In this work, we examine critically the relation between orbital magnetic dipole (scissors mode) ... more 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 →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 f p 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 quadrupole collectivity of orbital magnetic dipole transitions. We find that the large orbital B(M 1) strength in 44 Ti relative to 46 Ti and 48 Ti cannot be explained by simple deformation arguments.
The $6^{+}_{1}$ and $6^{+}_{2}$ in $^{48}$Ti form a nearly degenerate doublet. In a single j shel... more The $6^{+}_{1}$ and $6^{+}_{2}$ in $^{48}$Ti form a nearly degenerate doublet. In a single j shell calculation with the matrix elements from experiment the $6^{+}_{1}$ changes sign under the interchange of protons and neutron holes (odd signature) while the $6_{2}^{+}$ does not (even signature). As a consequence the calculated B(E2) $6_{1}^{+}\to 4_{1}^{+}$ is much stronger than the $6_{2}^{+}\to 4_{1}^{+}$ and the Gamow-Teller matrix element to the $6_{2}^{+}$ state vanishes. When using some popular interaction e.g. FPD6 in single j shell the ordering of the even signature and odd signature states gets reversed, so that the Gamow-Teller matrix element to the $6^{+}_{1}$ state vanishes and the $6_{2}^{+}\to 4_{1}^{+}$ E2 transition is the strong one. When configuration mixing is introduced, the E2 transition $6_{2}^{+}\to 4_{1}^{+}$ persists in being large. However the Gamow-Teller strengths reverse, with the large matrix element to the $6_{1}^{+}$ state in agreement with experiment...
Lawson has shown that one can obtain sensible wave functions even in the weak deformation limit o... more 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 [Formula: see text] factors of excited states of select even–even Ar isotopes with emphasis on the comparison of [Formula: see text]Ar and [Formula: see text]Ar. These [Formula: see text] factors are compared with the values that are obtained by matrix diagonalization in the same space using the WBT residual interaction.
The g-factor of the very short-lived (τ = 0.12(1) ps) 2 + 1 state at 3.290 MeV in 36 S, g(2 + 1) ... more The g-factor of the very short-lived (τ = 0.12(1) ps) 2 + 1 state at 3.290 MeV in 36 S, g(2 + 1) = +1.3(5), has been determined for the first time by utilizing the Coulomb excitation of 36 S beams in inverse kinematics combined with the transient field technique. In addition, for the 3 − 1 state (with τ = 0.9(1) ps) at 4.192 MeV, a value of g(3 − 1) = +0.8(5) was obtained. In the same experiment, the g-factor of the 2 + 1 state in 40 Ar has been remeasured using the α-transfer reaction on the same carbon target. Lifetimes of several excited states in both 36 S and 40 Ar were also redetermined using the Doppler-shift-attenuation method. The new experimental g-factors and deduced B(E2) values were compared with results of shell-model calculations with the WBT effective interaction and the sdfp shell model space. Comparisons with previous results for 32,34,38,40 S and 36,38,40 Ar shed light on how the properties of the Ar and S isotopes change as the N = 20 shell closure is approached, crossed, and left behind.
Abstract Effective forces are derived from realistic two-body forces, and renormalizations coming... more Abstract Effective forces are derived from realistic two-body forces, and renormalizations coming from excitation of closed shells are shown to be important in achieving agreement with empirical binding energies.
Shell-model calculations of B(E2)'s and quadrupole moments in the even-even nuclei in the g 9/2 s... more Shell-model calculations of B(E2)'s and quadrupole moments in the even-even nuclei in the g 9/2 shell region below 100 Sn are hampered by the fact that the inclusion of the g 7/2 configuration will lead to model spaces that are too large to handle. Understanding the impact of specific orbit wave functions in large-scale shell-model (LSSM) calculations helps to shed light on the validity of the results that are obtained. We therefore examine lighter nuclei in the beginning of the fp region where one can normally include all the orbitals, f 7/2 , p 3/2 , p 1/2 , and f 5/2 , for both protons and neutrons. We perform such calculations but then take a step back and exclude the f 5/2 orbital. By comparing the results of the two calculations we can hope to get insight into the importance of the missing spin-orbital partner in other regions.
Motivated by the problem of the relative importance of J = 1+ T = 0 pairing versus the better est... more Motivated by the problem of the relative importance of J = 1+ T = 0 pairing versus the better established J = 0+ T = 1 pairing in nuclei, we here address the problem of the number of np pairs of a given angular momentum in selected nuclei, 44Ti, 46Ti, and 48Ti. The number of pairs is obviously relevant to np pickup reactions such as (p,3He). One can address also the np transfer reaction (3He,p) and indeed there is a proposal by the Berkeley group to do the reaction 44Ti(3He,p)46Sc. In this work we will consider the single j shell model for the Ti isotopes. In the near future more elaborate calculations are planned.
Sum rules are obtained for orbital M1, spin M1 and GT operators for the ICP interaction. There is... more Sum rules are obtained for orbital M1, spin M1 and GT operators for the ICP interaction. There is a striking difference between the behavior of these quantities depending upon whether we use the ICP or Q.Q interactions. Consider nuclei with J=0^+ T=1 ground states e.g. ^10Be, ^22Ne and consider the case of degenerate single particle energies ( ε(p_3/2)=ε(p_1/2) and ε(d_5/2)=ε(2s_1/2)=ε(d_3/2)).
In this work we examine empirical deficiencies of the core-renormalized realistic effective inter... more In this work we examine empirical deficiencies of the core-renormalized realistic effective interactions and seek simple corrective potentials. We note the inability of the current realistic interactions to account for the energies of isobaric analog states; likewise they are unable to reproduce the changes in the single-particle energies, as one goes from one closed shell to another. We note that
1. Electric spin excitations 219 1.4. Conclusion to chapter 1 231 1.1. Introduction 219 2. Non-sp... more 1. Electric spin excitations 219 1.4. Conclusion to chapter 1 231 1.1. Introduction 219 2. Non-spin-flip collective magnetic excitations 232 1.2. Electric spin dipole resonances 219 2.1. Ml transitions 232 1.2.1. The 2C case 219 2.2. Higher multipoles 236 1.2.2. The 160 case 221 2.3. A rotational approach 239 1.2.3. Coexistence of S = 0 and S = I states 223 2.4. Configuration mixing calculations 241 1.2.4. The 40Ca case 225 2.5. The effective interaction dependence of the B(M1) 1.2.5. 208Pb 228 strengths 242 1.3. Electric spin quadrupole resonances 228 2.6. Concluding remarks to chapter 2 244 1.3.1. Energy of S = 1 quadrupole states 228 Note added in proof 245 1.3.2. Coexistence of giant S = 1) and S = 1 quad-References 246 rupole states 230
Energy-weighted sum rules are obtained for the spin part, both isoscalar and isovector, of the ma... more Energy-weighted sum rules are obtained for the spin part, both isoscalar and isovector, of the magnetic dipole operator. The same interaction is used to evaluate the relevant double commutator as is used to induce ground-state correlations. First a spin-dependent delta interaction is used; then a tensor interaction is considered.
We address the general question of the predominance of prolate shapes among deformed nuclei. For ... more We address the general question of the predominance of prolate shapes among deformed nuclei. For that purpose, we show that the kinetic energy of a deformed intrinsic state is a minimum when the deformation is such that the mean square components of the momentum are isotropically distributed. In this way we generalize Mottelson's self-consistency equation to apply to a general class of deformed shell-model wave functions. The generalized equation is then used to show that states of maximum deformation compatible with the Pauli principle have lowest kinetic energy and, as a consequence, prolate shapes are preferred over oblate shapes.
In this work, we examine critically the relation between orbital magnetic dipole (scissors mode) ... more 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 →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 f p 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 quadrupole collectivity of orbital magnetic dipole transitions. We find that the large orbital B(M 1) strength in 44 Ti relative to 46 Ti and 48 Ti cannot be explained by simple deformation arguments.
The $6^{+}_{1}$ and $6^{+}_{2}$ in $^{48}$Ti form a nearly degenerate doublet. In a single j shel... more The $6^{+}_{1}$ and $6^{+}_{2}$ in $^{48}$Ti form a nearly degenerate doublet. In a single j shell calculation with the matrix elements from experiment the $6^{+}_{1}$ changes sign under the interchange of protons and neutron holes (odd signature) while the $6_{2}^{+}$ does not (even signature). As a consequence the calculated B(E2) $6_{1}^{+}\to 4_{1}^{+}$ is much stronger than the $6_{2}^{+}\to 4_{1}^{+}$ and the Gamow-Teller matrix element to the $6_{2}^{+}$ state vanishes. When using some popular interaction e.g. FPD6 in single j shell the ordering of the even signature and odd signature states gets reversed, so that the Gamow-Teller matrix element to the $6^{+}_{1}$ state vanishes and the $6_{2}^{+}\to 4_{1}^{+}$ E2 transition is the strong one. When configuration mixing is introduced, the E2 transition $6_{2}^{+}\to 4_{1}^{+}$ persists in being large. However the Gamow-Teller strengths reverse, with the large matrix element to the $6_{1}^{+}$ state in agreement with experiment...
Lawson has shown that one can obtain sensible wave functions even in the weak deformation limit o... more 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 [Formula: see text] factors of excited states of select even–even Ar isotopes with emphasis on the comparison of [Formula: see text]Ar and [Formula: see text]Ar. These [Formula: see text] factors are compared with the values that are obtained by matrix diagonalization in the same space using the WBT residual interaction.
The g-factor of the very short-lived (τ = 0.12(1) ps) 2 + 1 state at 3.290 MeV in 36 S, g(2 + 1) ... more The g-factor of the very short-lived (τ = 0.12(1) ps) 2 + 1 state at 3.290 MeV in 36 S, g(2 + 1) = +1.3(5), has been determined for the first time by utilizing the Coulomb excitation of 36 S beams in inverse kinematics combined with the transient field technique. In addition, for the 3 − 1 state (with τ = 0.9(1) ps) at 4.192 MeV, a value of g(3 − 1) = +0.8(5) was obtained. In the same experiment, the g-factor of the 2 + 1 state in 40 Ar has been remeasured using the α-transfer reaction on the same carbon target. Lifetimes of several excited states in both 36 S and 40 Ar were also redetermined using the Doppler-shift-attenuation method. The new experimental g-factors and deduced B(E2) values were compared with results of shell-model calculations with the WBT effective interaction and the sdfp shell model space. Comparisons with previous results for 32,34,38,40 S and 36,38,40 Ar shed light on how the properties of the Ar and S isotopes change as the N = 20 shell closure is approached, crossed, and left behind.
Abstract Effective forces are derived from realistic two-body forces, and renormalizations coming... more Abstract Effective forces are derived from realistic two-body forces, and renormalizations coming from excitation of closed shells are shown to be important in achieving agreement with empirical binding energies.
Shell-model calculations of B(E2)'s and quadrupole moments in the even-even nuclei in the g 9/2 s... more Shell-model calculations of B(E2)'s and quadrupole moments in the even-even nuclei in the g 9/2 shell region below 100 Sn are hampered by the fact that the inclusion of the g 7/2 configuration will lead to model spaces that are too large to handle. Understanding the impact of specific orbit wave functions in large-scale shell-model (LSSM) calculations helps to shed light on the validity of the results that are obtained. We therefore examine lighter nuclei in the beginning of the fp region where one can normally include all the orbitals, f 7/2 , p 3/2 , p 1/2 , and f 5/2 , for both protons and neutrons. We perform such calculations but then take a step back and exclude the f 5/2 orbital. By comparing the results of the two calculations we can hope to get insight into the importance of the missing spin-orbital partner in other regions.
Motivated by the problem of the relative importance of J = 1+ T = 0 pairing versus the better est... more Motivated by the problem of the relative importance of J = 1+ T = 0 pairing versus the better established J = 0+ T = 1 pairing in nuclei, we here address the problem of the number of np pairs of a given angular momentum in selected nuclei, 44Ti, 46Ti, and 48Ti. The number of pairs is obviously relevant to np pickup reactions such as (p,3He). One can address also the np transfer reaction (3He,p) and indeed there is a proposal by the Berkeley group to do the reaction 44Ti(3He,p)46Sc. In this work we will consider the single j shell model for the Ti isotopes. In the near future more elaborate calculations are planned.
Sum rules are obtained for orbital M1, spin M1 and GT operators for the ICP interaction. There is... more Sum rules are obtained for orbital M1, spin M1 and GT operators for the ICP interaction. There is a striking difference between the behavior of these quantities depending upon whether we use the ICP or Q.Q interactions. Consider nuclei with J=0^+ T=1 ground states e.g. ^10Be, ^22Ne and consider the case of degenerate single particle energies ( ε(p_3/2)=ε(p_1/2) and ε(d_5/2)=ε(2s_1/2)=ε(d_3/2)).
In this work we examine empirical deficiencies of the core-renormalized realistic effective inter... more In this work we examine empirical deficiencies of the core-renormalized realistic effective interactions and seek simple corrective potentials. We note the inability of the current realistic interactions to account for the energies of isobaric analog states; likewise they are unable to reproduce the changes in the single-particle energies, as one goes from one closed shell to another. We note that
1. Electric spin excitations 219 1.4. Conclusion to chapter 1 231 1.1. Introduction 219 2. Non-sp... more 1. Electric spin excitations 219 1.4. Conclusion to chapter 1 231 1.1. Introduction 219 2. Non-spin-flip collective magnetic excitations 232 1.2. Electric spin dipole resonances 219 2.1. Ml transitions 232 1.2.1. The 2C case 219 2.2. Higher multipoles 236 1.2.2. The 160 case 221 2.3. A rotational approach 239 1.2.3. Coexistence of S = 0 and S = I states 223 2.4. Configuration mixing calculations 241 1.2.4. The 40Ca case 225 2.5. The effective interaction dependence of the B(M1) 1.2.5. 208Pb 228 strengths 242 1.3. Electric spin quadrupole resonances 228 2.6. Concluding remarks to chapter 2 244 1.3.1. Energy of S = 1 quadrupole states 228 Note added in proof 245 1.3.2. Coexistence of giant S = 1) and S = 1 quad-References 246 rupole states 230
Energy-weighted sum rules are obtained for the spin part, both isoscalar and isovector, of the ma... more Energy-weighted sum rules are obtained for the spin part, both isoscalar and isovector, of the magnetic dipole operator. The same interaction is used to evaluate the relevant double commutator as is used to induce ground-state correlations. First a spin-dependent delta interaction is used; then a tensor interaction is considered.
We address the general question of the predominance of prolate shapes among deformed nuclei. For ... more We address the general question of the predominance of prolate shapes among deformed nuclei. For that purpose, we show that the kinetic energy of a deformed intrinsic state is a minimum when the deformation is such that the mean square components of the momentum are isotropically distributed. In this way we generalize Mottelson's self-consistency equation to apply to a general class of deformed shell-model wave functions. The generalized equation is then used to show that states of maximum deformation compatible with the Pauli principle have lowest kinetic energy and, as a consequence, prolate shapes are preferred over oblate shapes.
Maimonides Responsa about prayer for converts was not addressed to
Obadiah the Norman Proselyte b... more Maimonides Responsa about prayer for converts was not addressed to Obadiah the Norman Proselyte but rather to a later Obadiah, probably a Muslim.. The Norman was born in 1070 but Maimonides was not born until 1135 or a bit later. WIkipedia, by connecting 2 passages lead to the wrong conclusion on this matter.
Maimonides Responsa about prayer for converts was not addressed to
Obadiah the Norman Proselyte b... more Maimonides Responsa about prayer for converts was not addressed to Obadiah the Norman Proselyte but rather to a later Obadiah, probably a Muslim.. The Norman was born in 1070 but Maimonides was not born until 1135 or a bit later. WIkipedia, by connecting 2 passages lead to the wrong conclusion on this matter. But in the top line "In his letter to Obadiah the Proselyte, Maimonides relied on Genesis" If one clicks on Obadiah the Proselyte , one gets the bottom paragraph where it says incorrectly that this was Obadiah, the Norman Proselyte..
This is an update with new material.
A surprisingly number of high ranking Hebrew scholars have s... more This is an update with new material. A surprisingly number of high ranking Hebrew scholars have said that Maimonides issued a Responsa to Obadiah the Norman Proselyte concerning whether a convert to Jadaism could pray the same way as a Jew. But this turns out not to be true. The Norman was born in 1070, converted in1102 but Maimonides was not born until 1135 to 1140. Maimonides Responsa was to a later Obadiah, probably a muslim
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Obadiah the Norman Proselyte but rather to a later Obadiah, probably a Muslim.. The Norman was born in 1070 but Maimonides was not born until 1135 or a bit later.
WIkipedia, by connecting 2 passages lead to the wrong conclusion on this matter.
Obadiah the Norman Proselyte but rather to a later Obadiah, probably a Muslim.. The Norman was born in 1070 but Maimonides was not born until 1135 or a bit later.
WIkipedia, by connecting 2 passages lead to the wrong conclusion on this matter.
But in the top line
"In his letter to Obadiah the Proselyte, Maimonides relied on Genesis"
If one clicks on Obadiah the Proselyte , one gets the bottom paragraph where it says incorrectly that this was Obadiah, the Norman Proselyte..
A surprisingly number of high ranking Hebrew scholars have said that Maimonides issued a Responsa to Obadiah the Norman Proselyte concerning whether a convert to Jadaism could pray the same way as a Jew.
But this turns out not to be true. The Norman was born in 1070, converted in1102 but Maimonides was not born until 1135 to 1140.
Maimonides Responsa was to a later Obadiah, probably a muslim