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An event generator for inelastic proton-nucleus interactions at
intermediate energies
Persram, D.
https://publications-cnrc.canada.ca/fra/droits
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NRC Publications Record / Notice d'Archives des publications de CNRC:
https://nrc-publications.canada.ca/eng/view/object/?id=18719f1b-eab4-4864-94ee-500ffc2da7c6 https://publications-cnrc.canada.ca/fra/voir/objet/?id=18719f1b-eab4-4864-94ee-500ffc2da7c6
proton-nucleus interactions at intermediate
energies
Declan Persram
March, 2006
PIRS-1000
Ionizing Radiation Standards
Institute for National Measurement Standards
National Research Council
Ottawa, Ontario K1A 0R6 Canada
In this work, we seek to develop a numerically fast method of determining multiplic-ities and double-differential cross sections of secondaries produced in the intermediate energy inelastic interaction of protons with atomic nuclei. We accomplish this by cou-pling global parameterisations of the relevant quantities of interest derived from BUU simulations to the GEM2 evaporative decay model. Issues associated with the mean-field representation of the incident proton are addressed by treating the spectrum of nuclear excitation levels as an equa-probable continuum. In general, we find good agreement with the ICRU63 results for incident energies in excess of 100 MeV. Below this, it is likely that a more careful treatment of the nuclear excitation spectrum is necessary.
Contents
1. Introduction 1
2. BUU dynamical model 6
3. GEM statistical model 17
4. BUU parameterisations 19
5. Pre-equilibrium event generation 32
6. An alternative formulation 39
7. Results & Discussion 59
8. Conclusions 61
List of Figures
1 Nuclear matter optical potential . . . 8
2 Nuclear binding energy per nucleon . . . 9
3 Free space nucleon-nucleon elastic cross section . . . 11
4 Nucleon effective mass in nuclear matter . . . 12
5 Binary nucleon-nucleon collisional centre of mass energy . . . 13
6 Nuclear monopole and quadruple moments . . . 15
7 Nuclear charge and mass density profiles . . . 16
8 Time evolution of momentum-space quadruple moment . . . 20
9 Pre-equilibrium nucleon multiplicity . . . 22
10 Projectile survival probability . . . 23
11 Quasi-elastic angular deflection function . . . 25
12 Pre-compound angular deflection function . . . 27
13 Pre-compound nucleon kinetic energy spectra . . . 28
14 Parameterisation of the pre-compound kinetic energy spectra . . . 30
15 Impact parameter weighting . . . 33
16 BUU event-generator light particle multiplicity . . . 37
17 BUU event generator angle-integrated secondary neutron and proton kinetic energy spectra . . . 38
18 Alternate event generator angle-integrated secondary neutron and proton ki-netic energy spectra . . . 41
19 BUU and Kalbach angular deflection functions . . . 42
20 Secondary neutron double-differential cross section . . . 43
21 Secondary proton double-differential cross section . . . 44
22 Alternate event generator light particle multiplicity . . . 46
23 Alternate event generator angle-integrated secondary neutron and proton ki-netic energy spectra . . . 47
24 Impact parameter dependence of light particle multiplicities . . . 48
25 Universal generator neutron and proton multiplicities . . . 49
26 Universal generator composite particle charge multiplicity . . . 51 LIST OF FIGURES
27 Universal generator secondary neutron and proton angle-integrated kinetic energy spectra from 250 MeV incident protons . . . 52 28 Universal generator secondary neutron and proton angle-integrated kinetic
energy spectra from 200 MeV incident protons . . . 53 29 Universal generator secondary neutron and proton angle-integrated kinetic
energy spectra from 100 MeV incident protons . . . 54 30 Universal generator secondary neutron and proton angle-integrated kinetic
energy spectra from 80 MeV incident protons . . . 55 31 Universal generator secondary neutron and proton angle-integrated kinetic
energy spectra from 50 MeV incident protons . . . 56 32 Universal generator secondary neutron and proton angle-integrated kinetic
energy spectra from 30 MeV incident protons . . . 57 33 Isobar production from protons incident on Al at 180 MeV . . . 58
1
.
Introduction
The study of the interaction of energetic protons with matter brings with it, besides basic scientific interest, many physics-based applications. These applications range from space ra-diation protection (of both human occupants1, 2and electronic devices3), residual isotope
pro-duction from solar cosmic rays,4 cancer therapy via the irradiation of tumors,5, 6 to spallation and accelerator driven transmutation for sub-critical reactor assemblies and transmutation of radioactive waste.7, 8 In these systems, one must address the interaction of intermediate
energy protons with condensed matter. The physics encompassed in these includes, but is not limited to, electromagnetic and nuclear physics. For the latter, one requires information regarding the interaction of protons with atomic nuclei. In this report, we aim to study the intermediate energy inelastic interaction of protons with nuclei in an attempt to produce a
fast event generator algorithm. We note that the term intermediate in this work is meant to
imply energies above both single-particle and collective nuclear excitations (of several MeV) and below the inelastic nucleon-nucleon scattering threshold of several hundreds of MeV. We do not attempt to calculate the full reaction cross section,9–11 but the production of light
secondaries (n, p, d, t, 3He, α) once an inelastic interaction has been initiated.
It is well-known that the many-body strong interaction contains many complexities12, 13
that to this day are not fully understood. In addition, the added complexities inherent in the application of strong interaction physics to finite nuclear systems, with both its single-particle and collective degrees of freedom, are clear. Therefore, much of the research in this area is concentrated on phenomenology and semi-classical results obtained in partially relativistic theories in an attempt to delineate gross features of interacting nuclear systems. Semi-classical theories are attractive due to their relative computational ease as compared with more detailed physical theories. We adopt this modus operandi here.
The many possible outcomes of a collision of an intermediate energy proton with a nucleus include absorption, few nucleon expulsion, and fragmentation.14 The dynamics of
such reactions range from potential dominated effects at low energy where the Pauli exclu-sion principle forbids intra-nuclear colliexclu-sions between the constituent nucleons, to colliexclu-sion dominated dynamics where potential effects are reduced. The transition between these two intermediate extremes is expected to occur near the fermi energy (ǫf ∼37 MeV). This has
been partially resolved through the comparison of various models, both theoretical and com-putational, with observables such as direct and elliptic flow, and linear momentum transfer
in heavy ion collisions.15 The importance of the so-called direct inelastic interaction (that
is, scattering from a nucleon within the nucleus without the formation of a compound16
nucleus) in the collision of protons with nuclei has been highlighted in intra-nuclear cascade codes.17, 18 In addition, consideration of the nuclear level spacing19 (∼1 MeV) in
compari-son to the proton kinetic energy implies that many nuclear excitation levels can be active during a collision (direct or not) resulting in many available final-state channels. While this tends to be problematic for a fundamental theory describing the dynamics, it also lends to semi-classical interpretations which are almost always considerably more tractable, both theoretically and computationally.
Early studies of nucleon induced nuclear reactions reactions focused mainly on evap-oration from excited nuclei.20, 21 Here it was assumed that the nuclear decay channels are
independent of the formation process,22 with statistical processes governing the dynamics.
While the predictive capabilities of such models and their modern counterparts23–25 are far
reaching in the low energy portion of the spectrum of emitted nucleons, it has been shown that such models are unable to reproduce the high-energy tail observed in some reactions.26
Theories which attempted to describe the nuclear dynamics in a cascade picture were par-tially successful at resolving the high-energy part of the spectrum, but were less successful elsewhere. Here the nucleon dynamics are treated in a sequential, binary intra-nucleus nucleon-nucleon scattering picture.27 These theories often employ no -or a static- nuclear
potential and a low-energy (< ǫf) cascade cutoff.28, 29 Typically, at low bombarding
ener-gies, potential effects are important and at high energies cascade effects are important. The need for binary collision based theories has been shown to be necessary to describe reactions down to lab bombarding energies of 100 MeV.30 At higher energies (180 MeV), it has been
demonstrated that intra-nuclear cascade (INC) codes alone were insufficient at describing the dynamics.31 Thus, at the energies of interest in this work, both cascade and nuclear-potential
effects are important.
In general, our current understanding of the reaction mechanisms points to the need for hybrid approaches entailing both equilibrium and pre-equilibrium effects where surface and volume emission of nucleons (and composites) are obtained from both INC and statistically driven models. Timescale separation of these physical processes is clear, where the surface effects are expected to occur on nuclear transit times of τ ∼ 10−23s and volume effects are
expected to occur on timescales of τ ∼ 10−16→ 10−19s. Typically, but not exclusively, one
processes.
A often-used model which attempts to include in some way aspects contained in both the fast and slow component is the exciton model,32–34which considers the excitation of
particle-hole pairs and transports them through the nuclear medium. Parameter tuning utilising the nuclear optical potential has been used in some recent exciton model calculations.34 Exciton
model studies indicate the importance of surface and collective effects,35 and attempts to
in-clude them in some manner or other are popular. Often, the models used to describe the fast stage of the reaction are able to give multiplicities and kinetic energy spectra only. Here, one typically resorts to parameterised angular distributions36 based on phenomenology, which
have recently been supported by theoretical arguments.37 Although a theory encapsulating
more fundamental physical principles such as the quantum mechanical model of Feshbach, Kerman, and Koonin (FKK38) are able to produce an angular distribution of emitted
sec-ondaries, these model are quite complex and accordingly, require substantial computational resources.
Models employing various theoretical inputs often come with a myriad of switches and options. Often, it difficult to isolate various effects. Obtaining reliable global parame-ter sets is then difficult. For instance, recently, a nucleon-nucleus event generator used in the GEANT439 framework was tested for its ability to reproduce experimentally measured
quantities. This model has a low energy limit of 100 MeV and takes cross sections from GHEISHA40 (previously a calorimeter simulator). Problems with conservation laws and in
the inability to reproduce the so-called quasi-elastic peak41, 42 in addition to difficulties in
ob-taining a global parameter set were encountered.39 Additionally, recent comparisons between
MCNPX,43 Fluka,44 and TALYS45 indicated that Fluka under-predicts neutron multiplicity
and TALYS over-predicts proton multiplicity. Improved single-nucleon multiplicities have been obtained with MCNPX using the INCL446 cross sections, however neither the shape
nor magnitude of the composites were correctly predicted. MCNPX simulations utilising the cross sections calculated with GNASH47 (in place of INCL4) did not provide satisfactory
results. Interestingly, it is the GNASH code that has been used to produce tabulated stan-dards for the International Commission on Radiological Units and Measurements (ICRU) reaction cross sections.48 We note that within the seemingly bewildering sets of models
and parameter spaces (hybrid and other), some codes seem to consistently produce reliable results, such as CEM2k49 (which is a polygamous combination of INC, exciton, and
benefit from some parameter tuning. Recent studies with this code also indicate differences between various evaporation and fragmentation algorithms25, 50, 51 that have yet to have been
experimentally verified.52 Another model based on the partitioning of phase space53, 54 looks
promising but is relatively young and could stand some more benchmarking.
A microscopic approach often used in simulations of heavy ion collisions is the Quantum Molecular Dynamics (QMD) model.55, 56 Here, the nucleons are evolved by calculating the
pair-wise force each nucleon experiences due to the presence of all other nucleons. Such a model was implemented for pre-equilibrium studies of nucleon-induced nuclear reactions from 100 MeV→3 GeV.57 The implementation in that work gives good nuclear ground state
properties and reasonable energy conservation. The evaporative part of the calculation was handled by the GEM2 evaporation code. Favourable results were obtained. It was confirmed within this model that high energy secondaries originate predominantly from direct surface interactions while the low energy secondaries originated from processes consistent with evaporation from an excited nucleus. Other QMD calculations incorporating low energy corrections can be found in the literature. Typically, the potentials used in these codes include density, symmetry, coulomb and Pauli blocking effects.58–60
In general, except for the division between the high and low energy part of the spectrum associated with direct and compound processes respectively, it is often the case that the many ingredients in the nuclear interaction process are difficult to disentangle, and often difficult to implement. We address this here by implementing a model which treats the pre-equilibrium phase of the (proton-nucleus) reaction in a microscopic semi-classical approach that includes contributions from density dependent nuclear and isospin terms, Coulomb and surface contributions, and a momentum-dependent nuclear term derived in the Hartree-Fock approximation as well as an in-medium nucleon-nucleon cross section consistent with this momentum dependence. The last two ingredients are essential in terms of understanding reaction dynamics in heavy ion collisions61, 62 and, as such, we include them in this study. The
model we use here for the pre-equilibrium dynamics goes by the name Boltzmann-Uehling-Uhlenbeck (BUU63, 64) and is described in some detail in section 2.. In short, the BUU model
treats the nuclear potential just discussed in a mean field approximation and includes a nucleon-nucleon scattering term corrected for in-medium effects. This is calculated in a fully self-consistent manner. In addition, the transition between potential and hard-scattering dominated dynamics and collective effects are handled naturally in this theory. Due to the large separation of timescales inherent in this problem we confine the BUU results to the
pre-equilibrium phase of the reaction. To describe evaporation from an equilibrated source, we use the the GEM2 decay code. If successful, this method of modeling the proton-nucleus reaction should be easily extended to handle incident neutrons and composite projectiles of energies in excess of those of interest here assuming the proper provisions to handle particle production are accounted for. Note that complex particles in the approach we use here can only be produced through evaporative channels. Dynamical production of composites (deuterons) is possible in BUU-type calculations,65 but we do not implement this here.
Ultimately, we wish to use the pre-equilibrium predictions of the BUU model such that one can quickly obtain the results of a proton-nucleus inelastic interaction without the need to resort to the full (and lengthy) numerical computation. To this end, we seek to parameterise the multiplicity, kinetic energy, and angular distributions of free nucleons as predicted by the pre-equilibrium part of the BUU simulation.
In section 2. we describe the BUU model in some detail. A short description of the GEM2 evaporation code is presented in section 3.. The reader is encouraged to review the supplied references as a full description of the evaporation code is beyond the scope of this work. Section 4. is devoted to a detailed discussion of the BUU parameterisation procedure. In section 5. some results from the combined BUU+GEM2 evaporation scheme is compared to recent results from the ICRU6348report. The data from this report was generated from a
combination of both theoretical models and experimental data. The parameter sets there-in have often been adjusted to match existing experimental data. However, due to a lack of the latter for many entrance channels, this tuning is not done for all systems examined in that report. In the interest of brevity and simplicity, we benchmark all results obtained here with the ICRU63 results only. In section 6. we address some of the shortcomings of the model developed here and offer some alternatives appropriate for reproducing the (ICRU63) data. Finally, we provide a short discussion and conclusion in section 7. and section 8. respectively.
2
.
BUU dynamical model
In this work we adopt a microscopic model of the proton-nucleus interaction that includes aspects of the interacting system that have been shown to be important. For instance, attempts at modeling the dynamics in these systems have indicated the importance of the nuclear surface.27, 31, 32, 66 It has also been demonstrated that the kinetic energy spectra of
emitted nucleons show distinct volume and surface contributions.67 In addition, low-lying
collective nuclear excitations35 have been shown to have noticeable signatures in the final
emission spectra obtained with the exciton model.66 It is also clear that Coulomb effects
will play a role. Experimental evidence from nucleus-nucleus collisions strongly favours a nuclear potential that depends on both density and momentum.68 Theoretical arguments
also favour these dependencies. Inclusion of an isospin dependence of the nuclear potential is also desirable, unfortunately there are many uncertainties as to its functional form. These range from surface effects,69 density dependent effects,70 and more recently, to
momentum-dependent effects.71, 72 Ideally, one would adopt a nuclear potential that contains terms
corresponding to all of the above, that is, density, momentum, Coulomb, surface, and isospin dependencies. We adopt this philosophy here. We note that due to the largely unknown behaviour of the isospin potential in interacting nuclear systems, we utilise an often used and computationally simple parameterisation in terms of neutron and proton densities.73 In
addition to the effects arising from the nuclear many-body potentials, individual nucleons can interact with each other via elastic and inelastic two-body (three- and higher-body processes are possible, but unlikely) scattering processes. To address this, one often uses the free-space nucleon-nucleon cross section. However, these collisions are complicated by many-body effects and generally lead to an in-medium cross section that is different from that in free-space.74 It is thus desirable to include such effects in a model of proton-nucleus
interactions. Utilisation of a nuclear potential with momentum-dependent terms augmented with phase-space and flux arguments naturally leads to an in-medium cross section which is in general smaller than that of the free-space value.74 We utilise an in-medium cross section
consistent with the momentum dependence of the nuclear optical potential in this work. To the author’s knowledge, no attempts at modeling the proton-nucleus interaction that simultaneously includes all of the above ingredients exist in the literature. To this end, we adopt the Boltzmann-Uehling-Uhlenbeck (BUU) microscopic transport model which includes both intranuclear nucleon-nucleon collisions and mean field effects derived from the nuclear potential. Furthermore, we adopt the Lattice Hamiltonian method75 (suitably
augmented to handle momentum-dependent potentials76) to evolve the mean field. For
completeness, we show below the formal structure contained within this model. In Lattice Hamiltonian simulations for nuclear systems, one typically assigns a collection (in this work we use Nens=100) of “test-particles” to each nucleon. The positions and momenta of all
test-particles are evolved in fixed time steps according to Hamilton’s equations of motion, where the Hamiltonian is calculated on a configuration space cubic lattice of spacing δx. At the time-step nodes, one allows for individual nucleon-nucleon collisions according to the in-medium cross section. The Pauli exclusion principle is simulated by including a statistical blocking factor dependent upon the phase-space distribution at a given time. In this work, for all nuclei studied, we obtain a 96% blocking efficiency in simulations of numerically cold isolated nuclei. The position and momentum of each test-particle “i” are determined by:
˙
ri = +∇piH p˙i = −∇riH (1) where, for a system of A nucleons, H is the total Hamiltonian:
H =XA×Nens i q p2 i + m2− m + Nens(δx)3 X α Vα. (2)
Here, Vα is the total potential energy density (per real nuclear system) at lattice site α.
In this work, Vα includes nuclear density (Skyrme77–79) and momentum dependent80 terms,
surface,75Coulomb, and isospin73terms. With the single particle phase space density f (r, p),
the corresponding single particle potential uα ≡ δVα/δf (r, p) reads:
uα(pi, τi) = A ρα ρ0 ! + B ρα ρ0 !σ +2C ρ0 X j6=i R (rα− rj) 1 +pi−pj Λ 2 (3) + As ρ2/30 ∇ 2 ρα ρ0 ! + (τi+ 1/2) 4πǫ0 (δx)3 X β6=α ρ(p) α |rα− rβ| + τi 2D ρ0 ρ(n) α − ρ(p)α .
In the above, ρα = ρ(n)α + ρ(p)α and ρα =PiR (rα− ri) is the nucleon density at lattice site α.
The index τiassumes the value of 1/2 (-1/2) for protons (neutrons). The R (rα− rj) function
in the above assigns a finite configuration space width to each test-particle. It has been determined that a spline of order three provides for good energy and momentum conservation in simulations of colliding nuclei.76 This is also true of particle-hydrodynamic codes such
terms from equation 3 are removed. The test particles are assumed to have well-defined momenta. The five parameters (A, B, σ, C, Λ) are fixed to reproduce the experimentally extracted optical potential for nuclear matter as shown in figure 1, nuclear matter binding energy per nucleon of 16 MeV at saturation density ρ0, equilibrium at ρ0, and a nuclear
incompressibility82 of K=210 MeV. Note that for nuclear matter, the surface, Coulomb, and
isospin terms vanish. In this work, values of Etot from figure 1 of concern to us range from
-50 MeV→250 MeV. For the strength of the isospin potential, we adopt the often-used73, 83
value of D=+32 MeV. In order to obtain the strength of the surface interaction, we require a reasonable fit to the binding energy per nucleon in finite nuclei as shown in figure 2. We obtain this fit with As =-18 MeV. Note that with this potential we are well able to describe
the binding energy of stable and neutron deficient nuclei. Neutron rich nuclei are over-bound as shown in figure 2.
Figure 1: Nuclear matter optical potential as a function of total energy taken from nucleon-nucleus scattering data. Open (solid) circles are for proton (neutron) beams incident on a variety of heavy nuclei ( 28Si, 40Ca, 48Ca, 58Ni, 90Zr and 208Pb ),84 solid diamonds are for
p+( 40Ca, 48Ca, 58Ni and 90Zr ),85 and open squares are for p+ 40Ca.86 The potential used
in this work corresponds to the dark dashed line and was calculated at zero temperature and saturation density ρ0.
Figure 2: Binding energy per nucleon in finite nuclear matter. We adopt a finite range potential with As= −18 MeV in this work. In the left panel, the Weizs¨acker mass formula87
and Audi & Wapstra mass tables88 are shown by the shaded line and black solid circles
respectively. The lines in the right panel are from the mass tables.88 The BUU results (open
So far, the connection with the Boltzmann equation has remained hidden. It is revealed in the lattice spacing limit δx → 0 in the Lattice Hamiltonian method. Here, the single particle phase space density satisfies the integro partial-differential BUU transport equation:
∂f (r, p, t) ∂t + ∇ph · ∇rf (r, p, t) − ∇rh · ∇pf (r, p, t) (4) = Z dp1dΩ1 vrel∗ × dσ∗ dΩ ! f′f1′f ¯¯f1− f f1f¯′f¯1′ , where h = t + u is the single particle Hamiltonian (h ≡ δH/δf (r, p)), and:
fi ≡ f (r, pi, t) (5)
fi′ ≡ f (r, p′i, t)
¯
fi ≡ 1 − f (r, pi, t).
The collision integral89 on the right hand side of the BUU equation is solved in the parallel
ensembles technique utilising a statistical Pauli blocking function as already discussed.64
In the above, we introduce the in-medium relative velocity v∗
rel and in-medium
nucleon-nucleon elastic scattering cross section σ∗. At zero temperature (and equilibrium), the latter
reads:74, 76 σ∗ = m∗ m 2 σ, (6)
where the free space cross section σ is obtained from a parameterisation90 as shown in
figure 3. The effective mass m∗ is defined as:
v∗ rel= p m∗ = p m + ∇pu, (7)
where u is given by equation 3. We note that during the course of a nucleon-nucleus in-teraction, the nucleon effective mass will vary with momentum, density, and temperature. Figure 4 illustrates these dependencies. At the energies of concern in this work, we do not expect a large departure from T=0 and ρ=ρ0. Thus, (the square root of the) values of
the ratio of the in-medium to free-space cross section do not stray too far from the lowest curve in the left panel of figure 4. Note that as expected, in the high temperature, high momentum, and low density limits the in-medium cross section approaches the free-space value. Attention should be given to the fact that for the energies of interest in the work (that is, near ǫf), the momentum dependence of the mean field potential is steepest. Thus,
the in-medium cross section is varying most rapidly here. Also note that recent detailed Dirac-Brueckner-Hartree-Fock calculations91 show excellent agreement with the σ∗ used in
this work near ρ0 and for e(0)k : 20 → 200 MeV. This roughly corresponds to k : 1.5 → 4.8 in
figure 4. At the energies of concern in this work, use of the elastic nucleon-nucleon
scatter-Figure 3: Parameterisation of the free-space elastic nucleon-nucleon scattering cross sec-tion.90 In this work, a maximum 150 mb cut-off is employed.
ing cross section is justified. This is demonstrated in figure 5, where the average centre of mass energies obtained from unblocked collisions of nucleons inside the nucleus is plotted as a function of time. As the figure indicates, we do not expect inelastic collisions to play a role until e(0)k ∼500 MeV. The nuclear dynamics as realised in the BUU approach solve both the nuclear potential and in-medium cross section self-consistently. In addition, collective effects are a natural consequence of this (BUU) model. In general, mean field effects dominate below ǫf with collisions becoming increasingly important above ǫf.
Figure 4: Nucleon effective mass in nuclear matter as a function of momentum at saturation density ρ0 = 0.16fm−3 (left panel) and of density at the fermi surface kf = 1.33fm−1 (right
panel) at zero and finite temperature for an equilibrium nuclear matter distribution.62
In order to simulate proton-nucleus interactions, the nuclei generated in the BUU model should be stable over timescales consistent with pre-equilibrium dynamics (t∼300 fm/c). Typically, for initial conditions, one chooses a stationary solution of the BUU equation. The inclusion of momentum-dependence in the nuclear potential, however, considerably complicates this scheme.92 Furthermore, inclusion of the finite-range surface term can lead
to large and unphysical density oscillations at the centre of the nucleus.93 In order to alleviate
this problem, the initial nuclear profiles are taken to be step functions in coordinate space (of radii R = R0A1/3, with R0=1.14 fm). These profiles are then evolved in time while
slowly ramping up the surface potential from zero to As in equation 3. After this time
(∼ 50 → 100 fm/c), any expelled test-particles are re-sampled back into the nucleus. The reassigned positions are determined from the density profile thus generated. The momenta are reassigned using a local Thomas-Fermi procedure. Stability of these nuclei are then satisfactory as displayed in figure 6. It is only after this time that the binding energies as displayed in figure 2 are calculated. In figure 7 we show the nuclear profiles thus generated. These initial conditions are similar to recent QMD spallation simulations.94 We note that the
Figure 5: Centre of mass collision energies from unblocked binary nucleon-nucleon collisions as a function of time from BUU simulations of p +16O collisions at laboratory incident
energies of 50, 100, 200, and 500 MeV. The horizontal dotted line indicates the π production threshold.
density oscillations observed after the resampling procedure (c.f. top panel of figure 6) result from the temporary disruption of the position-momentum correlations generated during the ramp-up procedure. These oscillations are small however, and we do not believe they result in serious problems.
Finally we note some characteristics of the mean field description of the proton. Consider an isolated proton as simulated by the model presented here. According to equations 1 each test particle making up the proton responds to the mean field generated by all the other (pro-ton) test-particles. The result is a non-zero average internal kinetic energy (< ǫ >∼5 MeV). Upon Lorentz boosting to the desired laboratory bombarding energy, the ensuing momentum spread (which increases with the boost energy) probes unphysical regions of phase space (if we ignore the fact that the proton is a composite particle, this momentum spread should be a delta function at the input beam momentum). We will return to this point in section 4. This is also discussed in terms of the Li`ege intranuclear cascade code.46 Note that this
effect is specific to single nucleon projectiles (neutron and proton). For composite projec-tiles, the internal kinetic energy generated by the BUU model is in line with theoretical and experimental expectations for A >4.
Figure 6: Coordinate space r.m.s. radii and coordinate and momentum space quadruple moments for isolated light and medium weight nuclei obtained with the BUU model. Shown are the averages of 4,000 ensembles after the ramp-up procedure as described in the text.
Figure 7: Nuclear charge (red and magenta) and matter (black and magenta) density profiles averaged over 4,000 ensembles and from t : 0 → 300 fm/c obtained with the BUU model. The magenta curve in the lower right panel shows the mean field version of a proton. The solid dashed curves are experimentally measured charge densities.95
3
.
GEM statistical model
The BUU model as described in the previous section is used to characterise the dynamical behaviour of a proton-nucleus interaction up to near equilibrium configurations. Excited nuclear fragments at this stage are then handled by an evaporation code. Here we adopt the Generalised Evaporation Model25 (GEM2) to describe the sequential emission of nucleons and composites from low excitation energy pre-fragments. By pre-fragment we mean to describe near equilibrium excited nuclei that have not yet undergone an evaporative stage.
The GEM2 is based on earlier works20, 21 (“Weisskopf-Ewing”) that considered the
sta-tistical evaporation of nucleons from excited nuclei. In this work, where the pre-fragment excitation energy ǫ∗ is much greater than the nuclear level spacing, a large number of active
and overlapping energy levels results. The Weisskopf-Ewing evaporation scheme uses de-tailed balance arguments to describe the evaporation process. This invokes the need for an inverse cross section. For example, for neutron evaporation (of energy ǫk) from a nucleusAZX,
the inverse cross section for neutron absorption on A−1Z X at ǫk is required. Early attempts
used the black sphere approach where the inverse cross section is taken to be the geometrical cross section. Clearly, this is an oversimplification if one, for instance, considers Coulomb effects on protons. In addition, it was pointed out that the inverse cross section on A−1Z X should really be for A−1Z X∗, where the latter is an excited state of A−1
Z X.26 The
determi-nation of the inverse cross section remains problematic to this day. Another ingredient in this model is the nuclear level density. This last quantity also remains under intense debate and its value differs significantly from work to work.26, 96, 97 This is especially true when one
considers excited nuclei. The GEM2 model addresses some of these failings by including im-proved inverse cross sections for nucleon26 and composite projectile98 reactions. In addition,
provisions are made for evaporation of 66 different types of ejectiles (from single nucleons up to 24Mg) rather than just nucleons as considered in previous works. Nuclear masses
and binding energies are taken from recent tabulations.88 An improved level density99, 100
(“GCCI”) which includes pairing corrections and energy dependent spacing is utilised in this model. Combined with an INC scheme, GEM2 has shown reasonable success in isotope production in proton induced reactions on a variety of targets {O, Al, Fe, Nb , and Ag} at incident bombarding energies from 10 MeV→3 GeV.25
There are several switches in the GEM2 code that allow for different level densities, inverse cross sections, and numerical approximations. As an exhaustive study of the
ef-fects of these is beyond the scope of this work. We adopt the default parameter set which considers light ejectiles only (n, p, d, t,3He, α), uses the GCCI level density parameters, and
the Dostrovsky26 and Matsuse98 inverse cross sections for light and complex ejectiles
respec-tively. Note that fission processes, although built-in to the GEM2 evaporation scheme, are not relevant at the energies of interest in this work.
4
.
BUU parameterisations
As discussed in section 1., the main goal here is to obtain parameterisations of specific (pre-equilibrium) quantities from full BUU simulations of p+X interactions in order to be able to quickly reproduce the results of such runs for any impact parameter and any energy (within the range of interest in this report) without the need to resort to running the full simulation each time. That is, we wish to obtain global parameterisations of the nucleon multiplicity, kinetic energy, and angular deflection distributions in terms of the incident bombarding energy, impact parameter, and target charge and mass. We repeat that the use of the BUU model is limited to the pre-equilibrium stage of the reaction. A necessary condition for equilibrium is the vanishing of the momentum-space quadruple moment Qzz(p).
In microscopic transport simulations of nuclear collisions, one typically uses this quantity to determine the point in time when pre-equilibrium processes cease.46, 101 It is often the case
that other observables, such as multiplicities and linear momentum transfer also saturate with the vanishing of Qzz(p).46 We follow this line of thought and use the Qzz(p) to determine
when equilibrium has been reached. The BUU simulation is stopped at this point. We illustrate this in figure 8 for protons incident on a variety of light- and medium-weight targets at energies within the region of interest. Once Qzz(p) has dropped to the “background” level,
we assume equilibrium has been reached. Further emission is to be handled by the GEM2 evaporation code. We note that previously, configuration space separation of fragments (produced in a QMD simulation for example) have been used to determine the cessation of dynamical effects.102 Figure 8 clearly indicates that the stopping time increases with
decreasing incident energy. This increase is particularly problematic for energies below ǫf
where large amplitude oscillations in momentum (and configuration) space are initiated. At these energies, due to the Pauli blocking effect, the dissipative nature of the in-medium nucleon-nucleon collisions is inaccessible. Unfortunately, there is no way around this, and we are thus forced to run the BUU simulation for long times. The longest simulation time in this work is 300 fm/c for the p +63Cu system at e(0)
k =12 MeV at all impact parameters. For
the panels with e(0)k > ǫf in that same figure, the amplitude of Qzz(p) beyond the stopping
time does not exceed that of the background.
For the four systems displayed in figure 8, BUU simulations were performed at 13 impact parameters (with a ˆb2 weighting such that each impact parameter contributes equally to the
density-weighted geometrical cross section) and 11 incident energies of 12, 18, 25, 37, 50, 75, 100, 125, 150, 200, 250 MeV. For the lightest (heaviest) system, ∼4,000 (∼500) ensembles
Figure 8: Time evolution of the momentum space quadruple moment for protons incident on16O,27Al,40Ca, and63Cu at laboratory bombarding energies of 25, 50, 100, and 200 MeV
obtained with the BUU model (red). The blue points shown the evolution of Qzz(p) for an
isolated nucleus also obtained in the BUU approach. The Qzz(p) is calculated for bound
were generated per incident energy and impact parameter. The final state consisted of an excited near-target-mass pre-fragment, and a collection of free neutron and proton test-particles. Any test-particles belonging to the pre-fragment were removed from the analysis (from here-on, when we speak of the final-state, we refer to the free nucleons only, that is, nucleons with a positive single particle energy ǫ = ǫk+ u > 0). The free nucleons were
then boosted to the lab frame where the rest of the analysis was performed. In addition to the above simulations, runs were performed with each isolated target up to the maximum simulation time (for the lowest incident energy). This was necessary in order to address numerical evaporation which is unavoidable in a simulation of this type. The numerical evaporation was observed to have a characteristic temperature of 1.5→3.0 MeV for both neutrons and protons and remained roughly constant over the timescales investigated in this work. The multiplicity and kinetic energy spectra from these isolated nuclei were treated as background contamination and were subtracted from the collision simulations.
Nucleon multiplicity, kinetic energy, and angular deflection distributions of the free nucleons were then calculated for each combination of target, incident energy, and impact parameter. Before discussing the global parameterisations thus produced, we digress to examine some qualitative features of the final state configuration. In general, the latter consisted of a near-beam-rapidity (forward peaked) proton component. From here-on we refer to this as the quasi-elastic component. The remaining pre-equilibrium nucleons were found to be distributed in energy space from near-zero to near beam energies, located, in general, at large scattering angles. From here-on we refer to this component as the pre-compound component. Separation of these two components was achieved by classifying the nucleons according to the number of binary nucleon-nucleon collisions experienced during the reaction. That is, the quasi-elastic peak contained nucleons that suffered no collisions, while the pre-compound nucleons suffered at least one collision.
Total pre-equilibrium nucleon multiplicity was calculated by subtracting the background as described above. Various fitting functions were then applied to produce a smooth surface function for each target nucleus. We show the surface thus generated for p +16O collisions
in figure 9. As expected, the largest multiplicity was obtained for the most central collisions at the highest energy. This trend was also observed in QMD simulations.103 Furthermore,
we note that above ∼ ǫf the free nucleon multiplicity was at least unity. This is consistent
with results obtained with the hybrid and geometry-dependent model.104 In addition to
Figure 9: Surface parameterisation of the pre-equilibrium nucleon multiplicity for p +16O collisions as a function of incident energy e(0)
k and normalised impact parameter
b/bmax. This figure was generated via interpolation of BUU simulations performed at
up in the quasi-elastic component. The surface thus generated for the p +16O system is
displayed in figure 10. Here we note that the survival probability of the quasi-elastic peak
Figure 10: Same as figure 9 but for quasi-elastic proton multiplicity only.
increases with impact parameter. We also note that for ˆb ∼0.6→0.7, the projectile survival probability is nearly independent of the incident energy. The exception is at the lowest energy investigated where it is expected that fusion of the projectile and target not unlikely. At the other extreme, we see that for high-energy central collisions, the projectile essentially buries itself (along with its kinetic energy) in the target, resulting in a relatively large
pre-equilibrium nucleon multiplicity as shown in figure 9. With these two surfaces we are able to quickly look-up the total pre-equilibrium nucleon multiplicity and fraction of nucleons which belong to the quasi-elastic protons for all impact parameters and for all incident energies between 12 and 250 MeV for each of the targets studied thus far.
We now consider the angular deflection of the quasi-elastic and pre-compound nucle-ons. As the angular distribution of the background nucleons is isotropic, the method of background subtraction is omitted from this part of the analysis. We first consider the quasi-elastic contribution. As the goal of this work is to obtain global parameterisations we seek a relatively simple parameterisation that encapsulates the gross features of the angular distribution. To this end, it was determined that a Gaussian distribution in θlab was
appro-priate. Thus, two parameters were required to characterise the angular deflection: centroid µ and width σ. In figure 11 these two parameters are displayed for two impact parameter slices through the surface parameterisation, one for semi-central and one for peripheral impact pa-rameters. The angular deflection in this plot is investigated for light- and medium-weight nuclei. Looking at the top panel of that figure, it is clear that for semi-central and peripheral impact parameters the quasi-elastic peak is strongly forward peaked at high energy. As the energy decreases, particularly for the peripheral impact parameter, the quasi-elastic peak experiences a large angular deflection. This is consistent with the partial-orbiting of the projectile-target system. Note that the deflection angles in that figure represent negative
angle scattering. As we assume axial symmetry, the sign of this scattering angle is
irrele-vant. In the bottom panel of figure 11 we found that at high energies for semi-central and peripheral impact parameters the angular deflection of the quasi-elastic peak is strongly fo-cused. As the incident bombarding energy decreases, the angular deflection peak becomes much broader. This is attributed to the longer interaction times of the proton-target system where the incident projectile spends more time in the nuclear mean field, thus populating a larger region of phase space and obtaining a stronger signature of the momentum distribu-tion of the nucleons inside the nucleus in the process. We note that ǫf serves to delineate
gross features of the angular deflection.
For the pre-compound nucleons, it was determined that a Gaussian parameterisation was also useful to globally characterise the angular deflection of the pre-compound proton component. We note that the neutron angular deflection (not shown) was problematic and in the interest of choosing a simple parameterisation, a linear fit function was used. In figure 12 we display the Gaussian parameters for the same surface slices as shown in figure 11. The
Figure 11: Centroid (top) and width (bottom) of the Gaussian parameterisation of the quasi-elastic proton angular deflection peak. Shown are the results for light (16O) and medium
general character of the quasi-elastic angular deflection is reproduced here except that the magnitudes are larger. For instance, as indicated in the top panel of figure 12 it is evident that at high incident energies, the pre-compound protons emerge at relatively small angles as compared to that at low incident energy. With decreasing incident energy, the mean scattering angle approaches 90o for both semi-central and peripheral impact parameters.
The width of the pre-compound protons displays the same behaviour as the quasi-elastic protons. Presumably, this too is attributable to the increased time spent in the nuclear mean field at low incident energies. We note that both the mean scattering angle and the angular spread is greater for pre-compound protons than for quasi-elastic protons. We also note that the slight increase of the mean scattering angle with energy for the semi-central p +16O system is an artefact of the surface interpolation procedure. This increase (of ∼2o)
is not considered to be significant. Similar to the quasi-elastic proton, ǫf serves to delineate
gross features of the pre-compound proton angular distribution.
The analysis of the pre-equilibrium nucleon kinetic energy distribution was also sepa-rated into quasi-elastic and pre-compound components. For the latter, and in the interest of counting statistics, we do not differentiate between neutrons and protons (see text below). We first consider the pre-compound nucleons. Here, the nucleons are observed to take on kinetic energies up to near beam energies. As the background extends up to ∼ 3
5ǫf,
subtrac-tion of this quantity is essential in order to obtain clean kinetic energy spectra. In figure 13 we show the raw (red), background (blue) and corrected (black) kinetic energy distribution obtained at a single impact parameter for two incident energies and two targets. We note that, as already stated, the background exhibits an exponential shape characteristic of evap-oration from a hot source. Once the background has been subtracted, fits are performed on the corrected energy spectra.
Characterisation of the pre-compound nucleon kinetic energy distribution from collisions involving nuclei remains an open problem. The contributions from surface and volume emis-sion overlap and can have quite different shapes.67, 105 Typically one fits the pre-compound
kinetic energy spectra with a modified exponential in order to reflect the partially evapo-rative nature of the spectra. In this work, global fits to the pre-compound nucleon spectra were achieved by using a sum of two exponentials with a pre-exponential factor of e2
k: 1 e2 k dNp.c. dek ∝ e−ek/τ1 + Ae−ek/τ2. (8) Here, as before the parameters τ1, τ2, and A are functions of e(0)k , ˆb and target. That is,
Figure 13: Pre-compound nucleon kinetic energy spectra from the raw (red) and corrected (black) BUU simulations performed at 25 and 125 MeV incident bombarding energies for light- and medium-weight targets respectively. The blue points show the nucleon kinetic energy spectra obtained from a single isolated target nucleus. See text for details. For clarity, the error bars on the corrected data points are omitted.
for each incident energy, impact parameter, and target the parameters in equation 8 are specified by fitting the (cleaned) pre-compound nucleon kinetic energy distribution. Note that for protons, the Coulomb potential experienced by each at the end of the simulation is added to its kinetic energy. In the bottom panel of figure 14, we show the fits thus obtained (solid curves) for both light- and medium-weight targets at 25 and 125 MeV. As described above for the nucleon multiplicity, surfaces for each of the three parameters in equation 8 were obtained for each target nucleus. This procedure necessarily introduces some parameter drift. That is, the parameters obtained from the global surface fits do not exactly match those obtained from the individual fits. This parameter drift is illustrated by the dashed lines in figure 14. The differences in the fit functions displayed in this figure are typical of all the other targets, energies, and impact parameters. We do not consider the difference to be significant.
Characterisation of the kinetic energy distribution of the quasi-elastic peak presents some difficulties. At the end of section 2. it was mentioned that the Lorentz boosting of the mean field proton probes unphysical regions of phase space. Again, this is due to the internal kinetic energy arising from the mean field description of the proton. This effect is clearly demonstrated in the top panel of figure 14 at both low and high incident bombarding energies. We see that the resulting spread in kinetic energy results in a finite fraction of the quasi-elastic peak residing at energies higher than the input beam energy. Clearly, this result is unsatisfactory. Keeping this in mind, it was determined that a Gaussian provided for a suitable global fit function. The original fits along with the fit function taken from the surface parameterisation are shown by the sold and dashed lines respectively in the top panel of figure 14. As with the pre-compound nucleons, we do not think the parameter drift to be significant. In order to address the unphysical spread in kinetic energy seen in the figure, we rescale the Gaussians thus found to zero-width delta functions. Thus, the kinetic energy of the quasi-elastic peak is parameterised as:
dNq.e.
dek
= δ(ek− µ(e(0)k , ˆb)), (9)
where µ is the centroid of the Gaussian fitting function. Unfortunately, this rescaling omits any fraction of the quasi-elastic peak falling below the centroid of the Gaussian that may have a physical origin.
To summarise, we have performed BUU simulations of both isolated targets of 16O, 27Al, 40Ca, 63Cu and of protons incident on those targets at impact parameters ˆb : 0 → 1.05
Figure 14: Parameterisation of the pre-equilibrium nucleon kinetic energy spectrum for p +16O at e(0)
k = 25 MeV (blue) and p +40Ca at e (0)
k = 125 MeV (red). The top panel is
for the quasi-elastic proton peak and the bottom panel is for pre-compound nucleons. The solid curves indicate the fits obtained directly form the (corrected) BUU simulations. The dashed curves indicate the fit parameters obtained from the (global) surface parameterisation as described in the text.
numerical noise, the total pre-equilibrium nucleon multiplicity, quasi-elastic proton survival probability, angular deflection of quasi-elastic protons, pre-compound protons and neutrons, and kinetic energy distribution of the quasi-elastic peak and pre-compound nucleons were parameterised as functions of e(0)k and ˆb, for a total of 11 surface parameterisations per target nucleus. We are now in a position to characterise the pre-equilibrium stage of an inelastic proton-nucleus interaction, as predicted by the BUU simulations, for the four targets listed above at any impact parameter (that can initiate an inelastic reaction) and any incident energy from 12 MeV to 250 MeV. This is the subject of the following section.
5
.
Pre-equilibrium event generation
In the previous section parameterisation schemes were developed in order to characterise the pre-equilibrium dynamics of a proton-nucleus interaction in the BUU model. As discussed in section 1., further (statistical) decay of the pre-fragment is then handled by the GEM2 evaporation code. Similar methods have been developed in the QMD model where the latter is used to generate the pre-equilibrium nucleon multiplicity.57, 102 In that work, it was found
that for beam energies in excess of 50 (180) MeV, the pre-equilibrium nucleon multiplicity was < m >=2 (3). These multiplicities are larger than the results obtained here, but the trend is in general agreement with figure 9. The importance of multiple (that is, greater than unity) pre-equilibrium nucleon multiplicity has been discussed previously.30
In this section we describe how we generate the pre-equilibrium state from the parame-terisations presented in section 4. for a given set of initial conditions. These are the incident proton (laboratory) kinetic energy and target charge and mass. The pre-equilibrium state is generated by drawing from the surface parameterisations thus presented. First however, we must select an impact parameter. Selection of this quantity is obtained in a random fashion while taking into account the nuclear profile (as generated by the BUU model shown in figure 7) and geometrical scaling effects. Selection in this manner leads to the distribution shown in figure 15. This in effect slices through each of the surface parameterisations at a single impact parameter.
Two possible non-exclusive (initial) mechanisms that contribute to the pre-equilibrium configuration follow. The incident proton can scatter off of the target nucleus (AZX) via a
binary collision with a (loosely) bound target nucleon. This is interpreted as the quasi-elastic peak already discussed. No compound nucleus is generated in this scenario. Also, as this is an inelastic process, the initial target is left in an excited state (ǫ∗ = e(0)
k − e q.e. k ),
where eq.e.k is the final state kinetic energy of the proton. (Note that we have ignored any recoil kinetic energy. This is addressed once removal of all the pre-equilibrium nucleons is complete.) A second reaction mechanism can occur where the incident proton is absorbed by the target, thus creating a compound nucleus of excitation energy ǫ∗ = e(0)
k − EB, where
EB is the energy which binds the proton to the compound nucleus (A+1Z+1Y∗).32 We use the
proton separation energy Sp(A+1Z+1Y) calculated from mass tables88 for this quantity. Again,
any recoil considerations are left until the end of the pre-equilibrium phase. In both of the scenarios mentioned above an excited pre-compound nucleus results.
Figure 15: Impact parameter selection taking into account the BUU nuclear profiles and geometrical scaling effects.
In order to determine which of the above initial processes is initiated, we first select the total number of pre-equilibrium nucleons (mp.e.) from figure 9 (for the appropriate target
nucleus). For example, a value of mp.e.=1.2 gives us 1 nucleon 80% of the time and 2
nucleons 20% of the time. This choice is obtained by drawing one random number. Results obtained with QMD simulations are comparable.102 Once m
p.e. is determined, a second
random number is compared to the projectile survival probability shown in the surface plot in figure 10. If a quasi-elastic proton results, the first of the above two mechanisms is initiated by selecting a final-state proton (laboratory) kinetic energy according to equation 9. Scattering angles are sampled from the distributions presented in figure 11 for the appropriate target, incident energy, and impact parameter. We allow for a departure from the BUU sampling scheme as discussed above in the determination of the quasi-elastic scattering process. If such a process has been selected, we further allow for charge exchange reactions where the outgoing (quasi-elastic) nucleon is a neutron.104, 106 In this case, the target nucleus excitation
energy is reduced by Sn(A, Z) − Sp(A, Z + 1). Selection of this outgoing nucleon is based on
the ratio: Rp = σ∗ pp→pp+ σ∗pn→pn σ∗ pn→pn . (10)
Note however, in this work, the in-medium nucleon-nucleon cross section does not depend on isospin as there is no momentum-dependence in the isospin potential.71, 72 Thus the effect
on pp → pp and pn → pn are indistinguishable. For the σpp→pp and σpn→pn, we use the
parameterisation presented in figure 3.90 This leads to an almost energy-independent ratio
of Rp∼ 2.5, which is in qualitative agreement with measurements performed on the p + Cu
system.107
In the event that a quasi-elastic proton does not result, an excited compound nucleus
A+1
Z+1Y∗ is formed. In either case, an excited pre-compound nucleus results which may or may
not emit nucleons. This is what is meant by the processes not being mutually exclusive. In order to draw additional nucleons (to satisfy mp.e.) from this system, we must choose the
isospin of the emitted nucleon. In two-nucleon removal from Cd at 400 MeV it has been demonstrated, however, that the ratio of emitted protons to neutrons is not correlated with the neutron content of the target30, 108 and that this ratio can fluctuate substantially from
nucleus to nucleus.109 Nonetheless, in view of these difficulties, and in the interest of
obtain-ing a simple numerical scheme, we bias the choice of outgoobtain-ing nucleon by Rp ∝ Z∗/A∗, where
Z∗ and A∗ are the charge and mass content of the excited nucleus. One could use the BUU
shown that this quantity can depend significantly on both the density-dependence110 and the
momentum-dependence of the isospin potential.71 In this work, as we are using a simplified
isospin potential that depends on density only, we are not fully confident that the BUU pro-ton to neutron ratio obtained is satisfactory. We thus use the ratio Rp ∝ Z∗/A∗ as presented
above. The kinetic energy of the outgoing nucleon (ep.e.k ) is sampled from the distribution shown in the bottom panel of figure 14, for the appropriate combination of target, impact parameter and initial bombarding energy. We then assume that this nucleon impinges on a potential barrier of height Sn(Z∗, A∗) for neutrons and Sp(Z∗, A∗) + Vcoul(Z∗, A∗) for protons.
For the latter, the barrier penetration calculation can be quite complex.26 We employ a
simple 1D barrier penetration calculation to determine if the selected nucleon escapes from the pre-compound nucleus. If the nucleon escapes from the nucleus, the scattering angle is sampled from the appropriate distribution as displayed in figure 12 for protons. For neutrons, we sample from the linear fit (not shown). The (new) pre-compound nucleus excitation en-ergy is then updated: ǫ∗(A − 1) = ǫ∗(A) − (S
τ(A) + ep.e.k ), where the label τ represents the
isospin of the outgoing nucleon. This process is iterated until either mp.e. is satisfied, or
ǫ∗ < S
τ. Note that we do not consider final state interactions in this model. An illuminating
schematic representation of the ideas presented here can be found in the work of Nieckarz.30
Energy and momentum conservation are then enforced by adjusting the kinematics of the pre-fragment. Any interactions that violate energy and momentum conservation are rejected. This rejection rate is largest for the lighter systems (∼ 7% for 12C) and small-est for the heavier systems (< 1% for 208Pb). The rejection rates increase slightly with
incident bombarding energy. Finally, the excited pre-fragment (and its newly assigned ki-netic and excitation energy) is then fed to the GEM2 evaporation code where generation of {n, p, d, t,3He, α} is commenced. We then examine the multiplicities and spectra of the
products thus generated. We confine our focus to light particle multiplicities, neutron and proton double differential cross sections, and isotope generation only.
Note that the rescaling in equation 9 is problematic. Due to the difficulties associated with parameterising the quasi elastic kinetic energy spectrum, we perform simulations ac-cording to the above scheme with the quasi-elastic peak selection turned on and off. In figure 16 we display the multiplicities thus realised. We note that without the (high energy) quasi-elastic process (circles), the final multiplicities are higher. This is easily understood if one considers that the mechanism for high energy removal is now absent, thus the pre-fragment is able to attain higher excitation energies and evaporate more (relatively low
energy) particles. Except for the heaviest system (63Cu), we find qualitative agreement with
the results from ICRU6348 (solid lines). The kinetic energy distributions however, are less
satisfactory. We show in figure 17 the resulting spectra obtained with and without the quasi-elastic scattering peak. We note that neither with nor without the quasi-quasi-elastic contribution are the the mid-rapidity portions of the proton kinetic energy spectra adaquetly reproduced. This may be due in part to the quasi-elastic rescaling as presented in equation 9. Alterna-tively, one may, for instance, opt to include the full spread of the quasi-elastic peak shown in the top panel of figure 14 for ek< µ only. We do not pursue this course in this work. We do
note that with the quasi-elastic process included in the event generation procedure we are better able to reproduce the spectrum endpoints. Thus indicating the need for high-energy processes. This is true for all systems displayed in that figure, and especially for the neutron spectra obtained for the27Al and 63Cu targets. This is due to the inclusion of the exchange
process already mentioned. At the low energy portion of the kinetic energy spectrum, we find qualitative agreement with the ICRU63 results. We attempt to salvage some the the BUU parameterisation in the following section.
Figure 16: Particle multiplicities for inelastic p + {16O,27Al,40Ca,63Cu} interactions as a
function of incident bombarding energy. The continuous lines are taken from the ICRU63 report.48 The stars and circles result from the event generator as described in the text
Figure 17: Neutron and proton kinetic energy spectra corresponding to the multiplicities presented in figure 16 at an incident laboratory bombarding energy of 80 MeV. The lines and points have the same meaning as in figure 16. In each panel, from top to bottom, the curves are shown for16O,27Al,40Ca, and 63Cu targets respectively. In addition, from top to
6
.
An alternative formulation
The results of the previous section indicate that the method of drawing from the BUU pa-rameterisations as presented here are inadequate in terms of describing the kinetic energy distribution of final state nucleons. This, presumably, is partly due to the rescaling of the quasi-elastic peak. We thus look to replace this part of the parameterisation. Considering that the many excitation levels populated during the de-excitation phase are largely un-known,111–113 and at the energies of interest in this work, the nuclear excitation energy ǫ∗
is expected to be much larger than the nuclear level spacing,20 we make the assumption
that the many closely spaced and partially overlapping nuclear excitation energies are repre-sentable as a continuum. Furthermore, as in the exciton32 and hybrid model,114 we assume
that all excitation energies of the target are equa-probable. With this ansatze, selection of the quasi-elastic nucleon kinetic energy is computationally trivial. We choose the kinetic energy (eq.e.k ) of the outgoing quasi-elastic peak randomly from ∼0 up to the near beam energies (note that we employ an exponential cutoff corresponding to the lowest nuclear ex-citation level). Just as before, this leaves the target with an exex-citation energy of e(0)k − eq.e.k . Unfortunately, the localised quasi-elastic (energy) peak is now absent. In figure 18 (open circles) we show the kinetic energy spectra thus obtained with this random sampling for the quasi-elastic peak. The agreement with the ICRU63 results show substantial improvement over the results obtained with the full BUU parameterisation. Furthermore, in the interest of consistency, we also show the resultant kinetic energy spectrum when we replace the BUU parameterisation for all pre-equilibrium nucleons with the random sampling just described. This is shown by the stars in the same figure. The agreement with the ICRU63 data in this case is also fair. In the interest of simplicity and taking into account the general agreement with the ICRU63 data obtained in this manner, we use the random sampling technique for all pre-equilibrium nucleons from here-on. Note that now, the only difference between drawing nucleon kinetic energies for both disassembly processes is in the endpoint energy. For the quasi-elastic events, the maximum allowed kinetic energy is near the beam energy (recall that we exponentially suppress excitation energies below the first level), for nucleons ejected from the excited nucleus it is reduced by both the binding energies of the incident proton and the ejected nucleon to the excited (compound) nucleus. We note that the workload in obtaining the surface parameterisations is now substantially reduced as there is no longer a need for parameterising the kinetic energy distributions from the BUU simulations. This comes at the expense of loosing structure in the kinetic energy spectrum predicted by the
BUU model.
So far, we have deferred any discussion of the double-differential cross section. Our focus was to first obtain reasonable multiplicities, and then reasonable angle-integrated kinetic energy spectra. We now consider the BUU parameterisations of the angular deflection of the pre-equilibrium nucleons. We further note that substantial work has been done on parameterising the angular distribution of nucleons from nucleon-nucleus collisions.36, 115 The
approach chosen there was to look for relatively simple phenomenology to describe global angular distributions. To this end, over 900 systems for which experimental data exist were investigated and it was determined that the shape of the angular distribution of emitted nucleons depends on the outgoing kinetic energy of each. The result was that the angular distribution function can be parameterised in terms of exponentials of the cosine of the scattering angle. The energy dependence appears in part as multipliers to the cosine of the scattering angle. We have implemented this phenomenology115 in our scheme to compare
with the BUU results. In figure 19, we compare these two sampling functions for collisions of protons with 16O at 30 and 100 MeV for both neutrons and protons. We find qualitative
agreement between the two. We investigate the implications of these two sampling functions by examining the neutron and proton double differential cross section for protons incident on40Ca at 80 MeV in figure 20 and figure 21 respectively. We find close agreement between
the two parameterisations with the Kalbach systematics providing a slightly better match to the ICRU63 results at backward scattering angles. This is true for both neutrons and protons. We also note that the agreement obtained at other incident energies and targets is similar to the ones presented in the above figures. In the spirit of reducing the work required to parameterise the BUU results, we adopt the Kalbach systematics as a replacement to the BUU systematics. The computational cost associated with sampling the angular deflection in the event generator (up to a factor of 1000) can be substantial. We also note that for a portion of the ICRU63 results, the Kalbach systematics are employed, so the agreement is not too surprising.
After employing the random kinetic energy sampling and the Kalbach systematics for the angular distribution the only dependencies on the BUU parameterisations are the im-pact parameter selection of figure 15 and the total pre-equilibrium nucleon multiplicity and projectile survival probability of figures 9 and 10. We now go on to test the sensitivity of the multiplicity and angle-integrated kinetic energy distributions to the impact parameter selection procedure. In figure 22 we show the multiplicities thus obtained using ˆb sampling
