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2) Consider a linear chain of N +1 atoms, each of mass m. with harmonic forces (which might be modelled by N springs of spring constant K) acting between nearest neighbours. Taking the atoms at each end of the chain to be fixed, find the dispersion relation for longitudinal waves on the chain. For

2) Consider a linear chain of N +1 atoms, each of mass m. with harmonic forces (which might be modelled by N springs of spring constant K) acting between nearest neighbours. Taking the atoms at each end of the chain to be fixed, find the dispersion relation for longitudinal waves on the chain. For

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2) Consider a linear chain of N +1 atoms, each of mass m. with harmonic forces (which might be modelled by N springs of spring constant K) acting between nearest neighbours. Taking the atoms at each end of the chain to be fixed, find the dispersion relation for longitudinal waves on the chain. For the case when N is very large, find the density of states and an expression for the internal energy at temperature T. Evaluate the internal energy explicitly in the high- and low-temperature limits.
Transcribed Image Text:2) Consider a linear chain of N +1 atoms, each of mass m. with harmonic forces (which might be modelled by N springs of spring constant K) acting between nearest neighbours. Taking the atoms at each end of the chain to be fixed, find the dispersion relation for longitudinal waves on the chain. For the case when N is very large, find the density of states and an expression for the internal energy at temperature T. Evaluate the internal energy explicitly in the high- and low-temperature limits.
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