Condensed Matter II
Problem set #4
Spring 2023
1 Free electrons in Na
Figure 1: Energy bands in sodium along symmetry directions. Chang and Callaway, PRB
11, 1324 (1975)
1.1 Qualitative study
(i) With the help of the computed band structure of Na in Fig. 1, suggest what the
structure of the empty BCC lattice must look like (assuming isotropic effective mass
m0).
(ii) How high above the bottom of the conduction band is the Fermi energy located in
this model?
(iii) Justify that Na may be considered “the simplest of the simple metals”.
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1.2 Quantitative study
(i) Use Bloch’s theorem to give the expression of the energy El(k) in a generic 3D
crystal, where l ≡ (l1, l2, l3) is the band index, and k is a wave vector.
(ii) A schematic of the Brillouin zone of the BCC lattice is provided in Fig. 2. Use
the previously obtained result, and derive the expression of the energy levels at the
special points Γ, H, P, and N in the empty BCC lattice.
Figure 2: Brillouin zone for the body-centered cubic lattice. M. Dresselhaus, Group theory
- Application to the Physics of Condensed Matter, Fig. C.5.b, p 511.
Figure 3: Empty lattice.
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