The atoms of Na and Cl as free atoms (not bounded together) have the electronic configurations $1s^2,2s^2,2p^6,3s^1$ and $1s^2,2s^2,2p^6,3s^2,3p^5$ and in the crystalline state (when they make together the NaCl) have the configurations $1s^2,2s^2,2p^6$ and respectively $1s^2,2s^2,2p^6,3s^2,3p^6$.

a) Explain why the NaCl crystal is a dielectric

b) If the crystal is made up by N molecules of NaCl, find the number of energetic states from the conduction band, respectively from the valence band.

The figure below shows simplified how the energetic bands arise in the NaCl crystal when the atoms of Na and Cl are brought close together until the equilibrium distance $r_0$ between the ions.Initially when the atoms of Na and Cl are far apart there is one single energy level for 3p and 3s for each atom. When the atoms are brought close together , the same levels (3p, respectively 3s) of many Na and Cl atoms overlap and make a band extended in energy. The distance $r_0$ is the equilibrium distance between atoms of Na and Cl in the crystal.

a)

As one can observe from the figure the 3s band of the Na is found above the 3p band of the Cl and all the electrons from the 3s band will go into the 3p band. Because the number of electrons that come

into the 3p band is equal to the number of free states, it follows that the band 3s (conduction band) will be completely free and the the 3p band (valence band) will be completely occupied. The gap between the two bands is $E_g =7 eV$ and therefore the crystal of NaCl is a typical dielectric (semiconductors gap is around 1 eV, above 3 eV the crystal is perfect dielectric).

b)

The conduction band contains 2N states and the valence band has 6N states that are completely filled with the 6N valence electrons of Na and Cl atoms.