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Real Solids - more than one atom per unit cell • Molecular vibrations – Helpful to classify the different types of vibration • Stretches; bends; frustrated rotations etc. • Same is true of vibrations in solids – But to understand the possibilities need to look at a more complex model solid A linear chain with 2 atoms per unit cell • Must have 2NAvo vibrational modes per mole of substance (2R heat cap at hi T) • Vibrations divide into two classes – Atoms in unit cell move in-phase; known as an acoustic mode (b) – Atoms in unit cell move in antiphase; known as an optical mode (a) A three dimensional solid • Get longitudinal and transverse waves. The heat capacity • For a solid with p atoms per primitve unit cell, there will be (per mole of primitive cells) – 3NAvop normal modes – 3NAvo acoustic modes – 3NAvo (p-1) optical modes • And a hi T heat capacity of 3pR • Optical modes tend to be of a high frequency – Einstein model – Not excited at “low” T • Acoustic modes vary in frequency from 0 to max. – Debye model – Contribute even at low T freq Measurement of vibrations in solids • Infra-red absorption – Excites optical modes where these give range to a change in dipole moment • Inelastic neutron scattering – Use thermal neutrons – Undergo energy loss/gain when they are scattered from a material – Energy exchange represents the phonon energy – More favourable selection rules than IR absorption Thermal conduction • Metals conduct heat via the conduction electrons, but some insulators are even better. • Heat is carried by the phonons, which can travel unimpeded through a perfect crystal. • Thermal resistance arises from – Scattering by imperfections – Phonon-phonon collisions • According to simple theory depends on the – heat cap. (C) – phonon vel. (v) – phonon mean free path (l) • At low T, l= const=size of crystal. So K varies as T3 (debye) • At hi T, C= constant and l proportional to no.of phonons ie 1/T • Diamond is a very good thermal conductor because of a. high sound velocity. b. high Debye T K 1 3 Cv The electronic heat capacity • Peculiar observation in metals – Electrical conduction “a free electron gas” – Heat capacity - very small electronic heat capacity • Arises because electrons are too light to follow Maxwell-Boltzmann laws • Instead get a Fermi-Dirac distribution • At T=0, all the states up to Ef are full. • At T>0, only a small number of electrons close to Ef can be excited. only a fract ion of electrons T/T f take up energy U thermal T 3 NkT (classical energy fraction) 2 T f C elec v dU 3RT T (exact = 12 2 R ) dT Tf Tf •Tf=Ef/k=20,000 K typically. •So at room T, Celec is about 0.01 of the expected classical value • At low T, lattice vibrations are small enough to see the electronic term Cv AT 3 BT Cv 2 AT B T