faktor hamburan atomik

8/18/2019 faktor hamburan atomik

1/6

478 R E L A T I O N S B E T W E E N T H E P H A S E S O F S T R U C T U R E F A C T O R S

r e s p o n d i n g t o F i g . 3( c) , o n e c a n n o t d e d u c e f ro m t h e m

e n o u g h f u r t h e r c o r r e c t s i g n s t o p r o d u c e F i g . 3 ( d) .

O n l y a s h a r p e n i n g o f t h e o r i g i n a l m a x i m u m r e s u l t s ,

a n d n o t i t s re s o l u t i o n in t o i n d i v i d u a l m a x i m a , a s i s

s h o w n i n F i g . 3 (e ). S u c h c o n s i d e r a t io n s m u s t b e k e p t

i n m i n d i n a n y u s e o f s ig n r e l a ti o n s o r p h a s e r e l a t io n s

w h e n t h e d a t a a r e n o t s u f f i c ie n t t o r e s o lv e i n d i v i d u a l

a t o m s .

I a m g r a t e f u l t o m y c o l le a g u e s D r F . H . C r i c k a n d

D r M . M . W o o l f s o n f o r t h e h e l p f u l a d v i c e t h e y h a v e

g i v e n i n t h e c o u r s e o f d i s c u s s i o n s o n t h i s t o p i c .

R e f e r e n c e s

BOKHOVEN, C., SCHOONE, J. C. ~; BIJVOE T, J. i~. (1951 ).

Acta Cryst. 4, 275.

COCHRA~, W . & WOOLFSON, M. M. (1955) . Acta Gryst.

8 1 .

~A UPT ~A _~, H . & ~ , J . ( 19 53 ). Solution of the

Phase Problem. I. The Centrosymmetric Crystal. A .C .A .

M o n o g r a p h N o . 3 . W i l m i u g t o n : T h e L e t t e r S h o p .

SAYP.E, D. (1952 ). Acta Gryst. 5, 60.

WATSOn, G. N. (1922) .

A Treatise on the Theory of Bessel

2 unctions. C a m b r i d g e : U n i v e r s i t y P r e s s .

WrLSO~¢, A. J . C. (19 49). Acta Cryst. 2, 318.

WOOLFSON, M. M . (1954). Acta Cryst. 7, 61.

Acta Cryst. (1955) . 8 , 478

N e w C a l c u l a t io n s o f A t o m i c S c a tt e ri n g F a c t o r s

B Y J . B E R G H U IS IJB E R T H A M . I-IA A~TAPPEL AND M . POTT~.RS

Mathemat ica l Cen t re , Ams terdam, The Nether lands

~ D B . O . LOOPSTRA *, CA ROLIN E H . 1VLkCGTTJLAVRY AND A . L. VEENENDAA L

Labora tory fo r General and Inorgan ic Chem is t ry , Un ivers i t y o f Ams terdam , The Nether lands

(Received

17

February

1955)

S c a t t e r in g f a c to r s f o r t w e n t y - t h r e e a t o m s h a v e b e e n c a l c u l a te d f r o m H a r t r e e a n d H a r t r e e - F o c k

r a d i a l w a v e f u n c t i o n s . T h e r e s u l ts a r e c o m p a r e d w i t h J a m e s & B r i n cl l ey ' s v a l u e s a n d w i t h t h o s e

r e c e n t l y o b t a i n e d b y o t h e r a u t h o r s .

1

S i n c e t h e c a l c u l a ti o n s o f s c a t t e r i n g f a c t o rs b y J a m e s

& B r i n d l e y ( 1 93 1 ) m a n y n e w d a t a o n e l e c tr o n d i s t r ib u -

t i o n s i n a to m s , c o m p u t e d b y t h e s e l f -c o n s i s te n t f i e ld

m e t h o d , h a v e b e c o m e a v a i la b l e . T h e w o r k o f J a m e s

& B r i n d l e y ( q u o t e d a s J B i n t h e f o l lo w i n g ) w a s s o m e

y e a r s a g o e x t e n d e d b y V i e r v o l l & 0 g r i m ( 19 4 9) , w h o

i n c l u d e d e l e c t r o n d i s t r i b u t i o n - d a t a o n N a + , K + , a n d

C u + a n d e x t e n d e d t h e s i n 0/~ t r a n g e . V i e r v o ll & 0 g r i m

w e r e p r i m a r i l y i n t e r e s t e d i n t h e f v a l u e s a t h i g h e r

s i n 0 /~ t; t h e y t h e r e f o r e a p p l i e d o n l y w a v e f u n c t i o n s

c a l c u l a t e d w i t h o u t e x c h a n g e , a s t h e c o r r e c t i o n s f o r

e x c h a n g e w o u l d p r e s u m a b l y h a v e o n l y s m a l l e ff e ct s

o n t h e s c a t t e r i n g f a c t o r s i n t h i s r e g i o n .

R e c e n t l y , n e w v a l u e s f o r s c a t t e ri n g f a c t o r s h a v e

b e e n c a l c u l a t e d b y : ( a) M c W e e n y ( 19 5 1) ( M c W ) , f o r

a l l a t 0 m ~ l i g h t e r t h a n h Ta f ro m w a v e fu n c t i o n s g i v e n

b y D u n c a n s o n a n d C o u l s o n ; ( b) H o e r n i & Ib e r s ( 19 5 4)

( H I ) f o r C , 1 ~ a n d 0 f r o m n e w e r a v a i l a b l e s e lf - c o n -

s i s t e n t f i e l d d a t a w i t h e x c h a n g e ; (c ) H e n r y ( 1 95 4 ),

f o r A u + a n d H g + +, f r o m s e l f - c o n s i s t e n t f i e l d d a t a

w i t h o u t e x c h a n g e .

I t i s e v i d e n t f r o m c o m p a r i s o n o f t h e r e s u l ts o f H I

a n d J B t h a t t h e t a k i n g i n t o a c c o u n t o f e x c h a n g e g i v e s

e f f e c t s w h i c h m a k e t h e c o r r e s p o n d i n g e f f e c t s o n c a l -

c u l a t e d s t r u c t u r e f a c t o r s o u t s i d e t h e e r r o r l i m i t f o r

* Now at J .E .N.E .R. , Kje l ler per Lf l les t rom, Norway.

s t ru c t u r e d e t e r m i n a t i o n b y a c c u r a t e m o d e r n m e t h o d s .

A l so , t h e i n t e r v a l i n s i n 0 /~ t c h o s e n b y J B i s r a t h e r

t o o l a r g e f o r s u i t a b l e i n t e r p o l a t i o n i n t h e l o w s i n 0 /~ t

r e g i o n .

I n v i e w o f t h e p r e p a r a t i o n o f V o l u m e 3 o f t h e n e w

e d i t i o n o f t h e

In terna t iona l TaS les fo r X- ray Crys ta l -

Zography, i t w a s d e c i d e d t o r e c a l c u l a t e a t o m i c s c a t-

t e r i n g f a c t o r s f r o m a l l t h e n e w e s t a v a i l a b l e s e lf -

c o n s i s t e n t f i e l d d a t a . T h i s d e c i s i o n h a d , i n f a c t , b e e n

t a k e n b e f o re w e w e r e a w a r e o f t h e w o r k b e i n g d o n e b y

H I . W e t h e r e fo r e i n c l u d e d C , N a n d 0 i n o u r co m -

p u t a t i o n s , w h i c h h a s t h e a d v a n t a g e o f g i v i n g t h e f

v a l u e s f or t h e s e a t o m s a t t h e s a m e i n t e r v a l s a s f o r t h e

o t h e r a t o m s , a n d , m o r e o v e r , a f f o r d s t h e p o s s i b i l i t y o f

a n e x t r a c h e c k .

I n t h i s p a p e r w e g i v e o u r r e s u l t s o n a ll a t o m s , u p t o

R b+ ~ f o r w h i c h t h e r e q u i r e d e l e c tr o n d i s t r i b u t i o n s a r e

a v a i l a b l e . W e o m i t t e d H a n d H e , h o w e v e r , fo r w h i c h

t Bes ides the l i t e ra ture ment ioned a t the foot of Table 1 ,

we found references to ca lcula t ions on Sc and T i (Hancock,

1934), and on N i (Gray & Manning, 1941) . We w ere unable

t o ob t a i n t he f o r mer ; t he l a t t e r i s a shor t communi ca t i on

w i t hou t numer i ca l da t a .

Ma nning & Goldberg (1938) gave da ta o n Fe . Th ese , how-

ever , a re on a logar i thmic scale of r , whereas the data for the

other a tom s were on a l inea r r sca le . I t i s therefore imposs ible

t o hand l e t he Fe a t om i n t he s ame mechan i ca l w ay a s t he

others . We are now comput ing the scat ter ing fac tor of Fe

separa te ly and we intend to repor t on i t in due t ime.


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J . BERGHUIS , I JBERTHA : M: . HAANAP P EL AND M . P OTTERS

e t a l .

479

Table 1 .

A t o m i c s c a t t e r in g f a c t o r s

N S :

Electron densit ies calculated for the normal state.

V S :

Electron densit ies calculated for the valence state.

X: The elect ron d is t r ibu t ion has been calcu lated wi th exchange.

Li Be

sin 0/~t

N S X 1 N S X 2

0-00 3.000 4.000

0.05 2.710 3.706

0-10 2.215 3-065

0 15 1.904 2.462

0-20 1.741 2.059

0.25 1.627 1.827

0.30 1.512 1.693

0.35 1-394 1.600

0.40 1.269 1.520

0.50 1-032 1.362

0.60 0.823 1-195

0-70 0.650 1.030

0.80 0.513 0.877

0.90 0.404 0.739

1.00 0.320 0.621

1.10 0.255 0.521

1-20 0.205 0-438

1.30 0.164 0.369

C N O F F- Ne Na

V S X 3 N S X 4 N S X 5 N S ( X 2 p ) 6 N S ( X 2 p ) e N S ( X 2 p ) 6 N S X r

6 000 7.000 8.000 9-000 10. 000 10.000 11.00

5.764 6.781 7.796 8.790 9-630 9.812 10.56

5.141 6.203 7.250 8-208 8.733 9.295 9.76

4.362 5.420 6.482 7.396 7.656 8.546 9-02

3.612 4.600 5.634 6.501 6.597 7-665 8-34

3 003 3.856 4.814 5.625 5.643 6-768 7.62

2.538 3.241 4.094 4-837 4.820 5.905 6.89

2.212 2.760 3.492 4-160 4.129 5.128 6-16

1.983 2.397 3.010 3.598 3.566 4.454 5.47

1.707 1.944 2.338 2- 769 2.751 3.403 4-29

1.548 1.698 1.944 2.252 2.237 2-692 3.40

1.423 1.550 1-714 1-926 1-921 2-234 2-76

1.313 1.444 1.566 1.725 1.723 1.934 2.31

1.202 1.350 1.462 1.587 1.583 1.737 2.00

1.096 1.263 1.374 1.484 1.485 1.601 1.78

0 992 1.175 1-296 1-404 1.406 1.496 1-63

0.896 1.083 1.220 1.333 1-334 1.418 1.52

0.802 1.005 1.144 1.263 1.264 1.345 1.44

Mg2+ A13+ Si4+

s i n O~ 2 N S X s N S 8 N S X 1 °

0.00 10.00 10.00 10.00

0.05 9.91 9.93 9.95

0.10 9.66 9-72 9.79

0-15 9-26 9.38 9-54

0.20 8.75 8.94 9.20

0.25 8-15 8.42 8.79

0.30 7.51 7-85 8.33

0.35 6.85 7.26 7.83

0-40 6-20 6.65 7.31

0.50 4.99 5.51 6.26

0.60 4.03 4-53 5.28

0.70 3.28 3.72 4.42

0.80 2.71 3.10 3.71

0.90 2.30 2.62 3.13

1.00 2.01 2.27 2.68

1.10 1.81 2.01 2.33

1.20 1.65 1.82 2.06

1.30 1.54 1.68 1.86

Cu + Zn Ga

sin 0/~ N S X 15 NS 16 N S 16

0.00 28.00 30.00 31.00

0.05 27.67 29.30 30.12

0-10 26.71 27.63 28.29

0-15 25.30 25.67 26.35

0.20 23.59 23.74 24.48

0.25 21-76 21.88 22.67

0.30 19.92 20.11 20.94

0.35 18.14 18.41 19.28

0.40 16.50 16.83 17.70

0.50 13 66 14.05 14.88

0.60 11.45 11-84 12.54

0.70 9.80 10.15 10.72

0.80 8.61 8.90 9.35

0.90 7.75 7.99 8-34

1.00 7.12 7.32 7.59

1.10 6-64 6.81 7-02

1.20 6.24 6-40 6-57

1.30 5-89 6.04 6-20

1: Foc k & Pet ras hen , 1935.

2: t ta rtre e & Hart ree, 1935.

3 : J uc ys , 1947.

4: Hart ree & Hartr ee, 1948.

5: Hart ree, H artr ee & Swirles, 1939.

6: Brow n, 1933.

C1- A K + Ca Cr 2+

N S X 1 1 2 ¥ S X 1 2 NS X T 2 ~ S X 13 N S I 4

18.00 18.00 18.00 20.00 22.00

17.33 17.54 17.65 19-09 21.65

15.68 16.30 16.68 17.33 20.67

13.74 14.65 15.30 15.73 19.27

11.97 12.93 13-76 14.32 17.67

10.57 11.42 12.27 12.98 16.04

9.51 10.20 10.96 11-71 14.50

8.74 9.25 9.89 10.59 13.10

8.15 8.54 9.04 9-64 11.87

7.30 7.56 7.86 8.26 9.93

6.60 6.86 7.11 7.38 8.60

5.91 6.23 6.51 6-75 7.69

5.24 5.61 5-94 6.21 7.06

4.60 5.01 5.39 5.70 6.56

4.01 4.43 4.84 5.19 6-13

3.49 3.90 4.32 4.69 5.72

3.06 3.43 3.83 4-21 5.31

2.69 3 03 3.40 3.77 4-91

Ge As

N~ rl7 N~.~16

32.00 33-00

30-98 32-11

28.91 30.06

26.95 27.81

25.09 25.76

23-31 23.96

21.67 22-30

20.08 20.75

18.53 19.27

15.71 16.50

13.29 14.06

11.34 12.03

9.85 10-40

8.71 9-15

7.86 8.21

7.24 7.50

6.74 6-95

6.35 6.51

7: Ha rt re e & I-Iartree, 1948.

8: Yost, 1940.

9: Hart ree, 1935.

10: Hart ree, H artr ee & Manning, 1941c.

11: Har tree & Hartr ee, 1936a.

12: H ar tre e & Har tre e, 1938b.

Rb +

N S ~

36-00

35-35

33.61

31.28

28.85

26-61

24-68

23.02

21.57

19.07

16.82

14.75

12.88

11.28

9.96

8.89

8-05

7.40

13: H artr ee & Hartr ee, 1938a.

14: Mo one y, 1939.

15: H ar tre e & Har tre e, 1936b.

16: Har tre e, I-Iartree & Manni ng, 1941a.

17: t tar tree , Hart ree & Manning, 1941b.

33*


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48 0 N E W C A L C U L A T I O N S O F A T O M I C S C A T T E R I N G F A C T O R S

x

3

I ~ T

0 0 - 2 0 - 4

0 " 6 0 - 8 1 . 0 1 " 2

S i n # / / / ,

N e

F _

Fig . 1 . Atomic sca t te r ing fac tor s for Ne , F- , F , O, N and C, ca lcu la ted by d if fe ren t au thors .

[ ] : Hoern i & Iber s . × : McWeeny. A : Vie rvoll & Ogr im. • : Jam es & Br indley . O : Our ca lcu la tions .

Ar rows J ', J~ ind ica te th a t the symbol co inc ides w ith th e curve , b u t has been drawn of f the curve to av oid c rowding .

a t o m s w e c o n s i d e r t h a t M c W e e n y ' s d a t a a r e c o m -

p l e t e l y a d e q u a t e . I f n e c e s s a r y , t h e e l e c t r o n d i s t r i b u -

t i o n s w e r e s m o o t h e d o u t s p h e r i c a l l y , a s w a s a l s o d o n e

b y H I .

A p o i n t o f c o n si d e r a t io n h a s b e e n t h e c o m p u t a t i o n

f o r a t o m s i n d i f f e r e n t s t a t e s o f i o n i z a t i o n . T h e n e c e s -

s a r y d a t a a r e a v a i l a b le o n l y f o r L i a n d L i + ( F o c k &

P e t r a s h e n , 1 9 3 5 ); O - , O , O + , O + + ( H a r t r e e , H a r t r e e

& S w i r l es , 1 9 3 9) ; F a n d F - ( B r o w n , 1 9 3 3 ) ; N a a n d

N a + ( H a r t r e e & H a r t r e e , 1 94 8) a n d C a a n d C a + +

( H a r t r e e & H a r t r e e , 1 9 3 8 a) . T h e s e h a v e a l l b e e n

c a l c u l a t e d w i t h e x c h a n g e . I t h a s b e e n r e a l i z e d ( J a m e s ,


4/6

J . B E R G H U I S , I J B E R T H A M . H A A N A P P E L A N D M. P O T T E R S

et al.

481

. 2

2 0 4 0 6 0 8 1 : 0 1

S i n 0 / ~

B e

Fig. 2. Atomic scatte ring factors for Li and Be.

× : McWe eny. /k : Viervoll ~)grim. • : flames Brin dley . C) : Our calcu lation s.

1 9 4 8 ; B i j v o e t & L o n s d a l e , 1 9 5 3) , t h a t t h e s c a t t e r i n g

f a c t o r c h a n g e s v e r y l i t t l e w i t h t h e s t a t e o f i o n i z a ti o n ,

e x c e p t , o f c o u r s e , f o r v e r y s m a l l v a l u e s o f s in 0 / ;t .

W e h a v e c h e c k e d a n d c o n f i r m e d th i s fo r F a n d F -

( se e T a b l e 1 )'. F o r t h e o t h e r a t o m s w e h a v e o n l y c o m -

p u t e d f f o r t h e n o r m a l s t a t e , u n l e s s d a t a f o r t h i s s t a t e

w e r e n o t a v a i l a b l e .

T h e c a l c u l a t i o n s h a v e b e e n p e r f o r m e d o n I B M m a -

c h i n e s i n t h e M a t h e m a t i c a l C e n t r e , A m s t e r d a m . F o l -

l o w i n g J B , w e h a v e c a l c u l a t e d t h e s e p a r a t e c o n t r i b u -

t i o n s o f i n d i v i d u a l e l e c t r o n o r b i t a l s i n o r d e r t o b e

a b l e t o i n v e s t i g a t e t h e p o s s i b i l i t y o f i n t e r p o l a t i o n t o

t h e a t o m s o m i t t e d f r o m o u r l is t . W e w i ll r e p o r t o n

t h i s l a t e r . A s i n t e r v a l s i n s i n 0 / 2 w e h a v e c h o s e n :

0-00 (0.05) 0 .40 (0.1) 1 .30 A -x,

t h a t i s, t o t h e e n d o f t h e M o K d i f f r a c t i o n r eg i o n .

B e c a u s e o f t h i s c h o i c e o f i n t e r v a l , o u r r e s u l t s a r e

d i r e c t l y c o m p a r a b l e w i t h t h e J B v a l u e s .

T h e i n t e g r a l s

lP~(r)

s in

r/ lurdr

w h e r e g = 4 z s i n 0 / 2, w e r e u n i f o r m l y c a l c u l a t e d :

( a ) t o t h e s a m e u p p e r l i m i t rz = 1 3 a l l, w h e r e a r t

i s t h e ' a t o m i c u n i t ' , 0 . 5 2 9 1 7 A ,

( b ) i n t h e s a m e s t e p s

r/aH=O

(0.005) 0 .3 (0-05) 13.0.

F o r t h e l a t t e r p u r p o s e , t h e d a t a f r o m t h e l i t e r a t u r e

w e r e i n t e r p o l a t e d t o t h e r e q u i r e d i n t e r v a l s .

T h e i n t e g r a t i o n s w e r e c a r r i e d o u t a c c o r d i n g t o

S i m p s o n , f o l l o w i n g a p r o c e d u r e d e s c r i b e d i n t h e r e p o r t

M R 1 5 o f t h e M a t h e m a t i c a l C e n tr e , A m s t e r d a m .

T h e i n t e g r a t i o n s w e r e c h e c k e d :

( a ) b y t h e n o r m a l i z i n g i n t e g r a l

IilP~(r) dr;

( b ) b y t h e i n t e g r a l

nP~(r) (s in ~tir//~ir

d r ,

0 i

w h e r e t h e s u m e x t e n d s o v e r a l l v a l u e s o f / x i , i .e .

of sin 0/2~;

(c ) b y p e r f o r m i n g a l l 2 0 33 i n t e g r a t i o n s t w i c e , u s i n g

d i f f e r e n t c o u n t e r s a n d t y p e - b a r s .

T h e d a t a o f T a b l e 1 a r e g i v e n to t h r e e d e c i m a l p l a c e s


5/6

482 N E W C A L C U L A T I O N S O F A T O M I C S C A T T E R I N G F A C T O R S

2 8 - °

2 4 °

A

•

2 0

e

1 6

1 2

8

4 -

0

I I I / I I I - -

0 0 . 2 1 0 1 . 2

I I I I I I

0 4 0 6 0 8

S in 8 A

Fig. 3. Scattering factor for Cu+.

O: Our calculations for Cu+ with exchange. A : V iervoll & (3grim for Cu withou t exchange.

• : James & Brindley for Cu from Thom as-Ferm i field.

u p t o n e o n a n d t o t w o d e c i m a l p l a c e s f o r a t o m s o f

h i g h e r a t o m i c n u m b e r ; t h e l a s t p l a c e , o f c o u r s e , d o e s

n o t h a v e a n y p h y s i c a l s i g n i f i c a n c e , b u t i t f a c i l i t a t e s

i n t e r p o l a t i o n .

I t i s se e n f r o m F i g . 1 t h a t o u r r e s u l t s a g r e e c o m -

p l e t e l y

with those of HI . The agreement wi th

M e W e e n y ' s v a l u e s , a l t h o u g h c a l c u l a t e d f r o m d i f f e r e n t

e l e c t r o n - d i s t r i b u t i o n a p p r o x i m a t i o n s , i s i n g e n e r a l v e r y

g o o d ( F i g s . 1 a n d 2 ) e x c e p t f o r F a n d N e .

A c o m p a r i s o n w i t h J a m e s & B r i n d l e y ' s v a l u es

s h o w s t h e f o l l o w i n g p o i n t s :

( a) F - , A 1 3 + a n d R b + h a v e e v i d e n t l y b e e n c o m -

p u t e d f r o m t h e s a m e e l e c t r o n - di s t r ib u t i o n d a t a .

T h e d e v i a t i o n s a r e i n g e n e r a l s m a l l e r t h a n 1 % ,

a n d s o a r e i n s i g n i f i c a n t .

(b ) L i , O , F , Na , S i4 +, C 1 - a n d K + : A p p a r e n t l y ,

c)

d)

d i f f e r e n t s e ts o f e l e c t r o n - d i s t r i b u t i o n d a t a h a v e

b e e n u s e d . T h e o n e s w e u s e d w e r e a l l c o m p u t e d

t a k i n g e x c h a n g e i n t o a c c o u n t ( in t h e c a s e o f F

o n l y f o r t h e 2 p - e l e c tr o n s ) . E x c e p t f o r I A ( F i g . 2 ) ,

t h e f i t i s m u c h w o r s e t h a n i n t h e f o r m e r g r o u p ,

d i f fe r e n c e s g o i n g u p t o 1 0 % ; d i s c r e p a n c i e s a r e

p a r t i c u l a r l y l a r g e f o r F a n d f o r O . T h i s s h o w s

t h e c o n s i d e r a b l e i n f l u e n c e o f e x c h a n g e o n t h e

s c a t t e r i n g f a c t o r s .

B e , N , N e , M g ~+ a n d A h a v e b e e n c o m p u t e d

b y J B b y i n t e r p o la t i o n . T h e d i s c re p a n ci e s w i t h

o u r v a l u e s a r e i n g e n e r a l n o t l a r g e r t h a n i n t h e

f o r m e r g r o u p . A g a i n , t h e a g r e e m e n t i s g o o d f o r

B e , t h e l i g h t e s t a t o m o f t h e g r o u p ( F i g . 2 ).

C a , C r 2+, C u+, Zn , Ga , Ge a nd As : F o r these

e l e m e n t s , J B g a v e o n l y f v a l u e s o b t a i n e d f r o m

T h o m a s - F e r m i e l e c tr o n d i s tr i b u ti o n s . T h e a g r e e -

m e n t o f t h e s e v a l u e s w i t h o u r s i s i n g e n e r a l

wor se th an in t he cases (b) and ( c ) . (C a ~+, ca l -


6/6

J . BERGHUI S, I JBERT HA M. HAANAPP EL AND M. POTTERS et al. 483

culated by JB from Hartree wave functions

without exchange, fits our curve from sin 0/A =

0.5 onwards.)

Finally, Fig. 3 gives a comparison between the scat-

tering factor of Cu+, calculated by us from wave func-

tions with exchange, and that for Cu.calculated by

Viervoll & 0grim without exchange. The discrepancies

are appreciable throughout the Cu K range. Likewise,

Viervoll & 0grim's values for Ca and Cr are consider-

ably smaller than ours at low sin 0/~ values. I t ap-

pears, therefore, that more calculations for moderately

heavy elements are very desirable.

We wish to than k the Directors of the Mathematical

Centre, Amsterdam, for their ver y helpful cooperation

and the Netherlands Organization for Pure Research

(Z. W. 0.) for generous support. Our than ks are also

due to Prof. D. R. Hartree for stimulating advice and

for his interest in this work.

R e f e r e n c e s

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FOCK, V. PETRASHEN, M. J . (1935). Phys Z Sowjet

8, 547.

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475.

HANCOCK, S. T. R. (1934). M. Sc. Thesis, Manchester.

HARTREE, D. R. (1935). Proc. Roy. Soc. A, 151, 96.

HARTREE,

A , 150 ,

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A , 155 ,

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HA_RTREE,

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A G e n e r a l i z e d T r e a t m e n t o f C o ld W o r k i n P o w d e r P a t t e r n s *

B r B . E . WARRE~

Massachusetts Insti tute of Technology, Cambridge, Massachusetts, U.S . A.

Received 14 February 1955)

Calculations of the effect of particle-size and cold-work distortion in the broadening of powder-

pattern lines have been simplified in several previous treatments by considering the reflections as

00l for orthor homb ic axes. By a s uitable tran sfor mati on of variables and axes, it is possible to car ry

through the calculation for the general hkl reflection for a crystal of any system. The general result

obtained is identical to t hat previously obtained with th e simplifying assumptions. The result is

expressed in terms of particle-size and disto rtion Fourier coefficients which are o btain ed from the

experimental peak shapes.

1 . I n t r o d u c t i o n

To develop a Fourier treatment of the broadening of

X-ray powder pattern lines by cold-work distortion,

Stokes & Wilson (1942, 1949) and Warren & Averbach

(1950) have assumed a transformation of axes such

that the reflection could be considered as 001 for

* Research sponsored by th e U.S. Atomic Energy Commission.

orthorhombic axes. This simplifies considerably the

mathematical treatment, but for the general hkl re-

flection from crystals of low symmetry such a trans-

formation is not possible. Nevertheless, the result ob-

tained b y this simplified treatme nt appears to be quite

general, suggesting that the transformation to ortho-

rhombic axes is not really necessary. It is the purpose

of this paper to give a generalized treatment for any

faktor hamburan atomik

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