NASA/TM-1998-208722

S

Parameterizations of Pion Energy Spectrum

in Nucleon-Nucleon Collisions

Francis A. Cucinotta and John W. Wilson

Langley Research Center, Hampton, Virginia

John W. Norbury

University of Wisconsin, Milwaukee, Wisconsin

October 1998

https://ntrs.nasa.gov/search.jsp?R=19990047477 2018-05-11T17:28:37+00:00Z

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NASA/TM- 1998-208722

Parameterizations of Pion Energy Spectrum

in Nucleon-Nucleon Collisions

Francis A. Cucinotta and John W. Wilson

Langley Research Center, Hampton, Virginia

John W. Norbury

University of Wisconsin, Milwaukee, Wisconsin

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23681-2199

October 1998

Available from the following:

NASA Center for AeroSpace Information (CASI)

7121 Standard Drive

Hanover, MD 21076-1320

(301) 621-0390

National Technical Information Service (NTIS)

5285 Port Royal Road

Springfield, VA 22161-2171

(703) 487-4650

Abstract

The effects of pion Oz) production are expected to play an important role in radiation exposures in the upper atmosphere or on the

Martian surface. Nuclear databases for describing pion production

are developed for radiation transport codes to support these studies.

We analyze the secondary energy spectrum of pions produced in

nucleon-nucleon (NN) collisions in the relativistic one-pion exchange

model. Parametric formulas of the isospin cross sections for one-pion

production channels are discussed and are used to renormalize the

model spectrum. Energy spectra for the deuteron related channels

(NN --> dx) are also described.

1. Introduction

A convenient representation of the differential cross section in energy of particles created in the

interactions of space and atmospheric radiation with materials is required for radiation transport comreactions, an abundance of mesons is produced through the nuclear force. High-energy reactions

increase in relative importance in the upper atmosphere due to the Earth's magnetic field, reducing

lower energy ion components which have insufficient energy to overcome the threshold for meson production. The one-pion production channels dominate the meson production for energies up to about

1.5 GeV/amu and represent an important contribution at higher energies. The energy region below

1.5 GeV/amu extends above the galactic cosmic ray (GCR) peak at 0.2 to 0.6 GeV/amu. Also, most of

the energy range of particles seen in solar particle events (SPE) are dominated by the one-pion exchange

interactions. The scattering of high-energy protons or neutrons on hydrogen is important because protons and neutrons represent over 90 percent of the particle flux in materials and also because hydrogenous materials in tissue and material structures are important. The description of the one-pion

production mechanism in nucleon-nucleon (NN) collisions is also needed for modeling the pion production mechanism in inclusive proton and neutron collisions on target atoms in applying reaction theory or

in Monte Carlo simulations (refs. 4 and 5).

In this report, we describe the calculation of the secondary energy spectrum in proton-proton (pp)

and neutron-proton (np) reactions. The one-pion production channels are modeled by using the relativistic one-pion exchange (OPE) model. Parametric models for the isospin components in NN reactions and

xN reactions are discussed. The one-pion channels discussed here can be appended with parameterizations of high-energy models (refs. 6 through 8) to provide the energy spectrum for inclusive pion production in NN collisions at overall energies of interest for space radiation studies.

2. One-Pion Production Cross Sections

The cross sections for production of a single pion in nucleon-nucleon (NN) collisions may be

written in terms of four independent cross sections by applying isospin conservation. For a transition

from an initial isospin state I. to a final isospin state It the cross section is denoted _I / • For formation

l J if

of a deuteron d in the final state, a superscript d is used. In table 1 we list the isospin and masses of the

the various reaction channels in NN collisions. Various authors have considered parameterizations of

these cross sections: cyl0, _01, IJll' and cyd0. The fits of Wilson and Chun (1988) are useful because

they extend over all energies. The work of VerWest and Arndt (ref. 9) is significant because they have

reconsidered discrepancies in older data sets, resulting in a large reduction in the isospin zero cross

section o01 over previous estimates and are in good agdreement with the more recent measurements

(ref. 9). The formula of VerWest and Arndt (ref. 9) for el0 is (in units where -h = c = 1)

old(s) = _ (Pr] _

S--_o_

2p (,_o)

m_F 2

+

(2.1)

where

2 2

p = s/4 - m N

2

s = 4m N + 2mNT L

2

SrtN = (.S-mN)

p_(s) = [s - (m d - taro 2] [s - (m d + mn) 2]

4s

2 2

Po = So4- mN

so = (m N +mo )2

with m N the nucleon mass, m N = 0.939 GeV, mn the pion mass, and mn

in reference 9 for Olilf is

-m l2+m F2

where s* = (M) 2,

= 0.138 GeV. The formula used

(2.2)

p_(s) =

2

qr (s)=

[s-(m N- (M))2][s-(mN + (M))2I

4s

[s*- (m N-mn)2l[s *- (m N + mrt) 21

4s

(M(s)) = M o + (arctan Z+ -

arctan Z_)-I(1 + Z2+/

2

where

Z+ = (/s - m N - Mo)(2/Fo)

Z_ = (m N + mn- Mo)(2/Fo)

with M o = 1.22 GeV and F o = 0.12 GeV. The parameters of the model are listed in table 3. The formulas of equations (2.1) and (2.2) are valid only for T L < 1.5 GeV.

Wilson and Chun have considered the following parameterizations of the isospin cross sections for

all energies:

0.435T L

2.563 e - 17.47 e -6"044TL

cYlo(d) = J7.531 e2"82TL+ 44.8 e -5"69TL

| 0.0885 T L - 1.754 T L

[0.22 e - 1.96 e

( TL < 0.6 GeV) ]

/

(0.6< TL < 1.3 GeV)I

/

(1.3< TL )J

(2.3)

36 - 0"7(TL-Tth )1"6 :

Ol0(nP) = _ 1 -e

T L

(2.4)

001

7.2

=_-_l

T L

1-e I I(TL Tth)14 II

-2(TL-Tth )2 ]

-e i

i

J

(2.5)

O'll =

5 i _3.75(TL_Tth)2

:l-e

T0.522

(2.6)

where Tth is the single-pion production threshold and TL is the laboratory nucleon energy.

Comparisons of the fits described above are shown in figure 1. Large discrepancies exist, especially

for c01 and o10. Because the model of VerWest and Arndt (ref. 9) is more accurate at lower energies,

we will use this model below 1.3 GeV. The resulting fit is shown in figure 2, and comparisons to data

(ref. 10) for It+ and It0 production in proton-proton (pp) collisions are shown in figure 3. The result of

joining equations (2.1) and (2.2) with equations (2.3) to (2.6) will represent the one-pion production

data quite accurately over the energy range of interest for radiation transport codes and will be used to

renormalize the model spectra described in the next section.

3. Pion Energy Spectrum

The deuteron production channels are two-body final states and are therefore much easier to parameterize than the other pion production channels. By isospin conservation (neglecting Coulomb effects),

we have the relationship

1 1 d

o(pn ---)ItOd) = r2o(pp --_ It+d) = ,_o10 (3.1)

3

Theangulardistributionisparameterizedas

d(Ildo

dr2 - (I10N10{,AI0 d d f d + cos20- BI0d cos40 I

(3.2)

d •

where NI0 is a normalization constant and 0 is the cm scattering angle. The energy dependent parameters Ado and Bid0 are

AI0 = 0.27 l+0.13cos 0._ (3.3)

and

}

BIO = _(TL + 0.4) ]

0.6 1 - exp _373

(T L < 0.4 GeV)

( T L > 0.4 GeV)

(3.4)

Comparisons of equations (3.1) to (3.4) to experimental data+(refs. 11 and 12) are shown in figure 4(a)

for the _ d _ pp reaction and m figure 4(b) for the pp _ rcd reaction.

The energy distribution in the laboratory frame of the final deuteron is related to the center of mass

(c.m.) angular distribution by

do 27ts dod0

' d_

dTd "( 2 2"_{ 2 2) m N ,,,"_, s, mN, mN)l_s , md, m_

(3.5)

where the function _.is defined

_.(s, A, B) = (s - A - B) 2 - 4AB

and s is the Mandelstam variable given by s =( pN l +PN2) 2-

In applying equation (3.5), we use

(3.6)

COS 0 =

2 2

m d - m n - 2mNm d + 2EnEp - 2mNT d

2pp PA

(3.7)

4

where barred quantities are the c.m. values given by

2 2

s + m d - m_t

. ,/'s

(3.8)

E d -

2 2

s +mrt-m d

r

2 /'S

(3.9)

/S

Ep = -_- (3.10)

The kinematical limits on the kinetic energy of the deuteron Td are found from equation (3.7) by observing that Icos 01-< 1.

The energy distribution in the laboratory reference frame for the pion is

d

do 2rcs dcYl0

dT n ,'..... d_2

mN,,,,,,,'_,,' ( s, raN,2 m 2)( N _, s, md,2 m 2) n

(3.11)

with

2 2

mTt - m d - 2mNm _ + 2EdErc - 2mNTTt

cos 0 = (3.12)

2pp Pd

and a similar expression is found for the deuteron spectrum. Calculations of energy spectra of secondary

pions and deuterons for several beam energies are shown in figures 5 and 6, respectively.

The NN ---) NNrt reactions are assumed to proceed through the formation and decay of the A resonance. The mechanism is described by figure 7(a). The A forms an isospin quartet (I = 3/2), and it is useful to consider the coupling of the nN system wave functions in isospin space to understand the A decay

properties. Denoting the nucleon isospin by IN and the pions by I_, we have

INrt) = £ lllz)(II z I ININzlr_Inz) (3.13)

1, 1 Z

Equation (3.13) is used to obtain the components of the A wave function with the result

IA++) = Ir_+p) (3.14a)

IA+ ) = l[Irt+n) + 42 Irt°p)l (3.14b)

,,,"3

5

[A°) = 1[,,_2 [n°n) + In-p)] (3.14c)

f"3 •

IA-) = 17t-p) (3.14d)

Branching ratios for the formation of various pion species in NN collisions are obtained by using

equations (3.14).

The invariant differential cross section for NN --_ NA has been evaluated in the relativistic onepion exchange (OPE) model. We use this model and the assumption of isotropic decay of the A in its

rest frame to obtain the momentum distribution and energy spectrum of pions and nucleons in the

NN _ NNr_ channels. The Mandelstam variables for the reaction are defined in terms of the four

momentum vectors of the various particles defined in figure 7 and given by

s = (K 1 + K2 )2 (3.15)

t = (K 1-K3 )2 (3.16)

u = (K2-K3)2 (3.17)

The cross-section distribution in t is written in terms of the matrix element M as

d__ 1 IM 2 1 (3.18)

dt 64rt 412

where

IF = ...... /K-'K "( 1 2)-mN 2 2 (3.19)

The matrix element M is decomposed as

IMi 2 = IMdirl 2 + :iMexl 2 + !MINTi 2 (3.20)

The direct amplitude is given by (ref. 13)

(fn f;I 2 F4(t)

2

t[t- (m_-mN)2] [(m A+ mN )2-t]

3m_

(3.21)

where fn and fn are the coupling constants with values 1.008 and 2.202, respectively, and mrt and

m A are the pion-fixed A mass with values 0.139 GeV and 1.232 GeV, respectively. In equation (3.21),

F(t) is the form factor for the off-shell meson which is parameterized as

6

A 2 - 2

m_

F(t) - (3.22)

A 2 - t

where the value of the parameter A will be discussed below. The exchange term in equation (3.20) is

equivalent in form to equation (3.21) with the replacement (u _ t).

The interference term is given by

1 2 = (f. f'n]. F2(t)F2(u) 12

2 2

IM'NTI _,m= )(,_mnXu_mn)em a

2 2 4 4 r 2 2 1

+ tu-(m2-m2)(t+u)+(mN+mA) 4 tu-mN(mA-mN)(m2-m2) }

(3.23)

In order to account for the finite width of the A resonance, a mass distribution p(g2) is introduced

through

do - A(s, t, u)p(I.t 2) (3.24)

dt dg 2

where g2 = (K 4 + Kn)2 and A(s, t, u) is the invariant cross section of equation (3.18) with the fixed

A mass m A replaced by g. The mass distribution is parameterized in terms of the elastic pion-nucleon

cross section and the A width as (ref. 13)

2

K A (YrtN

p(g2) _

8rt2mAF

(3.25)

where K A is given by

The width is parameterized as

I.... 2

2 _i'[ LI'2+ m2 - mn 2

KA(bt 2, m n) = ,' -m N 2

'.,, 4bt

2 2 3

,i2'm:/

(3.26)

(3.27)

7

with the form factor Z given by

2(2 2) K2

Z(g 2,m2n) = Ka ma, m n +

2 2 <. m:)+2

(3.28)

and parameter values F o = 0.12 GeV, and _: = 0.2 GeV. By neglecting Coulomb effects, the shape

of the secondary energy spectrum for the components of the isospin triplet of pions is largely determined by the r_N cross section in equation (3.25), with the remaining kinematic factors in

equations (3.18) to (3.28) being identical for each pion species. The description of the rtN cross

sections is given in the appendix.

The decay of the A (A _ Nrt) is assumed to be isotropic in the A rest frame such that the pion

distribution from the decay is

de2 * - lrtA(s, t, u)p(!a 2)

dt dg d_ n

(3.29)

and the nucleon distribution from the decay is

d_2 * - I_A(s, t, u)p(bl. 2)

dt dg dff2 N

(3.30)

where starred quantities are in the A rest frame. A second contribution to the nucleon spectrum comes

from the nondecaying nucleon line in figure 7, as discussed below. The A mass is found as

and using

dg2 0l -t2 *

= _ dE n (3.32)

3E n

allows the invariant momentum distribution of the pion to be written

,.-. (PlP3_Og 2 ..

_ 1 I asL3/---7-/----Tats' t, u)p(g 2)

4_2 _" Pn )3En

(3.33)

8

with a similar expression for the decay nucleon spectrum. The energy spectrum of the pion in the laboratory system is then

do_ l fda3df_rtp n ____',A(s,t,u)p(g2) (3.34)

dEn 4x2 \ Pn] bE_

The laboratory energy spectrum of the decay nucleon is

d(y D

dE N

,,-, (PlP3_ b_.t2 ,,

_ 1 f d_3 axzNPNl'-'-7--| -""7"ats' t, u)p(g 2)

4x2 _"PN ) bEN

(3.35)

The energy spectrum of the recoil nucleon is given by (ref. 14)

max 2 2

d(yR - I0N d(cos ON) rnN_t2 PNA(s, po t, u)p(t.t 2)

Xm A

(3.36)

where Po is the laboratory momentum of the incident nucleon.

A comparison of equation (3.24) to experimental data for the pp _ nA ++ reaction (refs. 10 and 15)

is shown in figure 8, with good agreement found. The parameter A in the form factor of equation (3.22)

has a strong effect on the calculations. As noted by Jain and Santra (ref. 14), the inclusion of distorted

waves effects, the value of A chosen, and a value of - 1 GeV provides the best fit to data. Calculations of

energy spectrum for x + and x ° production in pp collisions and rt+, x °, x- production in neutron-proton

(np) collisions are shown in figures 9 and 10, respectively. The variations in the A _ xN decay vertices

due to the isospin dependence provide only a modest change in shape between the various production

channels. Coulomb effects which are not treated herein will provide further dependence on the pion

charge.

4. Inclusive Pion Production Spectrum

As the kinetic energy of nucleons increases, the threshold for 2, 3... pion production is reached and

so is production of heavier mesons such as the kaon. The threshold energies for several production processes are listed in table 4. The production threshold is also dependent on charge conservation when

individual species of mesons in the reaction are produced. Inclusive meson production data have been

parameterized in convenient forms by several authors (refs. 6 through 8). The spectrum in one-pion,

two-pion, and other channels will be somewhat distinct due to the mechanisms involved. The two-pion

production channels have been considered by Sternheimer and Lihdenbaum (ref. 16) using a purely

kinematic form of the isobar model. There are two distinct mechanisms for two-pion production which

are through the excitation of two A's:

N + N--_ A 1 +A2--_N+N+x+x (4.1)

or through the excitation of higher mass nucleon resonances:

N + N--_ N + N*---> N + N +x + x (4.2)

9

whereN* is a nucleon resonance of higher mass than the A, which is assumed to decay through the

emission of two pions. The mechanisms of equations (4.1) and (4.2) are illustrated in figures 7(b)

and 7(c), respectively. The mechanism of equation (4.2) will have a cross-section structure similar to

the one-pion model, NN --_ NNn, described above. However, with A _ Nn, the vertex is replaced by

the N* _ Nrcn vertex. The mechanism of equation (4.3) will contain the mass distribution of two A's

with a structure such as

d(y

2 2

dt dg I dP2

2 2

(4.3)

thus requiring one additional numerical integral to obtain the pion energy spectrum.

The inclusive pion spectrum can be represented as a sum over the spectrum for each multiplicity at

pion as

do _ do + d_

dE n dEnl _ + ""

(4.4)

d(y

where _ represents the spectrum described by equation (3.34). The inclusive spectrum can be represented by the one-pion channel plus the high-energy model of Schneider, Norbury, and Cucinotta

(ref. 7) renormalized to exclude the one-pion contributions, or alternatively, by including the two-pion

production channels separately, as described previously.

5. Concluding Remarks

One-pion production channels will dominate pion production for a significant fraction of galactic

cosmic ray (GCR) and solar particle event (SPE) exposures in free space, in the upper atmosphere, or on

the Martian surface. In this report, the one-pion production cross sections were discussed, and a convenient formula for their numerical representation was found. These models will be appended with highenergy models to span all energies of importance in GCR studies. The formula described here also can

be used to model pion and nucleon production spectra in nucleon-nucleus and nucleus-nucleus

reactions.

NASA Langley Research Center

Hampton, VA 23681-2199

September 1, 1998

10

Appendix

Pion-Nucleon Scattering

1

The rcN scattering amplitude is written to represent the pion isospin I n and nucleon isospin _x N as

1 1

fnN = _(fl/2 + 2f3/2) + _(f3/2-fl/2)ln ''_N (A1)

wherefl/2 and f3/2 are amplitudes for total isospin 1/2 or 3/2, respectively. Introducing the total isospin

of the rtN system,

1

T = i n + _x N (A2)

and rtN wave functions

and noting

ITTz) = Z(lnlz'CN'CZ [ TT Z) I lnlzXNXz)

1z

(A3)

(A4)

leads to the following relations for elastic scattering:

(rt+P ] f_N [ rt+P) = f3/2 (A5)

1

(rr°P I f TtN [ lt°P) = _(fl/2 + 2f3/2) (A6)

- 1

(g-P I frtN I 71; p) = _(f3/2 + 2fl/2) (A7)

( rt+nl fnN I g+n) = 1_(f3/2 + 2fl/2) (A8)

1

(rr'°n [ fr_N [ It°n) = _(fl/2 + 2f3/2) (A9)

(_-n I f rtN I_-n) = f3/2

with similar relations found for charge-exchange matrix elements.

(A10)

11

The total cross sections can be found from the optical theorem. Numerous measurements exist for

+

the total n p and n-p reactions. From equations (A5) to (A10), we can find solutions for the other

nN collision pairs (neglecting Coulomb effects) by using equations (A11) through (A14).

aTop ---- 21 I (Yl_+p T + (_l"C-p T -- OEX( n-p ) (A11)

T T

+ = _J _ (A12)

nn np

T T

c o = _ o (A13)

_n 7tp

T T

= _ + (A14)

/t-n 7_ p

12

References

2.

.

.

.

6.

Wilson, John W.; Townsend, Lawrence W.; Schimmerling, Walter; Khandelwal, Govind S.; Khan, Ferdous;

Nealy, John E.; Cucinotta, Francis A.; Simonsen, Lisa C.; Shinn, Judy L.; and Norbury, John W.: Transport

Methods and Interactions for Space Radiations. NASA RP- 1257, 1991.

Cucinotta, Francis A.: Calculations of Cosmic-Ray Helium Transport in Shielding Materials. NASA TP-3354,

1993.

Cucinotta, F. A.; Townsend, L. W.; Wilson, J. W.; Shinn, J. L.; Badhwar, G. D.; and Dubey, R. R.: Light Ion

Components of the Galactic Cosmic Rays: Nuclear Interactions and Transport Theory. Adv. Space Res.,

vol. 17, no. 2, 1996, pp. 77-86.

Bertini, H. W.: Nonelastic Interactions of Nucleons and Pi Mesons With Complex Nuclei at Energies Below

3 GeV. Phys. Rev. C, third ser., vol. 6, Aug. 1972, pp. 631-659.

Lemaire, M. C.: Medium Effects in Pion Production. Nucl. Phys. A, vol. 489, no. 4, 1988, pp. 781-802.

Badhwar, G. D.; Golden, R. L.; and Stephens, S. A.: Analytic Representation of the Proton-Proton and

Proton-Nucleus Cross-Sections and Its Application to the Sea-Level Spectrum and Charge Ratio of Muons.

Phys. Rev. D, vol. 15, Feb. 1977, pp. 820-831.

Pion and Kaon Production in Proton-Proton Collisions. Astrophys. J. Suppl. Ser., vol. 97, no. 2, Apr. 1995,

pp. 571-574.

8. Choudhur, R. K.; Bhattach, S.; and Bhattach, D. R: Lepton-Hadron Relation and the Inclusive Low-PT

Charged Secondaries at ISR Energies in pp Scattering. Nuov. Cim. A, vol. 50, no. I, 1979, pp. 133-154.

9. VerWest, B. J.; and Amdt, R. A.: NN Single Pion Production Cross Sections Below 1500 MeV. Phys. Rev. C,

vol. 25, no. 4, 1982, pp. 1979-1985.

10. Kubota, Y.; Koiso, H.; Sai, F.; Sakamoto, S.; and Yamamoto, S. S.: Measurement of the pp Cross Sections in

the Momentum Range 0.9-2.0 GeV/c. Nucl. Phys. A, vol. 386, no. 2, 1982, pp. 571-588.

11. Richard-Serre, C.: Study of the Reaction Positive Pion Plus Deuteron Yields Proton Plus Proton for Pion

Energy Range Between 143 and 264 MeV CERN-68-40, European Org. Nuci. Res., Dec. 1968.

12. Heinz, R. M.; Overseth, O. E.; Pellett, D. E.; and Perl, M. L.: Differential Cross Sections for p + p --->d + _+

From 1 to 3 BeV. Phys. Rev., vol. 167, no. 5, 1968, pp. 1232-1239.

13. Dmitriev, V.; Sushkov, O.; and Gaarde, C.: A-Formation in the IH (3He, 3H) A++ Reaction at Intermediate

Energies. Nucl. Phys. A, vol. 459, nos. 3-4, 1986, pp. 503-524.

14. Jain, B. K.; and Santra, A. B.: Study of the p(p, n) A++ Reaction in Nuclei. Nucl. Phys., vol. A519, 1990,

pp. 697-720.

15. Wicklund, A. B.; Arenton, M. W.; Ayres, D. S_ Diebold, R.; Kramer, S. L.; May, E. N.; Modulman, L. J.; and

Sauer, J. R.: Study of the Reaction pp --_ prt n With Polarized Beams From 3 to 12 GeV/c. Phys. Rev. D,

vol. 34, no. 1, 1986, pp. 19-41.

16. Sternheimer, R. M.; and Lihdenbaum, S. J.: Extension of the Isobaric Nucleon Model for Pion Production 1:

Pion-Nucleon, Nucleon-Nucleon, and Anti-Nucleon-Nucleon Interactions. Phys. Rev., vol. 123, no. 1, 1961,

pp. 333-370.

13

Table1.IsospinsandMassesofParticles

Particle

P

//

7[+

7[0

7[-

m ++

A+

A °

Ad

Isospin

1/2

1/2

1

1

1

3/2

3/2

3/2

3/2

0

Isospin projection

1/2

-1/2

1

0

-1

3/2

1/2

-1/2

-3/2

0

1

1

Mass, GeV

0.9383

.9396

.1396

.1350

• 1396

1.232

1.232

1.232

1.232

1.876

Table 2. Isospin Components of One-Pion Production Channels in

Nucleon-Nucleon (NN) Collisions

Channel

p p --4 7[+d

pp ---) ppn °

+

p p ---) pnT[

n p --) 7[°d

n p --_ npT[ °

+

np ---) nnT[

np --_ ppT[-

Isospin cross section (_lilt '

d

_10

(311

_11 + glO

1 d

2(_10

(¢_10 + (_01)

((_11 + (_01)

_(Oll +O01)

Table 3. Isospin Cross-Section Parameters

[From ref. 9]

Parameter l _0 i _11 ] _10

_.._.. ...... 6 030 / 3.772 ] 15.28 /

1_.............. 1 700 ] 1 262 0

m o, MeV . 1203 _ 1188 I 1245 I

F, MeV . ....... _ 134.3 ] 99.02 l 137 4

_01

146.3

0

1472

26.49

Table 4. Meson Production Thresholds

channel - Threshoidl Gev I

NN _ NNn 0.29

NN --, NNT[n . .602

NN --_ NNT[7[n .934

NN ---) KA 1.12

14

15

"_ 10

d

O

"___,

¢)

2

r..) 5 d

°10

_10

s _

tl t _ _

e I

/ Wilson and Chun

..... VerWest and Arndt

I

I

I!

!

_11

• I

tt • •

......,, aO1 ...

/'1 %_ st

/,.;,., "-. . .-"

.5 1.0 1.5 2.0

Laboratory kinetic energy, Tla b, GeV

Figure 1. Comparison of isotopic cross sections o for one-pion production versus laboratory kinetic energy of

incident nucleon.

6

d

O

15,

101

Connected formula

..... VerWest and Arndt

0

10-1

d

°10

o10

s

t

t

I

r

I

ss

I

I

t

l

/

I I

a

I

, gll

I

c

I p•

s t s_ e •

Is

is s

, , °01 Z_'. " "

100 101

Laboratory kinetic energy, Tla b, GeV

Figure 2. Comparison of isotopic cross sections _ for one-pion production versus laboratory kinetic energy of

incident nucleon using connection of low- and high-energy fit equations.

15

201

15

6

O

•= I0

0

l

0

f Tpp --) pp_°

pp --_ png+

,, pp --_ dtc+ L?

f!

,/

/

[

/

/

I j!

/

]

.5 1.0 1.5

Laboratory kinetic energy, Tlab, GeV

collisions versus laboratory nucleon kinetic energy.

16

E

d

B

O

,o

4;

3

.: z  ̧:, .:'

2 '_

1

{7{

Trt = 143 MeV (Tp = 572)

T_ = 188 (Tp = 662 MeV)

Trc = 264 (Tp = 815 MeV)

-.8

, . , , ,

....................... J I _ • 1

-.6 -.4

cos 0, cm

(a) _+ + d --+ p + p reactions.

I

--.2

4,-

J J J

0

.O6

d

"6.04

=-

O

ej

£

-_ .02

!

t

Tp = 1.0 GeV

,_ Tp = 1.3 GeV

t Tp=I.5GeV

! 4-

Sh

oi

-I.0

L._ . • i _ L_l t _: I __

-.8 -.6 -.4

cos 0, cm

(b) p + p --+ _+ + d reactions.

L J L .L J t

--.2 0

Figure 4. Comparisons of model fits to experimental data (ref. 11) for angular distributions; Tn is kinetic energy of

pion; Tp is kinetic energy of proton.

17

;>

,.Q

.2

10-1

10-21

10-3 //

,.

,,,,, ,, /

/

/

/

I

, ,, %% / , •• •

"'.., .....""

7

Ep, GeV

.4

...... .5

.7

........... .8

1.0

i

10"-4i! L_Lt.:l:tl_LJLLL:_aJ.t±_lll.l[.JJ.tt[..laJ:t_tl

0 100 200 300 400 500

Pion kinetic energy, Ex, MeV

Figure 5. Secondary x + spectrum in p + p --->x + + d reactions for several proton beams.

;>

,.Q

_5

e-,

o

10-1 .

10-3

Ep, GeV

.4

...... .5

,,,_ ,. / .6

• • _0 • .7

", \, / .8

...... , , / /" 1.0 • s

Y. ,,'6-/ ./

""".. \,, ,/ .. ..........

"'°'""....." • ° ° •

°"... ..... ...-"

10--4 l t _ L .... 1 I L _ I

0 200 400 600

Deuteron kinetic energy, Ed, MeV/amu

Figure 6. Secondary deuteron d spectrum in p + p --->x+ + d reactions for several proton beams.

18

N

N

i I

i N

N

A

4 I

!,1

(a) NN -4 NA -4 NNn.

N

N

N

[

,, A

/'"1

N

(b) NN -4-4AA -4 NNnn.

N

N

I N

,, /'

?c

s

N*

N

(c) NN -4 NN* -4 NNnn.

Figure 7. Diagrams for one-pion production through A or nucleon resonance N* formation and decay in nucleonnucleon (NN) collisions.

19

106

E

2_

E

O

E

L"

O

©

£

£

Ok,

105

104

103

102

101

i]

I]

I h

\

PL' GeV/c

2.23

3.00

r l 4.00

6.00

r

i

100 [ LI_±•__!_I_J L_Ll±:llJ _LIL±': JlJ_:• ..... I

0 1 2 3 4

Mandelstam t parameter, -t, (GeV/c) 2

Figure 8. Comparison of model to experiments (refs. 10 and 15) for invariant distribution in pp _ nA ++ reactions.

2O

>

,.Q

E

¢.)

e_O

0.)

".y_.

cO

102 Ep, GeV

0.5

...... .8

i

1.2

2.0

•"''..'?.._'-_ .. •.......... 3.0

loll .,/ .\-- ,

F-" .......... ...."-_--_-,

.......... "...... I"'\_'.

100' ' I

I \

t I

• I

I I

' !1 ] :

10-1 J t _ I iJJlJ I J'J Ll_l .! J J J _,_.l

10-2 10-1 100 101

Pion kinetic energy, Err, GeV

+

(a) p + p--> _ +n+p.

>

e_

etD

.-&

te102

101

100:

I

10-1i

10-2

Ep, GeV

.>..- 0.5

" " "'.'/ _" "-'x _, ----- 1.0

/ ' " 2.0

"" _" "" ........ ----'_-- _."x ........... 3.0

, 4.."_.

_ .......... ........... .','" ', _'...

""" I \" .- _.

............ i \"

I | "'

!

I I

I ,

, I

I 'I

1 L . _ I i tL] _ ] o] L ] i]! ] ] 1 ] ] tXt

10-1 100 l01

Pion kinetic energy, Err GeV

o

(b) p + p--_ rt +p+p.

beam energies.

21

>

E

r,)

o

o

e.,.

%

..=)

e=,

e=,

O

102

101

En, GeV

0.5

...... .8

----- 1.0

2.0

........... 3.0

100

- ,f

_ ......... •.

°"'''''lo-,, o.,.

.,

I i-\-

I "

II I I

II I | "

10-1' t t _ _J_.! [: t _'i J J'l_[ [ _"

10-2 I0-1 100

Pion kinetic energy, E_, GeV

..i-

(a) n + p ---)/t +n+n.

101 •

>

E

s- _ ' \",

"-J / I , .'. ,,"" '_-

L.-" I..." i _.. '.... _; .............

..... \..

.,.." I I" 00 [ ,"."

1

I

i '

10-I ! J J LJ_, 1 ,L :_tJ ,

10--2 10-1 100

Pion kinetic energy, E_t, GeV

o

(b) n + p ---) rc +n+p.

Figure 10. Calculations of pion (one-pion contribution) energy spectrum in

several beam energies.

En, GeV

0.5

...... .8

1.2

2.0

........... 3.0

neutron-proton (np) collisions at

22

>

_D

E

E

2

_D

e"

*.a

O

102:

101 --

100

......... .l.°"

II

10-1 _ . j j_ I

10-2 10-1

En, GeV

0.5

...... ,8

----- i.2

2.0

........... 3.0

\ "_',._.

i

|'°

" I I

"" I I

I

i ',

i_ _ tJ,:tl I L:

100

Pion kinetic energy, Ere, GeV

(c) n+p---_Tt +p+p.

Figure 10. Concluded.

23

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1. AGENCY USE ONLY (Leave blank) 2. October REPORT DATE 1998 3. Technical REPORT TYPEMemorandum AND DATES COVERED

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Parameterizations of Pion Energy Spectrum in Nucleon-Nucleon Collisions

6. AUTHOR(S)

Francis A. Cucinotta, John W. Wilson, and John W. Norbury

7. PERFORMING ORGANIZATIONNAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

WU 199-45-16-11

8. PERFORMING ORGANIZATION

REPORT NUMBER

L-17650

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NAS A/TM- 1998-208722

11. SUPPLEMENTARY NOTES

Cucinotta: Langley Research Center, Hampton, VA (now at Lyndon B. Johnson Space Center, Houston, TX);

Wilson: Langley Research Center, Hampton, VA; Norbury: University of Wisconsin, Milwaukee, WI.

12a. DISTRIBUTIONIAVAILABILITY STATEMENT

Unclassified-Unlimited

Subject Category 73 Distribution: Standard

Availability: NASA CASI (301) 621-0390

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

The effects of pion (rt) production are expected to play an important role in radiation exposures in the upper atmosphere or on the Martian surface. Nuclear databases for describing pion production are developed for radiation

transport codes to support these studies. We analyze the secondary energy spectrum of pions produced in nucleonnucleon (NN) collisions in the relativistic one-pion exchange model. Parametric formulas of the isospin cross

sections for one-pion production channels are discussed and are used to renormalize the model spectrum. Energy

spectra for the deuteron related channels (NN --_ dTt) are also described.

14. SUBJECT TERMS

Space radiation; Atmospheric radiations; Pion production; Transport radiation

17. SECURITY CLASSIFICATION

OF REPORT

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NSN 7540-01-280-5500

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15. NUMBER OF PAGES

28

16. PRICE CODE

A03

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