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66 Physics Formulary by ir. J.C.A. Wevers 2. J< 0 : S i and S j become antiparallel: the material is an antiferromagnet. Heisenberg’s theory predicts quantized spin waves: magnons. Starting from a model with only nearest neigh- bouring atoms interacting one can write: U = - 2 J ± S p · ( ± S p - 1 + ± S p +1 ) ± μ p · ± B p with ± B p = - 2 J B ( ± S p - 1 + ± S p +1 ) The equation of motion for the magnons becomes: d ± S dt = 2 J ¯ h ± S p × ( ± S p - 1 + ± S p +1 ) From here the treatment is analogous to phonons: postulate traveling waves of the type ± S p = ± u exp( i ( pka - ω t )) . This results in a system of linear equations with solution: ¯ h ω =4 JS (1 - cos( ka )) 12.5 Free electron Fermi gas 12.5.1 Thermal heat capacity The solution with period L of the one-dimensional Schr¨odinger equation is: ψ n ( x )= A sin(2 π x/ λ n ) with n λ n =2 L . From this follows E = ¯ h 2 2 m ± n π L ² 2 In a linear lattice the only important quantum numbers are n and m s . The Fermi level is the uppermost ±lled
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