arXiv subjects
Publications and source records attributed to Larry Zamick.
We consider the most negative and most positive value of the spin operator for a many nucleon system. In the usual Nilsson model and in the single nucleon model this value has a narrow range.In the Nilsson model with an excited intrinsic state we can get values of the spin operator greater than one.
The first even-even nucleus for which there are J=1+ T=0 states in a single j shell configuration is 48Cr. In this limited model space all T=0 to T=0 M1 matrix elements vanish.
We study quadrupole and monopole core polarization using harmonic oscillator wave funtions but with different length parameters for the valence particle as compared to the core. We use perturbation theory with a delta interaction. The results also hold for a density dependent delta interaction [1]. We study how the amount of core polarization varies with the distance of the valence particle from the core.
We find the largest negative value of the expectation of the spin operator for a many nucleon system.
We find the most negative and most positive values of the spin operator for a system of several nucleons. We us an ls basis to make the analysis.
Using the LS coupling we find the most positive and negative values of the spin operator sigma.These theoretical limits are much wider than the "empirical" single particle limits.A comparison of the Nilsson and Schmidt models is made for isoscalar moments.
We have compiled some sections of works by L. Zamick and collaborators which we hope will be of interest to the reader.
It is noted that the staggering parameters used to describe even-odd effects for isotope shifts can in some cases exhibit very rapidly varying behavior as a function of neutron number. On the other hand a three parameter formula (3P) with fixed coefficients can explain the same behaviour.
We study quadrupole and monopole core polarization using harmonic oscillator wave functions but with different length parameters for the valence particle as compared to the core. We use perturbation theory with a delta interaction. The results also hold for a density dependent delta interaction [1]. We study how the amount of core polarization varies with the distance of the valence particle from the core
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 $B(E2)_{0 \to 2}$. 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.
We present analytic proofs of the properties of four particle states in the g9/2 shell which have seniority v=4 and angular momentum I=4 or 6.We show in particular that the number of pairs with angular momentum I is equal to one for these states.
For each of the (9/2), (11/2) and (13/2) single j shells we have only one state with J=j V=3 for a five particle system. For four identical particles there can be more than one state of seniority four. We note some ``ratio'' relations for the coefficients of fractional parentage for the four and five identical particle systems.
We find a relationship between coefficients of fractional parentage (cfp) obtained on the one hand from the principal parent method and on the other hand from a seniority classification. We apply this to the Redmond recursion formula which relates $n \to n+1$ cfp's to $n-1 \to n$ cfp's where the principal parent classification is used. We transform this to the seniority scheme. Our formula differs from the Redmond formula inasmuch as we have a sum over the possible seniorities for the $n \to n+1$ cfp's, whereas Redmond has only one term.
Calculations of the spectra of various even-even nuclei in the fp shell ({44}Ti, {46}Ti, {48}Ti, {48}Cr and {50}Cr) are performed with two sets of two-body interaction matrix elements. The first set consists of the matrix elements of the FPD6 interaction. The second set has the same T=1 two-body matrix elements as the FPD6 interaction, but all the T=0 two-body matrix elements are set equal to zero (T0FPD6). Surprisingly, the T0FPD6 interaction gives a semi-reasonable spectrum (or else this method would make no sense). A consistent feature for even-even nuclei, e.g. {44,46,48}Ti and {48,50}Cr, is that the reintroduction of T=0 matrix elements makes the spectrum look more rotational than when the T=0 matrix elements are set equal to zero. A common characteristic of the results is that, for high spin states, the excitation energies are too high for the full FPD6 interaction and too low for T0FPD6, as compared with experiment. The odd-even nucleus {43}Ti and the odd-odd nucleus {46}V are also discussed. For {43}Sc the T=0 matrix elements are responsible for staggering of the high spin states. In general, but not always, the inclusion of T=0 two-body matrix elements enhances the B(E2) rates.
We note that the same mathematical results apply to problems involving quasispin and isospin, but the problems per se are different. In the quasispin case, one deals with a system of identical fermions (e.g. neutrons) and address the problem of how many seniority conserving interactions there are. In the isospin case, one deals with a system of both neutrons and protons and the problem in question is the number of neutron-proton pairs with a given total angular momentum. Other companion problems are also discussed.
In this report we examine topics relating to previous single j shell model results. A common thread in the two problems is that they both involve the isospin variable in an important way.
Calculations of the spectra of various even-even nuclei in the fp shell ($^{44}$Ti, $^{46}$Ti, $^{48}$Cr, and $^{50}$Cr) are performed with two sets of two-body interaction matrix elements. The first set consists of the matrix elements of the FPD6 interaction. The second set have the same T=1 two-body matrix elements as the FPD6 interaction, but all the T=0 two-body matrix elements are set equal to zero. Despite the drastic differences between the two interactions, the spectra they yield are very similar and indeed it is difficult to say which set gives a better fit to experiment. That the results for the yrast spectra are insensitive to the presence or absence of T=0 two-body matrix elements is surprising because the only bound two nucleon system has T=0, namely the deuteron. Also there is the general folklore that T=0 matrix elements are responsible for nuclear collectivity. Electric quadrupole transition rates are also examined. It is found that the reintroduction of T=0 matrix elements leads to an enhancement of B(E2)'s for lower spin transitions but in some cases for higher spin transitions one gets another surprising result that there is a small suppression.