Questions tagged [density-of-states]
The density-of-states tag has no usage guidance.
211
questions
1
vote
0
answers
26
views
How to derive elliptic integral of the first kind from $\int_{-\pi}^{\pi}\frac{1}{\sqrt{(\eta^{2}/2 - 3/2 -\cos 2\theta)^2+\cos^2\theta}}\ d\theta$ [migrated]
Spent several days already trying to figure out how to convert the given integral to this: $$\frac{8}{\sqrt{(\eta-1)^3(\eta+3)}}K\left(\sqrt{\frac{16\eta}{(\eta-1)^3(\eta+3)}}\right)$$
where
$$
K\left(...
- 11
0
votes
0
answers
18
views
Why do defect energy states appear only in the band-gap?
It has been shown that defects (due to doping for example) in a semiconductor cause a "tail" to appear in the density of states.
Why do these states appear in the bandgap,and how is the ...
- 1
0
votes
1
answer
51
views
Density of states of Fermi gas derivation
I'm going over this book. While deriving the gensity of states for a gas of fermions the author makes the following argument:
Remember that we are treating the
gas as having a set of states that can ...
- 304
2
votes
1
answer
136
views
Density of final states in photon absorption/emission by a hydrogen atom
Consider a hydrogen atom in an electromagnetic field. The Hamiltonian is of the form
$$\hat{H}=\underbrace{\frac{\hat{p}^2}{2m}+V(r)}_{\text{atom}}+\underbrace{\sum_{\vec{k},\sigma}\hbar cka^{\dagger}...
- 1,515
0
votes
1
answer
52
views
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\...
- 392
1
vote
0
answers
27
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 ...
1
vote
1
answer
36
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 ...
- 121
0
votes
0
answers
25
views
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 ...
- 384
1
vote
1
answer
34
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 ...
- 1,535
0
votes
2
answers
72
views
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)= \...
- 31
6
votes
2
answers
446
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 ...
- 1,386
1
vote
0
answers
52
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 ...
- 129
0
votes
2
answers
143
views
Phasespace density of $N$ harmonic oscilators
For one classical harmonic oscillator with Hamiltonian
$$H = \frac{p^2}{2m}+\frac{m\omega^2}{2}x^2$$
the density of states can be calculated as by calculating the number of states with Energy smaller ...
- 31
1
vote
0
answers
31
views
How does the energy gap in the all-to-all random Ising (Sherrington-Kirkpatrick) model scale with system size?
Recently, I asked the question Must spin glasses really have an exponential density of states close to the ground state?. Here, I give a related, more specific question, whose answer may give steps ...
- 1,360
2
votes
0
answers
40
views
Must spin glasses really have an exponential density of states close to the ground state?
I'm a complete beginner to spin glasses. I'm not even sure of the definition; I've mostly seen examples, like Sherrington-Kirkpatric with all-all pairwise normally distributed Ising interactions. ...
- 1,360
0
votes
0
answers
17
views
Wavevector and the thermal excitation of magnons
Please refer to page 334 of introduction to solid state physics, 8th edition, by Charles Kittel.
Here is a screenshot of the portion of the text where my doubts lie:
Equation (25):
$\bar{h}\omega = (...
- 867
0
votes
0
answers
98
views
The importance of the Debye frequency for superconductivity and the relation of it to critical temperature and the superconducting gap
I am preparing for an exam in superconductivity at the moment and one of the questions to prepare for is the one mentioned in the title. My preliminary answer would be that the Debye frequency $\...
0
votes
1
answer
197
views
Is the occupation number and density of states equation correct?
The relationship between occupation number (which is the number of particles at a certain energy level) and the density of states is as follows:
$$n(E) = D(E)F(E)$$
where $D(E)$ is the DOS and $F(E)$ ...
- 183
0
votes
0
answers
45
views
How to calculate two-particle spectrum/density of states
In quantum many body theory, there is a convenient process for calculating the single particle density of states using the imaginary-time Green's function $$\mathcal{G}(k,i\omega)= \langle \psi(k,i\...
- 154
1
vote
0
answers
65
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 ...
- 37
0
votes
1
answer
65
views
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 ...
- 413
0
votes
0
answers
42
views
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
$$<p|\hat{x}|\psi>...
- 149
0
votes
0
answers
253
views
Degeneracy of Photons
The density of states for a photon gas is defined by,
$$D(\epsilon)=\frac{g}{2\pi^2}\frac{\epsilon^2}{(\hbar c)^2} $$ where g is the number of independent internal states for a photon. The question is ...
- 381
0
votes
1
answer
77
views
Motivation for density of states [duplicate]
I have not found a good book on statistical mechanics that explains the quantity density of states well. The books I have read so far make the continuum limit approximation, which does not make much ...
- 370
1
vote
0
answers
131
views
Density of states of a finite potential well
Considering a finite square potential well.
The solution of it gives the isolated bound states (below zero) and continuous scattering states (above zero). Here the isolated and continuous are the ...
- 31
0
votes
0
answers
28
views
Density states of phonons with different speed at different directions:
I was trying to get the density of states $g(v)$ for phonons which have $2$ polarizations transversally, and $1$ polarization longitudinally.
My approach:
$$dN = \frac{1}{8} 4 \pi n^2 dn, \lambda = c/...
- 588
4
votes
1
answer
312
views
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 ...
- 743
0
votes
1
answer
56
views
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 ...
- 461
0
votes
1
answer
62
views
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 ...
- 1
1
vote
1
answer
90
views
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 ...
- 13
0
votes
0
answers
94
views
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}{\...
- 187
1
vote
2
answers
265
views
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 ...
- 221
1
vote
0
answers
210
views
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}$ (...
- 29
5
votes
0
answers
145
views
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 ...
- 69
2
votes
1
answer
189
views
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 ...
- 962
0
votes
1
answer
896
views
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 ...
- 1,154
2
votes
1
answer
61
views
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 ...
- 21
0
votes
1
answer
303
views
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,$$
where $n$ is the number density of atoms and $\lambda_T$ the thermal de ...
- 1,746
1
vote
1
answer
221
views
Going from 2D dispersion relation to density of states
If one was to have a 2D dispersion say: $$\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)$$
...
- 55
0
votes
2
answers
325
views
Counting the number of allowed energy states of a particle in a 3D box
I am reading a book on Modern Physics by Thornton and Rex. I am looking at a particle in a 3-dimensional infinite square well potential (particle in a box). If the particle is a photon, its energy can ...
- 1,128
2
votes
0
answers
91
views
Single particle density of states for non-free particle
I am trying to find the single particle density of states in terms of the energy, for a system with the single particle 2D Hamiltonian:
$$H=\frac{p^2}{2 m}+\alpha x \text { with } 0<y<L, x>0$$...
- 21
1
vote
1
answer
113
views
Partition Function from Dispersion Relation in Molecular Dynamics
I’ve seen there are ways to compute the dispersion relation of a crystal from molecular dynamics. An example of how to do this is discussed in this question: Computing phonon dispersion from molecular ...
- 11
5
votes
4
answers
2k
views
Average value of energy in statistical mechanics
I haven't taken any classes in Statistical Mechanics, but in studying Structure of Matter I found some ideas I'm not very familiar with, related with the average value of energy ($E$). Given a $p(E)$ ...
- 1,515
3
votes
1
answer
278
views
Energy of a Free electron gas in D dimensions
I am trying to calculate the internal energy of a free electron bas in a box in $D$ dimensions. To calculate the density of states, I used the following formula:
$$g(E) = \int \frac{d^Dk}{(2\pi)^D} \...
user261609
0
votes
1
answer
73
views
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,...
- 83
2
votes
1
answer
1k
views
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
$$
DOS(E) = \sum_{k \in BZ, ...
- 57
0
votes
1
answer
225
views
How can the total number of particles can be sum over all Density of states? [duplicate]
It is known that Density of states of a lattice structure is the probability distribution function of energy (for reference). This means that the sum/Integral of DOS over all energies is $1$. However, ...
- 965
2
votes
0
answers
29
views
Why is the conductance of a metal-insulator-metal tunnel junction parabolic?
For bias voltages below the tunnel junction barrier heights (and below the Fowler-Nordheim limit), tunnel junctions have a parabolic conductance as a function of bias. Is this due to the metallic ...
- 135
2
votes
3
answers
334
views
Density of states misunderstanding in Statistical Mechanics
In the simple model of a box filled with an ideal gas, one may write the total energy as the sum of kinetic energies of all particles
$$E = \sum_{i=1}^N\frac{\vec{p}_i^2}{2m}$$
and so if you construct ...
- 625
0
votes
1
answer
16
views
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 ...