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# Atomic Spectroscopy - Spectral Lines

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### 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")
("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

$$\epsilon_{\rm line} = (4\pi)^{-1} h\nu A_{ki}\, N_k ~ ,$$

(13)

where Aki is the atomic transition probability and Nk 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

$$\begin{eqnarray*} I_{\rm line}=\epsilon_{\rm line}l&=& \int_0^{+\infty} \, I(\lambda){\rm d}\lambda\\ &~& \\ &~& (4\pi)^{-1} \, (hc/\lambda_0) \, A_{ki}\, N_kl ~ ,\end{eqnarray*}$$

(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

$$W(\lambda)=[I(\lambda)-I^\prime(\lambda)]/I(\lambda)\quad ,$$

(15)

is used, where I(λ) is the incident intensity at wavelength λ, 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

$$W_{ik} = \int_0^{+\infty} \, W(\lambda){\rm d}\lambda = \frac{e^2}{4\epsilon_0\,m_{\rm e}\,c^2} ~ \lambda_0^2 \, N_i \, f_{ik}\,l\quad ,$$

(16)

where fik is the atomic (absorption) oscillator strength (dimensionless).

### Line Strengths

Aki and fik 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):

$$S=S(i,k) = S(k,i)= |R_{ik}|^2\quad,$$

(17)

$$R_{ik}=\langle\psi_k| \, P ~ |\psi_i\rangle\quad,$$

(18)

where ψi and ψk are the initial- and final-state wave functions and Rik is the transition matrix element of the appropriate multipole operator P (Rik 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

$$A_{ki}=\frac{2\pi e^2}{m_{\rm e}c\epsilon_0\lambda^2} ~ \frac{g_i}{g_k} f_{ik}= \frac{16\pi^3}{3h\epsilon_0\lambda^3 g_k} ~S\quad,$$

(19)

Numerically, in customary units (A in s-1, λ in Å, S in atomic units),

$$A_{ki}=\frac{6.6702\times10^{15}}{\lambda^2} \frac{g_i}{g_k} f_{ik} = \frac{2.0261\times10^{18}}{\lambda^3 g_k} S\quad ,$$

(20)

and for S and ΔE in atomic units,

$$f_{ik} = \frac{2}{3} (\Delta E/g_i) S\quad.$$

(21)

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

$$\begin{eqnarray*}\bar g_{i(k)} &=& \sum_{i(k)} (2J_{i(k)} + 1)\\ &=& (2L_{i(k)} + 1) (2S_{i(k)} + 1) \quad . \end{eqnarray*}$$

(22)

The Aki 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 Aki 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: $$a_0^4 e^2 = 2.013 × 10^{-79} m^4 C^2$$ (electric quadrupole), $$e^2 h^2/16\pi^2 m_{\rm e}^2 = \mu_{\rm B}^2 = 8.601 × 10^{-47} J^2 T^{-2}$$ (magnetic dipole). µB 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 1P10 - 1s3d 1D2 (allowed) transition in He I at 6678.15 Å: gi = 3; gk = 5; Aki = 6.38 × 107 s-1; fik = 0.711; S = 46.9 a02 e 2.]

### 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)

$$f_M=(\bar\lambda \bar{g}_i)^{-1} \sum_{J_k J_i} g_i\lambda(J_i, J_k) f(J_i, J_k) \quad .$$

(24)

$$\bar\lambda$$ is the weighted ("multiplet") wavelength in vacuum:

$$\bar\lambda=n\bar\lambda_{\rm air} = hc/\overline{\Delta E} \quad,$$

(25)

where

$$\overline{\Delta E}=\overline{E_k}-\overline{E_i}= (\bar g_k)^{-1} \sum_{J_k} g_k E_k-(\bar g_i)^{-1} \sum_{J_i} g_i E_i\quad ,$$

(26)

and n is the refractive index of standard air.

### Relative Strengths for Lines of Multiplets in LS Coupling

This table lists relative line strengths for frequently encountered symmetrical (P → P, D → D) and normal (S → P, P → D) multiplets in LS coupling. The strongest, or principal, lines are situated along the main diagonal of the table and are called x1, x2, etc. Their strengths normally diminish along the diagonal. The satellite lines yn and zn are usually weaker and deviate more from the LS values than the stronger diagonal lines when departures from LS coupling are encountered. The total multiplet strengths SM are also listed in this table. A discussion of their normalization as well as more extensive tables are given in Ref. [33].

### Relative Strengths for Lines of Multiplets in LS Coupling

Normal multiplets S - P, P - D, D - F, etc. Symmetrical multiplets P - P, D - D, etc.
Jm Jm - 1 Jm - 2 Jm - 3 Jm - 4   Jm Jm - 1 Jm - 2 Jm - 3
Jm - 1 x1 y1 z1     Jm       x1 y1
Jm - 2   x2 y2 z2   Jm - 1 y1 x2 y2
Jm - 3     x3 y3 z3 Jm - 2   y2 x3 y3
Jm - 4       x4 y4 Jm - 3     y3 x4

Multiplicity

Multiplicity
1 2 3 4 5   1 2 3 4 5
S - P D - D
SM = 3 6 9 12 15 SM = 25 50 75 100 125
x1 3.00 4.00 5.00 6.00 7.00 x1 25.00 28.00 31.11 34.29 37.50
y1   2.00 3.00 4.00 5.00 x2   18.00 17.36 17.29 17.50
z1     1.00 2.00 3.00 x3     11.25 8.00 6.25
x4       5.00 1.25
P - P
SM = 9 18 27 36 45 y1   2.00 3.89 5.71 7.50
y2     3.75 7.00 10.00
x1 9.00 10.00 11.25 12.60 14.00 y3       5.00 8.75
x2   4.00 2.25 1.60 1.25 y4         5.00
x3       1.00 2.25
D - F
y1   2.00 3.75 5.40 7.00 SM = 35 70 105 140 175
y2     3.00 5.00 6.75
x1 35.00 40.00 45.00 50.00 55.00
P - D x2   28.00 31.11 34.29 37.50
SM = 15 30 45 60 75 x3     21.00 22.40 24.00
x4       14.00 14.00
x1 15.00 18.00 21.00 24.00 27.00 x5         7.00
x2   10.00 11.25 12.60 14.00
x3     5.00 5.00 5.25 y1   2.00 3.89 5.71 7.50
y2     3.89 7.31 10.50
y1   2.00 3.75 5.40 7.00 y3       5.60 10.00
y2     3.75 6.40 8.75 y4         7.00
y3       5.60 6.75
z1     .11 .29 .50
z1     .25 .60 1.00 z2       .40 1.00
z2       1.00 2.25 z3         1.00
z3         3.00

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Created October 3, 2016, Updated March 21, 2022