The Quantum Theory.—Ac cording to the quantum theory (q.v.), as applied to the problem of the structure of the atom with planetary electrons circu lating about a nucleus, the angu lar momentum of an orbital elec tron can only assume certain dis crete values. According to Bohr's original form of the theory, the angular momentum p was restricted to such values that k being an integer, the azimuthal quantum number, and h Planck's constant (6.55 X erg X sec.).
Many of the difficulties in connection with the interpretation of spectra (see SPECTROSCOPY) and the upbuilding of atoms (see ATOM) have been simply correlated by supposing that the elec tron itself has an intrinsic spin (the angular momentum being half the Bohr unit, that is -1 • with which is associated an 27 \ intrinsic magnetic moment (equal to the Bohr unit, that is eh ), 4 irmc as was suggested by S. Goudsmit and G. E. Uhlenbeck (2925). The magnetic moment of an atom depends on the resultant of both the orbital and the spin moments of the constituent electrons. The atom as a whole in its normal state has a definite angular momentum, and the quantum theory suggests that, in the presence of a magnetic field, the resolved angular momentum (and the associated resolved magnetic moment) in the field direction, owing to "spatial quantization," can only assume certain discrete values; in other words, only certain definite orientations are pos sible. The general validity of the quantum theory conclusions has been demonstrated most directly by experiments on the magnetic deviation of atomic rays described in a later section. These have definitely established the occurrence of discrete orientations, and have shown that for a number of atoms of specially simple electronic structure (from a magnetic point of view), such as hydrogen (in which there is one electron), and sodium and silver (in which the resultant magnetic moment of all the electrons but one vanishes), the magnetic moment is equal to one Bohr mag neton.
Possible values for resolved magnetic moments of atoms can be more generally determined, on the basis of the quantum theory, from an analysis of the observations on the "splitting" of spectral lines in a magnetic field. It was, in fact, in connection with a quantum theory interpretation of the Zeeman effect that the of assuming discrete orientations was first realised. The frequency of lines in atomic spectra is proportional to the differ ence in energy of the atom in different quantum states. Since the
energy of an atom with a magnetic moment depends on its orientation in a magnetic field, from a study of the way spectral lines are influenced by a magnetic field in the Zeeman effect (q.v.) the possible resolved values of atomic magnetic moments de pending on the possible orientations in a field may be determined.
In the Langevin treatment of a paramagnetic it was assumed that any orientation of the magnetic "carriers" (which might be atoms, molecules or ions) in a field was possible, and an incorrect value for the magnetic moment will be deduced in the application of the theory. If j is the quantum number corresponding to the angular momentum of an atom as a whole, taking into account the discrete orientation, it may be shown that the magnetic moment expressed in Weiss magneton units, p (calculated in the usual way), should be given by where g is a factor (for which Lande has given a general formula which is confirmed by Zeeman effect observations) giving the ratio of the magnetic to the angular moment of the atom. In the simplest cases, when the atoms are in a so-called spectroscopic S state, where the magnetic moment of the atom is due solely to the intrinsic spins of the constituent electrons, this leads to a series of values of p corresponding to integral Bohr magneton values shown in the following table :— The order of magnitude of the magneton values calculated and their sequence from one to five Bohr magnetons are in fair agree ment with the results found for the ions with from 19 to 23 elec trons, suggesting that the quantum theory modification to the classical treatment of paramagnetism is on the right lines. Com plications and difficulties are, however, introduced by a more detailed consideration of the problem. In general the contribution to the magnetic moment of an atom or ion due to the orbital motion of the electrons does not vanish; for atoms in S, P, D states (using the spectroscopic nomenclature), for example, the contribution is o, I, 2 (in Bohr units). If the spectroscopic state of the atom or ion is definitely and completely known, the mag netic moment and p value can be calculated. In this way the p values for the rare earth ions have been calculated by F. Hund (Zeitsch. fur Phys., 1925), with results in remarkable agreement with those shown in the curve of fig. 31. For the first transition group of ions (fig. 3o), however, the agreement is much less satis factory. This may be due to magnetic carriers not in general being free to change their orientation as a whole in a field. Thus, the quantum theory has enabled the magnetic and spectroscopic properties of electronic systems to be correlated in a general way, but many problems as to details remain to be solved. The moments deduced for the paramagnetic molecules NO and 02 and 14.1) are in good agreement with those which may be deduced from an analysis of the band spectra.
The elements which give paramagnetic ions occupy a special position in the periodic table (in which the elements are grouped according to their atomic numbers) the significance of which will be briefly considered in relation to Bohr's theory of the structure of atoms in the last section.