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Equation for the multiplicity of a set of planes

Physics Asked by Joseph Knight on January 12, 2021

The multiplicity (m) of lattice planes counts the number of planes related to (hkl) by symmetry. For example, the multiplicity of the {100} planes would be 6 because the following planes are all identical for a cubic crystal system;

begin{equation}
(100) : (010) :(001) :(bar100):(0bar10):(00bar1)
end{equation}

For planes not involving 0 such as {hkl}, {hhk} or {hhh} I think the formula below based on permuting h, k and l as entities and correcting for repetitions works:

begin{equation}
m=frac{3!times 2^3}{n!}
end{equation}

where n represents the number of repetitions (i.e n=0 for {hkl}, 2 for {hhk} and 3 for {hhh}).

How can I generalise this for cases involving zeros? The problem I see is that: begin{equation}
0=bar0
end{equation}

Below is a table showing some more values for different sets of planes:

enter image description here

One Answer

begin{equation} m=frac{3!times 2^{(3-N)}}{n!} end{equation}

N counts the number of zeroes present in (hkl)

n counts the number of repeated items (including zeroes) as explained originally

Answered by Joseph Knight on January 12, 2021

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