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Chapter 4
Spectra
Charles Cowley, Wolfgang L. Wiese,
Jeffrey Fuhr, and Ludmila A. Kuznetsova
4.1
4.1
Online Database . . . . . . . . . . . . . . . . . . . . . .
53
4.2
Terminology for Atomic States, Levels, Terms, etc. .
54
4.3
Electronic Configurations . . . . . . . . . . . . . . . .
57
4.4
Spectrum Line Intensities . . . . . . . . . . . . . . . .
60
4.5
Relative Strengths Within Multiplets . . . . . . . . . .
65
4.6
Wavelengths and Wave Numbers . . . . . . . . . . . .
68
4.7
Atomic Oscillator Strengths for Allowed Lines . . . .
69
4.8
Nuclear Spin and Hyperfine Structure:
Low-Level Hyperfine Transitions . . . . . . . . . . . .
78
4.9
Forbidden Line Transition Probabilities . . . . . . . .
79
4.10
Spectra of Diatomic Molecules . . . . . . . . . . . . .
83
4.11
Energy Levels . . . . . . . . . . . . . . . . . . . . . . .
85
4.12
Transitions . . . . . . . . . . . . . . . . . . . . . . . . .
87
4.13
Selection Rules: Dipole Radiation . . . . . . . . . . .
89
ONLINE DATABASE
Extensive data and references are available online through the Internet [1]. A comprehensive, critically
evaluated database, whose address is given below, is maintained by the National Institute of Standards
and Technology (NIST). Files of special relevance to atomic spectroscopy are the Atomic Spectroscopic
53
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54 / 4
Page 54
S PECTRA
Database by J.R. Fuhr, W.C. Martin, A. Musgrove, J. Sugar, and W.L. Wiese, Bibliographic Database
on Atomic Transition Probabilities by J.R. Fuhr and H.R. Felrice, and Program of the NIST Atomic
Data Centers by W.L. Wiese and W.C. Martin. The uniform resource locator, or URL, is currently
http://physics.nist.gov/PhysRefData/contents.html. File names and locations are subject to change.
The above files might be found by first “opening” http://physics.nist.gov/ and following the appropriate
links, or simply by doing a network search for the keywords “NIST atomic data.”
4.2
TERMINOLOGY FOR ATOMIC STATES, LEVELS, TERMS, ETC.
The angular momenta of atoms are vector quantities describing the orbital angular momenta (l, L),
the spin (s, S), and the sum of the two (j, J). Lowercase letters are
used for individual electrons,
and uppercase letters refer to corresponding sums (e.g., L =
l). The magnitudes of these
vectors√are specified by quantum numbers usually written with lightface italic symbols. For example,
|l| = l(l + 1)h̄. Spectroscopists often interchange the meaning of the vector quantities and the
associated quantum numbers, and say, for example, that L is the sum of the l’s, although the relation is
only valid for the vector quantities. This loose usage is convenient and is followed here. Spectroscopic
levels are typically described by quantum numbers based on L S (Russell–Saunders) coupling. For
other coupling schemes, see [2] and [3]. Often, levels are expressed as mixtures of L S terms, where
the leading component is the single L S term that best describes the level.
Orbital angular momentum (or azimuthal quantum number), L = vector sum of orbital angular
momenta l of individual electrons. The unit is h/2π ≡ h̄, and the designations are
L (or l)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Designation (L)
Designation (l)
S
s
P
p
D
d
F
f
G
g
H
h
I
i
K
k
L
l
M
m
N
n
O
o
Q
q
R
r
T
t
U
u
Spin angular momentum, S = vector sum of s for individual electrons. The multiplicity of terms
= 2S + 1.
The effects of the atomic nucleus on atomic structure, including nuclear spin I , are treated in a
separate section below.
Total angular momentum quantum number, J =
vector sum L +S (in L S coupling). In j j coupling,
j = vector sum l + s for each electron, and J =
j. The total angular momentum J is said to be a
“good quantum number,” independent of the coupling scheme.
Electron shells are described by the principal quantum number n as follows:
n
1
2
3
4
5
6
7
Shell designation
K
L
M
N
O
P
Q
Only the magnitude of an angular momentum (e.g., |L|) and one of its components (e.g., L z )
are observables. The z component is chosen arbitrarily. Quantum numbers corresponding to these z
components are designated, for example, by m l , M L , or M J = M. If the atom is in a magnetic field, it
is convenient to choose the z direction along the field, so the m’s have been called magnetic quantum
numbers.
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4.2 T ERMINOLOGY FOR ATOMIC S TATES , L EVELS , T ERMS ,
ETC .
/ 55
Maximum values of various quantum numbers are limited as follows:
s = 12 ;
l ≤ n − 1;
J ≤ S + L;
S≤
1
2 na ;
M L ≤ L;
2
D
1 12
0
M ≤ J;
L ≤ l1 + l2 + · · · + ln a ,
where there are n a electrons in open shells.
Interpretation of a typical symbol for an atomic level, e.g., 2 p 3
2
p3
M S ≤ S;
2D 0 :
1 12
Principal quantum number of outer electrons = 2; i.e., L shell.
Three outer electrons with l = 1.
Multiplicity = 2, whence S = 12 .
Orbital momentum L = 2.
J = 1 12 , whence statistical weight g = 2J + 1 = 4.
The level is odd (omitted when level is even).
Possible J values for given L and S:
Singlets
Doublets
Triplets
Quartets
Quintets
Sextets
1 S , 1 P , 1 D , 1 F , 1G , 1 H , . . . .
0
1
2
3
4
5
2S , 2 P
2
2
2
1
1 1, D 1 1, F 1 1, G 1 1,....
1 2 ,2 2
2 2 ,3 2
3 2 ,4 2
2
2 ,1 2
3S , 3 P
3
3
3
1
0,1,2 , D1,2,3 , F2,3,4 , G 3,4,5 , . . . .
4S , 4 P
4
4
1 1 1 , D1 1 1 1 , F 1 1 1 1 , . . . .
1 12
1 2 ,2 2 ,3 2 ,4 2
2 ,1 2 ,2 2
2 ,1 2 ,2 2 ,3 2
5S , 5 P
5
5
5
2
1,2,3 , D0,1,2,3,4 , F1,2,3,4,5 , G 2,3,4,5,6 , . . . .
6S , 6 P
6
6
2 12
1 12 ,2 12 ,3 12 , D 12 ,1 12 ,2 12 ,3 12 ,4 12 , F 12 ,1 12 ,2 12 ,3 12 ,4 12 ,5 12 ,
6G
Septets
1 12 ,2 12 ,3 12 ,4 12 ,5 12 ,6 12 , . . . .
7S , 7 P
7
7
7
3
2,3,4 , D1,2,3,4,5 , F0,1,2,3,4,5,6 , G 1,2,3,4,5,6,7 , . . . .
The magnetic quantum numbers are usually not indicated unless the level is split by a magnetic
field. In the absence of such a perturbation, the energies of all levels with a given J are the same, and
are therefore (2J + 1)-fold degenerate.
Classical atomic spectroscopists have used the following hierarchial scheme to describe energy
states, combinations thereof, and transitions among such states, as given in Table 4.1.
Table 4.1. Hierarchy of designations.
Atomic
division
Specification
Statistical
weight g
Transition
State
Level
Specified by L, S, J , M, or L, S, M L , M S
Specified by L, S, J , e.g., 4S1 1
1
2J + 1
Component (of line)
Line
Term
Group of levels specified by L, S
2
(2S + 1)(2L + 1)
Multiplet
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56 / 4
S PECTRA
Table 4.1. (Continued.)
Atomic
division
Polyad
Configuration
Statistical
weight g
Specification
Group of terms from one parent term, and
with same multiplicity or S
Specified by n and l of all electrons
Transition
Supermultiplet
See text
Transition array
Nowadays, spectroscopists rarely use the term polyad. Very complicated level structures arise with
the filling of the 3d (iron group), 4d (palladium group), 5d (platinum group), 4 f (lanthanides), and 5 f
(actinides) subshells. Johansson refers to a subconfiguration for all of the levels that result from the
addition of an electron (nl) to a parent term. For example, if we use S p and L p to designate the spin
and orbital angular momentum of the parent, 3d 2 ( S p L p )nl has five subconfigurations corresponding to
the five allowed terms from d 2 . Similarly, 3d 4 ( S p L p )nl would have 16 subconfigurations. He uses the
term supermultiplet to mean all transitions between levels belonging to subconfigurations of opposite
parity [4].
4.2.1
Terms from Various Configurations
Table 4.2 gives the multiplicities and orbital angular momenta of the various terms arising in L S
coupling from the configurations listed [2, 3]. When a term can appear more than once, the number of
possible terms is written below the symbol. Complete shells s 2 , p 6 , d 10 , f 14 , etc., give rise to only 1 S
terms. They need not be considered for possible terms due to outer electrons.
Electrons with the same n and l are said to be equivalent. Terms arising from complementary
numbers of equivalent electrons are the same; e.g., terms from p 2 and p 4 are the same, since six
electrons complete the p shell.
The total weight of an electron configuration may be written [2]
g(l1w1 l2w2 · · · lnwn ) =
n 4li + 2
.
wi
i=1
Here, wi is the number of (equivalent) electrons with angular momentum li . A number of examples
are given below. If a single electron with angular momentum l is added to a parent with L p and S p ,
the total weight of the resulting terms of both resulting multiplicities is
g = (4S p + 2)[(b + 1)2 − a 2 ],
where b = |L p + l| and a = |L p − l|.
Often, atomic energy levels are not well described by a single electronic configuration. In such
cases, configuration interaction or configuration mixing is said to occur.
Table 4.2. Allowed terms for equivalent electrons.
Configuration
Terms
Total weight
Equivalent s electrons
s
s2
2S
1S
2
1
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4.3 E LECTRONIC C ONFIGURATIONS / 57
Table 4.2. (Continued.)
Configuration
Terms
Total weight
Equivalent p electrons
2 P0
p5
p4
p
p2
1SD
p3
2 P D0
4 S0
Equivalent d electrons
2D
d9
d8
d7
d
d2
d3
6
15
20
3P
1 S DG
10
45
120
3PF
2 P DFGH
4PF
2
d4
d6
1SDFGI
22
d5
3 P DFGH
2
2
210
5D
2
4 P DFG
2SP DFGH I
252
6S
3 2 2
f 13
f 12
f 11
f
f2
f3
2 F0
1 S DG I
14
91
364
3PFH
2 P D F G H I K L0
4SDFGI0
2 2 22
f4
f 10
1SDFGH I K L N
2 4
f5
f9
423
3 P DFGH I K L M
2
2 P D F G H I K L M N O0
4 S P D FG H I K L M0
4 57 67 553 2
f6
f8
f7
3 P DFGH I K L M N O
6 4 8 47 34 2 2
5SP DFGH I K L
6 5 9 7966 3 3
2 S P D F G H I K L M N O Q0
3 003
7F
3 2 322
4SP DFGH I K L M N0
2 5 71010 9 9 7 5 4 2
4.3
2 002
6 P F H0
2 34 4 332
1SP DFGH I K L M N Q
4
1 001
5SDFGI
32 4 3 422
6 P DFGH I0
3 432
8 S0
22 6 5 7 5 533
ELECTRONIC CONFIGURATIONS
Tables 4.3 and 4.4 give the electronic configurations for ground-level atoms [5]. The inner core of
electrons is not explicitly shown for heavier elements. Extensive tabulations of energy levels are
available [6, 7].
Table 4.3. Ground-level configurations.
Atom
1s
H 1
1
He 2
2
M
L
K
2s 2 p
O
3s 3 p 3d 4s 4 p 4d 5s
Ground
level
2S
1/2
2S
1/2
Li 3
2
1
Be 4
B 5
2
2
2
2
1
C 6
N 7
2
2
2
2
2
3
O 8
2
2
4
F
2
2
5
9
N
2 P0
1/2
4 S0
11/2
2 P0
11/2
1S
0
Atom
N
O
4f
5s 5 p 5d 5 f
Ag 47
1
Cd 48
2
In 49
2 1
1S
0
Sn 50
Sb 51
2 2
2 3
3P
0
Te 52
I 53
2 4
2 5
3P
1
Xe 54
2 6
Cs 55
2 6
P
Q
6s 6 p 6d 7s
Ground
level
2S
1/2
2 P0
1/2
4 S0
11/2
2 P0
11/2
1
2S
1/2
1S
0
3P
0
3P
2
1S
0
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58 / 4
S PECTRA
Table 4.3. (Continued.)
M
L
K
Atom
1s
2s 2 p
Ne 10
2
2
6
Na 11
2
2
6
2 1
Ne core
2 P0
1/2
4 S0
11/2
2 4
2 5
2 P0
11/2
2 6
2
2
6
2 6
Ca 20
Sc 21
Ti 22
V 23
18
Cr 24
Mn 25
Ar core
1
2
2
2
3
2
2
5
5
1
2
Fe 26
6
2
Co 27
7
2
Ni 28
8
2
Cu 29
2
2
6
2 6
10
Zn 30
Ga 31
28
6S
21/2
4F
41/2
2 2
2 3
Se 34
Br 35
2 4
2 5
2
6
2 6
10
2 6
2 6
1
Sr 38
Y 39
1
2
2
Zr 40
2
2
Nb 41
36
4
1
Mo 42
Tc 43
Kr core
5
5
1
2
Ru 44
Rh 45
7
8
1
1
Pd 46
10
2S
1/2
2D
11/2
6D
4F
41/2
8
Ce 58
1
2 6
1
2
1
2
4
5
2
2
3P
2
Sm 62
Eu 63
6
7
2
2
1S
0
Gd 64
7
1
Tb 65
9
2
Dy 66
Ho 67
10
11
2
2
3F
2
Er 68
Tm 69
12
13
2
2
7S
3
Yb 70
Lu 71
14
14
1
2
2
Hf 72
14 2 6
2
2
Ta 73
3
2
W 74
4
2
3F
4
Re 75
46 + 22
5
2
1S
0
Os 76
Ir 77
6
7
2
2
3P
0
Pt 78
9
1
Au 79
14 2 6 10
1
3P
2
Hg 80
Tl 81
1S
0
Pb 82
Bi 83
1S
0
Po 84
At 85
2 4
2 5
3F
2
Rn 86
2 6
7S
3
5F
5
1S
0
1S
0
2D
1 G0
4
4I0
41/2
6 H0
21/2
8 S0
31/2
46 + 32
14 2 6 10
Ra 88
Ac 89
46 + 32
5I
4
7F
0
9D
2
6 H0
71/2
5I
4I0
71/2
2 F0
31/2
2D
8
3H
6
1S
0
11/2
3F
2
4F
11/2
5D
0
5D
4
3D
3
6S
21/2
4F
11/2
2S
1/2
2
2 1
Fr 87
11/2
2
1S
0
4
Ground
level
2
Nd 60
Pm 61
1/2
6S
21/2
Ba 56
Q
6s 6 p 6d 7s
2
4 S0
11/2
2 P0
11/2
5s 5 p 5d 5 f
3
2S
1/2
2 P0
1/2
4f
Pr 59
5D
1
As 33
2
11/2
4F
11/2
2
2 1
Ge 32
Kr 36
Rb 37
2D
O
3P
0
2S
1/2
1
P
N
La 57
1S
0
2 2
2 3
Ar 18
K 19
Atom
2S
1/2
1
Al 13
S 16
Cl 17
Ground
level
1S
0
2
10
O
3s 3 p 3d 4s 4 p 4d 5s
Mg 12
Si 14
P 15
N
2 P0
1/2
2 2
2 3
4 S0
11/2
2 P0
11/2
2 6
1
2
2
2D
4K
2
2
1
2
2
U 92
3
1
2
3P
0
3P
2
1S
0
2S
1/2
1
Th 90
Pa 91
1S
0
1S
0
11/2
51/2
3F
2
5 L0
6
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4.3 E LECTRONIC C ONFIGURATIONS / 59
Table 4.4. Transuranic elements.
O
Atom
P
Q
5f
6s
6p
6d
7s
1
2
Np
93
4
2
6
Pu
Am
94
95
6
7
2
2
6
6
Cm
Bk
96
97
7
9
2
2
6
6
Cf
Es
98
99
10
11
2
2
6
6
2
2
Fm
Md
100
101
12
13
2
2
6
6
2
2
No
Lr
102
103
14
14
2
2
6
6
2
2
2
2
1
2
2
1
Ground
level
6L
51/2
8 S0
31/2
6H0
71/2
4I0
71/2
2 F0
31/2
2D
7F
0
9 D0
2
5I
8
3H
6
1S
0
31/2
Table 4.5 of first ions (Sc II, etc.), is restricted to those ions whose ground levels differ from those
of the preceding atom. Table 4.5 gives outer and incomplete shells only.
Table 4.5. First ions.
Element
Configuration
Sc
Ti
3d4s
3d 2 4s
V
Cr
3d 4
3d 5
Mn
Fe
3d 5 4s
3d 6 4s
Co
Ni
3d 8
3d 9
Cu
3d 10
Zr
4d 2 5s
Nb
4d 4
Mo
4d 5
Tc
4d 5 5s
Ru
4d 7
Rh
Pd
4d 8
4d 9
Ground
level
Element
Configuration
La
Ce
5d 2
4 f 5d 2
0
Pr
Nd
4 f 3 6s
4 f 4 6s
7S
3
Pm
Sm
4 f 5 6s
4 f 6 6s
3F
4
Eu
Gd
4 f 7 6s
4 f 7 5d6s
1S
0
Tb
4 f 9 6s
Dy
4 f 10 6s
Ho
4 f 11 6s
Er
4 f 12 6s
Tm
4 f 13 6s
Yb
4 f 14 6s
3D
4F
11/2
5D
6S
21/2
6D
2D
41/2
1
21/2
4F
11/2
5D
6S
21/2
4F
41/2
2D
21/2
0
7S
3
3F
4
Ground
level
4 H0
31/2
6I
31/2
8F
1/2
10 D 0
21/2
6I
81/2
4H
61/2
2S
1/2
Element
Configuration
Ta
W
5d 3 6s
5d 4 6s
6D
5I0
4
Re
Os
5d 5 6s
5d 6 6s
6D
7 H0
2
Ir
Pt
5d 7 6s
5d 9
2D
9 S0
4
Au
5d 10
7 H0
8
Th
6d 2 7s
Pa
5 f 2 7s 2
U
5 f 3 7s 2
Np
5 f 5 7s
Pu
5 f 6 7s
Am
5 f 7 7s
3F
2
5I0
8
3 F0
4
Ground
level
5F
1
1/2
7S
3
41/2
21/2
4F
11/2
4I0
41/2
8F
1/2
5F
5
1S
0
3H
4
7 H0
2
9 S0
4
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4.4
4.4.1
Page 60
S PECTRA
SPECTRUM LINE INTENSITIES
Definitions
We use the symbol =
D to mean “dimensionally equal to” or “has dimensions of”; =
D 0 means
“dimensionless.”
g = (dimensionless) statistical weight for a level = 2J + 1. Subscripts denote levels.
f = (dimensionless) oscillator strength, or simply f value. Unless otherwise stated, this
is the absorption oscillator strength f abs , related to the emission oscillator strength
f em (which is often taken to be negative) by g1 f abs = −g2 f em . Here, g1 and g2 are
the statistical weights of the lower level and upper level, respectively.
g f = weighted oscillator strength = g1 f 12 = −g2 f 21 . g f is symmetrical between
emission and absorption.
A = Einstein’s A =
D s−1 ; spontaneous transition probability (for a downward transition).
B12 , B21 = Einstein’s B; induced transition probability upward and downward. Bu ν =
probability of transition where u ν is the radiation energy density at the frequency
D s−1 . The B coefficients are sometimes defined with
ν of the transition. Then Bu ν =
specific intensity Iν , whence B Iν =
D s−1 .
S = line strength. Sum of the matrix elements of the electric dipole operator =
D e2 |x|2 .
Also used for higher-order radiation (see below).
= s−1 ). γcl is the full width at half maximum (FWHM)
γcl = classical damping constant (D
in units of circular frequency (ω = 2π ν) of an absorption line due to a classical
oscillator.
γ2 = reciprocal
mean
2
life of level = 1 A21 + 1 B21 u(ν21 ) + 3 B23 u(ν23 )+ collision terms, where level 1 is below
and level 3 is above 2.
γ = damping constant = γ1 + γ2 for transition 1 → 2. It is convenient to define damping
constants γν and γλ , for use when profiles are expressed in frequency or wavelength
units. Then γν /ν = γλ /λ = γ /ω.
= cm2 ) near an absorption line. Note: σλ = σν . Traditionally,
σν = atomic cross section (D
σν is written in terms of γ , not γν . Often aν or αν is used for atomic cross sections.
N1 = number of atoms per unit volume in level 1 (the lower level).
κ = N1 σν , the line absorption coefficient, which must be corrected for stimulated
emission: κcorr = κ[1 − exp(−hν/kT )].
ν0 = frequency at line center.
Ri , Rf = Initial and final radial wave functions of the active electron. For bound levels, Ri,f =
D
cm−3/2 . Commonly, r Ri,f ≡ Pi,f , where r = radius.
σ = proportional to radial transition moment (see below), not related to σν or σλ .
S = multiplet strength, scale as given in Table 4.9.
E = energy emitted due to spontaneous, bound–bound transitions in all directions, per
unit volume and time.
εRy = photon energy in rydbergs (e2 /2a0 = 2π 2 m e e4 / h 2 ).
n ∗ = effective principal quantum number; describes the energy of an atomic level.
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4.4 S PECTRUM L INE I NTENSITIES / 61
4.4.2
Formulas
For a spectral line that arises from a transition between levels αL S J and α L S J , the line strength for
a dipole transition is defined as
S=
|αL S J M|exq |α L S J M |2
(4.1)
M M q
=
M M q
|αL S J M|erCq(1) |α L S J M |2 .
(4.2)
Here, α and α stand for unspecified quantum numbers. q runs from 1 to 3 for the three components of
the position vector of the active electron, or equivalently, the three components of the spherical tensor
(1)
of rank 1, rCq . The C’s are proportional to the spherical harmonics of corresponding order:
4π
(1)
(1)
(1)
(l)
Cm ≡
Y m;
x 2 + y 2 + z 2 = |rC−1 |2 + |rC0 |2 + |rC+1 |2 .
2m + 1 l
Consider a simple electronic transition, where there is a single active optical electron (L p l
→ L p l ), where L p stands for the orbital angular momentum of the parent. The greater of l and l is usually written l> . In a nonrelativistic, single-configuration approximation, the line strength can be
written with the help of two Wigner 6 − j symbols [2]:
S = (2J + 1)(2J + 1)(2L + 1)(2L + 1)
2 2 ∞
2
Lp l L
L S J
2
× l>
Rir Rfr dr .
J 1 L
1 L l
0
The line strength S is often taken to be in atomic units (e = a0 = m e = 1), but that is not the case in
the following relations (the B’s used here are defined with energy density; we use m ≡ m e ):
g2 A21 = g2
8π hν 3
8π hν 3
64π 4
B
=
g
B
=
S12 or 21
21
1
12
c3
c3
3hλ3
= 3γcl g1 f 12 = −3γcl g2 f 21 =
γcl =
8π 2 e2 ν 2
8π 2 e2
=
,
3mc3
3mcλ2
g f = g1 f 12 = −g2 f 21 =
g1 B12 = g2 B21 =
= N2
σν dν =
κν =
mhν
8π 2 mν
g
B
=
S12 ,
1
12
π e2
3he2
8π 3
S12 ,
3h 2
E = N2 A21 hν =
8π 2 e2 ν 2
g1 f 12 ,
mc3
N2 8π 2 e2 hν 3
8π 2 e2 hν 3
g
f
=
N
(− f 21 )
1
12
2
g2
mc3
mc3
8π 2 e2 h
(− f 21 ),
mλ3
πe2
f abs ,
mc
γ
N1
πe2
=
D cm−1 ,
f abs 2
mc
4π (ν − ν0 )2 + (γ /4π )2
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62 / 4
Page 62
S PECTRA
κλ =
λ20 N1
γλ
πe2
=
f
D cm−1 ,
abs
2π (λ − λ0 )2 + (γλ /2)2
mc2
κν0 =
4 π e2
f abs N1 ,
γ mc
κλ 0 =
2e2 λ20
N1 f abs ,
mc2 γλ
πe2
π e2 2
f abs N1 ,
κλ dλ =
λ f abs N1 ,
mc
mc2 0
∞
2 ∞
2
nl 2
3
(Rn l ) =
Ri Rfr dr =
Pir Pf dr ,
κν dν =
0
(Rnnl l )2
=
(1)
(l> P ll )2
0
(notation of [2]),
1
(Rnnl l )2 ,
2 −1
4l>
χH
=Z
.
χion + χ p − χnl
σ2 =
n ∗nl
The effective principal quantum number n ∗ may be defined for each level with excitation χnl . The
core, or parent excitation, χ p , if present, must be added to the ionization energy χion . For example, the
2s 2 2 p 2 (1 D)3s level of N I at 12.36 eV (99 663 cm−1 ) is built on an excited parent in N II. Therefore,
in calculating n ∗ , one must add χ p = 1.90 eV to the (first) ionization energy 14.53 eV of N I.
4.4.3
Numerical Relations
The following relations are based on the above formulas, which are derived from an approximate,
nonrelativistic radiation theory. The numerical factors are given only to four figures. Physical constants
are from [8]. Note that the line strength S is in atomic units in the following:
g f = 303.8S/λ = 1.499 × 10−16 g2 Aλ2
(λ in Å, A in s−1 ),
S = 0.003 292g f λ = 4.936 × 10−19 g2 Aλ3
(λ in Å, A in s−1 ),
3
g2 A = 2.026 × 1018 S/λ3 = 2.678 × 109 εRy
S
= 0.6670 × 1016 g f /λ2
πe2 /mc = 0.026 54 cm2 s−1 ,
(λ in Å, A in s−1 ),
π e2 /mc2 = 8.853 × 10−13 cm,
γcl = 2.223 × 1015 /λ2 s−1 (λ in Å),
8π hν /c3 = 8π h/λ3 = 0.1665/λ3 (λ in Å),
a02 e2 = 6.460 × 10−36 cm2 esu2 .
3
4.4.4 Forbidden Transitions: Electric Quadrupole (E2)
and Magnetic Dipole (M1)
In astronomical usage, a line is called “forbidden” when it violates the rules for an electric dipole
(E1)-induced transition. The lines are designated with a bracket notation, e.g., [O III] for transitions
among the low-level, even-parity states of doubly ionized oxygen. E1 transitions with S = 0 occur
frequently and in the astronomical literature are often written with a single bracket. For example,
Sp.-V/AQuan/1999/10/29:07:01
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4.4 S PECTRUM L INE I NTENSITIES / 63
the transition 2s 2 1 S0 –2s2 p 3 P10 at λ1909 is written C III]. Such a transition is sometimes called
semiforbidden, or spin forbidden. In complex spectra the rule against intercombination of multiplicities
is violated so frequently that this notation is not particularly useful, and it is rarely employed. In the
= 0), and σ is the wave number of the photon (D
= cm−1 ).
formulas below, α is the fine structure constant (D
The gyromagnetic ratio of the electron spin has been assumed to be 2.000 [2]. For magnetic dipole
radiation,
4π 2 he2 σ 3 |γ J M|Jq(1) + Sq(1) |γ J M |2
3m 2 c2 q M M = 2.697 4 × 10−11 σ 3
|γ J M|Jq(1) + Sq(1) |γ J M |2 .
g2 A21 =
q M M
For electric quadrupole radiation, we show the explicit sum over i electrons. These sums are
(1)
(1)
implicit in the symbols Jq and Sq above. In practice, only one electron is important. We have
64π 4 e2 a04 σ 5 |γ J M|ri2 Cq(2) (i)|γ J M |2
15h
q M M i
−22 5
= 1.120 0 × 10 σ
|γ J M|ri2 Cq(2) (i)|γ J M |2 .
g2 A21 =
q M M i
4.4.5
Selection Rules
Selection rules for atomic transitions are summarized in Table 4.6, including rules for L S coupling.
When levels are not accurately described by single values of L and S, rules involving these quantum
numbers are no longer valid. However, even in complex atoms it is often the case that transitions that
violate the L S selection rules are weak. Configuration interaction can cause the selection rule on l
to be violated. An example is found in Si I, λ5621.61 of multiplet 17.01 [9]. This appears to be a jump
from 3 p4s to 3 p4 f (l = 3). The transition occurs because the 3 p4s configuration is mixed with
3 p3d.
Table 4.6. Selection rules for atomic transitions.a
Electric Dipole (E1)
l = ±1, parity change
n arbitrary
J = 0 ± 1, J = 0 ↔ J = 0
L = 0, ±1, L = 0 ↔ L = 0
S = 0
Magnetic Dipole (M1)
J = 0, ±1, J = 0 ↔ J = 0
M = 0, ±1
l = 0, n = 0, for all electrons
S, L = 0
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64 / 4
S PECTRA
Table 4.6. (Continued.)
Electric Quadrupole (E2)
J = 0, ±1, ±2, 0 ↔ 0, 12 ↔ 12 , 0 ↔ 1
l = 0, ±2, no parity change
n arbitrary
M = 0, ±1, ±2
S = 0, L = 0, ±1, ±2, L = 0 ↔ L = 0, 1
Note
a Rules for L and S hold for L S coupling, while those
for J are independent of the coupling conditions.
4.4.6
Radial Integrals and Related Calculations
The Coulomb approximation [10, 11] to the radial integral for a single electron is still of heuristic
interest. It uses the effective principal quantum numbers n ∗ . Let Z be the charge seen by the active
electron at large distances from the nucleus. Z = 1 for a neutral species, 2 for a first ion, etc. Set
a = (Z /n ∗ ). The normalizations of the wave functions in the Coulomb approximation are
1
Z
N= ∗
,
n
(n + l + 1)(n − l)
where is the gamma function. We shall use numbers 1 and 2 to distinguish upper and lower levels in
the relations below (as above, l> means the greater of the two values, l1 and l2 ):
σ =
1
2 −1
4l>
N1 (n ∗1 , l1 )N2 (n ∗2 , l2 )
∗
∗
max
(2a1 )n 1 (2a2 )n 2 (a1 + a2 )
n ∗1 +n ∗2 +2
Gp
p=0
p
C p−q (1)Cq (2).
q=0
The coefficients G and C are easily obtained from recurrence relations:
C0 = 1,
Ck (i) = −
(−n i∗ − li + k − 1)(li − n i∗ + k) a1 + a2
Ck−1 (i),
k
2ai
G 0 = (n ∗1 + n ∗2 + 2),
Gk =
i = 1, 2,
G k−1
.
n ∗1 + n ∗2 + 2 − k
For integral n ∗ ’s the coefficients C are 0 for k above n−l−1. Then, if the sum includes all nonvanishing
terms, the results are identical with those well known for hydrogen and hydrogenlike ions. The
Coulomb approximation usually gives good results when n ∗nl > l + 1 with max < n ∗1 + n ∗2 − 1.
Useful tables are given in [11].
Kurucz and Bell [12] have made extensive calculations of radial integrals for complex atoms using
scaled Thomas–Fermi–Dirac potentials. Results from the international “Opacity Project” are becoming
available [13].
4.4.7
Sum Rules
The Kuhn–Thomas–Reiche f -sum rule states
f 21 +
f 23 = z,
1
3
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4.5 R ELATIVE S TRENGTHS W ITHIN M ULTIPLETS / 65
where the summations are for level 1 below the selected level 2, and 3 above that level (including
an integral over continuum).
z is the number of atomic or ionic electrons. f 21 is negative and hence
for upward transitions 3 f 23 ≥ z. The rule is rigorous for nonrelativistic quantum mechanics, but
the sum includes physically unrealistic states. Restricted and approximate forms of the sum rule are
of more practical importance, as for more complex spectra where the lines concerned are mainly the
lowest members of their series and contain most of the total oscillaator strength.
The Wigner–Kirkwood rule for a one-electron jump [2] is
1 l(2l − 1)
for l → l − 1,
3 2l + 1
1 (l + 1)(2l + 3)
f =
for l → l + 1
3
2l + 1
f =−
(l is the orbital quantum number); for example,
f = − 19 ,
f = 1,
f = − 25 ,
f = 10
9 ,
p → ns,
s → np,
d → np,
p → nd.
The above rule may sometimes be used for complex spectra, but it applies precisely for hydrogen.
The J file and J group sum rules refer to a transition array, e.g., sp ↔ pp. A J file refers to
all transitions that begin or end on a specified level. Let all line strengths S(γ L S J , γ L S J ) within
a transition array be entered in an i × f matrix, with i being the number of initial levels and f the
number of final levels. A J file is any single row or column in this matrix. The J file sum rule states
that
S(γ L S J , γ L S J ) ∝ 2J + 1
J
and
S(γ L S J , γ L S J ) ∝ 2J + 1.
J
These two rules are independent of the coupling conditions, but apply only to simple transition arrays,
where either the moving electron has no equivalent congeners or the electron configuration with the
summed J or J values does not contain equivalent electrons.
A J group consists of all lines in a transition array connecting a level with a given J (e.g., initial)
with one with a given J (e.g., final). The J group sum rule states that the sum of the strengths of the
lines in a J group are independent of the coupling conditions.
4.5
RELATIVE STRENGTHS WITHIN MULTIPLETS
Table 4.9 gives the relative strengths of lines in multiplets. The notation used here is for L S-coupling
multiplets for a transition L S J → L S J . It is important to note that the relative strengths apply
much more generally to any case where two angular momenta, say j1 , and j2 , couple to a third
j3 , where j2 commutes with the dipole operator er [2]. As a result of this generality, these same
relative intensity tables may be used for lines in a hyperfine “multiplet” by the following substitution
of quantum numbers: J → F, L → J , and S → I , where F is the total angular momentum including
the nuclear spin I . Similarly, the relative intensities of what were once called “supermultiplets” may
Sp.-V/AQuan/1999/10/29:07:01
66 / 4
Page 66
S PECTRA
be computed by making the following exchange: J → L, L → l, and S → L p . Here, we assume a
single optical electron with angular momentum l that couples to a parent core with angular momentum
L p . It therefore turns out that tables [14] giving relative multiplet strengths are unnecessary.
The entries are all proportional to
L
(2J + 1)(2J + 1)W (L L J J ; 1S) = (2J + 1)(2J + 1) J
2
S
1
J
L
2
,
Table 4.7. Normal multiplets S P, P D, D F, etc.
Jm
Jm
Jm
Jm
−1
−2
−3
−4
Jm
Jm − 1
Jm − 2
x1
y1
x2
z1
y2
x3
Jm − 3
Jm − 4
z2
y3
x4
z3
y4
where the W is a Racah coefficient, and the symbol on the right is a Wigner 6 − j symbol [2]. We
normalize so that the sum of the entries for a given multiplet is S = (2S + 1)(2L + 1)(2L + 1). The
entries are therefore proportional to the line strengths as defined above and do not contain wavelengthdependent factors. Therefore, they are only approximately proportional to relative line intensities in
real (L S-coupling) multiplets.
The following qualitative rules describing the intensities in L S multiplets are of practical value.
The most intense lines are those where L and J change in the same sense, for example, J → J + 1
while L → L + 1. These strong lines are called the principal, or sometimes diagonal, lines of the
multiplet. In Tables 4.7 and 4.8, their intensities are called x1 , x2 , x3 , . . . . The intensity of the strongest
line on the (principal) diagonal is called x1 , and it belongs to the line involving the largest J value,
called Jm below. With a few exceptions that may be seen in Table 4.9, the intensities diminish down
the diagonal. Lines that fall off the main diagonal are called satellite, or off-diagonal, lines. There
are two kinds of multiplets to consider, the symmetrical ones (P → P, D → D, etc.), and “normal
multiplets” (L → L + 1, such as S → P or P → D). Since the line strength factors are independent
of which level is upper and which is lower, we are free to choose Jm to belong to the largest L. For
the symmetrical multiplets, we call the intensities of the lines for which Jm → Jm − 1, y1 , those for
which Jm − 1 → Jm − 2, y2 , etc. Lines with identical intensities fall on the complementary side
of the diagonal, as shown below. In normal multiplets, there are a second series of satellites with
Jm − 1 → Jm − 2, etc., which are designated z 1 , etc. We remark that in a breakdown of L S coupling,
the weaker lines typically deviate more strongly from the L S intensities, so that a calculation in L S
coupling may yield reasonable results for a line on the main diagonal, but could be badly off for a
satellite.
Table 4.8. Symmetrical multiplets P P, D D, etc.
Jm
Jm − 1
Jm − 2
Jm − 3
Jm
Jm − 1
x1
y1
y1
x2
y2
Jm − 2
Jm − 3
y2
x3
y3
y3
x4
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4.5 R ELATIVE S TRENGTHS W ITHIN M ULTIPLETS / 67
Table 4.9. Intensities in LS-coupling multiplets.
Multiplicity
1
2
3
4
5
6
7
8
9
10
11
SP
S=
x1
y1
z1
3
6
9
12
15
18
21
24
27
30
33
3.00
4.00
2.00
5.00
3.00
1.00
6.00
4.00
2.00
7.00
5.00
3.00
8.00
6.00
4.00
9.00
7.00
5.00
10.00
8.00
6.00
11.00
9.00
7.00
12.00
10.00
8.00
13.00
11.00
9.00
9
18
27
36
45
54
63
72
81
90
99
9.00
10.00
4.00
11.25
2.25
12.60
1.60
1.00
14.00
1.25
2.25
15.43
1.03
3.60
16.88
0.88
5.00
18.33
0.76
6.43
19.80
0.67
7.87
21.27
0.61
9.33
22.75
0.55
10.80
2.00
3.75
3.00
5.40
5.00
7.00
6.75
8.57
8.40
10.13
10.00
11.67
11.57
13.20
13.13
14.73
14.67
16.25
16.20
15
30
45
60
75
90
105
120
135
150
165
15.00
18.00
10.00
21.00
11.25
5.00
24.00
12.60
5.00
27.00
14.00
5.25
30.00
15.43
5.60
33.00
16.88
6.00
36.00
18.33
6.43
39.00
19.80
6.88
42.00
21.27
7.33
45.00
22.75
7.80
2.00
3.75
3.75
5.40
6.40
5.00
7.00
8.75
6.75
8.57
10.97
8.40
10.13
13.13
10.00
11.67
15.24
11.57
13.20
17.33
13.13
14.73
19.39
14.67
16.25
21.45
16.20
0.25
0.60
1.00
1.00
2.25
3.00
1.43
3.60
6.00
1.88
5.00
9.00
2.33
6.43
12.00
2.80
7.88
15.00
3.27
9.33
18.00
3.75
10.80
21.00
PP
S=
x1
x2
x3
y1
y2
PD
S=
x1
x2
x3
y1
y2
y3
z1
z2
z3
DD
S=
x1
x2
x3
x4
x5
25
50
75
100
125
150
175
200
225
250
275
25.00
28.00
18.00
31.11
17.36
11.25
34.29
17.29
8.00
5.00
37.50
17.50
6.25
1.25
40.74
17.88
5.14
0.22
2.22
44.00
18.38
4.37
5.00
47.27
18.94
3.81
0.14
8.00
50.56
19.56
3.38
0.49
11.11
53.85
20.21
3.03
0.95
14.29
57.14
20.89
2.75
1.50
17.50
y1
y2
y3
y4
S=
x1
x2
x3
x4
x5
y1
y2
y3
y4
y5
2.00
3.89
3.75
5.71
7.00
5.00
7.50
10.00
8.75
5.00
9.26
12.86
12.00
7.78
11.00
15.63
15.00
10.00
12.73
18.33
17.86
12.00
14.44
21.00
20.63
13.89
16.15
23.64
23.33
15.71
17.86
26.25
26.00
17.50
35
70
105
140
175
210
245
280
315
350
385
35.00
40.00
28.00
45.00
31.11
21.00
50.00
34.29
22.40
14.00
55.00
37.50
24.00
14.00
7.00
60.00
40.74
25.71
14.40
6.22
65.00
44.00
27.50
15.00
6.00
70.00
47.27
29.33
15.71
6.00
75.00
50.56
31.20
16.50
6.11
80.00
53.85
33.09
17.33
6.29
85.00
57.14
35.00
18.20
6.50
2.00
3.89
3.89
5.71
7.31
5.60
7.50
10.50
10.00
7.00
9.26
13.54
13.89
11.38
7.78
11.00
16.50
17.50
15.00
10.00
12.73
19.39
20.95
18.29
12.00
14.44
22.24
24.30
21.39
13.89
16.15
25.06
27.58
24.38
15.71
17.86
27.86
30.80
27.30
17.50
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68 / 4
S PECTRA
Table 4.9. (Continued.)
1
2
3
z1
z2
z3
z4
z5
4
5
6
7
0.11
0.29
0.40
0.50
1.00
1.00
0.74
1.71
2.40
2.22
8
9
10
11
1.00
2.50
4.00
5.00
5.00
1.27
3.33
5.71
8.00
10.00
1.56
4.20
7.50
11.11
15.00
1.85
5.09
9.33
14.29
20.00
2.14
6.00
11.20
17.50
25.00
FF
S=
x1
x2
x3
x4
x5
x6
x7
49
98
147
196
245
294
343
392
441
490
539
49.00
54.00
40.00
59.06
41.17
31.11
64.17
42.67
28.90
22.40
69.30
44.36
27.56
17.50
14.00
74.45
46.20
26.74
14.40
7.62
6.22
79.62
48.12
26.25
12.25
4.37
0.87
84.81
50.13
25.98
10.67
2.50
3.50
90.00
52.18
25.88
9.45
1.36
0.49
7.88
95.20
54.28
25.90
8.48
0.67
1.60
12.60
100.41
56.41
26.00
7.70
0.26
3.06
17.50
11.38
17.50
19.25
17.50
13.13
7.00
13.19
20.68
23.33
22.00
17.50
10.50
15.00
23.82
27.30
26.25
21.39
13.13
16.80
26.92
31.18
30.33
25.00
15.40
18.59
30.00
35.00
34.30
28.44
17.50
2.00
3.94
3.89
5.83
7.50
5.60
7.70
10.94
10.50
7.00
9.55
14.26
15.00
12.60
7.78
63
126
189
252
315
378
441
504
567
630
693
63.00
70.00
54.00
77.00
59.06
45.00
84.00
64.17
48.21
36.00
91.00
69.30
51.56
37.50
27.00
98.00
74.45
55.00
39.29
27.00
18.00
105.00
79.62
58.50
41.25
27.50
16.88
9.00
112.00
84.81
62.05
43.33
28.29
16.50
7.50
119.00
90.00
65.63
45.50
29.25
16.50
6.88
126.00
95.20
69.23
47.73
30.33
16.71
6.60
133.00
100.41
72.86
50.00
31.50
17.06
6.50
2.00
3.94
3.94
5.83
7.62
5.79
7.70
11.14
10.94
7.50
9.55
14.55
15.71
13.71
9.00
11.38
17.88
20.25
19.25
15.63
10.13
13.19
21.15
24.62
24.38
21.21
16.00
10.50
15.00
24.38
28.88
29.25
26.25
20.63
13.13
16.80
27.57
33.04
33.94
30.95
24.69
15.40
18.59
30.74
37.14
38.50
35.44
28.44
17.50
0.06
0.17
0.21
0.30
0.56
0.50
0.45
1.00
1.29
1.00
0.62
1.50
2.25
2.50
1.88
0.81
2.05
3.33
4.29
4.50
3.50
1.00
2.63
4.50
6.25
7.50
7.87
7.00
1.20
3.23
5.73
8.33
10.71
12.60
14.00
1.41
3.86
7.00
10.50
14.06
17.50
21.00
y1
y2
y3
y4
y5
y6
FG
S=
x1
x2
x3
x4
x5
x6
x7
y1
y2
y3
y4
y5
y6
y7
z1
z2
z3
z4
z5
z6
z7
4.6
WAVELENGTHS AND WAVE NUMBERS
Angstrom units (Å) and microns (micrometers, µm) are used for wavelengths in the tables presented
in the following sections. Astronomers often indicate wavelengths in angstrom units by the λ symbol.
Wavelengths may be truncated after the last unit of an angstrom or they may be rounded off. We have
Sp.-V/AQuan/1999/10/29:07:01
Page 69
4.7 ATOMIC O SCILLATOR S TRENGTHS FOR A LLOWED L INES / 69
tried to follow the latter procedure here, but there is no uniformity in the literature. Thus a line of Ca I
at 4 226.73 Å might be called either λ4226 or λ4227. Wave numbers are almost always given in units
of cm−1 , although reciprocal microns are occasionally used. Common symbols for wave numbers are
ν, ν̃, and σ .
Many workers use the SI unit nanometer (nm) for wavelengths, 1 nm = 10 Å. Wavelengths here
are given “in air” for (air) wavelengths greater than 2 000 Å. Air and vacuum wavelengths are related
by the index of refraction of air, n: λvacuum = nλair . An extensive tabulation [15] is based on Edlen’s
formula for n,
n = 1 + 6 432.8 × 10−8 +
2 949 810
25 540
+
,
8
2
146 × 10 − σ
41 × 108 − σ 2
where σ is the wave number in cm−1 . This formula suffices for conversions from air to vacuo when no
more than eight-figure accuracy is desired [16]. For shorter wavelengths reciprocal wave numbers, or
“vacuum wavelengths” are used. With the advent of space astronomy, some workers have suggested
the exclusive use of vacuum wavelengths, but this has not been adopted here.
Reader and Corliss [17] give a modern table of wavelengths of the chemical elements. They include
lines that are suitable for use in calibration of most spectrographs. Extensive references to wavelength
standards are given by Wiese and Martin [18].
4.7
ATOMIC OSCILLATOR STRENGTHS FOR ALLOWED LINES
Atomic hydrogen is considered separately here, in the nonrelativistic approximation in Table 4.10.
Exact numerical values have been available from the early days of quantum mechanics [19]; they
remain of heuristic as well as of practical value. For most cases of astrophysical interest, it is
permissible to ignore the electron spin in hydrogen. Each level with a given l is then (2l + 1)- rather
than 2(2l + 1)-fold degenerate, and the weight of all states belonging to a principal quantum number
n is n 2 rather than 2n 2 . The corresponding partition function at low temperatures is then 1 and not 2;
this must be used in the Boltzmann and Saha formulas when absorption or emission coefficients are
calculated. Authors sometimes use a notation that is valid if spin is ignored with statistical weights that
take spin into account [19, 20]. Transitions may be designated nl → n l , e.g., 1s → 2 p for Lyman α
n l )2 . The values for S in [20] and [21]
in absorption. If spin is ignored, then the line strength S = l> (Rnl
allow for the spin degeneracy and are twice this value. The Einstein coefficients, line strengths, and f
values must be defined in such a way that the intensities or equivalent widths of lines do not depend
on whether spin is included in the level-counting scheme. For example, consider the equivalent width
Wλ of a weak hydrogen absorption line when light passes through a uniform slab of thickness H . If
we use the Boltzmann formula to express the population of the lower level Ni as a function of the total
population of neutrals N T , we have
Wλ =
π e2 2
gi f ik
λ0 N T
[1 − exp(hν/kT )]e−χi /kT H.
2
u(T )
mc
Table 4.10. Radial integrals and absorption oscillator strengths for hydrogen.
Line
Lα
Lβ
Lγ
Transition
Wavelength (Å)
1s–2 p
1s–3 p
1s–4 p
1 215.67
1 025.72
972.54
n l )2
(Rnl
1.664 79
0.266 968
0.092 771
f abs
0.416 2
0.079 10
0.028 99
Sp.-V/AQuan/1999/10/29:07:01
Page 70
70 / 4
S PECTRA
Table 4.10. (Continued.)
n l )2
(Rnl
Line
Transition
Wavelength (Å)
f abs
Hα
Hα
Hα
2s–3 p
2 p–3s
2 p–3d
6 562.74
6 562.86
6 562.81
9.393 1
0.880 6
22.543 4
0.434 9
0.013 59
0.695 8
Hβ
Hβ
Hβ
2s–4 p
2 p–4s
2 p–4d
4 861.29
4 861.35
4 861.33
1.644 4
0.146 2
2.923 1
0.102 8
0.003 045
0.121 8
The factor [1 − exp(hν/kT )] allows for stimulated emission.
Spin doubles the value of all statistical weights and the partition function u(T ). Therefore, the sum
of the g f ’s for transitions including spin must be double the corresponding sum of the g f ’s with spin
ignored (gl = 2l + 1), in order to keep Wλ the same.
We use the convention that when a double subscript is written for an f or A value, the first subscript
belongs to the initial level. A few authors follow a convention from atomic spectroscopy that the lower
level is written first. Spin is ignored in calculating the absorption f values in Table 4.10 condensed
from [21].
It is also possible to ignore the l degeneracy of hydrogen, so that only transitions of the form n ↔ n are considered. Let n and l be the initial levels and let n and l be the final ones. Then one defines
average values of A as follows:
Ann = (1/n 2 )
(2l + 1)Anl→n l .
ll For example,
A32 = 19 (1A3s→2 p + 3A3 p→2s + 5A3d→2 p ).
A similar definition holds for the absorption f nn , but with the weights for the initial, lower level. Thus:
f 23 = 14 (1 f 2s→3 p + 3 f 2 p→3s + 3 f 2 p→3d ).
Data for the major series in hydrogen from [21] are given in Table 4.11.
Table 4.11. Average Einstein A’s and absorption f ’s.
Line
Transition
Wavelength (Å)
Lα
Lβ
Lγ
L limit
Hα
Hβ
Hγ
Hδ
H
H8
Hlimit
Pα
Pβ
Pγ
Plimit
1–2
1–3
1–4
1–∞
2–3
2–4
2–5
2–6
2–7
2–8
2–∞
3–4
3–5
3–6
3–∞
1 215.67
1 025.72
792.54
911.8
6 562.80
4 861.32
4 340.46
4 101.73
3 970.07
3 889.05
3 646
18 751.0
12 818.1
10 938.1
8 204
A (s−1 )
f abs
4.699 × 108
5.575 × 107
1.278 × 107
0.416 2
7.910 × 10−2
2.899 × 10−2
4.410 × 107
8.419 × 106
2.530 × 106
9.732 × 105
4.389 × 105
2.215 × 105
0.640 7
0.119 3
4.467 × 10−2
2.209 × 10−2
1.270 × 10−2
8.036 × 10−3
8.986 × 106
2.201 × 106
7.783 × 105
0.842 1
0.150 6
5.584 × 10−2
Sp.-V/AQuan/1999/10/29:07:01
Page 71
4.7 ATOMIC O SCILLATOR S TRENGTHS FOR A LLOWED L INES / 71
Table 4.11. (Continued.)
Line
Bα
Bβ
Bγ
Blimit
Transition
4–5
4–6
4–7
4–∞
Wavelength (Å)
40 512.0
26 252.0
21 655.0
14 584
A (s−1 )
f abs
2.699 × 106
1.038
0.179 3
6.549 × 10−2
7.711 × 105
3.041 × 105
In Table 4.12 for L α and Hα , the additional states and lines due to electron spin are shown explicitly.
The levels are designated with quantum numbers n, l, s, L, S, and J (appropriate to L S coupling).
Wavelengths and levels are from [22]. The f and A values were generated from the expression
for the line strength given above, using L p = 0 in the appropriate 6 − j symbol, and the numerical
constants from [8]. Use of these constants accounts for small differences with other tabulated values.
For example, our sum of the g f values for the two L α lines is 0.8321, while twice the value for 1s–2 p
given above is 0.8324.
In Table 4.13, multiplet numbers are mostly from [23]. They are labeled with u when the ultraviolet
table [24] is used. The values of log(g f ) given without explicit references were derived from [25].
Asterisks preceding the wavelengths indicate blends, in which case the g f is for the blend as a whole.
Accuracy assessments are indicated by letters [21]. References for the table are collected separately at
its end. Uncertainties in the range of 25%–50% are indicated by the letter D, those from 10%–25% by
a C, 3%–10% by B, 1%–3% by A, and within 1% by AA. The letter E is used for accuracies below
50%. The same scheme is followed for other sources when accuracy estimates are available.
Table 4.12. L α and Hα transitions with doublet structure.
χ1 (cm−1 )
χ2 (cm−1 )
g1 f 12
2 p 2 P1/2
2 p 2 P3/2
0
0
82 258.913
82 259.279
0.277 4
0.554 7
3 p 2 P1/2
3 p 2 P3/2
3s 2 S1/2
3s 2 S1/2
3d 2 D3/2
3d 2 D3/2
3d 2 D5/2
82 258.949
82 258.949
82 258.913
82 259.279
82 258.913
82 259.279
82 259.279
97 492.205
97 492.313
97 492.215
97 492.215
97 492.313
97 492.313
97 492.349
0.289 8
0.579 6
0.027 17
0.054 34
1.391
0.278 2
2.504
Wavelength (Å)
Lower
Upper
1 215.673 7
1 215.668 3
1s
1s
2S
1/2
2S
1/2
2s 2 S1/2
2s 2 S1/2
2 p 2 P1/2
2 p 2 P3/2
2 p 2 P1/2
2 p 2 P3/2
2 p 2 P3/2
6 562.272 0
6 562.725 6
6 562.752 0
6 562.909 9
6 562.710 1
6 562.867 5
6 562.852 0
g2 A21 × 10−8 (s−1 )
12.51
25.03
0.448 5
0.897 0
0.042 04
0.084 08
2.153
0.430 5
3.875
Table 4.13. Atomic oscillator strengths for allowed lines.
Multiplet
No.
Ji − Jk
λ (Å)
log(g f )
Atom
Transition
He II,
Li III,
Be IV,
B V, etc.
Hydrogen-like ions have nearly the same f values as those for hydrogen.
See discussion in [1] and [2] for Sc XXI–Ni XXVIII for higher-order effects.
He I
1s 2 –1s2 p
1s 2 –1s3 p
1s 2 –1s4 p
2u
3u
4u
Designation
Line
1 S–1 P 0
1 S–1 P 0
1 S–1 P 0
0–1
0–1
0–1
584.33
537.03
522.21
−0.558 8
−1.134 1
−1.524 9
Accuracy
AA
AA
AA
Reference
[3]
[3]
[3]
Sp.-V/AQuan/1999/10/29:07:01
72 / 4
Page 72
S PECTRA
Table 4.13. (Continued.)
Multiplet
Atom
Transition
He I
(Cont.)
1s2s–1s2 p
No.
1
Ji − Jk
3 S–3 P 0
1–0, 1, 2
0–1
1–0, 1, 2
0–1
1–0, 1, 2
0–1
0, 1, 2–1
1–0
0, 1, 2–1
1–0
0, 1, 2–1, 2, 3
1–2
0, 1, 2–1, 2, 3
1–2
0, 1, 2–1, 2, 3
1–2
1–0, 1, 2
0–1
1–0, 1, 2
0–1
1 S–1 P 0
1s2s–1s3 p
1s2s–1s4 p
1s2 p–1s3s
1s2 p–1s4s
1s2 p–1s3d
1s2 p–1s4d
1s2 p–1s5d
2
4
3
5
10
45
12
47
11
46
14
48
18
51
3 S–3 P 0
1 S–1 P 0
3 S–3 P 0
1 S–1 P 0
3 P 0 –3 S
1 P 0 –1 S
3 P 0 –3 S
1 P 0 –1 S
3 P 0 –3 D
1 P 0 –1 D
3 P 0 –3 D
1 P 0 –1 D
3 P 0 –3 D
1 P 0 –1 D
3 S–3 P 0
1s3s–1s3 p
1 S–1 P 0
3 S–3 P 0
1s3s–1s4 p
1 S–1 P 0
Li I
Be II
CI
C III
λ (Å)
log(g f )
Accuracy
*10 830
20 581
*3 889
5 016
*3 188
3 965
*7 065
7 281
*4 713
5 048
*5 876
6 678
*4 472
4 922
*4 026
4 388
*42 947
74 355
*12 527
15 084
0.208 8
−0.424 3
−0.713 5
−0.820 0
−1.111 9
−1.308 5
−0.203 7
−0.837 3
−1.021 6
−1.587 3
0.739 7
0.328 5
0.043 6
−0.442 7
−0.373 7
−0.886 6
0.427 0
−0.203 2
−0.823 4
−0.841 9
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
Reference
2s–2 p
2s–2 p
1 2 S–2 P 0
1 2 S–2 P 0
1/2–11/2
6 708
6 708
−0.001 2
−0.302 3
AA
AA
[3]
[3]
2s–2 p
2s–2 p
1 2 S–2 P 0
1 2 S–2 P 0
1/2–11/2
3 130
3 131
−0.177 2
−0.478 3
AA
AA
[3]
[3]
10 691
945.6
1 561
1 657
1 278
8 335
4 772
5 380
5 052
4 932
0.345
−0.118
−0.521
−0.285
−0.403
−0.437
−1.866
−1.615
−1.304
−1.658
B
C+
A
A
B−
B+
C
B
B
B
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
11/2–11/2
11/2–1/2
11/2–21/2
11/2–1/2
11/2–21/2
1/2–11/2
11/2–1/2
11/2–21/2
21/2–31/2
904.1
1 037
1 336
858.6
687.3
6 578
3 921
7 236
4 267
0.224
−0.310
−0.341
−1.284
0.082
−0.026
−0.232
0.298
0.717
B
B
B
B
B
B
B
B
C+
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
0–1
0–1
1–2
977.0
386.2
4 647
−0.120 0
−0.634
0.070
A+
B
B+
[3]
[3]
[3]
1
31u
3u
2u
u7
10
6
11
12
13
3 P 0 –3 D
2s 2 2 p–2s2 p 2
2s 2 2 p–2s2 p 2
2s 2 2 p–2s2 p 2
2 p–3s
2 p–3d
3s–3 p
3 p–4s
3 p–3d
3d–4 f
1u
2u
3u
4u
5u
2
4
3
6
2 P 0 –2 P
2s 2 –2s2 p
2s 2 –2s3 p
2s3s–2s3 p
1u 1 S–1 P 0
2u 1 S–1 P 0
1 3 S–3 P 0
2 p3s–2 p3 p
2 p 2 –2s2 p 3
2 p 2 –2 p3s
2 p 2 –2 p3d
2 p3s–2 p3 p
2 p3s–2 p4 p
2 p3s–2 p4 p
C II
Line
Designation
3 P–3 S 0
3 P–3 D 0
3 P–3 P 0
3 P–3 D 0
1 P 0 –1 S
3 P 0 –3 P
1 P 0 –1 P
1 P 0 –1 D
1 P 0 –1 S
2 P 0 –2 S
2 P 0 –2 D
2 P 0 –2 S
2 P 0 –2 D
2 S–2 P 0
2 P 0 –2 S
2 P 0 –2 D
2 D–2 F 0
1/2–1/2
1/2–1/2
2–3
2–1
2–3
2–2
2–3
1–0
2–2
1–1
1–2
1–0
Sp.-V/AQuan/1999/10/29:07:01
Page 73
4.7 ATOMIC O SCILLATOR S TRENGTHS FOR A LLOWED L INES / 73
Table 4.13. (Continued.)
Multiplet
Atom
Transition
C IV
2s–2 p
2s–2 p
2s–3 p
3s–3 p
No.
1u
1u
2u
1
Line
Designation
Ji − Jk
2 S–2 P 0
1/2–11/2
2 S–2 P 0
2 S–2 P 0
2 S–2 P 0
1/2–1/2
1/2–1/2, 11/2
1/2–11/2
λ (Å)
log(g f )
1 548 −0.419
1 551 −0.721
*312.4 −0.391
5 801 −0.194
−0.189 1
Accuracy
Reference
A
A
A−
A
[3]
[3]
[3]
[3]
AA
[3]
CV
1s 2 –1s2 p
1 S–1 P 0
0–1
NI
2 p 3 –2 p 2 3s
4 S 0 –4 P
11/2–21/2
21/2–31/2
11/2–11/2
21/2–11/2
1 199.6 −0.285
8 680
0.346
8 629
0.075
4 152 −1.981
B+
B+
B
C+
[3]
[3]
[3]
[3]
0–1
1–1
2–3
1–2
3–4
915.6
1 085
5 679.6
3 995
500.2
−0.782
−1.071
0.250
0.215
0.592
B+
B+
A
B+
C+
[3]
[3]
[3]
[3]
[3]
B
B
B
B
[3]
[3]
[3]
[3]
2 p 2 3s–2 p 2 3 p
2 p 2 3s–2 p 2 4 p
N II
N III
N IV
NV
2s 2 2 p 2 –2s2 p 3
2s 2 2 p 2 –2s2 p 3
2 p3s–2 p3 p
2u
1 4 P–4 D 0
8 2 P–2 P 0
6 4 P–4 S 0
3 P–3 P 0
2 p3 p–2 p3d
2u
1u
3
12
19
2s 2 2 p–2s2 p 2
2s 2 2 p–2s2 p 2
3s–3 p
2s2 p3s–2s2 p3 p
1u
1u
1
3
2 P 0 –2 D
2s 2 –2s2 p
2s 2 –2s3 p
2s3s–2s3 p
2s3 p–2s3d
1u
2u
1
3
1 S–1 P 0
1 P 0 –1 D
0–1
0–1
1–0, 1, 2
1–2
2s–2 p
2s–2 p
2s–3 p
3s–3 p
1u
1u
2u
1
2 S–2 P 0
1/2–11/2
3 P–3 D 0
3 P 0 –3 D
1 P 0 –1 D
3 D–3 F 0
2 P 0 –2 D
2 S–2 P 0
4 P 0 –4 D
1 S–1 P 0
3 S–3 P 0
2 S–2 P 0
2 S–2 P 0
2 S–2 P 0
11/2–21/2
11/2–11/2
1/2–11/2
21/2–31/2
1/2–1/2
1/2–1/2, 11/2
1/2–11/2
40.27
991.6 −0.357
991.5 −1.317
4 097 −0.057
4 515
0.221
765.1 −0.214 0 A+
247.2 −0.486
B
*3 481
0.238
B
4 058 −0.088
B
[3]
[3]
[3]
[3]
1 239 −0.505
1 243 −0.807
*209.3 −0.321
4 604 −0.278
A
A
A
A
[3]
[3]
[3]
[3]
N VI
1s 2 –1s2 p
1 S–1 P 0
0–1
28.79
−0.171 2
AA
[3]
OI
2 p 4 –2 p 3 3s
2u
5u
1
4
5
10
3u
1u
1
3
20
3 P–3 S 0
2–1
2–3
2–3
1–2
1–2
3–4
11/2–21/2
11/2–21/2
21/2–31/2
21/2–11/2
21/2–31/2
1 302
988.8
7 772
8 446
4 368
6 158
430.2
834.5
4 649
3 749
4 119
−0.585
−0.634
0.369
0.236
−1.983
−0.409
−0.139
−0.268
0.307
−0.105
0.433
A
B
A
B
C
B+
B+
B+
B+
B+
B+
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
[3]
3 P–3 D 0
2 p3s–2 p3 p
2 p3 p–2 p3d
1u
2u
2
14
2–3
2–2
2–3
2–3
835.3
703.9
3 760
3 715
−0.358
−0.293
0.162
0.149
A
A
C+
C+
[3]
[3]
[3]
[3]
2 p–3d
2s 2 2 p–2s2 p 2
2s2 p3s–2s2 p3 p
5u 2 P 0 –2 D
1u 2 P 0 –2 D
3 4 P 0 –4 D
11/2–21/2
11/2–21/2
21/2–31/2
238.6
790.2
3 386
0.258
−0.401
0.148
B
B
B
[3]
[3]
[3]
2 p 3 3s–2 p 3 3 p
O II
2 p 3 3s–2 p 3 4 p
2 p 3 3 p–2 p 3 4d
2 p 3 –2 p 2 3d
2s 2 2 p 3 –2s2 p 4
2 p 2 3s–2 p 2 3 p
2 p 2 3 p–2 p 2 3d
O III
O IV
2s 2 2 p 2 –2s2 p 3
3 P–3 D 0
5 S 0 –5 P
3 S 0 –3 P
3 S 0 –3 P
5 P–5 D 0
4 S 0 –4 P
4 S 0 –4 P
4 P–4 D 0
4 P–4 S 0
4 P 0 –4 D
3 P–3 P 0
3 P 0 –3 D
3 P–3 D 0
Sp.-V/AQuan/1999/10/29:07:01
74 / 4
Page 74
S PECTRA
Table 4.13. (Continued.)
Multiplet
Atom
Transition
OV
2s 2 –2s2 p
2s 2 –2s3 p
2 p3s–2 p3 p
2 p3 p–2 p3d
2s–2 p
2s–2 p
2s–3 p
3s–3 p
O VI
O VII
1s 2 –1s2 p
Ne I
2 p 5 3s–2 p 5 3 p
Ne II
2 p 4 3s–2 p 4 3 p
Line
Designation
Ji − Jk
λ (Å)
1u
2u
4
11
1 S–1 P 0
3 S–3 P 0
0–1
0–1
2–3
1–2
629.7 −0.290 5 A+
172.2 −0.407
B
4 124 −0.066
B
4 159 −0.356
B
[3]
[3]
[3]
[3]
1u
1u
2u
1
2 S–2 P 0
1/2–11/2
1/2–11/2
1 032 −0.576
1 038 −0.879
150.1 −0.451
3 811 −0.349
A
A
A−
A
[3]
[3]
[3]
[3]
0–1
21.60
−0.158 4
AA
[3]
No.
1 S–1 P 0
3 P 0 –3 D
2 S–2 P 0
2 S–2 P 0
2 S–2 P 0
1 S–1 P 0
1
1
1/2–1/2
1/2–11/2
log(g f )
Accuracy
Reference
2–3
6 402
0.345
B
4 P–4 P 0
21/2–21/2
3 694
0.09
D
11/2–21/2
122.7
0.313
D
[3]
0–1
465.2
−0.410
C
[3]
[3]
Ne VI
2 p–3d
2 P 0 –2 D
Ne VII
2s 2 –2s2 p
1 S–1 P 0
Ne VIII
2s–2 p
2 S–2 P 0
1/2–11/2
770.4
−0.689
B+
Ne IX
1s 2 –1s2 p
1 S–1 P 0
0–1
13.45
−0.141
A
[3]
3s–3 p
3s–3 p
3s–4 p
3 p–4s
3 p–5s
3 p–6s
3 p–3d
3 p–4d
3 p–5d
1
1
2
3
5
8
4
6
9
2 S–2 P 0
1/2–11/2
A
A
C
C
C
C
C
C
C
[3]
[3]
1/2–11/2
11/2–1/2
11/2–1/2
11/2–1/2
11/2–21/2
11/2–21/2
11/2–21/2
5 890
0.104
5 896 −0.197
3 302 −1.736
11 404 −0.163
6 161 −1.23
5 153 −1.732
8 195
0.51
5 688 −0.46
4 983 −0.962
1 S–1 P 0
3s3 p–3 p 2
1u
2
6
4
3
7
6u
0–1
2–1
1–0
2–1
2–3
1–2
0, 1, 2–0, 1, 2
2 852
5 184
11 828
3 337
3 838
8 807
*2 780
0.29
−0.158
−0.27
−1.10
0.414
−0.08
0.73
D
B
D
C
B
D
C
3s–3 p
3 p–4s
3 p–3d
3d–4 f
4s–4 p
1u
2u
3u
4
1
2 S–2 P 0
2 S–2 P 0
1/2–11/2
11/2–1/2
11/2–11/2
11/2, 21/2–21/2, 31/2
1/2–11/2
2 796
2 937
2 798
*4 481
9 218
0.09
−0.23
−0.43
0.973
0.26
C
C
D
C
C
2s 2 –2s2 p
1 S–1 P 0
0–1
368.1
−0.493
B
2s–2 p
2s–3 p
2 S–2 P 0
1/2–11/2
1/2–11/2
609.8
57.88
−0.775
−0.377
B
B
1s 2 –1s2 p
1 S–1 P 0
0–1
9.169
−0.128
B
3 p–4s
4s–5 p
3 p–3d
2 P 0 –2 S
11/2–1/2
3 962
6 696
3 093
−0.34
−1.343
0.263
C
C
C
Na I
Mg I
3s 2 –3s3 p
3s3 p–3s4s
3s3 p–3s5s
3s3 p–3s3d
Mg II
Mg IX
Mg X
Mg XI
Al I
2 S–2 P 0
2 S–2 P 0
2 P 0 –2 S
2 P 0 –2 S
2 P 0 –2 S
2 P 0 –2 D
2 P 0 –2 D
2 P 0 –2 D
3 P 0 –3 S
1 P 0 –1 S
3 P 0 –3 S
3 P 0 –3 D
1 P 0 –1 D
3 P 0 –3 P
2 P 0 –2 S
2 P 0 –2 D
2 D–2 F 0
2 S–2 P 0
1
5 2 S–2 P 0
3 2 P 0 –2 D
1/2–1/2
1/2–11/2
11/2–21/2
[4]
[4]
[4]
Sp.-V/AQuan/1999/10/29:07:01
Page 75
4.7 ATOMIC O SCILLATOR S TRENGTHS FOR A LLOWED L INES / 75
Table 4.13. (Continued.)
Multiplet
Atom
Transition
Al II
3s 2 –3s3 p
3s3 p–3s4s
Al III
3s–3 p
4s–4 p
Line
Ji − Jk
λ (Å)
2u 1 S–1 P 0
4u 3 P 0 –3 S
0–1
2–1
1 671
0.263
1 862 −0.192
B
B
1u 2 S–2 P 0
2 2 S–2 P 0
1/2–11/2
1/2–11/2
1 855
5 696
B
B
332.8
No.
Designation
Al X
2s 2 –2s2 p
1 S–1 P 0
0–1
Si I
3 p 2 –3 p4s
1u
43u
3
3u
4
5
6
3 P–3 P 0
2–2
2–1
0–1
2–3
2–3
2–2
2–1
4s–4 p
3d–4 f
3s 2 3 p–3s3 p 2
3s 2 3 p–3s3 p 2
3 p–3d
3 p–4s
3 p–4d
2
3
1u
5u
4u
2u
6u
2 S–2 P 0
Si III
3s 2 –3s3 p
3s4s–3s4 p
Si IV
3s–3 p
3s–4 p
4s–4 p
3s 2 3 p 2 –3s3 p 3
3 p4s–3 p4 p
Si II
log(g f )
0.047
0.235
−0.55
Accuracy
Reference
[4]
C
2 516 −0.241
2 882 −0.151
3 906 −1.092
2 217 −0.55
12 031
0.41
10 827
0.16
10 585 −0.19
C
C
C
C
D
D
D
1/2–11/2
21/2–31/2
1/2–11/2
11/2–11/2
11/2–21/2
11/2–1/2
11/2–21/2
6 347
0.23
4 131
0.463
1 808 −2.14
1 195
0.49
1 265
0.52
1 534 −0.28
992.7 −0.15
C
C
D
D
D
C
D
2u 1 S–1 P 0
2 3 S–3 P 0
4 1 S–1 P 0
0–1
1–2
0–1
1 207
4 553
5 740
0.22
0.292
−0.16
B
C
D
[5]
1u 2 S–2 P 0
2u 2 S–2 P 0
1 2 S–2 P 0
1/2–11/2
0.01
−1.34
0.195
B
D
B
[5]
1 D–1 P 0
1 S–1 P 0
3 P–3 D 0
3 P 0 –3 D
3 P 0 –3 P
3 P 0 –3 S
2 D–2 F 0
2 P 0 –2 D
2 P 0 –2 P
2 P 0 –2 D
2 P 0 –2 S
2 P 0 –2 D
[4]
[4]
[4]
1/2–11/2
1 394
457.8
4 089
Si XI
2s 2 –2s2 p
1 S–1 P 0
0–1
303.3
−0.576
C
Si XII
2s–2 p
2 S–2 P 0
1/2–11/2
499.4
−0.845
B
[6]
SI
3 p 3 4s–3 p 3 4 p
1
5 S0 – 5 P
2–3
9 213
0.42
D
[4]
S II
3s 2 3 p 3 –3s3 p 4
1u
4 S 0 –4 P
11/2–21/2
1 260
−1.31
C
2 P 0 –2 S
11/2–1/2
554.1
−0.425
C
786.5
*661.5
0.165
0.802
B
B
1/2–11/2
1/2–11/2
7 665
4 044
0.135
−1.915
B
C
0–1
2–1
2–1
2–3
2–3
2–3
2–2
3–3
4 227
6 162
3 974
4 455
3 644
3 362
4 303
5 589
0.243
−0.089
−0.906
0.26
−0.306
−0.578
0.276
0.21
B
C
C
C
C
C
C
D
S IV
3 p–4s
5u
SV
3s 2 –3s3 p
3s3 p–3s3d
1u 1 S–1 P 0
3u 3 P 0 –3 D
KI
4s–4 p
4s–5 p
Ca I
4s 2 –4s4 p
4s4 p–4s5s
4s4 p–4s6s
4s4 p–4s4d
4s4 p–4s5d
4s4 p–4s6d
4s4 p–4 p 2
3d4s–3d4 p
1 2 S–2 P 0
3 2 S–2 P 0
2
3
6
4
9
11
5
21
1 S–1 P 0
3 P 0 –3 S
3 P 0 –3 S
3 P 0 –3 D
3 P 0 –3 D
3 P 0 –3 D
3 P 0 –3 P
3 D–3 D 0
1/2–11/2
0–1
0, 1, 2–1, 2, 3
[4]
Sp.-V/AQuan/1999/10/29:07:01
76 / 4
Page 76
S PECTRA
Table 4.13. (Continued.)
Multiplet
Atom
Transition
Ca II
4s–4 p
3d–4 p
4 p–5s
4 p–4d
Sc I
Ti I
3d 2 4s–3d 2 4 p
3d 3 4s–3d 3 4 p
3d 3 4s–3d 2 4s4 p
Ti II
3d 2 4s–3d 2 4 p
3d 3 –3d 2 4 p
VI
3d 4 4s–3d 4 4 p
3d 4 4s–3d 3 4s4 p
3d 3 4s4 p–3d 3 4s5s
3d 3 4s4 p–3d 3 4s4d
Line
Designation
Ji − Jk
λ (Å)
log(g f )
Accuracy
1
2
3
4
2 S–2 P 0
1/2–11/2
21/2–11/2
11/2–1/2
11/2–21/2
3 934
8 542
3 737
3 179
0.135
−0.365
−0.15
0.51
C
C
C
C
12
14
15
16
6
4 F–4 G 0
41/2–51/2
41/2–31/2
31/2–41/2
31/2–31/2
21/2–11/2
5 672
4 744
5 521
5 482
4 082
0.49
0.42
0.29
0.27
−0.57
D
D
D
D
C
38
42
145
12
24
110
5 F–5 G 0
5–6
5–5
3–4
4–4
4–5
3–4
4 982
4 533
4 617
3 999
3 371
5 036
0.504
0.476
0.389
−0.056
0.13
0.130
C+
C+
C+
B
C
C+
[1]
[1]
[1]
[1]
[1]
[1]
1 4 F–4 G 0
2 4 F–4 F 0
7 4 F–4 F 0
31/2–41/2
41/2–41/2
41/2–41/2
3 361
3 235
3 323
0.28
0.336
−0.183
C
C+
C+
[1]
[1]
[1]
6 D–6 P 0
41/2–31/2
41/2–51/2
41/2–41/2
61/2–61/2
41/2–51/2
41/2–51/2
41/2–31/2
31/2–41/2
51/2–51/2
61/2–71/2
4 460
4 379
4 112
4 269
4 545
3 185
3 704
4 091
5 193
3 695
−0.15
0.58
0.408
0.65
0.45
0.69
0.18
0.33
0.29
0.97
C−
C
B
C−
C−
C
C−
C
C−
C−
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
No.
21
22
27
88
109
14
29
41
125
114
2 D–2 P 0
2 P 0 –2 S
2 P 0 –2 D
4 F–4 D 0
2 F–2 G 0
2 F–2 F 0
2 D–2 P 0
5 F–5 F 0
5 P–5 D 0
3 F–3 F 0
3 F–3 G 0
3 F–3 G 0
6 D–6 F 0
6 D–6 D 0
4 H –4 H 0
4 F–4 G 0
4 F–4 G 0
6 D–6 P 0
4 D–4 F 0
6 F 0 –6 F
6 G 0 –6 H
Reference
[4]
V II
3d 3 4s–3d 3 4 p
11 3 P–5 D 0
5 3 F–3 D 0
25 5 P–5 D 0
2–3
4–3
3–4
3 903
3 557
4 202
−0.89
−0.17
−1.75
B
B
D
[1]
[1]
[1]
Cr I
3d 5 4s–3d 5 4 p
1
7
38
22
4
43
7 S–7 P 0
3–4
2–3
6–7
4–5
3–4
6–6
4 254
5 208
3 964
4 351
3 579
3 744
−0.114
0.158
0.67
−0.44
0.409
0.318
B
B
D−
C
B
B
[1]
[1]
[1]
[1]
[1]
[1]
5
6
2
1u
6 D–6 D 0
41/2–41/2
41/2–51/2
21/2–31/2
21/2–31/2
4 041
3 807
4 031
2 795
0.285
0.19
−0.47
0.53
C+
B
C+
C
[1]
[1]
[1]
[1]
20
23
41
42
43
45
5 F–5 D 0
5–4
5–6
4–5
4–5
4–4
4–3
3 820
3 581
4 384
4 272
4 046
3 816
0.119
0.406
0.200
−0.164
0.280
0.237
B+
B+
B+
B+
B+
B
[2]
[2]
[2]
[2]
[2]
[7]
3d 4 4s 2 –3d 4 4s4 p
3d 5 4s–3d 4 4s4 p
Mn I
3d 6 4s–3d 6 4 p
3d 5 4s 2 –3d 5 4s4 p
Fe I
3d 7 4s–3d 7 4 p
5 S–5 P 0
5 G–5 H 0
5 D–5 F 0
7 S–7 P 0
5 G–5 G 0
6 D–6 F 0
6 S–6 P 0
6 S–6 P 0
5 F–5 G 0
3 F–5 G 0
3 F–3 G 0
3 F–3 F 0
3 F–3 D 0
Sp.-V/AQuan/1999/10/29:07:01
Page 77
4.7 ATOMIC O SCILLATOR S TRENGTHS FOR A LLOWED L INES / 77
Table 4.13. (Continued.)
Multiplet
Line
Atom
Transition
No.
Designation
Ji − Jk
λ (Å)
log(g f )
Accuracy
Fe I
(Cont.)
3d 6 4s 2 –3d 6 4s4 p
4
5
68
152
5 D–5 D 0
4–4
4–5
3–4
5–5
3 860
3 720
4 529
4 260
−0.710
−0.431
−0.822
0.077
B+
B+
B+
B
[2]
[2]
[2]
[7]
3d 7 4s–3d 6 4s4 p
3d 6 4s4 p–3d 6 4s5s
5 D–5 F 0
5 P–5 D 0
7 D 0 –7 D
Reference
Fe II
3d 6 4s–3d 6 4 p
27 4 P–4 D 0
38 4 F–4 D 0
21/2–31/2
41/2–31/2
4 233
4 584
−2.00
−2.02
C
D
[2]
[2]
Co I
3d 8 4s–3d 8 4 p
22
23
35
5
28
4 F–4 G 0
41/2–51/2
41/2–41/2
31/2–31/2
41/2–51/2
31/2–41/2
3 454
3 405
3 569
3 466
4 121
0.38
0.25
0.37
−0.70
−0.32
C+
C+
C
C
C
[2]
[2]
[2]
[2]
[2]
19
35
7
25
111
17
130
143
162
194
3 D–3 F 0
3–4
2–3
4–5
3–4
5–5
3–3
2–2
4–5
3–4
3–4
3 415 −0.06
3 619 −0.04
3 233 −0.90
3 051 −0.12
5 018 −0.08
3 374 −1.76
4 855
0.00
5 081
0.13
5 084
0.03
5 081
0.30
C
C
C
C+
D
C
D
D
D−
D−
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
3d 7 4s 2 –3d 7 4s4 p
3d 8 4s–3d 7 4s4 p
Ni I
3d 9 4s–3d 9 4 p
3d 8 4s 2 –3d 8 4s4 p
3d 9 4s–3d 8 4s4 p
3d 8 4s4 p–3d 8 4s5s
3d 8 4s4 p–3d 8 4s4d
3d 9 4 p–3d 9 4d
4 F–4 F 0
2 F–2 F 0
4 F–4 G 0
2 F–2 G 0
1 D–1 F 0
3 F–3 G 0
3 D–3 F 0
5 F 0 –5 F
3 D–5 F 0
3 P 0 –3 P
3 F 0 –3 G
3 D 0 –3 F
1 F 0 –1 G
Cu I
4s–4 p
3d 9 4s 2 –3d 10 4 p
4 p–4d
1 2 S–2 P 0
2 2 D–2 P 0
7 2 P 0 –2 D
1/2–11/2
21/2–11/2
11/2–21/2
3 248
5 106
5 218
−0.056
−1.50
0.26
C
D
D
Zn I
4s4 p–4s4d
4s4 p–4s4d
4 3 P 0 –3 D
6 1 P 0 –1 D
2–3
1–2
3 345
6 362
0.30
0.158
B
C
Sr I
5s 2 –5s5 p
2
1 S–1 P 0
0–1
4 607
0.283
C
2 S–2 P 0
1/2–11/2
11/2–1/2
4 078
4 306
0.151
−0.11
C
D
1 S–1 P 0
0–1
5 536
0.215
C
2 S–2 P 0
Sr II
5s–5 p
5d–6s
1
3 2 P 0 –2 S
Ba I
6s 2 –6s6 p
2
Ba II
6s–6 p
5d–6 p
6 p–6d
1
2 2 D–2 P 0
4 2 P 0 –2 D
1/2–11/2
21/2–11/2
11/2–21/2
4 554
6 142
4 131
0.163
−0.08
0.441
C
D
C
Hg I
6s6 p–6s7s
1 3 P 0 –3 S
1
1
0–1
1–2
2–1
4 047
4 348
5 461
−0.81
−0.92
−0.185
D
D
C
Pb I
6 p 2 –6 p7s
1
3 P–3 P 0
2–1
4 058
−0.18
D
References
1. Martin, G.A., Fuhr, J.R., & Wiese, W.L. 1988, J. Phys. Chem. Ref. Data, 17, Suppl. 3
2. Fuhr, J.R., Martin, G.A., & Wiese, W.L. 1988, J. Phys. Chem. Ref. Data, 17, Suppl. 4
3. Wiese, W.L., Fuhr, J.R., & Deters, T.M. 1996, J. Phys. Chem. Ref. Data Monograph, 7; and other data to be published
4. Wiese, W.L., Smith, M.W., & Miles, B.M. 1969, Atomic Transition Probabilities, Sodium Through Calcium, NSRDSNBS, 22
5. Morton, D.C. 1991, ApJS, 77, 119
6. Wiese, W.L., Smith, M.W., & Glennon, B.M. 1966, Atomic Transition Probabilities, H Through Ne, NSRDS-NBS, 4
7. O’Brian, T.R., Wickliffe, M.E., Lawler, J.E., Whaling, W., & Brault, J.W. 1991, J. Opt. Soc. Am., B8, 1185
Sp.-V/AQuan/1999/10/29:07:01
78 / 4
Page 78
S PECTRA
4.8 NUCLEAR SPIN AND HYPERFINE STRUCTURE:
LOW-LEVEL HYPERFINE TRANSITIONS
The angular momentum, or spin, of the ground levels of nuclei [26] can be of importance in atomic
spectra and structure. Nonzero spins result from unpaired nucleons and occur for some isotopes of most
elements. In elements with odd Z , the most abundant isotope will have a nonzero spin, so hyperfine
structure is most important for these species. However, secondary (odd-N ) isotopes of even-Z elements
may make a significant contribution to the overall line shape.
If the spin I is taken into account, the total angular momentum of an atomic level is F = J + I.
The vectors J and I are added using the same rules as when L and S are added to form J. The quantum
numbers F and I play analogous roles to J and S. Thus, for a given J and I there are 2I + 1 values of
F if J > I , and 2J + 1 if I > J . The number of elementary states belonging to a level with a given
F is 2F + 1, corresponding to the number of possible values of M F , the projection of F on the z axis
in units of h̄. When there should be no ambiguity, M F may be written without the subscript M.
Nuclear spin broadens spectral lines and adds 2I + 1 additional states to an atomic system. The
first factor, known as hyperfine splitting, may usually be ignored if the resultant width is much smaller
than that due to other broadening mechanisms, such as pressure or Doppler broadening. The additional
atomic states cancel in the Boltzmann and Saha formulas and usually are not accounted for explicitly.
The splitting of atomic levels due to nuclear spin (E M ) may be augmented (E Q ) if the nucleus
has an electric quadrupole moment [27]:
E M = 12 A[F(F + 1) − J (J + 1) − I (I + 1)] ≡ 12 AC,
E Q = B[C(C + 1) − 43 J (J + 1)I (I + 1)].
Values of A and B are given by [28].
Nuclear mass effects may be treated as follows. Let P be the momentum of the nucleus with mass
M, and let pi be the momentum of the ith electron. The kinetic energy is then
E=
p2
P2
i
+
.
2M
2m
i
We can eliminate P using P + i pi = 0, whence
1
1 2
1 E=
+
pi +
pi · p j .
2M
2m
M i, j>i
i
The first term is called the normal mass shift and gives rise to well-known displacements of lines
in very light elements [29]. The second term, called the specific mass shift, is difficult to calculate [27]
but may be measured in the laboratory. It can be significant even for heavy atoms [30].
Finally, nuclear volume, field effects, or isotope shifts occur because the potential at small r
departs from a pure 1/r dependence due to the finite size of the nucleus. Astrophysically important
consequences have been documented [31].
While the hyperfine width is difficult to calculate, the relative intensities of lines in a hyperfine
multiplet follow readily from the quantum theory of angular momentum. The relative line strengths
are written simply with a Wigner 6 − j symbol:
2
J I F
S(J I F ↔ J I F) ∝ (2F + 1)(2F + 1)
.
F 1 J The relative intensities are identical to those discussed for L S coupling, and the tables of Sec. 4.5
may be used with the substitutions J → F, S → I , and L → J .
Sp.-V/AQuan/1999/10/29:07:01
Page 79
4.9 F ORBIDDEN L INE T RANSITION P ROBABILITIES / 79
The celebrated 21-cm line in atomic hydrogen is an example of a pure magnetic dipole transition.
Similar transitions occur in ionized 3 He, as well as in deuterium. Results are summarized in Table 4.14,
with 1986 constants and transition frequencies from [32]. The formula for magnetic dipole radiation
simplifies in this case to
g F A(F , F) =
4π 2 he2 ge2 ν 3 |sn se F M |Sq(1) |sn se F M|2 .
3m 2 c5
MM q
(1)
(1)
Here, Sq is a spherical tensor, analogous to Cm , which operates in electron spin space. Quantum
numbers sn and se describe the spin states of the nucleus and electron. For 1 H I and 3 He II, g F = 3,
while for 2 H I it is 4. The sums over M, M , and q are 3/4 for 1 H I and 3 He II and 4/3 for 2 H I.
The numerical coefficient is 4.01367 × 10−42 ν 3 (cgs). We have neglected the magnetic moment of
the nucleus. The ground state orbital functions are not indicated, since they contribute only a trivial
multiplicative factor of unity.
4.9
FORBIDDEN LINE TRANSITION PROBABILITIES
Most of the lines in Table 4.15 are forbidden in the sense that they involve no change in parity. A
few intersystem lines are included. Both magnetic dipole (M1) and electric quadrupole (E2) lines are
possible at the same wavelength in many cases. The dominant radiation is indicated, but the A value
is for the sum over all mechanisms, including electric dipole radiation (for intersystem lines). When
both magnetic dipole and electric quadrupole transitions are permitted by their selection rules, the
Einstein A coefficient for the magnetic dipole will usually dominate for optical transitions. Generally,
Am /Aq ≈ 3 × 1011 /σ 2 , where σ is the wave number of the transition. Typical A values for electric
dipole transitions are 105 times larger than their magnetic dipole congeners. Accuracy estimates
from [33–36] are indicated where available. The notation is the same as in Sec. 4.7.
Table 4.14. Hyperfine transitions.
F
F
ν (Hz)
A21 (s−1 )
1
11/2
1
0
1/2
0
1.420 405 752 × 109
3.273 843 523 × 108
8.665 649 867 × 109
2.876 × 10−15
4.695 × 10−17
6.530 × 10−13
I
1H I
2H I
3 He II
1/2
1
1/2
Table 4.15. Forbidden and intercombination lines.
Atom
Array
He I]
[C I]
C II]
C III]
[N I]
1s 2 –1s2 p
2 p2
2s 2 2 p–2s2 p 2
2s 2 –2s2 p
2 p3
Designation
lower–upper
1 S–3 P 0
1 D–1 S
2 P 0 –4 P
1 S–3 P 0
4 S 0 –2 D 0
4 S 0 –2 D 0
Ji –Jk
0–1
2–0
11/2–21/2
0–1
11/2–11/2
11/2–21/2
λ (Å)
591.4
8 727
2 325.4
1908.7
5 198
5 200
A (s−1 )
Accuracy
1.76 × 10+2
0.634
52.6
114
2.26 × 10−5
5.77 × 10−6
B
B
B+
C
B
M1
or E2
E2
M1
E2
Reference
[1]
[2]
[2]
[2]
[2]
[2]
Sp.-V/AQuan/1999/10/29:07:01
Page 80
80 / 4
S PECTRA
Table 4.15. (Continued.)
Atom
Array
N II]
2 p2
Designation
lower–upper
1 D–1 S
3 P–1 S
3 P–1 S
3 P–1 D
3 P–1 D
3 P–1 D
3 P–3 P
3 P–3 P
3 P–3 P
N III]
N IV]
[O I]
2s 2 2 p 2 –2s2 p 3
2s 2 2 p 2 –2s2 p 3
2s 2 2 p–2s2 p 2
2s 2 –2s2 p
2 p4
3 P–5 S 0
3 P–5 S 0
2 P 0 –4 P
1 S–3 P 0
1 D–1 S
3 P–1 S
3 P–1 S
3 P–1 D
3 P–1 D
3 P–1 D
3 P–3 P
3 P–3 P
3 P–3 P
[O II]
2 p3
2 D 0 –2 P 0
2 D 0 –2 P 0
2 D 0 –2 P 0
2 D 0 –2 P 0
4 S 0 –2 P 0
2 S 0 –2 P 0
4 S 0 –2 D 0
4 S 0 –2 D 0
[O III]
2 p2
1 D–1 S
3 P–1 D
3 P–1 D
3 P–1 D
3 P–3 P
3 P–3 P
3 P–3 P
O III]
O IV]
O V]
[F IV]
[Ne III]
2s 2 2 p 2 –2s2 p 3
2s 2 2 p 2 –2s2 p 3
2s 2 p–2s2 p 2
2s 2 –2s2 p
2 p2
2 p4
3 P–5 S 0
3 P–5 S 0
2 P 0 –4 P
1 S–3 P 0
3 P–1 D
1 D–1 S
3 P–1 S
3 P–1 S
3 P–1 D
3 P–1 D
3 P–1 D
3 P–3 P
3 P–3 P
3 P–3 P
Ji –Jk
2–0
2–0
1–0
1–2
2–2
0–2
1–2
0–2
0–1
2–2
1–2
11/2–21/2
0–1
2–0
2–0
1–0
2–2
1–2
0–2
1–0
2–0
2–1
21/2–11/2
11/2–11/2
21/2–1/2
11/2–1/2
11/2–11/2
11/2–1/2
11/2–21/2
11/2–11/2
2–0
1–2
2–2
0–2
1–2
0–2
0–1
2–2
1–2
11/2–21/2
0–1
2–2
2–0
2–0
1–0
2–2
1–2
0–2
1–0
2–0
2–1
λ (Å)
5 755
3 071
3 063
6 548
6 583
6 527
121.8 µm
76.45 µm
205.3 µm
2 143
2 139
1 749.7
1 486
5 577
2 958
2 972
6 300
6 364
6 392
145.5 µm
44.06 µm
63.19 µm
7 320
7 331
7 319
7 330
2 470
2 470
3 729
3 726
4 363
4 959
5 007
4 931
51.81 µm
32.66 µm
88.18 µm
1 666
1 661
1 401.2
1 218.3
4 060
3 342
1 794
1 815
3 869
3 967
4 012
36.02 µm
10.86 µm
15.55 µm
A (s−1 )
Accuracy
1.17
1.40 × 10−4
3.15 × 10−2
9.20 × 10−4
2.73 × 10−3
5.45 × 10−7
7.40 × 10−6
9.69 × 10−13
2.07 × 10−6
1.27 × 10+2
5.49 × 10+1
3.08 × 102
1.02 × 10+3
1.26
2.42 × 10−4
7.54 × 10−2
5.65 × 10−3
1.82 × 10−3
8.60 × 10−7
1.75 × 10−5
1.34 × 10−10
8.91 × 10−5
9.91 × 10−2
5.34 × 10−2
5.19 × 10−2
8.67 × 10−2
5.22 × 10−2
2.12 × 10−2
3.06 × 10−5
1.78 × 10−4
1.71
6.21 × 10−3
1.81 × 10−2
2.41 × 10−6
9.76 × 10−5
3.17 × 10−11
2.62 × 10−5
5.48 × 10+2
2.20 × 10+2
1.47 × 10+3
3.68 × 10+3
9.25 × 10−2
2.72
1.88
2.02
0.159
4.92 × 10−2
9.60 × 10−6
1.15 × 10−3
2.19 × 10−8
5.97 × 10−3
B
B
B
B
B
B
B
C
B
B−
B−
C+
B
B+
C+
B+
B+
B+
B+
B+
C+
B+
B
B
B
B
C+
C+
C
C
B
B
B
C+
B+
B+
B+
B
B
B
B
B
B
A
A
A
B+
B+
B
A
M1
or E2
E2
E2
M1
M1
M1
E2
M1
E2
E2
E2
E2
M1
M1
M1
E2
M1
E2
M1
E2
E2
E2
E2
M1
M1
E2
M1
E2
M1
M1
E2
M1
E2
M1
M1
E2
E2
M1
M1
M1
E2
M1
E2
M1
Reference
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[3]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
Sp.-V/AQuan/1999/10/29:07:01
Page 81
4.9 F ORBIDDEN L INE T RANSITION P ROBABILITIES / 81
Table 4.15. (Continued.)
Atom
Array
[Ne IV]
2 p3
Designation
lower–upper
2 D 0 –2 P 0
2 D 0 –2 P 0
2 D 0 –2 P 0
2 D 0 –2 P 0
4 S 0 –2 P 0
4 S 0 –2 P 0
4 S 0 –2 D 0
4 S 0 –2 D 0
2 D 0 –2 P 0
[Ne V]
2 p2
1 D–1 S
3 P–1 D
3 P–1 D
3 P–1 D
3 P–3 P
3 P–3 P
3 P–3 P
Ne V]
Si III]
[S I]
[S II]
2s 2 2 p 2 –2s2 p 3
2s 2 2 p 2 –2s2 p 3
3s 2 –3s3 p
3 p4
3 p3
3 P–5 S 0
3 P–5 S 0
1 S–3 P 0
1 D–1 S
2 D 0 –2 P 0
2 D 0 –2 P 0
2 D 0 –2 P 0
2 D 0 –2 P 0
4 S 0 –2 P 0
4 S 0 –2 P 0
4 S 0 –2 D 0
4 S 0 –2 D 0
[S III]
3 p2
1 D–1 S
3 P–1 S
3 P–1 S
3 P–1 D
3 P–1 D
3 P–1 D
3 P–3 P
3 P–3 P
3 P–3 P
S III]
3s 3 p 2 –3s3 p 3
3 P–5 S 0
3 P–5 S 0
[Cl III]
[Cl IV]
3 p3
3 p2
4 S 0 –2 D 0
1 D–1 S
3 P–1 D
3 P–1 D
[Ar III]
3 p4
1 D–1 S
3 P–1 S
3 P–1 S
3 P–1 D
3 P–1 D
3 P–1 D
3 P–3 P
3 P–3 P
3 P–3 P
Ji –Jk
λ (Å)
A (s−1 )
Accuracy
21/2–11/2
11/2–11/2
21/2–1/2
11/2–1/2
11/2–11/2
11/2–1/2
11/2–21/2
11/2–11/2
11/2–1/2
2–0
2–2
1–2
0–2
1–2
0–2
0–1
2–2
1–2
0–1
2–0
21/2–11/2
11/2–11/2
21/2–1/2
11/2–1/2
11/2–11/2
11/2–1/2
11/2–21/2
11/2–11/2
2–0
2–0
1–0
2–2
1–2
0–2
1–2
0–2
0–1
2–2
1–2
11/2–11/2
2–0
1–2
2–2
2–0
2–0
1–0
2–2
1–2
0–2
1–0
2–0
2–1
4 714
4 724
4 716
4 726
1 602
1 602
2 424
2 422
4 726
2 973
3 426
3 345
3 300
14.33 µm
9.008 µm
24.25 µm
1 146
1 137
1 892
7 725
10 320
10 287
10 370
10 336
4 069
4 076
6 716
6 731
6 312
3 797
3 722
9 531
9 069
8 830
18.71 µm
12.00 µm
33.48 µm
1 729
1 713
5 538
5 323
7 531
8 046
5 192
3 005
3 109
7 136
7 751
8 036
21.83 µm
6.369 µm
8.992 µm
0.380
0.421
0.105
0.372
1.23
0.499
4.12 × 10−4
5.76 × 10−3
0.372
2.89
0.351
0.126
2.44 × 10−5
4.59 × 10−3
5.12 × 10−9
1.28 × 10−3
6.06 × 10+3
2.37 × 10+3
1.67 × 10+4
1.53
0.179
0.133
7.79 × 10−2
0.163
0.225
9.06 × 10−2
2.60 × 10−4
8.82 × 10−4
2.22
1.05 × 10−2
0.796
5.76 × 10−2
2.21 × 10−2
5.82 × 10−6
2.07 × 10−3
4.61 × 10−8
4.72 × 10−4
7.32 × 103
2.66 × 103
4.83 × 10−3
2.80
7.23 × 10−2
0.179
2.59
4.17 × 10−2
3.91
0.314
8.23 × 10−2
2.15 × 10−5
5.17 × 10−3
2.37 × 10−6
3.08 × 10−2
B
B
B
B
B
B
C+
C
B
B
B
B
B
A
B+
A
M1
or E2
M1
M1
E2
M1
M1
M1
E2
M1
M1
E2
M1
M1
E2
M1
E2
M1
E2
E2
M1
E2
E2
M1
M1
E2
E2
E2
E2
M1
M1
M1
E2
M1
E2
M1
M1
E2
M1
M1
E2
E2
M1
M1
M1
E2
M1
E2
M1
Reference
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[1]
[2]
[2]
[2]
[2]
[2]
[2]
[2]
[1]
[1]
[3]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[3]
[3]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
Sp.-V/AQuan/1999/10/29:07:01
Page 82
82 / 4
S PECTRA
Table 4.15. (Continued.)
Atom
Array
[Ar IV]
3 p3
Designation
lower–upper
2 p5
2 P 0 –2 P 0
21/2–11/2
11/2–11/2
21/2–1/2
11/2–1/2
11/2–11/2
11/2–1/2
11/2–21/2
11/2–11/2
2–0
2–0
1–0
2–2
1–2
0–2
1–2
0–2
0–1
11/2–1/2
2p
3 p4
2 P 0 –2 P 0
1/2–11/2
2 D 0 –2 P 0
2 D 0 –2 P 0
2 D 0 –2 P 0
2 D 0 –2 P 0
4 S 0 –2 P 0
4 S 0 –2 P 0
4 S 0 –2 D 0
4 S 0 –2 D 0
[Ar V]
3 p2
1 D–1 S
3 P–1 S
3 P–1 S
3 P–1 D
3 P–1 D
3 P–1 D
3 P–3 P
3 P–3 P
3 P–3 P
[Ar X]
[Ar XIV]
[K IV]
1 D–1 S
3 P–1 D
3 P–1 D
[Ca V]
[Ca XII]
[Ca XIII]
[Ca XV]
3 p4
2 p5
2 p4
2 p2
3 P–1 D
2 P 0 –2 P 0
3 P–3 P
3 P–3 P
3 P–3 P
[Fe II]
3d 6 4s–3d 7
6 D–4 P
6 D–4 F
3d 6 4s–3d 5 4s 2
6 D–6 S
6 D–6 S
3d 7 –3d 6 4s
4 D–2 P
[Fe III]
3d 6
5 D–3 F
[Fe IV]
[Fe V]
3d 5
4 G–4 F
3d 4
5 D–3 P2
[Fe VI]
3d 3
4 F–4 P
[Fe VII]
3d 2
3 F–3 P
4 F–4 G
5 D–3 P
5 D–3 F2
4 F–2 G
3 F–1 D
[Fe X]
[Fe XI]
3 p5
3 p4
2 P 0 –2 P 0
3 P–1 D
3 P–3 P
[Fe XIII]
3 p2
3 P–3 P
3 P–3 P
3 P–1 D
[Fe XIV]
[Fe XV]
Ji –Jk
3p
3s3 p
2 P 0 –2 P 0
3 P 0 –3 P 0
2–0
1–2
2–2
2–2
11/2–1/2
2–1
1–2
0–1
31/2–21/2
41/2–41/2
41/2–21/2
31/2–21/2
1/2–1/2
41/2–51/2
4–4
3–2
51/2–41/2
3–2
4–4
41/2–21/2
41/2–41/2
4–2
2–2
3–2
11/2–1/2
1–2
2–1
0–1
1–2
2–2
1/2–11/2
1–2
λ (Å)
7 237
7 171
7 331
7 263
2 854
2 868
4 711
4 740
4 626
2 786
2 691
7 006
6 435
6 133
7.903 µm
4.928 µm
13.09 µm
5 533
4 412
4 511
6 795
6 102
5 309
3 328
4 087
5 446
5 694
4 890
4 416
4 287
4 359
5 528
4 244
4 658
5 270
4 907
3 895
3 891
5 677
5 176
5 276
5 721
6 087
6 375
3 987
7 892
10 747
10 798
3 389
5 303
7 059
A (s−1 )
Accuracy
0.598
0.789
0.119
0.603
2.11
0.862
1.77 × 10−3
2.23 × 10−2
3.29
5.69 × 10−2
6.55
0.476
0.204
3.50 × 10−5
2.72 × 10−2
1.24 × 10−6
7.99 × 10−3
1.06 × 10+2
B
1.04 × 10+2
A
3.18
0.203
0.838
1.90
4.87 × 10+2
3.19 × 10+2
7.9 × 10+1
9.4 × 10+1
0.36
0.46
1.5
1.1
0.12
0.90
0.44
0.40
0.32
0.71
0.74
0.052
0.62
0.050
0.36
0.58
69.2
9.5
4.36 × 10+1
1.4 × 10+1
9.86
7.5 × 10+1
6.01 × 10+1
3.80 × 10+1
E
E
E
E
E
E
D
D
E
D
D
E
D
E
D
D
B
D−
C+
C+
C+
E
C+
C+
M1
or E2
Reference
M1
M1
E2
M1
M1
M1
E2
M1
E2
E2
M1
M1
M1
E2
M1
E2
M1
M1
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[1]
[4]
M1
E2
M1
M1
M1
M1
M1
M1
M1
M1
M1
E2
E2
M1
E2
M1
M1
E2
M1
M1
E2
M1
E2
M1
M1
M1
M1
M1
M1
M1
M1
M1
M1
[4]
[1]
[4]
[4]
[1]
[4]
[4]
[4]
[4]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
Sp.-V/AQuan/1999/10/29:07:01
Page 83
4.10 S PECTRA OF D IATOMIC M OLECULES / 83
Table 4.15. (Continued.)
Atom
Array
[Ni II]
3d 9 –3d 8 (3 F)4s
3d 9 –3d 8 (3 P)4s
3d 8
3 p5
3 p4
[Ni III]
[Ni XII]
[Ni XIII]
Designation
lower–upper
2 D–2 F
2 D–4 P
3 F–3 P
2 P 0 –2 P 0
3 P–1 D
3 P–3 P
3 p2
[Ni XV]
3 P–3 P
3 P–3 P
[Ni XVI]
3p
2 P 0 –2 P 0
Ji –Jk
21/2–21/2
21/2–21/2
4–2
11/2–1/2
1–2
2–1
0–1
1–2
1/2–11/2
λ (Å)
6 668
4 326
6 000
4 231
3 637
5 116
6 702
8 024
3 601
A (s−1 )
Accuracy
0.099
0.35
0.050
2.37 × 10+2
1.8 × 10+1
1.57 × 10+2
5.65 × 10+1
2.27 × 10+1
1.92 × 10+2
E
E
E
B
E
C+
C+
C+
C+
M1
or E2
Reference
E2
E2
E2
M1
M1
M1
M1
M1
M1
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
[5]
References
1. Mendoza, C. 1983, in Planetary Nebulae, edited by D.R. Flower, IAU Symposium No. 103 (Reidel, Dordrecht),
p. 143
2. Wiese, W.L., Fuhr, J.R., & Deters, T.M. 1996, J. Phys. Chem. Ref. Data Monograph, 7; and other data to be published
3. Morton, D.C. 1991, ApJS, 77, 119
4. Kaufman, V., & Sugar, J. 1986, J. Phys. Chem. Ref. Data Ser., 15, 321
5. Fuhr, J.R., Martin, G.A., & Wiese, W.L. 1988, J. Phys. Chem. Ref. Data Ser., 17, Suppl. 4
4.10
4.10.1
SPECTRA OF DIATOMIC MOLECULES
General Remarks
Realistic calculations of astronomical spectra today involve the use of extensive databases such
as HITRAN [37], RADEN [38], or the material assembled by Kurucz [39]. The proceedings of
IAU Commission 14 [40, 41] describe these sources and contain additional material, also covering
polyatomic molecules. Recent texts [42, 43] treat diatomic molecules.
4.10.2
Approximate Wave Function
It is often assumed that the total wave function of a diatomic molecule may be written as a product
containing electronic, vibrational, rotational, and nuclear spin components: ψ = ψe ψv ψr ψn . A more
general situation is considered below. Traditionally, electronic spin is included in ψe , but the nuclear
spin wave functions are written separately. In the simplest cases, ψv and ψr are the functions describing
the quantum oscillator and rotator. The latter are spherical harmonics. Sophisticated treatments of ψv
use realistic potential functions. In general, the rotational function ψr may include electronic angular
momentum. In this case, ψr is described by symmetrical top wave functions [43, 44]. For the rotational
functions to have the proper behavior with respect to parity operations, it is often necessary to use linear
combinations of symmetrical top functions.
4.10.3
Quantum Numbers and Notation
Angular momentum vectors L and S have the same meanings as for atoms. These, and other (e.g., J)
angular momenta, are often loosely referred to by the associated quantum numbers (L, S, J ).
R or O = angular momentum of nuclear (end over end) rotation. R = 0, 1, . . . .
N = total angular momentum apart from spin; formerly called K .
Sp.-V/AQuan/1999/10/29:07:01
84 / 4
Page 84
S PECTRA
S = total electron spin; (2S + 1) is given as a pre-superscript.
= projection of S on internuclear axis (can be positive or negative).
J = total angular momentum exclusive of nuclear spin.
= component of electron orbital angular momentum along one internuclear axis,
symbolized by ( = 0), ( = 1), ( = 2), . . . .
= | + |. + is used as a term subscript (e.g., 4 −1/2 , 4 3/2 ).
I = total nuclear spin.
F = total angular momentum including nuclear spin [not F(J ); cf. below].
M = projection of vector J (M J ) or F (M F ) on the z axis of the laboratory
coordinate system.
F(J ) = rotational energy in cm−1 , F1 , F2 , . . . .
A = spin-coupling constant; tabulated by [45] in footnotes.
Y = A/Bv describes intermediate coupling; small |Y | ⇒ case (b).
v = vibrational quantum number, v = 0, 1, . . . .
Te = equilibrium electronic energy (or “term value”) in cm−1 .
G(v) = vibrational energy in cm−1 .
ν00 = wave number of the 0–0 band of a band system.
+, − describe the parity of electronic wave functions of states, viz., + and − ,
with respect to reflection in plane of nuclei.
g, u
+, −
describe the parity of electronic wave functions in homonuclear diatomic molecules
with respect to inversion of electronic coordinates.
describe the total parity of ψe ψv ψr for rotational levels with respect to inversion of
all coordinates in the laboratory frame.
s, a
4.10.4
describe the parity of ψe ψv ψr ψn of homonuclear molecules with respect to exchange
of two nuclei.
Angular Momenta and Hund’s Cases [42–44, 46, 47]
The quantum numbers , , and all derive from the projection of vectors and are similar in nature to
the numbers M L , M S , and M J of atoms. In the nomenclature of molecular spectroscopy, only positive
values of these projections are commonly used. However, just as in the atomic case, positive and
negative projections occur, and it is often necessary to employ both signs in the theoretical description
of a molecular state.
Case (a): J = L + S + R. The projection of L, whose absolute magnitude is called , is well defined,
as is the projection of S, called . Unlike atoms, molecules have their full multiplicity, and ± is
written as a subscript, e.g., 4 5/2 , 4 3/2 , 4 1/2 , 4 −1/2 .
Case (b): L + R = N (formerly called K). N + S = J. Rotational levels, which may be labeled by
the quantum number N , are split into 2S + 1 sublevels if N > S, and 2N + 1 sublevels if S > N .
Case (c): L + S = Ja . The quantum numbers and are not “good,” but the projection of Ja on
the internuclear axis, , is well defined. N + Ja = J, the total angular momentum. Case (c) is common
for heavier molecules.
Sp.-V/AQuan/1999/10/29:07:01
Page 85
4.11 E NERGY L EVELS / 85
Case (d): L + R = N as in case (b), but the energy splitting due to spin and orbital angular
momentum is very small. The vector J = S + N does not differ significantly from N, and energy
levels are proportional to Bv R(R + 1).
4.11
ENERGY LEVELS
Approximate energy levels (in cm−1 ) may be calculated from the following formulas:
T = Te + G(v) + F(J ),
G(v) = ωe (v + 12 ) − ωe xe (v + 12 )2 + · · · ,
F(J ) = J (J + 1)Bv − J 2 (J + 1)2 Dv ,
Bv = Be − αe (v + 12 ) + · · · ,
Dv ≈ De .
For accurate work it is necessary to consult relations specialized for individual molecules (see [45]).
Electron spin manifests itself on molecular energy levels in a variety of ways that are not easily
described by general formulas (see [44]). The splitting of 2 levels due to spin, for example, may be
approximately described by the formulas below. Here, F1 and F2 refer to the levels with J = N + 12
and J − 12 , respectively. Y = A/B, as above.
F1 (J ) = Bv (J + 12 )2 − 2 − 12 4(J + 12 )2 + Y (Y − 4)2 + · · · ,
F2 (J ) = Bv (J + 12 )2 − 2 + 12 4(J + 12 )2 + Y (Y − 4)2 + · · · .
Levels with > 0 are twofold degenerate (±|M L |). Rotation can lift this degeneracy, giving rise
to -doubled pairs of levels with opposite parity. See [42, 48] for additional comments and notation
(a, b, c, d, e, f) used to describe rotational levels.
4.11.1
Molecular Constants
Tables 4.16 and 4.17 give the more important constants for selected electronic states of some common
diatomic molecules of astrophysical interest. These constants are sufficient for approximate and
heuristic work. For example, one may use them to locate lower-order bands and define their character
(red or violet degredation). Accurate work would require the use of more elaborate formulas than can
be written with these constants alone. Higher-order constants may be found in the papers cited.
Table 4.16. Selected constants for diatomic molecules.a
State
Te
ωe
ωe xe
Be
αe
De
re (Å)
1.599
1.115
3.053
1.994(−2)
1.656(−2)
4.644(−2)
1.03
1.29
0.74
1 H , D 0 = 4.478 07 eV
2
5
0
C 1 u 2 pπ
B 1 u+ 2 pσ
X 1 g+ 1sσ 2
100 089.8
91 700.0
0.0
2 444.66
1 357.19
4 402.93
65.58
20.15
123.07
31.324
19.984
60.847
Sp.-V/AQuan/1999/10/29:07:01
86 / 4
Page 86
S PECTRA
Table 4.16. (Continued,)
State
Te
ωe
ωe xe
.
αe
De
re (Å)
0.019 07
0.1175
0.016 816
0.012 55
0.016 969
0.016 312
0.016 662 5
0.018 01
6.72(−6)
6.86(−6)
6.319(−6)
6.517(−6)
6.509(−6)
6.196(−6)
6.463(−6)
6.964(−6)
1.27
1.38
1.39
1.21
1.32
1.37
1.31
1.24
1.162
1.170
0.013
0.016
7(−6)
1.50
1.49
1.968 79
1.715 62
1.899 783 2
0.019 96
0.017 12
0.017 372
6.58(−6)
6.129(−6)
6.406(−6)
1.15
1.23
1.17
0.018 92
0.019 04
0.017 5
6.41(−6)
6.36(−6)
6.121(−6)
1.352 3
1.205 74
1.128
0.724 2
19.38(−4)
0.969 66
Be
12 C , D 0 = 6.29 eV
2
6
0
d 3 g
B 1 1 g+
B 1 g
c 3 u+
A 1 u
b 3 g−
a 3 u
X 1 g+
20 024.597
15 409.139
12 082.336
9 124.212
8 391.408
6 435.736
718.318
0.0
1 788.222 0
1 424.119
1 407.465
2 085.899
1 608.199
1 470.415
1 641.329 59
1 855.014
16.457 4
2.571 1
11.479 4
18.623
12.060
11.155
11.651 95
13.555
1.755 523
1.481 0
1.463 685
1.921
1.616 628
1.498 64
1.632 365
1.820 10
12 C 14 N, D 0 = 7.74 eV
0
D 2 i
b 4 i
a 4+
B 2+
A 2 i
X 2+
a 3 +
a 3 r
X 1+
54 486.3
44 317
(36 400)
25 753.22
9 243.308
0.0
1 004.71
1 148
(1 400)
2 160.38
1 813.235
2 068.648
8.78
18.1
(20)
17.74
12.751
13.097
12 C 16 O, D 0 = 11.108 eV
0
55 825.49
48 686.70
0.0
1 228.60
1 743.41
2 169.814
10.468
14.36
13.288 3
1.344 6
1.691 24
1.931 3
16 O 1 H, D 0 = 4.392 eV
0
X 2 i
0.0
3 737.761
84.8813
18.9108
Note
a Units are cm−1 except as indicated. The power of ten to be applied to the entry for D is shown in parentheses.
e
References: H2 [1–4]; C2 [5–15]; CO [1,16]; CN [17–21].
References
1. Huber, K.P., & Herzberg, G. 1979, Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules
(Van Nostrand, New York)
2. Dabrowski, I. 1984, Can. J. Phys., 62, 1639
3. Abgrall, H., Roueff, E., Launay, F., Roucin, J.-Y., & Subtil, J.-L. 1993, J. Mol. Spectrosc., 157, 512
4. Balakrishnan, A., Smith, V., & Stoicheff, B.P. 1992, Phys. Rev. Lett., 68, 2149
5. Douay, M., Nietmann, R., & Bernath, P.F. 1988, J. Mol. Spectrosc., 131, 250, 261
6. Prasad, C.V.V., & Bernath, P.F. 1994, ApJS, 426, 812
7. Davis, S.P., Abrams, M.C., Phillips, J.G., & Rao, M.L.P. 1988, J. Opt. Soc. Am., B5, 2280
8. Galehouse, D.C., Brault, J.W., & Davis, S.P. 1980, ApJ, 42, 241
9. Simard, B., & Hackett, P.A. 1991, J. Mol. Spectrosc., 148, 128
10. Phillips, J.G. 1973, ApJS, 26, 313
11. Hocking, W.H., Gerry, M.C.L., & Merer, A.J. 1979, Can. J. Phys., 57, 54
12. Veseth, L. 1975, Can. J. Phys., 53, 299
13. Urdahl, R.S., Bao, Y., & Jackson, W.M. 1991, J. Chem. Phys. Lett., 178, 425
14. Amiot, C., Chauville, J., & Maillard, J.-P. 1979, J. Mol. Spectrosc., 75, 19
15. Davis, S.P., Abrams, M.C., Sandalphon, X.X., Brault, J.W., & Rao, M.L.P. 1988, J. Opt. Soc. Am., B5, 1838
16. Eidelsberg, M., Roncin, J.-Y, LeFloch, A., Launay, F., Letzelter, C., & Rostas, J. 1987, J. Mol. Spectrosc., 121, 309
17. Ito, H., Ozaki, Y., Suzuki, K., Kondow, T., & Kuchitsu, K. 1992, J. Chem. Phys., 96, 4195
18. Huang, Y., Barts, S.A., & Halpern, J.B. 1992, J. Phys. Chem., 96, 425
19. Ito, H., Ozaki, Y., Nagata, T., Kondow, T., & Kuchitsu, K. 1984, Can. J. Phys., 62, 1586
20. Prasad, C.V.V., & Bernath, P.F. 1992, J. Mol. Spectrosc., 156, 327
21. Kotlar, A.J., Field, R.W., Steinfeld, J.I., & Coxon, J.A. 1980, J. Mol. Spectrosc., 80, 86
Sp.-V/AQuan/1999/10/29:07:01
Page 87
4.12 T RANSITIONS / 87
Table 4.17. Selected constants continued: TiO.a
State
T0
ωe
ωe xe
αe
De
0.489 888
0.063 062
6.627(−7)
1.69
0.506 223
0.003 18
6.97(−7)
1.67
0.003 145
6.918(−7)
1.66
Be
re (Å)
48 Ti 16 O, D 0 = 6.87 eV
0
C 3 3
C 3 2
C 3 1
B 3 2
B 3 1
B 3 0
b 1
A3 4
A3 3
A3 2
E 3 2
E 3 1
E 3 0
d 1+
a 1
X 3 3
X 3 2
X 3 1
[19 536.63]
[19 441.47]
[19 341.68]
[16 266.797]
[16 247.951]
[16 255.986]
[14 721.14]
[14 365.60]
[14 193.69]
[14 019.43]
[12 016.13]
[11 925.26]
[11 840.15]
[5 667.10]
[3 448.32]
[202.617 7]
[97.817 7]
0.0
838.256 7
4.759 2
[863.563]
919.759 3
867.779 9
4.279 9
3.942 2
0.507 390
924
5.1
[0.515 5]
1 023.058 5
1 018.273
1 009.169 7
4.893 5
4.521
4.564 0
0.549 320
0.537 602
0.535 431
1.65
0.003 348
0.002 916
0.003 022
6.337(−7)
5.9(−7)
6.32(−7)
1.60
1.62
1.62
Note
a Units are cm−1 except as indicated. The power of ten to be applied to the entry for D is shown in
e
parentheses. For TiO the square brackets indicate that T0 is given rather than the usual Te . These apply
to the v = 0 vibrational level. The constants ωe , etc., are the same for the levels split by spin–orbit
interaction. References: TiO [1–4].
References
1. Gustavsson, T., Amiot, C., & Vergès, J. 1991, J. Mol. Spectrosc., 145, 56
2. Hildebrand, D.L. 1976, Chem. Phys. Lett., 44, 281
3. Merer, A.J. 1989, Annu. Rev. Phys. Chem., 40, 407
4. Brandes, G.R., & Galehouse, D.C. 1985, J. Mol. Spectrosc., 109, 345
4.12
TRANSITIONS
The upper level is written first, for both absorption and emission. Symbols describing the upper level
have a single prime, while a double prime is used for the lower level.
4.12.1
Rotation and Vibration
Rotational transitions in emission or absorption are assigned to P, Q, and R branches designated as
follows for dipole radiation:
P:
Q:
R:
J ↔ J = J − 1,
J ↔ J = J ,
J ↔ J = J + 1.
Transitions forbidden for electric dipole radiation can give rise to lines in an O branch (J ↔ J =
J − 2) and an S branch (J ↔ J = J + 2).
In case (b), when the spin splitting is small with respect to the rotational separation of the energy
levels, one can have P-, Q-, and R-form branches whose nomenclature depends on N and N . For
Sp.-V/AQuan/1999/10/29:07:01
88 / 4
Page 88
S PECTRA
example, a line in a P-form Q branch would arise when J ↔ J , but N ↔ N = N − 1. It
would be labeled P Q. The branch labels also contain subscripts. The symbol Q R12 would designate a
transition in a Q-form R branch from a lower level labeled F2 to an upper F1 . See [49] for additional
notation.
A common designation of rotational lines uses the J value of the lower level. Thus R(0) arises
in transitions between J = 1 and J = 0 (in absorption from J = 0 and in emission from
J = 1). Since J = 0 ↔ J = 0 is forbidden for electric dipole radiation, Q(0) does not occur.
The corresponding wave number is, however, called the band origin, ν00 or ν̃00 .
Vibrational transitions are designated by the corresponding quantum numbers. For example, the
(0–0) band means a transition from v = 0 to v = 0. The quantum number for the upper vibrational
state is written first.
4.12.2
Electronic Transitions
In the spectra of diatomic molecules line strengths are defined in the same way as for atomic transitions,
by a sum over the degenerate elementary states of both the upper and lower levels, which are labeled
by M and M :
µ|ψ M |2 .
S J J =
|ψ M |µ
M M This “line strength” is symmetrical in the upper and lower levels. The electric dipole moment,
here written as µ , is the sum of the electric moments (charge times displacement) of the electrons and
nuclei. This vector must be in the fixed or laboratory frame. For convenience, it is transformed to
the frame of the molecule with the help of Euler angles. In practice, for a given transition, only one
electron is important.
The line strength for an electronic transition may be written as a product of three factors [50, 51]
S J J = |Re |2 qv v S J J .
The quantity Re is called the electronic transition moment. Its definition, consistent with the Hönl–
London factor S (see below), is such that
|Re | = | S |z| S |,
= 0,
√ = S (x ± iy)/ 2 S ,
= ±1,
S J J = (2 − δ0, δ0, )(2S + 1)(2J + 1).
This normalization [50] holds for absorption or emission. In the former case, the value of J on
the right-hand side is J , while in the latter it is J . The Kronecker δ functions are zero if or is not equal to zero, and are unity otherwise. Consider a given J (J or J ). It is necessary to sum
the rotational strengths S for all allowed transitions from the (2 − δ0, )(2S + 1) levels with a given
J for which transitions are allowed. Thus the sum extends over more than one energy level in general
and includes lines with the same J that arise from doubling. If doubling is present in both upper
and lower levels, the number of allowed lines is exactly twice that which would result if there were
no degeneracy. However, if only one of the upper or lower levels is doubled, the resulting number of
allowed lines is the same as if neither upper nor lower were doubled because of the selection rule on
parity. The sum of these strengths may not equal the theoretical value for low levels where the full spin
multiplicity (of levels) has not developed [43].
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Page 89
4.13 S ELECTION RULES : D IPOLE R ADIATION / 89
The recommended normalization follows naturally if the rotational strengths are written with n − j
symbols [47, 52]. Thus for Hund’s case (a), we have
2
1
J J
S = (2J + 1)(2J + 1)
.
− −
The symbol in the large parentheses is a Wigner 3 − j symbol. This formula also holds for cases
(c) and (d), in the latter instance with the replacement of by . These cases are of less importance
for molecules of astrophysical interest. For case (b), it is necessary to decouple the electron spin, and
this introduces a 6 − j (curly bracket) symbol:
2 2
N
1
N N 1 N S = (2J + 1)(2N + 1)(2J + 1)(2N + 1) .
J
S J
− −
Pure Hund cases are only approximations to the more general description of molecular levels by
intermediate coupling. Intensity formulas have been given by various authors, e.g., [53–55], and
Whiting [56] has published a program for the S’s consistent with the above summation rules. We
recommend use of the Whiting code for all but – transitions, which are inherently case (b).
It is often useful to have guides to the rotational structure of electronic transitions. In addition to
the basic reference [44] useful diagrams may be found in [49, 51, 57].
Oscillator strengths and Einstein coefficients are related to S J J by the same formulas as for atoms
(Sec. 4.4).
4.13
SELECTION RULES: DIPOLE RADIATION
Many selection rules for diatomic molecules can be inferred from the properties of the n − j symbols of
Sec. 4.12; the relevant 3 − j or 6 − j symbol will vanish for the forbidden transition. For example, we
can infer for electric dipole radiation J = 0, ±1 with J = 0 ↔ J = 0. Similarly, we have = 0,
±1 for case (a) and = 0, ±1 for case (b). Case (b) also has N = 0, ±1 and N = 0 ↔ N = 0.
The 3 − j symbol vanishes if N = N while = = 0; consequently, N = 0 is forbidden for
↔ transitions [case (b)]. Similarly J = 0 for = 0 ↔ = 0 [case (a)].
The total spin operator commutes with the dipole moment; consequently, S is forbidden for
electric dipole radiation. In case (a), where is well defined, we also have = 0.
Symmetry of the electronic wave functions prevent + from combining with − , while symmetry
of the overall wave functions prevent positive–positive and negative–negative transitions. For
homonuclear molecules gerade–gerade and ungerade–ungerade transitions are prohibited, while
symmetric–antisymmetric rotational transitions cannot occur.
4.13.1
Parameters for Selected Electronic Transitions
Table 4.18 gives parameters for line-strength calculations in a few diatomic band systems of
astrophysical interest. The material is primarily for heuristic use. For detailed calculations it is
necessary to consult the sources cited. Entries are primarily from the RADEN database [38]. The
first three columns identify the systems and give wavelength ranges, following [58]. A very useful
table of persistent band heads is given in [59]. The fourth column contains the band origin for the 0–0
band. The following columns provide information relevant to line-strength calculations. Entries are
for r -centroid [Re (rv v )] and ab initio [Re (r )] calculations. The former are used with Franck–Condon
factors (qv v ) while the latter involve an integration of Re (r ) over the vibrational wave functions. The
final column of the table gives square of the transition moment for the 0–0 band, by the two methods,
2 = R 2 (r )q , while b indicates
with vibration included. A superscript a in this column indicates R00
e 00 00
2
2
R00 = |v = 0|Re (r )|v = 0| .
Swan
d 3 g –a 3 u
Deslandres–
d’Azambuja
C 1 g – A 1 u
A 2 – X 2 C2
4th positive
A 1 – X 1 +
CO
MgH
NO
NH
N+
2
N2
LaO
H2
H2
23 217.5
9 117.38
25 797.84
64 748.48
λλ4 314–4 890
λλ4 370–15 050
λλ3 440–4 600
λλ1 115–1 544 (abs.)
λλ2 006–2 785 (emiss.)
17 837.8
19 278.4
29 671.0
λλ5 015–6 450
λλ4 700–6 100
λλ2 680–5 450
A 2 – X 2 +
λλ2 800–5 900
λλ3 020–3 680
λλ1 950–3 400
First
negative
B 2 u+ – X 2 g+
A 3 – X 3 −
γ system
A 2 + – X 2 44 080.5
44 200.2
29 776.76
25 566.04
99 120.17
λλ1 028–1 239
Second
positive
C 3 u – B 3 g
90 203.55
λλ955–1 674
20 407.6
25 969.19
λλ3 390–4 110
λλ3 080–8 500
19 378.44
λλ3 400–7 850
A 2 – X 2 +
Lyman
B 1 u+ – X 1 g+
Werner
C 1 u – X 1 g+
B 2 + – X 2 +
Comet tail
Violet
B 2 + − X 2 +
CN
CO+
Red system
A 2 – X 2 +
CN
ν00 (cm−1 )
Approx. range (Å)
1.109 1
0.999 9
1.055 9
0.163 9
1.099 5
0.663 6
1.842 4
0.944
1.727 0
0.454 5
1.184 3
1.178 3
0.003 7
0.962 4
0.121 7
0.893 7
0.860 4
1.180 6
0.042 2
1.165 8
0.1150
1.206 2
0.918 0
1.291 3
0.990 7
1.131 3
0.495 7
0.724 4
1.293 8
0.544 5
q00
r00 (Å)
Re (rv v ) = (1.86 ± 0.17)(1–0.51rv v )
for rv v = 1.05–1.35 Å.
Re (r ) = 0.887 exp[−3.30(r − 0.95)]
for r = 0.95–1.40 Å.
Re (rv v ) = (12.12 ± 0.01)(1–1.631rv v +0.70rv v ) for rv v = 0.97–1.16 Å.
Re (r ) = 1.051 + 0.203 3r –0.4646r 2
for r = 0.85–2.65 Å.
Re = 0.210 ± 0.006.
See [7] for Re (r ), r = (1.25–20.0)a0 .
Re (rv v ) = (26.8 ± 1.5)(1–2.898 6rv v +2.749 9r 2 –0.8597r 3 )
vv
vv
for rv v = (1.0–1.20)a0 .
See [8] for Re (r ), r = (1.6–2.4)a0 .
See [2] for Re (r ),
r = (1.0–12)a0 .
See [6] for Re (r ),
r = (1.0–10.0)a0 .
Re (rv v ) = 348(−1 + 1.742 75rv v −0.99636r 2 + 1.880 3r 3 )
vv
vv
for rv v = 1.6–2.1 Å.
Re = 1.22
Re = 0.280 ± 0.008.
See [2] for Re (r ), r = (1.3–4.0)a0 .
Re (rv v ) = (0.19 ± 0.03)(1 + 0.571rv v )
for rv v = 1.05–1.27 Å.
See [3] for Re (r ), r = (1.4–4.0)a0 .
Re (rv v ) = (0.72 ± 0.02)(1–0.03rv v )
for rv v = 0.95–1.35 Å.
See [3] for Re (r ), r = (1.6–4.0)a0 .
Re (rv v ) = (2.94 ± 0.15)(1–0.68rv v )
for rv v = 1.0–1.3 Å.
See [4] for Re (r ), r = (1.8–8.0)a0 .
See [5] for Re (r ),
r = (1.4–3.1)a0 .
Re (rv v ) = (2.380 ± 0.28)(1 − 0.52rv v )
for rv v = 1.12–1.50 Å.
See [1]. Figure of Re (r )
for r = (2.0–3.5)a0 .
Recommended electronic
transition moment (ea0 )
0.002 6b
0.044 1a
0.046 3b
0.003 1a
0.326b
0.325a
0.25b
0.24a
1.42a
2.23a
0.078 5b
0.005 49b
0.037 1b
0.001 58b
0.427b
0.038 6a
0.042 6b
0.442a
0.078a
0.084b
0.051 1a
0.54b
0.441a
2 (ea )2
R00
0
90 / 4
CH
System
Molecule
Table 4.18. Parameters for molecular transition strengths.
Sp.-V/AQuan/1999/10/29:07:01
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S PECTRA
λλ4 200–5 600
λλ5 110–7 600
α system;
d 3 –a 3 γ system;
b 3 –a 3 21 631.48
21 548.46
21 536.36
16 033.81
15 741.31
15 426.78
1.744 8
1.756 4
0.973
1.644 8
0.995 0
1.794 0
0.313 0
1.633 1
0.672
1.659 0
0.915 2
1.631 3
0.719 1
1.008 0
0.993 2
1.545 9
0.409 2
For band (15,0)
0.000 272
1.310 7
0.906 7
q00
r00 (Å)
See [10], for Re (r )
for r = (2.8–3.8)a0 .
See [10], for Re (r ),
r = (2.8–3.8)a0 .
Re = 0.52.
Re = 1.83.
Re = 2.7.
Re = 2.25.
Re (rv v ) = (102 ± 25) exp(−2.57rv v )
for rv v = 1.58–1.72 Å.
Re (rv v ) = (0.42 ± 0.01)(1.0–0.75rv v )
for rv v = 0.8–1.2 Å.
See [9] for Re (r ), r = (1.3–4.4)a0 .
Re = 0.25 ± 0.03.
Re (rv v ) = 1.86–0.806 9rv v for rv v = 1.30–2.16 Å.
Recommended electronic
transition moment (ea0 )
References
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ZrO
λλ4 400–7 000
C 4 − – X 4 −
VO
λλ5 700–8 650
λλ5 700–6 800
λλ4 900–5 800
β system;
c 1 –a 1 γ system;
A 3 –X 3
A 2 – X 2 +
14 163.00
14 095.88
14 019.43
16 722.75
16 294.72
17 420.2
λλ4 050–6 300
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C 3 – X 3 TiO
YO
19 334.03
19 343.66
19 341.68
17 840.6
λλ3 863–4 278
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24 193.04
32 402.39
λλ2 608–4 107
OH
49 358.15
ν00 (cm−1 )
λλ1 750–5 350 (bands)
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O2
Approx. range (Å)
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Runge
B 3 u− – X 3 g−
A 2 + – X 2 System
Molecule
Table 4.18. (Continued.)
2.63b
0.97b
0.085a
3.33a
5.24a
4.63a
0.084a
0.010 9b
0.062 1a
0.009 53a
For (15,0) band
0.000 175a
2 (ea )2
R00
0
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92 / 4
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S PECTRA
ACKNOWLEDGMENTS
One of our authors (C.R.C.) thanks the following for advice and help of various kinds: P.F. Bernath,
T.M. Dunn, K.T. Hecht, Sveneric Johansson, R.L. Kurucz, W.C. Martin, D.C. Morton, R.W. Nicholls,
L.S. Rothman, R.L. Sears, R.H. Tipping, and E.E. Whiting.
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