Parameters spin—orbit

In the realistic case where the potential is spin-dependent, the spin-orbit method is in trouble (should the spin-orbit parameter be calculated with the up or the down spin potential ). The present formalism allows for the use of spin-dependent potential and wavefunctions. 

There is an interesting similarity in the character of the solution absorption spectra of the isoelectronic ions Np3+ and Pu1 even though the absorption bands in Pu1 + are all shifted toward higher energies due to increases in both the electrostatic (Fk) and spin-orbit ( ) parameters, Table VI. We have also examined the spectra of complex alkali-metal Pu(IV)... 

The electrostatic and spin-orbit parameters for Pu + which we have deduced are similar to those proposed by Conway some years ago (32). However, inclusion of the crystal-field interaction in the computation of the energy level structure, which was not done earlier, significantly modifies previous predictions. As an approximation, we have chosen to use the crystal-field parameters derived for CS2UCI6 (33), Table VII, which together with the free-ion parameters lead to the prediction of a distinct group of levels near 1100 cm-. Of course a weaker field would lead to crystal-field levels intermediate between 0 and 1000 cm-1. Similar model calculations have been indicated in Fig. 8 for Nplt+, Pu1 "1 and Amlt+ compared to the solution spectra of the ions. For Am t+ the reference is Am4" in 15 M NHhF solution (34). 

Reminder The one-electron spin-orbit coupling coefficient, is intrinsically positive. The many-electron spin-orbit parameter X is defined by... 

Here the sum is over all Z values for which there is a known R(Z) value. As expected, the values of the parameter q, obtained in this way, as a rule are very close to the corresponding quantities, following from the hydrogenic approximation q = 1 for the integrals of the electrostatic interaction, 2 for average energies, 4 for spin-orbit parameter and q = —k for mean values of rk. 

The curves in Fig. 4 demonstrate the dependence of the normalized critical current / Jc° s (0) on the spin-orbit parameter Aa for asymmetric (a)... 

Fig.4. Potential hypersurfaces of T, (3P0) and r4 (3P,) levels in the e (Q2, Q3) subspace calculated from the Fukuda coupling matrix with the vibronic coupling parameter Are t = 4x 10 9 J/m from Eq. (5) (denoted by b in [84]), the electron (exchange) repulsion parameter G and spin-orbit parameter are both set equal to 0.3 eV...

The operator [157] is a phenomenological spin-orbit operator. In addition to being useful for symmetry considerations, Eq. [157] can be utilized for setting up a connection between theoretically and experimentally determined fine-structure splittings via the so-called spin-orbit parameter Aso (see the later section on first-order spin-orbit splitting). In terms of its tensor components, the phenomenological spin-orbit Hamiltonian reads... 

For the sign of the spin-orbit parameter Aso, the following conventions apply ... 

To determine the first-order splitting pattern of an atomic state in terms of the phenomenological spin-orbit parameter Aso [Eq. 203], we utilize... 

Unfortunately, the Lande rule is not obeyed very well in heavy atoms extraction of an atomic spin-orbit parameter from purely experimental data may be tricky in these cases or even impossible. 

Typically, Hund s case (b)161 applies to molecules with A = 0, to which spin-orbit effects do not contribute in first order. For states with a nonzero A value and a rotational parameter Bv of the same order-of-magnitdue as the spin-orbit parameter Av, rotational coupling cannot be neglected. In this case, it... 

W = —/A-H /A-(vXE) = /A (pX r)jE/mcr for a spherically symmetric field E = Er/r. Since L = — (p X r) and fi = — Spa, equation 8 follows immediately.] For the transition metals of the iron group, the spin-orbit parameter X is of the order of 100 to 1000 cm 1. Goudsmit (231) has shown that X reverses sign on going from a less than half to a more than half filled shell. The physical origin of this sign reversal lies in the fact that an electron spin interacts more strongly with its own orbital momentum. From Hund s rule it follows that if a shell is less than half filled, the individual electrons have their spins parallel to the net spin if the shell is more than half filled, the individual electrons responsible for a multiplet have their spins antiparallel to the net spin. 

The rank k can take values 0, 1 and 2 by the triangle rule. Of these, the scalar term with k = 0 has no A dependence and hence does not affect the relative positions of the ro-vibrational energy levels. It just makes a small contribution to the electronic energy of the state r], A). The first-rank term produces a second-order contribution to the spin orbit interaction because it is directly proportional to the quantum number A from the 3-j symbol in the first line of (7.119). The contribution to the spin-orbit parameter A(R) which arises in this way is given (in cm-1) by... 

Lichten [3 5] studied the magnetic resonance spectrum of the para-H2, N = 2 level, and was able to determine the zero-field spin-spin and spin-orbit parameters we will describe how this was done below. Before we come to that we note, from table 8.6, that in TV = 2 it is not possible to separate Xo and X2. Measurements of the relative energies of the J spin components in TV = 2 give values of Xo + fo(iX2, and the spin-orbit constant A the spin rotation constant y is too small to be determined. In figure 8.18 we show a diagram of the lower rotational levels for both para- and ortho-H2 in its c3 nu state, which illustrates the difference between the two forms of H2. This diagram does not show any details of the nuclear hyperfine splitting, which we will come to in due course. 

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