(p.459) Appendix 6 Systematic absences (extinctions) in Xray diffraction and double diffraction in electron diffraction patterns
(p.459) Appendix 6 Systematic absences (extinctions) in Xray diffraction and double diffraction in electron diffraction patterns
A6.1 Systematic absences
In Xray diffraction (and also, except for the complication of double diffraction, in electron diffraction) the intensities of reflections from certain planes with Laue indices hkl are zero. Such reflections are said to be forbidden or extinguished or, better, systematically absent since they arise from the centring of the unit cell and /or the presence of translational symmetry elements—glide planes and screw axes. The identification of systematic absences is very useful since it provides the ‘first step’ in crystal structure determination.
The subject may be best introduced by considering the diffraction of light from a wide slit diffraction grating (Sections 7.4 and 13.3). In the diffraction of light extinctions or ‘missing orders’ occur for certain ratios of the slit spacing a and the slit width d. For example, referring to Fig. 7.7 or 13.7, if the slit width d is set equal to onethird of the slit spacing a, then the zero order minimum of the central peak from a single slit coincides with the angle of the third order diffracted peaks $n=\pm 3$. These peaks (and subsequently the sixth, ninth etc. order peaks) are therefore of zero intensity (i.e. extinguished). Other sets of systematic absences may be derived for other combinations of a and d—the most important being that when $d=\frac{1}{2}a$ (see Exercise 7.2).
Systematic absences in crystals may be derived in two ways: either by a consideration of the geometrical consequences arising from the use of centred, rather than primitive, unit cells and the presence of glide planes and screw axes, or by application of the structure factor equation. The former approach is the proper one, since systematic absences arise solely as a consequence of the architecture of crystals—but the latter approach is a useful means of gaining practice and familiarity with the structure factor equation. We will start with both approaches for an fcc crystal.
Consider Fig. A6.l which shows the unit cell vectors a, b, c for the conventional fcc unit cell and the unit cell vectors A, B, C for (one variant) of the primitive rhombohedral unit cell. The origin is drawn at the front lefthand corner to help make clear the shape of the rhombohedral cell.
(p.460) Writing down the transformation equations for unit cell vectors or axes (see Section 5.8):
Now, as shown in Section 5.8, Miller indices transform in the same way as unit cell vectors. Hence if (HKL) and (hkl) are the Miller indices of a plane referred to the primitive and fcc unit cells, respectively:
Now H, K, L are all integers, odd or even; hence 2H,2K,2L are all even integers, from which it follows that ($h+k)$, ($h+l)$, ($k+l)$ are also all even integers.
An even integer is either the sum of two even integers, or two odd integers, but not mixed. Hence, for each identity, h, k, l are either all odd or all even integers. In other words the lattice planes from which reflections occur are those for which h, k, l are all even or all odd integers. When h, k, l are mixed (some odd, some even) the Miller index (hkl) describes a set of planes which are not lattice planes since they do not all pass through lattice points.
Now let us derive the same result using the structure factor equation applied to an fcc crystal with an identical atom (atomic scattering factor $f)$ at each of the four lattice points in the cell. The (${u}_{n}\phantom{\rule{thinmathspace}{0ex}}{v}_{n}\phantom{\rule{thinmathspace}{0ex}}{w}_{n})$ fractional coordinate values of these atoms are:
(p.461) Hence,
Simplifying, and remembering that since the cell is centrosymmetric the sine components in the exponents vanish:
Now, cos$\mathrm{\pi}$ (even $\text{integer)}=+1$ and cos$\mathrm{\pi}$ (odd $\text{integer)}=1$. The value of ${F}_{hkl}$ will depend on whether h, k, l are all even, all odd or mixed. It is easily seen that if they are all even or all odd ${F}_{hkl}=4f$ and if they are mixed ${F}_{hkl}=0$.
The conditions for systematic absences from body and basecentred lattices can be derived in the same ways: either by referring the lattice to a primitive unit cell or by applying the structure factor equation. For example, for a bodycentred cubic crystal with identical atoms at (000) and $\left(\frac{1}{2}\frac{1}{2}\frac{1}{2}\right)$
Hence when ($h+k+l)$ = even integer ${F}_{hkl}$ = 2f and when ($h+k+l)$ = odd integer ${F}_{hkl}$ = 0.
Systematic absences for glide planes and screw axes may be derived in the same way; an example of each will suffice to show the principles involved.
Figure A6.2 shows a (monoclinic) crystal with a cglide plane in the xz plane or (010) plane (the plane of the paper). The action of the cglide is to translate any atom, atomic scattering factor f, with fractional coordinates (uvw) to $\left(u\stackrel{\u02c9}{v}\left(\frac{1}{2}+w\right)\right)$. Substituting (p.462) these two values in the structure factor equation:
Consider the h0l reflections (k = 0), i.e. the reflections in the zone whose axis is perpendicular to the glide plane (010)
When $l=\text{odd}$ integer, exp $\mathrm{\pi}\text{i}l=1$ and ${F}_{h0l}=0$. This then gives the condition for systematic absences for the h0l reflections; systematic absences occur when l equals an odd integer and, correspondingly, reflections occur when l equals an even integer. Clearly, for aglide in the (010) plane reflections occur when h equals an even integer.
Figure A6.3 shows a (monoclinic) crystal with a screw diad axis, 2${}^{\text{l}}$ along the yaxis, the action of which is to translate an atom with fractional coordinates (uvw) to $\left(\stackrel{\u02c9}{u}\left(\frac{1}{2}+v\right)\stackrel{\u02c9}{w}\right).$ Proceeding as before:
consider the 0k0 reflections ($h=l$ = 0) i.e. reflections arising from planes perpendicular to the screw axis.
When $k=\text{odd}$ integer, exp $\mathrm{\pi}\mathit{\text{i}}k=1$ and ${F}_{0k0}=0$ and reflections only occur from 0k0 planes when k equals an even integer. This result can also be seen intuitively: the action of the screw diad axis repeats the atomic layers at spacings equal to half the b lattice repeat distance.
(p.463) Table A6.1 lists the extinction criteria for lattices and translational symmetry elements. n and d are the symbols for diagonal glide planes where the translations are along two axes in the plane, each of which are half the unit cell repeat distance for an n glide plane and a quarter of the unit cell repeat distance for a d glide plane (d stands for ‘diamond glide’). In summary, Table A6.1 shows that lattices affect all reflections, glide planes affect only reflections which lie in the zone whose axis is perpendicular to the glide plane and screw axes affect only the reflections from planes perpendicular to the screw axis.
Table A6.1 Extinction criteria for lattices and symmetry elements
Lattice or symmetry 
Symbol 
Class of 
Condition for 

element type 
reflections 
presence 

Lattice type: 
hkl 

primitive 
P 
none 

bodycentred 
I 
$h+k+l=2n$ 

centred on the C face 
C 
$h+k=2n$ 

centred on the A face 
A 
$\phantom{+h\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}}k+l=2n$ 

centred on the B face 
B 
$h\phantom{+k\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}}+l=2n$ 

$h,k,l$ 

centred on all faces 
F 
$\text{all}=n\text{(odd)}$ 

$\text{or~all}=2n\text{(even)}$ 

rhombohedral, obverse 
R 
$h+k+l=3n$ 

rhombohedral, reverse 
R 
$hk+l=3n$ 

Glide plane $$ (001) 
a 
$hk0$ 
$h=2n$ 
b 
$k=2n$ 

n 
$h+k=2n$ 

d 
$h+k=4n$ 

Glide plane $$ (100) 
b 
$0kl$ 
$k=2n$ 
c 
$\phantom{h+\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}k}l=2n$ 

n 
$\phantom{+h\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}}k+l=2n$ 

d 
$\phantom{+h\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}}k+l=4n$ 

Glide plane $$ (010) 
a 
h0l 
$h=2n$ 
c 
$l=2n$ 

n 
$h+\phantom{\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}k}l=2n$ 

d 
$h+\phantom{\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}k}l=4n$ 

Glide plane $$ (1$\stackrel{\u02c9}{1}$0) 
c 
hhl 
$l=2n$ 
b 
$h\phantom{+k+l\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}}=2n$ 

n 
$h+k\phantom{\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}k}=2n$ 

d 
$\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2h+\phantom{\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}k}l=4n$ 

Screw axis $\text{}c$ 
${2}_{1},{4}_{2},{6}_{3}$ 
$00l$ 
$l=2n$ 
${3}_{1},{3}_{2},{6}_{2},{6}_{4}$ 
$l=3n$ 

${4}_{1},{4}_{3}$ 
$l=4n$ 

${6}_{1},{6}_{5}$ 
$l=6n$ 

Screw axis $\text{}a$ 
${2}_{1},{4}_{2}$ 
$h00$ 
$h=2n$ 
${4}_{1},{4}_{3}$ 
$h=4n$ 

Screw axis $\text{}b$ 
${2}_{1},{4}_{2}$ 
$0k0$ 
$k=2n$ 
${4}_{1},{4}_{3}$ 
$k=4n$ 

Screw axis $\text{}[110]$ 
${2}_{1}$ 
$hh0$ 
$h=2n$ 
(n = odd integer, 2n = even integer etc.)
Table A6.2 Reflecting planes ($\surd $) in cubic P, I, F and diamondcubic crystals.
N 
hkl 
cubic P 
cubic I 
cubic F 
Diamond cubic 

1 
100 
$\surd $ 

2 
110 
$\surd $ 
$\surd $ 

3 
111 
$\surd $ 
$\surd $ 
$\surd $ 

4 
200 
$\surd $ 
$\surd $ 
$\surd $ 

5 
210 
$\surd $ 

6 
211 
$\surd $ 
$\surd $ 

7 
— 

8 
220 
$\surd $ 
$\surd $ 
$\surd $ 
$\surd $ 
9 
300/221 
$\surd $ 

10 
310 
$\surd $ 
$\surd $ 

11 
311 
$\surd $ 
$\surd $ 
$\surd $ 

12 
222 
$\surd $ 
$\surd $ 
$\surd $ 

13 
320 
$\surd $ 

14 
321 
$\surd $ 
$\surd $ 

15 
— 

16 
400 
$\surd $ 
$\surd $ 
$\surd $ 
$\surd $ 
17 
410/322 
$\surd $ 

18 
330/411 
$\surd $ 
$\surd $ 

19 
331 
$\surd $ 
$\surd $ 
$\surd $ 

20 
420 
$\surd $ 
$\surd $ 
$\surd $ 
Note 1. There are no squares of three numbers which add up to 7, 15 etc. This means that there are ‘gaps’ in the sequence of reflections for cubic P crystals, i.e. ‘line 7’, ‘line 15’ etc. are missing. Note 2. Cubic I crystals show a uniform sequence of reflections (with no ‘gaps’ as in cubic P crystals). Note 3. Cubic F crystals show a characteristic sequence of lines: ‘twotogether’, ‘oneonitsown’, ‘twotogether’ etc.
Table A6.2 lists the reflecting planes in cubic P, I, F and diamondcubic crystals in order of decreasing d_{hkl}spacing. For cubic crystals ${d}_{hkl}=a/\sqrt{N}$ where $a=\text{the}$ lattice parameter and $N=({h}^{2}+{k}^{2}+{l}^{2})$. The conditions for reflection for a diamondcubic crystal are h, k, l are all odd or all even integers (since a diamondcubic crystal has an fcc lattice) with the additional condition that ($h+k+l)$ is either an odd integer or an integer which is an even multiple of 2 (see Exercise 9.5, p. 241).
F.2 Double diffraction
In electron diffraction the presence of face or bodycentring lattice points also gives rise to systematically absent reflections just as in the case of Xray diffraction (the first group in Table A6.1). However, reflections which are systematically absent in Xray diffraction as a result of the presence of translational symmetry elements (glide planes and screw axes, Table A6.1) may, and usually do, occur in electron diffraction patterns. This is known as double diffraction and occurs because the intensities of the diffracted beams may be comparable with that of the direct or undiffracted beam—a consequence of the dynamical interactions between the direct and diffracted beams. The effect, in terms of the geometry of the electron diffraction pattern, is that a strong diffracted beam can behave, as it were, as the direct beam and the whole pattern is in effect shifted so as to be centred about the diffracted beam. In the case of electron diffraction patterns from centred lattices all the spots are coincident and no new ones arise; but in most other cases new ones arise in positions in which the diffraction spots should be systematically absent. A simple example—that of electron diffraction from the hcp structure—will make this clear.
In the hcp structure the relevant translational symmetry element is the screw hexad ${6}_{3}$ (Fig. 4.10) which describes the symmetry of the sequence of A and B layer atoms (Fig. 1.5(b)). Consider the conditions for constructive interference for Bragg reflections from the A layers of atoms, i.e. the (0001) planes with interplanar spacing ${d}_{0001}=c$. When the path difference (PD) is 1λ (constructive interference), the path difference between the interleaving A and B layers of atoms, interplanar spacing ${d}_{0002}=c$/2 is $\frac{1}{2}\mathrm{\lambda}$, which is the condition for destructive interference; hence the 0001 reflection is (systematically) absent. Secondorder reflection from the (0001) planes (PD = 2$\mathrm{\lambda})$ corresponds to the firstorder reflection from the (0002) planes ($\text{PD}=1\mathrm{\lambda})$. Continuing in this way it turns out that 000l reflections where l is odd are systematically absent, as shown in Table A6.1. The same result may be found by applying the structure factor equation to the (0001), (0002), (0003), etc. planes, as in Example 4, Section 9.2: when l is odd the atomic scattering factors for the A and B layer atoms are equal and opposite.
(p.464) (p.465) (p.466) Now let us draw a commonly observed hcp electron diffraction pattern from the $[\stackrel{\u02c9}{1}2\stackrel{\u02c9}{1}0]$ zone (Fig. A6.4), showing the systematically absent reflections 0001, $000\stackrel{\u02c9}{1}$, 0003, $000\stackrel{\u02c9}{3}$ etc. in the row through the origin.^{1} Now suppose that the (strong) $10\stackrel{\u02c9}{1}0$ reflected beam acts as it were as the direct beam: the whole pattern will, in effect, be shifted from the origin 0000 to $10\stackrel{\u02c9}{1}0$ as shown by the arrow and spots will now appear in the systematically absent positions. The same result will be obtained for other choices of strong diffracted beams—as can easily be seen by making a tracing paper overlay in Fig. A6.4 and shifting the origin to the positions of different diffracted spots.
Double diffraction can also occur in situations in which the electron diffraction spots of two (or more) crystals are present in the pattern. In such cases the doublediffracted spots which appear may correspond to no dspacing in either crystal—and are probably the biggest cause of the frustrated electron microscopists’ ‘unindexable’ patterns. Figure A6.5 indicates the geometry involved. Here, two (high order) patterns are shown (p.467) consisting of widely separated rows (indicated by filled and open circles). The strong spots in the rows through the origin may act in effect as the direct beam, generating additional double diffraction spots in the positions marked by the letter d. The pattern appears to be from a relatively simple loworder zone—but of course it is not, and is either unindexable or (worse) may be indexed incorrectly. An example of the latter case is that of the electron diffraction patterns from twinrelated plates or laths of ferrite (bcc) which occur in the microstructures of some structural steels.
Figure A6.6 shows the spots (indicated by filled and open circles) for two bcc [110] zones twinrelated in the (112) plane. Double diffraction between these twinrelated zones gives rise to the spots marked d (i.e. ${\mathbf{\text{d}}}_{110}^{\ast}+{\mathbf{\text{d}}}_{{110}_{T}}^{\ast}={\mathbf{\text{d}}}_{\text{d}}$ where (110)${\text{T}}_{}$ is the twinspot of (110)). The presence of these additional spots indicates a pattern in which the spots outline an ‘elongated’ hexagon, as indicated, which is precisely the same shape as for an fcc $\u27e8110\u27e9$ zone—the four bcc 110 spots closer to the centre spot being identified as fcc 111 spots and the two doublediffraction spots further away being identified as fcc 200 spots. Since in steels d_{110} (ferrite) is closely similar to d_{111} (austenite), then this pattern may be misinterpreted as a $\u27e8110\u27e9$ austenite zone—leading to the erroneous conclusion that this phase is present at the ferrite twin boundary.
(p.468) A note on double diffraction in Xray diffraction
Double diffraction can also occur in Xray diffraction when two (or more) sets of crystal planes are simultaneously at the Bragg diffracting condition (i.e. the reciprocal lattice points lie simultaneously on the Ewald reflecting sphere). The effect was first described by M. Renninger in 1937 and can lead to violations of the systematic extinction criteria listed in Table A6.1.
Notes:
(^{1}) Systematic absences also occur in every third row (not shown in Fig. A6.4). These arise because the Alayer and Blayer atoms are in special equivalent positions in the hexagonal unit cell (see Section 4.6). This leads to the conditions for hkil: if $hk=3n$, $l=2n$, i.e. in the third row the 30$\stackrel{\u02c9}{3}$1,30$\stackrel{\u02c9}{3}$3, etc. reflections are systematically absent. These conditions are given in the table in Fig. 4.14, space group P6${3}_{}$/mmc, for the special equivalent positions for hcp structures denoted by the Wyckoff letters c and d.