Half-widths of the Stark levels

We shall next give an explicit formula for the half-width T on the energy scale of a not too broad Stark level. To this purpose we write (5.17b) with the use of the approximate version of (5.18a) as u +1 

For a not too broad Stark level an adequate approximate formula for T is obtained when one neglects the change with energy of u over the width of the level. Thus one finds from (5.51) that (Q /Q )2 assumes half of its maximum value when 

The half-width expressed in terms of the variable v is thus 2arcsin [l + exp(—2ifT)]1/4/2 — [1 + exp(—2if )] 1/4/2. By multiplying this quantity by dE/dv, i.e., by dvfdE 1, which according to the approximate version of (5.18b) is equal to d(L + (f)/2)/dE 1, we obtain the half-width F, which is thus 

Procedure for Transformation of the Phase-Integral Formulas into Formulas Involving Complete Elliptic Integrals 

The phase-integral quantities in the formulas obtained in Chapter 5 can be expressed in terms of complete elliptic integrals. One thereby achieves the result that well-known properties of complete elliptic integrals, such as for instance series expansions, can be exploited for analytic studies. Furthermore, complete elliptic integrals can be evaluated very rapidly by means of standard computer programs. 

---