By Arthur H. Kruse
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Extra info for Souslinoid and Analytic Sets in a General Setting
With this in mind and introducing again lower-state broadening and second-order quadrupole terms (using the same estimates for the atomic matrix elements as in the fourth-order dipole term but multiplying by a factor of 2 to account for different intermediate states), φ can now be written in terms of parabolic wave functions as Φ = ~ ^ Ν ( — ) 3 /// + + 2 \ma0/ f-f(v) J v Γ ( Σ Γ , Κ ' ) · L iff Y± Pmin Σ r, | * " > · < * " I *« In — Σ r , | * " > · < * " I r/ l n - ^ - 1 , kff^iff (i"\ri-2zizf O ÛH / Pmin kffytfff (110) PminJ with the lower p cutoff generalized to Pmin ~ (tt;2 - nf2)h/Zmv.
Except for fourth and higher (even) order terms, any entanglement disappears then on the average over various (statistically independent) perturbers, and it is thus actually sufficient to calculate [u(s + As, s) — 1] for single perturbers, adding their contributions afterwards. ) Moreover, it is necessary for the usual impact approximation (in contrast 37 I I . 3 . 4d) to assume that the duration of a single collision is much shorter than As. , from time s) [ti(e + As, s) - l ] a v « Σ / i Δβ exp[(t/ft)ffe] | (ih)"1 f " dx U/(x) + W"2 ί+"ώ f dy U/(x) U/(y) + · · ·] •'—oo ·'—oo J X exp[-(i/Ä)ffs] = βχρ[(ί/Α)Η«][Σ/,(5, - l)]exp[-(t/Ä)H«]Ae i = exp[(t7&)i7s] φ exp[— (i/h)Hs] As = (ζ'/ί)-1 exp[(i/Ä)Äe] 5C e x p [ ~ (*'/Α)ΙΓβ] As, (92) where j stands for all parameters of a given type of collision which occurs with frequency / , · , and where Sy is the corresponding scattering (S) matrix.
This leaves φ as the most likely source of errors (in addition to those stemming from the two basic approximations) in actual calculations, since W(F) values should normally be accurate to within a few percent unless there is strong turbulence and except near F = 0. Almost all calculations of φ rely on the classical path approximation for perturbing electrons (straight for hydrogen or hyperbolic for ionized helium). The frequency of collisions occurring within impact parameter and velocity intervals p, p + dp, and v, v + dv is then simply df = 2rNf(v) pv dp dv, (102) which replaces / , in Eq.
Souslinoid and Analytic Sets in a General Setting by Arthur H. Kruse