On the significance of two new Patterson functions for disordered crystals

Abstract

AuszugEs werden zwei neue Patterson-Funktionen definiert: die eine, AP 0T (r) = [P 0 (r) -P r (r)] · s Zelle (r), für den thermisch bewegten Kristall und die andere, ΔΡ ΟΑ {Γ) = [P B (r)-P(r)] · Szelle (r), für den ungeordneten Kristall. Diese Funktionen ergeben zwei neue FourierBeziehungen :Es ist möglich, aus den neuen Differenz-Pattersonfunktionen die stetigen Funktionen der diffusen Streuung, die von der Wärmebewegung oder der Lagenunordnung herrühren, ohne Kenntnis der Kristallstruktur zu berechnen. AbstractOn account of the Fourier-transform reciprocity theorem it is possible to go back and forth from physical space to reciprocal space. One mechanism involves Fourier summations of sampled values in one space, leading to a continuous, periodic function in the other. Two new Patterson functions are defined, one for a thermally agitated crystal, ZlP 0I ,(r) = [P 0 (r) -Pj-(r)] · s cel] (r), and another ΔΡ»ΑΓ) = [JVr) -P(Γ)] • *«"(r), 242 MABISA CANUT-AMOROS for a disordered crystal. These new functions yield to two new Fourier mates Τ Τ-1 and τ ^wW^Wf*)· τ~ι The new difference Patterson functions allow one to calculate the continuous diffuse-scattering functions arising from thermal and positional disorder, respectively, without any previous knowledege of the crystal structure. The quasi periodic Q functions of the thermally agitated or disordered molecular crystals are analyzed in terms of only two motifs calculated in terms of adequate Patterson functions.It is a well known fact that, in real crystals, at least thermal agitation disturbs the ideal periodicity. Wide use in crystal-structure analysis of Fourier series with coefficients l-i^il 2 (i-e · the Patterson series) implies, however, an infinite and strictly periodic structure, that is to an hypothetical structure whose electron density is identical in all unit cells, and which corresponds to the electron density of the real crystal under study, but averaged over time and space. The Patterson synthesis obtained from experimental data thus corresponds to the Patterson function of the hypothetical average crystal.On the other hand, the intensity diffracted by the real crystal is basically of two kinds: the Bragg intensities located at the lattice points of the reciprocal lattice, and a diffuse-scattering intensity. While the Patterson function corresponds to the inverse Fourier transform of the Bragg intensities only, the Q function corresponds to the inverse Fourier transform of the total intensity diffracted by the real crystal. HOSEMANN and BAGCHI (1962) pointed out that, instead of being ideally periodic as the Patterson function, the Q function, is only quasi-periodic.It was pointed out first by EWALD (1940), that it is not generally realized that the reciprocal lattice is only an incomplete representation of the Fourier transform of the crystal, and that much clearness of discussion can be gained by making full use of the conception of the Fourier transform. In order to successfully apply Fourier-transform theory to real crystals, it is necessary to...

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