598
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Acta Cryst. (1954). 7, 598
C o m m e n t s on p r o b a b i l i t y d i s t r i b u t i o n s for i n t e r a t o m i c v e c t o r s and a t o m i c c o o r d i n a t e s .
R. K. B u ~ o u ~ H a n d I). W. J. C~U~CKS~A~-~, School of Chemistry, The University, Leeds 2, England
By
(Received 19 May 1954)
R e c e n t papers b y H a u p t m a n a n d Karle ( H a u p t m a n &
Karle, 1952; K a r l e & H a u p t m a n , 1954; hereafter referred
to as I, I I respectively) have described a p r o b a b i l i t y
approach to the determination of interatomic vectors
a n d atomic coordinates. The purpose of the present note
is to p o i n t out t h a t t h e principal results of I a n d I I are
essentially equivalent to familiar P a t t e r s o n a n d Fourier
methods.
B y I(58), t h e p r o b a b i l i t y distribution of interatomic
vectors between atoms of t h e i t h a n d j t h kinds is
P ( x , y , z ) ----K a I I
a,k,z
\ a~ l+Biicos2r~(hx+ky÷lzj/.
1 +Bii cos 2rc(hx+ky+lz)
I n a n y practical case Bii = 2fij~/a~ < 1 (e.g. for 20 like
a t o m s in t h e u n i t cell Bii = ~ ) . I t is t h u s possible to
e x p a n d log P(x, y, z) in powers of Bii, whence
log P(x, y, z)
-~ log K a + ~ (-- 1 ) n - ~
n=~
--
~,k,~ \ a2
B~ cos n 2 n h " x ,
(1)
n/
where h - x - ~ hx+lcy+lz. When t h e scattering factors
fi can be w r i t t e n fi ~ - Z J , for all i, where Zi is indep e n d e n t of h, It, l, Bii is i n d e p e n d e n t of h, k, 5; in t h e
general case, Bii, in its dependence on h,/c, l, acts m u c h
as a small t e m p e r a t u r e factor. Treating the former case,
for simplicity,
log P(x, y, z) --log K a
'R'
oo
= 2 ; (-1)~-'B~2: {~hkz_
n=l
1)
h , k , l \ (72
cos~ 2 ~ h " x .
(2)
W h e n n = 1 the t e r m in the r.h.s, is
Bii'~ (R~'k':--l)
a9
c°s 2rch'x
(3)
a 'sharpened' P a t t e r s o n function w i t h origin p e a k removed a n d scaled b y t h e factor Bi# The P a t t e r s o n in
(3) is t h e n 'corrected' b y other distributions scaled b y
factors B2pBaj, etc. F o r B i j , - , ~ only the first two
corrections need be considered. T h a t for n ---- 2 is
--ii ~ / ~
h,k,l \
cos ~ 2 ~ h , x .
(4)
2
The P a t t e r s o n in (4) is of order B~jl2 a n d negative
w i t h respect to t h a t in (3). Similarly t h e t e r m for n ---- 3
yields a P a t t e r s o n on one-third the scale (in x, y, z)
w i t h a n origin p e a k of two-thirds height, a c o n s t a n t
t e r m a n d a correction
~B~, . ~ (R~k,~
h, k,l \
11 cos 2 ~ h " x
(72
(7)
to be added to (3). The n a t u r e a n d size of these corrections to (3) show t h a t log P(x, y, z) is simply a sharpened
P a t t e r s o n w i t h origin p e a k almost absent, together w i t h
minor peaks n o t ordinarily coinciding w i t h those for
interatomic vectors. Since t h e m a x i m a of exp {f(x)}
coincide w i t h those of f(x), t h e m a x i m a of P(x, y, z)
coincide w i t h the m a x i m a of a sharpened P a t t e r s o n .
F o r a finite set of (h, lc, 5) a sharpened P a t t e r s o n p e a k
has finite width, t h e w i d t h for convenience being t a k e n
between first zeros on either side. Defining p e a k w i d t h
in P(x, y, z) as t h e w i d t h between first values of P(x,y,z),
on either side of a peak, equal to u n i t y , P(x, y, z) has
p e a k w i d t h s identical w i t h those of a sharpened P a t t e r son of like terms, except in so far as t h e w i d t h s in P(x, y, z)
are modified b y a slight dependence of Bii on h,/c, 5 or
b y chance coincidences between t h e P a t t e r s o n s on scales
(in x, y, z) 1, ½, ½, etc. Thus P(x, y, z) shows no improvem e n t in resolving power over the sharpened P a t t e r s o n . *
P a p e r I suggests, however, t h a t t h e choice of Bii
determines I(58) as a distribution for vectors between
atoms of the k i n d i a n d t h e k i n d j only, so t h a t only a
few of the whole set of interatomic vectors appear in
a n y particular distribution. T h a t this is n o t the case is
apparent, once the relationship between. I(58) a n d t h e
P a t t e r s o n is realized. E v e n w h e n Bii is v e r y different
from Bpq, say, all P a t t e r s o n peaks appear in either
distribution. The m e t h o d s suggested in the 'Additional
procedures for analysis' of I are t h e n s i m p l y t h e superposition of two or more P a t t e r s o n s relatively displaced
b y vector distances. The resulting distributions h a v e
m a n y m a x i m a of which the most p r o m i n e n t result from
p e a k coincidences. F o r example, the suggested m e t h o d
for resolving the a m b i g u i t y between (x, y, z) a n d ( l - - x ,
l - - y , l - - z ) to obtain a consistent point set, appears to
correspond closely to the superposition m e t h o d of Clastre
& G a y (1950a, b).
II(5) gives the p r o b a b i l i t y distribution for t h e m a g n i tudes of structure factors, given the coordinates of atom 1,
for a c e n t r o s y m m e t r i c crystal,
The expression (4) can be w r i t t e n
P(Ahl . . . . . Ahm) ---- K exp
h,k,~\ a~
(l+cos2~h-2x),
~
. (8)
(5)
a n d (4) is t h u s a sharpened P a t t e r s o n on half the scale
(in x, y, z) of (3), w i t h a n origin p e a k h a l v e d in height,
together with a constant term
Reference to II(2) t h e n yields t h e distribution of
the
* An attempt to improve on the resolution by including
terms other than the first in the probability distribution,
2 B~j,
3 etc.
I(10), merely gave further terms of order Bi~,
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~lhp as 11(5). L e t this distribution be Q. Q m a y be interpreted as a function of x, y, z giving t h e most probable
coordinates of a t o m 1, a n d
m (Ah/ _flh/~ ~lhp)2
log Q = log K - -
=1
(9)
~V/n
2m~ 2 j ~ h ~
i=2
Put
~V/n
then
log Q
=_
~
IA~v
p=l [21:hp
_
2fm~Ah~m~
~2 ~s "
+j~h~ ~ , [
....2~h/, .J "
2Vh/~
- ~/~=1 ~h--e~
(2+~,~h~),
2Vh~
which has large negative peaks a t 0 or ½; a n d the second
t e r m is a Fourier series sharpened b y the factors flh,/~h,,.
The distribution o b t a i n e d b y a p r o b a b i l i t y appr'oac'h
w h e n t h e phases of the structure factors are k n o w n is
t h u s v e r y similar to a sharpened Fourier series.
W h e n the signs of t h e Ah~ are n o t known, K a r l e &
H a u p t m a n suggest the use of II(8),
Q
~
- (IAh,[ --flh, ~lh~)~
-{- exp (--([Ahpl +flh/~ ~lhp)2~
\
2Vh~
]/
A
_
2
2
2~h~
2
/
×
/
.
Using the
series expansion of log cosh x a n d
co 29s (22s-1)-s=l 2S (2S) !
(12)
inter-
B~s ~ (flh~[Ah~,l~lh~)__9s,
~=1 \
(13)
~h~
where t h e B2s are Bernouilli numbers. Since (13) contains
only even powers of IAh~] it m u s t be of the P a t t e r s o n
type. I n d e e d t h e corrections obtained for s ---- 2, 3, etc.,
to t h e P a t t e r s o n obtained b y t a k i n g s---- 1, are even
less significant t h a n those m a d e b y (4) a n d (7) to (3),
since w i t h all like a t o m s (13) becomes, in P 1 ,
co 29s (2 ~ s - 1)
,~V~ ( 2 s ~
B~
]Ah/~]s
(N-2)----~ p = l
•
}s
(2 + ~1, 2hp)
(14)
Thp
the successive coefficients for the s u m m a t i o n s over p
being a p p r o x i m a t e l y 1/(2N),--1/(121V2), 1/(45~V3), etc.
for s = 1, 2, 3 etc. T h u s the p r o b a b i l i t y distribution for
coordinates w h e n the structure-factor signs are n o t k n o w n
is essentially a sharpened P a t t e r s o n , which is on half scale
in x, y, z, a n d t h u s has some peaks coincident w i t h
atomic positions.
Similar discussions can be given for t h e distributions
in I I for non-centrosymmetric crystals. If the position
of one a t o m is a r b i t r a r i l y fixed a n d the phase angles are
assumed u n k n o w n , t h e resulting Patterson-like distrib u t i o n is of n o r m a l scale in x, y, z w i t h t h e fixed a t o m as
origin; t h u s again some peaks coincide w i t h the atomic
positions.
I t m a y be r e m a r k e d t h a t II(5) can be w r i t t e n down
directly b y appealing to t h e central limit t h e o r e m
(Cram~r, 1946) on the assumption of the independence
of the ~ih , P = 1,
, m, for each set i The assumption
of independence is clearly equivalent to t a k i n g the first
t e r m alone in the series referred to in I I . I t does not seem
possible, however, to obtain better t h a n the sharpened
P a t t e r s o n if this a s s u m p t i o n is made.
"
(111
Reference above shows t h a t only terms in A h ~ l h ~ are
i m p o r t a n t , so t h a t the structure-dependent p a r t of
log (Q/K) is
~ log eosh (/lha 'Aha[ ~lha/ .
v= 1
"Ch~
/
changing t h e order of summations, this t e r m becomes
(10)
The first t e r m on the r.h.s, is a c o n s t a n t ; in the space
group P i , the t h i r d becomes
599
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'
References
CLASTRE, J. & GAY, R. (1950a). C. R. Acad. Sci., Paris,
230, 1976.
CT.ASTRE, J. & GAY, R. (1950b). J. Phys. Radium, 11,
75.
CRAM~R, H . (1946). Mathematical Methods of Statistics.
P r i n c e t o n : U n i v e r s i t y Press.
HAVPT~A_W, H . & KARLE, J. (1952). Acta Cryst. 5, 48.
KARr.E, J. & I-IAUPrMA~, H . (1954). Acta Cryst. 7, 375.
Acta Cryst. (1954). 7, 599
Conversion
of Norelco fluorescent
spectrograph
J. J. DEMARCO a n d G. WEREMCHUX, Ordnance Materials
U.S.A.
to
an X-ray
diffractometer.
B y R. J.W~.iss,
Research Office, Watertown Arsenal, Watertown, Mass.,
(Received 21 December 1953 and in revised form 25 February 1954)
The N o r t h American Philips Fluorescent A t t a c h m e n t
consists of a n OEG-50-Machlett t u n g s t e n tube, which is
r u n a t 50 kV., 50 mA. a n d sprays X - r a y s on to a sample.
A n elemental analysis of the sample is m a d e b y analyzing
the characteristic K a n d /5 fluorescent lines b y means
of a single crystal a n d a goniometer.
B y utilizing a pure element such as cobalt one obtains
the characteristic K a a n d Kfl lines practically free of
continuous background. A filter will essentially eliminate
t h e Kfl. To m a k e use of this plane source of monochromatic r a d i a t i o n for powder diffraction one m u s t
create a focal spot so as to utilize the geometrical focusing
conditions necessary for high resolution. This is done b y
use of a slit between the plane source of fluorescent radiation a n d the powder specimen. I n addition, the plane
source of Co radiation is ti]ted u p w a r d so as to obtain