2.2 Landau’s Phenomenology
45
Δ trs s =
αa
2
3
4a
2
4
(2.48)
resulting in the presence of the latent heat Δ trs h = T trs Δ trs s. Further cooling causes
the disappearance of the local minimum at φ = 0 at T L = T 0 , below which thermodynamic stability of the S phase is lost. In summary, Eq. 2.45 predicts a discontinuous
(first-order) phase transition. The existence of the lower limit of supercooling (T L )
and the upper limit of superheating (T U ) is also predicted.
A first-order transition is not limited to the case of the presence of the third-order
term in the expansion of thermodynamic potential. In the case where the expansion
up to the fourth order (Eq. 2.26) is insufficient because of a 4 < 0 even with a 3 = 0,
we need to treat
ΔF(φ, τ ) = a 2 φ
2
− a 4 φ
4
+ a 6 φ
6
.
(2.49)
A similar analysis to the case with a 3 = 0 yields a first-order transition taking place
at
T trs =
a
2
4
4αa 6
+ T 0
(2.50)
with a jump in the order parameter to
√
a 4 /2a 6 from 0. The limits of the supercooling
and superheating are T L = T 0 and T U = a
2
4 /(3αa 6 ) + T 0 , respectively.
It seems natural to ask what happens when a 4 = 0 because a continuous transition
occurs for a 4 > 0 in contrast to the present (a 4 < 0) case.
7 On approaching null from
the negative side (a 4 < 0), the transition temperature (Eq. 2.50) also approaches to
T 0 . Thus, the transition point with a 4 = 0 is T 0 . The transition point T 0 with a 4 =
0 (assuming a 6 > 0), where a discontinuous (first-order) changes to a continuous
(second-order) transition (or vice versa), is called a tricritical point. In this case, the
order parameter grows as
φ 0 (τ ) =
−
α(T − T 0 )
3a 6
1/4
(2.51)
below T 0 . A tricritical behavior has experimentally been observed in some systems
[15] with varying intensive parameters (e.g., pressure or composition in multicomponent systems) other than temperature.
2.2.5 Plural Equivalent Order Parameters
It is often the case that the symmetry of the symmetric phase forces to take plural
yet equivalent order parameters into account. Namely, the order parameters to be
considered belongs to an irreducible representation (of the symmetry group) with
7 The ignorance of the sixth-order term with a 6 > 0 for a 4 > 0 has no essential effect.
45
Δ trs s =
αa
2
3
4a
2
4
(2.48)
resulting in the presence of the latent heat Δ trs h = T trs Δ trs s. Further cooling causes
the disappearance of the local minimum at φ = 0 at T L = T 0 , below which thermodynamic stability of the S phase is lost. In summary, Eq. 2.45 predicts a discontinuous
(first-order) phase transition. The existence of the lower limit of supercooling (T L )
and the upper limit of superheating (T U ) is also predicted.
A first-order transition is not limited to the case of the presence of the third-order
term in the expansion of thermodynamic potential. In the case where the expansion
up to the fourth order (Eq. 2.26) is insufficient because of a 4 < 0 even with a 3 = 0,
we need to treat
ΔF(φ, τ ) = a 2 φ
2
− a 4 φ
4
+ a 6 φ
6
.
(2.49)
A similar analysis to the case with a 3 = 0 yields a first-order transition taking place
at
T trs =
a
2
4
4αa 6
+ T 0
(2.50)
with a jump in the order parameter to
√
a 4 /2a 6 from 0. The limits of the supercooling
and superheating are T L = T 0 and T U = a
2
4 /(3αa 6 ) + T 0 , respectively.
It seems natural to ask what happens when a 4 = 0 because a continuous transition
occurs for a 4 > 0 in contrast to the present (a 4 < 0) case.
7 On approaching null from
the negative side (a 4 < 0), the transition temperature (Eq. 2.50) also approaches to
T 0 . Thus, the transition point with a 4 = 0 is T 0 . The transition point T 0 with a 4 =
0 (assuming a 6 > 0), where a discontinuous (first-order) changes to a continuous
(second-order) transition (or vice versa), is called a tricritical point. In this case, the
order parameter grows as
φ 0 (τ ) =
−
α(T − T 0 )
3a 6
1/4
(2.51)
below T 0 . A tricritical behavior has experimentally been observed in some systems
[15] with varying intensive parameters (e.g., pressure or composition in multicomponent systems) other than temperature.
2.2.5 Plural Equivalent Order Parameters
It is often the case that the symmetry of the symmetric phase forces to take plural
yet equivalent order parameters into account. Namely, the order parameters to be
considered belongs to an irreducible representation (of the symmetry group) with
7 The ignorance of the sixth-order term with a 6 > 0 for a 4 > 0 has no essential effect.