Meaning of soulution of Central Equation: Nearly free electron model

[mhsd91]

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,

$(\lambda_k - \epsilon)C_k + \Sigma_G U_G C_{k-G} = 0$



where $\lambda_k = (\hbar^2 k^2) / (2m_e)$, $G$ is the set of possible reciprocal lattice vectors and $C_k$ is det Fourier coefficients corresponding to the solution of the Schrödinger eq.:

$\psi_k = \Sigma_k C_k e^{ikx}$.


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 $\epsilon_\pm = \lambda_k \pm U_0$ for some $k$. Then I am told the energy gap, $\epsilon_{gap} = \epsilon_+ - \epsilon_-$, between two energy bands for this $k$ 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 $C_k$, how? .. Assuming we know the periodicity of the potential and $k$.

 

[moj20062001]

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,

$(\lambda_k - \epsilon)C_k + \Sigma_G U_G C_{k-G} = 0$



where $\lambda_k = (\hbar^2 k^2) / (2m_e)$, $G$ is the set of possible reciprocal lattice vectors and $C_k$ is det Fourier coefficients corresponding to the solution of the Schrödinger eq.:

$\psi_k = \Sigma_k C_k e^{ikx}$.


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 $\epsilon_\pm = \lambda_k \pm U_0$ for some $k$. Then I am told the energy gap, $\epsilon_{gap} = \epsilon_+ - \epsilon_-$, between two energy bands for this $k$ 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 $C_k$, how? .. Assuming we know the periodicity of the potential and $k$.

yes we can;
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)-