NIST: Atomic Spectros. - Spectral Lines

17. Spectral Lines: Selection Rules, Intensities, Transition Probabilities, Values, and Line Strengths

**Selection rules for discrete transitions**
	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)

Emission Intensities (Transition Probabilities)
The total power ε radiated in a spectral line of frequency ν per unit source volume and per unit solid angle is

(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 λ₀ the wavelength at line center.
Absorption f values
In absorption, the reduced absorption

(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)

where f_ik is the atomic (absorption) oscillator strength (dimensionless).
Line Strengths
A_ki and f_ik are the principal atomic quantities related to line intensities. In theoretical work, the line strength S is also widely used (see Atomic, Molecular, & Optical Physics Handbook, Chap. 21, ed. by G.W.F. Drake (AIP, Woodbury, NY, 1996):

(17)

(18)

where ψ_i and ψ_k are the initial- and final-state 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).

Relationships between A, f, and S

The relationships between A, f, and S for electric dipole (E1, or allowed) transitions in SI units (A in s^-1, λ in m, S in m² C²) are

(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)

g_i and g_k are the statistical weights, which are obtained from the appropriate angular momentum quantum numbers. Thus for the lower (upper) level of a spectral line, g_i(k) = 2 J_i(k) + 1, and for the lower (upper) term of a multiplet,

equation 22 (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).

**Conversion relations between S and *A_ki* for the most common forbidden transitions**
	SI units ^a	Numerically, in customary units ^b

Electric quadrupole	$A_{ki}=\frac{16\pi^5}{15h \epsilon_0 \lambda^5 g_k} S$	$A_{ki}=\frac{1.1199\times 10^{18}{g_k \lambda^5} S$

Magnetic dipole	$A_{ki}=\frac{16\pi^3 \mu_0}{3h \lambda^3 g_k} S$	$A_{ki}=\frac{2.697\times 10^{13}{g_k \lambda^3} S$

^a	A in s^-1, λ in m. Electric quadrupole: S in m⁴ C². Magnetic dipole: S in J² T^-2.
^b	A in s^-1, λ in Å. S in atomic units: = 2.013 × 10^-79 m⁴ C² (electric quadrupole), $$e^2 h^2/16\pi^2 m_{\rm e}^2 = \mu_{\rm B}^2$ = 8.601 × 10^-47 J² 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 ¹P super 0 sub
1 - 1s3d ¹D₂ (allowed) transition in He I at 6678.15 Å: g_i = 3; g_k = 5; A_ki = 6.38 × 10⁷ s^-1; f_ik = 0.711; S = 46.9 a₀² e².]

Relationships between Line and Multiplet Values
The relations between the total strength and f value of a multiplet (M) and the corresponding quantities for the lines of the multiplet (allowed transitions) are

$S_M=\sum S_{\rm line} ~ ,$ (23)

(24)

is the weighted ("multiplet") wavelength in vacuum:

(25)

where

(26)

and n is the refractive index of standard air.

	(17)
	(18)

$S_M=\sum S_{\rm line} ~ ,$	(23)
	(24)

17. Spectral Lines: Selection Rules, Intensities, Transition Probabilities, Values, and Line Strengths

Emission Intensities (Transition Probabilities)

Absorption f values

Line Strengths

Relationships between A, f, and S

Relationships between Line and Multiplet Values