Periodic particle arrangements using standing acoustic waves

Abstract

We determine crystal-like materials that can be fabricated by using a standing acoustic wave to arrange small particles in a non-viscous liquid resin, which is cured afterwards to keep the particles in the desired locations. For identical spherical particles with the same physical properties and small compared to the wavelength, the locations where the particles are trapped correspond to the minima of an acoustic radiation potential which describes the net forces that a particle is subject to. We show that the global minima of spatially periodic acoustic radiation potentials can be predicted by the eigenspace of a small real symmetric matrix corresponding to its smallest eigenvalue. We relate symmetries of this eigenspace to particle arrangements composed of points, lines or planes. Since waves are used to generate the particle arrangements, the arrangement’s periodicity is limited to certain Bravais lattice classes that we enumerate in two and three dimensions.

1. Introduction

We are interested in characterizing the possible periodic or crystal-like materials that can be fabricated by using ultrasound directed self-assembly [1,2]. In this fabrication method, a liquid resin containing small particles is placed in a reservoir that is lined by ultrasound transducers. By operating the transducers at a fixed frequency, a standing acoustic wave is generated in the liquid and drives the particles to certain locations. For example, when the particles are neutrally buoyant and less compressible than the surrounding fluid, the particles tend to go to the wave nodes (zero amplitude locations), as we explain later. Once the particles are in the desired positions, the resin is cured (i.e. hardened with light, a curing agent, etc.) and we obtain a material with inclusions placed in a periodic fashion.

To give a rough idea, one could use this technique to fabricate a crystal-like material (essentially a grating) that selectively reflects millimetre waves, i.e. electromagnetic waves with wavelengths of the order of 1 mm to 1 cm. Indeed, one could achieve this by placing subwavelength metallic particles periodically inside a dielectric resin matrix. If we assume the speed of sound in the resin matrix is 1500 m s⁻¹, we can expect the particles to be separated by between 5 μm and 50 μm provided the transducer operating frequency is between 150 kHz and 1.5 MHz. This rough estimate is based on a spacing of half an ultrasound wavelength that we rigorously justify later. A simple motivation, for now, is to observe that in one dimension, there are actually two nodes of the wave per wavelength. We emphasize though that our analysis is valid in the case where the particles do not cluster about the wave nodes and holds in two or three dimensions.

The use of acoustic waves for manipulating particles has been widely studied, especially for applications to microfluidics [3,4] and for use as acoustic tweezers [5–7], which have particular appeal as a biomedical tool [8]. Other applications include single-cell patterning using surface acoustic waves [9], tissue engineering [10] and fabrication of laminates [11]. Additionally, acoustically configurable crystals were already considered in [12,13]. Our work shows the theoretical limits of manipulation of particles with standing acoustic waves in the particular case where the desired particle patterns are periodic. We derive explicit control strategies to obtain different crystallographic symmetries and give easy to check criteria to predict whether the acoustic traps occur on isolated points, lines or planes.

For our study, we assume that acoustic waves at a fixed frequency f propagate in a fluid. The pressure and fluid velocity fields have the form $\tilde{p}(\mathit{x},t)=\mathrm{\Re }(\exp [-\text{i}\omega t]p(\mathit{x}))$ and $\tilde{\mathit{v}}(\mathit{x},t)=\mathrm{\Re }(\exp [-\text{i}\omega t]\mathit{v}(\mathit{x}))$ at position $\mathit{x}\in {\mathbb{R}}^d$ (d = 2 or 3) and time t. Here, ω = 2πf is the angular frequency and $\mathrm{\Re }$ denotes the real part of a complex quantity. The time-domain pressure and fluid velocity are related by

$$\begin{array}{rl}{\tilde{p}}_t+{\kappa }_0\mathrm{\nabla }\cdot \tilde{\mathit{v}}& =0\\ \text{and}{\rho }_0{\tilde{\mathit{v}}}_t+\mathrm{\nabla }\tilde{p}& =0,\end{array}\}$$ 1.1

where κ₀ is the compressibility of the fluid and ρ₀ its mass density. By using the expressions for $\tilde{p}$ and $\tilde{\mathit{v}}$ in (1.1), we see that the frequency domain pressure p is a complex valued solution to the Helmholtz equation Δp + k² p = 0, where the wavenumber is k = ω/c, and $c=\sqrt{{\kappa }_0/c_0}$ is the velocity of propagation of acoustic waves in the fluid (e.g. [14]). Note that the pressure and velocities in the frequency domain are related by $\mathit{v}={(\text{i}\omega {\rho }_0)}^{-1}\mathrm{\nabla }p$.

(a). The acoustic radiation potential

Small (compared to the wavelength) particles in a fluid are subject to an acoustic radiation force [15–18]. It is convenient to study the average of this force over a time period T = 1/f. To be more precise, the time average over a period of T-periodic function g is

$$⟨g⟩=\frac{1}{T}{\int }_0^{\text{T}}g(t)\hspace{0.17em}\text{d}t.$$

The net acoustic radiation force experienced by a small particle and averaged over a time period is proportional to $\mathit{F}=-\mathrm{\nabla }\psi$, where ψ(x) is the so-called acoustic radiation potential. For small spherical particles of fixed size, the acoustic radiation potential is given by¹

$$\psi ={\mathfrak{f}}_1\frac{{\kappa }_0}{2}⟨|\tilde{p}{|}^2⟩-{\mathfrak{f}}_2\frac{3{\rho }_0}{4}⟨|\tilde{\mathit{v}}{|}^2⟩,$$ 1.2

where ${\mathfrak{f}}_1=1-({\kappa }_p/{\kappa }_0)$ and ${\mathfrak{f}}_2=2({\rho }_p-{\rho }_0)/(2{\rho }_p+{\rho }_0)$ are non-dimensional constants depending on the compressibilities of the particle and the fluid (κ_p and κ₀, respectively) and on their mass densities (ρ_p and ρ₀, respectively).

Since the net forces point in the direction of the negative gradient of the potential ψ, the particles tend to cluster at the minima of the potential ψ. The acoustic radiation potential can be rewritten in terms of the frequency domain pressure as follows:²

$$\psi =\mathfrak{a}|p{|}^2-\mathfrak{b}|\mathrm{\nabla }p{|}^2={\left[\begin{array}{c}p\\ \mathrm{\nabla }p\end{array}\right]}^{\ast }\left[\begin{array}{cc}\mathfrak{a}& \\ & -\mathfrak{b}{\mathsf{I}}_d\end{array}\right]\left[\begin{array}{c}p\\ \mathrm{\nabla }p\end{array}\right],$$ 1.3

where $\mathfrak{a}={\mathfrak{f}}_1{\kappa }_0/4$, $\mathfrak{b}={\mathfrak{f}}_{2}3/(8{\rho }_0{\omega }^2)$, ${\mathsf{I}}_d$ is the d × d identity matrix and * denotes the conjugate transpose. The key observation here is that for fixed x, we can think of the acoustic radiation potential as a quadratic form in p and $\mathrm{\nabla }p$. We make no particular assumption on the signs of $\mathfrak{a}$ and $\mathfrak{b}$, as they depend on the physical properties of the particles and the fluid.

Remark 1.1. —

In general, the acoustic radiation potential depends also on the size and shape of the particles, thus by using (1.2) to predict where the particles cluster, we are neglecting the effect of the particle’s size and shape.

Remark 1.2. —

Optical forces analogous to the acoustic radiation force can be used to trap particles in crystal-like formations called optical lattices, which can be obtained by interfering laser beams. The trapped particles can either be larger or smaller than the wavelength. The case that is the closest to the acoustic setup we consider here is that of particles that are smaller than the wavelength (Rayleigh regime). For example, the net optical forces acting on non-absorbing dielectric particles that are much smaller than the wavelength can also be written as the gradient of a potential similar to (1.3) with $\mathfrak{b}=0$ (e.g. [19,20] and see [21] for an explicit comparison of radiation pressure in acoustics and optics). A rigorous treatment of how the theoretical results we present here apply to optical trapping is left for future studies.

Some of the two- and three-dimensional crystallographic symmetries that can be achieved using lasers have already been explored (e.g. [22–27]). To the best of our knowledge, there is no complete characterization of the possible geometries like the one we consider for acoustic trapping. We point out in particular that the study in [26] claims that all 14 Bravais lattices in three dimensions can be realized using 4 lasers. This would seem to be a more general result than the one we present in §3. Note, however, that the linear combinations of plane waves that are proposed in [26] to realize the 14 possible Bravais lattices are not periodic in general, and thus cannot be accurately described as Bravais lattices.

(b). Spatially periodic acoustic waves

To get a periodic arrangement of particles we take the pressure field p to be periodic. This can be achieved by taking a superposition of plane waves with wavevectors k₁, …, k_d in dimension d, that is

$$p(\mathit{x};\mathit{u})=\sum _{j=1}^d{\alpha }_j\exp [\text{i}{\mathit{k}}_j\cdot \mathit{x}]+{\beta }_j\exp [-\text{i}{\mathit{k}}_j\cdot \mathit{x}],$$ 1.4

where u = [α₁, …, α_d, β₁, …, β_d]^T are complex amplitudes, the wavenumber is k = |k_j| = 2π/ℓ and the wavelength is ℓ = c/f. If the wavevectors k_j form a basis of ${\mathbb{R}}^d$, which is assumed hereinafter, we can define the lattice [28] vectors a₁, …, a_d through the relations

$${\mathit{a}}_i\cdot {\mathit{k}}_j=2\pi {\delta }_{ij},i,j=1,\dots ,d.$$ 1.5

Here, δ_ij is the Kronecker delta with δ_ij = 1 if i = j and 0 otherwise. The relation between the lattice vectors and the wavevectors can be written concisely by introducing the d × d real matrices $\mathsf{A}\equiv [{\mathit{a}}_1,\dots ,{\mathit{a}}_d]$ and $\mathsf{K}\equiv [{\mathit{k}}_1,\dots ,{\mathit{k}}_d]$ which should be related by

$$\mathsf{A}=2\pi {\mathsf{K}}^{-T}.$$ 1.6

The field p is $\mathsf{A}$-periodic since³

$$p(\mathit{x}+\mathsf{A}\mathit{n};\mathit{u})=p(\mathit{x};\mathit{u})\text{ for any }\mathit{x}\in {\mathbb{R}}^d,\mathit{n}\in {\mathbb{Z}}^d\text{and}\mathit{u}\in {\mathbb{C}}^{2d}.$$ 1.7

Experimentally, we expect that an acoustic pressure field similar to the real part of (1.4) can be obtained far away from planar ultrasound transducers with normal orientations k_j by using the complex amplitudes u to determine the amplitudes and phases of the voltage driving the transducers (see figure 1 for an illustration). Because of this, we call u transducer parameters. We do not include the relation between u and an actual transducer’s operating voltages as it is out of the scope of the present study.

Figure 1.

A possible arrangement of ultrasound transducers (in blue) to generate fields close to (1.4) in two dimensions. (Online version in colour.)

Remark 1.3. —

We emphasize that our study is limited to d plane wave directions in dimension d, as this guarantees that the resulting acoustic fields are periodic. Having n > d plane wave directions could break the periodicity and so we can no longer use two- or three-dimensional crystallographic symmetries (or Bravais lattices) to describe the possible particle arrangements. Similarly, if we take the restriction to a plane of a three-dimensional periodic acoustic field, we may get a field that is not periodic. For example, take wavevectors ${\mathit{k}}_1={[\sqrt{2},0,\sqrt{10}]}^{\text{T}}$, ${\mathit{k}}_2={[0,\sqrt{3},3]}^{\text{T}}$ and ${\mathit{k}}_3={[\sqrt{5},\sqrt{7},0]}^{\text{T}}$ which have identical length equal to $2\sqrt{3}$. The restriction to the plane x₃ = 0 of the field p(x) = exp [ik₁ · x] + exp [ik₂ · x] + exp [ik₃ · x] is not periodic, even if p(x) itself is periodic in ${\mathbb{R}}^3$. This is because the numbers $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$ and $\sqrt{7}$ are rationally independent [29] which makes the functions f(t) = u(tv₀) quasi-periodic [30] in t for any non-zero choice of the vector v₀. We recall that the numbers α₁, …, α_n are said to be rationally independent if the only rational coefficients ω₁, …, ω_n for which ω₁α₁ + · · · + ω_nα_n = 0 are ω₁ = · · · = ω_n = 0.

Since p is $\mathsf{A}$-periodic, its associated acoustic radiation potential ψ must also be $\mathsf{A}$-periodic. In §2, we show that the extrema of the acoustic radiation potential can be predicted by the maximum and minimum eigenvalues of a 2d × 2d real symmetric matrix. We show that the level-sets of the acoustic radiation potential at values equal to the eigenvalues of this matrix are determined by the associated eigenspace. Note that the relation between the acoustic radiation potential and an eigendecomposition was already exploited in [1,2] to minimize the acoustic radiation potential at a set of points. Here, we are in a special case where we can find the minimizers explicitly and we give sufficient conditions for these to be isolated points, lines or planes arranged $\mathsf{A}$-periodically. The lattice vectors of the possible particle patterns must have reciprocal vectors with the same length. This limits the possible classes of crystallographic symmetries (Bravais lattices) that can be achieved with this method. We explore this limitation in two and three dimensions in §3. We summarize our findings and questions that were left open in §4.

2. Study of the acoustic radiation potential

The key to our study is to write the acoustic radiation potential as a quadratic form of the amplitudes u driving the plane waves (§2a). We observe that spatial shifts are equivalent to a similarity transformation of the matrix defining the quadratic form. Therefore, it suffices to study the acoustic radiation potential at the origin (§2b), where we can write the eigendecomposition of the associated matrix explicitly. Its eigenvalues give simple bounds on the acoustic radiation potential at constant power. In particular, the global minimum values must correspond to the smallest eigenvalue of the associated matrix (§2c). Then in §2d we study the possible level-sets of the acoustic radiation potential (at constant power) that are equal to one of the eigenvalues of the associated matrix. These particular level-sets may be composed of lattices (with anywhere between 2 and 2^d points per primitive cell), lines or planes, as is summarized in theorem 2.3.

(a). The acoustic radiation potential as a quadratic form

For a superposition of plane waves of the form (1.4), the acoustic radiation potential at a point x can be written as

$$\psi (\mathit{x};\mathit{u})={\mathit{u}}^{\ast }\mathsf{Q}(\mathit{x})\mathit{u},$$ 2.1

where the 2d × 2d Hermitian matrix $\mathsf{Q}(\mathit{x})$ is defined by

$$\mathsf{Q}(\mathit{x})=\mathsf{M}{(\mathit{x})}^{\ast }\left[\begin{array}{cc}\mathfrak{a}& \\ & -\mathfrak{b}{\mathsf{I}}_d\end{array}\right]\mathsf{M}(\mathit{x}),$$ 2.2

and we have used the (d + 1) × 2d complex matrix $\mathsf{M}(\mathit{x})$ that is given by

$$\mathsf{M}(\mathit{x})=[{\mathsf{M}}_{+}(\mathit{x})\hspace{0.17em}{\mathsf{M}}_{-}(\mathit{x})]\text{and}{\mathsf{M}}_{\pm }(\mathit{x})=\left[\begin{array}{c}\exp [\pm \text{i}{\mathit{x}}^{\text{T}}\mathsf{K}]\\ \mathsf{K}\hspace{0.17em}\text{diag}(\pm \text{i}\exp [\pm \text{i}{\mathit{x}}^{\text{T}}\mathsf{K}])\end{array}\right],$$ 2.3

where the exponential of a vector is understood componentwise and diag([a₁, …, a_n]) is the matrix with diagonal elements a₁, …, a_n.

We remark that a translation in x is equivalent to a unitary similarity transformation of $\mathsf{Q}(\mathit{x})$. To see this consider a point $\mathit{x}\in {\mathbb{R}}^d$, and write x = x₀ + ε. Note that

$$\psi ({\mathit{x}}_0+\mathit{\epsilon };\mathit{u})=\psi ({\mathit{x}}_0;\exp [\text{i}{[\mathsf{K},-\mathsf{K}]}^{\text{T}}\mathit{\epsilon }]\odot \mathit{u}),$$ 2.4

where $\odot$ is the componentwise or Hadamard product of two vectors. Indeed, write⁴ u = [α;β], $\mathit{\alpha },\mathit{\beta }\in {\mathbb{C}}^d$, and observe that a spatial shift ε is equivalent to changing the phase of α_j by k_j · ε and of β_j by −k_j · ε. Therefore, $\mathsf{Q}({\mathit{x}}_0+\mathit{\epsilon })$ and $\mathsf{Q}({\mathit{x}}_0)$ are related by the similarity transformation

$$\mathsf{Q}({\mathit{x}}_0+\mathit{\epsilon })=\text{diag}{(\exp [\text{i}{[\mathsf{K},-\mathsf{K}]}^{\text{T}}\mathit{\epsilon }])}^{\ast }\mathsf{Q}({\mathit{x}}_0)\text{diag}(\exp [\text{i}{[\mathsf{K},-\mathsf{K}]}^{\text{T}}\mathit{\epsilon }]).$$ 2.5

A practical consequence of (2.5) is that we can study the acoustic radiation potential at a particular point x₀ and use (2.4) or (2.5) to deduce its behaviour everywhere else. For simplicity, we take x₀ = 0.

(b). The acoustic radiation potential at the origin

Setting x = 0 in (2.2), we get that

$$\mathsf{Q}(\mathbf{0})=\mathfrak{a}\mathbf{1}{\mathbf{1}}^{\text{T}}-\mathfrak{b}\left[\begin{array}{c}\mathsf{K}\\ -\mathsf{K}\end{array}\right]{\left[\begin{array}{c}\mathsf{K}\\ -\mathsf{K}\end{array}\right]}^{\text{T}},$$ 2.6

where 1 is a vector of all ones of appropriate dimension. The following lemma 2.1 gives the eigendecomposition of $\mathsf{Q}(\mathbf{0})$ explicitly in terms of that of $\mathsf{K}{\mathsf{K}}^{\text{T}}$.

Lemma 2.1. —

Let σ₁ ≥ σ₂ ≥ · · · ≥ σ_d > 0 be the eigenvalues of $\mathsf{K}{\mathsf{K}}^{\text{T}}$ sorted in decreasing order and including multiplicity, and let {u₁, …, u_d} be a corresponding real orthonormal basis of eigenvectors.⁵ Let {z₁, …, z_d−1} be a real orthonormal basis for 1^⊥, the space of vectors orthogonal to 1. Then $\mathsf{Q}(\mathbf{0})$ admits the eigendecomposition $\mathsf{Q}(\mathbf{0})=\mathsf{U}\mathit{\Lambda }{\mathsf{U}}^{\text{T}},$ where Λ is the 2d × 2d diagonal matrix

$$\mathit{\Lambda }=\text{diag}([2\mathfrak{a}d,0,\dots ,0,-2\mathfrak{b}{\sigma }_1,\dots ,-2\mathfrak{b}{\sigma }_d])$$ 2.7

and $\mathsf{U}$ is the real 2d × 2d orthonormal matrix

$$\mathsf{U}=\frac{1}{\sqrt{2}}\left[\begin{array}{ccccccc}\mathbf{1}/\sqrt{d}& {\mathit{z}}_1& \dots & {\mathit{z}}_{d-1}& {\mathit{u}}_1& \dots & {\mathit{u}}_d\\ \mathbf{1}/\sqrt{d}& {\mathit{z}}_1& \dots & {\mathit{z}}_{d-1}& -{\mathit{u}}_1& \dots & -{\mathit{u}}_d\end{array}\right].$$ 2.8

Before proceeding to the proof of lemma 2.1, it is useful to introduce the decomposition ${\mathbb{C}}^{2d}=H_{+}\oplus H_{-}$, where $H_{\pm }=\{[\mathit{\alpha };\pm \mathit{\alpha }]\mid \mathit{\alpha }\in {\mathbb{C}}^d\}$. The unitary matrices

$${\mathsf{V}}_{\pm }=\frac{1}{\sqrt{2}}\left[\begin{array}{c}{\mathsf{I}}_d\\ \pm {\mathsf{I}}_d\end{array}\right]$$ 2.9

are such that $H_{\pm }=\text{range}({\mathsf{V}}_{\pm })$. Orthogonal projectors onto H_± are given by ${\mathsf{P}}_{\pm }={\mathsf{V}}_{\pm }{\mathsf{V}}_{\pm }^{\text{T}}$.

Proof. —

From our assumption that the wavevectors k₁, …, k_d form a basis, we deduce that the smallest eigenvalue of $\mathsf{K}{\mathsf{K}}^{\text{T}}$ should be positive. Indeed if u were a 0-eigenvector of ${\mathsf{K}}^{\text{T}}\mathsf{K}$, we would get $0={\mathit{u}}^{\text{T}}{\mathsf{K}}^{\text{T}}\mathsf{K}\mathit{u}=|\mathsf{K}\mathit{u}{|}^2$. Since $\mathsf{K}$ is invertible, we get the contradiction u = 0. Now note that we have

$${\left[\begin{array}{c}{\mathsf{V}}_{+}{\mathsf{V}}_{-}\end{array}\right]}^{\text{T}}\mathsf{Q}(\mathbf{0})\left[\begin{array}{c}{\mathsf{V}}_{+}{\mathsf{V}}_{-}\end{array}\right]=\left[\begin{array}{cc}2\mathfrak{a}\mathbf{1}{\mathbf{1}}^{\text{T}}& \\ & -2\mathfrak{b}\mathsf{K}{\mathsf{K}}^{\text{T}}\end{array}\right].$$ 2.10

The eigendecomposition of the matrix on the right-hand side of (2.10) can be obtained from that of 11^T and $\mathsf{K}{\mathsf{K}}^{\text{T}}$. To get the eigendecomposition of $\mathsf{Q}(\mathbf{0})$ from that of the right-hand side of (2.10), it is enough to solve for $\mathsf{Q}(\mathbf{0})$ using that $[{\mathsf{V}}_{+}{\mathsf{V}}_{-}]$ is an orthonormal matrix. ▪

(c). Bounds on the acoustic radiation potential

A simple consequence of the acoustic radiation potential being a quadratic form (2.1) is that for any transducer parameters $\mathit{u}\in {\mathbb{C}}^{2d}$ and positions $\mathit{x}\in {\mathbb{R}}^d$, we have the bounds

$${\mathrm{\lambda }}_{min}(\mathit{x})|\mathit{u}{|}^2\le \psi (\mathit{x};\mathit{u})\le {\mathrm{\lambda }}_{max}(\mathit{x})|\mathit{u}{|}^2,$$ 2.11

where ${\mathrm{\lambda }}_{max,min}(\mathit{x})={\mathrm{\lambda }}_{max,min}(\mathsf{Q}(\mathit{x}))$ are the maximum and minimum eigenvalues of $\mathsf{Q}(\mathit{x})$. However, we observed in (2.5) that for arbitrary x, $\mathsf{Q}(\mathit{x})$ is unitarily similar to $\mathsf{Q}(\mathbf{0})$, thus the eigenvalues of $\mathsf{Q}(\mathit{x})$ do not depend on x. By using λ_min(x) = λ_min(0) and λ_max(x) = λ_max(0) in (2.11), we get the bound

$${\mathrm{\lambda }}_{min}(\mathbf{0})|\mathit{u}{|}^2\le \psi (\mathit{x};\mathit{u})\le {\mathrm{\lambda }}_{max}(\mathbf{0})|\mathit{u}{|}^2$$ 2.12

for any $\mathit{x}\in {\mathbb{R}}^d$ and $\mathit{u}\in {\mathbb{C}}^{2d}$. Thus we can achieve the smallest possible acoustic radiation potential at a particular position x₀ by choosing transducer parameters u_min within the eigenspace of $\mathsf{Q}({\mathit{x}}_0)$ corresponding to λ_min(0) = λ_min(x₀). Hence u_min gives an explicit solution to the minimization

$$\underset{|\mathit{u}{|}^2=1}{min}\psi ({\mathit{x}}_0;\mathit{u}),$$ 2.13

as was remarked in [1,2]. Since the power to generate the plane waves (1.4) is proportional to |u|², the constraint in (2.13) means that we look only for transducer parameters that require the same power. The bound (2.12) guarantees that with transducer parameters u_min, x₀ is a global minimum of ψ(x;u_min), as a function of x. Note that this choice does not rule out the existence of local minima of ψ(x;u_min), where the particles could also be trapped. Also because ψ(x;u_min) is $\mathsf{A}$-periodic in x, the potential ψ(x;u_min) must also have global minima for x on the lattice

$$\{{\mathit{x}}_0+\mathsf{A}\mathit{n}\mid \mathit{n}\in {\mathbb{Z}}^d\}.$$ 2.14

A natural question is whether the global minima are limited to this lattice. This is answered negatively in the next section.

(d). Level-sets of the acoustic radiation potential

The set of all positions x for which the acoustic radiation potential has the same value γ is

$$L_{\gamma ,\mathit{u}}=\{\mathit{x}\in {\mathbb{R}}^d\mid \psi (\mathit{x};\mathit{u})=\gamma \},$$ 2.15

for particular transducer parameters u. To simplify the exposition, we restrict ourselves to the case |u| = 1, i.e. constant power. From the bound (2.12), we see that taking u to be an eigenvector of $\mathsf{Q}(\mathbf{0})$ associated with ${\mathrm{\lambda }}_{min}(\mathsf{Q}(\mathbf{0}))$ guarantees that the acoustic radiation potential has a global minimum at the origin. In fact the level-sets associated with any eigenpair of $\mathsf{Q}(\mathbf{0})$ are determined by the associated eigenspace.

Lemma 2.2. —

Let λ, u be an eigenpair of $\mathsf{Q}(\mathbf{0})$ with |u| = 1. Then

$$L_{\mathrm{\lambda },\mathit{u}}=\{\mathit{x}\in {\mathbb{R}}^d\mid \exp [\text{i}{[\mathsf{K},-\mathsf{K}]}^{\text{T}}\mathit{x}]\odot \mathit{u}\in \mathrm{\lambda }-\text{eigenspace of }\mathsf{Q}(\mathbf{0})\}.$$

Proof. —

Let x such that $\exp [\text{i}{[\mathsf{K},-\mathsf{K}]}^{\text{T}}\mathit{x}]\odot \mathit{u}\in \mathrm{\lambda }$-eigenspace of $\mathsf{Q}(\mathbf{0})$. Then from (2.4) and (2.1), we see that $\psi (\mathit{x};\mathit{u})=\psi (\mathbf{0};\exp [\text{i}{[\mathsf{K},-\mathsf{K}]}^{\text{T}}\mathit{x}]\odot \mathit{u})=\mathrm{\lambda }$. On the other hand, if x ∈ L_λ,u then $\exp [\text{i}{[\mathsf{K},-\mathsf{K}]}^{\text{T}}\mathit{x}]\odot \mathit{u}$ must be a λ-eigenvector of $\mathsf{Q}(\mathbf{0})$. This is because the Rayleigh quotient of a Hermitian matrix is equal to an eigenvalue if and only if it is evaluated at one of the corresponding eigenvectors. ▪

Note that by lemma 2.1, we may always be able to pick a λ-eigenvector of $\mathsf{Q}(\mathbf{0})$ that is of the form u = [v; ± v], where $|\mathit{v}|=1/\sqrt{2}$. We now give conditions on the symmetries of the λ-eigenspace that allow us to decide whether the periodic patterns consist of points, lines or planes. The conditions boil down to checking whether the λ-eigenvector u remains a λ-eigenvector after certain sign changes. The results are summarized in the following theorem. We emphasize that the results in theorem 2.3 do not only apply to the global minimum level-sets of the acoustic radiation potential but also to the level-sets corresponding to the other eigenvalues of $\mathsf{Q}(\mathbf{0})$. Nevertheless, our results do not say anything about local minima different from the global ones.

Theorem 2.3. —

Let λ, u be an eigenpair of $\mathsf{Q}(\mathbf{0})$ with u real and |u| = 1. Then the acoustic radiation potential level-set L_λ,u satisfies the following.

• —If any of the entries of u is zero, then the λ-level-set of the acoustic radiation potential contains lines or even planes (lemma 2.4). This corresponds to turning off one or more of the transducers.
• —If u has no zero entries and the λ-eigenspace of $\mathsf{Q}(\mathbf{0})$ is all contained within either H₊ or H₋ then the λ-level-set of the acoustic radiation potential is composed of between 2 and 2^d isolated points per primitive cell (lemma 2.5). The precise number of points is equal to the number of sign changes of u for which the resulting vectors remain in the λ-eigenspace.
• —If u has no zero entries and the λ-eigenspace of $\mathsf{Q}(\mathbf{0})$ straddles H₊ and H₋ then the λ-level-set of the acoustic radiation potential may contain lines (lemma 2.7) or even planes (lemma 2.9).

The first result applies to the situation where one or more of the transducers is off, i.e. when we pick eigenvectors u that have zero entries in the λ-eigenspace of $\mathsf{Q}(\mathbf{0})$. In this case, the level-set L_λ,u contains subspaces of the lattice vectors, which could be either lines or planes, depending on the dimension d.

Lemma 2.4. —

Let u = [v; ± v] be a real unit norm eigenvector of $\mathsf{Q}(\mathbf{0})$ associated with eigenvalue λ. Let Z = {j| v_j = 0}. If any entry of v is zero, i.e. $Z\ne \mathrm{\varnothing }$, then

$${\{{\mathit{k}}_j\mid j\notin Z\}}^{\perp }=\text{span}\{{\mathit{a}}_j\mid j\in Z\}\subset L_{\mathrm{\lambda },\mathit{u}}.$$ 2.16

Thus the λ-level-set of the acoustic radiation potential with parameters u must contain any linear combination of the lattice vectors associated with the zero entries in v.

Proof. —

Let $\mathit{z}\in {\{{\mathit{k}}_j\mid j\notin Z\}}^{\perp }$, then for $j\notin Z$, k_j · z = 0. Or in other words for $j\notin Z$ we have ${(\exp [\text{i}{\mathsf{K}}^{\text{T}}\mathit{z}])}_j=1$. Hence, we must have that $\exp [\text{i}{\mathsf{K}}^{\text{T}}\mathit{z}]\odot \mathit{v}=\mathit{v}$ since

$${(\exp [\text{i}{\mathsf{K}}^{\text{T}}\mathit{z}])}_j{v}_j=\{\begin{array}{ll}0& \text{for }j\in Z\text{and}\\ v_j& \text{for }j\notin Z.\end{array}$$

We conclude that $\exp [\text{i}{[\mathsf{K},-\mathsf{K}]}^{\text{T}}\mathit{z}]\odot [\mathit{v};\pm \mathit{v}]=\mathit{u}$ is still an eigenvector of $\mathsf{Q}(\mathbf{0})$ associated with λ and ψ(z;u) = ψ(0;u) = λ by lemma 2.2. The expression in terms of the a_j follows from (1.6). ▪

When all the entries of the eigenvector u in the λ-eigenspace of $\mathsf{Q}(\mathbf{0})$ are non-zero (i.e. all the transducers are activated), we can guarantee that the level-set L_λ,u is reduced to up to 2^d points per primitive cell, where the number of points is determined by whether, upon changing the signs of the entries of u, the new vector remains in the λ-eigenspace of $\mathsf{Q}(\mathbf{0})$.

Lemma 2.5. —

Let u = [v; ± v] ∈ H_± be a real unit norm eigenvector of $\mathsf{Q}(\mathbf{0})$ associated with eigenvalue λ, such that v_j ≠ 0 for all j = 1, …, d. Assume that the eigenspace associated with λ is a subset of H_±. Consider the set⁶

$$T_{\mathrm{\lambda },\mathit{u}}=\{\mathit{s}\in {\{0,1\}}^d\mid [{(-1)}^{\mathit{s}}\odot \mathit{v};\pm {(-1)}^{\mathit{s}}\odot \mathit{v}]\in \mathrm{\lambda }-\text{eigenspace of }\mathsf{Q}(\mathbf{0})\},$$ 2.17

where ${(-1)}^{\mathit{s}}\equiv [{(-1)}^{s_1},\dots ,{(-1)}^{s_d}]$. Then L_λ,u is a union of lattices given by

$$L_{\mathrm{\lambda },\mathit{u}}=\bigcup _{\mathit{s}\in T_{\mathrm{\lambda },\mathit{u}}}\{\mathit{x}(\mathit{n};\mathit{s})=\mathsf{A}(\mathit{n}+\mathit{s}/2)\mid \mathit{n}\in {\mathbb{Z}}^d\}.$$ 2.18

Proof. —

We show that ψ(x;u) = ψ(0;u) = λ if and only if x belongs to one of the lattices in (2.18).

Let us first assume that x belongs to one of the lattices in (2.18). Then we can find s ∈ {0, 1}^d such that [( − 1)^sv; ± ( − 1)^sv] is in the λ-eigenspace of $\mathsf{Q}(\mathbf{0})$ and $\mathit{x}=\mathsf{A}(\mathit{n}+\mathit{s}/2)$ for some $\mathit{n}\in {\mathbb{Z}}^d$. By (1.6), we have that $\exp [\text{i}{\mathit{k}}_j\cdot \mathit{x}]=\exp [\text{i}2\pi (n_j+s_j/2)]={(-1)}^{s_j}$ and so we also have

$$\exp [\text{i}{[\mathsf{K},-\mathsf{K}]}^{\text{T}}\mathit{x}]=\exp [-\text{i}{[\mathsf{K},-\mathsf{K}]}^{\text{T}}\mathit{x}]=[{(-1)}^{\mathit{s}};{(-1)}^{\mathit{s}}].$$ 2.19

We conclude that the vector $\exp [\text{i}{[\mathsf{K},-\mathsf{K}]}^{\text{T}}\mathit{x}]\odot [\mathit{v};\pm \mathit{v}]$ belongs to the λ-eigenspace of $\mathsf{Q}(\mathbf{0})$. In other words, x ∈ L_λ,u since by (2.4) we have

$$\psi (\mathit{x};\mathit{u})=\psi (\mathbf{0};\exp [\text{i}{[\mathsf{K},-\mathsf{K}]}^{\text{T}}\mathit{x}]\odot [\mathit{v};\pm \mathit{v}])=\psi (\mathbf{0};\mathit{u})=\mathrm{\lambda }.$$

Now let us assume that x ∈ L_u,λ, i.e. ψ(x;u) = λ. By using (2.4) again we have that $\exp [\text{i}{[\mathsf{K},-\mathsf{K}]}^{\text{T}}\mathit{x}]\odot [\mathit{v};\pm \mathit{v}]$ must be in the λ-eigenspace of $\mathsf{Q}(\mathbf{0})$. Moreover, the former vector can be split into components in H_± and $H_{\mp }$ as follows:

$$\exp \left[\text{i}\left[\begin{array}{c}{\mathsf{K}}^{\text{T}}\\ -{\mathsf{K}}^{\text{T}}\end{array}\right]\mathit{x}\right]\odot \left[\begin{array}{c}\mathit{v}\\ \pm \mathit{v}\end{array}\right]=\underset{\in H_{\pm }}{\underbrace{\cos \left[\left[\begin{array}{c}{\mathsf{K}}^{\text{T}}\\ -{\mathsf{K}}^{\text{T}}\end{array}\right]\mathit{x}\right]\odot \left[\begin{array}{c}\mathit{v}\\ \pm \mathit{v}\end{array}\right]}}+\text{i}\underset{\in H_{\mp }}{\underbrace{\sin \left[\left[\begin{array}{c}{\mathsf{K}}^{\text{T}}\\ -{\mathsf{K}}^{\text{T}}\end{array}\right]\mathit{x}\right]\odot \left[\begin{array}{c}\mathit{v}\\ \pm \mathit{v}\end{array}\right]}}.$$ 2.20

Since we assumed that the λ-eigenspace of $\mathsf{Q}(\mathbf{0})$ is a subspace of H_± we must have that $\exp [\text{i}{[\mathsf{K},-\mathsf{K}]}^{\text{T}}\mathit{x}]\odot [\mathit{v};\pm \mathit{v}]\in H_{\pm }$ and its $H_{\mp }$ component must be zero. By the decomposition (2.20), this means that $\exp [\text{i}{[\mathsf{K},-\mathsf{K}]}^{\text{T}}\mathit{x}]\odot [\mathit{v};\pm \mathit{v}]$ must be real. Using the definition (2.17) of the set T_λ,u, we see that there must be an s ∈ {0, 1}^d such that (2.19) holds. This imposes that x must be part of the lattice (2.18) with s ∈ T_λ,u. ▪

To better illustrate lemma 2.5, let us define the primitive cell

$$C=\{\mathsf{A}\mathit{\alpha }\mid \mathit{\alpha }\in {[0,1]}^d\}.$$ 2.21

Points within the primitive cell can be identified by their ‘atomic coordinates’ α ∈ [0, 1]^d. Thus, we have the following extreme cases:

• (i)If λ is a simple eigenvalue then T_λ,u = {0, 1} and there are two points per primitive cell in L_λ,u, namely 0 and 1/2, in atomic coordinates, provided u is a unit length eigenvector of $\mathsf{Q}(\mathbf{0})$ with non-zero entries.
• (ii)If λ is an eigenvalue of multiplicity d, then T_λ,u = {0, 1}^d and there are exactly 2^d points in L_λ,u per primitive cell, namely the points with atomic coordinates s/2, where s ∈ {0, 1}^d. This is of course provided u is a unit length eigenvector of $\mathsf{Q}(\mathbf{0})$ with non-zero entries and the λ-eigenspace of $\mathsf{Q}(\mathbf{0})$ is all within either H₊ or H₋.

Example 2.6 (Eigenvalue of multiplicity 2). —

Consider a two-dimensional example. Choose the wavevectors so that $\mathsf{K}={\mathsf{I}}_2$ and ℓ = 2π. Setting $\mathfrak{a}=\mathfrak{b}=1$ in the acoustic radiation potential and using (2.6) we get

$$\mathsf{Q}(\mathbf{0})=\mathbf{1}{\mathbf{1}}^{\text{T}}-\left[\begin{array}{c}\mathsf{K}\\ -\mathsf{K}\end{array}\right]{\left[\begin{array}{c}\mathsf{K}\\ -\mathsf{K}\end{array}\right]}^{\text{T}}=\left[\begin{array}{cccc}0& 1& 2& 1\\ 1& 0& 1& 2\\ 2& 1& 0& 1\\ 1& 2& 1& 0\end{array}\right].$$ 2.22

We choose the eigendecomposition $\mathsf{Q}(\mathbf{0})=\mathsf{U}\mathit{\Lambda }{\mathsf{U}}^{\text{T}}$ with

$$\mathsf{U}=\left[\begin{array}{cccc}1/2& -1/2& 1/\sqrt{2}& 0\\ 1/2& 1/2& 0& 1/\sqrt{2}\\ 1/2& -1/2& -1/\sqrt{2}& 0\\ 1/2& 1/2& 0& -1/\sqrt{2}\end{array}\right],$$ 2.23

and diag(Λ) = {4, 0, − 2, − 2} such that it conforms to H₊ and H₋, as in lemma 2.1. Note that the ( − 2)-eigenspace of $\mathsf{Q}(\mathbf{0})$ is identical to H₋. Thus a real unit length ( − 2)-eigenvector u with no zero entries must have the form u = [u₁, u₂, − u₁, − u₂]. Note that by changing the sign of u₁ and/or u₂ we get a vector that remains inside H₋. We deduce that T_−2,u = {0, 1}² (see lemma 2.5 for the definition of this set). Hence lemma 2.5 predicts acoustic radiation potential minima at the union of lattices

$$\bigcup _{\mathit{s}\in {\{0,1\}}^2}\{\mathit{x}(\mathit{n};\mathit{s})=2\pi (\mathit{n}+\frac{\mathit{s}}{2})\mid \mathit{n}\in {\mathbb{Z}}^d\}.$$ 2.24

Figure 2 shows that the acoustic radiation potential for this example with eigenvector u = [1, 1, − 1, − 1]^T/2 has four minimum points per primitive cell.

If we consider the acoustic radiation potential for an eigenvector with zero entries, we no longer have strict minima, by lemma 2.4. This is illustrated in figure 3, which shows the acoustic radiation potential for the same example, but with eigenvector $\mathit{u}={[1,0,-1,0]}^{\text{T}}/\sqrt{2}$.

Figure 2.

Acoustic radiation potential (from example 2.6) resulting in a tetragonal lattice arrangement of minima when the eigenvector has no zero entries. A primitive cell is outlined in white. The points in the lattice (2.24) are shown in red. Since the minimum eigenvalue of $\mathsf{Q}(\mathbf{0})$ has multiplicity 2, there are 4 minimum points per primitive cell. The black arrows indicate the directions normal to the transducers. (Online version in colour.)

Figure 3.

The acoustic radiation potential defined in example 2.6 results in lines of minima if the eigenvector used to compute the acoustic radiation potential has zero entries. The lines where minima lie are the spans of the lattice vectors specified by (2.16) and are indicated in red. A primitive cell is outlined in white. The black arrows indicate the directions normal to the transducers. (Online version in colour.)

Lemma 2.7. —

Let u = [v; ± v] ∈ H_± be a real unit norm eigenvector of $\mathsf{Q}(\mathbf{0})$ associated with eigenvalue λ. Consider the set $T_{\mathrm{\lambda },\mathit{u}}^{\pm }$ defined by

$$T_{\mathrm{\lambda },\mathit{u}}^{\pm }=\{\mathit{s}\in {\{0,1\}}^d\mid [{(-1)}^{\mathit{s}}\odot \mathit{v};\mp {(-1)}^{\mathit{s}}\odot \mathit{v}]\in \mathrm{\lambda }-\text{eigenspace of }\mathsf{Q}(\mathbf{0})\}.$$ 2.25

Note that if the set $T_{\mathrm{\lambda },\mathit{u}}^{\pm }$ is not empty, the λ-eigenspace of $\mathsf{Q}(\mathbf{0})$ straddles between H₊ and H₋, which is a possibility predicted by lemma 2.1. Then we can guarantee that L_λ,u contains the lines

$$\{\mathit{x}(\mathit{n};\theta )={\mathsf{K}}^{-T}(\theta {(-1)}^{\mathit{s}}+2\pi \mathit{n})\mid \theta \in \mathbb{R}\}\text{for }\mathit{n}\in {\mathbb{Z}}^d\text{and}\mathit{s}\in T_{\mathrm{\lambda },\mathit{u}}^{\pm }.$$ 2.26

Proof. —

Assume that x belongs to one of the lines (2.26), we show that ψ(x;u) = ψ(0;u) = λ. Then ${\mathsf{K}}^{\text{T}}\mathit{x}=\theta {(-1)}^{\mathit{s}}+2\pi \mathit{n}$, and we have that $\exp [\text{i}{\mathit{k}}_j\cdot \mathit{x}]=\exp [\text{i}\theta {(-1)}^{s_j}]$. The following decomposition in terms of a vector in H_± and in $H_{\mp }$ holds:

$$\exp \left[\text{i}\left[\begin{array}{c}{\mathsf{K}}^{\text{T}}\\ -{\mathsf{K}}^{\text{T}}\end{array}\right]\mathit{x}\right]\odot \left[\begin{array}{c}\mathit{v}\\ \pm \mathit{v}\end{array}\right]=\cos \theta \left[\begin{array}{c}\mathit{v}\\ \pm \mathit{v}\end{array}\right]+\text{i}\sin \theta \left[\begin{array}{c}{(-1)}^{\mathit{s}}\\ {(-1)}^{\mathit{s}}\end{array}\right]\odot \left[\begin{array}{c}\mathit{v}\\ \mp \mathit{v}\end{array}\right].$$ 2.27

By the definition of the set $T_{\mathrm{\lambda },\mathit{u}}^{\pm }$, the vector $[{(-1)}^{\mathit{s}}\odot \mathit{v};\mp {(-1)}^{\mathit{s}}\odot \mathit{v}]$ must be a λ-eigenvector of $\mathsf{Q}(\mathbf{0})$. Hence, $\exp [\text{i}[{\mathsf{K}}^{\text{T}};-{\mathsf{K}}^{\text{T}}]\mathit{x}]\odot [\mathit{v};\pm \mathit{v}]$ is also a λ-eigenvector of $\mathsf{Q}(\mathbf{0})$. Thus by (2.4), x remains in the λ-level-set of ψ, i.e. ψ(x;u) = λ. ▪

Example 2.8 (Lines of minima). —

Consider a two-dimensional example. Choose the wavevectors so $\mathsf{K}={\mathsf{I}}_2$, ℓ = 2π and set $\mathfrak{a}=1,\mathfrak{b}=0$ in the acoustic radiation potential. By (2.6), we have $\mathsf{Q}(\mathbf{0})=\mathbf{1}{\mathbf{1}}^{\text{T}}$. We choose the eigendecomposition $\mathsf{Q}(\mathbf{0})=\mathsf{U}\mathit{\Lambda }{\mathsf{U}}^{\text{T}}$ with diag(Λ) = {4, 0, 0, 0} and

$$\mathsf{U}=\frac{1}{\sqrt{2}}\left[\begin{array}{cccc}1/\sqrt{2}& -1/\sqrt{2}& 1& 0\\ 1/\sqrt{2}& 1/\sqrt{2}& 0& 1\\ 1/\sqrt{2}& -1/\sqrt{2}& -1& 0\\ 1/\sqrt{2}& 1/\sqrt{2}& 0& -1\end{array}\right].$$ 2.28

This eigendecomposition conforms with H₊ and H₋ and is consistent with lemma 2.1. Note that the 0-eigenspace of $\mathsf{Q}(\mathbf{0})$ straddles over H₊ and H₋, and that this eigenspace contains all of H₋. The real unit length vector u = [ − 1, 1, − 1, 1]^T/2 (the second column of $\mathsf{U}$) belongs to both H₊ and the 0-eigenspace of $\mathsf{Q}(\mathbf{0})$. Using definition (2.25), we can verify that $T_{0,\mathit{u}}^{\pm }={\{0,1\}}^2$. Thus lemma 2.7 predicts acoustic radiation potential minima along the families of lines

$$\{\mathit{x}(\mathit{n};\theta )={\mathsf{K}}^{-T}(\theta {(-1)}^{\mathit{s}}+2\pi \mathit{n})\mid \theta \in \mathbb{R}\},\text{for }\mathit{n}\in {\mathbb{Z}}^d\text{and}\mathit{s}\in {\{0,1\}}^2.$$ 2.29

These lines are indicated in figure 4 and coincide with minima of the acoustic radiation potential.

Figure 4.

Acoustic radiation potential of example 2.8 resulting in lines of minima. A primitive cell is outlined in white. The lines predicted by (2.29) are shown in red. The black arrows indicate the directions normal to the transducers. (Online version in colour.)

Lemma 2.9. —

Let u = [v; ± v] ∈ H_± be a real unit norm eigenvector of $\mathsf{Q}(\mathbf{0})$ associated with eigenvalue λ. Consider the set $R_{\mathrm{\lambda },\mathit{u}}^{\pm }$ defined by

$$\begin{array}{rl}{R}_{\mathrm{\lambda },\mathit{u}}^{\pm }& =\{(\mathit{s},\mathit{r})\in {({\{0,1\}}^d)}^2|\left[\begin{array}{c}{(-1)}^{\mathit{s}}\\ {(-1)}^{\mathit{s}}\end{array}\right]\odot \left[\begin{array}{c}\mathit{v}\\ \mp \mathit{v}\end{array}\right],\left[\begin{array}{c}{(-1)}^{\mathit{r}}\\ {(-1)}^{\mathit{r}}\end{array}\right]\odot \left[\begin{array}{c}\mathit{v}\\ \mp \mathit{v}\end{array}\right],\\ & \left[\begin{array}{c}{(-1)}^{\mathit{s}+\mathit{r}}\\ {(-1)}^{\mathit{s}+\mathit{r}}\end{array}\right]\odot \left[\begin{array}{c}\mathit{v}\\ \pm \mathit{v}\end{array}\right]\in \mathrm{\lambda }-\text{eigenspace of }\mathsf{Q}(\mathbf{0})\}.\end{array}$$ 2.30

Note that if the set $R_{\mathrm{\lambda },\mathit{u}}^{\pm }$ is not empty, the λ-eigenspace of $\mathsf{Q}(\mathbf{0})$ straddles between H₊ and H₋. Then we can guarantee L_λ,u contains the sets

$$\{\mathit{x}(\mathit{n};\theta ,\varphi )={\mathsf{K}}^{-T}(\theta {(-1)}^{\mathit{s}}+\varphi {(-1)}^{\mathit{r}}+2\pi \mathit{n})\mid \theta ,\varphi \in \mathbb{R}\},$$ 2.31

for $\mathit{n}\in {\mathbb{Z}}^d$ and $(\mathit{s},\mathit{r})\in R_{\mathrm{\lambda },\mathit{u}}^{\pm }$. The sets in (2.31) are guaranteed to be planes when the vectors ( − 1)^r and ( − 1)^s are linearly independent or equivalently ${(-1)}^{\mathit{r}+\mathit{s}}\notin \{-\mathbf{1},\mathbf{1}\}$.

Proof. —

Assume x belongs to one of the planes (2.31), we show that ψ(x;u) = ψ(0;u) = λ. Since ${\mathsf{K}}^{\text{T}}\mathit{x}=\theta {(-1)}^{\mathit{s}}+\varphi {(-1)}^{\mathit{r}}+2\pi \mathit{n}$, we must have that $\exp [\text{i}{\mathit{k}}_j\cdot \mathit{x}]=\exp [\text{i}(\theta {(-1)}^{s_j}+\varphi {(-1)}^{r_j})]$. Then we obtain the following decomposition in terms of vectors in H_± and $H_{\mp }$:

$$\begin{array}{rl}\exp \left[\text{i}\left[\begin{array}{c}{\mathsf{K}}^{\text{T}}\\ -{\mathsf{K}}^{\text{T}}\end{array}\right]\mathit{x}\right]\odot \left[\begin{array}{c}\mathit{v}\\ \pm \mathit{v}\end{array}\right]& =(\cos \theta \cos \varphi \left[\begin{array}{c}\mathbf{1}\\ \mathbf{1}\end{array}\right]-\sin \theta \sin \varphi \left[\begin{array}{c}{(-1)}^{\mathit{s}+\mathit{r}}\\ {(-1)}^{\mathit{s}+\mathit{r}}\end{array}\right])\odot \left[\begin{array}{c}\mathit{v}\\ \pm \mathit{v}\end{array}\right]\\ & +\text{i}(\sin \theta \cos \varphi \left[\begin{array}{c}{(-1)}^{\mathit{s}}\\ {(-1)}^{\mathit{s}}\end{array}\right]+\cos \theta \sin \varphi \left[\begin{array}{c}{(-1)}^{\mathit{r}}\\ {(-1)}^{\mathit{r}}\end{array}\right])\odot \left[\begin{array}{c}\mathit{v}\\ \mp \mathit{v}\end{array}\right].\end{array}$$ 2.32

By definition of the set $R_{\mathrm{\lambda },\mathit{u}}^{\pm }$, each vector in the decomposition must be a λ-eigenvector of $\mathsf{Q}(\mathbf{0})$. Hence $\exp [\text{i}[{\mathsf{K}}^{\text{T}};-{\mathsf{K}}^{\text{T}}]\mathit{x}]\odot [\mathit{v};\pm \mathit{v}]$ is also a λ-eigenvector of $\mathsf{Q}(\mathbf{0})$. By (2.4) this implies that ψ(x;u) = λ. ▪

Example 2.10 (Planes of minima). —

Consider a three-dimensional example. Choose the wavevectors so $\mathsf{K}={\mathsf{I}}_3$. Setting $\mathfrak{a}=1,\mathfrak{b}=0$ in the acoustic radiation potential gives $\mathsf{Q}(\mathbf{0})=\mathbf{1}{\mathbf{1}}^{\text{T}}$. An eigendecomposition $\mathsf{Q}(\mathbf{0})=\mathsf{U}\mathit{\Lambda }{\mathsf{U}}^{\text{T}}$ is chosen such that diag(Λ) = {6, 0, 0, 0, 0, 0} and

$$\mathsf{U}=\frac{1}{\sqrt{2}}\left[\begin{array}{cccccc}1/\sqrt{3}& 1/\sqrt{2}& 1/\sqrt{6}& 1& 0& 0\\ 1/\sqrt{3}& -1/\sqrt{2}& 1/\sqrt{6}& 0& 1& 0\\ 1/\sqrt{3}& 0& -2/\sqrt{6}& 0& 0& 1\\ 1/\sqrt{3}& 1/\sqrt{2}& 1/\sqrt{6}& -1& 0& 0\\ 1/\sqrt{3}& -1/\sqrt{2}& 1/\sqrt{6}& 0& -1& 0\\ 1/\sqrt{3}& 0& -2/\sqrt{6}& 0& 0& -1\end{array}\right],$$ 2.33

conforming to H₊ and H₋ (as in lemma 2.1). The 0-eigenspace of $\mathsf{Q}(\mathbf{0})$ straddles over H₊ and H₋, and contains all of H₋. The real unit length vector u = [1, − 1, 0, 1, − 1, 0]^T/2 (the second column of $\mathsf{U}$) belongs to both H₊ and the 0-eigenspace of $\mathsf{Q}(\mathbf{0})$. Using definition (2.30) we can verify that

$$R_{0,\mathit{u}}^{\pm }=\{(\mathit{s},\mathit{r})\in {\{0,1\}}^3\times {\{0,1\}}^3\mid {(-1)}^{s_1+r_1}={(-1)}^{s_2+r_2}\}.$$ 2.34

Thus lemma 2.9 predicts acoustic radiation potential minima along the sets given in equation 2.31, some of them being the planes indicated in figure 5. We have verified numerically that they coincide with the minima of the acoustic radiation potential.

Figure 5.

The minima of the acoustic radiation potential defined in example 2.10 appear on planes. The planes are displayed on a few unit cells (a) and for more clarity on a unit cell (b). The black arrows indicate the directions normal to the transducers. (Online version in colour.)

Example 2.11. —

Using the same matrix in (2.33) but choosing the unit-length 0-eigenvector $\mathit{v}=[1,1,-2,1,1,-2]/2\sqrt{3}\in H_{+}$ we get

$$R_{0,\mathit{v}}^{\pm }=\{(\mathit{s},\mathit{r})\in {\{0,1\}}^3\times {\{0,1\}}^3\mid {(-1)}^{s_1+r_1}={(-1)}^{s_2+r_2}={(-1)}^{s_3+r_3}\}.$$ 2.35

Now all the corresponding sets in (2.31) are lines because we have that ( − 1)^s+r ∈ { − 1, 1} for $(\mathit{s},\mathit{r})\in R_{0,\mathit{v}}^{\pm }$. These lines are displayed in figure 6.

Figure 6.

The minima of the acoustic radiation potential in example 2.11 appear on lines. The lines are displayed on a few unit cells (a) and for more clarity on a unit cell (b). The black arrows indicate the directions normal to the transducers. (Online version in colour.)

3. Achievable Bravais lattice classes

In earlier sections, we have taken the wavevectors k₁, …, k_d to be fixed. Now, we address the question of how to choose these vectors in order to obtain an arrangement of particles on a particular Bravais lattice. The position of the particles is dictated by the acoustic radiation potential ψ(x;u) which is $\mathsf{A}$-periodic. Since $\mathsf{A}=2\pi {\mathsf{K}}^{-T}$, and all the columns of $\mathsf{K}$ have the same length k, there are limitations to the possible Bravais lattices that we can obtain. To explore these limitations, we introduce the following definition.

Definition 3.1. —

We say that a Bravais lattice class is achievable if out of all class members with reciprocal lattice vectors k₁, …, k_d having same length, i.e.

$$|{\mathit{k}}_1|=\cdots =|{\mathit{k}}_d|,$$ 3.1

there is at least one member that does not belong to any of the other Bravais lattice classes.

In other words, a Bravais lattice class cannot be achieved if under the constraint of definition 3.1, all Bravais lattices within a class happen to belong to another class. For a two-dimensional example, note that tetragonal lattices are members of the orthorhombic lattice class. However, when we place the same length constraint on the reciprocal lattice vectors for an orthorhombic lattice, we end up with a tetragonal lattice. Thus we say that the orthorhombic lattice class cannot be achieved. We remark that definition 3.1 is irrespective of the particular particle arrangement inside a primitive cell, which could include isolated points, lines or planes (theorem 2.3).

Our results are summarized in tables 1 and 2 for two and three dimensions, respectively. We found that out of the five Bravais lattice classes in two dimensions (e.g. [28]) only three are achievable. In three dimensions, there are 14 Bravais lattice classes (e.g. [28]) but we found that only six are achievable.

Table 1.

Two-dimensional Bravais lattice classes that are achievable using standing acoustic waves. The reciprocal lattice vectors we give satisfy |g₁| = |g₂|, so they need to be rescaled so that their length is k in order to be interpreted as the wavevectors needed to obtain a particular Bravais lattice. The angle γ is the angle between the two primitive vectors in the lattice.

Bravais lattice class	reciprocal lattice vectors	implied symmetry
monoclinic	${\mathit{g}}_1=(1,-\cot \gamma )$	orthorhombic centred
	${\mathit{g}}_2=(0,\csc \gamma )$
orthorhombic	g1 = (1, 0)	tetragonal
	g2 = (0, 1)
orthorhombic centred	${\mathit{g}}_1=(\csc (\gamma )\sin (\gamma /2),\csc (\gamma /2)/2)$
	${\mathit{g}}_2=(\csc (\gamma )\sin (\gamma /2),-\csc (\gamma /2)/2)$
hexagonal	${\mathit{g}}_1=(1,1/\sqrt{3})$
	${\mathit{g}}_2=(0,2/\sqrt{3})$
tetragonal	g1 = (1, 0)
	g2 = (0, 1)

Table 2.

Three-dimensional Bravais lattice classes that are achievable using standing acoustic waves. The parameters a, b and c are the lengths of the sides of the conventional unit cell for the Bravais lattice when they can be adjusted. The reciprocal lattice vectors we give satisfy |g₁| = |g₂| = |g₃| and need to be rescaled so that they have length k in order for them to be the wavevectors needed to realize a particular Bravais lattice. The angle γ is the angle between two of the sides of the conventional unit cell.

Bravais lattice class	reciprocal lattice vectors	implied symmetry
triclinic primitive	|g1| = |g2| = |g3|
monoclinic primitive	g1 = ( − cosγ, − sinγ, 0)	cubic primitive (if cosγ = 0) tetragonal body-centred (if cosγ ≠ 0)
	g2 = (1, 0, 0)
	g3 = (0, 0, 1)
monoclinic base-centred	${\mathit{g}}_1=(-\cot \gamma ,-1,0)$	tetragonal body-centred
	${\mathit{g}}_2=\frac{1}{ac}(c\csc \gamma ,0,-a)$
	${\mathit{g}}_3=\frac{1}{ac}(c\csc \gamma ,0,a)$
orthorhombic primitive	g1 = (0, − 1, 0)	cubic primitive
	g2 = (1, 0, 0)
	g3 = (0, 0, 1)
orthorhombic base-centred	g1 = (b, − a, 0)	tetragonal body-centred (if a ≠ b) cubic primitive (if a = b)
	g2 = (b, a, 0)
	${\mathit{g}}_3=(0,0,\sqrt{a^2+b^2})$
orthorhombic body-centred	g1 = (1, 0, 1)	cubic body-centred
	g2 = (0, − 1, 1)
	g3 = (1, − 1, 0)
orthorhombic face-centred	g1 = (1/a, 1/b, 1/c)
	g2 = ( − 1/a, − 1/b, 1/c)
	g3 = (1/a, − 1/b, − 1/c)
tetragonal primitive	g1 = (1, 0, 0)	cubic primitive
	g2 = (0, 1, 0)
	g3 = (0, 0, 1)
tetragonal body-centred	g1 = (0, 1, 1)	cubic body-centred
	g2 = (1, 0, 1)
	g3 = (1, 1, 0)
trigonal primitive	g1 = (0, − 2/(3a), 1/(3c))
	${\mathit{g}}_2=(1/(\sqrt{3}a),1/(3a),1/(3c))$
	${\mathit{g}}_3=(-1/(\sqrt{3}a),1/(3a),1/(3c))$
hexagonal primitive	${\mathit{g}}_1=(1/(\sqrt{3}),-1,0)$	tetragonal body-centred
	${\mathit{g}}_2=(2/(\sqrt{3}),0,0)$
	${\mathit{g}}_3=(0,0,2/(\sqrt{3}))$
cubic primitive	g1 = (1, 0, 0)
	g2 = (0, 1, 0)
	g3 = (0, 0, 1)
cubic face-centred	g1 = ( − 1, 1, 1)
	g2 = (1, − 1, 1)
	g3 = (1, 1, − 1)
cubic body-centred	g1 = (0, 1, 1)
	g2 = (1, 0, 1)
	g3 = (1, 1, 0)

To generate tables 1 and 2, we took known reference tables associating Bravais lattice classes to the typical form of their reciprocal vectors. Then we imposed the condition that all the reciprocal vectors have the same norm.⁷ A general form for the reciprocal vectors is given in the second column of these tables. If the third column has an entry, then the particular class cannot be achieved and the class that is implied by the same norm reciprocal vector constraint is indicated. If the third column has no entry, it means that at least one representative belonging exclusively to the class can be achieved. Of course, tables 1 and 2 can be used to design lattices by taking the transducer normal orientations to be those in the second column. The reference tables we based our study on can be found in [31, table 3.3] for three dimensions. The two-dimensional reciprocal vectors can be derived from e.g. [28, fig. 1.7] and (1.6). We also illustrate in figures 7 and 8 representatives of the classes of Bravais lattices that are achievable using standing acoustic waves, in the particular case of isolated particle arrangements.

Figure 7.

Representatives of the three Bravais lattice classes that are achievable in two dimensions. The classes are: (a) orthorhombic centred (γ = π/4), (b) hexagonal and (c) tetragonal. A primitive cell and unit cell are shown in black (note the primitive cell and the unit cell are identical for the tetragonal case). The coloured regions represent areas where the acoustic radiation potential is less than λ_min + 0.1(λ_max − λ_min) and λ_min (resp. λ_max) are the minimum (resp. maximum) eigenvalues of $\mathsf{Q}(\mathbf{0})$. The blue regions are the expected locations of minima due to the prescribed minimum location (the origin). The red regions are other minima that appear in the process. In all panels, the acoustic radiation potential parameters are $\mathfrak{a}=\mathfrak{b}=1$. The black arrows indicate the directions normal to the transducer surfaces. (Online version in colour.)

Figure 8.

Representatives of the six achievable three-dimensional Bravais lattice classes. The classes are: triclinic primitive (g₁ = (1, 2, 7), g₂ = (8, 3, 5), g₃ = (1, 3, 5)), orthorhombic face-centred (a = 1, b = 2, c = 3), trigonal primitive (a = 1, c = 2), cubic primitive, cubic face-centred and cubic body-centred. The primitive cell is shown in black. The blue regions are the expected locations of minima due to the prescribed minimum location (the origin). The red regions are other minima that appear in the process. In all panels, the acoustic radiation potential parameters are $\mathfrak{a}=\mathfrak{b}=1$. The black arrows indicate the directions normal to the transducer surfaces. (Online version in colour.)

4. Summary and perspectives

We have shown that the behaviour of a periodic acoustic radiation potential in two or three dimensions can be characterized by a 4 × 4 or 6 × 6 real symmetric matrix, whose eigendecomposition can be found explicitly. Our main result is to use symmetries of the corresponding eigenspaces to predict whether the global minima of the acoustic radiation potential are limited to points, lines or planes. It is still an open question whether the global minima for periodic acoustic radiation potentials can be more general curves or surfaces. We also identify classes of Bravais lattices that can be achieved using a periodic acoustic radiation potential. We are currently working on extending this work to quasi-periodic arrangements of particles, which correspond to the case where we use a number of transducer directions that is larger than the dimension d. Note that the acoustic radiation potential derivation assumes the particles are spherical. Thus it is also interesting to see how the particle shape influences the possible periodic arrangements of particles. We also plan to characterize the scattering off gratings for millimetre waves that can be fabricated using ultrasound directed self-assembly.

Data accessibility

Matlab code for generating figures 2–8 is available at https://github.com/fguevaravas/crystals and has been tested with Matlab versions 2014b and 2018b on Mac OS X.

Authors' contributions

F.G.V. and C.M. contributed to designing, organizing and proving the theoretical results. The design of the numerical experiments is due to F.G.V. and C.M. The numerical experiments were written and performed by C.M. Both authors contributed equally to writing and revising the final manuscript.

Competing interests

We declare we have no competing interests.

Funding

The authors acknowledge support from Army Research Office Contract No. W911NF-16-1-0457.