- Email: [email protected]
The optical potential and nuclear structure
THE OPTICAL POTENTIAL AND NUCLEAR STRUCTURE
Bikash SINHA Wheatstone Laboratory, King’s College, London WC2R 2LS, UK. and The Niels Bohr Institute, Copenhagen, Denmark
(p1~C
NORTH-HOLLAND PUBLISHING COMPANY
—
AMSTERDAM
PHYSICS REPORTS (Section C of Physics Letters) 20, no. 1 (1975) 1—57. NORTH-HOLLAND PUBLISHiNG COMPANY
THE OPTICAL POTENTIAL AND NUCLEAR STRUCTURE Bikash SINHA Wheat stone Laboratory, King’s College, London WC2R 2LS, U.K. and The Niels Bohr Institute, Copenhagen, Denmark Received February 1975 Abstract: The optical potential for single and composite particle projectiles has been derived and discussed in detail for incident energy range 25 a~E ~ 90 MeV. The real central part has been calculated by folding an effective two-body interaction with the nucleonic density distribution. The importance of using an effective interaction with the correct saturation properties has been emphasized and discussed in detail. The possibility of identifying the optical potential as a l-Iartree—Fock single particle potential has also been explored. The imaginary potential has been discussed both from a second-order perturbation formalism and a simple forwardscattering amplitude approximation. The utility of the later approach for straightforward data fitting has been discussed in detail. Using the Blin-Stoyle prescription of the spin-orbit potential a three parameter optical model has been developed. Such models have been extended for inelastic scattering, composite particle scattering and heavy-ion scattering.
Contents: 1. Introduction 1.1. The historical background 1.2. Purpose of the optical model analysis 2. Foundation and formalism of the optical potential 2.1. The formalism 2.2. The optical potential 2.3. The spin—orbit potential 24. The phenomenological optical model 3. The microscopic optical potential 3.1. The folding model 3.2. Effective interactions and the optical potential 3.3. The exchange potential 3.4. Microscopic optical model analysis
3 3 4 4 4 8 8 12 12 19 25 30
3.5. Hartree—Fock single particle potential in the continuum 3.6. Second-order optical potential 4. The imaginary potential 4.1. Energy-conserving transitions 4.2. Forward-scattering amplitude approximation 5. Application to other fields 5.1. Composite particles 52. Inelastic scattering and effective interactions 5.3. Heavy-ion scattering 6. Summary and conclusions Acknowledgement References
Single orders for this issue
PHYSICS REPORTS (Section C of Physics Letters) 20, no. 1(1975)1—57 Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check.
L
Single issue price Dfl. 20.—, postage included.
35 38 40 40 42 48 48
so 54 55 56
B. Sinha, The optical potential and nuclear structure
3
1. Introduction 1.1. The historical background The scattering of particles as a means to probe the structure of a target has been customary experimental practice since the early days of electron diffraction experiments [1]. The experiments on nucleon scattering off nuclei in the early fifties were motivated along similar lines although in this case a curious feature, namely the broad resonances in the neutron scattering data [2], started off a new field of research that is, the use of an optical model to describe the scattering of the incident nucleons. Historically, the concept of the optical model was initiated by Bethe [3]; Fernberg et al. and later Feshbach et a!. developed it to include an imaginary potential to explain —
absorption.
The absorption of light waves was explained by using a complex refractive index. Similarly, an imaginary potential, in conjunction with a real potential, was needed to predict these broad neutron scattering resonances [3]. Better resolution and an increase in the energy of the incident projectile, from keV to MeV regions, made the scattering data look more promising by the middle fifties and consequently the concept of the optical potential went through a gradual process of refinement. Quite a few papers were published treating the scattering problem either (i) from a many-level dispersion theory point of view such as by Brown [4], and/or (ii) as a many-body problem of an A + I system where A is the atomic number of the target, such as by Feshbach [5] and Kerman eta!. [6]. Most of these works were pioneering in the sense that for the first time attempts were being made to understand a complex many-body scattering problem in terms of a one-body potential, namely, the optical potential. To be able to express a many~bodyinteraction in terms of an equivalent one-body potential is appealing not only for its theoretical elegance but also because it provides a reasonably reliable and simple method of relating experimental data to theory. Along with these theoretical approaches, however, a new wave of phenomenology seemed to have hit this field of activity, a remnant of which is still vibrant in some quarters. Not all of the phenomenological work, however, was just curve fitting with a set of parameters. With the increase in the precision of the experimental data it became possible to distinguish between the various components of the optical potential. By 1964 [7] it was established without doubt that the phenomenological optical potential needed three nuclear components, the real central, imaginary and the spin-orbit potential, the last term arising from the coupling between the spin and the orbital angular momentum of the incident particle. The detailed discussion of these components can be found in the following sections. The early sixties in the history of the optical potential unfortunately turned out more like a blind man’s survey of the parameterjungle and quite a lot of the work, although useful as a straightforward guide-line for the prediction of the scattering data, did not serve any fruitful purpose in furthering understanding of the underlying theoretical concepts. In 1968 however, Greenlees, Pyle and Tang (GPT [8]) motivated by the work of Drell [9], suggested a model, which, although mostly taken from the theoretical ideas of Drell sparked off a new method of analysis of the scattering data. Drell’s theory which was not used to any extent to analyse data was moulded into a suitable form by GPT. The GPT mode! although by no means the
4
B. Sinha, The optical potential and nuclear structure
first microscopic model used to analyse scattering data (see for example, ref. [23] for such models for ~I-Ic) for incident nucleus at least, did provide more insight into underlying physical principles. They showed that it was indeed possible to relate nuclear strUcttlre quantities, such as the shape and size of the nucleus and the range of the two-body interaction to the scattering data via the optical potential. The work that will be described in this report starts from this point the pedagogical foundation is, we hope, reasonably complete. The rest of the report is a detailed discussion on the new developments on that foundation and its application to other fields. —
1.2. Purpose of the optical model anal sis It is important to state the motivation for an optical model analysis, described in this report. The emphasis of this paper will be (i) to investigate the geometrical properties of the nucleonic distributions of the target, (ii) to test and investigate the applicability of two-body effective interactions used to calculate properties of a nucleus, and (iii) to try to understand the dynamical processes such as excitation of target states, responsible for both elastic and inelastic scattering. With increase in incident energy the projectile will have an associated wave-length comparable to the internucleonic distance, and therefore short range correlations between nucleons, embedded within a nucleus, will become more and more important. The scattering at high energy will however be mostly in the forward direction and thus the detail of nuclear structure will be of lesser significance. For low incident energy, on the other hand, one expects the optical potential to be mLlch deeper and the excitation of target states becomes more plausible; in the extreme low energy region compound nuclei can also be formed.
2. Foundation and formalism of the optical potential
2.]. The Jbrmalism There are several papers on the derivation of the optical potential from a many body point of view [4—6]. In this section we shall present a brief resumé of the derivation and indicate the relevance of the assumptions we shall make throughout the rest of the paper. The Schrödinger equation we have to solve for elastic or inelastic scattering of a nucleon is given by,
(Ho
-
E)~i —V~.
(2.l.1
where H0
HA + K0, HA being the Hamiltonian of the target nucleus, K0 the kinetic energy operator of the incident particle, and V~v(r0,r~)
(2.1.2)
is the sum of the two-body interactions between the incident particle at (r0) and the target nucleons (re), A being the total number of nucleons in the target nucleus. The quantity E is given by E = K0 + e~,where e~is the energy of the target ground state. For simplicity, it is usually taken to be zero.
B. Sinha, The optical potential and nuclear structure
5
The Schrödinger equation (2.1.1) in integral form is given by, 1 V~, E —H0+ie
(2.1.3)
where the second term in (2.1.3) includes the boundary condition of outgoing scattered waves only, and x0 is the initial state, consisting of an incident plane wave and the ground state of the target nucleus. The transition probability of the system depends on the scattering matrix, T = I VI~i),known popularly as the T matrix. T obeys the equation T=V+VGT
(2.l.4a)
=V+VGV+VGVGV+...
(2.l.4b)
where G is the propagator given by G = (E
—
H0
+
ie~’.The perturbation series represented by eq.
(2.1 .4b) is usually known as the Born series for the T-matrix.
In the following we shall concentrate only on elastic ground state scattering and for the present neglect the antisymmetrisation between the target nucleons and the incident nucleon. This means that we wish to obtain an integral equation for T of the form [12] Tei
=
(2.1.5)
U+ UGTe1,
where G contains no excited states of the nucleus, U is the optical potential and Tei is diagonal with respect to the nuclear ground state. Let n = 0 be the nuclear ground state and define the projection operators
P10>(0I;
Q= ~ n*O
with P + Q
1 and Tei = PT. If we operate P and Q on eq. (2.1 .4a), we obtain
PT=PV+PVG(P+ Q)T QT=QV÷QVG(P+Q)T. From these two equations we get, PT= [PV+PVG(l
—
QVG)’QV][l
+
GPT],
where T0~= PT
and the optical potential, using eq. (2.1.5) is U=PV+PVG(l —QVG)’QV
(2.1.6)
to first order U’ —PV,
(2.1.7) (2.1.8)
2
PV ÷PVGQV.
to second order U’ ÷U Eqs. (2.1.7) and (2.1.8) are the guiding equations for the work described in the following section. 2.2. The optical potential As indicated in the section 2.1 the optical potential to second-order is given by
6
B. Sinha. The optical potential and nuclear structure
~PV+PVGQV.
(‘opt
The
fIrst
(2.2.1)
term in eq. (2.2. 1) is always real and in coordinate space it can he simply written as
U~1(r0)~fP(r)v(Ir_r0J)d3r.
(2.2.2)
where p(i-) is the nucleonic density distribution and v is the two-body interaction defined in eq. (2.1.2). Eq. (2.2.2) is a familiar form of what is often referred to as the folding model. The folding model in the form of eq. (2.2.2) can however be only approximate: for example, if one uses a hard-core interaction eq. (2.22) goes to infinity. As discussed later v for a hard-core interaction is approximated by the long-range part of the interaction. The second-term PVGQ V is usually non-local and complex. In the limit e 0 -~
G
=
~ (E --H0 )‘
+
iir~(E
(2.2.3a)
—
and
VGQV
=
V~(E—H0)-’ QV + i7rVö(E
—
H0)QV,
(2.2.3b)
so that the principal value part of G, ~(E H0 )‘ which is real, contributes to the real part of the optical potential and the imaginary part contributes to the imaginary part of the potential. The real part of the second-order term arises from energy non-conserving virtual excitations of the target whereas the imaginary part refers to the energy conserving actual transitions. The derivation of the imaginary potential from the second-order term is discussed in section 4. 1. The contribution of the second-order term to the real potential clearly, is not taken into account by folding in a jree two-body interaction with the density, eq. (2.2.2). The essential philosophy of the work described in this paper is to assume that one can account for the second-order contributions in an approximate way by folding in a bound-state t~ffectivetwo-body interaction with the nucleon density. This means that v should be replaced by a Brueckner bound-state matrix say KB, so that ~-
K~=V+vGBKB,
(2.2.4)
~v+vGBv+vGBvGBv+...
(2.2.5)
The bound-state propagator GB is given by GB Q~/e,where Qp is the Pauli operator that inhibits transition to occupied states and e is the energy denominator. The propagator GB is different from
the propagator G used in eqs. (2.1.4). Whereas for scattering the energy denominator can go through singularities giving rise to energy conserving transitions, for the bound-state case the denominator is always bounded away from zero. However, for the real part of the optical potential we shall neglect the difference and assume that Re{VGQV}= ~ v(r0. r~)~(E—H0)’Qv(r0, t~) ~ v(r0, r~)GBv(r0, r1)
~ v(r0, r~)—-v(r0, r~)
1=1
iS!
.
e
Using eqs. (2.2.6) and (2.2.1) the real part of the second-order potential is given by
(2.2.6)
B. Sinha, The optical potential and nuclear structure
U2 =P~v(r0,r1)~v(r0,r1).
7
(2.2.7)
In effect, the second-order potential is therefore approximately calculated by folding in a boundstate estimate of vGBv with the nucleon density distributions. Kidwai [141 and also Kidwai and Rook [1 5] defended the use of a bound-state matrix rather than a T matrix for the calculation of the real part of the potential from a slightly different point of view. They show that in the absence of higher than two-body correlations the matrices do indeed behave similarly. Apart from the folding expression, eq. (2.2.2), representing the first-order term, there exists a first-order expression of the optical potential in momentum space which is very useful at higher energies and is given by U(q)M(q)F(q),
(2.2.8)
where q is the momentum transfer for a collision process, q = k’ —k, k’ and k being the final and the initial momenta3rinand a collision. F(q)two-body is the Fourier-transform of thegiven nucleonic M(q) is the scattering amplitude by, density, F(q) = fp(r) exp (—iq• r) d M(q) = — 2ir2m (k’ItIk), (2.2.9) where t is the free two-body transition matrix defined by, t=v+vgt, where g
=
(E
—
(2.2.10)
K
0 — K, + ie)’ and K, is the kinetic energy operator of a target nucleon. Once again, if eq. (2.2.10) is expanded in a Born series, t
v + vgv ÷ vgvgv
+
the first-order Born approximation of M(q) is given by 2~~ (k’IvIk), (2.2.11) M(q)— and U(q) in coordinate space is exactly reduced to the folding integral, eq. (2.2.2). Two important assumptions are however implicit in the discussion above. First, the impulse approximation which makes it possible to use a t matrix which is an approximation to a r-matrix, the latter having a propagator that involves the full target Hamiltonian HA. The spirit of the impulse approximation, originally suggested by Chew and Goldberger [13], is to suppress the detailed nuclear properties enabling the system to behave as free. The second assumption implied in eq. (2.2.9) is that the two-body scattering matrix is always on the energy shell; i.e. Ik’I = IkI. In the case of scattering of a nucleon from another nucleon, immersed in a nuclear medium the relation between k’ and k is fixed by the kinematics of the nucleon—nucleus system and so in the event of excitation of any target state the system is off the two-body energy shell, i.e. Ik’j ~ ki any more. It is expected that at sufficiently high energy the on-shell assumption is more likely to be valid but for low or medium energy scattering offshell effects become increasingly important. A more detailed discussion on this point is presented in section 4.2 where the imaginary potential is estimated using eq. (2.2.6).
B. Sinha, The opticalpotential and nuclear structure
8
In conclusion therefore it is important to state clearly that the real part of the optical potential is calculated by folding in a two-body pseudo potential with the target density distribution. Whereas a realistic two nucleon potential has hard cores etc. the pseudo potential is assumed to be the same as a bound state effective interaction, used for calculating the ground state properties of a nucleus. 2.3. The spin--orbit potential
So far we have considered only the real central and the imaginary part of the optical potential. By 1964 [71with the nieasurements of polarization data, it became evident that a spin-—orbit coupling between the spin and the angular momentum plays not a negligible role in the final optical potential. A microscopic expression has been derived by Blin-Stoyle [1 61 using a Gaussian interaction chosen a priori such that the final form-factor looks like Fso(ro)fvI5(ir_ro~)[~.~_l]p(r)d3r.
(2.3.!)
v15 is usually a Gaussian interaction of a short range. A further discussion of the spin—-orbit term can be found in section 3.1.
where
The origin and the reasons for the short-range nature of the spin— orbit potential are still uncertam. It was proposed [1 7 I that heavy vector mesons p, w or ~ are probably responsible for such an interaction and the short-range nature of the interaction is because of the large mass of these mesons. Serious comment on the origin of the spin—orbit potential is beyond the scope of this report. Blin-Stoyle prescription will be used throughout. 2.4. The phenomenological optical model
A detailed discussion of the phenomenological optical model can be found in the review article of Hodgson [7]. In this section we shall briefly survey the phenomenological model in order to show how much realistic information one can extract from it. The real-central part is represented by a local Saxon—Woods form factor with a depth parameter UR, a radius parameter rR and diffuseness dR. It is assumed implicitly that such a local potential is a phenomenological representa-
tion of both the first and second terms in (2.2.1): UR(rO)
UR/[1 +exp {(r0 —rRA’)/al{}l,
(2.4.1)
where A is the mass number of the target nucleus. The imaginary potential is usually represented by an algebraic sum of a Saxon- Woods and a Saxon—-Woods derivative form with geometrical parameters, either coupled (i.e. same for both the form factors) or uncoupled and two depth parameters W~,and W~so that, 113)/av.~I W1 (r0) W~/ [1 + exp ((r0 rvA + 4a 3)/a 5 W~ l/[ 1 + exp {(r0 r5A~ 5}l]. (2.4.2) --
--
r0 The factor 4a5 is used to normalise the form factor to unity at the peak. The spin—orbit potential is given by
B. Sin/ia, The optical potential and nuclear structure
Uso(ro)=U~ ±~._[l/[1+exp{(ro_rsoA1~)/aso}]]X1a,
9
(2.4.3)
where 1 and a are the angular momentum and the spin vectors, respectively, rso is a radius parameter and U~0a depth parameter. For the coupled imaginary potential form factor, the model thus has ten parameters. Fricke et al. [18] with the motivation of finding out standard values for the optical model geometrical parameters analysed (elastic) scattering (and polarization) data for incident protons. Standard parameters are extremely useful for prediction of results in scattering experiments and also, more important, for Distorted Wave Born Approximation predictions of inelastic scattering cross-section and polarizations. The distorted waves are generated by using the optical model obtained from such an analysis. In table 1, the results of Fricke’s analysis are shown for standard geometrical parameters which are as follows 6,aR = 0.75, r,(= r~=r~)=1.37, a rR l.l 1(avas) 0.64, r~0= 1.064 and a~0= 0.74. 2 column is not universal throughout quality of the fits obtained, as can be seen from the to x attribute any significance to such a theThe periodic table; it would be however somewhat reckless variation in the quality of the fits. It is of some interest to investigate the energy and isospin dependence of UR, the depth parameter. Perey [19] pointed out that before investigating the isospin dependence of UR one must first subtract the variation of UR with A arising from the variation of the Coulomb potential with Z. On the basis of the simple assumption that UR varies linearly with incident energy, he found the Coulomb correction term to be 0.4Z/A’/3. Fricke et al. [18] and also others workers such as Calderbank et al. [20]. Thomas and Burge [21] using the difference function method and Sinha et al. [22] using an additional Saxon—Woods derivative form factor in the real central part attempted to extract any possible isospin dependence of the depth parameter UR. For incident protons, the interaction of the protons with the neutrons of the target nucleus is stronger than with the protons. Therefore for N> Z nuclei, because of the isospin dependence of the two-body interaction, one expects some sort of dependence on the so-called asymmetry parameter e = (N Z)/A. Such a possible dependence was predicted by Lane [23] on theoretical grounds. Later in this report a more microscopic reasoning will be given for (N Z)/A dependence of the optical potential. A detailed discussion of this problem of extraction of any e dependence from a phenomenological point of view is given by Satchler et al. [24]. Taking into account the isospin dependence, energy dependence and Coulomb potential dependence, it is generally believed, although with reservations, that UR, the depth parameter, can be written empirically as (given by Perey [19] on the analysis of data around 1 5 MeV) —
—
UR
=
53.3
—
0.55 E
+
[0.4 ~
+
27 (N_ Z)]
,
(2.4.4)
so that, a symmetry potential of 27 MeV and an energy dependence of 0.5SE is obtained where E is expressed in MeV. Although quite useful for general survey, one cannot rely on such an empirical equation with any confidence. Fricke et al. [181 for example, found an energy dependence of 0.22E around 40 MeV. A more detailed discussion on energy dependence is given by Hodgson [7].
The next question regarding the e dependence was the shape of the isospin potential. Does it
10
B. Sinha, The optical potential and nuclear structure Table 1
l2~
UR(MeV) Wv(MeV) W~(MeV)
38.29 8.72 1.18
Fits obtained using Fricke’s standard geometry, 40 MeV incident proton 28Si 40Ca 54Fe 58Ni 59Co 60Ni 68Zn 90Zr
120Sn
208Pb
42.38 4.00 1.96
43.22 1.21 4.52
45.79 6.89 1.14
45.05 6.63 1.22
45.71
43.8 22.3 630
20.2 18,3 841
6.6 66.1 1015
6.4 37.4 1046
3.1 5.7 1088
5.68
45.74 5.47
2.5
46.60 6.70 2.46
47.76 4.69 3.46
48.77 4.72 4.62
52.76 6.12 4.31
2.08
7.7 18.9 1118
10.7 7.0 1250
35.1 8.2 1410
3.4 9.3 1691
56.5 9.4 2116
x~/N 0 4/N,0 OR(mb)
57.1 244.2 316
follow the same volume shape as the real central term or is it surface peaked? If the neutrons tend to “stick out on the nuclear surface”, one can expect the isospin potential to be surface peaked. Satchler [251 however showed that a surface peaked isospin potential can be simulated by just a Saxon—Woods form factor with a slightly increased radius parameter. Expanding any form-factor functionf(R + ~5R) f(r) ÷ ~R df(R)/dR one can take into account surface terms with an increased radius parameter R + ~R. This kind of argument, however, is on the tacit assumption that the radius parameter used for the isospin potential is identical with the spin isospin independent part of the potential, an assumption not necessarily valid for all the cases under consideration. Sinha and Edwards [26], with the motivation of exploring the possible208Pb shapewith and an structure of additional the isospin potential first analysed the 30 and 40 MeV scattering data of Saxon—Woods derivative term in the real central part. The parameters, chosen to describe such a term had three additional parameters, V 5, as and rs. The large number of the parameters in the model obviously made the analysis somewhat cumbersome and any “one to one” correspondence between a theory and the magnitude of the parameters could be misleading. Fortunately for 208 there exist (p, n) cross-section results due to Batty et al. [27] and one can test any form factor for the iso-spin potential by calculating (p, n) cross-sections using such a potential. In the (p, n) reaction, unlike elastic scattering, the isospin potential is wholly responsible for transition to the isobaric analogue of the ground state of the target nucleus. The programme of Sinha and Edwards was thus a two-stage programme. First using a Saxon—Woods derivative term in the real central part fits were obtained to both cross-section and polarization data. If the quality of the fits improved substantially the parameters found for the derivative term were then used to generate the (p, n) cross-section, on the assumption that the derivative term is wholly due to the isospin potential. This implies that the neutron excess distribution Pne = Pn Pp when folded in with isospin the two-body interaction is and identical to the derivative 208Pbthe (30 and 40dependent MeV) thepart fits of obtained for elastic scattering polarization improved term. For by a factor of two or three using an additional derivative term. The (p, n) cross-section prediction for 30 and 49.4 MeV was found (fig. 1) also to be quite reasonable, particularly in view of the fact that standard phenomenological potential fits are quite poor [271. They also compared the results with the Coulomb displacement energy obtained for 208Pb by Nolen et al. [281. The Coulomb displacement energy, 1~E~ is the Coulomb energy difference between the analogue state and its parent. ~ in its simplest form can be obtained by folding in the Coulomb potential with the neutron excess distribution, Pne~To estimate i~E~ ( 4’~refpnev(r)r2dr)it was assumed [26] that the neutron excess distribution Pne is identical —
B. Sin/ia, The optical potential and nuclear structure
11
200Pb (p,n)208B~
30.2MeV x 0.02 0.01
0.00~ 0
I
10
20
I
30 40 50 CM angle, degrees
I’
60
70
Fig. 1. A comparison between the experimental 208Pb (p,n)208Bi differential cross sections at 30 and SO MeV and theoretical predictions obtained by representing the symmetry potential with the best fit derivative terms found in the elastic scattering analyses.
to the derivative potential obtained for the best fit 30 MeV proton scattering data on 208Pb The result, 18.9 MeV, is close to the experimental value of 19.0 MeV. However, this close agreement is fortuitous since various corrections, such as exchange effects, to the simple folding model of were ignored. Further Pne = Pn p,, and the derivative potential has slightly different shapes because of the “smearing” arising from the finite range of the two-body interaction. However, since the Coulomb displacement energy is sensitive mainly to the r.m.s. radius (r2 )IYe2 of neutron excess and is only slightly affected by other details of the shape Pne, it is instructive to compare the values obtained for (r2)~ for this moment with that of Nolen et al. [28]. Indeed the value, 5.70 fm for (r2)~found by Nolen et al. agreed remarkably well with the value, 5.63 fm found by Sinha and Edwards. All this evidence collected together tends to suggest that indeed, for 208Pb the isospin potential is probably surface peaked. What is important as far as phenomenological models are concerned, is to try to relate the parr’neters obtained to well-defined physical properties. The large number of parameters makes this increasingly difficult but this problem is ameliorated to a large extent by microscopic analyses which relate to a much wider range of physical phenomena. —
12
B. Sinha, The optical potential and nuclear structure
3. The microscopic optical potential 3. 1. The ,thlding model By 1968 an exhaustive set of elastic scattering data for a wide variety of projectiles over a wide range of energies was available. The phenomenological optical model discussed in the last section has been used to analyse such data and reasonable systematics were obtained. Greenlees et al. [81 (GPT) developed a model to analyse the 30 MeV data with the motivation of extracting nuclear structure quantities such as the mean square radii and also study the density distribution of the target nucleons.
Although they developed the model from a variational calculation, the essential result that was used is in fact the first term eq. (2.2.1), i.e. in coordinate representation eq. (2.2.2). The antisymmetrisation between the coordinates of the incident nucleon and those of the target nucleons was neglected. For the two-body interaction GPT used a free two-body interaction of the following form v(r0)
=
Vd (r0) + v~(r0)r~ ‘r + v0(r0
) U.
fJ +
v01(r0 ~
+ [v,(ro)+v~1(ro)rt.TI S,2+ v,5(r0)~ [(r0
—
U
r~) (p0- p~) (ao+~~)I X
(3.1.1)
where i in the equation refer to the target nucleon. For nuclei with total angular momentum zero. the tensor term in first-order does not contribute, the terms which contribute are due to the first, second and the last term in eq. (3.1.1).
The terms are as follows,
(3.1.2)
3r+f[pfl(r)—pp(r)]ur~r-.roIc13r, UR(ro)Jpm(r)vd~r--roId giving rise to the real central potential, and U 5o(r0)
~
2,i
4ir n~i
(2n+l)!
d
2(,i
-
1) d2
~ ~ [~7-~
~
d2~2
-
~Pn~Pm]JVIs(2~2~}10(313)
giving rise to the spin orbit potential. Care should be exercised in the summation of eq. (3.1.3) because it is an asymptotic seiies. The real-central part has a volume term which follows the nuclear density but is smeared by Vd. The second part clearly is the iso-spin potential which depends on the difference between neutron and proton distributions and the isospin dependent part of the two-body interaction. The eq. (3.1.3) is reduced to the Blin-Stoyle prescription [161 for ii = I. For the form factor of the free two-body interaction, GPT usedis aan pseudo Yukawa potential with a mean square around 2. The pseudo potential average potential which is in agreement withradius the phase shift 2.5 fm of free two particle scattering. In a subsequent publication [291 the mean square radius analysis
was changed to approximately 4.0 fm2 using a Gaussian shape. They point out that a Yukawa shape overestimates the binding between the interacting nucleons at small distances. GPT [8] further assumed that p~,= (N/A)pm and p,, = (Z/A)pm so that the only difference between the neutron and the proton density distributions is in the normalization and not in the shape. Moreover, assuming v 7
=
~Vd,
one simply gets, for the real central potential,
3r, UR(rO)=f(1
+e~)pm(r)vdIr
--
r0Id
(3.1.4)
13
B. Sinha, The optical potential and nuclear structure
where ~ = 0.48 and e = (N Z)/A, which is quite simple to integrate after calculating the angular part of the integral analytically for a Yukawa shape. Several interesting results can be obtained from eq. (3.1.4). The volume integral of UR (r0), ~RS is related to the ~d, the volume integral of the two-body in—
teraction by the simple relation JRS/A =(l ~~flJd (3.1.5) 3r is the mass number of the target. Secondly, the mean square radius of UR(rO), where A —fpm(r)d
o~r2)Ris given by, (r2)R (r2)m ÷~T)d’
(3.1.6)
where (r2)m is the mean square radius of the nuclear matter density and (r2)d, the mean square radius of the two-body interaction. It should also be noted that (r2 )m is related to neutron and proton mean square radii by, N
Z
_~(r2)~ +—.(r2)~ (3.1.7) By an optical model analysis (see below) for a nucleus (r2 )m is obtained by searching on rm and am for the optimum fit to elastic scattering data. Since (r2)m is related to (r2)~and (r2)n by eq. (3.1.7), knowing (r2)~from say electron scattering experiments one can estimate (r2 GPT [81took a Saxon—Woods form-factor for the matter distribution with geometrical parameters rm am and assumed that the spin—orbit potential form-factor terminates after n = 1 in the polynomial. They used a phenomenological imaginary potential, a combination of a Saxon—Woods and a Saxon—Woods derivative term. The depth of the real central potential was varied so that various difficiencies in the model, such as exchange and higher order effects could be partly taken into account. In effect, therefore, the number of parameters was reduced from ten to eight. To test the sensitivity of the model to the mean square radius of the two-body interaction the best x2 was determined as a function of (r2)d. The details of the fits are given in ref. [81. A typical fit with such a model is shown in fig. 2 for elastic scattering of protons at 30.3 MeV. The x2 ‘s obtained with such a model were certainly comparable to those obtained by phenomenological analyses. Fig. 3 shows the variation of x2 with r2)d and as can be seen the GPT analysis pins down the (r2)d to within a reasonable range; although the later work of Greenlees et al. [29] showed that (r)d is not as well known as fig. 3 suggests. Table 2a shows the best-fit parameters found for such an analysis. They further showed that the two quantities ~RS IA and (r2 >d are well determined for the energy range concerned (table 2b). Any possible variation of JRSIA with energy could not be established with any confidence since the range of energy over which the analysis was carried out was not exhaustive enough. As will be shown later neither of these conclusions is strictly valid. The success of GPT with whatever drawbacks the model may have (to be discussed later in the section) shows quite clearly that a simple folding model can reproduce data reasonably well. The nuclear structure quantities such as the matter mean square radius, (r2)m ~ [1 + ~ (iram /Rm)2] with Rm = rmA”3 obtained for various nuclei were found to be quite reasonable in the light of other experimental results. However two most striking results of GPT caused some anxiety. The first problem was regarding (r2)m
.
),.~.
,
14
B. Sin/ia, The optical potential and nuclear structure
I
58
.
~
11111
~/
~ f I 7
::
~-r—
—r~r
59Co
30.3MeV
10
—
I
H
Ni 2 0
I
I-S- -U—-r—1- -
\
/\60Ni ~ 00
“~
~
~
~1 0 ~08 0.6
208 PbioTf\~~~~
0
20
40
60
80
100
120
140
160
180
CM. ANGLE (degrees) 11g. 2a. Fits to 30.3 MeV cross-section data for various elements. Solid lines are the GPT model predictions.
the difference 6 = Kr2 >~I2 Kr2 ~/2 of the neutron and proton root mean square radii. Kr2 >,1,/2 can be estimated from the empirical relation given by Acker et al. [30] from electron scattering experiments. Knowing Kr2 >m, the values of 6 obtained by using eq. (3. 1 .7) varied between 0.6 and 0.7 fm. Such a difference obviously indicates that the neutrons stick out by quite a substantial amount in N> Z nuclei, in agreement with the K mesic work of Burhop [311. It should be noted that since kaons get absorbed in the 10% density region the K-mesic data do not determine the r.m.s. radii and therefore such a comparison is not strictly valid. However this so-called neutron halo as predicted by GPT did not agree with the experimental predictions of Nolen et al. based on the measurement of the Coulomb energy differences [28] or with the more elaborate Hartree— Fock calculations of Negele [321 using a local density dependent interaction. It was pointed out that such large neutron skin thickness could be caused by the kind of two-body interaction used. A free two-body interaction as used by GPT does not have the correct saturation properties [32]. —
’.
B. Sin/ia, The optical potential and nuclear structure I
I
I I
I
I
I
15
I
-Ii ~AI~]Jr\\ .4
~
\~i\j
‘~‘~
~ -0.2
~
~:
+
29.0MeV
0.6
58
10
\I;f\’\
[\ ~\~J\ \i ‘p”
~
60N1
—
~ 1.0
___________________________
~vJ~f\/\~08p~8P’
20
40
60
80
100
120
140
160
160
C.M. ANGLE (degrees) Fig. 2b. Fits to the 29.0 MeV polarization data for various elements. Solid lines are the GPT model predictions.
In a subsequent publication however Greenlees et al. [291 showed that the root mean square difference need not be so great if one uses a Gaussian interaction and parameterises the neutron distribution keeping the proton distribution fixed according to the empirical prescription of Ackeretal. [30]. The second source of anxiety was the e dependence of the real potential. Greenlees et al. [29] showed that it is not possible to determine unambiguously the e dependence of the real part of the optical potential from analysis of elastic scattering data, as shown in fig. 4. The doubts that the e dependence of the optical potential cannot be found by analysing elastic scattering data were thus substantiated by these authors. Friedman [34] later showed that the inconsistency of the magnitude of 6 between the results of Coulomb energy difference and that of GPT can be resolved if one uses a density dependent interaction rather than a free two-body interaction, although his conclusions depended very much
16
B. Sinha, The optical potential and nuclear structure
I
I
Cc59 30.3
-
50-
MeV -.
-X~
x
—
ioo~N/
5.0
opi
--
o,OL___
I
0
2.0
30
4,0
-—
I 50
_i _..L.. 60 70
d frn2 Fig. 3. Variation of best-fit x2 values for cross section and polarization data with the two-body interaction mean square radius (OPT model). Table 2a Best-fit parameters OPT, (r2)d
l’able 2b =
2.25
J
fin2
1~5/..1(or t8Ni
~~(o
6~Ni
120 Sn
208 l’h
Element
Energy 1Mev)
~Rs/~1
14.5 30.3 40.0
421 409 377
~.
-~o3
60Ni -
-
14.5
4)1 455
30.3 14 S
390 4 4
14.5 40.0 40.0 303 400
445 . 0 ± - 23 11 390±11 411 * 1~ 371 14
Element (~Rd~kV) 52.53 W~(MeV) 4.50 141s (MeV) 3.85 r 1(fm) 1.357 a1(fm) 0.520 rUso(MCV) 5.16 01(fm) 1.159 a05(fm) 0.597 5.47
o,t(mb) o~(expt)
53.05 3.84 4.58
52.16 3.81 4.71
51.13 3.43 6.68
53.26 3.45 6.53
1.320 0.614 5.75 1.162 0.579
1.3 11 0.609 5.29 1.171 0.582
1.332 0.636 5.82 1.206 0.534
1.317 0.752 5.49 1.206 0.497
5.39
4.50
2.97
0.84
7.78 9.87 17.62 5.58 3.48 1085 1038±43 1153 1169±39 1154 1053±51 1643 1638±68 1986 1865±98
(r21,~(fin)
58Ni
-r
120Sn 208Pb
çmb)
4.12
4.11
4.16
5.02
5.84
Various elements
MeV
tin
± 16 ± 20
)
(G1’T) r~~° --- (r2~ tim)
0.71
± 0.14
064 0 . 70
*
± 15
10 - 11 ±
± 11
01 0 10
—
10 -
1)72 067
*
0.20 0 70
—
0.64
±0.40
B. Sin/ia, The optical potential and nuclear structure
I
58 150
208pb
I
17
I
~0~M~V
6
ZETA(~ 2 values (GPT model) with the isospin dependent part of the two-body interaction. Any value of ~ Fig. 4. Variation of best-fit x between —0.5 and 1.0 produces equally acceptable fits indicating the insensitivity of the data to l~.
on the choice of the parameters. It is known [38] that a properly parameterised density dependent interaction has adequate saturation properties, unlike a free two-body interaction and therefore, as speculated by Negele [32] it is seen that the high value of neutron skin thickness, 6, obtained by GPT is because of the interaction they use. Sinha [33], on the other hand, following the work of Friedman [34] showed that using a linear density dependence in the two-body interaction, the absence of an explicit c dependence can be explained. He found JR
(3.1.8)
5/A—(1+adSm)Jd+EJr(l+arSm),
Ud, a~are parameters associated with the density-dependent and a simple 2a~/R~).interaction ~d and J~ areSm theisvolume algebraic function Of Rm and am Sm (3am/Rm )(l ~ ~ integral of the spin, isospin independent term and the isospin dependent term in the two-body interaction. JRS/A in eq. (3.1.8) is not an explicit function of e because of Sm. Fig. 5 .shows how an improved dependence can be found by plotting JRs/A(l ÷adSm) against e~= e(l +arSm)/ (1 +adSm).
where
,
—
18
B. Sinha. The optical potential and nuclear structure
310r
40.0 MeV 290 270 ~oz,
‘~250-
303MeV
>340 a)
Jd=304~12MeV-F3
~ 320
(I)
V
300 380 14.5 MeV
~k360
~~d=337~30
340
MeV F3 —
320
60Nt
;~
63~
=
5ePd~e
420
-
400
-
eli
E
+
q~Sm)/(1.s. a~Sm)
38C
I I
90z,
58N~
360
206Pb
-
1
~
~°°L~~
~ 420
tss Co
60N1
.-~4 ~3806C
!~6
t,
20sn
20$pb
1440
420-
I,6s,,
63
Co
“SI,
eSNI
-
~°Fe
480
0
0.05
0.10
0.15
0.20
0.25
‘C
Fig. 5. Plot of ~JRS/A )c/(’ + adSm) as a function of ~ compared to the plot 0fJRS/A as a function of e. The improved linear dependence of the first plot is apparent.
B. Sin/ia, The optical potential and nuclear structure
19
k\~
Fig. 6. A schematic representation of a “two-particle—one hole” excitation.
Fig. 7. A schematic representation of a “two-particle—twohole” excitation.
One can generally conclude at this stage that a density dependent two-body interaction can explain some of the drawbacks encountered in the GPT model. Myers [93] has shown from a purely geometrical point of view that a two-body interaction with the correct saturation property is essential, a free two-body interaction being inadequate. In the following sections we shall investigate how consistent it is to use effective interactions with correct saturation properties instead of a free two-body interaction for constructing the optical model. 3.2. Effective interactions and the optical potential In calculating a potential to describe elastic scattering, 3 main types of contribution must be considered. The most important one comes from the direct ground-state to ground-state process in which the projectile and target interact once only. The second process involves virtual transitions to energetically-unreachable intermediate states, followed by a return to the initial state. Both of these processes are essentially elastic events and contribute to the real part of the optical potential. The third process involves actual transitions to energetically-reachable states. Such events, which include inelastic scattering and rearrangement collisions, have the effect of removing flux from the elastic channel and, thus, give rise to the imaginary part of the optical potential. Now, U0~~ of eq. (2.2.1) between momenta k and k’ can be written as, 0CC
(k’IU0~8Ik)= ~(k’,
(~
UflOCC
~
+
~
ulmn)Kmnlu
.
m,n E+j_Cm_En+l~
) ~,
i)
.
(3.2.1)
The states designated Im), In) and Ii) are the single-particle Hartree—Fock states, 1k’) denote a plane wave and v is the two-body interaction. The c’s denote the single-particle energies and the 2 k2 /2m. incident wave number k is given by E = h Sums taken over occupied or unoccupied states in the ground state wave function are indicated by the notation occ or unocc respectively. The second term in eq. (3.2.1) has a principal value contributing to the real term and an imaginary part for energy conserving transitions for which E + = + e,~,and this term is shown schematically in fig. 6. The incident nucleon initially in the state 1k) goes to state rn) in the continuum and excites a particle from below the Fermi level to above the Fermi level creating a hole state Ii) and a particle state In). The (2p—lh) states are depicted in fig. 6,
B.
20
Sin/ia. The optical potential and nuclear structure
the indicating interactions. For elastic scattering, it should be noted that k’l although the angle between the vectors may be non-zero. Higher order terms, among others, can constitutc (2p 2h) states, shown in fig. 7. 1-lowever, eq. (3.2. 1) can be looked upon as an indication of using a re-defined effective state-dependent two-body interaction, different from a free two-body interaction, v such that -
-
unoc C VeffV+
vI,nn) Knznlv -~
~,
n
E
-~-
+ ~
. -—
.
(3.~.2)
~, + l~
In two regions of Em ,e,~the second term can be important; when the denominator is small or zero or when the matrix elements of v are large. In the first case, when the propagator goes through a singularity, Veff gives rise to an imaginary potential. In the event of the denominator being quite small, on the other hand, the contribution to the real term becomes significant. A formal discussion on the derivation of the second term for the real part will be given later (section 3.6)~the calculation of the imaginary potential on this basis will also be discussed (section 4). One can now compare 0eff with the corresponding bound-state case where one obtains a boundstate effective interaction in a similar fashion [36] unoce
V~f=
v+
vlnin) Kmnlv —--—--~-—---——
in, 0
+ Ej
.
3.2.3)
-
where i and j label unoccupied orbitals. In the case of scattering mn) represents a “two-particle-one-hole” state but in the bound state calculation mn) represents a “two-particle—two-hole” state. The spirit of using a bound state effective interaction for scattering lies in the assumption that the difference between the two types of interaction is negligible and that one can use the same methods to take into account the strong repulsion and the correlations between nucleons at short distances as used for the bound state interaction. For hard-core interactions the separation distance method can be used to take into account the strong repulsion at short distances. This approximation of using, in effect, only the long—range part of the two—body interaction and cutting off the shortrange part by using a suitable separation distance is expected to be valid for elastic scattering where only the long-range part is expected to he important~the short-range behaviour of the interaction is not explored enough by the incident projectile. A criticism and discussion of this procedure is given in section 3.6. Generally, the effective interactions to be used are of two kinds. First, an interaction with a hard-core, the core radius being approximately 0.4 fni. In fig. 8, the interaction v is plotted as a lunction of the two-body separation r. The repulsion tends to infinity as the two-body separation approaches zero. This repulsion is the root cause of saturation within the nucleus. We shall take into account the repulsion by balancing it against the attractive part shown by the shaded area. This is known as the separation distance method of Moszkowski and Scott [391. it is generally energy or state dependent. With the increase in relative momenta of the colliding nucleons. one expects an increase in the magnitude of the separation distance. (For a detailed discussion see [66]). In effect the part of the two-body interaction that one uses is the thick line shown in fig. 8. It should also be noted that the separation distance method can only be used for the even components of the interactions. Odd state interactions which are entirely repulsive cannot be tackled by the separation distance method as discussed here. The second term in eq. (3.2.3) or eq. (3.2.2) is treated in many ways for hard-core interactions.
B. Sin/ia, The-optical potential and nuclear structure
21
POT
DISTANCE IN
fm
A
~LONG RANGE PART
Fig. 8. Two-body interaction and the separation distance.
One of the interactions used by Slanina and McManus [40] is the Kuo—Brown effective interaction [41], where the second term is suitably approximated by an effective energy denominator eeff and closure is used to simplify the calculation further. They (Kuo and Brown) noted that the second-order terms are large in the case of 3S1 states and are mainly due to tensor component of 4ISlvTQ(QP/e)vT211
the two-body interaction. Therefore, terms such as( 1s) in analogy with the second term in eq. (3.2.3) (where VTQ represents the long range part of the tensor force and Q~, known as the Pauli operator which suppresses any excitation to occupied states) become as important as the first-order term ~ IvI~15).It was found to be so, although handling of the Pauli operator Q~, because of its angular dependence, proved difficult. However, the situation becomes quite simple for the following two reasons [411: (1) The intermediate states e,~, and c,~can be considered essentially as free. (2) The remarkable property of the second-order exictation is that only the intermediate states of rather high energy —P220 MeV are of major importance. In fact, the matrix elements, if plotted as a function of relative momentum of the intermediate states, peak around 220 MeV. With these observations and on the basis of some approximations Kuo—Brown applied closure for the matrix
elements, such that, (UTI
(Q~/e) VTQ)
v4~/eetr
(3.2.4)
-
Taking into account the spin and isospin of the interacting nucleons, they found that the additional interaction from the second-order term is of the form ~ in the triplet even state. The state dependence of E + e,, was however neglected (discussed later in detail). Thus, recalling the folding integral and neglecting exchange the real central part of the optical potential for an incident proton is —
U~(r0)—jp~(r)vP~Ir-—roI d~r+fp0(r)v0~Ir—roId~r,
~
(3.2.5)
(p, n refer to a proton and a neutron respectively) where ~ and
Vnp
(3.2.6)
~~se j(3t)t.e.
+Vs.e.)
(3.2.7)
22
B. Sin/ia, The optical potential and nuclear structure
-
-
©
x —x—
V~ .~i V1~
( ~
X—
~
)
x~x
~
~
~
x
x___x___x—
-6
~
©
_x_—~x~~
10
©20
t-
—8 .25
.49
.74
I 0
.98
1.23
I
I
1
13
1.44
I 2)
2
1 72’—k1 I
1
3
k~(f Fig. 9. Dependence of Kuo—Brown tensor matrix on k~.,the square of the local Fermi-momentum.
where Vte and Vs.e. refer to the triplet even and singlet even part of the interactions. For these interactions the odd-state component is assumed to be negligibly small. It should be noted that there is no a priori reason for this being so; it depends on how the effective interaction is construct-
ed. For example, the soft-core potentials discussed later in the section usually have a singlet—odd interaction the strength of which is certainly not negligible. For the particular case of the Kuo—Brown [41] interaction, Vse(X)
t4’~~(x) forx> dse
(3.2.8)
forx~d 50
where HJ refer to the Hamada— Johnston [42] interaction between two free nucleons and dse is the separation distance discussed before. Similarly, forx> dte
(3.2.9)
for x~
B. Sinha, The optical potential and nuclear structure
23
tion due to Green [44], the density-dependence in effect mocks up the state dependence of the interaction. This is an empirical interaction, in the sense that the parameters for the density dependence were chosen by fitting the singlet-even and triplet-even contributions to the binding energy of nuclear matter. Although extremely reliable for correct binding energy prediction etc., it is not possible in the case of the Green interaction to distinguish clearly between the first and secondorder terms of eq. (3.2.3). The Green interaction [44] has the general form,
u8,5(x)
=
c51(l ~
forx> d~,1
(3.2.10)
forx~d~~ where s,t refer to the singlet-even and triplet-even part of the interactions, c~5and ~ are sets of parameters which can be found in ref. [44] and ~ is the Kallio—Kollveit interaction [45]. The density function Pm is evaluated theinto centre of mass coordinate thedependence two interacting nucleons. type 3 of dependence, to at take account empirically thefor state of the effectThe p~~’ ive interaction was first suggested by Bethe [46] from a general nuclear structure point of view. Kuo [47] later calculated the Kuo—Brown matrix elements taking into account the state dependence of Q~,the Pauli operator in eq. (3.2.3). Fig. 9 shows the dependence of the matrix element (VTQ(Q/e)VTQ) as a function of k~the local Fermi momentum, defined by k~= 1.57r2p~in a Thomas—Fermi approximation. The graph is interesting, because it predicts a linear dependence on k~.,i.e. p~3,as suggested by Bethe [46]. Taking into account this state dependence explicitly, V4Q becomes a function of r and Pm such that 14Q(r,pm)=
VIQ(TO,Pmo)
[1
—
a~(p~/p~o)213} ,
(3.2.11)
where Pmo is the central density, and a~is a proportionality constant the value of which depends on the quantum numbers of the wave functions in the matrix elements [471. An average value of 0.13 will suffice. Dependence of (VTQ(QP/e)vTQ) on the magnitude of c, and j was also investigated by Kuo [47]. This dependence will be ignored for the present discussion. The second group of interactions, used for calculating the real part of the optical potential are known as soft-core interactions. One distinct advantage of the soft-core interactions is the straightforward convergence of the matrix elements within the nuclear medium. In recent years quite a few Hartree—Fock calculations have been performed using soft-core interactions. Particularly successful applications can be found in ref. [32] where Negele uses a finite-range density dependent repulsion term. Among others, the interaction due to Pandharipande [48] is also a soft-core interaction, although he uses a zero-range repulsion term. Both these interactions have been used by Thomas, Sinha and Duggan [49] (TSD) for an extensive analysis. A detailed discussion will follow later. Another successful and rather simple interaction has recently been used by Vautherin and Brink [50] and later by Negele and Vautherin [51] which is due to Skyrme [52]. The remarkable property of the Skyrme interaction is that with the help of five parameters, the Hartree—Fock singleparticle field becomes a local function of Pm , the nucleonic density function, r, the quantum mechanical kinetic energy density and J, the spin density. Dover and Giai [53] have calculated the optical potential using the Skyrme interaction, which will be discussed in more detail in section 3.4.
B. Sinha, The optical potential and nuciear structure
24
-
60r
-
-
30
Ca
-i
N.
20L
--
H
L~
--
10
r~)
Fig. 10. A comparison of theoretical re,.~ - . , ,tentials obtained by using the density-dependent interactions with the phenomenological form-factors (Slanina and McManus).
Slanina [121 and Slanina and McManus [40] first attempted to calculate the gross properties of the real part of the optical potential using an effective interaction. They used the Kuo—Brown [41] interaction and both the strong and weak versions of the density-dependent interactions of Green. Table 4 shows the volume integral of the optical potentials obtained by Slanina and McManus along with the effective mean square radius of the interaction, the quantities being defined as follows: ~Rs
(3.2.12)
47rfuR(ro)r~drO ,
and (r2)err = (r2)
2),.,,, (3.2.13) 0~~ Kr where Kr2 )opt fUR (r 0 )r~dr0/fUR (r0 )r~dr0 (3.2.14) 2 )eff in eq. (3.2.13) stresses the point that for density-dependent interactions The definition of Kr which depend on the centre-of-mass coordinate for the two interacting nucleons, the only way —
.
B. Sinha, The opticalpotential and nuclear structure
25
one can estimate Kr2 )eff is by an indirect method using eq. (3.2.13). The (N—Z)/A dependence of the optical potential depth found by Slanina and McManus has a depth which varies approximately between 13 to 30 MeV, generally agreeing with the results of Fricke et al. [18]. They noted however that the geometry of the folding process can enhance the dependence substantially at times. The agreement of their results with phenomenology and Greenlees et al. is quite good. Fig. 10 shows the comparison of the potential form-factors. It is interesting to note that the general shapes of the form-factors are not very different from each other, although individually they are derived using different interactions. The work of Slanina and McManus therefore confirms the view that using an effective interaction the potentials obtained are similar to those obtained by phenomenology, at least for the
gross properties. The finer details of the optical potential, one expects, can only be understood by fitting data, or by using a somewhat more elaborate model as discussed in the following sections. 3.3. The exchange potential Before going on to discuss how efficiently the folding models reproduce data it is necessary to take into account the exchange term arising from the antisymmetrization between the coordinates of the incident nucleon and those of the target nucleons. It is recognised [71 that most of the energy dependence of the optical potential arises from the exchange term, although variation of the separation distance from the effective interaction (used to construct the optical potential) with the energy of the incident projectile for example can also be important [55]. Including exchange, the terms of the form UR (r 0 )11i(r0) are replaced by non-local 3rin the Schrodinger equation [56] where p(r, r terms like fp(r, ro)vexlr— r0Ii~’(r)d 0)is the mixed density and
Vex ~S the
exchange component of the two-body interaction. For the interactions which are
of the Serber type, i.e. no odd-state term, the exchange and the direct components are identical. However, in the presence of the odd-state interaction, the odd-state part of the interaction changes sign for the exchange component of the interaction [33], enhancing the exchange effect considerably [33, 46]. It should be noted at this point that the incident projectile cannot distinguish between the exchange and the direct part of the interaction, it is the resultant potential that is of importance. Therefore, the relative strength of the exchange and the direct term are not of crucial importance as often suggested [561; it really depends on the particular kind of interaction used. Slanina and McManus [40] calculated an equivalent local potential for the exchange term using the method of Perey and Saxon [73] for transforming the non-local potential to an equivalent local potential. They concluded that almost 80% of the energy dependence of the optical potential arises from the 2) exchange term. Both the volume integral per nucleon ~RS /A and the mean square radius (r 0~1increase somewhat by the inclusion of the exchange term. Owen and Satchler [56] calculated the non~localexchange term using the Kuo—Brown interaction. To take into account the odd-state interaction they used a Gaussian repulsive potential which fits the two-body P phase shifts. The theoretical prediction of scattering cross-section using the non-local exchange term along with a direct term and a phenomenological imaginary and spin—orbit potential was subjected to a conventional optical model analysis. Although procedures of this kind may in fact cloud the reality somewhat the conclusions they arrived at are rather interesting. For details the reader is referred to ref. [56] but the most important conclus-
B. Sin/ia, The optical potential and nuclear structure
26
sions are that exchange (i) enhances the asymmetry effect (arising from the neutron excess), (ii) increases iRs/A by as much as 40% at 10MeV and (iii) contrary to the work of Slanina and McManus, reduces ~ However, the conclusions depend on whether the odd-state interaction is included or not, as pointed out before. Thomas et al. [49] and Sinha [55] rather than emphasizing the non-local nature of the exchange term calculated the equivalent local potential by using a plane-wave approximation. It was found that the phenomenological properties of the energy dependence of the optical potential can
be well reproduced by such an approximation. The derivation of an equivalent local potential discussed below is motivated by the work of Negele and Vautherin [51]. The term such as fp(r, r0 ) vlr r0 I i,D(r)d3 r can be transformed to a local term if i~(r)can be expressed in some way in terms of
111(r
—
0).
The integrand can then be calculated explicitly without
solving the Schrodinger equation each time. This can be done by taking advantage of the fact that the two-body interaction is of short range, so that expressing r = r0 + s where s Ir r0 and ex—
panding 111(r) in a Taylor expansion one gets ili(r)
exp (s ‘7~)i,Li(r~).
(3.3.1)
Such an expansion is only possible because for any large value of s, u(s) cuts the integrand to zero. Performing the angular integral over the direction of s, essentially in the same manner as in ref. [511the expansion becomes, ~(r)
=
[(j
=
F(—V~)11i(r~)
=
F(k
}
~)]~(r~)
d(cos ~) exp (s V0) /fd(cos .
(3.3.2)
2) ili(r 0)
2) where F(k and sIsI
=
(3.3.3)
sin (ks)/ks
(3.3.4)
and it has been assumed that k2 —vi. In the calculation of exchange potential we have used a local approximation to evaluate k, so that, EEcm
UR(rO)— Vc(ro)]
,
(3.3.5)
where UR is the real part of the optical potential and V~,the Coulomb potential for a charged incident projectile. Thus the non-locality is suppressed by transforming l/.l(r) in terms of 111(r 0) so that the local equivalent to the non-local exchange term becomes, U~x(ro)fp(r,ro)vexIr_roIF(k2)d3r.
(3.3.6)
What should one use for the mixed density p(r, r0)? The obvious and the most simple choice would be to use the Slater mixed density for infinite nuclear matter, assuming that the -effects due to the boundary of the nucleus are not very important [57]
27
B. Sin/ia, The optical potential and nuclear structure
Pb
Pb
~50. -
—-~.
-
0
40 —.
- -
N - —WDDE _~DDE
.‘—~._-.
I
I
I
20
30
40
-
PROTON ENERGY (MeVI Fig. 11. Energy dependence of the optical potential depth for
40Ca and 208Pb (TSD).
B. Sin/ia, The optical potential and nuclear structure
28
SERBER TYPE INTERACTION 7.0
60
NEGELE —VAUTHERIN -
5.0
- —
SLATER
-
MIXED DENSITY
MIXED DENSITY
CaI/.OMeVI
-
0
30
20 NUCL
RADIUS IN
~5O tm
Fig. 12. The exchange potential using a Serber-type of interaction.
p(r, r0)
3.0 =
p(R)pSt(sk~) = p(R
)—---
sk r
ii (sk~),
(3.3.7)
where2Pm. kF is the P~is local Fermi momentum by the approximations as, of nu= Here the nuclear density,given or some sortThomas—Fermi of an average value for each type 1.5TT cleon i = p, n, such that
k~= 1.57r2(p 1(r)+p1(r0)).
(3.3.8)
Negele and Vautherin [511 on the other hand, by using a suitable averaging method, recently oh tamed the following expression ~
,
—s kF
(3.3.9)
2, 11k, where r(R) is the quantum mechanical kinetic energy density given by r(R) ~ V11i0(R)1 being the single-particle wave functions. The great advantage of an expression such as eq. (3.3.9) is that it takes into account the surface effects through the second term and it can be easily calculated once r(R) is known. What sort of prediction in terms of energy dependence and absolute magnitude does one obtain using such a prescription for the exchange potential? We shall investigate in what follows these questions with the simplest assumptions first, i.e. with F(k2) j 0(ks) and p(r, r0) = p(R)p~(sk1,-). It should be noted that expanding Jo (I..-s) in a power series of (ks), one obtains first the linear ener-
B. Sin/ia, The optical potential and nuclear structure
(EVEN
29
ODD) TYPE INTERACTION
•
70
NEGELE -VAUTHERIN MIXED DENSITY
50 — — — —
SLATER MIXED DENSITY
50 40
40Ca (40MeV) —
—
-------5.
uJ30. ‘S
‘S W
20-
‘‘S
‘S ‘S
10 .5. 5.-
I
1.0
2.0
I
I
3.0
4.0
NUCL. RADIUS IN
Fig. 13. The exchange potential using both even
50
tm +
odd type of Negele interaction
1321.
gy dependence, reminiscent of the phenomenology, from terms like (k2s2) (eq. (3.3.5)), and then higher order terms of E from (k4s4) etc. Fig. 11 shows the energy dependence obtained by Thomas et al. [49] using these prescriptions for F and p(r, r 0), for various interactions N (Negele [30]), WDDE, SDDE (Green [44]), and P (Pandharipande [48]). The dependence is linear for the energy range considered, thus validating the phenomenological prediction. The coefficient of linearity obtained in this work turned out to be 0.26. Sinha et al. [54] later showed, that taking into account the energy dependence of the separation distance the coefficient is reduced to 0.22, almost identical to the value obtained by Frickeetal. [18]. What happens if the eq. (3.3.8) is used for p(r, r0)? Fig. 12 shows the form-factor obtained for the exchange using eq. (3.3.6) for an effective Kuo—Brown interaction at 40 MeV incident proton energy. As can be seen, the difference is most significant in the surface region. However, considering that the exchange potential for a Serber type of interaction is only 15% of the total potential, the difference is not substantial. For interactions which have odd components, such as due to Negele [30] the difference could be somewhat more important as shown in fig. 13 but without any serious consequence since, in the final optical potential (direct plus exchange), these differences will not be of any numerical significance. Indeed, it has been found that the crosssection predictions are almost identical [55] using either the Slater or Negele and Vautherin prescriptions.
B. Sinha, The optical potential and nuclear structure
30
10
-
-
R=6O’~
\
\‘~
54.
10
‘~ ‘S
~\
6789
~ Ni N
\N’~ N
“!‘S.
\
‘S..
“
‘S.
‘S
‘~-..
‘
X
N
r
~S.
~
O 2
-
\
j
0(ks)
\
—
2
3
\ 0
1
‘5..
~
~
\
r I
‘S
08
R(F)
‘S
—
—
————
QUADRATIC SLATER
TRUNCATED
2
3
EXPANSION
S(F)
l-ig. 14. The Negele—Vautherin mixed density at the point j0(ks) goes through zero. For explanation of the error bars see ref. 511.
Why is this so? The explanation lies in fig. 14, where an almost accurate truncated expansion of (Negele and Vautherin [51]) is compared with the Slater prediction. 2) Although terms dropthey off differ to substantially for almost allafter values shouldinbethe noted thatthus the cutting u(s) F(koff the mixed density zero almost immediately 1.0 of fmRasit shown figure, p(r, r0)
contribution beyond s = 1.5 fm. The points discussed above are only true of course for elastic scattering. For inelastic scattering or for scattering where surface effects are comparatively more important the merits of eqs. (3.3.4) and (3.3.9) will be both tested and revealed. At present this point is being pursued [581 for inelastic scattering. We end this section by concluding that a local expansion of the generally non-local exchange term is possible without incurring any significant error and that this is possible only because of the short range nature of the two-body interaction. 3.4. Microscopic optical model analysis
So far, the method of calculation of the real central part of the optical potential using a boundstate effective interaction has been discussed. The need for the use of an effective interaction as opposed to a free two-body interaction has also been emphasized. Furthermore, the exchange term arising from the antisymmetrization of the coordinates of the incident nucleon with those of the target nucleons has also been calculated using a local expansion. The real central part is thus,
B. Sin/ia, The optical potential and nuclear structure
UR(ro)Ud(ro)+Uex(ro),
(3.4.1)
where Ud(rO) = i=p,n ~ fp,(r) vefflr
—
r01 d3r
(3.4.2)
2 ) d~r
and Uex (r 0)
=
31
2I~ fpi(r, r0) u~,,Ir
i=
—
(3.4.3)
r0 I F(k
the various terms having been explained in the previous section. How efficiently can such a model predict the data? The optical model used to do so is due to Thomas et al. [49]. They used the effective interactions due to Green [44], Negele [32], Pandharipande [48] and also a Gaussian interaction with a variable range parameter. The Green interaction has already been discussed, the Negele interaction is comprised of three components: short range (s) and long range (d) direct terms and an exchange term (e). In order to simplify the radial dependence average interactions between like (L), (p,p or n,n) and unlike particles (U) (n, p) are taken, i.e.
u,~~(x) = ~
[Vs~i,e(X)
+ V~d,e(X)1.
(3.4.4)
The functions V~j,e(X)are density dependent and are of the form vav(x)=v~(x)+vr~(x)k~. ,
(3.4.5)
where kF is the local Fermi momentum given by k~= l.57r2p~
(3.4.6)
,
and a = 3 for the s component and 1 for d- and e-parts. The radial variations in Va%’(X) have been calculated from Negele’s radial functions for kF = 1 .0 and 1.4 fm~ using his prescriptions 1.4°v(k~= l.0,x) u(kF = 1.4,x) _______________________________ —
=
1.4° 1 —
(3.4.7)
v(kF = 1.4, x) v(kF = 1.0, x) ________________________ 1.4° —
v~(x)
=
—
To take into account the different strengths of the like and unlike interactions they are written as, =
~
V~,’2u~”(x)k~. ,
(3.4.8)
where the constants V 0,1 are given in ref. [32]. The “adjusted” Negele interactions are used in
which the short-range average interactions at 1.0 and 1.4 fm~are multiplied respectively by 0.66903 and 0.83702. In the Pandharipande interaction [48]
31+3V33)g(x) u~(x)~V(V u~,,(x)~V(3V’3+V31+V~+3V33)g(x) v~~(x) = V(V3’ .~
u,,(x)
—
3V33)g(x)
~-V(3V’3+V31— V11
—
3V33)g(x)
B. Sinha, The optical potential and nuclear structure
32
I
I
I
I
5.
I
58N1
20
40
60
80
100
120
I
I
I
I
I
30MeV
140
160
180
e cm l’ig. 15. Fits to the t8Ni cross-section data C~SDmodel).
where V~correspond to the singlet and triplet odd or even components of the interaction and the values of the constants along with the strength V are given in ref. [48]. The range of the Yuka-
wa form factorg(x) is 1.6 fn~1.The short-range repulsion reducing the strength of the attraction is given by a soft zero-range density-dependent term Vdd(X) also given in ref. [481. The optical-model used by Thomas et al. [49] = S
X UR (r 0)
+
d
+
iW~—f(r0, R1, a1) ur0
iWvf(r0, R1, a1)
.
+
I /1\2 ~
U~0 -
ii,
f50(r0) 1 a
,
(3.4.10)
-
where UR(rO)is defined in eqs. (3.4.l)—--(3.4.3) and S is a scaling parameter. The form-factors for the imaginary potentials used [49] were phenomenological, described in sectiOn 2.4 with depth parameters W~,,W~and geometrical parameters r1, a1. (In this sense, the model of Thomas et al.
B. Sin/ia, The optical potential and nuclear structure I
I
I
I
I
I
I
/7
1/ —~ ~
/
\.\
‘--r
“.
111,
//
\.
\\
/‘
,‘
-~
\
\il.4 T ,\l
i,.’
I
‘.
~
Ii”
‘I”\,
ii -I
-
/ -‘
i f’/i
\\r
\ ~,
\‘
IT rv I~/~’
i~’ “~r’..i
0.5
/
I L~-’~’~i
I \ /.r~\T
•1~
~I
I
/ ~~~-~—1——-i ‘jI T
~1~ II
Ii,
I
33
tI\I i
LY
~
-‘~
~/ -‘
-~—
!1’
‘~~_
‘I
.-~‘
•\lb
5
I.~
\\\
I~
‘S. ‘S
.\\“S.i’,J.
\\Ij
-.
II -0.5 10
1.0
I
20
30
I
I
I
I
40
I
I
I
I
I
50
60
70
80
90
I
I
I
I
:
.t.._1
_I
I
I
110
120
~
—
~
0.5
I
.~‘ .
I~.
I
130
.
“
~
\
I
I
100 0Cm
I
140
I
I
150
160
170
I_Z~,
~
,‘‘
‘
II
//
— — I.”
S
S
_i
I.
I’
I
!‘
‘
/ z 0 4 NI
5!
~
.5.
“~‘•
.
I
~...../ \•• ‘S
I ‘
., _______ -
•.
S
\
\
/
\
/
\.
4OCQ)45.5)MeV
-0.5 —.....
/
NEG WDDE
•
~55.
0 10
I
I
I
I
I
I
I
I
I
20
30
40
50
60
70
80
90
100
I
110
I
I
I
I
I
120
130
140
150
160
170
0cm Fig. 16. Fits to the 40Ca polarization data (TSD model).
[49] is not wholly microscopic but only a first attempt.) The spin—orbit term is identical to that of Greenlees et al. [8] described in section 3.1. It was found in a previous analysis [43] for 40Ca
B. Sin/ia, The optical potential and nuclear structure
34
Best-fit parameters for Energy
Interaction
Sx UR (MeV)
Table 3 TtNi for various interactions (TSD)
W~(MeV)
)VS(MCV)
rj(fm)
a 1(fm)
U50(MeV)
x~-
30.3
\VDDE SDDE N P
0.93 x 0.83 x 0.99 x 1.17X
56.7 61.4 48.5 36.2
4.62 3.46 4.82 2.96
0.00 4.02 6.92 9.68
1.68 1.20 1.20 1.12
0.63 0.83 0.47 0.47
102.7 125.7 137.3 113.9
1287 746 1665 4441
40.0
WDDE SDDE N P
0.93 0.79 0.90 1.07
x x x x
55.1 59.8 45.7 33.8
5.88 6.43 6.84 5.16
2.00 0.37 2.62 5.80
1.37 1.56 1.32 1.26
0.65 0.54 0.52 0.55
152.2 151.2 142.5 114.6
1729 1670 2502 6130
Table 4 2> Examples of the variation of (r 6ff and ~RS with type of interaction Nucleus
Energy 1MeV)
160
30.3
40Ca
40.0
WDDI
SDDF
N
I’
3.54
3.97
4.82
3.78
6346
6375
7315
7164
3.91 14415
4.53 14735
5.81 15483
4.86 14870
3.98
4.67
6.26
5.31
~RS
20801 4.80 33725
21014 5.71 35730
21945 6.84 36579
22509 6.20 31891
~‘2~eft
6.12
7.50
8.15
7.94
Jp~
80544
82354
85055
71775
2~eff ~r Kr2l~t-i
J~ ttNi
40.0
r2? 6ff
90Zr
208Pb
40.0 40.0
~RS (r2)et~f
proton scattering data that provided the nuclear density distribution reproduced satisfactorily the electron scattering data, nucleon separation and binding energies per nucleon, the model for a given interaction is unable to differentiate between nucleonic density functions obtained using different schemes. Thomas et al. exploited this fact and therefore restricted the analysis to one type of nucleon density function for each nucleus analysed. For the nuclei ~ ~°Ca,90Zr and 208Pb the neutron and proton distributions used were obtained by Negele [32] using a doubly
self-consistent Hartree Fock calculation. For 58Ni the proton distribution of Batty and Greenlees [59] and the neutron distribution of Zaidi et al. [601. The data analysed were for incident protons, 160 (30 and 50 MeV), 40Ca (30, 35.8, 40 and 45.8 MeV), 208Pb (30 and 40 MeV), 90Zr (40 MeV) and 58Ni (30, 40 and 50 MeV) with the six parameter optical model.
In fig. 15, a typical fit for 58Ni cross-section data is shown using Negele (N), weak density dependent Green (WDDF) and a Gaussian (G) interaction. Fig. 16 shows a typical fit for the polarization data for 40Ca. Table 3 shows the best fit parameters and x2 obtained for 58Ni. The results for other nuclei are very similar except ~ this particular case being discussed later in this section.
B. Sin/ia, The optical potential and nuclear structure
35
The scaling parameters as can be seen in table 3 come out very close to 1, indicating the validity of the model. It is clear however that (i) No systematics can be obtained for W~ and W~. (ii) The fits obtained by using Negele (N) and Pandharipande (P) interactions, which are soft core, are inferior in quality to those obtained by using hard-core interactions such as Green (WDDE or SDDE, the weak and strong versions). Table 4 shows the (r2 >eff = (r2 >opt (r2 >m and ~RS with various type of interactions. It seems that (r2>eff conjectured to be well defined, is no longer a well determined quantity. The fits obtained by SDDE for 208Pb were found to be similar in quality to those obtained by using a Gaussian (G) and yet (r2)ert changed by —3.0 fm2. For (N) and (P) interactions (r2)ecr is persistently large. The dependence of (r2 )eff on A number is probably a manifestation of the density dependence of the effective interaction. It is thus correct to conclude that (i) bound-state effective interactions can be used to generate optical potentials, (ii) analysing the scattering data, although it is not possible to distinguish between nucleonic density distributions obtained by different methods it is in fact possible to detect the difference between different interactions, (iii) soft-core interactions-are probably not very good for scattering (this conclusion can only be qualitative), and (iv) although ~RS is a well determined quantity (r2)ett is model dependent. The energy dependence of ~RS is still not very significant within the energy range concerned. For a wider energy range investigation see ref. [49]. Another interesting feature of the analysis is the worsening in quality of the fits towards lower A, fig. 17. It is conjectured that off-shell effects, assumed in this model to be negligible can in fact become important for light nuclei and also the non-locality in both the imaginary and the exchange potential in the real part can also become important. Furthermore, it is possible that the very concept of single-particle local optical potential may indeed become somewhat tenuous for light nuclei, where cluster effects such as a-clusters have to be taken into account explicitly. —
3.5. Hartree---Fock single particle potential in the continuum The preceeding discussions tend to indicate the validity of the basic idea of linking the detailed theoretical knowledge about the nucleus as a whole to the problem of scattering without introducing a great deal of additional approximation. In this section, we shall try to explore the possibility of expressing the Hartree—Fock single particle potential as an equivalent optical potential. It should be noted right at the beginning, that the Hartree—Fock approximations obviously overlooks a very special effect, relevant for scattering, namely the polarization of the target nuclear field induced by the incident projectile. The virtual excitation term, described previously, is ignored in this method. It is taken into account only very approximately in view of the fact that one uses an effective interaction, which is usually density dependent. Mackintosh [61] has shown that for incident energies in the range 20 ~ E ~ 60 MeV the contribution to the optical potential arising from the coupling to the inelastic levels is not of any significant magnitude so that the dynamical degrees of freedom responsible for such a coupling may be assumed to be of no importance thus justifying the previous argument for neglecting polarization. (This, in effect, is yet anotherjustification for using a bound-state matrix for description of scattering.)
B.
36
Sinha, The optical potential and nuclear structure
The work of Dover and Giai [53] is probably most useful in elucidating the utility of the Hartree---Fock single-particle field for scattering. They use a Skyrme 152] interaction given by a two-body part V12 and a three-body contact
term V123 2 k~V12~k’ = V123
=
+
t0(l
x0P0)
+ ~ t1
t3ö (r~ r~) ~(r2 ----
+
k’2 )+ t
(k
2 k k’ .
+ iw0(a1 + a2)~ (k
X
k’)
(3.5.1)
r3),
where t0, x~and w0 are adjustable constants [52], k and k’ are two-nucleon relative momenta and P0 = J~(1 + a1 a2) is the spin exchange operator. The Hartree—-Fock single-particle equation is given by, t1,
t2,
t3,
r
/12
1fl~r(’~) L
I--u~+
1(1+1) r —
where the index a 3.3). e
=
1 d rL2/n~(r)J /12 1 dr u0 + U~(r)n~ = e~u~ , (3.5.2) ~nljr}, T is the Z-component of the nucleon spin (not r defined in section -----
--
---
---—--------
0 is the energy eigenvalue and m~(r) is the nucleon effective mass defined by [531 /12
/12 =
------
2tn
..in~(r)
+ ~ (t~+ t2 )p(r) + ~ (t2
---
t1
)pT(r),
(3.5.3)
the single-particle potential U~(r)is a sum of central, spin—-orbit and Coulomb contributions expressed as a local function of three density functions, p~(r),the nucleon densities, T~(r)the kinetic energy densities and JT(r) the spin densities: p7(r)
=
T~(r)
~I~j(r)I2.
=
Vøj(r)~2: ./~(r)=
~-
(3.5.4)
~Ø7(r)l. a~1(r).
The exact expression for U~can be found in ref. [53]. The eq. (3.5.3) involves only the discrete eigenstates of the A-particle system. In the H F approximation, excitations of the (A + 1) particle system correspond to unoccupied states of the A-particle system. Thus, one can envisage the scattering problem as follows first, the H—-F equations of the A-particle system are solved in order to find U~(r)and ,n(r), then, the same equation is solved with the eigenvalue ea, replaced by the continuous variable E, the incident energy, using the potentials U~(r)and in~(r)previously obtained. As pointed out before, this picture essentially neglects the polarization effect induced by the incident projectile. In practice, one simply has to recast eq. (3.5.2) into a typical scattering equation 2 r---d2 1(1+ 1) 1 + V ~h —---i- u~(r)+ ~—----u~(r) 7(r, E)u~(r)= Eu~(r) (3.5.5) 2/nLdr r and identify V~(r,E) with the optical potential. For this particular case Dover and Giai [53] found ---
.
that a transformation u~(r)
rm(r)1 I
I
L
/fl
J
1/2
u~(r),
does in fact recast eq. (3.5.4) into the form eq. (3.5.5) so that
(3.5.6)
B. Sin/ia, The optical potential and nuclear structure
37
Fig. 17a. The real central potential using the Skyrme interaction for various elements.
208
Pb
3“I
‘S
/ ‘S
-
——..
5.,
‘S.
/
/
‘I
S S I 5
rUm)
Fig. 17b. The isospin potential using the Skyrme interaction for various elements.
___ m L
2
m(r) r
d — 2
dr2
/
(
/12
\
I—
\2m(r)J
m(r)
d ~—
2/12
/
I
~2
\
-I
dr \2m(r) /
/ m~(r)\ 1+11— T ~E. J \ m(r) / (3.5.7)
21
Consistency is maintained because (m~(r)/m)~ 1 so that u~(r)~ u~(r)and therefore the phase-shifts for scattering will be the same since both Ua and u~have the same asymptotic limit for large r. The same authors L531 noted that the role of the term m(r)/m is to dampen the non-local wave function relative to the local one, found first by Perey and Buck [621. A detailed study of V~(r,E) revealed that most of the gross properties of the optical potential such as energy dependence,
B. Sin/ia, The optical potential and nuclear structure
38
spin-orbit dependence, isospin dependence etc. are reasonably well predicted by eq. (3.5.7), some ot’ which is shown in fig. 17. It should be noted however, that because such an approach derives its results from the HartreeFock calculation for the bound state properties of the nucleus it does not shed any light on the imaginary potential, which is by far the most interesting term in the optical potential.
--
3. 6. Second-order optical potential
In applying the bound-state effective interaction for the description of the scattering potential, the second-order terms for scattering have been assumed to behave like the corresponding boundstate terms. Apart from the difference in the energy denominator (see section 3.2) the Pauli-operator Q~ in the Kuo--- Brown scheme [41] may behave differently for scattering, such that an angle-averaging
procedure may not be quite accurate. Undoubtedly all these possibilities make the estimation of the second-order term rather difficult. Considering a two-particle—-one-hole excitation scheme (fig. 6), the second-order optical potential can be written using a perturbation formalism as Rkn-,Ivlij)12 (innIvljk)~2 +
112 =
!fl,fl>k~j
ern+E 0-J
j~.
.
--—--------
i,jk~
p~l+Ek_~~Ej
(3.6.1)
~
Both c~and e~the single-particle energies, as pointed out before (section 3.2) must be greater than F, the Fermi energy. Ek is the incident energy. There can be an exchange term which represents “blocking” by the presence of a particle in state 1k), of the virtual fluctuations of the core in which particles in states i, j go to states k, in. Quite clearly the contribution to the real-central part comes from the principal value of L/2 since the denominator can go through a singularity giving rise to the energy-conserving transitions, responsible for the imaginary potential. Sinha [63] using a single-particle energy scheme, planewave intermediate states, and a Fermi gas model of the nucleus obtained a local expression for U2, using both the tensor and the central terms. Johnson and Martin [64] using Watson formalism [65] also estimated the second-order term, suitable more for high energy scattering. The second-order term, obtained by Johnson and Martin is however highly non-local, making it unsuitable for general comparison with any local optical model. The real part of the second-order potential, U~calculated using plane wave intermediate states was compared with the results obtained using the Kuo—Brown prescription for the second-order
term. The real part of the second-order-optical potential for the latter case can be calculated by substituting for the energy denominator, Em + ~., e~the Kuo--Brown estimate of 240 MeV. The last approximation is in the same spirit as the “closure approximation”. The comparison will give us some idea as to the accuracy of using a bound-state effective interaction, such as the Kuo—Brown interaction, for estimating second-order effects (see section 3.4). In fig. 18 the second-order term U~’obtained using the closure approximation is compared with U~,as expressed in eq. (3.6.1). As is clear, the effective energy denominator approximation overestimates the second-order term by as much as a factor of two and also the form-factor is quite different from U~.Examining the energy denominator of eq. (3.6.1) where Ek ~ 0 (so that the final value of the denominator is reduced in numerical magnitude) one concludes that the overestimation of U~can —
39
B. Sin/ia, The opticalpotential and nuclear structure
— 70
16
‘
s
—)U
~ 9TT~-~~ I
1(r,).U1(r1))
~
~,
-
~
7O~
I
~
— U2(r~) —
~.
-(Ul(rI).U~(~))
10 20 3.0 4.0
80
—-—
5.0
60 70
- ~
1.0
20 3.0
0 20
16 14
40ca
~
¶6
- - -
30 40 50 60 7080
x05
~°ca
~5...,,
70
60
-......
-—-_.~ ~
10 20
I
4.0 5.0 60 70
ID 20
30 40 50
6.0 70
80
5.0 6.0 70
8.0
16
14 2 — ..~,,
58~
3.0 4.0 50
—~ —T-~---~..
60
70
0 16
~
58NI
10 20
3.0
40
2O8p~
5.”
0
1020
304.0506.07.0
Nuci radius in fm
—~
0
1020304.05.06.0708.0 Nuci radIus
Ifl
fm
—.
Fig. 18. Second-order optical potential as a function of nuclear radius.
only be ascribed to either the error of using an angle averaged Pauli operator such as used in the Kuo—Brown interaction or to the fact that m + ~ probably is much larger than 220 MeV for scattering. Indeed, it has since been pointed out by Brown [41] that the energy denominator should be 420 MeV so that U~now becomes numerically closer to U~,although the shape is still likely to be different. Obviously the calculation of U~Lhas many drawbacks. A single-particle scheme has been used whereas collective representations of states may be more realistic. Further, plane-wave intermediate states may be an oversimplification for scattering. The Fermi gas model used to describe the nucleus is known to be inaccurate on the surface of the nucleus, a region which is likely to be quite important. All these inadequacies of the model may indeed lead to an overestimation of U~and therefore the substantial difference between U~and U~Lmay also be exaggerated. Nevertheless, the discrepancy between U~and U~Ldoes tend to indicate that estimates of the second-order terms using bound state interactions may be erroneous.
B. Sin/ia, The optical potential and nuclear structure
40
4. The imaginary potential 4. 1. Lnergy-conser~ingtransitions
It has already been pointed out that the imaginary part of the optical potential arises from the energy-conserving transitions in the second-order term of the optical potential expressed in eq. (2.2. 1). In other words W, the imaginary potential in operator form is given by I-i’
=
iirPV~(E H0 )Q V. --
(4. 1. 1)
However, W expressed in this way, exhibits rapid fluctuation as the energy E Is varied over the energy intervals of the order of the width of the compound nuclear states. One of the methods for “averaging” the compound elastic effects is to replace E byE + 11. The effects of various avel’aging intervals I are given in ref. [66] and more recently in ref. [671. O’Dwyer et al. [67] calculated W for low energy neutron scattering E < 14 MeV in this fashion. For the energy-range (25 ~ £ ~ 90 MeV) under consideration the compound elastic contributions are neglected and only the direct inelastic channels will be considered. Usually W is both non-local and energy dependent. The first systematic attempt to calculate W from such a second-order perturbation formalism was by Shaw [101. He used the simplest possible approximations namely plane wave intermediate states, energy-conserving transitions to which give rise to the imaginary potential. He also represented the incident projectile by a plane wave. The results obtained by Shaw. although likely to be unreliable because of the oversimplified assumptions, do in tact predict the general properties of H! known phenomenologically [7], in particular the surface bump associated with the Saxon--
Woods derivative term. More recently calculation of the imaginary potential has been perfbrmed using a plane wave propagator but taking into account the actual inelastic levels as the intermediate states. The properties of the collective states particularly the low lying ones (which are likely to be important) are known quite well from experiments on inelastic scattering, so conceptually there is not much ambiguity in the problem. Slanina [121, Vinh-Mau [681 and Bruneau and Vinh-Mau [69] have calculated H!, using
such a scheme. The imaginary potential H!, obtained by Slanina [121 is non-local and energydependent. Pick-up channels such as (p, d) are neglected in Slanina’s calculation. The set of intermediate states was truncated so as to include only the low lying collective states. These low-lying collective states include the effects of long-range correlations. The effect of long-range correlations on the imaginary potential was studied by Terasawa [70] and he found that including “pairing correlations” enhanced the potential by a factor of three. Thus the strongly excited T = 0 states are most important and these were the states considered by from Slanina. state 12Ca, 40Ca) were taken the The worklow of lying Gillet collective and Sanderson wave functions for some nuclei ( [71]. To estimate the strengths of the higher lying collective states of a given inultipole a sum rule due to Lane [72] was used. The local equivalent potential obtained by using the method of Percy and Saxon [73] is shown in fig. 19 for 40Ca at 20 and 40 MeV. Considering the various approximations involved, the prediction is quite remarkable. Rao et al. [74] recently noted that the use of a free particle propagator is not correct. The Hamiltonian H 0 in the propagator G (section 2. 1) now includes an interaction potential V. pre-
B. Sin/ia, The optical potential and nuclear structure
GRAY ETAL
40Ca
41
—
THEORETICAL- — FRICKEETAL——
IMAG POT. 1MeV] 11
THEORETICAL—-—
10
20MeV
9
40MeV
8 7
5 /
S
/
5
/
/
3
—.5
—
/
,~,—.
\\
-~
~
~/
-
—.-----~—
2
‘S
I
- -
0
1
-
RADIUS IN fm
—,
“~. .‘~
I
I
I
I
2
3
4
5
~
6
Fig. 19. The imaginary potential as a function of nuclear radius. Theoretical predictions are compared with phenomenological form-factors.
viously neglected, so that H 0 = HA + T + V where HA is the Hamiltonian of the target nucleus, T the kinetic energy of the projectile and V is the interaction potential, in effect the optical potential. Rao et al. [74] however neglected the interchannel couplings, i.e. the off-diagonal parts of the propagator and calculated G only for the diagonal parts of V. It should be noted that a complex potential V in the propagator is more appropriate and its effect can be important. The wave function representing the projectile state is now a distorted wave (as opposed to a plane wave such as used by Slanina) generated by the real part of V, UR (r0), so that the general form of W is now W(r,r’)
V0~(r)G,~(r,r’)V10(r’) ,
~
i
(4.1.2)
0
where the propagator G~(r,r’) is given by G~(r,r )
=
(2ir)
—
3Jr
d3k
such that /12 k~/2p= E
—
X(*(k,r)X(_)(k,r1) (h2/2p)(k~—k2) +ie
(4.1.3)
,
E, E
1 being the energy associated with the energy states of the target nucleus and x’s are the distorted waves. Slanina [12] and Vinh-Mau [68] essentially used a particle—hole description for the matrix element V01. Cugnon [75] also developed a microscopic model to describe the inelastic states. Rao et al., on the other hand, mostly for reasons of simplicity used a collective or deformed potential model (reasonably well-established by now) to describe the collective states such that
V10(r)
=
(i, XpI VI0,00) =
aUR (2X+l)~~3~(i)R
~
Y~(?),
(4.1.4)
where the spin and its projection of the ith state are X, p and ~ the deformation parameter.
B. Sin/ia, The optical potential and nuclear structure
42
The advantage of such a procedure is that at the expense of throwing out some microscopic details of the inelastic states, one can relate H-’ directly to the energy and the deformation parameter of the inelastic states. Rather than trying to use a local equivalent of highly non-local W, they used a non-local code to calculate the scattering from a non-local potential. The somewhat detailed calculation of Rao et al. eventually led to the following conclusions, (1) A plane-wave propagator is probably wrong and a distorted wave propagator should be used to describe the incident channel. (2) Taking into account just the inelastic channels does not account for the experimentally observed total absorption cross-section. Further, they found that the calculated angular distribution of the cross-section data differ significantly from the measured ones as a function of the scattering angle. The rather pessimistic situation of (2) led them to postulate that pick-up reactions such as (p. dl)
and (p, pn) can contribute a significant part to the imaginary potential, particularly from the region just outside the nuclear surface. Furthermore it was noted that too little absorption for low partial waves can be due to the lack of excitation processes in the nuclear interior: it is tempting, they remarked, to include them as compound nucleus formation by allowing bound-states embedded in the continuum as intermediate states.
The conclusion, one can reach from the speculation of possible pick-up contribution is that such an effect will shield an incident proton from inelastic transitions since the pick-up “takes place” first, as it is on the very edge of the nuclear surface.
In conclusion to this section, it is suggested that despite quite detailed calculation to account for the dynamics of excitation, no consistent picture seems to emerge. It seems now to be impossible even to predict what types of excitation channels are important. However, the failure of a consistent picture makes W rather an interesting function to compute. In the next section, leaving this detailed picture behind we shall try to compute the imaginary potential from a very simple but “frivolous” point of view. Curiously enough, such a model, as will be seen is in reasonable agreement with the data and phenonienology.
4.2. forward-scattering amplitude approxuna tion In the nineteen-fifties, a number of authors [76] calculated the imaginary potential by assuming that “absorption” arises entirely from the forward scattering amplitude of nucleon—nucleon scat-
tering, sometimes referred to as the “frivolous model”. A simple kinetic theory of gases argument can also lead to the same results as obtained using the forward scattering amplitude approximation.
Recalling eq. (2.2.8) one can write out the first-order optical potential within the framework of the impulse approximation as, 2
U~(r0)-—-~-—-J’5d~qM(q)I;(q)exp(iqr0),
(4.2.1) 2ir in 11(q) is the two-body scattering matrix. F(q) the form-factor for nucleon where, recapitulate, i density to function p(r) andq is the momentum transfer, q = k’ k, k’ and k being respectively the final and initial momenta in a collision. It has already been pointed out in section 2 how the folding integral for the real part can be derived from such an equation. To derive the imaginary potential from eq. (4.2.1), we assume that only the forward scattering —
B. Sin/ia, The optical potential and nuclear structure
43
amplitudeM(0) is important, which also implies Ik’] = so that, the two-body matrices are necessarily always on the energy shell. The justification of such an assumption can be seen in eq. (4.2.1). For heavy nuclei one knows that F(q) is a function which peaks at the origin and falls off rapidly with q while M(q) changes much more slowly [5]. For lighter nuclei where the non-zero q values become important, the argument is less valid. Using the forward scattering amplitude approximation the imaginary part of eq. (4.2.1) is reduced to 2
W(r0)
=
Im {M(0)}p(r0)
—
(4.2.2)
m
by using the definition of F(q), the Fourier transform of the density p(r0), and it has been assumed that the effective mass, m* m (see below). Using the optical theorem, the imaginary part of the forward scattering amplitude can be shown to be as follows, —
Im {M(0)}
k =
—
4ir
(a)
(4.2.3)
,
where k is the wave number of the incident nucleon inside the nucleus and (a> is the average twobody total cross-section inside the nucleus. Thus the imaginary potential is given by, W(r0)
=
~1i V~,
~
p~(r0)(a>p1
(4.2.4)
where V~is the velocity inside the nucleus Vp
=
/2 V(~m.~ UR(rO)— Vc(ro)) m
(4.2.5)
.
is the centre-of-mass incident energy; m can be generahised to a radially dependent effective mass, m* [10,76], although it was found that such a change did not alter the results significantly, justifying the assumption m~ m, mentioned above. It should be noted however that originally an effective mass m* 0.7 m was used [76] although a radially dependent effective mass, m* is more appropriate. Thus one calculates UR(rO), the real part first and then knowing V~from eq. (4.2.5), W(r0) is calculated as a function of r0 in this way, some sort of self consistency is maintained. The average total nucleon—nucleon cross-section, (U)pj has been calculated by Clementel and Villi [78]. They used a Fermi gas model for the nucleus and the local Fermi momentum was 2p estimated using the Thomas—Fermi approximation such that ~ = 3ir 1. Energy conserving single particle transitions are considered beyond the Fermi level so as not to violate the Pauli principle. The energy conserving excitation of the target nucleon beyond the Fermi level is the process of absorption [79]. Green, Sawada and Saxon [771 and also Sinha and Duggan [80] however observed that it is implicit in using eq. (4.2.2) that a zero-range interaction has been used. This can be seen by recalling that M(q) is proportional to t(q), the two-body scattering matrix and in coordinate space, t(r) can be expressed as a Born series of two-body interaction, v(r) (see sec. 2.2). The zero-range approximation can however be improved upon by using an effective density, instead of density, p, as defined below [77, 80] Ecm
-~
—~
—
44
B. Sin/ia, The opticalpotential and nuclear structure
10
~
T1_~~
~
“Cu Es LO Mel.’ (I
~/~S/i
—
FOLDED MODEL
— — UNFOLDED ¶-~DEL — — - — 10 PARAMETERS 10
20
30405060
708090
1~110 120130140 150160 170
8cm
Fig. 20a. Fits obtained to the cross-section data using a three parameter modal.
B. Sin/ia, T/ie optical potential and nuclear structure
45
Z°~PbC. £0 61eV
0.5
___ ib
io
-
~
~o~o
~
‘°Zy
10
-0.5
20
~
30
40
51)
60
70
80
90
100
110
120
130
E•
£0 61eV
11.0
ISO
‘°CQ E .4361eV
— 20 30 obtained 40 60to the 60 polarization -70 80 data 90 using 100a three 110 parameter 120 130 model. l~3 ¶50 Fig. 20b. Fits
10
160
i~o
B. Sin/ia, The optical potential and nuclear structure
46
Table 5 Volume integral/A using a three-parameter model
—
Nucleus
Energy
(r
2)R
Kr2 ~eff
JRS/A
J
(tm2)
(tm2)
(MeV fm3)
(MeV 1/A fm3)
J1/A parameters TSD (MeV fm
208Pb
40
37.81
7.45
406.0
111.21
101.89
30
37.91
7.56
440.5
109.69
111.63
90Zr
40
23.77
5.69
386.47
105.43
102.99
40Ca
40
16.05 16.05
4.66 4.66
427.34 393.14
94.20 98.82
99.71 104.88
50
11.34
4.05
515.85
117.34
48.45
40
11.32
4.03
455.8
103.23
(MeV)
30 160
Polarization
0
‘
—.
I,. -
1-.
--
3)
~
~
•
~O E. 39.7 MeV
-0.5 FOLDED MODEL —
—
UNFOLDED
10
MODEL
PARAMETERS
10
20
30
40
50
60
70
SD
90
100
110
120
130
140
150
160
e cm lig. 21. I-its to the 160 polarization data, indicating the deterioration of the fit at backward angles.
P~ff=
Ir
{fPirveff
3r.
r
(4.2.6)
0~d3r)/fveff(r) d
The expression for the imaginary potential now becomes, W(r 0)
=
4h
~
V~
i~p, n
p~ff(r0)(U)pj.
(4.2.7)
Eq. (4.2.6) describes a kind of “averaging” of p with respect to Veff and a detailed derivation of eq. (4.2.7) can be found in ref. 180]; suffice it to say now that the quality of the fits to the scat-
47
B. Sin/ia, The opticalpotential and nuclear structure
KB, kuOS8 KB. k rD
‘°Ca(40MeV)
—
IMAGINARY POTENTIAL
—
—
16.0
12.0
-- —
-.
-
—
~--~
OS
~0
15
20
~ .~
-
--—-
—
kp,BRUECKNER et ol kF, NV DEFINITION
.._
- -
ZERO FORCE
- -
PHENOMENOLOGY
/
—
25 ~35 NUCL. RADIUS INfm
45
5~5
Fig. ~22. Imaginary potential form-factor as a function of nuclear radius
tering data improved greatly using eq. (4.2.7) rather than using eq. (4.2.4) for estimating the imaginary potential. Calculating the imaginary potential as shown in eq. (4.2.7), using the folding model for the real central potential, discussed in the last section and a Blin-Stoyle prescription of the spin—orbit potential, Sinha and Duggan [80] developed a three parameter model with three variable scaling parameters for the real central, imaginary and the spin—orbit potentials. Fig. 20(a,b) shows a typical fit obtained for the cross-section and polarization data using such a model. As is clear the quality of the fits obtained is quite satisfactory and comparable with the fits obtained by using a standard phenomenological model. Table 5 shows the volume integrals and the mean square radii. It is interesting to note that J1/A like JRs/A for the real part also has a reasonably constant value. The fits however deteriorated for backward angle scattering data and also for lighter elements, fig. 21. For backward angles the non-zero momentum transfers can be important and also the twobody matrices are most likely to go off the energy shell [811. In the case of the lighter nuclei both these effects are expected to be more important [61. The non-zero momentum transfers can be taken into account approximately. Expanding M(q) around q = 0 and taking2M( into) account only the even powers [61, one obtains M(q)M(0)+~2 a ~ 0 + (4.2.8)
[
B. Sin/ia, The optical potential and nuclear Structure
48
However V2p(r0)
3q, F(q) exp (iq. r0) d so that substituting eq. (4.2.9) in eq. (4.2.1) gives =
—
—h2
W(r 0)
(4.2.9)
fq2
2
2ir ni
M(0)
Generalising p(r0) to W(r0)~ Vp
32M( )
1
Petf(”O),
~ =
1
5) ,I1
(U>pj(l
-—
v~
[-~~]
2M(0)
aq
p(r~).
(4.2.10)
q0
one gets +~V~)p~ff(rØ),
(4.2.11)
where g using the result of KMT [6], was found to be K 0.5833. Using the modified version of W(r0), the fits to the backward angle improved slightly [821. The importance of the imaginary potential in the nuclear surface, however, throws serious doubt on the validity of the use of the Thomas—Fermi approximation for the local Fermi momentum k~ since it is known that the T—F approximation breaks down in the nuclear surface. Fig. 22 shows the breakdown point of the T---F approximation. The curves which tend to rise sharply are obtained by using modified definitions of k1,- which take into account the inhomogeneity of the nuclear density on the surface of’ the nucleus [82]. It is fortunate that, although the T—F approximation breaks down, a direct comparison with the potentials of phenomenology shows that the imaginary potential is expected to he very small at the point of breakdown. In other words, the validity of the T---F approximation is narrowly saved in this case. The success of the “frivolous model” although rather surprismg seems to indicate that the forward scattering approximation, which is essentially a high energy approximation is valid even for the medium energies under consideration. The local optical potential does not require a great deal of a priori knowledge of the nucleus apart from its density and therefore can be used generally for analysis of elastic scattering data.
5. Application to other fields
In this section we shall discuss how the ideas developed so far can be applied to i) composite particle scattering, ii) inelastic scattering, and iii) heavy-ion scattering. A considerable amount of theoretical work has been done in all the three fields and an exhaustive survey of the theoretical work is not the aim of this paper. The discussion presented below will be concentrated essentially on the folding model, with the motivation of testing the sensitivity and the applicability of the bound-state effective interaction in such a model. 5.1. Composite particles
A considerable quantity of scattering cross-section data has accumulated for incident composite particles such as helium-3, tritons and deuterons. In this section we shall try to explore the possibility of extending the nucleon—nucleus optical model to composite particle scattering.
49
B. Sin/ia, The opticalpotential and nuclear structure
This type of approach is different from treating each type of composite projectile individually and studying the optical potential in terms of a (3He/t/d)-(t)Z) interaction. We feel that in so far as nuclear physics at this energy range is concerned, the fundamental building-brick should be the (tX —9Z) interaction rather than the (composite projectile—nucleon) interaction. The Watanabe [831 model was probably the first attempt to formulate the composite-particle projectile potential in terms of a nucleon—nucleus potential. The scheme is quite straightforward the nucleon—nucleus optical potential is folded in with the density function of the composite projectile and then multiplied by the number of nucleons constituting the projectile. In operator form, this would mean —
(5.1.1)
~
where uC is the internal wave function of the composite projectile and n is the total number of particles constituting the composite projectile (c). Secondary effects are expected to change the structure of U~~ somewhat. 6 These include break-up of the projectile while in the nuclear field or second-order excitations to higher excited states, not accounted for in the nucleon case, or even the distortion of the composite particle wave function due to the nuclear field. Samaddar et al. [85] and Mukherjee [861 developed a scheme where the composite projectile propagator Gc(E,R) for incident energy E was expanded in terms of the nucleon propagator GN (E, r) by an approximate series. The algebra is somewhat complicated but the essential results can be summarised in the following manner: (i) The Watanabe terms taken together represent the first-order term. (ii) There is a correction term to the first-order term which depends strongly on the secondorder terms in the nucleon—nucleus potential. It turned out that in effect the second-order terms are multiplied by factors twice as large as the first-order terms. This indicates an enhancement of the sensitivity of the composite particle optical potential to the second-order effects; in the specific case of the Kuo—Brown interaction one would expect an increased effect from the r~ term [411. It suggests therefore that composite particle scattering is one way in which the various approximations about the second-order term can be tested. (iii) The imaginary potential, being essentially a second-order contribution, must therefore give rise to second-order contribution to the final optical potential. (iv) The relative motion of the constifuent nucleons in the incident particle gives rise to kinetic energy correction terms in the final optical potential, neglected in the Watanabe model. 3He optical potential starting from a Recently attempts were made to calculate [87] the nucleon—nucleus optical potential, calculated theoretically as discussed in section 3. The real central part of the nucleon optical potential was calculated using the Kuo—Brown effective interaction. The first and second-order contributions were separated out, the second-order term being solely due to the tensor term — ~ V~ 1[411 in the Kuo—Brown interaction. The exchange term was also calculated as discussed in section 3.3. A forward-scattering amplitude approximation was used to calculate imaginary potential. should noted that the 3He isthemore likely to be valid It than the be corresponding caseforward-scattering of a single nucleonapproximation for at the same energy — the wavelength associated with 3He is three times smaller. Folding in with the 3He wave function using the formalism of Samaddar et al. [85] the fits obtained to data were quite reasonable [871. It turned out that the kinetic energy terms, arising from the relative motion of the constituent nucleons of 3He were relatively small. This in effect justifies the “adiabatic” approximation of Johnson and Soper [84] who neglect such a motion. —
B. Sinha, The optical potential and nuclear structure
50
In the case of the deuteron, because of its low binding energy it is likely to be distorted once inside the nuclear field and a simple folding model may not be the correct method to estimate the potential. A detailed discussion on deuterons, beyond the scope of this paper can be found in ref. [841 and references therein. The 4He potential is discussed in section 5.3. 5.2. Inelastic scattering and effective interactions
The Distorted Wave Born Approximation has been very successful in formulating a simple theory of inelastic scattering. The coupled channels method has also proved to be a very useful tool to describe inelastic scattering. Theories, more microscopic than simple phenomenological collective models, have been developed over the years, especially by Satchier and co-workers. In this section
we shall not present a survey of this type of work but rather concentrate on the new development in the construction of transition matrices for inelastic scattering using bound-state two-body effective interactions described before. The most simple method to investigate the applicability of effective interactions for inelastic scattering is probably to use the folding model. The transition potential in the DWBA matrix element is given by, 17 V
—/t
AB — \WB
¶
V
WA’
‘
where ~.‘B and I~1A are the final and iniiial target states respectively and v is the two-body interaction. The DWBA transition amplitude is given by the matrix elements of VAB taken between Xh and Xa’ the corresponding distorted waves for the outgoing and incoming projectile respectively. The transition potential, VAB is estimated in a folding model, first by deforming the target density and then folding it in with the two-body effective interaction, expanded in multipoles. Such an approach was considered by Hnizdo and’Lowe [89] and also by Edwards and Sinha [88]. In the case of for example a Saxon—Woods type of density distribution, one can deform the radius parameter such that R
=
R 0 (i
+
~
Y~(O,Ø)~
(5.2.2)
and fold in with
v(s)
v1,~(r1, r2) Y15,1 (~~) Yirn (p2).
=
(5.2.3)
in’
for a chosen effective interaction v(s). It was noted that the form factor for VAB obtained in this way is different from the form-factor VAB, obtained using the standard phenomenological DWBA approach. (To recapitulate, the phenomenological DWBA approach, VAB is identified with the deformed phenomenological optical model potential.) Satchler [90] however pointed out that such a procedure can lead to erroneous results if one neglects the exchange contribution, which is expected to be important for inelastic scattering. Furthermore, it was shown [90] that depending on how one chooses to deform the density, one gets drastically different results. The matter becomes really quite controversial as one looks deeper into the problem of self-consistency. The potentials used to generate the density function should in fact be used to calculate the deformed po-
tential. The choice of an arbitrary effective interaction may thus plague the final result with problems of double counting, inconsistency etc. The considerable discrepancy noted by Edwards and
B. Sin/ia, The optical potential and nuclear structure
51
Sinha [881 between standard phenomenological DWBA and the deformed folding model may thus be misleading because of the consistency problem, if for no other reason. This nagging shadow of inconsistency can be overcome, to some extent, by using the Skyrme interaction [52]. Looking at the approximate Hartree—Fock expression for the optical potential derived by Dover and Giai [53] one finds that, for the Skyrme interaction, the real part is solely a function of density if the kinetic energy density r is approximated by the Thomas—Fermi expression. A good deal more physics can be incorporated into the H—F optical potential if one uses the Negele—Vautherin [51] prescription to derive the H—F single-particle wave equation. The exchange term in that case can be clearly seen, unlike with the empirical Skyrme interaction. Symbolically this means U0~1 Dover, Giai
F
(~~kY -
i
; p, r)
=~
F(~I~, p1(r, r’); p,
Negele —Vautherin
i
where the set of Skyrme interaction parameters ~ are now expressed as a function of density, mixed density, etc. Once again approximating the quantum-mechanical kinetic energy density by the T—F approximation one gets ~ as a function of density. The great advantage of such a procedure is that the deformation of density is directly related to the deformation of U, so that inconsistency cannot plague the results. Such an approach is being pursued. A very similar type of calculation has been made by Davies and Satchler [91]. Instead of using the momentum dependent Skyrme interaction, they used an equivalent potential by observing that the momentum-dependent terms can be regarded as the lowest terms in a short-range expansion of a finite-range central potential which includes Majorana space exchange. They 40Ca. Theconsidered transition the excitation by than 17 and 30 constructed MeV protonsfrom to the lowest model, 3 andwere 5 states of by using RPA potentials rather being a folding estimated hole—particle wave functions of Gillet and Sanderson [711. No imaginary part of the interaction was included. The fits to the data were remarkably good, see ref. [91]. In conclusion Davies and Satchler [91] noted that the density dependent components of the Skyrme interaction are very important. Without this component the resultant cross-section could be an order of magnitude too large. A simpler but equally appealing method was adopted recently by Satchler et al. [92] for analysis of proton inelastic scattering data on 40Ca and 208Pb. The long-range part of the Hamada— Johnston interaction along with the wave-functions provided by Gillet and Sanderson [71] was used to construct the transition potential VAH, eq. (5.2.1). A separation distance of 1.05 fm was used to “simulate” the hard-core of the two-body interaction. They also used the Kallio— Kolltveit interaction [45] but found that the cross-section increased considerably. This is probably because, as Satchler et al. noted [92], Kallio—Kolltveit interaction does not have the correct saturation property. The knock-on exchange term was calculated by using a pseudo zero-range potential [921 and it was found that the exchange effects enhance the cross-section appreciably. However, the pseudo zero-range potential [911 used to calculate the exchange effects may lead to overestimation of the cross-section. A detailed calculation of such knock-on exchange term starting from eq. (3.3.6) is now in progress [94]. Tedder et al. [94], first deform the mixeddensity and then fold it in with the two-body effective interaction u(s), which is expanded in
52
B. Sin/ia, The optical potential and nuclear structure
multipoles. Two kinds of interactions are used: the Kuo—Brown interaction [411 and the Green interaction [44]. Preliminary results indicate that cross-section predictions, using both the interactions are almost identical in magnitude and compare rather well with results obtained by Satchler et al. [921. It was found that the density-dependent part of the interaction (which is a
Kallio—Kolltveit interaction plus a density-dependent term, see section 3.4) plays a very important role. This confirms the speculation of Satchler et al. [92] who suggested that the overestimation of cross-section using the Kallio—Kolltveit interaction is due to absence of any kind of density-dependence. There is obviously need for detailed microscopic calculation such as done by Satchler et al. [92] using realistic wave functions which are calculated self-consistently but the folding model does
predict results reasonably well and has the distinct appeal of simplicity and yet being more meaningful than an ordinary phenomenological deformed potential. For a general application this kind of approach is probably very useful. 5.3. Heavy-ion scattering A knowledge of the ion—ion optical potential is of importance for the understanding of nuclear reaction induced by colliding ions. The strength and the geometrical properties of the potential is expected to play a crucial role in deciding whether the colliding ions for a particular
orbital angular momentum will fuse leading to a compound nucleus or scatter away elastically, inelastically, or induce direct reactions. However, to use the very concept of an optical potential when two colliding ions tend to overlap significantly and dynamical distortions start to take place is almost certainly wrong. The time dependent behaviour of such a process also makes the oversimplified static picture of a potential as a function of the separations between the centres of mass also very dubious. It is only in the tail of the nuclear interaction, when the colliding ions “struggle” between Coulomb repulsion and nuclear attraction that the optical potential is relevant. The interaction energy of ion—ion collision has been estimated by several authors [95—991. The most simple approach, taken by Broglia and Winther [95] and also by Brink and Rowley [95], is to fold in a phenomenological nucleon—nucleus optical potential with the projectile density. in this procedure each nucleon of the projectile is treated essentially as free and therefore the saturation property of the two-body interaction, which prevents the nuclear density from increasing beyond a certain magnitude, is ignored. Thus the potential calculated in this way is expected to be overestimated when the overlap of the densities of the colliding nuclei is significant but should be of the right order of magnitude at or near the touching radius when the overlap is negligible. For incident 4He, however, this single folding procedure gave reasonable results, as shown by Batty and Greenlees [59 I. Wilczynski [96], on the other hand, has derived the interaction potential by assuming that the derivative of the surface energy represents the force acting between two spherical liquid drops, without taking account of the diffuseness of the nuclear surface. He went on to calculate the critical angular momentum for fusion of two nuclei using such a potential and it was found that the energy independence obtained from his model does not agree with experiment [971. Galin et al. [97] and recently Brink and Stancu [98] have taken the “sudden approximation” approach of Brueckner et al. to estimate the ion—ion potential. The nuclear interaction energy, in this approach, is given by
B. Sin/ia, The optical potential and nuclear structure
800 With Saturation Without Saturation
-
700
600
________ -
-
5~55
Target
~
40Ca
\
500-
700
-
600
-
Target 208Pb
\
500-
\~
\~is
\ 400
53
~
400
-
-
-u 300 -t
~l6O -U,inMeV
300
inMeV
-
\
\
\
—--..
200
200
_-_......
N
-
10:
1.0
2.0
3.0
4.0 5.0 6.0
7.0 8.0 9.0 —‘
1.0 2.0 3.0 4.0
5.0
6.0
7.0
8.0
9.0
10.0 11.0
12.0
Nuct. Radius R in fm
Fig. 23. The ion—ion potential as a function of R, the distance between the centres of the colliding nuclei. The saturation term arises from the density-dependent part of the effective interaction.
3r (5.3.1) 1(lR—rI)+p2(r)}—~C{p1(r)} —~C{p2(r)}]d where ~C{p(r)} is the energy-density functional and Pi and P2 are the density distributions of the two nuclei. The energy density functional when integrated over all space gives the energy of a nucleus. It should be emphasized that in such a calculation it is assumed that the structure of each nucleus is entirely conserved during the contact and nuclear-matter densities overlap in a reversible process without any rearrangements. Therefore, the potential, U~~~(R) will certainly not describe the interaction after the nuclei have fused. Whereas Galin et al. [97] use a phenomenological energy-density functional Brink and Stancu [98] use a microscopic ~C{p(r)} derived by Vautherin and Brink [50] using the Skyrme interaction. They observed [98] that the “powerful” repulsion produced by Pauli exchange effects as the U~<~(R)f[lC{p
54
B. Slit/ta. The
optical potential and nuclear structure
overlap of the two nuclei becomes significant, tends to produce a very shallow potential of’ ---10 MeV. Thus the saturation properties, ‘‘bLsilt into’’ the formalism of Brink and Stancu [98] inhibited the nuclei from coming close. Recently [99] the interaction potential for ion- ion collision was also calculated by using a density-dependent interaction and then doubly folding it in with the densities of the target and the projectile. The density-dependent pai’t of the two-body interaction, as is well known, takes into account the saturation property of the two-body interaction. The importance of using an effective interaction such as a density-dependent interaction has been amply demonstrated for a nucleon--nucleus optical potential in this paper. In an ion--ion interaction, however, each nucleon in the projectile is embedded in a nuclear medium thus departing from a free nucleon case. The local density in the two-body interaction therefore should he computed taking into account the contribution of the density from the target and the projectile. It was found [99] that the density-
dependent part contributes most significantly to the lowering of the attraction as the total density increases, thus producing saturation. The “saturation correction” terms, arising from the density-dependent part of the interactions for ~1-Ic, 4He, 160 on 40Ca and 208Pb are shown in fig. 23. Clearly, the role of the saturation term became more important i) with increase in mass number of the projectile, and ii) with more overlap of the density of the colliding nuclei. Application of these results in related fields, such as fusion and direct reactions of colliding nuclei will obviously be rather interesting. The imaginary potential also for ion—ion collision has been calculated by Alexander and Malik [1001 using essentially the forward scattering amplitude approximation. Although open to serious drawbacks such a procedure which is essentially an extension of the nucleon potential can throw some light on the heavy-ion potential, at least in the tail region. Efforts along this direction are however at the speculative stage.
6. Summary and conclusions The real central part of’ the nucleon--nucleus optical potential can be calculated by folding in a two-body effective interaction with the nucleonic density distribution. A free two-body interaction is not appropriate. It has been found that the effective interaction, used for such a folding can be the same as that used for the bound-state calculation. This approximation of using a boundstate interaction is valid however only within certain limits, mostly depedent on the energy of the incident projectile. It has been shown that in the energy range 20 ~ E ~ 90 MeV, use of the hardcore effective interactions reproduces data with reasonable efficiency. The anti-symmetrisation between the coordinates of the incident projectile and those of the target nucleons gives rise to the exchange term, mostly responsible for the energy dependence of the optical potential. The cxchange term is non-local. A local equivalent potential has been derived and it has been shown that
the local potential adequately describes the exchange effects and predicts accurately the energy dependence found from phenomenological analysis. It is misleading to quote the relative strength of the exchange effects generally: the magnitude of’ the exchange potential depends on the relative strength of the spin and iso-spin mixture of the interacting nucleons. Including odd-state interactions, for example, enhances the exchange effects considerably. Tl1is model of the real central part, it is conjectured, can be used generally to find out the neu-
B. Sin/ia, The optical potential and nuclear structure
55
tron distribution of nuclei by fitting the scattering data, assuming that the proton distribution is reasonably well predicted by the empirical relation given by Acker et al. [301 from electron scattering experiments. The imaginary potential has been discussed both using a second-order perturbation theory and a forward scattering amplitude approximation. It has been found that a “proper” dynamical calculation of the imaginary potential can be performed by taking into account energy-conserving transitions to low-lying collective states. The importance of (p, d) pick-up has also been considered [74]. It was shown that such a calculation although very useful to understand the underlying “physics” of the origin of the imaginary potential, is not however suitable for any data fitting procedure. A simple forward scattering amplitude approximation, on the other hand, seems to be able to predict the data with reasonable efficiency. The importance of a finite range two-body interaction as opposed to a zero-range interaction has been emphasized. It is concluded that a finite range interaction is essential. Non-zero momentum transfers have been taken into account by an approximate expansion of the scattering matrix around the forward direction [80]. The imaginary potential derived in this fashion, although having several shortcomings, seems to be satisfactory enough for a general optical model analysis. The spin—orbit interaction potential has been briefly discussed. An expression derived originally by Blin-Stoyle [16] has been used throughout and shown to be quite adequate. The possible extension of such a model to heavier projectiles or even to ion—ion potentials has also been discussed. The possibility of a semi microscopic analysis of inelastic scattering data starting from such a model has been indicated. Finally, it is felt that within the energy range 20 E ~ 90 MeV, to visualize scattering essentially as an A + 1 system with the incident projectile in the continuum and interacting with a “bound-state” effective interaction is a valid picture. On the other hand, for the imaginary potential the so-called high energy approximation of forward scattering seems to be tenable even in medium energy range, a result which may be somewhat surprising but nevertheless valid. ~
Acknowledgement It is indeed a pleasure to thank Professor G.E. Brown for his continuous interest in a “coherent” account of the optical potential which, I hope, has been fulfilled not too incoherently. The enthusiasm, encouragement and the physical insight that I could share with Drs. Richard Griffiths and Victor Edwards is gratefully acknowledged. I would especially like to thank Dr. Richard Griffiths, Kevin Malloy and Dr. D.P. Weet for critically reading the manuscript and helping me to resist the temptation to succumb to rash conclusions, which probably still plague the report but hopefully make it more exciting nonetheless. Discussions with Feroze Duggan and Chris Webb have proved rewarding throughout. The manuscript was written while on a visit to the Niels Bohr Institute, Copenhagen. A travel grant from the Royal Society of England is gratefully acknowledged. The many stimulating discussions in the Niels Bohr Institute particularly with Professor Ben Mottelson and Dr. Philip J. Siemens have, I feel, enriched my understanding of the optical potential. Last but not least I would like to thank Joy Cohler for typing the manuscript with such patience.
B. Sin/ia, T/ie optical potential and nuclear structure
56
References [1
C. Davisson and L.H. Germer, Phys. Rev. 30 (1927) 705;
G.P. Thomson, Proc. Roy. Soc. 117 (1928); 119 (1928). 1-I.H. Barschall, Phys. Rev. 86 (1952) 431. [31 HA. Bethe, Phys. Rev. 47 (1935) 747; H. Feshbach, CE. Porter and V.F. Weisskopf, Phys. Rev. 96(1954)448; S. Fernbach, R. Serber and TB. Taylor, Phys. Rev. 75 (1949) 1352. 14] G.E. Brown, Revs. Mod. Phys. 31(1959) 893. [51 H. Feshbach, Ann. Rev. Nuci. Science 8 (1959) 49. [61 AK. Kerman, H. McManus and R.M. Thaler, Ann. Physics 8 (1959) 551. 171 P.E. Hodgson, Ann. Rev. NucI. Science 17 (1967) 1; Nuclear reaction and nuclear structure (Clarendon Press, Oxford, 1971). [8] G.W. Greenlees, G.J. Pyle and Y.C. Tang, Phys. Rev. 171 (1968) 1115. [91 S.D. Drell, Phys. Rev. 100 (1955) 97. [101 G.L. Shaw, Ann. Physics 8 (1959) 509. [11] D.J. Thouless, Ph.D. thesis, Cornell Univ., New York 1958;Phys. Rev. 112 (1958) 906. [12] D. Slanina, Ph. D. thesis, Michigan State University, East Lansing, 1969, unpublished. [13] G.F. Chew and ML. Goldberger, Phys. Rev. 87 (1952) 778. [14] H.R. Kidwai, Ph.D. thesis, University of Oxford, 1969, unpublished. [151 l-I.R. Kidwai and J.R. Rook, Nuci. Phys. A169 (1971) 417. [161 R.J. Blin-Stoyle, Phil. Mag. 46 (1955) 973. [17] G. Breit, Phys. Rev. 120 (1960) 287; R.A. Bryan and B.L. Scott, Phys. Rev. 135 (1964) B434; J.J. Sakurai, Phys. Rev. 119 (1960) 1784. [181 M.P. Fricke and G.R. Satchler, Phys. Rev. 139 (1965) B567; M.P. Fricke et al., Phys. Rev. 156 (1967) 1207. [191 F.G. Perey, Phys. Rev. 131 (1963) 745. [20] M. Calderbank et al., Nuci. Phys. A105 (1967) 601. [211 G.L. Thomas and E.J. Burge, Nuci. Phys. A128 (1969) 545. [22] B.C. Sinha et at., Nucl. Phys. A183 (1972) 401. [231 AM. Lane, Nucl. Phys. 35 (1962) 676. [24] G.R. Satchier, R.M. Drisko and RI-I. Bassel, Phys. Rev. 136 (1964) B637 125] G.R. Satchler, Iso-spin in nuclear physics (North-Holland, 1969) p. 391. [261 B.C. Sinha and V.R.W. Edwards, Phys. Lett. 31B (1970) 273. [271 C.J. Batty et al., NucI. Phys. A116 (1968) 643. [28] J.A. Nolen, J.P. Schiffer and N. Williams, Phys. Lett. 27B (1968) 1. [29] G.W. Greenlees, W. Makofsky and G.J. Pyle, Phys. Rev. Cl (1970) 1145. [30] H.L. Acker et al., Nuel. Phys. 87 (1966) 1. [31] E.H. Burhop, NucI. Phys. B! (1967) 438. [32] J.W. Negele, Phys. Rev. 1C (1970) 1260. [33] B.C. Sinha, Phys. Lett. 33B (1970) 279. [34] E. Friedman, Phys. Lett. 29B (1969) 213. 135] R.J. Griffiths, private communication. 1361 A.D. MacKellar et al., Phys. Rev. C3 (1971) 460. [37] G.E. Brown, Comm. on Nuclei and Particles. [38] P.J. Siemens, Nucl. Phys. A141 (1970) 225. [39] S.A. Moszkowski and B.L. Scott, Ann. Phys. 11(1960) 65. [40] D. Slanina and H. McManus, Nucl. Phys. A116 (1968) 271. 141] T.T.S. Kuo and G.E. Brown, Nucl. Phys. 85 (1968) 40; G.E. Brown, Nuclear models and forces (3rd ed.) p. 253. [42] T. Hamada and ID. Johnston, NucI. Phys. 34 (1962) 382. [43] G.L. Thomas and B.C. Sinha, Phys. Rev. Lett. 26 (1971) 325. [44] AM. Green, Phys. Lett. 24B (1967) 384. [45] A. Kallio and K. Kolltveit. Nucl. Phys. 53 (1964) 87. [46] H.A. Bethe, Proc. lot. Conf. on Nuclear structure, Tokyo (1970) p. 56. [47]T.T.S. Kuo, Nucl. Phys. A103 (1967) 71.
121
B. Sinha, The opticalpotential and nuclear structure [48] V.R. Pandharipande, Nucl. Phys. A135 (1969) 419; A166 (1971) 317. [491G.L. Thomas, B.C. Sinha and F. Duggan, Nucl. Phys. A203 (1973) 305. [501 D. Vautherin and D.M. Brink, Phys. Rev. CS (1972) 626. [51] J.W. Negele and D. Vautherin, Phys. Rev. CS (1972) 1472. [521 T.H.R. Skyrme, Phil. Mag. 1(1956)1043. [53] C.B. Dover and NV. Giai, Nuci. Phys. A190 (1972) 373. [54] B. Sinha, D.K. Srivastava and N.K. Ganguly, Phys. Lett. 43B (1973) 113. [55]B. Sinha, King’s College, unpublished report (1973). [56] L.W. Owen and G.R. Satchler, Phys. Rev. Lett. 25 (1970) 1720. [57]V.R.W. Edwards, private communication. [58] P.W. Tedder et al., to be published and Rutherford Lab. report (1975). [591C.J. Batty and G.W. Greenlees, NucI. Phys. A133 (1969) 673. [60] S.A.A. Zaidi and S. Darmodjo, Phys. Rev. Lett. 19 (1967) 1446. 1611 R.S. Mackintosh, Nod. Phys. A164 (1971) 398. [62] F. Perey and B. Buck, Nuci. Phys. 32 (1962) 353. [63] B. Sinha, Nucl. Phys. A203 (1973) 473. [641 R.C. Johnson and D.C. Martin, Nucl. Phys. A192 (1973) 496. [65] K.M. Watson, Phys. Rev. 105 (1957) 1388. [66] G.E. Brown, Unified theory of nuclear models (North-Holland, 1967). [67] T.F. O’Dwyer, M. Kawai and G.E. Brown, Phys. Lett. 41B (1972) 259. [681 N. Vinh Mau, mt. Course on Nuclear theory, Trieste (1969). [69] M. Bruneau and N. Vinh Mau, Proc. 60th birthday of Prof. RE. Peierls, Birmingham (1967). [70] T. Terasawa, NucI. Phys. 39 (1962) 563. [711 V. Gillet and E.A. Sanderson, Nucl. Phys. 54 (1964) 472. [72] A.M. Lane, Nuclear theory (W.A. Benjamin Inc., 1964). [731 F.G. Perey and D.S. Saxon, Phys. Lett. 10 (1964) 107. [74] C.L. Rao, M. Reeves and G.R. Satchler, Nucl. Phys. A207 (1973) 182. [75]J. Cugnon, Nucl. Phys. A165 (1971) 393. [76] A.M. Lane and C.F. Wandel, Phys. Rev. 98 (1955) 1524; K. Harada and N. Oda, Prog. Theor. Phys. 21(1959) 260. [77] A.E.S. Green, T. Sawada and D.S. Saxon, The nuclear independent particle model (Academic Press, 1968) pp. 304, 306. [78] E. Clemental and C. Villi, Nuovo Cim. 1(1955)176. [79] V.F. Weisskopf, Science 113 (1951) 101. [80] B. Sinha and F. Duggan, Nucl. Phys. A226 (1974) 31; B. Sinha and F. Duggan, Upsalla Conf., Sept. 1973. [811 K. Mailoy and G.L. Thomas, to be published. [821 B. Sinha and F. Duggan, Phys. Lett. 47B (1973) 389. [83] S. Watanabe, Nucl. Phys. 8 (1958) 484. [84] R.C. Johnson and P.J.R. Soper, Phys. Rev. Cl (1970) 976. [85] S.K. Samaddar, R.K. Satpathy and S. Mukherjee, Nucl. Phys. AlSO (1970) 655. [86] S. Mukherjee, NucI. Phys. A118 (1968) 423. [87] B. Sinha, F. Duggan and R.J. Griffiths, Phys. Left. to be published. [88] V.R.W. Edwards and B.C. Sinha, Phys. Lett. 37B (1971) 225. [89] V. Hnizdo and J. Lowe, Phys. Lett. 35B (1971) 35. [90] G.R. Satchier, Phys. Lett 39B (1972) 492. [91] K.T.R. Davies and G.R. Satchler, Nucl. Phys. A222 (1974) 13. [92] G.R. Satchler et al., Z. Physik 260 (1973) 209; Nucl. Phys. A233 (1974) 265. [93] William D. Myers, Lawrence Berkeley Lab. report 1259. [94] P.W. Tedder et al., Nucl. Phys. to be published. [95] R.A. Broglia and A. Winther, Phys. Reports 4C (1972) 153; D.M. Brink and N. Rowley, Nucl. Phys. A219 (1974) 79. [96] J. Wilczynski, Phys. Lett. 47B (1973) 484; Nuci. Phys. A216 (1973) 386. [97] J. Galin et a!., Phys. Rev. C9 (1974) 1018. [981 D.M. Brink and F. Stancu, Nucl. Phys. Lab report, Oxford 59/74. [99] B. Sinha, Phys. Rev. Lett. 33 (1974) 600; Phys. Rev. C, to be published (May 1975). [1001 DR. Alexander and F.B. Malik, Phys. Lett. 42B (1972) 412.
57
Bikash SINHA Wheatstone Laboratory, King’s College, London WC2R 2LS, UK. and The Niels Bohr Institute, Copenhagen, Denmark
(p1~C
NORTH-HOLLAND PUBLISHING COMPANY
—
AMSTERDAM
PHYSICS REPORTS (Section C of Physics Letters) 20, no. 1 (1975) 1—57. NORTH-HOLLAND PUBLISHiNG COMPANY
THE OPTICAL POTENTIAL AND NUCLEAR STRUCTURE Bikash SINHA Wheat stone Laboratory, King’s College, London WC2R 2LS, U.K. and The Niels Bohr Institute, Copenhagen, Denmark Received February 1975 Abstract: The optical potential for single and composite particle projectiles has been derived and discussed in detail for incident energy range 25 a~E ~ 90 MeV. The real central part has been calculated by folding an effective two-body interaction with the nucleonic density distribution. The importance of using an effective interaction with the correct saturation properties has been emphasized and discussed in detail. The possibility of identifying the optical potential as a l-Iartree—Fock single particle potential has also been explored. The imaginary potential has been discussed both from a second-order perturbation formalism and a simple forwardscattering amplitude approximation. The utility of the later approach for straightforward data fitting has been discussed in detail. Using the Blin-Stoyle prescription of the spin-orbit potential a three parameter optical model has been developed. Such models have been extended for inelastic scattering, composite particle scattering and heavy-ion scattering.
Contents: 1. Introduction 1.1. The historical background 1.2. Purpose of the optical model analysis 2. Foundation and formalism of the optical potential 2.1. The formalism 2.2. The optical potential 2.3. The spin—orbit potential 24. The phenomenological optical model 3. The microscopic optical potential 3.1. The folding model 3.2. Effective interactions and the optical potential 3.3. The exchange potential 3.4. Microscopic optical model analysis
3 3 4 4 4 8 8 12 12 19 25 30
3.5. Hartree—Fock single particle potential in the continuum 3.6. Second-order optical potential 4. The imaginary potential 4.1. Energy-conserving transitions 4.2. Forward-scattering amplitude approximation 5. Application to other fields 5.1. Composite particles 52. Inelastic scattering and effective interactions 5.3. Heavy-ion scattering 6. Summary and conclusions Acknowledgement References
Single orders for this issue
PHYSICS REPORTS (Section C of Physics Letters) 20, no. 1(1975)1—57 Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check.
L
Single issue price Dfl. 20.—, postage included.
35 38 40 40 42 48 48
so 54 55 56
B. Sinha, The optical potential and nuclear structure
3
1. Introduction 1.1. The historical background The scattering of particles as a means to probe the structure of a target has been customary experimental practice since the early days of electron diffraction experiments [1]. The experiments on nucleon scattering off nuclei in the early fifties were motivated along similar lines although in this case a curious feature, namely the broad resonances in the neutron scattering data [2], started off a new field of research that is, the use of an optical model to describe the scattering of the incident nucleons. Historically, the concept of the optical model was initiated by Bethe [3]; Fernberg et al. and later Feshbach et a!. developed it to include an imaginary potential to explain —
absorption.
The absorption of light waves was explained by using a complex refractive index. Similarly, an imaginary potential, in conjunction with a real potential, was needed to predict these broad neutron scattering resonances [3]. Better resolution and an increase in the energy of the incident projectile, from keV to MeV regions, made the scattering data look more promising by the middle fifties and consequently the concept of the optical potential went through a gradual process of refinement. Quite a few papers were published treating the scattering problem either (i) from a many-level dispersion theory point of view such as by Brown [4], and/or (ii) as a many-body problem of an A + I system where A is the atomic number of the target, such as by Feshbach [5] and Kerman eta!. [6]. Most of these works were pioneering in the sense that for the first time attempts were being made to understand a complex many-body scattering problem in terms of a one-body potential, namely, the optical potential. To be able to express a many~bodyinteraction in terms of an equivalent one-body potential is appealing not only for its theoretical elegance but also because it provides a reasonably reliable and simple method of relating experimental data to theory. Along with these theoretical approaches, however, a new wave of phenomenology seemed to have hit this field of activity, a remnant of which is still vibrant in some quarters. Not all of the phenomenological work, however, was just curve fitting with a set of parameters. With the increase in the precision of the experimental data it became possible to distinguish between the various components of the optical potential. By 1964 [7] it was established without doubt that the phenomenological optical potential needed three nuclear components, the real central, imaginary and the spin-orbit potential, the last term arising from the coupling between the spin and the orbital angular momentum of the incident particle. The detailed discussion of these components can be found in the following sections. The early sixties in the history of the optical potential unfortunately turned out more like a blind man’s survey of the parameterjungle and quite a lot of the work, although useful as a straightforward guide-line for the prediction of the scattering data, did not serve any fruitful purpose in furthering understanding of the underlying theoretical concepts. In 1968 however, Greenlees, Pyle and Tang (GPT [8]) motivated by the work of Drell [9], suggested a model, which, although mostly taken from the theoretical ideas of Drell sparked off a new method of analysis of the scattering data. Drell’s theory which was not used to any extent to analyse data was moulded into a suitable form by GPT. The GPT mode! although by no means the
4
B. Sinha, The optical potential and nuclear structure
first microscopic model used to analyse scattering data (see for example, ref. [23] for such models for ~I-Ic) for incident nucleus at least, did provide more insight into underlying physical principles. They showed that it was indeed possible to relate nuclear strUcttlre quantities, such as the shape and size of the nucleus and the range of the two-body interaction to the scattering data via the optical potential. The work that will be described in this report starts from this point the pedagogical foundation is, we hope, reasonably complete. The rest of the report is a detailed discussion on the new developments on that foundation and its application to other fields. —
1.2. Purpose of the optical model anal sis It is important to state the motivation for an optical model analysis, described in this report. The emphasis of this paper will be (i) to investigate the geometrical properties of the nucleonic distributions of the target, (ii) to test and investigate the applicability of two-body effective interactions used to calculate properties of a nucleus, and (iii) to try to understand the dynamical processes such as excitation of target states, responsible for both elastic and inelastic scattering. With increase in incident energy the projectile will have an associated wave-length comparable to the internucleonic distance, and therefore short range correlations between nucleons, embedded within a nucleus, will become more and more important. The scattering at high energy will however be mostly in the forward direction and thus the detail of nuclear structure will be of lesser significance. For low incident energy, on the other hand, one expects the optical potential to be mLlch deeper and the excitation of target states becomes more plausible; in the extreme low energy region compound nuclei can also be formed.
2. Foundation and formalism of the optical potential
2.]. The Jbrmalism There are several papers on the derivation of the optical potential from a many body point of view [4—6]. In this section we shall present a brief resumé of the derivation and indicate the relevance of the assumptions we shall make throughout the rest of the paper. The Schrödinger equation we have to solve for elastic or inelastic scattering of a nucleon is given by,
(Ho
-
E)~i —V~.
(2.l.1
where H0
HA + K0, HA being the Hamiltonian of the target nucleus, K0 the kinetic energy operator of the incident particle, and V~v(r0,r~)
(2.1.2)
is the sum of the two-body interactions between the incident particle at (r0) and the target nucleons (re), A being the total number of nucleons in the target nucleus. The quantity E is given by E = K0 + e~,where e~is the energy of the target ground state. For simplicity, it is usually taken to be zero.
B. Sinha, The optical potential and nuclear structure
5
The Schrödinger equation (2.1.1) in integral form is given by, 1 V~, E —H0+ie
(2.1.3)
where the second term in (2.1.3) includes the boundary condition of outgoing scattered waves only, and x0 is the initial state, consisting of an incident plane wave and the ground state of the target nucleus. The transition probability of the system depends on the scattering matrix, T = I VI~i),known popularly as the T matrix. T obeys the equation T=V+VGT
(2.l.4a)
=V+VGV+VGVGV+...
(2.l.4b)
where G is the propagator given by G = (E
—
H0
+
ie~’.The perturbation series represented by eq.
(2.1 .4b) is usually known as the Born series for the T-matrix.
In the following we shall concentrate only on elastic ground state scattering and for the present neglect the antisymmetrisation between the target nucleons and the incident nucleon. This means that we wish to obtain an integral equation for T of the form [12] Tei
=
(2.1.5)
U+ UGTe1,
where G contains no excited states of the nucleus, U is the optical potential and Tei is diagonal with respect to the nuclear ground state. Let n = 0 be the nuclear ground state and define the projection operators
P10>(0I;
Q= ~ n*O
with P + Q
1 and Tei = PT. If we operate P and Q on eq. (2.1 .4a), we obtain
PT=PV+PVG(P+ Q)T QT=QV÷QVG(P+Q)T. From these two equations we get, PT= [PV+PVG(l
—
QVG)’QV][l
+
GPT],
where T0~= PT
and the optical potential, using eq. (2.1.5) is U=PV+PVG(l —QVG)’QV
(2.1.6)
to first order U’ —PV,
(2.1.7) (2.1.8)
2
PV ÷PVGQV.
to second order U’ ÷U Eqs. (2.1.7) and (2.1.8) are the guiding equations for the work described in the following section. 2.2. The optical potential As indicated in the section 2.1 the optical potential to second-order is given by
6
B. Sinha. The optical potential and nuclear structure
~PV+PVGQV.
(‘opt
The
fIrst
(2.2.1)
term in eq. (2.2. 1) is always real and in coordinate space it can he simply written as
U~1(r0)~fP(r)v(Ir_r0J)d3r.
(2.2.2)
where p(i-) is the nucleonic density distribution and v is the two-body interaction defined in eq. (2.1.2). Eq. (2.2.2) is a familiar form of what is often referred to as the folding model. The folding model in the form of eq. (2.2.2) can however be only approximate: for example, if one uses a hard-core interaction eq. (2.22) goes to infinity. As discussed later v for a hard-core interaction is approximated by the long-range part of the interaction. The second-term PVGQ V is usually non-local and complex. In the limit e 0 -~
G
=
~ (E --H0 )‘
+
iir~(E
(2.2.3a)
—
and
VGQV
=
V~(E—H0)-’ QV + i7rVö(E
—
H0)QV,
(2.2.3b)
so that the principal value part of G, ~(E H0 )‘ which is real, contributes to the real part of the optical potential and the imaginary part contributes to the imaginary part of the potential. The real part of the second-order term arises from energy non-conserving virtual excitations of the target whereas the imaginary part refers to the energy conserving actual transitions. The derivation of the imaginary potential from the second-order term is discussed in section 4. 1. The contribution of the second-order term to the real potential clearly, is not taken into account by folding in a jree two-body interaction with the density, eq. (2.2.2). The essential philosophy of the work described in this paper is to assume that one can account for the second-order contributions in an approximate way by folding in a bound-state t~ffectivetwo-body interaction with the nucleon density. This means that v should be replaced by a Brueckner bound-state matrix say KB, so that ~-
K~=V+vGBKB,
(2.2.4)
~v+vGBv+vGBvGBv+...
(2.2.5)
The bound-state propagator GB is given by GB Q~/e,where Qp is the Pauli operator that inhibits transition to occupied states and e is the energy denominator. The propagator GB is different from
the propagator G used in eqs. (2.1.4). Whereas for scattering the energy denominator can go through singularities giving rise to energy conserving transitions, for the bound-state case the denominator is always bounded away from zero. However, for the real part of the optical potential we shall neglect the difference and assume that Re{VGQV}= ~ v(r0. r~)~(E—H0)’Qv(r0, t~) ~ v(r0, r~)GBv(r0, r1)
~ v(r0, r~)—-v(r0, r~)
1=1
iS!
.
e
Using eqs. (2.2.6) and (2.2.1) the real part of the second-order potential is given by
(2.2.6)
B. Sinha, The optical potential and nuclear structure
U2 =P~v(r0,r1)~v(r0,r1).
7
(2.2.7)
In effect, the second-order potential is therefore approximately calculated by folding in a boundstate estimate of vGBv with the nucleon density distributions. Kidwai [141 and also Kidwai and Rook [1 5] defended the use of a bound-state matrix rather than a T matrix for the calculation of the real part of the potential from a slightly different point of view. They show that in the absence of higher than two-body correlations the matrices do indeed behave similarly. Apart from the folding expression, eq. (2.2.2), representing the first-order term, there exists a first-order expression of the optical potential in momentum space which is very useful at higher energies and is given by U(q)M(q)F(q),
(2.2.8)
where q is the momentum transfer for a collision process, q = k’ —k, k’ and k being the final and the initial momenta3rinand a collision. F(q)two-body is the Fourier-transform of thegiven nucleonic M(q) is the scattering amplitude by, density, F(q) = fp(r) exp (—iq• r) d M(q) = — 2ir2m (k’ItIk), (2.2.9) where t is the free two-body transition matrix defined by, t=v+vgt, where g
=
(E
—
(2.2.10)
K
0 — K, + ie)’ and K, is the kinetic energy operator of a target nucleon. Once again, if eq. (2.2.10) is expanded in a Born series, t
v + vgv ÷ vgvgv
+
the first-order Born approximation of M(q) is given by 2~~ (k’IvIk), (2.2.11) M(q)— and U(q) in coordinate space is exactly reduced to the folding integral, eq. (2.2.2). Two important assumptions are however implicit in the discussion above. First, the impulse approximation which makes it possible to use a t matrix which is an approximation to a r-matrix, the latter having a propagator that involves the full target Hamiltonian HA. The spirit of the impulse approximation, originally suggested by Chew and Goldberger [13], is to suppress the detailed nuclear properties enabling the system to behave as free. The second assumption implied in eq. (2.2.9) is that the two-body scattering matrix is always on the energy shell; i.e. Ik’I = IkI. In the case of scattering of a nucleon from another nucleon, immersed in a nuclear medium the relation between k’ and k is fixed by the kinematics of the nucleon—nucleus system and so in the event of excitation of any target state the system is off the two-body energy shell, i.e. Ik’j ~ ki any more. It is expected that at sufficiently high energy the on-shell assumption is more likely to be valid but for low or medium energy scattering offshell effects become increasingly important. A more detailed discussion on this point is presented in section 4.2 where the imaginary potential is estimated using eq. (2.2.6).
B. Sinha, The opticalpotential and nuclear structure
8
In conclusion therefore it is important to state clearly that the real part of the optical potential is calculated by folding in a two-body pseudo potential with the target density distribution. Whereas a realistic two nucleon potential has hard cores etc. the pseudo potential is assumed to be the same as a bound state effective interaction, used for calculating the ground state properties of a nucleus. 2.3. The spin--orbit potential
So far we have considered only the real central and the imaginary part of the optical potential. By 1964 [71with the nieasurements of polarization data, it became evident that a spin-—orbit coupling between the spin and the angular momentum plays not a negligible role in the final optical potential. A microscopic expression has been derived by Blin-Stoyle [1 61 using a Gaussian interaction chosen a priori such that the final form-factor looks like Fso(ro)fvI5(ir_ro~)[~.~_l]p(r)d3r.
(2.3.!)
v15 is usually a Gaussian interaction of a short range. A further discussion of the spin—-orbit term can be found in section 3.1.
where
The origin and the reasons for the short-range nature of the spin— orbit potential are still uncertam. It was proposed [1 7 I that heavy vector mesons p, w or ~ are probably responsible for such an interaction and the short-range nature of the interaction is because of the large mass of these mesons. Serious comment on the origin of the spin—orbit potential is beyond the scope of this report. Blin-Stoyle prescription will be used throughout. 2.4. The phenomenological optical model
A detailed discussion of the phenomenological optical model can be found in the review article of Hodgson [7]. In this section we shall briefly survey the phenomenological model in order to show how much realistic information one can extract from it. The real-central part is represented by a local Saxon—Woods form factor with a depth parameter UR, a radius parameter rR and diffuseness dR. It is assumed implicitly that such a local potential is a phenomenological representa-
tion of both the first and second terms in (2.2.1): UR(rO)
UR/[1 +exp {(r0 —rRA’)/al{}l,
(2.4.1)
where A is the mass number of the target nucleus. The imaginary potential is usually represented by an algebraic sum of a Saxon- Woods and a Saxon—-Woods derivative form with geometrical parameters, either coupled (i.e. same for both the form factors) or uncoupled and two depth parameters W~,and W~so that, 113)/av.~I W1 (r0) W~/ [1 + exp ((r0 rvA + 4a 3)/a 5 W~ l/[ 1 + exp {(r0 r5A~ 5}l]. (2.4.2) --
--
r0 The factor 4a5 is used to normalise the form factor to unity at the peak. The spin—orbit potential is given by
B. Sin/ia, The optical potential and nuclear structure
Uso(ro)=U~ ±~._[l/[1+exp{(ro_rsoA1~)/aso}]]X1a,
9
(2.4.3)
where 1 and a are the angular momentum and the spin vectors, respectively, rso is a radius parameter and U~0a depth parameter. For the coupled imaginary potential form factor, the model thus has ten parameters. Fricke et al. [18] with the motivation of finding out standard values for the optical model geometrical parameters analysed (elastic) scattering (and polarization) data for incident protons. Standard parameters are extremely useful for prediction of results in scattering experiments and also, more important, for Distorted Wave Born Approximation predictions of inelastic scattering cross-section and polarizations. The distorted waves are generated by using the optical model obtained from such an analysis. In table 1, the results of Fricke’s analysis are shown for standard geometrical parameters which are as follows 6,aR = 0.75, r,(= r~=r~)=1.37, a rR l.l 1(avas) 0.64, r~0= 1.064 and a~0= 0.74. 2 column is not universal throughout quality of the fits obtained, as can be seen from the to x attribute any significance to such a theThe periodic table; it would be however somewhat reckless variation in the quality of the fits. It is of some interest to investigate the energy and isospin dependence of UR, the depth parameter. Perey [19] pointed out that before investigating the isospin dependence of UR one must first subtract the variation of UR with A arising from the variation of the Coulomb potential with Z. On the basis of the simple assumption that UR varies linearly with incident energy, he found the Coulomb correction term to be 0.4Z/A’/3. Fricke et al. [18] and also others workers such as Calderbank et al. [20]. Thomas and Burge [21] using the difference function method and Sinha et al. [22] using an additional Saxon—Woods derivative form factor in the real central part attempted to extract any possible isospin dependence of the depth parameter UR. For incident protons, the interaction of the protons with the neutrons of the target nucleus is stronger than with the protons. Therefore for N> Z nuclei, because of the isospin dependence of the two-body interaction, one expects some sort of dependence on the so-called asymmetry parameter e = (N Z)/A. Such a possible dependence was predicted by Lane [23] on theoretical grounds. Later in this report a more microscopic reasoning will be given for (N Z)/A dependence of the optical potential. A detailed discussion of this problem of extraction of any e dependence from a phenomenological point of view is given by Satchler et al. [24]. Taking into account the isospin dependence, energy dependence and Coulomb potential dependence, it is generally believed, although with reservations, that UR, the depth parameter, can be written empirically as (given by Perey [19] on the analysis of data around 1 5 MeV) —
—
UR
=
53.3
—
0.55 E
+
[0.4 ~
+
27 (N_ Z)]
,
(2.4.4)
so that, a symmetry potential of 27 MeV and an energy dependence of 0.5SE is obtained where E is expressed in MeV. Although quite useful for general survey, one cannot rely on such an empirical equation with any confidence. Fricke et al. [181 for example, found an energy dependence of 0.22E around 40 MeV. A more detailed discussion on energy dependence is given by Hodgson [7].
The next question regarding the e dependence was the shape of the isospin potential. Does it
10
B. Sinha, The optical potential and nuclear structure Table 1
l2~
UR(MeV) Wv(MeV) W~(MeV)
38.29 8.72 1.18
Fits obtained using Fricke’s standard geometry, 40 MeV incident proton 28Si 40Ca 54Fe 58Ni 59Co 60Ni 68Zn 90Zr
120Sn
208Pb
42.38 4.00 1.96
43.22 1.21 4.52
45.79 6.89 1.14
45.05 6.63 1.22
45.71
43.8 22.3 630
20.2 18,3 841
6.6 66.1 1015
6.4 37.4 1046
3.1 5.7 1088
5.68
45.74 5.47
2.5
46.60 6.70 2.46
47.76 4.69 3.46
48.77 4.72 4.62
52.76 6.12 4.31
2.08
7.7 18.9 1118
10.7 7.0 1250
35.1 8.2 1410
3.4 9.3 1691
56.5 9.4 2116
x~/N 0 4/N,0 OR(mb)
57.1 244.2 316
follow the same volume shape as the real central term or is it surface peaked? If the neutrons tend to “stick out on the nuclear surface”, one can expect the isospin potential to be surface peaked. Satchler [251 however showed that a surface peaked isospin potential can be simulated by just a Saxon—Woods form factor with a slightly increased radius parameter. Expanding any form-factor functionf(R + ~5R) f(r) ÷ ~R df(R)/dR one can take into account surface terms with an increased radius parameter R + ~R. This kind of argument, however, is on the tacit assumption that the radius parameter used for the isospin potential is identical with the spin isospin independent part of the potential, an assumption not necessarily valid for all the cases under consideration. Sinha and Edwards [26], with the motivation of exploring the possible208Pb shapewith and an structure of additional the isospin potential first analysed the 30 and 40 MeV scattering data of Saxon—Woods derivative term in the real central part. The parameters, chosen to describe such a term had three additional parameters, V 5, as and rs. The large number of the parameters in the model obviously made the analysis somewhat cumbersome and any “one to one” correspondence between a theory and the magnitude of the parameters could be misleading. Fortunately for 208 there exist (p, n) cross-section results due to Batty et al. [27] and one can test any form factor for the iso-spin potential by calculating (p, n) cross-sections using such a potential. In the (p, n) reaction, unlike elastic scattering, the isospin potential is wholly responsible for transition to the isobaric analogue of the ground state of the target nucleus. The programme of Sinha and Edwards was thus a two-stage programme. First using a Saxon—Woods derivative term in the real central part fits were obtained to both cross-section and polarization data. If the quality of the fits improved substantially the parameters found for the derivative term were then used to generate the (p, n) cross-section, on the assumption that the derivative term is wholly due to the isospin potential. This implies that the neutron excess distribution Pne = Pn Pp when folded in with isospin the two-body interaction is and identical to the derivative 208Pbthe (30 and 40dependent MeV) thepart fits of obtained for elastic scattering polarization improved term. For by a factor of two or three using an additional derivative term. The (p, n) cross-section prediction for 30 and 49.4 MeV was found (fig. 1) also to be quite reasonable, particularly in view of the fact that standard phenomenological potential fits are quite poor [271. They also compared the results with the Coulomb displacement energy obtained for 208Pb by Nolen et al. [281. The Coulomb displacement energy, 1~E~ is the Coulomb energy difference between the analogue state and its parent. ~ in its simplest form can be obtained by folding in the Coulomb potential with the neutron excess distribution, Pne~To estimate i~E~ ( 4’~refpnev(r)r2dr)it was assumed [26] that the neutron excess distribution Pne is identical —
B. Sin/ia, The optical potential and nuclear structure
11
200Pb (p,n)208B~
30.2MeV x 0.02 0.01
0.00~ 0
I
10
20
I
30 40 50 CM angle, degrees
I’
60
70
Fig. 1. A comparison between the experimental 208Pb (p,n)208Bi differential cross sections at 30 and SO MeV and theoretical predictions obtained by representing the symmetry potential with the best fit derivative terms found in the elastic scattering analyses.
to the derivative potential obtained for the best fit 30 MeV proton scattering data on 208Pb The result, 18.9 MeV, is close to the experimental value of 19.0 MeV. However, this close agreement is fortuitous since various corrections, such as exchange effects, to the simple folding model of were ignored. Further Pne = Pn p,, and the derivative potential has slightly different shapes because of the “smearing” arising from the finite range of the two-body interaction. However, since the Coulomb displacement energy is sensitive mainly to the r.m.s. radius (r2 )IYe2 of neutron excess and is only slightly affected by other details of the shape Pne, it is instructive to compare the values obtained for (r2)~ for this moment with that of Nolen et al. [28]. Indeed the value, 5.70 fm for (r2)~found by Nolen et al. agreed remarkably well with the value, 5.63 fm found by Sinha and Edwards. All this evidence collected together tends to suggest that indeed, for 208Pb the isospin potential is probably surface peaked. What is important as far as phenomenological models are concerned, is to try to relate the parr’neters obtained to well-defined physical properties. The large number of parameters makes this increasingly difficult but this problem is ameliorated to a large extent by microscopic analyses which relate to a much wider range of physical phenomena. —
12
B. Sinha, The optical potential and nuclear structure
3. The microscopic optical potential 3. 1. The ,thlding model By 1968 an exhaustive set of elastic scattering data for a wide variety of projectiles over a wide range of energies was available. The phenomenological optical model discussed in the last section has been used to analyse such data and reasonable systematics were obtained. Greenlees et al. [81 (GPT) developed a model to analyse the 30 MeV data with the motivation of extracting nuclear structure quantities such as the mean square radii and also study the density distribution of the target nucleons.
Although they developed the model from a variational calculation, the essential result that was used is in fact the first term eq. (2.2.1), i.e. in coordinate representation eq. (2.2.2). The antisymmetrisation between the coordinates of the incident nucleon and those of the target nucleons was neglected. For the two-body interaction GPT used a free two-body interaction of the following form v(r0)
=
Vd (r0) + v~(r0)r~ ‘r + v0(r0
) U.
fJ +
v01(r0 ~
+ [v,(ro)+v~1(ro)rt.TI S,2+ v,5(r0)~ [(r0
—
U
r~) (p0- p~) (ao+~~)I X
(3.1.1)
where i in the equation refer to the target nucleon. For nuclei with total angular momentum zero. the tensor term in first-order does not contribute, the terms which contribute are due to the first, second and the last term in eq. (3.1.1).
The terms are as follows,
(3.1.2)
3r+f[pfl(r)—pp(r)]ur~r-.roIc13r, UR(ro)Jpm(r)vd~r--roId giving rise to the real central potential, and U 5o(r0)
~
2,i
4ir n~i
(2n+l)!
d
2(,i
-
1) d2
~ ~ [~7-~
~
d2~2
-
~Pn~Pm]JVIs(2~2~}10(313)
giving rise to the spin orbit potential. Care should be exercised in the summation of eq. (3.1.3) because it is an asymptotic seiies. The real-central part has a volume term which follows the nuclear density but is smeared by Vd. The second part clearly is the iso-spin potential which depends on the difference between neutron and proton distributions and the isospin dependent part of the two-body interaction. The eq. (3.1.3) is reduced to the Blin-Stoyle prescription [161 for ii = I. For the form factor of the free two-body interaction, GPT usedis aan pseudo Yukawa potential with a mean square around 2. The pseudo potential average potential which is in agreement withradius the phase shift 2.5 fm of free two particle scattering. In a subsequent publication [291 the mean square radius analysis
was changed to approximately 4.0 fm2 using a Gaussian shape. They point out that a Yukawa shape overestimates the binding between the interacting nucleons at small distances. GPT [8] further assumed that p~,= (N/A)pm and p,, = (Z/A)pm so that the only difference between the neutron and the proton density distributions is in the normalization and not in the shape. Moreover, assuming v 7
=
~Vd,
one simply gets, for the real central potential,
3r, UR(rO)=f(1
+e~)pm(r)vdIr
--
r0Id
(3.1.4)
13
B. Sinha, The optical potential and nuclear structure
where ~ = 0.48 and e = (N Z)/A, which is quite simple to integrate after calculating the angular part of the integral analytically for a Yukawa shape. Several interesting results can be obtained from eq. (3.1.4). The volume integral of UR (r0), ~RS is related to the ~d, the volume integral of the two-body in—
teraction by the simple relation JRS/A =(l ~~flJd (3.1.5) 3r is the mass number of the target. Secondly, the mean square radius of UR(rO), where A —fpm(r)d
o~r2)Ris given by, (r2)R (r2)m ÷~T)d’
(3.1.6)
where (r2)m is the mean square radius of the nuclear matter density and (r2)d, the mean square radius of the two-body interaction. It should also be noted that (r2 )m is related to neutron and proton mean square radii by, N
Z
_~(r2)~ +—.(r2)~ (3.1.7) By an optical model analysis (see below) for a nucleus (r2 )m is obtained by searching on rm and am for the optimum fit to elastic scattering data. Since (r2)m is related to (r2)~and (r2)n by eq. (3.1.7), knowing (r2)~from say electron scattering experiments one can estimate (r2 GPT [81took a Saxon—Woods form-factor for the matter distribution with geometrical parameters rm am and assumed that the spin—orbit potential form-factor terminates after n = 1 in the polynomial. They used a phenomenological imaginary potential, a combination of a Saxon—Woods and a Saxon—Woods derivative term. The depth of the real central potential was varied so that various difficiencies in the model, such as exchange and higher order effects could be partly taken into account. In effect, therefore, the number of parameters was reduced from ten to eight. To test the sensitivity of the model to the mean square radius of the two-body interaction the best x2 was determined as a function of (r2)d. The details of the fits are given in ref. [81. A typical fit with such a model is shown in fig. 2 for elastic scattering of protons at 30.3 MeV. The x2 ‘s obtained with such a model were certainly comparable to those obtained by phenomenological analyses. Fig. 3 shows the variation of x2 with r2)d and as can be seen the GPT analysis pins down the (r2)d to within a reasonable range; although the later work of Greenlees et al. [29] showed that (r)d is not as well known as fig. 3 suggests. Table 2a shows the best-fit parameters found for such an analysis. They further showed that the two quantities ~RS IA and (r2 >d are well determined for the energy range concerned (table 2b). Any possible variation of JRSIA with energy could not be established with any confidence since the range of energy over which the analysis was carried out was not exhaustive enough. As will be shown later neither of these conclusions is strictly valid. The success of GPT with whatever drawbacks the model may have (to be discussed later in the section) shows quite clearly that a simple folding model can reproduce data reasonably well. The nuclear structure quantities such as the matter mean square radius, (r2)m ~ [1 + ~ (iram /Rm)2] with Rm = rmA”3 obtained for various nuclei were found to be quite reasonable in the light of other experimental results. However two most striking results of GPT caused some anxiety. The first problem was regarding (r2)m
.
),.~.
,
14
B. Sin/ia, The optical potential and nuclear structure
I
58
.
~
11111
~/
~ f I 7
::
~-r—
—r~r
59Co
30.3MeV
10
—
I
H
Ni 2 0
I
I-S- -U—-r—1- -
\
/\60Ni ~ 00
“~
~
~
~1 0 ~08 0.6
208 PbioTf\~~~~
0
20
40
60
80
100
120
140
160
180
CM. ANGLE (degrees) 11g. 2a. Fits to 30.3 MeV cross-section data for various elements. Solid lines are the GPT model predictions.
the difference 6 = Kr2 >~I2 Kr2 ~/2 of the neutron and proton root mean square radii. Kr2 >,1,/2 can be estimated from the empirical relation given by Acker et al. [30] from electron scattering experiments. Knowing Kr2 >m, the values of 6 obtained by using eq. (3. 1 .7) varied between 0.6 and 0.7 fm. Such a difference obviously indicates that the neutrons stick out by quite a substantial amount in N> Z nuclei, in agreement with the K mesic work of Burhop [311. It should be noted that since kaons get absorbed in the 10% density region the K-mesic data do not determine the r.m.s. radii and therefore such a comparison is not strictly valid. However this so-called neutron halo as predicted by GPT did not agree with the experimental predictions of Nolen et al. based on the measurement of the Coulomb energy differences [28] or with the more elaborate Hartree— Fock calculations of Negele [321 using a local density dependent interaction. It was pointed out that such large neutron skin thickness could be caused by the kind of two-body interaction used. A free two-body interaction as used by GPT does not have the correct saturation properties [32]. —
’.
B. Sin/ia, The optical potential and nuclear structure I
I
I I
I
I
I
15
I
-Ii ~AI~]Jr\\ .4
~
\~i\j
‘~‘~
~ -0.2
~
~:
+
29.0MeV
0.6
58
10
\I;f\’\
[\ ~\~J\ \i ‘p”
~
60N1
—
~ 1.0
___________________________
~vJ~f\/\~08p~8P’
20
40
60
80
100
120
140
160
160
C.M. ANGLE (degrees) Fig. 2b. Fits to the 29.0 MeV polarization data for various elements. Solid lines are the GPT model predictions.
In a subsequent publication however Greenlees et al. [291 showed that the root mean square difference need not be so great if one uses a Gaussian interaction and parameterises the neutron distribution keeping the proton distribution fixed according to the empirical prescription of Ackeretal. [30]. The second source of anxiety was the e dependence of the real potential. Greenlees et al. [29] showed that it is not possible to determine unambiguously the e dependence of the real part of the optical potential from analysis of elastic scattering data, as shown in fig. 4. The doubts that the e dependence of the optical potential cannot be found by analysing elastic scattering data were thus substantiated by these authors. Friedman [34] later showed that the inconsistency of the magnitude of 6 between the results of Coulomb energy difference and that of GPT can be resolved if one uses a density dependent interaction rather than a free two-body interaction, although his conclusions depended very much
16
B. Sinha, The optical potential and nuclear structure
I
I
Cc59 30.3
-
50-
MeV -.
-X~
x
—
ioo~N/
5.0
opi
--
o,OL___
I
0
2.0
30
4,0
-—
I 50
_i _..L.. 60 70
l’able 2b =
2.25
J
fin2
1~5/..1(or t8Ni
~~(o
6~Ni
120 Sn
208 l’h
Element
Energy 1Mev)
~Rs/~1
14.5 30.3 40.0
421 409 377
~.
-~o3
60Ni -
-
14.5
4)1 455
30.3 14 S
390 4 4
14.5 40.0 40.0 303 400
445 . 0 ± - 23 11 390±11 411 * 1~ 371 14
Element (~Rd~kV) 52.53 W~(MeV) 4.50 141s (MeV) 3.85 r 1(fm) 1.357 a1(fm) 0.520 rUso(MCV) 5.16 01(fm) 1.159 a05(fm) 0.597 5.47
o,t(mb) o~(expt)
53.05 3.84 4.58
52.16 3.81 4.71
51.13 3.43 6.68
53.26 3.45 6.53
1.320 0.614 5.75 1.162 0.579
1.3 11 0.609 5.29 1.171 0.582
1.332 0.636 5.82 1.206 0.534
1.317 0.752 5.49 1.206 0.497
5.39
4.50
2.97
0.84
7.78 9.87 17.62 5.58 3.48 1085 1038±43 1153 1169±39 1154 1053±51 1643 1638±68 1986 1865±98
(r21,~(fin)
58Ni
-r
120Sn 208Pb
çmb)
4.12
4.11
4.16
5.02
5.84
Various elements
MeV
tin
± 16 ± 20
)
(G1’T) r~~° --- (r2~ tim)
0.71
± 0.14
064 0 . 70
*
± 15
10 - 11 ±
± 11
01 0 10
—
10 -
1)72 067
*
0.20 0 70
—
0.64
±0.40
B. Sin/ia, The optical potential and nuclear structure
I
58 150
208pb
I
17
I
~0~M~V
6
ZETA(~ 2 values (GPT model) with the isospin dependent part of the two-body interaction. Any value of ~ Fig. 4. Variation of best-fit x between —0.5 and 1.0 produces equally acceptable fits indicating the insensitivity of the data to l~.
on the choice of the parameters. It is known [38] that a properly parameterised density dependent interaction has adequate saturation properties, unlike a free two-body interaction and therefore, as speculated by Negele [32] it is seen that the high value of neutron skin thickness, 6, obtained by GPT is because of the interaction they use. Sinha [33], on the other hand, following the work of Friedman [34] showed that using a linear density dependence in the two-body interaction, the absence of an explicit c dependence can be explained. He found JR
(3.1.8)
5/A—(1+adSm)Jd+EJr(l+arSm),
Ud, a~are parameters associated with the density-dependent and a simple 2a~/R~).interaction ~d and J~ areSm theisvolume algebraic function Of Rm and am Sm (3am/Rm )(l ~ ~ integral of the spin, isospin independent term and the isospin dependent term in the two-body interaction. JRS/A in eq. (3.1.8) is not an explicit function of e because of Sm. Fig. 5 .shows how an improved dependence can be found by plotting JRs/A(l ÷adSm) against e~= e(l +arSm)/ (1 +adSm).
where
,
—
18
B. Sinha. The optical potential and nuclear structure
310r
40.0 MeV 290 270 ~oz,
‘~250-
303MeV
>340 a)
Jd=304~12MeV-F3
~ 320
(I)
V
300 380 14.5 MeV
~k360
~~d=337~30
340
MeV F3 —
320
60Nt
;~
63~
=
5ePd~e
420
-
400
-
eli
E
+
q~Sm)/(1.s. a~Sm)
38C
I I
90z,
58N~
360
206Pb
-
1
~
~°°L~~
~ 420
tss Co
60N1
.-~4 ~3806C
!~6
t,
20sn
20$pb
1440
420-
I,6s,,
63
Co
“SI,
eSNI
-
~°Fe
480
0
0.05
0.10
0.15
0.20
0.25
‘C
Fig. 5. Plot of ~JRS/A )c/(’ + adSm) as a function of ~ compared to the plot 0fJRS/A as a function of e. The improved linear dependence of the first plot is apparent.
B. Sin/ia, The optical potential and nuclear structure
19
k\~
Fig. 6. A schematic representation of a “two-particle—one hole” excitation.
Fig. 7. A schematic representation of a “two-particle—twohole” excitation.
One can generally conclude at this stage that a density dependent two-body interaction can explain some of the drawbacks encountered in the GPT model. Myers [93] has shown from a purely geometrical point of view that a two-body interaction with the correct saturation property is essential, a free two-body interaction being inadequate. In the following sections we shall investigate how consistent it is to use effective interactions with correct saturation properties instead of a free two-body interaction for constructing the optical model. 3.2. Effective interactions and the optical potential In calculating a potential to describe elastic scattering, 3 main types of contribution must be considered. The most important one comes from the direct ground-state to ground-state process in which the projectile and target interact once only. The second process involves virtual transitions to energetically-unreachable intermediate states, followed by a return to the initial state. Both of these processes are essentially elastic events and contribute to the real part of the optical potential. The third process involves actual transitions to energetically-reachable states. Such events, which include inelastic scattering and rearrangement collisions, have the effect of removing flux from the elastic channel and, thus, give rise to the imaginary part of the optical potential. Now, U0~~ of eq. (2.2.1) between momenta k and k’ can be written as, 0CC
(k’IU0~8Ik)= ~(k’,
(~
UflOCC
~
+
~
ulmn)Kmnlu
.
m,n E+j_Cm_En+l~
) ~,
i)
.
(3.2.1)
The states designated Im), In) and Ii) are the single-particle Hartree—Fock states, 1k’) denote a plane wave and v is the two-body interaction. The c’s denote the single-particle energies and the 2 k2 /2m. incident wave number k is given by E = h Sums taken over occupied or unoccupied states in the ground state wave function are indicated by the notation occ or unocc respectively. The second term in eq. (3.2.1) has a principal value contributing to the real term and an imaginary part for energy conserving transitions for which E + = + e,~,and this term is shown schematically in fig. 6. The incident nucleon initially in the state 1k) goes to state rn) in the continuum and excites a particle from below the Fermi level to above the Fermi level creating a hole state Ii) and a particle state In). The (2p—lh) states are depicted in fig. 6,
B.
20
Sin/ia. The optical potential and nuclear structure
the indicating interactions. For elastic scattering, it should be noted that k’l although the angle between the vectors may be non-zero. Higher order terms, among others, can constitutc (2p 2h) states, shown in fig. 7. 1-lowever, eq. (3.2. 1) can be looked upon as an indication of using a re-defined effective state-dependent two-body interaction, different from a free two-body interaction, v such that -
-
unoc C VeffV+
vI,nn) Knznlv -~
~,
n
E
-~-
+ ~
. -—
.
(3.~.2)
~, + l~
In two regions of Em ,e,~the second term can be important; when the denominator is small or zero or when the matrix elements of v are large. In the first case, when the propagator goes through a singularity, Veff gives rise to an imaginary potential. In the event of the denominator being quite small, on the other hand, the contribution to the real term becomes significant. A formal discussion on the derivation of the second term for the real part will be given later (section 3.6)~the calculation of the imaginary potential on this basis will also be discussed (section 4). One can now compare 0eff with the corresponding bound-state case where one obtains a boundstate effective interaction in a similar fashion [36] unoce
V~f=
v+
vlnin) Kmnlv —--—--~-—---——
in, 0
+ Ej
.
3.2.3)
-
where i and j label unoccupied orbitals. In the case of scattering mn) represents a “two-particle-one-hole” state but in the bound state calculation mn) represents a “two-particle—two-hole” state. The spirit of using a bound state effective interaction for scattering lies in the assumption that the difference between the two types of interaction is negligible and that one can use the same methods to take into account the strong repulsion and the correlations between nucleons at short distances as used for the bound state interaction. For hard-core interactions the separation distance method can be used to take into account the strong repulsion at short distances. This approximation of using, in effect, only the long—range part of the two—body interaction and cutting off the shortrange part by using a suitable separation distance is expected to be valid for elastic scattering where only the long-range part is expected to he important~the short-range behaviour of the interaction is not explored enough by the incident projectile. A criticism and discussion of this procedure is given in section 3.6. Generally, the effective interactions to be used are of two kinds. First, an interaction with a hard-core, the core radius being approximately 0.4 fni. In fig. 8, the interaction v is plotted as a lunction of the two-body separation r. The repulsion tends to infinity as the two-body separation approaches zero. This repulsion is the root cause of saturation within the nucleus. We shall take into account the repulsion by balancing it against the attractive part shown by the shaded area. This is known as the separation distance method of Moszkowski and Scott [391. it is generally energy or state dependent. With the increase in relative momenta of the colliding nucleons. one expects an increase in the magnitude of the separation distance. (For a detailed discussion see [66]). In effect the part of the two-body interaction that one uses is the thick line shown in fig. 8. It should also be noted that the separation distance method can only be used for the even components of the interactions. Odd state interactions which are entirely repulsive cannot be tackled by the separation distance method as discussed here. The second term in eq. (3.2.3) or eq. (3.2.2) is treated in many ways for hard-core interactions.
B. Sin/ia, The-optical potential and nuclear structure
21
POT
DISTANCE IN
fm
A
~LONG RANGE PART
Fig. 8. Two-body interaction and the separation distance.
One of the interactions used by Slanina and McManus [40] is the Kuo—Brown effective interaction [41], where the second term is suitably approximated by an effective energy denominator eeff and closure is used to simplify the calculation further. They (Kuo and Brown) noted that the second-order terms are large in the case of 3S1 states and are mainly due to tensor component of 4ISlvTQ(QP/e)vT211
the two-body interaction. Therefore, terms such as( 1s) in analogy with the second term in eq. (3.2.3) (where VTQ represents the long range part of the tensor force and Q~, known as the Pauli operator which suppresses any excitation to occupied states) become as important as the first-order term ~ IvI~15).It was found to be so, although handling of the Pauli operator Q~, because of its angular dependence, proved difficult. However, the situation becomes quite simple for the following two reasons [411: (1) The intermediate states e,~, and c,~can be considered essentially as free. (2) The remarkable property of the second-order exictation is that only the intermediate states of rather high energy —P220 MeV are of major importance. In fact, the matrix elements, if plotted as a function of relative momentum of the intermediate states, peak around 220 MeV. With these observations and on the basis of some approximations Kuo—Brown applied closure for the matrix
elements, such that, (UTI
(Q~/e) VTQ)
v4~/eetr
(3.2.4)
-
Taking into account the spin and isospin of the interacting nucleons, they found that the additional interaction from the second-order term is of the form ~ in the triplet even state. The state dependence of E + e,, was however neglected (discussed later in detail). Thus, recalling the folding integral and neglecting exchange the real central part of the optical potential for an incident proton is —
U~(r0)—jp~(r)vP~Ir-—roI d~r+fp0(r)v0~Ir—roId~r,
~
(3.2.5)
(p, n refer to a proton and a neutron respectively) where ~ and
Vnp
(3.2.6)
~~se j(3t)t.e.
+Vs.e.)
(3.2.7)
22
B. Sin/ia, The optical potential and nuclear structure
-
-
©
x —x—
V~ .~i V1~
( ~
X—
~
)
x~x
~
~
~
x
x___x___x—
-6
~
©
_x_—~x~~
10
©20
t-
—8 .25
.49
.74
I 0
.98
1.23
I
I
1
13
1.44
I 2)
2
1 72’—k1 I
1
3
k~(f Fig. 9. Dependence of Kuo—Brown tensor matrix on k~.,the square of the local Fermi-momentum.
where Vte and Vs.e. refer to the triplet even and singlet even part of the interactions. For these interactions the odd-state component is assumed to be negligibly small. It should be noted that there is no a priori reason for this being so; it depends on how the effective interaction is construct-
ed. For example, the soft-core potentials discussed later in the section usually have a singlet—odd interaction the strength of which is certainly not negligible. For the particular case of the Kuo—Brown [41] interaction, Vse(X)
t4’~~(x) forx> dse
(3.2.8)
forx~d 50
where HJ refer to the Hamada— Johnston [42] interaction between two free nucleons and dse is the separation distance discussed before. Similarly, forx> dte
(3.2.9)
for x~
B. Sinha, The optical potential and nuclear structure
23
tion due to Green [44], the density-dependence in effect mocks up the state dependence of the interaction. This is an empirical interaction, in the sense that the parameters for the density dependence were chosen by fitting the singlet-even and triplet-even contributions to the binding energy of nuclear matter. Although extremely reliable for correct binding energy prediction etc., it is not possible in the case of the Green interaction to distinguish clearly between the first and secondorder terms of eq. (3.2.3). The Green interaction [44] has the general form,
u8,5(x)
=
c51(l ~
forx> d~,1
(3.2.10)
forx~d~~ where s,t refer to the singlet-even and triplet-even part of the interactions, c~5and ~ are sets of parameters which can be found in ref. [44] and ~ is the Kallio—Kollveit interaction [45]. The density function Pm is evaluated theinto centre of mass coordinate thedependence two interacting nucleons. type 3 of dependence, to at take account empirically thefor state of the effectThe p~~’ ive interaction was first suggested by Bethe [46] from a general nuclear structure point of view. Kuo [47] later calculated the Kuo—Brown matrix elements taking into account the state dependence of Q~,the Pauli operator in eq. (3.2.3). Fig. 9 shows the dependence of the matrix element (VTQ(Q/e)VTQ) as a function of k~the local Fermi momentum, defined by k~= 1.57r2p~in a Thomas—Fermi approximation. The graph is interesting, because it predicts a linear dependence on k~.,i.e. p~3,as suggested by Bethe [46]. Taking into account this state dependence explicitly, V4Q becomes a function of r and Pm such that 14Q(r,pm)=
VIQ(TO,Pmo)
[1
—
a~(p~/p~o)213} ,
(3.2.11)
where Pmo is the central density, and a~is a proportionality constant the value of which depends on the quantum numbers of the wave functions in the matrix elements [471. An average value of 0.13 will suffice. Dependence of (VTQ(QP/e)vTQ) on the magnitude of c, and j was also investigated by Kuo [47]. This dependence will be ignored for the present discussion. The second group of interactions, used for calculating the real part of the optical potential are known as soft-core interactions. One distinct advantage of the soft-core interactions is the straightforward convergence of the matrix elements within the nuclear medium. In recent years quite a few Hartree—Fock calculations have been performed using soft-core interactions. Particularly successful applications can be found in ref. [32] where Negele uses a finite-range density dependent repulsion term. Among others, the interaction due to Pandharipande [48] is also a soft-core interaction, although he uses a zero-range repulsion term. Both these interactions have been used by Thomas, Sinha and Duggan [49] (TSD) for an extensive analysis. A detailed discussion will follow later. Another successful and rather simple interaction has recently been used by Vautherin and Brink [50] and later by Negele and Vautherin [51] which is due to Skyrme [52]. The remarkable property of the Skyrme interaction is that with the help of five parameters, the Hartree—Fock singleparticle field becomes a local function of Pm , the nucleonic density function, r, the quantum mechanical kinetic energy density and J, the spin density. Dover and Giai [53] have calculated the optical potential using the Skyrme interaction, which will be discussed in more detail in section 3.4.
B. Sinha, The optical potential and nuciear structure
24
-
60r
-
-
30
Ca
-i
N.
20L
--
H
L~
--
10
r~)
Fig. 10. A comparison of theoretical re,.~ - . , ,tentials obtained by using the density-dependent interactions with the phenomenological form-factors (Slanina and McManus).
Slanina [121 and Slanina and McManus [40] first attempted to calculate the gross properties of the real part of the optical potential using an effective interaction. They used the Kuo—Brown [41] interaction and both the strong and weak versions of the density-dependent interactions of Green. Table 4 shows the volume integral of the optical potentials obtained by Slanina and McManus along with the effective mean square radius of the interaction, the quantities being defined as follows: ~Rs
(3.2.12)
47rfuR(ro)r~drO ,
and (r2)err = (r2)
2),.,,, (3.2.13) 0~~ Kr where Kr2 )opt fUR (r 0 )r~dr0/fUR (r0 )r~dr0 (3.2.14) 2 )eff in eq. (3.2.13) stresses the point that for density-dependent interactions The definition of Kr which depend on the centre-of-mass coordinate for the two interacting nucleons, the only way —
.
B. Sinha, The opticalpotential and nuclear structure
25
one can estimate Kr2 )eff is by an indirect method using eq. (3.2.13). The (N—Z)/A dependence of the optical potential depth found by Slanina and McManus has a depth which varies approximately between 13 to 30 MeV, generally agreeing with the results of Fricke et al. [18]. They noted however that the geometry of the folding process can enhance the dependence substantially at times. The agreement of their results with phenomenology and Greenlees et al. is quite good. Fig. 10 shows the comparison of the potential form-factors. It is interesting to note that the general shapes of the form-factors are not very different from each other, although individually they are derived using different interactions. The work of Slanina and McManus therefore confirms the view that using an effective interaction the potentials obtained are similar to those obtained by phenomenology, at least for the
gross properties. The finer details of the optical potential, one expects, can only be understood by fitting data, or by using a somewhat more elaborate model as discussed in the following sections. 3.3. The exchange potential Before going on to discuss how efficiently the folding models reproduce data it is necessary to take into account the exchange term arising from the antisymmetrization between the coordinates of the incident nucleon and those of the target nucleons. It is recognised [71 that most of the energy dependence of the optical potential arises from the exchange term, although variation of the separation distance from the effective interaction (used to construct the optical potential) with the energy of the incident projectile for example can also be important [55]. Including exchange, the terms of the form UR (r 0 )11i(r0) are replaced by non-local 3rin the Schrodinger equation [56] where p(r, r terms like fp(r, ro)vexlr— r0Ii~’(r)d 0)is the mixed density and
Vex ~S the
exchange component of the two-body interaction. For the interactions which are
of the Serber type, i.e. no odd-state term, the exchange and the direct components are identical. However, in the presence of the odd-state interaction, the odd-state part of the interaction changes sign for the exchange component of the interaction [33], enhancing the exchange effect considerably [33, 46]. It should be noted at this point that the incident projectile cannot distinguish between the exchange and the direct part of the interaction, it is the resultant potential that is of importance. Therefore, the relative strength of the exchange and the direct term are not of crucial importance as often suggested [561; it really depends on the particular kind of interaction used. Slanina and McManus [40] calculated an equivalent local potential for the exchange term using the method of Perey and Saxon [73] for transforming the non-local potential to an equivalent local potential. They concluded that almost 80% of the energy dependence of the optical potential arises from the 2) exchange term. Both the volume integral per nucleon ~RS /A and the mean square radius (r 0~1increase somewhat by the inclusion of the exchange term. Owen and Satchler [56] calculated the non~localexchange term using the Kuo—Brown interaction. To take into account the odd-state interaction they used a Gaussian repulsive potential which fits the two-body P phase shifts. The theoretical prediction of scattering cross-section using the non-local exchange term along with a direct term and a phenomenological imaginary and spin—orbit potential was subjected to a conventional optical model analysis. Although procedures of this kind may in fact cloud the reality somewhat the conclusions they arrived at are rather interesting. For details the reader is referred to ref. [56] but the most important conclus-
B. Sin/ia, The optical potential and nuclear structure
26
sions are that exchange (i) enhances the asymmetry effect (arising from the neutron excess), (ii) increases iRs/A by as much as 40% at 10MeV and (iii) contrary to the work of Slanina and McManus, reduces ~ However, the conclusions depend on whether the odd-state interaction is included or not, as pointed out before. Thomas et al. [49] and Sinha [55] rather than emphasizing the non-local nature of the exchange term calculated the equivalent local potential by using a plane-wave approximation. It was found that the phenomenological properties of the energy dependence of the optical potential can
be well reproduced by such an approximation. The derivation of an equivalent local potential discussed below is motivated by the work of Negele and Vautherin [51]. The term such as fp(r, r0 ) vlr r0 I i,D(r)d3 r can be transformed to a local term if i~(r)can be expressed in some way in terms of
111(r
—
0).
The integrand can then be calculated explicitly without
solving the Schrodinger equation each time. This can be done by taking advantage of the fact that the two-body interaction is of short range, so that expressing r = r0 + s where s Ir r0 and ex—
panding 111(r) in a Taylor expansion one gets ili(r)
exp (s ‘7~)i,Li(r~).
(3.3.1)
Such an expansion is only possible because for any large value of s, u(s) cuts the integrand to zero. Performing the angular integral over the direction of s, essentially in the same manner as in ref. [511the expansion becomes, ~(r)
=
[(j
=
F(—V~)11i(r~)
=
F(k
}
~)]~(r~)
d(cos ~) exp (s V0) /fd(cos .
(3.3.2)
2) ili(r 0)
2) where F(k and sIsI
=
(3.3.3)
sin (ks)/ks
(3.3.4)
and it has been assumed that k2 —vi. In the calculation of exchange potential we have used a local approximation to evaluate k, so that, EEcm
UR(rO)— Vc(ro)]
,
(3.3.5)
where UR is the real part of the optical potential and V~,the Coulomb potential for a charged incident projectile. Thus the non-locality is suppressed by transforming l/.l(r) in terms of 111(r 0) so that the local equivalent to the non-local exchange term becomes, U~x(ro)fp(r,ro)vexIr_roIF(k2)d3r.
(3.3.6)
What should one use for the mixed density p(r, r0)? The obvious and the most simple choice would be to use the Slater mixed density for infinite nuclear matter, assuming that the -effects due to the boundary of the nucleus are not very important [57]
27
B. Sin/ia, The optical potential and nuclear structure
Pb
Pb
~50. -
—-~.
-
0
40 —.
- -
N - —WDDE _~DDE
.‘—~._-.
I
I
I
20
30
40
-
PROTON ENERGY (MeVI Fig. 11. Energy dependence of the optical potential depth for
40Ca and 208Pb (TSD).
B. Sin/ia, The optical potential and nuclear structure
28
SERBER TYPE INTERACTION 7.0
60
NEGELE —VAUTHERIN -
5.0
- —
SLATER
-
MIXED DENSITY
MIXED DENSITY
CaI/.OMeVI
-
0
30
20 NUCL
RADIUS IN
~5O tm
Fig. 12. The exchange potential using a Serber-type of interaction.
p(r, r0)
3.0 =
p(R)pSt(sk~) = p(R
)—---
sk r
ii (sk~),
(3.3.7)
where2Pm. kF is the P~is local Fermi momentum by the approximations as, of nu= Here the nuclear density,given or some sortThomas—Fermi of an average value for each type 1.5TT cleon i = p, n, such that
k~= 1.57r2(p 1(r)+p1(r0)).
(3.3.8)
Negele and Vautherin [511 on the other hand, by using a suitable averaging method, recently oh tamed the following expression ~
,
—s kF
(3.3.9)
2, 11k, where r(R) is the quantum mechanical kinetic energy density given by r(R) ~ V11i0(R)1 being the single-particle wave functions. The great advantage of an expression such as eq. (3.3.9) is that it takes into account the surface effects through the second term and it can be easily calculated once r(R) is known. What sort of prediction in terms of energy dependence and absolute magnitude does one obtain using such a prescription for the exchange potential? We shall investigate in what follows these questions with the simplest assumptions first, i.e. with F(k2) j 0(ks) and p(r, r0) = p(R)p~(sk1,-). It should be noted that expanding Jo (I..-s) in a power series of (ks), one obtains first the linear ener-
B. Sin/ia, The optical potential and nuclear structure
(EVEN
29
ODD) TYPE INTERACTION
•
70
NEGELE -VAUTHERIN MIXED DENSITY
50 — — — —
SLATER MIXED DENSITY
50 40
40Ca (40MeV) —
—
-------5.
uJ30. ‘S
‘S W
20-
‘‘S
‘S ‘S
10 .5. 5.-
I
1.0
2.0
I
I
3.0
4.0
NUCL. RADIUS IN
Fig. 13. The exchange potential using both even
50
tm +
odd type of Negele interaction
1321.
gy dependence, reminiscent of the phenomenology, from terms like (k2s2) (eq. (3.3.5)), and then higher order terms of E from (k4s4) etc. Fig. 11 shows the energy dependence obtained by Thomas et al. [49] using these prescriptions for F and p(r, r 0), for various interactions N (Negele [30]), WDDE, SDDE (Green [44]), and P (Pandharipande [48]). The dependence is linear for the energy range considered, thus validating the phenomenological prediction. The coefficient of linearity obtained in this work turned out to be 0.26. Sinha et al. [54] later showed, that taking into account the energy dependence of the separation distance the coefficient is reduced to 0.22, almost identical to the value obtained by Frickeetal. [18]. What happens if the eq. (3.3.8) is used for p(r, r0)? Fig. 12 shows the form-factor obtained for the exchange using eq. (3.3.6) for an effective Kuo—Brown interaction at 40 MeV incident proton energy. As can be seen, the difference is most significant in the surface region. However, considering that the exchange potential for a Serber type of interaction is only 15% of the total potential, the difference is not substantial. For interactions which have odd components, such as due to Negele [30] the difference could be somewhat more important as shown in fig. 13 but without any serious consequence since, in the final optical potential (direct plus exchange), these differences will not be of any numerical significance. Indeed, it has been found that the crosssection predictions are almost identical [55] using either the Slater or Negele and Vautherin prescriptions.
B. Sinha, The optical potential and nuclear structure
30
10
-
-
R=6O’~
\
\‘~
54.
10
‘~ ‘S
~\
6789
~ Ni N
\N’~ N
“!‘S.
\
‘S..
“
‘S.
‘S
‘~-..
‘
X
N
r
~S.
~
O 2
-
\
j
0(ks)
\
—
2
3
\ 0
1
‘5..
~
~
\
r I
‘S
08
R(F)
‘S
—
—
————
QUADRATIC SLATER
TRUNCATED
2
3
EXPANSION
S(F)
l-ig. 14. The Negele—Vautherin mixed density at the point j0(ks) goes through zero. For explanation of the error bars see ref. 511.
Why is this so? The explanation lies in fig. 14, where an almost accurate truncated expansion of (Negele and Vautherin [51]) is compared with the Slater prediction. 2) Although terms dropthey off differ to substantially for almost allafter values shouldinbethe noted thatthus the cutting u(s) F(koff the mixed density zero almost immediately 1.0 of fmRasit shown figure, p(r, r0)
contribution beyond s = 1.5 fm. The points discussed above are only true of course for elastic scattering. For inelastic scattering or for scattering where surface effects are comparatively more important the merits of eqs. (3.3.4) and (3.3.9) will be both tested and revealed. At present this point is being pursued [581 for inelastic scattering. We end this section by concluding that a local expansion of the generally non-local exchange term is possible without incurring any significant error and that this is possible only because of the short range nature of the two-body interaction. 3.4. Microscopic optical model analysis
So far, the method of calculation of the real central part of the optical potential using a boundstate effective interaction has been discussed. The need for the use of an effective interaction as opposed to a free two-body interaction has also been emphasized. Furthermore, the exchange term arising from the antisymmetrization of the coordinates of the incident nucleon with those of the target nucleons has also been calculated using a local expansion. The real central part is thus,
B. Sin/ia, The optical potential and nuclear structure
UR(ro)Ud(ro)+Uex(ro),
(3.4.1)
where Ud(rO) = i=p,n ~ fp,(r) vefflr
—
r01 d3r
(3.4.2)
2 ) d~r
and Uex (r 0)
=
31
2I~ fpi(r, r0) u~,,Ir
i=
—
(3.4.3)
r0 I F(k
the various terms having been explained in the previous section. How efficiently can such a model predict the data? The optical model used to do so is due to Thomas et al. [49]. They used the effective interactions due to Green [44], Negele [32], Pandharipande [48] and also a Gaussian interaction with a variable range parameter. The Green interaction has already been discussed, the Negele interaction is comprised of three components: short range (s) and long range (d) direct terms and an exchange term (e). In order to simplify the radial dependence average interactions between like (L), (p,p or n,n) and unlike particles (U) (n, p) are taken, i.e.
u,~~(x) = ~
[Vs~i,e(X)
+ V~d,e(X)1.
(3.4.4)
The functions V~j,e(X)are density dependent and are of the form vav(x)=v~(x)+vr~(x)k~. ,
(3.4.5)
where kF is the local Fermi momentum given by k~= l.57r2p~
(3.4.6)
,
and a = 3 for the s component and 1 for d- and e-parts. The radial variations in Va%’(X) have been calculated from Negele’s radial functions for kF = 1 .0 and 1.4 fm~ using his prescriptions 1.4°v(k~= l.0,x) u(kF = 1.4,x) _______________________________ —
=
1.4° 1 —
(3.4.7)
v(kF = 1.4, x) v(kF = 1.0, x) ________________________ 1.4° —
v~(x)
=
—
To take into account the different strengths of the like and unlike interactions they are written as, =
~
V~,’2u~”(x)k~. ,
(3.4.8)
where the constants V 0,1 are given in ref. [32]. The “adjusted” Negele interactions are used in
which the short-range average interactions at 1.0 and 1.4 fm~are multiplied respectively by 0.66903 and 0.83702. In the Pandharipande interaction [48]
31+3V33)g(x) u~(x)~V(V u~,,(x)~V(3V’3+V31+V~+3V33)g(x) v~~(x) = V(V3’ .~
u,,(x)
—
3V33)g(x)
~-V(3V’3+V31— V11
—
3V33)g(x)
B. Sinha, The optical potential and nuclear structure
32
I
I
I
I
5.
I
58N1
20
40
60
80
100
120
I
I
I
I
I
30MeV
140
160
180
e cm l’ig. 15. Fits to the t8Ni cross-section data C~SDmodel).
where V~correspond to the singlet and triplet odd or even components of the interaction and the values of the constants along with the strength V are given in ref. [48]. The range of the Yuka-
wa form factorg(x) is 1.6 fn~1.The short-range repulsion reducing the strength of the attraction is given by a soft zero-range density-dependent term Vdd(X) also given in ref. [481. The optical-model used by Thomas et al. [49] = S
X UR (r 0)
+
d
+
iW~—f(r0, R1, a1) ur0
iWvf(r0, R1, a1)
.
+
I /1\2 ~
U~0 -
ii,
f50(r0) 1 a
,
(3.4.10)
-
where UR(rO)is defined in eqs. (3.4.l)—--(3.4.3) and S is a scaling parameter. The form-factors for the imaginary potentials used [49] were phenomenological, described in sectiOn 2.4 with depth parameters W~,,W~and geometrical parameters r1, a1. (In this sense, the model of Thomas et al.
B. Sin/ia, The optical potential and nuclear structure I
I
I
I
I
I
I
/7
1/ —~ ~
/
\.\
‘--r
“.
111,
//
\.
\\
/‘
,‘
-~
\
\il.4 T ,\l
i,.’
I
‘.
~
Ii”
‘I”\,
ii -I
-
/ -‘
i f’/i
\\r
\ ~,
\‘
IT rv I~/~’
i~’ “~r’..i
0.5
/
I L~-’~’~i
I \ /.r~\T
•1~
~I
I
/ ~~~-~—1——-i ‘jI T
~1~ II
Ii,
I
33
tI\I i
LY
~
-‘~
~/ -‘
-~—
!1’
‘~~_
‘I
.-~‘
•\lb
5
I.~
\\\
I~
‘S. ‘S
.\\“S.i’,J.
\\Ij
-.
II -0.5 10
1.0
I
20
30
I
I
I
I
40
I
I
I
I
I
50
60
70
80
90
I
I
I
I
:
.t.._1
_I
I
I
110
120
~
—
~
0.5
I
.~‘ .
I~.
I
130
.
“
~
\
I
I
100 0Cm
I
140
I
I
150
160
170
I_Z~,
~
,‘‘
‘
II
//
— — I.”
S
S
_i
I.
I’
I
!‘
‘
/ z 0 4 NI
5!
~
.5.
“~‘•
.
I
~...../ \•• ‘S
I ‘
., _______ -
•.
S
\
\
/
\
/
\.
4OCQ)45.5)MeV
-0.5 —.....
/
NEG WDDE
•
~55.
0 10
I
I
I
I
I
I
I
I
I
20
30
40
50
60
70
80
90
100
I
110
I
I
I
I
I
120
130
140
150
160
170
0cm Fig. 16. Fits to the 40Ca polarization data (TSD model).
[49] is not wholly microscopic but only a first attempt.) The spin—orbit term is identical to that of Greenlees et al. [8] described in section 3.1. It was found in a previous analysis [43] for 40Ca
B. Sin/ia, The optical potential and nuclear structure
34
Best-fit parameters for Energy
Interaction
Sx UR (MeV)
Table 3 TtNi for various interactions (TSD)
W~(MeV)
)VS(MCV)
rj(fm)
a 1(fm)
U50(MeV)
x~-
30.3
\VDDE SDDE N P
0.93 x 0.83 x 0.99 x 1.17X
56.7 61.4 48.5 36.2
4.62 3.46 4.82 2.96
0.00 4.02 6.92 9.68
1.68 1.20 1.20 1.12
0.63 0.83 0.47 0.47
102.7 125.7 137.3 113.9
1287 746 1665 4441
40.0
WDDE SDDE N P
0.93 0.79 0.90 1.07
x x x x
55.1 59.8 45.7 33.8
5.88 6.43 6.84 5.16
2.00 0.37 2.62 5.80
1.37 1.56 1.32 1.26
0.65 0.54 0.52 0.55
152.2 151.2 142.5 114.6
1729 1670 2502 6130
Table 4 2> Examples of the variation of (r 6ff and ~RS with type of interaction Nucleus
Energy 1MeV)
160
30.3
40Ca
40.0
WDDI
SDDF
N
I’
3.54
3.97
4.82
3.78
6346
6375
7315
7164
3.91 14415
4.53 14735
5.81 15483
4.86 14870
3.98
4.67
6.26
5.31
~RS
20801 4.80 33725
21014 5.71 35730
21945 6.84 36579
22509 6.20 31891
~‘2~eft
6.12
7.50
8.15
7.94
Jp~
80544
82354
85055
71775
2~eff ~r Kr2l~t-i
J~ ttNi
40.0
r2? 6ff
90Zr
208Pb
40.0 40.0
~RS (r2)et~f
proton scattering data that provided the nuclear density distribution reproduced satisfactorily the electron scattering data, nucleon separation and binding energies per nucleon, the model for a given interaction is unable to differentiate between nucleonic density functions obtained using different schemes. Thomas et al. exploited this fact and therefore restricted the analysis to one type of nucleon density function for each nucleus analysed. For the nuclei ~ ~°Ca,90Zr and 208Pb the neutron and proton distributions used were obtained by Negele [32] using a doubly
self-consistent Hartree Fock calculation. For 58Ni the proton distribution of Batty and Greenlees [59] and the neutron distribution of Zaidi et al. [601. The data analysed were for incident protons, 160 (30 and 50 MeV), 40Ca (30, 35.8, 40 and 45.8 MeV), 208Pb (30 and 40 MeV), 90Zr (40 MeV) and 58Ni (30, 40 and 50 MeV) with the six parameter optical model.
In fig. 15, a typical fit for 58Ni cross-section data is shown using Negele (N), weak density dependent Green (WDDF) and a Gaussian (G) interaction. Fig. 16 shows a typical fit for the polarization data for 40Ca. Table 3 shows the best fit parameters and x2 obtained for 58Ni. The results for other nuclei are very similar except ~ this particular case being discussed later in this section.
B. Sin/ia, The optical potential and nuclear structure
35
The scaling parameters as can be seen in table 3 come out very close to 1, indicating the validity of the model. It is clear however that (i) No systematics can be obtained for W~ and W~. (ii) The fits obtained by using Negele (N) and Pandharipande (P) interactions, which are soft core, are inferior in quality to those obtained by using hard-core interactions such as Green (WDDE or SDDE, the weak and strong versions). Table 4 shows the (r2 >eff = (r2 >opt (r2 >m and ~RS with various type of interactions. It seems that (r2>eff conjectured to be well defined, is no longer a well determined quantity. The fits obtained by SDDE for 208Pb were found to be similar in quality to those obtained by using a Gaussian (G) and yet (r2)ert changed by —3.0 fm2. For (N) and (P) interactions (r2)ecr is persistently large. The dependence of (r2 )eff on A number is probably a manifestation of the density dependence of the effective interaction. It is thus correct to conclude that (i) bound-state effective interactions can be used to generate optical potentials, (ii) analysing the scattering data, although it is not possible to distinguish between nucleonic density distributions obtained by different methods it is in fact possible to detect the difference between different interactions, (iii) soft-core interactions-are probably not very good for scattering (this conclusion can only be qualitative), and (iv) although ~RS is a well determined quantity (r2)ett is model dependent. The energy dependence of ~RS is still not very significant within the energy range concerned. For a wider energy range investigation see ref. [49]. Another interesting feature of the analysis is the worsening in quality of the fits towards lower A, fig. 17. It is conjectured that off-shell effects, assumed in this model to be negligible can in fact become important for light nuclei and also the non-locality in both the imaginary and the exchange potential in the real part can also become important. Furthermore, it is possible that the very concept of single-particle local optical potential may indeed become somewhat tenuous for light nuclei, where cluster effects such as a-clusters have to be taken into account explicitly. —
3.5. Hartree---Fock single particle potential in the continuum The preceeding discussions tend to indicate the validity of the basic idea of linking the detailed theoretical knowledge about the nucleus as a whole to the problem of scattering without introducing a great deal of additional approximation. In this section, we shall try to explore the possibility of expressing the Hartree—Fock single particle potential as an equivalent optical potential. It should be noted right at the beginning, that the Hartree—Fock approximations obviously overlooks a very special effect, relevant for scattering, namely the polarization of the target nuclear field induced by the incident projectile. The virtual excitation term, described previously, is ignored in this method. It is taken into account only very approximately in view of the fact that one uses an effective interaction, which is usually density dependent. Mackintosh [61] has shown that for incident energies in the range 20 ~ E ~ 60 MeV the contribution to the optical potential arising from the coupling to the inelastic levels is not of any significant magnitude so that the dynamical degrees of freedom responsible for such a coupling may be assumed to be of no importance thus justifying the previous argument for neglecting polarization. (This, in effect, is yet anotherjustification for using a bound-state matrix for description of scattering.)
B.
36
Sinha, The optical potential and nuclear structure
The work of Dover and Giai [53] is probably most useful in elucidating the utility of the Hartree---Fock single-particle field for scattering. They use a Skyrme 152] interaction given by a two-body part V12 and a three-body contact
term V123 2 k~V12~k’ = V123
=
+
t0(l
x0P0)
+ ~ t1
t3ö (r~ r~) ~(r2 ----
+
k’2 )+ t
(k
2 k k’ .
+ iw0(a1 + a2)~ (k
X
k’)
(3.5.1)
r3),
where t0, x~and w0 are adjustable constants [52], k and k’ are two-nucleon relative momenta and P0 = J~(1 + a1 a2) is the spin exchange operator. The Hartree—-Fock single-particle equation is given by, t1,
t2,
t3,
r
/12
1fl~r(’~) L
I--u~+
1(1+1) r —
where the index a 3.3). e
=
1 d rL2/n~(r)J /12 1 dr u0 + U~(r)n~ = e~u~ , (3.5.2) ~nljr}, T is the Z-component of the nucleon spin (not r defined in section -----
--
---
---—--------
0 is the energy eigenvalue and m~(r) is the nucleon effective mass defined by [531 /12
/12 =
------
2tn
..in~(r)
+ ~ (t~+ t2 )p(r) + ~ (t2
---
t1
)pT(r),
(3.5.3)
the single-particle potential U~(r)is a sum of central, spin—-orbit and Coulomb contributions expressed as a local function of three density functions, p~(r),the nucleon densities, T~(r)the kinetic energy densities and JT(r) the spin densities: p7(r)
=
T~(r)
~I~j(r)I2.
=
Vøj(r)~2: ./~(r)=
~-
(3.5.4)
~Ø7(r)l. a~1(r).
The exact expression for U~can be found in ref. [53]. The eq. (3.5.3) involves only the discrete eigenstates of the A-particle system. In the H F approximation, excitations of the (A + 1) particle system correspond to unoccupied states of the A-particle system. Thus, one can envisage the scattering problem as follows first, the H—-F equations of the A-particle system are solved in order to find U~(r)and ,n(r), then, the same equation is solved with the eigenvalue ea, replaced by the continuous variable E, the incident energy, using the potentials U~(r)and in~(r)previously obtained. As pointed out before, this picture essentially neglects the polarization effect induced by the incident projectile. In practice, one simply has to recast eq. (3.5.2) into a typical scattering equation 2 r---d2 1(1+ 1) 1 + V ~h —---i- u~(r)+ ~—----u~(r) 7(r, E)u~(r)= Eu~(r) (3.5.5) 2/nLdr r and identify V~(r,E) with the optical potential. For this particular case Dover and Giai [53] found ---
.
that a transformation u~(r)
rm(r)1 I
I
L
/fl
J
1/2
u~(r),
does in fact recast eq. (3.5.4) into the form eq. (3.5.5) so that
(3.5.6)
B. Sin/ia, The optical potential and nuclear structure
37
Fig. 17a. The real central potential using the Skyrme interaction for various elements.
208
Pb
3“I
‘S
/ ‘S
-
——..
5.,
‘S.
/
/
‘I
S S I 5
rUm)
Fig. 17b. The isospin potential using the Skyrme interaction for various elements.
___ m L
2
m(r) r
d — 2
dr2
/
(
/12
\
I—
\2m(r)J
m(r)
d ~—
2/12
/
I
~2
\
-I
dr \2m(r) /
/ m~(r)\ 1+11— T ~E. J \ m(r) / (3.5.7)
21
Consistency is maintained because (m~(r)/m)~ 1 so that u~(r)~ u~(r)and therefore the phase-shifts for scattering will be the same since both Ua and u~have the same asymptotic limit for large r. The same authors L531 noted that the role of the term m(r)/m is to dampen the non-local wave function relative to the local one, found first by Perey and Buck [621. A detailed study of V~(r,E) revealed that most of the gross properties of the optical potential such as energy dependence,
B. Sin/ia, The optical potential and nuclear structure
38
spin-orbit dependence, isospin dependence etc. are reasonably well predicted by eq. (3.5.7), some ot’ which is shown in fig. 17. It should be noted however, that because such an approach derives its results from the HartreeFock calculation for the bound state properties of the nucleus it does not shed any light on the imaginary potential, which is by far the most interesting term in the optical potential.
--
3. 6. Second-order optical potential
In applying the bound-state effective interaction for the description of the scattering potential, the second-order terms for scattering have been assumed to behave like the corresponding boundstate terms. Apart from the difference in the energy denominator (see section 3.2) the Pauli-operator Q~ in the Kuo--- Brown scheme [41] may behave differently for scattering, such that an angle-averaging
procedure may not be quite accurate. Undoubtedly all these possibilities make the estimation of the second-order term rather difficult. Considering a two-particle—-one-hole excitation scheme (fig. 6), the second-order optical potential can be written using a perturbation formalism as Rkn-,Ivlij)12 (innIvljk)~2 +
112 =
!fl,fl>k~j
ern+E 0-J
j~.
.
--—--------
i,j
p~l+Ek_~~Ej
(3.6.1)
~
Both c~and e~the single-particle energies, as pointed out before (section 3.2) must be greater than F, the Fermi energy. Ek is the incident energy. There can be an exchange term which represents “blocking” by the presence of a particle in state 1k), of the virtual fluctuations of the core in which particles in states i, j go to states k, in. Quite clearly the contribution to the real-central part comes from the principal value of L/2 since the denominator can go through a singularity giving rise to the energy-conserving transitions, responsible for the imaginary potential. Sinha [63] using a single-particle energy scheme, planewave intermediate states, and a Fermi gas model of the nucleus obtained a local expression for U2, using both the tensor and the central terms. Johnson and Martin [64] using Watson formalism [65] also estimated the second-order term, suitable more for high energy scattering. The second-order term, obtained by Johnson and Martin is however highly non-local, making it unsuitable for general comparison with any local optical model. The real part of the second-order potential, U~calculated using plane wave intermediate states was compared with the results obtained using the Kuo—Brown prescription for the second-order
term. The real part of the second-order-optical potential for the latter case can be calculated by substituting for the energy denominator, Em + ~., e~the Kuo--Brown estimate of 240 MeV. The last approximation is in the same spirit as the “closure approximation”. The comparison will give us some idea as to the accuracy of using a bound-state effective interaction, such as the Kuo—Brown interaction, for estimating second-order effects (see section 3.4). In fig. 18 the second-order term U~’obtained using the closure approximation is compared with U~,as expressed in eq. (3.6.1). As is clear, the effective energy denominator approximation overestimates the second-order term by as much as a factor of two and also the form-factor is quite different from U~.Examining the energy denominator of eq. (3.6.1) where Ek ~ 0 (so that the final value of the denominator is reduced in numerical magnitude) one concludes that the overestimation of U~can —
39
B. Sin/ia, The opticalpotential and nuclear structure
— 70
16
‘
s
—)U
~ 9TT~-~~ I
1(r,).U1(r1))
~
~,
-
~
7O~
I
~
— U2(r~) —
~.
-(Ul(rI).U~(~))
10 20 3.0 4.0
80
—-—
5.0
60 70
- ~
1.0
20 3.0
0 20
16 14
40ca
~
¶6
- - -
30 40 50 60 7080
x05
~°ca
~5...,,
70
60
-......
-—-_.~ ~
10 20
I
4.0 5.0 60 70
ID 20
30 40 50
6.0 70
80
5.0 6.0 70
8.0
16
14 2 — ..~,,
58~
3.0 4.0 50
—~ —T-~---~..
60
70
0 16
~
58NI
10 20
3.0
40
2O8p~
5.”
0
1020
304.0506.07.0
Nuci radius in fm
—~
0
1020304.05.06.0708.0 Nuci radIus
Ifl
fm
—.
Fig. 18. Second-order optical potential as a function of nuclear radius.
only be ascribed to either the error of using an angle averaged Pauli operator such as used in the Kuo—Brown interaction or to the fact that m + ~ probably is much larger than 220 MeV for scattering. Indeed, it has since been pointed out by Brown [41] that the energy denominator should be 420 MeV so that U~now becomes numerically closer to U~,although the shape is still likely to be different. Obviously the calculation of U~Lhas many drawbacks. A single-particle scheme has been used whereas collective representations of states may be more realistic. Further, plane-wave intermediate states may be an oversimplification for scattering. The Fermi gas model used to describe the nucleus is known to be inaccurate on the surface of the nucleus, a region which is likely to be quite important. All these inadequacies of the model may indeed lead to an overestimation of U~and therefore the substantial difference between U~and U~Lmay also be exaggerated. Nevertheless, the discrepancy between U~and U~Ldoes tend to indicate that estimates of the second-order terms using bound state interactions may be erroneous.
B. Sin/ia, The optical potential and nuclear structure
40
4. The imaginary potential 4. 1. Lnergy-conser~ingtransitions
It has already been pointed out that the imaginary part of the optical potential arises from the energy-conserving transitions in the second-order term of the optical potential expressed in eq. (2.2. 1). In other words W, the imaginary potential in operator form is given by I-i’
=
iirPV~(E H0 )Q V. --
(4. 1. 1)
However, W expressed in this way, exhibits rapid fluctuation as the energy E Is varied over the energy intervals of the order of the width of the compound nuclear states. One of the methods for “averaging” the compound elastic effects is to replace E byE + 11. The effects of various avel’aging intervals I are given in ref. [66] and more recently in ref. [671. O’Dwyer et al. [67] calculated W for low energy neutron scattering E < 14 MeV in this fashion. For the energy-range (25 ~ £ ~ 90 MeV) under consideration the compound elastic contributions are neglected and only the direct inelastic channels will be considered. Usually W is both non-local and energy dependent. The first systematic attempt to calculate W from such a second-order perturbation formalism was by Shaw [101. He used the simplest possible approximations namely plane wave intermediate states, energy-conserving transitions to which give rise to the imaginary potential. He also represented the incident projectile by a plane wave. The results obtained by Shaw. although likely to be unreliable because of the oversimplified assumptions, do in tact predict the general properties of H! known phenomenologically [7], in particular the surface bump associated with the Saxon--
Woods derivative term. More recently calculation of the imaginary potential has been perfbrmed using a plane wave propagator but taking into account the actual inelastic levels as the intermediate states. The properties of the collective states particularly the low lying ones (which are likely to be important) are known quite well from experiments on inelastic scattering, so conceptually there is not much ambiguity in the problem. Slanina [121, Vinh-Mau [681 and Bruneau and Vinh-Mau [69] have calculated H!, using
such a scheme. The imaginary potential H!, obtained by Slanina [121 is non-local and energydependent. Pick-up channels such as (p, d) are neglected in Slanina’s calculation. The set of intermediate states was truncated so as to include only the low lying collective states. These low-lying collective states include the effects of long-range correlations. The effect of long-range correlations on the imaginary potential was studied by Terasawa [70] and he found that including “pairing correlations” enhanced the potential by a factor of three. Thus the strongly excited T = 0 states are most important and these were the states considered by from Slanina. state 12Ca, 40Ca) were taken the The worklow of lying Gillet collective and Sanderson wave functions for some nuclei ( [71]. To estimate the strengths of the higher lying collective states of a given inultipole a sum rule due to Lane [72] was used. The local equivalent potential obtained by using the method of Percy and Saxon [73] is shown in fig. 19 for 40Ca at 20 and 40 MeV. Considering the various approximations involved, the prediction is quite remarkable. Rao et al. [74] recently noted that the use of a free particle propagator is not correct. The Hamiltonian H 0 in the propagator G (section 2. 1) now includes an interaction potential V. pre-
B. Sin/ia, The optical potential and nuclear structure
GRAY ETAL
40Ca
41
—
THEORETICAL- — FRICKEETAL——
IMAG POT. 1MeV] 11
THEORETICAL—-—
10
20MeV
9
40MeV
8 7
5 /
S
/
5
/
/
3
—.5
—
/
,~,—.
\\
-~
~
~/
-
—.-----~—
2
‘S
I
- -
0
1
-
RADIUS IN fm
—,
“~. .‘~
I
I
I
I
2
3
4
5
~
6
Fig. 19. The imaginary potential as a function of nuclear radius. Theoretical predictions are compared with phenomenological form-factors.
viously neglected, so that H 0 = HA + T + V where HA is the Hamiltonian of the target nucleus, T the kinetic energy of the projectile and V is the interaction potential, in effect the optical potential. Rao et al. [74] however neglected the interchannel couplings, i.e. the off-diagonal parts of the propagator and calculated G only for the diagonal parts of V. It should be noted that a complex potential V in the propagator is more appropriate and its effect can be important. The wave function representing the projectile state is now a distorted wave (as opposed to a plane wave such as used by Slanina) generated by the real part of V, UR (r0), so that the general form of W is now W(r,r’)
V0~(r)G,~(r,r’)V10(r’) ,
~
i
(4.1.2)
0
where the propagator G~(r,r’) is given by G~(r,r )
=
(2ir)
—
3Jr
d3k
such that /12 k~/2p= E
—
X(*(k,r)X(_)(k,r1) (h2/2p)(k~—k2) +ie
(4.1.3)
,
E, E
1 being the energy associated with the energy states of the target nucleus and x’s are the distorted waves. Slanina [12] and Vinh-Mau [68] essentially used a particle—hole description for the matrix element V01. Cugnon [75] also developed a microscopic model to describe the inelastic states. Rao et al., on the other hand, mostly for reasons of simplicity used a collective or deformed potential model (reasonably well-established by now) to describe the collective states such that
V10(r)
=
(i, XpI VI0,00) =
aUR (2X+l)~~3~(i)R
~
Y~(?),
(4.1.4)
where the spin and its projection of the ith state are X, p and ~ the deformation parameter.
B. Sin/ia, The optical potential and nuclear structure
42
The advantage of such a procedure is that at the expense of throwing out some microscopic details of the inelastic states, one can relate H-’ directly to the energy and the deformation parameter of the inelastic states. Rather than trying to use a local equivalent of highly non-local W, they used a non-local code to calculate the scattering from a non-local potential. The somewhat detailed calculation of Rao et al. eventually led to the following conclusions, (1) A plane-wave propagator is probably wrong and a distorted wave propagator should be used to describe the incident channel. (2) Taking into account just the inelastic channels does not account for the experimentally observed total absorption cross-section. Further, they found that the calculated angular distribution of the cross-section data differ significantly from the measured ones as a function of the scattering angle. The rather pessimistic situation of (2) led them to postulate that pick-up reactions such as (p. dl)
and (p, pn) can contribute a significant part to the imaginary potential, particularly from the region just outside the nuclear surface. Furthermore it was noted that too little absorption for low partial waves can be due to the lack of excitation processes in the nuclear interior: it is tempting, they remarked, to include them as compound nucleus formation by allowing bound-states embedded in the continuum as intermediate states.
The conclusion, one can reach from the speculation of possible pick-up contribution is that such an effect will shield an incident proton from inelastic transitions since the pick-up “takes place” first, as it is on the very edge of the nuclear surface.
In conclusion to this section, it is suggested that despite quite detailed calculation to account for the dynamics of excitation, no consistent picture seems to emerge. It seems now to be impossible even to predict what types of excitation channels are important. However, the failure of a consistent picture makes W rather an interesting function to compute. In the next section, leaving this detailed picture behind we shall try to compute the imaginary potential from a very simple but “frivolous” point of view. Curiously enough, such a model, as will be seen is in reasonable agreement with the data and phenonienology.
4.2. forward-scattering amplitude approxuna tion In the nineteen-fifties, a number of authors [76] calculated the imaginary potential by assuming that “absorption” arises entirely from the forward scattering amplitude of nucleon—nucleon scat-
tering, sometimes referred to as the “frivolous model”. A simple kinetic theory of gases argument can also lead to the same results as obtained using the forward scattering amplitude approximation.
Recalling eq. (2.2.8) one can write out the first-order optical potential within the framework of the impulse approximation as, 2
U~(r0)-—-~-—-J’5d~qM(q)I;(q)exp(iqr0),
(4.2.1) 2ir in 11(q) is the two-body scattering matrix. F(q) the form-factor for nucleon where, recapitulate, i density to function p(r) andq is the momentum transfer, q = k’ k, k’ and k being respectively the final and initial momenta in a collision. It has already been pointed out in section 2 how the folding integral for the real part can be derived from such an equation. To derive the imaginary potential from eq. (4.2.1), we assume that only the forward scattering —
B. Sin/ia, The optical potential and nuclear structure
43
amplitudeM(0) is important, which also implies Ik’] = so that, the two-body matrices are necessarily always on the energy shell. The justification of such an assumption can be seen in eq. (4.2.1). For heavy nuclei one knows that F(q) is a function which peaks at the origin and falls off rapidly with q while M(q) changes much more slowly [5]. For lighter nuclei where the non-zero q values become important, the argument is less valid. Using the forward scattering amplitude approximation the imaginary part of eq. (4.2.1) is reduced to 2
W(r0)
=
Im {M(0)}p(r0)
—
(4.2.2)
m
by using the definition of F(q), the Fourier transform of the density p(r0), and it has been assumed that the effective mass, m* m (see below). Using the optical theorem, the imaginary part of the forward scattering amplitude can be shown to be as follows, —
Im {M(0)}
k =
—
4ir
(a)
(4.2.3)
,
where k is the wave number of the incident nucleon inside the nucleus and (a> is the average twobody total cross-section inside the nucleus. Thus the imaginary potential is given by, W(r0)
=
~1i V~,
~
p~(r0)(a>p1
(4.2.4)
where V~is the velocity inside the nucleus Vp
=
/2 V(~m.~ UR(rO)— Vc(ro)) m
(4.2.5)
.
is the centre-of-mass incident energy; m can be generahised to a radially dependent effective mass, m* [10,76], although it was found that such a change did not alter the results significantly, justifying the assumption m~ m, mentioned above. It should be noted however that originally an effective mass m* 0.7 m was used [76] although a radially dependent effective mass, m* is more appropriate. Thus one calculates UR(rO), the real part first and then knowing V~from eq. (4.2.5), W(r0) is calculated as a function of r0 in this way, some sort of self consistency is maintained. The average total nucleon—nucleon cross-section, (U)pj has been calculated by Clementel and Villi [78]. They used a Fermi gas model for the nucleus and the local Fermi momentum was 2p estimated using the Thomas—Fermi approximation such that ~ = 3ir 1. Energy conserving single particle transitions are considered beyond the Fermi level so as not to violate the Pauli principle. The energy conserving excitation of the target nucleon beyond the Fermi level is the process of absorption [79]. Green, Sawada and Saxon [771 and also Sinha and Duggan [80] however observed that it is implicit in using eq. (4.2.2) that a zero-range interaction has been used. This can be seen by recalling that M(q) is proportional to t(q), the two-body scattering matrix and in coordinate space, t(r) can be expressed as a Born series of two-body interaction, v(r) (see sec. 2.2). The zero-range approximation can however be improved upon by using an effective density, instead of density, p, as defined below [77, 80] Ecm
-~
—~
—
44
B. Sin/ia, The opticalpotential and nuclear structure
10
~
T1_~~
~
“Cu Es LO Mel.’ (I
~/~S/i
—
FOLDED MODEL
— — UNFOLDED ¶-~DEL — — - — 10 PARAMETERS 10
20
30405060
708090
1~110 120130140 150160 170
8cm
Fig. 20a. Fits obtained to the cross-section data using a three parameter modal.
B. Sin/ia, T/ie optical potential and nuclear structure
45
Z°~PbC. £0 61eV
0.5
___ ib
io
-
~
~o~o
~
‘°Zy
10
-0.5
20
~
30
40
51)
60
70
80
90
100
110
120
130
E•
£0 61eV
11.0
ISO
‘°CQ E .4361eV
— 20 30 obtained 40 60to the 60 polarization -70 80 data 90 using 100a three 110 parameter 120 130 model. l~3 ¶50 Fig. 20b. Fits
10
160
i~o
B. Sin/ia, The optical potential and nuclear structure
46
Table 5 Volume integral/A using a three-parameter model
—
Nucleus
Energy
(r
2)R
Kr2 ~eff
JRS/A
J
(tm2)
(tm2)
(MeV fm3)
(MeV 1/A fm3)
J1/A parameters TSD (MeV fm
208Pb
40
37.81
7.45
406.0
111.21
101.89
30
37.91
7.56
440.5
109.69
111.63
90Zr
40
23.77
5.69
386.47
105.43
102.99
40Ca
40
16.05 16.05
4.66 4.66
427.34 393.14
94.20 98.82
99.71 104.88
50
11.34
4.05
515.85
117.34
48.45
40
11.32
4.03
455.8
103.23
(MeV)
30 160
Polarization
0
‘
—.
I,. -
1-.
--
3)
~
~
•
~O E. 39.7 MeV
-0.5 FOLDED MODEL —
—
UNFOLDED
10
MODEL
PARAMETERS
10
20
30
40
50
60
70
SD
90
100
110
120
130
140
150
160
e cm lig. 21. I-its to the 160 polarization data, indicating the deterioration of the fit at backward angles.
P~ff=
Ir
{fPirveff
3r.
r
(4.2.6)
0~d3r)/fveff(r) d
The expression for the imaginary potential now becomes, W(r 0)
=
4h
~
V~
i~p, n
p~ff(r0)(U)pj.
(4.2.7)
Eq. (4.2.6) describes a kind of “averaging” of p with respect to Veff and a detailed derivation of eq. (4.2.7) can be found in ref. 180]; suffice it to say now that the quality of the fits to the scat-
47
B. Sin/ia, The opticalpotential and nuclear structure
KB, kuOS8 KB. k rD
‘°Ca(40MeV)
—
IMAGINARY POTENTIAL
—
—
16.0
12.0
-- —
-.
-
—
~--~
OS
~0
15
20
~ .~
-
--—-
—
kp,BRUECKNER et ol kF, NV DEFINITION
.._
- -
ZERO FORCE
- -
PHENOMENOLOGY
/
—
25 ~35 NUCL. RADIUS INfm
45
5~5
Fig. ~22. Imaginary potential form-factor as a function of nuclear radius
tering data improved greatly using eq. (4.2.7) rather than using eq. (4.2.4) for estimating the imaginary potential. Calculating the imaginary potential as shown in eq. (4.2.7), using the folding model for the real central potential, discussed in the last section and a Blin-Stoyle prescription of the spin—orbit potential, Sinha and Duggan [80] developed a three parameter model with three variable scaling parameters for the real central, imaginary and the spin—orbit potentials. Fig. 20(a,b) shows a typical fit obtained for the cross-section and polarization data using such a model. As is clear the quality of the fits obtained is quite satisfactory and comparable with the fits obtained by using a standard phenomenological model. Table 5 shows the volume integrals and the mean square radii. It is interesting to note that J1/A like JRs/A for the real part also has a reasonably constant value. The fits however deteriorated for backward angle scattering data and also for lighter elements, fig. 21. For backward angles the non-zero momentum transfers can be important and also the twobody matrices are most likely to go off the energy shell [811. In the case of the lighter nuclei both these effects are expected to be more important [61. The non-zero momentum transfers can be taken into account approximately. Expanding M(q) around q = 0 and taking2M( into) account only the even powers [61, one obtains M(q)M(0)+~2 a ~ 0 + (4.2.8)
[
B. Sin/ia, The optical potential and nuclear Structure
48
However V2p(r0)
3q, F(q) exp (iq. r0) d so that substituting eq. (4.2.9) in eq. (4.2.1) gives =
—
—h2
W(r 0)
(4.2.9)
fq2
2
2ir ni
M(0)
Generalising p(r0) to W(r0)~ Vp
32M( )
1
Petf(”O),
~ =
1
5) ,I1
(U>pj(l
-—
v~
[-~~]
2M(0)
aq
p(r~).
(4.2.10)
q0
one gets +~V~)p~ff(rØ),
(4.2.11)
where g using the result of KMT [6], was found to be K 0.5833. Using the modified version of W(r0), the fits to the backward angle improved slightly [821. The importance of the imaginary potential in the nuclear surface, however, throws serious doubt on the validity of the use of the Thomas—Fermi approximation for the local Fermi momentum k~ since it is known that the T—F approximation breaks down in the nuclear surface. Fig. 22 shows the breakdown point of the T---F approximation. The curves which tend to rise sharply are obtained by using modified definitions of k1,- which take into account the inhomogeneity of the nuclear density on the surface of’ the nucleus [82]. It is fortunate that, although the T—F approximation breaks down, a direct comparison with the potentials of phenomenology shows that the imaginary potential is expected to he very small at the point of breakdown. In other words, the validity of the T---F approximation is narrowly saved in this case. The success of the “frivolous model” although rather surprismg seems to indicate that the forward scattering approximation, which is essentially a high energy approximation is valid even for the medium energies under consideration. The local optical potential does not require a great deal of a priori knowledge of the nucleus apart from its density and therefore can be used generally for analysis of elastic scattering data.
5. Application to other fields
In this section we shall discuss how the ideas developed so far can be applied to i) composite particle scattering, ii) inelastic scattering, and iii) heavy-ion scattering. A considerable amount of theoretical work has been done in all the three fields and an exhaustive survey of the theoretical work is not the aim of this paper. The discussion presented below will be concentrated essentially on the folding model, with the motivation of testing the sensitivity and the applicability of the bound-state effective interaction in such a model. 5.1. Composite particles
A considerable quantity of scattering cross-section data has accumulated for incident composite particles such as helium-3, tritons and deuterons. In this section we shall try to explore the possibility of extending the nucleon—nucleus optical model to composite particle scattering.
49
B. Sin/ia, The opticalpotential and nuclear structure
This type of approach is different from treating each type of composite projectile individually and studying the optical potential in terms of a (3He/t/d)-(t)Z) interaction. We feel that in so far as nuclear physics at this energy range is concerned, the fundamental building-brick should be the (tX —9Z) interaction rather than the (composite projectile—nucleon) interaction. The Watanabe [831 model was probably the first attempt to formulate the composite-particle projectile potential in terms of a nucleon—nucleus potential. The scheme is quite straightforward the nucleon—nucleus optical potential is folded in with the density function of the composite projectile and then multiplied by the number of nucleons constituting the projectile. In operator form, this would mean —
(5.1.1)
~
where uC is the internal wave function of the composite projectile and n is the total number of particles constituting the composite projectile (c). Secondary effects are expected to change the structure of U~~ somewhat. 6 These include break-up of the projectile while in the nuclear field or second-order excitations to higher excited states, not accounted for in the nucleon case, or even the distortion of the composite particle wave function due to the nuclear field. Samaddar et al. [85] and Mukherjee [861 developed a scheme where the composite projectile propagator Gc(E,R) for incident energy E was expanded in terms of the nucleon propagator GN (E, r) by an approximate series. The algebra is somewhat complicated but the essential results can be summarised in the following manner: (i) The Watanabe terms taken together represent the first-order term. (ii) There is a correction term to the first-order term which depends strongly on the secondorder terms in the nucleon—nucleus potential. It turned out that in effect the second-order terms are multiplied by factors twice as large as the first-order terms. This indicates an enhancement of the sensitivity of the composite particle optical potential to the second-order effects; in the specific case of the Kuo—Brown interaction one would expect an increased effect from the r~ term [411. It suggests therefore that composite particle scattering is one way in which the various approximations about the second-order term can be tested. (iii) The imaginary potential, being essentially a second-order contribution, must therefore give rise to second-order contribution to the final optical potential. (iv) The relative motion of the constifuent nucleons in the incident particle gives rise to kinetic energy correction terms in the final optical potential, neglected in the Watanabe model. 3He optical potential starting from a Recently attempts were made to calculate [87] the nucleon—nucleus optical potential, calculated theoretically as discussed in section 3. The real central part of the nucleon optical potential was calculated using the Kuo—Brown effective interaction. The first and second-order contributions were separated out, the second-order term being solely due to the tensor term — ~ V~ 1[411 in the Kuo—Brown interaction. The exchange term was also calculated as discussed in section 3.3. A forward-scattering amplitude approximation was used to calculate imaginary potential. should noted that the 3He isthemore likely to be valid It than the be corresponding caseforward-scattering of a single nucleonapproximation for at the same energy — the wavelength associated with 3He is three times smaller. Folding in with the 3He wave function using the formalism of Samaddar et al. [85] the fits obtained to data were quite reasonable [871. It turned out that the kinetic energy terms, arising from the relative motion of the constituent nucleons of 3He were relatively small. This in effect justifies the “adiabatic” approximation of Johnson and Soper [84] who neglect such a motion. —
B. Sinha, The optical potential and nuclear structure
50
In the case of the deuteron, because of its low binding energy it is likely to be distorted once inside the nuclear field and a simple folding model may not be the correct method to estimate the potential. A detailed discussion on deuterons, beyond the scope of this paper can be found in ref. [841 and references therein. The 4He potential is discussed in section 5.3. 5.2. Inelastic scattering and effective interactions
The Distorted Wave Born Approximation has been very successful in formulating a simple theory of inelastic scattering. The coupled channels method has also proved to be a very useful tool to describe inelastic scattering. Theories, more microscopic than simple phenomenological collective models, have been developed over the years, especially by Satchier and co-workers. In this section
we shall not present a survey of this type of work but rather concentrate on the new development in the construction of transition matrices for inelastic scattering using bound-state two-body effective interactions described before. The most simple method to investigate the applicability of effective interactions for inelastic scattering is probably to use the folding model. The transition potential in the DWBA matrix element is given by, 17 V
—/t
AB — \WB
¶
V
WA’
‘
where ~.‘B and I~1A are the final and iniiial target states respectively and v is the two-body interaction. The DWBA transition amplitude is given by the matrix elements of VAB taken between Xh and Xa’ the corresponding distorted waves for the outgoing and incoming projectile respectively. The transition potential, VAB is estimated in a folding model, first by deforming the target density and then folding it in with the two-body effective interaction, expanded in multipoles. Such an approach was considered by Hnizdo and’Lowe [89] and also by Edwards and Sinha [88]. In the case of for example a Saxon—Woods type of density distribution, one can deform the radius parameter such that R
=
R 0 (i
+
~
Y~(O,Ø)~
(5.2.2)
and fold in with
v(s)
v1,~(r1, r2) Y15,1 (~~) Yirn (p2).
=
(5.2.3)
in’
for a chosen effective interaction v(s). It was noted that the form factor for VAB obtained in this way is different from the form-factor VAB, obtained using the standard phenomenological DWBA approach. (To recapitulate, the phenomenological DWBA approach, VAB is identified with the deformed phenomenological optical model potential.) Satchler [90] however pointed out that such a procedure can lead to erroneous results if one neglects the exchange contribution, which is expected to be important for inelastic scattering. Furthermore, it was shown [90] that depending on how one chooses to deform the density, one gets drastically different results. The matter becomes really quite controversial as one looks deeper into the problem of self-consistency. The potentials used to generate the density function should in fact be used to calculate the deformed po-
tential. The choice of an arbitrary effective interaction may thus plague the final result with problems of double counting, inconsistency etc. The considerable discrepancy noted by Edwards and
B. Sin/ia, The optical potential and nuclear structure
51
Sinha [881 between standard phenomenological DWBA and the deformed folding model may thus be misleading because of the consistency problem, if for no other reason. This nagging shadow of inconsistency can be overcome, to some extent, by using the Skyrme interaction [52]. Looking at the approximate Hartree—Fock expression for the optical potential derived by Dover and Giai [53] one finds that, for the Skyrme interaction, the real part is solely a function of density if the kinetic energy density r is approximated by the Thomas—Fermi expression. A good deal more physics can be incorporated into the H—F optical potential if one uses the Negele—Vautherin [51] prescription to derive the H—F single-particle wave equation. The exchange term in that case can be clearly seen, unlike with the empirical Skyrme interaction. Symbolically this means U0~1 Dover, Giai
F
(~~kY -
i
; p, r)
=~
F(~I~, p1(r, r’); p,
Negele —Vautherin
i
where the set of Skyrme interaction parameters ~ are now expressed as a function of density, mixed density, etc. Once again approximating the quantum-mechanical kinetic energy density by the T—F approximation one gets ~ as a function of density. The great advantage of such a procedure is that the deformation of density is directly related to the deformation of U, so that inconsistency cannot plague the results. Such an approach is being pursued. A very similar type of calculation has been made by Davies and Satchler [91]. Instead of using the momentum dependent Skyrme interaction, they used an equivalent potential by observing that the momentum-dependent terms can be regarded as the lowest terms in a short-range expansion of a finite-range central potential which includes Majorana space exchange. They 40Ca. Theconsidered transition the excitation by than 17 and 30 constructed MeV protonsfrom to the lowest model, 3 andwere 5 states of by using RPA potentials rather being a folding estimated hole—particle wave functions of Gillet and Sanderson [711. No imaginary part of the interaction was included. The fits to the data were remarkably good, see ref. [91]. In conclusion Davies and Satchler [91] noted that the density dependent components of the Skyrme interaction are very important. Without this component the resultant cross-section could be an order of magnitude too large. A simpler but equally appealing method was adopted recently by Satchler et al. [92] for analysis of proton inelastic scattering data on 40Ca and 208Pb. The long-range part of the Hamada— Johnston interaction along with the wave-functions provided by Gillet and Sanderson [71] was used to construct the transition potential VAH, eq. (5.2.1). A separation distance of 1.05 fm was used to “simulate” the hard-core of the two-body interaction. They also used the Kallio— Kolltveit interaction [45] but found that the cross-section increased considerably. This is probably because, as Satchler et al. noted [92], Kallio—Kolltveit interaction does not have the correct saturation property. The knock-on exchange term was calculated by using a pseudo zero-range potential [921 and it was found that the exchange effects enhance the cross-section appreciably. However, the pseudo zero-range potential [911 used to calculate the exchange effects may lead to overestimation of the cross-section. A detailed calculation of such knock-on exchange term starting from eq. (3.3.6) is now in progress [94]. Tedder et al. [94], first deform the mixeddensity and then fold it in with the two-body effective interaction u(s), which is expanded in
52
B. Sin/ia, The optical potential and nuclear structure
multipoles. Two kinds of interactions are used: the Kuo—Brown interaction [411 and the Green interaction [44]. Preliminary results indicate that cross-section predictions, using both the interactions are almost identical in magnitude and compare rather well with results obtained by Satchler et al. [921. It was found that the density-dependent part of the interaction (which is a
Kallio—Kolltveit interaction plus a density-dependent term, see section 3.4) plays a very important role. This confirms the speculation of Satchler et al. [92] who suggested that the overestimation of cross-section using the Kallio—Kolltveit interaction is due to absence of any kind of density-dependence. There is obviously need for detailed microscopic calculation such as done by Satchler et al. [92] using realistic wave functions which are calculated self-consistently but the folding model does
predict results reasonably well and has the distinct appeal of simplicity and yet being more meaningful than an ordinary phenomenological deformed potential. For a general application this kind of approach is probably very useful. 5.3. Heavy-ion scattering A knowledge of the ion—ion optical potential is of importance for the understanding of nuclear reaction induced by colliding ions. The strength and the geometrical properties of the potential is expected to play a crucial role in deciding whether the colliding ions for a particular
orbital angular momentum will fuse leading to a compound nucleus or scatter away elastically, inelastically, or induce direct reactions. However, to use the very concept of an optical potential when two colliding ions tend to overlap significantly and dynamical distortions start to take place is almost certainly wrong. The time dependent behaviour of such a process also makes the oversimplified static picture of a potential as a function of the separations between the centres of mass also very dubious. It is only in the tail of the nuclear interaction, when the colliding ions “struggle” between Coulomb repulsion and nuclear attraction that the optical potential is relevant. The interaction energy of ion—ion collision has been estimated by several authors [95—991. The most simple approach, taken by Broglia and Winther [95] and also by Brink and Rowley [95], is to fold in a phenomenological nucleon—nucleus optical potential with the projectile density. in this procedure each nucleon of the projectile is treated essentially as free and therefore the saturation property of the two-body interaction, which prevents the nuclear density from increasing beyond a certain magnitude, is ignored. Thus the potential calculated in this way is expected to be overestimated when the overlap of the densities of the colliding nuclei is significant but should be of the right order of magnitude at or near the touching radius when the overlap is negligible. For incident 4He, however, this single folding procedure gave reasonable results, as shown by Batty and Greenlees [59 I. Wilczynski [96], on the other hand, has derived the interaction potential by assuming that the derivative of the surface energy represents the force acting between two spherical liquid drops, without taking account of the diffuseness of the nuclear surface. He went on to calculate the critical angular momentum for fusion of two nuclei using such a potential and it was found that the energy independence obtained from his model does not agree with experiment [971. Galin et al. [97] and recently Brink and Stancu [98] have taken the “sudden approximation” approach of Brueckner et al. to estimate the ion—ion potential. The nuclear interaction energy, in this approach, is given by
B. Sin/ia, The optical potential and nuclear structure
800 With Saturation Without Saturation
-
700
600
________ -
-
5~55
Target
~
40Ca
\
500-
700
-
600
-
Target 208Pb
\
500-
\~
\~is
\ 400
53
~
400
-
-
-u 300 -t
~l6O -U,inMeV
300
inMeV
-
\
\
\
—--..
200
200
_-_......
N
-
10:
1.0
2.0
3.0
4.0 5.0 6.0
7.0 8.0 9.0 —‘
1.0 2.0 3.0 4.0
5.0
6.0
7.0
8.0
9.0
10.0 11.0
12.0
Nuct. Radius R in fm
Fig. 23. The ion—ion potential as a function of R, the distance between the centres of the colliding nuclei. The saturation term arises from the density-dependent part of the effective interaction.
3r (5.3.1) 1(lR—rI)+p2(r)}—~C{p1(r)} —~C{p2(r)}]d where ~C{p(r)} is the energy-density functional and Pi and P2 are the density distributions of the two nuclei. The energy density functional when integrated over all space gives the energy of a nucleus. It should be emphasized that in such a calculation it is assumed that the structure of each nucleus is entirely conserved during the contact and nuclear-matter densities overlap in a reversible process without any rearrangements. Therefore, the potential, U~~~(R) will certainly not describe the interaction after the nuclei have fused. Whereas Galin et al. [97] use a phenomenological energy-density functional Brink and Stancu [98] use a microscopic ~C{p(r)} derived by Vautherin and Brink [50] using the Skyrme interaction. They observed [98] that the “powerful” repulsion produced by Pauli exchange effects as the U~<~(R)f[lC{p
54
B. Slit/ta. The
optical potential and nuclear structure
overlap of the two nuclei becomes significant, tends to produce a very shallow potential of’ ---10 MeV. Thus the saturation properties, ‘‘bLsilt into’’ the formalism of Brink and Stancu [98] inhibited the nuclei from coming close. Recently [99] the interaction potential for ion- ion collision was also calculated by using a density-dependent interaction and then doubly folding it in with the densities of the target and the projectile. The density-dependent pai’t of the two-body interaction, as is well known, takes into account the saturation property of the two-body interaction. The importance of using an effective interaction such as a density-dependent interaction has been amply demonstrated for a nucleon--nucleus optical potential in this paper. In an ion--ion interaction, however, each nucleon in the projectile is embedded in a nuclear medium thus departing from a free nucleon case. The local density in the two-body interaction therefore should he computed taking into account the contribution of the density from the target and the projectile. It was found [99] that the density-
dependent part contributes most significantly to the lowering of the attraction as the total density increases, thus producing saturation. The “saturation correction” terms, arising from the density-dependent part of the interactions for ~1-Ic, 4He, 160 on 40Ca and 208Pb are shown in fig. 23. Clearly, the role of the saturation term became more important i) with increase in mass number of the projectile, and ii) with more overlap of the density of the colliding nuclei. Application of these results in related fields, such as fusion and direct reactions of colliding nuclei will obviously be rather interesting. The imaginary potential also for ion—ion collision has been calculated by Alexander and Malik [1001 using essentially the forward scattering amplitude approximation. Although open to serious drawbacks such a procedure which is essentially an extension of the nucleon potential can throw some light on the heavy-ion potential, at least in the tail region. Efforts along this direction are however at the speculative stage.
6. Summary and conclusions The real central part of’ the nucleon--nucleus optical potential can be calculated by folding in a two-body effective interaction with the nucleonic density distribution. A free two-body interaction is not appropriate. It has been found that the effective interaction, used for such a folding can be the same as that used for the bound-state calculation. This approximation of using a boundstate interaction is valid however only within certain limits, mostly depedent on the energy of the incident projectile. It has been shown that in the energy range 20 ~ E ~ 90 MeV, use of the hardcore effective interactions reproduces data with reasonable efficiency. The anti-symmetrisation between the coordinates of the incident projectile and those of the target nucleons gives rise to the exchange term, mostly responsible for the energy dependence of the optical potential. The cxchange term is non-local. A local equivalent potential has been derived and it has been shown that
the local potential adequately describes the exchange effects and predicts accurately the energy dependence found from phenomenological analysis. It is misleading to quote the relative strength of the exchange effects generally: the magnitude of’ the exchange potential depends on the relative strength of the spin and iso-spin mixture of the interacting nucleons. Including odd-state interactions, for example, enhances the exchange effects considerably. Tl1is model of the real central part, it is conjectured, can be used generally to find out the neu-
B. Sin/ia, The optical potential and nuclear structure
55
tron distribution of nuclei by fitting the scattering data, assuming that the proton distribution is reasonably well predicted by the empirical relation given by Acker et al. [301 from electron scattering experiments. The imaginary potential has been discussed both using a second-order perturbation theory and a forward scattering amplitude approximation. It has been found that a “proper” dynamical calculation of the imaginary potential can be performed by taking into account energy-conserving transitions to low-lying collective states. The importance of (p, d) pick-up has also been considered [74]. It was shown that such a calculation although very useful to understand the underlying “physics” of the origin of the imaginary potential, is not however suitable for any data fitting procedure. A simple forward scattering amplitude approximation, on the other hand, seems to be able to predict the data with reasonable efficiency. The importance of a finite range two-body interaction as opposed to a zero-range interaction has been emphasized. It is concluded that a finite range interaction is essential. Non-zero momentum transfers have been taken into account by an approximate expansion of the scattering matrix around the forward direction [80]. The imaginary potential derived in this fashion, although having several shortcomings, seems to be satisfactory enough for a general optical model analysis. The spin—orbit interaction potential has been briefly discussed. An expression derived originally by Blin-Stoyle [16] has been used throughout and shown to be quite adequate. The possible extension of such a model to heavier projectiles or even to ion—ion potentials has also been discussed. The possibility of a semi microscopic analysis of inelastic scattering data starting from such a model has been indicated. Finally, it is felt that within the energy range 20 E ~ 90 MeV, to visualize scattering essentially as an A + 1 system with the incident projectile in the continuum and interacting with a “bound-state” effective interaction is a valid picture. On the other hand, for the imaginary potential the so-called high energy approximation of forward scattering seems to be tenable even in medium energy range, a result which may be somewhat surprising but nevertheless valid. ~
Acknowledgement It is indeed a pleasure to thank Professor G.E. Brown for his continuous interest in a “coherent” account of the optical potential which, I hope, has been fulfilled not too incoherently. The enthusiasm, encouragement and the physical insight that I could share with Drs. Richard Griffiths and Victor Edwards is gratefully acknowledged. I would especially like to thank Dr. Richard Griffiths, Kevin Malloy and Dr. D.P. Weet for critically reading the manuscript and helping me to resist the temptation to succumb to rash conclusions, which probably still plague the report but hopefully make it more exciting nonetheless. Discussions with Feroze Duggan and Chris Webb have proved rewarding throughout. The manuscript was written while on a visit to the Niels Bohr Institute, Copenhagen. A travel grant from the Royal Society of England is gratefully acknowledged. The many stimulating discussions in the Niels Bohr Institute particularly with Professor Ben Mottelson and Dr. Philip J. Siemens have, I feel, enriched my understanding of the optical potential. Last but not least I would like to thank Joy Cohler for typing the manuscript with such patience.
B. Sin/ia, T/ie optical potential and nuclear structure
56
References [1
C. Davisson and L.H. Germer, Phys. Rev. 30 (1927) 705;
G.P. Thomson, Proc. Roy. Soc. 117 (1928); 119 (1928). 1-I.H. Barschall, Phys. Rev. 86 (1952) 431. [31 HA. Bethe, Phys. Rev. 47 (1935) 747; H. Feshbach, CE. Porter and V.F. Weisskopf, Phys. Rev. 96(1954)448; S. Fernbach, R. Serber and TB. Taylor, Phys. Rev. 75 (1949) 1352. 14] G.E. Brown, Revs. Mod. Phys. 31(1959) 893. [51 H. Feshbach, Ann. Rev. Nuci. Science 8 (1959) 49. [61 AK. Kerman, H. McManus and R.M. Thaler, Ann. Physics 8 (1959) 551. 171 P.E. Hodgson, Ann. Rev. NucI. Science 17 (1967) 1; Nuclear reaction and nuclear structure (Clarendon Press, Oxford, 1971). [8] G.W. Greenlees, G.J. Pyle and Y.C. Tang, Phys. Rev. 171 (1968) 1115. [91 S.D. Drell, Phys. Rev. 100 (1955) 97. [101 G.L. Shaw, Ann. Physics 8 (1959) 509. [11] D.J. Thouless, Ph.D. thesis, Cornell Univ., New York 1958;Phys. Rev. 112 (1958) 906. [12] D. Slanina, Ph. D. thesis, Michigan State University, East Lansing, 1969, unpublished. [13] G.F. Chew and ML. Goldberger, Phys. Rev. 87 (1952) 778. [14] H.R. Kidwai, Ph.D. thesis, University of Oxford, 1969, unpublished. [151 l-I.R. Kidwai and J.R. Rook, Nuci. Phys. A169 (1971) 417. [161 R.J. Blin-Stoyle, Phil. Mag. 46 (1955) 973. [17] G. Breit, Phys. Rev. 120 (1960) 287; R.A. Bryan and B.L. Scott, Phys. Rev. 135 (1964) B434; J.J. Sakurai, Phys. Rev. 119 (1960) 1784. [181 M.P. Fricke and G.R. Satchler, Phys. Rev. 139 (1965) B567; M.P. Fricke et al., Phys. Rev. 156 (1967) 1207. [191 F.G. Perey, Phys. Rev. 131 (1963) 745. [20] M. Calderbank et al., Nuci. Phys. A105 (1967) 601. [211 G.L. Thomas and E.J. Burge, Nuci. Phys. A128 (1969) 545. [22] B.C. Sinha et at., Nucl. Phys. A183 (1972) 401. [231 AM. Lane, Nucl. Phys. 35 (1962) 676. [24] G.R. Satchier, R.M. Drisko and RI-I. Bassel, Phys. Rev. 136 (1964) B637 125] G.R. Satchler, Iso-spin in nuclear physics (North-Holland, 1969) p. 391. [261 B.C. Sinha and V.R.W. Edwards, Phys. Lett. 31B (1970) 273. [271 C.J. Batty et al., NucI. Phys. A116 (1968) 643. [28] J.A. Nolen, J.P. Schiffer and N. Williams, Phys. Lett. 27B (1968) 1. [29] G.W. Greenlees, W. Makofsky and G.J. Pyle, Phys. Rev. Cl (1970) 1145. [30] H.L. Acker et al., Nuel. Phys. 87 (1966) 1. [31] E.H. Burhop, NucI. Phys. B! (1967) 438. [32] J.W. Negele, Phys. Rev. 1C (1970) 1260. [33] B.C. Sinha, Phys. Lett. 33B (1970) 279. [34] E. Friedman, Phys. Lett. 29B (1969) 213. 135] R.J. Griffiths, private communication. 1361 A.D. MacKellar et al., Phys. Rev. C3 (1971) 460. [37] G.E. Brown, Comm. on Nuclei and Particles. [38] P.J. Siemens, Nucl. Phys. A141 (1970) 225. [39] S.A. Moszkowski and B.L. Scott, Ann. Phys. 11(1960) 65. [40] D. Slanina and H. McManus, Nucl. Phys. A116 (1968) 271. 141] T.T.S. Kuo and G.E. Brown, Nucl. Phys. 85 (1968) 40; G.E. Brown, Nuclear models and forces (3rd ed.) p. 253. [42] T. Hamada and ID. Johnston, NucI. Phys. 34 (1962) 382. [43] G.L. Thomas and B.C. Sinha, Phys. Rev. Lett. 26 (1971) 325. [44] AM. Green, Phys. Lett. 24B (1967) 384. [45] A. Kallio and K. Kolltveit. Nucl. Phys. 53 (1964) 87. [46] H.A. Bethe, Proc. lot. Conf. on Nuclear structure, Tokyo (1970) p. 56. [47]T.T.S. Kuo, Nucl. Phys. A103 (1967) 71.
121
B. Sinha, The opticalpotential and nuclear structure [48] V.R. Pandharipande, Nucl. Phys. A135 (1969) 419; A166 (1971) 317. [491G.L. Thomas, B.C. Sinha and F. Duggan, Nucl. Phys. A203 (1973) 305. [501 D. Vautherin and D.M. Brink, Phys. Rev. CS (1972) 626. [51] J.W. Negele and D. Vautherin, Phys. Rev. CS (1972) 1472. [521 T.H.R. Skyrme, Phil. Mag. 1(1956)1043. [53] C.B. Dover and NV. Giai, Nuci. Phys. A190 (1972) 373. [54] B. Sinha, D.K. Srivastava and N.K. Ganguly, Phys. Lett. 43B (1973) 113. [55]B. Sinha, King’s College, unpublished report (1973). [56] L.W. Owen and G.R. Satchler, Phys. Rev. Lett. 25 (1970) 1720. [57]V.R.W. Edwards, private communication. [58] P.W. Tedder et al., to be published and Rutherford Lab. report (1975). [591C.J. Batty and G.W. Greenlees, NucI. Phys. A133 (1969) 673. [60] S.A.A. Zaidi and S. Darmodjo, Phys. Rev. Lett. 19 (1967) 1446. 1611 R.S. Mackintosh, Nod. Phys. A164 (1971) 398. [62] F. Perey and B. Buck, Nuci. Phys. 32 (1962) 353. [63] B. Sinha, Nucl. Phys. A203 (1973) 473. [641 R.C. Johnson and D.C. Martin, Nucl. Phys. A192 (1973) 496. [65] K.M. Watson, Phys. Rev. 105 (1957) 1388. [66] G.E. Brown, Unified theory of nuclear models (North-Holland, 1967). [67] T.F. O’Dwyer, M. Kawai and G.E. Brown, Phys. Lett. 41B (1972) 259. [681 N. Vinh Mau, mt. Course on Nuclear theory, Trieste (1969). [69] M. Bruneau and N. Vinh Mau, Proc. 60th birthday of Prof. RE. Peierls, Birmingham (1967). [70] T. Terasawa, NucI. Phys. 39 (1962) 563. [711 V. Gillet and E.A. Sanderson, Nucl. Phys. 54 (1964) 472. [72] A.M. Lane, Nuclear theory (W.A. Benjamin Inc., 1964). [731 F.G. Perey and D.S. Saxon, Phys. Lett. 10 (1964) 107. [74] C.L. Rao, M. Reeves and G.R. Satchler, Nucl. Phys. A207 (1973) 182. [75]J. Cugnon, Nucl. Phys. A165 (1971) 393. [76] A.M. Lane and C.F. Wandel, Phys. Rev. 98 (1955) 1524; K. Harada and N. Oda, Prog. Theor. Phys. 21(1959) 260. [77] A.E.S. Green, T. Sawada and D.S. Saxon, The nuclear independent particle model (Academic Press, 1968) pp. 304, 306. [78] E. Clemental and C. Villi, Nuovo Cim. 1(1955)176. [79] V.F. Weisskopf, Science 113 (1951) 101. [80] B. Sinha and F. Duggan, Nucl. Phys. A226 (1974) 31; B. Sinha and F. Duggan, Upsalla Conf., Sept. 1973. [811 K. Mailoy and G.L. Thomas, to be published. [821 B. Sinha and F. Duggan, Phys. Lett. 47B (1973) 389. [83] S. Watanabe, Nucl. Phys. 8 (1958) 484. [84] R.C. Johnson and P.J.R. Soper, Phys. Rev. Cl (1970) 976. [85] S.K. Samaddar, R.K. Satpathy and S. Mukherjee, Nucl. Phys. AlSO (1970) 655. [86] S. Mukherjee, NucI. Phys. A118 (1968) 423. [87] B. Sinha, F. Duggan and R.J. Griffiths, Phys. Left. to be published. [88] V.R.W. Edwards and B.C. Sinha, Phys. Lett. 37B (1971) 225. [89] V. Hnizdo and J. Lowe, Phys. Lett. 35B (1971) 35. [90] G.R. Satchier, Phys. Lett 39B (1972) 492. [91] K.T.R. Davies and G.R. Satchler, Nucl. Phys. A222 (1974) 13. [92] G.R. Satchler et al., Z. Physik 260 (1973) 209; Nucl. Phys. A233 (1974) 265. [93] William D. Myers, Lawrence Berkeley Lab. report 1259. [94] P.W. Tedder et al., Nucl. Phys. to be published. [95] R.A. Broglia and A. Winther, Phys. Reports 4C (1972) 153; D.M. Brink and N. Rowley, Nucl. Phys. A219 (1974) 79. [96] J. Wilczynski, Phys. Lett. 47B (1973) 484; Nuci. Phys. A216 (1973) 386. [97] J. Galin et a!., Phys. Rev. C9 (1974) 1018. [981 D.M. Brink and F. Stancu, Nucl. Phys. Lab report, Oxford 59/74. [99] B. Sinha, Phys. Rev. Lett. 33 (1974) 600; Phys. Rev. C, to be published (May 1975). [1001 DR. Alexander and F.B. Malik, Phys. Lett. 42B (1972) 412.
57