- Please Do A And B If Possible D Would Also Be Nice 1 (132.76 KiB) Viewed 15 times
please do a and b. if possible, d would also be nice
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answerhappygod
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please do a and b. if possible, d would also be nice
please do a and b. if possible, d would also be nice
are classical. They can move only along the direction of the chain. Assume the periodic boundary condition (a ring). As we derived, the acoustic modes of this system consist of two branches w+ (g) and w_(q) characterised by the following dispersion relation: M (m+M\2 4 w² (9) = k (m +1²) ± sin²(gao/2) mM (7) mM where w is a frequency, and q is a wave vector. (You can optionally derive this relation, but it is not necessary.) In the quantum framework, Eq. (7) describes the dispersion relation of phonons which obey Bose statistics. We assume that the chain of atoms has a finite length L, so that the wave vectors q are allowed to have discrete values qn = 2mn/L, where n is an integer number. The specific heat is defined as dE C = (8) dT where E is the total energy of the system: E(qn) E = Σ (9) expBT E(gn) 1 The summation extends over all possible wave vectors in the first Brillouin zone. The energy of the phonons is E(gn) = ħw+(an). For numerical calculations, use the following parameters: m = 1, M = 5, ħ= 1, L = 100, 1, k1, kg = 1. ao a.) Obtain the specific heat for this system numerically, using the equations above (you can do it with the help of Wolfram Mathematica, NumPy, or any other numerical computation tools). b.) Formulate the Debye model for this system and, on its basis, calculate the specific heat numerically as a function of temperature in Debye model. In order to apply the Debye model, you should obtain two characteristic parameters, the Debye frequency and the speed of sound c entering model's dispersion relation = cq. The speed of sound should be obtained as c = lim Əw_(q) Əq (10) 9-0 2 For this system, the Debye frequency corresponds to a maximal wave vector equal to twice the maximal wave vector of the first Brillouin zone. (Explain why.) c.) Obtain analytical approximation for the low-temperature behaviour of the specific heat in part (b). d.) Plot the results of (a) and (b) for the temperature range T€ [0, 15] in the same figure for comparison.
are classical. They can move only along the direction of the chain. Assume the periodic boundary condition (a ring). As we derived, the acoustic modes of this system consist of two branches w+ (g) and w_(q) characterised by the following dispersion relation: M (m+M\2 4 w² (9) = k (m +1²) ± sin²(gao/2) mM (7) mM where w is a frequency, and q is a wave vector. (You can optionally derive this relation, but it is not necessary.) In the quantum framework, Eq. (7) describes the dispersion relation of phonons which obey Bose statistics. We assume that the chain of atoms has a finite length L, so that the wave vectors q are allowed to have discrete values qn = 2mn/L, where n is an integer number. The specific heat is defined as dE C = (8) dT where E is the total energy of the system: E(qn) E = Σ (9) expBT E(gn) 1 The summation extends over all possible wave vectors in the first Brillouin zone. The energy of the phonons is E(gn) = ħw+(an). For numerical calculations, use the following parameters: m = 1, M = 5, ħ= 1, L = 100, 1, k1, kg = 1. ao a.) Obtain the specific heat for this system numerically, using the equations above (you can do it with the help of Wolfram Mathematica, NumPy, or any other numerical computation tools). b.) Formulate the Debye model for this system and, on its basis, calculate the specific heat numerically as a function of temperature in Debye model. In order to apply the Debye model, you should obtain two characteristic parameters, the Debye frequency and the speed of sound c entering model's dispersion relation = cq. The speed of sound should be obtained as c = lim Əw_(q) Əq (10) 9-0 2 For this system, the Debye frequency corresponds to a maximal wave vector equal to twice the maximal wave vector of the first Brillouin zone. (Explain why.) c.) Obtain analytical approximation for the low-temperature behaviour of the specific heat in part (b). d.) Plot the results of (a) and (b) for the temperature range T€ [0, 15] in the same figure for comparison.