Journal of Statistical Physics

, Volume 127, Issue 5, pp 915–934

# Distribution of Particles Which Produces a “Smart” Material

• A. G. Ramm
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

## References

1. 1.
H. Bateman and A. Erdelyi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).Google Scholar
2. 2.
V. Marchenko and E. Khruslov, Boundary-Value Problems in Domains with Fine-Grained Boundary (Naukova Dumka, Kiev, 1974). (Russian)Google Scholar
3. 3.
A. G. Ramm, Scattering by Obstacles, pp. 1–442 (D. Reidel, Dordrecht, 1986).Google Scholar
4. 4.
A. G. Ramm, Recovery of the potential from fixed energy scattering data, Inverse Probl. 4:877–886 (1988); 5:255 (1989).Google Scholar
5. 5.
A. G. Ramm, Stability estimates in inverse scattering, Acta Appl. Math. 28(N1):1–42 (1992).Google Scholar
6. 6.
A. G. Ramm, Stability of solutions to inverse scattering problems with fixed-energy data. Milan J. Math. 70:97–161 (2002).Google Scholar
7. 7.
A. G. Ramm, Inverse Problems (Springer, New York, 2005).Google Scholar
8. 8.
A. G. Ramm, Wave Scattering by Small Bodies of Arbitrary Shapes (World Science Publishers, Singapore, 2005).Google Scholar
9. 9.
A. G. Ramm, Distribution of particles which produces a desired radiation pattern, Commun. Nonlinear Sci. Numer. Simulation, doi:10.1016/j.cnsns.2005.11.001 (to appear).Google Scholar
10. 10.
A. G. Ramm, Inverse scattering problem with data at fixed energy and fixed incident direction, (submitted).Google Scholar
11. 11.
A. G. Ramm and S. Gutman, Computational method for acoustic focusing, Intern. J. Comput. Sci. Math. 1 (2007) (to appear).Google Scholar
12. 12.
A. G. Ramm, Many-body wave scattering by small bodies, J. Math. Phys. 48(N1): (2007).Google Scholar

## Authors and Affiliations

1. 1.Mathematics DepartmentKansas State UniversityManhattanUSA