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Questions tagged [density-of-states]

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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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16 views

Richardson-Dushman Equation using Momentum Space

How can we derive the Richard Dushman Equation using momentum space? J=B$_0$T$^2$e$^{-(\phi/kT)}$, B$_0$=4$\pi$qmk$^2$/h$^3$ Am$^{-2}$K$^{-2}$, T=Temperature in Kelvin, $\phi$ is work function of ...
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0answers
36 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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25 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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24 views

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
82 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
29 views

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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44 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
55 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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40 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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0answers
28 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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105 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
241 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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67 views

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
149 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
52 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
245 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
134 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
49 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
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1answer
93 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
377 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
443 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
159 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
52 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
238 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
1k 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
367 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 ...
1
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1answer
115 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
85 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
813 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
476 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
73 views

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
149 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
367 views

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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2answers
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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
732 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
78 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 ...
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1answer
109 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
305 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
150 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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2answers
313 views

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 ...
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0answers
114 views

Fermi's golden rule and the DoS of scattering states

Can the Fermi's golden rule $$\Gamma_{fi} ~=~ \rho(E_f) \frac{2\pi}{\hbar} |M_{fi}|^2$$ be applied for transitions of discrete states to scattering states? If yes, then what should the density of ...
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1answer
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What is the difference between these two expressions for the partition function, Z?

What is the difference between these two expressions given for the partition function, Z? $$Z = \sum_{i}e^{-\varepsilon_i/kT}$$ $$Z = \sum_{j} g_je^{-\varepsilon_j/kT}$$ where each energy level has ...
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0answers
130 views

Density of states from the Retarded Green's function for a rotating black hole

I have been studying the scattering of a scalar field around a rotating black hole in the near-horizon extremal limit. The radial solution provides the retarded Green's function, just by taking the ...
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0answers
221 views

Calculate 2D Effective mass from bulk effective mass

I am trying to create a self consistent Shrodinger Poisson Solver for various semiconductors. There is already one done by Professor Hu from UC Berkeley - QM CV Simulator. Looking at the code, they ...
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1answer
2k views

Density of states for arbitrary dispersion relation

If I have a 3D dispersion relation $E=E(k_x, k_y, k_z)$ I have an equation for the density of states, which is $D(E)=\frac{1}{\nabla_k E}\int\frac{dS}{(2\pi)^3}$ 1) I am confused about the ...
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1answer
3k views

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 ...