Nuclear Instruments and Methods in Physics Research B 308 (2013) 40–45

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Nuclear Instruments and Methods in Physics Research B

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Finite sum expressions for elastic and reaction cross sections

0168-583X/$ - see front matter Published by Elsevier B.V.http://dx.doi.org/10.1016/j.nimb.2013.05.003

⇑ Corresponding author.E-mail addresses: charles.m.werneth@nasa.gov, charles.werneth@gmail.com

(C.M. Werneth), khin.maung@usm.edu (K.M. Maung), lawrence.mead@usm.edu(L.R. Mead), steve.r.blattnig@nasa.gov (S.R. Blattnig).

Charles M. Werneth a,⇑, Khin Maung Maung b, Lawrence R. Mead b, Steve R. Blattnig a

a NASA Langley Research Center, 2 West Reid Street, Hampton, VA 23681, USAb University of Southern Mississippi, Department of Physics and Astronomy, 118 College Drive, Box 5046, Hattiesburg, MS, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 8 February 2013Received in revised form 2 April 2013Available online 16 May 2013

Keywords:Glauber theoryEikonal approximationMethod of partial waves

Nuclear cross section calculations are often performed by using the partial wave method or the Eikonalmethod through Glauber theory. The expressions for the total cross section, total elastic cross section, andtotal reaction cross section in the partial wave method involve infinite sums and do not utilize simplify-ing approximations. Conversely, the Eikonal method gives these expressions in terms of integrals but uti-lizes the high energy and small angle approximations. In this paper, by using the fact that the lth partialwave component of the T-matrix can be very accurately approximated by its Born term, the infinite sumsin each of the expressions for the differential cross section, total elastic cross section, total cross section,and total reaction cross section are re-written in terms of finite sums plus closed form expressions. Thedifferential cross sections are compared to the Eikonal results for 16Oþ 16O; 12Cþ 12C, and pþ 12C elasticscattering. Total cross sections, total reaction cross sections, and total elastic cross sections are comparedto the Eikonal results for 12Cþ 12C scattering.

Published by Elsevier B.V.

1. Introduction

The basic quantity that one measures in nuclear collision exper-iments is the cross section, which is defined as the ratio of the num-ber of the events of one particular type per unit time per unitscatterer to the relative flux of the incident particles on the target[1]. While the total cross section provides useful information aboutthe interaction between the colliding particles, more informationcan be obtained about the interaction from the differential cross sec-tion. The cross sections can be further distinguished by classificationaccording to the reaction types, such as elastic, inelastic, and varioussubdivisions of reaction cross sections. This article focuses on thecalculations of the elastic differential cross section, total elastic crosssection, total reaction cross section, and total cross section.

In order to calculate these cross sections, one usually solves theSchrödinger equation or a relativistic equation for the projectileand target for a given incoming energy. The method of partialwaves is often used to solve these equations, although the Eikonalapproximation through the Glauber framework is also utilized fre-quently [2]. This paper deals primarily with the partial wave meth-od but also includes comparisons to Eikonal results.

The partial wave method involves solving the Schrödingerequation in either position-space or momentum-space without

any approximation except for the numerical methods used. Inthe Eikonal method, the high energy approximation and small an-gle approximations are utilized [1]. With the partial wave method,the scattering amplitude is written as an infinite sum of ampli-tudes for each partial wave, and the differential cross section is ob-tained by taking the absolute magnitude square of the scatteringamplitude. The total elastic cross section is obtained by integratingthe differential cross section over the solid angle, and the opticaltheorem is used to relate the total cross section to the forwardscattering amplitude [1]. In either case, one has to deal with infi-nite sums of partial waves. At low energies, these partial wavesums converge rapidly, but as the incident energy increases, morepartial waves are needed. Therefore, a more efficient solutionmethod is desirable.

This article describes the method of partial waves and how aninfinite number of partial waves can be calculated using a finitenumber of terms. The final expression for the scattering amplitudeinvolves only finite sums. Also presented are expressions for calcu-lating elastic, reaction, and total cross sections where only finitesums are involved. The differential cross sections are comparedto the Eikonal results for 16Oþ 16O; 12Cþ 12C, and pþ 12C elasticscattering. Total cross sections, total reaction cross sections, andtotal elastic cross sections are compared to the Eikonal results for12Cþ 12C scattering.

The aim of the present work is not to explain experimental databut to show that the new summation formulas are more efficientthan the usual method of partial waves. The current work shows

C.M. Werneth et al. / Nuclear Instruments and Methods in Physics Research B 308 (2013) 40–45 41

that the results are reasonable for the set of parameters used for16Oþ 16O; 12Cþ 12C, and pþ 12C scattering.

This paper is organized as follows. In Section 2, the partial wavesolution method for the Lippman–Schwinger equation is reviewed,new finite summation formulas for the differential cross section,total elastic cross section, total cross section, and total reactioncross section are derived, and the nucleon–nucleus and nucleus–nucleus potentials are discussed. It is demonstrated in Section 3that the new finite summation formulas converge more rapidlythan the usual method of partial waves, and the conclusions are gi-ven in Section 4.

2. Theory

Section 2.1 is a review of the partial wave solution method [1,3–5] of the Lippman–Schwinger equation. First, the transition opera-tor for the Lippman–Schwinger scattering equation and solutionmethod through the R-matrix equation are reviewed. Next thetransition operator for the Lippman-Schwinger equation and theR-matrix are expanded in an infinite set of partial waves, whichare then related by the Heitler equation. In Section 2.2, a new setof finite summation expressions for the differential cross section,total elastic cross section, total cross section, and total reactioncross section are derived. The form of the potential used for the cal-culations that follow is given in Section 2.3.

2.1. Method of partial waves

The Lippmann–Schwinger equation is actually the Schrödingerequation with scattering boundary conditions, and the transition-matrix operator T is written in operator form as

T ¼ V þ VGþ0 T; ð1Þ

where V is the interaction operator between the colliding particles,Gþ0 is the free outgoing Green’s operator, and T is the T-matrix oper-ator. The quantity of interest is the T-matrix element,TðqÞ ¼ hk0 j T j ki, where q ¼ k0 � k is the momentum transfer. Forelastic scattering in the center of mass frame, jk0j ¼ jkj. The Lipp-mann–Schwinger equation in momentum-space form is then,

hk0 j T j ki ¼ hk0 j V j ki þZhk0 j V j k00iG0ðk00Þhk00 j T j kidk00; ð2Þ

where Gþ0 ðk00Þ � ðEk � Ek00 þ igÞ�1 is the free Green’s function withoutgoing boundary conditions. The scattering amplitude f ðhÞ is re-lated to the T-matrix by f ðhÞ ¼ �ð2pÞ2q=khk0 j T j ki, whereq ¼ k2dk=dE and the scattering angle h is the angle between theincoming wave vector k and the outgoing wave vector k0. The differ-ential cross section is given by dr=dX ¼ jf ðhÞj2, and the total elasticcross section is then obtained by integrating over the solid angle:rel

TOT ¼Rðdr=dXÞdX. One way of solving the Lippman-Schwinger

equation is through the R-matrix equation and is defined in opera-tor form as

R ¼ V þ VPG0R; ð3Þ

where P represents the principal value integral. The equation forhk0 j R j ki has the same form as the equation for T, but the integralis the principal value integral.

In order to solve the elastic scattering T-matrix element, oneuses the method of partial waves and expands

OðqÞ ¼ hk0 j O j ki ð4Þ

¼X1l¼0

Xl

m¼�l

Olðk0; kÞYml�ðk̂0ÞYm

l ðk̂Þ ð5Þ

¼X1l¼0

2lþ 14p

Olðk0; kÞPlðxÞ; ð6Þ

where the notation, k̂0, indicates the azimuthal and polar anglesassociated with k0 (similar notation is used for the azimuthal andpolar angles associated with k), Ym

l�ðk̂0Þ and Ym

l ðk̂Þ are the sphericalharmonics, PlðxÞ are the Legendre polynomials, and O representssome operator, which, in this case, may be T;R, or V. After integrat-ing out the spherical harmonics, the lth partial wave equation for Tis

Tlðk0; kÞ ¼ Vlðk0; k0Þ þ2l�h2

Z 1

0

Vlðk0; k00ÞTlðk00; kÞk2 � k00

2 þ igk00

2dk00; ð7Þ

where l is the reduced mass of the projectile-target system, and igis used to incorporate the outgoing scattering boundary conditions.The equation for Rlðk0; kÞ also assumes the same form, but now theintegral is a principal value integral:

Rlðk0; kÞ ¼ Vlðk0; kÞ þ2l�h2 P

Z 1

0

Vlðk0; k00ÞRlðk00; kÞk2 � k002

k002dk00: ð8Þ

The input for these integral equations are the lth components of thepotential Vlðk0; kÞ given by [4]

Vlðk0; kÞ ¼ 2pZ 1

�1hk0 j V j kiPlðcos hÞdðcos hÞ; ð9Þ

where VðqÞ ¼ hk0 j V j ki is the Fourier transform of the positionspace potential VðrÞ. Once the R-matrix equation is solved for theon-shell Rlðk; kÞ, the on-shell Tlðk; kÞ is obtained through the Heitlerequation [1]:

Tlðk; kÞ ¼Rlðk; kÞ

1þ ipqRlðk; kÞ: ð10Þ

The solution methods for these equations are well-known and willnot be repeated here. The interested reader will find an excellentdiscussion of solution methods in Ref. [1]. Once the lth partial wavecomponent of Tlðk; kÞ is solved, the elastic scattering T-matrix is ob-tained by summing over the infinite number of partial waves:

TðqÞ ¼ hk0 j T j ki ¼X1l¼0

2lþ 14p

Tlðk0; kÞPlðcos hÞ; ð11Þ

where Plðcos hÞ are the Legendre polynomials of order l. For elasticscattering in the center of mass frame, jk0j ¼ jkj ¼ k andq ¼ 2k sinðh=2Þ, thus

TðqÞ ¼X1l¼0

2lþ 14p

Tlðk; kÞPlðcos hÞ: ð12Þ

The infinite series contained in the above expression is oftenapproximated by its value up to some finite l. It is hoped that, fora large enough l, the T-matrix is small enough that it is negligible.However, very large l may be required for high incident energies,and the assessment of this convergence leads to additionalcomplications.

2.2. Finite sum expressions

In this section, finite sum expressions for the scattering ampli-tude and cross sections are derived. One way to approach thisproblem is to use the fact that for large l, the Born approximationbecomes sufficient [6]; that is, for some l P Lmax,

Tlðk; kÞ � Vlðk; kÞ: ð13Þ

Using the above approximation, the infinite sum for the T-matrix isrewritten as

42 C.M. Werneth et al. / Nuclear Instruments and Methods in Physics Research B 308 (2013) 40–45

TðqÞ ¼X1l¼0

2lþ 14p Tlðk; kÞPlðcos hÞ ð14Þ

¼XLmax

l¼0

2lþ 14p Tlðk; kÞPlðcos hÞ þ

X1l¼Lmaxþ1

2lþ 14p Tlðk; kÞPlðcos hÞ

¼XLmax

l¼0

2lþ 14p

Tlðk; kÞPlðcos hÞ þX1

l¼Lmaxþ1

2lþ 14p

Vlðk; kÞPlðcos hÞ

¼XLmax

l¼0

2lþ 14p

Tlðk; kÞPlðcos hÞ þX1

l¼Lmaxþ1

2lþ 14p

Vlðk; kÞPlðcos hÞ

þXLmax

l¼0

2lþ 14p

Vlðk; kÞPlðcos hÞ �XLmax

l¼0

2lþ 14p

Vlðk; kÞPlðcos hÞ

¼XLmax

l¼0

2lþ 14p

½Tlðk; kÞ � Vlðk; kÞ�Plðcos hÞ þ VðqÞ; ð15Þ

where the Born approximation has been used in the 3rd line of Eq.(14). After some l P Lmax; Tlðk; kÞ ¼ Vlðk; kÞ to some desired accu-racy. In the 4th and 5th lines of Eq. (14), a term was added and sub-tracted in order to rewrite the two middle sums as one, which isrecognized as VðqÞ. The functional form is known since it is the po-tential. Now, it can be seen that the infinite sum of the Tlðk; kÞ iswritten as the sum of the potential term VðqÞ plus the finite sumsof the Tlðk; kÞ and Vlðk; kÞ. These are summed only to Lmax, the valueof l at which Tlðk; kÞ equals Vlðk; kÞ within a pre-assigned value oftolerance. VðqÞ does not require any new calculation since it isthe momentum space potential.

Since the scattering amplitude f ðqÞ is related to TðqÞ by

f ðqÞ ¼ �ð2pÞ2 qk

TðqÞ; ð16Þ

the expression for the elastic scattering amplitude becomes

f ðqÞ ¼ �ð2pÞ2 qk

XLmax

l¼0

2lþ 14p

½Tlðk; kÞ � Vlðk; kÞ�Plðcos hÞ

� ð2pÞ2 qk

VðqÞ: ð17Þ

Note that this expression for the elastic scattering amplitude in-volves finite sums only. The finite summation idea for the scatteringamplitude was also observed by Picklesimer et al. [7].

The expression for the total cross section can be found by usingthe optical theorem, which relates the total cross section to theimaginary part of the forward scattering amplitude; rTOT ¼ 4p

kImf ðh ¼ 0Þ. Therefore,

rTOT¼ �ð2pÞ2qk

h i4pk

XLmax

l¼0

2lþ14p

Im½Tlðk;kÞ�Vlðk;kÞ�Plðcosðh¼0ÞÞ ð18Þ

þ4pk

Im �ð2pÞ2qk

Vðq¼0Þh i

¼�16p3 qk2

XLmax

l¼0

2lþ14p

Im½Tlðk;kÞ�Vlðk;kÞ�þrTOTðBornÞ; ð19Þ

where the last term in Eq. (19) is identified as the total cross sectionin the first Born approximation. Note that the latter is the expres-sion for the total cross section rTOT in terms of finite sums andthe total cross section obtained in the first Born approximation,rTOTðBornÞ, which is obtained by taking the imaginary part of themomentum space potential at the forward angle.

Next, a similar expression for the total elastic cross section relTOT

is derived. The total elastic cross section is given by

relTOT�

ZdrdX

� �dX ð20Þ

¼Zð2pÞ2 q

k

� �2jTj2dX

¼Zð2pÞ2 q

k

� �2Xl;l0

2lþ14p

� �2l0 þ1

4p

� �Tlðk0;kÞT 0lðk

0;kÞPlðxÞP0lðxÞdxd/;

where x � cos h. After performing integration over x and /, the Bornapproximation for the higher l-values (i.e. Tl � Vl) will be used, andafter adding and subtracting

PLmaxl¼0 ð2lþ 1Þ=4pjVlj2, the following is

obtained:

relTOT ¼

ð2pÞ4q2

k2

XLmax

l¼0

2lþ 14p

� �jTlj2 � jVlj2� �" #

þ relTOTðBornÞ; ð21Þ

where the last term is the total elastic cross section in the first Bornapproximation given by

relTOTðBornÞ �

X1l¼0

2lþ 14p

� �j �ð2pÞ2 q

kVlðk0; kÞ

� �j2

¼Zj � ð2pÞ2 q

kVðqÞj2dX: ð22Þ

Again, the above expression for the total elastic cross section in-volves finite sums and an integral over the known function jVðqÞj2.

An expression for the total reaction cross section, rreTOT, can be

found by using rTOT ¼ relTOT þ rre

TOT, which results in

rreTOT ¼ �

ð2pÞ3qk2

XLmax

l¼0

2lþ 14p

� �ImðTlðk0; kÞ � Vlðk0; kÞÞ

" #þ rTOTðBornÞ

� ð2pÞ4q2

k2

XLmax

l¼0

2lþ 14p

� �jTlj2 � jVlj2� �" #

� relTOTðBornÞ: ð23Þ

Note that all of these expressions involve only finite sums.

2.3. Potential

The partial wave analysis presented herein is used to predictdifferential cross sections, total cross sections, elastic cross sec-tions, and inelastic cross sections. In the previous section, it wasstated that that the momentum-space potential would be neededto compute the T-matrix. This section describes the form of the po-tential used in the calculations that follow in the results section.

For nucleon–nucleus scattering,

VðqÞ ¼ AT tnnðqÞqTðqÞ; ð24Þ

where AT is the number of nucleons in the target, tnn is the nucleon–nucleon transition amplitude, and qTðqÞ is the charge density of thetarget. The transition amplitude is given by [8]

tnnðqÞ ¼1

4prðeÞknn½aðeÞ þ i� exp

�BðeÞq2

2

� �; ð25Þ

where e is the nucleon–nucleon kinetic energy in the center of massframe, knn is the nucleon–nucleon wave vector, rðeÞ is nucleon–nu-cleon total cross section, aðeÞ is the real to imaginary ratio, and BðeÞis the slope parameter.

In the results that follow, the harmonic well charge densitymodel is used,

qðqÞ ¼ q0p32a3 1þ 3c

2

� �� ca2

4q2

� �exp½�q2s2�; ð26Þ

with

q0 ¼1

p32a3

1þ 3c2

� ��1

; ð27Þ

where a and c are nuclear charge parameters [8,9], ands2 ¼ a2=4� r2

n=6 ; r2n ¼ a2

prot � 0:11ðN=ZÞ, where N is the number ofneutrons, Z is the number of protons, and aprot � 0:87 fm is the pro-ton radius [8,10]. The charge density was converted to a matterdensity by dividing the charge density by the proton charge formfactor as shown in Eq. (27).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14θ

CM (degrees)

100

101

102

103

104

105

106

dσ/d

Ω (m

b/sr

)

dataEikonalL

max = 95 (Cut)

Lmax

= 95 (All)L

max = 120 (Cut)

Lmax

= 120 (All)

16O +

16O

TLab

= 1120 MeV

Fig. 2. Differential cross section for 16Oþ 16O scattering at 1120 MeV. Experimentaldata are taken from Khoa et al. [13]. See text for explanation.

Table 1Kinetic energy dependent (TLab) parameters for the nucleon–nucleon transitionoperator in Eq. (25).

Lab energy (reaction) r (fm2) B (fm2) a

704 MeV (16Oþ 16O) 11.93 0.24 0.98

1120 MeV (16Oþ 16O) 7.16 0.26 1.19

204.2 MeV (12Cþ 12C) 36.08 0.20 0.77

242.7 MeV (12Cþ 12C) 30.21 0.21 0.79

288.6 MeV (12Cþ 12C) 25.05 0.22 0.82

C.M. Werneth et al. / Nuclear Instruments and Methods in Physics Research B 308 (2013) 40–45 43

The nucleus–nucleus potential is similar to Eq. (24), though thenuclear charge density is used for the projectile as well; therefore[8]

VðqÞ ¼ APAT tnnðqÞqPðqÞqTðqÞ; ð28Þ

where AP is the number of projectile nucleons, and qPðqÞ is the den-sity of the projectile. Harmonic well nuclear charge densities havebeen used for both the projectile and target in this paper fornucleus–nucleus scattering.

The values of the slope parameter and aðeÞ for the 12C + 12Creaction are not given in Townsend et al. [10]. The authors of thecurrent work use Ref. [8] for the slope parameter, and the currentauthors have fit to experimental data in Ref. [11] to obtain an esti-mate of the isospin averaged aðeÞ for projectile energies between30 and 2200 MeV. Finally, the nucleon-nucleon cross sections areobtained from Ref. [12].

3. Results

In this section, the partial wave results are compared to the Eik-onal approximation and experimental data. The partial wave re-sults are computed with two distinct models—one model utilizesthe standard partial wave solution as described in Section 2.1 ofthe current work and documented in Ref. [1], and the other modeluses the finite sum expressions derived in Section 2.2. In addition,the Eikonal approximation described in Ref. [1] has been used. Thedifferential cross sections as predicted from the two distinct partialwave models are compared to the Eikonal results for16Oþ 16O; 12Cþ 12C, and pþ 12C elastic scattering. Total cross sec-tions, total reaction cross sections, and total elastic cross sectionsare compared to the Eikonal results for 12Cþ 12C scattering.

The aim of the current work is to show that the new summationformulas are more efficient than the usual method of partial waves.Therefore, Coulomb correction terms have not been included in thecurrent results. The interested reader will find a partial wave anal-ysis with Coulomb corrections in Ref. [10].

The differential cross sections for 16Oþ 16O scattering at twoprojectile energies are given in Figs. 1 and 2. The harmonic well nu-clear charge density parameters in Eq. (26) for 16O are a ¼ 1:83 fmand c ¼ 1:54 [9]. The transition amplitude parameters in Eq. (25)are given in Table 1. The experimental data are given as solid cir-cles with error bars and are taken from Khoa et al. [13].

Fig. 1 shows the differential cross section for an 16Oþ 16O colli-sion with a projectile kinetic energy of 704 MeV. Lmax ¼ 80 indi-

5 6 70 1 2 3 4 8 9 10 11 12 13 14θ

CM (degrees)

101

102

103

104

105

106

dσ/d

Ω (

mb/

sr)

dataEikonalL

max = 80 (Cut)

Lmax

= 80 (All)L

max = 95 (Cut)

Lmax

= 95 (All)

16O +

16O

TLab

= 704 MeV

Fig. 1. Differential cross section for 16Oþ 16O scattering at 704 MeV. Experimentaldata are taken from Khoa et al. [13]. See text for explanation.

cates that the cross sections are calculated by summing thepartial waves up to l ¼ 80: likewise, for Lmax ¼ 95. Cut representsthat the summation of partial waves is terminated at Lmax, whichis the usual method of calculation involving an infinite sum of par-tials waves for the T-matrix. All indicates that the finite summationformulas for the scattering amplitude have been used:dr=dX ¼ jf ðqÞj2, where f ðqÞ is given by Eq. (17). The red (Cut) andviolet (All) lines are obtained with Lmax ¼ 80, and the green (Cut)and blue (All) lines are produced with Lmax ¼ 95. Fig. 1 shows thatthe differential cross sections obtained from Lmax ¼ 80 (All),Lmax ¼ 95 (Cut), and Lmax ¼ 95 (All) are nearly identical; however,the Lmax ¼ 80 (Cut) case has yet to converge with the other results.This shows that the new finite summation formulation convergesmore rapidly than the case involving truncation of the infinite ser-ies for the same Lmax. Also note that the partial wave results are inagreement with the Eikonal approximation at smaller angles butbegin to diverge slightly at larger angles, as expected.

The differential cross section for an 16Oþ 16O collision with aprojectile kinetic energy of 1120 MeV is presented in Fig. 2. This re-sult shows that the higher energy case requires a larger Lmax forconvergence than the lower energy case. As with the lower energycase, the finite summation formula converges more rapidly thanthe usual method of truncating the infinite series for Lmax ¼ 95,which is seen from the convergence of the differential cross sec-tions that result from Lmax ¼ 95 (All), Lmax ¼ 120 (Cut), andLmax ¼ 120 (All). Again, it is observed that the partial wave resultsagree with the Eikonal approximation at smaller angles but beginto diverge at larger angles.

The differential cross sections for 12Cþ 12C scattering at projec-tile energies of 204.2, 242.7, and 288.6 MeV are given in Figs. 3–5.The harmonic well nuclear charge density parameters in Eq. (26)for 12C are a ¼ 1:692 fm and c ¼ 1:082 [10,9]. The transition

0 5 10 15 20 25 30 35 40θ

CM (degrees)

100

101

102

103

104

105

106

dσ/d

Ω (m

b/sr

)

dataEikonalL

max = 50 (Cut)

Lmax

= 50 (All)L

max = 80 (Cut)

Lmax

= 80 (All)

12C +

12C

TLab

= 204.2 MeV

Fig. 3. Differential cross section for 12Cþ 12C scattering at 204.5 MeV. Experimentaldata are taken from Refs. [10,14]. See text for explanation.

0 5 10 15 20 25 30 35 40θ

CM (degrees)

10-1

100

101

102

103

104

105

106

dσ/d

Ω (m

b/sr

)

dataEikonalL

max = 50 (Cut)

Lmax

= 50 (All)L

max = 80 (Cut)

Lmax

= 80 (All)

12C +

12C

TLab

= 242.7 MeV

Fig. 4. Differential cross section for 12Cþ 12C scattering at 242.7 MeV. Experimentaldata are taken from Refs. [10,14]. See text for explanation.

5 10 15 20 25 30 35 40θ

CM (degrees)

10-2

10-1

100

101

102

103

104

105

106

dσ/d

Ω (m

b/sr

)

dataEikonalL

max = 50 (Cut)

Lmax

= 50 (All)L

max = 80 (Cut)

Lmax

= 80 (All)

12C +

12C

TLab

= 288.6 MeV

Fig. 5. Differential cross section for 12Cþ 12C scattering at 288.6 MeV. Experimentaldata are taken from Refs. [10,14]. See text for explanation.

44 C.M. Werneth et al. / Nuclear Instruments and Methods in Physics Research B 308 (2013) 40–45

amplitude parameters in Eq. (25) are given in Table 1. The experi-mental data are given as solid circles with error bars and are takenfrom Refs. [10,14].

The differential cross sections given in Figs. 3–5 show the con-vergence behavior as illustrated in the previous example; that is,the finite sum formulas converge more rapidly than the standardmethod of partial waves, although both models converge rapidlyat the lowest energy. There is excellent agreement in the angularposition of the maxima and minima with experimental data. How-ever, there is noticeable difference between the magnitude of thedifferential cross section of the models and the experimental dataat the most forward scattering angles. The effect is the most pro-nounced for the lowest projectile energy, which is closest to theCoulomb barrier. The ostensible effect of the Coulomb interactionis the least noticeable at the highest energy, as expected. Overall,the differential cross sections are systematically under-predictedat the most forward scattering angles, as the current work doesnot include the Coulomb interaction. Again, the authors restatethat the purpose of this research is to show that the finite summa-tion formulas converge more rapidly than the standard method ofpartial waves, and that has been demonstrated here.

Table 2 indicates the total cross sections, total reaction crosssections, and total elastic cross sections for 12Cþ 12C reactions atprojectile kinetic energies of TLab = 204.2 MeV, 242.7 MeV, and288.6 MeV as calculated with the partial wave and Eikonal meth-ods. Experimental data are taken from Cole et al. [14]. The param-eters for the harmonic well nuclear charge density in Eq. (26) for12C are a ¼ 1:692 fm and c ¼ 1:082 [9,10], and the nucleon transi-tion amplitude parameters used for the 12Cþ 12C reactions arefound in Table 1. The apparent effect of the Coulomb interactionis noticeable in that that total cross sections are under-predicted;however, the convergence behavior is again demonstrated. In eachcase, for Lmax ¼ 50, the new finite summation formulas result incross sections that are converging to those produced withLmax ¼ 80, whereas the Lmax ¼ 50 (Cut) case has yet to convergeto the cross sections obtained with Lmax ¼ 80. This is further evi-dence that the finite summation formula converges more rapidlythan the partial wave method, which truncates the infinite seriesfor the T-matrix.

It is necessary to note that the to real to imaginary ratio of thetransition amplitude is undetermined, albeit the authors of thecurrent work use a fit to limited experimental data to predictaðeÞ. Nonetheless, the magnitude of the differential cross sectionalso depends on this parameter, as well as parameterizations ofthe isospin averaged nucleon–nucleon cross section and the slopeparameter. The parameters utilized in the current work result from

Table 2Total cross sections, total reaction cross sections, and total elastic cross sections for12Cþ 12C reactions as calculated with the partial wave and Eikonal methods.Experimental data are taken from Cole et al. [14]. See text for discussion.

TLab (MeV) rTOTðmbÞ rreTOTðmbÞ rel

TOTðmbÞ

204.2 Lmax ¼ 50 (Cut) 2393 1303 1090Lmax ¼ 50 (All) 2429 1331 1089Lmax ¼ 80 (All) 2429 1331 1089Eikonal 2447 1314 1133Experiment 2619� 80 1367� 40 –

242.7 Lmax ¼ 50 (Cut) 2328 1265 1063Lmax ¼ 50 (All) 2359 1296 1063Lmax ¼ 80 (All) 2359 1296 1063Eikonal 2386 1281 1104Experiment 2586� 80 1320� 40 –

288.6 Lmax ¼ 50 (Cut) 2260 1224 1036Lmax ¼ 50 (All) 2292 1257 1035Lmax ¼ 80 (All) 2295 1259 1035Eikonal 2320 1246 1074Experiment 2660� 80 1330� 40 –

0 5 10 15 20 25 30 35 40θ

CM (degrees)

10-4

10-3

10-2

10-1

100

101

102

103

104

dσ/d

Ω (m

b/sr

)

dataEikonalL

max = 20 (Cut)

p + 12

C

TLab

= 500 MeV

Fig. 6. Differential cross section for pþ 12C scattering at 500 MeV. Experimentaldata are taken from Ref. [15]. See text for explanation.

C.M. Werneth et al. / Nuclear Instruments and Methods in Physics Research B 308 (2013) 40–45 45

a global fit to experimental data and are not adjusted to obtain a fitto any particular reaction. Also, isospin averaging may result in dis-crepancy between theoretical prediction and experimental data.

As an example of a differential cross section with a projectileenergy well above the Coulomb barrier, consider the 500 MeVpþ 12C reaction in Fig. 6. At this energy, the differential cross sec-tion as predicted by the finite summation formula and the Eikonalmethod are in good agreement with experimental data [15]. Notethat the error bars indicate statistical uncertainty only.

4. Conclusions

A set of finite summation expressions is derived for the differ-ential cross section, total elastic cross section, total reaction crosssection, and the total cross section in the partial wave method.These expressions are derived by using the Born approximationfor all partial waves above some Lmax, where Lmax is determinedby the fact that Tl ¼ Vl to a desired accuracy at l ¼ Lmax. Since theexpressions contain only finite sums up to Lmax, the eventualnumerical inaccuracy of calculating Vl and Tl for the higher partialwaves is not a concern.

It is found that, for a given Lmax, the finite summation formulasconverge more rapidly to the results obtained with higher Lmax

than the usual partial wave method, which involves truncation ofan infinite series for the T-matrix. The reason for this differenceis that the new formulation of the cross section includes essentiallyall partial waves once Tl ¼ Vl. It is observed that more partialwaves are needed at higher energy than lower energy for conver-

gence. In addition, the partial wave results are found to agree withthe Eikonal approximation at smaller angles, but slight differencesare noted at larger angles.

It is noted that the differential and total cross sections were un-der-predicted for 12Cþ 12C reactions at low energies, and this isattributed to Coulomb effects, which were not included in the cur-rent model, and an undetermined real to imaginary ratio, aðeÞ, ofthe transition amplitude. As a result, the 500 MeV pþ 12C reactionwas chosen to illustrate the validity of the finite summation partialwave model for reactions well above the Coulomb barrier. Theparameters used in the current work result from a global fit toexperimental data and were not adjusted to obtain a fit to anyone particular reaction. Additional discrepancies between experi-mental data and the prediction of the theoretical models may bedue to isospin averaging.

It is not the goal of this research to explain experimental databut to show that the new finite summation formulas are more effi-cient than the usual partial wave method. This work shows that theresults are reasonable for the set of parameters used for16Oþ 16O; 12Cþ 12C, and pþ 12C scattering.

Acknowledgments

The authors thank Drs. Ryan Norman, Tony Slaba, FrancisBadavi, John Norbury, and Jonathan Ransom for reviewing this pa-per. This work was supported in part by NASA grants NNX09AE94A(University of Southern Mississippi) and NNX10AD18A (Universityof Tennessee).

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