emoram

to th_

pectator

Aipha-

...................................................................................-:;

race

(NASA-T_-42_,2) Cc?4Pt:C[I. UNS TO r HzcPARTICIPANT-SPECTaTOR MnE/EL OF HIGH-ENERGY

ALPHA-PARTICLF FRAGMENTATIL]N (NASA) 13 p

CSCI 20H Unclas

Hill3 0329510
+
NASA Technical Memorandum 4262

Corrections to the

Participant-Spectator Model

of High-Energy Alpha-Particle

Fragmentation

Francis A. Cucinotta, Lawrence W. Townsend,

and John W. Wilson

Langley Research Center

Hampton, Virginia

John W. Norbury

Rider College

Lawrenceville, New Jersey

National Aeronautics and

Space Administration

Office of Management

Scientific and Technical

Information Division

1991

Symbols and Notation

Particles and fragments:

a, b projectile fragments

N nucleon

n neutron

P projectile

p proton

T target

v virtual particle

X final target state

a alpha particle

Symbols:

Ai

B

Di

D (p, q)

FSI

fiT

i

K

Ki

Ks, gp

ki

kij

k, k I

mi

Pi

Pi

mass number of particle i

slope parameter

energy of particle i in center-ofmass frame

dispersion integral (eq. (20))

energy of particle i

final-state interactions

scattering amplitude of fragment i

imaginary number

phase space factor

4_momentum of virtual particle i

defined in equations (9) and (12)

momentum of virtual particle i

relative momentum of virtual

particles i and j

intermediate-state relative momenta

(eqs. (17) and (18))

mass of particle i

4-momentum of particle i

momentum of particle i

Pij

/)lab

q

Ts,rp

T ,Tp

tij

V

Y ,Yp

£S

Oi

Otab

#0

P

O"

+

¢

relative momentum of particles i

and j

laboratory momentum, MeV

total momentum transfer

momentum of particle i in center-ofmass frame

double-scattering contribution to

transition matrix

contributions to transition matrix

from spectator and participant

terms

distorted transition matrix

contributions

full transition amplitudes for

interaction between particles i and j

normalization volume

defined in equations (I0) and (13)

= -- 2#ab_ s

relative velocity between particles i

and j

infinitesimal energy

separation energy

emission angle of particle i

laboratory emission angle, deg

reduced mass of particles i and j

ratio of the real to imaginary parts

of the forward scattering amplitudes

cross section, mb

relative cluster momentum

distribution

distorted relative cluster momentum

distribution

overlap function

solid angle of emission of particle i

nl

PRE'CEDI['LG P/_GE t_LAHK NOT F:LMED

Abstract

The participant-spectator model of nuclear fragmentation is described in terms of pole graphs from direct reaction theory. Corrections to the model for more than one projectile fragment scattering on

the target are considered using a triangle graph model. Results for

alpha-particle fragmentation at 1 GeV/A indicate that corrections to

the participant-spectator model are significant, as indicated by the large

interference effects found between the pole and triangle graph terms in

the double- and single-differential cross sections.

Introduction

The description of biological damage from galactic cosmic rays (GCR) ultimately depends on the

track structure of energetic ions in tissue (refs. 1

and 2). Risk assessment for deep space missions requires accurate transport codes for determining the

differential flux of ions behind natural and protective

radiation shielding. Previous studies (refs. 3 and 4)

have indicated the importance of the nuclear fragmentation data base in developing such transport

codes. Nuclear fragmentation drastically alters the

composition of ion fields, and its proper description

is essential for track structure models or any fluencebased risk system.

For high-energy reactions, the participantspectator model (ref. 5) describes the dominant

peripheral channels where only a small number of

projectile fragments are produced. These move in the

forward direction with velocities near that of the projectile. The nuclear abrasion process occurs for large

impact parameters when the overlapping volumes of

the projectile and target nuclei, called participants,

are sheared off in the collision. The remaining portion of the projectile, the spectator, is assumed to

receive only a small momentum transfer in the collision. The spectator fragment may be in an intermediate excited state (prefragment stage), decaying

to the final fragment through particle evaporation in

the ablation step of the reaction. In contrast to the

peripheral breakup channels, there are central collisions when almost complete overlap of projectile and

target volumes leads to a multiplicity of fragments in

a wide cone of emission angles.

In previous work (refs. 6 and 7), we have considered the diagram approach to direct reaction theory

(ref. 8) for describing the abrasion step in terms of

dispersion pole diagrams. In this work, we consider

corrections to the pole diagrams for two-body dissociation in order to estimate contributions when

more than one projectile fragment interact strongly

(participate) in the reaction. The direct reaction

graphs with singularities closest to the physical values of the fragment variables give the dominant contributions to the cross sections. For the direct reaction approach to be useful, only a few dispersion

graphs should contribute over the kinematical region

of interest. In references 6 and 7, we showed that the

single-pole diagram corresponding to the participantspectator assumption saturates the production cross

section only if the mass of the fragment of interest is much larger than that of the participant fragments. For the lightest nuclei, and for some dissociation channels for heavier projectiles, fragments with

comparable" mass are produced in a single channel.

Rescattering corrections may then become important

and are investigated herein for 4He projectiles (alpha particles) fragmenting on 1H targets. For heavier systems, the multiple scattering approach considered here is expected to be modified by using the

high-energy optical model (refs. 9 and 10) to properly account for distortion and cascade effects and by

treating the ablation step according to the methods

in reference 11.

Pole Diagrams

Consider the two-body dissociation of a projectile

P fragmenting on a target T:

P+T-_a+b+X (1)

where X is the final target state, and a and b are the

projectile fragments. The transition matrix Tfi for

this reaction is related to the momentum distribution

for producing the fragment a by

dp---_ da V 3 /---- (27r)5/3pT dCt b KJT.fil 2 (2)

whe.re fl is the relative velocity in the initial state, K

is the phase space factor, and V is the normalization

volume; a summation over all final target states X is

implied. In the overall center-of-mass frame (CM),
PT PX

K v

Pb

Pp

ea

Figure 1. Spectator term for projectile fragmentation.

PT PX

Pp

K v

ea

Pb

Figure 2. Participant term for projectile fragmentation.

assuming azimuthal symmetry about the beam direction, we have

dEbEx (3)

K = pb(E b + EX ) + paEb cos(0a + Ob)

The pole diagram for the spectator contribution

(Serber term), where the observed fragment is assumed to avoid interaction with the target, is shown

in figure 1. A first correction to the spectator model

is to reverse the roles of the participant and spectator, with the observed fragment interacting with the

target as shown in figure 2. These terms have simple

poles at the value of the interacting fragment's mass

(ref. 8) and both contribute at small Pa if the masses,

ma and mb, are comparable (refs. 6 and 7). Here the

singularities of both graphs are relatively close to the

physical region. Similar conclusions are expected if

the Treiman-Yang criterion (ref. 12) is used to test

the spectator pole term. The spectator contribution

to the transition matrix is written

Aa

(4)

and the participant contribution is written

Ab

Tp = ¢ (--Pb + "_pPP) taT(Q) (5)

2

where ¢ is the overlap function for the virtual dissociation of the projectile, Q is the total momentum transfer, and tiT is the full transition amplitude

for the fragment-target interaction. In equations (4)

and (5), we are using the high-energy (on-shell) approximation, and all amplitudes are evaluated at the

initial energy. Note that the total momentum transfer is

Q = PT - PX = Pa + Pb - PP (6)

and the relative momentum of the fragments is

1

Pab = 7- (AbPa - AaPb) (7)

._p

The amplitudes appearing in the pole diagrams

are transformed to their proper CM frames using

relativistic kinematics and the Moller invariants such

that

-2_

Ts = ----V-- v/_sv/-_s ¢(Ra) fbT(RQ) (8)

where f is the scattering amplitude,

[ ETEbEx(Ep- Ea) ] [ flbTDbDx J

(9)

and

= .o)1'"

L EpEa(Ep - Ea) J (10)

For the participant term, we have

Tp= v/ v p ¢(-l b)LT(aQ) (11)

where

[ DTDaDx(D P_ Db) ] 1/2 [ RaT(D a + DX) ] 112

v , [ETEaEx(Ep_Eb) J [ flaTDaDx ]

(12)

and

G = [mpDb(mp - Db) 1/2 [ EpEb(Ep -_ Eb) ] (13)

In equations (8) through (13), D and R denote

energies and momenta in the proper CM frame,

which may differ for each amplitude.

The contribution of the pole terms to the momentum distribution is now written

dpa - nPT1 i d_b KI - _ _(1_ ) AT(I_ )

- g pYp LT(Rq)I

(14)
where we have defined the relative cluster momentum

distribution

¢(R)- J-ff

(2zr)3/2¢(R ) (15)

Final-State Interactions

The fragments a and b are expected to interact

following their separation, and their relative momentum vector is expected to have a relatively small

value. The diagrams for final-state interactions (FSI)

between projectile fragments are shown in figures 3

and 4. Following references 13 to 15, we use a separable potential model that incorporates orthogonality

between the bound and scattering states of the projectile fragments and that is appropriate for small

Pab. Note that orthogonality is violated if an optical potential is employed, since the same potential is

not employed to describe the bound and scattering

states.

For figure 3, we write

T_FSI = tbT(Q) 3 / dk'2"ob2O(k) _ k,2t.b(k, + i_k') (16) Pab

where # is reduced mass and we define the

intermediate-state relative momenta

1

k = _-- (Abka - Aakv)

_p

(17)

and

kp = 1 (Abk a _ Aakb) (18)

Following references 13 to 15, we use the separable

potential model for tab such that equation (16) is

reduced to

(19)

where the dispersion integral is defined in reference 14 as

f dk ¢(k + q/2) (a2 + k2) (I)(k) D(p, q) (20) J p2 _ k2 + i_

and a is related to the a-b separation energy es

through a 2 = --2#abes. The dispersion integral is

evaluated in analytic form for typical phenomenological forms of the overlap functions.

PT_ PX

K_ Pb

Kv

PP a

Figure 3. Spectator term with final-state interaction.

PT PX

Pa

Pp Kb Pb

Figure 4. Participant term with final-state interaction.

We now define a distorted momentum distribution given by

D(Rab, 2_p RQ)

_(Ra, Rab) = 4)(Ra) - _(Rab ) D(Rab, 0)

(21)

and the distorted spectator term

T, = T, + Ty SI

_ -2. (a.,Rab) hr(q) (22)

V

Similarly for the participant contribution

Tp = Tp + TpFsI

K_pYp (_ (Rb, Rab ) faT(Q) (23) V

Evaluation of these terms for model inputs is discussed subsequently. The pole model with FSI for

the fragment momentum distribution is now written as in equation (14) with the distorted terms discussed above replacing the relative cluster momentum distributions.

Double-Scattering Corrections

The corrections to the pole diagrams for scattering by a second projectile fragment on the target are shown in figures 5 and 6. Figures 7

and 8 show further contributions from FSI between a

and b. The contributions from the graphs in figures 7

and 8 axe expected to be difficult to evaluate, since
KX_ PX

Pa

Pb

Figure 5. Rescattering correction for spectator term for

projectile fragmentation.

KX

Pp

PT PX

Ka

ea

Pb

Figure 6. Rescattering correction for participant term for

projectile fragmentation.

K x .....

PX

a

P

Figure 7. Rescattering correction for spectator term with

final-state interaction.

Px

K_) Pb

Figure 8. Rescattering correction for participant term with

final-state interaction.

4

the complications of a three-body propagator cannot be avoided, even in the cluster model employed

herein. The contributions from the double-scattering

diagrams of figures 5 and 6 are estimated using the

high-energy propagator derived in reference 9.

Define the relative momenta for figure 5 to be

1

(AaPX - ATPa) (24) PaX -- AT + Aa

and

1

kaX = (Aak X - ATka)

AT + Aa

and the momentum transfers to be

(25)

ql = PT - kx (26)

and

(t2 = kx - PX (27)

with

kaX = PaX + q2 (28)

We write the double-scattering contribution to the

transition matrix of figure 5 as

T1 = _V f dkax 21_aTtbT(ql)____2.___ k'_.-:- _e¢(kab) taT(q2) (29) PaX -- aX 5" e

Ignoring the noninvariance of the amplitudes on the

right side of equation (29), we approximate this

expression by

T 1 _ -- dq2

_r V l_bT

×

_q2 2 - 2pax "q2 q- ie

(30)

Treating only the singularities of the propagator and

using contour integration, we reduce equation (30) to

where

T_ '_ 4rriPaX fOrr sin _ cos _ d_ fbT(Q -- x) VlZbT

x ¢(Pa Aa -x) - -_pPP faT(X) (31)

X = --2pax COS_ dI2 (32)

and (t2 is a unit vector.

The singularity structure of the overlap function is ignored here, since only first estimates of the

double-scattering corrections are considered. In a
similarmanner,wefind the double-scatteringcontributionto thetransitionmatrixof figure6:

T_ _ 47riPbX fo r sin _ cos _ d_ .faT(Q - Y) V#aT

where

and

× (-pb + 3-- p Ab pP + Y) fbT(Y) (33)

1

(AbPx - ATPb) (34)

PbX -- AT + Ab

y = --2Pb X cos ( c12 (35)

The approximations to the double-scattering

terms given by equations (31) and (33) are evaluated

numerically using the inputs described subsequently.

Results and Discussion

We now apply our model to 3He production from

1 GeV/A alpha particles scattering on 1H targets.

The treatment of the summation over target states

for composite targets is not discussed in this report.

General properties of the overlap functions for single

nucleon knockout have been reported by Berggren

(ref. 16). For the 3He-n overlap, a sum of Yukawa

terms is assumed to be

2

ai (36)

= Z ;2 + 2

i=1 °ti

with al = a, and the normalization is

f I(I)(p)]2dp = IZ[ 2 (37)

with

D(p,O)=47r_-_a_ajl____[l+ 32+p2 1

ai + aj (ai -- ip)(a a -- ip)

tj

(39)

We note that solutions of this dispersion integral

differ in references 13 through 15.

At high energies we use diffractive approximations to the a-T and b-T scattering amplitudes. For

neutron-proton scattering, this is

a(p + i)kNN e_l/2Uq2 (40)

fbr(q) -- 47r

with the NN scattering parameters listed in table 1.

For 3He-proton scattering, we use the Glauber approximation result (ref. 17):

faT(q) = Z3 (_)[ (--1)J 2_2-_22-B) a(l__ip) ]j-1

j=l

x a(1 - iP) e_Wjq2

2j

(41)

where

6B + (3 - j)R 2

wj = 12j

with the radius R = 1.51 fm.

(42)

Table 1. NN Parameters at 1 GeV

np

PP

a, mb B, fm2

43.7

47.6

0.26 --0.26

.24 -.09

where IZ[ 2 is the total probability of finding the two

fragments in the projectile. Note from reference 16

that IZ[ 2 < 1. For 3He-n, al = 0.846 fm -1 and from

reference 15, 32 = 1.12 fm -1, al = 1, and a2 -- -1.

For [Z[ 2, we use 0.9. Values in the literature for 1Z[ 2

range from 0.6 to 0.9 depending on the method of

determination. The dispersion integral is then found

to be

D(p,q) = q Zaiaj {tan -1 (q2----2 j2) i,j a i -- aj

+_ +

+ _ln

-t- (p + q/2) 2 _ (38)

+ (-p-- q/2) 2 J

Results for the double-differential cross section at

several laboratory angles are shown in figures 9 to 11.

The experimental data of reference 18 are shown in

figures i0 and 11. The dashed line is the spectator term, equation (22); the dotted line the participant term, equation (23); the dash-dot line the coherent sum of the spectator and participant terms;

and the solid line the coherent sum of participantspectator terms and the double-scattering terms of

equations (31) and (33). Single-scattering results include FSI. The double-scattering terms are observed

to contribute in a nonnegligible way at 0 °. This

makes a simple extraction using the distorted-waveborn-approximation of the overlap function from

small angle data, as was suggested in reference 19, invalid. The double-scattering approximations of equations (31) and (33) do not neglect the longitudinal
30-

.... ITs 12

A ....... ITpl 2

t ---- I_-s + _p12

/.i_,\t --ll- s + Tp + T_) + T_II 2

20 'i '_

d2o i '\,

/, ,,'............',;,,,\

o ..---..___ _'_ ..... i-"-'"" l "........ "'-/%'_............. "_-"_",, ,

4000 4500 5000 5500 6000

Plab' MeV

Figure 9. Double-differential cross section at 8la b --_ 0° for a + IH ---* 3He at 1.02 OeV/A.

d2_

dll dp

mb

MeV-sr

3O

20

10 -

0

4000

o

¢,

o

0

o

0 °

4500 5000 5500

Plab' MeV

.... ITs 12

.............ITpl 2

----- ITs + l'pl 2

--i_" s + Tp + T1 + T212

<> Experiment

I

6000

Figure 10. Double-differential cross section at _iab = 0.65° for a + 1H --_ 3He at 1.02 GeV/A. Experimental data from

reference 18.
d2_

dQ dp

mb

MeV-sr

.3

.2

.... i=rs12

o ............ i_-pl2

----- IT s + Tpl 2

--i rs+ + + 2

Experiment

/ f-',_

,'/_,

o_' _,° o

oooo / \i,

4500 5000 5500

Plab' MeV

0 t

4000 6000

Figure 11. Double-differential cross section at 9|a b ---- 4.07 ° for cx + ]H _ 3He at 1.02 GeV/A. Experimental data from

da

dQ

mb/sr

105

reference 18.

104

.... irrs 12

103

102

101

100

10-1

0

Figure 12. Angular distribution for c_ +

............ i=rpl2

Irrs + l"pt 2

"_:!', ,_ o ':T_;r::e:t TI+ T2'2

I I "" i """, 1 1 "11

1.5 3.0 4.5 6.0 7.5 9.0

elab, deg

tH -_ 3He at 1.02 GeV/A. Experimental data from reference 18.
momentumtransferand thereforeleadto a good

predictionof the positionof the peakin the cross

sectionwith increasingangle(asseenin fig. 11at

t_lab -----4.07°). We note that in the GIauber model of

alpha-particle breakup of reference 20, the downshift

in the momentum distribution must be treated in an

ad hoc manner.

Interference effects between the various scattering terms are found in all results. The spectator and

participant terms interfere constructively for all angles considered. The energy-dependent parameters

Pnp and ppp determine the interference effects to a

large degree. Prescription for the on-shell fragmenttarget interaction energy could then be used to study

the interference effects in more detail. The singularity structure of the overlap function and FSI

effects on the double-scattering terms need to be

studied in order to make conclusions about the magnitude and interference contributions of the doublescattering terms.

In figure 12, the angular distribution for 3He

production is shown in the laboratory system. The

importance of the participant and double-scattering

terms is seen at all angles. We expect that the small

differences between our calculations and the data at

the smallest angles between 1.5 ° and 4.5 ° could be

reduced if the phases between the various terms were

treated correctly. We underestimate the data at the

largest angles, which may be because of contributions

not treated here, such as intermediate-state deuteron

production, charge exchange, and pion production.

Results for the total production cross section are

given in table 2. Good agreement is found with

experiment when all scattering terms are included.

Table 2. 3He Production Cross Section

in _ + 1H Reaction at 1 GeV/A

Experiment (ref. 18) ....

2 ...........

f pj2

f s+ pl2 .........

+rp + 2 ....

or,mb

24.1 =t=1.9

12.2

3.3

20.7

22.7

Conclusions

A first approximation to the double-scattering

correction to the participant-spectator model of fragmentation was found to be important in describing

alpha-particle breakup. Interference effects between

single- and double-scattering graphs were found to

determine the overall magnitude and shape of the

double- and single-differential production cross sections. The prescription for the on-shell projectile

cluster-target amplitude energy should be considered to study the interference effects in more detail.

The singularity structure of the projectile dissociation overlap function should be considered to improve

the results given here. Good agreement with experiment for the total production cross section was found

for the single projectile energy considered. Extensions of this work are expected to contribute to the

development of a nuclear cross section data base for

galactic cosmic ray transport codes.

NASA Langley Research Center

Hampton, VA 23665-5225

March 12, 1991

References

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pp. 372-378.

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Nioradze, M. S.; Siemiarczuk, T.; Stepaniak, J.; Streltsov,

V. N.; Wilkin, C.; and Zielinski, P.: Final-State Interaction in the High-Energy Proton-Deuteron Break-Up

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9


Report Documentation Page

Nalic_I Aeronautics and

Space AdrninisIraT_on

1. ReportNAsANO.TM_4262 [2. Government Accession No. '3. Recipient's Catalog No.

4. Title and Subtitle

Corrections to the Participant-Spectator Model of High-Energy

Alpha-Particle Fragmentation

7. Author(s)

Francis A. Cucinotta, Lawrence W. Townsend, John W. Wilson,

!5. Report Date

May 1991

6. Performing Organization Code

8. Performing Organization Report No.

L-16883

10. Work Unit No.

199-04-16-11

11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

and John W. Norbury

9. Performing Organization Name and Address

NASA Langley Research Center

Hampton, VA 23665-5225

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, DC 20546-0001

15. Supplementary Notes

Francis A. Cucinotta, Lawrence W. Townsend, and John W. Wilson: Langley Research Center,

Hampton, Virginia.

John W. Norbury: Rider College, Lawrenceville, New Jersey.

16. Abstract

The participant-spectator model of nuclear fragmentation is described in terms of pole graphs from

direct reaction theory. Corrections to the model for more than one projectile fragment scattering

on the target are considered using a triangle graph model. Results for alpha-particle fragmentation

at 1 GeV/A indicate that corrections to the participant-spectator model are significant, as

indicated by the large interference effects found between the pole and triangle graph terms in

the double- and single-differential cross sections.

17. Key Words (Suggested by Author(s))

Nuclear fragmentation

Inclusive momentum distribution

Participant-spectator models

18. Distribution Statement

Unclassified--Unlimited

Subject Category 73

19. Security Cl_sif. (of this report) 20. Security Classif. (of this page) 121. No. of Pages] 22.

Unclassified Unclassified l 12 I

NASA FORM 1626 OCT 86

For sale by the National Technical Information Service, Springfield, Virginia 22161-2171

Price

A03

NASA-Langley, 1991