0.10 Sspd_chapter 1_part 12_quantum mechanical interpretation of  (Page 2/4)

Near the upper edge of the conduction band, electrons will suffer Bragg Reflection and this will be decided by the direction of orientation of the lattice plane from where the specular reflection takes place.

Lattice thermal vibration or/and lattice defects or/and lattice impurities cause the disorderliness and this disorderliness leads to change of direction of propagation or the change of direction of constructive interference. This change of direction is called scattering of electron. The change of direction is random and the distance over which this happens is statistically varying . The mean distance over which a straight line motion is maintained before scattering occurs is called the mean free path and the mean time taken to cover the mean free path is called mean-free time. The electron scattering within a real crystalline lattice which suffers from all defects and imperfections and which is at Room Temperature has been illustrated in Figure(1.69).

Figure 1.69. Scattering Phenomena of electron in a real crystalline lattice where lattice thermal vibration, lattice defect, lattice impurities and lattice boundaries are present.

This scattering of electron does not depend on the density of lattice centers but on the degree of disorderliness of the lattice centers and this degree of disorderliness depends on thermal vibration, lattice defects and lattice impurity density.

By increasing the temperature, the amplitude of lattice thermal vibrations is increased ;

During the growth of the crystal, lattice defects are increased;

During doping and diffusion, impurities are introduced in the crystal.

All these factors contribute towards disorderliness which in turn contribute towards the scattering of the conduction electrons. It is this scattering which creates resistance in a conducting solid. There are two parameters which describe the scattering phenomena:

Mean free path(<l>) and mean free time(τ).

As seen in Figure(1.69.b), at l ₁ , l ₂ , l ₃ , l ₄ ,……..distances the direction changes.

Therefore mean free path:

<l>=[∑ l n ]/N………………………………………… 1.100

If the time taken is t ₁ , t ₂ , t ₃ , t ₄ , …. along the randomly varying paths then the mean free time:

τ = [∑t n ]/N……………………………………………… 1.101

And<l>/τ = average thermal velocity= v thermal ………………………. 1.102

By Equipartition Law of Energy, every degree of freedom has an average thermal energy = (1/2)kT. Since conducting electron has three degrees of freedom(x, y, z) hence the total average thermal energy of a conducting electron is equal to (3/2)kT.

Therefore (1/2)m e * .v thermal 2 = (3/2)kT ……………………………. 1.103

Where k= Boltzman’s constant=1.38×10 ^-23 J/Kelvin = 8.62×10 ^-5 eV/Kelvin

And T is temperature at Kelvin scale. m _e ^* = effective mass of electron.

Temperature in Celsius scale is added to 273 to obtain temperature in Kelvin Scale. The zero of Kelvin scale occurs at -273°Celsius. At this temperature i.e. at 0 Kelvin the amplitude of thermal vibrations of lattice centers is zero and if there is no lattice defect and no doping then we have a perfect orderly lattice which will behave like a superconductor.