fulltext

Volume integrals and mean square radii of optical potentials

by inversion

A. Y. ABUL-MAGD(1) and M. H. SIMBEL(2)

(1) Department of Mathematics and Computer Science, Faculty of Science

United Arab Emirates University - P.O. Box 17551, Al-Ain, United Arab Emirates (2) Department of Physics, Faculty of Science, Zagazig University - Zagazig, Egypt (ricevuto il 18 Aprile 1997; approvato il 4 Agosto 1997)

Summary. — Within Glauber’s eikonal approximation, the volume integrals and

mean square radii of the real and imaginary parts of the optical potential are expressed as integrals of the parametrized phase shifts which can directly be obtained from comparison with scattering experiments. Evaluating these quantities using McIntyre-Wang-Becker S-matrix parametrizations for 12_C-12_{C scattering} shows that they are comparable with the corresponding quantities for the phenomenological Woods-Saxon potentials that fit the same data. Non-eikonal corrections are evaluated and found to be relatively small even at energies as low as 8 MeV/nucleon.

PACS 24.10.Ht – Optical and diffraction models. PACS 25.70.Bc – Elastic and quasielastic scattering.

1. – Introduction

The past decade witnessed a revival of interest in applying the strong absorption model (SAM) [1] to the analysis of angular distributions for scattering between heavy ions. The reason is the appearance of accurate measurements of the elastic scattering of several projectiles like12_{C and}16_{O at different energies in a wide range of scattering} angles [2-6]. These measurements yielded a wide variety of cross-sections ranging from the ones characterized by Fraunhofer diffraction alone up to those exhibiting rainbow scattering. The SAM represents the scattering matrix as a smooth function of angular momentum S(l) whose parameters are determined by fitting the data. However these parameters have not so far been related to the parameters of the nuclear densities or those of the nucleon-nucleon interaction. A natural extension of this model is to invert the SAM-type S-functions into the corresponding optical potentials. A substantial number of papers have been recently devoted to obtaining “optical potential by inversion” through the “two-step phenomenology” using various

parametrizations for the S-matrix and several inversion techniques (e.g. [7-14]). The most successful and yet simple functional form for S(l) is the one suggested by McIntyre, Wang and Becker (MWB) [15]. While inversion can be fully carried out quantum mechanically (e.g. [8, 9]), the WKB inversion scheme [10] has been found stable and accurate in defining the potential to small radii [11].

Unfortunately, the phenomenological optical potentials and the corresponding

S-functions obtained from direct solution of the relevant Schrödinger equations

happen to be not the same as the phenomenological S-functions resulting from the SAM analysis of the same data and the potentials obtained by solving the corresponding inverse scattering problem at fixed energy [7, 11, 12, 14]. This is due to several types of ambiguities, revealed and hidden, involved in both models [12, 13,16, 17]. Less sensitive to the optical-model ambiguities are the volume integrals and root-mean-square (rms) radii of the real and imaginary parts of the potential. Namely for this reason these quantities are considered to be particularly relevant for comparing different potentials that give equal fits to the data. Their energy dependence predicted by the dispersion relations has been frequently used to identify the “unique” optical potentials (e.g. [18]). To our knowledge, no quantitative or even qualitative analogous evaluation for the volume integrals and rms radii of the inversion potentials has been given. The present paper carries out this task for the purpose of comparison between the phenomenological potentials obtained in the optical-model analysis and those calculated by inversion of the SAM phase shifts. Section 2 uses the Glauber eikonal approximation [19] for the inversion step, to evaluate the volume integrals and rms radii of the potential in terms of the parametrized phase shifts. The Glauber formalism is known to be equivalent to the WKB approximation for the phase shifts when the interaction potential is much less than the incident energy. This condition is frequently satisfied in heavy-ion collisions. In sect. 3 we calculate the volume integrals and rms radii for the scattering of12_{C by}12_{C in the energy range from} 100 to 2400 MeV, when the scattering function S(l) is expressed by the MWB parametrization, and compare with the corresponding quantities obtained for the phenomenological optical potentials in sect. 4. In sect. 5, we estimate the first-order non-eikonal corrections and conclude this work by a summary and discussion of the results (sect. 6).

2. – Inversion using eikonal approximation

Glauber’s eikonal approximation yields a simple solution for the inverse scattering problem [19]. This has been used to obtain the optical potential by inversion of the phase shifts as early as in 1971 [20]. As a consequence of the recent revival of interest in heavy-ion potential and the success of the two-step phenomenology, new attempts to obtain optical potentials by inversion using the Glauber approximation have recently been done [21, 22]. The present paper is an extension of work in this direction. In this section, we obtain expressions for the volume integrals and root-mean-square (rms) radii of the real and imaginary parts of the optical potential in terms of integrals of the phase shifts over the impact parameter. In Glauber’s eikonal approximation [19], the amplitude for scattering from a spherically symmetric potential U(r) are expressed as

f (u) 42(ikO2p)



exp [iq Q b]

[

exp [ 2 ix(b) ] 21

]

d2_{b ,} (1)

where k is the wave number of the relative motion, q is the momentum transfer, and x(b) 4— 1 ˇ_n



0 Q U(

k

b2 1 z2) dz , (2)

in which n is the relative velocity. Equation (1) can be obtained from the WKB approximation for the scattering amplitude by using the semi-classical relation between the impact parameter b and the angular momentum l:

kb 4l1 1

2 , (3)

replacing the Legendre polynomial by its small-angle asymptotic expression, and approximating the summation over angular momenta by an integration over the impact parameter. Then, expanding the WKB expression for the phase shift d(l) in powers of

UOE, where E is the energy in the center-of-mass system, and neglecting terms of

order higher than the first yields

d(l) 4x(b) .

(4)

Expression (2) for x(b) can be regarded as an integral equation for U(r) whose solution is given by U(r) 4 ˇn 2 p 1 r d dr



r Q x(b) b db

k

b2_{2 r}2 . (5)

Equation (5) has been considered for expressing the optical potential in terms of the phase shifts obtained from the strong-absorption-model analysis of heavy-ion scattering data [20-22]. The potential thus obtained can be regarded as an approximation to the WKB inversion potential valid when the condition UOEb1 is satisfied, and the WKB approximation, as known [11], provides a stable and accurate inversion scheme.

The volume integrals per interacting nucleon pair, JR and JI, and rms radii r–R and

r–I, of the real and imaginary parts of the optical potentials, respectively, can directly be expressed as integrals of the phase shifts using the Glauber approximation. Indeed multiplying both sides of eq. (2) by 4 pb db and integrating from zero to infinity, we obtain after simple arrangements

JRf 2 1 ATAP



_{Re U(r) d}3 r 4 4 pˇn ATAP



0 Q Re x(b) b db , (6) and JIf 2 1 ATAP



_{Im U(r) d}3 r 4 pˇn ATAP



0 Q dNS(b)NOdb NS(b) N b2_{db ,} (7)

we obtain for the rms radii, in the case of spherical symmetry, r–R2f

s

r2Re U(r) d3 r

s

Re U(r) d3_r 4 6 pˇn ATAPJR



0 Q Re x(b) b3_{db ,} (8) and r–I2f

s

r2_{Im U(r) d}3_r

s

Im U(r) d3 r 4 3 pˇn 4 ATAPJI



0 Q dNS(b)NOdb NS(b) N b 4 db . (9)

The shape and parameter ambiguities are both present in the strong absorption model. This is, as recently discussed by Steward et al. [12], a consequence of the fact that the cross-section data are limited by scattering angle to span a finite range of momentum transfer. Extracting the phase shifts from the elastic-scattering differential cross-section would have been possible if the latter were known for all values of the momentum transfer. This can easily be seen from eq. (1), where the

S-function is obtained from the scattering amplitude by an inverse Fourier

transform: e2 ix(b) 4 1 2 1 2 pi



e 2iq Q b_{f (q) d}2_{q .} (10) 3. – MWB parametrization

In the following, we consider the MWB five-parameter representation for the

S-function [15] which has been widely used for the analysis of heavy-ion elastic

scattering at a variety of incident energies (see, e.g., [23]). It can be expressed for the modulus of scattering matrix element as

NS(l) N 4

y

1 1exp

y

lg2 l D

zz

21 (11)

and for the phase as

arg S(l) 42m

y

1 1exp

y

l 2l 8g D8

zz

21 . (12)

The MWB parametrization has been successfully used in ref. [7] to fit the elastic scattering of12_{C by}12_{C at incident energies ranging from 139.6 to 2400 MeV. The fits in} most cases are even better than the corresponding fits by an optical model with the six-parameter Woods-Saxon potential. We have expressed the best-fit values obtained in [7] for the angular momenta of the grazing waves lg and l 8g and the corresponding widths D and D8 in terms of the radius and diffusivity parameters R, R 8, d and d 8, respectively, through the following semiclassical relationships:

kR 4n1

k

n2_{1 (l}g1 1 O2 )2, and kd 4D , (13)

show a definite trend to decrease with increasing the incident energy, while those obtained for the diffusivity parameters show an opposite trend. However, the variation of these parameters with energy is not monotonous. We would have liked to represent the volume integrals and rms radii of the inversion potentials as smooth functions of energy in order to compare them with the corresponding quantities for the phenomenological potentials obtained in the analysis of other data as well. For this purpose, we have approximated the energy dependence of the parameters by linear functions of the relative wave number:

.

`

/

`

´

R 47.32120.1958k fm , d 40.626010.00862k fm , R 845.30320.0872k fm , d 840.558410.01879k fm , m 44.97820.1945k . (14)

Applying semiclassical relations of the type (3) to the angular momentum l, the grazing waves lg _{and l 8}g and the corresponding widths D and D8, we rewrite eqs. (11) and (12) as NS(b) N 4

k

1 1exp

k

R 2b d

ll

21 , (15) Re x(b) 4

k

1 1exp

k

b 2R 8 d 8

ll

21 . (16)

Substituting eqs. (15) and (16) into eqs. (6)-(9), we finally obtain

JR4 4 pˇnd 8 2_m ATAP



0 Q x dx 1 1ex 2R 8 Od 8 , (17) JI4 4 pˇnd2 ATAP



0 Q x2_dx 1 1ex 2ROd , (18) r–2 R4 3 2d 8 2



0 Q x3_dx 1 1ex 2R 8 Od 8

N



0 Q x dx 1 1ex 2R 8 Od 8 , (19) r–I24 3 4 d 2



0 Q x4dx 1 1ex 2ROd

N



0 Q x2dx 1 1ex 2ROd 8 . (20)

4. – Comparison with phenomenological potentials

We now compare the volume integrals and rms radii of the inversion potentials derived above for the MWB parametrization with the corresponding quantities for the phenomenological optical potentials obtained in the analysis of scattering of12_{C by}12_C

Fig. 1. – Volume integrals of the real (left) and imaginary (right) parts of the optical potentials of the scattering of 12_{C by} 12_{C obtained by inversion of phase shifts parametrized in the form} suggested by McIntyre et al. [15] without (solid lines) and with (dashed lines) non-eikonal corrections, compared with the corresponding quantities calculated for the phenomenological potentials obtained in optical-model analysis [3, 6, 11, 24, 25].

at incident energies ranging from 100 to 2400 MeV [3, 6, 11, 24, 25]. The results of the comparison are shown in figs. 1 and 2 by solid lines for the potential by inversion while the phenomenological values are shown as dots. We see from these figures that the volume integrals and rms radii of the empirical imaginary potentials are in good agreement with those obtained for the inversion of the scattering function. On the other hand, the volume integrals and rms radii of the real parts of the potentials obtained by inversion are different from those obtained for the phenomenological potentials. These conclusions agree with the results reported in [7, 11] for the comparison of the S(l) functions obtained from the SAM analysis with those calculated for potentials obtained in the optical-model analysis of the same data. In these papers, it is found that the quantity NS(l)N obtained for the MWB approach is somewhat different from the one deduced from the Woods-Saxon potential at incident energies of 240 MeV (fig. 5 of ref. [7]) and 360 MeV (fig. 1 of ref. [11]). However, the agreement improves at the higher energies 1016 MeV (fig. 7 of ref. [7]), 1449 and 2400 MeV (fig. 1 of ref. [11]). The same behavior is observed in the comparison between the imaginary parts of the inversion and phenomenological potentials (fig. 11 of ref. [7] and fig. 7 of ref. [11]). This is expected since NS(l)N is the main factor in the calculation of the imaginary volume integrals and rms radii

(

eqs. (7) and (9)

)

. On the other hand, the quantity arg S obtained from MWB parametrization severely disagrees with the corresponding quantity deduced from Woods-Saxon potential at incident energy of 240 MeV, while at 1016 MeV the disagreement is less severe (figs. 5 and 7 of ref. [7]). The same behavior is exhibited by Re V drawn for both models (fig. 10 of ref. [11]),

Fig. 2. – Root-mean-square radii of the real (left) and imaginary (right) parts of the optical potentials of the scattering of12_{C by}12_{C obtained by inversion of phase shifts parametrized in the} form suggested by McIntyre et al. [15] without (solid lines) and with (dashed lines) non-eikonal corrections, compared with the corresponding quantities calculated for the phenomenological potentials obtained in optical-model analysis [3, 6, 11, 24, 25].

again since arg S defines the real volume integral and rms radius in eqs. (6) and (8). The main outcome of this discussion is to emphasize the fact that the volume integrals and rms radii are the relevant quantities in comparing potentials deduced from several approaches with radically different starting assumptions.

5. – The non-eikonal corrections

Glauber’s theory has indeed been successfully used in describing the scattering and reaction between complex nuclei, especially when Coulomb as well as non-eikonal corrections have been introduced [26-37]. Corrections to the eikonal approximation have been available for a long time [26]. Essentially, they are given as a series expansion in powers of UOE. Recently, the Glauber model with first- and second-order non-eikonal corrections has been applied to elastic scattering of protons at intermediate energies by Faeldt et al. [33] and found to improve the agreement with experimental data. Carstoiu and Lombard [34] applied them to extend the calculation of the differential and total reaction cross-sections of heavy-ion collisions into the lower-energy regime, assuming that the optical potential has a Gaussian shape. Cha and Kim [35] extended these studies to nucleus-nucleus collisions by introducing Coulomb effects in the calculation of the phase shifts in all orders of the eikonal expansion. They found that the differential and total reaction cross-sections calculated from the first- and second-order eikonal phase shifts improved the agreement with

the experimental data and optical model results for intermediate-energy heavy-ion collisions.

This section presents an estimation for the first-order non-eikonal approximation to the volume integrals and rms radii of the real and imaginary parts of the inversion potential. We start by introducing the first-order non-eikonal correction to the phase shifts [26] by writing x(b) 42 1 ˇ_n

y



0 Q U(

k

b2_{1 z}2) dz 1 1 4 E

g

1 1b d db

h



0 Q U2

(

q

b2_{1 z}2

)

dz

z

. (21)

Multiplying both sides of this equation by ( 4 pˇnOATAP) b and integrating over b, we obtain, after solving the last integral by parts,

KR1 iKI4 4 pˇn ATAP



0 Q x(b) b db 42 1 ATAP



k

_{U(r) 2} 1 4 EU 2_(r)

l

_d3_{r .} (22)

The quantities KR and KI are defined so that, in the absence of the non-eikonal correction, they are equal to the volume integrals of the real and imaginary potentials expressed by eqs. (6) and (7), respectively.

We first consider the case when the optical potential has a Gaussian shape:

U(r) 42V exp [2r2

Oa2] 2iW exp [2r2 Ob2] . (23)

In this case, the integrals involved in eq. (22) can be calculated analytically. One then obtains KR4 1 ATAP

y

p3 O2Va3₁ (pO2) 3 O2 4 E (V 2 a3_{2 W}2b3)

z

(24) and KI4 1 ATAP

y

p3 O2Wb3₁ p 3 O2 2 E VWa3_b3 (a2 1 b2)3 O2

z

. (25)

We can also easily calculate the volume integral per interacting nucleon pair and the rms radii of the components of the potential (23). We then obtain

JR4 p3 O2_Va3 ATAP , JI4 p3 O2_Wb3 ATAP , r–R4k3 O2 a , and r – I4k3 O2 b . (26)

We use these relations to express the parameters of the optical potential in eqs. (24) and (25) in terms of the volume integrals and mean square radii, which can then be rewritten as JR4 KR2

g

3 p

h

3 O2 _ATAP 32 E

u

J2 R r–3R 2 J 2 I r–3I

v

(27)

and JI4 KI2

g

3 2 p

h

3 O2 _ATAP 2 E JRJI (r–2 R1 r –2 I)3 O2 . (28)

We can obtain similar relations for the rms radii. For this purpose, we define

b–2 Rf 6 pˇn ATAPKR



0 Q Re x(b) b3_{db ,} (29) and b–I2f 6 pˇn ATAPKI



0 Q Im x(b) b3_{db 42} 3 pˇn 4 ATAPKI



0 Q dNS(b)NOdb NS(b) N b 4 db . (30)

Substituting (21) and (23) into (29) and (30) and integrating, then again expressing the parameter of the optical potential in terms of its volume integrals and rms radii by means of eqs. (26), we obtain

JRr –2 R4 KRb –₂ R2

g

3 p

h

3 O2 _{3 ATAP} 64 E

u

J2 R r–R 2 J 2 I r–I

v

(31) and JIr–2I4 KIb –₂ I2

g

3 2 p

h

3 O2 _{3 ATAP} 2 E JRJIr–R2r–2I (r–2 R1 r–I2)5 O2 . (32)

We want to express the integrals of the potential JR , Iand r –

R , Iin terms of the integrals of the phase shifts KR , I and b–R , I which are defined so as to be equal to the corresponding integrals of the potential in the eikonal approximation. To do this, we solve eqs. (27), (28), (31) and (32) by iteration. The first iterative solution of this equation system is given by

JR4 KR2

g

3 p

h

3 O2 _ATAP 32 E

u

KR2 b–3 R 2 K 2 I b–3 I

v

, (33) JI4 KI2

g

3 2 p

h

3 O2 _A TAP 2 E

u

KRKI (b–2 R1 b –₂ I)3 O2

v

, (34) r–2 R4 b –₂ R2

g

3 p

h

3 O2 _{3 A} TAP 64 EKR

u

K2 R b–R 2 K 2 I b–I

v

, (35) and r–2 I4 b –₂ I2

g

3 2 p

h

3 O2 _{3 A} TAP 2 EKI KRKIb –₂ Rb –₂ I (b–2 R1 b –₂ I)5 O2 , (36)

Higher-order iteration will add terms of order 1 OE2 _{which will be important when} higher-order non-eikonal corrections will be taken into account.

Equations (33)-(36) are obtained for the case when the potential has the Gaussian shape (23). We have used them, however, to estimate the non-eikonal corrections for the integrals of the inversion potential which is not expected to be Gaussian. We believe that these estimates are useful as far as the corrections themselves are small and the interacting pair of nuclei are not heavy.

In figs. 1 and 2, we show the result of calculation of the volume integrals and root mean square radii of the potential obtained by means of eqs. (33)-(36) for the inversion of McIntyre’s S-function with parameters given above. We see from these figures that the corrections are not large even for the smallest energy considered where they contribute by about 10%.

6. – Summary and conclusions

We have applied the Glauber eikonal approximation to express the volume integrals and rms radii of the optical potential as integrals over the scattering function. We consider the parametric representation for S(l) suggested by McIntyre et al. [15]. Our calculations show that the real part of the inversion potential is more sensitive to this shape ambiguity of S(l) than the imaginary part. The volume integrals and rms radii of the imaginary parts of the potentials obtained for the scattering of 12C by 12C by the present two-step phenomenology are consistent with the corresponding quantities obtained in the optical-model analysis of the same data, as well as other data for the same system.

First-order non-eikonal corrections to the volume integrals and rms radii have been evaluated, first by considering an optical potential with Gaussian radial dependence and expressing the corrections in terms of the expressions obtained in the eikonal approximation, and then assuming that the resulting expressions are valid at least for the collision of the light nuclei under consideration. Calculation with the McIntyre parametrization shows that the non-eikonal corrections are about 10% at 100 MeV and decrease with increasing the incident energy and that they are practically negligible beyond 1000 MeV.

Coulomb corrections are also not expected to have strong effect on the results obtained in this paper. Following Vitturi and Zardi [29] for example, we consider the integration in eq. (2) for the phase shifts as being evaluated along the tangent line to the Coulomb trajectory at the distance of closest approach r0

I of the l-partial wave defined by

kr0

I4 n 1

k

n21 (l 1 1 O2 )2.

The impact parameter b in eq. (2) will be replaced by the distance of closest approach

r0

l . With this approach, Vitturi and Zardi [29] as well as several other authors have

been able to give a satisfactory description for the elastic scattering between heavy ions starting from the Glauber phase shifts. In this case, the inversion potential will still be calculated by means of eq. (5) with the notation for the integration variable, b, changed into r0

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