Physical Chemistry Third Edition

1166 28 The Structure of Solids, Liquids, and Polymers

For fairly large values of the integers, the number of sets of integers such thatslies

betweensands+dsis approximately equal to

Number of waves in the interval (s,s+ds)

πs^2

2

ds (28.2-14)

In a solid there can be longitudinal waves in which the oscillation is parallel to the

direction of the wave, and transverse waves in which the oscillation is perpendicular

to the direction of the wave. The result in Eq. (28.2-14) must be multiplied by a factor

of 3 for the two transverse waves (polarized at right angles to each other) and one

longitudinal wave. If the speed of sound depends on the frequency and if the longitudinal

and transverse waves do not move at the same speed we regard the constant value of

cas an average speed that can be different for each solid substance.

EXAMPLE28.3

Show that Eq. (28.2-14) is correct.

Solution

We construct a mathematical space in whichsx,sy, andszare plotted on three Cartesian axes.

The number of points inside a given region of this space corresponding to sets of integral

values is nearly equal to the volume of that region, since there is one such point per unit

volume. Consider a spherical shell of thicknessds. Only one octant of the coordinate system

is included, since all of the integers are positive. The volume of a spherical shell of radiuss

and thicknessdsis 4πs^2 ds. The volume in the first octant is 1/8 of this, or (πs^2 /2)ds.

Equations (28.2-13) and (28.2-14) can be combined:

(Number of waves inds)

3 π

2

(

2 Lν

c

) 2

ds



3 π

2

(

2 Lν

c

) 2

2 L

c

dν 12 π

V

c^3

ν^2 dνg(ν)dν (28.2-15)

whereV is the volume of the crystal. The functiong(ν) is called thefrequency

distribution:

g(ν) 12 π

V

c^3

ν^2 (28.2-16)

In a crystal ofNatoms the total number of vibrational modes is equal to 3N−6, which

we approximate by 3NsinceNis a large number. Debye chose a maximum frequency

νDsuch that he had the correct number of vibrational normal modes:

3 N

∫νD

0

g(ν)dν

12 πV

c^3

∫νD

0

ν^2 dν

4 πV ν^3 D

c^3

(28.2-17)

ν^3 D

3 Nc^3

4 πV

(28.2-18)

Debye’s frequency distribution of vibrational frequencies for copper is shown in

Figure 28.8 along with the experimental distribution.

0

Density of states

g(



)

/10^13 radians s^21

12345

D

Figure 28.8 The Debye Distribution

of Frequencies, with the Experimen-

tal Distribution of Frequencies for

Copper.The distribution is shown as a

function ofω 2 πν. From J. S. Blake-

more,SolidState Physics, 2nd ed.,

W. B. Saunders, Philadelphia, 1974,

p. 126.