Physical Chemistry Third Edition

28.2 Crystal Vibrations 1163

The Einstein Crystal Model

This model assumes that all of the vibrational normal modes act like harmonic

oscillators with the same frequency. It is used to represent a crystal of a monatomic

substance such as a solidified inert gas, a metal, or a network covalent crystal such as

diamond. If a monatomic crystal hasNatoms its vibrational energy is given by

EvibU 0 +

(^3) ∑N− 6
i 1
hνvi≈U 0 +

∑^3 N

i 1

hνvi (28.2-1)

whereνis the vibrational frequency andviis the quantum number for normal mode

numberi. The energy of the crystal in its ground vibrational state, including the zero-

point energy, is denoted byU 0. In the second equality we ignore the difference between

3 N−6 and 3N, sinceNis a large number.

The canonical partition function for the entire crystal is a sum over all values of the

quantum numbers:

Z

∑∞

v 1  0

∑∞

v 2  0

···

∑∞

v 3 N 0

exp

⎛

⎜

⎜

⎜

⎜⎜

⎝

−U 0 −

∑^3 N

i 1

hνvi

kBT

⎞

⎟

⎟

⎟

⎟⎟

⎠

(28.2-2)

The normal modes are distinguishable from each other, so the sums are independent,

and this expression can be factored with one factor for each normal mode. The factors

are all identical after summation, and we write

Ze−U^0 /kBT

∏^3 N

i 1

⎛

⎝

∑∞

vi 0

e−hνvi/kBT

⎞

⎠e−U^0 /kBT

(∞

∑

v 0

e−hνv/kBT

) 3 N

e−U^0 /kBTz^3 N (28.2-3)

where each sum in the product is denoted byz, which is a partition function for a

single vibration. The normal modes are assumed to be distinguishable from each other,

so there is no need to divide byN!. The functionzis the same as the vibrational

partition function in Eq. (25.4-18):

z

∑∞

v 0

e−hνv/kBT

1

1 −e−hν/kBT

(28.2-4)

so that the canonical partition function is:

Ze−U^0 /kBT

(

1

1 −e−hν/kBT

) 3 N

(28.2-5)

ln(Z)−

U 0

kBT

+ 3 Nln(z)−

U 0

kBT

+ 3 Nln

(

1

1 −e−hν/kBT

)

−

U 0

kBT

− 3 Nln

(

1 −e−hν/kBT

)

(28.2-6)