# Structure of the brunnian nucleus <sup>10</sup>C

N. Curtis, N.L. Achouri, N.I. Ashwood, H.G. Bohlen, W.N. Catford, N.M.

Clarke, M. Freer, P.J. Haigh, B. Laurent, N.A. Orr, et al.

► To cite this version:

N. Curtis, N.L. Achouri, N.I. Ashwood, H.G. Bohlen, W.N. Catford, et al.. Structure of the brunnian nucleus <sup>10</sup>C. 9th International Conference on Clustering Aspects of Nuclear Structure and Dynamics, 2007, Stratford upon Avon, United Kingdom. pp.012022, 10.1088/1742-6596/111/1/012022 . in2p3-00338056

HAL Id: in2p3-00338056

https://in2p3.hal.science/in2p3-00338056v1

Submitted on 19 May 2025

**HAL** is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>
OPEN ACCESS

Structure of the brunnian nucleus <sup>10</sup>C

To cite this article: N Curtis et al 2008 *J. Phys.: Conf. Ser.* 111 012022

View the [article online](#) for updates and enhancements.

You may also like

- Astrophysical reaction rates for <sup>6</sup>He and <sup>9</sup>Be production by electromagnetic radiative capture and four-body recombination
R de Diego, E Garrido, D V Fedorov et al.

- <sup>28</sup>Si + as an example of novel methods for elastic scattering experiments
M Norrby

- Study of internal structures of <sup>9,10</sup>Be and <sup>10</sup>B in scattering of <sup>4</sup>He from <sup>9</sup>Be
S M Lukyanov, A S Denikin, E I Voskoboynik et al.

<figure><img src="image_2.png" /></figure>
# Structure of the Brunnian Nucleus $^{10}$C

N Curtis<fnref n="1" />, NL Achouri<fnref n="2" />, NI Ashwood<fnref n="1" />, HG Bohlen<fnref n="3" />, WN Catford<fnref n="4" />,
NM Clarke<fnref n="1" />, M Freer<fnref n="1" />, PJ Haigh<fnref n="1" />, B Laurent<fnref n="2" />, NA Orr<fnref n="2" />, NP
Patterson<fnref n="4" />, N Soić<fnref n="5" />, JS Thomas<fnref n="4" /> and V Ziman<fnref n="1" />

E-mail: n.curtis@bham.ac.uk

**Abstract.** The 2α + 2p breakup and structure of the only known Brunnian nucleus, $^{10}$C, has been studied at 33.3 MeV/nucleon. Detection of the 2α + 2p breakup fragments using a highly segmented Si-Si-Si-CsI detector array has allowed the $^{10}$C → $^{9}$B + p, $^{9}$B → $^{8}$Be + p, $^{8}$Be → α + α and $^{10}$C → $^{6}$Be + α, $^{6}$Be → $^{5}$Li + p, $^{5}$Li → α + p decay channels to be reconstructed. Proton emission was seen to be favoured. Resonant breakup was observed from excited states at E$_{x}$ = 4.20, 5.31 and 6.74 MeV. The previously unobserved state at 4.20 MeV may correspond to a 0$^{+}$ α+2p+α cluster structure.

**1. Introduction**

In recent years there has been considerable interest in Borromean nuclei [1]. These are a class of
nuclei that can be described as having a three-body structure, with the additional property that
the removal of any one of the three constituents results in an unbound two-body subsystem.
One of the best known examples is $^{11}$Li, which may be described as having a $^{9}$Li+n+n structure
in its ground state. The two-body subsystems, $^{10}$Li ($^{9}$Li+n) and the di-neutron (n+n), are
both unbound, hence all three bodies are required to give an overall bound nuclear system. The
reproduction of the structure of Borromean nuclei therefore provides a stringent test of nuclear
few body models.

There is only one known example of a four-body nucleus that has Borromean properties. This
is $^{10}$C, which may be described as $\alpha+\alpha+p+p$. This nucleus has the property that the removal
of any one of the four constituents produces an unbound three-body system. The removal of a
proton from $^{10}$C produces $^{9}$B, which is unbound to $^{8}$Be+$p$ decay by 185 keV, and the removal of
an $\alpha$-particle leads to $^{6}$Be. Although $^{6}$Be is bound by -595 keV to $^{5}$Li+$p$ decay, this decay does
occur as the $^{5}$Li ground state is ~1.5 MeV wide. In addition, the removal of any two particles
from $^{10}$C results in an unbound two-body system. If both protons are removed a $^{8}$Be nucleus,
unbound by 92 keV to $\alpha+\alpha$ decay, is produced. The removal of one proton and one $\alpha$-particle
results in $^{5}$Li, which is unbound to $\alpha+p$ decay by 1967 keV, and the removal of both $\alpha$-particles
would produce the di-proton (unbound to $p+p$ decay by 93 keV). All four bodies must be present

<footnote n="1">¹ School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom.</footnote>

<footnote n="2">² LPC-ENSICAEN, IN2P3-CNRS et Université de Caen, F-14050 Caen Cedex, France.</footnote>

<footnote n="3">³ Hahn-Meitner-Institut GmbH, Glienicker Str. 100, 14109 Berlin, Germany.</footnote>

<footnote n="4">⁴ School of Electronics and Physical Sciences, University of Surrey, Guildford, Surrey, GU2 7XH, United Kingdom.</footnote>

<footnote n="5">⁵ Rudjer Bošković Institute, Department of Experimental Physics, Bijenička 54, HR-10000 Zagreb, Croatia.</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Example of a 4^th^ order Brunnian link.</figcaption></figure>

to give a bound system, and ¹⁰C may therefore be described as a “Super-Borromean” nucleus. In fact ¹⁰C is the only known nuclear example of a 4^th^ order Brunnian link [2], a nontrivial link in knot theory that becomes trivial if any component is removed. An example visualisation of a 4^th^ order Brunnian link is shown in Fig. 1. Three-body Borromean nuclei are examples of 3^rd^ order Brunnian links.

The aim of this experiment was to study the spectroscopy of ¹⁰C and to determine the relative strengths of the α and proton decay channels: ¹⁰C → ⁹B + p, ⁹B → ⁸Be + p, ⁸Be → α + α and ¹⁰C → ⁶Be + α, ⁶Be → ⁵Li + p, ⁵Li → α + p.

**2. Experimental Details**

The experiment was performed using a 33.3 MeV/nucleon ¹⁰C beam provided by the GANIL accelerator facility. This was produced following the fragmentation of a 60 MeV/nucleon ¹²C primary beam in a 3 mm thick Be production target. The secondary beam was purified to 98.6 % ¹⁰C using the LISE3 spectrometer and had an average intensity of 2.5 × 10⁴ particles per second. It was tracked onto the ¹²C reaction target (20, 45 and 95 mg/cm² thicknesses were used) on an event by event basis using a position sensitive drift chamber. A Ta beam stop, 5 mm thick and 10 mm in diameter, was positioned 106 mm downstream from the target. This protected the detector telescope, which was placed on the beam axis (at 0°), from direct and small angle scattered ¹⁰C beam. The detector array consisted of four separate elements, each (50 × 50) mm² in active area. The first two elements were silicon resistive strip detectors, placed at 123 and 130 mm from the target. Each was 500 µm thick and segmented into 16 independent 3 mm wide strips. Position information (with a resolution of ~ 1 mm) along the 50 mm strip length was provided by resistive charge division. The strips on the front detector were vertical and those on the second horizontal. The third element was a 1000 µm thick double sided silicon strip detector (DSSD), placed 148 mm from the target. The front face was segmented into 16 independent 3 mm wide horizontal strips, the back face having 16 similar, but vertical, strips. The final element was an array of 64 CsI scintillators, placed 160 mm from the target. Each crystal had an area of (6 × 6) mm² and was 20 mm thick. Together the 4 elements of the detector telescope provided ΔE-E particle identification with isotopic separation for H, He and Li ions, and the highly segmented nature of the array allowed all four particles from the decay of ¹⁰C (α+α+p+p) to be detected in coincidence. The telescope was calibrated using an α-particle source and secondary cocktail beams of mixed isotopes at five different magnetic rigidity settings of the spectrometer. The energy resolution was ~ 200 keV (FWHM) for the silicon detectors and ~ 1.5% (FWHM) for the CsI crystals.

**3. Experimental Results**

The multiplicity (M) in the DSSD was used to group events for analysis. Three categories were considered: M = 3 (ααp), M = 3 (αpp) and M = 4 (ααpp). The decay energy (E<sub>decay</sub>) for
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Decay energy spectra for (a) $^8$Be, (b) $^9$B, (c) $^5$Li and (d) $^6$Be. The arrows indicate the known ground state energies. The various lines in (c) are described in the text.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 3.** $E_x$ spectra for (a) $^9$B + *p* and (b) $^6$Be + *α* decay of $^{10}$C. The various lines are described in the text.</figcaption></figure>

each channel was reconstructed from the kinetic energies and momenta of the decay fragments: $E_{decay} = (\sum_{i=1}^{n} E_{fragment_i}) - E_{parent}$, where *n* is the number of decay fragments and $E_{parent} = P_{parent}^2 / 2m_{parent}$. $P_{parent}$ was obtained from the sum of the momenta of the *n* detected particles. $E_{decay}$ is related to the excitation energy ($E_x$) of the decaying nucleus via $E_x = E_{decay} - Q_n$, where $Q_n$ is the *n*-body decay Q-value.

In Fig. 2(a) the $E_{decay}$ spectrum for the two $\alpha$-particles in the M = 3 ($\alpha\alpha p$) channel is shown. The arrow indicates the expected position (92 keV) of the $^8$Be ground state (gs). A strong peak is seen at this energy. The $^8$Be$_{gs}$ events were then selected and, along with the proton in the M = 3 ($\alpha\alpha p$) channel, used to reconstruct $E_{decay}$ for $^9$B. This is shown in Fig. 2(b). The arrow again indicates the expected ground state position, $Q_3 = 278$ keV. In Fig. 2(c) the $E_{decay}$ spectrum for the decay of $^5$Li to $\alpha+p$ ($Q_2 = 1967$ keV) is shown. This was obtained from the M = 3 ($\alpha p p$) events. No peak is seen at the ground state decay energy. However, the Q-value for the decay of $^6$Be to $^5$Li+$p$ is -595 keV. Therefore the available decay energy for $^5$Li$_{gs} \rightarrow \alpha+p$, for $^5$Li nuclei populated following the decay of $^6$Be, is 1967 - 595 = 1372 keV. Such decays proceed via the tail of the ~1.5 MeV wide $^5$Li$_{gs}$. A calculation (indicated by the dashed line in Fig. 2(c)) of the decay phase space and an *l*=1 centrifugal barrier indicates the yield from the decay is shifted away from 1372 keV towards $E_{decay}(^5\text{Li}) = 0$ keV, with 96 % of the $^5$Li$_{gs}$ events being in the 0 to 900 keV range. Gating within this range (900 keV is indicated by the vertical dotted line in Fig. 2(c)) allowed the $^5$Li$_{gs}$ events resulting from the decay of $^6$Be to be selected. The $E_{decay}$ spectrum for $^6$Be decay, reconstructed from the $^5$Li$_{gs}$ events and the second proton in the M = 3 ($\alpha p p$) data, is shown in Fig. 2(d). A clear peak is seen at the expected ground state energy of $Q_3 = 1372$ keV (indicated by the arrow).

The $^{10}$C $E_x$ spectra, obtained from M = 4 ($\alpha\alpha p p$) events and $E_x = E_{decay} - Q_4$, where $Q_4 = -3.727$ MeV, are shown in Fig. 3. The dot-dashed lines indicate Monte Carlo predicted detection efficiencies. The $^9$B+$p$ decay channel was selected by gating on $^8$Be$_{gs}$ events in the $^8$Be $E_{decay}$
spectrum reconstructed from the two α-particles. This is not shown, but is very similar in form to that for the M = 3 (ααp) data (Fig. 2(a)). Fig. 3(a) therefore corresponds to the ¹⁰C → ⁹B + p, ⁹B → ⁸Be + p, ⁸Be → α + α decay channel. Events not decaying via the ⁸BeGS are shown in Fig. 3(b). An additional gate, on the ⁶BeGS, was applied to this data in order to minimize the contribution from decays to the ⁸Be excited state at 3.04 MeV. The ⁶Be E<sub>decay</sub> spectrum for the M = 4 (ααpp) events is very similar in form to that for the M = 3 (αpp) data shown in Fig. 2(d). The events in Fig. 3(b) therefore correspond to the ¹⁰C → ⁶Be + α, ⁶Be → ⁵Li + p, ⁵Li → α + p decay channel. The peaks observed in the spectra have been fitted using Gaussian line shapes and a polynomial background. In Fig. 3(a) two peaks are observed, at (4.20 ± 0.01) and (5.31 ± 0.03) MeV and with widths of (280 ± 10) and (450 ± 90) keV (FWHM) respectively. The peak at 4.20 MeV appears to be a new, previously unreported, state in ¹⁰C. The 5.31 MeV peak corresponds to the known states at 5.22 and 5.38 MeV [3]. In Fig. 3(b) one peak is seen at (6.74 ± 0.04) MeV with a width of (570 ± 110) keV. This corresponds to the known state at 6.58 MeV [3]. The fitted peaks are indicted by the dotted lines in Fig. 3, the background by the dashed lines and the overall fits by the solid lines. The uncertainties given for the centroid energies are statistical only. An additional 200 keV systematic uncertainty in E<sub>x</sub> is typical in breakup work of this kind. The cross-sections for the ⁹B+p and ⁶Be+α breakup of ¹⁰C are (9.2 ± 1.9) and (3.3 ± 0.7) mb respectively. These values were obtained by integrating the experimental yields over a range of excitation energies, 4.0 to 7.0 MeV in the case of the ⁹B+p decay and 5.1 to 12.0 MeV for the ⁶Be+α channel.

<figure><img src="image_6.png" /><figcaption>**Figure 4.** Angular distributions for the (a) 4.20 and (b) 5.31 MeV states in ¹⁰C. The various lines are described in the text.</figcaption></figure>

Angular distributions for the 4.20 and 5.31 MeV states observed in the ⁹B+p decay of ¹⁰C are shown in Fig. 4. The angle ψ corresponds to that between the relative velocity vector of the ⁹B and p decay fragments and the beam (z) axis. Monte Carlo predicted angular distributions, obtained assuming isotropic decay, are indicated by the dotted lines. The ψ distribution for the 4.20 MeV state, shown in Fig. 4(a), appears to have little structure, although there is some evidence that the distribution has a minima at ψ ≈ 90°. This would suggest an odd order Legendre Polynomial should be used to describe the data. The dashed line in Fig. 4(a) corresponds to a Monte Carlo efficiency corrected Legendre Polynomial of order 1, and the dot-dash line to an efficiency corrected order 3 Legendre Polynomial. Neither appears to provide a
good description of the experimental distribution. The angular distribution for the 5.31 MeV state (Fig. 4(b)) does exhibit a much clearer periodic structure, however, and is well described by a Monte Carlo corrected Legendre Polynomial of order 2 (dashed line).

## 4. Discussion and Conclusion

There are four states in the mirror nucleus $^{10}$Be in the 4 to 7 MeV region that the present work is sensitive to: 5.958 MeV [2$^{+}$], 5.960 MeV [1$^{-}$], 6.179 MeV [0$^{+}$] and 6.263 MeV [2$^{-}$]. The 6.58 MeV state in $^{10}$C has been previously assigned a tentative spin/parity of (2$^{+}$) [3], and as such may well be associated with the 5.958 MeV [2$^{+}$] state in $^{10}$Be. The angular distribution for the 5.22 and 5.38 MeV states in $^{10}$C (unresolved as a single peak at 5.31 MeV in the present work) is well described by an order 2 Legendre Polynomial (Fig. 4(b)). As the proton decay of a 1$^{-}$ (2$^{-}$) state in $^{10}$C would proceed through *l*-values of 0, 2 (0, 2, 4) there would appear to be an association between the 5.22 and 5.38 MeV states in $^{10}$C and the 5.960 MeV [1$^{-}$] and 6.263 MeV [2$^{-}$] states in $^{10}$Be. The proton decay of a 0$^{+}$ state in $^{10}$C would decay via *l*=1. This suggests a link between the 6.179 MeV [0$^{+}$] state in $^{10}$Be and the 4.20 MeV peak in $^{10}$C, as the angular distribution for the 4.20 MeV state appears to require an odd *l* Legendre Polynomial to be described. The 6.179 MeV [0$^{+}$] state in $^{10}$Be has recently been shown to possess a highly deformed $\alpha+2n+\alpha$ molecular structure [4]. If the newly observed 4.20 MeV state in $^{10}$C is indeed the analogue of this state then it may well have an $\alpha+2p+\alpha$ configuration.

In summary, the breakup of the only known Brunnian nucleus, $^{10}$C, into $2\alpha + 2p$ has been measured at 33.3 MeV/nucleon. The cross-sections obtained indicate that the dominant decay is $^{9}$B+$p$, rather than $^{6}$Be+$\alpha$. A new state has been observed in $^{10}$C at an excitation energy of $E_x = 4.20$ MeV. This may well be the $\alpha+2p+\alpha$ analogue of the 6.179 MeV [0$^{+}$] $\alpha+2n+\alpha$ molecular state in $^{10}$Be.

## Acknowledgments

The assistance of the GANIL and LPC technical staff is gratefully acknowledged. This work has been supported by the European Community FP6 - Structuring the ERA - Integrated Infrastructure Initiative - contract EURONS n° RII3-CT-2004-506065.

## References

[1] M.V. Zhukov, B.V. Danilin, D.V. Fedorov, J.M. Bang, I.J. Thompson and J.S. Vaagen, Phys. Rep. **231**, 151 (1993).

[2] H.Brunn, Über Verkettung, S.-B.Math.-Phys.Kl. Bayer Akad. Wiss. **22**, 77 (1892).

[3] D.R. Tilley, J.H. Kelley, J.L. Godwin, D.J. Millener, J.E. Purcell, C.G. Sheu and H.R. Weller, Nucl. Phys. **A745**, 155 (2004).

[4] M. Freer, E. Casarejos, L. Achouri, C. Angulo, N.I. Ashwood, N. Curtis, P. Demaret, C. Harlin, B. Laurent, M. Milin, N.A. Orr, D. Price, R. Raabe, N. Soić, and V.A. Ziman, Phys. Rev. Lett. **96**, 042501 (2006).