Questions tagged [density-of-states]
The density-of-states tag has no usage guidance.
2
votes
0answers
30 views
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 ...
0
votes
0answers
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 ...
0
votes
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 < \...
0
votes
0answers
25 views
Fourier transform involving Bessel functions
I need help finding the Fourier transform of the function
$$
\rho(\vec{r}) = \alpha \delta_{\vec{r},0} \left(\lambda\lambda' J_1
(\beta |\vec{r}|)Y_1(\beta |\vec{r}|) - \pi^2 J_0
(\beta |\vec{r}|)Y_0(...
0
votes
0answers
24 views
Density of states after scattering
I need some help with a probably simple question because I'm not sure whether my approach is correct.
Let the free Green's function of a system on a discrete lattice described by a massless Dirac ...
2
votes
1answer
82 views
Liouville's Theorem For Spacetime
Liouville's theorem states that the density of phase space governs the continuity equation.
$$\frac{\partial\rho}{\partial t}+\sum_{i=1}^n\Big(\frac{\partial(\rho\dot{q_i})}{\partial q_i}+\frac{\...
2
votes
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 ...
0
votes
0answers
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=...
2
votes
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 $\varepsilon=\frac{\hbar^2k^2}{2m}$:$$
D(\varepsilon)=\frac{L^2m}{2\pi\hbar}\,.$$
...
0
votes
0answers
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?
0
votes
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
$$g(\epsilon)=\begin{cases}\sqrt{\epsilon-\epsilon_0} & \text{for }\epsilon > \epsilon_0 \\ 0 &...
3
votes
0answers
105 views
Density of states for a system of $N$ particles (both interacting and non-interacting case)
The density of states $\rho(\epsilon)$ for a single particle (with $g$ number of internal states such as spin) confined in a volume $V$ can be calculated as $$\rho(\epsilon)=\frac{4\pi gV}{h^3}p^2\...
0
votes
0answers
14 views
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?
0
votes
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 ...
4
votes
0answers
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_\...
0
votes
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 ...
1
vote
0answers
39 views
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 ...
1
vote
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 ...
1
vote
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 = \...
0
votes
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 ...
0
votes
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
0
votes
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 ...
-2
votes
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{...
0
votes
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 ...
0
votes
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?
1
vote
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? ...
1
vote
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 ...
1
vote
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
votes
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
vote
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
votes
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
votes
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
votes
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 ...
3
votes
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+\...
0
votes
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) ...
1
vote
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 $\omega$.
$$ \rho(E) = \frac{m}{2\pi^2\hbar^3} \int d^3r \sqrt{2m(E-V(\vec{r}))}\Theta(...
-1
votes
2answers
1k views
Density of States in 2D Tight Binding Model
Hello I am trying to find the density of states for the dispersion relation:
$$E(k_x,k_y) =\cos(k_x a) -\cos(k_y a),$$
over an entire period, not simply around the minimum. For a crystal of length $L,$...
1
vote
2answers
732 views
Partition function for continuous energy
The partition function has usually two definitions: the first is for discrete microstates with energies $E_i$. In this case it is defined as
$$Z = \sum_{i} e^{-\beta E_i},$$
where $i$ ranges over ...
4
votes
0answers
421 views
Landau quantization: degeneracy of first level
In some books the degeneracy of one Landau level in a two-dimensional gas of free electrons is calculated in the following way:
Note: The electron spin is not considered.
Number of states of a free ...
0
votes
1answer
78 views
Practical way of expressing the $\delta$-function [closed]
I have got a problem in using the $\delta$-function. As we know, this function is often used to define a 'density'-related quantity. Such as the density of states or some correlation function. Take ...
0
votes
1answer
109 views
Density of states from $k$ to $E$
Speaking about Quantum mechanics, considering the "particle in a box" condition as an approximation of the electrons condition in a semiconductor, let the material be represented by a volume $V$ with ...
0
votes
1answer
305 views
Fermi Dirac distribution and degenerate energy states
In Quantum Mechanics and in semiconductor materials, the number of electrons $N$ in conduction band is usually computed as follows:
$$N = \int_{E_c}^{+\infty} g_c(E)f(E)dE$$
where $g_c(E)$ is the ...
2
votes
0answers
150 views
Density of States for a separable hamiltonian
There are $N$ non interacting electrons in a potential well:
\begin{align}
H&= -{1 \over 2 } \nabla^2 + U(x,y,z) \\
U(x,y,z)&={1\over2}\omega^2z^2 \; \mbox{for} \; (x,y) \in [0,L]\times [0,L]; ...
5
votes
2answers
313 views
Rigorous definition of density of states for continuous spectrum
For operators with pure point spectra it is clear how to count number of states corresponding to a given eigenvalue - one can just calculate the dimension of eigenspaces. I am wondering how to do it ...
0
votes
0answers
114 views
Fermi's golden rule and the DoS of scattering states
Can the Fermi's golden rule
$$\Gamma_{fi} ~=~ \rho(E_f) \frac{2\pi}{\hbar} |M_{fi}|^2$$
be applied for transitions of discrete states to scattering states? If yes, then what should the density of ...
1
vote
1answer
75 views
What is the difference between these two expressions for the partition function, Z?
What is the difference between these two expressions given for the partition function, Z?
$$Z = \sum_{i}e^{-\varepsilon_i/kT}$$
$$Z = \sum_{j} g_je^{-\varepsilon_j/kT}$$
where each energy level has ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
