arXiv subjects
Publications and source records attributed to L. Zamick.
With a certain approximation for the Coulomb matrix elements in a single j shell of protons and neutrons it is found that wave functions of states of odd angular momentum J in an even-even nucleus are not strongly affected by their presence,
Calculations in the sd and pf shells reported some time ago by Satuła\etal\ [Phys.~Lett.~B~407, 103 (1997)] are redone for an extended analysis of the results. As in the original work, we do calculations for one mass number in each shell and consider in each case the sequence of lowest energies for isospins 0, 2, and 4, briefly the symmetry spectrum. Following further the original work we study how this spectrum changes when parts of the two-nucleon interaction are turned off. The variation of its width is explored in detail. A differential combination $ε_\text{W}$ of the three energies was taken in the original work as a measure of the so-called Wigner term in semi-empirical mass formulas, and it was found to decrease drastically when the two-nucleon interaction in the channel of zero isospin is turned off. Our analysis shows that the width of the symmetry spectrum experiences an equally drastic decrease, which can be explained qualitatively in terms of schematic approximations. We therefore suggest that the decrease of $ε_\text{W}$ be seen mainly as a side effect of a narrowing of the symmetry spectrum rather than an independent manifestation of the two-nucleon interaction in the channel of zero isospin.
In the single j shell(f_{7/2}){}^{48} Cr is the first even-even nucleus for which there are T=0 (Isoscalar) J=1^{+} states and T=1J=0^{+}These states are here studied. This nucleus ,in the same model space, is mid-shell for both neutrons and protons and this leads to many selection rules.
We define the "quadrupole ratio" r_{Q}=\dfrac{Q_{0}(S)}{Q_{0}(B)} where Q_{0}(S) is the intrinsic quadrupole moment obtained from the static quadrupole moment of the 2_{1}^{+} state of an even-even nucleus and Q_{0}(B) the intrinsic quadrupole moment obtained from B(E2)_{0\rightarrow2} . In both cases we assume a simple rotational formula connecting the rotating frame to the laboratory frame. The quantity r_{Q} would be one if the rotational model were perfect and the energy ratio E(4)/E(2) would be 10/3. In the simple vibrational model, r_{Q} would be zero and E(4)/E(2) would be two. There are some regions where the rotational limit is almost met and fewer where the vibrational limit is also almost met. For most cases, however, it is between these two limits, i.e. 0<|r_{Q}|<1 . There are a few cases where r_{Q} is bigger than one, especially for light nuclei. In most cases the quadrupole ratio is positive but there are two regions with negative ratios. The first case is that of light nuclei and the second has certain nuclei close to ^{208} Pb.
We consider the expectation value of the quantity [3+ σ(1).σ(2)]/4 . This has a value +1 for 2 nucleons with spin S=1and zero for S=0. We show that for the jj coupling 2 particle configuration [j(1) j(2)]^{J} the expectation value has the structure A+B J(J+1) where A and B are constants. We then show that for a 2proton-2neutron configuration with total angular momentum I the expectation value per pair is independent of the details of the wave function and has a similar structure A' +B' I(I+1) with B'=B/6.
In this work we look at the low lying nuclear structure of several N=Z nuclei residing between the doubly magic nucei ^{40} Ca and ^{100} Sn. Using large shell model codes we calculate and discuus the systematics of enegies. We show energy levels, B(E2)'s, static quadrupule moments and g factors. In all cases we compare the results of 2 different interactions which yield significanly different occupation numbers. We compare with the simplest versions of the rotational and vibrational models. By examinnig B(E2)'s and static quadrupole moments we make associations with collective models find that in the model space here considered ,{}^{88} Ru is oblate . The quadruple moment of ^{92} Pd is very small consistent with the vibrational model.
We point out a simplicity that arises when we use an interaction in which only an energy with odd J is non-zero. The emphasis is on J=J_{\text{max}} and in particular J=9^{+} in the g_{9/2} shell. It is noted that high overlaps can be deceptive. In many cases a single set of unitary 9-j coefficients gives either an exact or a surprisingly good approximation to the wave function of a non-degenerate state. The many degeneracies that occur in these calculations are discussed and explained. As a counterpoint, we compare the results with an interaction in which both the J=0 and J=J_{\text{max}} two-body matrix elements are equal (and attractive). Comparisons with a more realistic interaction are also made.
We point out a simplicity that arises when we use an interaction in which only an energy with odd J is non-zero. The emphasis is on J= J_{max} and in particular J=9+ in the g_{9/2} shell. It is noted that high overlaps can be deceptive. In many cases a single set of U9-j coefficients gives either an exact or a very good approximation to the wave function of a yrast state.
We show how degeneracies, accidental or otherwise, can obscure some interesting physics. We further show how one can get around this problem.
It is noted that in single j-shell calculations certain odd-spin states in even--even nuclei lie in a narrow energy band; likewise certain states in odd--odd nuclei.
In calculations of isoscalar magnetic moments of odd-odd N=Z nuclei it was found that for medium to heavy mass nuclei large scale shell model calculations yielded results which were very close to much simpler single j shell ones. To understand this we compare isoscalar and isovector configuration mixing in first order perturbation theory using a spin dependant delata interaction.The isoscalar corrections are much smaller
In this work we examine two recent effective shell model interactions, JUN45 and JJ4B, that have been proposed for use in the $f_{5/2},p_{3/2}, p_{1/2}, g_{9/2}$ model space for both protons and neutrons. We calculate a number of quantities that did not enter into the fits undertaken to fix the parameters of both interactions. In particular we consider static quadrupole moments (Q's) of excited states of the even-even $^{70-76}$Ge isotopes, as well as the B(E2) values in these nuclei. (We have previously studied $^{70}$Zn isotopes using JJ4B.) Some striking disagreements between the JUN45 prediction and the experimental results had already been noted for the quadrupole moments of the $2_1^+$ states of these nuclei. We investigate whether these discrepancies also occur for the JJ4B interaction. Subsequently, we also apply both interactions to calculate the Q's of some more highly excited states and compare the two sets of predictions regarding the nature of the nuclear states under consideration. In order to gain insight into these more complex large-scale shell-model calculations, we examine the corresponding and much simpler single-j shell model calculations in the $g_{9/2}$ neutron shell.
We study the role of configuration mixing in the heavier even-even isotopes of Argon. We begin by limiting the configurations of the even-even Ar isotopes to $(d_{3/2}^2)_π$ $(f_{7/2}^n)_ν$. There, due to the particular location in this shell model space of $^{40}$Ar and $^{44}$Ar, we find that the spectra, B(E2)'s and magnetic moments of these two nuclei are identical. Any deviation from this equality is direct evidence of configuration mixing. In a larger shell model space there are significant differences between these two nuclei, with $^{44}$Ar being more collective. We also consider other even-even isotopes of Argon and study how their nuclear structure effects evolve with N. We compare in the full 0$\hbar ω$ space $(sd)_π$ $(fp)_ν$ the results of calculations with the WBT interaction and with the newer SDPF, denoted SDPF-U, interaction.
We consider T=0 states in even-even and odd-odd N=Z nuclei. The g factors that emerge are isoscalar. We find that the single j shell model gives simple expressions for these g factors which for even-even nuclei are suprisingly close to the collective values for K=0 bands. The g factors of many 2+ in even-even nuclei and 1+ and 3+ states in odd-odd nuclei have g factors close to 0.5.
Lawson has shown that one can obtain sensible wave functions even in the weak deformation limit of the Nilsson model as long as one projects out states of good total angular momentum. We apply this method to obtain wave functions and magnetic $g$ factors of excited states of select even-even Ar isotopes.
In the rotational model for a K=0 band in an even-even nucleus, there is a single parameter--Q_0, the intrinsic quadrupole moment. All B(E2)'s in the band and all static quadrupole moments are expressed in terms of this one parameter. In shell-model calculations, this does not have to be the case. In this work, we consider ground-state bands in {44}Ti, {46}Ti, {48}Ti, {48}Cr, and {50}Cr. We have two models. First, we use a Surface Delta Interaction with degenerate single-particle energies (SDI-deg). We compare this with results of a shell-model calculation using the standard interaction FPD6 and include the single-particle energy splitting. Neither model yields a perfect rotational I(I+1) spectrum, although the SDI-deg model comes somewhat closer. Overall, the simple rotational formula for B(E2)'s and static quadrupole moments hangs together very nicely.
In a previous work we considered the number of pairs of particles with angular momentum J12 for systems with angular momentum zero.Here we extend the work to states of any total angular momentum.We find simplified relations satisfied for a system of four nucleoons with isospin zero and two.
A full fp calculation is performed for states which were degenerate in a single-j-shell calculation in which isospin-zero two-body matrix elements were set to zero energy. Most of the splitting in a complete shell calculation (but not all) comes from the T=0 part of the interaction.