aa r X i v : . [ phy s i c s . a t o m - ph ] N ov Electric dipole moments of actinide atoms and RaO molecule
V. V. Flambaum
School of Physics, University of New South Wales, Sydney 2052,Australia and Institute for Advanced Study, Massey University (Albany Campus),Private Bag 102904, North Shore MSC Auckland, New Zealand (Dated: October 22, 2018)We have calculated the atomic electric dipole moments (EDMs) induced in
Pa and
Ac bytheir respective nuclear Schiff moments S . The results are d ( Pa) = − . · − [ S/ ( e · fm )] e · cm = − . · − η e cm ; d ( Ac) = − . · − [ S/ ( e fm )] e cm = − . · − η e cm . EDM of Pa is3 · times larger than Hg EDM and 40 times larger than
Ra EDM. Possible use of actinidesin solid state experiments is also discussed. The T,P-odd spin-axis interaction in RaO molecule is500 times larger than in TlF.
PACS numbers: PACS: 32.80.Ys,21.10.Ky
I. INTRODUCTION
Measurements of atomic EDM allows one to testCP violation theories beyond the Standard Model (see,e.g. [1, 2]). The best limits on atomic EDM have beenobtained for diamagnetic atoms Hg [3] and Xe [4]; thereis also limit on T,P-odd spin-axis interaction in TlFmolecule[5]. EDM of diamagnetic atoms and moleculesis induced by the interaction of atomic electrons withthe nuclear Schiff moment. Schiff moments produced bythe nuclear T,P-odd interactions have been calculated inRefs. [7, 8, 9, 10, 11, 12, 13]. 2-3 orders of magnitudeenhancement can exist in nuclei with octupole deforma-tion [14] or soft octupole vibrations [15]. This motivatednew generation of atomic experiments with
Ra (seee.g. [16]) and
Rn. Current status of EDM experi-ments and theory can be found on the website of theINT workshop [17]. Most accurate calculations of atomicEDM produced by the nuclear Schiff moments have beenperformed for Hg, Xe, Rn, Ra,Pu [18], Yb and He [19]atoms and TlF molecule [6, 18]. In this work we wouldlike to note that several orders of magnitude larger effectsappear in different systems.
II.
RaO
MOLECULE
Due to the octupole enhancement [14] the Schiff mo-ments of , Ra exceeds that of , Tl [7, 8] ∼ Z . There-fore , one may hope that experiments with RaO maybe up to 3 orders of magnitude more sensitive to thenuclear T,P-violating interactions than the experimentswith TlF. The results of molecular calculations are usu-ally expressed in terms of the following matrix element: X = − π h Ψ | [ ∇ · n , δ ( R )] | Ψ i , (1)where Ψ is the ground state wave function and n is theunit vector along the molecular axis. The T,P-odd spin- axis interaction constant can be expressed in terms of theSchiff moment S (see e.g. [18]): h Ψ | H W | Ψ i = 6 X S · n . (2)Here S is the Schiff moment vector. For TlF molecule X = 7475 atomic units (a.u.) [6, 18]. We use this resultto estimate X for RaO. One can view TlF molecule asan ion compound Tl + F − . The electronic configurationof Tl + is ... s . The electric field of F − produces s-p hy-bridization of Tl + orbitals and non-zero matrix element X in Eq.(1). The electronic configuration of Ra, ... s , issimilar to Tl + , and the T,P-odd spin-axis interaction inRaO is also due to the s-p hybridization. A simplest wayto estimate X for RaO is to use known Z dependenceof the Schiff moment effect: Z R ( Zα ) where R ( Zα ) isthe relativistic factor [7]. This gives X ( RaO ) /X ( T lF ) ≈ . S ( Ra ) /S ( T l ). A slightly more accurate result may beobtained using existing atomic EDM calculations (atomicEDM and molecular T,P-odd spin-axis interaction de-pend on the same matrix elements of the Schiff mo-ment field). Hg atom has the same electronic config-uration as Tl + . The ratio of Ra and Hg EDM wascalculated in [18]: d ( Ra ) /d ( Hg ) = 3 . S ( Ra ) /S ( Hg ).The larger value for Ra is due to higher nuclear charge: Z =88 for Ra, Z =80 for Hg and Z =81 for Tl. Using d ( Ra ) /d ( Hg ) = 3 . S ( Ra ) /S ( Hg ) we obtain an estimate X ( RaO ) /X ( T lF ) ≈ . S ( Ra ) /S ( T l ). As a final valuewe will use an intermediate result X ( RaO ) /X ( T lF ) ≈ . S ( Ra ) /S ( T l ) which is between the EDM estimate andthe relativistic factor estimate. This gives the T,P-oddspin-axis interaction in RaO h Ψ | H W | Ψ i = 1 . × ( S · n ) a.u. . (3)The Ra Schiff moment S is 200 times larger than the TlSchiff moment, altogether we obtain 500 times enhance-ment in RaO in comparison with TlF. Note that the errorof this number is probably dominated by the nuclear cal-culations of the Schiff moments. III. EDM OF ACTINIDE ATOMS
The largest Schiff moment was found for
Pa [14]where the atomic calculation of EDM is absent. Belowwe obtain the result for this EDM. For the first time theenhancement of P,T-violation in
Pa nucleus was foundby Haxton and Henley in Ref. [20]. This nucleus containsvery close excited level (220 eV) which has the same spinas the ground state level ( I =5/2) and opposite parity.These ground and excited states, 5 / + and 5 / − , canbe mixed by the nucleon P,T-odd interaction. Haxtonand Henley performed calculations in the Nilsson model(using single-particle orbitals for the quadrupole nucleardeformation) and found that nuclear EDM and magneticquadrupole moment are significantly enhanced. Unfor-tunately, they did not calculate Schiff moment. Calcu-lation of the Schiff moment was performed in Ref. [14]assuming different model (octupole nuclear deformation)which gives an additional enhancement due to the collec-tive nature of the Schiff moment in nuclei with octupoledeformation. A similar mechanism produces enhance-ment of the T,P-odd electric octupole moment [21]. Itis interesting that in Pa all four T,P-odd nuclear mo-ments (Schiff, EDM, magnetic quadrupole and octupole)contribute to atomic EDM. Let us start from the Schiffmoment which gives a dominating contribution in
Pa.The electron configuration of Pa is ... s f d . TheSchiff moment field is confined inside the nucleus. Thehigh-wave 6 d and 5 f electrons practically do not pene-trate inside the nucleus and have very small matrix ele-ments for Schiff moment field. If we neglect these smallmatrix elements, the atomic EDM comes from the Ra-like core ... s . In this approximation we may use theresult for Ra, d = − . · − [ S/ ( e f m )] e cm from Ref.[18], to calculate Pa EDM. The coefficient actually shouldbe slightly larger since the Pa charge Z = 91 is largerthan the Ra charge Z = 88. Another reference point isPu, Z=94, where d = − . · − [ S/ ( e · f m )] e · cm from Ref. [18]. Pu has the electron configuration ... s f where the contribution of 5 f electrons is notvery important (as explained above). The Pa atom, Z = 91, is exactly in between Ra, Z = 88, and Pu,Z=94. Therefore, we take the average value as Pa EDM, d = − . · − [ S/ ( e · f m )] e · cm . The accuracy of thisresult is about 20 % (see Ref. [18]).Now discuss the contributions of other T,P-odd mo-ments. Nuclear EDM contributes in combination withmagnetic hyperfine interaction between nucleus andatomic electrons [22]. However, this contribution has rel-atively slow increase with nuclear charge, ∼ Z , and maybe neglected. The contributions of magnetic quadrupole,electric octupole and Schiff moments increase faster than Z . Electric octupole and magnetic quadrupole induceatomic EDM only if electron angular momentum J is notzero (since the EDM vector d i can only be produce fromnuclear magnetic quadrupole tensor M ik as d i ∼ M ik J k or octupole third rank tensor as d i ∼ O ikj J k J j ). Theelectron angular momentum J = 11 / d and 5 f electrons. The matrix elements ofvery singular magnetic quadrupole and electric octupolefields for these orbitals are small since these matrix el-ement comes from small distances where the high-waveelectrons do not penetrate. If we neglect these small ma-trix elements, the atomic EDM comes from the Ra-likecore ... s which has zero electron angular momentumand no contributions from the magnetic quadrupole andelectric octupole. Atomic EDM in this approximationcomes entirely from the Schiff moment field which mixes s − p orbitals. There are additional arguments why wedo not need to include the electric octupole and magneticquadrupole contributions into our approximate calcula-tions. Without any enhancement (e.g. for spherical nu-clei), the electric octupole contribution to atomic EDM issubstantially smaller than the magnetic quadrupole andSchiff contributions (see comparison of the correspondingmatrix elements in Ref. [21]). The octupole deformation(or the soft octupole mode) gives the collective enhance-ment of the Schiff and octupole moments, however, itdoes not enhance the magnetic quadrupole (the smallnuclear energy denominator is a common factor for allthree contributions, so it does not influence the ratio ofthem). These arguments stress again importance of theSchiff moment contribution.The Schiff moment of Pa was calculated in Ref. [14]: S = 1 . · − e fm η where η is the dimensionlessstrength of the nucleon P, T − odd interaction in unitsof the Fermi constant. Substituting this value we obtainEDM of Pa atom: d = − . · − η e cm . (4)This value is 3 · times larger than Hg atomic EDM,4 · − η e cm , and 40 times larger than Ra EDM,2 . · − η e cm (this comparison is based on the Schiffmoments from Refs. [8, 14] and the atomic calculationsfor Hg and Ra from Ref. [18] ).A similar calculation for Ac ( Z = 89, atomic con-figuration ... s d , J = 3 /
2) gives d = − . · − [ S/ ( e f m )] e cm = − . · − η e cm . (5)We would like to suggest another possible applicationof actinides. Recently, the measurements of electronEDM and nuclear Schiff moments in the solid compoundscontaning rare-earth atoms (e.g. gadolinium) have beenproposed [23, 24, 26] (corresponding calculations havebeen performed in [25, 26]). The actinides are electronicanalogues of rare-earth atoms. Because of rapid increaseof atomic EDM with nuclear charge it may be worth con-sidering similar compounds with actinides. For example,uranium and thorium have isotopes which are practicallystable. Atomic EDM induced by the electron EDM in-creases with Z as [27] d ∼ Z ( j + 1)(4 γ − γ (6)where γ = (( j + 1 / − Zα ) / and j is the electronangular momentum (maximal contribution comes from j = 1 / Z = 92) with gadolinium( Z = 64) we see that in uranium compounds the effect is ∼ [1] V. F. Dmitriev, and I. B. Khriplovich, Physics Reports,
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