(27) |

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

(28) |

If only one branch (*i* ′) exists (or if all other branches
may be neglected), one obtains
*A _{ki ′} τ_{k}* = 1, and

(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

(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*R*_{M}):

(31)

(32)

(33)

(34)

(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

(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)*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)

### 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*^{-2}*E*=*E*_{0}+*E*_{1}*Z*^{-1}+*E*_{2}*Z*^{-2}+ ... ,(37) *Z*^{2}*S*=*S*_{0}+*S*_{1}*Z*^{-1}+*S*_{2}*Z*^{-2}+ ... ,(38) *f*=*f*_{0}+*f*_{1}*Z*^{-1}+*f*_{2}*Z*^{-2}+ ... ,(39) *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]

(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.