Insights into the rotational phase transition in the perovskite structure can be gained by treating the octahedra as nearly rigid objects. If each octahedron is assigned a single variable to describe its orientation, \(\theta\), two neighbouring octahedra have lowest energy if they rotate by exactly the same amount in opposite directions. Show that the harmonic energy between two neighbouring octahedra can be written as $$ E(i, j)=\frac{1}{2} K\left(\theta_{i}+\theta_{k}\right)^{2} $$ For a phase transition to occur, it can be presumed that each octahedron also experiences a local double-well potential of the form $$ E(\theta)=-\frac{\kappa_{2}}{2} \theta^{2}+\frac{\kappa_{4}}{4} \theta^{4} $$ Show that when the harmonic term is combined with the interactions between neighbours, a soft mode is obtained at \(\mathbf{k}=\left(\frac{1}{2}, \frac{1}{2}, 0\right)\).

Step by step solution

01

Understanding the Problem

Identify the given parameters and understand the provided energy equations. The harmonic energy between two neighboring octahedra is given by \[ E(i, j)=\frac{1}{2} K(\theta_{i}+\theta_{k})^{2} \]. Each octahedron also experiences a local double-well potential, given by \[ E(\theta)=-\frac{\kappa_{2}}{2} \theta^{2}+\frac{\kappa_{4}}{4} \theta^{4} \]. The goal is to combine these terms and show that a soft mode is obtained at \(\mathbf{k}=(\frac{1}{2}, \frac{1}{2}, 0)\).

02

Harmonic Energy Interaction

Identify and explain the harmonic energy term. The harmonic energy between two neighboring octahedra is given as: \[ E(i, j)=\frac{1}{2} K(\theta_{i}+\theta_{k})^{2} \] This indicates that the harmonic energy is minimized when neighboring octahedra rotate by equal and opposite amounts.

03

Local Double-Well Potential

Examine the double-well potential energy term for individual octahedra: \[ E(\theta)=-\frac{\kappa_{2}}{2} \theta^{2}+\frac{\kappa_{4}}{4} \theta^{4} \] This shows that each octahedron tends to adopt certain orientations described by phenomonological constants \( \kappa_{2} \) and \( \kappa_{4} \). Double-well potentials generally have two local minima in their energy landscapes.

04

Combined Energy Expression

Combine both energy terms to find the total energy considering both local potential and nearest neighbor interactions:\[ E_{tot} = \frac{1}{2} K (\theta_{i}+\theta_{k})^{2} - \frac{\kappa_{2}}{2} \theta^{2} + \frac{\kappa_{4}}{4} \theta^{4} \]

05

Developing the Soft Mode

Assume the system near the phase transition and expand in terms of Fourier components. If the octahedrals oscillate then it would imply varying wavenumbers \(\mathbf{k}\). Here the significant finding mode is obtained at: \[ \mathbf{k}=\left( \frac{1}{2}, \frac{1}{2}, 0 \right) \] This indicates that there is a modulation in the system where the relevant modes exist when phases vary from cell-to-cell indicating that the soft phonon mode captures the instability and shows the driving factor about phase transition scenario due to the potential energy surface due to double-well.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Harmonic Energy

Harmonic energy is a key concept in understanding the dynamics of systems with periodic or oscillating components, such as the octahedra in a perovskite structure. The harmonic energy between two neighboring octahedra is given by:
\( E(i, j)=\frac{1}{2} K(\theta_{i}+\theta_{k})^{2} \)
This equation signifies that the total energy is minimized when neighboring octahedra rotate equally but in opposite directions. For example:

• If \(\theta_i = -\theta_k\), the term \(\theta_i + \theta_k\) will be zero, minimizing the energy.

Harmonic potentials are typically quadratic in nature, meaning they have a form that depends on the square of the displacement, or in this case, the orientation angle \(\theta\). This form ensures the system tends to stabilize around the equilibrium position where the energy is at a minimum.

Double-Well Potential

A double-well potential is an important concept in phase transitions and has a significant impact on the energy landscape of octahedra in a perovskite structure.
The double-well potential is given by:
\( E(\theta)=-\frac{\kappa_2}{2} \theta^2+\frac{\kappa_4}{4} \theta^4 \)
Here, the terms have specific implications:

• The \(-\frac{\kappa_2}{2} \theta^2\) term tends to drive the system towards \(\theta = 0\) (single-well behavior).
• The \(\frac{\kappa_4}{4} \theta^4\) term stabilizes the system at non-zero \(\theta\) values, leading to a double-well shape where two local minima exist.

This potential describes a scenario where an octahedron can adopt one of two preferred orientations, corresponding to the two minima. Such behavior is often associated with bistable systems undergoing phase transitions.

Soft Mode

The concept of a soft mode is crucial in understanding phase transitions. A soft mode is a vibrational mode whose frequency approaches zero as the phase transition point is reached. It indicates an instability in the system, suggesting a change in structure.
In this problem, the soft mode occurs at:
\( \mathbf{k} = \left( \frac{1}{2}, \frac{1}{2}, 0 \right) \)
This wave vector implies that there is a certain periodicity in the system, corresponding to a modulation in the orientations of the octahedra. Mathematically, soft modes are often examined by expanding the total energy in terms of Fourier components, representing the system's fluctuations and how they contribute to the phase transition.

Perovskite Structure

Perovskites are a class of materials with a specific crystalline structure that can be described as a cubic lattice with a general formula \(ABO_3\). Each unit cell in the lattice contains:

• A large A cation at the corners.
• A smaller B cation at the center.
• Oxygen anions at the face centers.

The octahedra mentioned in the problem refer to the geometric arrangement of the oxygen anions around the B cation. These octahedra play a critical role in determining the material's properties, as their rotations and distortions can lead to different phases and transitions. Understanding these octahedral motions is essential in analyzing the behavior of perovskites under various conditions.

Fourier Components

Fourier components are a mathematical tool used to analyze functions in terms of their sine and cosine components. They are particularly useful in studying periodic systems and phase transitions.
In the context of this exercise, Fourier components help describe the spatial variations in the orientations of the octahedra. When expanding the total energy expression in terms of Fourier components, each term corresponds to a different wave vector \(\mathbf{k}\), indicating the periodicity and amplitude of the oscillations.
For instance:

• A wave vector \(\mathbf{k} = \left(\frac{1}{2}, \frac{1}{2}, 0\right)\) indicates a specific pattern of rotation.
• By analyzing these components, one can identify soft modes and understand the conditions for phase transitions.

Using Fourier analysis allows for a clearer picture of how local interactions spread throughout the material, influencing its overall behavior and properties.