Forbidden/Allowed Reflections/Diffraction
Spots in Diffraction Patterns
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Certain lattice, such as body centered cubic (Cubic I) and face centered cubic (Cubic F), have "kinematically" forbidden reflections. In other words, due to the arrangements of the atoms in the unit cell, these are reflections where the intensity of the scattered wave is zero, given by,

         Conditions of forbidden and allowed reflections (h k l) of common crystal structures ---------------------------- [3897]
where,
          xi, yi, and zi -- The positions of the atoms in the unit cell.

Those reflections are destructive and thus they do not show up in the diffraction patterns, however, the relevant crystalline planes certainly exist in the crystals.

Table 3897a lists the symmetry elements that cause systematic forbidden reflections.

Table 3897a. Symmetry elements that cause systematic forbidden reflections.
Symmetry Not existing systematic forbidden reflections Existing systematic forbidden reflections
Centers of inversion Yes  
Rotation axis Yes  
Mirror plane Yes  
Roto-inversion axis Yes  
Screw axis No Yes
Glide plane No Yes

The reflections that are kinematically forbidden due to the presence of screw axes or glide planes will appear in the diffraction pattern of a crystal that scatters dynamically. However, the kinematically forbidden reflections due to unit-cell centering [2] do not show in the dynamical scattering.

Table 3897b. Conditions of forbidden and allowed reflections (h k l) of common crystal structures (F: the Structure Factor).
Bravais Lattice
Forbidden reflections
Allowed reflections
F
Number of lattice points per cell
Example Compounds
Primitive Cubic
None Any h, k, l f 1 α-Po
fcc
h, k, l are mixed odd and even h, k, l are all odd or all even 4f 4 fcc metals, GaAs, NaCl-rocksalt, ZnS-zincblende
bcc
h + k + l is odd h + k + l is even 2f 2 bcc metals
fcc
h, k, l are mixed odd and even; or, all even and h + k + l ≠ 4n As fcc, but if all even and h + k + l ≠ 4n, then absent (n is integer)
    Si, Ge, Sn - diamond cubic
Base centered
  h, k and l all odd or all even 2f 2  

h + k + l is odd
      bct
Primitive Hexagonal
h + 2k = 3m and l is odd All other cases     Hexagonal closed packed (hcp) metals
Hexagonal close-packed (hcp)
      Reflection example  
  h + 2k = 3n with l odd 0 0001  
  h + 2k = 3n with l even 2f 0002  
  h + 2k = 3n ± 1 with l odd f3 01-11  
  h + 2k = 3n ± 1 with l even f 01-10  

For hcp crystals, the (0 0 0 l) reflections, e.g. for the case with 164 (P-3m1) space group, are forbidden when l is odd. However, those reflection positions often show diffraction intensity, which is probably caused by chemical order on the basal planes, or by double or multiple diffraction (scattering).

Table 3897c. Miller indices of diffracting planes, and allowed and forbidden reflections.
{hkl} Σ[h2 + k2 + l2 ] FCC Diamond cubic BCC
{100} 1 - - -
{110} 2 - - 110
{111} 3 111 111 -
{200} 4 200 200 200
{210} 5 - - -
{211} 6 - - 211
  7 - - -
{220} 8 220 220 220
  9 - - -
{310} 10 - - 310
{311}
11 311 331 -
{222} 12 222 - 222
  13 - - -
{321} 14 - - 321
  15 - - -
{400} 16 400 400 400
  17 - - -
{411} 18 - - 411
{330} 18 - - 330
{331} 19 331 331 -
{420} 20 420 - 420
  21 - - -
{332} 22 - - 332
  23 - - -
{422} 24 422 422 422
  25 - - -
{431} 26 - - 431
{511} 27 511 511 -
{333} 27 333 333 -
  28 - - -
  29 - - -
{521} 30 - - 521
  31 - - -
{440} 32 440 440 440

Table 3897d. Forbidden and allowed reflections (h k l) of some materials.
Lattice
Examples of forbidden reflections
Examples of allowed reflections
Example Compounds
Cubic perovskite structure
{1 0 0}   SrTiO3

For symmetry determination with both XRD and electron diffraction crystallography, we are looking for symmetry-related reflections. Then, the difference between the two techniques is mainly on systematically forbidden reflections:
        i) In XRD profiles, the forbidden reflections typically have absolutely zero intensity.
        ii) In SAED patterns, such forbidden reflections always have some degree of intensity due to double diffraction from multiple scattering.
Therefore, in SAED analysis, to minimize the multiple scattering, we need to use extremely thin specimens. Fortunately, precession electron diffraction (PED) gives an opportunity to provide closer kinematical conditions and are less dynamical than SAED. Given a large precession angle, the kinematical forbidden reflections can be identified [1]. However, in most cases, the forbidden reflections in PED patterns will not be fully absent since the patterns are produced as a sum of many misaligned electron diffraction patterns and screw axes can only be completely absent if the zone axis of the crystal is perfectly aligned.

Furthermore, for CBED patterns, we have:
         i) The odd order reflections in the direction of the axis will be forbidden if the glide plane is parallel to the electron beam.
         ii) For a screw axis or glide plane, if the projection of the unit cell in the beam direction has a symmetry, then the forbidden reflections would not be fully forbidden but would obviously be very weak.

 

 

 

 

[1] J. P. Morniroli, A. Redjaïmia, S. NIcolopoulos, Ultramicroscopy 107 (2007) 514.
[2] D. E. Sands. Introduction to crystallography. Dover Publications, 1993.

 

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