Simulation of Fiber Diffraction Patterns - American Chemical Society

4 Simulation of Fiber Diffraction Patterns E. SUZUKI, R. D. B. FRASER, T. P. M A C R A E , and R. J. ROWLANDS

Downloaded by CORNELL UNIV on September 21, 2016 | http://pubs.acs.org Publication Date: November 17, 1980 | doi: 10.1021/bk-1980-0141.ch004

Division of Protein Chemistry, CSIRO, 343 Royal Parade, Parkville, Victoria 3052, Australia

Many polymeric materials have a fibrous texture i n which elongated particles with an ordered internal structure are preferentially aligned p a r a l l e l to a particular direction termed the fiber a x i s . Diffraction patterns obtained from such materials contain information about both the particles and the matrix i n which they are embedded. This matrix may consist of amorphous polymer of the same or different composition to the p a r t i c l e or may be a l i q u i d . The factors which influence the diffraction pattern and about which it therefore provides information include the following: 1. Intraparticle (a) molecular structure (b) random and cumulative disorder (c) periodic distortions (d) p a r t i c l e dimensions 2.

Interparticle

3.

Matrix

(a) d i s t r i b u t i o n function for p a r t i c l e orientation (b) i n t e r p a r t i c l e ordering

(a) mean electron density (b) r a d i a l electron density d i s t r i b u t i o n . In the past, p r a c t i c a l d i f f i c u l t i e s of data c o l l e c t i o n have placed severe restrictions on the extraction of this information. However the recent development of a technique for calculating a quasi-continuous map of the specimen intensity transform (1) affords the p o s s i b i l i t y of r e a l i s i n g the full potential of fiber diffraction data. Comparison o f t h e observed specimen i n t e n s i t y t r a n s f o r m w i t h t h a t c a l c u l a t e d f o r a model o f t h e s t r u c t u r e o f t h e specimen p r o v i d e s a powerful t e s t o f t h e c o r r e c t n e s s o f the model. I n t h e p r e s e n t c o n t r i b u t i o n we d e s c r i b e some p r e l i m i n a r y a t t e m p t s t o s i m u l a t e f i b e r d i f f r a c t i o n p a t t e r n s . When t h e o b s e r v e d a n d simulated i n t e n s i t y transforms a r e d i s p l a y e d v i s u a l l y they provide a u s e f u l guide t o t h e progress o f a s t r u c t u r e refinement as w e l l

0-8412-05 89-2/ 80/47-141-061 $05.00/0 © 1980 American Chemical Society French and Gardner; Fiber Diffraction Methods ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

62

FIBER

DIFFRACTION

METHODS

as p r o v i d i n g h i n t s f o r f u r t h e r i m p r o v e m e n t s . S i n c e t h e i n f o r m a t i o n i s stored i n d i g i t a l form i t i s p o s s i b l e t o d e v i s e measures o f goodness o f f i t , t a i l o r e d t o t h e p a r t i c u l a r problem, w h i c h c a n be u s e d f o r a u t o m a t e d r e f i n e m e n t . The s p e c i m e n i n t e n s i t y t r a n s f o r m I i s a type of c o n v o l u t i o n p r o d u c t o f t h e p a r t i c l e i n t e n s i t y t r a n s f o r m Ip and t h e p a r t i c l e o r i e n t a t i o n density function (1,2). The p r o c e d u r e t h a t we h a v e u s e d t o s i m u l a t e Ip i n v o l v e s f i r s t l y t h e c a l c u l a t i o n o f t h e i n t e n s i t y t r a n s f o r m f o r an i n f i n i t e p a r t i c l e , w i t h a p p r o p r i a t e a l l o w a n c e s f o r random f l u c t u a t i o n s i n a t o m i c p o s i t i o n s and f o r matrix scattering. A mapping o f Ip i s t h e n c a r r i e d o u t w h i c h i n c l u d e s t h e e f f e c t s o f f i n i t e p a r t i c l e d i m e n s i o n s and o f i n t r a p a r t i c l e l a t t i c e d i s o r d e r , i f t h i s i s p r e s e n t . A mapping o f I i s t h e n o b t a i n e d from J by i n c o r p o r a t i n g t h e e f f e c t s o f i m p e r f e c t particle orientation. Q


s

p

Calculation

of I n t e n s i t y Transform of P a r t i c l e

(1^)

Two t y p e s o f d i f f r a c t i o n p a t t e r n s a r e e n c o u n t e r e d i n p r a c t i c e : i n the f i r s t , the r e f l e c t i o n s are d i s c r e t e i n d i c a t i n g that the i n t e r n a l s t r u c t u r e of the p a r t i c l e i s c r y s t a l l i n e , i n the second, there i s a continuous d i s t r i b u t i o n of i n t e n s i t y along d i s c r e t e layer l i n e s i n d i c a t i n g that the p a r t i c l e i s h e l i c a l . In a p a r t i c l e c o n t a i n i n g h e l i c e s arranged on a c r y s t a l l i n e l a t t i c e t h e d i f f r a c t i o n p a t t e r n w i l l o f c o u r s e be o f t h e f i r s t t y p e . In the p r e s e n t d e s c r i p t i o n we l i m i t c o n s i d e r a t i o n t o c a s e s where t h e n a t u r a l b r e a d t h o f r e f l e c t i o n s o r l a y e r l i n e s from i n d i v i d u a l p a r t i c l e s does n o t l e a d t o s i g n i f i c a n t o v e r l a p . C r y s t a l l i n e P a r t i c l e s . U s u a l l y , one o f t h e u n i t c e l l edges i s p r e f e r e n t i a l l y o r i e n t e d p a r a l l e l t o t h e f i b e r a x i s and t h e p a r t i c l e i n t e n s i t y transform corresponds to a s i n g l e - c r y s t a l r o t a t i o n p a t t e r n and t h e r e f l e c t i o n s a r e c o n f i n e d t o l a y e r l i n e s s p a c e d a t i n t e r v a l s o f l/o w h e r e a i s t h e d i m e n s i o n o f t h e u n i t c e l l p a r a l l e l to the f i b e r axis (2). I t i s c o n v e n i e n t t o u s e s p h e r i c a l p o l a r c o o r d i n a t e s (Dj0 ^) a t some s t a g e s o f t h e c a l c u l a t i o n a n d c y l i n d r i c a l p o l a r c o o r d i n a t e s (R ty Z) a t o t h e r s t o d e f i n e p o s i t i o n i n r e c i p r o c a l s p a c e ( F i g u r e 1 ) . The d i s t a n c e D c o r r e s p o n d s t o t h e r e c i p r o c a l o f the Bragg s p a c i n g . ^ V a l u e s o f \^y^j\ a r e c a l c u l a t e d and s t o r e d f o r e a c h l a y e r l i n e , where i s the structure factor f o r the r e f l e c t i o n with Miller indices h k and I. The e f f e c t s o f m a t r i x s c a t t e r i n g a r e approximated by a m o d i f i c a t i o n o f t h e atomic s c a t t e r i n g f a c t o r s according to the expression 9

s

3

9

2,3

2

f'(D) = f(D) - vp e x p (- vv D )

(1)

where f(D) i s t h e n o r m a l a t o m i c s c a t t e r i n g f a c t o r , f*(D) i s t h e m o d i f i e d f a c t o r , v i s t h e v o l u m e o f m a t r i x d i s p l a c e d b y t h e atom

French and Gardner; Fiber Diffraction Methods ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

4.

SUZUKI

Fiber Diffraction Patterns

ETAL.

63


and p i s t h e mean e l e c t r o n d e n s i t y o f t h e m a t r i x ( 4 ) . The e f f e c t s o f random d i s p l a c e m e n t s o f t h e atoms f r o m t h e i r l a t t i c e p o s i t i o n s are i n c o r p o r a t e d by f u r t h e r m o d i f y i n g t h e atomic s c a t t e r i n g f a c t o r s b y t h e i n c l u s i o n o f o n e o r more t e m p e r a t u r e f a c t o r s , f o l l o w i n g s t a n d a r d p r o c e d u r e s (5,60. The s p r e a d o f t h e r e f l e c t i o n s a r o u n d t h e r e c i p r o c a l l a t t i c e p o i n t s due t o t h e f i n i t e p a r t i c l e d i m e n s i o n s a n d t o c u m u l a t i v e l a t t i c e d i s o r d e r s o f the " i d e a l l y p a r a c r y s t a l l i n e " type i s c a l c u l a t e d a c c o r d i n g t o t h e f o r m u l a g i v e n b y Hosemann and W i l k e (7). They showed t h a t f o r a o n e - d i m e n s i o n a l c r y s t a l t h e i n t e g r a l b r e a d t h ( $ ) v a r i e s w i t h t h e M i l l e r i n d e x (h) o f t h e r e f l e c t i o n and c a n b e a p p r o x i m a t e d b y t h e e x p r e s s i o n 2 2 2 2

where (3^ i s t h e i n t e g r a l b r e a d t h i n t h e a b s e n c e o f a a pgaair a c r y s t a l l i n e d i s o r d e r , i s t h e mean l a t t i c e d i s t a n c e a n d i s t h e mean s q u a r e l a t t i c e d i s t a n c e . The v a l u e s a t t h e r e c i p r o c a l l a t t i c e p o i n t s a r e normalized so that t h e integrated intensity about each r e c i p r o c a l l a t t i c e p o i n t remains p r o p o r t i o n a l t o The c y l i n d r i c a l l y a v e r a g e d p a r t i c l e i n t e n s i t y t r a n s f o r m i s obtained by e v a l u a t i n g I

p

(R Z) y

J

FL

=

where I(R ^^Z)

1

(R^sZ).

i s the stationary

3

o>

(3)

particle intensity

transform.

H e l i c a l P a r t i c l e s . The i n t e n s i t y d i s t r i b u t i o n a l o n g t h e layer l i n e s i n the i n t e n s i t y transform o f a h e l i c a l p a r t i c l e i s continuous but i s completely defined by a s e t o f values t a b u l a t e d a t i n t e r v a l s o f 112d w h e r e d i s t h e d i a m e t e r o f a n e x s c r i b e d c y l i n d e r (8). V a l u e s o f t h e c y l i n d r i c a l l y averaged square o f t h e m o d u l u s o f t h e s t r u c t u r e f a c t o r f o r o n e p e r i o d o f t h e helix, a r e calculated a t these i n t e r v a l s according t o the expression = fJ 2

]i)

F(R ^l/c)F*(R,ty l/c).d$ 3

(4)

9

P r o c e d u r e s u s e d a r e d e s c r i b e d i n d e t a i l e l s e w h e r e ( 5 ) . The e f f e c t of m a t r i x s c a t t e r i n g i s a g a i n approximated by m o d i f y i n g t h e atomic s c a t t e r i n g f a c t o r s a c c o r d i n g t o e x p r e s s i o n ( 1 ) . Random f l u c t u a t ions i n atomic p o s i t i o n s p a r a l l e l t o the a x i s o f t h e h e l i x a r e a l l o w e d f o r b y m u l t i p l y i n g f'(D) b y t h e t e r m exp(-%B Z ) w h e r e B i s a Debye t e m p e r a t u r e f a c t o r . The e f f e c t s o f f i n i t e p a r t i c l e l e n g t h a r e a p p r o x i m a t e d b y s u p p o s i n g t h a t t h e l a y e r l i n e s have a G a u s s i a n p r o f i l e i n t h e Z d i r e c t i o n (9) s o t h a t : I (R,Z) p

=

I

(R l/o) 9

2

2

exp [ - i i p ( Z - Z / e ; ]


(5)

64

FIBER DIFFRACTION

Z

Journal of Applied Crystallography


Figure 1. Relationship between reciprocal space position vector D and the reciprocal space coordinates (1)

Figure 2. Intensity transforms for points uniformly distributed on a helix with radius r = 4.5 A, unit height h = 3 A, and unit twist t = 108°. (a) Particle intensity transform for p = 500 A; (b) particle intensity transform for p — 100 A; (c) specimen intensity transform for p = 100 A, a = 3°. 0


METHODS


SUZUKI

ET AL.


Figure 3. Comparison of (a) simulated specimen intensity transform calculated for a model of the structure of collagen with (b) a mapping of the observed specimen intensity transform derived from a diffraction pattern obtained with the specimen tilted at 15.75° to the normal to the x-ray beam. The parameters used in the simulation are those derived from the observed pattern (11). No allowance has been made for interparticle interference effects which are responsible for the sampling of the particle intensity transform along the equator in the observed specimen intensity transform.


66

FIBER DIFFRACTION METHODS

where p i s t h e p a r t i c l e l e n g t h and I

(R l/o)

=

s

(p/o) ^

(6)

The e f f e c t s o f p a r a c r y s t a l l i n e t y p e d i s o r d e r i n t h e z d i r e c t i o n c a n be a p p r o x i m a t e d by d e c r e a s i n g t h e v a l u e o f p w i t h i n c r e a s i n g Z (7). C a l c u l a t i o n o f I n t e n s i t y T r a n s f o r m o f Specimen ( J ^ )


I t h a s b e e n shown ( 1 , 2 ) t h a t f o r a n o r i e n t a t i o n d e n s i t y f u n c t i o n o f the type G(a) = e x p ( - a / 2 a ) / 2 T T a 2

2

2

(7)

where a i s t h e i n c l i n a t i o n o f t h e p a r t i c l e a x i s t o t h e f i b e r a x i s and a i s a p a r a m e t e r , t h e v a l u e o f t h e s p e c i m e n i n t e n s i t y t r a n s ? o r m f o r a p a r t i c u l a r v a l u e o f D a n d a c a n be c a l c u l a t e d from t h e e x p r e s s i o n : 2 , (a -a ) sina sina = - rl y * y p[- f - H v — V ^ > y V a 2a a o o o w h e r e i (x) = e x p ( - x ) I (x) and I fej i s a m o d i f i e d B e s s e l f u n c t i o n o f t h e s e c o n d S i n d o f o r S e r z e r o , and t h e s u f f i x e s s a n d p r e f e r r e s p e c t i v e l y t o sample and p a r t i c l e i n t e n s i t y t r a n s f o r m space. The s p e c i m e n i n t e n s i t y t r a n s f o r m i s mapped b y e v a l u a t i n g e x p r e s s i o n ( 8 ) n u m e r i c a l l y f o r e a c h e l e m e n t o f t h e a r r a y . The r e s u l t i s c o n v e n i e n t l y d i s p l a y e d u s i n g an O p t r o n i c s P h o t o w r i t e and e x a m p l e s i l l u s t r a t i n g some o f t h e p r o c e d u r e s d e s c r i b e d h e r e a r e g i v e n i n F i g u r e s 2 and 3. A l t h o u g h n o t i l l u s t r a t e d i n t h e s e examples t h e e x t e n s i o n o f t h e method t o i n c l u d e i n t e r p a r t i c l e i n t e r f e r e n c e e f f e c t s i s straightforward. The p r i n c i p l e s i n v o l v e d and t h e n e c e s s a r y f o r m u l a e a r e g i v e n b y James ( 1 0 ) . S i m i l a r l y t h e e f f e c t s o f s p e c i m e n a b s o r p t i o n and f i n i t e beam s i z e c a n r e a d i l y be incorporated i f required. I

(D

e x

f

)

s -%

r

s

i

r

n

r

8

r

Q

Literature Cited 1. 2. 3. 4. 5.

Fraser, R.D.B., MacRae, T.P., Miller, A . , Rowlands, R.J. J. Appl. Cryst. 1976, 9, 81-94. Holmes, K.C., Barrington-Leigh, J . Acta Cryst. 1974, A30, 635-638. Bunn, C.W. "Chemical Crystallography", Clarendon Press: Oxford, 1945. Fraser, R.D.B., MacRae, T.P., Suzuki, E. J . Appl. Cryst. 1978, 11, 693-694. Fraser, R.D.B., MacRae, T.P. "Conformation in Fibrous Proteins", Academic Press: New York, 1973.


)

4.

SUZUKI ET AL.


6.

67

"International Tables for X-ray Crystallography", Vol. III, Kynoch Press: Birmingham, 1962. 7. Hosemann, R., Wilke, W., Makromol. Chem. 1968, 118, 230-249. 8. Bracewell, R. "The Fourier Transform and Its Applications", McGraw-Hill: New York, 1965. 9. Stubbs, G.J. Acta Cryst. 1974, A30, 639-645. 10. James, R.W. "The Optical Principles of the Diffraction of X-rays", Bell and Sons: London, 1954. 11. Fraser, R.D.B., MacRae, T.P., Suzuki, E. J . Mol. Biol. 1979, 129, 463-481. 29,

1980.


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Simulation of Fiber Diffraction Patterns - American Chemical Society

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