A.— IVIATHEMATICAL AND PHYSICAL SCIENCES. 39 



contributions by Pauli,'" Goudsmit,^^ and Heisenberg,'" a general theory 

 of complex spectra has been developed in a practical form by Hund.^' 

 By a complex spectrum, from this point of view, is to be understood the 

 spectrum of an atom which contains more than one electron with k>l 

 in outer uncompleted n^ groups. Thus, while aluminium, with two of 

 the outermost electrons in 3i orbits and one in a 3., orbit, gives a ' simple ' 

 spectrum, the next element, silicon, with tivo electrons in 3^ orbits, gives 

 a ' complex ' spectrum. The theory enables the deeper spectrum terms 

 corresponding to any specified configuration of electrons to be determined 

 with considerable certainty. While it adds nothing to the theory of 

 simple spectra, the theory is clearly of great importance in relation to 

 spectra of greater complexity. It shows, for example, that deep-lying 

 terms which must be classed as F terms on account of combination 

 properties and Zeeman effects, are quite compatible with low values of 

 the angular momenta of the individual electrons. 



It is a fundamental feature of the new theory that, in a complex 

 spectrum, the quantum numbers which specify an electron orbit are quite 

 distinct from those which specify a spectroscopic term. The former are 

 five in number, viz., n, k^, k^, m^, m.^ The latter, which number three, 

 are represented by r, l,j. n is the principal quantum number as previously 

 defined ; k^ is equal to k — h (where k is the azimuthal quantum number) ; 

 jfcj may be either k^ — ^ or k^-i-^ ; and wij and m.^, which are expressible 

 in terms of k^ and ko, are the magnetic quantum numbers for weak and 

 strong fields respectively. The term quantum numbers are defined as 

 follows : r is half the multiplicity of the system ; I denotes the type of 

 term (1=^, |, f . . . for S, P, D . . . terms respectively) ; and j is the 

 inner quantum number which distinguishes the components of a given 

 I term. The theory consists of semi-empirical rules for deducing r, I, and j 

 for the deeper-lying terms from the quantum numbers of the electrons in 

 uncompleted groups. 



The assignment of five quantum numbers to each electron orbit is due 

 to Pauli, who supplemented it by a hypothesis — generally known as Pauli's 

 principle — which asserts that no two electrons in an atom can occupy 

 orbits having the same values for these five quantities. This principle 

 can be shown to lead immediately to the scheme of electron distribution 

 suggested by Main Smith and Stoner ; so that in deducing spectroscopic 

 terms from the orbital quantum numbers given above, it is consistent, 

 and even necessary, to deal with the particular orbits given by Main Smith 

 and Stoner's scheme. 



It is impossible here to give in detail the procedure to be followed 

 in deriving the terms ; when the rules are grasped it becomes mainly a 

 matter of arithmetic. There is a particular case, however, in which the 

 calculation is greatly simplified. If the normal state of an ionised atom 

 is known to be specified by values, r=R, Z=L, then the neutral atom, 

 formed by the addition of an electron in an n,. orbit (provided n, k are not 



»» Zeit.f. Phys., vol. 31, p. 765 (1925). 

 » Zeit.f. Phys., vol. 32, p. 794 (1925). 

 '2 Zeit.f. Phys., vol. 32, p. 841 (1925). 

 s' Zeit. f. Phys., vol. 33, p. 345 ; vol. 34, p. 296 (1925). 



