3.23 Sspd_chapter 1_part10_ continued_the theoretical basis of band  (Page 3/3)

Figure 1.42.a. The arrangement of two coupled springs.

Figure 1.42. The effect of two coupled springs and three coupled springs in splitting the individual energy levels and their natural frequencies of vibrations.

In Figure (1.42) we show two identical coupled spring loaded mass system and three identical coupled spring loaded mass system. The coupling is achieved through a massless and weak spring. Each individual spring loaded mass is constrained by frictionless guides. When the main spring loaded masses are uncoupled they exhibit identical frequencies of oscillation ν ₀ . In two spring loaded mass system coupling results in splitting of the oscillation frequencies into two closely spaced frequencies namely :

ν ₀ + ∆ν and ν ₀ -∆ν.

In three spring loaded coupled system the central frequency splits into three closely spaced frequencies and in N coupled system the central frequency splits into N closely spaced frequencies. If N is very large then this splitting results into a continuous band of frequencies.

If we consider the frequency of the loaded spring to be analogous to the ground energy state of the electron in a hydrogen atom then two interacting hydrogen atoms will lead to the discrete ground energy state splitting into two energy states and the same splits into three energy states when three atoms are mutually interacting. By corollary a discrete ground energy state splits into N closely spaced energy states when there is a linear array of N atoms close enough to be mutually interacting. This N closely spaced energy states appears as a band of permissible energy states.

Let us examine two isolated atoms each with a diameter of d Å. The symmetric matter wave of principal quantum number n =1 corresponds to E ₀ = h ² /(8md ² ).

The symmetric matter wave of each atom will combine to give a symmetric wave and anti–symmetric wave belonging to a combined potential well of dimension 2d Å. Hence the resultant symmetric wave will have energy E ₁ = h ² /(32md ² ) = E ₀ /4 and the anti-symmetric wave will have energy E ₂ = h ² /(8md ² )= E ₀ . Thus two atoms interaction result in the symmetric wave energy E ₀ splitting into (anti-symmetric)E ₀ and (symmetric)E ₀ /4 energy states.

Let us examine three isolated atoms. The lowest symmetric wave corresponds to E ₀ belonging to the principal quantum number n=1. When they are brought close enough to be interacting then the symmetric wave will combine to give three possible combinations: the lowest will be a symmetric wave of n=1, the anti-symmetric wave of n=2 and the next highest symmetric wave of n=3, all belonging to a combined potential well of dimension 3d Å. These three matter waves have energies :

E ₁ = h ² /(8m(3d) ² )= E ₀ /9; E ₂ = 4h ² /(8m(3d) ² ) =(4/9)E ₀ ; E ₃ =9 h ² /(8m(3d) ² )= E ₀ ;

Thus 3 atoms interaction cause the first energy state split into 3 energy states. By the same logic we can conclude that N atoms will interact to split the ground state E ₀ into N energy states where lowest energy level would be E ₀ /N and highest energy state would be E ₀ .

This simple model based on ‘electron in a potential well’ explains how a discrete energy state splits into N discrete energy states but this model completely fails to show as to how the continuous band results.

This model is too simplistic and does not correspond to the reality. We have assumed a flat bottomed potential well where as in fact it is an undulating surface due to periodic potential variation as shown in Fig(1.41.b). A more accurate model is Kronig Penny Model which is the topic of discussion in the next section