fulltext

The Coulomb-modified Glauber description of reaction cross-sections

of heavy ions with nuclei

A. Y. ABUL-MAGD(1) and M. TALIBALI-ALHINAI(2)

(1_{) Department of Mathematics and Computer Science, Faculty of Science} U.A.E. University - Al-Ain, United Arab Emirates

(2_{) Department of Mathematics and Statistics, College of Science, Sultan Qaboos University} Muscat, Sultanate of Oman

(ricevuto il 19 Giugno 1997; approvato l’1 Luglio 1997)

Summary. — Glauber’s theory is adopted to calculate the total nucleus-nucleus

reaction cross-section at high energies. The projectile nuclear density applied is Gaussian while that of the target is uniform. The average nucleon-nucleon cross-sections and the root-mean-square radii are the only ingredient parameters in the present treatment. At relatively low energies, Glauber’s total reaction cross-section is modified to take into account the Coulomb effects. In both energy regions and for the relevant nuclear densities, predictions fit reasonably with experiment.

PACS 25.70 – Low and intermediate energy heavy-ion reactions.

1. – Introduction

In nuclear reactions the total reaction cross-section is one of the most important observables required to fully describe strong interactions. It finds applications in various fields of research including shielding against heavy-ions originating from, for example, space radiation, and also against radiobiological effects resulting from clinical exposure [1]. It is not surprising, therefore, to find the total reaction cross-section, sR,

being a centre of attraction of experimental and theoretical studies (see [2-4] and references therein). High-energy nucleus-nucleus reactions are successfully treated in the framework of Glauber’s theory [5], which is mainly based on the independent individual nucleon-nucleon collisions in the overlap zone of the colliding nuclei. Even in the simplest cases in which some physical effects such as Coulomb effects, Fermi motion, Pauli blocking, etc., are ignored, yet, the Glauber approach provides reasonable agreements with experiment. Karol [6], within Glauber’s model and omitting all these effects, derived an analytical formula for the reaction cross-section for collisions between two Gaussian nuclei, which gave an excellent representation of the experimental reaction cross-section of light nuclei interacting at high energies. In order to extend the analysis to lower energies, Charagi et al. [7] succeeded in modify-ing Karol’s formulation to include the influence of the repulsive Coulomb interaction.

Most of the previous analysis using Glauber’s theory assumed Gaussian density distributions for both the target and projectile. This leads to a considerable simplification in the calculations of the nuclear transparency. However, while the Gaussian distribution may provide a reasonable description for the density of light nuclei, certainly it will be invalid for intermediate mass and heavy nuclei. In this work, using Glauber’s model we are going firstly to compute the total reaction cross-section for collisions of projectiles whose density distribution has a Gaussian radial dependence with target nuclei having uniform nuclear density distribution. Then, we consider low-energy nucleus-nucleus collisions, where the projectile straight line trajectory deviates, being affected by the Coulomb repulsive forces. We modify the standard Glauber reaction cross-section to account for the Coulomb effect, by relating the impact parameter to the distance of closest approach of the associated Rutherford trajectory. An analytical modified expression for the total reaction cross-section is established. The comparison of the present calculations with experiment is shown to be quite satisfactory.

2. – Reaction cross-sections for projectiles with Gaussian densities

At high energies where the Coulomb effects play no significant role, the standard Glauber form of the reaction cross-section is expressed as

sR4 2 p



defines the probability that at the impact parameter b , the high-energy projectile traverses the target without interaction. The function T(b) describes the transparency of the target towards the projectile, in which

s884

(ZPZT1 NPNT) spp1 (ZPNT1 ZTNP) spn

APAT

is the average energy-dependent nucleon-nucleon cross-section, while

x(b) 4









(

)

is the thickness function for the nuclear densities of the interacting nuclei, rP , Tare the

nuclear densities of the projectile and target nuclei, respectively, while f accounts for the finite range of the nucleon-nucleon interaction.

Karol derived an analytical expression for sR by assuming a zero-range

nucleon-nucleon effective interaction and a Gaussian shape for the nuclear-matter density distributions in both the target and the projectile,

ri(ri) 4ri( 0 ) exp

y

z

where ai and ri( 0 ) are the diffuseness and central nuclear density, respectively; both

are related to the root-mean-square radius, Rrms(i), through [8]

aP4

o

In this case, the transparency function becomes

T(b) 4exp

[

]

where x0is given by

x04 p2s–88rP( 0 ) rT( 0 ) aP3aT3O(aP21 aT2) .

Then, the integration over b in eq. (1) can be carried out analytically, and one obtains sR4 p(aP21 aT2)[ln x01 E1(x0) 1g] .

This equation is often used to analyse the reaction cross-section data for heavy-ion collisions at high energies. While the Gaussian density might provide a reasonable description for light nuclei, it is certainly invalid for intermediate- and heavy-mass nuclei. In this article, we consider the case when only the projectile has a nuclear density described by a Gaussian function. We also use a Gaussian form for the nucleon-nucleon range function [7],

f (b) 4 1 pr2 0 exp

y

z

where r0 is a parameter related to the slope of the nucleon-nucleon differential

scattering cross-section. This allows carrying out the integration over zP and bP

analytically, and thus obtaining

x(b) 4 2 a 3 PrP( 0 ) kp (aP21 r02) exp

y

z



y

z

g

h



where I0(j) is the modified Bessel function [9].

A simple expression can be obtained for the thickness function if we adopt the uniform distribution for the target nuclear density, which assumes the following form:

rT(rT) 4rT( 0 ) U(rT2 TR) ,

in which (rT2 RT) is the unit step function and

rT( 0 ) 43ATO4 pRT3

is the target’s central nuclear density with RT4k5 O3 Rrms( T ). In this case, the thickness

function becomes (7) _{x(b) 4} 2 a 3 PrP( 0 ) rT( 0 ) kp r02 exp

y

z



k

y

z

g

h

Fig. 1. – The total nucleus-nucleus reaction cross-section calculations. The present calculations are given by the solid curves, while the dashed curves show Karol’s predictions [6].

Inserting this expression combined with the transparency function—given by eq. (2)—into eq. (1), the total reaction cross-section is immediately obtained.

Figure 1 shows the comparison of the present reaction cross-section predictions (the dashed curves) with the corresponding experimental results cited in ref. [10] for

12_{C colliding with targets of different masses;} 20_Ne27_Al, 57_Fe,66_{Zn and}89_{Y. Also shown}

in fig. 1 are the results obtained by Karol’s treatment [6] (the solid curves). Clearly, if one is interested in collisions of light projectiles with light targets the present treatment is found to be an inadequate measure of the reaction cross-section. Such reactions are better treated by either Karol’s [6] calculations in which both the projectile and target densities are described by a Gaussian distribution. On the other hand, for interactions of light projectiles with heavy targets the best fit of the experimental data is achieved by the present formulation.

3. – Coulomb modification of Glauber’s reaction cross-section

At high energies, the Glauber model shows an excellent agreement with the experimental data. However, this model fails to reasonably describe collisions induced at relatively low energies. The disagreement obtained is due to the significant role played by the Coulomb repulsive potential whose effects are dominant in the low-energy range. Such Coulomb effect breaks the characteristic Glauber assumption that the projectile travels along straight-line trajectories. In this subsection we attempt to present a modified form of Glauber’s reaction cross-section which extends the scope of Glauber’s formalism to low-energy regions. Several attempts have been made to include the Coulomb effects into the Glauber formalism [11-13]. The most successful approach, which is based on the WKB approximation for the phase shifts, replaces the impact parameter b in eq. (1) in the transparency T(b) by the distance of closest approach b 8 of the Coulomb deviated projectile trajectory. Accordingly, the transpar-ency function is evaluated at impact distance b 8. The reaction cross-section sRin terms

of the transparency function T(b8) is then expressed as follows: sc R4 2 p



The distances b and b 8 are related to one another through [11]

b 84 1

k(h 1

k

2

1 k2b2_{) ,}

(9)

where k is the wave number and h 4ZPZTe2Oˇ2n2 is the Sommerfield parameter.

Equation (9) can be rewritten as b2 4

g

h

is the Coulomb potential at a distance b 8 from the centre of the target. Following the suggestion of ref. [7], we replace Vc(b 8) by its value Vcat the strong absorption radius,

so that eq. (10) becomes b2 4

g

h

where Ecm is the centre-of-mass energy. In eq. (8), changing the integration variable

into b 8 we arrive at sc R4 2 p

g

h



Comparing the last expression with Glauber’s total reaction cross-section, eq. (1), we

Fig. 2. – The Coulomb-modified proton-nucleus reaction cross-section predictions (the solid curves) compared with calculations using Glauber’s formula (the dashed curves).

arrive at sc R4

g

h

Before applying this equation to heavy-ion collisions, we check its validity on the reaction cross-sections for protons on12_C,40_{Ca and}208_{Pb, where data are available in a}

wide range of energy. We consider the target to have a Gaussian nuclear density. Then the reaction cross-section becomes

sc R4

g

h

₍

)

where x04 kp ar( 0 ) s–88, E1(x0) is the exponential integral and g 40.5772 is Euler’s

constant [9].

The results of calculations are displayed in fig. 2. As is seen from the figure, the reaction cross-section predictions without the inclusion of the Coulomb term (the dashed curves) overestimates the experimental data. Meanwhile, the comparison of the results of the modified expressions (14) (the solid curves) with the experiment shows reasonable agreement.

For low-energy nucleus-nucleus collisions, we confined ourselves, once more, to interactions between light projectiles and heavy targets. We note that Coulomb potential is evaluated at the interaction radius given by [12],

R 4rP1 rT1 fEcm21 O3(AP1 O31 AT1 O3) , where ri4

o

with rci4 1.15 Ai1 O3( 1 11.565Ai22O321.043 Ai24 O3) fm , r 40.76 fm and f41.16960.012.

Fig. 3. – The Coulomb-modified nucleus-nucleus reaction cross-section predictions. The dot-dashed curves correspond to the modified reaction cross-section for collisions involving Gaussian projectiles and uniform targets compared with the results given in ref. [7].

In fig. 3 the predictions of modified reaction cross-section given by eq. (13) are shown for nucleus-nucleus collisions induced in the energy range 100–300 MeV. We compare the results of eq. (13) in the case when the nuclear densities are Gaussian-uniform with the calculated results of Charagi et al. [7]. As seen, the present low-energy modification plus the relevant choice of nuclear densities bring the reaction cross-section predictions into close agreement with the experimental results.

3. – Conclusions

In the present article we presented simple expressions for the total reaction cross-section of a light nucleus by a heavy one based on Glauber’s multiple scattering theory in its optical limit. We assumed that the density of the light nucleus has a Gaussian radial dependence, while the heavy nucleus has a uniform density. This choice gave a satisfactory agreement with the experimental data of reactions involving heavy targets shot by light projectiles at energies above 100 MeV.

To extend the analysis towards lower energies we have to include the effective role played by the Coulomb repulsion for the calculation of the reaction cross-section. This has been done by evaluating the transparency of the target towards the projectile at the distance of closest approach of the deviated projectile trajectory. The derived Coulomb-modified reaction cross-section may be considered a generalized form of the one cited in ref. [5]. The present formula is found to be satisfactory in describing the experimental data of both proton-nucleus and nucleus-nucleus reactions. Such modification broadens the scope of applicability of Glauber model. Other physical effects not included in the present treatment, such as the internal Fermi motion of the nucleons in the nuclei and the Pauli “blocking” effects, will be considered in a forthcoming paper [14].

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