use
o f K(B). The group K(8) having been contracted, p r e s c r i p t i o n B c o n s i s t s
o f the f o l l o w i n g steps:
->• ( i ) Use equation (6.19) f o r the u n i t a r y elements o f K(E).
( i i ) Use equation (6.20) f o r the a n t i u n i t a r y elements (those which
contain 8, the t i m e - i n v e r s i o n operator) o f X(B).
The f o l l o w i n g o b j e c t i o n s can be made to t h i s procedure:
(a) C r a c k n e l l (1973) ob j e c t s t o K l e i n e r ' s procedure i n using the group
Jt(B) and st a t e s t h a t : i n using the group X(B) i n s t e a d o f G*, K l e i n e r
assumes t h a t only even terms i n 8 appear i n / / ( B ) , the Hamiltonian o f the
c r y s t a l . This i s not j u s t i f i e d , since the only p h y s i c a l reason f o r t h i s
assumption i s the modified Onsager r e l a t i o n
° ^ ( B ) = o. (-B) (6.21) i j j i
which means t h a t the diagonal components o f the e l e c t r i c a l c o n d u c t i v i t y
tensor are even f u n c t i o n s of B. This i s c e r t a i n l y a d i f f e r e n t c o n d i t i o n
from the assumption t h a t a l l the components be even f u n c t i o n s of B.
Furthermore Cracknell f i n d s the assumption " p r e s c r i p t i o n B leads t o
c o r r e c t s i m p l i f i c a t i o n s o f a t r a n s p o r t tensor p r o p e r t y of a c r y s t a l w i t h
symmetry described by one o f the magnetic p o i n t group, because the r e q u i r e
ment o f inv a r i a n c e o f the tensor under H[B) leads t o the r i g h t answer f o r
- 97 -
the generalized Onsager r e c i p r o c i t y r e l a t i o n u n j u s t i f i e d .
(b) We have the f u r t h e r o b j e c t i o n t o the procedure adopted by K l e i n e r
t h a t i n c o n s t r u c t i n g the magnetic p o i n t group G', 8 , the o p e r a t i o n o f
time-inversion,has been i d e n t i f i e d w i t h the ope r a t i o n o f magnetic moment
i n v e r s i o n o p e r a t i o n ( f o r more d e t a i l on why t h i s i d e n t i f i c a t i o n i s not
c o r r e c t see Section 6.4.1(c)).
6.4.3 P r e s c r i p t i o n C ; by C r a c k n e l l
C r a c k n e l l (1973) f o l l o w s B i r s s ' macroscopic approach t o the
problem, and he also assumes w i t h o u t any pr o o f t h a t the magnetic moment-
i n v e r s i o n operator can be i d e n t i f i e d w i t h 6, the t i m e - i n v e r s i o n operator.
Then i n de a l i n g w i t h the a n t i u n i t a r y operations which do i n v o l v e 8, he
adopts the f o l l o w i n g procedure:
I . Consider Ohm's law i n the absence o f any magnetic f i e l d
( i n t e r n a l or e x t e r n a l )
Ji = o . (6.22)
« -*• -*• •*• 8 w i l l reverse the sign of J , but w i l l leave E u n a l t e r e d ; t h a t i s J
i s c tensor ( i . e . a tensor which changes s i g n under the o p e r a t i o n o f 8)
o f rank one and E i s an i tensor ( i . e . a tensor which remains i n v a r i a n t
under 8 ) o f rank one. Therefore, from (6.22), o i s a c tensor o f
rank two,
9 0 i j = " °ij < 6 * 2 3 )
From t h i s C r a c k n e l l comes t o the conclusion t h a t 6 cannot be a symmetry
ope r a t i o n o f the complete c o n f i g u r a t i o n o f specimen-*-J + E. Then he
regards the specimen as a "black box" which contains some m a t e r i a l e n t i t i e s
and a magnetic f i e l d B, and considers a c r y s t a l f o r which 8 i s a symmetry
- 98 -
ope r a t i o n . Now since 6 i s a symmetry operation o f the black box, then
9 0 ^ = FFJLJ (6.24,
Then Cracknell claims t h a t equations (6.24) and (6.23) are not i n c o n f l i c t
because equation (6.24) a p p l i e s i f the symmetry o f the specimen i s des
c r i b e d by one o f the grey groups, and i f i t i s assumed t h a t Neumann's
p r i n c i p l e s t i l l holds f o r 6, w h i l e equation (6.23) r e f e r s t o the a p p l i c a
t i o n o f 8 t o a l a r g e r system o f which 8 i s not a symmetry o p e r a t i o n .
Therefore i f equation (6.24) holds, then equation (6.23) cannot h o l d ,
because would then be n u l l .
I I . Now i f B »* 0, t h a t i s , when the black box co n s i s t s o f a magnetically
ordered c r y s t a l or consists o f a non-magnetic c r y s t a l s i t u a t e d i n an
ap p l i e d magnetic f i e l d , 6 i s not a symmetry oper a t i o n on i t s own. But
there are c e r t a i n a n t i u n i t a r y operations o f the form 8S, where S i s a
s p a t i a l p o i n t group o p e r a t i o n , which are symmetry operations o f the black
box. Then Cracknell supposes t h a t Neumann's p r i n c i p l e s t i l l holds
(despite B i r s s ' argument) f o r these operations 8S. That i s ,
(8S) a±i (H) = (B) . (6.25)
Now t o make use of the t r a n s f o r m a t i o n p r o p e r t i e s o f the tensor under these
a n t i u n i t a r y operations, the r e s u l t o f the a p p l i c a t i o n o f 8 t o a (B) must
be determined. For t h i s C r a c k n e l l s t a t e s t h a t "the e f f e c t o f 8 on the
motions of a l l the p a r t i c l e s i n the black box w i l l be t o reverse the
d i r e c t i o n s o f t h e i r v e l o c i t i e s and, t h e r e f o r e , the Lorentz forces
p. j i Q e (v x. H); t h e r e f o r e , the Hamiltonian w i l l only remain i n v a r i a n t
i f H i s also reversed. Consequently, the c u r r e n t f l o w i n g i n a given
d i r e c t i o n as a r e s u l t o f the a p p l i c a t i o n of an e l e c t r i c f i e l d w i l l be
- 99 -
un a l t e r e d i f the d i r e c t i o n of B i s also reversed; t h a t i s
6 0 ^ (1) = 0 (-B) . (6.26)
Cr a c k n e l l describes equation (6.26) t o be simply a statement o f t h e f a c t
t h a t the compound opera t i o n o f ti m e - r e v e r s a l p l u s changing the sign o f
B i s a symmetry o p e r a t i o n o f the system i n the black box. Equations (6.26)
and (6.24) only apply when d i r e c t c u r r e n t i s under c o n s i d e r a t i o n . For
a l t e r n a t i n g c u r r e n t , C r a c k n e l l defines the f o l l o w i n g equations t o replace
equations (6.24) and (6.26) r e s p e c t i v e l y ,
6°ij = °ij ( 6 , 2 7 >
and 6 0 ^ (B) = a* (-B) (6.28)
Then s u b s t i t u t i n g from equation (6.28) i n t o equation (6.25) C r a c k n e l l
gives the f o l l o w i n g treatment ,
o 1 ; J ( 8 ) = 6S 0 (B) - S S o ^ t B )
S o j j (-B)
That i s ,
or
i . e . ,
V S ) = S i p S j q a p q [ S _ 1 ("S)J ( 6 ' 2 9 )
a i j ( 5 } = S i p S j q ° p q ( l ) ' ( 6 - 3 0 ) '
a..(8) = S. S. 0* (B) (6.31) i j i p j q pq
- 100 -
Cr a c k n e l l , t h e r e f o r e , a r r i v e s a t the f o l l o w i n g procedure, f o r f i n d i n g
t h e r e s t r i c t i o n s on the form o f magnetoconductivity tensor, imposed by
magnetic symmetry o f the c r y s t a l , (which he c a l l s p r e s c r i p t i o n C):
( i ) Use equations (6.5) and (6.8) based on Onsager's theorem,
( i i ) use
• R i 0 (6.3: i j i p Jq pq
based on Neumann's p r i n c i p l e f o r the u n i t a r y elements R o f G', the
magnetic p o i n t group, and
( i i i ) use equation (6.29), based on a modified form of Neumann's
p r i n c i p l e , f o r the a n t i u n i t a r y elements 0S, o f G'.
P r e s c r i p t i o n A and p r e s c r i p t i o n C w i l l always lead t o the same
r e s u l t s , except t h a t , as a r e s u l t o f p a r t ( i i i ) , p r e s c r i p t i o n C may lead
t o some f u r t h e r s i m p l i f i c a t i o n on the form o f beyond t h a t achieved
by p r e s c r i p t i o n A.
He have the f o l l o w i n g o b j e c t i o n s t o t h i s p r e s c r i p t i o n :
(a) I n p a r t ( i i ) o f p r e s c r i p t i o n C magnetoconductivity tensor com
ponents have been t r e a t e d l i k e f i eld-independent tensor components, which
i s n o t c o r r e c t (see Akgfiz and Saunders 1975a).
(b) I d e n t i f i c a t i o n o f the t i m e - i n v e r s i o n o p e r a t i o n w i t h the magnetic-
moment i n v e r s i o n o p e r a t i o n i s not c o r r e c t (see Section 6.3.1 f o r more
d e t a i l ) , and t h i s leads t o the f o l l o w i n g weaknesses i n Cracknell's d i s
cussion t o j u s t i f y h i s p r e s c r i p t i o n ; i n p a r t I o f h i s d i s c u s s i o n despite
Cracknell's claim we b e l i e v e t h a t equation (6.23) i s i n c o n f l i c t w i t h
equation (6.24), because equation (6.23) always holds and i s always
a second rank p o l a r c tensor (which changes s i g n under the opera t i o n o f
- 101 -
time- i n v e r s i o n ) and i f 8 i s the symmetry o p e r a t i o n o f the system i s
a n u l l tensor. Furthermore i n p a r t I I , we f i n d the discussion on the
e f f e c t o f 6 on (8) which leads t o equation (6.26) u n s a t i s f a c t o r y .
F i r s t l y because i t says t h a t as a r e s u l t of the o p e r a t i o n o f 6 on the
system, the v e l o c i t y v (and t h e r e f o r e the c u r r e n t d e n s i t y J) w i l l be
reversed, and f o r the Lorentz f o r c e , and t h e r e f o r e the Hamiltonian, t o
remain i n v a r i a n t under 6, B the magnetic f i e l d should also be reversed.
Then i t says t h a t as a r e s u l t o f reversing the d i r e c t i o n of 8, J the
c u r r e n t d e n s i t y w i l l remain unaltered. These two statements are con
t r a d i c t o r y . Then there i s the f u r t h e r o b j e c t i o n t o Cracknell's procedure
i n d e r i v i n g equation (6.31) t h a t : i n w r i t i n g equation (6.29) C r a c k n e l l ,
w i t h o u t any di s c u s s i o n , recognizes the need t o operate the symmetry
opera t i o n on the argument o f (g) as w e l l as on i t s components, b u t -1 •+ -V
i n w r i t i n g equation (6.30) he replaces S (-B) by (B) which may be
c o r r e c t f o r c e r t a i n symmetry operations, b u t i s not t r u e f o r others.
6.4.4 P r e s c r i p t i o n D ; Present work
A major weakness of the e a r l i e r p r e s c r i p t i o n i s t h a t they a l l -»•
s u f f e r from the l a c k o f t h e i r authors t o appreciate a^(B) i s a f i e l d -
dependent tensor. A new p r e s c r i p t i o n has t h e r e f o r e been developed i n
which the magnetoconductivity tensor components are t r e a t e d as the compon
ents o f a field-dependent tensor, and the generalized Neumann's p r i n c i p l e
(see Section 5.8) i s employed t o f i n d the symmetry and antisymmetry
r e s t r i c t i o n s on the form o f c ( 8 ) . Furthermore the antisymmetry operator
A, i s not i d e n t i f i e d w i t h 8 , the time i n v e r s i o n operator. To f i n d the
e f f e c t o f A on (B) we must ensure the i n v a r i a n c e o f Ohm's law under
t h i s o p e r a t i o n . Ohm's law f o r d i r e c t c u r r e n t i n the presence o f a magneti
f i e l d can be w r i t t e n as ,
- 102 -
J ± ( B ) = ai ; j ( B ) . (6.33)
Under the operation of magnetic moment i n v e r s i o n (the antisymmetry
o p e r a t i o n ) , t h i s becomes
A J±(B) = A o (B) A E_j . (6.34)
->•
To f i n d the e f f e c t of A on ° ij( B)» i f c i s r e q u i r e d t o know the e f f e c t of
A on J^(B) and E^. The e l e c t r i c f i e l d v ector E^ i s i n v a r i a n t under A,
A = . (6.35)
->• The e f f e c t o f A on B i s defined as ,
/A B = - B (6.36)
when the operator A acts on a system c o n t a i n i n g a magnetic moment and -»•
a c u r r e n t d e n s i t y J ^ B ) , the only e f f e c t i s t o a l t e r the d i r e c t i o n o f -*• B and so ,
A (S) = J M A f f ) = J i(-B) . (6.37)
For Ohm's law t o hol d i n the system under the op e r a t i o n o f magnetic
moment i n v e r s i o n , s u b s t i t u t i o n from (6.35) and (6.37) i n t o (6.34) leads
t o ,
A o (8) - a±i (-§) (6.38)
For the symmetry operations A iR , Neumann's p r i n c i p l e demands t h a t ,
AIR 0 ^ ( 8 ) = o ( A IR B) f (6.39)
or IRAQ (8) = o ( i R A B ) , (6.40) l j l j
- 103 -
since A IR = IR A, s u b s t i t u t i n g f o r /A o^.lB) from (6.38) i n t o (6.40),
we o b t a i n
IR o (-8) = o ( IR A 8 ) = o j L j (- IR B) .
Therefore
IR ° i j(B) = o i 3 < IR 8 ) . (6.41)
Thus we o b t a i n
V | , R | R l q V l < R | R 2 q S q ' R 3 q V = Rim Rjn °mn 'W'
(6.42)
Therefore, the a n a l y t i c a l form o f Neumann's p r i n c i p l e f o r a(B) (or
P^^(B)) applies t o c r y s t a l s belonging t o any o f the three types ( I , I I ,
I I I ) o f Heesch-Shubnikov p o i n t groups, f o r both kinds o f s p a t i a l symmetry
operations (the ones which are elements of the magnetic p o i n t group on
t h e i r own, and the ones which i n combination w i t h A, the op e r a t i o n o f
magnetic moment i n v e r s i o n , are elements o f the magnetic p o i n t group).
When H / 0 , the form of j (B) does not depend on whether the specimen
co n s i s t s of a non-magnetic c r y s t a l (paramagnetic and diamagnetic and some
ant i f e r r o m a g n e t i c c r y s t a l s ) i n an ap p l i e d magnetic f i e l d or o f a magneti
c a l l y ordered c r y s t a l . Furthermore since o i s a second rank p o l a r tensor
and B a f i r s t rank a x i a l tensor they are both i n v a r i a n t under the opera
t i o n o f i n v e r s i o n i n the space (see Table A.1). ->•
Therefore our p r e s c r i p t i o n D f o r f i n d i n g the forms o f o ( H ) f o r a
c r y s t a l belonging t o a magnetic p o i n t group i s as f o l l o w s :
( i ) Find the corresponding c l a s s i c a l p o i n t group G o f the magnetic
p o i n t group G*, noti n g t h a t G' depends upon the d i r e c t i o n o f B ,
- 104 -
( i i ) take the enantiomorphous p o i n t group o f t h i s c l a s s i c a l p o i n t group
and use i t s generating elements i n equation (6.42), which i s based on the
t r a n s f o r m a t i o n law f o r f i e l d dependent tensors and generalized Neumann's
p r i n c i p l e , t o d i s t i n g u i s h the non-zero components f o r a chosen magnetic
d i r e c t i o n , and
(iii) apply equation (6.8) based on Onsager's theorem.
Thus we have concluded t h a t the symmetry r e s t r i c t e d forms o f
a ^ ( B ) f o r c r y s t a l s belonging t o magnetic p o i n t group G' are i d e n t i c a l t o
those of the c r y s t a l s belonging t o the corresponding p o i n t group G. The -»•
l a t t e r have been l i s t e d f o r B d i r e c t e d along the major c r y s t a l l o g r a p h i c
axes by AkgiJz and Saunders (1975a).
6.5 Symmetry r e s t r i c t i o n s on the forms of other t r a n s p o r t p r o p e r t i e s
I n a magnetic c r y s t a l or a non-magnetic c r y s t a l i n an e x t e r n a l -+ ->
magnetic f i e l d the e l e c t r i c a l and heat c u r r e n t d e n s i t i e s J and q are
r e l a t e d t o the ele c t r o m o t i v e force E and the temperature g r a d i e n t VT by
the phenomenological l i n e a r t r a n s p o r t equations
E - P s i(H) J + a..(B) V. r (6.43)
q i = * i j < H ) J j " K i j ( 8 ) 7 j F (6.44)
where each component o f the second rank t r a n s p o r t tensors - the magneto--*• -y r e s i s t i v i t y P ^ ( 8 ) (the reverse of the magnetoconductivity a^(B)), the
-+ -> magnetothermoelectric power ^ (5) , the magneto-Peltier e f f e c t ^ j ^ ) and -*•
the magnetothermal c o n d u c t i v i t y k^^(B) - depends on the magnetic f i e l d .
Thus these t r a n s p o r t p r o p e r t i e s should be d e a l t w i t h as field-dependent
tensors. The s p a t i a l and magnetic symmetry r e s t r i c t i o n s on the forms o f
- 105 -
a l l these tensors are the same as those on the magnetoconductlvlty tensor.
The only difference i s the i n t r i n s i c symmetry dictated by Onsager's
theorem, which i s not the same for a l l these properties. That i s
° i j ( B ) = °ji (~ B ) ' (6.45)
P i j ( B > = P i j ( _ B ) ' (6.46)
K i j ( B ) = K i j ( " ^ ) ' (6.47)
and (1/r) TT±^ (8) = " ^ ( - 8 ) (6.48)
Therefore the magnetoresistivity tensor P ^ j ^ a n d the magnetothermal •+ -> conductivity tensor <^ (H) take the same forms as a (g). The forms of
-> the magnetothermoelectric power (Bj) and the magneto-Peltier e f f e c t
•>
^ i j ^ f ° r c r v s t a * 8 belonging to the magnetic point group G' are the same
as those of c r y s t a l s belonging to the corresponding c l a s s i c a l point group
G, i n a magnetic f i e l d ; they are di f f e r e n t from a ^ ( B ) . These forms
have been tabulated by AkgSz and Saunders (1975b).
6.6 The permittivity of a c r y s t a l -*•
In an anisotropic body, the magnetic f i e l d i n t e n s i t y H ( f i e l d
strength) and the magnetic induction B (flux density) are connected by
the r e l a t i o n (see Nye 1957) ,
B i = ^ i j H j ' ( 6 , 4 9 )
where i s c a l l e d the permittivity of the body and i s a second rank
polar tensor with respect to s p a t i a l transformations. But, as considered
i n Section (3.7), there are cases where we have to deal with more
- 106 -
complicated transformation operators of the kind (IR^ I iR J «v^} , with
IR operating only on the spin (magnetic) space coordinates. Therefore P the behaviour of p.^ has to be investigated i n t h i s space, and to do
t h i s the t e n s o r i a l behaviour of B and H i n that space have to be found.
Magnetic f i e l d i s a vector ( f i r s t rank tensor) i n the magnetic
subspace and so i s the magnetic induction. Therefore, using the quotient
law (Section 4.6), i t can be shown that permittivity must be a second
rank tensor i n the magnetic subspace as well as i n the physical subspace.
This means that i n the combined space of magnetic and physical subspaces
magnetic f i e l d and magnetic induction w i l l each have nine components
(each of the three components i n the physical, or geometrical, subspace
w i l l have three components i n the magnetic subspace). Therefore the
permittivity tensor w i l l have eighty one components i n t h i s combined
space. Equation (6.49) may now be written
(B ±) = (HJ (6.50) a
i 3 ap 3 B
where a and $ refer to the coordinate axis in the magnetic subspace and
i and j to the coordinate axis i n the geometrical subspace.
A l i n e a r relationship between the magnetic induction, B, and the ->•
magnetic f i e l d H does not hold for a l l the substances. There are some
cases where permittivity depends on the magnitude of the applied magnetic
f i e l d . In the ordinary physical subspace t h i s may be shown by
B± = ^ j ^ ) H j • (6.51)
In t h i s case permittivity can be dealt with as a f i e l d dependent tensor
f i e l d and every transformation operator w i l l operate on the components
of i t s argument (H) as well as on i t s own components.
- 107 -
APPENDIX I
SPATIAL SYMMETRY RESTRICTED FORMS OF FIELD DEPENDENT
TENSORS OF RANK ZERO IN THE FIELD SUBSPACE, WHERE
FIELD COMPONENTS ARE THEMSELVES TENSORS IN
GEOMETRICAL SUBSPACE
A.1 Introduction
I n dealing with the f i e l d dependent tensor properties of s o l i d s ,
the si t u a t i o n where the f i e l d components have some kind of t e n s o r i a l
dependence on the coordinate axes i n geometrical subspace i s frequently
encountered. The magnetoresistivity tensor i n magnetic c r y s t a l s i s an
example of such physical properties, where the components of r e s i s t i v i t y
tensor depend on the magnetic f i e l d which i t s e l f i s a f i r s t rank tensor
i n geometrical subspace. Here the transformation operations of the co
ordinate axes i n geometrical subspace w i l l a f f e c t the coordinate axes of
the f i e l d subspace too (see Section 5.9), unless otherwise stated.
Inserting symmetry operations of t h i s king i n Neumann's pri n c i p l e
(equation 5.18), the general forms of these f i e l d dependent tensors for
each of the 32 c l a s s i c a l crystallographic point groups have been obtained
i n t h i s appendix. Three-dimensional rotation groups have been employed
as the instrument for i d e n t i f i c a t i o n of the non-zero components. The
forms of f i e l d dependent tensors up to fourth rank i n the matter tensor
and second rank i n the f i e l d tensor (the argument of the f i e l d dependent
tensor) are tabulated a t the end of t h i s appendix.
To avoid writing long sentences we use ' f i e l d dependent tensor'
to mean " f i e l d dependent tensors of rank zero (scalar function) i n the
f i e l d subspace" i n the r e s t of t h i s appendix.
- 108 -
A.2 Application of Neumann's Prin c i p l e
Transformation law of a f i e l d dependent tensor (equation 3.20)
requires that both the tensor components and the f i e l d components (the
argument of the tensor) be transformed. Therefore Neumann's p r i n c i p l e
(equation 5.18) based on t h i s transformation law for an orthogonal trans-i
formation of coordinates (in the geometrical subspace) x± = can
be written as
c d B<4i. C|(71 o o ..F ) = I o | a. a. o. . ...B t (F ) (A.l) i j k . . . 1 1 mn op np 1 1 i r j s kt r s t . . . mo...
where a l l the indices refer to the coordinates i n geometrical subspace,
f i e l d components are denoted by F , o 's are the elements of the * mo... i j
transformation matrix, and |o|its determinant. Both the f i e l d dependent
tensor (matter tensor) components B and the arguments ( f i e l d tensor)
F can transform either as polar (zero c and d i n equation A.l) or
a x i a l (c and d equal one i n equation A.l) tensors. This r e s u l t s i n four
d i f f e r e n t kinds (labelled a , 0, y, &) of f i e l d dependent tensors which
can be described by the following equation for the a n a l y t i c a l form of
Neumann's p r i n c i p l e :
tensor type a - polar matter tensor (d = 0) dependent upon polar f i e l d
tensor (c - 0 ) .
B (o a . . . f ) - o. o. a , ^ . . . B (F ) (A.2) i j k . . . mn op np... i r j s kt r s t . . . mo....
tensor type 3 - polar matter tesnor (d = 0) dependent upon a x i a l f i e l d
tensor (c - 1)
B. .. (|o|o O ..F ) - 0. 0. a l A _ . . . B _ (F ) (A.3) i j k . . . 1 1 mn op np.. i r j s kt r s t . . . mo...
- 109 -
tensor type y - a x i a l matter tensor (d = 1) dependent upon polar f i e l d
tensor (c = 0)
B. (o a .. F ) «= p a o a B . (r ) (A.4) i j k . . . mn op np.. i r j s kt r s t mo...
tensor type 6 - a x i a l matter tensor ( d e l ) dependent upon a x i a l f i e l d
tensor (c = 1)
B^. (|o| a a ... F ) = lol o. o. o...... B (F ) i j k . . . ' 1 mn op np... 1 1 i r j s kt r s t . . . mo..
(A. 5)
The matter and f i e l d tensors can be further c l a s s i f i e d into those
of even and odd rank to give the following nomenclature:
tensor of kind a - even rank matter tensor and even rank f i e l d tensor»
tensor of kind b - odd rank matter tensor and even rank f i e l d tensor;
tensor of kind c - even rank matter tensor and odd rank f i e l d tensor;
tensor of kind d - odd rank matter tensor and odd rank f i e l d tensor.
Combination of t h i s nomenclature with the a, 3, y, 6 c l a s s i f i c a t i o n
leads to sixteen d i s t i n c t types of tensor l a b e l l e d oa, ...., fid.
To find the maximum si m p l i f i c a t i o n i n the forms of these sixteen
types of tensor the s e t of generating matrices used by B i r s s (1966)
(see Section 5.4) has been employed i n equations (A.2), (A.3), (A.4)
and (A.5). During t h i s substitution of generating matrices, certain
e q u a l i t i e s or relationships have been found between tensors of the same
rank for different point groups. These w i l l now be l i s t e d .
- no
1. The forms of the tensors for c r y s t a l s belonging to centro-
symmetrical point groups are affected by the f a c t that o^ 1' i s a generating
element; for
-1 i f i - p
0 i f l | i p
and the determinant |c/*M i s -1. I n general
B^, (±a o .. F ) » ±o.. o.. a . . .... B... (F ) i j k . .. mm oo mo.. i i j j kk i j k . . . mo...
(A.6)
where the + sign corresponds to polar and the - sign to a x i a l tensors.
This can be written as
B... (±(-1)" F ) - ± (-1)° B... <F ) (A.7) i j k . . . mo... i j k . . . mo...
where m and n are respectively the ranks of the matter and f i e l d tensors.
Thus four different sets of equations, which depend upon both the rank
and p o l a r i t y of the tensor, r e s u l t :
XjK.... mo.•• i j x . . . . mo....
for t h i s case, does not introduce any s i m p l i f i c a t i o n and therefore
the appropriate enantiomorphous point group generating elements are suf
f i c i e n t to produce the maximum s i m p l i f i c a t i o n of the tensor.
( i i ) B (F ) = -B (F ) = 0 (A.8b) l j J c . . . . mo. • • i j K . . . . mo....
hence the matter tensor i s n u l l .
- I l l -
( i i i ) B . (-F ) • B4 (F ) (A.8c) l j K . . . . mo... l j K . . . . mo...
the matter tensor components are even functions of the f i e l d tensor.,
components,
(iv) B (-F ) = -B, (F ) (A.8d) l j K . . . . mo... l j K . . . . mo...
the matter tensor components are odd functions of the f i e l d tensor com
ponents. Inspection of equations (A. 8) for each of the sixteen d i f f e r e n t
types of tensor leads to the general r e s u l t shown i n Table A.l.
2. For those point groups which contain only the generating elements
o ( 0 ) , o ( 2 ) , o ( 3 ) , o ( 6 ) , o ( 7 ) and o ( 9 ) whose determinants are unity, the
forms of a l l the f i e l d dependent tensors of the same rank for a given
point group are i d e n t i c a l for both the polar and a x i a l matter tensors;
t h i s i s independent of the polar or a x i a l nature of the f i e l d tensor.
(3) (5) (3) (5) 3. The generating matrices o and a are such that a = -a
and | a ( 3 ) | = 1 = - | o ( 5 ) | . Therefore
{ - l > n a . <3> o. ( 3> o <3> .... B (F ) = i r j s kt r s t . . . . mo...
= ±CJ ( 5 ) o 4 ( 5 ) o < 5 ) ... B ^ (F ) (A.9) i r j s kt r s t . . . . mo...
where n i s the rank of matter tensor. Thus for even rank polar or odd
rank a x i a l f i e l d tensors, the two generating elements o^ 3' and ^ have
the same ef f e c t on the even rank polar or odd rank a x i a l matter tensors.
A.3 Tabulation of the forms of the f i e l d dependent tensors
The equalities and other relationships between the f i e l d dependent
tensors between and within point groups are shown i n Table 2. Null tensors
- 112 -
are also shown where they occur. To make matters easier, the plan used
i n construction of the table i s based upon that used by B i r s s (1966),
i n h i s similar table (4a) for constant tensors. Where tensors are labelled
with a different c a p i t a l l e t t e r , t h e i r forms are d i s s i m i l a r (some c a p i t a l
l e t t e r s have bars on them. This i s to increase the supply of l e t t e r s and
has no other special meaning). The subscripts m (for even numbers) and
n (for odd numbers) show the rank of the matter tensors . (F ) and (F ) P <1
represent the f i e l d tensor, here p shows an even rank and q an odd rank
tensor; where they occur the superscripts o and e show that only odd or
even functions of the f i e l d components are present. The significance of
a dash on the l e t t e r s B or C representing the matter tensor i s that there
i s i n t h i s case a simple relationship between the pair s of forms B and B'
or C and C , namely that i f even or odd terms are present i n the dashed
l e t t e r they are absent i n the un-dashed one and vice versa. Thus, i f a
component of B i s an even function of the f i e l d components, then that
component of B" i s an odd function. Furthermore, i f a component of B
contains both odd and even terms, then that component of B' i s zero and
vice versa.
The complete forms of the tensors represented symbolically i n
Table A.2 are detailed i n Tables A.3 to A.14. The forms of the f i e l d
dependent tensors up to fourth rank (m=0, 2, 4; n = 1, 3) for the
matter ..tensor and up to second rank (p = 0, 2; q = 1) for the argument
have been obtained by systematic substitution of the generating matrices
for each point group into equations (A.2) - (A.5). The matter tensor
components are represented by X, Y, Z and of course for higher rank
transors i n combination such as XY. The f i e l d tensor components are
shown as x, y, z, also i n combinations such as xy for a second rank
f i e l d tensor. Each column shows a p a r t i c u l a r tensor component and a
- 113 -
row shows a l l the components of a p a r t i c u l a r tensor; certain components
e x i s t only as either an even or an odd function of the f i e l d tensor -
t h i s i s indicated d i r e c t l y . When an equality between components occurs
then i t i s shown by writing i n i t s place that term (or even or odd part
of the term) i n the same row to which i t i s equal. To reduce the s i z e
of the tables a system of permutation of the matter tensor component
s u f f i c e s ( s i m i l a r to that used by B i r s s (1966)) for constant tensors has
been adopted. Thus the notations of the type XZ(2) show permutations
which apply to every component i n the column and the number i n brackets
gives the number of d i s t i n c t components obtained. In addition the f i e l d
dependent tensors require a s i m i l a r permutation xy2 and yz2 of the
s u f f i c e s for the f i e l d tensor. The forms presented i n tables A.3 - A.14
provide the basis for studies of the anisotrppy of the f i e l d dependent
properties of c r y s t a l s .
A.4 Use of the tabulation to find T 2 (F ) for the point group 3m
To exemplify the use of these tables l e t us extract the form of
a second rank polar matter tensor which depends upon a f i r s t rank a x i a l
f i e l d tensor for the point group 3m. In Table A.2 t h i s type of tensor
occurs as T (F ) i n the tenth column i n the 3m row. This i s one of m q those tensors which obeys case (1)(equation A.8a) and so has the same
form i n t h i s chosen point group (3m) as i t does for the corresponding
enantiomcrphcus point group (32). The ranks of the tensor sought lead
to T^(F^) and the form can therefore be obtained from the section for
m • 2 and q = 1 i n Table A.7. In the even and odd terminology (see
for example, Akgbz and Saunders 1975 a, Akgoz and Saunders 1975 b,
SUmengen and Saunders 1972)
- 114 -
even odd
T (F ) 11 1 T (F ). 13 l '
T (F ) 22 V l '
, T 3 1 ( r i } ' T 3 3 ( V
T 1 2 ( F 1 )
T 2 1 ( F 1 } * T 2 3 ( F 1 }
T (F ) 32 1 r
T (F ) = i j 1 2'
T n ( F 2 ) . T 1 3 ( F 2 H
T 2 2 ( F 2 )
•T (F ) 11 V 2' T (F ) 13 1 2'
T ( F : ) 2 2 V r
L T 3 1 ( F 2 ) . T 3 3 ( F 2 r - T 3 1 ( F 2 ) . T 3 3 ( F 2 )
T (F ) " 1 1 1 3'
T U ( F 3 >
T 3 3 ( F 3 , J L
T 1 2 ( F 3 )
-T (p \
(A.12)
A. 5 Conclusion
The forms of the tensors shown i n Table A.3 have been obtained
by reference only to the symmetry r e s t r i c t i o n s imposed by the point
group operations. In addition a f i e l d dependent tensor may exhibit
further symmetry i n t r i n s i c to the physical nature of the property i t
describes. Then the tensor w i l l be si m p l i f i e d further.
- i l D -
O O O 9 9 o 9 9 a a a < < a < < a . a a 9 9 a 9 9
3 3 9 3 N . •i »i •i 9 9 0 •1 1 o •1 1 9 9 3 0 0 9 0 0 PC PC ft 9 3 PC 9 3 PC PC PC j r Hi •a Hi 9
a H« 0 o M- 9 tr Hi 0 0 Hi 9 9 rf 9 0 H - rf ^ H - 0 rf M rf 9 rf 9 rf a 9 a rt M 9 r-> rf
»1 0 a •1 a 0 rt rt 1 1 9 rf 9 rt rf rf 9 9 9 rt 9 9 9 rf 0 . 9 0 9 3 9 9 9 0 U O 9 0 U 0 3 •1 0 •1 0 0 0 O 0
•i O 1 i 1 0 1 •1
0 9 Hi 0 9 •1 0 w- •1 0
9 9 rf 9 9 rt M rf »-• rt 9 OQ 9 •0 a 0 9 Q . 9 4> • 9 •1 3 o <5
a 1 < 1 9 O 0 r-> r-> rf a rt 9 rf c 3 0 0 <B rf 9 9 rf /~\ 0 •a rf •1 •1 9 9 9 9 3 H -IB c 9 0 Hj 9 C n 0 oq 0 Hi 9 O 9 0 0 e 0 M 0 H 9 9 H - 0 •1 (i 0 •1 9 O M w 9 9 O Q 0 rf
rt »1 o 1 9 9 •1 M rt O H- Q rt rf 0 •1 u a 0 0 0 o 0 o 9 C tt 3" 9 9 o 9 o o 9 H- Hi rt O rt 9 IS 9 •o 9 9 0 H » Hi w c 9 rf o •o 0 0 •d O H - 9 0 3 9
0 o 9 o •» O oq 0 3 a» Hi 9 9 0 9 H * "2 O u 3 9 9 Hi 9 9 9 0 1 O rt 9 rf 9 M H -ID rf 0 rt rt 9 9
a 0 1 rf
9 TO 9 Hi 0 9 Hi 0 9 9 1 9 H» H. 0 H- •1 0 9 o 0 9 9 rt 9 9 rf rf c 9 rf M rf 0 U •a rf a O 9 a 9 0 X o
H' a 1 < •1 H » M /"» 0 m o rf a rt 9 0 0 9 1 9 9 9 rf 9 9 rf M 1 C O 9 9 O 3 Hi 9 9 0 M 0 9 •1 0 C 9 0 Hi 3 H | 3 t-> u 0 1 •o O 3 0 0 c to H - V
9 e 0 9 1 . H . O O •1 9 O TO 0 rf rt Hi rf O rf •1 O H. b-l rf 9 H I H » e O H - 0 rf a 0 9 H- f*» 9 in o 0 O o H- O •1 10 H- o oq a 3 o 9 O 0 rf o M- TJ •a 0 9 •a 3 9 <r rf
9 9 O o o to •o 3 S 9 M M* 3 O o 3 o 0 *J rf 9 3 9 H> 3 0 O 3 O to 1 r* 3
H> 9 3 Hi 9 O rt 3 rf 3
to rf 0 rt W to
H > 0 9 H I 0 9 9 cm 9 H * M 0 H » 0 9 *i 3 9 9 rf 9 9 rf 3 o 0 •o > M rt M rf rt d 3 o X a . O 9 a 9 9 0 •o rt M H -
a 1 < •1 H - 0 0 rf a rf 9 0 TO O 3 N . f-1 • (0 rt 9 3 rf *1 9 9 C O 3 Hi 9 9 9 9 3 O l_i 0) . Hi 9 (0 e 9 0 Hi 3 9 *1 M 0 H ' n O 9 0 O c U U •o 0 -<; 9 rf •1 o O 1 3 O c 0 3 1 rt M rf
rf •J n •1 Hi rf O 0 / - \ a 9 o f* O rt H I H - c 3 H - •1 0 O o o H - o H - 3 0 0 fJ- rf B 9 o 9 0 0 0 TO o ^ 9 rf
u 9 *S 3 9 H- •1 3 Q 0 "9 0 U •o 9 0 o 0 3 9 0 0 9 o 3 M H - O to 0 Hi 9 9 O 3 rf 9 3 0 9 9 3 Hi 9 i .-+ 1 f+ . 9 rf 3 0 rt 0 rt
0 0
0 oq 9 Hi 0 9 Hi 0 9 •1 9 H- 4 0 H - 1 0 9 0 0 9 0 rt 9 9 rf » > c 9 M rf f-1 rf X X 0 •o rt a 9 0 a 0 9 H 1 H "
e H- < •1 a •l 0 0 • H » oq 0 rt 9 rf a M M
Hi 9 9 3 9 3 rf 0 rf 9 0 C _ 3 9 3 Hi 0 Hi 3
o 9 4 M O to Hi 3 0 C 3 * ^ H - U •1 "3 M 0 O c W O 3 to 0 rf
9 0 3" JC •1 3 O 1 O O l-J rf 9 rt O rt * n H. rt h a A i+ H- c 9 -~> o rf o H - •1
9 u 3 H - o o o O n ri-oq 0 H - 9 o o 5 3 o 9 rf
•o o r 3 9 0 9 3 0 9 0 •1 c W V o r u 3 H * H- 3 O 3 0 o O to 9 9 9 o 3 9 Hi 3 -t c s rf 3 H, 0 3 9 9 rt 3 rf 3 9 « rt 0 rt rf 0 0 0
Table Al
- 116 -
u if
i i n E r s > If1 IT B ™*
5 5 5 f S' f 8 ? - * I I B S •
s"5
i ( 9
II Is
-S S . 8 "l IF
5 5 5 5 55 5 5 5 5 S55 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5'? 5 35 5 5 5'5 5 35 5 35 35 5 5 ?5 5 5 5 3 3 5 5 ? i 'H i
its
••• r
1*2
.•V . --! -!
f ?? .y .••
."• .»• -5 -5 -5 .«
5 35 5 . - • . i f * ^ *
35
355 35
555 35
555 5 5
35 5 35
i 55 i 5
i 5 »_ »_
3555355
3355555
5 5 5 5 5 5 5 .° . r A ." ." S 3 3 3 3 3 3 «_ V *- *_
553155
3 5
•i 5
1 5
3
35 5
s
I * *
3 5 5 ^ •• .* .« .5 _a .3
l I 3 5
33
*? 1 3
i l l
?
I 1
5
»"" L= •! <
a
5 5
5' 5' 5' 1 5 3 ?5 5 5 5'5 5
5 5 5 5 5 5 3
?55 5 3'53
35 5 5 3'5 3
1 5' 5' 5' 1 5 5
v -2 -z
55 .• 3
355 35 5 5 5 5 i 55 s .1 f ." »- 'Z '2 'Z
5 5 35 5
•J - i »- »- 5 5
355 35
< c < < v "Z *- *3 «_ »- •,
35 5 3 5
1 5 5 1 5
555 3 5
*'«•' 5 * 5 3 5" 5 5 5 5 5 1 5 5 1 5 5
•e s < i * { S »_ «_ »_ *. »_ 55 5 555 5 53 5 555 5 5 55555 5 5 5 5 a
35 5 ? 5
.» .» ." ."
5-5 5355 5 3 5 5 5
3 5 5
5 5 5 5 5 5
1 5
35
35
« 33
11
53
55
1L H
i n
:jl i i
- 117 -
Table A3
Tho f o n i B o f f i e l d dependant t e n x o r o o f z e r o t h r i n k fn>o> I n the u t t e r t a n x o r
= 0 X ' « l 'l = 1 X ' » ) X I I ) P a 2 X l x x ) X f y y ) X f t x ) X f x y ) X f y x ) X f x z 2 ) M y z t )
) 9
X f x ) W Xf l ) X f y J x m A fF_> o 2
Xf v i ) X f y y ) X f x i ) X f x y t X f v x ) ' X f x x ) X f y x )
) D X f x ) V V even e v e n X'7> B I F . ) o t X 'x>> X f y y ) x r « ) X f x y l X f y x ) e v e n oven
) D — rfo'V odd odd — V V -• - — - — odd odd
) 0
e v e n r I F , i o 1
X l > ) X*y\ e v e n W even e v e n e v e n e v e n e v e n X f x x ) X f y z )
0> X f i ) 0 <F,>
o 1 e von e v e n e v e n D f F i
a 2 X f x x l X f y y ) X f * x ) even e v e n e v e n e v e n
) 0
e v e n 1! I F ) ° 1
e v e n e v e n X ' z l V V e v e n e v e n e v e n XI xy ) X f y x ) e v e n e v e n
) 0 w — — odd V V — _ udd odd — —
> 9
add w — — — V V odd odd odd — — — e
> 9
X f « ) V V e \ e n e X f x t X f z l V V X l x x ) X f x x ) « i x i > X f x y ) X f - x y ) X i x x )
1 9
e v e n ' o ' V e v e n e X f x ) e v e n V V X l « » ) X ( - x x ) e v e n X f x y ) X f x y ) e v e n x7xx)
) 0
— v v e v e n -x7.) — v v X ( . « ) - X l x x ) — X f x y ) - X f - x y ) e v e n -x7xx)
) D
add V V e v e n e - X ( « ) odd V V Xf »x) -X I - X I ) odd X f x y ) - X f x y ) e v e n - X f x x )
) a
X ( x ) U f F . ) o 1 e v e n xTx) e v e n V V Xf xx> X f x x ) X f t i ) e v e n x7xy) e v e n x7xx)
> 0
e v e n H ( I " . ) o 1 e v e n x7x» X l x ) v v wvirn x f x x ) even X f x y ) X l - i y ) e v e n X*f I I )
) o
— S ( F . l o 1 — — odd V V — — — add - X ? i y l — —
) a
odd ° o ' V — — — V V odd "> odd add — — — —
) o
e v e n v v e v e n x7») oven V V Xf xx) X I - x x ) e v e n • v o n x7xy> e v e n x7xz)
o ' — v . 1 e v e n -x7.) — V V X t z x l - X f x x ) — e v e n -xVxy) e v e n -x7xz)
) o
odd e v e n - x f . ) — V V Xf xx) - X f - x x ) odd e v e n -x7xy) • v « n - X ? x z )
) o
X f x ) S o ' 1 ! ' X (< ) X l y ) X l r . ) V V Xf xx> X f y y ) X « z z ) X f x y ) X f y x ) X f x z ) x7yz)
) o
X f x ) V ' . ' e v e n X(>» e v e n V V X f u ) X f y y ) X f x x ) e v e n oven X f x x ) e v e n
o> even a 1
X ( « ) e v e n M i l V V even e v e n e v e n X t x y ) X f y x ) • v e n X f y z )
o> — V V odd - odd V V — — add odd — odd
o> odd v v — odd — V V odd odd add — — odd —
X f x ) V V e v e n e v e n X f z ) V V X f x x ) X l y y ) X f x x ) X f x y ) X f y x ) e v e n e v e n
> o
oven V V S i x ) \ f > 1 e v e n V V e v e n e v e n e v e n e v e n e v e n Xf i i > X f y z )
— V V odd odd — V V — — — — — odd odd
F ) o odd v v — — odd V V odd odd odd odd odd —
F ) o X f i ) V V e v e n even *i\tm V V Xf I X ) X i v y ) X f z i ) e v e n e v e n e v e n e v e n
F ) o
e v e n F f F , ) o 1 e v e n e v e n X I / I V V e v e n e v e n over. X f x y ) X f y x ) e v e n e v e n
F > o — C ( F . ) o 1 — — oild V V — — — add odd — —
F > o
add l'l f F . ) o 1 — — — V V add odd add — — — —
F ) a e v e n 1 f F ) o 1
e v e n Xf ; ' ) V V e v e n e v e n e v e n even oven Xf X f ) o v e n
F ) o — J f F ) o 1 — odd V V — — — - — udd —
F > o odd it ( F . )
o 1 — — V V add odd odd — — — —
F ) O X f i ) f. f F 1
o 1 oven x f . . r. I F >
o l Xf r v l X f XX 1 X f xx ) a von x7yx) X T i y )
F ) a
X f x ) i ( F ) o 1
e v e n x f x> \ V » V V X t x x ) Xf XX) X f x x ) even x7xy) x7xy) X?xy)
F ) o
even 5 ( F . ) o 1 e v e n x7») X?«) V V evon x7xx) x7xx) e v e n x7xy) X ? x y ) x7xy)
F 1 o — O ( F > o 1 — — — V V — — — e v e n - X ? x y > - X ? x y ) x ' fxy )
F o > oild P I F . )
o 1 V V odd xVxxl X?xx) e v e n - X ? x y ) - X 7 x y ) x7xy)
•->:. The forms o f f i e l d dependent t e n s o r s b i roriTTaie ( n = 1) f o r t h e n a t t e r t e n s o r and ranks z e r o ( p B O) and one (<\ = 1) f o r t h e f i e l d t e n s o r .
n = I n = 1 p = o X ( x ) V ( x ) Z ( x ) '1 = 1 X( x ) X ( y ) X ( z ) Y ( x ) Y(y) Y ( z ) Z ( x ) Z ( y ) Z ( z )
v v X ( x ) Y ( x ) Z ( x ) V V X ( x ) X ( y ) X<z) V ( x ) Y(y) Y ( z ) Z ( x ) Z ( y ) Z ( z )
v v — — Z ( x ) v v odd odd — odd odd — even even Z ( z )
B'l ( V X ( x ) V ( x ) — V v even even X ( z ) even even Y(z) odd odd —
v v odd odd even v v — - — odd — — odd Z ( x ) Z ( y ) oven
V V — — — vv odd — — — Odd — — — odd
V V — — odd v v — Odd — odd — — — — —
F l ( F o > — — Z ( x ) vv — odd — odd — — even even even
W — — even v v odd — — — odd — even even £(z)
V Fo> — — Z ( x ) v v odd odd — o
-X(y) X?x) — even Z?x) Z ( z )
V V — — even vv odd odd — X?y) -X?x) — even Z?x) even
V V — — — v v odd odd — X?y) -X?x) — even -Z?x) —
v v — — odd v v odd odd — -X?y) X°(x) — even -Z?x) odd
v v v v —
—
odd vv M 1 ( F 1 )
odd odd —
o -X(y)
X?x) —
—
— odd
V V — — Z ( x ) v v — odd — -X?y) — — even Z?x) even
°l ( Fo> — — even v v Odd — — — X?x) — even Z?x) Z ( z )
V V V V v v
— —
vv v v vv
odd odd
odd — — —
-X?x) -X?x)
X?x)
— .
— — Odd
v v — — Z ( x ) v v X ( x ) X ( y ) — Y(X) Y ( y ) — Z ( x ) z(y) Z ( z )
v v — — — v v odd — — even Y(y) — odd — odd
v v — — odd v v — Odd — Y ( x ) even — — odd —
vv — — 7.1*) vv even x(y) — odd — — even Z ( y ) even
v v — — even v v X ( x ) even — — odd — Z ( x ) even Z ( z )
v v — — /.(x) v v odd odd — odd odd — even even Z ( z )
v v — — 9 von v v Z ( x ) Z ( y ) even
V V — — — v v even even — even even — odd odd —
V V — — odd vv X ( x ) X ( y ) — Y(x) Y ( y ) — — — odd
v v — — — v v odd — — — odd — — — odd
V V — • odd vv — odd —
Odd — - — — — —
G.(F ) 1 u — Z(x) — odd —
odd — —
even even even
f l , ( F ) 1 o — — even
i i l < F l > Odd
— — — Odd
— even even Z ( z )
T,(F ) 1 o — — .-• ' Vv — — — — — — odd — —
J , ( F ) 1 o — — — v v — — — even even — odd — —
K,(F ) 1 o — — — v v odd — — even Y ( y ) — — — odd
L (K > 1 o — — — L l ' K l >
odd — — — X?x) — — — X?x)
M,(F ) 1 o — — — v v odd — — — X?x) — — — X?x)
N,(K ) 1 o — — — — — — — — — — — —
0,(F ) 1 o — — — v v — — — — — — — — —
1 o — — — vv odd — — — X?x) — — — X?x)
wo
X f « ) X<yy« X l x z ) X ' » > 1 » ( > « ) X ( x z z ) X ( v z J ) Y ( « l Y ( > y ) Y ( x z ) V ( « y ) Y ( y > ) V(vx2> Y ( y i 2 ) 7.f xx ) z«y>» Z ( x x ) Z f y a ) Z ( x z J ) Z f y z J l
/ l a x ) / ( « • > Z l«> ) / l i a l Z f x z ) Z1 > r )
Z l x x ) Z l y y ) Z f x x ) Z ( » y ) Z l v x ) O V O H e v e n
— — — — — odd odd
• Veil e v e n • v o n • Veil W e l l Zf x z l Z l v z )
— — — odd odd — —
odd odd odd — — — —
X l M ) Z«yy> Z f x z ) • v » n e v e n e v e n e v e n
ft i n n oven evon Z ( « y ) Z ( y > l e v e n oven
Z f x x ) Z l x x ) Z < z z ) Z ( x y ) Z f - x y ) oven z f x z )
/ r i m ) 7.1 - a x ) evon Z ( x v ) 7. (x> ) e v e n Z f z z )
Z l x x ) -7.<xx> — Z ( > y ) - 7 . 1 - x v ) e v e n - V . f z z )
/ . « « « ) oild Z . x y ) -7.1 x>> e* en - 7 ? z j >
— — — odd •77xi i — —
D i M Z ? x x ) odd — — — —
Z l a a ) Z l X X ) Z l r . » ) ev en z7xo f j VHII 7.?xr.)
f r i m i z ' / x x ) 1 Vf l l l X ( > y ) /.(»>•) nvon Z ? x z )
-• — — odd 7.f xy ) — —
— — — odd Z.f a y ) — —
— — — odd -7.fav> — —
Z { » « ) 7. ( v . ) Z < « > / f a y ) Z f y z ) 7 ( y « >
— - — odd odd — odd
oild odd odd — — odd —
/ ( « « ) 7.1 yy) Z ( » n • 1*11 •f ven 7 . (xz ) e v e n
even e v e n R V O I I 7.1 xy) Z f y x l o w n Z f y z )
Z ' « « ) /.(»•>•) 7 . ( zz ) Zf *>•> 7.(yx) e v e n e v e n
e v e n ev411 e v e n e v e n e i e n Z f x z ) Zf v z )
- — — — — odd odd
o<ld o.ld Odd odil odd — —
— — — odd odd — —
odd odd odd — — — —
Z . d i ) ?•'>•>> Z ( z x ) • v a n • von e v e n e v e n
e v e n OVOI I • von Z . x y ) Z ( v x ) e v e n e v e n
odd
o'ld
— — — odd O d d —
— — — x f y x ) X ? z y ) — —
— — — x f x y ) x f x y ) — —
— — — x f x y ) x f z y ) — —
— — — x f z y ) x f z y ) — —
— — — a
- X ( i y ) X ? l y ) —
X f x x ) X f v y ) X ( » ) X ( x y ) x r y x l X f x z )
— — — — — odd
X f x x ) X f y y ) X ( z z ) X f z v ) X f y a ) oven
odd odd odd odd odd —
X ( y z )
odd
— odd
— — odd
o d d
o d ' l
o i l d
o d d
m i d
u i h l
XI x x l X ' y i l
odd odd
X ( x x ) X ' t t )
even m e n
A l x v ) S f v x ) X ' x z )
odd odd —
— — o.lil
e \ v ' i e v e n S ' x z l
X f x y ) X ' y x l even
oild
mid
odd
odd
oilil
odd
odd
odd
odd
oilil
OiHI
X (v r.)
Ollll
« k m
H v z l
add
V l x x ) Y l v y ) Y ( Z £ ) Y f x y ) Y l v x > Y fe»> W v z )
— — — — — odd odd
Y I x x ) W v y ) Y ( z z ) Y(x>> Y l y x ) e v e n e v e n
odd odd odd odd odd — —
— — — — — odd —
— — — — — — odd
— — — — — - x f v z ) x f x z )
— — — — — x f y z ) - X f x x )
— — — — — X?v»> - X ? x « )
— • - - — — - - - x f v z ] X ? x l )
- — — — — - x f v z ) —
— — — - — x f x r )
— — — - X ? x z )
— — — — — x f v » ) —
u
_ — X<yz) —
— — — — !vf>z) —
— — — — — - x f v z ) —
W i l l i l . x ) Y f k i ) Y ( > I )
even evi-n VI xz ) e v e n
Y l z v ) V ( > « ) e v e n V ( y z )
H i l l W v » )
VI X X ) YI v V I
I 'VI ' l l »-Vf*ll
— — mill
odd odd
o id odd
X ( x x ) X»>y )
e v e n e v e n
odd odd
XI x v ) X '>>) e v e n
e v e n c \ c n \ r x r )
oild odd
W h X ) W > v )
mid — odd
— — n d d —
— — odd odd
odd oihl — —
Y f x y ) Y f v x ) e v e n even
• i \ en i-ven W x z ) Y f v z )
— — — — a d d —
— o d d —
— — odd
— — — — — — odd
— — — odd
odd odd —
odd odd —
u d i i
odd
- X ? z y )
X ? z > )
X ? z y )
- x f z y )
odd oild
W a x ) i I v y )
e v e n c . v n
— e v e n even even e v e n
xfxy) xfxy) X?«y) xfxy) Xfxy)
Table A6 m = 2 — — P = o XXfx) YVf x) Z Z f x ) XYfx) YXfx) X Z ' 2 ) f x ) YZf2
v v XXfx) YYf x) Z Z f x ) XYfx) YXfx) XZf x) YZf x
V V XXfx) YYfx) Z Z f x ) XYfx) YXfx) — —
" i ' V XZfx) YZfx
C fF ) 2 o even ever even even even odd odd
D.fF ) 2 O XXfx) YYfx) Z Z f x ) — — — —
w even even even odd odd — —
W — — — XYfx) YX'x) — —
V V Odd odd Odd even even — —
W XXfx} XXfx) Z Z f x ) XYfx) -XYfx) — —
V V XX(x) XXf-x) even XYfx) -XY(-x) — —
W XX ( x ) -XXfx) — XYfx) XYfx) — —
K (F ) 2 O XXfx) -XXf-x) odd XYfx) XYf-x) — —
v v XXfx) XXfx) Z Z f x ) — — — —
M 2 f F ^ even XX Px) even Odd -XY?x) — —
V V — — — XYfx) -XYfx) — —
v v odd XX?x) Odd even -XY?x) — —
V V XXfx) XXf-x) even — — — —
V V XXfx) -XX(x) — — — — —
V V XXfx) -XXf-x) odd — — — —
V V XXfx) XXfx) Z Z f x ) XYfx) -XYfx) — —
V V XXfx) XXfx) Z Z f x ) — — —
W even XX? x) even odd -XY?x) — —
V V — — — XYfx) -XYfx) — —
w 2 ( F o ) odd Odd odd even -XY?x) — —
V V XXfx) XXfx) Z Z f x ) XYfx) -XYfx> — —
V V even XX?x) even even -XY?x) — —
C„(F ) 2 o D (F )
a o
odd XX?x> odd odd -XY?x) — —
V V XXfx) XXfx) Z Z f x ) — — — —
V V even XX?x) even odd -XY?x) — —
v v — ' — — XYfx) -XYfx) — —
V V Odd XX?x) odd even -XY?x* _ V V even XX?x) even — — — —
V V — — — — — — ' —
V V odd XX?x) odd — — — —
V V XXfx) XXfx) XXfx) — — —
M„<K 1 2 o XXfx) XXfx) XX'x) — — — —
N,(F ) 2 o even xxVx) XX? s) — — — —
O fF ) 2 o — — — — — —
P„fF )
a o
Odd XX?x) XX?x) — — —
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S, 5, M H M
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3 3 I I
6 B M M H
i I I
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| | I I I
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i i i i
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w
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I s a | 3
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m u
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1 i i
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* & ! i I | | i 1 , , | | | i I t i } | | i | | I | | | | i i | | , , i | } | |
i i i i i i i i i i i | I | i i i 3 H i i i i i i i i i i i i i i i i = i 3 ' ' a a a a a a a ~ * i t
J J jL JL » I i i I I I I 2 f .. I 3 | 1 1 3 3 . 1 * 3 . . . 3. £ 3. 3 3. I I i I i { I 3
1 1 1 1 1 1 1 1 1 s 1 1 3 | . I | 3 1 i 3 i i . . 3, 3 i 1 3 3 « i I I i . | I 3 * M M M R
I I I I I I I I I I I
S. =„ I I I I 5. I I I i I .1 H. 3 i i i i i . . I I i i i i i i | i I 3 I 3 m m * H N a «
i I i I I I I I I | I I I I S 1 a 1 I I S i i i j | i i j " | | | 1 1 i i B I J 3 M H
I I I I • 1 1 1 1 1 ' ' i 2 I i I ! i i 5 i i i ! 1 i i 1 I i i 3 i i " I M I
I i i i I I i i H i i I I i | S B i I ! H I ! I 3 i i ! B 3 n I 1 i 8 ! ' H 3 "* u. s « • » • • », s a i l s a a
I I I I I ! I I } I i i i 1 2 I 3 I i I 3 i i I S. | i I J, 3. a, i . 1 f i i 3 I f 3 ^ , C« W W W * W W W W W w
Table A10
- 126 -
I I I I I
H * B •* 5 3 a 9 a a & 2* c £° c"
X K M M 0 3 3 3 s i C 5 ° 5 ° 5 "
I I I I I
I I I I 1
i i i i i i n m i a i i i U ' i i m . ! i i i M n
M M l i H • N M M M H H I I M
i s l l l a j i l S i i l B l l l l i i I I I I I I i i i i 5 1 8 i l l • a s
I I I S I I & i fi I i B i i i i 1 I. . i I I I ! i I I i i I I §
I I I I I i I
a s v a a ! 3 . So 5 , 2 S
I I * *
1 1 1 1
i i i I
i 2 2 !
n i l
a, a, a, i H I I H
M •
a. i » 4 —a » I I
1 1
3, a,
a, i M M
a, a.
E, I N H K H
3 3 M M
a. 3 M M M H 2 3 3 3 i ^ a i># X M »< M
M M M
5, 5, Hi H U M
I I K M 14 M
3,3 3 3 "6 t-H *>G N N N M
* g * M
a. a. s s ~ o - a
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: i i l § | i i i i | S S S o i ^ s i i B f i B B S ' r j ? ^ c 4 r ° "-o - o **« • -©
M N H H '
I I I I I I I I I I I I I I I I I I I I I I I
H
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m H "
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a • • • i l l
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3.3 3, 3 i f, 3 i 2 3 3 3 3 3 3
• • . f . ^ • O # *4
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I I I
I 3 3 i » i i
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i ' i i ? I 1 i
i i 3 | 3 3 3 3 £ £ » £ i
1 1 i S i 1 i i
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f • * • • *
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* H I I I * I t
i i i i 1 1 s. i - O A * « d « i «
»« X H '<
i i i i i i i i
a s , * * * *
I 3 3 s 3 3 H
M U S H H • H V M H
• i i i i n i % S i • s s
2 5 S 3 I i 3 3 5 3 5 3
i : i l V, I 5
i & I I i I s
i i I i i i s
I I i I S I I H M
i I i | M . 3 • • M o • M
i i i h n i 3 I \ I d s
i i I 1 i a
l I I S I | 3
i 9 i I I 3 3
I | " 5 a I 2 1 I 5 2 - S 5
i i i 3 I ' i 3 O M « • N N
* 3 1 I 1 3 s
i i i | 2 i i
• i n 1 i 3
M i l 1 3 3
' i ' 3 l i s
Table A10(Cont.)
- 127 -
3 S , •
1. 1 I I 1 t £> w *
' i •
i •
1 ! a pi »•*
• 1
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t j H 9 %
fi B •n •*! V H
1 1 •
M 3 P
M
3 <-« H
M N
2. K
1 M
3. p
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a i M P>
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M
5. • *
3 a a 3 i r a ? r • i
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s A a i
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E.-P
1 M
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P
M M
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n
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p p
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p
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a i p
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M •< 1. — • •
5 ii 3 S i n . i S S l i p • «• i i
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£* 3* ' i
1
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H M
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r i
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N
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P_ p
i i > 3 5 « K
Table A10(Cont.)
- 128 -
>• « <
M W H W H U M ** **
I I I I I I I I I I I I 1 5 1 D
l I j S ft l I l l l l l l I
I l l I l l I ' l l l I l | E i
I I I I I i I I i i i i | S I 5
I I I I I I i i I I l l { E 1
M M M W N)
C t« K U ~ S U t S U fct U L .
3 )3 3 5 5 * i
£ i a
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E ft
I I ' 4
I a i i i i i I i
I a I I I I I I I
I I I I I I I I I 1 ft I E | | I 1 2 i D S
1 1 I I 1 I i I I I I S { i 1 E { i
i E i i : i i
i E i i i i i I
ft E i i i i i |
i E i i i i ft |
ft S ft ft ft ft i 5
l l
I I
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\ * 1 * 1 * " l * . M N M '
1 1 1 1 ft E i a
K
B M Mi
1 1 1 1 ft 5 >
K X M
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B
m M
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5 •
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8 «< N
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ft 1 1 ft 1 0 0
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9
|
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I I i i i j 3 ft I i i i f 3 ft 5 f
i i ! I i | 3 ft i i i i { 3 £ a a
I I I l l | }
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a, 2, 3. 3. & 3 § • N x m H u
i i i i i i I
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I I I I I ! I
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n K * M M ft I 2» > i 2e ^ » I * f * S V
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» « i
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I ! I I ] 3 ft I
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j j j ft I ft I j B a 5 ft I ' s i
a s ft i ft i i n i B s I S i
1 5 ft i ft- i 1 S } B ft I £ !
N » S 2 I S £. M ! I
I I I I I
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3 i i i > i i i
ft 3 4
3 3 i i i i i 3 a
3 3 i i i i i I I
1 3 i i i i i
I i l I ft 3 i 3
l l l l ft 3 i 3
H H M H H
S o S . E > 5 . f . * i f
I 1
I I
I I
I I
I ft 3 i 3 | I
I ft 3 i 3 I I
I ft 3 i 3
ft 3 a
I . 3 I I I I 5 . 5 . I I I S . H o l M M M « B r •* M M M M M N M
ft ft I I i ft 3
I B I I I • N
S 8
1 8 1 I I R. 8 3 S B
» ft I E 8 I 1 ! | ft I § I g I M
9 tt M H M M
. . . N O . 0 N N
B | 8 ft I | 1 8 8 H M B h M M I I ' f If
I B I I I J a I i 3 ~ 5
I 8 a
O N O 4 0 N •
S e>- a & c. N « N a a c. N «
J N N N N N
I ! 5 I! S li S
I i >
a 9 I
V K f H
Table A l l
L t a. ?. i * i i y i
i i i i
i i i i
I I I I
i i i i
i i i i
i i i i
• i i i I ;
| ' M |
1 f 1
I I 1
s c t
i i
i i i i i 2 1
I I i i I jj i
i i i i
i i i i
i i i i
i i i i
i i i i
i i i i
i i i i
• t m m »• • |
a h a a. • * E B r M
_ • • i.
I I I I I I
! ! ! ! ' •
1 i
i i i i i i
£ M M M - t •
?. 1 3. > I . * » • • •
a 5 a ' 3 I 1 A r» & * • • •
t I I I I I
- 129 -
t i I j !: 3 j i I ;: !i S f! !i !• : : ; i ' j j i
i i i II j j B i i c i i i i j
i l l
I • * | ! | ! I *« 3 I • I ?. | i j S l l I l J s a l l l l i
E ! :
i J 3 t i l i } s 3 i i i i i
I I . 3 i " i I 3 3 3 i i I I i e t c c t : :
i j 5 I i i I 3 | 4 i i i i i 4 4 4
I j I I I i l 5 l 5 i i : i i
S 4-J J 1 4 f 4
i l i i l l
l i U l l h B i l l
l D t a
I S I
t I I
I I I
I I I
I I I
I I I
( I I
I I I
I I I
I I I
I 1 9 3 3 l
l I ? a
I I 3 c
I I 5 4
I 5
I I I I 3 I 3
I I } i I I i j 3 ; I I I I i r i ! I I I ! I i E. i i
I I I .
I i 3 } i l i t 3 j 3 i i i r a
I I | 3 I i I 3 I i 3 i i i i I i
l l i 3 I I j 3 I i 3 i i i 3 a a
l l l l B E 3. i B g g i S i i i i l l i t s S
t i l l I I J 3 1 i 3 . | i I 3 i i i
n i i i i i i 3 i • 3 i i * 3 i i i
l s 1 l 3 j i I j 3 I i a i i i • f i
i i i i i i t i i . i i } I s
i i i i i
i i i i i
i i i i i
i ; i i
i i i i i
I ! I l I 1 3 i 3
I I I I I I 3 i s
! I I I l I 3 i 3 t :
l l l I I ! 3 i 3 4 1
I l l I I I 3 i 3 5 ?
I l J S S J B J I I [ i I
I S I I . I j 8 f B 1 I S i i i | B I i E E B B j B I i \ I E 5 I £ I I £ f • £ I I 5. l i s
J S ! i I i ! B J B . i B i i i i B 1. i B B B B \ 8 ! i j i E a a " ? a ' 5 a a s a a i
} E i i l i j | ] B i i n i i i j B . . i n 2 B 3 } S l i } i l i s :
R { l J I l j ? a i I I n I I i B j I I 5 | i S ? j I i } I 3 4 4 4 4 4
5 5 i I I i i I I i i I E I I I 3 } i 1 S B B a B } i I j i B ? i I i i i i i ?. i i • i
j j . i i i n | B i . i . i . ; j i . i j j i i | . i g i \ i i i i & i
T a b l e A l l ( C b n t . )
- 130 -
I I I I I I i » I i s ' i i s i | } i i | i i i ! I i i { } ] } i } 3 S N I N
. - I ?
I I I I I I I i | 3 I i 1 i } 3 j 3 1 i 3 i i i 3. B I i 8 3 8 tl w w 3 ** " *S
5 2 3 5. B
I I I I I I I I | 3 1 i I i 3 3 3 3 ! i 3 i i i I 3 I i a H 3 a " 5 5 5 : r s - ~ ~
3 5 3 a a a
I I I i i I i I 3 J B i 3 i B. 3 3 H i 3 i i i 3. 3 3, i 3 3 3 3 | 3 I ^ • ok * S < 4 -N# o» " O os # 4 * * • «*4 ON o» ON ON 3 ON B S
L s s • r r
c n | B 9
I I 3 a
i I I I I I I | S
! I I I I 1 i | }
I I I I I I I I I
l I i l l l I i i
I l
l I
l I I
I I I
l l I I
1 1 l I
I I I I I I 1 I |
I } 3 3 } i n
• i i
S a a s S | n a n a f » **0 "-O ON ON* O-O .0% ON OH «4 <« I *«
M N M *« i f *4 I »4
H N •« H
I I I
I I - i 3 3 3 3 i I 3 3 I I a 3 i ft 3 3 a - - - - - -J 4 j J
3 3 3
i 3 ' 3 3 3 4 i
I | I 3 M . M
I i I 3
I | I 3 4
i i H i i 2 I i 3 i i i L i i i LL&l i 1 W H M «
i i I 3 i I i i M i " " 1.1 i i 8 .11 B &
B ? ; B
i l l I 3 3
I 3 I 3 s
l 3 I 3 i
I 3 S I i 3 3 I i fl I I i I B 1 B
I i I I I I I I I
i 3 3. I i 3 i i i i i i i i i i i i i i
i i S i fi
l i s t 5
I l l I I I I l I I l I l I l l l I l l l I i I i I 3 i 3
I | 3 i i 5 3 i i 3 I i I I I i i i i i i i i J ? 5 5
I 3 3 i i 3 J i I 3 I I I i i i i i i ii i i i i
i 1 3 I
I 8 H |
Table Al l (Cont . )
- 131 -
I I- I I I I I I I " 3 1 fi s B
I e l I I I I I I I I l l s
S H i l l I I 8 i I S | | I i B | I U 1 I I 1 I I I I I I 1 8 I I I I i I H - - - - - 3. 3 Ti C H h H M H
l I I I I 1 3 I i i i I 3 5 * l 8 l 1 5 « I l I i I l I ? £ B a
I I I I I I I I I 3 I 3 B a
I I I I I | 3 I i i i 1 i | s 1 8 i 1 S S i i i i i i I i i i i i i , i I B , §
I I I I I I I I I I I I I I I I I I I I I I i I I i i ] i i i i i i i i I S i a S B w w
I l I I I. I i I l I I 1 I 2 s I I I 3 ! 3 I I I l l I I I ! I i 3 i 3 3. 3. i 5 $
i I I I I i I I I i i I I § 8 i I 8 3 1 3 w £
I I I I I I I & B i a
i § e s 5 a s • a - , 1 S j i l l i ? a a a g a g , • a s a a B s • , g , s o M
S 3 H U M H H ». H
i i i i i i I i i i i ' i s i i i 8 3 I 3 i i i i B 3 i 3 3 S 3 i I l i r r ? ? s r
• O N
a a a I I a
a ii a
3 x M
3 a a
a 3 ? a M M
a a
3 a
i 3 § i i
i I I
3 3 3
3 3
a •
3 3
8 I 3
8 « 1
8 I !
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I . ; 3 I
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I 1 3 1
5 S . a M M M M M
a a a a s a s s
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I I M M M M O
a a a a s I s n R M M K N M M H 3
5 5 2 , • M M M K M
§ a s I § a a a a I i a a K M C
a a
M 8 I } a E •
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3 5 ' s a |
§ a i g a s a I '
I 3 I 1 s a | a l i s
X H
a a
a i
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a I
I i I.
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K M M M M O
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9 3
I s a
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& a M M M M
s a •o a I I
a J a a << S« M M H M M *
l a a ? a
I a 3 f I
I 3 a B B
a s 5 '
t i a a
3 i / -» 3 3
t | a a
9 3 3 3 3 I » I I I I • O K © • H
9 9 ' -
Table Al l (Con t . )
- 132 -
i I i I i i i i i i
M M M M M Q * . • O
5 a j 5 5 i i j I i * h I M •
ft • M • A • w a s* so
H l h i i j f i I » I * *«
M M M « S, t* M S# W
a a 3 a a i i a i J ? J 5 ' 8 * 1 * 3 5 >
a a a a i • i i i I a s s s M M M M ft ft B
a a a a i i i i i i W • N H 9
3 5 H M A O M
- a a a i i « 3 i i 5 . 4 i i I
a a 111 i i I i i s i st st s i
• 4 K M M M O S O
a a a a s a | a I i 5 3 3 5 ? A
a a a a a s i 3 I i s | s | s
a a a a a & i | I i » » ! S I
K M M M M ft B
a a a a a i i i i i ST St Sf st •« N N N N M
a a a a a i i » ! i M M N * M
M •< St s i St
a a a a a i * 3 \ • i i i " " *
K M M M M •
•a a a a a i i j i i ft M • If if
2 I 1 * 2 I i * a
i ! * & s*
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5 3 I
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2 2 I
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8
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M M M M M M N N N N < < • < • < - 4 • < ox * x *s ox ox
B M • K H If I t M M
M N M M M N M N N N
0 • • • •
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I | -3 3 8 3 a s I I I I I I I I t M M kt w G> s* x* s*
I I n 3 s i 1
H M s t
M K M X M N N N N B M M O ft M N M M M O
s 3 3 8 l 2 3 3 3 3 3 8
2 3 2 2 2 > 4
2 3 2 2 2 '
a 3 i * * i <s *s a a a B P
O M ft N M M H
s 3 | a a a s I •a 4 4 a. 4 H ft M M w H
s a i a a g a i M St s t Sj
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I a a
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m N
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fi B B K S 8.
B > 2 I i 5
ft < i « s S
B
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3 3 1 1 2 1 3 8
a a a
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B*^ * "IT 1? I T s* s t w w N
i j n n M st s t SI s» x s i st s t St s i
I 2* n 3 S 3 !
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S| St St St SJ H M M • K W S* So V SK
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8 i -3
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a -<• i F 3 3
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4
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a 3 i S os 3 3
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Table AlKCont . )
3 M M M M
a a a s
M M H M M O
6 I I 6 I
! 3 i N
: 3 3 «•••
I S 3
I i 8
- 133 -• •< -e M M £
I | s s a a a S N I *« I
m «* ««r
0 • •« S H
£ l i 5 I 2
• 5, 3 3 5 M M M Q
a a 3 I »* •« H H ** ** w w S| Vf „
I I 1 1 M M M M M M 3
a a a a a a i * » # * N «»*>
E M i m • i H
M M M • M
• -1 0 - I •<
I I B | I S 5 i
i i a i i a a
i S S . M M M . K M M M M M M M
i s a i a a a i a a a i a s a s a B w C S> w w
*W M A A M 3 3
s. 2. a, a. a. • ' ? i &
i . 1.3. a, 1 i i ! i i 9 s « fl .
M M M M M M M O . 3 - 4 3 •< <
a . i a . a a a a a & i f s f i s s M P M • H I P K P P M M N H M M M M P P w w w S w p w w W W
8 I i I a
a s i i f a
O - < 3 < M M M
I M | a a, a. a. I P M *4 ** •« H M H M M
H *• m n N • H 4 * * I *4
2. 5
t 4 w M
A 9
a, I w v
3 3 3 r n
1 g
I M M
S, 2 V. 1 •4 M W M
M M
3, a w w
3 2 3 i a <••« H
M M M M M
a 3 a a 5 *-*9 -*«>
•4 *4 *< M M p H P N *«J
I I 3 3
M M M M M . 3
a, 1 s. a. a, i i s " i s B a 8 i
a a I i ] i i
u a a a a I i a i i s ? s s e
3 3 3 3 3 §. i I 3 3 3 ; a 5 c r T
3. 3, 2. 2, 3 1 I I I *
3 B 3 ~»
i a M
i s
a 5 M 3 - * 3
B 3 B
O -4 •
S B 3 3
i 1 2
i i | a a s. " t i t
M M M
a, a, a. i M M M
i 3, a. 3. * s *
3. 3 3 i — • - « 3 *<3
t a B B B a B B B a U
i i i a a s 3 « 3 i s s g g s g s 9 <«» »>* m #•» *<0 » - •» * K
N N <4 *« M 4 *4 *« S *«l w w * 4 w * « t w w w * « w * « i w
& a | 3 I S 5 M N
» i i I I I I M M
* 3 . *
' l i s
i i a | i a a M M K
w w M w H w
M M M
i a. 3 &
I 5 ri
I 3 M
i 1 1 S n g, i §, g n 3 g i M M P I M I M M I * I P I N M B P P P P N P M M M p f i
M M M
M m •»!
i 5, N.. E, E, I S S B S S
« « #^ v
a. a, a, a. a, i ' » i i M H M H N H • « « « • « • < .**t M
M M M M M
2, 2, 3, 3, 2 —o ~*o t-a *~t
M M M K • «4 X I *< • < <4
«*> w w w w
M M M M M
8 a a a s I P I N *<
! i S " -M M M M M
a a a a s
i a i s
i
3 H
I 3
i i i
3 3 14
I l i a
I i i s f i i | a i •
i i >
i i i i i i | i
i i i 1 i • j v r 3 v i
I I 1 I I
• M l p H J N P P P P 3 w i , i * P P P P w w
M M M M M 3 O N 3
2 . 2 , 2 . 3 . 2 . 1 1 1 1 * 3 1 P N P P *« M
* • »< » « • « P *C
I l M M X M M 3 3
2, 5. 2. So 1 1 i 1 »
H I X X N 1 P M P • «
M M M M M 3 N
3, 2. 3. H. 3 . 1 • 3 l I i p p M «< p • • • P M P
3. i 1 i l i I 1 i i I i i H i i 1 3
I 1 2 I i i a a 5 3 3 i i 3 3 3 3 3 3 ! i ! a i i a a
i i M i a a M H P
P B P
I ! 3 | a s 3, 1 i 3 3. 3 3 3 3 3 i I B \ \ i a I " » M *
H ** **
M M : H
• i i a m a
1 3 1 2 3 3 3 ( 3 3 5 3 3 3 3
3 *C M M M .
h N N H I - S A A i , 1
S P H M <4 M P N N
A 4
B B a B 8 B i A 5 « 4 A A A 1
1 M M l S l i . 1 1 i i a . l l l ! 3 « ^ *-3 « -3 *a3 « - 0 « 4 « - 0 rWJ
1 " M 3 5 3 | .« i 1 § S 3 M I «tf *4 *< <<
3 M « N
I I a I I s S !
I | a B 3 5 3 £ i i 3 5 3 M I B p i • • n
M M P M H
I M M
A 2 1 A 2
-* 3
a i
i I
3 i 3 5 5 5 5 5 S : 5 5 3 5 H * —«D *» . <-« A M ^ <-0 #S M OH A H M P P i N i M P P I P I H * M P P H M M P S p p M M M w w w w p w p w w w p w p w
I a •< S S M K t< I M C M C C £ Q I
w w w w w w
' i M i J - i J H ! i s i i n i J i
3 N 3 M 1 3 3 I S * C s C
3 N 3 M
1 1 3 3 1 9 9 3
* K
I I a
W w w
8 N t 1 2 I
B * i i a a i n i i i a n U i
P P •* P p p P H M M
B B B i i B B B B B B , - * 3 ' « ' < a * - * *-a ^ -*» *
C H 3 3 f*
& a | | | a I 5.
3 1 3 1 3
3 3 3 3
I I S 3 * I ?
• • H s i 1 1 9 9
i '* i a s |
T a b l e A l l ( C o n t . )
- 1 3 4 -
_ i i ^ r . „ » i
C '2 «Z
O f K
8 I | " 3. 3 .
3. >
3 1 O l -J? T i l
• I * 3
° w ° V ° w QZ
3 3 3 M I * -
• H X I X •5 I •*
s i > i « <
H H "
M m m » » • 3 3 3 3 3 3
5 5 3 I 3 s g i is H 3 3 3 3 3 3 3 . 3
3 3 X X • * * • M X
3. 3 H M
* » * * * • , » ; V V * « * „ * „ * , » „ * _ ~
• o o o o «3 - » ° w ° w * » . ^ ° w ° w
i I I I H § s i ' i ! i 0 I I » ' i I i <
M • K
M U M
3 1 3 3 3 3
5. "• fi
l l • R • • « ^ w w
§ l 8 \ g S , g g , § \ B i | \ B i |. 1 E \ \ \ \ . | | 1 | § § I i i 3 j i
» ' 1 i i i I * ' i a » • i I i i C" - • • »
1 i n g i i i i i ! § I I
• * "1 * 3 3 3 2 . 2 3 , H U H • * » X X X
n R E s
3 3 3 3 3 3 H H H
I I I I I I I I
I I I I 1 t I I
i i X M Mj
A i 3 . 5 ? >
X X
3 3 X H
* • • 3 3 3
3 . 3 3 .
s a i l » M
3 3
M M H M
s s r • •> «« K •< X X X •« 3 3 i
f I i i j j i i • J j j | i j i i ! i
I l I I i I 3 i i a 3 3 1 X M M M 9
w v w w
i l l i
> > < < i > ' > i n i 1 1 » n i » i i * i < i § § H i i i > i i i 3
H M • M M M K N
I I I I I i I i I I I ! I | S I I | { § g | | | | | | | | | | | | | | | I | I E i • ( • 7 K
i i ' i ' i i I I ! i | 1 I i 1 i i I I i i i i ! | i 3
I I I I i I I I I I I I I I I I I | | | | | | | | | | | | | | |
1 1 ' ' 1 1 1 1 I l I I I I I I i i i i i i j I i I I I I I I I 5 i 5
T a b l e A 1 2
- 135 -
i i i s i § * i i 1 1 m 1 1 i a k i i 9 a a n * 1 1 s m 9 1 1 m i • m =
S 5 3 S | 5 S | S S S | S a 5 , e s 2 r s 5 3 , a s a s ' 2 i j , ' - < » n h 1 U 1 i H 1 i U 1 i 3 3 3 5 3 * J i 3 3 3 3 3 * I s g i
H 1 1 1 s 1 n 1 1 1 ft i i § ft i 1 1 a 1 1 i 1 1 i i s a 1 5 ft 1 1 1 1 i n j a j a a a a a £ a » £ £ £ £ £ £ £ £ £
1 9 3 ' i 1 1 1 i 1 1 1 1 1 1 i ! 1 > i ! ( ! ! a ft > i a a a a a ft i I a i i a a
a a a a a a a a a a a £ a a £ a ; £ • ~
3 3 3 3 3 H. 1 H . & 1 E, 5 I 1 E, H L 1 | M I I a L 1 I « S B S a * ' I • i W V <W H
H i f 1 1 1 i i ' u i 1 u i ' u 1 1 • 1 6 > i § h i i i z i t i & a a a a a a a ? a a a I a I a a a j a j a
i
: a : I s a i 1 3 3
i i
i ' ' ' • ' " i • i i i i i i i 3 3 i i i i i i i i i i i i i i i i i i i 3 i I t I f « « • -4
• I I I I I
1 1 1 1 1 1 ' i l i ' l l l ' l 1 1 1 ' 1 1 1 1
M U M
i • 1 ' ' i i i i i i 1 1 1 3 i i i i i i i i i i i i i i i i i i i i i i a i s a a a a a a a r 7
' 1 • • I f £ 1 1 1 1 y 1 1 a 1 1 n 1 1 1 1 1 1 1 1 1 . 1 1 1 1 1 1 1 1 1 1 £ • £ £ w £ S ° E ' * T ^ z
> 1 1 1 > 1 a a 1 1 1 1 g i § 1 s 1 H i 1 1 1 1 1 1 1 1 1 1 i i i § 1
C*?? 0 " So £« So S S S 2
1 1 1 1 1 1 i L 1 1 • 1 1 § i 1 i 1 i t i 1 1 1 > ' < • • 1 • 1 1 • » 1 1 1 • i
1 1 1 1 1 1 1 I 1 1 I 1 3 3 J 3 I 1 S 1 1 1 I E I
1 • 1 1 • 1 1 ! 1 n 1 * 1 n h 1 • « ' ' 1 j n » n n i s ^ > i > ; ? ? * a a , 2 a . 2 x , a a ? 1 2
T a b l e A 1 2 ( C o n t . )
- 136 -
ml 0 1 a I E I r l • ! t . i J*l - ° l . ^ l « l i n n l s i > i • < e <•» ^ * * * * * * *
"»4 " " l
O 5 L 3 * «*• *
• • • • i i a a a
i i i i i i i i i i j i 2 M 2 § I I 2 2 a i a i | J I l i S a " » » » *
i i i i i i S g f i l l i 1 H ' i I i H i i 1 i i i M i 1 1 i i
1 1 1 . I OS8 - 4
M M M M M
I % i % I 1 1 I 1
w ~ tS 2. it • 3 3 2 3 1 3 3
3 a 3 3 3 3 2. S S, 5 M H H N
' 1 1 1 I I ! i > i l l s I I I 1 1 i I 2 1 a a
a a a a & 3 £. a . " " * iT
a a a a a a a a a. a a a
i l i U ' 1
X 2, "* « 1 1 1 1 1 1 s
1 1 1 1 1 ' 1
Sj s i st SI VI
M M M M M
a a a a a 1 | a S. a. a» 2. " i 2. 2. "*
2 I I
I I
2 2
• • 2 2
3 3
l i I I 8 I 3 3. a. 8 ' ' 2 2 3 * I
! M 1 I 1 2 3 3 I 3 I ^ ~ ^ £ . -«B * •
SI SI st st st
1 1 a a a a a I I M K M >f M i * H «
•-•> —» o O —» M M M W H -1
I I « < 3 a I I • 3 3 a
1 ' 1 a a 1 a a
M M M
g a a M X M
St SJ s i
1 1 1 1 1 1 1
2 I l
1 I i St
* ° 2 I &
» * s t a a
I 2 2
I 3 3 3 3 8 3 3 i I 3 2 . i -•o *•« ~. M H M I I I I H H w W w M s r N V w
• 1 2 2 2 2 2 2 1 1 2 I g * H
' 1 l l U L l ' ' H I ' H H M H M M St
I I a \ 1 1 1 i 1 1 n i 1 • m m
' 1 1 1 ' ' 1 s ! 2 ! 1 1 1 I " 2 i i 1 1 I 1 • 1 2 I 1 1 i 2 i i 2 i 1 1 j *
i I 1 1 i i ! i I I ' 1 E *
2 2 1 *
£ I 2 a HUM
I i
1 " 1 a a
I I I I I I I I j . I I I i 1 3 { { a & 1 a I I 1
3 2»
i <
3.
I i
3 3 2 ~°
s a 2 3.
3 2*
a 3,
3 3 M M < • <
3 3
s s ^ s ' s i i s i a j i S a f i & a j . i I i i a m * a m
5 a 5 a
Table A13
- 137 -
• • • •
i 1 i i I " i i i i i i i i 1 j i i i i i i | I i i j • i i i | ! i i I
- I S £ *
i i i i i i i i 1 I i i 1 * l 1 I s * i g i i S 1 i i i i i 1 1 1 i i i i 1 g - - - i s
i i i i i"i i i 3 I i * * • ! 1 H ' * i i i i g 1 i i g l I i § i • i I ' i 3 2" M 'l? ? ? :? ^ ~ ^ — H H H H M
I i I I I I J ! j i i i 1 5 I i M
I i 1
•t *•*
ft 4
* 3 JT'
11 1 I f 1 H H
•I a B 8 8 ! J i s s j i ' 3
M I N I M ; i < < H
i i & i a
5 M
I 1 1 E 5 I « •
I 8 8 8 8 13 •3« ^ A ^ 3 S
N M M M
& 2P 3» & 7 Z i 1 i m m
I ' l l ** n n
S 8 i 5 > m N
1 1 M M S 5 a> s m n
M M M M M
B B :! K S « N M N H
8 8 8 8 8 3 I 3, - « • < < - • a » 1 I S I Z £
i i i i i i I I } ! > • | M 1 M •
I * 1
M
e M 8 3
M
1 1 1 3 i i i a a a 3 ; ! 1 * s, «. & « I
* * * * * * * *
i i f i i i i i 5 1 > • 1 8 3 1 **
i i 1
<* **
1 1 1 | " | 1 i s s a | i i s» > J> S» 3 s
H M tt m
O M
U i i 1 1 • 1 1 1 • l i • i n i • i l l u u • I s i i n i • u < n • & 5- * * £ £ £ £ E w i £ i E i - J i J S i * ?
i i i i l } | i i i i l S | i i i & g | j j i i i l i S i i i i l i i i i i f i £ £
I iiIiH ' « • l 8 8 } i I I , j 8 I I I ! I i i I ! I I I ; i i ; ; i £ ? c
' • ' i i i i i i i i r } g i i i i § j . g i i i i i i i i i i i i i i i £ £ '
I I I • S £ B
I I a i # tt K
B I
M PC M H M M M M M a 5> » i i
I i 8 E 8 8 I I I l 1 i 3 I 3 * & a» *? £ N
Table A13 (Cont.)
- 138 -
i i i ' i ' " ' ' * * ' I i ' * I 3 * ' a ' ' ' ' H i H g § G i I t i i j i a
I I i I I I * * I ' l I | l l i H i J 3 ! ! I I I i I I I I ! I i i i ! 3 , 3 S £ £ £ £
i • I I I I I I • & * l | | i I i I g | | i I i i I I I I I I 1 i I s i i | £ g
W M M M M N • B B - - - - ' ? ? r v j r i f
I I I I l. I i I I I g i i i i i j g B * s * * I I s * s I H I I ! I | ! g "
' ' I ' I ' I l l l l l l I I I I i I I i i I l ! I I I I I I I I I i i I B I H M M
n I i i s I t t B
& 7 U U i s 1 1 1 1 1 M i ' 11 i i 11 j i y i i | s i y
: : ? : £ £ • ? Ho 5. ?° £ ? ? !
' I 1 ' I 1 I 1 l i i l g j l I } E I .' | i i i i g j j i j j g g j j i i i i i f i t i S 1 1 ; ; ? £° ~° »° x
I i I ' I I I I I I I I I I r I I I I I I I I I I I I I I i I i I i i i i H i B
! i i i i i i i i i i i i I | i i i n | i
! ! M I i i | i i ! i i I § | i i \ s s i
• a i i ! i ' ' ! • 8 § | 5 • M 1 B M « M « 3 3 3
• . • • H O O K s. i i I s i i § i i " i i 9 a i 3
! i i ! I f f f ! i i I M I I I
B i B H B S
l 3
o M • • M M 2 3 J 3 " *s
3 3 M l H P
1 111 . 1 1 ' 1 1 1 N ' i n . i H I n 1 1 1 1 1 H i 1 1 1 i M M I
J . i s
I 1 1 M ' 1 1 11 ' i a 2 n 3 ' ' n s s n > ' M i 1 1 1 £ 5" ? ? S" S* 2. A S* S* M
& 5 3 9 M
w H M
Table A13(Cont.)
- 139 -
H 1 L i '• 1 1 1 1 1 1 ' 1 1 L i ' L I 11 H i ' L I L I H I ! 1 1 n , i J
i « § i 3 • 1 i • 1 i i 1 M i i n i § j i i i i i 1111 i i i i n i n a. 3. s« ? S* £ E £ E «
J, *•*
j m i ' 1 I ' ' i i ' 1 I I » ' i 3 I i i i i i i i i i I I i i M I I i s
i 9 i i I 1 1 i 1 1 I i s 1 i I 1 s N I I i i « s 8 i 8 8 8 8 i I M i > I I c - r a S ' S * £ 1 a a £ £ £ £ £ £ E E
| i 111 • < i ^ > ! i • M i 1 M i H i i 1 1 N i i i ! i i i i 8 M § a. "
U l U ' * B ' ' M ' K I i & > f 8 8 I I I i i i M i M ' i i M I M
11 i 11 i ' ! ' i I ? I ' i f i i 8 18 i i i i. M i s f 8 8 i &B i i , 8 8 i i i i * , 5> £ • • • • • • i ? «T "m
M i n i i ! i i M' • I I I i M M | | i 1 1 M i I U . 1 1 1 1 91 1 1 91
a B B = a . 2. 1 5. 1 1 1
h • m m m M i i ! i i I § i 1 i M I a l i i i l n i i 1 1 N « s i «
£ £ 2 S * ? s ? £ sr* « £ a £ i
H i l l ' 1 1 ' i l l ! ' I I 1 U U I I L > L l l l l ! I ' 1 1 1 i 1 I I
U J U ' " i i i 11 i » ! i ' 1 1 1 n § i . . m i i i • • M n n
3 3 I 15 I M . i n i i i n u i M M i l U » 1 -i I ! M !
Table Al3(Cont.)
- 140 -
am• • 1 1 1 N & I M t! I M C K M N N N I ' . N (S N N K S B * H C M M M M H S M M M M. M Ml E 2 " » • • • P m m <-d «
i g a ; : i s a 1 • a i i 1 a a
S a l s
I I I I I I l l I I i I = j I I § I I I § I I I I I I I I I I I ! I I e • 3 M » I j K a
' ' ' ' I ' l l l & i i [ j j i s | j j £ i ! j i i i i i i i i i i i i i i i i j i ' j j
I I I I I I I I I I I I I I I I I I I I I ! 5 i 3 •» • • • « mm • •
J § B g 1 I i i ! i • i I i i i n i S g i i i i i i i I I l 1 1 i i I i I I s 3 » E . 5 t - » «
• ' I I I I I I I I 8 I H 1 ' M 3 I i 2 i i i i i i i 1 I I i i i I i i | I |
> • I I I I M I I I I J J j I I ] I J I I I I I I I I I I I :l I I I I I a , i
1 1 1 1 s S , , i , i i g 1 i I i i i I | i i i i i i i I ! I I I s I I s I s s a
I l I I I I I I I 8 * I g j I I } J I I I I I I I , I ; | I I I , | 1 , , | O - t ft M
1 1 1 ' . I I $ I I ' I I 3 3 I I H I H ' ' I I I I I I I ! I » I 5 mm n m
1 1 I ' I I I I I * * l I \ i i I ! 5 | a i i i i i i i g a s 5 i I I i i ? 1 3 3 I I • •
3 1 : ! ! I s ! ! i s S i C " 1 ' 1 ' 1 1 1 S 3 2 " 5 M & r ? i:
. . s i : s i I I ft I « • n M
t s B F 5 3 ; . .. , , j 3 J . t • , . i a 3 - i i : = i « ' , « H ! i « i i j ^ ? ? • < ? 2 - »
' ' 1 I ' § s I I I ' I E g £ ' i S 8 5 g i i I I I I l l i I l I l I I I a I 3
Table A13(Cont.)
- 141 -
I I I I I | | I I I i I 3 ! ' * ' » 9 I- 2 I n 3 5 I i l l i : I i
S 1 H H i i i i i i § 1 < ^ i i r H i i i i i i i g§ g g i i ? t . i i £ £ £ £ §• £* ? p.
i i i i 1 I g
l I I I I g g I I I l l g g g i i E E I 2 1 I I l I I I ! I l l l 1 1 1 ^ a h a
i i i i i | l i i i I 1 1 } i i i i l \ I i i i i i i i 8 3 H i 1 1 1 1 i 1 I £ ff? ? £* £o 5* £
? a a ? £* 2 S £ £ p a S* * £ w W W w
s 6 • 1 1 1 1 m • • u u - • • • < > ' > < « 1 1 1 1 1 s § ' § & i • £ » , • » • £ ? J?
I B" I J I I I M I § 1 I " I 1 1 1 1 I ; 1 I I ! I 1 I i '. § ! I E — O M M
« i i i i i i i I i i ' • M I i g i ' s ' ' ' I i i i I i ? I i j i i s i \ \
I U U i i i n i s 1 1 1 n i 1 1 n n i 1 1 n s § i 1 1 1 i s s E ° C ° C P £ • £0 =• w • a ? * * r> ?> w ' 2 2 If M H H
1 I I.I j I ' ' * 5 i I 1 I I 2 I ' I E 3 ' ' \ \ H M I I jj I f a E £ c° E* E ^ s' ?
I I I I I I I I I I ' » 1 1 \ I ! s 1 ' I 1 i ' J i ' 1 \ \ s I ! ! «•» C Z . • 2 * m M «XI
I I N O • N N S 2 N 3 " • S
U h , s 1 I f c ' i ' M s H s s 1 . i 1 s ' 5 a « a • i s : ?• ? a * a £ 5 s - 5«r i l l ? * E* ? = ? r a
I a a
Table A13 (Cont.)
- 142 -
r*« m ui
I i 3 I M 1 I § 1 1 I § 3 I B a 1 I 1 a n • J w pi >* M M * n 'n
8 J
I 3 B I B S C S
a s B g
3 3>
3 M
3
a. 5 a
> 2 3 i . 3 i
I 3 M i • i i § § B 1 1 i i i ! i i i a » 5=
a
I I I i I I
I I I
3
5
i i H M
i 1 i
• H • • • . • H •
! 3 1 I I l s § i a M
5 : s i S
1 I *
B B
C S 5» &
* § ' ' I f I I | | a i S B B n a 7 ? ! ' ' i ! i M i i i i f I
S t I ' I i S I I I s a i i i i i i i i s s i i i i I § i f
i ? 1 8 I
i 3 i 9 •*•
1 i 3 3
3 3 M M ~ 5 S S E S T ?
- - a i
i I
S fi ' I'I 5 i e i i >< W W
J 3 *
5 3 S
a
i M § n 1 i n i
3 ? I!
B fi M X > 5, P p
I B B 3. B
B
a. B I 3 § B i l ? 3 i
2i ?* ? I I 9 3 • I I.
7 * > B B
S B ' fi 3 ' • •
I 8 I ; i
Table A14
- 143 -
s> a> s« I I i f M ' M M M M M
3. i . ? i a
M 4
Pi
fi_ M 3 3
i i i i I 3 3 & * ? § I l a s s e s s i i S I" ? I. a. 5- !• 1
! I
C 5 S S 5. i - S . 3- i l l i I a n i l ' ' m m - • i
« M N H N M S • M N H H
3 ! s n M i i g h is H M It II S S f l f i
* ' I H • d
i | i t «
a 3
N I 3 »
I 3
fi X
I 5 I I I I I i i
5. 5. i i a I H i i I ' 8 S i 8 8 H I I * B I I
s S B I i i | i> 2, 5. 1 3 1
s & I I I i S fi 5» •:•
H H i I 3 5 2 3 3 I S
i i I I • • • • . • a N o
s i i s ' i i i M
i i i i 1 r 1 | S. 5. S- 5- 5. I 8 jj. | 1 1 I ! H 8 I
I I I I I I I I s i I • M O
I I i B S I K IS
i i i i i i i i i s I I I I fi
Table A14(Cont.)
- 144 -
3 3 3 3 3 3 3 3 3 3
' ' ' ' ' 1 ' 1 i l l . 1 i ' 11 11 I ' I 1 1 1 I I s i i i l
a a 3 3 . r 7
H • H
a a ? a a a & 3. 3 2. 2 a. M N p M 1 N M It H SI K S
I 1 I 1 I I I I 5 I i I I i I g a i i I a 1 I I S
o o o o o 8 S ft » ft a a a A s I I c J I 1 S I j H c 5 I S 5 i s I i s a s I 8 i I
i i l f i i i I « . i s i i I i i i i I I 5 1 1 1 1 ' ' 8 ?
I i I I I I i i 1 f i i i I 8 | S f i ! a t i I f j I I | ! | ! 5 ? i I H 8 . S
I I I I I I i i i i i i i i n I i I i B ' ' < ! i I i a a s s j 8 1 e s a a i a 8 ' 1 I 6 1
' ' ' ' ' I ' I 1 g i ' I ' s H ? S I i I I I ! i l a a a a M K M M M M * M M *» V »?
1 a 8 I
I I I I I I I * ' * i i s n i , i i i § a a M M
8 8 8 8 8 g | | I I | 8 I 3 I a a
I I a a a a a a 1 i a a
Table A14(Cont.)
- 145 -
1 1 1 1 1 1 1 • 1 1 1 ' 1 1 § i r | i i g i i i [ i , , } i ] 11 ] ,
i i i i i i I 1 ' ' * H i I I i i I i i i i 1 i i i I i i ! ! i i 3 I i
i i i i i i i i | 3 i i i i
i i i i i | ' 2 3 *
3 i I i l | 8 S i 8 S E 3> 3
fl I
H w *e
' * i ! 3 i 3 1. i 3 i i i a g t i n g 6 i i i i E fl
2 3
I I I I I ! I a M M M M M I M I I I S 8 S fi a 1 a 1 1 I
M M I 15 S > a a> 5 3 I I 3
B K
I 3 j s I & 5 a> fi
' 1 I a I l * 3* I 8 S
i i i i i i i i i I i i i I 3 s | i 3 i i i
5>
I J
B &
B B
3» a»
B B B 3 ! I
I I I I I § § 1 i i M ! 3 1 ' I 3 ' ' < i i a» a> s fl i 3 I I
i i i i i i i i I 3 I 3 f I I I 3 3
5. 3
s a I 3
I ! ; ; i i ! I 3 3
i i i i i i i i a a a H X rt
5» £ 2» I a a 3 a I 3 I I I a a a
M M M
a a a a B B
I 3 4 I S 1 a tt 1 3
i a I 1 3 l a a M K 5> 5>
a a 2 ir>
B » M < 3 1
Table A14 (cont .)
- 146 -
i i | . s s 1 1 1 » 1 s 1 1 1 3 1 1 1 s 1 1 H i l ' u s s s n 1 1 i » n w w * « w w — M
1 1 1 1 1 1 1 ' M ' ' i I U ! 3 * ' I i i " i a ' i I U s ! ! i i M i
i i i- I i i 8 I M I i i 8 I M ! i M i i i 11 I I M M
H U M a i i i i a s
i i i i i I a ! i i i ' I B * ' e i s
I I I I
I l l I
I I I I
• s s s 1 I a 8
I ! i 1
i i i i i | i i i i
I I I I I I I I I i I I I I I
I I I I I I I I I ! I I I I I & a I
I I I
I I I
I | § 8 I s I j a. a i i i i i i i i i i i | i | |
i §
' i i i i I I i i i i i
i l l
I I
I S i i i i i i I I I I I I l s a |
O M « M I l l l i l l l l l I I I l I
S fi S 3 a i 9
! I 8 l I I a 1 i | I l M M . B B 9 N U
a a a o a ! I I I I I I 8 8 8 8 8 I I 8 I
l I l I l 1 1 i I 8 8 i g i i 8 i I g i i i ; l I I i i 8 8 i I 1 S I I j i s f
1 ' 1 ' ' I I I I I I | 8 I | | I i l i | | , | i i i i i - i i i ; I | i | B ? n m m
' 1 1 1 1 1 1 1 1 1 1 1 i I 8 J N I I 3 i I I I I l I i I . I I I I I I 3 i 3
' i i i i i i i i i i i i i i i } i i , 3 5 I i i i I l l l l l i i i I i i 3 i 3
Table A14(Cont.)
- 147 -
i i i i i 1 1 1 1 1 i i i 31 i n 1 1 1 1 i i i i i i 1 1 i i i i i 1 1 i 1 1 ? i 5 S 4
' I I I I I S I I I I I 1 | 1 I jj ! I I I I I I I I I I I I I I I i i I ! i I n M a K
1 I I I I I I I l i i l j j g i i | jj i l' ! i .1 | i i i I a jj H jj I 1 i l i } 7 7 7 * 3 s 3* JJ.
8 3 1 i S •}
1 9 . 1 U K 1 1 1 1 1 1 I i i * 1 i i - 1 H H M *• » f> f
I I I I I I I i a j J
M i i M e M M M M a M M M M C. M M M Mk f » i *
I I a i s s
I I I I I IB S I i i i f | I i I i | « g , , , , i , , , i , j , i , , o N O N O. N
1 1 " • 1 I i i i i i i I g I i I i I B E " i " i i i i i i i i i i i 11 I 5
1 1 1 1 1 1 i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i I 3
1 1 1 1 1 i 1 i i i i i n i I I i B f I i I I I I l I I l I I I I I I 1 a
I I < ' < 1 1 i i i i l 1 g I i i I B i I i i i i i i i i i i i i i
B I I I I I i 1 S 1 ! 8 U I I I I I 2 B I I I I I I I I I * I 8 I I I I i 1 a» a x« a» a> a. • •
1 1 1 1 ' ! 1 > i i i i B i i « i i g | » i i i i i i i i i i i i i i i i i O N e. :-J
< 1 > > > > i > 1 1 i i n 1 1 n i i B i i i i i i i i i 1 1 g i
i i i i i i i i 1 1 • NO I 5 a s i s i i i i i i i i i i i i | | l B
Table Al4(Cont.)
- 148 -
I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I | | I I I 9 I 3
i I I I I I S I I I I I i S f i S § i i i i i i i i i i i i i i i § t
i i i i I I i i i i i I i i i i i i i i i i I i I g i
i i i i i i i i i 6 I i I I S i i B i i i i i i i i i i t 1 n . « m w —
5> <0 <0 fi H 3 I " i 5.
I I I I I I I I 1 s c N H N N 5 a» B . S 5» 7 Hp a» « I I 1 I
I I ! I 3 3 ! i | B i. ' i
B B 3 3 i l 3 "•
5 ^ 3 3 i I 3 3 I I
3 I 3 3 3 ' ' I 2 I i 3 8 ' I 3 1 ' I s n i n s n i i M i i
1 ! ! I ! ' 1 ! 1 1 | i - i | | i ! | | I | | § i i 3 | j | j | i i j i ] i | 9 9
3 3 I i • • ! i M ! i i ! fi I i s s s : s s S ' • 3 I . 3 S 3 I i 5 3 I § 3 H i H I i i
3 3
3 3 ! fi ! I 3 ; 8 1 i 3 3 ! ? ; 1 • i 1 i s 3 s 1 1 i J 3
i i i 3 3 I I I I I i 1 3 I ! i i ; i i i i i i i i j 1 3
B S 3 3 I 3 I fi 3 3 3 3 I i 5 I
Table Al4(Cont.)
- 149 - i
C 5
ABB 5 3 3
6 j J ? V t z
fl a § § i J J J 4
A A a a B » 5 « * If II
& I
a . e a i 1 & £
I I
BBS 3. 3 S' a' 3P ;
3 3 M - w w
i I
i <
8 I §
* i E S 3 S i ? S 3 3 3 • 2 S 1 I j a ^ 'T <4 W W
a a s 2 3 1 i „ 3» * > I I
- U l - ! • I ' 1 • 1 5
?« 5 I O M & | 3 M M M > « < < * |
a 3 z J : : 5 5 : i
2. E
a s & I s 3 3." 1 §
I ! 3. a> s a. a i a a a > - ' 1 3 3 5
•4> A —i *
a a i 3 3, 1 5 5 a * ? a * ! M i i ! ! i a
- - - « -
a i » I I I 8 2 I I M PI A a 3.? t 3
a a 3 s 3
I 8
i ; E i * 1 1 a 1 1 n a a a i 1 4 ?
w *< «« w
i i 3 » 1 S 2 B 3 S 3. 5 * a> 5*
fi BA A B B , , I
I I I
I I
» a i S 3 s 3 * H * I i B B A i B B I i j
£ £ E
> . i * > 1 1 1 n i *-> i M m 1 1 i a i i c i
8 8 ' i s 8 i ! I § I I I I i i i I I I | t I I l
i i > a a
m U . U l l U i 1 I
J 3 8 I. B "
s a 8 i a
! I * i i I » ' H I H 3 .8 . i § B I I I 8 » i \\ I I I
1 M ' I S i 8 I i I 1 ! i i 8 | | | | | | . 8 | ! 1 i | |
8 B H ! I i i 8 \ \ BE B B i 8 | j ? -. s
1 1 1 1 4 ' 1 1 ' ' ! I i 8 | i i i | | g I | } i i | } | | | | , i | | g i
I I 5 I I I 1 1 I i ' 1 I i f M f ' ' ! ! ! 1 ! i • i M M \ \
Table A14 (Cont.)
- 150 -
a a 3 5 s a f a a 3 a
I ' i 1 1 E s I » ' n 8 S § n ' H M m k E S
| | 3 I g 1 l | I I j g s I i j a I \ g | g g | s I \ g j | j g i I
B 3 r
i S i a
i ii
s g c s s , i S i i j a i i i S i . t B S B S s i . s g c R a B S i j R ; • 3 3 3 3 3 ' I j » i g a » i j B " ' I a B c s s 8 , I B S B B B B 1 I S i 1
s n : S E B B S I . i | i f ! i i ! I i • i s i I H ! i I I • i \ \ i H I i I 1 i i
4 4 M M *
a m • 1 1 1 M M 1 1 1 1 1 { 1 1 1 1 1 1 1 1 r i ; 1 1 1 i j 11 i g £ *, M
1 U i I . 1 M 1 1 N > s 1 i s i [ H i M i i i ! ! i i i i > i i g M g
i • • « •
1 1 1 1 1 1 • i * ' n • • M * i i a n l H < U s H § s i i i n s * * * * M - r ? ! ! ; i 1 i.* ** "» —
j a s a - s 4 a w *4 w
1 a
5 fi 5 S fi 2 , ; a a a a a 8 . 1 I I-1 l I 3 I I n M 8 1 n I H I I 1 i l l s 1 1 8 « i n
a a a j | i • I I j f j I ?
I f f H ' H ! - ' H I ' H I I f f I ' M i n i ' " I "
8 H ! I 1
1 1 4 1 4 1 1 1 1 s n 8 • n 1 M i § i u 1 n i n M 1 1 n n n 1 * l i ' s i i s i i i i i i 1 |
5 S 5 5
i A £> i £
Table A14 (Cont.)
- 151 -
S EB a 3 • i ! i 2. 2. 3. 2. 1 1 t J! " » « «
1 i i • 1 1 i 1 s 1 1 1 K s . 1 1 l i t B L i ' i n s i n a i . I : ; E i s s i £ ; ^
! ) I I 1 1 1 J 1 1 1 1 * 1 s 1 1 • 1 3 M M 1 1 1 § 1 * i § s 1 i n ' i n ? 5 s 1 I £ i I I 2 i i i a _ J £ £ 2. I I I
H I I I 8 1 i 8 1 i § 8 1 1 § 8 1 i n j I I 1' § I n I I 1 1 s H ' 3 3 .? C J 3 B a a a s ~ ~ 7 5 ? ? ? ? ? ! s .? ? » g _
n i l i l ' i i ' l I l ' i l l ' l i l i j f i i ' i M i I 8 1 • i 3 i i I 3 ft - - i w w w w w w w w ~ g £ k w w w £ » « w w W W
i 1111 1 1 I | S 2 I 8 1 i § 1 8 3 1 § I I 1 1 s 8 1 j s 8 8 i I s i i • 3 3
i i l U l | l | M i ! | i ' I , l ! i 3 I U l l l U | l l | i 3 i a a
} } } } } ' T 1 " . 1 ' I ' " " ! ! } } } " M i l f f " 1 1 ! c & : a • s 3
I I I I I I I I I I I I i 3 1 I j S 1 I 2 1 1 1 1 1 1 1 1 1 ! 1 1 1 1 1 1 3 1 s * ? 5
i i i i i i i i i i i i | § i i j § I i § i i i i i i i i i i i i i i i i | i |
i i i i i i i i i i i i ^ | i . y | i J . i . . . i . i , M i n i i s i
I I I i '| i i | i i i i | | I i § 1 i I | i i i i i i i i i i " i i i i s | i |
i i i i i i I I i i i i J 3 I i § I i 1 § i i i i i i i i I i i I i I i * 3 1 3
T a b l e A14 ( C o n t . )
- 152 -
I I i I I i i I I I I I jj I i I | 5 * I § i I i i I I i I I I I i I I i i ! I I i l l I
I I I 1 S I i i l l l I j | I i 5 I i i j j i i i i i i i I 1 I I I i i I i | I 3
i I l I l l l l I i l I | jj I l f H I i | i i i i i i i i i | | | | | | i § | "
I I l I l l l l l i I I j jj I j | s I H I i i i i i i i i i i • i i i i I a i H 3 3 5 5
I I I ! I I I I I I I I j j i , | J i I j I I I I I I I I « I I I I I I a | a
I I ! I I I I I I I I I 1 | I I | | I I | I f I I I I I I I I I i I I I i a i a
I I I I I I i i I I I J H a I s | I s a I I I I I i I I I I I I I a I
I I I I I I I i i 8 I I g | i 1 f a I I a I I l i I l I I ! I I I 1 1 l l I I 3
s .g 8 I I i i I i i I g f i I g I i I a i i i i i i i 1 I ! I I i I <Q - ^ J 3 * «H —« ~>
I I ! I I I I I I I I I i : a i i a •
i i i i i i I i i i i i | s i i | a I i S | i i i i i
I l l I l l I I I I I I I
* » M I i s a
l a |
I ' I l I I ! i I I I I 3 k i I 3 i i l 2 I l I I I I I l l I I I I I ! M I m m H
i i i i i i I ! i i i i ? a i i H i I a I I I ! I I I I I I I I • i * a i a
Table A14 (Gont.)
- 153 -
H I i i i j g i i H I i a i I i I I I I I I I i i i i i
I I I l l I l l l i I l i I I I I | 1 i 3 ! i i i I I I I I 1 I i I 1 i
K 1 M
• O « 3
s» s. a» i 5 i 3.
i i i I I I I I ! i i I i | * a a
I I I I I I 3 I I I I I i B I I I f a % I l I I I I I I | l i | i i i
I I I I I \ I I I I I I I i i I * I \ l I l l I l l l I i I l l I I i I I 3 i 3
I I 1 1 1 i 1 1 1 1 • 1 i i i 1 i111 5 ! I ! I 1 I I I ! I I I I | I I a I
a ft n
i i i i i | i i i i i i i i * i i » I i s i i i i i i i i i i i i i i i
I I l I I | | | i i | i I | 1 l l i g i | i i i i i i i i i i i i i i i I g 5
? a * 5* ? ? s - s 5 S I I ft I <
•j ft I I ft I a l ! l l l l l l s B ? 9 l i l
* & 3.
i i i i 1 1 1 1 1 1 1 1 i i 1 1 1 M n i i I I I I a 5 I s i I 1 i * ? ? ** M M M
s a & s a » . S a j i s i 5 a a , , , , B
I I I I I I I I I I I 3 I
1 1 1 1 1 i 1 1 1 1 1 1 1 1 '* i & 1 1 n ' . 1 1 I I i I I I I I I I I s 3 I ;
i i i I I s j s I I I I s I M M I M m 5» M M Ji I I I I I I I I I I I I a i a x
5 1 s 5
Table A14 (Cont.)
- 154 -
1 1 1 I " i ! i i .i i i ? i l i i I i M i i i ' i i i i i i i i i i i a % a 5 ;
' 1 i ' i I ! i i i i i 1 | i i i I | ! | i i i i i i i i i i l % i |
II1111 i 1 1 1 1 5 a i ' S | ' i a t u i i s 1 1 s 1 i 1 a l l l l a • ! S B ? * £ * - S
1 1 1 ' i i. i I I I I l I | I i I 1 | f jj I I I I i I \ 2 jj, % 3 l I I l i i J H U M
_ a i i i
1 1 ' i i 1 | s i i i i ] | i i i i { g g i i i i , i i , l I I l l I I i I I
i I l ' l M 2 l l l I \ I I l I l 1 ] J l l l i i I I I I I i I i I I 1 g l 3
I l I l I i H 1 ' ' 1 « s g 1 s 1 s s s l I l l l I l I l ! I i I I l fi 3 i E < .
i i i i i i | i i I I I i 3 l l i a a a I l l l l l l I I l I l I i I i s I
I I I I I f j I I | | | | I I i | I I \ g i i | i ! i | i | i i I I I l I I , s
B B S M M M S> S> 3»
a c s M M at
M *C M «4 4 H
I I I 1 I i. n i i i ? » & ? t t * i
S 5.
I I I i l l ! i i ! I i I I I I 3 I 3 I I I I I I I M •( a | 1 i | | 5 W D M i ; - O • TJ ?
M N H H
I I I i I I I I | | I i I i | S I jj i i i | H § i y u « EJ X M >' K * ^ •* K • H K M M H N M
I N N
i i i I 1 1 1 1 i s i i i : | S { | l i S i i i ? • N O N N N M
M M M M -! 3 -8 I I I
Table A14 (Cont.)
- 155 -
I I I I I ! I I | S ! | 1 | & I I I I J C
M M M M M O
I I I I I 8 I a | i i i i S 8 H i >
B B B i
H & B a
I l l s
i
i 1111 1 1 1 I 1 ' 1 1 1 M S I I I B I I I s I I I g § g g J | i l f i I B
I I I I I I I ' i ! i ' i M ! M i " I i i i I i i i ? i i i i i i 3
• 1 i ' ' ' s i i i i i i i n n ' 2 § i i i { i i i f i | f { | , , H § | |
i i i i i i i i
i i l i l l i i | jj I I I i
5 I < i 1 1 ft H i n n n i f i i 3» s a» ~ *4 A A
I I I I I I I
B M M B B S I •« M M •< N S •
* £ i * *
' ft ft ft " A I ft ft ft 5 B I 2» 5 3> a 3i 2 3. 2 5.
I I I
I I I
B
I R B
A M
3 i
S B
n i h
B & I I I
I 1 1 B 8 a a
3 I 5 3 C 3 B » i ? 2> » s 4 M *t *C •
| I S 1 I
U ft I ! J ' I? I l i. I? i
' § ! I I I " ! S 3 ! I » § E s ft ft ft . | f 2 i J 1 ! I I 1
I I I . I I I I I j | I I i I jj I I I
<•* -» M «•» w S<
R R I E * 4. 3. i ' •
I f l l S s i l l 5 K 5 I
s a l 1 t « • l a s ' 1 8. 5, a a s
? « S. 3. s s i i I
Table A14 (Cont.)
- 156 -
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