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LATTICE VIBRATION - Coggle Diagram

- - - - Very heavy
        
        Forces between atoms are too large
    - - Atoms are not very heavy
        
        There is vibration around the equilibrium lattice position
        
        The static lattice models are valid only at zero temperatures
        
        The forces between atoms is not an infinite force
        
        Temperature>0, each atom has thermal energy
    - - Atoms or ions(physically linked together through bonding system), the vibrational is in general not isolated
        
        The basis of each lattice point & constituent atoms of each basis vibrate to the fixed lattice and with each other
        
        when wave propagate along the crystal, entire planes of atoms move phase with displacement either parallel or perpendicular
  - - - Amplitude of the lattice vibration is small.
        
        At higher amplitude some unharmonic effects occur.
        
        its motion can be calculated by a generalization of the method used to analyse a simple harmonic oscillator
      - In the linear region (the region of elastic deformation)
        
        Hooke’s Law
      - There are some effects of nonlinearity or ‘anharmonicity’ for larger atomic displacements.
        
        Anharmonic effects are important for interactions between phonons and photons.
  - - - is amount of heat (celsius or kelvin unit)
      - Heat capacity at constant Volume (CV)
      - Heat capacity at constant Pressure (CP)
  - - - The high temperature behavior is good(T=0)
        
        Low temperature is not very good
      - Einstein assumed that all the oscillators oscillate with a common frequency.
        
        Average thermal energy (1D)
        
        Cv
      - For 3D
    - - Same as Einstein model, recovers the Dulong-Petit law at high temperatures
        
        Max vibration frequency is determined by Debye frequency
      - At low temperature
    - - Boundary conditions are periodic boundary which impose the restriction that a wave function must be periodic on a certain Bravais lattice.
        
        Often applied in solid state physics to model an ideal crystal.
- - - - Total force = Force Right - Force Left)
        OR
        Eqn of motion of nth atom
    - - There is a minimum possible wavelength, given by twice the equilibrium separation a between atoms.
        
        Any wavelength<2*a* mapped onto a wavelength>2a
        
        Shorter wavelengths are physically identical with some longer wavelength (configuration of the systems look the same)
- - - - Two different atoms, two different mass
        
        Model with two atoms per unit cell
        
        Separation space = (1/2)a
      - The model is complicated due to 2 different type of atoms
        
        Eqn of motion same as monoatomic lattice
      - Linear Dispersion
        
        Dispersion relation have to branches (corresponds to +ve and -ve signs of dispersion relation)
        
        Optical Branch
        
        Upper branch is due to the +ve sign of the root
        
        Is a higher energy vibration (the frequency is higher, and certain amount of energy needed to excite this mode)
        
        Acoustical Branch
        
        Lower branch is due to the -ve sign of the root
        
        It gives rise to long wavelength vibrations (Speed of sound)
      - Transverse mode
        
        Amplitude of vibration is strongly exaggerated
        
        Optical Branch
        
        k=0, Max frequency
        
        K =(pi/a), min frequency
        
        So,
        
        Acoustical Branch
        
        k= (pi/a), mas frequency
- - - - If phonon interacts inelastically:
        
        Phonon is created
        
        Phonon is absorbed
    - - 1. Piezoelectric
        
        Electric field is applied to a piezoelectric field, it experiences strain
        
        EM generate an oscillating electric field
        
        Oscillating electric field & piezoelectric transducer at the same frequency
        
        Emits Phonon
        
        Inefficient conversion of phonon to phonon
        
        Frequency > 10GHz Not suitable
      - 2. Thermal Excitation
        
        Current flow causing e- temperature rise
        
        Hot e- releases energy to the surrounding (by emitting phonon and phonon)
        
        Probability to emit phonon>phonon
        
        Above the threshold frequency, only photons are produced
      - 3.Electron Tunnelling
        
        Thin layer of insulator is placed between two thin layers of metal (to form barrier for electron)
        
        e- can tunnel through the barrier at certain energy
        
        e- speed increases with additional kinetic energy eV
        
        addtional energy released in the form of phonon emission
        
        Phonon is emitted by the hot e- that losses energy when returning to equilibrium