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Journal of Statistical Physics

, Volume 127, Issue 5, pp 915–934 | Cite as

Distribution of Particles Which Produces a “Smart” Material

  • A. G. RammEmail author
Article

Abstract

If A q(β, α, k) is the scattering amplitude, corresponding to a potential \( q\in L^2(D) \), where D⊂ℝ3 is a bounded domain, and \( e^{ik\alpha \cdot x} \) is the incident plane wave, then we call the radiation pattern the function \( A(\beta):=A_q(\beta, \alpha, k) \), where the unit vector α, the incident direction, is fixed, β is the unit vector in the direction of the scattered wave, and k>0, the wavenumber, is fixed. It is shown that any function \( f(\beta)\in L^2(S^2) \), where S 2 is the unit sphere in ℝ3, can be approximated with any desired accuracy by a radiation pattern: \( ||f(\beta)-A(\beta)||_{L^2(S^2)}<\epsilon \), where ∊ >0 is an arbitrary small fixed number. The potential q, corresponding to A(β), depends on f and ∊, and can be calculated analytically. There is a one-to-one correspondence between the above potential and the density of the number of small acoustically soft particles D mD, 1≤ mM, distributed in an a priori given bounded domain D⊂ℝ3. The geometrical shape of a small particle D m is arbitrary, the boundary S m of D m is Lipschitz uniformly with respect to m. The wave number k and the direction α of the incident upon D plane wave are fixed. It is shown that a suitable distribution of the above particles in D can produce the scattering amplitude \( A(\alpha',\alpha)$, $\alpha',\alpha\in S^2 \), at a fixed k>0, arbitrarily close in the norm of L 2(S 2× S 2) to an arbitrary given scattering amplitude f(α ', α), corresponding to a real-valued potential qL 2(D), i.e., corresponding to an arbitrary refraction coefficient in D.

Keywords

scattering by small bodies scattering amplitude radiation pattern nanotechnology inverse scattering 

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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Mathematics DepartmentKansas State UniversityManhattanUSA

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