Isotope shift, non-linearity of King plot and search for nuclear island of stability V. V. Flambaum, A. J. Geddes and A. V. Viatkina 18th May 2017 FFK-2017, Warsaw, Poland Isotope shift π π + Ξπ βπ = πΌπ = ππ + πΉπ Normal mass shift (NMS) center of mass change (finite nuclear mass effect) Field shift nuclear electric charge distribution Specific mass shift (SMS) electron correlation R R+Ξ΄R Field shift single-electron approximation Point-like nucleus Uniformly charged spherical nucleus R Potential r V(r) R r V(r) ππ 2 π= π ππ 2 3 π2 π= β π 2 2π 2 Field shift single-electron approximation Point-like nucleus Uniformly charged spherical nucleus R Perturbation r πΏπΈ1 = π0 |πΏπ|π0 V(r) R r V(r) V |π0 π β 0 | β β πΏπ [1] G. Racah, Nature 129, 723 (1932). [2] J. E. Rosenthal and G. Breit, Physical Review 41, 459 (1932). V |π0 π β 0 | β ππππ π‘ πΏπ 1 12 π (π β πΎ) 2ππ πΏππ = × 2 π§π + 1 2 π + 1 2 π + 3 [Ξ 2πΎ + 1 ] ππ΅ π β nuclear charge π β (equivalent) nuclear radius π§π β ion charge π β Dirac quantum number π = -1 1 -2 2 π 2 β (πΌπ)2 πΌπ 3 πΏπ π π¦ π πΏπ = π 2 β π 1 πΌπ β ionization potential of the electron ππ΅ β Bohr radius π π¦ β Rydberg constant -3 β¦ π 1/2 π1/2 π3/2 π3/2 π5/2 β¦ πΎ= 2πΎ Island of stability β’ Isotopes with (theoretically predicted) magic neutron number N = 184 are not produced in laboratories. β’ Possibility: find them in astrophysical data? [1] β’ Calculate isotope shifts for N = 184 isotopes and add them to spectra of synthesized isotopes. [1] V. A. Dzuba, V. V. Flambaum, and J. K. Webb, arXiv:1703.04250 (2017) King plot πΌπ = ππ + πΉπ βπ1,π΄π΄β² = πΎ1 ππ΄π΄β² + πΉ1 πΏ mass shift π 2πΎ1 ππ΄π΄β² + πΊ1 πΏ 2πΎ2 ππ΄π΄β² × field shift 1 ππ΄π΄β² 2πΎ 2πΎ × π2,π΄π΄β² = πΎ2 + πΉ2 π₯π΄π΄β² + πΊ2 π¦π΄π΄β² [1] King, W. H. (1963) 1 1 = β ππ΄ ππ΄β² πΎ1 = π 1 2 β (πΌπ)2 π 1 = β1, 1 πΎ2 = π1,π΄π΄β² = πΎ1 + πΉ1 π₯π΄π΄β² + πΊ1 π¦π΄π΄β² βπ2,π΄π΄β² = πΎ2 ππ΄π΄β² + πΉ2 πΏ ππ΄π΄β²1 + πΊ2 πΏ ππ΄π΄β²2 π΄π΄β² π 2 2 β (πΌπ)2 π 2 = β2, 2, β3, 3, β¦ 1 ππ΄π΄β² π1 vs. π2 plot for π΄, π΄β² = π΄1 , π΄2 , π΄3 , β¦ is called King plot King plot non-linearity π2 π = π΄π΄3 β’ Sources of nonlinearities from higher-order contributions π π2,π΄π΄2 π2,π΄π΄1 β’ New Physics ? Long-range force with couplings not proportional to electric charge should contribute to King plot non-linearity. [1] π = π΄π΄2 π = π΄π΄1 π1,π΄π΄1 π1,π΄π΄2 π1 [1] J. C. Berengut, D. Budker, C. Delaunay, V. V. Flambaum et al. βProbing new light force-mediators by isotope shift spectroscopyβ, arXiv: 1704.05068 12 2ππ 11 12π (π π (π ββπΎ) πΎ) 2π πΏππ π = = × βπ × 2 2 +112πΎ2 π 1 12 π 3 3[Ξ[Ξ2πΎ2πΎ ++ 1 1] ] ππππ π§π§ππ + 2 π + + 2 π + + ion charge -1 1 -2 2 -3 β¦ π 1/2 π1/2 π3/2 π3/2 π5/2 β¦ πΎ= π 2 β (πΌπ)2 πΌπΌπ 3π 3 πΏπ 2πΎ π π π¦ π π¦ π non-linearities π β nuclear charge π β (equivalent) nuclear radius π§π β π = 2πΎ 2πΎ πΌπ β ionization potential of the electron ππ β Bohr radius π π¦ β Rydberg constant 12 2ππ 11 12π (π π (π ββπΎ) πΎ) 2π πΏππ π = = × βπ × 2 2 +112πΎ2 π 1 12 π 3 3[Ξ[Ξ2πΎ2πΎ ++ 1 1] ] ππππ π§π§ππ + 2 π + + 2 π + + non-linearities π β nuclear charge Liquid drop model π0 π΄1/3 , π0 =1.15 fmradius π π β=(equivalent) nuclear π§Rπ β π = ion charge -1 1 -2 2 -3 β¦ π 1/2 π1/2 π3/2 π3/2 π5/2 β¦ πΎ= π 2 β (πΌπ)2 A [1] Angeli, I., and K. P. Marinova. Atomic Data and Nuclear Data Tables 99.1 (2013): 69-95. [2] Fricke, G., Heilig, K. "Nuclear charge radii" (2004). 2πΎ 2πΎ πΌπΌπ 3π 3 πΏπ 2πΎ π π π¦ π π¦ π π from experiment [1,2] πΌπ β ionization potential of the electron ππ β Bohr radius π π¦ β Rydberg constant 1 12 π (π β πΎ) 2π βππ = × 2 π§π + 1 2πΎ 2 π + 1 2 π + 3 [Ξ 2πΎ + 1 ] ππ Liquid drop model π = π0 π΄1/3 , π0 =1.15 fm non-linearities 2πΎ π from experiment πΌπ 3 2πΎ π π π¦ Summary β’ Simple analytical formula for field shift derived. β’ Found estimated values for isotope shift in metastable super-heavy nuclei. β’ Estimated non-linearity of King plot is small, but for Yb+ and Hg+ above 1 Hz. Backup slides Element Transition IS, experiment (MHz) IS, theory (MHz) Ca (A=46-48) 3p6 4s2 β 3p6 4s 4p -25.3 ± 1.0 -31 Yb (A=174-176) 4f14 6s2 β 4f14 6s 6p 993 ± 250 1217 Hg (A=202-204) 5d10 6s2 β 6d10 6s 6p 5238 ± 11 4939 CI+MBPT No (A=259-286) 7s2 - 7s 7p -7.2 cm^-1 [1] Fricke, G., Heilig, K. "Nuclear charge radii" (2004). [2] V. A. Dzuba, V. V. Flambaum, and J. K. Webb, arXiv:1703.04250 (2017) -9.2 cm^-1 Backup slides π 2 5 2 = π 3
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