What is Density of states: Definition and 149 Discussions

In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the proportion of states that are to be occupied by the system at each energy. The density of states is defined as




{\displaystyle D(E)=N(E)/V}
, where


{\displaystyle N(E)\delta E}
is the number of states in the system of volume


{\displaystyle V}
whose energies lie in the range from


{\displaystyle E}


{\displaystyle E+\delta E}
. It is mathematically represented as a distribution by a probability density function, and it is generally an average over the space and time domains of the various states occupied by the system. The density of states is directly related to the dispersion relations of the properties of the system. High DOS at a specific energy level means that many states are available for occupation.
Generally, the density of states of matter is continuous. In isolated systems however, such as atoms or molecules in the gas phase, the density distribution is discrete, like a spectral density. Local variations, most often due to distortions of the original system, are often referred to as local densities of states (LDOSs).

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  1. UFSJ

    How to normalize the density of states in JDoS Wang-Landau?

    Hi guys. I want some help understanding how I can make the normalization of the JDoS density of states (Ω[E,m]) in the Wang-Landau algorithm. When I am working with DoS (Ω[E]) I use the knowledge that the value of the density of states in the ground states must be equal to Q (Q = 2 for the...
  2. Mohammad-gl

    A Can I calculate partial density of states using tight binding?

    I am studying a 2D material using tight binding. I calculated density of states using this method. Can I also calculate partial density of states using tight binding?
  3. R

    Phonon Density of States (PDOS) at Gamma Point

    Hello everyone! I'm trying to replicate phonon density of states (PHDOS) diagrams for some solids using Quantum Espresso. The usual way I do it is the following one: scf calculation at minima (pw.x) Calculation of dynamical matrix in reciprocal space with nq=1 or 2 (ph.x) Calculation of...
  4. Addez123

    Solving the Density of States: Understanding dn/dE

    $$n = \sqrt{n_x^2 + n_y^2 +n_z^2}$$ $$E = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$$ $$n = \sqrt{ \frac{2mL^2E}{\pi^2 \hbar^2} }$$ This is all given by the textbook. It's even as friendly as to say $$\text{differential number of states in dE} = \frac{1}{8}4 \pi n^2 dn$$ $$D(E) = \frac{...
  5. chikchok

    Phonon density of states and density of states of free electrons

    In the following pdf I tried to calculate the density of states of free electrons and phonons. First, I found the free electron DOS in 1D, it turns to be proportional to (energy)^(-1/2) and in 2D it is constant. However, I am not sure I found the DOS for phonons in the second part of the...
  6. patric44

    Derivation of the density of states?

    hi guys i have a question about the derivation of the density of states , after solving the Schrodinger equation in the 3d potential box and using the boundary conditions ... etc we came to the conclusion that the quantum state occupy a volume of ##\frac{\pi^{3}}{V_{T}}## in k space and to...
  7. A

    I Change of variables in the Density of States function

    I have a problem where I am given the density of states for a Fermion gas in terms of momentum: ##D(p)dp##. I need to express it in terms of the energy of the energy levels, ##D(\varepsilon)d\varepsilon##, knowing that the gas is relativistic and thus ##\varepsilon=cp##. Replacing ##p## by...
  8. tanaygupta2000

    What is the correct formula for the density of states in 2D for normal atoms?

    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 = 2πApdp/h^2 I think this is the formula for normal particles, and so for photons I need to...
  9. J

    I Accuracy of the Density of States

    I'm trying to understand the detailed concept of why the density of states formula is accurate enough to calculate the number of quantum states of an energy level, including degeneracy, within a small energy interval of ##dE##. The discrete energie levels are calculated by $$E = \frac{h^2 \cdot...
  10. snatchingthepi

    Partition function from the density of states

    I'm given the following density of states $$ \Omega(E) = \delta(E) + N\delta(E-\Delta) + \theta(E-\Delta)\left(\frac{1}{\Delta}\right)\left(\frac{E}{N\Delta}\right)^N $$ where $ \Delta $ is a positive constant. From here I have to "calculate the canonical partition function as a function of $$...
  11. dRic2

    I The density of states independent of Boundary Conditions

    Most undergrad textbook simply say that it is intuitive that boundary conditions should not play a role if the box is very large. Other textbooks suggest that this should be taken for granted since the number of particles at the surface are orders of magnitude smaller that the number of bulk...
  12. Diracobama2181

    Density of States: Proportional to Volume in k Space?

    The dubious assumption I am making is that the integral over the density of states is proportional to the volume in k space. Since $$\epsilon=\frac{(\hbar)^2k^2}{2m}$$ for part a, and $$\epsilon=(\hbar)\omega k$$ for part b, and $$V\propto k^d$$ for d dimensions in k space. So, $$\int...
  13. I

    I Where did the extra 8 come from in the derivation for density of states?

    I was looking for a derivation for the density of states and I came across this page: https://ecee.colorado.edu/~bart/book/book/chapter2/ch2_4.htm I followed the derivation and came up with: g(E) = (1/L3)dN/dE = (1/L3)L3/∏2*k2 * dk/dE =K2/∏2 * dk/dE =K2/∏2 * g(E) =...
  14. TIF141

    Relationship between k and E when deriving the density of states

    Number of states in that volume of k-space, ##n(k)dk## is: $$n(k)dk = (\frac{L^3}{4 \pi^3}) \cdot 4 \pi k^2 dk = \frac{L^3}{\pi^2}dk$$. Then the notes state that by defintion, ##n(k)dk = n(E)dE##, and hence $$n(E)d(E) = \frac{L^3}{\pi^2}dk$$. I don't quite see why this is true - isn't it the...
  15. astrocytosis

    Volume integral over a gradient (quantum mechanics)

    Homework Statement 1) Calculate the density of states for a free particle in a three dimensional box of linear size L. 2) Show that ##\int f \nabla g \, d^3 x=-\int g \nabla f \, d^3 x## provided that ##lim_{r \rightarrow \inf} [f(x)g(x)]=0## 3) Calculate the integral ##\int...
  16. S

    A Quantum Confinement Effect and Density of States

    I've been reading a bit about the quantum confinement effect on nanowires, particularly how it changes the band structure. I'm trying to find an explanation on why the density of states splits into sub-bands. At the moment all I'm running into is 'because of the quantum confinement effect' which...
  17. R

    I Density of states in the ideal gas

    The MB energy distribution is: MB_PDF(E, T) = 2*sqrt(E/pi) * 1/(kB*T)^(3/2) * e^(-E/(kB*T)) How do I arrive at the density of states which hides inside the expression 2*sqrt(E/pi) * 1/(kB*T)^(3/2) ? I've only seen DOS that have "h" in them.. I want it to contain only E, pi, kB and T.. This is...
  18. R

    I Do Simple 2D Ising Models Have Constant Density of States?

    Do simple 2D Ising models have constant density of states? How is it calculated?
  19. C

    I Density of States: 1-Dim Linear Phonons & Electrons Differences

    I'm to get the density of states of 1-dim linear phonons, with N atoms. I think it's a lot similar to that of 1-dim electrons, except that two electrons are allowed to be in one state by Pauli exclusion principle. To elaborate, ##dN=\frac{dk}{\frac{2π}{a}}=\frac{a}{2π}dk## for...
  20. A

    I Density of States -- alternative derivation

    I am trying to understand the derivation for the DOS, I get stuck when they introduce k-space. Why is it necessary to introduce k-space? Why is the DOS related to k-space? Perhaps if someone could come up to a slightly different derivation (any dimensions will do) that would help. My doubt ELI5...
  21. P

    A Localized states and density of states

    Hello, Let's suppose we have a two dimensional lattice which is periodic along certain direction, say x-direction, allowing us to define a quasi momentum k_x. The lattice is not periodic along the y-direction (perpendicular to x-direction). Therefore, we are able to obtain the band structure...
  22. M

    I 1D Phonon density of state derivation

    I'm having some trouble finding consistent results for the derivation of the 1D phonon density of state. I'm applying periodic boundary conditions to a 1D monatomic chain. I can work through and find that D(K)=L/(2π). This is the same result as given by Myers (1990, p. 127). Myers uses only...
  23. Kara386

    Calculating 3D Density of States for a Dispersion Relation | Homework Solution

    Homework Statement Calculate the single-particle density of states ##g(\epsilon)## for the dispersion relation ##\epsilon(k) = ak^{\frac{3}{2}}## in 3D. Use ##g(k) = \frac{Vk^2}{2\pi^2}##. Homework EquationsThe Attempt at a Solution This question is worth lots of marks. My solution is a few...
  24. A

    I Density of states at Fermi level for metal vs semiconductor

    We are doing spectroscopy on some semiconductors covered by a layer of Aluminium. My professor says it might be a challenge for to see the valence band structure of the semiconductor because the metal has a high density of states at the fermi level. Does this make sense to you? Does a metal have...
  25. 1

    I Why do we calculate the density of states using k-space?

    In statistical physics the calculation to obtain the density of states function seems to involve an integral over an eighth of a sphere in k-space. But why do we bother moving from n-space to k-space, if there's a linear relation between n and k i.e. n = (L/π)k ? Why don't we just integrate over...
  26. Arnd Obert

    I Density of states with delta function

    Hello, I'm stuck with this exercise, so I hope anyone can help me. It is to prove, that the density of states of an unknown, quantum mechanical Hamiltonian ##\mathcal{H}##, which is defined by $$\Omega(E)=\mathrm{Tr}\left[\delta(E1\!\!1-\boldsymbol{H})\right]$$ is also representable as...
  27. amjad-sh

    I Volume in K space occupied per allowed state

    An allowed state of a molecule in a gas that is in a box of length L can be represented by a point in 3 dimensional K-space, and these points are uniformly distributed.In each direction points are separated by a distance π/L. A single point in K-space occupies a volume (π/L)^3. The number of...
  28. E

    I Number of electrons in conduction band

    Hello! In order to obtain the number of actual electrons in the conduction band or in a range of energies, two functions are needed: 1) the density of states for electrons in conduction band, that is the function g_c(E); 2) the Fermi probability distribution f(E) for the material at its...
  29. M

    I Density of states from 3D to 2D

    Hi, I know how to calculate density of states for both cases, but it is not clearly to me how I can go from 3D case to 2D. I have energy from infinite potential well for 3D $$E=\frac{\hbar \pi^2}{2m}(\frac{n_x^2}{l_x}+\frac{n_y^2}{l_y}+\frac{n_z^2}{l_z})$$ let make one dimension very small...
  30. E

    I Fermi sphere and density of states

    Hello! When computing the density of states of electrons in a lattice, a material with dimensions L_x, L_y, L_z can be considered. The allowed \mathbf{k} vectors will have components k_x = \displaystyle \frac{\pi}{L_x}p k_y = \displaystyle \frac{\pi}{L_y}q k_z = \displaystyle \frac{\pi}{L_z}r...
  31. Dimani4

    A Fermi energy Ef changes with applied electric field?

    Hi people, I don't understand why when we apply the electric field to the metal Ef remains the same. Ef as translation energy of electrons remains the same but we accelerate the electrons with applied electric field so the translation energy increases too? In other hand according the formula...
  32. P

    Momentum density of states for pion decay.

    Homework Statement I am trying to calculate the ratio of the density of states factor, ##\rho(p)##, for the two decays: $$\pi^+\rightarrow e^++\nu_e~~$$ and $$\pi^+\rightarrow \mu^++\nu_{\mu}~~$$ Homework Equations ##\rho(p)~dp=\frac{V}{(2\pi\hbar)^3}p^2~dp~d\Omega## Which is the number...
  33. L

    How to change density of states to eV

    I know that if I multiply e^3/2 (e= 1.602x10^-19) to the unit below I can change it to eV Can somebody help me I don't know why
  34. Telemachus

    Density of states, and integral of the Sommerfeld type

    It is easy to show that when you have a quantum system, let's think for example in electrons in a metal, then there appears summation over electron states of the form, e.g. for the energy for a free electron gas at T=0K: ##E=2 \sum_{k\leq k_f} \frac{\hbar^2}{2m}k^2## Where ##k_f## denotes the...
  35. I

    How does doping affect the degeneracy of semiconductors?

    Hello, I am new to the forum, so I am directly stating my questions. 1)What determines the density of states of Phonons in a semiconductor? 2)Does degeneracy of semiconductors depend only on doping? Thanks
  36. J

    Surface States within band gap STM/STS

    Hi there people! So my question is why you can see localized surface states within the band gap of the material with an STM. How is a tunneling circuit being established?
  37. Coffee_

    Density of states in Fermi's golden rule

    Fermi's golden rule contains a term that is the density of the final states ##\rho(E_{final})##. For my problem we have no time depending potentials so that's the same as ##\rho(E_{initial})##. If I understand the definition of ##\rho## correctly, it's the number of states in an interval...
  38. 4piElliot0

    Density of Energy Levels - Strange Summation

    Homework Statement *This is not my whole problem, I am only stuck on how to interpret one part of the question. Put simply, I want to find the expression for the density of energy levels in a given energy band per unit volume (in some crystal structure). Say I have an infinitesimal interval of...
  39. C

    Relation between electronic band structure and Fermi energy

    I have some qualitative questions about the relation between band structure, density of states, and Fermi energy (or Fermi level). 1) Say you have a given electronic band structure (energy as a function of k) obtained by any method. How do you relate this to the Fermi energy (or Fermi level) ...
  40. O

    Volume Density of states (electrons)

    Homework Statement a)Find the densities of states 0.08 eV above the conduction band edge and 0.08 eV below the valence band edge for germanium. Be careful with units and be sure to give the units for your answer. b) Find the volume density of states (i.e. number of states per unit volume)...
  41. A

    How to find the density of states from IV plot.

    Is there any way to find the density of states from the IV graph
  42. P

    Density of States of a metal at Fermi level

    Why can not the density of states of a metal at Fermi level be zero? Thanks!
  43. Z

    What is the derivation for the vacuum density of states in the Purcell effect?

    The Purcell effect is when an atom placed inside a high finesse cavity with a very small mode volume gets an increase in the spontaneous emission rate. I've tried to find correct explanation for this effect, but it seems hard to find, except that it comes from an increase in the vacuum density...
  44. D

    Having trouble with Density of States

    I need to find the total number of states in a 1D monatomic lattice using the density of state equation g(ω), and I am having a hard time doing so. I'm fairly certain all I need to do ins integrate it, but this is proving to be a greater challenge than I thought it would...
  45. K

    Green's function and density of states

    Dear all, In his book chapter " Green’s Function Methods for Phonon Transport Through Nano-Contacts", Mingo arrives at the Green's function for the end atom of a one dimensional lattice chain (each atom modeled as a mass connected to neighbouring atoms through springs). He gives the green...
  46. hideelo

    Density of states in 2 dimensional box

    I am trying to calculate the density of energy states in a two dimensional box. The way my professor did this is by first calculating the amount of states with their energy less than some energy e and taking its derivative with respect to e. In order to see how many energy states there are with...
  47. Laplas

    Density of states of photonic crystal

    Hello! Does anybody know how to calculate the DOS as a function of Neff and λ/Λ? Thanks.
  48. Laplas

    Density of states, photonic crystal

    Hello! I know how to calculate band structure and density of states of photonic crystal (example is pic.1) Does anybody know how to plot such DOS maps? The second picture is from the article about photonic crystal fibers by Rodrigo Amezcua.
  49. C

    Trouble understanding definition of density of states

    According to my thermo textbook the density of states should really be called the density of orbitals because "it refers to the solutions of a one particle problem and not to the states of the N particle system". This makes perfect sense to me but now I'm confused about references to the density...