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Arab Journal of Nuclear Science And applications, 46(2), (115-127) 2013

115

Effect of Free and In- Medium Proton – Nucleon Total Cross

Sections on p 12 C Total Cross Section.

Samia S. A. Hassan

Mathematics and Theoretical Physics Dept., Nuclear Research Center,

Atomic Energy Authority, P.N.13759, Egypt.

Received: 27/3/2011 Accepted:4/5/2011

ABSTRACT

In the framework of optical limit of Glauber multiple scattering model, the

total cross section for proton scattering on 12C nucleus is calculated in the energy

range 30 ≤ Ep ≤ 1000 MeV. The effect of different experimental data and

phenomenological forms of free and in – medium proton – nucleon total cross

sections are studied. In addition, the slope parameter of proton – nucleon elastic

scattering amplitude is modified according to free and in –medium proton –

nucleon total cross section up to 200 MeV. Both relativistic mean field (RMF) and

extended relativistic mean field (E-RMF) densities are used to describe the matter

density of the target nucleus. The in – medium corrections reduce the values of

proton – nucleon total cross section, which in turn reduce the proton – nucleus total

cross section. It is found that the use of free- space scattering amplitude (ρ=ρο= 0)

provide results seems to be acceptable in comparison with the available

experimental data.

Key Words: Glauber Multiple Scattering Model / Total Cross Section / In-Medium

Proton- Nucleon Cross Sections.

INTRODUCTION

The scattering and propagation of low and intermediate proton energy from the nuclear matter

of stable target nucleus is a subject of much interest and activity. One of the fundamental observables

that can identify the mechanism of nuclear collision is the nuclear total cross section. The study of

nuclear cross section and other integrated cross sections are particularly important in several areas,

e.g., accelerator shielding, space radiation effect, medical application and transmutation of transuranic

wastes by spallation reaction.

Over the past two decades, the Glauber model (1) (GM) has been applied successfully by several

authors (2-9) to study the proton – nucleus scattering at different ranges of energy. Many of the

experimental measurements for proton – nucleus (pA) have been analyzed mostly by invoking the so –

called optical limit approximation (OLA) for the evaluation of the Glauber scattering amplitude via the

total nuclear phase shift functions (10-13). Although the OLA neglects entirely any corrections between

constituents in the projectile or target, it gives a reasonable satisfactory account of the experimental

pA data (14).

The Glauber model calculations require an additional phenomenological parameter: the ratio of

the experimental free nucleon – nucleon (NN) cross section to the in – medium NN cross section. This

means that the in – medium NN cross section should have some density dependence and may be less

than its free values, especially in the low and intermediate energy regions. This is reasonable, since the


116

medium posses further restrictions due to complicated effects like Pauli blocking and Fermi motion (15). There are many reported theoretical calculations of in – medium NN cross section by the use of G-

matrix theory, but it is desirable to determine it from the observed data of some suitable quantity in a

direct way like proton – induced reaction cross sections (16,17). M.Kohno et al. (18) calculated the in –

medium NN total cross section from reaction matrices of the non relativistic Brueckner approach. The

non relativistic cross sections are found to be reduced from free ones as observed in relativistic

Brueckner method. This reduction is ascribed to the flux renormalization represented by an effective

mass. On – shell and half off – shell in medium NN total cross section are determined within the

relativistic Brueckner – Hartree – Fock model (19). The resulting total cross sections are , however,

reduced by not more than about 25% compared to the on – shell values. On the other hand, using both

the double folding optical potential and OLA, the effect of in – medium NN total cross section and

finite range force on the reaction cross section for a deformed target nuclei are studied (20). In addition,

NN total cross sections in the nuclear medium with unequal densities of protons and neutrons are

calculated using Dirac – Brueckner – Hartree – Fock approach together with realistic NN potentials (21). The effect of asymmetry in neutron and proton concentrations is examined and concluded that the

mean free path of a nucleon could be affected in a significant way by the presence of isospin

asymmetry in the medium (21). Moreover, probing the in – medium NN total cross section in heavy ion

collisions has been investigated by means of the isospin – dependent quantum molecular dynamics (22).

The slope parameter NN of NN elastic scattering amplitude is not a very well determined

quantity (14). Some authors (23) in their analysis of elastic scattering differential cross section data,

assumed it to be zero for nucleon energy < 100 MeV. Furthermore, this parameter is treated as an

adjustable parameter in nucleus – nucleus scattering (24). Tag El-Din et al.(13) discussed different values

of NN in the analysis of proton – nucleus reaction cross section in the energy range from 30 to 2200

MeV.

Usually the calculations based on OLA of GM are performed using a one – body density of the

target nucleus, calculated from a nuclear wave function, in case of pA scattering. During the last years,

the relativistic mean field (RMF) theory has received a wide attention due to its successful description

of many nuclear ground – state properties, such as binding energy and nuclear radii in the entire

region(25). In this theory, the nucleons are treated as Dirac spinors interacting by the exchange of σ, ,

ρ mesons and photons (26). The inclusion of these mesons take into account the proton – neutron

asymmetry and give an impression that the theory can be used to nuclei far away from the valley of β

– stability.

An extension for RMF theory (E – RMF) (27) arises from the field theory motivated effective

Lagrangian approach. E – RMF formalism can be interpreted as a covariant formulation of density

functional theory as it contains all the higher terms in the Lagrangian, obtained by expanding it in

powers of the meson fields (28).

The goal in this work is to identify if the medium corrections of NN scattering total cross section

can modify appreciably proton – nucleus nuclear total cross section over a wide range of projectile

energy.


117

MATHEMATICAL FORMALISM

In the framework of GM, the nucleon– nucleus elastic scattering amplitude can be written as (1):

2

1 1

11

1( ) ...... ( ,..... ) 1 (1 ( )) ( )

2

A Aiq bcm

A A j j j

jj

ikF q dr dr e r r b s r

A

(1)

where .c m

k refers to the momentum of the incident nucleon in the center - of - mass system ( with

1 ), q is the momentum transfer from the projectile to the target nucleons ( )f iq k k= - , b is

the impact parameter vector , 1, ,L Ar r stands for the position vectors of the target nucleons with

respect

to the origin of the target nucleus, js are the projections of the target nucleon coordinates jr onto the

impact parameter plane, 1( , , )L Ar ry is the ground state wave function and A is the mass number of

the target nucleus. The Dirac delta function δ determines the center of mass constraint and ( )jj

b sg -

is the nucleon - nucleon nuclear profile function which is connected to the nucleon -nucleon elastic

amplitude ( )j

f q by the relation (1)

.( )2

0

1( ) ( )

2

jiq b s

j j jb s d q e f qik

(2)

ok is the proton momentum in the nucleon -nucleon (NN) center - of - mass system ( with 1 ). The

nucleon - nucleon profile function ( )jj

b sg - is related to both the total nuclear profile function

( )b and the total nuclear phase shift function ( )bc by the formula (1)

1

( )

(1 ( )) 1 ( )A

jj

j

i b

b s b

ec

g G=

ص - - = -

=

, (3)

According to the dynamical approximation of GMSM, where there is no overlap between the potentials

describe the interaction between the incident nucleon and the target nucleons, so ( )c b can be expanded

as (1)

(1) (2) (3)

( ) ( ) ( ) ( )c c c c= + + + Kb b b b (4)


118

where (1)

( )b symbolize to the single scattering process and (2)

( )b determine the double scattering

processes, and so on.

The total nuclear cross section for nucleon - nucleus scattering can be expressed by the relation (1)

t

c.m

4Im F(0) ,

k

ps = (5)

where Im F(0) is the imaginary part of the forward nucleon - nucleus scattering amplitude (

equation (1)) at the momentum transfer q = 0.

Since OLA " the leading term of equation(4)" has been successfully used as a convenient tool to

describe hadron – nucleus and nucleus – nucleus scattering (1), so, in this work, we will consider the

OLA nuclear phase shift function as:

(1)( ) ( ) ( )jOLA jb iA r b s d rc r g= - ٍ (6)

( )rr represents the one body ground state density of the target nucleus12C and is

described by both (RMF) and (E-RMF) as a sum of two Gaussian (27), where

( )2

2

1

x )) ,p( (ei i

i

r c a r dr Arr r=

= ه =- ٍ (7)

The coefficients ic and ia are in the units of fm3 and fm

2, respectively. For RMF :

1 -1.19229c , 2 1.4191c , 1 0.4315a and 2 0.36777a and their values for E-RMF are

1 -3.77056c , 2 3.96943c , 1 0.37809 a and 2 0.36006a . These values are adjusted to

obtain both the charge root- mean square radius and the binding energy of the 12C nucleus.

The usual form of nucleon – nucleon scattering amplitude is the three-parameters spin and iso-

spin independent formula, which is widely used in many applications for hadron- nucleus and nucleus

– nucleus scattering (11,12, 29, 30-32 ), where

2( ) 1

( ) exp4 2

t

NN NN

NNNN

k if q q

(8)

Here, t

NN , NN , and NN represent the average nucleon – nucleon(NN) total cross section, the

average ratio of real to imaginary part at forward NN amplitude ( 0q ) and the average slope

parameter which determines the fall-of the angular distribution of the NN elastic scattering

amplitude, respectively.


119

t t

pp pn

t t

pp pp pn pn

t t

pp pn

pp

t

N

N

p

N

N

n

N

N

z n

z n

z n

z n

z n

z n

(9)

where t

pp , t

pn are the proton-proton and proton- neutron total cross sections, pp ,

pn are the

proton-proton and proton- neutron ratio of real to imaginary parts at q=0, ,z n are the numbers of

protons and neutrons respectively , z , n are the numbers of protons and neutrons respectively and

finally pp , pn are the proton-proton and proton- neutron slope parameters.

Substituting from equation (8) into equation (2), we get

{ }2( )exp

4( ) ( ) / 2

t

NN NN

NN

NN

j jjb s b

is

s eb

pbg - = -

+- (10)

Using equation (7) and equation (10), the OLA nuclear phase shift function (equation (6)) has the form

2

(1) 2

1 3/ 2

1( ) ( ) exp( /(2 )).

12( ) (2 )

t

iNN NN NNOLA

ii

NNi

i

cAb i b

aa

a

(11)

The Parameters of NN Scattering Amplitude

In this study, several sources for the parameters of NN scattering amplitude (equation (8)) in the

energy range from 30 MeV to 1000 MeV are taken into consideration as follows:

(i) t

NN , NN and NN ( NN = 0 in the range Ep < 100 MeV ) values are taken from the

experimental data of free p – p and p – n scattering (27). These parameters are used to estimate

the nuclear total reaction cross section for nucleus – nucleus (27) and proton – nucleus scattering (13). These values are denoted by set 1.

On the other hand, using the same values of t

NN and NN as in set 1, Tag El-Din et al (13)

discussed the effect of the slope parameter NN on proton - nucleus total reaction cross – section


120

within two arguments. The first one NN is considered as a constant value (=0.432 fm2), which gave a

reasonable results of R for p – 3He over a wide range of energy (33), this set is described by set 2.

In the second argument, NN is determined by its relation to both

t

NN and NN at Ep≤ 300

MeV, This set of parameters is called set 3, where (34)

2

1

16

tNNNNNN

(12)

At Ep > 300 MeV , NN is taken from set 1. These three sets of parameters are presented in Table 1.

Table 1. The parameters of NN scattering amplitude (equation (9)).

E

(MeV)

tNN

(27)

(fm2)

eNN(27)

NNb (fm2)

Set 1(27) Set 2 (33) Set 3

30 19.6 0.87 0

0.423

0.685

38 14.6 0.89 0 0.521

40 13.5 0.9 0 0.486

49 10.4 0.94 0 0.39

85 6.1 1 0 0.243

94 5.5 1.07 0.51 0.234

100 5.29 1.435 0.51 0.322

120 4.72 1.38 0.535 0.273

150 3.845 1.245 0.575 0.195

200 3.28 0.93 0.62 0.121

325 3.03 0.305 0.31 0.31

343 2.84 0.26 0.31 0.31

425 3.025 0.36 0.24 0.24

500 3.62 0.04 0.0625 0.0625

550 3.62 0.04 0.0625 0.062

625 4.0 -0.095 0.08 0.08

800 4.26 -0.07 0.105 0.105

1000 4.32 -0.275 0.105 0.105

(ii) In the analysis of Pauli blocking and medium effects in nucleon knockout reactions, Bertulani and

Conti (35) developed new fits for the energy dependent free NN total cross sections, separated in

three energy intervals, by means of the expressions


121

2

2 7 3 10 4

pp

3 7 2 15 4

19.6 4253/ E 375/ E 3.86 10 E

(for E 280 MeV)

32.7 5.52 10 E 3.53 10 E 2.97 10 E

(for 280MeV E 840 MeV)

50.9 3.8 10 E 2.78 10 E 1.92 10 E

(for 840MeV E 5 GeV)

(13)

For proton – proton collisions, and

2

5 2 9 3

pn

3 6 2 10 3

89.4 2025/ E 19108/ E 43535/ E

(for E 300 MeV)

14.2 5436 / E 3.72 10 E 7.55 10 E

(for 300 MeV E 700 MeV)

33.9 6.1 10 E 1.55 10 E 1.3 10 E

(for 700MeV E 5 GeV)

(14)

for proton – neutron collisions. E is the projectile laboratory energy. The coefficients in the above

equations have been obtained by a least square fit to the NN total cross section experimental data over a

variety of energies ranging from 10 MeV to 5 GeV. The values of NN at Ep < 300 MeV is calculated

using equation (12), this set is denoted by set 4, while NN and NN (at Ep > 300 MeV ) are taken

from set 1. This set is described by set 4.

(iii) In the framework of Glauber model, Abu-Ibrahim et al. (36) used another experimental data for

p – p and p – n parameters in the energy range 40 ≤ Ep ≤ 1000 MeV. The average values of these

parameters are introduced as set 5 (see table 2).


122

Table 2. The parameters of NN scattering amplitude (36) .

E

(MeV)

tNN

(fm2)

eNN

NNb (fm2)

40 14.4 0.695 0.462

60 9.15 0.952 0.375

80 6.79 1113 0.325

100 5.515 1.184 0.281

120 4.74 1.184 0.239

140 4.235 1.134 0.202

160 3.89 1.067 0.172

180 3.64 0.968 0.146

200 3.45 0.878 0.126

240 3.21 0.684 0.096

300 3.06 0.444 0.074

425 3.01 0.348 0.074

550 3.47 0.037 0.097

650 3.93 -0.082 0.136

700 4.1 -0.121 0.14

800 4.235 -0.059 0.152

1000 4.255 -0.258 0.172

(iv) Based on the Born nucleon – nucleon interactive potential and Dirac – Brueckner approach for

nuclear matter, the in – medium NN total cross section were calculated by Li and Machleidt (LM) (37) for incident energies up to 300 MeV in a laboratory frame and for matter densities up to 2ρο

,where ρο is the saturation density of normal nuclear matter (0.15 ≤ ρο ≤ 0.19 fm-3 ). This

semiemperical formula can be written as

1.05 3

0.5 4 labpp lab 1.2

1.51 22.9

0.53 labpn lab 1.34

for proton - proton scattering and

for proton - neutron scattering

1 0.1667E23.5 0.00256(18.2 E ) ,

1 9.704

1 0.0034E31.5 0.092 20.2 E ,

1 21.55

(15)

This equation has been applied to study some features of nuclear reactions (38).

On the other hand, CaiXiangzhou et al. (38) combined the energy dependence of free – space NN

total cross section of Charagi and Gupta (39) with LM formula (equation (15)). So, a new

phenomenological formula for in – medium NN total cross section is proposed by the following

expression


123

0.06 1.481 2 4 lab

pp 1.46

0.04 2.021 2 lab

pn 1.90

1 7.772E13.73 15.04 8.76 68.67 ,

1 18.01

1 20.88E70.67 18.18 25.26 113.85 ,

1 35.86

(16)

lab

2

1 E1 , 1

931.5

where is the ratio of projectile velocity to light velocity. The coefficients in equation (16) are

obtained by a least square fit to experimental total cross section data over a wide incident energy range

from 10 MeV to 1 GeV. In both cases, namely equations(15) and(16), NN and NN (at Ep >300 MeV )

are considered from set 1, while NN (at Ep < 300 MeV ) are modified according to equation (12). This

is described by set 6 and set 7, respectively with ρ = ρo = 0 and ρ = ρo = 0.17 fm-3, respectively.

RESULTS and DISCUSSION

In the framework of OLA of Glauber model, proton – 12C nuclear total cross sections are

calculated at

30 ≤ Ep ≤ 1000 MeV using both RMF and E-RMF densities for the target nucleus. Various free and in –

medium nucleon – nucleon total cross sections and slope parameters are introduced.

The generally used Gaussian parameterization for ( )NNf q (equation (8)) is most suited for

intermediate and high energies where the small angles NN scattering cross section are mostly diffractive

and peaked in the forward direction so that the slope parameter NN can be determined with a fair

degree of certainty. Otherwise, at low energies this Gaussian parameterization becomes less satisfactory

where the scattering is non – diffractive and not many partial waves are included. Perhaps this is one of

the reasons that a cursory survey of the literature shows that very different values of NN have been

used in performance of proton – nucleus nuclear total cross section(13) and nucleus – nucleus scattering (

42).

Table 3 displays the nuclear total cross section using sets (1), (2) and (3) with both RMF and E-

RMF densities.


124

Table 3: Nuclear total cross section for proton – 12C scattering. The experimental data are from

refs. (40, 41), the value in parentheses represent the incident laboratory proton energy in

MeV.

It is apparent that t (E-RMF) > t (RMF) by about 2.5% in the whole energy range for each

set. This ratio remains the same for sets (4), (5), (6) and (7) (with ρ = ρo = 0). This led to that t (with

E-RMF) becomes more appropriate with the available experimental data. At ρ = ρo = 0.17 fm-3, this

ratio reduced to nearly 2%. For a quantitative discussion to elucidate the accommodation of each set in

fitting the experimental results, a difference factor (d) was introduced as (43)

( ) (exp)

( )

t t

t

OLAd

OLA

(17)

It is found that using E-RMF density and NN from experimental data, the difference factor (d)

for set (1) up to set (5) is 6%, 2%, 5%, 8% and 6%, respectively. This clarify that set (2) with NN =

0.423 fm2 is more appropriate NN input data for extracting the nuclear total cross section for p – 12C.

This confirms the previous work of the nuclear total cross reaction section for scattering of proton on

Li, B and Be targets in the energy range from 30 to 2200 MeV (13). These results are represented

graphically in Figure (1).

E(MeV)

t (mb)

Experimental

Data set (1) set (2) set (3)

RMF E-RMF RMF E-RMF RMF E- RMF

30 636.6 669.1 741.3 770.1 798.7 825.5 ----

38 577.7 605.5 664.5 688.5 682.5 705.8 ----

40 561.8 588.3 643.8 666.6 654.8 677.1 ----

49 508.1 530.3 574.3 592.8 570.0 588.8 ----

85 396.0 409.7 434.9 445.6 428.2 439.4 ----

94 414.8 423.7 408.8 418.2 394.5 405.0 ----

100 405.1 413.6 399.3 408.3 392.3 401.8 ----

120 377.6 384.8 371.3 379.0 362.1 370.6 353± 7(137)

150 330.9 336.0 323.4 329.2 312.8 319.5 324± 6(158)

200 295.4 299.4 289.0 293.5 282.3 287.5 306± 8

325 269.0 273.3 272.5 276.5 ------ ------ 292± 6(315)

343 256.4 260.3 259.6 263.2 ------ ------ 286± 6(348)

425 266.3 270.9 272.2 276.2 ------ ------ 286±14(414)

500 294.0 300.8 310.2 315.4 ------ ------ 315± 9

550 293.7 300.6 310.2 315.4 ------ ------ 324 ± 16(353)

625 314.6 322.4 332.3 338.5 ------ ------ 336± 3(650)

800 329.1 337.4 346.8 353.5 ------ ------ 352± 7

1000 332.0 340.4 350.1 356.9 ------ ------ 356± 8


125

102

103

200

400

600

800

(

mb

)

E(MeV)

Set1

Set2

Set3

Set4

Set5

Exp Data

Fig. 1: p- 12 C total cross section for sets from 1 to 5. The experimental data are

taken from (40, 41).

From equations (15) and (16), where it is generally believed that the in – medium NN total cross

section differs from the free – NN total cross section, mainly due to the Pauli blocking of the

intermediate and final states as well as the mean field. Applying equation (17), the difference factors for

sets (6) and (7) using E-RMF with ρ = ρo = 0 are 10 % and 7%, respectively. However, the calculations

show that introducing equations (15) and (16) with ρ = ρo = 0.17fm-3 with E-RMF density underestimate

the experimental data, as manifest in figure 2. The difference factor became 100% in the energy range

up to 200 MeV using set (6) and 30% over the whole energy range using set (7).

It is evident from equation (15) that NN (ρ = ρo = 0.17fm-3) decreases by about 57% from NN

(ρ = ρo = 0), while this ratio becomes around 23% using equation (16). This in turn led to t (p – 12C)

with both RMF and E-RMF densities decreases by about 41% and 15%, respectively. Therefore, the

formula (16) can be regarded more density dependent than formula (15). Moreover, at

ρ = ρo = 0.17fm-3, NN (equation (16)) exceeds the values from equation (15) by 38%, so, t (p – 12C)

increases by 33% in the energy range Ep < 300 MeV.


126

102

103

200

400

600

800

(m

b)

E(MeV)

Set6 (=0)

Set7 (=0)

Set6 (==0.17fm

-3)

Set7 (==0.17fm

-3)

Exp Data

Fig. 2: p- 12 C total cross section for sets 6 and 7. The experimental data are taken

from (40, 41).

In conclusion, the free NN total cross sections can predict proton – nucleus nuclear total cross

section over a wide range of energy within OLA. Introducing higher order terms of GM may slightly

improve the situation. On the other hand, proton – nucleus total cross section depends strongly on the

slope parameter of NN elastic scattering amplitude. Unfortunately, in – medium NN total cross section

gives unsatisfactory agreement with the available experimental data. The reason may be attributed to

that, it is difficult to use OLA to estimate the role of mean field, the Pauli blocking effect and NN

interaction simultaneously. Using some dynamical models, which incorporates these effects

simultaneously, like quantum molecular dynamic (QMD) can solve this problem.

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