Production cross section and decay study of $^{243}$Es and $^{249}$Md

In the study of the odd-$Z$, even-$N$ nuclei $^{243}$Es and $^{249}$Md, performed at the University of Jyv\"askyl\"a, the fusion-evaporation reactions $^{197}$Au($^{48}$Ca,2$n$)$^{243}$Es and $^{203}$Tl($^{48}$Ca,2$n$)$^{249}$Md have been used for the first time. Fusion-evaporation residues were selected and detected using the RITU gas-filled separator coupled with the focal-plane spectrometer GREAT. For $^{243}$Es, the recoil decay correlation analysis yielded a half-life of $24 \pm 3$s, and a maximum production cross section of $37 \pm 10$ nb. In the same way, a half-life of $26 \pm 1$ s, an $\alpha$ branching ratio of 75 $\pm$ 5%, and a maximum production cross section of 300 $\pm$ 80 nb were determined for $^{249}$Md. The decay properties of $^{245}$Es, the daughter of $^{249}$Md, were also measured: an $\alpha$ branching ratio of 54 $\pm$ 7% and a half-life of 65 $\pm$ 6 s. Experimental cross sections were compared to the results of calculations performed using the KEWPIE2 statistical fusion-evaporation code.


I. INTRODUCTION
Determining the boundaries of the nuclear chart, particularly in the region of super-heavy nuclei (SHN), is one of the key questions driving fundamental nuclear physics. The SHN owe their existence to shell effects, as without them the Coulomb repulsion would make the nuclei beyond Z = 104 unstable against fission [1]. In this context, detailed spectroscopy of very heavy nuclei (VHN) and SHN is of paramount importance to provide information on the nuclear landscape close to the high-A limit of the nuclear chart, as well as on the nature of the predicted island of stability. The challenge of these experiments is related to low production cross sections and, in odd- mass nuclei, to the complexity of spectra where various collective and single-particle excitations may lie close in energy. On the other hand, the studies of odd-mass nuclei are rewarded by the wealth of information regarding single-particle states, exceeding what can be obtained for even-even nuclei [2]. Regarding the known excited states of single-particle or collective nature, little data is available for Es (Z = 99) and Md (Z = 101) isotopes [2,3]. Before in-beam spectroscopy of these odd-Z nuclei can be attempted, feasibility studies are a prerequisite, in particular measurements of production cross sections. Such measurements also help to improve the description of the fusion-evaporation reaction mechanism, providing new constraints for the models.
In this paper, the production cross sections for 243 Es and 249 Md populated directly in the fusion-evaporation reactions 197 Au( 48 Ca,2n) 243 Es and 203 Tl( 48 Ca,2n) 249 Md are reported. The targets and projectiles were chosen as a compromise between the predicted production cross sections and the transmission in the separator. In particular, very asymmetric reactions using actinide targets were not considered, as in such cases (i) the large angular dispersion due to the low recoil velocity and neutron emission results in a poor transmission, (ii) the low recoil energy reduces the detection efficiency at the focal plane, both effects being not fully compensated by enhanced cross sections.
The present study also allowed the half-lives and decay properties of these nuclei to be updated, as well as those of 245 Es, populated by the α decay of 249 Md. It should be noted that α-decay branching ratios and, to a lesser extent, half-lives are needed to deduce production cross sections. Finally, the measured production cross sections for 243 Es and 249 Md are discussed in the context of the Z 100 region and compared to the predictions of the KEWPIE2 statistical fusion-evaporation code [4].

II. EXPERIMENTAL SETUP
The experiments were performed at the Accelerator Laboratory of the University of Jyväskylä (JYFL). The fusion-evaporation residues, including 243 Es and 249 Md, were separated from the fission fragments, the primary 48 Ca beam and the beam-and target-like reaction products using the Recoil Ion Transport Unit (RITU) gasfilled separator [5,6], which was operated at a He pressure of 0.4 -0.6 mbar. The RITU transmission is estimated to be approximately 30 % for the reactions considered here. The beam current was measured at regular intervals using a Faraday cup, and monitored using the detectors counting rate, thus allowing the beam dose to be deduced with an uncertainty of 20 %.
At the focal plane of RITU, the separated fusionevaporation residues were first detected in a positionsensitive multi-wire proportional counter (MWPC) and then implanted in two adjacent double-sided silicon strip detectors (DSSDs), both detectors being part of the Gamma Recoil Electron Alpha-Tagging (GREAT) spectrometer [7]. The MWPC provided a time of flight (ToF) and energy loss (∆E) measurement, allowing (i) selection of the fusion-evaporation residues using a ToF-∆E identification matrix (ii) correlations with the DSSD, which enable the recoiling residues (coincidence) to be discriminated from the decay products (anti-coincidence). Each DSSD is 300 µm thick and consists of 60 × 40 strips with a 1 mm strip pitch. The Y side of the DSSD was calibrated using an external mixed 239 Pu, 241 Am, and 244 Cm α source. An energy offset is applied to account for the energy loss of the α particle in the detector entrance window (in case of external source), and for the daughter nucleus recoil (decay from the detector after implantation), so that the resulting energy corresponds to the literature value for the nuclei studied in the present work. The X side was amplified with a higher gain to measure low energy conversion electrons, and calibrated using an external 133 Ba source. Signals from all detectors were processed by a trigger-less acquisition system known as the Total Data Readout (TDR) [8]. The recoil decay correlation analysis was performed using the software package Grain [9]: after a first selection using the ToF-∆E identification matrix, the fusion-evaporation residues (recoils) were identified using the energy of the α particles registered in the same pixel of the DSSD subsequent to the implantation of a recoil. The SAGE array [10] surrounded the target for the prompt gamma and conversion-electron detection, however data from this detector were not used in the present work.

A. Decay and half-life measurement
The 243 Es isotope was discovered in the 1970s by Eskola et al. using the 233 U( 15 N,5n) 243 Es reaction [11,12], and latter revisited in the 1990s by Hatsukawa et al., using the 233 U( 14 N,4n) 243 Es reaction [13]. A more recent study, performed with the SHIP separator at GSI by Antalic et al. [14], has shown that 243 Es decays to its daughter via an α-particle with an energy of 7893 ± 10 keV, with a half-life of T 1/2 = 23 ± 3 s and a α-decay branching ratio of 61 ± 6 % . An α-particle fine structure was tentatively observed with peaks at 7745 ± 20 and 7850 ± 20 keV. In the work of Antalic et al. 243 Es was populated in the decay of the mother nucleus 247 Md, while in the present study it was directly produced in the 197 Au( 48 Ca,2n) 243 Es reaction, with a 21 pnA 48 Ca beam at ∼ 210 MeV energy impinging on a 197 Au target. The 48 Ca + 197 Au reaction has already been studied in the 1990s by Gäggeler et al. [15], however, few spectroscopic data were available at that time, preventing the discrimination of fusion-evaporation residues from 2n and 3n channels. Fig. 1 presents the α-particle energy spectrum measured in the DSSD resulting from recoil-α correlations, with the decay of 243 Es clearly visible. The time distribution (∆T ) of the α decay with respect to the implantation, selecting the 243 Es α-decay energy, is presented in Fig. 2. In the inset, the time distribution is drawn as a function of ln(∆T ) using a maximum search time of 10 h. The peak at ln(∆T ) = 10.5 corresponds to the 243 Es decay, while that around ln(∆T ) = 16 is related to random correlations occurring at an average time interval of ≈ 5000 s. The spectrum in the main panel can be fitted using the function [16]: where λ is the decay constant of the nucleus of interest and r is the random correlation rate. Similarly, the spectrum in the inset can be fitted following the method described in Ref. [17]. As expected, both procedures give the same result, yielding the half-life of T 1/2 = 24±3 s, in agreement with the results of the experiment performed at SHIP [14].
∆T The inset of Fig. 2 demonstrates that the 243 Es decay events can be well separated from the background in the defined range ln(∆T ) < 12.5, which corresponds to a time window of 268 s after the recoil implantation. This search time is used in the next section in order to determine the number of events corresponding to the α decay of 243 Es.
The recoil-α-α correlations were used to search for the decay of 239 Bk following the 243 Es decay. The negative outcome of this search is again consistent with the results of the measurement at SHIP [14]. The decay properties of nuclei studied in the present work are summarized in Table I.

B. Production cross section
In order to study the production cross section for 243 Es using the fusion-evaporation reaction 197 Au( 48 Ca,2n) 243 Es, two different beam energies were used. The target used for this measurement was a 270 ± 13 µg cm −2 thick 197 Au self-supporting foil. The cyclotron delivered a 213 ± 1.0 MeV beam first passing  [23] through the 100 µg cm −2 carbon window of the SAGE electron spectrometer. The first part of the study was performed with a beam energy in the Middle of the Target (MoT) estimated to be 210.0 ± 1.0 MeV. Then a carbon degrader foil of 100 µg cm −2 was placed upstream to reduce the incident energy (MoT) to 208.0 ± 1.0 MeV. The spectrum presented in Fig. 1 corresponds to the total statistics, namely with and without the degrader. The number of counts attributed to the 243 Es α decay was obtained using a maximum search time of 268 s. The contribution from random correlations was estimated by integrating the random correlations component (second term in Eq. 1 in the case λ r) using this time window. After subtracting this background, the number of α particles stemming from 243 Es was determined to be 50 ± 7 (32 ± 6) without (with) the carbon degrader foil. The uncertainties were evaluated following the method described in Ref. [24]. In the present work, the statistics is large enough to consider standard normal distributions, therefore symmetric uncertainties are adopted.
During the acquisition time without and with the degrader, the number of 48 Ca nuclei that impinged on the 197 Au target was equal to (1.6 ± 0.3) × 10 16 and (1.2 ± 0.2) × 10 16 , respectively. Taking into account the 197 Au target thickness, the α-decay branching ratio of 61 ± 6 % [14], the α-detection efficiency of 55 %, and assuming a RITU transmission of 30 %, a production cross section σ( 243 Es) = 37 ± 10 nb was deduced for a beam energy of 210.0 ± 1.0 MeV (without degrader), and σ( 243 Es) = 32±9 nb for a beam energy of 208.0±1.0 MeV (with degrader). Only statistical uncertainties corresponding to the beam dose, number of α-particles and α-decay branching ratio are given. The RITU transmission of 30 % is actually a transmission × detection efficiency including the transmission through the separator, the time-of-flight and the DSSD detection efficiencies. The results are presented in Table II.
The odd-Z nucleus 249 Md was populated using the fusion-evaporation reaction 203 Tl( 48 Ca,2n) 249 Md in three different irradiation campaigns. The first campaign was focused on cross-section measurements at two different bombarding energies of 214.3 ± 1.1 and 212.7 ± 1.1 MeV. The results are reported in section IV B. The two subsequent campaigns aimed principally at the inbeam and decay spectroscopy of 249 Md, results of which will be reported in a forthcoming publication. The data collected in the three campaigns were used to derive the 245 Es and 249 Md half-life and α-decay branching ratios, as presented in the following section.
A. 249 Md and 245 Es decay and half-life measurement The α-particle energy spectra obtained using recoilα and recoil-α-α correlations, with the statistics of the three campaigns summed together, are presented in Fig. 3. A maximum search time of 10 min after the identification of an implanted recoiling nucleus was used. 249 Md features an electron capture (EC)/β + decay branch feeding 249 Fm. The α decay of the latter is observed using recoil-α correlations since the detection system is insensitive to the β + particle (see the upper panel of Fig. 3). The 245 Es α decay observed using recoil-α correlations corresponds to the events when the α particle emitted from 249 Md escapes from the DSSD without being detected. The α decay of 249 Fm is more clearly visible in Fig. 4, which represents the α-decay time in a logarithmic scale as a function of the α-particle energy. Using recoil-α-α correlations allows the mother, 249 Md and daughter, 245 Es α decays to be isolated as shown in middle and bottom panels of Fig. 3. From the literature, the α-particle energies are: E α ( 249 Md)= 8026 ± 10 keV [22], and E α ( 245 Es)= 7730 ± 1 keV [13]. The satellite peaks in the α decay of 249 Md at 7956 and 8087 keV, suggested in [22], are also tentatively observed in the present work. Figure 5 shows the time distribution of the 249 Md α decay with respect to the implantation time. The distribution plotted as a function of ln(∆T ) for a maximum search time of 24 h is shown in the inset. As shown in      Table I. The α-decay branching ratio of 249 Md is defined as the ratio of the α-decay branch to 245 Es, to the total decay strength, including the EC/β + branch to 249 Fm. The latter is evaluated using the number of events attributed to the 249 Fm α decay from Figs. 3 and 4, corrected for the 249 Fm α-decay branching ratio. A correction is also applied to take into account the fraction of 249 Fm nuclei that decay during the search time of 600 s. The 249 Fm half-life of 2.6 ± 0.7 min is taken from the evaluated data [25]. The 249 Fm α-decay branching ratio of 15.6 ± 1.0 % is taken from Hessberger et al. [26], which is more recent that the evaluation of Ref. [25] 1 . The resulting α-decay branching ratio deduced in the present 1 It should be noted that in Ref. [26], the half-life of 249 Fm has not been re-measured. The value adopted in this reference is actually that of the evaluation Ref. [25], i.e. 2.6 ± 0.7 min. In the most recent Nubase2016 evaluation [27], the α-decay branching ratio of 249 Fm is taken from Ref. [25] (33 ± 9 %) while for the half-life only the value from [28] (96 ± 6 s) is selected. work is b α ( 249 Md) = 75 ± 5 %. The evaluated value of b α ( 249 Md)> 60 % [25] corresponds to the measurement of Hessberger et al., which has been obtained in the study of the 257 Db decay chain [19]. A more recent value of b α ( 249 Md) = 75 %, quoted without uncertainty in the PhD thesis of B. Streicher [23], is in perfect agreement with our measurement; see also Table I. The α-decay branching ratio of 245 Es can be extracted in two distinct ways. The first possibility is to derive it as the ratio of the number of events corresponding to 249 Md obtained using recoil-α-α and recoil-α correlations, corrected for the DSSD efficiency for a full-energy measurement α = 55 %, under the condition that the recoil-αα correlations are obtained by gating on the full-energy peaks only:

T [s]
The second option is to obtain it as the ratio of counts corresponding to 245 Es and 249 Md in the total α-particle spectrum. Both methods lead to the same value of b α ( 245 Es) = 54 ± 7 %. For comparison, the previously reported values were b α ( 245 Es) = 40 ± 10 % (Eskola et al. [18]), b α ( 245 Es) = 80 +20 −50 % (Hessberger et al. [19]). The decay properties of 249 Md and 245 Es are summarized in Table I.

B. Production cross section
The fusion-evaporation reaction 203 Tl( 48 Ca,2n) 249 Md was studied at two different bombarding energies. The cyclotron delivered a 218 MeV beam first passing through the 100 µg cm −2 carbon window of the SAGE electron spectrometer. The 203 Tl target having a thickness of 318 ± 16 µg cm −2 was evaporated on a carbon foil of 20 µg cm −2 , and covered by a 10 µg cm −2 carbon protection layer. The resulting energy in the middle of the 203 Tl target was estimated to be 214.3 ± 1.1 MeV. Using in addition a 80 µg cm −2 carbon degrader foil resulted in an energy of 212.7 ± 1.1 MeV MoT.
The spectra were obtained using a search time of 207 s i.e. eight 249 Md half-lives. Contrary to the 243 Es case, the background was found to be negligible.
The total number of 48 Ca particles that impinged on the target was (1.8 ± 0.4) × 10 15 ((1.5 ± 0.3) × 10 15 ) for the measurement without (with) carbon degrader foil. Using a 203 Tl target thickness of 318 ± 16 µg cm −2 , an α branching ratio of 75 ± 5 %, a RITU transmission × detection efficiency of 30 % and a full-energy α-detection efficiency of 55 %, cross sections σ( 249 Md) of 300 ± 80 nb and 70 ± 40 nb are deduced for the incident energies of 214.3 and 212.7 MeV, respectively. Again, only statistical uncertainties are given. The results are summarized in Table III. In this section we discuss the new cross-section measurements for 243 Es and 249 Md. These results are placed in the context of experimental cross sections for cold fusion-evaporation reactions, 2n channel, for Z ≈ 100, presented in Fig. 7, and compared to new reactions dynamics calculations using the statistical fusionevaporation code KEWPIE2 [4].

A. 2n channel fusion-evaporation systematics
It is generally acknowledged that the fusionevaporation reactions can be described as three subsequent independent processes: capture, compoundnucleus formation, and survival of the residual nucleus. The description of the capture step is rather well controlled in terms of barrier penetration, with no rapid evolution as a function of mass and charge when using similar projectiles and targets. The formation step results in a sharp decrease of the cross section for projectiletarget combinations with Z p Z t 1600 − 1800, known as the fusion hindrance, which prevents the formation of a compound nucleus by leading the di-nuclear composite towards quasi-fission route. This effect starts to act in the region considered here, and it can account for the exponential decrease of the cross sections observed for larger Z values in Fig. 7. Consequently, only the survival step can account for the decrease of cross sections below Z ≈ 102. The global trend displayed by the cross sections presented in Fig. 7 may be explained by a combination of two effects. First, the four-fold magic character of the 48 Ca + 208 Pb → 256 No * reaction leads to a low Q value and therefore a higher survival probability in the evaporation and de-excitation processes. This enhancement is observed for 254 No and neighbouring residual nuclei. Second, the semi-magicity at Z = 100, N = 152 leads to higher shell corrections (higher fission barrier) and therefore higher survival probability around 252 Fm. Note that if the cross sections are plotted as a function of the mass or neutron number, they also display a bell-shaped behaviour.   [37] ( 256 Rf), [21] ( 257 Db), [38] ( 259 Sg), [22] ( 260 Sg), [39] ( 261 Bh), [40] ( 264 Hs).

B. Cross-section calculations
In the following, the fusion-evaporation cross sections illustrated with the new experimental results for 243 Es and 249 Md are discussed in terms of survival from the compound to the residual nucleus, with an emphasis on the effect of the fission barrier. The present measurements are performed in a mass region where the fusion hindrance is not yet significant. Consequently, the fusion process is modelled in the KEWPIE2 code by considering only the capture phase, which is computed using a proximity potential and the Wentzel-Kramers-Brillouin (WKB) approximation, see Ref. [4] for details.
The KEWPIE2 code [4] treats the competition between light-particle evaporation and fission, which occurs within an excited compound nucleus, using the statistical formalisms of Weisskopf [43] and Bohr-Wheeler [44], respectively. The entire set of default parameters used in the KEWPIE2 code is presented in Ref. [4]. In the following we will only focus on a few parameters, which are not well-defined either theoretically or experimentally in this mass region [45]. These parameters are the reduced friction parameter β, the shell-damping energy E d and the shell corrections ∆E sh . These parameters are related, respectively, to the viscosity of nuclear matter, the stability of shell corrections with temperature and the   Es obtained in the present paper, and the calculations of the 1n, 2n and 3n cross sections performed with the KEWPIE2 code using the default parameters (macroscopic part described by the Thomas-Fermi parametrization as proposed by Myers-Swiatecki [41], and the microscopic part based on the FRDM shell corrections [42]). Bottom: same for 249 Md.
fission-barrier height, following Eq. 3 for the latter: where B f is the fission-barrier height and B LDM the liquid-drop fission barrier. The default values used in the KEWPIE2 code are β = 2 × 10 21 s −1 , E d = 19 MeV, while the finite-range droplet model (FRDM) ∆E sh shell corrections are taken from Ref. [42]. It should be stressed that those parameters mainly affect the fission process that is known to be dominant for heavy and super-heavy nuclei. Indeed, a small variation of the fission parameters, such as the strength of the dissipation or the fissionbarrier heights, leads to a significant modification of the survival probability and, consequently, the related observables, in particular the production cross sections.   (Tables II  and III) compared to the calculations performed with the KEWPIE2 code using the default parameters. For 249 Md, the calculation reproduces the measured production cross sections well, while it underestimates them by a factor of 5 for the 243 Es case. The discrepancy for this latter case cannot be explained by a failure of the fusion model. Indeed, for a beam energy corresponding to the present measurement (E cm ≈ 169 MeV), the fusion model provides a fusion cross section σ f us = 55 mb in good agreement with the measurement σ f us = 42 mb of Ref. [46]. Moreover, a discussion of the fusion crosssection for the 48 Ca+ 208 Pb reaction, for which the WKB approximation provides a good description without fusion hindrance considerations, can be found in Ref. [4]. In Fig. 9, the fission-barrier heights or the reduced friction parameters have been increased in order to reproduce the measurements for the 2n evaporation channel. Concerning the fission-barrier heights, it is necessary to add 500 keV to the absolute value of the shell corrections (with the liquid-drop fission barrier kept unchanged, see Eq. 3), to obtain a good agreement between the calculations and the data. Furthermore, the reduced friction parameter has to be increased by a factor of three, i.e. to β = 6 × 10 21 s −1 , in order to obtain the same agreement. It should be stressed that these adjustments remain within the uncertainty intervals for these parameters, as discussed in Refs. [4,45]. Moreover, no theoretical model can presently predict the fission-barrier heights with an accuracy better than 0.5 − 1 MeV [47][48][49]. In the super-heavy nuclei region, differences between the models can be as large as 4 MeV [50]. Consequently, we cannot attribute the discrepancy observed for 243 Es (Fig. 8) to any specific parameters used in the KEWPIE2 code, neither to any inputs from other nuclear models, in particular those related to the fission process. Hence, the measured production cross sections for the 243 Es and 249 Md isotopes can be fully explained within the uncertainties in nuclear models and phenomenological parametrizations implemented in the KEWPIE2 code.
A way to provide constraints on the parameters used in the KEWPIE2 code would be to use more precise measurements in the very-heavy and super-heavy nuclei mass region for a whole set of different evaporation channels, including a large scan in excitation energy for each of them. Indeed, using relevant data can help to fix and/or eliminate the impact of a specific parameter. Such an approach based on the Bayesian inference is discussed in [4,51].

VI. CONCLUSION
The odd-Z 243 Es and 249 Md were produced in the 197 Au( 48 Ca,2n) 243 Es and 203 Tl( 48 Ca,2n) 249 Md fusionevaporation reactions, respectively. The half-life of 243 Es, 249 Md and its daughter 245 Es were measured and the results were found compatible with those obtained in previous measurements following α-decay of heavier nuclei. The precision of the half-lives of 249 Md and 245 Es was increased, as well as those of the α-decay branching ratios for those nuclei.
Production cross-sections of 243 Es and 249 Md have been measured for the first time using 48 Ca-induced reactions, and compared to the calculations performed with the KEWPIE2 code [4]; a good agreement was obtained using the standard parameters.