Resolving the 11B(p, ${\alpha_0}$) Cross Section Discrepancies between 0.5 and 3.5 MeV

TL;DR

This paper presents new measurements of the 11B(p, α0) reaction cross section to resolve existing discrepancies in previous data, using advanced silicon detectors and low beam current for improved accuracy.

Contribution

It provides the first precise measurement of the 11B(p, α0) cross section with modern detection technology to address prior inconsistencies.

Findings

Resolved normalization discrepancies in cross-section data

Achieved more accurate energy scale calibration

Provided reliable data for applications in nuclear physics

Abstract

The reaction 11B(p, 3${\alpha}$) is relevant for fields as diverse as material science, nuclear structure, nuclear astrophysics, and fusion science. However, for the channel proceeding via the ground state of 8Be, the available cross-section data shows large discrepancies of both normalization and energy scale. The present paper reports on a measurement of the 11B(p, ${\alpha_0}$) cross section using an array of modern large area segmented silicon detectors and low beam current on an enriched thin target with the aim of resolving the discrepancies amongst previous measurements.

Tables

Table 1. Table 1: Tabulated angle-integrated cross sections. The uncertainty is approximately 10   % times 10 percent 10\text{\,}\% for all data points except the lowest two where it is 12   % times 12 percent 12\text{\,}\% .

$E_p$ [keV]	${\sigma }_{\alpha 0}$ [mb]	$E_p$ [keV]	${\sigma }_{\alpha 0}$ [mb]
$500$	$2.5\left(3\right)$	$2150$	$8.4\left(9\right)$
$600$	$3.6\left(4\right)$	$2250$	$12.0\left(12\right)$
$683$	$4.3\left(5\right)$	$2300$	$15.5\left(16\right)$
$800$	$5.5\left(6\right)$	$2350$	$20\left(2\right)$
$850$	$5.3\left(6\right)$	$2400$	$24\left(2\right)$
$900$	$5.8\left(6\right)$	$2450$	$35\left(3\right)$
$950$	$6.1\left(6\right)$	$2500$	$44\left(4\right)$
$950$	$6.1\left(6\right)$	$2550$	$52\left(5\right)$
$1000$	$6.3\left(6\right)$	$2600$	$58\left(6\right)$
$1100$	$6.3\left(6\right)$	$2640$	$55\left(6\right)$
$1300$	$6.2\left(6\right)$	$2650$	$54\left(5\right)$
$1500$	$6.0\left(6\right)$	$2700$	$45\left(5\right)$
$1600$	$5.9\left(6\right)$	$2750$	$33\left(3\right)$
$1700$	$6.1\left(6\right)$	$2800$	$24\left(2\right)$
$1800$	$6.9\left(7\right)$	$2850$	$15.3\left(15\right)$
$1900$	$7.9\left(8\right)$	$2900$	$8.6\left(9\right)$
$1900$	$9.1\left(9\right)$	$2950$	$5.5\left(6\right)$
$1935$	$10.9\left(11\right)$	$3000$	$4.3\left(4\right)$
$1970$	$14.6\left(15\right)$	$3100$	$4.8\left(5\right)$
$2000$	$14.9\left(15\right)$	$3300$	$6.0\left(6\right)$
$2030$	$8.8\left(9\right)$	$3400$	$6.8\left(7\right)$
$2060$	$6.2\left(6\right)$	$3500$	$8.1\left(8\right)$
$2100$	$6.3\left(6\right)$

Equations

$$\frac{d\sigma}{d\Omega}(\theta)=\frac{\sigma}{4\pi}\Bigg{[}1+\sum_{i=1}^{4}a_{i}P_{i}(\cos\theta)\Bigg{]}.$$

Full text

††offprints: H. O. U. Fynbo (fynbo@phys.au.dk)

1 Introduction

The reaction $\isotope[11]{B}(p,3\alpha)$ is relevant for fields as diverse as material science, nuclear structure, nuclear astrophysics, and fusion science. In a sequential picture, this reaction can proceed via the ground state of \isotope[8]Be, $\isotope[11]{B}(p,\alpha_{0})\isotope[8]{Be}$, or via the first excited state, $\isotope[11]{B}(p,\alpha_{1})\isotope[8]{Be}^{*}$.

Boron depth profiling in bulk matter can be studied using the technique of nuclear reaction analysis (NRA) where a nuclear reaction on the element of interest produces a particle with high enough energy to escape the material, and the energy distribution of the emitted particles can be used to deduce the depth profile. Boron depth profiling using NRA requires an accurate cross section of the $\isotope[11]{B}(p,\alpha_{0})$ reaction in the energy range up to $34\text{\,}\mathrm{MeV}$ Kokkoris2010 .

The rate of the $\isotope[11]{B}(p,3\alpha)$ reaction is also relevant for understanding the astrophysical abundances of the light elements Li, Be, and B Boesgaard2005 ; Lamia2012 . Specifically, the abundances of these elements in stellar atmospheres can be depleted due to plasma mixing phenomena inside the stars at a rate which depends on fundamental nuclear cross sections.

The $\isotope[11]{B}(p,3\alpha)$ reaction is considered a candidate for fusion energy generators as an alternative to e.g. the most favored $t(d,n)\alpha$ reaction Moreau1977 . In order to evaluate the feasibility of the $\isotope[11]{B}(p,3\alpha)$ reaction accurate cross sections are required Sikora2016 .

Finally, the cross section of the $\isotope[11]{B}(p,3\alpha)$ reaction provides a direct probe for resonances in \isotope[12]C above the \isotope[11]B$+p$ threshold. For proton energies up to $2\text{\,}\mathrm{MeV}$ the excitation spectrum exhibits resonances due to three isospin $T=1$ states at 16.11, 16.62 and $17.76\text{\,}\mathrm{MeV}$, which have spin and parity of $2^{+}$, $2^{-}$ and $0^{+}$ and whose properties are well established. Additionally, a very broad $1^{-}$, $T=1$ resonance and sub-threshold resonances of unknown spin and parity are believed to contribute to the excitation spectrum in the region between the 2- and 0+ resonances Symons1963 ; Segel1965 ; Barker2002 .

A handful of measurements over extended ranges of beam energies and/or emission angles of the final state particles have been published, but these measurements differ significantly both in the energy scale and in the absolute normalization of the cross sections. Figure 1 shows cross section data from a set of measurements of the $\isotope[11]{B}(p,\alpha_{0})$ reaction which is the focus of the present paper. As is evident, there are significant deviations of the order of $100\text{\,}\mathrm{keV}$ in the energy scale and up to $70\text{\,}\%$ in the magnitude of the cross section between different measurements. Likely sources of such deviations can be accelerator energy calibrations and inaccurate evaluations of target thicknesses.

Some of us have recently published a re-measurement of the $\isotope[11]{B}(p,3\alpha)$ reaction at a proton energy corresponding to the $16.11\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ $2^{+}$ $T=1$ resonance, which produced a more precise and accurate determination of the cross section at that energy Munch2018 . Here we use the same experimental approach to study the $\isotope[11]{B}(p,\alpha_{0})$ reaction for proton energies in the range $0.53.5\text{\,}\mathrm{MeV}$.

2 Experiment

The experiment was conducted at the Aarhus University $5\text{\,}\mathrm{MeV}$ accelerator which provided a beam of protons with an energy between $0.5$ and $3.5\text{\,}\mathrm{MeV}$. The beam impinged on a thin boron target which was oriented $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ with respect to the beam axis. The beam current was measured $1\text{\,}\mathrm{m}$ downstream of the target position and the beam spot was collimated to be $1$\mathrm{mm}\times 1$$\mathrm{mm} by a set of horizontal and vertical slits.

The boron target was manufactured by evaporating $99\text{\,}\%$ enriched \isotope[11]B onto a thin $\mathord{\sim}$4\text{,}\mathrm{\SIUnitSymbolMicro g}\text{,}{\mathrm{cm}}^{-2}$$ natural carbon backing. In order to determine the target thickness the target was oriented at angles of $0\text{\,}\mathrm{\SIUnitSymbolDegree}$ and $180\text{\,}\mathrm{\SIUnitSymbolDegree}$ with respect to the beam axis and bombarded with $2\text{\,}\mathrm{MeV}$ $\alpha$ particles. The \isotope[11]B thickness could then be deduced from the energy shift of the carbon peak using the procedure described in Ref. Munch2018 . The result was $12.6(12)\text{\,}\mathrm{\SIUnitSymbolMicro g}\text{\,}{\mathrm{cm}}^{-2}$.

Charged particles were detected using an array of four Double Sided Silicon Strip Detectors (DSSD) in close geometry. These provided a simultaneous detection of energy and position of emitted charged particles. The array consisted of two annular DSSDs (S3 from Micron Semiconductors) placed $42\text{\,}\mathrm{mm}$ up- and downstream of the target. Additionally, two quadratic DSSDs (W1 from Micron Semiconductors) were placed approximately $36\text{\,}\mathrm{mm}$ from the target center at an angles of $85\text{\,}\mathrm{\SIUnitSymbolDegree}$ and $95\text{\,}\mathrm{\SIUnitSymbolDegree}$ with respect to the beam axis.

The trigger logic was handled by a GSI Vulom4b module VULOM4b running the TRLO II firmware Johansson2013 . The trigger logic was configured as a logical OR between all four detectors, with the downstream detector downscaled by a factor of eight. The firmware maintains scaler values for the accepted and total number of triggers for each detector. The deadtime for each run was determined from these.

The energy scan was performed in steps of $100\text{\,}\mathrm{keV}$ with smaller steps near well-known resonances. Data were acquired at each energy for approximately one hour with a beam current of 0.3–1 nA. Longer measurements were performed for the $0^{+}$ state at $17.76\text{\,}\mathrm{MeV}$ and the $3^{-}$ state at $18.35\text{\,}\mathrm{MeV}$. The result of these longer measurements will be reported separately Kirsebom2018 .

3 Data reduction

The analysis was performed event by event on list-mode type data. The detected particles were assumed to be $\alpha$ particles and the energies were corrected for losses in the target foil and the dead layers of the detectors, which are $\mathord{\sim}$500\text{,}\mathrm{nm} thick for the annular design and $\mathord{\sim}$100\text{\,}\mathrm{nm} for the quadratic design. The corrected energies and the angles were transformed to the beam-target center-of-mass (CM) system. An example of the resulting CM energy spectrum can be seen in fig. 3 with a clear peak at high energy corresponding to the $\alpha_{0}$ channel. At lower energies the broad distribution corresponding to $\alpha_{1}$ and its secondary particles is present. Peaks from the elastic scattering of the proton beam off boron, carbon, and small impurities of an element close to iron can also be seen at low energies. To select the $\alpha_{0}$ channel, a cut is placed on the $\alpha_{0}$ peak and the CM angles projected out. A selection of the resulting angular distribution can be seen in fig. 4. The distributions evolve slowly from forward focussed to approximately symmetric to strongly backward focussed. During the analysis, it was observed that some pixels in the forward detectors were partially shadowed by the target ladder. Conservatively these pixels were excluded from the analysis and simulation.

In order to determine the solid angle of the detection system, a Monte Carlo simulation was performed with the SimX tool Munch2018simx . A beam of protons with a $1$\mathrm{mm}\times 1$$\mathrm{mm} profile was generated and propagated to a random depth in the boron layer. At this location, an $\alpha_{0}$ was generated and emitted isotropically in the CM system. The $\alpha_{0}$ was propagated out of the target and into the detectors in a straight-line trajectory. Energy losses were calculated using the SRIM tables which approximately account for straggling SRIM . The structure of the simulation output was identical to the real data and the simulated data was subjected to the same analysis as the real data. The solid angle for a given CM angle can then be calculated as the ratio between the number of detected hits to the number of simulated events. The angular distribution is then given by the ratio of the data and simulation for each CM angle. The differential cross section was in this way determined by scaling the angular distribution with the integrated current and acquisition deadtime. The resulting angular resolved cross sections were then fitted with the lowest five Legendre polynomials

[TABLE]

4 Results and discussion

The Legendre coefficients as a function of beam energy are shown in the lower four panels of fig. 5 together with the coefficients obtained by Refs. Symons1963 ; Segel1965 ; Becker1987 . In addition, the position of the natural parity states, listed in the latest compilation Kelley2017 , are also indicated. For the two lowest-order terms, there is generally good agreement between all four datasets in the regions of overlap. For the third-order term, there is excellent agreement between the present data and Symons et al.(SY) while the agreement with Segel et al.(SE) is good above $\sim$1.5\text{,}\mathrm{MeV}, and a systematic deviation is observed below that energy. This deviation might be due to the experimental difficulties noted by SE for measurements below $1\text{\,}\mathrm{MeV}$ beam energy. For the fourth-order term, there is again good agreement with SY, but a systematic shift is seen at all energies when comparing to SE. We speculate that this is due to the latter only measuring at six different angles whereas SY measured either 8 or 13 angles. From the data, it appears that the $0^{+}$, $3^{-}$ and $2^{+}$ levels have an appreciable effect on the Legendre coefficients, while the effect of the $1^{-}$ levels appear negligible. This might simply be due to their width being as large as $\mathord{\sim}$1\text{\,}\mathrm{MeV}. With the present agreement with previous datasets, we are unable to discern between the two differing interpretations presented in Refs. Barker2002 ; Symons1963 ; Segel1965 for the region below $2\text{\,}\mathrm{MeV}$. The reader is thus referred to them for a detailed discussion of the angular coefficients.

The angle-integrated cross section can be seen in the first panel of fig. 5 and in tabular form in table 1. It shows the same behavior as observed in the previous measurements. In fig. 6 we compare data from the present experiment and the recent high-resolution differential cross section dataset of Kokkoris et al.(KO) for $\theta=$150\text{,}\mathrm{\SIUnitSymbolDegree}$$ Kokkoris2010 . The agreement is better than $5\text{\,}\mathrm{keV}$ which validates the overall energy calibration of the present dataset. A similar comparison shows good agreement between the present data and that of SY. Thus, we cannot verify the claim of KO of a discrepancy between the energy scales of KO and SY. However, the agreement between our data and those of SE requires the energy of SE to be shifted down by approximately $38\text{\,}\mathrm{keV}$.

As discussed in the introduction, there are several normalization issues between the different datasets Beckman1953 ; Symons1963 ; Segel1965 ; Davidson1979 ; Borchers1983 ; Becker1987 . In order to shed light on this, the ratios between the cross sections of other datasets and the present have been computed. This can be seen in fig. 7 where the median ratio and the 25–75 inter quartile ratio are written in the legend. Note that the spectrum of SE has been shifted in energy as discussed above. The first conclusion is that within error the present measurement is consistent with the measurement of Becker et al.(BE), while the difference compared to Davidson et al.is a constant scaling factor. There also seems to be simple scaling relations for the data of Beckman et al.below $\mathord{\sim}$1.3\text{,}\mathrm{MeV}$$ and the data of SE and SY in the range 1.5–$2.75\text{\,}\mathrm{MeV}$, while larger deviations occur outside these ranges. The present data only overlaps with the energy range measured by Börchers et al.(BO) in a limited range. However, within that range, the agreement is better than $2\sigma$.

Scaling all datasets except BE and BO with constant factors given in fig. 7 yields the cross sections seen in fig. 8, which confirms visually that the different data sets can be brought into agreement by a constant scaling. Considering the good agreement between our measurement and the two most recent experiments by BE and BO, we argue that the overall normalization is now better than $15\text{\,}\%$.

5 Conclusion

We have measured the $\isotope[11]{B}(p,\alpha_{0})$ reaction for proton energies in the range $0.53.5\text{\,}\mathrm{MeV}$. From an analysis of the angular distributions, total cross sections and energy dependent Legendre polynomial coefficients have been extracted from the data. Disagreements between previous measurements Beckman1953 ; Symons1963 ; Segel1965 ; Davidson1979 ; Borchers1983 ; Becker1987 on both the magnitude and energy dependence of the cross section have been resolved.

There is disagreement in the literature on the interpretation of the cross section and Legendre coefficients below $2\text{\,}\mathrm{MeV}$ in terms of resonances in \isotope[12]C. Symons et al.interprets this as ghosts of resonances below the proton threshold with spin and parity $2^{+}$ or $3^{-}$ Symons1963 , while Barker interprets this region as a broad $1^{-}$ resonance in addition to the high energy tail of a broad $0^{+}$ resonance below the proton threshold in \isotope[12]C Barker2002 .

We were unable to resolve this issue with a measurement of the $\alpha_{0}$ channel. Instead this issue requires a measurement of the $\alpha_{1}$ channel and probably also of the phase-space (Dalitz) distributions of the 3$\alpha$ emission of that channel. That will provide additional information on the spin and parity of the involved resonances in \isotope[12]C, which should be fed into a new R–matrix analysis similar to that performed by Barker.

Acknowledgement

We would like to thank Folmer Lyckegaard for manufacturing the target. We also acknowledge financial support from the European Research Council under ERC starting grant LOBENA, No. 307447. OSK acknowledges support from the Villum Foundation through Project No. 10117. MM acknowledges support from the Nustar Colaboration.