When two hydrogen atoms come close, their 1s level splits into two, the bonding and the antibonding orbital. In a solid the same thing happens among an enormous number of atoms. The atoms in a crystal are arranged regularly, and the orbitals of the outer electrons overlap with those of the neighboring atoms.
The upper part of the figure shows N identical wells at equal spacing. With one well there is one level, E_0. With two wells the level splits into two, E_0 - t and E_0 + t. Here t measures how strongly neighboring wells are coupled and is called the transfer integral. The more the orbitals of the two wells overlap, the larger t becomes.
If each well is coupled only to its neighbors, the levels of N wells are E_j = E_0 - 2t\cos\dfrac{j\pi}{N+1} for j = 1, 2, \ldots, N. For N = 2 this gives E_0 \mp t, the bonding and antibonding orbitals with the overlap integral neglected. For N = 3 there are three levels, E_0 - \sqrt{2}t, E_0 and E_0 + \sqrt{2}t.
There are as many levels as wells, but however large N becomes, the levels never spread beyond the range from E_0 - 2t to E_0 + 2t, a width of 4t. The more wells there are, the closer the levels are within that same width.
One cubic centimeter of a crystal contains more than 10^{22} atoms. With 10^{22} levels in a range a few eV wide, the spacing is about 10^{-22}\,\mathrm{eV}, far too small to measure. In effect the levels cover a continuous range of energy. This range is called an energy band. Each atomic level, 1s, 2s, 2p and so on, gives rise to one band.
When the number of atoms is enormous, the effect of the edges of the crystal can be ignored. The potential energy of an electron in the crystal then repeats with the same period a as the row of ions, V(x + a) = V(x). The stationary states in such a periodic potential have a definite form.
In 1928 Bloch showed that a wave function in a periodic potential can be written as \psi(x) = e^{ikx}u(x), where u(x) is a periodic function with u(x + a) = u(x). The wave function is a plane wave e^{ikx} with wave number k, multiplied by a function that has the same shape in every cell. This is Bloch's theorem, and waves of this form are called Bloch waves.
Put another way, \psi(x + a) = e^{ika}\psi(x). Moving to the next cell only multiplies the wave function by the phase e^{ika} and does not change its absolute value. So the probability density |\psi|^2 is the same in every cell. The electron is not near any particular atom; it is spread over the whole crystal in the same way.
The figure represents the attraction of the ions by delta functions at x = 0, a, 2a, \ldots and shows the Bloch wave with ka = 0.4\pi. The upper curve is the real part of the wave function, which moves to the right as time passes. It has sharp corners at the ions because its slope changes at each delta function. Taken as a whole, it is a wave with wavelength \dfrac{2\pi}{k} = 5a. The lower curve, |\psi|^2, repeats the same shape in every cell.
The wave number k is what tells the states of the electron apart. Changing k by \dfrac{2\pi}{a}, however, leaves e^{ika} unchanged, so all the states can be counted within the range -\dfrac{\pi}{a} < k \le \dfrac{\pi}{a}. This range is called the Brillouin zone.
A Bloch wave is a stationary state, so in a perfectly periodic crystal an electron keeps moving without being scattered by the ions. The electrical resistance of a metal comes from disturbances of the periodicity, such as the thermal vibration of the ions and impurities. This is why the resistance of a pure metal decreases when it is cooled and the thermal vibration becomes smaller.
Without the attraction of the ions, the electron would move freely and its energy would follow the parabola E = \dfrac{\hbar^2k^2}{2m}, the dashed line in the figure. The vertical axis is in units of E_1 = \dfrac{h^2}{8ma^2}, the energy of a free electron at k = \dfrac{\pi}{a}. This is the same as the lowest level of a square well of width a.
For a row of delta functions, Bloch's theorem together with the change of slope at the ions gives the relation \cos ka = \cos qa - P\dfrac{\sin qa}{qa}. Here q = \dfrac{\sqrt{2mE}}{\hbar} is the wave number the electron would have if it were free. P = \dfrac{m\alpha a}{\hbar^2} measures the strength of the attraction, with \alpha the strength of a single delta function. For P = 0 we get k = q and the parabola again. This is the model that Kronig and Penney studied in 1931, in the limit of narrow and deep wells.
The value of the right-hand side depends on E, and its absolute value can exceed 1. Since \cos ka can only take values from -1 to 1, no Bloch wave exists with such an energy. The shaded ranges of energy in the figure are these, and they are called gaps, or band gaps. The ranges of energy between the gaps, which the electron can have, are the bands.
The figure shows what happens as the attraction is increased from zero. While it is weak, the curve stays almost a parabola, but it is cut at k = \dfrac{\pi}{a}, \dfrac{2\pi}{a}, \dfrac{3\pi}{a}, and the pieces separate up and down. Gaps open only where k is a whole multiple of \dfrac{\pi}{a}.
To compare with the parabola of a free electron, the figure extends k beyond the Brillouin zone. The second band is drawn in \dfrac{\pi}{a} < |k| < \dfrac{2\pi}{a} and the third in \dfrac{2\pi}{a} < |k| < \dfrac{3\pi}{a}, but shifting k by a whole multiple of \dfrac{2\pi}{a} brings each of them into the Brillouin zone.
In the picture of wells in a row, the atomic levels spread out into bands. In the picture that starts from free electrons, the parabola is cut in places and the pieces become bands. The two pictures give the same result. The first suits inner electrons whose orbitals overlap little, and the second suits cases where the attraction of the ions has only a small effect, such as the valence electrons of a metal.
A free electron wave e^{ikx} moving through the crystal is partly reflected by each ion. At k = \dfrac{\pi}{a} the wavelength is 2a. Two waves reflected by neighboring ions differ in path by 2a there and back, exactly one wavelength, so all the reflected waves are in phase and reinforce each other. This is the same condition as for the Bragg reflection of X-rays.
Because the reflections reinforce each other, the wave moving right, e^{i\pi x/a}, and the wave moving left, e^{-i\pi x/a}, mix with equal size and form a standing wave. Two standing waves can be made: their sum gives \cos\dfrac{\pi x}{a} and their difference gives \sin\dfrac{\pi x}{a}. Making a sum and a difference of two waves is what we did with the two orbitals of the covalent bond.
The density of the first, \cos^2\dfrac{\pi x}{a}, is largest at the ions, while \sin^2\dfrac{\pi x}{a} is largest between them. The ions attract electrons, so the first, with more of the electron at the ions, has lower potential energy, and the second, with more of the electron between the ions, has higher potential energy. At the same k the energy splits into two, and the range between them is the gap.
The figure shows the densities of the two standing waves for the row of delta functions. The upper one has nodes at the ions, so it is not affected by the delta functions, and its energy stays at E_1, the same as for a free electron. The lower one has its largest density at the ions, and its energy drops to 0.29E_1. The difference, 0.71E_1, is the width of the first gap.
The same thing happens at k = \dfrac{2\pi}{a}. There the reflections from neighboring ions reinforce each other with a difference of two wavelengths, two standing waves form again, and the second gap opens. When the attraction is weak, the width of the n-th gap is twice the amplitude of the part of the ions' potential energy that oscillates with period \dfrac{a}{n}.
To remove the effect of the edges, imagine N atoms arranged in a ring. Going once around the ring brings us back to the same place, so \psi(x + Na) = \psi(x) must hold. By Bloch's theorem \psi(x + Na) = e^{ikNa}\psi(x), so e^{ikNa} = 1. That is, k must be a whole multiple of \dfrac{2\pi}{Na}. This is called the periodic boundary condition.
The Brillouin zone -\dfrac{\pi}{a} < k \le \dfrac{\pi}{a} contains exactly N whole multiples of \dfrac{2\pi}{Na}. So one band contains N states with different k. The figure marks the allowed values of k as dots on the band E = E_0 - 2t\cos ka for atoms coupled only to their neighbors. As the number of atoms goes from 4 to 8 to 16, the number of dots grows with it.
An electron has spin, and each k state can hold up to two electrons with opposite spins. With N atoms a band has N states, each holding up to two electrons, so one band holds 2N electrons, two per atom.
This count agrees with what we saw for wells in a row, where there were as many levels as wells. Whether the atoms form a ring or a row with ends, N atoms give N states. In a ring, those N states can be told apart by k.
How many electrons a band actually contains depends on how many electrons each atom has. Sodium has one 3s electron per atom, so the band formed from 3s holds only half as many electrons as it could. Whether a substance has a band that is only partly filled decides whether it conducts electricity. How this works is the subject of the article on metals, insulators and semiconductors.