Isotope shifts of the $1s^22s2p(J)$ -$1s^22s^2$ transition energies in Be-like thorium and uranium

TL;DR

This paper presents highly precise calculations of isotope shifts in Be-like thorium and uranium ions, accounting for relativistic, electron correlation, Breit, QED, nuclear deformation, and polarization effects.

Contribution

It introduces a comprehensive computational approach combining multiple advanced effects to accurately predict isotope shifts in heavy ions.

Findings

Calculated isotope shifts with high precision

Quantified relativistic and correlation effects

Included QED and nuclear corrections

Abstract

Precise calculations of the isotope shifts in berylliumlike thorium and uranium ions are presented. The main contributions to the field and mass shifts are calculated within the framework of the Dirac-Coulomb-Breit Hamiltonian employing the configuration-interaction Dirac-Fock-Sturm method. These calculations include the relativistic, electron-electron correlation, and Breit-interaction effects. The QED, nuclear deformation, and nuclear polarization corrections are also evaluated.

Tables

Table 1. Table 1: Individual contributions to the isotope shifts of the 1 ​ s 2 ​ 2 ​ s ​ 2 ​ p 1 / 2 ​ ( J ) − 1 ​ s 2 ​ 2 ​ s 2 1 superscript 𝑠 2 2 𝑠 2 subscript 𝑝 1 2 𝐽 1 superscript 𝑠 2 2 superscript 𝑠 2 1s^{2}2s2p_{1/2}(J)-1s^{2}2s^{2} transition energies in 232,230 Th 86+ , 238,236 U 88+ , and 238,234 U 88+ (in meV) with given values of δ ​ ⟨ r 2 ⟩ 𝛿 delimited-⟨⟩ superscript 𝑟 2 \delta\langle r^{2}\rangle .

	232,230Th86+	238,236U88+	238,234U88+
	${}_{}{}^{232,230}\delta ⟨r^2⟩$=0.2050 ${\mathrm{fm}}^2$	${}_{}{}^{238,236}\delta ⟨r^2⟩$=0.1676 ${\mathrm{fm}}^2$	${}_{}{}^{238,234}\delta ⟨r^2⟩$=0.334 ${\mathrm{fm}}^2$
	$1s^{2}2s2p_{1/2}\left(J=0\right)-1s^{2}2s^2$
Main contributions
Field shift	$-$112.4	$-$110.8	$-$220.7
Mass shift	0.1	0.1	0.2
QED
Field shift	0.6	0.6	1.2
Mass shift	0.4	0.4	0.9
Others
Nuclear polarization	1.6	1.1	2.3
Nuclear deformation	1.5	$-$2.2	$-$2.4
Total IS theorya	$-$108.2(22)	$-$110.8(31)	$-$218.5(32)
	$1s^{2}2s2p_{1/2}\left(J=1\right)-1s^{2}2s^2$
Main contributions
Field shift	$-$112.8	$-$111.0	$-$221.3
Mass shift	0.1	0.1	0.2
QED
Field shift	0.6	0.6	1.2
Mass shift	0.4	0.4	0.9
Others
Nuclear polarization	1.6	1.1	2.3
Nuclear deformation	1.5	$-$2.2	$-$2.4
Total IS theorya	$-$108.5(22)	$-$111.1(31)	$-$219.1(32)

Table 2. Table 2: Individual contributions to the isotope shifts of the 1 ​ s 2 ​ 2 ​ s ​ 2 ​ p 3 / 2 ​ ( J ) − 1 ​ s 2 ​ 2 ​ s 2 1 superscript 𝑠 2 2 𝑠 2 subscript 𝑝 3 2 𝐽 1 superscript 𝑠 2 2 superscript 𝑠 2 1s^{2}2s2p_{3/2}(J)-1s^{2}2s^{2} transition energies in 232,230 Th 86+ , 238,236 U 88+ and 238,234 U 88+ (in meV) with given values of δ ​ ⟨ r 2 ⟩ 𝛿 delimited-⟨⟩ superscript 𝑟 2 \delta\langle r^{2}\rangle .

	232,230Th86+	238,236U88+	238,234U88+
	${}_{}{}^{232,230}\delta ⟨r^2⟩$=0.2050 ${\mathrm{fm}}^2$	${}_{}{}^{238,236}\delta ⟨r^2⟩$=0.1676 ${\mathrm{fm}}^2$	${}_{}{}^{238,234}\delta ⟨r^2⟩$=0.334 ${\mathrm{fm}}^2$
	$1s^{2}2s2p_{3/2}\left(J=2\right)-1s^{2}2s^2$
Main contributions
Field shift	$-$124.6	$-$123.7	$-$246.4
Mass shift	0.3	0.3	0.6
QED
Field shift	0.9	0.9	1.8
Mass shift	0.4	0.4	0.8
Others
Nuclear polarization	1.7	1.2	2.6
Nuclear deformation	1.5	$-$2.4	$-$2.7
Total IS theorya	$-$119.8(22)	$-$123.2(32)	$-$243.2(33)
	$1s^{2}2s2p_{3/2}\left(J=1\right)-1s^{2}2s^2$
Main contributions
Field shift	$-$125.2	$-$124.3	$-$247.7
Mass shift	0.3	0.3	0.6
QED
Field shift	0.9	0.9	1.8
Mass shift	0.4	0.4	0.8
Others
Nuclear polarization	1.7	1.2	2.6
Nuclear deformation	1.5	$-$2.4	$-$2.7
Total IS theorya	$-$120.4(22)	$-$123.9(32)	$-$244.5(33)

Equations

$$\rho (r,R)=\frac{N}{1+\exp [(r-c)/a]},$$

$$V_N(r,R)=-4\pi \alpha Z\underset{0}{\overset{\infty }{\int }}d{r}^{\prime }{r}^{\prime 2}\rho (r^{\prime },R)\frac{1}{r_{>}},$$

$$\delta E_{FS}=F\delta ⟨r^2⟩,$$

$$F=⟨\psi \mid \sum _i\frac{d{V}_N(r_i,R)}{d⟨r^2⟩}\mid \psi ⟩,$$

$$H_M$$

$$\displaystyle\rho(r)=\frac{1}{4\pi}\int{d{\mbox{\boldmath$n$}}\rho({\mbox{\boldmath$r$}})}\,,$$

$$\rho({\mbox{\boldmath$r$}})=\frac{N}{1+{\rm{exp}}[(r-r_{0}(1+\beta_{20}Y_{20}(\theta)+\beta_{40}Y_{40}(\theta))/a]}\,,$$

Full text

I Introduction

Theoretical and experimental studies of the isotope shifts in highly charged ions can provide tests of the relativistic and QED theory of the nuclear recoil effect in nonperturbative regime and serve as a good tool for precise determination of the nuclear charge radius differences. First isotope shift measurements in highly charged ions were performed in Refs. ell96 ; ell98 ; sch05 . The measurements of the isotope shifts of the binding energies in B-like argon cre06 and in Li-like neodymium Brandau_2008 have provided the first tests of the relativistic theory of the nuclear recoil effect with highly charged ions. The latter experiment led also to determination of the nuclear charge radius difference for the 142,150Nd isotopes. The use of the dielectronic recombination technique Brandau_2008 ; bra10 ; bra13 at the GSI/FAIR facilities allows also the related experiments for heavy Be-like ions.

The main goal of the present work is to extend our calculations of the isotope shifts in Li- and B-like ions Zubova_2014 ; Zubova_2016 to Be-like thorium and uranium ions, which are of most interest for the experimental study. Previously, the calculations of the isotope shifts in Be-like ions have been performed in the range $Z=5-74$ with the use of the multiconfiguration Dirac-Fock method Naze_2014 . The calculations of the transition energies in these ions have been performed in Refs. Cheng_2018 ; Kaigorodov_2019 (see also Ref. Yerokhin_2015 ; Yerokhin_2014 and references therein).

The precision of the isotope shift measurements in heavy ions is approaching the level of the QED effects. Moreover, it is expected that at the FAIR facilities this precision will be improved by an order of magnitude. It means that the relevant theoretical calculations must be performed to the utmost accuracy. In the present paper, the dominant contributions to the isotope shifts are calculated employing the Dirac-Coulomb-Breit Hamiltonian. These calculations, which are based on the Dirac-Fock-Sturm method tup03 ; Tupitsyn_2018 , include the relativistic, electron-electron correlation, and Breit-interaction effects. Additionaly, we evaluate the QED, nuclear deformation, and nuclear polarization corrections which become rather large for heavy ions.

The relativistic units ($\hbar=c=1$) are used throughout the paper.

II Theory

II.1 Nuclear size effect

The finite nuclear size effect (the so-called field shift) is caused by the difference in the nuclear charge distribution of the isotopes. The main contribution to the field shift can be calculated in the framework of the Dirac-Coulomb-Breit Hamiltonian. The nuclear charge distribution is usually approximated by the spherically-symmetric Fermi model

[TABLE]

where the parameter $a$ is generally fixed to be $a=2.3/(4{\rm\ln}3)$ fm and the parameters $N$ and $c$ are determined using the given value of the root-mean-square ($\rm{rms}$) nuclear charge radius $R=\langle r^{2}\rangle^{1/2}$ and the normalization condition \int{d{\mbox{\boldmathr}}\rho({r},R)}=1. The potential induced by the nuclear charge distribution $\rho(r,R)$ is defined as

[TABLE]

where $r_{>}={\rm{{max}}}(r,r^{\prime})$. Since the finite nuclear size effect is mainly determined by the $\rm{rms}$ nuclear charge radius, the energy difference between two isotopes can be approximated as

[TABLE]

where $F$ is the field shift factor and $\delta\langle r^{2}\rangle$ is the mean-square charge radius difference. In accordance with this definition and the virial theorem, the $F$ factor can be also evaluated by

[TABLE]

where $\psi$ is the wave function of the state under consideration and the index $i$ runs over all atomic electrons.

II.2 Relativistic nuclear recoil effect

The fully relativistic theory of the nuclear recoil effect can be formulated only in the framework of quantum electrodynamics Shabaev_1985 ; Shabaev_1988 ; pac95 ; sha98 ; adk07 . However, to the lowest relativistic order (within the Breit approximation), the nuclear recoil effect can be taken into account using the effective recoil operator Shabaev_1985 ; Shabaev_1988 ; Palmer_1987 :

[TABLE]

This operator can be used for relativistic calculations of the nuclear recoil effect in ions and atoms (see, e.g., Refs. Zubova_2014 ; Zubova_2016 ; Naze_2014 ; tup03 ; Tupitsyn_2018 ; fis16 ; fil17 and references therein). The calculation is carried out by averaging the operator (5) with the eigenvectors of the Dirac-Coulomb-Breit Hamiltonian.

II.3 QED, nuclear deformation and nuclear polarization corrections

Since our consideration is restricted to very heavy ions, the independent-electron approximation can be used to evaluate the QED, nuclear deformation, and nuclear polarization corrections.

To calculate the self-energy and vacuum-polarization corrections to the field shift, one can use analytical formulas for these corrections derived for H-like ions in Ref. Milstein_2004 . In case of uranium, an approximate formula obtained by fitting the direct numerical calculations Yerokhin_2011 can be also employed.

The QED calculation of the one-electron recoil effect for $n=1,2$ states was performed in Refs. art95a ; sha98b ; adk07 ; mal18 . In addition, two-electron recoil contributions of zeroth order in $1/Z$ should be taken into account for the $1s^{2}2s2p(J)$ states. A detailed analysis of the relevant contributions for He-like ions was presented in Ref. mal18 .

To evaluate the nuclear deformation effect, one has to replace the standard spherically symmetric Fermi model for the nuclear charge distrubution by koz08

[TABLE]

where \rho({\mbox{\boldmathr}}) is the axially-symmetric Fermi distribution,

[TABLE]

consistent with the normalization condition \int{d{\mbox{\boldmathr}}\rho({\mbox{\boldmathr}})}=1. Here $Y_{20}(\theta)$ and $Y_{40}(\theta)$ are the spherical functions, $\beta_{20}$ and $\beta_{40}$ are the quadrupole and hexadecapole deformation parameters koz08 ; bem73 ; zum84 ; moe95 . The difference between the nuclear size effect obtained with the deformed model (6) and the standard spherically-symmetric Fermi model (1) at the same rms radius is ascribed to the nuclear deformation effect.

Finally, the interaction between the electrons and the nucleons causes the nucleus to make virtual transitions to excited states. This results in the increase of the binding energy of the electrons. To evaluate this effect, which is known as the nuclear polarization effect, one should consider the two-photon electron-nucleus interaction diagrams in which the intermediate nuclear states are excited. The calculations of this effect were performed in Refs. plu95 ; plu96 ; nef96 ; vol14 .

III Results and discussion

In Tables 1, 2 the individual contributions to the isotope shifts of the $1s^{2}2s2p(J)-1s^{2}2s^{2}$ transition energies in 232,230Th86+, 238,236U88+, and 238,234U88+ are presented. The nuclear charge radii and the $\delta\langle r^{2}\rangle$ differences have been taken from Ref. ang13 . The field shifts are evaluated within the framework of the Dirac-Coulomb-Breit Hamiltonian using the formulas (3)-(4). The calculations are performed using the conguration-interaction Dirac-Fock-Sturm method for an extended nucleus tup03 . The excited configurations are obtained from the basic configuration via single, double, and triple excitations of electrons. The accuracy of the calculations is defined by the stability of the results with respect to a variation of the basis size. The same method has been used to calculate the mass shifts within the approximation defined by the effective nuclear recoil operator (5).

The QED corrections to the field shifts have been evaluted in the one-electron approximation using the related formulas from Ref. Milstein_2004 ; Yerokhin_2011 . The QED effect on the mass shift was obtained as a sum of one- and two-electron contributions evaluated to zeroth order in $1/Z$. The one-electron terms have been taken from Ref. mal18 while the two-electron corrections have been calculated in this work. As one can see from the tables, the QED recoil contribution is even larger than the mass shift obtained within the framework of the Breit approximation. The nuclear deformation and polarization effects have been taken from the related calculations for Li-like thorium and uranium Zubova_2014 . The uncertainties of these effects determine the total theoretical uncertainties.

IV Conclusion

The isotope shifts of the $1s^{2}2s2p(J)-1s^{2}2s^{2}$ transition energies in Be-like thorium and uranium ions are calculated including the relativistic, electron-electron correlation, Breit, and QED contributions. The nuclear polarization and nuclear deformation corrections are taken into account within the framework of the independent-electron approximation. The QED effects contribute on the level of the total uncertainty which is mainly defined by the nuclear polarization and deformation effects.

Acknowledgements.

This work was supported by RFBR (Grant No. 18-32-00275), SPSU-DFG (Grants No. 11.65.41.2017 and No. STO 346/5-1), Ministry of Education and Science of the Russian Federation Grant No. 3.1463.2017/4.6, and DAAD Programm Ostpartnerschaften, TU Dresden. M.Y.K., A.V.M., and V.M.S. acknowledge also support from the Foundation for the advancement of theoretical physics and mathematics “BASIS”.