Use the results of the previous problem and Example 29-9 to...

Use the results of the previous problem and Example $29-9$ to verify the entries in the following table. \begin{array}{cl} \hline & \text { Cubic lattice type for which } \\ \text { Miller indices (hkl) } & \text { a reflection is observed } \\ \hline 100 & \mathrm{pc} \\ 110 & \mathrm{pc} & \\ 111 & \mathrm{pc} & \mathrm{fcc} & \\ 200 & \mathrm{pc} & \mathrm{fcc} & \mathrm{bcc} \\ 210 & \mathrm{pc} & & \\ 211 & \mathrm{pc} & & \mathrm{bcc} \\ 220 & \mathrm{pc} & \mathrm{fcc} & \mathrm{bcc} \\ 300 & \mathrm{pc} & & \\ 221 & \mathrm{pc} & & \\ 310 & \mathrm{pc} & & \mathrm{bcc} \\ 311 & \mathrm{pc} & \mathrm{fcc} & \\ 222 & \mathrm{pc} & \mathrm{fcc} & \mathrm{bcc} \\ 320 & \mathrm{pc} & & \\ 321 & \mathrm{pc} & & \mathrm{bcc} \\ 400 & \mathrm{pc} & \mathrm{fcc} & \mathrm{bcc} \\ \hline \end{array}

Use the results of the previous problem and Example $29-9$ to verify the entries in the following table.
\begin{array}{cl}
\hline & \text { Cubic lattice type for which } \\
\text { Miller indices (hkl) } & \text { a reflection is observed } \\
\hline 100 & \mathrm{pc} \\
110 & \mathrm{pc} & \\
111 & \mathrm{pc} & \mathrm{fcc} & \\
200 & \mathrm{pc} & \mathrm{fcc} & \mathrm{bcc} \\
210 & \mathrm{pc} & & \\
211 & \mathrm{pc} & & \mathrm{bcc} \\
220 & \mathrm{pc} & \mathrm{fcc} & \mathrm{bcc} \\
300 & \mathrm{pc} & & \\
221 & \mathrm{pc} & & \\
310 & \mathrm{pc} & & \mathrm{bcc} \\
311 & \mathrm{pc} & \mathrm{fcc} & \\
222 & \mathrm{pc} & \mathrm{fcc} & \mathrm{bcc} \\
320 & \mathrm{pc} & & \\
321 & \mathrm{pc} & & \mathrm{bcc} \\
400 & \mathrm{pc} & \mathrm{fcc} & \mathrm{bcc} \\
\hline
\end{array}

Key Concepts

Structure Factor Calculation

The structure factor is a mathematical description of how the scattering from atoms in a crystal interferes with each other. It is calculated by summing the contributions from all atoms in the unit cell, taking into account their positions and scattering amplitudes. This calculation determines whether a particular (hkl) reflection will be enhanced or cancelled (resulting in a systematic absence), and is central to predicting the diffraction pattern for different lattice types.

Systematic Absences

Systematic absences arise from the symmetry and centering of the crystal lattice, where certain diffraction peaks are forbidden due to destructive interference. These absences provide clues to the lattice type, since each type (pc, fcc, bcc) exhibits characteristic conditions under which reflections will or will not occur. Understanding these conditions is crucial to interpreting X-ray diffraction data and correlating it with the crystal structure.

Miller Indices

Miller indices are a set of three integers that denote the orientation of a crystal plane by taking the reciprocals of the intercepts that the plane makes with the crystallographic axes. They are a fundamental tool in crystallography for identifying specific planes within a crystal lattice and understanding diffraction patterns as they relate to the internal structure of the material.

Crystal Lattice Types

Different cubic lattice types, such as primitive cubic (pc), face-centered cubic (fcc), and body-centered cubic (bcc), have distinct arrangements of lattice points within the unit cell. These arrangements affect the distribution of atoms and, consequently, the diffraction pattern. Each lattice type has its own set of allowed reflections based on its symmetry and centering, which leads to systematic absences in certain directions.

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