Anomalous Hyperfine Structure of the $^{229} $ Th Ground-State Doublet in Muonic Atom

$Anomalous Hyperfine Structure of the $^{229} $ Th Ground-State Doublet in Muonic Atom$

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[16, nucleus the 1 8,tecoeaiesotnoseiso ik 9 nthe nuclei in excited solids [9] of Dicke of system emission a spontaneous properties cooperative as physicochemical the isomer the [8], thorium study use to chemical to probe the ability to the transition [8], and nuclear bridge environment the electronic the of sensitivity decay via level high the nuclear mention isomeric we the Finally, of interest. scientific erable parameter interaction h excited the 229 ae nmoi tm( atom muonic in gated ut tdigo smrcsaeb h pia methods optical the 12–15]. by [5, re- state a possible isomeric directly, As becomes of photons [10]. studying shell range electron sult, VUV the with the interaction nucleus no emit without the is and there dielectric, absorb Since such can in [11]. channel interest decay particular conversion of is gap band w s h ytmo units of system the use (we 1 n t xsec scnre n[] n h ground the and [2]) in confirmed is 5 existence its and [1] ffc ftevraino h n tutr constant structure fine the relative of The laser variation [6]. a the range and VUV of [3–5] the effect in time transition for nuclear is standard at This metrological new applications. impact a and significant development scien- a technological of have number on could a for that hope breakthroughs a us tific gives proximity range Its is optical transition. the reason the to The of energy physics. low anomalous of the areas different from specialists level S / nti okthe work this In h nqetasto ewe h o-yn isomeric low-lying the between transition unique The 2 1 hncesvateioei tt 1] h eairof behavior The [10]. state isomeric the via nucleus Th / + kblsnIsiueo ula hsc oooo ocwS Moscow Lomonosov Physics Nuclear of Institute Skobeltsyn nmlu antcHprn tutr fthe of Structure Hyperfine Magnetic Anomalous 2 (0 3 tmcobtcetsavr togmgei edat field magnetic strong very a creates orbit atomic / . 2 0) + ( tt nthe in state E 229 ssniiet h ieecsi h ensur hrerad charge mean-square the in differences the to sensitive is h ie ulvl with sublevels mixed the pnaeu ea ftegon tt oteioei tt t state isomeric the to state ground the of decay spontaneous fteioei tt xed 0%fo t au o point- a for le value of inversion its partial from (b) reversed), tha 100% shown is exceeds is sublevels It of state order neutron. isomeric unpaired collective the the for of of framework model Nilsson the the in of magnetization nuclear the of the pia ag o eti auso h intrinsic the of values certain for range optical ASnmes 32.v 61.e 32.10.Fn 36.10.Ee, 23.20.Lv, numbers: PACS states. ground and isomeric the between transition is ainlRsac ula nvriyMPI 149 Kashi 115409, MEPhI, University Nuclear Research National h antchprn MF pitn ftegon n low- and ground the of splitting (MHF) hyperfine magnetic The hncesisd ilcrc ihalarge a with dielectrics inside nucleus Th 7 = 229 ula aeyIsiueo A,BlsaaTlky 2 Mos 52, Tulskaya Bol’shaya RAS, of Institute Safety Nuclear . 229 hncesi uncao ( atom muonic in nucleus Th 8 ± hgon-tt obe sinvesti- is doublet ground-state Th m 0 229 µ 229 . 5 1 − q S hncesdasatninof attention draws nucleus Th / V iseeg smaue in measured is energy (its eV) h accelerated Th, 1 Λ / 2 QCD ~ 229 = Th c 7 r loo consid- of also are [7] ) 1 = ∗ F h uno the on muon The . 2 = n h strong the and ) α d h H pitn fthe of splitting MHF the (d) , dcyo the of -decay Dtd eray1,2018) 19, February (Dated: µ 1 − S uncAtom Muonic 1 / 2 .V Tkalya V. E. 229 e Th 2 ) g ∗ K a encluae osdrn h distribution the considering calculated been has peia oriaesse,and system, coordinate spherical h unDrcwv ucina h origin. the at function wave Dirac muon the σ Here the ovn h ia qain o h ailprso the of parts radial the for equations large, Dirac the solving where un respectively, muon, fti edi ie ytefruafrteFricontact Fermi the for formula the by interaction given is field this of ok rmsn nrgr oeprmna research. and experimental unusual to very regard is in situation promising This looks electric the etc. the between sublevels, transitions of MHF optical existence in and effect) possible transition penetration the monopole the mixing, (or states size the nuclear dynamic the finite of of role value important effect its an from nucleus, state point-like isomeric a the for of structure MHF of tion htteMFsltighsanme fnntiilfea- ( non-trivial of of case number the a in has tures splitting here MHF demonstrate the We that therein). references exam- and for (see [18–28] levels ple nuclear of splitting (MHF) hyperfine fncersbeesadsotnosdcyo h ground the of decay state spontaneous and sublevels nuclear of e ( tem ei edi h etro the of center the in field netic h uni the in muon The aeUiest,Lnnkegr,Mso 191 Russia 119991, Moscow gory, Leninskie University, tate atradrdcdpoaiiyo h nuclear the of probability reduced and factor h amplitude The h em otc interaction contact Fermi The r h al arxs and matrixes, Pauli the are eso the of vels 1 ∗ S 5 µ x io h obe ttsi osbebetween possible is states doublet the of ii xf xg g / 1 1 − m ( / 2 S = ula oe ihtewv functions wave the with model nuclear kspae c the (c) place, akes x ′ 2 ′ 1 + e ( ( ) / a h eito fMFstructure MHF of deviation the (a) t x x 229 r/R n small, and , 2 hl fmoi tmadthe and atom muonic of shell oteisomeric the to and ) so hse3,Mso,Russia Moscow, 31, shosse rskoe 3 229 2 + ) iencermgei ioe(the dipole magnetic nuclear like / energy − H hGon-tt obe in Doublet Ground-State Th 2 229 Th 0 + b µ where , o 111 usaand Russia 115191, cow m ( f hgon-tt obe and doublet ground-state Th smrcsaemyb nthe in be may state isomeric E = ) ψ (1 ( µ ∗ x µ 3 1 + − S + ) hc oss ftemo on on bound muon the of consist which µ r h asso h lcrnand electron the of masses the are / (0) µ 1 2 B 16 1 − / + 3 f S 2 − b (7 = r a ecluae ueial by numerically calculated be can π ) 1 ( ( / 1 x E V E . 2 m stemo oriaei the in coordinate muon the is 3 m 8 e/ ) 229 tt eut nasrn mag- strong a in results state / 0 ( opnnsof components , ± − µ e x 2 2 rniinwhich transition µ 0 + )) Th m 1 . B 229 ψ 5 xf tt,teaoaydevia- anomaly the state, − e e scnie h sys- the consider us Let . µ ) σ V eesin levels eV) 2 ∗ R (0) hnces h value The nucleus. Th steBh magneton, Bohr the is ( V h ata inversion partial the : | x ψ 0 ( 0 = ) µ x 1 = steapiueof amplitude the is (0) )) xg . | 2 , 2 229 A ( , x 1 ψ hnucleus. Th 0 = ) / µ 3 ( mi the is fm x ) . : (1) 2

5 contact interaction is very significant. However, since + rnH3ê2 @631 DL the muon density decreases quickly to the nuclear edge 4 the obtained values are grossly overestimated. rm rm The distributed magnetic dipole model. The influence 3 c + of the finite nuclear size on the MHF splitting was first rnH5ê2 @633 DL considered by Bohr and Weisskopf [30]. Later the ef- 2 fect of the distribution of nuclear magnetization on MHF 1 structure in muonic atoms was studied by Le Bellac [31]. According to their works, in the case of deformed nucleus 0 the energy of sublevels is given by Eq. (2), where 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 3 x=RêR0 Eint = d r j(r)A(r) (3) Z Figure 1: (color online). Dimensionless densities of muon + is the energy of interaction of the muon current j(r) = (ρµ), unpaired neutron (ρn) in the ground 5/2 [633] state + 0 + m eψ (r)αψµ(r) (α = γ γ, γ are the Dirac matrices) and isomeric 3/2 [631] state, ρc is the core magnetization. µ with− the vector potential of the electromagnetic field A(r) generated by the magnetic moment of the nucleus. average radius of the 229Th nucleus that has a form of For a system of “rotating deformed core (with the collec- ℜ charged sphere, b = mµR0, E and V (x) are, respectively, tive rotating angular momentum ) + unpaired neutron the muon binding and potential energies (in the units of (with the spin Sn)”, vector-potential is determined by the mµ) in the field produced by the nucleus protons. (For relation [30, 31] the lower muonic states, electron screening plays a neg- 1 ligible role [18, 27]. Therefore we neglect here the effects A(r)= d3R [ρ (R)g S + ρm(R)g ℜ] ∇ , n S n c R r r R due to the influence of the electron shell on the muon − Z × | − (4)| wave function.) where ρ (R) is the distribution of the spin part of the We assume that the proton density of the nucleus has n nuclear moment and ρm(R) is the distribution of the core the Fermi shape ρ (x)= ρ /[1 + exp((x 1)/χ)], where c p 0 magnetization, g is the spin g-factor, and g is the core χ = [0.449+ 0.071(Z/N)]/R is diffuseness− or the half- S R 0 gyromagnetic ratio. The distributions ρ (R) and ρm(R) density parameter of the proton density Fermi distribu- n c are normalized: d3Rρ (R)=1, d3Rρm(R)=1. tion [29]. The density is normalized by the condition n c ∞ Here we use the standard nuclear wave function ρ (x)x2dx = Ze, where Z is the nucleus charge. R R 0 p [32] ΨI = (2I + 1)/8π2DI (Ω)ϕ (R), where The muon wave function is normalized by the condi- MK MK K R ∞ I 2 DMK (Ω) is thep Wigner D-function of the Euler angles tion 0 ρµ(x)x dx =1, where ρµ(x) is the muon density 2 2 are denoted, collectively, by Ω, ϕK (R) is the wave func- ρµ(x)R = g (x)+ f (x). The result of calculation of the tion of external neutron coupled to the core, K is the muon density is presented in Fig. 1. To evaluate the mag- component of I along the symmetry axis of the nucleus, netic field one can use Eq. (1) with the value of the muon 3/2 and M is the component of I along the direction of mag- wave function given by ψµ(0) = Y00(ϑ, ϕ)g(0)/R0 , netic field. where Y00(ϑ, ϕ) is the spherical harmonic, and from cal- As follows from Eqs. (3–4), Eint consists of two parts. culations it follows that g(0) = ρµ(0) = 1.76. The first part is the interaction of the muon with the Thus, according to Eq. (1) thep magnetic field at the 229 external unpaired neutron and the second one is the in- center of the Th nucleus is about 23 GT. Interaction of teraction of the muon with the rotating charged nuclear point magnetic moments of the ground state (µgr =0.45) core. These energies are calculated in accordance with and isomeric state (µ = 0.076) with the magnetic field is − formulas from [31]. In our case of the muon interacts leads to a splitting of the nuclear levels. The energy of with the nucleus in the head levels of rotational bands the sublevels is determined by the formula (for such states we have K = I), and two contributions F (F + 1) I(I + 1) s(s + 1) take the following form: E = Eint − − , (2) 2Is n (core) I IgK ρn(y) 3 Eint = E0 m d y where Eint = µgr(is)µN Hµ is the interaction energy, I +1 gR hMi − Z ρc (y) − y µN = e/2Mp is the nuclear magneton (Mp is the proton 3 Θ(I,θ) mass), I is the nuclear state spin, s is the muon spin. The 1 x f(x)g(x)dx . (5) × Z0 − 1 quantum number F takes two values F = I 1/2 for the ± 2 2 ground and isomeric states and determines the sublevels Here, E0 = 2e Mp/[3(MpR0) ], gK is the intrinsic g − ∗ energy. The resulting energy values are given in Fig. 2. factor, ρn(y) = ϕK (y) ϕK (y), y = R/R0, Θ(I,θ) = The MHF splitting found in the model of the Fermi 4π/5Y20(θ)(2I + 1)/[I(2I + 3)]. The first term in the p 3

∞ - 229 * square brackets in Eq. (5), = f(x)g(x)dx = 3250 (µ Th) hMi 0 1S 0.1895, corresponds to the interactionR of the muon with 3200 F − a point nuclear magnetic dipole. The resulting energy 150 µ =-0.076 3 is 2 sublevels are close to the values calculated with Eq. (1). 100 + 5 eV 2 501 3/2 3 For the unpaired neutron the wave functions ϕK 1 0 + were taken from the Nilsson model. The structure 5/2 Inversion M1 229 -50 µ + of the intrinsic state ϕK of the Th ground state -100 = 0.45 1 E0 + π + gr 4 5/2 (0.0) is K [NnzΛ] = 5/2 [633]. The structure of -150 5 + + the isomeric state 3/2 (7.8 eV) is 3/2 [631] [33]. For (eV) Energy -200 Point Bare nuclear each of these states, the wave function has the form -250 nucleus magnetic 2 -300 dipoles ϕK = φΛ(ϕ)φΛ,nr (η)φnz (ζ) where the quantum num- Distributed Mixed ∆g ber nr = (N nz Λ)/2, the variables on the axes -35040 magnetic F=2 levels K − − -4005 dipoles ∆B(M1) ζ = R0 Mpωzycosθ, η = R0 Mpω⊥ysinθ, the energies of thep oscillatory quanta ωp = ω 1+2δ/3 and z 0 Figure 2: (color online). Magnetic hyperfine structure of the ω⊥ = ω 1 4δ/3, where ω = 41/Ap1/3 MeV is the 229 0 − 0 Th ground-state doublet in muonic atom in various models. harmonicp oscillator frequency, δ = 0.95β, and β is the The uncertainty range for the energy of the states due to + + parameter of the deformation of the nucleus defined in variations of parameters gK and BW.u.(M1; 3/2 → 5/2 ) terms of the expansion of the radius parameter R = (see text for details) is shown on the right. R0(1 + βY20(θ)+ ...). The constituent wave functions are as follows: φΛ(ϕ)= 2 of the states with the equal values of F ). From Fig. 3 iΛϕ −η /2 Λ (Λ) 2 e /√2π, φΛ,nr (η) = e η Lnr (η )/Nη, φnz (ζ) = 2 it follows that in the range 0.30 < gK < 0.29 the −ζ /2 (Λ) + − − e Hnz (ζ)/Nζ , where Lnr is the generalized Laguerre 3/2 [631] state has a nonzero magnetic moment, whereas polynomial, Hnz is the Hermite polynomial [34], Nη,ζ the MHF splitting is absent or very weak. Conversely, are the normalization factors. The density distributions the magnetic moment of the isomeric level equals to zero + of the unpaired neutron in the states 5/2 [633] and for gK 0.206, while the MHF splitting is relatively 3/2+[631] averaged over the angles θ and ϕ are shown large. The≈ − reason is the following. The magnetic field in Fig. 1. In our numerical calculations we took into ac- generated by the spin of the nucleon is sensitive to the count the asymmetry of the nucleon wave functions in non-sphericity of the wave functions ϕK . This leads to Eq. (5), but neglected the small difference between ωz the appearance of the additional factor Θ(I,θ) in the spin and ω⊥. part of the Eq. (5) [30, 31]. Averaging over the angles re- For the core magnetization we used the classical den- duces the spin contribution in respect to the orbital part. sity of magnetic moment, ρm x2/[(1 + exp((x 1)/χ)], A small imbalance emerged in the system leads to the vi- c ∝ − obtained from proton density ρp by averaging over the olation of the “fine tuning” between the spin and orbital angles. Such quadratic dependence was used in [19, 35]. parts of the magnetic moment and to the effect described m The normalized function ρc is shown in Fig. 1. above. This mechanism can also occur in other nuclei The resulting scheme of MHF splitting for with low energy (up to some kiloelectronvolts) levels. − (µ 229Th)∗ is shown in Fig. 2. For g-factors of Mixing of the sublevels with F = 2. To find the final 1S1/2 the ground state we have used values, which are ac- position of the sublevels we now consider the mixing of cepted nowadays: g = 0.309, g = 0.128 [36]. The the states with F = 2 [25]. The interaction energy, , R K E reduction of MHF structure in comparison with the of the nuclear and muon currents during the transition + model of point nuclear magnetic dipole is about 56% for between the 3/2 , F = 2 sublevel with the energy E1 +| i the 5/2+(0.0) state. and the 5/2 , F = 2 sublevel with the energy E2 can | i For calculation of the isomeric state we have taken be found from equations given in Refs. [39, 40]. They g = 0.309 and g = 0.29 which is obtained from generalize the static Bohr-Weisskopf effect for the case of R K − the mean value gK gR =0.60 (the values gK gR = nuclear excitation at the electron (muon) transitions in 0.59 0.14 and |0.61− 0.|10 were measured in| [38]).− The| the atomic shell. For M1 transition we obtain ± ± gyromagnetic ratio gR =0.309 0.016 from [36] is deter- + + ± = E0ξ (15/2)BW.u.(M1;3/2 5/2 ), mined with a much higher precision than gK gR for the E hMi → + | − | p + + −2 band 3/2 [631], and existing uncertainty in gK gR is where B (M1;3/2 5/2 )=3.0 10 is the re- | − | W.u. → × related exclusively with gK: gK =0.29 0.12. This leads duced probability of the nuclear isomeric transition in ± to uncertainty in the position of levels (see in Fig. 2). Weisskopf’s units [41], ξ is a factor that takes into ac- A somewhat paradoxical situation can take place be- count the dynamic effect of the nuclear size [40] or the cause of the complex structure of the magnetic moment penetration effect [42]. Calculation of the nuclear current of the isomeric state and the behavior of the muon wave with the neutron wave function in the Nilsson model gives function (currently we consider a variant without mixing the value of ξ =0.45. As a result, we have 150 eV. E ≃ 4

E +40 µ ∆ 0.02 is is = 0, E=0

+ )

E is 30 N 0.00 µ (

+ Eis 20 E + µ = 0, ∆E=0 3/2 is -0.02 is + , F= µ Eis 10 2 Eis 0 -0.04 E

Eis -10 ∆ -0.06 + =1 µ Eis -20 , F E 3/2 is -0.08 Eis -30 -0.10 Energy of sublevels(eV)

E -40

is Magnetic moment

E -50 -0.12 is -0.19 -0.21 -0.23 -0.25 -0.27 -0.29 -0.31 -0.33 gK Figure 4: (color online). Range of values of gK and B (M1; 3/2+ → 5/2+) at which the transitions between Figure 3: (color online). Imbalance of the MHF interaction W.u. the sublevels lie in the optical or VUV ranges. The dotted for the composed magnetic moment of the isomeric state in 229 lines show the areas where the sublevels have the same energy. Th: the energies of the sublevels relative to Eis and the magnetic moment µis as a function of the gyromagnetic factor gK in the absence of mixing of the states with F = 2 (see text for details). Accordingly, the component of the transition, which con- nects the state 5/2+, F = 3 with b 5/2+, F = 2 gives the main contribution| to thei transition| 2 in Fig. 2i. This The energies of the new sublevels with F = 2 are cal- transition occurs via a spin flip of the muon without culated according to the formulas [43]: changing nuclear state. + 2 2 The main decay channels of the 5/2 , F =3 sublevel E1′,2′ = [E1 + E2 (E1 E2) + (2 ) ]/2, ± − E is the transition to the 5/2+, F =| 2 ground statei sub- p | i where E1′(2′) are the energies of the new sublevels level (labeled as 1 in Fig. 2). The probability of the ′ 3/2+(5/2+), F = 2 . We emphasize that these ener- transition 1 calculated by means of formulas of Refs. | i −11 gies are valid for the most probable values of gK and [26, 44] is 2.8 10 eV. The transition is accompa- + + × BW.u.(M1;3/2 5/2 ). Variations of the parame- nied by the emission of conversion electrons. Muon in → − 229 ∗ ter gK in the range (gK = 0.29 0.12) and the re- (µ Th) is practically inside the thorium nucleus. ± 1S1/2 duced probability of the nuclear transition (currently Electronic shell perceives the system “muon + Thorium 3 10−3 B (M1;3/2+ 5/2+) 5 10−2 [41]) × ≤ W.u. → ≤ × nucleus” as the Actinium nucleus of charge 89. Therefore, gives a fairly large area of uncertainty (see in Fig. 2) in the internal conversion will take place in the electron shell the position of the levels. of the Ac atom. For the transition 1 the internal conver- 5 In Fig. 4 we reproduce values of gK and sion coefficient αM1 is equal to 6.6 10 (it have been + + × BW.u.(M1;3/2 5/2 ) with the energy dif- found using the code described in [8]) with the full width → −5 ference between the sublevels less than 10 eV. Γtot =1.8 10 eV. This means that the half-life of the The existing of the optical range for the tran- sublevel 5×/2+, F =3 is less than 2.5 10−11 s. I.e. the + + ′ sitions 5/2 , F = 3 3/2 , F = 2 and relaxation| of this leveli is completed prior× to the muon + | ′ i+ → | i 3/2 , F = 2 3/2 , F = 1 is an unusual fea- absorption ( 10−7 s) or the muon decay (2 10−6 s). | i → | − i 229 ∗ ture of the MHF structure in (µ Th) . It gives a ∼ 2 × 1S1/2 Taking into account the coefficient b , the radiation hope that advanced optical methods can be applied for width of the transition 2 is 1.1 10−14 eV and the to- −7 × 7 the study of this extraordinary nuclear state. tal width equals to 7.0 10 eV (αM1 = 6.0 10 ). Transitions between sublevels. Both sublevels of the Thus, the probability of× the isomeric state excitation× at + isomeric state 3/2 (7.8 eV) lie below the ground-state the decay of the ground state is 3-4%. Modern muon fac- + sublevel 5/2 , F = 3 . As a result, spontaneous transi- tories generate of 105 muonic atoms per second. Thus we | i 3 tions to the isomeric level accompanied by its population can expect the formation of the order of Nis 3 10 become possible. isomeric nuclei per second. From the measurements≃ × of Mixing of the sublevels with F = 2 significantly in- the corresponding conversion electrons one can hope to creases the probability of the transitions 2 and 4 in Fig. 2 identify experimentally the fast transitions 3, 4, and between the sublevels of the ground and isomeric states. 5. They are comparable in intensity with the transi- The wave functions of the new sublevels have the form tions 1 and 2. The measurement of the parameters 3/2+, F =2 ′ = √1 b2 3/2+, F =2 + b 5/2+, F =2 of the transitions can give information about gK and + + | + i′ − +| i |2 + i B(M1;3/2 5/2 ). 5/2 , F =2 = b 3/2 , F =2 + √1 b 5/2 , F =2 , → | i − | i − | i The value N 3 103 s−1 is a lower estimate. The is ≃ × where b = (E ′ E )/ (E ′ E )2 + 2 0.47 [43]. muon capture by atom is followed by a cascade of muon 1 − 1 1 − 1 E ≃ p 5 transitions in the atomic shell. The process of nonradia- [13] S. Stellmer, M. Schreitl, and T. Schumm, Sci. Rep. 5, tive nuclear excitation by means of direct energy transfer 15580 (2015). from the excited atomic shell to the nucleus via the vir- [14] J. Jeet, C. Schneider, S. T. Sullivan, et al., Phys. Rev. 114 tual X-photon is possible if the muon transition is close Lett. , 253001 (2015). [15] A. Yamaguchi, M. Kolbe, H. Kaser, et al., New J. Phys. in energy and coincides in type with the nuclear one (see 17, 053053 (2015). for example [24]). This effect was predicted by Wheeler [16] J. A. Wheeler, Rev. Mod. Phys. 21, 133 (1949). [16]. In the case of resonant excitation of the levels of the [17] Y. N. Kim, Mesic Atoms and Nuclear Structure (North- + 5/2 [633] rotational band the probability of the popula- Holland Publ. Comp., Amsterdam, 1971). tion of the 3/2+[631] isomeric state is estimated by 1-2%. [18] C. S. Wu and L. Wilets, Annu. Rev. Nucl. Sci. 19, 527 (This value corresponds to the probability of the isomer (1969). population at the α decay of 233U, which involves mainly [19] J. Johnson and R. A. Sorensen, Phys. Lett. B 28, 700 the levels of the 5/2+[633] band in 229Th.) However, a (1968). [20] R. Engfer and F. Scheck, Z. Phys. 216, 274 (1968). precise account of the isomer population in muonic tran- [21] R. Baader, H. Backe, R. Engfer, et al., Phys. Lett. B 27, sitions can be given only experimentally. 428 (1968). Another interesting consequence of the F = 2 states [22] V. Klemt, J. Speth, and K. Goke, Phys. Lett. B 33, 331 mixing is the possible existence of the E0 component at (1970). the transition 5 in Fig. 2. The E0 transition is sensitive [23] R. Link, Z. Physik 269, 163 (1974). to the differences in the mean-square charge radii R2 [24] A. Ruetschi, L. Schellenberg, T. Q. Phan, et al., Nucl. p 422 [45]. The probability of the transition depends onh thei Phys. A , 461 (1984). [25] S. Wycech and J. Zylicz, Acta Phys. Pol. B 24, 637 E0 transition strengths ρ(E0)2, which is proportional to 2 2 2 2 2 4 2 (1993). b (1 b )( R + R + ) /R . ρ(E0) =0 in the − h pi5/2 − h pi3/2 0 [26] F. F. Karpeshin, S. Wycech, I. M. Band, et al., Phys. framework of the simplified model for the charge distri- Rev. C 57, 3085 (1998). 2 354 bution ρp used in this work. In reality the radii Rp 3/2+ [27] D. F. Measday, Phys. Rep. , 243 (2001). 2 h i 112 and Rp 5/2+ can differ in magnitude and the detection [28] K. Beloy, Phys. Rev. Lett. , 062503 (2014). of theh E0i transition would be a step towards a better un- [29] W. M. Seif and H. Mansour, Int. J. Mod. Phys. E 24, 1550083 (2015). derstanding of the properties of the low-energy doublet 77 229 [30] A. Bohr and V. F. Weisskopf, Phys. Rev. , 94 (1950). in Th. [31] M. Le Bellac, Nucl. Phys. 40, 645 (1963). This research was carried out by a grant of Russian [32] A. Bohr and B. R. Mottelson, Nuclear Structure. Vol. II: Science Foundation (project No 16-12-00001). Nuclear Deformations. (World Scientific, London, 1998). [33] R. G. Helmer and C. W. Reich, Phys. Rev. C 49, 1845 (1994). [34] M. Abramowitz and I. A. Stegun, Handbook of Mathe- matical Functions (National Bureau of Standards, Wash- ∗ Electronic address: [email protected] ington, D.C., 1964). [1] B. R. Beck, J. A. Becker, P. Beiersdorfer, et al., Phys. [35] M. Finkbeiner, B. Fricke, and T. Kuhl, Phys. Lett. A Rev. Lett. 98, 142501 (2007). 176, 113 (1993). [2] L. von der Wense, B. Seiferle, M. Laatiaoui, et al., Nature [36] J. C. E. Bemis, F. K. McGowan, J. J. L. C. Ford, et al., 533, 47 (2016). Phys. Scr. 38, 657 (1988). [3] E. Peik and C. Tamm, Europhys. Lett. 61, 181 (2000). [37] A. M. Dykhne and E. V. Tkalya, JETP Lett. 67, 251 [4] C. J. Campbell, A. G. Radnaev, A. Kuzmich, et al., Phys. (1998). Rev. Lett. 108, 120802 (2012). [38] L. Kroger and C. Reich, Nucl. Phys. A 259, 29 (1976). [5] G. A. Kazakov, A. N. Litvinov, V. I. Romanenko, et al., [39] E. V. Tkalya, Nucl. Phys. A 539, 209 (1992). New J. Phys. 14, 083019 (2012). [40] E. V. Tkalya, JETP 78, 239 (1994). [6] E. V. Tkalya, Phys. Rev. Lett. 106, 162501 (2011). [41] E. V. Tkalya, C. Schneider, J. Jeet, and E. R. Hudson, [7] V. V. Flambaum, Phys. Rev. Lett. 97, 092502 (2006). Phys. Rev. C 92, 054324 (2015). [8] V. F. Strizhov and E. V. Tkalya, Sov. Phys. JETP 72, [42] E. L. Church and J. Weneser, Ann. Rev. Nucl. Sci. 10, 387 (1991). 193 (1960). [9] R. H. Dicke, Phys. Rev. 93, 99 (1954). [43] A. S. Davydov, Quantum Mechanics (Pergamon Press, [10] E. V. Tkalya, A. N. Zherikhin, and V. I. Zhudov, Phys. Oxford, England, 1965). Rev. C 61, 064308 (2000). [44] R. Winston, Phys. Rev. 129, 2766 (1963). [11] E. V. Tkalya, JETP Lett. 71, 311 (2000). [45] E. L. Church and J. Weneser, Phys. Rev. 104, 1382 [12] W. G. Rellergert, D. DeMille, R. R. Greco, et al., Phys. (1956). Rev. Lett. 104, 200802 (2010).