^{1,a)}, Maarten Verdult

^{2}, Marjolein Dijkstra

^{1}and René van Roij

^{2}

### Abstract

We employ the coupled dipole method to calculate the polarizabilitytensor of various anisotropic dielectricclusters of polarizable atoms, such as cuboid-, bowl-, and dumbbell-shaped nanoparticles. Starting from a Hamiltonian of a many-atom system, we investigate how this tensor depends on the size and shape of the cluster. We use the polarizabilitytensor to calculate the energy difference associated with turning a nanocluster from its least to its most favorable orientation in a homogeneous static electric field, and we determine the cluster dimension for which this energy difference exceeds the thermal energy such that particle alignment by the field is possible. Finally, we study in detail the (local) polarizability of a cubic-shaped cluster and present results indicating that, when retardation is ignored, a bulk polarizability cannot be reached by scaling up the system.

This work is part of the research programme of FOM, which is financially supported by NWO. Financial support by an NWO-VICI grant is acknowledged.

I. INTRODUCTION

II. FORMALISM OF THE CDM: POTENTIAL ENERGY, POLARIZABILITY, AND ORIENTATIONAL ENERGY

A. Introduction

B. Static polarizability and orientational energy of dipole clusters in an external electric field

C. Rotationally symmetric clusters

D. Fourfold rotationally symmetric clusters

E. Units of distance

III. DIELECTRIC RODS AND PLATELETS

A. Dielectric strings

IV. DIELECTRIC BOWLS AND DUMBBELLS

A. Bowls

B. Dumbbells

V. THE ENHANCEMENT FACTOR OF DIELECTRIC CUBES

A. Theoretical predictions

B. Numerical methods

1. Exploiting symmetry

2. The Gauss-Seidel method

C. Numerical results

1. Local enhancement factor

VI. CONCLUSIONS AND OUTLOOK

### Key Topics

- Electric fields
- 57.0
- Polarizability
- 36.0
- Polarization
- 27.0
- Dielectrics
- 14.0
- Eigenvalues
- 14.0

## Figures

Properties of an *l* × *l* × *L* cuboid-shaped cluster of atoms with atomic polarizability α_{0} on a simple cubic lattice with spacing . (a) The elements *f* _{ xx } (red) and *f* _{ zz } (blue) of the enhancement factor matrix as a function of the shape parameter *l*/*L*, for *L* = 10. Note that *f* _{ xx } = 1 at *l*/*L* ≈ 0.73, *f* _{ zz } = 1 at *l*/*L* ≈ 1.17, and *f* _{ xx } = *f* _{ zz } = 1.05687 at *l*/*L* = 1. (b) The difference Δ_{ f } = *f* _{ zz } − *f* _{ xx } of the enhancement factor elements as a function of *l*/*L*, for several values of *L*. (c) The energy difference Δ (in units of *k* _{ B } *T*) of turning the cuboidal rod from its least to its most favorable orientation in an external electric field, as a function of the number of atoms *N* in a rod, with shape parameters *l*/*L* = 0.10, 1/6 ≈ 0.17, and 1/3 ≈ 0.33. System parameters are given in the text, and the solid lines are linear fits to the data.

Properties of an *l* × *l* × *L* cuboid-shaped cluster of atoms with atomic polarizability α_{0} on a simple cubic lattice with spacing . (a) The elements *f* _{ xx } (red) and *f* _{ zz } (blue) of the enhancement factor matrix as a function of the shape parameter *l*/*L*, for *L* = 10. Note that *f* _{ xx } = 1 at *l*/*L* ≈ 0.73, *f* _{ zz } = 1 at *l*/*L* ≈ 1.17, and *f* _{ xx } = *f* _{ zz } = 1.05687 at *l*/*L* = 1. (b) The difference Δ_{ f } = *f* _{ zz } − *f* _{ xx } of the enhancement factor elements as a function of *l*/*L*, for several values of *L*. (c) The energy difference Δ (in units of *k* _{ B } *T*) of turning the cuboidal rod from its least to its most favorable orientation in an external electric field, as a function of the number of atoms *N* in a rod, with shape parameters *l*/*L* = 0.10, 1/6 ≈ 0.17, and 1/3 ≈ 0.33. System parameters are given in the text, and the solid lines are linear fits to the data.

(a) The diagonal elements *f* _{ xx } (red squares) and *f* _{ zz } (blue circles) of the enhancement factor matrix of a straight line of *L* dipoles along the *z* axis. The spacing between the atoms is (note that for *f* _{ ii }, no other parameters are needed to define this system). (b) The energy difference Δ associated with turning this string from its least to its most favorable orientation in an external electric field *E* _{0} = 100 V mm^{−1}, for three different values of the dimensionless interatomic distance, (red), (blue), and (yellow). Choosing atomic polarizability (silica), these values correspond to spacings of, respectively, *a* ≈ 3.04 Å,3.48 Å, and 5.21 Å. The temperature is *T* = 293 K (room temperature).

(a) The diagonal elements *f* _{ xx } (red squares) and *f* _{ zz } (blue circles) of the enhancement factor matrix of a straight line of *L* dipoles along the *z* axis. The spacing between the atoms is (note that for *f* _{ ii }, no other parameters are needed to define this system). (b) The energy difference Δ associated with turning this string from its least to its most favorable orientation in an external electric field *E* _{0} = 100 V mm^{−1}, for three different values of the dimensionless interatomic distance, (red), (blue), and (yellow). Choosing atomic polarizability (silica), these values correspond to spacings of, respectively, *a* ≈ 3.04 Å,3.48 Å, and 5.21 Å. The temperature is *T* = 293 K (room temperature).

Construction and definition of parameters for (a) bowl and (b) dumbbell. (a) The bowl diameter σ and the bowl thickness *d* completely define its shape, as follows. If *d* < σ/2, the shape of the bowl is defined by subtracting one sphere from another (the blue area). (See Ref. 21.) If *d* > σ/2, we let *d* → σ − *d* and use the same construction, but take the intersection of the spheres instead of the difference (the orange area). Note that in the latter case, the shape is no longer a “bowl” in the traditional sense of the word. (b) The shape of the dumbbell is constructed by adding two (overlapping) spheres. The sphere diameter σ and the distance between the sphere centers *L* completely define the shape of the dumbbell. Note that if *L* > σ, the “dumbbell,” in fact, consists of two separate spheres.

Construction and definition of parameters for (a) bowl and (b) dumbbell. (a) The bowl diameter σ and the bowl thickness *d* completely define its shape, as follows. If *d* < σ/2, the shape of the bowl is defined by subtracting one sphere from another (the blue area). (See Ref. 21.) If *d* > σ/2, we let *d* → σ − *d* and use the same construction, but take the intersection of the spheres instead of the difference (the orange area). Note that in the latter case, the shape is no longer a “bowl” in the traditional sense of the word. (b) The shape of the dumbbell is constructed by adding two (overlapping) spheres. The sphere diameter σ and the distance between the sphere centers *L* completely define the shape of the dumbbell. Note that if *L* > σ, the “dumbbell,” in fact, consists of two separate spheres.

Examples of the dipole setup for (a) a bowl (shape parameter *d*/σ = 0.275) and (b) a dumbbell (shape parameter *L*/σ = 0.55), both intersected with a simple cubic lattice. Each sphere corresponds to an inducible dipole (Lorentz atom).

Examples of the dipole setup for (a) a bowl (shape parameter *d*/σ = 0.275) and (b) a dumbbell (shape parameter *L*/σ = 0.55), both intersected with a simple cubic lattice. Each sphere corresponds to an inducible dipole (Lorentz atom).

Quantities associated with a bowl-shaped cluster of atoms on a simple cubic (sc) lattice with dimensionless lattice constant , shape parameter *d*/σ, and diameter σ [illustrated in Fig. 3(a)]. For σ/*a* = 20, panel (a) shows the elements α_{ xx } (red) and α_{ zz } (blue) of the polarizability tensor, and (b) the elements *f* _{ xx } (red) and *f* _{ zz } (blue) of the enhancement factor tensor, both as a function of *d*/σ. The light red and blue crosses in panel (b) indicate the enhancement factor elements of a hemisphere as calculated using continuum theory in Ref. 33. Panel (c) shows the energy difference of turning a bowl from its least to its most favorable orientation in an external electric field *E* _{0} = 100 V mm^{−1}, as a function of the number of atoms in the bowl, for *d*/σ = 0.25, 0.4, and 0.75. The atomic polarizability is 5.25 Å^{3} (yielding lattice constant *a* ≈ 3.48 Å) and the temperature is *T* = 293 K. Panel (d) shows the difference |Δ_{ f }| = |*f* _{ zz } − *f* _{ xx }| of the enhancement factor elements in the *z* and *x* directions, as a function of *d*/σ, for σ/*a* = 7.5, 10.5, 13, 15.5, and 18, showing a strong dependence on the shape parameter *d*/σ and a weak dependence on the size parameter σ/*a*.

Quantities associated with a bowl-shaped cluster of atoms on a simple cubic (sc) lattice with dimensionless lattice constant , shape parameter *d*/σ, and diameter σ [illustrated in Fig. 3(a)]. For σ/*a* = 20, panel (a) shows the elements α_{ xx } (red) and α_{ zz } (blue) of the polarizability tensor, and (b) the elements *f* _{ xx } (red) and *f* _{ zz } (blue) of the enhancement factor tensor, both as a function of *d*/σ. The light red and blue crosses in panel (b) indicate the enhancement factor elements of a hemisphere as calculated using continuum theory in Ref. 33. Panel (c) shows the energy difference of turning a bowl from its least to its most favorable orientation in an external electric field *E* _{0} = 100 V mm^{−1}, as a function of the number of atoms in the bowl, for *d*/σ = 0.25, 0.4, and 0.75. The atomic polarizability is 5.25 Å^{3} (yielding lattice constant *a* ≈ 3.48 Å) and the temperature is *T* = 293 K. Panel (d) shows the difference |Δ_{ f }| = |*f* _{ zz } − *f* _{ xx }| of the enhancement factor elements in the *z* and *x* directions, as a function of *d*/σ, for σ/*a* = 7.5, 10.5, 13, 15.5, and 18, showing a strong dependence on the shape parameter *d*/σ and a weak dependence on the size parameter σ/*a*.

The enhancement factor *f* of a cubic *L* × *L* × *L* cluster with atoms on a cubic lattice, as a function of the number of atoms along the rib *L*, in (a) for the dimensionless lattice spacings (blue), (red), (yellow), (green), (light blue), and (purple), and in (b) a zoom-in (along the vertical axis) for with the data points generated by different numerical methods (see text and Table II): SGS (dark yellow points and curve), NGS (red points), SSL (green points), SDL (bright yellow points), and NDL (light blue points). Note that SSL slightly underestimates *f* for *L* ≈ 40.

The enhancement factor *f* of a cubic *L* × *L* × *L* cluster with atoms on a cubic lattice, as a function of the number of atoms along the rib *L*, in (a) for the dimensionless lattice spacings (blue), (red), (yellow), (green), (light blue), and (purple), and in (b) a zoom-in (along the vertical axis) for with the data points generated by different numerical methods (see text and Table II): SGS (dark yellow points and curve), NGS (red points), SSL (green points), SDL (bright yellow points), and NDL (light blue points). Note that SSL slightly underestimates *f* for *L* ≈ 40.

The enhancement factor *f* of a 120 × 120 × 120 cube of atoms, as a function of the (dimensionless) lattice constant . In blue, the numerical results are plotted, while in red, the prediction as given by the Clausius-Mosotti relation, Eq. (25), is shown. The dashed green line is Eq. (22) of Ref. 32, where the enhancement factor is calculated using continuum theory. Note the excellent agreement between the latter and our result, despite the completely different approaches.

The enhancement factor *f* of a 120 × 120 × 120 cube of atoms, as a function of the (dimensionless) lattice constant . In blue, the numerical results are plotted, while in red, the prediction as given by the Clausius-Mosotti relation, Eq. (25), is shown. The dashed green line is Eq. (22) of Ref. 32, where the enhancement factor is calculated using continuum theory. Note the excellent agreement between the latter and our result, despite the completely different approaches.

(a) The orientation of the planes, cube, electric field, and coordinate system with respect to each other: the planes are cut through the middle of the cube, in the plane and the plane, while the electric field is applied in the *z* direction. The cube is oriented such that the ribs lie along the Cartesian directions. The (green) line along the *x* axis denotes the intersection of the two planes and is also represented in panels (b) and (c). (b) and (c) The local enhancement factor *f*′ (defined in the text) of two sheets of dipoles lying on perpendicular planes, cut through the middle of a 120 × 120 × 120 cube of dipoles on a cubic lattice, with lattice constant . Panel (b) corresponds to the blue and panel (c) corresponds to the red plane as depicted in panel (a). The yellow and green lines appearing in this figure also correspond to the directions along which we plot the local enhancement factor in Fig. 10.

(a) The orientation of the planes, cube, electric field, and coordinate system with respect to each other: the planes are cut through the middle of the cube, in the plane and the plane, while the electric field is applied in the *z* direction. The cube is oriented such that the ribs lie along the Cartesian directions. The (green) line along the *x* axis denotes the intersection of the two planes and is also represented in panels (b) and (c). (b) and (c) The local enhancement factor *f*′ (defined in the text) of two sheets of dipoles lying on perpendicular planes, cut through the middle of a 120 × 120 × 120 cube of dipoles on a cubic lattice, with lattice constant . Panel (b) corresponds to the blue and panel (c) corresponds to the red plane as depicted in panel (a). The yellow and green lines appearing in this figure also correspond to the directions along which we plot the local enhancement factor in Fig. 10.

The local enhancement factor *f*′ along a straight line in (a) the *x* direction (, illustrated in green in Fig. 9) and (b) the *z* direction (, illustrated in yellow in Fig. 9), through the middle of a cubic cluster of atoms on a simple cubic lattice with lattice spacing , for cube rib lengths *L* = 10, 20, 50, and 120. In both panels, the rib length was scaled out, but in the inset of panel (a), we also plot *f*′ as a function of the absolute *x*-coordinate.

The local enhancement factor *f*′ along a straight line in (a) the *x* direction (, illustrated in green in Fig. 9) and (b) the *z* direction (, illustrated in yellow in Fig. 9), through the middle of a cubic cluster of atoms on a simple cubic lattice with lattice spacing , for cube rib lengths *L* = 10, 20, 50, and 120. In both panels, the rib length was scaled out, but in the inset of panel (a), we also plot *f*′ as a function of the absolute *x*-coordinate.

## Tables

Lattice spacings *a*, atomic polarizabilities α_{0}, and dimensionless lattice spacings of some typical substances. (See Ref. 26.)

Lattice spacings *a*, atomic polarizabilities α_{0}, and dimensionless lattice spacings of some typical substances. (See Ref. 26.)

An overview of techniques used for calculating the polarizability of a cubic cluster of atoms on a simple cubic lattice, and their associated acronyms, as used in the caption of Fig. 7. In the LAPACK methods, we load the elements of the matrix in memory to the numerical precision specified in the “Precision” column and use the routines in the LAPACK package to solve the relevant set of linear equations. The Gauss-Seidel methods involve (re-)calculating the elements of the matrix on the fly and, starting from an initial guess, using 20 iterations of the Gauss-Seidel method to solve the set of linear equations. The “Symmetries” column refers to whether or not the symmetries of the dielectric cube were exploited. The “*L* _{ max }” column lists estimates for largest feasible rib lengths that each method can handle, given our available resources.

An overview of techniques used for calculating the polarizability of a cubic cluster of atoms on a simple cubic lattice, and their associated acronyms, as used in the caption of Fig. 7. In the LAPACK methods, we load the elements of the matrix in memory to the numerical precision specified in the “Precision” column and use the routines in the LAPACK package to solve the relevant set of linear equations. The Gauss-Seidel methods involve (re-)calculating the elements of the matrix on the fly and, starting from an initial guess, using 20 iterations of the Gauss-Seidel method to solve the set of linear equations. The “Symmetries” column refers to whether or not the symmetries of the dielectric cube were exploited. The “*L* _{ max }” column lists estimates for largest feasible rib lengths that each method can handle, given our available resources.

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