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Questions tagged [density-of-states]

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How does an object change their form? [on hold]

We know a composite object such as an apple or a rock change its shape in relation to its composition and what serve as the composition of a composite object are molecules, atoms, etc. which are ...
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1answer
33 views

Problem with finding the density of states of an $N$-body system

I am having problems solving a particular problem in my Statistical Mechanics course. We have a system that is composed of $N$ non-interacting particles each of mass $m$. The particles are bound to ...
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1answer
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Is this equation for the density of states of an elastic isotropic material an approximation?

The density of states of phonons can be calculated with $$ Z(\omega)=\frac{V}{(2\pi)^3}\int_{\omega=\text{const}}\frac{d\vec{f}_\omega}{|\vec\nabla\omega|} $$ where $\omega$ is the phonon frequency ...
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2answers
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Deriving density of states in different dimensions in k space

The results for deriving the density of states in different dimensions is as follows: 3D: $g(k)dk = 1/(2\pi)^3 4 \pi k^2 dk$ 2D: $g(k)dk = 1/(2\pi)^2 2 \pi k dk$ 1D: $g(k)dk = 1/(2\pi) 2 dk$ I get ...
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0answers
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Physical interpretation of Total and element resolved density of states

[ What can be the physical interpretation of Total and partial(element resolved) density of states as given in picture. Here redline is representing the fermi level. How we can relate it to the ...
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68 views

Density of states of bosons in function of momentum with energy $\epsilon = cp$

I am working out an average number N of bosons of spin $S = 0$ connected to a two-dimensional domain with surface A. The gas is ultrarelativistic with a single particle energy $\epsilon = cp$. The ...
2
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1answer
81 views

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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When can we say $x$ and $p$ are “independent variable”, in order to find the Vlasov equation?

I have a question about "independent variable" in physics, and especially variable in Lagrangian or Density Function. I read several questions about it in this forum and although I have the feeling I ...
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49 views

Density of states as a function of dimension: what happens between 3D and 4D?

Consider a parabolic dispersion $\varepsilon_{\boldsymbol q} = \frac{\boldsymbol |q|^2}{2m}$. The two-particle density of states $\rho_2(\boldsymbol k, \varepsilon)$ is zero for $\varepsilon < \...
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29 views

Fourier transform involving Bessel functions

I need help finding the Fourier transform of the function $$ \rho(\vec{r}) = \alpha \delta_{\vec{r},0} \left(\lambda\lambda' J_1 (\beta |\vec{r}|)Y_1(\beta |\vec{r}|) - \pi^2 J_0 (\beta |\vec{r}|)Y_0(...
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0answers
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Density of states after scattering

I need some help with a probably simple question because I'm not sure whether my approach is correct. Let the free Green's function of a system on a discrete lattice described by a massless Dirac ...
2
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1answer
98 views

Liouville's Theorem For Spacetime

Liouville's theorem states that the density of phase space governs the continuity equation. $$\frac{\partial\rho}{\partial t}+\sum_{i=1}^n\Big(\frac{\partial(\rho\dot{q_i})}{\partial q_i}+\frac{\...
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0answers
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The formula for the average number of fermions $\langle N \rangle$

In the context of Fermi gases (or fluids in general), one would typically in the grand-canonical formalism use the formula $\langle N \rangle = -\frac{\partial \psi}{\partial \mu}$, where $\psi$ is ...
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0answers
64 views

Density of States General Formula

So I have this formula regarding the density of states in 3D and 2D respectively are given by a function of the dispersion relation: $N=\frac18 \frac{4\pi}{3}(V^{\frac13}\frac{k(\omega)}{\pi})^3$ $N=...
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2answers
146 views

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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79 views

Band structure and Density of states (DOS)

Can someone explain how these two plots are related? How are the peaks in the right are associated with the left figure?
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31 views

Finding the chemical potential of a system

For a system of non interacting electrons at temperature T the density of states is given by $$g(\epsilon)=\begin{cases}\sqrt{\epsilon-\epsilon_0} & \text{for }\epsilon > \epsilon_0 \\ 0 &...
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0answers
143 views

Density of states for a system of $N$ particles (both interacting and non-interacting case)

The density of states $\rho(\epsilon)$ for a single particle (with $g$ number of internal states such as spin) confined in a volume $V$ can be calculated as $$\rho(\epsilon)=\frac{4\pi gV}{h^3}p^2\...
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How dispersion relation is used when calculating the density of available states in a metal crystal? [duplicate]

found this question somewhere and still do not know how is it useful for calculation of density of state?
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2answers
319 views

Calculating density of states for $N$ identical Fermions subject to a potential

Given a system with N identical Fermions, with spin $\frac{1}{2}$ and mass $m$, subject to the potential: $V(\vec{r}) = \frac{1}{2}m\omega^{2}(x^{2}+y^{2}+z^{2})$ and the single particle energy ...
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Density of states from band structure

Let the density of states be given by $$ g(\epsilon) = \int \frac{\mathop{d^3 q}}{4\pi^3} \delta(\epsilon - \epsilon(\vec{q})), $$ where $\epsilon(\vec{q}) = \frac{\hbar^2}{2m}q_\perp^2 + h_\pm(q_\...
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1answer
217 views

Calculating density of states given energy levels and degeneracy

In my statistical mechanics class, my professors did a problem in which he calculated the density of states, however I am having trouble justifying his approach. I did the problem beforehand in an ...
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0answers
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Density of States and Quantum Jumps

The specific question that I'm working on is "If I have a particle in the bound state of a 1-D delta function potential at $t = - \infty$, and I apply a harmonic perturbation $V(x,t) = V_0xcos(\omega ...
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1answer
98 views

Are there two kinds of boundary conditions for Density of States of electron in 1D/2D/3D bulk?

when I'm going through the online Density of States(DOS) deriving courses, I find that there seem to be 2 set of boundary condition which will lead to the same result. Note: K is the one in p=(h/2π)K ...
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1answer
316 views

Density of states for free particle [closed]

I am asked to find the density of states for a free particle as a function of $|p|$, $\Delta|p|$ and $\Delta V$. I also have the expression for the number of states $\Delta\Omega(p) = g(p)\Delta p = \...
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1answer
152 views

Density of states of electrons

The problem is as follows: Let the density of states of the electrons in some sample be assumed to be a constant D for $\epsilon > 0$ ($D=0$ for $\epsilon<0$) and the total number of electrons ...
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1answer
59 views

Calculate total number of electrons from density of states? [closed]

Can someone tell me how to calculate the total number of electrons from partial density of states (projected on each atom)? Thanks
0
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1answer
137 views

Phonon Density of State

Suppose there is a material such as graphane which has two atoms, Carbon and Hydrogen (I mean that it has two or more atoms). How can I calculate phonon Density of State (DoS) for such system? Can I ...
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2answers
508 views

Probability density in Quantum mechanics and mass of particle

Is there any relationship between the probability density in QM given by \begin{equation} P\left(t\right) = \int\limits_{V} \psi^{*}(\mathbf{x},t)\psi(\mathbf{x},t)\mathrm{d}^{3}\mathbf{x} \end{...
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1answer
595 views

Physical interpretation of density of states

The following pictures shows some typical examples of density of states: and for superconductor: Here is my interpretation: Energy level zero represents Fermi energy level. Positive energy ...
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1answer
204 views

Is there a relation between density of states (DOS) and carrier mobility in semiconductors?

By changing DOS, mobility how to change? What is the relationship between DOS and mobility, if there is?
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1answer
56 views

Clarification on how to calculate the density of states

I'm a little confused on how this works. The question asks to calculate the density of states $$\nu(E) = \int_0^{\infty}dp\,\delta(E-E_p)$$ where $$E_p = \frac{p^2}{2m}$$ This should be simple right? ...
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0answers
290 views

Effective mass for density of states calculation and for conductivity calculation

In silicon, for the effective mass for density of states calculation, electron mass (1.08) is more than hole mass (0.81). Whereas, the effective mass for conductivity calculation, hole mass (0.386) is ...
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1answer
2k views

The fermi gas model of a nucleus

In my nuclear physics lecture we learned about the "fermi gas modell" of a nucleus with which I have some problems. First the potential for the nucleons is shown in the picture below and it makes ...
2
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1answer
437 views

Getting tight binding density of states more accurately

I calculated numerically the density of states (DoS) for the 3-D tightbinding dispersion $\epsilon(k_x,k_y,k_z)=-2t\,(\cos k_x + \cos k_y + \cos k_z)$ and obtained the following plot [$t=1$ has been ...
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1answer
141 views

density of states of bogoliubov quasiparticles

For a simple fermionic system the formula for calculating the density of states(DOS) is $N(E) = \sum_{n}\delta(E-E_{n})$ where $\{E_{n}\}$ is the set of eigenvalues obtained after diagonalizing the ...
3
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1answer
106 views

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 ...
2
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1answer
1k views

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} \...
2
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1answer
591 views

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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0answers
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Passing from a discrete summation to a continous integal

I'm having trouble understanding the math behind a step in an explanation of BCS theory. At one point the superconductor gap $\Delta$ is defined as \begin{equation} 1 = V \sum_q \frac{1}{\xi_q^2+\...
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1answer
189 views

Number of states $\Omega(E)$ derivation (Reif problem 2.4)

Problem 2.4 from Reif's "Fundamentals of Statistical and Thermal Physics": Consider an isolated system consisting of a large number $N$ of virtually non-interacting localized (not translating) ...
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0answers
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Calculate density of states of particle in a 3D-harmonic oscillator

Problem: Calculate the density of states of a particle with mass $m$ in a 3D-harmonic oscillator with frequency $\omega$. $$ \rho(E) = \frac{m}{2\pi^2\hbar^3} \int d^3r \sqrt{2m(E-V(\vec{r}))}\Theta(...
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Density of States in 2D Tight Binding Model

Hello I am trying to find the density of states for the dispersion relation: $$E(k_x,k_y) =\cos(k_x a) -\cos(k_y a),$$ over an entire period, not simply around the minimum. For a crystal of length $L,$...
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2answers
955 views

Partition function for continuous energy

The partition function has usually two definitions: the first is for discrete microstates with energies $E_i$. In this case it is defined as $$Z = \sum_{i} e^{-\beta E_i},$$ where $i$ ranges over ...
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Landau quantization: degeneracy of first level

In some books the degeneracy of one Landau level in a two-dimensional gas of free electrons is calculated in the following way: Note: The electron spin is not considered. Number of states of a free ...
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1answer
81 views

Practical way of expressing the $\delta$-function [closed]

I have got a problem in using the $\delta$-function. As we know, this function is often used to define a 'density'-related quantity. Such as the density of states or some correlation function. Take ...
0
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1answer
115 views

Density of states from $k$ to $E$

Speaking about Quantum mechanics, considering the "particle in a box" condition as an approximation of the electrons condition in a semiconductor, let the material be represented by a volume $V$ with ...
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1answer
376 views

Fermi Dirac distribution and degenerate energy states

In Quantum Mechanics and in semiconductor materials, the number of electrons $N$ in conduction band is usually computed as follows: $$N = \int_{E_c}^{+\infty} g_c(E)f(E)dE$$ where $g_c(E)$ is the ...
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0answers
172 views

Density of States for a separable hamiltonian

There are $N$ non interacting electrons in a potential well: \begin{align} H&= -{1 \over 2 } \nabla^2 + U(x,y,z) \\ U(x,y,z)&={1\over2}\omega^2z^2 \; \mbox{for} \; (x,y) \in [0,L]\times [0,L]; ...
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Rigorous definition of density of states for continuous spectrum

For operators with pure point spectra it is clear how to count number of states corresponding to a given eigenvalue - one can just calculate the dimension of eigenspaces. I am wondering how to do it ...