Number of Possible Primitive Vectors and Primitive cells in a Bravais Lattice According to my Book Solid State Physics by Ashcroft and Mermin, given a 3-dimensional Bravais lattice, there are infinite sets of 3 primitive vectors that we can come up with that will span the lattice. This also means for a given Bravais lattice there are also infinitely different parallelepiped primitive cells one can come up with. I can image construct 2 different sets of 3 vectors that span a lattice and give 2 different primitive cells. However, the claim that there are infinitely many different primitive vectors and cells is hard to imagine. Is there proof or explanation to check that this is true?
 A: Consider any lattice with primitive vector triplet ($\vec{x},\vec{y}, \vec{z})$. Any lattice vector $\vec{v}$ can be written as
$\vec{v} = v_x \vec{x} + v_y \vec{y} + v_z \vec{z}$
where $v_x, v_y, v_z$ are integers
For any primitive vector triplet, following two conditions should be true.

*

*For a primitive lattice vector $\vec{v}$, $gcd(v_x,v_y,v_z)=1$


*

Consider any vector triplet ($\vec{a},\vec{b},\vec{c}$).
Magnitude of their determinant
\begin{vmatrix}
a_x & a_y & a_z \\ 
b_x & b_y & b_z \\ 
c_x & c_y & c_z
\end{vmatrix}
should be 1.
In fact, first condition is necessary condition for second condition to be true.
Now consider an example of simple cubic lattice.
Consider this inifinite family of primitive vector triplet
($\vec{x},\vec{y}, m\vec{x} + n\vec{y} + \vec{z}$) for all integral values of $m$ and $n$.
In fact, this can be generalised to generate infinite families of primitive vector triplets. Pick any primitive vector triplet($\vec{v_1},\vec{v_2},\vec{v_3}$). Then ($\vec{v_1},\vec{v_2},\vec{v_3} + m\vec{v_1} + n\vec{v_2}$) is an infinite family of primitive vector triplets for all integral values of $m$ and $n$.
Matrix with determinant magnitude being 1 is called unimodular matrix. So, the problem of finding primitive lattice vector triplets is equivalent to find all unimodular matrices (of size 3 for lattice of dimension 3).
Unimodular matrices form special linear group over integers and the generators of these groups are elementary matrices.
