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Density of States, Photoemission and Integrating the Number of States

I'm reading Fowler's theory on photoemission. I'm stuck on a part which Fowler helpfully identifies as "obvious". Fowler sets up the free electron model, suggesting that electrons need a ...
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Density of states - as a function of electron velocity

A condensed matter textbook states that in a 3D gas of electrons obeying the Fermi-Dirac statistics the number of electrons per unit volume having velocity components in the ranges $u, u+d u$, $v, v+d ...
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What causes bands to shift in energy or become narrow?

This is the density of states for pure palladium in bulk : This is the density of states of palladium hydride : clearly the d bands are shifting and becoming narrow. The narrowing of bands is said ...
Ajaykrishnan R's user avatar
2 votes
1 answer
56 views

Relationship between Density of States and the Fermi level

My understanding is that given the DOS of a material we find the fermi level by filling electrons into those energy levels and when we run out of electrons we reach close to the fermi level (or ...
Ajaykrishnan R's user avatar
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Change in internal chemical potential of electrons in the 2DEG

I was reading about quantum capacitance and came across the following formula: $$\Delta \mu = \frac{N}{\rho}$$ where N is the number of electrons moved from the metal to the low-density-of-states ...
Blackwidow's user avatar
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Volume occupied by mechanical states, and density of states

Suppose we express the phase space volume occupied by all mechanical states $(\mathbf{q},\mathbf{p})$ with energy equal to or less than mechanical energy $E$ as: \begin{equation}\Phi(E,\lambda)=\int\...
Adrien-Marie Deschamps's user avatar
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Is the inverse relation between the DOS and the energy gradient in $k$-space only valid in 1D systems?

In my solid state course, I was taught that Van Hove singularities can be explained because the DOS follows: $$\mathrm{DOS} \propto \frac{1}{\nabla_k E} .$$ However, the DOS of a 3D system is ...
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Recover non-relativistic density of state

The density of state of a non-relativistic particle ($E = \hbar^2k^2/2m$) in 3D is: $$\rho_{class}(E) = \dfrac{V}{4\pi^2}\left(\dfrac{2m}{\hbar^2}\right)^{3/2}E^{1/2}.$$ The density of state of an ...
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What does energy density state for holes in valence band mean and how is this equation applicable there?

Equation 1 is the energy density function for free particles confined in a 3D box. Now we could apply this equation to conduction band of a semiconducting material taking the electrons present there ...
AYM Shahriar Rahman's user avatar
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Density of states for non-interacting Bosons

I am tasked with calculating the density of states in terms of the angular frequency given the dispersion relation. But I couldn't help but think: why can't we calculate the density of states by ...
SAMS's user avatar
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The dimension of Spectral Function

For a free electron gas model with volume $V$, the spectral function is $$A(k,\epsilon)=2\pi\delta(\epsilon-\epsilon_k)$$ so the density of states, $$D(\epsilon)=\frac{1}{V}\sum_k\delta(\epsilon-\...
steven's user avatar
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How to compute the Auger transition rate?

I've been working on computing the Auger transition rate for an exotic atom, one where an electron has been replaced by a particle of arbitrary mass and (integer) charge. I've been closely following ...
Gabriel's user avatar
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1 answer
74 views

Boundary condition for density of state in momentum space

I am working on fermions in metal, where you need $\frac{2}{V}\int_{0}^{P_{f}}\frac{L^{3}}{(2\pi\hbar)^{3}}dP$ to get the total number of occupied states, where the coefficient $\frac{L^{3}}{(2\pi\...
QFT's user avatar
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2 votes
1 answer
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Quasiparticle Density of States

In this paper we have the following: The corresponding quasiparticle density is given by the equation $$n_{qp} = 4N_0 \int_\Delta ^\infty dE \frac{E}{\sqrt{E^2 - \Delta^2}} f(E),$$ where $N_0$ is the ...
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Degeneracy in a 2D planar cavity (box potential)

Assume a finite 2D planar cavity. One can write the energy of a photon in this cavity as $$ \begin{equation} E(k_x, k_y)=\hbar c \sqrt{k_x^2+k_y^2+k_z^2}, \end{equation} $$ where $k_z$ is fixed (hence ...
Andris Erglis's user avatar
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1 answer
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Computation of Single-particle Density of States for a particle in a container [closed]

I have a system of N particles stored in a cylindrical container (radius $R$, height $L$). I need to determine the single-particle density of states, which we've defined as: $$ g(\epsilon) = \frac{1}{...
Claudio Menchinelli's user avatar
-1 votes
1 answer
39 views

Density of states for fermions - Statistical Mechanics [closed]

I'm studying about the density of states $g(k)$ of fermions at the moment and its equations, but I'm confused about the following: This is an equation given in the notes for the course, where $N$ is ...
user374355's user avatar
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Density of states; when states are discrete

Consider simple fermi Dirac distribution $f(\epsilon)$, where to get the total number of particles($T=0$), we need to do the following integration $\int_0^{e_f}f(\epsilon)\mathcal{D}(\epsilon)\,d\...
Debamalya Dutta's user avatar
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Does the number of electrons in a material affect the density of states in this material?

The Fermi-Dirac distribution, given by $$f(E) = \frac{1}{1 + \exp\left(\frac{E - E_{\text{F}}}{k_{\text{B}} T}\right)}$$ ​ describes the probability that a state with energy $E$ is occupied by an ...
Nicolas Schmid's user avatar
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Integral of the square of the spectral density in a quantum field theory

The quantity of interest is $$ \int_0^\infty dE \, \rho(E)^2 $$ where $\rho(E)$ is the spectral density in a Quantum Field Theory. I am wondering whether it has any physical meaning, and it can be ...
knuth's user avatar
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What (which) is the correct expression for the occupation probability in the time-dependent Anderson-Newns model?

First things first. Let us consider a system consisting of a particle (atoms, molecules) interacting with the electrons of a metal substrate described by a Hamiltonian \begin{equation} H = H_{metal} + ...
ae_chan's user avatar
4 votes
4 answers
498 views

DOS in Fermi Golden Rule

I was reading second chapter of Introductory Nuclear Physics by Kenneth S.Krane, and in that chapter he was giving about the logic of why there must be a factor of $\rho(E_{f})$ in the decay ...
Anshul Sharma's user avatar
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Question about Debye's and Einstein's Heat Capacity Theory

I am reading the phonon part of Omar's textbook, but I am a bit confused over the way through which the distribution of the frequencies of oscillators are determined. In Einstein's model all ...
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Density of states for a 1D Fermi gas in the gravitational field

We are considering a 1D Fermi gas in a gravitational field. The energy levels are given as $E_n = mgh_0n^{2/3}$ and we are asked to calculate the density of states $D(E)$ for the case that $E_0=mgh_0 \...
bcserven's user avatar
1 vote
1 answer
62 views

Phase Space Contributions to Decays with Fixed Momentum

Neglecting helicity, consider the decays $π^-\rightarrow e^-\bar{ν_e}$ and$π^-\rightarrow μ^-\bar{ν}_μ$. I understand that from Fermi's golden rule, the decay probability should be proportional to the ...
Poo2uhaha's user avatar
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1 answer
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Confusion in calculation of density of states

So, when we calculate density of states of the electrons/ phonons, or while during a similar calculation of no. of modes per frequency in the Blackbody radiation, we assume Standing wave solutions: $...
Shikhar Chamoli's user avatar
0 votes
1 answer
181 views

Degeneracy in 3 dimensional infinite well and Number of states between $E$ and $E + e$

I was reading Statistical Mechanics by Mcquarrie and met a function that describes the number of states of a single particle that has equal of less energy than $E$ in a 3d infinite well. The book ...
sasha kirilova's user avatar
3 votes
0 answers
951 views

Bethe ansatz and density of states for XXX spin chain

Consider the 1 dimensional Heisenberg antiferromagnet with Hamiltonian $$ H = J\sum_{i=1}^L \vec S_i \cdot\vec S_{i+1}$$ and periodic boundary conditions. I understand that this can be solved exactly ...
proteus7's user avatar
1 vote
1 answer
288 views

What is the relation between degeneracy and density of states?

The definition of degeneracy and density of states (DOS) looks resembling to each other, what is the relation and difference between these two concepts? And is there a mathematical relation or formula ...
Gaelthorn's user avatar
2 votes
0 answers
121 views

Density of States and the Spectral Measure

The spectral theorem states that for any self adjoint operator $H$ on some Hilbert space $\mathcal{H}$, there exists a projection-valued measure $E_H$ such that $$H= \int_{\mathbb{R}} \lambda \mathrm{...
Connor Mooney's user avatar
3 votes
2 answers
260 views

Conditions for a sum to become an integral using the density of states?

In most basic statistical physics/condensed matter discussion the density of states is used to convert a discrete sum to a continuous integral $$\sum_{\alpha} \mapsto \int d\epsilon \ g(\epsilon).$$ ...
user246795's user avatar
3 votes
1 answer
196 views

Dirac delta of matrix argument - Matrix model path integral vs Hilbert space

Assume a Hamiltonian $H$ with $N$ orthonormal eigenstates $\{\vert n\rangle\}$ of energies $\epsilon_n$. One can define a density of states, \begin{align} \rho(E)&=\mathrm{tr}\,\hat{\delta}(E-\hat{...
vrata's user avatar
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1 answer
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Can I calculate partial density of states (PDOS) using tight binding approximation? [closed]

I am using tight binding approximation for a 2D material by 2*2 Hamiltonian, and I have ploted the density of states correctly. Can I also calculate the partial density of states using tight binding? ...
Mohammad Mortezaei Nobahari's user avatar
1 vote
1 answer
114 views

Interpretation of negative density of states and fourth type Van Hove singularity?

If I have the energy of a free electron gas in three dimensions centered at energy $E_0$, $$E_n(k)=E_0+\frac{\hbar^2}{2}\left(\frac{k_x^2}{m_x}+\frac{k_y^2}{m_y}+\frac{k_z^2}{m_z}\right),$$ I can ...
Juan Pedro Martinez's user avatar
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0 answers
140 views

Why do defect energy states appear only in the band-gap?

It has been shown that defects (due to doping for example) in a semiconductor cause a "tail" to appear in the density of states. Why do these states appear in the bandgap,and how is the ...
ajaxd9's user avatar
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0 votes
1 answer
126 views

Density of states of Fermi gas derivation

I'm going over this book. While deriving the gensity of states for a gas of fermions the author makes the following argument: Remember that we are treating the gas as having a set of states that can ...
Sgg8's user avatar
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2 votes
1 answer
364 views

Density of final states in photon absorption/emission by a hydrogen atom

Consider a hydrogen atom in an electromagnetic field. The Hamiltonian is of the form $$\hat{H}=\underbrace{\frac{\hat{p}^2}{2m}+V(r)}_{\text{atom}}+\underbrace{\sum_{\vec{k},\sigma}\hbar cka^{\dagger}...
Mr. Feynman's user avatar
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1 vote
1 answer
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Compute density of states in momentum space and in phase space

In the book Cosmology by Daniel Baumann, I encountered the following claim: Solving the Schrödinger equation with periodic boundary condicions gives: $$\vec{p}=\dfrac{h}{L}(r_1\hat{x}+r_2\hat{y}+r_3\...
Wild Feather's user avatar
1 vote
0 answers
98 views

The DOS effective mass

If we consider the spin-orbit coupling in semiconductors, it is known that the degeneracy of the valance band is lifted up and we got 2 sub-bands the light hole and the heavy hole that are still ...
Ayoub EL-Amrani's user avatar
1 vote
1 answer
98 views

Calculating the One-Particle Density of States of the ripplon Gas

I'm trying to understand an example that I found in my notes but I don't understand the difference in my results compared to how my teacher did it. It's from Statistical Physics and it seems that ...
bsaoptima's user avatar
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DOS for arbitrary volume

One can easily derive for the Density of States (DOS) of photons $D(\omega)=\frac{V\omega^2}{\pi^2c^3}$ by assuming that the volume is a cube. Is it possible to apply this formula also for different ...
Silas's user avatar
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1 answer
71 views

Angular-momentum resolved density of states

It is well known that in 3d, for a non-relativistic, free particle the density of states scales as $D(E) \propto E^{1/2}$. The problem is, we can classify the eigenstates according to the total ...
poisson's user avatar
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2 answers
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Integral to express the density of states in the microcanonical ensemble

I was exploring a problem in the microcanonic ensemble, when we have to systems that exhange heat and in the solutions I came across two possible ways to express the total density of states. $D(E)= \...
Ana Branco's user avatar
6 votes
2 answers
773 views

On the statistical meaning of density of states (DOS)

According to the so-called law of the unconscious statistician: The expected value $\langle \cdot \rangle$ of a measurable function of ${\displaystyle X}$, ${\displaystyle g(X)}$, given that $X$ has ...
ric.san's user avatar
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1 vote
0 answers
73 views

Microcanonical density of states of the two body system

For a two body system with hamiltonian $H=\frac{P^2}{2M}+\frac{p^2}{2\mu}-\frac{Gm^2}{r}$ and assuming minimal distance between particles r>a, and some large volume V containing the system, I am ...
Nitzan R's user avatar
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1 vote
2 answers
1k views

Phase space density of $N$ harmonic oscillators

For one classical harmonic oscillator with Hamiltonian $$H = \frac{p^2}{2m}+\frac{m\omega^2}{2}x^2$$ the density of states can be calculated as by calculating the number of states with Energy smaller ...
Physics101's user avatar
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68 views

How does the energy gap in the all-to-all random Ising (Sherrington-Kirkpatrick) model scale with system size?

Recently, I asked the question Must spin glasses really have an exponential density of states close to the ground state?. Here, I give a related, more specific question, whose answer may give steps ...
user196574's user avatar
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2 votes
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Must spin glasses really have an exponential density of states close to the ground state?

I'm a complete beginner to spin glasses. I'm not even sure of the definition; I've mostly seen examples, like Sherrington-Kirkpatric with all-all pairwise normally distributed Ising interactions. ...
user196574's user avatar
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1 vote
1 answer
743 views

Is the occupation number and density of states equation correct?

The relationship between occupation number (which is the number of particles at a certain energy level) and the density of states is as follows: $$n(E) = D(E)F(E)$$ where $D(E)$ is the DOS and $F(E)$ ...
Ajaykrishnan R's user avatar
2 votes
1 answer
109 views

How is differential momentum assigned in multiparticle system of QFT?

I've been following Schwartz's book on quantum field theory, and got stuck at page 59 on Section 5.1 'cross section' of the book which argues that the region of final state momenta is the product of ...
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