The radiative lifetime τ_{k} of an atomic level k is related to the sum of transition probabilities to all levels i lower in energy than k:
The branching ratio of a particular transition, say to state i ′, is defined as
If only one branch (i ′) exists (or if all other branches may be neglected), one obtains A_{ki ′} τ_{k} = 1, and
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).
The nonrelativistic energy of a hydrogenic transition [Eqs. (1), (10)] is
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 R_{M}):
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
Data for some lines of the main spectral series of hydrogen are given in the table below.
Transition ^{ } | Customary^{ }name ^{a} | λ ^{b}_{ }(Å) | g_{i} ^{c} | g_{k}^{ } | A_{ki}^{ }(10^{8} s^{-1}) | |
---|---|---|---|---|---|---|
1-2 | (L_{α }) | 1 215. | 67 | 2 | 8 | 4.699 |
1-3 | (L_{β }) | 1 025. | 73 | 2 | 18 | 5.575(-1) ^{d} |
1-4 | (L_{γ }) | 972. | 537 | 2 | 32 | 1.278(-1) |
1-5 | (L_{δ }) | 949. | 743 | 2 | 50 | 4.125(-2) |
1-6 | (L_{ε }) | 937. | 80 | 2 | 72 | 1.644(-2) |
2-3 | (H_{α }) | 6 562. | 80 | 8 | 18 | 4.410(-1) |
2-4 | (H_{β }) | 4 861. | 32 | 8 | 32 | 8.419(-2) |
2-5 | (H_{γ }) | 4 340. | 46 | 8 | 50 | 2.530(-2) |
2-6 | (H_{δ }) | 4 101. | 73 | 8 | 72 | 9.732(-3) |
2-7 | (H_{ε }) | 3 970. | 07 | 8 | 98 | 4.389(-3) |
3-4 | (P_{α }) | 18 751. | 0 | 18 | 32 | 8.986(-2) |
3-5 | (P_{β }) | 12 818. | 1 | 18 | 50 | 2.201(-2) |
3-6 | (P_{γ }) | 10 938. | 1 | 18 | 72 | 7.783(-3) |
3-7 | (P_{δ }) | 10 049. | 4 | 18 | 98 | 3.358(-3) |
3-8 | (P_{ε }) | 9 545. | 97 | 18 | 128 | 1.651(-3) |
Nonrelativistic atomic quantities for a given state or transition in an isoelectronic sequence may be expressed as power series expansions in Z^{ -1}:
where E_{0}, f_{0}, and S_{0} are hydrogenic quantities. For transitions in which n does not change (n_{i} = n_{k}), f_{0} = 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]
$$\begin{eqnarray*}
(nl-n^\prime l^\prime)_{\rm Li}&\rightarrow& \left[ (n+1)l-(n^\prime+1)
l^\prime \right]_{\rm Na}\\
&\rightarrow& \left[ (n+2)l-(n^\prime+2) l^\prime \right]_{\rm Cu}~\rightarrow~\ldots \quad .
\end{eqnarray*}$$
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.