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📌How to Derive the Hamiltonian for Phonons in a Crystal Lattice

27 Mar

Understanding the dynamics of atoms in a crystal lattice is crucial for exploring material properties such as thermal and electrical conductivity, elasticity, and more. This post will walk you through the derivation of the Hamiltonian for phonons—quantized sound waves or vibrations in a crystal lattice—using a step-by-step approach. Let’s dive into the mathematical framework that allows us to describe these vibrations in terms of phonon modes using Fourier transforms.

1. Starting with Classical Dynamics

In a crystal, atoms are arranged in a repeating pattern and can oscillate around their equilibrium positions. These oscillations can be modeled using classical mechanics principles.

Classical Kinetic Energy: For a lattice with atoms of mass mmm, the kinetic energy is given by: \boldsymbol{Kinetic Energy = \frac{1}{2m} \sum_{n} p_n^2}.​ Here, \boldsymbol{p_n}​ is the momentum of the n-th atom.

Classical Potential Energy: For nearest-neighbor interactions, the potential energy, assuming a spring-like interaction, is: \text{Potential Energy} = \frac{1}{2} \sum_{n} K (u_{n+1} - u_n)^2, where K is the spring constant between neighboring atoms.

2. Transitioning to Reciprocal Space (Fourier Transform)

To simplify the problem, we use Fourier transforms to express both displacement and momentum: \boldsymbol{u_n = \sum_k A_k e^{ikna}} and \boldsymbol{p_n = \sum_k P_k e^{ikna}}, where \boldsymbol{A_k} and \boldsymbol{P_k}​ are the Fourier components of displacement and momentum, respectively, and \latex \boldsymbol{a}$ is the lattice constant.

3. Expressing Energy in k-space

Kinetic Energy in k-space: By substituting the Fourier expansions into the kinetic energy expression and using the orthogonality of the exponential terms, we get: \text{Kinetic Energy} = \frac{N}{2m} \sum_k P_k P_{-k} where N is the total number of lattice sites.

Potential Energy in k-space: Similarly, substituting into the potential energy and simplifying using the properties of exponential sums leads to: \text{Potential Energy} = \frac{N}{2} \sum_k 2K \sin^2\left(\frac{ka}{2}\right) A_k A_{-k}.​

4. Constructing the Hamiltonian

Combining these energy expressions, the Hamiltonian for the system in k-space becomes: H = \frac{1}{2} \sum_k \left(\frac{N}{m} P_k P_{-k} + m \omega_k^2 A_k A_{-k} \right), where: \omega_k = \sqrt{\frac{2K}{m} \sin^2\left(\frac{ka}{2}\right)}. This expression encapsulates the total energy (kinetic plus potential) of the lattice vibrations.

5. Quantum Mechanical Formalism

In quantum mechanics, we promote the A_k, A_{-k}, and P_kP_{-k}​ to operators and introduce phonon creation and annihilation operators to quantize the Hamiltonian. This framework allows us to study the properties of phonons and their interactions within the crystal.

Conclusion

This derivation illustrates how classical mechanics concepts, combined with Fourier analysis and quantum mechanics, provide a comprehensive description of lattice dynamics. The resulting Hamiltonian for phonons enables physicists to predict various material properties and understand deeper phenomena in solid-state physics.

This blog post is intended for educational purposes, offering a step-by-step approach to understanding complex physical concepts in material science and condensed matter physics. Whether you are a student or a researcher, mastering these principles is crucial for further exploration in the field.

Detailed derivation here:

Step 1: Expressing Kinetic Energy in k-space

Classical Kinetic Energy:

The kinetic energy of a lattice in real space is given by: \text{Kinetic Energy} = \frac{1}{2m} \sum_{n} p_n^2​

where p_n​ is the momentum of the n-th atom. Transforming this momentum to k-space: p_n = \sum_k P_k e^{ikna}.

Substitute into Kinetic Energy Expression:

\text{Kinetic Energy} = \frac{1}{2m} \sum_{n} \left(\sum_k P_k e^{ikna}\right) \left(\sum_{k'} P_{k'} e^{ik'na}\right)

Using Orthogonality of Exponential Terms:

We employ the orthogonality of exponential terms to simplify the kinetic energy expression, leveraging the properties of the discrete Fourier transform and the periodic boundary conditions of the lattice:

\sum_{n} e^{i(k+k')na} = N \delta_{k,-k'}

where \delta_{k,-k'}​ is the Kronecker delta, which is non-zero only when k' = -k. This simplification leverages the periodic boundary conditions and the properties of the discrete Fourier transform. Substituting back, we get: \text{Kinetic Energy} = \frac{N}{2m} \sum_k P_k P_{-k}

Step 2: Expressing Potential Energy in k-space

Classical Potential Energy:

The potential energy in a linear chain with nearest-neighbor interactions can be described as:

\text{Potential Energy} = \frac{1}{2} \sum_{n} K (u_{n+1} - u_n)^2

This equation accounts for the energy due to the displacement differences between adjacent atoms in the chain, where K is the spring constant.

Transforming Displacements into k-space

To simplify the analysis and to handle complex systems more efficiently, we use Fourier transforms to convert the displacements from real space to reciprocal space (k-space). The displacement of the n-th atom and its next neighbor in the chain are represented as follows:

u_n = \sum_k A_k e^{ikna}
u_{n+1} = \sum_k A_k e^{ik(n+1)a} = \sum_k A_k e^{ika} e^{ikna}

Potential Energy in k-space

Substituting these Fourier transformed displacements into the potential energy expression, we get:

\text{Potential Energy} = \frac{1}{2} \sum_{n} K \left( \sum_k A_k e^{ika} e^{ikna} - \sum_k A_k e^{ikna} \right)^2

This expands to:

\text{Potential Energy} = \frac{1}{2} \sum_{n} K \left( \sum_k (e^{ika} - 1) A_k e^{ikna} \right)^2

Simplifying the Expression Using Orthogonality

To simplify further, apply the orthogonality properties of the exponential functions:

\text{Potential Energy} = \frac{1}{2} K \sum_n \sum_k \sum_{k'} (e^{ika} - 1)(e^{-ika} - 1) A_k A_{k'} e^{i(k-k')na}

Using the identity:

\sum_n e^{i(k-k')na} = N \delta_{k,k'}

This reduces the expression for potential energy in k-space:

\text{Potential Energy} = \frac{N}{2} K \sum_k |e^{ika} - 1|^2 A_k A_{-k}
\text{Potential Energy} = \frac{N}{2} \sum_k K (2 - 2\cos(ka)) A_k A_{-k}

The final formula correctly represents the potential energy due to lattice vibrations analyzed in k-space, offering a framework for examining dynamics and thermodynamic properties of crystal lattices in solid-state physics.

Substituting Displacements into the Potential Energy Expression

After transforming the displacements into k-space, substitute these into the potential energy equation for nearest-neighbor interactions. The transformation and subsequent squaring of the difference between adjacent displacements yield:

Transformed Potential Energy Expression:

\text{Potential Energy} = \frac{1}{2} K \sum_n \left( \sum_k A_k e^{ika} e^{ikna} - \sum_k A_k e^{ikna} \right)^2

This expands to:

\text{Potential Energy} = \frac{1}{2} K \sum_n \left( \sum_k (e^{ika} - 1) A_k e^{ikna} \right)^2

Breaking Down the Expanded Expression

Expanding the squared term and applying the Fourier transform properties, the potential energy expression becomes:

\text{Potential Energy} = \frac{1}{2} K \sum_n \sum_k \sum_{k'} (e^{ika} - 1) A_k (e^{-ika} - 1) A_{k'} e^{i(k-k')na}

Simplifying Using Orthogonality of Exponential Functions

Utilize the orthogonality property of the exponential terms, where the sum over all lattice points n introduces a Kronecker delta \delta_{k,k'}​, simplifying the double summation over wave vectors k and k′ by collapsing it when k \neq k':

\sum_n e^{i(k-k')na} = N \delta_{k,k'}

Thus, the potential energy in k-space simplifies to:

\text{Potential Energy} = \frac{N}{2} K \sum_k |e^{ika} - 1|^2 A_k A_{-k}

This is further simplified using the identity |e^{ika} - 1|^2 = 2 - 2\cos(ka)|:

\text{Potential Energy} = \frac{N}{2} \sum_k K (2 - 2\cos(ka)) A_k A_{-k}

Final Expression for Potential Energy in k-space

The final expression for potential energy, reflecting the contributions of phonon modes characterized by their wave vectors in the crystal lattice, is:

\text{Potential Energy} = \frac{N}{2} \sum_k 2K \sin^2\left(\frac{ka}{2}\right) A_k A_{-k}

This formatted sequence provides a clear step-by-step transformation of the potential energy from real space to reciprocal space, highlighting the mathematical manipulation and simplifications used in solid-state physics to handle complex crystal lattice dynamics effectively.

Using Orthogonality of Exponential Terms:

Start by applying the orthogonality property of the exponential terms, which simplifies the expression significantly:

\sum_n e^{i(k-k')na} = N \delta_{k,k'}

This relationship indicates that the sum over all sites n is non-zero only when k equals k′′, which is a fundamental property used in Fourier analysis, simplifying the sums over wave vectors.

Transforming Potential Energy into k-space:

With this simplification, the potential energy in the crystal lattice, which was initially expressed in terms of real-space displacements, can now be fully expressed in k-space:

\text{Potential Energy} = \frac{1}{2} K N \sum_k |e^{ika} - 1|^2 A_k A_{-k}

Breaking Down the Potential Energy Formula:

The term |e^{ika} - 1|^2| can be expanded using Euler’s formula, where e^{ix} = \cos(x) + i\sin(x), thus:

|e^{ika} - 1|^2 = (e^{ika} - 1)(e^{-ika} - 1) = (e^{ika} - 1)(e^{-ika} - 1) = 2 - 2\cos(ka)

Therefore, substituting back into the potential energy expression gives:

\text{Potential Energy} = \frac{N}{2} \sum_k K (2 - 2\cos(ka)) A_k A_{-k}

Final Expression for Potential Energy in k-space:

Simplifying further using the trigonometric identity for cosine, the potential energy expression becomes:

\text{Potential Energy} = \frac{N}{2} \sum_k 2K \sin^2\left(\frac{ka}{2}\right) A_k A_{-k}

This final expression reflects the potential energy contribution from all phonon modes in the lattice, characterized by their wave vector k, and encapsulates the effects of lattice vibrations in terms of their amplitude A_k​ and the wave vector-dependent sinusoidal term.

This method of transforming and simplifying potential energy into k-space is crucial for analyzing the dynamics and thermodynamic properties of materials in the field of solid-state physics. The use of k-space representations allows physicists to handle complex interactions within the crystal lattice effectively, paving the way for deeper understanding and innovative material science applications.

Step 3: The Hamiltonian in k-space

After obtaining the expressions for kinetic and potential energies in terms of the Fourier components of the displacements and momenta, we can construct the Hamiltonian for the system. The Hamiltonian, which represents the total energy (kinetic plus potential) of the system, is a crucial tool in quantum mechanics and solid state physics for studying the dynamics of lattice vibrations.

Hamiltonian Expression:

The Hamiltonian in k-space is given by:

H = \frac{N}{2} \sum_k \left(\frac{1}{m} P_k P_{-k} + 2K \sin^2\left(\frac{ka}{2}\right) A_k A_{-k} \right)

This expression combines the transformed kinetic and potential energy terms:

  • Kinetic Energy Term: \frac{N}{m} P_k P_{-k} represents the kinetic energy in kkk-space, where PkP_kPk​ and P−kP_{-k}P−k​ are the momentum components for wavevector kkk and its negative, reflecting the preservation of momentum in the lattice.
  • Potential Energy Term: 2K \sin^2\left(\frac{ka}{2}\right) A_k A_{-k} encapsulates the potential energy due to displacements from equilibrium positions, characterized by the sinusoidal square term that varies with the wavevector kkk.

Simplified Hamiltonian with Angular Frequency:

We can introduce the angular frequency \omega_k​ for the phonon modes to simplify the expression:

\omega_k = \sqrt{\frac{2K}{m} \sin^2\left(\frac{ka}{2}\right)}

Thus, the Hamiltonian simplifies to:

H = \frac{1}{2} \sum_k \left(\frac{N}{m} P_k P_{-k} + m \omega_k^2 A_k A_{-k} \right)

This form of the Hamiltonian clearly indicates the relationship between the wavevector-dependent angular frequency \omega_k​ and the kinetic and potential energies of the system, providing a comprehensive description of the energy distribution among the phonon modes in the lattice.

Conclusion

The Hamiltonian in k-space effectively describes how the total energy of the crystal lattice is distributed among various phonon modes, each characterized by a distinct wavevector k. This framework is essential for exploring physical phenomena such as heat capacity, thermal conductivity, and other dynamic properties of materials at the microscopic level. By understanding these principles, researchers can predict material behavior under various conditions, facilitating advancements in material science and solid-state physics.

This derivation combines classical lattice dynamics with the properties of Fourier transforms under periodic boundary conditions, allowing the expressions for kinetic and potential energy in terms of momentum and displacement in kkk-space to be derived step by step.

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