Pseudoscalar

Clearly, the pseudoscalar term vanishes at these points so the ci character at the roots is maintained, no matter whether there are or are not A-i terms. Also, the vanishing of Ai terms will not lead to new ci s.) On the other hand, by circling over a large radius path q oo,so that all ci s are enclosed, the dominant term in Eq. (84) is the last one and the acquired Berry phase is —4(2tc)/2 = —4ti. 

During the p decay process, there exists anapole moment along the spin axis of the parent nuclei [1]. The anapole moment presents a new kind of dipole moment which is invariant under time reversal and odd under parity. A pseudoscalar p( V x H. ct) exists between the anapole moment and the spin of the emitted electrons, where p is the interaction strength. This interaction breaks parity conservation. 

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. 

Abstract. The —> um° decay is studied using the method of phenomenological chiral Lagrangians. Obtained in the framework of this method the expression of weak hadronic currents between vector and pseudoscalar mesons has been checked and it is shown that this decay channel proceeds only due to the — p - mixing diagram. 

Here, we consider the (ft —> unr° decay by the method of phenomenological chiral Lagrangians(PCL s)(Weinberg,1967). Studies of this decay channel is of interest in this model for the following reasons First, this decay channel is a unique laboratory for verification of weak hadron currents between pseudoscalar and vector meson states which was obtained earlier (Nasriddinov, 1998) within the formalism of phenomenological chiral Lagrangians... 

According to the expression for weak hadronic currents between pseudoscalar and vector meson states (f), the Born amplitude of this decay is equal to zero, since the structure constants /i3j = fs3i = 0, (in other words, the current // responsible for the direct — um° decay (FIG.f) is zero). 

In the PCL the strong interaction Lagrangian of vector mesons with vector and pseudoscalar mesons has the form... 

The next diagram (FIG. 3) also does not contribute to the partial width for the 0 — W7r° decay. In this case the Lagrangian of the strong coupling of axial-vector mesons to vector and pseudoscalar mesons is derived in a similar way and has the form (Nasriddinov, 1994)... 

The NJL-model parameters are taken from two fits to the masses and decay constants of pseudoscalar mesons in the vacuum, performed by Rehberg et... 

By forming the maxima or minima respectively of these quantities, the extrema being taken over all r with zf 0, we get chirality numbers which are characteristic properties of the skeleton. It should be emphasized that the condition is zfV 0 and not zr 0 because the latter would only lead to trivial numbers which express the fact that a ligand partition is active for any achiral frame if it contains at least one chiral ligand. The nontrivial numbers we want should present information about the pseudoscalar properties of the particular molecular class in question. We find these chirality numbers from the following maxima and minima ... 

It is evident that methods analogous to the ones developed here could be applied to molecular properties which, instead of being pseudoscalar, belong to some other representation of the skeleton point group (vector, tensor, etc. properties). To treat such properties, one needs only to induce from a different representation of than the chiral one. 

Table 4 lists, for skeletons i—v, the -chirality functions obtained by the first procedure (%i) and the alternative version of the second procedure (xz) of Section IV, together with the representation of SA and of Young diagrams remain. The parameters A are scalar parameters, the x are pseudoscalar. Different parameters may be used for different representations. 

In diatomic molecules, T2 = 0, and thus the expectation value of C vanishes. This is the reason why this operator was not considered in Chapter 2. However, for linear triatomic molecules, t2 = / / 0, and the expectation value of C does not vanish. We note, however, that D J is a pseudoscalar operator. Since the Hamiltonian is a scalar, one must take either the absolute value of C [i.e., IC(0(4 2))I or its square IC(0(412))I2. We consider here its square, and add to either the local or the normal Hamiltonians (4.51) or (4.56) a term /412IC(0(412))I2. We thus consider, for the local-mode limit,... 

Helicity [26,42-45] is a pseudoscalar whose definition is based on the following integral relation ... 

Therefore (u1 is indeed a vector but ( w° also changes sign under inversion, and so it is a pseudoscalar. 

A quantity T that is invariant under all proper and improper rotations (that is, under all orthogonal transformations) so that T = T, is a scalar, or tensor of rank 0, written 7(0). If T is invariant under proper rotations but changes sign on inversion, then it is a pseudoscalar. 

Pseudoscalars with the property T T (where the positive sign applies to proper rotations and the negative sign applies to improper rotations) are also called axial tensors of rank 0, 7 (0)ax. A quantity T with three components 7 7 2 7) that transform like the coordinates x x2 x3 of a point P, that is like the components of the position vector r, so that... 

Here A2 symbolizes a pseudo-scalar of A2 symmetry, normalized to unity. The actual form of this pseudoscalar need not bother us. The only property we will have to use later on is that even powers of A2 are equal to +1. Now we can proceed by defining rotation generators f x, y,t 2 in the standard way, as indicated in Table 1 [10]. Note that primed symbols are used here to distinguish the pseudo-operators from their true counterparts in real coordinate space. Evidently the action of the true angular momentum operators t y, (z on the basis functions is ill defined since these functions contain small ligand terms. 

Note that we have chosen to use primed symbols to emphasize the shell-theoretical nature of these labels, as opposed to true spatial symmetry assignments. The corresponding tetragonal symmetries can easily be found from the standard D h j D4k subduction relations, keeping in mind that the results must be multiplied by the pseudoscalar Blg representation of D4h, One thus obtains ... 

Starting from this rotated set complex orbitals and (t)3 multiplet operators may be constructed in a way which is entirely analogous to the treatment of Sect. 2. Hence the multiplets in Table 2 can be used equally well for trigonal complexes, keeping in mind that the axis of quantization is now the z axis. This implies that the subduction rules for real components in Eq. 15 have to be replaced by the appropriate S03 j. O j D3 subduction rules. In order to obtain the real forms of the (t2)3 basis functions the resulting expressions have to be multiplied once again by the pseudoscalar quantity of A2 symmetry. The appropriate product rules have been given by Ballhausen [59], For the individual orbital functions one obtains ... 

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