Clebsch-Gordon

Clebsch-Gordon coefficients, 404 Clifford algebra, 520 Closed trajectory, 328 Closest approach, distance of, 6 Code... 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

That is, the character of the product representation is just the product of the characters. Hence the Clebsch-Gordon coefficients are... 

YkM is a vector coupled product of solid harmonics[69] given by the Clebsch-Gordon expansion,... 

From the properties of the Clebsch-Gordon coefficients we infer that the only non-zero terms in the sum (6-38) are those for which (c.f. )... 

Since the Gaunt Coefficient (2 k CfJ- > 2 k) may be expressed in terms of Clebsch-Gordon coefficients it is easy to show that the parameters Ejm and 22jm are proportional so that the reduction of the matrix elements of the static potential to combinations of radial integrals over the r/-orbital basis (potential parameters) has the same structure for both local and non-local potentials which can therefore be treated on the same footing. 

Evaluation of the Clebsch-Gordon coefficients then leads to the following results for the /4, ... 

Clebsch-Gordon coefficients, and the ratio of orbital and density function normalization coefficients (4 ),... 

For simplicity, we notate the Clebsch-Gordon coefficients as instead... 

The symmetry of the Clebsch-Gordon coefficients is again used in conjunction with Eq. (10) (02,m = 0.2,-m) resulting in the elimination of terms with Z - 1,3. For example, the terms with I = 1 have (where only k,j) = ( 2, T2) are shown explicitly)... 

Here y-/kj is the normalization constant, ( jm) denotes the Clebsch-Gordon coefficient, and Yfm is the spherical harmonics. The radial contribution into the mode functions (18) depends on the boundary conditions as follows [24]... 

Using the properties of the Clebsch-Gordon coefficients [71] and spherical harmonics [70], for the mode functions (18) in this limit we get... 

Taking into account the properties of spherical harmonics [70], Clebsch-Gordon coefficients [71], and spherical Bessel and Hankel functions [70], it is possible to show that the mode functions in (18) obey the following condition of symmetry ... 

Inserting numerical values for the Clebsch-Gordon coefficients gives... 

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