By Hammar G.W.

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**Additional resources for A Preliminary Report on the Magnetic Susceptibilities of Some Cases**

**Example text**

The second term is the exchange energy term. It is negative and arises from the antisymmetry of the wave function. The direct part arises from the q D 0 part of the Hamiltonian and cancels the H b C H e b terms as a result of charge neutrality. The exchange term is not the total that arises from the electron–electron interaction. All that is left out is called the correlation energy. The leading contribution to the correlation energy of the degenerate electron gas will be obtained using Feynman graph techniques.

If µ D 0 for T < T0 , the integral in Eq. 119) is less than N/V in Eq. 123) because the value of the denominator is increased relative to its value at T0 and the full value of N/V will not be reproduced. This can be rectiﬁed if we treat the system as follows. Below T0 , the system consists of two components: (1) particles occupying the zero momentum state with a mean occupation number N0 , and (2) particles occupying the excited state. 124) which gives N0 /V D (N/V )[1 (T/ T0 )3/2 ] for T < T0 .

Then, k 0 D k sin θ cos φ 0 , sin θ sin φ 0 , cos θ zO 0 D (sin ξ cos φ, sin ξ sin φ, cos ξ ) k z0 D k 0 zO D k sin θ sin ξ cos(φ 0 φ) C cos θ cos ξ . 117) Therefore, k z0 / k z D cos θ plus a term in cos(φ 0 φ). The latter term gives zero on integration over the azimuthal angle. So, we have e E d f 0 (ε k ) D „ dk Zπ Z2π d θ sin θ σ(k, θ ) n s v (k) f 1 (ε k ) 0 Â d φ0 1 0 k z0 kz Ã Zπ D d θ sin θ (1 2π n s v (k) f 1 (ε k ) c o s θ ) σ(k, θ ) . 119) 0 we have f 1 (ε k ) D e E Λ(k) d f 0 (ε k ) .