<p>Precise nuclear charge radii provide stringent constraints on nuclear structure and few-body nuclear calculations. Their extraction from isotope-shift measurements relies critically on an accurate evaluation of the atomic field-shift factors. In this work, we present high-precision calculations of the field-shift factors for the helium isotopes <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(^{3}\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mrow /> <mn>3</mn> </mmultiscripts> </math></EquationSource> </InlineEquation>He and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(^{4}\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mrow /> <mn>4</mn> </mmultiscripts> </math></EquationSource> </InlineEquation>He, including second-order relativistic perturbative corrections. For the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(2\,^3\!S-2\,^3\!P\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mmultiscripts> <mspace width="0.166667em" /> <mrow /> <mn>3</mn> </mmultiscripts> <mspace width="-0.166667em" /> <mi>S</mi> <mo>-</mo> <mn>2</mn> <mmultiscripts> <mspace width="0.166667em" /> <mrow /> <mn>3</mn> </mmultiscripts> <mspace width="-0.166667em" /> <mi>P</mi> </mrow> </math></EquationSource> </InlineEquation> transition, we achieve ppm-level accuracy and obtain field-shift factors of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(-1212.291(1)\ {{\mathrm{kHz}}/{\mathrm{fm}}^{2}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>1212.291</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mspace width="4pt" /> <mrow> <mi mathvariant="normal">kHz</mi> <mo stretchy="false">/</mo> <msup> <mrow> <mi mathvariant="normal">fm</mi> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(^{3}\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mrow /> <mn>3</mn> </mmultiscripts> </math></EquationSource> </InlineEquation>He and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(-1212.455(1)\, {{\mathrm {kHz}}/{\mathrm {fm}}^{2}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>1212.455</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <mrow> <mi mathvariant="normal">kHz</mi> <mo stretchy="false">/</mo> <msup> <mrow> <mi mathvariant="normal">fm</mi> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(^{4}\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mrow /> <mn>4</mn> </mmultiscripts> </math></EquationSource> </InlineEquation>He. Using these improved values, we re-evaluate the difference of the squared nuclear charge radii to <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(\Delta R^{2}=1.0736(21)\, {{\mathrm {fm}}^{2}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>1.0736</mn> <mrow> <mo stretchy="false">(</mo> <mn>21</mn> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <msup> <mrow> <mi mathvariant="normal">fm</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> based on the latest electronic-helium isotope-shift measurement. The resulting value is currently limited mainly by experimental uncertainty and shows a <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(2.7\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2.7</mn> <mi>σ</mi> </mrow> </math></EquationSource> </InlineEquation> tension with the determination from muonic-helium spectroscopy. Our results provide an improved nuclear-size benchmark for helium isotopes and highlight the importance of further high-precision nuclear and atomic studies to achieve a consistent determination of nuclear charge radii across different probes.</p>

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Field-shift factors and nuclear charge-radius difference in 3He and 4He

  • Fang-Fei Wu,
  • Li-Yan Tang,
  • Zong-Chao Yan,
  • Zhen-Xiang Zhong,
  • Han-Kui Wang,
  • Ting-Yun Shi,
  • Xiao-Qiu Qi

摘要

Precise nuclear charge radii provide stringent constraints on nuclear structure and few-body nuclear calculations. Their extraction from isotope-shift measurements relies critically on an accurate evaluation of the atomic field-shift factors. In this work, we present high-precision calculations of the field-shift factors for the helium isotopes \(^{3}\) 3 He and \(^{4}\) 4 He, including second-order relativistic perturbative corrections. For the \(2\,^3\!S-2\,^3\!P\) 2 3 S - 2 3 P transition, we achieve ppm-level accuracy and obtain field-shift factors of \(-1212.291(1)\ {{\mathrm{kHz}}/{\mathrm{fm}}^{2}}\) - 1212.291 ( 1 ) kHz / fm 2 for \(^{3}\) 3 He and \(-1212.455(1)\, {{\mathrm {kHz}}/{\mathrm {fm}}^{2}}\) - 1212.455 ( 1 ) kHz / fm 2 for \(^{4}\) 4 He. Using these improved values, we re-evaluate the difference of the squared nuclear charge radii to \(\Delta R^{2}=1.0736(21)\, {{\mathrm {fm}}^{2}}\) Δ R 2 = 1.0736 ( 21 ) fm 2 based on the latest electronic-helium isotope-shift measurement. The resulting value is currently limited mainly by experimental uncertainty and shows a \(2.7\sigma \) 2.7 σ tension with the determination from muonic-helium spectroscopy. Our results provide an improved nuclear-size benchmark for helium isotopes and highlight the importance of further high-precision nuclear and atomic studies to achieve a consistent determination of nuclear charge radii across different probes.