Multi-step processes in heavy-ion induced single-nucleon transfer reactions
aa r X i v : . [ nu c l - e x ] J u l Multi-step processes in heavy-ion induced single-nucleon transferreactions
N. Keeley, ∗ K. W. Kemper,
2, 3 and K. Rusek National Centre for Nuclear Research,ul. Andrzeja So ltana 7, 05-400 Otwock, Poland Department of Physics, Florida State University, Tallahassee, Florida 32306, USA Heavy Ion Laboratory, University of Warsaw,ul. Pasteura 5a, 02-093 Warsaw, Poland
Abstract
It was first noted during the 1970s that finite-range distorted wave Born approximation (FR-DWBA) calculations were unable satisfactorily to describe the shape of the angular distributionsof many single-proton (and some single-neutron) transfer reactions induced by heavy ions, withcalculations shifted to larger angles by up to ∼ ◦ compared with the data. These reactions ex-hibited a significant mismatch, either of the reaction Q value or the grazing angular momentumof the entrance and exit channels, and it was speculated that the inclusion of multi-step trans-fer paths via excited state(s) of the projectile and/or ejectile could compensate for the effect ofthis mismatch and yield good descriptions of the data by shifting the calculated peaks to smallerangles. However, to date this has not been explicitly demonstrated for many reactions. In thiswork we show that inclusion of the two-step transfer path via the 4.44-MeV 2 + excited state ofthe C projectile in coupled channel Born approximation calculations enables a good descrip-tion of the
Pb( C, B) Bi single-proton stripping data at four incident energies which couldnot be described by the FR-DWBA. We also show that inclusion of a similar reaction path forthe
Pb( C, C) Pb single-neutron pickup reaction has a relatively minor influence, slightlyimproving the already good description obtained with the FR-DWBA. ∗ Corresponding author: [email protected] . INTRODUCTION When reactions induced by heavy ions were first extensively studied in the 1970s, itwas found that while the single-neutron transfer data could usually be well described byfinite-range distorted wave Born approximation (FR-DWBA) calculations, in many systemsthe single-proton transfer data showed significant angular shifts compared to the calcula-tions. A good example of this phenomenon was seen in the C +
Pb system wherethe
Pb( C, C) Pb single-neutron pickup data were well described by FR-DWBA cal-culations, whereas similar calculations for the
Pb( C, B) Bi single-proton strippingshowed a progressively greater shift to larger angles (typically a few degrees) compared tothe data as successively higher-lying levels in
Bi were populated [1].Data for three near-barrier energies, 77.4, 97.9 and 116.4 MeV, are presented in Ref. [1].For the proton-stripping reaction the discrepancy between FR-DWBA calculations and dataincreases as the Q value becomes more negative, i.e. as more highly excited states of the Bi residual are populated, or the bombarding energy is reduced, while for the neutronpickup reaction the data are well described by FR-DWBA calculations for all states of the
Pb residual at all bombarding energies. Data for the same reactions at a bombardingenergy of 101 MeV exhibit similar behavior [2].The difference in behavior of the proton stripping and neutron pickup reactions in the C +
Pb system may be understood by means of the concept of “matching.” For systemsinvolving heavy ions as projectiles at energies close to the Coulomb barrier the Sommerfeldparameter is large and a semiclassical picture based on Rutherford trajectories should hold.Within this view a transfer reaction will only take place with appreciable probability if thetrajectories before and after the transfer are continuous. Imposing this condition leads tomatching criteria for the reaction Q value, see e.g. Buttle and Goldfarb [3]. A more detailedunderstanding of matching requirements may be obtained by considering continuity of linearand angular momenta in the initial and final trajectories and in a seminal article Brink [4]gave matching conditions for the angular momentum transfer and Q value based on thesecriteria which should be satisfied if the transfer probability is to be large.These matching conditions lead to the concept of the “Q window” for heavy ion reactions,whereby those levels with Q values within a few MeV of the optimum value are seen to bepreferentially populated. Transfer to a given level must also satisfy the angular momentum2atching requirements in that the difference between the grazing angular momenta of theincoming and outgoing trajectories should closely match the allowed angular momentumtransfers. In Ref. [1] it was noted that the discrepancy between the FR-DWBA calculationsand the data for the
Pb( C, B) Bi reactions was correlated with the degree of mis-match between the grazing angular momenta in the entrance and exit channels; in otherwords, the angular momentum mismatch tended to be larger for the population of levels in
Bi at higher excitation energies and lower incident C energies, exactly where the dis-crepancy between FR-DWBA calculations and data was largest. The
Pb( C, C) Pbreaction satisfies the matching requirements rather better, hence the FR-DWBA was ableto provide a good description of these data.It was speculated in both Refs. [1] and [2] that the shift in angle of the FR-DWBAcalculations compared to the
Pb( C, B) Bi data could be due to the effects of couplingsto inelastic channels, particularly of the lighter reaction partner. However, this has not todate been explicitly demonstrated. Given the renewed interest in reactions induced by lightheavy ions in connection with the availability of radioactive beams it seems timely to revisitthe existing data for stable systems where these problems occur using more sophisticatedreaction models. In this work we show that it is possible to obtain a good descriptionof the
Pb( C, B) Bi data at all four bombarding energies without the need for adhoc adjustments of the optical potentials by including the two-step transfer path via the4.44-MeV 2 +1 state of C in coupled channel Born approximation (CCBA) calculations.We also demonstrate that the inclusion of this reaction path in similar calculations for the
Pb( C, C) Pb reaction has a relatively small effect, slightly improving the alreadygood description of these data by FR-DWBA calculations.
II. CALCULATIONSA. The
Pb( C, B) Bi proton stripping reaction
The data sets of Refs. [1] and [2] were reanalyzed. A series of CCBA calculations of the
Pb( C, B) Bi proton stripping reaction including the two-step transfer path via the4.44-MeV 2 + excited state of C were performed with the code fresco [5], employing thepost form of the DWBA formalism and incorporating the full complex remnant term. Inputs3 C energy (MeV) V (MeV) r (fm) a (fm) W (MeV) r W (fm) a W (fm)77.4 52.2 1.256 0.56 5.02 1.256 0.79297.9 49.9 1.256 0.56 11.7 1.256 0.406101.0 51.1 1.256 0.56 13.3 1.256 0.415116.4 42.0 1.256 0.56 20.0 1.256 0.406TABLE I. Entrance channel C +
Pb Woods-Saxon potential parameters used in the CCBAcalculations. The heavy-ion radius convention, R i = r i × (A / p + A / t ) fm was used and r C = 1 . were similar to the FR-DWBA calculations of Ref. [1]. The most extensive data set is thatat a C bombarding energy of 97.9 MeV so this was used as a test case. In the entrancepartition, coupling to the 4.44-MeV 2 +1 state of C was included using standard rotationalmodel form factors for an oblate quadrupole deformation. The B ( E
2; 0 + → + ) was takenfrom Ref. [6] and the nuclear deformation length, δ = − .
40 fm, from Ref. [7]. The opticalpotential was of Woods-Saxon form and the parameters were adjusted so that the coupledchannel calculation gave the same elastic scattering angular distribution as an optical modelcalculation using the C +
Pb parameters listed in Table I of Ref. [1]. The resultingvalues are given in Table I. The five-parameter B +
Bi Woods-Saxon potential fromTable I of Ref. [1] was employed in the exit channel. This was obtained by fitting B +
Bi elastic scattering data at an appropriate energy (74.6 MeV). The B + p and Pb + p binding potentials were taken from Ref. [1].Two sets of spectroscopic amplitudes for the h C(0 + ) | B + p i and h C(2 + ) | B + p i overlaps were tested, those of Cohen and Kurath [8] and those of Rudchik et al. [9]. Thevalues are given in Table II. The spectroscopic amplitudes for the h Bi | Pb + p i overlapswere adjusted to give the best description of the stripping cross sections. The small 1 p / component of the h C(2 + ) | B + p i overlap predicted by Cohen and Kurath [8] was omittedfrom the calculations since tests found that it had no discernible effect on the results. SinceCohen and Kurath [8] give values for the spectroscopic factors, which are positive definite,and we require spectroscopic amplitudes, which may be positive or negative, for our CCBAcalculations, both positive and negative relative signs of the spectroscopic amplitudes for4 C x nℓ j S x C B p p / C ∗ .
44 11 B p p / − . a [8], − .
505 [9]1 p / ± . b [8], 0 .
505 [9] a sign determined empirically b Not used, sign not determined
TABLE II. Spectroscopic amplitudes for the (cid:10) C | B + p (cid:11) overlaps used in the CCBA calcu-lations. Amplitudes S x for the A = C + x overlaps are given, where the nℓ j are the quantumnumbers for the relative motion of x about the core C. Note that the signs of the amplitudes fromRef. [9] are given here according to the phase convention used in the fresco code [5]. These differfrom the signs given in Ref. [9] by the following factor: ( − J C + j − J A , where J C is the spin of thecore, C, J A the spin of the composite, C + x = A , and j is the total relative angular momentumof x with respect to C . the h C(0 + ) | B + p i and h C(2 + ) | B + p i overlaps were tested. However, there was aclear preference for a negative relative sign, since a positive sign was found to shift the peaksin the calculated stripping angular distributions to larger angles, worsening the agreementwith the data; a negative sign shifted the calculated peaks to smaller angles. Rudchik et al. [9] give the spectroscopic amplitudes so these were used directly, although since the phaseconvention adopted in Table 2 of Ref. [9] is different from that of fresco the signs first hadto be converted. The signs in Table II follow the fresco convention, and the conversionfactor is given in the caption.Different choices of B +
Bi optical potential parameters, e.g. those of Ref. [10], didnot improve the agreement between the CCBA calculations and the stripping data. Thespectrum presented in Ref. [1] shows that transitions to levels of
Bi leaving the B in its2.12-MeV 1 / − excited state were weakly populated—almost on the level of background—and inelastic couplings between the levels of B in the exit partition were therefore omittedfrom our calculations.The calculations at the other energies used identical inputs with two exceptions: theentrance channel optical potential parameters and the spectroscopic amplitudes for the h Bi | Pb + p i overlaps. The entrance channel optical potential parameters at 77.4 MeV5ere obtained by fitting the 76.5 MeV C +
Pb elastic scattering data of Rudakov et al. [11] and at 116.4 MeV by fitting the 118 MeV elastic scattering data of Friedman et al. [12].At 101 MeV the entrance channel potential was obtained by fitting the elastic scatteringangular distribution calculated using the optical model and the parameters of Ref. [1] witha coupled channel calculation. The resulting parameters are given in Table I. The spectro-scopic amplitudes for the h Bi | Pb + p i overlaps were adjusted at each energy to givethe best description of the relevant stripping data. B. The
Pb( C, C) Pb neutron pickup reaction
The data sets of Refs. [1] and [2] for this reaction were also reanalyzed by means of CCBAcalculations including the two-step transfer path via the 4.44-MeV 2 + excited state of C,and employing the prior form of the DWBA formalism with the full complex remnant term.Inputs were similar to the corresponding FR-DWBA calculations of Ref. [1] and the entrancechannel potentials and coupling strengths were identical to those described in the previoussubsection. The five-parameter C +
Pb Woods-Saxon potential of Table I of Ref. [1]was used in the exit channel. This was obtained by fitting C +
Pb elastic scatteringdata at an appropriate energy, in this case 86.1 MeV. The C + n and Pb + n bindingpotentials were as used in Ref. [1].Two sets of spectroscopic amplitudes for the h C | C(0 + ) + n i and h C | C(2 + ) + n i overlaps were tested, those of Cohen and Kurath [8] and those of Ziman et al. [13]. Thevalues are given in Table III. The relative signs of the Cohen and Kurath [8] spectro-scopic amplitudes for the h C | C(0 + ) + n i and h C | C(2 + ) + n i overlaps were deter-mined empirically, as described above for the corresponding spectroscopic amplitudes for the Pb( C, B) Bi proton stripping reaction. In this case the best description was obtainedwith a positive sign, a negative sign being found to destroy the already good description ofthe neutron pickup data obtained with the FR-DWBA. Ziman et al. [13] give values for thespectroscopic amplitudes which were used directly, after converting the signs to the phaseconvention used by fresco , as described in the previous subsection. The spectroscopicamplitudes for the h Pb | Pb + n i overlaps were adjusted at each incident C energy togive the best description of the relevant pickup data.6 C x nℓ j S x C C n p / C C ∗ . n p / a [8], 1.124 [13] a sign determined empirically TABLE III. Spectroscopic amplitudes for the (cid:10) C | C + n (cid:11) overlaps used in the CCBA calcu-lations. Amplitudes S x for the A = C + x overlaps are given, where the nℓ j are the quantumnumbers for the relative motion of x about the core C. Note that the signs of the amplitudes fromRef. [9] are given here according to the phase convention used in the fresco code [5]. These differfrom the signs given in Ref. [9] by the following factor: ( − J C + j − J A , where J C is the spin of thecore, C, J A the spin of the composite, C + x = A , and j is the total relative angular momentumof x with respect to C . III. RESULTSA. The
Pb( C, B) Bi proton stripping reaction
In Figs. 1 and 2 we compare the results of CCBA calculations employing the cou-pling scheme described in section II A with the data for the
Pb( C, B) Bi reactionat 77.4 MeV [1], 97.9 MeV [1], 101 MeV [2], and 116.4 MeV [1]. The calculations usingthe C → B + p spectroscopic amplitudes of Cohen and Kurath [8] are denoted by thesolid curves and those using the corresponding spectroscopic amplitudes of Ref. [9] by thedashed curves. For most levels the results are virtually indistinguishable, exceptions beingstripping to the 3.12-MeV 3 / − and, to a lesser extent, the 2.84-MeV 5 / − state, wherethe spectroscopic amplitudes of Ref. [9] give an improved description of the data. Whilethe goal of this work is to investigate the proton stripping reaction mechanism rather thanextract proton spectroscopic factors for the h Bi | Pb + p i overlaps we note that thevalues obtained using the projectile-overlap spectroscopic amplitudes of Ref. [9] show onlyminor variations from those obtained with the Cohen and Kurath amplitudes. Also shownare the results of DWBA calculations using the parameters of Ref. [1]. The agreement of theCCBA calculations with the data is much better than that of the DWBA at all energies andfor all levels of Bi. Depending on the bombarding energy and the particular level of the7 -1 -1 -1 -1 -1 d σ / d Ω ( m b / s r) -1 -1 -1
50 60 70 80 90 100 θ c.m. (deg) -1
20 30 40 50 60 7010 -1 (a)(b)(c)(d)(e) (f)(g)(h)(i)(j) FIG. 1. Calculations for the
Pb( C, B) Bi reaction at bombarding energies of 77.4 MeV(left) and 97.9 MeV (right) compared with the data of Ref. [1]; (a) to (e) denote stripping to the0.0-MeV 9 / − , 0.90-MeV 7 / − , 1.61-MeV 13 / + , 2.84-MeV 5 / − , and 3.12-MeV 3 / − states of Bi at a bombarding energy of 77.4 MeV and (f) to (j) at a bombarding energy of 97.9 MeV.The solid curves denote the results of CCBA calculations using projectile-overlap spectroscopicamplitudes derived from Cohen and Kurath [8]. The dotted curves denote the results of DWBAcalculations using the parameters of Ref. [1]. The dashed curves denote the results of CCBAcalculations using the projectile-overlap spectroscopic amplitudes of Ref. [9].
Bi residual involved, the inclusion of the two-step transfer path in the CCBA calculationsimproves the shape of the angular distribution and/or shifts the peak to smaller angles. Theoverall agreement of the CCBA calculations with the data is good, with the exception ofthat for population of the 3.12-MeV 3 / − state at all bombarding energies, the angular shiftof the calculated peaks being not quite enough to match the data for stripping to this state.8 -1 -1 -1 d σ / d Ω ( m b / s r) -1
20 30 40 50 θ c.m. (deg)
20 30 40 50 60 70 θ c.m. (deg) -1 (a)(b)(c)(d)(e) (f)(g)(h)(i) FIG. 2. Calculations for the
Pb( C, B) Bi reaction at bombarding energies of 101 MeV (left)and 116.4 MeV (right) compared with the data of Refs. [2] and [1], respectively; (a) to (e) denotestripping to the 0.0-MeV 9 / − , 0.90-MeV 7 / − , 1.61-MeV 13 / + , 2.84-MeV 5 / − , and 3.12-MeV3 / − states of Bi at a bombarding energy of 101 MeV and (f) to (i) at a bombarding energyof 116.4 MeV. The solid curves denote the results of CCBA calculations using projectile-overlapspectroscopic amplitudes derived from Cohen and Kurath [8]. The dotted curves denote the resultsof DWBA calculations using the parameters of Ref. [1]. The dashed curves denote the results ofCCBA calculations using the projectile-overlap spectroscopic amplitudes of Ref. [9].
A comparison of the spectroscopic amplitudes for the various h C | B + p i overlaps inTable II reveals that while the values for the overlap linking the ground states of the coreand composite nuclei do not differ significantly between the two calculations, those linkingthe 4.44-MeV 2 + excited state of the C composite with the ground state of the B coreexhibit a substantial difference in the 1 p / components; that of Cohen and Kurath [8] beingnegligible whereas that of Ref. [9] is the same magnitude (but opposite sign) as that of9he 1 p / component. Nevertheless, both sets give equivalent descriptions of the ( C, B)proton stripping data, with a slight preference for the values of Ref. [9]. Therefore, althoughit is clear that the two-step transfer path via the 4.44-MeV 2 + state of C is essential to agood description of the proton stripping data the reaction mechanism is not sensitive to theprecise nature of the associated projectile-like overlap.
B. The
Pb( C, C) Pb neutron pickup reaction
The results of CCBA calculations employing the coupling scheme described in sectionII B are compared with the data for the
Pb( C, C) Pb neutron pickup reaction at 77.4MeV [1], 97.9 MeV [1], 101 MeV [2], and 116.4 MeV [1] in Figs. 3 and 4. The calculationsemploying the C → C + n spectroscopic amplitudes of Cohen and Kurath [8] are denotedby the solid curves and those employing the values of Ziman et al. [13] by the dashed curves.Also shown are the results of FR-DWBA calculations using the parameters of Ref. [1],denoted by the dotted curves. The inclusion of the two-step transfer path via the C 4.44-MeV 2 + state in all cases improves the already good description of the data by the FR-DWBAcalculations, broadening slightly the peaks of the angular distributions, although the effectis by no means as important as for the proton stripping reaction. Both sets of C → C + n spectroscopic amplitudes give similar results, although use of the Ziman et al. [13] valuesconsistently leads to larger spectroscopic amplitudes for the h Pb | Pb + n i overlaps,unlike for the Pb( C, B) Bi reaction where both sets of C → B + p spectroscopicamplitudes yielded similar values for the Bi → Pb + p amplitudes. This merely reflectsthe fact that for the Pb( C, B) Bi reaction the spectroscopic factors (the squaresof the spectroscopic amplitudes given in the tables) for the h C(0 + ) | B + p i overlap ofRefs. [8] and [9] are almost identical, see Table II, whereas for the Pb( C, C) Pbreaction the spectroscopic factor for the h C | C(0 + ) + n i overlap of Cohen and Kurath[8] is approximately a factor of 1.7 greater than that of Ziman et al. [13], see Table III.Both sets of spectroscopic amplitudes give equivalent descriptions of the data so that itis not possible to choose between them, although the Cohen and Kurath amplitudes yieldvalues for the Pb → Pb + n spectroscopic factors that are more consistent with otherdeterminations using light ion reactions, for example.10 -1 -1 d σ / d Ω ( m b / s r) -1 -1 -1
50 60 70 80 90 100 θ c.m. (deg)
20 30 40 50 60 7010 -1 (a)(b)(c)(d)(e) (f)(g)(h)(i)(j) FIG. 3. Calculations for the
Pb( C, C) Pb reaction at bombarding energies of 77.4 MeV(left) and 97.9 MeV (right) compared with the data of Ref. [1]; (a) to (e) denote pickup to the 0.0-MeV 1 / − , 0.57-MeV 5 / − , 0.90-MeV 3 / − , 1.63-MeV 13 / + , and 2.34-MeV 7 / − states of Pbat a bombarding energy of 77.4 MeV and (f) to (j) at a bombarding energy of 97.9 MeV. The solidcurves denote the results of CCBA calculations using projectile-overlap spectroscopic amplitudesderived from Cohen and Kurath [8]. The dotted curves denote the results of DWBA calculationsusing the parameters of Ref. [1]. The dashed curves denote the results of CCBA calculations usingthe projectile-overlap spectroscopic amplitudes of Ref. [13].
IV. SUMMARY AND CONCLUSIONS
In common with many heavy-ion induced proton transfer reactions the data for the
Pb( C, B) Bi proton stripping reaction could not be satisfactorily described with FR-DWBA calculations without ad hoc adjustments to the exit channel optical potentials [1, 2],the calculations tending to peak at larger angles than the data. It was remarked [1] that11 -1 -1 -1 d σ / d Ω ( m b / s r) -2 -1
20 30 40 50 θ c.m. (deg)
20 30 40 50 60 70 θ c.m. (deg) -1 (a)(b)(c)(d)(e) (f)(g)(h) FIG. 4. Calculations for the
Pb( C, C) Pb reaction at bombarding energies of 101 MeV(left) and 116.4 MeV (right) compared with the data of Refs. [2] and [1], respectively; (a) to(e) denote pickup to the 0.0-MeV 1 / − , 0.57-MeV 5 / − , 0.90-MeV 3 / − , 1.63-MeV 13 / + , and2.34-MeV 7 / − states of Pb at a bombarding energy of 101 MeV and (f) to (h) the 0.0-MeV1 / − , 0.90-MeV 3 / − , and 2.34-MeV 7 / − states of Pb at a bombarding energy of 116.4 MeV.The solid curves denote the results of CCBA calculations using projectile-overlap spectroscopicamplitudes derived from Cohen and Kurath [8]. The dotted curves denote the results of DWBAcalculations using the parameters of Ref. [1]. The dashed curves denote the results of CCBAcalculations using the projectile-overlap spectroscopic amplitudes of Ref. [13]. the discrepancy increased as the Q value became more negative (i.e. higher lying statesin Bi were populated) or the bombarding energy was reduced, that is, as the mismatchbetween the grazing angular momenta in the entrance and exit channels increased. It wasspeculated [1, 2] that the shifts in the angular distributions might be due to coupling toinelastic channels, specifically those of the projectile and/or ejectile, but to date this has12ever been demonstrated.In this work we have shown that CCBA calculations including two-step transfer via the4.44-MeV 2 + excited state of C provide a good description of the available data sets. Cou-plings to the ground state reorientation and excitation of the 2.12-MeV 1 / − excited stateof B had a relatively minor influence. The influence of the two-step path becomes moreimportant as the excitation energy of the residual
Bi increases and/or the bombardingenergy decreases, suggesting that this coupling is indeed compensating for the increasedangular momentum mismatch. Two sets of shell model spectroscopic amplitudes for the h C | B + p i overlaps were employed, one derived from Cohen and Kurath [8] and theother taken from the work of Rudchik et al. [9], and both gave equivalent descriptions of thestripping data, with a slight preference for the values of Rudchik et al. . While the amplitudesfor the overlap linking the ground state of the C composite with the ground state of the B core were not significantly different, this was not the case for those linking the excitedstate of the composite with the ground state of the core, which have completely differentweightings for the 1 p / component in the two calculations. We thus conclude that whileinclusion of the two-step transfer path via the C 4.44-MeV 2 + state is essential for a gooddescription of the data the calculations are not sensitive to the details of the accompanyingprojectile-like overlap.It was also demonstrated that similar CCBA calculations for the Pb( C, C) Pbreaction were able to improve slightly the already good description of the data by theFR-DWBA [1, 2], although the influence of the two-step transfer via the 4.44-MeV 2 + excited state was significantly smaller than for the proton-stripping case. Two sets of shellmodel spectroscopic amplitudes for the h C | C + n i overlaps [8, 13] were tested, bothgiving equivalent descriptions of the pickup data, although the values of Ziman et al. [13]yielded somewhat larger spectroscopic amplitudes for the h Pb | Pb + n i overlaps. Thisis commensurate with the ratio of the spectroscopic factors (the squares of the spectroscopicamplitudes) for the h C | C(0 + ) + n i overlap for the two sets, see Table III. We againconclude that the calculations are not sensitive to the details of the projectile-like overlap,with the exception that the relative positive sign of the amplitudes for the h C | C(0 + ) + n i and h C | C(2 + ) + n i overlaps is firmly established; a relative negative sign shifts thetransfer peaks to smaller angles by 1–2 ◦ , destroying the agreement with the data.In Figs. 5 and 6 we compare the optimum angular momentum transfer, L opt , with the13llowed angular momentum transfers and plot the values of Q − Q opt for each of the statespopulated in the Pb( C, B) Bi and
Pb( C, C) Pb reactions, respectively. Valuesof Q opt for the A ( a, b ) B transfer processes were calculated according to the Brink matchingrules [4, 14]: Q opt = ( Z b Z B − Z a Z A ) e /R − mv , (1)where the charge on nucleus i is denoted by Z i e , the relative velocity of the two nuclei inthe region of interaction (separated by distance R ) by v , and the mass of the transferredparticle by m . The relative velocity v may be calculated as [15]: v = [2 ( E c . m . − E B ) /µ ] / , (2)where E B and µ are the Coulomb barrier and reduced mass of the projectile-target system,respectively. The optimum angular momentum transfers were calculated as the differencebetween the grazing angular momenta of the incoming ( L a ) and outgoing ( L b ) trajectories[14]: L opt = | L a − L b | . (3)The grazing angular momenta were taken as those values for which the elastic scattering S matrix has | S ( L a ) | ≈ | S ( L b ) | ≈ / √
2, obtained by smooth interpolation of the S matri-ces calculated using the appropriate optical potentials used in the FR-DWBA calculations.Following Ref. [14] the value of R in eqn. (1) was taken as the strong absorption radius,defined as the distance of closest approach for a Rutherford trajectory with angular momen-tum L = L a . For the Coulomb barrier E B in eqn. (2) we took a value of 57.4 MeV, thebarrier height calculated using the real part of the entrance channel optical potential usedin the FR-DWBA calculations.Figures 5 and 6 show that while the Pb( C, C) Pb neutron pickup is somewhatbetter Q-matched than the
Pb( C, B) Bi proton stripping all the transfers consideredfall within the likely Q windows, with the possible exception of proton stripping populatingthe 2.84-MeV 5 / − and 3.12-MeV 3 / − states of Bi at an incident C energy of 77.4 MeV,see Fig. 5 (n) and (o), respectively. This also applies to transfers via the 4.44-MeV 2 + excitedstate of C (plotted as the dashed curves) where R and v in eqn. (1) were calculated usingthe appropriate lower value of E c . m . . In order to satisfy the angular momentum matchingconditions the value of L opt for transfers leading to a particular final state of the residual14
36 -202036 -4-2048 L op t -4-20 Q - Q op t
369 -6-4-2080 100E lab (MeV)0369 80 100E lab (MeV) 80 100E lab (MeV)-6-4-20(a)(b)(c)(d)(e) (f)(g)(h)(i)(j) (k)(l)(m)(n)(o)
FIG. 5. For the
Pb( C, B) Bi reaction: In (a) to (e) the solid curves plot the optimumangular momentum transfer ( L opt ) values as a function of C bombarding energy for strippingfrom the C ground state and population of the 0.0-MeV 9 / − , 0.90-MeV 7 / − , 1.61-MeV 13 / + ,2.84-MeV 5 / − , and 3.12-MeV 3 / − states of Bi, respectively. The hatched bands denote therange of allowed L transfers for each transition. In (f) to (j) the dashed curves plot the L opt valuesas a function of C bombarding energy for stripping from the C 4.44-MeV 2 + excited state. Thehatched bands denote the range of allowed L transfers when the stripped proton is in the 1 p / level (as for stripping from the C ground state) and the dotted bands when the stripped protonis in the 1 p / level. In (k) to (o) the solid curves plot Q − Q opt for stripping from the ground stateof C and population of the 0.0-MeV 9 / − , 0.90-MeV 7 / − , 1.61-MeV 13 / + , 2.84-MeV 5 / − ,and 3.12-MeV 3 / − states of Bi, respectively while the dashed curves plot Q − Q opt for strippingfrom the C 4.44-MeV 2 + excited state. nucleus should correspond closely to the allowed L transfers for the transition in question.These latter depend on both ℓ and j of the transferred particle with respect to the projectile-like and target-like cores. For proton stripping from the C ground state the proton is ina pure 1 p / level, while for stripping from the 4.44-MeV 2 + excited state it is either in apure 1 p / level (Cohen and Kurath [8]; we ignored the small 1 p / level component of their15
36 -20203 -2004 L op t -202 Q - Q op t
036 -4-2080 100E lab (MeV)03 80 100E lab (MeV) 80 100E lab (MeV)-4-20(a)(b)(c)(d)(e) (f)(g)(h)(i)(j) (k)(l)(m)(n)(o)
FIG. 6. For the
Pb( C, C) Pb reaction: In (a) to (e) the solid curves plot the optimumangular momentum transfer ( L opt ) values as a function of C bombarding energy for pickup tothe C ground state and population of the 0.0-MeV 1 / − , 0.57-MeV 5 / − , 0.90-MeV 3 / − , 1.63-MeV 13 / + , and 2.34-MeV 7 / − states of Pb, respectively. The dotted bands denote the rangeof allowed L transfers for each transition. In (f) to (j) the dashed curves plot the L opt values as afunction of C bombarding energy for pickup to the C 4.44-MeV 2 + excited state. The hatchedbands denote the range of allowed L transfers for each transition. In (k) to (o) the solid curvesplot Q − Q opt for pickup to the ground state of C and population of the 0.0-MeV 1 / − , 0.57-MeV5 / − , 0.90-MeV 3 / − , 1.63-MeV 13 / + , and 2.34-MeV 7 / − states of Pb, respectively while thedashed curves plot Q − Q opt for pickup to the C 4.44-MeV 2 + excited state. wave function in our CCBA calculations) or a mixture of 1 p / and 1 p / levels (Rudchik etal. [11]). For neutron pickup the transferred neutron is in a 1 p / level ( C in its groundstate) or a 1 p / level ( C in its 4.44-MeV 2 + excited state). The resulting ranges of allowed L transfers have been plotted on Figs. 5 and 6 as the hatched (1 p / ) and dotted (1 p / )bands. While these accurately reflect the range of allowed values it should be recalled thatthese are in fact quantized, not continuous.Figure 5 only partially confirms the interpretation of Ref. [1] of the poor description16f the proton stripping data by the FR-DWBA as being due to poor angular momentummatching, since stripping from the C ground state is seen to be well matched at all incident C energies for transitions leading to the 0.0-MeV 9 / − and 1.61-MeV 13 / + states of Bi. However, for transitions to the remaining three levels which are poorly L matchedfor stripping from the ground state of C it is seen that stripping from the 4.44-MeV 2 + excited state is either well matched or better matched, at least partially explaining the muchimproved description of the proton stripping data by the CCBA calculations. The situationis rather less clear cut for the neutron pickup, see Fig. 6. Pickup for C in its ground stateis reasonably well L matched for all transitions except that to the 1.63-MeV 13 / + state of Pb but the description of the neutron pickup by the FR-DWBA is noticeably better thanthe proton stripping even for cases where the L matching is better for the latter. Also, whilefor some transitions pickup for C in its 4.44-MeV 2 + excited state is better L matched thanfor the ground state there appears to be little correlation between this and the importanceof the two-step transfer path. We thus conclude that while the matching conditions providea useful aid to an understanding of the reaction mechanisms for heavy ion transfer data theydo not tell the full story. Nevertheless, the matching concept remains an important tool inhelping to explain why the DWBA fails for certain reactions.Similar discrepancies between the results of FR-DWBA calculations and the data existfor other proton transfer reactions, e.g. the Pb( Li, He)
Pb reaction at 52 MeV [16].In this case the Li projectile has strong ground state reorientation and coupling to itsbound first excited state which may influence the transfer reaction in a similar mannerto the C coupling included in this work. However, the phenomenon is not restricted toproton transfers, since the
Pb( B, B) Pb data of Ref. [14] show a similar angulardisplacement compared to FR-DWBA calculations. This system may well be a case whereinelastic couplings in the ejectile are important, since B is strongly deformed. In lightof our conclusions concerning the spectroscopic amplitudes for the overlaps involving the4.44-MeV excited state of C such analyses should provide fruitful ground for collaborationwith structure theorists, since the apparent lack of sensitivity to the details of the overlapwould seem to require the use of theoretical spectroscopic amplitudes for firm conclusionsto be drawn.In summary, we have demonstrated the importance of two-step transfer via the 4.44-MeV excited state of C in describing the
Pb( C, B) Bi single proton stripping and17onfirmed its relatively minor influence on the
Pb( C, C) Pb single neutron pickupreaction. While the concepts of angular momentum and Q matching proved to be usefulaids to understanding this difference it was demonstrated that they do not provide a com-plete explanation, so that while they can point to cases where multi-step reaction paths areimportant they will not necessarily be able to provide an a priori conclusion as to the partic-ular paths required. Many similar cases exist where the reaction is angular momentum—orpossibly also Q —mismatched and inelastic excitations of the projectile and/or the ejectilemay be important in obtaining a good description of the transfer data. Such cases may besuccessfully handled within the framework of CCBA or coupled reaction channels theoryprovided the necessary spectroscopic amplitudes are available. [1] K. S. Toth, J. L. C. Ford, Jr., G. R. Satchler, E. E. Gross, D. C. Hensley, S. T. Thornton,and T. C. Schweizer, Phys. Rev. C , 1471 (1976).[2] W. von Oertzen, H. Lettau, H. G. Bohlen, and D. Fick, Z. Phys. A , 81 (1984).[3] P. J. A. Buttle and L. J. B. Goldfarb, Nucl. Phys. A , 299 (1971).[4] D. M. Brink, Phys. Lett. B , 37 (1972).[5] I. J. Thompson, Comput. Phys. Rep. , 167 (1988).[6] S. Raman, C. W. Nestor, and P. Tikkanen, At. Data Nucl. Data Tables , 1 (2001).[7] J. Cook, M. N. Stephens, K. W. Kemper, and A. K. Abdallah, Phys. Rev. C , 915 (1986).[8] S. Cohen and D. Kurath, Nucl. Phys. A , 1 (1967).[9] A. T. Rudchik, A. Budzanowski, V. K. Chernievsky, B. Czech, L. G lowacka, S. Kliczewski,A. V. Mokhnach, O. A. Momotyuk, S. E. Omelchuk, V. M. Pirnak, K. Rusek, R. Siudak, I.Skwirczy´nska, A. Szczurek, and L. Zem lo, Nucl. Phys. A , 51 (2001).[10] A. Shrivastava, S. Kailas, P. Singh, A. Chatterjee, A. Navin, A. M. Samant, V. Ramdev Raj,S. Mandal, S. K. Datta, and D. K. Awasthi, Nucl. Phys. A , 411 (1998).[11] V. P. Rudakov, K. P. Artemov, Yu. A. Glukhov, S. A. Goncharov, A. S. Dem’yanova, A. A.Ogloblin, V. V. Paramonov, and M. V. Rozhkov, Bull. Rus. Acad. Sci. Phys. , 57 (2001).[12] A. M. Friedman, R. H. Siemssen, and J. G. Cuninghame, Phys. Rev. C , 2219 (1972).[13] V. A. Ziman, A. T. Rudchik, A. Budzanowski, V. K. Chernievsky, L. G lowacka, E. I.Koshchy, S. Kliczewski, M. Makowska-Rzeszutko, A. V. Mokhnach, O. A. Momotyuk, O. . Ponkratenko, R. Siudak, I. Skwirczy´nska, A. Szczurek, and J. Turkiewicz, Nucl. Phys. A , 459 (1997).[14] J. L. C. Ford, Jr., K. S. Toth, G. R. Satchler, D. C. Hensley, L. W. Owen, R. M. DeVries, R.M. Gaedke, P. J. Riley, and S. T. Thornton, Phys. Rev. C , 1429 (1974).[15] C. Olmer, M. Mermaz, M. Buenerd, C. K. Gelbke, D. L. Hendrie, J. Mahoney, D. K. Scott,M. H. Macfarlane, and S. C. Pieper, Phys. Rev. C , 205 (1978).[16] A. F. Zeller, D. C. Weisser, T. R. Ophel, and D. F. Hebbard, Nucl. Phys. A , 515 (1979)., 515 (1979).