An impurity atom sitting at a fixed site in a crystal has three non-degenerate states

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Statistical foundations and the equilibrium ensembles · Statistical foundations and the equilibrium ensembles

Thermal Physics, Ch The Boltzmann factor, the canonical ensemble and the partition function

Problem

An impurity atom sitting at a fixed site in a crystal has three non-degenerate states, of energies

0

\varepsilon

and

3\varepsilon

, with

\varepsilon = 6.00\times10^{-21}\,\mathrm{J}

. The crystal acts as a reservoir at

T = 300\,\mathrm{K}

, and the coupling is weak, so the atom's energy levels are those listed. Take

k_B = 1.381\times10^{-23}\,\mathrm{J\,K^{-1}}

and give numerical answers to three significant figures. ((a)) Evaluate the partition function

Z

of one impurity atom. ((b)) Find the probability of each of the three states. ((c)) Find the mean energy

U

of one impurity atom, in joules. ((d)) A sample contains

N = 5.0\times10^{20}

such impurity atoms, each at its own fixed site and independent of all the others. Find the total mean energy of the impurities, in joules.

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