MATTER LogoIntroduction to Electrons in Crystals

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Density of States > 3 Dimensions and k-Space

Description of Flash Animation.

This animation shows states plotted on a 3-D momentum diagram.

This resource requires Flash Player 9 or above available to download from

Δp is small enough to be considered as a differential and in one dimension dpx = h/2L where the x indicates that we are considering the positive x direction.

In 3-D there will be similar equations for the y and z directions. The volume of a state in 3-D in a cube of size L is dpx dpy dpz, which is simply (h/2L)(h/2L)(h/2L) or h3/8L3.

k-space or momentum space
States are plotted on a 3-D momentum diagram (see box). The energy of each state depends on its momentum p, via E = p2/2m.

The electrons in a solid will adopt the lowest possible set of p values, that is the shortest vectors in k-space. This is achieved if they all lie within a sphere known as the Fermi sphere.

After Enrico Fermi. A fermi sphere is shown in cyan on the diagram.

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