Reflection Multiplicity

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The high-symmetry section of the table is reference material as indicated.

Reflection Multiplicity

In contrast to single-crystal diffraction in which symmetry-equivalent reflections can be measured individually and checked for equality, the one-dimensional nature of powder diffraction results in the exact superposition of reflections with the same d spacing. In calculating the intensity of a peak in a powder diffraction pattern, it is more efficient to calculate the intensity of a single reflection, hkl, and multiply it by j, the number of symmetry-equivalent reflections contributing to the single observed peak. This is called the multiplicity of the reflection. Multiplicity only occurs in powder diffraction, and so is not commonly discussed in connection with single-crystal data.

What determines the multiplicity of a reflection? There are two factors that need to be considered: firstly, the Laue class of the sample; and secondly, whether the reflection lies on one of the symmetry elements within the particular Laue class. Symmetry-equivalent reflections may be generated by applying the symmetry operators of the Laue class point group to a particular reflection. As an example, let us consider the monoclinic crystal system whose Laue symmetry is 2/m. The symmetry operators for this point group are x,y,z; -x,y,-z; -x,-y,-z; and x,-y,z. Applying these to a general reflection with indices hkl gives the following symmetry equivalent reflections hkl, -hk-l, -h-k-l, h-kl, respectively. The table below provides a non-exhaustive list of symmetry-equivalent reflections for several crystal systems: in particular, the trigonal and hexagonal systems have been omitted owing to their complexity.

Crystal System	Laue Class	Symmetry-Equivalent Reflections	Multiplicity
Triclinic	-1	hkl ≡ -h-k-l	2
Monoclinic	2/m	hkl ≡ -hk-l ≡ -h-k-l ≡ h-kl	4
Orthorhombic	mmm	hkl ≡ h-k-l ≡ -hk-l ≡ -h-kl ≡ -h-k-l ≡ -hkl ≡ h-kl ≡ hk-l	8
Tetragonal	4/m	hkl ≡ -khl ≡ -h-kl ≡ k-hl ≡ -h-k-l ≡ k-h-l ≡ hk-l ≡ -kh-l	8
	4/mmm	hkl ≡ -khl ≡ -h-kl ≡ k-hl ≡ -h-k-l ≡ k-h-l ≡ hk-l ≡ -kh-l ≡ khl ≡ -hkl ≡ -k-hl ≡ h-kl ≡ -k-h-l ≡ h-k-l ≡ kh-l ≡ -hk-l	16
Cubic	m-3	hkl ≡ -hkl ≡ h-kl ≡ hk-l ≡ -h-k-l ≡ h-k-l ≡ -hk-l ≡ -h-kl ≡ klh ≡ -klh ≡ k-lh ≡ kl-h ≡ -k-l-h ≡ k-l-h ≡ -kl-h ≡ -k-lh ≡ lhk ≡ -lhk ≡ l-hk ≡ lh-k ≡ -l-h-k ≡ l-h-k ≡ -lh-k ≡ -l-hk	24
	m-3m	hkl ≡ -hkl ≡ h-kl ≡ hk-l ≡ -h-k-l ≡ h-k-l ≡ -hk-l ≡ -h-kl ≡ klh ≡ -klh ≡ k-lh ≡ kl-h ≡ -k-l-h ≡ k-l-h ≡ -kl-h ≡ -k-lh ≡ lhk ≡ -lhk ≡ l-hk ≡ lh-k ≡ -l-h-k ≡ l-h-k ≡ -lh-k ≡ -l-hk ≡ khl ≡ -khl ≡ k-hl ≡ kh-l ≡ -k-h-l ≡ k-h-l ≡ -kh-l ≡ -k-hl ≡ lkh ≡ -lkh ≡ l-kh ≡ lk-h ≡ -l-k-h ≡ l-k-h ≡ -lk-h ≡ -l-kh ≡ hlk ≡ -hlk ≡ h-lk ≡ hl-k ≡ -h-l-k ≡ h-l-k ≡ -hl-k ≡ -h-lk	48

The above table lists multiplicities for general reflections hkl. It is important to realize that certain reflections will not have this multiplicity: for example, reflections which lie on the symmetry elements of the point group have a lower value for the multiplicity. Considering again the monoclinic crystal system and applying the symmetry operators to the reflection h0l gives only two symmetry-equivalent reflections, namely h0l and -h0-l. Thus the multiplicity for this class of reflection is only 2 (and not 4 as for the general reflection). The table below lists the multiplicities of various classes of reflection for the lower-symmetry crystal systems:

Computer programs that generate hkl lists for use in powder diffraction normally generate multiplicity values also.

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