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...
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...
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...
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...
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...
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...
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...
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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...
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...
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...
Hi people,
I dont 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...
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?
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...
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) ...
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...
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.
Dear All:
I'm very confusing with the relationship between photonic local density of states and the field intensity. In thermal equilibrium, the field intensity should be proportional to the photon's number (or squared) and also be proportional to the local density of states. We know that this...
Adopted from my lecture notes, found it a little fishy:
Shouldn't ##\frac{dp}{dE} = \frac{E}{p}## given that ##p = \sqrt{E^2 - m^2}##. Then the relation should be instead:
\frac{dp}{dE} = \frac{E}{p} = \frac{E}{\sqrt{E^2 - m^2}}
I'm a beginner in quantum optics. I always become confusing when the material's refractive index is complex. This time is about the photonic density of states.
We know that if the material is not absorbing or dissipative, meaning the refractive index is a real number, the local photonic density...