Clebsch-Gordan coefficients table

Table C.l presents the numerical values of some Clebsch-Gordan coefficients. Table C.2 contains formulae for the, often necessary, Clebsch-...

The authors are hopeful that the book will serve as a practical manual for the researcher, both theorist as well as experimentalist, in atomic, molecular or chemical physics. With this in mind, we have paid special attention to the comprehensive information necessary to follow all calculations through to their conclusions. We also hope that the material in the appendices will take the place of a short handbook with formulae on vector calculus, spherical functions, Wigner D-functions, Clebsch-Gordan coefficient tables, etc. We fully realize that one may find in various pa-... 

From (4.56) and Table 4.3, we derive the relative intensity ratios 3 2 1 1 2 3 for the hyperfine components of a Zeeman pattern of a powder sample. The transition probability for the case of the polar angle 6 = Oq can readly be calculated by integrating (4.56) only over the azimuthal angle (j). One obtains a factor (1 + cos 0o)/2 and sin 0o for m = 1 and m = 0, respectively, which are multiplied by the square of the Clebsch-Gordan coefficients. As a consequence of the angular correlation of the transition probabilities the second and fifth hyperfine components (Fig. 4.17) disappear if the direction k of the y-rays and the magnetic field H are parallel (0q = 0). 

As we can see from Eq. (6.8), the Wigner coefficients are much more symmetrical than the Clebsch-Gordan coefficients. Therefore, usually their tables are presented (see, for example, in [11] the tables for 71+72+73 < 16). 

The symmetry of a number of atomic quantities (wave functions, matrix elements, 3n./-coefficients etc.) with respect to certain substitution groups or simply substitutions like l — — / — 1, L — —L — 1, S — — S — 1, j — — j— 1, N 41 + 2 — N, v —y 41 + 4 — v leads to new expressions or helps to check already existing formulas or algebraic tables [322, 323]. Some expressions are invariant under such transformations. For example, Eq. (5.40) is invariant with respect to substitutions S — —S — 1 and v — 41 + 4 — v. Clebsch-Gordan coefficients in Table 7.2 are invariant under transformation j — —j — 1. However, applying this substitution to the coefficient in Table 7.2, we obtain the algebraic value of the other coefficient. 

Table 14.6. Clebsch-Gordan coefficients for the inner direct products of irreducible co-representations.

My aim has been to give in these tables only the most commonly required information. For character tables for n > 6, Cartesian tensor bases of rank 3, spinor bases, rotation parameters, tables of projective factors, Clebsch-Gordan coefficients, direct product... 

Table 7.1. Some Clebsch-Gordan coefficients (values which are unity according to equ. (7.36a) or can be calculated from the given values using equs. (7.39) have been omitted).

If this condition is not fulfilled, the Clebsch-Gordan coefficients vanish, otherwise they have certain numerical values (see Table 7.1). The Clebsch-Gordan coefficients are related to the Wigner coefficients [Wig51], also called 3j symbols jt = a, h = b, j3 = c,m1 = a, m2 = P,m3 = y), defined by... 

The electric-dipole transition is determined by the symmetry properties of the initial-state and the final-state wave functions, i.e., their irreducible representations. In the case of electric-dipole transitions, the selection rules shown in table 7 hold true (n and a represent the polarizations where the electric field vector of the incident light is parallel and perpendicular to the crystal c axis, respectively. Forbidden transitions are denoted by the x sign). In the relativistic DVME method, the Slater determinants are symmetrized according to the Clebsch-Gordan coefficients and the symmetry-adapted Slater determinants are used as the basis functions. Therefore, the diagonalization of the many-electron Dirac Hamiltonian is performed separately for each irreducible representation. 

From the form of Eq. (2.33) it becomes understandable why the anisotropy of polarization 7Z is sometimes called the degree of alignment. From the point of view of the determination of the magnitude of the polarization moments bPo the measurement of 71 is preferable, as compared with that of V, all the more so if one bears in mind that the population bPo appears only as a normalizing factor for all other bPQ and does not influence the shape of the probability density p(B,multipole moment dependence of V and 71 for various types of radiational transition (A = 0, 1) can be obtained using the numerical values of the Clebsch-Gordan coefficient from Table C.l, Appendix C. 

This fact complicates the measurement of the relaxation rate of the orientation Ti for details see Section 4.2. The numerical values of the Clebsch-Gordan coefficients entering into (2.37)-(2.39) are given in Table C.l. C values for particular cases are presented in Table 3.6. 

In particular, in the case of linearly polarized excitation through bPo> bPo-> and according to (3.19), three ground state polarization moments may emerge with k = 0,2 and 4. Indeed, substituting coefficients from Table 2.1 and Clebsch-Gordan coefficients from Table C.l into (3.19) we obtain... 

Following Sugano, Tanabe and Kamimura [1] the vector coupling coefficients (otherwise known as Clebsch-Gordan coefficients) shown in Table 1 inform us how the decomposition products are constructed from the initial functions. 

Determinantal representation of CSFs. Expansion of CSFs in the unitary group approach in terms of spin-orbital determinants. The coefficients are determined as products of factors,/k, determined from the Clebsch-Gordan coefficients and phase factors of Table II. The coefficient sparseness of the determinants is predictable and corresponds to the allowed area principle. 

The Clebsch-Gordan coefficients and the angular dependence functions needed to calculate the relative intensities of the hyperline lines for a f -> i transition with Ml multipolarity have already been given in Table 3.2 (p. 67) and discussed in detail in Section 3.7. In this appendix are listed other useful sets of coefficients for the cases... 

Just this matrix is generated from the previous generation of the coupling matrix U(12) and the new set of Clebsch-Gordan coefficients listed in Table 1.12. The resulting transformation matrix is... 

The basis functions in O can be written as certain linear combinations of the basis functions in O. In analogy with the Clebsch-Gordan coefficients or the 3-j s5unbols there exist tables of coupling coefficients for the finite point groups (27, 34). A convenient notation for a ket in O is... 

The probability of a yray transition occurring depends on the product of an angular-independent term (the square of the appropriate Clebsch-Gordan coefficient, see Condon and Shortley The Theory of Atomic Spectra Cambridge University Press, 2nd edn 1953) and an angular-dependent term where 6 is the angle between the direction of the y-tay and the principal axis (z axis) of the magnetic field or e.f.g. tensor. The probabilities are listed in Table 1 for Fe(57) transitions. 

The 85th Edition includes updates and expansions of several tables, such as Aqueous Solubility of Organic Compounds, Thermal Conductivity of Liquids, and Table of the Isotopes. A new table on Azeotropic Data for Binary Mixtures has been added, as well as tables on Index of Refraction of Inorganic Crystals and Critical Solution Temperatures of Polymer Solutions. In response to user requests, several topics such as Coefficient of Friction and Miscibility of Organic Solvents have been restored to the Handbook. The latest recommended values of the Fundamental Physical Constants, released in December 2003, are included in this edition. Finally, the Appendix on Mathematical Tables has been revised by Dr. Daniel Zwillinger, editor of the CRC Standard Mathematical Tables and Formulae it includes new information on factorials, Clebsch-Gordan coefficients, orthogonal polynomials, statistical formulas, and other topics. 

The following explicit expressions for the wave-functions representing a one-electron atom may be obtained by using the Clebsch-Gordan coefficients given in Table 3.3 ... 

For certain mathematical functions and operations it is necessary for the physicist to know their context, definition and mathematical properties, which we treat in the book. He does not need to know how to calculate them or to control their calculation. Numerical values of functions such as sinx have traditionally been taken from table books or slide rules. Modern computational facilities have enabled us to extend this concept, for example, to Coulomb functions, associated Legendre polynomials, Clebsch—Gordan and related coefficients, matrix inversion and diagonali-sation and Gaussian quadratures. The subroutine library has replaced the table book. We give references to suitable library subroutines. 

To find the total-angular-moraentura eigenfunctions, one must evaluate the coefficients in (11.33). These are called Clebsch-Gordan or Wigner or vector addition coefficients. For their evaluation, see Merzbacher, Section 16.6. For tables of them, see Anderson, Introduction to Quantum Chemistry, pages 332-345. 

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