6j-symbols

Secondly, due to the smallness of the rotational temperature for the majority of molecules (only hydrogen and some of its derivatives being out of consideration), under temperatures higher than, say, 100 K, we replace further on the corresponding summation over rotational quantum numbers by an integration. We also exploit the asymptotic expansion for the Clebsch-Gordan coefficients and 6j symbol [23] (JJ1J2, L > v,<0... 

The conditions determined by the selection rules for Clebsch-Gordan coefficients and 6j symbols provide the following block-diagonal structure of the operator... 

In the case of k = 0 (like the kinetic energy + electron-nuclear attraction operator in atoms, H(1)), the 6j-symbol collapses to a number... 

For algebraic evaluation of the 3j- and 6j-symbols the Racah formulae are useful (Table 58). Various vector coupling coefficients met so far are reviewed in Table 59. In point groups analogous coefficients occur (Table 60). 

RBK67] Radeloff J, Buttler N, Kesternich W and Bodensedt E 1967 Nucl. Instrum. Meth. 47 109. [RBM59] Rotenberg M, Bivins R, Metropolis N and Wooten JK 1059 The 3j and 6j-symbols (Technology Press, Massachusetts Institute of Technology). 

In all cases of non-linear light absorption the amount of calculations can be reduced if one applies the symmetry properties of the coefficients KA K>, KAX , Kj Xk npXK jn most simple way these properties can be written for P, Q and P transitions in the case of the first two coefficients which do not depend on the value of the lower index. Thus omitting this index, from the properties of the 6j-symbols [379] we obtain [27]... 

The result obtained reflects linear approximation, i.e. manifestation of the usual linear (observable signal is proportional to Tp) Hanle effect (Section 4.2). This contention becomes obvious if we substitute the polarization moments obtained into the expression for fluorescence intensity (5.34) and find the form of the degree of polarization V = (Iy — Ix)/(Iy + Ix) (Fig. 5.8). For instance, for a radiative transition (J = 1) -> (J" = 1), and using Table C.3 for numerical values of 6j-symbols and for values of from Table 2.1, we obtain... 

As mentioned in the preceding section, the law of conservation of angular momentum in the process of optical pumping of molecules is reflected in the coefficients kFXk, kFxk, KA K, K Ax and CK in Eqs. (5.28), (5.29), (5.30), (5.31) and (5.32). For optical transitions between states with large angular momentum quantum numbers J, J" the explicit form of these coefficients is considerably simplified [15, 39]. These simplifications are due to the possibility of employing asymptotic expressions of the 6j-symbols presented in Appendix C. Applying Eq. (C.12) for the coefficients KA K and K Ax we obtain... 

It is frequently useful to employ formulae for sums of 3 -symbols, connecting them with 6j-symbols (given in curly brackets, ) ... 

Table C.3. Numerical values of 6j-symbols for integral indices not exceeding 2...

Gordon coefficients C lj3 with 6 = 1. Table C.3 contains the numerical values of 6j-symbols for integral indices < 2. More complete tables may be found in [379]. 

The magnitude of the 6j-symbols does not change at permutation of columns, as well as at permutation of any two elements of the top line with two elements of a lower line positioned under the former. 

In cases where some of the angular momenta entering into the 6 j-symbols become large, one may apply asymptotic formulae for the 6j-symbols. 

The sum over S can be carried out analytically by relating the Wigner 6j symbols to the Wigner 9j symbol,... 

The summation over Jr, is then completed using the orthogonality of the Wigner 6j symbols,... 

The 3j and 6j symbols as well as the doubly reduced matrix elements needed in the calculation of crystalline field energies have been programmed for us by Gelio Alves and Chamindra Williams at Florida Atlantic University. 

In the second step we are left with the problem of evaluating, in the local basis, matrix elements of the various Casimir operators and functions thereof. This is basically a trivial task for diagonal operators. However, the Majorana operator deserves more computational skill, for obvious reasons. To accomplish this task, one needs standard routines for the evaluation of 6j symbols. It is then necessary to arrange the basis states in such a way that the block diagonal form of the Majorana operator is conveniently emphasized. 

Some of the consequences of the spherical symmetry of the hamiltonian of an atomic system for the Green s function are explored in this section. A number of results concerning tensor operators are invoked and the reader are assumed to be familiar with the Wigner 3j and 6j symbols. 

The Wigner 3j and 6j symbols, their basic properties uid their fundamental relations are discussed in a great many different sources. 

The U and V coefficients are simple products of 6j symbols with U12 and V12 representing interference terms between the transition amplitudes t( ) and t( ) (e.g. between Ml and E2) Un and Vn, and U22 and V22 result solely from the individual t( ) and t( ) components of the mixed transition. We will examine the form of these angular coefficients in the next section. Here, however, we note the existence of certain sum rules which can be anticipated by switching off the hfs interaction causing the spectrum to degenerate into a single component, viz,... 

Pound RV, Rebka GA Jr (1959) Phys Rev Lett 3 554 Pound RV, Rebka GA Jr (1960) Phys Rev Lett 4 337 Raimes S (1961) The vfuve mechanics of electrons in metals. North Holland, Amsterdam Rotenberg M, Bivins R, Metropolis N, Wooten JK Jr (1959) The 3j and 6j symbols. MIT Press, Cambridge Ruby SL (1974) J Phys 35 C6-209 Sano H (1972) Mossbauer spectroscopy the chemical applications. Kodansha, Tokyo Sawicki JA, Tyliszczak T, Gzowski O (1981) Nucl Instr Meth 190 433... 

The matrix elements are calculated from the reduced matrix elements of Nielson and Koster (1964) and the 3j and 6j symbols (Rotenberg et al., 1959) as described by Wybourne (1975). 

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