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Electric Quadrupole Moments of the D States of Alkaline-Earth-Metal Ions

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Electric Quadrupole Moments of the
D
States of Alkaline-Earth-Metal Ions
Chiranjib Sur, K.V.P. Latha, Bijaya K. Sahoo, Rajat K. Chaudhuri, and B.P. Das
Non-Accelerator Particle Physics Group, Indian Institute of Astrophysics, Bangalore - 560 034, India
D. Mukherjee
Department of Physical Chemistry, Indian Association for the Cultivation of Science, Kolkata - 700 032, Indiaand Jawaharlal Nehru Centre for Advanced Scientiﬁc Research, Bangalore 560064, India
(Received 1 August 2005; published 18 May 2006)The electric quadrupole moment for the
4
d
2
D
5
=
2
state of
88
Sr
; one of the most important candidatesfor an optical clock, has been calculated using the relativistic coupled-cluster theory. This is the ﬁrstapplication of this theory to determine atomic electric quadrupole moments. The result of the calculationis presented and the important many-body contributions are highlighted. The calculated electric quadru-pole moment is
2
:
94
0
:
07
ea
20
, where
a
0
is the Bohr radius and
e
the electronic charge while themeasured value is
2
:
6
0
:
3
ea
20
. This is so far the most accurate determination of the electric quadrupolemoment for the above mentioned state. We have also calculated the electric quadrupole moments for themetastable
4
d
2
D
3
=
2
state of
88
Sr
and for the
3
d
2
D
3
=
2
;
5
=
2
and
5
d
2
D
3
=
2
;
5
=
2
states of
43
Ca
and
138
Ba
,respectively.
DOI: 10.1103/PhysRevLett.96.193001 PACS numbers: 31.15.Ar, 06.30.Ft, 31.15.Dv, 32.10.Dk
The current frequency standard is based on the groundstate hyperﬁne transition in
133
Cs
which is in the micro-wave regime and has an uncertainty of one part in
10
15
[1].However, demands from several areas of science and tech-nology have led to a worldwide search [2] for even moreaccurate clocks in the optical regime. Some of the promi-nent candidates that belong to this category are
199
Hg
[3],
88
Sr
[4,5],
171
Yb
[6],
43
Ca
[7],
138
Ba
, etc. A series of measurements of the electric quadrupole moments havebeen performed on the metastable
D
states of most of theseions fairly recently [8–11] in connection with the inves-tigations for accurate optical frequency standards. Thesemeasurements provide an excellent opportunity for newtests of relativistic atomic many-body theories. The hyper-ﬁne structure constants and the electric quadrupole mo-ments are sensitive to the regions near and far from thenucleus, respectively, but they both are inﬂuenced by elec-tron correlation. Therefore, the knowledge that one wouldacquire from calculations of atomic electric quadrupolemoments iscomplementary tothat acquired fromhyperﬁneinteractions. However, calculations of atomic electricquadrupole moments have received relatively little atten-tion so far. The most rigorous calculation to date has beenperformed by Itano [12] by using the relativistic conﬁgu-ration interaction (RCI) method with a multiconﬁgurationDirac-Fock—extended optimized level (MCDF-EOL) or-bital basis. In this Letter, we use the relativistic coupled-cluster (RCC) theory to calculate the electric quadrupolemoments of
88
Sr
,
43
Ca
, and
138
Ba
. This is the ﬁrstapplication of this theory to calculate atomic electric quad-rupole moments. We place special emphasis on our
Sr
calculation as this ion is one of the leading candidates inthe search for an optical frequency standard [3–7].The RCC theory is equivalent to all-order relativisticmany-body perturbation theory. The details of the applica-tion of this theory have been discussed earlier in theliterature [13–15]. Here we shall only give a brief outline.Treating the
N
-electron closed shell Dirac-Fock state
j
i
as the reference state, the exact wave function in RCCtheory can be expressed as
j
i
exp
T
j
i
;
(1)where
T
is the core electron excitation operator. In thecoupled-cluster singles and doubles (CCSD) approxima-tion,
T
can be expressed as the sum of one- and two-bodyexcitation operators, i.e.,
T
T
1
T
2
, and can be writtenin the second quantized form as
T
T
1
T
2
X
ap
a
y
p
a
a
t
pa
14
X
abpq
a
y
p
a
y
q
a
b
a
a
t
pqab
;
(2)where
t
pa
and
t
pqab
are the amplitudes of the singles anddouble excitation operators, respectively. For a single va-lence system we deﬁne the reference state as,
j
N
1
v
i
a
y
v
j
i
(3)with the particle creation operator
a
y
v
. The wave functioncorresponding to this state can be written as
j
N
1
v
i
exp
T
f
1
S
v
gj
N
1
v
i
:
(4)Here,
S
v
S
1
v
S
2
v
X
v
p
a
y
p
a
v
s
pv
12
X
bpq
a
y
p
a
y
q
a
b
a
v
s
pqvb
(5)corresponds to the excitation operator in the valence sectorand
v
stands for valence orbital;
s
pv
and
s
pqvb
are the ampli-tudes of single and doubles excitations, respectively.Details concerning the evaluation of the closed and openshell amplitudes have been discussed earlier [16]. TriplePRL
96,
193001 (2006) PHYSICAL REVIEW LETTERS
week ending19 MAY 2006
0031-9007
=
06
=
96(19)
=
193001(4) 193001-1
©
2006 The American Physical Society
excitations are included in our open shell RCC amplitudecalculations in an approximate way [CCSD(T)] [17,18].The expectation value of any operator
O
with respect tothe state
j
N
1
i
is given by
h
O
i h
N
1
j
O
j
N
1
ih
N
1
j
N
1
i h
N
1
jf
1
S
y
g
O
f
1
S
gj
N
1
ih
N
1
jf
1
S
y
g
exp
T
y
exp
T
f
1
S
gj
N
1
i
;
(6)where
O
exp
T
y
O
exp
T
.The ﬁrst few terms in the above expectation value can beidentiﬁed as
O;
OS
1
;
OS
2
;S
y
1
OS
1
, etc.; are referred to asdressed Dirac-Fock (DDF), dressed pair correlation (DPC)[Fig. 1(a)] and dressed core polarization (DCP) [Fig. 1(b)],respectively. We use the term ‘‘dressed’’ because the op-erator
O
includes the effects of the core excitation operator
T
. Among the above, we can identify few other termswhich play crucial role in determining the correlationeffects. One of those terms is
S
y
1
OS
1
c
:
c
:
[Fig. 1(c)]which is called as dressed higher order pair correlation(DHOPC)since it directly involves the correlation betweena pair of electrons. In Table II individual contributionsfrom these diagrams are listed.The orbitals used in the present work are expanded interms of a ﬁnite basis set comprising of Gaussian typeorbitals using a two parameter Fermi nuclear distribution[19]. The interaction of the atomic quadrupole momentwith the external electric-ﬁeld gradient is analogous to theinteraction of a nuclear quadrupole moment with the elec-tric ﬁelds generated by the atomic electrons inside thenucleus. In the presence of the electric ﬁeld, this givesrise to an energy shift by coupling with the gradient of theelectric ﬁeld. Thus the treatment of the electric quadrupolemoment is analogous to the nuclear counterpart. The quad-rupole moment
of an atomic state
j
;J;M
i
is de-ﬁned as the diagonal matrix element of the quadrupoleoperator with the maximum value
M
J
, given by
h
JJ
j
zz
j
JJ
i
:
(7)Here,
is an additional quantum number which distin-guishes the initial and ﬁnal states. The electric quadrupoleoperator in terms of the electronic coordinates is given by
zz
e
2
X
j
3
z
2
j
r
2
j
;
where the sum is over all the electrons and
z
is the coor-dinate of the
j
th electron. To calculate the quantity weexpress the quadrupole operator in its single particle formas
2
m
X
m
q
2
m
(8)and the single particle reduced matrix element is expressedas [20]
h
j
f
k
q
2
m
k
j
i
i h
j
f
k
C
2
m
k
j
i
i
Z
drr
2
P
f
P
i
Q
f
Q
i
:
(9)In Eq. (9), the subscripts
f
and
i
correspond to the ﬁnal andinitial states, respectively;
P
and
Q
are the radial part of thelarge and small components of the single particle Dirac-Fock wave functions, respectively,
j
i
is the total angularmomentum for the
i
th electron. The angular factor is givenin by
h
j
f
k
C
k
m
k
j
i
i
1
j
f
1
=
2
2
j
f
1
q
2
j
i
1
q
j
f
2
j
i
1
=
2 0 1
=
2
!
l;k;l
0
;
(10)where
l;k;l
0
1
if
l
k
l
0
even
0
otherwise
;
l
and
k
being the orbital angular momentum and the rank,respectively.Finally using the Wigner-Eckart theorem we deﬁne theelectric quadrupole moment in terms of the reduced matrixelements as
h
j
f
j
2
m
j
j
i
i
1
j
f
m
f
j
f
2
j
i
m
f
0
m
f
h
j
f
k
2
k
j
i
i
:
(11)We focus on the clock transition
5
s
2
S
1
=
2
–
4
d
2
D
5
=
2
in
88
Sr
shown in Fig. 2. The electric quadrupole shift in theenergy levels arises due to the interaction of atomic electricquadrupole moment (EQM) with the external electric-ﬁeldgradient. The electric quadrupole moment in the state
4
d
2
D
5
=
2
was measured experimentally by Barwood
et al.
at the National Physical Laboratory, UK [8]. Since theground state
5
s
2
S
1
=
2
does not posses any electric quadru-pole moment, the contribution to the quadrupole shift forthe clock frequency comes only from the
4
d
2
D
5
=
2
state.In Table I we have presented the details of the basisfunctions used in this calculation and in Table II contri-butions from different many-body terms. The value of
in the
4
d
2
D
5
=
2
state measured experimentally is
qvv
oS
1
p
o
aO S
1
(a)
S
2
vvO S
2
(b) (c) O SS
1 1
qpvv
S
1
S
1
o
FIG. 1. The diagrams (a) and (c) are subsets of DPC diagrams.Diagram (b) is one of the direct DCP diagram.
PRL
96,
193001 (2006) PHYSICAL REVIEW LETTERS
week ending19 MAY 2006
193001-2
2
:
6
0
:
3
ea
20
[8], where
e
is the electronic charge and
a
0
is the Bohr radius. Our calculated value for the
4
d
2
D
5
=
2
stretched state is
2
:
94
0
:
07
ea
20
. To test the convergenceof our calculation we have performed more than ﬁvecalculations with different basis sets of increasing dimen-sions and by extrapolating the results to an inﬁnite basisset. We have estimated the error incurred in our presentwork, by taking the difference between our RCC calcula-tions with singles, doubles as well as the most importanttriple excitations [CCSD(T)] and only single and doubleexcitations (CCSD). We have also estimated the effect of Breit interaction which turns out to be 0.3% and liesbetween the error of our calculation. A nonrelativisticHartree-Fock determination resulted in
3
:
03
ea
20
[8].A subsequent calculation based on RCI with MCDF-EOLorbital basis yielded
3
:
02
ea
20
[10]. In that calculation,correlation effects arising from a subset of the terms
S
1
,
T
1
for single excitations and
S
2
and
T
2
for double excitationswere considered, where
S
1
and
S
2
are the cluster operatorsrepresenting single excitations from the valence
4
d
orbitalto a virtual orbital and double excitations from the valence
4
d
and the core
f
4
s;
4
p;
3
d
g
orbitals, with at most oneexcitation from the core, respectively. In our calculation,in addition to these effects, the effects arising from thenonlinear terms like
T
22
,
T
1
T
2
, etc., have been included. Inthe framework of CCSD theory, the single and doubleexcitations have been treated to all orders in electroncorrelation including excitations from the entire core.This amounts to a more rigorous treatment of electroncorrelation in comparison to the previous calculation per-formed using the RCI method. We have also determinedthe EQM for the metastable
4
d
2
D
3
=
2
state of
88
Sr
. Thevalue we obtained is
2
:
12
ea
20
, whereas Itano’s calculationyields the value
2
:
107
ea
20
[12].It is clear from Table II that the DDF contribution is thelargest. The leading correlation contribution comes fromthe DPC effects and the DCP effects are an order of magnitude smaller. This can be understood from the DPCdiagram [Fig. 1(a)] which has a valence electron in the
4
d
5
=
2
state. Hence the dominant contribution to the electricquadrupole moment arises from the overlap between vir-tual
d
5
=
2
orbitals and the valence, owing to the fact that
S
1
is an operator of rank 0 and the electric quadrupole matrixelements for the valence
4
d
5
=
2
and the diffuse virtual
d
5
=
2
orbitals are substantial. On the other hand, in the DCPdiagram [Fig. 1(b)], the matrix element of the same opera-tor could also involve the less diffuse
s
or
p
orbitals.Hence, for a property like the electric quadrupole moment,whose magnitude depends on the square of the radialdistance from the nucleus, this trend seems reasonable,whereas for properties like hyperﬁne interaction which issensitive to the near nuclear region, the trend is just theopposite for the
d
states [15]. As expected, the contributionof the DHOPC effect, i.e.,
S
y
1
OS
1
[Fig. 1(c)] is relativelyimportant as it involves an electric quadrupole matrixelement between the valence
4
d
5
=
2
and a virtual
d
5
=
2
orbital. Unlike many other properties, particularly the hy-perﬁne interactions, an all-order determination of this dia-gram is essential for obtaining an accurate value of theelectric quadrupole moment. One of the strengths of RCC
TABLE I. Number of basis functions used to generate the even-tempered Dirac-Fock orbitals and the corresponding value of
0
and
used for
Sr
.
s
1
=
2
p
1
=
2
p
3
=
2
d
3
=
2
d
5
=
2
f
5
=
2
f
7
=
2
g
7
=
2
g
9
=
2
Number of basis 38 35 35 30 30 25 25 20 20
0
10
5
525 525 525 425 425 427 427 425 425
2.33 2.33 2.33 2.13 2.13 2.13 2.13 1.98 1.98Active holes 4 3 3 1 1 0 0 0 0Active particles 10 10 10 11 11 8 8 6 6
5s S21/24d D24d D5/223/2674 nm"clock transition"
14555.9014916.24
FIG. 2. Diagram indicating the clock transition in
88
Sr
.Excitation energies of the
4
d
2
D
3
=
2
and
4
d
2
D
5
=
2
levels aregiven in
cm
1
.TABLE II. Contributions of the electric quadrupole momentfor the
4
d
2
D
5
=
2
state of
88
Sr
in atomic units from differentmany-body effects in the CCSD(T) calculation. The terms likeDDF, DCP, DPC, DHOPC are explained in the text. The remain-ing terms in Eq. (6) are referred to as ‘‘others.’’ The numbergiven in the column ‘‘Triples’’ corresponds to the contributionfrom approximate perturbative triple excitations.DDF DPC DCP DHOPC Others Triples Total3.4963
0
:
4306
0
:
0642
0.0353
0
:
0944
0
:
07
2.94
PRL
96,
193001 (2006) PHYSICAL REVIEW LETTERS
week ending19 MAY 2006
193001-3
theory is that it can evaluate such diagrams to all orders inthe residual Coulomb interaction.We have also calculated the EQMs of the metastable
2
D
3
=
2
;
5
=
2
states of
43
Ca
and
138
Ba
and they are given inTable III. We have also estimated the error for thesecalculations similar to
88
Sr
and they vary from 0.5% to1.8%. Our coupled-cluster calculations of EQMs for themetastable
D
states of these two ions agree well with theother calculations [12]. We ﬁnd that the correlation effectsare similar to
88
Sr
. Our analyses reveal that for both the
2
D
3
=
2
and
2
D
5
=
2
states the largest contribution is at theDDF level, followed by DPC and DCP, respectively. Thelatter two are opposite in sign. More speciﬁcally, the Dirac-Fock values overestimate the EQMs and that is the reasonwhy the correlation effects are so important in these cases.This is demonstrated in the case of the
88
Sr
4
d
2
D
5
=
2
statein Table II.In conclusion, we have performed an
ab initio
calcula-tion of the electric quadrupole moment for the
4
d
2
D
5
=
2
state of
88
Sr
to an accuracy of less than 2.5% using theRCC theory. In addition we have also determined theEQM of
4
d
2
D
3
=
2
state of the same ion and the EQMsof the metastable
2
D
3
=
2
;
5
=
2
states of
43
Ca
and
138
Ba
.Evaluation of the correlation effects to all orders as well asthe inclusion of the dominant triple excitations in ourcalculation were crucial in achieving this accuracy. Thisis the ﬁrst application of RCC theory to determine theelectric quadrupole moment of atomic systems. Our RCCcalculation of the EQM for the
4
d
2
D
5
=
2
state in
88
Sr
isthe most accurate determination of this property to date. Italso highlights the potential of the RCC theory to deter-mine the atomic properties at large distances from thenucleus with unprecedented accuracy.This work was supported by the BRNS for ProjectNo. 2002/37/12/BRNS. The computations were carriedout partly using the Xeon and Opteron supercomputingcluster at IIA and in Param Padma Supercomputer, CDAC,Bangalore. We are grateful to Dr. Wayne Itano andDr. Geoffrey Barwood for helpful discussions.
[1] http://tf.nist.gov/cesium/atomichistory.htm.[2] L. Hollberg
et al.
, J. Phys. B
38
, S469 (2005).[3] R. Rafac
et al.
, Phys. Rev. Lett.
85
, 2462 (2000).[4] J.E. Bernard
et al.
, Phys. Rev. Lett.
82
, 3228 (1999).[5] H.S. Margolis
et al.
, Phys. Rev. A
67
, 032501 (2003).[6] J. Stenger
et al.
, Opt. Lett.
26
, 1589 (2001).[7] C. Champenois
et al.
, Phys. Lett. A
331
, 298 (2004).[8] G.P. Barwood
et al.
, Phys. Rev. Lett.
93
, 133001 (2004).[9] P. Dube
et al.
, Phys. Rev. Lett.
95
, 033001 (2005).[10] W.H. Oskay, W.M. Itano, and J.C. Bergquist, Phys. Rev.Lett.
94
, 163001 (2005).[11] C. Roos (private communications).[12] W.M. Itano, Phys. Rev. A
73
, 022510 (2006).[13] U. Kaldor, in
Microscopic Quantum Many-Body Theoriesand their Applications
, edited by J. Navarro and A. PollsLecture Notes in Physics (Springer-Verlag, Berlin, 1998),p. 71, and references therein.[14] B.P. Das
et al.
, J. Theor. Comp. Chem.
4
, 1 (2005), andreferences therein.[15] C. Sur
et al.
, Eur. Phys. J. D
32
, 25 (2005).[16] G. Gopakumar
et al.
, Phys. Rev. A
66
, 032505 (2002).[17] U. Kaldor, J. Chem. Phys.
87
, 467 (1987).[18] U. Kaldor, J. Chem. Phys.
87
, 4693 (1987).[19] R.K. Chaudhuri, P.K. Panda, and B.P. Das, Phys. Rev. A
59
, 1187 (1999).[20] I.P. Grant, J. Phys. B
7
, 1458 (1974).TABLE III. Electric quadrupole moments (in
ea
20
) for themetastable
2
D
3
=
2
;
5
=
2
states of
43
Ca
and
138
Ba
.
43
Ca
138
Ba
States
3
d
2
D
3
=
2
3
d
2
D
5
=
2
5
d
2
D
3
=
2
5
d
2
D
5
=
2
Present work 1.338 1.916 2.309 3.382Others [12] 1.338 1.917 2.297 3.379
PRL
96,
193001 (2006) PHYSICAL REVIEW LETTERS
week ending19 MAY 2006
193001-4

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