Inner quantum number

Soon after Bohr developed his initial configuration Arnold Sommerfeld in Munich realized the need to characterize the stationary states of the electron in the hydrogen atom by. means of a second quantum number—the so-called angular-momentum quantum number, Bohr immediately applied this discovery to many-electron atoms and in 1922 produced a set of more detailed electronic configurations. In turn, Sommerfeld went on to discover the third or inner, quantum number, thus enabling the British physicist Edmund Stoner to come up with an even more refined set of electronic configurations in 1924. 

Petrashen , A.G., Rebane, V.N. and Rebane, T.K. (1973). Relaxation of electron multipole moments and collisional depolarization of resonance fluorescence of an atomic ensemble in the state with an inner quantum number j = 2, Optika i Spektroskopiya, 35, 408-416. [Opt. Spectrosc. (USSR), 35, 240-244]. 

After Sommorfeld, j is called the inner quantum number it represents the total mechanical moment of the atom. It too must of course be quantised. Since s — there are only the two possibilities,... 

The case of hydrogen is peculiar in one respect. Experiment gives distinctly fewer terms than are specified in the term scheme of fig. 8 for = 2 only two terms are found, for n = 3 only three, and so on. The theoretical calculation shows that here (by a mathematical coincidence, so to speak) two terms sometimes coincide, the reason being that the relativity and spin corrections partly compensate each other. It is found that terms with the same inner quantum number j but different azimuthal quantum numbers I always strictly coincide, for instance, the ns and the np, term, the p. , and the d, term, and so on such pairs of terms are drawn close together in fig. 8. For the value of the terms a formula was given by Sommerfeld (1916), even before the introduction of wave mechanics the same formula is also obtained when the hydrogen atom is calculated by Dirac s relativistic (E908) 11... 

In parhelium, then, the total spin moment s is 0 hence the total orbital moment is identical with the total angular momentum j = 1. This implies that the whole of the terms of parhelium are singlets, i.e. that to every azimuthal quantum number I there belongs only a single term with the inner quantum number j equal to 1. 

Here n denotes tte principal quantum number, which can take all values from 1 upwards I is the azimuthal quantum number, which runs from 0 to n—l j is the inner quantum number, and is only capable of the two values stated m is the projection of j on the specially distinguished direction and, according to the rules for quantisation of direction, runs through the 2j + 1 values between —j and - j. Alternatively, instead of j and m we may use the projections of I... 

Box 1.5 Angular momentum, the inner quantum number, /, and spin-orbit coupling... 

Finally, we have the resultant inner quantum number, J, also called the total angular momentum quantum number since the total angular momentum is given by ... 

The liighest value of the resultant inner quantum number, /= (L + S) =8... 

The existence of the doublet systems in the alkali metal elements requires another quantum number. This number was called the inner quantum number by Sommerfeld and is designated by the letter J and is used to distinguish the components of doublets. The S terms all have a J index of the P terms of and I and the D terms of f and f. Table 2-2 lists the J values for doublet terms. Analysis of the spectra of the alkali metals leads to the selection rule that AJ = 0 or 1, but also that the transition from J = 0 to J = 0 is excluded. 

J inner quantum number (total angular momentum)... 

The principal quantum number n defines the shell in which the electron is located the maximum number of electrons taken up in a shell is 2 . Within the shell the electrons reside in orbitals of different symmetry, described by the angular momentum quantum number I, which can take values of 0, 1, 2,..., n. An orbital can accommodate up to two electrons having opposite spins. Each of the electrons is characterized by the inner quantum number / that can take values of/ 1/2. 

The young Stoner, undaunted by these theoretical problems, reexamined the experimental evidence on optical as well as X-ray spectra of atoms. Based on his studies, he suggested that the number of electrons in each completed level should equal twice the inner quantum number of that particular shell. This produced the scheme shown in table 7.4 for ascribing electrons to sheik. 

And finally, the chemist Main Smith, drawing on the same X-ray data as the atomic physicist Stoner, as well as chemical evidence, was able to arrive at the conclusion that the number of electrons in each subgroup was twice the value of the inner quantum number. He thus obtained the same electron subgrouping several months before Stoner, whose discovery of the same concept is far better acknowledged by historians of science. 

We now want to take into account the magnetic moments of the electrons. The situation of case (b) which must be presented now is realised for all terms of the light elements (up to about Ar) and for all S-terms. Following II, we can say immediately that when the partition of the term (3) or (3a) consists of 2 -coupled spins and n—2z = 2r uncoupled spins (n is the number of electrons in the molecule) that for a term with I < n — z = r, terms with the 2/ -f-1 inner quantum numbers j r — IjV — I, r I result for a term with I > — z = r 2r 1 terms... 

In case (b), and so for light elements and all S-terms, the fine structure is determined by the electronic angular momentum A, which takes on the values 0,0 1,2,3,..., by the azimuthal quantum number Z, which can take on only integer values 0,1,2,..., by the positive or negative character respectively and by the inner quantum number j, which can take on the values [/ — r, Z — r + 1,..., Z + r. The multiplicity of the term is indicated by 2r + 1, so that j is half-integer for... 

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