# Crystallography, Ashcroft&Mermin

1. Feb 17, 2007

### Feynmanfan

1. The problem statement, all variables and given/known data

I'm having nightmares with this problem, apparently simple (A&M Chapter 7 Prob 2).

I have to show that a trigonal primitive lattice, depending on its angle, can represent fcc or bcc. This I kind of figured it out intuitively. But the serious problem is number b), where I have to prove that a trigonal lattice of basis a_i with angle 60^0, and a two point basis +-\frac{1}{4}(a_1+a_2+a_3) can represent a simple cubic lattice.

2. Relevant equations

A trigonal primitive cell set of vector can be
$a_1=\frac{a}{2}(x+y)$
$a_2=\frac{a}{2}(z+y)$
$a_3=\frac{a}{2}(x+z)$

The definition of trigonal is, same angle between vectors and same length.
A simple cubic lattice is clear that it can be represented by a (x,y,z)

3. The attempt at a solution

Well... I added a1+a2+a3 and divided by 4 as the problem said but i can't understand why this represents a simple cubic lattice. In addition I am asked to guess what it represents if I divide by 8 instead of 4.

I guess I don't know what "two point basis means".