Occupation modulation

Janssen, T.; Janner, A.; Looijenga-Vos, A.; Wolff, P. M. de

doi:10.1107/97809553602060000624

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

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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-Vos^b and P. M. de Wolff^c ^‡

^a Institute 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). For a simple sinusoidal modulation [m = ±1 in (9.8.1.29)], there are only main reflections and first-order satellites (m = ±1). One may introduce an additional coordinate t and generalize (9.8.1.27) 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.

References

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