Diffraction is a phenomenon in which scattered waves constructively interfere when waves are incident on a crystal under specific conditions (Bragg conditions). However, satisfying the Bragg conditions does not necessarily mean that diffraction will occur. When a crystal possesses symmetry operations such as compound lattice translations, screw axes, or glide planes, the intensities of diffracted waves for specific Miller indices are extinguished1, even when the Bragg conditions are satisfied. This page summarizes what types of reflection conditions are imposed on what types of crystal planes when the crystal has symmetry operations of compound lattice translations (lattice types), glide planes, or screw axes.
The type of reflections is classified by the type of Miller indices. For example, \(h0l\) refers to crystal planes where “\(h\) and \(l\) are arbitrary but \(k\) is \(0\).” If written as \(hkl\), all crystal planes are the target. If written as \(hhl\), crystal planes where \(h=k\) are the target. The notation \(l =2n\) in reflection conditions means that the reflection appears when that value is even. Note that this is not a condition for extinction.
Reflection Conditions Due to Lattice Type
| Type of reflections | Reflection condition | Centering of cell | Translation vector | Centering symbol | |||
|---|---|---|---|---|---|---|---|
| \(hkl\) | None | Primitive | \(\bf a\), \(\bf b\), \(\bf c\) | \(P\), \(R\) (rhombohedral axes) | |||
| \(h + k = 2n\) | \(C\)-face centered | \(\bf a\), \(\bf b\), \(\bf c\), \(({\bf a}+{\bf b})/2\) | \(C\) | ||||
| \(h + k = 2n\) | \(A\)-face centered | \(\bf a\), \(\bf b\), \(\bf c\), \(({\bf b}+{\bf c})/2\) | \(A\) | ||||
| \(h + k = 2n\) | \(B\)-face centered | \(\bf a\), \(\bf b\), \(\bf c\), \(({\bf c}+{\bf a})/2\) | \(B\) | ||||
| \(h + k +l = 2n\) | Body centered | \(\bf a\), \(\bf b\), \(\bf c\), \(({\bf a}+{\bf b}+{\bf c})/2\) | \(I\) | ||||
| \(h + k = 2n,\) \(k + l= 2n,\) \(h + l = 2n\) | All-face centered | \(\bf a\), \(\bf b\), \(\bf c\), \(({\bf a}+{\bf b})/2\), \(({\bf b}+{\bf c})/2\), \(({\bf c}+{\bf a})/2\) | \(F\) | ||||
| \(-h + k + l = 3n\) | Rhombohedrally centred | obverse setting (standard) | \(\bf a\), \(\bf b\), \(\bf c\), \((2{\bf a}+{\bf b}+{\bf c})/3\), \(({\bf a}+2{\bf b}+2{\bf c})/3\) | \(R\) (hexagonal axes) | |||
| reverse setting | \(\bf a\), \(\bf b\), \(\bf c\), \(({\bf a}+2{\bf b}+{\bf c})/3\), \((2{\bf a}+{\bf b}+2{\bf c})/3\) | ||||||
Reflection Conditions Due to Glide Planes
| Type of reflections | Reflection condition | Glide plane | Crystallographic coordinate system to which condition applies | |||||
|---|---|---|---|---|---|---|---|---|
| Orientation of plane | Glide vector | Symbol | ||||||
| \(0kl\) | \(k = 2n\) | \((100)\) | \({\bf b}/2\) | \(b\) | \(e\) | Monoclinic (a unique), Tetragonal | Orthorhombic, Cubic | |
| \(l = 2n\) | \({\bf c}/2\) | \(c\) | ||||||
| \(k + l = 2n\) | \(({\bf b}+{\bf c})/2\) | \(n\) | ||||||
| \(k + l = 4n\) \((k, l = 2n)\) | \(({\bf b}\pm{\bf c})/4\) | \(d\) | ||||||
| \(h0l\) | \(l = 2n\) | \((010)\) | \({\bf c}/2\) | \(c\) | \(e\) | Monoclinic (b unique), Tetragonal | Orthorhombic, Cubic | |
| \(h = 2n\) | \({\bf a}/2\) | \(a\) | ||||||
| \(l + h = 2n\) | \(({\bf c}+{\bf a})/2\) | \(n\) | ||||||
| \(l + h = 4n\) \((l, h = 2n)\) | \(({\bf c}\pm{\bf a})/4\) | \(d\) | ||||||
| \(hk0\) | \(h = 2n\) | \((001)\) | \({\bf a}/2\) | \(a\) | \(e\) | Monoclinic (c unique), Tetragonal | Orthorhombic, Cubic | |
| \(k = 2n\) | \({\bf b}/2\) | \(b\) | ||||||
| \(h + k = 2n\) | \(({\bf a}+{\bf b})/2\) | \(n\) | ||||||
| \(h + k = 4n\) \((h, k = 2n)\) | \(({\bf a}\pm{\bf b})/4\) | \(d\) | ||||||
| \(h\,\bar{h}\,0\,l\) \(0\,k\,\bar{k}\,l\) \(h\,0\,\bar{h}\,l\) | \( l = 2n\) | \((11\bar{2}0)\) \((\bar{2}110)\) \((1\bar{2}10)\) | \(\{11\bar{2}0\}\) | \({\bf c}/2\) | \(c\) | Hexagonal | ||
| \(h\,h\,\overline{2h}\,l\) \(\overline{2h}\,h\,h\,l\) \(h\,\overline{2h}\,h\,l\) | \(l = 2n\) | \((1\bar{1}00)\) \((01\bar{1}0)\) \((\bar{1}010)\) | \(\{1\bar{1}00\}\) | \({\bf c}/2\) | \(c\) | Hexagonal | ||
| \(hhl\) \(hkk\) \(hkh\) | \(l = 2n\) \(h = 2n\) \(k = 2n\) | \((1\bar{1}0) \) \((01\bar{1})\) \((\bar{1}01)\) | \(\{110\}\) | \({\bf c}/2\) \({\bf a}/2\) \({\bf b}/2\) | \(c, n\) \(a, n\) \(b, n\) | Rhombohedral | ||
| \(hhl, h\bar{h}l\) | \(l = 2n\) | \((1\bar{1}0),(110)\) | \({\bf c}/2\) | \(c , n\) | Tetragonal | Cubic | ||
| \(2h + l = 4n\) | \(({\bf a}\pm{\bf b}\pm{\bf c})/4\) | \(d\) | ||||||
| \(hkk, hk\bar{k}\) | \(h = 2n\) | \((01\bar{1}),(011)\) | \({\bf a}/2\) | \(a, n\) | ||||
| \(2k + h = 4n\) | \(({\bf a}\pm{\bf b}\pm{\bf c})/4\) | \(d\) | ||||||
| \(hkh, \bar{h}kh\) | \(k = 2n\) | \((\bar{1}01),(101)\) | \({\bf b}/2\) | \(b, n\) | ||||
| \(2h + k = 4n\) | \(({\bf a}\pm{\bf b}\pm{\bf c})/4\) | \(d\) | ||||||
Reflection Conditions Due to Screw Axes
| Type of reflections | Reflection condition | Screw axis | Crystallographic coordinate system to which condition applies | |||
|---|---|---|---|---|---|---|
| Direction of axis | Screw vector | Symbol | ||||
| \(h00\) | \(h = 2n\) | \([100]\) | \({\bf a}/2\) | \(2_1\) | Monoclinic (a unique), Orthorhombic, Tetragonal | Cubic |
| \(4_2\) | ||||||
| \(h = 4n\) | \({\bf a}/4\) | \(4_1, 4_3\) | ||||
| \(0k0\) | \(k = 2n\) | \([010]\) | \({\bf b}/2\) | \(2_1\) | Monoclinic (b unique), Orthorhombic, Tetragonal | Cubic |
| \(4_2\) | ||||||
| \(k = 4n\) | \({\bf b}/4\) | \(4_1, 4_3\) | ||||
| \(00l\) | \(l = 2n\) | \([001]\) | \({\bf c}/2\) | \(2_1\) | Monoclinic (c unique), Orthorhombic | Cubic |
| \(4_2\) | Tetragonal | |||||
| \(l = 4n\) | \({\bf c}/4\) | \(4_1, 4_3\) | ||||
| \(000l\) | \(l = 2n\) | \([001]\) | \({\bf c}/2\) | \(6_3\) | Hexagonal | |
| \(l = 3n\) | \({\bf c}/3\) | \(3_1, 3_2, 6_2, 6_4\) | ||||
| \(l = 6n\) | \({\bf c}/6\) | \(6_1, 6_5\) | ||||