Parity conservation

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. 

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. 

Example Problem 241 Am is a long-lived a emitter that is used extensively as an ionization source in smoke detectors. The parent state has a spin and parity of and cannot decay to the + ground state of 237Np because that would violate parity conservation. Rather, it decays primarily to a excited state (85.2%, E = 59.5 keV) and to a f higher lying excited state (12.8%, E = 102.9 keV). Estimate these branching ratios and compare them to the observed values. 

In Chapter 1, we introduced the concept of parity, the response of the wave function to an operation in which the signs of the spatial coordinates were reversed. As we indicated in our discussion of a decay, parity conservation forms an important selection rule for a decay. Emission of an a particle of orbital angular momentum / carries a parity change (— l/ so that 1+ —0+ or 2 0+ a decays are forbidden. In general, we find that parity is conserved in strong and electromagnetic interactions. 

Figure 8.6 Schematic diagram of the Wu et al. apparatus. (From H. Frauenfelder and E. M. Henley, Subatomic Physics, 2nd Edition. Copyright 1991 by Prentice-Hall, Inc. Reprinted by permission of Pearson Prentice-Hall.) A polarized nucleus emits electrons with momenta pt and P2 that are detected with intensities Ii and 72. The left figure shows the normal situation while the right figure shows what would be expected after applying the parity operator. Parity conservation implies the two situations cannot be distinguished experimentally (which was not the case).

The names parity-favoured and parity-unfavoured should not be confused with parity conservation in the present case they just describe the partition into the classes by comparing (— 1)A with the parity (— l/ of the photoelectron (see equ. (8.31)). 

In order to proceed further, and to evaluate the remaining matrix element in (8.233), we must now take note of the fact that A is a signed quantity and therefore use the parity-conserving combinations introduced earlier ... 

We follow the conventions described by Brown, Kaise, Kerr and Milton [115] in order to form parity-conserved functions, as discussed in detail in, section 6.9. Parity is related to the behaviour of a state or function under the space-fixed inversion operator... 

Our procedure now is to calculate the matrix elements of the effective Hamiltonian in the primitive basis set, and then to reformulate these elements in the parity-conserved set (8.360). 

We now make use of the results derived above, using the parity-conserved basis functions defined in equations (8.360). For states of positive parity we have the following... 

These results are the same as those obtained by Freund, Herbst, Mariella and Klemperer [112] except for the. /-dependent phase factors in our matrices. These arise because of our specific definitions of the parity-conserved basis function and are necessary if the energies of the A-doublet components are to alternate with J. If we know the values of the five molecular constants appearing in these matrices, we can calculate the energies of the levels, of both parity types, for each value of J. In practice, of course, it was the task of the experimental spectroscopists to solve the reverse problem of determining the molecular parameters from the observed transition frequencies. 

The magnetic hyperfine interaction terms were given in equation (8.351) and the electric quadrupole interaction in equation (8.352). We extend the basis functions by inclusion of the 7Li nuclear spin I, coupled to J to form F the value of / is 3/2. We deal with each term in turn, first deriving expressions for the matrix elements in the primitive basis set (8.353), and then extending these results to the parity-conserved basis. All matrix elements are diagonal in F, and any elements off-diagonal in S and / can of course be ignored. 

This is an important result because, as we shall see when we deal with the parity-conserved basis functions, matrix elements with Ay = 0, 1, 2 are significant. The complete expression for the matrix elements of the dipolar interaction is obtained by combining (8.376) with (8.379) to yield ... 

It is now a simple matter to use the above results for the primitive basis functions to generate matrix elements for the parity-conserved basis. For the positive-parity states the hyperfine matrix is as follows. 

Complete matrices for the parity-conserved 2 If fine-structure states (exclusive of nuclear spin terms) may now be constructed by combining the A-doubling matrices given above with the spin-orbit and rigid body rotation matrices given in our discussion of the LiO spectrum. The matrix representation is block diagonal for each value of J and each parity. The results for the positive and negative parity states are as follows. 

This is, in essence, the result needed to construct figure 8.47. There is more to be done, however, because it is necessary to use (8.430) to derive matrix elements for the parity-conserved functions, and then to take note of the rotational distortion which mixes the fine-structure states. This mixing can be represented by an effective 12 value, which is designated 12 eff in table 8.28, where the results of the Stark experiments are listed. 

Figure 8.50. Stark energies of the A -doublet levels for Q =2, J = 2. On the left-hand side, in zero field, the wave functions are the parity-conserved combinations given in equation (8.432). On the right-hand side, in strong field, the wave functions are the simple combinations shown, with parity not conserved.

Now, we recall, we must construct parity-conserved basis functions from the primitive functions, and these are... 

This particular form of the spin orbit interaction, due to Fontana [27] and Chiu [28], was discussed in chapter 8. It is also important to remember that the wave functions are -doublets, so that parity-conserved combinations should be employed ... 

Lee TD, Yang CN (1956) Question of parity conservation in weak interactions. Phys Rev 104 254... 

Wu CS, Ambler E, Hayward RW et al (1957) Experimental test of parity conservation in beta... 

The effects of parity conservation are not relevant for classical mechanics. Classical systems are in fact arrangements of mixed parity, so that no new information is obtained by taking their mirror images. 

Entities that move in the interface are achiral and massless. A virtual photon consists of a virtual particle/anti-particle pair. The vector bosons that mediate the weak interaction are massive and unlike photons, distinct from their anti-particles. The weak interaction therefore has reflection symmetry only across the vacuum interface and hence /3-decay violates parity conservation. 

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