Parity in atoms

The second factor in (2.66) describes quite generally the transition probability for all possible two-photon transitions such as Raman scattering or two-photon absorption and emission. Figure 2.30 illustrates schematically three different two-photon processes. The important point is that the same selection rules are valid for all these two-photon processes. Equation (2.66) reveals that both matrix elements D,- and Dkf must be nonzero to give a nonvanishing transition probability A,/. This means that two-photon transitions can only be observed between two states i) and I/) that are both connected to intermediate levels fe) by allowed single-photon optical transitions. Because the selection rule for single-photon transitions demands that the levels i) and A ) or A ) and /) have opposite parity, the two levels i) and I/) connected by a two-photon transition must have the same parity. In atomic two-photon spectroscopy s s or s d transitions are allowed, and in diatomic homonuclear molecules Eg Eg transitions are allowed. 

The Hamiltonian considered above, which connmites with E, involves the electromagnetic forces between the nuclei and electrons. However, there is another force between particles, the weak interaction force, that is not invariant to inversion. The weak charged current mteraction force is responsible for the beta decay of nuclei, and the related weak neutral current interaction force has an effect in atomic and molecular systems. If we include this force between the nuclei and electrons in the molecular Hamiltonian (as we should because of electroweak unification) then the Hamiltonian will not conuuiite with , and states of opposite parity will be mixed. However, the effect of the weak neutral current interaction force is mcredibly small (and it is a very short range force), although its effect has been detected in extremely precise experiments on atoms (see, for... 

Hence, = I + 1 if k > 0 and = I — 1 if k < 0. Consequently, in the Dirac-Pauli representation and have definite parity, (—1) and (—1) respectively. It is customary in atomic physics to assign the orbital angular momentum label I to the state fnkm.j- Then, we have states lsi/2, 2si/2) 2ri/2, 2p3/2, , if the large component orbital angular momentum quantum numbers are, respectively, 0,0,1, ,... while the corresponding small components are eigenfunctions of to the eigenvalues 1,1,0,2,. 

Abstract. Investigation of P,T-parity nonconservation (PNC) phenomena is of fundamental importance for physics. Experiments to search for PNC effects have been performed on TIE and YbF molecules and are in progress for PbO and PbF molecules. For interpretation of molecular PNC experiments it is necessary to calculate those needed molecular properties which cannot be measured. In particular, electronic densities in heavy-atom cores are required for interpretation of the measured data in terms of the P,T-odd properties of elementary particles or P,T-odd interactions between them. Reliable calculations of the core properties (PNC effect, hyperfine structure etc., which are described by the operators heavily concentrated in atomic cores or on nuclei) usually require accurate accounting for both relativistic and correlation effects in heavy-atom systems. In this paper, some basic aspects of the experimental search for PNC effects in heavy-atom molecules and the computational methods used in their electronic structure calculations are discussed. The latter include the generalized relativistic effective core potential (GRECP) approach and the methods of nonvariational and variational one-center restoration of correct shapes of four-component spinors in atomic cores after a two-component GRECP calculation of a molecule. Their efficiency is illustrated with calculations of parameters of the effective P,T-odd spin-rotational Hamiltonians in the molecules PbF, HgF, YbF, BaF, TIF, and PbO. 

After discovery of the combined charge and space parity violation, or CP-violation, in iT°-meson decay [7], the search for the electric dipole moments (EDMs) of elementary particles has become one of the most fundamental problems in physics [6, 8, 9, 10, 1]. A permanent EDM is induced by the weak interaction that breaks both the space symmetry inversion and time-reversal invariance [11]. Considerable experimental effort has been invested in probing for atomic EDMs induced by EDMs of the proton, neutron and electron, and by P,T-odd interactions between them. The best available restriction for the electron EDM, de, was obtained in the atomic T1 experiment [12], which established an upper limit of de < 1.6 X 10 e-cm, where e is the charge of the electron. The benchmark upper limit on a nuclear EDM is obtained in atomic experiment on i99Hg [13], ]dHgl < 2.1 X 10 e-cm, from which the best restriction on the proton EDM, dp < 5.4 x 10 " e-cm, was also recently obtained by Dmitriev Sen kov [14] (the previous upper limit on the proton EDM was obtained in the TIE experiment, see below). 

The P,T-parity nonconservation parameters and hyperfine constants have been calculated for the particular heavy-atom molecules which are of primary interest for modern experiments to search for PNC effects. It is found that a high level of accounting for electron correlations is necessary for reliable calculation of these properties with the required accuracy. The applied two-step (GRECP/NOCR) scheme of calculation of the properties described by the operators heavily concentrated in atomic cores and on nuclei has proved to be a very efficient way to take account of these correlations with moderate efforts. The results of the two-step calculations for hyperfine constants differ by less than 10% from the corresponding exper-... 

The prediction of a heavy boson has received preliminary empirical support [92,96] from an anomaly in Z decay widths that points toward the existence of Z bosons with a mass of 812 GeV 1 33j [92,96] within the SO(l) grand unified field model, and a Higgs mechanism of 145 GeV4gj3. This suggests that a new massive neutral boson has been detected. Analysis of the hadronic peak cross sections obtained at LEP [96] implies a small amount of missing invisible width in Z decays. The effective number of massless neutrinos is 2.985 0.008, which is below the prediction of 3 by the standard model of electroweak interactions. The weak charge Qw in atomic parity violation can be interpreted as a measurement of the S parameter. This indicates a new Qw = 72.06 0.44, which is found to be above the standard model pre-... 

The parity of atomic states is important in spectroscopy. A radial function is an even function [see (1.113)] the spherical harmonic Y(m is found to be an even or odd function of the Cartesian coordinates according to whether / is an even or odd number. For a many-electron atom, it follows that states arising from a configuration for which the sum of the / values of all the electrons is an even number are even functions when 2,/, is odd, the state has odd parity. 

For homonuclear diatomic molecules, the electronic wave functions have definite parity (g or w), and since del is of odd parity, we must have a change in parity of f/el (corresponding to the Laporte rule in atoms) ... 

As was pointed out in Chapter 4, division of the radiation into electric and magnetic is connected with the existence of two types of multipoles, characterized by the parities (—l)fc and (—l)fc+1, respectively. The first ones we have studied quite thoroughly in Chapters 24-26. Here let us consider in a similar way the M/c-transitions. Again, as we have seen in Chapter 4, the potential of the electromagnetic field in this case does not depend on gauge. Therefore only one relativistic expression (4.8) was established for the probability of M/c-radiation, described by the appropriate operator (4.9). The probability of non-relativistic M/c-transitions (in atomic units) is given by formula (4.15), whereas the corresponding non-relativistic operator has the form (4.16). 

Extremely accurate calculations of electronic transitions may contribute to explaining parity non-conserving effects in atoms [50, 252-254]. 

I. Shimamura, H. Wakimoto, A. Igarashi, Resonance states of unnatural parity in positronic atoms, Phys. Rev. A 80 (2009) 032708. 

Latal, H. Parity Violation in Atomic Physics, in Janoschek, R. Ed., Chirality From Weak Bosons to the a-Helix Springer-Verlag Berlin, 1991, pp. 1-17. 

Let us now consider the second mechanism, namely, the appearance of the electronic contribution gj due to the interaction with the paramagnetic electronic states. In particular, the singlet terms 1II and of one parity (either u u or g - g) interact because of the non-zero matrix elements of the electron-rotation operator [—l/(2/iro)](J+L- + J L+), where // is the reduced mass, ro is the internuclear distance (in atomic units) and the cyclic components of the vectors are defined in the same way as in [267] = Lx iLy, = Jx iJy connecting the x and y... 

LigPrep of Schrodinger can generate stereoisomers consistent with specified stereochemical information (e.g. parities in SD files) by varying the chirahties of the atoms for which chirahties are missing. Alternatively, the chirahties of aU chiral atoms may be varied. 

From this second point of view the many-body problem is simply a nuisance, and in fact hydrogen or hydrogenlike ions are generally considered to be the best places to search for new physics. The new physics that will be treated in this chapter is that of the weak interactions, which lead to parity nonconserving (PNC) transitions in atoms. While this effect has... 

Parity Nonconservation in Atoms Status of Theory and Experiment, E. N. Fortson and L. Wilets... 

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