11 th International Conference on Nucleus-Nucleus Collisions May 29, 2012

Derivation of Correction Terms to the Eikonal Expansions in the Abrasion-Ablation Model for High Energy Heavy Ion

Collisions

11th International Conference on Nucleus-Nucleus CollisionsMay 29, 2012

S. Bhatt, L. W. Townsend, M. PourArsalanThe University of Tennessee, Knoxville, TN 37996-2300

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Overview:

Introduction Abrasion-Ablation Model Eikonal Corrections to the Phase Function Optical Potential Derivation Results Conclusions and Future Work

Introduction• Interaction of cosmic radiation with space craft and crew members during long duration

flights and safety concerns.

• Although nucleon component is the dominant mode of secondary particle production a significant amount of the secondary particle production also due to primary helium and high energy heavy ion (HZE) particles

• Models for characterizing transport of secondary particles produced from HZE reactions extremely important

– Accurate and precise transport models and relevant nuclear cross section databases– Better radiation dose estimates and efficient shielding design

• The Abrasion-Ablation Model has been widely used to describe the fundamental physics of secondary particle production in HZE interactions

The Abrasion-Ablation Model

•The abrasion-ablation model was initially formulated by Bowman, Swiatecki and Tsang and extended by Hufner, Schaffer, and Schurmann [3] and also several others researchers at NASA.

•The abrasion part of the model is based on Glauber multiple scattering theory which applies the expansion of scattering terms using eikonal differential approximations.

• Whereas the ablation model is based on classical evaporation model by Weisskopf-Ewing.

•The abrasion-ablation model has been further extended to the evaluation of momentum distributions for nucleon production in heavy ion collisions.

The Abrasion Process

• Based on Eikonal approximations in the formulation of scattering amplitude given by the Glauber Multiple scattering series

– considered in terms of the small angle approximations, scattering strictly in forward direction in the plane of incident momentum

– Assumes the total Eikonal phase is equal to sum of individual Eikonal phases from each scattering

• Glauber multiple scattering:– Fourier-Bessel representation of the scattering amplitude by replacing

Legendre polynomials with zero order Bessel functions

Motivation

• Although the Eikonal approximations based on Glauber model have been widely used

– Validity of the small angle approximations for 3 or more particles in the final state in unclear

– May not apply at lower energies where the medium effects come into play

• Further, Light ion production (z ≤ 2) is inherently a three dimensional problem and is not properly described by the small angle approximation

Motivation

• Various different approaches to the correction of the Eikonal approximations

• Theoretically the number of possible variants of Eikonal approximations is unlimited– The set of rays representing the scattering wave can be imagined to

propagate through unlimited ways

• Most approximations require a small angle approximation of some sense– No compelling argument that any method is valid at all large scattering

angles.

Current Work

• We have used correction to the Eikonal expansions developed by Wallace by including higher order terms to the phase function

• Wallace converts the partial wave series into a Fourier-Bessel expansion, based on the Legendre polynomials

– Creates an infinite series– Leading term is same as the Glauber Model– Expansion is exact in the sense that no small angle approximation is

made.

Current Work

• Formulation of four higher order correction terms to the phase function

• Comparison of contribution of each higher order correction term – Total abrasion cross section– Differential cross section– Stability of correction terms

• Comparison of the double differential cross sections with correction terms to the experimental data

Abrasion Formalism• The cross section for abrading n projectile nucleons is given by:

where is the binomial coefficient and

gives the nucleon non-removal probability. Thus is the probability of a nucleon being removed due to abrasion.

2 1Fn AP

n

AP b P b bdb

n

PAn

2Imexp

P

bP b

A

1 ( )P b

Abrasion Formalism

• The abrasion cross section can be thus expressed as

• Above equation treats all nucleons as identical objects. To differentiate between the protons and neutrons, above equation can be replaced by

2Im 2 Im2 1 exp exp

Fn A

Pn

P P

b bAbdb

n A A

2Im 2Im2 1 exp exp

Pn z A n z

P Pnz

P P

b bN Zbdb

n z A A

Abrasion Formalism

• The phase function in above equations is expressed in Glauber’s Model as

and the elastic scattering amplitude being expressed in terms of impact parameter representation as

with being the zero order Bessel function expressed in terms of momentum transfer q and impact parameter b

( )2

( ) m V r dzk

b

00

( ) ( ) 1 exp ( )f ik J qb i b b db

0J qb

Ablation Formalism• In the ablation process the prefragment nuclei are in the excited

state and give up their excess energy by evaporating nucleons, light ion clusters and gamma rays.

• In this work, the ablation process is described using works by Cucinotta et.al , and Kikuchi & Kawai.

• The nucleon emission spectrum from the ablation process is given by the Weiskopf-Ewing statistical decay model.

• Strength of the ablation state is calculated using the total abrasion cross section.

Ablation Formalism• In this model the probability function for emission of a particle is

given as :

where is the reduced nucleon mass, is the formation cross section, represents the level density of the residual nuclei , is the statistical weight and is the excitation energy

• In the above equation Pi (j,Ei) represents the probability that a

prefragment j emits a particle with energy Ei

*

*

0

0

( )

2 ( )( , )

( , )l j

i i CN j ii i E S

l

ig w E EP j E

P j E dE

E

i CN

0w ig*jE

Derivation of Correction Terms

• Wallace uses direct conversion of Fourier Bessel series using expansion of Legendre polynomials in terms of Bernoulli Polynomials

• Thus deriving an impact parameter representation of scattering amplitude.

• The expansion allowed for the phase shifts to be written in terms of Eikonal phase functions, creating an infinite series where leading term is same as Glauber’s series and additional terms are correction terms.

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• In the series mentioned above the individual terms can be given as:

and the total series can be given in terms of single expression as

1

1( ) 1( )( 1)!

nnn

n

bb V r dzk n k b k k

12

11 2 2( ) 1( ) ( )

( 1)!

nnn

nbb V b z dz

k n k b k k

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• One of the concerns in the above given Eikonal series is is flux conservation.

• Secondly, it is of importance to discuss the convergence of the series and also validity of results with regards to quantum mechanical results.

• Carstoiu et al conclude that the series shows good convergence properties and is in reasonable agreement with quantum mechanical results.

• Further, they also conclude that these expansions can be used for calculations of total cross section even at low energies


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Optical Potential Term

• The optical potential, V(r), can be expressed as.

where AP and AT are projectile and target mass numbers,

N = AP + AT

and are projectile and target single particle nuclear densities and

is the two particle transition amplitude

2 2

3 32( ) ,P TT P

mA AV r d z z d y x y z t e yN

P x y z T z

,t e y

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Transition Amplitude• The two particle transition amplitude can be expressed as: .

where = energy in two body CM frame , with = energy dependent nucleon-nucleon cross section = ratio of real to imaginary parts for forward scattering amp.

= slope parameter (range of interaction)

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2, ( ) 2 ( ) exp2 ( )

e yt e y e e i B em B e

e 212

vP T

NkvmA A

e

( )e

( )B e

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Results: Total Abrasion Cross section

Energy(MeV/Nucleon ) (mb) (mb) (mb) (mb) (mb)

100 784.32 1368.40 1640.73 1690.05 1688.81

200 774.21 1074.81 1119.02 1122.45 1122.45

300 773.91 1022.38 1050.22 1051.92 1051.92

500 780.40 1035.56 1060.73 1062.08 1062.82

1000 783.19 1004.09 1019.68 1020.28 1020.28

3000 781.38 864.69 866.73 866.76 866.76

5000 781.60 833.82 834.58 834.59 834.59

10000 781.97 805.01 805.15 805.15 805.15

1

0i

i

2

0i

i

3

0i

i

4

0i

i

1i

i

Table 2: Total Abrasion Cross Section with Correction Terms for 12C on 27Al at 10 Degrees

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Results: Total Abrasion Cross section

Energy(MeV/Nucleon ) (mb) (mb) (mb) (mb) (mb)

100 1476.63 2259.92 2577.92 2635.33 2633.68

200 1259.47 1604.47 1654.65 1658.55 1658.55

300 1251.13 1529.00 1559.52 1561.36 1561.36

500 1329.27 1604.52 1630.31 1631.64 1631.64

1000 1395.25 1623.86 1638.73 1639.27 1639.27

3000 1343.94 1421.68 1423.27 1423.29 1423.29

5000 1346.69 1390.02 1390.49 1390.50 1390.50

10000 1348.77 1364.36 1364.43 1364.43 1364.43

Table 2: Total Abrasion Cross Section with Correction Terms for 20Ne on 16O at 20 Degrees

1

0i

i

2

0i

i

3

0i

i

4

0i

i

1

ii

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Results: Double Differential Cross Section

50 100 150 200 250 3000

5

10

15

Energy (MeV)

Dou

ble

Diff

eren

tial (

mb/

MeV

/Sr/i

on)

Proj. 0 CProj. 2 CProj. 4 CTarg. 0 CTarg. 2 CTarg. 4 C

Figure 12: Comparison of differential cross section from abrasion process for 100 MeV/nucleon 12C on 20Ne at 200 scattering angle for 0, 2 and 4 correction terms.

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Results: Double Differential Cross Section

Figure 12: Comparison of differential cross section from ablation process for 100 MeV/nucleon 12C on 20Ne at 200 scattering angle for 0, 2 and 4 correction terms.

50 100 150 200 250 3000

1

2

3

4

5

6

Energy (MeV)

Dou

ble

Diff

eren

tial (

mb/

MeV

/ion/

Sr)

Proj. 0 CProj. 2 CProj. 4 CTarg. 0 CTarg. 2 CTarg 4 C

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Comparison to Experimental Data

• Comparison of double differential cross sections for the secondary neutron to the experimental data

– 290 MeV/nucleon 12C beam colliding on 12C target

– 400 MeV/nucleon 14N beam colliding on 12C target

– 400 MeV/nucleon 20Ne beam colliding on 64Cu target

• Experimental Data from works by Nakamura and Heilbronn [3] from experiments at Heavy Ion Medical Application Center (HIMAC), Japan

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290 MeV/nucleon 12C beam colliding on 12C Target

100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

Energy (MeV)

Dou

ble

Diff

eren

tial (

mb/

MeV

/Sr/i

on)

No Correction TermsTwo Correction TermsExp. Data with Error

100 200 300 400 500 600 700 800 90010-3

10-2

10-1

100

101

102

103

Energy (MeV)

Dou

ble

Diff

eren

tial (

mb/

MeV

/Sr/i

on)


100 200 300 400 500 600 700 80010-2

10-1

100

101

102

Energy (MeV)

Dou

ble

Diff

eren

tial (

mb/

MeV

/Sr/i

on)


50 100 150 200 250 300 350 400 450 50010-3

10-2

10-1

100

101

Energy (MeV)

Dou

ble

Diff

eren

tial (

mb/

MeV

/Sr/i

on)


5 Degrees 10 Degrees


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400 MeV/nucleon 14N beam colliding on 12C Target

100 200 300 400 500 600 700 800 900 100010-2

10-1

100

101

102

103

Energy (MeV)

Dou

ble

Diff

eren

tial (

mb/

MeV

/Sr/i

on)


100 200 300 400 500 600 700 800 900 100010-2

10-1

100

101

102

Energy (MeV)

Dou

ble

Diff

eren

tial (

mb/

MeV

/Sr/i

on)


100 200 300 400 500 600 700 800 900 100010-2

10-1

100

101

102

Energy (MeV)

Dou

ble

Diff

eren

tial (

mb/

MeV

/Sr/i

on)


100 200 300 400 500 600 700 80010-3

10-2

10-1

100

101

102

Energy (MeV)

Dou

ble

Diff

eren

tial (

mb/

MeV

/Sr/i

on)




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Surface Plot of 400 MeV/nucleon 20Ne beam on 64Cu Target

5

10

15

20

25

30

35

40

0200

400600

8001000

1200

10-4

10-2

100

102

(degree)

Energy (MeV)

Dou

ble

Diff

eren

tial (

mb/

MeV

/Sr/i

on)

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Surface Plot of 290 MeV/nucleon 12C beam on 12C Target

0

10

20

30

40

50

60

0 100 200 300 400 500 600 700 800 900

10-2

100

102

Energy (MeV)

(Degree)

Dou

ble

Diff

eren

tial (

mb/

MeV

/Sr/i

on)

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Conclusions• Development of higher order correction terms to relax the small angle

approximation in the abrasion-ablation formulation

• Comparison of contributions of higher order correction terms at different energies– Major contributions from first two correction terms, contributions of

third and fourth order terms comparatively very small.

• Comparison to experimental data for double differential cross sections– Over predicts at smaller scattering angles and under predicts at larger

angles– Over all shape matches fairly good – Significant increase with addition of the correction terms

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Future work

• Implementation of the higher order correction terms to the HZE transport codes and fragmentation models

• Incorporation of the isobar formulation and decay model in the current formalism to improve energy predictions below beam energy

• Further investigation of neutron momentum distribution models and approximation models for single particle nuclear densities to improve cross section predictions at larger angles.

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QUESTIONS ???

11 th International Conference on Nucleus-Nucleus Collisions May 29, 2012

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