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FKK 2010, St. Petersburg, 10.12.10
Precision calculations of the hyperfine structure in highly charged ions
Andrey V. Volotka, Dmitry A. Glazov,
Vladimir M. Shabaev, Ilya I. Tupitsyn, and Günter Plunien
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Introduction and Motivation
# Heavy few-electron ions provides possibility to test of QED at extremely strong electric fields
Interelectronic interaction ~ 1 / Z
QED ~ α
=> high-precision calculations are possible!
However, in contrast to light atoms,the parameter αZ is not small
=> test of QED to all orders in αZ
In U92+: αZ ≈ 0.7
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FKK 2010, St. Petersburg, 10.12.10
Introduction and Motivation
# Investigations of the hyperfine structure and g factor in heavy ions provide
high-precision test of the magnetic sector of bound-state QEDin the nonperturbative regime(hyperfine splitting and g factor of H-, Li-, and B-like heavy ions)
Fundamental physics
independent determination of the fine structure constantfrom QED at strong fields(g factor of H- and B-like heavy ions)
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eV)8(0840.5exp E
# Measurements of the ground-state hyperfine splitting in H-like ions
Klaft et al., PRL 1994209Bi82+
eV)6(1646.2exp E
Crespo López-Urrutia et al., PRL 1996; PRA 1998
165Ho66+
eV)18(7190.2exp E185Re74+
eV)18(7450.2exp E187Re74+
eV)2(2159.1exp E
Seelig et al., PRL 1998207Pb81+
eV)25(21351.3exp E
Beiersdorfer et al., PRA 2001203Tl80+
eV)29(24409.3exp E205Tl80+
Hyperfine structure in heavy ions
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# Basic expression for the hyperfine splitting
3p
3
3)(
1
1
)12)(1(
12)(
Mmlj
I
m
g
n
ZE
aa
I
a
a
)( ZA – relativistic factor δ – nuclear charge distribution correctionε – nuclear magnetization distribution correction
ZZB /)( – interelectronic-interaction correction of first-order in 1/Z2/),( ZZZC – 1/Z2 and higher-order interelectronic-interaction correction
SQEDx – screened QED correction
QEDx – one-electron QED correction
Hyperfine structure in heavy ions
)( ZA )1( )1( QEDx ZZB /)( 2/),( ZZZC SQEDx
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FKK 2010, St. Petersburg, 10.12.10
Hyperfine structure in heavy ions
# Ground-state hyperfine splitting in H-like ions
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Hyperfine structure in heavy ions
# Ground-state hyperfine splitting in Li-like ions
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# Bohr-Weisskopf correction
)(),(~ rKrKε LS
,
)()(
)()(
)(
0
0
rdrfrg
rdrfrg
rK
r
S
0
03
3
)()(
)()(1
)(
rdrfrg
rdrfrgrr
rK
r
L
Bohr-Weisskopf correction depends linearly on the functions KS(r) and KL(r)
[Shabaev et al., PRA 1998]
Hyperfine structure in heavy ions
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FKK 2010, St. Petersburg, 10.12.10
)()(
)(
)(
)()1(
)2(
)1(
)2(
)1(
)2(
Zfε
ε
rK
rK
rK
rKs
s
sL
sL
sS
sS
0)(nucl
rf
rgrVεmσ
r
κσ
dr
diσ zxy
For a given κ the radial Dirac equations are the same in the nuclear region
[Shabaev et al., PRL 2001]
)(2
)(nuclnucl
nucl2
2
RVR
Z
n
Zε
aa
# Bohr-Weisskopf correction
=> the ratio of the Bohr-Weisskopf corrections is very stable with respect to variations of the nuclear models
Hyperfine structure in heavy ions
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)1()2( ξ ss EEE
Hyperfine structure in heavy ions
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# Vacuum-polarization correction
[Sunnergren, Persson, Salomonson, Schneider, Lindgren, and Soff, PRA 1998]
[Schneider, Greiner, and Soff, PRA 1994]
[Artemyev, Shabaev, Plunien, Soff, and Yerokhin, PRA 2001]
[Sapirstein and Cheng, PRA 2001]
Second-order terms in perturbation theory expansion
Hyperfine structure in heavy ions
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# Self-energy correction
[Persson, Schneider, Greiner, Soff, and Lindgren, PRL 1996]
[Shabaev, Tomaselli, Kühl, Artemyev, and Yerokhin, PRA 1997]
[Blundell, Cheng, and Sapirstein, PRA 1997]
Second-order terms in perturbation theory expansion
Hyperfine structure in heavy ions
[Yerokhin and Shabaev, PRA 2001]
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Hyperfine structure in heavy ions
# Screened QED correction: effective potential approach
)()()()( scrnucleffnucl rVrVrVrV
[Glazov, Volotka, Shabaev, Tupitsyn, and Plunien, PLA 2006]
[Volotka, Glazov, Tupitsyn, Oreshkina, Plunien, and Shabaev, PRA 2008]
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FKK 2010, St. Petersburg, 10.12.10
Hyperfine structure in heavy ions
# Screened self-energy correction: effective potential approach
rdr
rρrV c
0
scr
)()(
Different screening potential have been employed
core-Hartree potential
Kohn-Sham potential
– density of the core electrons)(rρc
3/1
20
scr )(32
81
3
2)()(
rrρr
rdr
rρrV t
t
– total electron density)(rρt
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# Screened vacuum-polarization correction
Third-order terms in perturbation theory expansion
32 diagrams
Hyperfine structure in heavy ions
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# Screened self-energy correction
Third-order terms in perturbation theory expansion
36 diagrams
Hyperfine structure in heavy ions
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# Screened self-energy correction Derivation of the formal expressions Regularizations of the divergences Ultraviolet divergences: diagrams (A), (B), (C), (E), and (F) Infrared divergences: diagrams (C), (D), and (F)
Calculation Angular integrations Evaluation of regularized zero- and one-potential terms in momentum-space Contour rotation: identification of the poles structure Integration over the electron coordinates and the virtual photon energy
Verification Angular integrations: analytical and numerical 2 different contours for the integration over the virtual photon energy Different gauges: Feynman and Coulomb Comparison with results obtained within screening potential approx.
Hyperfine structure in heavy ions
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Numerical results
# Screened self-energy correction xSQED(SE) in the Feynman and Coulomb gauges for the Li-like 209Bi80+
Feynman CoulombA, irr 0. 001544 0. 001555B, irr - 0. 000380 - 0. 000398C, irr 0. 001928 0. 001952D, irr - 0. 000936 - 0. 000945E, irr 0. 000028 0. 000028F, irr - 0. 000174 - 0. 000172G, red - 0. 001298 - 0. 001307H, red 0. 000331 0. 000331I, red 0. 000066 0. 000066
Total 0. 001109 0. 001109
Kohn-Sham screening 0. 0012core-Hartree screening 0. 0013
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Numerical results
# Specific difference between hyperfine splitting in H- and Li-like bismuth in meV
ξΔE (1s ) ΔE (2s ) Δ'EDirac value 876. 638 844. 829 - 31. 809QED - 5. 088 - 5. 052 0. 036Screened QED 0. 194( 6) 0. 194( 6) local potential approx. 0. 21( 4) 0. 21( 4)Interel. 1/Z - 29. 995 - 29. 995Interel. higher orders 0. 25( 4) 0. 25( 4)
Theory: - 61. 32( 4)
;ξ )1()2( ss EEE for Z=83 we obtain ξ=0.16886
=> possibility for a test of screened QED on the level of few percent
Remaining uncertainty ≈ 0.005 – 0.010 meV
[Volotka, Glazov, Shabaev, Tupitsyn, and Plunien, PRL 2009][Glazov, Volotka, Shabaev, Tupitsyn, and Plunien, PRA 2010]
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Numerical results
# Bohr-Weisskopf corrections for H-, Li-, and B-like bismuth
)5(0148.0)1(
NS
)1(exp
)1(QED
)1(NS)1(
s
ssss
E
EEEε
)3(0782.1)1(
)2(
s
s
ε
ε
The ratio of the Bohr-Weisskopf corrections
Knowing 1s hyperfine splitting from experiment, the Bohr-Weisskopf correction can be obtained
)2(295.0)1(
)2(
s
p
ε
ε
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ΔE(2s) ΔE(2p) Dirac value (point nucl.) 958. 5 296. 35Nuclear size - 113. 7( 7) - 9. 84( 5)Bohr-Weisskopf - 13. 1( 7) - 1. 11( 5)Interelectronic interaction - 29. 7 - 27. 31( 22)QED - 4. 8 - 0. 25( 2)
Total theory: 797. 22( 15) 257. 83( 22)
Exp.: 820( 26) * 791( 5) **
Numerical results
# Hyperfine splitting in Li- and B-like bismuth in meV
*Beiersdorfer et al., PRL 1998
**Beiersdorfer et al., unpublished
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# Two-photon exchange correction
Outlook
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Summary
# Conclusion
the most accurate theoretical prediction for the specific differencebetween hyperfine structure values in H- and Li-like Bi has been obtained
rigorous evaluation of the complete gauge-invariant set of the screenedQED corrections has been performed