Free electron model of atom and energy bands in solids

  • In atoms electrons orbits round the nucleus in their respective stable orbits.
  • Coulomb force due to nucleus on outermost electrons known as valence electrons is negligible.
  • These valence electrons are not bound with any particular atom and they are free to bind with any other atom in the crystal lattice.
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Superconductivity fact file

  • Bulk superconductor in a week magnetic field will act as a perfect diamagnet , with zero magnetic induction in the interior.
  • Nonmagnetic impurities have no marked effect on the SC transition temperature.
  • A sufficiently strong magnetic field will destroy SC. At critical temperature critical field is zero HC(TC)=0
  • Values of HC are always low for type I superconductors.
  • For a given HC the area under magnetization curve is same for type II SC as for type I SC.
  • In all SC entropy decreases markedly on cooling below transition temperature.
  • Superconducting state is the more ordered state.
  • Contribution to the heat capacity in the SC state is an exponential form with an argument proportional to -1/T
  • In SC the important interaction is electron-electron interaction which orders the electrons in K space with respect to the fermi gas of the electrons.
  • The argument of the exponential factor in the electronic heat capacity of a SC is found to be -Eg/2kT
  • The transition in zero magnetic field from the superconducting state to the normal state is the second order phase transition, not involving any latent heat but discontinuity in heat capacity.
  • Energy gap decreases continuity to zero as the temperature is increased to transition temperature.
  • For photons of energy less than energy gap , the resistivity of the superconductor vanishes at absolute zero.
  • As the temperature is increased not only does the gap decreases , but the resistivity for photon with energy below the energy gap no longer vanishes except at zero frequency.

Comparison between insulators and conductors

(1) Insulators
  • Insulators have very wide forbidden energy gap nearly of the order of 5eV or more.
  • Because of this very high energy gap it becomes impossible for electrons present in valence band to cross the gap and reach to the conduction band and this makes electrical conduction a practical impossibility in insulators at room temperature.
  • However at very high temperatures or with very high voltage applied across the ends of the insulator , it may conduct and this is termed as breakdown of an insulator.
(2) Conductors

  • Conduction band and valence band overlaps in case of a conductor.
  • Value of forbidden energy gap is zero for conductors in other words it does not exists at all.
  • For conductors or metals , valence band energies are same as conduction band energies and an valence electron can very easily become conduction electron (or, free electron) without any supply of heat energy.
  • This is why metals contain large number of free electrons even at room temperature and are good conductor of electricity.

Continuous spectrum and characteristic X-Rays?

When energetic electrons bombard atoms in a metal target (for ex. tungsten) an electron may be ejected from innermost K-shell, the atom then is in exited state and is unstable. If an electron from L-shell now moves to vacancy in K-shell , the energy of atom is decreased and simultaneously there is emission of radiation. If E is the change in energy when electron moves from L-shell to K-shell then ,
E=hν where h is the plank's constant and ν is the frequency of radiation.
Thus , ν=E/h and for high energies , wavelength of the radiation is short and is of the order of 10-8cm for X-Rays.
When we study X-Rays from a target it is observed to be a continuous spectrum with intense lines. These intense lines depends on the metal used as target and these are called characteristic X-Rays. The continuous spectrum depends on applied potential difference , current flowing in the filament and atomic number of target.

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