Papers by Mohamed Slim Fayache
The large basis no-core shell-model approach allows one to perform ab initio calculations of the ... more The large basis no-core shell-model approach allows one to perform ab initio calculations of the properties of nuclei up to mass A=16. This technique is based on a separation between the model space and the excluded space at the two-body cluster level. At this level the matrix elements of the effective two-body interaction (or Hamiltonian) can be calculated exactly using the matrix elements of the transformation operator connecting these two spaces. So far, results converged to very high accuracy have been obtained with this procedure for the A= 3 and 4 systems, allowing us also to determine the effective three-body and four-body clusters. For the A=3 system we have also performed calculations including a two-pion-exchange three-nucleon interaction. We have developed, as well, simplified multi-nucleon forces and used them to study the A=4 system. We are now applying our ab initio shell-model approach to p-shell nuclei and have obtained a good approximation to the low-lying propertie...
The magnetic dipole strength and energy-weighted strength distribution is calculated in 8 Be, as ... more The magnetic dipole strength and energy-weighted strength distribution is calculated in 8 Be, as well as the separate orbit and spin parts. All ∆N =2 excitations over and above (and including) the configuration 0s 4 0p 4 are included. The interaction has a central, two-body spin-orbit and a tensor part. The energy-independent and energyweighted orbital strength distribution is remarkably insensitive to the presence or absence of the spin-orbit or tensor interaction-not so the spin strength. The energy-weighted strength distribution can be divided into a low enegy and a high energy part. The high energy orbital part is somewhat less but close to the low energy part, in fair agreement with a prediction that they be equal by de Guerra and Zamick and by Nojarov. There is a wide plateau separating the low energy part from the high energy part.
Working in the framework of the ab-initio no-core shell model, we derive two-body effective inter... more Working in the framework of the ab-initio no-core shell model, we derive two-body effective interactions microscopically for specific harmonic-oscillator basis spaces from the realistic Argonne V8' nucleon-nucleon potential. However, our two-body effective interaction is only the dominant part of the A-body (A >= 3) effective interaction, which also contains three-, four- and, in general, up to A-body components. We model these multi-body components as a simple two-parameter, basis-space-dependent interaction. We then investigate the coupling needed to provide accurate descriptions of selected nuclear observables. We successfully model the multi-body interaction as a zero-range two-body operator times the number operator Nsps that measures the harmonic-oscillator quanta of the spectator nucleons.
Using the interaction $Q \cdot Q ~+~ xV_{S.O.}$ where $V_{S.O.}$ is a two-body spin-orbit interac... more 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 $\hbar \omega$ 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) $\mu(2^+_1)$.
The magnetic dipole strength and energy-weighted strength distribution is calculated in 8 Be, as ... more The magnetic dipole strength and energy-weighted strength distribution is calculated in 8 Be, as well as the separate orbit and spin parts.
We compare Elliott's SU model in a 0hbar ω space with an extended calculation in a (0+2) hbar ω s... more We compare Elliott's SU model in a 0hbar ω space with an extended calculation in a (0+2) hbar ω space. We use the hamiltonian p^2/2 m + 1/2 m ω^2 r^2 - \chi/2 sum_i,jQ(i) \cdot Q(j). In order to mix major shells we take for Q simply r^2Y2 i.e. we do not include the momentum terms of Elliott. In a 0hbar ω space the diagonal part of Q \cdot Q is a one body potential - 5 \chi/(8 π) r^4. In the 0d-1s shell it is necessary to have a certain single particle splitting between 0d and 1s in order to get Elliott'ts SU(3) results. The above r^4 potential gives us only 2/3 of the necessary splitting. The remaining 1/3 comes from the exchange interaction of the 0d valence nucleon with the 0s core. On a related topic, in the 0p shell (ΔN=0) calculations of ^10Be with Q \cdot Q there are many supermultiplet degeneracies but also some other degeneracies, e.g. of the 2^+1 and 2^+2 states, and of the [330] and [411] orbital configurations. When ΔN=2 excitations are allowed in lowest order perturbation theory (using the direct p-h matrix element of Q \cdot Q), \chi merely gets renormalized and hence the above degeneracies are not removed. However in a shell model diagonalization the higher order effects come into play and these other degeneracies are removed by a substantial amount while the supermultiplet degeneracies are maintained.
The B(M1) rate can be written as |√B(l) + √B(s)e^iφ|^2 where e^iφ = +1 or -1. Here B(l) is the va... more The B(M1) rate can be written as |√B(l) + √B(s)e^iφ|^2 where e^iφ = +1 or -1. Here B(l) is the valueof B(M1) when only the orbital part of the operator is taken into account; B(s) only the spin part. We perform calculations in the sd shell of B(M1) transitions from the ground states of N=Z nuclei (T=0,J=0) to J=1^+ states with isospin T=0 (isoscalar) and T=1 (isovector). We consider the summed strength SB(M1) and define the angle θ such that SB(M1)=SB(l)+SB(s)+2√SB(l)√SB(s)cosθ. For isoscalar transitions one can showθ=180^o i.e. maximum destructive interference. For isovector transitions in N=Z nuclei we find θ is close to 90^o (cosθ=0) which indicates a randomness of sign e^iφ. Indeed, in the SU(3) limit θ is equal to 90^o because SB(s) is zero. For a certain realistic interaction which contains central,spin-orbit and tensor parts the values of θ for the N=Z nuclei ^20Ne, ^24Mg, ^28Si, ^32S and ^36Ar are respectively 90^o, 90.4^o, 92.3^o, 93.3^o and 95.4^o. We are closer to the SU(3) limit for the lower part of the sd shell, consistent with other analyses. For the NneqZ nucleus ^22Ne the value of θ for T=1 -> T=1 is 86.0^o; for T=1 -> T=2 it is 102.5^o, while for T=1 -> all T it is 88.75^o. This means we are close to randomness of sign if we choose all transitions, but there is much less randomness if we consider separate final isospins.
We allow up to $4 \hbar\omega$ excitations in $^8Be$ relative to the basic configuration $(0s)^4(... more We allow up to $4 \hbar\omega$ excitations in $^8Be$ relative to the basic configuration $(0s)^4(0p)^4$ in order to see if there are low-lying intruder states in $^8Be$. We use three interactions $-\chi Q \cdot Q, -\chi Q \cdot Q$ + spin-orbit, and a realistic (x,y) interaction, where for $x=1,y=1$ one gets values for the spin-orbit and tensor interactions which correspond closely to those of a non-relativistic $G$ matrix such as Bonn A. For the three interactions considered, if we allow up to $2 \hbar \omega$ excitations we get the energy of the lowest $J=0^+$ intruder state to be at 32.1 MeV, 30.1 MeV and 33.8 MeV, respectively. If we allow up to $4 \hbar\omega$ excitations the corresponding values are 26.5 MeV, 26.5 MeV and 28.7 MeV. We thus support the statement made by E. Warburton in his 1986 $R$ matrix analysis of the $\beta^{\mp}$-delayed alpha spectra from the decay of $^8Li$ and $^8B$ that ``satisfactory fits are obtained without introducing intruder states below 26-MeV excitations'' (Phys. Rev. C 33, 303(1986)). In $^{10}Be$ however, with the first two ``$Q \cdot Q$'' interactions, we get (with up to $2 \hbar \omega$ excitations) low-lying intruder states at 9.7 MeV and 11.9 MeV respectively. Thus the presence of low-lying intruder states in $^{10}Be$ (the $0^+_2$ state at 6.179 MeV may or may not be an intruder) does not imply that there are low-lying intruder states in $^8Be$.
Shell model calculations are performed for magnetic dipole excitations in 8 Be and 10 Be in which... more Shell model calculations are performed for magnetic dipole excitations in 8 Be and 10 Be in which all valence configurations plus 2hω excitations are allowed (large space). We study both the orbital and spin excitations. The results are compared with the 'valence space only' calculations (small space). The cumulative energy weighted sums are calculated and compared for the J = 0 + T =0 to J = 1 + T =1 excitations in 8 Be and for J = 0 + T =1 to both J = 1 + T =1 and J=1 + T =2 excitations in 10 Be. We find for the J = 0 + T =1 to J = 1 + T =1 isovector spin transitions in 10 Be that the summed strength in the large space is less than in the small space. We find that the high energy energy-weighted isovector orbital strength is smaller than the low energy strength for transitions in which the isospin is changed, but for J = 0 + T =1 to J = 1 + T =1 in 10 Be the high energy strength is larger. We find that the low lying orbital strength in 10 Be is anomalously small, when an attempt is made to correlate it with the B(E2) strength to the lowest 2 + states. On the other hand a sum rule of Zheng and Zamick which concerns the total B(E2) strength is rea-sonably satisfied in both 8 Be and 10 Be. The Wigner supermultiplet scheme is a useful guide in analyzing shell model results. In 10 Be and with a Q · Q interaction the T = 1 and T = 2 scissors modes are degenerate, with the latter carrying 5 3 of the T = 1 strength.
The energies and transition rates to $J=1^{+} T=1$ and $J=2^{+} T=0,1$ states in ^{12}C with matr... more The energies and transition rates to $J=1^{+} T=1$ and $J=2^{+} T=0,1$ states in ^{12}C with matrix elements fitted to realistic $G$ matrix elements obtained in non-relativistic approaches are studied. Then the effects of varying the strengths of the two-body tensor and spin-orbit interactions are also considered. The calculations are done in both a small space (0p) and a large space (0p + $2\hbar\omega$). In the small space the B(M1) from ground to the $J=1^{+} T=1$ is enhanced and gets closer to experiment if the strength of the spin-orbit interaction is increased and/or if that of the tensor interaction is made weaker. In a large space the spin B(M1) gets reduced by almost a factor of two. A `self-weakening' mechanism for the tensor interaction which succeeded in explaining anomalies in other nuclei does not seem to work for this case.
Shell model calculations are performed for magnetic dipole excitations in $^8{Be}$ and $^{10}{Be}... more Shell model calculations are performed for magnetic dipole excitations in $^8{Be}$ and $^{10}{Be}$ in which all valence configurations plus $2\hbar\omega$ excitations are allowed (large space). We study both the orbital and spin excitations. The results are compared with the `valence space only' calculations (small space). The cumulative energy weighted sums are calculated and compared for the $J=0^+$ $T$=0 to $J=1^+$ $T$=1 excitations in $^8{Be}$ and for $J=0^+$ $T$=1 to both $J=1^+$ $T$=1 and $J$=$1^+$ $T$=2 excitations in $^{10}{Be}$. We find for the $J=0^+$ $T$=1 to $J=1^+$ $T$=1 isovector {\underline {spin}} transitions in $^{10}{Be}$ that the summed strength in the {\underline {large}} space is less than in the {\underline {small}} space. We find that the high energy energy-weighted isovector orbital strength is smaller than the low energy strength for transitions in which the isospin is changed, but for $J=0^+$ $T$=1 to $J=1^+$ $T$=1 in $^{10}{Be}$ the high energy strength is larger. We find that the low lying orbital strength in $^{10}{Be}$ is anomalously small, when an attempt is made to correlate it with the $B(E2)$ strength to the lowest $2^+$ states. On the other hand a sum rule of Zheng and Zamick which concerns the total $B(E2)$ strength is reasonably satisfied in both $^8{Be}$ and $^{10}{Be}$. The Wigner supermultiplet scheme is a useful guide in analyzing shell model results. In $^{10}Be$ and with a $Q \cdot Q$ interaction the T=1 and T=2 scissors modes are degenerate, with the latter carrying 5/3 of the T=1 strength.
The spin-orbit splittings in the spectra of nuclei with mass numbers 5, 15 and 17 are studied wit... more The spin-orbit splittings in the spectra of nuclei with mass numbers 5, 15 and 17 are studied within the framework of shell-model configuration mixing calculations including 2$\hbar \omega$ excitations. The contributions of the two-body spin-orbit and tensor components of the nucleon-nucleon interaction are studied in various model spaces. It is found that the effects of the two-body spin-orbit interaction are dominant and quite sensitive to the size of the model-space considered. The effects of the tensor interaction are weaker. The correlations effects which are included in the larger (0+2) $\hbar \omega$ shell-model space reduce the spin-orbit splitting in the case of A=5 by 20%, and enhance it for A=15 by about the same 20%. However, it is found that the correlations have a very small effect on the $d_{3/2} - d_{5/2}$ splitting in A=17.
Shell model calculations are carried out for isovector M1 transitions J=0^+ arrow J=1^+ in even-e... more Shell model calculations are carried out for isovector M1 transitions J=0^+ arrow J=1^+ in even-even nuclei, first using a non-relativistic G matrix. The single particle energies are calculated with the same interaction used for the valence nucleons. For several nuclei e.g. ^12C, ^20Ne and ^22Ne the M1 rates are much too weak. We can increase the M1's and get better agreement with experiment by increasing the strength the spin-orbit interaction and (to a lesser extent) by decreasing the strength of the tensor interaction. We note that in the Dirac phenomenology and in some three body force models the spin-orbit interaction does get enhanced. A subtle point is that in the 1s-0d shell the spin-orbit interaction that we require is substantially larger than one which fits the experimental ``d_3/2 - d_5/2'' splitting in ^17O of 5.2 MeV. The orbital and spin M1 strengths are separately studied.
We perform shell model calculations in odd-odd nuclei using a quadrupole-quadrupole interaction a... more We perform shell model calculations in odd-odd nuclei using a quadrupole-quadrupole interaction and single-particle splittings chosen so as to obtain the SU(3) results. Elliott had shown that such an interaction gives rotational bands for which the energies go as I(I+1). This is certainly true for eve-even and for odd-even or even-odd nuclei with $K\neq 1/2$. We have looked at odd-odd nuclei e.g $^{22}Na$ and found somewhat different behaviour. In $^{22}Na$ the $I=1^+_1 T=0$ and $I=0^+_1 T=1$ states are degenerate, and a rotational band built on the $I=0^+_1 T=1$ state behaves in a normal fashion. For the $I+1^+_1 T=0$ band however, we find that the energy is given by $E(I)-E(1^+_1)=AI(I+1)$. This differs from the `normal' behaviour which would be $E(I)-E(1^+_1)=AI(I+1)-2A$.
The main thrust of this work is the search for observables in nuclear structure calculations whic... more The main thrust of this work is the search for observables in nuclear structure calculations which give a clear indication as to whether any medium modifications of the nucleon-nucleon tensor interaction are required to obtain a microscopic understanding of nuclear structure data. We study this mainly in the context of nuclear structure (chapters 1-4), but we also include a chapter on nucleon -nucleus scattering and polarization transfer (chapter 8). In the context of nuclear structure (e.g. the splitting of the J^pi = 0^- T = 0 and T = 1 states in ^{16}O), it is found that the inclusion of certain higher -shell configurations in shell model calculations has the same effect as the renormalization of the effective nucleon -nucleon interaction-namely to make the tensor interaction appear to get weaker inside the nucleus. Thus, simple configuration mixing in a large enough shell model space can be thought of as a mechanism for the modification of the nucleon-nucleon intraction within the nuclear medium. On the other hand, the search for such an effect in the context of a proton-nucleus polarization transfer reaction ^{28}Si(vec p,vec p ^')^{28}Si^{6 -} indicates that, in first-order perturbation theory, the effect is negligible and that it may be necessary to include higher-order terms to get a noticeable nuclear medium effect on the polarization observables. In another venue, we study nuclear transitions involving the magnetic dipole operator M1 in the light nuclei ^4He, ^8Be and ^{10 }Be. We analyze the M1 transitions in the spin and orbital channels and find that an interesting low-energy symmetry holds in ^4He for several realizations of the nucleon-nucleon interaction. We also examine the energy distribution of the M1 strength, and for ^8Be and ^{10 }Be we examine a linear energy-weighted sum rule relating the isovector orbital M1 strength (the so -called scissors-modes) to the electric quadrupole (E2) strength. The symmetry properties of the scissors-mode excitations are then studied using the Wigner supermultiplet scheme.
We address some properties of the quadrupole-quadrupole ($Q \cdot Q$) interaction in nuclear stud... more We address some properties of the quadrupole-quadrupole ($Q \cdot Q$) interaction in nuclear studies. We first consider how to restore $SU(3)$ symmetry even though we use only coordinate and not momentum terms. Using the Hamiltonian $H=\sum_i (p^2/2m + m/2 \omega^2 r_i^2) -\chi \sum_{i < j}Q(i) \cdot Q(j) - \chi/2 \sum_i Q(i) \cdot Q(i)$ with $Q_{\mu}=r^2 Y_{2,\mu}$, we find that only 2/3 of the single-particle splitting ($\epsilon_{0d}-\epsilon_{1s}$) comes from the diagonal term of $Q \cdot Q$ -the remaining 1/3 comes from the interaction of the valence nucleus with the core. On another topic, a previously derived relation, using $Q \cdot Q$, between isovector orbital $B(M1)$ (scissors mode) and the ``difference'' ($B(E2, isoscalar)-B(E2, isovector)$) is discussed. It is shown that one needs the isovector $B(E2)$ in order that one get the correct limit as one goes to nuclei sufficiently far from stability so that one subshell (neutron or proton) is closed.
The effect of the "B" term in the interaction −χQ · Q(1 + B τ (1) · τ (2)), which was previously ... more The effect of the "B" term in the interaction −χQ · Q(1 + B τ (1) · τ (2)), which was previously considered in the 0p shell, is now studied in a larger space which includes ∆N = 2 excitations. We still get a collapse of low-lying states below the conventional J = 0 + ground state when B is made sufficiently negative.
For a $Q \cdot Q$ interaction the energy weighted sum rule for isovector orbital magnetic dipole ... more For a $Q \cdot Q$ interaction the energy weighted sum rule for isovector orbital magnetic dipole transitions is proportional to the difference $\sum B(E2, isoscalar) - \sum B(E2, isovector)$, not just to $\sum B(E2, physical)$. This fact is important in ensuring that one gets the correct limit as one goes to nuclei, some of which are far from stability, for which one shell (neutron or proton) is closed. In $0p$ shell calculations for the even-even Be isotopes it is shown that the Fermion SU(3) model and Boson SU(3) model give different results for the energy weighted scissors mode strengths.
The interaction -\chi Q\cdot Q(1+Bτ\cdot τ(2)) is used in a shell model calculation for ^10Be. Wh... more The interaction -\chi Q\cdot Q(1+Bτ\cdot τ(2)) is used in a shell model calculation for ^10Be. Whereas for B=0 the 2_1^+ state is two-fold degenerate, introducing a negative B causes an `isovector' 2^+ state to come down to zero energy at B = -0.67 and an S = 1, L = 1 triplet (J = 0^+, 1^+, 2^+) to come down to zero energy at B = -0.73. These are undesirable properties, but a large negative B is apparently needed to fit the energy of the isovector giant quadrupole resonance. The calculations are being extended to include 2hbar ω excitations beyond the 0p shell. The isovector quadrupole components of realistic interactions are also being analyzed.
... It was shown by B. Jancovici and I. Talmi [30] (see also DR Inglis [31]) that one needs a ten... more ... It was shown by B. Jancovici and I. Talmi [30] (see also DR Inglis [31]) that one needs a tensor interaction in the (0p) shell space in order to get a vanishing matrix element. ... A 506, 1 (1990). [2] MS Fayache, P. von Neumann-Cosel, A. Richter, YY Sharon and L. Zamick, preprint. ...
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Papers by Mohamed Slim Fayache