Crystals and Energy Bands

You may recall from high school chemistry that a model that is very successful in explaining the electronic composition of stable atomic species is called the shell model. In this model, very similar to the planetary model, the atom is thought to be made of shells of different types. Each shell-type can accommodate a certain number of electrons. For example, an "s-shell" can be occupied by no more than two electrons, whereas p-shells can have up to six electrons in them, etc. To "make stable atoms" we start by filling electrons in these shells starting in "level" 1, then level-2, etc., following the specified hierarchy: 1s; 2s, 2p; 3s, 3p; 4s, 3d, 4p; 5s, 4d, 5p; etc. (Remember that the chemistry of materials is due to electromagnetic interactions alone, so for this we only need to concern ourselves with electrons.) We start with the innermost shell, an s-shell. Our first atom-type has one electron in its s-shell; that's of course the hydrogen atom, H. The next atom has two electrons in its s-shell; that's He. The third atom on our list (see the Periodic Table) has three electrons, two in its 1 s-shell and then one in its 2 s-shell; and so on. According to this model, which incidentally is very well supported by the more rigorous and fundamental laws of Quantum Mechanics, atoms prefer to fill their outer most shell. Inert gases, for example, all (except for He) have a completely filled outer p-shell. Those atoms that manage this are energetically very stable. Those that don't would then prefer to interact with another atom of the correct type to allow an exchange of electrons so that they come closer to fulfilling this desire for closed shells in order to reduce their overall energy. A bond between two types of atoms by this give-and-take is strongest when both atoms get the most fulfillment - both end up with closed shells.

A good example of this is the case of table-salt: NaCl. Sodium atoms, Na, are hydrogenic (hydrogen-like); i.e. they have one electron in their outermost shell, which is an s-shell just outside an inert gas shell. So, to emulate the rare gases, all they need to do is to get rid of this electron. They cannot do this on their own (why?), but they can bind with an atom that wants an additional electron to complete its outer most shell. This happens to be true for the chlorine atom that has 5 electrons in its outermost shell, a p-shell. By sharing this electron with chlorine, the sodium atom still remains neutral, but now it, as well as the Cl, both end up with "filled" outer shells. In more ways than one, this is a happy marriage. Of course, in a grain of salt there are many Na and Cl atoms. (One grain of salt has roughly about 1 mg of mass. One proton is about 10-27 kg. How many atoms are then in one grain?) These atoms arrange themselves in a very organized pattern, called a crystal. These are rows and columns of atoms, as in a three-dimensional array, with interchanging Na and Cl species. In a salt crystal, a Na atom shares its electron with several Cl atoms, and yet because of the geometry of the crystal each Cl atom ends up with a net total of one extra and each Na atom with only one fewer electron (how?).

The electric attraction of the electrons to the atom's positively charged nucleus is the primary force that holds the atom together as a whole. So, in our shell model of the atom we must also account for the difference in attraction of electrons in different shells as well as the repulsion among the electrons themselves. Electrons in outer shells are further away from the nucleus, so their force of attraction is weaker than the inner shell electrons. In this respect, those electrons that belong to the unfilled outermost shell are the most loosely bound electrons in the atom. These electrons are the ones that are mostly responsible for the atom's interaction with another atom, so they are given a special name: the valence electrons.

How is the shell model related to our picture of atomic energy levels? It is primarily the valence electrons that connect these two pictures. Each atom species has its own unique energy levels, but its shell structure is not unique. When it is in its ground state, then its shells are filled according to the shell model's recipe, mentioned above. But once the atom is excited, then its electron configuration will change. The first excited state, for example, could require the valence electron to jump into the next higher shell. Further excitation of the atom could lead to even more drastic changes in its shell structure. But each of the excited energy levels corresponds to a different occupation of the accessible shells by the atom's electrons.

If atoms were made of (stationary) electrostatic charges, then most atomic species would have no reason to get together to make molecules or solids. But fortunately for us, this is not the case. Two neutral atoms can attract each other simply by changing their charge distribution so that they each become polar, i.e. the electron cloud of charges shifts over so that its center is not at the positive nucleus (see the figure below). In this sense even atoms of the same species find it energetically favorable to form a molecule. Because of this two totally neutral atoms of nitrogen prefer to form an N2 molecule even in the gas form. See the model drawing, below, which shows three time frames. The top frame shows two atoms with spherically symmetric charge distributions. These atoms do not initially interact with each other. But a momentary polarization of one atom can cause the other atom to polarize as well. This is the picture in the middle frame. The bottom frame the shows that two polarized atoms attract and form a molecule in which the electronic clouds of the two atoms overlaps.

When we cool a gas, the ease with which it makes a liquid has a great deal to do with how easily the atoms in the gas form a molecule. Things get even more interesting when the liquid is further cooled to make a solid. The form, i.e. atomic structure, in the solid has not only to do with the atoms wanting to share electrons, but also with a totally different aspect of energy for a large collection of atoms. This relates to the probability of arrangements, or the so called entropy. What concerns us here is that for some species of atoms the lowest possible energy is reached when the atoms arrange themselves is a very regular and repeatable array of positions, called a crystal.

In every-day terminology the word "crystal" usually refers to a glassy and transparent object. But in our sense of the "regular network" a crystal may be totally opaque. In fact, most metals are crystalline in structure and are not at all transparent (at least in the visible range). Why should atoms form a regular array and make up a crystal? Why shouldn't they just clump together? In most substances atoms clump randomly, forming a solid. But in some special situations atoms crystallize to make very regular network structures. These networks can have different geometrical regularities. For example, in some crystals, the network structure is a simple cube that repeats in all directions. One atom sits at each corner of the cube, making up a simple cubic crystal. In a variation of this, in addition to the simple cubic structure each face of the cube has an atom at its center; this is called a face centered cubic crystal, etc. The salt crystal of NaCl has the simple cubic structure in which Cl and Na atoms occupy alternate corners of the cube, shown below:

This network structure in the crystal is called a lattice. The atoms in the lattice are not motionless. They jiggle and oscillate, but on average remain at a fixed site. The higher the temperature of the solid, the more vigorous is the jiggling oscillation of the lattice atoms. In the case of metals, which are electric conductors, the electron clouds of the lattice atoms overlap allowing the valence electrons to jump from lattice site to site when an electric potential (or voltage) is applied to the metal. So conductors allow these electrons to move freely - they are called "free electrons" - and make up an electric current, or flow of electric charge. Insulators, on the other hand, do not share their electrons and they are not free to migrate around. These materials, such as glass, wood, rubber, most plastic, and pure water, do not support electric currents under normal circumstances. What happens to the quantized energy levels of an atom when it is influenced by the presence of all the other lattice atoms in the crystal? It turns out that the idea of quantized energy levels still remains valid, but in a totally different picture. Instead of each atom having its own set of energy levels the solid as a whole can be described by a set of so called energy bands. In this picture the bands represent the quantized energy of the whole crystal. This is represented by the number of the energy bands, their energy widths, the gap between the bands, and whether the band is full, partially full, or empty. So, in a way these bands in the crystal play a similar role to the shells in the case of individual atoms. Just as in the atomic shell model we "fill" energy bands in a solid with electrons. Depending on the atomic species that make up the crystal, these bands get filled or not. For example, in the case of the sodium atom that we examined above, the corresponding energy band picture for its solid form is shown below:

Again it should be stressed that this band picture is really a combination of the energy-level-diagram and the occupancy picture of the shell model. So, here the large gap between the 1s and 2s bands represents the relatively large amount of energy that is necessary to promote an electron in a single atom of sodium from the 1s to the 2s shell. Also the filled 1s, 2s, and 2p bands indicate Na's shell structure. At the same time, the unfilled 3s band shows that these valence electrons in solid sodium can continuously change their energy within this band. This is totally due to the huge number of atoms in the solid structure and was not allowed in the quantized single atom.