Quantum_ Einstein, Bohr and the Great Debate About the Nature of Reality - Manjit Kumar [87]
If each electron in an atom is assigned the quantum numbers n, k, m, and each unique set of numbers labels a distinct electron orbit or energy level, then according to Stoner, the number of possible energy states for, say, n=1, 2 and 3 would be 2, 8 and 18. For the first shell n=1, k=1 and m=0. These are the only possible values the three quantum numbers can have and they label the energy state (1,1,0). But according to Stoner, the first shell is closed when it contains 2 electrons, double the number of available energy states. For n=2, either k=1 and m=0 or k=2 and m=–1,0,1. Thus in this second shell there are four possible sets of quantum numbers that can be assigned to the valence electron and the energy states it can occupy: (2,1,0), (2,2,-1), (2,2,0), (2,2,1). Therefore, the shell n=2 can accommodate 8 electrons when it is full. The third shell, n=3, has 9 possible electron energy states: (3,1,0), (3,2,-1), (3,2,0), (3,2,1), (3,3,-2), (3,3,-1), (3,3,0), (3,3,1), (3,3,2).29 Using Stoner's rule, the n=3 shell can contain a maximum of 18 electrons.
Pauli had seen the October issue of the Philosophical Magazine, but ignored Stoner's paper. Not known for his athleticism, Pauli ran to the library to read it after Sommerfeld mentioned Stoner's work in the preface to the fourth edition of his textbook Atomic Structure and Spectral Lines.30 Pauli realised that for a given value of n, the number of available energy states, N, in an atom that an electron could occupy was equivalent to all the possible values that the quantum numbers k and m could take, and was equal to 2n2. Stoner's rule yielded the correct series of numbers 2, 8, 18, 32 … for the elements in the rows of the periodic table. But why was the number of electrons in a closed shell twice the value of N or n2? Pauli came up with the answer – a fourth quantum number had to be assigned to electrons in atoms.
Unlike the other numbers n, k, and m, Pauli's new number could have only two values, so he called it Zweideutigkeit. It was this 'two-valuedness' that doubled the number of electron states. Where there had previously been a single energy state with a unique set of three quantum numbers n, k, and m, there were now two energy states: n, k, m, A and n, k, m, B. These extra states explained the enigmatic splitting of spectral lines of the anomalous Zeeman effect. Then the 'two-valued' fourth quantum number led Pauli to the exclusion principle, one of the great commandments of nature: no two electrons in an atom can have the same set of four quantum numbers.
The chemical properties of an element are not determined by the total number of electrons in its atom but only by the distribution of its valence electrons. If all the electrons in an atom occupied the lowest energy level, then all the elements would have the same chemistry.
It was Pauli's exclusion principle that managed the occupancy of the electron shells in Bohr's new atomic model and prevented all of them from gathering in the lowest energy level. The exclusion principle provided the underlying explanation for the arrangement of the elements in the periodic table and the closing of shells with chemically inert rare gases. Yet despite these successes, Pauli admitted in his paper, 'On the Connection between the Closing of Electron Groups in Atoms and the Complex Structure of Spectra', published on 21 March 1925 in Zeitschrift für Physik: 'We cannot give a more precise reason for this rule.'31
Why four quantum numbers, and not three, were needed to specify the position of electrons in an atom was a mystery. It had been accepted since the seminal work of Bohr and Sommerfeld that an atomic electron in orbital motion around a nucleus moves in three dimensions and therefore