Questions tagged [density-of-states]

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Derivation of density of states (free electrons)

I am reading Condensed matter physics from M.Marder. This is the derivation for the density of states for free electrons. $\begin{aligned} D(\mathcal{E}) &=\int[d \vec{k}] \delta\left(\mathcal{E}...
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287 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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325 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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141 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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137 views

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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Validity of formula for density of states for free electron gas

I believe the formula for density of states given by $\frac{\hbar^2}{2m}{(\frac{3\pi^2N}{V})}^{\frac{2}{3}}$ is a good approximation to actual count only when $E_f$ is much larger than energies given ...
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19 views

Calculating density of state from a certain dispersion relation analytically

This used to be a really frustrating question when I was taking the solid state physics. It is still a knotty one for me now. I wish to figure it out completely. If one reads the definitions or ...
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11 views

What is a pseudo wave and how to calculate it?

In a description of the density of states in https://wiki.fysik.dtu.dk/gpaw/documentation/pdos/pdos.html I find the term pseudo (partial) wave (in the section Molecular Orbital PDOS). What is it and ...
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124 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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202 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 ...
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181 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 ...
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67 views

Density Of states derivation

In the aspect of density of state derivation or simply assuming the frequency of a solid as a continuous distribution we have to come up with an equation expressing the density of states. Its derived ...
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22 views

Density of states LL in graphene [closed]

I am using the Kernel Polynomial Method to determine the spectral density of a 2DEG system that has been sujected to a perpendicular magnetic field B. I wish to determine (a) What the amplitudes of ...
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56 views

1D density of states help

I am given a 1D band whose energy is $$E(k)=E_0-t\cos(ak)$$ Then I have to compute the DOS relative to that band. Here is my calculation: $$g(E)=\dfrac{1}{L}\dfrac{dN}{dk}\dfrac{dk}{dE}$$ where $\...
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21 views

Density of states for fermions for temperatures other than absolute zero

Let's say I wanted to calculate the density of states for $\mathrm{2D}$ Fermions with an energy dispersion of $E_F=ck^6$ at $T=0$ The process would be quite mathematically simple. I'd take the ...
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3answers
6k views

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 ...
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371 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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Density of states, $\pi$ vs $2\pi$ [duplicate]

When calculating density of states for an electron, some arrive that k (the wavenumber) is $2\pi n x /L $ and some say it is $\pi n x /L$? Where L is the dimension of the well I’ve heard vague ...
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24 views

Density of states in 1D semiconductor

I'm given the dispersion relation for the energy band of my semiconductor: $E(k)=\alpha+\beta\cos(ka)$ where $a, \alpha, \beta$ are know parameters and I must obtain the density of states from that. ...
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35 views

Momentum in breit wigner cross section

The Breit Wigner cross section derived in my lecture notes is $\sigma = \frac{g\pi}{p_i^2}\frac{\Gamma_{Z\rightarrow i}\Gamma_{Z \rightarrow f}}{(E-E_0)^2+\frac{\Gamma^2}{4}}$ where $g$ is the spin ...
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Confusion about average KE in free electron model via Density of States (DOS)

Assume a low temperature regime in which levels up to the Fermi Level, $E_F$, are populated. I have evaluated the density of states in energy space as $$D(E)=\frac{L^3}{\pi^2\hbar^3}(2m_e^2E)^{1/2},$...
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575 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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241 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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62 views

Whats the function of density of states equation?

There was a question in Statistical Mechanics 3rd ed by RK Pathria and PD Beale, section 3.8, asking to show that the harmonic oscillator obeys the equipartition theorem. It was well proven. But at ...
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Why do the $C_v$ of gapless systems have a power law behaviour?

The functional dependence of the heat capacity $C_v$ of systems with gapless excitations (e.g., lattice with acoustic phonons, Heisenberg ferromagnet with spin waves etc) is like a power law $$C_v\sim ...
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62 views

Calculate phonon density of states

I need to calculate phonon density of states for a cuprate superconductor. I know there is a general formula for the calculation of phonon density of states by Einstein models like this $$D(\omega)=(\...
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27 views

Distribution of photons emitted by atoms

I am currently revising quantum gases, and a small but confusing thought experiment has been bugging me for a while. I understand the bookwork stuff on photons and how a photon gas in a blackbody ...
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65 views

Weighted histogram analysis method (WHAM) equations

I am struggling in deriving the WHAM equations. Among others, I follow this paper by Kumar et al. In the appendix, in eq. 24 they write the density of states as a weighted sum over the "measured" ...
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1answer
107 views

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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36 views

What is the density of states $SiO_2$?

We build the model by the finite element method. In our model here is silicon dioxide (SiO2). To carry out calculations, it is necessary to know the density of states and the effective mass. Question:...
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164 views

Density of states for a tight binding model

So we have been given a dispersion relation of the form: $$ E=6-2(\cos k_xa+\cos k_ya) $$ and asked to calculate the density of states. The equation for the density of states is (eq 2.48 from here ...
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Is the DOS (density of states) wrong for degenerated case?

The density of states (DOS) is defined as $$\mathcal{N}\left(\lambda\right)=\sum_{n=1}^{M}\delta\left(\lambda-\lambda_{n}\right).$$ We can then get $$\int d\lambda\mathcal{N}\left(\lambda\right)=M,$$ ...
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92 views

Summing over quantum states

For a system of $N$ identical particles we deal in quantum mechanics with wave functions $\langle \{\mathbf{r}_i \} \mid \Psi \rangle=\Psi(\mathbf{r}_1,\dots,\mathbf{r}_N)$ from which determine the ...
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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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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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58 views

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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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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834 views

Direct definition of density of states

I've been studying statistical mechanics and in the book there's something the author calls density of states which he introduced in a kind of indirect way. Basically, the author argues that if we ...
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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 ...
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1answer
406 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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103 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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350 views

How does the density of states for black-body radiation change with geometry?

If I have a hollow conducting cylinder with another conducting cylinder inside it (as with a coaxial cable), would the density of states of the photons/radiation between the two cylinders be any ...
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39 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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589 views

Why is the density of states in $k$-space constant?

Why are the allowed states in $k$-space equidistant in every direction? As a consequence of this, the density of states for phonons in 3D is $$\frac{V}{(2\pi)^3}$$ while for electrons it is $$2 \frac{...
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406 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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187 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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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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577 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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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 &...