Hammond C. The Basics of Crystallography and Diffraction

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394 Appendix 6

Hence,

hkl

= f exp 2 πi(h0 +k0 +l0) + f exp 2πi



+ k

+ l0



+f exp 2 πi



+ k0 +l



+ f exp 2πi



h0 + k

+ l



Simplifying, andremembering that since the cell is centrosymmetric the sine components

in the exponents vanish:

hkl

= f (1 +cos π(h + k) + cos π(h + l) + cos π(k + l)).

Now, cosπ (even integer) =+1 and cosπ (odd 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 base-centred 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 body-centred cubic crystal

with identical atoms at (000) and





hkl

= f exp 2 πi(h0 +k0 +l0) + f exp 2πi



+ k

+ l



= f (1 +cos π(h + k +l)).

Hence when (h + k + l) = even integer F

hkl

=2f and when (h + k + l) = odd integer

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 c-glide plane in the xz plane or (010)

plane (the plane of the paper). The action of the c-glide is to translate any atom, atomic

scattering factor f , with fractional coordinates (uvw)to



u¯v



+ w



. Substituting

Fig. A6.2.

Appendix 6 395

these two values in the structure factor equation:

hkl

= f exp 2πi(hu +kv + lw) + f exp 2π i



hu +k ¯v + l



+ w



Consider the h0l reﬂections (k = 0), i.e. the reﬂections in the zone whose axis is

perpendicular to the glide plane (010)

h0l

= f exp 2πi(hu +lw) +f exp 2πi(hu + lw) exp 2πi

= f exp 2πi(hu + lw)[1 +exp π il].

When l = odd integer, exp π il =−1 and F

h0l

= 0. This then gives the condition for

systematic absences for the h0l planes; systematic absences occur when l equals an odd

integer and, correspondingly, reﬂections occur when l equals an even integer. Clearly,

for a-glide 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

along the y-

axis, the action of which is to translate an atom with fractional coordinates (uvw)to



¯u



+ v



¯w



. Proceeding as before:

hkl

= f exp 2πi(hu +kv + lw) + f exp 2π i



h¯u + k



+ v



+ l ¯w



consider the 0k0 reﬂections (h = l = 0) i.e. reﬂections arising from planes perpendicular

to the screw axis.

0k0

= f exp 2πi(kv) + f exp 2π i



+ kv



= f exp 2πi(kv)[1 +exp π ik].

When k = odd integer, exp π ik =−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.

Fig. A6.3.

396 Appendix 6

TableA6.1 lists theextinctioncriteria for lattices and translationalsymmetry 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 A6.2 lists the reﬂecting planes in cubic P, I, F and diamond-cubic crystals in

order of decreasing d

hkl

-spacing. For cubic crystals d

hkl

= a/

√

N where a = the lattice

parameter and N = (h

+ k

+ l

). The conditions for reﬂection for a diamond-cubic

crystal are h, k, l are all odd or all even integers (since a diamond-cubic 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).

A6.2 Double diffraction

In electron diffraction the presence of face- or body-centring lattice points also gives rise

to systematically absent reﬂections just as in the case of X-ray diffraction (the ﬁrst group

in TableA6.1). However, reﬂections which are systematically absent in X-ray 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 interactionsbetween the directand diffractedbeams. The effect, in termsof 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

(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

λ, which is

the condition for destructive interference; hence the 0001 reﬂection is (systematically)

absent. Second-order reﬂection from the (0001) planes (PD = 2λ) corresponds to the

ﬁrst-order reﬂection from the (0002) planes (PD = 1λ). 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.

Now let usdrawa commonly observed hcp electron diffraction pattern from the[

10]

zone (Fig. A6.4), showing the systematically absent reﬂections 0001, 000

1, 0003, 000

Appendix 6 397

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

body-centred Ih+ k + l = 2n

centred on the C face Ch+ k = 2n

centred on the A face Ak+ l = 2n

centred on the B face Bh+ l = 2n

h, k, l

centred on all faces F all = n(odd)

or all = 2n(even)

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

rhombohedral, reverse Rh−k + l = 3n

Glide plane || (001) ahk0 h = 2n

bk= 2n

nh+ k = 2n

dh+ k = 4n

Glide

plane || (100) b 0kl k = 2

cl= 2n

nk+ l = 2n

dk+ l = 4n

Glide plane || (010) ah0lh= 2n

cl= 2n

nh+ l = 2n

dh+ l = 4n

Glide plane || (110) c hhl l = 2n

bh= 2n

nh+ l = 2n

d 2h + l = 4n

Screw axis || c 2

00ll= 2n

l = 3n

l = 4n

l = 6n

Screw axis || a 2

h00 h = 2n

h = 4n

Screw axis || b 2

0k0 k = 2n

k = 4n

Screw axis || [110] 2

hh0 h = 2n

(n = odd integer, 2n = even integer etc.)

398 Appendix 6

Table A6.2 Reﬂecting panes (

√

) in cubic P, I , F and diamond-cubic crystals.

N hkl cubic P cubic I cubic F Diamond cubic

1 100

√

2110

√√

3 111

√√√

4 200

√√√

5 210

√

6211

√√

7—

8 220

√√√ √

9 300/221

√

10 310

√√

11 311

√√√

12 222

√√√

13 320

√

14 321

√√

15 —

16 400

√√√ √

17 410

√

18 330/411

√√

19 331

√√√

20 420

√√√

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: ‘two-together’, ‘one-on-its-own’,

‘two-together’ etc.

0000 0002 0004

1010

1011 1012 1013 1014

Diffraction pattern from h c p metal, [1210] zone

Fig. A6.4.

Appendix 6 399

etc. in the row through the origin.

Now suppose that the (strong) 10

10 reﬂected beam

acts as it were as the direct beam: the whole pattern will, in effect, be shifted from the

origin 0000 to 10

10 as shown by the arrowandspots 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 double-diffracted

spots which appear may correspond to no d -spacing in either crystal—and are proba-

bly the biggest cause of the frustrated electron microscopists’ ‘unindexable’ patterns.

Figure A6.5 indicates the geometry involved. Here, two (high order) patterns are shown

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 low-order zone—but of course it is not, and is

either unindexable or (worse) may be indexed incorrectly. An example of the latter case

Fig. A6.5.

Systematic absences also occur in every third row (not shown in Fig. A6.4). These arise because the

A-layer and B-layer atoms are in special equivalent positions in the hexagonal unit cell (see Section 4.6). This

leads to the conditions for hkil:ifh − k = 3n, l = 2n, i.e. in the third row the 30

31,30

33, etc. reﬂections

are systematically absent. These conditions are given in the table in Fig. 4.14, space group P6

/mmc, for the

special equivalent positions for hcp structures denoted by the Wyckoff letters c and d.

400 Appendix 6

Fig. A6.6.

is that of the electron diffraction patterns from twin-related 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 twin-related in the (112) plane. Double diffraction between these twin-related

zones gives rise to the spots marked d (i.e. d

∗

110

∗

110

= d

where (110)

is the twin-

spot 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



110



zone—the four bcc {110} spots closer to the centre spot being identiﬁed

as fcc {111} spots and the two double-diffraction 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



110



austenite zone—leading to the erroneous

conclusion that this phase is present at the ferrite twin boundary.

Answers to exercises

Chapter 1

1.1 Figure E1.1 shows a tetrahedron of height h and edge-length equal to the interatomic

spacing a. The ‘B layer’ atom lies above the centre of the triangle formed by three ‘A layer’

atoms. The distance AC is

a cos 30 = a/

√

3. Hence from a right-angled triangle ABC

= a

−

whence h =

√

2/3a. Hence c/a = 2h/a = 2

√

2/3 = 1.633. The centre

of the tetrahedron above the triangular base (denoted by X in Fig. E1.1), is obtained by

projecting a bisector from any edge, e.g. AB to X . cos x = h/a and BX = a/2.

cos x

a/2 a/h = a

/2h and since a =

√

3/2 h, BX =

2.2h

h or CX = 1/4 h: i.e. the centre

of a tetrahedron is at 1/4 of its height.

1.2 As can be seen in Figure 1.13(d), the tetrahedron of four A atoms is not regular but has

four shorter edges of length a

√

3/2 and two longer edges of length a. However, unlike

the octahedral interstitial site all four A atoms are equidistant from the interstitial X atom

at the centre of the tetrahedron. The geometry is best appreciated with reference to a plan

view (Fig. E1.2) showing two bcc cells and one tetrahedral interstitial site which lies, by

symmetry, at height

a in the common cell face. The distance from the centre of the

tetrahedron to any corner is



(a/2)

+ (a/4)

= a

√

5/4 = r

. The closest distance

betweenAatoms isalong thedirection of the body-diagonal of the cube, i.e. 2r

= 1/2

√

3a.

Hence r

+ r

= a

√

5/4 = r

√

5/3 = 1.291r

whence r

= 0.291r

1.3 The interstitial site is at the centre of a triangular prism of height and side-length 2r

(Fig. E1.3). The interstitial atom X at height r

is equidistant from all six corners of the

prism. Hence, proceeding as in Exercise 1.1, AC =

cos 30

◦

= r

√

3, and AX



4/3r

+ r

= r

√

7/3.

Hence AX = r

+ r

= 1.528r

whence r

= 0.528r

a/2

Fig. E1.1.

402 Answers to exercises

√

2=a3

Fig. E1.2.

Fig. E1.3.

1.4 (a) In the ccp structure there are four sets of close-packed planes and six close-packed

directions and inthe hcp structureonly one setof close-packed planesand three close-packed

directions.

(b) In the bcc structure there are six sets of closest-packed planes and four close-packed

directions.

1.5 In theccp structure eachclose-packed plane containsthree close-packeddirections, i.e. three

slip systems share a common close-packed plane. As there are four sets of close-packed

planes in the ccp structure there are therefore twelve slip systems. In the bcc structure each

closest-packed plane contains two close-packed directions, i.e. two slip systems share a

Answers to exercises 403

common closest-packed plane. As there are six sets of closest-packed planes in the bcc

structure there are therefore also twelve slip systems.

1.6 The atomic coordinates are (000) (for all the layer’ atoms situated at the corners of the unit

cell), and either









for the ‘B layer’ or the ‘C layer’ atom within the cell,

depending upon the arbitrary choice made between these two possible (B or C) sites.

1.7 For B6 (carborundum II) the stacking sequence is ABCACBA … giving a six-layer unit

cell repeat distance. In the Frank notation this is ∇∇∇…, i.e. an inversion every

three layers (rather than an inversion every layer in B4 and an inversion every two layers

in B5).

For B7 (carborundum I) the stacking sequence isABACBCACBABCBACA… giving a

ﬁfteen-layer unit cellrepeatdistance. In theFrank notation thisis∇∇∇∇∇∇∇∇∇

…, i.e. inversions every two and three layers alternately.

1.8 Fault anA layer to a B layer as in Fig. 1.20(a)–(b). Then fault not the C layer in Fig. 1.20(b)

but the A layer above which becomes a B layer; all the layers above are re-labelled B → C

and C → A as before. Again, fault the next A layer above to become a B layer and repeat

as before. The stacking sequence is clearly CBCB …, i.e. the hcp stacking sequence.

1.11 The sequence is: ∇∇∇∇ (a four-layer repeat).

Chapter 2

2.1 There are twice as many mirror lines in p3m1 and the triads lie at the intersections of all of

them; in p31m half the triads are not intersected by mirror lines.

2.2 The lattice type is p3m1 (refer to your drawings or Question 2.1).

2.3 The lattice type is p2—i.e. the pattern has identical symmetry to Fig. 2.7; it is based on the

oblique plane lattice, not a rectangular plane lattice (primitive or centred) because there are

no mirror lines of symmetry present.

2.4 The lattice type is pg. The glide lines run vertically through the back heels of the black men

and the raised hands of the white men.

2.5 A 4



axis with vertical and horizontal m



lines and diagonal m lines (see Fig. 2.11).

2.6 The one-dimensional lattice types (Fig. 2.23) are:

(a) p112 (b) p112 (c) p112

(d) p112 (e) pmm2 (d) pmm2

(g) p1a1 (h) p111 (i) p112.

2.7 Figure 2.24(b) is counterchange plane group p2g



2.8 There are three such counterchange patterns based upon pma2(p2mg) as shown

in Fig. E2.8.

Chapter 3

3.1 (a) isorthorhombic C; (b) is not a Bravais lattice because the points do not allhave anidentical

environment [see the two-dimensional examples, Figs 2.2(b) and (c)]; (c) is orthorhombic

P (two primitive cells are drawn together).

3.2 (a) is orthorhombic P (relocate the origin at x



); (b) is orthorhombic I (relocate the origin

at 0

0); (c) is orthorhombic C (relocate the origin at 00z



; the motif is two atoms, one of

each type).

3.3 For α>90

◦

it is a ‘distorted’truncated octahedron (i.e. rectangular, rather than square faces

and unequal-sided hexagonal faces) and for α<90

◦

it is a ‘distorted’rhombic dodecahedron

(i.e. with 12 faces of unequal-shaped diamonds). See Fig. 3.7(d) and (e).