Our first use of the HF code on parallel computers was the study of identical superdeformed (SD) rotational bands in the Hg region. The GAMMA system, a prototype of the DELTA system, consists of 64 nodes connected by an 8-dimensional hypercube architecture.
Chapter 1
Nuclear Hartree-Fock Method
- Introduction
- Effective interactions
- Constrained Hartree-Fock equations
- Hartree-Fock energy
- Hartree-Fock equations
- Pairing energy
- Constraint
- Numerical procedures
- Chapter 2
These are summarized in table 1.1. The Hartree-Fock equations for Skyrme force are obtained by making the total Hartree-Fock energy stationary with respect to individual variation of single-particle states c1>i, with the condition that q>i is normalized. where vlei is a Lagrange multiplier. where the effective mass m~ and potential field Uq and Wq are given by. where the Vc is given by. The boundary conditions of the system are chosen such that the wave functions vanish outside the box.
Implementation on Parallel Computers
- Introduction
- Decomposition method
- Optimizing the communication speed
- Performance and efficiency
- Conclusion
- Chapter 3
For this particular system we use 90 (Nn) neutron wave functions and 60 (Np) proton wave functions. The additional wave functions are needed to enable possible crossings and pairs. The number of operations required to solve the Poisson equation is independent of the number of wave functions and is so small that it can be ignored. Orthonormalization accounts for about 58% of the total operations in the 208Pb case and the number of operations grows quadratically with the number of wave functions.
However, the amount of communication for this is independent of the number of wave functions and can be ignored for more than a few wave functions. In the original sequential code, the derivatives of the wave functions are stored to save computation time. To see how the speed of the code scales with the number of nodes from 1 to 128, we performed calculations with 4°Ca using 20 wave functions.
Cranked Hartree-Fock Study of Superdeformed Rotational Bands
Introduction
In Section 3.2, we discuss the experimental background of identical bands and why they come as a surprise to nuclear theorists. In this chapter, we will try to study such a phenomenon using the cranked Hartree-Fock (CHF) method. In Section 3.4 , we apply the CHF method to study identical superdeformed bands in the Hg region, and in Section 3.5 we apply the method to bands in the Dy region.
Identical bands
Some caution, we assume here that each pair of compared transitions has the corresponding spins (I+ 1/2, I), but the spins of the SD bands were not measured. However, two features distinguish the bands in this region from those near A= 150: .. many identical bands appear in pairs separated by two mass units, .. a large number of bands may be associated with the SD band in 192Hg, which appears to it serves as a dual magic core in this region. Since the unexpected discovery of duplicated SD bands, it has become clear that such duplicated bands have already been found in rare earth and actinide regions under normal deformation, although this was not explicitly pointed out in the original data analysis.
Both studies conclude that the contribution to the moment of inertia of the single-particle orbitals depends greatly on both distortion and on the Nilsson quantum numbers of the orbitals, some orbitals making almost no contribution to the moment of inertia. Another attempt to explain this phenomenon [33] relies on the strong coupling limit of the particle rotor model. This last result assumes that the moment of inertia of the even nucleus is not affected by the extra particle, a hypothesis that is not easy to justify.
Cranked Hartree-Fock method
Due to the breaking of the time-varying symmetry, we have to solve the HF equations for both members of the signature partners. where h' is the one-particle Routhian h' = h -wJz · Effective mass m~ and spin-orbit W q are the same as in Eq. 1.15), while the spin scalar Uq, the spin vector Vq and the current Cq are given by. For a rotating nucleus, the time-reserved state .- of i is no longer a Ruthian eigenstate of h'. It is shown that the change of the moment of inertia of the SD bands with spin is certainly related to the gradual disappearance of the spin-coupling correlation.
Furthermore, the strength of the pair force is not well known at such large deformations: calculations based on the Woods–Saxon potential and the seniority pairing interaction require an ad hoc renormalization of the pair strength in order to reproduce the experimental increase in the dynamic moment of inertia :12. Finally, coupling is known not to significantly affect much of the mean field, especially near the shell closure. We expect that the merger will not change the fundamentals of the mean field shape of 192Hg, which is magical when superdeformed so that it will form.
A first possible explanation of our results could be that the different behaviors observed for the four A = 194 bands are related to some properties of the orbitals of single particles added to the 192Hg nucleus. Indeed, as expected, the relative values of the moments of inertia of the rigid body follow the same trends as the quadrupole moments. Although small, these negative contributions are not negligible, representing 1.5% of the total angular momentum.
This shows that the main contribution to the angular momentum of one particle comes from the variation of the mean field with angular momentum. The angular momentum of the 194Pb and 194Hg states obtained from that of 192Hg (from Table 1) and the contribution from the two additional orbits (from Table 2) are smaller than the angular momentum of the 192Hg core. In our study, twinning is the result of a precise balance between the changes in the mean field and the behavior of single particle states.
However, our calculation highlights the importance of the dependence of the mean field on rotation, an effect that was not included in previous calculations. However, the [301Jt orbitals do not meet the criteria for orbitals leading to twins found in the previous section, since they have large characteristic splittings and one of the orbitals has a positive alignment along the rotation axis. Such a display suggests that all three Tb rotation bands can be considered identical to the 152Dy band at Jz ~ 40n.
As discussed above, it seems acceptable to designate the experimental 151 Tb twin band as one of the three configurations considered here. Our calculations suggest that it is more favorable for the twin band to be one of the K = ±~ bands rather than the K = 1/2 band as suggested by pseudo SU(3). Due to the almost zero splitting of the signature of [303H orbitals, compared to the significant splitting of the signature of [301 H.
Conclusion
SD band built on the form isomer or SD minimum, whereas in 194Pb it is an excited band. Although the individual particle level schemes calculated with the Skyrme interaction SkM* and with the most frequently used Woods Saxon parametrization are very similar [20], the quadrupole deformation where 62 and. The inclusion of a seniority pairing interaction, although it reduces the quadrupole deformations of the SD bands, does not change the ordering of 194Pb single-particle orbitals, at least at zero spin.
If we assume the validity of our calculations, it is possible that the SD band built on the SD minimum of 194Pb has not yet been observed. This would challenge the assumption often made by researchers that the most intense band observed is the lowest.
Chapter 4
Atomic parity nonconservation and neutron radii in cesium isotopes
Introduction
The atomic PNC is ordered by the effective binding interaction between the electron and the nucleus (if we take only the part that remains after averaging the hyperfine components) of the shape. Below we will separate out the effects of finite kernel size (ie, effects associated with deviations of qn,p from unity); these terms will be introduced by correcting the nuclear structure to the weak charge. The nuclear structure correction Q]);-c(N, Z) describes part of the PNC effect caused by the finite size of the nucleus.
The uncertainties in these quantities, or equivalently, in the differences of the mean square radii of the neutrons o(~(r2)N,N'), then ultimately limit the accuracy with which the fundamental parameters, such as sin2 ew,. Furthermore, we estimate the uncertainty in these quantities, respectively in their differences, because they represent the ultimate limitations for the interpretation of the PNC measurements. Finally, in section IV we calculate the corrections of the nuclear structure to the weak charges Qwc(z = 55, N and their uncertainties and discuss the corresponding limiting uncertainties in the determination of the fundamental parameters of the standard model.
Nuclear Hartree-Fock calculations
We believe that such a separation is very useful because, as noted above, f (r) in Eq. 4.5), are essentially independent of the atomic structure.). Since some of the cesium isotopes discussed here are deformed, it is very important to consider the deformational degrees of freedom. The method of solving the HF+BCS equations by discretizing the wave functions on a rectangular grid enables arbitrary uniform multipole deformation.
Comparison with experiment
Experimental binding energies and isotopic shifts o(r;) are also listed for comparison. Binding energies are in MeV, all radial momenta in fm.) Experimental isotopic shifts are from Ref. We include this shape fluctuation effect using the quantities (/32) extracted from the measured transition matrix elements B(E2, o+ - t 2+) and the relation. We take the average B(E2) of the corresponding Xe and Ba isotopes with neutron numbers N = 78-84 and correct the 133Cs-139Cs radii accordingly, as shown in fig.
In a completely consistent calculation, a similar correction would also have to be made for the distorted cesium isotopes. With such a B(E2) the correction is numerically the same as in the semimagical 137Cs.) We believe that this shortcoming explains the slightly worse agreement in the distorted cesium isotopes. The following notations are used: experimental isotope shift 0, spherical HF isotope shifts +, HF including equilibrium distortion D, and corrected for zero-point vibrations x.
Neutron number N
Estimated uncertainties in PNC effects
It is now easy to assess the uncertainty in the factors Qn,p given the coefficients h , f4. Substituting the corresponding values, we find that the uncertainty is 6qn,p = 4.6 x 10-4, caused almost entirely by the uncertainty in the mean square radii (r~,p). More importantly, they are independent of the nuclear structure and cancel out the differences 6.qn,p·.
The assumed uncertainty in the root mean square radius shifts, and hence in the changes in the Qn,p factors results in a relative uncertainty 6Qw/Qw of 5x10-4. This uncertainty therefore represents, within the nuclear model we use, the "final" constraint on nuclear structure in Standard Model tests of PNC atomic experiments involving several isotopes. In atomic PNC experiments involving a single isotope, the uncertainty in the mean square neutron radius is larger, and 1 fm2 seems to be a reasonable choice.
Conclusion
Bibliography
