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# THEORY OF SOLIDS PHYS 567

UW

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This 11 page Class Notes was uploaded by Dr. Simeon Wiza on Wednesday September 9, 2015. The Class Notes belongs to PHYS 567 at University of Washington taught by Staff in Fall. Since its upload, it has received 37 views. For similar materials see /class/192455/phys-567-university-of-washington in Physics 2 at University of Washington.

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Date Created: 09/09/15

Lecture 1 Lattice Vibrations phonons and thermal properties of dielectrics Consider a periodic array of atoms ions which contains q units per elementary cell The coordinate of an elementary cell is characterized by a vector n with integer components 111712 and 713 which correspond to translations along the primitive vectors 31732 and a3 a1 3233 y 0 The position rnj of the j th atom in n th elementary cell is given by expression rnj an unj a1n1 agng 13713 pj unj 1 Here an is the equilibrium position of j th atom in n th elementary cell While pj is its position in the cell at the origin Vector unj denotes a displacement of the atom n j from its equilibrium position Vibration modes Since the potential energy has an equilibrium at rnj an the classical Hamiltonian of the lattice as a function of mo menta pm and displacements unj could be written as a power series in displacements p2 397 u Z n n 11 n7 1 a a E EDDjjAjj n nunju j 1 g ZBgfnflKn 11711 I1Hu ju jugnjn The system is invariant with respect to discrete translation along the Bravais lattice Therefore the force matrices A B 7 etc depend not on n n or n themselves but on their difference n n n n7 etc The equation of motion follows from Eq 2 1 pm Z figgm n u E n7j nanjN Mjunj Pnr 4 Since the translation of whole crystal along an arbitrary distance d u gt u j da does not change its energy ByljJ nfl ugj ugnjn a 11139 and generates no force the force matrices A B obey the conditions 2 AZ w If 07 5 n7j Eggjln n n n 0 6 nntjN We forget for the time being about the higher powers of the displacements and restrict ourselves to harmonic ap proximation A certain asymmetry of Eqs 3 4 caused 3 by inequality of the masses Mj could be removed by the substitution 1 a a unjt 7 leM Dn n n7 J which leads to the equations of motion of the form 7123425 D m nuw 8 The Fourier transform of the real variable wnt wnj Z lw kleiim lkrquot 711139 Hamilkw wjk wjl k k lt9 leads to the equation of motion in the form of the eigenvalue problem for a BL X 3g matrix Dk wgw k Z D kw k 10 The solution of this eigenvalue problem gives 3g branches of the vibration spectrum w3k with 3g eigenvectors ejk 5 which obey ortho normalization and completeness condi tions Ze k 5 jak 5 SSAk k 11 J39 kZe k m k 5 51176 12 Therefore the displacement unj can be expressed through the contributions of the eigenmodes of the crystal 1 unj Ak8ejk8eiiwsktikrnAltk8 jk8eiw5ktiikrn 13 4 with partial complex amplitudes Aks For zero quasi momentums k 0 Eq 10 takes the form Aq n n 2 a J W w k 0ek 05 E e k 05 14 7 my MMjM 7 Multiplying Eq14 by AMj summing over j and using Eq 5 we arrive at the condition w k 0 ZxMjejk 07 8 0 15 j which clearly shows that either ZxMjejk 078 7t 0 J and then w3k 0 or Zquejk 08 0 j and possibly w3k y 0 This means that 3 branches of the lattice vibrations correspond to the displacement of the centre of mass of the elementary cell acoustic modes and their frequency vanishes as k gt 0 In the rest of the vi bration modes to the center of mass does not move optic modes and their frequencies do not vanish as k gt 0 Phonons The momentum pnj Man can be expressed through the amplitudes A3k as pnjt z39 I Mwsk Aksejkse i 5ktikrquot 00 15 5 Therefore the Hamiltonian 2 can be rewritten in a diag onal form HA A Z w k Ak5tks Altk5Ak5 17 ks while the quantum commutation relations lp j7u jl 415501 nM w 5 15 18 take the form A3k7A kl MAkM UkN 0 19 A3k k l 2hw5ltkgtAkms 20gt Eqs 1920 provoke the introduction the creation annihilation operators altkgt 2 Mk asltkgt Asltkgt7 21gt lak7 05le 03007 akl 07 03007 a k Ami3 mm h The displacement unj can now be written as we omit its dependence on time t uni k2 askejkseikrquot ak Jkse ikrquotl 7 23 while the Hamiltonian and its energy levels are H k altkgtasltkgt asltkgtaltkgt 7 24gt Eltnsltkgtgt k lt2nsltkgt 1 7 mm altkgtasltltmgt Thermodynamics The calculation of thermodynamic functions should begins with the partition function ei hw5k2 Z Z 2 exp l 5Enl l Em 26 ks nk5l ks 6 5 The mean values of different quantum operators could be found by summation over occupation numbers 113k For instance the mean occupation number is given by the Plank formula 713k Z1231 anexp 1 exp hwsk l 27 hw8k n This shows that the chemical potential of phonons is equal to zero and therefore the grand canonical thermodynamic potential 9 coincides with free energy h S k FT Tan 2 ks Tln lt1 EMMA 7 28gt where the rst term in the sum does not depend on temper ature and stands for contribution of zero point fluctuations while the second one gives the contribution of thermal uc tuations The entropy S and speci c heat CV are the rst and the second derivative of free energy airway 8T 8T 8T ks T2 6621015004 29 ova Tlt Using the phonon spectral density 9W Z5 w w3k7 30 ks we can rewrite Eq 29 in the form JrT 1de 90 W 31 Density of states is an additive value over partial contribu tions of different modes At low frequencies w lt w only three acoustic modes contribute to the density of states Their frequencies wk are linear with quasi momentum k but sound velocities strictly speaking depend on its direc tion even for highly symmetric cubic crystals lf neverthe less to neglect this elastic anisotropy the density of states takes the form gltwgtlwo gm mkmo i go 1 2 i go i 2W2 30 30 i w 3v07 Where c and ct are velocities of longitudinal and transverse sounds respectively wD N w is the so called Debye fre quency and 110 is the volume of the unit cell The integral of the density of states over frequencies is equal to total number of modes per unit cell This gives the sum rule 0 dwgw 33 WW czk 260 cm 32 8 Combination of Eq 31 Eq 32 and Eq 33 gives two asymptotes for speci c heat 3 Tth3 T lt th CT h 34 v0 q 7 th This presents the low temperature Debye law C olt T3 and the high temperature Dulong Petit law of constant speci c heat1 Thermal expansion Thermal expansion occurs due to anharrnonicity It is easy to see that on a simple model of a diatomic molecule in which the nuclei move in the potential schematically rep resented in Fig l as a function of distance R between the nuclei Near its minimum at R R0 the potential energy can be represented as M w 2 In the framework of classical statistical mechanics the mean distance of nuclei RT is given by the following ratio deRRdemMBURN deemM U l 1Pay attention to small numerical coefficient at correction to hightemperature mm m Rw gm R mn w asymptote This explains why the classical Dulong Petit law is valid not only at T gt Map but up to T 2 thT 2The sign of the coefficient 7 in the cubic term in this formula 7 gt 0 corre sponds to convexity of the curve UR at large Rt R0RT Figure 1 Potential energy of a diatomic molecule as a function of the distance R between the nuclei It is seen that the average separation increases with increasing the temperature Expanding the exponentials in Eq 36 in cubic terms in oscillation amplitudes R R0 we arrive at the expression 1 L 6T 2M 2w47 which couples the thermal expansion linear in temperature T to the sign of anharmonic constant 7 gt 1 Thermal expansion of crystals originates from the same ba sic source the dependence of the free energy F T V and the Gibbs potential ltIgt T on both temperature and vol ume pressure For instance at low temperature T lt th RT R0 R R04gt R0 T 37 127Wv60 127L6ji vohgwf hg 8p m mpagt vmm 1 vow gt 10 At high temperature T Z th d 1 V T V 3 TV 7 7 39 lt 7p m q odp lt gt Finally the coei cient of thermal expansion oz has the fol lowing asymptotes a i d 12Tglvohgwi 9VR T lt hWDS V 8T p dp 215 T 2 th 40gt Equation 40 shows that the thermal expansion is the re sult of the dependence of the phonon frequencies on pres sure and volume ie it is the result of anharmonicity At high temperature the thermally expansion does not depend on temperature While at low temperature it decreases as T3 and vanishes at zero temperature Therefore the depen dence of the coei cient of thermal expansion aT on tem perature is very similar to that of the speci c heat C If a single parameter 9 N th can be introduced as an energy scale in the temperature dependence of the Gibbs potential Mp mm W 41gt then Tb 92 Cap gr 42gt and nally am 1 d9 11 Therefore the temperature dependence of the ratio aC can indicate Whether the phonon system might be charac terized by a single parameter of the dimension of energy

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