10.4.1 Speed of Sound
The speed of sound provides some of our best information about the inner structure
of the earth. It is also relevant to the properties of many synthetic materials, such as
thermoelectrics used for direct conversion of heat to electricity. NRVS turns out to
be an excellent tool for obtaining the speed of sound on small samples and
(if necessary) under extreme conditions such as the high pressures and high temperatures relevant to the earth’s interior. Furthermore, in the acoustic modes involved in
sound propagation, the unit cell moves as a whole, and all of the atoms have the same
displacements. This also allows one to determine the speed of sound for a material
that does not have a natural NRVS isotope by using a dilute probe nucleus.
To relate the NRVS-derived PVDOS to the Debye model total DOS, we make the
approximation that the intermolecular and intramolecular modes are decoupled. In
the simplest case of an isotropic and/or polycrystalline solid, the Debye model yields
for the speed of sound v D :
D E
ð Þ ¼
e
m
m
E
2
2π 2 n ħ
3 v
3
D
ð10:23Þ
where e
m is the mass of the “probe” nucleus, m is the average atomic mass in the
material, and n is the number density of atoms. This can be rearranged to give an
expression for the Debye speed of sound v D :
v D ¼
e
m
m
E
2
2π 2 n ħ
3 D E
ð Þ
! 1=3
ð10:24Þ
or in terms of wave numbers:
v D ¼
4π m X c
3
ρ
n
2
D n
ð Þ
! 1=3
ð10:25Þ
Although one could in principle use Eq. 10.24 or Eq. 10.25 at a single energy to
derive the speed of sound, in practice it is better to use a range of data to check for
consistency. One approach is to fit the low-energy portion of the spectrum with a
parabola according to Eq. 10.23 and use the region over which there is a good match.
In the same vein, one can make a plot of D(E) vs. E
2 and extract ν D from the slope.
Finally, one can even plot D(E)/E
2 vs. E and use the range over which this appears
constant. In any case, it is important to only use the range of data over which the
density of states D(E) exhibits Debye-like behavior (Fig. 10.10 and Table 10.3).
270
10 Nuclear Resonaynce Vibrational Spectroscopy
The speed of sound provides some of our best information about the inner structure
of the earth. It is also relevant to the properties of many synthetic materials, such as
thermoelectrics used for direct conversion of heat to electricity. NRVS turns out to
be an excellent tool for obtaining the speed of sound on small samples and
(if necessary) under extreme conditions such as the high pressures and high temperatures relevant to the earth’s interior. Furthermore, in the acoustic modes involved in
sound propagation, the unit cell moves as a whole, and all of the atoms have the same
displacements. This also allows one to determine the speed of sound for a material
that does not have a natural NRVS isotope by using a dilute probe nucleus.
To relate the NRVS-derived PVDOS to the Debye model total DOS, we make the
approximation that the intermolecular and intramolecular modes are decoupled. In
the simplest case of an isotropic and/or polycrystalline solid, the Debye model yields
for the speed of sound v D :
D E
ð Þ ¼
e
m
m
E
2
2π 2 n ħ
3 v
3
D
ð10:23Þ
where e
m is the mass of the “probe” nucleus, m is the average atomic mass in the
material, and n is the number density of atoms. This can be rearranged to give an
expression for the Debye speed of sound v D :
v D ¼
e
m
m
E
2
2π 2 n ħ
3 D E
ð Þ
! 1=3
ð10:24Þ
or in terms of wave numbers:
v D ¼
4π m X c
3
ρ
n
2
D n
ð Þ
! 1=3
ð10:25Þ
Although one could in principle use Eq. 10.24 or Eq. 10.25 at a single energy to
derive the speed of sound, in practice it is better to use a range of data to check for
consistency. One approach is to fit the low-energy portion of the spectrum with a
parabola according to Eq. 10.23 and use the region over which there is a good match.
In the same vein, one can make a plot of D(E) vs. E
2 and extract ν D from the slope.
Finally, one can even plot D(E)/E
2 vs. E and use the range over which this appears
constant. In any case, it is important to only use the range of data over which the
density of states D(E) exhibits Debye-like behavior (Fig. 10.10 and Table 10.3).
270
10 Nuclear Resonaynce Vibrational Spectroscopy