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Parameterizations of Pion Energy Spectrumin Nucleon-Nucleon Collisions

Francis A. Cucinotta and John W. WilsonLangley Research Center, Hampton, VirginiaJohn W. Norbury

University of Wisconsin, Milwaukee, Wisconsin

October 1998

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NASA/TM-1998-208722

Parameterizations of Pion Energy Spectrumin Nucleon-Nucleon Collisions

Francis A. Cucinotta and John W. WilsonLangley Research Center, Hampton, VirginiaJohn W. Norbury

University of Wisconsin, Milwaukee, Wisconsin

National Aeronautics andSpace AdministrationLangley Research Center

Hampton, Virginia 23681-2199

October 1998

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Abstract

The effects of pion (π) production are expected to play an impor-tant role in radiation exposures in the upper atmosphere or on theMartian surface. Nuclear databases for describing pion productionare developed for radiation transport codes to support these studies.We analyze the secondary energy spectrum of pions produced innucleon-nucleon(NN) collisions in the relativistic one-pion exchangemodel. Parametric formulas of the isospin cross sections for one-pionproduction channels are discussed and are used to renormalize themodel spectrum. Energy spectra for the deuteron related channels(NN→dπ) are also described.

1. Introduction

A convenient representation of the differential cross section in energy of particles created in theinteractions of space and atmospheric radiation with materials is required for radiation transport com-puter codes to be efficient for engineering design and risk assessment (refs. 1 through 3). In high-energyreactions, an abundance of mesons is produced through the nuclear force. High-energy reactionsincrease in relative importance in the upper atmosphere due to the Earth’s magnetic field, reducinglower energy ion components which have insufficient energy to overcome the threshold for meson pro-duction. The one-pion production channels dominate the meson production for energies up to about1.5GeV/amu and represent an important contribution at higher energies. The energy region below1.5GeV/amu extends above the galactic cosmic ray (GCR) peak at 0.2 to 0.6 GeV/amu. Also, most ofthe energy range of particles seen in solar particle events (SPE) are dominated by the one-pion exchangeinteractions. The scattering of high-energy protons or neutrons on hydrogen is important because pro-tons and neutrons represent over 90 percent of the particle flux in materials and also because hydroge-nous materials in tissue and material structures are important. The description of the one-pionproduction mechanism in nucleon-nucleon (NN) collisions is also needed for modeling the pion produc-tion mechanism in inclusive proton and neutron collisions on target atoms in applying reaction theory orin 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 relativis-tic one-pion exchange (OPE) model. Parametric models for the isospin components in NN reactions andπN reactions are discussed. The one-pion channels discussed here can be appended with parameteriza-tions of high-energy models (refs. 6 through 8) to provide the energy spectrum for inclusive pion pro-duction in NN collisions at overall energies of interest for space radiation studies.

2. One-PionProductionCrossSections

The cross sections for production of a single pion in nucleon-nucleon (NN) collisions may bewritten in terms of four independent cross sections by applying isospin conservation. For a transitionfrom an initial isospin stateIi to a final isospin stateIf the cross section is denotedσIiIf. For formationof a deuterond in the final state, a superscriptd is used. In table 1 we list the isospin and masses of theparticles we consider. In table 2 we list the relationship between the four independent cross sections andthe various reaction channels in NN collisions. Various authors have considered parameterizations ofthese cross sections:σ10,σ01,σ11,andσd10. The fits of Wilson and Chun (1988) are useful becausethey extend over all energies. The work of VerWest and Arndt (ref. 9) is significant because they have

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reconsidered discrepancies in older data sets, resulting in a large reduction in the isospin zero crosssectionσ01over previous estimates and are in good agreement with the more recent measurements

–(ref.9). The formula of VerWest and Arndt (ref. 9) forσd10 is (in units whereh=c=1)

β

moΓ pr dπ

-σ10(s)=--------α ----- ------------------------------------------------22p o 2222p

sπN–mo +moΓ

22

(2.1)

where

p=s/4–mNs=4mN+2mNTLsπN=(–mN)

2

2

2

2

2

2

[s–(md–mπ)][s–(md+mπ)]2

pr(s)=----------------------------------------------------------------------------------4spo=so/4–mNso=(mN+mo)

2

2

2

withmN the nucleon mass,mN = 0.939 GeV,mπ the pion mass, andmπ = 0.138 GeV. The formula usedin reference 9 forσII is

if

q/q mΓpo o π r

σπ(s)=--------α ----- ---------------------------------------------22

2 222p po

s*–mo+moΓ

β

223

(2.2)

wheres* = M ,

[s–(mN– M )][s–(mN+ M )]2

pr(s)=-------------------------------------------------------------------------------------------4s

2

qr(s*)

2

2

2

[s*–(mN–mπ)][s*–(mN+mπ)]=-------------------------------------------------------------------------------------------4s

2

22

qo=q mo

1+Z2 –1+

M(s) =Mo+(arctanZ+–arctanZ_) ----------------

2

_ +1Z

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where

Z+=(–mN–Mo)(2/Γo)Z_=(mN+mπ–Mo)(2/Γo)

withMo=1.22 GeV andΓo=0.12GeV. The parameters of the model are listed in table 3. The for-mulas of equations (2.1) and (2.2) are valid only forTL<1.5 GeV.

Wilson and Chun have considered the following parameterizations of the isospin cross sections forall energies:

LL

2.563e–17.47e 2.82TL–5.69TL

σ10(d)= 7.531e+44.8e

0.0885TL–1.754TL

–1.96e 0.22e

0.435T–6.044T

(TL≤0.6 GeV)

(0.6<TL≤1.3 GeV)

(1.3≤TL)

1.6

(2.3)

–0.7(TL Tth)36

σ10(np)=---------1–e

1.2TL

–1.1(TL Tth)7.2

σ01=---------1–e

1.1TL

1.4

(2.4)

1–e

–2(TL Tth)

2

(2.5)

–3.75(TL Tth)5

σ11=--------------1–e

0.522TL

2

(2.6)

whereTth is the single-pion production threshold andTL is the laboratory nucleon energy.

Comparisons of the fits described above are shown in figure 1. Large discrepancies exist, especiallyforσ01andσ10. 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) forπ+ andπ0 production in proton-proton (pp) collisionsare shown in figure 3. The result ofjoining equations (2.1) and (2.2) with equations (2.3) to (2.6) will represent the one-pion productiondata quite accurately over the energy range of interest for radiation transport codes and will be used torenormalize the model spectra described in the next section.

3. PionEnergySpectrum

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

0+11d

σ(pn→πd)=--σ(pp→πd)=--σ10

22

(3.1)

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The angular distribution is parameterized as

dσ10dddd24-----------=σ10N10 A10+cosθ–B10cosθ

d

d

d

(3.2)

whereN10 is a normalization constant andθis the cm scattering angle. The energy dependent parame-d

tersAd10 andB10 are

TL

=0.271+0.13cos -------------

0.182

dA10

(3.3)

and

0 d

B10= –(TL+0.4)

0.6 1–exp---------------------------- 0.3

(TL<0.4GeV)

(TL≥0.4GeV)

(3.4)

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

++

for theπd→pp reaction and in figure 4(b) for thepp→πd reaction.

The energy distribution in the laboratory frame of the final deuteron is related to the center of mass(c.m.) angular distribution by

dσ102πsdσ

--------------------=-----------d dTd

22 22 mNλs,mN,mNλs,md,mπ

where the functionλ is defined

λ(s,A,B)=(s–A–B)–4AB

ands is the Mandelstam variable given bys= pN+pN .

12

In applying equation (3.5), we use

md–mπ–2mNmd+2πp–2mNTd

-cosθ=----------------------------------------------------------------------------------2p2

2

22

d

(3.5)

(3.6)

(3.7)

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

s+md–mπ

π=------------------------2s+mπ–md

d=------------------------2p=-2

2

22

2

(3.8)

(3.9)

(3.10)

The kinematical limits on the kinetic energy of the deuteronTd are found from equation (3.7) by observ-ing thatcosθ≤1.

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

dσ102πsdσ

--------------------=-----------d dTπ

22 22 mNλs,mN,mNλs,md,mπ

with

mπ–md–2mNmπ+2dπ–2mNTπ

cosθ=-------------------------------------------------------------------------------------2pd

2

2

d

(3.11)

(3.12)

and a similar expression is found for the deuteron spectrum. Calculations of energy spectra of secondarypions and deuterons for several beam energies are shown in figures 5 and 6, respectively.

TheNN→NNπ reactions are assumed to proceed through the formation and decay of the reso-nance. The mechanism is described by figure 7(a). The forms an isospin quartet (I= 3/2), and it is use-ful to consider the coupling of theπN system wave functions in isospin space to understand the decayproperties. Denoting the nucleon isospin byIN and the pions byIπ, we have

|Nπ =

I,IZ

∑|IIZ IIZ|ININ

Z

IπIπ

Z

(3.13)

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

| =|πp

+1+

| =--[|πn +|πop ]

++

+

(3.14a)(3.14b)

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oo 1

| =--[|πn +|πp ]

(3.14c)

| =|πp (3.14d)

Branching ratios for the formation of various pion species in NN collisions are obtained by usingequations (3.14).

The invariant differential cross section forNN→N has been evaluated in the relativistic one-pion exchange (OPE) model. We use this model and the assumption of isotropic decay of the in itsrest frame to obtain the momentum distribution and energy spectrum of pions and nucleons in theNN→NNπ channels. The Mandelstam variables for the reaction are defined in terms of the fourmomentum vectors of the various particles defined in figure 7 and given by

s=(K1+K2)t=(K1–K3)

2

(3.15)(3.16)(3.17)

2

u=(K2–K3)

2

The cross-section distribution int is written in terms of the matrix elementMas

211dσ

-M--------------=--------264πdt4IF

(3.18)

where

IF=

The matrix elementMis decomposed as

M

2

(K1K2)–mN

(3.19)

=Mdir+Mex+MINT

222

(3.20)

The direct amplitude is given by (ref. 13)

4 fπf*2F(t)t[t–(m –mN)][(m +mN)–t]π

-Mdir= -------------- ---------------------------------------------------------------------------------------------------------------22m π 23m

t–mπ

2

2

2

2

(3.21)

wherefπ andf*π are the coupling constants with values 1.008 and 2.202, respectively, andmπ and

m are the pion-fixed 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

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Λ–mπ

F(t)=-------------------2Λ–t

22

(3.22)

where the value of the parameterΛ will be discussed below. The exchange term in equation (3.20) isequivalent in form to equation (3.21) with the replacement(u t).

The interference term is given by

22 fπf*112π F(t)F(u)

--MINT= -------------- --------------------------------------------------24 mπ 2 2 6m

t–mπu–mπ

224422m –mN (t+u)– m +mN tu+mN(mN+m ) m –mN × tu+

2222 4

m –mN (t+u)+(mN+m )tu–mN(m –mN) m –mN +tu–

(3.23)

In order to account for the finite width of the resonance, a mass distributionρ(µ) is introducedthrough

2dσ

----------------=A(s,t,u)ρ(µ)

2dtdµ

2

2

2

(3.24)

whereµ=(K4+Kπ) andA(s,t,u) is the invariant cross section of equation (3.18) with the fixed massm replaced byµ. The mass distribution is parameterized in terms of the elastic pion-nucleoncross section and the width as (ref. 13)

K σπN2

ρ(µ)=--------------------2

8πm Γ

whereK is given by

µ+mN–mπ2---------------------------------–mN

24µ

2

2

2

2

(3.25)

K (µ,

The width is parameterized as

2

2mπ)

=(3.26)

22

K µ,mπ 22

-Z µ,mπ Γ=Γo----------------------------- 22 K m ,mπ

3

(3.27)

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with the form factorZ given by

K m ,mπ +κ 22

Z(µ,mπ)=------------------------------------------2222K µ,mπ +κ

2

2

2

2

(3.28)

and parameter valuesΓo=0.12 GeV, andκ=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 deter-mined by theπN cross section in equation (3.25), with the remaining kinematic factors inequations(3.18) to (3.28) being identical for each pion species. The description of theπN crosssections is given in the appendix.

The decay of the ( →Nπ) is assumed to be isotropic in the rest frame such that the piondistribution from the decay is

21dσ

-A(s,t,u)ρ(µ)---------------------------=-----*24πdtdµd π

(3.29)

and the nucleon distribution from the decay is

21dσ

-A(s,t,u)ρ(µ)----------------------------=-----*24πdtdµd N

(3.30)

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

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

**

µ= E4+Eπ

2

2

(3.31)

and using

* µ

-dEπdµ=--------* Eπ2

2

(3.32)

allows the invariant momentum distribution of the pion to be written

dσ1Eπ----=--------2dπ

4π

13 µ22

- ---------A(s,t,u)ρ(µ)d3 ----------- p* E*ππ

∫

(3.33)

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

1dσ

---------=--------2dEπ

4π

∫

13 µ22

- ---------A(s,t,u)ρ(µ)d3d πpπ ----------- p* E*ππ

(3.34)

The laboratory energy spectrum of the decay nucleon is

1dσ

----------=--------2dEN

4π

D

∫

13 µ22

- ----------A(s,t,u)ρ(µ)d3d NpN ----------- p* E*NN

(3.35)

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

dσ

----------=dEN

R

θN

max

∫o

mNµpN2

-------A(s,t,u)ρ(µ)d(cosθN)-------------2

πm po

22

(3.36)

wherepo is the laboratory momentum of the incident nucleon.

A comparison of equation (3.24) to experimental data for thepp→n reaction (refs. 10 and 15)is shown in figure 8, with good agreement found. The parameterΛin the form factor of equation (3.22)has a strong effect on thecalculations. As noted by Jain and Santra (ref. 14), the inclusion of distortedwaves effects, the value ofΛchosen, and a value of～1 GeV provides the best fit to data. Calculations ofenergy spectrum forπ+ andπoproduction in pp collisions andπ+,πo,π production in neutron-proton(np) collisions are shown in figures 9 and 10, respectively. The variations in the →πN decay verticesdue to the isospin dependence provide only a modest change in shape between the various productionchannels. Coulomb effects which are not treated herein will provide further dependence on the pioncharge.

++

4. Inclusive Pion Production Spectrum

As the kinetic energy of nucleons increases, the threshold for 2, 3 . . . pion production is reached andso is production of heavier mesons such as the kaon. The threshold energies for several production pro-cesses are listed in table 4. The production threshold is also dependent on charge conservation whenindividual species of mesons in the reaction are produced. Inclusive meson production data have beenparameterized 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-pionproduction channels have been considered by Sternheimer and Lihdenbaum (ref. 16) using a purelykinematic form of the isobar model. There are two distinct mechanisms for two-pion production whichare through the excitation of two ’s:

N+N→ 1+ 2→N+N+π+π

or through the excitation of higher mass nucleon resonances:

N+N→N+N*→N+N+π+π

(4.2)(4.1)

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whereN* is a nucleon resonance of higher mass than the , which is assumed to decay through theemission of two pions. The mechanisms of equations (4.1) and (4.2) are illustrated in figures 7(b)and7(c), respectively. The mechanism of equation (4.2) will have a cross-section structure similar tothe one-pion model,NN→NNπ, described above. However, with →Nπ, the vertex is replaced bytheN*→Nππ vertex. The mechanism of equation (4.3) will contain the mass distribution of two ’swith a structure such as

22dσ

----------------------------=A(s,t,u)ρ1 µ1 ρ2 µ2

22

dtdµ1dµ2

(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 atpion as

dσdσdσ

---------=-----------+-----------+…

dEπdEπdEπ

12

(4.4)

dσ

where----------- represents the spectrum described by equation (3.34). The inclusive spectrum can be repre-dEπ

1

sented 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-pionproduction channels separately, as described previously.

5. Concluding Remarks

One-pion production channels will dominate pion production for a significant fraction of galacticcosmic ray (GCR) and solar particle event (SPE) exposures in free space, in the upper atmosphere, or onthe Martian surface. In this report, the one-pion production cross sections were discussed, and a conve-nient formula for their numerical representation was found. These models will be appended with high-energy models to span all energies of importance in GCR studies. The formula described here also canbe used to model pion and nucleon production spectra in nucleon-nucleus and nucleus-nucleusreactions.

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Appendix

Pion-Nucleon Scattering

1

TheπN scattering amplitude is written to represent the pion isospinπ and nucleon isospin--N as

2

11

fπN=--(f1/2+2f3/2)+--(f3/2–f1/2)π N

33

(A1)

wheref1/2 andf3/2 are amplitudes for total isospin 1/2 or 3/2, respectively. Introducing the total isospin

of theπN system,

1

=π+--N

2

andπN wave functions

|TTZ =

(A2)

∑ IπIZτNτZ|TTZ |IπIZτNτZ

IZ

(A3)

and noting

211

π N= T–-----

4

(A4)

leads to the following relations for elastic scattering:

πp|fπN|πp =f3 2

oo1

πp|fπN|πp =--(f1 2+2f3 2)

3 1

πp|fπN|πp =--(f3 2+2f1 2)

3++1

πn|fπN|πn =--(f3 2+2f1 2)

3oo1

πn|fπN|πn =--(f1 2+2f3 2)

3

+

+

(A5)

(A6)

(A7)

(A8)

(A9)

πn|fπN|πn =f3 2

with similar relations found for charge-exchange matrix elements.

(A10)

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

+

the totalπpandπpreactions. From equations (A5) to (A10), we can find solutions for the otherπN collision pairs (neglecting Coulomb effects) by using equations (A11) through (A14).

σT

1TTπop=-2-σπ+p+σ

π p

–σEX(πp)σ

Tπ+

n=σ

Tπ

pσTπon=σTπopσTπ n

=σTπ+p

(A11)

(A12)

(A13)

(A14)