(b)\u00a0\u00a0(5 pts) Prove that in two dimensions the free electron density of levels\u00a0g(<\/em>e<\/em>)<\/em>\u00a0is a constant independent of\u00a0e\u00a0for\u00a0e\u00a0> 0, and 0 for\u00a0e\u00a0< 0.\u00a0\u00a0What is the constant?<\/p>\n 2.\u00a0\u00a0A two-dimensional metal has one atom of valence one in a simple rectangular primitive cell of a1 = 2\u00a0and a2 = 4.<\/p>\n (a)\u00a0\u00a0(5 pts)\u00a0Draw\u00a0the first and the second\u00a0Brillouin\u00a0zones.<\/p>\n (b)\u00a0\u00a0(5 pts)\u00a0Calculate\u00a0the radius of the free electron Fermi sphere and draw this sphere to scale on the drawing of the\u00a0Brillouin\u00a0zones.<\/p>\n (c)\u00a0\u00a0(5 pts) Draw the Fermi surface in the reduced zone scheme and show schematically the effect of a weak crystal potential.<\/p>\n 3.\u00a0\u00a0The\u00a0Wannier\u00a0functions of a band are defined in terms of the Block functions of the same band by<\/p>\n where\u00a0r<\/strong>n<\/sub><\/em>\u00a0is a lattice point.<\/p>\n (a) (5 pts)\u00a0Prove\u00a0that\u00a0Wannier\u00a0functions about different lattice points\u00a0n<\/em>,\u00a0m<\/em>\u00a0are orthogonal:<\/p>\n This\u00a0orthogonality\u00a0property makes the functions often of greater use than atomic\u00a0orbitals\u00a0centered on different lattice sites, because the latter are not generally orthogonal.<\/p>\n (b)\u00a0\u00a0(5 pts) The\u00a0Wannier\u00a0functions are peaked around the lattice sites.\u00a0\u00a0Show that for<\/p>\n![\"\"](\"http:\/\/pages.charlotte.edu\/vasily-astratov\/wp-content\/uploads\/sites\/152\/2012\/11\/image005.jpg\" "\\"image005\\"")
<\/a><\/p>\n<\/a><\/p>\n