Tensor parts

Anet F A L and O Leary D J 1992 The shielding tensor. Part II. Understanding its strange effects on relaxation Concepts Magn. Reson. 4 35-52... 

It is worth mentioning that expressions (15.75) and (15.76), apart from terms that are scalar with respect to quasispin, also include terms that contain a second-rank tensor part in quasispin space. Substituting (15.75) and (15.76) into the expressions for the Casimir operators of appropriate groups (5.34), (5.29) and (5.33) yields... 

This expression can be rewritten in a form where the same diagonal (scalar) part, Fn, and the off-diagonal (tensor) part, Vjj, occurs... 

Finally, let us note that the tensor part HLJ is itself gauge invariant. As we said, one has to choose a gauge (i.e., a coordinate system) in order to solve the equations. It is well-known that the vector modes are generally irrelevant for cosmology, so that most of the coordinate systems which are used only deal with scalar perturbations. Let us review some of the most used coordinate systems. 

Abstract. We have calculated the scalar and tensor dipole polarizabilities (/3) and hyperpolarizabilities (7) of excited ls2p Po, ls2p P2- states of helium. Our theory includes fine structure of triplet sublevels. Semiempirical and accurate electron-correlated wave functions have been used to determine the static values of j3 and 7. Numerical calculations are carried out using sums of oscillator strengths and, alternatively, with the Green function for the excited valence electron. Specifically, we present results for the integral over the continuum, for second- and fourth-order matrix elements. The corresponding estimations indicate that these corrections are of the order of 23% for the scalar part of polarizability and only of the order of 3% for the tensor part... 

Here 7q (nLJ) corresponds to the scalar part of hyperpolarizability, 72 (nLJ) corresponds to the rank 2 tensor part of hyperpolarizabihty, 74 (nLJ) determines the tensor part of the rank 4. As the held strength F increases, the sphtting of the level may reach values comparable to the distance between levels of the same parity (the components of the hne structure of an atomic multiplet) and, therefore, the level shift AEnLJM can be found by solving the secular equation... 

In the perturbation theory for degenerate states the resonant hyperpolarizability is determined by the tensor part of polarizability [9] and may be extracted out of the fourth-order terms self-consistently in the case of nondegenerate perturbation theory the resonant part appears for separate sublevels of an atomic multiplet. The numerical results are listed in Table 2. 

C3 [kHz] tensor part of the nuclear spin-spin interaction constant... 

Employing the resulting approximate relation Cy Cj + C - 0.586Cf Cf as given by Chao [15], and the integral theorem (E.6a), the second collisional pressure tensor part (E.28) might be expressed as ... 

---