Notizen
Mapping Parts of the Electron Density Distribution from X-Ray Bragg Scattering Intensities
(Lambda Technique) Karl F. Fischer
F a c h r i c h t u n g 17.3 — K r i s t a l l o g r a p h i e , U n i v e r s i t ä t des Saarlandes, Saarbrücken
Z . N a t u r f o r s c h . 3 6 a , 1 2 5 3 - 1 2 5 4 ( 1 9 8 1 ) ; received October 8, 1981
B y a p p r o p r i a t e selection o f t w o or t h r e e wavelengths, intensity differences can be used for o b t a i n i n g d i r e c t l y t h e electron density d i s t r i b u t i o n (i.e. t h e a r r a n g e m e n t o f a t o m s ) for parts o f a crystal s t r u c t u r e . A p p l i c a t i o n t o macro- molecules a n d amorphous b i n a r y substances appear fea- sible.
X-rays taken from a synchrotron source can be tuned over a wide range in A. For using anomalous scattering effects [1] close to K or L absorption edges Ac, special conditions for intensity collection can be met. The method briefly described below uses symmetry conditions imposed on the real (a) and imaginary part (b) of the x-ray atomic scattering factor defined by
a = ( f0 + f')T and b = f" T
with T = Debye-Waller "temperature factor".
A first possibility consists in selecting two wave- lengths Ai < 2.2 on both sides of Ac of one species of atoms (called "edge atoms") in a crystal structure such that
«ei = «e2, bei ^ be 2 ( l a )
holds with
e denoting the edge atoms (of which k are as- sumed to be in the unit cell),
1, 2 denoting Ai and X2.
If A2 — Ai is small enough, the corresponding a and b for the "normal scatterers" (m — k in the unit cell and denoted by the subscript n) follow
an2 > bnig^bn2<bei. ( l b ) From the well-known general expression for | F (h) |2
m m
\F(h)\2= 2 2 ( ^ v + bßbv)cosh(rv-rß) + (aß bv — bß av) sin h (rv — rß)
R e p r i n t requests to P r o f . D r . K . F . Fischer, F a c h b e r e i c h 17 der U n i v e r s i t ä t des Saarlandes, K r i s t a l l o g r a p h i e , D - 6 6 0 0 Saarbrücken.
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it follows via straight-forward arguments that the scaled intensity differences
, ... \Fn(h)\*-\FMh)\2
A12 ( A ) = r 7 , ( 2 ) Oel — 0e2
permit computing a Fourier transform J5*
L12 (u) = & [A12 (h)] = L\2c (u) + i L12S (u) . (3) The real part L\2c(U) of (3) yields the vectors be- tween the anomalous scatterers only [2]t. The imaginary part L\2s, (u) contains k(m — k) vectors from each e-atom to all n-atoms. They represent k parallel "images" of the n-atom structure as
"seen" from each e-atom [3]. These images are, however, disturbed by equivalent inverse and negative images caused by the anti-centrosym- metry of L\2b (lO- This will partly or (for centro- symmetric structures) completely erase these images.
In this case one can make use of a second sym- metry condition by selecting a third wavelength A3 such that the role of be and ae in ( l a ) is inter- changed :
ae3 =1= ®e2 , &e3 — ,
«n3 ^ ®n2 , Ön3 ^ bn2 . (4)
In full analogy to (2) and (3), the Fourier trans- form
L23 (U) = ^23c (u) + i L2ss (U) (5) is computed. Addition of parts of (2) and (3) ac-
cording to
L2sc(u) + Li2s(w) — LI2c(U) k m
= 2 2 2 9nv(r)*ßeAr) (6)
compensates the anti-centrosymmetry mentioned above by the centrosymmetry oi L2$c(u) [4].
The right-hand sidet of Eq. (6) with
Qn(?) electron density distribution (including ther- mal vibration) of the n-atoms,
ße (r) sharp, spherical density distribution for each of the e-atoms (smeared out by its thermal vibration), and
* denoting convolution,
t These statements are exact i f a l l bn — 0 a n d 6ei 4= / (sin 0) a n d &e3 — &e2 4= / (sin 0). T h e y are close a p p r o x i m a t i o n s under t h e conditions (1) a n d (4).
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provides undisturbed images (compared with
•^I2s iV'))- It can be solved to obtain the n-atom arrangement (step 2) — either by deconvolution or by other techniques —, if the arrangement of the e-atoms has been found from Lizc{u) (step 1).
Step 2 (which can also be applied to Za2s (u) alone) yields the true symmetry of the complete structure, including enantiomer or polarity. This step com- prises a partial structure analysis without "know- ing" phases. If only one e-atoms exists, step 1 is unnecessary and step 2 trivial, with results
m
similar to [5]. In this case ^ <?n(f) follows directly from the diffraction intensity differences AizQi) and yl23 (b) (or A 12(h) alone) without knowledge of phasest. A computer program has been imple- mented which performs step 2 (from Eq. (6) or from Li2s(u)).
[1] A n o m a l o u s S c a t t e r i n g , e d i t e d b y S. R a m a s e s h a n a n d S. C. A b r a h a m s , M u n k s g a a r d l t d . , Copenhagen 1975.
[ 2 ] T . S a k a m a k i , S . H o s o y a , a n d T . F u k a m a c h i , A c t a Cryst.
A 86, 183 (1980).
[3] K . F . Fischer, papers g i v e n a t a) H a m b u r g / D E S Y - H A S Y L A B opening meeting, J a n . 1981; b) K a r l s r u h e / A G K r - M e e t i n g , M a r c h 1981 ( A b s t r a c t b y K . Fischer, A . R i b b e n s , J . S p i l k e r , a n d G . Schäfer, Z . K r i s t . 156, 3 ( 1 9 8 1 ) ; c) O t t a w a / I U C r - M e e t i n g , A u g . 1981, paper
Applications of this method (called "lambda technique" in our laboratory) may be envisaged for determination of crystal structures which can- not be solved by present routine methods, e.g. also pseudosymmetric and/or super-structures. Because the n-atoms need not be treated as individuals, a density map of unresolved (and perhaps not resolv- able) n-atoms can be obtained. This may be helpful in the investigation of positionally disordered ma- cromolecules (provided that collecting a double or triple data set appears possible). I n amorphous binary substances, partial pair distribution func- tions [6] of the two constituents e and n can be separated in (e, e)- and (e, n)-distributions.
I wish to express cordial thanks to Mrs. A.
Ribbens, Mr. J. Spilker, and Dr. G. Schäfer from our laboratory for a number of contributions to this topic which will be published in detail elsewhere.
N o . 15.4-04 ( A b s t r a c t - b o o k , p. C - 3 0 8 ) , t o be published i n A c t a C r y s t .
[4] A . R i b b e n s , D i p l o m a r b e i t (Master's Degree Thesis), S a a r b r ü c k e n , A u g u s t 1980.
[5] Y . O k a y a , Y . Saito, a n d R . P e p i n s k y , P h y s . R e v . 9 8 , 1857 (1955).
[6] C. N . J . W a g n e r a n d H . R u p p e r s b e r g , A t o m i c E n e r g y R e v i e w , Suppl. N o . 1 ( 1 9 8 1 ) , p p . 1 0 1 - 1 4 2 , I A E A , W i e n 1981.
