How was this dynamical matrix solved?

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In summary, the author uses different symmetry points to calculate the frequencies at different points in the Brillouin zone. The calculation of the frequencies is changed when using different symmetry points because the coordinates of the points plugged into the frequencies equations are different. The best way to solve the eigenvalue problem is to use the formula ##D - \omega^2 I = 0## and plug in the coordinates of the desired symmetry point to obtain the frequencies at that point.
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nova215
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Solving this matrix for its eigenvalues produced the phonon spectrum of graphene - what was the method for finding the eigenvalues?
Starting on page 11 of this paper on lattice dynamics, the phonon spectrum of graphene is calculated. I do not really understand how the author used the matrix they created in order to calculate the spectrum. Thanks!
 

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He first defines the force-constants in real space (that is, the second derivative of the potential, see eq 1) and labels them ##\phi_{ij}(n, m)##. Once all the force-constants are defined, he uses the definition of the dynamical matrix (eq 4) to obtain the matrix D (which is just the "Fourier transform" of the dynamical matrix ##\phi_{ij}(n,m)##).

Finally, once you know the form of D, it is just a matter of solving the eigenvalue problem:
$$D - \omega^2 I = 0$$
(where ##I## is the identity matrix). He does this final step at the end of page 12 to obtain the values in eq (26-29)
 
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Thank you so much for the response!

Why does he vary his calculations for the different symmetry points and how is the calculation of the various frequencies changed when he uses different symmetry points? Also, what is the best way to solve that eigenvalue problem in your response?

I really appreciate the help!
 
  • #4
I don't understand your question. Once you solve the eigenvalue problem ##D - \omega^2 I = 0## (see the previous answer), you find four frequencies as a function of your wave-vector ##\mathbf k##:
$$\omega_i = \omega_i(\mathbf k) = \omega_i(k_x, k_y)$$
since this a two-dimensional problem, i.e. ##\mathbf k = (k_x, k_y)##. Now, if you are interested in the frequencies at a particular point of the Brillouin zone, you just plug in the coordinates of that point: for example, if you want to calculate the frequencies at M, which has coordinates ##(2 \pi/\sqrt{3}a, 0)##, you just evaluate ##\omega_i(2 \pi/\sqrt{3}a, 0)##.
If you want to plot the dispersion along the direction ##\Gamma \rightarrow M##, since both ##\Gamma## and ##M## have ##k_y = 0##, you simply need to calculate ##\omega_i(k_x, 0)## and, as in the paper, you'll find 4 mathematical expression as functions of ##k_x## (see eq. 26-29).
 

1. How does the dynamical matrix work?

The dynamical matrix is a mathematical tool used to describe the vibrational behavior of a physical system. It is a matrix of second derivatives of the potential energy with respect to the atomic coordinates, and it provides information about the frequencies and modes of vibration of the system.

2. What is the process for solving a dynamical matrix?

The process for solving a dynamical matrix involves several steps. First, the potential energy surface of the system is calculated using quantum mechanical methods. Then, the second derivatives of the potential energy with respect to the atomic coordinates are calculated and arranged into a matrix. This matrix is diagonalized to obtain the eigenvalues and eigenvectors, which correspond to the frequencies and modes of vibration of the system. Finally, the results are interpreted and used to understand the vibrational behavior of the system.

3. What are the limitations of solving a dynamical matrix?

One of the main limitations of solving a dynamical matrix is that it requires accurate calculations of the potential energy surface, which can be computationally expensive. Additionally, the dynamical matrix assumes that the system is in thermal equilibrium, so it may not accurately describe non-equilibrium systems. It also does not take into account the effects of anharmonicity, which can be important for some systems.

4. How is the dynamical matrix used in materials science?

The dynamical matrix is an important tool in materials science as it can provide information about the mechanical, thermal, and electronic properties of materials. It is used to understand the vibrational behavior of materials, such as phonon dispersion and thermal conductivity. It can also be used to predict the stability and phase transitions of materials, as well as to design new materials with desired properties.

5. Can the dynamical matrix be solved for any system?

The dynamical matrix can be solved for any system as long as the potential energy surface can be calculated and the system is in thermal equilibrium. However, the accuracy of the results may vary depending on the complexity of the system and the computational resources available. Additionally, the dynamical matrix may not be suitable for non-equilibrium systems or those with strong anharmonicity.

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