Majorana couplings

Crystal anapole moment is composed of the atomic magnetic moments which array in anapole structure [3]. It has the same intrinsic structure as Majorana neutrino [2], If we plant a p decay atom into this anapole lattice, the crystal anapole moment will couple to the nuclear anapole moment of the decaying nuclei. So the emitted electron will be given an additional pseudoscalar interaction by the presence of the crystal anapole moment. Then the emission probability will be increased. This is a similar process to that assumed by Zel dovich [1], The variation of the decay rate may be measured to tell whether the crystal anapole moment has an effect on the p decay or not. 

As the anapole interaction is the candidate which directly breaks parity conservation in electromagnetic interaction [1], it is very desirable to test whether the anapole moment could couple to the p decay or not. This experiment can be performed by solid state detectors as well asby a magnetic spectrometer. There are also other choices for the crystal samples [3] and p sources. Since the anapole moment has the same intrinsic structure as for Majorana neutrinos, its coupling is valid to both p decay and p+ decay. 

The off-diagonal elements of the Majorana operator in Eq. (4.29) illustrate, for the first time, the appearance of a nonlinear resonance (Section 3.4) within the algebraic approach. For two identical modes, an m = (1,-1) resonance is expected to be very important. Mx2 indeed couples such [nearly degenerate, cf. Figure 4.1 or Eq. (3.28)] states whose quantum numbers differ by m. Note furthermore that the 0 (2) quantum number is conserved, va + vb = const, as expected for a 1,-1 resonance because m = (1,1) is orthogonal to m = (1,-1), cf. Eq. (3.31). 

We have already discussed in Section 4.5 the local-to-normal transition for two coupled oscillators. The situation is quite analogous for two coupled rovibrators. The local-to-normal transition can be described by combining the operators of the local chain with those of the normal chain. It is convenient to introduce the Majorana operator... 

Figure 4.12 Representation of the couplings induced by the Majorana operator in the first multiplet, n = 1. N and give the order of magnitude of the couplings.

For the first multiplet, n = 1, and similar expressions for the other multiplets. The states of the second multiplet, n = 2, which are coupled by the operator Mn, are (04°0), (12°0), (02° 1) (20°0), (10°1), (00°2). From the structure of the matrices, one can see that the Majorana operator does two things simultaneously. It produces the local couplings that are needed to go from local to normal situations, and it introduces, when viewed from the normal-mode basis, Dar-ling-Dennison (1940) couplings of the type < v v 2, v3IVIv( -F 2, v , v3 2 >. 

Note that there is a duality that stems from the two different ways one can view the Hamiltonian (4.67) (Lehmann, 1983 Levine and Kinsey, 1986). As written, the Majorana operator serves to couple the local-mode states. But the Majorana operator is [cf. Eq. (4.66)] the Casimir operator of U(4) and is a leading contributor to the Hamiltonian, Eq. (4.56) describing the exact normal-... 

The total matrix, including both Majorana and Fermi couplings, is then... 

Normal behavior is induced by Majorana couplings. The situation is similar to that discussed in Section 4.17 for linear molecules, the Hamiltonian being... 

We consider next Majorana-type couplings. These are introduced, as in the previous case of triatomic molecules (cf. Section 4.17), by the operators... 

In the same way as discussed in the preceding sections, one can include Majorana operators, M13 and M23. Since these are in the wrong coupling scheme, one must use the recoupling techniques of Section 2. The matrix elements of Mn and M23 are given by... 

Once this calculation is completed, one can then examine each spectral region bounded by intervals of energy of the order of AE = 100 cm-1 and couple the states of a given species that fall into that region. Table 6.7 shows, for example, states up to three quanta of vibration of total species E]u that fall in the region 5950-6050 cm-1. These states are subsequently coupled by residual interblock interactions of the Majorana type [Eq. (6.16)]. A complete account of this type of calculations is given in Iachello and Oss (1993). 

These matrices have been discussed in detail in Section 4.17. As a result of these Majorana couplings there is a modification to the shape parameters of Eq. (7.165). Also when Fermi couplings are present, there is a modification to the shape parameters of Eq. (7.165). We do not report these modifications here but rather show in Table 7.2 how good the l/N method is in determining the vibrational frequencies (and thus the range parameters pt and p2). As one can see from the table, the error is of the order is 1%. Since N is of order 100, this error reflects the l/N accuracy mentioned. 

These matrix elements are equivalent to those of Eq. (3.124), apart from anharmonic contributions of the order of v/N. So we see that the extended Majorana operator has the required effect on the states involved in the resonance mechanism. At the same time > SB does not preserve the coupled 65 (2) symmetry in other words, + Vg is not conserved anymore. Consequently, the block-diagonal structure of the Hamiltonian operator is destroyed and the numerical diagonalization of... 

It is, of course, possible to extend this calculation to obtain, in closed analytical form, the first excited polyad, Vj = 2. The result is shown schematically in Fig. 34. In particular, we notice the direct coupling between pairs of stretching modes in light of the selection rule (for Ub = const) A(u -I-Uj) = 0 and Au, Au = 0, 1. This means that the (initially degenerate) stretches 100), 010) now mix and split under the effect of Mj2. Due to the symmetry under bond exchange, we obtain either symmetric or antisymmetric wavefunctions, as discussed for the one-dimensional case. The difference here is the presence of the bending mode, which is also involved in the coupling scheme induced by the Majorana operator. We can see in both Fig. 34 and Eq. (4.45) that... 

As discussed previously for bent molecules, the local model (4.54) is a poor approximation when intermode coupling occurs, so we now need to introduce the Majorana operator. The explicit analysis of this problem is perfectly analogous to the previous one, apart from the different conversion law between algebraic and vibrational quantum numbers. Moreover, in a linear molecule we expect to obtain vibrational wavefunctions... 

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