Band Theory of Metals: The Elements by Simon L. Altmann

By Simon L. Altmann

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Since the curvature is small, the properties of any small part of this object must be practically indistinguishable from those of a straight one. On the other hand, it is clear that if we go repeatedly round the circle we satisfy exactly the same periodicity conditions as are valid along the straight line of Fig. 13. The periodic boundary conditions 9(0) =

As in the elementary theory, the velocity of this wave or phase velocity is given by (51) or, which is the same, by 1 (58) vp = vk' . In order to define the group velocity we first add u p two waves of the same amplitude: Metals [Ch. 1] Acos2n(kx-vt), (59) Acos2n(k'x-v'ty (60) Band Theory of 26 = Ψ Ψ' = In forming the sum Φ we use the trigonometric relation cos ( α + β ) = cos a 2 cos K - ß) Φ = a M +ß) · , , , , 2 ^ C O S T T [ ( a : - A : ) A : - ( V - V ) / ] C O S 7 7 [ ( A : H - Â : ) X - ( ^ + I' Wf If the difference between the two waves is very small, k—k ν— v' = Δ ν are small and we can write approximately Φ = [2A cos n(xkk - ί Δν)] cos 2ΤΓ(/ΓΛ; - vi).

Two functions that satisfy this property are said to be Method. (82) orthogonal. On account of the hermitian property, jf*Wn)dr = j(Rfm)*fndr. (83) Band Theory of 34 Metals [Ch. 1] F r o m (81), take εη out of the integral on the left-hand side and, from the complex conjugate equation of (80), e* on the right. F r o m exercise (1) «£ =

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