Unit tensor operators

As for any one-electron operator, the (orbital) unit tensor operator has the following reduced matrix elements ... 

Thus any one-electron operator has reduced matrix elements equal to those of the unit tensor operator times its one-electron reduced matrix element (Z 1 1 l). Other results are as follows ... 

In the complete basis set of ATs PvLSMlMs) referring to an ln configuration, the matrix elements of the CF operator (as for any one-electron operator) can be expressed with the help of the unit tensor operator... 

The reduced matrix elements of a many-electron operator can be expressed through the unit tensor operator... 

The unit tensor operator of the first rank is as simple as ... 

Table 49 terms a Non-zero reduced matrix elements of the unit tensor operators between dn- ... 

The (orbital) unit tensor operator of rank k for one electron is introduced as having the reduced matrix element equal to unity, i.e. 

The unit tensor operators are irreducible-tensor operators with reduced matrix elements of unity. They are a valid choice to use as a basis in order to express any arbitrary tensor operator as a linear combination, since they are linearly independent. Attention is restricted to these for the sake of simplicity. Hence the definition of unit tensor single-particle operator U (aK, a L, r) is... 

The operator being a single particle tensor operator, may be expanded as a linear combination with the unit tensor operators as a basis. It follows from the antiunitary nature of CL(= 0()) that... 

In equation (8.60) 7 represent phenomenological parameters which connect the ground state (f"irj and the final state fnir J through the matrix of the square of the unit tensor operators U(X. These quantities are sensitive to the accuracy of the oscillator strength and the nature of transitions used in their computation. 

Judd (13) has shown that the oscillator strength of an induced electric dipole transition may be related to the energy of the transition (v, in cm. ") and the square of the matrix elements of the unit tensor operators connecting the initial and final states via three phenomenological parameters T (A = 2, 4, and 6) according to Equation 2. 

In these formulae the symmetry adaptation coefficients, 3/- and 6y-symbols occur along with the set of crystal field parameters Dq, Ds, Dt, Da and Dt. In addition, there are the reduced matrix elements of the n-electron unit tensor operators, evaluable with the help of the coefficients of fractional parentage (dn lvSL dnvSL) as follows... 

Although the use of any one of these two methods completely solves the problem of calculating the matrix elements of Hi for configurations, several short-cuts can be achieved corresponding to various special cases. These also are due to Racah (1943, 1949). He defined unit tensor-operators such that... 

Here is a dimensionless unit tensor operator acting in the 4f-electrons. It transforms under rotations like an angular momentum state with total angular momentum A and z-component fi and under spin rotations like an angular momentum X with z-component v. 

The whole theoretical procedure applied to derive the famous expression of the J-O Theory is becoming clearer when the inter-shell unit tensor operators are introduced. They are defined by the reduced matrix elements in the following way ... 

In the terms of unit tensor operators, Vcryst has the following form ... 

The tensorial form of each product of the unit tensor operators in the terms contributing to the transition amplitude (10.11) can be simplified by coupling the objects to a tensorial product following the relation. 

In equation (10.17), in addition to the angular term, radial integral and matrix element of the unit tensor operator there is the crystal field parameter This is the stmctural factor from the multipole expansion of the crystal field potential the radial part from equation (10.6) is included in the radial term, as shown in equation (10.19). The limitation of the... 

When the intermediate coupling scheme is used for identification of the energy levels of the lanthanide ion, the matrix element of unit tensor operator U > in the expression for the transition amplitude (10.17) (or the line strength) exists only if the triangular condition is satisfied, namely... 

In order to derive a relativistic version of the elecfiic-dipole / <—) f transitions the concept of Sandras and Beck [59] is applied to include new effects in an effective way. This means that every unit tensor operator analyzed in the non-relativistic approach has to be replaced prior to the partial closure by a double unit tensor operator that acts within the spin (k) - orbital (k) space. The transformation is as follows. 

The effective operators defined in (10.35) extend the standard Judd-Ofelt effective operators by the interactions via the crystal field potential within the spin part of the space. The reduction to the non-relativistic case is easily seen when all of the ranks of operators acting within the spin space are equal to zero. Indeed, setting ki = k2 = 3 = 0 results in = 1 k2 = t and k = k. In such a situation, the rank of effective unit tensor operator is even, the... 

The unit tensor operators of a standard presentation are formally replaced in equation (10.36) by double tensor operators with the zero rank for the spin part of the space. 

The results of the discussion presented in the previous section indicate however that the semi-empirical approach based on equation (10.36) would neglect the purely relativistic effects represented by the unit tensor operators In order to include these effects in... 

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