42
2 Phase Transitions
This phase transition is continuous in φ 0 with a transition temperature of T 0 . Thermodynamic quantities exhibit anomalies at T 0 . By inserting Eq. 2.31 into Eq. 2.26,
the thermodynamic potential in the equilibrium at T is obtained as
Δf (T ) = −
α
2
4a 4
(T − T 0 )
2
.
(T < T 0 ).
(2.32)
Hereafter, we use f and other small letters for the free energy minimized regarding
the order parameter(s) and corresponding quantities. Thus, excess entropy Δs(T )
and heat capacity Δc(T ) as functions of temperature (T < T 0 ) are
Δs(T ) = −
∂Δf (T )
∂T
=
α
2
2a 4
(T − T 0 )
(2.33)
Δc(T ) = T
∂Δs
∂T
=
α
2
2a 4
T.
(2.34)
Although thermodynamic potential (Δf (T )) and entropy (Δs(T )) are continuous,
heat capacity (Δc(T )) exhibits a discontinuous step at T 0 by α
2 T 0 /2a 4 (downward on
heating) because Δc = 0 for T > T 0 . Thus, the phase transition described by Eq. 2.26
is a second-order phase transition. Note that a similar result is obtained while taking
the temperature dependence of a 4 into account as far as it remains positive.
Landau’s phenomenology predicts larger heat capacity for the B phases irrespective of its relative location to the S phase. Namely, the heat capacity exhibits a stepped
anomaly upward on heating if the B phase is a high-temperature phase. Indeed, there
exist such examples [12–14].
If the order parameter is (spontaneous) polarization, the linear response to the
external electric field is analyzed in the following way. A linear term expressing the
effect of external field h is added to thermodynamic potential as
ΔF(φ, τ , h) = a 2 φ
2
+ a 4 φ
4
− φh.
(2.35)
Even in the symmetric “phase,” φ is no longer zero but finite. The equation determining its magnitude is
∂ΔF(φ, τ , h)
∂φ
= 2a 2 φ + 4a 4 φ
3
− h = 0.
(2.36)
For the S phase, the smallness of φ allows us to neglect the third-order term.
2a 2 φ − h = 0.
(2.37)
Thus, the (small) polarization of the S phase under the field is given by
2 Phase Transitions
This phase transition is continuous in φ 0 with a transition temperature of T 0 . Thermodynamic quantities exhibit anomalies at T 0 . By inserting Eq. 2.31 into Eq. 2.26,
the thermodynamic potential in the equilibrium at T is obtained as
Δf (T ) = −
α
2
4a 4
(T − T 0 )
2
.
(T < T 0 ).
(2.32)
Hereafter, we use f and other small letters for the free energy minimized regarding
the order parameter(s) and corresponding quantities. Thus, excess entropy Δs(T )
and heat capacity Δc(T ) as functions of temperature (T < T 0 ) are
Δs(T ) = −
∂Δf (T )
∂T
=
α
2
2a 4
(T − T 0 )
(2.33)
Δc(T ) = T
∂Δs
∂T
=
α
2
2a 4
T.
(2.34)
Although thermodynamic potential (Δf (T )) and entropy (Δs(T )) are continuous,
heat capacity (Δc(T )) exhibits a discontinuous step at T 0 by α
2 T 0 /2a 4 (downward on
heating) because Δc = 0 for T > T 0 . Thus, the phase transition described by Eq. 2.26
is a second-order phase transition. Note that a similar result is obtained while taking
the temperature dependence of a 4 into account as far as it remains positive.
Landau’s phenomenology predicts larger heat capacity for the B phases irrespective of its relative location to the S phase. Namely, the heat capacity exhibits a stepped
anomaly upward on heating if the B phase is a high-temperature phase. Indeed, there
exist such examples [12–14].
If the order parameter is (spontaneous) polarization, the linear response to the
external electric field is analyzed in the following way. A linear term expressing the
effect of external field h is added to thermodynamic potential as
ΔF(φ, τ , h) = a 2 φ
2
+ a 4 φ
4
− φh.
(2.35)
Even in the symmetric “phase,” φ is no longer zero but finite. The equation determining its magnitude is
∂ΔF(φ, τ , h)
∂φ
= 2a 2 φ + 4a 4 φ
3
− h = 0.
(2.36)
For the S phase, the smallness of φ allows us to neglect the third-order term.
2a 2 φ − h = 0.
(2.37)
Thus, the (small) polarization of the S phase under the field is given by