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Coulomb phase shift

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1: 33.25 Approximations
§33.25 Approximations
Cody and Hillstrom (1970) provides rational approximations of the phase shift σ 0 ( η ) = ph Γ ( 1 + i η ) (see (33.2.10)) for the ranges 0 η 2 , 2 η 4 , and 4 η . …
2: 33.13 Complex Variable and Parameters
The quantities C ( η ) , σ ( η ) , and R , given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that
3: 33.2 Definitions and Basic Properties
33.2.9 θ ( η , ρ ) = ρ - η ln ( 2 ρ ) - 1 2 π + σ ( η ) ,
33.2.10 σ ( η ) = ph Γ ( + 1 + i η ) ,
σ ( η ) is the Coulomb phase shift. … Also, e i σ ( η ) H ± ( η , ρ ) are analytic functions of η when - < η < . …
4: 33.10 Limiting Forms for Large ρ or Large | η |
σ 0 ( η ) = η ( ln η - 1 ) + 1 4 π + o ( 1 ) ,
σ 0 ( η ) = η ( ln ( - η ) - 1 ) - 1 4 π + o ( 1 ) ,
5: 33.5 Limiting Forms for Small ρ , Small | η | , or Large
33.5.7 σ 0 ( η ) - γ η , η 0 ,
6: 33.23 Methods of Computation
§33.23 Methods of Computation
7: 5.20 Physical Applications
Rutherford Scattering
In nonrelativistic quantum mechanics, collisions between two charged particles are described with the aid of the Coulomb phase shift ph Γ ( + 1 + i η ) ; see (33.2.10) and Clark (1979). …
8: Bibliography C
  • C. W. Clark (1979) Coulomb phase shift. American Journal of Physics 47 (8), pp. 683–684.
  • W. J. Cody and K. E. Hillstrom (1970) Chebyshev approximations for the Coulomb phase shift. Math. Comp. 24 (111), pp. 671–677.
  • 9: 33.11 Asymptotic Expansions for Large ρ
    33.11.1 H ± ( η , ρ ) e ± i θ ( η , ρ ) k = 0 ( a ) k ( b ) k k ! ( ± 2 i ρ ) k ,
    10: 33.6 Power-Series Expansions in ρ
    33.6.5 H ± ( η , ρ ) = e ± i θ ( η , ρ ) ( 2 + 1 ) ! Γ ( - ± i η ) ( k = 0 ( a ) k ( 2 + 2 ) k k ! ( 2 i ρ ) a + k ( ln ( 2 i ρ ) + ψ ( a + k ) - ψ ( 1 + k ) - ψ ( 2 + 2 + k ) ) - k = 1 2 + 1 ( 2 + 1 ) ! ( k - 1 ) ! ( 2 + 1 - k ) ! ( 1 - a ) k ( 2 i ρ ) a - k ) ,