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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. Plunien Technical University of Dresden, Germany St.-Petersburg, 2010 1. Production of polarized HCI beams 1.1 Radiative polarization: simple estimates Radiative polarization occurs via radiative transitions 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 Spin-flip transition rates for electrons (lab.system) W spin-flip = 64 (3 ћc3)-1 │μ0│5 H3 γ5 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 FAIR at GSI with H ≈ 6 T, γ ≈ 23 → TP ≈ 103 hours → too long ! 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 H-like 151Eu ion (I = 5/2). E FZee g J F(F 1) J(J 1) - I(I 1) μ 0 , 2F(F 1) 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. MF ' 1s1/2 gJ - electron g-factor. F' = 3 * A. Prozorov, L. Labzowsky, D. Liesen and F. Bosch 1s1/2 Phys. Lett. B574 (2003) 180 F=2 MF 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. This leads to the partial polarization of the 1s1/2 F' = 3 state. 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). 1.3 Description of polarization The polarization state of an ion after i-th "cycle" (switching on the laser) is described by the density matrix: ρF(i) = ΣMF nFMF(i) ψFMF* ψFMF . Normalization condition: ΣMF nFMF(i) = 1, ψFMF are the wavefunctions, nFMF(i) the occupation numbers F, MF the total angular momentum and projection of an ion Degree λ of polarization is defined as: λF (i) = F-1 ΣMF nFMF(i) MF Nonpolarized ions: nFMF = (2F + 1) -1, λF = 0 Fully polarized ions: nFF = 1, λF = 1 1.4 Dynamics of polarization The occupation numbers are defined with the recurrence relations via the M1 transition probabilities: n n (i) FM F (i) F' M' F 1 (i -1) n FM F δ M' F M F 2 1 W(F' M'F FMF ) (i) 1 (i-1) n F'M'F n FM F 2 M'F M F ,M F 1 (F' M'F ) 2 (F' M'F ) W(F' M'F FMF ) MF width of the sublevel F’M’F Uniform initial population λ F(0) = 0, n FMF(0) = (2F+1)-1 After first cycle: λ F(1) = 0.1667 After 40 cycles: λ F(40) = 0.9993 Opposite initial population λ F(0) = -1, n F-F(0) = 1 After first cycle: λ F(1) = - 0.6667 After 40 cycles: λ F(40) = 0.9986 λ, nFMF λ 1 0 The polarization time for +2 λF(40) = 0.999 TP = 40 · 10.9 ms = 0.44 s +1 N 0 10 40 1.5 Nuclear polarization Nuclear polarization density matrix ρI = < ψFMF │ρF│ψFMF >el (integration over electron variables) ψFMF = ΣMIMJ CFMF IJ (MIMJ) ψIMI ψJMJ ψIMI , ψJMJ nuclear, electronic wave functions ρI = ΣMI nIMI ψIMI* ψIMI ; nIMI = ΣMJMF nFMF [CFMF (MIMJ)]2 Degree λ of nuclear polarization: λI = I-1 ΣMI nIMI MI Maximum nuclear polarization for the case of full electron polarization nFF = 1 (F = 2) in Eu ions: λI max = 0.93 1.6 Polarization of one- and two-electron ions Polarization in He-like ions with total electron angular momentum equal to zero (2 1S0, 2 3P0) is nuclear polarization. In polarized one-electron HCI the nuclei are also polarized, due to the strong hyperfine interaction (hyperfine splitting 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 1S0, 2 3P0) the nuclear polarization remains unchanged. 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 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 longitudinal polarization of the ions via selective laser excitation. 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. The movement of an ion in a magnetic system of a ring can be described classically with the equation of motion: dv/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 timedependent field. The spin dynamics which is influenced by the transitions between hyperfine and Zeeman sublevels we describe quantum - mechanically. 2.2 Spin dynamics and the instantaneous quantization axis (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)│s ζ(t)│χS(t) > = 0 (∗∗) From (∗) and (∗∗) follows the equation for IQA: ∂ζ/∂t = μ0 (H(t) x ζ(t)) (∗∗∗) 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 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 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. 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: ζx(0) = 0, ζy(0) = 0, ζz(0) = 1 The magnetic field H is oriented along the vertical (x) axis: Hx = H(t), Hy = Hz = 0 Solution of the Schrödinger equation reads: 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) with the timedependent frequency ω(t) = φ(t) / t 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 -4 μ0 By comparing eqs. (A) and (B) we conclude that the rotation angle for IQA after passing one bending magnet amounts to about 104 π. Thus, it will be extremely difficult to fix the direction of polarization before the start of the PNC experiment. 2.4 Solution of the problem: "Siberian Snake" A Siberian Snake rotates the polarization (IQA) by an angle π 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: Siberian Snake beam 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. 3. Diagnostics of polarization 3.1 The hyperfine quenching (HFQ) of polarized twoelectron ions in an external magnetic field The HFQ transition probability for the polarized ion in an external magnetic field: 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 case of the 2 1S0 – 1 1S0 HFQ, the coefficient Q1 is: 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 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 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. Stöhlker and S. Tashenov, Phys. Rev. Lett. 94 (2005) 203202 The formula tan 2χ ~ λ F was confirmed experimentally (for λ 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. 3.3 Linear polarization of X-ray photons in HFQ transitions in polarized ions Photon density matrix I 1 P3 k , λ ρ̂ γ k , λ' 2 P1 iP2 P1 iP2 1 - P3 k is the photon momentum: k =𝜔𝜈, 𝜔 is frequency 𝜆 , 𝜆‘ are the helicities: 𝜆 = sph 𝜈 =± 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 3.4 Stokes parameters P1 I 0 I 90 I 0 I 90 P2 I 45 I135 I 45 I135 P3 - circular polarization Iα – intensity of the light, polarized along the axis α. Stokes parameters via photon density matrix: k 1 ρ̂ γ P1 k 1 ρ̂ γ k 1 k 1 ρ̂ γ k 1 k 1 ρ̂ γ k 1 ρ̂ γ P2 i k 1 ρ̂ γ k 1 k 1 k 1 k 1 ρ̂ γ k 1 k 1 ρ̂ γ k 1 k 1 Schematic position of the axes in the X-ray polarization observation experiment 3.5 Rotation of the photon density matrix 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‘ ν k, λ, j ρ̂ γ k, λ, j' Const D0μ (θθ νμ C LL ν0 ( M L ' M L )C LL νμ mm' i L-L (1) mm' LL'n jm LML LM L λ' λ (λλ) jm αA L'M L ' j' m' jm αA LML j' m' * 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 3.6 Application to the 21S0→11S0 HFQ transition (magnetic dipole photons) P Const n FM F M 1 MF 3M 2F F(F 1) (2F 1)F(F 1)(2F 3) F – total angular momentum of an ion; nFMF – occupation numbers Nonplarized ions: nFMF = const: PM1=0; PM2=0 independent on the polarization. Hence, the photons are nonpolarized if they are emitted by nonpolarized by nonpolarized ions. For 21S0 state of 15163Eu61+ : F=I=5/2, n5/2 5/2 = 5/6, n5/2 3/2 = 1/6 𝜆F = 𝜆I = (1/F) ΣMF nFMF MF = 0.93 PM1 = -0.4, PM2=0 3.7 Polarization and alignment Thus, one cannot extract the degree of polarization 𝜆F from the Stokes parameters Stokes parameter PM1 defines „the degree of alignment“ which can be defined as aF =ΣMF nFMF MF2 - a0F where a0F =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,MF and amaxF = 1/3 F(2F - 1) 3.8 Stokes parameters 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. 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. If it would be possible, the probability should contain a pseudeoscalar term, constructed from the vectors 𝜻 and e (for electric photons) or 𝜻 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 𝜻, cannot be built, and only quadratic in 𝜻 terms like (𝜻e)2 or (𝜻(e×k))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 WHFQ = WHFQ0 [1 + Q2 (𝜻 sph)] sph = i (e*×e) photon spin 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 = direction of photon emission sph = photon spin ζ = direction of ion polarization h = direction of external magnetic field (unit vector) 4.2 Parity violating coefficients 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 = state admixed to state i by HW R2 = λR1 (λ = degree of ion beam polarization) R3*= Re [(< i│μH│i > + < a│μH│a >) (Ei - Ea - i Γ/2) -1] R1 μ = 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 4.3 Parity conserving coefficients Q1 = λ Re [ (< i │μH │i > + < b│μH│b >) · (Ei - Ea - i Γ/2)-1 (Wbf / Wif)1/2 ] b = level closest to level i of the same parity, admixed by the magnetic field H Q2 = a λ, a ≈ 1 4.4 He-like HCI: level crossings ΔE/E 5·10-3 δ (2 3P1) δ (2 3P0) 10-3 δ (2 3P1) δ (2 3P0) 110 δ(23P0) = [E(21S0) – E(23P0)] / E(21S0) δ(23P1) = [E(21S0) – E(23P1)] / E(21S0) Z Data from: A.N. Artemyev, V.M. Shabaev, V.A. Yerokhin, G. Plunien and G. Soff, Phys.Rev. A71 (2005) 062104 4.5 PNC effects in He-like HCI: a survey of proposals V.G. Gorshkov and L.N. Labzowsky Zh. Eksp. Teor. Fiz. Pis' ma 19 (1974) 30 21S0 - 23P1 crossing Z = 6, 30, nuclear spin-dependent weak constant, R =10 -4 A. Schäfer, G. Soff, P. Indelicato and W. Greiner Phys. Rev A40 (1989) 7362 2 1S0 – 2 3P0 crossing, Z = 92, two-photon laser excitation G. von Oppen Z. Phys. D21 (1991) 181 2 1S0 – 2 3P0 crossing, Z = 6, Stark-induced emission, R = 10 -6 V.V. Karasiev, L.N. Labzowsky and A.V. Nefiodov Phys. Lett. A172, 62 (1992) 2 1S0 – 2 3P0 crossing in U (Z = 92), HFQ decay R ~ 10-4 R.W. Dunford Phys. Rev. A54 (1996) 3820(1974) 30 2 1S0 – 2 3P0 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 Phys. Rev A63 (2001) 054105 21S0 – 23P0 crossing, Z = 63, hyperfine quenching with polarized ions, R = 10 -4 A.V. Nefiodov, L.N. Labzowsky, D. Liesen, G. Plunien and G. Soff Phys. Lett. B534 (2002) 52 21S0 – 23P1 crossing, Z = 33, nuclear anapole moment, polar. ions, R = 0.6·10 -4 G.F. Gribakin, E.F. Currell, M.G. Kozlov and A.I. Mikhailov Phys. Rev. A72, 032109 (2005) 2 1S0 – 2 3P0 crossing Z = 30 – Z = 48, dielectronic recombination, polarized incident electrons, R ~ 10-8 A.V. Maiorova, O.I. Pavlova, V.M. Shabaev, C. Kozhuharov, G. Plunien and Th. Stoelker J. Phys. B 42 205002 (2009) 2 1S0 – 2 3P0 crossing, Z = 90, 64 radiative recombination linear X-ray polarization, polarized electrons, R ~ 10 -8 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 157Gd : I =3/2, g I = - 0.3398 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 151Eu : I =5/2, g I = + 3.4717 4.8 PNC effect in He-like polarized HCI Basic hyperfine-quenched (HFQ) transition: │1s2s 1S0 > + 1/ΔES<1s2s 1S0 │H hf│ 1s2s 3S1> │1s2s 3S1 > → │1s2 1S0 > + γ (M1) where Hhf = hyperfine interaction Hamiltonian, ΔES = [ E(2 3S1) – E(2 1S0) ] PNC - allowed transition: │1s2s 1S0 > + 1/ΔESP<1s2s 1S0 │H W│1s2p 3P0> 1/ΔEP<1s2p 3P0 │H hf│ 1s2p 3P1> · │1s2p 3P1 > → │1s2 1S0 > + γ (E1) where ΔESP = [ E(2 3P0) – E(2 1S0) ], ΔEP = [ E(2 3P1) – E(2 3P0) ] , R2 = λ [ W HFQ + PNC (E1) / W HFQ (M1)]1/2 4.9 Evaluationt of the coefficient R2 One-electron polarized ions: dWjj’ = dW(0)jj‘ + dW(PNC)jj‘ dW(0)jj‘ = Σλ <k,λ,njl │ργ│ 1s2s 3S1> │ k,λ,n’j’l’ > Parity nonconservation: │ njlm > → │ njlm > + [ En’’jl’’ – Enjl]-1<njlm│H W│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 cos θ = (𝜻ν) and after summation over the angular momentum projections we obtain the following result 4.10 Basic magnetic dipole transition (l = l‘) for one-electron ions 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 4.11 He-like Eu: 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) R2 λ] 1 3 2 S Ĥ 2 P1 R ΔE 0 HFS 3G 2s,2p S E1 R2 1.14 104 ΔE SP (I 1) 21S0 Ĥ HFS 23 S1 R M1 ΔE P dWHFQ0 = WHFQ0 /4π angular independent part 4.12 Possible determination of the degree of alignment aF The term containing aF 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 (𝜻ν)2, (𝜻×ν)2 in the expression for the probability. It also vanishes when the polarization is absent, since then aF = 0. For defining aF one has to measure dWHFQ for two different angles: dWHFQ (θ=0) - dWHFQ (θ=π/2) / dWHFQ 0 = - (18/35) aF 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 ΔE (Eu) = - 0.224 ± 0.069 eV Z = 64 Z = 63 Re (ΔE – i Γ/2) -1 = ΔE (ΔE2 + Γ2/4) -1; Γ(Gd) = 0.0016 eV (HFQ E1 23P0→11S0) Lifetime (s) Z 64 63 2 3P0 (HFQ E1) 4 · 10 -12 4 · 10 -13 Lifetime (s) R(max) / λ R(min) / λ 2 1S0 (2 E1) 1.0 · 10 -12 1.2 · 10 -12 0.052 (ΔE = Γ) 1.0 · 10 -4 0 (ΔE = 0) 0.6 · 10 -4 Disadvantage of Gd: Lifetime of 2 3P0 longer than lifetime of 2 1S0 HFQ (E1) transition 2 3P0 → 1 1S0 unresolvable 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 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 L12 subshell. Next the PNC experiment can start: observation of the asymmetry (𝜻ν) in the HFQ probability of decay 21S0→11S0. 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 Statistical losses: Not enough! After the dressing of ions and the PNC experiments the He-like ions leave the ring. The ring should be filled again! 4.15 Scheme of the 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 4.16 Observation time for the PNC effect Observation time to fix the PNC effect tobs (fix) Number of events necessary to fix the PNC effect N(fix) = 108 Revolution time trev = 10-6 s 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 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. Semertzidis Proc. of the Workshop on Frontier Tests of Quantum Electrodynamics and Physics of the 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. Silenco and E.J. Stephenson Phys. Rev. Lett. 93, 052001 (2004) 5.2 Spin precession of the particle in the external magnetic field H : Lab. frame: g – gyromagnetic ratio (g=2 for leptons), q – charge Rest frame: ωT - frequency of Thomas precession (ds/dt)rest = s ×Ωμ a = ½ g -1 For leptons Bargmann-MichelTelegdi (BMT) equation a ≈ α/π ≈ 10-3 5.3 Precession around the direction of the particle velocity Frequency: Field compensation: ωp = 0. 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: 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 5.5 EDM spin precession for H-like HCI for any particle For H-like HCI Frequency of the EDM precession: EDM: If de ≈ 10-28 e cm, η ≈ 10-17 5.6 EDM spin rotation angle z Hm z Hm E E φ y -φ IQA IQA y Hc x x IQA rotation in the plane xy due to the IQA rotation in the plane xy due to the motional magnetic field Hm=β×E motional magnetic field neglecting electron EDM. In the + IQA rotation in the plane yz due to the electron EDM. absence of E the IQA is directed along y axis. φ – electron EDM rotation angle (in the plane yz) averages to zero due to the Hm rotation; compensating magnetic field Hc is necessary. 5.7 Observation of the EDM effect in storage rings A. β – active bare nuclei or HCI with β – active nuclei and closed electron shells: Decay process: N* → N + e- + ν͞ e B. muon: Decay process: μ- → e- + ν͞ e + νμ Observation: asymmetry ζne ζ – polarization of the nuclei (muon) ne – direction of the decay electron emission Both processes are P – violating. However they are due to the weak interaction. Therefore no additional smallness 5.8 Observation of the electron EDM with H-like HCI in storage ring Laser excitation of the HF excited level 1s1/2 F=2 → 1s1/2 F=3 for 15163Eu 62+ Decay is observed; decay time ~ 11 ms, then again excitation sph – photon spin; νph - direction of the photon emission (sph νph) = ± 1 chirality, or circular polarization ζ – ion polarization vector; λ – degree of polarization Qc = const; for 1s1/2 F=3 → 1s1/2 F=2 transition Qc = ½. Asymmetry observed: ζνph ; if circular polarization is fixed, there is no P-violation. Summation over circular polarizations (±) gives zero 5.9 Scheme of the electron EDM experiment z E d snake φ φ Φ y d First revolution x field on z d snake M -Φ d -φ x Second revolution φ field off E 2φ -φ d x y z d snake Third revolution M M y field on M – magnetic system of the ring, d – photon detectors that fix circular polarization, φ – EDM rotation angle grows linearly with time 5.10 Estimates for the observation time Asymmetry A = λ Qc λ = F sinφ ≈ F φ (φ « 2π); F=3 for Eu62+ φ = | ωd | tobs p, p is the part of the ring where electric field is applied ωd is the frequency of the EDM caused spin precession Numbers: E ≈ 105 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); Hc ≈ 300 gauss These fields E, Hc applied within pL, do not disturb essentially the trajectory | ωd | ≈ η 1010 s-1 a) Let A ~ 10-5 ; η ~ 10-17 , de ~ 10-28 e cm; then tobs ~ 105 ~ 30 hours b) Let A ~ 10-6 ; η ~ 10-19 , de ~ 10-30 e cm; then tobs ~ 106 ~ 12 days 6. CONCLUSIONS • • • Polarization: for 15163Eu62+ polarization time for the 100% polarization tpol = 0.44 s Nuclear polarization, corresponding to 100% ion polarization: 93% PNC experiment Time necessary for the observation of the PNC effect tobs PNC ~ 0.1 s Time necessary for the measurement of the PNC effect with accuracy 0.1% (higher than in neutral Cs) tobsPNC ~ 30 hours Time necessary to observe electron EDM at the level 10-28 e cm: tobsEDM ~ 30 hours Time necessary to observe electron EDM at the level 10-30 e cm: tobsEDM ~ 12 days