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Atomic Spectroscopy - Atomic Lifetimes

18.   Atomic Lifetimes

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).

19.   Regularities and Scaling

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 [34]

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

(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 December 23, 2019