3.04 MB

Atomic Theory of Few-Electron Systems for Nuclear
Charge Radii Determination
Zong-Chao Yan
Department of Physics
University of New Brunswick
Canada
[email protected]
(Established 1785)
March 19-21, 2012, Bad Honnef
Collaborators:
Gordon Drake (Univ. of Windsor)
Li-Ming Wang (Univ. of New Brunswick)
Hao-Xue Qiao (Wuhan University)
W. Nörtershäuser GSI team
Supports:
NSERC, SHARCnet, ACEnet
P
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J
E
C
T
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Li-11 nuclear charge radius by W. Nörtershäuser:
R. Sánchez, G. Ewald, D. Albers, J. Behr, P. Bricault,
B. A. Bushaw, A. Dax, J. Dilling, M. Dombsky, G. W. F.
Drake, S. Götte, R. Kirchner, H.-J. Kluge, Th. Kühl,
J. Lassen, C. D. P. Levy, M. R. Pearson, E. J. Prime,
V. Ryjkov, A. Wojtaszek, Z.-C. Yan, C. Zimmermann
Be-11 nuclear charge radius by W. Nörtershäuser:
D. Tiedemann, M. Záková, Z. Andjelkovic, K. Blaum M. L.
Bissell, R. Cazan, G. W. F. Drake, Ch. Geppert, M.
Kowalska, J. Krämer, A. Krieger, R. Neugart, R. Sánchez,
F. Schmidt-Kaler, Z.-C. Yan, D. T. Yordanov, C.
Zimmermann
Be-12 nuclear charge radius by W. Nörtershäuser:
A. Krieger, K. Blaum, M. L. Bissell,N. Frömmgen, Ch.
Geppert,M. Hammen, K. Kreim, M. Kowalska,J. Krämer,
T. Neff,R. Neugart, G. Neyens, Ch. Novotny, R.
Sánchez, and D. T. Yordanov
Measurement of nuclear charge radii
Nuclear physics: nuclear model-dependent
Atomic theory & measurement: model-independent (Drake, 1980’s)
∆Enuc
2π Rnuc
=
3
2
∑ δ (r )
i
i
Etheory = Enr + α 2 Erel + α 3 EQED +  + ∆Enuc
Eexperi = Etheory (Rnuc ) ⇒ Rnuc determined
Test low energy nucleon-nucleon interaction potential
Review article: Halo Nuclei in Laser Light, in Lecture Notes in Physics,
745 131-153 (Springer-Verlag, Berlin, 2008).
W. Nörtershäuser, et al. PRA , 83, 012516 (2011).
Theoretical background
For low-Z systems, we use perturbation theory:
H = H 0 + α H rel + α H QED + 
2
3
H 0Ψ 0 = E0Ψ 0
Etot = E0 + α 2 Ψ 0 H rel Ψ 0 + α 3 Ψ 0 H QED Ψ 0 + 
Variational principle:
Ψ tr H 0 Ψ tr
Etr ≡
Ψ tr Ψ tr
then
Etr ≥ E0
Etr
E0
Relativistic and QED corrections
H rel = −
α2
8
∑∇ +
i
∑ δ (ri ) [lnα −2 ]− β (nLS ) +
4
3
∆ EQED = α 3 Z
Q=
1
4π
1

1
(
)
r
r
∇
⋅
∇
+
⋅
⋅
∇
∇

∑
i
j
ij
i
j  +
3 ij
rij
2 i > j  rij

α2
4
i
∑ lim r (a )
−3
i> j
β (nLS ) =
a →0
∑
n
ij
0pn
∑
2
19  14
164 
+  lnα +

30  3
15 
∑ δ (rij ) −
i> j
14
Q
3
+ 4π (γ + lna )δ (rij )
(En − E0 ) ln En − E0
0pn
2
( E n − E0 )
n
The Bethe logarithm β (nLS ) is very difficult to calculate.
For Rydberg states of two-electron ions (Goldman and Drake):
4
(
0.316205(6 )
Z − 1)
β (nLS ) = β (1s ) + 4 3 β (nL ) +
6
Z n
Z
r −4
nL
Derive the me/mp ratio
(Paris, Dusseldorf, Amsterdan)
Ground state of lithium
Computational features
• Quadrupole precision real*16 (32 digits)
• Parallelization
a) matrix elements ∝ N
b) power method ∝ N
• Use QD (32*2=64-digit arithmetic by Bailey) to
check and find no loss of precision.
2
3
Wang, Yan, Qiao, Drake, PRA (submitted)
2
2
1s 3d D
QED corrections
g≈10
Difficulty:
α
~1
α + β +γ
Chun Li solved this problem (brand new
recursion relations, unpublished)
Thanks