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Physics 463, W12 Name (First, Last): _____________________ VI. Energy Bands in Solids (due Mar. 29) Reading: Kittel’s Chap 7, 8 (1sthalf) & 9: p161-204, p221-235, 242-255. Ashcroft & Mermin Chap 8-10 & 12. 1.Periodic functions. (15 pts) A function 𝑉(𝒓�⃗)has the same periodicity of a Bravais lattice. Prove that 𝑉(𝒓�⃗)can be expanded in the form: 𝑉(𝒓�⃗) =∑ 𝑉𝑮��⃗𝑒𝑖𝑮��⃗⋅𝒓�⃗𝑮��⃗, where 𝑮��⃗are reciprocal lattice vectors. 2.1D electron band for a weak potential. (30 pts) Electrons of mass 𝑚are confined to one dimension. A weak periodic potential of period 𝑎, 𝑉(𝑥) =𝑉0𝑐𝑜𝑠(2𝜋𝑥/𝑎), is applied. a.Under what conditions will the nearly free-electron approximation work? Assuming that the condition is satisfied, sketch the three lowest energy bands in the first Brillouin zone. Number the energy bands (starting from one at the lowest band). b.Calculate (to first-order in 𝑉0) the energy gap between the first and second band, and at 𝑘=𝜋/𝑎