Create the groups which are given as the vector space (or module) over integers.

[Hongyi Zhao]

For crystallographic space groups, say, the diamond structure, we have the following information, as described here [1].

The primitive cell lattice vectors can be defined as follows:

 a1 = (0, 1/2, 1/2), a2 = (1/2, 0, 1/2), a3 = (1/2, 1/2, 0)

The translation vectors, can be selected as follows:

t0 = (0, 0, 0), t1 = (0, 1/2, 1/2), t2 = (1/2, 0, 1/2), t3 = (1/2, 1/2, 0)

The above-mentioned translation vectors can be used to extend the primitive cell to the larger conventional cell, which has the following lattice vectors:

b1 = (1, 0, 0), b2 = (0, 1, 0), b3 = (0, 0, 1)

It is well known that the above specific set of vectors can be studied by the group theory method, say by lattice [2], or by the groups of vector space (or module) over integers

Having said that, I still feel that using such group tools to study the problem described here is quite difficult. Any hints/tips will be highly appreciated.

[1] https://en.wikipedia.org/wiki/Unit_cell

[2] https://en.wikipedia.org/wiki/Lattice_(group)

Regards,

HZ

[Alan Stafford]

Burgers vector - Wikipedia

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