1、1. Brillouin zones of rectangular latticeSet the primitive translation vectors of the rectangular lattice areThe unit vector perpendicular to the x-y plane is The axis vectors of the reciprocal lattice areThe reciprocal lattice1xyO1st BZ2nd BZ22. Brillouin zone, rectangular lattice(a) Set the primit

2、ive translation vectors of the rectangular lattice areThe axis vectors of the reciprocal lattice areThe first Brillouin zonexyO3(b) For the 2d metal with one atom of valence electron, the first Brillouin zone is half filled, i.e. the space inside the Fermi sphere is half of the first Brillouin zone.

3、thereforexyO4(c) The periodic zone sketchWith the Harrison construction of the Fermi surface, one has5(c) The first energy band6(c) The second energy band73. Hexagonal close-packed structureFor a simple hexagonal lattice, the primitive translation vectors areThe reciprocal lattice vectors are8Consid

4、ering a basis with m atoms, the crystal potential is the sum over all atoms in the basiswhere the Fourier component9The shortest reciprocal lattice vector parallel to the c axis isThenFor the hcp structure, there are two atom in a basis:(0, 0, 0) and (a/2, 31/2a/6, c/2)10For the crystal made up of d

5、ivalent atoms in a simple hexagonal lattice, the number of electrons exactly equals to the number of the orbitals in first BZ. If the energy gap is big enough, this crystal will insulator. For the crystal made up of monovalent atoms in hcp structure, the number of the electrons equals to the number of the orbitals in first BZ. However energy gap along Gc direction is zero, i.e. there is no ener