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16 1 16.1. Classical (Atomistic) Theory of Heat Capacity Average energy of one-dimensional harmonic oscillator E kBT Average energy per atom ( three-dimensional harmonic oscillator ) E 3kBT It was shown that average kinetic energy of a particle is expressed as Ekin 2 3 k BT 2 (from previous chapter) Average potential energy of vibrating atom has the same average magnitude as the kinetic energy. So the total energy of vibrating atom is expressed as Etot=Epot.+Ekin. Namely, the total internal energy per mole E 3N0 kBT Finally, the molar heat capacity is E Cv 3N 0 k B T v Inserting numerical values for N0 and kB Cv 25 J/mol K 5.98 cal/mol K Simple model can readily explain the experimentally observed heat capacity except that the calculated heat capacity turned out to be temperature independent - need QM explanation. 3 16.2. Quantum Mechanical Considerations-The Phonon 16.2.1 Einstein Model The energy of the i th energy level of a harmonic oscillator 1 i i hv 2 h = Planck's constant of action v = frequency of harmonic oscillator In 1907, Einstein postulated that the energies of the above-mentioned classical oscillators should be quantized, i.e., he postulated that only certain vibrational modes should be allowed – these lattice vibration quanta were called phonons. Phonon waves propagate through the crystal with the speed of sound. They are elastic waves vibrating in a longitudinal and/or in a transverse mode. 4 The allowed energies of a single oscillator, En n The average number of phonons at a given temperature is given as 1 N ph exp( k BT (Bose-Einstein statistics) ) 1 Eosc N ph exp( Boson: any particle with an integral number of spins. Photon and phonon are boson particles k BT E 3 N 0 Eosc 3N 0 exp( ) 1 k BT The average energy of an isolated oscillator. ) 1 exp( ) 2 k BT E Cv ( )v 3 N 0 k B (Einstein model) T k BT (exp 1) 2 k BT 5 Heat capacity at a constant volume U hv / kT 2 hv hv / kT Cv 3 nhv ( e 1) e kT 2 T v 2 e hv / kT hv 3nk hv / kT 1) 2 kT (e For large temperatures, e x 1 x applies, which yields Cv 3nk - follows classical Dulong-Petit value At low temperatures, Einstein theory predicts an exponential reduction which is conradictory with the experimental results which show by T 3 Einstein temperature is hv E k So, e E / T E cv 3R E / T 1) 2 T (e 6 2 16.2.2 Debye Model We now refine the Einstein model by taking into account that the atoms in a crystal interact with each other oscillators are thought to vibrate interdependently. Einstein model considered only one frequency of vibration D. When interactions between the atoms occur, many more frequencies are thought to exist, which range from about the Einstein frequency down to the acoustical modes of oscillation. Debye modified the Einstein equation by replacing the 3N0 oscillators of a single frequency with the number of modes in a frequency interval, d, and by summing up over all allowed frequencies. Applying periodic boundary conditions over N3 primitive cells within a cube of side length L, vs k (velocity of sound is constant: = s ) exp[i ( K x x K y y K z z )] k exp[i ( K x ( x L) K y ( y L) K z ( z L)] whence, L 3 4 3 V3 N ( ) k 2 3 2 3 6 vs dN V2 2 4 D( ) K x , K y , K z 0; ; ; ... d 2 2 vs 3 L L Therfore, there is one allowed value of K per unit volume of 3 V L 2 8 3 7 16.2.2 Debye Model Total energy of vibration for the solid Eosc = the average energy of one oscillator E Eosc D( )d E 3V 2 2 3 s e / kT 1 1) kT 3V 2 D( ) 2 3 (in 3-D) 2 s (exp 3 D 0 = d Heat capacity at a constant volume E 3V 2 Cv ( ) v 2 3 2 T 2 s kT D 0 4e (e / kT / kT 1) 2 d Or indicating with Debye temperature D Cvph 8 T 9kN 0 D 3 D /T 0 x 4e x d x 2 (e 1) 12 4 T Cv nk ( )3 5 D x kT hv , kT D D k hvD k D debye frequency 16.3. Electronic Contribution to The Heat Capacity Thermal energy at given temperature Ekin 3 3 kBTdN k BTN ( EF )kBT 2 2 (consider electrons as non-interacting particles) The Heat capacity of the electrons E 2 Cvel 3 k BTN ( EF ) T v Population density at E For E < EF 3N * electrons N ( EF ) 2 EF J N * number of eletrons which have an energy equal to or smaller than EF * 9 N *kB2T E 2 3N C 3kBT 2 EF 2 EF T v el v F V 2m 3 2 1 2 ( ) E (6.8) 2 2 2 * 2 2 2 N 3 EF (3 ) (6.11) V 2m N (E) J K So far, we assumed that the thermally excited electrons behave like a classical gas. 9 In reality, the excited electrons must obey the Pauli principle. So, C el v 2 N *kB2T 2 = EF 2 2 N *kB T TF EF k BTF If we assume a monovalent metal in which we can reasonably assume one free electron per atom, N* can be equated to the number of atoms per mole. C el v 2 N0 kB2T 2 EF = 2 2 N 0 kB T TF J K mol Below the Debye temperature, the heat capacity of metals is sum of electron and phonon contributions. Cvtot Cvel Cvph T T 3 Cvtot T 2 T 3k B2 N ( EF ) 1 9kn D 10 3 D /T 0 x 4e x d x 2 (e 1) Thermal effective mass mth* (obs.) m0 (calc.) 11