The Periodic Table

From 1920 to 1923, Bohr applied his ideas to explain features of the periodic table; Bohr's work in this vein was corrected and extended by Pauli in 1924 (with an assist from Stoner).

Suppose we treat the states in the term scheme for hydrogen as slots to hold electrons. Let us hypothesize that each slot can hold at most two electrons. We start with hydrogen, and keep adding electrons, adding new electrons always to the lowest slot that has a vacancy. (Bohr christened this the Aufbauprincip, the building-up principle.) The energy levels of the slots may shift somewhat as a result of interactions between the electrons. This simple model (extremely naive from the modern viewpoint) serves to explain a remarkable number of chemical regularities. For example, energy level $n$ can hold $2n^2$ electrons. We therefore get completely filled energy levels for 2 electrons, 2+8 electrons, 2+8+18 electrons, etc.-- which correspond exactly to the noble gases.

Bohr was able to explain some properties of the rare earth elements using his version of this model⁷; he even predicted correctly that element 72 (not yet discovered) would not be a rare earth, but instead would resemble zirconium (contrary to what some chemists expected). Students of Bohr then discovered element 72 in zirconium ore samples, and element 72 was named hafnium in honor of the Latin name for Copenhagen.

Why two electrons per slot? Pauli proposed adding a new quantum number to the triple $(n,l,m)$; nowadays we use $s$ for this number, and recognize that it stands for the spin of the electron. So $s$ is restricted to the values $\pm \frac{1}{2}$, and each state (or slot) in the term scheme above is really two states. In other words, our Hilbert space must be enlarged to a space with basis $\{v_{nlms}\}$. Pauli also proposed his famous exclusion principle: a state can hold at most one electron.

A historical note: Bohr made no detailed orbit calculations for multi-electron atoms. He did use some general intuitive principles stemming from a classical picture, e.g., a electron in a state of large $n$ will be far away from the nucleus, and electrons with smaller $n$ will partly ``screen'' the charge of the nucleus from the far-off electron. Nevertheless, the Bohr-Sommerfeld hydrogen model had a comforting, semi-classical feel to it: we compute the classical orbits, then decree that only some are permitted. For multi-electron atoms, Bohr dropped this connection. The term scheme becomes almost a combinatorial device.

Physicists (such as Born and Heisenberg) who did perform detailed orbit calculations found utter disagreement between theory and experiment-- even for helium, a mere two electrons. No surprise, since the classical picture leaves out spin, the exclusion principle, and other purely quantum interactions between the electrons. Bohr intuited just how far to push the classical picture; he picked those approximation schemes that survive (reinterpreted) in quantum mechanics.

© 2001 Michael Weiss

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