Various coupling schemes

11 Classification of energy levels 11.2 Various coupling schemes 

Zero-order energy of the central field approximation, described by the central symmetric part of the potential, does not contain interaction of the momenta. Therefore, in zero-order approximation all states of a given configuration differing from each other by quantum numbers m, m, i.e. by different orientation of orbital and spin momenta 1, and s,-, have the same energy, and the corresponding level is degenerated (4/ + 2) times. 

Interaction of the momenta is contained in a non-spherical part of the Coulomb interaction and in the spin-orbit interaction. The value of the energy of the interaction of two momenta depends on the angle between them, therefore, in such a case the definite inter-orientation of all one-electronic momenta is settled. Differently oriented states have different energy, i.e. the zero-order level splits into sublevels and its degeneracy disappears. 

Let us illustrate this method in the special case of two non-equivalent electrons ni/ir fe, described by four momenta two orbital li, I2 and two spin si, S2- Having in mind the commutativity of the addition as well as the fact that the interaction of the orbital momentum of a given electron with its own spin momentum is much stronger than that with 

In all cases (11.2)—(11.5) we have the same resultant (total) momentum J and two intermediate ones, used to denote the coupling scheme obtained. These momenta may be considered as additional quantum numbers, allowing unambiguous classification of the energy levels. 

The total number of levels with given J is the same for all coupling schemes. If there is only one level with given J value, then all coupling schemes are equally valid for it. The level with maximal J value may serve as such an example. In order to obtain such a level we have all momenta directed parallel, i.e. we can couple them only in one fashion. 

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The theory of transformation matrices is described in detail in [9]. In [11] universal and efficient graphical methods of operating with Clebsch-Gordan coefficients and transformation matrices are described. Their use allows one easily to find expressions for the most complex matrix elements of the operators, corresponding to physical quantities, in various coupling schemes, and to carry out their processing if necessary. Here we shall present only the minimal data on transformation matrices and methods of their evaluation. 

Classification of the energy levels using various coupling schemes... 

Usually, the first way is utilized in practice. This is due to the well developed mathematical technique necessary, by the presence of the expressions for both the matrix elements of the energy operator and of the electronic transitions in various coupling schemes. However, the second method is much more universal and easier to apply, provided that there are known corresponding transformation matrices. Now we shall briefly describe this method. 

Let us notice that momenta of each shell may be coupled into total momenta by various coupling schemes. Therefore, here, as in the case of two non-equivalent electrons, coupling schemes (11.2)—(11.5) are possible, only instead of one-electronic momenta there will be the total momenta of separate shells. To indicate this we shall use the notation LS, LK, JK and JJ. Some peculiarities of their usage were discussed in Chapters 11 and 12 and will be additionally considered in Chapter 30. Therefore, here we shall restrict ourselves to the case of LS coupling for non-relativistic and JJ (or jj) coupling for relativistic wave functions. We shall not indicate explicitly the parity of the configuration, consisting of several shells, because it is simply equal to the sum of parities of all shells. 

We shall not present here complete sets of selection rules for all transitions in question. For these transitions they become much more complex, e.g. already for transition (25.24), the pentagon conditions can occur. Interested readers may find them in [173]. Selection rules differ considerably for the various coupling schemes. They change if we account for the relativistic corrections to the non-relativistic E/c-transition operators. However, all selection rules involving intermediate momenta are more or less approximate. Due to the presence of interaction between coupled momenta a certain intermediate coupling scheme takes place, and these selection rules are violated. 

As we have seen in Chapter 11, the energy levels of atoms and ions, depending on the relative role of various intra-atomic interactions, are classified with the quantum numbers of different coupling schemes (11.2)— (11.5) or their combinations. Therefore, when calculating electron transition quantities, the accuracy of the coupling scheme must be accounted for. The latter in some cases may be different for initial and final configurations. Then the selection rules for electronic transitions are also different. That is why in Part 6 we presented expressions for matrix elements of electric multipole (Ek) transitions for various coupling schemes. 

With the introduction of electronic angular momentum, we have to consider how the spin might be coupled to the rotational motion of the molecule. This question becomes even more important when electronic orbital angular momentum is involved. The various coupling schemes give rise to what are known as Hund s coupling cases they are discussed in detail in chapter 6, and many practical examples will be encountered elsewhere in this book. If only electron spin is involved, the important question is whether it is quantised in a space-fixed axis system, or molecule-fixed. In this section we confine ourselves to space quantisation, which corresponds to Hund s case (b). 

Figure 12.18. Various coupling schemes for absorbent recovery, (u) Use of steam or inert gas stripper. (i ) Use of reboiled stripper, (c) Use of distillation.

Lane, A. M., and R. G. Thomas The Theory of Nuclear Reactions and the Compound Nucleus. Rev. Mod. Phys. to be published in 1957. — A very full treatment of the dispersion theory of nuclear reactions, particularly valuable for its treatment of reaction constants in various coupling schemes. Comparison with experimental information is made also Lane, A. M. The reduced widths of nuclear energy levels, A.E.R.E. Report T/R1289 (1954). (United Kingdom Atomic Energy Authority.)... 

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