Finding transition temperature of Landau ferroelectric

In summary, the conversation discusses the process of finding the temperature at which a phase transition occurs, represented by ##T_{\theta}^{'}##. The conversation also mentions using a cubic equation to solve for ##T_{\theta}^{'}##.
  • #1
Homework Statement
In terms of Landau theory, a “proper ferroelectric” is completely analogous to a ferromagnet, where the order parameter is the polarization ##P## (dipole moment per volume). We can write a free energy for the ferroelectric phase transition as ##F_P = \frac 1 2 \alpha (T - T_P)P^2 + \frac 1 4 g_4 P^4 + ...##.
For a molecular crystal a structural phase transition can occur where the molecular axis tilts by an angle ##\theta## with respect to the crystal axis. The associated free energy is ##F_{\theta} = \frac 1 2 a_{\theta}(T - T_{\theta}) \theta^2 + \frac 1 4 b_{\theta} \theta^4 + ... ##.
Under certain circumstances (the molecule is chiral, i.e., lacks inversion symmetry), the dipole moment on the molecule can couple to the molecular tilt, with a coupling term ##F_{P \theta} = -t \theta P ## where t is the coupling constant.

##T_{\theta} > T_P## and a tilt phase transition will occur first on cooling. A net polarization can exist due to coupling.

a. In the tilted phase ##T_P < T < T_{\theta}##, determine ##\theta## and ##P## in terms of ##\alpha##, ##a_{\theta}##, ##b_{\theta}##, ##T_P##, and ##T_{\theta}##. You can neglect the ##P^4## term.
b. Because of the coupling, the tilt transition does not occur at ##T_{\theta}##, but some other temp ##T_{\theta}^{'}##. Determine ##T_{\theta}^{'}## and the critical exponent ##b## for order parameter ##\theta##.
Relevant Equations
##\frac {\partial F} {\partial \xi} = 0 ##
So for part a, I separately minimized F wrt ##\theta## and ##P## and got the following.

$$\frac {\partial f} {\partial \theta} = a_{\theta}(T-T_{\theta})\theta + b_{\theta}\theta^3 - tP = 0$$
$$ \frac {\partial f} {\partial P} = \alpha(T-T_P)P -t\theta$$
$$ P = t\theta \alpha (T-T_P) $$

Then subbing this into the expression for θ gives me...

$$ \theta \left[ a_{\theta}(T-T_{\theta}) + b_{\theta}\theta^2 - t^2\alpha(T-T_P) \right] = 0 $$

which I can solve for ##\theta = 0## or ##\theta = \sqrt {\frac {t^2} { \alpha (T - T_P) } }- a_{\theta}(T - T_{\theta})##. And plugging this in gives ##P = \pm \frac {t} {\alpha (T - T_P)} \sqrt {\frac {t^2} { \alpha (T - T_P) } }- a_{\theta}(T - T_{\theta})##.

Now my problem is with b, because I don't think I have the proper approach of finding the temperature at which the phase transition occurs, ##T_{\theta}^{'}##. My idea was that maybe ##T_{\theta}^{'}## is the temperature ##T## when ##\theta## goes to 0.

That is
$$\pm \sqrt {\frac {t^2} { \alpha (T_{\theta}^{'} - T_P) } }- a_{\theta}(T_{\theta}^{'} - T_{\theta}) = 0 $$
$$ \frac {t^2} { \alpha (T_{\theta}^{'} - T_P) } = (T_{\theta}^{'} - T_{\theta})(T_{\theta}^{'} - T_P) $$

which seems a bit difficult to solve for ##T_{\theta}^{'}##. The hint I have from class is that I should somehow re-write the expression for P in terms of ##T_{\theta}^{'}##.
 
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  • #2
It is a cubic equation of ##T'_{\theta}## whose solution we know by formula.
 

1. What is the Landau ferroelectric phase transition?

The Landau ferroelectric phase transition is a phenomenon in which a material undergoes a change in its electric polarization at a specific temperature, known as the transition temperature. This transition is caused by a change in the symmetry of the material's crystal structure, resulting in the formation of spontaneous electric dipoles.

2. How is the transition temperature of Landau ferroelectric determined?

The transition temperature of Landau ferroelectric is typically determined through experimental techniques such as differential scanning calorimetry or dielectric measurements. These methods involve measuring the changes in heat capacity or electric permittivity of the material as it is heated or cooled, respectively.

3. What factors affect the transition temperature of Landau ferroelectric?

The transition temperature of Landau ferroelectric is influenced by various factors, including the composition, crystal structure, and external conditions (e.g. applied electric field, pressure, etc.) of the material. Additionally, the presence of impurities or defects can also affect the transition temperature.

4. Can the transition temperature of Landau ferroelectric be controlled?

Yes, the transition temperature of Landau ferroelectric can be controlled by altering the composition or crystal structure of the material, as well as by applying external conditions such as electric fields or pressure. This allows for the development of materials with tailored transition temperatures for specific applications.

5. What are the applications of Landau ferroelectric materials?

Landau ferroelectric materials have a wide range of applications, including in electronic devices, energy storage and conversion systems, and sensors. They are also used in memory devices, such as ferroelectric random-access memory (FeRAM), due to their ability to retain polarization states even when the electric field is removed.

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