You have already met this in Part IA Materials and Mineral Sciences. For X-rays, electrons and neutrons, the structure factor for hkl reflections, Fhkl, is given by
Fhkl = fn exp 2
{
πi hu(
n +kvn+lwn)
}
n atoms
∑
where fn is the scattering factor of atom n. Hence, it follows from this definition that for a face- centred lattice, systematic absences occur when h, k and l are mixed odd and even, and for a body- centred lattice systematic absences arise when (h + k + l) is odd.
In addition, the motif can also make reflections have zero structure factors. For example, in h.c.p. metals, F001 = 0 because the atoms at height z = 1/2 interfere destructively with those at z = 0.
Silicon, used as a standard for X-ray diffraction, has zero structure factors when h + k + l = 4n + 2, so that the N (= h2 + k2 + l2) values of the first few lines to be seen are 3, 8, 11, 16, 19 and 24.
RbBr and KCl, both of which have the NaCl structure, have absent reflections when h, k and l are all odd for all practical purposes because the scattering factors of the anions and cations in these two examples are almost identical.
Note that the c.c.p. crystal structure has a reciprocal space b.c.c. lattice, and visa-versa. Ewald sphere
By definition the radius of the Ewald sphere is 1/λ. An initial undiffracted X-ray can be represented by a point on the sphere. Diffraction occurs when the X-ray is scattered through an angle of 2θ relative to the forward direction of the X-ray by the hkl set of crystal planes. Since this is an elastic event, the diffracted X-ray and initial X-ray have the same magnitude of their wave number (=1/λ), and so the diffracted X-ray can also be represented by a point on the surface of the sphere.
The angle between these two points subtended at the centre of the circle is 2θ, and the chord between these two points is the magnitude of the reciprocal lattice vector ghkl. This is 1/dhkl, as in the diagram below:
Hence, it follows that sinθ = 1/ 2dhkl
1/λ , i.e., we derive the Bragg equation λ = 2dhklsinθ.
Electron diffraction patterns typically have θ ˜ 1°, since for 100 kV electrons λ = 0.037 Å and interplanar spacings are typically in the range 1-2 Å.
In addition, for electrons, spots close to the Ewald sphere have finite intensity because of the thin ~ 1000 Å crystals through which the electrons beam is transmitted to form the diffraction pattern. Conventional ‘spot’ electron diffraction patterns are formed by a parallel beam of electrons as the incident beam. These can have intense double diffraction: intensity can be scattered from one beam to another because of the small angles required for Bragg diffraction. This is a consequence of the dynamicalnature of electron diffraction, more of which in C20, Part III.
A consequence of double diffraction is that reflections in electron diffraction patterns which have zero structure factors because of the motif, can have finite, strong intensities when electron beams are suitably oriented.
Thus, for example, in electron diffraction it is common for 002 spots to be strong in the silicon [110 ] zone through double diffraction from the 111 and 111 spots., since vectorially 111 + 111 = 002, and the 111 and 111 reflections have non-zero structure factors. The diagram below demonstrates this.
000 002
111
111
Conversely, in the silicon [100] zone, 002 reflections will be systematically absent, as there is no double diffraction route capable of generating intensity at the 002 position in reciprocal space.
Convergent beam electron diffraction (CBED) patterns
These are what the terminology implies: in these patterns the electron beam is defined by a convergence angle, α, and so the electron beam is no longer a parallel beam. This gives rise to discs rather than spots in the Zero Order Laue Zone (ZOLZ), i.e. the zone for which hu + kv + lw = 0 for the hkl spots in this zone when the electron beam direction is [uvw], as in conventional spot patterns.
C6H9 - 3 - C6H9 Within the discs, as in the example below from silicon, information from higher angles is present as Higher Order Laue Zone (HOLZ) lines. These lines contain information about the lattice parameter, the accurate electron beam direction and any strain present in the crystal. The lines arise from inelastically scattered Bragg electrons which are subsequently Bragg diffracted by HOLZ planes. CBED patterns are particularly useful in symmetry determination, strain determination and composition analysis (see for example, Williams and Carter, Transmission Electron Microscopy).
Symmetry in electron diffraction patterns
Convergent beam electron diffraction (CBED) can be used to demonstrate that <111> axes in silicon have three fold symmetry, rather than the six fold symmetry which could be inferred from an examination of conventional selected area ‘spot’ electron diffraction patterns (i.e. ones obtained without using convergent beam). The Higher Order Laue Zone (HOLZ) lines exhibit the three fold symmetry clearly: note the equilateral triangle in the centre of the 000 disc formed by some of them.
Expanded Zero Order Laue Zone (ZOLZ) region of a CBED pattern of silicon taken with the electron beam parallel to [111].
Point group symmetries of CBED patterns can be determined as in the four examples below. More advanced techniques enable space groups of crystal structures to be determined – see for example discussion in D.B. Williams and C.B. Carter, Transmission Electron Microscopy (Plenum 1996).
C6H9 - 5 - C6H9
For relatively small ( < 500 mm) camera lengths convergent beam patterns show higher order Laue zone (HOLZ) rings, as in the example below.
Experimental CBED pattern showing the first order Laue zone clearly.
The terminology to label the centre of the pattern and the rings is as follows:
ZOLZ – Zero order Laue zone: hu + kv + lw = 0 for the hkl spots in this zone when the electron beam direction is [uvw].
FOLZ – First order Laue zone: hu + kv + lw = 1 for the hkl spots in this ring. SOLZ – Second order Laue zone: hu + kv + lw = 2 for the hkl spots in this ring.
A straightforward calculation in reciprocal space shows that the radius of the nth HOLZ ring, rn*, is given by the formula
rn* = k2 – k – nt*2 1/2 - 2knt*1/2
where k = 1/λ and t* is the reciprocal of the magnitude of the lattice vector [uvw]. In making this calculation, use is made of the result that for the nth ring, hu + kv + lw = n = |p*| |uvw|, where p* is the projection of the vector ha* + kb* + lc* lying in the HOLZ onto the direction [uvw]. Hence, |p*| = n / |uvw| ≡ nt*.