<figure><img src="image_59.png" /><figcaption>FIG. 9. (Color online) Single-particle energies for neutron and proton states in $^{252}$No at axial equilibrium deformation of the HFB energy, where the charge quadrupole moment is $Q_0 = 13.75$ e b. The labels $[Nn_z\Lambda]$ are assigned by analogy with a Nilsson diagram.</figcaption></figure>

with $\hat{Z}$ and $\hat{N}$ as the proton and neutron number operators, respectively, and $\lambda_Z$ and $\lambda_N$ Lagrange multipliers.

The two-quasiparticle excitations are sought for using blocking calculations also performed in the same HFB framework. Here, a trial state $|\Phi'_{ij}\rangle = \eta_i^+ \eta_j^+ |\Phi_{ij}\rangle$ is defined, in which $\eta_i^+$ is a quasiparticle creation operator. The index $i$ (respectively $j$) refers to the set of quantum numbers of a one-quasiparticle orbital labeled $i$ (respectively $j$) and located in the vicinity of the Fermi energy of the unblocked HFB solution $|\Phi\rangle$. Then the equation

$$ \delta\langle\Phi'_{ij}|\hat{H} - \lambda_N \hat{N} - \lambda_Z \hat{Z}|\Phi'_{ij}\rangle = 0 \quad (9) $$

is solved, and the two-quasiparticle excitation energy is obtained as the difference [22]:

$$ E_{2qp}^{ij} = \langle \Phi'_{ij} | \hat{H} | \Phi'_{ij} \rangle - \langle \Phi | \hat{H} | \Phi \rangle. \quad (10) $$

If time-reversal symmetry is assumed, the signature partner pairs with angular momentum projections on the z axis, $K_- = K_1 - K_2$ and $K_+ = K_1 + K_2$, and parity $\pi = \pi_1 \cdot \pi_2$ are degenerate in energy.

The wave functions in the rotating frame actually are subject to fulfilment of spacial symmetries, namely, (i) parity operator $\hat{\Pi}$, (ii) z signature $\hat{S}_z = i\hat{R}_3(\pi)$ with $\hat{R}_3$ as the operator for rotation around the z axis, and (iii) the product $\hat{K} \cdot \hat{\Pi}_2$ of operators, where $\hat{K}$ stands for time-reversal symmetry and $\hat{\Pi}_2$ for reflection with respect to the x-z plane. These symmetries form a set of self-consistent symmetries which are commuting only for the vacuum HFB solutions of a system rotating around the z axis [50]. Furthermore, two-quasiparticle excitations rotating along the x axis do break the signature symmetry here taken along the z axis and defined as

$$ \hat{S}_z = ie^{-i\pi\hat{J}_z}, \quad (11) $$

where $\hat{J}_z$ is the projection of angular momentum onto the z axis. The z-signature is broken because $\hat{J}_x$, the projection of

angular momentum onto the x-axis, and $\hat{S}_z$ do not commute [51].

Within the HFB theory expressed in the rotating frame, the aligned two-quasiparticle internal structure of a nucleus with total angular momentum $I$, neutron number $N$, and proton number $Z$ is determined by solving the equation for the dynamical quasiparticle vacuum $|\Phi'_{ij,+}\rangle$:

$$ \delta\langle\Phi'_{ij,+}|\hat{H} - \omega\hat{J}_x - \lambda_N\hat{N} - \lambda_Z\hat{Z}|\Phi'_{ij,+}\rangle = 0, \quad (12) $$

with the dynamical constraint on the x component $J_x$ of the total angular momentum $I$ defined by

$$ J_x = \langle \Phi'_{ij,+} | \hat{J}_x | \Phi'_{ij,+} \rangle = \sqrt{I(I+1) - \langle J_z \rangle^2}, \quad (13) $$

where the indexes $ij$ stand for the blocked orbitals and the plus sign indicates aligned configurations.

In $^{250}$Fm and $^{252}$No, the two-quasiparticle signature partner bandheads with $\langle J_z \rangle^\pi = K_+^\pi = 8^-$ and $\langle J_z \rangle^\pi = K_-^\pi = 1^-$ quantum numbers are built from the neutron 9/2$^-$ and 7/2$^+$ levels closest to the Fermi energies (see Fig. 9). For both nuclei, breaking time-reversal symmetry removes the energy degeneracy of the signature partner $K_+^\pi = 1^-$ and $K_-^\pi = 8^-$ head levels. The two head levels display intrinsic excitation energies which differ by 200 keV. The aligned $K^\pi = 8^-$ two-quasiparticle head levels are lowest in energy. Excitation energies of the $K_-^\pi = 1^-$ and $K_+^\pi = 8^-$ solutions are as follows: $E_x(K_-^\pi = 1^-) = 1.138$ MeV, $E_x(K_+^\pi = 8^-) = 0.920$ MeV and $E_x(K_+^\pi = 1^-) = 1.195$ MeV, $E_x(K_+^\pi = 8^-) = 0.982$ MeV, for $^{250}$Fm and $^{252}$No, respectively. The arithmetic mean energies of these signature-partner pairs are $E_{m,2qp} = 1.029$ MeV and $E_{m,2qp} = 1.031$ MeV for $^{250}$Fm and $^{252}$No, respectively. Both mean energies are close to those obtained previously in blocking calculations without breaking time-reversal symmetry, namely, 1.010 MeV ($^{250}$Fm) and 1.070 MeV ($^{252}$No) [22]. These results are in qualitative agreement with expectations based on the Gallagher rule [43]. After correcting for zero-point energy $E_0 = 8\hbar^2/2\mathcal{J}^{(1)}$, that is, $E_0 = 49$ keV (51 keV) for $^{252}$No ($^{250}$Fm) as inferred from the kinematic moments of inertia $\mathcal{J}^{(1)}$ shown in Fig. 9, the calculated $I^\pi = 8^-$ head level energies are $E_x(I^\pi = 8^-) = 0.971$ MeV and $E_x(I^\pi = 8^-) = 1.031$ MeV for $^{250}$Fm and $^{252}$No, respectively. These calculated excitation energies are reasonably close to those found in experiments, namely, $E_x = 1.199$ MeV and $E_x = 1.254$ MeV for $^{250}$Fm and $^{252}$No, respectively.

# IV. MOMENT OF INERTIA

## A. General trend

The moments of inertia provide a tool to quantify the robustness and rigidity of the K isomer and the rotational band. The experimental kinematic and dynamic moments of inertia $\mathcal{J}^{(1)}$ and $\mathcal{J}^{(2)}$, respectively, are shown in Fig. 6 for the ground-state band (dashed black lines) and the $K^\pi = 8^-$ bands in the $^{250}$Fm and $^{252}$No (red triangles) isotones. Blue triangles represent theoretical results using the D1S Gogny force. Excellent agreement has been found for the ground-state bands [22] using the same theoretical framework. It is worth
noting that a good degree of accuracy is also obtained for the $K^\pi = 8^-$ two-quasiparticle isomeric band as far as the moment of inertia and the excitation energies are concerned. The agreement is in particular very good at low frequency, especially for $^{250}$Fm. One should again note some irregularities in the moment of inertia that will be discussed in Sec. IV B.

In general, the moment of inertia of the ground-state band increases with increasing frequency under the influence of the Coriolis antipairing force, which slowly aligns the angular momentum along the rotation axis. On the other hand, the two-quasiparticle configuration of the excited band blocks the neutron pairing, increasing the kinematic moment of inertia already at low frequencies. At $\hbar\omega \sim 0.12$ MeV, the gain is ~10% for $^{250}$Fm and reaches ~30% for $^{252}$No. This effect is well known in odd-mass nuclei compared to their even-even neighbors. More generally, the moment of inertia is expected to increase with the number of quasiparticles involved. It is important to stress that the difference in the moment of inertia between the ground-state and the $K^\pi = 8^-$ bands reflects a change in pairing energy, not in deformation. Indeed, the neutron pairing energy has been found null in the present blocking HFB calculations while a quadrupole moment of 13.75 e b is predicted for the ground state close to 13.84 e b for the $K^\pi = 8^-$ isomeric state. In $^{250}$Fm, the values are 13.39 e b (ground state) and 13.47 e b ($K^\pi = 8^-$).

In the $K^\pi = 8^-$ bands, the rotor is not only more rigid but also more stable with the rotation than the ground-state band. This can be inferred from the comparison of $\mathcal{J}^{(1)}$ and $\mathcal{J}^{(2)}$. Indeed, (i) the kinematic moment of inertia is almost constant as a function of the rotational frequency, and (ii) the difference between $\mathcal{J}^{(1)}$ and $\mathcal{J}^{(2)}$ is small (at least at low rotational frequency). In other words, there is little contribution from the unpaired nucleons as soon as the nucleus rotates. The alignment of the angular momentum along the rotation axis is small because of the robustness of the high-$K$ configuration.

## B. Anomaly in the moment of inertia

As mentioned above, the regularity of rotational bands changes at $\hbar\omega \sim 0.19$ MeV for $^{252}$No but also for $^{250}$Fm at a slightly higher frequency: the moment of inertia decreases and seems to recover its initial trend at higher frequency. The theoretical moments of inertia also display irregularities but they are less pronounced.

Irregularities in the moment of inertia often result from the crossing of bands having the same spin and parity. When the bands get closer in energy, they repel mutually, inducing some perturbations of the level energies, displacements that are magnified in the dynamic moment of inertia. One of the bands displays a bump in the moment of inertia while the interacting partner displays a dip. As another consequence of the adiabatic interaction, the two bands exchange their single-particle configuration along the crossing. In a more general context, the time-dependent interaction of quantum systems is known as a Landau-Zener crossing [52]. Many examples of band crossings are found along the nuclear chart, for instance in superdeformed bands [53].

<figure><img src="image_60.png" /><figcaption>FIG. 10. (Color online) Quasiparticle energies calculated vs spin $I$ for neutrons. The color code is for the occupation probabilities. Occupied orbitals are shown in red and empty ones in blue. States are labeled with quantum numbers $\pi$ (parity) and $\Omega$ (projection onto the $x$ axis of quasiparticle angular momentum).</figcaption></figure>

## 1. Theoretical analysis

To test the hypothesis of band crossing the investigation of Routhian properties has been carried out. The neutron quasiparticle energies in the rotating frame (Routhians) are shown in Fig. 10. They are labeled with quantum number $\pi$ (parity) and $\Omega$ (projection onto the $x$ axis of quasiparticle angular momentum at $I = 8\hbar$). Let us first note that the occupied quasiparticle orbitals labeled with the quantum numbers $\Omega^\pi = 9/2^-$, $7/2^+$ display energies that are almost constant as a function of spin at least up to spin $I = 25\hbar$. This is consistent with the robustness of the $K^\pi = 8^-$ quasiparticle configuration as a function of the rotation. Also, these calculations did not show any orbital crossing among occupied orbitals with $\pi = -$.

The irregularities in the calculated moments of inertia should be correlated to the evolution of another parameter. Inspection of the proton pairing energy as a function of spin (see Fig. 11) does not reveal any changes or a pairing collapse that could induce the anomalies in the theoretical moments of inertia.
<figure><img src="image_61.png" /><figcaption>FIG. 11. (Color online) Proton pairing energy for $^{250}$Fm and $^{252}$No.</figcaption></figure>

As customary in nuclear structure calculations, the axial and triaxial quadrupole deformations $\beta$ and $\gamma$ are defined as functions of the mass quadrupole moments $Q_{20}$ and $Q_{22}$, respectively, as

$$ \beta = \sqrt{\frac{\pi}{5}} \frac{\sqrt{Q_{20}^2 + 3Q_{22}^2}}{A\langle r^2 \rangle} \quad (14) $$

and

$$ \gamma = \arctan \sqrt{\frac{Q_{22}}{Q_{20}}}, \quad (15) $$

where $A$ and $\langle r^2 \rangle$ are the nuclear mass numbers and mean square radius of the mass distribution, respectively, and where

$$ \langle r^2 \rangle = \frac{3}{5} (r_0 A^{1/3})^2 \quad (16) $$

with $r_0 = 1.2$ fm. The deformation parameters have here been calculated using the above equations for $Q_{20}$ and $Q_{22}$ values calculated for each spin value from $I = 8\hbar$ to $I = 24\hbar$. Over this spin sequence $\beta$ and $\gamma$ values remain constant to less than 3% and 1%, respectively, for $^{252}$No and $^{250}$Fm. These features suggest no shape evolution through the bands, which may explain the irregular pattern displayed by the calculated dynamic moments of inertia (see Fig. 6).

## 2. Experimental band crossing

We next investigate whether the irregularities observed in the $J^{(2)}$ measurements stem from band crossing. One of the possible candidates to cross the $K^\pi = 8^-$ band at $\hbar\omega \sim 0.19$ MeV is the side band $K^\pi = 2^-$ . It is not excluded that the $K^\pi = 8^-$ band could instead cross an unobserved band such as $K^\pi = 7^-$ predicted by Delaroche et al. [22] and observed in the isotone $^{248}$Cf [54] at 1.5 MeV. Returning to the first hypothesis for which we have experimental information, we see that both bands have the same $\pi = -$ parity and similar bandhead excitation energies (see Fig. 4). To illustrate such a scenario we have plotted in Fig. 12 experimental excitation energies as a function of spin for the ground-state band (black

<figure><img src="image_62.png" /><figcaption>FIG. 12. (Color online) Energy vs spin for the ground-state rotational band (black dots), $K^\pi = 2^-$ excited band (blue triangles), and isomeric rotational band (red dots) for $^{252}$No. The dashed line is for an extrapolation of the observed $K^\pi = 2^-$ band to spins higher than $I = 7\hbar$.</figcaption></figure>

squares), the $K^\pi = 8^-$ band (red dots), the $K^\pi = 2^-$ band (blue triangles), as well as a smooth extrapolation of this latter band to spins higher than $I = 8\hbar$ (dashed line). As can be seen, the dashed curve crosses the red curve in the vicinity of the spin value $I = 17\hbar$, which is close to the rotational frequency $\hbar\omega \sim 0.19$ MeV where the shallow minima in the $J^{(2)}$ values is observed. Due to the lack of statistics and to hindered feeding, the higher frequency portion of the $K^\pi = 2^-$ band is not observed. It is therefore impossible to check whether this band would display a bump in $J^{(2)}$ values at approximately the same frequency where $J^{(2)}$ values of the $K^\pi = 8^-$ band are showing a bump. One should remember also that the $K^\pi = 2^-$ band is observed at low spin because it is fed by a decay-out branch of the $K^\pi = 8^-$ level. If the assumption of band crossing holds correct, the observed $K^\pi = 8^-$ band crosses the $K^\pi = 2^-$ band at $\hbar\omega \sim 0.19$ MeV, and the band configurations will get exchanged: the $K^\pi = 2^-$ configuration becomes energetically favored and preferentially fed at higher frequencies. Such an interpretation holds too for the excited bands in $^{250}$Fm where the crossing takes place at slightly different spin and frequency, namely, $I \sim 16\hbar$ and $\hbar\omega \sim 0.21$ MeV. Although $K$ is no longer a good quantum number at higher spins, our scenario would require almost degenerate unperturbed states. Hence the suggested scenario will remain tentative until new data collected in higher statistic measurements are available.
### 3. Discussion

The presence of side bands with spin and parity $2^{-}$ seems to be a common feature in $N = 150$ isotones (e.g., $^{252}$No, $^{250}$Fm [20], $^{248}$Cf [19], and $^{246}$Cm [16,18]). The similar and low excitation energy ($E_{x} \sim 1$ MeV) of the $K^{\pi} = 2^{-}$ bandhead level suggests a collective character (namely, octupole correlations). On the theoretical side, structure models are predicting $\pi = -$ levels at low excitation energy for this mass region [24,55]. Within the RPA the low-energy $K^{\pi} = 2^{-}$ level is calculated for $^{252}$No at $E_{x} = 0.998$ MeV [24], close to the experimental value. This study did not provide the major components of the $K^{\pi} = 2^{-}$ state wave function. Such structure information is available for $^{246}$Cm, another $N = 150$ isotone, for which the two-neutron $9/2^{-}[734]_{v} \otimes 5/2^{+}[622]_{v}$ and two-proton $3/2^{-}[521]_{\pi} \otimes 7/2^{+}[633]_{\pi}$ quasiparticle excitations form the dominant components of the lowest $K^{\pi} = 2^{-}$ state [55]. More recently, projected shell-model calculations performed for $N = 150$ isotones (including $^{252}$No) have focused on nonaxial octupole correlations [56]. It is interesting to note that the excitation energy range $E_{x} \sim 2.7-3.2$ MeV for states with spins from $I^{\pi} = 16^{-}$ to $I^{\pi} = 18^{-}$ in the calculated $K^{\pi} = 2^{-}$ band of $^{252}$No is in qualitative agreement with the energy $E_{x} \sim 2.7$ MeV where crossing between the $K^{\pi} = 2^{-}$ and $K^{\pi} = 8^{-}$ bands is expected to occur (see Fig. 12).

Obviously, the next generation of HFB calculation in the rotating frame would consist in breaking time-reversal symmetry, z-signature, as well as left-right symmetry. This task is beyond the scope of the present work.

If the interpretation as an octupole character for the $K^{\pi} = 2^{-}$ band and interpretation of the band crossing are correct, it would induce an interesting feature at high angular momentum. The $K^{\pi} = 0^{-}$ octupole vibration predicted close to the $K^{\pi} = 2^{-}$ component [24] contains only the $\alpha = 1$ signature sequence. The Coriolis mixing between $\alpha = 1$ states in $K^{\pi} = 0^{-}$ and $K^{\pi} = 2^{-}$ bands induces an odd-even signature splitting (see, for instance, [57]). As discussed previously the top of the $K^{\pi} = 8^{-}$ isomeric band should mutate to a $K^{\pi} = 2^{-}$ character with therefore an odd-even signature splitting feature. Evidence for such a staggering has been found in both $^{250}$Fm and $^{252}$No but the statistical uncertainties are too large to draw any conclusions. More statistics and/or extension of measurements to higher spins would clarify this assumption and more generally the anomalies in the moments of inertia.

### V. SUMMARY AND CONCLUSION

In this paper we reported on a detailed investigation of nonyrast states in $^{252}$No. In-beam γ-ray spectroscopy was performed at the University of Jyväskylä using the recoil-decay tagging technique, with an emphasis on collective states built on the two-quasiparticle $K^{\pi} = 8^{-}$ isomer. The study of this rotational band and decay toward the ground state provides valuable information on the single-particle structure, collectivity, and K isomerism. The large reduced hindrance factor and the strong stability of the rotational band support high purity of the K isomeric state, which persists with increasing rotational frequency. The single-particle configuration component in the $K^{\pi} = 8^{-}$ level is probed by the interband

$M1$ and intraband $E2$ experimental γ-ray intensity ratios that provide an indirect measurement of the $g_{K}$ gyromagnetic factor. As deduced from the data a two-quasiparticle neutron state $7/2^{+}[624]_{v} \otimes 9/2^{-}[734]_{v}$ is suggested for the $K^{\pi} = 8^{-}$ band. The $K^{\pi} = 8^{-}$ isomers in other $N = 150$ isotones with lower masses, i.e., $^{246}$Cm, $^{248}$Cf, and $^{250}$Fm, are thought to be built on the same two-neutron configuration.

The $J^{(2)}$ values inferred from γ-ray measurements for the $^{252}$No and $^{250}$Fm $K^{\pi} = 8^{-}$ bands were compared with HFB calculations in the rotating frame that break time-reversal and z-signature self-consistent symmetries. A good overall agreement between measurements and calculations is obtained for bandhead energies and moments of inertia. However, the calculations failed in reproducing $J^{(2)}$ values which display a shallow minimum observed at frequency $\hbar\omega \sim 0.18-0.22$ MeV for $^{252}$No. This feature is tentatively interpreted as a crossing between the $K^{\pi} = 8^{-}$ and $K^{\pi} = 2^{-}$ bands, the latter band not being observed for spins beyond $I = 7\hbar$. We do not exclude the possibility that the $K^{\pi} = 8^{-}$ band might instead cross another unobserved band. Measurements with higher statistics would be a valuable asset to clarify the issue. The systematic occurrence of low-energy $K^{\pi} = 2^{-}$ excitations in $N = 150$ isotones is strongly suggestive of octupole correlations. Such correlations are ignored in the present microscopic model. However, they could be handled by breaking one more self-consistent symmetry, namely, the left-right symmetry. The hope is that extended cranking HFB calculations would (i) shed light on the interpretation of irregularity observed in the $J^{(2)}$ moment of inertia of the $K^{\pi} = 8^{-}$ band at frequency $\hbar \approx 0.18-0.22$ MeV and (ii) provide a guide to infer quantum numbers of the quasiparticle components of the $\pi = -$ crossing band. Such calculations are under consideration. It would also be a valuable task to challenge predictions based on quasiparticle random phase approximation (QRPA) calculations [58], provided that the theory is extended to the rotating frame. Such an approach proved successful previously for the interpretation of back-bending phenomena observed in $\pi = -$ superdeformed bands of Hg isotopes [59,60]. Running QRPA calculations in the rotating frame with the D1S force as sole input would be a formidable task that most likely is manageable with the availability of algorithms and computers of the next generation.

Finally, it would be interesting to extend the present measurement of $K^{\pi} = 8^{-}$ band at high spin in other $N = 150$ isotones, e.g., $^{246}$Cm and $^{248}$Cf, and obviously in the heavier $^{254}$Rf nuclide. Extension of the $K^{\pi} = 8^{-}$ rotational bands in $^{250}$Fm and $^{252}$No to spins higher than $I^{\pi} = 22^{-}$ would also help to clarify the anomalies in the moment of inertia.

### ACKNOWLEDGMENTS

We thank the accelerator staff at the University of Jyväskylä for the excellent performance of the ion source and accelerator. H. Goutte is acknowledged for involvement in developing the cranking HFB code. This work has been supported by the EU-FP6-I3 Project EURONS No. 506065, the Academy of Finland [CoE Nuclear and Accelerator Based Physics Programme at JYFL, grants to PTG (111965) and CS (209430)], the EC "Marie Curie Actions" grant, the UK STFC, and the US
Department of Energy (Contract No. DE-AC02-06CH11357). MV appreciates support from the Slovak grant agency VEGA (Contract No. 2/0105/11). SA was supported by the Slovak Research and Development Agency Contract No. APVV-0105-10 and VEGA Contract No. 1/0613/11. We thank the

UK-France (STFC/IN2P3) Loanpool and the GAMMAPOOL European Spectroscopy Resource for the loan of detectors for JUROGAM. This work has benefited from the use of TNT2 cards, developed and financed by CNRS/IN2P3 for the GABRIELA project.

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7.5 Musett: A segmented Si array for Recoil-Decay-Tagging studies at VAMOS
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Musett: A segmented Si array for Recoil-Decay-Tagging studies at VAMOS

<figure><img src="image_65.png" /></figure>

CrossMark

Ch. Theisen<sup>a,*</sup>, F. Jeanneau<sup>a</sup>, B. Sulignano<sup>a</sup>, F. Druillole<sup>a</sup>, J. Ljungvall<sup>b</sup>, B. Paul<sup>a</sup>, E. Virique<sup>a</sup>, P. Baron<sup>a</sup>, H. Bervas<sup>a</sup>, E. Clément<sup>c</sup>, E. Delagnes<sup>a</sup>, A. Dijon<sup>c,1</sup>, E. Dossat<sup>a</sup>, A. Drouart<sup>a</sup>, F. Farget<sup>c</sup>, Ch. Flouzat<sup>a</sup>, G. De France<sup>c</sup>, A. Görgen<sup>a,2</sup>, Ch. Houarner<sup>c</sup>, B. Jacquot<sup>c</sup>, W. Korten<sup>a</sup>, G. Lebertre<sup>c</sup>, B. Lecornu<sup>c</sup>, L. Legeard<sup>c</sup>, A. Lermitage<sup>d</sup>, S. Lhenoret<sup>a</sup>, C. Marry<sup>c</sup>, C. Maugeais<sup>c</sup>, L. Menager<sup>c</sup>, O. Meunier<sup>a</sup>, A. Navin<sup>c</sup>, F. Nizery<sup>a</sup>, A. Obertelli<sup>a</sup>, E. Rauly<sup>d</sup>, B. Raine<sup>c</sup>, M. Rejmund<sup>c</sup>, J. Ropert<sup>c</sup>, F. Saillant<sup>c</sup>, H. Savajols<sup>c</sup>, Ch. Schmitt<sup>c</sup>, M. Tripon<sup>c</sup>, E. Wanlin<sup>d</sup>, G. Wittwer<sup>c</sup>

ARTICLE INFO

Article history:

Received 18 July 2013

Received in revised form

6 February 2014

Accepted 6 February 2014

Available online 17 February 2014

Keywords:

Recoil separator

Alpha spectroscopy

Silicon detector

Recoil decay tagging

Wien filter

ABSTRACT

A new segmented silicon-array called MUSETT has been built for the study of heavy elements using the Recoil-Decay-Tagging technique. MUSETT is located at the focal plane of the VAMOS spectrometer at GANIL and is used in conjunction with a γ-ray array at the target position. This paper describes the device, which consists of four 10 × 10 cm<fnref n="2" /> Si detectors and its associated front-end electronics based on highly integrated ASICs electronics. The triggerless readout electronics, the data acquisition and the analysis tools developed for its characterization are presented. This device was commissioned at GANIL with the EXOGAM γ-ray spectrometer using the fusion-evaporation reaction ¹⁹⁷Au(²²Ne, 5n)²¹⁴Ac. Additionally, the performance of the VAMOS Wien filter used during the in-beam commissioning is also reported.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

The study of the heaviest elements having extreme mass and charge is a major topic in nuclear physics. These elements are artificially produced using fusion-evaporation reactions and suffer from low production cross-sections, which decrease dramatically with atomic number. While decay studies have been performed for decades, prompt γ-ray or conversion-electron spectroscopy of transfermium elements (atomic number larger than 100) is more difficult and the associated challenges have been overcome since 1998, with the pioneering study of ²⁵⁴No at the Argonne National Laboratory (USA) and at the University of Jyväskylä, Finland (see Ref. [1] for a recent review).

These studies require highly efficient and selective devices to detect and unambiguously identify the rare isotopes produced in an overwhelming background of unwanted reactions. The optical elements of a zero degree separator or spectrometer provide the basic selection needed. Since the rejection of projectiles and parasitic reactions is never ideal, the focal plane (FP) detection setup provides the additional identification of the transmitted reaction products. Prompt spectroscopy experiments are possible thanks to the Recoil-Tagging (RT) and Recoil-Decay-Tagging (RDT) techniques [2]. In the former case, the prompt transitions emitted around the target are filtered when a recoil having the expected characteristics (for instance energy, Time-of-Flight (ToF), energy loss, etc.) is identified at the FP of the spectrometer/separator. The RDT technique provides an additional and in most cases univocal identification, exploiting the decay characteristics of implanted nuclei. This technique uses time and position correlations between the implanted nuclei and its subsequent decays (e.g. α-decay).

At GANIL, the VAMOS (VAriable MOde Spectrometer) high acceptance ray-tracing spectrometer was initially designed for the study of exotic nuclei produced with SPIRAL1 radioactive

<footnote><sup>a</sup> CEA, Centre de Saclay, IRFU, F-91191 Gif-sur-Yvette, France</footnote>

<footnote><sup>b</sup> CSNSM, F-91405 Orsay Campus, France</footnote>

<footnote><sup>c</sup> GANIL, Boulevard Henri Becquerel, BP 55027 - F-14076 CAEN Cedex 05, France</footnote>

<footnote><sup>d</sup> IPNO, 15 rue Georges Clémenceau, F-91406 Orsay, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: christophe.theisen@cea.fr (Ch. Theisen).</footnote>

<footnote n="1">¹ Present address: CEA Bruyères-le-Châtel, DIF/DPTA/SPN, F-91297 Arpajon, France.</footnote>

<footnote n="2">² Present address: Department of Physics, University of Oslo, Oslo, Norway.</footnote>
beams, for a wide range of reactions such as transfer or deep-inelastic [3–5]. A zero degree operation for fusion–evaporation reaction studies was not a primary focus of VAMOS. However, a separation between the fusion–evaporation residues and the direct beam can be obtained either with its Wien Filter (WF), or in a gas-filled mode [6].

VAMOS can be coupled to the prompt γ-ray spectrometer EXOGAM [7,8]. EXOGAM is made of germanium clover detectors assembled in a close geometry around the target. The 12-clover configuration compatible with VAMOS provides a large efficiency of ~11% at 1.33 MeV, and is ideal for detecting medium-spin cascades emitted by heavy nuclei. In the near future the coupling with the γ-tracking array AGATA [9] will provide a further increase in efficiency and sensitivity. The coupling of EXOGAM and AGATA scheduled in 2014–2016 is expected to reach a photopeak efficiency of ~15% at 1.33 MeV along with an improved Doppler correction. The VAMOS zero degree mode combined with EXOGAM/AGATA will be suited and efficient for spectroscopic studies using fusion–evaporation reactions. A series of in-beam tests have been consequently conducted to characterize VAMOS for RT and RDT studies.

A first test of VAMOS as a zero degree separator was performed using the reaction $^{208}\text{Pb}(^{18}\text{O}, 3-4n)^{222-223}\text{Th}$. The α decay of $^{223,223}\text{Th}$ and their daughters could be measured. The known level scheme of the $^{222}\text{Th}$ octupole band could be built using γ-γ coincidences.

A second characterization of VAMOS and its WF was performed, using the idea<fnref n="3" /> to direct the beam “straight through” thus allowing a better control of the beam (see details in Section 4.2). During this test the transmission for the fusion–evaporation reaction $^{197}\text{Au}(^{22}\text{Ne}, 5n)^{214}\text{Ac}$ was measured.

In parallel, the VAMOS gas-filled mode was successfully tested [6]. Unprecedented transmissions have been deduced for the $^{40}\text{Ca} + ^{152}\text{Sm}$ reaction (80% for α xn and above 95% for xn yp channels).

Finally, the FP array MUSETT (MUr de Silicium pour l'Etude des Transfermiens par Tagging - Silicon wall for the study of transfermium by tagging) segmented Si wall built for RDT studies at VAMOS was commissioned. The segmented Si wall and its new electronics and data acquisition system were tested at VAMOS (using the WF mode) coupled to EXOGAM using mainly the $^{197}\text{Au}(^{22}\text{Ne}, 5n)^{214}\text{Ac}$ reaction.

This article describes the technical aspects of the MUSETT Si wall and the results of the in-beam commissioning. It is organized as follows: MUSETT is described in the next section. The test-bench and characterization are then detailed in Section 3. Section 4 is devoted to the in-beam commissioning, including the performance of the VAMOS WF. Finally, a summary and perspectives are given in Section 5. Supplementary material related to this article can be found online.

# 2. The MUSETT Si array

## 2.1. Specifications and constraints

MUSETT is originally designed for the VAMOS FP and has therefore to be adapted as much as possible to its ion optics, which is dispersive in the horizontal plane. Since the fusion–evaporation residues produced in the most asymmetric reactions carry low kinetic energy (typically 0.05 MeV/A or less), the implantation depth is shallow (a few µm). The dead layer at the entrance of the detector should therefore be minimized. As mentioned earlier, using the RDT technique requires energy, time

and position measurement of the recoil and its subsequent decay. This is usually achieved using Double-sided Silicon Strip Detectors (DSSDs).

The large number of electronics channels required (for about 1000 strips) rules-out the choice of Font-End Electronics (FEE) preamplifiers using discrete components not only because of the volume constraints, but also due to the large number of corresponding feedthroughs. ASICs with multiplexed output are the only viable solution.

The energy resolution is a critical requirement and is necessary to resolve α-decay fine structures (i.e. better than 1%). On the other hand, the time resolution is not a critical issue in the present application. In the case of RDT experiments, the electronic dead-time is not only constrained by the average detection rate (typically a maximum of a few kHz) but a more severe constraint is related to the minimum lifetime of the nuclear states of interest. We would like to measure the times between the implantation and subsequent decay in the same detector element as low as few tens of µs. In the case of RDT experiments, events separated by long periods of time, up to minutes or more need to be correlated in time. This can be achieved using a universal clock.

It should be pointed out that the above-mentioned specifications overlap to a large extent with that of the MUST II array [10] designed mainly for direct reaction studies at GANIL. MUSETT capitalized extensively on the MUST II developments.

## 2.2. Technical description

We present below the silicon detectors and their FEE. The infrastructure needed and the integration performed to operate the detector modules is then described.

### 2.2.1. Si detectors

The MUSETT silicon array is made of four modules of ~10 × 10 cm² each, to obtain a total detection area of ~40 × 10 cm², which covers the recoil implantation profile in both VAMOS WF and gas-filled modes. The detectors are to a large extent based on the MUST II detectors with substantial improvements to cope with the implantation and detection of very heavy nuclei and their subsequent decay.

Each n-type, DC coupled DSSD is 300 µm thick and has 128 strips on each side (X-Y readout) : see Fig. 1. The crystal orientation is (100). X (Y) strips correspond to the front junction (back ohmic) side of the detector. Particles enter the detector from the junction side. The junction p+ implantation is made of a shallow Boron implantation of less than 0.25 µm (typically 0.1 µm). The front side segmentation is made with an aluminium contact grid 3000 Å thick whose total area is of 2.73% coverage only. There is no other metallic or oxide layer on the front side in order to reduce the energy loss for the incoming heavy nuclei. On the back ohmic side, the contacts consist of Al strips. The strip width on both sides is 700 µm with a pitch of 760 µm. The strip separation (interstrip) is therefore 60 µm. The thickness of the $\text{SiO}_2$ layer between the p+ front strips is 1 µm. The isolation between the n+ back strips is made with p-stop structures. The strip length is 97.220 mm for each side. The Si detector is mounted on a 2.4 mm epoxy (FR4) printed circuit board. The guard rings are kept floating. The Si detectors were provided by Micron Semiconductors, UK [13] and are known as “TTT3 design” in the catalogue.

The high voltage (negative) is applied on the front junction face of the Si detectors, which is therefore AC coupled to the FEE through polarization circuits. The depletion voltage of all detectors is less than –60 V for which the leakage current does not exceed 1 µA at room temperature for a new detector. Fig. 2 shows examples of capacitance as a function of the bias voltage (courtesy of Micron Semiconductors). A typical value is ~4 nF above the

<footnote n="3">³ Suggested by M.R.</footnote>
<figure><img src="image_66.png" /><figcaption>**Fig. 1.** Close view of the front p+ face (X strips) of a MUSETT DSSD taken with a microscope.</figcaption></figure>

<figure><img src="image_67.png" /><figcaption>**Fig. 2.** Capacitance as a function of the bias voltage for two MUSETT detectors having a depletion voltage of –54 V (serial number 2577-10, top) and –30 V (serial number 2501-16, bottom). Arrows indicate the nominal depletion voltages. Results provided by Micron Semiconductors [13].</figcaption></figure>

depletion voltage. Fig. 3 shows examples of leakage currents as a function of the bias voltage measured for different temperatures.

### 2.2.2. ATHED ASICs and COFEE board

Given the high number of electronics channels needed and the space constraints, highly integrated ASICs electronics have been used. The ATHED chip (ASICs for Time and High Energy Deposit) is the latest version of the MATE chip [12] developed for the MUST II project. The ASICs that operate 16 strips are grouped by 4 units on the COFEE<fnref n="4" /> mother boards (designed by IPN Orsay), which provide

<figure><img src="image_68.png" /><figcaption>**Fig. 3.** Leakage current as a function of the bias voltage for two MUSETT detectors (the same detectors as in Fig. 2: serial number 2577-10 on the top, serial number 2501-16 on the bottom) for three different temperatures. The measurement for detector 2501-16 was performed before and after an in-beam measurement (fluence of approximately 10⁶ heavy-ions/cm², mainly ²²Ne ions in the energy range 10–110 MeV). Arrows indicate the nominal depletion voltages.</figcaption></figure>

the power supply, input/output and slow control interfaces. Consequently, four COFEE boards are needed to operate a DSSD.

ATHED delivers three types of analogue information per channel: time, energy and the leakage DC detector current (see Fig. 4). ATHED also gives a trigger logic signal corresponding to the crossover above an adjustable threshold value.

The first stage is a Charge Sensitive Amplifier (CSA), based on a single-ended folded cascode architecture designed to convert both polarities (for both sides of the DSSD) of the input signal. The Rf specific block in parallel to the feedback capacitor Cf defines the value of a 60 MΩ equivalent feedback resistor, and the DC voltage level of the amplifier output. Four different values of the feedback capacitance can be selected, leading to different energy ranges. The most used ranges are ±45 MeV (Cf = 2.6 pF) and ±0.5 GeV (Cf = 27.6 pF). The output signal is processed in two separate blocks: energy and timing. The energy stage is composed of a CR-RC filter with two different possible values of the peaking time (1 or 3 µs). The amplitude of the shaped signal is memorized on the external hold signal managed by the readout card, via a track and hold stage.

The timing stage incorporates a fast shaper, a discriminator and a Time to Amplitude Converter (TAC). The shaper has a differential structure, with a CR-RC filter for a 22 ns peaking time. The threshold voltage is given by an internal 8 bit programmable Digital to Amplitude Converter, common to the 16 channels. The discriminator output delivers the start signal to the TAC. The OR between the 16 individual start signals alerts the readout card that an event has been detected. The TAC is stopped by an external stop signal. Two TAC ranges can be selected: 300 or 600 ns. Any of the 16 channels can be independently inhibited.

In the readout process, the analogue energy and timing information of the 16 channels are multiplexed and transmitted serially at a rate of 2 MHz through a voltage to current converter to the external Analogue to Digital Converter (ADC) inputs (analogue bus, differential signals). The triggering of the ATHED readout is common to the COFEE board (64 channels): all channels are read-out if any of the channels initiated the sequence (request mode). It is also possible to trigger the read-out by an external source, namely the readout electronics without any discriminator response. This so-called "Trig event" mode is used e.g. in pedestal calibration, leakage current measurement, and detection chain characterization. The slow control of ATHED is defined and carried on a standard serial I2C bus. ATHED is manufactured using the AMS BICMOS 0.8 µm technology. The die area is 42 mm² for a number of ~16,000 transistors.

<footnote n="4">⁴ COFEE stands for COulex Front-End Electronics; besides MUSETT, one of our goals is to use this multipurpose electronics with annular Si detectors for Coulomb excitation studies.</footnote>
<figure><img src="image_69.png" /><figcaption>Fig. 4. Schematic view of ATHED.</figcaption></figure>

<figure><img src="image_70.png" /></figure>

<figure><img src="image_71.png" /><figcaption>Fig. 5. (a) Picture of the MUSETT silicon array consisting of 65,536 pixels with the cabling for the slow control, voltages and acquisition signals. (b) View of the tile structure with kapton cables, the electronics and the copper cooling structures.</figcaption></figure>

The slow control of each ASIC also allows controlling a multi-
plexer, selecting the inspection line of a channel, and controlling
an internal generator to test all inputs from the ASICs. Each ASIC is
read out on a separate analogue bus to reduce the transfer time to
16 µs. Since only the junction side of the Si detector is biased, only
two of the four COFEE boards need to be AC coupled to the
detector. This function is realized with daughter board polarization
circuits. The value of the coupling capacitances is 10 nF. In the
Saclay test bench environment, each COFEE board is connected to
a CVM card to manage the configuration and the data, while in the
GANIL environment, four COFEE boards are connected to a MUVI
VXI card (see Sections 3.1.1 and 4.1).

### 2.2.3. Detector module and integration

The Si wall has been designed as a set of four identical detector
modules (see Fig. 5). The idea of identical detector modules is to
simplify the wall integration and to have interchangeable modules.
All modules include the components needed to operate the Si
detector: the FEE, the connectors, the mechanics, etc. The frame of
a module is made of a copper structure in which a cooling fluid
circulates to dissipate the ASICs power and to keep it at a constant
temperature. The Si detector performance also improves due to
the cooling. The connection between the detector and the COFEE
electronics is made by kapton flex cables (see Fig. 5). In order to

reduce the dead zones (Si detector printed circuit board), the
detectors are arranged in a tile geometry.

### 2.2.4. Infrastructure of the Si array

The MUSETT array needs to be as far as possible remotely
controlled since access to the experimental room is not possible
during in-beam experiments. The high and low voltages are
provided by a CAEN [11] SY 2527 system. The COFEE FEE require
5 low voltage lines (± 2.5 V, ± 5 V, +3.3 V) supplied by four CAEN
floating low voltage A1517B and A1518B boards. These boards
provide the voltage stability using remote sensing lines. The high
voltage is supplied by a A1837N CAEN module. A LAUDA low-
temperature thermostat RP845 [14] is used for the detectors and
FEE cooling. The WINTHERM® Plus software [15] is used to control
and monitor the cooling system.

## 3. Characterization

This section describes the environment developed to charac-
terize the Si detector modules. Besides the FEE described pre-
viously in Section 2.2.2, the test bench (Section 3.1) installed at
Saclay using VME electronics (Section 3.1.1) with its DAQ system
(Section 3.1.2) is described. Analysis procedures and examples of
detector characterization are given in subsections 3.1.3 and 3.2.

3.1. Test bench

3.1.1. Readout: the CVM boards

The CVM board “Carte VME Multi-Application” (multi-applica-
tion VME Board) has been designed as a multipurpose and “light”
electronics for characterization, integration or eventually in-beam
experiments. The idea is to have a unique hardware but different
firmwares adapted to the various operating modes.

The architecture of the laboratory test bench is sketched in
Fig. 6. One MUSETT module is read-out by 4 COFEE boards each
including 4 ATHED. Each COFEE board is controlled and acquired
by one CVM board. Hence, the full MUSETT test bench is made of
four slave CVM boards plus one CVM board in the VME master
mode. A common trigger is not used, therefore the four CVM slave
cards acquire data independently. A windows-based PC controls
the entire system through the USB interface of the CVM master
card, namely the STUC ("Sonde de Test USB Configurable" - Config-
urable USB Test Probe) generic USB daughter card interface.

A CVM is connected to the four serial analogue buses and digitizes the energy and time information from each ASIC. The most complex function on the board consists of four 14 bit ADC channels corresponding to the 4 ASICs analogue bus of a COFEE board. After digitization at a 2 MHz rate (16 µs for the 16 energy and time amplitudes), CVM data are pushed into a FIFO.

Data are time-stamped using a common 48 bit 50 MHz clock.
The CVM master VME interface is connected to the PC via the USB
link; it interfaces with the slave modules, which communicate
with the master board through the VME bus.

3.1.2. DAQ

To control the CVM system, a data acquisition system has been
developed using a client/server architecture: see Fig. 7. The
experiment control and monitoring is made using the LabVIEW
[16] framework. It includes the hardware slow control (CVM,
COFEE, ATHED) and the monitoring part of the data flow. The data
communication is made through a daemon server that waits for
the LabVIEW client requests, and converts the commands into USB
functions for the hardware configuration.

The multithread daemon also handles the data flow. It scans
whether the CVM slave boards have available data in their FIFO
<figure><img src="image_72.png" /><figcaption>Fig. 6. Layout of the MUSETT test bench.</figcaption></figure>

<figure><img src="image_73.png" /><figcaption>Fig. 7. Test bench DAQ architecture.</figcaption></figure>

and if so, initiates a direct memory access sequence to read out the data. An event corresponds, at this level, to the digitized signals from 64 strips with its associated time-stamp. Part of the data flow can be sent to the LabVIEW program for monitoring.

### 3.1.3. Data analysis

Data analysis tools have been developed on linux using the ROOT [17] framework. The analysis is performed in different steps using as an initial input the data provided by the DAQ: raw data decoding, pedestal calibration, time ordering, $\alpha$ calibration, gain matching, and event building. Since the number of detector channels is relatively large, automatic procedures have been implemented for pedestal alignment, $\alpha$ calibration, data quality diagnostics, etc. Since the four CVM slave boards are triggerless, an event-builder based on time correlations has been implemented. Diagnostics are also provided to characterize the system e.g. missing channels, anomalous pedestal or width of an $\alpha$ peak. Examples of detector performance resulting from the data analysis are given in the next section.

## 3.2. Detector performance

We summarize in this section the representative characteristics of the detectors and electronics using the Saclay test bench. Measurements for only one detector (serial number 2501-16, depletion voltage $V_{dep} = -30$ V) are presented in this section (other detectors behave in a similar manner). All measurements presented in this section were obtained using the ASICs preamplifier

with a feedback capacitance $Cf=2.6$ pF ($\pm$ 45 MeV range). All source measurements were performed in the vacuum at a pressure less than $10^{-5}$ mbar.

### 3.2.1. Energy response, noise level

The noise of the whole chain can be evaluated measuring the pedestal width. This measurement is performed using the “trigger mode” i.e. the electronics readout is triggered using a logic pulse generator, without the need of any detector response. Without a detector, the pedestal width is ~1.0 bit rms. With a detector, the value is ~1.2 bits rms with only small changes as a function of temperature (from 6 to 24 °C), bias voltage or peaking time for a fully depleted detector but not over-biased (up to –120 V for the detector presented in this section). Thus, the intrinsic chain resolution is ~20 keV FWHM. The minimum threshold measured at 6 °C with the MUSETT detection chain is ~200 keV (~300) keV for the X (Y) detector side.

The detector resolution has been measured using a triple $\alpha$ source ($^{239}$Pu, $^{241}$Am, $^{244}$Cm) illuminating the junction side. The source was positioned ~10 cm above the detector. Measurements were performed as a function of the temperature, bias voltage, for both 1 and 3 µs peaking times. Again, small variations are observed as a function of the bias voltage (for a given temperature and peaking time) for a fully depleted detector (the parallel noise is negligible). For a very high voltage (or current), not only does the resolution increase because of the parallel noise, but also some strips begin to malfunction with e.g. asymmetric peaks, spurious peaks, and coupling between neighbouring strips.

Table 1 summarizes the resolutions at a bias voltage of $2V_{dep} = -60$ V, $T=6$ and 24 °C with 1 and 3 µs peaking times for the $^{244}$Cm $\alpha$ energy at 5804.8 keV. In the analysis, the peaks corresponding to the transitions of $^{239}$Pu at 5105.5, 5144.3 and 5156.6 keV, of $^{241}$Am at 5442.8 and 5485.6 keV, and of $^{244}$Cm at 5762.6 and 5804.8 keV were fitted simultaneously with 7 Gaussian curves. Single spectra were used without any coincidence condition or cut. The X junction side is superior compared to the Y ohmic (back) side. The resolution is better for a peaking time of 3 µs, where the serial noise is lower. This is true only if the detector current is low: when the current is above ~3 µA, the parallel noise increases and becomes the main contribution. In this case, it is better to use a 1 µs peaking time.
<caption>**Table 1**</caption>

Resolution obtained at 5804.8 keV for different temperatures and peaking times, at a bias voltage of $2V_{dep} = -60$ V.

<table><thead><tr><th>Peaking time</th><th>$T=24$ °C</th><th>$T=6$ °C</th></tr></thead><tbody><tr><td rowspan="2">$\tau = 1$ µs</td><td>$FWHM(X) = 41.4$ keV</td><td>$FWHM(Y) = 32.6$ keV</td></tr><tr><td>$FWHM(Y) = 44.1$ keV</td><td>$FWHM(Y) = 34.8$ keV</td></tr><tr><td rowspan="2">$\tau = 3$ µs</td><td>$FWHM(X) = 33.6$ keV</td><td>$FWHM(X) = 27.3$ keV</td></tr><tr><td>$FWHM(Y) = 37.1$ keV</td><td>$FWHM(Y) = 31.3$ keV</td></tr></tbody></table>

<figure><img src="image_74.png" /><figcaption>**Fig. 8.** (a) Example of α spectra. Full (dotted) lines correspond to the X (Y) detector side. The measurement was performed at $T=6$ °C and 3 µs peaking time. (b) FWHM strips resolution for the $^{244}$Cm peak at 5805 keV. Blue triangles: $T=24$ °C, 1 µs peaking time. Red circles: $T=6$ °C, 1 µs peaking time. Green squares: $T=6$ °C, 3 µs peaking time. All measurements were performed at a bias voltage of $2V_{dep} = -60$ V. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)</figcaption></figure>

As shown in Fig. 8b a small dispersion between strips is obtained. For a peaking time of 3 µs and at $T=6$ °C, the resolution worsens at one edge of the detector (large Y strip number). This is actually due to the kapton cables, which are the longest for these strips, increasing the capacitance and therefore both 1/f and serial noises. This effect is lower when using a 1 µs peaking time and unobservable when the detector current is larger. It is worth noting that all strips perform properly. The total α spectra for both sides ($T=6$ °C, $HT=-60$ V, $\tau=3$ µs) are shown in Fig. 8a, where one clearly sees the fine structures of the α decays.

Tests were also performed with a prototype detector from the S3 collaboration (Super Separator Spectrometer [18]). This detector is known as “TTT5” in the Micron Semiconductors catalogue. It is inspired to a large extent from MUSETT with a lower capacitance and lower leakage current. With a peaking time of 3 µs, an average resolution of 25.5 keV is obtained on the X side, while the best strips have a resolution of ~22 keV. Detailed results will be reported elsewhere.

### 3.2.2. Cross-talk

A cross-talk corresponds to the influence of a particular strip on its neighbouring strips and/or neighbouring electronics channels.

With the MUSETT FEE, neighbouring strips do not necessarily correspond to neighbouring FEE channels. This dissimilarity between strips and electronics channels was made to facilitate the COFEE card rooting. This feature allows us to disentangle the strips and electronics cross-talk as well as to deduce the cross-talks from α-source measurements.

The electronics cross-talk is found only between a responding channel *n* and the next one (channel *n*+1) of each ASIC. It is interpreted as a multiplexing residue in the serial analogue bus. In the worst case a value of 0.4% has been measured. Should it be necessary, electronics cross-talk can be corrected off-line. It is important to point-out that the electronics channels *n* and *n*+1 are not supposed to respond simultaneously in physical events.

The strips cross-talk corresponds to the pedestal shift of the two neighbours of the firing strip. In contrast to the electronics cross-talk, this effect is symmetric and affects equally both neighbours. The strip cross-talk is negligible on the Y ohmic back side, and is measured to be 0.5% of the main amplitude on the X junction front side, which indicates that it is related to the AC coupling mode on the junction side: the collected charge is split between the decoupling and interstrip capacitances. The pedestal shift depends neither on the temperature, nor on the detector, since the detector capacitances are geometric features that do not change significantly as soon as the detector is depleted. Note again the small cross-talk level (~0.5%). Moreover, it influences neighbours that in most cases do not respond simultaneously, except in the case of interstrip events discussed in the following section.

### 3.2.3. Interstrip events

When electron-hole pairs are generated in the X or Y interstrip regions, signals can be collected on adjacent strips. The detector response for these events can be visualized using the two dimensional plot (E1, E2), where E1 and E2 are the energies detected in the two neighbouring strips: see Fig. 9. Such events have been discussed in several papers e.g. [19–26] as charge carrier sharing and/or trapping in the interstrip SiO$_2$ region, but with somehow different interpretations or formalisms. Basically, electron-hole pairs are separated by the electric field and consequently move toward the electrodes. The signal amplitude and shape results from the migration of both electrons and holes. The observed interstrip patterns (Fig. 9) result actually from two effects. First, charge carriers can be collected on adjacent electrodes (charge sharing). Second, the front and/or back signals amplitude can be lowered if electrons and/or holes trapping occurs. Charge trapping corresponds to losses close to the Si–SiO$_2$ interface, resulting from a high-density electron accumulation layer. Details of the trapping and consequences on the signals are given in Ref. [26]. In the following, we outline briefly the main features of interstrip events and their consequences on the data analysis.

On the X side, the plots are characteristic of incomplete charge collection. Region (1) shown in Fig. 9 corresponds to hits near the edge of the interstrip. Holes are collected on the nearest strip, but electrons are partially trapped close to the Si–SiO$_2$ interface, which reduces the signal on the adjacent X strips by $-\Delta Q_t$ (remember that the total signal results from both holes and electrons moving to the strips). One therefore measures $(Q_{particle} - \Delta Q_t, -\Delta Q_t)$. This region extends in principle up to $(Q_{particle}/2, -Q_{particle}/2)$ when all electrons are trapped. It should be noted that only a fraction of the total energy is measured on the Y side for those events, which corroborates electron trapping. When increasing the bias voltage, the fraction of events in region (1) decreases as shown in Fig. 10, which corresponds to less electron trapping. This trend is also discussed in Ref. [26]. Fig. 10 corresponds to measurements performed at $T=6$ °C and 1 µs peaking time. Counts in the different regions of Fig. 9 were obtained setting 2D gates on the
<figure><img src="image_75.png" /><figcaption>Fig. 9. Interstrip events for the X and Y side for three different bias voltages. See text for the explanation of the region labels.</figcaption></figure>

<figure><img src="image_76.png" /><figcaption>Fig. 10. Fraction of interstrip events. See text for details.</figcaption></figure>

(E1, E2) plots. Very close figures were obtained at other temperatures or with a 3 µs peaking time.

In Refs. [20,19,22], the L1 or L2 lines were observed and interpreted as hits in the middle of the interstrip. In both L1 and L2 cases, holes diffuse to neighbouring X strips (charge sharing) with all electrons moving toward the rear side (L2) or fully trapped in the region of the Si–SiO₂ interface (L1). Note that in Refs. [20,19] only the L1 line is observed, while mainly L2 is observed in Ref. [22]. This can be related to the nature of particles illuminating

the interstrip: α particles in the first case and 59.5 keV photons in the second case. The half-thickness for the 59.5 keV photons absorbtion in Si is ~8 mm therefore electron-hole pairs can be created deeply in the detector without consequent electron trapping. On the contrary, α particles have a range of 25–30 µm therefore electron-hole pairs are created close to the Si–SiO₂ interface with consequently electron trapping prevailing. In the case of MUSETT detectors we are in an intermediate configuration between the L1 and L2 extreme cases (region (2) in Fig. 9) with electrons partially trapped. Region (2) moves towards the L2 line with increasing bias voltage, the fraction of these events being almost constant as a function of the bias voltage (see Fig. 10). Also, the energy is only partially detected on the back side with an average signal increasing as a function of the bias voltage when the number of electrons trapped decreases.

When electrons drift to the ohmic back side, the transverse size of the electrons cloud increases with the depth. Therefore the track can overlap between two strips: the signal is shared between these strips with $E_1+E_2 = E_{particle}$ (if the hit did not occur simultaneously in the X interstrip region). It corresponds to region (3) in Fig. 9. Charge sharing in silicon strip detectors has been discussed in Ref. [21] considering the diffusion and drift in the electric field. More exhaustive interpretations include the diffusion and drift in the plasma column resulting from the large electron-hole pair concentration: see e.g. Refs. [23,25,24]. As shown in Fig. 10, the fraction of region (3) events decreases when the bias voltage increases, since the transverse cloud size is smaller for a larger
electric field. The fraction should also decrease with the implantation depth, increase with the particle energy and temperature.

Charge sharing on the Y side may also occur when electrons are partially trapped on the X interstrip (regions (1) and (2) discussed above). Such events correspond to region (4) in Fig. 9. It should be noted that the probability to have electron trapping in the X interstrip (regions (1) and (2)) and at the same time Y charge sharing results in region (4). Indeed, one can check that $Fraction(4) = (Fraction(1)+Fraction(2)) \times Fraction(3)$. Therefore, the total fraction of interstrip events is not the sum of the fractions in the four regions. As another consequence, the fraction of region (4) decreases with the bias voltage.

From the above discussion, it turns out that the deposited energy is only partially detected in regions (2) and (4) for which it cannot be recovered. On the contrary, shared charges on the Y side (region (3)) can be added to recover the total energy; see Fig. 11. Because signals are summed, the resolution is only ~45 keV compared to ~32 keV for a single hit. In region (1) the total energy can be recovered using $E_1-E_2$ for the left bottom branch and $E_2-E_1$ for the right top branch. This holds if all holes are collected on a single strip regardless of the fraction of electrons trapped. The reconstruction has however a poor resolution of ~120 keV and is shifted downward by about 110 keV (see Fig. 11), which corresponds approximately to the energy loss of α particles in the $SiO_2$ layer.

All effects taken into account, it turns out that the full particle energy is better detected on the Y side, although the resolution is slightly worse compared to the junction side. Contrary to Ref. [20], we do not suggest to reject off-line the events for which two adjacent strips exhibit a non-zero charge. The fraction of such events is, as shown above, significant and cannot be ignored at least in low statistics experiments such as heavy elements studies.

It should be noted that we have performed a similar study with the S3 detector prototype, which has among characteristics a reduced strip separation on the junction side (30 µm compared to 60 µm for MUSETT). In brief, the fraction of events in region (1) decreases by a factor about 3. As expected, the interstrip Y charge sharing is very close because the strip width and pitch are

unchanged on this side. Detailed results will be published elsewhere.

Finally, it should be noted that we have not observed with the MUSETT detector time-dependent effects like in Ref. [27]. This is probably because the measurements were performed after reaching stable conditions. On the other hand, time-dependent effects were noticed with the S3 prototype, and are being investigated.

## 4. In-beam commissioning

This section is devoted to the in-beam commissioning of MUSETT performed at GANIL. VAMOS in the WF mode is an important aspect of the experiment therefore Section 4.2 discusses the WF performance. The MUSETT in-beam performance is reported in Section 4.3.

### 4.1. Experimental conditions

Although references to the two first in-beam tests mentioned in Section 1 are made, we concentrate here on the experimental conditions of the MUSETT commissioning.

The $^{22}$Ne$^{10+}$ beam at an energy of 114 MeV was delivered by the CIME cyclotron with an intensity of 4.5 pnA (about $3 \times 10^{10}$ particles per second). A $^{197}$Au target (250 µg/cm$^2$) followed by a $^{12}$C equilibrium foil (30 µg/cm$^2$) was used. This last foil is added to ensure that the ions reach a charge state close to their equilibrium value : see e.g. Ref. [28] and references therein. The targets were installed in the center of the EXOGAM array consisting during this experiment of 7 clover detectors with a target to crystal distance of 11.4 cm. VAMOS was used in its Q-Q-WF optics mode (two electric quadrupoles followed by the Wien Filter). The magnetic dipole after the WF was not used except for specific measurements, e.g. a scan of the beam charge states. The position of MUSETT was chosen according to ions optics calculations maximizing the transmission of the recoils deflected at an angle of 6° by the WF. The flight path between the target and MUSETT was 7.4 m for central trajectories. More details of the WF and optics are given in Section 4.2.

Although all DSSDs were available, we used MUSETT in a 2-detector configuration in order to (i) minimize the radiation damage and (ii) simplify the complexity of the electronics and acquisition chain. MUSETT was operated with a cooling fluid temperature of ~20°. The beam was dumped at 0° in a Faraday cup whose current was monitored and recorded by the acquisition system.

The standard GANIL electronics and data acquisition system were used. MUSETT was read-out using the MUVI VXI cards [29] initially designed by the MUST II collaboration. These cards with their associated softwares and firmwares were upgraded to operate MUSETT in a triggerless mode. Data from the EXOGAM, VAMOS and MUSETT subsystems were time-stamped using a common 100 MHz clock. Upgrades of the DAQ were also made: new implementation of the data flow based on the NARVAL framework [30], of the GANIL Central Run Control, and of the Graphical User Interfaces. The on-line data control and off-line data analysis were made using the GRU (GANIL Root Utilities) and ViGRU (Visualization of GRU) packages [31], which are to a large extent based on the ROOT framework [17].

### 4.2. The VAMOS Wien filter

With a WF, the separation between the ions of interest from the beam and other unwanted reactions is based on their velocity. In very asymmetric reactions, the velocity difference between the recoil and the beam is large. This is therefore an ideal for the

<figure><img src="image_77.png" /><figcaption>**Fig. 11.** Top: Spectra for the X front side. Full lines correspond to single strip events. The dashed lines correspond to the difference $E_i-E_j$ when charges are induced on two neighbouring strips. Bottom: Spectra for the Y back side, full lines corresponding to single strip events. The dashed lines correspond to the sum energy of two neighbouring strips. The interstrip dashed spectra have been scaled for better visibility. Data are taken at $T=6$ °C, 3 µs peaking time, -60 V bias voltage.</figcaption></figure>
specificities of the VAMOS WF. The reaction $^{197}$Au($^{22}$Ne, $5n$)$^{214}$Ac is relevant in that respect with a beam (recoil) velocity of ~3.17 cm/ns (~0.31 cm/ns).

This section summarizes the VAMOS WF characteristics and performance determined using ion optics simulations compared with experimental results. The simulations and optimization of the electric quadrupole settings have been performed using the Zgoubi package [32,33]. We do not intend to show here fully realistic simulations since many parameters like the recoil angular distribution after the target, the charge state distribution, scattering inside the spectrometer, etc., were not calculated using a realistic physics generator. Our goal is to show the main characteristics relevant for the MUSETT commissioning at VAMOS and to a certain extent for future heavy element studies.

The VAMOS WF has inner dimensions of length × width × height=1 m × 1 m × 0.15 m. The WF is dispersive in the horizontal plane. The maximum fields available are E=300 kV/m and B=0.2 T. A safe electric field of 250 kV/m was used to avoid electric discharges. The electric and magnetic fields are oriented such that an accidental voltage breakdown after e.g. a spark would move the beam in a direction opposite to the detectors.

Requiring a maximum residues transmission, a WF is traditionally tuned with $E/B = v_0(\text{residues})$, driving the recoils at 0° independent of their charge state: see Fig. 12(a). However, as soon as the ions do not fulfil the condition $v=E/B$, a WF is not charge-state independent anymore, and thus it is dispersive in the horizontal plane. This traditional mode therefore disperses the beam, which potentially increases the probability of beam scattering into the detectors. We investigated an unusual WF mode tuned with $E/B = v_0(\text{beam})$, thus driving the beam at 0° regardless of its charge state, to allow a better focussing of the beam onto the beam dump: see Fig. 12(b). This mode may be of great interest in the case of radioactive beams where dumping radioactive species is a key experimental issue. Ion optics simulations show that the recoil transmissions are similar in both modes. Fig. 13 shows a simulation of a $^{214}$Ac recoil distribution at the focal plane of VAMOS. The transmission has been optimized for the $10^+$ charge state.

It should be noted that when the VAMOS optics is tuned for a given magnetic rigidity (or charge state), it defocuses all other magnetic rigidities. This is true, not only for the recoils, but also for the beam. As an example, the ion optics optimized for $^{214}$Ac$^{10+}$ focuses also $^{22}$Ne$^{10+}$ which has a similar $B\rho$, but disperses other beam and recoil charge states.

### 4.2.1. Effective length

While the geometric length of the WF is 1 m, the electric field acts on a smaller length. An effective length of 0.90 m has been determined in the first in-beam test using a $^{18}$O beam where the position was precisely measured using secondary electron

<figure><img src="image_78.png" /><figcaption>**Fig. 12.** Wien filter modes. (a) Traditional WF with nuclei of interest driven at 0°. (b) Beam "straight through" mode tried for the first time in this work.</figcaption></figure>

detectors [34,35]. All results of simulations shown in this paper have been obtained with an effective length of 0.9 m, the MUSETT array being centered around the $^{214}$Ac central trajectories.

### 4.2.2. Transmission

The transmission is calculated as the ratio of the number of recoils detected at the focal plane to the predicted number of recoils produced at the target. Experimentally, the number of recoils is obtained using their $\alpha$ decay. Note that the transmission, using the above definition, is a function of the detector size and efficiency (it is in fact a transmission-detection efficiency). The transmission has been measured for the $^{197}$Au($^{22}$Ne, $5n$)$^{214}$Ac reaction. At the beam energy of interest, a value of the cross-section of 2.87 mb was taken from Ref. [36].

A transmission of 36 ± 2% and $21 \pm 2\%$^5 was deduced in the second and third in-beam experiments, respectively. The difference is due to the spacial coverage of the detection system that was different between the two experiments. Our simulations using an optics optimized for $^{214}$Ac$^{10+}$ transmission lead on the other hand to transmissions of ~66% and ~45%, respectively. Comparison between our measurements and simulations has to be taken with care. There is indeed a factor of ~2 between the measurements and the simulation, which suggests a systematic error that may be ascribed to two different sources:

* Recoil velocity and angular distribution: These distributions arise from two effects: particle evaporation (neutrons) and straggling in the target, which can be estimated using e.g. the PACE [37] and SRIM [38] codes, respectively. While the contribution from neutron emission can easily be estimated, less confidence can be associated to the contribution arising from the straggling in the target. The latter part is estimated to be the dominant contribution with $\Delta v/v \sim 15\%$ and $\Delta\theta \sim 100$ mrad (compared to $\Delta v/v \sim 6\%$ and $\Delta\theta \sim 50$ mrad from the neutron emission).

* The uncertainty in the cross-section taken from Ref. [36] is ±40%.

The deduced transmission should therefore be taken as an estimate and not as an accurate value. We should nonetheless stress that a transmission of ~40% for very asymmetric reactions such as $^{197}$Au($^{22}$Ne, $5n$)$^{214}$Ac is larger than the value obtained with other recoil separators coupled to a large efficiency Ge array: see for instance [1,39] and references therein. To give another example, our simulations indicate a transmission of ~50% for the fusion-evaporation reaction $^{238}$U($^{22}$Ne, $5n$)$^{255}$No.

### 4.2.3. Rejection

The beam rejection factor can be obtained from the ratio of beam intensity impinging on the target to the event rate (excluding the fusion-evaporation events) detected at the focal plane.

A rejection of ~$10^7$ has been obtained in both recoil or beam straight through WF modes. As shown in Fig. 14, both direct (full energy) and scattered beam are detected with MUSETT.

While the presence of the full energy beam contamination does not overlap with the recoil energy, nor their decay, and therefore is at an acceptable level, the major part is made of a continuum spectrum interpreted as beam scattered inside the VAMOS spectrometer. To our knowledge, such events would be difficult to simulate. Also, our simulations predict that no direct beam should reach the focal plane detection. At this stage, we do not have yet a convincing explanation for the origin of the background sources. As a comparison, a beam suppression of about $2 \times 10^{10}$ has been measured for the LISE WF [40] for the $^{209}$Bi($^{48}$Ca, $2n$)$^{255}$Lr

<footnote n="5">5 Statistical error only; the error on the reference cross-section is not included.</footnote>
<figure><img src="image_79.png" /><figcaption>**Fig. 13.** Simulation of the <sup>214</sup>Ac residues detected at the focal plane of VAMOS. Top: recoil distribution (X,Y) at the focal plane. The rectangle corresponds to the area of the 2-MUSETT detector configuration used during the in-beam commissioning. Bottom left: projection of the recoil distribution on the X axis. Bottom right: same if detected in the two MUSETT detectors.</figcaption></figure>

<figure><img src="image_80.png" /><figcaption>**Fig. 14.** ToF (arbitrary units) versus energy (on the X front side) identification matrix. Note that the time increases from right to left.</figcaption></figure>

fusion-evaporation reaction [41]. The LISE WF has however a total length of 12 m and the beam is stopped at mid filter ~8 m far away from the focal plane detection.

To summarize this section, the study of very asymmetric fusion-evaporation reactions at the mb level is feasible according to the measured rejection of the VAMOS WF. The main advantage of VAMOS in the context of fusion-evaporation reactions is its transmission, especially large for very asymmetric reactions.

### 4.3. RDT using the <sup>22</sup>Ne+<sup>197</sup>Au reaction

Finally, we present in this section representative results obtained during the MUSETT in-beam commissioning.

Fig. 14 shows a ToF versus energy identification matrix. The ToF measured between MUSETT (start) and EXOGAM (stop) is given in arbitrary units and increases from right to left. Using the X (front side), the direct beam saturates the preamplifiers hence the calibration is incorrect for these events. Direct beam (full energy) corresponds to the area on the top right part of the matrix. The horizontal line starting from this area corresponds to a direct beam randomly correlated with EXOGAM. The parabola portion on the left corresponds to scattered

<figure><img src="image_81.png" /><figcaption>**Fig. 15.** Energy spectra. The total X front-side spectrum is obtained without any condition on position and time. The spectrum resulting from position and time correlations is scaled by a factor 5. See text for details.</figcaption></figure>

beam. Scattered beam can also be random correlated with EXOGAM (mostly associated with the wrong cyclotron beam pulse): it corresponds to the background filling the plot and overlapping with the recoils of interest. It is possible to avoid such loci using a beam repetition period longer than the ToF inside the spectrometer. Finally, evaporation residues correspond to the region around a ToF ~6500 channels and $E_X=5$ MeV. Note that the energy is not corrected for any pulse height defect.

The total X (front side) energy spectrum is shown in Fig. 15. The α decay of nuclei populated in the fusion-evaporation reaction and of their daughters grows on a large background of scattered beam and to a smaller extent of fusion-evaporation residues. The spectrum resulting from recoil-α correlations is also shown in Fig. 15. Recoils were selected using a 2D gate set on the ToF versus energy identification matrix. A decay was then
searched in the same pixel within an interval of 1 min. This procedure reduces the background dramatically and removes long-lived daughter nuclei and grand-daughter nuclei like $^{210}$Rn.

In $^{214}$Ac, the energy of the most intense $\alpha$ peak at 7208 ± 3 keV corresponds to the ground-state feeding of $^{210}$Fr (energy of 7215 ± 3 keV reported in [42]). We find that the peak on its left is a doublet with energies of 7083 ± 9 keV and 7128 ± 6 keV. The 7083 keV peak corresponds to the 7082 ± 4 keV line of [42] and 7080 ± 15 keV line of [43], feeding a level at 128 keV in $^{210}$Fr. The 7128 keV line is not reported in Ref. [42], but interpreted in Ref. [43] ($E_{\alpha} \approx 7125$ keV) as resulting from the summing of the 7083 keV line with conversion electrons. Note that the summed intensity of the 7083 and 7128 keV lines (46 ± 3%) fits well with the 42 ± 2% intensity reported in Ref. [42] for the 7082 keV transition.

Half-lives can be measured using the recoil-$\alpha$ correlation procedure. A plot representing the decay time (in natural logarithm scale) versus the $\alpha$-decay energy is shown in Fig. 16. The $^{215}$Ra $\alpha$ peak (p4n fusion-evaporation channel) with a half-life of ~1.67 ms can clearly be seen, while it was hardly visible in Fig. 15. The $^{209,210}$Fr decays are present in Fig. 16. Since the plot results from recoil-$\alpha$ correlations, it turns out that these two nuclei can be populated directly by fusion-evaporation ($\alpha$2n and $\alpha$n channels). The $^{209,210}$Fr $\alpha$-decay peaks shown in the total spectrum of Fig. 15, result therefore from the direct feeding after fusion-evaporation but also from the $^{213,214}$Ac $\alpha$ decay.

As an example, the decay curve of $^{214}$Ac is shown in Fig. 17. The half-lives are deduced fitting the decay plot with two exponential taking into account the random correlations: see e.g. Ref. [44].

The measured half-lives are summarized in Table 2 where they are compared with values from the literature. The agreement is excellent for $^{213}$Ac and $^{215}$Ra with the most recent results of Heßberger et al. [43]. Evaluated half-lives for $^{214}$Ac and $^{215}$Ac are about 45 years old [45]. We re-confirmed the $^{215}$Ac half-life. We have improved the accuracy for $^{214}$Ac and find a slightly lower value. Tabulated half-lives for $^{209}$Fr and $^{210}$Fr were obtained 40–50 years ago and display large dispersions. Because of the low statistics and uncertainty due to the long lifetime, we were unable to improve the accuracy. Our value for $^{210}$Fr is compatible with those of Griffioen and Macfarlane [46] and Valli et al. [47] within $1\sigma$. On the other hand, we find a value for $^{209}$Fr lower compared to Griffioen and Macfarlane [46], Hornshøj et al. [48] and Valli et al. [47].

<figure><img src="image_82.png" /><figcaption>Fig. 16. Matrix representing the decay time in natural logarithm scale as a function of the $\alpha$-decay energy.</figcaption></figure>

<figure><img src="image_83.png" /><figcaption>Fig. 17. Decay plot of $^{214}$Ac.</figcaption></figure>

<table><caption>Table 2

Half-lives measured in this work compared with data in the literature.</caption><thead><tr><th>Isotope</th><th>Half-life, this work</th><th>Half-life, literature</th></tr></thead><tbody><tr><td><sup>209</sup>Fr</td><td>40 ± 6 s</td><td>54.7 ± 1 [46]<br>50 ± 0.3 s [48]<br>52 ± 2 s [47]</td></tr><tr><td><sup>210</sup>Fr</td><td>163 ± 30 s</td><td>159 ± 5 [46]<br>191 ± 4 s [47]</td></tr><tr><td><sup>213</sup>Ac</td><td>723 ± 23 ms</td><td>731 ± 17 ms [43]</td></tr><tr><td><sup>214</sup>Ac</td><td>7.56 ± 0.11 s</td><td>8.2 ± 0.2 s [45]</td></tr><tr><td><sup>215</sup>Ac</td><td>175 ± 20 ms</td><td>170 ± 10 ms [45]</td></tr><tr><td><sup>215</sup>Ra</td><td>1.65 ± 0.16 ms</td><td>1.67 ± 0.01 ms [43]</td></tr></tbody></table>

Recoil-Decay-Tagged $\gamma$-ray spectra were obtained for the most intense fusion-evaporation channels: see Fig. 18 for $^{213,214}$Ac. A level scheme has been proposed for $^{213}$Ac by Heßberger et al. [49], based on $^{217}$Pa decay studies. Using prompt $\gamma$-ray spectroscopy, we also observed the proposed transitions at 613.0 and 820.5 keV. Heßberger et al. observed in $^{214}$Ac a transition at 91.8 keV after $\alpha$ decay of $^{218}$Pa [43]. We are unable to confirm this transition but we have found several new transitions. Because of the modest statistics and complexity of odd-odd nuclei, an improved level scheme cannot be suggested. Gamma-rays were observed to deexcite the $29/2^+$ isomer of $^{215}$Ac, previously studied using the fusion-evaporation reaction $^{204}$Pb($^{15}$N,$4n$)$^{215}$Ac [50]. Several tentative transitions were observed in the present study, but none of them corresponds to the proposed decay of the $29/2^+$ isomer.

To conclude this section, our data representing a few hours of beam time provide new results and point out the need for high-statistics and high-quality data sets in the region of neutron-deficient isotopes near the magic number $N=126$.

## 5. Summary and perspectives

The segmented Si wall MUSETT has been developed for Recoil-Decay-Tagging studies at VAMOS, GANIL. The array includes a Front End Electronics based on ASICs. A test bench has been developed at Saclay for tests and characterization of the detectors. Multipurpose VME cards, a data acquisition system and analysis tools have been developed. An energy resolution as low as 25 keV has been measured at 5.8 MeV. The detector and test bench developments have been made with a wider perspective. For instance, the test bench is used to characterise Double-sided Silicon Strip Detectors and preamplifier prototypes for the S3 collaboration.
<figure><img src="image_84.png" /><figcaption>Fig. 18. Recoil-decay-tagged γ-ray spectra of $^{213,214}$Ac.</figcaption></figure>

The MUSETT array has been commissioned at GANIL with VAMOS in the Wien Filter vacuum mode, in conjunction with EXOGAM and using the fusion-evaporation reaction $^{22}$Ne + $^{197}$Au. The Recoil-Decay-Tagging technique has been successfully tested. Alpha-tagged prompt γ-ray spectra from $^{213-215}$Ac were obtained.

Performance of the VAMOS Wien Filter was studied for very asymmetric fusion-evaporation reactions. A large transmission has been deduced and the beam rejection allows studies at the mb level. For less asymmetric reactions, the VAMOS gas-filled mode recently tested provides a superior performance in terms of transmission and rejection [6].

The coupling of VAMOS with EXOGAM, its electronics upgrade (EXOGAM2), AGATA [9] or PARIS [51] will cover a large physics program: spectroscopy of heavy elements, study of exotic shapes, reaction mechanism studies, etc. Radioactive beams from SPIRAL2 (see e.g. [52]) foreseen by the end of the decade will open even greater perspectives.

## Acknowledgments

MUSETT has been funded by an ANR (Agence Nationale de la Recherche) grant (ANR-05-BLAN-0150 MUSETT). This work has been supported by the EU-FP5 EXOTAG (contract HPRI-CT-1999-50017) and EU-FP6 INTAG (contract RI13-CT-2004-506065) projects. We are grateful to the GANIL staff for the support during the experiments: accelerator division, G. Frémont for making the targets. M. Kmiecik, A. Lopez-Martens, R. Lozeva, A. Maj, W. Męczyński, M. Ziębliński and M. Zielińska are acknowledged for their help during data taking.

## Appendix A. Supplementary material

Supplementary data associated with this article can be found in the online version, at http://dx.doi.org/10.1016/j.nima.2014.02.016.

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7.6 Magnetism of an Excited Self-Conjugate Nucleus:
Precise Measurement of the g Factor of the $2_1^+$
State in $^{24}$Mg
Magnetism of an Excited Self-Conjugate Nucleus: Precise Measurement of the *g* Factor of the 2<sup>+</sup> State in <sup>24</sup>Mg

A. Kusoglu,¹,² A. E. Stuchbery,³,\* G. Georgiev,¹ B. A. Brown,⁴,⁵ A. Goasduff,¹ L. Atanasova,⁶,† D. L. Balabanski,⁷
M. Bostan,² M. Danchev,⁸ P. Detistov,⁶ K. A. Gladnishki,⁸ J. Ljungvall,¹ I. Matea,⁹ D. Radeck,¹⁰
C. Sotty,¹,‡ I. Stefan,⁹ D. Verney,⁹ and D. T. Yordanov⁹,¹¹,¹²

¹CSNM, CNRS/IN2P3; Université Paris-Sud, UMR8609, F-91405 Orsay-Campus, France

²Department of Physics, Faculty of Science, Istanbul University, Vezneciler/Fatih, 34134 Istanbul, Turkey

³Department of Nuclear Physics, RSPE, Australian National University, Canberra, Australian Capital Territory 2601, Australia

⁴National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824, USA

⁵Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA

⁶Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, BG-1784 Sofia, Bulgaria

⁷ELI-NP, Horia Hulubei National Institute of Physics and Nuclear Engineering, 077125 Magurele, Romania

⁸Faculty of Physics, St. Kliment Ohridski University of Sofia, 1164 Sofia, Bulgaria

⁹IPN, Orsay, CNRS/IN2P3, Université Paris-Sud, F-91406 Orsay Cedex, France

¹⁰Institute for Nuclear Physics, University of Cologne, Zülpicher Straße 77, D-50937 Köln, Germany

¹¹Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany

¹²CERN European Organization for Nuclear Research, Physics Department, CH-1211 Geneva 23, Switzerland

(Received 4 November 2014; published 12 February 2015)

A precise measurement of the *g* factor of the first-excited state in the self-conjugate (*N* = *Z*) nucleus
<sup>24</sup>Mg is performed by a new time-differential recoil-in-vacuum method based on the hyperfine field of
hydrogenlike ions. Theory predicts that the *g* factors of such states, in which protons and neutrons occupy
the same orbits, should depart from 0.5 by a few percent due to configuration mixing and meson-exchange
effects. The experimental result, *g* = 0.538 ± 0.013, is in excellent agreement with recent shell-model
calculations and shows a departure from 0.5 by almost 3 standard deviations, thus achieving, for the first
time, the precision and accuracy needed to test theory. Proof of the new method opens the way for wide
applications including measurements of the magnetism of excited states of exotic nuclei produced as
radioactive beams.

DOI: 10.1103/PhysRevLett.114.062501

PACS numbers: 21.10.Ky, 21.60.Cs, 23.20.En, 27.30.+t

The *g* factor is an important observable in the study
of the quantum mechanics of nuclear excitations, being
sensitive to single-particle aspects of the wave function.
Because alternative effective interactions in the shell-model
approach can describe excitation energies equally well but
predict significantly different configuration mixing in the
wave functions and, hence, different *g* factors, measure-
ments of nuclear magnetism play a critical role in building
an accurate understanding of nuclear structure. The *g* factor
*g* and magnetic moment *μ* are related by *μ* = *g*I where *μ* has
the units of nuclear magnetons and the angular momentum
*I* is in units of *ħ*.

For many years, the *g* factors of the first-excited states
of even-even nuclei with equal numbers of protons and
neutrons (*N* = *Z*) were expected to depart little from
*g* = 0.5 [1]. This behavior occurs for self-conjugate nuclei
because protons and neutrons occupy the same orbits and

the intrinsic-spin moments of the nucleons largely cancel,
leaving the orbital motion of the protons to produce the
nuclear magnetism. More recent shell-model calculations,
however, predict departures from *g* = 0.5 by up to 10%
for the first-excited 2⁺ states in the *N* = *Z* sd-shell nuclei
from <sup>20</sup>Ne to <sup>36</sup>Ar [2]. These departures stem from three
mechanisms. First, configuration mixing in the shell-model
basis space does not fully quench the spin contributions
to the nuclear moment. Second, the Coulomb interaction
between protons leads to isospin mixing, which introduces
isovector contributions to the nuclear moment. Third,
within the nucleus, meson exchange and higher-order
configuration mixing contributions modify the magnetic
dipole operator from that of a free nucleon.

On the experimental side, the predicted departures from
g = 0.5 have not previously been observed. The excited
states in question are short lived, having lifetimes of a few
picoseconds. Their g factors must be measured via the spin
precession of the nucleus in an extremely strong magnetic
field, of the order of 10 kT or more. Such fields can be
produced at the nucleus only by hyperfine interactions.
Experimental precision and accuracy for these measure-
ments has been limited, in part, because the short nuclear

<footnote>Published by the American Physical Society under the terms of
the Creative Commons Attribution 3.0 License. Further distri-
bution of this work must maintain attribution to the author(s) and
the published article's title, journal citation, and DOI.</footnote>
lifetimes require the measurement of small differences in count rate. A more fundamental limitation, however, has stemmed from the use of ions with complex atomic configurations, for which the net strength of the hyperfine field is an uncertain superposition of many components.

This Letter reports a new measurement of the g factor of the first-excited state in the N = Z nucleus $^{24}$Mg (excitation energy $E_x$ = 1.369 MeV, mean lifetime $\tau$ = 1.97 ps [3]) based on hyperfine fields of hydrogenlike Mg ions. By the use of these well-defined hyperfine fields, together with efficient particle and $\gamma$-ray detection, the new measurement achieves the accuracy and precision needed to test the predicted departures from g = 0.5.

The experimental method is based on the observation of the precession of the nuclear moment as hydrogenlike $^{24}$Mg ions fly through vacuum. As illustrated in Fig. 1, excited nuclei emerge from a target foil as ions carrying one electron. The nuclear spin I is aligned by the reaction whereas the atomic spin J is oriented randomly. The hyperfine interaction couples the atomic spin to the nuclear spin, and together they precess about the total $F = I + J$ with a frequency proportional to the nuclear g factor. Thus, the orientation of the nuclear spin is periodically reduced and restored during the flight through vacuum. As a consequence, the angular intensity pattern of the $\gamma$ rays emitted by the nuclei varies periodically, in step with the orientation of the nuclear spin. In the traditional recoil-in-vacuum, or “plunger,” technique [4], the ions travel a set distance through vacuum before being stopped in a thick stopper foil, which immediately quenches the hyperfine interaction and freezes the orientation of the nuclear spin. The nuclear precession frequency is determined by observing changes in the radiation pattern as the flight time is varied by changing the distance between the target and stopper foils.

Here, we report the first use of a new time-differential recoil-in-vacuum (TDRIV) method. Proposed by Stuchbery, Mantica and Wilson [5] as a method suited for radioactive beams, its novel feature is to replace the thick stopper foil by a thinner foil that simply resets the electron configuration. For radioactive beams, this change allows projectile-excitation experiments in which the radioactive beam ion is

<figure><img src="image_85.png" /><figcaption>FIG. 1. Sketch of experiment. The “stopper” of the traditional plunger technique is replaced by a thin foil that resets the electron configuration of H-like ions. The particle detector, with segmentation around the beam axis, is located downstream of the $\gamma$-ray detectors.</figcaption></figure>

detected at forward angles out of the view of the $\gamma$-ray detectors. In the present application to $^{24}$Mg, the method enables experiments on high-velocity ions ($v/c \sim 0.1$) for which the optimal charge-state distribution of about 50% H-like can be achieved. The previous measurement [6], by the conventional TDRIV method following the $^{12}$C($^{16}$O, $\alpha\gamma$)$^{24}$Mg reaction, achieved a Mg recoil velocity of only $v/c \sim 0.056$ so that the H-like fraction was around 15%; most Mg ions carried three or four electrons.

A beam of $^{24}$Mg at an energy of 120 MeV (5 MeV/nucleon) from the ALTO accelerator facility at IPN Orsay was excited in glancing collisions on a stretched foil of $^{93}$Nb, 2.4 mg/cm$^2$ thick. Excited projectiles emerged from this target with $\sim 93$ MeV, corresponding to a velocity of $v/c = 0.0915(5)$. This velocity and its uncertainty were determined from experimental Doppler shifts and by evaluation of the reaction kinematics, taking into account the energy loss of the beam in the target. A 1.7 mg/cm$^2$ thick $^{197}$Au foil served as the movable, stretched “reset” foil.

The experimental setup was comprised of the ORGAM hyperpure germanium (HPGe) detector array surrounding the Orsay plunger [7], on which the stretched foils were mounted, and an eightfold segmented plastic scintillation detector, located inside the beam line, 61 mm downstream from the target. Each segment had an azimuthal opening of $\Delta\phi_p = 30^\circ$ and a polar opening angle from $\theta_p = 33^\circ$ to $\theta_p = 38^\circ$. The flight time of the excited ions T is related to the target-reset foil separation D by $T = D/(v \cos \theta_p)$, where $\langle v \cos \theta_p \rangle$ represents an average over the angular acceptance of the particle detector.

ORGAM was populated with 13 HPGe detectors at the polar angles $\theta = 46.5^\circ, 72.1^\circ, 85.8^\circ, 94.2^\circ, 108.0^\circ, 133.6^\circ$, and $157.6^\circ$, relative to the beam axis. Gamma-ray detection angles near $90^\circ$ were favored as these show the strongest anisotropy around the $\phi$ direction.

Data were taken in event-by-event mode, recording the arrival time and amplitude of the detected radiation from each particle and $\gamma$ detector. Twenty-four target-reset foil distances from (near) the touching point of the foils to about 100 $\mu$m separation were measured. The beam intensity was about 0.3 pA, and the running time was approximately 2 h for each distance.

Coincidence events corresponding to a $\gamma$-ray detection in the ORGAM array and a beam-particle detection in the plastic scintillator were sorted from the event data. Random coincidences were subtracted. An example of a resultant $\gamma$-ray spectrum is shown in Fig. 2. The intensity of the peak corresponding to the $2^+ \rightarrow 0^+$ transition of $^{24}$Mg was determined for all particle-$\gamma$ combinations.

In the presence of vacuum deorientation, the time-dependent particle-$\gamma$ angular correlation takes the form (see e.g., Ref. [8] and references therein)

$$W(\theta_p, \theta_\gamma, \Delta\phi, t) = \sum_{kq} a_{kq}(\theta_p) G_k(t) D_{q0}^{k*}(\Delta\phi, \theta_\gamma, 0), \quad (1)$$
<figure><img src="image_86.png" /><figcaption>FIG. 2. Random-subtracted γ-ray spectrum collected at 80 µm plunger separation, showing the $^{24}$Mg 2+ → 0+ 1368-keV photopeak. Data for all γ-ray detectors in coincidence with one particle detector segment are shown.</figcaption></figure>

where $\theta_p$ and $\theta_\gamma$ are the polar detection angles for particles and γ rays, respectively; $\Delta\phi = \phi_\gamma - \phi_p$ is the difference between the corresponding azimuthal detection angles. $a_{kq}(\theta_p) = B_{kq}(\theta_p)Q_k F_k$, where $B_{kq}(\theta_p)$ is the statistical tensor, which defines the spin orientation of the initial state. $F_k$ represents the F coefficient for the γ-ray transition, and $Q_k$ is the attenuation factor for the finite size of the γ-ray detector. $D_{q0}^{k*}(\Delta\phi, \theta_\gamma, 0)$ is the Wigner-D matrix. For E2 excitation, $k=0, 2, 4$ and $-k \le q \le k$. The attenuation coefficients $G_k(t)$ specify the time-dependent vacuum deorientation effect. For H-like $J=1/2$ configurations, the $G_k(t)$ are cosine functions with a frequency determined by the nuclear g factor.

We refer to ions that decay between the target and the reset foil as “fast” and those that decay after the reset foil as “slow.” The TDRIV method does not require that the γ-rays emitted from the fast and slow ions be separated in the observed energy spectrum. Decays of slow ions beyond the reset foil oscillate as $G_k(T)\tilde{G}_k(\infty)$, where T is the flight time and $\tilde{G}_k(\infty)$ is the average integral attenuation coefficient for slow ions that decay beyond the reset foil [5]. The fast component, however, is an average over decays taking place between the target and reset foils, so a range of precessions angles contribute and the oscillations are washed out [5]. Because the fast and slow components of the γ-ray line are not resolved, the net angular correlation shows damped oscillations, with the rate of damping determined by the nuclear lifetime.

With eightfold segmentation of the particle detector and 13 detectors in ORGAM, there are 104 individual particle-γ combinations. To analyze the data, the 104 time-dependent angular correlations were evaluated based on Eq. (1) and ordered according to the amplitude of the oscillations and whether the γ-ray intensity should initially increase, $W^\uparrow(T)$, or decrease, $W^\downarrow(T)$, with time. Forty-nine particle-γ combinations increase in magnitude initially. The remaining 55 particle-γ combinations initially decrease. Ratios of the coincidence γ-ray intensity corresponding to $W^\uparrow/W^\downarrow$ were formed in order, beginning with the pairing of the case showing strongest increase with the case of strongest decrease. These ratios were then formed into a geometric average

$$R(T) = \left( \prod_{i=1}^{n} \frac{W_i^\uparrow(T)}{W_i^\downarrow(T)} \right)^{1/n} \quad (2)$$

where n is the number of $W_i^\uparrow/W_i^\downarrow$ ratios included. The experimental geometric averages $R(T)$ largely factor out the detection efficiency for both γ-rays and particles.

Sensitivity is lost if $W^\uparrow/W^\downarrow$ ratios showing small amplitude oscillations are averaged with ratios showing large amplitude oscillations. The data set was therefore analyzed by forming geometric ratios in three groups, two of which are shown in Fig. 3. The $n=14$ combinations showing the largest amplitude oscillations are labeled “strong,” while the $n=17$ ratios showing a moderate amplitude are labeled “intermediate.” A further $n=18$ pairs show a small amplitude. Because of the symmetry of the particle- and γ-detector arrays, certain particle-γ detector combinations should show the same angular correlation at all times. Ratios of such combinations should show a null effect. An example is shown in Fig. 3, labeled “null.”

The g factor was determined from fits to the experimental data, as shown in Fig. 3. Fitting was performed using a computer code [9] that models the experimental conditions in detail based on Coulomb-excitation calculations, the formulas in Ref. [5], and Eq. (1) and then assembles $R(T)$ ratios in the same way that the experimental data are combined. The fitting procedures were broadly similar to those of Horstman et al. [6], the main difference being that the H-like K-shell hyperfine field is dominant in our measurement. Its value, $B_{1s}(0) = 29.09$ kT, was evaluated

<figure><img src="image_87.png" /><figcaption>FIG. 3 (color online). $R(T)$ ratio data, Eq. (2), and fits based on detailed modeling of the experiment [9]. The distance is the separation of target and reset foils (22.4 µm = 1 ps flight time). The frequency of the oscillation determines the g factor.</figcaption></figure>
with the General Relativistic Atomic Structure Package, GRASP2K [10]. Relativistic effects are of order 1%; the uncertainty in $B_{1s}(0)$ is negligible, which underpins the accuracy of the experimental g factor.

Results of the fits to the $R(T)$ data having strong, intermediate, and weak amplitude oscillations were $g = 0.538(13)$, 0.539(24), and 0.54(3), respectively, where the uncertainties are statistical only. The weighted average is $g = 0.538(11)$ (statistical error).

Systematic errors were evaluated as (i) $\delta g = \pm 0.0045$ from an uncertainty of $\pm 1.5$ mm in the distance from the target to particle-detector face, (ii) $\delta g = \pm 0.0040$ from uncertainty in lifetime, $\tau = 1.97(5)$ ps [3], (iii) $\delta g = \pm 0.0035$ from the uncertainty in $v/c$, and (iv) $\delta g = \pm 0.0010$ from uncertainties in the distribution of hyperfine fields. The experimental g factor is therefore $g = 0.538 \pm 0.011(\text{statistical}) \pm 0.007(\text{systematic})$ or $g = 0.538(13)$, in reasonable agreement with, but more precise than, the previous measurement, $g = 0.51(2)$ [6]. The improvement stems in part from better statistical precision; however, systematic errors are also reduced. Uncertainty in the distribution of hyperfine fields has a small influence on the present measurement but was an important source of uncertainty in the previous measurement.

The first-excited state of $^{24}$Mg is an isospin $T = 0$ state in a nuclide with $N = Z$. As such, it is useful to write the magnetic moment in terms of the isoscalar and isovector matrix elements

$$ \mu = gI = g_{\ell_0} \langle \ell_0 \rangle + g_{\ell_1} \langle \ell_1 \rangle + g_{s_0} \langle s_0 \rangle + g_{s_1} \langle s_1 \rangle, \quad (3) $$

where $\ell$ and $s$ represent the orbital and spin operators, and the subscripts 0 and 1 represent isoscalar and isovector, respectively. $I = \langle \ell_0 \rangle + \langle s_0 \rangle$. The free-nucleon values for the g factors are $g_{\ell_0} = (g_{\ell p} + g_{\ell n})/2 = 0.5$, $g_{\ell_1} = (g_{\ell p} - g_{\ell n})/2 = 0.5$, $g_{s_0} = (g_{sp} + g_{sn})/2 = 0.880$, and $g_{s_1} = (g_{sp} - g_{sn})/2 = 4.706$. (See Refs. [11–13] for further details.)

We first consider the sd shell-model space with isospin conserving Hamiltonians for which the isovector terms are zero: $\langle \ell_1 \rangle = \langle s_1 \rangle = 0$. Thus, if $\langle s_0 \rangle = 0$, $g(2^+) = g_{\ell_0} = 0.5$ for the bare M1 operator. However, the sd shell model gives small but nonzero values for $\langle s_0 \rangle$ [1]. For the $^{24}$Mg case, $\langle s_0 \rangle = 0.069$ is obtained with the universal sd-shell interaction USDB. The USDA and USDB interactions with 30 and 56 parameters, respectively, update the universal sd-shell Hamiltonian USD to include additional data on neutron-rich nuclei [14]. USDB gives a slightly better rms deviation; however, there is little difference in the wave functions of stable nuclides. The following discussion is based on USDB, making reference to USD and USDA to give an indication of the theoretical uncertainty in the effective Hamiltonian. As will become evident below, this uncertainty affects the g factor at the level of $\pm 0.001$. Taking USDB wave functions and bare nucleon values for $g_{\ell_0}$ and $g_{s_0}$ gives $g(2^+) = 0.513$, which falls short of our experimental result.

Next we evaluate the effect of isospin mixing. In $^{24}$Mg, the dominant contribution comes from mixing with the lowest $T = 1$, $I^\pi = 2^+$ state at $E_x \sim 10$ MeV. The isovector matrix elements were evaluated with the isospin non-conserving Hamiltonian of Ormand and Brown [15], obtaining $\langle \ell_1 \rangle = 0.020$ and $\langle s_1 \rangle = 0.0012$. Thus, with the addition of isospin mixing, $g(2^+) = 0.521$, which still falls short of the experimental value at the level of 1 standard deviation. The results with the USDA and USD interactions are 0.522 and 0.520, respectively.

It is well known that there are corrections to all of the matrix elements in Eq. (3) from mesonic exchange currents and higher-order configuration mixing. These corrections have been evaluated for the $d_{5/2}$ orbit at $A = 17$ by Towner and Khanna [11] and Arima et al. [12]. Because the magnetic moment of the predominantly $T = 0$ first-excited state in $^{24}$Mg is dominated by the isoscalar orbital term, it is most sensitive to the corrections to $g_{\ell_0}$, denoted $\delta g_{\ell_0}$. The contribution to this correction coming from higher-order configuration mixing is $\delta g_{\ell_0} = 0.010$ according to Ref. [11] and $\delta g_{\ell_0} = 0.011$ according to Ref. [12], but there is disagreement for the mesonic-exchange contribution with Ref. [11] giving essentially zero and Ref. [12] giving $\delta g_{\ell_0} = 0.013$ (see Table 7.2 in Ref. [12]). Nevertheless, the resulting values of $g(2^+) = 0.531$ and 0.544, evaluated with the USDB Hamiltonian plus isospin nonconserving contributions and $\delta g_{\ell_0}$ corrections from Refs. [11,12], respectively, are both within the range of the experimental uncertainty.

An alternative approach is to determine the M1 operator empirically by performing a global fit to a wide range of data [2,13]. Our experimental g factor is shown in Fig. 4 along with previous results for $N = Z$ nuclei in the sd shell and NUSHELLX [16] calculations in the sd model space with the USDA and USDB interactions and the corresponding empirical M1 operators [2]. As is evident from Fig. 4, the new measurement is in very good agreement with these calculations; USDB gives $g(2^+) = 0.544(17)$. An uncertainty of about $\pm 0.017$ in these theoretical g factors comes mainly from the $\delta g_\ell$ terms in the empirical M1 operator.

<figure><img src="image_88.png" /><figcaption>FIG. 4 (color online). Present result (open red circle) and previous results (filled green circles) [6,17–19] compared to shell-model calculations with USDA (dashed brown line) and USDB (solid blue line) calculations.</figcaption></figure>
Thus, shell-model calculations consistently predict that the *g* factor of the first-excited state in the *N* = *Z* nucleus <sup>24</sup>Mg is increased from *g* = 0.5, and our experiment confirms these predictions for the first time.

We have validated a new method for measuring the *g* factors of excited nuclear states with lifetimes in the picosecond regime. Measurements on stable isotopes like <sup>24</sup>Mg can reach new levels of precision and test nuclear model calculations in ways that were not previously possible. Moreover, as the method was designed for applications to radioactive beams, the present work prepares the way for a future measurement on the neutron-rich nucleus <sup>32</sup>Mg in the “island of inversion” [20].

The authors thank the technical staff of the ALTO facility for providing the beam and for assistance during the preparation and execution of the experiment. We also thank the UK/France (STFC/IN2P3) Loan Pool and GAMMAPOOL European Gamma-Ray Spectroscopy Pool for the loan of the detectors for ORGAM. This work was supported in part by the 2214-TUBITAK (Scientific and Technological Research Council of Turkey) Programme, the Scientific Research Projects Coordination Unit of Istanbul University under Project No. 46811, the European Community FP7 Capacities-Integrated Infrastructure Initiative-Contract ENSAR No. 262010, P2IO (Physique des 2 Infinis et des Origines) laboratory of excellence, Bulgarian NSF Grant No. DID-02/16, the Australian Research Council Grant No. DP0773273, and the U.S. National Science Foundation, NSF Grant No. PHY-1404442.

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<footnote>*Corresponding author.
andrew.stuchbery@anu.edu.au</footnote>

<footnote>†Present address: Department of Medical Physics and Biophysics, Medical University-Sofia, 1431 Sofia, Bulgaria.</footnote>

<footnote>‡Present address: KU Leuven, Instituut voor Kern-en Stralingsfysica, 3001 Leuven, Belgium.</footnote>
7.7 Measurement of lifetimes in <sup>62,64</sup>Fe, <sup>61,63</sup>Co, and <sup>59</sup>Mn
Measurement of lifetimes in $^{62,64}$Fe, $^{61,63}$Co, and $^{59}$Mn

M. Klintefjord,¹,* J. Ljungvall,² A. Görgen,¹ S. M. Lenzi,³ F. L. Bello Garrote,¹ A. Blazhev,⁴ E. Clément,⁵ G. de France,⁵ J.-P. Delaroche,⁶ P. Désesquelles,² A. Dewald,⁴ D. T. Doherty,⁷ C. Fransen,⁴ A. Gengelbach,⁸ G. Georgiev,² M. Girod,⁶

A. Goasduff,⁹ A. Gottardo,¹⁰ K. Hadyńska-Klęk,¹ B. Jacquot,⁵ T. Konstantinopoulos,² A. Korichi,² A. Lemasson,⁵ J. Libert,¹⁰ A. Lopez-Martens,² C. Michelagnoli,⁵ A. Navin,⁵ J. Nyberg,⁸ R. M. Pérez-Vidal,¹¹ S. Roccia,² E. Sahin,¹ I. Stefan,¹⁰

A. E. Stuchbery,¹² M. Zielińska,⁷ D. Barrientos,¹³ B. Birkenbach,⁴ A. Boston,¹⁴ L. Charles,¹⁵ M. Ciemala,¹⁶ J. Dudouet,¹⁷ J. Eberth,⁴ A. Gadea,¹¹ V. González,¹⁸ L. Harkness-Brennan,¹⁴ H. Hess,⁴ A. Jungclaus,¹⁹ W. Korten,⁷ R. Menegazzo,²⁰

D. Mengoni,²⁰,³ B. Million,²⁰ A. Pullia,²¹,²² D. Ralet,²,²³,²⁴ F. Recchia,²⁰,³ P. Reiter,⁴ M. D. Salsac,⁷ E. Sanchis,¹⁸

O. Stezowski,¹⁷ Ch. Theisen,⁷ and J. J. Valiente Dobon⁹

¹Department of Physics, University of Oslo, N-0316 Oslo, Norway

²CSNSM, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, 91405 Orsay, France

³Dipartimento di Fisica e Astronomia dell'Università and INFN, Sezione di Padova, I-35131 Padova, Italy

⁴Institut für Kernphysik, Universität zu Köln, D-50937 Köln, Germany

⁵GANIL, CEA/DRF-CNRS/IN2P3, Bd. Henri Becquerel, BP 55027, F-14076 Caen, France

⁶CEA, DAM, DIF, F-91297 Arpajon, France

⁷Irfu, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France

⁸Institutionen för fysik och astronomi, Uppsala Universitet, SE-751 05 Uppsala, Sweden

⁹INFN Laboratori Nazionali di Legnaro, Viale dell'Università, 2, I-35020 Legnaro, Italy

¹⁰Institut de Physique Nucléaire, CNRS/IN2P3-Université Paris-Sud, F-91406 Orsay, France

¹¹Instituto de Física Corpuscular, CSIC-Universidad de Valencia, E-46071 Valencia, Spain

¹²Department of Nuclear Physics, Australian National University, Canberra ACT 0200, Australia

¹³CERN, CH-1211 Geneva 23, Switzerland

¹⁴Oliver Lodge Laboratory, The University of Liverpool, Liverpool, L69 7ZE, United Kingdom

¹⁵IPHC, UNISTRA, CNRS, 23 rue du Loess, 67200 Strasbourg, France

¹⁶The Henryk Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, 31-342 Kraków, Poland

¹⁷Université de Lyon, Université Lyon-1, CNRS/IN2P3, UMR5822, IPNL, 4 Rue Enrico Fermi, F-69622 Villeurbanne Cedex, France

¹⁸Departamento de Ingeniería Electrónica, Universitat de Valencia, Burjassot, Valencia, Spain

¹⁹Instituto de Estructura de la Materia, CSIC, Madrid, E-28006 Madrid, Spain

²⁰INFN Sezione di Padova, I-35131 Padova, Italy

²¹University of Milano, Dept. of Physics, I-20133 Milano, Italy

²²INFN Milano, I-20133 Milano, Italy

²³Institut für Kernphysik, Technische Universität Darmstadt, Darmstadt D-64289, Germany

²⁴GSI Helmholtzzentrum für Schwerionenforschung GmbH, Planckstraße 1, D-64291 Darmstadt, Germany
(Received 16 November 2016; revised manuscript received 3 January 2017; published 9 February 2017)

Lifetimes of the $4_1^+$ states in $^{62,64}$Fe and the $11/2_1^-$ states in $^{61,63}$Co and $^{59}$Mn were measured at the Grand Accélérateur National d'Ions Lourds (GANIL) facility by using the Advanced Gamma Tracking Array (AGATA) and the large-acceptance variable mode spectrometer (VAMOS++). The states were populated through multinucleon transfer reactions with a $^{238}$U beam impinging on a $^{64}$Ni target, and lifetimes in the picosecond range were measured by using the recoil distance Doppler shift method. The data show an increase of collectivity in the iron isotopes approaching $N = 40$. The reduction of the subshell gap between the $v2p_{1/2}$ and $v1g_{9/2}$ orbitals leads to an increased population of the quasi-SU(3) pair ($v1g_{9/2}, v2d_{5/2}$), which causes an increase in quadrupole collectivity. This is not observed for the cobalt isotopes with $N < 40$ for which the neutron subshell gap is larger due to the repulsive monopole component of the tensor nucleon-nucleon interaction. The extracted experimental $B(E2)$ values are compared with large-scale shell-model calculations and with beyond-mean-field calculations with the Gogny D1S interaction. A good agreement between calculations and experimental values is found, and the results demonstrate in particular the spectroscopic quality of the Lenzi, Nowacki, Poves, and Sieja (LNPS) shell-model interaction.

DOI: 10.1103/PhysRevC.95.024312

<footnote>*malin.klintefjord@fys.uio.no</footnote>
# I. INTRODUCTION

A cornerstone of nuclear structure physics is the existence of “magic” nuclei with increased binding due to large gaps in the single-particle shell structure. Spectroscopy of exotic nuclei away from the line of $\beta$ stability has shown that the magic numbers are not universal throughout the nuclear chart but depend on the ratio of neutron to proton numbers; that is, they may vary as a function of isospin [1]. Far from stability, the relative strength of the different terms in the nuclear force may vary and cause a modification of the shell structure. The theoretical description of the changing shell structure is challenging and requires appropriate valence spaces and effective interactions for shell-model calculations and improved energy functionals for mean-field-based models.

Regions in the nuclear chart with rapid changes in collectivity as a function of Z or N are of particular interest because they allow stringent testing of theoretical models. The neutron-rich nuclei close to $^{68}$Ni show a rapid variation in collectivity, which is understood in the shell model as the combination of the effect of the monopole part of the central and tensor force [2,3], leading to a reduced shell gap at N = 40 as protons are removed, and quadrupole correlations arising from the multipole part. This favors a substantial gain in correlation energy via the excitation of neutrons across the N = 40 gap to the quasi-SU(3) partner orbitals $v_{g9/2}$ and $v_{d5/2}$. It should also be noted that, for protons, this is accompanied by an increase in excitations across the Z = 28 gap to the $p_{3/2}$ orbital, which is the quasi-SU(3) partner of the $f_{7/2}$ orbital. This mechanism has been theoretically investigated and compared with available experimental data for cobalt, iron, manganese, and chromium isotopes. It was concluded that the arising collective structures can only be reproduced by including the $v_{d5/2}$ together with the $v_{g9/2}$ orbital in the model space as one approaches N = 40 [4–7]. It was shown that the wave functions of the low-lying states contain a large contribution from these intruder states, while the configurations that correspond to “normal” shell filling lie higher in energy. This is the same mechanism that causes the disappearance of the N = 8 and N = 20 shell closures for neutron-rich nuclei, giving rise to so-called islands of inversion not only at the N = 8 and N = 20, but also at the N = 40 harmonic-oscillator shell closure [8].

A substantial amount of experimental data exist for the region, starting with $^{68}_{28}$Ni$_{40}$ with a high excitation energy of the $2_{1}^{+}$ state and a small $B(E2; 2_{1}^{+} \rightarrow 0_{1}^{+})$ value, suggesting a closed-shell nucleus [9,10]. On the other hand, no strong N = 40 shell gap is seen in mass measurements [11,12], and the weak $B(E2; 2_{1}^{+} \rightarrow 0_{1}^{+})$ value in $^{68}$Ni is understood as being due to a strong neutron component in the $2_{1}^{+}$ excitation [13]. Spectroscopic studies of neutron-rich cobalt isotopes have shown that the N = 40 subshell gap inhibits the development of low-lying collective structures up to $^{65}_{27}$Co$_{38}$. Here, increasing neutron occupation of orbitals above the N = 40 gap manifests itself through the coexistence of spherical and deformed structures at low excitation energy. Even in $^{67}$Co the yrast states can be reasonably well described as a proton particle (hole) weakly coupled to the $2_{1}^{+}$ state in the corresponding iron (nickel) isotopes [14–18].

Spectroscopy experiments following $\beta$ decay [19–21], multinucleon transfer [22,23], and knock-out reactions [24] have found a sudden drop in the excitation energy of the $2_{1}^{+}$ states in neutron-rich iron isotopes from N = 38. This is consistent with lifetime measurements [8,25] and intermediate-energy Coulomb excitation experiments [26] that show a strong increase of the $B(E2)$ values. Recent measurements performed at RIKEN have extended the systematics of excitation energies in chromium and iron isotopes towards the N = 50 shell gap [27], indicating a continuation of the N = 40 island of inversion toward N = 50. Extensive spectroscopic studies of manganese isotopes up to $^{63}$Mn indicate an increase in collectivity already at lower neutron numbers [28,29]. Recent collinear laser spectroscopy of odd-even manganese isotopes up to $^{65}$Mn and the measurement of the g factor of the ground state indicate that there is an increasing fraction of neutron excitation across the N = 40 subshell gap for manganese isotopes with N $\ge$ 36 [30].

The enhanced collectivity of even-even nuclei around $^{66}$Fe and $^{64}$Cr was also the subject of theoretical investigations using mean-field-based approaches [31–34]. Hartree–Fock–Bogoliubov (HFB) calculations with the Gogny D1S interaction found the N = 40 subshell gap to be almost constant in size between Z = 20 and Z = 40 [35]. The potential-energy surfaces for $^{64}$Cr and $^{66}$Fe were found with a spherical minimum that is soft toward prolate deformation. Applying the generator coordinate method with a five-dimensional collective Hamiltonian (5DCH) yielded $B(E2; 2_{1}^{+} \rightarrow 0_{1}^{+})$ values for $^{64}$Cr and $^{66}$Fe that are in relatively good agreement with the experimental values, whereas the excitation energies were significantly overestimated [35].

While a coherent picture of nuclear structure around $^{68}$Ni starts to emerge, there are still only few electromagnetic transition probabilities known in the region. Experimental $B(E2)$ values for the neutron-rich iron isotopes are only known for the $2_{1}^{+} \rightarrow 0_{1}^{+}$ transitions [8,25,26], but not for higher-lying transitions. For the cobalt isotopes, lifetimes were reported for the $9/2^{-}$ and $11/2^{-}$ states in $^{61}$Co [36], the $3/2^{-}$ and $9/2^{-}$ states in $^{63}$Co [16], the $11/2^{-}$ state in $^{63}$Co [18], and the $11/2^{-}$ and $13/2^{-}$ states in $^{65}$Co [18]. For the manganese isotopes some transition strengths are known from Coulomb excitation of $^{61}$Mn [37], and lifetimes were very recently reported for the $7/2^{-}$, $9/2^{-}$, and $11/2^{-}$ states in $^{63}$Mn [38]. No transition strengths have been reported for $^{59}$Mn so far. As theoretical descriptions of this region, in particular recent shell-model calculations [27], are showing an impressive agreement for excitation energies over a large range of isotopes, it is of great interest to challenge these models with experimental transition strengths, which are more sensitive to details of the wave functions than excitation energies alone. The measurements of additional experimental electromagnetic transition probabilities therefore provide important benchmarks for both shell-model and beyond-mean-field calculations.

The recoil-distance Doppler shift (RDDS) technique has been successfully applied in recent years in combination with multinucleon transfer reactions and the identification of reaction products in magnetic spectrometers [39]. Here, we report on the first RDDS lifetime measurement with AGATA
coupled to VAMOS++ at GANIL. New lifetimes were measured in $^{62,64}$Fe, $^{61,63}$Co, and $^{59}$Mn by using multinucleon transfer reaction between $^{238}$U projectiles and $^{64}$Ni target nuclei. The article is organized as follows: Experimental details and the data analysis are described in Secs. II and III, respectively. The results are presented in Sec. IV and discussed and compared with theoretical calculations in Sec. V.

## II. EXPERIMENTAL DETAILS

Excited states of nuclei in the $A \approx 60$ region were populated in multinucleon transfer-reactions from a 6.5A MeV $^{238}$U beam with an intensity of 0.2–0.5 pNA impinging on a 1.25-mg/cm$^2$-thick $^{64}$Ni target at GANIL. The VAMOS++ spectrometer [40–42] was used to detect and identify the target-like transfer products. It was positioned at 45° with respect to the beam axis, close to the grazing angle of the reaction. During this experiment the detection system of VAMOS++ consisted of a set of dual position-sensitive multiwire proportional counter detectors at the entrance of the spectrometer used for the time-of-flight measurement and giving the direction of the ions for Doppler correction, two sets of drift chambers used to determine the trajectory of the ions after the dipole magnet followed by multiwire proportional chambers to give the time of flight, and finally ionization chambers for measuring the total energy and energy loss of the ions at the focal plane. This focal plane detection in VAMOS++ allows for the reconstruction of the particle trajectories and the magnetic rigidity. In this experiment two settings of the magnetic rigidity were used with $B\rho = 0.97$ T m and $B\rho = 0.92$ T m.

Gamma rays were detected in the AGATA germanium $\gamma$-ray tracking array [43], which at the time of the experiment consisted of 19 crystals placed 23.5 cm from the target position. The detectors covered the backward angles from 145° to 180° with respect to the spectrometer axis. Count rates in each crystal were about 6–10 kHz depending on current beam intensity, and trapezoidal shaping with a rise time of 2.5 µs and a peak time of 3 µs was used. A $\gamma$-particle coincidence rate between AGATA and VAMOS++ of a few hundred hertz allowed for pulse-shape analysis of the signals from the segmented AGATA detectors to be performed online. The $\gamma$-ray tracking procedure was performed after each run (typically 12 hours of data taking) during the experiment reading data from disk. The high position resolution of AGATA, together with the reconstruction of the velocity vector for the ions in VAMOS++, resulted in a $\gamma$-ray resolution of 5.6 keV for the Doppler corrected 1345 keV $\gamma$-ray line in $^{64}$Ni. The data, including the digitized germanium detector signals, were written to disk for further improved analysis [44].

The Orsay universal plunger system (OUPS) [45] was used for the lifetime measurements using the RDDS method. Particles recoiling out of the target were slowed down by a 3.0-mg/cm$^2$-thick $^{24}$Mg degrader before entering the VAMOS++ spectrometer with a mean recoil velocity after the degrader of 10.7% of the speed of light. Target and degrader foils were mounted orthogonal to the entry axis of VAMOS++. Data were taken with six different target-degrader distances, $d_0 + 10.1(4)$ µm, $d_0 + 19.9(2)$ µm,

$d_0 + 40.0(2)$ µm, $d_0 + 59.9(3)$ µm, $d_0 + 80.3(5)$ µm, and $d_0 + 206(4)$ µm, for about 24 h per distance. The offset $d_0$ was determined by using known lifetimes; see Sec. III. For the five shortest distances an active feedback system was used to control the distance.

## III. DATA ANALYSIS

The target-like reaction products were identified in mass, charge, and atomic number with the VAMOS++ spectrometer. Nuclei with the same atomic number Z were identified by their characteristic energy loss from the $\Delta E - E$ energy spectrum, shown in Fig. 1. The mass-over-charge ratio of the ions was determined from the time of flight, the path through the spectrometer, and the magnetic rigidity. Combined with the total energy of the ions the mass can be determined, and the mass distributions for the manganese, iron, and cobalt chains are shown in Fig. 2. Gamma rays detected with AGATA in coincidence with $^{62,64}$Fe, $^{61,63}$Co, and $^{59}$Mn, summed over all six distances, are shown in Fig. 3.

Gamma rays from the decaying excited states in the recoiling nuclei are emitted in flight. Due to the energy loss in the degrader, Doppler shifts are different for decays before and after the degrader, respectively. The spectra were incremented with $\gamma$ rays that were Doppler corrected event by event by using the velocity vector of the recoiling ions after the degrader measured with the VAMOS++ spectrometer. Gamma rays emitted before the degrader have a higher velocity than those used in the Doppler correction and appear at lower energies when observed under backward angles. In the cases where the lifetime is comparable to the time of flight between the target and degrader, this procedure results in two distinct peaks for each decay, with decays occurring before the degrader contributing to the *shifted* peak (at lower energy) and decays occurring after the degrader contributing to the *unshifted* peak.

The effect of $\gamma$ rays emitted during deceleration of the ions in the degrader foil was investigated for the different distances by using the AGATA Monte Carlo simulations package [46] and found negligible compared with the statistical and other

<figure><img src="image_89.png" /><figcaption>FIG. 1. Energy loss of the target-like reaction products in VAMOS++ as a function of total energy detected, with cuts on manganese (dotted), iron (dash-dotted), cobalt (dashed), and nickel (solid) isotopes.</figcaption></figure>
<figure><img src="image_90.png" /><figcaption>FIG. 2. Mass distributions for the manganese, iron, and cobalt isotopic chains.</figcaption></figure>

systematic errors, except for the 2+ state in $^{64}$Ni, where systematic uncertainties from this effect dominates the statistical ones, and are of the order of 10%.

<figure><img src="image_91.png" /><figcaption>FIG. 3. γ-ray spectra in coincidence with ions identified as $^{59}$Mn, $^{62,64}$Fe, and $^{61,63}$Co, summed over all six distances.</figcaption></figure>

<figure><img src="image_92.png" /><figcaption>FIG. 4. Schematic level scheme illustrating the feeding and decay of the level of interest. Unobserved feeding is accounted for by introducing a fictive level (marked with v).</figcaption></figure>

Decay curves $R(x)$ for the level of interest and for observed feeding transitions, $R_f(x)$, were constructed according to

$$R^{(\text{expt})}(x) = \frac{I_{u}^{(\text{expt})}(x)}{I_{s}^{(\text{expt})}(x) + I_{u}^{(\text{expt})}(x)}, \qquad (1)$$

$$R_f^{(\text{expt})}(x) = \frac{1}{\epsilon_{\text{rel.}}^{(\text{expt})}} \frac{I_{u,f}^{(\text{expt})}(x)}{I_s^{(\text{expt})}(x) + I_u^{(\text{expt})}(x)},$$

where $I_s^{(\text{expt})}(x)$ and $I_u^{(\text{expt})}(x)$ are the measured shifted and unshifted intensities, respectively, of the decay transition from the level of interest, $I_{u,f}^{(\text{expt})}(x)$ is the intensity of the unshifted component of the observed feeder, and $\epsilon_{\text{rel.}}^{(\text{expt})}$ is the relative detection efficiency for the feeding transition relative to the decay transition.

Lifetimes were fit to the observed γ decays by solving the coupled linear differential equations arising from the decay of the level of interest and its feeders, as illustrated in Fig. 4. The model for the level schemes used in this work consists of the level of interest, any observed transitions from feeder states, and *unseen feeding*, which is treated by introducing a fictive level approximating possible delayed feeding to the level of interest from other levels. More complicated feeding schemes have been tried, giving compatible results but with a higher $\chi^2/N_{\text{dof}}$ by a factor two or more. Therefore, the simplest scheme has been used. Given the proportion of decays coming from the feeders, $N_{0,f}$ and $N_{0,v}$, and the lifetimes, $\tau$, $\tau_f$, and $\tau_v$, the fitted decay curves are

$$R(t) = e^{-t/\tau} + N_{0,f}\tau_f \frac{e^{-t/\tau_f} - e^{-t/\tau}}{\tau_f - \tau} \\ + N_{0,v}\tau_v \frac{e^{-t/\tau_v} - e^{-t/\tau}}{\tau_v - \tau}, \\ F(t) \equiv 1 - R(t), \qquad (2)$$

$$R_f(t) = N_{0,f}e^{-t/\tau_f},$$

where F is called the flight curve. Note that $N_{0,v}$ only includes the delayed unseen feeding. Any prompt unseen feeding, given by $1 - N_{0,f} - N_{0,v}$, does not influence the lifetime measurement.

Experimental data were obtained for $N_{\text{dist}} = 6$ different distances $d_n^{(\text{expt})}$ between target and degrader. The corresponding time of flight to the degrader is

$$t_n^{(\text{expt})} = \frac{d_n^{(\text{expt})} - d_{\text{off.}}^{(\text{expt})}}{v^{(\text{expt})}}, \qquad (3)$$

where $v^{(\text{expt})}$ is the velocity of the particles before the degrader. The velocity $v^{(\text{expt})}$ was obtained from the VAMOS++
spectrometer and corrected for the energy loss in the degrader by using the Doppler shift of the γ transition under study. The offset $d_{\text{off}}^{(\text{expt})}$ was not measured during the experiment. Therefore, a value for the offset was obtained from the well-constrained fit of the lifetime of the $2_1^+$ state in $^{64}$Ni [47], resulting in $d_{\text{off}}^{(64\text{Ni})} = -16.5 \pm 0.7 \ \mu\text{m}$.

The intensities need to be normalized to allow for a direct comparison with experimental intensities. This is done by fitting normalization constants $I_{s+u,n}$, one for each distance of the degrader. By using both the shifted and unshifted components from the level of interest, the normalization constants $I_{s+u,n}$ are constrained by the total number of detected γ decays from this level; namely, $I_{s,n}^{(\text{expt})} + I_{u,n}^{(\text{expt})}$.

The lifetimes that best describe the data are found by using a nonlinear least-square minimization,

$$ \chi^2(\tau, \tau_f, \tau_v, N_{0,f}, N_{0,v}, \vec{I}_{s+u,v}, \epsilon_{\text{rel}}, d_{\text{off}}, \vec{d}) = \sum_{n=1}^{N_{\text{dist}}} \left[ \begin{aligned} & \left( \frac{I_{u,n}^{(\text{expt})} - I_{s+u,n} R(t_n)}{\sigma_{I_{u,n}^{(\text{expt})}}} \right)^2 + \left( \frac{I_{s,n}^{(\text{expt})} - I_{s+u,n} F(t_n)}{\sigma_{I_{s,n}^{(\text{expt})}}} \right)^2 \\ & + \left( \frac{I_{u,n,f}^{(\text{expt})} - \epsilon_{\text{rel.}} I_{s+u,n} R_f(t_n)}{\sigma_{I_{u,n,f}^{(\text{expt})}}} \right)^2 \\ & + \left( \frac{v^{(\text{expt})} - v}{\sigma_{v^{(\text{expt})}}} \right)^2 + \left( \frac{\epsilon_{\text{rel.}}^{(\text{expt})} - \epsilon_{\text{rel.}}}{\sigma_{\epsilon_{\text{rel.}}^{(\text{expt})}}} \right)^2 + \left( \frac{d_{\text{off}}^{(64\text{Ni})} - d_{\text{off.}}}{\sigma_{d_{\text{off}}^{(64\text{Ni})}}} \right)^2 \\ & + \sum_{n=1}^{N_{\text{dist}}} \left( \frac{d_n^{(\text{expt})} - d_n}{\sigma_{d_n^{(\text{expt})}}} \right)^2 \end{aligned} \right] \quad (4) $$

where $t_n = (d_n - d_{\text{off}})/v$. By fitting the velocity, distances, distance offset, γ efficiency and normalization coefficients, the systematic uncertainties from these quantities were taken into account in the fitting procedure. Thus, the uncertainties in the obtained lifetimes incorporate all of these sources of experimental uncertainties. This also makes it possible to determine how much the different sources of experimental uncertainties influence the values and uncertainties of the obtained lifetimes, which we demonstrate in the following section. Here statistical errors refer to the errors extracted from the fit using Eq. (4) excluding the four last terms whereas the statistical errors are extracted from the increase in error on the lifetime using the full expression.

## IV. RESULTS

The analysis procedure was tested by applying it to the lifetime of the first $2^+$ state in $^{64}$Ni. The resulting lifetime of 1.537(76)stat(150)sys ps agrees well with 1.570(50) ps obtained in Ref. [48] and the adopted value of 1.469(75) ps from Ref. [47]. However, it is in disagreement with the value of 1.287(52) ps from a recent Coulomb excitation experiment by Allmond et al. [49].

Because the lifetime of the $2^+$ state in $^{64}$Ni was used to constrain the offset parameter $d_{\text{off}}$ for the distance, it was investigated how much the choice of its value influences the results for other nuclides. Using the newer value of Allmond et al. to determine the offset parameter yields shorter lifetimes for the other nuclides, but differences are within the error

bars and would not change any conclusions; e.g., for $^{62}$Fe the lifetime of the $4_1^+$ states goes from 0.86(25) to 0.58(16) ps. For other nuclei the changes are similar. Constraining the distance parameter with the longer lifetime value of 1.537(76)stat(150)sys ps results in a lower $\chi^2$ ($\chi^2/N_{\text{dof}} = 9$ vs $\chi^2/N_{\text{dof}} = 37$) of the fit and was therefore chosen. Approximately 8% of the uncertainty for the $2^+$ state in $^{64}$Ni originates from the measured intensities of the shifted and unshifted transitions, 68% from the statistical uncertainty in the velocity, and 19% from the distance offset. Applying the analysis method to the $4_1^+$ state in $^{60}$Fe results in a lifetime of 1.20(30) ps, with $\chi^2/N_{\text{dof}} = 1.29$ and 21% population from the seen $6_1^+ \to 4_1^+$ and $5_1^- \to 4_1^+$ transitions, with an effective lifetime of 15(15) ps. In the fit, 24% of the decay occurred via the long-lived (>9 ps) unseen feeder. In this case the statistical uncertainty is 98% of the total error. The obtained lifetime is in agreement with the result in Ref. [50].

For $^{62}$Fe, the shifted and unshifted components of the $4_1^+ \to 2_1^+$ transition at 1299 keV are shown in Fig. 5. For this nucleus, the statistics allowed the inclusion of the $6_1^+ \to 4_1^+$ transition and the $5_1^- \to 4_1^+$ transition as seen feeding transitions, while unobserved feeding was taken into account by introducing a fictive feeding state. The corresponding decay curves are presented in Fig. 6. The fit yields a lifetime of $\tau_{4_1^+} = 0.86(25)$ ps with $\chi^2/N_{\text{dof}} = 1.72$, where $N_{\text{dof}} = 11$. In the fit, 27% of the intensity of the $4_1^+$ state comes via the $6_1^+ \to 4_1^+$ transition with an effective lifetime of 9.5(24) ps, 25% via the $5_1^- \to 4_1^+$ transition with an effective lifetime of 58(50) ps, and 14% is attributed to unseen transitions with an effective lifetime of 75(75) ps. In this fit 97% of the error is statistical.

In $^{64}$Fe, the lifetime of the $4_1^+$ state was obtained from the shift of the $4_1^+ \to 2_1^+$ γ-ray transition at 1017 keV, shown in Fig. 5. The unresolved transitions at 1079 keV (from $6_1^+ \to 4_1^+$) and at 1078 keV (from $5_1^- \to 4_1^+$) were included as seen feeding in the fit, and the decay curves are shown in Fig. 6. As the $5_1^-$ state is more strongly populated than the $6_1^+$ state and has a substantial lifetime it will be the main contributor to long-lived feeding. Subtracting the observed feeding, in combination with poor statistics, results in large uncertainties and hence very weak constraints on the fitting parameters. With unobserved feeding included in the fitting procedure, a short lifetime of 0.19(98) ps was obtained for the $4_1^+$ state with $\chi^2/N_{\text{dof}} = 0.63$. The low $\chi^2$ suggest a under-constrained fit, and such a short value lies outside the sensitivity range of the experiment. Assuming no unobserved feeding will always lead to a longer lifetime for the state of interest, providing only an upper limit for the lifetime. In case of the $4_1^+$ state in $^{64}$Fe an upper limit of $\tau < 1.8$ ps was found and adopted. In this case the fit yields $\chi^2/N_{\text{dof}} = 1.15$ with $N_{\text{dof}} = 9$, and 57% of the feeding was observed with an effective lifetime of 32(29) ps.

The γ-ray spectra gated on $^{63}$Co, shown in Fig. 5, were used to extract the lifetime from the $11/2_1^- \to 7/2_1^-$ transition at 1674 keV γ-ray energy. Figure 6 shows the best fit of the decay curve with an unseen feeder of the order of 7 ps, resulting in $\tau_{11/2_1^-} = 0.55(19)$ ps with $\chi^2/N_{\text{dof}} = 1.6$ and $N_{\text{dof}} = 3$. Here, the relative strength of the unobserved feeding is 62%. This result is consistent, within the joint error bars, with the result found in Ref. [18]. In $^{61}$Co, the $11/2_1^- \to (9/2)_1^-$ transition
<figure><img src="image_93.png" /><figcaption>FIG. 5. The spectra used to extract the lifetime of the $11/2^−$ state in $^{59}$Mn and $^{61,63}$Co and the $4^+$ in $^{62,64}$Fe for the different distances $d$ (in $\mu$m). Shown as color coded areas are the intensities used for determining the decay curves. The area to the left is the shifted component while the right area is the unshifted component.</figcaption></figure>

at 1664 keV (see Fig. 5) was used to investigate the lifetime of the $11/2^−$ state. Seen feeding from the $(13/2)^− \rightarrow 11/2^−$ transition was included in the fit. Similar to the case of $^{64}$Fe, the statistics were not sufficient to extract a precise value of the lifetime, and only an upper limit of $\tau < 2$ ps is found assuming no unobserved feeding. The $\chi^2/N_{dof}$ for this fit is 2.3 with nine degrees of freedom.

Finally, the lifetime from the $11/2^− \rightarrow 7/2^−$ transition [28] at 1189 keV $\gamma$-ray energy in $^{59}$Mn was extracted from the spectra shown in Fig. 5. Fitting the lifetime with an assumed long-lived unobserved feeder ($\tau_v > 7$ ps), where 17% of the decay proceeds via this feeder, resulted in a lifetime of $\tau_{11/2^−} = 2.63(40)$ ps with $\chi^2/N_{dof} = 0.39$ and $N_{dof} = 3$. The fit to the decay curve is shown in Fig. 6.

V. DISCUSSION

In Fig. 7 experimental and theoretical values for the reduced transition strengths in the iron isotopes $^{56–68}$Fe are shown for the $2^+_1 \rightarrow 0^+_1$ and $4^+_1 \rightarrow 2^+_1$ transitions. The new data points

from this work are shown with stars. A small increase in the $B(E2)$ value for the $4^+_1 \rightarrow 2^+_1$ transition from $N = 34$ to $N = 36$ is observed. The upper limit for the lifetime of the $4^+_1$ state in $^{64}$Fe indicates a larger increase in the $B(E2)$ value from $N = 36$ to $N = 38$. Experimentally, the iron isotopes with $R_{42} = E(4^+)/E(2^+)$ ratios between the harmonic vibrator limit of 2 and the rotational limit of 3.3 [51] while having $B(E2; 4^+_1 \rightarrow 2^+_1)/B(E2; 2^+_1 \rightarrow 0^+_1)$ ratios smaller than 2 can be characterized as soft rotors, as already suggested by Daugas *et al.* [52].

Large-scale shell-model calculations using the modified LNPS interaction [4,27], also shown in Fig. 7, accurately reproduce the $B(E2; 4^+_1 \rightarrow 2^+_1)$ values within the experimental errors. For the effective charges, we have adopted those deduced in Ref. [53]: $e_n = 0.46$ and $e_p = 1.31$. This increase in collectivity can be explained by a decrease of the subshell gap at $N = 40$, driven by the monopole proton-neutron part of the effective interaction, favoring neutron excitations into the quasi-SU(3) partners $\nu 1g_{9/2}-\nu 2d_{5/2}$. The occupation number for the $\nu 1g_{9/2}$ orbital increases from 1.14 for the $4^+_1$ state
<figure><img src="image_94.png" /><figcaption>FIG. 6. The decay curves used to extract the lifetime of the $11/2^−$ states in $^{59}$Mn and $^{61,63}$Co and the $4^+_1$ state in $^{62,64}$Fe. Note that, for $^{62}$Fe, two observed feeder states have been included in the fit. For $^{64}$Fe the fit used to extract the upper limit on the lifetime is shown.</figcaption></figure>

in $^{62}$Fe to 2.13 in $^{64}$Fe. In relative terms the increase of occupation in the $v2d_{5/2}$ orbital from $N=36$ to $N=38$ is even larger, going from 0.1 to 0.35 for the $4^+_1$ states. On the proton side this is accompanied by an increase in the occupation of the $2p_{3/2}$ orbital, which is a quasi-SU(3) partner of the $1f_{7/2}$ orbital. The gain in energy due to the quadrupole correlations largely exceeds the energy needed to excite nucleons to the higher-lying orbitals.

We have also performed microscopic calculations based on constrained Hartree-Fock-Bogoliubov (CHFB) theory using the Gogny D1S interaction [57,58] and mapping to the five-dimensional collective Hamiltonian (5DCH). The method is described in detail in Ref. [59]. Worth noticing is that the CHFB + 5DCH calculations contain no free parameters except for those specifying the phenomenological D1S interaction, which is used across the entire nuclear chart. As can be seen in Fig. 7, these calculations give a relative increase in the $B(E2; 4^+_1 \rightarrow 2^+_1)$ values with neutron number that is very similar compared with the increase found in the shell-model calculations, but the absolute values are approximately twice as large. While the calculated $B(E2; 4^+_1 \rightarrow 2^+_1)$ value for $^{64}$Fe is consistent with the experimental limit, the value for $^{62}$Fe is too large. The $B(E2; 2^+_1 \rightarrow 0^+_1)$ values obtained with the 5DCH approach, on the other hand, are in good agreement with both experiment and shell-model calculations. Based on the transition strengths the results of the 5DCH calculations indicate a vibrational rather than rotational character for the neutron-rich Fe isotopes. The occupation numbers for the

<figure><img src="image_95.png" /><figcaption>FIG. 7. $B(E2 \downarrow)$ values for even iron isotopes for transitions from the first $2^+$ and the first $4^+$ state. The solid green lines with circles are shell-model calculations and the blue dashed lines with diamonds are D1S calculations, both a part of the present work. Previous experimental results with error bars are from Refs. [8,25,26] and Nuclear Data Sheets (NDS) [54–56], while the red stars are the new experimental results presented here.</figcaption></figure>

$\pi 1 f_{7/2}$ and $\pi 2 p_{3/2}$ orbitals are constant for the iron isotopes, with the latter remaining small, showing no increase of proton excitations across the $Z=28$ shell gap, contrary to the shell-model predictions. Furthermore, at the mean-field level there are no signs of a weakening of the $N=40$ subshell gap [35,52]. A similar result was found for neutron-rich zinc isotopes, where the 5DCH calculations also overestimated the $B(E2; 4^+_1 \rightarrow 2^+_1)$ values by a factor of two to three [60]. The 5DCH calculations provide dynamical deformed shapes for the ground state, $2^+$, $4^+$, and $6^+$ yrast levels that are not strong enough. These discrepancies could be due to the missing of an explicit tensor term in the effective $NN$ interaction in the Gogny D1S force. Work on implementation of such a term is currently in progress [61]. It should also be mentioned that the collective behavior assumed in the generator coordinate method (GCM) with the Gaussian overlap approximation (GOA) may not be valid close to the $Z=28$ shell closure. The potential-energy surface obtained from the CHFB calculations shows softness in both $\beta$ and $\gamma$ deformation parameters, indicating no deformed minima in the iron region.

In neutron-rich cobalt isotopes a set of low-lying states has been interpreted as the $7/2^−$ proton hole weakly coupled to the $2^+$ excitation in the corresponding nickel isotope; see, e.g., Refs. [16,17] and references therein. In particular it is expected that the first $11/2^−$ state belongs to this multiplet and, therefore, the $B(E2; 11/2_1^- \rightarrow 7/2_1^-)$ values in the chain of cobalt isotopes should closely follow the $B(E2; 2_1^+ \rightarrow 0_1^+)$ values in the respective nickel isotones. In Fig. 8 the reduced transition strengths for the decay of the first $2^+$ state in nickel isotopes are compared with that of the decay of the first $11/2^−$ state in the cobalt isotopes. Also shown are large-scale shell-model calculations based on the modified LNPS
<figure><img src="image_96.png" /><figcaption>FIG. 8. $B(E2 \downarrow)$ values for odd cobalt isotopes for transitions from the first $11/2^-$ to the ground state. The green solid line with circles represents shell-model calculations performed within this work. Previous experimental results with error bars are from Refs. [18,36] and the compilations [47,62] while the red stars are the new experimental results presented here. For comparison, the $B(E2 \downarrow)$ values for the first $2^+$ state in the even nickel isotopes are shown also.</figcaption></figure>

interaction [4,27]. The new data points in this work are marked with stars. Although it was only possible to extract a lower limit for the transition probability in $^{61}$Co ($N = 34$), it is clear that the result is incompatible with the older measurement from the work of Regan *et al.* who deduced $10^{+10}_{-3}$ ps [36]. Taking this into consideration, the hypothesis that the first $11/2^-$ state in the neutron-rich cobalt isotopes belongs to the multiplet generated by the coupling of the $7/2^-$ proton hole to the $2^+$ state in the nickel isotopes is a good first-order understanding of their structure. The shell-model calculations reproduce very well the experimental data. Analyzing them in detail shows that both $^{61}$Co and $^{63}$Co lie outside the island of inversion below $^{68}$Ni, where an increased population of quasi-SU(3) partner orbitals ($\nu 1g_{9/2} - \nu 2d_{5/2}$) generates quadrupole correlations. The occupation numbers for the $\nu 1g_{9/2}$ orbital remain small, around 0.1 for $^{61}$Co and 0.5 for $^{63}$Co for the first $11/2^-$ states. The $\nu 2d_{5/2}$ orbital is very weakly populated as well. Calculations restricted to a *fp* model space reproduce the excitation energies, and to a lesser extent also the transition strengths. This suggests that $^{61,63}$Co can be viewed as being on the limit of the *fp* space, as was already concluded by Recchia *et al.* [17].

Only few transition strengths are known for the manganese isotopes, in particular for the $11/2_1^- \rightarrow 7/2_1^-$ transition. The systematics of the $11/2_1^-$ level in the neutron-rich manganese isotopes ($A = 55-61$) shows rather constant energies with no signs of rapid changes in its structure. In $^{55}$Mn the $B(E2; 11/2_1^- \rightarrow 7/2_1^-)$ has been determined to be $167(20) \ e^2 \text{fm}^4$ [63], to be compared with $110^{+15}_{-12} \ e^2 \text{fm}^4$ in $^{59}$Mn as measured in this work. Our calculations overestimate the $B(E2; 11/2_1^- \rightarrow 7/2_1^-)$ value in $^{59}$Mn, but remain smaller than the $B(E2; 11/2_1^- \rightarrow 7/2_1^-)$ value in $^{55}$Mn. Hence, neither the experimental nor the theoretical result show any sign of enhanced collectivity for $^{59}$Mn. This is consistent with recent collinear laser spectroscopy studies of Babcock *et al.* [30],

<table><caption>TABLE I. Summary of experimental and theoretical lifetimes and $B(E2)$ values obtained in this work. Previously measured lifetimes are also given with corresponding reference.</caption><thead><tr><th rowspan="2">Nuc.</th><th rowspan="2">State</th><th rowspan="2">γ energy<br>[keV]</th><th colspan="2">τ [ps]</th><th colspan="3">B(E2 ↓) [e<sup>2</sup> fm<sup>4</sup>]</th></tr><tr><th>Expt.</th><th>Prev.</th><th>Expt.</th><th>SM</th><th>D1S</th></tr></thead><tbody><tr><td><sup>59</sup>Mn</td><td>11/2<sub>1</sub><sup>-</sup></td><td>1189</td><td>2.63(40)</td><td></td><td>111<sup>+21</sup><sub>-15</sub></td><td>152</td><td></td></tr><tr><td><sup>62</sup>Fe</td><td>4<sub>1</sub><sup>+</sup></td><td>1299</td><td>0.86(25)</td><td></td><td>256<sup>+105</sup><sub>-58</sub></td><td>326</td><td>515</td></tr><tr><td><sup>64</sup>Fe</td><td>4<sub>1</sub><sup>+</sup></td><td>1017</td><td>&lt;1.8</td><td></td><td>&gt;420</td><td>441</td><td>599</td></tr><tr><td><sup>61</sup>Co</td><td>11/2<sub>1</sub><sup>-</sup></td><td>1664</td><td>&lt;2</td><td>10<sup>+10</sup><sub>-3</sub> [36]</td><td>&gt;24</td><td>43</td><td></td></tr><tr><td><sup>63</sup>Co</td><td>11/2<sub>1</sub><sup>-</sup></td><td>1672</td><td>0.55(19)</td><td>1.0(3) [18]</td><td>104<sup>+58</sup><sub>-28</sub></td><td>88</td><td></td></tr></tbody></table>

which suggest an increase of collectivity due to a weakening of the $N = 40$ subshell closure beyond $N = 36$.

## VI. SUMMARY

In the present work, lifetimes of excited states in moderately-neutron-rich cobalt, iron, and manganese isotopes have been measured by using the recoil distance Doppler shift method. The nuclei $^{61,63}$Co, $^{62,64}$Fe, and $^{59}$Mn were produced in an experiment at GANIL via multinucleon transfer reactions of a $^{238}$U beam on a $^{64}$Ni target, identified on an event-by-event basis in the large-acceptance spectrometer VAMOS++, and γ rays were detected using the AGATA γ-ray spectrometer. It was possible to determine the lifetime of the $4_1^+$ state in $^{62}$Fe and give an upper limit for $^{64}$Fe. For the odd-Z neighbor isotopes, lifetimes for the $11/2^-$ states in $^{59}$Mn and $^{63}$Co were determined, whereas for $^{61}$Co again only an upper limit could be given. Theoretical calculations have been performed and compared with the experimental results, as summarized in Table I. Large-scale shell-model calculations using the modified LNPS interaction [4,27] give a very good description of the low-energy nuclear structure in the region. They show an increase in quadrupole collectivity when approaching $N = 40$. Quadrupole correlations induce the excitation of neutrons to the $\nu 1g_{9/2}$ and $\nu 2d_{5/2}$ orbitals. For the even-even iron isotopes, a rotational-like behavior emerges as $N = 40$ is approached. In the case of $^{59}$Mn, the predicted transition probability for the $11/2_1^- \rightarrow 7/2_1^-$ transition is larger than the measured one. It should be pointed out that our shell-model calculations are capable of reproducing both excitation energies and transition probabilities for nuclei both inside and outside of the island of inversion found at $N = 40$.

Beyond-mean-field CHFB + 5DCH calculations using the D1S interaction were also compared with the experimental data. Although the overall trend of increasing collectivity with increasing neutron number for the iron isotopes is clearly seen, the beyond-mean-field calculations suggest a vibrational-like structure and no clear change in structure close to $N = 38$. This might originate from the lack of an explicit tensor term in the force.

## ACKNOWLEDGMENTS

We are grateful for the help from Johan Goupil and the GANIL technical staff. The authors would also like to thank M. Loriggiola, Laboratori Nazionali di Legnaro, and
G. Fremont, GANIL, for preparing the target and degrader foils. This project received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement n°654002. Support from the Norwegian Research Council, project grant 213442, is acknowledged. The project has received funding from the Swedish Research Council. This

work is supported by the Spanish Ministerio de Economía y Competitividad under contract FPA2014-57196-C5-4-P. One of the authors, A. Gadea, was partially supported by MINECO and Generalitat Valenciana, Spain, under the grants FPA2014-57196-C5 and PROMETEOII/2014/019, and by EU FEDER funds.

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7.8 Experimental determination of reference pulses for highly segmented HPGe detectors and application to Pulse Shape Analysis used in gamma-ray tracking arrays
Experimental determination of reference pulses for highly segmented HPGe detectors and application to Pulse Shape Analysis used in γ-ray tracking arrays

H.J. Li<fnref n="1" />, J. Ljungvall<fnref n="2" />,ª, C. Michelagnoli<fnref n="3" />,<fnref n="1" />, E. Clément<fnref n="1" />, J. Dudouet<fnref n="2" />,<fnref n="4" />, P. Désesquelles<fnref n="2" />, A. Lopez-Martens<fnref n="2" />, and G. de France<fnref n="1" />

Received: 26 January 2018 / Revised: 25 September 2018

Published online: 26 November 2018

© Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2018

Communicated by C. Ur

**Abstract.** For the first time, bases of signals delivered by highly segmented HPGe detectors, for identified hit locations, have been determined *in situ*, that is in the actual accelerator-target-detection system conditions corresponding to data acquisition during a physics experiment. As a consequence, these bases include all the genuine features and alterations of the signals induced by the experimental setup, *e.g.* diaphony, electronic response, specificity of individual crystals. The present pulse shape bases were constructed using calibration source data taken at the beginning of the AGATA campaign at GANIL. An experiment performed at GANIL using the AGATA γ-ray detector together with the VAMOS spectrometer was used to validate the bases. The performance of the bases when used for pulse-shape analysis has been compared to the performance of the standard bases, composed of pulse shapes generated by a computer simulation used for AGATA. This is done by comparing the Doppler correction capability. The so-called Jacobian method used to generate the *in situ* bases also produces correlations that can be applied to locate in a direct way (no search algorithm) the location where a γ-ray interacted given that only one segment is hit. As about 50% of all pulse-shape analysis is performed on crystals with only one segment hit this will allow for a large reduction in the needed computer power. Different ways to improve the results of this prospective work are discussed.

# 1 Introduction

A new generation of γ-ray tracking spectrometers, the Advanced Gamma Tracking Array (AGATA) [1] in Europe and the Gamma Ray Energy Tracking Array (GRETA) [2] in the USA, open new possibilities for nuclear structure studies with their excellent energy resolution, good peak-to-total and high efficiency for γ-ray energies in the range of tens of keV up to ten MeV. The concept of γ-ray tracking has motivated a long effort of Research and Development worldwide. In Europe the AGATA spectrometer is under construction to reach (in the next decades) an angular coverage of $4\pi$. Meanwhile several AGATA campaigns have been performed with up to 35 High-Purity Segmented Germanium crystals at the laboratories of Legnaro National Laboratory [3], GSI [4], and GANIL [5]. AGATA has shown its advantage of high energy resolution

at large recoil velocities as compared to the traditional Germanium detector array like CLARA [6], EXOGAM [7] or the RISING setup [8].

The performance of γ-ray tracking arrays depends on the γ-ray tracking algorithm, which requires an accurate determination of the energies and positions of the γ-ray interactions. The positions and energies are given by Pulse Shape Analysis (PSA) on the digitized signals from each AGATA HPGe detector crystal. Each crystal has 36 segments and one central contact for a total of 37 output signals. Currently, AGATA uses the Adaptive Grid Search (AGS) [9] method. The pulse shape basis used for each detector has to be an accurate representation of the response of the detectors to achieve high position accuracy and resolution. Pulse-shape bases presently used within the AGATA collaboration are created using the AGATA Detector Library (ADL) [10]. The code calculates the electric field in the detector and generates signals by transporting

<footnote n="1">¹ GANIL, CEA/DRF-CNRS/IN2P3, BP 55027, 14076 Caen Cedex 5, France</footnote>

<footnote n="2">² CSNSM, Université Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, 91405 Orsay, France</footnote>

<footnote n="3">³ Institut Laue-Langevin, B.P. 156, F-38042 Grenoble Cedex 9, France</footnote>

<footnote n="4">⁴ Université Lyon 1, CNRS/IN2P3, IPN-Lyon, F-69622 Villeurbanne, France</footnote>

<footnote>ª e-mail: joa.ljungvall@csnsm.in2p3.fr</footnote>
the charge carriers in this field and calculating the signals induced on the segments and central contact using the Shockley-Ramo theorem. The ADL takes into consideration the detector geometry, the impurity distribution, and the charge carrier mobilities. A considerable effort has been put into the experimental determination of physical parameters of importance to the pulse-shape generation in the AGATA detectors [11–13], reducing the uncertainties in a significant way as compared to the situation at the beginning of the AGATA project. Cross talk between electric channels is corrected using experimentally measured values on an event-by-event basis [14,15]. The response of the electronics is modeled in a realistic way but not on a channel by channel basis. This work has been successful and the PSA of AGATA gives an interaction position resolution of 5 mm Full Width at Half Maximum (FWHM) [16–18]. This is the needed resolution in order to construct a γ-ray tracking array with good performance [19].

However, the position resolution as determined from in-beam data using Doppler broadening [16] or from imaging methods [17,18,20] are not in accordance with the position resolution that would maximize the performance of γ-ray tracking. This is observed, when using the Orsay Forward Tracking algorithm [21], by the need to increase the parameter corresponding to the position resolution by a factor of 3 as compared to the value used for simulations where a position resolution of 5 mm FWHM was assumed. As the actual shape of the distribution of the errors of the PSA is not known it is not possible to make a direct connection between this factor 3 and the position resolution. A tempting interpretation is that a non-Gaussian position error distribution would require a larger effective position resolution for γ-ray tracking, while still giving a FWHM of Doppler corrected γ-ray peaks corresponding to a Gaussian position error distribution. If the bases were perfect the error distribution should be close to Gaussian and the error estimates from direct methods looking at the first interaction point and that from the parameter in the γ-ray tracking algorithm should coincide. Moreover, the hit-position distributions given by pulse-shape analysis performed on highly segmented HPGe detectors tend to show strong clustering effects, not reflecting the homogeneity of the γ-ray radiation field. There is no other mechanism than PSA that can generate an apparent clustering of γ-ray interactions. Using computer simulations of the pulse shapes to calculate both the signal basis and signals based on interaction position given by Geant4 simulations it is possible to verify that the clustering effect is not an artefact of the chosen PSA algorithms. This effect is therefor attributed to the fidelity of the basis. It is possible to build a complete experimental basis using 3D position scanning of the detectors [22–29], which should give a very accurate basis. However, as the electronic response function in principle changes with a specific set of electronics used for a detector and considering possible effects from the aging (neutron damage and annealing of the crystal) of the detector, the experimental data bases would in principle have to be redone more frequently than is practically possible if these effects are to be included in full detail.

A basis generation method that circumvents these problems was proposed by Désesquelles in refs. [30–32]. The idea is that a correlation is constructed between the known theoretical distribution of interaction points in a detector and experimental raw estimators of the positions valid for events with a single γ-ray interaction in the crystal. This could allow to build an experimental basis of pulse shapes in the same manner as energy calibrations are prepared. Before the experiment, a $^{60}$Co source centered at the target position is used to acquire data and build experimental data bases for the experiment. As an additional bonus, correlations between interaction positions and raw estimators can be used for online PSA for events with only one segment having a net charge, under the assumption of a single γ-ray interaction point. The validity of this method has been evaluated using both five sets of simulated signals [31] and experimental pulses from the scanning system at the University of Liverpool [32]. As the real interaction position is known for data from a scanning table the mean error in position determination could be estimated to be less than 5 mm, hence fulfilling the resolution condition needed for γ-ray tracking. In the present paper, the method proposed by Désesquelles is applied using data taken for energy calibration with a non collimated γ source and tested on real in-beam data using the AGATA electronics chain and detectors in experimental conditions. To the knowledge of the authors this has not been done before. The work reported on in this paper had as goals to investigate the feasibility and performance of this method.

An additional important problem faced by γ-ray tracking arrays is the correct determination of the number of γ-ray interactions in a detector segment. This problem is outside the scope of the present work, and the number of segments that has been hit is identified with the number of γ-ray interactions. Work performed within the AGATA and GRETA/GRETINA Collaborations underlines the need for high fidelity bases in order to improve on this determination [33].

The paper is organized as follows: in sect. 2 the method, referred to in this paper as in situ, is described in detail and the choices of the experimental raw estimators are presented together with examples of database pulses generated with this method. Comparisons are made in sect. 3 between the PSA done with experimental and with calculated data bases. The comparison is based on the Doppler correction capabilities using experimental in-beam data from an AGATA experiment performed at GANIL. Using PSA based on the experimental correlations directly without passing by the Adaptive Grid Search is also explored. Conclusions and outlooks are given in sect. 4.

## 2 In situ method for pulse shape basis construction and hit location

In the following, we make the usual approximation that the signal in a segment is induced by a single γ-ray interaction. This is often not the case for γ rays with an energy
<figure><img src="image_97.png" /></figure>

**Fig. 1.** (Color online) Schematic diagram of one AGATA crystal with the three different coordinate systems used in this paper shown in different colors. The top system (dashed lines) shows the coordinates used for all but the first ring. Note that the origin of the coordinate system is at the front of the crystal, not at the back as could be suggested by the figure. For the first ring, a system with $r_1$, $\phi_1$, and $z$ is used (solid lines), having an origin at (0, 0, 30) mm in the crystal reference system shown with $x'$, $y'$, and $z'$ axes (dash-dotted lines). The latter system is also the coordinate system used for Geant4 simulations.

above a few hundred keV. Geant4 simulation performed within the scope of this work of a $^{60}$Co source, however, shows that if an energy condition of $E_{segment} > 200$ keV is used together with the condition that only one segment has been hit the energy weighted average position of the interactions in this segment is on average located within 0.8 mm of the position of the largest single interaction. The largest interaction also contains, on average, 85% of the deposited energy. These distances are smaller than the expected sensitivity showing that the identification between the number of hit segments and number of $\gamma$-ray interactions does not render the method presented in the paper a priori invalid. The conditions of a hit segment multiplicity of one and a minimum energy $E_{segment} > 200$ keV have been applied to both the simulated and experimental data used for the construction of the data bases.

Both the construction of an experimental basis using the *in situ* method as well as the application to real-time pulse shape analysis require the construction of good correlations between the theoretical $\gamma$-ray interaction point distributions, e.g. $r$, $\phi$, and $z$ in fig. 1, and experimental raw estimators of the same quantities. Note the use of two different coordinate systems, depending on the ring number, for the experimental estimators, this because the

<figure><img src="image_98.png" /></figure>

**Fig. 2.** Scheme describing the *in situ* method to produce an experimental basis of pulse shape. For details, see text.

electric field gradient should be as parallel as possible to the $r$ coordinate to simplify the experimental raw estimator. The theoretical $\gamma$-ray interaction distributions are given by AGATA Geant4 simulations [34] with a source at the same position as used to collect the experimental source data. Raw estimators, described in previous work by Désesquelles *et al.* [31,32] and later in this paper, are then calculated using the signals from a $^{60}$Co source centered at the target position. The raw estimators use both the net-charge signal from the segment in which the $\gamma$-ray interacted and the transient signals from neighboring segments. For example, if a $\gamma$-ray interaction is in segment A4, see the schematic diagram in fig. 1, the pulses in segments A3, A4, A5, B4, and F4 are used to produce the raw estimators.

The steps needed to construct an experimental basis (which gives the correlations needed for real-time PSA as a part of the process) are shown in fig. 2. The simulations should give an accurate description of the geometry used for the source measurements, i.e. all detectors should be included in the simulation. The Geant4 simulation used in this work is capable of reproducing the measured efficiency within 3% [35]. A total of 3 billion events were simulated to generate a statistics of 30 million selected events in a detector. Calibrated energies $E$ and traces for all the segments were recorded for a $^{60}$Co source centered at the target position during the measurement. For this work, data was collected for a total of 12 hours with an event rate of about 400 Hz per crystal, taken at the beginning of the AGATA Campaign at GANIL, with the AGATA detectors at their nominal position. The experimental signals used have been processed in the same way as is done in AGATA before the signals are used for PSA, e.g. they are
gain-matched and time aligned. The segment multiplicity has been determined as done by the AGATA preprocess- ing codes [1]. No further processing was done. Only data where one segment has been hit by the γ ray can be used. This condition is applied on both experimental and simu- lated data. For each interaction, the energy deposited by the γ -ray and the three spatial Cartesian coordinates (see the green coordinates in fig. 1) are obtained from the simu- lation. In the case of multiple interactions in one segment, the energy-weighted average position is used. This is the standard procedure to emulate the effects of pulse-shape analysis in simulations within the γ -ray tracking commu- nity. The extraction of the correlations between the ex- perimental traces and the hit averaged positions is done in three steps. First r is defined as the distance between the hit position and the center line of the core contact in the xy -plane for the last five rings ( i.e. , rings 2–6), see the cylindrical coordinate system (dashed lines) in fig. 1, i.e.

$$
r = \sqrt{x^2 + y^2}. \tag{1}
$$

This definition is used for backwards rings (2 to 6) only. Because of the special shape of the electric field lines in the first ring (A1-F1), r is redefined to be

$$
r_1 = \sqrt{x^2 + y^2 + (z - 30)^2}, \quad (2)
$$

which is the distance between the hit position and a point at (0, 0, 30) mm, see the coordinate system with solid lines in fig. 1. The raw estimator for r , noted f$_{r}$ , which was op- timized using simulated detector pulses as a part of this work, is given by a combination of the estimators sug- gested in ref. [31],

$$
f_r = \frac{\sum_i (i - \langle i \rangle)^3 s_i}{\sum_i s_i} - 400 \times \frac{\sum_i S_{ti} + S_{bi} + S_{li} + S_{ri}}{\sum_i s_i}, \quad (3)
$$

using both the net-charge signal and the transient signals from the neighboring segments in the source measurement. Here, i is the index of the signal trace, S is the signal from the hit segment ( S$_{i}$ is one sample of the signal), s$_{i}$ = ( S$_{i}$$_{+1}$ − S$_{i}$$_{−}$$_{1}$ ) / 2 is the derivative of the signal and

$$
\langle i \rangle = \sum_i s_i / \sum_i s_i
$$

is the average sample of the derivative of the pulse with ∑ $_{i}$s$_{i}$ = E the calibrated energy of the signal. Finally S$_{t}$ , S$_{b}$ , S$_{i}$ and S$_{r}$ are signals from the four direct neighbors of the hit segment. Using the previous example, with a γ -ray interaction in segment A4, its top S$_{t}$ , bottom S$_{b}$ , left S$_{i}$ and right neighbor S$_{r}$ are A5, A3, B4, and F4, respectively. Then, for each segment, the distributions of both r and its raw estimator f$_{r}$ are divided into intervals with the condition that each interval has the same probability, e.g. interval r$_{i}$ and f$_{ri}$ both hold X % of the total number of counts. This procedure is illustrated in fig. 3. Finally, using this so-called Jacobian correspondence, one can deduce, from the measurement of f$_{r}$ , an estimate of r . The new version of the f$_{r}$ estimator was developed using signals of the ADL base, and gives better results.

<figure><img src="image_99.png" /><figcaption>Fig. 3. Example showing how the r - f$_{r}$ correlation is con- structed for segment A1 in detector A002. In the top left corner the distribution of the experimental raw estimator f$_{r}$ is shown, with the raw estimator on the y axis and the number of counts on the inverted x axis. The simulated r distribution is shown at the bottom with an inverted y axis. A correlation is con- structed by identifying the coordinate value and raw estimator value that corresponds to the same fraction of the integral of each distribution, respectively. Explicit examples are given for 10% and 90%.</figcaption></figure>

The second step is to calculate ϕ and its raw estimator f$_{ϕ}$ . ϕ is the azimuthal angle with origin along the sym- metry plane of the segment, as shown in fig. 1 with ϕ$_{1}$ for the first ring and ϕ for the other rings. Hence, ϕ is an angle in the range between around − 30 ◦ and 30 ◦ , but the exact values vary with detector type and segment. The raw estimator of ϕ we have used is

$$
f_{\varphi} = \frac{(\max S_1 - \min S_1) - (\max S_r - \min S_r)}{(\max S_1 - \min S_1) + (\max S_r - \min S_r)}. \quad (4)
$$

It is calculated from the transient signals of the direct neighbors in the same ring. This estimator was found to perform better than the one used in ref. [31]. Then, the Jacobian correspondence between ϕ and its raw estimator f$_{ϕ}$ is calculated as previously, except that the calculation is not done in the whole segment but inside small r$_{i}$ ± Δ r intervals (corresponding to f$_{ri}$ ± Δ f$_{r}$ for the raw estima- tor), see fig. 4. Inside each r$_{i}$ interval, the length of the ϕ steps varies roughly as ∝ r . This is to keep the volume of the voxels close to constant inside the detector volume.
<figure><img src="image_100.png" /><figcaption>**Fig. 4.** (Color online) First row: simulated $r-\varphi$ distribution for segments A1 (a), A3 (b) and A6 (c). Second row, raw estimator $f_r-f_\varphi$ distribution for segments A1 (d), A3 (e) and A6 (f). For each pair of simulated and experimental data (i.e. (a)–(d), (b)–(e), and (c)–(f)) the squares show the used divisions in $r$, $f_r$ and $\varphi$, $f_\varphi$ and each pair of squares contains the same fraction of events thus allowing the correspondence between the raw estimators and the coordinates. The third row explicitly shows the correlations between the raw estimators $f_z$ and the $z$ coordinates when constrained to the white filled squares shown in the first two rows. That is, the upper-left histogram in figure (g) is filled when $f_r$ and $f_\varphi$ are within the white square in figure (d), the bottom histogram of figure (g) is filled when $r$ and $\varphi$ in the simulations is within the white square in figure (a). The line shows the correlation between the raw estimator $f_z$ and $z$ coordinate for this $r$, $\varphi$ grid point. This is repeated for segments A3 and A6 in figure (h) and (i), respectively.</figcaption></figure>