Blackbody radiation shift in 87Rb frequency standard

Abstract

The operation of atomic clocks is generally carried out at room temperature, whereas the definition of the second refers to the clock transition in an atom at absolute zero. This implies that the clock transition frequency should be corrected in practice for the effect of finite temperature of which the leading contributor is the blackbody radiation (BBR) shift. Experimental measurements of the BBR shifts are difficult. In this work, we have calculated the blackbody radiation shift of the ground-state hyperfine microwave transition in 87Rb using the relativistic all-order method and carried out detailed evaluation of the accuracy of our final value. Particular care is taken to accurately account for the contributions from highly-excited states. Our predicted value for the Stark coefficient, k_S=-1.240(4)\times 10^{-10}\text{Hz/(V/m)}^{2} is three times more accurate than the previous calculation [1]. Comment: 7 pages

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arXiv:1007.0462v1 [physics.atom-ph] 3 Jul 2010
Blackbody radiation shift in87Rb frequency standard
M. S. Safronova
Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716 and
Joint Quantum Institute, University of Maryland Department of Physics and
National Institute of Standards and Technology, College Park, MD 20742
Dansha Jiang
Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716
U. I. Safronova∗
Physics Department, University of Nevada, Reno, Nevada 89557
(Dated: July 6, 2010)
The operation of atomic clocks is generally carried out at room temperature, whereas the definition
of the second refers to the clock transition in an atom at absolute zero. This implies that the clock
transition frequency should be corrected in practice for the effect of finite temperature of which the
leading contributor is the blackbody radiation (BBR) shift. Experimental measurements of the BBR
shifts are difficult. In this work, we have calculated the blackbody radiation shift of the ground-
state hyperfine microwave transition in87Rb using the relativistic all-order method and carried out
detailed evaluation of the accuracy of our final value. Particular care is taken to accurately account
for the contributions from highly-excited states.
kS = −1.240(4) × 10−10Hz/(V/m)2is three times more accurate than the previous calculation [1].
Our predicted value for the Stark coefficient,
PACS numbers: 06.30.Ft, 31.15.ac, 31.15.ap, 32.70.Cs
I.INTRODUCTION
The present definition of the second in the Interna-
tional System of Units (SI) is based on the microwave
transition between the two hyperfine levels (F = 4 and
F = 3) of the133Cs ground state and refers to the clock
transition in an atom at absolute zero. The relative stan-
dard uncertainty of the Cs microwave frequency standard
is 4×10−16[2] at the present time. In 2006, the Interna-
tional Committee for Weights and Measures (CIPM) rec-
ommended [3] that the ground-state hyperfine microwave
transition in87Rb [4, 5] be used as secondary represen-
tation of the second, along with four optical transition
frequencies.
The operation of atomic clocks is generally carried out
at room temperature implying that the clock transition
frequency should be corrected for effects of finite temper-
ature, of which the leading contributor is the blackbody
radiation (BBR) shift. The BBR shift at room temper-
ature effecting the Cs microwave frequency standard has
been calculated to high accuracy (0.35% and 1%, respec-
tively) in Refs. [6, 7] implying a 6 ×10−17fractional un-
certainty. These calculations are in agreement with a
0.2% measurement [8].
The BBR shift contributes to Rb frequency standard
at 10−14level (see [9] for the review of the present status
of BBR shift uncertainties for all atomic clocks). The
most recent value of the BBR shift in Rb microwave
∗Permanent address: Institute of Spectroscopy, Russian Academy
of Science, Troitsk, Moscow, Russia
frequency standard is accurate to 1% [1]. As a result,
ultimate relative uncertainty induced by the BBR shift
in87Rb frequency standard was significantly larger than
that of the133Cs frequency standard. We note that we
refer to uncertainty of the scalar Stark coefficient. Ac-
tual experimental uncertainty will also include error due
to temperature stabilization. The calculation of Ref. [1]
also disagreed with the old 1975 theoretical calculation of
Ref. [10] by 2.5%. As a result, more accurate calculation
of Rb BBR shift is in order.
In this work, we calculated the blackbody radiation
shift of the ground-state hyperfine microwave transition
in87Rb using the relativistic all-order method and eval-
uated the accuracy of our final value.
value of the scalar Stark coefficient, kS = −1.240(4) ×
10−10Hz/(V/m)2is accurate to 0.3%. Our calculation
reduced the uncertainty in Rb frequency standard due to
BBR shift to the level of accuracy similar to that of the
Cs case.
Our predicted
Another motivation for the present work was to pro-
vide a systematic approach to evaluation of theoretical
uncertainty using the calculation of BBR shift in Rb as
an example. Modern applications of theoretical atomic
calculations frequently require some knowledge of the ac-
curacy of theoretical numbers. With new advances in
theoretical methods and in computational power, it is
essential to develop consistent strategies to evaluate the
accuracy of theoretical data. Such evaluations are diffi-
cult but very beneficial both to specific applications and
benchmark comparisons of theory and experiment. In
this work, we described the evaluation of uncertainties of
the electric-dipole matrix elements, hyperfine matrix ele-
ments, and remainders of various sums in sufficient detail

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2
TABLE I: Selection of the “best set” values for the 5pj − ns, 6pj − ns, and 7pj − ns electric-dipole reduced matrix elements.
See text for details. Absolute values of the lowest-order DHF and SD all-order values in a.u. and their relative difference in %
are given in columns 2-4.
Transition
5p1/2− 5s
5p1/2− 6s
5p1/2− 7s
5p1/2− 8s
5p1/2− 9s
5p3/2− 5s
5p3/2− 6s
5p3/2− 7s
5p3/2− 8s
5p3/2− 9s
DHF
4.8189
4.2564
0.9809
0.5139
0.3380
SD∆(SD-DHF)
14.2%
3.3%
2.8%
2.0%
1.6%
Final
4.2310
4.1458
0.9527
0.5022
0.3314
Source
Expt
SDsc
SDsc
SDsc
SDsc
Unc. (%)
0.07%
0.66%
0.17%
0.30%
0.36%
Unc. source
Expt.
SDsc-SD
SDsc-SD
SDsc-SD
SDsc-SD
“Best set”
4.231(3)
4.146(27)
0.953(2)
0.502(2)
0.331(1)
4.2199
4.1187
0.9543
0.5037
0.3326
6.8017
6.1865
1.3925
0.7265
0.4771
5.9551
6.0135
1.3521
0.7098
0.4677
14.2%
2.9%
3.0%
2.4%
2.0%
5.9780
6.0472
1.3497
0.7077
0.4662
Expt
SDsc
SDsc
SDsc
SDsc
0.08%
0.56%
0.18%
0.29%
0.34%
Expt
SDsc-SD
SDsc-SD
SDsc-SD
SDsc-SD
5.978(5)
6.047(34)
1.350(2)
0.708(2)
0.466(2)
6p1/2− 5s
6p1/2− 6s
6p1/2− 7s
6p1/2− 8s
6p1/2− 9s
6p3/2− 5s
6p3/2− 6s
6p3/2− 7s
6p3/2− 8s
6p3/2− 9s
0.3825
10.2856
9.3594
1.9219
0.9702
0.3335
9.6839
9.1896
1.8532
0.9364
14.7%
6.2%
1.8%
3.7%
3.6%
0.3248
9.7450
9.2092
1.8616
0.9364
SDsc
SDpT
SDpT
SDpT
SD
2.69%
0.63%
0.21%
0.45%
0.50%
SDsc-SD
SD-SDpT
SD-SDpT
SD-SDpT
0.5%
0.325(9)
9.745(61)
9.209(20)
1.862(8)
0.936(5)
0.6055
14.4575
13.5514
2.7047
1.3583
0.5409
13.5918
13.3529
2.6001
1.3056
11.9%
6.4%
1.5%
4.0%
4.0%
0.5276
13.6804
13.3755
2.6129
1.3056
SDsc
SDpT
SDpT
SDpT
SD
2.51%
0.65%
0.17%
0.49%
0.50%
SDsc-SD
SD-SDpT
SD-SDpT
SD-SDpT
0.5%
0.528(13)
13.680(89)
13.376(23)
2.613(13)
1.306(7)
to demonstrate specific approaches that were used. The
methods of the uncertainty evaluation outlined in this
paper can be used for various other calculations.
II. METHOD
The electrical field E radiated by a blackbody at tem-
perature T and described by Planck’s law induces a non-
resonant perturbation of atomic transitions at room tem-
perature [11]. The average electric field radiated by a
blackbody at temperature T is
?E2? = (831.9 V/m)2
?T(K)
300
?4
.(1)
The frequency shift of an atomic state due to such an
electrical field can be related to the static electric-dipole
polarizability α(0) by [12]
δν = −1
2(831.9 V/m)2
?T
T0
?4
α(0)
?
1 + ǫ
?T
T0
?2?
,
(2)
where ǫ is a small dynamic correction due to the fre-
quency distribution, and T0is usually taken to be 300K.
The dynamic correction ǫ was evaluated in Ref. [1] and
was found to be small, ǫ = 0.011, for Rb microwave fre-
quency standard. Therefore, we do not recalculate it in
this work.
In the case of the optical transitions, the lowest (sec-
ond) order polarizabilities of the clock states are differ-
ent. In the case of the ground-state hyperfine microwave
frequency standards, the lowest (second) order polariz-
abilities of the clock states are identical and the lowest-
order BBR shift vanishes. Therefore, the Stark shift of
the ground state87Rb hyperfine interval (F = 2−F = 1)
is governed by the static third-order F-dependent polar-
izability α(3)
F(0).
In this work, we evaluate the scalar Stark coefficient
kS,
kS= −1
2
?
α(3)
F=2(0) − α(3)
F=1(0)
?
. (3)
The expression for the α(3)
F(0) is given by [6]:
α(3)
F(0) =
1
3
gIµn(−1)F+I+jv(2T + C + R),
?
(2I)(2I + 1)(2I + 2)
?jv I F
I jv 1
?
×
(4)
where gI is the nuclear gyromagnetic ratio, µn is the
nuclear magneton, I = 3/2 is the nuclear spin, and jv=
1/2 is the total angular momentum of the atomic ground
state. The F-independent sums T, C, and R for the
ground state of Rb, |v? ≡ |5s?, are given by [6]:
T =
?
?
m,n?=5s
AT?5s?D?m??m?D?n??n?T(1)?5s?
(Em− E5s)(En− E5s)
,(5)
C =
m,n?=5s
AC?5s?D?m??m?T(1)?n??n?D?5s?
(Em− E5s)(En− E5s)
,
R =
1
2?5s?T(1)?5s?
??
m∈val
−
?
m∈core
?
|?5s?D?m?|2
(Em− E5s)2,

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3
TABLE II: Selection of the “best set” values for diagonal and off-diagonal matrix elements of the magnetic hyperfine operator
T(1)in 10−8a.u. Absolute values of the lowest order DHF, all-order SD, and all-order SDpT values are given in columns 2-4.
DHF
22.0830
11.4126
7.3042
5.1907
3.9328
SD SDpT
34.6801
16.8497
10.6061
7.4786
5.6404
Expt.
34.6810
16.8602
10.6086
7.4855
5.6563
Final
34.6810
16.8602
10.6086
7.4786
5.6404
Source
Expt.
Expt.
Expt.
SDpT
SDpT
Unc. (%)
0.00%
0.06%
0.02%
0.09%
0.28%
Unc. source
Expt.
Expt-SDpT
Expt-SDpT
Expt-SDpT
Expt-SDpT
“Best set”
34.681
16.860(10)
10.609(2)
7.479(7)
5.640(16)
5s − 5s
5s − 6s
5s − 7s
5s − 8s
5s − 9s
36.1633
17.4008
10.9262
7.6957
5.8004
5p1/2− 5p1/2
5p1/2− 6p1/2
5p1/2− 7p1/2
5p1/2− 5p3/2
5p1/2− 6p3/2
5p1/2− 7p3/2
5p3/2− 5p3/2
5p3/2− 6p3/2
5p3/2− 7p3/2
5p3/2− 6p1/2
5p3/2− 7p1/2
2.4023
1.4218
0.9681
0.3835
0.2273
0.1550
4.3197
2.4431
1.6390
0.3396
0.1946
0.1312
4.1460
2.3582
1.5853
0.3274
0.1886
0.1272
4.12234.1223
2.3582
1.5853
0.3274
0.1886
0.1272
Expt.
SDpT
SDpT
SDpT
SDpT
SDpT
0.2%
0.6%
0.6%
1%
1%
1%
Expt.
from 5p1/2
from 5p1/2
from 5p3/2
from 5p3/2
from 5p3/2
Expt.
from 5p3/2
from 5p3/2
from 5p3/2
from 5p3/2
4.122(8)
2.358(14)
1.585(10)
0.327(3)
0.189(2)
0.127(1)
1.3496
0.8000
0.5453
0.2269
0.1545
2.7786
1.5755
1.0583
0.1905
0.1275
2.6682
1.5212
1.0241
0.1845
0.1236
2.72292.7229
1.5483
1.0412
0.1845
0.1236
Expt.0.065%
1%
1%
1%
1%
2.723(2)
1.548(15)
1.041(10)
0.185(2)
0.124(1)
av. SD, SDpT
av. SD, SDpT
SDpT
SDpT
where ?i?D?j? are electric-dipole reduced matrix ele-
ments and ?i?T(1)?j? are the matrix elements of the mag-
netic hyperfine operator T(1). The quantities ATand AC
are the angular coefficients given in our case by
AT =
(−1)jm+1/2
2
AC = (−1)jm−jn
?1 1/2 1/2
1 jm
jn
?
.
The sums are made finite with the use of finite B-spline
basis set in a spherical cavity. The sum over the complete
finite basis set is equivalent to the sum over the bound
states and integration over the continuum.
complete set of DHF wave functions on a nonlinear grid
generated using B-splines constrained to a spherical cav-
ity. A cavity radius of 220 a0is chosen to accommodate
all ns and np valence orbitals up to n = 12. The basis
set consists of 70 splines of order 11 for each value of the
relativistic angular quantum number κ.
Sums over m and n run over all possible states allowed
by the selection rules and limits of the sums. There-
fore, three distinct sets of matrix elements are needed for
the present calculations: electric-dipole matrix elements
between ns and mpj states, ?mpj?D?ns?, and diagonal
and off-diagonal matrix elements of the magnetic hyper-
fine operator for both ns and np states: ?ns?T(1)?5s?,
?mpj1?T(1)?npj2?. Therefore, the calculation of the BBR
shift reduces to the evaluation of the electric-dipole and
magnetic hyperfine matrix elements.
In this work, we use atomic units, in which, e, me,
4πǫ0and the reduced Planck constant ¯ h have the numer-
ical value 1. Polarizability in a.u. has the dimension of
volume, and its numerical values presented here are thus
expressed in units of a3
0, where a0≈ 0.052918 nm is the
Bohr radius. The atomic units for α can be converted
We use a
to SI units via α/h[Hz/(V/m)2] = 2.48832×10−8α[a.u.],
where the conversion coefficient is 4πǫ0a3
constant h is factored out.
We start our calculation by evaluating all three terms
in Dirac-Hartree-Fock (DHF) approximation.
sulting DHF values for the T, C, and R terms in atomic
units are
0/h and Planck
The re-
2TDHF= 2.376 × 10−3,
RDHF= 3.199 × 10−3.
CDHF= 6.111× 10−6,
Then, we replace all dominant matrix elements by the
“best set” values that have been evaluated for their ac-
curacy and replace corresponding energies by their ex-
perimental values [13, 14]. We refer to the terms where
such replacements have been made as “main” terms, and
refer to the remaining terms calculated in the DHF ap-
proximation as remainders.
We note that it is essential not to mix DHF and high-
precision data within a single contribution. For example,
experimental energies should not be combined with DHF
matrix elements in any of the terms. In the present calcu-
lations, all data in main terms are high-precision theory
or experiment values and all data in the remainders and
in core terms are taken to be DHF values. Mixing val-
ues of significantly different accuracy leads to fictitious
changes in the final results, in particularly in Term T. We
carried out numerical tests that support this statement,
and we attribute this issue to the violation of the finite
basis set completeness.
We note that while we use the experimental values of
the energies in the main terms, the accuracy of our all-
order theoretical energy values is very high. We made ex-
tensive comparison of removal energies calculated using
the SD all-order method and experimental values [13, 14]
for the (5 − 11)s, (5 − 10)pj, (4 − 10)dj, and (4 − 7)fj

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4
TABLE III: Absolute values of the electric-dipole reduced matrix elements used in the calculation of the BBR shift and their
uncertainties in atomic units (ea0).
Transition
5s − np1/2
6s − np1/2
7s − np1/2
8s − np1/2
9s − np1/2
10s − np1/2
11s − np1/2
n = 5
4.231(3)
4.146(27)
0.953(2)
0.502(2)
0.331(1)
0.243(1)
0.189(1)
n = 6
0.325(9)
9.75(6)
9.21(2)
1.862(8)
0.936(5)
0.607(3)
0.442(2)
n = 7
0.115(3)
0.993(7)
16.93(9)
16.00(2)
3.00(2)
1.474(7)
0.942(5)
n = 8
0.060(2)
0.388(5)
1.856(9)
25.9(1)
24.5(1)
4.40(2)
2.12(1)
n = 9
0.037(1)
0.222(3)
0.751(8)
2.95(2)
36.7(2)
34.8(2)
6.05(3)
n = 10
0.026(1)
0.148(2)
0.430(6)
1.20(1)
4.25(2)
49.4(2)
46.8(2)
n = 11
0.020(1)
0.109(2)
0.289(4)
0.69(1)
1.73(2)
5.78(3)
63.9(3)
5s − np3/2
6s − np3/2
7s − np3/2
8s − np3/2
9s − np3/2
10s − np3/2
11s − np3/2
5.978(5)
6.047(34)
1.350(2)
0.708(2)
0.466(2)
0.341(1)
0.266(1)
0.528(13)
13.68(9)
13.38(2)
2.61(1)
1.306(7)
0.845(4)
0.614(3)
0.202(4)
1.53(1)
23.7(1)
23.19(2)
4.19(2)
2.04(1)
1.302(7)
0.111(3)
0.621(7)
2.82(2)
36.3(2)
35.5(2)
6.13(3)
2.92(2)
0.073(2)
0.363(5)
1.18(1)
4.45(2)
51.2(3)
50.3(3)
8.40(4)
0.053(2)
0.246(4)
0.68(1)
1.85(2)
6.39(3)
68.9(3)
67.7(3)
0.040(2)
0.182(3)
0.465(7)
1.08(2)
2.66(3)
8.65(4)
89.2(4)
states. Additional first and second-order Breit contribu-
tions, Lamb shift, and third-order Coulomb correlation
correction not accounted for by the SD approximations
were also included into the calculation. Our values agree
to better than 10 cm−1for all levels with exception of the
5s, 6s, 7s, 4d3/2, and 4d5/2levels, where the differences
are 27, 25, 12, 32, and 29 cm−1, respectively. We note
that the ground state energy is -33691 cm−1, making the
agreement better than 0.1%.
III. ”BEST SET” MATRIX ELEMENTS AND
THEIR UNCERTAINTIES
The “best set” consists of our all-order high-precision
results and several experimental values. The following
128 matrix elements have been replaced by the all-order
or experimental values:
?mpj?D?ns?, m = 5 − 12, n = 5 − 12;
?ns?T(1)?5s?, n = 5 − 12;
?mpj1?T(1)?npj2?, m = 5 − 7, n = 5 − 7.
The all-order calculation of Rb matrix elements has been
described in detail in [15].
We illustrate the selection of the “best set” values of
the electric-dipole matrix elements and determination of
their uncertainties in Table I, where we list a few exam-
ples. The complete table is given in Ref. [16]. The ab-
solute values in atomic units (ea0) are given in all cases.
We list the lowest-order DHF results, all-order SD val-
ues, and their relative differences in percent in columns
2 - 4 of Table I. The relative differences of DHF and
single-double (SD) all-order numbers give a good esti-
mate of the size of the correlation correction. In general,
the smaller the correlation correction, the more precise
our theoretical values are. The final values used in our
“best set” are listed in column 5. The next column iden-
tifies the source of these values for each of the matrix
elements. The 5s−5pjmatrix elements are experimental
values from [17]. All other E1 matrix elements are from
all-order calculation that included SD, SDpT, or SDsc
values. The SDsc values include additional corrections
added to SD ab initio results by means of the scaling pro-
cedure described in Ref. [18] and references therein. The
SDpT label refers to ab initio all-order calculations that
include single, double, and partial triple excitations. The
selection of the particular value as final is determined by
the study of the dominant correlation correction terms
(because the scaling procedure is only applicable for cer-
tain classes of terms) and accuracy requirements. In the
present calculation, very high accuracy is not needed for
matrix elements with high values of principal quantum
numbers. In such cases, SD values are sufficiently accu-
rate for E1 matrix elements.
Evaluation of theoretical uncertainties is a very diffi-
cult problem since it essentially involves evaluation of the
quantity that is not known beforehand. Several strate-
gies can be used in evaluating the uncertainties of the
all-order results, including the study of the breakdown of
the various all-order contributions, identification of the
most important terms, and semi-empirical determination
of important missing contributions. Our uncertainty es-
timates are listed in percent in column labeled “Unc.”.
The method for determining uncertainty is noted in the
next column labeled “Unc. source”. Where the scaling
was performed, it is expected to estimate the dominant
missing correlation correction (see Ref. [18] and refer-
ences therein for explanation). Therefore, it is reasonable
to take the difference of the ab initio and scaled results
as the uncertainty. This is indicated by SDsc-SD note in
the “Unc. source” column. We note that this procedure
is expected to somewhat overestimate the uncertainty.
In some cases, where such high accuracy was not re-
quired but the same correlation terms were dominant,
we carried out ab initio SDpT calculation (i.e. partially
included triples) instead and took these values as final.

Page 5

5
The uncertainties were estimated at the differences of
the SD and SDpT numbers in those cases. We note that
numerous tests were conducted in the past that demon-
strated that the above-mentioned procedures of the un-
certainty estimates are valid (see Ref. [18] for review of
the all-order method and its applications). In the cases
of transitions with high values of the principal quantum
numbers (for example, np − 10s transitions) where only
rough estimates of uncertainties were needed, we used
uncertainty estimate from the previous transition. For
example, we use 0.5% as uncertainty estimate for the
6pj−9s transitions since the uncertainty for the 6pj−8s
ones was 0.5%. Since relative correlation correction gen-
erally decreases with n, such procedure can overestimate
the uncertainty, but should not underestimate it. The fi-
nal results and their uncertainties are summarized in the
last columns of Table I.
The best set values for the electric-dipole matrix el-
ements and their uncertainties are summarized in Ta-
ble III.
Selection of the “best set” values for diagonal and off-
diagonal matrix elements of the magnetic hyperfine oper-
ator T(1)in 10−8a.u is illustrated in Table II. The com-
plete table is given in Ref. [16]. To convert the diagonal
matrix element in atomic units to hyperfine constants in
MHz, one multiplies the values in Table II by
6.5797× 109gI
?jv(jv+ 1)(2jv+ 1),
where the nuclear gyromagnetic ratio gI = 1.83416 for
87Rb and jvis total angular momentum of the electronic
state. Triple corrections are large for hyperfine matrix
elements and have to be included. Scaling procedure can
not be applied here since the terms that it corrects are
generally not dominant unlike the cases of the ns − n′p
matrix elements above. The remaining columns in Ta-
ble II are the same as in the E1 matrix element tables.
Most of the diagonal hyperfine matrix elements are
taken from the experiment.
ties are listed where experimental data are used. Off-
diagonal hyperfine matrix elements between the s-states
?ns?T(1)?n′s? can be also evaluated from experimental
hyperfine constants using the formula
Experimental uncertain-
|?ns?T(1)?n′s?| =
?
?ns?T(1)?ns??n′s?T(1)?n′s?, (6)
that is useful for the cases where accurate values of the
hyperfine constants A are available. We list such val-
ues for the off-diagonal matrix elements as experimen-
tal. Since a large number of high-precision experimental
values is available for matrix elements in Table II, the
remaining uncertainties for off-diagonal matrix elements
are assigned based on the differences of the theory val-
ues for the most relevant diagonal matrix elements with
experiment. For example, the entry “from 5p1/2” in the
“Unc. source” column indicates that the difference of
the theoretical 5p1/2hyperfine constant with the exper-
imental value was used to assign the uncertainty of the
TABLE IV: Absolute values of the diagonal and off-diagonal
matrix elements of the magnetic hyperfine operator T(1)in
10−8a.u. See text for conversion of diagonal matrix elements
in atomic units to hyperfine constants in MHz.
Matrix element
5s − 5s
5s − 6s
5s − 7s
5s − 8s
5s − 9s
5s − 10s
5s − 11s
Value
34.681
16.86(1)
10.609(2)
7.479(7)
5.64(2)
4.45(1)
3.63(1)
Matrix element
5p1/2− 5p1/2
5p1/2− 5p3/2
5p1/2− 6p1/2
5p1/2− 6p3/2
5p1/2− 7p1/2
5p1/2− 7p3/2
Value
4.122(8)
0.327(3)
2.36(1)
0.189(2)
1.59(1)
0.127(1)
5p3/2− 5p3/2
5p3/2− 6p1/2
5p3/2− 6p3/2
5p3/2− 7p1/2
5p3/2− 7p3/2
2.723(2)
0.185(2)
1.55(2)
0.124(1)
1.04(1)
6p1/2− 6p1/2
6p1/2− 6p3/2
6p1/2− 7p1/2
6p1/2− 7p3/2
1.3453(3)
0.108(1)
0.902(2)
0.073(1)
6p3/2− 6p3/2
6p3/2− 7p1/2
6p3/2− 7p3/2
0.889(1)
0.072(2)
0.58(1)
7p1/2− 7p1/2
7p1/2− 7p3/2
7p3/2− 7p3/2
0.6020(3)
0.049(1)
0.4034(3)
TABLE V: Comparison of the DHF values for the main con-
tributions??12
m=5
?to term T with the final “best set” values.
n refers to terms of the ns sum. The relative difference be-
tween the two values is given in the last column.
n
6
7
8
9
10
11
12
DHF
0.0016114
0.0002277
0.0000787
0.0000378
0.0000217
0.0000141
0.0000104
FinalDif.
-6.3%
-5.6%
-4.1%
-3.7%
-3.7%
-3.9%
-5.0%
0.0015159(83)
0.0002156(18)
0.0000756(7)
0.0000365(5)
0.0000209(4)
0.0000135(4)
0.0000099(4)
off-diagonal matrix element. We note that contributions
of the np−np′matrix elements to total uncertainty of the
static Stark coefficient kSis very small, and approximate
estimate of uncertainties is sufficient.
The best set values for the hyperfine matrix elements
and their uncertainties are summarized in Table IV.
IV. BBR SHIFT UNCERTAINTY
The total uncertainty of the main terms of the static
Stark coefficient is obtained by adding uncertainties from
all contibutions in quadrature. The uncertainties in the
remainders are evaluated separately for each term.
Term T contains two sums, over ns and over mpj.
First, we study the the remainder of the mpj sum,
(m > 12) for each of the first few ns terms, i.e.
break down each ns term as
we
?
ns


12pj
?
2pj
[...] +
Npj
?
13pj
[...]

.