Molecular Sieves - ACS Publications

2 Zeolite Structure Refinement KARL F. FISCHER

Downloaded by DICLE UNIV on November 7, 2014 | http://pubs.acs.org Publication Date: June 1, 1973 | doi: 10.1021/ba-1973-0121.ch002

Institut fuer Kristallographie, Universitaet des Saarlandes, Saarbruecken, Germany

Zeolite structures pose unconventional problems for crystal structure refinement. These problems arise from positional disorder, pseudo-symmetry, twinning, high mobility of some atoms, and (sometimes) the inaccessibility of single-crystal data. Methods are discussed for investigating split atoms,Si-Aldistribution, pseudo-symmetry, and for dealing with parameter correlation and limited data sets. Some additional techniques which have not been applied to zeolite structures are mentioned.

T o u r i n g t h e l a s t 15 or so years, t h e m a j o r i t y of zeolite c r y s t a l s t r u c t u r e s (almost a l l zeolite m i n e r a l s as w e l l as a n u m b e r of s y n t h e t i c zeolites) h a v e been d e t e r m i n e d b y x - r a y a n a l y s i s , a n d m o s t of t h e m h a v e been refined u s i n g F o u r i e r a n d / o r least-squares techniques. T o o b t a i n a clear p i c t u r e of t h e v a l u e a n d v a l i d i t y of these results, one has t o r e m e m b e r t h a t c r y s t a l d e t e r m i n a t i o n b y n o r m a l x - r a y (or n e u t r o n ) d i f f r a c t i o n leads, after t h e s o l u t i o n of t h e so-called " p h a s e p r o b l e m , " t o t h e e l e c t r o n (or nuclear) d e n s i t y d i s t r i b u t i o n p(x,y,z) i n t h e e l e m e n t a r y c e l l , a v e r a g e d o v e r t i m e a n d space. I t is c o m p u t e d b y " F o u r i e r s y n t h e s i s " f r o m t h e s t r u c t u r e factors F ba(hkl), whose m a g n i t u d e (but n o t phase) is o b t a i n e d d i r e c t l y f r o m d i f f r a c t i o n i n t e n s i t y measurements. T i m e average essent i a l l y means t h a t t h e c o m p u t e d d e n s i t y d i s t r i b u t i o n is d e r i v e d f r o m t h e d i s t r i b u t i o n f u n c t i o n of t h e e l e c t r o n (or n u c l e a r ) d e n s i t y of a l l t h e a t o m s of t h e c e l l ( w h i c h are k e p t fixed i n t h e i r places, t h u s g i v i n g a p o i n t f u n c t i o n for t h e n u c l e u s ) . T h i s f u n c t i o n is t h e n m o d i f i e d b y i n d i v i d u a l " s m e a r i n g " functions resulting from the t h e r m a l vibrations. Generally, a Gaussian d i s t r i b u t i o n is used for t h i s purpose. Space average means t h e s u p e r p o s i t i o n of a l l e l e m e n t a r y cells of t h e c r y s t a l u n d e r i n v e s t i g a t i o n (or m o r e e x a c t l y of one mosaic b l o c k of t h i s c r y s t a l , a s s u m i n g t h a t a l l b l o c k s are identical other t h a n minor orientational disorder). Q

F o r c r y s t a l - c h e m i c a l discussion of t h i s d e n s i t y d i s t r i b u t i o n , each d e n s i t y p e a k m u s t be i n t e r p r e t e d as a n a t o m of d i s t i n c t c h e m i c a l species, associated w i t h p a r a m e t e r s d e s c r i b i n g i t s p o s i t i o n , v i b r a t i o n a l c h a r a c t e r istics, a n d t h e l i k e . F o r a n o r m a l c r y s t a l s t r u c t u r e of a w^ell-defined 31 In Molecular Sieves; Meier, W., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.

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c o m p o u n d w i t h u n a m b i g u o u s s y m m e t r y , t h i s assignment of a n a t o m i c species t o a peak is no p r o b l e m . Zeolites, u n f o r t u n a t e l y , are i n m o s t cases n o t n o r m a l b u t " p a t h o l o g i c a l " c r y s t a l s i n m a n y respects; e q u i p o i n t s for H 0 a n d cations m a y n o t be f u l l y o c c u p i e d , a n d t h e c o r r e s p o n d i n g l y s m a l l electron d e n s i t y peaks c a n n o t a l w a y s be c l e a r l y l a b e l e d for a specified a t o m . T h e p-peaks m a y be u n u s u a l (e.g., elongated ellipsoidal) i n shape, indicating highly anisotropic v i b r a t i o n amplitudes a n d / o r positional disorder. T h e d i s o r d e r of S i a n d A l a p p e a r i n g i n t h e t e t r a h e d r a l (T-) sites a n d t h e c o r r e s p o n d i n g l y s m e a r e d p-peaks of T - a t o m s (as w e l l as of t h e d i r e c t l y b o n d e d O-atoms) m a y be caused b y t w i n n i n g , w h i c h also c a n l e a d t o t h e a s s u m p t i o n of a w r o n g s y m m e t r y of the s t r u c t u r e . T w i n n i n g is i n t u r n f a v o r e d b y t h e p s e u d o - s y m m e t r y of m a n y f r a m e w o r k s . Downloaded by DICLE UNIV on November 7, 2014 | http://pubs.acs.org Publication Date: June 1, 1973 | doi: 10.1021/ba-1973-0121.ch002

2

Justification of Refinement Is extensive zeolite s t r u c t u r e refinement j u s t i f i e d b e y o n d m e r e s c i e n tific c u r i o s i t y ? A n a n s w e r c a n r e a d i l y be g i v e n : a n o r m a l , unrefined zeolite s t r u c t u r e m a y p r o v i d e sufficient k n o w l e d g e a b o u t t h e f r a m e w o r k b y defining m a x i m u m c h a n n e l a p e r t u r e a n d cage size. I t w i l l n o t , h o w e v e r , p r o v i d e a clear p i c t u r e of t h e d i s t r i b u t i o n of a t o m s i n s i d e t h e pores, n o r w i l l t h e S i - A l d i s t r i b u t i o n i n t h e f r a m e w o r k be k n o w n i n m o s t cases; t h i s f r a m e w o r k s t r o n g l y influences t h e c a t i o n d i s t r i b u t i o n as w e l l as t h e v a r i a t i o n of t h e (electric) p o t e n t i a l i n the c a v i t i e s . T h e i n a d e q u a t e k n o w l e d g e o b t a i n e d b y c o n v e n t i o n a l m e t h o d s of a b o u t 15 years ago is i l l u s t r a t e d b y t h e c o n t r a s t between refinements presented i n t h i s v o l u m e a n d t h e c o n t e n t of a p a p e r b y t h e a u t h o r o n g m e l i n i t e w h i c h describes a n average s t r u c t u r e (or s y m m e t r i z e d s t r u c t u r e , see b e l o w ) . S i - A l d i s t r i b u t i o n appears t o be r a n d o m ; u n u s u a l l y h i g h t e m p e r a t u r e factors i n d i c a t e t h e s u p e r p o s i t i o n of structures w i t h probably higher ordering a n d lower s y m m e t r y ; low electron d e n s i t y peaks, often w i t h p o o r shape a n d i m p o s s i b l e distances, p r e v e n t a discussion of cage a n d c h a n n e l filling as l o n g as t h e t r u e s y m m e t r y is u n k n o w n . U n c o n v e n t i o n a l refinement m e t h o d s give d e t a i l e d results of present d a y zeolite s t r u c t u r e . T h e t w o m a i n refinement methods are F o u r i e r synthesis of p(x,y,z) (or m o d i f i c a t i o n s thereof) a n d least-squares c o m p u t a t i o n s w h i c h d e t e r m i n e t h e best n u m e r i c a l p a r a m e t e r s t o describe p. T h e y are o b t a i n e d b y minimizing £ w(hkl)-(\F (hkl)\ hkl ohB

-

\F i(hkl\)* oa

= ZwA

2

w h e r e \/w(hkl) is t h e square of t h e s t a n d a r d d e v i a t i o n of each o b s e r v e d B o t h m e t h o d s are t h e o r e t i c a l l y e q u i v a l e n t a n d r e l a t e d b y a w e l l k n o w n m a t h e m a t i c a l expression (1). T h e s t a n d a r d procedures of c r y s t a l s t r u c t u r e refinement are t r e a t e d i n n u m e r o u s b o o k s . T h e r e f o r e , we c o n -

In Molecular Sieves; Meier, W., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.

2.

FISCHER

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Structure Refinement

s i d e r o n l y a few a d d i t i o n a l p o i n t s , w h i c h m a y be h e l p f u l i n s t u d y i n g structure details. Positional Disorder vs. Temperature Vibration


I n a single c r y s t a l i t is easy t o decide w h e t h e r a n elongated electron d e n s i t y peak is best described b y one f u l l y o c c u p i e d p o s i t i o n w i t h one h i g h v i b r a t i o n a m p l i t u d e or b y t w o r o u g h l y h a l f - o c c u p i e d e q u i p o i n t s w h i c h are n o t v e r y close [e.g., 1 A , (2)]. T h i s is because t h e c o n v o l u t i o n of t h e e l e c t r o n d e n s i t y of one a t o m w i t h a w i d e G a u s s i a n d i s t r i b u t i o n (corres p o n d i n g t o a large r o o t m e a n square v i b r a t i o n a m p l i t u d e ) gives a different d i s t r i b u t i o n of electron d e n s i t y i n space (p-peak shape) c o m p a r e d w i t h t h e o v e r l a p (addition) of t w o separate a t o m s w i t h h a l f o c c u p a n c y a n d a n a r r o w G a u s s i a n s m e a r i n g f u n c t i o n . C l e a r l y , t h e p e a k shape is w e l l defined for converged F o u r i e r series o n l y , w h i c h means t h a t a l l a v a i l a b l e F data m u s t be i n c l u d e d . I n general, t h e reflections o b t a i n e d i n t h e M o - K a range of r e c i p r o c a l space are sufficient. D a t a l i m i t e d b y C u - K a f r e q u e n t l y are n o t enough for close s p l i t a t o m s because of series t e r m i n a t i o n errors. T h e closer t h e distance between t h e s p l i t a t o m s , t h e h i g h e r are t h e r e q u i r e m e n t s for t h e d a t a . W i t h p r e s e n t - d a y a c c u r a c y of i n t e n s i t i e s , i t seems unsafe t o recognize s p l i t H 0 w ith distances m u c h b e l o w 0.5 A . F o r peaks elongated f r o m t w i n n i n g , see M e i e r ' s c h a p t e r (3). o b s

2

T

I f different s t r u c t u r e models—e.g., different a r r a n g e m e n t s of cations a n d / o r H 0 m o l e c u l e s — h a v e t o be considered, each s h o u l d be refined u n t i l convergence. T h e n , t h e models c a n be c o m p a r e d o n t h e basis of t h e i r w e i g h t e d R factors : 2

u s i n g a s t a t i s t i c a l test devised b y H a m i l t o n (4).

T h e result indicates not

o n l y w h i c h m o d e l is t o be preferred, b u t i t also p r o v i d e s a p r o b a b i l i t y measure (significance level) for r a n k i n g one m o d e l over a n o t h e r . Si-Al Distribution S o m e t i m e s i t is dangerous t o estimate t h e s t a t i s t i c a l A l c o n t e n t of a t e t r a h e d r a l ( T — ) site o n t h e basis of t h e a v e r a g e d T - 0 distance for t h i s t e t r a h e d r o n . T - 0 varies w i t h T - O - T angle a n d t h e c a t i o n c o o r d i n a t i o n (5) a n d p o s s i b l y other factors s u c h as h y d r o g e n bonds. F o r b o n d lengths w h i c h are s y s t e m a t i c a l l y shortened b y i d e a l i z e d h i g h s y m m e t r i e s see M e i e r ' s discussion (3). D e s p i t e t h e s i m i l a r i t y of t h e electron densities of S i a n d A l a t o m s , t h e i r s c a t t e r i n g factor curves / are s t i l l different enough t o a l l o w a refinem e n t of t h e S i - A l d i s t r i b u t i o n of a T - s i t e a c c o r d i n g t o : / T = w-/si + (1 -

m)-/ i A



34

MOLECULAR SIEVES

m b e i n g refined as a d d i t i o n a l p a r a m e t e r f o r each T - s i t e . T h e p r o c e d u r e (6) i s successful f o r essentially o r d e r e d s t r u c t u r e s (m ^ 0.2 o r m ^ 0.8) i f t h e f o l l o w i n g c o n d i t i o n s a r e m e t . I n t e n s i t y measurements m u s t be done w e l l b e y o n d s i n 0/A = 0.9 ( M O - K a d a t a r e q u i r e d ) ; one scale f a c t o r f o r a l l d a t a (or a t least n o l a y e r - l i n e o r s i n 0/X dependence of scale f a c t o r s ) ; q u a l i t y of d a t a sufficient t o d e t e r m i n e a n i s o t r o p i c t e m p e r a t u r e p a r a m eters; t r u e single c r y s t a l (see b e l o w ) . P r o g r a m s exist f o r r e f i n i n g i n dependent m p a r a m e t e r s as w e l l as f o r m ' s c o n s t r a i n e d b y t h e c h e m i c a l a n a l y s i s of t h e s a m p l e . T h e y c a n also b e a p p l i e d t o o t h e r m i x e d p o p u l a t i o n s (e.g., cations) w i t h a l a r g e r difference of t h e a t o m i c s c a t t e r i n g powers u n d e r less rigorous c o n d i t i o n s (e.g., 7, 8). I n a l l cases, h o w e v e r , t h e c o r r e l a t i o n effects between m a n d t h e t e m p e r a t u r e p a r a m e t e r s ( a n d sometimes also t h e scale factor) m u s t be considered c a r e f u l l y . F o r solutions t o t h i s p r o b l e m for S i - A l d i s t r i b u t i o n , where i t is serious, see R e f s . 6,9. Neutron Diffraction T h e s m a l l r e l a t i v e difference i n x - r a y s c a t t e r i n g p o w e r of S i a n d A l is r o u g h l y d o u b l e d f o r n e i i t r o n d i f f r a c t i o n (b = 0.42 a n d 0.35 r e s p e c t i v e l y ) , w h i c h t h u s s h o u l d be preferred for i n v e s t i g a t i n g S i - A l d i s t r i b u t i o n s . I t also offers t h e p o s s i b i l i t y of l o c a t i n g H (or D ) atoms, t h u s p r o v i d i n g i n f o r m a t i o n o n H bridges, etc. T h e l i m i t e d n u m b e r of n e u t r o n d i f f r a c t i o n studies o n zeolite s t r u c t u r e s u n t i l n o w (owing t o t h e l o w flux of t h e n e u t r o n sources a n d consequently t h e necessity of l o n g measurements o n large crystals) w i l l c e r t a i n l y be increased since h i g h flux reactors are n o w b e i n g u s e d (e.g., i n G r e n o b l e , F r a n c e , where also t h e n e w diffractometer t y p e " h e d g e h o g " (10) w i l l a l l o w u s t o measure several d o z e n s c a t t e r i n g intensities f r o m a s m a l l c r y s t a l a t t h e same t i m e ) . O n e slight d i s a d v a n t a g e of n e u t r o n d i f f r a c t i o n comes f r o m t h e i n dependence of s c a t t e r i n g power of s i n 6 ( w h i c h is otherwise q u i t e useful). D i s r e g a r d i n g p o s i t i o n a l disorder, a l l n u c l e a r d e n s i t y peaks are s h a p e d b y t h e r m a l v i b r a t i o n o n l y a n d therefore c o n t a i n n o i n f o r m a t i o n a b o u t t h e n a t u r e of t h e a t o m (besides t h e t r i v i a l f a c t t h a t t h e p e a k cannot be a t t r i b u t e d t o a nucleus of l o w e r s c a t t e r i n g cross section). Pseudosymmetry-Twinning P s e u d o s y m m e t r y creates some of t h e most difficult p r o b l e m s i n crystal structure work. T h e pseudo-symmetric crystal m a y be a true single c r y s t a l w i t h s m a l l d e v i a t i o n s f r o m a higher s y m m e t r y (case 1) or (case 2) a t w i n p r o d u c i n g a " s y m m e t r i z e d s t r u c t u r e " (3) w i t h a s i m u l a t e d higher s y m m e t r y . I n b o t h cases, t h e t r u e s t r u c t u r e of l o w s y m m e t r y [ H - s t r u c t u r e (3, 11)] m a y g i v e t h e same reflections w i t h o n l y slight i n t e n s i t y v a r i a t i o n s (a), or a d d i t i o n a l reflections (b) exist w h i c h are weak, so


2.

35


FISCHER

t h a t a r o u t i n e s o l u t i o n of t h e i r phase p r o b l e m i s n e a r l y hopeless f o r zeolite structures.

( I n contrast t h e pseudo-structure

of h i g h s y m m e t r y [ A -

s t r u c t u r e (3,11)] c a n n o r m a l l y be s o l v e d b y t h e u s u a l techniques.) A m e t h o d f o r i n v e s t i g a t i n g a l l f o u r t y p e s of p r o b l e m s (of w h i c h 2 a i s t h e most c o m p l i c a t e d a n d l b is t h e least c o m p l i c a t e d ) has been d e r i v e d b y M e i e r a n d V i l l i g e r (12).

T h e distance least squares ( D L S ) procedure

uses t h e w e l l - k n o w n distances D f o r S i - O , A l - O , 0 - 0 ( a n d e v e n t u a l l y others) of t h e f r a m e w o r k a n d refines a t o m i c p a r a m e t e r s b y a least-squares procedure m i n i m i z i n g M

D

= 2w(D, l a

Dknown)

2

where w corresponds t o w j i n R e f . 12 (p. 4 1 3 ) . T h e s u m m a t i o n i s done over a l l t h e independent distances of t h e s t r u c t u r e u n d e r t h e s y m m e t r y r e s t r i c t i o n of t h e assumed space g r o u p . T h e p o s i t i o n a l i n p u t p a r a m e t e r s for t h e first c o m p u t a t i o n of Z> i a r e t a k e n f r o m t h e A - s t r u c t u r e . A f t e r convergence of t h e D L S refinement, t h e final a t o m i c p o s i t i o n s p r o v i d e a s t a r t i n g set f o r t h e u s u a l s t r u c t u r e f a c t o r least-squares or F o u r i e r refinem e n t . I n t h e case of m o r e t h a n one possible space g r o u p f o r t h e H s t r u c t u r e , a n d therefore different s y m m e t r y r e s t r i c t i o n s , t h e best fit (lowest M ) indicates t h e m o s t p r o b a b l e s y m m e t r y . ( I n d o u b t f u l cases, t h e H a m i l t o n test (4) c a n also b e a p p l i e d t o t h e D L S results.) T h e m e t h o d has been successfully u s e d i n t h e d e t a i l e d i n v e s t i g a t i o n of zeolites of t y p e a a n d b , o n t h e basis of single c r y s t a l a n d p o w d e r d i f f r a c t i o n d a t a : (13, H, 15, 16, 17). A m o r e d e t a i l e d t r e a t m e n t of p s e u d o - s y m m e t r y a n d t w i n n i n g i n c l u d i n g n e w a p p l i c a t i o n s of t h e D L S m e t h o d i s g i v e n i n another c h a p t e r (3). T h e D L S p r o g r a m (18) exists also i n a v e r s i o n u s i n g a d d i t i o n a l r e s t r i c t i o n s : D L S R (14) •


2

ca

D

Parameter Correlation P s e u d o - s y m m e t r i c structures a r e k n o w n t o cause h i g h c o r r e l a t i o n coefficients r between parameters (28)—e.g., p* a n d p i n p a r t i c u l a r ( b u t n o t necessarily o n l y ) between those r e l a t e d b y p s e u d o - s y m m e t r y o p e r a t i o n s ; r is l i m i t e d t o tj

jf

tj

-

1 ^ Tit $ 1

a n d is a p p r o x i m a t e l y zero for a l l ij p a i r s i n n o r m a l least-squares refinements. T h e r c a n be c o m p u t e d d u r i n g f u l l - m a t r i x least-squares r u n s . I f one or m o r e r are near — 1 o r + 1 , t h e y l e a d a t best t o serious u n d e r e s t i m a t i o n of t h e s t a n d a r d d e v i a t i o n s f o r t h e parameters i n q u e s t i o n ; i n o t h e r cases the least-squares c o m p u t a t i o n fails t o w o r k a t a l l . T h e s e h i g h c o r r e l a tions are caused b y t h e close p r o p o r t i o n a l i t y of tj

tj


36

MOLECULAR SIEVES

for the majority of the F(hkl), producing an unusually high off-diagonal element dFhki w r— E hkl Opi hk

dFhu — Opj

of the least-squares matrix. (This proportionality is sometimes exact for all F's of a symmetrized structure. Thus, it is impossible to find the deviations from the symmetrized A-structure (which lead to the true H-structure) by structure-factor least-squares techniques. A n r of +1 or —1 corresponds to a dependence between the two parameters p and pj which does not change the quantity 2w • A and causes a singularity in the leastsquares matrix.) Therefore, one should compute a least-squares cycle with a full matrix program to detect possible correlations. (If diagonal or block-diagonal programs must be used because of many parameters and/or unsufficient storage space in the computer, at least all the positional parameters should be combined in one full matrix block for this test run.) tj

t


2

Serious correlations can sometimes be reduced by a careful investigation of the corresponding dF/dp and their dependence on different classes of hkl, regions in reciprocal space, etc. and a critical elimination of those reflections for which both derivatives are nearly proportional to one another. This method was successful for a highly pseudo-symmetric structure of type l a with several |r | ranging between 0.80 and 0.98 (19). Reducing the numbers of observations increases the estimated standard deviations of the parameters. Values of \r \ < 0.4 or 0.5 are not serious. itj

y

tj

Powder Diffraction Data One serious disadvantage of powder data is accidental peak overlap. It can be partially overcome by careful measurement of the peak profiles— e.g., by step scanning. The experimental profile is then analyzed in terms of parameters for a normal profile. It is defined by a function, generally consisting of symmetric and asymmetric parts, and parameters vary systematically with 6. The gross intensity of a multiple peak can thus be split into its components. Of course, the higher the difference in 0 is, the better defined will be the individual intensity of the components; this has to be taken into account in weighting the data (see e.g., Ref. 17). A versatile program for analyzing powder diffraction profiles has been written by Thoni (20). For the analytical representation of peak profiles for cubic powder data see Ref. 21. One advantage of powder data compared with single crystal intensities is that secondary extinction is greatly reduced for powder diffraction. Since secondary extinction is worse for reflections of high intensities and low glancing angles (where peak overlap in the powder diagram is less frequent), it is sometimes worthwhile to supplement single crystal data with these few structure factors derived from powder measurements.


2.

FISCHER


37

Additional Remarks


N e a r l y l i q u i d - l i k e b e h a v i o r of exchangeable cations a n d sorbed m o l e cules i n a zeolite c a n be i n v e s t i g a t e d . S i m p s o n a n d S t e i n f i n k (22) f o r m u l a t e d l i q u i d s c a t t e r i n g functions for r a n d o m o r i e n t a t i o n of molecules a n d different types of u n i f o r m d i s t r i b u t i o n of b o t h molecules a n d cations i n s i d e a large cage. T h e y h a v e been successfully a p p l i e d t o v a r i o u s f a u j a s i t e - t y p e structures (21, 22, 23, 24). L i q u i d s c a t t e r i n g (like s c a t t e r i n g f r o m other deviations f r o m a n o r m a l c r y s t a l structure) also appears i n t h e b a c k g r o u n d (see b e l o w ) . N e u t r o n a n d x - r a y d i f f r a c t i o n d a t a c a n be c o m b i n e d for least-squares refinement (25) i n s t e a d of r e f i n i n g b o t h d a t a separately a n d c o m p a r i n g t h e results. T h i s j o i n t refinement has some a d v a n t a g e s : T h e increased a m o u n t of d a t a w i l l a l l o w refinement to l o w e r s t a n d a r d d e v i a t i o n s of t h e parameters. B e c a u s e of t h e difference i n s c a t t e r i n g m e c h a n i s m , t h e h y p e r p l a n e representing 2 w . A a b o v e t h e p a r a m e t e r space a p p a r e n t l y has a different t o p o l o g y . T h u s , no h i g h correlations exist between x - r a y a n d n e u t r o n p o s i t i o n a l parameters despite t h e i r close s i m i l a r i t y (25). Therefore, one s h o u l d check w h e t h e r a j o i n t refinement w o u l d reduce some of t h e severe c o r r e l a t i o n effects a r i s i n g f r o m p s e u d o - s y m m e t r y . Also, split atoms s h o u l d be m o r e easily detected t h a n b y n e u t r o n or x - r a y measurem e n t alone. F i n a l l y , j o i n t refinement c a n help i d e n t i f y a t o m s w i t h l o w s t a t i s t i c a l o c c u p a n c y . W i t h present-day n e u t r o n d i f f r a c t i o n e q u i p m e n t , i t w i l l be n o p r o b l e m t o measure b o t h x - r a y a n d n e u t r o n d a t a f r o m t h e same c r y s t a l . 2

T h e opposite signs for t h e n e u t r o n s c a t t e r i n g p o w e r of h y d r o g e n a n d d e u t e r i u m (—0.38 a n d + 0 . 6 5 ) offers the p o s s i b i l i t y for i n v e s t i g a t i n g (slow) self-diffusion between different w a t e r sites a n d / o r l o c a l i z a t i o n of w a t e r molecules w i t h different m o b i l i t y if d i f f r a c t i o n experiments are c a r r i e d out for a s a m p l e where D 0 is exchanged i n steps vs. H 0 . 2

2

I n D L S c o m p u t a t i o n s , constant a t o m i c distances are used a n d h a v e been v e r y useful. Therefore, i t appears w o r t h w h i l e t o c o n s t r a i n t h e p o s i t i o n a l parameters of the f r a m e w o r k w i t h respect t o t h e k n o w n distances for S i - O , A l - O , a n d 0 - 0 for n o r m a l s t r u c t u r e f a c t o r least-squares c o m p u t a tions. C o n s t r a i n e d refinement essentially reduces t h e n u m b e r a n d / o r v a r i a b i l i t y of t h e p a r a m e t e r s a n d c a n be h e l p f u l for w o r k w i t h l i m i t e d d a t a sets (e.g., for p o w d e r diffraction). C o n s t r a i n e d refinement has been d i s cussed b y P a w l e y (26, 27). So far, o n l y n o r m a l d i f f r a c t i o n ( w i t h s h a r p m a x i m a at t h e p o i n t s of r e c i p r o c a l lattice) has been m e n t i o n e d . I n v e s t i g a t i o n of b a c k g r o u n d s c a t t e r i n g c a n p r o v i d e extended i n f o r m a t i o n o n v a r i o u s k i n d s of c r y s t a l defects (well b e y o n d s t a c k i n g f a u l t s a n d t h e like) as has been d e m o n s t r a t e d e.g. o n metal a n d feldspar structures.


38

MOLECULAR SIEVES

Acknowledgment It is a pleasure to thank W. M . Meier for valuable discussions and for advance information on material prior to publication (3). I also thank J. B. Uytterhoeven for providing literature data (21). Literature Cited 1. Cochran, W., Acta Cryst. (1948) 1, 138. 2. Fischer, K. F., Scramm, V., ADVAN. CHEM. SER. (1971) 101, 250.


3.

Meier, W. M., ADVAN. CHEM. SER. (1973) 121, 39.

4. Hamilton, W., "Statistics in Physical Sciences," pp. 157-162, Table V, Ronald Press, New York, 1964; Acta Cryst. (1965) 18, 502. 5. Brown, G. E., Gibbs, G. V., Ribbe, P. H., Amer. Mineral. (1969) 54, 1044. 6. Fischer, K., Tschermaks Mineral. Petrograph. Mitteilungen (1965) 10, 203. 7. Fischer, K., Z. Krist. (1968) 127, 110. 8. Fischer, K., Amer. Mineral. (1966) 51, 814. 9. Fischer, K., Zehme, H., Schweiz. Mineral. Petrograph. Mitteilungen (1967) 47, 163. 10. Klar, B., International Union of Crystallography, Kyoto Congress (1972) Paper XXIV-22. 11. Katz, L., Megaw, H. D., Acta Cryst. (1967) 22, 639. 12. Meier, W. M., Villiger, H., Z. Krist. (1969) 129, 411. 13. Barrer, R. M., Villiger, H., Z. Krist. (1969) 128, 352. 14. Gramlich, V., Meier, W. M., Z. Krist. (1971) 133, 134. 15. Schramm, V., Fischer, K. F., ADVAN. CHEM. SER. (1971) 101, 259. 16. Baerlocher, Ch., Meier, W. M., Z. Krist. (1972) 135, 339. 17. Baerlocher, Ch., Barrer, R. M., Z. Krist. (1972) 136, 245. 18. Villiger, H., DLS Program and Manual, E T H Zuerich, 1969. 19. Schollmayer, R., Master's Thesis, Saarbruecken (1973). 20. Thöni, W., CUFIT Program and Manual, ETH Zuerich, 1972. 21. Mortier, W. J., Ph.D. Thesis, Leuwen (1972). 22. Simpson, H. D., Steinfink, H., Acta Cryst. (1970) A26, 158. 23. Simpson, H. D., Steinfink, H., J. Amer. Chem. Soc. (1969) 91, 6225. 24. Simpson, H. D., Steinfink, H., J. Amer. Chem. Soc. (1969) 91, 6229. 25. Duckworth, J. A. K., Willis, B. T. M., Pawley, G. S., Acta Cryst. (1969) A25, 482. 26. Pawley, G. S., in "Advances in Structure Research by Diffraction Methods," Vol. 4, Pergamon, Vieweg, 1972. 27. Pawley, G. S., Advanced Study Institute, Aarhus University, Denmark, Lecture Notes, 1972. 28. Geller, S., Acta Cryst. (1961) 14, 1026. RECEIVED February 15, 1973.


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