International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 9.8, p. 913

Section 9.8.1.5. Occupation modulation

T. Janssen,a A. Janner,a A. Looijenga-Vosb and P. M. de Wolffc

aInstitute for Theoretical Physics, University of Nijmegen, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands,bRoland Holstlaan 908, NL-2624 JK Delft, The Netherlands, and cMeermanstraat 126, 2614 AM, Delft, The Netherlands

9.8.1.5. Occupation modulation

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Another type of modulation, the occupation modulation, can be treated in a way similar to the displacive modulation. As an example consider an alloy where the positions of the basic structure have space-group symmetry, but are statistically occupied by either of two types of atoms. Suppose that the position r is occupied by an atom of type A with probability [p({\bf r})] and by one of type B with probability [1-p({\bf r})] and that p is periodic. The probability of finding an A atom at site [{\bf n}+{\bf r}_j] is [P_A({\bf n}+{\bf r}_j)=p_j[{\bf q}\cdot({\bf n}+{\bf r}_j)], \eqno (9.8.1.27)]with [p_j(x)=p_j(x+1)]. In this case, the structure factor becomes [\eqalignno{\qquad S_{\bf H} &=\textstyle\sum\limits_{\bf n}\textstyle\sum\limits_j\big[\big(\,f_A\,p_j[{\bf q}\cdot({\bf n}+{\bf r}_j)]+f_B\{1-p_j[{\bf q}\cdot({\bf n}+{\bf r}_j)]\}\big) \cr &\quad\times\exp[2\pi i{\bf H}\cdot({\bf n}+{\bf r}_j)]\big], & (9.8.1.28)}]where [f_A] and [f_B] are the atomic scattering factors. Because of the periodicity, one has [p_j(x)=\textstyle\sum\limits_mw_{jm}\exp(2\pi imx). \eqno (9.8.1.29)]Hence, [\eqalignno{ S_{\bf H} &=\textstyle\sum\limits_j\bigg\{f_B\Delta({\bf H})\exp(2\pi i{\bf H}\cdot{\bf r}_j)+(\,f_A-f_B)\textstyle\sum\limits_m\Delta({\bf H}+m{\bf q})w_{jm} \cr &\quad \times\exp[2\pi i({\bf H}+m{\bf q})\cdot{\bf r}_j]\bigg\}, & (9.8.1.30)}]where Δ(H) is the sum of δ functions over the reciprocal lattice of the basic structure: [\Delta({\bf H})=\textstyle\sum\limits_{h_1h_2h_3}\delta\left({\bf H}-\textstyle\sum\limits^3_{i=1}h_i{\bf a}^*_i\right).]Consequently, the diffraction peaks occur at positions H given by (9.8.1.7)[link]. For a simple sinusoidal modulation [m = ±1 in (9.8.1.29)[link]], there are only main reflections and first-order satellites (m = ±1). One may introduce an additional coordinate t and generalize (9.8.1.27)[link] to [P_A({\bf n}+{\bf r}_j, t) = p_j[{\bf q}\cdot({\bf n}+{\bf r}_j)+t], \eqno (9.8.1.31)]which has (3 + 1)-dimensional space-group symmetry. Generalization to more complex modulation cases is then straightforward.








































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