Total spin angular momentum quantum number

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... 

Now the total spin-angular momentum quantum number S is given by the number, n, of unpaired electrons times the spin angular momentum quantum number s for the electron, that is, S = nil. Substitution of this relationship into Eq. (5.11) yields an alternative form of the spin-only formula. 

Fio. 2-11.—The interaction of the spin angular momentum vectors of two electrons to form a resultant total spin angular momentum vector, corresponding to the value of the total spin quantum number S = 0 or S = 1. 

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. 

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...

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... 

As in the case of the orbital angular momentum, the upper case 5 here signifies the total spin angular momentum of aU the unpaired electrons in the atom outside of closed shells. The filled, inner subshells contribute nothing to 5. It follows that if the configuration has only a single electron that is not in a filled subsheU, the 5 quantum number is simply the quantum number for the single electron. 

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

When there are two electrons in a molecule, their total spin angular momentum quantum number S is given by Equation 1.9. 

The product mn = 2SA + 1X25 + 1) is the number of possible total spin angular momentum quantum numbers S of the encounter complex that are allowed by combination of the quantum numbers SA and SB of the reaction partners as defined by Equation 2.30. 

The same algebra holds true for the total spin angular momentum S, its projection Sz and the corresponding quantum numbers S and Ms. 

S = total spin angular momentum quantum number... 

Before moving to further examples, we must address the interaction between the total angular orbital momentum, L, and the total spin angular momentum, 5. To do so, we define the total angular momentum quantum number, J. Equation 21.11 gives the relationship for the total angular momentum for a multi-electron species. 

If the coupling between the total orbital angular momentum, L, and the total spin angular momentum, S, has to be taken into account, the total angular momentum J should be used. Where J is the vector sum of L and S. The quantum number of the vector J is again restricted to certain discrete amounts. 

Up = ordinary Hall coefficient R, = spontaneous Hall coefficient R, = extraordinary Hall coefficient 5 = spin quantum number S - total spin angular momentum 5j = spin operators T = absolute temperature T = tesla... 

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