A semi-classical model for breakup reactions of light nuclei: 4He(p, pn)3He

Nuclear Physics A516 (1990) 77-107 North-Holland

A SEMI-CLASSICAL

MODEL

LIGHT Lars BILDSTEN,

FOR BREAKUP

NUCLEI:

REACMONS

OF

4He(p, pn)‘He

Ira WASSERMAN

and Edwin E. SALPETER

Center for Radiophysics and Space Research, Space Sciences Building, Cornell University, Ithaca, NY 14853, USA

Received 18 January 1990 Abstract: We introduce a new model for single-nucleon knockout reactions on light nuclei that includes multiple-scattering effects. We focus on 4He since the energy and angular distributions of the neutrons and 3He from the inelastic reaction 4He(p, pn)3He are of astrophysical interest in the problems of accreting neutron stars, solar flares, hot (kT - MeV) plasmas, and cosmic-ray interactions with the interstellar medium. We have performed a semi-classical, Monte Carlo calculation for proton energies ranging from just above the reaction threshold (25.72 MeV) to just below the free nucleon-nucleon ?r production threshold (280 MeV). Our model involves only two free parameters, one that suppresses the nucleon-nucleon scattering cross sections, and another that weights the occurrence of multiple scattering. These two free parameters are fixed by comparison to a subset of the experimental data. Once fixed, the model provides total cross sections and detailed energy-angle distributions that are in excellent agreement with the available experimental data. We show that excluding multiple scattering between the outgoing nucleons and the residual nucleus leads to strong disagreement with the experimental data. We describe how to implement these results in the context of a larger problem.

1. Introduction 1.1. ASTROPHYSICAL

SETTING

The energy and angular spectra of neutrons and 3He produced in the spallation of 4He by protons over a large energy range is relevant in at least four different astrophysical environments: accreting neutron stars, solar flares, hot (MeV) plasmas, and cosmic ray interactions in the interstellar medium. For astrophysical applications an accurate model is needed that produces the distributions over a large proton energy range. For neutron star accretion, the range is from threshold to 100-300 MeV, whereas for cosmic-ray interactions and solar flares, the range is up to several GeV. The range of interest for hot plasmas is from threshold to less than a hundred MeV. At relatively low energies (near and within several times the threshold energy), the standard distorted-wave impulse approximation (DWIA) determination of these reactions is inaccurate due to multiple scattering effects. Thus, we have developed a new semi-classical model that, we feel, incorporates the most important physical effects. This Monte Carlo simulation is similar to intra-nuclear cascade codes ‘) in that it follows nucleon-nucleon scatters resulting from the original proton impact, 0375-9474/90/$03.50 @ 1990 - Elsevier Science Publishers B.V. (North-Holland)

L. Bildsten et al. / Breakup reactions

78

but

is somewhat

Numerical

see, the results Ultimately, spallation kinetic

simpler

simulations

for the light

of the model

agree quantitatively we plan

on accreting energy

of order

to apply neutron

nuclei

of greatest

can be performed

astrophysical

quickly,

and,

very well with the (fairly sparse) our results stars.

100-300 MeV

to detailed

Material per

accreting

nucleon,

interest. as we shall

available

simulations onto

depending

data.

of nuclear

a neutron on the

star has surface

redshift ‘). At least some of this kinetic energy may be thermalized in the neutron star atmosphere via Coulomb collisions with the atmospheric electrons 3*4). When the surface magnetic field is strong (B b lOi* G), Coulomb stopping lengths become quite long (320 g * cm-*), and direct nuclear reactions between the infalling nuclei and the atmospheric protons or nuclei become probable 5Y6).For incoming (atmospheric)

4He, spallation

reactions

with atmospheric

(infalling)

protons

produce

free

neutrons, protons and other products (D, 3H, 3He). As has been pointed out by Schvartsman ‘) and Brecher and Burrows *), those free neutrons that are not absorbed by 3He (via ‘He(n, P)~H) eventually “recombine” with a proton, resulting in the emission of a 2.2 MeV -y-ray photon. Because nuclear reaction lengths are much greater than Compton scattering lengths (2.5 gr . cme2), Compton scattering will generally degrade the y-ray line photons. In general, we expect line emission to be spread over a broad energy range via scattering, and the actual energy spectrum of the escaping y-rays will depend crucially on the distribution in atmospheric depth of the line photon emission. production as a function of through the atmosphere that angle distribution of neutrons tions, particularly since we

The y-ray emissivity, in turn, depends on the neutron depth, as well as on the subsequent neutron diffusion culminates in recombination with protons. The energyat birth is an important input to the diffusion calculaexpect the neutrons to be preferentially downward-

beamed in the accretion stream net neutron flux). In addition,

(although diffusion 3He, an important

will inevitably smooth out the by-product of the spallation

reactions, strongly absorbs neutrons via 3He(n, P)~H [refs. ‘**)I, so its distribution in the atmosphere must also be calculated. Thus the detailed energy-angle distribution of both n and 3He are needed. The astronomical observation of -y-rays from accreting

neutron

stars would

serve as a useful

diagnostic

of the accretion

flows,

and of the surface properties of neutron stars. With the launch of the Gamma-Ray Observatory in 1990, there is some hope that such y-ray emission will be seen 9*10). The mechanism of particle acceleration (primarily protons) during solar flare events is still not completely understood [for an overview on this problem, see Ramaty and Murphy “)I. Measurements of the neutrons produced in spallation reactions of these protons with atmospheric 4He serve as an indirect probe of the accelerated proton distribution. The neutrons have been directly observed both at the Earth and by the Solar Maximum Mission. In addition, neutron capture on protons in the solar atmosphere produces a 2.2 MeV y-ray line, which has been observed. From these observations, one would like to place constraints on the original population of accelerated particles. Clearly, this is a complicated task and a first step is knowing the energy and angular distribution of the produced neutrons.

L. Bildsien et aL J Breakup reactions

19

Current models make simple assumptions about the neutron distribution ‘*). Our more detailed formulation will modify these results. In addition, recent work on thermal breakup of 4He in hot plasmas (kT- MeV) has shown that the neutrons produced are important to the evolution of the plasma, since they contribute to other nuclear processes 13-15). 1.2. QUALITATIVE

OVERVIEW

Our calculation is a semi-classical, Monte Carlo simulation of the breakup process for proton lab energies in the range 25.72-280 MeV. This energy range is appropriate for spallation in accreting neutron stars, where the energy per nucleon for the infalling material is characteristically 60.3m,c2 = 280 MeV on entry into the neutronstar atmosphere ‘). Because accreting ions decelerate when they enter the atmosphere, we need to determine the reaction at a continuum of proton energies up to the value on entry. Spallation begins when the incident proton strikes one of the constituent 4He nucleons in an elastic collision, possibly transferring enough energy for the struck nucleon to escape the nucleus. After the initial collision, the “intermediate nuclear state” in our model consists of the interloping proton, struck nucleon, and a localized 3He- or 3H-like remnant “nucleus”. Both the incoming proton and the struck nucleon are allowed to elastically interact with the remnant as they attempt to exit the nuclear region. From our model, we compute energy-angle distributions for both outgoing neutrons and the 3He or 3H remnant, and the reaction cross section a(E). We consider neither the pickup reaction 4He(p, d)‘He nor the higher energy regime where r production becomes possible. As we shall see, our model contains two adjustable parameters. The first parameter, F, which scales the nucleon-nucleon scattering cross sections from their vacuum values to their values in nuclear matter, is determined from comparisons of our results to the reaction cross section measurements of Nicholls et al. 16) at E, = 141 MeV, and to experimental cross sections compiled by Meyer “). The other parameter, p, which weights the importance of secondary interactions, is not rigorously or precisely determined from comparison to any one data set. However, we found that p - 1 gave excellent agreement with detailed neutron energy-angle distributions measured by Nicholls et al. 16) for 4He(p, pn)3He at E, = 141 MeV, and detailed p and 3H energy-angle distributions measured from the mirror reaction 4He(n, np)3H at E,, = 90 MeV by Tannenwald **). Once these two parameters are chosen, we test our model by independent comparison with other available data, in particular, proton energy spectra measured by Wesick et al. 19) for proton bombardment of 4He at incident proton energies of 100 and 150 MeV. That these independent comparisons yield fairly close agreement between our models and experiment encourages confidence in the applicability of our numerical results. The outline of this paper is as follows. In sect. 2, we discuss previous models for spallation reactions. In sect. 3 we describe our model for the breakup reaction and discuss the nuclear physics data that is utilized. In sect. 4, we give results for the

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L. Bildsten et al. / Breakup reactions

cross section for a range of initial proton energies for the astrophysically interesting 4He(p, pn)3He reaction, and by comparing to experiment fix one of the free parameters. We also provide fitting formulae for both the (p, pn) and (p, 2p) reaction cross sections. In addition, the energy and angular distributions of the neutrons and 3He are compared to the existing data, thus determining the second free parameter. It is shown that our model provides an excellent determination of both the total cross sections and the neutron and 3He energy-angle spectra. In sect. 5, we discuss the implementation of our results in an astrophysical context. Appendix A discusses nucleon-nucleon scattering and provides some simple fitting formulae for n-p and p-p elastic scattering.

2. Earlier work on breakup reactions Nuclear reactions involving three or more particles in the final state have been treated in two different ways. The intra-nuclear cascade (INC) method yields a very accurate description in good agreement with experimental data for multi-nucleon knockout processes. For single-nucleon knockout reactions (e.g. (p, 2p), (p, pn)), the distorted-wave impulse approximation (DWIA) works well at relatively high incident proton energies (Einc b 200 MeV), and provides a prescription for experimentally measuring the constituent nucleon momentum distribution. Intra-nuclear cascade (INC) codes were written in the 1950’s and 1960’s [refs. 1,20*21)]based on Serber’s **) suggestion that a high-energy nucleon-nucleus collision could be modelled by following the classical path of the incident nucleon through the nucleus, treating the overall interaction as a series of two-body interactions between the incoming nucleon and the constituent nucleons (which have a Fermi gas momentum distribution initially). This approach should be valid for incident nucleon energies B 100 MeV, high enough so that nucleon-nucleon scattering occurs on a time short compared to the time it takes the nuclear potential to modify the nucleon momentum. In INC codes, struck nucleons may interact repeatedly with other consitutent nucleons, causing a cascade. Since the cross sections for elastic nucleon-nucleon interactions were well determined, this process could be carried out rather directly. [In practice, of course, a number of factors conspire to lengthen nucleon mean-free-paths in a nucleus, principally Pauli blocking of final-nucleon momentum states and collective effects that alter the nucleon effective mass inside a “nuclear medium”, see Negele and Yazaki 23), Fantoni et al. ‘“).I The INC codes determine the spectra of the nucleons emitted during the cascade, which occurs on a timescale of order the nucleon crossing time of the nucleus. The residual nucleus will generally be highly excited and unstable to particle emission, and will shed its excess energy by emitting low-energy nucleons and gamma-rays. To our knowledge, this method has not been applied to the various breakup channels of 4He.

L. Bildsien et al f Breakup reactions

81

A different approach to the breakup process is usually taken for the single-particle knockout reactions (p, pn) and (p, 2~). These are treated as single scattering events between the incoming nucleon and a constituent nucleon in a direct reaction in which the incident nucleon gives the constituent nucleon enough momentum to escape the nucleus. This “quasi-free” scattering is calculated using free nucleonnucleon scattering cross sections in a plane-wave impulse approximation (PWIA) at energies high enough ( Eincb 300 MeV) so that the incident and departing nucleons are not substantially distorted by the nuclear potential well. At low energies the distortion of the incident and departing nucleons due to the nuclear potential well becomes important, requiring a distorted-wave impulse approximation (DWIA). These approaches have proven quite successful at high incident energies (E B 200 MeV), where the impulse approximation is unquestionably valid [see Jacob and Maris 25) for a review]. DWIA has been applied extensively to the knockout reaction 4He(p, 2p)‘H and compared to experimental results at lab-frame proton energies of 65, 85, and 100 MeV by Pugh et al. 26), at 156 MeV by Frascaria et al. 27), and at 250,350 and 500 MeV by van Oers et al. 28). Much less work has been done on the reaction 4He(p, np)3He, with most of it concentrating on understanding the final-state interactions between the two outgoing nucleons 29). Thus, this work only considered a limited part of phase space, where the two outgoing nucleons had a low relative energy. In the DWIA, the differential scattering cross section factorizes into the free nucleon-nucleon scattering matrix element and the constituent-nucleon momentum distribution. If this is the only reaction mechanism occurring, an experimental measurement of the differential cross section determines the constituent-nucleon momentum distribution. A comparison can be made to theory by assuming a form of the wavefunction and theoretically calculating the differential cross section. A successful comparison then provides justification for assuming the prevalence of this single reaction mechanism. In the case of 4He(p, ~P)~H at high incident proton energies (E b 300 MeV), the agreement between the experimental and theoretical differential cross-section is reasonable given the uncertainties in the optical-model parameters that are used in determining the distortion of the incident and outgoing plane waves 28). Pugh et al. 26) compared their experimental results for (p, 2p) reactions on D, 3He, and 4He at low (S 100 MeV) proton energies to PWIA calculations showing that the theoretical differential cross sections must be reduced by an energy dependent factor to agree with experiment. For 4He, the discrepancy was as large as a factor of 50. Haracz and Lim”) attributed this discrepancy to multiple scattering effects between the exiting nucleons and the residual nucleus. Using a semi-classical model for the rescattering, they explained both the energy dependence of the discrepancy and its dependence on the identity of the residual nucleus. Roos 31) showed that by performing a DWIA for the 4He(p, 2p)3He reaction, agreement with the experimental data of Pugh et al. 26) could be obtained by reducing the magnitude of the theoretically determined differential cross sections by an

82

L. Bildsten et al. / Breakup reactions

energy-independent factor of 2. Thus, at low energies, there is still some discrepancy. More recent measurements by Wesick et al. 19) of the inclusive proton spectra from 100 and 150 MeV protons also show the effects of multiple scattering at these incident energies. From these experiments it is clear that secondary interactions play an important role at low energies. These experiments also demonstrated that the multiple-scattering effects became more important as the size of the residual nucleus increases. This is due to the larger area presented to the exiting nucleons by the residual nucleus. 3. Description of the model We are primarily interested in the reaction 4He(p, pn)3He at a wide range of incident proton energies, including energies near threshold (25.72 MeV) as well as much higher energies (-100-300 MeV). To cover the low-energy and high-energy regimes with equal accuracy, we have formulated a new model that is a hybrid of INC and the simple impulse approximation. The main distinguishing feature of the 4He breakup reaction is that it only involves 5 nucleons. As a result, it is generally a bad approximation to adopt a smooth model for the nucleus, as in the standard INC described in the previous section. We treat the unperturbed 4He nucleus as a “bag” of four discrete nucleons, whose radius is r, = 1.67 fm. The mean free path for the incident proton to interact with one of the constituent nucleons can be rather short. For example, the mean free path for an incident 100 MeV proton is h = 1.0 fm (using vacuum cross sections for proton scattering off protons and neutrons at rest), and it is still shorter at lower energies. (Including the momenta of the constituent nucleons lowers this estimate in general.) Since the incident proton typically loses energy in its first collision within the nucleus (except at very low energies) its mean free path for subsequent collisions is smaller than when it entered. Moreover, the struck nucleon produced in the first collision may also interact with the remaining, unscattered constituent nucleons. Thus, secondary scattering within the disturbed 4He nucleus is clearly important. From these considerations, the following semi-classical picture emerges. At very high energies -200-300 MeV, the incident proton interacts only once before departing, and the struck nucleon suffers little or no distortion due to the residual nucleus while leaving. At the lower energies the mean-free path for the incident proton is smaller than the nuclear radius, and the primary collision tends to occur close to the point of entry. Afterwards, both the struck nucleon and the incident proton have short mean free-paths and thus will interact with unscattered constituent nucleons as they attempt to exit the nuclear region. Fig. 1 displays the sequence of events in the “lab” frame where the 4He is initially at rest. The four panels display different portions of the sequence. We treat the bound nucleons as point particles with equal kinetic energies moving on radial trajectories in the nucleus. (These simplifying assumptions may easily be relaxed.)

L. Bildsten et al. / Breakup reactions

83

Classical Proton Path __ ______.---_.---

Proton Neutron (al

( c)

Initial

No Secondary

(b)

Setup

Interactiam

(P=O)

(d

Rmary

) Secondary

Interaction

Interactions

(P *

0)

Fig. 1. Cartoon displaying the series of interactions in our model. Part (a) shows the initial setup with the proton (hollow circle) approaching at an impact parameter b. The constituent neutrons (filled circles) and protons (hollow circles) are on radial orbits with momenta pi. Part (b) displays the nucleus after the first interaction. The dashed line shows the proton’s path into the nucleus. The residual nucleus (enclosed by the dotted curve) has a total momentum -p,,. Excluding secondary interaction (p = 0) leads to part (c), where the three bodies go their separate ways if there was adequate energy given to the neutron to escape. Including secondary interactions (p > 0) allows for momentum transfer between any of the two nucleons and the 3He (part (d)). This shows the case where only the neutron interacts again.

The incoming proton is treated as a point particle with an energy-dependent interaction cross section with the constituent nucleons. The nuclear potential well gives the incident proton a momentum kick upon entering the nucleus, and refracts the proton to smaller impact parameters. The calculation consists of first choosing the position in the nucleus where the primary collision occurs and then exchanging momentum between the two nucleons. After the primary interaction, both nucleons are allowed to elastically interact once more with the residual nucleons, which are treated as a single “cohesive unit”, i.e. an off -shell nuclear state. (Further interactions could also be included, but will generally result in additional fragmentation.) A scattering event actually occurs only if the incident proton is unbound in the nuclear potential well following all interactions. The struck nucleon may also escape if it has sufficient kinetic energy to do so. (Final states with an unbound incident proton and bound struck nucleons correspond to elastic scatters, since there are no bound excited states of 4He.)

L. Bildsten et al. / Breakup reactions

84

As we shall see, only two free parameters will be introduced for overall normalization of the results. These are fixed by comparison to a subset of the experimental data, and are held fixed at these values for all further comparisons. We j?nd that once these free parameters have been fixed our model provides accurate predictions of d~~erentiul cross sections at a large range of incident proton energies.

3.1. NUCLEAR

PHYSICS

INPUT

The ground-state structure of the 4He nucleus has been studied in great detail over the years. For our semi-classical breakup model, we need to specify the momentum distribution of the constituent nucleons in the unperturbed 4He nucleus. The momentum dist~bution of the constituent nucleons is well-dete~ined below the Fermi momentum PFermi- 100 MeV/c [ref. 32)], but is less certain at larger momenta where the short-range NN interaction (and the resulting correlation) is important. Although we could easily adopt a more complicated-and precise distribution, for simplicity we assign each nucleon a radially directed momentum pi of equal magnitude (with 1 pi = 0, of course). For 4He, the binding energy per nucleon is roughly 7 MeV (see table l), implying a classical kinetic energy of -33 MeV in an assumed potential well depth of 40 MeV. In our simple scheme, we therefore assign constituent nucleons momenta of magnitude lpi/ = J2m x 33 MeV = 250 MeV/c. Since the total angular momentum vanishes for radially directed momenta, this setup crudely models the predominantly S-wave 4He nucleus. We take the 4He radius to be the charge radius measured by elastic electron scattering, r, = 1.67 f 0.03 fm [ref. 33)]. Our admittedly crude momentum distribution adequately incorporates the characteristic nuclear kinetic energies and total nuclear spin for 4He; it does not include the spread in the actual kinetic energies and non-S-state admixtures. We regard it as a reasonable but simplified treatment of the effects of internal motions of the 4He constituents, effects that are especially important at low (< 100 MeV) incident proton energies. To realize our breakup model we also need the free p-n and p-p elastic scattering differential cross section in the proton energy range O< E < 650 MeV. The semiphenomenological fitting formulae that we use to describe the nucleon-nucleon scattering are described in detail in appendix A. These cross sections apply to scattering in vacuum. In practice, collective effects alter nucleon-nucleon cross sections in nuclear matter relative to their free (vacuum) values 23,24).To mimic this effect, we lower all nucleon-nucleon cross sections by a single, energy independent fucfor F. We discuss the dete~ination of F for our models in sect. 4.1. The possible Gnal states for the proton bombardment of the 4He nucleus, along with the Q-values and threshold energies are shown in table 1. Pion producing reactions in the p-4He system can occur at proton energies lower than the free nucleon-nucleon pion production threshold when one nucleon in the 4He is carrying a large fraction of the momentum, an intrinsically unlikely circumstance which we ignore.

L. Bildsten et al. / Breakup reactions

85

TABLE 1 Inelastic reactions and Q-values for p + 4He Reaction 4He(p, “He(p, 4He(p, 4He(p, 4He(p, 4He(p,

Q NV)

d)3He 2p)3He pn)3He pd)D ppn)D ppnn)H

“1

Threshold (MeV) 22.94 24.71 25.12 29.81 32.59 35.31

-18.354 -19.815 -20.578 -23.848 -26.072 -28.297

“) Ref. 34). 3.2. SAMPLING

OF INITIAL

EXTERIOR

IMPACT

PARAMETERS

We focus on the reaction 4He(p, pn)3He for the rest of the paper. Breakup into ~+P+~H will not be discussed here, except in sect. 4.2 for comparison with data at E,= 100 and 150 MeV. The Monte Carlo simulation consists of colliding individual protons with the nucleus at fixed impact parameters. Since the NN elastic cross section at E - 100 MeV is comparable to m’, = 91 mb, incoming protons with impact parameters greater than the 4He radius can interact with the nucleus and must be included in the simulation. A given incident proton can (1) pass directly through the nucleus without any interaction, (2) interact with the nucleus without depositing enough energy to cause it to break up, or (3) initiate breakup into ~+n+~He. The reaction cross section is determined by the number of breakup events. For the remainder of this paper, all lengths will be expressed in units of the 4He radius, r,. Coulomb effects are explicitly ignored in both the initial and final states. This is a good approximation in the initial state, since the barrier energy (2e2/r, = 1.7 MeV) is much less than the threshold energy (25.7 MeV), but is relatively poor in the final state near threshold where Coulomb effects will reduce the cross section. Incident protons with impact parameters b < 1 are sampled uniformly in surface area. However, protons with b > 1 are sampled according to the amount of overlap between a cylinder that contains the incident proton and the nucleus. Fig. 2 illustrates the scattering geometry for protons with b > 1. An incoming proton with b > 1 occupies a cylinder of effective dimensionless “scattering” radius

(1) for interacting with any neutron inside the 4He nucleus, where r, is the 4He radius, ~,r( E,.,,) is the neutron-proton elastic scattering cross section for a center-of-mass energy EC.,.,, and F is the cross-section reduction parameter. The constituent neutrons in the nucleus are on radial orbits, so there is a simple relation between the neutron’s angular position and EC,,., E c.m.=;(E,+E,-2p&$%),

(2)

86

L. Bildsten et al. / Breakup reactions

Fig. 2. Geometry for proton collisions with impact parameters greater than the 4He radius. The proton interacts with a neutron in the nucleus at a relative angle B with respect to the velocity. Since the neutrons are on radial orbits inside the 4He, they must lie on the cone with half-angle 6.

where p is the cosine of the angle between the constituent neutron and proton velocities, which we call the “cone angle”. The radius r, is largest center-of-mass energies, that is, for neutrons moving away from the incident An incoming proton will overlap with a constituent neutron at an angle provided its dimensionless impact parameter b is within (see fig. 2) LX.(CL) =m+

G(@) *

incident for low proton. cos-’ p,

(3) Since the radius r, is largest for neutrons moving parallel to the proton, the largest possible impact parameter at a given incident proton energy usually occurs for y = 1 at low energies, where r,(p) dominates in eq. (3). At higher energies, where r&) < 1, the maximum occurs for /J = 0, where neutrons are moving perpendicular to the proton. Plots of b,,,(y) for a range of incident proton energies are shown in fig. 3. For a given incident proton energy Ep, we define b,,,(E,) to be the largest possible impact parameter for all F. For incident protons of energy E, at b > 1, impact parameters are chosen randomly in equal area bins between 1 and b,,, (E,). To determine whether a particular proton can possibly interact with a constituent neutron, we consider the geometric overlap of the proton cylinder with the conical surfaces occupied by neutrons in our model 4He nucleus. For a proton incident at a particular impact parameter, P( b, p) is the probability of overlap with a neutron moving at direction cosine CL.To proceed, we choose p from a uniform distribution C-1, 11. The neutron is confined to lie somewhere on the cone with opening angle 0 = cos-’ p, and the fraction of the cone’s surface area that is inside the scattering cylinder of radius r,(p) is the probability for overlap. After some algebra, we find that the overlap probability is 0

P(z) =

1

z>l,

cos-’ z-(l/a)zJi?

l>z>O,

1-(lf~)COS-‘~Z~+(l/71)~Z~~ 1

-l
(l/T)

L. Bildsten et al. / Breakup reactions

87

F=l 4

-1

-.5

0

.5

1

CL

Fig. 3. Maximum impact parameter for the proton to interact with a neutron at angle 0 = cos-1 CL,for F = 1. The impact parameter is in units of the “He radius. Each line is for a different incoming proton energy E,.

where z = (b - r,)/Jl -CL’. (Note that for z > 1, b > b,,,(p).) A random number is chosen uniformly between zero and one to determine if the overlap occurred. For those protons that overlap, the neutron location is chosen from the distribution of allowed positions on the cone. Knowing this position and CL,we determine an effective impact parameters (b,,< 1) at which the proton actually approaches the nucleus. After these protons have been assigned an effective impact parameter, &, we (1) discard the neutron position found above, (2) discard the initial impact parameter that was greater than one, and (3) treat this incident proton for all subsequent purposes as incident from infinity with impact parameter beff< 1. After this assignment has been made, the proton is treated in an identical manner as those protons that originated with impact parameters less than one. This prescription was adopted for computational simplicity, as it allows a uniform treatment of all impact parameters, b < 1 or b > 1. Although unusual, it should include the important physical effects when averaged over all angles and all allowed impact parameters larger than 1. In a given simulation the inclusion of incident protons with initial b > 1 adds an additional number of protons bmx(I”)

P(b, cl)b db,

(5)

1

where P(b, p) is given by eq. (4), and with initial impact parameters less than is to increase the reaction cross section to disagreement with the experimental

N b 1 would lead data.

88

L. Biidsten et al. / Breakup reactions

3.3. REFRACTION AND DETERMINATION

OF SCA7TERING

LOCATION

The

proton approaches the nucleus with an exterior impact parameter b < 1 (or an effective impact parameter b,,< l), and energy E,, and receives a momentum kick in the radial direction upon “entering” the nucleus. For a nuclear potential of depth V,, the magnitude of the radial velocity kick is, taking nuclear recoil into account, Av. -==j(&b2+5Vo/4Ep-di??), pi

(6)

where u is the initial proton velocity. The entrance of the proton into the nucleus increases the kinetic energy of the system by V,. The proton energy and impact parameter inside the ~~c~eu~ in the new 4He rest frame (since it received an equal and opposite kick) are then E=E,+;V,,

(3

b “=(1+5v*/4E,)1/2’

(8)

and

respectively. The momentum kick given to the 4He does not change its internal energy, but is important in its effect on the lab distributions ofthe eventually emitted particles. To determine the scattering location within the nucleus, we need to tabulate the number of mean free paths to neutron scattering in a traversed length s, the “optical depth”,

where E+ and E- are the n-p center-of-mass energies for the interaction of the incoming proton with the radially inward and outward moving neutrons, and n = 3f& is the number density of nucleons in the 4He. The path through the nucleus is determined by the refracted impact parameter b’. Fig. 4 shows the optical depth through the nucleus as a function of incident impact parameter b for four different incident proton energies. Note that r> 1 for typical proton energies and impact parameters. Receding neutrons provide the largest contribution to T since they yield lower center-of-mass energies. Because of refraction, the optical depth is large even for b = 1. The point of collision is determined by randomly sampling from the dist~bution exp(-r), which is the probability that a proton penetrates to depth T without scattering a neutron. If a depth 7 is chosen that is larger than the maximum value for that impact parameter, then the proton has traversed the nucleus without interacting. For those choices of 7 inside the nucleus, the positions and velocities of the colliding proton and neutron are determined. Since the location of the 4He

L. Bildsten et al. / Breakup reactions

.2

.4

Impact

.6

Parameter

89

.6

1

b

Fig. 4. Optical depth r of the nucleus for protons entering at impact parameter b for various incoming proton energies, E,, and F = 1. The refraction and kinetic energy increase of the proton upon entering the nucleus is included in this calculation.

center-of-mass and its velocity are known, the position and velocity of the residual 3He nucleus are also determined.

3.4. SCATTERING

KINEMATICS

AND

THE

BREAKUP

CONDITION

The neutron-proton elastic scattering rearranges momenta, giving momentum to the constituent neutron, and conserves the total kinetic energy in the system. We use the free neutron-proton scattering kinematics, ignoring nuclear medium effects on kinematics and the Pauli exclusion principle. Before collision, but after refraction, the internal kinetic energy of the 4He can be written as the sum of the neutron and 3He kinetic energies in the 4He centre-of-mass,

where p,, = -p3 is the constituent neutron momentum, and m is the neutron mass. The proton collides with a neutron, giving it a momentum kick Ap, which is determined by sampling from the n-p differential scattering cross section (described in appendix A). In the new 4He rest frame, moving relative to the previous frame at u = Ap/4m, the 4He internal kinetic energy is E! _ P; ; P; ; 3 AP’ I in - AP lnt

2m

6m

8 m

m

’

(11)

Define the excess of energy above threshold as E

=3 AP*; P~*AP exceSS 8

m

-+0, m

(12)

90

L. Bildsten et al. / Breakup reactions

where Q = M(4He) - M - M(3He) = -20.58 MeV. If E,,,,,, < 0, then the momentum transferred to the neutron was insufficient for escape. If no further scattering is allowed, such events do not produce breakup, contributing instead to other processes that are discarded in our simulation. It is possible that further scattering occurs between the three particles, thus modifying the excess energy. This effect is discussed in the next section. Since the proton lost kinetic energy to the nucleus, there is no guarantee, a priori, that it will be unbound in the final state. Using the same static potential V, as before, the condition for the proton to escape is Ek > 2 V,, where EL is the proton energy in the rest frame of the highly excited 4He. Bound final states are not allowed, because the corresponding A = 5 nuclei do not exist. (If a bound final state is realized in the model, we ignore this scattering event, and propagate the incident proton further through the nucleus, thus giving it another opportunity to interact.) The unbound proton can now depart. Since the 4He is in a highly excited and generally nonspherical state at this point, there is no obvious way to describe the spatial variation of the potential that the exiting proton encounters. Thus, we assume that the proton simply encounters a potential gradient that is anti-parallel to its velocity and gives the proton an energy reducing velocity kick (13) where EL = imvr2. This is a subtraction of the kinetic energy that was added into the system upon the proton’s entrance into the nucleus. The unbroken 4He receives a momentum kick in the opposite direction, and maintains the same excitation energy. After the proton has exited the nucleus, the proton energy in the new 4He rest frame is EE = EL -iv,. At this point, the 4He is moving in the lab frame at a velocity that is a result of the direct n-p scattering and the refraction of the incident and exiting proton. After the proton has left the nucleus, the neutron is allowed to exit if E,,,,,, > 0. The excess excitation energy will convert to kinetic energy after the breakup, thus E

P: =-+p: exceSS 6m 2m ’

(14)

where IpA is the momentum of the neutron and 3He in the current 4He rest frame. Combining eq. (14) with eq. (12) gives the final momentum squared as pf=~(Qm+~Ap2+p;Ap).

(15)

The neutron escapes the nucleus and leaves with this as the final momentum in the 4He center-of-mass. In this escape, we assume that the potential gradient that the neutron encounters on exit is antiparallel to its momentum, thus reducing the

L. Bildsten et al. / Breakup reactions

91

momentum, and not refracting it. We make this simple, energy-conserving assumption since it is clear that the nucleus will not retain its unperturbed potential shape after these scattering events. Energy is conserved in this series of interactions to within the machine accuracy, and was explicitly checked for each event. In addition, the threshold determined in the simulation is at the correct proton energy of 25.7 MeV. For each successful event, the sequence of velocity kicks is recorded so that the proton, neutron and 3He velocities are known. These constitute the most important output of the code, and determine the differential cross sections. 3.5. SECONDARY

INTERACTIONS

After the primary collision, we allow the struck neutron and proton to interact elastically with the residual ‘He nucleus, before the actual breakup or the proton

exit from the nucleus. We do not allow for inelastic scattering, which could further fragment the residual nucleus (and would contribute to other interesting reactions, such as 4He(p,ppn)D or 4He(p,ppnn)H). This second, elastic collision further rearranges momentum among the neutron, proton, and ‘He, giving them extra kicks, AP”, %, and Ap3, respectively. These momentum transfers sum to zero, and conserve the total kinetic energy of the system. In the Monte Carlo simulation, the struck nucleon and the proton are localized in the nucleus after the primary interaction. The residual ‘He nucleus position and velocity are also determined since the location of the 4He center-of-mass and its velocity are known. To find the probability of a second interaction, we introduce an energy-independent nucleon-nucleon scattering cross section u = rp2ri, where p is a free parameter, and constrain the other three nucleons to lie within a sphere of radius r, about the residual 3He nucleus center-of-mass. The optical depth for subsequent interactions is then determined by tracing the neutron or proton path through this “nucleus”. The end result is either (1) no scattering, (2) p-3He scattering, (3) n-3He scattering, or (4) three-body p-n-3He scattering. All secondary collisions are assumed to be elastic and to conserve angular momentum, thus only allowing a rotation of the momentum vectors about the two-body angular momentum axis in the center-of-mass. The angle of rotation is chosen uniformly between zero and 27r. Three-body events proceed via two consecutive two-body interactions, each of which separately conserves angular momentum and energy. The order of these interactions is specified by a time-of-flight condition. Given the location of the two different scattering events, the time it takes the n(p) to reach the scattering site, t,(t,), is known. Whichever nucleon reaches its scattering site first is assumed to interact first. This works well, provided the other nucleon does not leave the nuclear region during this time. In the event that this occurs, the interaction times are artificially halved. It is important to have some reasonable criterion to determine the order of events, since the overall momentum exchange depends on the ordering.

92

L. Bildsten et al. / Breakup reactions

In reality, the experimentally determined optical-model parameters (or, even better the phase shifts) for pPHe or n-3He scattering could have been used to determine the secondary scattering. This procedure however, is somewhat uncertain, since the p-3He(n-3He) scattering here is from a virtual, highly excited state that need not resemble the on-shell nucleus studied experimentally. In addition, our assumption of isotropic scattering in the center-of-mass overestimates the backscattering relative to what the optical model would predict. The frequency of secondary interactions depends on /3, and may be considerable. Fig. 5 shows the fraction of breakups that occurred with each type of secondary interaction as a function of /3 for an incoming proton energy of 100 MeV. The dashed (dot-dashed) line is the fraction of events where only the proton (neutron) interacted with the 3He. The solid line is the fraction of breakups in which both the neutron and proton interacted with the 3He. At low p, secondary interactions are infrequent because the residual nucleus is optically thin. As /3 is increased, it is clear that three-body secondary interactions dominate. The total sum saturates around p - 2 since the residual nucleus is then optically thick; the final outcomes in our breakup model are independent of p for /3 > 2, as we ignore tertiary and all higher order multiple scattering by the residual nucleus. Note that the fraction of events involving secondary interactions is less than one at saturation because the exit paths followed by escaping nucleons may not even intersect the 3He. (This effect is most pronounced at high energies, for which primary collisions occur fairly uniformly throughout the nucleus.) Our preferred value of p will be determined in sect. 4.2 through direct comparison to data. After the secondary interaction, the extra momentum kicks, Ap,,, App, and Ap3 have been determined. We do not allow for further interaction after this point. As described earlier, the proton will escape the nuclear region if it has adequate kinetic

-:

E,=lOO MeV, F=0.66 n and p both

--:

Fig. 5. Fraction the three types

p only only

:n

of secondary interaction events that occur as a function of p. The three curves represent of secondary interactions: n and p both interact with the 3He (solid), n interacts with the ‘He (dot-dashed), and p interacts with the 3He (dashed).

L. Bildsten et al. / Breakup reactions

93

energy. If it is bound, and cannot escape, we ignore these scattering events. This secondary scattering changes the excitation energy of the 4He, modifying the final momentum equation (eq. 15)) to read p: = f[ Qm + :Ap* + p,, * ( Ap + $Ap,, + +Ap,) + Ap,, - ( Ap + ;Ap,,) + AP, - C&b, + h,

+

(16)

;A~,11,

which reduces to the earlier eq. (15) for Ap, = Ap,, = 0. The total kinetic energy is conserved in these re-arrangements and was explicitly verified for each event. There are two important effects of the secondary interactions. By allowing the proton to interact again with the residual nucleus, the ‘He receives a forward kick, which serves to reduce the excitation energy in the original system, and to give the ‘He a larger energy in the lab, since it is not just a “spectator.” The neutrons that emerge are seen in a broader angular region, since primarily forward moving neutrons are scattered to larger angles when they encounter the residual nucleus. Both of these effects are necessary for qualitative agreement between our model and experimental data. Primary collisions alone - i.e. /3 = 0 - are generally inconsistent with experiment, as we shall see in sect. 4.2 below. 4. Comparison to experimental

data

Our calculation yields both the total cross section and the differential energy and angular distributions given the values for two free parameters. One is a reduction factor F in the nucleon-nucleon scattering cross section that accounts for the experimental fact that the nucleon mean-free path in a nucleus is longer than would be calculated from the vacuum nucleon-nucleon cross sections even when Pauli exclusion effects are incorporated 23). The other free parameter is /3, which weights the importance of secondary interactions. By comparing to the data, we fix these free parameters. Once these parameters are fixed, the model provides the absolute neutron and 3He distributions that can be directly compared to the experimental data.

4.1. REACTION

CROSS

SECTIONS:

THE VALUE

OF F

Experimental measurements of the reaction cross section alone are sufficient to fix F Because the total cross section is mainly determined by the probability that primary collisions occur, it only depends on p weakly. Fig. 6a shows our calculated cross section for F = 0.68 and 1, with /3 = 1. The experimental data points are from Nicholls et al. 16) at 141 MeV and the compilation of Meyer 17). Clearly, the F = 1 curve (dashed line) disagrees with all of the data. The F-value was varied to determine the best possible fit to the experimental cross sections. The solid line shows our best fit curve with F = 0.68, resulting in X*/(d.o.f.) = 0.5. Experimental data near threshold were excluded from this fitting since we have ignored Coulomb effects. Recent work on the mean free path of a nucleon in a nucleus suggests F - 0.6

94

L. Bildsren et al. / Breakup reactions 150

, .

-:

100 Proton

,

,

,

,

,

,

,

,

,

,

,

,

,

,

et al.

,

,

,

034

(p.pn) F=0.68

200

Lab Energy

,

: Sourkes

0 20

300

I’ lI,.,I1,,,II,,,,I”” 30 40

Proton

(&V)

Lab Energy

50

60

(MeV)

Fig. 6. (a) Calculated cross sections for “He(p, pn)‘He and 4He(p, ~P)~H compared to experiment. The filled circles are the experimental data for the (p, pn) reaction, while the dashed line is our calculated (p, pn) cross section for F = 1.0, and the solid line is for F = 0.68. The open circles are the experimental data of Nicholls et al. 16) at E, = 141 MeV and the semi-empirical values of Meyer I’) for the (p, pp) reaction, while our calculation is given by the dot-dashed line for F = 0.68. Table 2 shows the direct comparison between theory and experiment. (b) Total inelastic cross section at low energies. The solid line is the sum of (p, pn), (p, 2p), (p, d), while the dashed line is the sum of only the first two, which are computed in our model. The dot-dashed line is the fit of Gould 13) for (p, d). The solid triangles are the experimental data of Sourkes et al. 37) and the solid circles are from Meyer 16). The thin vertical dashed lines show the thresholds for (from the left) (p, d), (p, 2p), (p, pn).

TABLE 2 Comparison

between

theory

and experiment

E, (MeV

uexpbb)

a,,, (mb)

28 30 50 53 90 141 300

4.8zt1.3 10*5 43*8 49*8 46*8 36k1.5 27”:’

(P, pn) 9.6 18.7 52.6 53.8 47.4 35.7 29.6

Wickersham 35) Griffiths/Harbison 36) Griffiths/Harbison 36) Meyer “) Tannenwald I*) Nicholls et al. 16) Meyer “)

28 53 90 141 300

8.9+ 1.0 17*3 17*3 10.8 * 0.6 16? 4

(P. PP) 5.0 18.2 18.1 19.3 23.0

Wickersham 3S) Meyer “) Meyer “) Nicholls et al. 16) Meyer 17)

Ref.

L. Bildsten et al. / Breakup reactions

95

[refs. 23’4)]. Table 2 compares our calculated cross sections for F = 0.68 with the experimental values at several incident proton energies. At energies near the threshold, our cross section is high; including Coulomb suppression in the exit channel would result in closer agreement with the experimental values. An accurate analytic fit to our calculated (p, pn) cross section is for F = 0.68 and p = 1 Q&Xl)(E) = 58.04

2 1+exp[(Eth-E)/6.8]-1

1

1 1 + 6.73 exp (-E/32.25)

)I 3

mb,

(17)

where E is the proton lab energy (in MeV) and Et,, = 25.72 MeV is the threshold energy for the reaction. This fit has an average deviation of 1.4% from our numerical results. This cross section approaches u(,,~,) (CO)= 29.5 mb at high energies. We have also determined the cross section for the inelastic channel 4He(p, ~P)~H in our model for F = 0.68; this is shown as the dot-dashed curve in fig. 6a, and is compared with experimental data in table 2. The data points for (p, 2p) are taken from Meyer 17) at E, = 53,90, and 300 MeV, and result in a x2/(d.o.f.) = 0.75 in comparison to our results. Meyer’s 17) cross sections are not experimental measurements, but rather semi-empirical inferences from other cross section measurements at the same energies. Well above threshold, our model is in greatest disagreement with the value at 141 MeV, but Nicholls et al. 16) suggest that their value may actually be an underestimate. An accurate analytic fit to our model is 2

q,,,,)(E) = 18.54 l+exp [(Efh-E)/5.84]-1 O.l4(E/llO-1) ‘+l+exp[(70-E)/5.266]

1 1’ mb

(18)

where Et,.,= 24.77 MeV is the threshold for the 4He(p, 2p)3He reaction. This fit has an average deviation of 1% from the numerical results. This cross section increases with energy at high energy due to the slight increase of the p-p elastic scattering cross section. The total reaction cross section was measured by Sourkes et al. 37) for incident proton energies from threshold to 48 MeV. In this energy range, the dominant inelastic channels are (p, d), (p, 2p), and (p, pn). A convenient parametrization of the (p, d) cross section was given by Gould 13) including Coulomb effects. Adding Gould’s 13) cross section to our results, we can directly compare to the experimental data, as is shown in fig. 6b. The solid line is the sum of (p, d), (p, 2p) and (p, pn), the dashed line is (p, 2p) + (p, pn), and the dot-dashed line is the fit by Gould 13) for (p, d). Our cross sections were computed without Coulomb suppression, resulting in a slight overestimate in this low-energy range (particularly between 30 and

L. Bildsten et al. / Breakup reactions

96

40 MeV). The measured excited

inelastic

cross section

state of ‘Li in the (p, d) reaction

ref. 38) for a discussion energy discrepancy, inelastic

complete

phase-shift

our model agrees remarkably

cross section

4.2. DIFFERENTIAL

and

is high near threshold

that was not included analysis].

because

in Gould’s Aside

of an fit [see

from this low

well with the experimental

p-4He

at E < 50 MeV.

CROSS

SECTIONS:

THE

VALUE

OF /3

Comparisons of the detailed energy-angle distributions with available data can now be used to fix /3. The experimental neutron energy and angular spectra at E, = 141 MeV are shown in figs. 7a and 7b as triangles. These data are from Nicholls et al. 16), who measured a total of 842 (p, pn) events. They measured the neutron energy spectra for five different angular ranges up to to 50”. We summed these energy spectra over angles, to obtain da/d& which is shown in fig. 7a for all neutrons with lab angles less than 50’. The vertical error bars represent m deviations. Horizontal error bars should also be shown, since some of their neutron energies are uncertain by as much as 30 MeV, but are ignored for simplicity. Our numerical results are shown for /3 = 1 (filled circles, connected for p = 0 (open circles). To incorporate the large experimental

by a solid line), and uncertainties in the

energies of the neutrons, we also assigned equivalent fractional to our results at the same percentage as the experimental data, in the comparison. (The displayed points do not include this The dashed line is the spectrum frequently used in solar flare

energy uncertainties with little difference slight modification.) models [Murphy ef

al. 39), see determination

2

,,,,,,,

,,,

/,,

,,I

B]. Simulations

with either

p = 0 or p = 1

,,,

,,,1,,,1,,,1,,,1,,,1,,I:

0

0

20

40

Neutron Fig. 7. angles results are for

in appendix

80

Lab Energy

80

100

(i&V)

120

0

20

Neutron

40

Lab Angle

60

80

(Degrees)

(a) Energy spectra for neutrons produced at a proton beam energy of E,= 141 MeV with less than 50”. The triangular points are the data from Nicholls et al. 16). The open circles are of our calculation for no second interactions (/3 = 0). The filled circles connected by a solid fi = 1. The dashed line is the Murphy, Dermer and Ramaty 39) model. (b) Angular spectra neutrons produced at a proton energy of E,= 141 MeV.

lab the line for

L. Bildsfen et al. / Breakup reactions

91

generally agree better with the experimental points in fig. 7a than the model of Murphy et al. 3g). Fig. 7b shows the experimental angular spectra and the comparison to our results; clearly the /3 = 0 points (open circles) are too forward peaked. Secondary interactions broaden the angular distribution, and the p = 1 results in fig. 7b agree with the data better than the /3 = 0 results. However, we cannot unambiguously choose /3 from this comparison, as different non-zero /3 - 1 agree with the data more or less equally well. To further pin down the value of /3, we use da/dE and da/d0 for the protons and 3H from the reaction 4He(n, np)3H at an incident neutron energy of 90 MeV measured by Tannenwald 18). Since this reaction is simply the mirror of 4He(p, pn)3He simulated here, we can use it in a direct comparison by identifying protons with neutrons and the 3H as 3He. (The difference in the threshold energy for these two reactions, 1.8 MeV, is unimportant at the incident energy of 90 MeV.) Fig. 8a shows the angular spectra for the produced protons. Once again, the p = 1 curve (solid line) agrees better with the experimental data than either p = 0 or the Murphy et al. 3g) model. The same can be said for fig. 8b, which shows the agreement between theory and experiment for the proton energy spectra. Tannenwald’s data on the 3H provides a very striking example of the importance of secondary interactions. Fig. 8c shows the angular spectra for the 3H. For p = 0 backscattering is much more pronounced than in the data, but for p = 1, simulation and experiment agree rather well. Fig. 8d compares the 3H energy distribution. The solar flare model (dashed line) does relatively poorly, whereas we pick up the low-energy peak to a greater degree. 4.3. FURTHER

COMPARISONS

WITH

EXPERIMENT

By comparing to the experimental data 16*18),we have determined F = 0.68, and chosen /3 = 1 as our “best” model values. The choice p = 1 is not rigorous in the sense of a minimum x2, but should instead be viewed as a reasonable guess. Our model is now complete and has no additional adjustable parameters.

Thus, for an

arbitrary incident proton energy, we can provide doubly diferential cross sections that need no additional normalization. Now that our model parameters are fixed, we can compare to additional experimental data. A much larger amount of experimental data is available on the related inelastic reaction 4He(p, ~P)~H. Since the primary goal of this work was to obtain the constituent proton momentum distribution in 4He, only a limited fraction of phase space was measured for most of these experiments. However, the recent experiment by Wesick et al. I’) determined the inclusive proton spectra from proton bombardment of D, 3He, and 4He separately at incident energies of E, = 98.7 MeV and E, = 149.3 MeV. Proton energy spectra were measured at four angles (17.5”, 30”, 45”, and 60”) at 98.7 MeV and just the two lowest angles at 149.3 MeV. Since these proton energy spectra contain contributions from all inelastic processes we

L. Bildsten et al. / Breakup reactions

98

.fJ

,

,

,

,

,

,

,

,

,~.._...____,

0 0

100

50

150

I 0

Proton Lab Angle (Degrees) .5

,\I,,

,,,@

,,,,

I

I

I

I

20

50

100

I

I

I

,

,

,

Exp. Data (U /3=1.0 @=O.O Murphy et al. _ (F&=QO arev)

I

I

40

I

>\,I

I

60

80

Proton Lab Energy (MeV) ’

,,

I

(C)T .: Exp. Data e &?=l.O 0: #=O.O -m: Murphy et al.&=90 YeV) 1

0

I

,

. : *: 0 : -~:

150

I

,

I

I

I

. : Exp. Data (d) 1 *: f3=1.0 0 Murphy et al.&=90 YeV) _

: #9=0.0 --:

20

0

40

60

“H Lab Energy (MeV)

3H Lab Angle (Degrees)

Fig. 8. (a) Angular spectra of protons produced at a neutron beam energy of 90 MeV in the reaction 4He(n, np)3H. The triangular points display the experimental data of Tannenwald I*). Labels are the same as in fig. 7. (b) Energy spectra for protons in the lab frame for a beam energy of E, = 90 MeV. (c) Angular spectra of 3H produced at a beam energy of E, = 90 MeV. (d) Energy spectra of 3H produced at a beam energy of E, = 90 MeV.

cannot

perform

a direct comparison.

Instead,

we determined

the total contribution

to the proton spectra from the two reactions 4He(p, ~P)~H and 4He(p, pn)3He that can be simulated in our model. These results should then always lie below the experimental curve since they exclude the contributions from the other inelastic processes. This is a nontrivial test of our model. Figs. 9a-9d compare the model versus the experimental data for /I = 0 (open circles) and p = 1 (solid line) at Ep = 98.7 MeV. Fig. 9a shows proton energy spectra at the lab angle 17.5”, and comparisons at larger angles are shown in figs. 9b-9d. Figs. 10a and lob are for the incident proton energy of 149.3 MeV. (We summed our numerical results over an angular bin width of A cos 0 = 0.04, which is slightly larger than the experimental width.) Clearly, p = 0 is in complete disagreement with the data at 17.5”, as the model values exceed most of the experimental ones, but /3 = 1 is consistent. While

L. Bildsien et al. / Breakup reactions 1.2

, -

1 _

,

I

I

1

.9,=17.5*, q=ee.7 Filled: ~=l.O.

1

I

,

I

I

99

,

YeV

.

Open:fl=o.o

m

>,: -5 _

.

/’

.B -

I-

J

2 -

o - (a) vt 6

,

I

I

I

I

I

I

20

Proton .5

La? Energy

/ , , , , _ 0,=45’, E,=W3.7

.4 _

I

I

I

Filled:

@=l.O.

, , ,

80

,

I

20

(MeV)

,

.3,,,

I

I

I

40

Proton

, , ,

I

I-80

60

Lab Energy

,,I

I

I

,,,

(MeV)

,

,,,

f3=0.0 ys

,“,

,,‘i ‘.._I ’

\\\\

-,,,’

(b)

YeV

Open:

,____^_--

o

,--I

‘60

- s k

.3 -

-2 -2 L

2 -

9

-

%

-2 -

.l -

20

40

Proton

60

Lab Energy

80

(MeV)

%

20

60

Proton ?ab

Finer:;

(l&V)

Fig. 9. Inclusive proton spectra at lab angles (a) 17.5”, (b) 30”, (c) 45”, and (d) 60” for a proton bombarding energy of E, = 98.7 MeV. The dashed line displays the experimental data from Wesick et al. 19), which includes protons from all inelastic processes. The filled circles connected by a solid line show our determination of the contributions to the inclusive cross sections from (p, 2p) and (p, pn) reactions for B = 1.0. The open circles display the results for B = 0.0.

it is difficult to obtain a quantitative assessment of how well our theory compares to the experiment because of the unknown contributions from the other inelastic channels (4He(p,pd)D, 4He(p, ppn)D, etc.), we feel that our model is in better agreement than the DWIA calculations 19) which require energy and angle-dependent normalization factors to obtain satisfactory agreement with the data and, moreover, show quasi-free spectral peaks that are absent in both the data and our simulated spectra. 5. Implementation of the results The results of these calculations are needed for use in astrophysical applications. Useful fitting formulae for the total cross sections were given in sect. 4.1, but finding

100

L. B&&ten et al. / Breakup reactions

.6-

I”

’ I”

4=17.5’.

Filled:

‘I”

E,=149.3

@=l.O.

II”

HeV

’ I( “I

I”

I I ” - 0,=30’,

h

Open: fl=0.0

1 1 ‘1 E,=149.3

_ Filled:

p=i.O.

I’

I ’ / 1”

I”’

Open:

,/‘;

f3=0.0

,/’ _A ,f _/’ ,___ ,___, _,_c-~d Ias----e

P

20

m

m

=

:

Protor%b

120

140

,/’

m

I,,

; I

m \

,,,I,,,

20

Lrp:y!kv,

,**’

-

/II/I 40

1

YeV

120 P4rq7ton

&

Gne?gy

(h$

Fig. 10. Inclusive proton spectra at lab angles (a) 17.5”, and (b) 30” for a proton bombarding energy of ED= 149.3 MeV. Labels are the same as in fig. 9.

simple representations of the energy and angular distributions is more difficult. Since this reaction is just a small part of a much larger problem, we need a simple recipe for incorporating the resulting distributions. We feel that it is most accurate and economical to simply tabulate the numerical data, and then interpolate from these stored tables. Eventually, we intend to implement these results in a Monte Carlo code, so we need a fast way to sample neutron (or 3He) energies and angles from the known distribution for a given incident proton energy. (Since we are interested in large numbers of reactions, it is sufficient to sample neutron and 3He distributions in an uncorrelated way.) It proves convenient to sample in the center-of-mass frame, where the allowed n and 3He energies are limited kinematically (see derivation in appendix B). The neutron (or 3He) energy and angle in the lab frame can be obtained by a galilean boost. We express the neutron and 3He energies in the center-of-mass in terms of the maximum allowed energies, i.e. x, = En/E,, and x3 = 2 E3 /E,, where E,,, is the maximum neutron energy possible after the collision. (The maximum 3He energy is half of the maximum neutron energy.) Since the collision is azimuthally symmetric, we need only consider the polar angle 8 = cos-’ /-L.We define a probability distribution h(x, p) so that h(x, p) dx dp is the probability that a final-state particle (n or 3He) has an energy between x and x +dx and an angle between p and ~1+d+ We can always rewrite this as h(x, p) = hO(~)hl(xI~), where

J

1

ho(P)

=

h(x,

~1

dx.

(19)

0

ho(p) dp is the probability that a particle has an angle between p and p + dp, irrespective of its energy. The other distribution, h,(xI,u} dx, is the probability that a particle has energy between x and x + dx for a given angle ,u. To randomly choose an angle from this distribution, we choose a random number rP (which is unifo~ly Thus,

L. Bildsten et al. / Breakup reactions

101

distributed between 0 and 1) and equate it to the cumulative angular distribution, P r,=g&)= _, h&) dF. (20) I Fig. 11 shows g,,(p) for neutrons (solid) and ‘He (dashed) for a proton energy of 100 MeV. The slope of these curves give the probability distribution, h&). The steep slope for neutrons as p -, 1 shows that neutron emission is forward peaked in the center-of-mass. The opposite is true for 3He, which is peaked in the backwards direction. Thus the collision dynamics preserve to some extent the original direction of relative motion. The scaled distribution g,,(p) is rather similar at different incident proton energies, but is not exactly energy-independent. In practice, we store the function go(p) in 40 equal probability bins, which means that each bin has a different width dp, depending on the slope (h,(p)) at that point. Once p has been determined, we find the dimensionless energy x by choosing a random number r,, and solving r,=h,(x)=

X Wx’b) I0

dx’ ,

(21)

for x. For each of the 40 angular bins, we store the neutron energy spectrum in 40 equal probability bins. Figs. 12a and 12b show h,(x) for 9 of the 40 angular bins. From the different curves in figs. 12a and 12b, we see that 40 bins gives a fairly good representation of the variability of h(xIp). Note that the forward beamed particles tend to have higher average energy than the backward directed particles. This is because backward directed neutrons arise mostly due to secondary interactions, and thus have lost energy to the 3He. Figs. 13a and 13b show h,(x) for 3He. The functions go(p ) and h,(x) are tabulated at incident proton energies separated by equal 10 MeV intervals between 30 and 280 MeV. To sample at energies between

-

: Neutrons

.a - --: ke

Fig. 11. The cumulative

angular distribution in the center-of-mass of the neutrons (dashed) for an incident proton energy .EP= 100 MeV.

(solid)

and 3He

102

L. Bildsten et ai. / Breakup reactions 1

1

,,,,,,,,,",'ll,"* Neutrons, E,=lOOMeV

',,/,"/,,,,'1',"* Neutrons, Ep=lOO HeV

0.

b: -0.017+<0.075 A: -0.453+<-0.360 0: -0.82O
.*** .69 ' I:".' _.q'"a : _*mo _a. LMo* ,*

IADDl* .- nL(**

“0

2

.6

.4

.6

1

0

.2

.a

.4

1

x,

%

Fig. 12. The cumulative energy distributions of the neutrons for different angular bins. Part (a) shows tive bins for angles in the forward direction, while part (b) shows four bins in the backward direction. The bombarding proton energy was Er = 100 MeV.

points on our energy grid, we linearly interpolate between the two nearest tables. This is done by choosing an x and CLfor each of the neighboring incident proton energies (using the same set of random numbers) and linearly inte~olating to obtain x and ,u for the desired incident proton energy. With this method, the neutron and ‘He distributions can be obtained for any desired proton energy between 30 and 280 MeV. To give a concrete example that shows the accuracy of this method, we generated 100 000 neutrons and 3He particles for an incident proton energy of 150 MeV through an interpolation from the tables are 140 and 160 MeV. A table was constructed at 150 MeV from these data and compared to the table made directly from the numerical

.8

.6

.4

.2

0 “0

.2

.6

.4 xs

.6

1

0

.2

.B

.4

.6

x3

Fig. 13. The cumulative energy dist~butions of the %e for different angular bins. Part (a) shows five bins for angles in the forward direction, while part (b) shows four bins in the backward direction. The bombarding proton energy was E,, = 100 MeV.

1

L. Bildsten et al. / Breakup reactions

103

simulation. The absolute differences in the entries between these two tables provides a measure of the interpolation error. The maximum deviation was 0.05, with an average deviation of 0.01. Thus, the average deviation in energy or angle determination due to interpolation is roughly 1%. Since we store tables in 10 MeV intervals, in practice we will never interpolate over an energy range this large, so our implementation errors should be even smaller. Since these distributions are of interest to the astrophysical community, the tables and the FORTRAN code needed to implement them are available from the authors upon request. 6. Conclusions We have calculated detailed neutron and 3He energy-angle spectra for the inelastic reaction 4He(p, pn)3He in a semi-classical model with only two free parameters. Our results agree well with the currently available experimental data for particular values of the two free parameters. This calculation provides a new description of the low-energy spallation from light nuclei and can be extended to nuclei other than 4He. Our semi-classical picture incorporates most of the relevant physics and allows for a rough determination of the importance of multiple scattering effects through a direct comparison to the experimental data. Now that our free parameters are fixed, further comparisons to new data can be made directly, with no modifications allowed. The reaction studied here is of astrophysical importance in solar flares and accreting neutron stars, where these new results will be applied. The main results reside in sect. 4.1, where we provide convenient fitting formulae for the reaction cross sections for 4He(p, pn)3He and 4He(p, ~P)~H, and in sect. 4.2 where we compare directly to the experimental data. The method that we use to implement these results in a larger simulation is described in sect. 5. The tables described therein are available to the reader upon request. In addition, appendix A contains new fitting formulae for n-p and p-p elastic scattering cross sections that might be of some general use. This work was supported in part by NASA grant NAGW-666 and NSF grant AST89-13112. L.B. thanks the Fannie and John Hertz Foundation for fellowship support. Appendix A NUCLEON-NUCLEON

SCATTERING

complete phase shifts are available for n-p and p-p scattering up to 1 GeV [ref. “)I. These would provide the most accurate parameterization possible, but their implementation would be too time-consuming in a larger simulation. Instead, The

104

L. Bildsten et ai. / Breakup reactions

with a limited sacrifice of accuracy, we use semi-phenomenological fitting formulae to generate the total cross sections. We r 280 MeV in the lab) are also needed. The experimental nucleonnucleon scattering data used to perform the fits were taken from the compilations of Hess 4’), Barashenkov and Maltsev 42), and Wilson 43). [More recent data are compiled (though not explicitly displayed) in refs. 44”5).] Gammel 46)performed an empirical fit to the n-p experimental data below 40 MeV using an analytic form based on effective range theory. The n-p elastic cross section is %,,(E) =

3a l,206E+(-l.86~0.~415E+0.00013~E2~2 +

7r

l.206E+(0.4223+0.13E)2

b,

64.1)

for a neutron target at rest, where E is the proton lab energy in MeV, For higher energies we performed a linear, least squares fit to the experimental data. In our code, we use the following formulae for n-p elastic scattering for neutrons initially at rest as a function of incident proton (lab) energy E in MeV. Below 38.22 MeV, Gammel’s fit is used (eq. (A.l)), whereas between 38.22~ E < 111.511 MeV we use 5673.25 + 113412 c&E ) = 4.323 + ~ ---mb, E E2 and for 111.511
(A-2)

use

CT&?) = 28.1451+-

1431.74+297627 -mb. E E2

(A-3)

The resulting x’/(d.o.f.) for data points above 10 MeV was -2. The n-p differential cross section in the center-of-mass is purely s-wave at energies below - 10 MeV, and involves higher partial waves at higher energies. Knowing the phase shifts, the differential cross section can be easily generated. Instead, we use a simple empirical form introduced by Gammel 46), for the differential cross section in the center-of-mass (which he admits has “practically no theoretical basis”), do,, CT,&?) l-i- ii cos* B l+;d ’ da - 4~ where

(A.4)

105

L. Bildsten et al, / Breakup reactions

and E is the lab frame proton energy. When used with our fit for a,,(E), this form agrees with the data given in Wilson 43) quite well, even at energies as high as 200 MeV. Except for upturns at low angles due to Coulomb scattering, the p-p differential cross section is essentially flat in the center-of-mass over a large energy range 47). This is surprising since many partial waves contribute to the differential cross section. In our simulations, we assume that the nuclear part of p-p scattering is isotropic in the center-of-mass. At low energies (E < 10.54 MeV), we used an effective range determination of the nuclear part of the p-p scattering that incorporates the Coulomb barrier. At very low energies (E < 1 MeV), it becomes highly unlikely that the protons can tunnel to a separation small enough to see the nuclear force. Thus, at low energies, the nuclear cross section goes to zero. Fitting the form generated through the effective range theory gave 229.510

6596.46

3920.61

1

-(E+0.62)+(E+0.62)2-(E+0.62)3 mb' where E technique difference scattering found for

L4.6)

is the proton lab energy in MeV. At higher energies we used the same to fit the data as we did for the high energy n-p scattering. The only is that at energies above the pion-production threshold, the p-p elastic cross section seems to increase somewhat. The best fitting formulae we energies above 10.54 MeV were

a,,(E) = -2.0331+-

2690.66 + 6498.86 -mb,

E

(A-7)

E2

for 10.541~ E < 42.738 MeV, 454.414+65 760.5 upp( E) = 17.8465 +-mb,

E

(A8)

E2

for 42.738 < E < 158.555 MeV, and app(E)=11.7386+0.0189E+-

1362.11

E

mb,

(A.9)

for 158.555 < E < 650 MeV. These formulae were determined by a least squares fit to the experimental data in different energy ranges. They were then connected at their intersections. Between 10 and 300 MeV, the resulting X2/(d.o.f.) was 3.8 for our fit. Appendix B SIMPLE

NEUTRON

AND ‘He ENERGY-ANGLE

DISTRIBUTION

Murphy, Dermer, and Ramaty 39) made the following assumptions to obtain the neutron energy-angle distribution in the p-4He center-of-mass: (1) isotropic angular

L. Bildsten et al. / Breakup reactions

106

distribution, and (2) flat energy distribution up to a maximum neutron energy E,,,. This maximum energy occurs when the neutron moves anti-parallel to the proton and ‘He in the center-of-mass, and is &I3= W,+

Q) t

(B.1)

where E,, is the proton lab energy. The double differential cross section in the p-4He c.m. is then d2a =LT O(E,,,--EL), dE,!,dp; 2Em

(B-2)

where 0 is the Heaviside step function, which is 1 for a positive argument and zero otherwise. The lab-frame distribution is obtained by transforming into the lab frame. This results in

03.3)

where E,, is the neutron energy in the lab, and p,, = cos 8 is the cosine of the angle between the neutron and the proton beam in the lab. At a particular angle in the lab frame, there is then a maximum neutron energy which can appear, which is given by E y=

E,+&E,(2pz-~)+$LE,~E,,‘E,,+&(~~-~).

03.4)

Similar relations can be derived for the 3He distribution. The maximum 3He c.m. energy is half of that of the neutron. If we make the same assumptions about its energy and angle distribution in the cm., the resulting lab distribution is 1/Z u E3 -=_d2v > dE, dpu, E, ( E3+&Ep-&m x @(SE,-

E,+&E,,+$,m).

distributions are used in the paper for the purpose of comparison data and our results. These

03.5)

to both

References 1) H.W. Bertini, Phys. Rev. 131 (1963) 1801 2) S.L. Shapiro and S.A. Teukolsky, Black holes, white dwarfs, and neutron stars (Wiley, New York, 1983) 3) Ya. Zel’dovich and N. Shakura, Sov. Astron-A. J. 13 (1969) 175 4) M. Alme and J. Wilson, Astrophys. J. 186 (1973) 1015 5) A.K. Harding, P. Mtszlros, J.G. Kirk and D.J. Galloway, Astrophys. J. 278 (1984) 369 6) G.S. Miller, E.E. Salpeter and I. Wasserman, Astrophys. J. 314 (1987) 215 7) V.F. Schvartsman, Astrophysics 6 (1972) 56 8) K. Brecher and A. Burrows, Astrophys. J. 240 (1980) 642 9) J.D. Kurfess et af, in Gamma-Ray Observatory Science Workshop, 1989, ed. W.N. Johnson, Goddard Space Flight Center Special Publication, p. 3-35

L. Bildsren et al. / Breakup reactions

107

10) V. SchSnfelder, in Gamma-Ray Observatory Science Workshop, 1989, ed. W.N. Johnson, Goddard Space Flight Center Special Publication, p. 3-24 11) R. Ramaty and R.J. Murphy, Space Sci. Rev. 45 (1987) 213 12) X.-M. Hua and R.E. Lingenfelter, Solar Physics 107 (1987) 351 13) R.J. Gould, Nucl. Phys. B266 (1986) 737 14) N. Guessoum and R.J. Gould, Astrophys. J. 345 (1989) 356 15) N. Guessoum and C.D. Dermer, AIP Conf. Proc., no. 170, Nuclear spectroscopy of astrophysical sources, ed. N. Gehrels and G.J. Share (American Institute of Physics, New York, 1988) p. 332 16) J.E. Nicholls et al. Nucl. Phys. A181 (1972) 329 17) J.P. Meyer, Astr. Astrop. Suppl. 7 (1972) 417 18) P.E. Tannenwald, Phys. Rev. 89 (1953) 508 19) J.S. Wesick et al. Phys. Rev. C32 (1985) 1474 20) M.L. Goldberger, Phys. Rev. 74 (1948) 1269 21) N. Metropolis et al. Phys. Rev. 110 (1958) 185 22) R. Serber, Phys. Rev. 72 (1947) 1114 23) J. Negele and K. Yazaki, Phys. Rev. Lett. 47 (1981) 71 24) S. Fantoni, B.L. Friman and V.R. Pandharipande, Phys. Lett. BlO4 (1981) 89 25) G. Jacob and T. Maris, Rev. Mod. Phys. 38 (1966) 121 26) H.G. Pugh et al. Phys. Lett. B46 (1973) 192 27) R. Frascaria et al. Phys. Rev. Cl2 (1975) 243 28) W.T.H. van Oers et al. Phys. Rev. C25 (1982) 390 29) T. Sawada, G. Paic, M.B. Epstein and J.G. Rogers, Nucl. Phys. A141 (1970) 169 30) R.D. Haracz and T.K. Lim, Phys. Rev. Cl0 (1974) 431 31) P.G. Roos, in AIP Conf. Proc., No. 36, Momentum wavefunctions, ed. D.W. Devins (American Institute of Physics, New York, 1976) p. 32 32) A.N. Antonov, P.E. Hodgson and I.Zh. Petkov, Nucleon momentum and density distributions in nuclei (Oxford Univ. Press, New York, 1988) 33) J.S. McCarthy, I. Sick and R.R. Whitney, Phys. Rev. Cl5 (1977) 1396 34) S. Fiarman and W.E. Meyerhof, Nut. Phys. A206 (1973) 1 35) A.F. Wickersham, Phys. Rev. 107 (1957) 1050 36) R.J. Griffiths and S.A. Harbison, Astrophys. J. 158 (1969) 711 37) A.M. Sourkes et al. Phys. Rev. Cl3 (1976) 451 38) G.R. Plattner, A.D. Bather and H.E. Conzett, Phys. Rev. CS (1972) 1158 39) R.J. Murphy, C.D. Dermer and R. Ramaty, Astrophys. J. Suppl. 63 (1987) 721 40) R.A. Arndt et al. Phys. Rev. D28 (1983) 97 41) W.N. Hess, Rev. Mod. Phys. 30 (1958) 368 42) V.S. Barashenkov and V.M. Maltsev, Fortschr. der Phys. 9 (1961) 549 (BM61) 43) R. Wilson, The nucleon-nucleon interaction (Wiley, New York, 1963) 44) R.A. Arndt, R.H. Hackman and L.D. Roper, Phys. Rev. Cl5 (1977) 1002 45) R.A. Arndt, R.H. Hackman and L.D. Roper, Phys. Rev. C9 (1974) 555 46) J.L. Gammell, in Fast neutron physics, part 2, ed. J.B. Marion and J.L. Fowler (Wiley, New York, 1963) p. 2185 47) P. Marmier and E. Sheldon, Physics of nuclei and particles, vol. 2 (Academic Press, New York, 1969) p. 1076