5.5 Related Issues and Examples
105
To make such analyses on non-simple solids such as molecular solids, the assumption of the number of degrees of freedom is necessary. If three is assumed, the limiting value to the absolute zero, lim T →0 Θ D (T ), reflects the (averaged) sound velocity,
which can be compared with simple solids. On the other hand, the assumption of 6 (3
translational and 3 rotational degrees of freedom) yields the measure of the highest
frequency of the lattice vibration. If the interest is solely on the demonstration of
anomalous behavior, the number of degrees of freedom may be chosen arbitrarily,
as in the example shown above.
5.5.2 Debye–Waller Factor
Standard diffraction crystallography practically utilizes the so-called Debye–Waller
factor to describe the intensity of diffracted radiation (X-ray, neutron, or electron)
relative to the ideal intensity expected for the complete crystal. The factor has originally been introduced to describe the effect of thermal motion [15, 16]. Nowadays,
however, the factor is experimentally obtained as the parameter that reflects the random distribution of an atom around the ideal (i.e., averaged) position. When all
atoms independently exhibit isotropic displacement around the averaged position,
the Debye–Waller factor has the following form:
exp
−
1
2
|q|
2
σ
2
.
(5.107)
Here, q is the scattering vector, and σ
2 is the mean squared displacement. Although
the distribution of atomic displacement may be anisotropic in reality, the factor
reflects the mean squared displacement. In practice, U = σ
2 and B = 8π
2 U are
reported as the temperature parameter or mean squared displacement parameter.
In the quantum mechanical treatment of a harmonic oscillator with a mass m and
the potential energy function
1
2
kx
2 , the expectation value of the squared displacement
x
2
n associated with the nth energy-eigenstate is given by
x
2
n =
ω
k
n +
1
2
(n = 0, 1, 2 . . .),
(5.108)
where ω =
√
k/m is the angular frequency. The above expression is equivalent to the
energy of the nth state E n divided by k. That is, x
2
n = E n /k. This equality indicates
that the equipartition of the total energy to the kinetic and potential energies holds
for each energy-eigenstate. Thus, the same equality holds for the thermal average:
x
2
=
1
k
E
105
To make such analyses on non-simple solids such as molecular solids, the assumption of the number of degrees of freedom is necessary. If three is assumed, the limiting value to the absolute zero, lim T →0 Θ D (T ), reflects the (averaged) sound velocity,
which can be compared with simple solids. On the other hand, the assumption of 6 (3
translational and 3 rotational degrees of freedom) yields the measure of the highest
frequency of the lattice vibration. If the interest is solely on the demonstration of
anomalous behavior, the number of degrees of freedom may be chosen arbitrarily,
as in the example shown above.
5.5.2 Debye–Waller Factor
Standard diffraction crystallography practically utilizes the so-called Debye–Waller
factor to describe the intensity of diffracted radiation (X-ray, neutron, or electron)
relative to the ideal intensity expected for the complete crystal. The factor has originally been introduced to describe the effect of thermal motion [15, 16]. Nowadays,
however, the factor is experimentally obtained as the parameter that reflects the random distribution of an atom around the ideal (i.e., averaged) position. When all
atoms independently exhibit isotropic displacement around the averaged position,
the Debye–Waller factor has the following form:
exp
−
1
2
|q|
2
σ
2
.
(5.107)
Here, q is the scattering vector, and σ
2 is the mean squared displacement. Although
the distribution of atomic displacement may be anisotropic in reality, the factor
reflects the mean squared displacement. In practice, U = σ
2 and B = 8π
2 U are
reported as the temperature parameter or mean squared displacement parameter.
In the quantum mechanical treatment of a harmonic oscillator with a mass m and
the potential energy function
1
2
kx
2 , the expectation value of the squared displacement
x
2
n associated with the nth energy-eigenstate is given by
x
2
n =
ω
k
n +
1
2
(n = 0, 1, 2 . . .),
(5.108)
where ω =
√
k/m is the angular frequency. The above expression is equivalent to the
energy of the nth state E n divided by k. That is, x
2
n = E n /k. This equality indicates
that the equipartition of the total energy to the kinetic and potential energies holds
for each energy-eigenstate. Thus, the same equality holds for the thermal average:
x
2
=
1
k
E