In-beam gamma-ray and electron spectroscopy of 249,251 Md
R. Briselet, Ch. Theisen, B. Sulignano, M. Airiau, K. Auranen, D. M. Cox, F. Déchery, A. Drouart, Z. Favier, B. Gall, T. Goigoux, T. Grahn, P. T. Greenlees, K. Hauschild, A. Herzan, R.-D. Herzberg, U. Jakobsson, R. Julin, S. Juutinen, J. Konki, M. Leino, A. Lopez-Martens, A. Mistry, P. Nieminen, J. Pakarinen, P. Papadakis, P. Peura, P. Rahkila, E. Rey-Herme, J. Rubert, P. Ruotsalainen, M. Sandzelius, J. Sarén, C. Scholey, J. Sorri, S. Stolze, J. Uusitalo, M. Vandebrouck, A. Ward, M. Zielińska, B. Bally, M. Bender, W. Ryssens
IIn-beam γ -ray and electron spectroscopy of , Md R. Briselet, Ch. Theisen,
1, a
B. Sulignano, M. Airiau, K. Auranen, D. M. Cox,
2, 3, b
F. D´echery,
1, 4
A. Drouart, Z. Favier, B. Gall, T. Goigoux, T. Grahn, P. T. Greenlees, K. Hauschild, A. Herzan,
2, c
R.-D. Herzberg, U. Jakobsson,
2, d
R. Julin, S. Juutinen, J. Konki,
2, e
M. Leino, A. Lopez-Martens, A. Mistry,
3, f
P. Nieminen,
2, g
J. Pakarinen, P. Papadakis,
2, 3, h
P. Peura,
2, i
E. Rey-Herme, P. Rahkila, J. Rubert, P. Ruotsalainen, M. Sandzelius, J. Sar´en, C. Scholey,
2, j
J. Sorri,
2, k
S. Stolze,
2, l
J. Uusitalo, M. Vandebrouck, A. Ward, M. Zieli´nska, B. Bally,
1, m
M. Bender, and W. Ryssens Irfu, CEA, Universit´e Paris-Saclay, F-91191 Gif-sur-Yvette, France University of Jyvaskyla, Department of Physics, P.O. Box 35, FI-40014 Jyvaskyla, Finland University of Liverpool, Department of Physics,Oliver Lodge Laboratory, Liverpool L69 7ZE, United Kingdom Institut Pluridisciplinaire Hubert Curien, F-67037 Strasbourg, France Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France IP2I Lyon, CNRS/IN2P3, Universit´e Claude Bernard Lyon 1, F-69622, Villeurbanne, France Center for Theoretical Physics, Sloane Physics Laboratory,Yale University, New Haven, Connecticut 06520, USA (Dated: July 8, 2020)The odd- Z Md nucleus was studied using combined γ -ray and conversion-electron in-beamspectroscopy. Besides the previously observed rotational band based on the [521]1 / − configura-tion, another rotational structure has been identified using γ - γ coincidences. The use of electronspectroscopy allowed the rotational bands to be observed over a larger rotational frequency range.Using the transition intensities that depend on the gyromagnetic factor, a [514]7 / − single-particleconfiguration has been inferred for this band, i.e., the ground-state band. A physical backgroundthat dominates the electron spectrum with an intensity of (cid:39)
60% was well reproduced by simulatinga set of unresolved excited bands. Moreover, a detailed analysis of the intensity profile as a functionof the angular momentum provided a method for deriving the orbital gyromagnetic factor, namely g K = 0 . +0 . − . for the ground-state band. The odd- Z Md was studied using γ -ray in-beamspectroscopy. Evidence for octupole correlations resulting from the mixing of the ∆ l = ∆ j = 3[521]3 / − and [633]7 / + Nilsson orbitals were found in both , Md. A surprising similarity ofthe
Md ground-state band transition energies with those of the excited band of
Lr has beendiscussed in terms of identical bands. Skyrme-Hartree-Fock-Bogoliubov calculations were performedto investigate the origin of the similarities between these bands. a [email protected] b Present address: University of Lund, Box 118, 221 00 Lund,Sweden c Present address: Institute of Physics, Slovak Academy of Sci-ences, SK-84511 Bratislava, Slovakia d Present address: Laboratory of Radiochemistry, Department ofChemistry, P.O. Box 55, FI-00014 University of Helsinki, Finland e Present address: CERN, CH-1211 Geneva 23, Switzerland f Present address: GSI Helmholtzzentrum f¨ur Schwerionen-forschung GmbH, 64291 Darmstadt, Germany g Present address: Fortum Oyj, Power Division, P.O. Box 100,00048 Fortum, Finland h Present address: STFC Daresbury Laboratory, Daresbury, War-rington WA4 4AD, United Kingdom i Present address: International Atomic Energy Agency, Vienna,Austria j Present address: The Manufacturing Technology Centre, PilotWay, Ansty, CV7 9JU, United Kingdom k Present address: Sodankyl¨a Geophysical Observatory, Universityof Oulu, 90014 Oulu, Finland l Present address: Physics Division, Argonne National Labora-tory, 9700 South Cass Avenue, Lemont, Illinois 60439, USA m Present address: Departamento de F´ısica Te´orica, UniversidadAut´onoma de Madrid, E-28049 Madrid, Spain
I. INTRODUCTION
Despite significant and steady advances in the synthe-sis of the heaviest elements, reaching the predicted su-perheavy island of stability is still a distant objective,because of the ever-decreasing cross sections. Neverthe-less, nuclear spectroscopy, mass measurements and laserspectroscopy of the heaviest nuclei have shown their ef-fectiveness by providing information on the quantum na-ture of extreme mass nuclei [1–5], without which the nu-clei would no longer be bound beyond Z (cid:39) Z =114,120, or 126 and neutron number N =172 or 184 [6–10]. The validity of these predictions in a region wherethe models are extrapolated is hence questionable, as isthe concept of magic numbers in this region [11]. It istherefore essential to compare predictions to comprehen-sive, reliable and relevant spectroscopic data, in partic-ular for deformed midshell nuclei where a large diversityof orbitals are accessible, some of which are involved inthe structure of heavier spherical nuclei, i.e., placed justabove and below the predicted superheavy spherical shellgaps. a r X i v : . [ nu c l - e x ] J u l The present study of the odd- Z , Md nuclei isan integral part of this approach by providing inputs interms of both proton single-particle and collective prop-erties. We report on the previously unobserved ground-state (g.s.) band of
Md, assign its single-particle con-figuration, and deduce the gyromagnetic factor. We alsodiscuss the most intense transition observed in
Mdusing both γ -ray and conversion-electron spectroscopy,and in Md using γ -ray spectroscopy alone, as beingcompatible with octupole correlations. Finally, a com-parison of Md with the
Lr nucleus revealed un-expected similarities between transition energies. Themechanism leading to these identical bands has beentested with Hartree-Fock-Bogoliubov (HFB) calculationsusing a Skyrme functional and several parametrizationsof pairing correlations.
II. EXPERIMENTAL DETAILS
The experiments were performed at the AcceleratorLaboratory of the University of Jyv¨askyl¨a. The
Mdnuclei were populated using the fusion-evaporation reac-tion
Tl( Ca,2 n ) Md. The Ca beam was providedat 218 MeV, resulting in an energy at the middle of thetarget of 214 MeV, at which the fusion-evaporation crosssection is about 760 nb [12]. An average beam inten-sity of ≈ ≈
230 h ofdata taking. The
Tl targets, 99.45 % enrichment, ≈ µ g/cm thick, were sandwiched between a C backingof 20 µ g/cm and a C protection layer of 10 µ g/cm .The experimental setup is schematically representedin Fig. 1. The Md nuclei were separated from thebeam and other unwanted products using the Recoil IonTransport Unit (RITU) gas-filled recoil separator [13, 14]operated at a He gas pressure of 0.4 mbar. The recoilingnuclei were detected using the Gamma Recoil ElectronAlpha Tagging (GREAT) focal plane spectrometer [15].After passing through a multiwire proportional cham-ber (MWPC), ions were implanted in a set of two side-by-side 300- µ m-thick double-sided silicon strip detectors(DSSDs). Each DSSD had a size of 60 ×
40 mm with 1-mm strips pitch in both X and Y directions. The totalDSSD counting rate was approximately 250 Hz. The am-plification on the X side was set to a high gain in order tooptimize the detection of low-energy conversion electronsin a range of approximately 50-600 keV. The Y side wasamplified using a lower gain in order to cover energies upto approximately 15 MeV. Besides the RITU filter, anadditional selection of the ions of interest was made us-ing a contour gate on the the energy-loss ∆ E measuredin the MWPC versus the time-of-flight (ToF) measuredbetween the MWPC and the responding DSSD. The tun-nel detectors and the planar and Clover Ge focal-planedetectors were operated during the experiment but notused in the present analysis. The combined transmissionand detection efficiency for the Md residues was esti-mated at ≈ α decay (recoil-decay tagging) was not effec-tive due to the low α -decay branching ratio of 9.5% for Md [16]. Therefore only the recoil-tagging techniquewas used, which is adequate since only the reaction chan-nel of interest is open.Gamma-rays and conversion-electrons emitted at thetarget position were detected using an array known asSilicon And GErmanium (SAGE) [17]: γ -rays were de-tected using Compton-suppressed HPGe detectors (20coaxial and 24 clovers) having a total γ -ray photopeakefficiency of ≈
10% at 200 keV and an average energyresolution of 2.8 keV FWHM at 1 MeV. A stack of 0.5-mm-thick Cu and 0.1-mm-thick Sn absorbers were placedin front of the Ge detectors to reduce the contributionof fission-fragments x-rays. The detection threshold wasapproximately 20 keV. The maximum counting rate ofeach coaxial (clover) crystal was kept below (cid:39)
30 (20)kHz. After being transported by a solenoid placed up-stream the target and tilted 3.2 ◦ with respect to the beamaxis, electrons were detected in a 90-fold segmented Sidetector with a thickness of 1 mm and an active diam-eter of 48 mm. The electron detection efficiency peaksat ≈
6% for an energy of 120 keV with an average en-ergy resolution of 6.5 keV FWHM in the 50- to 400-keVenergy range. Low-energy atomic electrons were partlysuppressed using an electrostatic barrier biased at -35 kV.The separation between the He gas-filled region and theupstream beam line, including the electrostatic barrierregion, was made using two 50- µ g/cm C foils. The max-imum counting rate of each segment was kept below 15kHz. The detection threshold was approximately 30 keV.Contour gates constraining SAGE time versus the ToF,and SAGE energy versus SAGE time were used to cleanthe spectra. These gates were left wide enough to favorthe statistics when using γ - γ coincidences (Secs. III Aand IV). For an analysis that requires intensity measure-ment, the gates were tightened to favor cleanliness (Secs.III B and III C). This can lead, however, to a systematicerror in the relative intensities of conversion electrons ver-sus γ rays that was estimated at 20 %. Digital signal pro-cessing was used for the SAGE array (100 MHz, 14 bits)while signals from the MWPC and the DSSDs were pro-cessed using standard analog electronics and peak sensinganalog to digital converters.The experimental conditions for the study of Mdwere similar and are detailed in Ref. [18]. In brief,the nuclei of interest were produced using the fusion-evaporation reaction
Tl( Ca,2 n ) Md. The
Tltargets, 97.08 % enrichment, ≈ µ g/cm thick, weresandwiched between a C backing of 20 µ g/cm and a Cprotection layer of 11 µ g/cm . The Ca beam was de-livered at a beam energy of 219 MeV, resulting in anenergy of (cid:39)
215 MeV in the middle of the target. Datawere taken during (cid:39)
80 h with a beam intensity of (cid:39) γ rays were collected.Data were handled using the triggerless Total DataReadout system [19] and sorted using the grain softwarepackage [20]. The theoretical conversion coefficients were FIG. 1. Schematic view of the experimental set-up at the Accelerator Laboratory of the University of Jyvskyl, with from leftto right the SAGE array for in-beam spectroscopy, the RITU gas-filled separator, and the GREAT focal-plane detectors. Seethe text for details. calculated using the bricc code [21].
III. ROTATIONAL BANDS IN
MDA. γ -ray spectroscopy The γ -ray singles spectrum resulting from the recoil-tagging technique is shown in Fig. 2(a). The previouslyobserved rotational band structure, interpreted as be-ing built on the proton Nilsson orbital [521]1 / − [12],is shown in Fig. 2(b). The spectrum was created froma sum of gates on the peaks of interest, projected froma matrix of recoil-gated γ - γ coincidences. Compared tothe previous work, we cannot confirm the proposed tran-sition at the highest rotation frequency with an energy of483(1) keV for which only six counts were observed [12].We instead suggest a transition at 478(2) keV for whicheight counts were collected in the present work. Thereis also evidence for an additional transition with an en-ergy of 513(1) keV ( (cid:39)
12 counts). A structure with a γ -ray spacing of about half of the former has been found,and is therefore consistent with ∆ I = 2 E γ - γ analysis has been found in mutualcoincidence with at least four other transitions in thesame band. It should be noted that the transition at (cid:39)
334 keV is a doublet with the 335 keV transition of the K π = 1 / − band. For the sake of clarity, a partial levelscheme summarizing the results of this work is providedin Fig. 3.The single-particle configurations considered in thefollowing for the previously unobserved band are those predicted at low energy by the macroscopic-microscopicmodels used in Ref. [22–25] and the self-consistent mod-els used in this work (see Sec. V B) as well as thoseobserved by decay spectroscopy in the neighboring Mdisotopes [22, 26], which reduces the alternatives to the[514]7 / − ground-state and to the [633]7 / + single-particle configurations.Considering therefore a band-head angular momentum I = K = 7 / J (1) calculatedusing different angular momenta I i → I f hypotheses forthe transitions (or in other words the number of unob-served transitions), with the predictions of Chatillon etal. [12], He et al. [27] and Zhang et al. [28]. The bestagreement has been obtained using I i = 13 / → / J (1) mo-ment of inertia at lower rotational frequencies yields anenergy of 133 keV for the unobserved 11 / → / / → / et al. [3], which provides an energy of62 keV for the first member of the 7 / − rotational g.s.band.As shown in the inset of Figs. 2(b) and 2(c), the two ro-tational bands are in coincidence with γ rays around 582,596, 608, and 860 keV and 595 and 630 keV, respectively.Although there are other candidate peaks visible in thisregion, only those listed here produce coincidences withthe rotational band. In the even-even actinide nuclei,transitions in this energy range are typically observed inthe de-excitation of vibrational states, e.g. 2 − states in Cm [29],
Fm [30], or
No [31]. Also, in the odd-proton
Lr, de-excitation of high- K rotational bandsproceed via transition in this energy range [32]. How- C oun t/ k e V . ( ) . ( ) . ( ) . ( ) . ( ) ( ) . ( ) ( ) ( ) . ( ) . ( ) . ( ) ( ) . ( ) ( ) ( ) [ ( )] [ ( )] ×1/5 K α K β ( ) / → /
100 200 300 400 500
Energy (keV) C oun t/ e V / → /
500 600 700 800 90005 582 596608 860600500 700 800 90005 630595 (a)(b)(c)[521]1/2 - [514]7/2 - FIG. 2. (a) γ -ray spectra of Md resulting from recoil tagging. The spectra (b) and (c) were projected from a sum of gateson the peaks of interest using recoil-gated γ - γ coincidence data. The transitions in mutual coincidence are shown with dottedblue ([521]1 / − ) and dashed red (previously unobserved band) lines. ever, in our case, coincidences did not allow us to makethe link with a collective structure at higher energy. B. Electron spectroscopy
Turning to in-beam conversion electron spectroscopy,the analysis was based only on the total recoil-gatedspectrum due to the paucity of γ -electron coincidencedata. The experimental spectrum shown in Fig. 4(a)was obtained by subtracting the random-correlated back-ground using time gates before and after events in promptcoincidence. L and M components from the 195-keV(L,M195) and 244.3-keV (L244) transitions belonging tothe K π = 1 / − band are clearly apparent. Clearly visi-ble are the L and M conversion lines of a transition at 144keV, which fits well with the extrapolated energy for the13 / − → / − E γ rayoverlaps with the Kβ x-ray line, which explains whyit cannot be evidenced using γ -ray spectroscopy alone.The most intense transition at 389 keV will be discussedfurther in Sec. IV.We also observed peaks below 100 keV with no counter-part in the γ -ray spectrum. In order to fully understandthe electron spectrum and possibly constrain the single-particle configurations, we have performed a simulationof the conversion-electron spectra. We have adopted a purely analytical approach in our simulations. Comparedto a Monte Carlo approach, this was possible becausethe number of ingredients was limited and because theresponse of the electron detector was quite simple. Theadvantage is that the simulations were fast and that allparameters easily controlled. The physics inputs for therotational bands are described below. As soon as thetransitions to be simulated have been listed (energy, rel-ative intensity, multipolarity, and mixing ratio), the ra-diative and converted intensities were calculated usingthe conversion coefficients. The intensity was then cor-rected from the detector efficiency. For the electrons,the spectrum was simply simulated as Gaussian havingthe experimental resolution, with no background. Thissimplistic approach is justified by the fact that there isalmost no background due to electron (back-)scattering.This was checked using nptool [33] (a simulation andanalysis framework for low-energy nuclear physics experi-ments based on geant4 [34]) in the 50-to 500-keV energyrange: More that 85% of the electrons were indeed fullyabsorbed in the Si detector. The less than 15% remain-ing set of electrons contribute to a background which ismostly concentrated below 150 keV and which resemblesthe physical background that will be described in the fol-lowing. Since the physical background dominates the ma-jority of the spectrum, this implies that this backgroundis overestimated by about 15%, which is of the same or- (133)189.0238.8 (61)289.0334376 161.8214.8263.8311.2358(40) 87144195.3244.3291.8335376.8415447(478)(513) Md [514]7/2 - (g.s.)[521]1/2 - FIG. 3. Partial level scheme of
Md resulting from thiswork. All spins and parities are tentative. The band-beadenergies are taken from Ref. [16]. der of magnitude as the systematic experimental errors.As a matter of fact, the conclusions regarding discretetransitions will not be altered by our assumptions. Onlythe interpretation of the physical background is slightlybiased by this hypothesis, but as we will see later, theanalysis of this background is essentially qualitative. Asfar as γ rays are concerned, the same approach has beenchosen, but in this case, a background-less detector re-sponse is no longer justified. Simulated γ -ray spectraare therefore not presented in this study. The K x-rayemission after internal conversion was also included inthe simulations. The x-ray energies and intensities weretaken from Ref. [35]. The code has been implementedusing the root framework [36].The ingredients for the K π = 1 / − band were the ex-perimental energies and intensities. The band was sub-sequently extrapolated at lower energies using a smoothmoment of inertia resulting in transition energies of87 keV (9 / − → / − ) and 40 keV (5 / − → / − ),the former being evidenced in the electron spectrum[Fig. 4(a)]. These energies are strongly similar to those ofbands based on the same single-particle configuration inthe neighboring Bk and
Es [37]. The transition in-tensity within the band was deduced from the γ -ray spec-tra, corrected for the internal conversion, and assumedconstant for the transitions at 40, 87, and 144 keV. Thecorresponding electron intensities, after taking into ac-count the detector response, are shown with green circlesin Fig. 4(b). Electron Energy (keV) (b) - [521]1/2 I=1 M1/E2 D - [514]7/2 I=2 E2 D - [514]7/2 Simulation
Experiment - Simulation including [514]7/2 + Simulation including [633]7/2Physical background L M L M L M L K L (a) C oun t s / k e V I n t en s i t y FIG. 4. (a) The experimental
Md electron spectrum (solidblue line) is compared to simulations including a physicalbackground (dotted black line) and discrete peaks. The sim-ulation (solid red line with the propagated uncertainties inshaded red) corresponds to the sum of the transition at 389keV, the K π = 1 / − band, the previously unobserved rota-tional band using the [514]7 / − configuration and the physi-cal background. A similar simulation assuming the [633]7 / + configuration for this band is shown with a dashed red line.The peaks are labeled with the transition energies preceded bythe electronic shell. (b) The simulated transition intensitiescorrected according to the experimental detection efficiencyare shown with green circles for the K π = 1 / − band. Forthe other band, ∆ I = 1 (∆ I = 2) transitions are shown withblue squares (red triangles) assuming the [514]7 / − configu-ration. A different approach has been adopted for the otherrotational band: In that case, the transition intensi-ties were simulated using the electromagnetic properties(electric quadrupole and magnetic dipole moments) as aninput assuming either the [514]7 / − or [633]7 / + single-particle configuration. The rotational band was first ex-trapolated at higher angular momenta using a smoothmoment of inertia. The total experimental intensity pro-file (converted plus radiative) as a function of the angularmomentum, N ( I ), was fitted using a Fermi distribution N ( I ) = a/ { I − b ) /c ] } . This prescription providedthe normalization factor a , the average angular momen-tum entry point b = 14 (cid:126) , and the diffuseness c = 3 (cid:126) . Itis interesting to note that the intensity profile inferredhere corresponds remarkably well to that measured for No at similar conditions of excitation energy and an-gular momentum for the compound nucleus [38]. Thetransition rates of the stretched E I = 1 E /M B ( M
1) and B ( E T γ ( M
1) and T γ ( E
2) : B ( M
1) = 34 π ( g K − g R ) K (cid:104) I i K | I f K (cid:105) [ µ N ] , (1) B ( E
2) = 516 π e Q (cid:104) I i K | I f K (cid:105) [ e fm ] , (2) T γ ( M
1) = 1 .
76 10 E B ( M
1) [s − ] , (3) T γ ( E
2) = 1 .
59 10 E B ( E
2) [s − ] , (4)with g R (cid:39) Z/A being the rotational gyromagnetic factor, µ N being the Bohr magnetron, and Q being the electricquadupole moment. The orbital gyromagnetic factor hasbeen approximated as g K = ( g s Σ + g l Λ) /K , where g s ( l ) is the nucleon spin (orbital) gyromagnetic factor. Forthe protons, we adopted g s = 5 .
59 and g l = 1. A reduc-tion of the spin gyromagnetic factor g eff s = 0 . g free s wasused, a value generally adopted for heavy nuclei. Therelations above result in g K = 0 .
66 and g K = 1 .
34 forthe [514]7 / − and [633]7 / + configurations, respectively.The transition rates, corrected for internal conversionand according to the intensity profile N ( I ) were subse-quently used to calculate the transition intensities alongthe rotational band. Actually, the intensity calculationscan be performed purely analytically for the entire ro-tational band. In practice, the calculations were madefrom the top of the band, the intensities being calculatedfor all the transitions steeping downwards.The resulting simulated electron intensities after tak-ing into account the detector response for the [514]7 / − hypothesis are shown in Fig. 4(b) with red triangles forthe stretched E I = 1 transitions. As a result, the electron spectrum isdominated by ∆ I = 1 transitions. The complexity of thespectrum is obvious along with a fragmented intensitypattern. However, the simulation of this band essentiallygenerates tails in the peaks of the K π = 1 / − band. Thisis due to two factors: (i) most of the conversion-electronenergies of this band are often found close to some of the K π = 1 / − band and (ii) the intensities of theses bandtransitions is lower than those of the K π = 1 / − band.Experimentally resolving all peaks was not possiblegiven the detector resolution and the collected statistics.The simulated intensity assuming the [633]7 / + configu-ration has similar features but with a larger contributionof M | g K − g R | value). Since thetotal transition intensity was normalized from the γ -rayspectrum, the resulting electron intensities would exceedthose of the [514]7 / − case by a factor of ≈
3, which isclearly not compatible with the measurement, and thisrules out the [633]7 / + configuration.After summing the [521]1 / − and [514]7 / − contribu-tions, there is still a large background in the electronspectrum peaking at ≈
80 keV. In Fig. 4(a), a simulationcorresponding to a set of rotational bands with K and themoment of inertia randomly sampled is shown with a dot-ted black line. More precisely, each band configurationhas been chosen either as a proton 1qp excitation with1 / ≤ K ≤ / ≤ K qp ≤ K rotationalbands in the Fm-Lr region. We have arbitrarily taken afraction of 50% proton 1qp excitations, 25% proton 3qpexcitations and 25% 1qp proton ⊗ Md population. It should be reminded here thata background-less response of the electron detector wasassumed. As discussed above, this leads to an underes-timated simulated background and therefore the presentbackground analysis should be regarded as qualitative.Our conclusion is, however, entirely consistent with thoseof Butler et al. in the case of
No, for which an elec-tron background has been well reproduced using a set of K = 8 rotational bands with 40% intensity [41]. Thedecay spectroscopy of No unambiguously confirms thepresence of a K π = 3 + K π = 8 − K π = 8 − isomericstate is fed with an intensity ratio of 28 (2) %. Thus,most of the unresolved background identified by Butler et al. arises from the de-excitation of a band based on ahigh K π = 8 − state. Using in-beam γ -ray spectroscopy,a rotational band based on a K π = 8 − isomeric state wasalso observed in Fm [30] and in
No [31], with an iso-meric ratio of 37 (2) % [46] and (cid:39)
30 % [47], respectively.A similar situation is thus expected in odd-mass nucleiwith the presence of high- K K isomer has been recently evidencedin Md [48]. It is therefore realistic to interpret theobserved electron background in
Md as correspondingto several unresolved bands, built either on high- K / + or [521]3 / − . However, band intensities areexpected to be more fragmented compared to even-evennuclei due to the presence of several single-particle statesat low excitation energy.We have also simulated the γ -ray counterpart of thisphysical background, which, interestingly, also repro-duces well the γ -ray background shape in the 150- to 600-keV energy range. This must be considered, however, asa qualitative consideration since the response of the Gedetectors was not fully included in our simulations. Withregard to x-rays, their intensity turns out to be very sen-sitive to K , as expected from the strong K dependenceof the B ( M
1) rates [Eq.(1)], but no definitive conclusioncan be drawn from the present work.Finally, the total simulated spectrum ([521]1 / − and[514]7 / − configurations, transition at 389 keV plus thephysical background) is shown using a solid red line withthe envelope corresponding to the uncertainty propaga-tion [Fig. 4(a)]. The global shape is well reproduced ex-cept at (cid:39)
40 keV, interpreted as the energy tail of δ electrons that were not cut by the electrostatic barrier orthe electronic threshold. The same simulation assumingthe [633]7 / + configuration is shown with a red dashedline for comparison, clearly overestimating the measuredintensity and again ruling-out this configuration. C. Intensity profile
The experimental intensity profile of stretched E K π = 7 / − g.s. band is shown inFig. 5. The intensities were taken from the γ rays in the189.2-keV ( I i = 15 /
2) to 311.3-keV ( I i = 25 /
2) range,corrected from the internal conversion. Only statisticaluncertainties were considered here since a systematic er-ror would simply scale the distribution. Although associ-ated with large error bars, oscillations around I i = 21 / g K = 0 .
66 is shown with blue squares. h ( i E2 transition initial spin I02004006008001000120014001600 S t r e t c hed E t o t a l i n t en s i t y ( a r b . un i t s ) Exp. - [514]7/2 = 0.66 k g + [633]7/2 = 1.34 k g k g ) k - l n L ( g -0.16+0.19 = 0.69 K g FIG. 5. The experimental intensity profile for the K = 7 / g K = 0 .
66 (blue squares) or g K = 1 .
34 (green triangles). The experimental gyromag-netic factor was deduced using the likelihood estimator L , − ln L ( g K ) being plotted in the inset. The solid line was ob-tained using the entire experimental profile, the two higherspin transitions being ignored to draw the dashed line. This simulated profile reproduces the experimentaldata remarkably well, notably the oscillations and an intensity jump between the states with I i = 23 / /
2. Below I i = 23 /
2, the ∆ I = 1 transition ener-gies are lower than the K electron-binding energy (145.6keV) while they are higher above. This change results ina sharp decrease of the M (cid:39) I = 1 transition rate below I i = 23 /
2. The other im-pact is an increased stretched E I i = 23 /
2, clearly visible in the simulation and evidencedin the experimental data. For comparison, the intensityprofile assuming the [633]7 / + configuration ( g K = 1 . g K can be deduced using themaximum likelihood technique. For convenience, theopposite of the logarithm of the likelihood estimator, − ln L ( g K ), is used since one has simply − ln L = 1 / χ .This estimator is plotted in the inset of Fig. 5. The esti-mator, the most likely g K value and its uncertainty werederived as follows. The intensity profile was simulated fordifferent g K values in the 0 ≤ g K ≤ . g K cor-responds to the minimum − ln L min . The uncertaintieswere obtained using − ln L ( g K + σ + ) = − ln L ( g K − σ − ) = − ln L min + 0 .
5, as depicted in the inset of Fig. 5. Themethod allowed us to deduce the most likely gyromag-netic factor g K = 0 . +0 . − . . This value is in remarkableagreement with the value g K = 0 .
66 estimated abovefor the [514]7 / − configuration using the particle-plus-rotor model. In contrast, for the [633]7 / + configuration, g K = 1 .
34 deviates by 2 . σ from the most likely value.It should be noted that the estimator is symmetric withrespect to g R = Z/A and therefore the value g K = 0 . g K = 0 . +0 . − . , remarkably close to the value ob-tained using seven experimental points. The estimator − ln L ( g K ) using these five experimental values is plottedwith a dashed line in the inset of Fig. 5. Both curves arevery similar around the most likely value since the lasttwo experimental points have a higher uncertainty andtherefore a lower statistical weight. The second reasonis that the model reproduces the two last experimentalpoints very well for the most likely g K value, contribut-ing only marginally to the estimator around its minimum.In contrast, the two experimental points that contributemost to the estimator are those who delineate the abruptjump in the intensity profile ( I i = 23 / / − configuration. IV. EVIDENCE FOR OCTUPOLECORRELATIONS IN , MD The most intense
Md peak in both γ -ray andconversion-electron spectra corresponds to a transitionat 389 keV. Although this transition collects about 16 %of the de-excitation flow (compared to (cid:39)
10% and (cid:39) K π = 1 / − and K π = 7 / − bands, respectively),no coincident transition has been observed with a suf-ficient confidence level. Its experimental conversion co-efficient α K = 1 . ± . ± . M α L = 0 . ± . ± . α L is again compatible with an M M ≈
23 ns and therefore out ofthe view of the SAGE detection. A rate enhancement is,however, possible if the initial and final states are coupledvia a 2 − octupole phonon. Several examples of octupole-vibration coupling have been observed in the actinide re-gion. First examples were reported in the , Cm and , Cf by Yates et al. using transfer reactions [49, 50].A more recent example has been provided in
Fm [51].The occurrence of octupole correlations arise whenorbitals with ∆ l = ∆ j = 3 are close in energy. In the Z ∼ , N ∼
152 region, this is realized on the neu-tron side for the { [512]5 / + ( g / ) , [734]9 / − ( j / ) } orbits. For the protons, the pair candi-dates are { [512]5 / − ( f / ) , [624]9 / + ( i / ) } and { [521]3 / − ( f / ) , [633]7 / + ( i / ) } ; all these pairs favora K π = 2 − octupole phonon as the lowest collectiveexcitation. Although the [521]3 / − , [512]5 / − , and[633]7 / + orbitals have not been observed in Md yet,the second pair is a better candidate since both orbitalshave been predicted at lower energy compared to thefirst pair [16, 24, 25]. In that case, the 3 / − state wouldhave a [633]7 / + ⊗ − component and therefore pusheddown compared to pure single-particle predictions.The 389-keV transition can be hence tentatively as-signed to a 3 / − → / + transition that could be, be-cause of the coupling with an octupole phonon, of mixed E /M δ mixing ratio fora E /M δ ( E /M
2) = T γ ( E T γ ( M
2) = α K ( M − α K (Exp.) α K (Exp.) − α K ( E . (5)Because of the large experimental uncertainties, our mea-surement is compatible with no mixing. However, an up- per limit, within one σ , of δ ( E /M ≤ E Md as shown in Fig. 6.This transition is the most intense in the recoil-taggedspectrum apart from a contamination of
Tl Coulex[Fig. 6(a)]; its intensity decreases using γ - γ coincidences[Fig. 6(b)], consistent with only few radiative transi-tions in coincidence as in Md, and finally the tran-sition is still present when tagging on the Md α de-cay [Fig. 6(c)]. The energy and characteristics similarto Md suggests a transition of similar nature in bothisotopes. This is consistent with the fact that several cal-culations predict a very similar single-particle structureof
Md and
Md [16, 23–25].
Recoil-Tagging Tl K α K β (a) C oun t/ e V Recoil-Tagging γ−γ (b)
100 200 300 400 500 600
Energy (keV)
Recoil-Decay Tagging(c)
FIG. 6. γ -ray spectra of Md resulting from (a) recoil-tagging, (b) sum of γ - γ recoil-tagged coincidences, and (c)recoil-decay tagging using a maximum correlation time of200 s between the recoil implantation and the subsequent α decay. It should be noted that the statistics was not sufficientto establish rotational structures on a solid basis. V. COMPARISON OF
MD WITH
LRA. Similarity of transition energies: A possiblecase of identical bands
An unexpected feature of the K π = 7 / − g.s. rota-tional band of Md is its resemblance to the excitedband of
Lr [52], both based on the [514]7 / − orbital.The transition energies are indeed identical within theuncertainties up to I i = 23 /
2, beyond which the dif-ference slowly increases (Table II ). On the other hand, It should be noted that no spins were suggested for the tran-sitions in Ref. [52]. The spins proposed in the evaluation [53]
TABLE I. Internal conversion coefficient for a transition proceeding via K or L atomic shell with an energy of 389 keV. Thetheoretical coefficients [21] for E M E M E
3, and M E M E M E M α K − − − − ± . ± . α L − − − − − ± . ± . I i for Md (this work) and
Lr [52]. Tentative transitions are written in brackets.[514]7 / − [521]1 / − I i ( (cid:126) ) Md Lr I i ( (cid:126) ) Md Lr13/2 161.8 (3) 9/2 [87]15/2 189.0 (7) 189 (1) 13/2 144 (1)17/2 214.8 (5) 215 (1) 17/2 195.3 (3) 196.6 (5)19/2 238.8 (4) 239 (1) 21/2 244.3 (3) 247.2 (5)21/2 263.8 (3) 264.6 (5) 25/2 291.8 (10) 296.2 (5)23/2 289 (1) 288.4 (5) 29/2 335 (1) 342.9 (5)25/2 311.2 (6) 314.0 (5) 33/2 376.8 (4) 387 (1)27/2 334 (1) 338 (1) 37/2 415 (2) 430 (1)29/2 358 (1) 359 (1) 41/2 447 (2)31/2 376 (1) 384 (1) 45/2 [478 (2)]33/2 49/2 [513 (1)] rotational bands based on the [521]1 / − configuration donot exhibit similarities at such a level of precision. Such aphenomenon of identical rotational bands (IBs) was firstobserved in a pair of superdeformed bands (SD) of Tb(first excited band) and
Dy (yrast band) [54], laterconfirmed in numerous cases and which turned out to beemblematic of SD bands. The phenomenon of IBs hasalso been observed at intermediate and normal deforma-tions, with bands having a variable degree of similarity,and sometimes with IBs for nuclei that differ substan-tially in mass (Ref. [55] and references therein).It is worth reminding that, at moderate deformation,the classical moment of inertia of a rigid homogeneousbody is proportional to A / (1 + 0 . β ) [56]. Phraseddifferently, the transition energies of such rotating rigidbodies scale with A − / for the same deformation. There-fore, for the Md-
Lr pair, an energy difference of (cid:39)
3% between the transition energies of the two bands isexpected in such a purely macroscopic framework, sig-nificantly higher than the observation. For the pair ofbands based on the 7/2 − single-particle state, five tran-sitions are identical within one keV, which could be con-sidered as not very impressive at first sight compared toe.g. the phenomenon in SD bands. As we will discuss inthe following, however, this case turns out to be uniquein the transuranium region. It is also interesting to note should be excluded since the lowest unobserved transitions, be-cause they were highly converted, were ignored in the evaluatedlevel scheme. that the transition energies for the K π = 1 / − band arelarger in Lr than in
Md, while the opposite effectis expected according to the A − / scaling of rotationalenergies at fixed angular momentum and deformation.Numerous mechanisms have been advocated to explainthe IB phenomenon such as (i) the spin alignment of spe-cific orbitals along the rotation axis in the strong couplinglimit of the particle-rotor model, (ii) the role of symme-tries and in particular the pseudo-SU(3) scheme, (iii) therole of orbitals not sensitive to the rotation, in particu-lar, those having a high density in the equatorial plane(low number of nodes n z in the plane perpendicular tothe symmetry axis), (iv) the role of time-odd terms, etc.;see Ref. [55] and references therein. None of them werefully satisfactory since they were neither predictive norcapable of identifying the underlying mechanism. Someglobal analyses using mean-field approaches suggest thatthe mechanism is not as simple as a quantum alignmentor purely related to single-particle properties, but resultsfrom a cancellation of several contributions (deforma-tion, mass, pairing), resulting in the identical bands (e.g.Refs. [57–60]).Identical bands were previously reported in even-eventransuranium nuclei [61]: The three or four first ground-state band transitions in Pu, , Cm, and
Cf areidentical within 2 keV. The more recent improved spec-troscopy of
Pu [62],
Cm and
Cf [63] has shownthat the transition energies deviate significantly above I f = 8. More impressive are the ground-state bands of , U that are identical up to spin I f = 22 + within 2keV [61]. In this reference, this has been interpreted inthis region of midshell nuclei as the filling of orbitals driv-ing small deformation changes that counteract the massdependence. In any event, even if these bands cannotall be qualified as being identical, these cases recall thatthe systematics of moments of inertia can locally deviatevery strongly from the overall scaling with A / [56].To establish whether other identical bands are presentin the transuranium region, we have inspected all bandshaving at least eight measured transitions, which repre-sent 30 cases in even-even nuclei and 29 bands in odd- N or - Z nuclei (odd-odd nuclei were not considered). Thedata were taken from the Evaluated Nuclear StructureData File (ENSDF) and from more recent publicationsor unpublished works [46, 63, 64]. This survey revealedtwo other even-even pairs that are identical within 2 keVfor the four lowest spin transitions: They are respectively Pu and
Pu, and
Fm and
No. The equality ofthe transitions, however, is verified over a few transitionsonly. We also mention the case of the (
Pu-
No) pair0whose transition energies are identical within 2 keV upto I f = 6. This case can be hardly explained as these nu-clei differ by eight protons and two neutrons, and may beconsidered as an accidental degeneracy. It is worth men-tioning that the general trend of the 2 + collective statein the N = 152, Z = 100 region can be well explainedby a change of the moment of inertia because of a reduc-tion or increase in pairing correlations when approachingor leaving the deformed shell gaps (see the discussion,e.g. in Ref. [2]). Several nuclei, however, deviate fromthis trend (in particular the Cf isotopes). No explanationthat would convincingly explain all cases has been foundso far. Also, octupole correlations have been evidencedin the − Pu [62, 65] isotopes. Clearly, beyond-meanfield effects have to be taken into account in that case.Except the
Md-
Lr pair, we have not identified othercases of odd-mass nuclei having identical transitions be-tween them, or having identical transitions with one oftheir neighboring even-even nuclei.An intriguing fact in the IBs discussed here is thatthe nuclei differ by four mass units, more precisely an α particle. In the rare-earth-metal region, an α chainof even-even nuclei with bands identical up to I f = 12has been identified: Dy,
Er,
Yb,
Hf,
Os(and
W to a lesser extent) [66]. An interpretationbased on the algebraic interacting boson model was pro-posed in the same reference. The quadrupole moment islinked to the N p N n product, N p ( N n ) being the numberof valence protons (neutrons) with respect to the nearestmagic shell. This product is similar for all these nucleiand moreover in the language of the interacting bosonmodel, these nuclei form a F -spin multiplet having thesame number of valence particles N p + N n ; i.e., they arepredicted to have a similar structure. However, the ma-jor drawback of this approach is that it overpredicts theoccurrence of identical bands. Deficiencies to reproducethe gyromagnetic factor for these nuclei have also beennoticed [67]. FIG. 7. Schematic proton Nilsson diagram for
Md and
Lr inspired from calculations using a Woods-Saxon poten-tial (see, e.g. Ref. [68]).
The nuclear structure and deformation can be ex-pected to change when filling deformation-driving or-bitals (either down- or up- sloping as a function of thedeformation driving the nucleus toward larger or lower deformation, respectively). In this respect, it is inter-esting to note that because of the sequential filling ofproton levels, the ground-state and the first excited stateconfigurations of
Md versus
Lr are interchanged.Depending on its characteristics, the additional occupiedpair of orbits in
Lr can lead to subtle deformationchanges between the two nuclei; see Fig. 7.In the ground state of
Md, one level out of the pairof up-sloping [514]7 / − orbitals is filled, while in Lrthe 7 / − state corresponds to the same configuration plusa pair of filled [521]1 / − down-sloping orbitals, the latterdriving the nucleus toward slightly larger deformations.The deformation-driving effect in Lr for this configu-ration goes therefore in the same direction as the overall A / macroscopic dependence of the rigid moment of in-ertia. The filling of orbitals as such therefore cannot ex-plain the experimental finding of identical 7 / − bands,quite on the contrary: Based on this simple argument,the two bands should be even more different than canbe expected from the global scaling of moments of in-ertia. On the other hand, the [514]7 / − pair is filledfor the 1 / − Lr ground state, therefore driving thenucleus towards a lower deformation compared to the
Md 1 / − excited state, which goes in the opposite di-rection as the A / trend: A mechanism consistent withidentical bands for the [521]1 / − configuration, again incontradiction with the experimental finding.Furthermore, there is a pair of neutron orbits thatis filled to pass from Md to
Lr, namely in the[734]9 / − Nilsson orbital. According to calculations us-ing a Woods-Saxon potential (see, e.g. [68]) or calcula-tions presented below (see the self-consistent Nilsson di-agram in Fig. 9), this level is not sloping around theground-state deformation, which justifies ignoring neu-tron levels at the present level of discussion.There is also the experimental observation to considerthat the transition energies for the K π = 1 / − bandschange in the opposite direction to that expected fromthe A − / scaling. Moreover, from a purely macroscopicpoint of view, the deformation of Lr should decreasefrom β (cid:39) . (cid:39) A / term) and lead to the same energies as Md.Therefore, the mass-deformation compensation mecha-nism discussed above does not have the correct orderof magnitude since only small deformation changes areexpected and cannot explain simultaneously the largertransition energies for the K π = 1 / − Lr band andIBs for the K π = 7 / − bands, unless one assumes thatthere is an additional mechanism that decreases the mo-ment of inertia in Lr. If the mechanism is the same forboth configurations, then it just has the right size for the7 / − bands to make them identical, but “overshoots” forthe 1 / − bands.The mass-deformation compensation mechanism re-sulting from the filling of levels is therefore unable toexplain the experimental findings. There clearly haveto be additional compensation effects, for example, fromchanges in pairing correlations or the alignment of single-1particle states as proposed in Refs. [57, 61, 69–71] forthe observation of identical bands found for pairs of rare-earth-metal nuclei.In this respect, it is interesting to note that the Mdand
Lr nuclei are the neighbors of
Fm and
Norespectively with one additional proton. As already men-tioned, the yrast bands of these two even-even nucleiare also identical for the first four transitions. For thesame reasons discussed above for the case of
Md and
Lr, the mass-deformation compensation mechanismalso cannot explain the similarity between the yrast bandof
Fm and
No. With
Fm being proton magicdeformed and
No neutron magic deformed, a simpleexplanation of the change in moment of inertia in termsof a change in pairing correlations is also not straightfor-ward. The rotational K π = 1 / − bands of Md and
Lr can be phenomenologically described by the cou-pling of a proton in the K π = 1 / − orbit to the ground-state band of Fm and
No, respectively. In the mostbasic version of such a model [56], one automatically ob-tains identical K π = 1 / − bands in Md and
Lras well. However, this is not observed, indicating thatthere are additional changes that are not the same whenpassing from
Fm to
Md and from
No to
Lr,respectively. For the K π = 7 / − bands the situation iseven more complicated since for Md the K π = 7 / − band could be interpreted as a proton in the K π = 7 / − orbit coupled to the ground-state band of Fm, whereasfor
Lr the K π = 7 / − band corresponds a 2p-1h exci-tation relative to the ground-state band of No.
B. Self-consistent mean-field analysis
To better understand the conditions for the emer-gence of identical bands for the nuclei studied here,we performed microscopic cranked self-consistent-mean-field calculations for the K π = 1 / − and K π = 7 / − bands in Md,
Md, and
Lr. The calculationswere made with the coordinate-space solver
MOCCa [72, 73] that is based on the same principles as thecode used for the Skyrme-HFB calculations reported inRefs. [10, 12, 16, 52]. We employ the recent SLy5s1parametrization of the Skyrme energy density functional(EDF) [74] that was adjusted along similar lines as thewidely-used SLy4 parametrization [75] used in those ref-erences, but with a few differences in detail, the mostimportant one being a constraint on the surface energycoefficient that leads to a much better description of fis-sion barriers of heavy nuclei [76]. As pairing interactionwe choose a so-called “surface pairing” with cutoffs asdefined in Ref. [77].For further discussion, it is important to recall thatSLy5s1 does not reproduce the empirical deformed shellclosures at Z = 100 and N = 152 [2–4], a property thatit shares with SLy4 and almost all other available nu-clear EDFs that have been applied to the spectroscopyof very heavy nuclei so far [10, 78]. Instead, SLy5s1 gives prominent deformed proton gaps at Z = 98 and Z = 104,and an additional deformed neutron gap at N = 150; seeFigs. 8 and 9. For the Md and Lr isotopes discussedhere, the Fermi energy is in the direct vicinity of theseshell closures, which has some influence on the calculatedproperties of their rotational bands.We observe that, for a given Skyrme interaction, thesimilarity of in-band transition energies for different nu-clei depends sensitively on the details of the treatmentof pairing correlations. To illustrate this finding, fourdifferent options will be compared. The first one is theHFB + Lipkin-Nogami (HFB+LN) scheme as defined inRef. [77] with a pairing strength of − forprotons and neutrons that was adjusted to describe su-perdeformed rotational bands in the neutron-deficient Pbregion. This prescription has also been used in Refs. [12,16, 52]. The second option is a HFB+LN scheme with areduced pairing strength of − that was ad-justed to reproduce the kinematic moment of inertia of Fm at low spin when used with SLy5s1 [79]. While theLN scheme is a popular prescription to avoid the break-down of HFB pairing correlations in the weak-pairinglimit, it is known to have some conceptual problems, themost prominent one not being variational. As an alter-native, Erler et al. [80] proposed a small modificationthat can be applied to any pairing interaction and thatprevents the breakdown of pairing when being insertedinto a standard HFB calculation. Their fully variationalstabilized HFB scheme was used as a third pairing op-tion, again with a surface pairing interaction of strength − . As the fourth option we use the stan-dard HFB scheme as the most basic reference case, againwith a pairing strength of − . ǫ F ǫ F FIG. 8. Nilsson diagram of proton single-particle levelsaround the Fermi energy for mass quadrupole deformations β as defined in Ref. [76] around those of the ground state forfalse vacua of Md and
Lr, calculated with SLy5s1 andstabilized HFB pairing. The K π = 1 / − and K π = 7 / − lev-els are highlighted in color. The Fermi energy (cid:15) F is indicatedby a dashed line in each panel. Independent of the pairing option chosen, we find a2 ǫ F ǫ F FIG. 9. Same as Fig. 8, but for neutrons. calculated 1 / − ground state for Md and
Md, buta 7 / − ground state for Lr. In each case, the otherstate is a low-lying excitation at less than 160 keV. Thisresult is at variance with experimental data, for whichthe relative order of these levels is the other way round[16]. This finding is intimately connected to the incorrectdeformed gaps found in the Nilsson diagram of Fig. 8: inorder to obtain the correct level sequence, the K π = 1 / − level has to be pushed up relative to the other levelssuch that it is above the K π = 7 / − level at all relevantdeformations. This would open up a gap at Z = 100 andsignificantly reduce the Z = 104 gap; see the detaileddiscussion of this point in Ref. [16]. Similar problemsfor the relative position of these two levels were foundfor virtually all widely used nuclear EDFs [10, 78]. Itis noteworthy that the UNIDEF1 SO parametrization ofRef. [81] for which the spin-orbit interaction has beenfine-tuned to give deformed Z = 100 and N = 152 shellgaps does not improve on the relative position of thesetwo levels. In addition, it predicts that the 9 / + [624]level is nearly degenerate with them, which is difficultto reconcile with the systematics of band heads in thisregion.The Nilsson diagrams of Figs. 8 and 9 have been calcu-lated for false vacua, meaning HFB states that have thecorrect odd particle number on average, but no blockedquasiparticles. It is noteworthy that the relative posi-tions of many neutron and proton levels visibly changewhen going from Md to
Lr: Filling a further pairof neutron and proton orbits changes all other levelsthrough self-consistency. Such self-consistent rearrange-ment of deformed shells seems to be a general featureof heavy deformed nuclei when calculated within self-consistent models [78].The rotational levels in each band have been con-structed by solving the cranked HFB equations with aconstraint on the collective angular momentum I z = (cid:104) ˆ J z (cid:105) such that J ( J + 1) = I z + K , with K held fixed at 7 / /
2, respectively. The odd particle can be put eitherinto the orbit with + K or − K , which leads to two dif- E ( M e V ) K π = 7 / − E ( M e V ) J J
FIG. 10. E K π = 7 / − band calcu-lated with SLy5s1 and the pairing options for the three nucleias indicated. Calculated values are plotted in color as indi-cated, whereas experimental values are plotted with smallergray and black symbols for Md and
Lr, respectively.Full symbols indicate transitions in the favored band, andopen symbols indicate transitions in the nonfavored band. ferent solutions of the HFB equations that we identifywith the states in the two signature-partner bands thatcan be observed experimentally [82]. With increasingspin I z , one finds a signature splitting between the twocalculated partner bands into an energetically favoredand non-favored band. For the calculated and observed K π = 7 / − bands, the signature splitting is too small tobe resolved on the plots. For the calculated K π = 1 / − band, however, it is quite substantial. As there are noexperimental data for the non-favored K π = 1 / − band,we will not discuss its properties here.The resulting E K π =7 / − bands are displayed in Fig. 10. It is immediatelyvisible that the calculated energies depend significantlyon the pairing option. To understand the origin of thedifferences between pairing options and nuclei, Fig. 11displays the corresponding dispersion of particle number (cid:104) (∆ N ) (cid:105) = (cid:104) N (cid:105) − (cid:104) N (cid:105) . The latter is a measure for theamount of pairing correlations.Within a given pairing scheme, all calculated bandsare very similar at low spin. There are, however, visibledifferences between the actual transition energies whencomparing the four pairing schemes. For these nuclei thatall are in the weak-pairing limit for either protons or neu-trons or both, using stabilized HFB or HFB+LN insteadof pure HFB reduces the moment of inertia when the cal-culations were done with the same pairing strength, asthese schemes tend to enhance pairing correlations. For3 h ( ∆ N ) ih ( ∆ N ) i J K π = 7 / − J FIG. 11. Dispersion of neutron number (filled symbols) andproton number (open symbols) of states obtained when block-ing the favored orbit for the K π = 7 / − band, calculated withSLy5s1 and the pairing options for the three nuclei as indi-cated. the lowest transitions, the best agreement between thebands in different nuclei is found for HFB, but at higher J the bands visibly differ for that scheme, in particularthe one of Md. This is a consequence of the breakdownof neutron pairing with increasing spin, which quickly in-creases the moment of inertia for this nucleus. Preventingthe collapse of pairing with any of the other three pair-ing schemes brings the transition energies much closertogether over the entire band. It is to be noted that thebreakdown of neutron pairing at high spin in
Md is anartifact of the too large N = 150 gap at the Fermi en-ergy visible in Fig. 9. Similarly, the breakdown of protonpairing in the HFB calculation of Lr is an artifact ofthe too large Z = 104 gap. Assuming that the deformedgaps were at N = 152 and Z = 100 instead, the relativeamount of pairing correlations would be quite different:Protons should be more paired in Lr than in the twoMd isotopes, while neutrons should be less paired in
Lrthan the Md isotopes.Figure 12 displays the transition energies between lev-els in the K π = 1 / − band of the same three nuclei, andFig. 13 displays the corresponding dispersions of particlenumber. The overall trends are very similar to what isfound for the K π = 7 / − bands. Again, the very closeagreement of transitions in HFB at low spin is spoiledwhen neutron pairing breaks down at higher spin, an ef-fect that is visibly reduced when using stabilized HFB orthe LN scheme, in particular at high pairing strength. Itis noteworthy that the similarity of the three calculated K π = 1 / − bands is slightly better than the agreementbetween the three calculated K π = 7 / − bands, whilefor data this is the other way round. Differences betweenthe transition energies between same levels in the differ- ent K π = 1 / − bands are nevertheless still larger thanwhat is found in experiment by about a factor of 2. E ( M e V ) K π = 1 / − E ( M e V ) J J
FIG. 12. Same as Fig. 10, but blocking the favored orbit forthe K π = 1 / − band. h ( ∆ N ) ih ( ∆ N ) i J K π = 1 / − J FIG. 13. Same as Fig. 11, but blocking the favored orbit forthe K π = 1 / − band. In spite of the wrong relative order of the 1 / − and7 / − proton levels, the down-sloping 1 / − levels are al-most empty in the calculated excited 7 / − band of theMd isotopes, while they are almost completely filled for Lr as one would expect if the level sequence were theone suggested by experiment as depicted in Fig. 7. Sim-ilarly, the up-sloping 7 / − levels are almost empty forthe 1 / − band of the Md isotopes, whereas they are al-4most completely empty for Lr as would be expectedfrom the empirical shell structure. As proton pairing isweak for these odd- Z nuclides anyway, the blocked pro-ton configurations are therefore not much affected by theimperfections of the single-particle spectrum.All states in the calculated rotational bands havea dimensionless quadrupole deformation as defined inRef. [76] of β (cid:39) .
3, with differences on the few per-cent level that depend on the nucleus, spin, blocked state,and pairing option used. With increasing spin J , the de-formation of all configurations is slowly decreasing. Inparallel, all configurations become slightly triaxial, withthe γ angle remaining below 2 deg. Comparing bands,we observe some systematic differences in quadrupole de-formation that can be attributed to differences in the fill-ing of single-particle levels near the Fermi energy. The β value of the 7 / − band of the two Md isotopes isslightly smaller by about 0.003 than the β value of the7 / − band of Lr for all pairing options but HFB+LN.This is a consequence of the two additionally filled down-sloping, and therefore deformation-driving, 1 / − levelsas already discussed for the schematic Nilsson diagramof Fig. 7. The enhanced proton pairing correlations pro-duced by the HFB+LN scheme reduce this effect andlead to a near-identical deformation of the 7 / − bandfor all three nuclei. Similarly, the calculated deformationof the 1 / − band of the Md isotopes is systematicallylarger than the deformation of the 7 / − band. The dif-ference ∆ β is as large as 0.006 for the HFB option butremains much smaller for the standard HFB+LN scheme.This can be attributed to the filling of the deformation-driving 1 / − level, while the up-sloping 7 / − is almostempty. The deformation of the 1 / − bands of the Mdisotopes is also larger than the deformation of the 1 / − band of Lr because the filled up-sloping 7 / − levels inthe latter drive the shape to smaller deformations. Theeffect is again largest with a ∆ β of about 0.012 when us-ing the HFB option that does not produce proton pairingcorrelation for Lr such that the change in the fillingof orbits is largest. Using the other pairing schemes, the7 / − level is always partially filled to a varying degree,such that the change in deformation is reduced to abouthalf that size.The self-consistent calculations thereby confirm theschematic analysis of Fig. 7 concerning deformationchanges, including the finding that deformation cannotbe the sole explanation for the experimentally found re-duction of the moment of inertia of both bands when go-ing from Md to
Lr, as it only brings a change intothe right direction for the 1 / − bands. Changes in pair-ing correlations also have to be an important factor. Firstof all, with increasing pairing correlations this simple pic-ture of deformation changes driven by proton levels beingfilled or empty becomes blurred. Second, a reduction ofpairing correlations in general reduces in-band transitionenergies [56, 82]. As shown in Figs 11 and 13, the calcu-lated pairing correlations are lower in Lr compared to
Md, which should lead to an increase (decrease) of the moment of inertia (transition energies) in
Lr while theopposite trend is needed to reconcile the contradictionsmentioned above.To summarize the discussion, the similarity of calcu-lated transition energies in spite of sizable differences inthe other properties discussed above points to accidentalcancellation effects between the changes in shell struc-ture, deformation, and pairing as ingredients of the iden-tical K π = 7 / − bands and near-identical K π = 1 / − bands in Md and
Lr. However, it is difficult toquantify the changes brought by these effects, such thatan additional mechanism might be at play that leads toa universal reduction of the moment of inertia of
Lrcompared to
Md. Even if such a yet unidentifiedmechanism is needed, it is qualitatively described by thecranked HFB calculations, at least at low spin. Withincreasing spin, the differences between the calculatedbands become larger, as is the case for experiment. Thecalculations predict that the respective band of
Mdwill also be very similar to what was found for
Mdand
Lr, again in spite the large differences betweendeformation and pairing. The sensitivity of the calcu-lated transition energies to details of the pairing schemealso suggests that obtaining identical bands to a precisionthat is comparable with experiment is essentially a fine-tuning problem. Using the SLy4 parametrization insteadof SLy5s1 produces slightly different results but leads tothe same conclusions.All of these conclusions have to remain qualitative,though, as it should not be forgotten that finding iden-tical bands at the 1-keV level is beyond the limits ofwhat can be expected for the systematic errors of thecranked HFB method as such. It is also difficult to assessthe possible role of octupole correlations, whose presenceis hinted by the present data as discussed in Sec. IV,on the values for transition energies, as the coupling ofstates with octupole phonons is outside of the scope ofany pure mean-field model. As a first step in that direc-tion, exploratory beyond-mean field calculations includ-ing particle-number and angular-momentum projectionson top of (parity-conserved) triaxial one-quasiparticlestates were recently performed for
Md, using a vari-ant of the Skyrme EDF designed for this particular pur-pose [83]. Although these calculations yield moments ofinertia that are too small, they appropriately predict a K π = 7 / − ground state as well as the correct orderingof the levels in the signature partner bands. VI. SUMMARY AND CONCLUSION
To summarize, this work provides the detailed prop-erties of two rotational bands in the odd- Z Md in-terpreted as built on the [514]7 / − and [521]1 / − Nils-son orbitals, the former being the g.s. band. Conver-sion electron spectroscopy allowed the rotational bandsto be extended to lower rotational frequencies for theband based on the [521]1 / − Nilsson orbital. The con-5version electron intensity was also used to constrain thesingle-particle configuration for the K = 7 / / + configuration. It was also shownthat the band intensity profile in the presence of largeinternal conversion oscillates, providing a method to de-duce the gyromagnetic factor. The most intense transi-tion in both , Md has been tentatively interpretedas a 3 / − → / + M Md-
Lr pair isthe only case identified so far for odd-mass transuraniumnuclei, which moreover differ by four mass units. Argu-ments based on a mass-deformation-pairing compensa-tion fail to explain the experimental similarities (7 / − )and differences (1 / − ) between Md and
Lr. Anadditional and unexplained mechanism reducing the mo-ment of inertia in
Lr, that is probably independent ofthe filling of specific level, would explain simultaneouslyIBs for the 7 / − configuration and even larger changesof the moment of inertia for the 1 / − bands. HFB calcu-lations suggest there is not a simple mechanism leadingto identical bands in the A = 250 mass region. There-fore, the similarity can be hence considered as accidental.While the collective properties are generally well repro-duced by the present calculations, our study of the par-ticular case of similar bands points to the high sensitivity of the model to its ingredients and in particular to pairingcorrelations. From both an experimental and theoreticalpoint of view, the present work provides a step towarda better description of the super-heavy nuclei region andthe still speculative island of stability. ACKNOWLEDGEMENTS
We acknowledge the accelerator staff at the Univer-sity of Jyv¨askyl¨a for delivering a high-quality beam dur-ing the experiments. Support has been provided by theEU 7th Framework Programme Integrating Activities -Transnational Access Project No. 262010 (ENSAR), bythe Academy of Finland under the Finnish Centre ofExcellence Programme (Nuclear and Accelerator BasedPhysics Programme at JYFL, Contract No. 213503),and by the UK STFC. We thank the European Gamma-Ray Spectroscopy pool (Gammapool) for the loan of thegermanium detectors used in the SAGE array. B.B. ac-knowledges the support of the Espace de Structure et der´eactions Nucl´eaire Th´eorique (ESNT) at CEA in France.The self-consistent mean-field computations were per-formed using HPC resources of the computing center ofthe IN2P3/CNRS. W.R. gratefully acknowledges supportby U.S. DOE grant No. de-sc0019521 [1] M. Block, D. Ackermann, K. Blaum, C. Droese,M. Dworschak, S. Eliseev, T. Fleckenstein, E. Haettner,F. Herfurth, F. 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