Sampling points and reciprocal lattice vectors

To specify the periodicity vectors (generally called lattice vectors), add the following statement immediately following the statement specifying S and before the statements computing M and N,

R=diag([6; 6; 6])

Note that this means that we'll be working with a cubic unit cell of side length 6 bohr.

Next, add a code block to the bottom of your ``setup.m'' file to form the lists of real-space sample points r and reciprocal lattice vectors G from the formulae derived in lecture,

$${\bf r} = {\bf M} \left({\mbox{\bf Diag\,}}(S)\right)^{-1} {\bf R}^T$$ (2)

$${\bf G} = 2 \pi {\bf N} {\bf R}^{-1}$$ (3)

Finally, add a statement to create a column vector G2 containing the square-magnitude of each G-vector.

Hint: For this final part, note that

G.^2

produces the squares of each element of G and that

sum(A,d)

sums the matrix A along its d-th dimension.

To check your final results from ``setup.m'', you should copy the two files ``slice.m'' and ``view.m'' from Appendix A and Appendix B, respectively, into your octave directory. (To understand more how the view() and slice() functions work, you may enter ``help slice'' and ``help view'' at the octave prompt after you have copied the files.) Once you have these functions in your directory, change the size specification at the top of ``setup.m'' to that of a $20\times25\times30$ problem (`` $S=[20; 25; 30]$''), rerun ``setup.m'', and then enter ``view(G2,S)'' at the octave prompt. This will then generate plots of slices of $G^2$ in the n1=0 plane, n2=0, and n3=0 planes, respectively. In each case, you should see parabolic surfaces centered on the origin.

Finally, check your result for r by running ``view(r(:,k),S)'' for k=1,2,3 and verifying that you see the correct behavior.