What is Dispersion relation: Definition and 65 Discussions

In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the dispersion relation, one can calculate the phase velocity and group velocity of waves in the medium, as a function of frequency. In addition to the geometry-dependent and material-dependent dispersion relations, the overarching Kramers–Kronig relations describe the frequency dependence of wave propagation and attenuation.
Dispersion may be caused either by geometric boundary conditions (waveguides, shallow water) or by interaction of the waves with the transmitting medium. Elementary particles, considered as matter waves, have a nontrivial dispersion relation even in the absence of geometric constraints and other media.
In the presence of dispersion, wave velocity is no longer uniquely defined, giving rise to the distinction of phase velocity and group velocity.

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  1. deuteron

    I Dispersion Relation in Different Media

    In my lectures, we have derived the dispersion relation $$ |\vec k|^2 = \frac {n^2 \omega^2}{c^2}$$ by substituting in a plane wave solution for the electromagnetic wave, into the wave equation derived from the Maxwell equations $$\Delta\vec E= \mu_0\epsilon_0 \frac {\partial^2 \vec...
  2. patric44

    I What is the significance of the dispersion relation?

    hi guys i would like to know what is the physical significance of the dispersion relation , i know that it relates the energy and momentum vector and correspondingly the energy and momentum with each other , but what does this tell me about the system , and why should i care that the dispersion...
  3. F

    What happens to σ (conductivity) when the permitivity is negative?

    We are using the textbook by Jin, "Theory and Comutation of Electromagnetic Fields". In the section on metamaterialshe derives the dispersion relationship. He shows that when ε'= -ε & μ' = μ then the dispersion equation γ =\sqrt{jωμ( jωε +σ )} = α + jβ goes to γ = α = ω \sqrt{μ'ε'}...
  4. M_Abubakr

    Infinite-Extent 2D Mass-Spring System vibration and dispersion relation

    Hi, I am trying to find equation of motion and its solutions for a 2D infinite lumped mass spring system as depicted in figure. All the masses are identical, All the springs are identical, and even the horizontal and vertical periodicity is the same n=a. I need to try find dispersive relation...
  5. Kaguro

    Group velocity and the dispersion relation

    After noting w=vk and differentiating with respect to k, and lots of simplifying, I get: Vg = c/n +(2*pi*0.6)/(k*n) This doesn't correspond to any numerical value though...
  6. K

    How to calculate this dispersion relation

    I have no idea how this dispersion relation was deduced, and also what's the meaning of including plus and minus in the formula.
  7. confusedius

    Solid State Physics: Draw the Dispersion Relation from the Fermi Surface

    Homework Statement ln the figure below you (b, which is taken from Jenö Sólyom Fundamentals of the physics of solids. Volume 2 chapter 19) see the Fermi sphere of radius k_F inside one section in two dimensions of the Brillouin zone of Na. Draw the dispersion relation E(k) from the I point in...
  8. P

    I Relativistic Energy Dispersion Relation: Explained

    I'm in the process of learning special relativity (SR), and I'm a bit confused as to why the relativistic energy dispersion relation ##E^{2}=m^{2}c^{4}+p^{2}c^{2}## gives the energy for a free particle? I get that it is the sum of (relativistic) kinetic energy plus the rest mass term (a...
  9. Pushoam

    Dispersion relation for a surface wave of a pool of water

    Homework Statement Homework EquationsThe Attempt at a Solution ## v = \frac { \omega } k ## ## \omega = \sqrt{ kg \tanh (k) } ##I have no idea to guess the graph. I put g = 9.8 and tried to calculate ## \omega ## for different values of k. ## \omega (0 ) = 0, \omega (30) =...
  10. S

    A Taylor expansion of dispersion relation - plasma physics

    Hello, I may working through attached paper and really need help with deriving equation in appendix - A4 to give A10. http://iopscience.iop.org/article/10.1088/0004-637X/744/2/182/pdf Any help would be greatly appreciated. thanks, Sinéad
  11. M

    Nonlinear Schrodinger Equation Dispersion Relation

    The Nonlinear Schrodinger Equation (NSE) is presented as: $$i\frac{∂A}{∂z} = \frac{1}{2}β_2\frac{∂^2A}{∂t^2}-\gamma|A^2|A$$ The steady state solution $$A(z)$$ Can be derived as an Ansatz given by: $$ A(z) = \rho(z)e^{i\phi(z)}$$ By substituting and solving the ODE, the steady state...
  12. victor94

    A Classical gas with general dispersion relation

    i'm trying to understand the solution to this problem: http://physweb.bgu.ac.il/COURSES/StatMechCohen/ExercisesPool/EXERCISES/ex_2065_sol_Y13.pdf (link to the problem and the solution of it) All my questions come from the partition function: 1) From where the term (2*pi)^d comes from?, I...
  13. T

    I Dispersion relation in tight binding model

    Hamiltonian of tight binding model in second quantization is given as H = -t \sum_{<i,j>} a_i^{\dagger} a_j After changing basis it is H = \sum_{\vec{k}} E_{\vec{k}} a_{\vec{k}}^{\dagger} a_{\vec{k}} where E_{\vec{k}} = -t \sum_{\vec{b}} e^{i \vec{k} \cdot \vec{b}} where \vec{b} is a nearest...
  14. adamaero

    Dispersion relation ~ modern phyics (solid state physics)

    Homework Statement Which cannot be the structure of two acoustic branches, nor three acoustic branches? Simple cubic, FCC, BCC, diamond cubic, NaCl lattice Homework Equations N/A http://solid.fizica.unibuc.ro/cursuri/solid_en/curs_solid_EN.pdf#page=61...
  15. otaKu

    Solid State A good book to learn about phonon dispersion relation

    So I see them in the books labelled as accoustic and optical phonons but I don't seem to find a comprehensive treatment of the matter for a beginner who doesn't know a thing about the dispersion curves. I'd prefer not to dwell too much into the mechanical treatment if possible since I just need...
  16. E

    A Selection rules in IXS and EELS

    Hello everybody, since I have to plan an experiment to map the energy-momentum dispersion of a bosonic excitation, I have a question related to the difference between the excitations probed by Inelastic X-ray Scattering (IXS) and Electron Energy Loss Spectroscopy (EELS). Both the techniques are...
  17. M

    I Dispersion relation of a transmission line - questions

    Hi, I have transmission line with dispersion relation ω=sin(kx), so then means that for one value of ω I have two values of k. I apply voltage with some frequency with is allowed to move in the line. First question is, how can I influence what k will be generated inside the line. The another...
  18. Rrrenata

    The energy dispersion relation for sc, bcc and fcc?

    Homework Statement I need to calculate the energy dispersion relation in the tight binding for simple cubic, base centered cubic and face centered cubic crystals. There are no values given, they just need the result depending on the lattice constant a. Homework Equations E (k) = alpha + beta *...
  19. D

    Dispersion relation for non-relativistic quantum particles

    In class I learn that we can get the dispersion relation for particles by using E=hbar*w and p=hbar*k. The calculated phase velocity is w/k = hbar*k/2m, while the group velocity is dw/dk=hbar*k/m. All these make sense to me, except one thing: I always thought that E=hbar*w=hf is only applicable...
  20. W

    Dispersion relation for diatomic linear chain.

    Hi. Here's the dispersion relation for a diatomic linear chain, where the distance is a/2 between each atom. My issue here is that if you set m_1=m_2=m, i.e. set both atoms equal to each other, it doesn't automatically reduce to the old acoustic dispersion relation as the ± term doesn't...
  21. C

    Dispersion relation for the free electron model

    I'm trying to get my head around what this means exactly. I've plotted the graph to help verse me with the functions that I've derived. From the free electron model, the wavefunctions are treated as planewaves of the form \psi_\mathbf{k}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} Due to...
  22. U

    Dispersion relation and their origins & meaning

    Hi Everyone, I'm trying to understand dispersion relations in general. I know that for a simple wave like a light wave there is a 'constant phase' so the dx/dt is equal to the ratio of the angular frequency (omega) by the wave vector (k). However what does a 'constant phase' mean? How can I...
  23. N

    Lattice wave dispersion relation

    Hi. A very quick question. Why is it impossible for a wave to travel on a linear one-atomic chain if its wavelength equals the lattice constant? I.e. the lattice points vibrate with a wavelength equal to the distance between them? Here's what I mean...
  24. Robsta

    Distance to pulsar with plasma dispersion relation

    Homework Statement Pulsars are stars that have suffered gravitational collapse. They rotate rapidly and emit a narrow beam of radiation. The pulse lengths, at the earth, are ∼1ms and the periods are ∼1s. Within a few months of the discovery of pulsars distance estimates were obtained by...
  25. H

    Dispersion relation (particle in a box)

    I am learning some basic solid state physics idea, like density of state ...etc. For particle in a 1D box, E = n^2 (pi)^2 (h_bar)^2 / 2mL^2 But why it is written as E = (h_bar)^2 k^2 /2m does it means that energy eigenvalue E is related to momentum k ? I guess it is not because momentum is...
  26. R

    Nonlinear Dispersion Relation with Imaginary Part

    Homework Statement I've determined the dispersion relation for a particular traveling wave and have found that it contains both a real and an imaginary part. So, I let k=\alpha+i\beta and solved for \alpha and \beta I found that there are \pm signs in the solutions for both \alpha and \beta...
  27. T

    Heat capacity using Debye dispersion relation

    Homework Statement Using the Debye dispersion approximation, calculate the heat capacity of a harmonic, monatomic, 1D lattice. Next, find the temperature dependence in the low temperature limit. (Assume that the longitudinal mode has spring constant CL = C, and the two transverse modes both...
  28. T

    Mathematica Inverting dispersion relation with Mathematica

    Hi guys, I'm currently working on a project related to Faraday instability, and I of course came across the dispersion relation for capillary-gravity waves, i.e. ω² = tanh(kh)(gk + σk³/ρ) . Now I would need to numerically solve this relation for the wavenumber k as a function of depth h...
  29. M

    To calculate Fermi energy from arbitrary dispersion relation

    Homework Statement problem statement is attached as problem.pdf Homework Equations eqn are given in the pdf file The Attempt at a Solution I have tried in vain to connect Fermi energy with dispersion relation. I just don't have any clue ,I also tried to determine the effective...
  30. G

    Dispersion relation in cylindrical plasma column

    Hi. is there anyone who is familiar with surface wave plasma discharges? I want to solve dispersion equation of surface waves in cylindrical plasma column,numerically,to obtain "phase diagram" and "attenuation diagram" from dispersion equation solving. But this dispersion equation include...
  31. A

    Understanding Phonon Dispersion Relation

    Hi Guys, I am learning some solid state physics. I see a lot of pictures with Phonon Dispersion Relation, with \omega (\vec{k}) on the y-axis and \Gamma, X, M, \Gamma, R on the x-axis. I don't understand, why the angular frequenzy \omega (\vec{k}) is important. Or why is this...
  32. C

    Debye model and dispersion relation

    Homework Statement I have seen case studies of the 3D Debye model where the vibrational modes of a solid is taken to be harmonic with dispersion relation \omega = c_sk. It is said that for temperatures much less than the Debye temperature, the heat capacity at constant volume C_V\sim T^3...
  33. M

    Dispersion Relation: Beads and String

    When making the transition from the dispersion relation for a beaded string to the relation for a continuous string, I'm confused about the following issue. Take a to be the spacing between beads, m the mass of each bead, and T the tension in the string. We assume these to be constant. For the...
  34. matt_crouch

    Calculate the spin wave dispersion relation for a ferromagnetic heisenberg model

    Homework Statement Calculate the spin wave dispersion relation Ek for the ferromagnetic Heisenberg model with jtot = 1/2 Assume a 1d square lattice and interactions of strength J between nearest neighbours and zero elsewhere Homework Equations H|k> = [E0 +2jtot\sum J(r)(1-Exp(ik.r) ]...
  35. K

    Dispersion relation for (100) and (001)?

    May I know what is the difference between the dispersion relation for 100 and 001 on the E-K diagram? Can i say 001 has lesser dispersion? But why is it so?
  36. P

    Why is the dispersion relation of magnons quadratic in k for small wavelengths?

    Hello, Can anyone explain to me why, concept wise (not from calculations that I get it), is the dispersion relation of magnons (spin waves) quadratic in k for small wavelengths? Also, can you give me other examples where such behavior appears? Thank you
  37. T

    Free electron dispersion relation, help?

    Hi there, Could anybody explain how the free electron dispersion relation would be modified by the presence of a periodic potential..? I'm struggling to get my head around it. Thanks!
  38. Y

    Calculating 2D Dispersion Relation with Different Atom Types

    I have derived 2D dispersion relation which has the same atoms. But I also need to calculate this 2D dispersion relation with two different atoms. One atom is located at the center and the other type of atoms surrounds this atom. But I am not sure how ı should calculate it because only...
  39. A

    Phonon and dispersion relation

    hello I am new in this forum.. and i would like to ask first this is statement that i confused about 'At low values of k (i.e. long wavelengths), the dispersion relation is almost linear, and the speed of sound is approximately ω a, independent of the phonon frequency. As a result, packets of...
  40. A

    Trying to find dispersion relation

    Homework Statement \imath\frac{\partial u}{\partial t} + \frac{\partial^2 u}{\partial x^2}=0 \left(x,t\right) = \int^{\infty}_{-\infty}A\left(k\right)e^{\imath\left(kx-wt\right)}dk u\left(x,0\right)=\delta\left(x\right) Homework Equations Not sure how to get w(k) The...
  41. A

    How do I Find the Dispersion Relation for a PDE?

    \imath\frac{\partial u}{\partial t} + \frac{\partial^2 u}{\partial x^2}=0 \left(x,t\right) = \int^{\infty}_{-\infty}A\left(k\right)e^{\imath\left(kx-wt\right)}dk u\left(x,0\right)=\delta\left(x\right) This is what I am working with. I am supposed to find the dispersion relation. So far...
  42. C

    Dispersion relation and group velocity

    Hello there PF readers, The group velocity for example of electron waves is given by the derivative of the dispersion relation: \frac{dE}{dp}=v (this is for free electrons) ^{1}. Now the Heisenberg's uncertainty principle has two forms, one for position and momentum and the other for energy...
  43. M

    Deriving the Phonon Dispersion Relation using Quantum-mechanical approach

    I'm doing a literature review on dispersion relations, and I've been told that if i can derive the phonon dispersion relation, it would help my review. So i was wondering if anybody could help me with the derivation either through Quantum-mechanical approach or Semi-classical treatment of...
  44. K

    Dispersion relation of amorphous materials

    Hi, I am trying to find dispersion relation of amorphous material (using data obtained by molecular dynamics simulation). As it is not periodic system one can not find it by standard method of diagonalizing the force constant matrix. I think one can do it by taking Fourier transform of...
  45. I

    Langmuir waves dispersion relation

    I was just looking at an expression (a dispersion relation, omega^2 = ...) similar to that of warm electron's in a plasma http://en.wikipedia.org/wiki/Plasma_oscillation expect with an extra imaginary term, which I think comes out from the full derivation of the dispersion relation for warm...
  46. D

    Phonon dispersion relation : meaning of mode crossings

    Hello all, What does it mean when two phonon branches get crossed in phonon dispersion relation ? Also, in certain high symmetry direction, one can find only one transverse branch and other direction two separate branches, what do these response mean? (Ref. Solid state physics by...
  47. X

    Group velocity and Dispersion Relation

    Hi there So I was looking into group velocity and related matters and found myself quite confused. So now I have a few questions which I feel I need to understand (primarily the first one). Any help with these would be awesome and I would be very grateful... 1) Why is the group...
  48. DaTario

    Dispersion relation and Photonic Energy and Momentum Equations in QM

    Hi All, In quantum mechanics, related to photon, we have these two equations as valid ones: E = h x f p = h / lambda But we have in vacuum the dispersion relation c = lambda x f. 1) How these relations change when the dispersion relation change ? 2) Is it universally correct to...
  49. S

    Energy spectrum from dispersion relation E(k)

    Hi. What I'm trying to do is to obtain the energy spectrum from the following dispersion relation: E^4-A·E^3+B·E^2-C·E+D-F·E^2·cos(k·a_0)^2+G·E·cos(k·a_0)^2-H·cos(k·a_0)^2 = 0 where E is the energy, k is the wave vector and a0 the distance between adjacent neighbors in a 1-Dimensional...