Kittel figure 9.12 (I attached it)

  • Thread starter Thread starter ehrenfest
  • Start date Start date
  • Tags Tags
    Figure
Click For Summary

Homework Help Overview

The discussion revolves around a figure from Kittel that illustrates energy gradients in the context of band theory and electron orbits. Participants express confusion regarding the implications of energy gradients pointing inwards and the meaning of filled orbits in relation to the First Brillouin zone.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the concept of energy gradients in band structures, questioning how negative gradients can occur and how orbits are filled. There are inquiries about the physical meaning of "hole orbits" and the relationship between energy and wave vector (k).

Discussion Status

The discussion is active, with participants providing insights into the filling of bands and the behavior of energy versus k diagrams. Some guidance has been offered regarding the interpretation of the figure, but multiple interpretations and questions remain unresolved.

Contextual Notes

Participants are grappling with assumptions about the nature of orbits and energy levels, particularly in relation to the First Brillouin zone and the behavior of electrons in different band structures. There is also a focus on the dimensionality of the problem and its implications for the dispersion relation.

ehrenfest
Messages
2,001
Reaction score
1

Homework Statement


This figure is driving me insane. How can the energy gradient point inwards! For the hole orbit. The shading represents filled orbits (I am pretty sure), so how can you possibly have orbits filled everywhere except the center of the First Brillioun zone. That is outrageous! What does a hole orbit physically mean??


Homework Equations





The Attempt at a Solution

 

Attachments

  • kittel_figure_9_12.jpg
    kittel_figure_9_12.jpg
    68.2 KB · Views: 427
Physics news on Phys.org
Consider the 1D case. Your dispersion relations are in bands --- so you can get negative gradients even for +ve k. As the Fermi level goes up, you can start filling a band from high k's going towards k=0. It's hard to describe without drawing...
 
genneth said:
Consider the 1D case. Your dispersion relations are in bands --- so you can get negative gradients even for +ve k.

What is "+ve k"?

genneth said:
As the Fermi level goes up, you can start filling a band from high k's going towards k=0.

How?
 
See http://venables.asu.edu/qmms/band.gif as an example (it was the first reasonable one I could find with google images)

Now imagine filling up from the lowest energies. As the Fermi level crosses the first band, you start filling from low k to high k. But when it gets to the 2nd band it starts from high k --- at the zone boundary.
 
Last edited by a moderator:
Just to be sure, you are referring to the energy versus k diagram on the right and using the fact that the first band is concave up and the second band is concave down? Although, they did not show it, I assume k=0 is below the minimum of the first band? Also, is k on the x-axis equal to the magnitude of the vector k or one of its components?
 
Yes to all, and k is on the x-axis since we are considering a 1D situation. For higher dimensions, the dispersion relation is a surface through some k-E space, but the general idea holds --- within a band, the edge of the BZ can be lower in energy than the middle.
 
OK. Referring to the attachment, can you explain why the sense of the orbit is opposite in a) and b)? It is an electron that is doing the orbit in both cases, right?
 

Similar threads

Kittel Chapter 7: Explaining Energy Gaps with First Brillioun Zone
  • · Replies 1 ·
Replies
1
Views
3K
Solving Central Force Motion: Sketch Equivalent Elliptic & Hyperbolic Orbits
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad I need some support, please, for a gravity assist analysis
  • · Replies 86 ·
3
Replies
86
Views
9K
Undergrad Did I figure-out z-translation of 2D-coordinates correctly?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Condition for Centering Starlight in a Water-Filled Telescope?
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Non-locality by memory (Jung, 2020) applied to stochastic mechanics
  • · Replies 5 ·
Replies
5
Views
2K
Graduate How Do You Calculate the Correct Launch Window for an Inclined Orbit Around Io?
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Hall effect in P-type semiconductors: electron-centric heuristic?
  • · Replies 0 ·
Replies
0
Views
3K
Solve N Ground State in (2s+1)(L)(J) Format
  • · Replies 4 ·
Replies
4
Views
5K
Is my fictional Solar System stable and realistic?
  • · Replies 21 ·
Replies
21
Views
6K