The forbidden magnetic dipole (M1) and electric quadrupole (E2) transition properties of ions are of great importance in the diagnostic processes of the plasmas, due to the fact that the increment of populations in metastable states is caused by the low rates of collisional de-excitations in the upper states of the forbidden transitions. This is vital to infer the plasma temperature and dynamics by observing the transitions between the fine-structure levels of the ground state of HCIs in the laboratory tokamak.^[1,2] The intensities of the forbidden transitions are also used in the identification of the observed spectra, which are important to acquire the abundances of the atomic elements in astronomical objects.^[3,4] Since the spectral range is wide, the accurate calculations of the atomic parameters along the isoelectronic sequences are imperative. On the other hand, the HCIs are proposed as the candidates for making new high-precision atomic clocks with accuracy below 10⁻¹⁹, which will be useful in the applications on the time service, navigation, communication, fundamental physics, and so forth.^[5–9] The breakthrough of the cooling and trapping techniques for the HCIs has shed light on the realization of the new ultra-precise optical frequency standards.^[10] However, finding a suitable ion is still under investigation, which is compatible with the experimental feasibility, simplicity, and high index.^[11]

With the development of the theoretical methods in atomic physics and the capacity of the available computers, the transition properties in different isoelectronic sequences have been widely studied for the astrophysical and laboratory interests in recent years (see Refs. [12]–[18] and the references therein). For the K-like ions, Biémont et al. have calculated the energy levels and transition properties of the 3d configuration with .^[19] Similarly, Ali et al. have presented their results with in the framework of the Dirac–Fock single-configuration approximation in the late 1980’s.^[20] Charro et al. performed calculations on the M1 and E2 transition probabilities of the transition in the K-like sequences with by using the relativistic quantum defect orbital method.^[21] The K-like isoelectronic sequence ions, such as Zn¹¹⁺, Se¹⁵⁺, have already been produced and studied in the laboratory plasmas.^[20] The M1 transition rate of the transition in Kr¹⁷⁺ has been measured in two experiments using the electron beam ion trap (EBIT).^[22,23] This result was not reproducible by an ab initio calculation, but combining experimental energies with the calculated transition matrix elements, the results were matching with the measured values within the 1% level.^[23] In contrast to K and Ca⁺ of the K-like isoelectronic sequence, higher charged ions have the first two low-lying states as states. This fine-structure splitting becomes larger with increasing in atomic number (Z). Hence, these ions with the forbidden transitions of fine-structure splitting could serve as potential candidates for the HCI clocks. As has been demonstrated in Ref. [9], the monovalent (ns) (np) or divalent (np)² ions are suitable for clock transitions. The above fine structure splitting with longer lifetime of the upper state can be considered for atomic clocks. The obvious advantages are the low degeneracy and the simplicity of their structures, with respect to other candidates of the nf valence electrons.^{[6–8,24–28]} Following this idea, Yu et al. carried out the calculations on various systematics for the clock transitions in the Al-like ions with . These transitions were predicted to provide the quality factors .^[11] Therefore, it would be desirable to learn whether the K-like ions can be apt for the clock transition in the optical spectral region with higher quality factors.

In this work, we carry out the calculations on the M1 and E2 transition probabilities of the fine-structure splitting of the transition along the K-like sequence with employing the MCDHF method. We analyze contributions from the electron pair correlations and choose an appropriate correlation model for the calculations. Contributions from the Breit and the leading-order QED interactions are also estimated. The lifetimes, linewidth and quality factors are given.

In order to verify the validity of the computational model, we show the calculated excitation energy and forbidden transition probabilities of Kr¹⁷⁺ as the functions of the computational model in Table 1. Here, . Comparing the results obtained from DHF and MR-rSD calculations, and are not quite sensitive to the electron correlations, leading to changes less than 0.6% and 1.8% on the bases of the DHF results, respectively. Although is smaller than by about 5 orders of magnitude, considerations of the electron correlations make the changes about 17% on it. Comparing the results obtained from the 7rSD and MR-rSD calculations, it can be found that the higher-order correlations cause the relative changes of about 0.07% on the excitation energy, 0.2% on , 0.5% on in the Coulomb gauge and 0.2% on in the Babushkin gauge, respectively. The differences between the DHF/DHF+X, 7rSD/7rSD+X, and MR-rSD/MR-rSD+X (X = B, Q) results show that the contribution of the leading QED effects on the transition properties is as small as about 0.15% on the excitation energy, 0.5% on , 0.4% on , and 0.7% on , respectively. Contrary to the QED effects, the Breit interaction influences the results considerably on the excitation energy about 6%, 16% on , 31% on and 25% on . It is noted that the line strength of M1 transition, , is insensitive to the electron correlations, the Breit interaction and the leading QED effects. It implies that the reliability of can be determined by the accuracy of the calculated .

Table 1.

Table 1.

Table 1. Excitation energies in unit cm−1, M1 transition probabilities in unit s−1 with the corresponding line strength in unit a.u. and E2 transition probabilities in unit s−1 of the transition for Kr17+ as the functions of the computational model. The notations “C” and “B” represent the results of in the Coulomb and Babushkin gauge, respectively. represents the difference between the (B) and (C). The number in square brackets represents the power of 10. . Items (C) (B) DHF 16452.8 47.9607 2.3954 3.7079[−4] 3.6708[−4] 9.99[−3] DHF+Q 16477.4 48.1758 2.3954 3.7245[−4] 3.6983[−4] 7.04[−3] DHF+B 15527.5 40.3151 2.3954 3.1168[−4] 2.7483[−4] 1.18[−1] DHF+BQ 15552.1 40.5067 2.3954 3.1316[−4] 2.7701[−4] 1.15[−1] 4 16510.6 48.4677 2.3954 3.3941[−4] 3.0520[−4] 1.01[−1] 5 16514.4 48.5007 2.3954 3.1213[−4] 3.0608[−4] 1.94[−2] 6 16519.6 48.5471 2.3954 3.0244[−4] 3.0603[−4] 1.17[−2] 7 16521.6 48.5638 2.3954 3.0891[−4] 3.0623[−4] 8.66[−3] 7rSD 16539.9 48.7259 2.3954 3.0840[−4] 3.0821[−4] 6.36[−4] 7rSD+Q 16564.6 48.9442 2.3954 3.0968[−4] 3.1047[−4] 2.54[−3] 7rSD+B 15653.9 41.3075 2.3954 2.1407[−4] 2.3425[−4] 8.62[−2] 7rSD+BQ 15678.6 39.7743 2.2956 2.7789[−3] 2.6322[−4] 9.05[−1] MR-rSD 16550.7 48.8211 2.3954 3.0689[−4] 3.0768[−4] 2.57[−3] MR-rSD+Q 16575.4 49.0399 2.3954 3.0814[−4] 3.0994[−4] 5.80[−3] MR-rSD+B 15665.3 41.3981 2.3954 2.1304[−4] 2.3394[−4] 8.93[−2] MR-rSD+BQ 15690.1 41.5943 2.3954 2.1396[−4] 2.3576[−4] 9.25[−2] Experiments NIST[39] 15695a Träbert et al.[22] 44.1(2.0) Guise et al.[23] 40.8(5) Calculations Urrutia et al.[40] 15656.8 41.7, 42.0& Guise et al.[23] 15617.0d 41.4b, & 41.6c, & 41.02d, 41.6d, & Träbert et al.[22] 41.89 2.60[−4] Charro et al.[21] 15969.6 43.9, 41.7& 2.21[−4] Ali et al.[20] 15575.6 40.7, 41.6& 2.79[−4] Biémont et al.[19] 15703.0 41.9, 41.9& 2.93[−4] Ali et al.[37] 15671e a the semiempirical result by Kaufman et al.;[38]. b FAC result. c lowest-order RMBPT result. d GRASP2K result. e the result in the multi-configuration approximation. & the adjusted results presented in Ref. [23].

Table 1. Excitation energies in unit cm−1, M1 transition probabilities in unit s−1 with the corresponding line strength in unit a.u. and E2 transition probabilities in unit s−1 of the transition for Kr17+ as the functions of the computational model. The notations “C” and “B” represent the results of in the Coulomb and Babushkin gauge, respectively. represents the difference between the (B) and (C). The number in square brackets represents the power of 10. .

The calculated excitation energies of Kr¹⁷⁺ in Table 1 using 7rSD+BQ and MR-rSD+BQ models differ from the NIST value^[39] by about 0.1% and 0.03%. It is implied that, although some of the first-order correlations are neglected, the 7rSD+BQ model can still produce accurate results. The MR-rSD+BQ model can effectively capture the main contributions of the higher-order correlations. As for , our result is in agreement with the recent experimental values by Guise et al.^[23] at the level of 1%, and almost equals the adjusted values in their work. It is reasonable that the adjusted values of are extrapolated by using the experimental energy values,^[38] while our calculated excitation energy has a good agreement with it. In Table 1, the results of other theoretical calculations are listed. Our results on the excitation energy agree with other calculations, with the differences less than 1%. The transition rates and are also consistent with other calculations.

The results for the excitation energies and the forbidden transition probabilities of the transition in the K-like ions with are presented in tables 2 and 3. Comparing the results of different computational models, it can be found that, among the contributions to the transition properties, the Breit interaction is the most important one. The influences of the electron correlations to the excitation energies are smaller than the Breit interaction, but it is still significant. The leading-order QED effects are quite small. It is shown that the excitation energies obtained using the MR-rSD+BQ model are in good agreement with the NIST data^[39] and the differences are within 0.4%. The relative contribution of QED effects is around 0.15% to the excitation energies. The transition rates and increase rapidly along the sequence, and the M1 transition is the dominant one. It can be said that for all the ions considered in this work are around 2.40 a.u. and they are insensitive to the electron correlations, Breit interaction and the leading QED effects. The electron correlations have more influences on than .

Table 2.

Table 2.

Table 2. Excitation energies in unit cm−1 and the M1 transition probabilities in unit s−1 of the transition for the K-like ions with . For convenience, the notations “D” and “M” are used to represent the models “DHF” and “MR-rSD”, respectively. The number in square brackets represents the power of 10. . Items Fe7+ Co8+ Ni9+ Cu10+ Zn11+ Ga12+ Ge13+ As14+ Se15+ Br16+ Kr17+ D 1975.6 2621.8 3399.3 4324.1 5413.1 6684.0 8155.7 9847.5 11779.9 13974.3 16452.8 D + Q 1978.0 2625.1 3403.7 4329.8 5420.4 6693.3 8167.2 9861.6 11797.0 13994.9 16477.4 D + B 1793.9 2396.1 3123.7 3992.5 5019.0 6220.7 7615.8 9223.5 11063.9 13158.0 15527.5 D + BQ 1796.4 2399.4 3128.1 3998.2 5026.4 6229.9 7627.3 9237.6 11081.0 13178.5 15552.1 M + B 1829.9 2436.9 3171.2 4047.3 5082.1 6293.2 7698.8 9318.2 11171.6 13280.0 15665.3 M + BQ 1832.5 2440.3 3175.7 4053.2 5089.6 6302.5 7710.4 9332.5 11188.9 13300.8 15690.1 NIST[39] 1836 2451 3178 4057 5095 6308 7714 9337a 11193a 13306a 15696 D 8.3141[−2] 1.9431[−1] 4.2344[−1] 8.7148[−1] 1.7095 3.2181 5.8452 10.2882 17.6086 29.3915 47.9607 D + Q 8.3449[−2] 1.9505[−1] 4.2509[−1] 8.7495[−1] 1.7164 3.2314 5.8699 10.3324 17.6855 29.5216 48.1758 D + B 6.2249[−2] 1.4832[−1] 3.2858[−1] 6.8599[−1] 1.36276 2.5941 4.7595 8.4537 14.5889 24.5356 40.3151 D + BQ 6.2503[−2] 1.4893[−1] 3.2998[−1] 6.8895[−1] 1.3686 2.6057 4.9391 8.4925 14.6567 24.6510 40.5067 M + B 6.6075[−2] 1.5602[−1] 3.4281[−1] 7.1460[−1] 1.4147 2.6859 4.9168 8.7168 15.0191 25.2246 41.3981 M + BQ 6.6348[−2] 1.5667[−1] 3.4528[−1] 7.1774[−1] 1.4210 2.6979 4.7811 8.7568 15.0888 25.3430 41.5943 Ref. [19] 6.70[−2] 1.60[−1] 3.48[−1] 7.25[−1] 1.43 2.72 4.98 8.83 15.2 25.6 41.9 Ref. [20] 6.67[−2] 1.59[−1] 3.46[−1] 6.95[−1] 1.38 2.62 4.81 8.54 14.7 24.8 40.7 Ref. [21] 9.87[−2] 2.09[−1] 4.25[−1] 8.36[−1] 1.59 2.95 5.32 9.35 16.0 26.8 43.9 a Kaufman et al.[38].

Table 2. Excitation energies in unit cm−1 and the M1 transition probabilities in unit s−1 of the transition for the K-like ions with . For convenience, the notations “D” and “M” are used to represent the models “DHF” and “MR-rSD”, respectively. The number in square brackets represents the power of 10. .

Table 3.

Table 3.

Table 3. The E2 transition probabilities in unit s−1 of the transition for the K-like ions with . The same notations “D” and “M” in Table 2 are adopted. is the relative difference between and in the model “M+BQ”. The number in square brackets represents the power of 10. . Items Fe7+ Co8+ Ni9+ Cu10+ Zn11+ Ga12+ Ge13+ As14+ Se15+ Br16+ Kr17+ (C) D 1.1320[−7] 3.2541[−7] 8.6509[−7] 2.1545[−6] 5.0754[−6] 1.1386[−5] 2.4479[−5] 5.0665[−5] 1.0135[−4] 1.9658[−4] 3.7079[−4] D + Q 1.1362[−7] 3.2665[−7] 8.6846[−7] 2.1631[−6] 5.0961[−6] 1.1433[−5] 2.4583[−5] 5.0883[−5] 1.0179[−4] 1.9745[−4] 3.7245[−4] D + B 8.4753[−8] 2.4839[−7] 6.7129[−7] 1.6960[−6] 4.0459[−6] 9.1786[−6] 1.9933[−5] 4.1631[−5] 8.3968[−5] 1.6411[−4] 3.1168[−4] D + BQ 8.5099[−8] 2.4942[−7] 6.7414[−7] 1.7033[−6] 4.0635[−6] 9.2195[−6] 1.2948[−5] 4.1822[−5] 8.4358[−5] 1.6488[−4] 3.1316[−4] M + B 4.6614[−8] 1.3963[−7] 4.0477[−7] 1.0418[−6] 2.5307[−6] 5.8414[−6] 1.2892[−5] 2.7331[−5] 5.5925[−5] 1.1078[−4] 2.1304[−4] M + BQ 4.6815[−8] 1.4024[−7] 4.0651[−7] 1.0463[−6] 2.5416[−6] 5.8666[−6] 2.0023[−5] 2.7448[−5] 5.6166[−5] 1.1126[−4] 2.1396[−4] (B) D 1.0162[−7] 2.9970[−7] 8.1173[−7] 2.0498[−6] 4.8788[−6] 1.1036[−5] 2.3876[−5] 4.9665[−5] 9.9747[−5] 1.9411[−4] 3.6708[−4] D + Q 1.0225[−7] 3.0160[−7] 8.1701[−7] 2.0634[−6] 4.9120[−6] 1.1112[−5] 2.4045[−5] 5.0022[−5] 1.00474[−4] 1.9554[−4] 3.6983[−4] D + B 6.2735[−8] 1.9107[−7] 5.3190[−7] 1.3756[−6] 3.3435[−6] 7.7054[−6] 1.6953[−5] 3.5802[−5] 7.2901[−5] 1.4367[−4] 2.7483[−4] D + BQ 6.3163[−8] 1.9240[−7] 5.3567[−7] 1.3855[−6] 3.3680[−6] 7.7628[−6] 1.4538[−5] 3.6076[−5] 7.3466[−5] 1.4479[−4] 2.7701[−4] M + B 5.5923[−8] 1.6728[−7] 4.6089[−7] 1.1830[−6] 2.8609[−6] 6.5718[−6] 1.4431[−5] 3.0446[−5] 6.1983[−5] 1.2219[−4] 2.3394[−4] M + BQ 5.6304[−8] 1.6844[−7] 4.6414[−7] 1.1914[−6] 2.8817[−6] 6.6201[−6] 1.7081[−5] 3.0676[−5] 6.2455[−5] 1.2313[−4] 2.3576[−4] 1.69[−1] 1.67[−1] 1.24[−1] 1.22[−1] 1.18[−1] 1.14[−1] 1.47[−1] 1.05[−1] 1.01[−1] 9.64[−2] 9.25[−2] Ref. [19] 7.09[−8] 2.16[−7] 5.84[−7] 1.51[−6] 3.64[−6] 8.34[−6] 1.83[−5] 3.84[−5] 7.80[−5] 1.53[−4] 2.93[−4] Ref. [20] 7.04[−8] 2.14[−7] 5.80[−7] 1.41[−6] 3.41[−6] 7.85[−6] 1.73[−5] 3.69[−5] 7.41[−5] 1.46[−4] 2.79[−4] Ref. [21] 7.04[−8] 2.05[−7] 5.34[−7] 1.34[−6] 2.74[−6] 6.74[−6] 1.45[−5] 3.04[−5] 6.05[−5] 1.17[−4] 2.21[−4]

Table 3. The E2 transition probabilities in unit s−1 of the transition for the K-like ions with . The same notations “D” and “M” in Table 2 are adopted. is the relative difference between and in the model “M+BQ”. The number in square brackets represents the power of 10. .

The transition properties for the ions are summarized in Table 4. Here, the linewidth , , and ν is the transition frequency. Since the contribution of the E2 channel to the total transition rate A is quite small, comparing to the M1 channel, we neglect it in the estimation of the lifetime of the upper states. One can see that A increases rapidly along the Z, so the lifetimes (τ) of the state decrease nearly three orders of magnitude from Fe⁷⁺ to Kr¹⁷⁺, and the transition linewidths (Γ) increase with the same orders. According to the definition of Q, we calculated the values and found that it decreases slowly along the sequence. The Q values of all ions are larger than . For the promising candidates of HCI clocks, it would be better that the clock transition locates in the optical regime with a high Q value. Considering the manipulation process of the atomic clock experiments, the longer lifetime of the upper state of the clock transition is preferable. Although the fine-structure splitting in the ²D state of Kr¹⁷⁺ is 637 nm, which is in the optical region, the lifetime of the state of it is only 24 ms, which is not good enough for the clock experiment. The state of Fe⁷⁺ has the long lifetime as 15 s and the Q value is also large, but the transition wavelength is 5457 nm, which is not easy to build proper lasers for experiments. As the practical and balanced choice, Cu¹⁰⁺ and Zn¹¹⁺ are recommended. The transitions of both ions are in the near infrared region and the lasers could be available. The lifetimes of the state are about 1 s, which is long enough for experiments.

Table 4.

Table 4.

Table 4. The wavelength λ, total transition probabilities A, lifetimes τ, linewidth Γ, and quality factor Q of the transition for the K-like ions with . The number in square brackets represents the power of 10. . Items λ/nm A/s−1 τ/s Γ/Hz Q Fe7+ 5457.15 0.07 15.07 0.0106 5.20[15] Co8+ 4097.91 0.16 6.38 0.0249 2.93[15] Ni9+ 3148.87 0.35 2.90 0.0550 1.73[15] Cu10+ 2467.20 0.72 1.39 0.1142 1.06[15] Zn11+ 1964.79 1.42 0.70 0.2262 6.75[14] Ga12+ 1586.66 2.70 0.37 0.4294 4.40[14] Ge13+ 1296.95 4.78 0.20 0.7861 2.94[14] As14+ 1071.53 8.76 0.11 1.3937 2.00[14] Se15+ 893.74 15.09 0.07 2.4015 1.40[14] Br16+ 751.83 25.34 0.04 4.0335 9.89[13] Kr17+ 637.34 41.59 0.02 6.6199 7.11[13]

Table 4. The wavelength λ, total transition probabilities A, lifetimes τ, linewidth Γ, and quality factor Q of the transition for the K-like ions with . The number in square brackets represents the power of 10. .

For the interests of the plasma diagnostic processes and the candidates of the HCI clocks, we have investigated the M1 and E2 transition properties of the fine-structure splitting in ²D state for the K-like ions for with the MCDHF method. The results of the excitation energies have good agreements with NIST data and the differences are within 0.4%. The contributions of the electron correlations, Breit interaction, and the leading QED effects are analyzed. It is shown that the good treatment of electron correlations and inclusion of the Breit interaction are essential for the reliable prediction of the transition properties for these ions. In choosing candidates for HCI clocks from the K-like ions in , Cu¹⁰⁺ and Zn¹¹⁺ are recommended for the balanced consideration of the Q factor, the lifetime of the clock state and the feasibility of the lasers.