Inelastic Proton Scattering Research Articles

Relativistic Dirac analyses of proton scattering from $$^{92}{\text {Zr}}$$

Relativistic coupled channel analyses based on the Dirac equation are presented for polarized 800-MeV proton inelastic scatterings from an axially symmetric deformed nucleus $$^{92}{\text {Zr}}$$ by using an optical potential model and the first-order collective model. The sequential iteration method using a computer program is employed to solve the complicated Dirac coupled channel equations. The optical potential parameters of the Woods–Saxon shape and the deformation parameters of several low-lying excited states are determined phenomenologically and analyzed. The coupled channel effect between the excited states that belong to the ground-state rotational band is investigated.

Dirac Analyses of Proton Inelastic Scatterings from 206Pb

Polarized 650-MeV proton inelastic scatterings from 206Pb are analyzed phenomenologically in the relativistic Dirac formalism using an optical potential model. The first-order collective model is used to obtain the transition potentials by considering several excited states at the inelastic scatterings. By reducing the Dirac equation to the Schrodinger-like second-order differential equation, we obtain and analyze the effective central and spin-orbit optical potentials. The deformation parameters for several excited states are calculated and compared with those obtained from nonrelativistic calculations.

Dirac Phenomenological Analyses of 1.047-GeV Proton Inelastic Scatterings from 62Ni and 64Ni

Unpolarized 1.047 GeV proton inelastic scatterings from Ni isotopes, $^{62}$Ni and $^{64}$Ni are analyzed phenomenologically employing an optical potential model and the first order collective model in the relativistic Dirac coupled channel formalism. The Dirac equations are reduced to the Schr\"{o}dinger-like second-order differential equations and the effective central and spin-orbit optical potentials are analyzed by considering mass number dependence. The multistep excitation via $2^+$ state is found to be important for the $4^+$ state excitation in the ground state rotational band at the proton inelastic scatterings from the Ni isotopes. The calculated deformation parameters for the 2$^+$, the 4$^+$ states of the ground state rotational band and the first 3$^-$ state are found to agree pretty well with those obtained in the nonrelativistic calculations.

Dirac phenomenological analyses for the proton scatterings from nickel isotopes

Relativistic Dirac coupled channel analyses are performed phenomenologically using an optical potential model for the intermediate energy proton inelastic scatterings from nickel isotopes, 58Ni and 60Ni. The first-order rotational collective model is used for the transition optical potentials to describe the low-lying excited collective states of the ground state rotational band. The complicated Dirac coupled channel equations are solved phenomenologically by varying the optical potential and the deformation parameters to reproduce the experimental data, using a computer program that uses a sequential iteration method. The channel-coupling effects of the multistep transition process for the excited states of the ground state rotational band are found to be strong and lead the calculation results to better agreement with the experimental data when the channel coupling between the excited states is added in the calculation. The Dirac equations are reduced to the second-order differential equations to obtain the Schrödinger equivalent effective central and spin–orbit optical potentials, and the obtained effective potentials are analyzed.

Spectroscopy of Ti54 and the systematic behavior of low-energy octupole states in Ca and Ti isotopes

Excited states of the $N=32$ nucleus $^{54}$Ti have been studied, via both inverse-kinematics proton scattering and one-neutron knockout from $^{55}$Ti by a liquid hydrogen target, using the GRETINA $\gamma$-ray tracking array. Inelastic proton-scattering cross sections and deformation lengths have been determined. A low-lying octupole state has been tentatively identified in $^{54}$Ti for the first time. A comparison of $(p,p')$ results on low-energy octupole states in the neutron-rich Ca and Ti isotopes with the results of Random Phase Approximation calculations demonstrates that the observed systematic behavior of these states is unexpected.

Relativistic Dirac analyses of polarized proton scatterings to the 2+ gamma vibrational band in 24Mg and 26Mg

Relativistic Dirac analyses are performed phenomenologically for the high-lying excited states that belong to the 2+ gamma vibrational band at the 800 MeV polarized proton inelastic scatterings from the s-d shell nuclei, 24Mg and 26Mg. Optical potential model is used and scalar and time-like vector potentials are considered as direct potentials. First-order vibrational collective models are used to obtain the transition optical potentials to accommodate the high-lying excited vibrational collective states. The complicated Dirac coupled channel equations are solved phenomenologically to reproduce the differential cross section and analyzing power data by varying the optical potential and deformation parameters. It is found that the relativistic Dirac coupled channel calculation could describe the high-lying excited states of the 2+ gamma vibrational band at the 800 MeV polarized proton inelastic scatterings from s-d shell nuclei 24Mg and 26Mg reasonably well, showing mostly better agreement with the experimental data compared to the results obtained from the nonrelativistic calculations. Calculated deformation parameters for the excited states are analyzed and compared with those of nonrelativistic calculations.

Dirac Calculations for Proton Inelastic Scattering at Intermediate Energies

Relativistic proton inelastic scattering from different targets (16O, 24Mg, 28Si, 40Ca, 54Fe, 58Ni, 90Zr, 154Sm, 176Yb, and 208Pb) at intermediate energies is analyzed in the framework of phenomenological optical potentials based on the Dirac formalism. Parameters of the Dirac phenomenological potential with Woods Saxon (WS) shape are obtained. The first order vibrational collective model with one phonon is used to calculate the transition optical potentials to the first low-lying excited state (2+) of the investigated target nuclei. Also, the variation of deformation length (δ) with energy and mass number is studied. It is noticed that the deformation length increases slightly with energy at intermediate range.

Collective excitations ofRu96by means of(p,p′γ)experiments

Background: One-phonon mixed-symmetry quadrupole excitations are a well-known feature of near-spherical, vibrational nuclei. Their interpretation as a fundamental building block of vibrational structures is supported by the identification of multiphonon states resulting from a coupling of fully-symmetric and mixed-symmetric quadrupole phonons. In addition, the observation of strong M1 transitions between low-lying 3(-) and 4(+) states has been interpreted as an evidence for one-phonon mixed-symmetry excitations of octupole and hexadecapole character. Purpose: The aim of the present study is to identify collective one-and two-phonon excitations in the heaviest stable N = 52 isotone Ru-96 based on a measurement of absolute M1, E1, and E2 transition strengths. Methods: Inelastic proton-scattering experiments have been performed at the Wright Nuclear Structure Laboratory (WNSL), Yale University, and the Institute for Nuclear Physics (IKP), University of Cologne. From the acquired proton-gamma and gamma gamma coincidence data we deduced spins of excited states, gamma-decay branching ratios, and multipole mixing ratios, as well as lifetimes of excited states via the Doppler-shift attenuation method (DSAM). Results: Based on the new experimental data on absolute transition strengths, we identified the 2(+) and 3(+) members of the two-phonon mixed-symmetry quintuplet (2(1,ms)(+) circle times 2(1,s)(+)). Furthermore, we observed strong M1 transitions between low-lying 3(-) and 4(+) states suggesting one-phonon symmetric andmixed-symmetric octupole and hexadecapole components in their wave functions, respectively. The experimental results are compared to sdg-IBM-2 and shell-model calculations. Conclusions: Both the sdg-IBM-2 and the shell-model calculations are able to describe key features of mixed-symmetry excitations of Ru-96. Moreover, they support the one-phonon mixed-symmetry hexadecapole assignment of the experimental 4(2)(+) state.

Dirac coupled-channel analyses of the high-lying excited states at 22Ne(p,p′)22Ne

Dirac phenomenological coupled channel analyses are performed using an optical potential model for the high-lying excited vibrational states at 800 MeV unpolarized proton inelastic scatterings from $^{22}$Ne nucleus. Lorentz-covariant scalar and time-like vector potentials are used as direct optical potentials and the first-order vibrational collective model is used for the transition optical potentials to describe the high-lying excited vibrational collective states. The complicated Dirac coupled channel equations are solved phenomenologically using a sequential iteration method by varying the optical potential and the deformation parameters. Relativistic Dirac coupled channel calculations are able to describe the high-lying excited states of the vibrational bands in $^{22}$Ne clearly better than the nonrelativistic coupled channel calculations. The channel-coupling effects of the multistep process for the excited states of the vibrational bands are investigated. The deformation parameters obtained from the Dirac phenomenological calculations for the high-lying vibrational excited states in $^{22}$Ne are found to agree well with those obtained from the nonrelativistic calculations using the same Woods-Saxon potential shape.

Dirac coupled channel analyses of proton inelastic scattering from the 40Ar nucleus

0.8 GeV proton inelastic scatterings from $^{40}$Ar nucleus are analyzed using an optical potential model in the Dirac coupled channel formalism. A rotational collective model is used to obtain the transition optical potentials for the low lying excited states of the rotational band. The optical potential parameters of a Woods-Saxon shape and the deformation parameters of the excited states are searched phenomenologically to reproduce the experimental differential cross-section data by using the sequential iteration method. The effect of a multistep process is investigated by including the channel coupling between two excited states in addition to the couplings between the ground state and the excited states. The calculated deformation parameters of the excited states are compared with those obtained by using the nonrelativistic calculations. The results of the Dirac coupled channel calculations are found to show pretty good agreements with the experimental data of the ground state and the low-lying excited states. The multistep excitation process via the channel coupling with the second $2^+$ state is found to be important for the excitation of the $4^+$ state at the inelastic scatterings from the deformed nucleus, $^{40}$Ar.

Dirac phenomenological analyses of unpolarized proton inelastic scattering from 22Ne

Unpolarized 800 MeV proton inelastic scatterings from an s-d shell nucleus $^{22}$Ne are analyzed using phenomenological optical potentials in the Dirac coupled channel formalism. The first-order rotational collective model is used to obtain the transition optical potentials for the low lying excited collective states that belong to the ground state rotational band of the nucleus. The optical potential parameters of Woods-Saxon shape and the deformation parameters of the excited states are varied phenomenologically using the sequential iteration method to reproduce the experimental differential cross section data. The effective central and spin-orbit optical potentials are obtained by reducing the Dirac equations to the Schr\"odinger-like second-order differential equations and the surface-peaked phenomena are observed at the real effective central potentials when the scattering from $^{22}$Ne is considered. The obtained deformation parameters of the excited states are compared with those of the nonrelativistic calculations and another s-d shell nucleus $^{20}$Ne. The deformation parameters for the $2^+$ and the $4^+$ states of the ground state rotational band at the nucleus $^{22}$Ne are found to be smaller than those of $^{20}$Ne, indicating that the couplings of those states to the ground state are weaker at the nucleus $^{22}$Ne compared to those at the nucleus $^{20}$Ne. The multistep channel coupling effect is confirmed to be important for the $4^+$ state excitation of the ground state rotational band at the proton inelastic scattering from the s-d shell nucleus $^{22}$Ne.

Study of mixed-symmetry excitations in 96Ru via inelastic proton-scattering

Mixed-symmetry states of octupole (L = 3) and hexadecapole (L = 4) character have been recently proposed in the N = 52 isotones 92 Zr and 94Mo, based on strong M1 transitions to the lowest-lying 3− and 4+ states, respectively. In order to investigate similar excitations in the heaviest stable N = 52 isotone 96Ru, two inelastic proton-scattering experiments have been performed at the Wright Nuclear Structure Laboratory (WNSL), Yale University, USA and the Institute for Nuclear Physics, University of Cologne, Germany. From the combined data of both experiments, absolute E1, M1, and E2 transition strengths were extracted, allowing for the identification of candidates for MS octupole and hexadecapole states. The structure of the low-lying 4+ states is investigated by means of sdg-IBM-2 calculations.

Dirac Coupled Channel Analyses of Proton Inelastic Scatterings from s-d Shell Nuclei

Proton inelastic scatterings from several s-d shell nuclei are analyzed using optical potential model and collective model in Dirac coupled channel formalism. The optical potential parameters obtained phenomenologically for the scatterings from the s-d shell nuclei are compared with each other for systematic Dirac analysis. Dirac equations are reduced to the second-order differential equations in order to obtain the Schroedinger equivalent effective central and spin-orbit optical potentials, and the surface-peaked phenomena are observed at the real effective central potentials for the scatterings from 22Ne, 20Ne and 24Mg. By analyzing the obtained effective spin-orbit potentials, it is confirmed that the spin-orbit interaction is a surface-peaked interaction. The first-order rotational collective models are used to describe the low-lying excited states of the ground state rotational bands in the s-d shell deformed nuclei, and the obtained deformation parameters are analyzed by comparing with each other, and compared with those obtained by using the nonrelativistic calculations. The obtained deformation parameters of Dirac phenomenological calculations for the s-d shell nuclei are found to agree pretty well with those of the nonrelativistic calculations using the same Woods-Saxon potential shape, even though the theoretical bases are quite different.

Mixed-symmetry octupole and hexadecapole excitations inN=52isotones

In addition to the well-established quadrupole mixed-symmetry states, octupole and hexadecapole excitations with mixed-symmetry character have been recently proposed for the N = 52 isotones 92Zr and 94Mo. We performed two inelastic proton-scattering experiments to study this kind of excitations in the heaviest stable N = 52 isotone 96Ru. From the combined experimental data of both experiments absolute transition strengths were extracted.

Mixed-symmetry octupole and hexadecapole excitations in theN=52isotones

Background: Excitations with mixed proton-neutron symmetry have been previously observed in the $N=52$ isotones. Besides the well established quadrupole mixed-symmetry states (MSS), octupole and hexadecapole MSS have been recently proposed for the nuclei $^{92}$Zr and $^{94}$Mo. Purpose: The heaviest stable $N=52$ isotone $^{96}$Ru was investigated to study the evolution of octupole and hexadecapole MSS with increasing proton number. Methods: Two inelastic proton-scattering experiments on $^{96}$Ru were performed to extract branching ratios, multipole mixing ratios, and level lifetimes. From the combined data, absolute transition strengths were calculated. Results: Strong $M1$ transitions between the lowest-lying $3^-$ and $4^+$ states were observed, providing evidence for a one-phonon mixed-symmetry character of the $3^{(-)}_2$ and $4^+_2$ states. Conclusions: $sdg$-IBM-2 calculations were performed for $^{96}$Ru. The results are in excellent agreement with the experimental data, pointing out a one-phonon hexadecapole mixed-symmetry character of the $4^+_2$ state. The $\big< 3^-_1||M1||3^{(-)}_2\big>$ matrix element is found to scale with the $<2^+_{\mathrm{s}}||M1||2^+_{\mathrm{ms}}>$ matrix element.

Dirac coupled channel analyses of the 2− gamma vibrational band excitation in 20Ne

Dirac coupled channel analyses are performed using optical potential model for the high-lying excited states that belong to the 2$^-$ gamma vibrational band at the 800 MeV unpolarized proton inelastic scatterings from $^{20}$Ne. The first order vibrational collective models are used to obtain the transition optical potentials to describe the high-lying excited vibrational collective states and Lorentz-covariant scalar and time-like vector potentials are used as direct optical potentials. The complicated Dirac coupled channel equations are solved phenomenologically to reproduce the differential cross sections data by varying the optical potential and deformation parameters using minimum chi-square method. It is found that relativistic Dirac coupled channel calculation could describe the excited states of the 2$^-$ gamma vibrational band in $^{20}$Ne much better than the nonrelativistic coupled channel calculation, especially for the 2$^-$ and 3$^-$ states of the band. It is shown that the multistep excitation process via channel coupling with the $3^-$ state is essential to describe the $2^-$ state excitation and pure direct transition from the ground state is dominant for the 3$^-$ state excitation of the 2$^-$ gamma vibrational band in $^{20}$Ne.

Dirac analyses of proton inelastic scatterings from an s-d shell nucleus 20Ne

Proton inelastic scatterings from an s-d shell nucleus 20Ne are analyzed using an optical potential model in the Dirac coupled channel formalism. Lorentz covariant scalar and vector optical potentials are included in the calculation, and the first-order rotational collective model is used to obtain the transition potentials for the low-lying excited collective states of the ground state rotational band of the nucleus. Dirac coupled channel equations are solved phenomenologically to reproduce the differential cross section and analyzing power data by varying the optical potential and deformation parameters. The multistep process via coupled channels turns out to be important in describing the 4+ excited state of the ground state rotational band. The calculated deformation parameters of the excited states are compared with those of nonrelativistic calculations based on the Schrodinger equations. The Dirac equations are reduced to obtain Schrodinger-like second-order differential equations, and the obtained effective central and spin-orbit optical potentials are analyzed and compared with those of other s-d shell nuclei.

SYSTEMATIC DIRAC ANALYSES OF PROTON INELASTIC SCATTERINGS FROM AXIALLY SYMMETRIC DEFORMED NUCLEI

Relativistic Dirac coupled channel analyses using optical potential model are performed for the 800 MeV proton inelastic scatterings from 26 Mg and the results are compared with those from several other axially symmetric deformed nuclei for the systematic Dirac analyses. Employing scalar-vector model, scalar and time-like vector optical potentials in Lorentz covariant form are calculated phenomenologically by solving Dirac coupled channel equations using sequential iteration method. Dirac equations are reduced to second-order differential equations to obtain Schrödinger equivalent effective central and spin-orbit optical potentials and it is found that the heavier deformed nucleus has the larger effective central potential strength. Using the first-order rotational collective model to describe the low-lying excited states of ground state rotational band in the deformed nuclei, deformation parameters for the excited states are calculated and it is observed that the lighter deformed nucleus has the larger deformation parameter for the lowest lying excited 2+ state at the 800 MeV proton inelastic scattering, indicating the stronger coupling to the ground state compared to that of heavier nucleus.

Relativistic Analysis of Intermediate Energy Proton Inelastic Scatterings from Heavy Deformed Nuclei

A relativistic coupled channel analysis based on the Dirac equation is performed for inelastic scattering of an intermediate energy proton from a heavy, deformed nucleus by using an optical potential model. In order to describe the low-lying excited states of the deformed nucleus, we used a rotational collective model to get the transition optical potentials. The Dirac coupled-channel equations are solved numerically to calculate the di erential cross sections by using the sequential iteration method. Employing Dirac phenomenology, we vary the optical potential parameters and deformation parameters to reproduce the experimental observables. The theoretical results are observed to agree reasonably well with the inelastic scattering experimental data of 800 MeV p+176Yb, and the importance of the multistep process for inelastic scatterings at a deformed nucleus is con rmed. The e ective Schr odinger-equivalent central and spin-orbit potentials are calculated by reducing the Dirac equation to a Schr odinger-like second-order di erential equation and by comparison them with those of a Schr odinger calculation in which the spin-orbit potential is not included, showing the relatively large strength of a real central potential. Surface-peaked phenomena are not observed for e ective central potentials when the heavy, deformed nucleus of 176Yb is considered.

Quadrupole collectivity in silicon isotopes approaching neutron number [formula omitted

Quadrupole deformation parameters, | β 2 , ( p , p ′ ) | , have been deduced for 36,38,40Si from measured inelastic proton-scattering cross sections. Due to the strong Z = 14 subshell gap, low-lying quadrupole collectivity in these nuclei is attributed to the excitation of valence neutrons. Enhanced collectivity at N = 26 indicates a reduced N = 28 shell gap at large neutron excess in this chain of isotopes. Data are compared to large-scale shell-model calculations and prior Coulomb excitation measurements on 36,38Si.