September 2014 

NASA/TP–2014-218529 

Elastic Differential Cross Sections

Charles M. Werneth 

Langley Research Center, Hampton, Virginia 

Khin M. Maung and William P. Ford 

University of Southern Mississippi, Hattiesburg, Mississippi 

John W. Norbury 

Langley Research Center, Hampton, Virginia 

Michael D. Vera 

University of Southern Mississippi, Hattiesburg, Mississippi 
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September 2014 

NASA/TP–2014-218529 

Elastic Differential Cross Sections 

Charles M. Werneth 

Langley Research Center, Hampton, Virginia 

Khin M. Maung and William P. Ford 

University of Southern Mississippi, Hattiesburg, Mississippi 

John W. Norbury 

Langley Research Center, Hampton, Virginia 

Michael D. Vera 

University of Southern Mississippi, Hattiesburg, Mississippi 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 

Acknowledgments 

The authors would like to thank Drs. Steve Blattnig, Ryan Norman, Jonathan Ransom, 

and Francis Badavi for reviewing this manuscript. This work was supported by the 

Human Research Program under the Human Exploration and Operations Mission 

Directorate of NASA and NASA grant number NNX13AH31A. Khin Maung Maung 

would like to thank Alexander Maung for his helpful conversations. 
Contents

1 Introduction 1

2 Theory 4

2.1 Lippmann-Schwinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Multiple Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Matrix Element of Optical Potential . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Optimum Factorization . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Eikonal Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 Nucleon-Nucleus Optical Potential . . . . . . . . . . . . . . . . . . 16

2.4.2 Nucleus-Nucleus Optical Potential . . . . . . . . . . . . . . . . . . . 16

2.4.3 Optical Potential in Momentum-Space . . . . . . . . . . . . . . . . 18

2.5 Lippmann-Schwinger Partial Wave Solution Method . . . . . . . . . . . . . 20

2.6 Lippmann-Schwinger 3D Solution Method . . . . . . . . . . . . . . . . . . 21

3 Numerical Methods 23

3.1 Eikonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Partial Wave Lippmann-Schwinger . . . . . . . . . . . . . . . . . . . . . . 24

3.3 3D Lippmann-Schwinger . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4 Localized Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5 Equal Mass Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Results 30

5 Conclusions 32

6 Appendix A 32

7 Appendix B 34

8 Appendix C 35

8.1 Non-relativistic Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

8.2 Relativistic Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

iii
Nomenclature

AA Nucleus-Nucleus

CM Center of Mass

LI Lorentz Invariant

LS Lippmann-Schwinger

LS3D Three Dimensional Lippmann-Schwinger

MST Multiple Scattering Theory

NA Nucleon-Nucleus

NN Nucleon-Nucleon

NR Non-Relativistic

PW Partial Wave

REL Relativistic

iv
List of Figures

1 Transition amplitude for AA scattering . . . . . . . . . . . . . . . . . . . . 40

2 Optimum factorization for AA scattering . . . . . . . . . . . . . . . . . . . 40

3 Vectors for position space optical potential . . . . . . . . . . . . . . . . . . 41

4 Optical potential for low energy p + 4He reaction . . . . . . . . . . . . . . 42

5 Optical potential for high energy p + 4He reaction . . . . . . . . . . . . . . 42

6 Optical potential for low energy 4He + 20Ne reaction . . . . . . . . . . . . 43

7 Optical potential for medium energy 4He + 20Ne reaction . . . . . . . . . . 43

8 Optical potential for low energy 4He + 4He reaction . . . . . . . . . . . . . 44

9 Optical potential for high energy 4He + 4He reaction . . . . . . . . . . . . 44

10 Optical potential for low energy 12C + 12C reaction . . . . . . . . . . . . . 45

11 Optical potential for high energy 12C + 12C reaction . . . . . . . . . . . . . 45

12 Kinematic functions for 4He + 4He reaction . . . . . . . . . . . . . . . . . 46

13 Alternate view of kinematic functions for 4He + 4He reaction . . . . . . . . 46

14 Ratio of kinematic functions for 4He + 4He reaction . . . . . . . . . . . . . 47

15 Kinematic functions for 56Fe + 56Fe reaction . . . . . . . . . . . . . . . . . 47

16 Alternate view of kinematic functions for 56Fe + 56Fe reaction . . . . . . . 48

17 Ratio of kinematic functions for 56Fe + 56Fe reaction . . . . . . . . . . . . 48

18 Elastic differential cross sections for p + 16O reactions . . . . . . . . . . . . 49

19 Elastic differential cross sections for p + 56Fe reactions . . . . . . . . . . . 50

20 Elastic differential cross sections for 4He + 16O reactions . . . . . . . . . . 51

21 Elastic differential cross sections for 12C + 56Fe reactions . . . . . . . . . . 52

22 Elastic differential cross sections for 4He + 4He reactions . . . . . . . . . . 53

23 Elastic differential cross sections for 20Ne + 20Ne reactions . . . . . . . . . 54

24 Elastic differential cross sections for 56Fe + 56Fe reactions . . . . . . . . . . 55

25 Comparisons to experimental data . . . . . . . . . . . . . . . . . . . . . . . 56

v
The eikonal, partial wave (PW) Lippmann-Schwinger, and three-dimensional

Lippmann-Schwinger (LS3D) methods are compared for nuclear reactions that

are relevant for space radiation applications. Numerical convergence of the

eikonal method is readily achieved when exact formulas of the optical potential

are used for light nuclei (A ≤ 16) and the momentum-space optical potential is

used for heavier nuclei. The PW solution method is known to be numerically

unstable for systems that require a large number of partial waves, and, as a

result, the LS3D method is employed. The effect of relativistic kinematics is

studied with the PW and LS3D methods and is compared to eikonal results.

It is recommended that the LS3D method be used for high energy nucleonnucleus reactions and nucleus-nucleus reactions at all energies because of its

rapid numerical convergence and stability.

1 Introduction

The space radiation environment is composed of solar particle emissions and ions produced

from supernovae distributed throughout the galaxy[1, 2]. Solar particle events, including

both coronal mass ejections and solar flares, are composed of mostly protons with energies

that can exceed several hundred MeV. Galactic cosmic rays originate from the shock waves

of supernovae and consist of protons and heavier ions with energies that reach hundreds of

GeV per nucleon. Radiation transport codes are used to describe the transport of ions, and

secondary particles produced from nuclear collisions, from the space radiation environment

through shielding materials. Space radiation transport codes require cross sections for

the numerous nuclear reactions that occur as a result of collisions of nuclei in the space

radiation environment with nuclei in the shield. NASA’s deterministic transport code,

HZETRN [3–5], currently transports all ions up to nickel—where, thereafter, incident

particle fluxes are negligible [6]—with energies that extend from MeV to hundreds of

GeV per nucleon through shielding materials. Efficient, accurate codes are needed for the

computation of nuclear cross sections due to the large number of nuclear reactions that

occur at these energies.

The Lippmann-Schwinger (LS) equation is an expression for the scattering transition

amplitude [7]. Scattering amplitudes can be obtained by either solving the LS equation

or by employing some approximation, such as the eikonal method. The elastic differential

cross section is computed from the absolute square of the scattering amplitude, and the

total cross section is related to the imaginary part of the forward scattering amplitude.

The elastic cross section is obtained by performing the angular integration of the elastic differential cross section, and the reaction cross section is found from the difference

between the total and elastic cross sections.

For elastic scattering, it is convenient to use the ground state and excited state projectors to write an equivalent set of coupled equations for the LS equation: the elastic

1
scattering equation and the defining equation for the optical potential [8, 9]. This is useful for the study of elastic scattering because all of the complications that arise from the

intermediate excited states are not explicitly found in the elastic scattering equation. The

input into the elastic scattering equation is the optical potential, which can be expressed

in an infinite series of nucleon-nucleon (NN) transition amplitudes, tNN. If the transition

matrix is written for ground states of the projectile and target, then, in the factorization

approximation, the first order optical potential is proportional to tNN and the nuclear

densities of the projectile and target [10–12]. The model of tNN used in the present work

is parameterized to NN total cross sections, slope parameters, and the real to imaginary

ratios of the transition amplitude. Nuclear charge density distributions are obtained from

electron scattering experiments [13, 14]. Matter densities of nuclei are found from nuclear charge densities by factoring out the charge distribution of the proton. The internal

charge structure of the proton is not taken into account in this analysis; instead, nucleons

are treated as point particles. Harmonic well densities are typically used for lighter nuclei

because of the Gaussian-like decay of the nuclear charge density as a function of radial

distance. Wood-Saxon densities, also known as two-parameter and three-parameter Fermi

densities, are better suited for heavier nuclei, where the nuclear charge density is relatively

constant before decreasing to zero at larger radial distances.

The two most common ways of solving the LS equation are to use the eikonal approximation or the method of partial wave (PW) decomposition. The eikonal approximation

was first introduced by Moliere and systematically developed by Glauber in the treatment

of many-body nuclear reactions with a quantum collision theory of composite objects [15].

The eikonal approximation can be derived by assuming high energy and small angle scattering, which leads to a linearized propagator in the LS equation from which the eikonal

scattering wave function may be obtained [7]. The scattering amplitude is determined

from the eikonal phase factor, which is a function of the the optical potential [5, 16–20].

Besides being an approximation, a drawback of the eikonal approximation is that it

may be numerically inefficient for the evaluation of the cross sections for a given optical potential. In the position-space representation, the optical potential, U(r), is given

by a 6-dimensional integration for heavy ion collisions [5]. Therefore, the eikonal phase

factor depends on a 6-dimensional integral in the position representation of the optical

potential and an additional integration variable over a coordinate in the scattering plane.

The numerical integration over 7-dimensions in the position space representation is inefficient when an analytic expression of the optical potential is not known. It is desirable

to use exact formulas for the optical potential when analytic expressions of the optical

potential can be found. The current work implements expressions of the optical potential

for nucleon-nucleus (NA) and nucleus-nucleus (AA) scattering utilizing harmonic well nuclear matter densities for light nuclei (A ≤ 16), and the optical potential is expressed in

momentum space for cases where no analytic expression can be found (A > 16).

The present work expresses the optical potential as a function of momentum transfer

and reduces the 7-dimensional integration of the eikonal phase function to 2-dimensions,

2
which significantly increases numerical efficiency. An additional advantage of this formulation of the eikonal phase function is that the momentum space representation of the

optical potential may be used for any nuclear density and transition amplitude, provided

the Fourier transforms may be evaluated or the explicit forms of the the densities are

known in momentum-space.

The LS equation may also be solved via the method of partial wave decomposition

[7, 21], where the transition amplitude is expanded in an infinite series of functions of

relative momenta and angular dependent spherical harmonics or Legendre polynomials.

After projecting out the angular dependence, the transition amplitude is solved for a given

partial wave. Once the partial wave solutions are found, the full solution for the transition

amplitude is found by re-summing the series, which is terminated when some pre-defined

tolerance of precision is reached.

The PW method is known to become numerically unstable for reactions that require

many partial waves [7], which is not only limited to high energy NA reactions (GeV/n)

but also includes AA reactions at relatively low energy per nucleon (hundreds of MeV/n).

The numerical instability can be traced back to highly oscillating Legendre polynomials in

the PW expansion and large on-shell momenta for elastic reactions, where contributions

to the transition amplitude tend to be localized.

Although there are numerical limitations associated with the PW method, the full

three-dimensional Lippmann-Schwinger (LS3D) solution method circumvents the necessity of using highly oscillating Legendre polynomials [22–26]. Most of the LS3D studies

have consisted of NN interactions [22, 23, 25] with the exception of Rodriguez–Gallardo

et al. [24] who studied NA and AA reactions at relatively low energies and Liu et al. [26]

who studied three-body reactions. This demonstrates the validity of the method and can

be compared to results generated with the PW method since few partial waves are needed

for such reactions. The LS3D method is compared to the PW and eikonal methods for

NA and AA reactions with energies extending from 150 MeV/n to 20 GeV/n.

The eikonal solution method is a non-relativistic approximation; however, when energies become sufficiently high, relativistic effects will be manifested in the elastic differential

cross section. Relativistic kinematics are needed for high energy reactions and are easily

incorporated into the momentum-space representation of the PW and LS3D equations,

where the momentum is simply a number instead of a spatial derivative operator, as in

the position-space representation. At relativistic energies, the PW and LS3D models will

agree if convergence of the partial wave solution is reached, but both methods should

differ from the eikonal results. In the low energy limit, the eikonal method should break

down and begin to diverge from the PW and LS3D results because of the approximations

inherent in the eikonal method. To examine the effect of kinematics, model results are

compared for various nuclear reactions at relativistic and non-relativistic energies.

At relativistic energies, the inner structure of the nucleons may be probed. The multiple scattering theory (MST) upon which the model of interaction is based and the NN

transition amplitude do not account for the inner structure of the nucleons. The compli3
cations associated with the inner structure of the nucleons are assumed to be included in

the parameterizations to experimental NN transition amplitudes.

In this paper, exact formulas are presented for the optical potential in the position

space representation for light nuclei. A momentum-space representation of the eikonal

approximation is presented and used for heavier nuclei. Both methods are utilized in

the present work and are significantly more efficient than performing the usual eikonal

calculation in the position-space representation. The PW and LS3D methods are solved

with non-relativistic and relativistic kinematics, and comparisons of the models are made

for reactions that are relevant to space radiation. Additional comparisons to experimental

data are presented. Based on the results presented herein, it is recommended that the

LS3D method be used for high energy NA reactions and AA reactions at all energies

because of its rapid numerical convergence and stability. The effect of the kinematics for

projectiles and targets with equal masses and extensive comparisons to experimental data

will be communicated in subsequent manuscripts.

The present work is organized as follows. In section 2, a theoretical overview of the

LS equation, MST, the elastic scattering equation, and the optical potential are reviewed.

This is followed by a discussion of the eikonal, PW, and LS3D methods. The numerical

solution methods are discussed in section 3, and model comparisons are given in section

4. The conclusions are stated in section 5.

2 Theory

2.1 Lippmann-Schwinger Equation

The LS equation is an expression for the scattering transition operator—the fundamental

quantity that is used to evaluate the elastic differential, elastic, reaction, and total cross

sections for nuclear reactions—and is given as

T = V + V G+

0 T, (1)

where V is the sum of residual two-body interactions (potentials) for the projectile-target

system, and G

+

0

is the unperturbed two-body propagator [27]. In momentum-space, the

transition matrix element may be written

hk

0

|T|ki = hk

0

|V |ki + hk

0

|V G+

0 T|ki, (2)

where k and k

0 are the initial and final relative momenta of the projectile-target system

in the center of momentum (CM) frame, respectively. Using the notation T(k

0

, k) ≡

4
hk

0

|T|ki, the momentum-space integral equation representation of the LS equation is

T(k

0

, k) = V (k

0

, k) + ZV (k

0

, k

00)G+

0

(k

00, k)T(k00

, k)dk

00

, (3)

where k = |k| and completeness over momenta has been used,

ˆ1 = Z

dk

00|k00ihk00|. (4)

For elastic scattering, the scattering amplitude, f(θ), is related to the transition matrix

by [7]

f(θ) = (−2π)

2ρ

k

T(k, θ), (5)

where k = |k|, θ is the scattering angle, the density of states, ρ, is given by

ρ = k

2

dk/dE (6)

and E is the energy. For non-relativistic (NR) kinematics, E = k

2/2μ, where μ =

(mPmT )/(mP + mT ) is the reduced mass, mP is the mass of the projectile, and mT is

the mass of the target. When using relativistic (REL) kinematics, E =

p

k

2 + m2

p

P +

k

2 + m2

T

.

The total cross section is proportional to the imaginary part of the forward scattering

amplitude, σtot = (4π/k)Im [f(θ = 0)], and the elastic differential cross section is found

from taking the absolute square of the scattering amplitude dσ/dΩ = |f(θ)|

2

. One simply

integrates the differential cross section over the solid angle to obtain the elastic cross

section, σel =

R

(dσ/dΩ)dΩ. The reaction cross section, σre, can be found by subtracting

the elastic cross section from the total cross section, σre = σtot − σel.

2.2 Multiple Scattering Theory

The fundamental theory which describes the interaction is MST, which was developed

because the Born series proved to be inadequate due to the strong nature of the nuclear

force. Watson showed that the unperturbed Hamiltonian can be separated from the sum

of nucleon-nucleon interactions (residual interaction) between the projectile and target

such that the leading term in the Born series can be expressed as the sum of pseudo twobody Watson-τ operators [28]. For AA scattering, the transition amplitude from equation

(1) is written [27, 29]

T =

X

AP

i=1

X

AT

j=1

Tij , (7)

5
where

Tij = τij + τijG

+

0

X

AP

k6=i

X

AT

l6=j

Tkl. (8)

AP and AT are the number of nucleons in the projectile and target, and the Watson-τ

operator is

τij ≡ vij + vijG

+

0

τij . (9)

The objective of the MST is to express the transition amplitude in terms of the free

NN transition amplitude, tij = vij + vijgtij , where g is the free two body propagator. To

that end, the pseudo two-body operator is written [27, 29]

τij = tij + tij (G

+

0 − g)τij . (10)

The impulse approximation is often used and is based on the idea that the projectile

has sufficiently high energy such that the target nucleon is treated as if it were free; that

is, τij ≈ tij , which leads to the following transition amplitude [27, 29]:

Timpulse =

X

AP

i=1

X

AT

j=1

tij +

X

AP

i=1

X

AT

j=1

tijG

+

0

X

AP

k6=i

X

AT

l6=j

tkl + · · · . (11)

The propagator G

+

0

includes the ground states and all intermediate excited states of the

projectile and target. As a result, equation (11) is difficult to solve.

If elastic differential cross sections are sought, then it is convenient to separate the

LS equation into an equivalent set of coupled equations with the ground state (P) and

excited state (Q) projectors [8, 9]. The elastic scattering equation is

T = U + UP G+

0 P T, (12)

and the optical potential is given by

U = V + V QG+

0 QU. (13)

All of the intermediate excited states have been included in the optical potential. With

that in mind, the Watson series for the optical potential may be written in a form similar

to equations (7) and (8) [29],

U =

X

AP

i=1

X

AT

j=1

Uij , (14)

and

Uij =  ̃τij +  ̃τijQG+

0 Q

X

AP

k6=i

X

AT

l6=j

Ukl, (15)

6
with

τ ̃ij ≡ vij + QG+

0 Qτ ̃ij . (16)

Again,  ̃τ may be expressed in terms of the free two-body amplitude [27, 29]

τ ̃ij = tij + tij (QG+

0 Q − g) ̃τij , (17)

where the tilde notation,  ̃τ , indicates that the pseudo two-body operators have been

modified by the excited state projector, Q. If the impulse approximation is used ( ̃τij ≈ tij ),

the Watson series for the optical potential becomes [29]

Uimpulse =

X

AP

i=1

X

AT

j=1

tij +

X

AP

i=1

X

AT

j=1

tijQG+

0 Q

X

AP

k6=i

X

AT

l6=j

tkl + · · · . (18)

Finally, if the higher order terms are neglected, and only single scattering is considered

[27, 29]:

U

1st

impulse ≈

X

AP

i=1

X

AT

j=1

tij . (19)

At higher energies, it is assumed that higher order terms do not contribute significantly

more than the single scattering term.

In all of the results that follow, U ≡ U

1st

impulse, and the elastic scattering equation

becomes

T(k

0

, k) = U(k

0

, k) + Z

U(k

0

, k

00)T(k00

, k)

E(k) − E(k

00) + i

dk

00

, (20)

where E is the energy and i ensures outward scattering boundary conditions.

2.3 Matrix Element of Optical Potential

The explicit form of the optical potential must be defined before the solution of the LS

equation can be found. The matrix element for the optical potential is written [10]

U(k

0

, k) = hk

0

; φ

AP

0 φ

AT

0

|U|φ

AP

0 φ

AT

0

; ki =

X

AP

i=1

X

AT

j=1

hk

0

; φ

AP

0 φ

AT

0

|tij |φ

AP

0 φ

AT

0

; ki, (21)

where |φ

AP

0

i and |φ

AT

0

i are the ground states of the projectile and target, respectively.

Summing over individual NN interactions, the matrix element becomes

U(k

0

, k) = APAT hk

0

; φ

AP

0 φ

AT

0

|t|φ

AP

0 φ

AT

0

; ki. (22)

Single scattering between a projectile and target nucleus is shown in Fig. 1. k and k

0

7
are, respectively, the initial and final momenta of the projectile nucleus in the AA CM

frame. Likewise, −k and −k

0 are the initial and final momenta of the target nucleus in

the AA CM frame. The internal virtual nucleon momenta over which the integration is

performed are denoted p1 and p2. PP and PT are the momenta of the core of the AP − 1

virtual projectile and AT − 1 virtual target nuclei, respectively. Inserting completeness

over the initial and final internal nucleon momenta and using conservation of momenta

in equation (22) gives

U(k

0

, k) = APAT

Z

hk

0

; φ

AT

0 φ

AP

0

|p

0

2p

0

1

ihp

0

2p

0

1

|t|p2p1ihp2p1|k; φ

AT

0 φ

AP

0

i (23)

× dp

0

1dp

0

2dp1dp2δ(p

0

1 − p1 − q)δ(p

0

2 − p2 + q),

where q = k

0 − k is the momentum transfer.

After evaluating the delta functions, equation (23) becomes,

U(k

0

, k) = APAT

Z

hφ

AT

0

|p2 − qihp2|φ

AT

0

ihφ

AP

0

|p1 + qihp1|φ

AP

0

i (24)

× hk

0

; p2 − q, p1 + q|t|k; p2, p1idp1dp2.

The notation above includes initial and final relative momenta of the projectile/target

system in the CM frame, k and k

0

, and the internal nucleon momenta relative to the CM

of the (projectile or target) nucleus, denoted pi, with i = 1, 2. These momenta (pi) are

associated with the relative kinetic energy of the nucleons within the projectile (or target).

The semicolon is used to show that the relative internal momenta are in a different space

than the relative momenta. In the discussion that follows, k and k

0 do not change and

are included for the convenience of later manipulation.

2.3.1 Optimum Factorization

Next, the optimum factorization for NA reactions in the CM frame [12, 30–32] is extended

to AA reactions in the CM frame and is shown in Fig. 2, where the symmetric coordinates

have been chosen for time reversal invariance [12, 30] and to allow for easier calculations.

The initial momentum of the projectile nucleon is p1 − q/2, and the final momentum is

p1 + q/2. Similarly, the initial momentum of the struck target nucleon is p2 + q/2, and

the final momentum is p2 − q/2. It is important to note that the momenta were chosen

such that q = k

0 − k and is the reason why the signs are reversed for the momentum

transfer of the target nucleon.

The wave functions will eventually be related to the nuclear charge densities. Thus,

the arguments of the wave functions will be written as arguments of the nucleon momenta

relative to the cores of the AP − 1 and AT − 1 nuclei. Let P

P

be the momentum of the

projectile nucleon relative to the core of the AP − 1 nucleus; likewise, P

T

is used for the

target nucleons. The initial momentum is denoted i and the final momentum is f. In

8
addition, if K ≡ (k + k

0

)/2 and momentum conservation from Fig. 2 are used, then the

core momentum of the Ap −1 virtual projectile nucleus can be expressed as PP = K−p1.

It also follows that PT = −K − p2.

The relative momentum between particles a and b with momenta pa and pb and masses

ma and mb is defined P ≡ (mapb − mbpa)/(ma + mb). The initial relative momentum of

the projectile nucleon can be found by inspection of Fig. 2. Let pa be the momentum

of the core of the AP − 1 nucleus (pa = PP ), and let pb be the initial momentum of the

projectile nucleon (pb = p1 −q/2) with ma = AP −1 and mb = 1. This leads to an initial

relative momentum of the projectile nucleon relative to core of the AP − 1 nucleus of

P

P

i =

 

AP − 1

AP

!

h

p1 −

q

2

i

−

PP

AP

= p1 −

 

AP − 1

AP

!

q

2

−

K

AP

. (25)

Similarly, the other relative momenta may be found,

P

P

f =

 

AP − 1

AP

!

h

p1 +

q

2

i

−

PP

AP

= p1 +

 

AP − 1

AP

!

q

2

−

K

AP

, (26)

P

T

i =

 

AT − 1

AT

!

h

p2 +

q

2

i

−

PT

AT

= p2 +

 

AT − 1

AT

!

q

2

+

K

AT

, (27)

and

P

T

f =

 

AT − 1

AT

!

h

p2 −

q

2

i

−

PT

AT

= p2 −

 

AT − 1

AT

!

q

2

+

K

AT

. (28)

9
The optical potential becomes

U(k

0

, k) = APAT

Z

ρ ̃T (P

T

f

, P

T

i

) ̃ρP (P

P

f

; P

P

i

) (29)

× hk

0

; p2 −

q

2

, p1 +

q

2

|t|k; p2 +

q

2

, p1 −

q

2

idp1dp2,

where  ̃ρ is the one-nucleon (proton or neutron) density matrix of the nucleus and is related

to the nuclear charge density, ρ, by [12, 30]

ρ(q) ≡ ρ ̃

 

A − 1

A

q

!

=

Z

ρ ̃

 

y −



A − 1

A



q

2

, y +



A − 1

A



q

2

!

dy, (30)

where y is a dummy integration variable.

It is convenient to make a change of variables by defining the following: P1 ≡ p1 −

K/AP and P2 ≡ p2 + K/AT . Then the matrix element of the optical potential becomes

U(k

0

, k) = APAT

Z

ρ ̃T

 

P2 −



AT − 1

AT



q

2

, P2 +



AT − 1

AT



q

2

!

dP1dP2 (31)

× ρ ̃P

 

P1 +



AP − 1

AP



q

2

, P1 −



AP − 1

AP



q

2

!

× hk

0

; P2 −

q

2

−

K

AT

, P1 +

q

2

+

K

AP

|t|k; P2 +

q

2

−

K

AT

, P1 −

q

2

+

K

AP

i,

where the transition amplitude is evaluated in the AA CM frame. The transition amplitude in equation (31) can be written

hk

0

; P2 −

q

2

−

K

AT

, P1 +

q

2

+

K

AP

|t|k; P2 +

q

2

−

K

AT

, P1 −

q

2

+

K

AP

i (32)

= hk

0

; k

0

2

, k

0

1

|t|k, k2; k1i,

10
where the initial of final projectile and target nucleon momenta in the AA CM frame are

k1 = P1 −

q

2

+

K

AP

(33)

k2 = P2 +

q

2

−

K

AT

(34)

k

0

1 = P1 +

q

2

+

K

AP

(35)

k

0

2 = P2 −

q

2

−

K

AT

(36)

The current work uses parameterizations of NN amplitudes to predict NA and AA

differential cross sections. Consequently, the quantity that allows for the transformation

of the transition amplitude from the AA CM frame the NN CM frame, known as the

M ̈oller factor (η) [7, 33], must be found.

The Moller transformation factor can be found by examining the Lorentz invariant

(LI) cross section, which is written [7, 12]

dσ =

Z

dk

0

1

E0

1

dk

0

2

E0

2

δ(E − E

0

)δ(P − P

0

)

(2π)

4

B

|M12|

2

, (37)

with

M12 =

p

E0

1E02 T12p

E1E2, (38)

where E and P are the initial total energy and momentum of the system, E

0 and P0 are

the final total energy and momentum of the system, E1 and E2 are the total energies of

the two particles before the collision, E

0

1

and E

0

2

are the total energies of the two particles

after the collision, k1 and k2 are the initial momenta of the two particles, k

0

1

and k

0

2

are

the final momenta of the two particles, and B =

1

2

p

λ(s, m2

1

, m2

2

), where

λ(s, m2

1

, m2

2

) = (s − m2

1 − m22

)

2 − 4m2

1m22

. (39)

Since, dki/Ei, δ(E − E

0

)δ(P − P0), B, and dσ are LI, then M12 is LI. Thus,

[M12]AA = [M12]NN . (40)

The initial and final relative momenta of projectile and target nucleons in the NN CM

frame can be expressed as

k ̃ ≡ −

1

2

" P2 +

q

2

−

K

AT

!

−

 

P1 −

q

2

+

K

AP

!# (41)

11
and

k ̃0 ≡ −

1

2

" P2 −

q

2

−

K

AT

!

−

 

P1 +

q

2

+

K

AP

!# . (42)

Using equation (38), the AA CM frame is related to the NN CM frame by

p

EN(k

0

2

)EN(k

0

1

)hk

0

; k

0

2

, k

0

1

|t|k; k2, k1iAA

p

EN(k2)EN(k1) (43)

=

q

EN(k ̃0)EN(−k ̃0)hk

0

; k ̃0, −k ̃0|t|k; k ̃, −k ̃iNNqEN(k ̃)EN(−k ̃) .

Therefore, < t >AA= η < t >NN, where

η =

s

EN(k ̃0)EN(−k ̃0)EN(k ̃)EN(−k ̃)

EN(k2

0

)EN(k1

0

)EN(k2)EN(k1)

, (44)

The notation can be simplified by defining

Y ≡

1

2

(k ̃ + k ̃0) = 1

2



K

AT

− P2 +

K

AP

+ P1



(45)

such that

hk

0

; k

0

2

, k

0

1

|t|k; k2, k1iAA = ηtNN(q; −2Y, 2Y). (46)

and

U(k

0

, k) = APAT

Z

ρ ̃T

 

P2 −



AT − 1

AT



q

2

, P2 +



AT − 1

AT



q

2

!

(47)

×ρ ̃P

 

P1 +



AP − 1

AP



q

2

, P1 −



AP − 1

AP



q

2

!

ηtNN(q; −2Y, 2Y)dP1dP2.

The NN transition amplitude may be written as a function of q and Y(P1, P2) without

loss of generality.

Since the size of the nucleus is much larger than the range of the NN interaction, the

transition amplitude varies more slowly with P1 and P2 when compared to the one nucleon

density matrix for the nucleus [12, 30]. For this reason, the NN transition amplitude is

evaluated at fixed values of P1 and P2, namely P1 = P10 and P2 = P20. The optimum

values of P10 and P20 are estimated by expanding the transition amplitude in a Taylor

series and requiring second-order terms (and higher) to be zero. After the optimum values

are found, the optical potential for elastic scattering is stated for central potentials.

12
The transition amplitude may be expanded as

tNN(q; 2Y) ≈ tNN(q;

K

AT

− P20 +

K

AP

+ P10) (48)

+

∂

∂P20

tNN(q;

K

AT

− P20 +

K

AP

+ P10)(P2 − P20)

+

∂

∂P10

tNN(q;

K

AT

− P20 +

K

AP

+ P10)(P1 − P10) + θ(P1, P2),

where θ represents higher order terms. Using equation (30) and the Taylor series expansion

leads to the following matrix element of the optical potential:

U(k

0

, k) = APAT ηtNN(q;

K

AT

− P20 +

K

AP

+ P10)ρT (q)ρP (q) (49)

+



APAT

∂

∂P20

tNN(q;

K

AT

− P20 +

K

AP

+ P10)



×

Z

(P2 − P20) ̃ρT (P2, q) ̃ρP (P1, q)dP1dP2



+



APAT

∂

∂P10

tNN(q;

K

AT

− P20 +

K

AP

+ P10)



×

Z

(P1 − P10) ̃ρT (P2, q) ̃ρP (P1, q)dP1dP2



+ . . . .

The optimum values for P20 and P10 can be estimated by requiring the second order

terms in the Taylor series expansion to be zero [12]; that is,

Z

(P2 − P20) ̃ρT (P2, q) ̃ρP (P1, q)dP1dP2 = 0 (50)

and

Z

(P1 − P10) ̃ρT (P2, q) ̃ρP (P1, q)dP1dP2 = 0. (51)

Equation (50) can be written as

P20ρT (q)ρP (q) = ZρP (q) ̃ρT (P2, q)P2dP2. (52)

The single particle density matrix for the nucleus is derived from the ground state wave

function, which is a symmetric function of its coordinates and invariant under the parity

transformation (P2 → −P2) [12], since the strong interaction preserves parity. Likewise,

the entire integral is invariant under a parity transformation and evaluates to zero; thus,

P20 = 0. The same argument follows for the second term of the matrix element of the

13
optical potential and P10 = 0. Using these values for P10 and P20 leads to

U(k

0

, k) = APAT ηtNN "

q;

 

AP + AT

APAT

!

K

#

ρT (q)ρP (q). (53)

Only central potentials have been considered; that is, the effects of spin are ignored.

As a result, the interaction must be a function of the possible scalar products in equation

(53): q · q, q · K, and K · K. Using |k| = |k

0

| for elastic scattering, one can show that

U(k

0

, k) = APAT ηtNN(eNN, q)ρT (q)ρP (q), (54)

where eNN is the NN CM kinetic energy. The current work uses an NN transition amplitude that is evaluated at the energy available per nucleon that is related to the fixed

beam energy [11].

In the next section, the eikonal approximation is discussed, and exact formulas for the

optical potential are given in position-space with the transition amplitude and harmonic

well densities described in Appendices A and B.

2.4 Eikonal Solution Method

The eikonal approximation is used for high energy, small angle scattering to calculate

elastic, reaction, total, and elastic differential cross sections [7, 34]. To compute cross

sections with the eikonal method, one solves for the eikonal scattering amplitude, f(θ),

which is given as [7]

f(θ) = k

i

Z∞

0

J0(2k sin(θ/2)) e

iχ(k,b) − 1



b db, (55)

where k is the relative momentum of the projectile-target system in the CM frame, J0 is

the ordinary cylindrical Bessel function, θ is the scattering angle in the CM frame, b is

the impact parameter, and χ(k, b) is the eikonal phase shift function, which is obtained

by integrating over the optical potential, U(b, z) [7]:

χ(k, b) = −

1

2k

Z∞

−∞

U(b, z)dz. (56)

The z-integration is taken to be in the same direction as the initial wave vector of the

incident projectile. The elastic differential cross section is found by taking the absolute

square of the scattering amplitude, dσ/dΩ = |f(θ)|

2

.

Since the eikonal approximation satisfies the optical theorem, the total cross section

14
is proportional to the imaginary part of the forward scattering amplitude,

σtot =

4π

k

Imf(θ = 0) = 4π

Z∞

0

[1 − e

−Imχ

cos(Re χ)]b db. (57)

The total elastic cross section, σel, is obtained by integrating the elastic differential cross

section [7],

σel =

Z

dσ

dΩ

dΩ = 4π

Z∞

0

[1 − e

−Imχ

cos(Reχ)]b db

− 2π

Z∞

0

[1 − e

−2Imχ

]b db,

and the reaction cross section, σre, is found by using σre = σtot − σel; thus,

σre = 2π

Z∞

0

[1 − e

−2Imχ

] b db. (58)

All of the cross sections described above are functions of the eikonal phase function

and depend on the optical potential, as shown in equation (56). For AA scattering, the

optical potential may be expressed as [5, 16]

U(r) = APAT

Z

tNN(|rNN|)ρP (|rP |)ρT (|rT |) drT drNN, (59)

where A is the number of nucleons, P represents the projectile, T represents the target,

tNN is the NN transition amplitude, and ρ is the nuclear matter density. rNN is the vector

between a nucleon in the projectile and a nucleon in the target, rP is the vector that

extends from the center of the projectile nucleus to a nucleon in the projectile, rT is

the vector between the center of the target nucleus to a nucleon in the target, r is the

relative distance between the centers of the projectile and target nuclei, R = r + rT is

the distance from the center of the projectile to a nucleon in the target. The distance

from the center of the projectile nucleus to a nucleon in the projectile may be expressed

as rP = r + R = r + rT + rNN, which, when substituted into equation (59), leads to

U(r) = APAT

Z

tNN(|rNN|)ρP (|r + rT + rNN|)ρT (|rT |)drT drNN. (60)

(Note that |r| =

√

b

2 + z2

in the cylindrical coordinate system.) The vectors used in this

15
discussion of the AA optical potential are illustrated in Fig. 3.

In the next two sections, analytic expressions for the optical potential are presented

for nuclei with A ≤ 16, where harmonic-well nuclear matter densities have been used.

Harmonic well densities [5] are given by

ρ

HW(r) = (α + βr2

) exp 

−r

2

4s

2



, (61)

and the NN transition amplitude is [5]

tNN(r) = τ exp 

−r

2

2B(e)



. (62)

(See Appendices A and B for a full description of the densities and transition amplitudes

used in the current work.)

2.4.1 Nucleon-Nucleus Optical Potential

For NA collisions, the single projectile nucleon charge density is taken as a Dirac delta

function, and the harmonic-well nuclear matter density is used for the target. Thus, the

exact optical potential from equation (60) may be expressed in the following form with

AP = 1:

U(r) = (C0 + C1r

2

) exp[−C2r

2

] (63)

with

C0 = τAT

 

π

μ1 + μ2

!3/2



αT +

3βT

2(μ1 + μ2)



, (64)

C1 =

τ βTAT μ

2

1π

3/2

(μ1 + μ2)

7/2

, (65)

and

C2 = μ1 −

μ

2

1

(μ1 + μ2)

, (66)

where

μ1 =

1

2B

and μ2 =

1

4s

2

T

. (67)

2.4.2 Nucleus-Nucleus Optical Potential

The calculation of the optical potential is repeated for AA collisions with harmonicwell nuclear charge densities for both the projectile and the target, which results in the

following formula for the optical potential:

16
U

(r) = (A

0

+A

1

r

2

+A

2

r

4

) exp[−A

3

r

2

], (68)

where

A

0

=

2πNA

P

A

T

θ

r

πθ

(69)

×



α

T

Λ

1

2

+

34θ

(β

T

Λ

1 + Λ2

α

T

)

+

15Λ

2

β

T

8θ

2



,

A

1

=

2πNA

P

A

T

θ

r

πθ

(70)

×



α

T

Λ

2

2

+

14θ

(3Λ

2

β

T

−4α

T

Λ

2

δ)

+

δ

2θ

2

(β

T

Λ

1

δ + Λ

2

α

T

δ−5β

T

Λ

2

)

+

5δ

2

Λ

2

β

T

2θ

3



,

A

2

=

2πNA

P

A

T

θ

r

πθ

(71)

×



Λ

2

β

T

δ

2

2θ

2

−

β

T

Λ

2

δ

3

θ

3

+

Λ

2

β

T

δ

4

2θ

4



,

and

A

3

=δ−

δ

2θ

, (72)

where

N=

2πτκ

r

πκ

, (73)

κ=

1

4s

2P

+

1

2B

, (74)

θ=

1

4s

2T

+ δ, (75)

17
δ =

1

4s

2

P

−

1

16s

4

P κ

, (76)

Λ1 =

αP

2

+

3βP

4κ

, (77)

and

Λ2 =

βP

2

+

βP

32κ

2s

4

P

−

βP

4κs2

P

. (78)

These results have been derived independently by Townsend and Bidasaria [35, 36]. In

the next section, the optical potential is written as a function of momentum transfer,

which leads to more efficient evaluation of cross sections when exact expressions for the

optical potential are not known.

2.4.3 Optical Potential in Momentum-Space

In section 2.2, it was shown that the phase function is obtained by integrating the optical

potential in the position-space representation. In this section, the phase shift function is

written such that the optical potential is expressed as a function of momentum transfer

q. By doing so, the number of integration dimensions will be significantly reduced.

To begin, the NN transition amplitude and nuclear charge densities in equation (60)

are replaced with their Fourier transforms

U(r) = APAT

(2π)

6

Z

drT drNN dq1 dq2 dq3 [tNN(|q1|)ρT (|q2|)ρP (|q3|) (79)

× e

−iq1·rNNe−iq2·rT e−iq3·(r+rT +rNN)

],

where the Fourier transforms are given by

tNN(r) = 1

(2π)

3

Z

tNN(q)e

−iq·r

dq, (80)

and

ρ(r) = Zρ(q)e

−iq·r

dq. (81)

The normalizations for the Fourier transforms of ρ(r) and tNN(r) differ because of the

traditional normalization of nuclear charge densities. Next, integration over rT and rNN

is performed with the Dirac delta function,

δ(A) = 1

(2π)

3

Z

e

−iA·B dB, (82)

18
which results in

U(r) = APAT

Z

tNN(|q1|)ρT (|q2|)ρP (|q3|)δ(q1 − q3)δ(q1 + q2)e

−iq3·r

dq1 dq2 dq3. (83)

After evaluating the delta functions, the optical potential is reduced to integration over

the momentum transfer,

U(r) = ZU(q)e

−iq·r

dq, (84)

where U(q) = APAT tNN(|q|)ρP (|q|)ρT (|q|).

Next, the Fourier transform of the optical potential (84) is substituted into the expression for the eikonal phase shift function from equation (56),

χ(k, b) = −

1

2k

Z∞

−∞

dz ZU(q)e

−iq·(z+b)

dq (85)

= −

1

2k

Z∞

−∞

Z

dzdq U(q)e

−iq·b

e

−iqz cos θ

,

where r = b+z in cylindrical coordinates has been used. The integration of z is performed,

which results in

χ(k, b) = −

π

k

Z

1

q

δ(cos θ)U(q)e

−iq·b

dq, (86)

where the following delta function has been used,

2π

q

δ(cos θ) = Z∞

−∞

e

−iqz cos θ

dz. (87)

Using dq = q

2dq sin θdθdφ and evaluating the delta functions leads to the final form for

the eikonal phase function,

χ(k, b) = −

π

k

Z∞

0

dq Z

2π

0

q U(|q|)e

−iqb cos φ

dφ. (88)

The advantage of equation (88) is that the optical potential is in the momentum-space

representation, and the z-integration need not be performed. Instead, the 7-dimensional

integral for χ has been reduced to 2-dimensions over the magnitude of the momentum

transfer, q, and the angle, φ, between the momentum transfer and the impact parameter.

This result significantly increases the efficiency for the numerical evaluation of χ.

19
2.5 Lippmann-Schwinger Partial Wave Solution Method

The LS equation is often solved with the method of partial waves, a well-known method

that is described in standard texts [7, 37]. An overview of the method is presented here.

Equation (20) is solved with the partial wave method using the optical potential from

equation (54). Any function, Z, can be expanded in a complete orthonormal set of basis

functions,

Z(q) = Z(k

0

, k) = X∞

l=0

X

l

m=−l

Zl(k

0

, k)Y

m∗

l

(

ˆk

0

)Y

m

l

(

ˆk) (89)

=

X∞

l=0

2l + 1

4π

Zl(k

0

, k)Pl(x), (90)

where q = k

0 − k is the momentum transfer, Ym∗

l

(

ˆk

0

) and Y

m

l

(

ˆk) are spherical harmonics,

ˆk and ˆk

0

represent the polar and azimuthal angles associated with k and k

0

, respectively;

Pl(x) are the Legendre polynomials, and x = cos(θkk0), where θkk0 is the angle between

k and k

0

. After expanding T and U, one integrates out the angular dependence, which

leads to the PW LS equation

Tl(k

0

, k) = Ul(k

0

, k) + Z∞

0

Ul(k

0

, k00)Tl(k

00, k)

E(k) − E(k

00) + i

k

002

dk00, (91)

where

Ul(k

0

, k) = 2π

Z

1

−1

U(q)Pl(x)dx. (92)

A common way of solving the above equation is through the R-matrix, which consists

of a principal value integral. When expanded in partial waves, Rl has a form similar to

Tl above,

Rl(k

0

, k) = Ul(k

0

, k) + P

Z∞

0

Ul(k

0

, k00)Rl(k

00, k)

E(k) − E(k

00)

k

002

dk00, (93)

where P represents the principal value integral. The Heitler equation may be used to

obtain Tl(k

0

, k),

Tl(k

0

, k) = Rl(k

0

, k)

1 + iπρRl(k

0

, k)

, (94)

20
where ρ is given in equation (6). The final step is to re-sum the infinite series,

T(q) = hk

0

|T|ki =

X∞

l=0

2l + 1

4π

Tl(k

0

, k)Pl(x). (95)

It was recently demonstrated that elastic scattering finite sum expressions converge

more quickly than the infinite series above [21]. The derivation is based on the idea that

the Born approximation becomes sufficient for some large angular momentum; that is, for

l > lmax, Tl(k, k) ≈ Ul(k, k). Using this idea, the finite sum expression for the scattering

amplitude is [10, 12, 21, 38]

f(θ) = −(2π)

2

ρ

k

X

lmax

l=0

2l + 1

4π

[Tl(k, k) − Ul(k, k)]Pl(x) − (2π)

2

ρ

k

U(q). (96)

The total cross section is proportional to the imaginary part of the forward scattering

amplitude [21],

σtot = −16π

3

ρ

k

2

X

lmax

l=0

2l + 1

4π

Im[Tl(k, k) − Ul(k, k)] + σ

Born

tot , (97)

where σ

Born

tot is the total cross section in the Born approximation. The elastic cross section

is derived from taking the absolute square of the elastic differential cross section [21],

σel =

(2π)

4ρ2

k

2

"X

lmax

l=0

2l + 1

4π



|Tl|

2 − |Ul

|

2

)

#

+ σ

Born

el , (98)

where

σ

Born

el =

Z

| − (2π)

2

ρ

k

U(q)|

2

dΩ, (99)

and the reaction cross section is obtained by subtracting the elastic cross section from the

total cross section.

2.6 Lippmann-Schwinger 3D Solution Method

The LS3D solution method avoids the numerical difficulties associated with the PW

method and has been used for relatively low energy reactions [22–26]. This section reviews the LS3D method as it has been described in the work of Elster et al. [22] and

Rodriquez-Gallardo et al. [24].

If one considers only central potentials in equation (3), then both T and V are scalar

functions; that is, f(k

0

, k) = f(k

0

, k, ˆk

0

·

ˆk) for some function f, where ˆk (ˆk

0

) represents

21
the angular unit vectors associated with k (k

0

). The possible scalar products of the LS

equation are as follows [22, 24]:

x

0 ≡ ˆk0

·

ˆk (100)

x

00 ≡ ˆk00

·

ˆk

y ≡ ˆk

00

·

ˆk

0

.

The incoming momentum, k, is taken to be in the direction of the z-axis, and the azimuthal

angle between k and k

0

is set to zero: φ

0 = 0; therefore, y may be expressed as a function

of x

0

, x

00, and φ00 [22, 24],

y = x

0x00 +

p

1 − x

02

p

1 − x

002

cos φ

00

. (101)

Note that only U(k

0

, k00, y) is dependent on φ

00; therefore [22],

U ̄(k

0

, x0, k00, x00) ≡

Z

2π

0

U(k

0

, k00, y)dφ00, (102)

and the LS equation becomes [22]

T(k

0

, k, x0) = U(k

0

, k, x0) (103)

+

Z∞

0

k

002

dk00 Z

1

−1

dx00U ̄(k

0

, x0, k00, x00)T(k

00, k, x00)

E(k) − E(k

00) + i

.

The propagator may be expressed as [37]

lim

η→0

1

E(k) − E(k

00) + i

= P

 

1

E(k) − E(k

00)

!

− iπδ(E(k) − E(k

00)), (104)

where P is the principal value. The LS equation becomes (after suppressing the on-shell

momentum in T and V )

T(k

0

, x0) = U(k

0

, x0) + P

Z

U ̄(k

0

, x0, k00, x00)T(k

00, x00)k002

dk00dx00

E(k) − E(k

00)

(105)

− iπ Z

U ̄(k

0

, x0, k00, x00)T(k

00, x00)k002

dk00dx00δ(E(k) − E(k

00)).

Next, the delta function is evaluated, and the kinematic factors are specified. The NR

energy is E = k

2/2μ, where μ is the reduced mass of the projectile-target system, and

22
k is the relative momentum of the projectile-target system in the AA CM frame. The

REL energy is E = EP + ET, where EP =

p

m2

P + k

2 and ET =

p

m2

T + k

2

. P denotes

projectile, and T denotes target. The delta function may be expressed as

δ(E(k) − E(k

00)) = δ(k − k

00)

|

dE

dk |(k

00=k)

=

μ

kin

k

δ(k − k

00), (106)

where negative momenta have been disregarded, kin represents non-relativistic (NR) or

relativistic (REL) kinematics, and

μ

kin =







mPmT

mP+mT

for non-relativistic kinematics

EPET

EP+ET

for relativistic kinematics.

(107)

After evaluating the delta function, equation (105) becomes

T(k

0

, x0) = U(k

0

, x0) + P

Z

U ̄(k

0

, x0, k00, x00)T(k

00, x00)

E(k) − E(k

00)

k

002

dk00dx00 (108)

− iπμkin ZU ̄(k

0

, x0, k, x00)T(k, x00)kdx00.

Equation (108) is solved with the numerical methods described in the next section.

3 Numerical Methods

Thus far, three different solution methods for the LS equation have been presented, including the eikonal appproxiation, the PW, and LS3D solution methods. This section

describes the numerical techniques used to implement the three methods.

3.1 Eikonal

Exact formulas for the optical potentials in equations (63) and (68) are used for light nuclei reactions (A ≤ 16). For reactions with heavier nuclei (A > 16), the momentum-space

representation of the eikonal phase function described in equation (88) is implemented. Although the momentum space method for the eikonal phase function is much more efficient

than the position space calculation, additional interpolation over the impact parameter

and momentum transfer was performed for additional numerical efficiency. Convergence

of the total elastic cross sections was used to establish the number of Gaussian quadrature

points used for integration. The number of Gaussian quadrature points for the eikonal

solution method was increased to a maximum of 100 points such that the total elastic

23
cross sections changed less than 1% for all reactions with energies up to 100 GeV/n.

3.2 Partial Wave Lippmann-Schwinger

As discussed in the last section, one may first solve for the lth PW of the R-matrix,

equation (93), and then find Tlfrom the Heitler equation. The transition matrix is found

by performing the sum in equation (95). To solve for Rl, an even number of Gaussian

quadrature points, N, is used to perform the integration of the momentum; thus, equation

(93) becomes

Rl(k

0

, k) = Ul(k

0

, k) + X

N

k

00=1

Ul(k

0

, k00)Rl(k

00, k)k002

wk

00

E(k) − E(k

00)

, (109)

where k

00 now represents a discrete Gaussian quadrature transformed index, and wk

00 is

the corresponding weight for the integrated variable. Note that a principal value singularity will occur in the propagator when k

00 = k and may be removed by implementing

a subtraction scheme [37] or the method of Sloan [39]. Sloan’s method requires that the

pole be placed at the center of an even number of Gaussian quadrature points when integrated. Since the pole occurs at k

00 = k, the limits of integration are divided into two

parts: [0, ∞) → [0, 2k] and [2k, ∞). By placing the pole in the center of the interval

[0,2k], and by using an even number of Gaussian quadrature points, the principal value

singularity is removed.

To isolate Rl, the above equation is written

X

N

k

00=1

Λ(k

0

, k00)Rl(k

00, k) = Ul(k0

, k), (110)

where

Λ(k

0

, k00) = δk

0

,k00 −

Ul(k

0

, k00)k

002

wk

00

E(k) − E(k

00)

. (111)

If the k

0

index is allowed to span the same discrete indices as k

00, then the above matrix

equation can be solved, and Rl(k

00, x00) can be found. Observe, however, that k00 are

transformed integration points that are not necessarily equal to the on-shell momentum,

k. Since the main focus of the current work is elastic scattering, Rl(k, k) is ultimately

sought. To accomplish this, an additional point is added to the sum for the on-shell

momentum, which is not an integration point: k

00 = k with weight wk

00 = 0 for the N + 1

point. Consequently, the matrix equation becomes

N

X

+1

k

00=1

Λ(k

0

, k00)Rl(k

00, k) = Ul(k0

, k), (112)

24
and the solution of Rl(k, k) can be found.

The number of partial waves needed for an acceptable tolerance of convergence is not

known a priori. Partial waves were generated until |Tl − Ul| ≤ 10−4%.

3.3 3D Lippmann-Schwinger

The numerical implementation of the LS3D method proceeds in the same manner as the

PW method, but there are now two additional integration variables over azimuthal and

polar angles. The azimuthal dependence only occurs in the potential and is integrated

with 80 Gaussian quadrature points. As was seen with the PW method, the principal

value integral over momenta is handled with Sloan’s method [39], and an additional onshell momentum is introduced such that k

00 = k with wk

00 = 0 for the N + 1 Gaussian

point; thus, equation (108) becomes,

T(k

0

, x0) = U(k

0

, x0) +

N

X

+1

k

00=1

X

N

x00=1

U ̄(k

0

, x0, k00, x00)T(k

00, x00)k002

wk

00wx00

E(k) − E(k

00)

(113)

− iπμkin

N

X

+1

k

00=1

X

N

x00=1

U ̄(k

0

, x0, k00, x00)T(k

00, x00)k00wx00δk,k00.

Equation (113) can be written as a matrix equation,

N

X

+1

k

00=1

X

N

x00=1

Λ(k

0

, x0, k00, x00)T(k

00, x00) = U(k0

, x0), (114)

with

Λ(k

0

, x0, k00, x00) = "δk

00,k0δx00,x0 −

U ̄(k

0

, x0, k00, x00)k

002

wk

00wx00

E(k) − E(k

00)

(115)

+ iπμkinU ̄(k

0

, x0, k00, x00)k

00wx00δk,k00#

.

In order to solve the above matrix equation, k

0

is allowed to span the same discrete values

of k

00; likewise, x0 uses the same discrete points as x00

.

The system of equations can be solved to obtain T(k

00, x00), where the solution is given

for the on-shell and off-shell momenta at the Gaussian transformed angles, which are

not evenly spaced. The transition matrix can be evaluated at any angle by substituting

T(k

00, x00) into equation (113).

In order to solve equation (113), the summing indices are reduced to single indices,

25
(k

0

, x0) → i and (k

00, x00) → j, by using the following relations

i = x

0 + N(k0 − 1) (116)

and

j = x

00 + N(k00 − 1), (117)

where x

0

, x

00

, k

0

, k

00 represent discrete Gaussian transformed integration points (not continuous variables). N integration points have been used for the angular integration and

N + 1 points have been used for the integration over momenta. Equation (114) reduces

to the following complex system of linear equations:

N(

X

N+1)

j=1

Λ(i, j)T(j) = U(i). (118)

Since the index i is allowed to vary over the same range as j, N(N + 1) systems of equations must be solved (in contrast to N + 1 systems of equations for the PW method).

Convergence of the total elastic cross sections was used to establish the number of Gaussian quadrature points used for integration. The number of Gaussian quadrature points

for the LS3D solution method was increased to a maximum of N = 44 points such that

the total elastic cross sections changed less than 1% for all reactions with energies up to

100 GeV/n.

3.4 Localized Contributions

It has been observed that the transition amplitude for reactions with large on-shell

momenta—including high energy NA reactions and AA reactions at every energy—do

not converge efficiently if the integration ranges of both momenta and polar angles are

not restricted to regions that give significant contributions to the LS equation. The

momenta which give non-zero contributions are estimated from the range of the optical

potential and tend to be localized near the on-shell momentum, k.

The range of the potential is determined from the NN transition amplitude, nuclear

matter densities, and kinematic factors. Consider a high energy collision of a projectile

nucleon with a nucleus, such as 4He, which is modeled with a harmonic-well nuclear matter

density (Gaussian). The transition amplitude and nuclear density are Gaussian functions

of momentum transfer, |q| = k

2 +k02−2kk0

cos(θ). The greatest contributions occur when

k

0 ≈ k at forward scattering angles. Therefore, the range of the potential is localized near

the on-shell momentum and drops off quickly thereafter. At lower nucleon projectile

energies, the on-shell momentum is smaller, and the range of the optical potential drops

off more slowly. If one considers the collision of two light nuclei, then the range of

the optical potential is further reduced because of the additional harmonic well density,

26
which is equivalent to multiplying with another q dependent Gaussian. Hence, even at

low energies, the momenta that give non-zero optical potentials for AA reactions tend to

be localized near the on-shell momentum.

The range of interaction may be estimated with a high degree of certainty using a

forward scattering angle approach. The maximum range of the potential, Rmax, may be

estimated at the forward scattering angle by moving outward from the on-shell momentum

until the potential becomes effectively zero. As a measure of localization, a localization

factor, μloc, is defined,

μloc ≡

|k − Rmax|

k

, (119)

where k is the on-shell momentum. Small values are indicative of potentials that decay

rapidly from the on-shell momentum. In the examples that follow, forward scattering has

been used for the potentials, and the kinetic energy of the projectile in the laboratory

frame has been used.

Fig. 4 shows the real part of the optical potential for a p + 4He reaction at 100 MeV.

At this energy, the NR and REL potentials are nearly equal, although there is a slight

REL shift of the maximum toward a higher on-shell momentum. The localization factor

for this reaction is μloc ≈ 90% and is an example of a potential that is not localized near

the on-shell momentum.

In contrast to the low energy NA reaction, a p + 4He reaction at 100 GeV is shown

in Fig. 5. There is a significant shift in the REL on-shell momentum as compared to the

NR k. It is also obvious that the maximum magnitude of the potential at k is reduced

in the REL case. The momenta are highly localized for this reaction with a localization

factor of μloc ≤ 5%.

Figs. 6 and 7 show the real part of the optical potential for 4He + 20Ne reactions at

100 MeV/n and 500 MeV/n. At 100 MeV/n, the localization factor is μloc ≈ 22%, which

is more localized than the NA reaction at the same energy. When the energy is increased

to 500 MeV/n, the localization factor is reduced to μloc ≈ 10%. This trend shows that the

momenta become more localized near the on-shell momenta as the energy is increased.

In addition, localization of momenta near k occurs at lower energies per nucleon for AA

reactions than for NA reactions.

Figs. 8-11 illustrate the localization of momenta for equal mass systems. The real part

of the optical potential is shown for low and high energy 4He + 4He reactions in Figs. 8

and 9. Similar results are shown for 12C + 12C reactions in Figs. 10 and 11. As expected,

there is a slight difference between the NR and REL potentials at low energy. Differences

become more obvious as the maximum of the potential decreases with increasing energy.

There is no relativistic shift of the on-shell momentum due to kinematics for equal mass

systems. In each case, the potential dies off rapidly from the on-shell momentum, and

localization of momentum near k is clearly observed.

A small percentage (10−8 %) of the maximum value of the elastic scattering optical

27
potential is used to estimate the range of the potential, Rmax, and the range of momenta

(δR) from the on-shell value that give non-zero contributions to the LS equation, which the

authors define as δR ≡ Rmax −k. If k −δR > 0, then the limits of integration are changed:

k

00 → [k − δR, k + δR]. Note that the pole occurs at k00 = k, but if the integration is

handled with Sloan’s method [39], then the principal value singularity is removed because

the pole is placed at the center of an even number of Gaussian quadrature points. When

k − δR ≤ 0, the integration proceeds in the usual way: [0, 2k] and [2k, ∞).

High energy reactions tend be forward scattering, so it is important to carefully specify

the angular integration range such that convergence can be achieved efficiently. The

elastic scattering optical potential was used to estimate the maximum range of the polar

integration angle, θmax, and was used for all reactions presented in this document.

3.5 Equal Mass Kinematics

There are several kinematic factors that may change according to the kinematics selection:

the on-shell momentum, k; the elementary NN on-shell momentum, kelm; the elementary

(NN) reduced mass, μelm; the factor that transfers the transition amplitude to the scattering amplitude, k(dk/dE); and the propagator. The relativistic (REL) on-shell momentum

reduces to the non-relativistic (NR) on-shell momentum for equal mass systems; that is,

k

REL = kNR =

p

(mTLab)/2, where m is the mass of the projectile or target, and TLab is

kinetic energy of the projectile in the lab frame. Likewise, it follows that k

NR

elm = k

REL

elm .

To examine the effect of kinematics for equal mass systems, the LS equation is written

as an infinite series,

T = V + V G+

0 T = V + V G+0 V + V G+0 V G+0 V + . . . . (120)

Using completeness of over momenta, the matrix element of the transition amplitude may

be written,

hk

0

|T|ki =hk

0

|V |ki +

Z

hk

0

|V |k1ihk1|G

+

0

|k2ihk2|V |kidk1dk2 (121)

+

Z

hk

0

|V |k1ihk1|G

+

0

|k2ihk2|V |k3ihk3|G

+

0

|k4ihk4|V |kidk1dk2dk3dk4 + . . . .

After operating with the propagator, hki|G

+

0

|kj i = G

+

0

(k, ki)δ(ki − kj ), the transition

amplitude becomes,

hk

0

|T|ki =hk

0

|V |ki +

Z

hk

0

|V |k1iG

+

0

(k, k1)hk1|V |kidk1 (122)

+

Z

hk

0

|V |k1iG

+

0

(k, k1)hk1|V |k2iG

+

0

(k, k2)hk2|V |kidk1dk2 + . . . ,

28
where the momenta have been relabeled for clarity.

The only kinematic factors that appear in the potential are kelm and μelm, but, as

previously stated, the elementary on-shell momenta are equivalent for NR and REL kinematics. Next, the scattering amplitude is obtained from the transition matrix, but only

those factors that may lead to kinematic differences between the REL and NR scattering

amplitudes are stated explicitly:

f(θ) ∝ k

dk

dE "X∞

n=1

1

μ

n

elm

nY−1

i=1

1

[E(k) − E(ki)]#

, (123)

where the momenta integration variables are ki and the integration symbols have been

suppressed. The NR scattering amplitude is expressed as

f

NR(θ) ∝

X∞

n=1

2

n−1An

nY−1

i=1

1

[k

2 − k

2

i

]

, (124)

and the REL scattering amplitude is

f

REL(θ) ∝

X∞

n=1

A

n

nY−1

i=1

[1 + h(ki, k)]

[k

2 − k

2

i

]

, (125)

where

h(ki, k) = s

k

2

i + m2

k

2 + m2

, (126)

and A is the number of nucleons in the projectile or target. It can now be seen that, for

equal mass systems, the REL scattering amplitude reduces to the NR amplitude when

h(ki, k) → 1. (127)

For cases in which ki ≈ k, this condition is satisfied.

To illustrate the effect of the kinematics, the first couple of terms of the scattering

amplitude are examined,

f

NR(θ) ∝ A +

2A2

k

2 − k

2

i

+ . . . . (128)

and

f

REL(θ) ∝ A +

A2[1 + h(ki, k)]

k

2 − k

2

i

+ . . . . (129)

29
The relevant terms for comparison are

f1 ≡

2

k

2 − k002

(130)

and

f2 ≡

[1 + h(k

00, k)]

k

2 − k002

, (131)

where k

00 represents the integration variable.

The kinematic functions, f1 and f2, for a 4He + 4He reaction at TLab = 3.3 GeV/n

(k ≈ 5 GeV) are shown in Figs. 12-14. In Fig. 12, it appears that f1 and f2 are identical

for all integration values, k

00. However, upon closer inspection in Fig. 13, differences

are more clearly seen when k

00 is farthest from the on-shell momentum. In Fig. 14, the

black line shows the ratio f2/f1, and the intersecting red, dashed lines show that the two

functions are identical at the on-shell momentum. Fig. 12 clearly shows that the greatest

contribution occurs at the pole. Similar results are shown in Figs. 15-17 for a 56Fe + 56Fe

reaction at TLab = 100 GeV/n.

In this section, it has been demonstrated that the relevant kinematic factors lead to

NR and REL scattering amplitudes which are equal at the on-shell momentum for equal

mass systems. Although differences between the scattering amplitudes can be observed

when integration momenta are farthest away from the on-shell momentum, the greatest

contributions occur near the pole. As a result, very little difference between scattering

amplitudes will be observed for equal mass systems.

4 Results

In the results that follow, each model uses the same set of fundamental parameterizations

for nuclear matter densities and the NN transition amplitude. Harmonic well and twoparameter Fermi (Wood-Saxon) nuclear charge data are taken from references [13, 14]

and are normalized to matter densities as described in reference [5]. When data are not

available for the two-parameter Fermi densities, a nuclear droplet model [40] is used for

parameter estimates. Nuclei are assumed to be near the beta stability curve. The NN

transition amplitude used in the current work is described in Appendix A and depends

on parameterizations of the NN cross sections, slope parameter, and real to imaginary

ratio of the transition amplitude. The NN cross sections are taken from reference [41].

Parmeterizations of the slope parameters and real to imaginary ratios of the data in

reference [42] are given in Appendix B.

In Figs. 18-24, NA and AA elastic differential cross sections are shown at energies that

are relevant to space radiation applications, including p + 16O, p + 56Fe, 4He + 4He, 4He

+ 16O, 12C + 56Fe, 20Ne + 20Ne, and 56Fe+56Fe reactions at lab projectile kinetic energies

of 150, 500, 1000, and 20,000 MeV/n. The projectile energy of 150 MeV/n represents the

30
lower bound on model applicability. Results are indicated non-relativistic by (NR) and

relativistic by (REL). LS3D (REL) results are given as a solid red line; a dashed, black

line is used for eikonal results, denoted (Eik); a green square represents the PW (NR)

results; a solid blue circle indicates LS3D (NR) results; and a violet asterisk is for PW

(REL) results. Note that the Coulomb interaction has not been included in this analysis.

Excellent agreement between PW and LS3D results are seen in Figs. 18-21 for each

kinematic selection for energies greater than 150 MeV. The p + 16O and p + 56Fe reactions at 150 MeV/n in Figs. 18 and 19 show slight disagreements between NR PW

and LS3D codes and eikonal results. This is likely the result of the forward scattering

approximation used in the eikonal method, since very light projectiles may deviate from

forward scattering at low energy. The slight disagreement between the eikonal and NR

PW and LS3D codes is not observed for the heavier nuclei in Figs. 20 and 21, where the

small angle scattering approximation is more appropriate.

The next obvious feature is that of the relativistic shift observed in Figs. 18-21. The

magnitude of the differential cross section is larger at smaller angles as compared to the

NR cases. The effect is more pronounced at higher energies, as expected, but is also driven

by projectile and target mass differences. A comparison of Figs. 18 and 20 shows that

the relativistic effect is more pronounced for the p + 16O reaction, which has larger mass

difference than the 4He + 16O system. Ultimately, the relativistic effects can be tracked

back to kinematic differences in the relative on-shell momentum.

Elastic scattering for equal mass systems is depicted in Figs. 22-24. Fig. 22 shows the

elastic differential cross section for 4He + 4He. At the lowest energies, the eikonal model

appears to accurately describe the scattering at small angles, but begins to deviate from

the other models at higher angles. The most obvious difference between Fig. 22 and Figs.

18-21 is that there is no shift in the differential cross section at higher energies in Fig. 22.

Moreover, there is no relativistic shift observed in Figs. 23 and 24 for the 20Ne + 20Ne

and 56Fe + 56Fe reactions. It was demonstrated in section 3.5 that the REL scattering

amplitude is equal to the NR amplitude at the on-shell momentum for equal mass systems

and that the largest contributions near the pole. Therefore, there is no differential cross

section shift associated with the REL kinematics for equal mass systems.

All of the reactions presented in Figs. 18-24 have been extended to 100 GeV/n. The

results at this energy are consistent with the 20 GeV/n reactions and have not been

shown here, although, as expected, there is a larger shift in the differential cross section

for unequal mass systems.

As an example of the LS3D method and illustration of the relativistic shift, comparisons to experimental data [43–46] are performed. Fig. 25 shows the elastic differential

cross sections of the following reactions: (a) p + 32S at TLab = 1 GeV [43] (b) p + 40Ca at

TLab = 500 MeV [44] (c) p + 58Ni at TLab = 1 GeV [45] and (d) 4He + 40Ca at TLab = 347

MeV/n [46]. NR results are indicated as a solid red line, and REL results are indicated

with a solid blue line. In each case, there is better agreement with experiment when

relativistic kinematics are used. Since the fundamental parameterizations are based on

31
small-angle scattering data, the results are in better agreement with the measured differential cross section data at forward scattering angles. Also note that spin-dependence

and medium effects have not been included, which may account for differences between

the model and the experimental data in Fig. 25 (d).

5 Conclusions

The eikonal, PW, and LS3D methods have been compared for NA and AA reactions for

reactions relevant for space radiation applications. Numerical convergence of the eikonal

method is readily achieved when the formulas presented herein are used for light nuclei

(A ≤ 16) and the momentum-space representation of the optical potential is used for

heavier nuclei. The LS formalism has an advantage over the eikonal method in that relativistic kinematics are easily included. The PW solution method is numerically unstable

for reactions that have large on-shell momenta, including both high energy reactions and

relatively low energy AA reactions, due to the highly oscillatory Legendre polynomials

needed for convergence of these systems. To circumvent this difficulty, the 3D solution

method was implemented. Convergence of the LS3D equation can be achieved quickly

after identifying the integration range for momenta and polar angles that give non-zero

contributions to the LS equation. This numerical method is also useful for obtaining convergence for the partial wave analysis; however, numerical instability still exists because

of the Legendre polynomial oscillations.

It was shown that the NR PW and NR LS3D methods agree with the eikonal method,

except at very low energies for projectile nucleons, where the eikonal method is not wellsuited. As the lab energy is increased, relativistic effects are seen as a shift in differential

cross section resonances toward higher magnitudes and lower angles. There was no observed relativistic shift of the differential cross section for equal mass systems. It was

shown in section 3.5 that the NR and REL on-shell momentum are equal for equal mass

systems and that the largest contributions to the LS equation occur near the pole, k.

Based on the results presented herein, it is recommended that the LS3D method be

used for high energy NA and AA reactions at all energies because of its rapid numerical

convergence and stability. Extensive comparisons between models and experimental data

are underway. The results of that research and a verification and validation analysis of

the models will be elucidated in a future manuscript.

6 Appendix A

NN transition amplitude and nuclear matter densities

In this appendix, the position space representation and Fourier transforms of the

NN transition amplitude and nuclear matter densities are given. Usually, harmonic-well

32
nuclear matter densities are used for A ≤ 16, and Wood-Saxon matter densities are used

for A > 16 [5, 16]. The harmonic-well matter density in position space is [5]

ρ

HW(r) = (α + βr2

) exp 

−r

2

4s

2



, (132)

where

α =

ρ

HW

0 a

3

8s

3



1 +

3γ

2

−

3γa2

8s

2



(133)

and

β =

ρ

HW

0 a

3

8s

3

γa2

16s

4

, (134)

The normalization factor is

ρ

HW

0 =

1

π

3/2a3

[1 + 3

2

γ]

, (135)

where γ and a are parameters given in references [13, 14], s

2 = a2/4 − r2

p

/6, and rp is the

proton radius [5, 16]. The Fourier transform of the harmonic-well nuclear matter density

is given by [5]

ρ

HW(q) = ρHW

0 π

3/2

a

3

[(1 + 3

2

γ) −

a

2γ

4

q

2

]e

−q

2s2

. (136)

The Wood-Saxon nuclear matter density is given as [5]

ρ

WS(r) = ρWS

0

1 + e

r−R

cA

, (137)

where the normalization is

ρ

WS

0 =

3

4π



1

R3 + π

2c

2

AR



. (138)

R is the half density radius, and cA is related to the surface diffuseness, c, by [5]

cA =

2rp

√

3

ln 3β − 1

3 − β

−1

(139)

and

β = exp 

rp

c

√

3



, (140)

where rp is the proton radius. The parameters, R and c, are given in references [13, 14].

The Fourier transform of the Wood-Saxon matter density is [12]

ρ

WS(q) = 4π

q

ρ

WS

0 φ(q), (141)

33
where

φ(q) = πcAR



− cos(qR)

sinh(qcAπ)

+

πcA

R

sin(qR) coth(qcAπ)

sinh(qcAπ)

(142)

−

2cA

πR

X∞

n=1

(−1)n nqcA

[(qcA)

2 + n2

]

2



.

The NN transition amplitude is [5]

tNN(r) = τ exp 

−r

2

2B(eNN)



(143)

with

τ = −

r

eNN

m

σ(eNN)

[2πB(eNN)]3/2

[κ(eNN) + i] , (144)

where eNN is the kinetic energy of the NN system in the CM frame, mp is the proton

mass, B(eNN) is the slope parameter, σ(eNN) is the NN cross section, and κ(eNN) is the

real to imaginary ratio of the NN cross section.

The Fourier transform of the NN transition amplitude is given as [5]

tNN(q) = −1

(2π)

2

1

μ

kσ(eNN)

4π

[κ(eNN) + i]e

−B(eNN)q

2/2

, (145)

where μ is the reduced mass of the NN system, and k is the relative momentum in the

NN CM frame.

7 Appendix B

Real to imaginary ratios and slope parameters for the proton-proton (pp) and protonneutron (pn) cross sections are given in this section and are parameterized to the data

listed in reference [42]. The parameterizations are a function of proton kinetic energy in

the lab frame, , with energy in units of GeV.

The pp real to imaginary ratio is

κpp =

(

c0 + c1, for 0 ≤  ≤ 0.2 GeV

a+c+e2+g3

1+b+d2+f 3

, for  < 0.2 ≤ 100 GeV,

(146)

where c0 = 1.0736, c1 = −0.8370 GeV−1, and a = 1.3204, b = 1.0713 GeV−1, c = −0.9240

GeV−1, d = 1.0478 GeV−2, e = −0.3279 GeV−2, f = 0.01601 GeV−3, and g = 0.0025

GeV−3.

34
The np real to imaginary ratio is given by

κnp =

(

c0 + c1, for 0 ≤  ≤ 0.15 GeV

a+c

1+b+d2

, for  < 0.15 ≤ 100 GeV,

(147)

where c0 = 0.7763, c1 = 2.2886 GeV−1, and a = 3.2411 × 108, b = 1.1305 × 109 GeV−1,

c = −7.8409 × 108 GeV−1, d = 8.0467 × 107 GeV−2.

The slope parameter for the pp reactions is

Bpp =

a + c + e2

1 + b + d2 + f 3

(148)

where a = 0.4665 fm2, b = 0.4319 GeV−1, c = 0.2897 fm2 GeV−1, d = 5.5596 ×10−3 GeV−2,

e = 4.5448 ×10−3fm2 GeV−2, and f = 3.2662 ×10−6 GeV−3. For the np reactions, the

slope parameter is

Bnp =

a + c + e2 + g3

1 + b + d2 + f 3

, (149)

where a = 0.4622 fm2, b = -0.1776 GeV−1, c = -0.0139 ×10−3fm2 GeV−1, d = 0.0207

GeV−2, e = -6.5369 ×10−3fm2 GeV−2, f = 6.9016 ×10−3 GeV−3, g = 4.9074 ×10−3fm2

GeV−3.

8 Appendix C

Kinematics

The relative on-shell momentum in the CM frame is needed as input for the LS equation. This appendix describes the NR and REL kinematics for reactions of the form,

A + B → C + D, (150)

where the relative momentum is written as a function of the kinetic energy of the projectile

in the laboratory frame.

8.1 Non-relativistic Kinematics

With NR kinematics, the relative momentum may be expressed as

Pi =

mBpA − mApB

mA + mB

, (151)

35
where mA is the mass of the projectile, pA is the momentum of the projectile, mB is the

mass of the target, pB is the momentum of the target. Pi = pACM = −pBCM, where piCM

is the momentum of particle i in the CM frame. The momentum of particle B in the lab

frame is pBLab = 0, and the relative momentum can be written as

Pi =

mB

mA + mB

pALab, (152)

where pALab is the momentum of particle A in the lab frame. The kinetic energy available

in the CM frame is

TCM = P

2

i

/2μ = TLab 

mB

mA + mB

!

, (153)

where TLab = p

2

ALab

/2mA and μ = mAmB/(mA + mB). If the kinetic energy of the

projectile in the lab frame is given, then the relative on-shell momentum may be obtained

from

k ≡ |Pi| =

p

2μTCM =

r

2μmBTLab

mA + mB

. (154)

8.2 Relativistic Kinematics

In this section, the relativistic momentum is expressed as a function of TLab. The fourmomentum is denoted with a superscript, μ, to avoid confusion with previous discussions

in this document. In the lab frame, the four-momenta of particles A and B are

p

μ

ALab

= (EALab, pALab) and p

μ

BLab

= (mB, 0), (155)

and in the CM frame,

p

μ

ACM

= (EACM, pACM) and p

μ

BCM

= (EBCM, pBCM), (156)

where EiLab is the relativistic energy for particle i in the lab frame, EiCM is the relativistic

energy for particle i in the CM frame, piLab is the three-momentum for particle i in the

lab frame, and piCM is the three-momentum for particle i in the CM frame.

The Mandelstam variable, s, is an invariant defined by the four-vector’s length, s =

(p

μ

A + p

μ

B

)

2

. In the CM frame, s = (EA + EB, 0)

2

, and the total energy is √

s. In the lab

frame,

s = (p

μ

ALab

+ p

μ

BLab

)

2 = m2

A + m2B + 2mBEALab. (157)

The kinetic energy is found by subtracting the rest mass from the total energy, TALab =

EALab − mA, so

s = (mA + mB)

2 + 2mBTALab. (158)

36
The Mandelstam variable can be used to express the total energy as a function of the

relative three-momentum by using the CM frame,

s = (EA + EB)

2 =

 q

P

2

i + m2

A +

q

P

2

i + m2

B

!2

, (159)

where Pi = pACM = −pBCM. After performing the square and regrouping terms,

|Pi| =

1

2

√

s

q

[s − (m2

A + m2B

)]2 − 4m2

Am2B

. (160)

Substituting equation (157) into equation (160) leads to the final result for the relative

momentum,

k ≡ |Pi| =

mB

√

s

q

TALab [TALab + 2mA]. (161)

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39
k

φ

AP

0

(AP )

PP

(AP − 1)

k

0

(AP )

φ

AP

0

−k

0

(AT )

φ

AT

0

(AT − 1)

PT

φ

AT

0

−k

(AT )

p1

p

0

1

p p2

0

2

t

Figure 1: Transition amplitude for AA scattering in the AA CM frame. k (k

0

) is the

initial(final) momentum of the projectile nucleus in the AA CM frame. −k (−k

0

) is the

initial(final) momentum of the target nucleus in the AA CM frame. The internal nucleon

momenta are denoted p1 and p2. PP and PT are the momenta of the core of the AP − 1

virtual projectile and AT − 1 virtual target nuclei, respectively. See text for explanation.

k

φ

AP

0

(AP )

PP

(AP − 1)

k

0

(AP )

φ

AP

0

−k

0

(AT )

φ

AT

0

(AT − 1)

PT

φ

AT

0

−k

(AT )

p1 + q/2 p1 − q/2

p p2 + q/2 2 − q/2

t

Figure 2: Optimum factorization for AA scattering in the AA CM frame. k (k

0

) is the

initial(final) momentum of the projectile nucleus in the AA CM frame. −k (−k

0

) is the

initial(final) momentum of the target nucleus in the AA CM frame. The internal nucleon

momenta are denoted p1 and p2. PP and PT are the momenta of the core of the AP − 1

virtual projectile and AT − 1 virtual target nuclei, respectively. q is the momentum

transfer for the AA system. See text for explanation.

40
rP

r

R

rT

rNN

Figure 3: Illustration of the vectors used for the nucleus-nucleus optical potential. The

distance from the center of the projectile nucleus to a nucleon in the projectile nucleus is

rP . Likewise, rT is the distance from the center of the target nucleus to a nucleon in the

target. The center to center distance between nuclei is r, and rNN is the distance between

a nucleon in the projectile to a nucleon in the target. R = r + rT is the distance from the

center of the projectile to a nucleon in the target.

41
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

k" (GeV)

0.1

0.2

0.3

0.4

0.5

U(k",k) (GeV)

NR

REL

TLab = 100 MeV/n (p + 4He)

Figure 4: Real part of optical potential for a p + 4He reaction with TLab = 100 MeV (NR

k ≈ 346 MeV and REL k ≈ 349 MeV) at θCM = 0.

9 10 11 12 13 14 15 16

k" (GeV)

0

0.1

0.2

0.3

0.4

0.5

0.6

U(k",k) (GeV)

NR

REL

TLab = 100 GeV/n (p + 4He)

Figure 5: Real part of optical potential for a TLab = 100 GeV (NR k ≈ 10.96 GeV and

REL k ≈ 13.63 GeV) p + 4He reaction at θCM = 0.

42
0.75 1 1.25 1.5 1.75 2 2.25 2.5

k" (GeV)

0

2

4

6

8

10

U(k",k) (GeV)

NR

REL

TLab = 100 MeV/n (4He + 20Ne)

Figure 6: Real part of optical potential for a TLab = 100 MeV/n (NR k ≈ 1.44 GeV and

REL k ≈ 1.46 GeV) 4He + 20Ne reaction at θCM = 0.

2 2.5 3 3.5 4 4.5 5

k" (GeV)

0

0.5

1

1.5

2

2.5

3

U(k",k) (GeV)

NR

REL

TLab = 500 MeV/n (4He + 20Ne)

Figure 7: Real part of optical potential for a TLab = 500 MeV/n (NR k ≈ 3.23 GeV and

REL k ≈ 3.39 GeV) 4He + 20Ne reaction at θCM = 0.

43
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

k" (GeV)

0

0.5

1

1.5

2

2.5

3

U(k",k) (GeV)

NR

REL

TLab = 100 MeV/n (4He + 4He)

Figure 8: Real part of optical potential for a TLab = 100 MeV/n (k ≈ 0.87 GeV) 4He +

4He reaction at θCM = 0.

4 4.5 5 5.5 6

k" (GeV)

0

1

2

3

4

U(k",k) (GeV)

NR

REL

TLab = 3.3 GeV/n (4He + 4He)

Figure 9: Real part of optical potential for a 4He + 4He reaction with TLab = 3.3 GeV/n

(k ≈ 5 GeV) at θCM = 0.

44
2 2.5 3 3.5 4 4.5

k" (GeV)

0

3

6

9

12

15

18

U(k",k) (GeV)

NR

REL

TLab = 150 MeV/n (12C + 12C)

Figure 10: Real part of optical potential for a 12C + 12C reaction at 150 MeV/n (k ≈ 3.18

GeV) with θCM = 0.

25 25.5 26 26.5 27

k" (GeV)

0

5

10

15

20

25

30

35

40

45

50

U(k",k) (GeV)

NR

REL

TLab = 10 GeV/n (12C + 12C)

Figure 11: Real part of optical potential for a 12C + 12C reaction at 10 GeV/n (k ≈ 26

GeV) with θCM = 0.

45
0 1 2 3 4 5 6 7 8 9 10

k" (GeV)

-4

-3

-2

-1

0

1

2

3

4

Function (GeV-1

)

f

1

f

2

Kinematic Functions (4He + 4He)

TLab = 3.3 GeV/n

Figure 12: Kinematic functions, f1 and f2, for a 4He + 4He reaction at TLab = 3.3 GeV/n

(k ≈ 5 GeV). The integration variable is denoted k

00

.

0 1 2 3 4 5 6 7 8 9 10

k"(GeV)

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Function (GeV-1

)

f

1

f

2

Kinematic Functions (4He + 4He)

TLab = 3.3 GeV/n

Figure 13: Alternate view of kinematic functions, f1 and f2, for a 4He + 4He reaction

with on-shell momentum k ≈ 5 GeV. The integration variable is denoted k

00

.

46
0 1 2 3 4 5 6 7 8 9 10

k" (GeV)

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

f

2/f

1

Ratio of Kinematic Functions (4He + 4He)

TLab = 3.3 GeV/n

Figure 14: Ratio of kinematic functions, f2/f1. The two functions are identical at the

on-shell momentum, shown with intersecting red, dashed lines. The integration variable

is denoted k

00

.

0 200 400 600 800

k" (GeV)

-6×10-4

-4×10-4

-2×10-4

0

2×10-4

4×10-4

6×10-4

Function (GeV-1

)

f

1

f

2

Kinematic Functions (56Fe + 56Fe)

TLab = 100 GeV/n

Figure 15: Kinematic functions, f1 and f2, for a 56Fe + 56Fe reaction at TLab = 100

GeV/n (k ≈ 384 GeV). The integration variable is denoted k

00

.

47
0 200 400 600 800

k" (GeV)

-5e-05

-4e-05

-3e-05

-2e-05

-1e-05

0

1e-05

2e-05

3e-05

4e-05

5e-05

Function (GeV-1

)

f

1

f

2

Kinematic Functions (56Fe + 56Fe)

TLab = 100 GeV/n

Figure 16: Alternate view of kinematic functions, f1 and f2, for a 56Fe + 56Fe reaction

with on-shell momentum k ≈ 384 GeV. The integration variable is denoted k

00

.

0 200 400 600 800

k" (GeV)

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

f

2/f

1

Ratio of Kinematic Functions (56Fe + 56Fe)

TLab = 100 GeV/n

Figure 17: Ratio of kinematic functions, f2/f1. The two functions are identical at the

on-shell momentum, shown with intersecting red, dashed lines. The integration variable

is denoted k

00

.

48
0 10 20 30 40 50 60 70 80 90

θ

CM (degrees)

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 150 MeV/n (p + 16O)

(a)

0 10 20 30 40 50 60

θ

CM (degrees)

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 500 MeV/n (p + 16O)

(b)

0 5 10 15 20 25 30 35 40

θ

CM (degrees)

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 1 GeV/n (p + 16O)

(c)

0 0.5 1 1.5 2 2.5 3 3.5 4

θ

CM (degrees)

10-2

10-1

100

101

102

103

104

105

106

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 20 GeV/n (p + 16O)

(d)

Figure 18: Elastic differential cross sections for p + 16O reactions for projectile lab kinetic energies of 150, 500, 1000, 20000 MeV. Eik. represents eikonal, LS3D represents

three-dimensional Lippmann-Schwinger, and PW represents partial wave. Non-relativistic

results are denoted (NR) and relativistic results are denoted (REL).

49
0 10 20 30 40 50 60 70 80 90

θ

CM (degrees)

10-3

10-2

10-1

100

101

102

103

104

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 150 MeV/n (p + 56Fe)

(a)

0 5 10 15 20 25 30 35 40

θ

CM (degrees)

10-4

10-3

10-2

10-1

100

101

102

103

104

105

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 500 MeV/n (p + 56Fe)

(b)

0 5 10 15 20 25 30 35

θ

CM (degrees)

10-4

10-3

10-2

10-1

100

101

102

103

104

105

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 1 GeV/n (p + 56Fe)

(c)

0 0.5 1 1.5 2 2.5 3 3.5 4

θ

CM (degrees)

10-4

10-3

10-2

10-1

100

101

102

103

104

105

106

107

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 20 GeV/n (p + 56Fe)

(d)

Figure 19: Elastic differential cross sections for p + 56Fe reactions for projectile lab

kinetic energies of 150, 500, 1000, 20000 MeV. Eik. represents eikonal, LS3D represents

three-dimensional Lippmann-Schwinger, and PW represents partial wave. Non-relativistic

results are denoted (NR) and relativistic results are denoted (REL).

50
0 5 10 15 20

θ

CM (degrees)

100

101

102

103

104

105

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 150 MeV (4He + 16O)

(a)

0 2 4 6 8 10 12 14

θ

CM (degrees)

10-4

10-3

10-2

10-1

100

101

102

103

104

105

106

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 500 MeV (4He + 16O)

(b)

0 1 2 3 4 5 6 7 8 9 10

θ

CM (degrees)

10-4

10-3

10-2

10-1

100

101

102

103

104

105

106

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 1 GeV (4He + 16O)

(c)

0 0.4 0.8 1.2 1.6 2 2.4

θ

CM (degrees)

10-2

10-1

100

101

102

103

104

105

106

107

108

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 20 GeV (4He + 16O)

(d)

Figure 20: Elastic differential cross sections for 4He + 16O reactions for projectile lab

kinetic energies of 150, 500, 1000, 20000 MeV/n. Eik. represents eikonal, LS3D represents

three-dimensional Lippmann-Schwinger, and PW represents partial wave. Non-relativistic

results are denoted (NR) and relativistic results are denoted (REL).

51
0 1 2 3 4 5 6 7 8 9 10 11 12

θ

CM (degrees)

100

101

102

103

104

105

106

107

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 150 MeV/n (12C + 56Fe)

(a)

0 1 2 3 4 5

θ

CM (degrees)

10-1

100

101

102

103

104

105

106

107

108

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 500 MeV/n (12C + 56Fe)

(b)

0 1 2 3 4

θ

CM (degrees)

10-1

100

101

102

103

104

105

106

107

108

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 1 GeV/n (12C + 56Fe)

(c)

0 0.1 0.2 0.3 0.4 0.5

θ

CM (degrees)

102

103

104

105

106

107

108

109

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 20 GeV/n (12C + 56Fe)

(d)

Figure 21: Elastic differential cross sections for 12C + 56Fe reactions for projectile lab

kinetic energies of 150, 500, 1000, 20000 MeV/n. Eik. represents eikonal, LS3D represents

three-dimensional Lippmann-Schwinger, and PW represents partial wave. Non-relativistic

results are denoted (NR) and relativistic results are denoted (REL).

52
0 10 20 30 40 50 60

θ

CM (degrees)

10-4

10-3

10-2

10-1

100

101

102

103

104

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 150 MeV/n (4He + 4He)

(a)

0 5 10 15 20 25 30 35

θ

CM (degrees)

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 500 MeV/n (4He + 4He)

(b)

0 2 4 6 8 10 12 14 16 18 20 22

θ

CM (degrees)

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

105

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 1 GeV/n (4He + 4He)

(c)

0 1 2 3 4 5

θ

CM (degrees)

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

105

106

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 20 GeV/n (4He + 4He)

(d)

Figure 22: Elastic differential cross sections for 4He + 4He reactions for projectile lab

kinetic energies of 150, 500, 1000, 20000 MeV/n. Eik. represents eikonal, LS3D represents

three-dimensional Lippmann-Schwinger, and PW represents partial wave. Non-relativistic

results are denoted (NR) and relativistic results are denoted (REL).

53
0 2 4 6 8 10 12 14

θ

CM (degrees)

10-2

10-1

100

101

102

103

104

105

106

107

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 150 MeV/n (20Ne + 20Ne)

(a)

0 1 2 3 4 5 6

θ

CM (degrees)

10-2

10-1

100

101

102

103

104

105

106

107

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 500 MeV/n (20Ne + 20Ne) 

(b)

0 1 2 3 4 5

θ

CM (degrees)

10-4

10-3

10-2

10-1

100

101

102

103

104

105

106

107

108

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

Tlab = 1 GeV/n (20Ne + 20Ne)

(c)

0 0.2 0.4 0.6 0.8 1

θ

CM (degrees)

10-1

100

101

102

103

104

105

106

107

108

109

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

Tlab = 20 GeV/n (20Ne + 20Ne)

(d)

Figure 23: Elastic differential cross sections for 20Ne + 20Ne reactions for projectile lab

kinetic energies of 150, 500, 1000, 20000 MeV/n. Eik. represents eikonal, LS3D represents

three-dimensional Lippmann-Schwinger, and PW represents partial wave. Non-relativistic

results are denoted (NR) and relativistic results are denoted (REL).

54
0 0.5 1 1.5 2 2.5 3

θ

CM (degrees)

103

104

105

106

107

108

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 150 MeV/n (56Fe + 56Fe)

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

θ

CM (degrees)

101

102

103

104

105

106

107

108

109

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 500 MeV/n (56Fe + 56Fe)

(b)

0 0.2 0.4 0.6 0.8 1 1.2

θ

CM (degrees)

101

102

103

104

105

106

107

108

109

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 1 GeV/n (56Fe + 56Fe) 

(c)

0 0.05 0.1 0.15 0.2 0.25

θ

CM (degrees)

103

104

105

106

107

108

109

1010

d

σ/d

Ω (mb/sr)

LS3D (REL)

Eik.

PW (NR)

LS3D (NR)

PW (REL)

TLab = 20 GeV/n (56Fe + 56Fe)

(d)

Figure 24: Elastic differential cross sections for 56Fe + 56Fe reactions for projectile lab

kinetic energies of 150, 500, 1000, 20000 MeV/n. Eik. represents eikonal, LS3D represents

three-dimensional Lippmann-Schwinger, and PW represents partial wave. Non-relativistic

results are denoted (NR) and relativistic results are denoted (REL).

55
0 2 4 6 8 10 12 14 16 18

θ

CM (degrees)

10-1

100

101

102

103

104

105

d

σ/d

Ω (mb/sr)

Reference [43]

LS3D (NR)

LS3D (REL)

TLab = 1 GeV (p + 32S)

(a)

0 5 10 15 20 25 30

θ

CM (degrees)

10-2

10-1

100

101

102

103

104

d

σ/d

Ω (mb/sr)

Reference [44]

LS3D (NR)

LS3D (REL)

TLab = 500 MeV (p + 40Ca)

(b)

0 2 4 6 8 10 12 14 16 18 20

θ

CM (degrees)

10-1

100

101

102

103

104

105

d

σ/d

Ω (mb/sr)

Reference [45]

LS3D (NR)

LS3D (REL)

TLab = 1047 MeV (p + 58Ni)

(c)

0 2 4 6 8 10 12

θ

CM (degrees)

10-1

100

101

102

103

104

105

106

d

σ/d

Ω (mb/sr)

Reference [46]

LS3D (NR)

LS3D (REL)

TLab = 342 MeV/n (4He + 40Ca)

(d)

Figure 25: Elastic differential cross sections for (a) p + 32S at TLab = 1 GeV [43] (b) p +

40Ca at TLab = 500 MeV [44] (c) p + 58Ni at TLab = 1 GeV [45] and (d) 4He + 40Ca at

TLab = 347 MeV/n [46].

56
REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

2. REPORT TYPE 

Technical Publication

 4. TITLE AND SUBTITLE

Elastic Differential Cross Sections 

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Werneth, Charles M.; Maung, Khin M.; Ford, William P.; Norbury, John 

W.; Vera, Michael D.

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

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National Aeronautics and Space Administration

Washington, DC 20546-0001

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 REPORT NUMBER

L-20452

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NASA

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Availability: NASA CASI (443) 757-5802

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14. ABSTRACT

The eikonal, partial wave (PW) Lippmann-Schwinger, and three-dimensional Lippmann-Schwinger (LS3D) methods are 

compared for nuclear reactions that are relevant for space radiation applications. Numerical convergence of the eikonal method 

is readily achieved when exact formulas of the optical potential are used for light nuclei (A <= 16) and the momentum-space 

optical potential is used for heavier nuclei. The PW solution method is known to be numerically unstable for systems

that require a large number of partial waves, and, as a result, the LS3D method is employed. The effect of relativistic 

kinematics is studied with the PW and LS3D methods and is compared to eikonal results. It is recommended that the LS3D 

method be used for high energy nucleon-nucleus reactions and nucleus-nucleus reactions at all energies because of its rapid 

numerical convergence and stability. 

15. SUBJECT TERMS

Elastic differential Cross section; Relativistic kinematics

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01- 09 - 2014