Nuclear structure aspects of Schiff Moments N.Auerbach Tel Aviv University and MSU
T-P-odd electromagnetic moments In the absence of parity (P) and time (T) reversal violation the T P-odd moments for a quantum particle (system) will be zero. For example, the electric dipole moment: A direction in space is defined by the spin j. The electric dipole d=dj. But d=er, is a polar vector while j is an axial vector, so one changes sign under a parity transformation, the other not. Because of parity, d=0. But also under time reversal d does not change sign, but j does. In order to have a non-zero d both P and T must be violated.
EDM Limits as of summer 2005 Particle Exp. Limit [10-27 e cm] SM [factor to go] Possible New Physics [factor to go] e (Tl) < 1.6 10 11 1 μ < 1.05 *10 9 10 8 200 τ < 3.1 * 10 11 10 7 1700 n < 63 10 4 60 Tl (odd p) < 10 5 10 7 10 5 Hg (odd n) < 0.21 10 5 various back
References to initial work Parity Mixing and Time Reversal Violation in Octupole Deformed Nuclei V.Spevak and N.Auerbach, Physics Letters B359, 254, 1995 Collective T and P odd electromagnetic moments in nuclei with octupole deformations N.Auerbach, V.V.Flambaum,, and V.Spevak Phys.Rev.Lett.76:4316-4319,1996 4319,1996 Enhanced T odd P odd electromagnetic moments in reflection asymmetric nuclei V.Spevak, N.Auerbach,, and V.V.Flambaum Phys.Rev.C56:1357-1369,1997 1369,1997
NH3 molecule
The ammonia molecule and parity doublets ψ ± = 1 2 ( a ± b )
Pear shaped nuclei and time reversal violation (or how to improve the limits on time reversal violation by a factor of 1K without investing $100000K)
Parity doublets when the P-T-odd force is zero: + - Ψ + d int Ψ + = 0 ΔE Ψ = ( + )/ 2 Ψ + = ( + + )/ 2 when the P-T odd force is not zero: Ψ = Ψ + +αψ α = Ψ + V PT Ψ ΔE So, in the lab frame we see: d z = 2α d int I I +1
Parity Doublets + and are not completely orthogonal. Each tunnels through the barrier dividing the left and right potenials giving rise to to a non zero overlap ε = + - and the wave functions should be written more precisely as : Ψ ± = 1 ( ) ( + ± ) ± ε 2 1 The energy splitting ΔE = E + - E - is proportional to ε. Higher is the barrier, smaller is the overlap ε, and therefore the energy splitting between the members of the doublet is smaller.
Mixing of parity doublets Ψ r d ± int and r S = S int 1 = 2 r = d n or 1 Ψ = 2 int int both are zero. Introducing a Ψ = Ψ α = Ψ S z Ψ E + + Ψ ( JMK ± JM K ) v n + αψ parity and time reversal odd potential - V [( 1+ α ) JMK + ( 1 α ) JM K ] V E PT Ψ + J = 2α S J + 1 int PT one obtains :
Skyrme Hartee-Fock calculations of Ra isotopes M.Bender et al.
Enhancements in Radium some Ra nuclei
Schiff moment The Schiff-Purcell-Ramsey- Hellman-Feynman theorem. The nuclear dipole moment causes the atomic electrons to rearrange themselves so that they develop a dipole moment opposite that of the nucleus. In the limit of non-relativistic electrons and a point nucleus the electrons dipole moment exactly cancels the nuclear moment, so that the net atomic dipole moment vanishes! For a finite size nucleus the screening is not complete and one is left with a vector called the Schiff moment r S 1 r r 2 5 2 = r d r d 10 3Z
Measuring Atomic EDMs A TRV component in the N-N interaction EDMs of of individual nucleons EDM+Schiff moment of the nucleus TRV in the electric field Atomic EDM Experiment External E and B
Main results Intrinsic Schiff moments and laboratory frame Schiff moments, as well as induced atomic dipole moments are presented for several actinide nuclei; the values for 199Hg, 129Xe, and 133Cs are given for comparison. 223Ra 225Ra 223Rn 221Fr 223Fr 225Ac 229Pa S intr (e fm 3 ) 24 24 15 21 20 28 25 S (10 8 ge fm 3 ) 400 300 1000 43 500 900 12000 d at (10 25 ge cm) 2700 2100 2000 240 2800 199Hg 129Xe 133Cs S (10 8 ge fm 3 ) -1.4 1.75 3 d at (10 25 ge cm) 5.6 0.47 2.2 g η
The origin of the enhancement 1. The quadrupole +octupole deformation leads to large intrinsic Schiff moments. 2. The parity doublets in the reflection asymmetric nuclei are very close in energy. 3. At the same time the parity and time reversal violating matrix element between the members of the doublet is relatively large. 4. The nuclei that are quadrupole+octupole deformed are also heavy, having large Z. 5. Atomic physics in these atoms is of help.
Conclusions a) In a reflection-asymmetric nucleus enhanced collective T,P-odd electric moments appear if T,P-odd terms are present in the nuclear Hamiltonian. b) The T,P-odd Schiff moments in heavy nuclei with intrinsic reflection asymmetry are typically enhanced by more than two orders of magnitude in comparison with reflection-symmetric deformed or spherical nuclei. c) Due to the atomic structure effects, atomic electric dipole moments are enhanced compared to the lighter analogs. For atoms of nuclei with Z around 90, the atomic enhancement is a factor of 8, in comparison with analogous atoms with Z around 55. d) The atomic dipole moments induced by T,P-odd hadron-hadron interaction in the nuclei studied are typically enhanced 400-1000 times in comparison with Hg and Xe nuclei, for which the best experimental upper limits on atomic electric dipole moments are obtained. Recent detailed microscopic, studies by Bender, Engel, Dobaczewski and others have confirmed these findings. This opens new experimental possibilities of studying time reversal violation.
Recent developments It was suggested that soft octupole vibrations observed in some regions of the nuclear chart more frequently than static octupole deformation may produce a similar enhancement of the Schiff moment. Estimates of the Schiff moment generated in nuclei with a quadrupole deformation and soft octupole mode showed that the resulting Schiff moments are indeed enhanced. A related idea was explored recently. It is known that some nuclei are soft with respect to both quadrupole and octupole modes. The light isotopes of Rn and Ra are spherical but with a soft quadrupole mode. The spectra of the nuclei display quasi-vibrational bands based on the ground state and on the octupole phonon, with positive and negative parity, respectively. These bands are connected via low-energy electric dipole transitions. This situation seems a-priori to be favorable for the enhancement of T,P-odd effects.
Particle+vibrations The interaction between the odd particle with spin j and vibrations of the even core will induce an admixture of the quadrupole phonon into the ground state : g. s. x = = a 0 j = J [ j 3 ] 2 + [ j 2 ] An opposite parity state with the same spin J can be formed by coupling an octupole phonon to the particle j : J + a A P - T - odd interaction will mix the above two states. J
In the realistic cases we calculated, there is a large fragmentation of s.p. strength (especially for the positive parity), so that the final results for the Schiff moment are similar to the ones obtained for Hg or Xe.
Octupole deformations The transition of a spherical nucleus to an octupole deformed state was discussed in the RPA framework in the past. It was found that for realistic interactions of the lowest 3 The stability of the nucleus against variations of a certain variableq can be expressed in Hartree- Fock energy terms of a static polarizability defined for instance through the dependenceof a constrained ( E / A) 2 1 P = 2 2 α α = 0 In the RPA, this quantity can be calculatedas, m 1 P = 2 A by relating it the inverse energy weighted sum rule, m 1 = n Q 0 with ω being the RPA energies corresponding to excitation from the ground state n n ω to excited states n RPA root collapsein certain areas of n 2 E / A on the constraining termαq, with quantum numbersdictated by thesymmetry of As the polarizability increases the system becomesless stable against deformation characterized by the collective variableq ; a negative value of P indicates a situation in which a collapse has ocurred (in the RPA framework). the periodic table. the energies 0 the operator Q.
Particle + phonon Only when the quadrupole vibration is almost degenerate with the spherical ground state, the strong enhancement occurs. In the table we show scaled QRPA results for the Schiff moment S as a function of the low-lying frequency of collective oscillations in the even core. The scaling is introduced by multiplying the actual quadrupole and octupole excitation energies by the factor y. Nucleus y S 219Ra 1.0 0.3 0.1-0.2 0.01 6.2 221Ra 1.0-0.1 0.1 6 0.01 560 We see that very large enhancements by orders of magnitude occur when the even-even system is close to the onset of a phase transitions and the nucleus acts essentially as a statically deformed system.
Limitations of the RPA The QRPA calculations with realistic parameters do not predict small energy denominators E+ E which would provide a strong enhancement of the effect. In the present calculations, the spacing between the opposite parity states of the same total spin is of order 1 21 MeV rather than few tens of kev as it is in deformed nuclei [20,21]. In the deformed case, the appearance of closely spaced parity doublets is inherent in the model, well understood and confirmed by experiments. In the present calculations, the parity doublets do not emerge naturally, so that the spacing between the opposite parity states is very sensitive to the details of single-particle and phonon spectra as well as to their interactions. Any proximity of levels is quite accidental. In general, the spectra of odd-neutron nuclei in the case of soft transitional even core cannot be reliably reproduced by the standard version of the QRPA.
Particle phonon coupling. Conclusions. The approach used was based on the QRPA that allows one to define microscopically the structure of collective modes and coupling of the odd particle to quadrupole and octupole phonons. We should conclude that the truly deformed pear shaped nucleus cannot be mimicked by a simple particle+core model involving only one-phonon states. We confirmed the effect of enhancement of the Schiff moment for very small quadrupole and octupole frequencies. The even-even system close to onset of deformation acts essentially similarly to the statically deformed system, where the enhancement was established earlier.
Collaborators V.F. Dmitriev,, V.V. Flambaum, R.A.Sen kov kov, V.Spevak, V.G. Zelevinsky