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Magnetic space groups. Guccione, Rosalia Giuseppina 1963

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MAGNETIC SPACE GROUPS by ROSALIA GUCCIONE L a u r e a , U n i v e r s i t a ' d i Pale r m o , 1957 M . S c , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1960 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department o f PHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1963 -In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of j the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia,, I agree that the L i b r a r y s h a l l make i t freely-a v a i l a b l e fo.r reference and study. I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying, or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of P k y <, lcs> The U n i v e r s i t y of B r i t i s h Columbia,. Vancouver 8, Canada. Date A pr.'.L 2 ? y I 9f3  PUBLICATION Guccione, R. , Tosi, M.P. , Asdente, M Migration Barriers for Cations and Anions i n A l k a l i Halide Crystals, J Phys. Chem. Solids _10, 162 (1959). The U n i v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY • of ROSALIA GUCCIONE Laurea, U n i v e r s i t a 1 d i Palermo, 1957 M.Sc, The U n i v e r s i t y of B r i t i s h Columbia, 1960 FRIDAY, APRIL 26, 1963, at 9:30 A.M. IN ROOM 452, BUCHANAN BUILDING COMMITTEE IN CHARGE Chairman: F.H. Soward C. Froese G„De" Bv Robinson J. Grindlay L.G. de Sobfino W. Opechowski G.M. Volkoff P. R a s t a l l External Examiner: G.F. Koster Massachusetts I n s t i t u t e of Technology Cambridge, Mass. MAGNETIC SPACE GROUPS ABSTRACT Magnetic space groups (MSGs) were f i r s t introduced (under a di f f e r e n t name) by Heesch more than 30 years ago, and a l i s t of a l l of them was published by Belov, Neronova and Smirnova in 1955. However, no mathematically rigorous d e r i v a t i o n of MSGs can be found i n the e x i s t i n g l i t e r a t u r e , although an outline of a method for obtaining a large class of MSGs was published by Zamorzaev i n 1957. In this thesis a systematic rigorous method for constructing MSGs i s described in d e t a i l , and a proof that the method in fact gives a l l the MSGs i s presented,, The method also leads i n a natural way to a new c l a s s i f i c a t i o n of MSGs which i s useful for a systematic study of the arrangements of magnetic moments i n ferro-magnetic, ferrimagnetic and antiferromagnetic c r y s t a l s . GRADUATE STUDIES F i e l d of Study: T h e o r e t i c a l S o l i d State Physics Electromagnetic Theory Theory of Measurements Nuclear Physics Elementary Quantum Mechanics Advanced Quantum Mechani cs G.M. Volkoff J„R„ Prescott 'J„B0 Warren FoA. Kaempffer F.A„ Kaempffer D i e l e c t r i c s and Magnetism M„ Bloom Special R e l a t i v i t y Theory P„ R a s t a l l Group Theory Methods i n Quantum Mechanics W. Opechowski Physics of the S o l i d State R. Barrie Other Studies; D i f f e r e n t i a l Equations C„ Froese Numerical Analysis I T.E. H u l l i i i ABSTRACT Ma g n e t i c space groups (MSGs) were f i r s t i n t r o d u c e d (under a d i f f e r e n t name) by Heesch more than 30 y e a r s ago, and a l i s t o f a l l o f them was p u b l i s h e d by B e l o v , Neronova and Srairnova i n 1955. However, no m a t h e m a t i c a l l y r i g o r o u s d e r i v a t i o n o f MSGs can be found i n the e x i s t i n g l i t e r a t u r e , a l t h o u g h an o u t l i n e o f a method f o r o b t a i n i n g a l a r g e c l a s s o f MSGs was p u b l i s h e d by Zaraorzaev i n 1957. I n t h i s t h e s i s a s y s t e m a t i c r i g o r o u s method f o r c o n s t r u c t i n g MSGs i s d e s c r i b e d i n d e t a i l , and a p r o o f t h a t the method i n f a c t g i v e s a l l t h e MSGs i s p r e s e n t e d . The method a l s o l e a d s i n a n a t u r a l way t o a c l a s s i f i c a t i o n o f MSGs whi c h i s u s e f u l f o r a s y s t e m a t i c s t u d y o f the arrangements o f s p i n s i n f e r r o m a g n e t i c , f e r r i m a g n e t i c and a n t i f e r r o m a g n e t i c c r y s t a l s . The f i r s t and the l a s t c h a p t e r o f the t h e s i s d e a l w i t h the p h y s i c a l a s p e c t s o f the problem, the r e m a i n i n g c h a p t e r s w i t h p u r e l y m a t h e m a t i c a l a s p e c t s o f i t . i i ACKNOWLEDGMENTS I have been f o r t u n a t e i n h a v i n g a r e s e a r c h s u p e r v i s o r , P r o f e s s o r W. Opechowski, as a c t i v e l y i n t e r e s t e d i n t h i s work as I . I w i s h t o thank the N a t i o n a l R e s e a r c h C o u n c i l o f Canada f o r f i n a n c i a l h e l p i n the form o f a S t u d e n t s h i p . i v TABLE OF CONTENTS Acknowledgments i i A b s t r a c t i i i Ch a p t e r 1 I n t r o d u c t i o n and Summary 1 Ch a p t e r 2 Some P r o p e r t i e s o f Ma g n e t i c Groups and the Idea of a Method o f C o n s t r u c t i n g Them. 13 S e c t i o n 2.1 13 S e c t i o n 2.2 15 S e c t i o n 2.3 17 Notes t o Cha p t e r 2 20 Chap t e r 3 D e r i v a t i o n o f R u l e s f o r C o n s t r u c t i n g A l l Subgroups o f Index 2 o f A l l Space Groups. 22 S e c t i o n 3.1 23 S e c t i o n 3.2 47 S e c t i o n 3.3 62 S e c t i o n 3.4 64 Notes t o Cha p t e r 3 73 Ch a p t e r 4 Ma g n e t i c Space Groups and T h e i r P r o p e r t i e s 75 S e c t i o n 4.1 75 S e c t i o n 4.2 78 S e c t i o n 4.3 87 V C h a p t e r 4 ( c o n t i n u e d ) S e c t i o n 4.4 89 S e c t i o n 4.5 95 Notes t o C h a p t e r 4 99 C h a p t e r 5 I n v a r i a n t Arrangements o f S p i n s 100 S e c t i o n 5.1 100 S e c t i o n 5.2 108 Notes t o C h a p t e r 5 111 L i s t o f Symbols 112 L i s t o f M a g n e t i c Space Groups 113 B i b l i o g r a p h y 133 1 CHAPTER 1 INTRODUCTION AND SUMMARY T h i s t h e s i s i s m a i n l y d e v o t e d t o a group t h e o r e t i c a l d e r i v a t i o n and c l a s s i f i c a t i o n o f magnetic space groups (MSGs). The r e a s o n f o r s t u d y i n g the MSGs i s t h a t t h e y a r e groups of inhomogeneous l i n e a r s p a c e - t i m e t r a n s f o r m a t i o n s w h i c h l e a v e a l l p o s s i b l e t h r e e - d i m e n s i o n a l p e r i o d i c a l a r r a n g e -ments o f magnetic moments i n v a r i a n t and hence a r e o f importance f o r the t h e o r y of m a g n e t i c a l l y o r d e r e d c r y s t a l s , t h a t i s , f e r r o m a g n e t i c , a n t i f e r r o m a g n e t i c and f e r r i m a g n e t i c c r y s t a l s . By a " p o s s i b l e arrangement" o f magnetic moments we mean an arrangement o f magnetic moments w h i c h would c e r t a i n l y o c c u r i n n a t u r e i f i t had o n l y t o s a t i s f y r e q u i r e m e n t s o f i n v a r i a n c e under some of the s p a c e - t i m e t r a n s f o r m a t i o n s j u s t mentioned, but w h i c h may a c t u a l l y not o c c u r i f i t s e x i s t e n c e v i o l a t e s some d y n a m i c a l law. By " p e r i o d i c a l " arrangement o f magnetic moments we mean an arrangement o f magnetic moments which has t h e t r a n s l a t o r y symmetry of a l a t t i c e . The problem o f d e r i v i n g a l l MSGs i s s i m i l a r t o t h a t o f d e r i v i n g t h e groups w h i c h l e a v e a l l p o s s i b l e t h r e e -d i m e n s i o n a l p e r i o d i c a l arrangements o f atoms i n v a r i a n t . As i s w e l l known the s o l u t i o n t o t h i s problem i s g i v e n by the t h e o r y o f space groups. O b v i o u s l y the two problems a r e not i n d e p e n d e n t , s i n c e , i n a magnetic c r y s t a l , i n d i v i d u a l magnetic moments a r e l o c a t e d a t p o s i t i o n s o f magnetic atoms. 2 One i s then n a t u r a l l y l e d t o ask t h i s q u e s t i o n : how many d i f f e r e n t p o s s i b l e p e r i o d i c a l arrangements o f magnetic moments a r e t h e r e ? The answer o b v i o u s l y depends on what one means by " d i f f e r e n t " arrangements o f magnetic moments. I f an arrangement of magnetic moments i s i n v a r i a n t under a group o f s p a c e - t i m e t r a n s f o r m a t i o n s , and i f a second arrangement o f magnetic moments i s i n v a r i a n t under a second group o f s p a c e - t i m e t r a n s f o r m a t i o n s , t h e n the two arrangements w i l l be c a l l e d " d i f f e r e n t " i f t h e r e e x i s t s no inhomogeneous space t r a n s f o r m a t i o n w h i c h t r a n s f o r m s one group i n t o t h e o t h e r . A c c o r d i n g t o t h i s c o n v e n t i o n a l l t he magnetic arrangements w h i c h a r e l e f t i n v a r i a n t under groups t r a n s -f o r m a b l e i n t o one a n o t h e r by means o f an inhomogeneous l i n e a r space t r a n s f o r m a t i o n a r e not d i f f e r e n t . Thus t o t h e arrangements o f magnetic moments w h i c h a r e not d i f f e r e n t i n our sense t h e r e c o r r e s p o n d s a c l a s s o f s p a c e - t i m e t r a n s f o r m a t i o n groups w h i c h c a n be t r a n s f o r m e d i n t o one a n o t h e r by means o f an inhomogeneous l i n e a r space t r a n s f o r m a t i o n . The answer t o the q u e s t i o n f o r m -u l a t e d above i s then t h a t t h e r e a r e as many d i f f e r e n t p o s s i b l e p e r i o d i c a l arrangements as t h e r e a r e c l a s s e s o f MSGs. I f we r e f e r t o a c l a s s o f MSGs as a MSG then we say t h a t t h e r e a r e as many d i f f e r e n t p o s s i b l e arrangements o f magnetic moments as t h e r e a r e MSGs. In the c o u r s e o f t h i s t h e s i s we use the term MSG i n b o t h s e n s e s , t h a t i s , a t t i m e s we use i t t o i n d i c a t e a p a r t i c u l a r MSG, a t o t h e r t i m e s we use i t t o i n d i c a t e a c l a s s of MSGs. However, i t s h o u l d be c l e a r from the c o n t e x t i n wh i c h o f the two senses t h e term i s used. 3 Although what we have s a i d about MSGs so f a r contains i m p l i c i t l y t h e i r d e f i n i t i o n , we have to s t a t e more p r e c i s e l y some p r o p e r t i e s of ma g n e t i c a l l y ordered c r y s t a l s before we can formulate that d e f i n i t i o n e x p l i c i t l y . As i s well-known a l l m a g n e t i c a l l y ordered c r y s t a l s can be d i v i d e d i n t o three c l a s s e s ( f o r s i m p l i c i t y , we consider here the case of zero temperature): ferromagnetic c r y s t a l s , where a l l the magnetic moments of the i n d i v i d u a l atoms are p a r a l l e l and, hence, the macroscopic magnetic moment has the maximum value compatible with the magnitude of the i n d i v i d u a l magnetic moments; anti f e r r o m a g n e t i c c r y s t a l s , where the macroscopic magnetic moment i s zero, d e s p i t e the f a c t that the i n d i v i d u a l magnetic moments are not o r i e n t e d at random} f e r r i m a g n e t i c c r y s t a l s , where the macroscopic magnetic moment i s n e i t h e r zero, nor has the maximum value compatible w i t h the magnitude of the i n d i v i d u a l magnetic moments. Obviously the c r y s t a l s of the three c l a s s e s have a common f e a t u r e , a non-vanishing average magnetic moment of 4 i n d i v i d u a l magnetic atoms. T h i s f e a t u r e w i l l s u g g e s t a n a t u r a l d e f i n i t i o n of MSGs. I t i s g e n e r a l l y a c c e p t e d t h a t the S c h r o e d i n g e r e q u a t i o n o f a p e r f e c t c r y s t a l c o n s i d e r e d as a system o f n u c l e i and e l e c t r o n s i s i n v a r i a n t under the o p e r a t i o n s o f t h e d i r e c t p r o d u c t group F x A, where F i s the space group o f the c r y s t a l and A i s the t i m e - r e v e r s a l group c o n s i s t i n g o f the i d e n t i t y E and t h e t i m e - r e v e r s a l operation E ' ( t - * - t ). (As we a r e i n t e r e s t e d o n l y i n the s t a t i o n a r y s t a t e s o f the c r y s t a l , we d i s r e g a r d t i m e - t r a n s l a t i o n s , ) ( S t r i c t l y s p e a k i n g t h e S c h r o e d i n g e r e q u a t i o n i s not i n v a r i a n t under the group F x A, but i t i s i n v a r i a n t under th e group o f s t a t e - v e c t o r t r a n s f o r m a t i o n s w h i c h c o r r e s p o n d t o the s p a c e - t i m e t r a n s f o r m a t i o n s o f F x A. However as no quantum m e c h a n i c a l a p p l i c a t i o n of MSGs w i l l be d i s c u s s e d i n t h i s t h e s i s , and the quantum m e c h a n i c a l language w i l l not be used e x c e p t i n the nex t few p a r a g r a p h s , i t would be e x a g g e r a t e d t o i n t r o d u c e a t t h i s s t a g e t h e con c e p t o f double magnetic space g r o u p s , analogous t o t h a t of doub l e space g r o u p s . ) I n the case o f a magnetic c r y s t a l , i n the absence o f an e x t e r n a l magnetic f i e l d , the S c h r o e d i n g e r e q u a t i o n i s s t i l l i n v a r i a n t under the group F x A and hence i n p a r t i c u l a r under t i m e - r e v e r s a l . However t h i s t i m e - r e v e r s a l i n v a r i a n c e i s i n c o m p a t i b l e w i t h t h e n o n - v a n i s h i n g of the average magnetic moment o f the i n d i v i d u a l atoms. I n f a c t , i f m i s 5 the magnetic moment operator for an i n d i v i d u a l atom and i s an eigenstate of the system, the non-vanishing of the average i n d i v i d u a l magnetic moment implies that the sum being over a l l the eigenstates belonging to a given energy eigenvalue of the system. On the other hand the time-reversal invariance implies because both the eigenstate |yu.> and i t s time-reversed \/^>Q are eigenstates belonging to the same energy eigenvalue of the system and m anticommutes with the time-reversal operator 0 . One has then to conclude that although the symmetry group of the Schroedinger equation of a magnetic c r y s t a l contains the time-reversal operator, an energy eigenstate and the corresponding time-reversed eigenstate cannot coexist i f the c r y s t a l i s magnetically ordered. In other words, the tr a n s i t i o n to the magnetically ordered state implies an apparent reduction of the symmetry of the problem: the physical properties of a magnetically ordered c r y s t a l are as i f 6 t h e symmetry group o f t h e S c h r o e d i n g e r e q u a t i o n was a subgroup o f F x A (a) w h i c h does n ot c o n t a i n the i d e n t i t y o f F combined w i t h t i m e - r e v e r s a l as an ele m e n t , and (b) w h i c h i s i s o m o r p h i c t o F or t o a space group w h i c h i s a subgroup o f F. Each such subgroup M o f F x A i s c a l l e d magnetic space group. I n g e n e r a l , we d e f i n e an MSG as b e i n g any subgroup o f F x A: (a) w h i c h does n ot c o n t a i n the i d e n t i t y o f F combined w i t h t i m e - r e v e r s a l as an element, (b) w h i c h i s i s o m o r p h i c t o some space group. S i n c e t h e r e i s o n l y one a b s t r a c t group o f o r d e r 2, the problem o f c o n s t r u c t i n g a l l MSGs i s c l e a r l y independent o f the p h y s i c a l meaning a s s i g n e d t o t h e element E* o f the group A. I n f a c t , MSGs were f i r s t c o n s i d e r e d under a d i f f e r e n t name and i n q u i t e a d i f f e r e n t c o n n e c t i o n more t h a n 30 y e a r s ago by Heesch (1930)} he r e f e r s t o them as " f o u r - d i m e n s i o n a l groups o f t h e t h r e e - d i m e n s i o n a l s p a c e " . The analogous two-d i m e n s i o n a l problem ( t h a t i s the case i n whi c h F i s a t w o - d i m e n s i o n a l space group) has been c o n s i d e r e d and s o l v e d even e a r l i e r by Hermann ( 1 9 2 8 ) , A l e x a n d e r and Herrman ( 1 9 2 8 ) , Weber (1929) and Heesch (1929). The m o t i v a t i o n f o r i n t r o -d u c i n g such new groups was a t t h a t t ime 7 e i t h e r p u r e l y m a t h e m a t i c a l o r c r y s t a l l o g r a p h i c a l . The element E' o f A was i n t e r p r e t e d e s s e n t i a l l y as meaning the change o f v a l u e of a c o o r d i n a t e c a p a b l e of assuming two v a l u e s , or as a change of c o l o u r i n a p l a n e ornament (assuming t h a t e x a c t l y two c o l o u r s were a v a i l a b l e ) . Of c o u r s e , no r e f e r e n c e t o the problem of magnetic o r d e r i n g was p o s s i b l e a t t h i s e a r l y s t a g e o f the h i s t o r y o f t h e problem. Heesch (1930) was not o n l y t h e f i r s t t o c o n s i d e r the MSGs i n the sense j u s t i n d i c a t e d , and t o g i v e a l i s t o f MSGs o f t h e t r i c l i n i c and m o n o c l i n i c s y s t e m , but a l s o seems t o have been the f i r s t t o r e a l i z e the im p o r t a n c e o f h i s groups from the p h y s i c a l p o i n t o f view. He s a y s " H i e r d u r c h e m p f i e h l t s i c h d i e s e K o o r d i n a t e ( t h a t i s the c o o r d i n a t e mentioned above) a l s V e r t r e t e r e i n e r j e d e n E i g e n s c h a f t des homogenen ebenen (but he has i n mind a l s o the t h r e e - d i m e n s i o n a l case as i s c l e a r from the r e s t of h i s p aper) D i s k o n t i n u u m s , d i e s i c h auf d i e Fo r m e l " p l u s -minus" b r i n g e n l a s s t , d.h. f u r j e d e " p o l a r e " E i g e n s c h a f t z.B. B e s e t z u n g m i t z w e i v e r s c h i e d e n e n Ionen, E l e k t r o n e n -s p i n , e l e k t r i s c h e D o p p e l s c h i c h t , magnetisches B l a t t , D u r c h s e t z u n g m i t e i n e r D i p o l v e r t e i l u n g , r e e l l e s and v i r t u e l l e s B i l d , v o r h e r - n a c h h e r , usw." P o s s i b l y the " v o r h e r - n a c h h e r " c o u l d be i n t e r p r e t e d as a r e f e r e n c e t o t i m e - r e v e r s a l , but no f u r t h e r mention o f t h i s p o s s i b i l i t y c a n be found i n h i s paper. 8 Heesch a l s o gave the complete l i s t of what we w i l l c a l l t he magnetic p o i n t groups ( t o be d e f i n e d i n Ch a p t e r 2 ) , and he e s t i m a t e d the number o f magnetic space groups t o be c l o s e t o 1800 ( a c t u a l l y t h e r e a r e 1421 o f them). The complete l i s t o f MSGs has been p u b l i s h e d o n l y i n 1955 by B e l o v , Neronova and Smirnova (1955). These a u t h o r s do not even mention Heesch*s work, nor do th e y make any r e f e r e n c e t o the problem o f magnetic o r d e r i n g and t o the imp o r t a n c e of t i m e - r e v e r s a l i n t h i s c o n n e c t i o n . The same i s t r u e o f Zamorzaev who has a l s o g i v e n a l i s t of MSGs i n an u n p u b l i s h e d t h e s i s (1953) u n a v a i l a b l e t o u s , and who p u b l i s h e d (1957) a s h o r t paper i n w h i c h he o u t l i n e s ( w i t h o u t p r o o f s ) h i s method o f d e r i v i n g t h o s e MSGs w h i c h a r e i s o m o r p h i c t o the symroorphic space groups. What we c a l l MSGs i s c a l l e d by B e l o v , Neronova and Smirnova, and by Zamorzaev v "Shubnikov g r o u p s " , a name now w i d e l y u s e d , not o n l y i n R u s s i a , but h a r d l y j u s t i f i e d from the h i s t o r i c a l p o i n t o f view. A c l e a r r e a l i z a t i o n t h a t MSGs a r e o f importance f o r a s y s t e m a t i c c l a s s i f i c a t i o n and d i s c u s s i o n o f m a g n e t i c a l l y o r d e f e d c r y s t a l s , and t h a t the element E' o f A must be i n t e r p r e t e d i n t h i s c o n n e c t i o n as t i m e - r e v e r s a l , i s due t o 015-1) Landau and L i f s c h i t z . I n f a c t the term "magnetic space A gro u p s " has been i n t r o d u c e d by them. The a c t u a l a p p l i c a t i o n s o f the i d e a o f Landau and L i f s c h i t z t o s p e c i f i c magnetic problems a r e v e r y r e c e n t . The f i r s t t o a p p l y MSGs t o the problem of i n t e r -p r e t a t i o n of magnetic n e u t r o n d i f f r a c t i o n d a t a were Donnay, 9 C o r l i s s , Donnay, E l l i o t t and H a s t i n g s (1958) and Le C o r r e ( 1 9 5 8 ) . Le C o r r e a l s o d i s c u s s e s some o t h e r a p p l i c a t i o n s o f MSGs. As has been shown by R i e d e l and Spence (1960) and Van d e r Lugt (1961) n u c l e a r magnetic resonance s p e c t r a i n s i n g l e c r y s t a l s can a l s o be i n t e r p r e t e d i n terms o f MSGs. A d d i t i o n a l h i s t o r i c a l and b i b l i o g r a p h i c a l remarks w i l l be found a t the end o f each c h a p t e r o f t h i s t h e s i s (see a l s o Donnay ( 1 9 6 1 ) ) . A l t h o u g h a l i s t o f a l l MSGs i s a v a i l a b l e ( B e l o v , Neronova and Smirnova, 1955, 1957) and some i n d i c a t i o n s how t o o b t a i n i t a r e g i v e n i n the above-quoted papers by Heesch ( 1 9 3 0 ) , B e l o v , Neronova and Smirnova (1955) and Zamorzaev ( 1 9 5 7 ) , no r i g o r o u s and complete m a t h e m a t i c a l method o f c o n s t r u c t i n g a l l MSGs has been d e s c r i b e d i n t h e e x i s t i n g l i t e r a t u r e . Such a method i s d e s c r i b e d i n t h i s t h e s i s , s t a r t i n g o u t from the d e f i n i t i o n of MSGs g i v e n above. ( S t r a n g e l y enough, not even t h i s e x p l i c i t d e f i n i t i o n can be fo u n d i n the l i t e r a t u r e . ) We p r e s e n t the method i n such a way t h a t i t s completeness i s e v i d e n t , i n o t h e r words, we prove t h a t t h e r e e x i s t s no MSG which i s not o b t a i n a b l e by u s i n g our method. In p a r t i c u l a r , i n C h a p t e r 2, we d i s c u s s some immediate consequences o f our d e f i n i t i o n of MSG. We show t h a t the problem o f c o n s t r u c t i n g a l l MSGs i s e a s i l y r e d u c e d t o t h e problem o f f i n d i n g a l l subgroups D o f i n d e x 2 o f a l l 10 space groups F. T h i s r e s u l t i s i m p l i c i t l y c o n t a i n e d i n Zamorzaev's paper ( 1 9 5 7 ) , and i n a paper by Tavger and Z a i t s e v (1956) w h i c h d e a l s w i t h the s i m i l a r but much e a s i e r problem of c o n s t r u c t i n g a l l magnetic p o i n t g r o u p s ) . I n C h a p t e r 3 we deduce r u l e s f o r f i n d i n g the D's o f a l l syramorphic and non-symmorphic space g r o u p s , and we show t h a t a l l t h e B's a r e i n f a c t o b t a i n a b l e from our r u l e s . I n c o n n e c t i o n w i t h the d e r i v a t i o n o f the r u l e s i t t u r n s out t o be u s e f u l t o c l a s s i f y a l l the D's i n t o t h r e e d i f f e r e n t k i n d s . C o r r e s p o n d i n g l y i n C h a p t e r 4 we show t h a t t h e r e a r e t h r e e d i f f e r e n t k i n d s o f MSGs. I n t h a t c h a p t e r we a l s o g i v e t h e d e f i n i t i o n and t h e p r o p e r t i e s of the magnetic l a t t i c e and the magnetic p o i n t group b e l o n g i n g t o an MSG. A co m p l e t e l i s t o f magnetic l a t t i c e s and magnetic p o i n t groups i s g i v e n (which i s not new). F i n a l l y we show t h a t some MSGs a r e o b t a i n e d more than once by means o f our r u l e s , and we i n d i c a t e ways o f d e a l i n g w i t h t h i s redundancy. W h i l e i n C h a p t e r s 2, 3 and 4 the p h y s i c a l meaning o f the element E' of A does not p l a y any r o l e , i n C h a p t e r 5, where we b r i e f l y d i s c u s s magnetic moment arrangements i n c r y s t a l s , i t becomes e s s e n t i a l t o i n t e r p r e t t h a t element as t i m e - r e v e r s a l . I n p a r t i c u l a r , we i n t r o d u c e i n C h a p t e r 5 the u s u a l d i s t i n c t i o n between MSGs w h i c h may l e a v e f e r r o m a g n e t i c c r y s t a l s i n v a r i a n t and th o s e w h i c h may n o t . We g i v e an e x p l a n a t i o n based s o l e l y on symmetry, why f e r r o m a g n e t i c space 11 groups whose magnetic p o i n t groups c o n t a i n elements o f o r d e r h i g h e r t h a n two cannot l e a v e n o n - c o l l i n e a r f e r r i m a g n e t i c c r y s t a l s i n v a r i a n t . We a l s o d i s c u s s the p h y s i c a l s i g n i f i c a n c e o f the con c e p t o f a " f a m i l y of MSGs" i n c o n n e c t i o n w i t h the i d e n t i f i c a t i o n o f t h e MSG of a g i v e n arrangement o f magnetic moments. C o r r e s p o n d i n g l y , as an example o f the c l a s s i f i c a -t i o n o f MSGs i n t o f a m i l i e s we l i s t t he MSGs o f t r i c l i n i c , m o n o c l i n i c , t r i g o n a l , h e x a g o n a l and c u b i c systems a r r a n g e d i n t h a t p a r t i c u l a r way (which i s d i f f e r e n t from t h a t used i n the t a b l e s o f B e l o v , Neronova and Smirnova (1957D a t the end of the t h e s i s . The t h e o r y o f MSGs i s , o f c o u r s e , not ex h a u s t e d by the c o n t e n t s o f t h i s t h e s i s . An e s s e n t i a l p a r t o f the t h e o r y i s t h a t w h i c h d e a l s w i t h the c o r e p r e s e n t a t i o n s ( i n t h e sense o f Wigner A) o f the MSGs. The c o r e p r e s e n t a t i o n s o f th e MSGs have been d i s c u s s e d r e c e n t l y by Dimmock and Wheeler (1962a, 1962b). We have d e v e l o p e d a more complete t r e a t m e n t o f the c o r e p r e s e n t a t i o n s but i t w i l l n o t be i n c l u d e d i n t h i s t h e s i s . F i n a l l y , we would l i k e t o mention t h a t t h e MSGs as d e f i n e d by us c l e a r l y do not l e a v e i n v a r i a n t n o n - p e r i o d i c a l arrangements o f magnetic moments l i k e t he s p i r a l arrangements, 12 w h i c h does not mean, however, t h a t t h e MSGs a r e of no i m p o r t a n c e i n t h i s c a s e . C h a p t e r s 2, 3 and 4 f o r m , e x c e p t f o r minor changes, an e s s e n t i a l p a r t o f a monograph "Magnetic Groups" t o be p u b l i s h e d j o i n t l y w i t h P r o f . W. Opechowski who has s u p e r v i s e d t h i s work. 13 CHAPTER 2 SOME PROPERTIES OF MAGNETIC GROUPS AND THE IDEA OF A METHOD  OF CONSTRUCTING THEM, SECTION 2.1 I n c h a p t e r 1 we have i n t r o d u c e d the d e f i n i t i o n o f a magnetic space group (MSG) w h i c h we r e p e a t here f o r c o n v e n i e n c e i n a somewhat d i f f e r e n t but e q u i v a l e n t form. (2.1.1) A magnetic space group M i s any subgroup o f any F x A w h i c h s a t i s f i e s the two f o l l o w i n g c o n d i t i o n s : (a) i t does not c o n t a i n the element E E ' , (b) i t i s i s o m o r p h i c t o F. Here F i s any space group ( e l e m e n t s : F-^  = E ( i d e n t i t y ) , F2» F 3 » # , * ) » A i s the group c o n s i s t i n g o f two elements A^ = E ( i d e n t i t y ) , = E' ( t i m e - r e v e r s a l ) , and the elements of the d i r e c t p r o d u c t F x A a r e p a i r s F A , an element o f F b e i n g a l w a y s w r i t t e n on the l e f t and an element o f A on the r i g h t . However, as a l r e a d y mentioned i n C h a p t e r 1, we may r e g a r d t h e group A as the a b s t r a c t group o f o r d e r 2 as l o n g as we do not a p p l y the arguments and r e s u l t s o f t h i s c h a p t e r t o p h y s i c s . S i n c e F i n d e f i n i t i o n (2.1.1) i s an a r b i t r a r y space g r o u p , and each space group i s a subgroup o f the g e n e r a l inhomogeneous l i n e a r group <|, d e f i n i t i o n (2.1.1) can be r e p l a c e d by the f o l l o w i n g e q u i v a l e n t one: 14 (2.1.2) A magnetic space group M i s any subgroup o f G x A w h i c h s a t i s f i e s c o n d i t i o n s (a) and (b) s t a t e d i n ( 2 . 1 . 1 ) . I n p a r t i c u l a r , i t f o l l o w s from ( 2 . 1 . 1 ) , or ( 2 . 1 . 2 ) , t h a t e v e r y space group i s an MSG. We s h a l l o c c a s i o n a l l y c a l l MSGs w h i c h a r e not s i m p l y space groups " n o n - t r i v i a l " MSGs. I t i s c o n v e n i e n t t o d e f i n e magnetic r o t a t i o n g r o u p s , and magnetic l a t t i c e s i n an analogous manner: (2.1.3) L e t R be the group o f a l l p r o p e r and i m p r o p e r r o t a t i o n s ( i n the t h r e e - d i m e n s i o n a l E u c l i d e a n s p a c e ) j a magnetic r o t a t i o n group RJJ i s then any subgroup o f R x A w h i c h does not c o n t a i n the element EE*. (2.1.4) I f a magnetic r o t a t i o n group i s i s o m o r p h i c t o any one o f the c r y s t a l l o g r a p h i c p o i n t groups i t w i l l be c a l l e d a magnetic p o i n t group. (2.1.5) L e t T be a l a t t i c e ( i . e . a group o f p r i m i t i v e t r a n s l a t i o n s i n the t h r e e - d i m e n s i o n a l E u c l i d e a n space);, a magnetic l a t t i c e Tjj i s then any subgroup o f T x A w h i c h does not c o n t a i n the element EE', and which c o n t a i n s t h r e e l i n e a r l y independent t r a n s l a t i o n s . I t f o l l o w s from t h e s e d e f i n i t i o n s t h a t " o r d i n a r y " r o t a t i o n groups and " o r d i n a r y " l a t t i c e s a r e s p e c i a l c a s e s o f magnetic r o t a t i o n groups and magnetic l a t t i c e s . We see from the above d e f i n i t i o n s t h a t magnetic groups o f a l l t h r e e v a r i e t i e s ( i . e . magnetic space g r o u p s , magnetic r o t a t i o n groups and magnetic l a t t i c e s ) a r e always 15 subgroups o f a d i r e c t p r o d u c t group o f the form (J x A, where G i s d i f f e r e n t i n each o f the t h r e e c a s e s but A i s always the same group o f two e l e m e n t s . That i s why, i n S e c t i o n 2.2 we d e f i n e a m a t h e m a t i c a l analogue o f magnetic groups f o r t h e c a s e t h a t G i n G x A i s a r b i t r a r y , and d i s c u s s some o f the consequences o f t h a t more g e n e r a l d e f i n i t i o n . I n S e c t i o n 2.3 we s p e c i a l i z e the r e s u l t s o f S e c t i o n 2.2 t o the case o f magnetic g r o u p s . SECTION 2.2 L e t G x A be the d i r e c t p r o d u c t o f an a r b i t r a r y group G, whose elements w i l l be denoted by G, and t h e group A o f o r d e r 2, whose elements w i l l be denoted by A, o r , more s p e c i f i c a l l y , by E ( t h e i d e n t i t y ) and E*. The elements of t h i s d i r e c t p r o d u c t a r e p a i r s GA, an element o f G b e i n g always w r i t t e n on the l e f t and an element o f A on the r i g h t . We s h a l l c a l l an element GA o f G x A "pri m e d " i f A = E*, and "unprimed" i f A = E. I n p a r t i c u l a r , we s h a l l c a l l t h e element EE* t h e "primed i d e n t i t y " . We s h a l l o f t e n use a s i m p l i f i e d n o t a t i o n : G f o r GE, and G' f o r GE 1. From t h e s e d e f i n i t i o n s i t f o l l o w s i m m e d i a t e l y t h a t : (2.2.1) The p r o d u c t of any two unprimed elements i s unprimed. (2.2.2) The p r o d u c t o f any two primed elements i s unprimed. 16 (2.2.3) The p r o d u c t o f a primed element w i t h an unprimed one ( i n e i t h e r o r d e r ) i s primed. We now d e f i n e f o r each g i v e n , a r b i t r a r y G a c l a s s o f groups w h i c h we s h a l l c a l l ro-groups; the l e t t e r "m" i n "m-groups" i s supposed t o i n d i c a t e t h a t , f o r an a p p r o p r i a t e c h o i c e o f G, m-groups become magnetic groups o f one of the t h r e e k i n d s i n t r o d u c e d i n S e c t i o n 2.1. (2.2.4) An m-group o f G x A ( o r b e l o n g i n g t o G x A) i s any subgroup o f G x A wh i c h does not c o n t a i n the primed i d e n t i t y EE'. O b v i o u s l y , the m-groups a r e of two k i n d s : t h o s e /w is/ w h i c h c o n s i s t e n t i r e l y o f unprimed elements o f G x A, and t h o s e w h i c h c o n t a i n some primed e l e m e n t s . The former w i l l be c a l l e d " t r i v i a l " m-groups, the l a t t e r " n o n - t r i v i a l " m-groups. (2.2.5) By o m i t t i n g the primes i n the elements o f an m-group one o b t a i n s a subgroup o f G wh i c h i s i s o m o r p h i c t o the m-group. T h i s theorem f o l l o w s i m m e d i a t e l y from the d e f i n i t i o n (2.2.4) o f m-groups and ( 2 . 2 . 1 ) , ( 2 . 2 . 2 ) , 2.2.3). (2.2.6) The unprimed elements o f a n o n - t r i v i a l m-group c o n s t i t u t e a subgroup of i n d e x 2. The p r o o f o f (2.2.6) f o l l o w s from the f o l l o w i n g theorem v a l i d f o r any group K: (2.2.7) A subgroup L o f a group K i s a subgroup o f i n d e x 2 i f and o n l y i f the p r o d u c t o f two a r b i t r a r y elements 17 o f K not b e l o n g i n g t o L i s an element b e l o n g i n g t o L. I n f a c t , s i n c e , a c c o r d i n g t o (2.2.1), t h e p r o d u c t o f any two unprimed elements o f an m-group i s an unprimed e l e m e n t , the unprimed elements form a subgroup o f i t . That th e subgroup i s of i n d e x 2 f o l l o w s i m m e d i a t e l y from (2.2.7) i n c o n j u n c t i o n w i t h ( 2 . 2 . 2 ) . Theorems (2.2.5) and (2.2.6) t a k e n t o g e t h e r s i m p l y mean t h a t e v e r y m-group i s i s o m o r p h i c t o some subgroup H of 8 and t h a t H t h e n n e c e s s a r i l y has a subgroup o f i n d e x 2. Hence, c o n v e r s e l y , one w i l l o b t a i n a l l n o n - t r i v i a l m-groups of G x A i f one a p p l i e s the f o l l o w i n g r u l e : (2.2.8) F i n d a l l t h o s e subgroups H o f G w h i c h have subgroups D H o f i n d e x 2. For each D H o f each H combine the elements o f w i t h the i d e n t i t y E of A and the elements o f the c o s e t H - D** w i t h the element E* o f A. The s e t o f a l l elements o f 5 % and (H - D^E' w i l l t h e n n e c e s s a r i l y c o n s t i t u t e an m-group of G x A (because the s e t i s o b v i o u s l y a subgroup o f S x A, and does not c o n t a i n the element E E ' ) . An m-group o b t a i n e d i n t h i s way from a g i v e n H and D H w i l l be denoted by M(6H). SECTION 2.3 I n t h i s s e c t i o n we s p e c i a l i z e t h e g e n e r a l r e s u l t s o f S e c t i o n 2.2 t o t h e c a s e o f magnetic g r o u p s . Take the group G o f S e c t i o n 2.2 t o be the g e n e r a l inhomogeneous l i n e a r group <L Then, i n view o f (2.2.5) and 18 o f the f a c t t h a t a l l space g r o u p s , a l l r o t a t i o n groups and a l l l a t t i c e s a r e subgroups o f C^ , the d e f i n i t i o n s , g i v e n i n S e c t i o n 2.1, of the t h r e e t y p e s o f magnetic groups can be r e f o r m u l a t e d as f o l l o w s : (2.3.1) M a g n e t i c space groups M a r e t h o s e m-groups of Cj, x A wh i c h are i s o m o r p h i c t o space groups F. (2.3.2) M a g n e t i c r o t a t i o n groups R M a r e tho s e m-groups of C, x A wh i c h a r e i s o m o r p h i c t o r o t a t i o n groups R. (2.3.3) ; M a g n e t i c l a t t i c e s T M a r e tho s e m-groups of C^x A which a r e i s o m o r p h i c t o l a t t i c e s T. The r u l e (2.2.8) can t h e n be d i r e c t l y a p p l i e d t o the t h r e e c a s e s . I n the case o f MSGs the subgroups H are the space g r o u p s , and t h e s e a r e known. The problem o f f i n d i n g a l l MSGs has thus been r e d u c e d t o the problem of f i n d i n g a l l subgroups D*1 o f i n d e x 2 o f a l l space groups F. (2.3.4) A space group F and the MSGs o b t a i n e d by means o f r u l e (2.2.8) from a l l the 13 F's o f F w i l l be s a i d t o form the " f a m i l y " o f F. As i s well-known ( s e e , f o r example, Lomont ( 1 9 5 9 ) ) , two space groups a r e r e g a r d e d as b e l o n g i n g t o the same c l a s s o f space groups and a r e denoted by the same c r y s t a l l o g r a p h i c symbol i f and o n l y i f they a r e c o n j u g a t e subgroups o f the g e n e r a l inhomogeneous l i n e a r group S i m i l a r l y , two MSGs w i l l be r e g a r d e d as b e l o n g i n g t o the same c l a s s o f MSGs, and denoted by the same c r y s t a l l o -g r a p h i c symbol ( t h e n o t a t i o n used w i l l be e x p l a i n e d i n 19 C h a p t e r 4) i f and o n l y i f t h e y a r e c o n j u g a t e subgroups o f the d i r e c t p r o d u c t group C^x A. Two space groups w h i c h do not b e l o n g t o the same c l a s s o f space groups w i l l be o c c a s i o n a l l y r e f e r r e d t o as " p r o p e r l y d i f f e r e n t " . S i m i l a r l y , two MSGs whi c h do not b e l o n g t o t h e same c l a s s o f MSGs w i l l be o c c a s i o n a l l y r e f e r r e d t o as " p r o p e r l y d i f f e r e n t " . Perhaps i t s h o u l d be mentioned i n p a s s i n g t h a t M(D^ F) and M (D2 F ) may v e r y w e l l be p r o p e r l y d i f f e r e n t even though D i F and D 2 F a r e not p r o p e r l y d i f f e r e n t . T h i s p o i n t and s i m i l a r q u e s t i o n s w i l l be d i s c u s s e d l a t e r on ( S e c t i o n 4.3). I n the f o l l o w i n g c h a p t e r s we s h a l l need a more e x p l i c i t n o t a t i o n f o r the elements o f a space group, and of an MSG. We s h a l l d e n o t e , as i s customary, an element of a space group F by (R|t(ft),+ t ) where R i s a p r o p e r o r improper r o t a t i o n , r (R) the n o n - p r i m i t i v e t r a n s l a t i o n ( w hich i s always z e r o f o r the symmorphic space groups) b e l o n g i n g t o R, and t a p r i m i t i v e t r a n s l a t i o n . The elements o f an MSG a r e then o f the form ( R J T (Rt)i- fc;) £ and ( R j t C ^ + t j E ' , where (RJ xr(RJ +tj b e l o n g s t o D F, and ( RJtfR,) • t j b e l o n g s t o the c o s e t F - D F. I n most c a s e s the s i m p l i f i e d n o t a t i o n , i n t r o d u c e d a t the b e g i n n i n g of S e c t i o n 2 . 2 , w i l l be used: ( R J T(R ) + t j f o r an unprimed element o f an MSG, and ( R J T ( R ^ + bt)' f o r a primed element. 20 NOTES TO CHAPTER 2 The p r o c e d u r e summarized i n theorem (2,2.8) i s a s t r a i g h t f o r w a r d g e n e r a l i z a t i o n o f the well-known p r o c e d u r e t o f i n d a l l subgroups o f the group R o f a l l p r o p e r and improper r o t a t i o n s w hich do not c o n t a i n space i n v e r s i o n . (See, f o r example, Weyl ( 1 9 5 2 ) , Appendix B ) . The group R i s a d i r e c t p r o d u c t R p r x I where R p r i s the group o f a l l p r o p e r r o t a t i o n s and I i s the group c o n s i s t i n g o f i d e n t i t y and space i n v e r s i o n . Most of the theorems p r o v e d i n t h i s c h a p t e r can be found s t a t e d , i n a l e s s g e n e r a l form and w i t h o u t p r o o f s , i n Zamorzaev (1957). As has a l r e a d y been mentioned i n C h a p t e r 1, Zamorzaev speaks o f Shubnikov groups r a t h e r than MSGs and r e g a r d s h i s method o f c o n s t r u c t i n g Shubnikov groups as b e i n g " e s s e n t i a l l y t he Shubnikov method" (Shubnikov,1951). However, Shubnikov has not c o n s i d e r e d the t h r e e - d i m e n s i o n a l c a s e . Zamorzaev's theorem 3 says e s s e n t i a l l y the same as our d e f i n i t i o n (2.1.1) o f MSG. Zamorzaev's d e f i n i t i o n of Shubnikov groups i s c o n t a i n e d i n the f o l l o w i n g two paragraphs (we quote from the E n g l i s h t r a n s l a t i o n ) : "We s h a l l d e f i n e a Fedorov group as a group o f s p a t i a l symmetry t r a n s f o r m a t i o n s (which we s h a l l c a l l homologous) w i t h t h e f o l l o w i n g p r o p e r t i e s : 1) t h e r e e x i s t s a sphere o f s u f f i c i e n t l y l a r g e r a d i u s R such t h a t no m a t t e r where i t i s l o c a t e d i n space t h e r e l i e w i t h i n i t homologous images o f any a r b i t r a r y p r e v i o u s l y g i v e n p o i n t of space (homogeneity) and 2) t h e r e i s a t l e a s t one p o i n t i n the space w h i c h has no homologous images a r b i t r a r i l y c l o s e t o i t ( d i s c r e t e n e s s ) . 21 * • • L e t us now a s s i g n p o s i t i v e (+) o r n e g a t i v e (-) s i g n s t o the p o i n t s of space and d e f i n e a Shubnikov group as a group o f symmetry and a n t i s y m m e t r y t r a n s f o r m a t i o n s , w h i c h we s h a l l c a l l homologous and a n t i h o m o l o g o u s , r e s p e c t i v e l y , w h i c h f u l f i l l s t h e f o l l o w i n g r e q u i r e m e n t s : p r o p e r t y 1) (homogeneity) i s now f u l f i l l e d by b o t h homologous and antihomologous images o f a p o i n t , and p r o p e r t y 2) ( d i s c r e t e n e s s i s m a i n t a i n e d i n i t s p r e v i o u s form." We g i v e here a b r i e f d i c t i o n a r y of terms as used by Zamorzaev and o t h e r R u s s i a n a u t h o r s , and by o u r s e l v e s . The d i c t i o n a r y o f terms i s as f o l l o w s : A = t i m e - r e v e r s a l group = a n t i s y m m e t r y group = c h a n g e - o f - c o l o r group. M(^ F) = n o n - t r i v i a l MSG = b l a c k and w h i t e space group = minor Shubnikov group. F x A = g r e y space group = major Shubnikov group. F = o r d i n a r y s p a c e group «= Fedorov group = c o l o u r l e s s space group. 22 CHAPTER 3 DERIVATION OF RULES FOR CONSTRUCTING ALL SUBGROUPS OF INDEX 2  OF ALL SPACE GROUPS. I n t h i s c h a p t e r we d e s c r i b e a s y s t e m a t i c and e x h a u s t i v e method o f c o n s t r u c t i n g a l l the subgroups D F o f i n d e x 2 o f a l l space groups F. I n our d i s c u s s i o n o f t h i s problem we t a k e t h e knowledge o f the p r o p e r t i e s o f space groups f o r g r a n t e d . F o r the b a s i c d e f i n i t i o n s and p r o p e r t i e s we r e f e r t h e r e a d e r t o K o s t e r (1957) and Lomont (1 9 5 9 ) ; a d e t a i l e d d e s c r i p t i o n o f a l l space groups i s g i v e n i n " I n t e r n a t i o n a l T a b l e s f o r X-Ray C r y s t a l l o g r a p h y " (1952). In S e c t i o n 3.1 we s y s t e m a t i c a l l y d e r i v e the n e c e s s a r y c o n d i t i o n s f o r s e t s o f elements o f F t o be subgroups D F; t h i s e n a b l e s us t o f o r m u l a t e r u l e s f o r c o n s t r u c t i n g a l l s uch s e t s . I n many c a s e s the c o n d i t i o n s a r e a l s o s u f f i c i e n t , but not a l w a y s . The s u f f i c i e n t c o n d i t i o n s a r e d e r i v e d i n S e c t i o n 3.2. The r u l e s o f S e c t i o n 3.1 and 3.2 r e q u i r e , f o r most space groups F, the knowledge o f a l l subgroups R D o f i n d e x 2 o f t h e i r p o i n t g r o u p s , and o f a l l subgroups T° of i n d e x 2 o f t h e i r l a t t i c e s . S e c t i o n 3.3 i s d e v o t e d t o a s u r v e y o f the subgroups R D and T°, I n S e c t i o n 3.4 examples a r e g i v e n o f the c o n s t r u c t i o n o f a l l I ^ ' s f o r a few F's u s i n g the r u l e s o f S e c t i o n 3.1 and 3.2, and the data o f S e c t i o n 3.3. The f o l l o w i n g A w i l l o f t e n be used i n d e n o t i n g the 23 elements o f the v a r i o u s groups c o n s i d e r e d i n t h i s c h a p t e r (and i n the f o l l o w i n g c h a p t e r s ) : L e t H be a group and one o f i t s subgroups o f i n d e x 2. An element o f H w h i c h b e l o n g s t o D H w i l l be l a b e l e d w i t h a L a t i n s u b s c r i p t , e.g. H a, H b , H e , e t c . ( " L a t i n e l e m e n t s " ) ; an element o f H w h i c h does not b e l o n g t o D** ( i t b e l o n g s t h e n t o the c o s e t H - D H) w i l l be l a b e l e d w i t h a Greek s u b s c r i p t , e.g. , , H y e t c . ("Greek e l e m e n t s " ) . SECTION 3.1 L e t (R I f (R)+t) be the g e n e r a l element o f a space group F. The l e t t e r R denotes a p r o p e r o r improper r o t a t i o n o f the p o i n t group R b e l o n g i n g t o F. The symbol r ( R ) d e n o t e s a n o n - p r i m i t i v e t r a n s l a t i o n b e l o n g i n g t o R o f the form f (R) =» y, vt<x.t + v scx^ , Here a^, ag, ag a r e t h e t h r e e b a s i c p r i m i t i v e t r a n s l a t i o n s o f the l a t t i c e T o f F, and V, , v t ; ^» a r e numbers whose a b s o l u t e v a l u e s a r e s m a l l e r t h a n u n i t y . F i n a l l y the l e t t e r t denotes a p r i m i t i v e t r a n s l a t i o n of T. L e t ( R x |o*(R x)+t x) be the g e n e r a l element o f a sub-group gF o f i n d e x 2 o f F. F o r s i m p l i c i t y , from now on, we w r i t e D i n s t e a d o f D F, as no c o n f u s i o n can a r i s e . Here R x denotes a p r o p e r or improper r o t a t i o n o f the p o i n t group R x b e l o n g i n g t o D; ff"(Rx) denotes a n o n - p r i m i t i v e t r a n s l a t i o n c o r r e s p o n d i n g t o R x and t x denotes a p r i m i t i v e t r a n s l a t i o n o f the l a t t i c e T x o f D. 24 O b v i o u s l y R x and T x a r e ( p r o p e r o r imp r o p e r ) subgroups o f R and T r e s p e c t i v e l y . I f t he p o i n t group R x i s a p r o p e r subgroup of R we s h a l l denote the subgroup D by D T J i f R x i s the p o i n t group R i t s e l f t h e subgroup D w i l l be denoted by D R . Subgroups Dip. We f i r s t s t a t e two theorems c o n c e r n i n g DIJ>: ( 3 . 1 . 1 ) The l a t t i c e o f D T i s n e c e s s a r i l y i d e n t i c a l w i t h t h e l a t t i c e T of F. ( 3 . 1 . 2 ) The p o i n t group o f D T i s n e c e s s a r i l y a subgroup R D o f i n d e x 2 o f the p o i n t group R o f F. P r o o f o f ( 3 . 1 . 1 ) f o l l o w s from the o b s e r v a t i o n t h a t i f the l a t t i c e o f D T were a p r o p e r subgroup o f T the n D>j< would be a subgroup o f F o f an i n d e x h i g h e r than 2 . To prove ( 3 . 1 . 2 ) c o n s i d e r any two elements o f the c o s e t F-Diji o f D T, (R]^ | f ( R J+t^) and ( R 2 l'c(R2)+t2). Here R j and R 2 b e l o n g t o R-R x and t ^ and t 2 b e l o n g t o T. S i n c e F i s a space group, i t s l a t t i c e T i s i n v a r i a n t under R, t h a t i s R i t 2 e T f o r any c h o i c e o f R 4 € R - R x and t 2 £ T. S i n c e D T i s a subgroup o f i n d e x 2 o f F a c c o r d i n g t o theorem ( 2 , 2 . 7 ) t h e p r o d u c t (RxR2 | t - ( R i ) + R i tr (R2)+ti+Rit2) b e l o n g s t o D T . T h i s i s the case o n l y i f R1R2& R x o r , u s i n g theorem ( 2 . 2 . 7 ) a g a i n , i f R x - R D. From ( 3 . 1 . 2 ) i t f o l l o w s : ( 3 . 1 . 3 ) A space group F has no subgroups o f the type D^ i f i t s p o i n t group has no subgroups o f i n d e x 2 . T 25 I f the p o i n t group R o f F has subgroups R D o f i n d e x 2 t h e n theorems (3.1.1) and (3.1.2) g i v e i m m e d i a t e l y t h e f o l l o w i n g r u l e f o r c o n s t r u c t i n g a l l subgroups D T o f F: (3.1.4) Take the s e t P o f a l l t h o s e elements ( R a l T ( R f t ) + t ) o f F f o r w h i c h Rft b e l o n g s t o some s p e c i f i e d R D and t i s any t r a n s l a t i o n of the l a t t i c e T o f F. The s e t o b v i o u s l y c o n s t i t u t e s a D T. By t a k i n g a l l t he R D , s i n t u r n one o b t a i n s a l l the D T's o f F. Next we c o n s i d e r the q u e s t i o n , under w h i c h c o n d i t i o n s a subgroup D T w i l l be symmorphic. I f F i s a symmorphic space group, t h e n o b v i o u s l y e v e r y subgroup D T of i t i s a symmorphic space group t o o . I t s g e n e r a l element i s ( R a l t ) w i t h R a b e l o n g i n g t o a subgroup R D o f R and t b e l o n g i n g t o T, I n a non-symmorphic space group F t h e elements w i t h no n o n - p r i m i t i v e t r a n s l a t i o n s form o b v i o u s l y a subgroup o f F. I f the p o i n t group b e l o n g i n g t o such a subgroup o f F i s a subgroup o f i n d e x 2 of the p o i n t group R of F, t h e n o b v i o u s l y t h e space group F has a symmorphic subgroup D T. O t h e r w i s e any subgroup Dip of a non-symmorphic space group F i s a non-symmorphic space group t o o . I t s g e n e r a l element i s (R f t I t ( R a ) + t ) where a g a i n R a e R D, t <= T. I t s h o u l d be n o t i c e d t h a t i n t h i s c a s e o"(R a) = r ( R a ) . Subgroups D R. The p o i n t group o f D R i s by d e f i n i t i o n i d e n t i c a l w i t h t h e p o i n t group o f F; hence, o b v i o u s l y : 26 ( 3 . 1 . 5 ) The l a t t i c e T D o f D R i s a subgroup o f i n d e x 2 o f the l a t t i c e T o f F. From ( 3 . 1 . 5 ) i t does not f o l l o w , however, t h a t t r a n s -l a t i o n s w hich b e l o n g t o T - T D cannot o c c u r as t r a n s l a t i o n s i n t he elements o f D R . I t o n l y f o l l o w s t h a t i f t h e y do o c c u r t h e y must be n o n - p r i m i t i v e t r a n s l a t i o n s o f D R . Hence t h e subgroups D R may be o f two k i n d s : t h o s e , denoted by D R O , i n whose elements no t r a n s l a t i o n s of T - T° o c c u r j and t h o s e , denoted by D 0 , i n whose elements the t r a n s l a t i o n s o f tia T - T D do o c c u r . O b v i o u s l y , one can always choose as one o f the t h r e e b a s i c p r i m i t i v e t r a n s l a t i o n s o f T a t r a n s l a t i o n b e l o n g i n g t o T - T D. L e t the t r a n s l a t i o n so chosen be t a > S i n c e T D i s a subgroup o f i n d e x 2 o f T, e v e r y t r a n s l a t i o n o f T b e l o n g i n g t o T - T° can be e x p r e s s e d as a sum of t and a t r a n s l a t i o n b e l o n g i n g t o T D. ( 3 . 1 . 6 ) An element o f D R i s o f the form: ( R | C(R,) + ) where R € R j t a e T s and i f D = D P so i f D -• x> . R Ho* . > t a ( R ) may be e i t h e r z e r o o r e q u a l t o t a ; the l a t t e r p o s s i -b i l i t y must o c c u r f o r a t l e a s t one element o f i f . (we assume 27 h e r e t h a t by no c h o i c e of the c o o r d i n a t e system can t be a t r a n s f o r m e d away from a l l the elements o f D R a; f o r i f t h i s was p o s s i b l e the group i n q u e s t i o n would not be D R a but D R o.) S i n c e D R i s a space group i t s l a t t i c e must be i n v a r i a n t under any r o t a t i o n of i t s p o i n t group. Or, more p r e c i s e l y , f o r any R b e l o n g i n g t o the p o i n t group R o f D and any t a b e l o n g i n g t o the l a t t i c e o f D R, one must have (3.1.7) *^=h where t b b e l o n g s t o T°. But t h e p o i n t group R o f D R i s a l s o the p o i n t group o f t h a t F of w h i c h D R i s a subgroup of i n d e x 2. Hence: (3.1.8) The l a t t i c e T° o f D R and the l a t t i c e T o f F have the same h o l o h e d r y , the " h o l o h e d r y " o f a l a t t i c e b e i n g , a c c o r d i n g t o the u s u a l d e f i n i t i o n , the l a r g e s t p o i n t group w h i c h l e a v e s the l a t t i c e i n v a r i a n t . (Of c o u r s e , the p o i n t group R need not be the h o l o h e d r y o f the two l a t t i c e s . ) Subgroups D R o. To c o n s t r u c t a l l the subgroups D R o o f a g i v e n space group F one has t h u s , i n view o f ( 3 . 1 . 6 ) , the f o l l o w i n g r u l e : (3.1.9) Take the s e t A of a l l t h o s e elements (R | t ( R ) + t a ) o f F f o r w h i c h t a b e l o n g s t o some s p e c i f i e d T D w h i c h has the same h o l o h e d r y as T and R i s any r o t a t i o n of the p o i n t group R of F. I f the s e t A forms a group t h e n i t w i l l be n e c e s s a r i l y a subgroup D R o. By t a k i n g a l l s u ch T D , s i n t u r n one o b t a i n s a l l the D R o ' s of F. The c o n d i t i o n s 28 f o r the s e t L t o form a group a r e d i s c u s s e d l a t e r on i n S e c t i o n 3.2. The l i s t of a l l subgroups T° ( t h e r e a r e a t most 7 of them) o f a g i v e n l a t t i c e T i s g i v e n i n S e c t i o n 3.3. I t i s o b v i o u s from (3.1.9) t h a t D R o i s or i s not symmorphic, a c c o r d i n g as F i s or i s not symmorphic. Subgroups Dft a. I n o r d e r t o f o r m u l a t e the r u l e s f o r c o n s t r u c t i n g a l l t h e Dp^'s o f a g i v e n space group F, i t i s c o n v e n i e n t t o d i s c u s s s e v e r a l p o s s i b l e c a s e s s e p a r a t e l y . We s h a l l d i s c u s s t h e s e v e r a l p o s s i b l e c a s e s ( t h e y o b v i o u s l y e x h a u s t a l l p o s s i b i l i t i e s ) i n the f o l l o w i n g o r d e r : A) F whose p o i n t group R has subgroups R D: A l ) symmorphic F; A2) non-symmorphic F whose p o i n t group R does not c o n t a i n elements o f o r d e r h i g h e r than 2 j A3) non-symmorphic F whose p o i n t group R c o n t a i n s elements o f o r d e r h i g h e r t h a n 2. B) F whose p o i n t group R has no subgroups R D. Case A l (symmorphic F ) . Here we have the f o l l o w i n g theorem: (3.1.10) Those elements o f D R a o f a symmorphic F f o r w h i c h t f l ( R ) = 0 c o n s t i t u t e a subgroup Q o f i n d e x 2 o f DR a, and, hence, Q i s a subgroup o f i n d e x 4 o f F. To prove (3.1.10) we f i r s t o b s e r v e t h a t Q i s a group*, i n f a c t the p r o d u c t o f two elements w i t h t a ( R ) = 0 g i v e s an element w i t h t a ( R ) = 0 because ( s i n c e D R a i s a space group) 29 R t a = t b . To show t h a t Q i s a subgroup o f i n d e x 2 o f D j ^ l e t us m u l t i p l y any two elements o f D p o (, (R^ 1 t^+t^) a n d (R2 I t a + t b ) . The p r o d u c t i s always o f t h e form (R3R2 I t c ) because R t b = t c and R t f l = t g ( t h e l a s t e q u a t i o n f o l l o w s from t h e f a c t t h a t i f R t f l = t d one would have t f l = R ' ^ t ^ w h i c h would v i o l a t e the c o n d i t i o n t h a t the l a t t i c e o f D j ^ i s i n v a r i a n t w i t h r e s p e c t t o R ) , o r i n o t h e r words, the p r o d u c t b e l o n g s t o Q. Hence, a c c o r d i n g t o theorem ( 2 . 2 . 7 ) , the subgroup Q o f D R a i s of i n d e x 2. From the above p r o o f o f (3.1.10) i t a l s o f o l l o w s : (3.1.11) The l a t t i c e o f Q i s i d e n t i c a l w i t h t h e l a t t i c e T D o f D R o < and the p o i n t group o f Q i s a subgroup R D o f i n d e x 2 o f R. I n o t h e r words: the g e n e r a l element o f Q i s o f the f orm (R a I t a ) where R ae R D and t a € T D . Theorems (3.1.10) and (3.1.11) i m p l y t h a t the elements o f F can be a r r a n g e d i n t o l e f t c o s e t s r e l a t i v e t o Q as f o l l o w s : 0 t h c o s e t j S t c o s e t (3.1.12) 2 n d c o s e t 3 r d c o s e t <RJ V< RalV From (3.1.12) i t i s i m m e d i a t e l y a p p a r e n t t h a t (3.1.13) the 0 t h and t h e 3 r d c o s e t c o n s t i t u t e the group Dp w from w h i c h we have s t a r t e d o u t | (3.1.14) the 0 t h and t h e 2 n d c o s e t a l s o c o n s t i t u t e a g r o u p , w h i c h i s o f the k i n d D R o; 30 (3.1.15) the 0 t n and the 1 s t c o s e t a l s o c o n s t i t u t e a gr o u p , w h i c h i s o f the k i n d Df. The f o l l o w i n g r u l e f o r c o n s t r u c t i n g a l l t he Dp^'s o f a g i v e n symmorphic F w i l l t hen be v a l i d . (3.1.16) Take t h e s e t A o f a l l t h o s e elements ( R a I t a ) of the symmorphic space group F f o r which R a b e l o n g s t o some s p e c i f i e d BP and t a b e l o n g s t o some s p e c i f i e d T^ o f T wh i c h has the same h o l o h e d r y as T. The s e t A then w i l l form a group w h i c h w i l l be n e c e s s a r i l y a subgroup o f i n d e x 4 o f F, and e v e r y subgroup Q o f any subgroup Dp^ o f F w i l l be i d e n t i c a l w i t h one of the groups o b t a i n e d i n t h i s way by t a k i n g a l l s u ch p a i r s BP} T D. A l s o the s e t c o n s i s t i n g o f the elements o f the 0 t h and 3 r d c o s e t s w i l l n e c e s s a r i l y form a group. T h i s i s e i t h e r a subgroup Dp^ o r D R o o f F. The l a t t e r c a s e o c c u r s when a l l t h e t ^ can be t r a n s f o r m e d away, o r , i n o t h e r words, when the group, formed by the 0 t n and 3 r d c o s e t s o f F r e l a t i v e t o Q i s not p r o p e r l y d i f f e r e n t (see S e c t i o n 2.3) from t h e group formed by t h e 0 t n and 2 n d c o s e t s o f F r e l a t i v e t o Q. That A forms a group f o l l o w s i m m e d i a t e l y by c o n s i d e r i n g the p r o d u c t o f two a r b i t r a r y e l ements o f A . ( R a l t a ) ( R b | t b ) « ( R a | t a + R a t b ) . I n f a c t , R a t b = t c i n v i e w o f ( 3 . 1 . 8 ) . That the s e t c o n s t i t u t e d by the 0 t n and 3 r d c o s e t s i s a group f o l l o w s s i m i l a r l y by c o n s i d e r i n g the p r o d u c t o f two a r b i t r a r y elements o f t h e s e t . T h i s p r o d u c t n e c e s s a r i l y b e l o n g s t o t h e s e t , a g a i n because o f (3.1.8) w h i c h i m p l i e s R ^ t ^ = t g , 31 Ra*<y = *y ( a f a c t a l r e a d y used i n the p r o o f o f (3.1.10)). 9 D i g r e s s i o n . R u l e (3.1.16) can be s l i g h t l y e x t e n d e d so as t o become a r u l e f o r o b t a i n i n g a l l t h e subgroups D o f i n d e x 2 o f a g i v e n symmorphic space group F i f i t s p o i n t group R has subgroups R D. T h i s p o s s i b i l i t y r e a d i l y f o l l o w s from (3.1.14) and ( 3 . 1 . 1 5 ) , and the r u l e s (3.1.4) and (3.1.9) s p e c i a l i z e d t o the case o f a symmorphic F. The extended r u l e (3.1.16) i s : (3.1.17) Take the s e t A o f a l l t h o s e e l e m e n t s ( R a | t a ) o f t he symmorphic space group F f o r w h i c h R a b e l o n g s t o some s p e c i f i e d R D and t a b e l o n g s t o some s p e c i f i e d T D o f F, whi c h has t h e same h o l o h e d r y as T. The s e t A i s a group, w h i c h w i l l be n e c e s s a r i l y a subgroup Q o f i n d e x 4 o f F, and the elem e n t s o f F can be a r r a n g e d i n t o c o s e t s as i n t h e ( 3 . 1 . 1 2 ) . The elements o f the 0 t n and I s * c o s e t w i l l t h e n n e c e s s a r i l y form a subgroup D-p o f F. S i m i l a r l y the elements o f the 0 t n and 2 n d c o s e t w i l l form a subgroup D R o and t h o s e o f t h e 0 t n and 3 r d c o s e t w i l l form a subgroup D R a. By t a k i n g a l l p o s s i b l e p a i r s R D, T D o f F, one w i l l o b t a i n i n t h i s way a l l p o s s i b l e subgroups D f , D R o and D R a of a symmorphic space group F ( i f i t s p o i n t group R has subgroups R D ) . Case A2 and A3 (non-symmorphic F ) . The p r o c e d u r e f o r c o n s t r u c t i n g t h e Spy's i n Case A l i s based on the v a l i d i t y o f theorem ( 3 . 1 . 1 0 ) . T h i s theorem i s not a lways v a l i d i n the case o f non-symmorphic space groups. 32 C o n s e q u e n t l y , the p r o c e d u r e cannot be always u s e d f o r non-symmorphic space groups w i t h o u t some m o d i f i c a t i o n . I f a non-symmorphic space group F has a p o i n t group w h i c h does not c o n t a i n elements o f o r d e r h i g h e r t h a n 2 theorem (3.1.10) i s s t i l l v a l i d a l t h o u g h i t s p r o o f i s somewhat more c o m p l i c a t e d . F o r a non-symmorphic space group F whose p o i n t group c o n t a i n s some elements o f o r d e r h i g h e r than 2 theorem (3.1.10) may o c c a s i o n a l l y be v a l i d but i s not v a l i d i n g e n e r a l . That i s why we d i s t i n g u i s h between Case A2 and A3. B e f o r e c o n s i d e r i n g t h e two c a s e s s e p a r a t e l y we w i l l r e p e a t some v e r y well-known p r o p e r t i e s o f the elements o f a non-symmorphic space group. L e t (R | r ( R ) + t ) be an element o f some F. I f R i s a p r o p e r r o t a t i o n o r a r e f l e c t i o n i n a p l a n e a n o n - p r i m i t i v e t r a n s l a t i o n v ( R ) a l o n g the r o t a t i o n a x i s i n t h e f i r s t c a s e , and p a r a l l e l t o t h e p l a n e o f r e f l e c t i o n i n the second c a s e , cannot be t r a n s f o r m e d away by s h i f t i n g the o r i g i n o f the c o o r d i n a t e 1 s y s t e m . I n a l l o t h e r c a s e s t h e r e i s always a t r a n s l a t i o n by means o f wh i c h T (R) can be t r a n s f o r m e d away. I t w i l l be c o n v e n i e n t t o say t h a t v (R) i s a " t r u e " n o n - p r i m i t i v e t r a n s l a -t i o n when i t cannot be t r a n s f o r m e d away. D i f f e r e n t elements o f a space group may have d i f f e r e n t t r u e T ( R ) . I n such a case the n o n - p r i m i t i v e t r a n s l a t i o n o f an element R w i l l n o t i n o n l y g e n e r a l c o n s i s t o f the t r u e tr ( R ) A b u t a l s o o f an a d d i t i o n a l n o n - p r i m i t i v e t r a n s l a t i o n w h i c h c o u l d be t r a n s f o r m e d away from t h a t p a r t i c u l a r element a l t h o u g h not from a l l t he o t h e r elements 33 o f t h e group. We w i l l c a l l t h i s a d d i t i o n a l n o n - p r i m i t i v e t r a n s -l a t i o n t h e " a p p a r e n t " n o n - p r i m i t i v e t r a n s l a t i o n o f R. Case A2 (non-symmorphic F b e l o n g i n g t o the m o n o c l i n i c o r  d r t h o r h o r o b i c system). The analogue o f theorem (3.1.10) i n t h i s c a s e i s as f o l l o w s : (3.1.18) L e t F be a non-symmorphic space group whose p o i n t group has no elements o f o r d e r h i g h e r t h a n 2. Those e l e m e n t s (R I r ( R ) + t a ( R ) + t b ) o f a subgroup D R w o f F f o r w h i c h tgt(R) = 0 form a subgroup o f i n d e x 2 o f Dp^, and, hence a subgroup Q X o o f i n d e x 4 o f F. The p r o o f o f (3.1.18) c o n s i s t s o f two p a r t s . F i r s t we show t h a t t (R) cannot be e q u a l t o t ^ / 2. I n f a c t , i f t r ( R ) were a t r u e n o n - p r i m i t i v e t r a n s l a t i o n and were e q u a l t o t w / 2 then e q u a t i o n s R t a - t b and R t ^ = T Q would i m p l y (R l ^ ( R ) + t a ( R ) + t a ) 2 - (E I t B ) , w h i c h would c o n t r a d i c t the f a c t t h a t the l a t t i c e o f Dp^ i s TD, and hence does not c o n t a i n t g . On the o t h e r hand, r (R) cannot be an apparent n o n - p r i m i t i v e t r a n s l a t i o n e q u a l t o t ^ / 2 because i f i t d i d t h i s would i m p l y the p r e s e n c e i n DR(X. o f some o t h e r element w i t h a t r u e n o n - p r i m i t i v e t r a n s l a t i o n e q u a l t o t ^ / 2 . We c o n c l u d e t h a t V (R) can o n l y be z e r o o r e q u a l t o a 34 h a l f o f a p r i m i t i v e t r a n s l a t i o n o f tu. The r e s u l t i m p l i e s n e x t t h a t the p r o d u c t o f two elements o f 0 * i |^ ( R i J + t ^ ( R i ) + t a J and (R2l^(R2)+ t«(R2) + tb) i s o f t h e f o r m ( R l R 2 l r ( R l R 2 i + t c J i f b o t h t^CRi) and t w ( R 2 ) a r e z e r o o r b o t h d i f f e r e n t from z e r o . The f i r s t p a r t o f t h i s s t a t e m e n t shows t h a t the elements w i t h t^CR) = 0 form a group, the second p a r t t o g e t h e r w i t h theorem (2.2.7) shows t h a t the group i s a subgroup o f i n d e x 2 o f D R ( X, and, hence, a subgroup Q T ( ? o f i n d e x 4 o f F. O b v i o u s l y , (3.1.19) the l a t t i c e o f Q T ( J i s i d e n t i c a l w i t h the l a t t i c e T° of D R k and the p o i n t group o f Q v o i s a subgroup R D o f i n d e x 2 o f R. In o t h e r words the g e n e r a l element o f S i s o f the form ( R a | f ( R a ) + t a where R a € RD, t a <= TD. From (3.1.18) and (3.1.19) i t f o l l o w s t h a t the elements o f F can be a r r a n g e d i n t o l e f t c o s e t s r e l a t i v e t o Q r as f o l l o w s : . . » 0 t h c o s e t ( E t O ) ( R a | ' c r ( R a ) + t a ) 1 s t c o s e t (E | t 0 < ) ( R a | t r ( R a ) + t a ) (3.1.20) 2 n d C O S e t (R^ | T r ( R 0 c ) ) ( R a | ^ ( R a ) + t a ) 3 r d c o s e t ( R a |T(R J + t ^ ) ( R a | r ( R a ) + t a ) . The 0 t n and the 3^d c o s e t c o n s t i t u t e the group DR<X from which we have s t a r t e d o u t ; the 0 t n and t h e 2 n d c o s e t c o n s t i t u t e a group o f the k i n d D R o j the 0 t n and the 1 s t c o s e t c o n s t i t u t e a group o f t h e k i n d D*p. 3 5 Hence, j u s t as i n the case of a symmorphic space group F, we have the following r u l e for constructing a l l the DR<x*s of a given non-symmorphic space group F whose point group does not contain elements of order higher than 2 : ( 3 . 1 . 2 1 ) Take the set 0 of a l l those elements (R a l t ( R a ) + t a ) o f F f o r which R a belongs to some f i x e d subgroup R D of R and t a belongs to some subgroup T D of T which has the same holohedry as T. If 0 i s a group, then i t w i l l be necessarily a subgroup of index 4 of F and every subgroup Q 7 0 of any subgroup DRoti of F w i l l be i d e n t i c a l with one of the groups obtained i n that way by considering a l l such pairs R D and T D. Arrange the elements of F into cosets r e l a t i v e to 9 = q v o as i n ( 3 . 1 . 2 0 ) . I f the elements of the 0 T H and 2 N D coset form a group t h i s group i s a subgroup D R o of F, i f the elements of the 0 T H and 3 R D coset form a group t h i s group i s a subgroup D R { / of F. In the case of symmorphic space groups we could generalize the analogous rule ( 3 . 1 . 1 6 ) for constructing a l l the D R ( X 's so as to obtain the extended rule ( 3 . 1 . 1 7 ) for constructing a l l the D's of a given symmorphic Ft In the present case such an extension i s not possible. The reason for t h i s difference i s that while every symmorphic D T can always be regarded as consisting of the 0 T H and 1 S T coset i n ( 3 . 1 . 1 2 ) , t h i s i s not always so i n the case of a non-symmorphic Df. Hence, i n the present case of a non-symmorphic F, the analogue of the extended rule need not give a l l the 36 Dip's f o r some F' s , and, i n f a c t , i t does n o t , as w i l l be shown i n S e c t i o n 3.2. Of c o u r s e , t h i s does not i n v a l i d a t e t h e c o m p l e t e n e s s o f our g e n e r a l p r o c e d u r e , a c c o r d i n g t o w h i c h a l l D^'s o f a g i v e n F can be o b t a i n e d by u s i n g ( 3 . 1 . 4 ) . Case A3 (non-symmorphic F b e l o n g i n g t o the t e t r a g o n a l ,  t r i g o n a l , h e x a g o n a l o r c u b i c s y s t e m ) . We w i l l c o n s i d e r i n t u r n non-symmorphic space groups F b e l o n g i n g t o the f o u r systems j u s t enumerated. T e t r a g o n a l s y s t e m ; F whose l a t t i c e i s p l a i n ( P ) . Here we have t o d i s t i n g u i s h between two c a s e s : E i t h e r the l a t t i c e T° o f a D R w o f F i s such t h a t the b a s i c p r i m i t i v e t r a n s l a t i o n t ^ b e l o n g i n g t o T - i s p e r p e n d i c u l a r t o the 4 - f o l d a x i s , o r the l a t t i c e T D i s such t h a t tK i s p a r a l l e l t o th e 4 - f o l d a x i s . j; I n t h e former c ase the p r o c e d u r e d e v e l o p e d f o r Case A2 w i l l o b v i o u s l y be v a l i d w i t h o u t any change. I n o t h e r words, a theorem anal o g o u s t o (3.1.18) which e n s u r e s t h e e x i s t e n c e o f a subgroup Q r o w i l l be v a l i d , and a l l subgroups D R c < o f t h i s k i n d w i l l be o b t a i n e d from t h e r u l e ( 3 . 1 . 2 1 ) . I n t h e l a t t e r c a s e , t h a t i s i n the case o f D^^ 's i n w h i c h t ^ i s p a r a l l e l t o the 4 - f o l d a x i s , t h e p r o c e d u r e must be m o d i f i e d . L e t us c a l l the subgroups DR0< o f t h i s k i n d t h e " s t a n d a r d " subgroups o f F. A s t a n d a r d subgroup D R W has thu s the p r o p e r t y t h a t the b a s i c p r i m i t i v e t r a n s l a t i o n o f i t s l a t t i c e lP a l o n g the d i r e c t i o n o f the 4 - f o l d a x i s i s t w i c e as l a r g e as t h a t o f T a l o n g the same d i r e c t i o n . 37 L e t (R lt : ( R ) + t _ ( R ) + t Q ) be an element o f a s t a n d a r d subgroup Djja o f F. I f R i s o f o r d e r 2 t h e n a t r u e n o n - p r i m i t i v e t r a n s l a t i o n must be o f t h e form t b / 2 whereas an a p p a r e n t non-p r i m i t i v e t r a n s l a t i o n can be o f the form t^/2 p r o v i d e d t h a t the element f o r wh i c h i t i s a t r u e n o n - p r i m i t i v e t r a n s l a t i o n i s o f o r d e r 4. I f T (R4) o f an element R4 o f o r d e r 4 has a component^(R4) wh i c h i s a t r u e n o n - p r i m i t i v e t r a n s l a t i o n o f R4 the n o r where >w i s e i t h e r 0 or 1. Hence: (3.1.22) On l y t h o s e t e t r a g o n a l space groups F w h i c h c o n t a i n t h e element 4 o r the element 4g can have s t a n d a r d subgroups D R ( X . (Here we have u s e d , f o r c o n v e n i e n c e , the n o t a t i o n o f the " I n t e r n a t i o n a l T a b l e s " , and we s h a l l do t h a t o c c a s i o n a l l y i n the r e m a i n d e r o f t h i s s e c t i o n . ) I t i s easy t o see t h a t i f F c o n t a i n s the element 4 a theorem a n a l o g o u s t o (3.1.18) w h i c h e n s u r e s the e x i s t e n c e o f a subgroup Q r o a g a i n w i l l be v a l i d , and a l l subgroups Dp^ w i l l be o b t a i n e d from the r u l e ( 3 . 1 . 2 1 ) . I f , on the o t h e r hand, F c o n t a i n s the element 4g then t h i s element must become e i t h e r 4-^  o r 4% i n a s t a n d a r d subgroup 3 8 D R c x o f F. I t w i l l be 4± o r 4 3 depending on whether t ^  (R4) = O or t a (R4) = t a . One has the n the two f o l l o w i n g theorems: ( 3.1 . 2 3 ) I f F c o n t a i n s t h e element 4 2 t h e n those e l e m e n t s o f a s t a n d a r d subgroup D R ^ of F f o r wh i c h t ^ (R) = 0 do n ot form a group j i n o t h e r words, subgroups Q.^0 do not e x i s t i n t h i s c a s e . The v a l i d i t y o f t h i s theorem f o l l o w s i m m e d i a t e l y from the f a c t t h a t , s i n c e v ( R 4 ) = t ^ / 2 , (R4 l ^ ( R 4 ) + t a J 2 a ( R 4 2 I w h i c h c o n t r a d i c t s the r e q u i r e m e n t t h a t the sq u a r e o f each element of Qxo must b e l o n g t o Q ^ . ( 3.1 . 2 4 ) I f F c o n t a i n s the element 4 2 t h e n each s t a n d a r d subgroup D R o ( o f F has subgroups o f i n d e x 2 w i t h e l ements o f the form (R a|'F (R A )+t ( R a ) + t a ) where t ^ ( R A ) = t ^ f o r some R A . The v a l i d i t y of t h i s theorem f o l l o w s from the f a c t t h a t t h e p o i n t group R o f F has by as s u m p t i o n subgroups o f i n d e x 2 , and from the f a c t t h a t the p r o d u c t o f any two elements o f the form (R^ | T ( R 0 ( ) + t ( X ( R c / ) + t a ) i s n e c e s s a r i l y o f t h e form ( R a | T r ( R a ) + t O T ( R a ) + t b ) . From ( 3.1 . 2 4 ) f o l l o w s : ( 3.1 . 2 5 ) The l a t t i c e o f a subgroup Q R < X o f a group D R o i i s i d e n t i c a l w i t h t h e l a t t i c e T D of D R O ( and t h e p o i n t group o f Q _ i s a subgroup R ° o f i n d e x 2 o f R . Theorem ( 3.1 . 2 4 ) a l s o i m p l i e s t h a t the elements o f an F c o n t a i n i n g the element 4 2 can be a r r a n g e d i n t o l e f t c o s e t s 39 r e l a t i v e t o Q as f o l l o w s : 0 t h c o s e t (E I 0 ) ( R a l ^ ( R a ) + t ( A ( R a ) + t a } 1 s t c o s e t (E| t^JCRal^CRaJ+t^CRaJ+ta) (3.1.26) 2 n d c o s e t <RjTr<Rj)(Rj ' c<R a)+t a(R a)+t a) 3 r d c o s e t ( R j ' s C R j + t^ K R a l ^ C R a J + V C R a J + t a ) -The 0 t h and t h e 3 r d c o s e t , and the 0 t h and the 2 n d c o s e t c o n s t i t u t e groups D R o l} the 0*n and t h e I s * c o s e t c o n s t i t u t e a group o f the k i n d D T» Hence, we have the f o l l o w i n g r u l e f o r c o n s t r u c t i n g a l l th e s t a n d a r d Dp^'s o f a g i v e n non-symmorphic F whic h c o n t a i n s 4 2 « (3.1.27) Take the s e t Z o f elements ( R a | r ^ ( R a ) + t < x ( R a ) + t a ) o f F suc h t h a t R a b e l o n g s t o some f i x e d subgroup R D o f R, t a b e l o n g s t o some subgroup TP o f T w h i c h has t h e same h o l o h e d r y as T and whose b a s i c p r i m i t i v e t r a n s l a -t i o n a l o n g t h e 4 - f o l d a x i s i s t w i c e the b a s i c p r i m i t i v e t r a n s l a t i o n o f T a l o n g the same d i r e c t i o n and t^CR^p) =* t ^ . I f X i s a group, t h e n i t w i l l be n e c e s s a r i l y a subgroup o f i n d e x 4 o f F, and e v e r y subgroup Q r a o f any subgroup D R o< of F w i l l be i d e n t i c a l w i t h one o f the groups o b t a i n e d i n t h a t way by c o n s i d e r i n g a l l p o s s i b l e p a i r s o f R D and T°. Arrange the e l e m e n t s o f F i n t o c o s e t s r e l a t i v e t o 27 = <3 as i n (3.1.26). I f t h e elements o f the 0 t h and 2 n d c o s e t form a group t h i s group i s a subgroup D R ^ o f F. S i m i l a r l y i f the elem e n t s o f the 0 t n and 3 r d c o s e t form a group t h i s group i s 40 a subgroup DRo- of F. Tetragonal system: F whose l a t t i c e Is body-centered ( I ) . As Is w e l l known i n these space groups besides the elements of the second order there can be only the elements 4 or 4^ and t h e i r powers. Because of the condition (3.1.8) requiring T and T^ to have the same holohedry, only one kind of T° has to be considered i n t h i s case (for the proof of t h i s statement see Section 3,3), namely a p l a i n l a t t i c e T° whose elements are the translations here Q. _ v b • c . ou+ b-c n . Ou - b f C 5 * < V ? — > ° S - 5 — and + b+ c « 2 <»/ I f F contains the element 4, the s i t u a t i o n i s the same as i n the case of a tetragonal F with a p l a i n l a t t i c e discussed before: a subgroup Q r o of D R^ e x i s t s , and rule (3.1.21) can be used for constructing a l l D R (/s. If F contains the element 4^ then t h i s element must be either 4^ or 4 3 i n a subgroup D R ( X of F. I t w i l l be 4^ or 4 3 depending on whether t (R4) « 0 or t (R 4) - . 41 One then p r o v e s as i n t h e c a s e o f a t e t r a g o n a l F w i t h a p l a i n l a t t i c e t he v a l i d i t y o f two theorems w h i c h d i f f e r from (3.1.23) and (3.1.24) o n l y by r e p l a c i n g " 4 2 " by " 4 x " and o m i t t i n g t h e word " s t a n d a r d " , and hence shows t h a t the elements o f F c a n be a r r a n g e d as i n ( 3 . 1 . 2 6 ) . We th e n o b t a i n t h e f o l l o w i n g r u l e f o r f i n d i n g a l l the D R c / s o f an F h a v i n g a t e t r a g o n a l bodyr c e n t e r e d l a t t i c e and c o n t a i n i n g the element 4^: (3.1.28) Take the s e t X o f elements ( R a | T r ( R a ) + t 0 < ( R a ) + t a ) o f F s u c h t h a t R a b e l o n g s t o some f i x e d subgroup R° o f R, t a b e l o n g s t o t h e subgroup T D o f T whi c h has th e same h o l o h e d r y as T and t ^ (R4 2) = t ^ . I f X i s a group, t h e n i t w i l l be n e c e s s a r i l y a subgroup o f i n d e x 4 o f F, and e v e r y subgroup Q^ ^ o f any subgroup BRo( o f F w i l l be i d e n t i c a l w i t h one o f t h e groups o b t a i n e d i n t h a t way by c o n s i d e r i n g a l l p o s s i b l e p a i r s o f R D and T°. Arra n g e t h e elements o f F i n t o c o s e t s r e l a t i v e t o X = $ as i n ( 3 . 1 . 2 6 ) . I f t h e elements o f the 0 t n and 2 n d c o s e t form a group t h i s group i s a subgroup Dp^ o f F. S i m i l a r l y , i f t h e ele m e n t s o f the O t n and 3 r d c o s e t form a group t h i s group i s a subgroup D R o ( o f F. T r i g o n a l s ystem and h e x a g o n a l system. The p r o c e d u r e f o r f i n d i n g a l l the DRo( 's o f an F of either o f t h e s e two systems i s the same as i n the case o f t h e t e t r a g o n a l s y stem e x c e p t f o r some o b v i o u s m o d i f i c a t i o n s . T r i g o n a l s y s t e m ; F whose l a t t i c e i s h e x a g o n a l . I n t h i s c a s e because o f the h o l o h e d r y c o n d i t i o n (3.1.8) 42 a l l subgroups D R of F have the same l a t t i c e w h i c h has a l o n g the 3 - f o l d a x i s a b a s i c p r i m i t i v e t r a n s l a t i o n t w i c e as l a r g e as t h a t o f T a l o n g the same d i r e c t i o n . ( A l l D R c < 's a r e thus " s t a n d a r d subgroups" i n t h e sense i n wh i c h t h i s term has been used i n the t e t r a g o n a l c a s e . ) L e t us c a l l t ^ t h e b a s i c p r i m i t i v e t r a n s l a t i o n of T a l o n g the t r i g o n a l a x i s , t h a t o f T D i n the same d i r e c t i o n w i l l be zt^  . As i s w e l l known i n a space group F o f the t r i g o n a l s y s t e m the t r u e n o n - p r i m i t i v e t r a n s l a t i o n f o r a r o t a t i o n Rg o f o r d e r 3 can o n l y be 0, 1/3 o r 2/3 o f the b a s i c p r i m i t i v e t r a n s l a t i o n a l o n g the a x i s ( t h e c o r r e s p o n d i n g elements o f F a r e 3, 3^ and 3 2 ) . Thus i n a subgroup D R < 3 ( o f a space group F the t r u e n o n - p r i m i t i v e t r a n s l a t i o n f o r t h e r o t a t i o n o f o r d e r 3 w i l l be 0, ± * ^ , o r 1. 2 t , and 0 , _ L 1 ^, Jt 1 c, = _L 11 , f o r i t s square r e s p e c t i v e l y . Hence: 1 01 2> (3.1.29) A space group o f the t r i g o n a l system w h i c h c o n t a i n s the element 3 can o n l y have subgroups D R ( X which c o n t a i n t h a t element. A space group which c o n t a i n s 32 has o n l y subgroups D R o ( w h i c h c o n t a i n the element 3^. A space group w h i c h c o n t a i n s 3^ has o n l y subgroups D R o ( w h i c h c o n t a i n t h e element 32* From (3.1.29) i t f o l l o w s t h a t , i f F c o n t a i n s the element 3, a theorem analogous t o (3.1.18) e n s u r i n g the e x i s t e n c e o f a subgroup Q r o w i l l be v a l i d , and the r u l e AX (3.1.21) can be used f o r o b t a i n i n g a l l D R a 's i n t h i s c a s e , j u s t as i n t h e analogous t e t r a g o n a l c a s e . 4 3 A g a i n , because o f ( 3.1 . 2 9 ) , the two f o l l o w i n g theorems, analogous t o ( 3.1 . 2 3 ) and ( 3.1 . 2 4 ) can be proved i n t he same way as i n t h e t e t r a g o n a l c a s e : ( 3 . 1 . 3 0 ) I f F c o n t a i n s 3^ o r 3 2 t h o s e elements o f DRo(. * o r w h i c h t ^ ( R ) sx o do not form a group} i n o t h e r words subgroups Q^ 0 do not e x i s t i n t h i s c a s e . ( 3 . 1 . 3 1 ) I f F c o n t a i n s 3^ o r 3 2 then each subgroup D R ( * has subgroups Q T O < . By the same argument as i n the t e t r a g o n a l c a s e we have t h e n the f o l l o w i n g r u l e f o r c o n s t r u c t i n g a l l the D R < 5 ( 's f o r the t r i g o n a l c a s e : ( 3 . 1 . 3 2 ) Take the s e t tp- o f elements (Ra l ^ ( R a ) + t o ( ( R a ) + t a ) o f F such t h a t R a b e l o n g s t o some f i x e d R D o f R, t a b e l o n g s t o the subgroup T D o f T whic h has the same h o l o h e d r y as T and t ^ ( R a ) = t ^ f o r R3 o r f o r i t s i n v e r s e . I f i s a group t h e n i t w i l l be n e c e s s a r i l y a subgroup o f i n d e x 4 o f F , and e v e r y subgroup o f any subgroup D ^ R C X o f F w i l l be i d e n t i c a l w i t h one o f the groups o b t a i n e d i n t h a t way by c o n s i d e r i n g a l l p o s s i b l e p a i r s o f R D and T^. Arrange th e elements o f F i n t o c o s e t s r e l a t i v e t o = Q r ( X a s i n ( 3 . 1 . 2 6 ) . I f the elements o f the 0 T H and 2 N D c o s e t form a gro u p , t h i s group i s a subgroup DJJ,^ o f F . S i m i l a r l y , i f the elements o f the 0 * N and 3 R D c o s e t form a group t h i s group i s a subgroup D R ( X o f F . T r i g o n a l s y s t e m ; F whose l a t t i c e i s rhombohedral. I n t h i s case i t t u r n s out (see S e c t i o n 3 . 2 ) t h a t the c o n d i t i o n s f o r the e x i s t e n c e o f D R ' s o f a non-symmorphic F 44 a r e never s a t i s f i e d ; i n o t h e r words, a t r i g o n a l non-symmorphic F w i t h a rhombohedral l a t t i c e does not have subgroup Dp. Hexagonal system I n t h i s c a s e , because o f the h o l o h e d r y c o n d i t i o n (3.1.8), a l l subgroups Dp o f an he x a g o n a l space group F have the same l a t t i c e T° w h i c h has the b a s i c p r i m i t i v e t r a n s l a t i o n a l o n g the 6 - f o l d a x i s t w i c e as l a r g e as t h a t o f T a l o n g the i same d i r e c t i o n . A r o t a t i o n Rg o f o r d e r 6 can have a n o n - p r i m i t i v e t r a n s l a t i o n e q u a l t o 0, 1/6, 2/6, 3/6, 4/6, 5/6 o f t h e b a s i c p r i m i t i v e t r a n s l a t i o n a l o n g the d i r e c t i o n o f i t s a x i s . The i c o r r e s p o n d i n g elements o f F a r e : 6, 6 i , 62, 63, 64, 65. I f 2t o < i s the b a s i c p r i m i t i v e t r a n s l a t i o n a l o n g the 6 - f o l d a x i s o f a Dp, the p o s s i b l e t r u e n o n - p r i m i t i v e t r a n s -l a t i o n s o f R 6 i n D R a r e : 0 , 1 2 ^ , 1 2 ^ , 1 2 ^ »f 2t<X » Jilb.. Hence: (3.1,33) I f F c o n t a i n s tho element ( R 6 0 ) a subgroup D R ^ o f F can c o n t a i n e i t h e r the element (Rg|0) o r the element ( R g l t ^ ) = (Rg | JL ^  t^), t h a t i s , i n the f i r s t c a s e D p^ c o n t a i n s the element 6, i n the second case c o n t a i n s the element 63. I f F c o n t a i n s (Rg \z. t^), a subgroup Dp ^  o f F can c o n t a i n e i t h e r (R f i \± 2^) o r ( R e | ^ t k + tw ) = ( R g | ^ 2 f e ) . I» the f i r s t c a s e D p a c o n t a i n s the element 6^, i n the second case c o n t a i n s the element 64. F i n a l l y i f ¥ c o n t a i n s (Rg I A. t^), a subgroup D R ^ o f F may c o n t a i n e i t h e r (Rg \2g 2^) o r (Rg = ( R g [_£ 2 ^  ). I n t h e f i r s t c a s e DR(j( c o n t a i n s 45 i n the second case D R ( X c o n t a i n s 6 5 . From (3.1.33) i t f o l l o w s t h a t I f F c o n t a i n s the element 6 a Q. w i l l e x i s t j u s t as i n the analogous case o f t e t r a g o n a l and t r i g o n a l s y s t e m , and the r u l e (3.1.21) can be used f o r o b t a i n i n g a l l the D R ( X 's o f F. I n a l l o t h e r c a s e s two theorems a n a l o g o u s t o ( 3 . 1 . 2 3 ) , (3.1.24) and ( 3 . 1 . 3 0 ) , (3.1.31) w i l l be v a l i d , and e i t h e r a Q w i l l e x i s t o r no D p w i l l e x i s t a t a l l . From th e d i s c u s s i o n g i v e n i n the f o l l o w i n g s e c t i o n i t f o l l o w s t h a t no D R can rs* r^-f e x i s t i f F c o n t a i n s elements 6 ^ , 6 3 o r 6 5 . I f F c o n t a i n s 6 2 o r 6 4 t h e n a r u l e f o r c o n s t r u c t i n g a l l D R c / s of F can be f o r m u l a t e d . I t w i l l be analogous t o ( 3 . 1 . 2 8 ) . C u b i c system. The problem o f c o n s t r u c t i n g a l l D R < x ' s o f a n F i n t h i s c a s e can be e a s i l y s e t t l e d u s i n g the r u l e s d e r i v e d p r e v i o u s l y , and the d i s c u s s i o n o f t h i s c a s e can be v e r y b r i e f . P o i n t groups o f the space groups o f t h e c u b i c system always have elements o f o r d e r 3 o r o f o r d e r 3 and 4 b e s i d e s elements o f o r d e r 2. The r o t a t i o n s o f o r d e r 3 never have t r u e non-p r i m i t i v e t r a n s l a t i o n s . I f the l a t t i c e o f a space group i s p l a i n a r o t a t i o n o f o r d e r 4 may have a t r u e rion-primitive t r a n s l a t i o n e q u a l t o 0, 1/4, 2/4 o r 3/4 o f the b a s i c p r i m i t i v e t r a n s l a t i o n a l o n g the 4 - f o l d a x i s . The c o r r e s p o n d i n g e l e m e n t s o f F a r e 4, 4 ^ 42, 4 3 . i f t h e l a t t i c e i s body-c e n t e r e d a r o t a t i o n o f o r d e r 4 may appear i n the space group 46 o n l y as 4 o r 4^, Thus what has been s a i d f o r space groups o f the t e t r a g o n a l system h o l d s f o r space groups o f the c u b i c s ystem w h i c h c o n t a i n 4 f 4^, 4 2 o r 4g. F o r space groups o f the c u b i c system which do not c o n t a i n e lements o f o r d e r 4 t h e t h e o r y i s e s s e n t i a l l y the same as f o r the case o f non-symmorphic space groups w i t h no el e m e n t s o f o r d e r h i g h e r t h a n 2. B) F whose p o i n t group R has no subgroups R° There a r e o n l y two p o i n t groups w h i c h do not have subgroups R D. They a r e : 3 and 23. The symmorphic space groups w i t h t h e s e p o i n t groups have o n l y subgroups D R o w h i c h can be c o n s t r u c t e d by means o f r u l e ( 3 , 1 . 9 ) . The space group F23 does n ot have any subgroup o f i n d e x 2 a t a l l . The non-symmorphic space groups a r e P 3 j , P 3 2 , P 2 j 3 , 12^3. Of them P2^3 does n ot have subgroups D R because the c o n d i t i o n s (see S e c t i o n 3.2) under w h i c h the elements (R | t ( R ) + t a ) o f P2^3 would form a group a r e not s a t i s f i e d . Of t h e o t h e r s , P3^ has a subgroup P 3 2 ,and P 3 2 has a subgroup P 3 1 # B o t h subgroups have a l a t t i c e w h i c h d i f f e r s from t h a t o f the o r i g i n a l space group because i t s b a s i c p r i m i t i v e t r a n s l a t i o n a l o n g the 3 - f o l d a x i s i s t w i c e as l a r g e as t h a t o f the o r i g i n a l l a t t i c e i n t h a t same d i r e c t i o n . B o t h subgroups c o n t a i n elements w i t h t ^ (R) = t ^ . I n f a c t , P3^ c o n t a i n s t h e element ( R 3 | T ( R 3 ) + t ) w i t h R 3 a r o t a t i o n o f o r d e r 47 3 a n d r ( R 3 ) = ^ ^ and i t s subgroup D R ^ c o n t a i n s ( R 3 | T ( R 3 ) + t o ( + t a ) m ( R 3 | | 2 ^ + t a ) ; P 3 2 c o n t a i n s ( R 3 | 7 r ( R 3 ) + t ) w i t h r ( R 3 ) =-| ^  and i t s subgroup c o n t a i n s ( R 3 i ^ t * +ta) and C R g 2 ^ + V t a ) = ( R 3 2 : | £ 2 t« + t f t ) . The group I 2 j 3 has as a subgroup D R o t h e space group P2^3. SECTION 3.2 We have p r o v e d q u i t e g e n e r a l l y i n S e c t i o n 3.1 t h a t a l l t h e subgroups Dip, D R o and D R ( X o f a symmorphic space group F can be o b t a i n e d from the subgroups Q o f F o f the form ( R a I ^cJ e x c e p t i n the case o f an F whose p o i n t group does not have subgroups o f i n d e x 2 and t h a t a l l t he Q's can be c o n s t r u c t e d by means o f the r u l e ( 3 . 1 . 1 6 ) . I n the e x c e p t i o n a l c a s e s j u s t mentioned the subgroups o f i n d e x 2 o f F a r e n e c e s s a r i l y o f the k i n d D R o o r D R ( X and can be a l l o b t a i n e d u s i n g r u l e (3.1.9) i n the symmorphic c a s e , and have been a l l enumerated i n the non-symmorphic c a s e . We have a l s o p r o v e d t h a t a l l the subgroups D R o and D R ^ o f a non-symmorphic space group F whose p o i n t group does n o t c o n t a i n elements o f o r d e r h i g h e r than 2 can be o b t a i n e d from the subgroups Q_ o f F o f t h e form (R„ | ^ ( R Q ) + t Q ) and t h a t a l l t he subgroups D R o and D R o^ o f a non-symmorphic space group F whose p o i n t group c o n t a i n s elements of o r d e r h i g h e r t h a n 2 as w e l l as elements o f o r d e r 2 can be o b t a i n e d from th e subgroups Q ^ and Q~.^  o f F . However, i n the case o f non-symmorphic space groups F we have o n l y g i v e n n e c e s s a r y 48 c o n d i t i o n s f o r t h e s e t s A , 0, Zf o f elements o f v a r i o u s <v rs ' F t o form subgroups Q^ o o r Q^. I n t h i s s e c t i o n we show what the n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s a r e f o r t h e s e s e t s t o form groups Q or Q r o < > and a l s o what the necessary and s u f f i c i e n t c o n d i t i o n s a r e f o r the p a i r s o f c o s e t s r e l a t i v e t o Q r o o r Q^^ t o form groups D R o o r D R o ( , We w i l l d e a l f i r s t (Case A2) w i t h non-symmorphic space groups whose p o i n t groups do not c o n t a i n elements o f o r d e r h i g h e r than 2. We w i l l d e a l (Case A3) w i t h the non-symmorphic space groups whose p o i n t groups c o n t a i n elements o f o r d e r h i g h e r than 2 l a t e r on. Case A2 Here we have t h e f o l l o w i n g theorem: (3.2.1) The s e t Q ( d e f i n e d i n (3.1.21)) o f e l e m e n t s ( R a l T ( R a ) + t a ) o f a g i v e n space group F i s a group i f and o n l y i f (3.2.2) Kt^T* , (3.2.3) • V * ^ * kc , f o r any c h o i c e o f R a, R b and t a . These c o n d i t i o n s s i m p l y f o l l o w from the f a c t t h a t 0 i s n e c e s s a r i l y a group i f the p r o d u c t o f any two elements o f 0, ( R a | T ( R a ) + t a ) and ( R b | f ( R b ) + t b ) , b e l o n g s t o 6> . I t s h o u l d be n o t i c e d t h a t the p r o d u c t j u s t mentioned would a l s o be an element o f 0 i f the f o l l o w i n g c o n d i t i o n s 49 were s a t i s f i e d : f o r any c h o i c e o f R a, R b and t b . However, th e s e c o n d i t i o n s c a n not be s a t i s f i e d f o r an a r b i t r a r y c h o i c e o f R a, R b and t b . (Take, f o r example, t b = 0.J From c o n d i t i o n s (3.2.2) and (3.2.3) one o b t a i n s : (3.2.4) I f 0 i s a group t h e n i t i s a space group whose p o i n t group i s a R D and whose l a t t i c e i s a T ° , and, as a subgroup o f F, i t i s a subgroup Q 1 r o o f i n d e x 4. B e f o r e we s t a t e the c o n d i t i o n s under w h i c h the 0 t n and 2 n d c o s e t o f (3.1.21) form a g r o u p , we s h a l l f i r s t have t o see o f w h i c h form t h e elements o f the 2 n d c o s e t a r e i n the c a s e o f a non-symmorphic space group whose p o i n t group has no elements o f o r d e r h i g h e r t h a n 2. (3.2.5) S i n c e t h e 2 n d c o s e t can be w r i t t e n as (R^ ^ ( R ^ ) ) ( R a | T ( R a ) + t a ) , i t s g e n e r a l element can be e i t h e r (R p |T(Rp)+t b) or ( R ? ftr (Rp)+tp) depending on whether t h e c o n d i t i o n s (3.2.6) (3.2.7) 50 o r the c o n d i t i o n s (3.2.8) R A 6 ^ * > (3.2.9) T ( ^ ' R j = T r ( R ( , ) + R 0 < t ( R a ) + tv f a r e s a t i s f i e d f o r any c h o i c e o f R a, R ^  and t a . However, i t i s o b v i o u s t h a t c o n d i t i o n (3.2.8) i s not s a t i s f i e d f o r any t a . (Take, f o r example, t a = 0 . ) I t f o l l o w s t h a t the elem e n t s o f the 2 n d c o s e t can o n l y be o f t h e form (RplTCRpJ+tb). R e q u i r i n g t h a t the p r o d u c t o f any two elements o f th e s e t c o n s i s t i n g o f the 0 t n and 2 n d c o s e t i s a g a i n an element o f t h e s e t one o b t a i n s : (3.2.10) The 0 t h c o s e t ( R a | T ( R a ) + t a ) and the 2 n d c o s e t (R I'CCR-J+t*,) form a group i f and o n l y i f p P (3.2.11) TO^PJ --^*>V^%) +boL , (3.2.12) T ( R ^ ) = T ( * T ) + RYT:0y) + ^ > f o r an a r b i t r a r y c h o i c e o f R a, R 3 and R^. Combining ( 3 . 2 . 6 ) , ( 3 . 2 . 7 ) , (3.2.11) and (3.2.12) one o b t a i n s : (3.2.13) The 0 t h and 2 n d c o s e t form a group i f and o n l y i f (3.2.14) & b o . 6 ^ D , (3.2.15) T?(R 1Rj = Tr(Rj + R 1TT(«J +t k , f o r any R, R i and R 2 b e l o n g i n g t o R and any t a b e l o n g i n g t o C o n d i t i o n (3.2.14) i s n o t h i n g e l s e but the c o n d i t i o n (3.1.8) w h i c h s t a t e s t h a t T D must have t h e same h o l o h e d r y as T. Hence (3.2„13) can be r e f o r m u l a t e d as f o l l o w s ; (3.2.16) The 0 t h and 2 n d c o s e t form a group i f and o n l y i f T and Tp have the same h o l o h e d r y , and (3.2.17) ^ R j s t f M + V e M A , f o r any R i and R 2 b e l o n g i n g t o R. C o n s i d e r now th e 0 t n and 3 r d c o s e t . (3.2.18) The elements o f t h e 3 r d c o s e t (R^ I T ( R ^ J + t ^ ) ( R a | T ( R a ) + t a ) can be e i t h e r o f th e form (RB It O R (RB I17 C R B ) + t b ) depending on whether the c o n d i t i o n s (3.2.19) -R t Q T D t (3.2.20) t ( R ^ ) = ^ ( *R«)*^( RJ * b c , o r t h e c o n d i t i o n s (3.2.21) R A e f - ? > > (3.2.22) ^ V j ^ o ^ V ^ J ^ c , 52 a r e s a t i s f i e d f o r an a r b i t r a r y c h o i c e o f R , R , and t Q . However i t i s o b v i o u s t h a t c o n d i t i o n (3.2.21) i s not always s a t i s f i e d . (Take f o r example, t ^ = 0 . ) . I t f o l l o w s t h a t the e lements o f the 3 r d c o s e t can o n l y be o f t h e form (RB | T ( R B ) + t B ) . U s i n g t h e same arguments as i n t h e p r e v i o u s c a s e one o b t a i n s : (3.2.23) The 0 t h c o s e t ( R a | T ( R a ) + t a ) and t h e 3 r d c o s e t ( R B |T(Rg)+tg) form a group i f and o n l y i f (3.2.24) T ^ R ^ t f M + R ^ ) ^ , (3.2.25) *(^%) =^)+Y^) ^  ; f o r an a r b i t r a r y c h o i c e o f R a, Rg and R ^ . Combining ( 3 . 2 . 1 9 ) , ( 3 . 2 . 2 0 ) , (3.2.24) and (3.2.25) one o b t a i n s : (3.2.26) The 0 t h and 3 r d c o s e t form a group i f and o n l y i f (3.2.27) K t ^ T * (3.2.28) = > f o r an a r b i t r a r y c h o i c e o f R, R^ and R 2 b e l o n g i n g t o R and t a b e l o n g i n g t o T D. 53 U s i n g a g a i n the con c e p t o f h o l o h e d r y , (3.2.26) can be r e f o r m u l a t e d as f o l l o w s : (3.2.29) The 0 t h and 3 r d c o s e t form a group i f and o n l y i f T and T D have the same h o l o h e d r y , and (3.2.30) ^(Kj + K^iK)^ , f o r any R i and R2 b e l o n g i n g t o R. As one can see from (3.2.13) and (3.2.26) t h e c o n d i -t i o n s under which the 0* n and 2 n d c o s e t , and the 0 t n and 3 r d c o s e t form groups a r e i d e n t i c a l . Hence: (3.2.31) I f f o r a g i v e n Q V o o f a space group F whose p o i n t group does not have elements o f o r d e r h i g h e r than 2 a subgroup D R o o f F e x i s t s , t h e n t h e r e e x i s t s a l s o a sub-group D B o k o f F w i t h the same l a t t i c e HP as D R o. C o n v e r s e l y : (3.2.32) I f a iL w i t h a g i v e n T° does not e x i s t , RO t h e n t h e DRoi, w i t h the same l a t t i c e does not e x i s t e i t h e r . Case A3 We now c o n s i d e r t h o s e non-symmorphic space groups F whose p o i n t groups c o n t a i n elements o f o r d e r h i g h e r t h a n 2 as w e l l as elements o f o r d e r 2. I f the subgroup T D o f T i s such t h a t the b a s i c p r i m i t i v e t r a n s l a t i o n t ^ b e l o n g i n g t o T but not t o T D i s p e r p e n d i c u l a r t o the a x i s o f h i g h e r o r d e r t h e n i n o r d e r t o 54 c o n s t r u c t the D's o f an F one has t o c o n s i d e r the s e t W of ( 3 . 2 . 1 ) , and t h e t h e o r y o f Case A2 a p p l i e s w i t h o u t any change. However i f the t r a n s l a t i o n t ^ d e f i n e d above i s p a r a l l e l t o t h e a x i s o f h i g h e r o r d e r one has t o c o n s i d e r the s e t <£> o f e l e m e n t s (R a |"E(Ra)+to<.(Ra)+ta) o f F w h i c h w i l l be the s e t 2 o f (3.1.27) i f F has t h e p l a i n l a t t i c e and c o n t a i n s the element 4 2 , the s e t X o f (3.1.28) i f F has t h e b o d y - c e n t e r e d l a t t i c e and c o n t a i n s the element 4^ t e t c . (3.2.33) A s e t $> o f elements (R a|^ ( R a ) + t o ( ( R a ) + t a ) f o r an a r b i t r a r y c h o i c e o f R a, R b and t a . These c o n d i t i o n s f o l l o w from the f a c t t h a t i s n e c e s s a r i l y a group i f t h e p r o d u c t o f any two elements o f $ , ( R a | T ( R a ) + t 0 < ( R a ) + t a ) and ( R b ^ ( R ^ + t ^ ( R b ) + t b ) , b e l o ngs o f F i s a group i f and o n l y i f (3.2.34) ' Cn> OL/ > (3.2.35) t o (|> . A l s o t h e y a r e the o n l y c o n d i t i o n s under w h i c h <p i s a group. I n f a c t , the c o n d i t i o n s (3.2.35) (3.2.36) 55 under which $ would be a group t o o , a r e not s a t i s f i e d f o r an a r b i t r a r y c h o i c e o f R a, R b and t a . (3.2.37) I f i s a group i t i s a space group whose p o i n t group i s a R D and whose l a t t i c e i s a T D , and as a subgroup of F. i s o f i n d e x 4, We now c o n s i d e r the p a i r s o f 0 t n and 2 n d c o s e t , and o f 0 t n and 3 r d c o s e t . However, b e f o r e d o i n g t h a t we w i l l have t o see o f w h i c h form the elements o f the 2 n d and o f t h e 3 r d c o s e t a r e . We know t h a t the 2 n d c o s e t c o n s i s t s o f the elements ( R ^ | ^ ( R ^ ) ) ( R a | T ( R a ) + t ^ ( R a ) + t a ) . I t s g e n e r a l element can be e i t h e r ( R B | t ( R B ) + t 0 ( ( R B ) + t b ) o r (Rg | V (Rg) ^ ( R g ) + t g ) depending on whether the c o n d i t i o n s (3.2.38) H a ^ e f * , (3.2.39) T R J + h^\)-z(• R j t f R J * t, (R ) j + t f l , or the c o n d i t i o n s (3.2.40) ^ t ^ ^ T - T 6 } (3.2.41) Tfoc RJ + t R j = t ( R J +R< [tf R J + * K , a r e s a t i s f i e d f o r an a r b i t r a r y c h o i c e of R a, R w , and t a . However, i t i s o b v i o u s t h a t (3.2.40) i s not always s a t i s f i e d . (Take, f o r example, t a = 0 ) . 56 (3.2.42) The 0 t h c o s e t ( R a |^(R a ) + t 0 ( ( R a ) ) and the 2 n d c o s e t ( R e l ^ R g J + t ^ R g J + t t j ) a group i f and o n l y i f ( 3 . 2 . 4 3 ) «t(ua^f) -^vy = -Wig+ypg (3.2.44) v(R^ + y R YR p) r 1^ } + -tj R^) +»1[t(R(l)+WR(i)]+^ , f o r an a r b i t r a r y c h o i c e o f R a, R g and R^ , . Combining ( 3 . 2 . 3 8 ) , ( 3 . 2 . 3 9 ) , (3.2.43) and (3.2.44) one o b t a i n s : (3.2.45) The 0 t h and 2 n d c o s e t form a group i f and o n l y i f ( 3 . 2 . 4 6 ) K r ^ T * ; (3.2.47) T ( R 4 R 2 W 1 ^ ( 1 ^ ) = f o r any R, R-^  and R 2 b e l o n g i n g t o R and any t a b e l o n g i n g t o TP. A g a i n one can r e f o r m u l a t e ( 3 . 2 . 4 5 ) , (3.2.46) and (3.2.47) as f o l l o w s : 57 (3.2.48) The 0 t h and 2 n d c o s e t form a group i f and o n l y i f T and T° have the same h o l o h e d r y and i f (3.2.47) h o l d s f o r an a r b i t r a r y c h o i c e o f R^ and R 2 o f R. C o n s i d e r now t h e 0* n and 3 r d c o s e t . U s i n g the same arguments as i n the p r e v i o u s case one can e a s i l y show t h a t t h e g e n e r a l element o f t h e 3 r d c o s e t i s o f the form (RglT ( R g ) + t o ( ( R B ) + t g ) and t h a t t h e 0 t h and 3 r d c o s e t form a group i f and o n l y i f (3.2.49) S ^ T * (3.2.50) r (R 4 R2) + ^ ( k , 1^  = Tr (Rj + Rj f o r an a r b i t r a r y c h o i c e o f R, R^ and R 2 b e l o n g i n g t o R and t a b e l o n g i n g t o TP o r , i n o t h e r words: (3.2.51) The 0 t h and 3 r d c o s e t form a group i f and o n l y i f T and T D have t h e same h o l o h e d r y and i f (3.2.50) h o l d s . (3.2.52) I f the 0 t h and 2 » d c o s e t and the 0 t h and 3 r d c o s e t form g r o u p s , t h e s e a r e space groups and as sub-groups o f F th e y a r e b o t h D R ^ . S i n c e t h e 0 t h and 2 n d c o s e t , and the 0 t h and 3 r d c o s e t form groups under t h e same c o n d i t i o n s , i f t h e f i r s t p a i r o f c o s e t s c o n s t i t u t e s a group t h e second one does t o o . 58 C o n v e r s e l y , i f the f i r s t p a i r o f c o s e t s i s not a group, the second p a i r i s not a group e i t h e r . C o n d i t i o n (3.2.50) i s never s a t i s f i e d f o r non-symmorphic space groups w h i c h have the p l a i n l a t t i c e and the elements 4 ^ , 4 3 , 6 ^ , 6 3 , 6 5 , t h a t i s why i n S e c t i o n 3.1 we s a i d t h a t t h e s e space groups do not have s t a n d a r d D R o ( . S i m i l a r l y , c o n d i t i o n (3.2.30) i s never s a t i s f i e d f o r non-symmorphic t r i g o n a l space groups w i t h a rhombohedral l a t t i c e ) t h a t i s why we have not d i s c u s s e d t h i s c a s e i n S e c t i o n 3.1. T h i s c ompletes the d i s c u s s i o n o f t h e problem o f f o r m u l a t i n g and p r o v i n g the r u l e s f o r c o n s t r u c t i n g a l l subgroups o f i n d e x 2 o f a l l space g r o u p s . D i g r e s s i o n Our method o f c o n s t r u c t i n g the subgroups D o f F i s based on t h e o b s e r v a t i o n t h a t , a p a r t from e x c e p t i o n a l c a s e s , t h e s e subgroups have i n t u r n subgroups o f i n d e x 2 w h i c h are subgroups o f i n d e x 4(Q, Q T o o r Q ) o f F, and t h a t t h e s e subgroups of i n d e x 4 o f F can be v e r y e a s i l y found ( r o u g h l y s p e a k i n g , by t a k i n g a " h a l f " o f t h e p o i n t group R of F and a " h a l f " o f t h e l a t t i c e T o f F ) . The e x c e p t i o n a l c a s e s a r e , as we have s e e n , o f two k i n d s . The p o i n t group R may not have subgroups R D o f i n d e x 2; t h i s s i t u a t i o n has been d e a l t w i t h by means o f r u l e ( 3 . 1 . 9 ) , and by c o n s i d e r i n g the f o u r e x c e p t i o n a l non-symmorphic F's s e p a r a t e l y (end o f S e c t i o n 3.1): i n b o t h 59 c a s e s no r e f e r e n c e i s made t o t h e subgroups o f i n d e x 4 o f F. The o t h e r k i n d o f e x c e p t i o n a l s i t u a t i o n a r i s e s f o r some non-symmorphic F ' s j here some D*", whi c h can always be o b t a i n e d from r u l e ( 3 . 1 . 4 ) , cannot be o b t a i n e d from any or Q^^ as p o i n t e d out e x p l i c i t l y i n S e c t i o n 3.1 i n c o n n e c t i o n w i t h t h e c a s e o f t h e non-symmorphic F's o f t h e m o n o c l i n i c and o r t h o r h o m b i c systems. We s h a l l examine here somewhat c l o s e r the r e a s o n s f o r t h i s b e i n g s o . C o n s i d e r f i r s t t h e case o f t h e m o n o c l i n i c and o r t h o -rhombic systems (F whose p o i n t group does not have elements o f o r d e r h i g h e r t h a n 2 ) . I f F i s symmorphic, no d i f f i c u l t y a r i s e s : (3.2.53) A l l subgroups D T o f a symmorphic F can be o b t a i n e d from some Q o f F. The v a l i d i t y o f t h i s theorem i s an immediate conse-quence o f t h e f o l l o w i n g theorem w h i c h i s a r e s u l t o f the s u r v e y o f l a t t i c e s i n S e c t i o n 3.3: (3.2.54) With the e x c e p t i o n o f the f a c e - c e n t e r e d c u b i c l a t t i c e e v e r y l a t t i c e T has a t l e a s t one subgroup T D o f i n d e x 2 w i t h the same h o l o h e d r y as T. I n f a c t , from ( 3 . 2 . 5 4 ) , i t i m m e d i a t e l y f o l l o w s : (3.2.55) W i t h the e x c e p t i o n o f t h e f a c e - c e n t e r e d c u b i c l a t t i c e e v e r y l a t t i c e T has a t l e a s t one subgroup o f i n d e x 2 w h i c h i s i n v a r i a n t under a l l t h e subgroups o f i n d e x 2 o f the h o l o h e d r y o f T and i t s subgroups. Hence: (3.2.56) Each symmorphic F o f the m o n o c l i n i c and 60 o r t h o r h o m b i c system has a t l e a s t as many subgroups Q o f i n d e x 4 as t h e r e a r e subgroups RD o f the p o i n t group R. I n o t h e r words f o r e v e r y RD t h e r e i s a t l e a s t one subgroup Q of F; '"V I f F i s non-symmorphic ( f o r any syste m , not o n l y mono-c l i n i c o r o r t h o r h o m b i c ) we have t o t a k e i n t o a c c o u n t an a d d i t i o n a l c i r c u m s t a n c e : (3.2.57) A n e c e s s a r y c o n d i t i o n f o r a s e t Ii o f elem e n t s ( R a ^ ( R a ) + t a ) o f a non-symmorphic space group F t o be a group i s f o r a l l the elements o f the s e t . Here R a i s a pr o p e r o r improper r o t a t i o n o f o r d e r n, and R a k s t a n d s f o r t h e k t n power o f the r o t a t i o n R a. There a r e o n l y a few ca s e s when f o r a g i v e n R D t h e r e a r e no subgroups T D such t h a t (3.2.57) i s not s a t i s f i e d . O b v i o u s l y t h i s happens when t h e elements o f Ii have t r u e non-p r i m i t i v e t r a n s l a t i o n s i n a l l t h r e e d i r e c t i o n s o f t h e edges o f the n o n - p r i m i t i v e u n i t c e l l o f T. More p r e c i s e l y : (3.2.58) I f i n a non-symmorphic space group F w i t h no elements o f o r d e r h i g h e r t h a n 2 t h e r e a r e two p e r p e n d i c u l a r d i a g o n a l g l i d e p l a n e r e f l e c t i o n s ( e . g . Pnnm), o r two pe r p e n -d i c u l a r diamond g l i d e p l a n e r e f l e c t i o n s ( e . g . F d d d ) , o r t h r e e screw r o t a t i o n s o f o r d e r 2 about p e r p e n d i c u l a r axes ( e . g . Pnma) o r a screw r o t a t i o n o f o r d e r 2 and a d i a g o n a l g l i d e p l a n e r e f l e c t i o n p e r p e n d i c u l a r t o i t , t h e n (3.2.3) i s not s a t i s f i e d 61 f o r a 0 c o n t a i n i n g such p a i r s o f o p e r a t i o n s whatever may be. Hence we c o n c l u d e : (3.2.59) Not a l l t h e subgroups Dip o f e v e r y non-symmorphic group F b e l o n g i n g t o t h e m o n o c l i n i c o r o r t h o r h o m b i c system can be o b t a i n e d from some Q T o . However, we r e p e a t , t h i s i s not a d i f f i c u l t y s i n c e a l l t h e subgroups o f any F can be o b t a i n e d by means o f the r u l e ( 3 . 1 . 4 ) . I n the case o f the t r i g o n a l , h e x a g o n a l and c u b i c systems t h e s i t u a t i o n i s q u i t e s i m i l a r . I f F i s symmorphic, no d i f f i c u l t y a r i s e s , and (3.2.53) r e m a i n s t r u e . I n t h e case o f non-symmorphic F a d i f f i c u l t y a r i s e s because c o n d i t i o n (3.2.57) when a p p l i e d t o subgroups becomes (3.2.60) 0 ~ > ( R J + t * ( R ^ C > ^ ^ ^ and t h i s c o n d i t i o n i s not s a t i s f i e d by a l l t h o s e elements o f o r d e r 2 whose t r u e n o n - p r i m i t i v e t r a n s l a t i o n has a component a l o n g the a x i s o f h i g h e r o r d e r . ( T h i s o c c u r s , e.g. i n the groups P4 2cm, P 3 c l ) . Hence: (3.2.61) For a non-symmorphic space group whose p o i n t group c o n t a i n s elements o f o r d e r h i g h e r t h a n 2 t h e r e i s not 62 n e c e s s a r i l y a Q M f o r e v e r y o f R. Hence: (3.2.62) Not a l l t h e subgroups D T o f a non-symmorphic space group F whose p o i n t group c o n t a i n s e l e m e n t s o f o r d e r rs h i g h e r t h a n 2 can be o b t a i n e d from some Q and t h e r e f o r e Vol. t h e y have t o be o b t a i n e d by means o f ( 3 . 1 . 4 ) . SECTION 3.3 From t h e p r e v i o u s s e c t i o n i t i s c l e a r t h a t i n o r d e r t o c o n s t r u c t the subgroups D o f a g i v e n space group F one has t o f i n d a l l the d i f f e r e n t subgroup T° o f the l a t t i c e T o f F and the subgroups R° o f i t s p o i n t group R. A complete l i s t o f subgroups R D o f each o f the 32 p o i n t groups i s g i v e n i n "The I n t e r n a t i o n a l T a b l e s f o r X-Ray C r y s t a l l o g r a p h y " ( 1 9 52). We now t u r n t o the problem o f f i n d i n g a l l t h e d i f f e r e n t subgroups T D o f a g i v e n l a t t i c e T. Two l a t t i c e s a r e c a l l e d " d i f f e r e n t " i f t h e y do not c o n s i s t of the same s e t o f p o i n t s w h i c h a r e t h e e n d - p o i n t s o f p r i m i t i v e t r a n s l a t i o n v e c t o r s . As i s w e l l known, two s e t s o f t h r e e b a s i c p r i m i t i v e t r a n s l a t i o n s g e n e r a t e the same l a t t i c e i f and o n l y i f t h e r e e x i s t s an i n t e g r a l u n i m o d u l a r s q u a r e m a t r i x w h i c h t r a n s f o r m s one s e t i n t o the o t h e r . Hence, i f su c h a m a t r i x does n ot e x i s t the two s e t s o f b a s i c p r i m i t i v e t r a n s l a t i o n s g e n e r a t e d i f f e r e n t l a t t i c e s . U s i n g t h i s c r i t e r i o n i t i s easy t o enumerate a l l the d i f f e r e n t subgroups 63 T D of a g i v e n T; t h e r e a r e a t most seven o f them. L e t a^, &2 and a 3 be t h e b a s i c p r i m i t i v e t r a n s l a t i o n s o f the l a t t i c e T. O b v i o u s l y the group c o n s i s t i n g o f the p r i m i t i v e t r a n s l a t i o n s t a = 2 n ^ a i + n 2 a 2 + n 3 a 3 w i t h n l f n 2 and n 3 i n t e g e r s c o n s t i t u t e s a subgroup o f i n d e x 2 o f T. F i v e o t h e r subgroups can be o b t a i n e d from i t by p e r m u t a t i o n o f a^, a 2 and a 3 but o n l y two o f them a r e d i f f e r e n t among th e m s e l v e s and d i f f e r e n t from T D j . They are the group T 1 D 2 c o n s i s t i n g o f the p r i m i t i v e t r a n s l a t i o n s t a = n^a-j+2n 2a 2+n 3a 3 and the group T D_ c o n s i s t i n g o f the p r i m i t i v e t r a n s l a t i o n s o t a = n 1 a 1 + n 2 a 2 + 2 n 3 a 3 . The group T D 4 c o n s i s t i n g o f the p r i m i -t i v e t r a n s l a t i o n s t a = 2n^a^+n 2(a^+a 2)-(-n 3a 3 i s a l s o a subgroup of i n d e x 2 o f T and a g a i n f i v e o t h e r subgroups T 1 0 can be o b t a i n e d from i t by a l l p o s s i b l e p e r m u t a t i o n s o f a^, a 2 and a 3 . Of them o n l y two a r e d i f f e r e n t among them s e l v e s and d i f f e r e n t from T D 4 . They can be chosen as the group T D 5 c o n s i s t i n g o f the p r i m i t i v e t r a n s l a t i o n s t a = 2 n 1 a 1 + n 2 a 2 + n 3 ( a 1 + a 3 ) and the group T ° 6 c o n s i s t i n g o f t h e p r i m i t i v e t r a n s l a t i o n s t a = n ^ a + 2 n 2 a 2 + n 3 ( a 2 + a 3 ) . F i n a l l y the group iPj c o n s i s t i n g o f t h e p r i m i t i v e t r a n s l a t i o n s t a = 2 n j a i + n 2 ( a ^ + a 2 ) + n 3 ( a ^ + a 3 ) i s a subgroup o f i n d e x 2 o f T d i f f e r e n t from the s i x a l r e a d y d e f i n e d . There a r e no o t h e r subgroups o f i n d e x 2 o f T. In f a c t any o t h e r s e t o f p r i m i t i v e t r a n s l a t i o n s o f T f o r m i n g a group i s a subgroup o f T o f i n d e x h i g h e r than 2. 6 4 We l i s t here the seven subgroups of index 2 of a l a t t i c e T whose basic primitive translations are a^, a 2 and a 3 : T » ! - (2a x, a 2 , a 3) , 1 2 ~ (a-j^ 2a 2, a g) , mD _ 3 ( a l f a 2 , 2a 3) ; T D 4 = ( 2 a l f a 1 + a 2 , a 3 ) , mD _ 1 5 (2a^, a 2 , a^+a3) ; T D 6 = (ap, 2a 2, a 2+a 3) ) mD _ 7 ( 2 a l f a 1 + a 2 , a 1 + a 3 ) If there are r e l a t i o n s between the vectors a^, a 2 , a 3 (which i s always the case for systems d i f f e r e n t from the t r i c l i n i c system), the number of d i f f e r e n t subgroups Tp of a given T w i l l be less than seven. This point i s discussed further i n Chapter 4 . Of course, not every subgroup T 0^ ( i = 1, 2...7) of the l a t t i c e T of a given space group F s a t i s f i e s condition ( 3.1 . 8 ) which require to have the same holohedry as T. SECTION 3 . 4 In t h i s section we s h a l l give examples of the theory explained i n the present chapter. We s h a l l construct a l l the 6 5 subgroups D of the space groups P4, P4 1 # p4 2 and P4g of the tetragonal system. A l l these space groups have a plain la t t i c e P with the basic primitive translations a-^ , a 2 and a 3 which in this case are also the edges of the parallelepiped unit c e l l of P, that i s , a^ = a, ag = b, a 3 = c. Also a = b and a = B = y - 90°. From these relations i t follows that the only subgroups of index 2 of the plain tetragonal lattice which have the same holohedry as the original l a t t i c e are of the type Tpg, T ° 4 and T ° 7 . A l l four space groups considered above have the point group 4. This group consists of the elements E - identity - rotation about z-axis by 90°, C 1 - rotation about z-axis by 180°, C 3 - rotation about z-axis by 270°. The elements E and C ^ constitute the only subgroup of index 2 of the point group 4. We now consider each space group separately. The space group P4 i s symmorphic and consists of the following elements (we shall repeatedly use this phrase instead of a more correct phrase "has the following 66 g e n e r a t i n g e l e m e n t s " ) : L e t us c o n s i d e r t h e s e t s o f elements ( E l ^ a , +/y\ib ,(C U*| 0) , ( £ " 1 2 ^ ^ ^ 1 , ( 0 . + b ) + ^ £ ) ; ( C A j | 0 ) , ( e | f c ^ t o , + ^ ( o . + b)*'v\a(a>e)), (Ck ? z | 0) < Each o f t h e s e s e t s i s a group and a subgroup Q o f P4. C a l l them Q^, Q 2 a n d Q3 r e s p e c t i v e l y . We t h e n c o n s i d e r t h e 1 s t , 2 n d and 3 r d c o s e t o f P4 r e l a t i v e t o each Q, and see w h i c h o f them form a group w i t h Q. C o n s i d e r f i r s t . The c o s e t s r e l a t i v e t o a r e 0 t h c o s e t ( E | 0 Iff K , a , ^ ^ J . b 4 ^ 0 ) , ( £ | o ) ( C * l | o ) , 1 s t c o s e t f e l c K E l ' v t . o ^ + ^ ^ b+^ac) ,(£|c)(Cu^|0) , 2 n d c o s e t ( C ^ | 0 ) ( F K l o u + /vx l.t+2^o) 1(C u »|o)(C tJ"|o) / 3 * " d c o s e t (C / ( ¥|c)(E|^ )a.+ ' W ^ + 2 ^ 3 o ) 1 ( C ( v i | c ) ( C t t \ |o) . 67 rs 4 - u The subgroup Q^, wh i c h i s the 0 c o s e t , t o g e t h e r w i t h the 1 s t c o s e t , c o n s t i t u t e s a group which i s the space group P2. T h i s i s a subgroup D T o f P4 because i t has t h e same l a t t i c e as P4 and because i t s p o i n t group i s a Fp o f 4, The subgroup Q l t t o g e t h e r w i t h the 2 n d c o s e t , c o n s t i -t u t e s a group w h i c h i s the space group P4. T h i s i s a subgroup D R o o f P4 because i t has the same p o i n t group as P4, a l a t t i c e w h i c h i s a T D o f th e l a t t i c e P o f P4 and c o n t a i n s no elements w i t h t ^ which i n t h i s case i s c. F i n a l l y t j j , t o g e t h e r w i t h the 3 r d c o s e t , forms a group w h i c h i s t h e space group P 4 2 . T h i s i s a subgroup D R < X o f P4 because i t has t h e same p o i n t group as P4, a l a t t i c e w h i c h i s a o f P and elements w i t h t ^ = c. rs rs C l e a r l y i n t h i s c a s e the space groups D R o and D R o < a r e p r o p e r l y d i f f e r e n t ( i n the sense d e f i n e d i n S e c t i o n 2 . 3 ) , o r as we s h a l l o f t e n s ay f o l l o w i n g t he more customary t e r m i n o l o g y , " i n e q u i v a l e n t " . S i m i l a r l y t h e c o s e t s o f p4 r e l a t i v e t o Q 2 a r e 0 t h c o s e t (f|o)(E|2/^ 1ou + ^ J > (a, + lo) + A » 3 c ) > j J | o ) ( c 6 j | o ) 1 s t c o s e t (E|fc)(E|2"vA, +^ t(cX+b)+M 4c) , (c-|c)(C^11 o ) 2nd C O S e t {Cui\o){e{2^o^^(0s^) +M J C),\o) (C u l \o) 3 r d c o s e t ( C ^ I ^ E ^ / v v ^ ^ ^ a + b J r ^ t j ^ C ^ l c J t C ^ l o ) . 68 The O t n coset, which i s Q 9, together with the 1 S T t 2 i coset gives the space group P2, together with the 2 N D , or the 3 r d coset gives the space group P4. In this case D R q and D N , are equivalent (that i s , not properly d i f f e r e n t ) . K ot. rs F i n a l l y the cosets of P4 r e l a t i v e to Q3 are 0 T H coset (f|o)(£|2^lA/+^(oo1-MW^+^)),(e|o)(Ci<ll 0) 1 S T coset |2^,<h-+^(o'+t) + / M j(oi.+ c))J(>E|c)(cu||o) 2 N D coset ( ^ \ o ) { E \ 2 ^ ( x ^ J o . ^ ) ^ ^ o ) ) A c k ^ \ o ) l C h \ o ) 3 r d coset (Ci.a.1 c)(e|a )),Cc„|c)(cu^|o) . The 0 T N and 1 S T coset give the space group P2, the 0 T N and 2 N D and the 0 T H and 3 r d give the space group 14. We now consider the space group P 4 1 # It consists of the elements ( E K ^ . ^ b . ^ c J X C ^ I t ) , ^ ^ ^ ) , (C^| iS ) 69 L e t us c o n s i d e r the s e t s o f elements (E 12^, a, + ™ L(cu + b) ^ J C ) I ( C^^j •% ) ^ (E 12^,0. + a* + W) + /v\3(oui-c)), (C^\\) . O n l y the second s e t i s a group, a subgroup Q o f P4^. The c o s e t s o f p4-^ r e l a t i v e t o t h i s subgroup Q ^  a r e 0 t h c o s e t ( t | 0 ) ( £ | l A , , c u +™t(ft,+b) +/* 3c), (F|o)(c a| 1 s t c o s e t 2 n d c o s e t (E|a,Xe|2^,a.*^(<x, + b)wv>jC,) ,(F|(x,)(C A^| %) 3 r d c o s e t ( c ^ K - ^ ^ f l z ^ . o u + ^ i l o L ^ ^ ^ c ^ ^ i l ^ + Q - X c ^ 1 ! ^ ) . The 0 t n and I s * c o s e t g i v e the space group P 2 j , the O t n and 2 n d and the 0 t h and 3 r d g i v e the space group P4. There a r e no o t h e r subgroups D o f P4^. 70 The space group P 4 2 c o n s i s t s o f the elements (E |/v\,a/+^fck4.^ac),(cw»|i),(c^l|0) ,{Cul Ii) • C o n s i d e r t h e s e t s o f elements (E l ^ . f t y + ^ l o + 2 . ^ 4 c ) , ( G U i l | c ) , (E U / M , A, + ^ t(ou + W J T ^  c ) , (Cki | 0 ) (E I I •V\,l5u + v t J & , + b) j(Cu»llo) They a r e a l l g r o u p s . The f i r s t and t h e t h i r d a r e subgroups QToi o f P 4 2 t t h e second i s a subgroup Q r o o f P 4 2 > The c o s e t s o f P 4 2 r e l a t i v e t o the f i r s t subgroup Q T ( X a r e 0 t h COSet (l=|0)(EH,Cu+/viLb+2^ic,),CL r|ci)CCaj |c) 1** c o s e t ( t k)(e|^ 1(Cu+'wLW+Z'v. ie,),(ek)fcu»llc) 2 n d c o s e t (C 4»|f )(eKcx. v^\ov z - . c j ^ c ^ l % i ^ ^ l c ) 71 The 0 t h and 1 s t coset give the space group P2j the O t n and 2 n d coset give the space group P4JJ the 0 t n and 3 r d coset give the space group P4 3 # The cosets r e l a t i v e to the second subgroup Q <r are 0 t h coset (fl°)(e|2'v»1a. + 'M1>(on.U)T'vvifc)J(Fjo)(c<,i1'|o) Cc 4*|%)(e|2'M 1cv+^ l(a. + l 0)+^ 3c) J( LC l t»|^)(c^ i ,-|o) ; The 0 T n and 1 s t coset give the space group P2j the Oth and 2 n d coset give the space group P4 2j the 0*° and 3 r d coset give the space group P4 2 again. The cosets of P4 2 r e l a t i v e to the t h i r d subgroup of index 4 of P4 2 are 0 t h COSet [E |o)(tr|2'v» lO, + /^ l.(a*-b)+/v» s[ou + c ) ) ; ( f l o ^ C ^ ^ l c ) 1 s t COset | 0 u)(F |2'M 1 0/ + ^ t(on-b) + /v»a(ou + c)) , ( f | o ^ ) ( c ^ j |c) 1st coset 2 o d coset 3 r a coset 2 n d coset 72 3 c o s e t [ckl | ^ + 1 » / ) ( E | 2 /V I I O V + ^ \ 1 ( O U + \ O ) t / K ^ o - v c ) ) ; ( C ^ | \ + ^ ) ( c ^ The 0*h and l s " t c o s e t g i v e t h e space group P 2 j the 0 t h and t h e 2 n d and the 0 t h and 3 r d c o s e t g i v e b oth the space group 14^. There a r e no o t h e r subgroups D o f P42. F i n a l l y we have t o c o n s i d e r the non-symmorphic space group P 4 3 . T h i s c o n s i s t s o f t h e elements (E |/Vv,fty-^ vlo4 ^ 3 c ) , ( C ^ | ^ ) , ( C u ^ | _ | ^ ( C ^ c , ) . C o n s i d e r the f o l l o w i n g s e t s o f ele m e n t s : (E |^,a^ + ^ v b + z , ( c i , *H i ) * (E | 2 ^ , a.+ 0^  + ^ ) 4 . ^ ( 0 ^ 4 - 0 ) ) , (c u* l| % ) Of them o n l y the second s e t i s a group, a subgroup Q o f P4«. The c o s e t s o f P4o r e l a t i v e t o t h i s Q^ a r e 0 t h c o s e t ( E | o ) ( e"|2Ai l lPu + ^ f o u + - b ) ^ A i 3 c ) > ( t \ o ) { c U i . L i ^ j 1 s t c o s e t ( L r | c o ) ( L - l 2 ' v i l ^ t ^ 1 ( 0 L + b ) i - ^ 3 c ) , ( f j o ^ J f c ^ * 1 ! £.) ; 73 2 n d c o s e t (^\^\2^^^y)^c)Xc^\^)(c^\^) ) 3 r d COSet ( c ^ l ^ + . c v ^ t - l ^ o . + ^ o ^ b ^ o ) , [cu a | i c * a,) ( c ^ l | s.) • The 0 t n and 1 s t c o s e t g i v e the space group P 2 ^ j the 0 t n and 2 n d and the 0 t n and 3 r d c o s e t g i v e the space group P 4 3 . The space group P 4 g has no o t h e r subgroups D. NOTES TO CHAPTER 3 A s y s t e m a t i c d i s c u s s i o n o f the problem o f f i n d i n g and c l a s s i f y i n g the subgroups (not n e c e s s a r i l y o f i n d e x 2) of a g i v e n space group was g i v e n many y e a r s ago by Hermann (1929). In p a r t i c u l a r , he has i n t r o d u c e d the d i v i s i o n o f a l l subgroups o f a g i v e n space group i n t o two c l a s s e s : " K l a s s e n g l e i c h e U n t e r g r u p p e n " ( w h i c h , i n the case o f t h e subgroups o f i n d e x 2, c o r r e s p o n d t o our ff^'s) and " Z e l l e n g l e i c h e U n t e r g r u p p e n " (which c o r r e s p o n d t o our Sip's). However, Hermann was not s p e c i f i c a l l y i n t e r e s t e d i n the subgroups o f i n d e x 2, and he has not c o n s i d e r e d the d i s t i n c t i o n between D R o ' s and D R o C s w h i c h we have i n t r o d u c e d i n S e c t i o n 3.1, nor has he n o t i c e d the Important r o l e p l a y e d by subgroups of i n d e x 4 (our Q, §T _ 0 and ). The i d e a o f c o m b i n i n g a " h a l f " o f the elements o f the p o i n t group o f a space group w i t h a " h a l f " o f the elements o f i t s l a t t i c e f o r t h e c a s e o f symmorphic space groups ( i . e . c o n s t r u c t i n g our subgroup Q) i s i m p l i c i t i n Zamorzaev ( 1 9 5 7 ) , who a p p a r e n t l y has used the i d e a t o o b t a i n the magnetic space 74 groups f o r t h e case o f symmorphic space groups. As no r u l e s f o r c o n s t r u c t i n g the D's are f o r m u l a t e d i n h i s p a p e r , and no p r o o f s a r e g i v e n , i t i s d i f f i c u l t t o s a y t o what e x t e n t our method d i f f e r s from h i s . The i m p o r t a n t case o f non-symmorphic space groups i s d i s c u s s e d by Zamorzaev i n a few s e n t e n c e s . The seven subgroups o f i n d e x 2 o f a l a t t i c e a r e l i s t e d i n Zamorzaev*s paper. I t i s easy t o see t h a t the problem o f f i n d i n g a l l the subgroups DH o f an a r b i t r a r y group H i s e q u i v a l e n t t o t h e problem o f f i n d i n g a l l a l t e r n a t i n g r e p r e s e n t a t i o n s o f H. I n f a c t , the f a c t o r group H/DH has o n l y two r e p r e s e n t a t i o n s : the i d e n t i c a l r e p r e s e n t a t i o n , and t h e a l t e r n a t i n g r e p r e s e n t a t i o n . Hence, each subgroup D H o f H w i l l "engender" ( t h e meaning o f t h i s term i s e x p l a i n e d i n Lomont ( 1 9 5 9 ) , page 234) one a l t e r n a t i n g r e p r e s e n t a t i o n o f H. C o n v e r s e l y , knowing a l l a l t e r n a t i n g r e p r e s e n t a t i o n s o f H, one can f i n d a l l b^'s: f o r each a l t e r n a t i n g r e p r e s e n t a t i o n one f i n d s the group £H which engenders i t by s i m p l y p i c k i n g out from H a l l t h o s e elements t o w h i c h +1 c o r r e s p o n d s i n t h e a l t e r n a t i n g r e p r e s e n t a t i o n i n q u e s t i o n . The e q u i v a l e n c e of t h e two problems has been n o t i c e d f o r the case i n which H i s a p o i n t group or a space group by Indenbom (1959) and N i g g l i ( 1 9 5 9). 75 CHAPTER 4 MAGNETIC SPACE GROUPS AND THEIR PROPERTIES — — — — f — 1 — — — — — i i SECTION 4.1 I n t h i s c h a p t e r we s h a l l be d e a l i n g o n l y w i t h non-t r i v i a l MSGs, Fo r c o n v e n i e n c e we s h a l l r e p e a t here r u l e (2.2.8) s p e c i a l i z e d t o t h e case o f MSGs. (4.1.1) I n o r d e r t o o b t a i n a l l the n o n - t r i v i a l MSGs one has t o c o n s i d e r a l l t he space groups F v/hich have subgroups D o f i n d e x 2. Then f o r each D of each F one has t o combine the elements o f D w i t h the i d e n t i t y E o f A and the elements o f the c o s e t F - D w i t h the element E* o f A. The s e t o f a l l e l e m e n t s o f DE and (F - D)E' w i l l t h e n n e c e s s a r i l y c o n s t i t u t e a n o n - t r i v i a l MSG. An MSG o b t a i n e d i n t h i s way from a g i v e n F and D w i l l be denoted by M(D). As we have s a i d i n S e c t i o n 3.1 i n g e n e r a l a space group F has subgroups o f i n d e x 2 o f t h r e e k i n d s : ftp, 6R Q A N D 6R W . From t h i s and from (4.1.1) we c o n c l u d e : (4.1.2) The MSGs o b t a i n e d from a g i v e n F a r e i n g e n e r a l o f t h r e e k i n d s : M T = M ( D T ) , M R o = M ( D R o ) and MR<* ~ ^ ( ^ R o J * S i n c e a subgroup D T o f F c o n s i s t s o f t h e elements ( R a | r ( R a ) + t ) , where R & b e l o n g s t o some R D o f R and t be l o n g s t o T, the c o r r e s p o n d i n g MSG c o n s i s t s o f t h e unprimed elements ( R a l ^ ( R a ) + t ) and o f t h e primed elements (R^ ^ ( R ^ J + t ) 7 6 Similarly, since a subgroup I ) R o of F consists of the elements (R|T(R)+ta) where R belongs to It and t belongs to a subgroup T D of T with the same holohedry as T, the corresponding MSG M R o consists of the unprimed elements (R|T-(R)+ta) and of the primed elements (R|"C(R)+tg)'. Finally, since a subgroup DRflC of rs* F consists of the elements (Ral r(Ra^ + ta^ a n d (R<x |£'(R0<)+to<+ta) where R a belongs to some RD of R, R^ belongs to £ - R D , t a belongs to some T u which has the same holohedry as T and t^ belongs to T - T°, the corresponding MSG MR(^ consists of the unprimed elements (R a|f(R a)+t a) and (R^ ^ (R^)+t^+t a) and of the primed elements (Ra |t(R a )+tg)» and (R^ ^ (Rj+t^+tg)'. In Section 2.1 we have defined magnetic lattices and magnetic point groups independently of MSGs. We w i l l now define the magnetic lattice and the magnetic point group "belonging to" a given MSG and we w i l l also briefly discuss their properties. (4.1.3) We c a l l the set of a l l primitive translations of an MSG M the magnetic lat t i c e "belonging to" M (or the magnetic lattice "of" M). Obviously this set is a group and hence a subgroup of M. Let us introduce a useful convention: c a l l "primed" those rotations or translations which appear only in primed elements of an MSG, and c a l l "unprimed" those rotations or those translations which appear both in primed and unprimed elements of an MSG. Then we can say, according to (4.1.2): (4.1.4) The magnetic lattice of an M,p consists of the unprimed translations (E|t); the magnetic lattice of an M R o or an MR^ consists of the unprimed translations (E|t a) and 77 o f the primed t r a n s l a t i o n s (E |t ) ' . (4.1.5) The magnetic l a t t i c e o f an MSG M i s an i n v a r i a n t subgroup o f M. T h i s f o l l o w s from ( 2 . 2 . 1 ) , ( 2 . 2 . 2 ) , (2.2.3) and from the f a c t t h a t R t a b e l o n g s t o T° and R t ^ be l o n g s t o T - T D f o r any c h o i c e o f R, t a and t ^ . (4.1.6) We c a l l t he l a r g e s t magnetic p o i n t group w h i c h l e a v e s a magnetic l a t t i c e T M i n v a r i a n t h o l o h e d r y o f T^. We now come t o the d e f i n i t i o n o f magnetic p o i n t group " b e l o n g i n g t o " a g i v e n MSG M. (4.1.7) The magnetic p o i n t group " b e l o n g i n g t o " an MSG M i s the s e t o f e i t h e r unprimed, o r unprimed and primed r o t a t i o n s , w h i c h a r e t h e r o t a t o r y p a r t o f t h e elements o f M. From ( 4 . 1 . 2 ) , i t f o l l o w s : (4.1.8) The magnetic p o i n t group b e l o n g i n g t o an MSG M T c o n s i s t s o f a subgroup o f i n d e x 2 o f unprimed elements and a c o s e t o f primed e l e m e n t s . The magnetic p o i n t group b e l o n g i n g t o an MSG M R o o r M R c o n s i s t s o f unprimed elements o n l y . A complete l i s t of a l l p r o p e r l y d i f f e r e n t MSGs w i t h the e x c e p t i o n o f t h o s e b e l o n g i n g t o t h e o r t h o r h o m b i c system i s g i v e n a t the end o f t h i s t h e s i s ( t h e l i s t f o r the o r t h o -thombic system i s i n p r e p a r a t i o n ) . The MSGs a r e a r r a n g e d i n t o f a m i l i e s . I n t h i s r e s p e c t our l i s t d i f f e r s from t h a t g i v e n by B e l o v , Neronova, Smirnova (1957). The c l a s s i f i c a t i o n 78 o f MSGs i n t o f a m i l i e s seems t o be more u s e f u l from the p h y s i c a l p o i n t o f view The symbols used t o denote the v a r i o u s p r o p e r l y d i f f e r e n t MSGs a r e tho s e o f B e l o v , Neronova, Smirnova. Each symbol c o n s i s t s o f two p a r t s (as i n t h e case o f the i n t e r n a t i o n a l symbols f o r o r d i n a r y space g r o u p s ) : the f i r s t p a r t i n d i c a t e s the magnetic l a t t i c e o f the MSG i n q u e s t i o n , the second p a r t i t s magnetic p o i n t group and the n o n - p r i m i t i v e t r a n s l a t i o n s . The symbols f o r magnetic l a t t i c e s and magnetic p o i n t groups a r e e x p l a i n e d i n S e c t i o n s 4.2 and 4.3. I t s h o u l d be mentioned t h a t two f a m i l i e s o f MSGs c o n s i s t o f t r i v i a l magnetic space groups o n l y : t h e f a m i l y o f F23, and the f a m i l y o f P2^3. These a r e t h e o n l y two space groups which do not have subgroups o f i n d e x 2 a t a l l . Each o f the r e m a i n i n g 228 f a m i l i e s c o n s i s t s o f a t l e a s t two members, but i n g e n e r a l , o f more than two. There a r e a l t o g e t h e r 1491 p r o p e r l y d i f f e r e n t magnetic space groups. SECTION 4.2 As we have s a i d i n S e c t i o n 2.1, the o r d i n a r y o r B r a v a i s l a t t i c e s a r e a s p e c i a l case o f the magnetic l a t t i c e s . We d e f i n e a c l a s s o f magnetic l a t t i c e s i n a n a l o g y t o the u s u a l d e f i n i t i o n of a c l a s s o f B r a v a i s l a t t i c e s ( s e e , f o r example, Lomont ( 1 9 5 9 ) , page 198): (4.2.1) Two magnetic l a t t i c e s ( w i t h a l a t t i c e p o i n t a t the o r i g i n ) b e l o n g t o the same c l a s s i f one can be t r a n s -formed i n t o the o t h e r by means o f a homogeneous l i n e a r 79 t r a n s f o r m a t i o n which a l s o t r a n s f o r m s the h o l o h e d r y o f the one i n t o the h o l o h e d r y o f the o t h e r , the term h o l o h e d r y b e i n g used i n the sense o f ( 4 . 1 . 6 ) . There a r e 22 c l a s s e s o f n o n - t r i v i a l magnetic l a t t i c e s ( t h a t i s , magnetic l a t t i c e s w h i c h a r e not o r d i n a r y l a t t i c e s ) , s o t h a t a l t o g e t h e r t h e r e a r e 36 c l a s s e s o f magnetic l a t t i c e s . The r e a s o n f o r the e x i s t e n c e o f o n l y 22 new magnetic l a t t i c e s i s t h a t a l t h o u g h , as we have s a i d , an o r d i n a r y l a t t i c e T has i n g e n e r a l seven d i f f e r e n t subgroups T° o f i n d e x 2, r e l a t i o n s among the t h r e e b a s i c p r i m i t i v e t r a n s l a t i o n s o f T and the c o n d i t i o n f o r e v e r y T D t o have the same h o l o h e d r y as T reduce the number o f T^'s o f a g i v e n T from which magnetic l a t t i c e s a r e d e r i v e d . I n the case o f an o r d i n a r y l a t t i c e a p r i m i t i v e u n i t c e l l i s d e f i n e d as a p a r a l l e l e p i p e d w i t h edges g i v e n by t h r e e b a s i c p r i m i t i v e t r a n s l a t i o n s a^, a 2 , The same d e f i n i t i o n h o l d s f o r a n o n - t r i v i a l magnetic l a t t i c e , e x c e p t t h a t some, or a l l , b a s i c p r i m i t i v e t r a n s l a -t i o n s become primed. I n b o t h c a s e s ( o r d i n a r y l a t t i c e and n o n - t r i v i a l m agnetic l a t t i c e ) the whole l a t t i c e can be r e p r o d u c e d by t r a n s l a t i o n o f the p r i m i t i v e u n i t c e l l t h r o u g h p r i m i t i v e t r a n s l a t i o n s . However, i n the case of a n o n - t r i v i a l magnetic l a t t i c e t he p r i m i t i v e t r a n s l a t i o n s w i l l be e i t h e r unprimed ( f o r an MSG Mj) o r unprimed and primed ( f o r an MSG M R ) . T h i s d e f i n i t i o n o f the p r i m i t i v e u n i t c e l l i s d i f f e r e n t i n t he case o f MSGs M R from the one o f t e n used i n p h y s i c a l 80 a p p l i c a t i o n s . The u s u a l d e f i n i t i o n i s chosen such t h a t t h e whole l a t t i c e can be o b t a i n e d by t r a n s l a t i o n o f the p r i m i t i v e u n i t c e l l t h r o u g h unprimed p r i m i t i v e t r a n s l a t i o n s . Hence the volume o f the p r i m i t i v e u n i t c e l l d e f i n e d i n t h i s way may be a m u l t i p l e o f the p r i m i t i v e u n i t c e l l a c c o r d i n g t o our d e f i n i t i o n . As i n the case o f o r d i n a r y l a t t i c e s i t i s p o s s i b l e t o d e f i n e a symmetric u n i t c e l l , but we s h a l l n o t need t h i s d e f i n i t i o n h e r e . To d e s c r i b e t h e v a r i o u s c l a s s e s o f magnetic l a t t i c e s we s h a l l use n o n - p r i m i t i v e u n i t c e l l whose edges a, b, c w i l l be t a k e n a l o n g the c o n v e n t i o n a l c o o r d i n a t e axes o f each o f the seven c r y s t a l l o g r a p h i c s y s t e m s ; a, b, c a r e pri m e d o r unprimed p r i m i t i v e t r a n s l a t i o n s . A t t h e end o f t h i s s e c t i o n we g i v e a l i s t of the 36 c l a s s e s o f magnetic l a t t i c e s a r r a n g e d i n t o " f a m i l i e s " ; a " f a m i l y " o f magnetic l a t t i c e s T M c o n s i s t s o f an o r d i n a r y l a t t i c e T and o f a l l the magnetic l a t t i c e s o b t a i n e d from T by means o f r u l e (2.2.8) (H i n the r u l e i s T i n the p r e s e n t c a s e ) , and h a v i n g the same h o l o h e d r y as T. Each f a m i l y i s a r r a n g e d i n t h i s way: f i r s t the o r d i n a r y l a t t i c e , t h e n the magnetic l a t t i c e s ( i n g e n e r a l , one from each c l a s s ) d e r i v e d from i t l i s t e d i n the same o r d e r i n w h i c h the seven subgroups lP^ o f a l a t t i c e ¥ a r e a r r a n g e d i n S e c t i o n 3.3. An o r d i n a r y l a t t i c e i s r e p r e s e n t e d by means o f i t s b a s i c p r i m i t i v e t r a n s l a t i o n s e x p r e s s e d i n terms o f a, b and c; a magnetic l a t t i c e i s r e p r e s e n t e d by means o f t h r e e independent 81 p r i m i t i v e t r a n s l a t i o n s o f i t s subgroup o f unprimed elements (see l i s t of subgroups T 1 ^ of a l a t t i c e f i n S e c t i o n 3.3). The symbols i n t r o d u c e d by B e l o v , Neronova and Smirnova (1955) a r e a l s o g i v e n . These symbols a r e r e a l l y not v e r y s u i t a b l e i f the p r i n c i p l e o f a c l a s s i f i c a t i o n i n t o f a m i l i e s i s adopted. Moreover two d i f f e r e n t symbols sometimes denote the same c l a s s o f magnetic l a t t i c e s ( e . g . C l a s s 12 and C l a s s 1 9 ) . L i s t of c l a s s e s o f magnetic l a t t i c e s  T r i c l i n i c system: a ^ b ^ c , a? B / y * 90° . 1. (a,b,c) P 2. (a,b,2c) P s O b v i o u s l y a l l the n o n - t r i v i a l magnetic l a t t i c e s t h a t one can d e r i v e from the l a t t i c e s o f the c l a s s P b e l o n g t o the c l a s s P g . M o n o c l i n i c system: a / b ^  c, a = y = 90° 5* B. 3. (a,b,c) P 4. (2a,b,c) P a 5. (a,2b,c) P b 6. (2a,a+b,c) C a 82 7. (Odde , b, c) c S i n c e we have t a k e n t h e y - a x i s t o be the unique a x i s , t h e magnetic l a t t i c e s ( a , b , 2 c ) b e l o n g t o the c l a s s P a , and the magnetic l a t t i c e s ( a , 2b, b+c) b e l o n g t o the c l a s s C a. The r e m a i n i n g d i f f e r e n t magnetic l a t t i c e s d e r i v e d from a l a t t i c e P or C b e l o n g t o the c l a s s P c o f t h e t r i c l i n i c s ystem. Orthorhombic system: a ^ b ^ c , a ~ B = y - 90° . 10. 11. (2a,b,c) P, 12. ( 2 c ^ c v + ^ c ) C ( a j l b , b + c j A 13. (za-, tx+b cv+cj F, 14. fcc^t , b, c J 15. ( l ^ > , c ) P. 16. (£^k f b . l c ) C 17. (2 c y b ; b ) o-+b + c J I s a c s C c c 83 1 8 ((X+b Ci+r b-t-c 1 1 Z ' 2 ^ ' 2 i 19.  ' ft>tb , o,+ b cx+r, , b + c ) Ac t 2 ; 2 2 ' 2. / * foi-f-b ^ cy+c P~+C + b t r ) C A I 2 ' 2. ' 2 2 1 20. f X + b tc Cn-b - C (X- b - C ) I 1 2 ; i z I 2 1 . h CX-t-b-t-C 0u+ b + c , CVV b - c On- b + c , cv-b-o\ P T r 2 ' 2 2 ; 2 ~ + 1 I 1 The mag-ae*±c l a t t i c e s G-^ belong- to- the- cJ ras&-A c. i t The magnetic l a t t i c e s d e r i v e d from an o r t h o r h o m i c l a t t i c e w h i c h do not b e l o n g t o any o f t h e c l a s s e s l i s t e d above, b e l o n g t o the c l a s s P s o f the t r i c l i n i c system. T e t r a g o n a l system: a = b ^  c, a = B = y = 9 0 ° . 22. ( C L ; b / c J P 23 . (a,, b 2 c ) P c 24. (°-O~,0n-b,cj P c 25. (,2.eu/0L+la>,t> + c) l c 26. / n+b + c (X + b-c cx-b-c \ I 27 /o Ou±ki±£_ ft-b-t-c . a + b^ c CX4- b r e • CC-b- c) p ' * l r 2 ' 2. 2 ^ 2. 2 ' * 84 The magnetic l a t t i c e s ( 2 a , b , c ) , (a,2b,c) b e l o n g t o the c l a s s P s o f t h e o r t h o r h o m b i c system. The magnetic l a t t i c e s (2a,b,a+c), (a,2b,b+c) b e l o n g t o the c l a s s C a o f the o r t h o r h o m b i c system. A l l the magnetic l a t t i c e s d e r i v e d from I w i t h the e x c e p t i o n o f P j b e l o n g t o the c l a s s P g o f the t r i c l i n i c system, T r i g o n a l system: a = b = c, a = B-- y = 120° , 28. ( o - . b . c ) R 29. ( 1 a , o^ + k>/ cx> +oj A l l the o t h e r magnetic l a t t i c e s d e r i v e d from a l a t t i c e R b e l o n g t o t h e c l a s s P g o f the t r i c l i n i c s ystem. Hexagonal system: a = b / c t a = B = 90° , y = 120° . 30. (cu, b ; c j P 31. ( o , ; b , Z c j P | c A l l the o t h e r magnetic l a t t i c e s d e r i v e d from a l a t t i c e P b e l o n g t o the c l a s s P g o f t h e t r i c l i n i c s ystem. C u b i c system: a = b = c, a = B = y = 90° . 32. ( a , b , c j P 33. (2 o o ( cx + b / C L + C ) F s 85 34. / Ct + b cx-t-c b+-c \ F ^ 2 ) Z ' 2. / 35 / cx+b + c Cu-t-b-c C i - b - c \ I V 2 ' 2. v 2. / 36. on-W+c cv + bn-c , cx+-b- c (V-t-b+c , a - b - c ) p_ \L z > z 4 Z ] Z + 2 / 1 The magnetic l a t t i c e s ( 2 a , b , c ) , ( a , 2 b , c ) , (a,b,2c) b e l o n g t o the c l a s s P„ o f t h e t e t r a g o n a l systems t h e magnetic l a t t i c e s (2a,a+b,c), (2a,b,a+c), (a,2b,b+c) b e l o n g t o the c l a s s PQ o f t h e t e t r a g o n a l system. The magnetic l a t t i c e s ( I $±±k a±_k + Q±±^_ b ) , 2 2 2 ' 2. (2 ^M^-, and (a=±k ,2 a,+ c , + j ^ c _ j b e l o n g L. i_ L. c. 2 2 . 2 . t o the c l a s s A„ o f the o r t h o r h o m i c system. c * * A l l o t h e r magnetic l a t t i c e s d e r i v e d from F and I b e l o n g t o the c l a s s P g o f the t r i c l i n i c system. Meaning of symbols used by B e l o v . Neronova and Smirnova. P l a i n l a t t i c e s P. P s a primed t r a n s l a t i o n a l o n g one edge, P a " " " " an edge i n the a d i r e c t i o n , P b " " " " an edge i n the b d i r e c t i o n , P c " " " " an edge i n the c d i r e c t i o n , C a " " " " edges i n the a and b d i r e c t i o n f o r the m o n o c l i n i c and o r t h o -rhombic s y s t e m , P " " " " edges i n the a and b d i r e c t i o n s f o r the t e t r a g o n a l s y s t e m , 86 primed t r a n s l a t i o n a l o n g edges i n the b and c d i r e c t i o n s , " " " edges i n the a, b and c d i r e c t i o n s i n the t e t r a g o n a l s y s t e m , " " " edges i n t h e a, b and c d i r e c t i o n s i n t h e o r t h o r h o m b i c and c u b i c s y s t e m , P l a i n l a t t i c e s R primed t r a n s l a t i o n a l o n g edges i n t h e a, b and c d i r e c t i o n s . B a s e - c e n t e r e d l a t t i c e s C primed t r a n s l a t i o n a l o n g an edge i n the c d i r e c t i o n , " " " the d i a g o n a l s o f the C f a c e , " " " the d i a g o n a l s o f the C f a c e and t h e edge i n the c d i r e c t i o n i n the o r t h o r h o m b i c s y s t e m , F a c e - c e n t e r e d l a t t i c e s F primed t r a n s l a t i o n a l o n g the d i a g o n a l s o f t h e A and B f a c e , " " '» t h e d i a g o n a l s o f the B and C f a c e . B o d y - c e n t e r e d l a t t i c e s I primed t r a n s l a t i o n a l o n g the body d i a g o n a l . 87 SECTION 4.3 I n S e c t i o n 2.1 we have g i v e n the d e f i n i t i o n [(2.1.4JJ o f a magnetic p o i n t group. There a r e 90 o f s u c h g r o u p s ; 32 o f them a r e the o r d i n a r y c r y s t a l l o g r a p h i c p o i n t groups. The p r o p e r t i e s o f the magnetic p o i n t groups have been g i v e n i m p l i c i t l y i n C h a p t e r 2. Here we w i l l o n l y mention a p r o p e r t y w h i c h i s p e c u l i a r t o magnetic p o i n t groups ( o r , more g e n e r a l l y , p e c u l i a r t o a l l magnetic r o t a t i o n g r o u p s ) : A r o t a t i o n Rg o f o r d e r 3 cannot be a primed element of a magnetic p o i n t group. I n f a c t i f i t was then ( R 3 1 ) 3 = E' would b e l o n g t o t h e magnetic p o i n t group as w e l l , w hich i s not c o m p a t i b l e w i t h the d e f i n i t i o n o f magnetic p o i n t group. I t s h o u l d be mentioned t h a t a r o t a t i o n R3 never appears i n the c o s e t s o f a p o i n t group R r e l a t i v e t o i t s subgroups I P so t h a t the magnetic p o i n t groups a r e o b t a i n e d by c o n s i d e r i n g a l l the 32 o r d i n a r y p o i n t g roups. We now g i v e a l i s t of the 90 magnetic p o i n t g r o u p s , a r r a n g e d i n t o f a m i l i e s ; a " f a m i l y " of magnetic p o i n t groups c o n s i s t s o f a t r i v i a l magnetic p o i n t group H ( t h a t i s , an " o r d i n a r y " p o i n t group) and o f a l l magnetic p o i n t groups d e r i v e d from H by means of r u l e 2.2.8. Each l i n e i n the l i s t g i v e s groups of one f a m i l y . The n o t a t i o n used i n the l i s t has been i n t r o d u c e d by B e l o v , Neronova and Smirnova ( 1 9 5 5 ) , and i s a s t r a i g h t f o r w a r d g e n e r a l i z a t i o n o f t h e u s u a l i n t e r n a t i o n a l n o t a t i o n f o r the 88 o r d i n a r y p o i n t g roups. The i n t e r n a t i o n a l symbol f o r an o r d i n a r y p o i n t group i n d i c a t e s the g e n e r a t i n g elements of the group. I n the symbol f o r a magnetic p o i n t group, t h o s e g e n e r a t i n g elements w h i c h a r e primed have a dash as a s u p e r -s c r i p t . By o m i t t i n g the dashes i n the symbol o f a magnetic p o i n t group, one o b t a i n s t h e symbol o f the o r d i n a r y p o i n t group t o whose f a m i l y t h e magnetic p o i n t group i n q u e s t i o n b e l o n g s . L i s t of magnetic p o i n t groups 1 T T' 2 2 m m' 2/m 2'm 2/ra 1 2 V m 1 222 2 f2'2 mm2 m'm2' m' m»2 m'm'm m'm'm' 4 4* 4/m 4 Vra 422 4 ,22» 4mm 4'm1m 4/m' 4Vm' 42*2' 4m'm' 89 42m 4'2'm 4'2m' 42»m' 4/mmm 4/m'mm 4'/mm'm 4'/m'm'm 4/mm'm* 4/m'm'm' 3 3 3°' 32 32' 3m 3m' 3m 3~'m 3~'m' 3m' 6 6' 6 6' 6/m 6*/m 6/m' 6'/m' 622 6'2'2 62'2' 6mm 6'm'm 6m*m' 6m2 6'm'2 6i'm2' 6m'2' 6/mmm 6/m'mm 6'/mm'm e'/ra'm'm 6/mm'ra' 6/m'm'm' 23 m3 m'3 432 4»32' 43m 4'3m' m3m m'3m m3m' m'3m* SECTION 4.4 Our method of d e r i v i n g a l l MSGs has a weakness which i t s h a r e s w i t h a l l s y s t e m a t i c methods o f d e r i v i n g a l l o r d i n a r y space groups: i t o c c a s i o n a l l y g i v e s the same MSG more t h a n 90 once. Here the phrase " t h e same MSG" i s used i n the f o l l o w i n g sense: Mj_ and M 2 a r e " t h e same" = M-j_ and M 2 b e l o n g t o t h e same " c l a s s o f MSGs" = M^ and M 2 a r e not " p r o p e r l y d i f f e r e n t " = M"j and M 2 a r e " e q u i v a l e n t " . The d e f i n i t i o n s o f the terms " p r o p e r l y d i f f e r e n t MSGs" and "a c l a s s o f MSGs" have been g i v e n i n S e c t i o n 2.3. I n the p r e s e n t s e c t i o n we f o r m u l a t e a c e r t a i n number o f g e n e r a l theorems w h i c h e n a b l e us t o d e c i d e whether o r not two MSGs a r e e q u i v a l e n t . We do not g i v e t h e p r o o f s o f t h e s e theorems because t h e s e p r o o f s a r e a l l s i m p l e consequences o f the b a s i c p r o p e r t i e s o f MSGs, and of the immediate i m p l i c a t i o n s o f the d e f i n i t i o n s o f su c h well-known c o n c e p t s as isom o r p h i s m , automorphism, and e q u i v a l e n c e o f s e t s o f m a t r i c e s . I t w i l l be c o n v e n i e n t (but not n e c e s s a r y ) t o i n t e r p r e t , from now on, an MSG as a group o f m a t r i c e s o f f o u r columns and f o u r rows as i s o f t e n done i n the case o f o r d i n a r y space groups. Such an i n t e r p r e t a t i o n i s p o s s i b l e because t h e one-to-one c o r r e s p o n d e n c e 91 ( R |^(R)^L-) < — » R 0 0 0 1 (4.4.1) R| 7 :0)4-) ' <- >-R 0 0 0 1 i s o b v i o u s l y an isomorphism ( t h e 3 x 3 - m a t r i x R i s the ( p r o p e r o r improper) r o t a t i o n m a t r i x , the "C^'s and t ^ ' s , k = 1,2,3, a r e components o f T and t a l o n g the c o o r d i n a t e a x e s ) . U s i n g t h i s i n t e r p r e t a t i o n t he e q u i v a l e n c e o f two MSGs becomes s i m p l y the e q u i v a l e n c e o f two c o r r e s p o n d i n g m a t r i x g r o u p s , the e q u i v a l e n c e t r a n s f o r m a t i o n b e i n g a 4x4 m a t r i x S r e p r e s e n t i n g an element of the g e n e r a l inhomogeneous l i n e a r group ^ . More p r e c i s e l y : (4.4.2) M^ L and M 2 a r e e q u i v a l e n t MSGs i f and o n l y i f t h e r e e x i s t s a m a t r i x S such t h a t SM^S - 1 = M 2 < Hence: (4.4.3) I f M(D^) and M(D~2) a r e e q u i v a l e n t , then S D i S " 1 = D 2 S G K b ^ - D j ^ S " 1 = M(D 2)-D 2; 92 and the c o n v e r s e s t a t e m e n t i s a l s o t r u e (4.4.4) I f i s a subgroup o f i n d e x 2 o f F^ and i s a subgroup o f i n d e x 2 o f F 2 t h e n M(D^) and 81(1)2) a r e e q u i v a l e n t i f and o n l y i f t h e r e e x i s t s a m a t r i x S such t h a t S ( F 1 - D 1 ) S " 1 = F 2-D 2. From (4.4.4) we c o n c l u d e : (4.4.5) Two space groups from w h i c h two e q u i v a l e n t MSGs a r e d e r i v e d a r e n e c e s s a r i l y e q u i v a l e n t . (4.4.6) Two e q u i v a l e n t MSGs n e c e s s a r i l y b e l o n g t o the same f a m i l y . (4.4.7) Two MSGs M(D^) and M(D 2) a r e n e c e s s a r i l y noi^equivalent i f 6^ and D2 a r e not e q u i v a l e n t . A l e s s t r i v i a l theorem a l s o f o l l o w s i m m e d i a t e l y from ( 4 . 4 . 4 ) : (4.4.8) I f the subgroups and 62 of two MSGs M(Di) and M(D 2) d e r i v e d from a space group F a r e e q u i v a l e n t but the isomorphism i m p l i e d by t h i s e q u i v a l e n c e i s not i n d u c e d by an automorphism o f F, t h e n M(D^) and M(Dg) a r e no t e q u i v a l e n t . T h i s automorphism o f F must be an o u t e r automorphism as b o t h 5^  and 62 b e i n g i n v a r i a n t subgroups o f F a r e i n v a r i a n t under any i n n e r automorphism o f F. However: 93 (4.4.9) I f the subgroups and D 2 of two MSGs M(D^) and M(D 2) d e r i v e d from a space group F a r e e q u i v a l e n t and the isomorphism i m p l i e d by t h i s e q u i v a l e n c e i s i n d u c e d by an automorphism of F, M(D^) and M(D 2) need not be e q u i v a l e n t . (To d e c i d e whether t h e y a r e e q u i v a l e n t o r not one has t o compare the t r a c e s o f a l l m a t r i c e s M(D^) and M(D 2) w h i c h c o r r e s p o n d t o one a n o t h e r under the isomorphism between M(D^) and M(D 2) i n d u c e d by the automorphism of F.) We s h a l l now g i v e a c e r t a i n number o f r u l e s c o n c e r n i n g the e q u i v a l e n c e o f the subgroups D of a space group F and the e q u i v a l e n c e o f t h e MSGs d e r i v e d from them. (4 . 4.10) A subgroup D T o f a g i v e n space group F can nev e r be e q u i v a l e n t t o a subgroup D R of F. (4.4.11) An MSG M(D T) and an MSG M(D R) d e r i v e d from a space group ¥ a r e never e q u i v a l e n t . (4.4.12) Two subgroups Sj, o f a space group F may be e q u i v a l e n t o r n o t . I f t h e y a r e not e q u i v a l e n t t h e c o r r e s -p onding MSGs a r e n e c e s s a r i l y not e q u i v a l e n t . I f t h e y a r e e q u i v a l e n t the c o r r e s p o n d i n g MSGs may s t i l l be e q u i v a l e n t o r n o t . (4.4.13) L e t D T^ and D T 2 be two subgroups o f F and l e t R D^ and R D 2 be t h e i r p o i n t g r o u p s . I f R 0^ and R ° 2 a r e not i s o m o r p h i c , t h e n D^ ,^  and a r e n e o e s s a r - i l y . n o t e q u i v a l e n t and hence M(D T^) and M(D,p2) a r e n o t e q u i v a l e n t e i t h e r . I f R D^ and R D 2 a r e i s o m o r p h i c and e q u i v a l e n t t h e n 94 DFJIJ and Drp2 a r e e q u i v a l e n t o n l y i f the 3x3 m a t r i x w h i c h t r a n s -forms R D J L i n t o R ° 2 t r a n s f o r m s the l a t t i c e T i n t o a l a t t i c e o f the same c l a s s t o wh i c h T be l o n g s and the n o n - p r i m i t i v e t r a n s l a t i o n o f e v e r y element o f R D ^ i n t o t h e n o n - p r i m i t i v e t r a n s l a t i o n o f i t s c o r r e s p o n d i n g element i n R D 2 « The isomorphism between D^^ and D i m p l i e d by t h e i r e q u i v a l e n c e i s t h e n i n d u c e d by an automorphism o f F, and the MSGs 0 (6 ,^ ) and M(D,j,2) may but need not be e q u i v a l e n t (compare 4.4.9). (4.4.14) The subgroups B R o and D R a o f a space group F d e r i v e d from a subgroup Q or Q r o f i n d e x 4 o f F a r e e q u i v a l e n t i f the t r a n s l a t i o n t whi c h appears i n some elements of D_ can be t r a n s f o r m e d away from a l l o f them. Hoc I n f a c t D R o and Dp^ have the same p o i n t group and the same l a t t i c e and i f the e q u i v a l e n c e t r a n s f o r m a t i o n S such . t h a t D R o = SD R aS~^ e x i s t s i t i s a pure t r a n s l a t i o n and hence l e a v e s the p o i n t group and t h e l a t t i c e o f Dp^ unchanged. (4.4.15) Two MSGs M(D_ ) and M(D D ) d e r i v e d from a KO Kef. space group F a r e e q u i v a l e n t i f D R o and D R^ a r e e q u i v a l e n t . I n t h i s c a s e the e q u i v a l e n c e o f D R o and D R f f i s a s u f f i c i e n t c o n d i t i o n f o r the e q u i v a l e n c e o f M ( D R o ) and M(D R f f) • a-nd 1 W < ^ \ ^ (4.4.16) Two subgroups D R o l A a n d D R f l l o f a space group F a r e e q u i v a l e n t i f t h e i r l a t t i c e s T ^ and f D 2 b e l o n g t o t h e same B r a v a i s c l a s s and i f the m a t r i x w h i c h t r a n s f o r m s 95 T0-^ i n t o T°2 l e a v e s the n o n - p r i m i t i v e t r a n s l a t i o n s ( i f any) of t h e elements o f the p o i n t group R unchanged. (4.4.17) I f the isomorphism i m p l i e d by the e q u i v a -l e n c e o f two subgroups 6 R q 1 and D R o 2 o r D R f l l and D R a 2 of a space group F i s i n d u c e d by an automorphism of F, the n the two c o r r e s p o n d i n g MSGs may but need n ot be e q u i v a l e n t . (4.4.18) A subgroup D R q and a subgroup D j ^ o f a space group F d e r i v e d from two d i f f e r e n t subgroups Q r o f i n d e x 4 o f F can be e q u i v a l e n t o n l y i f the t r a n s l a t i o n t can be t r a n s -formed away from i f the l a t t i c e s o f the two space groups b e l o n g t o the same B r a v a i s c l a s s and i f the m a t r i x w h i c h t r a n s f o r m s one of the two l a t t i c e s i n t o t he o t h e r t r a n s f o r m s the n o n - p r i m i t i v e t r a n s l a t i o n o f an element o f the f i r s t group i n t o t he n o n - p r i m i t i v e t r a n s l a t i o n o f th e c o r r e s p o n d i n g element i n the second group. A g a i n t h e i s o -morphism between the two groups may or may not be i n d u c e d by an automorphism o f F and hence the two c o r r e s p o n d i n g MSGs may o r may not be e q u i v a l e n t . SECTION 4.5 In S e c t i o n 3.4 we have d e r i v e d a l l t he subgroups o f in d e x 2 o f the space groups P4, P 4 l t p 4 2 and P 4 3 . We w i l l now g i v e the c o r r e s p o n d i n g MSGs a r r a n g e d i n t o f a m i l i e s . As we have s a i d i n S e c t i o n 3.4 the space group P4 has o n l y one subgroup D T. To t h i s subgroup t h e r e c o r r e s p o n d s the 9 6 MSG P 4 * w h i c h c o n s i s t s o f t h e elements To the subgroup D R q o f P 4 w h i c h i s o b t a i n e d from (See S e c t i o n 3 . 4 ) t h e r e c o r r e s p o n d s the MSG P c 4 which c o n s i s t s o f the elements (E|/- 1a...^b + Z ^ 4(C (, l|0),(C, i l|0 ) r ( C^|0) ;(F|c)' To the subgroup D R F L o f P 4 which i s o b t a i n e d from the same t h e r e c o r r e s p o n d s t h e MSG P c 4 2 w n * c n c o n s i s t s o f t h e elements ( f K 1c . + ^ k + 2^o) l(C J |,|c) l(C^ l|0) l(C < i;|c),(E|c)' O b v i o u s l y P c 4 and P c 4 2 a r e not e q u i v a l e n t . To t h e subgroups D K q and D R A o f P 4 w h i c h a r e e q u i v a -l e n t and o b t a i n e d from Q 2 t h e r e c o r r e s p o n d s one Msg PQ 4 which c o n s i s t s o f the elements To the subgroups D R O and D R A o f P 4 w h i c h a r e e q u i v a l e n t and o b t a i n e d from § 3 t h e r e c o r r e s p o n d s one MSG I c 4 c o n s i s t i n g o f the elements ( E 1 2 / ^ * ^ 0 ^ 4 W a + 4 ^ 97 o f Thus the f a m i l y P4 c o n s i s t s o f P4, P4» , P c 4 , P c 4 2 , P C 4 , I c 4 . The space group P4^ has o n l y one subgroup D T t o which t h e r e c o r r e s p o n d s the MSG P4^'. T h i s group c o n s i s t s o f t h e elements (G K a, + ^ b ^ c ) , (c J , ( C ^ | £), (c^ 13g) The group P4^ has a l s o o n l y one subgroup D R o and Sj^. They a r e e q u i v a l e n t and t o them t h e r e c o r r e s p o n d s one MSG P c ^ i which c o n s i s t s o f Thus the f a m i l y o f P 4 1 c o n s i s t s o f P 4 1 , P 4 , 1 , P c 4 l * S i m i l a r l y , from t h e space group P 4 2 the f o l l o w i n g MSGs a r e d e r i v e d : P 4 2 ' w h i c h c o n s i s t s o f the elements P c 4 ^ w h i c h c o n s i s t s o f the elements 98 P c 4 3 w h i c h c o n s i s t s o f the elements P C 4 2 w n i c n c o n s i s t s °f "the elements (E\l^,^^[c ^ ) i - ^ ) . ( c ^ | ^ ) J ( C « l | 0 ) ( ( C „ a | f ) ; l F H ' * c 4 l w n i c n c o n s i s t s o f the elements Thus t h e f a m i l y of P 4 2 c o n s i s t s o f P 4 2 , P 42*t pc 4l» p c 4 3 » P C 4 2 » I c 4 l -From the space group P 4 3 the f o l l o w i n g MSGs a r e d e r i v e d : P4 3» which c o n s i s t s o f t h e elements P C 4 3 w n i c h c o n s i s t s o f t h e elements Thus the f a m i l y o f P 4 g c o n s i s t s o f P 4 3 , P 4 3 ' , Pc 4 3» 99 NOTES TO CHAPTER 4. G r a p h i c a l r e p r e s e n t a t i o n o f a l l b l a c k - a n d - w h i t e l a t t i c e s (magnetic l a t t i c e s ) can be found i n a paper by B e l o v , Neronova and Smirnova ( 1 9 5 5 ) , and i n a r e v i e w a r t i c l e by Le C o r r e (1959). A l i s t i n w h i c h a l l b l a c k - a n d - w h i t e l a t t i c e s seem t o be a r r a n g e d i n t o f a m i l i e s i s g i v e n by Zamorzaev (1957). The u n c e r t a i n t y a r i s e s from the l a c k o f any comments on the p a r t o f the a u t h o r , and from some minor i n c o n s i s t e n c i e s w h i c h may be due t o m i s p r i n t s . M a g n e t i c p o i n t groups were f i r s t l i s t e d by Heesch (1929) and more r e c e n t l y a g a i n by Tavger and Z a i t s e v (1956) who do not quote Heesch. A l i s t i n w h i c h magnetic p o i n t groups a r e a r r a n g e d i n t o f a m i l i e s i s g i v e n by Le C o r r e (1959). F i n a l l y i t s h o u l d be mentioned t h a t t h e r e e x i s t s an e n t i r e l y d i f f e r e n t method t o d e r i v e a l l MSGs. I t i s a s p e c i a l c a s e o f a v e r y g e n e r a l method r e c e n t l y proposed by B i e n e n s t o c k and Ewald (1962) t o i n v e s t i g a t e s y s t e m a t i c a l l y the symmetries o f t h e r e c i p r o c a l o r F o u r i e r space o f a c r y s t a l . 100 CHAPTER 5 INVARIANT ARRANGEMENTS OF SPINS In S e c t i o n 5.1 we c o n s i d e r MSGs i n t h e i r r e l a t i o n t o th e i n v a r i a n t arrangements o f (average) magnetic moments i n f e r r o m a g n e t i c , f e r r i m a g n e t i c and a n t i f e r r o r a a g n e t i c c r y s t a l s . I n S e c t i o n 5.2 we d e s c r i b e a pr o c e d u r e f o r d e t e r m i n -i n g t he MSG whi c h l e a v e s i n v a r i a n t a g i v e n arrangement o f magnetic moments. Fo r s i m p l i c i t y , we w i l l c a l l an arrangement o f magnetic moments an "arrangement of s p i n s " , t h a t i s , the average magnetic moment o f an i n d i v i d u a l magnetic atom w i l l be s i m p l y c a l l e d a " s p i n " . W h i l e i n C h a p t e r s 2, 3 and 4 the p h y s i c a l meaning a t t a c h e d t o the element E' o f A c o u l d be l e f t u n s p e c i f i e d , i n t h e p r e s e n t c h a p t e r i t becomes e s s e n t i a l t o i n t e r p r e t E* as t i m e - r e v e r s a l . SECTION 5.1 We b e g i n w i t h s t a t i n g e x p l i c i t l y t h e well-known 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 s p i n ( t h e y were a l r e a d y used i n i n t r o d u c i n g the d e f i n i t i o n o f MSGs i n Cha p t e r 1 ) : (5.1.1) F o r p r o p e r and improper r o t a t i o n s t h e s p i n i s an a x i a l v e c t o r , f o r t i m e - r e v e r s a l i t i s a p o l a r v e c t o r ; the o n l y e f f e c t o f t i m e - r e v e r s a l on the s p i n i s t o change i t s " s e n s e " , the " d i r e c t i o n " o f the s p i n r e m a i n i n g the same. 101 Some s i m p l e r u l e s f o l l o w i m m e d i a t e l y from 5.1.1: (5.1.2) A s p i n o f a g i v e n d i r e c t i o n and sense i s t r a n s f o r m e d i n t o a n o t h e r s p i n o f t h e same d i r e c t i o n and sense under any t r a n s l a t i o n , under a p r o p e r o r improper r o t a t i o n about an a x i s p a r a l l e l t o the s p i n (hence under a r e f l e c t i o n i n a p l a n e p e r p e n d i c u l a r t o the d i r e c t i o n o f the s p i n ) and under space i n v e r s i o n . (5.1.3) I f a s p i n i s a l o n g an a x i s o f p r o p e r r o t a t i o n s , o r i t s o r i g i n i s i n a r e f l e c t i o n p l a n e p e r p e n d i -c u l a r t o t h e s p i n o r a t a c e n t e r o f i n v e r s i o n , i t i s l e f t i n v a r i a n t under the p r o p e r r o t a t i o n , the r e f l e c t i o n o r the space i n v e r s i o n r e s p e c t i v e l y . I f a s p i n i s a l o n g an a x i s o f an improper r o t a t i o n but i t s o r i g i n i s not a t the c e n t e r o f i n v e r s i o n , t h e n i t i s t r a n s f o r m e d by the improper r o t a t i o n i n t o a n o t h e r s p i n o f the same d i r e c t i o n and s e n s e . (5.1.4) A s p i n o f a g i v e n d i r e c t i o n and sense i s t r a n s f o r m e d i n t o a n o t h e r s p i n of the same d i r e c t i o n and o p p o s i t e sense by a p r o p e r o r improper r o t a t i o n t h r o u g h 180° about an a x i s p e r p e n d i c u l a r t o the d i r e c t i o n o f the s p i n . (5.1.5) A s p i n i s t r a n s f o r m e d i n t o a n o t h e r s p i n of t h e same d i r e c t i o n but o p p o s i t e sense under a primed t r a n s l a t i o n , under a primed p r o p e r o r improper r o t a t i o n about an a x i s p a r a l l e l t o the s p i n and f i n a l l y under t h e c o m b i n a t i o n o f space i n v e r s i o n w i t h t i m e - r e v e r s a l ( primed space I n v e r s i o n ) when the o r i g i n o f the g i v e n s p i n i s not a t t h e c e n t e r o f i n v e r s i o n . ( T h i s f o l l o w s from (5.1.1) and ( 5 . 1 . 2 ) ) . 102 ( 5 . 1 . 6 ) No s p i n can be a l o n g an a x i s o f a primed p r o p e r r o t a t i o n , can have i t s o r i g i n a t a c e n t e r o f a primed space i n v e r s i o n o r can be p e r p e n d i c u l a r t o a p l a n e o f prime d r e f l e c t i o n and have i t s o r i g i n on the p l a n e . These r e s t r i c t i o n s a r i s e from the f a c t t h a t no two s p i n s o f o p p o s i t e sense can be a t t h e same p o i n t i n t h e c r y s t a l . Thus t h e y do not h o l d any l o n g e r i f the primed p r o p e r r o t a t i o n , t h e p rimed space i n v e r s i o n and t h e primed r e f l e c t i o n mentioned above a r e f o l l o w e d by n o n - p r i m i t i v e t r a n s l a t i o n s . ( 5 . 1 . 7 ) A s p i n o f a c e r t a i n d i r e c t i o n and sense i s t r a n s f o r m e d i n t o a n o t h e r s p i n o f t h e same d i r e c t i o n and sense under a primed p r o p e r o r improper r o t a t i o n o f o r d e r 2 about an a x i s p e r p e n d i c u l a r t o the d i r e c t i o n o f the s p i n . ( T h i s f o l l o w s from ( 5 . 1 . 1 ) and ( 5 . 1 . 4 ) J . ( 5 . 1 . 8 ) I n p a r t i c u l a r , i f a s p i n has i t s o r i g i n on an a x i s of a primed p r o p e r o r improper r o t a t i o n of o r d e r 2 i t i s l e f t i n v a r i a n t under the r o t a t i o n . An i l l u m i n a t i n g g r a p h i c a l r e p r e s e n t a t i o n of the above r u l e s has been g i v e n by Donnay, C o r l i s s , Donnay, E l l i o t t and H a s t i n g s ( 1 9 5 8 ) i n F i g . 1 o f t h e i r paper. An e s s e n t i a l p a r t o f the c o n t e n t s of r u l e s ( 5 . 1 . 2 ) -( 5 . 1 . 8 ) can be f o r m u l a t e d as f o l l o w s : ( 5 . 1 . 9 ) The d i r e c t i o n and sense o f a s p i n a r e l e f t 0v,ly i n v a r i a n t under the group .£2. _L- and hence under any o f A /VWv /^J W i t s s ubgroups. Here ^ . _ i — i - s t a n d s f o r any r o t a t i o n about i l l an a x i s p a r a l l e l t o t h e d i r e c t i o n of the s p i n , 1—%- s t a n d s / v w 1 1 103 f o r a r e f l e c t i o n i n a p l a n e p e r p e n d i c u l a r t o the d i r e c t i o n o f the s p i n , -L iL 4- and - i _ J - f o r two primed r o t a t i o n s 1 1 1 1 1 1 1 o f o r d e r 2 about two axes p e r p e n d i c u l a r t o the d i r e c t i o n o f t h e s p i n and t o each o t h e r r e s p e c t i v e l y . J_ _L X and J—'L J_. f o r p r i m e d r e f l e c t i o n s i n two p l a n e s p e r p e n d i c u l a r t o the two p r e v i o u s l y d e f i n e d a x e s . C o n s i d e r now a magnetic c r y s t a l b o t h as an arrangement o f s p i n s and an arrangement o f atoms. (5.1.10) We s h a l l s a y t h a t a magnetic c r y s t a l i s i n v a r i a n t under an MSG M i f i t i s i n d i s t i n g u i s h a b l e b o t h as an arrangement o f atoms and an arrangement o f s p i n s from t h e c r y s t a l o b t a i n e d from i t by a p p l y i n g a l l unprimed and primed elements o f M. (5.1.11) I f a magnetic c r y s t a l as an arrangement o f atoms i s i n v a r i a n t under a space group F, t h e n , as an arrangement o f s p i n s , i t i s e i t h e r i n v a r i a n t under an MSG o f the f a m i l y o f F, i n p a r t i c u l a r under F i t s e l f , o r under a member o f t h e f a m i l y of a subgroup o f F. (5.1.12) C o n v e r s e l y , i f a magnetic c r y s t a l as an arrangement o f s p i n s i s i n v a r i a n t under an MSG M i t i s i n v a r i a n t , as arrangement o f atoms, under t h e space group F t o whose f a m i l y M b e l o n g s . (The s t a t e m e n t i s t r i v i a l i f M c o i n c i d e s w i t h F, and i s e a s i l y p r o v e d i n o t h e r c a s e s i f one remembers the way i n which an MSG M(F) i s d e r i v e d from a space group F j . We a r e now i n a p o s i t i o n t o d i s c u s s the MSGs i n 104 t h e i r r e l a t i o n t o magnetic c r y s t a l s . F i r s t o f a l l we s h a l l d i s t i n g u i s h between MSGs w h i c h can l e a v e f e r r o m a g n e t i c c r y s t a l s i n v a r i a n t and MSGs whi c h c a n n o t . We s h a l l c a l l them " f e r r o -m a g n e t i c " and " n o n - f e r r o m a g n e t i c " space groups r e s p e c t i v e l y . More p r e c i s e l y : (5.1.13) We s h a l l s a y t h a t an MSG M i s " f e r r o m a g n e t i c " i f t h e r e e x i s t s a t l e a s t one d i r e c t i o n s u c h t h a t a s p i n arrangement i n w h i c h a l l the s p i n s have t h a t d i r e c t i o n and a g i v e n sense i s i n v a r i a n t under a l l primed and unprimed elements o f M. As f e r r o m a g n e t i c c r y s t a l s have a non-zero m a c r o s c o p i c m a g n e t i c moment i n some w e l l - d e f i n e d d i r e c t i o n , t he magnetic p o i n t group b e l o n g i n g t o an MSG M which l e a v e s a f e r r o m a g n e t i c c r y s t a l i n v a r i a n t has t o be, a c c o r d i n g t o ( 5 . 1 . 9 ) , a subgroup o f 22_ -L- - i - There a r e 31 o f such subgroups and t h e i r l i s t i s g i v e n a t the end o f t h i s s e c t i o n . There we a l s o i n d i c a t e t h e d i r e c t i o n t h a t a s p i n must have t o be t r a n s f o r m e d i n t o a s p i n o f the same d i r e c t i o n and sense under each o f t h e 31 subgroups o f %L, — O b v i o u s l y the t r a n s l a t i o n s o f an MSG M which l e a v e s a f e r r o m a g n e t i c c r y s t a l i n v a r i a n t must be o r d i n a r y t r a n s l a t i o n s , and hence M i s e i t h e r an o r d i n a r y space group o r an MSG o f the k i n d M^. Hence: (5.1.14) An MSG M i s f e r r o m a g n e t i c i f and o n l y i f i t s p o i n t group i s a subgroup o f sss. JJL _±L . and i f i t i s e i t h e r an o r d i n a r y space group o r o f the k i n d M^. 105 There a r e 275 f e r r o m a g n e t i c space groups. They a r e marked w i t h an a s t e r i s k i n t h e l i s t o f MSGs a t t h e end o f t h i s t h e s i s . We have c o n s i d e r e d so f a r o n l y f e r r o m a g n e t i c s p i n arrangements. We want now t o make a few remarks about f e r r i m a g n e t i c and a n t i f e r r o m a g n e t i c s p i n arrangements. A l l t h e s e remarks a r e e s s e n t i a l l y based on the f a c t t h a t a c c o r d i n g t o (5.1.^5) the s p i n s can o n l y occupy t h o s e s i t e s i n a c r y s t a l whose symmetry groups a r e subgroups o f oo 1 / ?' . I n v a r i a n t f e r r i m a g n e t i c s p i n arrangements a r e o f c o u r s e c o m p a t i b l e w i t h f e r r o m a g n e t i c space groups o n l y . And, c o n v e r s e l y , f o r each o f the 275 f e r r o m a g n e t i c space groups t h e r e e x i s t s an i n v a r i a n t f e r r i m a g n e t i c s p i n arrangement. However, i n view o f ( 5 . 1 . 1 4 ) , t h e s e f e r r i m a g n e t i c arrangements cannot be o f a c o l l i n e a r type i f t h e r e i s o n l y one s p i n per u n i t c e l l . I n o r d e r t o have c o l l i n e a r t y pe o f f e r r i m a g n e t i c arrangements t h e r e must be a t l e a s t two s p i n s per u n i t c e l l . I n a n o n - c o l l i n e a r f e r r i m a g n e t i c c r y s t a l s p i n s cannot occupy p o s i t i o n s whose symmetry group i s d i f f e r e n t from 1, 1, 2', m», 2 ,/m ,, 2,mra'. I n f a c t , i f the symmetry o f a s i t e where a s p i n i s s i t u a t e d i s a subgroup o f -eo. JLL iJ. / d i f f e r e n t from the groups j u s t enumerated, the s p i n can o n l y be i n one f i x e d d i r e c t i o n w h i c h i s l e f t unchanged by the magnetic p o i n t group of the f e r r o m a g n e t i c space group b e i n g c o n s i d e r e d . 106 I n p a r t i c u l a r i f the p o i n t group b e l o n g i n g t o t h e space group o f a c r y s t a l c o n t a i n s r o t a t i o n s o f o r d e r h i g h e r t h a n 2, s p i n s cannot occupy s i t e s whose symmetry groups c o n t a i n elements o f o r d e r h i g h e r than 2. I t seems t h a t atoms a r e p r e f e r e n t i a l l y s i t u a t e d a t s i t e s o f h i g h e r symmetry i n a c r y s t a l , i t f o l l o w s t h a t the f e r r o m a g n e t i c space group w h i c h l e a v e s a n o n - c o l l i n e a r f e r r i m a g n e t i c c r y s t a l i n v a r i a n t c a nnot c o n t a i n elements o f o r d e r h i g h e r than 2. E v e r y MSG l e a v e s some a n t i f e r r o m a g n e t i c arrangements o f s p i n s i n v a r i a n t . I f an MSG i s o f the k i n d M T but not f e r r o m a g n e t i c then an a n t i f e r r o m a g n e t i c s p i n arrangement l e f t i n v a r i a n t by i t must be n o n - c o l l i n e a r . T u r n i n g now t o the MSGs o f the k i n d M R, we have t o d i s t i n g u i s h here between t h o s e MSGs M R whose p o i n t group i s a subgroup o f _ L l iL w i t h no primed e l e m e n t s , and the r e m a i n i n g MSGs M R. I n the former case o n l y c o l l i n e a r a n t i f e r r o m a g n e t i c s p i n arrangements a r e p o s s i b l e ; i n the l a t t e r c a s e o n l y n o n - c o l l i n e a r ones. These s t a t e m e n t s presuppose t h a t t h e r e i s o n l y one s p i n per u n i t c e l l o r , more p r e c i s e l y , t h a t the whole s p i n arrangement can be o b t a i n e d by a p p l y i n g a l l the o p e r a t i o n s o f the MSG t o a s i n g l e s p i n . 107 L i s t o f the 31 subgroups o f A-^ ^ W y /yv\J /Vw' A l l o w e d d i r e c t i o n o f s p i n s 1 1 Any d i r e c t i o n 2 1 Any d i r e c t i o n 3 2 A l o n g the a x i s 4 2' P e r p e n d i c u l a r t o the a x i s 5 ro P e r p e n d i c u l a r t o the p l a n e 6 m' Any d i r e c t i o n i n the p l a n e 7 2/m A l o n g the a x i s 8 2'/m' P e r p e n d i c u l a r t o t h e a x i s 9 22'2' A l o n g the unprimed a x i s 10 2m'm' A l o n g the a x i s 11 2'mm' P e r p e n d i c u l a r t o the a x i s 12 mm'ra' P e r p e n d i c u l a r t o t h e unprimed p l a n e 13 4 A l o n g the a x i s o f h i g h e r o r d e r 14 42' " 15 4/m " 16 4m,m' » 17 4/mm,m, " 18 4° 19 4 2 ^ * 20 3 " 21 32' 108 22 3ra' A l o n g the a x i s o f h i g h e r o r d e r 23 3 24 3m 1 " 25 6 26 ^ ' 2 ' 27 6 28 62» " 29 6/m 30 era'm' 31 e/mm'ra' " SECTION 5.2 I n t h i s s e c t i o n we make some g e n e r a l remarks c o n c e r n i n g the d e t e r m i n a t i o n o f the MSGs o f a magnetic c r y s t a l assuming t h a t i t s s t r u c t u r e and the arrangement of i t s s p i n s a r e known. As has a l r e a d y been s t a t e d , the symmetry of the s i t e s o c c u p i e d by magnetic atoms must be such t h a t a s p i n l o c a t e d i n any of t h o s e s i t e s i s l e f t unchanged by the elements of the symmetry groups o f the s i t e s . Hence, as we have seen b e f o r e , the symmetry groups o f the s i t e s o c c u p i e d by magnetic atoms must be subgroups o f _ss2_ 2-' j~L . I t f o l l o w s t h a t i f the magnetic atoms o f a c r y s t a l i n v a r i a n t under a space group F occupy s i t e s whose symmetry groups a r e p o i n t groups whose f a m i l i e s c o n t a i n subgroups o f jQ£2_ 2J- the MSG which l e a v e s t h e magnetic c r y s t a l 109 i n v a r i a n t may b e l o n g t o the f a m i l y of Fj i n the o p p o s i t e case i t must n e c e s s a r i l y b e l o n g t o t h e f a m i l y o f a subgroup of F. I n o r d e r t o accommodate the magnetic atoms i n t h e i r s i t e s , t h i s subgroup of F must have some s p e c i a l s i t e s i n common w i t h F, and t h e s e s i t e s must have symmetry groups which s a t i s f y a g a i n the c o n d i t i o n s j u s t f o r m u l a t e d . T h i s subgroup of F may or may not b e l o n g t o the same system as F. I f i t does th e n the magnetic and the o r d i n a r y l a t t i c e must have the same h o l o h e d r y . I f i t does not t h e n the two l a t t i c e s have d i f f e r e n t h o l o h e d r y . C o n s i d e r , f o r example, a c r y s t a l whose space group be l o n g s t o t h e c u b i c system. Suppose i t i s f e r r o m a g n e t i c . There a r e no f e r r o m a g n e t i c space groups b e l o n g i n g t o the c u b i c system. Thus the MSG o f the c r y s t a l must b e l o n g e i t h e r t o t h e rhombohedral, t e t r a g o n a l , o r t h o r h o m b i c ^ m o n o c l i n i c o r t r i c l i n i c system and the magnetic l a t t i c e does not b e l o n g t o the c u b i c system. S i m i l a r c o n s i d e r a t i o n s a p p l y t o the case o f f e r r i m a g n e t i c and a n t i f e r r o m a g n e t i c c r y s t a l s . I n p a r t i c u l a r , i n the case o f a n t i f e r r o m a g n e t i c c r y s t a l s i t i s p o s s i b l e t o e x c l u d e many MSGs as i n c o m p a t i b l e w i t h a g i v e n arrangement o f s p i n s on the b a s i s o f g e n e r a l r e l a t i o n s h i p s between s p i n arrangements and MSGs d e s c r i b e d a t the end o f S e c t i o n 5.1. 110 As an example o f what has been s a i d i n t h i s c h a p t e r we s h a l l b r i e f l y d i s c u s s the magnetic s t r u c t u r e o f MnO. MnO i s an a n t i f e r r o m a g n e t i c c r y s t a l . We assume t h a t i t s space group i s Fra3m, ( t h e r e i s some doubt about the c o r r e c t n e s s o f t h i s a s s u m p t i o n ) . The Mn atoms a r e i n the s i t e s The group m3m i s not a subgroup o f — _lL nor does any such subgroup b e l o n g t o the f a m i l y o f m3ro. I t f o l l o w s t h a t t h e MSG o f t h e c r y s t a l must b e l o n g t o the f a m i l y o f a subgroup o f Fm3m. The l a t t i c e o f Fm3m i s a f a c e - c e n t e r e d c u b i c l a t t i c e , t h u s the n o n - p r i m i t i v e p a r a l l e l e p i p e d u n i t c e l l o f t h e c r y s t a l • i s a cube. From n e u t r o n d i f f r a c t i o n e x p e r i m e n t s i t i s known t h a t p a r a l l e l s p i n s l i e i n ( l l l ) - p l a n e s o f the c u b i c l a t t i c e , and t h a t s p i n s l y i n g i n two c o n s e c u t i v e ( l l l ) - p l a n e s a r e a n t i p a r a l l e l ; however the d i r e c t i o n o f the s p i n s i n the p l a n e s i s not known. As the magnetic u n i t c e l l o f MnO one u s u a l l y t a k e s a cube whose volume i s e i g h t t i m e s t h a t o f the c u b i c u n i t c e l l d e f i n e d above. I f £ i s the edge o f the p a r a l l e l e p i p e d u n i t c e l l o f Fm3m, 2 t i s the edge o f the magnetic p a r a l l e l e p i p e d u n i t c e l l . One can e a s i l y c o n v i n c e o n e s e l f t h a t t h i s arrangement o f s p i n s i s c o m p a t i b l e w i t h t h e magnetic l a t t i c e C c of the m o n o c l i n i c s y s t e m ; the C f a c e b e i n g p a r a l l e l t o the ( l l l ) - p l a n e and the c - a x i s b e i n g a l o n g the d i a g o n a l [1101. I l l I n t he new l a t t i c e C the magnetic atoms a r e a t Q0O,0O^ ,^4° ' T I T There a r e two MSGs w i t h l a t t i c e C c and t h e f o u r p o s i t i o n s j u s t l i s t e d . They a r e C 2/m and C 2/c. I f a,b,c a r e t h e t h r e e edges o f the p a r a l l e l e p i p e d v u n i t c e l l o f C, C c2/m c o n s i s t s o f ( E K ^ + ^ * 2 ^ ) i ( C a y | 0 ) y ( l | 0 ) , ( f | c / whereas C Q2/c c o n s i s t s o f In C 2/m the symmetry group o f t h e s i t e s of the Mn c atoms i s 2/m, i n C c2/c i t i s 2,/m'. I t f o l l o w s t h a t i f s p i n s i n the ( l l l ) - p l a n e a r e p a r a l l e l t o the b - d i r e c t i o n t he MSG i s C c2/m. I f s p i n s a r e p a r a l l e l t o the a - d i r e c t i o n then the MSG i s C_2/c. S h o u l d s p i n s be i n n e i t h e r o f th e s e d i r e c t i o n s c t h e MSG would be P ST, t h a t i s , an MSG o f the t r i c l i n i c system, NOTES TO CHAPTER 5 The l i s t of the 31 subgroups o f J2S. XL 2L was f i r s t /Vw OW /V*\> g i v e n by Tavger (1958). The l i s t o f the 275 f e r r o m a g n e t i c groups was f i r s t g i v e n by Neronova and B e l o v (1960). 112 LIST OF SYMBOLS i— g e n e r a l inhomogeneous l i n e a r group F space group, o r c l a s s o f e q u i v a l e n t space groups A a b s t r a c t group o f two elements E ( i d e n t i t y ) and E'; a l s o t h e t i m e - r e v e r s a l group. rr 1 subgroup o f i n d e x 2 (due) o f t h e group H D = D*" subgroup o f i n d e x 2 o f a space group F D.j. D*" w h i c h has the same l a t t i c e T as F D R which has t h e same p o i n t group R as F D R o and D R f l two k i n d s o f D R T l a t t i c e ( t h a t i s , group of p r i m i t i v e t r a n s l a t i o n s ) T D = D1, R p o i n t group RD = 5R OT subgroup o f i n d e x 4 ( q u a t t r o ) o f a symmorphic 0Tr subgroup o f i n d e x 4 o f a non-symmorphic F Q r o and Q r a two k i n d s o f Q r M magnetic space group, or c l a s s o f magnetic space groups M(D) magnetic space group whose unprimed elements form the group 8 M"T = M"(Drp) STR - M(D R) % o = W R O ) % a = M ( D R a ) T"M magnetic l a t t i c e o f M S M magnetic p o i n t group o f &" 113 LIST OF MAGNETIC SPACE GROUPS (MSGs o f the or t h o r h o r a b i c system a r e not i n c l u d e d . ) N o t a t i o n and arrangement a r e e x p l a i n e d i n C h a p t e r 4. T r i c l i n i c C a2 P am system *P1 P a 2 Pv,2 P b 2 1 P bm •PI C am *P2 *P2^ T P 2 *Pc a i *P2* m *Pm *Pm' *Pc P I ' *C2 P ST *C2' C c 2 M o n o c l i n i c s ystem PQ2 2 p C 2 l *Cm *P2 p a c P b c •Cm' C cm P cm *Cc •Cc' C c c P C C *P2/m P 2 V m P2/m t *P2,/m* P a2/m P b2/m C a2/m P b2 1/m P c 2 / c *P2 1/m P2-I/m P 2 1 / r a , *P2^/m' Pa 2l/» P c 2 1 / c *C2/m C2'/m C2/IB , Cc2/m P c2/ra P c2 1/m P A 2 / c P A 2 1 / C •P2/c P2'/c P2/c' *P2'/c 1 P a 2 / c P b 2 / c C a2/c P b 2 l / c *P2 1/c P2^/c P 2 1 / c ' *P2*/c• PaV c •C2/c C2»/c C2/c' *C2 f/c * C c2/c P c 2 / c P c 2 1 / c P c4 P c 4 3 P C 4 T e t r a g o n a l _ system I c 4 *P4 3 * P 4 1 C ' l *p4 P 4 1 *l¥ 3 P4' P c 4 3 14' P c 4 P j 4 *I4 V 14' 4/m 1 4 c *P4/m P j 4 PC4 2 c 2 P4'/m P I 4 2 P4/m t * I 4 1 P4'/ra' 14' P c 4 / m P„4, 1 P I 4 1 P C 4 / r a * p 4 2 p I 4 3 ^Y" P 4 2 P c4 2/m 4 P 4, P r 4 / n c 1 * p 4- C •P42/m P4 2/m P4 2/m f P4 2/m* P c4 2/n» •p4/n P4'/n P4/n' P4 Vn» P c4/n P c 4 2 / n *P4 2/n P 4 2 / n P 4 2 / n t P4 2/n» I c V a • I 4/m 14'/ra 14/m* H'/m1 Pj4/m P j 4 2 / m P j 4 / n P j 4 2 / n l 4 ' / a 1 1 4 ^ ' 14^/a' 422 P422 P4'22' *P42»2 * 116 P4»2'2 P c422 P c422 I c 4 2 2 P C 4 2 2 2 V 2 ! 2 P42 12 P4'2 ±2» *P42^2» P4*2*2 P c 4 2 l 2 P c 4 2 2 l 2 P 4 X 2 2 P4^22» * P 4 ] 2 , 2 ' P4*2 ,2 1 V i 2 2 pc 4i 2 2 P 4 ^ 2 1 2 ' * P 4 1 2 , 2 ' P4^2^2 P 4 2 2 2 P 4 2 2 2 ' * P 4 2 2 , 2 ' P 4 22'2 P c 4 l 2 2 P c 4 2 2 2 P C 4 2 2 1 2 P c 4 3 2 2 I c 4 l 2 2 P 4 2 2 l 2 p 422 l 2 ' •P42212* P4 22^2 P c 4 l 2 l 2  P c 4 3 2 l 2 P4g22 P4^22' *P4 32'2' P4^2'2 P c 4 3 2 2 P C 4 3 2 1 2 P 4 3 2 l 2 P 432 l 2 ' * p 4 3 2 l 2 ' P 4 3 2 l 2 1422 117 I 4 ' 2 2 f *I42V 14 2 2 Pj 4 2 2 P I 4 2 1 2 P j 4 2 2 2 P I 4 2 2 1 2 I 4 1 2 2 I 4 J 2 2 ' • 1 4 ^ ' 2 ' . t 1 I 4 1 2 2 P!4 X22 P I 4 1 2 1 2 P j 4 3 2 2 P I 4 3 2 1 2 4mm P4mra P4 m m . » i P4 mm t i *p4m m Pc4mm P^ ,4mm I c4mm P c4bm P c 4 2 c m P c 4 c c P c4 2mc I c4cm P4bm p4 b m p4'bm * *p4b'm* P c4bm P c4 2nm P c 4 n c P 4 9 b c P4 2cm i « P 4 2 c m P4 2cm' *p4 2c m P c 4 2 c m P c4 2nm P4 2nm P 4 2 n m M » t P4 2nm f » *P4 2n m P4cc t i P4 c c i » P4 cc i i *P4c c 118 P c 4 c c P 4nc p4nc . ' t P4 n c A % » P4 nc i t *p4n c P4 2mc .» » P 4 2 m c . t » P4 2mc *p4 2m c P c4 2mc P c 4 2 b c I 4,md c I c 4 l C d P 4 2 b c P4*b'c P 4 2 b c ' I ^md P c42Q *p4 2b'c' l ^ m ' d P c 4 2 l m i i 1 4 ^ 14mm , , P 4 2 c *I4 m d I4 ,m ,m P4 2 c 14 mm I 4 1 c d P4 2c *l4m m I 4 1 c d *P42 c Pj4mm l 4 i c d ' P c 4 2 c P T4 Qnm *I4 c V I 42d I « X c P j 4 n c p C 4 2 i c P_4 Qmc / 1 2 P j 4 2 c m P j 4 c c 42m P42 ±m P42ro —t t I4cm p 4 2i™ , . P 4 V m , 14 c m P 4 P4'2m' — , , 14'cm* * P 4 2 i m *P4~2' m' — • I ^ c V p c 4 2 l m P 42m _ Pj4bm p c 4 2 l c P c42m I c42m P 4 2 1 c — « I P4 2 x c —t t P4 2 1 c •P42 - J C P4m2 t P4 m 2 —t i P4 m2 *P4m'2' P c4m2 P c4m2 I c4m2 P c 4 c 2 P c4b2 P4c2 —' i P4 c 2 P4" ,c2 t •P4c'2 * P c 4 c 2 P c4n2 P4b2 — i i P4 b 2 _ t t P4 b2 — i i *p4b 2 P 4b2 c P 4n2 c I 4c2 c P4n2 — i i P4 n2 *P4 n' 2' I4m2 14* 10*2 —t i 14 m2 *l4m'2' 120 P][4m2 Pj4n2 I4c2 —t t 14 c 2 — i » 14 C 2 T ' ' * l 4 c 2 P j 4 c 2 P ] [4b2 I42m —t » 14 2 m —t t 14 2m *l4Vm' P j 4 2 c P j ^ j m F I 4 2 1 C I42d l4Vd — i i 14 2d * l 4 V d ' 4/mmm p4/mmm p4/m'mm P4'/mm'm p4'/mmm, p4'/m,m'm *p4/mm ra * * t P4 /m mm • i i p4/m m m Pc4/mmm P^4/mmm I 4/mmm c P 4/racc P 4/nbm PQ4/mbm Pc4/nmm P c4 2/mmc P c4 2/mcm p4/mcc p4/m'cc P4 1/mc'c P4 /mcc' P4 /m c c *P4/mc 1c' i t t P4 /m c c • t • p4/m c c P c4/mcc I c4/mcm P 4/nnc Pr,4/mnc 121 P c 4 / n c c P4/nbm i p4/n bm p4'/nb'm t t p4 /nbm t » » P4 /n b m t • *p4/nb m p4 /n bm t t • p4/n b m P 4/nbm c P c4/nnc p4/nnc P4/n'nc P4 /nn c P4 /nnc P4 /n n c *p4/nn'c' ~A * / ' ' P4 /n nc p4 / n ' n ' c ' p4/mbm p4/m'bm p4'/mb'm t t P4 /mbm » t • P4 /m b m '*p4/mb,m' P4 /m bm p4/m b m Pc4/mbm P c4/mnc p c 4 2 / n b c P c4 2/mbc P c4 2/nnm P c4 2/mnm p4/mnc p4/m'nc P4 1/mn'c t i P4 /mnc A » / « * P4 /m n c t t *p4/mn c t i « P4 /m nc p4/m'n'c' p4/nmm i p4/n mm i i p4/nm m i t p4 /nmm t ' • p4 /n m m *p4/nm m P4 /n mm p4/n m m Pc4/nmm 122 P 4/ncc c P c4 2/nmc P 4 0/ncm c * p4/ncc t p4/n cc P4'/nc'c p4'/ncc' t t i p4 /n c c t t *p4/nc c t , t P4 /n cc • i t p4/n c c p42/mmc i p4 2/m mc t t p4 /mm c p42/mmc t t i p4 2/m m c *P42/mm * c * P42/m mc A , » ' ' P4 2/m m c Pc4 2/mmc p C 4 2 / m b c I c4 i/amd I c 4 1 / a c d P c 4 2 / n b c p4 2/mcm P4 2/m*cm P4 2/mc'm P4 2/mcm' P4 2/m ,c*m *P42/mc'm' p4 2/m cm P42/m*c ,m t Pc4 2/mcm Pc42/mnm P c4 2/nnm P c 4 2/nmc P C 4 2 / n C m P 4 2/nbc i P4 2/n be p4 2/nb c » , P4 2/nbc » t i P4 /n b c * P 4 2 / n b , c ' t t p 4 2 / n be t , t p 4 2 / n b c P4 2/nnm P 4 2 / n *nm p 4 2 / n n m P4l/nnm* 123 A ' / * « p 4 2 / n n m t i *P4 2/nn m p4 2/n'nm' p 4 2 / n n m p4 2/mbc t p4 2/m be p4 2/mb'c t t p4 2/mbc P4 f/m ,b*c 2 t t *p4 2/mb c P4 2/m be » i i p4 2/m b c p42/mnm p42/m'nm t i P4 /mn m t i P4 /mnm „' / ' • P4 2 /m n m *p4 2 /mn'm' P42/m'nm' A , 1 » ' p4 2 /m n m p42/nmc P4g/n'mc p42/nm f c p42/nmc' p42/n m'c * p 4 2 / n m V p42/n'IDC , P 4 2 / n , m , c ' p42/ncm t P 4 2 / n cm p42/nc 'm p42/ncm' p 4 0 / n c m *p4 2 /nc 'm p 4 2 / n cm i » » p 4 2 / n c m I4/mmm i I4/m mm 14 /mm m t t I4 /mmm i i » 14 /m m m t t *l4/mm m i « i 14 /m mm 1 4 / m ' m V Pj4/mmm Pj4/nnc Pj4/mnc Pj4/nmm Pj4 2 /nnm P,40/mnm 124 P ^ / n m c 14/mcm t I4/m cm l4*/mc 'm I4'/mem 14 /m c m » i *l4/mc m 14 /m cm • t t 14/m c m Pj4/mcc Pj4/nbm Pj4/mbm Pj4 / n c c Pj4 2 /mmc Pj42/mcm P j 4 2 / n b c Pj4 2 /mbc Pj4 2 / n c m I4 1/amd T r i g o n a l *R3 system * — i 14 /a md R3 A 3 l4J/am*d *P3 R j 3 14^/amd' P c 3 14 / a m d 1 P312 i * p 3 n *l4../am d' £ x *P312 ^ / a ' m d ' **** i * * 14^/a m d I 4 i / a ' c d 14^/ac d 14^/a'cd' 14.^/a c d * P 3 2 —t P3 P c312 P321 P c 3 l 14 /acd c 1 — ± *P32 1 *R3 P c 3 2 1 R j 3 I4 1/acd P3J12 iV/a'c'd 3* *P3 112' *P3 * l 4 1 / a c , d * P c 3 2 1 2 . . . P c 3 P 3 1 2 1 P c 3 2 2 1 P 3 2 1 2 *P3 212' P C 3 X 1 2 P 3 2 2 1 * P 3 2 2 ' l P c 3 ! 2 1 R32 *R32' Rj 3 2 P3ml *P3m'l P c 3 m l P c 3 c l P31m *P31m' P c31m P c 3 1 c R3m *R3m' Rj3m R j 3 c R3c *R3c' P31ro P3 ' l m p F l m ' *P3~lm' P„31m 126 P c 3 1 c P31c p S ' l c —t t P3 l c *P31c * P3ml P3 - ,ml W ' B ' I •PSm'l P c 3 m l P c 3 c l P 3 c l P3"'cl P ^ ' c ' l *P3c»1 127 R3m *P6i 6/m t i R3'm P6 — i t R3 m *R3m Rj3m R T 3 c I R3c R3'c —t t R3 c *R3c Hexagonal system P6* '5 *P62 P 6 2 P c 6 5 P c 6 3 P 6 ' p c 6 *P6/m P6'/m *P6c P6/m' P6* P6'/m' P 6/m c P 6„/m c 3 P c 6 1 *P6 3/m P c 6 4 p 6 3 / m - , P63/m' P63/m P64 P c 6 2 622 6 P622 *P6 P6V2 ^ P6'22* *P62*2' P 622 c P 6o22 C «5 P 6 X 2 2 P6]2'2 P6^22' * P 6 1 2 , 2 ' P 6 5 2 2 P 6 g 2 2 P6g22* *P6g2'2 * P 6 2 2 2 P6 22'2 P 6 2 2 2 ' *P6 22'2' P c 6 l 2 2 P c 6 4 2 2 P6 422 P642'2 P6 422' *P6 42'2' P c 6 2 2 2 P c 6 5 2 2 P6 322 P 6 3 2 ' 2 i pe'22* 3 *P6 32 ,2* 6mm P6mm „ i i P6 m m t * P6 mm *P6m'm' Pc6mm 128 P c 6 c c P c6 3cm P 6.,mc c o P6cc P6 c c t i P6 cc *P6c c P6 cm o i i P6gC m P6 3cm *P63c'm' P6 3mc » i P6 3m c P6 3mc * •Pe^m'c* P6ro2 P 6 " V 2 *P6va12 t P c6m2 P c 6 c 2 P6c2 P6'c'2 P6*c2' *P6c'2' P62m pie^'m p e^m' *p"62 'TO' P c?2m P c 6 2 c P62c — t 1 P6 2 c P6*2c' *P6 2'c' 6/mmm P6/mmm P6/m ram P6 /mm m P6'/mmm' pe'/m'm ro P6 /m mm *P6/mm m t I I P6/m m m P„6/mmm P c6/mcc P c6 3/rocm P c6 3/mmc 129 P6/mcc P6/m c c 1 1 P6 /mc m P6*/nicc' P6 /m c c P6 /m cc *P6/mc'c' „ , ' » » P6/m c c P6 /mem P6 3/m'cra P63/mc'm P6 3/mcm' P6g/m c m P6g/m cm *P6 3/mc m' P6 3/m'c * m 130 P63/mmc P 2 1 3 Im3 P63/m*mc Im'3 . , I 2 1 3 P6 /mm c P m3 •J I p I 2 i 3 P63/mmc A P6 3/m m c m3 P6 3/ra mc' Pm3 *P63/mm * c' Pm'3 P6 3/m m c F s m 3 C u b i c Pn3 system Pn'3 P23 F s d 3 Fm3 F23 Fm*3 123 Fd3 P r 2 3 Fd'3 Pa3 P a f 3 Ia3 Ia'3 P.^3 432 F s 2 3 P432 t i P4 32 F s 4 3 2 P 4 2 3 2 P 4 2 3 2 F 4,32 s A F432 F4'32' F 4 x 3 2 „ t t F 4 1 3 2 1432 I 4 , 3 2 ' P j 4 3 2 P j 4 2 3 2 P4 g32 P 4 g 3 2 P 4 1 3 2 P 4 i 3 2 * 1 4 ^ 2 I 4 J 3 2 ' P I 4 1 3 2 P I 4 3 3 2 P43m pl ' sm* F s43m F g 4 3 c F43m FT' 3m' I43m I4'3m f P ^ m P T 4 3 n P43n P4*3n' F43c F 4 ' 3 C ' 143 d 14*3d* m3m Pm3m Pm*3m Pm3m' Pm'sm* Fgm3m F_m3c s Pn3n Pn*3n Pn3n i t Pn 3n Pm3n Pm'3n Pm3n' Pm'sn' Pn3m t Pn 3m Pn3m' i i Pn 3m F sd3m F s d 3 c Fm3m » Fm 3m Fm3m' i t Fm 3m Fm3c Fm'3c Fm3c' t t Fm 3c Fd3m i Fd 3m Fd3m' » t Fd 3m Fd3c Fd'3c Fd3c' • t Fd 3c Im3m t Im 3m Im3m' « t Im 3m 132 P m3m I P j n 3 n Pj.m3n Pjn3m Ia3d l a ' 3 d I a 3 d ' I a ^ d ' 133 BIBLIOGRAPHY A l e x a n d e r , E. and Herrmann, K., (1928) Z . K r i s t a l l o g r . 69. 295. A l e x a n d e r , E. and Herrmann, K., (1929) Z. K r i s t a l l o g r . 70. 328. B e l o v , N.V., Neronova, N.N. and Smirnova, T.S., (1955) Trudy I n s t . K r i s t . Akad. Nauk. S.S.S.R. 11, 3 3 . B e l o v , N.V., Neronova, N.N. and Smirnova, T.S., (1957) K r i s t a l l o g r a f i y a 2, 3. E n g l i s h t r a n s l a t i o n : (1957) S o v i e t Phys. C r y s t . J2, 311. B i e n e n s t o c k , A. and Ewald, P.P., (1962) A c t a C r y s t . 15, 1253. Oimmock, J . 0. and Wheeler, R.G., (1962a) J . P h y s . Chem. S o l i d s 23, 729. Diramock, J . 0. and Wheeler, R.G., (1962b) Phys. Rev. 127. 391. Donnay, J . D. H., (1961) A c t a C r y s t . 14. Donnay, G., C o r l i s s , L.M., Donnay, J.D.H., E l l i o t t , N., and H a s t i n g s , J.M., (1958) Phys. Rev. 112. 1917. D z i a l o s h i n s k i i , I.E.,(1957) Zh. eksp. t e . F i z . S.S.S.R. 32, 1547, E n g l i s h t r a n s l a t i o n : (1957) J . Exp. Theor. Phys. U.S.S.R. 5_, 1259. Heesch, H., (1929) Z. K r i s t a l l o g r . 71, 95. Heesch, H., (1930) Z. K r i s t a l l o g r . 73, 325. Hermann, C., (1929) Z. K r i s t a l l o g r . 69, 546. Indenbom, V.L., (1959) K r i s t a l l o g r a f i y a 4, 619. E n g l i s h t r a n s l a t i o n : (1959) S o v i e t Phys. C r y s t . 4_, 578. I n t e r n a t i o n a l T a b l e s f o r X-Ray C r y s t a l l o g r a p h y , (1952) v o l . 1 . ) Kynoch P r e s s , Birmingham. 134 K o s t e r , G.F. ( 1 9 5 7 ) S o l i d S t a t e P h y s i c s , j>, Academic P r e s s , New York. Landau, L.L., and L i f s c h i t z , E.M., ( 1 9 5 1 ) S t a t i s t i c a l P h y s i c s GITTL ( S t a t e Tech. L i t . P r e s s ) . E n g l i s h t r a n s l a t i o n : ( 1 9 5 8 ) S t a t i s t i c a l P h y s i c s , Addison-Wesley, R e a d i n g , Mass. Le C o r r e , Y., ( 1 9 5 8 ) J . Phys. Radium 1 9 , 7 5 0 . Lomont, J.S., (1959) A p p l i c a t i o n s o f F i n i t e Groups, Academic P r e s s , New York. Neronova, N.N. and B e l o v , N.V., ( 1 9 5 9 ) K r i s t a l l o g r a f i y a £, 8 0 7 . E n g l i s h t r a n s l a t i o n : ( 1 9 6 0 ) S o v i e t Phys. C r y s t . 4_, 7 6 9 , N i g g l i , A., ( 1 9 5 9 ) Z. K r i s t a l l o g r . I l l , 2 8 8 . R i e d e l , E.P. and Spence s R.D., ( 1 9 6 0 ) P h y s i c a 2 6 , 1 1 7 4 . S h u b n i k o v , A.V. ( 1 9 5 1 ) The Symmetry and Antisymmetry o f F i n i t e F i g u r e s i n R u s s i a n (not examined) Acad, S c i , U.S.S.R. P r e s s , Moscow* T a v g e r , B.A, and Z a i t s e v , V.M., ( 1 9 5 6 ) Zh, eksp. t e . F i z . S.S.S.R. 3 0 , 5 6 4 , E n g l i s h t r a n s l a t i o n : ( 1 9 5 6 ) J . Exp. Theor. Phys, U.S.S.R. 3_, 4 3 0 , T a v g e r , B.A., ( 1 9 5 8 ) K r i s t a l l o g r a f i y a 3_, 3 4 0 . E n g l i s h t r a n s l a t i o n : ( 1 9 5 8 ) S o v i e t Phys. C r y s t . 3_, 3 4 1 . Van d e r L u g t , W., ( 1 9 6 1 ) T h e s i s , U n i v . of Leyden, i n Dutch. Weber, L. ( 1 9 2 9 ) Z. K r i s t a l l o g r . 7 0 , 3 0 9 , W e y l t H . , ( 1 9 5 2 ) Symmetry, P r i n c e t o n U n i v , P r e s s , P r i n c e t o n , Appendix B, Wigner, E., ( 1 9 5 9 ) Group Theory and i t s A p p l i c a t i o n s t o the Quantum Mechanics o f Atomic S p e c t r a , Academic P r e s s , New York. 135 Zamorzaev, A.M., (1953) "A G e n e r a l i z a t i o n o f the Fedorov Groups", T h e s i s i n R u s s i a n (not examined). Zamorzaev, A.M., (1957) K r i s t a l l o g r a f i y a 2_, 15. E n g l i s h t r a n s l a t i o n : (1958) S o v i e t Phys. C r y s t . 3_, 401. 

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