Transcript Parity violation in He-like Gadolinium and Europium Leonti

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 MF ,MF 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 ρˆ 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  I90
I 0  I90
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 ρˆ γ
P1  
k  1 ρˆ γ


k  1  k  1 ρˆ γ


k  1  k  1 ρˆ γ

k  1 ρˆ γ
P2  i 
k  1 ρˆ γ

k 1

k 1


k  1  k  1 ρˆ γ


k  1  k  1 ρˆ 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 ρˆ γ k, λ, j'  Const D0μ
(θθ
νμ
C
LL
ν0
( M L ' M L )C
LL
νμ

i L-L (1)mm'  LL'n jm 
mm' LM LM 
L
L
  λ'
λ
(λλ) 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
3M2F  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
ˆ
2
S
H
2
P1 R ΔE
0
HFS
3G2s,2p
S
E1
R2  
 1.14104
3
ˆ
ΔESP (I  1) 21 S0 H
2
S1 R M1 ΔE P
HFS
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