bei48482_FM

Einstein’s Formula

In 1907 Einstein discerned that the basic flaw in the derivation of Eq. (9.51) lies in

the figure of kTfor the average energy per oscillator in a solid. This flaw is the same

as that responsible for the incorrect Rayleigh-Jeans formula for blackbody radiation.

According to Einstein, the probability f( )that an oscillator have the frequency is

given by Eq. (9.39), f( ) 1 (ehvkT1). Hence the average energy for an oscillator

whose frequency of vibration is is

h f( ) (9.52)

and not kT. The total internal energy of a kilomole of a solid therefore becomes

E 3 N 0

  (9.53)

and its molar specific heat is

cV

V

 3 R 

2

(9.54)

We can see at once that this approach is on the right track. At high temperatures,

h kT, and

eh^ kT 1 

since ex 1 x

Hence Eq. (9.52) becomes^  h (h kT)kT, which leads to cV 3 R, the

Dulong-Petit value, as it should. At high temperatures the spacing h between

possible energies is small relative to kT, so is effectively continuous and classical

physics holds.

As the temperature decreases, the value of cVgiven by Eq. (9.54) decreases. The

reason for the change from classical behavior is that now the spacing between possi-

ble energies is becoming large relative to kT, which inhibits the possession of energies

above the zero-point energy. The natural frequency for a particular solid can be

determined by comparing Eq. (9.54) with an empirical curve of its cVversus T. The

result in the case of aluminum is 6.4  1012 Hz, which agrees with estimates

made in other ways, for instance on the basis of elastic moduli.

Why is it that the zero-point energy of a harmonic oscillator does not enter this analy-

sis? As we recall, the permitted energies of a harmonic oscillator are (n ^12 )h , n0,

1, 2,.... The ground state of each oscillator in a solid is therefore  0 ^12 h , the zero-

point value, and not  0 0. But the zero-point energy merely adds a constant,

temperature-independent term of  0 (3N 0 )(^12 h ) to the molar energy of a solid, and

this term vanishes when the partial derivative (ET)Vis taken to find cV.

x^3



3!

x^2



2!

h



kT

eh^ kT



(eh^ kT1)^2

h



kT

E



T

Einstein specific

heat formula

3 N 0 h



eh^ kT 1

Internal energy

of solid

h



eh^ kT 1

Average energy

per oscillator

322 Chapter Nine

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