THE NUCLEAR SURFACE

HAL Id: jpa-00216354

https://hal.archives-ouvertes.fr/jpa-00216354

Submitted on 1 Jan 1975

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

THE NUCLEAR SURFACE

D. Jackson

To cite this version:

D. Jackson. THE NUCLEAR SURFACE. Journal de Physique Colloques, 1975, 36 (C5), pp.C5-1-C5-

6. �10.1051/jphyscol:1975501�. �jpa-00216354�

THE NUCLEAR SURFACE

D. F. JACKSON

Dept. of Physics, University of Surrey, Guildford, U.K.

Abstract. — Nuclear scattering and reactions which give information on the nuclear surface are described. These processes include elastic scattering of electrons and of high energy protons, pion reactions with nuclei, and scattering of alpha-particles and heavy ions. The interpretation of the experimental data is examined in order to show which regions of the nucleus and which moments of the density distribution are determined by a particular method. The results are compared with theories of the nuclear surface.

1. Introduction. — The study of the nuclear surface has two purposes : (i) to provide parameters which may be used in calculations of more fundamental properties or in analyses of other data; (ii) to test theories of the nuclear ground state.

The one-particle density distributions p

p

(r), pjr), for the protons and for the neutrons are density dis- tributions for point particles in the nucleus. The matter distribution is

with the normalization

It is not easy to define the nuclear surface precisely.

In terms of the simple fermi distribution with para- meters c, a we may take the surface to be the transition region t = 4.39 a between the 90 % and 10 % density points and we may also take the extreme surface or periphery of the nucleus to be the region beyond the halfway radius R = cA

1/3

. For distributions derived from models which incorporate the effects of shell structure it is more difficult to define a sur- face region precisely because of the fluctuations in the inner region. However, when these distribu- tions are folded with the nucleon charge distribution to give the nuclear charge distribution p

^ch

or are folded with an effective interaction to form an optical potential, the amplitude of the fluctuations are damped. The folding also has the effect of increas- ing the diffuseness and the r.m.s. radius.

There are several methods of obtaining these dis- tributions.

(i) Phenomenological methods in which we choose a functional form with certain parameters, e.g. the fermi distribution, parabolic fermi, etc.

(ii) Model-dependent methods in which the distri- bution is generated using a suitable nuclear model, e.g. single-particle model with a local state-inde- pendent potential, single-particle model with state- dependent or non-local potential, self-consistent model such as HF or D D H F , energy density formalism.

Once the parameters of* the model have been deter- mined, the density distribution of the nucleus is predicted without any free parameters.

(iii) Model-independent methods. Both of the preced- ing methods suffer from the defect that different regions of the distribution are linked through the functional form or the model calculation. Conse- quently, it is not always possible to decide whether a particular feature is necessary to reproduce the data or whether it arises merely from the model.

During recent years, methods have been developed to treat electron scattering and muonic X-ray data in a model-independent way [1, 2, 3, 4].

2. Proton distribution in the nuclear surface. — Electromagnetic methods provide very direct infor- mation on the nuclear charge distribution. The prin- ciple source of uncertainty in electron scattering, apart from that arising from the experimental measu- rement, is due to the restriction of the experiment to a certain maximum value of the momentum transfer.

If we consider the Born approximation, for simplicity,

JOURNAL DE PHYSIQUE

Colloque C5, supplément au n° 11, Tome 36, Novembre 1975, page C5-1

Résumé. — On décrit la diffusion et les réactions nucléaires qui nous renseignent sur la surface des noyaux. Ces événements comprennent la diffusion élastique des électrons et protons de haute énergie, les réactions des pions, et la diffusion des particules alpha et des ions lourds. L'interpréta- tion des mesures expérimentales est analysée pour montrer quelles régions du noyau et quels moments de la distribution de matière nucléaire ou de charge sont à déterminer par chaque méthode. On compare les résultats aux prédictions des calculs théoriques.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1975501

C5-2 D.

F.

JACKSON

the cross-section is proportional to the square of the form factor Fch(q) which is given by

Thus the experiment provides information on the Fourier components of pch with wavelength

Higher momentum components and shorter wave- lengths are not measured. In the model-dependent methods these components are specified by the model prediction for p,,. The model-independent methods are based on the concept that there is limited reso- lution in r-space, so that if p,, is zero beyond r

=

R the space can be divided into N resolved regions of width Ar where

Within Ar the experiment gives no information about the smoothness or otherwise of pch so that it may be represented by expansions of suitable functions. An example of the results of this type of analysis of electron scattering [5] is shown in figure 1. The data fitted have qmax= 2.74 fm-l.

FIG. 1. - Nuclear charge distribution for 208Pb obtained from a model independent analysis of electron scattering data with

qmax = 2.74 fm-' [5].

In analyses of this type the moments of the charge distribution, defined as

are also determined together with the uncertainties.

Results obtained [I] for '08Pb are shown in figure 2.

In general, electron scattering up to -- 250 MeV determines the low moments of the charge distribution.

In particular, ( r2 ) is determined if reasonable assumptions are made about the form factor for

FIG. 2. - Moments M(k) of the nuclear charge distribution for '08Pb obtained from a model-independent analysis of electron

scattering data with q,,, = 1.6 fm-' [I].

q < qmin. However, ( r2 ) is determined more preci- sely for light and medium mass nuclei by muonic X-ray measurements [6].

The model-independent method is lengthy and difficult. Consequently, relatively few analyses have been carried with emphasis on spherical nuclei.

Nevertheless, this method reveals that the surface region is quite well-defined and that single-particle and HF models reproduce the surface behaviour quite well. There is essentially no knowledge of -the very low density region and evidence of failure of HF theories in the interior region [5, 71. With the variety of methods now used, it becomes difficult to quote parameters for the surface thickness, but parameters of charge distributions have recently been tabu- lated [6, 81.

It is generally assumed that the determination of p,, immediately gives p,, but in certain cases such as the calcium isotopes, the contribution to the charge distribution from. the electron-neutron interaction can be important [9].

3. Matter distribution in the nuclear surface.

-

The nuclear matter distribution can be investigated through the elastic scattering and reactions of a variety' of strongly interacting projectiles.

For medium energy nucleon scattering, up to

- 60 MeV, it is now clear that scattering determines the low moments of the potential, i.e. the volume integral and the r.m.s. radius, but the connection between these quantities and the properties of the matter distribution is difficult to establish owing to uncertainty about the effective interaction and the contribution of exchange terms.

The simplest information available consists of reac-

tion and total cross-sections for high energy pro-

jectiles. Pion reaction cross-sections in the energy

region 0.5-2.0 GeV behave very accurately as A2I3 [lo],

which suggests that the interaction occurs in the

nuclear surface. A student of mine has recently exa-

mined this A-dependence in more detail [I I], using

the high energy approximation and an optical poten-

tial for n* scattering from nuclei of the fo'rm

THE NUCLEAR SURFACE C5-3

where the (~(np) are the pion-proton total cross-

sections averaged over the fermi momentum of a proton in the nucleus. In view of the results of Allar- dyce et al. [lo], discussed in section 4, we took pn(r)

^E

pp(r). In order to reproduce the A-dependence of the reaction cross-sections it is essential to use a matter distribution which has a diffuse surface. Results for a ferrni distribution with fixed parameters are shown in figure 3. Othef pairs of fermi parameters c, a

FIG. 3.- A-dependence of the n+ reaction cross-section at 1 .O GeV/c calculated with a fixed parameter fermi distribution [l 11.

were found which fitted the reaction cross-sections

.

on 12C and '08Pb and the moments of these distri- butions were calculated. Figure 4 shows that these distributions have a common moment which differs for light and heavy nuclei. The parameters of the distributions for '08Pb are given in Table I.

Parameters of the fermi distributions for '08Pb whose moments are shown in figure 4

We also found several other distributions for '08Pb which gave agreement with the reaction cross-

FIG. 4. - Moments M(k) of nuclear matter distributions of fermi shape which fit the n+ reaction cross-sections for '08Pb and 12C at 1.0 GeV/c [ll]. The parameters of the distributions for '08Pb

are given in Table I.

section. These are shown in figure 5. Curve A is the fermi distribution with a

=

0.55 fm, curve B differs from it only inside 2 fm and gives almost identical moments. Curve C gives higher values for the low moments up to k

=

5 ; its important characteristic is a smaller effective diffuseness and not the removal of the interior. Curve D gives moments which are consistently smaller than those for the fermi distri- bution by about 4 % and the values do not converge.

Thus, as has been noted [12] in the case of muonic X-rays, plots of moments can be misleading if the calculations are restricted to fermi distributions. By varying the distributions at large distances we found that changes beyond 8 fm in '08Pb were quite inef- fective but changes beyond 7 fm introduced a substan- tial change in the cross-section. We conclude that pion reaction cross-sections in this energy range are sensitive only to the transition region.

Neutron total cross-sections in the energy range 1-30 GeV vary approximately as This suggests that these data may be a little more sensitive to the interior, and an analysis by Franco [13] suggests that in heavy nuclei the main contribution comes from the region around R-a.

Total cross-sections for photon absorption by nuclei in the energy region 2-20 GeV behave as [14].

Coherent photoproduction of rho mesons has been

studied in the vector dominance model using, the

a rho-nucleus optical potential of the form

C5-4 D. F. JACKSON

deduce a slightly lower value of a,,. The nuclear r.m.s.

radii obtained by this method exceed the values by other methods by - 0.35 fm for C and 0.15-0.25 for Pb. In view of the many simplifying features of the theory it would seem more profitable to insert known matter distributions and obtain a more reliable estimate for o(pN).

In a simple impulse approximation with a high energy approximation Kolbig and Margolis 1181 have shown that the cross-section for coherent particle production

FIG. 5. - Nuclear matter distributions for '08Pb which fit the nf reaction cross-section at 1.0 GeV/c [l 11. See text for details.

where /? is the ratio of the real to imaginary part of the rho-nucleon forward scattering amplitude and of(pN) is the total rho-nucleon cross-section modified to take account of the double-scattering contri- bution [15]. Alvensleben et al. [16] have taken p,(r) to be a fermi distribution with a

=

0.545 fm-I and

k

⁼

cA1J3 for all nuclei. By varying c to fit the data on thirteen nuclei between Be and U they find c

=

1.12 f 0.02 fm and a,, - 26 mb. Codding- ton et al. [17] use the same parameters for p, but

0

5

10

b ( f m )

FIG. 6 . -The integrand of the effective nucleon number N(A, o,, uc) for '08Pb calculated with a single-particle model for the nuclear

matter distribution [19].

is proportional to the square of the effective nucleon number N(A, 4 o,, 4 a,) while the lowest term in the cross-section for incoherent production is propor- tional to N(A, a,, o,) where a,, a, are the total cross- sections for the particle-nucleon interactions and N(A, o,, a,)

=

(o, - go)-' S [e-aaT(b)

^-

^e-~cT(b)] - ^{d7-b (9)}

Figure 6 shows the integrand for N(A, o,, o,) as calculated by Auger and Lombard [19] for '08Pb using a single-particle model for p,. This shows that photoproduction, a,

=

0, probes the whole volume of the nucleus while other production process are restricted to the outer regions.

There is clearly a limit to what can be learnt from total cross-sections, and particle production in parti- cular is subject to several uncertainties. The new data at 1 GeV which include both differential cross-section and polarization measurements should provide more detailed information. It appears that known distri- butions reproduce the data [20], but it would be very interesting to see a model-independent analysis at this energy to determine precisely what region of the nucleus is probed.

4. Neutron-proton differences in the surface.

-

The data on pion reaction cross-sections have been used to study the differences between p, and p, in the surface [lo]. The ratio of the

z-

to x+ reaction cross-sections has been calculated using the optical potential defined in eq. (6) together with a real part.

The distributions which give agreement with the data have very small or zero differences in the r.m.s.

radii for the proton and neutron distributions. For light nuclei this result is consistent with the predic- tions of DDHF but for heavy nuclei there is- a slight discrepancy. Figure 7 shows the results for various distributions for 208Pb and Table I1 gives the r.m.s.

radii of these distributions.

Microscopic studies of the charge-exchange reac-

THE NUCLEAR SURFACE C5-5

R.m.s. radii of the proton and neutron distributions for '08Pb used for the calculations shown inJigure 7 ( r2 ):I2 ( r2 ):I2 ⁽ r2 ):I2 - ( r2 ):I2 ( r2 ):I2/( r2 ):I2

Distribution (fm) (fm) (fm> (fm)

- - -

F 5.42 5.42

HYD 5.44 5.39

ZD 5.44 5.43

NEG 5.37 5.60

BG 5.44 6.06

CT ( l l - 1 arises from different choices for the effective inter-

FIG. 7. - The ratio of the reaction cross-section for n- and n+ on 208Pb. The moments of the distributions are given in Table 11.

tion have also been used to compare p, and p, because the form factor for the transition is given by

where U, is the isospin-dependent part of the nucleon- nucleon interaction. Schery et al. [21] have obtained a very accurate angular distribution for the '08Pb (p, n) reaction with 25.8 MeV protons. They analyse the data using fermi distributions and find a,

=

0.44 fm,

R,

=

7.02 fin and ( r2 ),I( r2 ),

=

1.07 + 0.03 which is not consistent with the results from pions. However, good agreement with the same datacan be obtained [22]

with ( r2 ),

=

( r2 ), if a complex form factor is derived .from a density-dependent interaction. This indicates yet again the uncertainty arising in analyses of this type because the effective interactions are not precisely known.

Alpha-particle scattering has been used by many authors [23-281 to deduce values for ( r2 ), - ( r2 ),.

Results are given in Table 111. The variation probably

action or different treatment of the energy depen- dence of this interaction.

Diflerences in r.m.s. radii of neutron and proton distributions obtained fvom a-particle scattering

Target mass Reference

number ⁽ rZ );I2

-

( r2 ):I2

5. The extreme surface.

^-

It is by now well- known that the scattering of a-particles and heavier projectiles is sensitive to the region of nucleus beyond the halfway radius. For energies near the Coulomb barrier the essential parameters are the barrier radius and the nuclear potential at that radius [29, 301.

Medium energy scattering determines the strong absorption radius and the potential at this radius [29, 311, or alternatively the radii at which the poten- tial has some small fixed value [32]. Radii defined in this fashion can be expressed as proportional to

All3

with coefficients which are much larger than the one for the halfway radius of the matter distribu- tion [29, 321. Nevertheless, the results can be repro- duced with microscopic folding models using known forms for the matter distribution [33]. Because of the folding procedure, contributions to the poten- tial at the strong absorption radius come from regions of the nucleus at somewhat smaller distances but not from the interior [25]. For this reason, it is possibly not very meaningful to discuss the comparison of neutron and proton distribution by this method in terms of the r.m.s. radii, as is done here in Table 111.

The very low density region may be studied most

effectively through the properties of hadronic atoms

but there are still many problems to be solved in the

construction of the hadron-nucleus potential.

C5-6 D. F. JACKSON

6. Conclusion.

-

The regions of the nucleus probed

15 ,Electron

-

by various processes are now understood in many

scatferlng

cases. These regions are indicated in figures 8 and 9

<--Total C~OS;-secttons react~on

-

I

^[ectron scattering ⁴

a p a r t i c l e + scattering

\

t K a o n i c - - a t oms

<--Kaon~c--

atoms

/

FIG. 9. - The ratio of neutron density to proton density predicted by a single-particle model for lZOSn. The region of the nucleus

probed by various processes is indicated.

in comparison with single-particle density distribu- tions which are in reasonable agreement with the data. These figures show that it is possible to have ( r2

)p =

< ^r2

ⁿ⁾

and still have a large neutron excess

0 2 4 6 8

in the extreme surface region, as required by the data

r Ifml

on hadronic atoms. Hence there need not be any

FIG. 8. - The nuclear matter distribution for lZ0Sn predicted

conflict between the interpretation of different pro-

from a single-particle model. The region of the nucleus probed ^CeSSeS

but we must regard the they

by various processes is indicated.

provide as complementary.

References

[l] FRIEDRICH, J. and LENZ, F., Nucl. Phys. A 183 (1972) 523.

[2]- SICK, I., Phys. Lett. 44B (1973) 62; Nucl9hys. A 218 (1974)

509. ^-

[3] BORYSOWICZ, J. and HETHERINGTON, J. H., Phys. Rev. C 7 (1973) 2293.

[4] FRIAR, J. L. and NEGELE, J.'w., Nucl. Phys. A 117 (1968) 575.

[5] DREHER, B., FRIEDRICH, J., MERLE, K., ROTHAAS, H. and LUHRS, G., Nucl. Phys. A235 (1974) 219.

[6] BARRETT, R. C. and JACKSON,-. F., Nuclear Sizes and Struc- ture (Oxford University Press) to be published, Chapter 6.

[7] TURCK, S., BELLICARD, J. B., FROIS, B., HUET, M., LECONTE, Ph., PHAN XUAN HO and SICK, I., Contribution to this Conference.

[8] DE JAGER, C. W., DE VRIES, H. and DE VRIES, C., Atomic and Nuclear Data Tables 14 (1974) 479.

[9] BERTOZZI, W., FRIAR, J. L., HEISENBWG, J. and NEGELE, J. W., Phys. Lett. 41B (1972) 408.

[lo] ALLARDYCE, B. W. et al. (Birmingham-Rutherford-Surrey Collaboration), Nucl. Phys. A 209 (1973) 1.

[Ill EARLE, N., University of Surrey B. Sc. Project (1975), unpu- blished.

[12] BARRETT, R. C., Phys. Lett. 33B (1970) 388.

[13] FRANCO, V., Phys. Rev. C 6 (1972) 748.

[14] WEISE, W., Phys. Rev. Lett. 31 (1973) 773.

[I51 MONIZ, E. J. and DIXON, G. D., Phys. Lett. 30B (1969) 393.

[16] ALVENSLEBEN, H. et al., Phys. Rev. Lett. 24 (1970) 786 and 792.

[17] CODDINGTON, P., ATKISS, M., BRODBECK, T. J., LOCKE, D. H., MORRIS, J. V., NEWTON, D. and SLOAN, T., preprint.

[18] KOLBIG, K. S. and MARGOLIS, B., Nucl. Phys. B 6 (1968) 85.

[19] AUGER, J. P. and LOMBARD, R. J., Phys. Lett. 45B (1973) 115.

[20] THIRION, J., Proceedings of the Uppsala Conference on High

Energy Physics and Nuclear Structure (ed. G. Tibell) (1973) 168.

ALKHAZOV, G. D. et al., ibid. 176.

[21] SCHERY, S. D., LIND, D. A. and ZAFIRATOS, C. D., Phys. Rev.

C 9 (1974) 416.

[22] FRIEDMAN, E., Phys. Rev. C 10 (1974) 2089.

[23] MAILANDT, P., LILLEY, J. S. and GREENLEES, G. W., Phys.

Rev. Lett. 28 (1972) 1075.

[24] BATTY, C. J., FRIEDMAN, E. and JACKSON, D. F., NUCI. Phys.

A 175 (1971) 1.

[25] BATTY, C. J. and FRIEDMAN, E., Phys. Lett. 34B (1971) 7.

[26] BARNETT, A. and LILLEY, J. S., Phys. Rev. C 8 (1974) 2010.

[27] BERNSTEIN, A. M. and SEWLER, W. A., Phys. Lett. 34B (1971) 569; Phys. Lett. 39B (1972) 583.

[28] TATISCHEFF, B., BRISSAUD, I. and BIMBOT, L., Phys. Rev. C 5 (1972) 234.

[29] BLAIR, J. S., Lectures in Theoretical Physics, Vol VIIIc (1966) 343.

[30] GOLDRING, G., SAMUEL, M., WATSON, B. A., BERTIN, M. C.

and TABOR, S. L., Phys. Lett. 32B (1970) 465.

BERTIN, M. C., TABOR, S. L., WATSON, B. A., EISEN, Y. and GOLDRING, G., Nucl. Phys. A 167 (1971) 216.

GUTBROD, H., WINN, W. G. ~ ~ & B L A N N , M., P h p . Rev. Lett.

30 (1973) 1259.

[31] JACKSON, D. F. and MORGAN, C. G., Phys. Rev. 175 (1968) 1402.

[32] FERNANDEZ, B. and BLAIR, J. S., Phys. Rev. C 1 (1970) 523.

BADAWAY, I., BERTHIER, B., CHARLES, P., EERNANDEZ, B. and GASTEBOIS, J., Contribution to this Conference.

[33] BRINK, D. M. and ROWLEY, N., Nucl. Phys. A 219 (1974) 79.

JACKSON, D. F., Phys. Lett. 32B (1970) 232.