Crystal structure | Kronig Penney Model | Brillouin zones

Kronig – Penney Model

The model illustrates the behavious of an electron in a periodic potential. It assumes that the potential energy of an electron in a linear array of positive nuclei has a form of periodic square potentials. The potential consists of an infinite row of rectangular potential wells separated by barrier width b and space periodicity a.

The model potential is,

The Schroedinger wave equations for two regions are,

         We have assumed that E<Vo

Due to the behavior of electron in a periodic potential, the above wavefunctions must be of Bloch form, hence,

Substituting above value and putting

u₁(x) instead of Uk(x) in Region-I

u₂(x) instead of Uk(x) in Region-II

We get from equation 3 and 4

For the solution of equation 5, let

For the solution of equation 6

Hence the solutions of equation 5 and 6

Where A, B, C, D are constants.


To evaluate the above constants, we have the following boundary conditions,

For non-trivial solution, the determinant of coefficients of A,B,C,D must be equal to zero :

Solving the determinant, we get

Above equation is very complicated. To get more convenient equation Kronig and Permay considered the case when

Then equation 13 reduces to,


When P→0 , then from equation 14, we see that

1. Since Cos ka lies between +1 and -1, the L.H.S of equation 14 should take up only those values of αa for which its value lie between +1 and -1

For such values of αa, the wave solutions,

are allowed values and other values of αa are not allowed values.

The energy spectrum consists of an infinite number of allowed energy bands (shown in thick line) and separation between two allowed energy bands there are no energy levels, called forbidden bands (shown in dotted line).

The boundaries of allowed energy bands corresponds to   Coska = ±1

                                                         оr,    ка = nπ

                                                         or       k = nπ/a

2. When αa increases the term PSinαa/αa decreases so width of allorwed energy band increases and hence forbidden energy regions become narrower.

(3) With increasing binding energy of the electrion, p increases, hence width of allowed energy bands decreases. When  p→∞, width of the allowed energy bands become infinitely narrow and are independent of k i.e spectrum becomes a line spectrum.

When p→∞

The energy levels in this case are discrete and the electron is completely bound. This case applies to crystals where the electrons are tightly bound to their nuclei.

When P→0 (no barrier), the electron can be considered. to be moving freely through the potn. barrier well. This case applies to crystals where the electrons are almost free of their nuclei


For P = ∞, the energy spectrum is a line spectrum.

for, P=0, the energy spectrum is quasi-continuous.


Brillouin zones

When P→0

If N be the primitive cell in the crystal of length L,

Substituting in equation 4 , we get

According to Pauli’s exclusion principle, each wave function can be occupied by at the most of two electrons i.e there are 2N electrons in a band.

For spherical case,, number of electrons in the first Brillouin zone

                            = 2N x 2N x 2N

                        = 8N3

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