12.1.4: Extinctions

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12.29.png — Figure 12.29: Destructive interference and extinction

Sometimes, due to destructive interference, planes containing many atoms do not produce diffraction. We term this phenomenon a systematic absence, or an extinction. Figure 12.29a shows a two-dimensional drawing of a structure in which atoms occupy the corners (grey atoms) and the center (red atoms) of each unit cell. As shown in Figure b, when 010 diffraction occurs at angle θ, beam X and Y are completely in phase. If the diffraction is 1st order, beam Y travels exactly 1λ farther than beam X.

However, the constructive interference cannot prevail when we account for the (red) atoms between the (010) planes. X-rays scattered by these atoms travel ¹/₂λ more or less than those scattered by the adjacent (010) planes. This produces waves that are out-of-phase and we get perfect destructive interference, shown in Figure c. No X-rays will be found at the angle θ although it satisfies Bragg’s Law for d₀₁₀. This effect is extinction, and we would say the 010 diffraction peak is extinct.

Although (010) planes do not diffract, (020) planes do. But the smaller d₀₂₀ compared with d₀₁₀ results in the larger diffraction angle shown in Figure d. The atomic arrangement we saw earlier in this chapter in Figure 12.25 also produces extinctions. 100 diffraction is absent because additional atoms occur halfway between the (100) planes.

Figures 12.25 and 12.29 show two-dimensional examples of structures that result in extinctions due to end-centered and body-centered unit cells. In three dimensions, end-centered, body-centered, and face-centered arrangements also produce extinctions. We often describe extinctions using arithmetical rules. For body centering, the rule is that an hkl peak will only be present if h + k + l is an even number.

Besides centering, screw axes and glide planes also cause extinction because they, like centering, result in planes of atoms between other planes. The systematic extinction of certain X-ray peaks, then, is one way we figure out space group symmetry. Box 12-2 gives more examples of extinction rules.

Extinction Rules

Lattice centering can cause X-ray extinctions. Screw axes and glide planes do too. Arithmetical rules involving h, k, and l describe which peaks can be present. (Although called extinction rules, the rules usually describe which peaks are present, not which peaks are extinct.)

The table below lists the affected peaks and gives rules for their presence. This table includes only some of the many possible screw axes. Rules for other screw axes can be inferred by analogy with the ones listed here.

For lattice centering, extinction rules affect all hkl reflections. But screw axes and glide planes only affect some. For example, as listed in the table below, a 2₁ screw axis parallel to the a-axis causes extinction of h00 peaks if h is an odd number. If h is an even number, the peak will be present, and other (not h00) peaks are not affected at all.

Symmetry Element	Affected Reflection	Condition for Reflection to Be Present
Lattice Centering
primitive lattice (P) body-centered lattice (I) end-centered lattice (A) end-centered lattice (B) end-centered lattice (C) face-centered lattice (F)	hkl hkl hkl hkl hkl hkl	always present h + k + l = even k + l = even h + l = even h + k = even h, k, l all odd or all even
Screw Axes
2-fold screw, 2₁ parallel to a 4-fold screw, 4₂ along parallel to a 6-fold screw, 6₃ parallel to c 3-fold screw, 3₁ or 3₂ parallel to c 6-fold screw, 6₂ or 6₄ parallel to a 4-fold screw, 4₁ or 4₃ parallel to a 6-fold screw, 6₁ or 6₅ parallel to c	h00 h00 00l 00l h00 h00 00l	h = even h = even l divisible by 3 l divisible by 3 h divisible by 4 h divisible by 4 l divisible by 6
Glide Plane Perpendicular to the B-axis
a glide c glide n glide d glide	h0l h0l h0l h0l	h = even l = even h + l = even h + l divisible by 4

To see the importance of extinctions, consider almandine, a garnet. Almandine is cubic and its crystals have symmetry ⁴/_m3²/_m. But the atomic arrangement has a body centered unit cell, a screw axis (4₁), and two glide planes (a and d); it belongs to space group = I 4₁/a 3 ²/_d.

For cubic minerals we can calculate d-values :

So, we can calculate d-values for all combinations of h, k, and l. If we only consider combinations that could give X-ray peaks for 2θ < 90^o, we get 726 possibilities. That is a lot. However, when we apply the extinction rules for body centering, the screw axis, and the two glide planes, most reflections go away. We are left with only 51 possibilities! And some of these correspond to planes that contain no, or few, atoms – so they produce no reflections.

The diagram below (Figure 12.30) shows a standard reference pattern for almandine. It contains fewer than 15 visible peaks. Some appear spaced regularly, but gaps (missing peaks) correspond to extinctions.

12.30.jpg — Figure 12.30: Diffraction pattern for almandine