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Caesium Chloride assumes a 'body centred cubic structure' in which each caesium ion is surrounded by 8 chlorine ions located at the vertices of a cube, and vice-versa.
Mathematically, we can choose cartesian coordinates such that the ions lie on the integer lattice comprising the points (l, m, n) with l, m,n integers. The caesium ions are located at points with l+m+n even and the chlorine ions at points with l-m-n odd.
To start, make sure that you understand why the lattice points represent correctly a body centred cubic structure.
I can transform the points {\bf v} in a lattice by multiplying by a constant matrix M or adding a constant vector {\bf c} through
{\bf v} \rightarrow {\bf v} +{\bf c}\text{ or } {\bf v} \rightarrow M{\bf v}
Which of the following vectors and matrices preserve exactly the structure of the caesium chloride when they transform the lattice?
M =\pmatrix{1&0&0\cr 0&0&-1\cr 0&1&0}\,,\pmatrix{3&0&0\cr 0&1&0\cr 0&0&2}\,,\pmatrix{1&1&-1\cr -1&1&1\cr 1&-1&1}
{\bf c} = \pmatrix{1\cr 0\cr 0}\,,\pmatrix{1\cr 1\cr 0}\,,\pmatrix{2\cr 2\cr 2}\,,\pmatrix{4\cr -2\cr -8}
In each case prove why the crystal structure is preserved, or explain what goes wrong.
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