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Lattice Dynamics of Crystals Giuseppe Grosso, Giuseppe Pastori Parravicini, in Solid State Physics (Second Edition), 2014 Einstein Model In the Einstein model, the actual frequencies of the normal modes are replaced by a unique (average) frequency w (Einstein frequency). If N is the total number of atoms, Eq. (9.34) for the heat capacity at constant volume becomes C₁(T) = 3N The behavior of Cy ə ħwe aat hw/k₂T e -ħw /k₂T → e and Cv → 3N KB a Cv (T) = 3N k. -1 for for hw 2 e a Bk T hw (e e we/k₂T hw₂/kg-1) T 2 in the low and high temperature limits is k₂T « hw kgT » ħwe a B In the high temperature limit the Einstein model recovers the Dulong and Petit value 3N_k. In the low temperature limit, the Einstein model predicts for C₁(T) an exponentially vanishing behavior, contrary to the 7³ experimental law. The origin of this discrepancy is the presence in crystals of the phonon acoustic branches, which cannot be mimicked by a unique Einstein frequency, and actually need a more realistic description.

Lattice Dynamics of Crystals Giuseppe Grosso, Giuseppe Pastori Parravicini, in Solid State Physics (Second Edition), 2014 Einstein Model In the Einstein model, the actual frequencies of the normal modes are replaced by a unique (average) frequency w (Einstein frequency). If N is the total number of atoms, Eq. (9.34) for the heat capacity at constant volume becomes C₁(T) = 3N The behavior of Cy ə ħwe aat hw/k₂T e -ħw /k₂T → e and Cv → 3N KB a Cv (T) = 3N k. -1 for for hw 2 e a Bk T hw (e e we/k₂T hw₂/kg-1) T 2 in the low and high temperature limits is k₂T « hw kgT » ħwe a B In the high temperature limit the Einstein model recovers the Dulong and Petit value 3N_k. In the low temperature limit, the Einstein model predicts for C₁(T) an exponentially vanishing behavior, contrary to the 7³ experimental law. The origin of this discrepancy is the presence in crystals of the phonon acoustic branches, which cannot be mimicked by a unique Einstein frequency, and actually need a more realistic description.

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Please tell me the solution process why the value comes out at low temperature.

Lattice Dynamics of Crystals Giuseppe Grosso, Giuseppe Pastori Parravicini, in Solid State Physics (Second Edition), 2014 Einstein Model In the Einstein model, the actual frequencies of the normal modes are replaced by a unique (average) frequency w (Einstein frequency). If N is the total number of atoms, Eq. (9.34) for the heat capacity at constant volume becomes C₁₂(T) = Cv hw Ə aoT hw/kT e = 3N The behavior of → e -Cv (T) -ħw / KT and Cv → 3N kB a = 3N k. - 1 for for hw 2 e a Bk T ħw / kg T hw₂/kg-1) e 2* in the low and high temperature limits is k₂T « hw kgT » ħwe In the high temperature limit the Einstein model recovers the Dulong and Petit value 3N kg. In the low temperature limit, the Einstein model predicts for C₁(T) an exponentially vanishing a B behavior, contrary to the T experimental law. The origin of this discrepancy is the presence in crystals of the phonon acoustic branches, which cannot be mimicked by a unique Einstein frequency, and actually need a more realistic description.
Transcribed Image Text:Lattice Dynamics of Crystals Giuseppe Grosso, Giuseppe Pastori Parravicini, in Solid State Physics (Second Edition), 2014 Einstein Model In the Einstein model, the actual frequencies of the normal modes are replaced by a unique (average) frequency w (Einstein frequency). If N is the total number of atoms, Eq. (9.34) for the heat capacity at constant volume becomes C₁₂(T) = Cv hw Ə aoT hw/kT e = 3N The behavior of → e -Cv (T) -ħw / KT and Cv → 3N kB a = 3N k. - 1 for for hw 2 e a Bk T ħw / kg T hw₂/kg-1) e 2* in the low and high temperature limits is k₂T « hw kgT » ħwe In the high temperature limit the Einstein model recovers the Dulong and Petit value 3N kg. In the low temperature limit, the Einstein model predicts for C₁(T) an exponentially vanishing a B behavior, contrary to the T experimental law. The origin of this discrepancy is the presence in crystals of the phonon acoustic branches, which cannot be mimicked by a unique Einstein frequency, and actually need a more realistic description.
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