Calculation of neutron-He scattering up to 30 MeV
Microscopic calculations of four-body collisions become very challenging in the energy regime above the threshold for four free particles. The neutron-He scattering is an example of such process with elastic, rearrangement, and breakup channels.
We aim to calculate observables for elastic and inelastic neutron-He reactions up to 30 MeV neutron energy using realistic nuclear force models.
We solve the Alt, Grassberger, and Sandhas (AGS) equations for the four-nucleon transition operators in the momentum-space framework. The complex-energy method with special integration weights is applied to deal with the complicated singularities in the kernel of AGS equations.
We obtain fully converged results for the differential cross section and neutron analyzing power in the neutron-He elastic scattering as well as the total cross sections for inelastic reactions. Several realistic potentials are used, including the one with an explicit isobar excitation.
There is reasonable agreement between the theoretical predictions and experimental data for the neutron-He scattering in the considered energy regime. The most remarkable disagreements are seen around the minimum of the differential cross section and the extrema of the neutron analyzing power. The breakup cross section increases with energy exceeding rearrangement channels above 23 MeV.
pacs:21.45.-v, 21.30.-x, 25.10.+s, 24.70.+s
Experimentally, four-nucleon () physics is studied most extensively through proton-He () and deuteron-deuteron () reactions tilley:92a (), i.e., charged particle beams and non-radioactive targets. The system is simpler since it involves three protons and one neutron and, furthermore, only elastic and breakup channels exist. Theoretically, below breakup threshold scattering constitutes a single channel problem, making it also simpler to calculate. Indeed, accurate numerical calculations for low-energy elastic scattering have been performed using several rigorous approaches, i.e., the hyperspherical harmonics (HH) expansion method viviani:01a (); kievsky:08a (), the Faddeev-Yakubovsky (FY) equations yakubovsky:67 () for the wave function components lazauskas:04a (), and the Alt, Grassberger and Sandhas (AGS) equations grassberger:67 () for transition operators deltuva:07a (); deltuva:07b (). The latter method uses the momentum-space framework, while the former two are implemented in the coordinate space framework. All these methods were benchmarked in Ref. viviani:11a () below breakup threshold for observables in neutron- () and elastic scattering; good agreement between calculations was found, confirming their reliability.
However, the physics of reactions in coupled proton- (), neutron- () and systems is more rich. The scattering process in these systems resembles a typical nuclear reaction where, depending on the available energy, elastic, charge exchange, transfer, and breakup reactions may take place simultaneously. At the same time such reactions are more difficult to calculate. Indeed, the coordinate space methods are limited so far to processes up to threshold lazauskas:09a (); viviani:10a (). In contrast, the momentum-space calculations are available for all elastic and rearrangement , , and reactions below three-cluster breakup threshold deltuva:07c (); deltuva:10a (). However, the extension to higher energies constitutes a major difficulty, since the asymptotic boundary conditions in coordinate space become highly nontrivial due to open two-, three- and four-cluster channels. In the momentum-space framework this is reflected in a very complicated structure of singularities in the kernel of the integral equations. Formally, these difficulties can be avoided by rotation to complex coordinates lazauskas:11a (); lazauskas:12a () or continuation to complex energy efros:94a (); kamada:03a () that lead to bound-state like boundary conditions and nonsingular kernels. However, further technical complications arise in practical calculations, especially when using realistic nuclear force models. Although the solution of FY equations for scattering with modern potentials using complex scaling is underway lazauskas:pc (), at present the only realistic calculations of scattering above threshold are performed using the momentum-space AGS equations deltuva:12c (); deltuva:13c (), but limited to and elastic scattering. The complex energy method kamada:03a (); uzu:03a () was used to deal with the complicated singularities in the four-particle scattering equations; its accuracy and practical applicability was greatly improved by a special integration method deltuva:12c ().
In the present work, following the ideas of Refs. deltuva:12c (); deltuva:13c (), we calculate elastic and inelastic scattering over a wide range of neutron beam energies up to MeV. The Coulomb interaction is included using the method of screening and renormalization taylor:74a (); alt:80a (); see Refs. deltuva:05a (); deltuva:07b () for more details on the practical implementation. Within this method the standard AGS scattering equations for short-range potentials are applicable. Compared to our previous scattering calculations above breakup threshold deltuva:13c (), an additional complication for reactions is the presence of the rearrangement channels and the mixing of total isospin and 1 states. On the other hand, calculations are somehow simpler than those for or , since there is no long-range Coulomb interaction in the asymptotic state, making the convergence of the partial-wave expansion slightly faster.
Ii 4N scattering equations
We treat protons and neutron as identical particles in the isospin formalism and therefore use the symmetrized version of the AGS equations deltuva:07a () that are integral equations for the four-particle transition operators , i.e.,
For scattering the initial two-cluster partition that is of type is labeled as , whereas corresponds to the partition. They are chosen as (12,3)4 and (12)(34), respectively; in the system of four identical particles there are no other distinct two-cluster partitions. The transition operators for these 3+1 and 2+2 subsystems are obtained from the respective integral equations
The pair (12) transition matrix is derived from the corresponding two-nucleon potential that, beside the nuclear part, includes also the screened Coulomb potential for the pair. The screening function is taken over from Refs. deltuva:07b (); deltuva:07c () but the dependence on the screening radius is suppressed in our notation. The permutation operators of particles and and their combinations and together with a special choice of the basis states ensure the full antisymmetry of the four-nucleon system. The basis states must be antisymmetric under exchange of two particles in the subsystem (12) for the partition and in (12) and (34) for the partition. All transition operators acquire their dependence on the available energy through the free resolvent
with the complex energy and the free Hamiltonian .
Although the physical scattering process corresponds to the limit, the AGS equations are solved numerically at a complex energy with finite positive . This way we avoid the very complicated singularity structure of the kernel and are faced with quasisingularities, that can be accurately integrated over using a special integration method developed in Ref. deltuva:12c (). The singularities (quasisingularities for finite ) of the AGS equations correspond to open channels. In addition to elastic and three- and four-cluster breakup channels, present in the and scattering deltuva:12c (); deltuva:13c (), in the reaction there are the rearrangement channels and . They are treated in the same way as the elastic channel. The limit needed for the calculation of scattering amplitudes and observables is obtained by the extrapolation of finite results. Previous calculations uzu:03a (); deltuva:12c () employed the point method schlessinger:68 (). In the present work, as an additional accuracy check, we use also the cubic spline extrapolation with a nonstandard choice of boundary conditions, namely, the one ensuring continuity of the third derivative chmielewski:03a (). These two different methods lead to indistinguishable results confirming the reliability of the extrapolation procedure. We use ranging from 1 to 2 MeV at the lowest considered energies and from 2 to 4 MeV at the highest energies. About 30 grid points for the discretization of each momentum variable are used.
As mentioned, the potential for the pair must include both the nuclear and the screened Coulomb potential ; see Refs. deltuva:07b (); deltuva:07c () for more details. The limit is calculated separately for each value of the Coulomb screening radius and the renormalization procedure deltuva:07b (); deltuva:07c () is performed subsequently. Thus, the scattering amplitude connecting the initial state with any two-cluster state is given by
Here, are the Faddeev amplitudes of the initial (i) or final (f) channel states , whereas and are the weight factors resulting from the symmetrization. Note that the channel state requires explicit symmetrization under the exchange of two deuterons, since the employed basis states do not obey this symmetry. The initial and final bound state energies , relative two-cluster momenta and reduced masses obey the on-shell relation . The renormalization factor is defined as in Refs. deltuva:07b (); deltuva:07c (); it is simply 1 for the state which is not distorted by the long-range Coulomb interaction. In the present calculations we use fm which is fully sufficient for convergence.
The spin-averaged differential cross section for the transition to the , or final state is
where the summation runs over the initial and final spin projections and , and is the number of initial spin states for two spin particles. The total cross section for a given reaction is obtained by integrating Eq. (5) over the solid angle for and final states and for final state.
The breakup amplitudes can be obtained from the half-shell matrix elements of as described in Refs. deltuva:12a (); deltuva:13a (). However, in the present work we only calculate the total three- and four-cluster breakup cross section as the difference between the total and all two-cluster cross sections. The total cross section is obtained using the optical theorem as
The AGS equations are solved by us in the momentum-space partial-wave framework. We define the states of the total angular momentum with projection as for the configuration and for the . In the literature they are called sometimes as K-type and H-type basis states, respectively. Here and are the four-particle Jacobi momenta in the convention of Ref. deltuva:12a (), , , and are the associated orbital angular momenta, and are the total angular momenta of pairs (12) and (34), is the total angular momentum of the (123) subsystem, and are the spins of nucleons 3 and 4, and , , and are channel spins of two-, three-, and four-particle system.
With respect to isospin, there are important differences as compared to previous and calculations. Two types of isospin states are used in the present calculations for the configuration, and . They are related by a simple unitary transformation with Clebsch-Gordan coefficients . Here is the isospin of the pair (12), are the isospins of nucleons 3 and 4, is the isospin of the (123) subsystem, and is total isospin of the system, with , , and being the respective projections. For the configuration the isospin states are , with being the isospin of the pair (34).
The eigenstates of the total isospin are more convenient to calculate the action of the permutation operator and transformations between the K- and H-type states, since these operations conserve . In contrast, the channel states mix the total isospin but have fixed values of and , i.e., for and for . Furthermore, , , and depend on , implying that also the location of quasi-singularities of and the special integration weights deltuva:12c () depend on . Thus, the calculation of and is done using the isospin basis. The two-nucleon transition matrix is different for , , and pairs. It preserves but depends on its projection , i.e., . This gives rise to the coupling between and states, i.e., the nonvanishing components are
Abbreviating by , in terms of , , and transition operators we obtain
In the configuration the two-nucleon transition matrix couples the states with different but preserves the other isospin quantum numbers, i.e., the nonvanishing components are
The above operator, abbreviated by , can be expressed through as
The nondiagonal isospin coupling in Eqs. (8) and (10) is due to the charge dependence of the underlying interaction, with the Coulomb repulsion yielding the dominant contribution. However, the and thereby also components resulting from this charge dependence in the and channel states are very small, of the order of 0.01%. The channel state is pure state. The breakup channel state is limited to and 1, and solely the channel state may have moderate component. In fact, the leading contribution of states is of first order in the charge dependence for the four-cluster breakup amplitude but of second order, i.e., much smaller, for all other amplitudes. Thus, states can be safely neglected in the solution of the AGS equations if the four-cluster breakup amplitude is not explicitly calculated. This is in close analogy with scattering where the total isospin states can be safely neglected when calculating elastic scattering and total breakup cross section, but are important in particular kinematic configurations of breakup deltuva:05d (). We therefore include only and 1 states in the present calculations of scattering.
The results are well converged in terms of angular momentum states. At the highest considered neutron beam energy MeV we include four-nucleon partial waves with , , , and . The most demanding observables are the transfer and breakup cross sections. The convergence for elastic and charge exchange reactions is faster. The number of partial waves can be reduced at lower energies and in lower states.
We study the scattering using several models of realistic high-precision potentials: the inside-nonlocal outside-Yukawa (INOY04) potential by Doleschall doleschall:04a (); lazauskas:04a (), the charge-dependent Bonn potential (CD Bonn) machleidt:01a (), and its coupled-channel extension CD Bonn + deltuva:03c (). The latter allows for an excitation of a nucleon to a isobar and thereby yields mutually consistent effective three- and four-nucleon forces (3NF and 4NF). The () binding energy calculated with INOY04, CD Bonn, and CD Bonn + potentials is 7.73, 7.26, and 7.53 MeV (8.49, 8.00, and 8.28 MeV), respectively; the experimental value is 7.72 MeV (8.48 MeV). We therefore use INOY04 for predictions at all considered energies since this potential yields nearly the experimental value for the binding energy. Other potentials are used at fewer selected energies to investigate the dependence of predictions on the force model. The calculations with the CD Bonn (CD Bonn + ) potential are performed only at neutron energies of 6, 8, 12, and 22 MeV (12 and 22 MeV).
In Fig. 1 we show the differential cross section for elastic scattering as a function of the center of mass (c.m.) scattering angle . The neutron energy ranges from 6 to 30 MeV, the highest energy at which, to the best of our knowledge, exclusive data for elastic scattering exist. The differential cross section decreases with the increasing energy and also changes the shape; the calculations describe the energy and angular dependence of the experimental data fairly well. There are disagreements between different data sets, in particular, between haesner:exfor () and seagrave:60 () at MeV, between drosg:74a () and antolkovic:67 () at MeV, and between haesner:exfor () and drosg:74a () at MeV. Only at MeV it is quite obvious that the data antolkovic:67 () is inconsistent with other measurements and calculations.
It is interesting to compare the present results with the ones for elastic scattering deltuva:13c (), as there are several differences. First, the energy dependence is slower for . Second, below 10 MeV the data are well described at but slightly underpredicted at larger angles, while the data are described well at with slight underprediction at smaller angles. On the other hand, both and data are well reproduced by the theory between 12 and 22 MeV in the whole angular regime, but the minimum of around gets underpredicted above 23 MeV. This may indicate a need to include an additional 3NF, as in the case of the nucleon-deuteron scattering witala:98a (); nemoto:98c (). The sensitivity to the potential model is similar in both and cases. It is insignificant beyond the minimum of that roughly scales with the binding energy; a weaker binding corresponds to a deeper minimum. At MeV the CD Bonn and CD Bonn + results are lower than those of INOY04 by 14 % and 8 %, respectively. At MeV this correlation is violated amounting to 18 % reduction for both CD Bonn and CD Bonn + potentials. This may be due to an almost complete cancellation of two competing -isobar contributions, the effective 3NF and the NN dispersion. While the former increases at the minimum by 15 %, the latter decreases it by roughly the same amount. A partial cancellation between two-baryon dispersive and 3NF effects is a characteristic feature of the CD Bonn + model, seen also in previous studies deltuva:08a (); deltuva:13c ().
In Fig. 2 we show the neutron analyzing power for the elastic scattering at neutron energies ranging from 8 to 22 MeV. The qualitative reproduction of the experimental data by our calculations is reasonable, except for the data sets busser (); busse () that are incompatible also with other data lisowski:76a (); klages:85 (). Some discrepancies, decreasing as the energy increases, exist around the minimum and the maximum. The sensitivity to the nuclear force model and the energy dependence are quite weak. In all these respects, the behavior of the in the elastic scattering is qualitatively the same as observed for the proton analyzing power in the elastic scattering deltuva:13c ().
To the best of our knowledge, there are no experimental data for other spin observables in the elastic scattering. Nevertheless, we calculated various spin correlation and spin transfer coefficients. In all studied cases we found only small sensitivity of the predictions to the force model. As a characteristic example in Fig. 3 we present results for target analyzing power , spin correlation coefficient , and neutron spin transfer coefficient at and 22 MeV. Comparing with the corresponding observables in the elastic scattering deltuva:13c () we observe that some of them like and , exhibit a very different angular and energy dependence. This is not surprising given the fact that scattering involves both total isospin and 1 states while is restricted to . On the other hand, this indicates that accurate measurements of spin correlation and/or spin transfer coefficients that differ significantly from previously studied observables may test the nuclear interaction in a novel way, in particular the proper mixing of isospin and 1 states.
Next we consider rearrangement reactions initiated by collisions. We present here only two examples for and processes measured at MeV in Ref. antolkovic:67 (), since most experiments are performed for the time reversed reactions and that will be studied elsewhere. In Fig. 4 we show the differential cross section for the charge exchange reaction at MeV. The theoretical predictions agree with the data antolkovic:67 () only at forward and backward angles. On the other hand, the data drosg:78 () transformed from the time reversed reaction at MeV is in a considerably better agreement with our predictions. In particular, the shape of the observable with two local minima is well described by the theory, as found in our preliminary calculations for the reaction deltuva:14a (). Thus, very likely the data points from Ref. antolkovic:67 () at intermediate angles are inaccurate.
In Fig. 5 we show the differential cross section for the transfer reaction at MeV. The observable is symmetric with respect to and peaks at forward and backward directions. The overall agreement between theoretical calculations and the data antolkovic:67 () is fair, given the large errorbars and, possibly, further inaccuracies in the data antolkovic:67 (), especially at intermediate angles where is small and has several local extrema. To draw a more definite conclusion on transfer reactions, calculations and analysis of and reactions need to be accomplished.
Finally, in Fig. 6 we show the energy dependence of the total and partial cross sections for all open channels, i.e., elastic, charge-exchange, transfer, and breakup. This extends our previous results deltuva:14a () up to MeV. The theoretical predictions are below the data in the regime MeV where several resonant states exist and whose location is not well predicted by the underlying force models as discussed in Refs. deltuva:07c (); fonseca:02a (). On the contrary, the agreement is nearly perfect at higher energies up to 22 MeV, but moderate discrepancies arise in and cross sections above MeV. The total breakup cross section, including both three- and four-cluster channels, increases rapidly with energy and above MeV exceeds for all other inelastic channels. The experimental data for the total breakup cross section drosg:74a () are in agreement with theoretical predictions, although the data point at MeV is inconclusive owing to very large error bars.
We considered neutron- scattering at neutron energies ranging from 6 to 30 MeV. We solved the Alt, Grassberger, and Sandhas equations for the symmetrized four-nucleon transition operators in the momentum-space framework. We included the Coulomb force and used several realistic potentials. The complicated singularities in the kernel of AGS equations above breakup threshold were treated by the complex energy method with special integration weights. Fully converged results were obtained not only for elastic scattering, but also for inelastic reactions. Furthermore, total cross sections for all reaction channels were calculated, showing the importance of breakup at higher energies.
The overall agreement between the theoretical predictions and the experimental data is good. Few moderate discrepancies exist in the extrema of elastic analyzing power and differential cross section, similar to the case of elastic proton- scattering. The charge exchange and transfer reactions will be analyzed in more detail through time reverse processes and ; the respective calculations are in progress.
- (1) D. R. Tilley, H. Weller, and G. M. Hale, Nucl. Phys. A541, 1 (1992).
- (2) M. Viviani, A. Kievsky, S. Rosati, E. A. George, and L. D. Knutson, Phys. Rev. Lett. 86, 3739 (2001).
- (3) A. Kievsky, S. Rosati, M. Viviani, L. E. Marcucci, and L. Girlanda, J. Phys. G 35, 063101 (2008).
- (4) O. A. Yakubovsky, Yad. Fiz. 5, 1312 (1967) [Sov. J. Nucl. Phys. 5, 937 (1967)].
- (5) R. Lazauskas and J. Carbonell, Phys. Rev. C 70, 044002 (2004).
- (6) P. Grassberger and W. Sandhas, Nucl. Phys. B2, 181 (1967); E. O. Alt, P. Grassberger, and W. Sandhas, JINR report No. E4-6688 (1972).
- (7) A. Deltuva and A. C. Fonseca, Phys. Rev. C 75, 014005 (2007).
- (8) A. Deltuva and A. C. Fonseca, Phys. Rev. Lett. 98, 162502 (2007).
- (9) M. Viviani, A. Deltuva, R. Lazauskas, J. Carbonell, A. C. Fonseca, A. Kievsky, L. E. Marcucci, and S. Rosati, Phys. Rev. C 84, 054010 (2011).
- (10) R. Lazauskas, Phys. Rev. C 79, 054007 (2009).
- (11) M. Viviani, R. Schiavilla, L. Girlanda, A. Kievsky, and L. E. Marcucci, Phys. Rev. C 82, 044001 (2010).
- (12) A. Deltuva and A. C. Fonseca, Phys. Rev. C 76, 021001(R) (2007).
- (13) A. Deltuva and A. C. Fonseca, Phys. Rev. C 81, 054002 (2010).
- (14) R. Lazauskas and J. Carbonell, Phys. Rev. C 84, 034002 (2011).
- (15) R. Lazauskas, Phys. Rev. C 86, 044002 (2012).
- (16) V. D. Efros, W. Leidemann, and G. Orlandini, Phys. Lett. B 338, 130 (1994).
- (17) H. Kamada, Y. Koike, and W. Glöckle, Prog. Theor. Phys. 109, 869L (2003).
- (18) R. Lazauskas, private communication (2014).
- (19) A. Deltuva and A. C. Fonseca, Phys. Rev. C 86, 011001(R) (2012).
- (20) A. Deltuva and A. C. Fonseca, Phys. Rev. C 87, 054002 (2013).
- (21) E. Uzu, H. Kamada, and Y. Koike, Phys. Rev. C 68, 061001(R) (2003).
- (22) J. R. Taylor, Nuovo Cimento B 23, 313 (1974); M. D. Semon and J. R. Taylor, Nuovo Cimento A 26, 48 (1975).
- (23) E. O. Alt and W. Sandhas, Phys. Rev. C 21, 1733 (1980).
- (24) A. Deltuva, A. C. Fonseca, and P. U. Sauer, Phys. Rev. C 71, 054005 (2005).
- (25) L. Schlessinger, Phys. Rev. 167, 1411 (1968).
- (26) K. Chmielewski, A. Deltuva, A. C. Fonseca, S. Nemoto, and P. U. Sauer, Phys. Rev. C 67, 014002 (2003).
- (27) A. Deltuva, Phys. Rev. A 85, 012708 (2012).
- (28) A. Deltuva and A. C. Fonseca, Phys. Rev. C 87, 014002 (2013).
- (29) A. Deltuva, A. C. Fonseca, and P. U. Sauer, Phys. Rev. C 72, 054004 (2005).
- (30) P. Doleschall, Phys. Rev. C 69, 054001 (2004).
- (31) R. Machleidt, Phys. Rev. C 63, 024001 (2001).
- (32) A. Deltuva, R. Machleidt, and P. U. Sauer, Phys. Rev. C 68, 024005 (2003).
- (33) B. Haesner, W. Heeringa, H. O. Klages, H. Dobiasch, G. Schmalz, P. Schwarz, J. Wilczynski, B. Zeitnitz, and F. Käppeler, in EXFOR Database (NNDC, Brookhaven, 1982).
- (34) J. D. Seagrave, L. Cranberg, and J. E. Simmons, Phys. Rev. 119, 1981 (1960).
- (35) M. Drosg, D. K. McDaniels, J. C. Hopkins, J. D. Seagrave, R. H. Sherman, and E. C. Kerr, Phys. Rev. C 9, 179 (1974).
- (36) B. Antolkovic, G. Paic, P. Tomaš, and D. Rendic, Phys. Rev. 159, 777 (1967).
- (37) H. Witała, W. Glöckle, D. Hüber, J. Golak, and H. Kamada, Phys. Rev. Lett. 81, 1183 (1998).
- (38) S. Nemoto, K. Chmielewski, S. Oryu, and P. U. Sauer, Phys. Rev. C 58, 2599 (1998).
- (39) A. Deltuva, A. C. Fonseca, and P. U. Sauer, Phys. Lett. B 660, 471 (2008).
- (40) P. Lisowski, T. Rhea, R. Walter, C. Busch, and T. Clegg, Nucl. Phys. A 259, 61 (1976).
- (41) H. O. Klages, W. Heeringa, H. Dobiasch, B. Fischer, B. Haesner, P. Schwarz, J. Wilczynski, and B. Zeitnitz, Nucl. Phys. A443, 237 (1985).
- (42) F. Büsser, H. Dubenkropp, F. Niebergall, and K. Sinram, Nucl. Phys. A 129, 666 (1969).
- (43) W. Busse, B. Efken, D. Hilscher, H. Morgenstern, and J. Scheer, Nucl. Phys. A 187, 21 (1972).
- (44) M. Drosg, Nucl. Sci. Eng. 67, 190 (1978); in EXFOR Database (NNDC, Brookhaven, 1978).
- (45) A. Deltuva and A. C. Fonseca, Phys. Rev. Lett. 113, 102502 (2014).
- (46) B. Haesner, W. Heeringa, H. O. Klages, H. Dobiasch, G. Schmalz, P. Schwarz, J. Wilczynski, B. Zeitnitz, and F. Käppeler, Phys. Rev. C 28, 995 (1983).
- (47) M. E. Battat et al., Nucl. Phys. 12, 291 (1959).
- (48) J. H. Gibbons and R. L. Macklin, Phys. Rev. 114, 571 (1959).
- (49) A. C. Fonseca, G. Hale, and J. Haidenbauer, Few-Body Syst. 31, 139 (2002).