Application of the Rietveld Crystal Structure Refinement Method to

Chapter 4

Application of the Rietveld Crystal Structure Refinement Method to Cellotetraose 1

2

3,4

3

A. Sakthivel , A. D. French , B. Eckhardt , and R. A. Young 1

School of Textile Engineering, Georgia Institute of Technology, Atlanta, GA 30332 Southern Regional Research Center, U.S. Department of Agriculture, P.O. Box 19687, New Orleans, LA 70179 School of Physics, Georgia Institute of Technology, Atlanta, GA 30332

2

Downloaded by CORNELL UNIV on October 17, 2016 | http://pubs.acs.org Publication Date: June 22, 1987 | doi: 10.1021/bk-1987-0340.ch004

3

The Rietveld method of structure refinement from x-ray powder d i f f r a c t i o n data was applied to cellotetraose, which gives a diffraction pattern similar to that of cellulose II. Unit cell dimensions were consistent with earlier work. The space group was shown to be P1 rather than P2 . Models, r i g i d except for the positions of the 0(6) atoms, were used to test the effects on the calculated diffraction patterns of parallel and antiparallel packing modes, 0(6) positions, and the presence of hydrogen atoms. The hydrogen atoms had a negligible effect on the c a l c u l a t e d x-ray d i f f r a c t i o n pattern. The intensities of specific individual Bragg reflections were s u f f i c i e n t l y affected by the 0(6) positions so that they may help to indicate those positions, even though the net effect on the overall pattern-fitting was s m a l l . Results from refinements of an antiparallel model against intensities calculated from a parallel model indicated that differentiation may be d i f f i c u l t on the basis of x-ray powder diffraction patterns. Preliminary results based on simple models with tetramer symmetry close to 2 suggested that the two tetramers in the unit cell are slightly inclined to the c axis. 1

1

I t h a s l o n g b e e n known t h a t c e l l o t e t r a o s e , t h e β 1 , ^ - l i n k e d t e t r a m e r o f D - g l u c o s e , y i e l d s a powder d i f f r a c t i o n pattern e x h i b i t i n g more c r y s t a l l i n i t y b u t o t h e r w i s e s i m i l a r t o t h a t o f c e l l u l o s e I I , the a l l o m o r p h r e s u l t i n g from treatment of c e l l u l o s e I i n s t r o n g l y a l k a l i n e s o l u t i o n or from r e g e n e r a t i o n from s o l u t i o n . I n the quest t o o b t a i n s t r u c t u r a l i n f o r m a t i o n a p p l i c a b l e t o c e l l u l o s e I I , p r e v i o u s workers, using t i n y s i n g l e c r y s t a l s , e.g., 0.2 χ 0.1 χ 0.01 mm " u s u a l l y o f p o o r q u a l i t y b u t 3

4

Current address: Fachbereich Physik, Universität Bremen, Bibliotekstrasse, Postfach 330440, 2800 Bremen 33, Federal Republic of Germany 0097-6156/87/0340-0068$06.00/0 © 1987 American Chemical Society

Atalla; The Structures of Cellulose ACS Symposium Series; American Chemical Society: Washington, DC, 1987.


4.

Rietveid Crystal Structure Refinement Method

SAKTHIVEL ET AL.

69

w i t h w e l l developed (001) f a c e s " (1_), s u g g e s t e d (2) t h a t the space g r o u p o f c e l l o t e t r a o s e was P1 and d e t e r m i n e d t h e u n i t c e l l o f cellotetraose accordingly. Further single-crystal diffraction work has, however, been t h w a r t e d by x - r a d i a t i o n damage o c c a s i o n e d by t h e l o n g e x p o s u r e s r e q u i r e d b e c a u s e o f t h e s m a l l s i z e o f t h e crystals. R i e t v e i d r e f i n e m e n t (3>4) p e r m i t s s t r u c t u r a l study with powder d i f f r a c t i o n d a t a . The e n t i r e d i f f r a c t i o n p a t t e r n i s c a l c u l a t e d from a model c o n s i s t i n g of the u n i t c e l l p a r a m e t e r s , c r y s t a l l i t e s i z e ( l i n e w i d t h ) , p r o p o s e d a t o m i c p o s i t i o n s and thermal parameters. The v a l i d i t y o f t h e p r o p o s e d m o d e l i s measured by c o m p a r i s o n of the o b s e r v e d and c a l c u l a t e d i n t e n s i t y a t each s t e p - s c a n i n c r e m e n t , i n c l u d i n g background c o r r e c t i o n , and the model p a r a m e t e r s a r e s y s t e m a t i c a l l y r e f i n e d i n o r d e r to p r o v i d e the b e s t agreement i n a l e a s t squares sense. I n I m m i r z i s v e r s i o n (5), the m o l e c u l e i s d e s c r i b e d i n terms of g e n e r a l i z e d c o o r d i n a t e s so t h a t the model can be r e f i n e d when the d a t a a r e l i m i t e d and the m o l e c u l e i s complex. This i s similar t o t h e a n a l y s e s o f f i b e r d i f f r a c t i o n d a t a i n w h i c h t h e monomers are kept r i g i d e x c e p t f o r r o t a t i n g s i d e groups. However, powder d i f f r a c t i o n d a t a a r e l e s s l i k e l y t o be a f f e c t e d by e r r o r s i n c o l l e c t i o n , c o r r e c t i o n and r e d u c t i o n t h a n f i b e r data. E r r o r s from p r e f e r r e d o r i e n t a t i o n produced i n sample p r e p a r a t i o n a r e p o s s i b l e , but can be t e s t e d f o r . A l t h o u g h the R i e t v e i d method has been used s u c c e s s f u l l y w i t h hundreds of m a t e r i a l s , i n c l u d i n g some p o l y m e r s s u c h as p o l y p r o p y l e n e (6), i t i s b e l i e v e d t h a t t h i s i s the f i r s t a t t e m p t to p e r f o r m a thorough s t r u c t u r a l s t u d y of an o l i g o s a c c h a r i d e w i t h powder x-ray d i f f r a c t i o n data. In t h i s paper, we d e s c r i b e some of t h e d e t a i l s o f t h e R i e t v e i d m e t h o d , as m o d i f i e d by I m m i r z i f o r polymers and f u r t h e r m o d i f i e d l o c a l l y , and d i s c u s s our assessments of v a r i o u s a s s u m p t i o n s used i n s u c h a s t u d y . A preliminary s t r u c t u r a l r e s u l t i s a l s o presented. ?

EXPERIMENTAL PROCEDURE The s a m p l e s o f c e l l o t e t r a o s e s t u d i e d h e r e i n w e r e p r o v i d e d by Dr. F r e d P a r r i s h , S o u t h e r n R e g i o n a l R e s e a r c h C e n t e r , Dr. R o s s Brown, U n i v e r s i t y o f F l o r i d a , and Dr. J o h n V e r c e l l o t t i , V - L a b s , C o v i n g t o n , LA. T h e y w e r e made by h y d r o l y s i s o f c e l l u l o s e i n h y d r o c h l o r i c a c i d and s e p a r a t e d from the o t h e r r e s u l t i n g o l i g o m e r s through column chromatography. The s t e p - s c a n n e d , w i d e - a n g l e d i f f r a c t i o n d a t a were o b t a i n e d ( r e f l e c t i o n mode) w i t h a s t a n d a r d Θ-2Θ x - r a y powder d i f f r a c t o m e t e r u s i n g a d i f f r a c t e d beam monochromator and CuKa r a d i a t i o n from a s t a n d a r d s e a l e d - o f f x - r a y tube. The s c a n n i n g range was from 7° t o 43.40° 2Θ; no c l e a r l y r e c o g n i z a b l e p e a k s w e r e o b s e r v e d b e y o n d 43.40°. The s t e p w i d t h was 0.04° and t h e c o u n t i n g t i m e was 250 seconds per step. S i n c e l i t t l e improvement i n the d a t a r e s u l t e d from m a i n t a i n i n g the specimen near l i q u i d n i t r o g e n t e m p e r a t u r e s , a l l r e s u l t s r e p o r t e d h e r e w e r e o b t a i n e d f r o m room t e m p e r a t u r e data. The a b s o r p t i o n f a c t o r exp(-ut) (equal to ïtransmitted^incident ? n o r m a l l y i n c i d e n t beam) was measured as 76.4$. The w i d e - a n g l e d i f f r a c t i o n p a t t e r n i s shown i n F i g u r e 1. o r

a


70

THE STRUCTURES OF CELLULOSE

S m a l l - a n g l e d a t a were c o l l e c t e d w i t h a s m a l l a n g l e x - r a y d i f f r a c t o m e t e r equipped w i t h a p o s i t i o n s e n s i t i v e d e t e c t o r p l a c e d 25.0 cm away from the sample ( t r a n s m i s s i o n mode). The r e s u l t i s shown i n F i g u r e 2. RIETVELD REFINEMENT METHOD Given the a t o m i c p o s i t i o n s i step i s t

from

a model,

the

i n t e n s i t y y^

at

the

n

=

*i y

i

>

+

yi.b

Σ

*i,H 2

= G(e e )

H

r

|F | P T (LP)

H

H

H

H

i


where t

n

^ = background at the i step Η ' = M i l l e r i n d i c e s h , k , l f o r a Bragg r e f l e c t i o n G ( e i ~ 6 ) = B r a g g r e f l e c t i o n p r o f i l e f u n c t i o n (a P e a r s o n V I I f u n c t i o n was u s e d , s e e r e f e r e n c e s 4 and 6 f o r more d e t a i l s ) Q± = s c a t t e r i n g a n g l e at the i step F = structure factor P = m u l t i p l i c i t y of Η (LP)j = L o r e n t z and p o l a r i z a t i o n f a c t o r at s t e p i T = exp(-P(a ) ), preferred orientation function Ρ = p r e f e r r e d o r i e n t a t i o n parameter ot = a c u t e a n g l e between the p r e f e r r e d o r i e n t a t i o n d i r e c t i o n and the r e c i p r o c a l l a t t i c e v e c t o r f o r Η H

t

n

H

H

2

H

H

H

The Y

background = b

i f b

i n t e n s i t y (y^

) i s calculated

+ (b -b )(29 -2e )/

1

2

1

1

i

as

fe

(2θ -2θ ) + b 1

2

3

*

G(2e -28 ) s

1

where 20! and 2 θ are the l i m i t s of the s c a n n i n g range, G ( 2 6 - 2 0 ) i s a Pearson VII f u n c t i o n w i t h 3 r e f i n a b l e parameters, and b , b , b , 2 0 are a l s o parameters t h a t are r e f i n e d . The e q u a t i o n g i v e n above f o r y i s e m p i r i c a l and a g r a p h i c a l r e p r e s e n t a t i o n of i t can be found i n r e f e r e n c e 6. The s t r u c t u r e f a c t o r F , i s c a l c u l a t e d as f o l l o w s : 2

3

x

2

i

3

3

i b

H

F

H

j Mj fj Xj,yj,Zj B^ d λ H

= I

Mjfjexp{-2πi(hXj+kyj + l Z j ) } e x p { - B j ( d

9 t H

2

/2) }

= r a n g e s from 1 to no. of atoms i n the u n i t = s i t e occupancy (of the j atom s i t e ) = atomic s c a t t e r i n g f a c t o r f o r the j atom = f r a c t i o n a l c o o r d i n a t e s of the atom j = temperature f a c t o r =2 sin0/X = wavelength o f the r a d i a t i o n used

The d i f f e r e n c e between the i s measured by the r e s i d u a l ,

t

t

observed

cell

n

and

n

calculated

patterns



4.

SAKTHIVEL ET AL.

β

1

9

ΐΌ

11

12

A A Α

71


ΐ'β

Λ A ΐ!» À>

2!

22

1

23

2*4

Α

2(5 2^ 2^ 29 &

31 32 33 34

A A

1

3?

Α

39

Α

41 42

SCATTERING ANGLE

F i g u r e 1. Wide a n g l e x - r a y powder d i f f r a c t i o n p a t t e r n o f c e l l o t e t r a o s e a t room t e m p e r a t u r e . CuK radiation.

1.00 0.96

0.24 0.20 016 0.12 0.08 0.04 0.00

1

1

2

1

3

1

4

5*

6*

7*

β'

9*

ΐΌ l'l

12 13 1*4

A

SCATTERING ANGLE

F i g u r e 2. Small-angle p o w d e r - d i f f r a c t i o n p a t t e r n of c e l l o t e t r a o s e a t room temperature ( t r a n s m i s s i o n mode). CuK radiation and a p o s i t i o n s e n s i t i v e d e t e c t o r a t 25.0 cm from the sample were used.


Α

72


χ

2

=

w

Σ

i(yi,obs-yi,calc

) 2

( 1 )

w h e r e t h e sum e x t e n d s o v e r a l l s c a t t e r i n g a n g l e s 2Θ a t w h i c h a measurement of the d i f f r a c t e d i n t e n s i t y y^ , was made. Wj i s a weight f a c t o r , u s u a l l y chosen t o be 1/y ' . A minimum i n χ is sought. A n e c e s s a r y c o n d i t i o n f o r t e r m i n a t i o n of r e f i n e m e n t i s t h a t the g r a d i e n t of χ w i t h r e s p e c t t o each o f the M r e f i n e d parameters, x , has vanished. The r e s u l t i n g equations are n o n l i n e a r i n t h e x s and c a n n o t be s o l v e d a n a l y t i c a l l y f o r t h e parameter s h i f t s dx which w i l l minimize χ . Therefore, the c a l c u l a t e d i n t e n s i t i e s a r e e x p a n d e d i n t h e x * s and o n l y l i n e a r t e r m s a r e k e p t . L e t t i n g b r a c e s d e n o t e m a t r i c e s and p a r e n t h e s e s denote v e c t o r s , one ends up wih an e q u a t i o n of the form o b s

2

i

0

bs

2

k

f

k

2

k


k

{A}

(dx)

= b

(dx)

= {A}

(2)

from which,

The

various

A

quantities w

= I

j k

i

-1

(b)

(3)

are: 5 y

i,calc

6 v

i,calc

6xj

w

i

6 v

i,calc

6 X

dx

k

= x

o

l

d

k

6x

( v

... (4)

k

v

i,obs~ i,calc

)

J

- x

n

e

w

k

(6)

T h i s c l e a r l y shows s e v e r a l i m p o r t a n t f e a t u r e s of t h e R i e t v e i d r e f i n e m e n t p r o c e d u r e . F i r s t l y , i t i n v o l v e s i n v e r s i o n o f an M*M matrix. S e c o n d l y , i t works o n l y i f the s t a r t i n g v a l u e s are a l r e a d y c l o s e t o a minimum, because of the l i n e a r a p p r o x i m a t i o n i n the d e r i v a t i o n of e q u a t i o n (2). T h i r d l y , w h a t e v e r minimum i s o b t a i n e d , i t i s not n e c e s s a r i l y the g l o b a l minimum one i s l o o k i n g f o r b u t may be a l o c a l m i n i m u m , s i n c e t h e m e t h o d i s a l o c a l one. Finally, because of f i n i t e w i d t h of every r e f l e c t i o n , the c a l c u l a t e d i n t e n s i t y at each s t e p c o n t a i n s c o n t r i b u t i o n s from several neighboring r e f l e c t i o n s . I n our c a s e up t o 45 p o s s i b l e r e f l e c t i o n s can c o n t r i b u t e t o the i n t e n s i t y c a l c u l a t e d at a s i n g l e p o i n t i n the d i f f r a c t i o n p a t t e r n . The program used f o r R i e t v e i d r e f i n e m e n t i n t h i s work i s an e x t e n s i v e l y m o d i f i e d v e r s i o n o f REFIN (FORTRAN V), w r i t t e n by A. I m m i r z i (5). I n o r d e r to a s s e s s the agreement between the c a l c u l a t e d and o b s e r v e d p a t t e r n , s e v e r a l n u m b e r s can be c a l c u l a t e d . Most commonly used a r e R and R. w p

p


4.

SAKTHIVEL ET AL.


73

1/2

(7)

wp obs

and

^ Rp

=

y

y

I i,obs~ y

i,calcl (8)

i,obs

On a s t r i c t l y s t a t i s t i c a l b a s i s R i s the p r e f e r r e d i n d i c a t o r b e c a u s e i t s numerator i s the f u n c t i o n t h a t i s b e i n g m i n i m i z e d ( e q u a t i o n 1). Note t h a t t h e s e R v a l u e s are n o r m a l l y l a r g e r than t h o s e r e p o r t e d f o r f i b e r and s i n g l e - c r y s t a l s t u d i e s because they a r e b a s e d on t h e i n t e n s i t i e s a t a l l s t e p s i n t h e p a t t e r n and n o t j u s t on i n d i v i d u a l l y o b s e r v e d Bragg r e f l e c t i o n i n t e n s i t i e s . Another i n d i c a t o r can be c a l c u l a t e d w h i c h i s more a k i n t o the c o n v e n t i o n a l R v a l u e w i t h w h i c h some r e a d e r s w i l l be f a m i l i a r from t h e l i t e r a t u r e on s i n g l e c r y s t a l s t r u c t u r e r e f i n e m e n t :


w p

f

1

Σ R

B

( I

i,"obs"

"

I

i,calc

)

-

(9) ί

I

i,"obs"

In i t s f o r m u l a t i o n , R i s comparable to the c o n v e n t i o n a l R v a l u e based on i n t e n s i t i e s which, f o r a normal d i s t r i b u t i o n o f e r r o r s , i s l a r g e r t h a n t h e most quoted R v a l u e based on s t r u c t u r e f a c t o r s by a f a c t o r of /2. In e q u a t i o n 9, I i i b t i i s w r i t t e n w i t h q u o t a t i o n marks because i t i s not a c t u a l l y o b s e r v e d d i r e c t l y . Rather, t h e t o t a l i n t e n s i t y observed f o r a s e t of o v e r l a p p i n g r e f l e c t i o n s i s a p p o r t i o n e d among the r e f l e c t i o n s i n the r a t i o s of the c a l c u l a t e d i n t e n s i t i e s (3fJ0. R i s , t h e r e f o r e , b i a s e d i n f a v o r of the model b e i n g used. However, i t i s u s e f u l because i t i s r e l a t i v e l y i n s e n s i t i v e to f e a t u r e s , such as p r o f i l e shape e r r o r s , which produce an i n f l a t i o n of Rp and R without c r y s t a l s t r u c t u r a l s i g n i f i c a n c e . B

0

S

B

w p

COMPUTER MODEL BUILDING The m o l e c u l a r f o r m u l a f o r c e l l o t e t r a o s e , the Β 1 , 4 - l i n k e d t e t r a m e r of D - g l u c o s e , i s ^2^4^42^21 · S* the c o n t r i b u t i o n s of the h y d r o g e n a t o m s t o t h e x - r a y p a t t e r n may be s a f e l y d i s r e g a r d e d ( F i g u r e 3), s t r u c t u r a l d e t e r m i n a t i o n c o n s i s t s of d e f i n i n g the x, y, and ζ c o o r d i n a t e s ( p l u s any t h e r m a l p a r a m e t e r s ) f o r each o f the r e m a i n i n g 45 atoms i n each t e t r a m e r . Even though 910 data p o i n t s were taken, no more t h a n a b o u t 30 i n t e n s i t y " p e a k s " c a n be observed, most of w h i c h are c o m p o s i t e s of s e v e r a l B r a g g reflections. T h e r e f o r e , o n l y a v e r y l i m i t e d number o f p a r a m e t e r s i n t h e model can be m e a n i n g f u l l y r e f i n e d . n c e

The c o o r d i n a t e s of the atoms i n the monomer were assumed t o be known from s i n g l e c r y s t a l s t u d i e s of g l u c o s e (7). Each monomer was t r e a t e d as a r i g i d body w i t h 3 d e g r e e s , e a c h , o f r o t a t i o n a l and t r a n s l a t i o n a l freedom.



74


The o r i g i n o f the c o o r d i n a t e system ( f o r example, c o i n c i d i n g w i t h t h e c e n t e r o f t h e f i r s t monomer) c a n be f r e e l y c h o s e n . The f i r s t monomer then has o n l y 3 r o t a t i o n a l degrees of freedom. The o t h e r t h r e e monomers a r e d e s c r i b e d by 3 t r a n s l a t i o n a l and 3 a n g u l a r p a r a m e t e r s , each. One parameter f o r each monomer d e s c r i b e s the o r i e n t a t i o n o f the oxygen of the p r i m a r y h y d r o x y l group (0(6)), w i t h r e s p e c t t o t h e C(4) - C(5) bond. W i t h t h i s s e l e c t i o n o f p a r a m e t e r s , the t e t r a m e r i s d e s c r i b e d i n terms of 25 parameters. S p e c i f i c a t i o n of i t s o r i e n t a t i o n w i t h r e s p e c t t o the u n i t c e l l r e q u i r e s another t h r e e a n g l e s . ( I n a s t r i c t l y m a t h e m a t i c a l sense, t h e s e t h r e e p a r a m e t e r s a r e redundant. Hence, t h e y a r e never r e f i n e d s i m u l t a n e o u s l y w i t h the o r i e n t a t i o n p a r a m e t e r s of the monomers). Thus, t h e t o t a l number of p a r a m e t e r s f o r t h e f i r s t t e t r a m e r i s 28. I t i s t h e n e a s y t o d e s c r i b e many t e t r a m e r s per u n i t c e l l ; a l l o n e h a s t o do i s t o t a k e 28 p a r a m e t e r s f o r the f i r s t t e t r a m e r and 28 p a r a m e t e r s p l u s t h r e e f o r the a d d i t i o n a l t r a n s l a t i o n a l d e g r e e s o f f r e e d o m f o r each additional tetramer. F o r two t e t r a m e r s t h i s adds up t o 59 parameters. I n most of our r e f i n e m e n t s , t h e number of p a r a m e t e r s a c t u a l l y r e f i n e d i n any one c y c l e was l i m i t e d t o about 20. O n l y β - c e l l o t e t r a o s e has been m o d e l e d i n t h i s s t u d y . The c e l l o t e t r a o s e sample c o u l d a l s o be a m i x t u r e o f a - and Bc e l l o t e t r a o s e s o r e v e n a l l a. In the case of m i x t u r e s , the a n o m e r s may o c c u r i n t h e same c r y s t a l s as i n t h e c a s e o f a-B m a l t o s e (8), or the powder sample may c o n t a i n two t y p e s of s i m i l a r c r y s t a l s t h a t d i f f e r a t t h e a n o m e r i c c a r b o n . T h i s i s among t h e t o p i c s t h a t w i l l be e x a m i n e d d u r i n g t h e s e c o n d p h a s e o f t h i s study. RESULTS AND

DISCUSSION

Space Group and U n i t c e l l p a r a m e t e r s . A c c o r d i n g t o P o p p l e t o n and M a t h i e s o n (]_), the space group f o r c e l l o t e t r a o s e i s e i t h e r P1 and The f a i r l y s t r o n g 001 r e f l e c t i o n o b s e r v e d a t 3.90° ( F i g u r e 2) e l i m i n a t e s t h e P 2 space group. S t a r t i n g from those i n the l i t e r a t u r e (2), the l a t t i c e parameters r e f i n e d to 1

a = 8.98 α

A

= 94.31°

b = 8.01

A,

Β = 89.27°

c = 22.34A Ύ =

116.45°

These p a r a m e t e r s agree w e l l w i t h t h e r e p o r t e d l a t t i c e p a r a m e t e r s (2), e x c e p t f o r c, w h i c h i s s m a l l e r by 0.25A i n o u r s t u d y . The d e n s i t y c a l c u l a t e d f r o m o u r p a r a m e t e r s i s 1.55 g / c m , t h a t f r o m (2) i s 1.52 g / c m and t h e m e a s u r e d d e n s i t y i s r e p o r t e d (1_) t o be 1.49 g / c m . The l a r g e , a s y m m e t r i c (P1) u n i t c e l l p e r m i t s 338 p o s s i b l e r e f l e c t i o n s i n the a n g u l a r range s t u d i e d . Most peaks are composed of many u n r e s o l v a b l e r e f l e c t i o n s . For i n s t a n c e , the major peaks a t 12.3, 19.9, 22.0, and 4 0 . 8 ° ( t o w h i c h 1T0, 1 10, 200, and 310, r e s p e c t i v e l y , a r e t h e major c o n t r i b u t o r s ) c o n t a i n 6, 9, 10 and 40 reflections, respectively. I n the R i e t v e i d method, the i n t e n s i t y at any p o i n t i n the c a l c u l a t e d d i f f r a c t i o n p a t t e r n i s the sum of c o n t r i b u t i o n s from n e i g h b o r i n g r e f l e c t i o n s . In the d i f f r a c t i o n 3

3

3


4.

SAKTHIVEL ET AL.


75

p a t t e r n s c a l c u l a t e d i n t h i s work, t h e r a n g e o f i n f l u e n c e o f any r e f l e c t i o n was l i m i t e d t o t w i c e t h e w i d t h of the p r o f i l e at h a l f the maximum h e i g h t . Number o f T e t r a m e r s

per U n i t

Cell.


A: T r i a l s w i t h one t e t r a m e r p e r c e l l . We f i r s t t r i e d s e v e r a l m o d e l s c o n t a i n i n g one t e t r a m e r p e r u n i t c e l l ( o f h a l f t h e c e l l volume f i n a l l y used), as proposed by P o p p l e t o n and M a t h i e s o n (1_). However, we c o u l d not produce t h e r o u g h l y e q u a l i n t e n s i t i e s of the two v e r y s t r o n g p e a k s a t 20° and 22° a l o n g w i t h a s t r o n g one a t 12°. We t h e r e f o r e c o n c l u d e d t h a t the number of t e t r a m e r s per u n i t c e l l i s g r e a t e r than one. B: T r i a l s w i t h two t e t r a m e r s p e r c e l l . As d e s c r i b e d by G a r d n e r and B l a c k w e l l f o r c e l l u l o s e ( 9 ) , two t e t r a m e r s c a n be p a c k e d i n the u n i t c e l l i n 3 ways: P a r a l l e l - u p , a n t i p a r a l l e l , and p a r a l l e l down. I n p a r a l l e l - u p m o d e l s , z - c o o r d i n a t e s o f t h e a t o m s a t t h e r e d u c i n g ends of both the t e t r a m e r s are g r e a t e r than t h o s e o f the n o n r e d u c i n g e n d s , and c o n v e r s e l y f o r t h e p a r a l l e l - d o w n m o d e l s . A n t i p a r a l l e l models c o n t a i n one "up" and one "down" t e t r a m e r . In t h e i r s t u d i e s , Sarko and M u g g l i (1_0) have used a and b axes w h i c h a r e i n t e r c h a n g e d compared t o ours, t h o s e i n the B l a c k w e l l - G a r d n e r system, and t h o s e i n r e f e r e n c e s (1,2). T h i s r e s u l t s i n the c a x i s p o i n t i n g i n a d i r e c t i o n o p p o s i t e t o t h a t i n the B l a c k w e l l - G a r d n e r system. T h i s has c a u s e d some c o n f u s i o n i n t h a t a p a r a l l e l - u p m o d e l i n one s y s t e m comes t o be c a l l e d a p a r a l l e l - d o w n m o d e l i n the o t h e r system and v i c e v e r s a . We have r e f i n e d s e v e r a l p a r a l l e l - u p and a n t i p a r a l l e l models a g a i n s t the d i f f r a c t i o n d a t a and the b e s t agreements between t h e c a l c u l a t e d and the e x p e r i m e n t a l d a t a are shown i n F i g u r e s 4 and 5. The ab and ac p r o j e c t i o n s of the c o r r e s p o n d i n g models are shown i n F i g u r e s 6 and 7. The R v a l u e s f o r the b e s t models a r e g i v e n i n T a b l e I. Since

a v a i l a b l e models

T a b l e I . R v a l u e s f o r the b e s t

Model

R

P

R

wp

R

Bragg

Parallel-up

0.1 41

0.190

0.185

Antiparallel

0.171

0.226

0.235

the R v a l u e s a r e f a i r l y low ( f o r x - r a y R i e t v e i d r e f i n e m e n t ) , e s p e c i a l l y f o r the p a r a l l e l up model, we b e l i e v e t h a t the g e n e r a l f e a t u r e s of the m o d e l s a r e c o r r e c t . B e f o r e a f i n a l m o d e l can be p r o p o s e d , h o w e v e r , f a c t o r s s u c h as t h e p a r a l l e l down m o d e l , t h e 0(6) p o s i t i o n , t h e p o s s i b i l i t y o f t h e α a n o m e r i c f o r m , and t h e c o n f o r m a t i o n a n g l e s between monomers i n t h e t e t r a m e r s must be more t h o r o u g h l y examined. The c a r t e s i a n and the f r a c t i o n a l c o o r d i n a t e s



76


l't

Α A

1*4

Α Λ Λ A A A

2*1

A A

A A A Ai

24 SCATTERMO ANGLE

F i g u r e 3. C a l c u l a t e d x - r a y powder d i f f r a c t i o n p a t t e r n s f o r models w i t h o r w i t h o u t the hydrogen atoms. The p a t t e r n f o r the model not i n c l u d i n g hydrogen i s shown i n the same f i e l d , and t h e i r d i f f e r e n c e i s shown i n the lower f i e l d .

lO l'l

12

A

14

A

16

Λ A A

2(3 2*1 22 23 2*4 2^ 26 2*7 26^ 29 SCATTERING ANGLE

A

31 32

A

34

A A jV A

39

A

41 42

A

F i g u r e 4. C a l c u l a t e d and o b s e r v e d p a t t e r n s f o r the b e s t p a r a l l e l model. The o b s e r v e d p a t t e r n i s shown by d o t s w i t h v e r t i c a l e r r o r b a r s based on c o u n t i n g s t a t i s t i c s . The c a l c u l a t e d p a t t e r n i s shown by a c o n t i n u o u s c u r v e i n the same f i e l d and the d i f f e r e n c e between the o b s e r v e d and the c a l c u l a t e d p a t t e r n s i s shown i n the lower f i e l d . R f o r t h i s model i s 0.19. wp


4.


SAKTHIVEL ET AL.


•

ι A

. _ β'

9

u

1

ΐΌ l'i

12 13 14

hA Α

16

Λ

IB 1*9

Α

21 22 23 24 2^ 26 2*7 28 29 3(5 31 3*2 33 3*4 SCATTERING ANGLE

Α

1

36

Z?

38 3*9 *!θ 4*1 42 43

F i g u r e 5. C a l c u l a t e d and o b s e r v e d p a t t e r n s f o r the b e s t a n t i p a r a l l e l model. The o b s e r v e d p a t t e r n i s shown by d o t s w i t h v e r t i c a l b a r s based on c o u n t i n g s t a t i s t i c s . The c a l c u l a t e d p a t t e r n i s shown by a c o n t i n u o u s c u r v e i n the same f i e l d and the d i f f e r e n c e between the o b s e r v e d and the c a l c u l a t e d p a t t e r n s i s shown i n the lower f i e l d . R f o r t h i s model i s 0.226. wp

F i g u r e 6. Best p a r a l l e l model o b t a i n e d from R i e t v e i d r e f i n e m e n t a g a i n s t o b s e r v e d p a t t e r n (R = 0.19). a) be p r o j e c t i o n and b) ab p r o j e c t i o n . A l l the cErbon and the oxygen atoms a r e shown f o r one c o r n e r and one c e n t r a l m o l e c u l e o n l y , i n the u n i t cell. Only the oxygen atoms a r e shown f o r o t h e r m o l e c u l e s .


77


78

of the best a v a i l a b l e p a r a l l e l - u p and a n t i p a r a l l e l models a r e g i v e n i n T a b l e s I I and I I I , r e s p e c t i v e l y . I n a l l t h e models, t h e m o l e c u l a r axes o f the t e t r a m e r s are s l i g h t l y i n c l i n e d w i t h r e s p e c t to the c a x i s .


0(6)

position.

The p o s i t i o n o f t h e 0(6) i s d e s c r i b e d by t h e c o n f o r m a t i o n o f t h e C(6)-0(6) bond w i t h r e s p e c t t o t h e C ( 5 ) - 0 ( 5 ) a n d t h e C ( 4 ) - C ( 5 ) bonds. F o r e x a m p l e , i f t h e C(6)-0(6) bond i s t r a n s t o C ( 5 ) 0(5)and gauche t o C(4)-C(5), t h e n t h e p o s i t i o n o f 0(6) i s d e s c r i b e d as " t g " . The p o s i t i o n o f 0(6) atoms i n c e l l u l o s e s t r u c t u r e s i s a t t h e center of s e v e r a l c o n t r o v e r s i e s . I n an e x p l o r a t o r y way, we have e x p l o i t e d t h e c o n t i n u o u s dependence o f the f e a t u r e s of the powder d i f f r a c t i o n p a t t e r n on c r y s t a l s t r u c t u r a l d e t a i l s t o i n v e s t i g a t e t h e s e n s i t i v i t y o f t h e d i f f r a c t i o n p a t t e r n t o c h a n g e s i n 0(6) position. I n t h e work r e p o r t e d i n t h e f o l l o w i n g paragraphs, a l l R p values a r e f o r comparison of p a r a l l e l - u p models with the o b s e r v e d p a t t e r n . The p o s i t i o n o f 0(6) was f i x e d as g g , t g , g t by s p e c i f y i n g t h e t o r s i o n a n g l e d e s c r i b i n g i t s p o s i t i o n as 0, -120, and 120°, r e s p e c t i v e l y . A l s o , t h e i n d i v i d u a l t e t r a m e r s were c l o s e t o h a v i n g a 2 - f o l d screw symmetry. W

A: " t g " vs. "gt" models. A powder d i f f r a c t i o n p a t t e r n f o r a model w i t h a l l 0(6) a t o m s l o c a t e d i n t h e " t g " p o s i t i o n was c a l c u l a t e d . R was 0.20. T h i s v a l u e i s s l i g h t l y h i g h e r than t h a t o f the b e s t p a r a l l e l m o d e l b e c a u s e i n t h e b e s t p a r a l l e l m o d e l t h e 0(6)'s a r e a l l p o s i t i o n e d s l i g h t l y away from " t g " . W i t h a l l o t h e r p a r a m e t e r s i n the model kept unchanged, a l l t h e 0(6) p o s i t i o n s w e r e c h a n g e d t o " g t " . R was 0.22, w h i c h i n d i c a t e s t h a t the o v e r a l l change i n t h e p a t t e r n was s m a l l . The c a l c u l a t e d p a t t e r n s f o r both t h e " t g " and the "gt" models are g i v e n i n F i g u r e 8. The p a t t e r n s a r e s i m i l a r but a n o t i c e a b l e d i f f e r e n c e o c c u r s i n t h e p e a k s a r i s i n g f r o m t h e 004, 10*4 a n d 104 r e f l e c t i o n s ( a t 15.9°, 19.1° a n d a t 1 9 . 6 ° , r e s p e c t i v e l y ) . The s e n s i t i v i t y of t h e c a l c u l a t e d p a t t e r n t o changes i n i n t e r n a l c o n f o r m a t i o n , s u c h a s t h e 0(6) p o s i t i o n , makes i t seem p r o b a b l e t h a t a s y s t e m a t i c s t u d y o f t h e v a r i o u s 0(6) p o s i t o n s and o t h e r i n t e r n a l changes w i l l l e a d t o lower R values. w p

w p

w p

n g ^ v s " t g , g t , t g , g t " m o d e l s . I n t h e n e x t m o d e l , t h e 0(6) c o n f o r m a t i o n s were t g , g t , t g , g t ( f r o m the n o n - r e d u c i n g t o t h e r e d u c i n g end). A g a i n , the o v e r a l l e f f e c t on t h e p a t t e r n was s m a l l ( R p = 0.22) b u t t h e r e w e r e p e r c e p t i b l e c h a n g e s i n n o n - z e r o " £ " r e f l e c t i o n s ( F i g u r e 9), e.g., 002 h a s v a n i s h e d . I t i ssmall but s i g n i f i c a n t changes such as t h i s t h a t may u l t i m a t e l y l e a d t o t h e c o r r e c t model. W

C. " t g " v s " t g , gg, t g , gg" m o d e l s . F i g u r e 10 shows t h a t t h e v e r y s t r o n g 110 and 200 r e f l e c t i o n s have changed c o n s i d e r a b l y from t h e " t g " model. R f o r t h i s model was 0.25. These c o m p u t a t i o n a l e x p e r i m e n t s show t h a t changes i n t h e 0(6) p o s i t i o n produce changes i n the r e f l e c t i o n s a t s m a l l 2Θ which a r e w p



4.

SAKTHIVEL ET AL.


F i g u r e 7. Best a n t i p a r a l l e l model o b t a i n e d from R i e t v e i d r e f i n e ment a g a i n s t o b s e r v e d p a t t e r n (R = 0.226). a) be p r o j e c t i o n b) ab p r o j e c t i o n . A l l the carbon^and the oxygen atoms a r e shown f o r one c o r n e r and one c e n t r a l m o l e c u l e o n l y , i n the u n i t cell. Only the oxygen atoms a r e shown f o r o t h e r m o l e c u l e s . i.oo 0.96 0.92 0.88 0.84 0.80 0.76 0.72 0.68 0.64 0.60 0.56 0.52 0.48 X

0.44 0.40 0.36 0.32 0.28 0.24 0.20 0.16 0.12 0.08 0.04 0.00

lO l'l

7 7 , 12 13 14 15 16 17

, ,

19 2*0 2*1 22 23

> 26 27 28 29 30 3*1 32 33 34 3i> 36 !

3V 38

39 40

SCATTERING ANGLE

F i g u r e 8. Calculated d i f f r a c t i o n patterns with d i f f e r e n t C(6)0(6) c o n f o r m a t i o n s . The p a t t e r n f o r the " t g " model i s g i v e n by d o t s . The p a t t e r n f o r the " g t " model i s g i v e n by the continuous curve. The d i f f e r e n c e i s shown i n the lower f i e l d .



T a b l e I I . F r a c t i o n a l and c a r t e s i a n c o o r d i n a t e s o f t h e b e s t p a r a l l e l model. S p e c i e s 1 i s oxygen and s p e c i e s 2 i s carbon. The f i r s t 45 atoms a r e i n t h e c o r n e r m o l e c u l e and the r e s t a r e in t h e c e n t e r molecule. I n each t e t r a m e r t h e f i r s t three monomers c o n s i s t o f 6 c a r b o n atoms and 5 oxygen atoms and t h e monomer a t t h e r e d u c i n g end has 6 carbons and 6oxygens No. S p e c i e s 1 2 3


4 5 6

I

9 10 11

12 1

?

14 15 16 1

I 18

19

20

21 22 ?

2

24 25 26 2

Z 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45

Fractional

1 2 2 2 2 2 2

-.121

1 1 1 1 2 2 2 2 2 2 1 1 1

.032 .035 .139

\ 2

2 2 2 2 2 1 1 1 1 1 2 2 2 2 2 2

1

-.001

-.095 -.024 .074 .029 .039 -.060

.127 .013 .108

.029

-.068 -.016

-.019 .079 -.032 -.035

-.125 -.095 .025 -.069 .002 .100

.054

.065 -.035

.057

.061 .165

.152 .039 .133

.055 -.042 .010

.006 .104

1

-.006

i

-.009 -.099

1 1

.083

Coordinates

.297 .164

.278 .020 -.087

-.170 -.024 .064 .116

-.236 -.246 -.291 -.153 -.266

-.014 .093 .182

.041 -.047 -.116 .237 .258 .320 .187 .301

.043

-.064

-.147

-.001

.087

.139 -.213

-.223 -.268 -.130 -.243 .009 .116

.205 .064 -.024 -.093 .260

.281 .162

-.076 -.008 -.014 -.060 -.048 .013 .063 .045 -.113 -.094 .028 .138 .215 .201 .169 .189 .252 .295 .271 .113 .149 .274 .386 .454 .447 .401 .413 .474 .524 .506 .348 .368 .489 .599 .676 .662 .630 .650 .714 .757 .733 .574 .611 .735 .810

Cartesian

-2.142 -.587 -1.841 -.282 .974 -'.lie -.130 1.154 2.123

2.175 .664 1.916 .314 -.938 -.787 -.321 .874 .130 -1.157 -2.040 -1.993 -.440 -1.693 -.134 1.122 1 .010 .586 -.619 .017 1.302

2.270 2.322 .811 2.064 .462 -.790 -.640 -.174 1.022 .277 -1.010 -1.892 .165

Coordinates

2.123 1.170 1.987 .142 -.621 -1.213 -.172 .457 .832 -1.689 -1.757 -2.082 -1.092 -1.905 -.102 .662 1.300

.296 -.337 -.832 1.697 1.845 2.287 1.334 2.151 .306 -.457 -1.050 -.009 .621 .995 -1.524 -1.592 -1.918 -.927 -1.740 .062 .826 1.464 .460 -.172 -.667 1.861 2.008 1.160

-1 .879 .273 .500 -1 ,362 -1 .009 .385 1.417 .960 -2.593 -1.947 .780 3.268 4.892 4.652 3.783 4.152 5.521 6.572 6.098 2.580 3.186 5.940 8.415 10.022 9.795 8.933 9.285 10.682 11.712 11.257 7.702 8.347 11.077 13.565 15.189 14.949 14.077 14.447 15.817 16.867 16.393 12.874 13.481 16.237 18.012


SAKTHIVEL ET AL.


Table I I . Continued

No.

Species

46

1 2 2 2 2 2 2 1 1 ί 1 i 2 2 2 2 2 2 1 ί 1 1


^Z

48 49 50 51 52 53

54 55 56 57 58 59 60 61 62 6

3

64 65 66 6

Z

68 69 70 71 72 73 74 75 76 78 79 80 81 82 8

3

84 85 86 87 88 89 90

2 2 2 2 2 2 1 i 1

\

2 2 2 2 2 2 1 i ί 1

Fractional

.576 .555 .656 .566 .462 .511 .512 .616 .509 .490 .396 .528 .539 .444 .524 .622 .570 .572 .474 .585 .590 .679 .563 .541 .643 .553 .449 .498 .499 .603 .496 .477 .383 .515 .526 .431 .511 .609 .557 .559 .461 .572 .577 .666 .483

Coordinates

-.003 .266 .158

.389 .492 .598 .473 .387 .272 .624 .671 .900 .657 .765 .503 .404 • 337

.492 •5Z

1

.586 .245 .269 .136 .406 .298 .529 .632 .737 .613 .526 .412 .763 .810 1.040 .797 .904 .642 .543 .476 .631 .710 .725 .385 .408 .552

.107 .156 .148 .105 .120 .181 .229 .210 .051 .076 .198 .327 .383 .368 .335 .355 .418 .462 .437 .280 .314 .440 .569 .618 .609 .567 .582 .643 .691 .672 •513

.538 .660 .788 .845 .830 .797 .817 .879 .924 .899 .742 .776 .901 .976

Cartesian

5.183 4.031 5.325 3.695 2.398 2.456 2.912 4.153 3.598 2.183 1.165 1.539 2.500 1.261 2.916 4.145 3.917 3.387 2.220 3.168 4.424 5.143 4.569 3.417 4.711 3.082 1.785 1.843 2.299 3.540 2.985 1.570 .551 926 887 647 303 532 . 3 Ο Ο

13

Λ A

A

16 17 IB 19 2(3 21 22 23 24 26 27 28 29 30 31 SCATTERING ANGLE

33 3*4 3*5 36 37 38 39 40 41 42

43.

F i g u r e 9. C a l c u l a t e d d i f f r a c t i o n p a t t e r n s f o r models w i t h d i f f e r e n t C(6)-0(6) conformations. The p a t t e r n f o r the " t g " model i s g i v e n by d o t s . The p a t t e r n f o r the " t g , g t , t g , g t " model i s g i v e n by the c o n t i n u o u s c u r v e . The d i f f e r e n c e i s shown i n the lower f i e l d .

SCATTERING ANGLE

F i g u r e 10. C a l c u l a t e d d i f f r a c t i o n p a t t e r n s w i t h d i f f e r e n t C(6)-0(6) conformations. The p a t t e r n f o r the " t g " model i s g i v e n by d o t s . The p a t t e r n f o r the " t g , gg, t g , gg" model i s g i v e n by t h e c o n t i n u o u s c u r v e i n the same f i e l d . The d i f f e r e n c e i s shown i n the lower f i e l d .



4.

SAKTHIVEL ET AL.


F i g u r e 11. C a l c u l a t e d d i f f r a c t i o n p a t t e r n s f o r p a r a l l e l and a n t i p a r a l l e l models r e f i n e d a g a i n s t each o t h e r . The p a t t e r n f o r the p a r a l l e l models i s g i v e n by d o t s . The p a t t e r n f o r the a n t i p a r a l l e l model i s g i v e n by the c o n t i n u o u s c u r v e i n the same field. The d i f f e r e n c e i s shown i n the lower f i e l d .


85


86

p o t e n t i a l l y a n a l y z a b l e even models a r e f a i r l y c l o s e .

though

the R

Refinement of P a r a l l e l a g a i n s t A n t i p a r a l l e l

f w p

s

f o r a l l of

these

Models.

I n an a t t e m p t t o l e a r n how much i n f o r m a t i o n r e g a r d i n g c h a i n p o l a r i t y i s a v a i l a b l e from powder d i f f r a c t i o n d a t a such as ours, we r e f i n e d an a n t i p a r a l l e l model a g a i n s t i n t e n s i t i e s c a l c u l a t e d from a p a r a l l e l model. N e x t , we r e f i n e d t h e p a r a l l e l m o d e l a g a i n s t t h e i n t e n s i t i e s c a l c u l a t e d f r o m t h a t a n t i p a r a l l e l mode. The R was o n l y 1 H% i n b o t h c a s e s . The s i m i l a r i t y o f t h e s e two c a l c u l a t e d p a t t e r n s ( F i g u r e 11) u n d e r l i n e s t h e d i f f i c u l t y o f c h o o s i n g a model based on d i f f r a c t i o n d a t a alone. P a c k i n g energy c a l c u l a t i o n s , a l o n g w i t h s p e c t r o s c o p i c d a t a and hydrogen bonding p o s s i b i l i t i e s s h o u l d be u s e f u l i n m a k i n g a f i n a l c h o i c e . In particular, packing energy m i n i m i z a t i o n w i l l be h e l p f u l to e l i m i n a t e the poor c o n t a c t s p r e s e n t i n the model proposed here.


w p

CONCLUSIONS X - r a y powder d i f f r a c t i o n d a t a i n d i c a t e t h a t t h e u n i t c e l l f o r c e l l o t e t r a o s e i s t r i c l i n i c (space group P1) w i t h two t e t r a m e r s per unit cell. Although there i s a s m a l l preference f o r a p a r a l l e l s t r u c t u r e at t h i s s t a g e , s i m i l a r i t i e s between p a t t e r n s c a l c u l a t e d f o r p a r a l l e l and a n t i p a r a l l e l models make i t d i f f i c u l t t o choose one o v e r t h e o t h e r . In a d d i t i o n t o the r a t h e r i n s e n s i t i v e R v a l u e s , the d i f f e r e n c e s i n s p e c i f i c r e f l e c t i o n s i n the o b s e r v e d and t h e c a l c u l a t e d p a t t e r n s f o r d i f f e r e n t m o d e l s may be u s e d as c r i t e r i a i n s e l e c t i o n o f a model. F o r f a i r l y c r y s t a l l i n e p o w d e r s , s u c h as c e l l o t e t r a o s e , t h e R i e t v e i d method seems t o have a power c o m p a r a b l e t o f i b e r d i f f r a c t i o n f o r a t y p i c a l c e l l u l o s e samples. Peak o v e r l a p i s even m o r e s e v e r e , b u t t h e d a t a a r e known w i t h g r e a t e r c o n f i d e n c e because o f l e s s p r o b a b l e e r r o r i n c o l l e c t i o n and c o r r e c t i o n o f the d a t a . T h e r e f o r e , i t i s l i k e l y t o be a w o r t h w h i l e c o m p l e m e n t t o the f i b e r method f o r h i g h l y c r y s t a l l i n e samples, even i f o r i e n t e d samples are a v a i l a b l e . Acknowledgments We t h a n k P r o f e s s o r A.F. T u r b a k f o r h e l p f u l d i s c u s s i o n s and a d v i c e . T h i s m a t e r i a l i s b a s e d upon work s u p p o r t e d i n p a r t by t h e U.S. D e p a r t m e n t o f A g r i c u l t u r e u n d e r C o - o p e r a t i v e A g r e e m e n t No. 58-7b30-3-564.

Literature Cited 1. Poppleton P.J., Mathieson A.McL., Nature., 219,1046-1049, (1968) 2. Poppleton P.J., Gatehouse B.M., J. Polym. Sci., A2,375-376, (1972) 3. Rietveld H.M., J. Appl. Cryst., 2, 65-71, (1969) 4. Young R.Α., Wiles D.Β., Advances in X-ray Analysis, 24,1-23, (1981) 5. Immirzi Α., Acta Cryst., B36,2378-2385, (1980)


4.

SAKTHIVEL ET A L .

6. 7. 8. 9. 10.

Rietveid Crystal Structure Refinement Method 87

Young R.A., Lundberg J.L., Immirzi Α., ACS Symposium Series, No. 141, 69-91, Fiber Diffraction Methods, Eds. French A.D., Gardner K.H., (1980) Chu S.S.C., Jeffrey G.A., Acta Cryst., B24, 830-838, (1968) F. Takusagawa and R.A.Jacobson, Acta Cryst., B34, 213, (1978) Gardner K.H., Blackwell J., Biopolymers, 13, 1975-2001, (1974) Sarko Α., Muggli R., Macromolecules, 7,486-494, (1974) 1987


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Application of the Rietveld Crystal Structure Refinement Method to

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