Questions tagged [density-of-states]
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The importance of the Debye frequency for superconductivity and the relation of it to critical temperature and the superconducting gap
I am preparing for an exam in superconductivity at the moment and one of the questions to prepare for is the one mentioned in the title. My preliminary answer would be that the Debye frequency $\...
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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 $F(E)$ ...
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How to calculate two-particle spectrum/density of states
In quantum many body theory, there is a convenient process for calculating the single particle density of states using the imaginary-time Green's function $$\mathcal{G}(k,i\omega)= \langle \psi(k,i\...
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How is differential momentum assigned in multiparticle system of QFT?
I've been following Schwartz's book on quantum field theory, and got stuck at page 59 on Section 5.1 'cross section' of the book which argues that the region of final state momenta is the product of ...
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Determining the Number of Points in the $n$-Space
Electron gas is a collection of non-interacting electrons. If these electrons are confined to certain volume (for example, cube of metal), their behavior can be described by the wavefunction which is ...
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Density of states equal to 1?
I have found a problem with my quantum mechanics background theory, more specifically regarding the identity operator and the momentum density of states.
The following equation
$$<p|\hat{x}|\psi>...
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Degeneracy of Photons
The density of states for a photon gas is defined by,
$$D(\epsilon)=\frac{g}{2\pi^2}\frac{\epsilon^2}{(\hbar c)^2} $$ where g is the number of independent internal states for a photon. The question is ...
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Motivation for density of states [duplicate]
I have not found a good book on statistical mechanics that explains the quantity density of states well. The books I have read so far make the continuum limit approximation, which does not make much ...
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Density of states of a finite potential well
Considering a finite square potential well.
The solution of it gives the isolated bound states (below zero) and continuous scattering states (above zero). Here the isolated and continuous are the ...
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Density states of phonons with different speed at different directions:
I was trying to get the density of states $g(v)$ for phonons which have $2$ polarizations transversally, and $1$ polarization longitudinally.
My approach:
$$dN = \frac{1}{8} 4 \pi n^2 dn, \lambda = c/...
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On $k$-space density of states and semiclassical transport
I am reading Chapter 12 of Ashcroft and Mermin and I have a great many questions, but one sticks out in particular.
As background, we note that it can be shown quite generally (by applying Born-von ...
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Allowed energy levels in an $E$-$k$ diagram
For a particle confined in an infinite potential well in 1D, the $k$ value is quantized as $k=nπ/a$, where $a$ is the length of the region where $V(x)=0$. However, the $E$-$k$ diagram derived from ...
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Why discontinuous density of states will cause isolated energy pole when perturbation is applied to it?
Recently I'm struggling with Green's functions. It is said that when G(E) diverges around E_0 ,the density of states at ...
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Why are single-electron states occupied at $T=0$ in superconductors?
Some textbooks show a density of states where the single-electron states below the Fermi energy are occupied at $T=0$ (see picture). However, I thought that at $T=0$ all electrons are paired. Hence ...
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Density of States of Free Electrons in a Magnetic Field
In lectures we have been shown that the density of states (DOS) of 3D free electrons in a magnetic field is:
$$D(E) = \frac {m^\frac32}{\sqrt{2}\pi^2\hbar^2} \sum^{v_{max}}_{0} \frac {\hbar\omega_c}{\...
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What is "Density of States" and how does one generally find it?
I'm walking out of a graduate quantum midterm kicking myself because I was asked to compute density of states as a function of energy for a spin $1/2$ particle of mass $m$ in a hard wall box of length ...
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Density of states: Ashcroft Mermin, Chapter 9 question no2
Ashcroft Mermin chapter 9 number 2
Hey, I'm just sutck on the problem 2 of the Ashcroft Mermin. I did prove the density of state but I'm having a hard time finding $k_\text{min}$ and $k_\text{max}$ (...
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Justification and interpretation of Fermi Golden Rule (second order) in Resonance Energy Transfer (RET)
I hope someone can give me some new insight to understand this.
Fermi's golden rule is wildly used to calculate the rate for RET. I have some difficulties in understanding its justification and ...
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Density of states in microcanonical ensemble for discrete and continuum energy spectrum
I'm introducing myself to statistical mechanics using two books: Introduction to Statistical Physics by S. Salinas, and Statistical Physics of Particles, by Mehran Kardar. Both textbooks work on an ...
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Derivation of Density of states in 3D
In our lecture today, the professor introduced the concept of density of states. We found the expression of it, for the 3D case, but no steps were shown and also we did not specify the system at hand ...
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How could one "empirically" calculate density of states?
I've been studying statistical mechanics and recently came across an interesting optional challenge.
The Einstein model and the Debye model of solids are common ways of describing the heat capacities ...
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Phase space density for Bose-Einstein condensation
A figure of merit for Bose-Einstein condensation is the phase space density which can be defined as
$$\rho=n\lambda_T^3,$$
where $n$ is the number density of atoms and $\lambda_T$ the thermal de ...
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Going from 2D dispersion relation to density of states
If one was to have a 2D dispersion say: $$\varepsilon(k)=k_x^2-k_y^2$$
We know the dispersion relation generally can be written as:$$ D(\varepsilon)=\sum_{k_x}\sum_{k_y}\delta(E-\varepsilon(k_x,k_y)$$
...
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Counting the number of allowed energy states of a particle in a 3D box
I am reading a book on Modern Physics by Thornton and Rex. I am looking at a particle in a 3-dimensional infinite square well potential (particle in a box). If the particle is a photon, its energy can ...
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Single particle density of states for non-free particle
I am trying to find the single particle density of states in terms of the energy, for a system with the single particle 2D Hamiltonian:
$$H=\frac{p^2}{2 m}+\alpha x \text { with } 0<y<L, x>0$$...
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Partition Function from Dispersion Relation in Molecular Dynamics
I’ve seen there are ways to compute the dispersion relation of a crystal from molecular dynamics. An example of how to do this is discussed in this question: Computing phonon dispersion from molecular ...
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Average value of energy in statistical mechanics
I haven't taken any classes in Statistical Mechanics, but in studying Structure of Matter I found some ideas I'm not very familiar with, related with the average value of energy ($E$). Given a $p(E)$ ...
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Energy of a Free electron gas in D dimensions
I am trying to calculate the internal energy of a free electron bas in a box in $D$ dimensions. To calculate the density of states, I used the following formula:
$$g(E) = \int \frac{d^Dk}{(2\pi)^D} \...
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Is density of states temperature dependent? [duplicate]
I have learning about the free electron gas in 3 dimensions is a box of size $L$ and volume $L^3$ at zero temperature. For calculations at zero temperature, I have been using the density of states ...
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Is quark-quark interaction possible under extreme heat and pressure circumstances?
The neutron-rich core of neutron stars, underneath extreme pressure and heat, undergoes a phase transition to quark-gluon plasma. If both heat and pressure are increased to exceed beyond the TOV limit,...
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Numerical calculation of density of states
I am trying to figure out the numerical interpretation of density of states for a fermionic system under a periodic potential.
The equation for the density of states reads
$$
DOS(E) = \sum_{k \in BZ, ...
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How can the total number of particles can be sum over all Density of states?
It is known that Density of states of a lattice structure is the probability distribution function of energy (for reference). This means that the sum/Integral of DOS over all energies is $1$. However, ...
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Why is the conductance of a metal-insulator-metal tunnel junction parabolic?
For bias voltages below the tunnel junction barrier heights (and below the Fowler-Nordheim limit), tunnel junctions have a parabolic conductance as a function of bias. Is this due to the metallic ...
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Density of states misunderstanding in Statistical Mechanics
In the simple model of a box filled with an ideal gas, one may write the total energy as the sum of kinetic energies of all particles
$$E = \sum_{i=1}^N\frac{\vec{p}_i^2}{2m}$$
and so if you construct ...
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Recent review on high density equation of state?
Can anyone suggest a recent review on the equation of state of the matter at high (nuclear and above) densities?
I would like this review to contain both astrophysical applications, mainly for neutron ...
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Relation between critical temperature and density of states
The BCS theory predicts that the critical temperature of the superconducting transition is given by
$$
T_c \approx \theta \exp \left (- \frac{1}{U D(\epsilon_F)} \right )
$$
where $\theta$ is the ...
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Double and triple sum into integral with density of states
I asked this on mathematical forum section.
I'm triying to expand the results of certain calculation, where the author has the following kind of sums:
$$
\sum_{j} A(\omega_j) n(\omega_j),
$$
where $A$ ...
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Normalization of phonon density of states
Analogous to electronic structure calculations, we can solve for dispersion band structure of phonons for lattices using harmonic lattice approx. And we can find the so-called phonon density of states ...
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Canonical ensemble: what if the phase space density is not known?
In canonical ensemble, the probability is defined as
\begin{equation}
P(E)=\frac{g(E)\exp(-E/T)}{Z},
\end{equation}
and the partition function is defined as
\begin{equation}
Z(T)=\int_0^{\infty}dE\,g(...
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Constructing the density of states of multiple independent harmonic oscillators
I have a system of $N$ uncoupled 1D quantum harmonic oscillators, each with its own frequency $\omega_i$. The density of states for a single quantum harmonic oscillator shall be defined as
$$
\rho(E) =...
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Density of states and boundary conditions: how the density of states is physical if it depends on box size
This question is closely related to this one: Why is the density of states required conceptually? Should it be seen as a mathematical trick related to Fourier series?
But it was suggested that I ask ...
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Why is it valid to always use the particle in a box model in density of states calculations?
Suppose we would like to calculate the density of states of some 3-D system given the dispersion relation $\omega = f(k)$. In every such example I have come across (for instance, with phonon ...
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Why is the density of states required conceptually? Should it be seen as a mathematical trick related to Fourier series?
[edit]: My misunderstanding is more precisely asked here: Density of states and boundary conditions: how the density of states is physical if it depends on box size :it was suggested to open a new ...
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How to compute density of states for very large numbers of atoms?
I have a code to compute density of states of an Hamiltonian $H$. I'd like to compute it for a very large number of atoms (Currently I'm performing simulations with 1000 atoms, with polarization the ...
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Problem understanding what densities of states represent
I understand that the one particle partition function for a particle in a box can be written as: $$Z_1 = \sum_{k_x} \sum_{k_y} \sum_{k_z} (2s+1) e^{- \beta \epsilon( \vec k) } $$
My first question is ...
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Why don't we consider the negative values of $k_x, k_y$ and $k_z$ when we compute the density of states of a 3D infinit square well?
I watched some derivations of the density of states in a square box of length L with potential $V=0$ for the points inside the box and $V=\infty$ outside the box.
Using separation of variable one can ...
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How to do Wang-Landau sampling when the energy configurations are unknown?
My question is about estimating the density of states for boolean formula. Where the problem is that given a boolean formula $F$, with $n$ variables the energy is a function $E:\{0,1\}^{n}\rightarrow \...
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Density of quantum states allowed
For a 3D particle in a box, the density of states (or the number of allowed states with a wave vector whose magnitude lies between $k$ and $k + dk$ is) is given by: $$g(k) dk = \frac{V k^2 dk}{2 \pi^2}...
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Density of states of one classical harmonic oscillator
I have to determine the density of states of one tridimensional harmonic oscillator. I have to prove that the expression is the following $D(E) = aE^2$, a is a constant.
I know this is a 6-dimensional ...
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Longitudinal conductivity from density of states (DOS)
It is well-known that using the so-called Streda formula, the transversal conductivity $\sigma_{xy}$ and thus the Hall conductivity in a two-dimensional material is given as the derivative of the ...
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