An optical model analysis of neutron scattering

AN OPTICAL MODEL ANALYSIS OF NEUTRON SCATfERING

J. RAPAPORT Ohio University, Athens, Ohio 45701, USA

PHYSICS REPORTS (Review Section of Physics Letters) 87, No. 2 (1982) 25—75. North-Holland Publishing Company

AN OPTICAL MODEL ANALYSIS OF NEUTRON SCATFERING J. RAPAPORT Ohio University, Athens, Ohio 45701, USA Received 22 January 1982 Contents: 1. Introduction 2. Experimental results 2.1. Apparatus and procedure 2.2. Results 3. Formulation oftheOMP 3.1. Standard OMP 3.2. The reformulated optical model. A folding model 3.3. Energy dependence 4. Analysis ofthe data 4.1. Nucleon elastic scattering from self-conjugate nuclei

27 29 29 31 36 37 41 42 43

4.2. Nucleon scattering from isobars 5. The isospin dependence 5.1. Analysis in terms of potential depths 5.2. Analysis in terms of volume integrals per nucleon 6. The OMP for nucleon energies higher than 60 MeV 7. The spin-orbit potential 8. Inelastic neutron scattering 9. Conclusions References

49 49 49 54 69 69 70 71 73

43

Abstract: Neutron elastic scattering data in the range A > 16, 7< E < 30 MeV are analyzed using astandard localopticalmodeLpotential. The obtained parameters are compared with similar optical model parameters derived from proton elastic scattering. Empirical values for the Coulomb correction term, isospin dependence and energy dependence are obtained. The results axe compared with theoretical calculations.

Single ordersfor this issue PHYSICS REPORTS (Review Section of Physics Letters) 87, No. 2 (1982) 25—75. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publishers. Orders must be accompanied by check. Single issue price Dfl. 27.00, postage included.

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J. Rapaport, An optical model analysis of neutron scattering

27

1. Introduction The study of nuclear structure and nuclear shape by means of particle scattering, dates as far back as the beginning of this century when Rutherford El] published his results of scattering of a and j3 particles by matter. The experimental results indicated the existence of an atomic nucleus. Later, experiments in the early fifties done with electrons and nucleons were motivated along similar lines. Nuclear sizes were determined by electron scattering data and by the measurement of high energy neutron total cross section. The regularities observed in the total cross sections for neutrons up to 3 MeV [21 were explained by Feshbach, Porter and Weisskopf [3] introducing a model initiated by Bethe [4]. This model consisted in replacing the nucleus by a one-body complex potential which acts upon the incident nucleon. The real part represents the average potential in the nucleus while the imaginary part caused an absorption which described the formation of the compound nucleus. This model was the beginning of what we know today as the optical model potential (OMP). In the middle fifties with better resolution accelerators and an increase in the bombarding energy, the study of the scattering of charged particles, especially protons, led to a gradual refinement of the optical potential. It should be distinguished what is meant by “optical potential” because several different approaches may be taken to derive it. One obvious distinction is that experimental physicists talk about an “empirical optical potential” which is a local potential obtained by the analysis of scattering data differential cross section, polarization, total cross section and neutron strength function. Theoretical physicists prefer to speak about a “theoretical optical potential” which is a nonlocal potential which may be derived in various different ways. Preferentially starting from basic interactions. The two approaches are quite different but, of course, related. Several years ago Hodgson [51 indicated that the two types of analyses should be regarded as complementary. The theoretical optical model can give the overall form of the potential but phenomenological analyses are required to make them more precise. The aim of the theoretical optical potential is to replace all the interactions between the incident single nucleon and all the other nucleons by a simple one-body potential. The scattering problem was at first treated from a many-level dispersion theory point of view [6] or from a many-body problem of (A + 1) particles where A is the atomic number of the target [7,81. The idea is to understand the complex many-body scattering problem in terms of a one-body OMP. Many of the original methods and procedures used in these studies are described by Hodgson [91. More recently, several microscopic derivations of the theoretical optical potential have been performed. One formulation is the so called “nuclear matter approach” of Jeukenne, Lejeune and Mahaux [10,111 and Brieva and Rook [12] and another is the “nuclear structure approach” of Vinh Mau and Bouyssy [13,141 and Bernard and Van Giai [15]. Madsen, Osterfeld and Wambach [161 have suggested a method which combines both approaches and which can be used for elastic and inelastic scattering. As an alternative formulation to the standard Schrodinger equation, Arnold and Clark [17] have developed an optical model based on the Dirac equation. This article does not attempt to review the theoretical optical potential. The emphasis is placed on the empirical optical potential, especially in its relation to elastic neutron scattering data and the comparison with similar proton scattering analyses. In the earlier sixties the initial phenomenological work was based on fitting elastic scattering differential cross section with potentials defined by set of parameters. One of the most successful —

28

1. Rapaport, An optical model analysis of neutron scattering

of these potentials was obtained by Perey [181 from an analysis of many proton elastic scattering angular distributions for elements heavier than Al, with incident proton energies between 9 and 22 MeV. By using a phenomenological nuclear local optical potential with three components, real central, imaginary central and real spin-orbit, the aim was to find smooth trends in the parameters as a function of mass and energy. At the same time Perey and Buck [19] using an energy independent non-local optical potential were able to obtain good fit to neutron elastic scattering data up to 24 MeV. The energy dependence was accounted for purely in terms of the Gaussian non-locality term of the separable potential form factor. Another survey study that should be mentioned is that of Rosen, Beery and Goidhaber [201. They obtained a fixed set of parameters except for the energy dependence of the real depth which gave a good account of proton polarization measurement at 10.5 and 14.5 MeV incident energies on a range of target nuclei with A from 16 to 120. Several other surveys of more limited scope were also reported in the last sixties [2 1,22,23]. Simultaneous fitting to a large number of data sets for different target nuclei and different energies was made possible with fast computers giving rise to global optical model analyses. A comprehensive study of this kind was made by Becchetti and Greenlees [24]. A representative group of proton scattering data, 40 differential cross section data sets, 28 polarization sets and 8 reaction cross section measurements (mostly from isotopically pure targets) were used. Targets with 40< A < 210 and energies between 10 and 40 MeV were represented to obtain global proton OMP parameters. The quality of the available neutron elastic scattering data was substantially less than that for protons. The authors included 30 sets of differential elastic scattering, 4 polarization sets and 28 total cross section measurements for neutron energies below 24 MeV. The neutron data were used primarily as a check of features revealed by the proton data. Greenlees, Pyle and Tang [251 developed a reformulation of the optical model in which the real parts of the potential are obtained using a folding model from nuclear matter distributions and the nucleon—nucleon force. They also indicated that the volume integral per nucleon and the mean square radius of each component of the potential are the well determined quantities obtained from the analysis of elastic scattering data. Ambiguities between the potential depths and geometries [9] were thus avoided. Several authors have discussed the OMP. Hodgson in his book on this subject [9] may guide the reader to the extensive literature in this field. Also should be mentioned the review articles of Satchler [26], Sinha [27], Perey [28] and recent conferences on the subject [29,30]. The study of the OMP is important not only for its intrinsic interest but also for the extremely important role it plays in the description of nuclear reactions. Elastic scattering represents the largest of all partial cross sections when a nucleon interacts with ‘nuclei. It is also very important in nuclear applications where no experimental data are available. This paper presents an analysis of precise neutron scattering data in terms of an OMP, data which primarily have been obtained at Ohio University with monoenergetic neutrons up to 26 MeV. The comparison of such an analysis with similar analyses reported on proton scattering data permits the direct evaluation of Coulomb and isospin terms in the OMP.

J. Rapaport, An optical model analysis of neutron scattering

29

2. Experimental results 2.1. Apparatus and procedure During the last few years Ohio University has been involved in a high energy neutron scattering program. Several precise measurements and analyses have been reported [31—36]. The data have been obtained with the Ohio University 11 MeV T-shape tandem, a high current accelerator. A diode source produces by direct extraction either protons or deuterons up to 80 keY of energy which are injected through a 30°inflection magnet to the low energy beam extension of the beam transport system. There the beam may be chopped and bunched at a repetition rate of 5 MHz or lower. The pulsed beam after acceleration is focussed on a gas cell either at the center of a large room (target area A) where a flight path up to 7 m between 00 and 160°may be used or it may be focused through a recently installed beam swinger [381 to a similar gas cell (target area B). The new detector system may be located up to 30 m away from the target in a well shielded underground tunnel. Scattering angles between —4°and 160°are achieved. The accelerator provides pulsed protons and deuterons that are used in the production of monoenergetic neutrons. Pulses of about 600—800 psec FWHM with a 5 MHz rep. rate may be obtained with currents up to 5 iA on target. Monoenergetic neutrons are produced via the 3H(p,n)3He, 2H(d,n)3He and 3H(d,n)4He reaction. The 0°yields of the monoenergetic neutrons per sr, per pA, per sec, per 10 keV energy loss in the gas cell are shown in fig. I versus neutron energy obtained in the reaction. A minimum of 2 MeV and a maximum of 11 MeV have been assumed for the incident protons or deuterons. Usually a 3 cm long gas target cell [371 is filled with deuterium gas to a pressure of about 2.7 atm. or a similar cell is filled with tritium gas to a pressure of about 1.7 atm. Entrance window foils of either 5 mg/cm2 molybdenum or recently 4 mg/cm2 tungsten have been used. Both provide adequate low neutron background [39] and may be used up to charges in excess of 2 Coulomb with currents up to 6 pA (protons), 3—4 pA (deuterons). The

1o~ T(D.n)

i07.

I

•d ~~d,n)

1,

(MeV) Fig. 1. Comparison of zero-degree neutron yields for the indicated source reactions. A minimum projectile energy of 2 MeV and a maximum of 11 MeV have been assumed. An energy loss of 10 keV is assumed in the target.

30

J. Rapaport, An optical model analysis of neutron scattering

entrance foil and gas pressure introduce an energy spread of the outgoing neutrons, generally less than 100 keY. A small NE224 liquid scintillator (5.0 cm in diameter and 5.0 cm deep) has been used to monitor the neutron flux in target area A. It is usually located at an angle of —25° and a flight path of about 6 m. Scattered neutrons are observed by “Colorado” style [40] NE224 liquid scintillation detectors. The liquid is housed in a plastic container 18.4 cm in diameter and 5.0cm deep mounted on an RCA4522 photomultiplier tube. A similar detector that combines good resolution (about 1 nsec) and high efficiency [41] was used to detect 26 MeV neutrons. Detector thresholds are set such that for each case they are approximately half the maximum neutron scattered energy. Pulse shape discrimination is used in both the monitor and the detector to eliminate 7-ray events from the time of flight spectra. Data were taken every 5°in the interval 15° to 1 55° with 4 m flight paths for 7 MeV incident neutrons to 6.6 m for all other neutron energies. Data have been obtained at neutron energies of 7.0 ±0.04, 9.0 ±0.11, 11.0 ±0.09, 20.0 ±0.11, 24.0 ±0.13 and 26.0 ±0.06 MeV. The overall energy resolution varies between 280 keY (FWHM) for the 7.0, 9.0 and 11.0 MeV runs to 750 keY for the 20, 24 and 26 MeY experiments. A typical set up (target area A) is schematically presented in fig. 2. A properly situated copper shadow bar attenuates the direct neutron flux from the gas cell in the detector direction. The scattering samples are in general cyclindrical in shape, about 2 cm in diameter and 2.0 cm in height, weight approximately 1/3 to 1/2 of a mole, and have an enrichment greater than 90%. The differential cross sections have been normalized by rotating the detector to 0°with the scattering sample removed and measuring the incident neutron flux per monitor count. —

GAS- CELL

—

\~ \ SCATTERI~.,~ -~

j~~—MONITOR

—

-

__

SCINTILLATOR—

1

WATER

LEADI~

/

PARAFFIN & IRON

Fig. 2. Geometrical array and shielding used in the measurements of differential neutron elastic cross sections (target area A).

J. Rapaport, An optical modelanalysis of neutron scattering

31

In target area B the monitor is located at d = 1 m and 0 = —35°,supported by an arm that rotates with the beam swinger. A small stilbene crystal 2.5 cm in diameter and 2.5 cm thick is used as monitor. Pulse shape discrimination is employed to eliminate 7-ray events from the time of flight spectra. A complete description of this new facility is presented in ref. [381. 2.2. Results A list of the scattering samples and the energies at which elastic and inelastic data have been taken is shown in table 1. Data for the differential elastic cross section measurements were accumulated so as to have yield statistics better than 3%. This restriction had to be lifted in some cases, but yield statistics better than 10% were always achieved. The sample-out background spectrum that was obtained in each case was subtracted channel by channel after monitor normalization from the sample-in spectra. Experimental yields were calculated by fitting the peaks of interTable 1 Ohio University neutron scattering data Neutron energy [Mcvi Sample

7.0

9.0

11.0

20.0

24.0

26.0

XIn

16,180

x

Mg,26Mga) 21Al Si S Ca sly 55Mn Fe 59Co 58’60N1, a) Ni 88Sr 89y 90’92Zr

X XIn X XIn X X X X X In Xlii XIn XIn

XIn Xlii XIn XIn

X XIn Xlii XIn

X

X X In

x 92’96’98’100MO In

X

X

116,118,120,l22,l24Snta)

165Ho 181Ta 206’208Pb a)

X

Pb

In = Inelastic measurements. Total cross section measurements between 5—11; 20—26 MeV. a) Not all isotopes at all indicated energies.

X In X XIn X X X In X X

X

X XIn

X

X In

32

.1. Rapaport, An optical model analysis of neutron scattering

est with Gaussian line shapes superimposed on a residual linear background. For skewed peaks at forward angles, a superposition of a Gaussian line shape and up to eight Hermite polynomials were used [42]. Monitor and detector yields were corrected for dead time, corrections being generally less than 1%. The extracted elastic yields were corrected for the following factors: source anisotropy, relative detector efficiency, finite geometry, neutron flux attenuation, multiple scattering and compound nucleus contributions. The source anisotropy correction amounted to less than 7% and is due to the fact that the solid angle between sample and neutron source is larger than the solid angle between neutron source and detector at zero degree. The code DETEFF [43] was used to calculate the relative energy dependence of the detector efficiency. The correction factor amounted to a maximum value of 1.05 for some elastic groups whereas for some inelastic transitions to states about 5 MeY in excitation that correction factor was less than I 10. The measurement at zero degrees provided the absolute normalization. The Monte Carlo code MULTISCAT [44] was used to calculate finite geometry, neutron flux attenuation and multiple scattering corrections. Compound nuclear contributions were evaluated using the computer code HELENE [45] which performs a Hauser—Feshbach calculation together with a Porter—Thomas width fluctuation correction. This contribution was .

Ex(MeV)

6

4

I

‘

I

2 I

‘

0I

8I

60Ni(n,n’) II MeV

Ex(MeV) 4I

0I

60N1(n,r/) 24MeV 30 0

: 2

___ H sic

sio

~ CHANNEL

ri~

sic

~ic

sir

sic

e4s

air

CHANNEL

Fig. 3. Typical subtracted spectra for the scattering of 11 and 24 MeV neutronsof 60Ni.

ale

sic

.1. Rapaport, An optical model analysis of neutron scattering

I

,

—

I : ~I:’—•

I

33

—

~

102

~

-~

o3-\

S

-

\~~I03-\

2-

-

io

71 I 25°

-

I 65°

I 105°

I 145°

io2

-

°5Nb

,,l I I I 250 650

-

I

105

I

145

Fig. 4. Angular distributions for 11 MeV neutron elastic scattering [31]. The solid lines represent OMP calculations with parameters obtained in individual searches.

found significant only for light nuclei in the scattering of 7, 9 and 11 MeY neutrons, reaching up to 15% of the measured cross section values in the back angles for Mg at II MeY. At the first minimum of the diffraction pattern this correction for 92Mo at 7 MeY was less than 5%. The calculated compound—nucleus cross sections were subtracted from the measured values in those cases that amounted to values larger than a few percent, but in general at almost all angles for all the samples this correction was negligible (see refs. [3 1—36] for specific cases). The overall uncertainties in the relative angular distributions were generally less than 10% although a few measurements at backward angles had errors as large as 20%. The overall normalization uncertainty was of the order of 5%. A breakdown of the factors contributing to the overall uncertainties is reported in ref. [311. Typical time of flight spectra are shown in fig. 3. The sample-in minus sample-out spectra for the scattering of 11 and 24 MeY neutrons of 60Ni are shown as a function of excitation energy. Some of the observed angular distributions reported in refs. [3 1—361 are presented in figs. 4—8. The solid lines represent OMP calculations with individual searches. We do not present our data in tabular form; all the values have been communicated to the National Nuclear Data Center, Brookhaven National Lab, where they may be obtained upon request.

1

I

I

104—

I

I

I

f

-I

9Mo

-

Io3l_~”\\

1~ 1

1

iO~’ oaf

‘RHo

O2L\~T0

102

~

U,

~

I02~

\

~-

IO2f

IOOMO

-

~ ~ ‘a IO2~ 10

I~ Pb

I

~-

02j1~

\r>~

103\ 102

-

10’

-

~

I0~

-

10~ 0’ -I

_______________________________________________

‘25°

65°

05° 45°

I

25°

I

I

I

65°

I’

I

05° 45°

acM

Fig. 5. Angular distributions for 11 MeV neutron elastic scattering [311. The solid lines represent OMP calculationswith parameters obtained in individual searches.

10~.

iO~

102

~

~l02

~

i0~ 102

s~

102

io~

l0~

02 10~

0~ ~

I

,-a’ io~

102

102

:.-

102 100

15

‘•‘~

1 0~ 102 ~

Cfl(NN

~

....

100

~5

95

135

100

15

55

95

135

15

55

95

I~5

e~(degrees) Fig. 6. Neutron elastic differential cross sections parameters obtained in individual searches.

at E~=

11, 20 and 26 MeV [32]. The solid lines represent OMP calculations with

::\\‘‘~_°~

I0~

Q.o

0’

10’ ~24s~ 0’

•

U)

221N

E.24.OMeV

-

‘

U

120.N

to’ 02

102

IISIN

02

10

‘65N

10’

~

55

15

95

35

IS eCM

55

95

135

(degrees)

Fig. 7. Angular distributions for the elastic scattering of 11 and 24 Mev neutrons from even isotopes of Sn. The solid lines are OMP calculations with optimum values parameters [35].

I 0~

1 ~3

102

102

10~

l0~

1 02 -o U

208pB(N,N~

\v-\\I.~.\\

‘‘~, 4

iO~

1 ~0

iO~ 26

102

E~lIMEv

102

101

101

10°

100

15

55

95

135

15

~5

95

l~5

eCM (degrees) 2O8p~[33]. The solid lines are OMP calculationswith optimum Fig. 8.parameters. Differential cross sections for neutron elastic scattering of values

36

.1, Rapaport, An optical model analysis of neutron scattering

3. Formulation of the OMP When performing optical model analyses the data, in general, are fitted by using a simple equivalent potential. If the data used are for specific nuclei in a limited mass region “regional” OMP parameter sets are obtained. On the other hand data sets for a large number of nuclei and energies are used to obtain “global” OMP. The features of this “global” potential are based mostly on physical intuition and parameters are adjusted accordingly. As indicated by Perey [281 these potentials represent an average behavior and thus no attempt is made to reproduce with them the shape resonances in scattering found at low energies. The parameters of the OMP are expected to vary in a systematic and regular fashion with energy from nucleus to nucleus, which makes OMP analyses of light nuclei and low nucleon energy more difficult. However, the striking success of OMP analyses [24,25,28] with global parameters has made us forget to some extent that the individual structure of each nucleus must affect the nucleon— nucleus interaction. This point has raised difficulties concerning the concept of the empirical OMP. Should it be obtained from a good overall fit to many sets of data [46] or should it be adjusted to fit each set of data as accurately as possible? In the latter case the “local” OMP parameters are unique to the energy and nucleus from which they were obtained reproducing the data quite adequately. However, it is almost impossible to study energy and/or mass dependence from such sets of “local” OMP. —

~

~ 1

9MEV

~

~iO

iOo

lQ~J

15

55

95

135

~

15

55

95

135

(degrees> Fig. 9. Elastic neutron scattering of 100Mo [34]. The solid lines represent coupled channel calculations while the dotted lines represent OMP calculations with the same parameters used in the CCBA calculations. Individual best fit OMP calculations reproduce the data as well asthat shownwith the solid lines but an anomaly in the vanationof W with mass is obtained

J. Rapaport, An optical model analysis of neutron scattering

37

With a simultaneous search for best parameters to many data sets, some of the “global” OMP parameters (i.e. geometrical parameters) are the same for all nuclei. Thus, it is easy to study the dependence of energy, isospin and mass. Of course, the quality of the fit is not necessarily as good as it was in the individual case. A comparison of the two types of potentials may bring out “anomalies” that have a physical basis. In particular, individual neutron OMP studies of several even—even Mo isotopes [34] revealed an anomaly in the variation of the imaginary potential depth W with mass, if compared with the “global” trend. This anomaly was eliminated by introducing a coupled channel analysis and recovering the mass dependence of W of the “global” set of potentials (fig. 9). Other similar cases are described in ref. [46]. 3.1. Standard optical model potential In what follows the general notation will be used: U(r, E) = V(r, E) + iW(r, E) where U is the OMP with a real term V(r, E) and an imaginary term W(r, E). Using the same notation as in ref. [24] the OMP may be expressed as a sum of terms: -U=UN-Vc-U~, where V~is the Coulomb potential for an appropriate charge distribution, U~is the spin orbit potential and UN, the nuclear part on which we will concentrate our analysis: UN(r, E)

=

VN(r, E) + iWN(r, E).

The real central potential VN (r, E) is expected to follow roughly the shape of the matter distribution in the nucleus. Of course, this shape is not well known and furthermore the potential shape is expected to be affected by a density dependence of the nucleon—nucleon effective interaction. Several parameterizations of the real potential shape have been tried without a clear indication of a preference for a particular form. A convenient radial dependence, which has been conventional to use for both the nuclear potential and density distribution is the Woods-Saxon potential: VN(r, E) = V(E)(ex + l)’,

x

=

(r

—

R)/a

with the radius R increasing proportional to A’13. The surface thickness or diffuseness parameter, a, is roughly constant for all nuclei characterizing the charge distribution. A recent paper [47] modifies this shape by introducing a central depression to study the p + ‘2C scattering at 200 MeY. This central depression is predicted by microscopic theories. The r.m.s. radius (i?2>h/2, for protons, of V~(r)is found to be approximately 1 fm larger than the corresponding extension of the charge density as determined from the electron scattering analysis [25]. This difference has been interpreted in terms of the finite range of the nucleon—nucleon interaction [8]. An empirical evidence of this effect has also been reported by Holmqvist and

38

/.

Rapaport, An optical model analysis of neutron scattering

Wiedling [48] from neutron elastic scattering. However, the interpretation of the results of OMP analyses of nucleon scattering data to indicate that neutron distributions extend beyond proton distribution [25,48] may be doubtful. Friedman [49] has pointed out that by including a density dependence to the nucleon—nucleon interaction the calculations suggest that a neutron halo does not exist in heavy nuclei. It is more difficult to predict the radial dependence of the imaginary potential, which accounts for the loss of the incident nucleons due to inelastic scattering and other reactions. It is thought that it should occur throughout the nucleus, but factors such as the Pauli principle [9] particularly at low nucleon energy enhance the surface contributions. Also at low energyit is expected to preferentially excite surface modes that could enhance the surface absorption. As the incident nucleon energy increases these surface effects decrease and the imaginary term is assumed to be distributed more uniformly throughout the nucleus. A theoretical optical model developed by Manweiler [50] from nuclear matter theory and effective interactions shows a radial dependence for the absorptive potential as a function of nucleon energy in agreement with the above ideas. The empirical analyses also confirm these ideas and a predominant surface absorption potential taken to be either of a Gaussian shape or a Woods-Saxon derivative is needed at low energies. At higher energies a mixture of surface and volume terms are generally used. The standard form of UN used to analyze the neutron data is UN(r, E)

=

VR(E)f(r) + i[W~(E)g(r)

—

4WD(E) dg(r)/dr]

,

(3.1)

with the radial dependences taken to be of the Woods-Saxon shape f(x)=g(x)

=

(1 +ex)_~, x,

=

(r— r~A”3)/aj .

(3.2)

The assumption that the radius of the OMP may be written as R 3 is just an a priori as= r,A” sumption introduced to constrain the number of search parameters 1but as pointed out by Hodgson [9,51] may complicate the interpretation of the A, Z or N dependence of the empirical OMP (see also ref. [521). The Coulomb potential V~is usually taken to be that due to a uniform spherical charge distribution of radius R~= rcA’!3:

Vc(r)

(zZe2/2R~)[3 —(r/R~)2J

r~
(zZe2/r)

r>R~

=

where z = 1 for protons, z = 0 for neutrons. The OMP analyses are rather insensitive to the choice of Coulomb radius R~and give the same electron and proton cross sections as a more realistic potential calculated from the empirical nuclear charge distribution provided the r.m.s. radii are made equal [24]. The semiempirical formula given by Elton [53] is generally used to calculate Rc, R~= l.149A”3 + l.788A113

—

1.163/A

The radial dependence of the spin-orbit term of the OMP is usually takento besimilarinform

J. Rapaport, An optical model analysis of neutron scattering

39

to the Thomas spin-orbit coupling term found in the case of electrons: U~0= 2(V~+jW~)~—f~(r)L’e. 2in fm2. At incident The factor 2 is 50 theMeY: conventional pion Compton wave found lengthfor factor (fl/m~c) energies below no definite evidence has been non-zero W~([54,120]; but see ref. [132]). The nucleon—nucleon force is isospin dependent, thus one expects an isospin dependence in the OMP as it has been suggested by Lane [551. The review article by Satchler [261 discusses this in detail. Generally the isospin dependence is written decomposing the nuclear part of the OMP in isoscalar U 0(E) and isoveçtor U,(E) terms: UN(E,r)

(3.3)

U0(E,r)+~U,(E,r)tT.

In the above equation t, T are the isospin of the incident nucleon and target respectively. The isospin interaction splits the radial part of the potential into two diagonal terms which are responsible for the proton and neutron scattering and a non-diagonal term responsible for the (p, n) quasi elastic scattering. The latter is given by U~~(r, E) = 2~J~7~ U,(r, E), where e = (N

—

(3.4)

Z)/A represents the nuclear asymmetry coefficient. The diagonal terms are written:

UN(r, E) = U0(E)f(r) ±eU1(E)f1(r) + L~Ucf2(r),

(3.5)

where the +(—) sign is for protons (neutrons). The last term I~U~f2(r) represents the Coulomb correction term first introduced by Lane [56]. The OMP search code GENOA [571was used to fit the experimental data. The code allows both individual and global fitting, with prescribed constraints on the parameters. The standard form of the OMP defined above and parameterized in terms of isoscalar and isovectordefined terms was as: used. An objective and quantitative measure of the goodness ofthe fit is provided by

x2

N

X

2

=

2

E [Uth(O,) —Uexp(Oi)~ 0i) ~1

1 N

1, [

~°exp(

where N is the number of data points in the distribution being fitted, ath(Ol) and Gexp(O,) are the calculated and experimental values respectively, at angle O~,and ~a 05~(01) is the corresponding experimental uncertainty. For the search procedure the experimental in 0cxg(Oi) were increased to reflect the uncertainty in the scattering angle absolute estimateduncertainties to be less than 0.5 In order to reduce the size of the search space and thus the number of variable parameters and because no polarization data were fitted, the spin-orbit part of the potential was assumed to be real [54,58] and the strength and geometry were kept constant to the values of ref.[24]. Other .

.1. Rapaport, An optical model analysis of neutron scattering

40

values [28,58,59] were also tried but, as expected, the choice did not appreciably change the results. The multidimensional search was guided to avoid ambiguities [9] and some constraints given by physical arguments [9] were imposed. The search of individual optimum parameters was done in three stages, each stage differing from the one before by the number of OMP parameters which were allowed to vary. The following sequence was generally used: i) V, WD, (Wv)~ ii) V,aR, WD,(WV)~,rI iii) V,rR,aR, WD, (W~)~,r,,a1. The initial parameters for the succeeding steps in the sequence are the final values obtained in the preceding step. Any parameter not varied in a step remained fixed at its previous or initial value. This approad allows the search toproceedin- acontrolledmanner. Yalues-for--the individ~ ual OMPparanietersare reported in refs. [31—361. An average geometry for the OMP parameters may be obtained for each nucleus assuming it to be independent of energy. At this stage one may proceed to a “glob~”OMP analysis, introducing If not kept equal to zero. Table 2

Neutron7~En~ global OMP 30MeV) parameters * ( Real term:

V rR aR

(54.19 — 0.33 E) = =

—

e(22.7 —0.19 ~

[MeV]

1.198fm 0.663 fm

Imaginary term: a)E ~15MeV

4.28 +O.4E—e12.8

WD = W~= 0

b)E WD

=

Wi,r

= =

14.0

—

0.39 E — elO.4

—4.3 + 0.38 E 1.295 fm

a 1

0.59fm

Spin-orbit potential: t

Vso6.2MeV; W~~0 rSO

=

1.01 fm

aso = 0.75 fm *

From ref. [61];set A.

t Kept fixed during the search; values from ref. [24].

[MeV]

[MeV] [MeV]

J. Rapaport, An optical model analysis of neutron scattering

41

a common geometry which is normally taken to be the overall average geometry. A-n example of this procedure is presented in ref. [61]. The results of a neutron “global” OMP parameter set are shown in table 2. 3.2. The reformulated optical model. A folding model The measured observables (differential cross section, polarization or total cross section), do not determine the OMP uniquely and in the literature one finds several potentials that give comparable fits to the experimental data. In the multi-dimensional search of OMP parameters, one finds many local minima and it is difficult to identify the correct one. Some of the ambiguities are continuous such as VRr~ constant, WDa, ~ constant [9,26] and others are discrete. However, the discussion of the energy and isospin dependence of the OMP can be made less dependent on the choice of geometrical parameters, if it is discussed within the realm of the reformulated OMP. Early in 1958 Feshbach [71 suggested the use of the volume integral of the potential to study systematic trends because it incorporates contributions of both the well depth and its geometry. Later Greenlees and collaborators [25] reformulated the OMP by expressing the potentials in terms of the nuclear matter distribution and the nucleon—nucleon force. In this approach, the significant quantities to be compared are the volume integral per nucleon, J/A, and the root-mean-square (r.m.s.) radius. The reformulated model of Greenlees, Pyle and Tang [25,62,63] uses a folding-model analysis to obtain the real part of the potential. The nuclear proton p~(r)and neutron p~(r)density distributions are folded in with the appropriate components of the nucleon—nucleon potential. Thus, it is written as: U(r1) = fp(r2)u(r,2) dr2 The central isoscalar part of the potential VR is obtained by using p(r) = pn(r) + p~(r)normalized to the number of nucleons in the respective nucleus, and u = v0 the central spin and isospin independent term of the nucleon—nucleon effective interaction. The central isovector part uses p(r) = p~(r) p0(r) and u = ~ the isospin non spin-flip term of the effective interaction [63]. With the assumption of a spherical density distribution and a local density-independent effective interaction, Greenlees et al. [25] show that the volume integrals are as follows: a) Isoscalar part —

J0=AJ~, where J0

=

(3.6)

f V0(r) dr and J~= f p0 dr.

b) Isovector part J,

=

A~J~ ,

where ~ = ~T/~0’ assuming V~and v0 have the same shape. The m.s. radius is given by the general expression 2> = ff(r)r2 dr/f f(r) dr, (r

(3.7)

J.

42

Rapaport, An optical model analysis of neutron scattering

which takes for instance for the isoscalar part the form (r>R

(r2)~+ (r2)m

=

(3.8)

,

where (r>~,(r2)~and (r2>m are the m.s. radii of the isoscalar real potential VR, the v 0 term of the two-body potential and the matter distribution respectively. It is also pointed out [25] that within these assumptions, values of the volume integral per nucleon of the real term of empirical optical potentials (includes both isoscalar and isovector terms) plotted as a function of e = (N Z)JA should result in a straight line with a slope equal to ~andinte-rceptequaitoJ~: —

JR/A

=

J~, ±~J~(N

—

Z)/A

(+ for protons,

—

for neutrons),

(3.9)

where ~R /A is the volume integral of the real central term of the OMP. A similar analysis for the spin-orbit part of the OMP, V~,indicates [25] that the form factor is quite different from that of the real central term, more of a surface effect. Greenlees et al. [63]have analyzed 30.3 MeV proton elastic scattering data and obtained was em3 independent of A (or (N Z)/A )~No explanation pirical forapparent ~R IA 400 MeY fm with the above relation, unless ~ = 0 (see also ref. [641). offeredvalues for this contradiction TVth&~liiiiI~ uuile~ ãk~d TaT en~ii~i~ taken to ~e Woods-Saxon and Woods-ET1 w495 404 Saxon derivative types respectively, the volume integrals per nucleon J/A = (l/A)f U(r) dr are given approximately to order (aIR)2 by —

(J/A)~

3 V~ 01 ~irr

21

(3.10)-

,

01[ 1 +(ira/R)

where V~

3 the radius of the volume

01is the volume potential depth, a the diffuseness and R = rA” term of the OMP and (J/A)D~16(irR/A)2aVD[l+~(7ra/R)2],

(3.11)

where VD is the surface potential depth, a the diffuseness and R = rA”3 the radius of the surface term of the OMP. The m.s. radius for a Woods-Saxon radial dependence may be otaijed ir~ithc ~pr~~4c~ = ~(3R2+ 7~’2a2)where R = rA”3 and a are the radius and diffuseness parameters of the OMP. The above expression is only correct to order (aIR)2. 3.3. Energy dependence Early empirical analyses of nucleon elastic scattering showed that the real term of the OMP decreases as the incident energy increases. This can be identified to a dependence of the potential on the kinetic energy of the nucleon inside the nucleus. For nucleon energies less than 60 MeY the central real potential depth decreases linearly with energy at a rate of about 0.3 MeV~[241 or in terms of the volume integral per nucleon at a rate of about 2—3 (MeY fm3) per MeY [65]. In order to inc~dea large energy rae;’4OtOOO~M1u~~itlunivesiegydependenceofThe~ form V(E) = V(E = 0) ~ in E is needed [54,651. —

1. Rapaport, An optical model analysis of neutron scattering

43

The empirical evaluation of this energy dependence over a wide range of energies is important to determine its nature. As it has been indicated the empirical OMP is assumed to be a local one and as such has a trivial energy dependence due to the transformation from a non-local potential. Another possibility is the intrinsic energy dependence of the effective two-nucleon force. Owen and Satchler [661 and Van Oers and Haw [651 have found that there is a decrease in the magnitude of the energy coefficient for increasing mass number. Owen and Satchler [66] calculated explicitly the exchange effects which give rise to a non-local potential and their results implied that an intrinsic energy dependence was still required for the real OMP. Slanina and McManus [67] using an equivalent local potential for the exchange term reached a similar conclusion. The OMP analyses of proton scattering data need the knowledge of the Coulomb correction term and its energy variation in order to extract the energy dependence of the nucleon term. Thus, it is preferable to have neutron data to extract these quantities. Such analysis has been done by Bowen et a!. [681 using total and absorption neutron cross section to determine the energy variation of the real and imaginary potentials in the 10—100 MeV energy range. A linear energy dependence (a 0.25) was obtained for the real potential depth. Passatore [69] has evaluated the energy dependence of the empirical OMP by means of a dispersion relation and non-locality effects. He obtained good agreement with experimental data from 10 MeY to 1 GeV which showed a logarithmic energy dependence. The same type of analyses also gives the energy variation of the imaginary potential. The situation, however, is not as clear as for the real term because of the energy dependent form factor. At low energies (E 15 MeV) a surface absorption is generally used, while at energies larger than 60 MeV a volume absorption is needed. At energies between 15—60 MeY both surface and volume terms are frequently used [70]. It has become quite clear that a theoretical guidance about the energy dependence of the absorptive form factor would help considerably empirical OMP studies. This paper is concerned with neutrons up to 30 MeV. A linear energy dependence on real and imaginary potential depths is assumed. It has been indicated that the OMP has isoscalar and isovector terms (eq. (3.3)) which may have different energy dependences. It is proper, therefore, to study first nuclei with only an isoscalar OMP; these are self-conjugate nuclei. The study will be extended to other nuclei to determine the energy dependence of the isovector term. 4. Analysis of the data 4.1. Nucleon elastic scattering from self-conjugate nuclei In the case of self-conjugate nuclei (T = 0) the nuclear OMP has only an isoscalar term and may be written: UN(r, E)

=

U0(E)f(r) +

where the last term represents the Coulomb correction term. Physically, this corresponds to the fact that it is the kinetic energy of the nucleon inside the nucleus that determines the strength of the average potential [56,7 1]. An empirical study of neutron and proton elastic scattering [72] on T = 0, T = 1/2 and T = 2 targets showed to be in agreement with the above observations.

44

/.

Rapaport, An optical model analysis of neutron scattering

It has been shown by Perey [18]and by Satchier [26]that if the neutron OMP for T = 0 nuclei is written as: Un(r,E)= U0(E)f(r),

(4.1)

then the proton OMP at the same incident nucleon energy, E, may be written to lowest order in the local energy approximation as: U~(E) U0(E)f(r)

—

V~(r)aU/aE,

(4.2)

where Vc(r) is the Coulomb interaction potential. Because V~(r)does not vary rapidly within the nucleus an average value V~is normally used, V~= 1.73 Z/Rc MeV. 4.1.1. The real component of the OMP Assuming same radial shapes for the isoscalar and Coulomb correction term, the above expression (eq. (4.2)) reduces to a correction £~V1~ to the proton dependence OMP well-depth Forcoefficient a uniform 3 and an energy with aV0. linear charge distribution of radius R~ = 1.3 A” dV/dE = —0.3 the Coulomb correction has been given [181 as z~V~ = 0.4 (Z/A 1/3) MeV. It is clear from eqs. (4.1) and (4.2) that a comparison of optical model analysis of neutron and proton elastic scattering on the same T = 0 nucleus yields empirical information on the Coulomb correction term ~ Ucf 40Ca [32,73]. 2(r). Such analysis has been reported for

o

Ca (pip)

0

500

~

Ca (nan)

400

300

I

20

1;c

-60

E Fig. 10. Values of the volume integral per nucleon for the real term of the OMP for 40Ca (p, p) [74] and for Ca (n, n) [32,75].

I. Rapaport, An optical model analysis of neutron scattering

45

• 28Sl(p~p) •

SIln,n)

500

400

00

300

‘I

-

I

20

40

-

E (MeV)

60

Fig. 11. Values of the volume integral per nucleon for the real term of the OMP for 28S1(p, p) [76] and for Si(n, n) [32].

We present in figs. 10, 11, 12 the volume integrals per nucleon of the real term of the optical potential for nucleon scattering on 40Ca, 28Si and 32S, all self-conjugate nuclei. A comparison in terms of the volume integral is, in principle, independent of the choice of the OMP geometrical parameters. The proton data for 40Ca (fig. 10) has been obtained from the analysis of Van Oers [741while the neutron data at 11, 20 and 26 MeV are from ref. [32]and 30 and 40 MeV data points are from the work by DeVito et al. [75]. A linear relationship may be obtained between (JR/A) and E for proton energies between 26 and 76 MeY. Values below 26 MeV were not included in the least square fit because they are quite scattered probably due to the high (p, n) Q-value (Q = —15.1 MeY) and coupled reaction channel effects which affect the OM analysis. Assuming an uncertainty of ±1MeY in the determination of the optical model real potential depth the result in terms of volume integrals are: (JRIA)P

=

498

—

(2.9 ±0.2)E~ [MeY’fm3],

and a straight line with the above slope drawn through the points representing the neutron elastic scattering analysis is parameterized as: (1Rb4)~ =

473 — (2.9 ±0.2)En

[MeV fm3].

Thus the Coulomb contribution to the volume integral may be evaluated [73,75]:

J. Rapaport, An optical model analysis of neutron scattering

46

0

500

~

a

“OO

N

oN

N~

0

__________

300 I

I

20

I 40

I

I

60

E (MeV)

32S(p, p) [77] and for S(n, n) [32].

Fig. 12. Values of the volume integral per nucleon for the real term of the OMP for

(~vc/4)=(JR/14)p —(JR/A)fl(25± 3)

[Mel/fm3].

A valueM(~= (3.0±0.04)MeV is obtained for thepotentialdepthcontributionwhickifpa~ rameterized in the usual way (same radial dependence as isoscalar term) gives a value i~V~ = (0.5 ±0.07)Z/A’13. This is slightly higher than~th~value (0.46 ±0.07)Z/A’/3 reported in ref. [32];the present analysis includes the 30 and 40 MeYneutron data from ref~1751, notincuded~previously. We present in fig. 11 a similar analysis for 28Si. The proton elastic scattering between 14—40 MeV have been reported by De Leo et al. [76]. The neutron data [321 is also presented. The same least square linear dependence as in fig. 10: (JR/A)P

=

498 — (2.9 ±0.2)E~ [MeV fm3],

has been drawn through the proton data points while the linear dependence (JRIA)fl

=

484 — (2.9 ±0.2)E~ [MeY fm3],

has been obtained from the neutron data points. Thus the contribution to the Coulomb correction may be expressed as (14 ±5) MeV• fm3 which corresponds to a parameterization of the form = (0.33 ±0.12)Z/A’/3 MeY. This value is lower than the one obtained for 40Ca and it may reflect the influence of coupling effects between g.s. and excited states. The first excited state in 28Si (E~= 1.78 MeV) has a J’T = 2~and a large deformation value ~ ~ ~4f76Jhas beeird~duced~This indicategastrongcou-

J. Rapaport, An optical model analysis of neutron scattering

47

pling between the g.s. and the first excited state. Empirical evidence suggests that when searching for potential depths in a coupled channel (CC) formalism, both real and imaginary components are changed from the OMP values. We present in fig. 12 similar data but for 32S. The proton elastic scattering between 15 and 35 MeV have been reported by De Leo et al. [77]. Their optical model analysis has been done within a CC formalism. They assumed 32S to be described as a rotational nuclei and given only a quadrupole deformation. Elastic scattering and inelastic scattering to the 2.23 MeV J7~ 2~state and some analyzing power data were included in the analysis. Their results with a fixed energy-averaged geometrical OMP parameter for the volume integral per nucleon of the real term of the OMP are presented in fig. 12. Again a least square linear dependence of the form: (JR/A)P

=

500 —(2.9 ±0.2)E~ [MeV~fm3]

fits the empirical points quite well. A similar CC analysis was done for the neutron data [321 using the same proton geometry for the OMP. The results are indicated in fig. 12. The linear dependence may be expressed by: (JR/A)fl

=

480 —(2.9 ±0.2)E~ [Mel/fm3].

Thus the Coulomb correction contribution is (20 ±6) MeY fm3 which corresponds to a parameterization of the form ~Vc (0.45 ±0.13)Z/A”3 MeY. A recent analysis [781 of proton scattering on 160 in the energy range 20—50 MeY suggests a linear energy dependence of JR/A = (521 2.9 E) Mel/fm3 in good agreement with the values obtained for 40Ca, 325 and 28Si. The 24 MeY neutron scattering of 16Ø [30] gives an average value (JR/A)fl = 437 MeV fm3 which compared with the calculated 24 MeYprotonvalue (JR/A)P = 451 MeV’ fm3 gives a value (14 ±6) MeV• fm3 for the Coulomb correction term. All the results are summarized in table 3. —

Table 3 Coulomb correction contributions to the volume integral per nucleon (real term) from neutron and proton OMP analyses on

nuclei Nuclei

Energy range [MeV[

(J~v~/A) MeV.fm3]

160

E : 20—50 E~:24 E : 17—40

14±6

0.48±0.2

288i

14 ±5~

0.33

32~

E~: 11,20,26 E : 15—35 E~:11,20,26

20±6

0.45±0.13

E : 26—76 E~: 11,20,26,30,40

25

0.5

40Ca

±3.5

Defined by: AV~= a Z/A”3. (Assumed same form factor as isoscalar term in OMP.) t Values may reflect coupled channel effects not considered in the analyses. *

a *

±

±0.07

T= 0

48

J. Rapaport, An optical model analysis of neutron scattering

Excluding the 28Si results, a weighted average value for the coefficient a defined by L~ = aZ/A”3, a = (0.48 ±0.1), is obtained in good agreement with the previous empirical value [321 and in excellent agreement with the theoretical calculations of Jeukenne et al. [11,79,801 and of Arnold and Clark [17]. It should be indicated that the parameterization indicated here for ~ is only valid if the assumption of a linear energy dependence of V is satisfied. If this is not the case, the general expression ~ = — U/aE should be considered.

a

4.1.2. The absorptive component of the OMP As indicated from eqs. (4.1) and (4.2) if the nuclear OMP is energy dependent then a Coulomb correction term should be included in protons OMP. An empirical estimation of its value requires the knowledge of the energy dependence of the OMP. In the case of the real potential, the energy dependence is negative, dV/dE < 0, and thus the Coulomb correction, ~ is algebraically added (eq. (4.2)) to the real potential depth V 0. For the imaginary term the energy dependence may be obtained from the evaluation of the absorptive volume integral per nucleon versus energy. As it has been indicated the form factor of the absorptive term is energy dependent and thus the evaluation of the volume integral is less dependent of the choice of the form factors. Empirical determination of (J~/A) [731 indicates a positive coefficient for the energy dependence. Thus, empirically, an energy dependent Coulomb correction term L~W~ has to be subtracted from the absorptive potential depth for protons. This correction 1~W~ is of the opposite sign than the Coulomb correction term for the real term, ~ We present in fig. 13 the volume integral per nucleon for the absorbing part of the OMP for 40Ca.

• Ca(n,n) 0 Ca(p,p)

120

•

-

-

_—‘.

--1~~ -

-

I

0

-

80

j

.~

~

a (Jw/*),,-(,Jw/a)~

—(J~W~/a)

•

~O—4

I

I

tO

20

I

.

30

40

50

E (MeV) Fig. 13. Values for the volume integral per nucleon of the absorptive term of the OMP for 40Ca(p, p) [74] and for Ca(n, n) [32, 751.

J. Rapaport, An optical model analysis of neutron scattering

49

The proton data of refs. [74]and [83]are presented together with the neutron data of ref. [32]and ref. [75]. A value = (J~/A)~ (J~/A)~ = —(43 ±7)[l —

—

(0.028 ±0.005)E] MeY fm3

is obtained [73]. Assuming that g(r), the radial dependence of i~W~(r) is the same as the surface (derivative Woods-Saxon) radial dependence of W, the above expression may be written as: I~Wc(r,E) —[1 —(0.028 ±0.005)E](0.5 ±0.1) —~‘--g(r)MeV. A”3 A similar analysis for 28Si has been done by De Leo and Sterrenburg [84] obtaining a result (‘Tiiiwc/’1)

=

(57 ±l3)[l — (0.025 ±0.01 l)E] MeV fm3

in good agreement with the above 40Ca results. The slight larger value obtained for 28Si may be indicative of the intrinsic large deformation ((3 0.4) of 28Si not included in the OMP analysis. These empirical results are in good agreement with the theoretical evaluations of Jeukenne, Lejeune and Mahaux [80]. 4.2. Nucleon scattering from isobars As pointed out previously the search in a multiparameter space for parameters in the empirical OMP is frequently reduced by introducing the a priori assumption that the OMP radius varies as A 1/3, i.e. RR = rRA113 etc. This assumption definitely helps to constrain the number of search parameters but may complicate the interpretation of the A, Z or N dependence of the other parameters. Hodgson [51] has pointed out that the variation of the empirical OMP with nuclear asymmetry e = (N Z)/A may have partly a geometrical and partly an isospin origin. These may arise because of the OMP parameter ambiguities and, therefore, a deviation from an assumed A dependence for the radius parameter will be compensated by a variation in the potential depths with A, which in turn can be interpreted as a dependence on the asymmetry, e. However, in the case of isobars the simultaneous OMP analysis of nucleon elastic scattering should reflect the proper dependence on e. The A = 92 isobars, 92Zr and 92Mo were chosen for this study using the available neutron data (table 1) and proton elastic and analyzing-power data obtained at 30 MeV on 92Zr and 92Mo [85] and the 15 MeV proton elastic scattering data on 92Mo [86]. This analysis has been presented in ref. [87]. The conclusions, even though indicating values consistent with previous analyses, were not conclusive because of the energy dependence of the form factors to be used and the energy dependence of the potential depths. —

5. The isospin dependence 5.1. Analysis in terms of potential depths The nucleon—nuclear OMP has been written eq. (3.5), as

50

1. Rapaport, An optical model analysis of neutron scattering

U~(r,E)L1~(E)f(r)±eU1(E)f1(r)±~U~f2(r),

-

where the +(—) sign is for protons (neutrons). The previous sections 4.1.1 and 4.1.2 indicated an empirical method to obtain i~U~ f2(r). The real component ~ was found to be 15—20% larger than the value previously used in many proton elastic scattering analyses. The imaginary component ~ assumed previously to be zero, was found negative and energy dependent. It is, therefore, important to study the isovector term of the OMP in a manner consistent with these new results. Several methods have been used to determine the isospin term in the OMP. Based on the above eq. (3.5) the analyses of proton elastic scattering at the same energy on several nuclei should give the desired dependence. A linear graph of (U~— ~ U~)in terms of e = (N — Z)/A gives both the intercept, isoscalar potential depth U0 and the slope or isovector potential depth, U1. Satchler [26] summarizes the analyses of proton scattering from ranges of nuclei with The proton 13 + eV,. reenergies 9—61.4 MeY assuming a real well depth of the form V0(E) + 0.4 Z/A’ ported values of V, range from (10 ±8) to (38 ±8) MeY with no apparent energy dependence and an average value of approximately 24 MeV. It should be noted that the large spread in the reported values may be due to the different geometries used in the OMP analyses. No systematic behavior was determined for the absorptive term of the OMP, thus no value of W 1 was reported. The analysis of neutrons at the same energy does not present the problem of having to know the Coulomb correction term but still it is sensitive to the geometry used in the different analyses. Such an analysis of 11 MeY neutron elastic scattering [31] resulted in a value of V, = 22.5 MeV. It should be realized that for a potential depth of about V 50 MeV, typical for nucleon elastic scattering, the isoscalar term is about V0 = 48 MeY, the isovector term eV, 2 MeV and the Coulomb correction term for a medium mass nucleus about ~ 4MeV (AVe = 0 for neutrons). The analysis of either neutrons or protons at the same energy in order to obtain the isospin dependence will depend on differences between asymmetry values e = (N — Z)/A, which are not large to begin with. Thus, the accuracy of the method depends on how accurate the values of the OMP depths have been determined. Because of OMP parameter ambiguities [9] it is difficult to determine the radial dependence of U1(r, E) from scattering data alone. As indicated above, the real component eV1,is only about 4% of the main part of the potential. Satchler [26] has indicated that 2small changes in V Thus and Ramay is kept constant. surbe made without changing the scattering significantly provided yR face peaked term would not be easily distinguished from a change in well depth. It is usually assumed that V, has a volume radial shape equal to the isoscalar radial dependence. The imaginary term W, is generally assumed to be surfaced peaked with a radial dependence given by a derivative Woods-Saxon with same OMP parameters as the surface absorptive term. Another method suggested to study the isospin dependence is to maximize the difference between asymmetries by comparing proton and neutron OMP analyses on the same nucleus. This comparison may be done at the same lab nucleon energy, E~= E~,adjusting the proton potential by the Coulomb correction term; another comparison may be done at the same nucleon energy inside the nucleus. This may be accomplished if E~= E 0 + L~ECwhere i~E~ is the Coulomb displacement energy of the isobaric analog state of the target nucleus [881. In the first method we have

—

U~(E) U0(E)

I. Rapaport, An optical model analysis of neutron scattering

=

51

2eU1(E) ÷

where it has been assumed that the form factors for the isoscalar, isovector and Coulomb corrections are the same. Such a method was first used by Hodgson [89] and later by Erramuspe [90] and Velkley et al. [91]. Values for V1 = (23 ±4) MeV were reported. In the second method we have

—

U~(E~) U~(E0)= 2eU1(E0) with the same assumption for the form factors. Both methods were used 51V, in ref. [92].59Co, First‘81Ta a simultaneously 11 MeV neutron and 56Fe, and 209Bi wasanalysis done. Aofreanalysis of the same proton scattering 51V andelastic 59Co data using of the new value for the Coulomb correction term has been done with the results shown in fig. 14. Values V 1 = (18.5 ±1.5) MeV and W1 = (15 ±2) MeV were obtained as compared with the values V1 = 21.3 MeV and W1 = 12 MeV reported in ref. [92] which were deduced with ~ values from ref. [18] and L~W~ 0. In the and second method a comparison 93Nb =and ‘20Sn proton elastic scattering atofa 11 MeV neutron scattering data on Fe, higher shifted lab elastic energyE~=E 0 + z~E~ has also been reported [92]. Proton data reported in the literature at approximately the desired energy were chosen. Values of V1 = 22 MeV and W1 = 14 MeV were obtained.

E~=HMeV V(n,n)

E~=IIMeV ~ V(p.p)

:\~ 59Co(n.n)

59Co(p,p)

IC~%~fr&~\JI

(e)CM

(d.gr..s)

Fig. 14. Comparison of nucleon OMP analyses at E~ E~ 11 MeV.

J. Rapaport, An optical model analysis of neutron scattering

52

A third method to study the isovector interaction which seems more direct is by studying X(p, n)Y transitions between analog states. In this case the transition matrix element is given by (eq.(3.4))

and the isovector interaction instead of being just a small term to the main potential as in the previous two cases is the entire interaction. A number of studies adopting this viewpoint have been reported. Much of the early work may be found in ref. [261. A more recent analysis of quasielastic (p, n) reactions at 22.8 MeV has been presented by Carlson et a!. [93]. The real term of the isovector interaction V1 was found to have a value of approximately 17 MeV while the imaginary part of U1 was found to be different from the Becchetti and Greenlees [24] description; the suggested radial dependence of W1 was not that of a simple Woods-Saxon derivative and thus no potential depth W1 was given. Using the same method and in an effort to obtain the energy dependence of the isovector potential, al. [94] analyzed quasielastic (p, n) scattering at 25,OMP 35 and 45 MeV 48Ca, 90Zr,Patterson 120Sn andet208Pb targets. Of their several analyses, the preferred parameter seton if an isovector energy dependence is desired is given by V 1

(21.9

—

0.18E) MeV,

W1

=

(15.2— 0.18E) MeV,

with a form factor for V1 equal to the isoscalar form factor and a surface derivative Woods-Saxon form factor for the absorptive term W1. Another preferred set had no energy dependence on V1 (V1 = 17.7 MeV) while W1 = (18.1 —0.31 E) MeV. The difficulty in these analyses arises from the fact that because of the high negative Q-value for the (p, n) quasielastic scattering the incident proton energy is about 10—20 MeV higher than the outgoing neutron energy. The energy dependence of the form factor for the absorptive term in the OMP, thus may be quite different from both nucleons. At lower energies (outgoing neutrons) a pure surface term may be used while at higher energies (incident protons) both a surface and a volume term are needed. The ~ potential which in a Lane model consistent approach [55] may be interpreted as a difference between U~,and U~will, therefore, have a radial dependence for W1 different from a simple derivative Woods-Saxon. As indicated in eq. (3.4), the cross section for the isobaric analog (IA) transition scales as (N Z), the neutron excess. However, several empirical studies with incident proton energies up to 30 MeV on Ti isotopes [811 and Mo isotopes [82] have indicated a considerable departure from the above Lane-model predictions. Madsen et al. [1291 have studied the effects of inelastic coupling on 0~analog transitions. In particular for the Mo isotopes, the coupling to the inelastic excited J~= 2~and JW = 3 states and their analogs result in destructive interference of the dominant one-step mechanism with three in phase three-step processes. This effect reduces the calculated DWBA cross section; the CC calculations account for the observed deviation of linearity of the IA ground state cross section with neutron excess. The CC analysis is generally done with potential parameters adjusted from spherical OMP parameters. Thus the interpretation of isovector potential depths obtained from studies of (p, n) IA —

I. Rapaport, An optical model analysis of neutron scattering

53

transitions at bombarding energies where multistep processes are important should be cautiously done. The question of the energy dependence of the multistep process relative to the one-step process has been recently examined by Madsen and Brown [130]. Their results indicate that at low energies (several tens of MeV) multistep inelastic transitions play a very important role in IA transitions. At higher energies (E 100 MeV) the multistep amplitudes fall off more rapidly than the one-step and become less important. At 135 MeV, the 0~IA is essentially free of multistep inelastic effects. A fourth method that has been used to study the isovector term is by a simultaneous analyses of several nuclei at several energies. Global OMP analyses have been reported in the literature for protons [18,241 and neutrons [61]. Neutron elastic scattering data (see table 1) in the energy range 7—26 MeV were selected [61] on nuclei considered to be spherical 40Ca, 90Zr, 92Mo, “6’1248n and 208Pb. In this form possible complications in the description of elastic scattering data because of strong coupling between the ground state and excited states were avoided. The results (table 2) indicate that the isovector real term was given by V 1 = (22.7—0.19 E) MeV and W1 13 MeV for E~ 15 MeV and W1 10 MeV for E~ 15 MeV. This energy dependence of V1, a = —0.19, is similar to the one obtained in the (p, n) quasielastic scattering of Patterson et a!. [941,a = —0.18. Theoretical OMP calculations have reported values for the linear coefficient a = —0.1 [95], a = —0.17 [96] and a = —0.1 [801. A value a = —0.12 ±0.06 has also been reported [971 obtained from empirical analysis of

2~ —.

0 0

0T

T

r.

~

£

O.~

0

j.

—.~

0

T

7

—.

0 00

00

0

•1 O-~ 1

0

0

8

I

I

I

I

10

30

E

E~~Ec

I 50

(MeV)

Fig. 15. Values of V~.for 1.0 fm Yukawa interaction. The line represents the energy dependence of the real term of the isovector potential V1 obtained from a neutron OMP analysis (see table 2).

J. Rapaport, An optical model analysis of neutron scattering

54

100 MeV proton elastic scattering data on several nuclei and its comparison with similar analyses at lower energies. The analysis of quasi-elastic (p, n) transitions may be done both in a macroscopic Lane model as indicated above or in a microscopic formalism. The first one tries to determine the isovector part of the OMP, U1(r), while the second method is used to determine the isospin independent term VT of the effective nucleon—nucleon interaction. In the absence of exchange forces, Madsen has shown [98] for zero-range, density independent force, that V1 and V~are related by the expression: U1(r)T t/A

=

V~E (N1

—

Z,)R1(r)/27rsJN_ Z,

where R1 is the radial wave function for the bound particle and (N1) and (Z1) are the expectation values of neutron and proton numbers in the f-shell. In a naive way the energy dependence of VT should correspond to the energy dependence of the isovector strength of the OMP. Values of V~ obtained by Do~ringét al. [991 at 25, 35 and 45MeVicidentprotowenergies;at22~30an&4O MeV by Jolly et al. [100] and values reviewed by Austin [101] up to 50 MeV are indicated in fig. 15. All values have been normalized to a Yukawa radial dependence with a range of 1.0 fm and are shown versus nucleon energy inside the nucleus. The straight line corresponds to the energy variation of V1 as obtained in the neutron global analysis [61] presented in table 2 (see also refs. [102,1031). A fair agreement is observed. 5.2. Analysis in terms of volume integrals per nucleon The analysis of the isovector dependence of the OMP as discussed in the previous section is subject to the intrinsic ambiguities in the values of geometrical parameters and potential depths. 2>R of the real However, volume per nucleon (JR/A)less and the mean-square part of thetheOMP may integral be determined with much ambiguity [25]. A radius similar(ranalysis for the imaginary term (J~/A)has also been suggested by Agrawal and Sood [104], by Hodgson [1051 and by Kailas and Gupta [106]. 5.2.1. The real term of the OMP We present in fig. 16 the volume integral per nucleon for the real part of the OMP obtained from the analysis of proton elastic scattering of 28Si [76], 40Ca [74] and 208Pb [601 in the energy range 15—80 MeV. All the data points seem to be clustered within ±3%uncertainty (shaded area) around a linear energy dependence given by JR/A

(498

—

2.9 E) MeV fm3

The solid curve corresponds to an energy dependence obtained by Nadasen et a!. [54] from the analysis of proton elastic scattering in the range 80—200 MeV. The reported expression JR/A

=(765

—

llOlnE)MeVfm3

indicates that the observed linear dependence does not hold for energies above 60 MeV as has also been previously noted [60,69,74].

I. Rapaport, An optical model analysis of neutron scattering

55

600

~:

500

o

p+

o

~+

28S1 40Co

p+208Pb (~98 - 2.9E)

———

(765- ibm

E)

‘V

‘1)

0

:

300

1-

I

I

20

I

I

L~0

I

I

I

60

E (MeV) Fig. 16. Values of the proton volume integral per nucleon for the real term of the OMP for 28Si(p, p) [76], 40Ca(p, p) [74] and 208Pb(p, p) [60]. The linear dependence is a least square fit while the In E dependence is from ref. [54].

The empirical data in fig. 16 shows that (~R/A) for protons has at a given energy a value which within 3% is independent of the mass A of the nucleus studied. To emphasize this effect we present in fig. 17 empirical values of ~ /A) to which the value 2.9 E has been added. Values obtained from the OMP analysis of proton elastic scattering from 64’66’68Zn [1071, all the data reported by Fricke et al. [108] at 40 MeV and all the data reported by Mani et a!. [109] at 49.35 MeV have been included together with the data shown in fig. 16. It is interçsting to note that out of about 100 data points including nuclei from Si to Pb only 16 points fall outside the range (500 ±10) MeV fm3 in fig. 17 and only 7 outside a 3% uncertainty. A similar graph but now using (JR/A) values obtained from the OMP analysis of neutron elastic scattering from Ca, 92Mo and 208Pb in the energy range 7—30 MeV is shown in fig. 18. Again the value (JR/A) ÷2.9 E is presented versus energy. The shaded area represents a 3% uncertainty, while the horizontal line at a value (JR/A) 500 MeV• fm3 represents the “proton-line” obtained from fig. 17. It is apparent from fig. 18 that the (JR/A) values deduced from the neutron OMP analyses do have a mass dependence. Kailas et al. [1101 based on low energy proton elastic scattering data (E~ 10 MeV) and the data between 80—200 MeV reported by Nadasen [54] have paranieterized the value of (JR/A )~ for protons as:

56

J. Rapaport, An optical model analysis of neutron scattering

0

500

-_______

(~)+29E

~

~+

0

p+ Ca 208Pb p’~ + 64’66’68Zn Fricke et al. (140 P1eV) Mani et al, (50 MeV)

• A

40

1400

I

I

I

E (MeV) Fig. 17. Values of the proton volume integral per nucleon (JpjA)~,plus the energy dependent term 2.9 E for ~S~(p, p) 40Ca(p,p)[74],Zn(p,p)[107],andvaluesatE~=40MeV(108]andatE~=5oMeV[1o9].

(JRIA)P

=

[761,

509 exp(—0.006 E)MeV• fm3

At low energies the above expression may be written as: (JR/A)P

‘~

(509

—

3.05 E) MeV fm3

in close agreement with the present results. The extrapolation to lower energies (E~~ 10 MeV) near the Coulomb barrier has been studied by Eck and Thompson [1111. They indicate that the data show a more rapid energy dependence of the real potential depth than the linear coefficient indicated above, and it is explained in terms of the non-locality of the nucleon—nucleus interaction. Within the reformulated model of Greenlees et a!. [251 with the assumption of a local effective interaction the values of JR/A are expressed (eq. (3.9)) as:

J. Ropaport, An optical model analysis of neutron scattering

+

500

57

2.9E

•_-.__

\ \

\ \ \ \®\

\ \

\‘

40~

n+

-—\—-~-~--~-~-—-

L45Q

-

92 -

-

n

+

n

+

Mo

208

4~ /~/o//°//////

I

I

2~

I

I

Pb

-

I

E (MeV) Fig. 18. Values of the neutron volume integral per nucleon (JR/A)fl plus the energy dependent term 2.9 E for 40Ca(n, n) [32], 92Mo(n, n) [34) and 208Pb(n, n) [33]. The dotted line represents the average values for protons.

JR/A

=J~± ~

where +(—) is for protons (neutrons) and ~ = ~ 0.48 is obtained from the parameterization of the two-body nucleon—nucleon potential. The empirical (~RIA )~ values for protons do not follow the above equation indicating values independent of A or e. However, when the real potential depths VR obtained in the same OMP analyses are plotted as a function of e, a positive slope is obtained and values for V 1 ~ 24 MeV have been obtained for the isovector potential. As pointed out by Jeukenne, Lejeune and Mahaux [80] the striking difference in the results may be at first sight ascribed to the bracketed term in the expression of the volume integral per nucleon (eq. (3.10)): 3)2]}. ~rVR{r~[l + (lra/rRA” However, they show [801 that this interpretation is really incorrect and that the explanation is (JR/A)

=

58

.1. Rapaport, An optical model analysis of neutron scattering

due to density dependent effects in the effective nucleon—nucleus interaction. Jeukenne et a!. [80] proceed to calculate the contributions to the real volume integral per nucleon due to the isoscalar term which shows a decrease with A, the Coulomb correction term and the isovector term, both of which increase with A. The sum of these three terms result in values of (JR/A) independent of A (for A ~40). The proton energy at which these values were calculated was 35 MeV and the resulting (~RIA) 400 MeV• fm3 is in excellent agreement with the empirical results shown in fig. 16. Satchler [1121 has suggested a simple approach to illustrate this density dependent effect. It does not replace, of course, the only reliable way to obtain it, that is to do a proper calculation as has been done in ref. [801. The nucleon elastic scattering is given approximately by an optical potential [25]

f~

u

E

u 01~Dd~,

where i~,(~)is a normalized wave function describing the ground state of the target nucleon and u01 is the nucleon—nucleon potential. Using nuclear matter density distributions p(r), the above folding integral may be written: U(r1)

fp(r2)u(r12) d”2~

in the usual local approximation and leads to the results in ref. [25]. If we now assume that the interaction is density dependent (see fig. 19): u

=

v[r12, p(?)]

then U(r1) = jp(r2)v[r12, P(~)1dr2 and the volume integral is given by

*

~r2

Fig. 19. The coordinates for the folding integral.

r

R

>

r

Fig. 20. Shape of a density distribution to show density dependent effects.

.1. Rapaport, An optical model analysis of neutron scattering

59

Jdr1 fdr2 p(r2)v[r12, pfr)]

J

If we chose ~~ r2, then u

=

v[r12, p(r2)]

and J4ir fr2 drp(r)J~[p(r)] where J~(p)= fv(p) r~2dr12 If we assume a spherical distribution with density p0 with an added skin (see fig. 20) such that p(r) in the surface is just ~(p0) and J~(p0)is replaced by J~(~p0), then to order (t/R) the folded potential has an integral per nucleon:

I 3tx—ll 113 JIMeVfm 3 J/A~J(p)Il+— v ° L 2r0 A whereR —r 113 andx = 0Aassume t ~ 2.5 fm, r If we next 0 ~ 1.2 fm andx 1.5 [80], then 3] = KJ~(p 3. (5.1) J/A J~(p0)[1 + 1.6/A” 0)MeV fm The values of K in eq. (5.1) represent in an approximate manner the effects of density dependent effects in the volume integral per nucleon. It should also be pointed out that the density dependent interaction is energy dependent, thus the ratio x in the above expression, is sensitive to the nucleon energy [801. The calculated values of the volume integral per nucleon are indicated in table 4 for A = 40, 90 and 208. Values for (JR/A) using eq. (3.10) have been calculated assuming a Woods-Saxon form factor with rR = 1.2 fm and aR = 0.65 fm. The approximate value for the density dependent effects, K in eq. (4.3) are given in the second row. The values of (JR/A) divided by K shown as (J’/A) values in table 4 seem to be independent of A. These new values are just proportional to the values of VR. Thus any dependence on energy and/or mass obtained from a study of potential depths is reflected on the values of(J’/A). These dependences are hidden in the values of(JR/A) because of density dependent effects. Thomas, Sinha and Duggan [131] have studied the effects of density-dependent effective interactions derived from nuclear-structure calculations in the proton OMP. They also included the exchange term in a simple local approximation. The results indicate that volume integral values and r.m.s. OMP radii are sensitive to the density-dependent assumptions in the effective interaction. Off-shell effects are apparent from the worsening of the quality of fit for light nuclei. Values of (~R/A )~displayed in fig. 21 as a function of the nuclear asymmetry e = (N — Z)/A

60

J. Rapaport, An optical modelanalysis of neutron scattering Table 4 Density dependence effects in (J/A) t

(JR/A) 3]

K=[1+1.6/A” f/A

=

(JR/A) . K’

A=40

A=90

A208

9.03 VR 1.47

8.28 VR 1.36

7.84 VR 1.27

6.14 VR

6.09 VR

6.17 VR

t Values in [MeV.fm3]. have been obtained in the OMP analysis of 11 MeV neutrons [31]. The experimental points are indicated as open circles. The solid squares are the theoretical calculations of ref. [80] using what the authors called an “improved local density approximation”. The agreement with the empirical results is excellent. The solid line is a least square fit which may be represented as: (JR/A)fl

=

(443 ±l0)[1 —(0.7 ±0.l8)e] MeV fm3

Exp (E~= ii MeV)

0

• Theory 500

U

~

400

•

o=..• 0

0

T ~

300

1:

I

0.05

I

0.10

I

N—Z

0.15

I

0,20

(-A-) Fig. 21. Values of the neutron volume integral per nucleon (JpjA)~obtained at if 0 = 11 MeV (circles) [311. The solid squares represent microscopiccalculations with a density dependent effective interaction [80].

J. Rapaport, An optical model analysis of neutron scattering

61

“a”

It should be noted that the slope = 0.7 ±0.18 cannot be simply equated (as suggested in ref. [25])to the ratio of (v~/v0).As shown above the density dependence of the effective interaction does not make possible this simple comparison. The values of (JR/A)fl displayed in fig. 22 have been obtained in the OMP analyses of 24 MeV and 26 MeV neutrons. The line is a least square fit which may be represented as: 90± l4)[l —(0.6±0.2)e] MeV’fm3 (JR/A)fl(3 The slight change of the slope with neutron energy is indicative of an energy dependence of the isovector term of the OMP. Because of density dependent effects, sets of values of (JR/A) for either protons or neutrons are not suitable to study empirically the isovector contribution to the OMP. However, with the assumption that density effects are the same for both neutrons and protons one may attempt to obtain the isovector contributions from a comparison of these (~RIA) values. To illustrate the case, (JR/A) values are presented in fig. 23 for p + 208Pb, ref. [60] for 20—80 MeV and ref. [111] for 10—15 MeV together with the values of n + 208Pb [33]. The proton values are best represented by:

500—

•

E~=26MeV En = 24 MeV

400 0 • —

• • —

300

—

I

0.05

I

I

I

0.10

0.15

0.20

I N-Z Fig. 22. Values of the neutron volume integral per nucleon (JR/A) obtained atE 0 = 24 and 26 MeV. An uncertainty of five percent is assumed in the experimentally derived values.

62

J. Rapaport, An optical model analysis of neutron scattering 0

500

0 0 0~

-

0

208Pb

p+

• n

+

400

~

~

300

-

(*)n

=

~)

+

200

100

~

(~v

1)

A I

I 20

I 40 E (P1eV)

I 60

I

80

208Pb. The Fig. open 23. Values circlesofrepresent nucleonploton volumedata integral [601 per while nucleon the solid (JR/A) circlesobtained representinneutron the OMP data analysis [331.Tofhenucleon calculated scattering contribution from ofthe isovector term are indicated at the bottom of the figure.

(JR/A)P

=

(490 ±8)

—

(2.65 ±0.36) E~[MeV. fm3],

while the neutron values by: (JRIA)fl

=

(392 ±5)— (2.2 ±0.36)E~[MeV fm3].

The difference between these two quantities is expressed by:

J. Rapaport, An optical model analysis of neutron scattering 1’R/~’4)p—

(‘

(~R/’4)~ =

63

(J.~~~/~4) + 2e(J~ 1/A).

With the empirical value obtained for the Coulomb correction term and assuming 3it has the same is estimated. geometry as the isoscalar part of the OMP, a value J±v~/A = (52 ± 8) MeV fm Thus, the isovector contribution is given by: (J~,/A)(110±25)—(l.2±0.8)E [MeV’fm3] or in terms of potential depth: V 1

(14±3)—(0.16± 0.l)E[MeV]

A similar analysis done by DeVito [75] gives equivalent results. The energy dependence of V, is in agreement with the value obtained with the global neutron OMP analysis (table 2). The value (14 ±3) MeV at zero energy is rather smaller than the value V, 22 MeV obtained from the global neutron analysis but is in better agreement the value 51V and with 59Co indicated (18.5 ± 1.5) MeV obtained from the simultaneous analyses of n and p on previously. It also agrees with the value V, 17 MeV reported by Carlson et al. [93] from quasielastic scattering analyses. Jeukenne, Lejeune and Mahaux [113] have done a microscopic calculation of the symmetry potential with Reid’s hard-core nucleon—nucleon interaction in the framework of the Brueckner—Hartree—Fock approximation. Their reported value is V 1 (11.5—0.1 E) 3 fm larger than the usual radius R 1.2 A”3 MeV with an isovector potential radius R 1.31 A” fm used for the isoscalar part of the OMP. Calculating the volume integral per nucleon for 208Pb the resulting expression is: -~

(J~,/A)~(113— l.OE)MeVfm3 in excellent agreement with the results indicated above. 5.2.2. The absorptive term of the OMP The empirical absorptive part of the OMP is assumed to have a surface or volume form or a sum of both types depending of the incident nucleon energy. As it was discussed in the case of T = 0 nuclei, the study of the energy and/or isospin dependences of the imaginary term of the OMP is complicated if done in terms of potential depths not only because of the well-known ambiguities associated with the OMP parameters but also because the radial dependence of W, which is energy dependent, is not empirically well-known. In general the imaginary potential is a characteristic of the nucleus to be studied and of the incident nucleon energy. It will depend on the number of open channels, and the density of states available. Recently Osterfeld, Wambach and Madsen [114] have presented a fully antisymmetric, microscopic formulation of the calculation of the imaginary optical potential for.n +40Ca using nuclear RPA wavefunctions. The reported calculated [114] shape of W for 30 MeV incident neutrons is similar in shape (volume) to the phenomenological shape [61] but somewhat weaker. Another microscopic calculation of the n + 40Ca imaginary OMP in the energy range 10—50 MeV has also been recently reported by Bouyssy, Ngo and Vinh Mau [115]. The theoretical formalism is close to that of ref. [1141 and is based on a nuclear structure approach. The calculated shape [1151 of W for 10 MeV neutrons is mainly surface in

64

J. Rapaport, An optical model analysis of neutron scattering

:

120

~

~

A

_______

-

~80

o p

+ 40

Ca x Fricke et al (40 MeV) 4’66’68Zn A ~ + ‘ • p+208Pb op+

-

40—

I

I 10

I 20

I

I

30 E (P1eV)

‘40

I 50

I

Fig. 24. Values of the proton absorptive volume integral per nucleon (J~/A).See fig. 17 for references.

agreement with the empirical results; however, the absorptive potential is also smaller than the empirical one. The calculated energy dependence of (J~/A)nin the 11—26 MeV range agrees well with the empirical one but it is a factor of about 2 too small. This type of calculations with indication of the energy dependence of the radial dependence of W should help to guide future empirical studies. We present in fig. 24 values of (JW/A)P for protons obtained from the references used to obtain values for (~R/A )~ in fig. 17. In spite of presenting several nuclei for which nuclear structure effects should be quite different, it is quite remarkable as it was the case for the volume integral per nucleon for the real term of the OMP, that the empirical values of (J~/A)~ (shaded area in fig. 24) are independent both of the target mass and incident proton energy between 30—60 MeV. In fig. 25 and fig. 26 we present empirical values for the surface derivative Woods-Saxon and volume Woods-Saxon volume integral per nucleon obtained from the same references. The scatter observed in the empirical values on each component volume and surface is much larger than that observed from the sum (fig. 24) indicating the lack of sensitivity of just empirical OMP analysis to the separation of the absorptive part in the two components. The values of (JwD/A) fig. 25 seem to increase linearly with proton energy for 10 E~~ 23 MeV. This behavior is physically expected because as the incident nucleon energy increases, more inelastic channels are opened which results in increased absorption. Above approximately 23 MeV incident proton energy the values of (JwD IA) seem to decrease linearly to a value close to zero at about 60 MeV. On the other hand the empirical values of (Jw~/A)which are empirically taken to be zero up to about 20 MeV seem to increase with energy according to an approximate expression -

J. Rapaport, An optical model analysis of neutron scattering

A A A

A

-

65

So

A SAD

0

/A•”\

/

A

0 •‘\~

/

80-

000\o

I—0

~~c1

/

o\

/

~

\

a

A 0

40~

°°

1~ I

I 20

1.1

10

I._..._1 ~0 E (P1eV)

I

I

40

A A A S

I

50

Fig. 25. Values of the surface proton absorptive volume integral per nucleon (JWD/A). See fig. 17 for references. The dotted line represents a linear energy dependence.

MeV fm3 (Jwv/A) ~ 100 [1 e_°~°4~~’8)] For incident proton energies in the range 60 ~ E ~ 200 MeV only a volume absorption is empirically used with a value (J~/A) 100 MeV• fm3, both independent of mass and energy [541. Leeb and Eder [116] have analyzed available proton elastic scattering data (75 sets) and proton polarization data (60 sets) where the bulk of the data is for proton energies less than 100 MeV, in order to obtain OMP parameter systematics from which to obtain meaningful physics information. They parameterize the energy dependence for the surface absorption according to —

~-

WD~(a+bE)Ee~~E where a, b and c are parameters to be obtained from the data fit. They also obtain an analytical expression for the volume absorption W,,,,, with parameters obtained from the total nucleon— nucleon cross section for nucleon energies above 100 MeV where the use of the impulse approximation is justified [117]. The reported expression extrapolated linearly for E 0 is -~

W~, [a 1

—

a2(1 ÷exp{(E

—

a3)/a4}y’]

The energy dependences for the volume integrals obtained from the above expressions have

66

J. Rapaport, An optical model analysis of neutron scattering

120

—

100 [1-exp (-0,04 ~E-18~)1 E

0

40

I

I

lOll

I

I

E (P1eV) Fig. 26. Values ofthe volume proton absorptive volume integral per nucleon (Jwv/A). See fig. 17 for re~erences.

similar shapes to the empirical data presented in figs. 25 and 26 except that the predictions are higher than the indicated values for 15 MeV. In figs. 27 the value of (J~/A)~obtained from neutron elastic scattering data are presented versus neutron energy. The T = 0 nuclei, 28Si, 32S and 4°Caseem to have similar values but the results for n + 208Pb seem to be much lower reflecting an isospin dependence as it was the case for the real term of the neutron OMP (fig. 18). The values of (JW/A)~ obtained from neutron scattering are presented in figs. 28 and 29 as a function of nuclear asymmetry, e. The data obtained with 11 MeV neutrons [31] are shown in fig. 28 while data at 24 MeV and 26 MeV are presented in fig. 29. In both cases a similar dependence with e is observed: (Jw/A)

3 0 (l05± 5) [1 —(2±0.3)e] MeVfm No apparent energy dependence is noted. The discussion presented previously about the mass (or isospin) independence observed in the empirical values of (~R/A )~for protons is also valid in this case. There are three contributions to the proton volume integral: a) an isoscalar absorptive part, b) a positive isovector contribution which increases with A, and c) a negative Coulomb correction term. The sum of these terms calculated using a local density approximation by Jeukenne, Lejeune and Mauhaux [801 for 35 MeV incident protons give a value of about 100 MeV fm3 for A 50 in good agreement with the em-

J. Rapaport, An optical model analysis of neutron scattering

67

0

Sl(n~n)

A ~ 0

120

~

~

-

I

180

I 10

I

•

(nan)

Ca(n~n)

208Pb(n~n)

1

I 20

I 30

I

I

I 40

I

I

E (MeV) Fig. 27. Values of the absorptive volume integral per nucleon

(Jw/A)~for neutrons. See fig. 18 for references.

pirical results. The density dependent effects are important and thus again the simple expressions for volume integral per nucleon [25] should be used cautiously. De Leo and Micheletti [118] using an analysis similar to the one presented in fig. 23 have indicated that the differences between the imaginary volume integral per nucleon felt by neutrons and protons on 208Pb show evidence for an energy decrease of the isovector imaginary term of the OMP. For the absorptive Coulomb correction part they used the empirical values obtained from ref. [731.The reported result is written as: J~,/A = (147 ±10)

—

(1.7 ±0.5)E

10

E

40

[MeV fm3]

If a derivative Woods-Saxon form factor is assumed for W 1, the potential depth is given as: W1(E) = (15 ±1) —(0.18 ±0.05)E

10

E

40

[MeV]

This expression gives values for W, in agreement with those obtained from the neutron global OMP analysis reported in ref. [61] (see table 2).

68

120 ‘ii 4> a)

0

40

I

0.05

I

I

0.10

0.15

0.20

Fig. 28. Values of(Jw/A)~for 11 MeV neutrons [31] versus nuclear asymmetry, e = (N

• En

=

26 P1eV

En

=

24 P1eV

o

—

Z)/A.

120

to-

a)

~

80

40

I

0.05

~

I

I

(NZ’t\

0.15

0.20

Fig. 29. Values of (JwIA)n for 24 MeV and 26 MeV neutrons versus nuclear asymmetry, e = (EV — Z)/A.

J. Rapaport, An optical model analysis of neutron scattering

69

6. The OMP for nucleon energies higher than 60 MeV Proton elastic scattering at energies higher than 60 MeV have been analyzed by several authors [54,60,65,69,74]. The recent analysis of Nadasen et al. [54]for protons up to 180 MeV indicate that the real volume integral of the OMP may be described as: (JR/A)P

(765±10)— 110 lnE~ 40E~

180

[MeV.fm3]

which has been displayed in fig. 16. No systematic target dependence is observed and the scatter in the individual values of (~R/A )~ is of the order of 5% about an energy-dependent average value. The values of (J~/A)are seen to scatter somewhat more (±20%)around a mean value of about 100 MeV fm3. The quoted value for the isovector strength [54] V 1~59(1 —0.18lnE~) (60E~ 180)

[MeV],

has the ambiguities associated with the treatment of the Coulomb correction term, and should be studied with more detail. Unfortunately, no high energy neutron elastic scattering data are available to do this study. Another method is to analyze (p. n) quasi-elastic differential cross section at intermediate energies. The macroscopic cross sections depend (eq. (3.4)) on U, = V, + iW, and the calculated shapes are rather insensitive to how much real and absorptive potential depth are used in the isovector potential. A possible alternative to determine the real and imaginary strengths is by measuring neutron asymmetries in isobaric analog transitions with a polarized proton beam.

7. The spin-orbit potential No observables directly related to the spin-orbit potential were measured in the neutron elastic scattering experiments presented in this paper. In the OMP analyses, parameters for this term of the potential were kept fixed to values obtained in the literature determined mainly from polarized proton elastic scattering data. Hodgson [9] has reviewed the earlier studies and more recently Nadasen et al. [54] have extended these studies up to 180 MeV. Cooper and Hodgson [1191 have used energies of bound single-particle states and the results of OMP analyses of nucleon scattering to obtain an energy dependence of the spin-orbit potential depth: =

(6.5 ±0.5)

—

(0.023 ±0.012)E

[MeV],

with no evidence for dependence on nuclear asymmetry. Brieva and Rook [120]have calculated the spin-orbit part (V~+ iW~)of the OMP starting from the Hamada—Johnston inter-nucleon potential. They find that the radial shape of V~,is energy independent and peaks in the nuclear surface, with a slow decrease of the strength with energy in agreement with the empirical parameterization. The energy dependence of W~is greater, W~/V~ —0.05 at E 20 MeV while W~/V~ ~ —0.2 at 200 MeV. Empirical values for V~and W~are reported in ref. [54] for energies between 80 and 180 MeV. -~

70

J. Rapaport, An optical model analysis of neutron scattering

8. Inelastic neutron scattering The elastic neutron cross section discussed previously represents the largest of all partial cross sections. The next largest usually occur for inelastic scattering to a few low-lying excited states, known as collective states due to their large electromagnetic transition rates. These transitions are generally described in the collective model as oscillations about a spherical mean (vibrations) or as the rotations of a statically deformed shape. The simplest way of calculating the inelastic cross section is to use perturbation theory in the form of the distorted-wave Born approximation (DWBA). In some cases a strong excitation of these collective states is observed which implies a strong coupling between the ground state and inelastic channels. Then, the above DWBA is not adequate and it is then necessary to solve the appropriate coupled channel equations. In this section the nucleon excitation of the first collective 2~state for closed shell nuclei (either N or Z) are presented. Within the DWBA the hadron excitation of a collective state [121] is approximately proportional to the square of a transition multipole matrix element M which may be written as [1221 M

=

bnMn +

where b~~) are the external field neutron (proton) multipole strengths and M~~) the neutron (proton) multipole matrix elements 2 dr. M~(~) = fp~P)(r) r~

The quantities p~’~ are the neutron (proton) transition densities. Ratios of b~/b~ are given by Bernstein, Brown and Madsen [122] for different probes used to excite these collective states. It is shown that for low energy (10—50 MeV) hadron excitation, b 0/b~ 3 for a proton probe and ~ 1/3 for neutron probes. Thus it is stated [122] that a comparison of neutron and proton inelastic scattering is sensitive 4~ to the neutron and proton probes in the 160—200 MeVtransition region. densities. A similar sensitivity is supposed to exist for r~within the DWBA formalism, the transition amplitude is proportional to For vibrational nuclei a scaling factor 6, or deformation length with 6 = j3R M in which $3 is the deformation parameter. In the usual collective model it is assumed that neutrons and protons move in phase with the same amplitude and thus M 6EM where the external = N/Z which turn impliesprotons. 6~’= ~‘A systematic = electromagnetic (EM) probe0/M~ interacts only withinthe nuclear deviation from the above ‘simple picture is expected due to shell-structure effects. These effects have been calculated by Brown and Madsen [123] and have been observed experimentally [124]. Therefore, in general M~/M~ * N/Z and the neutron or proton contributions to a given transition may be empirically determined by a comparison of deformation lengths obtained from hadron scattering. Such a comparison has been teported for excitation of the 2~in nuclei with N5ff[T25ffor nuclei withZ = 50 [126]for 180 [36] and fornuclei withN 28 orZ = 28 [1271. The OMP which is expected to follow the shape of the nuclear matter distributions has been extended to provide a macroscopic description of inelastic scattering. The effective transition amplitude 13h UN may in turn be decomposed [123]in isoscalar and isovector contributions: $3hUN$3oUo±$3,EU,

/.

Rapaport, An optical model analysis of neutron scattering N=50

0.15

71

Z=50

a)

-

S

a

0.10

0.0E

I

,

I

I

I

I

I I-

0.2-

0,1

-~

~

-~

-

-

ao,o II

a1

-0.1

T I

S

(nan’)

~

0

(pep’)

i7ca 1 I.J1 -0.2

I

88

I

I

90

92

116

I

I

I

I

118

120

122

124

A— Fig. 30. (a) Comparison of experimentally obtained deformation parameters for protons and neutron incident onN = 50 and Z = 50 (tin isotopes) targets. (b) Isoscalar (p0) and isovector (i1~)deformation parameters. The no-parameter schematic-model predictions [123] for (~iusing the experimental values of pa,, are shown with the dashed lines.

where $3h may be either ~ or $3~ for incident protons or neutrons. Values extracted for $3~ and ~ are reported in refs. [125], [1261 and [361 forN= 50, Z 50 and Z = 8 nuclei respectively. The N = 50 and Z = 50 results are presented in fig. 30. The values of $3, for Z = 50 are compared with the schematic-model predictions of Brown and Madsen [123, 128]. 9. Conclusions The analysis of neutron scattering differential cross section has permitted the evaluation of em-

72

J. Rapaport, An optical model analysis of neutron scattering

pirical OMP parameters and the study of the dependence on energy and mass. It has been shown that for T = 0 nuclei a comparison of neutron and proton OMP analyses leads to evaluate both the real and absorptive part of the Coulomb correction term which has been parameterized as =

(0.48 ±0.l)—~--f(r) + i[—(l A”3

—

0.028 E) (0.5 ±0.1) A’!3 g(r)] —~---

[MeV],

in the energy range 10 E 40 MeV. In the above expression f(r) represents a volume form factor identical to the isoscalar part of the potential and g(r) a derivative Woods-Saxon. It is indicated that empirical data show that for the real term of the OMP, values of the integral per nucleon for protons are roughly mass independent (A ~ 28) with a linear dependence with energy (10 E 80 MeV). This contradiction with the reformulation model of Greenlees et al. [25] is apparently due to density dependent effects as indicated in ref. [80]. Similar density dependent effects are noted in the values for the real volume integral per nucleon for neutrons. In the case of 208Pb a comparison between neutrons and protons JR/A values is used to obtain the isovector strength and its energy dependence: (J~,/A)—’(110±25)—(l.2±0.8)E, 10E40

tMeV’fm3]

or in terms of potential depth V 1

-~

(14 ±3)— (0.16 ±0.l)E

[MeV].

A similar energy dependence has been obtained from a neutron global OMP. Theoretical calculations of the OMP by Jeukenne, Lejeune and Mahaux [801 indicate a remarkable agreement with the above values. A similar study of the absorptive part of the OMP also shows dependent effects and 3 for protons in the density energy range 20—200 MeV anda rather constant value Again (J~/A)~ 100 MeV fm and proton comparison for (.J~/A)values, assumindependent of mass. for 208Pb a neutron ing the empirical Coulomb correction, is reported by De Leo and Micheletti [118] to obtain the imaginary isovector strength and its energy dependence: (J~,/A)~(l47±l0)—(l.7±0.5)E,

l0E40

[MeVfm3]

with apotentialdepth: W

1(E)

=

(15 ±1) —(0.18 ±0.05)E

[MeV]

The above values are in good agreement with the theoretical calculations reported in ref. [80]. A Lane-consistent OMP is used to analyze neutron and proton inelastic scattering to the first 2~collective state in single closed shell nuclei. The analysis is used to obtain isoscalar and isovector deformation parameters and to obtain the ratio of neutron to proton transition densities. The results are in agreement with calculations in the schematic model of Brown and Madsen [1231.

J. Rapaport, An optical model analysis of neutron scattering

73

Acknowledgments The author wishes to thank Professor R.W. Finlay and Dr. G. Randers-Pehrson for many fruitful comments. Many illuminating discussions and correspondence with G.R. Satchler are gratefully acknowledged. The fine typing of this manuscript by Ms. Cindy White is sincerely appreciated. The permission given by North-Holland Publishing Company to reproduce figs. 4—9 and 13 is sincerely acknowledged. This work was supported in part by the National Science Foundation. References [1] E. Rutherford, The London, Edinburgh and Dublin Philosophical Magazine and Journal ofScience 21(1911) 669. [2] H.H. Barschall, Phys. Rev. 86 (1952) 431. [3] H. Feshbach, C.E. Porter and V.F. Weisskopf, Phys. Rev. 96 (1954) 448. [4] H.A. Bethe, Phys. Rev. 47 (1935) 747. [5] P.E. Hodgson, Congr. Intern. de Physique Nucléaire, ed. P. Gugenberger (Editions du CRNS, Paris 1964) p. 309. [6] G.E. Brown, Revs. Mod. Phys. 31(1959)893. [7] H. Feshbach, Ann. Rev. Nucl. Sci. 8 (1959) 49. [8] A.K. Kerman, H. McManus and R.M. Thaler, Ann. Phys. 8 (1959) 551. [9] P.E. Hodgson, Nuclear reactions and nuclear structure (Clarendon Press, Oxford, 1971). [10] J.-P. Jeukenne, A. Lejeune and C. Mahaux, Phys. Rep. 25 (1976) 83. [11] C. Mahaux, Microscopic optical potentials, Lect. Notes in Phys. 89, ed. H.V. Geramb (Springer, Berlin, 1979) p. 1. [12] F. Brieva and J.R. Rook, Nucl. Phys. A307 (1978) 493. [13] N. Vinh Mau and A. Bouyssy, Nucl. Phys. A257 (1976) 189. [14] N. Vihn Mau, Microscopic optical potentials, Lect. Notes in Phys. 89, ed. H.V. Geramb (Springer, Berlin, 1979) p. 40. [15] V. Bernard and Nguyen Van Giai, Nucl. Phys. A327 (1979) 397. [16] V.A. Madsen, F. Osterfeld and J. Wambach, Microscopic optical potentials, Lect. Notes in Phys. 89, ed. H.V. Geramb (Springer, Berlin, 1979) p. 151; Phys. Rev. C23 (1981) 179. [17] L.G. Arnold, B.C. Clark and R.L. Mercer, Phys. Rev. C19 (1979) 917 and references therein. [181 F.G. Percy, Phys. Rev. 131 (1963) 745. [19] F. Percy and B. Buck, Nuci. Phys. 32(1962)353. [20] L.J. Rosen, G. Beery and A.S. Goldhaber, Ann. Phys. (N.Y.) 34 (1965) 96. [21] M.P. Fricke, E.E. Gross, B.J. Morton and A. Zucker, Phys. Rev. 156 (1967) 1207. [22] G.R. Satchler, Nucl. Phys. A92 (1967) 273. [23] P. Kossanyi-Demay, R. de Swiniarski and C. Glaushausser, NucI. Phys. A94 (1967) 513. [24] F.D. Becchetti and G.W. Greenlees, Phys. Rev. 182 (1969) 1190. [25] G.W. Greenlees, G.J. Pyle and Y.C. Tang, Phys. Rev. 171 (1968) 1115. [26] G.R. Satchler, Cli. 9 in Isospin in Nucl. Phys., ed. D.H. Wilkinson (North-Holland, 1969) p. 390. [27] B. Sinha, Phys. Reports C20 (1975) No. 1. [281 F.G. Perey, in: Nuclear spectroscopy and reactions Part B, ed. J. Cerny (Academic Press, 1974) p. 137. [29] Nuclear optical model potential, Lect. Notes in Phys. 55, eds. Boff and Passatore (Springer, 1976). [30] Microscopic optical potential, Lect. Notes in Phys. 89, ed. H.V. von Geramb (Springer, 1978). [31] J.C. Ferrer, J.D. Carison and J. Rapaport, Nucl. Phys. A275 (1977) 325. [32] J. Rapaport, J.D. Carlson, D. Bainum, T.S. Cheema and R.W. Finlay, NucL Phys. A286 (1977) 232. [33] J. Rapaport, T.S. Cheema, D.E. Bainum, R.W. Finlay and J.D. Carlson, Nucl. Phys. A296 (1978) 95. [34] J. Rapaport, T.S. Cheema, D.E. Bainum, R.W. Finlay and J.D. Carison, Nucl. Phys. A313 (1979) 1. [35] J. Rapaport, M. Mirzaa, H. Hadizadeh, D.E. Bainum and R.W~Finlay, Nucl. Phys. A341 (1980) 56. [36] P. Grabmayr, J. Rapaport and R.W. Finlay, NucI. Phys. A350 (1980) 167. [37] J.D. Carison, Nuci. Inst. 113 (1973) 541. [38] R.W. Finlay, C.E. Brieut, D.E. Carter, A. Marcinkowski, S. Mellema, G. Randers-Pehrson and J. Rapaport, Nuci. Inst. and Meth. to be published. [39] P. Grabmayr, J. Rapaport, R.W. Finlay, V. Kulkarm and S.M. Grimes, in: Proc. NucL Cross Sect, for Technology NBS SP94, eds. Fowler, Johnson and Bowman (U.S. Dept. of Commerce, 1980) p. 542.

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