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The parity violation effects via the
$ {\mathrm{5d6s\; {^3D_1} \to 6s^2 \; {^1S_0}}} $ transition have been extensively investigated in ytterbium atoms. However, the M1 transition between the excitation state$ {\mathrm{5d6s\; {^3D_1}}} $ and the ground state$ {\mathrm{6s^2 \; {^1S_0}}} $ , as well as the hyperfine-induced E2 transition, significantly affects the detection of parity violation signal. Therefore, it is imperative to obtain the accurate transition probabilities for the M1 and hyperfine-induced E2 transitions between the excitation state${\mathrm{ 5d6s\; {^3D_1} }}$ and the ground state$ {\mathrm{6s^2\; {^1S_0}}} $ . In this work, we use the multi-configuration Dirac-Hartree-Fock theory to precisely calculate the transition probabilities for the${\mathrm{ 5d6s \; {^3D_1} \to 6s^2 \; {^1S_0} }}$ M1 and hyperfine-induced${\mathrm{ 5d6s \; ^3D_{1,3} \to 6s^2 \; {^1S_0} }}$ E2 transitions. We extensively analyze the influences of electronic correlation effects on the transition probabilities according to our calculations. Furthermore, we analyze the influences of different perturbing states and various hyperfine interactions on the transition probabilities. The calculated hyperfine constants of the e$ {\mathrm{^3D_{1,2,3}}} $ and${\mathrm{ ^1D_2}} $ states accord well with experimental measurements, validating the rationality of our computational model. By combining experimentally measured hyperfine constants with the theoretically derived electric field gradient of the extra nuclear electrons at the nucleus, we reevaluate the nuclear quadrupole moment of the$ ^{173} $ Yb nucleus as$ Q = 2. 89(5) \;\rm {b} $ , showing that our result is in excellent agreement with the presently recommended value.-
Keywords:
- hyperfine induced transition/
- Yb atom/
- hyperfine constant/
- multi-configuration Dirac-Hartree-Fock method
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Models AO VO NCFs $ J = 0 $ $ J = 1 $ $ J = 2 $ $ J = 3 $ DHF 1 1 2 1 VV-1 {$ {\rm 5 d6 s} $;$ {\rm 6 s^2} $} {$ {\rm 7 s, 6 p, 6 d, 5 f, 5 g} $} 15 16 35 24 C5V-2 {$ {\rm 5 s^25 p^65 d6 s} $;$ {\rm 5 s^25 p^66 s^2} $} {$ {\rm 8 s, 7 p, 7 d, 6 f, 6 g, 6 h} $} 336 1954 4361 3213 C4V-3 $ \{{\rm 4 s^24 p^64 d^{10}4 f^{14}5 s^25 p^65 d6 s};{\rm 4 s^24 p^64 d^{10}4 f^{14}5 s^25 p^66 s^2}\} $ {$ {\rm 9 s, 8 p, 8 d, 7 f, 7 g} $} 2896 20054 49368 37668 C4V-4 $ \{\rm 4 s^24 p^64 d^{10}4 f^{14}5 s^25 p^65 d6 s; \rm 4 s^24 p^64 d^{10}4 f^{14}5 s^25 p^66 s^2 \} $ {$ {\rm 10 s, 9 p, 9 d, 8 f, 8 g, 8 h} $} 5058 35649 88596 68104 C4V-5 $ \{{\rm 4 s^24 p^64 d^{10}4 f^{14}5 s^25 p^65 d6 s}; {\rm 4 s^24 p^64 d^{10}4 f^{14}5 s^25 p^66 s^2}\} $ {$ {\rm 11 s, 10 p, 10 d, 9 f, 9 g, 9 h} $} 7822 55699 139251 107472 C4V-6 $ \{{\rm 4 s^24 p^64 d^{10}4 f^{14}5 s^25 p^65 d6 s}; {\rm 4 s^24 p^64 d^{10}4 f^{14}5 s^25 p^66 s^2} \}$ {$ {\rm 12 s, 11 p, 11 d, 10 f, 10 g, 9 h} $} 10681 76208 190245 146319 C4V-7 $ \{{\rm 4 s^24 p^64 d^{10}4 f^{14}5 s^25 p^65 d6 s}; {\rm 4 s^24 p^64 d^{10}4 f^{14}5 s^25 p^66 s^2}\} $ {$ {\rm 13 s, 12 p, 12 d, 11 f, 10 g, 9 h} $} 13213 93967 232975 177889 CC5-7 $ \cup ${$ 5{\mathrm{ s^25 p^65 d6 s}} ; \rm 5 s^25 p^6 s^2 $} {$ {\rm 13 s, 12 p, 12 d, 11 f, 10 g, 9 h} $} 154602 435843 643878 750192 MR-3 $ \cup \{{\rm 5 s^25 p^6 p^2} ; {\rm 5 s^25 p^25 d^2}; {\rm 5 s^2 5 p^46 s^26 d7 d} $ {$ {\rm 9 s, 8 p, 8 d, 7 f, 7 g, 7 h} $} 754484 2123833 3122817 3614260 $ {5 {\mathrm{s}}^25 {\mathrm{p}}^6 {\mathrm{s}}7{\mathrm{ s}}} ; {5 {\mathrm{s}}^25 {\mathrm{p}}^66 {\mathrm{d}}7 {\mathrm{s}}}; {5 {\mathrm{s}}^25 {\mathrm{p}}^45{\mathrm{ d}}6 {\mathrm{s}}^26{\mathrm{ d}}}; $ ${\rm 5 s^25 p^55 d6 s6 p}; {\rm 5 s^25 p^65 f6 p} ; {\rm 5 s^25 p^66 s6 d} $} 
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$ {\rm ^3 D_2 \to {^1 S_0}} $ $ {\rm ^1 D_2 \to {^1 S_0}} $ Models $ \Delta E $ $ \rm{RME_{\rm L}} $ $ \rm{RME_{\rm V}} $ $ R_{\rm L} $ $ R_{\rm V} $ $ \Delta E $ $ \rm{RME_{\rm L}} $ $ \rm{RME_{\rm V}} $ $ R_{\rm L} $ $ R_{\rm V} $ DHF 21114.02 0.05 0.05 0.001 0.001 28822.95 $ - $15.05 $ - $13.59 403.87 329.46 VV-1 25010.57 2.09 2.00 3.85 3.51 26254.24 $ - $15.26 $ - $14.84 238.18 225.08 C4V-7 22406.02 1.18 1.12 0.71 0.64 26208.26 $ - $11.67 $ - $11.26 150.96 140.41 CC5-7 23171.20 0.86 0.85 0.45 0.43 28126.76 $ - $13.55 $ - $12.75 289.61 256.70 MR-3 24685.75 1.21 1.15 1.20 1.10 28313.29 $ - $12.63 $ - $12.84 260.01 232.25 Breit+QED 24553.44 1.18 1.13 1.11 1.02 28206.64 $ - $12.61 $ - $11.94 254.33 228.31 Bowers等[36] 1.12(4) Expt.[36] 1.45(7) NIST[35] 24751.95 27677.67 DownLoad:
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$ ^{171} {\mathrm{Yb}}$ $ ^{173} {\mathrm{Yb}}$ Ref. A A B $ {\rm ^3 D_1} $ Expt. $ - $2040(2) 562.8(5) 337(2) [36] $ - $2047(47) [37] $ - $2032.67(17) [38] 563(1) 335(1) [39] Theory $ - $2349 648 249 [11] 596 290 [40] 597 [41] $ - $2119.3 583.79 338.46 This work $ {\rm ^3 D_2} $ Expt. 1315(4) $ - $363.4(10) 487(5) [36] $ - $362(2) 482(22) [39] Theory 1354 $ - $373 387 [11] $ - $351 440 [40] $ - $765 [41] 1314.62 $ - $362.13 491.39 This work $ {\rm ^3 D_3} $ Expt. $ - $430(1) 909(29) [39] Theory $ - $420 728 [40] $ - $477 [41] 1626.97 $ - $448.17 836.5 This work $ {\rm ^1 D_2} $ Expt. 100(18) 1115(89) [39] Theory 131 1086 [40] 465 [41] $ - $313.87 86.46 1053.44 This work DownLoad:
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Models $ {\rm ^3 D_1} $ $ {\rm ^3 D_2} $ $ {\rm ^3 D_3} $ EFG Q EFG Q EFG Q DHF 0.23 6.09 0.32 6.47 0.55 7.07 C4V-7 0.52 2.75 0.77 2.69 1.29 2.99 CC5-7 0.43 3.26 0.63 3.27 1.10 3.52 MR-3 0.51 2.79 0.74 2.77 1.27 3.04 DownLoad:
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$ {\rm (^3 D_2, {^3 D_1})} $ $ {\rm (^1 D_2, {^3 D_1})} $ $ F' $ $ \varepsilon_{1}^{{\rm A}} $ $ \varepsilon_{1}^{{\rm B}} $ $ \varepsilon_{1} $ $ \varepsilon_{2}^{{\rm A}} $ $ \varepsilon_{2}^{{\rm B}} $ $ \varepsilon_{2} $ $ ^{171} $Yb 3/2 $ - $1.54[$ - $4] 0 $ - $1.54[$ - $4] 7.1[$ - $6] 0 7.1[$ - $6] 7/2 $ - $7.36[$ - $5] $ - $5.47[$ - $6] $ - $7.91[$ - $5] 3.39[$ - $6] 4.04[$ - $8] 3.43[$ - $6] $ ^{173} $Yb 5/2 $ - $7.17[$ - $5] 3.99[$ - $6] $ - $6.77[$ - $5] 3.30[$ - $6] $ - $2.95[$ - $8] 3.27[$ - $6] 3/2 $ - $5.03[$ - $5] 7.47[$ - $6] $ - $4.28[$ - $5] 2.31[$ - $6] $ - $5.53[$ - $8] 2.26[$ - $6] $ {\rm (^3 D_2, {^3 D_3})} $ $ {\rm (^1 D_2, {^3 D_3})} $ $ F' $ $ \varepsilon_{1}^{{\rm A}} $ $ \varepsilon_{1}^{{\rm B}} $ $ \varepsilon_{1} $ $ \varepsilon_{2}^{{\rm A}} $ $ \varepsilon_{2}^{{\rm B}} $ $ \varepsilon_{2} $ $ ^{171} $Yb 5/2 5.28[$ - $5] 0 5.28[$ - $5] –1.37[$ - $5] 0 $ - $1.37[$ - $5] 9/2 2.35[$ - $5] 2.56[$ - $6] 2.61[$ - $5] $ - $6.13[$ - $6] $ - $3.78[$ - $8] $ - $6.17[$ - $6] 7/2 2.54[$ - $5] $ - $3.46[$ - $7] 2.51[$ - $5] $ - $6.61[$ - $6] 5.1[$ - $9] $ - $6.61[$ - $6] $ ^{173} $Yb 5/2 2.21[$ - $5] $ - $2.41[$ - $6] 1.97[$ - $5] $ - $5.76[$ - $6] 3.55[$ - $8] $ - $5.73[$ - $6] 3/2 1.61[$ - $5] $ - $2.84[$ - $6] 1.32[$ - $5] $ - $4.81[$ - $6] 4.19[$ - $8] $ - $4.14[$ - $6] 1/2 8.40[$ - $6] $ - $1.83[$ - $6] 6.57[$ - $5] $ - $2.19[$ - $6] 2.7[$ - $8] $ - $2.16[$ - $6] DownLoad:
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$ R_1 $ $ R_3 $ Total $ F' $ $ T_1 $ $ T_2 $ $ T_1 $ $ T_2 $ 3/2 1.09[$ - $8] 0 2.64[$ - $9] 0 2.42(23)[$ - $8] 7/2 2.48[$ - $9] 1.37[$ - $11] 6.00[$ - $10] 8.53[$ - $14] 6.13(60)[$ - $9] 5/2 2.35[$ - $9] 7.29[$ - $12] 5.69[$ - $10] 4.55[$ - $14] 4.82(47)[$ - $9] 3/2 1.16[$ - $9] 2.55[$ - $11] 2.80[$ - $10] 1.60[$ - $13] 2.05(20)[$ - $9] $ R_1' $ $ R_3' $ Total $ F' $ $ T_1 $ $ T_2 $ $ T_1 $ $ T_2 $ 5/2 6.41[$ - $10] 0 4.96[$ - $9] 0 9.16(89)[$ - $9] 9/2 1.27[$ - $10] 1.51[$ - $12] 9.85[$ - $10] 3.75[$ - $14] 1.94(18)[$ - $9] 7/2 1.48[$ - $10] 2.74[$ - $14] 1.15[$ - $9] 6.82[$ - $16] 2.10(20)[$ - $9] 5/2 1.12[$ - $10] 1.33[$ - $12] 8.70[$ - $10] 3.30[$ - $14] 1.50(14)[$ - $9] 3/2 5.92[$ - $11] 1.85[$ - $12] 4.58[$ - $10] 4.60[$ - $14] 7.58(74)[$ - $10] 1/2 1.62[$ - $11] 7.68[$ - $13] 1.25[$ - $10] 1.91[$ - $14] 2.02(19)[$ - $10] DownLoad:
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IF Ref. 1/2, 3/2 5/2, 3/2 5/2, 5/5 5/2, 7/2 $ E2_{\rm A} $ 6.43[$ - $4] $ - $3.63[$ - $4] 6.34[$ - $4] $ - $7.52[$ - $4] Kozlov[11] 1.62[$ - $4] $ - $0.53[$ - $4] 9.26[$ - $5] $ - $1.09[$ - $4] This work $ E2_{\rm B} $ 0 $ - $3.90[$ - $5] 2.10[$ - $5] 2.80[$ - $5] Kozlov[11] 0 $ - $7.88[$ - $6] 5.16[$ - $6] 8.16[$ - $6] This work $ E2_{{\rm tot}} $ 6.40(1.0)[$ - $4] $ - $4.00(60)[$ - $4] 6.60(1.0)[$ - $4] $ - $7.20(1.2)[$ - $4] Kozlov[11] 1.62(6)[$ - $4] $ - $4.50(20)[$ - $5] 9.76(41)[$ - $5] $ - $1.01(4)[$ - $4] This work DownLoad:
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