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Exchange contributions to nucleus-nucleus potentials deduced from RGM phase shifts using inversion
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A 589 (1995) 377-394
Exchange contributions to nucleus-nucleus potentials deduced from RGM phase shifts using inversion R.S. Mackintosh 1, S.G. Cooper 2 Physics Department, The Open University, Milton Keynes MK7 6AA, UK Received 5 January 1995; revised 3 April 1995
Abstract We have determined, for certain representative cases, the representation in terms of local potentials of the contribution to the nucleus-nucleus interaction of one-nucleon exchange, core exchange and intermediate exchange terms. We do this by applying Sl-to-V(r) inversion, using the IP method, to phase shifts from RGM calculations of Fujiwara and Tang, and LeMere et al. For ta at 18 MeV we find a shape parity dependence which is identical in nature to that found in other cases recently. For 3He + tr at 60 MeV, a strong parity shape dependence still occurs, but of slightly different form. For tr+160, the 4N exchange generates a small parity dependence of the same nature, but this seems to disappear in the full RGM calculation. In the last two cases, we show the characteristic forms of potentials due to the different exchange terms. These calculations indicate the scope of inversion as a means of extracting information from microscopic calculations concerning the nucleus-nucleus interaction.
1. Introduction Little is firmly established concerning the details of how exchange processes or reaction channels contribute to internuclear potentials. The fact that it is now straightforward to perform accurate St or Stj --~ V ( r ) inversion presents one means of studying these questions since one can determine the potential corresponding to Sl or Slj derived from any theory such as RGM, CRC, etc. Potentials derived in this way may then be directly 1E-mail: [email protected] 2E-mail: [email protected] Elsevier Science B.V. SSDI 0375 -9474 ( 95 ) 00171-9
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compared to phenomenological potentials relevant to scattering or bound-state studies. In a few cases, there exist well-determined empirical phase shifts allowing phenomenological potentials to be determined by inversion of such phase shifts and bound-state energies. The nucleon-alpha case provides an example: the empirical phase shifts have been shown [ 1] to entail a potential with a characteristic parity-dependent radial form. RGMderived phase shifts exhibit [2] the same parity-dependent radial form. In this way a consequence of RGM was confirmed. This would probably be impossible by directly fitting data owing to the currently necessary simplifications of RGM calculations. Re.f [2] also directly determined the consequences for the Na potential of variations in what reaction channels are included, and this could readily be extended to any details of the RGM calculation. The same parity-dependent geometry has been found at low energies for ta [ 3 ] and a 12C [4] interactions, suggesting a general phenomenon, and one intention of this paper is to explore this possibility. More generally, we wish to show how linking microscopic theories such as RGM to inversion provides a means of establishing microscopically derived information concerning nuclear potentials. To date, only qualitative discussions exist: for example, LeMere et al. [5,6] discuss, using a simplified inversion technique, how the Wigner or Majorana character of exchange potentials depends upon the inclusion of various classes of exchange term. There are also detailed predictions concerning the way parity dependence relates to the nuclear masses involved based on general arguments put forward by Baye [7]. The cases studied here exploit existing published RGM phase shifts, and include examples which allow us to link with previously published cases and confront varied predictions. We study a variety of exchange terms for two contrasting reactions, t~J60 at 18 MeV and 3He-c~ at 60 MeV. Baye [7] predicts contrasting properties of the parity dependence for these two cases. We then exploit a more recent RGM study of the triton-~ interaction at lower energy, 18 MeV. This allows us to compare the mass-3 + mass-4 interaction at a different energy and also determine an imaginary parity-dependent component resulting from the multichannel nature of the calculation. It also enables a comparison with previously established potentials for the same system at energies below the inelastic threshold. We then have results for the mass-7 system over a wide range of energies. That RGM calculations of nucleus-nucleus scattering can be made to yield general properties of the nucleus-nucleus interaction has been emphasized by Horiuchi (for a review see Ref. [8]). Wada and Horiuchi [9] developed an elegant WKB-based inversion procedure to establish potentials which yield the RGM elastic channel S-matrix, St. The overall procedure has become known as RGM-WKB. Recently, other procedures for St ~ V ( r ) inversion have been developed, including the iterative-perturbative (IP) method [ 10,11,14] which is used in this paper. Other applications of the method are published in Ref. [ 12,13]. The IP method is not subject to the inevitable uncertainties of the WKB method at low energies. A comparison between the two inversion methods has been made for the case of 160 + 160 at a wide range of energies [15] and differences appear in V ( r )
R.S. Mackintosh, S.G. Cooper~Nuclear Physics A 589 (1995) 377-394
379
1-N .....
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Fig. 1. Phase shifts in degrees for 18 MeV ot + 160 for RGM calculations of LeMere et al. The cases are one-nucleon (IN) exchange, 2N exchange, 4N exchange and full RGM; in each case the phase shifts for the direct calculation are subtracted. at lower energies. It is at low energies that many of the subtle signs of/-dependence appear, particularly when one attempts to achieve a unified model of scattering and bound-state phenomena. We present in Appendix A some definitions connected with the IP method to which we shall refer in the account below. In Sections 2 to 4 we discuss the three contrasting examples, Section 5 relates the present results to existing studies and Section 6 summarizes our findings.
2. The t~-160 interaction at 18 MeV CM Lemere et al. [5] tabulate 6t for 18 MeV a-particles scattering from 160. The Coulomb potential is omitted and no reaction channels are included leading to real 6t and V ( r ) . We are given ~t for the direct (folding) model, for 1N, 2N, 3N and 4N exchange, as well as for ( I + 2 ) N exchange and for the full R G M calculation. In this paper we have not considered the rather weak 3N exchange. Since we shall need to refer to various properties of the LeMere et al. phase shifts, we display those relevant to our study in Fig. 1 in which we have subtracted the "direct" phase shifts. One sees from Fig. 1 that the 4N exchange is weak and apparently of pure Majorana character, that 2N exchange is repulsive and of Wigner character, that 1N exchange is strongly attractive, and that it is considerably stronger than the direct potential (the omitted 3N exchange is weak Wigner). Note the small Majorana component in IN
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R.S. Mackintosh, S.G. Cooper~NuclearPhysics A 589 (1995) 377-394
exchange, and the seeming disappearance of Majorana character in the full RGM 61. The classification into IN, 2N exchange terms is not unique as emerges from the form of RGM applied by Baldock et al. [ 16]. We discuss this point further in Appendix B. Indications of the degree of consistency of the Tang scheme will emerge at various points in the paper. LeMere et al. do give a detailed discussion of how the different exchange processes relate to an effective local potential between the nuclei. It is, however, based on (planewave) Born approximation arguments. Such methods cannot be relied upon to establish details such as the parity dependence of the potential geometry. Moreover, the Born approximation analytic results rely upon the gaussian form of the nucleon-nucleon interaction. The following calculations demonstrate how any RGM calculations can be related to local potential phenomenology; in principle, fine details of how the local nucleus-nucleus potential is related to fine adjustments of the nucleon-nucleon interaction, or details of the calculations can be established. 2.1. The direct potential
Fig. 2 (top) presents V ( r ) found by inverting the "direct" phases shifts; two potentials from independent inversions (gaussian and Bessel function inversion bases) are presented but are indistinguishable on this scale. This assures the unambiguity of the potential, apart from discrete ambiguities of a kind discussed below in connection with the inversion of 1N exchange and full RGM t~l. The volume integral is 259 MeV.fm 3 with an uncertainty of about 1 or 2 MeV.fm 3. 2.2. One-nucleon exchange
This is the dominant exchange process, making the largest exchange contribution to the final RGM phase shifts. (It is also the one-exchange component represented - approximately - in the M3Y model [17].) This case is also interesting from the point of view of the inversion process. The reason is that the phase shift falls by more than ~- radians between l = 9 and 10. Consequently, the same set of St = exp2i6t correspond both to the phase shifts given by LeMere, and phase shifts which are less by 7r radians for l ~< 9. This is, in effect, a serious ambiguity problem faced by any inversion process which transforms St --, V ( r ) . Indeed, our first attempts at inverting Sj derived from the given 1N exchange phase shifts led to potentials with essentially the same volume integrals as the direct phase shifts. This is clearly wrong; 1N exchange generates a potential with an extra node in the wave function for l ~< 9. The IP method can handle this situation. By using a sufficiently deep SRP it was possible to find a reasonably smooth potential which fits LeMere et ai.'s IN exchange phase shifts (the "wrong" potential is, indeed, much less smooth.) This is shown in the second part of Fig. 2 which shows the direct potential for comparison. The fact that the potential is not entirely smooth results from a Majorana admixture in the predominantly Wigner potential.
R.S. Mackintosh, S.G. Cooper/Nuclear Physics A 589 (1995) 377-394
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The top panel presents two indistinguishable solutions for the direct phases shifts. The central panel compares a direct potential (solid line) with a potential reproducing all the IN exchange phase shifts (dashed). The
bottom panel compares the 1N exchange potential of the central panel, now with the direct potential subtracted, with the potentials representing odd (dotted) and even (dashed) IN exchange phase shifts.
It is possible to find smoother potentials if one fits ~t separately for odd-I and even-l; the resulting potentials, all with the direct potential subtracted, are shown in the bottom panel of Fig. 2 together with the all-/ potential of the centre panel. Not only are the odd-I and even-I curves smoother than the all-/curve, but there is clear evidence that the parity dependence of the potential is represented by the different shape of the two terms, rather than the overall strength; indeed we see the pattern seen elsewhere [ 1-4] whereby the odd-I term is longer in range but shallower at the nuclear centre than the even-I term. A measure of the strength of the 1N exchange is the fact that it increases the volume integral of the potential by 139 MeV.fm 3, to about 391 MeV.fm 3. 2.3. Two-nucleon exchange The upper diagram in Fig. 3 shows (dashed curve) the potential representing the inclusion of 2N exchange with the direct potential subtracted. This term appears to be
382
R.S. Mackintosh, S.G. Cooper~Nuclear Physics A 589 (1995) 377-394
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Fig. 3. For 18 MeV CM alphas on 160, potentials with direct term subtracted, as follows: upper panel, 1N (solid), 2N (dashes) and full RGM (dots); lower panel, 4N all-/ (solid), 4N odd-/ (dashes), 4N even-/ (dots). purely Wigner in character and repulsive. It is about one quarter the magnitude at r = 0 of the 1N contribution which is included (solid line) for comparison. It is of somewhat lesser range, so that the volume integral is reduced in magnitude by some 25 MeV.fm 3 by 2N exchange.
2.4. Four-nucleon exchange Fig. 1 shows that 4N exchange (core exchange) leads to small alternating changes in Sl. For l ~< 8 there is additional attraction for even-l; for l ~< 7 there is similar repulsion for odd-/. It is possible to find a single V(r) reproducing all l, but it has narrow oscillations. It is shown in the lower diagram of Fig. 3 (solid curve) with the direct potential subtracted. The oscillations suggest parity dependence [ 18]. Inverting from 8t for just even-/ or just odd-/yields the relatively smooth potentials also shown in the lower part of Fig. 3 with the direct potential subtracted. The tendency for the odd-/potential to be stronger in the surface region, is apparently reversed in this case. This reversal is far from obvious from the behaviour of 8t seen in Fig. 1 where the Majorana effect is of the same sign for 1N and 4N, although covering different/-ranges in the two cases.
R.S. Mackintosh, S.G. Cooper~NuclearPhysics A 589 (1995) 377-394
383
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Fig. 4. Potentials for 18 MeV CM alphas on 160. Upper panel compares full RGM potentials as follows: solid line, fit to S/ with wrong t~t (see text); dashed lines represents the "correct" solution with tr = 0.009, the basis of "subtracted" potentials in other figures. Lower panel: comparing the direct (solid), full RGM (dashed) and Michel et al.'s 32 MeV potential (dotted).
2.5. The full R G M potential
This case requires the same care in avoiding a discrete ambiguity as the IN exchange potential. The potential we find is included, with direct potential subtracted, as the dotted line in the top diagram of Fig. 3. The overall R G M exchange contribution can be seen from this figure to be very close to a sum of the IN and 2N contributions. In Fig. 4, top panel, we show the complete RGM potential, together with the shallow potential which gives the same St = exp2it3t but not the correct ~t (solid line). The inversion gives a moderately low ~r of 0.009 (see Appendix A) and the potential appears to be have small uncertainties, not very apparent on the scale of Fig. 4. Finally, the lower panel of Fig. 4 compares the direct and R G M potentials with the 32 MeV phenomenological potential of Michel et al. [ 19]. There is as much general agreement as could be expected. The relevant M3Y-model potential. [ 17] is very similar to the Michel potential for r ,-~ 4 fm, but is almost 200 MeV deep at r = 0. The M3Y model contains an approximate representation of IN exchange but no representation of 2N exchange.
384
R.S. Mackintosh, S.G. Cooper~Nuclear Physics A 589 (1995) 377-394
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Fig. 5. For 60 MeV 3He + 4He, RGM phase shifts with indicated exchange terms, with "direct" phase shifts subtracted. The short-dashed, long-dashed and dot-dashed lines represent IN, 2N and 3N exchange terms, respectively. The dotted line indicates ( I + 3 ) N exchange and the solid line indicates the full RGM phases. These are all from the paper of LeMere et al.
2.6. Does RGM lead to an 1-independent a+160 potential? The possibility of a Majorana component is a relevant phenomenological question (see Section 5). There can be no definitive answer, at least at 18 MeV, until it is understood why the purely Majorana 4N contribution and the Majorana term in the IN contribution, which Fig. 1 suggests have the same sense, have been lost in the overall RGM phase shifts. This may well be due to the problem discussed in Appendix B. What does seem clear is that, as might be expected from the RGM 6t of Fig. 1, the overall RGM potential is smooth, showing no signature of parity dependence.
3. The 3 H e - a interaction at 60 MeV CM
We exploit S(l)-to-V(r) inversion to establish precise potentials corresponding to elaborate RGM calculations of LeMere et al. [5]. These calculations omit the Coulomb potential. There is no representation of channel coupling so IS(l)l -- 1 and the potentials will be real. Lemere et al. present not only the RGM phase shifts corresponding to all exchange processes and phase shifts for the direct (folding model) case, but also phase shifts corresponding to the separate inclusion of one-, two- and three-nucleon exchange as well
R.S. Mackintosh, S.G. Cooper~Nuclear Physics A 589 (1995) 377-394
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as combined ( l + 3 ) - n u c l e o n exchange. In Fig. 5 we plot the phase shifts of LeMere et al. (in degrees) with the phase shifts for the direct (folding) model subtracted for clarity (the direct 80 "~ 130°). We note the evident parity dependence largely due to 3N exchange. For higher l, 3N exchange becomes more significant than the IN attraction and a larger part of the total R G M phase shift. The 3N exchange contribution is evidently a superposition of a Majorana term and a repulsive Wigner term. The 2N exchange contribution is small and confined to lower partial waves. The exchange terms add roughly, but not precisely, linearly, as can be seen if, for example, one compares the 3N exchange term with difference between the ( l + 3 ) N exchange and 1N exchange terms.
3.1. The direct potential
With a reasonably arbitrary SRP (see Appendix A) one easily finds a smooth, very gaussian-like potential. Alternative inversion bases gave the virtually indistinguishable potentials with V(0) ~ 70 MeV. This lends confidence in the overall R G M + inversion procedure. One of these is shown in the Fig. 6; the direct potential should, of course, be smooth. The volume integral is ,-~ 404 MeV.fm 3.
386
R.S. Mackintosh, S.G. Cooper~Nuclear Physics A 589 (1995) 377-394
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Fig. 7. Potentialsfor 60 MeV CM 3He + 4He. Top: direct (solid), 3N even-/(dashes) and 3N odd-/(dots). Centre: IN (solid), (I+3)N even-/ (dashes), (I+3)N odd-/ (dots). Bottom: IN (solid), full RGM even-/ (dashes), full RGM odd-/(dots). 3.2. 1N exchange
Fig. 6 also presents the well-determined potential representing phase shifts with 1N exchange included. The 1N exchange contribution is an attractive Wigner potential, larger in magnitude than the difference between the M3Y potential and the direct potential. The M3Y [ 17] model potential is included as a dotted line. The latter is only slightly deeper than the direct potential but similar in geometric properties. 3.3. 3 N exchange
This is the core exchange case. There is no smooth 3 potential corresponding to the highly staggered St of Fig. 5. However, there exist independent potentials reproducing the odd and even phase shifts with o" < 10 -12 in each case, i.e. exact inversions. These are compared with the direct V ( r ) (solid line) in the top panel of Fig. 7. The odd-parity V ( r ) (dotted) is both deeper and longer ranged than the even-parity V ( r ) (dashes), 3 Inversionfor all l leads to a V(r) with oscillationssome 30 MeV in amplitude.
Cooper~NuclearPhysics A 589 (1995) 377-394
R.S. Mackintosh, S.G.
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Fig. 8. Potentials for 60 MeV CM 3He + 4He, the top two panels with direct potential subtracted, and the bottom panel with the potential including IN exchange subtracted. Top: 2N all-/ (solid), 3N even-/(dashes), 3N odd-/ (dots). Centre: 2N even-/ (solid), 2N odd-/ (dashes). Bottom: ( l + 3 ) N even-/ (solid) and odd-/ (dashed) compared with full RGM even-/(dots) and odd-/ (dot-dashed).
and in this case there is no cross-over at smaller radii. Consistently with the fact that the 3N exchange ~t also exhibit (Fig. 5) a repulsive Wigner term, the difference between the even and direct V(r) is greater than the difference between the odd and direct V(r). The 3N exchange increases the odd-/volume integral by 82 MeV.fm 3 and reduces the even-/ volume integral by 106 MeV.fm 3. The linearity with which exchange terms contribute to the potential can be judged from the central panel of Fig. 7 in which we compare the 1N potential with the even and odd-parity ( l + 3 ) N potentials. The differences are similar but not identical to those in the top panel.
3.4. 2N exchange The 2N exchange makes a small contribution to RGM as can be deduced from the RGM parity-dependent potentials in the bottom panel in Fig. 7 which are very little different from the ( I + 3 ) N potentials in the central panel. The 2N exchange component is shown with the direct potential subtracted in the top and centre panels of Fig. 8. In
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R.S. Mackintosh,S.G. Cooper~NuclearPhysicsA 589 (1995) 377-394
the top panel we compare the result of a direct/-independent fit to the 2N phase shifts compared to the even and odd-parity terms induced by 3N exchange (shown unsubtracted in Fig. 7, top panel). The oscillation is quite typical of what we expect when fitting 6t which "really" correspond to a parity-dependent potential. We have noted before [ 1,18] that Majorana effects always present the alternative: either a smooth parity-dependent potential or an oscillatory /-independent potential. Fitting the 2N odd-/and even-/ 6t separately we find the smooth parity-dependent V(r) of the central panel of Fig. 8. It is typical that the parity-dependent representation should be of much shorter range than is required of a parity-independent V(r). 3.5. (l +3)N exchange, 2N exchange and linearity If the contributions of individual exchange terms are not too large then they should contribute in a roughly linear fashion when added, the two-potential formula of GellMann and Goldberger prohibiting exact linearity. We have already noted in connection with Fig. 7 that the system is rather linear. The bottom panel of Fig. 8 compares the final even and odd RGM potentials (direct subtracted) with the corresponding ( l + 3 ) N potentials with the potential including IN exchange subtracted. If 2N exchange contributed linearly, the differences in the lower panel would be just the 2N potentials of the central panel. The agreement is remarkably close. This argues for the precision and unambiguity of the inversion procedure. 3.6. The full RGM potential Figs. 7 and 8 have already presented the full RGM potential. It is unquestionably strongly parity dependent, with a geometric parity dependence in which the o d d - / V ( r ) is of longer range than the even-I V(r). Indeed, from 3 fm outwards, it is some 70% or more deeper. There is a cross-over near 1 fm, and the even-I potential is slightly greater at r = 0. In the surface, the M3Y model potential falls between the two RGM components, but overall is less deep. The volume integrals of the odd and even potentials are, respectively, ~ 546 and ~ 378 MeV.fm 3. The respective RMS radii are 2.93 and 2.59 fm (cf. 2.80 for the direct) reflecting the marked geometric parity dependence.
4. t a scattering at 20 MeV CM
Fujiwara and Tang [20] performed elaborate multiconfiguration RGM calculations for the ta system at 20 MeV, CM. Here, we present the results of inverting the St ("TC case") deduced from Fig. 14 of that reference, with the limitations consequent on this means of defining the St to be inverted. In Fig. 9, we present the real and imaginary potentials which give precise fits to Sl of Fujiwara and Tang. Because of possible uncertainties related to the way we obtained St, we repeated the inversion several times with alternative SRP and inversion bases
R.S. Mackintosh, S.G.
Cooper~NuclearPhysics A 589 (1995) 377-394
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to the RGM phase shifts o f Fujiwara and Tang. Upper: real part, o d d - / ( s o l i d ) and even-/o- = 0.043 ( d a s h e s ) and o" = 0.0007 ( d o t s ) . Lower, the imaginary potential with same line conventions.
to eliminate the possibility that the visible features in this figure were artifacts. Two alternative V(r) for even-/ give an indication of the uncertainty in V(r). The conspicuous feature is the strong geometric parity dependence in both the real and imaginary terms. The odd-/real term is of strikingly long range as measured not only against the even-/term, but also the M3Y model V(r) (not shown) which is almost identical to the even-/ V(r) in the surface (although somewhat weaker for r < 2.5 fro, being 78 MeV at r = 0.) The odd-parity imaginary term is of surface character, while the even-parity component is essentially of volume form. This may be the first indication of what form of parity dependence of the imaginary component in a nucleus-nucleus potential follows from calculated exchange processes.
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R.S. Mackintosh,S.G. Cooper~NuclearPhysicsA 589 (1995)377-394
5. Relation to other work
5.1. Systematics of 3He (or t) +4He potentials Cooper [ 3 ] showed by inversion of single-channel RGM Sty and bound-state energies that the 3He+4He potential below the inelastic threshold had a parity-dependent radial form in which the odd-parity component was of greater range and volume integral than the even-parity component. As noted in the introduction, this same property is found for both the microscopic and phenomenological nucleon-a potential. The results of Sections 3 and 4 above allow us to confirm that a similar parity dependence in the radial form is a general feature of RGM derived potentials for this system. We can also draw tentative conclusions concerning the energy dependence. The possible danger in doing this lies in the fact that the different RGM calculations included different terms. For example, Fujiwara and Tang's 20 MeV calculation included reaction channels generating an imaginary term 4. The higher-energy cases both omitted spin and Coulomb effects, excluding the possibility of deducing spin-orbit terms (readily possible with IP inversion) and eliminating any distinctions between tritons and helions. One apparent energy dependence emerges: at the lowest energy [ 3 ], the odd- and evenparity potentials cross over, so that the even-parity potential is much deeper near r = 0. However, this might be modified by distortion effects in the l = 0 channel. At 60 MeV, Figs. 6 and 7 show that there is little cross-over so the even-parity potential is just marginally deeper at r = 0; the 20 MeV case is intermediate with a cross-over near 2.25 fm, but the even-parity term is less than 10 MeV deeper than the odd-parity term at r = 0 . As we mentioned, the above survey includes both single-channel and coupled-channel RGM but at different energies. We have also inverted St for 3He on 4He at 35 MeV for both single-channel and coupled-channel RGM [21 ]. We find that the channels included have a much smaller effect on the odd and even potentials than the difference between the odd and even potentials so that none of the conclusions about parity dependence are modified. The resulting potentials were intermediate between the 20 and 60 MeV cases, there being a crossover near 1.4 fm, with the odd-parity potential considerably deeper than the real potential for r > 3 fm.
5.2. ol+160 potentials Published fits [ 19] to the wide angular range cross section data for a + 1 6 0 have xZ/N ~ 150. We can reduce xZ/N by an order of magnitude, but only with systematic "wiggly" features in the potentials. It is well known that one can find wiggly potentials which give the same St as smooth but /-dependent (not necessarily parity-dependent) potentials. The question then arises as to whether the local potentials which precisely 4 It would be very interesting to study the relationship between the real and imaginaryterms generated by different and cumulativereaction channels.
R.S. Mackintosh, S.G. Cooper~NuclearPhysics A 589 (1995) 377-394
391
fit the data can be interpreted in terms of/-dependence in the underlying interaction. Firm knowledge of the manner in which different exchange contributions contribute to the effective local potential is a necessary precondition to answering such questions. Although we have shown that individual exchange terms have Majorana components, the overall RGM phases shifts and consequent potential do not have this property for reasons we do not understand.
5.3. Comparison with predicted behaviour Baye [7] makes general predictions concerning how both the strength and sign of the parity dependence are related to the particular interacting nuclei. The fact that core exchange is expected to give a deeper potential for even partial waves in the case of o~ + 160 is borne out by Fig. 3, lower panel, although not at the nuclear centre. The opposite prediction for the mass-7 system is also clearly borne out as far as volume integrals and the behaviour at larger radii is concerned. Nevertheless, core exchange is not the full story as far as parity dependence is concerned. Moreover, it is very unclear how to interpret the consistent geometric parity dependence involving quite large radii and partial waves in view of the suggestion in Ref. [7] that parity-dependence effects are restricted to small/-values; we note that parity dependence related to core exchange persists up to l = 11 in the case of 60 MeV 3He + o~. Of course, it can be argued that the our geometry dependence is somehow an artifact of the requirement that we find a smooth V(r) to fit certain sets of t3l. On the other hand, we argue that it is after all smooth potentials which are relevant to phenomenology. Moreover, these smooth potentials are not forced. They generally correspond to very low values of tr and are not subject to ambiguities. Indeed, it is far from a priori obvious that such smooth parity-doublet potentials exist to fit the patterns of t3t predicted by RGM.
5.4. Ambiguity of nN decomposition In Section 2 and in Appendix B we make reference to the fact that decomposition into nN terms is ambiguous and in some degree controversial. Although only the noexchange and full-exchange results are fully unambiguous, the present work suggests that the conventional decomposition of Tang and his co-workers can be useful. It is striking that the full-exchange potential for 3He on 4He at 60 MeV is indeed a linear sum of the of nN terms, while the linearity is less complete for ot on 160. From our point of view there is interest in establishing nN exchange potentials since they throw light on most calculations of nucleus-nucleus potentials (e.g. M3Y) which omit all exchange terms except 1N exchange and always get purely Wigner potentials. In this light, the large 2N contribution for ot on L60, which does indeed appear to add linearly into the total exchange potential, is noteworthy, as is the fact that the 1N exchange is pure Wigner for 3He on 4He at 60 MeV. Conversely, the Majorana component visible in the 1N exchange for ce on 160 probably does reveal the limits of this decomposition.
392
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6. General conclusions; future work
We have demonstrated that accurate inversion can provide a way of unlocking the detailed information concerning nuclear interactions which resides in the elastic scattering phase shifts calculated by, for example, RGM. WKB methods and Born approximation methods cannot be relied upon at low energies. It is particularly in the low-energy regime where subtle threshold and partial-wave dependence effects occur. The potentials produced by inversion are immediately of a form that can be related to phenomenology. In this way phenomenology can be brought to bear on the qualitative features of RGM results. At present RGM must use somewhat schematic NN interactions making direct comparison with experiment problematical. The most immediate question raised by this work is the following: does the apparent lack of Majorana component in the overall RGM potential persist to very low energies? The future importance of these studies lies in the fact that RGM can, in principle at least, represent processes which are generally omitted in reaction calculations [22]. For example there is evidence (e.g. Ref. [23] ) that rearrangement channels contribute strongly to nucleon-nucleus scattering. But the difficulty of including exchange processes and non-orthogonality effects within the usual reaction theory framework make such rearrangement contributions hard to establish decisively. RGM plus inversion might thus contribute to understanding the single-nucleon interaction with nuclei; it is quite likely that the complex nucleon-nucleus interaction is /-dependent in a way quite different from a simple Majorana dependence and this should be established or refuted. The usefulness of one-body (optical model) potentials in nuclear physics, whether for nucleons or composite particles, has been a continuing surprise. One limit was apparently reached with cases where exchange processes made large contributions which did not seem naturally to fit into a single-particle picture (e.g. 12C + 13C [24] ). By generalizing to potentials with separate geometries for odd and even parity it seems that we have far from reached the limit of simple optical-type models. Specific findings of the present work concerning nuclear interactions include: (i) Further evidence for a systematic Majorana shape term in nucleus-nucleus potentials due to exchange. (ii) First indications of the energy dependence of the shape effect. (iii) First indications of a strong parity dependence in the radial form of the imaginary part generated by reaction channels. (iv) Confirmation of the strong attractive and short range nature of the 1N contribution to nucleus-nucleus potentials. (v) Evidence that the 2N exchange contributes repulsively to a-nucleus potentials. This term is omitted from M3Y and other folding models. (vi) Evidence for the degree of linearity with which exchange components contribute to the overall potential. For alpha + 160, there is indeed an exception to the general linearity and this is apparently related to alternative formulations of RGM, see Appendix B. Concerning inversion, we note that the IP method employed here has, as well as the
R.S. Mackintosh, S.G. Cooper~Nuclear Physics A 589 (1995) 377-394
393
specific advantages explained elsewhere, the ability to handle the case where the St do not contain all the information concerning the potential, specifically the fact that a group of the St may correspond to ~t with an extra zr-phase.
Acknowledgements We are most grateful to the EPSRC of the UK for grant GR/H00895 supporting Dr. Cooper. We are also grateful to Professor Tang for his support. We would like to thank the anonymous referee for bringing the work of Baldock et al. to our attention.
Appendix A. Summary of IP inversion The iterative perturbative method for S(l)-to-V(r) inversion has been described many times, e.g. Refs. [ 10,11,14], and we merely mention certain aspects of it to which we refer in the text. Being an iterative method, the inversion is initiated from some "starting reference potential", SRP; it is part of our procedure to ensure that the final results are stable against variations in the SRE The quality of the inversion is measured in terms of o-, the "phase shift distance", defined by o-2 = ~
I~ - SII2,
(A.1)
I
where ~ is the "target" S-matrix for which we seek the potential, and ~ is the S-matrix of the inversion potential.
Appendix B. Classification of exchange terms The classification of exchange terms adopted by Tang and co-workers is based on a formalism in which the complete antisymmetrization of the system is represented by the complete antisymmetry of the ket in the exchange integrals. The formalism of Baldock et al. antisymmetrizes both bra and ket with the consequence that IN, 2N, ... exchange terms have little correspondence with what Tang et al. designate in this way. As Tang [25] says " . . . in our discussion, the term nucleon exchange is defined through the kernel function while in the discussion of Baldock et al. it is defined through the wavefunction". The formalism of Baldock et al. renders characterization into Wigner and Majorana terms not very useful and may not offer much practical advantage of simplification [26]. However, the fact that a Majorana term appears in intermediate components and yet disappears in the complete potential in the Tang formalism is probably related to the instabilities that were the motivation of the approach of Baldock et al.
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References I 11 S.G. Cooper and R.S. Mackintosh, Phys. Rev. C 43 (1991) 1001. [2] S.G. Cooper, R.S. Mackintosh, A. Cs~o6 and R.G. Lovas, Phys. Rev. C 50 (1994) 1308. [3] S.G. Cooper, Phys. Rev. C 50 (1994) 359. 141 S.G. Cooper and R.S. Mackintosh, Nucl. Phys. A 517 (1990) 285. 151 M. LeMere, D.J. Studeba, H. Horiuchi and Y.C. Tang, Nucl. Phys. A 320 (1979) 449. [6] M. LeMere, Y. Fujiwara, Y.C. Tang and Q.K.K. Liu, Phys. Rev. C 26 (1982) 1847. I71 D. Baye, Nucl. Phys. A 460 (1986) 581. [8[ H. Horiuchi, in Clustering aspects of nuclear structure, eds. J.S. Lilley and M.A. Nagarajan (Reidel, Dordrecht, 1985). [91 T. Wada and H. Horiuchi, Prog. Theor. Phys. 80 (1988) 488. [ 101 S.G. Cooper and R.S. Mackintosh, Inv. Probl. 5 (1989) 707. I I 1 ] S.G. Cooper and R.S. Mackintosh, Users manual for IMAGO, Open University preprint OUPD9201. [12] A. Adachour, D. Baye and P. Descouvement, Nucl. Phys. A 579 (1994) 305. [ 13 ] F.R. Lichtenth~iler, A.C.C. Villari, A. LEpine-Szily and L.C. Gomes, Phys. Rev. C 44 (1991) 1152. [ 14] S.G. Cooper, M.A. McEwan and R.S. Mackintosh, Nucl. Phys. A 552 (1993) 401. 115] S. Ait-Tahar, R.S. Mackintosh, S.G. Cooper and T. Wada, Nucl. Phys. A 562 (1993) 101. [16] R.A. Baldock, B.A. Robson and R.E Barrett, Nucl. Phys. A 351 (1981) 157. [ 171 G.R. Satchler and W.G. Love, Phys. Reports 55 (1979) 183. [ 181 S.G. Cooper and R.S. Mackintosh, Z. Phys. A 337 (1990) 357. 1191 E Michel, J. Albinski, P. Belery, Th. Delbar, Gh. Gr6goire, B. T~siaux and G. Reidemeister, Phys. Rev. C 28 (1983) 1904. [201 Y. Fujiwara and Y.C. Tang, Phys. Rev. C 31 (1985) 342. [211 Y. Fujiwara and Y.C. Tang, Phys. Rev. C 28 (1983) 1869. [221 K. Wildermuth and Y.C. Tang, A unified theory of the nucleus (Vieweg, Braunschweig, 1977). [231 S.G. Cooper, R.S. Mackintosh and A.A. loannides, Nucl. Phys. A 472 (1987) 101. [24] W. von Oertzen, Nucl. Phys. A 148 (1970) 529. 125] Y.C. Tang, in Lecture notes in physics, vol. 145, eds. T.T.S. Kuo and S.S.M. Wong (Springer, Berlin, 1981) p. 572. [26 ] Y.C. Tang, private communication.
Nuclear Physics A 589 (1995) 377-394
Exchange contributions to nucleus-nucleus potentials deduced from RGM phase shifts using inversion R.S. Mackintosh 1, S.G. Cooper 2 Physics Department, The Open University, Milton Keynes MK7 6AA, UK Received 5 January 1995; revised 3 April 1995
Abstract We have determined, for certain representative cases, the representation in terms of local potentials of the contribution to the nucleus-nucleus interaction of one-nucleon exchange, core exchange and intermediate exchange terms. We do this by applying Sl-to-V(r) inversion, using the IP method, to phase shifts from RGM calculations of Fujiwara and Tang, and LeMere et al. For ta at 18 MeV we find a shape parity dependence which is identical in nature to that found in other cases recently. For 3He + tr at 60 MeV, a strong parity shape dependence still occurs, but of slightly different form. For tr+160, the 4N exchange generates a small parity dependence of the same nature, but this seems to disappear in the full RGM calculation. In the last two cases, we show the characteristic forms of potentials due to the different exchange terms. These calculations indicate the scope of inversion as a means of extracting information from microscopic calculations concerning the nucleus-nucleus interaction.
1. Introduction Little is firmly established concerning the details of how exchange processes or reaction channels contribute to internuclear potentials. The fact that it is now straightforward to perform accurate St or Stj --~ V ( r ) inversion presents one means of studying these questions since one can determine the potential corresponding to Sl or Slj derived from any theory such as RGM, CRC, etc. Potentials derived in this way may then be directly 1E-mail: [email protected] 2E-mail: [email protected] Elsevier Science B.V. SSDI 0375 -9474 ( 95 ) 00171-9
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compared to phenomenological potentials relevant to scattering or bound-state studies. In a few cases, there exist well-determined empirical phase shifts allowing phenomenological potentials to be determined by inversion of such phase shifts and bound-state energies. The nucleon-alpha case provides an example: the empirical phase shifts have been shown [ 1] to entail a potential with a characteristic parity-dependent radial form. RGMderived phase shifts exhibit [2] the same parity-dependent radial form. In this way a consequence of RGM was confirmed. This would probably be impossible by directly fitting data owing to the currently necessary simplifications of RGM calculations. Re.f [2] also directly determined the consequences for the Na potential of variations in what reaction channels are included, and this could readily be extended to any details of the RGM calculation. The same parity-dependent geometry has been found at low energies for ta [ 3 ] and a 12C [4] interactions, suggesting a general phenomenon, and one intention of this paper is to explore this possibility. More generally, we wish to show how linking microscopic theories such as RGM to inversion provides a means of establishing microscopically derived information concerning nuclear potentials. To date, only qualitative discussions exist: for example, LeMere et al. [5,6] discuss, using a simplified inversion technique, how the Wigner or Majorana character of exchange potentials depends upon the inclusion of various classes of exchange term. There are also detailed predictions concerning the way parity dependence relates to the nuclear masses involved based on general arguments put forward by Baye [7]. The cases studied here exploit existing published RGM phase shifts, and include examples which allow us to link with previously published cases and confront varied predictions. We study a variety of exchange terms for two contrasting reactions, t~J60 at 18 MeV and 3He-c~ at 60 MeV. Baye [7] predicts contrasting properties of the parity dependence for these two cases. We then exploit a more recent RGM study of the triton-~ interaction at lower energy, 18 MeV. This allows us to compare the mass-3 + mass-4 interaction at a different energy and also determine an imaginary parity-dependent component resulting from the multichannel nature of the calculation. It also enables a comparison with previously established potentials for the same system at energies below the inelastic threshold. We then have results for the mass-7 system over a wide range of energies. That RGM calculations of nucleus-nucleus scattering can be made to yield general properties of the nucleus-nucleus interaction has been emphasized by Horiuchi (for a review see Ref. [8]). Wada and Horiuchi [9] developed an elegant WKB-based inversion procedure to establish potentials which yield the RGM elastic channel S-matrix, St. The overall procedure has become known as RGM-WKB. Recently, other procedures for St ~ V ( r ) inversion have been developed, including the iterative-perturbative (IP) method [ 10,11,14] which is used in this paper. Other applications of the method are published in Ref. [ 12,13]. The IP method is not subject to the inevitable uncertainties of the WKB method at low energies. A comparison between the two inversion methods has been made for the case of 160 + 160 at a wide range of energies [15] and differences appear in V ( r )
R.S. Mackintosh, S.G. Cooper~Nuclear Physics A 589 (1995) 377-394
379
1-N .....
2-N 4-N full RGM
200-
n~
P~.lO0-
0-
I
s
lo g
Fig. 1. Phase shifts in degrees for 18 MeV ot + 160 for RGM calculations of LeMere et al. The cases are one-nucleon (IN) exchange, 2N exchange, 4N exchange and full RGM; in each case the phase shifts for the direct calculation are subtracted. at lower energies. It is at low energies that many of the subtle signs of/-dependence appear, particularly when one attempts to achieve a unified model of scattering and bound-state phenomena. We present in Appendix A some definitions connected with the IP method to which we shall refer in the account below. In Sections 2 to 4 we discuss the three contrasting examples, Section 5 relates the present results to existing studies and Section 6 summarizes our findings.
2. The t~-160 interaction at 18 MeV CM Lemere et al. [5] tabulate 6t for 18 MeV a-particles scattering from 160. The Coulomb potential is omitted and no reaction channels are included leading to real 6t and V ( r ) . We are given ~t for the direct (folding) model, for 1N, 2N, 3N and 4N exchange, as well as for ( I + 2 ) N exchange and for the full R G M calculation. In this paper we have not considered the rather weak 3N exchange. Since we shall need to refer to various properties of the LeMere et al. phase shifts, we display those relevant to our study in Fig. 1 in which we have subtracted the "direct" phase shifts. One sees from Fig. 1 that the 4N exchange is weak and apparently of pure Majorana character, that 2N exchange is repulsive and of Wigner character, that 1N exchange is strongly attractive, and that it is considerably stronger than the direct potential (the omitted 3N exchange is weak Wigner). Note the small Majorana component in IN
380
R.S. Mackintosh, S.G. Cooper~NuclearPhysics A 589 (1995) 377-394
exchange, and the seeming disappearance of Majorana character in the full RGM 61. The classification into IN, 2N exchange terms is not unique as emerges from the form of RGM applied by Baldock et al. [ 16]. We discuss this point further in Appendix B. Indications of the degree of consistency of the Tang scheme will emerge at various points in the paper. LeMere et al. do give a detailed discussion of how the different exchange processes relate to an effective local potential between the nuclei. It is, however, based on (planewave) Born approximation arguments. Such methods cannot be relied upon to establish details such as the parity dependence of the potential geometry. Moreover, the Born approximation analytic results rely upon the gaussian form of the nucleon-nucleon interaction. The following calculations demonstrate how any RGM calculations can be related to local potential phenomenology; in principle, fine details of how the local nucleus-nucleus potential is related to fine adjustments of the nucleon-nucleon interaction, or details of the calculations can be established. 2.1. The direct potential
Fig. 2 (top) presents V ( r ) found by inverting the "direct" phases shifts; two potentials from independent inversions (gaussian and Bessel function inversion bases) are presented but are indistinguishable on this scale. This assures the unambiguity of the potential, apart from discrete ambiguities of a kind discussed below in connection with the inversion of 1N exchange and full RGM t~l. The volume integral is 259 MeV.fm 3 with an uncertainty of about 1 or 2 MeV.fm 3. 2.2. One-nucleon exchange
This is the dominant exchange process, making the largest exchange contribution to the final RGM phase shifts. (It is also the one-exchange component represented - approximately - in the M3Y model [17].) This case is also interesting from the point of view of the inversion process. The reason is that the phase shift falls by more than ~- radians between l = 9 and 10. Consequently, the same set of St = exp2i6t correspond both to the phase shifts given by LeMere, and phase shifts which are less by 7r radians for l ~< 9. This is, in effect, a serious ambiguity problem faced by any inversion process which transforms St --, V ( r ) . Indeed, our first attempts at inverting Sj derived from the given 1N exchange phase shifts led to potentials with essentially the same volume integrals as the direct phase shifts. This is clearly wrong; 1N exchange generates a potential with an extra node in the wave function for l ~< 9. The IP method can handle this situation. By using a sufficiently deep SRP it was possible to find a reasonably smooth potential which fits LeMere et ai.'s IN exchange phase shifts (the "wrong" potential is, indeed, much less smooth.) This is shown in the second part of Fig. 2 which shows the direct potential for comparison. The fact that the potential is not entirely smooth results from a Majorana admixture in the predominantly Wigner potential.
R.S. Mackintosh, S.G. Cooper/Nuclear Physics A 589 (1995) 377-394
381
-50-
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direct
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6
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r/fm F i g . 2. F o r 18 M e V C M alphas on 160, potentials obtained by inversion of phases shifts of LeMere et al.
The top panel presents two indistinguishable solutions for the direct phases shifts. The central panel compares a direct potential (solid line) with a potential reproducing all the IN exchange phase shifts (dashed). The
bottom panel compares the 1N exchange potential of the central panel, now with the direct potential subtracted, with the potentials representing odd (dotted) and even (dashed) IN exchange phase shifts.
It is possible to find smoother potentials if one fits ~t separately for odd-I and even-l; the resulting potentials, all with the direct potential subtracted, are shown in the bottom panel of Fig. 2 together with the all-/ potential of the centre panel. Not only are the odd-I and even-I curves smoother than the all-/curve, but there is clear evidence that the parity dependence of the potential is represented by the different shape of the two terms, rather than the overall strength; indeed we see the pattern seen elsewhere [ 1-4] whereby the odd-I term is longer in range but shallower at the nuclear centre than the even-I term. A measure of the strength of the 1N exchange is the fact that it increases the volume integral of the potential by 139 MeV.fm 3, to about 391 MeV.fm 3. 2.3. Two-nucleon exchange The upper diagram in Fig. 3 shows (dashed curve) the potential representing the inclusion of 2N exchange with the direct potential subtracted. This term appears to be
382
R.S. Mackintosh, S.G. Cooper~Nuclear Physics A 589 (1995) 377-394
20---
-20
~
~
-
-40-
.-'""
1-N all-t~
-6o% >
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..... -80 3-
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0
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0
I
I
I
2
4
6
r/fro
Fig. 3. For 18 MeV CM alphas on 160, potentials with direct term subtracted, as follows: upper panel, 1N (solid), 2N (dashes) and full RGM (dots); lower panel, 4N all-/ (solid), 4N odd-/ (dashes), 4N even-/ (dots). purely Wigner in character and repulsive. It is about one quarter the magnitude at r = 0 of the 1N contribution which is included (solid line) for comparison. It is of somewhat lesser range, so that the volume integral is reduced in magnitude by some 25 MeV.fm 3 by 2N exchange.
2.4. Four-nucleon exchange Fig. 1 shows that 4N exchange (core exchange) leads to small alternating changes in Sl. For l ~< 8 there is additional attraction for even-l; for l ~< 7 there is similar repulsion for odd-/. It is possible to find a single V(r) reproducing all l, but it has narrow oscillations. It is shown in the lower diagram of Fig. 3 (solid curve) with the direct potential subtracted. The oscillations suggest parity dependence [ 18]. Inverting from 8t for just even-/ or just odd-/yields the relatively smooth potentials also shown in the lower part of Fig. 3 with the direct potential subtracted. The tendency for the odd-/potential to be stronger in the surface region, is apparently reversed in this case. This reversal is far from obvious from the behaviour of 8t seen in Fig. 1 where the Majorana effect is of the same sign for 1N and 4N, although covering different/-ranges in the two cases.
R.S. Mackintosh, S.G. Cooper~NuclearPhysics A 589 (1995) 377-394
383
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Fig. 4. Potentials for 18 MeV CM alphas on 160. Upper panel compares full RGM potentials as follows: solid line, fit to S/ with wrong t~t (see text); dashed lines represents the "correct" solution with tr = 0.009, the basis of "subtracted" potentials in other figures. Lower panel: comparing the direct (solid), full RGM (dashed) and Michel et al.'s 32 MeV potential (dotted).
2.5. The full R G M potential
This case requires the same care in avoiding a discrete ambiguity as the IN exchange potential. The potential we find is included, with direct potential subtracted, as the dotted line in the top diagram of Fig. 3. The overall R G M exchange contribution can be seen from this figure to be very close to a sum of the IN and 2N contributions. In Fig. 4, top panel, we show the complete RGM potential, together with the shallow potential which gives the same St = exp2it3t but not the correct ~t (solid line). The inversion gives a moderately low ~r of 0.009 (see Appendix A) and the potential appears to be have small uncertainties, not very apparent on the scale of Fig. 4. Finally, the lower panel of Fig. 4 compares the direct and R G M potentials with the 32 MeV phenomenological potential of Michel et al. [ 19]. There is as much general agreement as could be expected. The relevant M3Y-model potential. [ 17] is very similar to the Michel potential for r ,-~ 4 fm, but is almost 200 MeV deep at r = 0. The M3Y model contains an approximate representation of IN exchange but no representation of 2N exchange.
384
R.S. Mackintosh, S.G. Cooper~Nuclear Physics A 589 (1995) 377-394
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,
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.
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2.6. Does RGM lead to an 1-independent a+160 potential? The possibility of a Majorana component is a relevant phenomenological question (see Section 5). There can be no definitive answer, at least at 18 MeV, until it is understood why the purely Majorana 4N contribution and the Majorana term in the IN contribution, which Fig. 1 suggests have the same sense, have been lost in the overall RGM phase shifts. This may well be due to the problem discussed in Appendix B. What does seem clear is that, as might be expected from the RGM 6t of Fig. 1, the overall RGM potential is smooth, showing no signature of parity dependence.
3. The 3 H e - a interaction at 60 MeV CM
We exploit S(l)-to-V(r) inversion to establish precise potentials corresponding to elaborate RGM calculations of LeMere et al. [5]. These calculations omit the Coulomb potential. There is no representation of channel coupling so IS(l)l -- 1 and the potentials will be real. Lemere et al. present not only the RGM phase shifts corresponding to all exchange processes and phase shifts for the direct (folding model) case, but also phase shifts corresponding to the separate inclusion of one-, two- and three-nucleon exchange as well
R.S. Mackintosh, S.G. Cooper~Nuclear Physics A 589 (1995) 377-394
385
.//7
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r/fm Fig. 6. For 60 MeV CM 3He + 4He, potentials representing ~t of LeMere el al. The solid line represents the direct potential, the dashed line the potential representing the inclusion of IN exchange, and the dotted line is the potential for the M3Y folding model.
as combined ( l + 3 ) - n u c l e o n exchange. In Fig. 5 we plot the phase shifts of LeMere et al. (in degrees) with the phase shifts for the direct (folding) model subtracted for clarity (the direct 80 "~ 130°). We note the evident parity dependence largely due to 3N exchange. For higher l, 3N exchange becomes more significant than the IN attraction and a larger part of the total R G M phase shift. The 3N exchange contribution is evidently a superposition of a Majorana term and a repulsive Wigner term. The 2N exchange contribution is small and confined to lower partial waves. The exchange terms add roughly, but not precisely, linearly, as can be seen if, for example, one compares the 3N exchange term with difference between the ( l + 3 ) N exchange and 1N exchange terms.
3.1. The direct potential
With a reasonably arbitrary SRP (see Appendix A) one easily finds a smooth, very gaussian-like potential. Alternative inversion bases gave the virtually indistinguishable potentials with V(0) ~ 70 MeV. This lends confidence in the overall R G M + inversion procedure. One of these is shown in the Fig. 6; the direct potential should, of course, be smooth. The volume integral is ,-~ 404 MeV.fm 3.
386
R.S. Mackintosh, S.G. Cooper~Nuclear Physics A 589 (1995) 377-394
o
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Fig. 7. Potentialsfor 60 MeV CM 3He + 4He. Top: direct (solid), 3N even-/(dashes) and 3N odd-/(dots). Centre: IN (solid), (I+3)N even-/ (dashes), (I+3)N odd-/ (dots). Bottom: IN (solid), full RGM even-/ (dashes), full RGM odd-/(dots). 3.2. 1N exchange
Fig. 6 also presents the well-determined potential representing phase shifts with 1N exchange included. The 1N exchange contribution is an attractive Wigner potential, larger in magnitude than the difference between the M3Y potential and the direct potential. The M3Y [ 17] model potential is included as a dotted line. The latter is only slightly deeper than the direct potential but similar in geometric properties. 3.3. 3 N exchange
This is the core exchange case. There is no smooth 3 potential corresponding to the highly staggered St of Fig. 5. However, there exist independent potentials reproducing the odd and even phase shifts with o" < 10 -12 in each case, i.e. exact inversions. These are compared with the direct V ( r ) (solid line) in the top panel of Fig. 7. The odd-parity V ( r ) (dotted) is both deeper and longer ranged than the even-parity V ( r ) (dashes), 3 Inversionfor all l leads to a V(r) with oscillationssome 30 MeV in amplitude.
Cooper~NuclearPhysics A 589 (1995) 377-394
R.S. Mackintosh, S.G.
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2-N even-?' 2-N odd-?'
- 10-t
0
-
_.
5 t -q q 10-t4 ] [
0
~
. . . .
--~
I
1
. . . .
I
2
. . . .
- -- -..... - -I
3
. . . .
1+3-N even-?. 1+3-N odd-?' RGM e v e n - ? ' RGM o d d - ? ' I
4
. . . .
[
.5
.... 6
r/fm
Fig. 8. Potentials for 60 MeV CM 3He + 4He, the top two panels with direct potential subtracted, and the bottom panel with the potential including IN exchange subtracted. Top: 2N all-/ (solid), 3N even-/(dashes), 3N odd-/ (dots). Centre: 2N even-/ (solid), 2N odd-/ (dashes). Bottom: ( l + 3 ) N even-/ (solid) and odd-/ (dashed) compared with full RGM even-/(dots) and odd-/ (dot-dashed).
and in this case there is no cross-over at smaller radii. Consistently with the fact that the 3N exchange ~t also exhibit (Fig. 5) a repulsive Wigner term, the difference between the even and direct V(r) is greater than the difference between the odd and direct V(r). The 3N exchange increases the odd-/volume integral by 82 MeV.fm 3 and reduces the even-/ volume integral by 106 MeV.fm 3. The linearity with which exchange terms contribute to the potential can be judged from the central panel of Fig. 7 in which we compare the 1N potential with the even and odd-parity ( l + 3 ) N potentials. The differences are similar but not identical to those in the top panel.
3.4. 2N exchange The 2N exchange makes a small contribution to RGM as can be deduced from the RGM parity-dependent potentials in the bottom panel in Fig. 7 which are very little different from the ( I + 3 ) N potentials in the central panel. The 2N exchange component is shown with the direct potential subtracted in the top and centre panels of Fig. 8. In
388
R.S. Mackintosh,S.G. Cooper~NuclearPhysicsA 589 (1995) 377-394
the top panel we compare the result of a direct/-independent fit to the 2N phase shifts compared to the even and odd-parity terms induced by 3N exchange (shown unsubtracted in Fig. 7, top panel). The oscillation is quite typical of what we expect when fitting 6t which "really" correspond to a parity-dependent potential. We have noted before [ 1,18] that Majorana effects always present the alternative: either a smooth parity-dependent potential or an oscillatory /-independent potential. Fitting the 2N odd-/and even-/ 6t separately we find the smooth parity-dependent V(r) of the central panel of Fig. 8. It is typical that the parity-dependent representation should be of much shorter range than is required of a parity-independent V(r). 3.5. (l +3)N exchange, 2N exchange and linearity If the contributions of individual exchange terms are not too large then they should contribute in a roughly linear fashion when added, the two-potential formula of GellMann and Goldberger prohibiting exact linearity. We have already noted in connection with Fig. 7 that the system is rather linear. The bottom panel of Fig. 8 compares the final even and odd RGM potentials (direct subtracted) with the corresponding ( l + 3 ) N potentials with the potential including IN exchange subtracted. If 2N exchange contributed linearly, the differences in the lower panel would be just the 2N potentials of the central panel. The agreement is remarkably close. This argues for the precision and unambiguity of the inversion procedure. 3.6. The full RGM potential Figs. 7 and 8 have already presented the full RGM potential. It is unquestionably strongly parity dependent, with a geometric parity dependence in which the o d d - / V ( r ) is of longer range than the even-I V(r). Indeed, from 3 fm outwards, it is some 70% or more deeper. There is a cross-over near 1 fm, and the even-I potential is slightly greater at r = 0. In the surface, the M3Y model potential falls between the two RGM components, but overall is less deep. The volume integrals of the odd and even potentials are, respectively, ~ 546 and ~ 378 MeV.fm 3. The respective RMS radii are 2.93 and 2.59 fm (cf. 2.80 for the direct) reflecting the marked geometric parity dependence.
4. t a scattering at 20 MeV CM
Fujiwara and Tang [20] performed elaborate multiconfiguration RGM calculations for the ta system at 20 MeV, CM. Here, we present the results of inverting the St ("TC case") deduced from Fig. 14 of that reference, with the limitations consequent on this means of defining the St to be inverted. In Fig. 9, we present the real and imaginary potentials which give precise fits to Sl of Fujiwara and Tang. Because of possible uncertainties related to the way we obtained St, we repeated the inversion several times with alternative SRP and inversion bases
R.S. Mackintosh, S.G.
Cooper~NuclearPhysics A 589 (1995) 377-394
389
-20-
> o
/"
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/z
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odd-g -80-
even-~ even-~
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~=0.043 cr=O.O007
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. __ .'[
t%
/ /
-1,5/, -2.0
. . . .
0
Imag. V(r)
/ i
1
. . . .
i
2
. . . .
i
3
. . . .
~
4
. . . .
i
. . . .
5
r/fm Fig. 9. For 20 M e V CM tritons scattering from 4He, treated as spinless and chargeless, potentials corresponding
to the RGM phase shifts o f Fujiwara and Tang. Upper: real part, o d d - / ( s o l i d ) and even-/o- = 0.043 ( d a s h e s ) and o" = 0.0007 ( d o t s ) . Lower, the imaginary potential with same line conventions.
to eliminate the possibility that the visible features in this figure were artifacts. Two alternative V(r) for even-/ give an indication of the uncertainty in V(r). The conspicuous feature is the strong geometric parity dependence in both the real and imaginary terms. The odd-/real term is of strikingly long range as measured not only against the even-/term, but also the M3Y model V(r) (not shown) which is almost identical to the even-/ V(r) in the surface (although somewhat weaker for r < 2.5 fro, being 78 MeV at r = 0.) The odd-parity imaginary term is of surface character, while the even-parity component is essentially of volume form. This may be the first indication of what form of parity dependence of the imaginary component in a nucleus-nucleus potential follows from calculated exchange processes.
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R.S. Mackintosh,S.G. Cooper~NuclearPhysicsA 589 (1995)377-394
5. Relation to other work
5.1. Systematics of 3He (or t) +4He potentials Cooper [ 3 ] showed by inversion of single-channel RGM Sty and bound-state energies that the 3He+4He potential below the inelastic threshold had a parity-dependent radial form in which the odd-parity component was of greater range and volume integral than the even-parity component. As noted in the introduction, this same property is found for both the microscopic and phenomenological nucleon-a potential. The results of Sections 3 and 4 above allow us to confirm that a similar parity dependence in the radial form is a general feature of RGM derived potentials for this system. We can also draw tentative conclusions concerning the energy dependence. The possible danger in doing this lies in the fact that the different RGM calculations included different terms. For example, Fujiwara and Tang's 20 MeV calculation included reaction channels generating an imaginary term 4. The higher-energy cases both omitted spin and Coulomb effects, excluding the possibility of deducing spin-orbit terms (readily possible with IP inversion) and eliminating any distinctions between tritons and helions. One apparent energy dependence emerges: at the lowest energy [ 3 ], the odd- and evenparity potentials cross over, so that the even-parity potential is much deeper near r = 0. However, this might be modified by distortion effects in the l = 0 channel. At 60 MeV, Figs. 6 and 7 show that there is little cross-over so the even-parity potential is just marginally deeper at r = 0; the 20 MeV case is intermediate with a cross-over near 2.25 fm, but the even-parity term is less than 10 MeV deeper than the odd-parity term at r = 0 . As we mentioned, the above survey includes both single-channel and coupled-channel RGM but at different energies. We have also inverted St for 3He on 4He at 35 MeV for both single-channel and coupled-channel RGM [21 ]. We find that the channels included have a much smaller effect on the odd and even potentials than the difference between the odd and even potentials so that none of the conclusions about parity dependence are modified. The resulting potentials were intermediate between the 20 and 60 MeV cases, there being a crossover near 1.4 fm, with the odd-parity potential considerably deeper than the real potential for r > 3 fm.
5.2. ol+160 potentials Published fits [ 19] to the wide angular range cross section data for a + 1 6 0 have xZ/N ~ 150. We can reduce xZ/N by an order of magnitude, but only with systematic "wiggly" features in the potentials. It is well known that one can find wiggly potentials which give the same St as smooth but /-dependent (not necessarily parity-dependent) potentials. The question then arises as to whether the local potentials which precisely 4 It would be very interesting to study the relationship between the real and imaginaryterms generated by different and cumulativereaction channels.
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fit the data can be interpreted in terms of/-dependence in the underlying interaction. Firm knowledge of the manner in which different exchange contributions contribute to the effective local potential is a necessary precondition to answering such questions. Although we have shown that individual exchange terms have Majorana components, the overall RGM phases shifts and consequent potential do not have this property for reasons we do not understand.
5.3. Comparison with predicted behaviour Baye [7] makes general predictions concerning how both the strength and sign of the parity dependence are related to the particular interacting nuclei. The fact that core exchange is expected to give a deeper potential for even partial waves in the case of o~ + 160 is borne out by Fig. 3, lower panel, although not at the nuclear centre. The opposite prediction for the mass-7 system is also clearly borne out as far as volume integrals and the behaviour at larger radii is concerned. Nevertheless, core exchange is not the full story as far as parity dependence is concerned. Moreover, it is very unclear how to interpret the consistent geometric parity dependence involving quite large radii and partial waves in view of the suggestion in Ref. [7] that parity-dependence effects are restricted to small/-values; we note that parity dependence related to core exchange persists up to l = 11 in the case of 60 MeV 3He + o~. Of course, it can be argued that the our geometry dependence is somehow an artifact of the requirement that we find a smooth V(r) to fit certain sets of t3l. On the other hand, we argue that it is after all smooth potentials which are relevant to phenomenology. Moreover, these smooth potentials are not forced. They generally correspond to very low values of tr and are not subject to ambiguities. Indeed, it is far from a priori obvious that such smooth parity-doublet potentials exist to fit the patterns of t3t predicted by RGM.
5.4. Ambiguity of nN decomposition In Section 2 and in Appendix B we make reference to the fact that decomposition into nN terms is ambiguous and in some degree controversial. Although only the noexchange and full-exchange results are fully unambiguous, the present work suggests that the conventional decomposition of Tang and his co-workers can be useful. It is striking that the full-exchange potential for 3He on 4He at 60 MeV is indeed a linear sum of the of nN terms, while the linearity is less complete for ot on 160. From our point of view there is interest in establishing nN exchange potentials since they throw light on most calculations of nucleus-nucleus potentials (e.g. M3Y) which omit all exchange terms except 1N exchange and always get purely Wigner potentials. In this light, the large 2N contribution for ot on L60, which does indeed appear to add linearly into the total exchange potential, is noteworthy, as is the fact that the 1N exchange is pure Wigner for 3He on 4He at 60 MeV. Conversely, the Majorana component visible in the 1N exchange for ce on 160 probably does reveal the limits of this decomposition.
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6. General conclusions; future work
We have demonstrated that accurate inversion can provide a way of unlocking the detailed information concerning nuclear interactions which resides in the elastic scattering phase shifts calculated by, for example, RGM. WKB methods and Born approximation methods cannot be relied upon at low energies. It is particularly in the low-energy regime where subtle threshold and partial-wave dependence effects occur. The potentials produced by inversion are immediately of a form that can be related to phenomenology. In this way phenomenology can be brought to bear on the qualitative features of RGM results. At present RGM must use somewhat schematic NN interactions making direct comparison with experiment problematical. The most immediate question raised by this work is the following: does the apparent lack of Majorana component in the overall RGM potential persist to very low energies? The future importance of these studies lies in the fact that RGM can, in principle at least, represent processes which are generally omitted in reaction calculations [22]. For example there is evidence (e.g. Ref. [23] ) that rearrangement channels contribute strongly to nucleon-nucleus scattering. But the difficulty of including exchange processes and non-orthogonality effects within the usual reaction theory framework make such rearrangement contributions hard to establish decisively. RGM plus inversion might thus contribute to understanding the single-nucleon interaction with nuclei; it is quite likely that the complex nucleon-nucleus interaction is /-dependent in a way quite different from a simple Majorana dependence and this should be established or refuted. The usefulness of one-body (optical model) potentials in nuclear physics, whether for nucleons or composite particles, has been a continuing surprise. One limit was apparently reached with cases where exchange processes made large contributions which did not seem naturally to fit into a single-particle picture (e.g. 12C + 13C [24] ). By generalizing to potentials with separate geometries for odd and even parity it seems that we have far from reached the limit of simple optical-type models. Specific findings of the present work concerning nuclear interactions include: (i) Further evidence for a systematic Majorana shape term in nucleus-nucleus potentials due to exchange. (ii) First indications of the energy dependence of the shape effect. (iii) First indications of a strong parity dependence in the radial form of the imaginary part generated by reaction channels. (iv) Confirmation of the strong attractive and short range nature of the 1N contribution to nucleus-nucleus potentials. (v) Evidence that the 2N exchange contributes repulsively to a-nucleus potentials. This term is omitted from M3Y and other folding models. (vi) Evidence for the degree of linearity with which exchange components contribute to the overall potential. For alpha + 160, there is indeed an exception to the general linearity and this is apparently related to alternative formulations of RGM, see Appendix B. Concerning inversion, we note that the IP method employed here has, as well as the
R.S. Mackintosh, S.G. Cooper~Nuclear Physics A 589 (1995) 377-394
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specific advantages explained elsewhere, the ability to handle the case where the St do not contain all the information concerning the potential, specifically the fact that a group of the St may correspond to ~t with an extra zr-phase.
Acknowledgements We are most grateful to the EPSRC of the UK for grant GR/H00895 supporting Dr. Cooper. We are also grateful to Professor Tang for his support. We would like to thank the anonymous referee for bringing the work of Baldock et al. to our attention.
Appendix A. Summary of IP inversion The iterative perturbative method for S(l)-to-V(r) inversion has been described many times, e.g. Refs. [ 10,11,14], and we merely mention certain aspects of it to which we refer in the text. Being an iterative method, the inversion is initiated from some "starting reference potential", SRP; it is part of our procedure to ensure that the final results are stable against variations in the SRE The quality of the inversion is measured in terms of o-, the "phase shift distance", defined by o-2 = ~
I~ - SII2,
(A.1)
I
where ~ is the "target" S-matrix for which we seek the potential, and ~ is the S-matrix of the inversion potential.
Appendix B. Classification of exchange terms The classification of exchange terms adopted by Tang and co-workers is based on a formalism in which the complete antisymmetrization of the system is represented by the complete antisymmetry of the ket in the exchange integrals. The formalism of Baldock et al. antisymmetrizes both bra and ket with the consequence that IN, 2N, ... exchange terms have little correspondence with what Tang et al. designate in this way. As Tang [25] says " . . . in our discussion, the term nucleon exchange is defined through the kernel function while in the discussion of Baldock et al. it is defined through the wavefunction". The formalism of Baldock et al. renders characterization into Wigner and Majorana terms not very useful and may not offer much practical advantage of simplification [26]. However, the fact that a Majorana term appears in intermediate components and yet disappears in the complete potential in the Tang formalism is probably related to the instabilities that were the motivation of the approach of Baldock et al.
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References I 11 S.G. Cooper and R.S. Mackintosh, Phys. Rev. C 43 (1991) 1001. [2] S.G. Cooper, R.S. Mackintosh, A. Cs~o6 and R.G. Lovas, Phys. Rev. C 50 (1994) 1308. [3] S.G. Cooper, Phys. Rev. C 50 (1994) 359. 141 S.G. Cooper and R.S. Mackintosh, Nucl. Phys. A 517 (1990) 285. 151 M. LeMere, D.J. Studeba, H. Horiuchi and Y.C. Tang, Nucl. Phys. A 320 (1979) 449. [6] M. LeMere, Y. Fujiwara, Y.C. Tang and Q.K.K. Liu, Phys. Rev. C 26 (1982) 1847. I71 D. Baye, Nucl. Phys. A 460 (1986) 581. [8[ H. Horiuchi, in Clustering aspects of nuclear structure, eds. J.S. Lilley and M.A. Nagarajan (Reidel, Dordrecht, 1985). [91 T. Wada and H. Horiuchi, Prog. Theor. Phys. 80 (1988) 488. [ 101 S.G. Cooper and R.S. Mackintosh, Inv. Probl. 5 (1989) 707. I I 1 ] S.G. Cooper and R.S. Mackintosh, Users manual for IMAGO, Open University preprint OUPD9201. [12] A. Adachour, D. Baye and P. Descouvement, Nucl. Phys. A 579 (1994) 305. [ 13 ] F.R. Lichtenth~iler, A.C.C. Villari, A. LEpine-Szily and L.C. Gomes, Phys. Rev. C 44 (1991) 1152. [ 14] S.G. Cooper, M.A. McEwan and R.S. Mackintosh, Nucl. Phys. A 552 (1993) 401. 115] S. Ait-Tahar, R.S. Mackintosh, S.G. Cooper and T. Wada, Nucl. Phys. A 562 (1993) 101. [16] R.A. Baldock, B.A. Robson and R.E Barrett, Nucl. Phys. A 351 (1981) 157. [ 171 G.R. Satchler and W.G. Love, Phys. Reports 55 (1979) 183. [ 181 S.G. Cooper and R.S. Mackintosh, Z. Phys. A 337 (1990) 357. 1191 E Michel, J. Albinski, P. Belery, Th. Delbar, Gh. Gr6goire, B. T~siaux and G. Reidemeister, Phys. Rev. C 28 (1983) 1904. [201 Y. Fujiwara and Y.C. Tang, Phys. Rev. C 31 (1985) 342. [211 Y. Fujiwara and Y.C. Tang, Phys. Rev. C 28 (1983) 1869. [221 K. Wildermuth and Y.C. Tang, A unified theory of the nucleus (Vieweg, Braunschweig, 1977). [231 S.G. Cooper, R.S. Mackintosh and A.A. loannides, Nucl. Phys. A 472 (1987) 101. [24] W. von Oertzen, Nucl. Phys. A 148 (1970) 529. 125] Y.C. Tang, in Lecture notes in physics, vol. 145, eds. T.T.S. Kuo and S.S.M. Wong (Springer, Berlin, 1981) p. 572. [26 ] Y.C. Tang, private communication.