Systematic extinction

In crystallography, systematic extinction is understood to mean the systematic lack of reflections in X-ray or neutron diffraction on crystals, which is due to the symmetry- induced destructive interference of diffracted rays. The cause of the systematic extinctions is a symmetry element of the space group connected to a translation vector (gliding mirror plane, screw axis or a centering of the Bravais lattice ).

meaning

The systematic extinction is of great importance for the structure elucidation of crystals, since conclusions can be drawn from it about the symmetry properties of the crystal. With the help of the systematic extinctions, the space group of the crystal can be determined. Therefore, the laws of extinction are given in the International Tables for the respective space group.

A distinction is made between integral extinction, which affects reflexes in the entire reciprocal space , zonal extinction, in which only reflexes of one plane in the reciprocal space ( zone ) are affected, and serial extinction, in which only reflexes on a straight line in reciprocal space are missing.

description

Systematic extinction is described using Laue indices hkl, which correspond to Miller’s indices (hkl) but, in contrast, do not have to be coprime. On the one hand, the affected area of the reciprocal space is specified (e.g. hkl for an integral extinction, hk0 for a zonal extinction in the l = 0 plane, or 00l for a serial extinction in the h = k = 0 line). On the other hand, either an extinction condition (condition for the absence of reflections) or a reflex condition (condition for the presence of reflections) is given (e.g. h + k = 2n, the sum of h and k must be even).

Integral extinction

Integral cancellations arise from a centering of the Bravais lattice .

example

In a body-centered cell there is a symmetrically equivalent atom at any position x, y, z at x + 0.5, y + 0.5, z + 0.5 for every atom. The contribution of each of these atomic pairs to the structure factor is:

${\ displaystyle f_ {m} e ^ {2 \ pi i (hx + ky + lz)} + f_ {m} e ^ {2 \ pi i (h (x + 0.5) + k (y + 0.5) + l (z + 0.5))} =}$

${\ displaystyle f_ {m} e ^ {2 \ pi i (hx + ky + lz)} (1 + e ^ {\ pi i (h + k + l)})}$

${\ displaystyle 1 + e ^ {\ pi i (h + k + l)} = {\ begin {cases} 2 & {\ text {if h + k + l even}} \\ 0 & {\ text {if h + k + l odd}} \ end {cases}}}$

There are therefore reflexes only at the points where h + k + l is even.

The following table gives an overview of the relationship between the centerings occurring in the Bravais grids and the resulting reflex conditions.

centering	Affected reflexes	Reflex condition
Primitive	-	no extinction
A-centered	hkl	k + l = 2n
B-centered	hkl	h + l = 2n
C-centered	hkl	h + k = 2n
body centered	hkl	h + k + l = 2n
Face-centered on all sides	hkl	h + k = 2n, h + l = 2n, k + l = 2n
Rhombohedral	hkl	-h + k + l = 3n

Zonal obliteration

Zonal extinctions are caused by glide mirror planes in the crystal.

example

If there is an a gliding mirror plane in a crystal perpendicular to c, which goes through the origin, there is a symmetrically equivalent in x + 0.5, y, -z for every atom m in x, y, z. The contribution of each of these atomic pairs to the structure factor is:

${\ displaystyle f_ {m} (e ^ {2 \ pi i (hx + ky + lz)} + e ^ {2 \ pi i (hx + ky-lz)} e ^ {\ pi ih})}$

for l = 0 it follows: ${\ displaystyle f_ {m} (e ^ {2 \ pi i (hx + ky)} \, (1 + e ^ {\ pi ih}))}$

${\ displaystyle 1 + e ^ {\ pi ih} = {\ begin {cases} 2 & {\ text {if h is even}} \\ 0 & {\ text {if h is odd}} \ end {cases}}}$

There are reflexes in the zone l = 0 only if h is even.

The following table gives an overview of the relationship between the glide mirror planes occurring in the space groups and the resulting reflex conditions.

Glide plane	location	Slip vector	Affected reflexes	Reflex condition
a	(001)	a / 2	hk0	h = 2n
b	(001)	b / 2	hk0	k = 2n
n	(001)	a / 2 + b / 2	hk0	h + k = 2n
d	(001)	a / 4 ± b / 4	hk0	h + k = 4n, h = 2n, k = 2n
a	(010)	a / 2	h0l	h = 2n
c	(010)	c / 2	h0l	l = 2n
n	(010)	a / 2 + c / 2	h0l	h + l = 2n
d	(010)	a / 4 ± c / 4	h0l	h + l = 4n, h = 2n, l = 2n
b	(100)	b / 2	0cl	k = 2n
c	(100)	c / 2	0cl	l = 2n
n	(100)	b / 2 + c / 2	0cl	k + l = 2n
d	(100)	b / 4 ± c / 4	0cl	k + l = 4n, k = 2n, l = 2n

Serial erasure

Serial extinctions arise from screw axes in the crystal.

example

If there is a screw axis 2 ₁ parallel c through the origin in a crystal , then for an atom m in x, y, z there is a symmetrically equivalent in -x, -y, 0.5 + z. The contribution of each of these atomic pairs to the structure factor is:

${\ displaystyle f_ {m} (e ^ {2 \ pi i (hx + ky + lz)} + e ^ {2 \ pi i (-hx-ky + lz)} e ^ {\ pi il})}$

for h = k = 0 the following applies: ${\ displaystyle f_ {m} (e ^ {2 \ pi ilz} \, (1 + e ^ {\ pi il}))}$

${\ displaystyle 1 + e ^ {\ pi il} = {\ begin {cases} 2 & {\ text {if l is even}} \\ 0 & {\ text {if l is odd}} \ end {cases}}}$

There are reflexes in the 00l direction only when l is straight.

The following table gives an overview of the relationship between the screw axes occurring in the space groups and the resulting reflex conditions.

Screw axis	location	Affected reflexes	Reflex condition
${\ displaystyle 2_ {1}}$ ; ; ${\ displaystyle 4_ {2}}$ ${\ displaystyle 6_ {3}}$	// [100]	h00	h = 2n
${\ displaystyle 3_ {1}}$ ; ; ; ${\ displaystyle 3_ {2}}$ ${\ displaystyle 6_ {2}}$ ${\ displaystyle 6_ {4}}$	// [100]	h00	h = 3n
${\ displaystyle 4_ {1}}$ ; ${\ displaystyle 4_ {3}}$	// [100]	h00	h = 4n
${\ displaystyle 6_ {1}}$ ; ${\ displaystyle 6_ {5}}$	// [100]	h00	h = 6n
${\ displaystyle 2_ {1}}$ ; ; ${\ displaystyle 4_ {2}}$ ${\ displaystyle 6_ {3}}$	// [010]	0k0	k = 2n
${\ displaystyle 3_ {1}}$ ; ; ; ${\ displaystyle 3_ {2}}$ ${\ displaystyle 6_ {2}}$ ${\ displaystyle 6_ {4}}$	// [010]	0k0	k = 3n
${\ displaystyle 4_ {1}}$ ; ${\ displaystyle 4_ {3}}$	// [010]	0k0	k = 4n
${\ displaystyle 6_ {1}}$ ; ${\ displaystyle 6_ {5}}$	// [010]	0k0	k = 6n
${\ displaystyle 2_ {1}}$ ; ; ${\ displaystyle 4_ {2}}$ ${\ displaystyle 6_ {3}}$	// [001]	00l	l = 2n
${\ displaystyle 3_ {1}}$ ; ; ; ${\ displaystyle 3_ {2}}$ ${\ displaystyle 6_ {2}}$ ${\ displaystyle 6_ {4}}$	// [001]	00l	l = 3n
${\ displaystyle 4_ {1}}$ ; ${\ displaystyle 4_ {3}}$	// [001]	00l	l = 4n
${\ displaystyle 6_ {1}}$ ; ${\ displaystyle 6_ {5}}$	// [001]	00l	l = 6n

literature

Schwarzenbach D. Kristallographie Springer Verlag, Berlin 2001, ISBN 3-540-67114-5
Hahn, Theo (Ed.): International Tables for Crystallography Vol. A D. Reidel publishing Company, Dordrecht 1983, ISBN 90-277-1445-2