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# Atomic Spectroscopy - Atomic Lifetimes

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The radiative lifetime τk of an atomic level k is related to the sum of transition probabilities to all levels i lower in energy than k:

$$\tau_k=\left(\sum_i \, A_{ki}\right)^{-1}\quad.$$

(27)

The branching ratio of a particular transition, say to state i ′, is defined as

$$A_{ki\prime} \Big/ \sum_i \, A_{ki} = A_{ki\prime} \, \tau_k\quad.$$

(28)

If only one branch (i ′) exists (or if all other branches may be neglected), one obtains Aki ′ τk = 1, and

$$\tau_k=1/A_{ki\prime} \tau_k\quad.$$

(29)

Precision lifetime measurement techniques are discussed in Atomic, Molecular, & Optical Physics Handbook, Chaps. 17 and 18, ed. by G.W.F. Drake (AIP, Woodbury, NY, 1996).

### Transitions in Hydrogenic (One-Electron) Species

The nonrelativistic energy of a hydrogenic transition [Eqs. (1), (10)] is

$$(\Delta E)_Z=(E_k-E_i)_Z=R_M\, hc\,Z^2(1/n_i^2-1/n_k^2)\quad.$$

(30)

Hydrogenic Z scaling. The spectroscopic quantities for a hydrogenic ion of nuclear charge Z are related to the equivalent quantities in hydrogen (Z = 1) as follows (neglecting small differences in the values of RM):

$$(\Delta E)_Z=Z^2(\Delta E)_{\rm H}\quad,$$

(31)

$$(\lambda_{\rm vac})_Z=Z^2(\lambda_{\rm vac})_{\rm H}\quad,$$

(32)

$$S_Z=Z^{-2}\, S_{\rm H}\quad,$$

(33)

$$f_Z=f_{\rm H}\quad,$$

(34)

$$A_Z=Z^4 A_{\rm H}\quad,$$

(35)

For large values of Z, roughly Z > 20, relativistic corrections become noticeable and must be taken into account.

f-value trends. f values for high series members (large n′ values) of hydrogenic ions decrease according to

$$f(n,l\rightarrow n^\prime, l\pm1)\, \alpha(n^\prime)^{-3}\quad.$$

(36)

Data for some lines of the main spectral series of hydrogen are given in the table below.

Some transitions of the main spectral series of hydrogen
Transition    Customary  name a λ b  (Å) gi c gk  Aki  (108 s-1)
1-2 (Lα ) 1 215. 67 2 8 4.699
1-3 (Lβ ) 1 025. 73 2 18 5.575(-1) d
1-4 (Lγ ) 972. 537 2 32 1.278(-1)
1-5 (Lδ ) 949. 743 2 50 4.125(-2)
1-6 (Lε ) 937. 80 2 72 1.644(-2)
2-3 (Hα ) 6 562. 80 8 18 4.410(-1)
2-4 (Hβ ) 4 861. 32 8 32 8.419(-2)
2-5 (Hγ ) 4 340. 46 8 50 2.530(-2)
2-6 (Hδ ) 4 101. 73 8 72 9.732(-3)
2-7 (Hε ) 3 970. 07 8 98 4.389(-3)
3-4 (Pα ) 18 751. 0 18 32 8.986(-2)
3-5 (Pβ ) 12 818. 1 18 50 2.201(-2)
3-6 (Pγ ) 10 938. 1 18 72 7.783(-3)
3-7 (Pδ ) 10 049. 4 18 98 3.358(-3)
3-8 (Pε ) 9 545. 97 18 128 1.651(-3)
 a Lα is often called Lyman α, Hα = Balmer α, Pα = Paschen α, etc. b Wavelengths below 2000 Å are in vacuum; values above 2000 Å are in air. c For transitions in hydrogen, gi(k) = 2(ni(k))2, where ni(k), is the principal quantum number of the lower (upper) electron shell. d The number in parentheses indicates the power of 10 by which the value has to be multiplied.

### Systematic Trends and Regularities in Atoms and Ions with Two or More Electrons

Nonrelativistic atomic quantities for a given state or transition in an isoelectronic sequence may be expressed as power series expansions in Z -1:

Z -2E = E0 + E1Z -1 + E2Z -2 + ...   ,

(37)

Z 2S = S0 + S1Z -1 + S2Z -2 + ...   ,

(38)

f = f0 + f1Z -1 + f2Z -2 + ...   ,

(39)

where E0, f0, and S0 are hydrogenic quantities. For transitions in which n does not change (ni = nk), f0 = 0, since states i and k are degenerate.

For equivalent transitions of homologous atoms, f values vary gradually. Transitions to be compared in the case of the "alkalis" are 

$$\begin{eqnarray*} (nl-n^\prime l^\prime)_{\rm Li}&\rightarrow& \left[ (n+1)l-(n^\prime+1) l^\prime \right]_{\rm Na}\\ &\rightarrow& \left[ (n+2)l-(n^\prime+2) l^\prime \right]_{\rm Cu}~\rightarrow~\ldots \quad . \end{eqnarray*}$$

(Eq)

Complex atomic structures, as well as cases involving strong cancellation in the integrand of the transition integral, generally do not adhere to this regular behavior.

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Created October 3, 2016, Updated February 17, 2022