Question

A ferroelectric crystal has a free energy of the form:

$F = \frac{1}{2} \alpha (T-T_{c})P^{2} + \frac{1}{4}bP^{4} + \frac{1}{6}cP^{6} +DxP^{6} + \frac{1}{2}Ex^{2}$

where P is the electric polarization and x represents the strain applied to the crystal. Minimize the system with respect to x and determine the free energy at the minimum. Under what circumstance is there a first-order phase transition for this system?

Answer 1

F = 1/2 alpha (T - T_c) P^2 + 1/4 b P^4 + 1/6 c P^6 + D_x P^6 + 1/2 E x^2 Condition for minimization: dF/dx = 0 \r\n dF/dx = D P^6 + E x = 0 E x = -D P^6 = > x = -D P^6 /E therefore x_min = -D P^6 /E The energy of minimum is, F = 1/2 alpha (T - T_c) P^2 + 1/4 b P^4 + 1/6 C p^6 + D (-D p^6 /E) P^6 + 1/2 E (-D P^6 /E)^2 = 1/2 (T - T_c) P^2 + 1/4 b P^4 + 1/6 C P^6 - D^2 P^12 /E + 1/2 D^2 P^12 /E = 1/2 (T - T_c) P^2 + 1/4 b p^4 + 1/6 C P^6 - 1/2 D^2 P^12 /E In first order phase transition, there is a discontinuity in the first derivative of the free energy with respect to thermodynamic variable \r\n dF/dp = alpha (T - T_c) P + b P^3 = 0 (only