Energy band formation from N lithium atoms

An energy band is a group of very closely spaced allowed energy levels that forms when a large number of atoms come together to make a solid. It forms because the outer orbitals of neighbouring atoms overlap and, by the Pauli exclusion principle, each atomic level must split into many slightly different levels.

Lithium atom

Lithium (Z = 3) has the configuration $1s^2\,2s^1$. The 1s shell is full; the 2s orbital has one electron and room for one more.

Two Li atoms

When two Li atoms come close, their 2s orbitals overlap. The two atomic 2s orbitals combine (LCAO) into two molecular orbitals:

• a bonding orbital, with lower energy than the atomic 2s level, and
• an antibonding orbital, with higher energy.

So the single 2s level splits into 2 levels. The 1s orbitals lie close to the nuclei and hardly overlap, so they split very little.

N Li atoms

For $N$ atoms (about $10^{23}$ per cm³ in a solid) the 2s level splits into N separate levels. They are so close together (spacing of order $10^{-22}$ eV) that they form an almost continuous 2s band.

 Energy
   ^         separated atoms     solid (N atoms)
   |  2p .........................######  2p band
   |                       ######### (overlap)
   |  2s ..................#########  2s band
   |                       #########  (N levels)
   |
   |  1s ..........................-   1s band
   |                                  (narrow, full)
   +-----------------------|-----------------> r
                          r0     interatomic distance

• Each level holds 2 electrons (spin up/down), so the 2s band has $2N$ states.
• N Li atoms supply only $N$ valence electrons, so the 2s band is only half filled.
• The 1s band ($2N$ states, $2N$ electrons) is full and very narrow because the 1s orbitals overlap very little.
• At the equilibrium spacing $r_0$, the 2s and 2p bands overlap, giving one wide band with many empty states just above the filled ones.

Result

Because the highest band is partly filled, electrons can easily move into empty states just above them when a small field is applied. This is why lithium (and other alkali metals) are good conductors. The band width depends on how much the orbitals overlap: outer orbitals give wide bands, inner orbitals give narrow ones.