Questions tagged [density-of-states]

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Density of states (DOS) integral when surface is not closed

According to the density of states (DOS) formula $$\rho(\varepsilon)\propto \int_{\varepsilon=\text{const}}\frac{dS}{|\nabla_k \varepsilon_k|}.$$ Since there is an integral on the constant energy ...
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DOS of Van Hove singularity in 2D square lattice tight binding model

For the simplest example, 2D square lattice tight binding model gives the energy band as $$\varepsilon_k=-2t(\cos k_x+\cos k_y) \, .$$ We know that $\mathbf{k}=(0,\pi)$ and related momentum points are ...
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How to prove that sum converges to integral using density of states?

Essentially, I would like to prove $$\sum_k f(k) \to \int f(k) \rho dE \tag{1}$$ where $$\rho = \frac{dk}{dE} \tag{2}$$ is the density of states and $k \to \infty$. The model is that there is a ...
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Density of phonon states from dispersion relation

I have a dispersion relation $$\omega( \textbf{q} ) = \omega_0 \sqrt{ \sum_{j=1}^{D} \sin^2{\frac{q_ja}{2}}. }$$ Where D is the dimension D=1,2,3. And my excersise is to calculate (numericaly on a ...
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How to calculate density of states for different gas models?

There are a couple examples I'm trying to understand, all in a box/square of length $L$: For an ideal gas in 2-D with $\varepsilon=\frac{\hbar^2k^2}{2m}$:$$D(\varepsilon)=\frac{L^2m}{2\pi\hbar}\,.$$ ...
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What does it mean to divide by the degeneracy of the state in this textbook excerpt?

This section of Griffiths Introduction to Quantum Mechanics deals with Boltzmann, Fermi-Dirac, and Bose-Einstein distributions. I don't understand this line (highlighted in yellow): Let's talk only ...
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Photon density of states: Polarization/Helicity degree of freedom?

Sakurai's "Advanced Quantum Mechanics" states in Eq. (2.116) that the density of states of a single photon with $\vec k$ vector pointing into the solid angle $d\Omega$ is given by \begin{equation} \...
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Why phonon density of state depends on velocity autocorrelation?

We know that if we take the Fourier Transformation of velocity autocorrelation function, we will get the phonon density of state. But why phonon density of state depends on this? What is the physical ...
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Getting the density of states for photons

I know that the density of states $g(\epsilon)d\epsilon$ is the number of states in the energy range $[\epsilon, \epsilon + d\epsilon]$. I considered a system of non-interacting free photons in 3 ...
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Relation between band structure, dispersion, density of states, and the Fermi energy and Fermi level

Despite the long title, this question is mostly qualitative (although I am interested in quantitative results if possible). Say you have an electronic band structure (energy as a function of "k") for ...
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How does phonon scattering change the distribution function?

For a one-dimensional structure, we know that the modified distribution function has the following energy dependency in equilibrium: \begin{equation} Z(\varepsilon)\,f(\varepsilon) = \dfrac{N_\text{1D}...
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Density of states of 3D harmonic oscillator

Consider the following passage, via this image: 5.3.1 Density of states Almost all of the spin-polarized fermionic atoms that have been cooled to ultralow temperatures have been trapped by ...