QED calculation of the ground-state energy
and the ionization potential of Be-like ions
A. V. Malyshev, A. V. Volotka, D. A. Glazov,
I. I. Tupitsyn, V. M. Shabaev, G. Plunien, and N. A. Zubova
Department of Physics
Saint-Petersburg State University
August 25, 2015
Outline:
Introduction and Motivation
Theoretical Description and
Numerical Procedure
Field shift in Boron-like ions
Summary and Outlook
Motivation
Experiments:
H-like:
T. Stöhlker et al.,
PRL, 2000;
A. Gumberidze et al., PRL, 2005.
He-like:
A. Gumberidze et al., PRL, 2004.
Li-like:
J. Schweppe et al., PRL, 1991;
C. Brandau et al.,
PRL, 2003;
P. Beiersdorfer et al., PRL, 2005.
Be-like:
P. Beiersdorfer et al., PRA, 1998;
I. Draganić et al.,
PRL, 2003;
D. Bernhardt et al., JPB, 2015.
B-like:
I. Draganić et al.,
V. Mäckel et al.,
PRL, 2003;
PRL, 2011.
The ground-state energies of highly-charged ions are of the
great importance for mass spectrometry
J. Repp et al., Appl. Phys. B, 2012;
K. Blaum et al., Phys. Scr., 2013;
E. G. Myers, Int. J. Mass Spectrom., 2013.
2
QED in the Furry picture
Highly-charged few-electron ions:
Ne ≪ Z
Ne is the number of electrons,
Z is the nuclear charge number
To zeroth-order approximation:
[−iα · ∇ + βm + Vnuc (r)] ψn (r) = εn ψn (r)
Interelectronic interaction and QED effects:
Interelectronic interaction
1
∼
Binding energy
Z
QED
∼ α(αZ)2
Binding energy
In uranium: Z = 92, αZ ≈ 0.7
3
The extended Furry picture
Vscr describes approximately the electron-electron interaction effects
Vnuc (r) −→ Veff (r) = Vnuc (r) + Vscr (r)
To zeroth-order approximation:
[−iα · ∇ + βm + Veff (r)] ψn (r) = εn ψn (r)
The extended Furry picture is used to accelerate the convergence of
the perturbation series
4
Contributions
The interelectronic
interaction diagrams
First- and second-order
QED diagrams
Third- and higher-order electron-correlation contributions
within the Breit approx.
(the large-scale configuration-interaction Dirac-Fock-Sturm method)
One-electron two-loop diagrams
5
Contributions
Beyond external field approximation:
m
Nuclear recoil finite nuclear mass, ∼
M
Breit approx. (all orders in 1/Z)
QED recoil effect in zeroth order
Nuclear polarization (internal degrees of freedom)
excited −→
nucl. state
←− bound
electron
ground state −→
6
Uranium (Z = 92): individual contributions, in eV
Contribution
(0)
EDirac
(1)
Eint
(2)
Eint,Breit
(2)
Eint,QED
(>3)
Eint,Breit
LDF
KS
PZ
−320572.56
−6637.37
−20.87
−320871.02
−6338.62
−20.45
−321276.02
−5933.84
−23.97
2.04
1.96
2.05
3.03
−327225.74
2.39
−327225.74
6.05
−327225.74
(1)
618.61
620.18
618.97
(2)
0.73
−0.86
0.36
−2.92
616.41
0.60
0.56
−0.45
−326608.62
−2.92
616.40
0.60
0.56
−0.45
−326608.63
−2.92
616.41
0.60
0.56
−0.45
−326608.63
Eint,total
EQED
EScrQED
(2l)
EQED
EQED,total
ERec,Breit
ERec,QED
ENucl.Pol.
Etotal
Individual contributions to the binding energy of Be-like U (in eV)
7
Binding energies of Be-like ions, in eV
Nucl.
56
26 Fe
This work
−22102.960(45)
Other theory
−22103.37a
−22103.299
NIST
−22102.1(1.8)
b
−22102.98(8)c
132
54 Xe
−100972.921(85)
−100973.7a
−100963(4)
−100973.75b
184
74 W
238
92 U
−198983.71(32)
−326608.6(1.2)
−198984.71b
−326608.5
b
−198987(3)
−326600(300)
This work: A. V. Malyshev et al., PRA, 2014.
a
b
c
M. F. Gu, At. Data Nucl. Data Tables, 2005.
M. H. Chen and K. T. Cheng, PRA, 1997.
V. A. Yerokhin, A. Surzhykov, and S. Fritzsche, PRA, 2014.
8
Ionization potentials for Be-like ions, in eV
Nucl.
56
26 Fe
This work
Other theory
−1951.307(21)
−1950.4(1.8)a
−1951.4b
132
54 Xe
−9590.935(37)
−9581(4)c
−9591.4b
238
92 U
−32386.14(20)
−32374(300)c
This work: A. V. Malyshev et al., PRA, 2015.
a
E. Biémont, Y. Frémat, and P. Quinet, At. Data Nucl. Data Tables, 1999.
M. F. Gu, At. Data Nucl. Data Tables, 2005.
c
G. C. Rodrigues et al., At. Data Nucl. Data Tables, 2004. (uncertainty
prescribed by NIST)
b
9
Isotope shifts
Field-shift factor:
δEFS = F δhr 2 i
⇔
F =
dE(R)
dhr 2 i
R = hr 2 i is the root-mean-square nuclear radius.
C. Brandau et al., PRL, 2008:
Isotope shifts in Li-like
2s − 2p1/2
FS (th.)
QED FS (th.)
···
IS (exp.)
150,142
Nd57+ (in meV)
2s − 2p3/2
−42.57
−44.05
0.22
0.24
···
···
−40.2(3)(6)
−42.3(12)(20)
∼ 0.5%
10
Field-shift factor for the 2p3/2 − 2p1/2 transition
in B-like ions (in GHz/fm2)
Preliminary results
CH(1s2 )
CH(1s2 2s2 )
LDF
FSE
259
253
256
FScrSE
−54
−48
−51
FVP
−267
−260
−261
FScrVP
42
36
37
(1)
Fint,QED
42
39
43
Ftot,QED
22(6)
20(6)
24(6)
FCI
−20570
−20570
∼ 0.1%
−20570
Two-electron ScrSE and ScrVP diagrams ∼
1
Z
11
Summary
The calculations of the ground-state binding energies of
Be-like ions are performed in the range: 18 6 Z 6 96
The accuracy was significantly improved and the most precise
theoretical predictions for the ground-state binding energies
and ionization potentials in Be-like ions have been obtained
Achieved accuracy allows tests of bound-state QED with
Be-like ions on the level ∼ 0.2%, provided the corresponding
experiments are performed to the required precision
12
Outlook
High-precision QED calculations of
Ionization potentials for the valence electrons in the
1s2 2s2 2p1/2 and 1s2 2s2 2p3/2 states of B-like ions
Binding energies of excited states in Be-like ions
More comprehensive analysis of the QED filed shift
in B-like ions
Outlook
High-precision QED calculations of
Ionization potentials for the valence electrons in the
1s2 2s2 2p1/2 and 1s2 2s2 2p3/2 states of B-like ions
Binding energies of excited states in Be-like ions
More comprehensive analysis of the QED filed shift
in B-like ions
Thank You for Your attention!