Mass measurements for T z =−2 fp -shell nuclei 40 Ti, 44 Cr, 46 Mn, 48 Fe, 50 Co and 52 Ni
C. Y. Fu, Y. H. Zhang, M. Wang, X. H. Zhou, Yu. A. Litvinov, K. Blaum, H. S. Xu, X. Xu, P. Shuai, Y. H. Lam, R. J. Chen, X. L. Yan, X. C. Chen, J. J. He, S. Kubono, M. Z. Sun, X. L. Tu, Y. M. Xing, Q. Zeng, X. Zhou, W. L. Zhan, S. Litvinov, G. Audi, T. Uesaka, T. Yamaguchi, A. Ozawa, B. H. Sun, Y. Sun, F. R. Xu
aa r X i v : . [ nu c l - e x ] S e p Mass measurements for T z = − f p -shell nuclei Ti, Cr, Mn, Fe, Co and Ni C. Y. Fu, Y. H. Zhang,
1, 2, ∗ M. Wang,
1, 2, †
X. H. Zhou,
Yu. A. Litvinov,
K. Blaum, H. S. Xu, X. Xu,
1, 5
P. Shuai, Y. H. Lam,
1, 2
R. J. Chen, X. L. Yan, X. C. Chen, J. J. He,
6, 7
S. Kubono, M. Z. Sun, X. L. Tu,
1, 4
Y. M. Xing, Q. Zeng,
1, 8
X. Zhou,
W. L. Zhan, S. Litvinov, G. Audi, T. Uesaka, T. Yamaguchi, A. Ozawa, B. H. Sun, Y. Sun, and F. R. Xu CAS Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern Physics,Chinese Academy of Sciences, Lanzhou 730000, People’s Republic of China University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China GSI Helmholtzzentrum für Schwerionenforschung, Planckstraße 1, 64291 Darmstadt, Germany Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany School of physics, Xi’an Jiaotong University, Xi’an 710049, China Key Laboratory of Beam Technology of Ministry of Education, College of Nuclear Science and Technology,Beijing Normal University, Beijing 100875, People’s Republic of China Beijing Radiation Center, Beijing 100875, People’s Republic of China School of nuclear science and engineering, East China University of Technology, Nanchang 330013, People’s Republic of China CSNSM-IN2P3-CNRS, Université de Paris Sud, F-91405 Orsay, France RIKEN Nishina Center, RIKEN, Saitama 351-0198, Japan Department of Physics, Saitama University, Saitama 338-8570, Japan Insititute of Physics, University of Tsukuba, Ibaraki 305-8571, Japan School of Physics, Beihang University, Beijing 100191, People’s Republic of China School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China State Key Laboratory of Nuclear Physics and Technology, School of Physics,Peking University, Beijing 100871, People’s Republic of China (Dated: September 29, 2020)By using isochronous mass spectrometry (IMS) at the experimental cooler storage ring CSRe, masses of short-lived Cr, Mn, Fe, Co and Ni were measured for the first time and the precision of the mass of Tiwas improved by a factor of about 2. Relative precisions of δ m / m = ( − ) × − have been achieved. Detailsof the measurements and data analysis are described. The obtained masses are compared with the Atomic-MassEvaluation 2016 (AME ′
16) and with theoretical model predictions. The new mass data enable us to extract thehigher order coefficients, d and e , of the quartic form of the isobaric multiplet mass equation (IMME) for the f p -shell isospin quintets. Unexpectedly large d - and e -values for A =
44 quintet are found. By re-visiting theprevious experimental data on β -delayed protons from Cr decay, it is suggested that the observed anomalycould be due to the misidentification of the T = J π = + isobaric analog state (IAS) in V. PACS numbers: 23.20.En, 23.20.Lv, 27.60.+j
I. INTRODUCTION
Atomic masses are widely applied to investigations in manyareas of subatomic physics ranging from nuclear structure andastrophysics to fundamental interactions and symmetries de-pending on the mass precision achieved [1, 2]. The evolutionof nuclear shell structure, nucleon correlations and changes ofdeformation are often studied through observing systematictrends of one- and two-nucleon separation energies, which arededuced directly from the atomic masses involved [3]. For in-stance, precision mass measurements of exotic nuclei have ledto discoveries of the disappearance of the neutron magic num-ber at N =
20 [4] and the rise of the new sub-shell closure at N =
32 [5, 6]. The masses of extremely exotic nuclei are usedto determine the borders of nuclear existence, the drip-lines[7, 8], as well as new mass measurements provide valuable ∗ Corresponding author. Email address: [email protected] † Corresponding author. Email address: [email protected] ‡ Corresponding author. Email address: [email protected] benchmarks for nuclear theories [9]. In nuclear astrophysicsthe needed ground state properties of many nuclides involvedin the rapid neutron capture (r-process) or the rapid protoncapture (rp-process) processes still have to be measured [10].In β -decay experiments, the Fermi (F) and Gamow-Teller(GT) transition strengths are deduced from the measured β feedings as well as the decay Q -values [11]. The latter are de-termined via the mass differences of the corresponding parentand daughter nuclei. Accurate nuclear masses in the lighter Z = N region are often used to test the validity of the isobaricmultiplet mass equation (IMME) [12, 13], which is associatedwith isospin symmetry in particle and nuclear physics. If abreakdown of the IMME is found, this may offer a possibil-ity to study mechanisms responsible for the isospin-symmetrybreaking [14].Exotic nuclei of interest today are typically short-lived andhave tiny production rates. Therefore mass measurements ofsuch short-lived and rare nuclei inevitably require very sensi-tive and fast experimental techniques. One of such techniquesis isochronous mass spectrometry (IMS) applied to nucleistored in a heavy-ion storage ring [15]. In the past few years,the masses of a series of T z = − − / f p shell have been measured by employingIMS at the Cooler Storage Ring (CSRe) in Lanzhou [16–18].As a continuation of this work, we report here precision massmeasurements of T z = − f p -shell nuclei produced in the pro-jectile fragmentation of Ni. The paper is organized as fol-lows: Experimental details and data analysis are described inSec. II. The new results and their impact on nuclear structureare given in Sec. III and Sec. IV, respectively. Summary andconclusions are given in Sec. V.
II. EXPERIMENT AND DATA ANALYSIS
The experiment was performed at the HIRFL-CSR acceler-ation complex [19], which consists of a separated sector cy-clotron (SSC, K = K = B ρ = .
05 Tm. Hence, C + and U + ions can beaccelerated to energies of about 1 GeV/u and 500 MeV/u, re-spectively. The CSRe has a circumference of 128.80 m and amaximal magnetic rigidity B ρ = .
40 Tm [20].In this experiment, 468 MeV/u Ni + primary beams ofabout 8 × particles per spill were fast-extracted from theCSRm and were fragmented in a ∼
15 mm thick Be targetplaced in front of the RIBLL2. The reaction products fromthe projectile fragmentation of Ni emerged the target withrelativistic energies mostly as bare nuclei. They were selectedand analysed in flight with the RIBLL2. The CSRe was tunedinto the isochronous ion-optical mode [21, 22] with the tran-sition energy γ t = . B ρ = . Cr + ions fulfillthe isochronous condition of γ = γ t , where γ is the relativisticLorenz factor. The required energy of the primary beams andthe magnetic rigidity of the RIBLL2 were determined withLISE++ simulations [23] to achieve the maximum yield andthe optimal transmission for Cr + ions. Every 25 s, a freshprimary beam was fast extracted to produce the nuclides ofinterest. Only a small fraction of the projectile fragments ly-ing within the B ρ acceptance of ± .
2% of the RIBLL2-CSResystem were transmitted and stored in the CSRe. In each in-jection, up to 18 ions were selected and injected into the CSRefor a measurement of 300 µ s.A time-of-flight (TOF) detector [24] was installed inside theCSRe aperture to measure the revolution times of stored ions.The detector was equipped with a 19 µ g/cm carbon foil of 40mm in diameter and a micro-channel plate (MCP) with a fasttiming anode (see Fig. 1). Passages of swift ions through thecarbon foil caused secondary electrons released from the foilsurface. The number of such secondary electrons depends onthe electronic stopping power of the passing ion [25], dE/dx ,which at relativistic energies is roughly proportional to thesquare of its atomic number Z . Combined with the geometri-cal efficiency of the MCP, the overall detection efficiency ofthe TOF detector for a single ion passage varied from 7% to Carbon foilMCPAnode e - Heavy ion
PotentialPlatesEqualizing ring
FIG. 1: (Color online). Schematic view of the time-of-flight detec-tor [24] installed inside the CSRe aperture. The electric field is gen-erated by the potential plates and an equalising ring. The magneticfield is produced by Helmholtz coils (not shown) placed outside theultra-high vacuum environment of the CSRe.
80% depending on the ion species. Secondary electrons wereisochronously guided to the MCP by perpendicularly arrangedelectric and magnetic fields. The timing signals from the an-ode were directly recorded by a high-performance digital os-cilloscope Agilent DSO90604A (20 GS/s rate, 6 GHz analogbandwidth). Typical fall-times of the negative-voltage timingsignals were 250 −
500 ps. The time stamps were extracted byusing the CFD (Constant Fraction Discrimination) techniqueapplied to the digitised timing signals. Finally, the time se-quences, i.e. the passage times as a function of the revolutionnumber, were obtained for each individual ion. Only the timesequences containing more than 30 timestamps within a cir-culation time of more than 100 µ s were considered in the dataanalysis. The time sequences were fitted with a second or-der polynomial function. The revolution times were obtainedas a slope of the fit curve at the 35th revolution. More de-tails of the typical signal processing and data analysis can befound in Ref. [22]. Since the magnetic fields of the CSRemagnets slowly drifted during the experiment, the field-driftcorrection procedure developed in Refs. [17, 26] has been im-plemented. The corrected revolution times were put into ahistogram forming a revolution-time spectrum.A part of the corrected revolution-time spectrum in a timewindow of 603 ns ≤ t ≤
622 ns is shown in Fig. 2. Theparticle identification has been done according to the proce-dures described in Refs. [22, 26]. The inset of Fig. 2 showsthe zoomed time range around Cr and Ar, where onesees that Cr and Ar with nearly the same m / q values( △ ( m / q ) / ( m / q ) ≈ × − ) are well separated. The corre-sponding standard deviations (or equivalently the root meansquares, RMS) of each peak are shown in Fig. 3 and rangebetween 1 and 5 ps. The parabolic shape of the RMS val-ues versus revolution times is well-understood. The minimumRMS value is found around Cr, for which the isochronoustuning of the CSRe was done. The widths of the revolution-time peaks increase when moving away from Cr.
604 606 608 610 612 614 616 618 620 622110 Revolution time (ns) C oun t s p s / A r- A r - / C r - A l - S i - C a - P - N e - S - T i - M g - C - M n - K - F e - A l - ( g . s . + i s o . ) C a - C o - N i - S i - N - T i - P - V - S - + C r- O - N a - C l - C r- + A r- FIG. 2: (Color online). A part of the measured revolution-time spectrum. T z = − Cr and Ar illustrating the corresponding revolution-time peaks well resolved.
606 608 610 612 614 616 618 620 62201234567 S t anda r d de v i a t i on ( p s ) Revolution time (ns) A l - ( g . s . + i s o m e r) S - + C r- C l - A r- T i - C r- M n - F e - C o - N i - FIG. 3: (Color online). Obtained standard deviations (RMS) of therevolution-time peaks shown in Fig. 2. Red squares are the values for T z = − S and Cr nuclei.
Four series of nuclides with − ≤ T z ≤ − / T z = − T z = − m / q ( t ) = a + a · t + a · t + a · t , (1)where a , a , a and a are free parameters. The well-knownmasses of T z = − / , − / ≤ t ≤
619 ns for T z = − / , − / t ∼
619 ns a systematic deviation is observedfor T z = − / T z = − / −150−100−50050100150 M E e x p − M E A M E ' ( k e V )
600 604 608 612 616 620 624
Revolution time (ns) O - M g1 - B e7 - A l - S i - T i - P - V - O - C l - M n47 - F e49 - F - T = 1 nuclei− z as calibrants T = 1/2 nuclei−T = 1 nuclei−T = 3/2 nuclei− zzz P - N e18 - S - N a0 - M g2 - C l - A r - K - C a37 - S i - N - C - FIG. 4: (Color online). Experimental ME values determined in thiswork compared with the literature values from the latest Atomic-Mass Evaluation AME ′
16 [27] and Ref. [30] for P. The T z = − σ massuncertainty in AME ′
16 and in Ref. [30] for P. uncertainties. The mass of S compiled in AME ′
16 [27] isfrom the earlier work using S ( He , He ) S reaction [28].The deviation to the obtained mass value of S in this exper-iment has been reported and discussed in Ref. [29].In the final calibration all nuclides with known masses inthe time window 608 ns ≤ t ≤
619 ns have been used exceptfor P and S. The masses of T z = − f p -shell nuclei, i.e. Ti, Cr, Mn, Fe, Co and Ni, were determined andconverted [17, 26] into atomic mass excesses defined as ME =( m − Au ) c . III. EXPERIMENTAL RESULTS
Newly determined masses of T z = − f p -shell nucleiare given in Table I together with their literature valuesfrom AME ′
16 [27] and Ref. [30] for P. The new and re-determined masses are compared with the literature values in
TABLE I: Experimental ME -values obtained in this work and from the AME ′
16 [27]. The number of ions identified, N , and the standarddeviations of the time peaks in Fig. 2, σ t , are listed in the second and third columns, respectively. The deviations, δ ME = ME CSRe − ME AME ′ ,are given in the sixth column. The predicted ME s from the IMME are given in the last column.Atom N σ t ME CSRe ME AME ′ δ ME IMME(ps) (keV) (keV) (keV) (keV) Ti 5 1.39 − ( ) − ( ) − ( ) − ( ) Cr 8 1.11 − ( ) − ( ) a − ( ) a − ( ) Mn 9 1.92 − ( ) − ( ) a a − ( ) Fe 19 2.83 − ( ) − ( ) a − ( ) a − ( ) Co 14 3.23 − ( ) − ( ) a a − ( ) Ni 34 3.18 − ( ) − ( ) a − ( ) b − ( ) a Extrapolated values in AME ′
16 [27].
Fig. 5. The black filled symbols in this figure indicate the nu-clei used for the calibration. Each of the N c = ME -valuesof the reference nuclides was re-determined by using the othernine nuclides as calibrants. This technique is referred to asleave-one-out cross-validation method [22]. The normalised χ n defined as χ n = vuut N c N c ∑ i = [( mq ) i , exp − ( mq ) i , AME ] [ σ exp ( mq ) i ] + [ σ AME ( mq ) i ] , (2)was found to be χ n = .
14. This value is within the expectedrange of χ n = ± / √ N c = ± .
22 at 1 σ confidence level,thus indicating that no additional systematic uncertainty needsto be considered.To show the accuracy/reliability of the present results, thewell-known ME -values for nuclides lying outside the consid-ered fitting range, namely Ar, Si, Ca, P [30], Ti, P, V, and O, were as well re-determined by extrapolating thefit function. The obtained results are given in Fig. 5 and showgood agreement with the literature values [27, 30]. Due to along-range extrapolation, our mass of Ar agrees only within3 σ with the result in Ref. [31].The mass of Ti was previously measured in Ca( π + , π − ) Ti double-charge-exchange reactions [32],and the experimental result of ME ( Ti) = − ( ) keVhas been adopted in the AME ′
16 [27]. Our measurementyields ME ( Ti) = − ( ) keV which is in agreementwith the adopted value, though the precision is improved by afactor of 2.1.The masses of Cr, Mn, Fe, Co and Ni were mea-sured for the first time in this work. For the even-evennuclei, the ME -values given in Table I correspond to theground states. It is worth noting that the revolution timesof Cr and Ar are very close to each other (see the in-set in Fig. 2). The mass of Ar was re-determined to be ME ( Ar ) = − ( ) keV which is in excellent agreementwith the literature value [31]. This provides an additional evi-dence for the reliability of the present measurement of Cr.In the case of the odd-odd nuclei Mn and Co, the con-tamination by low-lying isomeric states cannot be excluded.An isomeric state with excitation energy E x = . ( ) keV and half-life T / = . ( ) s [33] is known in Sc,
606 608 610 612 614 616 618 620 622−400−300−200−1000100200300400 O - V - P - T i - N - S i - C a - K - N i - C o - F e - C - M n - C r- T i - S - S i - C a - N a - A r- M g - N e - C l - M E e x p − M E A M E ' ( k e V ) Revolution time (ns) P - FIG. 5: (Color online). Differences between experimental ME -values determined in this work and those from the Atomic-MassEvaluation AME ′
16 [27] and Ref. [30] for P. Mass values for eachof the ten reference nuclides (filled black circles) were re-determinedby using the other nine nuclides as calibrants. The red circles corre-spond to the newly determined masses when all ten reference nu-clides were employed for mass calibration. The grey shadowed areasindicate the 1 σ mass uncertainty in AME ′
16 and in Ref. [30] for P. which is the mirror nucleus of Mn. Taking into accountthe mirror symmetry, a low-lying isomer at E x ∼
150 keVmay exist in Mn. Although low-lying isomers have beenobserved in odd-odd Co and Co, no such isomers wereobserved in mirror nuclei Co and V. In the experimen-tal revolution-time spectrum, single peaks without obviousbroadenings were observed for Mn and Co. Furthermore,the extracted peak widths follow the expected systematic be-haviour (see Fig. 3), though the counting statistics is low.The masses of Mn and Co could be calculated by us-ing the isospin multiplet mass equation (IMME) [34, 35] ex-pressed as ME ( α , T , T z ) = a ( α , T ) + b ( α , T ) · T z + c ( α , T ) · T z , (3)where ME s are mass excesses of isobaric analog states (IAS)of a multiplet with fixed mass number A and total isospin T . The coefficients a, b, c depend on A , T and other quan-tum numbers such as the spin-and-parity J π , but are indepen-dent from T z . This mass equation is considered to be accu-rate within mass uncertainties of a few tens of keV. The ME - -150-100-50050100150 M E e x p - M E fi t ( k e V ) A = 46, T=2
Mn−46 -2 -1 0 1 2-150-100-50050100150 M E e x p - M E fi t ( k e V ) T z A = 50, T=2
Co−50
FIG. 6: (Color online). Comparison of the experimental ME -valuesof Mn and Co with the IMME predictions. values of three IAS [33] were used to determine the a, b, c coefficients. The calculated ME -values of Mn and Co aregiven in the last column of Table I and are compared to theexperimental results in Fig. 6. Excellent agreement withinone standard deviation is obtained. It is therefore suggestedthat the measured here ME -values correspond to the groundstates. An additional argument supporting this suggestion isthat any contamination by an unknown isomeric state wouldlead to lower mass excess values than those given in Table I. IV. DISCUSSIONA. Test of nuclear mass models
The nuclear masses measured in this work can be usedto test modern nuclear mass models. The accuracy of cur-rent theoretical models has been recently investigated inRefs. [9, 36]. Among the ten often-used models of var-ious nature, the macroscopic-microscopic models of Wangand Liu [37, 38] and of Duflo and Zuker (DZ28) [39] werefound to be the most accurate in various mass regions char-acterised by the smallest RMS values of 250 ∼
500 keV. Fig-ure 7 shows a comparison of the new experimental masses of T z = − T z = − ≤ A ≤
52 region ( p f -shell)can be well described by the WS4 calculations, confirmingthe high predictive power of the model.
16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76-3.0-2.0-1.00.01.02.03.0
WS4+RBFETFSIQ M E - M E F RD M ( M e V ) Mass number A
DZ28HFB24 T z = -2 nuclei FIG. 7: (Color online). Comparison of the new experimental massvalues with predictions of several mass models for T z = − ′
16 [27] arepresented with the filled and open black circles, respectively. Thefilled red circles indicate the new experimental masses of this work.
In contrary to the global mass models aiming at describingthe entire mass surface, local mass relations are often moreaccurate in near extrapolations into unknown masses. Suchrelations are, for example, the IMME [34, 35], the Audi-Wapstra systematics [45], the Garvey-Kelson (G-K) massrelations [46–50], the mass relations based on the residualproton-neutron interactions [51–53], and the mass formu-las connecting mirror nuclei based on the isospin conserva-tion [54, 55]. An improved approach of Refs. [54, 55] gives aRMS value as small as 93 keV [56].
20 24 28 32 36 40 44 48 52 56−450−300−1500150300450 Mirror-nuclei approach of Ref. [56]IMMEExperiment M E M E − G K ( k e V ) Mass number A FIG. 8: (Color online). Comparison of the new experimental ME -values of T z = − ′
16 [27, 33]. The literaturemasses [27, 33] are shown with black filled circles. The new massesfrom Table I are indicated with red symbols. The shaded areas rep-resent 1 σ uncertainty of the theoretical predictions. Taking the Garvey-Kelson predictions as reference, Fig-ure 8 shows the comparison of the new experimental resultswith predictions by the IMME and the G-K relations as wellas the mirror-nuclei approach [56]. The predictions were ob-tained by using the known-mass values from AME ′
16 [27].The newly measured ME -values are in good agreement withthe mass predictions from both the IMME and the mirror-nuclei approach [56], but are systematically higher than thosefrom G-K relations except for Ti.
B. Validity of the isospin multiplet mass equation
Although the quadratic form of the IMME, i.e. Eq. (3),is commonly considered to be accurate, precision mass mea-surements can be used for testing its validity [16]. Typicallyone adds to Eq. (3) extra terms such as d · T z or/and e · T z ,which provide a measure of the breakdown of the quadraticform of the IMME. By taking into account the second-orderCoulomb effects, 3-body interactions, and isospin mixing,the d and e coefficients have generally been expected to besmaller than a few keV [57–61]. Numerous measurementshave been performed investigating the validity of the IMME.Reviews and compilations of existing data can be found inRefs. [12, 13] and references cited therein. For T = A =
40 was the heaviest multiplet with all massesexperimentally known.By adding the new nuclear masses obtained in this work,experimental information for A = , , , and 52 T = p f shell. The data were used to extract d - and e -coefficients which are shown in Fig. 9 as empty squares. −20−1001020 d c oe c i en t ( k e V ) ffi |d|/σ= .8 0 |d|/σ=5.4|d|/σ=3.8|d|/σ=4.7
20 24 28 32 36 40 44 48 52-20-1001020 e c oe ffi c i en t ( k e V ) Mass number A | |/σ= .e 6 6| |/σ= .e 7 7 FIG. 9: (Color online). d - and e -coefficients obtained from the least-squares fitting using the quartic (empty squares) form of the IMME.Results of 4-parameter fitting with only d - or e -coefficient includedare shown with filled symbols. ME -values from the AME ′
16 (blacksymbols) [27, 33] and from this work (red symbols) were used. Forthe A =
36 quintet the mass value of Ca was taken from Ref. [62].
For A = ,
50 quintets, only four-parameters fitting with either d - or e -coefficient included was possible. The corre-sponding results for d - and e -coefficients are plotted in Fig. 9with filled symbols. Except for A =
44, all obtained coeffi-cients are compatible with zero at 2 σ confidence level. For the A =
44 quintet, the results are | d | / σ d = . | e | / σ e = . | d | / σ d is observed for the A =
32 quin-tet, the magnitude of the d -coefficient is small and can the-oretically be reproduced by taking into account the isospinmixing [63, 64]. Large d - and e -coefficients for the A = d -coefficient is modified from − . ( . ) for the5-parameter fit to + . ( . ) for the 4-parameter fit, the con-vergence of the fitting procedure is questionable. C. β -delayed proton decays of Cr and the isobaric analogstate in V A large d -coefficient has already been obtained by usingthe T =
2, IAS in V assigned via the observations of β + -delayed protons from Cr decay [65]. However, to restore thevalidity of the IMME, the authors of Ref. [65] suggested thatthe state in Ti at E x = E x = T = Cr. If thesuggested value is used to extract a , b , c coefficients in thequadratic form of the IMME, the calculated mass excess of Cr is − ( ) keV which is 198(51) keV ( ∼ σ ) smallerthan our new experimental value of − ( ) keV. Fur-thermore, if the 5-parameters fitting is performed, the corre-sponding d - and e -coefficients significantly deviate from zero( d = − . ( . ) , e = . ( . ) ). These results contradicttheoretical expectations in the framework of isospin symme-try [57–61].The masses of Ca, Sc, and Ti are known with highprecision [27]. The T = Sc was identified in severalexperiments and is located at E x = ( ) keV [66]. Thisvalue has been confirmed in a recent investigation through a ( p , n ) -type Ca( He,t) Sc reaction [67]. The first T = Z = N self-conjugate nucleus Ti was identified viathe isospin-allowed Ti ( p , t ) Ti reaction [68, 69]. It waslater assigned to E x = ( ) keV through γ -decay spec-troscopy in the Ca ( α , γ ) Ti reaction [70]. Further studieshave led to the identification of a close-lying J π = + statebelow the T = Ti [71]. This state has been placedat E x = ( ) keV through γ -decay measurements [72],thus forming an isospin-mixed doublet. Decay-width analy-ses including the γ -decay branching ratios revealed that themain T = T =
2, IAS at E x = ( ) keV with a state at E x = ( ) keV with T = T = ME -values of T = Ca, Sc, and Ti mentioned above and the quadratic form of the IMME,the ME -values of T = Cr and V were calculatedto be − ( ) keV and − ( ) keV, respectively.The former value is in excellent agreement with − ( ) keV obtained in this work, whereas the latter value is 114 keV(8 . σ ) larger than the value given in Ref. [65]. TABLE II: Compilation of β -delayed proton decay energies, Q p , γ -ray energies, E γ , as well as their branching ratios, I p and I γ , for the decayof Cr. The results from Refs. [65, 73] are converted to the centre-of-mass energies. The weighted-average values are adopted in the table.Ref. [65] Ref. [73] Weighted Average Proposed Assignment Q p (keV) I p (%) Q p (keV) I p (%) Q p (keV) I p (%) This work Ref. [65]1 759(26) 0.6(2) 759(26) 0.6(2) 1 + to ( / − ) in Tior IAS to ( / + ) in Ti2 910(11) 1.7(3) 917(53) 2.7(5) 910(11) 2.0(3) 1 + to ( / − ) in Ti IAS to g.s. in Ti3 1384(12) 1.1(3) 1371(62) 1.4(3) 1384(12) 1.3(3) 1 + to ( / − ) in Ti4 1741(15) 0.6(3) 1719(44) 0.5(2) 1739(14) 0.5(2) 1 + to ( / + ) in Ti E γ (keV) I γ (%) E γ (keV) I γ (%) E γ (keV) I γ (%)1 676.9(3) 59(5) 676.9(3) 59(5) V:1 + to 2 + (g.s.) Ti + p V Cr Sc ++ . ( )
0 27791 3014 0.154(8)1 3152 0 39 2 . ( ) +++++ ++++
B( )GT +
1 3641 0 274(14 . ) + E = ( ) p E = ( ) p E = ( ) p E = ( ) p IAS . ( ) I = ( ) % γ T =42.8(6) ms
Q =10386(51) keV β S =1776(11) keV p + logft B(GT) I
1 3828(18) 4.61(18) 0.093 0.5(2) +
1 4055 0 119 6 . ( ) + −
1 3711 0 148(8 . ) + −
1 3161(17) 4.24(7) 0.219 2.0(3) +
1 3635(18) 4.27(11) 0.204 1.3(3) + β FIG. 10: (Color online). (Left) Proposed partial decay scheme of Cr. (Right) The level structure of Sc, the mirror nucleus of V, identifiedin the Ca( He, t ) Sc reaction [67]. Only levels with B(GT) ≥ β feedings ( I β ) and log f t values of the Cr β -decaysare deduced from the branching ratios of β -delayed protons and Q -values suggested in this work. The β -decay B(GT) values are calculatedfrom Eq. (4). The levels shown in green colour indicate the excited T = , J π = + IAS. All energies are in keV.TABLE III: Compilation of the ME -values for the ground states of Cr and Ca, and for the lowest J π = + , T = V, Ti, and Sc. The data are taken from AME ′
16 and NUBASE ′ Cr, V, and Ti. The unperturbed level en-ergy of the IAS in Ti [72] is used. The weighted-average mass ofthe ground state of V measured in Refs. [17, 74] is adopted. Theparameters of the IMME fits are listed in Table IV.Atom T z ME (g.s.) E x ME (IAS)(keV) (keV) (keV) Cr − − ( ) ∗ − ( ) ∗ V − − ( ) ( ) ∗ − ( ) ∗ Ti 0 − . ( ) ( ) ∗∗ − . ( ) ∗∗ Sc + − . ( ) ( ) − . ( ) Ca + − . ( ) − . ( ) ∗ from this work and ∗∗ unperturbed level from Ref. [72] The T = J π = + IAS in V [65] was originally pro-posed on the basis of β -delayed protons ( β -p) from Cr de-cay. There, the strongest proton branch with the relevant centre-of-mass energy, E p = ( ) keV, was assigned asdecaying from the expected T = J π = + IAS in V tothe ground state of Ti. However, the branching ratio ofthis transition is only 1.7(3)%, which is much smaller thanthe theoretical estimation of 28% for the super-allowed β de-cay of Cr to the T = V [65]. As no γ transitionsde-exciting the proposed IAS were observed to balance the β -feeding branching ratio, this assignment shall be carefullychecked.Recently, highly-sensitive measurements of β -delayed pro-tons from Cr were conducted by employing an optical timeprojection chamber (OTPC), leading to the observation of alow-energy proton peak with a mean centre-of-mass energyof 759(26) keV [73]. The ground-state mass of V has beenmeasured at the CSRe [17] and by Penning-trap mass spec-trometry [74]. Meanwhile, detailed level structure of Scincluding the Gamow-Teller transition strengths, B(GT), isavailable [67]. These data enabled us to revisit the decayscheme of Cr and address the assignment of the T = V. TABLE IV: The coefficients obtained from the fitting by using quadratic, cubic, and quartic forms of the IMME. The corresponding mass dataare given in Table III.a b c d e χ n − . ± . − . ± . . ± . − . ± . − . ± . ± − . ± . − . ± . − . ± . ± − . ± . − . ± . − . ± . ± ± − . ± . Table II summarises the available experimental informationon β -delayed protons and γ -transitions from Cr decay. Allvalues are weighted-averages of the data from two measure-ments [65, 73]. Given on the right side of Fig. 10 are theenergy levels of Sc with B(GT)-values larger than 0.1 as ob-tained in the Ca( He,t) Sc reaction [67]. Except for theIAS at E x = ∆ L = + states. The weighted-average mass of V fromRefs. [17, 74] is used, giving the proton separation energy of S p ( V ) = ( ) keV.An interesting observation in the decay of p f -shell nu-clei was made that the proton decay branches from the IASin Co [75] and Co [76] are very weak or even non-observable. Therefore, the conventional way to assign thestrongest proton peak at a relevant centre-of-mass energy asbeing from the IAS may cause misidentification. The low ornon-observable proton-decay branch has been attributed to thevery small isospin mixing of the related IAS [76]. In the caseof the proton decay of the IAS in V, the decay energy is ex-pected to be less that 1 MeV, which is even smaller than inthe cases of Co [75] and Co [76]. As a consequence, thebarrier penetration could be more difficult, providing higherhindrance for the proton decay of the IAS. Furthermore, theassignment of the strongest 910-keV protons as being fromthe decay of the IAS in V to the ground state of Ti is notsupported by dedicated theoretical calculations [77]. Two sce-narios are proposed to understand the β -delayed proton emis-sions in Cr decay, which refer to the symmetry of the levelstructure in the mirror nuclei V and Sc.First, it is assumed that no protons from the decay ofthe T = V were observed in the two experi-ments [65, 74]. For this case, a partial decay scheme of Cr is proposed and shown in Fig. 10. The strongest 910-keV proton decay branch is assigned to be the V ( + , E x = ) → Ti ( / − , E x = ) transition rather than the V ( IAS , E x = ) → Ti ( / − , g . s . ) transition suggestedin Ref. [65]. We have estimated the log f t values [78] foreach individual β transition to the levels in V by using theexperimental β feedings, i.e. I β = I p . The B(GT) strengthswere determined from the f t values via the equation B j ( GT ) = K λ f j t , (4)where K = . ( ) [79], λ = − . ( ) [80], the index j represents the daughter state at the excitation energy E j . Onesees that the level structure of V, including the level spacingsand the deduced B(GT) strengths, is very similar to the analogstates in Sc. By adopting the assignment of the 910-keV protons discussed above, the experimental β -delayed protonspectrum in Ref. [65] agrees well with the β -decay spectrumdeduced from the Ca( He,t) Sc reaction [67].Second possibility is to assign the 759-keV protons as beingfrom the V ( IAS , E x = ( )) → Ti ( / + , E x = ) transition. If referring to the mirror symmetry, this assignmentgives more relevant excitation energies of the two IAS in Vand Sc (see Fig. 10). This assignment can be checked withthe IMME. Table III presents the ME -values of the T = A =
44 quintet. The mass data are fitted as in Refs. [12, 13]using quadratic, cubic, and quartic forms of the IMME, andthe obtained coefficients are listed in Table IV. The d - and e -coefficients are compatible with zero within 2 σ . This re-sult indicates that the quadratic form of IMME is still validif the location of the IAS in V as proposed here is adopted.The large d - and e -coefficients demonstrated in Fig. 9, the so-called breakdown of the IMME, are caused most probably bythe misidentification of the IAS in V [65]. To confirm thisconclusion, precision determination of the IAS through mea-surements of β -delayed γ emissions in Cr decay is highlydesired.
V. SUMMARY
Mass measurements of neutron-deficient f p -shell nucleiproduced in the projectile fragmentation of 468 MeV/u Nibeam were performed by using the isochronous mass spec-trometry at the cooler storage ring CSRe in Lanzhou. Themasses of Cr, Mn, Fe, Co, and Ni were measuredfor the first time with relative precisions of ( − ) × − , andthe mass precision for Ti was improved by a factor of 2. Thenew mass values were compared with predictions of globalmass models as well as local mass relations. It is found thatthe experimental masses can be well described by the WS4model with the radial basis function correction [44]. A sys-tematic deviation seems to exist if comparing to the predic-tions of the Garvey-Kelson mass relation, while good agree-ment was achieved with the IMME and the recent mirror-nuclei approach [56].By using the new mass values, experimental data for five T = p f -shell, namely for A = , , , and 52 quintets.The extracted d - and e -coefficients of the quartic form of theIMME are compatible with zero within 2 σ except for A = d - and e -values were addressed byrevisiting the experimental data on β -delayed protons from Cr decay. It is suggested that the strongest 910-keV pro-ton branch is not from the de-excitation of the IAS in V.It is concluded that the observed breakdown of the quadraticform of the IMME as well as the large d - and e -coefficientsexhibited in Fig. 9 most probably originate from the misiden-tification of the IAS in V [65]. To confirm this conclusion,precision determination of the IAS through measurements of β -delayed γ emissions in Cr decay is highly desired.
Acknowledgments
We thank the staff of the accelerator division of theIMP for providing the stable beam. This work was sup-ported in part by the National Key R&D Program of China(Grant No. 2018YFA0404400, No. 2016YFA0400504 and No. 2016YFA0400501), the Strategic Priority Re-search Program of Chinese Academy of Sciences (GrantNo. XDB34000000), the Key Research Program of Fron-tier Sciences of CAS (Grant No. QYZDJ-SSW-S), theNSFC grants 11905259, 11905261, 11975280, U1932206,11805032, 11775277, 11961141004, and the Helmholtz-CASJoint Research Group HCJRG-108. Y.A.L. acknowledgessupport from European Research Council (ERC) under the EUHorizon 2020 Research and Innovation Programme (GrantAgreement No. 682841 "ASTRUm"). T.U., T.Y., and A.O.are supported in part by JSPS and NSFC under the "Japan-China Scientific Cooperation Program". C.Y.F. and Y.M.X.are thankful for the support from CAS "Light of West China"Program. [1] K. Blaum, Phys. Rep. , 1 (2006).[2] D. Lunney, J. M. Pearson, and C. Thibault, Rev. 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