Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics
Подождите немного. Документ загружается.
Hyperfine Structure 16.1 Splittings and Intensities 255
J
α
, I
β
multiplet in first-order perturbation theory can
be described relatively simply in terms of 3– j and 6– j
symbols. The contribution of the term which is the scalar
product of electron and nuclear operators of multipole k
is
∆E
k
(JIF, JIF)
= (−)
J+I +F
JJk
IIF
×
Jk J
J 0 − J
Ik I
I 0 −I
−1
A
k
, (16.6)
where for k ≥ 1,
A
k
=
JJ
T
k
(e)
JJ
·
II
T
k
(N)
II
. (16.7)
The commonly used hfs coefficients A, B,etc.,are
related to the A
k
by
A = A
1
/IJ, B = 4A
2
, C = A
3
, D = A
4
.
(16.8)
The isotope shift A
0
is the matrix element of the reduced
monopole operator.
The pattern of the splitting depends on the total angu-
lar momentum F wholly through the 6– j symbol. Since
for k = 0 the value of the 6– j symbol is independent of
F, the monopole term shifts all levels of the hyperfine
multiplet equally, independent of the value of F.
The F-dependence of the dipole contribution can be
found from the fact that the same 6– j symbol would
appear for any scalar product of k = 1 operators, for
example J· I. But in this product space, with J, I,andF
all good quantum numbers, the diagonal matrix elements
of J· I are just
J· I=
1
2
[F(F + 1) − J( J + 1) − I(I + 1)] ,
(16.9)
so that
∆E
1
(JIF, JIF)
=
1
2
A[F(F + 1) − J(J + 1) − I(I + 1)] ,
(16.10)
where, in terms of reduced matrix elements according to
the convention of Brink and Satchler ([16.35, p. 152]),
(the first version given in Sect. 2.8.4)
A =[J(J + 1)]
−1/2
J
T
1
(e)
J
[I(I + 1)]
−1/2
×
I
T
1
(N)
I
. (16.11)
A is the magnetic dipole hyperfine structure constant for
the atomic level J and nuclear state I .M1hfsshows
the same pattern of splittings as spin-orbit fine structure,
described sometimes as the Landé interval rule.
Electric quadrupole hfs is described by the quad-
rupole hyperfine structure constant B.Ifwedefinethe
quantity K =[F(F + 1) − J(J + 1) − I(I + 1)],then
∆E
2
(JIF, JIF)
=
1
4
B
[3K(K + 1)/2 − 2J(J + 1)I(I + 1)]
J(2J − 1)I(2I − 1)
.
(16.12)
The constant B is related to the tensor operators by
1
4
B =[J(2J − 1)/(J + 1)(2J + 3)]
−1/2
×
J
T
2
(e)
J
× [I(2I − 1)/(I + 1)(2I + 3)]
−1/2
×
I
T
2
(N)
I
, (16.13)
For higher multipoles, see [16.36].
The multipole expansion is important because it is
valid for relativistic as well as nonrelativistic situations,
and for nuclear penetration effects (hyperfine anomalies
discussed in Sect. 16.3.3) as well as for normal hyper-
fine structure. Its limitation comes from its nature as
a first-order diagonal perturbation. Off-diagonal contri-
butions, even when small, can perturb the pattern, but,
more importantly, can lead to misleading values for the
A
k
coefficients, including the isotope shift.
16.1.3 Intensities
When hyperfine structure is observed as a splitting in an
optical transition between different atomic levels, there
are relations between the intensities of the components.
The general rule for reduced matrix elements of a tensor
operator operating in the first part of a coupled space is
([16.35, p. 152])
JI F
Q
λ
(e)
J
IF
= (−1)
λ+I +F
+J
(2F
+ 1)
1/2
FF
λ
J
JI
× (2J + 1)
1/2
J
Q
λ
(e)
J
.
(16.14)
For a dipole transition (λ = 1) connecting atomic states
J and J
, with fixed nuclear spin I , the line strength
S
FF
of the hyperfine component connecting F and F
is related to the line strength S
JJ
by
S
FF
= (2F + 1)(2F
+ 1)
FF
1
J
JI
2
S
JJ
.
(16.15)
Part B 16.1
256 Part B Atoms
16.2 Isotope Shifts
Two distinct mechanisms contribute to isotope shifts
in atomic energy levels and transition energies. First,
there are shifts due to the different mass values of the
isotopes; these mass shifts can again be separated into
two kinds, the normal mass shift and the specific mass
shift. Second, there are shifts due to different nuclear
charge distributions in different isotopes. Shifts of this
sort are called field shifts, and can be considered to be
the monopole part of the hyperfine interaction.
The usual convention is to describe an isotope shift
in a transition as positive when the line frequency is
greater for the heavier isotope.
16.2.1 Normal Mass Shift
The normal mass shift occurs already for one-electron
atoms, where the energy scale in the c.m. frame for the
electron-nucleus system is directly proportional to the
reduced mass of the system, µ = mM/(M + m),where
m is the mass of the electron and M that of the nu-
cleus. The two natural limits are those of an infinitely
heavy nucleus, where µ = m, and positronium, where
µ =
1
2
m. The normal mass shift applies to all levels
of all atomic systems. The fractional shift in frequency
between isotopes of mass M
H
and M
L
is given by
(ν
H
− ν
L
)/ν
H
= m(M
H
− M
L
)/M
H
(M
L
+ m),
(16.16)
or
(ν
H
− ν
L
)/ν
L
= m(M
H
− M
L
)/(M
H
+ m)M
L
.
(16.17)
The normal mass shift between deuterium and protonic
hydrogen is (ν
D
− ν
H
)/ν
H
= 2.721 × 10
−4
, amounting
to 4.15 cm
−1
(0.179 nm) for Balmer α. It decreases
rapidly for heavy elements, with ∆ν/ν ≈ (A
H
−
A
L
)/(1823A
H
A
L
),wheretheA-values are the atomic
mass numbers; for the pair
208
Pb–
206
Pb, ∆ν/ν is then
2.56 × 10
−8
, corresponding to a wavelength shift of
0.000 014 nm for a line at 550 nm.
16.2.2 Specific Mass Shift
When the system has more than two particles the sit-
uation is more complicated; in particular, the center of
mass of any particular electron and the nucleus is no
longer the center of mass of the whole system. When
c.m. motion is removed from the Hamiltonian, there
remains a set of mass polarization terms
H
mp
=
1
M
j<k
p
j
· p
k
, (16.18)
where M is the nuclear mass and p
j
is the momentum of
the jth electron. The matrix elements of these terms can
be strongly state-dependent, and the difference in their
contributions for isotopes of different mass is called the
specific mass shift (SMS). For a transition a → b,the
lowest-order SMS between isotopes A and A
is
∆ν(a, b; A, A
) = K
a,b
(M
A
− M
A
)/M
A
M
A
,
(16.19)
where
K
a,b
=
a
j<k
p
j
· p
k
a
−
b
j<k
p
j
· p
k
b
.
(16.20)
It was earlier thought that the SMS, like the normal
mass shift, was always very small for heavy atoms, but
that has turned out not to be the case.
Since the operator p
1
· p
2
resembles the product of
two dipole transition operators, the matrix elements of
H
mp
vanish between simple product-type wave func-
tions unless allowed by dipole selection rules. For
example, in the 1sn
1
L and
3
L states of helium, the
only nonvanishing diagonal terms are the = 1 exchange
terms ±1s(1)np(2)| p
1
· p
2
|np(2)1s(1), with (−)for
singlet states and (+) for triplet states. The resulting
Hughes–Eckart level shift [16.37] is positive for singlets
and negative for triplets. For other states, H
mp
acquires
a nonzero value due to electron correlation (configura-
tion mixing) effects, but the resulting level shifts are
correspondingly smaller. Detailed tabulations for many
states of helium, including second-order corrections, are
given by Drake and Ya n [16.38].
For other atoms, the diagonal matrix elements of
H
mp
within a single electron configuration are weighted
by the same coefficients as those that weight the Slater
exchange integral G
1
. Relativistic corrections have been
given by Stone [16.39, 40]. Bauche and Champeau
[16.11] provide a useful discussion of the SMS, as does
King [16.16].
Among recent results, we mention high-precision
experimental work on the difference in the splittings
2
3
S
1
–2
3
P
J
between
4
He and
3
He [16.41–43], which can
be compared with theoretical results [16.43,44], and an
extensive multi-isotope study in Sm II, in which isotope-
shift results are used to deduce structure information for
a number of levels [16.45].
Part B 16.2
Hyperfine Structure 16.2 Isotope Shifts 257
16.2.3 Field Shift
The field shift is due to different electric monopole
interactions within the nucleus for different nuclei.
Since the field shift is state-dependent, it contributes
to the isotope shift of electronic transition frequencies.
When field shifts occur between different isomeric lev-
els of the same nuclear species, they are called isomer
shifts.
Following Seltzer [16.46], (see also the summary in
Fricke et al.[16.47]), a level shift can be written as
∆E =−e
2
[ρ
N
(A) − ρ
N
(A
)]
×
r
−1
N
ρ
e
dτ
e
−
r
−1
e
ρ
e
dτ
e
dτ
N
.
(16.21)
The electronic factor inside the second bracket can
be fitted as an even power series in r
N
, starting with
r
2 j+1
N
. The result is that the field shift for a transition a
is
∆ν(a, AA
) = F
a
λ
AA
, (16.22)
with
λ
AA
= δ
r
2
AA
+ (C
2
/C
1
)δ
r
4
AA
+ (C
3
/C
1
)δ
r
6
AA
+··· , (16.23)
where δ
r
k
AA
=
r
k
A
−
r
k
A
.ThetermF
a
is an elec-
tronic factor proportional to the change in electron
density at the nucleus between the initial and final
electronic states; in the simplest perturbation approach
F
a
is equal to −(4π/6)Ze
2
∆ρ
e
(0),where∆ρ
e
(0)
is the electron density at the nucleus in the lower
atomic state minus that in the upper (see Sect. 90.1).
The ratios C
2
/C
1
etc., which weight the higher even
moments, depend only on Z and not on the particu-
lar transition; values are tabulated in [16.46, 48, 49],
as are F-factors for K X-ray transitions. This ap-
proach has been generalized and reformulated by
Blundell et al. [16.49].
Measurements of isotope shifts of optical lines
have long played an important role in the deter-
mination of nuclear rms radii. A more compre-
hensive picture of nuclear charge distributions can
be found by combining optical isotope shift re-
sults with those for X-ray lines and for transitions
in muonic atoms, together with results of elastic
electron scattering from nuclei; a compilation of
the results of such an analysis is to be found in
Fricke et al. [16.47].
16.2.4 Separation of Mass Shift
and Field Shift
From the measured isotope shift for a spectral line a
between a pair of isotopes A and A
, one can define
a residual isotope shift by subtracting the normal mass
shift. The residual shift is the sum of two terms, each
with an electronic factor and a nuclear factor
∆ν
a,AA
= K
a
(M
A
− M
A
)/M
A
M
A
+ F
a
λ
AA
.
(16.24)
If the field shift is assumed to be negligible,
the electronic factor K
a
can be extracted. Similar-
ly, if the SMS is assumed to be negligible,
from the ratios ∆ν(a, A
1
A
2
)/∆ν(a, A
3
A
4
), ratios of
λ(A
1
A
2
)/λ(A
3
A
4
) can be obtained, and the ratios
∆ν(a, A
1
A
2
)/∆ν(b, A
1
A
2
) give ratios of the electronic
factors F
a
/F
b
.
If both contributions have to be considered, one can
gain information from a King plot [16.16, 50]inwhich
a modified residual shift ∆ν
(a, AA
) for line a is defined
as
∆ν
(a, AA
) = ∆ν(a, AA
)M
A
M
A
/(M
A
− M
A
).
(16.25)
(Some authors modify the whole shift, without sub-
tracting the normal mass shift; since the isotopic mass
dependence is the same for the two mass-shift contri-
butions, the separation proceeds as before, but with an
altered definition of the constant K .) For each isotope
pair AA
, this defines a point in the King plot whose
ordinate is, e.g., ∆ν
(a, AA
) and whose abscissa is
∆ν
(b, AA
). If the assumed additivity is valid, one finds
a linear relationship between the modified shifts for line
a and those for another line b according to
∆ν
(a, AA
) =
F
a
/F
b
∆ν
(b, AA
) + K
a
− K
b
F
a
/F
b
.
(16.26)
From the slope of the line one has the ratio of F-factors
for the two lines, and from the intercept a relation be-
tween the K -factors. A sometimes useful variant of the
King plot involves choosing a pair of isotopes, say B
and B
as a reference pair [16.16, 47]. A reduced shift
∆ν
(a, AA
) is then defined as
∆ν
(a, AA
) =
∆ν(a, AA
)M
A
M
A
(M
B
− M
B
)
(M
A
− M
A
)M
B
M
B
,
(16.27)
and ∆ν
(a) is plotted vs. ∆ν
(b).
It is not in general possible to extract F-and
K-values for individual transitions from optical isotope
Part B 16.2
258 Part B Atoms
shifts alone, unless one has at least one calculated or oth-
erwise reliably known value. Information about nuclear
charge distributions, however, and isotopic changes in
them, is available from other sources, including X-ray
isotope shifts, muonic X-rays, and elastic electron scat-
tering from nuclei. Calculated electronic factors for K
X-ray transitions can be expected to be more reliable
than those for optical transitions. The effects of finite
nuclear size are much bigger in muonic than electronic
atoms, and screening effects much smaller. Combined
analyses of all four types of data lead to the best current
knowledge of changes in nuclear charge parameters, and
thus contribute to knowledge of the electronic factors F
a
and K
a
for optical transitions [16.47].
16.3 Hyperfine Structure
We discuss first normal hyperfine structure, in which
terms coming from penetration of electrons into the
nuclear charge and current distributions can be ne-
glected; such contributions, the anomalous hyperfine
structure, are briefly discussed in Sect. 16.3.3. See also
Sect. 21.7.2.
16.3.1 Electric Multipoles
The electric quadrupole tensor operators, when penetra-
tion is neglected, are
T
2
(N ) = ρ
N
(r
N
)r
2
N
C
(2)
, (16.28)
where ρ
N
is the nuclear charge density, and
T
2
(e
i
) =−er
−3
i
C
(2)
(i). (16.29)
The matrix element of T
2
(N) that occurs in the ex-
pansion coefficient A
2
is just e/2 times the nuclear
quadrupole moment Q:
eQ = 2
II
T
2
(N)|II
. (16.30)
Experimental values of Q are tabulated by Ragha-
van [16.32] and more recently discussed by
Pyykkö [16.51]. The matrix element of T
2
(e), summed
over all electrons, also has a simple semiclassical inter-
pretation [16.7];
JJ
T
2
(e)
JJ
= (e/2)q
J
= (1/2)
∂
2
V
∂z
2
JJ
, (16.31)
where V is the electrostatic potential at the nucleus due
to the electrons. The quadrupole splitting constant can
thus be expressed as
B = 4 A
2
= e
2
Qq
J
= eQ
∂
2
V
∂z
2
JJ
. (16.32)
Because the matrix elements of spherical harmonics
in lsj-coupled states are independent of l (aside from
the parity requirement that l +l
+ k be even), the one-
electron matrix elements of T
2
(e) are [16.36]
ls j
T
2
(e)
l
sj
=−e
ls j
C
(2)
l
sj
r
−3
gg
+ ff
dr ,
(16.33)
where g and f arethelargeandsmallr-multiplied Dirac
radial functions, and the right-hand reduced matrix el-
ement is equal to (−1)
k−1
C
jk j
1
2
0
1
2
with k = 2 ([16.35,
p. 153]).
Electric hexadecapole hyperfine structure follows
similar rules [16.36]. While there is considerable in-
direct evidence that many nuclear states have nonzero
E4 moments [16.52], their effect on hyperfine structure
has been identified only occasionally [16.53, 54].
16.3.2 Magnetic Multipoles
For the magnetic multipole modes, when penetration
effects are neglected, and when the the nuclear current
is taken to be a sum of the orbital-current and spin-
current contributions of individual nucleons, the nuclear
tensor operator can be written as the sum
T
k
(N) = (e /2m
p
c)
n
[k(2k − 1)]
1/2
r
k−1
n
×
2g
ln
(k + 1)
−1
C
(k−1)
L
n
(k)
+ g
sn
C
(k−1)
S
n
(k)
,
(16.34)
where e /2m
p
c is the nuclear magneton, and in the
extreme single-nucleon model the orbital g-factors g
ln
are 1 for protons and 0 for neutrons, while the spin
g-factors g
sn
are 5.585 694 701(56) for protons and
− 3.826 085 46(90) for neutrons. Similarly, for elec-
tron i, the atomic Mk operator can be written
T
k
(e
i
) = ie[(k + 1)/k]
1/2
r
−k−1
i
α
i
C
(k)
i
(k)
.
(16.35)
Part B 16.3
Hyperfine Structure References 259
For the magnetic dipole case, the nuclear operator of
(16.29) becomes
T
1
(N) = (e /2m
p
c)
n
(g
ln
L
n
+ g
sn
S
n
). (16.36)
The nuclear currents, however, are more complicated
in the strongly interacting and relatively dense environ-
ment of real nuclear systems; renormalized g-factors
can be used to take some such effects into account. For
the quenching of spin matrix elements, see Castel and
Towner [16.55]; for the M1 mode, the effective g-factor
is typically in the neighborhood of 0.5 or 0.6 of the
free-space value. It is also sometimes useful to con-
sider effective orbital g-factors, but the effective value
is smaller, with g
eff
/g
free
= 1.05−1.1 [16.56]. The ma-
trix element of T
1
(N) is the nuclear magnetic dipole
moment
µ =
II
T
1
(N)
II
. (16.37)
Experimental values of µ are tabulated by Raghavan
[16.32].
In the semiclassical picture where ∆ E =−µ · B(0),
the contributions to B(0) from the orbital and spin
magnetism of the ith electron are
B
l
(0) =−2µ
0
r
−3
i
l
i
, (16.38)
B
s
(0) = 2µ
0
r
−3
i
(10)
1/2
s
i
C
(2)
i
(1)
, (16.39)
while the “contact” term for s-electrons is
B
c
(0) =−(16π/3)µ
0
ρ
i
(0)s . (16.40)
The one-electron matrix elements of T
1
(e) are [16.36]
ls j
T
1
(e)
l
sj
=−e(κ + κ
)
ls j
C
(1)
l
x
sj
×
r
−2
fg
+ gf
dr ,
(16.41)
where for a given combination ls j , κ = (l − j)(2 j + 1),
and l
x
is the opposite-parity orbital quantum number of
the small Dirac component, l
x
= l + 1for j = l + 1/2,
l
x
= l − 1for j = l − 1/2. Recent high precision cal-
culations for the hyperfine structure of helium and
lithium, including relativistic corrections and second-
order effects, have recently been done by Pachucki
[16.57, 58].
Magnetic octupole hyperfine structure [16.59, 60]
follows similar rules [16.36, 61]. Systems studied in
recent years include Eu I excited states [16.62].
16.3.3 Hyperfine Anomalies
When the electron density is nonzero inside the nu-
cleus, the interaction with different kinds of nuclear
current density is in general different; there is, for ex-
ample, a sensitivity to differing radial distributions of
spin and orbital currents. The result is that the ratio
of A-values for two isotopes is not necessarily the
same as the ratio of nuclear g-factors [16.63]. The
anomaly for two nuclear species a and b is characterized
by
a
∆
b
= (A
a
/A
b
)(g
a
/g
b
) − 1 . (16.42)
∆ is seldom larger than a few per cent. The the-
ory for the M1 mode was worked out by Bohr
and Weisskopf [16.64]; there are relatively recent re-
views of theory and experiment [16.65, 66], and an
especially clear exposition of the semiclassical limit
[16.67]. Recent experimental results include transi-
tions in isotopes of La II [16.68], and the ground
state of hydrogenlike thallium [16.69]. Anomalies are
expected to be considerably smaller for the electric
mode.
References
16.1 H. B. G. Casimir: On the Interaction Between Atomic
Nuclei and Electrons (Teyler’s Tweede Genootschap,
Haarlem 1963)
16.2 H. B. G. Casimir: On the Interaction Between Atomic
Nuclei and Electrons (Freeman, San Francisco 1936)
16.3 L. Pauling, S. Goudsmit: The Structure of Line Spec-
tra (McGraw Hill, New York 1930)
16.4 G. Herzberg: Atomic Spectra and Atomic Structure,
1st and 2nd edn. (Prentice-Hall and Dover, New
York 1937 and 1944)
16.5 G. Herzberg: Molecular Spectra and Molecu-
lar Structure, I. Spectra of Diatomic Mol-
ecules, 2nd edn. (Van Nostrand, Princeton
1950)
16.6 H. Kopfermann: Nuclear Moments (Academic, New
York 1958)
16.7 N. F. Ramsey: Molecular Beams (Clarendon, Oxford
1956)
16.8 J. C. Slater: Quantum Theory of Atomic Structure,
Vol. II (McGraw-Hill, New York 1960)
16.9 I. I. Sobelman: Introduction to the Theory of Atomic
Spectra (Pergamon, London 1972)
16.10 H. G. Kuhn: Atomic Spectra, 1st and 2nd edn. (Aca-
demic, New York 1962 and 1969)
Part B 16
260 Part B Atoms
16.11 A. Corney: Atomic and Laser Spectroscopy (Claren-
don, Oxford 1977)
16.12 M. Weissbluth: Atoms and Molecules (Academic,
New York 1978)
16.13 I. I. Sobelman: Atomic Spectra and Radiative Tran-
sitions (Springer, Berlin, Heidelberg 1979)
16.14 H. Haken, H. C. Wolf: Atomic and Quantum Physics
(Springer, Berlin, Heidelberg 1984)
16.15 L. Armstrong, Jr.: Theory of the Hyperfine Structure
of Free Atoms (Wiley, New York 1971)
16.16 W. H. King: Isotope Shifts in Atomic Spectra
(Plenum, New York 1984)
16.17 I. Lindgren, J. Morrison: Atomic Many-Body Theory
(Springer, Berlin, Heidelberg 1982)
16.18 J. Bauche, R.-J. Champeau: Adv. At. Mol. Opt. Phys.
12, 39 (1976)
16.19 K. Heilig, A. Steudel: New developments of classical
optical spectroscopy. In: Progress in Atomic Spec-
troscopy, Part A, ed. by W. Hanle, H. Kleinpoppen
(Plenum, New York 1978) pp. 263–328
16.20 V. W. Hughes, B. Bederson, V. W. Cohen, F. M. J.
Pichanik (Eds.): Atomic Physics (Plenum, New York
1969)articles by H. M. Foley, pp. 509–522, and
H. H. Stroke, pp. 523–550
16.21 J. C. Zorn, R. R. Lewis, M. K. Weiss (Eds.): Atomic
Physics, AIP Conf. Proc. 233 (American Institute of
Physics, New York 1991)
16.22 H. Walther, T. W. Hänsch, B. Neizert: Atomic
Physics, AIP Conf. Proc. 275 (American Institute of
Physics, New York 1993)
16.23 J. Hüfner, F. Scheck, C. S. Wu: Muon Phys. 1,201
(1977)
16.24 R. C. Barrett: Muon Phys. 1, 309–322 (1977)
16.25 R. C. Barrett, D. Jackson: Nuclear Sizes and Structure
(Oxford Univ. Press, Oxford 1977)
16.26 K. Pachucki: Phys. Rev. A 63, 032508 (2001)
16.27 G. Backenstoss: Ann. Rev. Nucl. Sci. 20, 467 (1970)
16.28 J. Konijn: An improved parametrization of the op-
tical potential for pionic atoms. In: Pions in Nuclei,
ed. by E. Oset, M. J. Vicente Vacas, C. Garcia Recio
(World Scientific, Singapore 1992) p. 303
16.29 C. J. Batty: Nucl. Phys. A 508, 89C (1990)
16.30 G. Backenstoss: Comtemp. Phys. 30, 433 (1989)
16.31 V. I. Korobov, D. Bakalov: J. Phys. B 34,L519
(2001)
16.32 P. Raghavan: At. Data Nucl. Data Tables 42,189
(1989)
16.33 M. J. Martin, J. K. Tuli (Eds.): Nuclear Data Sheets
(Academic, New York 1995) Nos. 74–76; each issue
contains a guide to the most recent compilation
for each atomic mass number
16.34 C. M. Lederer, V. S. Shirley: Table of Isotopes, 7th
edn. (Wiley, New York 1978)
16.35 D. M. Brink, G. R. Satchler: Angular Momentum,2nd
edn. (Clarendon Press, Oxford 1968)
16.36 C. Schwartz: Phys. Rev. 97, 380 (1955)
16.37 D. S. Hughes, C. Eckart: Phys. Rev. 36, 694 (1930)
16.38 G. W. F. Drake, Z.-C. Yan: Phys. Rev. A 46, 2378
(1992)
16.39 A. P. Stone: Proc. Phys. Soc. Lond. 77,786(1961)
16.40 A. P. Stone: Proc. Phys. Soc. Lond. 81,868(1963)
16.41 J. R. Lawall. Ping Zhao, F. M. Pipkin: Phys. Rev. Lett.
66, 592 (1991)
16.42 D. Shiner, R. Dixson, V. Vedantham: Phys. Rev. Lett.
74, 3553 (1995)
16.43 F. Marin, F. Minardi, F. S. Pavone, G. W. F. Drake: Z.
Phys. D 32, 285 (1995)
16.44 G. W. F. Drake: High-precision calculations for the
Rydberg states of helium. In: Long Range Casimir
Forces: Theory and Recent Experiments on Atomic
Systems, ed. by F. S. Levin, D. A. Micha (Plenum,
New York 1993) pp. 196–199 For isotope shifts in
Li
+
, see also [16.70]
16.45 P. Villemoes et al.: Phys. Rev. A 51, 2838 (1995)
16.46 E. C. Seltzer: Phys. Rev. 188, 1916 (1969)
16.47 G. Fricke et al.: At. Data Nucl. Data Tables 60,177
(1995)
16.48 F. Boehm, P. L. Lee: At. Data Nucl. Data Tables 37,
455 (1987)
16.49 S. A. Blundell et al.: J. Phys. B 20, 3663 (1987)
16.50 W. H. King: J. Opt. Soc. Am. 53, 638 (1963)
16.51 P. Pyykkö: Z. Naturforsch. 47a, 189 (1992)
16.52 R. F. Casten: Nuclear Structure from a Simple
Perspective (Oxford Univ. Press, Oxford 1990)
pp. 296–300
16.53 W. Dankwort, J. Ferch, H. Gebauer: Z. Phys. 267,
229 (1974)
16.54 O. Becker et al.: Phys. Rev. A 48, 3546 (1993)
16.55 B. Castel, I. S. Towner: Modern Theories of Nuclear
Moments (Clarendon Press, Oxford 1990)
16.56 T. Yamazaki et al.: Phys. Rev. Lett. 25, 547 (1970)
16.57 K. Pachucki: J. Phys. B 34, 3357 (2001)
16.58 K. Pachucki: Phys. Rev. A 66, 062501 (2002)
16.59 V. Jaccarino, J. G. King, R. A. Satten, H. H. Stroke:
Phys. Rev. 94, 1798 (1954)
16.60 P. Kusch, T. G. Eck: Phys. Rev. 94, 1799 (1954)
16.61 M. Mizushima: Quantum Mechanics of Atomic
Spectra and Atomic Structure (Benjamin, New York
1970) Sect. 9–10
16.62 W. J. Childs: Phys. Rev. A 44, 1523 (1991)
16.63 F. Bitter: Phys. Rev. 76, 150 (1949)
16.64 A. Bohr, V. F. Weisskopf: Phys. Rev. 77, 94 (1950)
16.65 S. Büttgenbach: Hyperfine Interactions 20,1(1984)
16.66 G. Savard, G. Werth: Ann. Rev. Nucl. Part. Sci. 50,
119 (2000)
16.67 R. A. Sorensen: Am. J. Phys. 35, 1078 (1967)
16.68 H. Iimura et al.: Phys. Rev. C 68, 054328 (2003)
16.69 P. Beiersdorfer et al.: Nucl. Instrum. Methods
Physics Research B 205, 62 (2003)
16.70 E. Riis, A. G. Sinclair, O. Paulsen, G. W. F. Drake,
W. R. C. Rowley, A. P. Levick: Phys. Rev. A 49,207
(1994)
Part B 16
261
Precision Oscil
17. Precision Oscillator Strength
and Lifetime Measurements
The accuracy of oscillator strength and lifetime
measurements has improved greatly in the
past twenty years. Nevertheless, these high
accuracies have been achieved for only a restricted
number of lines belonging to a few elements and
ionization stages [17.1]. Large numbers of precision
measurements must still be made as improved
experimental oscillator strengths are needed, both
as tests of theoretical concepts, and for diagnostics
and engineering applications.
17.1 Oscillator Strengths.............................. 262
17.1.1 Absorption
and Dispersion Measurements .... 262
17.1.2 Emission Measurements............. 263
17.1.3 Combined Absorption, Emission
and Lifetime Measurements ....... 263
17.1.4 Branching Ratios in Highly
Ionized Atoms........................... 264
17.2 Lifetimes ............................................ 264
17.2.1 The Hanle Effect ........................ 265
17.2.2 Time-Resolved Decay
Measurements .......................... 265
17.2.3 Other Methods .......................... 267
17.2.4 Multiplexed Detection................ 267
References .................................................. 268
A spectral line arising from a radiative transition be-
tween atomic states i and k is characterized by its
wavelength λ
ik
, its intensity and its shape. In the limit
of free atoms, the intensity per atom is determined by
the emission transition rate A
ik
or absorption oscilla-
tor strength f
ki
, and the shape by the natural width
Γ
i
= /τ
i
,whereτ
i
is the lifetime of the excited state.
While classical spectroscopic methods provide precise
wavelength measurements (1 part in 10
8
or better;
Chapt. 10), it has only recently been possible to meas-
ure oscillator strengths and lifetimes to better than
a few percent, as discussed in this chapter. Examples
of applications which require an accurate knowledge
of these quantities are the interpretation of astrophys-
ical data (Chapt. 82), atmospheric physics (Chapt. 84),
combustion (Chapt. 88), the modeling and diagnosis
of thermonuclear plasmas (Chapt. 86), nonlinear op-
tics (Chapt. 72), isotope separation (Chapt. 16), and the
development of new types of lasers (Chapt. 71).
Precision measurements of oscillator strengths and
lifetimes also provide stringent tests of atomic structure
calculations. These quantities are very sensitive to the
wave functions and the approximations used, particu-
larly in cases where electron correlations (Sect. 23.2.1)
and relativistic effects (Sect. 22.1) are significant. They
provide experimental tests of fundamental theory;
for example, of quantum electrodynamic corrections
(Sect. 27.2), and of the nonconservation of parity pre-
dicted by the unified electro-weak theory (Sect. 29.1).
In applications where only modest precision is
required, semi-empirical parameterizations involving
quantum defects, charge screening and polarization al-
low a few precise measurements to be extrapolated
along isoelectronic, homologous, isoionic and Rydberg
sequences. Similar methods have been applied to the
atomic energy levels themselves (Sect. 10.13). They al-
low one to produce a very large data base of moderate
precision [17.2].
The spontaneous transition rate A
ik
(Sect. 10.16.2)
is the probability per unit time (s
−1
) for an atom in any
one of the g
i
states of the energy level i to make a transi-
tion to any of the g
k
states of the level k. The lifetime τ
i
is then given by 1/τ
i
=
k
A
ik
. The branching fraction
for the kth channel of the decay of level i is defined
as F
B
= τ
i
A
ik
, and the branching ratio (Sect. 10.17)be-
tween two decay channels is R
B
= A
ik
/A
ij
. Emission
and absorption rate constants differ by a factor of λ
2
ik
,
and the absorption oscillator strength f
ki
(Sect. 10.17)is
defined by
g
k
f
ki
= Cλ
2
ik
g
i
A
ik
, (17.1)
where C = (32π
3
αa
2
0
Ry)
−1
= 1.499 19 × 10
−14
nm
−2
s.
Because of this relationship, the words transition rate and
oscillator strength will be used almost interchangeably.
Part B 17
262 Part B Atoms
17.1 Oscillator Strengths
Oscillator strengths can be determined directly through
absolute emission, absorption, or dispersion meas-
urements, or through the combined measurement of
branching ratios and lifetimes. Direct measurements
compare different transitions at the same time, and re-
quire sample equilibrium, a knowledge of the absolute
number density, and an absolute intensity measure-
ment. These are in contrast to time-resolved lifetime
measurements, which compare relative intensities from
the same transition at different times, and require
no absolute measurements. However, lifetime meas-
urements yield oscillator strengths directly only in
cases where a single decay transition channel ex-
ists, such as the lowest excited level in an atom or
ion. Thus, combined measurements of lifetimes and
branching ratios are often used where high preci-
sion values for oscillator strengths are required. Both
absorption and dispersion measurements involve the
number density of the lower level of the transition,
whereas emission measurements involve that of the
upper le vel.
17.1.1 Absorption
and Dispersion Measurements
Absorption measurements involve placing a sample of
atoms (for example, in a gas cell, an atomic beam, an arc,
a shock tube, or within the vapor column in a furnace)
between a continuous light source and a spectrometer.
For an isolated spectral line, the absorption cross section
for a beam of photons of frequency ν passing through
the sample is
σ
ik
(ν) = πα( /m)g(ν) f
ik
, (17.2)
where g(ν) is the spectral distribution function per unit
frequency normalized so that
∞
0
g(ν) dν = 1. If there
are N atoms per unit volume, the absorption coefficient
is k
ν
= Nσ(ν). The integrated intensity lost after passing
through a distance L of the sample is
∞
0
∆I(ν) dν = I
0
∞
0
1 − exp
−σ
ik
(ν)NL
dν
I
0
πα( /m)NL f
ik
, (17.3)
where the second line applies if the sample is optically
thin. Otherwise, the integral can be calculated directly, if
g(ν) is known, to determine f
ik
by the curve-of-growth
method.
The Furnace Method
High precision absorption measurements have been
achieved by Blackwell and co-workers [17.3], who have
used the furnace method to study the astrophysically
important neutral iron spectrum. These measurements
have quoted accuracies of 0.5% on a relative scale, and
2.5% on an absolute scale. This accuracy was obtained
through the use of a stable and isothermal furnace, low-
noise spectral intensity recording techniques, and two
identical high resolution spectrometers for the simulta-
neous recording of pairs of absorption lines. By selecting
successive line pairs of a suitable oscillator strength ra-
tio and adjusting the temperature and vapor pressure in
the furnace, a large dynamic range of oscillator strengths
could be covered. Recently, corroborative studies of the
uncertainties quoted in these measurements have been
undertaken, including tests that are coupled to other
methods that use combinations of lifetime and branching
ratio measurements [17.3].
The Hook Method
The absorption measurements described above deter-
mine the oscillator strength from the line intensity. An
alternative absorptive approach is the anomalous disper-
sion or “hook” method, which determines the oscillator
strength from the index of refraction at wavelengths near
the edge of an absorption line [17.1]. The advantages
of this method are its large dynamic range, its insen-
sitivity to the line shape, and the fact that it does not
saturate. The absorbing gas is placed in one arm of an
interferometer and a compensator is placed in the other
arm. This leads to the formation of oblique interference
fringes with two characteristic hooks symmetric about
the center of an absorption line. The oscillator strength
is determined by the wavelength separation W between
the hooks. Defining K = λ
ik
N
f
,whereN
f
is the number
of fringes per unit wavelength, then
f
ik
=
πKW
2
λ
3
ik
α
2
a
0
NL
(17.4)
where a
0
is the Bohr radius.
Synchrotron Radiation
Storage ring synchrotron radiation facilities now pro-
vide a source of continuum radiation that can extend the
wavelength range of absorption measurements. A tech-
nique [17.4] has been developed and applied that utilizes
a hollow cathode discharge as an absorbing sample, syn-
chrotron radiation as a continuum source, and a CCD
Part B 17.1
Precision Oscillator Strength and Lifetime Measurements 17.1 Oscillator Strengths 263
array for vacuum ultraviolet (VUV) detection. Here
relative oscillator strengths were obtained using two
separate detection systems. However, VUV calibration
standards are presently lacking (Sect. 17.1.4), and will
be required to obtain independent tests (Sect. 17.1.3)of
sets of branching fractions.
17.1.2 Emission Measurements
Emission methods use, for example, a hollow cath-
ode, wall-stabilized arc, or shock tube to excite the
source. Recent developments in the methods of Fourier
transform spectrometry (FTS) [17.5, 6]offerseveral
advantages over grating spectroscopy, as discussed
next.
Fourier Transform Spectrometry
As opposed to the dispersive nature of a grating spectro-
graph, FTS uses interference effects from a Michelson
interferometer. All radiation admitted to the spectrom-
eter is thus incident on the detectors at all times, but
different wavelengths are distinguished by their spatial
modulation frequencies. The interferogram from all si-
nusoidal signals is sampled at a prescribed interval of
path length, and compared with a laser of known fre-
quency following the same optical path. The spectrum
is recovered from the interferogram by means of a fast
Fourier transform (Sect. 8.1.3). Thus, for an FTS instru-
ment, the spectral range is determined by the sampling
step size, whereas the resolution depends on the max-
imum path difference (this is in contrast to a grating
spectrometer where the sampling step determines the
resolution, and the scan length determines the spectral
range).
The FTS method provides a number of attractive
features in precision oscillator strength measurements.
The axial symmetry and the replacement of the slit
by a larger aperture can provide a throughput that is
two orders of magnitude greater than that of a grat-
ing instrument of the same resolution. The precision
and reproducibility are determined by the laser stand-
ard and the linearity of the wavenumber scale. The
resolution can be increased as necessary to resolve
a specific source line. The superior resolution allows
blending and self-absorption to be more readily de-
tected. The spectral range is limited only by detectors
and filters. Recently, uv FTS instruments have been con-
structed that operate from the visible down to 175 nm.
Since all wavelengths are observed at all times, errors
from drifts in source conditions during scanning are
reduced.
Hollow Cathode Lamps
Hollow cathode discharge lamps are used extensively
for emission branching fraction measurements [17.5].
These lamps can generate an emission spectrum of es-
sentially any element, and the relatively low collision
rates result in line profiles that are narrow and primar-
ily Doppler broadened. The narrow line width is a major
advantage when studying line-rich spectra. The low col-
lision rates also imply that the discharges are far from
local thermodynamic equilibrium (LTE). However, this
does not affect the determination of branching fractions,
where only the relative strengths of lines from a common
upper level are measured. These measurements can then
be put on an absolute basis if lifetime measurements are
available for all of the upper levels.
17.1.3 Combined Absorption, Emission
and Lifetime Measurements
By combining measurements obtained in emission with
those obtained in absorption (or dispersion) to obtain
branching ratios, and then incorporating lifetime meas-
urements, it is possible to use a scheme that requires
no knowledge of level populations [17.1]. The scheme
was originally proposed by Ladenburg in 1933, and its
various modern implementations are known as “leap-
frogging,” “linkage” and “bow ties”.
The principle of “leap-frogging” or “linkage” is
illustrated in Fig. 17.1a. The decay of level 1 is un-
branched, so the 1 → 2 transition rate is assumed
known (kn) from a lifetime measurement. This is used
to specify the 2 → 3 oscillator strength using relative
absorption (ab) measurements for the 2 → 1 : 2 → 3
oscillator strength ratio (and appropriate factors of the
wavelengths and degeneracies). In a similar manner, this
is subsequently combined with relative emission (em)
measurements of the 3 → 2 : 3 → 4 branching ratio to
determine the 3 → 4 transition rate.
The principle of “bow ties” is illustrated in
Fig. 17.1b. The two branching ratios for 1 → 2 : 1 → 4
and for 3 → 2 : 3 → 4 are measured in emission (em).
The two oscillator strength ratios for 2 → 1 : 2 → 3and
for 4 → 1 : 4 → 3 are measured in absorption (ab). Af-
ter correction for wavelength and degeneracy factors
between f and A values, these relationships are com-
bined into a quantity known as the “bow tie ratio,” which
would be unity for ideally accurate measurements. A sig-
nificant deviation from unity can be used to trace the
observations that are in error. Figure 17.1bisa“simple
bow tie” connecting two upper and two lower levels with
four transitions. Higher order sets of measurements can
Part B 17.1
264 Part B Atoms
a)
13
24
kn
ab
em
b)
13
24
em
ab ab
em
Fig. 17.1a,b Illustration of methods for the determination
of ratios of oscillator strengths (ab = absorption, em =
emission, kn = known)
be similarly coupled; for example, a set of transitions
between three lower levels and three upper levels can be
coupled by nine simple bow ties.
In cases where both lifetime and complete branch-
ing ratio measurements exist for the same upper level,
then the branching ratios can be normalized to branch-
ing fractions divided by the lifetime to obtain absolute
transition probabilities.
17.1.4 Branching Ratios
in Highly Ionized Atoms
In highly ionized atoms, many measurements of life-
times in the 1–5% accuracy range have been made
by ANDC (Sect. 17.2.2) analysis of beam–foil meas-
urements. However, little work exists for the precision
measurement of branching ratios in highly ionized
atoms, where an intensity calibration of the detection
equipment is particularly difficult. In beam-foil exci-
tation, for example, Doppler broadening and Doppler
shifts, polarization due to anisotropic excitation, and the
short wavelength (≤ 200 nm) nature of the radiation are
not well-suited for use with standard techniques for cali-
brating grating spectrometer and detection systems by
the use of calibrated lamps. Calibrations have been car-
ried out using synchrotron radiation, or through the use
of previously known branching ratios.
One way in which lifetime measurements are used
to determine branching ratios involves precision studies
of the lifetimes of the individual fine structure com-
ponents of a multiplet decay. If one fine structure level
has a decay channel that is not available to the other
levels (for example, a spin-changing transition or an
autoionization mode made possible by a J-dependent
intermediate coupling) then the transition rate of the
extra channel can be determined by differential lifetime
measurements.
The use of Si(Li) detectors in the measurement
of very short wavelength radiation in highly ionized
atoms also offers possibilities for branching ratio meas-
urements. Since these devices can specify the photon
energy from pulse height information without the need
for a spectrometer, they can be calibrated for detection
efficiency as a function of energy. This type of detection
was recently used to determine the branching ratio of the
magnetic dipole channel to the two-photon decay chan-
nel in the 2s 2S
1
2
state in one-electron krypton [17.7].
Although radiometric calibration standards are avail-
able for λ>280 nm, technical challenges exist for their
extension to shorter wavelengths. The urgent need for
these standards and their desired characteristics have
been discussed by Lawler et al. [17.4], and an opera-
tional prescription for combining sources of uncertainty
in their specification has been presented by Sikström
et al. [17.8]. A semi-empirical method for obtaining line
intensity standards in the VUV has also been proposed
[17.9, 10] that uses intermediate coupling (IC) ampli-
tudes deduced from measured energy lev el data to obtain
intensity ratios. This utilizes the observation that the
ns
2
np
2
and ns
2
np(n + 1)s configurations in the Si, Ge
and Sn isoelectronic sequences exhibit negligible config-
uration interaction, hence the intensity ratios within their
transition arrays are accurately prescribed by the IC am-
plitudes. For the neutral atoms the transitions occur in the
visible region, and measurements confirm the validity of
the empirical values. Thus, isoelectronic extensions can
yield VUV standards.
17.2 Lifetimes
The total transition rate summed over all decay channels
can be measured either through frequency-resolved stud-
ies of the level width, or through time-resolved studies
of the level lifetime. In order to determine the natu-
ral linewidth in a field-free spectroscopic measurement,
either the lifetime must be very short, or the Doppler,
pressure, and instrumental broadenings must be made
very small. Line widths have been determined using
Part B 17.2