<p>The <sup>229</sup>Th isotope is a promising candidate for nuclear clocks. However, the clock transition varies due to the electron-induced nuclear frequency shift. To achieve the accuracy required for developing a nuclear clock, this effect must be precisely determined. In this work, we employed a non-perturbative multi-configuration Dirac-Hartree-Fock (MCDHF) method, in contrast to the perturbation theory used previously, to resolve the electron-induced field shift effect. As a more internally consistent <i>ab initio</i> method, this calculation accounts for subtle differences in the nuclear potential while considering the <sup>229</sup>Th isotope in both its ground and isomeric states. Consequently, the nuclear clock transition frequency difference between <sup>229</sup>Th<sup>3+</sup> and <sup>229</sup>Th<sup>4+</sup> was determined to be −639 MHz with computational convergency down to 1 MHz. Given recently measured transition frequency of <sup>229</sup>Th<sup>4+</sup> in <sup>229</sup>Th-doped CaF<sub>2</sub> [Nature 633, 63 (2024)], here the transition frequency of isolated <sup>229</sup>Th<sup>3+</sup> is predicted to be <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11433_2025_2675_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="233" /> </InlineMediaObject> <EquationSource Format="TEX">\(2020407009(1)_{\text{comp.}}(77)_{\delta\langle{r}^{2}\rangle}(70)_{\text{ext.}}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mn>2020407009</mn> <mo stretchy="false">(</mo> <mn>1</mn> <msub> <mo stretchy="false">)</mo> <mrow> <mtext>comp.</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mn>77</mn> <msub> <mo stretchy="false">)</mo> <mrow> <mi>δ</mi> <mo fence="false" stretchy="false">⟨</mo> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mn>70</mn> <msub> <mo stretchy="false">)</mo> <mrow> <mtext>ext.</mtext> </mrow> </msub> </math></EquationSource> </InlineEquation> MHz, with brackets indicating uncertainties stemming from our atomic structure computations, the nuclear charge radius difference from the literature, and the influence of the crystal environment as reported in the literature. This provides valuable guidance for direct laser excitation of isolated <sup>229</sup>Th<sup>3+</sup> based on ion trap experiments.</p>

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Prediction of nuclear clock transition frequency difference between 229Th3+ and 229Th4+ via ab initio self-consistent field theory

  • Ran Si,
  • Chaofan Shi,
  • Nan Xue,
  • Xiangjin Kong,
  • Chongyang Chen,
  • Bingsheng Tu,
  • Yu-Gang Ma

摘要

The 229Th isotope is a promising candidate for nuclear clocks. However, the clock transition varies due to the electron-induced nuclear frequency shift. To achieve the accuracy required for developing a nuclear clock, this effect must be precisely determined. In this work, we employed a non-perturbative multi-configuration Dirac-Hartree-Fock (MCDHF) method, in contrast to the perturbation theory used previously, to resolve the electron-induced field shift effect. As a more internally consistent ab initio method, this calculation accounts for subtle differences in the nuclear potential while considering the 229Th isotope in both its ground and isomeric states. Consequently, the nuclear clock transition frequency difference between 229Th3+ and 229Th4+ was determined to be −639 MHz with computational convergency down to 1 MHz. Given recently measured transition frequency of 229Th4+ in 229Th-doped CaF2 [Nature 633, 63 (2024)], here the transition frequency of isolated 229Th3+ is predicted to be \(2020407009(1)_{\text{comp.}}(77)_{\delta\langle{r}^{2}\rangle}(70)_{\text{ext.}}\) 2020407009 ( 1 ) comp. ( 77 ) δ r 2 ( 70 ) ext. MHz, with brackets indicating uncertainties stemming from our atomic structure computations, the nuclear charge radius difference from the literature, and the influence of the crystal environment as reported in the literature. This provides valuable guidance for direct laser excitation of isolated 229Th3+ based on ion trap experiments.