Meaning of soulution of Central Equation: Nearly free electron modelby mhsd91 Tags: central equation 

#1
May2313, 04:35 AM

P: 12

Considering the Nearly Free Electron model of solids, where we assume the valence electrons of some one dimensional(!) solid to move in a weak, periodic (with respect to the solids lattice constant) potensial.
We may derive (which I assume you are familiare with, and will not do here) the central equation as an algebraic reformulation of the time independent Schrödinger eq. corresponding to the model/potential at hand, [itex] (\lambda_k  \epsilon)C_k + \Sigma_G U_G C_{kG} = 0 [/itex] where [itex] \lambda_k = (\hbar^2 k^2) / (2m_e) [/itex], [itex] G [/itex] is the set of possible reciprocal lattice vectors and [itex] C_k [/itex] is det fourier coefficients corresponding to the solution of the Schrödinger eq.: [itex] \psi_k = \Sigma_k C_k e^{ikx} [/itex]. My problem is that I do not understand what exactly we do find if we solve the central equation. Say for instance I solve it and find the energy [itex] \epsilon_\pm = \lambda_k \pm U_0 [/itex] for some [itex]k[/itex]. Then I am told the energy gap, [itex] \epsilon_{gap} = \epsilon_+  \epsilon_ [/itex], between two energy bands for this [itex]k[/itex] at hand. Please (dis)confirm!? ... and then WHICH two bands are this gap between? (If that makes sense). And is it possible to find values for [itex] C_k [/itex], how? .. Assuming we know the periodicity of the potential and [itex] k [/itex]. 



#2
Jun2513, 09:37 AM

P: 12

in this case C is equal to: +_sgn(U)C u can find the exact equation in,Solid State Physics By Ashcroft&Mermin.chapter9,equation (9.29) 


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