# Theory of Polarization of Highly Charged Ions in Storage Rings

This article discusses the theory of polarization for highly charged ions in storage rings and its production, preservation, and application to the search for violations of fundamental symmetries. The authors explore the work of researchers from St. Petersburg State University and GSI Darmstadt, Germany, who have made significant contributions to this field.

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1. Theory of the polarization of highly-charged ions in storage rings: production, preservation and application to the search for the violation of the fundamental symmetries Theory of the polarization of highly-charged ions in storage rings: production, preservation and application to the search for the violation of the fundamental symmetries A. Bondarevska y a A. Bondarevska y a A. Prozorov A. Prozorov L. Labzowsky , L. Labzowsky , St. Petersburg State University, Russia D. Liesen D. Liesen F. Bosch F. Bosch GSI Darmstadt, Germany G. Plunien G. Plunien Technical University of Dresden, Germany St.-Petersburg, 2010

2. 1. Production of polarized HCI beams 1. Production of polarized HCI beams 1.1 Radiative polarization: simple estimates Radiative polarization occurs via radiative transitions Radiative polarization occurs via radiative transitions between Zeeman sublevels in a magnetic field between Zeeman sublevels in a magnetic field first discussed in: A.A. Sokolov, I.M. Ternov, Sov. Phys.Dokl. 8 (1964) 1203 first realized in Novosibirsk for electrons: Ya.S. Derbenev, A.M. Kondratenko, S.T. Serednyakov, A.N. Skrinsky, G.M. Tumaikin, Ya.M. Shatunov, Particle accelerators 8 (1978) 115 recent development: S.R. Mane, Ya.M. Shatunov and K. Yokoya J.Phys.G 31 (2005) R151; Rep. Progr. Phys. 68 (2005) 1997

3. Spin-flip transition rates for electrons (lab.system) W spin-flip = 64 (3 c 3 ) -1 0 5 H 3 5 W spin-flip = 64 (3 c 3 ) -1 0 5 H 3 5 = Lorentz factor; H = magnetic field, 0 = Bohr magneton = Lorentz factor; H = magnetic field, 0 = Bohr magneton Polarization time T P = W -1 Polarization time T P = W -1 Electrons: H 1 T, 10 5 , T P 1 hour Protons: << 0 T P huge HCI: 0 , but even for FAIR at GSI with H 6 T, 23 T P 10 3 hours too long !

4. 1.2 Selective laser excitation of the HFS levels* 1.2 Selective laser excitation of the HFS levels* Schematic picture of the Zeeman splitting of the hyperfine sublevels of the electronic ground state for the (I = 5/2). Schematic picture of the Zeeman splitting of the hyperfine sublevels of the electronic ground state for the H-like 151 Eu ion (I = 5/2). g J - electron g-factor. * A. Prozorov, L. Labzowsky, D. Liesen and F. Bosch Phys. Lett. B574 (2003) 180 The solid lines denote M1 excitations at a laser frequency = E HFS + 2 0 H. E HFS = 1.513 (4) eV. The dashed lines show the decay channels for Zeeman sublevels . 1s 1/2 1s 1/2 F = 2 F' = 3 M F ' M F

5. Transition rate Transition rate W (F'=3 F=2) = 0.197 10 2 s -1 W (F' M F ' FM F ) = const [C FF' 1, MF-MF' (M F , M F ')] 2 , C FF' 1, MF-MF' (M F , M F ') are Clebsh-Gordan coefficients The selective laser excitation to the 1s 1/2 F' = 3 state is performed by a laser with frequency . This leads to the of the 1s 1/2 F' = 3 state. The selective laser excitation to the 1s 1/2 F' = 3 state is performed by a laser with frequency . This leads to the partial polarization of the 1s 1/2 F' = 3 state. After the laser is switched off, the spontaneous decay to the ground state leads to its partial polarization during (lifetime of the F' = 3 level). After the laser is switched off, the spontaneous decay to the ground state leads to its partial polarization during 10.9 ms (lifetime of the F' = 3 level).

6. 1.3 Description of polarization The polarization state of an ion after i-th "cycle" The polarization state of an ion after i-th "cycle" (switching on the laser) is described by the density matrix: (switching on the laser) is described by the density matrix: F (i) = MF n FMF (i) FMF * FMF . Normalization condition: MF n FMF (i) = 1, Normalization condition: MF n FMF (i) = 1, FMF are the wavefunctions, n FMF (i) the occupation numbers F, M F the total angular momentum and projection of an ion Degree of polarization is defined as: F (i) = F -1 MF n FMF (i) M F Nonpolarized ions: n FMF = (2F + 1) -1 , F = 0 Fully polarized ions: n FF = 1, F = 1

7. 1.4 Dynamics of polarization The occupation numbers are defined with the recurrence relations via the M1 transition probabilities: The occupation numbers are defined with the recurrence relations via the M1 transition probabilities: width of the sublevel FM F width of the sublevel FM F

8. Opposite initial population Opposite initial population F (0) = -1, n F-F (0) = 1 F (0) = -1, n F-F (0) = 1 After first cycle: F (1) = - 0.6667 After first cycle: F (1) = - 0.6667 After 40 cycles: F (40) = 0.9986 After 40 cycles: F (40) = 0.9986 The polarization time for F (40) = 0.999 T P = 40 10.9 ms = 0.44 s 40 1 0 10 +2 +1 0 Uniform initial population Uniform initial population (0) = 0, n (0) = (2F+1) -1 F (0) = 0, n FMF (0) = (2F+1) -1 After cycle: (1) = After first cycle: F (1) = 0.1667 After 40 (40) = After 40 cycles: F (40) = 0.9993 , n FMF N

9. 1.5 polarization 1.5 Nuclear polarization Nuclear polarization density matrix I = < FMF F FMF > el (integration over electron variables) I = < FMF F FMF > el (integration over electron variables) FMF = MIMJ C FMF IJ (M I M J ) IMI JMJ IMI , JMJ nuclear, electronic wave functions I = MI n IMI IMI * IMI ; n IMI = MJMF n FMF [C FMF (M I M J )] 2 I = MI n IMI IMI * IMI ; n IMI = MJMF n FMF [C FMF (M I M J )] 2 Degree of nuclear polarization: Degree of nuclear polarization: I = I -1 MI n IMI M I I = I -1 MI n IMI M I Maximum nuclear polarization for the case of full electron Maximum nuclear polarization for the case of full electron polarization n FF = 1 (F = 2) in Eu ions: I max = 0.93

10. of one- and two-electron ions 1.6 Polarization of one- and two-electron ions Polarization in He-like ions with total electron angular momentum equal to zero (2 1 S 0 , 2 3 P 0 ) is nuclear polarization . In polarized the nuclei are also polarized, In polarized one-electron HCI the nuclei are also polarized, due to the strong hyperfine interaction (hyperfine splitting due to the strong hyperfine interaction (hyperfine splitting in the order of ). Polarization time is about . in the order of 1 eV ). Polarization time is about 10 -15 s . The capture of the second electron by the polarized one-electron ion does not destroy the nuclear polarization: the capture time, defined by the Coulomb interaction, is much smaller than the depolarization time, defined by the hyperfine interaction. If the total angular momentum of the two-electron ion appears to be zero (2 1 S 0 , 2 3 P 0 ) the nuclear polarization remains unchanged .

11. 2. Preservation of the ion beam polarization in storage rings 2.1 Dynamics of the HCI in a magnetic system of a storage ring The magnetic system of a storage ring (GSI) consists of a number of magnets including magnets which generate field components to the ion trajectory, quadrupole magnets and the electron cooler magnet (solenoid). The magnetic system of a storage ring (GSI) consists of a number of magnets including bending magnets which generate field components orthogonal to the ion trajectory, focusing quadrupole magnets and the longitudinal electron cooler magnet (solenoid). The latter one was also proposed to be used for the of the ions via selective laser excitation. The latter one was also proposed to be used for the longitudinal polarization of the ions via selective laser excitation. The peculiarity of storing polarized HCI compared to stored electrons or protons is that the is defined by the whereas the is defined by the . The peculiarity of storing polarized HCI compared to stored electrons or protons is that the trajectory dynamics is defined by the nuclear mass, whereas the spin dynamics is defined by the electron mass .

12. The in a magnetic system of a ring can The movement of an ion in a magnetic system of a ring can be described with the equation of motion: be described classically with the equation of motion: d v /dt = k ( H x v ) k = -Ze/Mc , v is the ion velocity, M, Ze are mass and charge of the nucleus, H is the magnetic field In the rest frame of an ion the motion appears like in a time- dependent field. The spin dynamics which is influenced by the transitions between hyperfine and Zeeman sublevels we describe quantum - mechanically .

13. 2.2 Spin dynamics and the instantaneous quantization axis ( IQA ) Relativistic effects are neglected (at GSI ring 1) Relativistic effects are neglected (at GSI ring 1) Spin motion in the ion reference system is described by the Schrdinger equation: Spin motion in the ion reference system is described by the Schrdinger equation: [i / t + 0 H (t) s ] S (t) = 0 ( ) H (t) is the magnetic field, s is the spin operator The IQA, denoted as , we define via an equation: / t < S (t) s (t) S (t) > = 0 ( ) From ( ) and ( ) follows the equation for IQA : / t = 0 ( H (t) x (t)) ( )

14. Equation ( ) coincides with the for the spin motion, however the definition ( ) is convenient for the quantum-mechanical description of polarization. Equation ( ) coincides with the pure classical equation for the spin motion, however the definition ( ) is convenient for the quantum-mechanical description of polarization. It can be proved that the with respect to IQA remains in an arbitrary time-dependent field. It can be proved that the degree of polarization with respect to IQA remains constant in an arbitrary time-dependent field. It can be also proved that the with respect to IQA in the process of of the excited hyperfine sublevel, i.e. remains the same for the ground- and excited hyperfine sublevels. It can be also proved that the degree of polarization with respect to IQA does not change in the process of spontaneous decay of the excited hyperfine sublevel, i.e. remains the same for the ground- and excited hyperfine sublevels.

15. 2.3 Rotation of IQA in the magnetic field of a bending magnet at GSI ring The initial polarization is directed along the longitudinal (z) axis: The initial polarization is directed along the longitudinal (z) axis: x (0) = 0, y (0) = 0, z (0) = 1 The magnetic field is oriented along the vertical (x) axis: The magnetic field H is oriented along the vertical (x) axis: H x = H(t), H y = H z = 0 Solution of the Schrdinger equation reads: Solution of the Schrdinger equation reads: x (t) = 0, y (t) = sin (t), z (t) = cos (t) x (t) = 0, y (t) = sin (t), z (t) = cos (t) t t (t) = 0 / H(t') dt' (A) 0 The IQA rotates in the horizontal plane (yz) with the time- dependent frequency (t) = (t) / t

16. The trajectory rotation occurs due to the Lorentz force. Roughly we can write the rotation angle for the ion trajectory after passing one GSI bending magnet (60 0 = /3): t N / H(t') dt' = /3 (B) 0 where N = Zm 0 /M. For Eu ions N = 2.268 10 -4 0 By comparing eqs. (A) and (B) we conclude that the rotation angle for IQA after passing one bending magnet amounts to about 10 4 . Thus, it will be extremely difficult to fix the direction of polarization before the start of the PNC experiment.

17. 2.4 Solution of the problem: "Siberian Snake" A Siberian Snake the polarization (IQA) . If after one revolution of an ion in the ring the IQA will acquire a deviation from the longitudinal direction, the Siberian Snake will rotate it like: A Siberian Snake rotates the polarization (IQA) by an angle around the z-axis . If after one revolution of an ion in the ring the IQA will acquire a deviation from the longitudinal direction, the Siberian Snake will rotate it like: beam Siberian Siberian Snake Snake IQA IQA Then, after two revolutions, the deviation caused by any reason will be canceled. It remains to count the revolutions and to start a PNC experiment after an even number of revolutions.Counting the revolutions seems to be possible for a bunched beam.

18. The hyperfine quenching 3. Diagnostics of polarization 3.1 The hyperfine quenching ( HFQ ) of polarized two- electron ions in an external magnetic field The HFQ transition probability for the polarized ion in an external magnetic field: The HFQ transition probability for the polarized ion in an external magnetic field: W HFQ = W 0 HFQ [ 1 + Q 1 ( h )] W HFQ = W 0 HFQ [ 1 + Q 1 ( h )] where W 0 HFQ is the HFQ transition rate in the absence of the external field, and h = H / | H |. where W 0 HFQ is the HFQ transition rate in the absence of the external field, and h = H / | H |. In case of the 2 1 S 0 1 1 S 0 HFQ, the coefficient Q 1 is: In case of the 2 1 S 0 1 1 S 0 HFQ, the coefficient Q 1 is: 1 Q 1 = 2 < 2 1 S 0 H 2 3 S 1 > / < 2 1 S 0 H HF 2 3 S 1 > is the magnetic moment of an electron , H HF is the hyperfine interaction Hamiltonian

19. The net signal (after switching off the magnetic field) is: W HFQ = Q 1 W 0 HFQ too small to be observed! However, as we shall see this is the unique experiment which allows for the direct measurement of the degree of polarization in the HFQ transition However, as we shall see this is the unique experiment which allows for the direct measurement of the degree of polarization in the HFQ transition For He-like Eu (Z = 63) and H = 1 T Q 1 = - 10 -7

20. Employment of REC (Radiative Electron Capture) 3.2 Employment of REC (Radiative Electron Capture) Employment of REC for the control of polarization of HCI beams via measurement of linear polarization of X-rays was studied in: A. Shurzhikov, S. Fritzsche, Th. Sthlker and S. Tashenov, Phys. Rev. Lett. 94 (2005) 203202 The formula tan 2 ~ F was confirmed experimentally (for F = 0) by: S. Tashenov et al. PRL 97 (2006) 223202 We will study the possibility for the control of the HCI beam polarization via measurement of linear polarization of X-rays in HFQ transitions. We will study the possibility for the control of the HCI beam polarization via measurement of linear polarization of X-rays in HFQ transitions.

21. Linear polarization of X-ray photons in HFQ transitions in polarized ions 3. 3 Linear polarization of X-ray photons in HFQ transitions in polarized ions Photon density matrix k is the photon momentum: k = , is frequency , are the helicities: = s ph = 1 The photon spin s ph =i( e* e ) , i.e. is defined only for the circular polarization (complex e ). P i : (i = 1,2,3) are the Stokes parameters

22. 3. 4 Stokes parameters P 3 - circular polarization I intensity of the light, polarized along the axis . Stokes parameters via photon density matrix: Schematic position of the axes in the X-ray polarization observation experiment

23. of the photon density matrix 3. 5 Rotation of the photon density matrix IQA Choice of the quantization axis: along IQA (beam polarization). The photon density matrix is written with the quantization axis . necessary It is necessary to rotate this matrix by an angle . The result for the transition between two bound states with the total electron momentum j, j Here: A LM L - photon wave function, LM L photon angular momentum and projection, Dirac matrices n jm occupation numbers for the initial electron states (define electron polarization ) D 0 ( ) Wigner function; in our case =45 0

24. 2 1 S 0 1 1 S 0 3.6 Application to the 2 1 S 0 1 1 S 0 HFQ transition (magnetic dipole photons) F F total angular momentum of an ion; n FM F occupation numbers P M 1 =0 Nonplarized ions: n FM F = const : P M 1 =0 ; P M 2 =0 P M 2 =0 independent on the polarization. Hence, the photons are nonpolarized if they are emitted by nonpolarized by nonpolarized ions. 2 1 S 0 151 63 Eu 61+ For 2 1 S 0 state of 151 63 Eu 61+ : F=I=5/2, n 5/2 5/2 = 5/6, n 5/2 3/2 = 1/6 F = I = (1/F) M F n FM F M F = 0.93 P M 1 = -0.4 P M 2 =0 P M 1 = -0.4 , P M 2 =0

25. Polarization and 3.7 Polarization and alignment Thus, one cannot extract the degree of polarization F from the Stokes parameters the degree of alignment Stokes parameter P M 1 defines the degree of alignment which can be defined as a F = M F n FM F M F 2 - a 0 F a F = M F n FM F M F 2 - a 0 F where a 0 F = M F (2F+1)-1 M F 2 = 1/3 F(F+1) Then for the fully nonpolarized ions a F =0. However, using the value of a F (as extracted from P M 1 ) one can check whether the ion polarization has its maximum value. For the maximum polarization n FM F = F,MF and a max F = 1/3 F(2F - 1) a max F = 1/3 F(2F - 1)

26. parameters for the 2 3 P 0 1 1 S 0 HFQ transition (electric dipole photons) 3. 8 Stokes parameters for the 2 3 P 0 1 1 S 0 HFQ transition (electric dipole photons) 2 3 P 0 1 1 S 0 For the investigation of the PNC effects in He-like Eu and Gd ions it will be important to know also the Stokes parameters for electric photons (transition 2 3 P 0 1 1 S 0 ). P E 1 = + 0.4 P E 2 =0 For Eu ions: P E 1 = + 0.4 , P E 2 =0 P M,E 1 = 0.4 The result P M,E 1 = 0.4 means that 70% of ions, polarized along I 0 axis are electric ones, and 70% of ions, polarized along I 90 axis are magnetic ones.

27. to measure the degree of the ion polarization via linear polarization . 3.9 Impossibility to measure the degree of the ion polarization via linear X-ray polarization . There are general arguments why the beam polarization (i.e. the degree of polarization) cannot be defined via the linear polarization of emitted photons. pseudeoscalar If it would be possible, the probability should contain a pseudeoscalar term, constructed from the vectors and e (for electric photons) or and (e k) (for magnetic ones). Moreover, this term should be quadratic in e or (e k). It is easy to check that such constructions, linear in , cannot be built, and only quadratic in terms like ( e) 2 or ( (ek)) 2 can arise . From these quadratic terms one can define the alignment, but not the polarization . The only possibility to measure the beam polarization via X-ray polarization is to use the circular polarization. Then W HFQ = W HFQ 0 [1 + Q 2 ( s ph )] s ph = i (e*e) photon spin

28. 4 . PARITY NONCONSERVATION EFFECTS IN 4 . PARITY NONCONSERVATION EFFECTS IN HCI 4 .1 POSSIBLE PARITY NONCONSERVATION (PNC) EFFECTS IN ONE-PHOTON TRANSITIONS FOR ATOMS AND IONS W if = W if 0 [ 1 + ( s ph n )R 1 + ( n)R 2 + ( hn )R 3 W if = W if 0 [ 1 + ( s ph n )R 1 + ( n)R 2 + ( hn )R 3 + ( h )Q 1 + ( s ph )Q 2 ] + ( h )Q 1 + ( s ph )Q 2 ] n = direction of photon emission n = direction of photon emission s ph = photon spin s ph = photon spin = direction of ion polarization = direction of ion polarization h = direction of external magnetic field (unit vector) h = direction of external magnetic field (unit vector)

29. 4.2 Parity coefficients 4.2 Parity violating coefficients R 1 R 1 = Re [ -i < i H W a > (E i - E a - i /2) -1 (W af / W if ) 1/2 ] H W = effective PNC Hamiltonian H W = effective PNC Hamiltonian i,f = initial, final state i,f = initial, final state a = state admixed to state i by H W a = state admixed to state i by H W R 2 = R 1 ( = degree of ion beam polarization) R 2 = R 1 ( = degree of ion beam polarization) R 3 * = Re [(< i H i > + < a H a >) (E i - E a - i /2) -1 ] R 1 = magnetic moment of the electron; H = external magnetic field * Ya. A. Azimov, A. A. Anselm, A. N. Moskalev and R. M. Ryndin Zh. Eksp. Teor. Fiz. 67 (1974) 17

30. 4.3 Parity coefficients 4.3 Parity conserving coefficients Q 1 Q 1 = Re [ (< i H i > + < b H b >) (E i - E a - i /2) -1 (W bf / W if ) 1/2 ] b = level closest to level i of the same parity, b = level closest to level i of the same parity, admixed by the magnetic field H admixed by the magnetic field H Q 2 = a , a 1 Q 2 = a , a 1

31. Data from: A.N. Artemyev, V.M. Shabaev, V.A. Yerokhin, G. Plunien and G. Soff, Phys.Rev. A71 (2005) 062104 4.4 HCI: level crossings 4.4 He-like HCI: level crossings (2 3 P 0 ) = [E(2 1 S 0 ) E(2 3 P 0 )] / E(2 1 S 0 ) (2 3 P 1 ) = [E(2 1 S 0 ) E(2 3 P 1 )] / E(2 1 S 0 ) E/E 110 Z (2 3 P 1 ) 5 10 -3 10 -3 (2 3 P 0 ) (2 3 P 0 ) (2 3 P 1 )

32. 4.5 effects in HCI: a survey of proposals 4.5 PNC effects in He-like HCI: a survey of proposals V.G. Gorshkov and L.N. Labzowsky V.G. Gorshkov and L.N. Labzowsky Zh. Eksp. Teor. Fiz. Pis' ma 19 (1974) 30 Zh. Eksp. Teor. Fiz. Pis' ma 19 (1974) 30 2 1 S 0 - 2 3 P 1 crossing Z = 6, 30, nuclear spin-dependent weak constant, R =10 -4 2 1 S 0 - 2 3 P 1 crossing Z = 6, 30, nuclear spin-dependent weak constant, R =10 -4 A. Schfer, G. Soff, P. Indelicato and W. Greiner A. Schfer, G. Soff, P. Indelicato and W. Greiner Phys. Rev A40 (1989) 7362 Phys. Rev A40 (1989) 7362 2 1 S 0 2 3 P 0 crossing, Z = 92, two-photon laser excitation 2 1 S 0 2 3 P 0 crossing, Z = 92, two-photon laser excitation G. von Oppen G. von Oppen Z. Phys. (1991) 181 Z. Phys. D21 (1991) 181 2 1 S 0 2 3 P 0 crossing, Z = 6, Stark-induced emission, R = 10 -6 2 1 S 0 2 3 P 0 crossing, Z = 6, Stark-induced emission, R = 10 -6

33. V.V. Karasiev, L.N. Labzowsky and A.V. Nefiodov V.V. Karasiev, L.N. Labzowsky and A.V. Nefiodov Phys. Lett. A172 , 62 (1992) Phys. Lett. A172 , 62 (1992) 2 1 S 0 2 3 P 0 crossing in U (Z = 92), HFQ decay R ~ 10 -4 2 1 S 0 2 3 P 0 crossing in U (Z = 92), HFQ decay R ~ 10 -4 R.W. Dunford R.W. Dunford Phys. Rev. A54 (1996) 3820(1974) 30 Phys. Rev. A54 (1996) 3820(1974) 30 2 1 S 0 2 3 P 0 crossing Z = 92, stimulated two-photon emission, R = 3 10 -4 2 1 S 0 2 3 P 0 crossing Z = 92, stimulated two-photon emission, R = 3 10 -4 L.N. Labzowsky, A.V. Nefiodov, G. Plunien, G. Soff, R. Marrus and D. Liesen L.N. Labzowsky, A.V. Nefiodov, G. Plunien, G. Soff, R. Marrus and D. Liesen Phys. Rev A63 (2001) 054105 Phys. Rev A63 (2001) 054105 2 1 S 0 2 3 P 0 crossing, Z = 63, hyperfine quenching with polarized ions, R = 10 -4 2 1 S 0 2 3 P 0 crossing, Z = 63, hyperfine quenching with polarized ions, R = 10 -4 A.V. Nefiodov, L.N. Labzowsky, D. Liesen, G. Plunien and G. Soff A.V. Nefiodov, L.N. Labzowsky, D. Liesen, G. Plunien and G. Soff Phys. Lett. B (2002) 52 Phys. Lett. B 534 (2002) 52 2 1 S 0 2 3 P 1 crossing, Z = 33, nuclear anapole moment, polar. ions, R = 0.6 10 -4 2 1 S 0 2 3 P 1 crossing, Z = 33, nuclear anapole moment, polar. ions, R = 0.6 10 -4

34. G.F. Gribakin, E.F. Currell, M.G. Kozlov and A.I. Mikhailov G.F. Gribakin, E.F. Currell, M.G. Kozlov and A.I. Mikhailov Phys. Rev. A72 , 032109 (2005) Phys. Rev. A72 , 032109 (2005) 2 1 S 0 2 3 P 0 crossing Z = 30 Z = 48, dielectronic recombination , polarized incident electrons, R ~ 10 -8 2 1 S 0 2 3 P 0 crossing Z = 30 Z = 48, dielectronic recombination , polarized incident electrons, R ~ 10 -8 A.V. Maiorova, O.I. Pavlova, V.M. Shabaev, C. Kozhuharov, A.V. Maiorova, O.I. Pavlova, V.M. Shabaev, C. Kozhuharov, G. Plunien and Th. Stoelker G. Plunien and Th. Stoelker J. Phys. B 42 205002 (2009) J. Phys. B 42 205002 (2009) 2 1 S 0 2 3 P 0 crossing, Z = 90, 64 radiative recombination 2 1 S 0 2 3 P 0 crossing, Z = 90, 64 radiative recombination linear X-ray polarization , polarized electrons, R ~ 10 -8 linear X-ray polarization , polarized electrons, R ~ 10 -8

35. 4.6 Energy Level Scheme for 4.6 Energy Level Scheme for He-like Gd Numbers on the r. h. side: ionization energies in eV The partial probabilities of the radiative transitions: s -1 Numbers in parentheses: powers of 10 Double lines: two-photon transitions I, g I : nuclear spin, g-factor 157 Gd : I =3/2, g I = - 0.3398

36. 4.7 Energy Level Scheme for 4.7 Energy Level Scheme for He-like Eu Numbers on the r. h. side: ionization energies in eV The partial probabilities of the radiative transitions: s -1 Numbers in parentheses: powers of 10 Double lines: two-photon transitions I, g I : nuclear spin, g-factor 151 Eu : I =5/2, g I = + 3.4717

37. 4.8 effect in He-like HCI 4.8 PNC effect in He-like polarized HCI Basic (HFQ) transition: Basic hyperfine-quenched (HFQ) transition: 1s2s 1 S 0 > + 1 / E S <1s2s 1 S 0 hf 1s2s 3 S 1 > 1s2s 3 S 1 > 1s2s 1 S 0 > + 1 / E S <1s2s 1 S 0 H hf 1s2s 3 S 1 > 1s2s 3 S 1 > 1s 2 1 S 0 > + (M1) where H hf = hyperfine interaction Hamiltonian, E S = [ E(2 3 S 1 ) E(2 1 S 0 ) ] PNC PNC - allowed transition: 1s2s 1 S 0 > + 1 / E SP <1s2s 1 S 0 W 1s2p 3 P 0 > 1s2s 1 S 0 > + 1 / E SP <1s2s 1 S 0 H W 1s2p 3 P 0 > 1 / E P <1s2p 3 P 0 hf 1s2p 3 P 1 > 1s2p 3 P 1 > 1 / E P <1s2p 3 P 0 H hf 1s2p 3 P 1 > 1s2p 3 P 1 > 1s 2 1 S 0 > + (E1) where E SP = [ E(2 3 P 0 ) E(2 1 S 0 ) ], E P = [ E(2 3 P 1 ) E(2 3 P 0 ) ] , R 2 = [ W HFQ + PNC (E1) / W HFQ (M1)] 1/2

38. 4.9 Evaluationt of the 4.9 Evaluationt of the coefficient R 2 One-electron ions: One-electron polarized ions: dW jj = dW (0) jj + dW (PNC) jj dW jj = dW (0) jj + dW (PNC) jj dW (0) jj = < k , ,njl 1s2s 3 S 1 > k , ,njl > dW (0) jj = < k , ,njl 1s2s 3 S 1 > k , ,njl > Parity Parity nonconservation: njlm > njlm > + [ E njl E njl ] -1

39. 4.10 Basic magnetic dipole transition (l = l) for one-electron ions 4.10 Basic magnetic dipole transition (l = l) for one-electron ions dW njl,njl = dW M1 njl,njl [1 + R 2 ( ) ] dW njl,njl = dW M1 njl,njl [1 + R 2 ( ) ] R 2 = - 2 njl,njl R E1 (njl ;njl)/ R M1 (njl;njl), l = 2j - l R 2 = - 2 njl,njl R E1 (njl ;njl)/ R M1 (njl;njl), l = 2j - l njl,njl = G njl,njl / E njl E njl G njl,njl = - (G F /22) Q W [P njl (r)Q njl (r) Q njl (r)P njl (r)] N (r)r 2 dr upper and lower radial components of the Dirac wave function for the electron P njl (r), Q njl (r) upper and lower radial components of the Dirac wave function for the electron reduced matrix elements for the electric and magnetic dipole transitions R E1, R M1 reduced matrix elements for the electric and magnetic dipole transitions

40. 4.11 He-like Eu: basic HFQ transition 2 1 S 0 1 1 S 0 4.11 He-like Eu: basic HFQ transition 2 1 S 0 1 1 S 0 dW HFQ ( 2 1 S 0 1 1 S 0 ) =dW HFQ 0 ( 2 1 S 0 1 1 S 0 ) + dW HFQ PNC ( 2 1 S 0 1 1 S 0 )= dW HFQ ( 2 1 S 0 1 1 S 0 ) =dW HFQ 0 ( 2 1 S 0 1 1 S 0 ) + dW HFQ PNC ( 2 1 S 0 1 1 S 0 )= = dW HFQ 0 ( 2 1 S 0 1 1 S 0 ) [1 6/35 a F P 2 (cos ) + ( ) R 2 ] = dW HFQ 0 ( 2 1 S 0 1 1 S 0 ) [1 6/35 a F P 2 (cos ) + ( ) R 2 ] dW HFQ 0 = W HFQ 0 /4 angular independent part dW HFQ 0 = W HFQ 0 /4 angular independent part

41. 4.12 Possible determination of the degree of a F 4.12 Possible determination of the degree of alignment a F no smallness parity conserving vanishes The term containing a F gives the possibility to measure the degree of alignment (or to check whether the maximum polarization is achieved) in a most simple way. This term has no smallness compared to 1, provided that the polarization (and alignment) is of the order of 1. It is parity conserving and corresponds to the scalars of the type ( ) 2 , ( ) 2 in the expression for the probability. It also vanishes when the polarization is absent, since then a F = 0. For defining a F one has to measure dW HFQ for two different angles: dW HFQ ( =0) - dW HFQ ( = /2) / dW HFQ 0 = - (18/35) a F dW HFQ ( =0) - dW HFQ ( = /2) / dW HFQ 0 = - (18/35) a F

42. 4.13 PNC effect in He-like HCI: Gd versus Eu 4.13 PNC effect in He-like HCI: Gd versus Eu E = E(2 1 S 0 ) E(2 3 P 0 ) from Artemyev et al. 2005 E (Gd) = + 0.004 0.074 eV Z = 64 E (Eu) = - 0.224 0.069 eV Z = 63 Re ( E i /2) -1 = E ( E 2 + 2 /4) -1 ; (Gd) = 0.0016 eV (HFQ E1 2 3 P 0 1 1 S 0 ) Lifetime (s) Lifetime (s) R(max) / R(min) / Z 2 3 P 0 (HFQ E1) 2 1 S 0 (2 E1) 64 4 10 -12 1.0 10 -12 0.052 ( E = ) 0 ( E = 0) 63 4 10 -13 1.2 10 -12 1.0 10 -4 0.6 10 -4 Disadvantage of Gd: Lifetime of 2 3 P 0 longer than lifetime of 2 1 S 0 HFQ (E1) transition 2 3 P 0 1 1 S 0 unresolvable from HFQ + PNC (E1) transition 2 1 S 0 1 1 S 0 : Background 10 5 New, more accurate value for E (Gd) = 0.023 0.074 eV (Maiorova et al 2009) does not change our conclusions

43. 4.14 PNC experiments: estimates 4.14 PNC experiments: estimates Polarization time for H-like ions: t pol = 0.44 s ; total number of ions in the ring: 10 10 . After the time t pol the dressing target should be inserted to produce He-like Eu ions in 2 1 S 0 state with polarized nuclei. Statistical loss: 10 -1 assuming the homogeneous distribution of the population among all L 12 subshell. ( ) Next the PNC experiment can start: observation of the asymmetry ( ) in the HFQ probability of decay 2 1 S 0 1 1 S 0 . Statistical losses Statistical losses : Efficiency of detector: 10 -2 Branching ratio of the HFQ M1 decay to the main decay channel 2 1 S 0 1 1 S 0 + 2 (E1): 10 -4 Total statistical loss: 10 -7 Number of interesting events : 10 10 10 -7 = 10 3 = N int Not enough! After the dressing of ions and the PNC experiments the He-like ions leave the ring. The ring should be filled again! Not enough! After the dressing of ions and the PNC experiments the He-like ions leave the ring. The ring should be filled again!

44. 4.15 Scheme of the PNC experiment 4.15 Scheme of the PNC experiment H-like ion beam H-like ion beam He-like ion beam He-like ion beam Bending magnet Bending magnet dressing target excitation target x-ray detector x-ray detector storage ring storage ring spin rotator

45. 4.16 Observation time for the PNC effect 4.16 Observation time for the PNC effect Observation time to fix the PNC effect t obs (fix) Number of events necessary to fix the PNC effect N(fix) = 10 8 Revolution time t rev = 10 -6 s t obs (fix) N int / t rev = N(fix) 0.1 s t obs (fix) = N(fix) t rev / N int = 10 8 10 -6 /10 3 s = 0.1 s Observation time to measure the PNC effect with accuracy 0.1%: t obs (0.1%) Observation time to measure the PNC effect with accuracy 0.1%: t obs (0.1%) Number of events necessary to measure the PNC effect with accuracy 0.1%: N(0.1%) = 10 14 Number of events necessary to measure the PNC effect with accuracy 0.1%: N(0.1%) = 10 14 30 hours t obs (0.1%) = N(0.1%)t rev /N int = 10 14 10 -6 /10 3 s = 10 5 s 30 hours

46. 5. ELECTRIS DIPOLE MOMENT (EDM) OF AN ELECTRON IN H-LIKE IONS 5. ELECTRIS DIPOLE MOMENT (EDM) OF AN ELECTRON IN H-LIKE IONS IN STORAGE RINGS 5.1 EDMS OF THE MUONS AND NUCLEI AT STORAGE RINGS I.B. Khriplovich Phys. Lett. B 444 , 98 (1998) I.B. Khriplovich Phys. Lett. B 444 , 98 (1998) I.B. Khriplovich Hyperfine Interactions 127 , 365 (2000) I.B. Khriplovich Hyperfine Interactions 127 , 365 (2000) Y.K. Semertzidis Proc. of the Workshop on Frontier Tests of Y.K. Semertzidis Proc. of the Workshop on Frontier Tests of Quantum Electrodynamics and Physics of the Quantum Electrodynamics and Physics of the Vacuum, Sandansky, Bulgaria (1998) Vacuum, Sandansky, Bulgaria (1998) F.J.M. Farley, K. Jungmann, J.P. Miller, W.M. Morse, Y.F. Orlov, F.J.M. Farley, K. Jungmann, J.P. Miller, W.M. Morse, Y.F. Orlov, B.J. Roberts, Y.K. Semertzidis, A. Silenco and E.J. Stephenson B.J. Roberts, Y.K. Semertzidis, A. Silenco and E.J. Stephenson Phys. Rev. Lett. 93 , 052001 (2004)

47. 5.2 Spin precession of the particle in the external magnetic field H : 5.2 Spin precession of the particle in the external magnetic field H : Lab. frame: Lab. frame: g q g gyromagnetic ratio (g=2 for leptons), q charge Rest frame: Rest frame: T - frequency of Thomas precession (d s /dt) rest = s (d s /dt) rest = s Bargmann-Michel- Bargmann-Michel- Telegdi (BMT) Telegdi (BMT) equation equation a = g -1 For leptons a / 10 -3 a = g -1 For leptons a / 10 -3

48. 5. 3 Precession around the direction of the particle velocity 5. 3 Precession around the direction of the particle velocity Frequency: Frequency: Field compensation: Field compensation: p = 0 .

49. 5.4 Precession of the angular momentum of the H-like HCI in storage ring 5.4 Precession of the angular momentum of the H-like HCI in storage ring H-like ion: particle with mass M (mass of the nucleus), charge q=Ze and magnetic moment . (magnetic moment of the electron) Thomas precession can be neglected. BMT equation: BMT equation: Field compensation is not possible: for the vertical field 1 T the static radial electric field 10 7 V/cm is necessary. H-like ion with nuclear spin I : total angular momentum F Kinematics will be defined by F Dynamics will be defined by 0 BMT equation: BMT equation: Exact proof: Wigner-Echart theorem

50. 5.5 EDM spin precession for H-like HCI 5.5 EDM spin precession for H-like HCI for any particle For H-like HCI For H-like HCI Frequency of the EDM precession: EDM: d e 10 -28 10 -17 If d e 10 -28 e cm, 10 -17

51. 5.6 EDM spin rotation angle 5.6 EDM spin rotation angle IQA rotation in the plane xy due to the IQA rotation in the plane xy due to the motional magnetic field H m = E motional magnetic field H m = E neglecting electron EDM. In the neglecting electron EDM. In the absence of E the IQA is directed along absence of E the IQA is directed along y axis. y axis. electron EDM rotation angle (in the plane yz ) averages to zero due to electron EDM rotation angle (in the plane yz ) averages to zero due to the H m rotation; compensating magnetic field H c is necessary. the H m rotation; compensating magnetic field H c is necessary. z y x E H m IQA y x E H m IQA z - IQA rotation in the plane xy due to the IQA rotation in the plane xy due to the motional magnetic field motional magnetic field + IQA rotation in the plane yz due to the electron EDM. + IQA rotation in the plane yz due to the electron EDM. H c

52. 5.7 Observation of the EDM effect in storage rings 5.7 Observation of the EDM effect in storage rings A. active bare nuclei or HCI with active nuclei and A. active bare nuclei or HCI with active nuclei and closed electron shells: closed electron shells: Decay process: N* N + e - + e B. muon: B. muon: Decay process: - e - + e + Observation: asymmetry n e polarization of the nuclei (muon) n e direction of the decay electron emission Both processes are P violating. However they are due to the weak interaction. Therefore no additional smallness

53. 5.8 Observation of the electron EDM with H-like HCI in storage ring 5.8 Observation of the electron EDM with H-like HCI in storage ring Laser excitation of the HF excited level 1s 1/2 F=2 1s 1/2 F=3 for 151 63 Eu 62+ Decay is observed; decay time ~ 11 ms, then again excitation s ph photon spin; ph - direction of the photon emission (s ph ph ) = 1 chirality, or circular polarization ion polarization vector; degree of polarization Q c = const; for 1s 1/2 F=3 1s 1/2 F=2 transition Q c = . Asymmetry observed: ph ; Asymmetry observed: ph ; if circular polarization is fixed, there is no P-violation . if circular polarization is fixed, there is no P-violation . Summation over circular polarizations () gives zero Summation over circular polarizations () gives zero

54. - - 5.9 Scheme of the electron EDM experiment 5.9 Scheme of the electron EDM experiment M magnetic system of the ring, d photon detectors that fix circular M magnetic system of the ring, d photon detectors that fix circular polarization, EDM rotation angle grows linearly with time polarization, EDM rotation angle grows linearly with time y x E d z snake d M First revolution First revolution x d snake d M x E d 2 snake d M Second revolution Second revolution Third revolution Third revolution field on field off field on - z z y y

55. 5.10 Estimates for the observation time 5.10 Estimates for the observation time Asymmetry A = Q c = F sin F ( 2 ); F=3 for Eu 62+ Asymmetry A = Q c = F sin F ( 2 ); F=3 for Eu 62+ = | d | t obs p , p is the part of the ring where electric field is applied = | d | t obs p , p is the part of the ring where electric field is applied d is the frequency of the EDM caused spin precession d is the frequency of the EDM caused spin precession Numbers: E 10 5 V/cm, p = l/L; l is the length of the field region; Numbers: E 10 5 V/cm, p = l/L; l is the length of the field region; L is the ring length; p = 0.001 (L = 100 m, l = 10 cm); H c 300 gauss L is the ring length; p = 0.001 (L = 100 m, l = 10 cm); H c 300 gauss These fields E , H c applied within pL, do not disturb essentially the These fields E , H c applied within pL, do not disturb essentially the trajectory trajectory | d | 10 10 s -1 | d | 10 10 s -1 a) Let A ~ 10 -5 ; ~ 10 -17 , d e ~ 10 -28 e cm ; then t obs ~ 10 5 ~ 30 hours a) Let A ~ 10 -5 ; ~ 10 -17 , d e ~ 10 -28 e cm ; then t obs ~ 10 5 ~ 30 hours b) Let A ~ 10 -6 ; ~ 10 -19 , d e ~ 10 -30 e cm ; then t obs ~ 10 6 ~ 12 days b) Let A ~ 10 -6 ; ~ 10 -19 , d e ~ 10 -30 e cm ; then t obs ~ 10 6 ~ 12 days

56. 6. CONCLUSIONS 6. CONCLUSIONS Polarization : for 151 63 Eu 62+ polarization time Polarization : for 151 63 Eu 62+ polarization time for the 100% polarization t pol = 0.44 s for the 100% polarization t pol = 0.44 s Nuclear polarization, corresponding to Nuclear polarization, corresponding to 100% ion polarization: 93% 100% ion polarization: 93% PNC experiment PNC experiment Time necessary for the observation of the PNC effect Time necessary for the observation of the PNC effect t obs PNC ~ 0.1 s t obs PNC ~ 0.1 s Time necessary for the measurement of the PNC effect Time necessary for the measurement of the PNC effect with accuracy 0.1% (higher than in neutral Cs) t obs PNC ~ 30 hours with accuracy 0.1% (higher than in neutral Cs) t obs PNC ~ 30 hours Time necessary to observe electron EDM at the Time necessary to observe electron EDM at the level 10 -28 e cm: t obs EDM ~ 30 hours level 10 -28 e cm: t obs EDM ~ 30 hours Time necessary to observe electron EDM at the Time necessary to observe electron EDM at the level 10 -30 e cm: t obs EDM ~ 12 days level 10 -30 e cm: t obs EDM ~ 12 days