Hyperfine Induced Transitions 1S0-3D1 in Yb

1 3 Hyperﬁne induced transitions S0 – D1 in Yb

M. G. Kozlov Petersburg Nuclear Physics Institute of NRC “Kurchatov Institute”, Gatchina 188300, Russia and St. Petersburg Electrotechnical University “LETI”, Prof. Popov Str. 5, 197376 St. Petersburg

V. A. Dzuba and V. V. Flambaum School of Physics, University of New South Wales, Sydney, NSW 2052, Australia (Dated: September 17, 2018 — November 22, 2018)

2 1 Parity violation experiment in Yb is made on the strongly forbidden M1 transition 6s S0 → 3 3 3 5d6s D1. The hyperfine mixing of the 5d6s D1 and 5d6s D2 levels opens E2 channel, whose 3 amplitude differs for F -sublevels of the D1 level. This effect may be important for the experimental search for the nuclear-spin-dependent parity violation effects predominantly caused by the nuclear anapole moment.

A. Introduction

Up to now the largest parity violation (PV) effect in 2 1 atomic physics was observed in the transition 6s S0 → 3 5d6s D1 in ytterbium [1–4]. The accuracy of the latest experiment [4] has reached 0.5%, which allowed to detect isotope dependence of the PV amplitude for even isotopes and obtain the limits on the interactions of additional Z0 boson with electrons, protons and neutrons. At this level of accuracy it becomes possible to observe a nuclear- spin-dependent (NSD) PV amplitude, which is roughly two orders of magnitude smaller than the nuclear-spin- independent (NSI) PV amplitude. For heavy nuclei this amplitude is dominated by the contribution of the nuclear anapole moment [5–7]. Among several smaller contributions there is one from the weak quadrupole moment [8]. 2 1 3 The dominant NSI PV amplitude 6s S0 → 5d6s D1 was calculated in Refs. [1,9–11] and the NSD PV amplitude was calculated in Refs. [11–13]. Experimental detection of the anapole moment in this transition would require precision measurements of the PV amplitudes for 2 1 3 different hyperfine components of the 6s S0 → 5d6s D1 3 3 transition and comparison with the accurate theory. FIG. 1. Hyperfine mixings εI,F of the 5d6s D1 and 5d6s D2 171 173 The largest contribution to the experimentally ob- levels in odd isotopes Yb (I = 1/2) and Yb (I = 5/2). served PV signal comes from the interference term of the PV amplitude and the Stark-induced amplitude [3]. However, there are other smaller contributions from the −2.04 GHz. The offdiagonal matrix elements of the hy- interferences with the forbidden M1 transition and the perfine interaction between the levels of the same mul- hyperfine induced E2 transition. The former one was tiplet are not suppressed, so for the isotope 171 we measured in [14] and was found to be: can expect mixing between these levels on the order of 2 GHz/263 cm−1 ∼ 3 × 10−4. The quadrupole amplitude 3 2 1 −4 2 1 3 |h5d6s D1||M1||6s S0i| = 1.33(21) × 10 (µ0) , (1) 6s S0 → 5d6s D2 was measured in Ref. [15]: arXiv:1810.09327v2 [physics.atom-ph] 21 Nov 2018 3 2 1 2 where µ0 is Bohr magneton. The latter amplitude is |h5d6s D2||E2||6s S0i| = 1.45(7) (ea0), (2) not known, but it is expected to be not much smaller. Moreover, it can produce NSD effects by the interference where e is elementary charge and a0 is Bohr radius. The 3 3 with the main NSI PV amplitude. Here we present cal- hyperfine mixing of the levels D1 and D2 leads to the culations of the dominant contribution to this amplitude hyperfine induced (HFI) quadrupole transitions from the 3 3 3 from the hyperfine mixing between states D1 and D2, ground state to the state D1. Figure1 shows that for which lie only 263 cm−1apart (see Figure1). the isotope 171 there is only one such transition to the 3 3 3 The hyperfine structure of the D1 and D2 levels sublevel F = /2; we can estimate its amplitude to be ∼ −4 2 was measured by Bowers et al. [15]. For example, for 4×10 (ea0). According to this estimate the rate of this 171 3 the isotope Yb the constant A( D1) was found to be HFI transition is about one order of magnitude smaller 2

3 3 than the rate of the M1 transition (1). For the isotope e = 1. In these units ∆ = E D2 − E D1 = 0.001198. 173 there are three such HFI transitions. In this paper The hyperﬁne interaction includes magnetic dipole and we calculate amplitudes of these four HFI transitions. electric quadrupole parts, which can be written as [18]:

(2) (2) Hhf = HA + HB ≡ gI V · I + QI T · R , (4) B. Hyperﬁne mixing

where gI and QI are g-factor and quadrupole moment of the nucleus (see TableI); V and T (2) are irreducible (2) 171 173 electronic tensors of rank 1 and 2, respectively, and R TABLE I. Nuclear moments of isotopes Yb and Yb. is the second rank nuclear tensor: 171Yb 173Yb Ref. (2) 3IiIk + 3IkIi − 2I(2I + 1)δi,k Spin 1/2 5/2 Ri,k = √ . (5) gI 0.9838 −0.2710 [16] 2 6I(2I − 1) QI (bn) 2.80(4) [17] In the following we need the reduced matrix element of this operator: The hyperﬁne mixing coeﬃcients εI,F from Fig.1 be- 3 tween F -sublevels of the levels D1,2 for the isotope with s spin I are given by the expression: (I + 1)(2I + 1)(2I + 3) hI||R(2)||Ii = . (6) 4I(2I − 1) 1 ε = h3D ,I,F |H |3D ,I,F i . (3) I,F −∆ 2 hf 1 Using angular momentum theory [19, 20] we can write In the following discussion we use atomic units ~ = me = matrix elements of the operators HA and HB as:

0 FIJ hJ, I, F |H |J 0,I,F i = (−1)I+F +J pI(I + 1)(2I + 1)g hJ||V ||J 0i , (7) A 1 J 0 I I s 0 FIJ (I + 1)(2I + 1)(2I + 3) hJ, I, F |H |J 0,I,F i = (−1)I+F +J Q hJ||T (2)||J 0i . (8) B 2 J 0 I 4I(2I − 1) I

In the diagonal case J = J 0 these expressions have the form:

1 gI hJ||V ||Ji hJ, I, F |HA|J, I, F i = X · ,X = F (F + 1) − J(J + 1) − I(I + 1) , (9) 2 pJ(J + 1)(2J + 1) p (2) 3X(X + 1) − 4I(I + 1)J(J + 1) 2QI J(2J − 1)hJ||T ||Ji hJ, I, F |HB|J, I, F i = · . (10) 8I(2I − 1)J(2J − 1) p(J + 1)(2J + 1)(2J + 3)

Comparing Eqs. (9,10) with standard definitions of the TABLE II. Relation between hyperfine mixing coefficients hyperfine parameters A and B [21], we find: A B εI,F and εI,F and electronic reduced matrix elements. gI hJ||V ||Ji A = p , (11) I,F 1/2, 3/2 5/2, 3/2 5/2, 5/2 5/2, 7/2 J(J + 1)(2J + 1) A εI,F p (2) 3 3 +290.3 −164.0 −233.7 −240.0 2QI J(2J − 1)hJ||T ||Ji h D2||V || D1i B = . (12) B p εI,F (J + 1)(2J + 1)(2J + 3) 3 (2) 3 −667.6 −362.2 +496.0 h D2||T || D1i Experimental and theoretical values of these constants are discussed in SectionE. According to Eq. (4) the mixing coefficients εI,F (3) ilar to Eqs. (11,12), where hyperfine constants are ex- can be separated in two parts: pressed in terms of the diagonal reduced matrix elements. A B To this end we substitute Eqs. (7,8) in (3) and take into εI,F = ε + ε . (13) I,F I,F account (13). Respective results are summarized in Ta- A B A We can now express coefficients εI,F and εI,F in terms of bleII. Note that the mixings εI,F for both isotopes are the offdiagonal electronic reduced matrix elements, sim- comparable, because they are proportional to the nuclear 3 magnetic moment µnuc = gI I, rather than gI . Interaction (18) mixes levels of opposite parity. As a result, the E1 transitions may be observed between the levels of the same nominal parity. In particular, the levels 2 1 3 2 1 3 C. HFI transition amplitude 6s S0 → 5d6s D1 6s S0 and 5d6s D1 are mixed with odd-parity levels with J = 1, which we designate as n1o. The two main 1,3 The amplitude of the HFI quadrupole transition contributions come from the levels 6s6p P1 [13]. The 2 1 3 2 1 3 6s S0 → 5d6s D1 between hyperfine sublevels is given resultant NSD PV E1 amplitude 6s S0 → 5d6s D1 can by: be written as:

NSD 2F 3 1 0 0 F −M E1 ≡ h3D ,I,F ||E1||1S ,I,Ii = (−1) hgD1,I,F,M|E2q| S0,I,F = I,M i = (−1) PV g1 g0 r (I + 1)(2I + 1)(2F + 1) F 1 I 3 1 FI 1 × hgD1,I,F ||E2|| S0,I,Ii , (14) × AP , (19) −M q M 0 3I 1 1 I where tilde marks a mixed level. The reduced matrix 3 o o 1 element is non-zero only because of this mixing with the GF κ X h D1||VP ||n1 ihn1 ||E1|| S0i 3 A = √ level D2: P E3 − E o 2 n D1 n1 3 o o 1 3 1 h D ||E1||n1 ihn1 ||V || S i hgD1,I,F ||E2|| S0,I,Ii − 1 P 0 . (20) 1 o 3 1 E S0 − En1 = εI,F h D2,I,F ||E2|| S0,I,Ii . (15) In Eq. (19) we again mark mixed states with tilde, but The remaining reduced matrix element can be expressed this time the mixing is caused by the PV interaction (18). in terms of the respective reduced matrix element for Expressions (19) and (20) agree with Eq. (8) from Ref. even isotopes (2): [11] and differ by an overall sign from Ref. [13]. The difference in sign can be caused by another phase con- h3D ,I,F ||E2||1S ,I,Ii = (−1)2I 2 0 vention, for example, by another order of adding angular 0 II × p(2I + 1)(2F + 1) h3D ||E2||1S i momenta [19, 20], or by an error. The dependence of the F 2 2 2 0 NSD amplitude E1PV on the quantum number F is given by = (−1)F −I p(2F + 1)/5 h3D ||E2||1S i . (16) Eq. (19), while the amplitude AP has to be calculated 2 0 numerically. This was already done in Refs. [11–13]. Combining Eqs. (15) and (16) we get the final expression for the HFI amplitude: E. Numerical results and discussion 3 1 hgD1,I,F ||E2|| S0,I,Ii Ground state configuration of Yb is [Xe]4f 146s2. Most F −I p 3 1 = (−1) εI,F (2F + 1)/5 h D2||E2|| S0i . (17) of the low excited states correspond to the excitation of the 6s electron. However, there are also states with ex- Using the experimental result (2) and the values from citations from the 4f subshell. It is important to check TableII one can express all HFI amplitudes in terms whether these states can be neglected in the configura- of the two electronic matrix elements h3D ||V ||3D i and 2 1 tion mixing, reducing the problem to the one with two h3D ||T (2)||3D i (see Eq. (4)), which are to be calculated 2 1 electrons above closed shells. It was demonstrated in numerically. earlier calculations [24–26] that such mixing is strong for some low-lying odd-parity states. In particular, the 13 2 o 2 1 3 4f 5d5/26s (7/2, 5/2) state is strongly mixed with the D. NSD PV amplitude 6s S0 → 5d6s D1 1 14 1 o 4f 6s6p P1 state due to small energy interval between them, δE = 3789 cm−1. Reliable calculations for such Nuclear-spin-dependent PV interaction has the same states require treating the Yb atom as a 16-electron sys- tensor structure, as the magnetic dipole hyperfine inter- tem. This can be done with the CIPT method developed action [22, 23]: in Refs. [25, 26]. On the other hand, the mixing of the 14 3 o GF κ former state with the 4f 6s6p P1 state is small and can HP = √ VP · I , (18) be neglected. The energy interval in this case is 10865 2I cm−1. where GF is Fermi constant and VP is electronic vector In the present work we are interested in the even- 3 3 14 operator. The dimensionless constant κ is of the order parity states D1 and D2 of the 4f 6s5d configuration. of unity. It includes several contributions, the largest The lowest state of the same parity and total angular is from the nuclear anapole moment [6,7]. There are momenta J = 1, or J = 2 containing excitation from 13 several definitions of this constant in the literature; here the 4f subshell is the 4f 5d6s6p (7/2, 3/2)2 state at we follow Refs. [11, 13]. E=39880 cm−1. Corresponding energy interval is large, 4

∆E = 15129 cm−1, and the mixing in this case can 171 173 be safely neglected. Therefore, for the purposes of the TABLE III. Hyperfine constants of isotopes Yb and Yb present work we can treat Yb atom as a system with in MHz. Theoretical values are calculated for the nuclear moments from TableI. two valence electrons above closed shells and apply the standard CI+MBPT method (configuration interaction 171Yb 173Yb Ref. 3 + many-body perturbation theory) [24, 27]. A( D1) Exper. −2040(2) 562.8(5) [15] We use the V N−2 approximation [28] and perform ini- Theory −2349 648 this work tial Hartree-Fock (HF) calculations for the Yb III ion 596 [31] 3 with two 6s electrons removed. The single-electron ba- B( D1) Exper. 337(2) [15] sis states are calculated in the field of the frozen core Theory 249 this work using the B-spline technique [29, 30]. The effective CI 290 [31] A(3D ) Exper. 1315(4) −363.4(10) [15] Hamiltonian for two external electrons has a form 2 Theory 1354 −373 this work CI ˆ ˆ ˆ −351 [31] Hˆ = h1(r1) + h1(r2) + h2(r1, r2), (21) 3 B( D2) Exper. 487(5) [15] Theory 384 this work ˆ ˆ where h1(ri) is a single-electron operator and h2(r1, r2) 440 [31] is a two-electron operator:

ˆ 2 N−2 ˆ h1(r) = cαp + (β − 1)mc + V (r) + Σ1(r), (22) To check the accuracy of this approach we calculate 2 ˆ e ˆ magnetic dipole (A) and electric quadrupole (B) hyper- h2(r1, r2) = + Σ2(r1, r2). (23) 3 3 |r1 − r2| fine constants for the D1 and D2 states of the isotopes 171Yb and 173Yb and compare them with the experiment Here α and β are Dirac matrixes, V N−2 is the potential (see TableIII). One can see that the agreement with the of the Yb III ion including nuclear contribution, Σˆ 1 and experiment for the constants A is better, than for the Σˆ 2 are correlation operators which include core-valence constants B. For the former the difference between the- correlations by means of the MBPT (see Refs. [24, 27] ory and experiment is 3% and 15% respectively, while for for details). the latter it is about 30% for both states. These differ- To calculate transition amplitudes we use the random- ences are most likely due to such factors as neglecting phase approximation (RPA). The same V N−2 potential higher-order core-valence correlations, incompleteness of as in the HF calculations needs to be used in the RPA the basis, and neglecting hyperfine corrections to the Σˆ calculations. The RPA equations for the Yb III ion can operators [32]. The latter corrections were included in be written as calculation [31], where the hyperfine constants (but not the offdiagonal amplitudes) were calculated within the HF ˆ N−2 N (Hˆ − c)δψc = −(f + δV )ψc. (24) same CI+MBPT method using V approximation. As we will see below, the dominant mixing is caused by the Here Hˆ HF is the relativistic HF Hamiltonian (similar to magnetic hyperfine interaction, where theoretical errors ˆ the h1 operator in (22), but without Σˆ 1), index c numer- are 15%, or less. We conclude that the accuracy of our ates states in the core, fˆ is the operator of the external calculations is satisfactory for the purposes of the present field (in our case it is either the nuclear magnetic dipole work. Numerical values of the offdiagonal hyperfine matrix field, or the nuclear electric quadrupole field), δψc is the elements are: correction to the core single-electron wave function ψc N−2 induced by external field, δV is the correction to the 3 3 −6 self-consistent HF potential due to field-induced correc- h D2||V || D1i = −1.71(26) × 10 a.u., (27) 3 (2) 3 −8 tions to all core wave functions. h D2||T || D1i = −4.4(13) × 10 a.u.. (28) The RPA equations are solved self-consistently for all states in atomic core. As a result, the correction to the Here we assign 15% error bar to the magnetic dipole term core potential, δV N−2 is found. It is then used as a and 30% error bar to the quadrupole term. Comparing correction to the operator of the external field and the these values with the data from TableII we see that mag- transition amplitudes T are calculated as netic term dominates over the electric quadrupole term by roughly an order of magnitude. Using experimental ˆ N−2 Tab = ha|f + δV |bi. (25) value (2) we get the final values for the HFI amplitudes, which are listed in TableIV. Note that the signs of the Here the states |ai and |bi are two-electron states found amplitudes depend on the phase conventions and we as- by solving the CI+MBPT equations sume positive sign of the amplitude (2). The final errors in TableIV include experimental error CI (Hˆ − Ea)|ai = 0 , (26) for the amplitude (1) and theoretical errors for amplitudes (27) and (28). Note that the dominant part of these with the CI Hamiltonian given by (21), (22), and (23). errors is common for all hyperfine transitions and the ra- 5

amplitude was calculated in [11] to be: TABLE IV. Reduced matrix elements of the transitions 2 1 0 3 171 6s S0,I,F = I → 5d6s D1,I,F for the isotopes Yb NSI −9 173 E1PV = 1.85 × 10 . (30) (I = 1/2) and Yb (I = 5/2). The HFI quadrupole transition amplitudes (17) are in ea2 and PV E1 transitions (19) are 0 This value agrees with earlier calculations [1,9, 10]. in the units of AP , which was calculated in Refs. [11–13]. Sub- scripts A, B, and tot. correspond to the contributions from The rates of the NSD-PV and the HFI quadrupole the magnetic dipole and electric quadrupole mixings and the transitions depend on the quantum numbers I and F NSD sum of the two. (see TableIV). The amplitude E1PV is roughly two orders of magnitude smaller than (30). The rate of the I,F 1/2, 1/2 1/2, 3/2 5/2, 3/2 5/2, 5/2 5/2, 7/2 quadrupole HFI transitions is: 3 E2A ×10 0.0 +0.643 −0.363 +0.634 −0.752 5 3 (αω) 2 E2B ×10 0.0 0.0 −0.039 +0.021 +0.028 W (E2 ) = |E2 | , (31) I,F 25(2F + 1) I,F 3 E2tot. ×10 0.0 +0.64(10) −0.40(6) +0.66(10) −0.72(12) where E2 is given in TableIV. Putting numbers in E1NSD/A +0.667 +0.471 −0.660 +0.231 +0.667 I,F PV P Eqs. (29) and (31) we get following ratios for the square roots of the rates: tios of the amplitudes are accurate to 3% – 4%. These 1/2 1/2 W (M1) : W (E21/2,3/2) ratios are particularly important for the interpretation of NSI 1/2 the PV experiment. Numerical results in TableIV are in : W (E1PV ) = 263 : 78 : 1 . (32) a good agreement with the estimate made above, which was based on the values of the hyperfine constants of the We see that though M1 transition is the largest, the levels 3D and 3D . quadrupole HFI transition is not very much weaker. The 1 2 NSI TableIV also lists angular factors for the NSD PV am- parity non-conservation rate [22] P ≡ 2|E1PV /M1| ≈ NSD 7 × 10−3. plitude E1PV from Eq. (19), which agree with the factors presented in Ref. [11]1. It is clear that PV amplitude has very different dependence on the quantum numbers I and F than the HFI amplitude (17). This difference G. Conclusions is mainly explained by the difference in the respective 6j-coefficients in Eqs. (7) and (19). The hyperfine in- We calculated hyperfine mixing of the F -sublevels of 3 3 teraction mixes level J = 1 with the level J = 2, while the levels D1 and D2. We found that for both odd- the PV interaction mixes level J = 1 with the odd-parity parity isotopes of ytterbium this mixing is dominated levels J = 1. by the magnetic dipole term. Using experimentally 2 1 3 measured in Ref. [15], the 6s S0 → 5d6s D2 transition amplitude we found amplitudes for the hyperfine induced 2 1 3 F. Transition rates E2 transition amplitudes 6s S0,I,I → 5d6s D1,I,F . These amplitudes appear to be only one order of 2 1 3 magnitude weaker than the respective M1 amplitude Transition 6s S0,I,I → 5d6s D1,I,F may go as M1, or as E2HFI. The PV interaction opens two additional chan- (1). Their knowledge is important for the analysis of the NSI NSD on-going measurement of the parity non-conservation in nels, E1PV and E1PV . These four transitions have different multipolarity and, therefore, different dependence this transition [4]. These amplitudes can interfere with on the transition frequency and different angular depen- the Stark amplitude and mimic PV interaction in the dence [33]. Because of that we can not directly com- presence of imperfections. In particular, they must be pare respective amplitudes. Instead we can compare the taken into account to separate nuclear-spin-dependent square roots of the respective transition rates. parity violating amplitude and to measure anapole 171 173 The rates for the NSI PV amplitude and M1 amplitude moments of the isotopes Yb and Yb. This will do not depend of the quantum numbers I and F and are not only give us information about new PV nuclear determined by the expression: vector moments in addition to the standard magnetic moments, but will also shed light on the PV nuclear 2 W (A1) = (αω)3|A1|2 , (29) forces [6,7, 34–36]. 9 where A1 is the respective reduced amplitude. For M1 transition this amplitude is given by (1). The NSI-PV ACKNOWLEDGMENTS

We are grateful to Dmitry Budker and Dionysis An- 1 −10 Note that the units in Table II in Ref. [11] should be 10 (iea0), typas for stimulating discussions and suggestions. This −9 not 10 (iea0). work was funded in part by the Australian Research 6

Council and by Russian Foundation for Basic Research port from the Gordon Godfree Fellowship and thanks the under Grant No. 17-02-00216. MGK acknowledges sup- University of New South Wales for hospitality.

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