Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 17.3-4

QUANTUM FLUIDS

Physics — Statistical Mechanics · adsorption-vibrational-grand-canonical

Problem

Suppose the adsorption model to be augmented by assuming that two adsorbed molecules on the same site interact in a vibrational mode of frequency

\omega

. Thus the energy of an empty site is zero, the energy of a singly occupied site is

\varepsilon_1

, and the energy of a doubly occupied site can take any of the values

\varepsilon_2 + n'\hbar\omega

, with

n' = 0, 1, 2, \ldots

Calculate

a

) The grand canonical partition sum

b

) The grand canonical potential

c

) The mean occupation number, as computed directly from (a)

d

) The mean occupation number, as computed directly from (b)

e

) The probability that the system is in the state with

n = 2

and

n' = 3

Answer: Denoting

\varepsilon_j - \mu

\varepsilon_j'

(b) \quad \Psi = \tilde{N}k_BT\ln\left(1 + e^{-\beta\varepsilon_1'} + \frac{e^{-\beta\varepsilon_2'}}{1 - e^{-\beta\hbar\omega}}\right)

(c,d) \quad \bar{n} = \frac{(1 - e^{-\beta\hbar\omega})e^{-\beta\varepsilon_1'} + 2e^{-\beta\varepsilon_2'}}{(1-e^{-\beta\hbar\omega})(1 + e^{-\beta\varepsilon_1'}) + e^{-\beta\varepsilon_2'}}

(e) \quad f_{2,3} = e^{-\beta(\varepsilon_2' + 2\hbar\omega)}/g