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Determining the 7Li(n,gamma) cross section via Coulomb dissociation of 8Li

The applicability of Coulomb dissociation reactions to determine the cross section for the inverse neutron capture reaction was explored using the reaction 8Li(gamma,n)7Li. A 69.5 MeV/nucleon 8Li beam was incident on a Pb target, and the outgoing neutron and 7Li nucleus were measured in coincidence. The deduced (n,gamma) excitation function is consistent with data for the direct capture reaction 7Li(n,gamma)8Li and with low-energy effective field theory calculations.


I. INTRODUCTION
It is well established that Coulomb dissociation cross sections can provide nuclear structure information about neutron-rich nuclei [1,2]. In nuclear astrophysics, neutron capture cross sections (σ n,γ ) on radioactive nuclei are important in nucleogenesis, and a cross-section measurement of the inverse reaction, Coulomb dissociation [3,4], might be the only way to obtain the capture cross sections. Complementary indirect techniques to determine neutron-capture cross sections, such as so-called "surrogate" methods, have been utilized in other systems, but comparison to Coulomb dissociation as well as direct measurements is desirable to better understand the applicability of such approaches [5]. In Coulomb dissociation, the projectile is dissociated into a neutron and a remainder fragment by a "target" photon absorbed from the electric field of a high-Z target nucleus. First-order perturbation theory gives the relationship between the Coulomb dissociation function dσ CD /dE γ and the photo disintegration cross section σ γ,n , as [3] σ γ,n (E γ , Eλ) = E γ n(E γ , Eλ) where n(E γ , Eλ) is the number of virtual photons with energy E γ and multipolarity Eλ. The principle of detailed balance [6] then yields σ n,γ from σ γ,n . It is desirable to test the accuracy of perturbation theory by comparing σ n,γ values deduced from Coulomb dissociation with directly-measured values of σ n,γ . Coulomb dissociation has been extensively applied to extract proton capture cross sections; see Ref. [7] for an overview of the various reactions. In contrast, neutron capture cross sections have been deduced from Coulomb dissociation and compared with the direct process only for the system 14 C(n,γ) 15 C [8,9]. In this case the dominant process is the capture of p-wave neutrons. Nakamura et al. [9] have demonstrated that the cross section derived from Coulomb dissociation of 15 C agrees well with the directly measured capture cross section [10].
In the present work we report on the Coulomb dissociation of 8 Li in order to extract the neutron capture cross section for the inverse reaction 7 Li(n,γ) 8 Li.
A 69.5-MeV/nucleon secondary 8 Li beam bombarded a Pb target exciting projectiles by virtual photons. Excited unbound states subsequently decay by neutron emission to 7 Li which is stable. The neutron capture reaction on 7 Li can be directly measured so that the validity of the Coulomb dissociation method to deduce the capture cross section of the inverse reaction can be tested. Indeed, the excitation function of the reaction 7 Li(n,γ) 8 Li has been measured over a wide range of energies from a few meV up to 1 MeV [11][12][13][14][15][16]. Partial level schemes of 7 Li and 8 Li are shown in Figure 1. The excitation and decay during the Coulomb dissociation process of 8 Li are indicated by the blue dashed lines. When a virtual photon from the Pb target excites the 8 Li projectile to an excitation energy E x above the neutron separation energy S n of 2.032 MeV, 8 Li decays to 7 Li with a decay energy of E d = E x − S n . In the direct process, a neutron is captured with energy E d and a γ-ray of energy E x is emitted.
Before the (n,γ) cross sections derived by detailed balance from Coulomb dissociation data can be compared with the directly measured neutron capture cross section, several corrections have to be applied. They will be discussed in detail in sections III B and III C.

A. Setup
The experiment was carried out at the National Superconducting Cyclotron Laboratory at Michigan State University. A 120 MeV/nucleon 18 O beam from the Coupled Cyclotron Facility bombarded a 2850 mg/cm 2 9 Be target. The secondary 8 Li beam was selected by the A1900 fragment separator utilizing an 825 mg/cm 2 aluminum wedge degrader. The average beam intensity was ∼150,000/s, the mean energy was 69.5 MeV/nucleon, and the energy dispersion could be best described by a rounded rectangle with FWHM = 1.8 MeV/nucleon. The 8 Li particles then impinged on 56.7 mg/cm 2 lead and 28.8 mg/cm 2 carbon targets corresponding to energy losses of 2.3 MeV and 2.2 MeV, respectively. A schematic view of the experi-mental setup is shown in Figure 2. The 8 Li beam particles were tracked with a pair of Cathode Readout Drift Chambers (CRDC) [20] separated by 2.76 m through a quadrupole triplet magnet onto the reaction target. A 26.4 mg/cm 2 thin plastic scintillator positioned just before the target provided the start signal for time-of-flight (ToF) measurements.
Neutrons from the breakup of 8 Li were detected by the Modular Neutron Array (MoNA) [21,22]. MoNA was arranged in 9 vertical layers of 16 horizontal scintillator bars each. The front face of the first layer was placed at 8.27 m from the reaction target. For the present analysis, only the first 6 layers and the center 1.6 m of the 2-m length were used. The horizontal and vertical acceptances were ±2.8°and ±3.1°, respectively. Each bar has a photomultiplier tube mounted on each end. The time and position of an interaction in a bar are calculated by the mean value and the difference of the left and right signals, respectively. The geometric mean of the leftand right signal charges is approximately proportional to the deposited energy. The individual bars were gainmatched using γ rays from 88 Y (1611 keV) and 228 Th (2381 keV) radioactive sources. The energy threshold was set at 0.7 MeVee.
Charged fragments from the reaction were deflected by the sweeper magnet [25] into a suite of charged-particle detectors [26]. Two CRDCs, separated by 1 m, determined the trajectories of the projectile-like fragments. A thin scintillator served as the fragment trigger and provided an energy loss (dE) measurement. The fragments were then stopped in a thick scintillator which recorded the remaining energy (E). A beam blocker placed behind the sweeper magnet on the high-rigidity side stopped the unreacted 8 Li beam in order to limit the overall count rate in the detection system. The average 7 Li rate entering the detectors was ≈ 0.3/s.

B. Incoming beam parameters
The position and angle of the incoming 8 Li at the target are important for the determination of the overall acceptances of the 7 Li fragments. Due to space constraints it was not possible to measure these beam parameters directly in front of the target as the target was located very closely behind a quadrupole triplet magnet (see Figure  2). Thus, the position and angle of the incoming beam were measured event-by-event with two CRDC tracking detectors located in front of the triplet magnet. The x and y positions in the CRDCs were calibrated with masks to an accuracy of 0.7 mm. The angular straggling in the timing detector was calculated with the program Lise ++ [27] to be 0.29 mrad which was small compared to the 1.7 mrad angular spread of the fragments. The beam trajectories through the magnet to the target were then calculated event-by-event with the particle optics code Cosy Infinity [28]. They were validated by bending the 8 Li beam without a target through the sweeper mag- net into the second set of CRDCs. This method has been sucecssfully used in several previous experiments [29][30][31].

C. Reconstruction of neutron energy and momentum vector
In order to reconstruct the decay energy spectrum the energy and momentum vector of the neutrons had to be reconstructed. The neutron direction was deduced from the position of the interaction in a MoNA scintillator bar relative to the target. The position resolution was 7 cm FWHM in the horizontal direction and 10 cm full width in the vertical direction determined by the height of the bars. The magnitude of the momentum as well as of the energy were calculated from the flight time and the path length. The absolute time for the central bars of each layer was calibrated using prompt γ rays from the target. The other bars within each layer were then synchronized to the central bar with cosmic rays. A time resolution of 1.15 ns (FWHM) was achieved which was dominated by the uncertainty of the flight path due to the 10 cm thickness of the bars.
The neutron energy spectrum for the events of interest (E d < 1.5 MeV) 1 is shown in Figure 3 potential energy of the beam as it approaches the Pb target nuclei. The neutron is emitted near the distance of closest approach where the kinetic energy is reduced by about 38 MeV or close to 5 MeV/nucleon.
The apparent asymmetry of the peak arises from the fact that relative to the beam velocity forward emitted neutrons gain more energy than backward emitted neutrons lose energy. In addition the angular acceptance for forward emitted neutrons is larger than for backward emitted neutrons.

D. Reconstruction and identification of 7 Li
The momenta of the 7 Li fragments were calculated from the tracks through the magnetic field of the sweeper magnet and the positions measured in the CRDCs. A field map of the sweeper magnet had been generated in seven horizontal layers with a grid of 6,000 points each [26]. Within a horizontal layer the magnetic field at any point was linearly extrapolated from a triangular mesh. In the vertical direction, the field was determined with a spline interpolation from seven values along a vertical line crossing the layers. The fringe field was extrapolated with Enge functions [28]. During the experiment, the field was monitored with a Hall probe. An adaptive fifth-order Runge-Kutta method [32] was used to track the fragments through the magnetic field, solving the equation of motion numerically. Starting with the known trajectory after the magnet as measured by the positions in the two CRDCs, particles with different trial momenta were backtracked through the magnet until their position coordinates at the target matched the reaction point determined by the incoming particle (see Section II B). The end point of the matching trajectories yielded the momentum vector of the 7 Li fragments at the reaction point. The Sweeper Magnet field map and a typical particle trajectory is shown in Figure 4. This method of momentum reconstruction was validated by bending the 8 Li beam without a target through the sweeper magnet into the CRDCs. The reconstructed momenta could then directly be compared with the momenta of the incoming particles.
Lithium fragments were identified by the energy deposited while passing through the thin scintillator located after the CRDCs. Isotopic separation was achieved from a two-dimensional spectrum of energy loss versus momentum as shown in Figure 5. The energy spectrum of 7 Li fragments for events with E d < 1.5 MeV is shown in Figure 6. In contrast to the neutron energy spectrum shown in Figure 3 the 7 Li distribution peaks close to the energy per nucleon of the incoming 8 Li beam. As mentioned earlier, the reduction due to the neutron binding energy and the energy loss in the target is small and the potential energy lost by the 8 Li as it approaches the Pb nuclei is regained by the 7 Li fragment after the breakup. The fragment energy distribution is also symmetric and narrower than the neutron spectrum which is due to the smaller velocity imparted to the fragment in the breakup.

III. DATA ANALYSIS
A. Decay energy spectrum The decay energy (E d ) from the breakup of 8 Li can be calculated from the energy and momenta of the neutrons (E n , p n ) and 7 Li fragments (E7 Li , p7 Li ) as (2) where m n and m7 Li are the rest masses of the neutron and 7 Li, respectively. Before the decay energy spectrum can be converted to the Coulomb dissociation function and subsequently the photo disintegration cross section, the data have to be corrected for the efficiencies and acceptances of the detector systems.
The intrinsic efficiency of MoNA was simulated with Geant4 [33,34] using the Menate R package [35] as described in [36]. In addition to the properties of MoNA itself, the simulations included the window of the vacuum chamber (1/4-inch of stainless steel) and the 827 cm of air between the target and the front face of MoNA. The efficiency varied linearly for neutron energies of interest from 75.3(30)% at 50 MeV to 66.1(26)% at 80 MeV. For the Coulomb dissociation function below 1.5 MeV the efficiency did not vary as function the neutron energy so that the overall spectrum could be corrected by 70.5(28)% corresponding to the efficiency at the average neutron energy of 65 MeV.
The efficiencies of the charged particle detectors were determined by removing the target and bending the 8 Li beam into the center of the focal plane detectors. The combined efficiency for the two tracking CRDC detectors (84.4(8)%), the target timing scintillator (99.9(10)%), the focal plane CRDC detectors (95.0(10)%), dE and E plastic scintillators (99.7(10)% each) was 79.0(16)%.
The overall acceptance for the detection of the neutrons and 7 Li fragments is correlated and depends on the decay energy. Thus, a Monte Carlo event-by-event simulation was written which included the properties of the incoming beam, the reaction mechanism and the geometry of the detectors including the beam blocker. The trajectories of 7 Li fragments were simulated following Rutherford scattering, and the energy of the virtual photon was selected according to the description by Baur and Bertulani [37]. The breakup into a neutron and 7 Li was assumed to be isotropic in the rest frame of 8 Li. The results of the simulation are shown as the circles in Figure 7. The dashed line corresponds to a fit of the form (1−c)/(e a(E d −b) )+c which was used to correct the decay energy spectrum for the detector efficiencies and acceptances.
The uncertainty of this correction was calculated to be 4.5% by combining the solid angle uncertainties of MoNA and the uncertainties of the fragment detector system which were estimated to be 4% and 2%, respectively.

B. Coulomb dissociation function and photo disintegration cross section
The decay energy spectrum contains contributions from Coulomb dissociation as well as nuclear reactions. Peripheral Coulomb breakup increases with the charge of the projectile and the target while the cross section for peripheral breakup due to the short-range nuclear force depends on the radii of the projectile and target nuclei. In order to subtract the nuclear contribution from the decay energy spectrum, the breakup spectrum was measured with a carbon target. The relative contributions from Coulomb and nuclear reactions were then parameterized as a function of the mass number (A) and charge (Z) of the target as described in Ref. [8]: with r 0 = 1.2 fm and r8 Li = 2.4 fm. The parameters a(E d ) and b(E d ) were determined from the measured cross sections of the lead and carbon targets and were about 0.6 mb/(MeVfm) and 0.03 mb/MeV, respectively. The nuclear contribution to the decay energy spectrum for the lead target was between 3% and 5% for E d up to 1.5 MeV. Coulomb-nuclear interference effects were determined by a DWBA calculation to be more than two orders of magnitude smaller than the individual contributions. The above paramerization is very similar to the approach by Fukuda et al. [38] who measured the breakup cross sections of 11 Be on Pb and C targets and deduced the Coulomb dissociation cross section according to σ CD = σ(P b) − Γσ(C). The scaling factor Γ of 2.1(5) was extracted from angular distribution measurements. Converting our paramerization to Γ yields a value of ∼1.5. The angular acceptance was the same (< 6 • ) in both experiments. It should also be mentioned that only the relative decay energy of each neutron-7 Li coincident pair is determined and a given decay energy can thus result from decay to either the 3/2 − ground state or the 1/2 − first excited state. However, using a continuum coupled-channels calculation with states generated with a potential model [39] transitions to the excited state by Coulomb dissociation were determined to be more than three orders of magni- tude smaller than transitions to the ground state; thus these contributions are considered negligible. The Coulomb dissociation function, which is directly related to the corrected decay energy spectrum by the substitution E γ = E d + 2.032 MeV, is shown in Figure  8. The photo disintegration cross section (σ γ,n ) can then be calculated by the virtual photon method according to Eq. (1).

C. Neutron capture cross section
Finally, the neutron capture cross section for the direct reaction 7 Li(n,γ) 8 Li is derived by the principle of detailed balance given by where µ is the reduced mass and E c.m. is the energy in the center-of-mass system in the n + 7 Li capture process which is equivalent to E d in the γ-ray dissociation reaction. This cross section only corresponds to the neutron capture to the ground state of 8 Li (σ n,γ0 ). However, the direct capture reaction can also proceed via the bound first excited state (σ n,γ1 ). Thus, the Coulomb dissociation data have to be corrected for these contributions. The total cross section can be expressed as: σ n,γ = σ n,γ0 1 + σ n,γ1 σ n,γ0 .
The ratio σ n,γ1 /σ n,γ0 can be evaluated at thermal neutron energies (E d = 0) where the branching ratio is known and because both γ-ray decays are E1 transitions, the relative intensities scale as E 3 γ : The branching ratios (BR) and γ-ray energies for thermal capture (E γ ) are 9.800(344)% and 90.20(260)% [18,19] and 1.051 MeV (S n − E * 1 , see Figure 1) and 2.032 MeV for the first excited and ground state, respectively. The correction factor σ n,γ /σ n,γ0 increases from 1.109(5) at E d = 0 to 1.243 (13) Another correction results from the fact that s-wave neutron capture into 1 + and 2 + states results in γ rays of only E1 multipolarity while Coulomb dissociation can be induced by both E1 and E2 photons. For a quantitative estimate, we calculated σ γ,n with a modified version of RADCAP [40] and found that E2 transition strengths are at least a factor 10 6 smaller than E1 transition strengths. In addition, as shown in the right panel of Figure 1, the direct capture reaction can proceed via the 3 + resonance of 8 Li. In Coulomb dissociation, this resonance would have to be excited by M1 transitions from the 2 + ground state of 8 Li. However, the number of virtual M1 photons is negligibly small. Thus, to better than a few percent, the Coulomb dissociation measures only the direct component of the neutron capture reaction.
The final neutron capture cross section as extracted from the inverse Coulomb dissociation measurement is shown by the solid black circles in Figure 9. The data are compared with previous direct neutron capture measurements by Imhof et al. [11] (open blue circles), Nagai et al. [13] (solid green triangle), Heil et al. [15] (solid red square), and another more recent measurement by Nagai et al. [16] (open black squares). The 254-keV resonance corresponding to the 3 + second excited state in 8 Li is clearly visible in the direct capture data by Imhof et al. ciation data. Overall, the non-resonant data from the two different methods are in good agreement; while the present data essentially agree with the Imhof data above the 254-keV resonance (within less than 10%), at energies below the resonance they are systematically about 20-25% lower than the previous data, although still within the 15-20% uncertainties of the individual data points. At energies above ∼200 keV the data deviate from the expected inverse velocity (1/v) dependence for s-wave capture (red solid line) which has been well established at lower energies [14] as will be discussed in the next section.
In Figure 10 [42]), and Rupak and Higa (blue shortdashed line, [43]). All models predict the observed deviation from the 1/v dependence of the cross section towards higher energies. The potential models for direct radiative capture by Nagai et al. [16] and Wang et al. [41] slightly overpredict the magnitude of the deviation. Huang et al. described the radiative proton-and neutron-capture for all available data on light nuclei with a simplified twobody treatment of the capture process [42]. While their fit describes the present data above ∼200 keV, it overpredicts the available data at lower energies. The best description of the data over the whole energy range is given by the recent low-energy effective field theory cal-culation by Rupak and Higa [43]. Their calculation also fits the available data in the eV range [14] and at thermal energies [49] which are not shown in Figure 10.
The deviation from the 1/v dependence has previously been included in the parameterization of the cross section as [41,50] The solid black line in Figure 10 shows the best fit to the available non-resonant data in the range from 10 -1000 keV and where s 0 was fixed at 6.7 µb(MeV) 1/2 to fit the measured cross section at thermal neutron energies [49]. In order to describe the data at higher energy a higher-order polynomial description might be necessary. The value for s 0 from the unconstrained fit can be compared to the slope extracted from the direct measurement of σ n,γ for energies between ∼1 eV and ∼1 keV by Blackmon et al. [14]. After transforming to the laboratory system and correcting for the excited state contribution it translates to a slope of 6.2(3)×10 −3 b(eV) 1/2 which agrees well with the value of 6.3(3)×10 −3 b(eV) 1/2 quoted by Blackmon et al..

V. CONCLUSION
In conclusion, the Coulomb dissociation reaction 8 Li(γ,n) 7 Li was measured with a 69.5 MeV/nucleon 8 Li beam on a Pb target, and the decay energy spectrum was calculated by measuring the outgoing neutron and 7 Li nucleus in coincidence. From these data and the principle of detailed balance the neutron capture cross section 7 Li(n,γ) 8 Li was deduced for energies between 25 keV and 1.5 MeV. The good agreement with directly measured cross sections demonstrates that Coulomb dissociation is a reliable method to extract neutron capture cross sections. It represents the first time that neutron capture cross sections for s-wave neutrons were derived from Coulomb dissociation. The only other case where the (γ,n) Coulomb dissociation cross sections were compared with directly measured cross sections was the system 14 C(n,γ) 15 C which involved predominantly the capture of p-wave neutrons [9]. The anticipated deviation from the 1/v behavior at higher neutron energies was observed and could be fitted within the parameterization of Baye [50] and Wang et al. [41]. The data also agree well with the results of the low-energy effective field theory calculations by Rupak and Higa [43]. ACKNOWLEDGMENTS Support of the National Science Foundation under grant Nos. PHY01-10253, PHY03-54920, PHY04-56463, PHY06-06007, PHY11-02511, the Department of Energy under grant Nos. DE-FG02-08ER41533 and DE-FC02-07ER41457 (UNEDF, SciDAC-2), the Research Corporation, the Hungarian Research and Technology Fund, grant No. KTIA AIK 12-1-2012-0020, and the OTKA under grant Nos. T049837 and K101328 is gratefully acknowledged. The authors also would like to thank the anonymous reviewer for valuable comments and suggestions.