⇦Chinese Physics Letters, 2019, Vol. 36, No. 11, Article code 113101 Parameters of Isotope Shifts for $2s2p\,{}^{3,1}\!P_{1} \to 2s^{2}\,{}^{1}\!S_0$ Transitions in Heavy Be-Like Ions * Xiang Zhang (张祥)1, Jian-Peng Liu (刘建鹏)1, Ji-Guang Li (李冀光)2**, Hong-Xin Zou (邹宏新)1** Affiliations 1College of Liberal Arts and Science, National University of Defense Technology, Changsha 410073 2Institute of Applied Physics and Computational Mathematics, Beijing 100088 Received 1 July 2019, online 21 October 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 11874090, 11604385, 91536106 and 11204374, and the Research Project of National University of Defense Technology under Grant No ZK17-03-11.
**Corresponding author. Email: li_jiguang@iapcm.ac.cn; hxzou@nudt.edu.cn Citation Text: Zhang X, Liu J P, Li J G and Zou H X 2019 Chin. Phys. Lett. 36 113101    Abstract The field shift and mass shift parameters of the 2$s2p\,{}^{3,1}\!P_{1}\to 2s^{2}\,{}^{1}\!S_0$ transitions in Be-like ions ($70 \le Z \le 92$) are calculated using the multi-configuration Dirac–Hartree–Fock and the relativistic configuration interaction methods with the inclusion of the Breit interaction and the leading QED corrections. We find that the mass shift parameters of these two transitions do not change monotonously along the isoelectronic sequence in the high-$Z$ range due to the relativistic nuclear recoil effects. A minimum value exists for the specific mass shift parameters around $Z=80$, especially for the 2$s2p\,{}^{3}\!P_{1}\to 2s^{2}\,{}^{1}\!S_0$ transition. In addition, the field shifts and mass shifts of these two transitions are estimated using an empirical formula, and their contributions are compared along the isoelectronic sequence. DOI:10.1088/0256-307X/36/11/113101 PACS:31.15.V-, 31.15.am, 31.30.J-, 31.30.Gs © 2019 Chinese Physics Society Article Text The atomic isotope shift is often used as a sensor of nuclear properties. For example, it provides essential information to understand anomalies in nuclei, such as odd-even staggering of the nuclear charge radii along the heavy isotopic chain.[1–3] By combining the measurements of isotope shifts with the calculations, we are able to extract model-independent nuclear radii.[4–7] To analyze high-resolution solar and stellar spectra and understand nucleosynthesis mechanisms,[8–10] it is also necessary to include isotope shifts as well as hyperfine structures in modelling of spectra.[11,12] It is well-known that the isotope shift is composed of the field shift (FS) and mass shift (MS). In theory, the mass shift parameters for highly charged ions, despite of making a tiny contribution to the isotope shift, are not negligible. Meanwhile, the effects of electron correlations should be taken into account for achieving high accuracy. Many works have been carried out to analyze the isotope shift parameters of the low-lying states in highly charged ions such as H-like,[13–15] He-like,[16] Li-like[14–18] and B-like[19] ions in the whole range along the isoelectronic sequences. It was found that the relativistic nuclear recoil corrections make significant contributions to the mass shifts for heavy elements.[16,20] Li et al.[17] recently reported the isotope shifts for the 2$s$–2$p$ resonance doublet of the lithium-like ions using the multi-configuration Dirac–Hartree–Fock (MCDHF) method. To our knowledge, this was the first time that the relativistic nuclear recoil corrections were taken into account in the MCDHF framework, and that the effects of electron correlations on the relativistic mass shifts of the 2$s$–2$p$ resonance transitions in Li-like ions were investigated in detail. Later, Nazé et al.[21] estimated the isotope shift parameters of Be-like ions with $5 \le Z \le 74$ using the same method as Li et al.;[17] however, the results in the high-$Z$ range ($Z > 74$) have not yet been given. Consequently, we calculate the isotope shift parameters for the 2$s2p\,{}^{3,1}\!P_{1} \to 2s^{2}\,{}^{1}\!S$ transitions of Be-like ions in the range from $Z=70$ to $Z=92$ in the framework of the MCDHF method. The Breit interaction and the leading QED effects are also included. In practice, the GRASP2K package[22] and the RIS4 module[23] are employed for these calculations. According to the MCDHF method, the atomic state wave function (ASF) is a linear combination of configuration state wave functions (CSFs) with the same parity $P$, the total angular momentum $J$, and its component along the $z$-direction $M_{J}$. The ASF can be expressed as $$\begin{align} {\it \Psi} ({\gamma PJM_{J}})=\sum\limits_{i=1}^N {c_{i} {\it \Phi} ({\gamma_{i} PJM_{J}})},~~ \tag {1} \end{align} $$ where $c_{i}$ is the mixing coefficient, and $\gamma_{i}$ represents the additional quantum number to uniquely define the state. The CSFs are constructed as linear combinations of Slater determinants, each of which is a product of one-electron Dirac orbitals. These one-electron Dirac orbitals and mixing coefficients $c_{i}$ are optimized in the self-consistent field (SCF) procedure by applying the variation principle. The MCDHF calculations are followed by the relativistic configuration interaction (RCI) calculations, in which the Breit interaction and the leading QED effects are included. The isotope shift is divided into the mass shift and field shift. For a spectral line $k$ between two isotopes $A$ and $A'$ with masses $M$ and $M'$, respectively, the isotope shift can be given by $$\begin{alignat}{1} \delta \nu_{k}^{A,{A}'} \approx \Big({\frac{{M}'-M}{M{M}'}}\Big)\frac{\Delta K_{k}}{h}+\Delta F_{k} \delta \langle {r^{2}} \rangle^{A,{A}'},~~ \tag {2} \end{alignat} $$ where $\Delta K_{k} (=K_{\rm u} -K_{\rm l})$ is the MS parameter between the upper (u) and lower (l) levels involved in the line $k$, $\Delta F_{k} (=F_{\rm u} -F_{\rm l})$ is the FS parameter of the corresponding line $k$, and $\delta \langle {r^{2}} \rangle^{A,{A}'}$ is the difference of the nuclear mean-square charge radii. The MS operator including the lowest relativistic correction in order of $m/M$ is expressed as[24–26] $$\begin{align} H_{{\rm MS}} =\,&\frac{1}{2M}\sum\limits_{i,j} \Big[ \boldsymbol{p}_{i} \cdot \boldsymbol{p}_{j} -\frac{\alpha Z}{r_{i}}\Big(\boldsymbol{\alpha}_{i} \\ &+\frac{({\boldsymbol{\alpha}_{i} \cdot \boldsymbol{r}_{i}})\boldsymbol{r}_{i}}{r_{i}^{2}}\Big)\cdot \boldsymbol{p}_{j}\Big],~~ \tag {3} \end{align} $$ where $\boldsymbol{\alpha}_{i}$ represents a vector incorporating the Dirac matrices. The MS operator in Eq. (3) can be further divided into the normal mass shift (NMS) and the specific mass shift (SMS) operator, corresponding to the one-body ($i=j$) and two-body terms ($i\neq j$), respectively, $$\begin{align} H_{{\rm MS}} =\,&H_{{\rm NMS}} +H_{{\rm SMS}}, \\ H_{{\rm NMS}} =\,&\frac{1}{2M}\sum\limits_i \Big[ \boldsymbol{p}_{i} \cdot \boldsymbol{p}_{i} -\frac{\alpha Z}{r_{i}}\boldsymbol{\alpha}_{i} \cdot \boldsymbol{p}_{i} \\ &-\frac{\alpha Z}{r_{i}}\frac{({\boldsymbol{\alpha}_{i} \cdot \boldsymbol{r}_{i}})\boldsymbol{r}_{i}}{r_{i}^{2}}\cdot \boldsymbol{p}_{i} \Big], \\ H_{{\rm SMS}} =\,&\frac{1}{2M}\sum\limits_{i\ne j} \Big[ \boldsymbol{p}_{i} \cdot \boldsymbol{p}_{j} -\frac{\alpha Z}{r_{i}}\boldsymbol{\alpha}_{i} \cdot \boldsymbol{p}_{j} \\ &-\frac{\alpha Z}{r_{i}}\frac{({\boldsymbol{\alpha}_{i} \cdot \boldsymbol{r}_{i}})\boldsymbol{r}_{i}}{r_{i}^{2}}\cdot \boldsymbol{p}_{j}\Big].~~ \tag {4} \end{align} $$ Hence, the MS parameter in Eq. (2) is written, in terms of the expectation value of the MS Hamiltonian for the state $i$, as $$\begin{align} \tilde{{K}}_{i} =\frac{M}{h}\langle {{\it \Psi}_{i}|{H_{{\rm MS}}}|{\it \Psi}_{i}} \rangle.~~ \tag {5} \end{align} $$ Considering the expressions of the Hamiltonians in Eq. (4), the mass shift parameters can be decomposed into three corresponding parts; i.e., $$\begin{align} \tilde{{K}}_{{\rm NMS}} =\,&\tilde{{K}}_{{\rm NMS}}^{(1)} +\tilde{{K}}_{{\rm NMS}}^{(2)} +\tilde{{K}}_{{\rm NMS}}^{(3)}, \\ \tilde{{K}}_{{\rm SMS}} =\,&\tilde{{K}}_{{\rm SMS}}^{(1)} +\tilde{{K}}_{{\rm SMS}}^{(2)} +\tilde{{K}}_{{\rm SMS}}^{(3)},~~ \tag {6} \end{align} $$ where the $\tilde{{K}}^{(1)}$ term is generally referred to as the non-relativistic contribution and the sum of the last two terms, written as $\tilde{{K}}^{(2)+(3)}$, as the lowest-order relativistic correction. The FS parameter for the state $i$ in Eq. (2) reads[27,28] $$\begin{align} F_{i} =\frac{2\pi}{3h}Z\Big({\frac{e^{2}}{4\pi \varepsilon_{0}}}\Big)|{{\it \Psi} (0)}|_{i}^{2}~~ \tag {7} \end{align} $$ with the total probability density $|{{\it \Psi} (0)}|^{2}$ of electrons at the origin. Hence, the FS parameter in line $k$ between two isotopes $A$ and $A'$ can be expressed as $$\begin{align} \delta \nu_{k,{\rm FS}}^{A,{A}'} \approx\,&\Delta F_{k} \delta \langle {r^{2}} \rangle^{A,{A}'}\\ =\,&\frac{Z}{3\hslash}\Big({\frac{e^{2}}{4\pi \varepsilon_{0}}}\Big)\Delta| {{\it \Psi} (0)}|_{k}^{2} \delta \langle {r^{2}} \rangle^{A,{A}'}.~~ \tag {8} \end{align} $$
Table 1. MS parameters (in GHz u) and FS parameters (in GHz/fm$^{2})$ of the 2$s2p\,{}^{3,1}\!P_{1}\to 2s^{2}\,{}^{1}\!S_{0}$ transitions in Yb$^{66+}$ in different computational models.
Model $\Delta \tilde{K}_{\rm NMS}^{(1)}$ $\Delta \tilde{K}_{\rm NMS}^{(2)+(3)}$ $\Delta \tilde{{K}}_{{\rm NMS}}$ $\Delta \tilde{{K}}_{{\rm SMS}}^{(1)}$ $\Delta \tilde{{K}}_{{\rm SMS}}^{(2)+(3)}$ $\Delta \tilde{{K}}_{{\rm SMS}}$ $\Delta F$
2$s2p\,{}^{3}\!P_{1}\to 2s^{2}\,{}^{1}\!S_{0}$
DHF $-$7146$\times$10$^2$ 7171$\times$10$^2$ 2554 $-$1866$\times$10$^3$ 8473$\times$10$^2$ $-$1018$\times$10$^3$ $-$2144$\times$10$^1$
$3s3p3d$ $-$7037$\times$10$^2$ 6993$\times$10$^2$ $-$4414 $-$1782$\times$10$^3$ 8147$\times$10$^2$ $-$9672$\times$10$^2$ $-$2086$\times$10$^1$
$5s5p5d5f5g$ $-$7055$\times$10$^2$ 6996$\times$10$^2$ $-$5874 $-$1796$\times$10$^3$ 8182$\times$10$^2$ $-$9777$\times$10$^2$ $-$2086$\times$10$^1$
SD $-$7055$\times$10$^2$ 6997$\times$10$^2$ $-$5836 $-$1798$\times$10$^3$ 8200$\times$10$^2$ $-$9777$\times$10$^2$ $-$2086$\times$10$^1$
B $-$7035$\times$10$^2$ 6959$\times$10$^2$ $-$7603 $-$1803$\times$10$^3$ 8252$\times$10$^2$ $-$9776$\times$10$^2$ $-$2071$\times$10$^1$
B$+$Q $-$7069$\times$10$^2$ 6972$\times$10$^2$ $-$9680 $-$1795$\times$10$^3$ 8218$\times$10$^2$ $-$9729$\times$10$^2$ $-$2072$\times$10$^1$
Ref. [21] $-$9855 $-$9780$\times$10$^2$ $-$2073$\times$10$^1$
2$s2p\,{}^{1}\!P_{1}\to 2s^{2}\,{}^{1}\!S_{0}$
DHF $-$1347$\times$10$^3$ 1201$\times$10$^3$ $-$1458$\times$10$^2$ $-$1376$\times$10$^3$ 2335$\times$10$^2$ $-$1143$\times$10$^3$ $-$2267$\times$10$^1$
$3s3p3d$ $-$1327$\times$10$^3$ 1179$\times$10$^3$ $-$1475$\times$10$^2$ $-$1296$\times$10$^3$ 2027$\times$10$^2$ $-$1093$\times$10$^3$ $-$2200$\times$10$^1$
$5s5p5d5f5g$ $-$1331$\times$10$^3$ 1181$\times$10$^3$ $-$1508$\times$10$^2$ $-$1311$\times$10$^3$ 2069$\times$10$^2$ $-$1172$\times$10$^3$ $-$2203$\times$10$^1$
SD $-$1331$\times$10$^3$ 1181$\times$10$^3$ $-$1507$\times$10$^2$ $-$1312$\times$10$^3$ 2072$\times$10$^2$ $-$1114$\times$10$^3$ $-$2203$\times$10$^1$
B $-$1322$\times$10$^3$ 1173$\times$10$^3$ $-$1490$\times$10$^2$ $-$1325$\times$10$^3$ 2169$\times$10$^2$ $-$1109$\times$10$^3$ $-$2188$\times$10$^1$
B$+$Q $-$1326$\times$10$^3$ 1174$\times$10$^3$ $-$1512$\times$10$^2$ $-$1317$\times$10$^3$ 2133$\times$10$^2$ $-$1104$\times$10$^3$ $-$2189$\times$10$^1$
Ref. [21] $-$1513$\times$10$^2$ $-$1109$\times$10$^3$ $-$2190$\times$10$^1$
For the Be-like ions ($70 \le Z \le 92 $) in this work, the reference configurations are 2$s^{2}$ for the ground and 2$s2p$ for the excited states, respectively. The active space approach is applied to build the ASFs.[29] The 1$s$ orbital is treated as the core orbital, and the others are valence orbitals. The orbitals occupied in the reference configurations are optimized as spectroscopic orbitals. We start from the Dirac–Hartree–Fock (DHF) approximation, and the wavefunctions of the 2$s^{2}\,{}^{1}\!S$ and the 2$s2p\,{}^{3,1}\!P_{1}$ states are determined separately. In self-consistent field procedure, the CSFs are generated by single and restricted double (SrD) substitutions from the reference configurations; i.e., only a single substitution from 1$s$ orbital is allowed. The active set is enlarged by virtual orbitals layer by layer until the physical quantities under investigation converge. Each layer is labeled as $nl$, where $n$ represents the principal quantum number, although no physical meaning for virtual orbitals, and $l$ stands for the orbital angular momentum quantum number; i.e., $s$, $p$, $d$, and so on. The MCDHF calculations are followed by the RCI computations. The CSFs included in the RCI calculations, labeled as SD, are generated by single and double (SD) excitations from the reference configurations to the largest active set with $n \le 6$, and the results considering the Breit interaction and the leading QED effects are labeled as B and Q, respectively. To check the reliability of our computational model, the isotope shift parameters of the 2$s2p\,{}^{3,1}\!P_{1}\to 2s^{2}\,{}^{1}\!S_0$ transitions in Yb$^{66+}$ ions are given in Table 1. Our results are in reasonable agreement with Ref. [21]. The difference is controlled within around 2% and 0.05% for the MS and FS parameters in these two transitions, respectively. The relativistic and the non-relativistic terms of the mass shifts have the same order of magnitude but different signs, making the cancellation in $\Delta \tilde{{K}}_{{\rm MS}}$ to a large extent.
Table 2. NMS, SMS parameters (in GHz u) and FS parameters (in GHz/fm$^{2})$ of the 2$s2p\,{}^{3,1}\!P_{1}\to 2s^{2}\,{}^{1}\!S_0$ transitions of Be-like ions from $Z=70$ to $Z=92$ in B+Q computational model.
$Z$ 2$s2p\,{}^{3}\!P_{1}\to 2s^{2}\,{}^{1}\!S_0$ 2$s2p\,{}^{1}\!P_{1}\to 2s^{2}\,{}^{1}\!S_0$
$\Delta \tilde{{K}}_{{\rm NMS}}$ $\Delta \tilde{{K}}_{{\rm SMS}}$ $\Delta F$ $\Delta \tilde{{K}}_{{\rm NMS}}$ $\Delta \tilde{{K}}_{{\rm SMS}}$ $\Delta F$
70 $-$9680 $-$9729$\times$10$^2$ $-$2072$\times$10$^1$ $-$1512$\times$10$^2$ $-$1104$\times$10$^3$ $-$2189$\times$10$^1$
70$^{\rm a}$ $-$9855 $-$9780$\times$10$^2$ $-$2073$\times$10$^1$ $-$1513$\times$10$^2$ $-$1109$\times$10$^3$ $-$2190$\times$10$^1$
71 $-$7866 $-$9894$\times$10$^2$ $-$2291$\times$10$^1$ $-$1583$\times$10$^2$ $-$1132$\times$10$^3$ $-$2425$\times$10$^1$
71$^{\rm a}$ $-$7914 $-$9952$\times$10$^2$ $-$2290$\times$10$^1$ $-$1583$\times$10$^2$ $-$1137$\times$10$^3$ $-$2424$\times$10$^1$
72 $-$5614 $-$1005$\times$10$^3$ $-$2527$\times$10$^1$ $-$1654$\times$10$^2$ $-$1160$\times$10$^3$ $-$2679$\times$10$^1$
72$^{\rm a}$ $-$4980 $-$1012$\times$10$^3$ $-$2522$\times$10$^1$ $-$1648$\times$10$^2$ $-$1166$\times$10$^3$ $-$2674$\times$10$^1$
73 $-$3158 $-$1019$\times$10$^3$ $-$2793$\times$10$^1$ $-$1728$\times$10$^2$ $-$1188$\times$10$^3$ $-$2967$\times$10$^1$
73$^{\rm a}$ $-$1703 $-$1030$\times$10$^3$ $-$2780$\times$10$^1$ $-$1710$\times$10$^2$ $-$1198$\times$10$^3$ $-$2953$\times$10$^1$
74 $-$244 $-$1033$\times$10$^3$ $-$3082$\times$10$^1$ $-$1802$\times$10$^2$ $-$1216$\times$10$^3$ $-$3281$\times$10$^1$
74$^{\rm a}$ 1407 $-$1045$\times$10$^3$ $-$3067$\times$10$^1$ $-$1781$\times$10$^2$ $-$1227$\times$10$^3$ $-$3264$\times$10$^1$
75 3103 $-$1045$\times$10$^3$ $-$3401$\times$10$^1$ $-$1877$\times$10$^2$ $-$1244$\times$10$^3$ $-$3628$\times$10$^1$
76 7034 $-$1056$\times$10$^3$ $-$3748$\times$10$^1$ $-$1950$\times$10$^2$ $-$1272$\times$10$^3$ $-$4007$\times$10$^1$
77 1131$\times$10$^1$ $-$1065$\times$10$^3$ $-$4139$\times$10$^1$ $-$2026$\times$10$^2$ $-$1300$\times$10$^3$ $-$4434$\times$10$^1$
78 1624$\times$10$^1$ $-$1072$\times$10$^3$ $-$4568$\times$10$^1$ $-$2100$\times$10$^2$ $-$1327$\times$10$^3$ $-$4904$\times$10$^1$
79 2184$\times$10$^1$ $-$1078$\times$10$^3$ $-$5040$\times$10$^1$ $-$2173$\times$10$^2$ $-$1355$\times$10$^3$ $-$5424$\times$10$^1$
80 2839$\times$10$^1$ $-$1082$\times$10$^3$ $-$5550$\times$10$^1$ $-$2242$\times$10$^2$ $-$1382$\times$10$^3$ $-$5988$\times$10$^1$
81 3567$\times$10$^1$ $-$1083$\times$10$^3$ $-$6119$\times$10$^1$ $-$2310$\times$10$^2$ $-$1409$\times$10$^3$ $-$6618$\times$10$^1$
82 4390$\times$10$^1$ $-$1082$\times$10$^3$ $-$6746$\times$10$^1$ $-$2374$\times$10$^2$ $-$1435$\times$10$^3$ $-$7314$\times$10$^1$
83 5302$\times$10$^1$ $-$1078$\times$10$^3$ $-$7445$\times$10$^1$ $-$2435$\times$10$^2$ $-$1462$\times$10$^3$ $-$8094$\times$10$^1$
84 6321$\times$10$^1$ $-$1072$\times$10$^3$ $-$8223$\times$10$^1$ $-$2491$\times$10$^2$ $-$1488$\times$10$^3$ $-$8964$\times$10$^1$
85 7480$\times$10$^1$ $-$1063$\times$10$^3$ $-$9075$\times$10$^1$ $-$2540$\times$10$^2$ $-$1513$\times$10$^3$ $-$9922$\times$10$^1$
86 8902$\times$10$^1$ $-$1049$\times$10$^3$ $-$9942$\times$10$^1$ $-$2567$\times$10$^2$ $-$1538$\times$10$^3$ $-$1090$\times$10$^2$
87 1038$\times$10$^2$ $-$1033$\times$10$^3$ $-$1097$\times$10$^2$ $-$2596$\times$10$^2$ $-$1562$\times$10$^3$ $-$1207$\times$10$^2$
88 1206$\times$10$^2$ $-$1012$\times$10$^3$ $-$1209$\times$10$^2$ $-$2610$\times$10$^2$ $-$1586$\times$10$^3$ $-$1334$\times$10$^2$
89 1390$\times$10$^2$ $-$9876$\times$10$^2$ $-$1335$\times$10$^2$ $-$2612$\times$10$^2$ $-$1610$\times$10$^3$ $-$1477$\times$10$^2$
90 1609$\times$10$^2$ $-$9570$\times$10$^2$ $-$1469$\times$10$^2$ $-$2591$\times$10$^2$ $-$1631$\times$10$^3$ $-$1631$\times$10$^2$
91 1842$\times$10$^2$ $-$9218$\times$10$^2$ $-$1623$\times$10$^2$ $-$2559$\times$10$^2$ $-$1652$\times$10$^3$ $-$1809$\times$10$^2$
92 2117$\times$10$^2$ $-$8809$\times$10$^2$ $-$1784$\times$10$^2$ $-$2489$\times$10$^2$ $-$1673$\times$10$^3$ $-$1995$\times$10$^2$
$^{\rm a}$Theoretical work of Nazé et al.[21]
cpl-36-11-113101-fig1.png
Fig. 1. The NMS and SMS parameters of the 2$s2p\,{}^{3,1}\!P_{1}\to 2s^{2}\,{}^{1}\!S_0$ transitions in Be-like ions as a function of the nuclear charge number $Z$. The crosses represent the results of this work, and the others are taken from Ref. [21].
The isotope shift parameters of the 2$s2p\,{}^{3,1}\!P_{1}\to 2s^{2}\,{}^{1}\!S_0$ transitions in Be-like ions from $Z=70$ to $Z=92$ are presented in Table 2. It is found that the difference in the NMS parameters of the 2$s2p\,{}^{3,1}\!P_{1}\to 2s^{2}\,{}^{1}\!S_0$ transition between our results and those from Ref. [21] for $Z$=70–74 is relatively large, because the NMS parameters are too small. However, as can be seen from Fig. 1, the trends of the mass shift parameters in the range of $Z$=70–74 are in good agreement with each other. In the high-$Z$ range, it is worth noting that the mass shift parameters of the 2$s2p\,{}^{3,1}\!P_{1}\to 2s^{2}\,{}^{1}\!S_0$ transitions no longer change monotonically. The SMS or NMS parameters may have minimum values along the isoelectronic sequence. These trends cannot be reflected by the results of Nazé et al.[21] Figure 2 shows the competition between the relativistic correction terms and the non-relativistic terms of mass shifts. For the 2$s2p\,{}^{3}\!P_{1}\to 2s^{2}\,{}^{1}\!S_0$ transition, it is shown that these two terms change in the opposite directions and, furthermore, the relativistic correction terms become dominant in the high-$Z$ region. This brings about a minimum value for the SMS parameters. For the 2$s2p\,{}^{1}\!P_{1}\to 2s^{2}\,{}^{1}\!S_0$ transition, the relativistic corrections of the SMS parameters are the same order of magnitude as the non-relativistic terms, and both change monotonously with increasing $Z$, as shown in Fig. 1. The relativistic correction terms and non-relativistic terms of the NMS parameters also have opposite trends as $Z$ increases, and the former dominates the mass shift for heavy elements. In summary, the relativistic nuclear recoil corrections play a key role in the mass shifts of Be-like ions, especially in the high-$Z$ region.
cpl-36-11-113101-fig2.png
Fig. 2. Mass shift parameters and corresponding non-relativistic terms for 2$s2p\,{}^{3,1}\!P_{1}\to 2s^{2}\,{}^{1}\!S_0$ transitions in Be-like ions. Here $\Delta \tilde{{K}}^{(1)}$ represents the non-relativistic term, $\Delta \tilde{{K}}$ represents the total NMS or SMS parameter and is the sum of the non-relativistic term and the relativistic correction term.
Isotope shift is composed of the mass shift and field shift, and it is necessary to study competition between them along the isoelectronic sequence. To evaluate the mass and the field shifts of the 2$s2p\,{}^{3,1}\!P_{1}\to 2s^{2}\,{}^{1}\!S_0$ transitions for Be-like ions, the difference of the nuclear mass and the mean-square charge radius for a given isotope pair should be determined. For simplification, the mass number $A$ is obtained using the semi-empirical formula[30] $$\begin{align} Z=\frac{A}{1.98+0.015A^{2/3}},~~ \tag {9} \end{align} $$ for a stable isotope of a given atom with the atomic number $Z$ and rounding the mass number $A$ to an integer value. We use the relation of Johnson and Soff[31,32] $$\begin{alignat}{1} \langle {r^{2}} \rangle^{1/2}=({0.836A^{1/3}+0.570})\,{\rm fm},~~{\rm for}~A>9,~~ \tag {10} \end{alignat} $$ to estimate the root-mean-square charge radius for an isotope with mass number $A$. Taking the results from Table 2, we illustrate the absolute values of the mass shifts $|\delta \nu_{\rm MS}|$ and the field shifts $|\delta \nu_{\rm FS}|$ of the 2$s2p\,{}^{3,1}\!P_{1}\to 2s^{2}\,{}^{1}\!S_0$ transitions in Be-like ions for the isotope pair $(A, A+1)$. As shown in Fig. 3, the field shift increases rapidly in the high-$Z$ region, while the mass shift remains almost unchanged. Spectra measurement of highly charged ions has been employed to extract the nuclear structure because of its many advantages, for instance, high resolution and signal-to-noise ratio. On the other hand, compared with the neutral or near-neutral atomic systems, electron correlation effects are relatively weak in highly charged ions. Thus, one could achieve high accuracy in calculations of atomic properties.
cpl-36-11-113101-fig3.png
Fig. 3. The absolute values of the frequency-mass shifts $|\delta \nu_{\rm MS}|$ and frequency-field shifts $|\delta \nu_{\rm FS}|$ for the isotope pair $(A, A+1)$ as a function of the nuclear charge. The results are shown for the 2$s2p\,{}^{3,1}\!P_{1}\to 2s^{2}\,{}^{1}\!S_0$ transitions in Be-like ions.
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