Atomic Displacement Parameter Nomenclature

7/27/2019 Atomic Displacement Parameter Nomenclature

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7 7 0Acta Cryst . ( 1 9 9 6 ) . A 5 2 , 7 7 0 - 7 8 1

A t o m i c D i s p l a c e m e n t P a r a m e t e r N o m e n c l a t u r eR e p o r t o f a S u b c o m m i t t e e o n A t o m i c D i s p l a c e m e n t P a r a m e t e r N o m e n c l a t u r e t

K. N . TRUEBLOOD * (Cha i rman ) , H . -B. B1SIRGI,b H BURZLAFF,c J. D. DUNITZ, d C . M . GRAMACCIOLI, e H. H . SCHULZ,U. SHMUELI AND S. C. ABRAHAMS (ex officio, I U C r C o m m i s s i o n o n C r y s t a ll o g r ap h i c N o m e n c la t u r e)

aDep artment of Chemistry and Biochemistry, Universi ty o f Cali fornia, L os Angeles, CA 90095, USA , bLaboratoriumfa r chemisch e und miner alogische Kristallographie, Un iversi t i# Bern, Freiestrasse 3, CH-3012 Bern, Switzerland,Clnstitut fa r K ristallographie, Universi t~it, Bisma rckstrasse 10, D -91054 Erlangen, Germany, dLaboratorium fa rorganische C hemie, ET tt Zentrum , C H-8092 Ziirich, Switzerland, eDipartimento d i Scienze della Terra, via Bott icel l i23, 1-20133 Milano, I taly , f Inst i tut fa r Kristallogra phie u nd Mineralog ie, Universi ti it , T heresienstrasse 41, D-8033 3Miinchen, Germany, gSchool o f Chemistry, Tel Aviv Universi ty , 6997 8 Tel Aviv, Israel, an d hph ysics Department,South ern Oregon State College, Ashland, OR 97520, U SA. E-mail: knt@ chem .ucla.edu(Received 12 June 1995; accepted 2 May 1996)

A b s t r a c tM o d e r n X - r a y a n d n e u t r o n d i f f r a c t i o n t e c h n i q u e s c a ng i v e p r e c i s e p a r a m e t e r s t h a t d e s c r i b e d y n a m i c o r s t a t i cd i s p l a c e m e n t s o f at o m s i n c ry s t a ls . H o w e v e r , c o n f u s i n ga n d i n c o n s i s t e n t t e r m s a n d s y m b o l s f o r t h e s e q u a n t i -t i e s occur i n t he c rys t a l l ograph ic l i t e ra tu re . Th i s r epor td i scusses v a r ious fo rm s of t he se quan t i t ie s , de r ivedf r o m p r o b a b i l i t y d e n s i t y f u n c t i o n s a n d b a s e d o n B r a g gd i f f r a c t i o n d a t a , b o t h w h e n t h e G a u s s i a n a p p r o x i m a -t ion i s appropr i a t e and when i t i s no t . The focus i se s p e c i a l l y o n i n d i v i d u a l a t o m i c a n i s o t r o p i c d i s p l a c e m e n tp a r a m e t e r s ( A D P s ) , w h i c h m a y r e p r e s e n t atomic mot ionand poss ib l e static displacive disorder. The f i r s t o f t hef o u r s e c t i o n s g i v e s b a c k g r o u n d i n f o r m a t i o n , i n c l u d i n gd e f in i ti o n s . T h e s e c o n d c o n c e r n s t h e k i n d s o f p a r a m e t e rd e s c r i b i n g a t o m i c d i s p l a c e m e n t s t h a t h a v e m o s t o f t e nbeen used in c rys t a l s t ruc tu re ana lys i s and hence a remost commonly found in t he l i t e ra tu re on the sub jec t . I ti nc ludes a d i scuss ion o f g raph ica l r epre sen ta t ions o f t heG a u s s i a n m e a n - s q u a r e d i s p l a c e m e n t m a t r i x . T h e t h i r ds e c t i o n c o n s i d e r s t h e e x p r e s s i o n s u s e d w h e n t h e G a u s s -Jan approx ima t ion i s no t adequa te . The f ina l sec t iong i v e s r e c o m m e n d a t i o n s f o r s y m b o l s a n d n o m e n c l a t u r e .

1 . I n t r o d u c t i o n1.1. Organization o f this Rep ort

T h e r e a r e f o u r m a i n s e c ti o n s . T h e Introduction con t in -ues , a f t e r t he p re sen t subs ec t ion , w i th a d i scuss ion o f ther a t io n a l e f o r a n d g e n e s i s o f t h e S u b c o m m i t t e e t h a t w r o teth i s r epor t ( 1 .2 ), no t a t i on fo r pos i t i on and d i s p l acem entvec to rs ( 1 .3 ), a cons ide ra t ion o f t he s t ruc tu re fac to r and1" Established 15 Fe bruar y 1994 by the IUC r Commission on C rys-tallographic Nomenclature, with all members appointed by 7 June1994, under ground rule s outlined in Acta C ryst. (1979), A35, 1072.H. B urzlaff resigned from the Subcomm ittee 2 Janu ary 1995. FinalReport accepted 20 December 1995 by the IUCr Commission onCrystallographic Nomenclature. 1996 International Union of CrystallographyPrinted in Great Britain - all rights reserved

a t o m i c d i s p l a c e m e n t s ( 1 . 4 ) a n d c o m m e n t s a b o u t t h et e r m i n o l o g y t o b e u s e d ( 1 . 5 ) .T h e s e c o n d m a i n s e c t i o n , Displacement parametersbased on the Gaussian approximation, c o n c e r n s t h e

k i n d s o f p a r a m e t e r s d e s c r i b i n g a t o m i c d i s p l a c e m e n t stha t have most o f t en been used in c rys t a l s t ruc tu rea n a l y s i s a n d h e n c e a r e m o s t c o m m o n l y f o u n d i nthe l i t e ra tu re on the sub jec t . I t s subsec t ions a re : 2 .1 ,Anisotropic displacement parameters , w h i c h i n c l u d e sa d is c u s s i o n o f th e c o m m o n s y m b o l s u s e d ; 2 .2 ,Equ ivalent isotropic displac emen t param eters; and 2 .3 ,Graphical representations of the G auss ian mean-squaredisplacement matrix .

The th i rd ma in sec t ion , Beyond the Gauss ian ap-prox imat ion, c o n s i d e r s t h e e x p r e s s i o n s u s e d w h e n t h eG a u s s i a n a p p r o x i m a t i o n i s n o t a d e q u a t e . O u r R e c om -mendat ions fo r nomenc la tu re a re i n t he f i na l ma in sec -t ion.

1.2. BackgroundA n i s o t r o p i c d i s p l a c e m e n t p a r a m e t e r s ( A D P s ) ( s e e t h e

d i s c u s s i o n o f t e r m i n o l o g y i n 1 . 5 a n d o f s y m b o l s i n 2 . 1 ) h a v e t y p i c a l l y b e e n d e t e r m i n e d a n d r e p o r t e d i nthe c rys t a l l ograph ic l i t e ra tu re i n a va r i e ty o f ways ,s y m b o l i z e d s o m e t i m e s a s U ( as i n U 0 o r U ) , s o m e t i m e sas B (as in B ij or B eq) , and som et im es aseq/~ (as in /3iJ) .U n f o r t u n a t e l y , t h e s e t e r m s h a v e n o t a l w a y s b e e n u s e dcons i s t en t ly . D imens ion le ss quan t i t i e s , e .g . th e ~ij, h a v eb e e n m i s i d e n ti f i ed a s / f i J v a l u e s , w h i c h a r e o f d i m e n s i o n( l eng th) 2 . W hen these and re l a t ed quan t i t ie s were f i r s td e t e r m i n e d m o r e t h a n f o u r d e c a d e s a g o , t h e y w e r econs ide red a t be s t t o have qua l i t a t i ve s ign i f i cance andincons i s t enc i e s , when no t i ced , were o f t en d i s rega rded .C r u i c k s h a n k ' s ( 1 9 5 6 ) p a p e r s m a r k e d a t u r n i n g p o i n ttoward s quan t i t a t i ve i n t e rp re t a t i on in te rms o f s implep h y s i c a l m o d e l s . S i n c e t h e n , t e c h n i c a l i m p r o v e m e n t s ,b o t h i n t h e m e a s u r e m e n t o f d i ff r a c ti o n d a t a a n d i n

Acta Crystallographica Section AISSN 0108-7673 1996


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K . N . T R U E B L O O D et al. 771compu t ing , have l ed to increased phys ica l s ign i f i cancei n t h e e x p e r i m e n t a l l y d e t e r m i n e d A D P s . T h e f a c t t h a tthese quan t i t i es a re now more mean ing fu l and the l acko f u n i f o r m i t y w i t h w h i c h t h e y h a v e b e e n p r e s e n t e d h a v eb e e n d i s c u s s e d i n f o r m a l l y i n r e c e n t y e a r s b y m o r e t h a no n e I U C r C o m m i s s i o n b u t n o c o n s e n s u s h a s p r e v i o u s l yb e e n r e a c h e d o n a n a c c e p t a b l e f o r m o f p r e se n t a t io n . T h i ss i tua t ion was b rough t recen t ly to the a t t en t ion o f theC o m m i s s i o n o n C r y s t a l l o g r a p h i c N o m e n c l a t u r e . T h a tb o d y e s t a b l i s h e d t h i s S u b c o m m i t t e e , c h a r g i n g i t w i t he x a m i n i n g t h e m e r i t s o f a d o p t i n g a u n i f o r m a p p r o a c hto repo r t ing in s t ruc tu ra l papers quan t i t i es tha t descr ibea t o m i c d i s p l a c e m e n t a n d w i t h m a k i n g s u c h r e c o m m e n -d a t i o n s a s m i g h t b e d e e m e d a p p r o p r i a t e . D i s c u s s i o nw i t h i n t h e S u b c o m m i t t e e , c o n d u c t e d p r i n c i p a l l y t h r o u g he l e c t r o n i c c o r r e s p o n d e n c e , i d e n t i f i e d t w o m a j o r a r e a s ti n w h i c h u n i f o r m d e f i n i t i o n s , t e r m i n o l o g y a n d n o m e n -c la tu re wou ld be des i rab le . These inc lude an i so t rop icd i s p l a c e m e n t p a r a m e t e r s i n t h e G a u s s i a n a p p r o x i m a t i o nand in s i tua t ions in wh ich the Gauss ian app rox imat ioni s inadequa te . In cons ider ing the uncer ta in t i es o f exper i -m e n t a l l y d e t e r m i n e d A D P s , t h e r e a d e r i s r e f e r r e d t o t h eR e p o r t o f a W o r k i n g G r o u p o n t h e E x p r e s s i o n o f U n -c e r t a i n t y i n M e a s u r e m e n t ( S c h w a r z e n b a c h , A b r a h a m s ,H a c k , P r i n c e & W i l s o n , 1 9 9 5 ) , a p p o i n t e d b y t h e I U C rC o m m i s s i o n o n C r y s t a l l o g r a p h i c N o m e n c l a t u r e , w h i c hd i scusses the genera l concep t o f uncer ta in ty in the resu l to f a c rys ta l log raph ic measu remen t and i t s app l i ca t ion .T h e r e c o m m e n d a t i o n s a t t h e e n d o f t h is R e p o r t a r ep roposed fo r u se in a l l fu tu re pub l ica t ions o f the In te r -n a t i o n a l U n i o n o f C r y s t a l l o g r a p h y . A u t h o r s o f s t r u c t ur e -r e f i n e m e n t p r o g r a m s , p a r t i c u l a r l y t h o s e i n w i d e s p r e a duse , a re encouraged to b r ing the i r p rog rams in to fu l lc o n f o r m i t y w i t h t h e s e r e c o m m e n d a t i o n s .1 .3 . Nota t ion fo r pos i t ion an d d isp lacement vec tors

Some no ta t ion to be u sed in th i s repo r t fo r bas i sv e c t o r s a n d t h e i r c o m p o n e n t s i s s u m m a r i z e d h e r e . T h eloca l in s tan taneous a tomic pos i t ion vec to r i s deno ted byr + u , wi th r the mean a tomic pos i t ion vec to r and u thed i s p l a c e m e n t v e c t o r o f a n a t o m f r o m i t s m e a n p o s i t i o n .We use the symbo ls a* , b* and c* , as i s common , fo rthe l eng ths o f the rec ip roca l axes .Q uanti t ies re ferred to the direct lat t ice b asis a, b , c(or equ iva len t ly a l, a 2, a3):

Co mp one nts of r : x , y , z (or equ ival ent l y x I , x 2 , x3).C o m p o n e n t s o f u : Z k x , A y , A z ( o r e q u i v a l e n t l y~Xx~, ~ , ~Xx3).R e l a t e d a n i s o t r o p i c d i s p l a c e m e n t p a r a m e t e r : ~ij _( z : L d , ~ J ) ; d i m e n s i o n l e s s .Quant i t ie s re fe rred to the bas is a 'a , b ' b , c * c (o requiva len t ly a I a l, aZa2 aaa 3):Com pone n ts o f r : 4 , r /, ( (o r e qu iva le n t ly 4 ~ 4 z , 43 ).

'f R e c o m m e n d a t i o n s o n t h e n o m e n c l a t u r e o f t h e r m a l d i f f u s e s c a tt e r in g( e . g . Wi l l i s , 1993) a nd d i so rde r d i f fuse s c a t t e r i ng ( e . g . J a g o d z i n s k i &Fre y , 1993) we re c ons i d e re d a s ou t s i de t he s c ope o f t h i s re por t .

C o m p o n e n t s o f u : A 4 , A t / , A ( ( o r e q u i v a l e n t l yA41, A 42,/X43).Rela ted a n i so t rop ic d i sp lac eme n t parame te r : U / =(A4 iA 4J) ; ( l eng th ) ~ .Quant i t ie s re fe rred to the Car te s ian bas is (e I , e 2, e3):C o m p o n e n t s o f r : 4 c , r l c , ( c ( o r e q u i v a l e n t l yC o m p o n e n t s o f u : A ~ c , A t / c , A ( c ( o r e q u i v a l e n t l yR e l a t e d a n i s o t ro p i c d i s p l a c e m e n t p a r a m e t e r : ~ -

{A4/CA4jC); (len gth ) 2.

1 .4 . The s truc ture fac to r an d a tomic d isp lacementsThe s t ruc tu re fac to r o f re f l ec t ion h i s g iven in a fa i r ly

g e n e r a l f o r m b y t h e F o u r i e r t r a n s f o r m o f t h e a v e r a g edens i ty o f sca t t e r ing mat te rF ( h ) = f ( p ( r ) ) e x p ( 2 7 r i h . r ) d 3 r - ~ [ ( p ( r ) ) ] , (1 )

wi th the in teg ra t ion ex tend ing over the repea t ing s t ruc-tu ra l mo t i f , con f ined to a s ing le un i t ce l l . The b racke t sdeno te a doub le averag ing over the poss ib le d i sp lace-men ts o f the a toms f rom the i r mean p os i t ions - a t imeaverage over the a tomic v ib ra t ions in each ce l l , fo l lowedby a space avera ge tha t cons i s t s o f p ro jec t ing a l l thet i m e - a v e r a g e d c e l ls o n t o o n e a n d d i v i d i n g b y t h e n u m b e ro f ce l l s , h i s a d i f f rac t ion vec to r obey ing the Laueequa t ions and p ( r ) i s the s t a t ic dens i ty o f the mo t i f ,cons i s t en t wi th the in s tan taneous loca l con f igu ra t ion o fthe nuclei in a uni t cel l .To reduce the above genera l p ic tu re to what i s u sedin conven t iona l c rys ta l s t ruc tu re ana lys i s , we f i r s t a s -sume tha t the average dens i ty o f mat te r in (1 ) can beregarded as a superpos i t ion o f averaged a tomic dens i t i es .Th i s so -ca l l ed i so la ted -a tom approx imat ion i s es sen t i a l lye q u i v a l e n t t o a s s u m i n g i n d e p e n d e n t l y d i s p l a c e d a t o m s , afa i r in i t i a l app rox imat ion , a l though no t genera l ly va l id .The average dens i ty o f sca t t e r ing mat te r a t the po in t rin a un i t ce l l can then be app rox imated as

N(p (r )) ~ ~ n k f pk( r -- rk) pk( r k -- rko) d3rk . (2)k = lHere , N i s the num ber o f a tom s in the un i t ce l l , n ki s the occup ancy fac to r o f the kth a tom, pk (r - rk ) i sthe dens i ty (e lec t ron dens i ty fo r X-rays o r a 6 func t ionweig h ted wi th the sca t t e r ing l eng th b k fo r neu t rons ) dueto a tom k a t a po in t r when the nuc leus o f a tom k i sat r k and Pk (rk -- rko ) i s the p robab i l i ty dens i ty func t ion(p .d . f . ) co r respond ing to the p robab i l i ty o f hav ing a tomk d i sp laced by the vec to r r k - rko f rom i t s re fe rencepos i t ion rko in an average un i t ce l l , wh ich wi l l be themean pos i t ion i f Pk i s su f f i c ien t ly symmet r ica l . I t i si m p o r t a n t t o r e m e m b e r t h a t t h e a p p r o x i m a t i o n s i n ( 2 )i n c l u d e t h e a s s u m p t i o n t h a t a t o m s a r e n o t d e f o r m a b l e ,


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7 72 A T O M I C D I S P L A C E M E N T P A R A M E T E R N O M E N C L A T U R Eby bond ing o r o therwise , even though a t th i s s t age thestatic a t o m i c e l e c t r o n d e n s i t y , P k ( r - rk ) , has no t beena s s u m e d t o b e s p h e r i c a l l y s y m m e t r i c .

I f (2 ) i s now subs t i tu ted in to (1 ) and the o rder o f thesummat ion and in teg ra t ion i s in te rchanged , the s t ruc tu ref a c t o r b e c o m e s

NF ( h ) _ ~ E nkFk(h ) (3 )k= l

w i t hF k ( h ) = f [ f P k ( r - - r k ) P k ( r , - r , o ) d 3 r k ]

x exp (2 7 r ih , r ) d3 r . (4 )If the subst i tu t ions r = t + rko and r k = u + rko aremade , the in teg ra l in (4 ) becomes

F k ( h ) = { f [ f P , ( t - u ) p k ( u ) d 3 u ] e x p ( 2 7 r i h ' t ) d 3 t }x exp (27 rih , rko ) . (5)

The inner in teg ra l in (5 ) has the fo rm o f a conven t iona lconvo lu t ion o f the dens i ty o f a tom k wi th the p .d . f , fo ra d i sp lacemen t o f th i s a tom f rom i t s mean pos i t ion ; theou te r in teg ra l i s a Four ie r t rans fo rm o f th i s convo lu t ion .T h i s t r a n s f o r m i s m u l t i p l i e d b y a n e x p o n e n t i a l t h a td e p e n d s o n t h e m e a n p o s i t i o n , rko, o f a t o m k .B y t h e c o n v o l u t i o n t h e o r e m , t h e F o u r i e r t r a n s f o r mo f a c o n v o l u t i o n e q u a l s t h e p r o d u c t o f F o u r i e r t r a n s -f o r m s o f t he f u n c t i o n s i n v o l v e d . W h e n t h is t h e o r e mis app l i ed to the ou te r in teg ra l in (5 ) , we ob ta in thec o n v e n t i o n a l a p p r o x i m a t i o n f o r t h e s t r u c t u r e f a c t o r o f aBragg re f l ec t ion

NF ( h ) _ ~ E nk f k ( h ) Tk ( h ) e x p ( 2 7 r i h , rko ). (6)k= l

I f we l e t v = ( r - rk ) and (as befo re ) u = ( r k - rko ) ,then in (6 )fk (h ) = f p~(v ) exp (27 r ih v ) d3v (7 )

i s the sca t t e r ing fac to r o r fo rm fac to r o f a tom k ( fo rneu t rons th i s i s rep laced by the sca t t e r ing l eng th b k ) and

T k ( h ) = f p k ( u ) e x p ( 2 7 ri h u ) d 3 u (8 )i s the Four ie r t rans fo rm o f the p .d . f . , Pk, fo r the d i s -p l a c e m e n t o f t h e k t h a t o m f r o m i t s r e f e r e n c e p o s i t i o n ,r t, ,, . Th i s t e r m con ta in s the dep ende nce o f the s t ruc tu ref a c t o r o n a t o m i c d i s p l a c e m e n t s a n d h a s b e e n k n o w n b yt h e n a m e s ' a t o m i c D e b y e - W a l l e r f a c t o r ' a n d ' a t o m i ctempera tu re fac to r ' ( s ee 1 .5 ) . There a re no res t r i c t ionson the func t iona l fo rm o f the p .d . f , in the in teg rand o f( 8 ) .

Let u s now reca l l tha t the s t ruc tu re - fac to r equa t ionu s e d i n r o u t i n e r e f i n e m e n t o f a t o m i c p a r a m e t e r s i s f u r -ther s imp l i f i ed in two ways :F i rs t , fo r X-rays , the s t a t i c a tomic e lec t ron dens i ty i s

a s s u m e d t o h a v e s p h e r i c a l s y m m e t r y . T h i s r e d u c e s t h ea tomic sca t t e r ing fac to r to the fo rmO ~f k ( I h l ) = 47r f p k ( u ) [ s i n ( 2 7 r l h l u ) / 2 ~ l h l u ] u 2 du, (9)0

w h i c h h a s b e e n c o m p u t e d a n d e x t e n s i v e l y t a b u l a t e d f o ra l l t h e n e u t r a l e l e m e n t s a n d m a n y i o n s ( M a s l e n , F o x& O ' K e e f e , 1 9 9 2 ) . T h e s p h e r i c a l - a t o m a p p r o x i m a t i o nnecessar i ly removes f ine de ta i l s o f the (ca lcu la ted ) e lec -t ron dens i ty , bu t may be u sed rou t ine ly , and se rve as as ta r t ing po in t fo r more re f ined de te rmina t ions o f a tomicpos i t ions and s tud ies o f charge de ns i ty (e.g. C o p p e n s &Becker , 1992 ; Copp ens , 1993) .Second , the p .d . f , fo r a tomic d i sp lacemen t i s mos t f re -q u e n t l y a p p r o x i m a t e d b y a u n i v a r i a t e o r t r iv a r i a t e G a u s s -i a n , d e p e n d i n g o n w h e t h e r t h e a t o m i c d i s p l a c e m e n t s a r eassumed to be i so t rop ic o r an i so t rop ic , respec t ive ly . I f at r ivar ia te Gauss ian i s as sumed and the a tomic subscr ip tk i s omi t t ed , the resu l t ing exp ress ion fo r T(h ) f rom (8 ) i s

T ( h ) = e x p [ - 2 7 r Z ( ( h u ) 2 ) ] . ( 1 0 )Equa t ion (10 ) can be der ived f rom the theo ry o f l a t t i ced y n a m i c s i n t h e harmonic a p p r o x i m a t i o n , w h i c h c o n -s iders on ly the (always p r e s e n t ) c o n t r i b u t i o n o f m o t i o nt o t h e a t o m i c d i s p l a c e m e n t (e.g. Wil l i s & Pryo r , 1975) .H o w e v e r , t h i s e q u a t i o n m a y a l s o be a p p l i e d t o static dis-plac ive disorder . The f o r m o f t h e a to m i c D e b y e - W a l l e rf a c t or , T ( h ) , r e p r e s e n t e d i n ( 1 0) i s t h e m o s t c o m m o n o n ein s t andard s t ruc tu re re f inemen ts and wi l l be d i scussedi n 2 . V a r i o u s o t h e r a p p r o x i m a t i o n s h a v e b e e n p r o p o s e dfo r s i tua t ions in wh ich the Gauss ian fo rmal i sm i s no ta d e q u a t e , e.g. w h e n t h e a n h a r m o n i c c o n t r i b u t i o n t o t h ec r y s t a l d y n a m i c s i s s i g n i f i c a n t ; t h e m o s t c o m m o n a r ed i scussed in 3 .

W e p r e s e n t n o w a s h o r t d i s c u s s i o n o f c o m m o n v a r i-an t s o f (10 ) , wh ich can be rewr i t t en asT ( h ) = e x p [ - 2 7 r 2 ( ( u h / I h l ) Z ) l h l 2 ] . (11 )

Th is shows tha t the exponen t i s p ropor t iona l to minust h e m e a n - s q u a r e p r o j e c t io n o f t h e a to m i c d i s p l a c e m e n tu on the d i rec t ion o f the d i f f rac t ion vec to r h t ime s thes q u a r e d m a g n i t u d e o f h . I f w e d e n o t e t h e p r o j e c t io n o f uon the d i rec t ion o f h by u h and mak e use o f the re la t ionI h l = 2 ( s i n 0 ) / A , ( 11 ) b e c o m e s

T ( h ) = e x p [ - 8 7 r Z ( u Z ) ( s i n 2 o ) 1 2 ] . (12 )As long as the a tomic d i sp lacemen ts a re an i so t rop ic , thev a l u e o f t h e a v e r a g e i n ( 1 2 ) d e p e n d s o n t h e d i r e c t i o n o fh . T h i s i s t h e n t h e a n i s o t r o p i c G a u s s i a n D e b y e - W a l l e r


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K . N . T R U E B L O O D et al. 7 7 3fac tor , T(h) , which i s d i scussed in de ta i l in 2 . I f ,h o w e v e r , t h e a t o mi c d i s p l a c e me n t s a r e i s o t r o p i c , t h ea v e r a g e i n ( 1 2 ) i s a c o n s t a n t d e t e r mi n e d b y t h e s t r u c t u r ea l o n e , b u t p o s s i b l y d i f f e r e n t f o r n o n - e q u i v a l e n t a t o ms ,a n d t h e l e f t - h a n d s i d e o f t h is e q u a t i o n n o l o n g e r d e p e n d so n t h e d i r e c t i o n o f h b u t o n l y o n i t s ma g n i t u d e . T h i s i st h e n t h e a t o mi c i s o t r o p i c G a u s s i a n D e b y e - W a l l e r f a c t o r ,

T ( I h l ) = e x p [ - 8 7 r 2 ( u 2 ) ( s i n e 0 ) / A e ] . ( 1 3 )T h e l o w e s t - o r d e r a p p r o x i ma t i o n t o T ( h ) i s t h e o v e r a l li s o t r o p i c D e b y e - W a l l e r f a c t o r . I t h a s t h e s a me f o r m a s( 1 3 ) a n d p r e s u me s t h a t a l l t h e a t o ms h a v e t h e s a mei s o t r o p i c me a n - s q u a r e d i s p l a c e me n t , ( u 2 ) . T h e w h o l ec rys ta l s t ruc ture i s a ss igned , in th i s approxima t ion , as i n g l e d i s p l a c e me n t p a r a me t e r . T h i s a p p r o x i ma t i o n i sused in in i t ia l s tages o f c rys ta l s t ruc ture de te rmina t ionb y d i r e c t me t h o d s .W e c o n c l u d e t h i s s e c t i o n w i t h s o me r e ma r k s o n t h es t ruc ture f ac tor fo r e lec t ron d i f f r ac t ion by a c rys ta l . Thedens i ty o f sca t te r ing ma t te r , p , i s he re in te rpre ted a s thed is t r ibu t ion of e lec t ros ta t ic po ten t ia l wi th in the un i t ce l l .T h i s p o t e n t i a l is t h e n a p p r o x i ma t e d b y a s u p e r p o s i t i o n o fe l e c t r o st a t i c p o t e n t ia l s c o n t r i b u t e d b y i n d i v i d u a l a t o ms ,a n d t h e e f f e c t s o f mo t i o n a r e t a k e n i n t o a c c o u n t , a sf o r X - r a y s a n d n e u t r o n s , b y t h e c o n v o l u t i o n o f t h ep o t e n t i a l o f a n a t o m a t r e s t w i t h t h e p r o b a b i l i t y d e n s i t yf u n c t i o n d e s c r i b in g t h e a t o mi c mo t i o n (e.g. Va i n s h t e i n &Z v y a g i n , 1 9 9 3) . T h e a t o mi c ( s p h e r ic a l ) s c a t t e ri n g f a c t o rfor e lec t ron d i f f r ac t ion , f e l , k ( I h l ) , f o r a n a t o m a t r e s tand d i f f r ac t ion vec tor h , i s r e la ted to tha t fo r X- raysb y t h e M o t t f o r mu l a (e.g. Vainsh te in , 1964) , which hasthe form fe l , k( lhl) cx [Zk - f k ( I h l ) ] / I h l 2 , whe re Z k i st h e a t o m i c n u m b e r a n d A ( l h l ) i s t h e X - r a y f o r m f a c t o rof a tom k [ see (9 ) ] . Th is fo rmula , wi th the cor rec tp r o p o r t i o n a l i t y c o n s ta n t s , h a s b e e n u s e d a l o n g w i t h o t h e rt e c h n i q u e s i n e x t e n s i v e t a b u l a t i o n s o f s p h e r i c a l f o r mfac tor s fo r e lec t ron d i f f r ac t ion ( see e.g. Co w l e y , 1 9 9 2 ) .T h e D e b y e - W a l l e r f a c t o r , h e r e e x p r e s s i n g t h e ' s me a r i n go u t ' o f t h e e l e c t r o s t a t i c p o t e n t i a l , i s g i v e n b y t h e s a mee x p r e s s i o n a s t h a t q u o t e d a b o v e f o r X - r a y s a n d n e u t r o n s(e.g. Va i n s h t e i n , 1 9 6 4 ; Va i n s h t e i n & Z v y a g i n , 1 9 9 3 ) .The s t ruc ture f ac tor fo r e lec t ron d i f f r ac t ion i s the re foreana logous to tha t appea r ing in (6 ) bu t i s o f ten g iven ina d i f f e ren t no ta t ion .

1.5. C omme nt s abou t t e rmino logyT h e q u a n t i t y T t h a t o c c u r s in ( 6 ) h a s b e e n c o m mo n l yr e f e r r e d t o e i t h e r a s t h e D e b y e - W a l l e r f a c t o r o r t h et e mp e r a t u r e f a c t o r b e c a u s e D e b y e ( 1 9 1 3 ) a n d W a l l e r( 1 9 2 3 ) f i r s t u n d e r s t o o d a n d f o r mu l a t e d t h e e f f e c t t h a tt h e r ma l v i b r a t i o n s w o u l d h a v e o n t h e i n t e n s i t y o f X - r a ys c a t t e r i n g . I t h a s , h o w e v e r , l o n g b e e n r e c o g n i z e d , a sd i s c u s s e d i n 1 .4 a b o v e , t h a t s t a ti c d i s p l a c e m e n t s w o u l dhave a s imi la r e f f ec t . We the re fore avo id the te rm' t e mp e r a t u r e f a c t o r ' a n d r e c o mme n d t h a t o t h e r s d o s o

a l s o , i n p a r t b e c a u s e o f t h i s a mb i g u i t y a b o u t t h e o r i g i no f t h e a t o mi c d i s p l a c e me n t s t h a t c a u s e t h e d i mi n u t i o ni n s c a t t e r i n g . A n o t h e r r e a s o n f o r a v o i d i n g t h e p h r a s e' t e mp e r a t u r e f a c t o r ' i s t h e c o n f u s i o n c a u s e d b y t h e f a c tt h a t i t h a s n o t i n f r e q u e n t l y i n t h e p a s t b e e n u s e d f o rt e r ms i n t h e e x p o n e n t i n e x p r e s s i o n s l i k e t h a t o n t h er igh t s ides o f (12) and (13) , r a the r than for the en t i r ee x p o n e n t i a l mu l t i p l i c a t i v e f a c t o r .A d e t a i l e d t r e a t me n t o f t h e p h y s i c a l b a c k g r o u n d o fp o s s i b l e a t o mi c d i s p l a c e me n t s i s q u i t e b e y o n d t h e s c o p eo f t h i s r e p o r t . H o w e v e r , w e s h a l l t r y t o s u mma r i z ea n d d e s c r i b e b r i e f l y t h e mo s t i mp o r t a n t c o mp o n e n t s o ft h e d i s p l a c e me n t . T h e b e s t k n o w n i s t h e d i s p l a c e me n ta r i s i n g f r o m a t o mi c v i b r a t i o n s . W h e n t h e s e r e s u l t f r o mt h e mo t i o n o f mo l e c u l e s o r mo l e c u l a r f r a g me n t s (e.g.W i l l i s & P r y o r , 1 9 7 5 ) , t h e y a r e u s u a l l y c h a r a c t e r i z e db y r e l a t i v e l y l a r g e a mp l i t u d e s . I n c r y s t a l s c o n t a i n i n gr e l a t i v e l y s t r o n g l y b o n d e d a t o ms (e.g. m o l e c u l a r a n di o n i c c r y s t a l s ) , mu c h s ma l l e r d i s p l a c e me n t a mp l i t u d e sr e s u l t f r o m t h e e v e r - p r e s e n t i n t e r n a l v i b r a t i o n s , s u c ha s b o n d s t r e t c h i n g a n d b e n d i n g (e.g. W i l s o n , D e c i u s& Cr o s s , 1 9 5 4 ). A l l o f t h e s e mo t i o n s a r e t e m p e r a t u r ed e p e n d e n t , u n l e s s t h e t e mp e r a t u r e i s v e r y l o w . O t h e re f f e c t i v e d i s p l a c e me n t s f r o m t h e me a n p o s i t i o n ma ya r i se a s a r e su l t o f a va r ie ty o f poss ib le type s o f d i sorde r .T h e s e i n c l u d e s ma l l d e v i a t i o n s f r o m i d e a l p e r i o d i c i t y ,p re sen t in a l l r ea l c rys ta l s ; o r ien ta t iona l d i sorde r , p re sen ti n ma n y mo l e c u l a r c r y s t a l s ; d e n s i t y a n d d i s p l a c e me n tmo d u l a t i o n s ; a n d s h o r t - a n d l o n g - r a n g e d i s p l a c i v e c o r -r e l a t i o n s . M a n y t y p e s o f d i s o r d e r g i v e r i s e t o d i f f u s es c a t te r i n g , w h i c h c a n o f t e n b e a n a l y z e d (e.g. J a g o d z i n s k i& Fr e y , 1 9 9 3 ) . T h e r e a r e , i n a d d i t i o n , n u me r o u s o t h e rp o s s i b l e c o n t r i b u t i o n s t o apparent d i s p l a c e me n t s , o n e o ft h e mo s t i m p o r t a n t o f w h i c h i s u s e o f a n i n a d e q u a t em o d e l , e.g. i n a d e q u a t e a b s o r p t i o n c o r r e c t i o n , o r u s eo f a G a u s s i a n p r o b a b i l i t y d e n s i t y f u n c t i o n w h e n i t i sinappropr ia te .I n v i e w o f t h e l a r g e n u mb e r o f p o s s i b l e c a u s e s o f a na p p a r e n t a t o m i c d i s p l a c e m e n t , w e r e c o m m e n d e x p a n d -i n g t h e d e f i n i t i o n o f ' D e b y e - W a l l e r f a c t o r ' t o i n c l u d ed i s p l a c e me n t s a r i s i n g f r o m a n y s o u r c e . W e w i l l u s e t h et e r m ' D e b y e - W a l l e r fa c t o r ' w h e n w e m e a n t h e e n t ir ef a c t o r t h a t mu l t i p l i e s t h e s c a t t e r i n g f a c t o r o f a n a t o m a tr e s t , a n d r e c o mme n d t h a t t h i s t e r m b e u s e d w h e n w o r d sa r e w a n t e d t o r e f e r t o t h e q u a n t i t y T ( h ) , o r T , t h a t o c c u r sin equa t ions such a s (6 ) , (8 ) and (10) th rough (13) .T h e r e w a s c o n s i d e r a b l e d i s c u s s i o n i n o u r Su b c o mmi t -t e e c o n c e r n i n g t h e p r o p e r w o r d s t o u s e w h e n r e f e r r i n gt o t h e t e r ms i n t h e e x p o n e n t t h a t a r e v a r i a b l e s d u r i n ga typ ica l lea s t - squa res r e f inement to f i t a s t ruc tura lmo d e l t o i n t e n s i t y d a t a . T h e s e t e r ms a r e f o r mu l a t e da n d s y mb o l i z e d i n v a r i o u s w a y s , d i s c u s s e d i n d e t a i li n 2 .1 b e l o w . W e r e c o mme n d u n a n i mo u s l y t h e t e r m' d i s p l a c e m e n t p a r a me t e r s ' ( o f t e n ' a n i s o t r o p i c d i s p l a c e -me n t p a r a me t e r s ' o r A D Ps ) t o d e s c r i b e t h e s e q u a n t i t i e s .T w o o f u s i n i t ia l l y f a v o r e d ' d i s p l a c e m e n t c o e f f i c i e n t s ' ,b e l i e v i n g t h a t o n c e r e f i n e me n t i s c o mp l e t e d , t h i s t e r m i s


5/12

7 7 4 A T O M I C D I S P L A C E M E N T P A R A M E T E R N O M E N C L A T U R Emo r e a p p r o p r i a t e , b u t w e r e p e r s u a d e d t h a t c u r r e n t u s a g es t r o n g l y f a v o r s t h e r e c o m m e n d e d t e r m , ' d i s p l a c e m e n tp a r a m e t e r s ' .

2 . D i s p l a c e m e n t p a r a m e t e r s b a s e do n t h e G a u s s i a n a p p r o x i m a t i o n2 . 1. A n i s o t r o p i c d i s p l a c e m e n t p a r a m e t e r sA s d i s c u s s e d i n 1 .4 a b o v e , d i f f ra c t i o n s t u d i e s y i e l di n f o r ma t i o n n o t o n l y a b o u t me a n a t o mi c p o s i t i o n s b u ta l s o a b o u t t h e p r o b a b i l i t y d e n s i t y f u n c t i o n s ( p .d . f . s ) o fa t o mi c d i s p l a c e me n t s f r o m t h e s e me a n p o s i t i o n s . I f t h ea t o mi c p .d . f , i s a s s u me d t o b e a t r i v a r i a t e G a u s s i a n , t h echa rac te r i s t ic func t ion cor re sponding to th i s p .d . f . - byd e f i n it i o n , it s Fo u r i e r t r a n s f o rm - c a n b e d e s c r i b e d b y t h es e c o n d m o m e n t s o f t h e p .d .f . , w h i c h i n th e p r e s e n t c o n -t e x t a r e c a l l e d a n i s o t r o p i c me a n - s q u a r e d i s p l a c e me n t s .I f n ot , h i g h e r c u mu l a n t s o f a n o n - G a u s s i a n p .d . f , c a n , i np r i n c i p le , a l s o b e d e t e r m i n e d ; t h e s e a r e s i mp l e f u n c t i o n so f m o m e n t s ( e .g . Kenda l l & Stua r t , 1977) , bu t the re a red i f f i c u l t i e s . Fo r e x a mp l e , t h e s e h i g h e r t e r ms a r e o n l yl i k e l y t o b e i mp o r t a n t w h e n t h e s e c o n d mo me n t s o ft h e p .d . f. s a r e r e l a t i v e l y l ar g e . H o w e v e r , a s c a n b e s e e nf r o m t h e b a s i c e x p r e s s i o n f o r t h e i s o tr o p i c D e b y e - W a l l e rf a c t o r, T = e x p [ - 8 7 r E ( u 2 ) ( s i n 2 0 ) / / ~ 2 ] , t h e l a rge r thes e c o n d mo me n t , t h e mo r e r a p i d l y t h e s c a t t e r i n g f r o mt h e a t o mi c c e n t e r i n q u e s t i o n f a l l s o f f w i t h i n c r e a s e i nt h e s c a t t e ri n g a n g l e . T h u s , j u s t w h e n t h e h i g h e r t e r msb e c o me i mp o r t a n t , t h e y b e c o me d i f f i c u l t t o me a s u r e f o rl a c k o f c o n t r i b u t i o n b y t h e s c a t t e r i n g c e n t e r t o t h e Br a g gin tens i t ie s .

T h e me a n - s q u a r e d i s p l a c e me n t s , w h i c h d e f i n e t h ep .d . f , i n t h e v a r i o u s G a u s s i a n a p p r o x i ma t i o n s , u s e d t ob e k n o w n a s a t o m i c v i b r a t i o n p a r a m e t e r s o r t h e r m a lp a r a m e t e r s b u t h a v e r e c e n t l y b e e n d e s i g n a t e d a s a t o m i cd i s p l a c e m e n t p a r a m e t e r s , i so t rop ic o r an iso t rop ic , toa l l o w f o r t h e e f f e c t s o f s t a t i c d i s p l a c i v e d i s o r d e r a s w e l la s f o r t h o s e o f t h e a l w a y s - p r e s e n t a t o mi c mo t i o n . T h e r ee x i s t s a n e x t e n s i v e l i t e r a t u r e o n t h e i n t e r p r e t a t i o n o ft h e s e p a r a me t e r s ( e .g . D u n i tz , S c h o m a k e r & T r u e b l o od ,1988 , and r e fe rences c i ted the re in) .

T h e p u r p o s e o f t h i s s e c t i o n i s t o r e l a t e a l t e r n a t i v ef o r ms o f a n i s o t r o p i c d i s p l a c e me n t p a r a me t e r s ( A D Ps ) t ot h e e x p r e s s i o n f o r t h e D e b y e - W a l l e r f a c t o r t h a t i s v a l i dw i t h i n t h e f r a me w o r k o f t h e a s s u mp t i o n s u n d e r l y i n g t h eh a r m o n i c a p p r o x i m a t i o n ( e .g . W i l l i s & P r y o r , 1 9 7 5 ) .W e a l s o d i s c u s s a n i s o t r o p i c d i s p l a c e me n t p a r a me t e r s i nr e l a t io n t o d i f f e re n t c o o r d i n a t e s y s t e ms , o u t l i n e t h e t r a n s -f o r ma t i o n p r o p e r t i e s o f t h e r e s u l t i n g q u a n t i t i e s , p r e s e n ts e v e r a l f o r ms o f e q u i v a l e n t i s o t r o p i c d i s p l a c e me n t p a -r a me t e r s , a n d d e s c r i b e b r i e f l y g r a p h i c a l r e p r e s e n t a t i o n so f t h e G a u s s i a n m e a n - s q u a r e d i s p l a c e m e n t m a t r i x.The usua l express ion for T(h) i s [ (10) r e s ta ted]T(h) = e x p [ - 2 7 r 2 ( ( h u ) 2 )] ( 1 4 )

= e x p [ - 2 7 r 2 ( ( h . u ) ( u . h ) ) ] . ( 1 5 )

T h e s e f u n d a me n t a l e q u a t i o n s t a k e o n d i f f e r e n t f o r msa c c o r d i n g t o t h e b a s i s v e c t o r s t o w h i c h w e r e f e r t h ed i f f r a c t i o n a n d d i s p l a c e me n t v e c t o r s . I n c a r r y i n g o u tc o o r d i n a t e t r a n s f o r ma t i o n s i n t h e f o r ma l i s m o f t e n -s o r a l g e b r a , q u a n t i t i e s t h a t t r a n s f o r m l i k e d i r e c t b a s i sv e c t o r s a r e c a l l e d c o v a r i a n t a n d a r e i n d i c a t e d b y s u b -s c r i p t s , w h i l e q u a n t i t i e s t r a n s f o r mi n g l i k e r e c i p r o c a lb a s i s v e c t o r s a r e c a l l e d c o n t r a v a r i a n t a n d a r e i n d i c a t e db y s u p e r s c r i p t s . T h e d i r e c t a n d r e c i p r o c a l b a s e s a r e n o tn e c e s s a r i l y t h o s e o f t h e c o r r e s p o n d i n g l a t ti c e s ; t h e y m a yb e a n y p a i r o f d u a l b a s e s . L e t u s f i r s t a s s u me t h a t t h ed i f f ra c t i o n v e c t o r i s r e f e r r e d t o t h e b a s i s o f t h e r e c i p r o c a ll a t t i c e a n d t h e a t o mi c d i s p l a c e me n t v e c t o r t o t h e b a s i sof the d i r ec t la t t ice , a s fo l lows:

h = ha* + kb* + lc*= hi a l + h2 a2 + h3 a3

3= Y ~ hi a i ( 1 6 )i = la n d

u = z 3 ,x a + A y b + A z c- - A a) a 1 + z3a~a2 + z3~3a3

3= y ~ z L r J a j . ( 1 7 )j = lN o t e t h a t th e c o mp o n e n t s o f h a n d u a r e d i me n s i o n l e s s .T h e f i rs t s c a l a r p r o d u c t a p p e a r i n g o n t h e r i g h t - h a n d s i d eo f ( 1 5 ) c a n n o w b e e v a l u a t e d a s

a n d , s i m i l a r l y ,

h . u = ( i= ~ l h i a i) . I j= ~ ~ J a j )3 3= E E h iz ~ c , i~ ji = l j = l3= ~ h j z ]o d (18)j = l

3u - h = ~ z A xt hr ( 1 9 )/ = 1W e u s e d h e r e t h e d e f i n i t i o n o f t h e d u a l ( d i r e c t a n drec iproca l ) bases :

1 i f i = j (2 0)a i ' a j = ~ J = 0 i f i s ~ j "I f we inse r t (18) and (19) in to (15) , we ob ta in fo r T(h)

T ( h ) = e x p - 2 7 r 2 2 Y ~ h j (Z L r , z ~ t ) h lj = l t = l- exp - y~ ~ h j 3 J l h t (21)j = l / = 1


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K . N . T R U E B L O O D et al. 7 7 5w i t h /3jl = 271.2 ( z L r j z ~ I ) . (22)T h e q u a n t i t y / 3 j r d e f i n e d b y ( 2 1 ) a n d ( 2 2 ) i s o n e o ft h e f r e q u e n t l y e mp l o y e d f o r ms o f t h e a n i s o t r o p i c d i s -p l a c e m e n t p a r a me t e r ; n o t e t h e u s e o f superscripts fo r theind ices , s ince the co mp one nts o f 13 a re contravariant. F o ra n a t o m, e a c h c o mp o n e n t / 3 j r i s 2 r e2 t i me s a n a v e r a g e o fa p r o d u c t o f t w o c o m p o n e n t s o f a n a t o mi c d i s p l a c e me n tvec tor , when the la t te r i s r e fe r red to the bas is o f thed i rec t la t t ice .

We sha l l now re ta in h a s de f ined by (16) bu t r ede f ineu a s fo l lows:u = A ( a * a + A ~T b* b + A f c * c

A ~ l a l a l + A ~ 2 a 2 a 2 -k- A ~ 3 a 3 a 33= ~ A{JaJa / . (23)

j = l

T h e c o m p o n e n t s o f u i n t h i s r e p r e s e n t a t io n , A { J , h a v e d i -m e n s i o n l e n g th a n d th e b a s is v e c t o rs ( a ' a , b ' b , c ' c ) a r ed i me n s i o n l e s s ( s e e e.g. H i r s h f e l d & Ra b i n o v i c h , 1 9 6 6 ) .O n l y i n o r t h o r h o mb i c , t e t r a g o n a l a n d c u b i c c r y s t a l s y s -t e ms mu s t t h e s e b a s i s v e c t o r s b e mu t u a l l y o r t h o g o n a lun i t vec tor s , i.e. or thonorma l , s ince i t i s on ly in theses y s t e ms t h a t t h e e q u a l i t i e s a * = 1/a, b* = 1/b andc* = 1/c are n e c e s s a r i l y t r u e . T h e d e p a r t u r e s o f t h e s eb a s i s v e c t o r s f r o m o r t h o n o r ma l i t y i n o t h e r s y s t e ms a r eassoc ia ted wi th the depa r tu re s o f the ang le s c~ , /3 and "3 ,f r o m 9 0 . I f w e n o w r e p e a t t h e e v a l u a t i o n o f t h e s c a l a rp r o d u c t s i n ( 1 5 ) w i t h h g i v e n b y ( 1 6 ) a n d u g i v e n b y(23) , we ob ta in fo r T( 3 , )T = e x p - 2 7 r 2 E E hjaJ(A{JA{ t )atht

j = l l = l

- e x p - 2 r r 2 ~ Y ~ hjaJUYtalht ,j = l l = l (24)

w i t h uJt = ( A~ iA ~ t) = /3Jt/(27rZaJat) (25)a n o t h e r w e l l k n o w n f o r m o f t h e A D P. T h i s f o r m i s o ft e np r e f e r r e d b e c a u s e t h e e l e me n t s o f t h e t e n so r ~ f U h a v ed i me n s i o n ( l e n g t h ) 2 a n d c a n b e d i r e c t l y a s s o c i a t e d w i t ht h e me a n - s q u a r e d i s p l a c e me n t s o f t h e a t o m c o n s i d e r e di n t h e c o r r e s p o n d i n g d i r e c t i o n s . N o t e i n p a r t i c u l a r t h a tt h e me a n - s q u a r e d i s p l a c e me n t i n a n a r b i t r a r y d i r e c t i o nd e n o t e d b y t h e u n i t v e c t o r n , w h e n n i s r e f e r r e d t o u n i tt T o b e p r e c i s e , t h e s y m b o l s U a n d / 3 ( f o r a g i v e n a t o m ) d o n o t r e p re -s e n t d i f f e r e n t te n s o r s, j u s t d i f f e r e n t se t s o f c o m p o n e n t s o f th e a t o m i cq u a d r a t i c m e a n - s q u a r e d i s p l a c e m e n t t e n s o r . T h e y c a n b e i n t e r r e l a t e db y t r a n s f o r m a t i o n s c o n v e r t i n g s e t s o f c o m p o n e n t s o f a t e n s o r i n t oe a c h o t h e r (e .g. S p a i n , 1 9 5 6) . F r e q u e n t l y , h o w e v e r , t h e p h r a s e ' s e to f c o m p o n e n t s o f a t e n s o r ' i s a b b r e v i a te d a s ' t e n s o r '. T h i s p r a c t i c e isf o l l o w e d h e r e a n d U , / 3 a n d B a r e s o m e t i m e s r e f e r r e d t o a s t e n so r s .

vec tor s pa ra l le l to the reciprocal bas is vec tor s so tha t i t sc o m p o n e n t s a r e c o v a r i a n t , is g i v e n b y n r U n ( s e e 2 .3 .2 ).I n a n y e v e n t , t h e d i me n s i o n l e s s e l e me n t s o f / 3 a r e a l s oc o r r e c t l y a s s o c i a t e d w i t h t h e g e n e r a l e x p r e s s i o n f o r T ( h ) ,g i v e n b y ( 1 5 ) .A n o t h e r f o r m o f th e a n i s o t r o p i c d i s p l a c e me n t p a r a m -e t e r , w h i c h i s u s e d i n s o me c o n v e n t i o n a l r e f i n e me n tc a l c u l a t i o n s , e s p e c i a l l y i n b i o mo l e c u l a r c r y s t a l l o g r a p h y ,is B jt = 87r2U t (26)a n d t h e c o r r e s p o n d i n g e x p r e s s i o n f o r T b e c o m e s

3 3 aJBJtatht)T = e x p - a E E h jj= l 1---1 (27)

S i n c e B a n d U a r e e q u i v a l e n t , a p a r t f r o m a c o n s t a n tf a c t o r, a n d U h a s a m o r e d i r e c t p h y s i c a l s i g n i f i c a n c e th a nB , w e r e c o mme n d t h a t t h e u s e o f B b e d i s c o u r a g e d .A b r i e f d i s c u s s i o n o f t h e t r a n s f o r ma t i o n p r o p e r t i e s o f/3 a n d U ma y b e h e l p f u l . T h e c o r r e s p o n d i n g r e p r e s e n t a -t i o n s o f t h e a t o mi c d i s p l a c e me n t v e c t o r a r e

3u - - ~ A ~ a i ( 2 8 )i= 1a n d 3U = ~ A ~ i a i a i , ( 2 9 )

i= 1r e s p e c t i v e l y ( H i r s h f e ld & Ra b i n o v i c h , 1 9 6 6 ). I f th e b a s i so f t h e d i re c t l a t ti c e is c h a n g e d i n s o me ma n n e r , t h e n e wc o m p o n e n t s o f th e d i s p l a c e m e n t v e c t o r u a r e r e l a t e d tot h e o l d o n e s b y l i n e a r t r a n s f o r ma t i o n s , s a y

3z ~ c t i - - E R~ z~ dc j ( 3 0 )j = l

a n d 3A ~ ti - - ~ Q ~ A ~ . ( 3 1 )j = l

T h e e l e me n t s o f t h e t r a n s f o r ma t i o n ma t r i c e s d e p e n d o nt h e o l d a n d n e w b a s e s . I t f o l l o w s f r o m ( 2 2 ) a n d ( 2 5 )t h a t / 3 a n d U t r a n s f o r m a s p r o d u c t s o f t h e c o r r e s p o n d -i n g c o mp o n e n t s o f t h e d is p l a c e m e n t v e c t o r . H e n c e , t h et r a n s f o r ma t i o n r u l e s f o r / 3 a n d U b e c o me

3 3~ 3 t i n E E l~ ig ?n 4 j k" ' j " k ~ (32)j = l k = lan d 3 3u 'i " = E ~_, Q~Q'kkUjk (33)

j = l k = la n d t h u s c o n f o r m t o t h o s e v a l i d f o r te n s o r s o f t h e s e c o n do r d e r (e.g. Sp a i n , 1 9 5 6 ) . T h e t r a n s f o r ma t i o n ma t r i c e s R


7/12

7 76 A T O M IC D I S P L A C E M E N T P A R A M E T E R N O M E N C L A T U R Eand Q a re obv ious ly d i f fe ren t , s i nce the bas i s vec to r s t owhich u , i n i t s two repre sen ta t ions , i s r e fe r red dependin a d i f fe ren t manne r on the bas i s o f t he d i rec t l a t t i c e .Th i s t r ansforma t ion p rope r ty w i l l be i l l us t ra t ed in de t a i li n 2 . 1 . 2 b y t h e o r t h o g o n a l i z a t i o n o f / ~ a n d U .W e c o m m e n t f i n a ll y o n t h e f o r m o f t h e D e b y e - W a l l e rf a c t o r w h e n b o t h t h e d i f f r a c t i o n v e c t o r a n d t h e a t o m i cd i s p l a c e m e n t v e c t o r a r e r e f e r r e d t o t h e s a m e C a r t e s i a nbasis , say e L e 2, e 3. I t i s un ders too d that the use of thisrepre sen ta t ion i s usua l ly , i n c rys t a l l ograph ic p rac t i ce ,p r e c e d e d b y a p p r o p r i a t e t r a n s f o r m a t i o n s ( s e e b e l o w f o ra de t a i l ed example ) .The h and u vec to r s , i n t he Ca r t e s i an repre sen ta t ion ,a r e g i v e n b y

h = htCe, + h2Ce2 + hCe33= E hCie i (34)i= 1

an du : A,~C e, + A,~Ce2 + A~CCe

3: E A(C e j (35)j = l

Al l t he i nd i ces a re g iven he re a s subsc r ip t s , s i nce in t heCar t e s i an repre sen ta t ion the pos i t i on o f t he i nd i ces isi r re l evan t . Note t ha t t he com pon ent s o f h i n (34) have d i -m e n s i o n ( l e n g th ) - l . T h e s c a l a r p r o d u c ts i n ( 1 5 ) a r e n o wr e a d i l y e v a l u a t e d a n d w e o b t a i n f o r t h e D e b y e - W a l l e rf a c t o r

T = e x p -2 7 r 2 E E A A (/ C) hj = ! / = 1

- ex p -2 7r ~- Y~ ~ hj = i / = 1

(36)

w i t h = ( A ~ C A ( / c ) ( 3 7 )a n e l e m e n t o f a n a t o m i c m e a n - s q u a r e d i s p l a c e m e n tt ensor , w i th d im ens ion ( l eng th) 2 , r e fe r red to a Ca r t e s i anbas i s . Th i s r epre sen ta t ion avo ids t he haza rds a ssoc i a t edwi th ca l cu l a t i ons i n ob l ique coord ina t e sys t ems and i su s e d a l m o s t a l w a y s i n l a t t i c e - d y n a m i c a l s t u d i e s a n dt h e r m a l m o t i o n a n a l y s i s , a n d v e r y o f t e n i n c o n s t r a i n e dr e f i n e m e n t o f a t o m i c p a r a m e t e r s .2 . 1 .1 . R e l a t i o n s h i p s b e t w e e n th e a n i s o t r o p i c d i s p l a c e -m e n t p a r a m e t e r s. The d i sp l acem ent pa ram e te r s f li j, U 0and B/ j , g iven by (22) , (25) and (26) , a re so c lose lyre l a t ed tha t t he use o f a l l o f t hem in t he c rys t a l l ograph icl i t e r a t u r e i s n o t o n l y u n n e c e s s a r y b u t c o n d u c i v e t oc o n f u s i o n . T h e r e l a t i o n s h i p s n e e d e d a r e s h o w n i n a

s i n g l e e q u a t i o n , o b t a i n e d b y c o m p a r i s o n o f ( 2 1 ) , ( 2 4 )and (27) :U t : f l J l /27r2aJa l = B Jl /87r 2 . (38)

I t i s t rue t ha t f ewer a r i t hme t i c ope ra t ions a re needed tocom pute T f rom r j r t han f rom U r , bu t w i th t he adven to f c o m p u t e r s t h i s a d v a n t a g e h a s b e c o m e n e g l i g i b l e .2.1.2. C o n s t r u c t i o n o f C a r t e s i a n m e a n - s q u a r e d i s -p l a c e m e n t t e n s o r s . R e f e r r i n g a n A D P t e n s o r t o aC a r t e s i a n b a s i s i s s o m e w h a t l e s s s i m p l e . W e p r o c e e d t oshow how th i s i s done , bo th i n o rde r t o i l l us t ra t e t hea b o v e - o u t l i n e d t r a n s f o r m a t i o n o f t e n s o rs a n d t o p r o v id es o m e b a c k g r o u n d f o r t h e f o l l o w i n g s e c t i o n .C o n s t r u c t a C a r t e s i a n s y s t e m b y t a k i n g , f o r e x a m p l e ,e I a long a , e 3 a long c* , and e 2 a long the v ec to r p rodu c te 3 e I . T h e r e s u l t i n g v e c t o r s

e l = a / a , e 2 = e 3 x e I , e 3 = c * / c * ( 39 )c o m p r i s e a n a p p r o p r i a t e a n d c o m m o n o r t h o n o r m a l s e to f b a s i s v e c t o r s .

Equa t ions (35) and (17) a re bo th express ions fo r u .H e n c e ,

u ---- A~lCe + z3~Ce2 + z3C3Ce3 = z3 xa + A y b + Azc. (40)I f w e t a k e t h e s c a l a r p r o d u c t s o f t h e l e f t- h a n d a n d r i g h t-hand s ides o f (40) w i th e 1 , e 2 and e 3 , we ob ta in t h reel inea r equa t ions o r a ma t r ix eq ua t ion o f t he fo rm

A ,~ = e 2 a e 2 b e 2 . (41)A ~ e 3 a e 3 . b e 3

T h i s i s a t ra n s f o r m a t i o n o f t h e c o m p o n e n t s o f u , r e f e r re dto t he bas i s o f t he d i rec t l a t t i c e , t o i t s Ca r t e s i an compo-n e n t s . T h e t r a n s f o r m a t i o n m a t r i x c a n b e e v a l u a t e d o n c ethe Ca r t e s i an bas i s vec to r s a re de f ined ( e . g . a s above ) . I fwe adopt t he i ndex no ta t ion in t he second l i ne o f (17) ,(41) can be wr i t t en a s

3A ~ c = ~ Ai jZ3JcJ , i = 1, 2, 3, (42 )j = l

wi thA i j = e i . a j .

W e c a n s i m i l a r l y t r a n s f o r m a p r o d u c t o f c o m p o n e n ts o fu be tween the l a t t i c e and Car t e s i an bases , and f ina l ly ana v e r a g e of such a p roduc t :

3 3( A ~ C A ~ C ) - - E E A j m A l n ( z ~ m z ~ x n ) (43)m = l n = l

I f w e n o w m a k e u s e o f ( 2 2 ) a n d ( 3 7 ), w e o b t a i n3 3

: ( 2 7 r 2 ) - I E E a j m al n f l nm , ( 44 )m = l n = l


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K . N . T R U E B L O O D et al. 777which is the express ion for the t ransformat ion of /3 toCar tes ian coordina tes .The or thogona l iza t ion of U proceeds a long s imi la rl ines . The required vers ion of (40) i s now [cf (23)]

u = z3~lCel + z 3 ~ e 2 + z3~Ce3= A~ a * a + A~ /b*b + A ~ c *c (45)

and, fo l lowing the same procedure by which (41) i sobta ined, we a r r ive a t the t ransformat ione I a e I b e I c ' ~

= e 2 a e 2 b e 2 c )3 a % - b e 3 c( o O o )b* 0 At/ ,0 c* ale

(46)

which re la tes the components of u , re fe r red to t h e aiaibasis , to i t s Car tes ian components . Equa t ion (46) can bewr i t ten conc ise ly as

3A~iC = ~ DijA~J, i = 1, 2, 3, (47 )j = lwith

Dij = (e i aj)a j (48)an e lement of the matr ix produc t appear ing in (46) . Thedesi red t ransformat ion is obta ined an a logously to (44) as

3 3U~jt= E E DjmDt, Urn" (49)m = l n = lw i t h . ~ = ( A ~ C A ~ C ) a n d U r n " =

The expl ic i t form of the t ransformat ion matr ix ap-pearing in (41), for the specif ic Cartesian basis definedin (39), is( i b c o s 7 c c o s/ 3 )A = bs i n7 - c s in /3 c os a * . ( 50 )

0 1/c*Of course , a Car tes ian bas is assoc ia ted wi th the di rec tand/or rec iproca l bases can be chosen in an unl imi tednum be r o f wa ys . A m or e ge ne r a l d i sc uss ion o f t heconst ruc t ion of such Car tes ian bases i s g iven e lsewhere(Shmuel i , 1993) .2.2. Equivalent isotropic displacement parameters

I t was pointed out by Hamil ton (1959) and by Wil l i s& Pryor (1975) tha t for minor depar tures f rom isot ropicmot ion, or for anisot ropic d isplacement parametersdeemed to be physica l ly ins ignif icant , i t may bewor thwhi le to replace the s ix-parameter descr ip t ion

of anisot ropic mot ion by a s ingle quant i ty , whichshould descr ibe an isot ropic equiva lent to the weakly ordubiously anisot ropic case .The I UC r C om m iss ion on Jour na l s ( 1986) r e c om -mended tha t ' equiva lent i sot ropic d isplacement param-e te rs ' be computed f rom the express ions proposed byHamil ton (1959) and by Wil l i s & Pryor (1975) . How-ever , a number of d i f fe rent incorrect express ions havea lso been used (F ischer & T i l lmanns, 1988) and th is hasled to considerable confusion. W e f ir st review the properdef ini t ions and demonstra te the i r equiva lence .The f irst definition of the equivalent isotropic dis-placement parameter , a s g iven by Hamil ton (1959) andWil l i s & Pryor (1975) , i s

1U e q = ~ ( U ~ I 1 + U ~ 2 2 + U ~ 3 3 ) , ( 5 1 )with ~ an e lement of a mean-squ are displacementtensor, referred to a Cartesian basis [see (34)-(37)] .The t race of U c , as g iven on the r ight-hand s ide of(51) , i s equiva lent to the sum of the e igenva lues of th ismatr ix . These e igenva lues a re of ten computed, s ince ane igenva lue of the m atr ix U c represents the mean -squaredisplacement a long the cor responding e igenvec tor . Ther ight-hand s ide of (51) can then be in te rpre ted as amean-square displacement averaged over a l l d i rec t ions .Equa t ion (51) can thus be appl ied to the computa t ionof Ueq either by ta king the trace o f U c, w hich is obtainedf rom (44) or (49) , or by us ing the sum of the e igenva luesof U c . H owe ver , i t i s essent ia l to note tha t (51) h oldsonly for the Car tes ian displacemen t tensor U c . I t wi l lg ive incor rec t va lues of U i f U is re fe rred to obl ioueeqbasis vectors and its trace take n instead of that o f U ~7.

S ince the bas is vec tors of the Car tes ian sys tem havethe proper ty e i . e j = 60, a consideration of (35), (37)and (51) readily leads to

Ueq = < I u I = ) - ~ ( U . U ) . ( 5 2 )This equa t ion is a convenient s ta r t ing point for tes t ingthe equiva lence of var ious def init ions o f Ueq. The seconddefinition by Willis & Pryor (1975) is the f irst l ine ofthe next equa t ion:

1(271-2)-1U e q - - ~ t r (/ 3 g ) (53)3 3= (27r2) - ' Y~. 6 / Z /3 Jk gk ,

j , l = 1 k = 13 3= (27r2) - l ~ ~] flJk(aj, at, ) (54 )j=lk=l

3 3_ 1- .~ E 2 < ~ J ~ k > ( a j " a , )j = l k = l- < u . u )-- 3

with g the rea l - space m etr ic tensor . This shows tha t (51)and (53) , the two recom men ded def init ions of Ueq, a reequiva lent .


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7 78 A T O M I C D I S P L A C E M E N T P A R A M E T E R N O M E N C L A T U R EIf we make use of (54) and (38) , two addi t iona le x p r e s s i o n s f o r Ueq can be obta ined:

3 3Ue q = 1 E E U J t a J a l a j ' a t (55)j = l / = 1

a nd 3 3= l (87r2)-1 ~ ~ BJtaJa ta j .a t . (56)U eq j = l l = lThus , (51) , (54) , (55) and (56) a re equ iva len t r epre -sen ta t ions o f the equ iva len t i so t rop ic mean-squa re d is -p l a c e me n t p a r a me t e r U .~ , o b t a i n a b l e fr o m t h e c o mm o n l ye mp l o y e d a n i s o t r o p i c d i s p l a c e me n t p a r a me t e r s .We can a l so a r r ive a t (54) by d i r ec t ly combin ing (44)a n d ( 5 1 ) a n d ma k i n g u s e o f a k n o w n p r o p e r t y o f t h ema t r i x A . W e h a v e

U eq i U c- 5tr3 3:

j=l 1=13 3 3 3= 5 E E 6jr E EAj m a t n < ' ~ m z ~ x n )j = l 1=1 rn= l n= l

3 3 3: .~ (27 r2 ) - i ) - -~ ~ E A j m A j n ~ m nj = l m = l n= l3 3 3= g ( 2 7 r 2 ) - I ~ --~ ~ E a m j A j ni T m nj = l m = l n : l3 3= 3(27r2) -1 E E gmn/~mnm = l n = l3 3

= .~(2'a'2) 1 E E /3mnam "anm = l n : l (57)s ince 3

E TA m k A k n = g m n = a m " ank = l(e .g . Pr ince , 1982) . This de r iva t ion shows tha t the va lueof Ueq does no t depend on the pa r t icu la r fo rm of thema t r i x A , w h i c h t r a n s f o r ms t h e c o mp o n e n t s o f u f r o mthe la t t ice to the Ca r te s ian bas is .Ac t a C r y s t a l lo g r a p h i c a r equ i re s tha t publ i shed va lueso f U _ b e a c c o mp a n i e d b y a n e v a l u a ti o n o f t h e s ta n d a r ddevia t ion (now s tanda rd unce r ta in ty) in these quant i t ie s .The ca lcu la t ion of th i s e s t ima te i s desc r ibed in de ta i lb y Sc h o ma k e r & M a r s h ( 1 9 8 3 ) . A u s e f u l me a s u r e o ft h e a n i s o t r o p y o f t h e me a n - s q u a r e d i s p l a c e me n t t e n s o ri s t h e r a t i o o f i t s m i n i mu m a n d ma x i mu m e i g e n v a l u e s .W e r e c o m me n d t h a t p u b l i s h e d o r d e p o s i te d v a l u e s o f U eqb e a c c o mp a n i e d b y b o t h t h e s t a n d a r d u n c e r t a i n t i e s a n dt h e ra t i o o f t h e mi n i mu m t o t h e ma x i m u m e i g e n v a l u e so f t h e c o r r e s p o n d i n g a n i s o t r o p i c d i s p l a c e me n t t e n s o r s .Bo t h t h e u n c e r t a i n t y o f U ~ a n d t h e r a ti o ma y b e h e l p f u lin judg ing the ex ten t to w hich th e use of Ueq i s jus t i f ied .

2.3. Gr aph ica l represen ta t ions o f the Gauss ianm e a n - s q u a r e d i s p l a c e m e n t m a t r i xJus t a s in te ra tomic d is tances in c rys ta l s a re mos t con-v e n i e n t l y d i s c u s se d i n t e r ms o f a t o m i c c o o r d i n a t e s s u c has x J a j with d imens ion leng th r a the r than in te rms of thed i me n s i o n l e s s c o mp o n e n t s x j, the phys ica l in te rpre ta t iono f a t o mi c d i s p l a c e me n t p a r a me t e r s i s mo s t c o n v e n i e n t i n

t e r ms o f th e me a n - s q u a r e d i s p l a c e me n t m a t r ic e s U w i t he l e me n t s h a v i n g d i me n s i o n ( l e n g th ) 2 .2.3.1. El l ip so ids o f cons tan t p robab i l i t y . In the ab-s e n c e o f a n h a r mo n i c i t y , t h e a n i s o t r o p i c me a n - s q u a r ed i s p l a c e me n t ma t r i x U c a n b e r e g a r d e d a s t h e v a r i -a n c e - c o v a r i a n c e ma t r i x o f a t r i v a r i a t e G a u s s i a n p r o b -ab i l i ty d is t r ibu t ion wi th p robabi l i ty dens i ty func t ionp ( x ) = [ d e t ( U - l ) / ( 2 7 r ) 3 ] l / 2 e x p ( - x r U - l x / 2 ) . ( 58 )

H e r e , x i s t h e v e c t o r o f d i s p l a c e me n t o f t h e a t o m f r o mi ts mean pos i t ion and U-1 i s the inve r se o f the quant i tyde f ined by (25) . I f the e igenva lues o f U a re a l l pos i t ive ,then the sur faces o f cons tan t p robabi l i ty de f ined by thequadra t ic fo rms

x r U - I x = c o n s ta n t ( 5 9)a re e l l ipso ids enc los ing some de f in i te p robabi l i ty fo ra tomic d isp lacement . This i s the bas is fo r the O R T E P' v ib ra t ion e l l ipso ids ' ( Johnson , 1965) tha t a re used inso many i l lus t r a t ions o f c rys ta l s t ruc ture s . The leng thsof the p r inc ipa l axes o f the e l l ipso ids a re p ropor t iona lto the e igen va lues o f the m a t r ix U c e xpressed in thea p p r o p r i a t e Ca r t e s i a n s y s t e m a n d t h e d i r e c t i o n s o f t h epr inc ipa l axes cor re spond to the e igenvec tor s o f th i sma t r i x . T h i s r e p r e s e n t a t i o n c a n n o t b e u s e d w h e n U h a so n e o r mo r e n e g a t i v e e i g e n v a l u e s , b e c a u s e t h e r e s u l t i n gnon-c losed sur faces a re no longe r in te rpre tab le in te rmso f t h e u n d e r l y i n g p h y s i c a l mo d e l .2.3.2. M e a n - s q u a r e d i s p l a c e m e n t s u rf a ce . T h eme a n - s q u a r e d i s p l a c e me n t a mp l i t u d e ( m . s .d . a . ) i n ad i r ec t ion de f ined by a un i t vec tor n i s

(U2)n = nr U n (60)wi th n r e fe r red to the un i t vec tor s aJ /a j , j = 1 , 2 , 3 , par-a l le l to the r ec iproca l v e c t o r s a j , j = 1, 2, 3 , resp ectiv ely.T h e b a s e s (a l / a l , a 2 / a 2 , a 3 /a 3 ) a n d (a l a l , a 2 a 2 , a 3 a 3 )a re mutua l ly r ec iproca l (Hir shfe ld & Rabin ovich , 1966) .N o t e th a t , w h e r e a s x r u - l x i n ( 5 9 ) i s d i me n s i o n l e s s ,n r U n has d im ens ion ( leng th) 2 . As n va r ie s , the sur faceg e n e r a t e d b y n r U n i s n o t a n e ll i p s o id ; i t is u s u a l l y p e a n u tshaped .Su c h s u r f a c e s c a n b e c o n s t r u c t e d e v e n f o r n o n -pos i t ive -de f in i te tensor s and they a re the re fore pa r -t icu la r ly use fu l fo r inspec t ing d i f f e rence tensor s AUb e t w e e n e x p e r i me n t a l U t e n s o r s a n d t h o s e o b t a i n e d f r o mk i n e ma t i c o r d y n a mi c mo d e l s o f a t o mi c a n d mo l e c u l a rmot ion (Hummel , Rase l l i & Bi i rg i , 1990) .


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K . N . T R U E B L O O D e t a l . 7 7 9T h e d i s t i n c t i o n b e t w e e n t h e s u r f a c e s d e f i n e d b y ( 5 9 )a n d ( 6 0 ) h a s o f t e n p r o v e d p u z z l i n g . N o t e t h a t t h e r i g h t -hand s ide of (59) i s a cons tan t , the ( a rb i t r a r i ly chosen)equi -probabi l i ty leve l fo r de f in ing the O R T E P e l l ipso ids .W h e n t h e ma t r i x o f t h e me a n - s q u a r e d i s p l a c e m e n t t e n s o r

i s non-pos i t ive de f in i te , the quadra t ic sur face de f ined by(59) i s no longe r c losed , and no e l l ipso id can be p lo t ted .In con t ra s t , the r igh t -hand s ide of (60) i s the mean-s q u a r e d i s p l a c e me n t a mp l i t u d e ( m . s .d . a . ) i n a g i v e nd i r e c t i o n a n d v a r i e s a s n v a r i e s . O n l y p o s i t i v e v a l u e so f t h e q u a n t i t y d e f i n e d i n ( 6 0 ) a r e me a n i n g f u l f o r a ni n d i v i d u a l a t o m, b u t n e g a t i v e v a l u e s c a n b e me a n i n g f u lw h e n d i f f e r e n c e s in m.s .d .a , va lues a re ca lcu la ted . Them.s .d .a , sur faces can be p lo t ted wi th the a id o f thep r o g r a m P E A N U T o f H u m m e l e t a l . ( 1 9 9 0 ) ; n e g a t i v ev a l u e s a r e p l o t t e d a s d a s h e d c o n t o u r s .

3 . B e y o n d t h e G a u s s i an a p p r o x i m a ti o nThe s i tua t ion i s le ss s t r a igh t forward i f the d is t r ibu t ionf u n c t i o n i s n o t G a u s s i a n . A l a r g e v a r i e t y o f d i f f e r e n ta p p r o x i ma t i o n f o r ma l i s ms , a s w e l l a s d i f f e r e n t n o me n -c la ture fo r s imi la r fo rmu la t ions , i s found in the l i t e ra ture .S u m m a r i e s h a v e b e e n g i ve n b y J o h n so n & L e v y ( 1 9 7 4 ),Z u c k e r & Sc h u l z ( 1 9 8 2 ) , Co p p e n s ( 1 9 9 3 ) a n d K u h s( 1 9 9 2 ). B y v i r t u e o f ( 8 ) , o n e m a y e x p r e s s e i t h e r p .d . f. ( u )o r T ( h ) a s a s e r i e s e x p a n s i o n a n d o b t a i n t h e o t h e rq u a n t i t y b y Fo u r i e r t r a n s f o r ma t i o n .T h e mo s t w i d e s p r e a d a p p r o a c h e s a r e b a s e d o n f o r -ma l i s m s d e v e l o p e d i n s t a ti s ti c s to d e s c r i b e n o n - G a u s s i a nd is t r ibu t ions ( Johnson , 1969) They use a d i f f e ren t ia le x p a n s i o n o f t h e G a u s s s i a n p .d . f . T w o f o r mu l a t i o n sa r e f o u n d i n f r e q u e n t l y u s e d r e f i n e me n t p r o g r a ms , t h ec u m u l a n t o r E d g e w o r t h e x p a n s i o n *

T a w ( h ) = T h ( h ) e x p [ ( 2 7 r t ) 3 " Y J E ~ h j h k h J 3 ! 4 j k l m h h . .+ (27r,) ~EW h j h k t m / a ! + "] (61)

a n d t h e q u a s i - m o m e n t o r G r a m - C h a r l i e r e x p a n s i o n 3 jk l h hT cc (h ) = Th(h)[1 + (27r,) "/cc : / k h t / 3 !

4 j k l m !+ ( 2 7 r , ) ~ G C h j h k h t h m / 4 " - t - . . . ] , (62)w i t h Th ( h ) t h e G a u s s i a n D e b y e - W a l l e r f a c t o r ( se e ~ 1 .4and 2 .1) and 7 j k t , ~ j k l m , . . . the th i rd- , four th - , . . .o rde r ( anha rmonic ) tensor ia l coe f f ic ien ts . The re a re ing e n e r a l 1 0 c u b i c , 1 5 q u a r t i c . . . . t e r ms t h a t e n t e r i n t othe t r ea tment . In s ta t i s t ic s , they a re ca l led cumulan tsa n d q u a s i - mo me n t s , r e s p e c t i v e l y . T h e y c o n s t i t u t e t h ep a r a me t e r s o f t he r e f in e me n t . Va r i o u s s y mb o l s f o r th e s ecoe f f ic ien ts a re sca t te red th rough the l i t e ra ture . Greek* In (61) , (62) , a nd m a ny o f t he r e ma i n i ng e q ua t i ons i n t h i s s e c t i on ,th e s u m m a t i o n c o n v e n t i o n h a s b e e n u s e d . I t i s a s s u m e d t h a t s u m m a t i o noc c ur s o ve r i nd i c e s t ha t a re r e pe a t e d , su c h a s j , k , I a nd m i n t he t e rmson t he r i gh t -ha nd s i de o f (61) a nd (62) .

l e t t e r s a r e c h o s e n h e r e t o c o mp l y w i t h t h e / ~ J k s o f t h eG a u s s i a n c a s e , w h i c h ma y t h u s b e c o n s i d e r e d a s s e c o n d -orde r coe f f ic ien ts . For the same reason , the f ac tor s( 2 7 r ) N / ~ , w i t h N t h e o r d e r o f t h e t e n s o r , a r e i n c l u d e d ,a l s o t o f o l l o w s t a n d a r d p h y s i c a l n o t a t i o n , w h i c h u s e sq = 27rh a s the sca t te r ing vec tor . T he f ac tor s (27r ) Na n d / o r t h e f a c t o r s l / N ! . ( e . g . K u h s , 1 9 9 2 ) a r e s o me t i me so mi t t e d i n t h e l i t e r a t u r e . Fo r c o mp a r a b i l i t y o f f u t u r ere su l t s , i t i s the re fore p roposed tha t on ly coe f f ic ien ts de -f ined a s in (61) and (62) be publ i shed and tha t subsc r ip tsb e u s e d t o i n d i c a t e t h e t y p e o f e x p a n s i o n e mp l o y e d .T h e , , / j k t , ~ j k l m . . . . a r e d i me n s i o n l e s s q u a n t i t i e s . A sp r o p o s e d b y K u h s ( 1 9 9 2 ) , t h e y ma y b e t r a n s f o r me d t oq u a n t i t ie s o f d i me n s i o n ( l e n g th ) N b y

U j k l ' ' ' : ( N ] . ) Y k l ' " / [ ( 2 7 r ) N a J a k a l . . . ] (63)w i t h /3 t o b e re p l a c e d b y % 6 . . . . N o t e t h a t t h is i sa g e n e r a l i z a t i o n o f ( 3 8 ) . I t mu s t b e s t r e s s e d , h o w e v e r ,tha t the ~ j k l , ~ j k l m . . . . a r e s i mp l e e x p a n s i o n c o e f f i c i e n t sa n d ( i n g e n e r a l ) h a v e n o d i r e c t p h y s i c a l me a n i n g . T h et r a n s f o r ma t i o n ( 6 3 ) t h u s h a s n o s u c h me r i t s a s i n t h eG a u s s i a n c a s e a n d s o me r e a l - s p a c e i l l u s t r a t i o n s s h o u l da l w a y s b e g i v e n t o p e r mi t t h e r e s u l t s t o b e a p p r e c i a t e d .T h e b e s t w a y i s c e r t a i n l y t o p l o t t h e c o r r e s p o n d i n g p .d . f .,o b t a i n e d b y i n v e r s i o n o f ( 6 2 ) o r (8 ) . O n l y p r o g r a m s t h a tp r o d u c e s e c t i o n s o f th e p .d . f .s s e e m t o b e c u r r e n t l y a v a i l -a b l e , a l t h o u g h a t h r e e - d i me n s i o n a l v i s u a l i z a t i o n s i m i l a rto O R T E P w o u l d b e h i g h l y d e s i r a b l e . A n o t h e r w a y o fp r e s e n t i n g t h e r e s u l t s i s b y t e n s o r c o n t r a c t i o n ( K u h s ,1 9 9 2 ) . Fo r e v e n - o r d e r t e r ms , f u l l c o n t r a c t i o n y i e l d s a ninva r ian t sca la r ,

N I O - - g j k g l m " " " g p q ~ j k l m . . . p q . ( 6 4 )Fo r t h e G r a m- Ch a r l i e r s e r i e s , t h i s q u a n t i t y i n d i c a t e sf l a t n e s s ( f o r n e g a t i v e v a l u e s ) o r p e a k e d n e s s ( p o s i t i v eva lues) o f the p .d . f . The g j k a r e t h e c o m p o n e n t s o f t h erea l - space me t r ic tensor Note tha t 210 = U e ~ , i .e . (64) isa n e x t e n s i o n o f ( 5 3 ). S i mi l a r l y , v e c t o r i n v a n a n t s ma y b ec a l c u l a t e d f o r o d d - o r d e r t e r ms ,

N ~ = g k l " " " g p q ~ j k l . . . p q , (65)g i v i n g t h e d i r e c t i o n o f ma x i ma l s k e w n e s s . Pa r t i a l c o n -t r a c t i o n o f e v e n - o r d e r t e r ms ,

= g l m ' ' " g pq pjk lm . p q, ~ (66)r e v e a l s t h e d i r e c t i o n s o f f l a t n e s s a n d p e a k e d n e s s .Var ious d iscuss ions in the l i t e ra ture ( see e . g .Kuhs , 1992 , and r e fe rences the re in) ind ica te tha t theG r a m- Ch a r l i e r f o r ma l i s m i s t h e b e s t c h o i c e i n r o u t i n ec r y s t a l l o g r a p h i c w o r k . I n p a r t i c u l a r, i t h a s t h e a d v a n t a g et h a t t h e r e v e r s e Fo u r i e r t r a n s f o r ma t i o n ( 8 ) c a n b e c a r r i e do u t a n a l y t i c a l l y ,

j k l Ip.d.f .(u) - - p .d-f .harm(n)[1 + " ~ G c H j k l ( U ) / 3 ., g j k l m l - -I . . ,+ ~ ' GC " ' j k l m ( U ) / 4 ! -~- "] ( 6 7 )


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7 80 A T O M IC D I S P L A C E M E N T P A R A M E T E R N O M E N C L A T U R Ewith Hjkt... (u ) Herm i te p o lyno mia ls , an d P 'd 'f 'ha rm (U) theh a r mo n i c p a r t o f t h e p .d . f . T h e s e p o l y n o mi a l s a r e t a b u -l a t e d b y J o h n s o n & L e v y ( 1 9 7 4 ) u p t o t h e f o u r t h o r d e ra n d b y Z u c k e r & Sc h u l z ( 1 9 8 2 ) u p t o t h e s ix t h o r d e r [ se ea l s o Co p p e n s ( 1 9 9 3 )] . T h e u s e o f t h e G r a m- - Ch a r l ie r e x -p a n s i o n ( 6 2 ) i s t h e r e f o r e r e c o mme n d e d , a l t h o u g h o t h e rf o r ma l i s ms ma y s o me t i me s b e a d v a n t a g e o u s f o r s p e c i a lp r o b l e ms . I n a n y c a s e , t h e r e s u l t s s h o u l d a l w a y s b ec a r e f u l l y c h e c k e d , e s p e c i a l l y i f h i g h e r - o r d e r t e r ms a r eu s e d m e r e l y t o i mp r o v e t h e a g r e e me n t o f t h e f it . S t ro n gand ex tended nega t ive r eg ions in the p .d . f , ind ica tei n a d e q u a c y o f t h e r e s u l t s . O n e a l s o h a s t o r e me mb e rtha t , wi th anha rmonic r e f inements , the pos i t ions and /~ jkobta ined a re no t necessa r i ly f a i th fu l r epre sen ta t ions o fthe mean and va r iance of the p .d . f . , r e spec t ive ly . Thismu s t b e b o r n e i n mi n d i f b o n d d i s t a n c e s a n d G a u s s i a nd isp lacement e l l ipso ids a re to be de r ived f rom the r e f inedpa rame te r s . In some s i tua t ions , i t may be be t te r tou s e o n l y t h e G a u s s i a n a p p r o x i ma t i o n , e v e n t h o u g h t h ere su l t ing R fac tor s may be h ighe r .

A n o t h e r p o s s i b i l i t y i s t h e e x p a n s i o n o f t h e s o - c a l l e done -pa r t ic le po ten t ia l (OPP) V(u) , which in the c la ss ica ll i m i t [ kT >> V(u) ] i s r e la ted to the p .d . f , by Bol tzmannsta t is t icsp .d . f . (u ) = e x p [ - V ( u ) / k T ] / Z

= p . d . f . ( 0 ) e x p [ - V ( u ) / k T ] (68)wi th Z the pa r t i t ion func t ion . The second equa l i ty i sob ta ined by se t t ing V(0) = 0 .T h e l a t t e r a p p r o a c h w a s f o r mu l a t e d b y D a w s o n &Wil l i s (1967) and Wil l i s (1969) fo r cub ic po in t g roupsa n d l a t e r g e n e r a l i z e d f o r a n y s y mme t r y b y T a n a k a &M a r u mo ( 1 9 8 3 ) . T h e O PP i s w r i t t e n a sV O P P ( u ) o P P , v O P P , , j , , k , , I , ~ O P P , , j , a k , ,l , a m= V~arm(U) + 'jk~ " " " + vjktm . . . + ' ' ' '

( 69)with Vha~ t h e h a r mo n i c ( q u a d r a t ic ) O PP a n d , . y O PPOPP--~ " ~and 6 . -. the th i rd- and four th-orde r coe f f ic ien ts , e -j x t mspec t ive ly , which a re de f ined in a Ca r te s ian sys tem.Since app l ica t ion of (68) and (8) does no t lead toa n a n a l y t i c a l e x p r e s s i o n f o r T ( h ) , t h e a n h a r mo n i c p a r tVanh = V - Vha~m is approxim a ted in (68) by

e x p [ - o va o aPVanh ( u ) / k T ] ~ 1 - Va, h (u) /k T . (70)The f ina l express ions f o r T PP( h ) a r e r a t h e r l e n g t h ya n d ma y b e f o u n d i n T a n a k a & M a r u mo ( 1 9 8 3 ) . Re f i n -O P P O P Pa b l e p a r a me t e r s a r e t h e 7 - - a n d 6 . . O t h e r f o r mu l a -j m J r t "~)ppt i o n s w i t h s i mp l e r e x p r e s s i o n s f o r T ( h ) h a v e b e e ni n t r o d u c e d b y Co p p e n s ( 1 9 7 8 ) , K u r k i - Su o n i o , M e r i s a l o& Pe l t o n e n ( 1 9 7 9 ) a n d Sc h e r i n g e r ( 1 9 8 5 ). N o n e o f t h e sea p p r o a c h e s s e e ms t o h a v e b e e n u s e d mu c h i n c r y s t a l l o -g r a p h i c s t u d i e s a n d f i n a l r e c o mme n d a t i o n s mu s t a w a i tfu r the r deve lo pm ents in th i s f ie ld . I t should a l so be no ted

t h a t t h e O PP a p p r o a c h t r e a t s e a c h a t o m a s a n i n d i v i d u a l(E ins te in ) osc i l la to r , which i s a poor approxima t ion fort i g h t l y b o u n d a t o ms i n mo l e c u l e s .T h e O PP a p p r o a c h i s p h y s i c a l l y me a n i n g f u l o n l y f o rp u r e l y d y n a mi c d i s p l a c i v e d i s o r d e r ( g iv i n g , f o r e x a mp l e ,the d i r ec t ions o f weak and s t rong bond s) and i s l imi ted tor a t h e r s ma l l a n h a r mo n i c i t i e s t h r o u g h t h e a p p r o x i ma t i o n(70) . Occas iona l ly , spec ia l expans ions (e.g. s y m m e t r y -a d a p t e d s p h e r i c a l h a r mo n i c s ) o f p .d . f . (u ) o r T ( h ) h a v eb e e n u s e d f o r s p e c i a l p r o b l e ms (e.g. c u r v i l i n e a r mo -t i o n , mo l e c u l a r d i s o r d e r ) ; s e e J o h n s o n & L e v y ( 1 9 7 4 ) ,Pre ss & Hi i l le r (1973) and Prandl (1981) . Aga in , thesee x p a n s i o n s d o n o t s e e m y e t t o h a v e e n t e r e d r o u t i n ec r y s t a l l o g r a p h i c w o rk . I t s h o u l d b e r e me mb e r e d t h a t t h ec la ss ica l l imi t k T >> V(u) , which i s a ssumed in (68) , maybe f a r f rom the ac tua l s i tua t ion even a t r o o m t e m p e r a t u re .

4 . R e c o m m e n d a t i o n s1 . The te rm ani so t rop ic d i sp lacement parame ters ( ab-brev ia ted ADPs) should be used in r e fe r r ing to theind iv idua l a tomic coe f f ic ien ts in the exponent o f thefac tor tha t desc r ibes the e f fec ts o f a tomic mot ion ands ta t ic d i sp lacement .2 . T h e e l e me n t s o f th e t e n s o r s U a n d / 5 s h o u l d a l w a y sbe supe r sc r ip ted when the r e f inement i s r e fe r red to ac r y s t a l s y s t e m r a t h e r t h a n t o a Ca r t e s i a n s y s t e m. T h i sde f in i t ion fo l lows f rom the de f in i t ion of the e lementso f U a n d / 5 a s c o n t r a v a r i a n t t e n s o r c o mp o n e n t s ( s e e2 .1) . The f r equent use o f subsc r ip ts fo r the ADPs , andspec i f ica l ly fo r those no t r e fe r red to Ca r te s ian sys tems ,i s incons is ten t wi th the i r t ensor ia l p rope r t ie s .3 . W i t h t h e c o mmo n G a u s s i a n a p p r o x i ma t i o n , u s ee i the r the quant i t ie s U ij, w h i c h h a v e d i me n s i o n ( l e n g th ) 2 ,de f ined in (25) , o r the d imens ion le ss /3 / j , de f ined in (22) .4 . W h e n t h e G a u s s i a n a p p r o x i ma t i o n t o t h e p r o b a -b i l i ty dens i ty func t ion i s no t deemed va l id , the use o ft h e G r a m- Ch a r l i e r e x p a n s i o n o f ( 6 2 ) i s r e c o mme n d e d ,a l t h o u g h o t h e r f o r ma l i s ms ma y s o me t i me s b e a d v a n t a -geous for spec ia l p rob lems .5 . S t a n d a r d u n c e r ta i n t i e s o f A D Ps o b t a i n e d f r o m af u l l - ma t r i x r e f i n e me n t a r e v a l i d w i t h i n t h e s y s t e m i nw h i c h t h e r e f i n e me n t i s ma d e . I f A D Ps a r e t r a n s f o r me dto any o the r ax ia l sys tem, Ca r te s ian or no t , then theu n c e r t a i n t i e s ma y a l s o b e c a l c u l a t e d b y t r a n s f o r mi n gt h e o r i g i n a l v a r i a n c e - c o v a r i a n c e ma t r i x t o t h i s n e waxia l sys tem and tak ing the squa re roo ts o f i t sd i a g o n a l e l e me n t s , i.e. the v a r i a n c e s . T h e r e q u i r e dv a r i a n c e - c o v a r i a n c e ma t r i x i s u s u a l l y n o t a v a i l a b l ef o r A D Ps t a k e n f r o m t h e l i t e r a t u r e . H e n c e , a l t h o u g hADPs can s t i l l be t r ansformed, the i r unce r ta in t ie s c a n n o tb e . Ca l c u l a t i o n s i n v o l v i n g p u b l i s h e d A D Ps a n d t h e i r(publ i shed) unce r ta in t ie s should the re fore be r e fe r red tot h e s a me s y s t e m o f c o o r d i n a t e s a s t h e o r i g i n a l re f i n e me n tin o rde r to r e ta in the s ign i f icance of the publ i shedunce r ta in t ie s .


12/12

K . N . T R U E B L O O D e t a l . 7816 . A v o i d u s i n g t h e t e r m ' t e m p e r a t u r e f a c t o r ' , b o t h

b e c a u s e t h e p h e n o m e n o n r e p r e s e n t e d m a y n o t b e d u ee n t i r e l y t o t h e r m a l m o t i o n a n d b e c a u s e t h a t p h r a s e h a si n t h e p a s t b e e n u s e d i n s e v e r a l q u i t e d i s t i n c t s e n s e s(s ee 1 . 5 ) .

7 . A v o i d u s i n g t h e G a u s s i a n a n i s o t r o p i c p a r a m e t e r st h a t a r e n o w u s u a l l y s y m b o l i z e d a s B ij a n d a r e d e f i n e di n ( 2 6 ) . T h e s e q u a n t i t i e s a r e d i r e c t l y p r o p o r t i o n a l t o t h er e c o m m e n d e d U j, t he ra t io be in g 87r 2 .

8 . A v o i d u s i n g A D P s t h a t d o n o t re p r e s e n t m a t r i x e le -m e n t s . I n s o m e e a r l y r e f e r e n c e s a n d c o m p u t e r p r o g r a m s ,c r o s s te r m s w e r e s o m e t i m e s d o u b l e d i n m a g n i t u d e , b e i n gr e p r e s e n te d , f o r e x a m p l e , a s hk ~ 12 i n s t e a d o f 2 h k ~ 12 f o rp r o g r a m m i n g c o n v e n i e n c e . T h i s w a s p o s s i b le b e c a u s et h e m a t r i x r e p r e s e n t i n g t h e A D P i s s y m m e t r i c , w i t h o n l ys i x i n d e p e n d e n t t e r m s . T h i s p r a c t i c e i s n o t f o u n d i nm o d e r n c r y s t a l l o g r a p h i c s o f t w a r e .

9 . P u b l i s h e d v a l u e s o f U e q s h o u l d a l w a y s b e a c -c o m p a n i e d b y t h e i r s t a n d a r d u n c e r t a i n t i e s . T h e r a t i oo f th e m i n i m u m t o th e m a x i m u m e i g e n v a lu e s o f th ec o r r e s p o n d i n g a n i s o t r o p i c d i s p l a c e m e n t t e n s o r s s h o u l da l s o b e p u b l i s h e d , e i t h e r i n t h e p r i m a r y p u b l i c a t i o n i t s e l fo r i n t h e s e c o n d a r y ( d e p o s i t i o n ) p u b l i c a t i o n .

A u t h o r s o f c r y s ta l l o g r a p h i c s o f t w a r e a n d c r y s t a l lo g r a -p h e r s w h o m a i n t a i n t h e i r o w n s o f t w a r e a r e e n c o u r a g e dt o i n t r o d u c e t h e m i n o r m o d i f i c a t i o n s t h a t a r e r e q u i r e df o r t h e i m p l e m e n t a t i o n o f t h e se R e c o m m e n d a t i o n s .

W e a r e e s p e c i a l l y i n d e b t e d t o V e r n e r S c h o m a k e r f o rh i s c r it i c a l r e a d i n g a n d c o u n s e l o n m a n y d r a f t s o f t h i sr e p o r t , t o H a n s B o y s e n f o r h i s s p e c i a l h e l p w i t h 3 , a n dt o R i c h a r d M a r s h , E m i l y M a v e r i c k a n d T u l l i o P i l a t i f o rt h e i r a d v i c e .

R e f e r e n c e sCoppens, P. (1978). Electron and Magnetization Densities inMolecules and Crystals, edi ted by P. Becker, pp. 521-543.New York: Plenum Press .Coppens, P. (1993). International Tables for Crystallography,Vol. B, Reciprocal Space, edited by U. Shmueli, Section1.2.11. Dordrecht : Kluwer Academic Publ ishers .Coppens, P. & Becker, P. J . (1992). International Tables forCrystallography, Vol . C , Mathematical, Ph ysical and Chem-ical Tables, edited by A. J . C. Wilson, Ch. 8.7. Dordrecht:Kluwer Academic Publ ishers .Cowley, J . M. (1992). International Tables fo r Crystallogra-phy, V o l . C , Mathematical, Ph ysical and Ch emical Tables,edi ted by A. J . C. Wilson, pp. 223-245. Dordrecht : KluwerAcademic Publ ishers .Cruickshank, D. W. J . (1956). Acta Cryst. 9 , 747-754 ,754-757 , 757-758 , 915-923 , 1005-1009 , 1010-1011 .Dawson, B. & Willis , B. T. M. (1967). Proc . R. Soc. LondonSer. A, 298 , 307-315 .Debye, P. (1913). V erh. Dtsch. Phys. G es. 15, 738-752.Dun itz, J. D., Sch om aker, V . & Trueblood, K. N. (1988). J .Phys. Chem. 92, 856-867.

Fischer, R. X. & Tillmanns, E. (1988). Acta Cryst. C44,775-776 .Hamil ton, W. C. (1959). Acta Cryst. 12, 609-610.Hirshfe ld, F. L . & Rabinovich, D. (1966). Acta Cryst. 20 ,146-147.Hummel, W. , Rasel l i , A. & Btirgi , H. -B. (1990). Acta Cryst.B46, 683-692 .IUCr Commiss ion on Journals (1986). Acta Cryst. C42, 1100.

Jagodzinski , H. & Frey, F. (1993). International Tables forCrystallography, V o l . B , Reciprocal Space, edi ted by U.Shmueli , Ch. 4 .2 . Dordrecht : Kluwer Academic Publ ishers .Johnson, C. K. (1965). A Fortran Thermal El l ipsoid PlotPro gram fo r Crystal Structure Il lustrations. O R N L R e p o r tNo. 3794. Oak Ridge Nat ional Laboratory, Oak Ridge,Tennessee, USA.Johnson, C. K. (1969). Acta Cryst. A25, 187-194.Johnson, C. K. & Levy, H. A. (1974). International Tablesfor X-ray Crystallography, V ol. IV, pp. 311-336. Birming-ham: Kynoch Press . (Present dis t r ibutor Kluwer AcademicPublishers, Dordrecht.)Kendal l , M. & Stuart , A. (1977). The Advanced Theory ofStatistics, Vol. 1, 4th ed. London: Griffin.Kuhs, W. F. (1992). Acta Cryst. A48, 80-98 .Kurki-Suonio, K., Merisalo, M. & Peltonen, H. (1979). Phys.Scr. 19, 57-63.Maslen, E . N. , Fox, A. G. & O'Keefe , M. A. (1992). Inter-national Tables fo r Crystallography, Vol . C , Mathematical,Physical and Chemical Tables, edited by A. J . C. Wilson,Section 6.1.1. Dordrecht: Kluwer Academic Publishers.Prandl , W. (1981). Acta Cryst. A37, 811-818 .Press , W. & HUller , A. (1973). Acta Cryst. A29, 252-256 .Prince, E. (1982). Ma thema tical Techniques in Crystallographyand Materials Science. New York: Springer-Verlag.Scheringer, C. (1985). Acta Cryst. A41, 73-79 .Schomaker, V. & Marsh, R. E . (1983). Acta Crys t . A39,819-820 .Schwarzenbach, D. , Abrahams, S. C. , Flack, H. D. , Prince , E .& Wilson, A. J . C. (1995). Acta Cryst. A51, 565-569 .Shmueli , U. (1993). International Tables for Crystallography,Vol. B, Reciprocal Space, edited by U. Shmueli, pp. 2-9.Dordrecht : Kluwer Academic Publ ishers .Spain, B. (1956). Tensor Calculus, 2nd ed. , Ch. I. London:Ol ive r and Boyd .Tanaka, K. & Marumo, F. (1983). Acta Cryst. A39, 631-641 .Vainshtein, B. K. (1964). Structure Analysis by ElectronDiffraction. Oxford: Pergamon Press .Vainshtein, B. K. & Zvyagin, B. B. (1993). InternationalTables for Crystallography, V ol. B, Reciprocal Space , edi tedby U. Shmueli , pp. 310-314. Dordrecht : Kluwer Academic

Publishers.Waller, I. (1923). Z. Phys. 17, 398-408.Willis, B. T. M. (1969). Acta Cryst. A25, 277-300 .Willis, B. T. M. (1993). International Tables for Crystallogra-phy, Vol. B, Reciprocal Space, edited by U. Shmueli, Ch. 4.1.Dordrecht : Kluwer Academic Publ ishers .Willis , B. T. M. & Pryor, A. W. (1975). Thermal Vibrationsin Crystallography. Cambridge Univers i ty Press .Wilson, E. B. , Decius, J . C. & Cross, P. C. (1954). MolecularVibrations. New York : McGraw-Hi l l .Zucker, U. H. & Schulz, H. H. (1982). Acta Crys t . A38,568-576 .

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