Electric dipole (E1) ("allowed") 
Magnetic dipole (M1) ("forbidden") 
Electric quadrupole (E2) ("forbidden") 


Rigorous rules  1.  Δ J = 0, ± 1 (except 0 0) 
Δ J = 0, ± 1 (except 0 0) 
Δ J = 0, ± 1, ± 2 (except 0 0, 1/2 1/2, 0 1) 
2.  ΔM = 0, ± 1 (except 0 0 when Δ J = 0) 
ΔM = 0, ± 1 (except 0 0 when Δ J = 0) 
ΔM = 0, ± 1, ± 2  
3.  Parity change  No parity change  No parity change 

With negligible configuration interaction 
4.  One electron jumping, with Δl = ± 1, Δn arbitrary 
No change in electron configuration; i.e., for all electrons, Δl = 0, Δn = 0 
No change in electron configuration; or one electron jumping with Δl = 0, ± 2, Δn arbitrary 
For LS coupling only 
5.  ΔS = 0  ΔS = 0  ΔS = 0 
6.  ΔL = 0, ± 1 (except 0 0) 
ΔL = 0 Δ J = ± 1 
ΔL = 0, ± 1, ± 2 (except 0 0, 0 1) 

(13) 
where A_{ki} is the atomic transition probability and
N_{k} the number per unit volume (number density) of excited
atoms in the upper (initial) level k. For a homogeneous light source of
length l and for the optically thin case, where all radiation escapes,
the total emitted line intensity (SI quantity: radiance) is
(14) 
where I(λ) is the specific intensity at wavelength λ, and λ_{0} the wavelength at line center.
(15) 
is used, where I(λ) is the incident intensity at wavelength &ambda;, e.g., from a source providing a continuous background, and I′(λ) the intensity after passage through the absorbing medium. The reduced line intensity from a homogeneous and optically thin absorbing medium of length l follows as
(16) 
(17)  

(18) 
where ψ_{i} and ψ_{k} are the initial and finalstate wave functions and R_{ik} is the transition matrix element of the appropriate multipole operator P (R_{ik} involves an integration over spatial and spin coordinates of all N electrons of the atom or ion).
(19) 
Numerically, in customary units (A in s^{1}, λ in Å,
S in atomic units),
(20) 
and for S and ΔE in atomic
units,
(21) 
(22) 
The A_{ki} values for strong lines of selected elements are given. For comprehensive numerical tables of A, f, and S, including forbidden lines (see Sources of Spectroscopic Data).
Experimental and theoretical methods to determine A, f, or S values as well as atomic lifetimes are discussed in Atomic, Molecular, & Optical Physics Handbook, Chaps. 17, 18, and 21, ed. by G.W.F. Drake (AIP, Woodbury, NY, 1996).
SI units ^{a}  Numerically, in customary units ^{b}  

Electric quadrupole  
Magnetic dipole  
^{a}  A in s^{1}, λ in m._{ } Electric quadrupole: S in m^{4} C^{2}. Magnetic dipole: S in J^{2} T^{2}. 
^{b}  A in s^{1}, λ in Å. S in atomic units: = 2.013 × 10^{79} m^{4} C^{2} (electric quadrupole), = 8.601 × 10^{47} J^{2} T^{2} (magnetic dipole). µ_{B} is the Bohr magneton. 
Oscillator strengths f are not used for forbidden transitions, i.e., magnetic dipole (M1), electric quadrupole (E2), etc.
[Numerical example: For the 1s2p ^{1}P  1s3d ^{1}D_{2} (allowed) transition in He I at 6678.15 Å: g_{i} = 3; g_{k} = 5; A_{ki} = 6.38 × 10^{7} s^{1}; f_{ik} = 0.711; S = 46.9 a_{0}^{2} e^{2}.]
(23)  

(24) 
is the weighted ("multiplet") wavelength in vacuum:
(25) 
(26) 