Quantum numbers total orbital angular momentum

Figure 2.2 Demonstration of the two equivalent nomenclatures used for the description of inner-shell levels and X-ray transitions (also Auger transitions, see below). The vertical direction is regarded as the energy axis (but is not to scale here). On the left-hand side is given the notation which is frequently used in inner-shell spectroscopy, on the right-hand side the corresponding single-orbital quantum numbers with n, t and j being, respectively, the principal quantum number, the orbital angular momentum and the total angular momentum which includes the spin of the electron. Also shown are the main X-ray transitions with their spectroscopic notation (for a more complete plot which includes...

Flowever, the values of the total orbital angular momentum quantum number, L, are limited or, in other words, the relative orientations of f j and 2 are limited. The orientations which they can take up are governed by the values that the quantum number L can take. L is associated with the total orbital angular momentum for the two electrons and is restricted to the values... 

Previously we have considered the promotion of only one electron, for which Af = 1 applies, but the general mle given here involves the total orbital angular momentum quantum number L and applies to the promotion of any number of electrons. 

Here L, S, and J are the quantum numbers corresponding to the total orbital angular momentum of the electrons, the total spin angular momentum, and the resultant of these two. Hund predicted values of L, S, and J for the normal states of the rare-earth ions from spectroscopic rules, and calculated -values for them which are in generally excellent agreement with the experimental data for both aqueous solutions and solid salts.39 In case that the interaction between L and S is small, so that the multiplet separation corresponding to various values of J is small compared with kT, Van Vleck s formula38... 

For terms (i.e., states where the total spin S and the total orbital angular momentum L are good quantum numbers), the allowed transition are = 0 and... 

Here, the relevant angular momentum vectors and quantum numbers are L (7), the total orbital angular momentum of the atom, obtained as the vector coupling of those corresponding to the core and to the outer electron(s), S (S), the total spin, and J (7), the total angular momentum for a given atomic level [8] ... 

The value of J depends on the total orbital angular momentum quantum number, L, and the total spin angulur momentum quantum number, S (Appendix C). Some calculated and experimental magnetic moments for lanthanide complexes are shown in Table 11.25... 

L is the quantum number specifying the total orbital angular momentum for the term, 5 the total spin angular momentum. Each of these momenta has components in any chosen direction, z say, which take on the integral values Lz, from L to -L, or S. from S to -S, respectively. There are 1L + 1 values of L, and 2S + 1 values of Sz, each with appropriate wave functions. Consequently, a term specified by L and S is (2L + + l)-fold degenerate. 

In the extreme case where the spin-orbit interaction is much larger than electronic repulsion, total orbital angular momentum L and total spin angular momentum S are no longer good quantum numbers. Instead, states are defined by total angular momentum J, which is the vector sum of all the total angular momenta y values of the individual electrons ... 

For many-electron light atoms, the Russell-Saunders coupling rules prevail One combines the orbital angular momenta lt of each electron, treated as a vector, to form the total orbital angular momentum quantum number (and vector) L = h one similarly couples the spin angular momentum quantum numbers s, into a total spin angular momentum quantum number S = s > then one adds L and S to get the total angular momentum vector... 

Multiplicity Spin Multiplicity) The number of possible orientations, calculated as 2S -L 1, of the spin angular momentum corresponding to a given total spin quantum number (S), for the same spatial electronic wavefunction. A state of singlet multiplicity has S = 0 and 2S -i- 1 = 1. A doublet state has S = 1/2, 2S -i- 1 = 2, etc. Note that when S > L (the total orbital angular momentum quantum number) there are only 21 -t 1 orientations of total angular momentum possible. 

For an f electron, Z = 3, so that the magnetic quantum number mi can have any one of the seven integral values between - -3 and -3. Vectorial addition of the m/-vaiues for the f electrons for the multi-electron ion affords L, the total orbital angular momentum quantum number ... 

L = total orbital angular momentum quantum number S = total spin angular momentum quantum number J = total angular momentum quantum number... 

The total orbital angular momentum and the total spin, described by L and 5, may be coupled exactly as in the one-electron case (p. 56). Their resultant total angular momentum is described by a quantum number /, with possible values L + S, L + — 5 and for... 

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