The accuracy of atomic co-ordinates derived from Fourier series in X-ray structure analysis

Abstract

A brief introduction to the subject of X-ray structure analysis is followed by a discussion of various conjectures regarding the accuracy of derived atomic co-ordinates and the importance of the latter in the derivation of molecular theory. Representing the synthesis in the form D ( x , y , z ) = 1 V ∑ ∑ ∑ − H K L + H K L ⁡ | F ( h , k , l ) | cos 2 π h x a + k y b + l z c − x ( h , k , l ) , , (1) and neglecting errors of computation, two sources of inaccuracy are shown to occur: ( a ) experimental errors in the | F | values; ( b ) errors due to ( H, K, L ) being finite. Information as to the magnitude of the errors in the | F | values has recently become available, and a comparison of two sets of experimentally determined | F | values shows that the errors are independent of the magnitude of the structure factors and have a most probable value Δ e = ± 0 .6 . (2) Examination of the shape of the atomic peaks derived from a number of Fourier syntheses shows that the radial density distribution can be closely represented by the function d ( r ) = A e − p r 2 , (3) where A depends on the atomic number of the particular atom and p appears to be fairly constant over a number of atoms from carbon to sulphur, a mean value being p =4.69. (4) A combined analytical-statistical analysis leads to the relation ε < 90.8 Δ e / N V ( λ p ) 1 2 , (5) where ε is the most probable error in the co-ordinate, N is the atomic number of the particular atom, V is the volume of the unit cell in A 3 , and λ is the wave-length corresponding to the smallest spacings observed. Taking the values of Δe and p given in (2) and (4) and considering a carbon atom in a unit cell of volume V = 583A 3 , (6) equation (5) leads, when all the information obtainable with copper K α radiation is used, to the value, ε < 0.0027A. A formula is also given for the case in which errors are proportional to the order of their parent reflexions. The problem of finite limits of summation is dealt with in Part 2. For a simple system containing only two carbon atoms the errors, calculated as upper limits, are: ρ ε ( A ) 1.5 0.019 1.8 0.009 2.0 0.005 where ρ is the radius of the sphere containing the reciprocal points of all planes included in the summation. The polyatomic case cannot be given general expression since the atomic positions form a determined system and are not subject to statistical laws. In any given structure the errors are shown to be calculable by the following procedure. Having calculated the structure factors from the final atomic co-ordinates, a synthesis is computed using these calculated values as coefficients. Any terms not included in the original synthesis with experimental coefficients are similarly omitted in this new synthesis. The co-ordinates derived will, in general, deviate slightly from those used in calculating the F values, these deviations give the errors, with reversed signs, of the original co-ordinates. A trial on an actual structure shows them to have a value of about ε=±0.02A.(7) It is suggested that by applying these corrections to derived co-ordinates more accurate values of the latter may be obtained, having errors approximately those of experiment given by equation (5). Finally, the value of ‘ p ’ is related to the quantity ' B ' defined by Debye & Waller by means of the equation 1 p = a ( B + c ) (8) where a and c are constants. This relation enables the effect of thermal agitation to be examined.