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A. Bondarevska y a A. Prozorov L. Labzowsky , St. Petersburg State University, Russia

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. Prozorov L. Labzowsky , St. Petersburg State University, Russia D. Liesen

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A. Bondarevska y a A. Prozorov L. Labzowsky , St. Petersburg State University, Russia

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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 A. Bondarevskaya A. Prozorov L. Labzowsky,St. Petersburg State University, Russia D. Liesen F. Bosch GSI Darmstadt, Germany G. PlunienTechnical University of Dresden, Germany St.-Petersburg, 2010

  2. 1. Productionofpolarized HCI beams1.1 Radiative polarization: simple estimates Radiative polarizationoccurs via radiative transitions between Zeeman sublevels in a magneticfield firstdiscussed in: A.A. Sokolov, I.M. Ternov, Sov. Phys.Dokl. 8 (1964) 1203 firstrealized in Novosibirsk forelectrons: Ya.S. Derbenev, A.M. Kondratenko, S.T. Serednyakov, A.N. Skrinsky, G.M. Tumaikin, Ya.M. Shatunov, Particleaccelerators 8 (1978) 115 recentdevelopment: S.R. Mane, Ya.M. Shatunovand K. YokoyaJ.Phys.G31 (2005) R151; Rep. Progr. Phys. 68 (2005) 1997

  3. Spin-flip transition rates for electrons (lab.system) W spin-flip = 64 (3 ћc3)-1 │μ0│5 H3γ5 γ = Lorentz factor; H = magnetic field, μ0 = Bohr magneton Polarization time TP = W -1 Electrons: H ≈ 1 T, γ≈ 10 5, TP≈ 1 hour Protons: μ << μ0 →TP huge HCI: μ≈μ0, but even for FAIRat GSI with H ≈ 6 T, γ≈ 23 → TP ≈ 103 hours → too long !

  4. 1.2 Selectivelaserexcitation ofthe HFS levels* The solid lines denote M1 excitations at a laser frequency ω = ΔEHFS + 2 μ0H. ΔEHFS = 1.513 (4) eV. The dashed lines show the decay channels for Zeeman sublevels. Schematicpictureofthe Zeeman splittingofthehyperfinesublevelsofthe electronic groundstatefortheH-like151Eu ion (I = 5/2). gJ - electron g-factor. * A. Prozorov, L. Labzowsky, D. Liesen and F. Bosch Phys. Lett. B574 (2003) 180 MF' 1s1/2 F' = 3 1s1/2 MF F = 2

  5. Transition rate W (F'=3 → F=2) = 0.197· 102 s-1 W (F' MF' → FMF) = const [CFF'1, MF-MF' (MF, MF')]2 , CFF'1, MF-MF' (MF, MF')are Clebsh-Gordan coefficients The selective laser excitation to the 1s1/2 F' = 3 state is performed by a laser with frequency ω. Thisleadstothepartial polarizationofthe 1s1/2 F' = 3 state. After thelaserisswitched off, thespontaneousdecaytothegroundstateleadstoits partial polarizationduring10.9 ms(lifetimeofthe F' = 3 level).

  6. 1.3 Description of polarization The polarizationstateof an ion after i-th "cycle" (switching on thelaser) isdescribedbythedensitymatrix: ρF(i) = ΣMFnFMF(i) ψFMF* ψFMF . Normalization condition: ΣMFnFMF(i) = 1, ψFMFarethewavefunctions, nFMF(i)theoccupationnumbers F, MFthe total angular momentumandprojectionof an ion Degreeλofpolarizationisdefinedas: λF(i) = F-1ΣMFnFMF(i) MF Nonpolarizedions: nFMF = (2F + 1) -1, λF = 0 Fullypolarizedions: nFF = 1, λF = 1

  7. 1.4 Dynamics of polarization The occupationnumbersaredefinedwiththerecurrencerelations via the M1 transitionprobabilities: width of the sublevel F’M’F

  8. Uniform initialpopulation λF(0) = 0, n FMF(0) = (2F+1)-1 After firstcycle: λF(1) = 0.1667 After 40 cycles: λF(40) = 0.9993 Oppositeinitialpopulation λF(0) = -1, n F-F(0) = 1 After firstcycle: λF(1) = - 0.6667 After 40 cycles: λF(40) = 0.9986 The polarization time for λF(40) = 0.999 TP = 40 · 10.9 ms = 0.44 s λ, nFMF 1 λ +2 +1 0 N 0 10 40

  9. 1.5 Nuclearpolarization Nuclearpolarizationdensitymatrix ρI = < ψFMF │ρF│ψFMF >el(integrationoverelectron variables) ψFMF = ΣMIMJ CFMFIJ (MIMJ) ψIMIψJMJ ψIMI , ψJMJnuclear, electronic wavefunctions ρI = ΣMInIMIψIMI* ψIMI ; nIMI = ΣMJMFnFMF [CFMF (MIMJ)]2 Degreeλofnuclearpolarization: λI = I-1ΣMInIMIMI Maximum nuclearpolarizationforthecaseoffullelectronpolarizationnFF = 1 (F = 2) in Euions: λI max= 0.93

  10. 1.6Polarizationofone- andtwo-electronions Polarization in He-like ions with total electron angular momentum equal to zero (2 1S0, 2 3P0) is nuclear polarization. In polarizedone-electron HCIthenucleiare also polarized, due tothe strong hyperfineinteraction (hyperfinesplitting in the order of1 eV). Polarization time isabout10 -15 s. The capture of the second electronby 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 1S0, 2 3P0) 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 magneticsystemof a storage ring (GSI) consistsof a numberofmagnetsincludingbendingmagnetswhichgeneratefieldcomponentsorthogonaltotheiontrajectory, focusingquadrupolemagnetsandthelongitudinalelectron cooler magnet (solenoid). The latterone was also proposedtobeusedforthelongitudinal polarizationoftheions via selectivelaserexcitation. The peculiarityofstoringpolarized HCI comparedtostoredelectronsorprotonsisthatthetrajectorydynamicsisdefinedbythenuclearmass,whereasthespindynamicsisdefinedbytheelectronmass.

  12. The movementof an ion in a magneticsystemof a ring can bedescribedclassicallywiththeequationofmotion: dv/dt = k (H x v) k= -Ze/Mc, vistheionvelocity, M, Zearemassandchargeof thenucleus,Histhemagneticfield In therestframeof an ionthemotionappearslike in atime-dependentfield. The spindynamicswhichisinfluencedbythetransitionsbetweenhyperfineand Zeeman sublevelswedescribequantum - mechanically.

  13. 2.2 Spin dynamicsandtheinstantaneousquantizationaxis (IQA) Relativistic effects are neglected (at GSI ring γ≈ 1) Spin motion in the ion reference system is described by the Schrödinger equation: [i ∂/∂t + μ0H(t) s] χS(t) = 0 (∗) H(t) is the magnetic field, s is the spin operator The IQA, denoted as ζ, we define via an equation: ∂/∂t < χS(t)│sζ(t)│χS(t) > = 0 (∗∗) From (∗) and (∗∗) follows the equation for IQA: ∂ζ/∂t = μ0 (H(t) x ζ(t)) (∗∗∗)

  14. Equation (∗∗∗) coincides with the pure classical equationfor the spin motion, however the definition (∗∗) is convenient for the quantum-mechanical description of polarization. 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 degree of polarization with respect to IQA doesnot 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 initialpolarizationisdirectedalongthe longitudinal (z) axis: ζx(0) = 0, ζy(0) = 0, ζz(0) = 1 The magneticfieldHisorientedalongthevertical (x) axis: Hx = H(t), Hy = Hz = 0 Solution oftheSchrödingerequationreads: ζx(t) = 0, ζy(t) = sin φ(t), ζz(t) = cos φ(t) t φ(t) = μ0/ћ∫H(t') dt' (A) 0 The IQA rotates in the horizontal plane (yz) withthe time-dependentfrequencyω(t) = φ(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 (600 = π/3): t μN/ћ∫H(t') dt' = π/3 (B) 0 where μN = Zmμ0/M. For Eu ions μN = 2.268 · 10 -4μ0 By comparing eqs. (A) and (B) we conclude that therotation angle for IQA after passing one bending magnet amounts to about 104π. Thus, it will be extremely difficult to fix the direction of polarization before the start of the PNC experiment.

  17. 2.4 Solution oftheproblem: "SiberianSnake" ASiberian Snake rotates the polarization (IQA) by an angle πaround 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: Siberian Snake IQA beam IQA Then, aftertworevolutions, 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. 3. Diagnosticsofpolarization3.1 Thehyperfinequenching (HFQ) ofpolarizedtwo- electronions in an externalmagneticfield The HFQ transitionprobabilityforthepolarizedion in an externalmagneticfield: WHFQ = W0HFQ [ 1 + Q1(ζh)] where W0 HFQ is the HFQ transition rate in the absence of the external field, and h=H/|H|. In caseofthe 2 1S0 – 1 1S0 HFQ, thecoefficient Q1is: Q1= 2 λ < 2 1S0│μH│2 3S1 > / < 2 1S0│HHF│2 3S1 > μ is the magnetic moment of an electron, HHF is the hyperfine interaction Hamiltonian

  19. For He-like Eu (Z = 63) and H = 1 T→ Q1 = -10-7 The net signal (after switching off the magnetic field) is: Δ WHFQ = Q1 W0HFQ 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 𝜆 in the HFQ transition

  20. 3.2 EmploymentofREC (Radiative Electron Capture) EmploymentofRECforthecontrolofpolarizationof HCI beams via measurementof linear polarizationof X-rays was studied in: A. Shurzhikov, S. Fritzsche, Th. Stöhlker and S. Tashenov, Phys. Rev. Lett. 94 (2005) 203202 The formulatan 2χ~λFwas confirmed experimentally (forλF = 0) by: S. Tashenov et al. PRL 97 (2006) 223202 We will studythepossibilityforthecontrolofthe HCI beam polarization via measurementof linear polarizationof X-rays in HFQ transitions.

  21. 3.3Linear polarizationofX-rayphotons in HFQtransitions in polarizedions Photon density matrix kis the photon momentum: k =𝜔𝜈, 𝜔is frequency 𝜆 , 𝜆‘ are the helicities: 𝜆 = sph𝜈 =± 1 The photon spin sph =i(e*×e), i.e. is defined only for the circular polarization (complex e). Pi: (i = 1,2,3) are the Stokes parameters

  22. 3.4Stokes parameters P3 -circular polarization Iα– intensity of the light, polarized along the axis α. Stokes parameters via photon density matrix: Schematic position of the axes in the X-ray polarization observation experiment

  23. 3.5Rotationofthephotondensitymatrix Choice of the quantization axis: along IQA (beam polarization). The photon density matrix is written with the quantization axis ν. It is necessary to rotate this matrix by an angle 𝜃. The result for the transition between two bound states with the total electron momentum j, j‘ Here: A𝜆 LML- photon wave function, LML– photon angular momentum and projection, 𝛼 – Dirac matrices njm– occupation numbers for the initial electron states (define electron polarization) D𝜈0𝜇(𝜃) – Wigner function; in our case 𝜃=450

  24. 3.6 Application to the 21S0→11S0HFQ transition (magnetic dipole photons) F– total angular momentumof an ion; nFMF – occupationnumbers Nonplarizedions: nFMF= const: PM1=0; PM2=0independent on thepolarization. Hence, thephotonsarenonpolarizediftheyareemittedbynonpolarizedby nonpolarizedions. For21S0stateof15163Eu61+ : F=I=5/2, n5/2 5/2 = 5/6, n5/2 3/2 = 1/6 𝜆F = 𝜆I = (1/F) ΣMFnFMF MF = 0.93 PM1= -0.4,PM2=0

  25. 3.7 Polarizationandalignment Thus, one cannot extract the degree of polarization 𝜆Ffrom the Stokes parameters Stokes parameter PM1 defines „the degree of alignment“ which can be defined as aF =ΣMF nFMF MF2 - a0F wherea0F =ΣMF (2F+1)-1 MF2 = 1/3 F(F+1) Then for the fully nonpolarized ions aF=0. However, using the value of aF (as extracted from PM1) one can check whether the ion polarization has its maximum value. For the maximum polarization nFMF = 𝛿 F,MFand amaxF = 1/3 F(2F - 1)

  26. 3.8 Stokesparameters for the 23P0→11S0 HFQ transition (electric dipole photons) 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 23P0→11S0 ). For Eu ions: PE1= + 0.4,PE2=0 The result PM,E1= ∓ 0.4 means that 70% of ions, polarized along I0 axis are electric ones, and 70% of ions, polarized along I90 axis are magnetic ones.

  27. 3.9 Impossibilityto measure the degree of the ion polarization via linear X-raypolarization. There are general arguments why the beam polarization (i.e. the degree of polarization) cannot be defined via the linear polarization of emitted photons. If it would be possible, the probability should contain a pseudeoscalar term, constructed from the vectors 𝜻and e (for electric photons) or 𝜻and (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 𝜻, cannot be built, and only quadratic in 𝜻terms like (𝜻e)2or (𝜻(e×k))2can 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 WHFQ = WHFQ0 [1 + Q2 (𝜻 sph)] sph = i (e*×e) photon spin

  28. 4. PARITY NONCONSERVATION EFFECTS IN HCI 4.1 POSSIBLE PARITY NONCONSERVATION (PNC) EFFECTS IN ONE-PHOTON TRANSITIONS FOR ATOMS AND IONS Wif = Wif0 [ 1 + (sphn)R1 + (ζn)R2 + (hn)R3 + (ζh)Q1 + (ζsph)Q2 ] n = directionofphotonemission sph = photonspin ζ = directionofionpolarization h = directionofexternalmagneticfield (unitvector)

  29. 4.2 Parityviolatingcoefficients R1= Re [ -i < i │HW │a > (Ei - Ea - i Γ/2)-1 (Waf/ Wif)1/2 ] HW = effective PNC Hamiltonian i,f = initial, final state a = stateadmixedtostate i by HW R2 = λR1(λ = degreeofion beam polarization) R3*= Re [(< i│μH│i > + < a│μH│a >) (Ei - Ea - i Γ/2) -1] R1 μ = magneticmomentoftheelectron; H = externalmagneticfield * Ya. A. Azimov, A. A. Anselm, A. N. Moskalevand R. M. Ryndin Zh. Eksp. Teor. Fiz. 67 (1974) 17

  30. 4.3 Parityconservingcoefficients Q1 = λRe [ (< i │μH │i > + < b│μH│b >) · (Ei - Ea - i Γ/2)-1 (Wbf/ Wif)1/2 ] b = levelclosesttolevel i ofthe same parity, admixedbythemagneticfield H Q2 = a λ, a ≈ 1

  31. 4.4 He-like HCI: levelcrossings ΔE/E 5·10-3 δ (2 3P1) δ (2 3P0) 10-3 δ (2 3P1) δ (2 3P0) Z 110 Data from: A.N. Artemyev, V.M. Shabaev, V.A. Yerokhin, G. Plunien and G. Soff, Phys.Rev. A71 (2005) 062104 δ(23P0) = [E(21S0) – E(23P0)] / E(21S0) δ(23P1) = [E(21S0) – E(23P1)] / E(21S0)

  32. 4.5 PNCeffects in He-likeHCI: a surveyofproposals V.G. Gorshkovand L.N. Labzowsky Zh. Eksp. Teor. Fiz. Pis' ma19 (1974) 30 21S0 - 23P1crossing Z = 6, 30, nuclearspin-dependentweakconstant, R =10 -4 A. Schäfer, G. Soff, P. Indelicatoand W. Greiner Phys. RevA40 (1989) 7362 2 1S0 – 2 3P0crossing, Z = 92, two-photonlaserexcitation G. von Oppen Z. Phys. D21 (1991) 181 2 1S0 – 2 3P0crossing, Z = 6, Stark-inducedemission, R = 10 -6

  33. V.V. Karasiev, L.N. Labzowsky and A.V. Nefiodov Phys. Lett. A172, 62 (1992) 2 1S0 – 2 3P0crossing in U (Z = 92), HFQ decay R ~ 10-4 R.W. Dunford Phys. Rev. A54 (1996) 3820(1974) 30 2 1S0 – 2 3P0crossing Z = 92, stimulatedtwo-photonemission, R = 3 ·10 -4 L.N. Labzowsky, A.V. Nefiodov, G. Plunien, G. Soff, R. Marrusand D. Liesen Phys. RevA63 (2001) 054105 21S0 – 23P0crossing, Z = 63, hyperfinequenchingwithpolarizedions, R = 10 -4 A.V. Nefiodov, L.N. Labzowsky, D. Liesen, G. Plunienand G. Soff Phys. Lett. B534 (2002) 52 21S0 – 23P1crossing, 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 Phys. Rev. A72, 032109 (2005) 2 1S0 – 2 3P0crossing Z = 30 – Z = 48, dielectronicrecombination, polarizedincidentelectrons, R ~ 10-8 A.V. Maiorova, O.I. Pavlova, V.M. Shabaev, C. Kozhuharov, G. PlunienandTh. Stoelker J. Phys. B 42 205002 (2009) 2 1S0 – 2 3P0crossing, Z = 90, 64 radiative recombination linear X-raypolarization, polarizedelectrons, R ~ 10 -8

  35. 4.6 Energy Level SchemeforHe-likeGd 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 157Gd : I =3/2, g I = - 0.3398

  36. 4.7 Energy Level SchemeforHe-likeEu 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 151Eu : I =5/2, g I = + 3.4717

  37. 4.8 PNCeffect in He-likepolarized HCI Basic hyperfine-quenched(HFQ) transition: │1s2s 1S0 > + 1/ΔES<1s2s 1S0 │H hf│ 1s2s 3S1> │1s2s 3S1 > → │1s21S0 > + γ (M1) whereHhf = hyperfineinteractionHamiltonian, ΔES = [ E(2 3S1) – E(2 1S0) ] PNC - allowedtransition: │1s2s 1S0 > + 1/ΔESP<1s2s 1S0 │H W│1s2p 3P0> 1/ΔEP<1s2p 3P0 │H hf│ 1s2p 3P1> · │1s2p 3P1 > → │1s21S0 > + γ (E1) whereΔESP= [ E(2 3P0) – E(2 1S0) ], ΔEP= [ E(2 3P1) – E(2 3P0) ] , R2 = λ [ W HFQ + PNC (E1) / W HFQ (M1)]1/2

  38. 4.9 EvaluationtofthecoefficientR2 One-electronpolarizedions: dWjj’ = dW(0)jj‘ + dW(PNC)jj‘ dW(0)jj‘ = Σλ<k,λ,njl │ργ│ 1s2s 3S1> │k,λ,n’j’l’> Paritynonconservation: │njlm> → │njlm> + [ En’’jl’’ – Enjl]-1<njlm│HW│n‘‘jl‘‘m> │n’’jl’’m> H W = - GF/2√2 QWρN(r)γ5 ,QW = - N + Z (1 – 4sin2θW), GF - Fermi constant ,ρN(r)– charge density distribution in the nucleus After rotating the photon quantization axis to the direction of the IQA (ion beam polarization axis) and by an angle θ cosθ = (𝜻ν) and after summation over the angular momentum projections we obtain the following result

  39. 4.10 Basic magneticdipoletransition (l = l‘) forone-electronions dWnjl,n’jl = dWM1njl,n‘jl [1 + R2 (𝜻ν) λ] R2 = - 2ηnjl,n’’jl̄ RE1(n’’jl̄;n’j’l)/ RM1(njl;n’j’l), l̄ = 2j - l ηnjl,n’’jl̄ = Gnjl,n‘‘jl̄ /En’’jl̄ – Enjl Gnjl,n‘‘jl̄ = - (GF/2√2) QW ∫[Pnjl(r)Qn’’jl̄(r) – Qnjl(r)Pn’’jl̄(r)] ρN(r)r2dr Pnjl(r), Qnjl(r) – upper and lower radial components of the Dirac wave function for the electron RE1, RM1 – reduced matrix elements for the electric and magnetic dipole transitions

  40. 4.11 He-likeEu: basic HFQ transition 2 1S0 – 1 1S0 dWHFQ(21S0→11S0)=dWHFQ0 (21S0→11S0) + dWHFQPNC(21S0→11S0)= =dWHFQ0 (21S0→11S0) [1 – 6/35 aFP2(cosθ) + (𝜻ν) R2λ] dWHFQ0 = WHFQ0 /4πangular independent part

  41. 4.12 PossibledeterminationofthedegreeofalignmentaF The term containingaFgives 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 (𝜻ν)2, (𝜻×ν)2 in the expression for the probability. It also vanishes when the polarization is absent, since then aF = 0. FordefiningaFonehastomeasuredWHFQfor two different angles: dWHFQ (θ=0) - dWHFQ (θ=π/2) / dWHFQ0 = - (18/35) aF

  42. 4.13 PNC effect in He-like HCI: Gd versus Eu ΔE = E(21S0) – E(23P0) from Artemyev et al. 2005 ΔE (Gd) = + 0.004 ± 0.074 eV Z = 64 ΔE (Eu) = - 0.224 ± 0.069 eV Z = 63 Re (ΔE – i Γ/2) -1 = ΔE (ΔE2 + Γ2/4) -1; Γ(Gd) = 0.0016 eV (HFQ E1 23P0→11S0) Lifetime (s) Lifetime (s) R(max) / λR(min) / λ Z 2 3P0 (HFQ E1) 2 1S0 (2 E1) 64 4 · 10 -12 1.0 · 10 -12 0.052 (ΔE = Γ) 0 (ΔE = 0) 63 4 · 10 -13 1.2 · 10 -12 1.0 · 10 -4 0.6 · 10 -4 Disadvantage of Gd: Lifetime of 2 3P0longer than lifetime of 2 1S0 HFQ (E1) transition 2 3P0 → 1 1S0unresolvable from HFQ + PNC (E1) transition 21S0 → 11S0 : Background ≈ 105 New, more accurate value for ΔE (Gd) = 0.023 ± 0.074 eV (Maiorova et al 2009) does not change our conclusions

  43. 4.14 PNC experiments: estimates Polarization time for H-like ions: tpol = 0.44 s; total number of ions in the ring: 1010. After the time tpol the dressing target should be inserted to produce He-like Eu ions in 21S0 state with polarized nuclei. Statistical loss: 10-1 assuming the homogeneous distribution of the population among all L12subshell. Next the PNC experiment can start: observation of the asymmetry (𝜻ν) in the HFQ probability of decay 21S0→11S0. Statistical losses: Efficiency of detector: 10-2 Branching ratio of the HFQ M1 decay to the main decay channel 21S0→11S0 + 2γ(E1): 10-4 Total statistical loss: 10-7 Number of “interesting events” : 1010 ×10-7 = 103 = Nint 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 Schemeofthe PNC experiment Bending magnet x-ray detector dressing target H-like ion beam excitation target He-like ion beam spin rotator storage ring Bending magnet x-ray detector

  45. 4.16 Observation time forthe PNC effect Observation time to fix the PNC effect tobs (fix) Number of events necessary to fix the PNC effectN(fix) = 108 Revolution timetrev= 10-6s tobs(fix) · Nint / trev= N(fix) tobs(fix) = N(fix) · trev / Nint = 108 · 10-6 /103 s =0.1 s Observation time to measure the PNC effect with accuracy 0.1%: tobs (0.1%) Number of events necessary to measure the PNC effect with accuracy 0.1%: N(0.1%) = 1014 tobs(0.1%) = N(0.1%)·trev/Nint = 1014·10-6/103 s = 105 s ≈ 30 hours

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

  47. 5.2 Spin precessionoftheparticle in theexternalmagneticfieldH : Lab. frame: g – gyromagnetic ratio (g=2 for leptons), q – charge Rest frame: ωT - frequency of Thomas precession (ds/dt)rest = s ×Ωμ Bargmann-Michel- Telegdi (BMT) equation a = ½ g -1 For leptons a ≈ α/π ≈ 10-3

  48. 5.3Precession around the direction of the particle velocity Frequency: Field compensation: ωp = 0.

  49. 5.4 Precessionofthe angular momentumofthe 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: Field compensation is not possible: for the vertical field 1 T the static radial electric field 107 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 μ0 BMT equation: Exact proof: Wigner-Echart theorem

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

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