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In polar coordinates, the volume element dV = r2 sin θ dr dθ dφ. 32) While this may look rather complicated, it simplifies greatly, since the operator r becomes a multiplier and the function r can be multiplied. 36) 15 GENERAL INTRODUCTION and the exponential integral is a commonly occurring one in quantum mechanics. It can be easily evaluated by using the formula : ∞ ∫ xn e–bx dx = n! 37) In this case, n = 3 and b = 2a0. Therefore, we get : ∞ ∫ r3 e–2 r/a0 dr = 3! 38) so that we can write the ‘expectation value’ as : ⎛ 4π ⎞ 3!

For closed shells, the sum of the angular momentum and spins is zero. In solids, we have ‘open shells’ for transition metals and rare earths. 5 for expectation value). Apart from paramagnetism due to the electrons, we must also consider diamagnetism. The latter results from the induction of ‘eddy’ currents by an external magnetic field. According to the Lenz’s rule, the magnetic moment of these induced currents is opposed to the applied field. The susceptibility thereby acquires a negative diamagnetic contribution.

96) dA( x ) dB( x ) and B′(x) = . γ = 1. 93). It should be noted that the product Q1Q2 = 1. , for those corresponding to (α + δ)2 < 4 → The two roots Q1 and Q2 will be “complex”, and since Q1Q2 = 1, they will be “conjugates”. 90), they are the “Bloch Functions”. In other regions of energy E, where, if we put (α + δ)2 > 4, the two ‘roots’ are real and reciprocals of each other. The roots corresponding to the solution of Schrodinger equation are : ψ1(x) = exp(μx)u(x) and, ψ2(x) = exp(– μx)u(x) GENERAL INTRODUCTION 35 where, μ = a real quantity.

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