Thermodynamics, Statistical Physics, and Quantum Mechanics

228 SOLUTIONS

and

The condition implies that andtherefore

4.100 Two-Dimensional Debye Solid (Columbia,

Boston)

a) The number of normalmodes in the 2Dsolidwithin theinterval of a

wavevector may bewritten

In the 2D solidthere areonly two independentpolarizations of theexcita-
tions, one longitudinaland one transverse.Therefore,

where is the average velocity of sound. To find the Debye frequency

we use the standard assumption that the integralof (S.4.100.2)from 0 to

a certain cut-off frequency is equal to the total number of vibrational

modes; i.e.,

Therefore,

We can express through

Then (S.4.100.2) becomes