Extinction Rules for Diamond

Extinction rules for Diamond

The structure factor is defined as below where xj, yj, and zj are the coordinates in term of lattice vectors of each of the atoms in the basis.

For diamond, the cubic cell have an 8-atoms basis for which (xi,yi,zi) are

(0,0,0); (0,½,½); (½,0,½); (½,½,0); ( ¼ , ¼, ¼); (¼,¾,¾); (¾,¼,¾,); ( ¾,¾, ¼)

By evaluation in these points, the structure factor become

By rearranging the last three terms

Now taking as a common factor of the last 4 terms

Now taking the terms within the square bracket as common factors

The first of the multiplicative terms is the structure factor for an fcc lattice and it is zero unless the three indexes have the same parity (all odd or all even), thus for a diamond lattice peaks of planes whose indexes have a different parity will be missing.

Let’s consider the second term for those cases where the first term is NOT zero.

1)  If all the indexes are odd, then their sum is odd too.

with m=ν1+ ν2+ ν3

(m+1)/2 takes care of the sign of i. Example if m=1, then the factor is exp(-iπ/2)=-i, if m=3, then the factor is +i. and so on.

So when all the indexes are odd, the second term in the multiplication is not zero so this extinction rules applies to diamond too.

2)  If all the indexes are even,

where 2m= ν1+ ν2+ ν3

So when m is even or m=2n, that factor is 2 but when m is odd (meaning that the sum of indexes add up to a number multiple of 2 but not multiple of 4), that factor is zero.

So, in a diamond lattices planes with indexes all even that add up to a number which is NOT a multiple of 4 are extinct.

The combining all the rules, for a diamond, diffraction will occur of planes whose indexes are

1)  All odd

2)  All even AND ν1+ ν2+ ν3=4n

Otherwise, the diffraction peak is extinct.

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