precision calculations of the hyperfine structure in highly charged ions
DESCRIPTION
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. Introduction and Motivation. # Heavy few-electron ions provides possibility to test of QED - PowerPoint PPT PresentationTRANSCRIPT
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
FKK 2010, St. Petersburg, 10.12.10
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
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)
FKK 2010, St. Petersburg, 10.12.10
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
FKK 2010, St. Petersburg, 10.12.10
# 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
FKK 2010, St. Petersburg, 10.12.10
Hyperfine structure in heavy ions
# Ground-state hyperfine splitting in H-like ions
FKK 2010, St. Petersburg, 10.12.10
Hyperfine structure in heavy ions
# Ground-state hyperfine splitting in Li-like ions
FKK 2010, St. Petersburg, 10.12.10
# 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
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
FKK 2010, St. Petersburg, 10.12.10
)1()2( ξ ss EEE
Hyperfine structure in heavy ions
FKK 2010, St. Petersburg, 10.12.10
# 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
FKK 2010, St. Petersburg, 10.12.10
# 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]
FKK 2010, St. Petersburg, 10.12.10
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]
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
FKK 2010, St. Petersburg, 10.12.10
# Screened vacuum-polarization correction
Third-order terms in perturbation theory expansion
32 diagrams
Hyperfine structure in heavy ions
FKK 2010, St. Petersburg, 10.12.10
# Screened self-energy correction
Third-order terms in perturbation theory expansion
36 diagrams
Hyperfine structure in heavy ions
FKK 2010, St. Petersburg, 10.12.10
# 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
FKK 2010, St. Petersburg, 10.12.10
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
FKK 2010, St. Petersburg, 10.12.10
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]
FKK 2010, St. Petersburg, 10.12.10
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
FKK 2010, St. Petersburg, 10.12.10
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
FKK 2010, St. Petersburg, 10.12.10
# Two-photon exchange correction
Outlook
FKK 2010, St. Petersburg, 10.12.10
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