Show that F(h k l)=f++f- if h, k, and l are all even or just one o

step 1 chemical behavior shown by chlorine, bromine and iodine. A with hydrogen with hydrogen chlorine

Show that F(h k l)=f++f- if h, k, and l ...

Show that $$ \begin{array}{ll} F(h k l)=f_{+}+f_{-} & \begin{array}{l} \text { if } h, k, \text { and } l \text { are all even } \\ \text { or just one of them is even } \end{array} \\ =f_{+}-f_{-} & \begin{array}{l} \text { if all are odd } \\ \text { or just one is odd } \end{array} \end{array} $$ for the CsCl(s) crystal structure shown in Figure $29.18 \mathrm{~b} .$ Cesium bromide and cesium iodide have the same crystal structure as cesium chloride. Compare the expected diffraction patterns of cesium chloride and cesium iodide. Recall that $\mathrm{Cs}^{+}$and $\mathrm{I}^{-}$are isoelectronic.

Show that
$$
\begin{array}{ll}
F(h k l)=f_{+}+f_{-} & \begin{array}{l}
\text { if } h, k, \text { and } l \text { are all even } \\
\text { or just one of them is even }
\end{array} \\
=f_{+}-f_{-} & \begin{array}{l}
\text { if all are odd } \\
\text { or just one is odd }
\end{array}
\end{array}
$$
for the CsCl(s) crystal structure shown in Figure $29.18 \mathrm{~b} .$ Cesium bromide and cesium iodide have the same crystal structure as cesium chloride. Compare the expected diffraction patterns of cesium chloride and cesium iodide. Recall that $\mathrm{Cs}^{+}$and $\mathrm{I}^{-}$are isoelectronic.

Key Concepts

Structure Factor

The structure factor is a fundamental concept in crystallography that represents the amplitude and phase of the scattered X-rays from a crystal. It is computed as a sum over all atoms in the unit cell, where each atom contributes its atomic scattering factor multiplied by a phase term determined by its position. This concept is critical for understanding how the arrangement of atoms within the unit cell influences the resulting diffraction pattern.

Atomic Scattering Factors

Atomic scattering factors quantify the scattering power of individual atoms and depend on the electron density around each nucleus. In analyzing diffraction patterns, differences in the scattering factors of atoms present in a crystal, such as those contributing with positive or negative signs, lead to variations in the intensity of the observed reflections.

Miller Indices and Parity

Miller indices (h, k, l) characterize specific crystallographic planes and directly enter the expression for the structure factor through their influence on the phase factors. The parity (whether the indices are even or odd) can result in phase differences that cause constructive or destructive interference. This explains why, depending on whether the indices are all even or odd, the structure factor can be expressed as sums or differences of atomic scattering factors.

Diffraction Pattern Analysis

The intensity of diffraction peaks in a crystal is determined by the square of the modulus of the structure factor. By analyzing how the structure factor varies with Miller indices, one can predict and interpret the presence, absence, or relative intensities of reflections in the diffraction pattern. This analysis is essential to inferring the underlying crystal structure and the positions of atoms within the unit cell.

Isoelectronic Considerations

Isoelectronic species have the same number of electrons and often exhibit similar electronic distributions. In the context of diffraction, when ions or atoms are isoelectronic, their atomic scattering factors are very similar in magnitude. This leads to diffraction patterns that can be nearly identical, even when the chemical identities or sizes of the ions differ, thereby influencing how substitutions in the crystal structure affect the overall pattern.

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