Within each of the electron shells of an atom, electrons can occupy sub-shells of varying energy; these are designated by the letter s, p, d or f. Figure 1(b) shows these sub-shells represented as circles and the relative energies and ordering of the shells and sub-shells. Each sub-shell (circle) can hold a maximum of two electrons. The smallest K shell is only big enough to accommodate a single s sub-shell and therefore contains a maximum of two electrons. The larger L shell can accommodate one s sub-shell and three higher energy p sub-shells. Each p sub-shell is iso-energetic with the other two but is differentiated from its two partners by quantum numbers that we will not consider in this simplified discussion. The availability of s and p sub-shells allows the L shell to hold up to 8 electrons. The M shell includes one s sub-shell, three p sub-shells and five d sub-shells (as with the p sub-shells, the five d sub-shells are iso-energetic) and accommodates up to 18 electrons; higher energy shells hold correspondingly greater numbers of electrons. When the available sub-shells within a shell are all filled by electrons, then that shell is said to be "closed" and the electrons within that shell are no longer available for chemical interactions or electrical conduction. Silicon, shown in Figure 1(a), has 14 electrons. These electrons are sequentially accommodated in the available shells, starting at the lowest energy K shell. For silicon, the K and L shells are completely filled (10 electrons) and the M shell is unfilled, with only four electrons occupying the available nine sub-shells (the nine sub-shells can accommodate up to 18 electrons); these electrons are distributed in the lower energy s and p sub-shells. These four electrons in the outer or highest energy electron shell, are commonly termed the "valence" electrons; they determine silicon's chemical and electrical properties. An alternate way of depicting the energy shells and sub-shells in the silicon atom and the way that the sub-shells are filled is shown in Figure 2. Figure 2 clearly shows the "closed" shell core electrons in the silicon K and L shells and the valence electrons in the unfilled M shell.
Now, when two or more atoms bind together, the electrons in each atom's valence sub-shell combine to create molecular sub-shells that accommodate all of the electrons from the atoms' valence sub-shells; it is this combination that constitutes the interatomic bond. Consider two silicon atoms coming together to produce a hypothetical "Si2" molecule. Each silicon atom contributes four electrons from its valence shell (two electrons in the single 3s sub-shell and two electrons in two of the three available 3p sub-shells) to form the molecular bonds between the two atoms in the Si2 molecule with eight electrons occupying eight molecular sub-shells (Remember, the core electrons don't participate in the bonding or electrical conduction). This energy levels of the molecular orbitals produced by the combination of the two Si atoms' atomic orbitals are shown graphically in Figure 3.
So, when two identical atoms combine, the available energy levels for the electrons from those atoms are doubled (the situation is somewhat more complex when two dissimilar atoms combine). Similarly, if three identical atoms combine, then there are three sub-shells generated for each sub-shell in the original atom; with four atoms, there are four sub-shells for each sub-shell in the original atom, etc. For descriptive purposes, let's use the example of a chain of n bonded hydrogen atoms each with a single available energy sub-shell to contribute to the molecule (in this case one electron in the 1s sub-shell). As atoms are progressively added to our hypothetical linear poly-hydrogen molecule, additional energy sub-shells are created between the upper and lower energy limits of the 1s shell. With each additional atom, the energy increment between sub-shells becomes smaller and the energy gap between the shells becomes slightly smaller. By the time enough atoms are added to the assemblage for it to be considered an extended solid, the number of atoms can be considered as effectively infinite. In terms of the sub-shells available for electrons within the solid, the ever-decreasing energy increment between sub-shells produces a band of available energy levels within which an electron can freely move rather than the discrete levels characteristic of isolated atoms or molecules. The continuous band of energy levels created in our hypothetical poly-hydrogen molecule corresponds to the 1s sub-shell in the isolated hydrogen atom. This band contains the accumulated electrons contributed by all of the hydrogen atoms in the molecule, creating a half-filled 1s energy band, as shown in Figure 4.