On the Order Parameter of the Continuous Phase Transition in the Classical and Quantum Mechanical Limits

Abstract

The mean-field theory is revisited in the classical and quantum mechanical limits. Taking into account the boundary conditions at the phase transition and the third law of the thermodynamics, the physical properties of the ordered and disordered phases are reported. The equation for the order parameter predicts the occurrence of a saturation of $\Psi ^2$ ~1 near $\Theta _S$, and the temperature below the quantum mechanical ground state is reached. The theoretical predictions are also compared with high-resolution thermal expansion data of SrTiO$_{\text {3}}$ single crystals and other some previous results. An excellent agreement has been found suggesting a universal behavior of the theoretical model to describe continuous structural phase transitions.

Appendix

Taking the quantum mechanical model for the continuous transition predicted by Salje et al. [17], the free energy is given generically by

$$\begin{aligned} G = G_{\text {D}} + a \left[ \coth \left( \Theta _{\text {S}}/T\right) -\coth \left( \Theta _{\text {S}}/T_{\text {C}} \right) \right] \Psi ^4 + b\Psi ^4. \end{aligned}$$

(31)

Making

$$\begin{aligned} \frac{\partial G}{\partial \left( \Psi ^2\right) } = 0, \end{aligned}$$

(32)

provides

$$\begin{aligned} \Psi ^2 = 0, \end{aligned}$$

(33)

for $T > T_{\text {C}}$ (disordered phase), and

$$\begin{aligned} \Psi ^2 = - \frac{a\left[ \coth {\left( \Theta _{\text {S}}/T\right) } - \coth {\left( \Theta _{\text {S}}/T_{\text {C}} \right) }\right] }{2b}, \end{aligned}$$

(34)

for $T \le T_{\text {C}}$ (ordered phase).

Taking $\Psi ^2 = 1$ for T = 0, implies

$$\begin{aligned} \frac{2b}{a} = - \left[ 1 - \coth {\left( \Theta _{\text {S}}/T_{\text {C}}\right) }\right] , \end{aligned}$$

(35)

which is a normalization for $\Psi ^2$.

Thus

$$\begin{aligned} \Psi ^2 = \frac{\left[ \coth {\left( \Theta _{\text {S}}/T\right) } - \coth {\left( \Theta _{\text {S}}/T_{\text {C}} \right) }\right] }{\left[ \coth {\left( \Theta _{\text {S}}/T_{\text {C}}\right) } - 1\right] }, \end{aligned}$$

(36)

implies

$$\begin{aligned} G =&\; G_{\text {D}} - a \left[ \coth {\left( \Theta _{\text {S}}/T_{\text {C}}\right) } - \coth {\left( \Theta _{\text {S}}/T\right) } \right] \Psi ^2 \\& - a/2\left[ 1 - \coth {\left( \Theta _{\text {S}}/T_{\text {C}}\right) }\right] \Psi ^4, \end{aligned}$$

(37)

or

$$\begin{aligned} G = G_{\text {D}} - \frac{a}{2}\frac{\left[ \coth {\left( \Theta _{\text {S}} / T_{\text {C}}\right) }-\coth {\left( \Theta _{\text {S}} / T \right) }\right] ^2}{\coth {\left( \Theta _{\text {S}} / T_{\text {C}}\right) }-1}. \end{aligned}$$

(38)

Such as $G = G(T,P)$, then

$$\begin{aligned} \left( \frac{\partial G}{\partial T} \right) _P = - S, \end{aligned}$$

(39)

or

$$\begin{aligned} -S = -S_{\text {D}} + a\Psi ^2\left( \Theta _{\text {S}}/T^2\right) \text {csch}^2\left( \Theta _{\text {S}}/T\right) , \end{aligned}$$

(40)

which provides the following equation for the entropy of the disordered phase, making S = 0 and $\Psi ^2$ = 1 as $T \rightarrow$ 0,

$$\begin{aligned} S_{\text {D}} = a \left( \Theta _{\text {S}}/T^2\right) \text {csch}^2\left( \Theta _{\text {S}}/T \right) . \end{aligned}$$

(41)

Futhermore, when $\Theta _{\text {S}} \rightarrow$ 0, $S_{\text {D}} \rightarrow S_{\text {C}}$, and $a \rightarrow S_{\text {C}}\Theta _{\text {S}}$ due to the classical limit which can be noticed in Fig. 2a. Thus,

$$\begin{aligned} S_{\text {D}} = S_{\text {C}}\left( \Theta _{\text {S}}/T\right) ^2\text {csch}^2\left( \Theta _{\text {S}}/T \right) , \end{aligned}$$

(42)

but taking

$$\begin{aligned} \left( \frac{\partial G_{\text {D}}}{\partial T} \right) _P = - S_{\text {D}}, \end{aligned}$$

(43)

one can show that

$$\begin{aligned} G_{\text {D}} = G^0 - S_{\text {C}} \Theta _{\text {S}} \coth {\left( \Theta _{\text {S}}/T \right) }, \end{aligned}$$

(44)

where $G^0$ is a reference for the free energy and appears due to the integration constant.

Thus,

$$\begin{aligned} G =&\; G^0 - S_{\text {C}}\Theta _{\text {S}}\coth {\left( \Theta _{\text {S}}/T\right) } - S_{\text {C}}\Theta _{\text {S}}\biggl[ \coth {\left( \Theta _{\text {S}}/T_{\text {C}} \right) }\\& - \coth {\left( \Theta _{\text {S}}/T \right) } \biggr] \Psi ^2 - \frac{S_{\text {C}}\Theta _{\text {S}}}{2}\left[ 1 - \coth {\left( \Theta _{\text {S}} / T_{\text {C}}\right) } \right] \Psi ^4. \end{aligned}$$

(45)

which is the equation by Salje et al. [17], with pre-factors determined based upon the boundary conditions at the critical temperature and absolute zero.

Then,

$$\begin{aligned} G = G_{\text {D}} - \frac{S_{\text {C}} \Theta _{\text {S}}}{2}\frac{\left[ \coth {\left( \Theta _{\text {S}} / T_{\text {C}}\right) }-\coth {\left( \Theta _{\text {S}} / T \right) }\right] ^2}{\coth {\left( \Theta _{\text {S}} / T_{\text {C}}\right) }-1}, \end{aligned}$$

(46)

which provides

$$\begin{aligned} -S = -S_{\text {D}} + S_{\text {C}}\Theta _{\text {S}}\Psi ^2\text {csch}^2\left( \Theta _{\text {S}}/T\right) \end{aligned}$$

(47)

or

$$\begin{aligned} S = S_{\text {D}} - S_{\text {D}}\Psi ^2, \end{aligned}$$

(48)

that leads to

$$\begin{aligned} S = S_{\text {D}}\left( 1 - \Psi ^2\right) , \end{aligned}$$

(49)

which has the same format as the equation for the classical limit, but takes into account the $\Psi ^2$ saturation effect near $T = \Theta _{\text {S}}$.

With reagard to thermal expansion, it can be obtained using the following relation for $\Omega$ [31]

$$\begin{aligned} -\left( \frac{\partial S}{\partial P}\right) _T = \left( \frac{\partial V}{\partial T}\right) _P = V_{\text {C}}\Omega , \end{aligned}$$

(50)

which provides

$$\begin{aligned} - \left( \frac{\partial S}{\partial P}\right) _T = - \left( \frac{\partial S_{\text {D}}}{\partial P}\right) _T \left( 1-\Psi ^2\right) +S_{\text {D}}\left( \frac{\partial \Psi ^2}{\partial P}\right) _T, \end{aligned}$$

(51)

where

$$\begin{aligned} \left( \frac{\partial \Psi ^2}{\partial P}\right) _T = \frac{\Theta _{\text {S}}}{T_{\text {C}}^2} \frac{\left[ \coth {\left( \Theta _{\text {S}}/T\right) }-1\right] }{\left[ \coth {\left( \Theta _{\text {S}}/T_{\text {C}}\right) }-1\right] ^2} \text {csch}^2\left( \Theta _{\text {S}} / T_{\text {C}}\right) \left( \frac{d T_{\text {C}}}{d P}\right) _T, \end{aligned}$$

(52)

and $\Theta _{\text {S}}$ is assumed to be constant with pressure and dependent only of the temperature. Thus, one can write

$$\begin{aligned} V_{\text {C}}\Omega =&\; V_{\text {C}}\Omega _{\text {D}}\left( 1-\Psi ^2\right) +S_{\text {D}}\frac{\Theta _{\text {S}}}{T_{\text {C}}^2} \frac{\left[ \coth {\left( \Theta _{\text {S}}/T\right) }-1\right] }{\left[ \coth {\left( \Theta _{\text {S}}/T_{\text {C}}\right) }-1\right] ^2} \text {csch}^2\\&\left( \Theta _{\text {S}} / T_{\text {C}}\right) \left( \frac{d T_{\text {C}}}{d P}\right) _T, \end{aligned}$$

(53)

but at T = $T_{\text {C}}$, $\Omega$ = $\Omega _{\text {O}}$, providing

$$\begin{aligned} V_{\text {C}}\left( \Omega _{\text {O}}-\Omega _{\text {D}}\right) = S_{\text {D}}\frac{\Theta _{\text {S}}}{T_{\text {C}}^2} \frac{\text {csch}^2\left( \Theta _{\text {S}}/T_{\text {C}}\right) }{\left[ \coth {\left( \Theta _{\text {S}}/T\right) }-1\right] } \left( \frac{d T_{\text {C}}}{d P}\right) _T, \end{aligned}$$

(54)

which gives

$$\begin{aligned} \Omega =&\; \Omega _{\text {O}}\left( 1-\Psi ^2\right) + S_{\text {D}}\frac{\Theta _{\text {S}}}{T_{\text {C}}^2} \frac{\text {csch}^2\left( \Theta _{\text {S}}/T_{\text {C}}\right) }{\left[ \coth {\left( \Theta _{\text {S}}/T\right) }-1\right] } \left( \frac{d T_{\text {C}}}{d P}\right) _T \left( 1-\Psi ^2\right) \\& -S_{\text {D}}\frac{\Theta _{\text {S}}}{T_{\text {C}}^2} \frac{\left[ \coth {\left( \Theta _{\text {S}}/T\right) }-1\right] }{\left[ \coth {\left( \Theta _{\text {S}}/T_{\text {C}}\right) }-1\right] ^2} \text {csch}^2\left( \Theta _{\text {S}} / T_{\text {C}}\right) \left( \frac{d T_{\text {C}}}{d P}\right) _T. \end{aligned}$$

(55)

After an algebraic work it is possible to show that the terms multiplied by $S_{\text {D}}$ cancel each other, remembering that $\left( 1 - \Psi ^2\right)$ is given by

$$\begin{aligned} \left( 1-\Psi ^2\right) = \frac{\coth {\Theta _{\text {S}}/T}-1}{\coth {\Theta _{\text {S}}/T_{\text {C}}}-1}, \end{aligned}$$

(56)

thus

$$\begin{aligned} \Omega = \Omega _{\text {O}}\left( 1-\Psi ^2\right) . \end{aligned}$$

(57)

Finally, based upon the equations for $G_{\text {D}}$, G, $S_{\text {D}}$, and S, it is possible to find the temperature dependencies for heat capacity at constant volume ($C_v$) and the internal energy (U) (not shown).