Extension of the Schiff theorem to ions and molecules
aa r X i v : . [ phy s i c s . a t o m - ph ] D ec Extension of the Schiff theorem to ions and molecules
V. V. Flambaum ∗ and A. Kozlov † School of Physics, University of New South Wales, Sydney 2052, Australia (Dated: October 9, 2018)According to the Schiff theorem the nuclear electric dipole moment (EDM) is screened in neutralatoms. In ions this screening is incomplete. We extend a derivation of the Schiff theorem to ions andmolecules. The finite nuclear size effects are considered including Z α corrections to the nuclearSchiff moment which are significant in all atoms and molecules of experimental interest. We showthat in majority of ionized atoms the nuclear EDM contribution to the atomic EDM dominateswhile in molecules the contribution of the Schiff moment dominates. We also consider the screeningof electron EDM in ions. PACS numbers: 31.30.jp, 21.10.Ky, 24.80.+y
I. INTRODUCTION
Permanent electric dipole moment (EDM) of elemen-tary particle or atom violates both P and T invariance.The Kobayashi-Maskawa mechanism leads to extremelysmall values of the EDMs of the particles. It is also tooweak to explain the matter-antimatter asymmetry of theUniverse. On the other hand, most of the popular ex-tensions predict much larger EDMs which are within ex-perimental reach. Therefore, measurements of EDM pro-vide an excellent method to search for physics beyond theStandard Model. The measurements of EDM in atomicand molecular experiments are presented in Refs. [1–19].The EDM of an atom is mostly due to either electronEDM and T,P-odd electron-nucleon interactions in para-magnetic systems (with non-zero electron angular mo-mentum J ) or due to the T, P -odd nuclear forces in dia-magnetic systems ( J = 0; nuclear-spin-dependent e-Ninteraction contributes here too). The existence of T, P -odd nuclear forces leads to the
T, P -odd nuclear momentsin the expansion of the nuclear potential in powers of dis-tance R from the center of the nucleus. The lowest-orderterm in the expansion, the nuclear EDM, is unobservablein neutral atoms due to the total screening of the externalelectric field by atomic electrons [20]. It might be possiblehowever to observe the nuclear EDM in ions, where it isscreened incompletely (see e.g. [21–23]). The first non-vanishing terms which survive the screening in neutralsystems are the Schiff moment which was defined in Ref.[24] (see also Refs. [25, 26] where the contribution of theproton EDM was considered) and the electric octupolemoment (the latter vanishes in nuclei of experimental in-terest which have spin 1/2). More accurate treatment ofthe finite nuclear size in Ref. [27] has shown that theatomic EDM is actually produced by the nuclear Localdipole moment which differs from the Schiff moment bya correction ∼ Z α where Z is the nuclear charge and α ∗ Email:fl[email protected] † Email:o.kozloff@student.unsw.edu.au is the fine structure constant. Since all experiments dealwith heavy atoms this correction is significant.In the non-relativistic classical limit the screening for-mulas can be obtained in a very simple way. The secondNewton law for the ion and its nucleus in the electric fieldreads ( M N + N e m e ) a i = ( Z − N e ) eE (1) M N a N = ZeE N (2) m e a e = eE e , (3)where m e and M N are the electron and nuclear masses; a i , a N and a e are the ion, nucleus and electron averageaccelerations respectively, E is the external electric field, E N is the average electric field at the nucleus, E e is theaverage electric field at one of the ion electrons, e is theproton charge, N e is the number of electrons in the ion.Since system of particles moves altogether, the averagedaccelerations must be equal ( a i = a N = a e ), therefore E N = Z − N e Z E M N M N + N e m e ≈ (1 − N e /Z ) E (4) E e ≈ ( Z − N e ) m e M N E . (5)As we can see, the average electric field for electronsis suppressed by the ratio of masses m e /M N that isvery small for heavy atoms. It means that in the non-relativistic limit there is practically no effect related tothe electron EDM in heavy atoms and ions, − d e · E e ≈ d with the externalfield, − d · E N , is suppressed by the factor ( Z − N e ) /Z .The same approach can be used to determine the elec-tric field at the nucleus in a diatomic molecule:( M + M + N e m e ) a i = ( Z + Z − N e ) eE ,M a = Z eE N ,E N = Z + Z − N e Z M M + M + N e m e E . (6)Screening is stronger for diatomic molecules because ofthe factor M / ( M + M ) that contains both nuclearmasses. This indicates that the nuclear motion cannot be ignored. We also see that in neutral atoms andmolecules the field at the nucleus is zero, therefore the in-teraction of the nuclear EDM d with the screened electricfield vanishes, dE N =0. Similarly, E e = ( Z + Z − N e ) m e M + M + N e m e E . (7)The different screening laws of EDM in neutral atoms,ions and molecules raise a number of new questions. Forexample, is the screening term in the nuclear Schiff mo-ment different in neutral atoms and ions? Can nuclearmotion in molecules produce any additional effects whichdo not exist in a single atom? Are there any new effects ofthe electron density polarization in ions and molecules?Simple classical formulas presented above do not answerthese questions. This motivates us to revisit the quan-tum Schiff theorem [20] and extend it to the cases of ionsand molecules. We also derive a formula which moreaccurately takes into account the finite nuclear size andcalculate corrections to the nuclear Schiff moment.The present work is also motivated by new exper-iments. Effects of EDM in molecules are enhanced[25, 26, 28, 29]. This is why the molecular experimentsare so popular now. Recently the EDM experiment hasbeen started with molecular ions [18]. The EDM exper-iments with atomic ions in the storage rings have beenconsidered too [19]. II. SCREENING OF EDM IN ATOMIC IONSA. Nuclear EDM and Schiff moment
The charge distribution in a finite size nucleus can bewritten as ρ ( r ) = ρ ( r ) + δρ ( r ), where R ρ d r = 1, δρ ( r )is due to the P, T -odd interactions. The
P, T -odd termin charge density leads to the nonzero nuclear dipole mo-ment d = d I /I = Ze R d rδρ r , where Ze is the nucleuscharge, e is the proton charge. Let us define N e as thenumber of electrons. If N e = Z a system is an ion. In aneutral atom ( N e = Z ) our derivation is expected to givethe same results as the Schiff theorem [20] including theeffects of the finite nuclear size [24, 27].The Hamiltonian of a single atom in an external elec-tric field E can be written in the following form:ˆ H = ˆ T + ˆ V + ˆ V + ˆ U + ˆ W , (8) whereˆ T = N e X i − ¯ h m e ∂ ∂ R i − ¯ h M N ∂ ∂ q N , ˆ V = N e X i>j e | R i − R j | − Ze N e X i Z d r ρ ( r ) | R i − q N − r | , ˆ V = N e X i e R i E − Ze q N E , ˆ U = − Ze N e X i Z d r δρ ( r ) | R i − q N − r | , ˆ W = − dE . Here R i and q N are the radius-vectors of the electronsand nucleus correspondingly. The expression for ˆ U canbe expanded in powers of r/R i since the nuclei size issmall compared to the atomic scales. Let us keep thefirst two nonvanishing terms:ˆ U = − d e N e X i R i − q N | R i − q N | − π Ze Z d rδρr r N e X i ∇ i δ ( R i − q N ) . In the above expansion the octupole term was omittedsince it leads to the mixing of the states with high elec-tron angular momentum and its contribution to the totalatomic EDM is small [24].Following Schiff let us define the operatorˆ Q = d Ze ∂∂ q N . (9)It is easy to check that there is a relation between [ ˆ Q, ˆ V ]and ˆ U ˆ U = h ˆ Q, ˆ V i − πe S N e X i ∇ i δ ( R i − q N ) (10) S = 110 (cid:26) Ze Z d δρr r − d Z d rρ ( r ) r (cid:27) , (11)where the expression for the Schiff moment S has thesame form as for a neutral atom [24]. Substituting ex-pression for ˆ U and ˆ W = h ˆ Q, ˆ V i into Eq. (8) we obtainˆ H = ˆ H + h ˆ Q, ˆ H i − πe S N e X i ∇ i δ ( R i − q N ) , (12)where ˆ H = ˆ T + ˆ V + ˆ V is the Hamiltonian of the systemin the external electric field without P, T -odd terms. Thecalculation gives the following result for the commutator h ˆ Q, ˆ H i = − d Ze i ¯ h h ˆ H , P N i = − d Ze M N ˆ a N , (13)where ˆ a N is the nuclear acceleration operator. To obtainthe average value of the acceleration operator we can usethe Ehrenfest theorem: h ˆ a N i = h F i M N = ( Z − N e ) e E M N , (14)where F is the average force acting on the nucleus (seeEq. (1)). Substituting the above expression to Eq. (13)we obtain for the averaged commutator of ˆ Q and ˆ H thefollowing equation h h ˆ Q, ˆ H i i = − (cid:18) − N e Z (cid:19) dE . (15)Substituting this result into Eq. (12) we obtain the effec-tive Hamiltonian of the ion in the external electric field E :ˆ H = ˆ H − (cid:18) − N e Z (cid:19) dE − πe S N e X i ∇ i δ ( R i − q N ) . (16)Note that the derivation above is done in the adiabaticapproximation assuming that we can average over elec-tron motion when we calculate the nuclear motion, i.e.we assume m e ≪ M N . If the number of electrons N e = Z the EDM term in the above expression vanishes, as theSchiff theorem predicts. In the ion case the nuclear EDMinteracts with the average field E N = E (1 − N e /Z ) thatacts on the ion’s nucleus.The last term in Eq. (16),ˆ H w = − πe S N e X i ∇ i δ ( R i − q N ) , (17)induces the ion EDM directed along the nuclear spin(which is the direction of the nuclear Schiff moment S ),similar to the EDM of neutral atoms. This expressionis not applicable for heavy atoms where the Dirac equa-tion gives infinite results for the electron wave functionsat the point-like nucleus. Accurate account of the fi-nite nuclear size gives the following form for the cor-rected Schiff moment electrostatic potential (defined byˆ H w = − eϕ S ( R )): ϕ S ( R ) = − S ′ · R B ρ ( R ) , (18)where B = R ρ ( R ) R dR is the normalization constant.In the limit of the point-like nucleus the expression (18)agrees with Eq. (17). The corrected Schiff moment S ′ isgiven by the equation (see Appendix) S ′ = Ze
10 11 − Z α · (cid:26)(cid:20) h r r i − h r ih r i − h r i ih q ij i (cid:21) − Z α R N (cid:20) h r r i − h r ih r i − h r i ih q ij r i (cid:21)(cid:27) (19) where q ij is the quadrupole moment tensor. Here weomitted higher order terms which are proportional to asmall factor Z α /
9. Outside the nuclear radius R N thenuclear density ρ ( R ) = 0 and the potential (18) vanishesin agreement with the Schiff theorem. Near the origin ρ ( R ) = const and the potential (18) is a linear functionof R . Therefore, the gradient of the Schiff moment poten-tial (18) gives a constant electric field inside the nucleuswhich is directed along the nuclear spin. This electricfield polarizes the electron distribution and produces theatomic EDM. The calculations of the atomic EDM havebeen performed, for example, in Refs. [24, 30–32].Below we make rough estimates to compare the nuclearEDM and the Schiff moment contributions to the atomicEDM. In the case of a spherical nucleus the nuclear EDM d , the nuclear Schiff moment and the atomic EDM D A induced by the Schiff moment have been estimated inRef. [24]: d ∼ − ηe · cm , (20) D A ∼ ( Z/ · − ηe · cm , (21)where η is the strength constant of the nuclear P, T -oddinteraction (in units of the weak Fermi constant G ). As-suming the single ionization we get for the nuclear EDMscreening factor 1 − N e /Z = 1 /Z . As a result, for theionic EDM induced by the nuclear EDM we get the es-timate 1 /Z · − η | e | cm. Thus, for the spherical nu-clei the nuclear EDM contribution exceeds the nuclearSchiff moment contribution by at least one order of mag-nitude. However, in heavy ions containing nuclei withthe octupole deformation (e.g. Ra + and Rn + ) theSchiff moment contribution is enhanced by three ordersof magnitude [33, 34] and is comparable to the nuclearEDM contribution (which is also slightly enhanced inthese ions). B. Electron EDM
For neutral atoms the electron EDM problem was in-vestigated in [35] and further developed in [36]. TheHamiltonian of the nucleus and relativistic electrons inthe external electric field E can be presented asˆ H = ˆ H + ˆ H w , (22)ˆ H = − ¯ h △ N / M N − Ze q N E + N e X i − i ¯ hc α i ∇ i + β i mc − Ze | R i − q N | + e R i E + X j>i e | R i − R j | (23)ˆ H w = − d e N e X i β i Σ i E t , (24) Σ = (cid:18) σ σ (cid:19) where E t is the total electric field acting on the electronwhich includes the external field E , the nuclear fieldand the field of other electrons, α and β are the Diracmatrices. It is convenient to present H w as the sum oftwo terms ˆ H w = ˆ H d + ˆ H d , (25)ˆ H d = − d e N e X i Σ i E t , (26)ˆ H d = − d e N e X i ( β i − Σ i E t . (27)As it was pointed in [35] the first term H d gives nocontribution to atomic EDM in a neutral atom. In anion the H d contribution is suppressed by a small factor m e /M N . It can be shown using the commutator relationˆ H d = h ˆ Q, ˆ H i , (28)ˆ Q = − d e e N e X i Σ i ∂∂ R i . (29)Note that the matrix elements of the operators in thethe H d come from the atomic size area where valenceelectrons (which contribute to the atomic angular mo-mentum and EDM) are non-relativistic. To estimate theaverage value of the commutator h ˆ Q, ˆ H i the Erehnfesttheorem can be employed h h ˆ Q, ˆ H i i = d e e h X i Σ i d p i dt ih X i Σ i d p i dt i ≈ h X i Σ i F i i = − e h X i Σ i E e i (30)Substituting expression (5) for E e into above equationwe obtain for the average value of ˆ H d h ˆ H d i ≈ − d e m e M N ( Z − N e ) h X i Σ i E i (31)We see that the averaged value h ˆ H d i is suppressed bythe small mass ratio m e /M N . It means, that in the limitof heavy nucleus ˆ H d gives no contribution to EDM.The second perturbation term ˆ H d vanishes in the non-relativistic limit since the matrix ( β i −
1) acts on the lowercomponents of the Dirac 4-spinors only. The operatorˆ H d induces atomic EDM given by the same expressionas for neutral atoms, except for the sum in the matrixelements is taken over electron number N e < Z : D = d e h | X ( β i − Σ i | i +2 ed e X n h | P ( β i − Σ i E t | n ih n | P R i | i E − E n (32)In heavy atoms the major contribution to D comes fromthe second term ( D ∼ R rel Z α d e where R rel ∼ H d . Note that a similar equationwith the perturbation ˆ H d gives zero result due to exactcancellation between the first and second terms. Indeed,the zero and the first order corrections to the atomicEDM induced by ˆ H d give EDM D = d e h | X Σ i | i + e X n h | h ˆ Q, ˆ H i | n ih n | P R i | i E − E n + e X n h | P R i | n ih n | h ˆ Q, ˆ H i | i E − E n (33)The above expression can be simplified in the followingway. For the matrix elements of the commutators thefollowing relations are valid h n | h ˆ Q, ˆ H i | i = − ( E − E n ) h n | ˆ Q | i (34) h | h ˆ Q, ˆ H i | n i = ( E − E n ) h | ˆ Q | n i (35)Substituting these expressions into Eq. (32) and usingthe completeness condition P | n ih n | = ˆ1 we obtain D = e X n X i h h | ˆ Q | n ih n | R i | i − h | R i | n ih n | ˆ Q | i i + d e h | X Σ i | i = d e h | Σ i | i + X i e h | h ˆ Q, R i i | i (36)Using definition of the operator ˆ Q it is easy to show that h ˆ Q, R i i = − d e /e Σ i . Hence, the second term in the aboveequation cancels the first term, so the dipole moment D induced by H d equals to zero. In this derivation we as-sume that the electron states are stationary. This is validif we neglect the ion acceleration. Therefore, the resultis consistent with Eq. (31).We see that EDM of an ion induced by the electronEDM is given by the same equation (32) as for neutralatoms (up to corrections ∼ m e /M N ). A similar conclu-sion is also valid for molecular ions. III. NUCLEAR EDM AND SCHIFF MOMENTIN MOLECULAR IONS
Let us consider a molecular ion with N e electrons andtwo nuclei with charges Z e and Z e . We assume thatthe second nucleus has EDM d and Schiff moment S .The molecular Hamiltonian is equal to the sum of thefollowing terms:ˆ T = N e X i − ¯ h m e ∂ ∂ R i − ¯ h M ∂ ∂ q − ¯ h M ∂ ∂ q , ˆ V = N e X i>j e | R i − R j | − Z e N e X i Z d r ρ ( r ) | R i − q − r |− N e X i Z e | R i − q | + Z Z e Z d r ρ ( r ) | q − q − r | , ˆ V = N e X i e R i E − Z e q E − Z e q E , ˆ U = − Ze N e X i Z d r δρ ( r ) | R i − q − r | , + Z Z e Z d r δρ ( r ) | q − q − r | , ˆ W = − dE , where q and q are the coordinates of first and secondnuclei respectively. Using the operatorˆ Q = d Z e ∂∂ q (37)we can present the molecular Hamiltonian in the formsimilar to Eq. (12):ˆ H = ˆ H + h ˆ Q, ˆ H i (38) − πe S ( N e X i ∇ i δ ( R i − q ) − Z ∂∂ q δ ( q − q ) ) . To calculate the average value of the commutator ˆ Q andˆ H we can use the same algorithm as for a single atom. h ˆ Q, ˆ H i = − d Z e i ¯ h h ˆ H , P i = − d Z e M ˆ a (39)Since the molecule moves as a single body the averageaccelerations of all its particles is equal to the molecularacceleration, i.e. h ˆ a i = h F i M + M + N e m e ≈ ( Z + Z − N e ) e E M + M , (40) h h ˆ Q, ˆ H i i = − M M + M Z + Z − N e Z dE . (41)Finally, the effective Hamiltonian of the molecular ion isˆ H = ˆ H − M M + M Z + Z − N e Z dE (42) − πe S ( N e X i ∇ i δ ( R i − q ) − Z ∂∂ q δ ( q − q ) ) , Thus, in a molecular ion the EDM term experiencesthe extra suppression. As for the Schiff moment term,it is still described by the same operator as for a sin-gle atom, except for the extra term proportional to ∂ ( δ ( q − q )) /∂ q describing the interaction of the chargeof the first nucleus and the Schiff moment of the secondnucleus. The matrix elements of such interaction are ex-tremely small due to the Coulomb barrier. IV. ENHANCEMENT OF THE SCHIFFMOMENT CONTRIBUTION TO
P, T -ODDEFFECTS IN POLAR MOLECULES
Now we can compare the contributions of the nuclearEDM and Schiff moment to
P, T -odd effects in polarmolecular ions. Important difference between moleculesand single atoms is that the nuclear motion significantlyaffects induced
P, T -odd effects. The Schiff moment con-tribution in polar molecules is enhanced because of thestrong internal electric field [25]. Another interpretationof the enhancement is due to the small distance betweenthe opposite parity rotational levels [24, 29].The nuclear
P, T -odd effects are studied in themolecules with zero electron angular momentum. Afteraveraging Hamiltonian Eq. (42) over electron wave func-tion we obtain the effective Hamiltonian for the nuclearmotion:ˆ H = − ¯ h µ △ q + U e + µω q − q e ) + BJ ( J +1)+ ˆ H w (43)where q = q − q , q e is the equilibrium distance betweenthe nuclei in averaged potential, J is the rotational an-gular momentum of the molecule, U e describes the inter-action of the partially screened nuclear EDM, the Schiffmoment term ˆ H w can be presented as [24, 26]ˆ H w = 6 XS I · n I , (44)where S = S I /I , n is the unit vector along the molecularaxis, X is the constant that appears after averaging theperturbation over the electron wave function. In the firstorder of the perturbation theory the Schiff term leads tothe rotation state mixing ψ (1) = 6 XS I z I X J ′ = J h Jm | n z | J ′ m i E J − E J ′ | J ′ m i (45)where ψ (0) = | Jm i is the unperturbed rota-tional wave function. Since the energy difference E J − E J ′ = B { J ( J +1) − J ′ ( J ′ +1) } can be very small forrotation levels, the state mixing can be significant. Thismixing induces EDM in the rotational state D Sz = 2 h ψ (0) | D M n z | ψ (1) i (46)= 6 XSD M I z IB J ( J + 1) − m J ( J + 1)(2 J − J + 3) ≡ K m SI z /I . (47)Here D M = D M n is the internal EDM of the polarmolecule. This formula is valid for J = 0. For J = 0the induced EDM is D Sz = − XSD M I z IB ≡ K m SI z /I (48)There is also the screened nuclear EDM contribution D dz to P, T -odd molecular EDM ( see Eq. (42)). Combin-ing this contribution with the Schiff moment contribution D Sz we obtained the P, T -odd part of the interaction of amolecular ion with the external electric field E : V = − (cid:18) M M + M Z + Z − N e Z d − K m S (cid:19) IE I (49)This equation tells us that there is actually no enhance-ment of the electric field in the polar molecule sincethe electric field at the nucleus is suppressed 1 /Z timesrather than enhanced. However, there is huge enhance-ment of the Schiff moment contribution since the expres-sion for the coefficient K m contains in the denominatorthe rotational constant B which may be five orders ofmagnitude smaller than the interval between atomic lev-els of opposite parity.Note that we can derive Eq. (49) treating E as aperturbation. Therefore, the energy shift produced bythe Schiff moment in Eq. (49) is actually proportionalto the average polarization of the polar molecule in theelectric field E . In the small electric field it is linear in E , however, in the high field it tends to the constant.This determines the saturation effect in the energy shiftproduced by the Schiff moment if we go beyond the weakelectric field E approximation (see Eq. (44) where theaverage polarization n z <
1) .Using Eq. (49) we can compare molecular EDM in-duced by the screened nuclear EDM and the Schiff mo-ment. Consider, for example, molecule PbF + since it hasthe same number of electrons as a well studied moleculeTlF where the effect of the nuclear Schiff moment hasbeen measured. The screened EDM term for PbF + is D N ∼ − ηe · cm ( EDM of F and EDM of odd iso-tope of Pb give comparable contributions since values of M/Z are approximately the same). To obtain the Schiffmoment induced EDM in the ground state we need toestimate the constant K m , given by Eq. (48). Sincethe molecular parameters are unknown for the ion weassume them to be of the order of their values for theneutral molecule TlF: X ≈ B = 1 . · − a.u.and dipole moment D M = 1 .
65 a.u. for TlF are takenfrom [38]. Finally, substituting all the parameters intoEq. (48) we obtain K m = 5 · a.u. Assuming theSchiff moment value for an odd isotope of Pb equal to S = 10 − ηe · f m [24] we obtain the value for the Schiffmoment contribution D S ∼ − ηe · cm which is threeorders of magnitude larger than the nuclear EDM contri-bution D N ∼ − ηe · cm. As it was mentioned above,in the nuclei with the octupole deformation like Ra the Schiff moment is enhanced. Therefore, in molecular ions like RaF + the Schiff moment induced EDM will be5 orders of magnitude larger than the partially screenednuclear EDM. V. CONCLUSIONS
Accurate treatment of the electron EDM effects showsthat the T,P-odd EDM of atomic and molecular ions athigh Z are dominated by the Z enhanced relativisticcorrection effect, similar to neutral systems. The directcontribution of electron EDM is suppressed by the screen-ing factor ( m e /M ) where M is the ion mass.The situation is different for the nuclear EDM. Inatoms the nuclear EDM is screened by the factor Z i /Z where Z i is the ion charge. However, the nuclear EDMstill dominates over the Schiff moment induced atomicEDM (with exception of heavy ions which contain nu-clei with the octupole deformation like Ra and
Rnwhere the Schiff moment is strongly enhanced).In molecular ions the nuclear EDM screening is slightlystronger than in atomic ions, the screening factor is( M N /M )( Z i /Z ). At the same the Schiff moment con-tribution is enhanced ∼ M N /m e ∼ times due to themixing of the close rotational states of opposite parity.There is the additional Schiff moment enhancement insuch molecular ions like RaF + . As a result, the Schiffmoment contribution is 10 − times larger than thescreened nuclear EDM contribution.This combination of the large enhancement factorsmakes molecular ion experiments an attractive alterna-tive to the atomic EDM experiments. VI. APPENDIX
According to Eq. (16) in the limit of the point-like nu-cleus the Schiff moment potential and its matrix elementare given by ϕ S ( R ) = 4 π S · ∇ δ ( R ) (50) h s | − eϕ S | p i = 4 πe S · ( ∇ ψ † s ψ p ) R =0 (51)For the solutions of the Dirac equation ( ∇ ψ † s ψ p ) R → isinfinite for a point-like nucleus. Therefore, for relativis-tic electrons it is necessary to account for the finite sizeof the nucleus and introduce a finite-size Schiff momentpotential. An appropriate potential has been shown [27]to increase linearly inside the nucleus and vanish at thenuclear surface: ϕ S ( R ) = − S ′ · R B n ( R ) , (52)where B = R n ( R ) R dR ≈ R N / R N is the nuclearradius and n ( R ) is a smooth function which is 1 for R < R N − δ and 0 for R > R N + δ ; n ( R ) can be taken asproportional to the nuclear density ρ (note that we canchoose any normalization of n ( r ) since the normalizationconstant cancels out in the ratio n/B , see Eq. (52)).Below we will accurately derive expression for the cor-rected Schiff moment S ′ that corresponds to the potential(52).The P, T -odd part of the nuclear electrostatic potentialwith electron screening taken into account can be writtenin the following form (see e.g. [34] for the derivation): ϕ ( R ) = Z Z eρ ( r ) | R − r | d r + d · ∇ Z eρ ( r ) | R − r | d r (53)As it was shown in [27] the expansion of the Coulombpotential in (53) in terms of the Legendre polynomialsgives the following dipole term in the potential: ϕ (1) ( R ) = Ze R Z ∞ R (cid:18) h r i R − r R + r r + h r i i q ij r (cid:19) ρ ( r ) d r (54)We see that ϕ (1) ( R ) = 0 if R > R N (nuclear radius)since ρ ( R ) = 0 in that region. Therefore, correspond-ing matrix elements will depend on the electron wavefunctions behavior inside the nucleus. All the electronorbitals for l > s and p Dirac orbitals. We will use thefollowing notations for the electron wavefunctions: ψ ( R ) = (cid:18) f ( R )Ω jlm − i ( σ · n ) g ( R )Ω jlm (cid:19) (55)where Ω jlm is a spherical spinor, n = R /R , f ( R ) and g ( R ) are the radial functions. Using ( σ · n ) = 1 we canwrite the electron transition density as ρ sp ( R ) = ψ † s ψ p = Ω † s Ω p U sp ( R ) (56) U sp ( R ) = f s ( R ) f p ( R ) + g s ( R ) g p ( R ) = ∞ X k =1 b k R k (57)The expansion coefficients b k can be calculated analyti-cally [27]; the summation is carried over odd powers of k . Using Eqs. (54,56) we can find the matrix elementsof the electron-nucleus interaction, h s | − eϕ (1) ( R ) | p i = − Ze h s | n | p i · (cid:26)Z ∞ [( h r i − r ) · Z r U sp dR + (cid:18) r r + h r i i q ij r (cid:19) Z r U sp R dR (cid:21) ρd r (cid:27) = − Ze h s | n | p i · ( ∞ X k =1 b k k + 1 (cid:20) h r ih r k +1 i − k + 4 h r r k +1 i + k + 1 k + 4 h r i ih q ij r k − i (cid:21)(cid:27) , (58) where h s | n | p i = R Ω † s n Ω p dφ sin θdθ , h r n i = R ρ ( r ) r n d r .Note, that all vector values h r r n i are due to P, T -oddcorrection δρ to the nuclear charge density ρ , while h r n i are the usual P, T -even moments of the charge densitystarting from the mean-square radius h r i = r q for k = 1.We now set the matrix elements (58) of the true nuclear T, P -odd potential to be equal to the matrix elements ofthe equivalent potential (52) which are given by h s | − eφ ( R ) | p i = 15 e h s | n | p i · S ′ R N Z ∞ U sp R n ( R ) dR = 15 e h s | n | p i · S ′ ∞ X k =1 b k R k − N k + 4 , (59)where we have made approximation R n ( R ) R k dR ≈ R k +1 N / ( k + 4). Equating (58) and(59) we obtain S ′ = Ze
15 1 P ∞ k =1 b k b k +4 R k − N ∞ X k =1 b k b k + 1 (cid:20) k + 4 h r r k +1 i − h r ih r k +1 i − k + 1 k + 4 h r i ih q ij r k − i (cid:21) (60)Thus we have a possibility of separating the nuclearand electronic parts of the calculation of atomic EDMs.The nuclear calculation involves only the determinationof S ′ and the atomic calculation involves only the effectsproduced by the equivalent potential (52).Note that S ′ in eq. (60) is different from the Localdipole moment L defined in Ref. [27]: L does not containthe sum in the denominator. The reason for the differenceis that here we reduce the problem to the nuclear sizeeffective potential (52) while in Ref. [27] the problem wasreduced to the contact effective potential (50) located inthe center of the nucleus.In the non-relativistic case ( Zα →