Error Estimates for Phaseless Inverse Scattering in the Born Approximation at High Energies

The enhancement factor α D ≡ 1+ δα D is defined (in D dimensions) as the ratio of the particle density at the origin to the density far upstream in the incident beam. At high incident momentum p (and for regular potentials) the first Born approximation is known to be adequate in 3D and 1D, and is assumed to be adequate also in 2D; it entails δα 1 = ─δα 3 if U ( r ) in 3D equals U ( x ) in 1D when r = x . For central potentials U ( r ) with U (0) finite, previous work implies δα 3 ~ ─δα 1 ~ ─ mU (0)/ p 2 , and δα 2 = 0 to the same order h °. It is shown that if U ( r →0) ~ U (0) + r U' (0) + ..., then δα 2 ~ (2 m U' (0) h / p 3 )(π/16), the value of U (0) being irrelevant. If U ( r →0) ~ ─ C / r q , with 0 < q < 2, then δα 3 ~ ─δα 1 ~ (2 mC / h q p 2- q ) π ½ Γ(1-½ q )/2Γ(½+½ q ); and, with -1 < q < 2, δα 2 ~ (2 m C / h q p 2- q )π ½ Γ 2 (1-½ q )/2Γ(½ q )Γ(3/2-½ q ). The only mathematics needed in 3D and 1D is the standard asymptotic estimation of Fourier integrals; but in 2D one needs to develop corresponding methods for integrals where the sine or cosine has been replaced by a product J 0 Y 0 of two Bessel functions.

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The Scattering of Positrons by Helium: Total Cross Sections up to 270 eV

Canadian Journal of Physics ◽

10.1139/p75-122 ◽

1975 ◽

Vol 53 (10) ◽

pp. 962-967 ◽

Cited By ~ 30

Author(s):

B. Jaduszliwer ◽

A. Nakashima ◽

D. A. L. Paul

Keyword(s):

Energy Range ◽

Cross Section ◽

Cross Sections ◽

Born Approximation ◽

Reasonable Agreement ◽

Experimental Results ◽

Formation Cross Section ◽

Sum Rule ◽

Total Cross Sections ◽

High Energies

The total cross sections for the scattering of positrons by helium have been measured by the method of transmission in the 16 to 270 eV energy range. The experimental results are higher than those of Canter et al. but are in reasonable agreement with recent results of Griffith et al., and at high energies tend towards Born approximation calculations. The integral of the cross section over positron momentum is smaller than the sum rule estimate made by Bransden et al. A tentative value of (0.034 ± 0.017)πa02 is assigned to the positronium formation cross section at threshold.

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TWO VERSIONS OF QUASI-NEWTON METHOD FOR MULTIDIMENSIONAL INVERSE SCATTERING PROBLEM

Journal of Computational Acoustics ◽

10.1142/s0218396x93000123 ◽

1993 ◽

Vol 01 (02) ◽

pp. 197-228 ◽

Cited By ~ 16

Author(s):

SEMION GUTMAN ◽

MICHAEL KLIBANOV

Keyword(s):

Newton Method ◽

Inverse Scattering ◽

Born Approximation ◽

Scattering Problem ◽

Plane Waves ◽

Inverse Scattering Problem ◽

Circular Cylinders ◽

Iterative Sequence ◽

Scattering Field ◽

Quasi Newton

Suppose that a medium with slowly changing spatial properties is enclosed in a bounded 3-dimensional domain and is subjected to a scattering by plane waves of a fixed frequency. Let measurements of the wave scattering field induced by this medium be available in the region outside of this domain. We study how to extract the properties of the medium from the information contained in the measurements. We are concerned with the weak scattering case of the above inverse scattering problem (ISP), that is, the unknown. spatial variations of the medium are assumed to be close to a constant. Examples can be found in the studies of the wave propagation in oceans, in the atmosphere, and in some biological media. Since the problems are nonlinear, the methods for their linearization (the Born approximation) have been developed. However, such an approach often does not produce good results. In our method, the Born approximation is just the first iteration and further iterations improve the identification by an order of magnitude. The iterative sequence is defined in the framework of a Quasi-Newton method. Using the measurements of the scattering field from a carefully chosen set of directions we are able to recover (finitely many) Fourier coefficients of the sought parameters of the model. Numerical experiments for the scattering from coaxial circular cylinders as well as for simulated data are presented.

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