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Consider a two-dimensional (2D) lattice having N atoms with mass m. Assume that each atom interacts with only nearest neighbors with force constant k. Thus, take the dispersion relation as 4k sin V (). qa Wg = where a is the lattice constant. m a) In the long-wavelength limit, i.e., as q → 0, obtain the density of modes D(w) = dN/dw, that is the number of vibration modes per frequency interval dw. You should work in the Debye model. b) Calculate the total (internal) energy U of the lattice at high temperatures (kBT » hw). c) At high temperature limit, the average potential energy is equal to the average kinetic energy, and thus half the total energy. Find the mean square displacement V(r2) of an atom from its equilibrium position. Comment on the stability of 2D crystals.

Consider a two-dimensional (2D) lattice having N atoms with mass m. Assume that each atom interacts with only nearest neighbors with force constant k. Thus, take the dispersion relation as 4k sin V (). qa Wg = where a is the lattice constant. m a) In the long-wavelength limit, i.e., as q → 0, obtain the density of modes D(w) = dN/dw, that is the number of vibration modes per frequency interval dw. You should work in the Debye model. b) Calculate the total (internal) energy U of the lattice at high temperatures (kBT » hw). c) At high temperature limit, the average potential energy is equal to the average kinetic energy, and thus half the total energy. Find the mean square displacement V(r2) of an atom from its equilibrium position. Comment on the stability of 2D crystals.

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Consider a two-dimensional (2D) lattice having N atoms with mass m. Assume that each atom interacts with only nearest neighbors with force constant k. Thus, take the dispersion relation as 4k qa sin 2 where a is the lattice constant. m a) In the long-wavelength limit, i.e., as q → 0, obtain the density of modes D(w) = dN/dw, that is the number of vibration modes per frequency interval dw. You should work in the Debye model. b) Calculate the total (internal) energy U of the lattice at high temperatures (kgT > hw). c) At high temperature limit, the average potential energy is equal to the average kinetic energy, and thus half the total energy. Find the mean square displacement V(r²) of an atom from its equilibrium position. Comment on the stability of 2D crystals.
Transcribed Image Text:Consider a two-dimensional (2D) lattice having N atoms with mass m. Assume that each atom interacts with only nearest neighbors with force constant k. Thus, take the dispersion relation as 4k qa sin 2 where a is the lattice constant. m a) In the long-wavelength limit, i.e., as q → 0, obtain the density of modes D(w) = dN/dw, that is the number of vibration modes per frequency interval dw. You should work in the Debye model. b) Calculate the total (internal) energy U of the lattice at high temperatures (kgT > hw). c) At high temperature limit, the average potential energy is equal to the average kinetic energy, and thus half the total energy. Find the mean square displacement V(r²) of an atom from its equilibrium position. Comment on the stability of 2D crystals.
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