Lattice vibration in solid state physics

 Every atom has tendency to vibrate about their mean position at temperature above zero Calvin. 

What is lattice vibration? 

        The mean position is referred as the Bravais lattice site. These vibrations occur due to interatomic interaction. As the atoms are present at lattice site, so these vibrations are called lattice vibration.   

Lattice vibration in solid state physics

         The lattice vibrations are the result of absorption of heat energy. A lattice is an array of atoms connected with each other by elastic spring. 

   Here all the atom present at lattice site are coupled and the motion of one single atom is shared by the neighbouring atoms. Thus the crystal vibrates on a whole .

      The dynamic external forces which are responsible for the vibrations obey Hook's law. It is assumed that the amplitude of the vibrations are small as compare to the interatomic spacing. 

   Harmonic approximation is valid for lattice vibrations. Thus, every atom present at lattice site is considered to be as harmonic oscillator.

     According to quantum theory, the energy E of an elastic mode of vibration of angular frequency when The mode is excited to a quantum number n is given as 

          E = (n+1/2)wh/2pi

Here w = angular frequency 

   Here h/w4pi is the zero point energy and 

nhw/2pi   represent the quanta of energy which is called phonon energy. 

          To understand the lattice vibration, let us revise interatomic force near the neighbourhood of equilibrium separation about which the atom vibrates. 

     The interatomic forces may be attractive or repulsive depending upon the separation between the two atoms .

Lattice vibration

    At normal distance ro ,the two atoms are placed at equilibrium space. Any increase or decrease in their separation, needs some energy. 

    An increase in their interatomic separation r leads to attractive force and similarly decrease in interatomic separation r leads to repulsive force. 

 For NaCl crystal the normal distance r is approximately 1 Armstrong which is of same order as the distance from the centre of the atoms to its outermost electron.

    As the Na and Cl atoms exist in a regular array, when we try to deform the solid by external forces, the strong interatomic forces between the atoms resist compression or extension. 

     A regular lattice harmonic forces between the atom and normal modes of vibration are called lattice wave

       Lattice waves have frequency from low value to high values of the order 10 Hz . The wavelength corresponding to high frequencies are comparable to the interatomic spacing. 

  Due to short wavelengths the motion of the neighbouring atom is uncontrolled, with each atom vibrating about mean position in three dimensions with average vibrational energy. 

      The electrical resistance in conducting solids arises due to the intersection of lattice vibration with free particle. 

    The theories of lattice vibration are associated with specific Heat of solids. These vibrations influence several physical properties like - thermal expansion, elastic constant, thermal expansion, thermal conductivity, phonon spectra ect. 

      This is all about lattice vibration in solid state physics 

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