angular distribution of neutrons from heavy ion induced reactions in thick targets

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doi:10.1016/j.ni

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Nuclear Instruments and Methods in Physics Research A 556 (2006) 577–588

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Angular distribution of neutrons from heavy ion inducedreactions in thick targets

Moumita Maitia, Maitreyee Nandyb, S.N. Roya, P.K. Sarkarc,�

aDepartment of Physics, Visva Bharati, Santiniketan 731235, IndiabSaha Institute of Nuclear Physics, 1/AF, Bidhannagar, Kolkata 700064, India

cH.P. Unit, V. E. C. Centre, 1/AF, Bidhannagar, Kolkata 700064, India

Received 23 September 2005; received in revised form 22 October 2005; accepted 27 October 2005

Available online 21 November 2005

Abstract

Angular distribution of neutron emission from different heavy ion projectiles of energy up to 10AMeV incident on various thick

targets have been analyzed with the help of existing models and empirical relations. Owing to the complexity of heavy ion induced

reactions, it becomes difficult to obtain any empirical relation for neutron angular distribution involving parameters like projectile energy

as well as target and projectile masses. An exponential function of the emission angle has been found to satisfy the relation approximately

involving these parameters. The calculated results using our proposed exponential relation are compared with experimental data showing

reasonably good agreement. This empirical expression provides an easy, quick and fairly reliable tool to calculate angular distributions of

emitted neutrons from heavy ion induced reactions in thick targets, which are required for various applications.

r 2005 Elsevier B.V. All rights reserved.

PACS: 25.70.�z; 24.10.�i

Keywords: Neutron yield; Proton induced reactions; Thick target; Empirical formula

1. Introduction

In the last few decades low and intermediate energyheavy ion beams have found widespread use in areas likeindustry and basic research for nuclear and atomic studies,surface analysis, material modification, ion implantation,etc. They have also found applications in medical sciences,e.g., in radiation therapy, production of medically im-portant isotopes for diagnosis and treatment, etc. Theenergy and directional distribution of neutrons emittedfrom these heavy ion reactions constitute important studiesin neutron physics, radiation dosimetry and shield design.

Neutron emissions from heavy ion induced reactionshave been measured by several workers. Hubberd et al. [1]have measured the total neutron yield from thick-targetsbombarded by heavy ions of energies around 10MeV pernucleon as a function of projectile energy and target mass.

e front matter r 2005 Elsevier B.V. All rights reserved.

ma.2005.10.111

ing author. Tel.: +9133 23371230; fax: +91 33 2334681.

ess: [email protected] (P.K. Sarkar).

Clapier and Zaidins [2] have measured the total neutronyield. Guo et al. in 1987 [3] have measured neutron yieldand have proposed an exponential function to estimate theangular distribution of neutron dose rate for 12 differentthick targets bombarded by 14 different heavy ionprojectiles accelerated to 11 different energies. Festag [4]and Ohnesorge et al. [5] measured the angular distributionof the equivalent dose rate. Li et al. [6] have measured theangular distribution of the neutron fluence rate at differentemission energy ranges. In some of these works severalparameters like energy, fluence rate, secondary yield andemission rate of neutrons in the forward direction have alsobeen measured. Nevertheless, available experimental datafor double differential neutron spectra from heavy ioninduced reactions from thick targets are not adequate tostudy the systematics of angular distribution of emittedneutrons. In many cases the measured angular distributionis only for the neutron dose.For nucleon and a-induced reactions, Kalbach [7] has

formulated an empirical expression to describe the angular

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ARTICLE IN PRESSM. Maiti et al. / Nuclear Instruments and Methods in Physics Research A 556 (2006) 577–588578

distribution of emitted neutrons by fitting a large numberof experimental data. For heavy ion induced reactions,insufficient experimental data and complexity of thereaction mechanism involved makes it difficult to discernany overall trend in the angular distribution of neutronemissions from various target-projectile combinations. It isobserved in low energy heavy ion reactions that nearly allnormalized angular distributions are peaked in the forwarddirection. In this energy region the anisotropy in angulardistribution is large for high energy bombardment of lighttargets by heavy particles and small for low energy lightions incident on heavier targets. Neutron emission fromheavy ion reactions comprises direct, preequilibrium (PEQ)and compound nuclear evaporation processes. The directpart involves projectile break up, transfer reactions andemission of particles from the projectile and/or the targetbefore any interaction between the projectile and the targetnucleons takes place. There are experimental evidencesindicating a significant probability for the break up of theheavy ion projectile with transfer of the part of theprojectile to the target nucleus. Subsequent particleemission from the resulting excited nucleus, which movesbackwards in the centre-of-mass (c.m.) system results in abackward peaking in this frame [8].

For lighter ions such as Li and Be, even at energies aslow as the Coulomb barrier, about 30–50% of theprojectiles suffer direct breakup. Neutron emissions fromthese reactions are strongly forward peaked with mostprobable energies of 3–4MeV. Emitted neutrons from lightion induced reactions are a mixture of low energy isotropicgroup and a high energy forward peaked group. Theheavier projectiles react mainly through compound nucleusformation and neutron emissions are isotropic in the c.m.system with low average energy.

Below the Coulomb barrier though compound nucleusformation is not expected, neutron production risingrapidly with energy has been observed. Such emissionspresumably arise from sub-barrier penetration of theprojectile ions and have strong peaks in the forward andbackward directions.

Some of the earlier workers have given empiricalformalism to estimate the angular distribution of neutrondose in heavy ion reactions. Li et al. [9] proposed anexponential function of the angle to calculate the angulardistribution of the energy integrated neutron fluence rate.Clapier and Zaidins [2] have suggested an empiricalformulation to describe the angular distribution ofneutrons. There are some difficulties associated with thisformula, which is discussed later in this paper. It has beenfound that the Kalbach systematics [7] which reproduce theneutron angular distribution quite well in the case of lightion induced reactions do not succeed in the case of heavyion reactions.

Several theoretical and phenomenological models likeFermi-jet [10], quantum molecular dynamics (QMD)[11,12], nucleon exchange transport, HION have been usedto estimate double differential neutron emission cross-

sections for heavy ion induced reactions. But all thesemodels are complex and none of them is able to properlypredict neutron emissions at all angles and over the widemass and energy range of interacting projectiles. In thepresent work our aim is to evaluate the existing empiricalrelations with the help of available experimental data andthen to find a simple expression for angular distribution ofemitted neutrons from heavy ion induced reactionsinvolving different targets and projectiles with energies upto about 10AMeV. In order to establish such anapproximate but global expression, we have first made acomparative study of the existing empirical formalisms, asmentioned. We have then attempted to find out anempirical expression based on the systematic study ofavailable experimental data. Our proposed empiricalexpression is an exponential where if the energy distribu-tion of the emitted neutrons is known the angulardistribution can be obtained. In the next section, we brieflydescribe the model HION, the empirical relation given byClapier and Zaidins and the Kalbach systematics. InSection 3, we have described our formalism. In Section 4we give a comparison of the calculated results usingourformalism, the other models described in Section 2 andthe experimental data.

2. Review and analysis of other approaches

2.1. The two-body scattering kinematics

In the formalism adopted for the HION [13,14] code,angular distribution of emitted neutrons in heavy ionreactions has been determined from two-body nucleon–nucleon scattering kinematics inside the target+projectilecomposite system. The probability that a particle is emittedat a particular angle with a particular energy is obtainedfrom the joint probability distribution of particles insidethe composite system prior to emission. This jointprobability distribution changes with each scattering. Ateach stage of the scattering or relaxation process bothenergy and linear momentum of the interacting particlesare conserved taking into account particle emission.Different stages of the relaxation process are designatedby the number of collisions N which the interactingnucleons undergo. Finally, the excitation energy is sharedamong all the nucleons to reach equilibrium. Beforeequilibration the joint probability PNðE;oÞdE do of aparticle having energy between E and E þ dE moving inthe direction between o and oþ do after suffering N

collisions is given by the recursion relation,

PN ðE;oÞdE do ¼ZZ

DðE0ÞPN�1ðE0;o0ÞKN ðE

0;o0 ! E;oÞ�

�dE0 do0�dE do. ð1Þ

PN�1ðE0;o0ÞdE0 do0 is the probability that the particle has

energy between E0 and E0 þ dE 0 moving in the direction

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Table 1

List of systems studied

Projectile Projectile Target Composite MeV/nucleon

energy (MeV) mass of composite

12C 75 C 24 3.125012C 75 Al 39 1.923012C 75 Cu 75 1.012C 75 Pb 220 0.340912C 120 C 24 5.012C 120 Al 39 3.076912C 120 Cu 75 1.612C 120 Pb 220 0.545416O 153 C 28 5.464216O 153 Al 43 3.558116O 153 Cu 79 1.936716O 153 Pb 224 0.683016O 116 181Ta 197 0.5888

M. Maiti et al. / Nuclear Instruments and Methods in Physics Research A 556 (2006) 577–588 579

between o0 and o0 þ do0 after previous ðN � 1Þ two-bodyscatterings. DðE0Þ is the probability that the Nth scatteringtakes place without prior emission of the particle con-sidered. KN ðE

0;o0 ! E;oÞ is the scattering kernel, i.e., theprobability that the particle in the energy-angle bin ðE 0;o0Þreaches the energy-angle bin ðE;oÞ after the two-bodyscattering.

In the heavy ion reaction model mentioned here themomentum of nucleons are described by Fermi distribu-tions. To account for the excitation energy brought into thesystem by the projectile, the composite system is dividedinto two subsystems—a ‘hot spot’ which is described by afinite temperature Fermi distribution and a ‘cold spot’described by a zero temperature Fermi distribution. At anystage of the relaxation process two-body interaction cantake place between a particle in the hot spot and one in thecold spot or between two particles in the hot spot. ThusKNðE

0;o0 ! E;oÞ has two parts: KK�KN ðE0;o0 ! E;oÞ

which is the scattering probability between a nucleon ofthe hot spot with one in the cold spot (Kikuchi–Kawaiscattering) and KH

N ðE0;o0 ! E;oÞ the scattering probabil-

ity between two nucleons in the hot spot. The temperatureT of the hot spot is determined from the entropy of thesystem. At each stage of the relaxation process some of theexcited particles may also be emitted depending on theemission probability. The emission probability pnð�Þ istaken as

pnð�Þ ¼lnCð�Þ

lnCð�Þ þ lnt ð�Þ(2)

where lnCð�Þ is the rate of emission of the ejectile of type nwith an energy � and lnt ð�Þ is the two-body collision rate.The details of calculation of these rates are given in Ref.[13].

Crucial to this approach are the two scattering kernelsthat are ultimately responsible for the angular distributionof the emitted neutrons. At the initial stage the projectilenucleons which have Fermi motion superimposed on theirincident velocity undergo scattering with the target nucleonwhich are in Fermi motion. Subsequently, in the compositesystem formed, all the nucleons take part in two-bodyscattering.

Now in the case of heavy ion induced reactions theprojectile may be removed from the entrance channel andenter the reaction channel even before any two-bodycollision takes place. This is because as the two heavyions approach each other, redistribution of the particleenergies takes place due to potential overlap. As aresult, some of the particles move above the separationenergy and then have a finite probability of emission. Thus,in the case of heavy ion induced reaction the total PEQneutron spectrum is obtained by summing the neutronemission over all values of N starting from N ¼ 0 up to thestage when equilibration is reached. The evaporationspectrum is obtained by a standard Weisskopf–Ewingcalculation.

Using this formalism with necessary modification forthick targets [15] we have calculated angular distribution ofneutrons for all the systems described in Table 1.

2.2. The Clapier and Zaidins formalism

Clapier and Zaidins [2] surveyed a series of experimentaldata and suggested a functional form for angular distribu-tion of neutrons, yðy; gÞ from heavy ion induced reactionson thick targets at a particular emission energy � as,

yðy; gÞ /1

4p1

lnð1þ 1=gÞ

� �1

gþ sin2ðy=2Þ

" #(3)

where y is the angle of observation and g is a fittingparameter. By using a normalization factor equal to theangle integrated neutron yield at any emission energy,angular yield of neutrons at that energy can be estimatedwith this formulation when g is known. Clapier and Zaidinshave given some values of g for a few target-projectilecombinations. From Eq. (3), g can be estimated as,

g ¼1

2

yð90�Þ

yð0�Þ � yð90�Þ(4)

where yð90�Þ and yð0�Þ are the yield of neutrons emittedwith the same energy � at 90� and 0�, respectively. The maindifficulty with this estimation is that the values of yð90�Þand yð0�Þ have to be known to get g. This is difficult inpractice where we do not know the angular distribution ofemitted neutrons as we are interested to find out yðy; gÞusing this expression.

2.2.1. Estimation of g from experimental data

In the present work, we have studied the dependence of gon incident energy, emission energy, projectile mass andtarget mass. To do this we have fitted the experimental dataconsidered in this work (see Table 1) with Eq. (3) to obtain

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120 MeV C

Neutron energy [MeV]0 10 20 30 40

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

153 MeV O

Neutron energy[MeV]0 5 10 15 20 25 30

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

75 MeV C

Neutron energy [MeV]2 4 6 8 10 12 14

0.00

0.25

0.50

0.75

1.00

1.25

1.50

C targetAl targetCu targetPb target

(a) (b)

(c)

γγγ

Fig. 1. Variation of g with neutron emission energy from thick C, Al, Cu

and Pb targets for the projectiles: (a) 75MeV C, (b) 120MeV C and (c)

153MeV O.

M. Maiti et al. / Nuclear Instruments and Methods in Physics Research A 556 (2006) 577–588580

the value of g for different systems. In Fig. 1(a) we haveplotted the variation of g with neutron emission energy for75MeV C on C, Al, Cu and Pb targets. Similar plots aregiven in Fig. 1(b) for 120MeV C projectiles and in Fig. 1(c)for 153MeV O projectiles on the same targets. The plotsshow that g is dependent on emission energy, projectileenergy, projectile mass as well as target mass. It is alsoobserved that for heavier targets (Cu and Pb) and for lowemission energies the value of g is relatively large. A largervalue of g indicates closer to isotropic distribution. Lowenergy neutrons are emitted by the compound nuclearevaporation process. For low mass targets even theevaporation neutrons have larger anisotropy (forwardpeaked) compared to high mass targets as can be seenfrom smaller values of g. Beyond emission energies ofabout 15MeV the values of g remain low and more or lessconstant with increasing emission energy. This indicatesthat for emissions from PEQ and direct processes theanisotropy remains almost same with increasing emissionenergy.

We are unable to empirically fit the trend of g withemission energy to discern any systematic behaviors of gwith projectile energy, projectile mass and target mass.However, we have compared the results from calculationsusing Eqs. (3) and (4) with the experimental measurements[16] to find out whether experimental spectral shape isreproduced and to what extent. For this we have normal-ized the calculated neutron emission spectra with theexperimental data at 0�.

2.3. The Kalbach systematics

Kalbach [7] has studied a large number of experimentaldata of light ion induced reactions and has proposed anempirical formalism for angular distributions of emittedneutrons, which is the modified version of earlier proposedKalbach–Mann systematics [17]. In Kalbach systematics,cross-section has been divided into two parts: multi-stepdirect (MSD), that has an unbound particle in each stage ofreaction process and shows peak at forward angles; andmulti-step compound (MSC) that carries all the boundparticles in each stage of relaxation and shows angulardistributions which are symmetric about 90� in the c.m.frame. MSD and MSC angular distributions have beenestimated separately and by adding them together the totalangular distribution has been obtained as,

yð�; yÞ ¼yð�Þ

4pa

sinhðaÞ½coshða cos yÞ þ f MSD sinhða cos yÞ�

(5)

where yð�; yÞ is the neutron yield with energy � at an angle yand yð�Þ is the yield of neutron with energy �. Here ‘a’ is aslope parameter that depends on the emission energy � andf MSD is the fractional constant. These systematics havebeen obtained for light ion or nucleon induced reactions.Kalbach has proposed different empirical formulae for thea parameter for nucleon and a-induced reactions. For ainduced reactions the expression is given as,

a ¼ 0:04�þ 1:8� 10�6�3. (6)

It has earlier been observed [18,19] that the Kalbachsystematics predicts angular yield of neutrons very wellfrom light ion induced reactions. In the present work weintend to investigate whether this formalism can be usedfor heavy ion induced reactions also.

2.3.1. Estimation of the parameters from experimental data

We have attempted to fit the available experimental[15,16] data (see Table 1) with Eq. (5). To get yð�Þ we haveintegrated the experimentally measured neutron yielddistributions at different angles over the emission angleswhich we denote as yexpð�Þ. However, with this value ofyexpð�Þ, the angular distribution obtained from experimen-tal data could not be fitted with Eq. (5). We have then triedto fit the experimental data by keeping yð�Þ, f MSD and a asfree parameters in Eq. (5). For a typical case of 153MeV Oions incident on thick Cu target we get the fitted values forthese quantities, while in Fig. 2 we plot the experimentaldata (symbol) at different neutron emission energies andthe fitted curve (dash). The fitted values of yð�Þ, which wedenote by yfitð�Þ are quite different in shape and magnitudefrom yexpð�Þ. The ratio yexpð�Þ=yfitð�Þ are plotted in Fig. 3(a).As can be seen, the difference between the two yieldincreases with increasing emission energy. The values of thea parameter obtained from the fitting are plotted in Fig.3(b) along with those for a-induced reactions. Thoughmuch larger in magnitude, the increasing trend of the a

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153 MeV O+Cu

0 40 80 120 160

Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

10-9

10-8

10-7

10-6

10-5

5 MeV10 MeV15 MeV20 MeV25 MeVfitted curve

Angle, θ, θ°

Fig. 2. Plot of neutron angular distributions at different emission energies

for 153MeV O incident on a thick Cu [16] target where the symbols

represent experimental data and the dashes are the fitted curves using

Kalbach systematics.

(a)

yexp (

ε)/y

fit(ε

)

80

120

160

200

240

280

(b)

a

0.1

1.0

10.0

100.0

1000.0

(c) Neutron energy[MeV]

0 5 10 15 20 25 30

f MS

D

0.6

0.7

0.8

0.9

1.0

Fig. 3. For 153MeV O on a thick Cu target [16] plots against emission

energy of: (a) the ratio yexpð�Þ=yfitð�Þ, (b) fitted a parameters (circle) and a

parameters for a-induced reactions, and (c) fitted f MSD values.


parameter values obtained by fitting the present heavy ionexperimental data are somewhat similar to those for aprojectiles. The fitted values of f MSD as plotted in Fig. 3(c),show that f MSD increases with increasing emission energy.This trend is expected as for higher emission energies theangular distribution becomes more forward peaked.It may be noted that the trends of the parameters though

remain approximately similar for other target-projectilesystems, are in fact system dependent. We have notsucceeded in finding out any general trend of theseparameters depending on target mass, projectile mass andprojectile energy based on the available experimental data.

3. The present formulation of angular distribution

Analyzing the available experimental data (Table 1) forangular distribution of emitted neutrons from heavy ioninduced reactions in thick targets we have found that anexponential function of the emission angle describesadequately the angular distributions of neutrons forprojectile energies up to 10AMeV. Such exponentialdependence of neutron dose on the emission angle hasearlier been found by others [3,9]. In the present work weattempt to parameterize the double differential neutronyield distributions using this relation.The angular yield, yð�; yÞ, of neutrons with energy � at an

angle y, can be expressed as,

yð�; yÞ ¼ yð�; 0Þ expð�myÞ (7)

where yð�; 0Þ is the neutron yield at forward direction ð0�Þwith the same energy �. Here we find that, the parameter m

is a function of both the emission energy � and the energyper nucleon of the composite system. Fig. 4 shows a plot ofln½yð�; yÞ=yð�; 0Þ� against y for 120MeV C projectile incident

120 MeV C+Pb

0 20 40 60 80 100 120 140

ln [

y(ε

,θε,θ)

/ y(

ε,0)

]

-6

-5

-4

-3

-2

-1

0

1

5 MeV7 MeV10 MeV15 MeV20 MeV25 MeV30 MeV35 MeV

Angle, θ, θ°

Fig. 4. Plot of ln½yð�; yÞ=yð�; 0Þ� against y for 120MeV C incident on a

thick Pb target at different neutron emission energies. Solid lines are the

fitted curves through the experimental data represented by symbols [16].

ARTICLE IN PRESS

0.25

0.30

0.35


on Pb target at different neutron emission energies, �. Thevalue of the slope parameter m is estimated at different �for all the experimental systems under consideration (Table1). The value of m increases with the increase of �, but therate of increment slows down at the higher values of �.

(a)

p

0.10

0.15

0.20

(b) x (MeV / Nucleon)0 1 4

q (

x103 )

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

2 3 5 6

Fig. 6. Plots of the parameters: (a) p and (b) q against x (MeV/nucleon).

Solid lines represent fitted curves through the calculated data (solid

circles).

3.1. Parameterizing the angular distribution slope

The slope parameter m of the angular distribution isfound to be dependent on � as,

m ¼ p�

E

� �þ q

�

E

� ��1(8)

where E is the incident energy of the projectile. Arepresentative plot of m against �=E for a few target-projectile combinations is shown in Fig. 5. The solid curve(Fig. 5) is obtained by fitting the data. The systems (target-projectile combinations) that are studied here are listed inTable 1. Two parameters p and q are related to x, i.e.energy (in MeV) per nucleon of the composite system as,

p ¼ 0:3898� 0:4028x1=2 expð�0:4277xÞ (9)

q ¼ 3:9976� 10�4 � 0:0306 exp �xþ 4:9164

1:2684

� �(10)

where x ¼ E=ðAP þ ATÞ, AP and AT being, respectively, theprojectile and target mass. Fig. 6(a) and (b) show,respectively, the variation of p and q with x. The variationof the parameter p with x follows a Maxwellian distribu-tion, whereas q follows an inverted decay curve. Toevaluate yð�; 0Þ, we integrate Eq. (7) between the limits 0�

and 90�, which gives,

yð�; 0Þ ¼ f ð�Þcð�Þ (11)

ε ε / E0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

m

0.00

0.01

0.02

0.03

0.04

0.05

120MeV C+Pb120MeV C+Cu75MeV C+Cu153MeV O+Pbfitted curve

Fig. 5. Plot of m against �=E for different target-projectile systems. Solid

lines are the fitted curves through the calculated m values (symbols).

where

f ð�Þ ¼yð�Þ

4pm

1� expð�90mÞ(12)

and c is constant for a given �. To remove the � dependencefrom c, we have analyzed some measured data and it hasbeen seen that,

logyð�; 0Þ

f ð�Þ

� �� 1:94 (13)

for all the available target-projectile system. We thenexpress yð�; 0Þ as,

yð�; 0Þ ¼yð�Þ

4pm

1� expð�90mÞexpð4:47Þ. (14)

yð�Þ is the angle integrated energy spectrum of emittedneutrons. From the combination of Eqs. (7) and (14) theexpression of angular yield of neutrons takes the form as,

yð�; yÞ ¼yð�Þ

4pm

1� expð�90mÞexpð4:47�myÞ. (15)

To calculate angular yield of neutrons from heavy ioninduced reactions using this relation, one has to supplyonly the appropriate angle integrated energy spectrum,yð�Þ, which can be estimated using nuclear reaction modelcodes. To provide yð�Þ in the present work we have taken

ARTICLE IN PRESS

120 MeV C+Al

Neu

tron

yie

ld (

n M

eV-1

sr-1

pro

ject

ile-1

)

10-8

10-7

10-6

10-5

10-4

120 MeV C+Al

0 40 80 120 16010-9

10-8

10-7

10-6

6 MeV

10 MeV

15 MeV

20 MeV

25 MeV

35 MeV

Angle, θ, θ°

Fig. 7. Comparison of experimentally measured data (symbol) [16] with

calculated results using our exponential relation (solid line), formulation

proposed by Clapier et al. (solid dot) and the two-body scattering

kinematics formalism (solid dash) at different emission energies of

neutrons for 120MeV C on Al system.

120 MeV C+Cu

Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

10-7

10-6

10-5

10-4

120 MeV C+Cu

0 40 80 120 160

10-9

10-8

10-7

10-6

6 MeV

10 MeV

15 MeV

20 MeV

25 MeV

30 MeV

Angle, θ, θ°

Fig. 8. Same as Fig. 7 for 120MeV C on Cu system.


the angle integrated energy spectrum from experimentaldata set available for different target-projectile systems.

4. Results and discussions

4.1. General trends

From our analysis of the experimental data using Clapierand Zaidins formula, it is observed from the trend of theparameter g that for high mass targets (like Cu and Pb) thelow energy neutron emissions are more isotropic comparedto low mass targets (like C and Al). The isotropy in lowenergy neutron emission decreases with increase inprojectile energy per nucleon. At higher emission energies(above about 15MeV) the emissions are forward peakedfor all targets and the degree of anisotropy is not verydifferent. Nevertheless, it is observed that the anisotropy isslightly more for light mass targets. Also, for heavierprojectiles the anisotropy is observed to be more. From theKalbach systematics we have observed that neutronemissions from heavy ion reactions are more anisotropiccompared to those from a-induced reactions, because ofthe higher value of the a parameter in the former case.From our proposed exponential relation it is observed thatthe anisotropy in angular distribution is dependent ontarget mass, projectile mass and projectile energy pernucleon.

4.2. Comparison of calculated angular distribution from

different approaches with experimental data

Calculated results using the present exponential relationhave been compared with experimental data as well as thecalculated results from two-body scattering kinematics [13]and Clapier and Zaidins [2] formalism. The calculatedresults have been normalized to the experimental data atthe extreme forward angle (i.e., 0�) to study the trend ofneutron yield angular distributions. Though there is noneed of normalization for two-body scattering kinematicsformalism [13], we have done so to compare the trend withother approaches mentioned above.

Figs. 7–12 show the comparisons of calculated neutronyield against the emission angle y (degree) with experi-mental data for different combinations of targets andprojectiles at various neutron emission energies. For120MeV C projectile incident on Al target [16] (Fig. 7),two-body scattering kinematics formalism overpredicts theexperimental data throughout the whole angular range andthe deviation increases with the increase of neutronemission energy, � and angle, y, whereas Clapier et al.formalism reproduces the trend of experimental data verywell except at high emission energies and at back angles. Inthis case, calculations using our proposed formalism are ingood agreement with experimental data for all neutronemission energies. For 120MeV C projectile incident on Cutarget [16] (Fig. 8), two-body scattering kinematicsformalism reproduces the trend of experimental data at

low energies with slight overprediction, but, the discre-pancy increases with the increase of energy. Clapier et al.formalism evaluates almost exactly the experimental dataexcept some overpredictions at high energies and at backangles. Our calculations agree well with experimental datain this case also. In the case of 120MeV C projectile

ARTICLE IN PRESS

120MeV C+Pb

Angle, θ, θ°0 40 80 120 160

Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

10-9

10-8

10-7

10-6

10-5

10-4

6 MeV

10 MeV

15 MeV

20 MeV

Fig. 9. Same as Fig. 7 for 120MeV C on Pb system.

Angle, θ, θ°

153 MeV O+Al

0 40 80 120 160

Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

10-7

10-6

10-5

10-4

5 MeV

10 MeV

15 MeV

Fig. 10. Same as Fig. 7 for 153MeV O on Al system.

153 MeV O+Cu

0 40 80 120 160

Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

10-8

10-7

10-6

10-5

10-4

5 MeV

10 MeV

15 MeV

20 MeV

25 MeV

Angle, θ, θ°

Fig. 11. Same as Fig. 7 for 153MeV O on Cu system.

153 MeV O+Pb

0 40 80 120 160

Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

10-9

10-8

10-7

10-6

10-5

5 MeV

10 MeV

15 MeV

20 MeV

25 MeV

Angle, θ, θ°

Fig. 12. Same as Fig. 7 for 153MeV O on Pb system.


incident on Pb target [16] (Fig. 9), two-body scatteringkinematics, calculations using Clapier et al. and ourrelation are very close to each other, reproduce the trendof measured data at low energies, but, with the increase ofenergy two-body scattering kinematics and our calcula-tions remain closer to each other overpredicting theexperimental data at back angles. Clapier et al. formulationevaluates the trend of experimental data fairly well at highenergies.

For the case of 153MeV O projectile incident on Altarget [16] (Fig. 10), two-body scattering kinematics and

our calculations are very close to each other and theyoverpredict the experimental data 60� upwards except atlow emission energies while Clapier et al. results under-predict the measured data at forward angles and agreeclosely at angles 60� and above except at large back angles.For 153MeV O projectile incident on Cu target [16] (Fig.11), the trend of experimental data is very well reproducedby both two-body scattering kinematics and Clapier et al.though two-body scattering kinematics results overpredictthe measured data to some extent. The calculated resultsfrom the present formalism show good agreement with

ARTICLE IN PRESS

75 MeV C+Al

Neutron energy (MeV)0 5 10 15 20

Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

10-8

10-7

10-6

10-5

10-4

10-3

10-2

0° (x102)

60° (x10)

120°

Fig. 14. Same as Fig. 13 for 75MeV C on Al.

75 MeV C+Cu

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

10-6

10-5

10-4

10-3

0° (x102)


experimental data at forward angles in each case, but theyoverpredict the data from 60� onwards except at lowemission energies. For 153MeV O projectile incident on Pbtarget [16] (Fig. 12), two-body scattering kinematics andClapier et al. both follow the trend of experimental datavery well but at back angles two-body scattering kinematicsresults underpredict and Clapier et al. results follow moreor less exactly the measured data. Our calculations aremore or less in reasonable agreement with experimentaldata.

From the above comparisons, we can conclude that theexponential relation does not reproduce the exact trend ofthe angular distribution of emitted neutrons, it represents amean straight line in a semilog plot through the measureddata. Therefore, the present exponential relation can atbest be taken as a good overall approximation in theabsence of adequate data to decipher such overall trendusing other models. Though the formalism of Clapier et al.reproduces fairly well the trend of neutron angular yield,difficulties exist in the evaluation of the slope parameter, g.It is not possible to have angular yield using this formalismwithout knowing yð0�Þ and yð90�Þ or the value of g by someother means. Agreement of the results from two-bodyscattering kinematics varies system wise. It is observed thatfor light targets two-body scattering kinematics do notpredict the large anisotropy at higher energies. It isanticipated that the strong forward peaked behavior isdue to enhanced contributions from direct reactionmechanism for lighter targets. In this situation, theexponential formulation for neutron yield distributionsfinds importance, complete in some sense, and is ourrecommended empirical relation to estimate neutronangular distributions from heavy ion reactions up to about10AMeV.

75 MeV C+C

Neutron energy (MeV)0 5 10 15 20

Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

10-9

10-8

10-7

10-6

10-5

10-4

10-3

0°(x102)

60°(x10)

120°

Fig. 13. Comparison of measured energy distributions of neutron yield

for 75MeV C on C [16] (symbol) with calculated results from present

exponential relation (line) at 0�, 60� and 120� lab angles. Error bars are

shown when they exceed the symbol size.

Neutron energy (MeV)

0 5 10 15 20

Neu

tro

10-8

10-760° (x10)

120°

Fig. 15. Same as Fig. 13 for 75MeV C on Cu.

4.3. Comparison of calculated and experimental energy

distributions at different angles

In order to test our empirical formalism in predictingdouble differential distributions of thick target neutronyield from heavy ion induced reactions, we have comparedour calculated results with available experimental data. Inour calculation energy spectra have been supplied bytaking weighted average of angular distributions availablein experimental data set of different target-projectilecombinations and multiplying them by 4p. Experimentallymeasured angular distributions of neutron yield for 75 and120MeV C incident on C, Al, Cu and Pb targets [16],

ARTICLE IN PRESS

120 MeVC+C

Neutron energy (MeV)0 10 20 30 40 50 60

Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

10-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-210-1

100

101

120°

90° (x10)

60° (x102)

30° (x103)

0° (x104)


for 120MeV C on C [16] (symbol) with calculated results from present

exponential relation (line) at 0�, 30�, 60�, 90� and 120� lab angles. Error

bars are shown when they exceed the symbol size.

120 MeV C+Al

Neutron energy (MeV)0 10 20 30 40 50 60

120°

90° (x10)

60° (x102)

30° (x103)

0° (x104)10-2

Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-1

100

101

Fig. 18. Same as Fig. 17 for 120MeV C on Al.

75 MeV C+Pb


0 5 10 15 20

Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

10-9

10-8

10-7

10-6

10-5

10-4

0° (x102)

60° (x10)(

120°

Fig. 16. Same as Fig. 13 for 75MeV C on Pb.

120 MeV C+Cu


0 10 20 30 40 50 60

120°

90° (x10)

60° (x102)

30° (x103)

0° (x104)

Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

Fig. 19. Same as Fig. 17 for 120MeV C on Cu.


153MeV O incident on C, Al, Cu and Pb [16] as well as116MeV O incident on Ta [15] have been compared withthe calculated results. Comparisons have been made atdifferent angles ranging from 0� to a maximum of 135�.Figs. 13–25 show the plots of those comparisons. In generalcalculated angular distributions are fairly well reproducedfor C, Al and Cu targets. For 75MeV C incident on C [16](Fig. 13), our calculation slightly overpredicts the experi-mental data at 0�, but at other angles the agreement

between calculation and measured data is good. For75MeV C incident on Al, Cu and Pb (Figs. 14–16) ourcalculations reproduce the experimental data fairly wellexcept for very low energy neutrons at backward angles. Inthe case of 120MeV C incident on C target [16] (Fig. 17)our calculation slightly over predicts the measured data at0� for high energy neutrons. At 90� angle our calculationshows underprediction at high emission energy of neutrons.For 120MeV C on Al and Cu [16] (Figs. 18, 19) ourcalculations agree well with the measured data throughoutthe whole range of emission energy of neutrons. In the case

ARTICLE IN PRESS

120 MeV C+Pb


120°

90° (x10)

60° (x102)

30° (x103)

0° (x104)

0 10 20 30 40 50

10-2

Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-1

100

Fig. 20. Same as Fig. 17 for 120MeV C on Pb.

153 MeV O+C


135°

90° (x10)

60° (x102)

30° (x103)

15° (x104)

0° (x105)

0 10 20 30 40 50 60

10-2

Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

102

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-1

100

101


for 153MeV O on C [16] (symbol) with calculated results from present

exponential relation (line) at 0�, 15�, 30�, 60�, 90� and 135� lab angles.

Error bars are shown when they exceed the symbol size.

153 MeV O+Al


135°

90° (x10)

60° (x102)

30° (x103)15° (x104)

0° (x105)

0 10 20 30 40 50 60

10-2

Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

102

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-1

100

101

Fig. 22. Same as Fig. 21 for 153MeV O on Al.

153 MeV O+Cu


135°

90° (x10)

60° (x102)

30° (x103)15° (x104)

0° (x105)

0 10 20 30 40 50 60

10-2

Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

102

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-1

100

101

Fig. 23. Same as Fig. 21 for 153MeV O on Cu.


of 120MeV C on Pb [16] (Fig. 20), our calculation is moreor less good except at 30�, where we get some under-prediction. For 153MeV O on C [16] (Fig. 21) ourcalculations overpredict slightly the measured data at 0�

and at backward angles of 60� onwards. In the case of153MeV O incident on Al [16] (Fig. 22), at forward angles,we have larger discrepancy between our calculations andexperimental data at high emission energies. For 153MeVO on Cu and Pb [16] (Figs. 23, 24), our calculations agree

well with experimental data at forward angles and slightlyover predict the measured data at backward angles. For116MeV O incident on Ta target [15] (Fig. 25), ourestimation matches well with the measured data at allangles except at high energy end. In all these cases,experimental data is available from 4MeV onwards. Wehave extrapolated our calculation to 1MeV neutronemission energy.From the study of angular distribution of neutron from

heavy ion induced reaction it has been observed thatempirical formalism reported in this work predicts the

ARTICLE IN PRESS

153 MeV O+Pb

135°

90° (x10)

60° (x102)

30° (x103)

15° (x104)

0° (x105)

Neutron energy (MeV)0 10 20 30 40 50

10-2

Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-1

100

101

Fig. 24. Same as Fig. 21 for 153MeV O on Pb.

116 MeV O + Ta


0 5 10 15 20 25

0° (x102)

30° (x10)

60°Neu

tro

n y

ield

(n

MeV

-1 s

r-1 p

roje

ctile

-1)

10-10

10-9

10-8

10-7

10-6

10-5

10-4


for 116MeV O on Ta [15] (symbol) with calculated results from the

present exponential relation (line) at 0�, 30� and 60� lab angles. Error bars

are shown when they exceed the symbol size.


experimentally measured distribution more or less satis-factorily.

5. Conclusions

We have made an empirical parameterization of theneutron angular distribution from heavy ion induced

reactions on thick targets for projectile energies up to10AMeV. This simple exponential relation does notpredict the angular distribution very accurately. However,no other approach is applicable to neutron emissions froma variety of reactions where projectiles of different energiesand masses are incident on various targets. The presentanalysis reveals the complex dependence of the angulardistribution of emitted neutrons on the target-projectilesystem in heavy ion reactions. Inadequate experimentaldata prevents to discern more accurate systematics of suchangular distributions and physical interpretation.

Acknowledgements

This work has been carried out as a part of a project(No. AERB/29/05) sponsored by the Atomic EnergyRegulatory Board, India.

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angular distribution of neutrons from heavy ion induced reactions in thick targets

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