Fusion barrier distribution from measurement of quasielastic scattering at θ c.m. = 180°

Distribution of fusion barriers reveals the effect of structure of the collision partners on the dynamics of fusion between two heavy ions. Experimental barrier distribution can be extracted from precisely measured fusion excitation function. Quasielastic excitation function is related to fusion excitation function by conservation of incident flux. Hence, barrier distribution can also be extracted from quasielastic excitation function, which is usually measured at large angles, by detecting the back-scattered projectile-like ions. However, the reflectance coefficient is exactly complementary to the transmittance coefficient at θ c.m. = 180°, which corresponds to orbital angular momentum, ℓ = 0. We report here extraction of fusion barrier distribution for 16O+142 Ce from measurement of quasielastic scattering at θ c.m. = 180°, by detecting the target-like ions in the forward angles using a recoil mass spectrometer. Quasielastic excitation functions were also measured by detecting the projectile-like ions at two large angles. Barrier distributions extracted from both fusion and quasielastic measurements were found to be nearly identical. This work, thus, provided the first experimental verification of validity of the scaling property and the iso-centrifugal approximation in extracting barrier distribution from quasielastic excitation functions measured at large angles.


Introduction
Fusion between two light nuclei can be described in terms of simple potential models.On the other hand, static deformation and surface vibrations of the collision partners and nucleon transfer channels are known to influence the dynamics of fusion between two heavy nuclei [1,2].Coupling of the relative motion with other degrees of freedom leads to splitting of the single fusion barrier into a continuous distribution of barriers.Barrier distribution (D) had been defined as the second derivative of the energy-weighted fusion cross sections (σ fus ) [3], i.e., Here, E stands for energy available in the centre of mass (c.m.) frame of reference.D fus has been extracted from precisely-measured fusion excitation functions for a large number of reactions over decades.
Similar information about the fusion barrier can also be extracted from measurement of differential quasielastic scattering cross sections, as the two phenomena are related through the conservation of incident flux [4].In this case, barrier distribution is defined as the first derivative of the ratio of differential quasielastic and Rutherford cross sections Here θ c.m. is the scattering angle in the c.m. frame of reference.The barrier distribution, defined by Eq. 2, should be evaluated at θ c.m. = 180 • , which corresponds to orbital angular momentum, ℓ = 0.However, projectile-like ions can not be detected at the extreme backward angles because of practical reasons.One needs to note here that different θ c.m. correspond to different grazing angular momenta (ℓ gr ).Effect of ℓ gr can be corrected by shifting E by an amount equal to the centrifugal potential.Thus, D qel can be obtained at an effective energy 2 ) E by carrying out the measurement at energy E and angle θ c.m. .The difficulty in measuring projectile-like ions at the extreme backward angles can be circumvented by the novel use of a recoil separator [5].Noting the fact that a target-like ion moves in the forward direction in the laboratory frame of reference for each back-scattered projectile-like ion, the former can be detected at the focal plane of a recoil separator.Following the pioneering work of Betts et al. [5], a series of measurements exploited this technique to study multi-nucleon transfer channels [6,7,8,9,10].Recently, Tanaka et al. [11] extracted barrier distributions from measurement of quasielastic excitation functions at θ c.m. = 180 • for a few systems to estimate the optimum projectile energy for synthesizing unknown nuclei in the superheavy mass region.D qel extracted from quasielastic excitation function, measured for ℓ ∼ 0 can directly be compared with D fus extracted from fusion excitation function.The results of large-angle scattering need not be scaled with respect to ℓ in this case.It had been pointed out earlier [12] that the iso-centrifugal approximation, incorporated in coupled-channels calculations, worked well for heavy ion fusion reactions.Its validity in case of quasielastic scattering can only be ensured by carrying out measurements at very large backward angles which correspond to very small values of ℓ.
We carried out simultaneous measurement of fusion and quasielastic excitation functions for a system at ℓ ∼ 0. In addition, quasielastic excitation functions were also measured by detecting back-scattered projectile-like ions at two large angles.Comparison of barrier distributions, extracted from different data sets, offered a unique opportunity to verify validity of the scaling property and the iso-centrifugal approximation while extracting fusion barrier distribution from quasielastic measurements.

The experiment
The experiment was carried out using the Heavy Ion Reaction Analyzer (HIRA) [13].A pulsed beam of 16 O, with a pulse separation of 4 µs, was bombarded on an isotopically enriched 142 Ce film of thickness ∼ 120 µg/cm 2 [14], sandwitched between thin layers of natural carbon ( nat C).Energy of the projectile (E lab ) was varied between 48 and 76 MeV, in steps of 1 MeV.Two silicon detectors, placed at θ lab = 15 • inside the target chamber, were used for monitoring the beam and normalization of cross sections.To reset charge states of the reaction products to equilibrium distribution, a nat C foil of thickness ∼ 10 µg/cm 2 was placed 10 cm downstream the target.Two more silicon detectors were placed at θ lab = 150 • and 138 • inside the target chamber to detect back-scattered projectile-like ions.The HIRA was operated at θ lab = 0 • , with an opening aperture of 5 msr.A multi-wire proportional counter (MWPC) with dimensions of 150 mm × 50 mm was stationed at the focal plane of the HIRA.The MWPC was operated with isobutane gas at a pressure of 5 mbar.The HIRA was tuned alternately for detection of the evaporation residues (ERs) and the target-like recoils (TRs), originating from fusion and quasielastic reactions, at the focal plane at each E lab .Energy loss (∆E) information of the ions was obtained from the cathode of the MWPC.Two separate time-to-amplitude converters (TACs) were set up to get measures of time-of-flight (TOF) of the ERs and TRs through the HIRA.Timing pulse from the MWPC anode, indicating the instant of arrival of the ions at the focal plane, was used as the start signal, whereas the radiofrequency pulse used for generating pulsed beam was used as the stop signal of both TACs.Events of interest were identified at the focal plane of the HIRA from correlation plots between ∆E and TOF.Fig. 1 shows the ∆E -TOF spectrum of the target-like ions recorded at the focal plane at E lab = 64 MeV.

Results and discussion
Fusion cross sections were calculated from measured yield of ERs at the focal plane, following the procedure described in Ref. [15].Transmission efficiency of the HIRA for target-like ions was calculated by a semi-microscopic Monte Carlo code [16].The effective solid angle of the spectrometer was determined experimentally by detecting the target-like ions at the focal plane, at an E lab sufficiently below the Coulomb barrier.This was to ensure validity of Rutherford scattering and absence of transfer channels.Differential quasielastic scattering cross sections at θ c.m. = 180 • were extracted [17] from yields of TRs recorded at the focal plane.Yields of projectile-like ions, measured by the two silicon detectors kept at large backward angles, were also converted to differential quasielastic scattering cross sections following Ref.[18].Measured excitation functions, along with the corresponding barrier distributions, are shown in Fig. 2.
Coupled-channels calculations, using the code ccfull [19,20], were performed to interpret the results.The aim was to reproduce the four sets of data with the same potential parameters and coupling scheme.A real potential in Woods-Saxon form with depth V 0 = 77 MeV, radius parameter r 0 = 1.17 fm and diffuseness parameter a 0 = 0.55 fm was found to be the optimum.For the quasielastic cases, an imaginary potential in Woods-Saxon form was used with depth W = 50 MeV, radius parameter r w = 1.0 fm and diffuseness parameter a w = 0.4 fm.Two phonons of the 2 + vibrational state of 142 Ce with β 2 = 0.125 were coupled to two phonons of 3 − vibrational state of 16 O with β 3 = 0.73 to achieve excellent reproduction of experimental excitation functions and barrier distributions.
Barrier distributions obtained from fusion and quasielastic excitation functions, measured at three angles, matched quite well with each other.This work, thus, demonstrated validity of the scaling property and the iso-centrifugal approximation in determining fusion barrier distribution from quasielastic scattering cross sections measured at large angles.

Figure 1 .
Figure1.Scatter plot between ∆E and TOF of the ions at HIRA focal plane for16 O+ 142 Ce at E lab = 64 MeV.Target-like ions, with higher ∆E and closely-related TOF, are marked.