Alexander G. Ramm^1*

¹ Department of Mathematics, Kansas State University, Manhattan, KS 66506-2602, USA.

*Corresponding Author:Alexander G. Ramm, Department of Mathematics, Kansas State University, Manhattan, KS 66506-2602, USA.

Citation: Alexander G. Ramm (2024) Wave Scattering by Many Small Impedance Particles and Applications. Nano Technol & Nano Sci J 6: 158.

Received:  June 19, 2024; Accepted: June 26, 2024; Published: June 30, 2024.

Copyright: © 2024 Alexander G. Ramm, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Formulas are derived for solutions of many-body wave scattering problem by small impedance particles embedded in a homogeneous medium. The limiting case is considered, when the size α of small particles tends to zero while their number tends to infinity at a suitable rate. The basic physical assumption is α<<d<<λ, where d is the minimal distance between neighboring particles, λ is the wavelength, and the particles can be impedance balls Β(xm,α ) with centers xm located on a grid. Equations for the limiting effective (self-consistent) field in the medium are derived. It is proved that one can create material with a desired refraction coefficient by embedding in a free space many small balls of radius α with prescribed boundary impedances. The small balls can be centered at the points located on a grid. A recipe for creating materials with a desired refraction coefficient is formulated. It is proved that materials with a desired radiation pattern, for example, wave-focusing materials, can be created.

There is a large literature on wave scattering by small bodies, starting from Rayleigh’s work (1871), [1, 2, 36]. For the problem of wave scattering by one body an analytical solution was found only for the bodies of special shapes, for example, for balls and ellipsoids. If the scatterer is small then the scattered field can be calculated analytically for bodies of arbitrary shapes, see [5], where this theory is presented. The many-body wave scattering problem was discussed in the literature mostly numerically, if the number of scatterers is small, or under the assumption that the influence of the waves, scattered by other particles on a particular particle is negligible (see [3], where one finds a large bibliography, 1386 entries). This corresponds to the case when the distance d between neighboring particles is much larger than the wavelength , and the characteristic size α of a small body (particle) is much smaller than λ. Theoretically and practically the assumptions α<<λ, d>>λ  are the simplest and they allow to neglect multiple scattering. By k = 2π the wave number is denoted. In contrast, in our theory the basic assumption is α<<d<<λ, and the multiple scattering is of basic importance. We give references to our papers and monographs in which the theory of wave scattering by small bodies of arbitrary shapes was developed under the assumption α<<d<<λ, [4–34]. The novelty of the results in this paper is in the location of the small bodies: they are placed o

n a grid. This may be of practical interest. In [35] for the first time the scattering problem for 10 billions small particles is solved numerically and numerical results are presented. This paper is a presentation of the new results under simplifying assumptions: the small particles D_m=B(x_m,α), 1 ≤ m ≤ M, are impedance balls with prescribed boundary impedances ζ_m; the centers x_m of the balls are placed on a grid and are embedded in a homogeneous space in a bounded domain D, for example, in a box. The basic results of this paper consist of:

1. Solution to many-body wave scattering problem by small impedance particles, embedded in a homogeneous medium, under the assumptions α<<d<<λ, where d is the minimal distance between neighboring particles and λ is the wavelength in this medium.
2. Derivation of the equations for the limiting effective (self-consistent) field in this medium, in which many small impedance particles are embedded, when α → 0 and the number M = M (α) of the small particles tends to infinity at an appropriate rate.
3. Derivation of linear algebraic systems (LAS) for solving many-body wave scattering problems. These systems are not obtained by a discretization of boundary integral equations, and they give an efficient numerical method for solving many- body wave scattering problems in the case of small scatterers under the assumption α<<d<<λ.
4. Formulation of a recipe for creating materials with a desired refraction coefficient.
5. Formulation of a method for creating materials with a desired radiation pattern.
Our methods give powerful numerical methods for solving many-body wave scattering problems in the case when the scatterers are small (see [31]).
Let us formulate the wave scattering problems we deal with. Let D be a bounded domain in R3 with a sufficiently smooth boundary. The scattering problem consists of finding the solution to the problem:

It will be clear from Section 3 that the function h (x) can be determined by choosing a suitable boundary impedance ζ(x). When α → 0, the ζ_m and h (x_m) can be considered as continuous functions ζ(x) and h (x_m).
The many-body scattering problem (1)–(4) has a solution and this solution is unique, see [31] In Section 2 a method for solving this problem is given. In Section 3 a recipe for creating materials with a desired refraction coefficient is given. In Section 4 a recipe for creating materials with a desired radiation pattern is given.

SolutionofMany-BodyScatteringProblem

We look for the solution of the form

where σ_m (S) are unknown, Q_m := S σ_m (S) ds. One may think about σ_m as of charge densities on S_m and of Q_m as of total charge on the surface S_m. We prove that is negligible compared to as α → 0.

This estimate justifies our claim since α « d. It follows that asymptotically, as  → 0, one has for |x − x_m | ≥ α. Note that M = 0(α^x−2). Formula (11) allows one to calculate u(x) at any point x, if the numbers u_m, 1 ≤ m ≤ M, are known. One can use the following linear algebraic system (LAS) for finding u_m.

The order M = 0(α^x−2) of this system is large if α is small. One can reduce this order: consider a covering of D by nonintersecting small cubes Δ_þ, 1 ≤ þ ≤ P, such that d « diam(Δ_þ) « λ, um ∼ uþ, hm ∼ hþ for all xm ∈ Δ_þ . Then formula (12) can be written as by formula (6). As α → 0, diam(Δ_þ) → 0 and formula (13) yields in the limit the integral equation for u.

LEMMA 2. Eq. (14) has a solution, this solution is unique and it is a limiting value of the solution to the scattering problem (1)– (4).

Proof: Apply the operator  ∇2+k2 to equation (14) and get (∇2+k2) u = 4πh(x)u(x.)   

This is a Schrödinger equation with potential q(x):=  4πh(x); equations (2)– (3) hold. We assumed lmh ≤ 0. Therefore (15) has at most one solution. It is a Fredholm-type equation, so it has a solution. Lemma 2 is proved.

It follows from Lemma 2 that the LAS (13) for  u p is solvable and its solution is unique. Let us write Eq. (15) as

∇²u+k²n²(x)u = 0, n²(x): = 1- 4πk²h(x) 

Embedding small impedance balls B (xm, α) in D results in creating in D a new material with the refraction coefficient n(x) = (1- 4πk-2 h(x))1\2. If one wants to have a material with the refraction coefficient n(x), then one chooses by (17) the function h(x). If h(x) is chosen, then one knows the boundary impedance ζ (x) which generates the desired h(x). The practical problem is to prepare small particles with the desired boundary impedance.

Recipe for Creating Materials with A Desired Refraction Coefficient

Let us formulate a recipe for creating materials with a desired refraction coefficient. Formula (17) shows that if h(x) is chosen properly, then any n(x) can be obtained in D.

Recipe for creating materials with a desired refraction coefficient:

a) Calculate by formula (17) the function h(x);

b) Distribute small impedance balls in the domain D by the distribution law (6).

The boundary impedances of these balls are defined by the function h(x).

Theorem 1. The refraction coefficient of the resulting medium tends to the desired coefficient n(x) as α → 0.

Let us show that a practically negative refraction coefficient n(x) can be obtained by the above recipe. Denote b:= 4πk-2 > 0 and write (17) as n(x) = (1-bh(x))1/2 = |1-bh(x)| 1/2 eØ/2, where Ø is the argument of 1-bh(x). Since the operator in (14) is Fredholm, it remains Fredholm under small perturbations. Therefore, one can take h - i∈ , where ∈> 0 is sufficiently small and equation (14) will still have a unique solution.

By choosing h so that Re (1-bh) > 0  and lm(1-bh) < 0 and small, one gets the argument Ø = 2π - δ, where δ > 0 is arbitrarily small if ∈ is sufficiently small. Then n(x) will be nearly negative: its argument will be π - δ/2.

Creating Materials with A Desired Radiation Pattern

Let us define what we mean by radiation pattern. Consider the scattering problem for Eq. (15)

where ???? satisfies the radiation condition. Assume that k > 0  and α∈s 2 are fixed. Then the scattering amplitude A(β, α, K) = A(β), where the dependence on K, α is dropped since K and α are fixed. The formula for the scattering amplitude is known, see, e.g. [34],

We call  A(β) the radiation pattern. Consider an inverse problem (IP): Given an arbitrary f (β) ∈ L2 (S2) and an arbitrary small ∈ > 0,  can one find a q∈L 2 (D) such that

Theorem 2. For any f (β) ∈ L2 (S2) and an arbitrary small ∈ > 0 there is a q∈L 2 (D) such that (20) holds.

Since small perturbations of q result in small perturbations of A(β), there are infinitely many potentials q for which inequality (20) holds.

The conclusion of Theorem 2 follows from Lemmas 3 and 4.

Lemma 3. The set {∫De - ikβ.xh(x)dx}  ∇h∈L 2 (D) is dense in L 2 (S 2).

Corollary 1. Given f ∈ L2 (S2) and ∈ > 0, one can find h∈L 2 (D) such that

Lemma 4. The set {q(x)u (x, α)} ∀ q∈L 2 (D).

Corollary 2. Given h∈L 2 (D) and ∈ > 0, one can find q∈L 2 (D)  such that

Since the scattering amplitude depends continuously on h, the inverse problem IP is solved by Lemmas 3 and 4.

Proof of Lemma 3: Assume the contrary. Then ∃ψ∈L 2 (S 2) such that

If  q∈L 2 (D), then this q solves the problem, and u, defined in (21), is the scattering solution and If q is not in L 2 (D), then the null set N: = {X: X∈D, u(x) = 0} is non-void.

The condition |∇uj |l ≥ c > 0, j = 1, 2, implies that a tubular neighborhood of the line l, Nδ = {x: √ |μ1|2 +  |u2 | 2 ≤ δ}, is included in a region {x: |x| ≤ c 0 2 δ}. This follows from the estimates

Here ξ∈l, x is a point on a plane passing through ???? and orthogonal to l, p = |x - ξ|, and δ > 0 is sufficiently small, so that the terms of order  p 2 are negligible, c2 = maxξ∈l|∇u(ξ)|,c_{2 =} minξ∈l|∇u(ξ)|.

Claim 2, and, therefore, Lemma 4 are proved.

Therefore, Theorem 2 is proved.

1. H C van de Hulst (1961) Light Scattering by Small Particles. Dover, New York.
2. L Landau, L Lifschitz (1984) Electrodynamics of Continuous Media. Pergamon Press, Oxfor.
3. P Martin (2006) Multiple Scattering, Cambridge Univ. Press, Cambridge.
4. AG Ramm (1986) Scattering by Obstacles, D. Reidel, Dordrecht.
5. AG Ramm (2005) Wave Scattering by Small Bodies of Arbitrary Shapes, World Sci. Publishers, Singapore.
6. AG Ramm (2007) Scattering by many small bodies and applications to condensed matter physics. Europe.
7. AG Ramm (2007) Many-body wave scattering by small bodies and applications. J. Math. Phys 48: 103-511.
8. AG Ramm (2007) Wave scattering by small particles in a medium. Phys. Lett.  367: 156–161.
9. A G Ramm (2007) Wave scattering by small impedance particles in a medium. Phys. Lett. A 368: 164–172.
10. A G Ramm (2007) Distribution of particles which produces a desired radiation pattern, Communic. in NonlinearSci. and Numer. Simulation 12: 1115–1119.
11. A G Ramm (2007) Distribution of particles which produces a “smart” material. J. Stat. Phys 127:915–934.
12. A G Ramm (2007) Distribution of particles which produces a desired radiation pattern. Physica B 394:253–255.
13. A G Ramm (2008) Creating wave-focusing materials, LAJSS (Latin-American J. Solids Structures) 5: 119–127.
14. A G Ramm (2008) Electromagnetic wave scattering by small bodies. Phys. Lett. A 372: 4298–4306.
15. A G Ramm (2008) Wave scattering by many small particles embedded in a medium. Phys. Lett. A 372: 3064–3070.
16. A G Ramm (2009) Preparing materials with a desired refraction coefficient and applications, In the bookTopics in Chaotic Systems: Selected Papers from Chaos 2008 International Conference”, Editors C. Skiadas, I. Dimotikalis, Char. Skiadas, World Scientific Publishing 265–273.
17. A G Ramm (2009) Preparing materials with a desired refraction coefficient, Nonlinear Analysis: Theory Methods 70:186-190.
18. A G Ramm (2009) Creating desired potentials by embedding small inhomogeneities. J. Math. Phys 50: 123-525.
19. A G Ramm (2010) A method for creating materials with a desired refraction coefficient. Int. J. Mod. Phys B 24:5261–5268
20. A G Ramm (2010) Materials with a desired refraction coefficient can be created by embedding small particles into the given material. IJSCS 2:17–23.
21. A G Ramm (2011) Wave scattering by many small bodies and creating materials with a desired refraction Coefficient. Afrika Matematika 22:33–55.
22. A G Ramm (2011) Scattering by many small inhomogeneities and applications, in: Topics in Chaotic Systems: Selected Papers from Chaos 2010 International Conference, Editors C. Skiadas, I. Dimotikalis, Char. Skiadas, World Sci. Publishing. 41–52.
23. A G Ramm (2010) Collocation method for solving some integral equations of estimation theory. Int. J. Pure Appl. Math 62: 57–65.
24. A G Ramm (2010) A method for creating materials with a desired refraction coefficient. Int. J. Mod. Phys B 27: 5261–5268.
25. A G Ramm (2011) Electromagnetic wave scattering by a small impedance particle of arbitrary shape. Opt. Commun 284: 3872–3877.
26. A G Ramm (2011) Scattering of scalar waves by many small particles. AIP Advances 1: 022-135.
27. A G Ramm (2011) Scattering of electromagnetic waves by many thin cylinders. Results in Physics 1: 13–16.
28. A G Ramm (2012) Electromagnetic wave scattering by many small perfectly conducting particles of an arbitrary shape. Opt. Commun 18: 3679–3683.
29. A G Ramm (2013) Electromagnetic wave scattering by small impedance particles of an arbitrary shape. JAMC 43: 427-444.
30. A G Ramm (2013) Many-body wave scattering problems in the case of small scatterers, JAMC 41:473–500.
31. A G Ramm (2013) Scattering of Acoustic and Electromagnetic Waves by Small Bodies of Arbitrary Shapes. Applications to Creating New Engineered Materials, Momentum Press, New York.
32. A G Ramm (2020) Creating Materials with a Desired Refraction Coefficient, IOP Publishers, Bristol, UK.
33. A G Ramm (2020) How can one create a material with a prescribed refraction coefficient? Sun Text Review of Material Science 1:1-102.
34. A G Ramm (2017) Scattering by Obstacles and Potentials, World Sci. Publ, Singapore.
35. A G Ramm (2015) N. Tran, A fast algorithm for solving scalar wave scattering problem by billions of particles. J. Algorithms Optimization 3: 1–13.
36. J Rayleigh, Scientific Papers, Cambridge Univ. Press, Cambridge 1992.