Questions tagged [density-of-states]

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Why can we approximate arbitrary volumes $V$ with cubic boxes of volume $L^3=V$ in quantum mechanics?

I've been studying quantum mechanics for two years now and it seems that in every textbook authors like to work with a box of size $L^3$ rather than an arbitrary volume $V$. Now the reason why seems ...
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1answer
46 views

Hit rate of molecules on a wall

Reviewing my final from last semester to prep for comps: Question: A piston of mass M can move freely in a tube with cross-section area A filled with ideal monoatomic gas with molecular mass m ≪ M and ...
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1answer
33 views

State density in one dimension

For a phonon we took in our lectures the state density for a 3D crystal and in order to find the number of states with an energy value between $[0,E)$ we did the division between the volume of the ...
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1answer
42 views

Is the density of states Lorentz invariant?

This is something that has been confusing me. A system can have a multitude of quantum states, and the energy of each will change depending on the frame of reference. However, the number of states ...
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25 views

Best way to calculate local density of states of tight binding model

I am currently trying to analyze a system using a tight binding model. I have a quite complicated unit cell with more than nearest neighbor hopping, that is repeating in one dimension. I have a matrix ...
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17 views

Dyson equation in terms of retarded Green function

Dyson equation, schematically written as $ G_0^{-1}-G^{-1}=\Sigma,$ holds for the causal Green function, which is the object used to formulate perturbation theory. However, working in a lattice model, ...
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29 views

Limits of integration on density of states in semiconductor

The density of electron states in a 3D semiconductor is given by $\rho(E)=\frac{1}{2\pi^2}\left(\frac{2 m^*}{\hbar^2}\right)^{3/2}\sqrt{E}$, derived commonly as shown here. I'm trying to understand ...
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14 views

How to prove the flat region for density of states $D(\epsilon)$ in tight binding electron model in 3D

The density of states for tight binding electron, $D(\epsilon)$, with respect to $\epsilon$(electron energy) followed an inverse cosine, a flat region ,and inverse cosine function in 3D. It was easy ...
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34 views

Find the density of states in X points of Silicon

The problem statement is given verbatim In Si, the dispersion relation at the [001] X points is: $$E=\frac{\hbar^2}{2}\left(\frac{k_x^2+k_y^2}{m_t}+\frac{(k_z-G)^2}{m_l}\right)$$ where G is the ...
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23 views

Effective density of states $N_c$ at different temperature for $\rm Si$

For Silicon at room temperature, Nc = 2.8x10^19 per unit volume. For 300K, m*/m = 1.81 for Silicon. Now Nc is proportional to 1.5th power of both temperature and m* (effective mass). So, at any other ...
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20 views

DOS for anisotropic 2D electron gas dispersion

If we start with the simple 2D isotropic-parabolic dispersion, \begin{align} E\left(\textbf{p}\right) & \approx\tilde{\varepsilon}_{0}+\alpha p_{y}^{2}+\alpha p_{x}^{2}, \label{1} \end{align} ...
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1answer
21 views

Density of States of 1D (ideal) Fermi Gas discrepency - Missing factor?

I wanted to find the density of states of a 1D ideal, noninteracting Fermi gas. My workings are below: $$D(\epsilon) = \frac{1}{2\pi}\int_{0}^{\infty}\delta(\epsilon-\epsilon_k)dk \times2$$ $$\...
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1answer
77 views

The density of states for free electron in conduction band

In Introduction to Solid State Physics, eighth edition, by Kittel, page 141, eqs. (20,21), the density of states for electron in conduction in three dimensions is $$D(\epsilon)\equiv \frac{dN}{d\...
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49 views

Why is the local density of states related to the retarded Green function?

Consider a Hamiltonian $H$ acting on the single-particle Hilbert space $\mathscr{H}$ representing lattice sites, i.e., $|r\rangle$ forms an orthonormal basis of $\mathscr{H}$ where $r$ ranges over ...
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2answers
49 views

Density of states in deferent dimension

Why the density of states in 2D is constant? Or in 3D why DOS is related to E^1/2 and in 1D and 0D how we can explain the relations physically?
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1answer
79 views

What is known about the density of states of the Anderson model?

This question was posted a week ago on MathOverflow without an answer: https://mathoverflow.net/questions/369156/what-is-known-about-the-density-of-states-for-the-anderson-model The Anderson Model is ...
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What happens to weight when ice melts? [closed]

A block of ice is weighed in a container. Then, it is left out to melt. Would the weight of the water be greater, less than, or equal to the ice? I know that it has something to do with density and ...
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48 views

Density of states for free electron confined to a volume

I'm confused on the difference in results I'm seeing for the density of states for a free electron (for example, a conduction electron in a metal). For one textbook (Kittel), I'm seeing that the ...
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1answer
57 views

How to calculate the number of distinct energy levels below a certain energy level? [closed]

The energy levels of a infinite square well is given by : $$\epsilon=\frac{h^2}{8ml^2}(n_x^2+n_y^2+n_z^2)=\frac{h^2}{8ml^2}r^2$$ The number of energy levels below a certain energy level for large ...
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1answer
33 views

Density of states - interpreting graph

I am trying to correctly wrap my head around the density of states concept, wonder if anyone can help.... When looking at the classical graph that is used to describe this concept, we have density of ...
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15 views

Degeneracy in linear tetrahedron method

In the linear tetrahedron method for the calculation of density of states, how does one circumnavigate the infinity error that would arise if two or more k-vertices of the tetrahedron have the same ...
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1answer
59 views

Motivation behind definition of density of state

The definition of density of state per unit volume stated in Girvin and Yang's Modern Condensed Matter Physics is $$\rho(E)=\int \frac{d^3k'}{(2\pi)^3} \delta(E-E')$$ I would like to gain more ...
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32 views

Is the number of spin states necessary in the density of states function?

I'm studying how to calculate the density of states in the final configuration in order to apply Fermi golden rule. For free EM field the following expression is the starting point: $$d^3n=\frac V {(2\...
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26 views

Density of States of electrons with tight binding and a Zeeman term in the Hamiltonian

Should density of states for electrons always be symmetric? The Hamiltonian I am considering is: $$ H= \Sigma_{<i,j>,\sigma} -t(C^{\dagger}_{i\sigma}C_{j\sigma}+ C^{\dagger}_{j\sigma}C_{i\sigma})...
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Show that the effective carrier density for electrons is $2\left(\frac{{m_c}^*k_BT}{2\pi\hbar^2}\right)^{3/2}$ [closed]

The 3-dimensional free electron density of states (DOS) including spin degeneracy is: $$g(E)=\frac{1}{2\pi^2}\left(\frac{2m_e}{\hbar^2}\right)^{3/2}\sqrt{E}$$ where $m_e$ is the electron mass, and $E$ ...
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53 views

Density of States

For getting the density of states formula for photons, we simply multiply the density of states for atoms by 2 (due to two spins of photons). I am getting the 2D density of states formula as :- g(p)dp ...
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2answers
75 views

Generalisation of the density states of phonons

Is it possible to generalize de density of states for phonons $\left( \left(\frac{L}{2\pi} \right )^3 \int \frac{dS_\omega}{v_g}\right)$ to a density of states which is also applicable to Bloch ...
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1answer
46 views

Density of states: Debye phonons vs free electons

In the Debeye approximation the density of states goes with phonon-energy^2, while the density of states for free electrons goes with sqrt(energy of the electrons), why is that? (I use Introduction ...
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68 views

Density of States (DOS) to energy graph

I am trying to find the amount of electrons in a conduction band in Si (Silicon), all I've got is a graph similar to this one: I've tried to integrate like this: $$ N = \int_{1}^{\infty} \frac{1}{1+\...
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61 views

2 different density of states for Graphene and Bismuth Selenide

I will solve the problem below by $1)$ working in reciprocal (wavenumber space) and in $2)$ energy space ($\epsilon$). But first some contextual background: Graphene is a single sheet of carbon ...
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1answer
525 views

Phonon density of states

How can I easily calculate phonon density of states from phonon dispersion? I want to compare DOS of graphene and Si from phonon dispersion. Is there a better alternative to Debye DOS = $\frac{w^2}{2\...
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$dE$ stands in my way to know the density of states in bulk crystal, how to get rid of it?

In a book about semiconductors, I found the following formula for the density of states: $$D(E)dE=\frac{(2m)^{3/2}E^{1/2}}{2\pi^2\hbar^2}dE. \tag{1}$$ In that book, the important lesson from this ...
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32 views

Density of states for free electron in one-dimension

I am trying to find the density of states of a free electron in one-dimension. I know that the result is given by $D(\epsilon) = 2 \frac{dn}{d\epsilon}$. However, I am unsure where this factor of 2 ...
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68 views

Inaccuracies of an energy level's degeneracy

I know that the inaccuracy is negligible but I'm trying to understand how it can be considered negligible in more detail. The formula for the discrete energy levels in a box with equal dimensions $L$ ...
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112 views

Physical explanation of temperature dependence of chemical potential

I have recently started my first course on statistical mechanics and have been learning about the Fermion gas. I was calculating the temperature dependence of chemical potential for an electron gas ...
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11 views

Can you add the density of states of two mono-atomic gases?

Say I have a system of 2 gases with $N_1$ and $N_2$ particles, each with respective masses $m_1$ and $m_2$. Would I be able to find the density of states for this system of two mixed gases $p(E)$ by ...
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599 views

What is the difference between the joint density of states and the density of state?

I think I understood the density of states, but I didn't understand the joint DOS. What is the main difference? What is the exact definition of the joint DOS? When do we use the joint DOS and when do ...
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1answer
145 views

Van Hove singularity (calculating the logarithm)

I am trying for hours to understand this calculation, I hope someone can help me with it. In the paper of Van Hove himself (https://doi.org/10.1103/PhysRev.89.1189) he derived the logarithmic ...
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1answer
59 views

Periodicity of density of states with decreasing dimension

In my lecture notes, there is the following graphic: With the 3D "bulk" configuration, there is clearly a $1/2$ power law, which I am able to explain by myself just by deriving the density of states ...
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1answer
144 views

Where does the Density of States come from?

There is an existing question here, which asks about the propagator for a free particle and the difference in its form when expressed as an integral over $p$ or over $E$. The accepted answer points to ...
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1answer
57 views

Number density of phonons in superfluid

I'm reading Superfluidity and Superconductivity by Tilley & Tilley. In section 2.4, the argument is made that the normal component of superfluid helium consists of phonons and rotons. The ...
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1answer
55 views

Density of States for a quantum well: Derivation?

Consider a quantum well, where we have: $E_{k_x,k_y,n_z}=\frac{\hslash^2k_x^2}{2m}+\frac{\hslash^2k_y^2}{2m}+f(n_z)$ with $k_x$ and $k_y$ having widths of $\frac{2\pi}{L}$ and $n_z$ varing in ...
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1answer
238 views

Density of states in irregularly-shaped volumes

A very common result is that the density of momentum states in a cubic volume is $\displaystyle\frac{V}{(2\pi\hbar)^3}$ in momentum space. How does this result extend to arbitrary volumes? Are there ...
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Where does the superconducting density of states come from?

I'm going through Tinkham (2nd edition) and in section 3.7.1 he makes a claim which seems quite weird: $N_n(\xi)$ is the Normal density of states (normal mode), so how would it be that the density of ...
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1answer
81 views

Find the density of states of a bouncing ball

Imagine a ball falling from a maximum height of $h$ and colliding with the ground at $z=0$. The ball only moves in the z-axis and the collisions are elastic. My job is to show that the density of ...
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1answer
55 views

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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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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2answers
154 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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39 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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122 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 $\...