Scattering of 15 C on 208 Pb at energies near the Coulomb barrier: Study of the experimental device response via the 12 C + 208 Pb scattering

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Introduction: neutron halos
Nuclear systems such as 6 He, 11 Li, 11 Be, 14 Be are known to have an extended neutron distributions: the so-called neutron halos [1,2].This feature occurs when the separation energy of valence neutrons is very low and certainly much smaller than the average binding energy per nucleon in a nucleus, so the valence neutrons can tunnel out of the nuclear potential to large distances with sizable probability.It has been an intense experimental and theoretical activity dedicated to study the existence of halos and their dynamics in reaction processes.The neutron halos produces a pronounced maxima at low excitation energies in the Coulomb dipole strength B(E1), very narrow transverse momentum distributions and large interaction cross-sections when measured at high energies [3].
The dynamics of the halo nuclei scattering at low energies, around the Coulomb barrier, is dominated by the coupling between the elastic channel and collective excitations, neutron transfer and breakup.The angular distributions of the elastic cross section and the core fragments present large sensitivity to these coupling e↵ects, due to the halo configuration.This has been demonstrated in previous studies with light exotic beams of 6 He, 11 Li and 11 Be scattered on heavy targets [4][5][6].The angular distribution of the elastic channel shows strong absorption patterns where the nuclear and Coulomb interference completely disappears.
⇤ e-mail: vicente.garcia@csic.es The 15 C nucleus (T 1/2 = 2.449(5) s) has a low single neutron separation energy S n = 1218.1 (8) keV in comparison to the two-neutron separation energy S 2n = 9394.5(8) keV [7] and the neighbors 14 C (S n = 8176.43keV) 16 C (S n = 4250 keV).The spins and parities of the ground and first excited state at E = 740 keV of 15 C are known to be The halo structure of 15 C has been investigated at relatively high energies in several experiments.The reaction cross section at 83 MeV/u shows an enhancement respect to the neighboring 14,16 C isotopes [8], and the longitudinal momenta of the 14 C fragments after 1n-breakup present a FWHM distribution between 64-70 MeV/c depending of the target that it is narrower than that of the neighbour 14,16 C isotopes, ⇠ 200 MeV/c, but wider than the typical ⇠ 40-50 MeV/c found for the archetype cases [3].These properties have hinted the presence of a 1n-halo configuration in the 15 C nucleus that would be unique in the sense that it can be described with an almost pure s 1/2 ground state wavefunction [8].
On the other hand, the enhancement of the cross section observed at 83 MeV/u is not seen at 950 MeV/u [8].Also nuclear matter radii and density were revisited in [9] obtaining a value of  = R v /R c = 1.81 for the ratio between radius of the valence neutron and core radius distribution, where  > 2 for typical halo nucleus.
To complete the understanding of the role of the halo in 15 C, its dynamical response at energies close to the Coulomb barrier has been studied in IS619 experiment, where, according with [10,11], a scattering dominated by the competition of 1n-stripping and breakup is expected.
A 12 C beam at the same energy per nucleon was measured with the same 208 Pb targets for calibration of the experimental device.

Experimental Methodology
The study of 15 C + 208 Pb elastic scattering was performed at HIE-ISOLDE, CERN.Where the 15 C beam was produced from the impact of 1.4 GeV proton pulses on a CaO production target.The isotope of interest was extracted, mass-separated and post-accelerated to 4.37 MeV/u, which is very close to the Coulomb barrier of the system.
The radioactive beam impacted on a target of 208 Pb with purity higher than 98% and two di↵erent thicknesses; one of 2.1 and the other of 1.2 mg/cm 2 .Using the thicker target, the energy losses increased but favors a higher amount of reactions and more statistics.Using the thin target the statistics decreased but there was less straggling, and better energy resolution.
The experimental setup for IS619 was the global reaction array, called GLORIA [12].It consisted of 6 silicon telescopes, all tangent to a 6 cm radius sphere in which center the reaction target was placed.All telescopes were composed of a 40 µm E detector and a 1 mm E detector.Both detectors had 16x16 strips, which means that they worked as a set of 256 pixels each with an angular resolution between 2 and 3º.Dynamic ranges were adjusted to around 50 MeV to detect the scattered 15  C ions at ⇠ 65 MeV lab.energy.The system had an angular coverage from 15 to 165º in a continous way with some overlapping areas between pairs of telescopes.The geometric efficiency was 25% of 4⇡.The detector resolution was in the order of 30 keV FWHM.
A scheme of this configuration is presented in Fig. 1 as well as the nominal angular range covered by each detector in Tab. 1.The target ladder was placed with a 30º tilt respect to the beam direction, this avoids shadowing at 90º but introduces an asymmetry in the energy losses through the azimutal angle.The telescope C did not work during the experiment.

Analysis procedure
For single particle detection in double-sided detectors, the juntion (P-side) and the ohmic (N-side) are read separately.For good events, these signals correspond to the same deposited energy.Experimentally a certain deviation between both sides is expected, so the |E p E n |  2 condition has been included in the data processing, where value is obtained from the gaussian fit applied to |E p E n | distribution for each detector, either front detector, E, or back detector, E.
As it is explained in [13], the beam is not polarised so the reaction fragments follow axial symmetry and the same physics is expected to happen over the same dispersion angle ✓.Due to that symmetry, the scattering for each ✓-angle follows a conical surface.This property allow the grouping of all the pixels that belong to a determined angular sector ( ✓) (Fig. 2 left).
The E-E telescope configuration makes possible ion identification (Fig. 2 right), hence, those events can be selected by a gate in the corresponding range of the 15 C elastic scattering events.The border strips of both detectors of each telescope were removed due to the lower coincidence e ciency.C+ 208

Pb elastic scattering
The comparison performed between 15 C Rutherford cross section and the experimental cross section of 15 C elastic scattering on lead given in [13] shows large error bars hampering any conclusion about the dynamic behaviour of 15 C beam on heavy targets.In [13], the geometry optimization was done using the stable 15 N present in the beam.
In addition, the 12 C+ 208 Pb scattering was measured under the same conditions described in section 2. At these energies, a pure elastic scattering behavior is expected for 12 C.This can be used as an additional set of data to check the optimization of the geometry made in [13] and to normalize the distribution of the cross section obtained for 15 C+ 208 Pb by the elastic scattering of the stable isotope. 12C+ 208 Pb data were treated with the same procedure explained in sections 3.
The angular distribution of the ratio between 12 C+ 208 Pb and Ruth is presented in Fig. 3.With 12 C = I pixel /⌦ pixel and Ruther f ord = N/ sin 4 (✓ pixel /2), where I pixel is the integral of 12 C ions for each pixel, N is the normalization constant, ⌦ pixel and ✓ pixel are the solid angle and scattering angle subtended for each pixel respectively.
Despite the e↵orts dedicated to the alignment of the chamber and detectors, a small shift (⇠mm) on the reaction point or a tilt of the beam direction is always expected.This results on small deviations of the solid angles ⌦ and angles (✓, ) of each pixel of the detectors due to the very close geometry of the setup so, a cloud of points with small deviations around 1 was expected due to the pure elastic scattering reaction of 12  C.In this case, a wide distribution of the points is shown in Fig. 3 where a clear upward trend can be observed in the data-sets for each telescope, in addition to the lack of continuity for the intersection zones between telescopes and the lack of symmetry for mirror telescopes A and B. This behavior indicates the need for further geometry optimization to be performed.

Geometry optimization
To carry out the geometric optimization of the experimental setup, a program generating the 3D geometry has been built in python.It takes the spherical coordinates of the center of each detector, the position (X, Y, Z) of the reaction point and the beam orientation as inputs.From these data, it computes the central angle and the solid angle subtended by each pixel of each telescope.

Solid angle (⌦) optimization
In order to compute the solid angle (⌦), it is necessary to calculate the coordinates of three corners (A, B, C) for each pixel, using a coordinate system centered on the target.Each pixel is then divided into m ⇥ n smaller pixels with coordinates (A i j , B i j , C i j ), where i 2 0, ..., n and j 2 0, ..., m and m = n = 100, and these smaller pixels are treated as di↵erential areas.The coordinates of each smaller pixel can be calculated using equation 1.
(1) The normal vector (Eq.2) and center (Eq. 3) of each smaller pixel is computed using the coordinates obtained above.The vector going from the pixel center to the target center ( ! r i j = C i j T arget) is then calculated, along with the modulus of that vector (r i j = !r i j ), which is the distance between the target and pixel center.
To compute the area of each smaller pixel, the pixel area is multiplied by the cosine of the angle between the normal vector ( ! n i j ) and the vector going from the pixel center to the target center ( ! r i j ), as !n i j is not parallel to ! r i j .The e↵ective pixel area is given by equation 4: where ' i j is the angle between !n i j and !r i j , and can be calculated using the dot product of these two vectors.
The solid angle for each smaller pixel is calculated using equation 6.This equation involves computing ↵ i j and i j , which are the half-lengths of the sides of the smaller pixel divided by the distance between the target and the pixel center, ↵ i j = !A i j B i j 2r i j and i j = !A i j C i j 2r i j .Since detector pixels are very small, they are treated as surface di↵erentials, and so equation 6 can be approximated using only the first order term (equation 7).

Optimizer
Once all the variables are computed the program returns ✓, and ⌦ associated to the index of each pixel.From these data, the cross section and the ratio with respect to Rutherford are recalculated.The optimizer is based on Gradient Descent method with a loss function of normalized 2 to optimize the computing time.
The hit patterns of telescope A has been used to optimize the geometry due to the fact that the largest statistic was present in the two front telescopes (A and B) and telescope A seems the most centered one, as shown in Fig. 4. On the other hand, telescope B seems o↵-centered so its position has to be optimized independently.
The parameters obtained for the reaction point position and beam orientation are presented in Tab. 2. And the e↵ective angles are given in Tab. 1.The parameter space was constrained between ±5 mm for X and Y, and ±3º for ↵ and .

Electronic efficiency correction
One of the consequences of the high counting rate in front telescopes A and B was that telescopes F, E and D count less as expected due to the low trigger rate for detectors placed backwards.To calculate the electronic e ciency factor, the angular region that matches telescopes A and B with F (1.126), F with E (1.174) and E with D (1.426) has been used.The application of those factors in addition to the geometry optimization give the final distribution for the elastic scattering of 12 C on 208 Pb presented in Fig. 5.

Conclusions
We have optimized the GLORIA device for the present 15 C+ 208 Pb study using the 12 C+ 208 Pb scattering at 4.37 MeV/u energy well below the 12 C Coulomb barrier.We have determine the e↵ective position of the 208  Pb target in comparison with the geometrical center of GLORIA.We have also determine the electronic response that make the distribution typical of Rutherford scattering.This normalization parameters will be used to extract the di↵erential 15 C+ 208 Pb elastic cross section. 12C / Ruth angular distribution geometrically optimized and corrected by electronic e ciency.

Figure 1 .
Figure 1.Schematic representation of the telescope configuration used during the experiment.

Figure 2 .
Figure 2. On the left hand side, the pixels selected in the angular coverage 49º< ✓ < 56º are shown.On the right hand side, the E-E plot where the contaminant 15 N and the beam 15 C are clearly separated.

Figure 3 .
Figure 3. Di↵erential elastic scattering of 12C + 208 Pb.The data per pixel shows large dispersion angular distribution that indicates thar a geometrical optimization is needed.

Table 2 .
Parameters obtained after optimization of the geometry.Coordinates X and Y of the reaction point and rotations of the beam around X axis (↵) and Y axis ( ) using the right-hand rule.Telescope X (mm) Y (mm) ↵ (º)

Figure 4 .
Figure 4. Hit maps of two front-side telescopes, where the beam comes in the middle between them.It can be seen how Telescope A is well centered around back strip 9 while Telescope B needs a geometric adjustment.

Figure 5 .
Figure 5.12 C / Ruth angular distribution geometrically optimized and corrected by electronic e ciency.

Table 1 .
The angular coverage of the six telescopes is given.Center of the detectors, geometric angle range for each telescope is given assuming that beam and target were centered and the final e↵ective angular coverage.