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The Kronig-Penney Model
The essential behaviour of electrons may be studied by periodic rectangular well in one dimensional which was first
addressed by Kronig Penney
It is assumed that when an electron is near the positive ion site, potential energy is taken as zero. Whereas outside the
well, that is in between two positive ions' potential energy is assumed to be Vo.
The wave function associated with an electron when it is in its first state is derived as follows
According to Schrodinger time independent equation,
According to Bloch theorem the potential V=0 then,
Let us consider
Then the equation transforms to ----- Eq - (1) The wave function associated with the electron when it is in the second state is derived as follows. Here the conditions are Let us consider Hence the equation transforms as
Similarly by substituting this in equation (2)
---------- Eq - (4)
By writing the general solutions for equations 3 and 4 we get four constants A, B, C and D. To know the values of these constants we apply the boundary conditions, as x=0, x=a, x=-b. In evaluation, the constants A, B, C and D vanished and the final equation obtained is ---------- Eq - (5) This equation cannot describe the motion of an electron with periodic motion. In order to express the relation in a more simplified form. So let us consider Vo tends to infinity then, b=0 and Then the initial conditions of the constants α^2 and β^2 and
Let us consider Where P is the measure of the potential barrier between the two potential wells. When we plot the graph by taking This equation satisfies only for those values of αa between -1 to 1
We also know that Equating both the above equations we get, This expression shows the behaviour of the electrons supports quantum-free electron theory
- If P tends to zero, Then We know that
and Equating both equations we get, After a simple calculation, we get This expression shows all the electrons are free to move without any constraints. This supports classical free electron theory.
