# Distribution of Particles Which Produces a “Smart” Material

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## Abstract

If *A* _{q}(β, α, *k*) is the scattering amplitude, corresponding to a potential \( q\in L^2(D) \), where D⊂ℝ^{3} 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 ℝ^{3}, 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* _{m}⊂ *D*, 1≤ *m*≤ *M*, distributed in an a priori given bounded domain D⊂ℝ^{3}. 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 *q*∊ *L* ^{2}(*D*), i.e., corresponding to an arbitrary refraction coefficient in *D*.

## Keywords

scattering by small bodies scattering amplitude radiation pattern nanotechnology inverse scattering## Preview

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