6.1 Melting in Reality
123
the fraction of the defect x D is related to the chemical potential of formation, Δ f μ =
Δ f h − T Δ f s as
x D = exp
−
Δ f μ
k B T
= exp
Δ f s
k B
exp
−
Δ f h
k B T
.
(6.1)
This relation can be confirmed in some ways. One being the most intuitive is the
difference between the volume expansivities determined macroscopically (dilatometry) and microscopically (e.g., by X-ray diffraction). While the former is affected by
the formation of defects such as lattice vacancies or interstitial molecules, the latter
reflects only the expansions of the averaged lattice spacing. The former is always
larger than the latter, and the difference is due to the formation of defects. Assuming
the formation of vacancies, which is expected to appear more readily than interstitials, the difference is directly proportional to the fraction of the defects. The fraction
of the defect amounts to a few percents at the melting temperature.
Another way to sense the formation of defects is the temperature dependence
of the heat capacity. Experimental (or apparent) heat capacity usually exhibits a
notable tail on the low-temperature side. This tail comes from not only the effects
of impurities (because real compounds cannot be perfectly pure) but also defects
formed in the crystalline lattice. The excess heat capacity C excess is obtained as
C excess =
∂ x D Δ f h
∂ T
p
.
(6.2)
Assuming constancy of both Δ f h and Δ f s based on a narrow width of the temperature
range of the interest (ca. a few 10 K), the excess heat capacity C excess becomes
C excess =
(Δ f h)
2
k B T 2 exp
Δ f s
k B
exp
−
Δ f h
k B T
.
(6.3)
Thus, the plot of T
2 C excess against the inverse temperature T
−1 gives both Δ f h
and Δ f s (and x D ). In the case of crystals of rare gas elements, the resultant Δ f h is
comparable to the lattice energy [2, 3]. The formation of lattice vacancies rationalizes
this coincidence. On the other hand, a similar analysis yields smaller Δ f h for crystals
of a simple polyatomic molecule [21]. The smallness is attributed to the formation
of orientational defects, which can be more accessible with a small energetic penalty
to appear than the formation of a vacancy for molecular crystals
123
the fraction of the defect x D is related to the chemical potential of formation, Δ f μ =
Δ f h − T Δ f s as
x D = exp
−
Δ f μ
k B T
= exp
Δ f s
k B
exp
−
Δ f h
k B T
.
(6.1)
This relation can be confirmed in some ways. One being the most intuitive is the
difference between the volume expansivities determined macroscopically (dilatometry) and microscopically (e.g., by X-ray diffraction). While the former is affected by
the formation of defects such as lattice vacancies or interstitial molecules, the latter
reflects only the expansions of the averaged lattice spacing. The former is always
larger than the latter, and the difference is due to the formation of defects. Assuming
the formation of vacancies, which is expected to appear more readily than interstitials, the difference is directly proportional to the fraction of the defects. The fraction
of the defect amounts to a few percents at the melting temperature.
Another way to sense the formation of defects is the temperature dependence
of the heat capacity. Experimental (or apparent) heat capacity usually exhibits a
notable tail on the low-temperature side. This tail comes from not only the effects
of impurities (because real compounds cannot be perfectly pure) but also defects
formed in the crystalline lattice. The excess heat capacity C excess is obtained as
C excess =
∂ x D Δ f h
∂ T
p
.
(6.2)
Assuming constancy of both Δ f h and Δ f s based on a narrow width of the temperature
range of the interest (ca. a few 10 K), the excess heat capacity C excess becomes
C excess =
(Δ f h)
2
k B T 2 exp
Δ f s
k B
exp
−
Δ f h
k B T
.
(6.3)
Thus, the plot of T
2 C excess against the inverse temperature T
−1 gives both Δ f h
and Δ f s (and x D ). In the case of crystals of rare gas elements, the resultant Δ f h is
comparable to the lattice energy [2, 3]. The formation of lattice vacancies rationalizes
this coincidence. On the other hand, a similar analysis yields smaller Δ f h for crystals
of a simple polyatomic molecule [21]. The smallness is attributed to the formation
of orientational defects, which can be more accessible with a small energetic penalty
to appear than the formation of a vacancy for molecular crystals