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Homework #10 1. Square lattice, free electron energies.(a) Show for a simple square lattice (two dimensions) that the kinetic energy of a free electron at a corner of the first Brillouin zone is higher than that of an electron at the midpoint of a side face of the zone by a factor of 2. (b) What is the corresponding factor for a simple cubic lattice (three dimensions)? (c) What bearing might the result of (b) have on the conductivity of divalent metals? Solution:(a) At the corner of the first Brillouin zone, ⎟⎠⎞⎜⎝⎛=aakππ,r, thus 2222222222)1,1()11(22mamamkEππhhh=+==. At the midpoint of a side face, ⎟⎠⎞⎜⎝⎛=0,akπr, thus 22222222)0,1(2)01(2mamaEππhh=+=. Therefore 2/)0,1()1,1(=EE. (b) For a simple cubic lattice, at the corner of the first Brillouin zone, ⎟⎠⎞⎜⎝⎛=aaakπππ,,r, thus 222222222)1,1,1(23)111(2mamaEππhh=++=. At the midpoint of a side face, ⎟⎠⎞⎜⎝⎛=0,0,akπr, thus 22222222)0,0,1(2)01(2mamaEππhh=+=. Therefore 3/)0,0,1()1,1,1(=EE. (c) For a divalent metal, the electron density is 3/2an=. Then the Fermi energy is 223/22223/222253.1)/6(2)3(2mamanmEFhhh===πππ. Thus )1,1,1()0,0,1(EEEF<<. The Fermi sphere is greater than the first Brillouin zone in the  direction, but smaller than the first Brillouin zone in the  direction.