A good way to understand the impact of the exclusion principle on atomic structure is to do some hypothetical atom building. We start with a stock of bare nuclei, one of each charge—a proton, a helium nucleus, a lithium nucleus, and so on—and a collection of electrons. To each nucleus we add electrons one at a time and think about the states of motion that result. Before starting this Jovian task, we can number the nuclei as well as name them. The atomic number, Z, is defined as the total charge of the nucleus in units of the proton charge. It is the number of protons in the nucleus and the number of electrons in the neutral atom. If no elements have been overlooked, it also labels the position of the element in the periodic table. Sodium, for instance, with 11 electrons, and 11 protons in its nucleus, is the eleventh element in the periodic table. These facts, simple today, were startling and wonderful insights in 1913.
In order to build the world—or at least its constituent atoms—methodically, let us begin with hydrogen (Z = 1). An electron brought near to a proton will cascade through successive states of motion to end in the state of lowest energy, the ground state, the normal state of the atom. Here the electron’s quantum numbers are
n = 1, ℓ = 0, mℓ = 0, ms = ½ or –½ .
Since it has no orbital angular momentum in this state, the electron’s orbital orientation quantum number is zero. Its spin may be oriented in either of two directions.
Consider next helium (Z = 2). The first electron will fall into the lowest state, with quantum numbers the same as those of the ground state of hydrogen. According to the exclusion principle, a second electron cannot join the first with identical quantum numbers. However, there is available at the lowest energy level a second state of motion differing from the first only in spin orientation. The second electron follows the first to a state with principal quantum number n = 1, and orbital angular momentum ℓ = 0, but with opposite spin orientation. Since these two electrons use up the available two states of motion at n = 1, this situation is described as a closed shell. Helium is the first closed-shell atom and the first member of the rare-gas family. There are many unique things about helium, all arising from its tightly bound closed-shell structure. To name a few: It is chemically the most inert of all elements; it has the highest ionization energy of any atom; and it has the lowest boiling point of any substance, 4.2 K.
I should remark at this point that hydrogen too has a uniqueness apart from being the first element in the periodic table. It is the only atom that has a single electron in a shell and is at the same time one electron shy of a closed shell. In the periodic table it occupies its own corner niche.
Hydrogen and helium complete the first period. A second period of eight members begins with lithium (Z = 3) and ends with the second rare gas, neon (Z = 10). In this period electrons fill successively the eight states of motion available at the second principal quantum number, n = 2, as detailed in the table below.
Possible values of four quantum numbers for the lowest two states of motion of an electron about a central positively charged nucleus
The first two electrons added about a lithium nucleus can drop into the n = 1 state with oppositely directed spin. The third electron is excluded. The lowest state of motion not already occupied is at the n = 2 level. The third electron drops down to this level and can go no further. The power of the exclusion principle is evident. Because of it, lithium has one relatively loosely bound electron. The properties of the atom are in every way vastly different than they would be if the exclusion principle did not act to prevent this electron from joining the first two in the lowest state of motion.
At the n = 2 level, the ℓ = 0 state can accommodate two electrons and the ℓ = 1 state six more (three orbital orientations, each with two spin orientations). When these eight states of motion are fully occupied, another closed shell results, this one at Z = 10, neon. Sodium (Z = 11) begins the third period. Its first ten electrons occupy closed shells. Its eleventh electron, excluded from the n = 1 and n = 2 shells, can cascade downward only as far as the n = 3 level. This single eleventh electron can be considered the active agent in sodium. It is responsible for most of the chemical and physical properties of sodium, including its similarity to lithium.
Although it is unnecessary to pursue this atom-by-atom survey further, one important point about atom building remains to be clarified. This is the way in which the energy-level pattern available to one electron is distorted by the other electrons in a heavy atom. Consider again the eleventh electron in sodium. It experiences approximately a central force, but this is not a Coulomb (inverse-square) force. At a great distance from the nucleus, attracted by a net charge of +e, it experiences a force whose magnitude is
If it penetrates close to the nucleus, it experiences the full force of eleven protons,
In between, the form of the law of force varies gradually between these two limits. Because of this variation, states of motion that would have the same energy (be “degenerate”) in hydrogen are separated in energy in heavier atoms. Occupying the n = 3 level in sodium, the last electron could have zero, one, or two units of angular momentum. The state of maximum angular momentum corresponds to a circular orbit, the state of minimum angular momentum to a thin ellipse penetrating close to the nucleus. This correspondence is reflected in the wave amplitudes of the different states, as illustrated in the figure below.
Radial cross-sections of wave functions for three different orbital angular momenta at the n = 3 energy level. The lower the angular momentum, the more the electron wave penetrates the space near the nucleus. These three states have equal energy in hydrogen, but have different energies in the atoms of heavier elements.
The wave of the ℓ = 0 state penetrates close to the nucleus, into the region of stronger attraction. The wave of the ℓ = 2 state peaks well away from the nucleus, in a region of weaker attraction. Consequently the three states, all of the same principal quantum number, are spread apart in energy, the ℓ = 0 state being lowest (tightest binding), the ℓ = 2 state being highest (loosest binding). This energy distortion is pictured schematically in the figure below for the first four principal quantum numbers.
Distorted energy level diagram produced by stronger attractive force experienced by electrons in states of motion with small angular momentum. The distortion influences the shell structure. An n = 3 level, for instance, appears in the fourth shell. (This diagram is schematic only, not to scale.)
At n = 3, the distortion is sufficient to influence the shell structure in an important way. The “circular” state, with ℓ = 2, is so loosely bound that it is pushed effectively up into the fourth shell. Instead of the eighteen elements that might be expected in the third period, based on counting the total number of states of motion available at n = 3, there are only eight elements—from sodium (Z = 11) through argon (Z = 18). A period of 18 elements shows itself first in the fourth row of the periodic table.
In this distorted energy diagram, the individual energy levels are labeled by the largest number of electrons they can hold without violating the exclusion principle. These numbers explain at once the observed lengths of the first five periods of elements. For the remainder of the periodic table, the energy distortion complicates the pattern further. Nevertheless, a very well defined periodicity persists through all the known elements.
So the entire structure of the periodic table is dictated by two discoveries of the 1920s: electron spin and the exclusion principle. By itself, spin would have been an interesting discovery, since it is a property of a fundamental particle. In conjunction with the exclusion principle, it is more than interesting. It is momentous. By doubling the number of states of motion available to the electron, the apparently innocuous fact that the electron spins shapes the periodic table and thereby shapes the world.
1 The exclusion principle actually applies to more than just electrons. It governs all fermions—particles of half-odd-integer spin—and applies separately to each such particle. Thus protons obey the exclusion principle, as do neutrons (both being particles of spin ½). A proton may say to another proton in the nucleus, “Keep out of my state of motion,” but at the same time welcome a neutron to share that state. Nuclear structure is as fundamentally shaped by the exclusion principle as is atomic structure.