arXiv subjects

L. Zamick

Publications and source records attributed to L. Zamick.

At least 37 records · Page 2Linked to original sources

arXiv2005

Seniority conservation and seniority violation in the g_{9/2} shell

The g_{9/2} shell of identical particles is the first one for which one can have seniority-mixing effects. We consider three interactions: a delta interaction that conserves seniority, a quadrupole-quadrupole (QQ) interaction that does not, and a third one consisting of two-body matrix elements taken from experiment (98Cd) that also leads to some seniority mixing. We deal with proton holes relative to a Z=50,N=50 core. One surprising result is that, for a four-particle system with total angular momentum I=4, there is one state with seniority v=4 that is an eigenstate of any two-body interaction--seniority conserving or not. The other two states are mixtures of v=2 and v=4 for the seniority-mixing interactions. The same thing holds true for I=6. Another point of interest is that the splittings E(I_{max})-E(I_{min}) are the same for three and five particles with a seniority conserving interaction (a well known result), but are equal and opposite for a QQ interaction. We also fit the spectra with a combination of the delta and QQ interactions. The Z=40,N=40 core plus g_{9/2} neutrons (Zr isotopes) is also considered, although it is recognized that the core is deformed.

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arXiv2005

Shell-model test of the rotational-model relation between static quadrupole moments Q(2^+_1), B(E2)'s, and orbital M1 transitions

In this work, we examine critically the relation between orbital magnetic dipole (scissors mode) strength and quadrupole deformation properties. Assuming a simple K=0 ground state band in an even-even nucleus, the quantities Q(2^+_1) (i.e., the static quadrupole moment) and B(E2)_{0_1 \to 2_1} both are described by a single parameter--the intrinsic quadrupole moment Q_0. In the shell model, we can operationally define Q_0(Static) and Q_0(BE2) and see if they are the same. Following a brief excursion to the sd shell, we perform calculations in the fp shell. The nuclei we consider ({44,46,48}Ti and {48,50}Cr) are far from being perfect rotors, but we find that the calculated ratio Q_0(Static)/Q_0(BE2) is in many cases surprisingly close to one. We also discuss the collectivity of orbital magnetic dipole transitions. We find that the large orbital B(M1) strength in {44}Ti relative to {46}Ti and {48}Ti cannot be explained by simple deformation arguments.

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arXiv2005

Isospin relations for four nucleons in a single j shell

We had previously used techniques involving isospin to count the number of states for three identical fermions in a single j shell with total angular momentum I=j. We generalize this to all I, but the main thrust of this work is to consider now a 4-fermion system. As before, one evaluates the eigenvalues of the Hamiltonian \sum_{i<j}[a + bt(i)t(j)] both from an isospin point of view and an angular momentum point of view. In the 4-particle case, we get a more limited result than in the 3-particle case, namely the number of T=0 states minus twice the number of T=2 states, all of a given angular momentum I.

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arXiv2005

Degeneracies when only T=1 two-body interactions are present

In the nuclear f_7/2 shell, the nucleon-nucleon interaction can be represented by the eight values E(J)=<(f^2_{7/2})^J |V| (f^2_{7/2})^J>, J=0,1,...,7, where for even J the isospin is 1, and for odd J it is 0. If we set the T=0 (odd J) two-body matrix elements to 0 (or to a constant), we find several degeneracies which we attempt to explain in this work. We also give more detailed expressions than previously for the energies of the states in question. New methods are used to explain degeneracies that are found in {45}Ti (I=25/2- and 27/2-), {46}V (I=12^+_1 and 13^+_1, as well as I=13^+_2 and 15+), and {47}V (I=29/2- and 31/2-).

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arXiv2005

Wave functions in the f_{7/2} shell, for educational purposes and ideas

We use the spectrum of {42}Sc as input to calculate wave functions and energy levels of all nuclei in the single-j-shell model with j equal to f_{7/2}. The value of these results for educational purposes is pointed out. Also new insights emerge. For example, it is noted that for the (Z=4,N=4) nucleus {48}Cr, a good quantum number comes out: (-1)^{(v_p+v_n)/2}, where v_p and v_n are the seniority quantum numbers of the protons and neutrons, respectively.

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arXiv2005

Expressions for the number of J=0 pairs in even-even Ti isotopes

We count the number of pairs in the single j-shell model of {44}Ti for various interactions. For a state of total angular momentum I, the wave function can be written in terms of the probability amplitude D(Jp Jn) that the protons couple to Jp and the neutrons to Jn. For I=0 there are three states with (I=0,T=0) and one with (I=0,T=2). The latter is the double analog of {44}Ca. In that case (T=2), the magnitude of D(JJ) is the same as that of a corresponding two-particle fractional parentage coefficient. In counting the number of pairs with an even angular momentum J, we find a new relationship obtained by diagonalizing a unitary nine-j symbol. We are also able to get results for the `no-interaction' case for T=0 states, for which it is found, e.g., that there are less (J=1,T=0) pairs than on the average. Relative to this `no-interaction case', we find for the most realistic interaction used that there is an enhancement of pairs with angular momentum J=0,2,1 and 7, and a depletion for the others. Also considered are interactions in which only the (J=0,T=1) pair state is at lower energy, only the (J=1,T=0) pair state is lowered and where both are equally lowered, as well as the QQ interaction. We are also able to obtain simplified formulae for the number of J=0 pairs for the I=0 states in {46}Ti and {48}Ti by noting that the unique state with isospin |Tz|+2 is orthogonal to all the states with isospin |Tz|.

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arXiv2005

Interrelationship of Isospin and Angular Momentum

It is noted that the simple interaction in isospin variables $a (1/4 - t(i)\cdot t(j))$, in a single $j$ shell calculation, can also be written with angular momentum variables. For the configuration $(j^2) J_A$ for even $J_A$ the isospin is one; for odd $J_A$ it is zero. Hence the above interaction can also be written as $a (1 - (-1)^{J_A})/2$. For the I=0 state of an even-even Ti isotope with $n$ neutrons, the hamiltonian matrix element of this interaction is $\bra [J'J']_0 |H| [JJ]_0\ket/a = (n+1) δ_{JJ'} - (n+1) (j^n Jj|\} j^{n+1} j) (j^n J'j|\} j^{n+1} j)$. The eigenvalues of this interaction can be found by using the isospin form of the interaction. They are $(n+1)a$ for $T = |N-Z|/2$ and zero for $T = |N-Z|/2 + 2$. One can apply this to some extent to obtain the number of pairs of nucleons with given total angular momentum $J_A$ in a given Ti isotope.

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arXiv2005

Alternate Derivation of Ginocchio-Haxton relation [(2j+3)/6]

We address the problem, previously considered by Ginocchio and Haxton (G-H), of the number of states for three identical particles in a single j-shell with angular momentum J=j. G-H solved this problem in the context of the quantum Hall effect. We address it in a more direct way. We also consider the case J=j+1 to show that our method is more general, and we show how to take care of added complications for a system of five identical particles.

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arXiv2005

Nuclear Incompressibility in Asymmetric Systems at Finite Temperature and Entropy

The nuclear incompressibility $κ$ is investigated in asymmetric systems in a mean field model. The calculations are done at zero and finite temperatures and include surface, Coulomb and symmetry energy terms for several equations of state. Also considered is the behavior of the incompressibility at constant entropy $kappa_Q$ which is shown to have a very different behavior than the isothermal $kappa$. Namely, $kappa_Q$ decreases with increasing entropy while the isothermal $kappa$ increases with increasing $T$. A duality is found between the adiabatic $kappa_Q$ and the T=0 isothermal $kappa$. Analytic and also simple approximate expressions for $kappa$ are given.

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arXiv2005

Alternate derivation of the Ginocchio--Haxton relation [(2j-3)/6]

We have found an alternate way of deriving the Ginocchio-Haxton relation [(2j-3)/6], where the square brackets mean the largest integer that is less than or equal to what is inside them. Our derivation involves the calculation of the number of states with total angular momentum J=j for 3 identical particles (e.g., neutrons) in a j-shell.

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arXiv2005

The number of J=0 pairs in {44,46,48}Ti

In the single j-shell, the configuration of an even-even Ti isotope consists of 2 protons and n neutrons. The I=0 wave function can be written in terms of D(J,Jv)'s, which are the probability amplitudes for the 2 protons to couple to J and the n neutrons to couple to Jv (where J stands for the angular momentum and v for the seniority quantum number). There are several states with isospin Tmin=|(N-Z)/2|, but only one with Tmax=Tmin+2. By demanding that the Tmax wave function be orthogonal to the Tmin ones, we obtain the following simple expressions for the number of J=0 np pairs in these Ti isotopes: For T=Tmin, number of pairs(J12=0)=2|D(00)|^2/n For T=Tmax, number of pairs(J12=0)=2n|D(00)|^2=2n(2j+1-n)/(2j+1)(n+1) For 44Ti we have also the results for even J12: number of nn pairs=number of pp pairs=number of np pairs=|D(J12,J12)|^2.

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arXiv2004

Nuclear Incompressibility at Finite Temperature and Entropy

Features of the nuclear isothermal incompressibility $κ$ and adiabatic incompressibility $κ_Q$ are investigated. The calculations are done at zero and finite temperatures and non zero entropy and for several equations of state. It is shown that $κ_Q$ decreases with increasing entropy while the isothermal $κ$ increases with increasing $T$. A duality is found between the adiabatic $κ_Q$ and the T=0 isothermal $κ$. Our isothermal results are compared with a recent lattice Monte Carlo calculation done at finite $T$. The necessity of including correlations is shown if $κ$ is to have a peak with increasing $T$ as seen in the Monte Carlo calculations. A peak in $κ$ is linked to attractive scattering correlations in two nucleons channel in the virial expansion in our approach which are Pauli blocked at low $T$.

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arXiv2003

Counting pairs in $^{44}$Ti with various interactions

We count the number of pairs in the single $j-$shell model of $^{44}$Ti for various interactions. For a state of total angular momentum $I$, the wave function can be written as $Ψ=\sum_{J_P J_N} D(J_P J_N) [(j^2)_{J_P}(j^2)_{J_N}]_I$, where $D(J_P J_N)$ is the probability amplitude that the protons couple to $J_P$ and the neutrons to $J_N$. For I=0 there are three states with ($I=0, T=0$) and one with ($I=0, T=2$). The latter is the double analog of $^{44}$Ca. In that case (T=2), the magnitude of $D(JJ)$ is the same as that of a corresponding two particle fractional parentage coefficient. In counting the number of pairs with an even angular momentum $J_A$ we find a new relationship obtained by diagonalizing a unitary nine-j symbol. We are also able to get results for the 'no-interaction' case for T=0 states, for which it is found that there are less ($J=1, T=0$) pairs than on the average. Relative to this 'no-interaction case' we find for the most realistic interaction used that there is an enhancement of pairs with angular momentum $J_A=0,2,1$ and 7, and a depletion for the others. Also considered are interactions in which only the ($J=0, T=1$) pair state is at lower energy, only the ($J=1, T=0$) pair state is lowered and where both are equally lowered.

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arXiv2003

Description of single and double analog states in the f7/2 shell: The Ti isotopes

The excitation energies of single analog states in even-odd Ti isotopes and double analog states in even-even Ti isotopes are microscopically described in a single j-shell formalism. A projection procedure for generalized BCS states has been used. As an alternative description a particle-core formalism is presented. The latter picture provides a two-parameter expression for excitation energies, which describes fairly well the data in four odd and three even isotopes of Ti.

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arXiv2003

Charge exchange operators sum rules and proton-neutron T=0 and T=1 pairing interactions

We study sum rules for charge exchange operators of the Gamow-Teller and Fermi type and explore what can be learned from them about proton-neutron pairing in T=0 and T=1 channels. We consider both energy weighted (EWSR) and non-energy weighted (NEWSR) sum rules. We use a schematic Hamiltonian to stress the distinctive roles of T=1 and T=0 pair interactions and correlations in the sum rules. Numerical results for 44Ti are presented for both schematic and realistic interactions.

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arXiv2003

Non-Scissors-Mode Behaviour of Isovector Magnetic Dipole Orbital Transitions Involving Isospin Transfer

We study the response of isovector orbital magnetic dipole (IOMD) transitions to the quadrupole-quadrupole ($Q \cdot Q$) interaction, to the isospin-conserving pairing interaction (ICP) and to combinations of both. We find qualitatively different behaviours for transitions in which the final isospin differs from the initial isospin versus cases where the two isospins are the same. For $N=Z$ even-even nuclei with $J^π=0^+, T=0$ ground states such as $^8Be$ and $^{20}Ne$, the summed $T=0 \to T=1$ IOMD from the ground state to all the $J=1, T=1$ states in the $0 \hbar ω$ space does not vanish when the $Q \cdot Q$ interaction is turned off. The pairing interaction (ICP) alone leads to a finite transition rate. For nuclei with $J=0^+, T=1$ ground states such as $^{10}Be$ and $^{22}Ne$, the summed $T=1 \to T=1$ IOMD $does$ vanish when the $Q \cdot Q$ interaction is turned off, as is expected in a good scissors-mode behaviour. However this is not the case for the corresponding sum of the $T=1 \to T=2$ IOMD transitions. In $^{22}Ne$ (but not in $^{10}Be$) the sum of the $T=1 \to T=2$ IOMD transitions is remarkably insensitive to the strengths of both the $Q \cdot Q$ and the ICP interactions. In $^{22}Ne$ an energy weighted-sum is similarly insensitive. All our calculations were carried out in the $0 \hbar ω$ space.

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arXiv2002

Allowed Gamow-Teller Excitations from the Ground State of 14N

Motivated by the proposed experiment $^{14}N(d,{^2He})^{14}C$, we study the final states which can be reached via the allowed Gamow-Teller mechanism. Much emphasis has been given in the past to the fact that the transition matrix element from the $J^π=1^+ T=0$ ground state of $^{14}N$ to the $J^π=0^+ T=1$ ground state of $^{14}C$ is very close to zero, despite the fact that all the quantum numbers are right for an allowed transition. We discuss this problem, but, in particular, focus on the excitations to final states with angular momenta $1^+$ and $2^+$. We note that the summed strength to the $J^π=2^+ T=1$ states, calculated with a wide variety of interactions, is significantly larger than that to the $J^π=1^+ T=1$ final states.

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arXiv2000

Exploratory Calculations of the Effects of Higher Shell Admixtures on Static Electric Quadrupole and Magnetic Dipole Moments of Excited States

Using the interaction $Q \cdot Q ~+~ xV_{S.O.}$ where $V_{S.O.}$ is a two-body spin-orbit interaction, we study the effects of varying x on the static electric quadrupole and magnetic dipole moments of the $2^+_1$ and $2^+_2$ states of $^{10}Be$. This is done both in the valence space $s^4p^6$ and in a larger space in which 2 $\hbar ω$ excitations are allowed. In the former case, for x=0, we have the Wigner Supermultiplet limit in which the $2^+_1$ and $2^+_2$ states are degenerate and correspond to K=0 and K=2 rotational states, with equal and opposite static quadrupole moments. Turning on the spin-orbit interaction with sufficient strength in the valence space gives an energy splitting to the two $2^+$ states in accord with experiment. When higher shell admixtures are allowed, we get quite a different behaviour as a function of x than in the valence space. Of particular interest is a value of x for which $Q(2^+_1)$ and $Q(2^+_2)$ both nearly vanish, and so does (somewhat coincidentally) $μ(2^+_1)$.

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