# 12.1.4: Extinctions

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- 18385

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Sometimes, due to destructive interference, planes containing many atoms do not produce diffraction. We term this phenomenon a *systematic absence*, or an *extinction*. Figure 12.29*a* 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 1*λ* 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}/_{2 }*λ* 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 *d _{010}*. 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 *d _{020}* compared with

*d*results in the larger diffraction angle shown in Figure

_{010}*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 2_{1} 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 *h*00) 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) |
hklhklhklhklhklhkl |
always presenth + k + l = evenk + l = evenh + l = evenh + k = evenh, k, l all odd or all even |

Screw Axes |
||

2-fold screw, 2_{1} parallel to a4-fold screw, 4 _{2} along parallel to a6-fold screw, 6 _{3} parallel to c3-fold screw, 3 _{1} or 3_{2} parallel to c6-fold screw, 6 _{2} or 6_{4} parallel to a4-fold screw, 4 _{1} or 4_{3} parallel to a6-fold screw, 6 _{1} or 6_{5} parallel to c |
h00h0000 l00 lh00h0000 l |
h = evenh = evenl divisible by 3l divisible by 3h divisible by 4h divisible by 4l divisible by 6 |

Glide Plane Perpendicular to the B-axis |
||

a glidec gliden glided glide |
h0lh0lh0lh0l |
h = evenl = evenh + l = evenh + l divisible by 4 |

To see the importance of extinctions, consider almandine, a garnet. Almandine is cubic and its crystals have symmetry ^{4}/_{m3}^{2}/* _{m}*. But the atomic arrangement has a body centered unit cell, a screw axis (4

_{1}), and two glide planes (

*a*and

*d*); it belongs to space group = I 4

_{1}/

*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.