Conventional Distorted-wave Born Approximation Research Articles

The 50th anniversary of the coupled channels Born approximation (CCBA) and the coupled reaction channels (CRC) theories of nucleon transfer reactions (a unique interplay between theory, experiment and computer technology, conducted during the most tumultuous period in modern American society)

Nucleon transfer reactions have played a fundamental role in understanding the single-particle components, shell structure and collective properties of atomic nuclei. The conventional distorted wave Born approximation (DWBA) envisioned the nucleon transfer reaction as a one-step process, which proceeds directly from the ground state of the target nucleus to a state of the residual nucleus. The coupled channels Born approximation (CCBA) and coupled reaction channels (CRC) theories evolved because a number of nucleon transfer reaction cross sections could not be reconciled within the DWBA. These coupled channels models revealed that, in addition to the “one-step” process of the DWBA, “multi-step” nucleon transfer processes involving accessary pathways can participate in populating the final nuclear state. In the CCBA, the auxiliary pathways involved inelastic excitations of the target and/or residual nucleus, whereas, in the CRC, the pathways included sequential nucleon transfer passing through nuclear states of an intermediate partition. Coherent addition of contributions from one-step and multi-step nucleon transfer processes resulted in dramatic alterations in reaction cross sections, which were experimentally confirmed. The CCBA and CRC linked the structure of the nuclei participating in a reaction to modalities of nucleon transfer arising during the relative motion between the interacting ions. These complementary theories inexorably changed physicists’ interpretations of nucleon transfer reactions and, in doing so, heralded in the new field of direct heavy-ion reactions.

A multi-slice simulation algorithm for grazing-incidence small-angle X-ray scattering

Grazing-incidence small-angle X-ray scattering (GISAXS) is an important technique in the characterization of samples at the nanometre scale. A key aspect of GISAXS data analysis is the accurate simulation of samples to match the measurement. The distorted-wave Born approximation (DWBA) is a widely used model for the simulation of GISAXS patterns. For certain classes of sample such as nanostructures embedded in thin films, where the electric field intensity variation is significant relative to the size of the structures, a multi-slice DWBA theory is more accurate than the conventional DWBA method. However, simulating complex structures in the multi-slice setting is challenging and the algorithms typically used are designed on a case-by-case basis depending on the structure to be simulated. In this paper, an accurate algorithm for GISAXS simulations based on the multi-slice DWBA theory is presented. In particular, fundamental properties of the Fourier transform have been utilized to develop an algorithm that accurately computes the average refractive index profile as a function of depth and the Fourier transform of the portion of the sample within a given slice, which are key quantities required for the multi-slice DWBA simulation. The results from this method are compared with the traditionally used approximations, demonstrating that the proposed algorithm can produce more accurate results. Furthermore, this algorithm is general with respect to the sample structure, and does not require any sample-specific approximations to perform the simulations.

X-ray dynamical diffraction from multilayer Laue lenses with rough interfaces

A modeling approach for x-ray dynamical diffraction from multilayer Laue lenses (MLLs) with rough interfaces is developed. Although still based on the principle of the distorted-wave Born approximation (DWBA), this model is formulated from the perspective of the physical scattering process, very different from the conventional DWBA formalism. Using this model, one can study x-ray scattering from rough interfaces in the regime of Fresnel diffraction and in the case of absorptive samples, for example, x-ray dynamical diffraction from MLLs with rough interfaces, which is hard to handle in the framework of the conventional DWBA. Theoretical simulations for various MLLs with rough interfaces are conducted. It is found that interfacial roughness results in a decrease in the local diffraction intensity, where the attenuation factor is a function of the root-mean-square (rms) roughness versus the local zone width ratio. This study shows that if all zones possess an identical rms roughness value that is less than half of the outmost MLL zone width, the focal broadening effect due to roughness is almost unnoticeable, provided that the mean position of the interface does not deviate from the required zone plate law. A further study shows that uncorrelated interfacial roughness can be treated themore » same as interfacial diffusion, in which case a roughness factor similar to the ''Debye-Waller factor'' can be used, and the pseudo-Fourier coefficients of the susceptibility function for an MLL [H. F. Yan et al., Phys. Rev. B 76, 115438 (2007)] have to be multiplied by this factor.« less

Deuteron stripping and pick-up on halo nuclei

We present an approach to halo transfer in $(p,d)$ and $(d,p)$ reactions in which excitation and breakup effects are treated in a simple way. The method assumes that halo excitation energies are small compared with kinetic energies in the channel containing halo nucleus. It is shown that in the zero-range approximation to this approach the transfer amplitude has a formal resemblance to the conventional distorted wave Born approximation amplitude in which the distorted wave in the channel containing the halo nucleus is replaced by an effective distorted wave. Finite-range corrections are included using a generalization of the local-energy approximation. As a first application of the method, numerical calculations have been performed for the ${}^{16}\mathrm{O}{(d,p)}^{17}\mathrm{O}{,}^{10}\mathrm{Be}{(d,p)}^{11}\mathrm{Be},$ and ${}^{11}\mathrm{Be}{(p,d)}^{10}\mathrm{Be}$ reactions. Deuteron breakup was included within the adiabatic approach of Johnson and Soper. It was found that including recoil excitation and breakup of the ${}^{17}\mathrm{O}$ and ${}^{11}\mathrm{Be}$ nuclei increases the calculated cross sections and thus decreases the deduced spectroscopic factors. This effect is expected to increase with incident energy and decrease when the mass of the core of the halo nucleus increases relative to the mass of the halo.

Measurements of differential cross-section ratios for single-nucleon transfer reaction pairs near A = 25

Differential cross sections for the (d, p), ( 3He, d), (α, t), and (α, 3He) reactions involving seventy-one residual states in 23Na, 25Mg, 25Al, and 27Al have been measured at a forward angle with incident energies of 17.5, 20.2, 34.8, and 34.8 MeV, respectively. The ratio of cross-section pairs involving formation of the same residual state is determined for forty-five cases where both the angular momentum transfer and single-particle spectroscopic strength have been previously established. These are compared to values calculated with conventional distorted-wave Born approximation analyses, and the utility of this technique for identifying some levels which are possible s- or p-wave resonances is demonstrated and discussed for states in the vicinity of proton thresholds. An application is made involving proton threshold states in 27Al.

11B(d,n)12C reaction and the adiabatic approximation.

Differential cross sections were measured for the $^{11}\mathrm{B}$(d,n)$^{12}\mathrm{C}$ reaction at 79 MeV. Three different adiabatic model calculations as well as conventional distorted-wave Born approximation calculations were performed for transitions to the 0.0, 4.44, 9.64, 12.71, and 15.11 MeV states in $^{12}\mathrm{C}$. One of the adiabatic optical model parameter sets gave consistently better agreement than the rest with the experimental data. The spectroscopic factors obtained for the above states were compared with theory and with previous measurements.

Systematics of (p→, d) analyzing powers at 94 MeV

Analyzing-power angular distributions of the (p,d) reaction on /sup 60/Ni, /sup 86/Sr, and /sup 208/Pb have been measured at 94-MeV bombarding energy. A distinctive j dependence was observed for the pickup of neutrons from the p/sub 1/2/, p/sub 3/2/, f/sub 5/2/, and f/sub 7/2/ orbitals which is independent of target mass and of the number of nodes in the neutron bound-state radial wave function. These characteristic j = l +- (1/2) signatures may thus be used as experimental templates for reliable spin determinations over the whole periodic table at this bombarding energy. Conventional distorted-wave Born approximation calculations using standard optical model parameters describe the experimental analyzing power angular distributions and the associated j dependence rather poorly.

Three-body calculations of elastic scattering and stripping of deuterons on 16O

A calculation is presented in which a reaction of complex nuclei is approximated by a model three-body problem. The results of such a calculation, for elastic scattering and stripping of deuterons from 16O, are compared to the results obtained from conventional distorted-wave Born approximation (DWBA) calculations. Since severe limitations had to be imposed on the three-body calculation, for computational reasons, and the Coulomb force was neglected, the problem is to be considered a model problem. Experimental data were used to guide the selection of the two-body interactions, but the results cannot meaningfully be compared with data. A comparison of the three-body results to those given by the DWBA show qualitatively good agreement in the differential cross sections for both elastic scattering and stripping, but quantitative agreement is obtained only for the elastic scattering results. Polarization observables show no similarity between the two calculations. Several possible reasons for this discrepancy are discussed, but more work is needed to separate various contributing factors.

Energy Dependence of the ReactionPb208(O16,N15)Bi209

The elastic scattering and one-proton transfer reactions induced by 312.6-MeV $^{16}\mathrm{O}$ ions on $^{208}\mathrm{Pb}$ are investigated and compared with data at lower incident energies. A conventional distorted-wave Born approximation analysis can account for the qualitative features of the energy spectra but overestimates the energy dependence of the observed absolute cross sections by almost an order of magnitude between 140 and 312.6 MeV.

Distorted-Wave Born-Approximation Analysis ofAr36,38(d, p)to Neutron Resonances inAr37,39

$^{36,38}\mathrm{Ar}(d, p)$ angular distributions at 9- and 10-MeV incident energy, for 13 neutron-unbound states in $^{37,39}\mathrm{Ar}$, are described in terms of conventional distorted-wave Born-approximation theory using complex-energy eigenstates as form factors, and spectroscopic information is extracted which is consistent with previous studies of the bound neutron states of $^{37,39}\mathrm{Ar}$.

Systematic Distorted-Wave Born-Approximation Predictions for Two-Nucleon Transfers: Applications to (d,α) Experiments

In the case of angular-momentum-mismatch conventional distorted-wave Born-approximation (DWBA) calculations tend to give results which are strongly dependent on the opticalmodel parameters chosen and, to a lesser degree, on finite-range and nonlocality effects. We discuss reasons for this sensitivity and present systematic calculations for ($d,\ensuremath{\alpha}$) reactions on nondeformed targets ranging from $^{48}\mathrm{Ti}$ to $^{208}\mathrm{Pb}$. Satisfactory DWBA results could be obtained for the entire range of targets, provided that all potentials for the generation of scattered and bound wave functions were restrained to have nearly identical physically meaningful real well geometries and real depths of $V\ensuremath{\approx}n{V}_{0}$, where ${V}_{0}$ is the proton scattering potential and $n$ is the number of nucleons in the projectile. The use of well geometries with ${r}_{0}=1.2$ fm, $a=0.75$ fm, and retention of the basic DWBA requirement that the optical potentials should also correctly fit elastic scattering removes the familiar ambiguities for deuteron and $\ensuremath{\alpha}$ potentials. It is shown that with these parameter restrictions finite range effects are expected to be small so that a first-order correction procedure is adequate. Explicit calculations are compared with over 30 ($d,\ensuremath{\alpha}$) angular distributions of known angular momentum transfer for experimental bombarding energies ranging from 12 to 17 MeV. Consistent agreement with experiment was obtained.

ReactionAr38(d,p)Ar39

The level structure of $^{39}\mathrm{Ar}$ has been investigated via the stripping reaction $^{38}\mathrm{Ar}(d,p)^{39}\mathrm{Ar}$ at an incident deuteron energy of 10.064 MeV with an over-all experimental resolution of approximately 35 keV. Proton groups leading to 44 states of $^{39}\mathrm{Ar}$ with excitation energies up to 7.727 MeV have been identified. 30 of these states with excitation energies up to 6.488 MeV are neutron-bound. The orbital-angular-momentum-transfer values and the spectroscopic factors have been extracted for 28 of the observed neutron-bound states using zero-range distorted-wave Born-approximation (DWBA) calculations corrected for nonlocality of the optical potential and finite range in the local-energy approximation. One $l=0$, one $l=2$, twelve $l=3$, and fourteen $l=1$ transfer values are assigned. Spin assignments have been made on the basis of the conventional shell-model ordering of states, by comparison with the level structure of the isotonic nucleus $^{41}\mathrm{Ca}$ in the reaction $^{40}\mathrm{Ca}(d,p)^{41}\mathrm{Ca}$ and on the basis of the Lee-Schiffer effect. It is found that the conventional DWBA calculation with spin-orbit coupling reproduces the observed Lee-Schiffer effect for $l=1$ states remarkably well. The results of the present investigation are in fair agreement with those of recent shell-model calculations.

Inadequacies in a Conventional Distorted-Wave Born-Approximation Analysis of (p,t) Transitions in Titanium Isotopes atEp=27MeV

Some measurements of ($p,t$) differential cross sections are reported for which a conventional distorted-wave Born-approximation analysis does not work well, and where the findings suggest caution in the interpretation of ($p,t$) studies in the $A=50$ mass region at a bombarding energy of about 27 MeV.

ReactionC12(He3, d)N13and Stripping to Unbound States

Excitation functions of deuterons from the $^{12}\mathrm{C}(^{3}\mathrm{He}, d)^{13}\mathrm{N}$ reaction over the bombarding-energy range 12-19 MeV were measured at several angles for the ground-state deuterons and at a laboratory angle of 40\ifmmode^\circ\else\textdegree\fi{} for deuterons leading to the ground state, to the first excited state, and to the unresolved second and third excited states. Detailed angular distributions were measured at 16, 17, and 18 MeV for the ground state through the third excited state and at 19 MeV for the ground state through the seventh excited state. The ground-state angular distributions were analyzed with the conventional distorted-wave Born approximation. For the excited states, all of which are unstable with respect to proton decay, a number of alternative methods of calculation were attempted and the results were compared.