The energy dispersion relation for sc, bcc and fcc?

Rrrenata
Messages
2
Reaction score
0

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 * S * e^[ik(R-R')],
for alpha = the Coulomb integral, beta = the exchange integral, and S = the sum over the nearest neighbors of atoms at position R.

The Attempt at a Solution


I can see how S depends on the crystal structure, but is that it? Should I just keep the formula and only change the number of the nearest atoms for the three different structures? Also, is it relevant where you fix R? Thank you!
 
Physics news on Phys.org
Are R' the positions of those nearest neighbors?
I would expect that this exponential gets summed over, not a sum multiplied by an exponential.
 
mfb said:
Are R' the positions of those nearest neighbors?
I would expect that this exponential gets summed over, not a sum multiplied by an exponential.
Oh my God, yes, it is the sum over the exponential and i am stupid.
 

Thread 'Please tell me where I am going wrong in this integral'

##|\Psi|^2=\frac{1}{\sqrt{\pi b^2}}\exp(\frac{-(x-x_0)^2}{b^2}).## ##\braket{x}=\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dx\,x\exp(-\frac{(x-x_0)^2}{b^2}).## ##y=x-x_0 \quad x=y+x_0 \quad dy=dx.## The boundaries remain infinite, I believe. ##\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dy(y+x_0)\exp(\frac{-y^2}{b^2}).## ##\frac{2}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,y\exp(\frac{-y^2}{b^2})+\frac{2x_0}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,\exp(-\frac{y^2}{b^2}).## I then resolved the two...
View full post »

Thread 'Direction of the magnetic field of an oscillating charge'

Hello everyone, I’m considering a point charge q that oscillates harmonically about the origin along the z-axis, e.g. $$z_{q}(t)= A\sin(wt)$$ In a strongly simplified / quasi-instantaneous approximation I ignore retardation and take the electric field at the position ##r=(x,y,z)## simply to be the “Coulomb field at the charge’s instantaneous position”: $$E(r,t)=\frac{q}{4\pi\varepsilon_{0}}\frac{r-r_{q}(t)}{||r-r_{q}(t)||^{3}}$$ with $$r_{q}(t)=(0,0,z_{q}(t))$$ (I’m aware this isn’t...
View full post »

Thread 'Initial conditions for orbits around a wormhole'

Hi, I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem. Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$ Where ##b=1## with an orbit only in the equatorial plane. We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$ Ultimately, I was tasked to find the initial...
View full post »

Similar threads

I Creep mechanism (thermal and irradiation induced/enhanced) and embrittlement in fcc and bcc
Replies
5
Views
2K
[PoM] Cohesive energy in an argon crystal
Replies
12
Views
4K
Dispersion relation ~ modern phyics (solid state physics)
Replies
1
Views
4K
Debye model and dispersion relation
Replies
1
Views
5K
Nearest Neighbors in solid state, but with basis
Replies
1
Views
3K
Recurrence relation for harmonic oscillator wave functions
Replies
2
Views
2K
Finding nearest neighbour equilibrium distance
Replies
4
Views
3K
Calculate the ratio of the Fermi wavevector to the radius of the largest sphere
Replies
1
Views
4K
Quantum Mech Homework Help Urgent: FCC, SC Lattice, Graphene Heat Capac
Replies
3
Views
6K
Why is the volume of FCC and BCC different in reciprocal space?
Replies
1
Views
31K

Hot Threads

Recent Insights

Back
Top