Modeling Phase Transitions with Landau Theory

Hello! Welcome back to our study of Landau theory.

In the last lesson, we extended our phenomenological framework to include an external field conjugate to the order parameter. By adding a simple linear term, $-hm$, to the free energy, we derived the mean-field equation of state, $h=am+b{m}^3$. This single equation unified our understanding of the system's behavior above, below, and at the critical point, and allowed us to derive the full set of standard mean-field exponents.

So far, our treatment has been abstract, dealing with a generic scalar order parameter $m$. Today, we will bridge the gap between this abstract theory and real-world physical systems. Our learning outcome is to apply Landau theory to model a physical system, such as a cubic-to-tetragonal structural transition.

Structural phase transitions, where a crystal's lattice symmetry changes, are fundamental in condensed matter physics and materials science. Applying Landau theory to these transitions requires a more sophisticated approach. We must carefully identify an order parameter that captures the change in crystal structure and construct a free energy that accounts for the coupling between this order parameter and the crystal's elastic strain. This will demonstrate the true power and versatility of the Landau formalism.

1. Order Parameters and Strain in Structural Transitions

In a structural phase transition, the arrangement of atoms changes, leading to a change in the crystal's point group symmetry. The order parameter must quantify this symmetry breaking.

A crucial physical quantity in any solid is the strain tensor, ${ϵ}_{ij}$, which describes the deformation of the crystal lattice. Since a structural transition inherently deforms the lattice, strain is always involved. This leads to two main classes of structural transitions:

• Proper Ferroelastic Transitions: The primary order parameter is a component of the strain tensor itself. The transition is driven by the spontaneous softening of an elastic constant, leading to a spontaneous deformation of the unit cell.
• Improper Ferroelastic Transitions: The transition is driven by a primary order parameter that is not a strain component. This could be, for example, the tilt angle of atomic groups (like in perovskites), an ordering of electric dipoles, or the onset of charge order. This primary order parameter then couples to the strain, inducing a spontaneous strain as a secondary effect. The crystal deforms as a byproduct of the underlying ordering process.

In both cases, we must expand the Landau free energy not just in terms of the primary order parameter $Q$, but also in terms of the components of the strain tensor, $ϵ$, and their couplings.

2. The Landau-Devonshire Free Energy: Coupling Order and Strain

To model a structural transition, the free energy must account for all relevant degrees of freedom. This leads to what is often called a Landau-Devonshire free energy, which has the general form:

$$F(Q,ϵ)=F_{\text{Landau}}(Q)+F_{\text{coupling}}(Q,ϵ)+F_{\text{elastic}}(ϵ)$$

Let's break this down:

1. $F_{\text{Landau}}(Q)$: This is the standard expansion in the primary order parameter $Q$ that we have been using, e.g., $\frac{1}{2}a{Q}^2+\frac{1}{4}b{Q}^4$.
2. $F_{\text{elastic}}(ϵ)$: This is the elastic energy stored in the crystal due to strain. To a good approximation, this is given by Hooke's law: $F_{\text{elastic}}=\frac{1}{2}{C}_{ijkl}{ϵ}_{ij}{ϵ}_{kl}$, where $C_{ijkl}$ is the tensor of elastic constants.
3. $F_{\text{coupling}}(Q,ϵ)$: This is the crucial new part. It contains terms mixing the order parameter and strain, like ${\lambda }_{1}Qϵ$, ${\lambda }_2{Q}^2ϵ$, etc. The precise form of these terms is strictly determined by the symmetries of the crystal and the order parameter. A term is only allowed if it is invariant under all symmetry operations of the high-symmetry phase.

The equilibrium state of the crystal for a given temperature is found by minimizing this total free energy with respect to both $Q$ and all components of $ϵ$. A key consequence is that even if the crystal is unstressed externally, a spontaneous strain ${ϵ}_s$ will appear below $T_c$ when $Q\ne 0$. This spontaneous strain is found by solving $\frac{\partial F}{\partial {ϵ}_{ij}}=0$, and it will be a function of $Q$.

The following document provides an excellent overview of this formalism and its application to several real mineral systems, showing how experimental strain data is analyzed using Landau theory.

3. Case Study: The Cubic-to-Tetragonal Transition in Perovskites

Let's now apply this framework to the specific case mentioned in our learning outcome: a cubic-to-tetragonal transition. A classic example occurs in many ABO${}_3$ perovskite materials, which are central to modern condensed matter physics.

In the high-temperature phase, these materials have a cubic crystal structure. As they are cooled, they can undergo a transition where the BO${}_6$ octahedra tilt, lowering the symmetry to tetragonal.

• High-Symmetry Phase: Cubic ($a=b=c$, all angles 90°).
• Low-Symmetry Phase: Tetragonal ($a=b\ne c$, all angles 90°).

Identifying the Order Parameter and Strain

The primary driving force for this transition is the cooperative tilting of the BO${}_6$ octahedra. Therefore, the primary order parameter, $Q$, is the tilt angle $\varphi$. This is an example of an improper ferroelastic transition.

The structural change from cubic to tetragonal is captured by a specific combination of strain components. The key deformation is that the $c$-axis becomes distinct from the $a$ and $b$ axes. This is described by the tetragonal shear strain, $e_t$:

$$e_t=\frac{1}{\sqrt{6}}(2{ϵ}_{zz}-{ϵ}_{xx}-{ϵ}_{yy})$$

This combination is constructed to be zero in the cubic phase and non-zero in the tetragonal phase, making it a "symmetry-adapted" strain.

Constructing and Analyzing the Free Energy

Now we construct a simplified Landau-Devonshire free energy. We need terms for the order parameter $Q$ (the tilt angle), the tetragonal strain $e_t$, and their coupling.

1. $F_{\text{Landau}}(Q)$: A standard ${\varphi }^4$ expansion: $\frac{1}{2}a(T-T_c)Q^2+\frac{1}{4}b{Q}^4$.
2. $F_{\text{elastic}}(e_t)$: The elastic energy for this strain: $\frac{1}{2}C{e}_t^2$, where $C$ is the corresponding elastic stiffness constant (related to $C_{11}-C_{12}$ for a cubic crystal).
3. $F_{\text{coupling}}(Q,e_t)$: We need the simplest coupling term that is invariant under the cubic symmetry operations. For this transition, the lowest-order allowed term is of the form $\lambda Q^2{e}_t$. This is known as linear-quadratic coupling (linear in strain, quadratic in the order parameter).

Our total free energy is:

$$F(Q,e_t)=\frac{1}{2}a(T-T_c)Q^2+\frac{1}{4}b{Q}^4+\lambda Q^2{e}_t+\frac{1}{2}C{e}_t^2$$

To find the equilibrium state, we minimize $F$ with respect to both variables. Let's first minimize with respect to the strain $e_t$, as it adjusts quickly to the primary order parameter:

$$\frac{\partial F}{\partial e_t}=\lambda Q^2+C{e}_t=0$$

This gives the spontaneous strain that develops in the ordered phase:

$$e_s=-\frac{\lambda }{C}{Q}^2$$

This is a crucial result: the tetragonal distortion ($e_s$) is proportional to the square of the primary order parameter (the tilt angle). This is a hallmark of this type of improper ferroelastic transition.

Now, we substitute this equilibrium strain back into the free energy to get an effective free energy that depends only on $Q$:

$$F_{\text{eff}}(Q)=\frac{1}{2}a(T-T_c)Q^2+\frac{1}{4}b{Q}^4+\lambda Q^2\left(-\frac{\lambda }{C}{Q}^2\right)+\frac{1}{2}C{\left(-\frac{\lambda }{C}{Q}^2\right)}^2$$

Simplifying this expression gives:

$$F_{\text{eff}}(Q)=\frac{1}{2}a(T-T_c)Q^2+\left(\frac{1}{4}b-\frac{{\lambda }^2}{2C}\right)Q^4$$

This effective free energy has the familiar ${\varphi }^4$ form, but with a renormalized quartic coefficient: $b^{\prime }=b-\frac{2{\lambda }^2}{C}$.

This has a profound physical consequence. Even if the transition is inherently second-order (i.e., the bare coefficient $b$ is positive), a strong enough coupling to strain ($\lambda$) can make the effective coefficient $b^{\prime }$ negative. This would drive the transition to become first-order. This phenomenon, where coupling to a secondary degree of freedom alters the nature of a phase transition, is a key insight provided by Landau theory.

Test your understanding!

Consider a **proper** ferroelastic transition, where the tetragonal strain $e_t$ is itself the primary order parameter. For symmetry reasons, the lowest-order coupling term to another strain, like the volume strain $e_v = \epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}$, is typically of the form $\gamma e_v e_t^2$.

The simplified free energy is: $F(e_t,e_v)=\frac{1}{2}a(T-T_c)e_t^2+\frac{1}{4}b{e}_t^4+\gamma e_v{e}_t^2+\frac{1}{2}K{e}_v^2$, where $K$ is the bulk modulus.

1. Find the spontaneous volume strain $e_{v,s}$ as a function of the order parameter $e_t$.
2. Substitute this back into the free energy to find the effective potential for $e_t$. How is the transition temperature shifted by the coupling?

Show answer

1. Minimize with respect to the volume strain:
   $\frac{\partial F}{\partial e_v}=\gamma e_t^2+K{e}_v=0⟹e_{v,s}=-\frac{\gamma }{K}{e}_t^2$
   The spontaneous volume change is proportional to the square of the order parameter strain.

2. Substitute back into $F$:
   $F_{\text{eff}}(e_t)=\frac{1}{2}a(T-T_c)e_t^2+\frac{1}{4}b{e}_t^4+\gamma \left(-\frac{\gamma }{K}{e}_t^2\right)e_t^2+\frac{1}{2}K{\left(-\frac{\gamma }{K}{e}_t^2\right)}^2$
   $F_{\text{eff}}(e_t)=\left(\frac{1}{2}a(T-T_c)-\frac{{\gamma }^2}{2K}\right)e_t^2+\left(\frac{1}{4}b\right)e_t^4$
   Wait, the calculation is: $-{\gamma }^2/K+{\gamma }^2/(2K)=-{\gamma }^2/(2K)$. So the term is $(\frac{1}{4}b)e_t^4$. Oh, let me recheck the algebra. The last term is $+\frac{1}{2}K\frac{{\gamma }^2}{K^2}{e}_t^4=+\frac{{\gamma }^2}{2K}{e}_t^4$.
   The terms multiplying $e_t^4$ are $\frac{1}{4}b$ and $-\frac{{\gamma }^2}{K}+\frac{{\gamma }^2}{2K}=-\frac{{\gamma }^2}{2K}$. So the $e_t^4$ coefficient is $b^{\prime }=b-\frac{2{\gamma }^2}{K}$.

   Let me re-examine the $e_t^2$ term. The form seems off. Let me correct the problem statement. The coupling should be linear-linear or linear-quadratic. A $e_v{e}_t^2$ term is a valid coupling. Let's re-evaluate.
   The effective free energy becomes:
   $F_{\text{eff}}(e_t)=\frac{1}{2}a(T-T_c)e_t^2+\left(\frac{1}{4}b-\frac{{\gamma }^2}{2K}\right)e_t^4$
   The quadratic coefficient, $\frac{1}{2}a(T-T_c)$, is not changed by this coupling. The transition temperature $T_c$ remains the same. However, the quartic coefficient is renormalized, $b^{\prime }=b-2{\gamma }^2/K$, which can change the order of the transition, just as in our main example.

   Let's consider a different coupling allowed by symmetry in some cases: linear-linear, $\gamma e_v{e}_t$. The free energy would be $F=\frac{1}{2}a(T-T_c)e_t^2+\frac{1}{3}d{e}_t^3+...+\gamma e_v{e}_t+\frac{1}{2}K{e}_v^2$.
   Minimizing wrt $e_v$ gives $e_{v,s}=-\frac{\gamma }{K}{e}_t$.
   Substituting back gives $F_{eff}=\frac{1}{2}a(T-T_c)e_t^2-\frac{{\gamma }^2}{2K}{e}_t^2+...=\frac{1}{2}\left(a(T-T_c)-\frac{{\gamma }^2}{K}\right)e_t^2+...$.
   The effective quadratic term is now $\frac{1}{2}a(T-(T_c+\frac{{\gamma }^2}{aK}))$. The coupling shifts the transition temperature upwards by $\Delta T_c=\frac{{\gamma }^2}{aK}$.

   The answer depends on the symmetry-allowed coupling term. For the question as stated, the $T_c$ is not shifted, but the quartic term is. This shows how different couplings have different physical effects.

To see the full complexity and power of this method, the following resource provides the "master equation" for tilting transitions in perovskites, which involves multiple tilt systems and strain components.

Conclusion

Key Takeaways:

• Landau theory can be successfully applied to real physical systems like crystals undergoing structural phase transitions.
• This requires identifying a primary order parameter ($Q$) and accounting for its coupling to lattice strain ($ϵ$).
• The total free energy, $F(Q,ϵ)$, must include terms for the order parameter, the elastic energy, and their symmetry-allowed couplings.
• Minimizing this free energy reveals the relationship between the spontaneous strain that arises in the low-symmetry phase and the order parameter (e.g., $e_s\propto Q$ or $e_s\propto Q^2$).
• Coupling to strain can renormalize the coefficients of the Landau expansion, potentially shifting the transition temperature or even changing a continuous transition into a first-order one.

Preview of the Next Module:

We have now explored the core phenomenological ideas of Landau theory. However, this is a mean-field theory that neglects fluctuations. While it provides a powerful qualitative (and sometimes quantitative) picture, it fails to predict the correct critical exponents observed experimentally. To go beyond mean-field theory, we need to consider models that can be solved more rigorously. In the next module, we will turn our attention to the Ising Model, a cornerstone of statistical mechanics. We will start by solving it exactly in one dimension and see why it does not exhibit a phase transition at finite temperature, a foundational result that highlights the importance of dimensionality.