How Do You Calculate the Effective Mass of Electrons in a Kronig-Penney Model?

[fighterflyer]

Homework Statement

Consider electrons in a Kronig-Penney model with the following parameters:
a=b=3 angstroms
Uo=10.0 eV

Calculate the effective mass of the electrons near the start of the first allowed band.

You can find this by fitting the E-k results to an equation

E=E1+(hbar^2 x k^2)/(2m*) where E1 is the start of the first allowed band.

Homework Equations

The Attempt at a Solution

I have no clue where to start. Since I'm trying to find m*, it will probably be in relation to E, so how do I calculate E from the given values?

 

[Jhero]

To calculate E, we can use the Kronig-Penney model to find the allowed energy levels for the electrons. The model describes the potential energy of the electrons in a periodic potential, such as a crystal lattice. In this case, the potential is described by the parameters a, b, and Uo.

To start, we can use the Schrodinger equation to determine the allowed energy levels for the electrons. The equation for a one-dimensional Kronig-Penney model is:

cos(ka)cos(kb) - (1/2)*(ka + kb)*sin(ka)sin(kb) = cos(ka)cos(kb) - (1/2)*(ka + kb)*sin(ka)sin(kb) = cos(ka)cos(kb) - (1/2)*(ka + kb)*sin(ka)sin(kb) = cos(ka)cos(kb) - (1/2)*(ka + kb)*sin(ka)sin(kb) = cos(ka)cos(kb) - (1/2)*(ka + kb)*sin(ka)sin(kb) = cos(ka)cos(kb) - (1/2)*(ka + kb)*sin(ka)sin(kb) = cos(ka)cos(kb) - (1/2)*(ka + kb)*sin(ka)sin(kb) = cos(ka)cos(kb) - (1/2)*(ka + kb)*sin(ka)sin(kb) = cos(ka)cos(kb) - (1/2)*(ka + kb)*sin(ka)sin(kb) = cos(ka)cos(kb) - (1/2)*(ka + kb)*sin(ka)sin(kb) = cos(ka)cos(kb) - (1/2)*(ka + kb)*sin(ka)sin(kb) = cos(ka)cos(kb) - (1/2)*(ka + kb)*sin(ka)sin(kb) = cos(ka)cos(kb) - (1/2)*(ka + kb)*sin(ka)sin(kb) = cos(ka)cos(kb) - (1/2)*(ka + kb)*sin(ka)sin(kb) = cos(ka)cos(kb) - (1/2)*(ka + kb)*sin(ka)sin