High-precision mass measurements for the isobaric multiplet mass equation at A = 52

TL;DR

This study measured the masses of specific isotopes at A=52 with high precision, tested the isobaric multiplet mass equation, and provided new data crucial for nuclear physics and astrophysics, including proton decay and separation energies.

Contribution

First experimental determination of masses for $^{52}$Co and $^{52}$Co$^m$, and analysis of the IMME at A=52 with no significant breakdown observed.

Findings

$^{52}$Co and $^{52}$Co$^m$ are more bound than predicted.

No significant deviation from quadratic IMME form at A=52.

Precise $Q$ value for proton decay in $^{53}$Co obtained.

Abstract

Masses of $^{52}$Co, $^{52}$Co$^m$, $^{52}$Fe, $^{52}$Fe$^m$, and $^{52}$Mn have been measured with the JYFLTRAP double Penning trap mass spectrometer. Of these, $^{52}$Co and $^{52}$Co$^m$ have been experimentally determined for the first time and found to be more bound than predicted by extrapolations. The isobaric multiplet mass equation for the $T=2$ quintet at $A=52$ has been studied employing the new mass values. No significant breakdown (beyond the $3\sigma$ level) of the quadratic form of the IMME was observed ($\chi^2/n=2.4$). The cubic coefficient was 6.0(32) keV ($\chi^2/n=1.1$). The excitation energies for the isomer and the $T=2$ isobaric analogue state in $^{52}$Co have been determined to be 374(13) keV and 2922(13) keV, respectively. The $Q$ value for the proton decay from the $19/2^-$ isomer in $^{53}$Co has been determined with an unprecedented precision, $Q_{p} = 1558.8(17)$ keV. The proton separation energies of $^{52}$Co and $^{53}$Ni relevant for the astrophysical rapid proton capture process have been experimentally determined for the first time. \end{abstract}

Tables

Table 1. Table 1: Properties of the nuclides studied in this work. T 1 / 2 subscript 𝑇 1 2 T_{1/2} is the half-life, I π superscript 𝐼 𝜋 I^{\pi} the spin-parity and E x subscript 𝐸 𝑥 E_{x} the excitation energy of the isomeric state. The values estimated from isospin symmetry or from systematic trends from neighboring nuclides with the same Z 𝑍 Z and N 𝑁 N parities are marked by “#”. The values are based on Ref.  Dong and Junde ( 2015 ) unless stated otherwise.

Nuclide	$T_{1/2}$	$I^{\pi }$	$E_x$ (keV)
52Co	104(7) ms	$6^{+}\#$
52Com	104(11) ms Hagberg et al. (1997)111Authors in Ref. Hagberg et al. (1997) do not have specific evidence for 52Com.	$2^{+}$#	380(100)# Audi et al. (2012a)
52Fe	8.275(8) h	$0^{+}$
52Fem	45.9(6) d	${12}^{+}$	6958.0(4)
52Mn	5.591(3) d	$6^{+}$

Table 2. Table 2: The weighted average cyclotron frequency ratios, r 𝑟 r , and mass-excess values, M ​ E J ​ Y ​ F ​ L 𝑀 subscript 𝐸 𝐽 𝑌 𝐹 𝐿 ME_{JYFL} , determined in this work in comparison with the mass-excess values from AME12 Wang  et al. ( 2012 ) . The atomic mass excess value -55416.1(6) keV for 52 Cr Wang  et al. ( 2012 ) was taken to calculate the mass excesses of the studied nuclides from the frequency ratios. The mass-excess values from AME12 Wang  et al. ( 2012 ) are given in the fourth column. The differences between the JYFLTRAP and the AME12 mass values are given in the fifth column.

Nuclide	$r$	$M{E}_{JYFL}$ (keV)	$M{E}_{AME12}$ (keV)	JYFL-AME12 (keV)
52Co	1.00043584(14)	-34331.6(66)	-33990(200)#	-342(200)
52Com	1.00044356(23)	-33958(11)	-33610(220)#	-348(220)
52Fe	1.0001464894(28)	-48330.67(60)	-48332(7)	1(7)
52Fem	1.0002903590(56)	-41370.01(65)	-41374(7)	4(7)
52Mn	1.0000973119(13)	-50709.97(59)	-50706.9(19)	-3(2)

Table 3. Table 3: Excitation energies, E x , I ​ A ​ S subscript 𝐸 𝑥 𝐼 𝐴 𝑆 E_{x,IAS} , and mass-excess values, M ​ E I ​ A ​ S 𝑀 subscript 𝐸 𝐼 𝐴 𝑆 ME_{IAS} , for the J π = 0 + , T = 2 formulae-sequence superscript 𝐽 𝜋 superscript 0 𝑇 2 J^{\pi}=0^{+},T=2 isobaric analog states at A = 52 𝐴 52 A=52 . For 52 Fe and 52 Mn, the mass-excess values for the IAS are based on the mass-excess values from this work and excitation energies from Ref.  Dong and Junde ( 2015 ) . For 52 Co, the mass excess for the IAS is based on the mass of 52 Co m measured in this work and the γ 𝛾 \gamma -ray energies from Ref.  Orrigo  et al. ( 2016 ) .

Nuclide	$T_Z$	$E_{x,IAS}$ (keV)	$M{E}_{IAS}$ (keV)
52Ni	-2	0	-23474(700)# Wang et al. (2012)
52Co	-1	2922(13)	-31410(11)
52Fe	0	8561(5) Dong and Junde (2015)	-39769.7(50)
52Mn	1	2926.0(5) Dong and Junde (2015)	-47783.97(77)
52Cr	2	0	-55418.1(6) Wang et al. (2012)

Table 4. Table 4: Coefficients and the reduced χ 2 superscript 𝜒 2 \chi^{2} values for the quadratic and cubic IMME fits (in keV) for the T = 2 𝑇 2 T=2 quintet at A = 52 𝐴 52 A=52 .

	Quadratic	Cubic
a	-39777.1(30)	-39769.4(50)
b	-8192.9(46)	-8192.8(46)
c	186.2(16)	172.2(75)
d		6.0(32)
${\chi }^2/n$	2.4	1.1

Equations

$$M(A,T,T_Z)=a(A,T)+b(A,T)T_Z+c(A,T)T_Z^2,$$

Full text

I Introduction

Assuming a charge-independent nuclear force, the isobaric analogue states (IAS) in an isobaric multiplet are degenerate. Their mass differences are due to Coulomb interaction and the neutron-proton mass difference. According to the Isobaric Multiplet Mass Equation (IMME) Weinberg (1959), the masses of IASs in a mass multiplet with an atomic mass number $A$ and isospin $T$ should lie on a parabola:

[TABLE]

where the coefficients $a$, $b$ and $c$ are interpreted as being the scalar, vector and tensor Coulomb energies. High-precision Penning-trap mass measurements have offered new possibilities to investigate the validity of the IMME, and have revealed a breakdown in the quadratic form of the IMME in a few cases, such as for $A=8$ Charity et al. (2011), $A=9$ Brodeur et al. (2012), $A=21$ Gallant et al. (2014), $A=31$ Kankainen et al. (2016), $A=32$ Kankainen et al. (2010a); Kwiatkowski et al. (2009), and $A=35$ Yazidjian et al. (2007). In general, however, the IMME seems to describe well the masses of isospin multiplets, and it has therefore been widely used to predict the masses of the most exotic members of the multiplets.

Sometimes the quadratic form of the IMME (Eq. 1) is not sufficient to describe the masses in an isobaric multiplet but a cubic ($dT_{Z}^{3}$) or even a quartic coefficient ($eT_{Z}^{4}$) is required. The $T=3/2$ quartets have shown an interesting, increasing trend in the cubic IMME coefficients when entering into the $f_{7/2}$ shell MacCormick and Audi (2014a, b). On the other hand, the quadratic IMME at $A=53$ has been recently revalidated with a reduced $\chi^{2}$ of 1.34, and the cubic coefficient has been found to be rather small, $d=5.4(46)$ keV Su et al. (2016) compared to $d=39(11)$ keV obtained in Ref. Zhang et al. (2012).

The $T=2$ quintets have not been experimentally explored in the heavier mass region but could provide further insight into the possible trend in the cubic coefficients. In this paper, we have experimentally determined the masses for 52Co, 52Fe, and 52Mn, which are members of the $T=2$ isobaric quintet at $A=52$ together with 52Cr and 52Ni. Previous IMME evaluations for the quintet have suggested that a large non-zero cubic coefficient, $d=28.8(45)$ keV, might be required for the IMME MacCormick and Audi (2014a, b). However, the test was not very stringent due to the lack of experimental mass values for 52Co and 52Ni. Thus, the mass of 52Co, determined here experimentally for the first time, is pivotal for testing the IMME and investigating whether there is a trend towards larger cubic coefficients for nuclei in the $f_{7/2}$ shell forming $T=2$ quintets.

In addition to the ground states of 52Co, 52Fe, 52Mn, we have studied isomeric states in 52Co and 52Fe as summarized in Table 1. The isomeric state of 52Co is of special interest because it can be used to determine the mass of the $T=2$ IAS in 52Co. The current knowledge of the $T=2$ IAS in 52Co is based on $\beta$-decay studies of 52Ni Dossat et al. (2007); Orrigo et al. (2016). Two prominent $\beta$-delayed proton groups with center-of-mass energies of 1057(11) keV and 1349(10) keV Dossat et al. (2007) have been observed from the IAS. Similar proton energies at 1048(10) keV and 1352(10) keV have been determined in a more recent work Orrigo et al. (2016). The proton peaks have been attributed to the decay of the IAS to the ground state and first excited states in 51Fe known from in-beam $\gamma$-ray spectroscopy Ekman et al. (2000); Bentley et al. (2000). The excitation energy of the IAS can thus be determined as a sum of the observed proton energy and the proton separation energy of 52Co. On the other hand, the excitation energy of the IAS can be derived from the observed $\gamma$-$\gamma$cascade ($E_{\gamma 1}=2418.3(3)$ keV, $E_{\gamma 2}=142.3(1)$ keV Dossat et al. (2007)) from the IAS to the presumed $\beta$-decaying $2^{+}$ isomer in 52Co. However, a discrepancy was found between the IAS energies of 52Co derived from the proton and $\gamma$-decay data when tabulated mass values were applied for 52Co and 52Com in Ref. Dossat et al. (2007). The $\gamma$-$\gamma$-cascade to the isomeric state in 52Co resulted in an IAS about 600 keV higher than the proton data leading to the ground state of 51Fe. Therefore, it was proposed Dossat et al. (2007) that the ground state mass excess of 52Co might be too high in the Atomic Mass Evaluation Audi et al. (2003). With our direct mass measurements of 52Co and 52Com, we can now determine the excitation energy of the isomeric state, and therefore infer the excitation energy of the $T=2$ IAS.

The masses of 52Co and 52Fe discussed in this paper were measured in conjunction with a post-trap spectroscopy experiment dedicated to the study of proton radioactivity from the $19/2^{-}$ isomer in 53Co. It is the isomer from which the first observations of proton radioactivity were made about 45 years ago Jackson et al. (1970); Cerny et al. (1970, 1972). In this respect, the mass of 52Fe is important as when combined with the former, precise mass measurements of 53Co and 53Com Kankainen et al. (2010b), as it provides a precise, external calibration point for proton-decay spectroscopy.

The nuclei studied in this work are also relevant for studies of the astrophysical rapid proton capture ($rp$) process occurring, for example, in type I $X$-ray bursts Wallace and Woosley (1981); Schatz et al. (1998). The proton capture rates as well as their inverse photodisintegration reactions depend sensitively on the reaction $Q$ values Parikh et al. (2009). In particular, the ratios of 51Fe$(p,\gamma)^{52}$Co-52Co$(\gamma,p)^{51}$Fe and 52Co$(p,\gamma)^{53}$Ni-53Ni$(\gamma,p)^{52}$Co reactions affecting the route towards heavier elements have been studied with experimental $Q$ values for the first time.

II Experimental Method

The isotopes of interest were studied in two separate experiments at the Ion-Guide Isotope Separator On-Line (IGISOL) facility Moore et al. (2013): 52Co and 52Fe in April and 52Mn in August 2015. A 50-MeV proton beam from the K-130 cyclotron impinging into an enriched 1.8-mg/cm2-thick 54Fe target was used to produce 52Co and 52Fe via fusion-evaporation reactions, whereas for 52Mn a 40-MeV proton beam was applied. The reaction products were stopped in helium gas, extracted and guided towards the mass separator using a sextupole ion guide (SPIG) Karvonen et al. (2008) before acceleration to 30 kV. A good fraction of the ions are singly-charged, and the mass number $A=52$ could be selected using a $55^{\circ}$ dipole magnet. A gas-filled radio-frequency quadrupole cooler and buncher Nieminen et al. (2001) cooled the ions and converted the continuous beam into narrow ion bunches which were injected into the JYFLTRAP double-Penning-trap mass spectrometer Eronen et al. (2012). In the first trap, ions were cooled, centered and purified via a mass-selective buffer gas cooling technique Savard et al. (1991). The masses of ions with charge-to-mass ratio $q/m$ were measured in the second measurement trap by determining their cyclotron frequency $\nu_{c}=qB/(2\pi m)$ in a magnetic field strength $B$ via a time-of-flight ion cyclotron resonance (TOF-ICR) technique Gräff et al. (1980); König et al. (1995).

The measurements of the ions of interest were sandwiched by similar measurements of the reference ion 52Cr+, which were linearly interpolated to the time of the actual measurement of the ions of interest to determine the magnetic field strength. The atomic masses were derived from the cyclotron frequency ratio $r=\nu_{c,ref}/\nu_{c}$ between the reference ion 52Cr+ and the ion of interest via $m=r\cdot(m(^{52}$Cr$)-m_{e})+m_{e}$.

Ion-ion interactions were studied by performing count-rate class analysis Kellerbauer et al. (2003) for the determined frequencies, except for 52Co and 52Com, for which it was not necessary since most of the bunches contained only one ion. No significant differences were observed when the count-rate class analysed frequency ratios were compared with the results obtained by restricting the number of ions to one to five ions/bunch. Thus, limiting the number of ions to one to five ions/bunch is sufficient to avoid possible frequency-ratio shifts due to ion-ion interactions for the present uncertainty level.

The uncertainties due to temporal fluctuations in the magnetic field, $\delta B/B=8.18(19)\times 10^{-12}$/min$\times\Delta t$[min] Canete et al. (2016) were negligible compared with the statistical uncertainties of the measured frequency ratios. For mass doublets with the same $A/q$, the mass-dependent and systematic uncertainties resulting from field imperfections cancel in the frequency ratio Roux et al. (2013). The internal and external uncertainties of the measured frequency ratios Birge (1932) were compared and the ratio was found to be close to unity. The larger of the two values was used for the weighted mean of the frequency ratios.

For 52Co, the trap cycle was kept as short as possible due to the short half-lives of 52Co+ and 52Com+ (see Table 1). A single, 100-ms-long quadrupolar radiofrequency (RF) excitation applied in the second trap was sufficient to resolve the ground-state 52Co+ and isomeric-state 52Com+ ions as shown in Fig. 1. Although each 52Co measurement took around three hours, the uncertainty due to temporal fluctuations in the magnetic field was still much less than the statistical uncertainty of the frequency ratio ($\approx$1-2$\times 10^{-7}$).

For 52Fe+, 52Fem+, and 52Mn+, the following Ramsey excitation patterns Kretzschmar (2007); George et al. (2007) were applied: 25 ms (On) - 350 ms (Off) - 25 ms (On) for 52Fe+ and 52Fem+, and 25 ms (On) - 750 ms (Off) - 25 ms (On) for 52Mn+. The data for these nuclides were collected interleavedly Eronen et al. (2009): after one frequency scan for the reference ion, a few frequency scans were collected for the ions of interest. This pattern was repeated as long as required for sufficient statistics (typically for a few hours). Such interleaved scanning reduces the uncertainties due to time-dependent fluctuations in the magnetic field considerably. Because of the high excitation energy of the isomeric state 52Fem ($E_{x}=6958.0(4)$ keV Dong and Junde (2015)), it was possible to separate the ground and isomeric states already in the first trap and measure them separately in the second trap. Examples of TOF-ICR resonances for 52Fe+ and 52Fem+ are given in Fig. 2.

For the cyclotron frequency measurements of 52Mn, a Ramsey cleaning technique Eronen et al. (2008) was additionally applied to resolve the $6^{+}$ ground state and the $2^{+}$ isomeric state with excitation energy 377.749(5) keV Dong and Junde (2015). A dipolar excitation pulse with a Ramsey pattern 5 ms (On) - 25 ms (Off) - 5 ms (On) in the second trap excited the motion at the modified cyclotron frequency of unwanted, isomeric-state ions, but the ions of the 52Mn ground state were unaffected. Following this excitation step, only the ground-state 52Mn+ ions could pass through the 1.5-mm diaphragm back to the first trap for recooling and recentering before the actual mass measurement in the second trap.

III Results and Discussion

The results of the mass measurements are summarized in Table 2. Detailed discussion related to the masses of 52Co, 52Fe and 52Mn can be found from sections III.1, III.3, and III.5, respectively. In addition, the results for the excitation energies of the isomer 52Com and the $T=2$ IAS in 52Co are discussed in section III.2. The impact on the proton separation energy of 53Co is explored in section III.4. In section III.6, the IMME for the $T=2$ quintet at $A=52$ is studied in detail. Implications for the rapid proton capture process are briefly discussed in section III.7.

III.1 The masses of 52Co and 52Com

The mass of 52Co has never been measured before: only an extrapolated value was given for it in the Atomic Mass Evaluation 2012 (AME12) Wang et al. (2012). In this work, five cyclotron frequency ratios for the ground state and four ratios for the isomeric state were determined (see Fig. 3). The weighted means of the frequency ratios are $1.00043584(14)$ and $1.00044356(23)$ for 52Co+ and 52Com+, respectively. The first experimental mass-excess values for 52Co and 52Com, $-34331.6(66)$ keV and $-33958(11)$ keV, are 342(200) keV and 348(220) keV lower than the extrapolated values in the AME12, respectively. Thus, 52Co is more bound than predicted by the atomic mass evaluation. On the other hand, our experimental value for the 52Co ground state is significantly higher than the estimation based on mirror symmetry and known $\beta$-delayed proton data, $-34490(88)$ keV Tu et al. (2016).

III.2 Excitation energies for the isomer 52Com and the $T=2$ IAS in 52Co

Based on the ground and isomeric state masses measured in the experiment, an excitation energy of $E_{x}=374(13)$ keV was determined for the isomer 52Com. This agrees well with the extrapolated value in the NUBASE 2012 evaluation, $E_{x}=380(100)$# keV Audi et al. (2012a) which is simply taken from $E_{x}=377.749(5)$ keV Dong and Junde (2015) for the analog state in the mirror nucleus 52Mn.

The spins and parities for the lowest states in 52Co have not been experimentally verified. Thus, we performed large-scale shell-model calculations with the full $fp$ shell ($t=7$) to study the lowest levels in 52Co. The calculations were performed without isospin-symmetry breaking terms (ISB) for FPD6, GXPF1A and KB3G, as well as with ISB for KB3G (for details, see e.g. Ref. Rudolph et al. (2008)). All calculations are in line with a $6^{+}$ ground state and a $2^{+}$ first excited (isomeric) state (see Fig. 4). However, all models have problems with producing the observed energy split between the $6^{+}$ and $2^{+}$ states. On the other hand, the experimental mirror energy differences between the $2^{+}$ and $1^{+}$ states in 52Co and 52Mn determined in this work, $-4(13)$ keV and $-31(13)$ keV, respectively, are in good agreement with the KB3G calculations including ISB terms, which yield differences of zero keV and $-20$ keV, respectively.

The excitation energy of 52Com has important consequences for studying the $T=2$ isobaric multiplet at $A=52$. Namely, the excitation energy of the $T=2$ IAS in 52Co can be determined based on the $\gamma$-$\gamma$ cascade from the $T=2$ IAS to the isomer Dossat et al. (2007): $E_{IAS}(^{52}$Co$)=E_{\gamma 1}$(2418.3(3) keV)+$E_{\gamma 2}$(142.3(1) keV)+$E(^{52}$Co${}^{m})=2934(13)$ keV. However, a recent experiment performed at GANIL Orrigo et al. (2016) obtained a significantly different energy for the most intense gamma transition, $E_{\gamma 1}=2407(1)$ keV, and a slightly smaller energy for the second transition, $E_{\gamma 2}=141(1)$ keV. With these values, $E_{IAS}(^{52}$Co$)=2922(13)$ keV is obtained (see Fig. 5). For both cases, the $\gamma$-$\gamma$ cascade is taken to feed the $(2^{+})$ isomer, i.e. 52Com.

On the other hand, the energy for the $T=2$ IAS in 52Co can be determined based on the proton separation energy of 52Co and $\beta$-delayed protons observed from the IAS with center-of-mass energies of $E_{p,CM}=1349(10)$ keV Dossat et al. (2007) and $E_{p,CM}=1352(10)$ keV Orrigo et al. (2016). Our new ground-state mass for 52Co results in a proton separation energy $S_{p}(^{52}$Co$)=1418(11)$ keV, when the mass values for 51Fe and 1H are taken from Ref. Wang et al. (2012). Assuming the observed protons come from the IAS, the excitation energy should be $E_{IAS}(^{52}$Co$)=E_{p,CM}+S_{p}(^{52}$Co$)=2767(15)$ keV Dossat et al. (2007) or 2770(15) keV Orrigo et al. (2016). Thus, the obtained excitation energy is 167(20) keV Dossat et al. (2007) or 152(20) keV Orrigo et al. (2016) lower than that obtained via the $\gamma$-$\gamma$ cascade.

The difference of around $150$ keV between the excitation energies of the IAS is much smaller than the excitation energy of the first excited state in 51Fe, $E_{x}=253.5(5)$ keV Xiaolong (2006). A missed $\gamma$-transition of around 150 keV in 51Fe is also unlikely as it should have been observed in coincidence with the intense proton peaks. The discrepancy could be explained if the mass of 51Fe was around 150 keV off in AME12 Wang et al. (2012). However, it is known with a precision of 9 keV and is based on two independent measurements Mueller et al. (1977); Tu et al. (2011). Hence, the observed protons are most likely emitted from a state below the IAS (see Fig. 5).

Thus, we obtain $E_{IAS}$(52Co)=2934(13) keV Dossat et al. (2007) for the excitation energy of the IAS in 52Co, or $E_{IAS}$(52Co)=2922(13) keV, if more recent values from Orrigo et al. Orrigo et al. (2016) are used. Both values are much closer to the excitation energy of the $T=2$, $0^{+}$ mirror state in 52Mn, $E_{x}=2926.0(5)$ keV than what was obtained from the proton radioactivity data. The difference between Refs. Dossat et al. (2007) and Orrigo et al. (2016) is mainly due to the discrepancy between the observed $\gamma$-transition energies, 2418.3(3) keV Dossat et al. (2007) and 2407(1) keV Orrigo et al. (2016). There is no clear explanation for the difference and a new measurement of the $\gamma$-$\gamma$ cascade would be required to obtain a more accurate excitation energy for the IAS.

III.3 The masses of 52Fe and 52Fem

Altogether 36 cyclotron frequency ratios for the ground state 52Fe and seven ratios for the $12^{+}$ isomeric state 52Fem were determined (see Fig. 6). The weighted means of the frequency ratios, $1.0001464894(28)$ and $1.0002903590(56)$, yield atomic mass-excess values of $-48330.67(60)$ keV and $-41370.01(65)$ keV for the ground state and isomer, respectively. These are in good agreement with the literature values Wang et al. (2012). Previously, the ground-state mass of 52Fe has been mainly based on the $\beta^{+}$ decay of 52Fe and the 54Fe$(p,t)^{52}$Fe reaction $Q$ value (see Fig. 7). The excitation energy for the isomer, $E_{x}=6960.7(9)$ keV, determined in this work differs by 2.7(10) keV from the literature value $E_{x}=6958.0(4)$ keV Dong and Junde (2015) based on the observation of an $E4$ $\gamma$ transitions from this $12^{+}$ yrast trap in 52Fe Gadea et al. (2005). It should be noted that the relative uncertainties of the measured frequency ratios, $2.8\times 10^{-9}$ and $5.6\times 10^{-9}$ for 52Fe+ and 52Fem+, respectively, are much smaller than the relative uncertainty in the reference mass $\delta m/m(^{52}$Cr$)=1.2\times 10^{-8}$. Therefore, the precision in the determined mass-excess values could be further improved via a more precise measurement of the reference 52Cr.

III.4 Proton separation energy of 53Co and the energy of the protons emitted from 53Com

The mass of 52Fe is relevant for determining the proton separation energy of 53Co and, in particular, for the energy of the protons emitted from the $19/2^{-}$ high-spin isomer 53Com Jackson et al. (1970); Cerny et al. (1970, 1972). By combining the newly measured 52Fe mass-excess value with the earlier JYFLTRAP mass measurements of 53Co ($-42657.3(15)$ keV Kankainen et al. (2010b)) and 53Com ($-39482.9(16)$ keV Kankainen et al. (2010b)), a proton separation energy $S_{p}(^{53}$Co)=1615.6(16) keV and a center-of-mass energy $E_{p,CM}(^{53}$Com)=1558.8(17) keV for the protons from the high-spin isomer to the ground state of 52Fe is obtained. The proton separation energy is in a perfect agreement with the AME12 value $S_{p}(^{53}$Co)=1615(7) keV Wang et al. (2012) but about four times more precise. Our value for the energy of the protons emitted from 53Com is around 20 times more precise than obtained via proton-decay experiments $E_{p,LAB}=1530(40)$ keV Jackson et al. (1970), $E_{p,LAB}=1570(30)$ keV Cerny et al. (1970), and $E_{p,CM}=1590(30)$ keV Cerny et al. (1972)). Thus, we have demonstrated that Penning-trap measurements can provide precise calibration values for charged-particle spectroscopy.

III.5 The mass of 52Mn

A frequency ratio $r=1.0000973119(13)$ was derived as a weighted mean of 24 individual cyclotron frequency ratios for 52Mn+ (see Fig. 8). This yields an atomic mass-excess value of $-50709.97(59)$ keV for 52Mn. The JYFLTRAP value is 3(2) keV higher than the AME12 value ($-50706.9(19)$ keV Wang et al. (2012)) but three times more precise. The AME12 value is mainly based on a $Q$-value measurement of 54Fe$(d,\alpha)^{52}$Mn Jolivette et al. (1976) which gives a mass-excess value of $-50706.4(23)$ keV. In fact, 52Mn is an example of a nuclide close to stability whose mass has not been determined with modern techniques. The observed difference to the AME12 value shows that it is worthwhile to check such mass values which are based on measurements performed decades ago. Although our new value disagrees with Ref. Jolivette et al. (1976), it is in a rather good agreement with other, less precise experiments done on 52Mn (see Fig. 9). Of these, the $\beta^{+}$ decay of 52Mn Katoh et al. (1960) and the value based on the 52Cr$(^{3}$He$,t)$ reaction Becchetti et al. (1971) agree very well with the present value. A more precise mass value for the reference 52Cr would be beneficial for 52Mn as well since the relative uncertainty of the frequency ratio, $1.3\times 10^{-9}$, is nine times smaller than the relative uncertainty of the reference mass.

III.6 The IMME for the $T=2$ quintet at $A=52$

The IMME was studied for the $T=2$ quintet at $A=52$ using the mass values for 52Com, 52Fe, and 52Mn determined in this work together with the mass values of 52Cr and 52Ni adopted from AME12 Wang et al. (2012) as summarized in Table 3. Our mass-excess value for the isomer 52Com and the energies of the $\gamma$-$\gamma$ cascade observed from the IAS in Ref. Orrigo et al. (2016) (see Sect. III.2) were used for the mass excess of the IAS in 52Co. The excitation energy of the $T=2$ IAS in 52Fe, 8561(5) keV Dong and Junde (2015), is based on a study employing the 54Fe$(p,t)$ reaction Decowski et al. (1978), where a doublet of two $0^{+}$ levels separated by around 4 keV were observed. For 52Mn, the IAS at 2926.0(5) keV Dong and Junde (2015) has been identified in many experiments with the main contribution coming from a 52Cr$(p,n\gamma)$ study DelVecchio (1973) where 2379.5(5) keV $\gamma$-rays from the IAS to the $1^{+}$ state at 546.438(6) keV were observed.

The results for the error-weighted quadratic and cubic fits for the IMME are given in Table 4. The reduced $\chi^{2}$ of 2.4 for the quadratic fit is well above one. However, the cubic coefficient $d=6.0(32)$ keV is within the $\pm 3\sigma$ limit from zero and, thus, compatible with zero. We checked the quadratic fit also without 52Ni, which is based only on the extrapolation of the mass surface Wang et al. (2012). A slightly higher reduced $\chi^{2}$ value, $\chi^{2}/n=3.3$, and a cubic coefficient of $d=5.8(32)$ keV were obtained in this instance. If a quartic term $eT_{Z}^{4}$ is assumed instead of a cubic term, a coefficient $e=2.9(18)$ keV is obtained, again consistent with zero.

Previously, a non-zero coefficient $d=28.8(45)$ keV and unstable behavior of coefficient $c$ from quadratic to cubic fits were observed MacCormick and Audi (2014a). We can now confirm that these have been due to erroneous data used in the fits. Prior to this experiment, it was assumed that protons with $E_{p,CM}=1352(10)$ keV Orrigo et al. (2016) originate from the $T=2$ IAS in 52Co. Using the mass-excess values of 51Fe and 1H from AME12 Wang et al. (2012) together with the proton energy, a mass-excess value of $-31561(13)$ keV for the IAS in 52Co is obtained. This differs by 152(17) keV from the value determined in this work. For comparison, we performed similar IMME fits using the AME12 data for the ground-state masses and $-31564(13)$ keV for the IAS in 52Co. The reduced $\chi^{2}$ for the quadratic fit was $18.9$ and the cubic coefficient $d=29.3(48)$ keV. Figure 10 shows differences between the mass-excess values and the quadratic and cubic fits for the dataset determined in this work and with the AME12 values (assuming that the 1349-keV protons originate from the IAS). Clearly, a better agreement is achieved with our data. The fits of the new dataset also suggest that the mass-excess value for 52Ni might be higher, meaning it could be less bound than predicted in the AME12.

The cubic coefficients for the $T=3/2$ quartets and $T=2$ quintets have been plotted in Fig. 11. Earlier, a trend of increasing cubic coefficients after entering the $f_{7/2}$ shell has been observed. However, a recent observation of $\gamma$-rays from the IAS in 53Co following the $\beta$ decay of 53Ni showed that the IAS is lower than anticipated by $\beta$-delayed proton data from Ref. Dossat et al. (2007). With the new excitation energy, the cubic coefficient for the $A=53$ quartet is $d=5.4(46)$ keV Su et al. (2016), and thus does not suggest a breakdown in the IMME. In this work, we obtained a very similar cubic coefficient for the $T=2$ quintet at $A=52$, $d=6.0(32)$ keV, and confirmed that the intensive beta-delayed proton group observed in the beta-decay of 52Ni Orrigo et al. (2016); Dossat et al. (2007) does not originate from the IAS in 52Co but from a state below it. This is also understandable since the beta-delayed protons from the $T=2$ isobaric analogue state decaying to the ground state of 51Fe ($T=1/2$) are isospin-forbidden, and thus, the proton-branch from the IAS should be rather small as it is possible only via isospin mixing between the $T=1$ and $T=2$ states in 52Co or $T=1/2$ and $T=3/2$ states in 51Fe, respectively.

In conclusion, there is no evident change in the cubic coefficients after entering the $f_{7/2}$ shell. Although the coefficients are on the order of some keV, they are still compatible with zero within the $\pm 3\sigma$ limit. The mass determination of the most exotic member of the $T=2$ multiplet at $A=52$, 52Ni, would be crucial to provide a more stringent test of the IMME at the heavier mass region.

III.7 Implications for the rapid proton capture process

The $rp$ process proceeds along nuclei close to the $N=Z$ line mainly via proton captures and $\beta^{+}$ decays, resulting in a thermonuclear runaway and a sudden release in energy observed, for example, in type I X-ray bursts Wallace and Woosley (1981); Schatz et al. (1998). Proton-capture $Q$ values are essential for modeling the $rp$ process as they determine the path: the ratio of inverse photodisintegration reactions to the total proton-capture rate ($\lambda_{\gamma,p}/N_{A}\langle\sigma v\rangle$) depends exponentially on the proton-capture $Q$ values. Those can have a significant effect as demonstrated in Ref. Parikh et al. (2009).

With the new mass-excess value determined in this work for 52Co, proton-capture $Q$ values for 51Fe$(p,\gamma)^{52}$Co and 52Co$(p,\gamma)^{53}$Ni can be experimentally determined for the first time. The new values, $Q(^{51}$Fe$(p,\gamma)^{52}$Co)=1418(11) keV and $Q(^{52}$Co$(p,\gamma)^{53}$Ni)=2588(26) keV, differ significantly from the extrapolated values of 1077(196)# keV and 2930(197)# keV Wang et al. (2012), respectively. In other words, 52Co is around 340 keV more proton bound and 53Ni less proton bound than expected from the extrapolations of the mass surface in AME12.

In Fig. 12 we show the effect of the new $Q$ values on the photodisintegration versus proton capture rate ratio. With the new $Q$ values, the route via 52Co is more likely than before as 52Co is more proton-bound. For example, when the experimental $Q$ value is used instead of the extrapolated AME12 value, photodisintegration rates on 52Co are suppressed by a factor of around 50-3000 compared to the proton-capture rates on 51Fe at temperatures below 1 GK. Although more detailed $rp$-process calculations would be needed to find out the effect on the whole $rp$ process, the big change in the 52Co mass value significantly changes the calculations related to the proton captures and photodisintegration reactions involving 52Co.

IV Conclusions and Outlook

In this work, we have performed direct mass measurements of 52Co, 52Com, 52Fe, 52Fem, and 52Mn with the JYFLTRAP double Penning-trap mass spectrometer. The masses of 52Co and its isomer 52Com have been experimentally determined for the first time. The new mass value for 52Co is significantly lower than that obtained via extrapolations in the AME12 showing that it is more bound than expected. The obtained excitation energy of the isomer, $E_{x}=374(13)$ keV, is in good agreement with its analog state in 52Mn with $E_{x}=377.749(5)$ keV Dong and Junde (2015). Based on isospin symmetry and supported by shell-model calculations, we assume $I^{\pi}=2^{+}$ for the isomeric state.

An important consequence of the mass measurements of the 52Co ground and isomeric states is that the mass for the $T=2$ IAS can be determined using data from $\beta$ decay of 52Ni Dossat et al. (2007); Orrigo et al. (2016). We have found that the protons assumed to originate from the IAS in Ref. Dossat et al. (2007) must come from a state at around 2770(15) keV in 52Co, which is significantly lower than the excitation energy determined for the IAS in this work, $E_{x}=2922(13)$ keV, based on the observed $\gamma$-$\gamma$ cascade Dossat et al. (2007); Orrigo et al. (2016) from the IAS to the $(2^{+})$ isomeric state in 52Co. The new excitation energy for the IAS agrees well with the analogue state in the mirror nucleus 52Mn, 2926.0(5) keV. It is interesting that the IAS seems to decay only via $\gamma$ transitions since the proton decays are isospin-forbidden, whereas the state below it has a substantial proton branch but no observed $\gamma$ transitions. In the future, further experiments to confirm the state from which the observed protons come from are needed. In addition, the discrepancy between the measured $\gamma$-transition energies of 2418.3(3) keV Dossat et al. (2007) and 2407(1) keV Orrigo et al. (2016) should be studied to improve the accuracy of the $T=2$ IAS in 52Co.

The masses of 52Fe, 52Fem, and 52Mn have been determined with a much higher accuracy than in AME12. The precision in the 52Fe mass value has been improved by a factor of around twelve, which allows a precise determination of the proton separation energy of 53Co, $S_{p}(^{53}$Co)=1615.6(16) keV. In addition, the energy of the protons emitted from the high-spin isomer in 53Com to the ground state of 52Fe has been determined with unprecedented precision, $E_{p,CM}=1558.8(17)$ keV. Penning-trap measurements of 52Fe, 53Co, and 53Com at JYFLTRAP have therefore delivered an external calibration value for proton-decay experiments.

Whereas the masses of 52Fe and 52Fem agree well with AME12, 52Mn shows a deviation of $-3(2)$ keV. The observed deviation demonstrates the importance of measuring masses also closer to stability. Previous experiments may have been performed already several decades ago, and the mass accuracy can be improved considerably using Penning-trap mass spectrometry. It should be noted that a more accurate mass value for the reference 52Cr would improve the precision of the mass values for 52Fe, 52Fem, and 52Mn determined in this work. Presently, the mass of 52Cr is linked to 55Mn via 52Cr$(n,\gamma)^{53}$Cr$(n,\gamma)^{54}$Cr$(p,\gamma)^{55}$Mn Audi et al. (2012b), where 55Mn has been measured with respect to 85Rb at ISOLTRAP Naimi et al. (2012).

The first mass measurement of 52Co provides also first experimental proton separation energies for 52Co and 53Ni, 1418(11) keV and 2588(26) keV, respectively. These are also the proton-capture $Q$ values for the proton captures 51Fe$(p,\gamma)^{52}$Co and 52Co$(p,\gamma)^{53}$Ni, which affect $rp$-process calculations. Since 52Co has been found to be more bound than predicted in AME12, photodisintegration reactions on 52Co are not so dominant as previously predicted, thus making it more likely that the $rp$ process proceeds via 51Fe$(p,\gamma)^{52}$Co.

Finally, we have thoroughly studied the IMME for the $T=2$ quintet at $A=52$ using the new mass values determined in this work. The quadratic fit results in $\chi^{2}/n=2.4$, which corresponds to around 10 % probability that the quintet can be described with a parabola. However, the cubic coefficient, $d=6.0(32)$ keV does not support a breakdown in the IMME. The cubic coefficient is significantly lower than obtained in the previous IMME evaluation, $d=28.8(45)$ keV MacCormick and Audi (2014a), and close to the value recently determined for the $T=3/2$ quartet at $A=53$ Su et al. (2016). The new value does not suggest a trend of increasing cubic coefficients when entering the $f_{7/2}$ shell. In the future, a mass measurement of 52Ni would provide a possibility for a more stringent test of the IMME at $A=52$.

Acknowledgements.

This work has been supported by the Academy of Finland under the Finnish Centre of Excellence Programme 2012–2017 (Nuclear and Accelerator Based Physics Research at JYFL) and the Swedish Research Council (VR 2013-4271). A.K., D.N., and L.C. acknowledge support from the Academy of Finland under grant No. 275389.