Aspects of the momentum dependence of the equation of state and of the residual cross section, and their effects on nuclear stopping
With the semiclassical Landau-Vlasov transport model we studied the stopping observable , the energy-based isotropy ratio, for the Xe + Sn reaction at beam energies spanning 12 to 100 MeV. We investigated the impacts of the nonlocality of the nuclear mean field, of the in-medium modified nucleon-nucleon () cross section and of the reaction centrality. A fixed set of model parameters yields values that favorably compare with the experimental ones, but only for energies below the Fermi energy . Above agreement is readily possible, but by a smooth evolution with energy of the parameter that controls the in-medium modification of cross section. By comparing the simulation correction factor applied to the free cross section with the one deduced from experimental data [Phys. Rev. C 90, 064602 (2014)], we infer that the zero-range mean field almost entirely reproduces it. Also, in accordance with what has been deduced from experimental data, around a strong reduction of the free cross section is found. In order to test the impact of sampling central collisions by multiplicity an event generator (HIPSE) was used. We obtain that high multiplicity events are spread over a broad impact parameter range, but it turns out that this has a small effect on the observable and, thus, on as well.
pacs:25.70.-z, 21.30.-x, 24.10.Lx
The ratio between transverse and longitudinal components of kinematical observables is a measure of the conversion of the initial entrance channel motion into intrinsic degrees of freedom in heavy-ion collisions (HICs). Such an observable gives an insight on the rate of a system’s equilibration, of the dissipation of the available energy, as well as of HIC stopping power stroe83 (); bauer88 (). Thanks to such an observable, the FOPI Collaboration has evidenced partial nuclear transparency in HIC in the beam-energy range 0.1 – 1 GeV fopi (). More recently, by examining the ratio of transverse to longitudinal energy and linear momentum for the most violent HICs, the INDRA Collaboration has revealed a substantial reduction of the nuclear stopping power at 10 – 100 MeV lehaut (). This stopping observable reaches a minimum around the Fermi energy and stagnates or very weakly increases with the further increase of at least up to 100 MeV, the upper limit of the energy range available in this study. The above observation is valid for all (mass symmetric) systems studied, with system masses = 72 – 394 a.m.u. It is worth emphasizing that the fusion cross section normalized by the total reaction cross section exhibits an analogously rapid fall-off up to about eud13 (); eud14 (), a behavior especially evident for mass-symmetric systems (cf. Fig. 6 of Ref. eud14 ()).
In a recent publication lopez14 () the above observable was analyzed for the = 1 subset of the same INDRA data. The = 1 displays a slightly stronger increase with for the heavier systems lopez14 () relative to the values obtained in the previous study lehaut () which included light charged particles and fragments, but also was somewhat more stringent on the selection of the most central events. The authors of Ref. lopez14 () report a minimum of around , which is particularly enhanced when is normalized to the Fermi-gas-model prediction of the incoming value at a given . su13 ()
In Ref. lopez14 () it was assumed that protons are predominantly dynamically emitted during the early reaction phase, in accordance with Refs. eud97 (); had99 (). Such a hypothesis offers a possibility of extracting information on the in-medium correction for the free nucleon-nucleon () cross section . Following such an argument, starting from the experimental values in Ref. lopez14 (), with some basic assumptions about the effects owing to the Pauli-exclusion principle, the nucleon mean free path was extracted and an effective value of the in-medium cross section was deduced. In the process, a correction factor was obtained by which has to be multiplied at each to get a proper value. The authors found that (i) a significant reduction of is present in HICs below 100 MeV and (ii) this change of is strongly dependent upon . At the lowest energies the measured is compatible with the full stopping value ( 1) and the effective amounts to about 0.4 . One should keep in mind that the authors claimed a large uncertainty on the factor below 30 MeV, a subject for which they have announced a devoted publication lopez14 (). At incident energies around where attains its minimum, is reduced to less than one fifth the free value ( = 0.17) and then the effective steadily and regularly increases up to half of the free value ( 0.5) at = 100 MeV lopez14 () (see also Fig. 3 in the present work).
The stopping observables and/or have also been investigated in isospin-dependent quantum molecular dynamics (IQMD) su13 (); liu01 (); zhang (); kaur11 (); vin12 () and antisymmetrized molecular dynamics (AMD) zhao () model studies of HICs. All these works were carried out before publication of Ref. lopez14 (). Neither of simulation approaches predicts the remarkable in-medium reduction of found in Ref. lopez14 (). In the AMD study, specific attention has been paid to performing the analysis by meticulously following the experimental procedure for data handling zhao (). The simulation with in-medium due to Li and Machleidt li93-4 () (the free ) undershoots (overshoots) the data of lehaut (). An agreement with the data could only be reached at 80 MeV by doubling the theoretically established of li93-4 (). A systematic investigation of the impact of on , however, has not been performed yet. The intention of the present study is twofold:
1. by varying the nuclear equation of state (EOS) and the parametrization of , to investigate how the semiclassical Landau-Vlasov (LV) transport model of HIC gre87 (); seb89 () complies with the experimentally deduced dependence of the stopping observable on and
2. by varying a simple multiplicative factor of the free cross section, to compare thus obtained values for with those reported in Ref. lopez14 ().
Ii Model ingredients
Within the semiquantal extension of the Boltzmann transport theory, the highly nonlinear LV equation governs the spatio-temporal evolution of the one-body density distribution function :
which gives the probability of finding at the instant a particle in the phase-space point . stands for the Poisson bracket, whereas is the one-body Hamiltonian describing the Coulomb potential and the nuclear mean field. We present the results obtained with a soft nonlocal mean field labeled D1-G1 ( = 228 MeV, = 0.67) due to Gogny dech80 () and those obtained with the standard simplification of the soft zero-range Skyrme interaction due to Zamick ( = 200 MeV, = 1.0) zam73 (). The D1-G1 force is reputed to reproduce fundamental properties of nuclear matter as well as those of finite nuclei dech80 () while the Zamick parametrization of the EOS is, owing to its simplicity, of rather widespread use in a number of microscopic approaches. Details on both the nonlocal and the local parametrizations of the used EOS may be found in Tables I and III of Ref. had95 (), respectively. We have demonstrated that the LV model is able to correctly describe experimental observations in the intermediate energy regime eud97 (); had99 (); seb89 (); had95 (); lv_g (). The use of only a density dependent EOS is legitimated by the finding liu01 (); vin12 () that the isospin dependence of the mean field has a weak, if any, influence on isotropy ratios. Experimental ’s for a number of HICs between various xenon and tin isotopes corroborate this result; cf. Table I of Ref. lehaut ().
The function is expanded onto a moving basis of coherent states taken as normalized Gaussians () with frozen width () in () space:
is the system mass number and is the number of coherent states ( equals 60 in the present study). The widths and are chosen such as to best reproduce the nuclear ground state characteristics of the two colliding nuclei. The local density reads
Gaussians move in the self-consistent mean field and suffer hard scattering between them, controlled by the Uehling-Uhlenbeck collision integral accounting for the fermionic character of interacting particles ueh33 ():
which takes into account energy and momentum conservation as well as the Pauli exclusion principle. Here, denotes the nucleonic mass, is the occupation number with the spin-isospin degeneracy seb89 (), and ( and ) are initial (final) momenta of the scattering particle pair, is the single-particle energy, while is the in-medium nucleonic cross section. is scaled so that a Gaussian-averaged mean-free path is the same as for a nucleon. The cross section dependence on isospin has been reported as crucial for the study of stopping liu01 (). Thhis kind of parametrization was proposed by Chen et al. chen68 (), which hereafter we label . This phenomenological is based on the empirical isospin and energy dependence of the free scattering and has been used in both lopez14 () and liu01 (). due to Li and Machleidt li93-4 (), which accounts for the in-medium effects and is also isospin dependent, is tested too.
Iii Results and discussion
where summation runs over particles of those reaction events that satisfy certain selection criteria. For the LV simulation results, the summation index of Eq. (5) runs over the free Gaussians, i.e., those which are not bound in large (residue-like) fragment(s). Among the experimentally studied systems, the Xe + Sn reaction has been measured at by far the most abundant number of values lehaut (); lopez14 (). Consequently, in the present work only the simulation of this system is performed. To acquire stable values, the simulation is carried out up to 600 fm/ at the lowest = 12 – 32 MeV and up to 240 fm/ at the highest = 80 – 100 MeV. Beyond that time the calculation was continued until 8 000 fm/, considering only the Coulomb repulsion due to reaction residues. Special care was taken in order to perform our analysis of simulation data as close as possible to experimental conditions.
iii.1 A density-dependent cross section
In the experimental analysis lopez14 (), event selection is based on the charged particle’s multiplicity. The authors selected the most central events that are estimated, in cross-section units, to be equivalent to 50 to 150 mb lopez14 (). We adopt the median value of 100 mb for our analysis. This amount corresponds to about 2 % of the total reaction cross section and, in a geometrical sharp-cut approximation, to 2.0 fm. Consequently, in this subsection our simulation is limited to 2.0 fm.
Figure 1 displays obtained with the momentum-dependent D1-G1 EOS (upper panel) and with the zero-range Zamick EOS (lower panel) for several parametrizations of the in-medium corrected . For comparison, the experimental ’s are shown by filled circles with the corresponding errors lopez14 (). As a reference, the results obtained with the in-medium uncorrected empirical free scattering chen68 () are displayed by the heavy dotted curves. This empirical is used as an input for the in-medium modified suggested by Cugnon et al. cug87 (). In their Brueckner -matrix in-medium renormalization of the interaction, they obtained a set of parameters explicitly describing the dependence of on the local density cug87 (). These simulation results are displayed by the red curves and reddish zone in Fig. 1: the zone shows the range of the values limited by the impact parameters = 1 fm (dashed bordering curve) and = 2 fm (full bordering curve). ( = 0 fm has no weight and at most of energies is roughly the same for = 0 and 1 fm.) The heavy curve in each zone represents the -weighted value in the range = 0 – 2 fm and corresponds to 2 % of . For both EOS, the values with are very similar to those obtained with (dotted curves). Clearly, in the full energy range investigated here the in-medium effects of have rather weak impact on the observable. Consequently, as for , the compatibility of and for both EOS may be observed at the lowest only when the experimental errors are accounted for. In addition, for the Zamick EOS, Fig. 1b), the simulation strongly overshoots the data at the highest ’s.
A full ab initio microscopic study of based upon the Dirac-Brueckner approach to nuclear matter was performed by Li and Machleidt li93-4 (). Besides dependence on energy, isospin, and density of for this , we have added an explicit dependence on angle. In contrast to the scattering of neutrons, which is taken as isotropic, those between neutron and proton and between protons are anisotropic in accordance with the fit of Ref. seb07 (), which is given in detail in the Appendix. Similarly to above, the corresponding values of are displayed by the blue curves and bluish zone in Fig. 1. Again, the compatibility of with is unsatisfactory. Nevertheless, for the D1-G1 EOS and , Fig. 1(a), the slope of the isotropy ratio excitation function is correct but the simulation somewhat undershoots the experimental points: may be taken as compatible with the lower edges of experimental errors on . For the Zamick EOS, Fig. 1(b), the compatibility with exists at low and around 60 MeV, but the general features of the data are poorly reproduced. Manifestly, none of the above parametrizations of and EOS can account for the observed behavior of the stopping observable in the full energy range.
The parameters in the above are of a fixed value. By an expansion around the saturation value , Klakow et al. have suggested a simple parametrization for the dependence of on the evolving nuclear density klakow (),
where is evaluated locally according to Eq. (3), and is a free parameter assumed to reduce the cross section, thus it is strictly negative. As before, for the value taken is the empirical . The authors have recommended for the domain [–0.3, –0.1] klakow (). In our simulation is varied between –0.1 and –0.6. These are presented in Fig. 1 by the thinner dashed curves with variable dash size. They display a more or less regular dependence on both and . For the nonlocal EOS and –0.6 –0.5 the values at are well reproduced in both slope and absolute value; cf. Fig. 1(a). At energies higher than , however, for each another and regularly increasing value of the parameter is required such that, at the highest here considered, it should become positive, implying an in-medium enhancement rather than a reduction of at 80 MeV. Let us mention that with corresponds to that of with = –0.1 in the full range of considered and for both EOS. Simulation results with and D1-G1 EOS are compatible to with = –0.6 and 50 MeV. For the Zamick EOS of Fig. 1(b) one does not find a range of of stable value of the parameter that gives
’s compatible with either or those due to .
In conclusion, neither choice of allows for a unique description of experimental observation. One faces the fact that every model study, ours and previous su13 (); liu01 (); zhang (); kaur11 (); vin12 (); zhao (), fails to reproduce with a single set of parameters the INDRA experimental results in the full energy range studied lehaut (); lopez14 (). In particular, all models but zhao () predict steadily decreasing values of when increases, while experimental results display a break in the slope around the Fermi energy .
iii.2 Global modification of the free cross section
Being clearly unable to reproduce the experimental data with different parametrizations of the residual cross section, with or without momentum dependence of the force, let us concentrate on our second task that is, by following Ref. lopez14 (), to infer the multiplicative factor between the in-medium cross section and the free one:
As previously done and as in lopez14 (), we take chen68 (). Of course, this simple cross-section normalization factor cannot completely describe the rather complex modification of the free interaction occurring in the nuclear medium. In particular, such a is frozen during a reaction course and depends only indirectly on . Nevertheless, the prescription of Eq. (7) allows one to get an insight into the global in-medium effects on nuclear medium stopping properties and enables a comparison of the factor obtained in our simulation with of Ref. lopez14 ().
Figure 2 displays as a function of and the cross-section factor for the two effective interactions. In the D1-G1 EOS case, Fig. 2(a), the parameter takes values 0.2, 0.5, 0.8, 1.0, 1.2, and 1.5. For the Zamick EOS, Fig. 2(b), it is varied between 0.1 and 0.8 in steps of 0.1.111Simulation was also performed for the pure mean field, i.e., the Vlasov equation with zeroed right-hand-side of Eq. (1), which is equivalent to taking the parameter = 0. As in Fig. 1, are for central HIC with 2.0 fm, where = 1 fm (2 fm) results are represented by the thin dashed (full) curves that boarder the (colored) zone of each of the values. As before, the heavy curve in each zone shows the -weighted that corresponds to 2 % of . displays a regular dependence on and . In accordance with expectation and corroborating the results of Fig. 1, higher (larger ) implies higher stopping power of HICs. Unlike experimental and like our results of Fig. 1, as well as of a number of previous theoretical works su13 (); liu01 (); zhang (); kaur11 (); vin12 (); zhao (), the LV-simulation steadily decreases with for all without a minimum around . At the lowest ’s the mean field completely dominates the course of the collision, and for each value and both EOS is compatible with . For 45 MeV, is well reproduced by the = 0.5 curve [Fig. 2(a)] and by the = 0.1 one [Fig. 2(b)], respectively. Again, a single value of cannot reproduce experimental results. However, similarly to the case of the parameter of Eq. (6), by allowing to change with one may find a set of values to achieve an agreement between and . The behavior of both the parameter and the factor with corroborates the experimental finding lopez14 () that the effective in-medium cross section drastically changes with and that around there is a break in this dependence.
We take the -weighted as the starting point to infer information about the correction factor by which one would have to multiply to comply with . The procedure is evidenced in the inset of Fig. 3 in which the D1-G1 EOS at 50 MeV is shown as an example. The horizontal red line and reddish background zone display the value and its uncertainty, respectively, at 50 MeV. Blue circles joined by a broken line are the LV simulation as a function of at the same energy. The crossing of this broken line with the red line and the edges of the reddish zone give the most appropriate value for the factor of Eq. (7) and its uncertainty, respectively.
In the main panel of Fig. 3 we show, by the open circles and squares joined by dashed curves, the thus obtained values plotted against for the D1-G1 and Zamick EOS, respectively. Within experimental errors, the values for 20 MeV are roughly compatible with any value and are not reported. The LV model with the highly recommended nuclear interaction D1-G1 for the range of energies of the present study and with the empirical cross section predicts, for all energies studied, about twice higher values compared to those suggested by Fig. 10 of Ref. lopez14 (); these are presented in Fig. 3 as black filled circles, with the gray area showing their uncertainties. In contrast to this, when experimental and simulation uncertainties are accounted for, the zero-range (local) Skyrme interaction in the Zamick implementation is compatible with above 35 MeV. Let us underline that for both EOS display a minimum around . The minimum is relatively more pronounced than the one suggested by and it is somewhat shifted in energy. The Zamick EOS gives a that reduces the free at all while the D1-G1 EOS gives 1 at 80 MeV.
iii.3 Centrality versus multiplicity
The most evident difference between a simulation and an experimental data analysis is in the reliability of the assessments of reaction impact parameter . Experimental selection of the most central collisions is made by assuming that there is a biunivocal correspondence between the reaction violence, i.e., the multiplicity of particles in a reaction event, and the reaction centrality. In a simulation the centrality is an input variable, thus it is under full control. In comparing simulation results and the earlier INDRA study of and lehaut () it has been underlined that selecting events via multiplicity strongly mixes events of different impact parameters over a rather broad span in zhang (); bonnet (). Thus, let us examine the vs multiplicity relationship and its influence on the isotropy ratio. For that purpose we use the semidynamical general-purpose event-generating code HIPSE (Heavy-Ion Phase-Space Exploration) intended to describe HICs at intermediate energies hipse (). At each energy 100 000 events are generated in the range = 0 – 7 fm. Let us note that, according to the expression of in Ref. tripathi (), the above range in is equivalent to 0.27 – 0.30 , depending on . At = 50 MeV the simulation was performed in the full impact parameter range of the Xe + Sn reaction, i.e., = 0 – 13 fm, in order to verify that in the non-covered range ( = 7 – 13 fm) the high multiplicity events, in which we are interested, are not present. By passing the generated events through a sophisticated INDRA-device geometry and detection-acceptance filter paolo () we found that it has no appreciable effect on the = 1 values. Mostly, the change in due to this filter is below 0.5 %.
Selecting the range = 0 – 2 fm, in a geometrical sharp-cut approximation, corresponds to 1.82 % to 1.92 % of in the studied range, i.e., between 104 and 116 mb. These values fall in the middle of the cross section values of the selected subset of the most violent INDRA data events analyzed in Ref. lopez14 (). Ideally, the reactions with 2 fm should correspond to 8163 out of the total 100 000 generated events. In reality, there were on the average 8109 such events with a fluctuation up to 3 % from energy to energy. We denote this precise number of events to search for, in the full ensemble of 100 000 events, the subset of events with the highest multiplicity that is by number of events closest to . By we label both the lowest multiplicity of the thus selected subset as well as the subset of events itself at each .
Let us check the behavior of the most violent HIPSE events. As a kind of ”background”, in Fig. 4 we show by the thin black line the -distribution histogram of the full 100 000 event data set for each second studied . The distribution of the events is shown by the red-line yellow-filled histogram. These high-multiplicity events are generated in a large domain of values which extends up to 5 fm. To make the distribution better visible, it is enlarged to the full frame size by the hollow red-line histogram. A Gaussian fit to it clearly demonstrates that the normal-law of data statistics correctly reproduces the distribution of subset over ’s. These events are in minor part (3 % to 29 %) belonging to the 2 fm subset of the full data set (hatched part of black histogram). From the Gaussian fit one infers that the maximum of these high-multiplicity events is about 3 fm and that it slightly decreases with the increasing .
Finally, let us apply the HIPSE distribution to the LV simulation results. Taking the Gaussian fit values of Fig. 4 as the weights for the integer values of , the -averaged are obtained for each studied value of the factor of Eq. (7). By this method, for the = 1 case these are, in millibarn units, also equivalent to 0.02 . extracted from thus averaged is in Fig. 3 shown by dot-dashed curves and open triangles, upright and reversed, for the D1-G1 EOS and Zamick EOS, respectively. For the nonlocal D1-G1 EOS the two -averaging intervals give strictly the same for 50 MeV. At = 80 and 100 MeV the respective values differ by about 20 % but are mutually compatible when errors are accounted for. For the zero-range Zamick EOS in the full interval, two -averaging intervals give compatible predictions for the values although for the more stringent centrality results are in somewhat better agreement with the values.
Iv Summary and conclusions
The semiclassical Landau-Vlasov (LV) transport model was used to study the energy-based isotropy ratio of Eq. (5) for the Xe + Sn reaction in the wide incident energy range 12 100 MeV. The focus of the present work is twofold:
(1) the search for the set of model ingredients which most favorably describes the experimental values for the = 1 species of Ref. lopez14 () and
In approaching the above point 1 we investigated (i) the role of the dependence of the nuclear mean field on momentum, i.e., of the nonlocality of the interaction, and (ii) the impact of the residual interaction through varied parametrizations of . The success in reproducing the experimental isotropy ratios of Ref. lopez14 () is mixed: Below the Fermi energy , the LV model with the strongly in-medium reduced cross section of Refs. li93-4 (); klakow () and with the momentum-dependent D1-G1 EOS leads to a correct description of ; cf. Fig. 1(a). A similar result may be obtained with both the D1-G1 EOS and the zero-range Zamick EOS when the free cross section is strongly scaled down by a constant multiplicative factor of Eq. (7): for the D1-G1 EOS 0.5, Fig. 2(a), and for the Zamick EOS 0.1, Fig. 2(b). Above there is no unique set of model parameters which would lead to a favorable description of the experimental . Earlier studies of the observable su13 (); liu01 (); zhang (); kaur11 (); vin12 (); zhao () have also failed to reproduce the experimental results of Ref. lehaut (). We emphasize, however, that a smooth variation of the parameter that controls the in-medium value of would lead to a complete description of the experimental for both EOS. Should one draw a conclusion that none of the existing studies on the in-medium modifications of around the Fermi energy appropriately accounts for the physical reality? The local or the nonlocal character of the interaction does not elucidate this question.
Regarding the above point 2, we may summarize the outcome of our study as follows: the LV simulation with the local Zamick EOS predicts the cross-section correction factor which clearly supports the experimentally deduced lopez14 (). The model predicts
(1) an appreciable reduction of cross section all along the energy range of interest, as well as
(2) the appearance of a break in the slope of the multiplicative factor after a minimum located near .
However, the agreement or disagreement between the absolute values of and should be considered with some caution due to two possible causes. On one hand, the value of factor may be altered by reaction centrality. Accordingly, an investigation of with a quasidynamical event generator HIPSE hipse () was carried out. It reveals that the event selection based on multiplicity and the geometrical sharp-cut approximation is not a correct centrality selector. Indeed, corroborating earlier findings zhang (); bonnet (), we show that this selection approach strongly mixes events of different impact parameters over a rather broad span of values; cf. Fig. 4. When a properly weighted contribution of ’s involved in the high-multiplicity events is accounted for, the isotropy ratios calculated for the thus relaxed centrality requirement and those strictly central do not differ much. The thus extracted does not change much as well. On the other hand, the derived values are based on a number of strong assumptions that allowed the link between the stopping ratio and the in-medium cross section lopez14 (). Hence, besides further experimental and theoretical considerations of the stopping observable intended to disentangle the remaining ambiguities a study of other related observables may shed some fresh light on the subject. In addition, the experimentally observed strong and rapid change of the effective in-medium residual cross section beyond the Fermi energy urges for an ab initio theoretical analysis of this problem, the solution of which might lie in the way the exclusion principle is accounted for su13 () and/or by incorporating the recent observation of short-range correlations in nuclei subedi (); hen (). Their consequences for transport descriptions of heavy-ion reactions are of high interest and need to be investigated.
Acknowledgements.Z.B. gratefully acknowledges the financial support and the warm hospitality of the Faculté des Sciences of University of Nantes and the Laboratory SUBATECH, UMR 6457. This work has been supported in part by Croatian Science Foundation under Project No. 7194 and in part by the Scientific Center of Excellence for Advanced Materials and Sensors.
|26 35||26||mon ()||0||0||1||0.966||-0.426||1.372||9|
|35 45||35||ben ()||0.97||-0.426||2.372||0.32||0.35||-0.40||10|
|45 53||45||ben ()||1.29||-0.073||1.97||0.32||-0.127||-0.18||8|
|53 63||53||ben ()||1.61||-0.2||1.79||-0.04||0.51||0.16||10|
|63 73||63||ben (); sca (); kin ()||1.57||0.31||1.95||0.33||-0.59||-0.30||10|
|73 90||73||ben ()||1.9||-0.28||1.65||0.9||0.205||-0.16||17|
|90 130||90||sca (); chi ()||2.8||-0.075||1.49||1.2||-0.465||-0.094||40|
|130 319||129||mea ()||4.0||-0.54||1.396||-1.3||0.665||-0.81||189|
The angular dependence of the nucleon-nucleon () cross section is expressed as
where is the total elastic cross section due to Li and Machleidt li93-4 (). The dimensionless weighting factor is equal to unity for the scattering between neutrons ( 1) and is increasingly anisotropic as energy increases for neutron-proton scattering (the case), and especially becomes strongly forward-backward peaked for the scattering between protons (). The parametrization of the angular dependence of is defined as seb07 ()
where, for the purpose of the fitting, the coefficients at each energy are expressed by the following functional dependence:
|5 9.9||5||spp ()||5176.1||-8.91||100.0|
|9.9 19.7||9.9||ja1 ()||1795.6||-9.29||52.62|
|19.7 39.4||19.7||ja2 ()||1071.0||-12.0||24.95|
|39.4 68||39.4||joh ()||1382.2||-19.26||11.16|
|68 144||68||you ()||1880.5||-26.77||6.16|
Parameters , , and are fixed by fitting the experimental data at nine beam energies between 26 and 319 MeV, index running over energies. and are expressed in MeV units. Between these values the parameters are assumed to change linearly with . The values of these parameters are given in Table 1 and is shown in Fig. 5(a).
Adopting a very crude estimate, the polar angle dependence of is defined as seb07 ()
where coefficients are expressed by the following functional dependence:
Due to indistinguishability of particles, coefficients and are divided by 2. At each energy the limiting angle reads . The overall angular distribution normalization is given by the value of , and Eq. (11) is used to define, on Monte Carlo grounds, the angle into which a couple of charged Gaussians is scattered in a collision. The parameters are fixed by fitting experimental differential cross sections at six energies ranging from 5 to 144 MeV denoted by the index . As above, and are expressed in MeV units. Between these energies, parameters are assumed to change linearly with . The values of these parameters are given in Table 2 and is shown in Fig. 5(b).
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