Optical-potential description of antiproton-nucleus scattering at 46.8 MeV

Nuclear Physics @ North-Holland

A435 (1985) 708-716 Publishing Company

OPTICAL-POTENTIAL DESCRIPTION OF ANTIPROTON-NUCLEUS SCATTERING AT 46.8 MeV K.-I. KUBO and

H. TOKI

Department of Physics, Faculty of Science, Tokyo Metropolitan University, Tokyo 158, Japan

Fukasawa, Setagaya-ku,

and M. IGARASHI Depariment of Physics, Tokyo Medical College, Shinjuku, Tokyo 160, Japan and Institute for Nuclear Study, University of Tokyo, Tanashi 188, Japan Received 4 June 1984 (Revised 17 August 1984) Abstractr Optical-potential analyses of the p-‘*C scattering at 46.8 MeV are presented and the existence of the continuous-parameter ambiguity is demonstrated. The elastic p-polarization is calculated and compared with the p-polarization at the same energy. The impact parameter of the antiproton absorbed is obtained from the eikonal model using the NN two-body forward-scattering amplitude.

1. Introduction The first clean antiproton-nucleus

(PA) scattering data at 46.8 MeV have recently

been reported by Garreta et aI. ‘) using the LEAR facility at CERN. The angular distribution of antiproton elastic scattering shows a sharp diffraction minimum at very forward angles, whereas that of proton scattering at the same energy shows a smoothly decreasing behaviour. This comparison indicates a long-range and strongly-absorptive potential for the antiproton scattering. In fact, Garreta et al. have reported the potential set for which the imaginary part ( W = 61 MeV, rh = 1.2 fm, u’ = 0.51 fm) is deeper than the real part ( V = 25 MeV, r,, = 1.17 fm, a = 0.61 fm). These potential parameters were found in their coupled-channel (12C O+-2+) calculation, and the potential is thus called the bare potential. They also pointed out the existence of the so-called parameter ambiguity that is well known in composite particle scattering, typically a-scattering, from a nucleus ‘). In a recent p-atom data anslysis, such an ambiguity was also reported by Wong et al. ‘) who found two families of the PA potential. Pioneer works on the elastic p + A scattering had already been performed before the LEAR data came out 4,5).The strong annihilation character of the RN interaction at short distance was taken into account in their pN t-matrix calculation. The resulting p+A potential by folding the t-matrix with the nuclear density predicts a larger 708

709

K.-I. Kubo et al. 1 Optical-potential description

strength for the imaginary part compared with that of the real part. The real part of the potential decreases with energy in contrast with that of the p + A potential, due to the sign change of the 7~- and w-meson exchange contributions by the G-parity transformation. Elastic polarization was also discussed by noting the destructive contribution of the u- and w-meson exchanges in the I*S potential ‘), whereas they are constructive in the case of the p+A potential. All these results are very interesting and it is worthwhile to know what are the qualitative and possibly quantitative aspects of PA scattering. We have now the first PA data available in order to quantify more prcecisely the theoretical discussions. In this article, we shall report the resuits of optical-potential analyses of p-‘*C scattering and demonstrate the existence of the continuous-parameter ambiguity. The elastic p-polarization is also calculated and compared with the p-polarization at the same energy. The radius where the p is absorbed is discussed based on the eikonal model employing the RN two-body forward-scattering amplitude. 2. Potential-parameter

ambiguity

In the present parameter search, the bare potential parameters reported by Garreta et al. I), which had been found in their 12C(O+-2+) coupled-channel calculation for the experimental data existing up to 65”, were adopted as a starting parameter set. The optical-potential parameters obtained by the present x2 fit are given as the first numerical set in table 1. TABLE p+ “C optical-potential

46.3 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0 120.0 130.0 140.0 150.0 200.0 300.0

0.940 1.17 1.03 0.912 0.817 0.740 0.676 0.621 0.572 0.528 0.490 0.454 0.421 0.391 0.267 0.0985

0 608 0.493 0.562 0.614 0.649 0.674 0.69 1 0.705 0.715 0.724 0.73 1 0.738 0.743 0.747 0.764 0.78 1

1

parameters at incident energy 46.8 MeV

82.7 62.5 74.1 85.5 94.2 100.4 105.1 108.9 112.1 114.8 117.0 119.0 120.9 122.3 128.2 134.8

1.05 1.11 1.07 1.03 1.01 0.988 0.973 0.962 0.953 0.945 0.939 0.933 0.928 0.924 0.909 0.893

0.623 0.63 1 0.625 0.625 0.627 0.629 0.630 0.632 0.633 0.634 0.635 0.636 0.636 0.637 0.639 0.641

The Coulomb radius parameter r, is fixed at 1.3 fm for each case.

78.9 63.2 56.7 57.9 61.3 65.3 70.2 74.4 78.3 81.7 84.8 87,6 90.1 92.3 100.0 109.2

643 641 642 644 645 647 648 649 649 650 650 651 651 652 653 654

1028 1027 1029 1030 1030 1030 1032 1033 1033 1034 1034 1035 1035 1035 1036 1037

710

K.-I. Kubo et al. / Optical-potential description

The parameter ambiguities were then studied in the following way. First, the O”-180” elastic-scattering differential cross section was calculated by using the parameter set obtained above. Then an automatic parameter search was performed by fitting each calculation to this first result. The real depth parameter V was fixed to each value ranging from 30 to 300 MeV with a 10 MeV step and all other parameters were made free. The x2 minimum was quickly attained. The differential cross sections thus obtained are shown in fig. la by solid curves; there are sixteen curves shown, which are almost identical up to 180”. The strongly oscillating shape of the fi angular distribution is quite different from the smoothly-decreasing angular distribution of the p-scattering. The potential shapes are shown in fig. 2. The imaginary depth W increases with the real depth V. The strength and the range of the imaginary potential are larger than the corresponding ones of the real potential, which is consistent with the feature of strong annihilation in RN scattering. The parameter ambiguity thus found is that of the so-called continuous ambiguity typically seen in (Y+ A scattering ‘). The p+ ‘*C potential at the same energy is also shown in fig. 2, which is the best-fit parameter set found by Becchetti and Greenlees “)_ It is worthwhile to note that, as compared to the p+ “C potential, the range of the real part of p + ‘*C is shorter, while its imagina~ part extends over the samll density surface region. We shall study now the sensitivity of the inelastic scattering differential cross section to the above-obtained parameter sets. The cross section leading to the 4.43 MeV 2+ excited state in ‘*C was calculated in the framework of first-order DWBA with a complex collective form factor. The result is shown in fig. 3. The shapes of the angular dist~bution are again almost identical for the various parameter sets, but the magnitudes change by a factor of 2. It is interesting to compare the (fi, 0’) cross section to the (p, p’) cross section: (i) the (ii, a’) oscillates more sharply, (ii) the (p, p’) peak cross section is smaller only by a factor 2-3 in all over the angular range, and (iii) the p elastic and inelastic angular distributions show an out-of-phase relation in oscillation, which is already known as the Blair phase-rule in the case of cr + A scattering. The evaluated p and p cross sections are normalized using the same deformation parameter & = 0.6 as was found in the (p, p’) excitation analysis ‘) within first-order DWBA. The (p, p’) experiment ‘) shows a peak cross section 5.3 f 1.7 mb/sr at the angle 32.5”. It is then seen from fig. 3 that the j?~cross-section magnitudes thus normalized sit within the range of the experimental value. This means that the first-order DWBA treatment of the (p, p’) can provide consistent information in nuclear structure studies as does the (p, p’) DWBA. In ref. ‘), the coupled-channel (O+-2+) calculation was performed using the same pZ value as used here and very similar results were obtained. These facts indicate that the optical model and the first-order DWBA are enough to analyze the p- ‘*C elastic and 2+ inelastic scattering, although again too large a deformation parameter is found to be necessary as was argued in ref. ‘).

ELASTIC

SCATTERING

y

FROM 12C AT 46.8

10-3 t 0 30 100 -+--1-80

60

90

MeV

120

150

180

b

60 R

40

z 20 0 t ?I 0 E aj --20 I? -40

Fig. 1. Calculated elastic scattering of p and p from the ‘*C target at 46.8 MeV. The dot-dash cruve is for the p-scattering. (a) The differential cross sections. For the p-scattering, the sixteen results corresponding to the parameter sets given in table 1 are presented. The p experimental data are taken from ref. ‘). (b) The polarization angular distributions. The six different results are shown for the p-scattering, with three different optical-potential parameter sets (A: V = 30 MeV; B: V = 60 MeV; C: V = 100 MeV) and with two different spin-orbit strengths (solid curve: Vls = -6 MeV; dashed curve: V, = -3 MeV). The proton V, is 6 MeV.

K.-I. Kubo et aL / Opiical-potential description

712 10

-20 -30 -40 s

-50

z ”

-60

AI ++ -70 5 n

-80

m -90 .+ c -100 al c O-110 a -120

----------

p

imaginaly

-------

p

real

~~....~~~~~~~ p

-130

part

part

imaginaly

part

-140 -150 -160 -1 70

0

1

2

3

r

4

5

6

(fm>

Fig. 2. Real and imaginary potential shapes of the fifteen parameter sets ( V = 30-300 MeV) given in table 1. The p + ‘*C potential at 46.8 MeV is also shown, which is the best-fit set found in ref. 6).

The microscopic effective two-body interaction between an antiproton and a nucleon has been investigated by several authors *), and some marked differences of its spin and isospin dependence from proton scattering have been predicted. Hence, it is very interesting to perform inelastic excitations of various states in order to determine the effective pN interaction. In this respect, we want to stress here that one advantage of the p beams for nuclear interaction studies is no appearance of the exchange term, which usually makes the theoretical analysis very complex in case of proton scattering. The above-mentioned large differences beween p and p scattering are also remarkable in the polarization angular distribution. The calculated results are shown in

713

K.-I, Kubo et ai. / Optical-potential description

,(y3L'

I I

I I

II/_-

0 3o

&.M.g:degr;::

150

'*O

Fig. 3. Inelastic scattering leading to the 4.43 MeV 2+ excitation in 12C at 46.8 MeV. The collective excitation mode1 is employed with the complex form factor. The deformation parameter & = 0.6 is used for the normalization. The experimental data are taken from ref. ‘).

fig. lb. For the spin-orbit potential, the Thomas form was assumed with real central geometrical parameters. The following two cases were tested for spin-orbit strength by considering the destructive contribution of u- and o-meson exchanges: compared with the proton spin-orbit strength (6 MeV), (i) the same strength but opposite sign (-6 MeV), and (ii) the half-strength (-3 MeV). The results are shown for three different optical-potential parameter sets corresponding to V = 30,60 and 100 MeV in table 1. The p-polarization changes rapidly with angle, which is quite different from the p-polarization at the same energy, reflecting the sharp diffractive oscillation of the p elastic differential cross section. Note that the polarization becomes large at angles of 60”, 100” and 150”, where the p+ “C cross section shows a peak. Furthermore, these three angles are unchanged for different spin-orbit potentials and different sets of the optical potential. These three angles are then posible candidates for the p+ “C polarization measurement at the incident energy about 50 MeV.

714

K-I.

Kubo et al. / Optical-potential description

3. Antiproton absorption radius

We shall now try to see the physics of the antiproton-nucleus scattering. At low energy, the antiproton-nucleon cross section is dominated by strong annihilation. Hence, the FN scattering amplitude is almost pure imaginary. The essential cause of distortion on the antiproton by the nucleus is therefore absorption, where the loss of the antiproton flux, while going through the nuclear medium, provides the diffraction pattern. The most economical way to take into account this absorption effect is to assume a linear trajectory due to the negligible or, if any, weak real potential and to calculate the loss of the flux as a function of the impact parameter b. Since the essential assumption in the eikonal approximation is the use of linear trajectory for the incoming particle, we shall take the eikonal prescription, although the energy here is very low. Although the antiproton energy considered here is low, the momentum is about 300 MeVjc, which corresponds to the pion momentum in the resonance-energy region, the scattering of which with the nucleus has been demonstrated to be described well with the eikonal approach9). There might be some doubt regarding the validity of the eikonal approximation for this low-energy p-nucleus scattering, though we want to emphasize here that the purpose of this section is to get the physical view of antiproton-nucleus scattering at the forward scattering angles. The scattering amplitude F( 6) for antiproton-nucleus scattering is written in the eikonal approximation as lo) a0 F(8)=-&

db bS(b)~o(~~),

s0

(1)

where Jo is the zeroth-order Bessel function, q is the momentum transfer and k the antiproton wave number in the laboratory frame. The source function S(b) as a function of the impact parameter b is S(b)=exp(X(b))-1

,

(2)

where CD X(b)

= i

5 -m

dz [KMr))

- kl .

(3)

Here the momentum K in the nuclear medium is given by the refractive index theory, y

= 1

+$p(r)f(o) ,

where p(r) is the nuclear matter density at position I: f(O) is the forward p-nucleon scattering amplitude in the laboratory frame. At low energy, the absolute value of f(0) is very large and is essentially pure imaginary “). Hence exp (X(b)) = 0 for b > R (nuclear radius) and exp (X(b)) = 1 for b < R. Exp (X(b)) changes rapidly in the narrow range just outside of the

K-I. Kubo et al. / Optical-potential description

715

nucleus, where the antiproton gets scattered by the nucleus without being absorbed. We define the most interesting trajectory in terms of the impact parameter Rabs through the condition lexp (X(RA)]

=$ .

(5)

Assuming a Woods-Saxon density distribution with the parameters (central density) p0 = 0.17 fme3, (half-density radius) c = 2.56 fm and (diffuseness) a = 0.5 fm, and using the total cross section o,,, = 235 mb [refs. “,‘*)I, we find Rabs= 3.7 fm .

(6)

This is far beyond the half-density radius c, where the density p( Rabs) = O.l02p,,. Hence the antiproton elastic and inelastic scattering are only sensitive to the nuclear structure at large distance. Certainly, this large value for Rabs is related to the large imaginary part of the optical potential, which has a large radius as shown in the previous section. The eikonal description is further developed to provide the differential cross section and the total cross section lo):

~=(kR.b.)2[J1(qR.b.)/s+aJO(qRabr)10.21+ Y*)ll*, (7) u,,, = 27rRib, + 4raR,,,,{0.2 1+4 In ( 1 + Y*)} ,

(8)

where Y=[ReK(p)-k]/ImK(p).

(9)

The first term in each expression is due to the strong absorption of the antiproton by the nucleus, while the second term is caused by contributions from the diffused region. The second term contributes only by less than 10%. If we take Rabs obtained above and calculate the position of the first minimum from eq. (7) and the total cross section from eq. (S), they are e,.,.( 1st min) = 45”)

a,,,=917mb.

Empirically, the angle is 40” [fig. 1, ref. ‘)I. We find here qualitatively a satisfactory result with the eikonal approach. If we were to push further the validity of the eikonal approximation and try to extract the effective Rabs from the angle of the first minimum, it comes out to Rabs (eff) = 4.05 fm. It provides a,,, = 1030 mb, which is very close to the value obtained from the optical-model calculation (table 1). Although the validity of the linear-trajectory assumption needs to be quantified, it is interesting to conjecture that this slight increase of the absorption ‘radius’ by 0.35 fm might be partially caused by the finiteness of an antiproton of the order of 1 fm, where the eikonal approximation assumes structureless particles.

716

K.-I. Kubo et al. / Optical-potcnt~l description

4. Summary An extensive study of the antiproton-nucleus optical potential has been reported. Existence of the continuous parameter ambiguity has been demonstrated by showing the potential shape and the differential cross section for ij - 12Celastic and inelastic scattering. It has been suggested that the angles 60” and 100” may be candidates for the p elastic polarization to be measured around 50 MeV incident energies; these angles are unchanged for the different optical-potential parameters once they are fixed for the elastic differential cross section. It was pointed out that the first-order DWBA gives consistent information for the 12C2+ excitation with the (p, p’) DWBA and with the @coupled-channel calculation reported previously by Garreta et al. ‘). As the physics of the antiproton-nucleus scattering is demonstrated in terms of the eikonal approximation, the antiproton being scattered by a nucleus can only see far outside of the nucleus where the density is of the order of &pO. Hence, the low-energy antiproton can be used as a powerful tool to investigate nuclear structure at very low density.

The authors wish to thank K. Nakamura for communicating the experimental data. The numerical calculation was performed by using the FACOM M-380 computer facility at the Institute for Nuclear Study, University of Tokyo.

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