STRUCTURE OF SOLIDS:

1. Crystal Lattices and Reciprocal Lattices

a. What is a Bravais lattice?

b. Define a Wigner-Seitz primitive cell. 

c. Define a set of reciprocal lattice vectors (b1, b2, b3) in terms 
   of a set of primitive lattice vectors (a1, a2, a3). 

d. Sketch a simple cubic cell and it's primitive vectors.  
   Sketch the reciprocal lattice vectors for the simple cubic cell 

e. Now build a primitive cubic cell in Cerius2 
   Start by putting your favorite atom in the model window
   Build crystal
   Find symmetry (update model)  	
   Look in the crystal builder for the General Positions menu  
   Change lattice type to cubic and centering to primitive
   Display the 100 and 111 miller planes  (successively - not simulataneously..)
   Display 2 crystal cells in the a direction, etc...
   Display the Brillouin Zone (it's in the Castep menu, under geometry)
   Under View, Options, one can check "Show Axes" - are these the 
   directions of the primitive lattice vectors?
   Under Brillouin Zone Display one can check "Reciprocal Axes - 
   are these the directions of the reciprocal lattice vectors?	       

f. Do the same as in (e) but for the body centered cubic lattice. 

g. Sketch the first, second and third Brillouin zones for a two
   dimensional square Bravais lattice.  

2. Bloch's Theorem and k-point sampling. 

a. If a potential is periodic, the eigenstates of H may be chosen
   to be eigenstates of all the translation operators of the lattice. 
   Why is this true?

b. Car and Parinello-type algorithms use periodic boundary conditions 
   and make use of the idea of a reciprocal lattice.  In terms of 
   computational cost, what is the reason for this?  

c. If no approximations were made, how many k points would a CP type
   program have to handle in order to solve a problem over an ideal
   (infinte) crystal (periodic boundary conditions).