## Abstract

The discovery of carbon nanotubes and C_{60} fullerenes has created an enormous impact on possible new nanomechanical devices. Owing to their unique mechanical and electronic properties, such as low weight, high strength, flexibility and thermal stability, carbon nanotubes and C_{60} fullerenes are of considerable interest to researchers from many scientific areas. One aspect that has attracted much attention is the creation of high-frequency nanoscale oscillators, or the so-called gigahertz oscillators, for applications such as ultrafast optical filters and nano-antennae. While there are difficulties for micromechanical oscillators, or resonators, to reach a frequency in the gigahertz range, it is possible for nanomechanical systems to achieve this. This study focuses on C_{60}–single-walled carbon nanotube oscillators, which generate high frequencies owing to the oscillatory motion of the C_{60} molecule inside the single-walled carbon nanotube. Using the Lennard-Jones potential, the interaction energy of an offset C_{60} molecule inside a carbon nanotube is determined, so as to predict its position with reference to the cross-section of the carbon nanotube. By considering the interaction force between the C_{60} fullerene and the carbon nanotube, this paper provides a simple mathematical model, involving two Dirac delta functions, which can be used to capture the essential mechanisms underlying such gigahertz oscillators. As a preliminary to the calculation, the oscillatory behaviour of an isolated atom is examined. The new element of this study is the use of elementary mechanics and applied mathematical modelling in a scientific context previously dominated by molecular dynamical simulation.

## 1. Introduction

Carbon nanotubes and fullerene molecules have generated considerable impact on nanotechnology and particularly for the creation of possible new nanodevices. Their small size together with their special and unique mechanical properties make both carbon nanotubes and fullerenes potential materials for many uses in nanomechanical systems. One aspect that has attracted much attention is the creation of nanoscale oscillators or the so-called gigahertz oscillators. We note that while there are difficulties for micromechanical oscillators, or resonators, to reach frequencies in the gigahertz range, it is possible for nanomechanical systems to achieve this. Cumings & Zettl (2000) experimented on multi-walled carbon nanotubes, where they removed the cap from one end of the outer shell and attached a moveable nanomanipulator to the core in a high-resolution transmission electron microscope. By pulling the core out and pushing it back into the outer shell, they report an ultra low-frictional force against the intershell sliding, which is also confirmed by Yu *et al*. (2000). Cumings & Zettl (2000) also observed that the extruded core, after release, quickly and fully retracts inside the outer shell owing to the restoring force resulting from the van der Waals interaction acting on the extruded core. These results led Zheng & Jiang (2002) and Zheng *et al*. (2002) to study the molecular gigahertz oscillators, where the sliding of the inner shell inside the outer shell of a multi-walled carbon nanotube can generate oscillatory frequencies up to several gigahertz.

Based on the results of Zheng *et al*. (2002), the shorter the inner core nanotube, the higher the frequency. As a result, Liu *et al*. (2005) investigated the case where, instead of using multi-walled carbon nanotubes, the high frequency is generated by a fullerene C_{60} oscillating inside a single-walled carbon nanotube. A fullerene C_{60}, commonly known as a buckyball, comprises 60 carbon atoms bonded to approximately form a sphere. We refer the reader to Dresselhaus *et al*. (1996) for details and properties of fullerenes. Further, in contrast to the multi-walled carbon nanotube oscillator, the C_{60}–nanotube oscillator appears not to suffer the rocking motion, which is associated with the high frictional effect. While Qian *et al*. (2001) and Liu *et al*. (2005) use the molecular dynamical simulation approach to study this problem, this paper employs elementary mechanical principles utilizing the continuum approximation arising from the assumption that the discrete atoms can be smeared across the surface, to provide a model for the C_{60}–single-walled carbon nanotube oscillator.

Further, we adopt Newton's second law for the oscillatory motion of the C_{60} molecule of radius *b* and oscillating in a carbon nanotube of radius *a* and length 2*L*, namely(1.1)where *Z* is the distance between the centres of the fullerene C_{60} and the carbon nanotube; *m*_{f}, the total mass of the fullerene C_{60}; *F*_{vdW}(*Z*), the van der Waals restoring force; and *F*_{r}(*Z*), the total sliding resistance force experienced by the C_{60} molecule. Since experiments have shown for multi-walled carbon nanotube oscillators that *F*_{r}(*Z*) is extremely small when compared with the restoring force, Zheng & Jiang (2002) neglect *F*_{r}(*Z*) in their calculation. However, Zheng *et al*. (2002) incorporate the sliding resistance force in the model of Zheng & Jiang (2002), where *F*_{r}(*Z*) is assumed to depend on the chirality of both shells in the intershell sliding.

In this paper, we study both cases with and without the frictional effect. Owing to the symmetry of the problem, we are only concerned with the force in the axial direction. In §3*b*, we show that for *b*<*a*≪2*L*, the axial van der Waals force , equation (3.5), behaves like two equal and opposite Dirac delta functions at the tube extremities *Z*=−*L* and *Z*=*L*. Accordingly, *F*_{vdW}(*Z*) used in Newton's second law can be approximated by(1.2)where *W* is the constant pulse strength. In the case where the frictional force is neglected, the C_{60} molecule oscillates inside the carbon nanotube with a constant velocity, agreeing with the result of Qian *et al*. (2001), and the instantaneous forces operating at the extremities of the tube serve only to change the direction of the fullerene. Further, this model also predicts that the oscillatory frequencies of the fullerene C_{60} should be in the gigahertz range. In §3*b*, it is shown that the approximation (1.2) is able to capture the essential mechanisms of a C_{60}–nanotube oscillator and provide reasonable agreement to the molecular dynamical studies of Qian *et al*. (2001) and Liu *et al*. (2005). We further emphasize that the major contribution of this paper is the use of elementary mechanics and mathematical modelling techniques in this context, where prior studies predominantly involve molecular dynamical simulation. In addition, we note that the modelling presented here differs from that given for the C_{60}–carbon nanotube oscillators (Qian *et al*. 2001; Liu *et al*. 2005) and other proposed models for the multi-walled carbon nanotube oscillators (Zheng & Jiang 2002; Zheng *et al*. 2002; Rivera *et al*. 2003) in two salient aspects. Firstly, we identify the resulting van der Waals force as approximately two delta functions and from which we obtain the constant velocity of the C_{60} travelling inside the carbon nanotube. Secondly, we model the frictional term for the spherical C_{60} as that arising from a ring of contact of a certain prescribed length rather than the full surface contact area as is done for the oscillating cylinder.

In the following section, we introduce the Lennard-Jones potential and the continuum approach of assuming an average surface density of carbon atoms. In §2, we also determine the location of the minimum potential energy of an offset atom and an offset C_{60} molecule inside a carbon nanotube. In §3, we first study the idealized situation of an atom moving along the carbon nanotube axis. A similar approach is applied in the second part of §3 since the C_{60} fullerene also assumed moving along the axis. The force generated from the Lennard-Jones potential is used as the restoring force for both the atom and the C_{60} molecule, causing the oscillation of the particle between both ends of the carbon nanotube. Although we neglect the frictional effect in §3, following Zheng *et al*. (2002) a periodic frictional force is introduced in §4 for the case of the oscillating fullerene. Finally, some conclusions are presented in §5.

## 2. Energy potentials for offset atoms and C_{60} fullerenes

Generally speaking and depending on the relative dimensions, a single atom and, to a certain extent, a buckyball will tend to have a preferred position located approximately one inter-atomic length from the surface of the carbon nanotube. In this section, we formally confirm this by examining the Lennard-Jones potential energy for an offset atom and an offset buckyball.

In continuum approximation, where carbon atoms are assumed to be uniformly distributed over the surface of molecules, the non-bonded interaction energy can be obtained from(2.1)where *n*_{g} and *n*_{f} represent the mean surface density of carbon atoms on a carbon nanotube and a buckyball, respectively, and *r* denotes the distance between two typical surface elements d*Σ*_{g} and d*Σ*_{f} on the two different molecules. The potential function adopted here is the classical Lennard-Jones potential given by(2.2)where *A* and *B* are the attractive and the repulsive constants, respectively. The Lennard-Jones potential has been used in different configurations, including the interactions between two identical parallel carbon nanotubes (Girifalco *et al*. 2000), carbon nanotube bundles (Henrard *et al*. 1999), a carbon nanotube and a C_{60} molecule (both inside and outside the tube; Girifalco *et al*. 2000) and two C_{60} molecules (Girifalco 1992).

### (a) An offset atom inside a single-walled carbon nanotube

Here, we determine the preferred position of an offset atom with reference to the cross-section of a carbon nanotube. This position is where the atom has its minimum potential energy. In an axially symmetric cylindrical polar coordinates, without the loss of generality, we may assume that the atom is located at (*ϵ*, 0, 0), as shown in figure 1, and that the carbon nanotube of infinite extent with a parametric equation of (*a* cos *θ*, *a* sin *θ*, *z*). We note that *ϵ* is the assumed distance of the offset atom from the central axis of the tube, *a* is the tube radius, −*π*≤*θ*≤*π* and −∞<*z*<∞. In this case, the distance *ρ* from the atom to the wall of carbon nanotube is given by(2.3)

Thus, from equation (2.1), the potential energy *E* for the offset atom, which interacts with the entire carbon nanotube, is of the form(2.4)

Now if we let , then *ρ*=(*λ*^{2}+*z*^{2})^{1/2}. By using the substitution *z*=*λ* tan *ψ* we obtain(2.5)

For a positive integer *m*, we know(2.6)where !! is the double factorial operator, defined as and , which upon using equation (2.5) becomes(2.7)

Further, if we define the integrals *J*_{n} as(2.8)where *n* is a positive integer, *α*=*a*^{2}+*ϵ*^{2} and *β*=2*aϵ*, noting that since *α*−*β*=(*a*−*ϵ*)^{2} and *a*>*ϵ* then *α*>*β*≥0, and equation (2.7) becomes(2.9)

In appendix A, we show that the integrals *J*_{n} can be evaluated either in terms of elliptic integrals or in terms of hypergeometric functions.

In figure 2, we plot the potential energy *E* as given by equation (2.9) with respect to *ϵ* for an atom inside the carbon nanotube (6, 6) (*a*=4.071 Å) and (10, 10) (*a*=6.784 Å). It can be seen that the minimum energy of the atom inside (6, 6) occurs at *ϵ*=0 and 3.291 Å for (10, 10) measured from the tube axis. This equates to a distance between the atom and the wall of 4.071 and 3.494 Å, respectively. We observe numerically that as the tube radius gets larger, the atom is likely to be closer to the tube wall. We note that the constants used in the numerical calculations throughout this paper are given in table 1, and *A* and *B* are the approximate values arising from those for the C_{60} and graphene interaction.

### (b) An offset C_{60} molecule inside a single-walled carbon nanotube

In this section, we determine the location of the minimum potential energy of a C_{60} molecule with reference to the cross-section of a carbon nanotube. In axially symmetric cylindrical polar coordinates, we assume the buckyball of radius *b* is located at (*ϵ*, 0, 0), as shown in figure 3, and in a carbon nanotube of infinite extent with a parametric equation (*a* cos *θ*, *a* sin *θ*, *z*). We note that *ϵ* is the distance between the centre of the offset molecule and the central axis of the tube, *a* is the tube radius, −*π*≤*θ*≤*π* and −∞<*z*<∞. From figure 3, the distance from the centre of C_{60} molecule to the wall of carbon nanotube is given by(2.10)

The interaction between the C_{60} molecule and the carbon nanotube in the continuum approximation is obtained by averaging over the surface of each entity. By performing the surface integral of the Lennard-Jones potential over the sphere, we find that the potential energy for a typical surface element on the tube interacting with the entire spherical C_{60} molecule of radius *b* is given by(2.11)where *ρ* is the distance between the tube surface element and the centre of the buckyball, as shown in figure 3. The detail of the derivation of equation (2.11) is given in appendix A of part I. Thus, the potential energy between the C_{60} molecule and the entire carbon nanotube is obtained by performing the surface integral of equation (2.11) over the carbon nanotube, thus(2.12)

Here, we rewrite equation (2.11) in the form(2.13)where , which gives . Again, we employ the substitution *z*=*λ* tan *ψ*, so that with the use of(2.14)and equation (2.8), where and *β*=2*aϵ*, the potential energy (2.12) becomes(2.15)where *J*_{n} are the integrals defined by equation (2.8) and evaluated in appendix A.

As shown in figure 4, the preferred location of the C_{60} molecule inside the carbon nanotube (10, 10) is where the centre of the buckyball lies on the tube axis. In the case of (16, 16) (*a*=10.856 Å), we obtain *ϵ*=4.314 Å (see the dash line in figure 4). These results are equivalent to the distance between the centre of the buckyball and wall of the nanotube of 6.784 and 6.542 Å, respectively. Furthermore, we observe that as the radius of the tube gets larger, the location where the minimum energy occurs, tends to be closer to the nanotube wall. These results agree with the work of Girifalco *et al*. (2000).

## 3. Oscillation of an atom or a C_{60} fullerene inside a single-walled carbon nanotube

Here, we adopt Newton's second law to describe the oscillatory motion of a particle (an atom or a fullerene C_{60}) in a carbon nanotube, namely(3.1)where *Z* is the distance between the centres of the particle and the carbon nanotube; *M*, the mass of the particle; *F*_{vdW}(*Z*) and *F*_{r}(*Z*), the restoring force and the frictional force, respectively. In part I of this paper, we determine the suction force that is sufficient to attract the particle (an atom or a C_{60} molecule) into the tube and begin oscillating, and we also prescribe the magnitude of the energy imparted to the particle by this force. Section 3*a* illustrates the analysis necessary in the case of the oscillating atom. Subsequently, we employ similar techniques to study the oscillation of a fullerene C_{60} inside a single-walled carbon nanotube.

### (a) Oscillation of a single atom inside a single-walled carbon nanotube

In an axially symmetric cylindrical polar coordinate system (*r*, *z*), an atom is assumed to be located at (0, *Z*) inside a carbon nanotube of length 2*L*, centred around the *z*-axis and of radius *a*, as shown in figure 5. Here, we assume that the atom oscillates along the *z*-axis. This assumption is valid for the carbon nanotube (6, 6), where the atom is likely to be on the *z*-axis owing to the minimum potential energy, as shown in §2*a*. From the symmetry of the problem, we are only concerned with the force in the axial direction, and on neglecting the frictional force *F*_{r}(*Z*), we have from equation (3.1)(3.2)where *m*_{0} is the mass of a single atom and is the total axial van der Waals interaction force between the atom and the carbon nanotube length 2*L*, given by(3.3)and we refer the reader to part I for details of this derivation. In figure 6, we plot as given by equation (3.3), for the case of the carbon nanotube (6, 6) (*a*=4.071 Å). We note that (6, 6) with *a*=4.071 Å satisfies the condition stated in part I of this paper, where both ends of the tube can generate the necessary attractive force to suck the atom inside, and the atom oscillates and never escapes the carbon nanotube. Throughout this paper, we note that the unit of potential energy is given by electron volt (1.602×10^{−19} N m), the unit of length and force are Å and eV/Å=1.602×10^{−9} N, respectively.

### (b) Oscillation of a fullerene C_{60} inside a single-walled carbon nanotube

In an axially symmetric cylindrical polar coordinate system (*r*, *z*), we assume a fullerene C_{60} is located inside a carbon nanotube of length 2*L*, centred around the *z*-axis and of radius *a*. As shown in figure 7, we also assume that the centre of the C_{60} molecule is in the *z*-axis. Again, this is justified for the carbon nanotube (10, 10), as previously described in §2*b*.

From the symmetry of the problem, we only need to consider the force in the axial direction. As a result, from Newton's second law, again neglecting the frictional force, we have(3.4)where *m*_{f} is the total mass of a C_{60} molecule and is the total axial van der Waals interaction force between the C_{60} molecule and the carbon nanotube, given by(3.5)where *P*(*ρ*) is the potential function given by equation (2.11), and . Again we refer the reader to part I for details of this derivation. Here, we assume that the effect of the frictional force may be neglected, which is reasonable for certain chiralities and diameters of the tube. For example, the preferred position of the C_{60} molecule inside the carbon nanotube (10, 10) is where the centre is on the *z*-axis. Thus, the molecule tends to move along the axial direction and not to suffer a rocking motion. However, in §4 the frictional force is included in this model.

In figure 8, we plot as given by equation (3.5), for the case of the carbon nanotube (10, 10) and we observe that the force is very close to zero everywhere except at both ends of the tube, where there is a pulse-like force which attracts the buckyball back towards the centre of the tube. For *b*<*a*≪2*L*, we find that can be estimated using the Dirac delta function and thus, equation (3.4) reduces to give(3.6)where *W* is pulse strength or the work (energy) of the C_{60} molecule, given by . Therefore, from equation (3.5), we find(3.7)where . We refer the reader to part I of this paper for the derivation of equation (3.7) (see eqn (4.11) in part I). For a double-walled carbon nanotube oscillator, we note that as the inner tube gets smaller to the order of the diameter of a buckyball, the van der Waals interaction force as shown in the molecular dynamics simulation of Legoas *et al*. (2003) becomes a peak-like force operating at both ends of the outer tube. This is similar to the behaviour of the C_{60}–nanotube oscillator as shown here and this similarity is also observed by Baowan & Hill (in press).

Now, we consider equation (3.6) and multiplying d*Z*/d*t* on both sides of equation (3.6), we obtain(3.8)

From d*H*(*x*)/d*x*=*δ*(*x*), where *H*(*x*) is the usual Heaviside step function, equation (3.8) can be written as(3.9)

By integrating both sides of equation (3.9) with respect to *t*, from *t*=0 (assuming the buckyball is at infinity) to any time *t*, where the ball is located at *Z*, and upon using that *H*(*Z*+*L*)−*H*(*Z*−*L*)=1 for −*L*≤*Z*≤*L* and zero elsewhere, we obtain(3.10)where *v*_{0} is the initial velocity that the ball is fired on the *z*-axis towards the open end of the carbon nanotube in the positive *z*-direction. We note that the initial velocity *v*_{0} is introduced for the case, where the C_{60} molecule is not sucked into the carbon nanotube owing to the strong repulsion force. From equation (3.10) for −*L*≤*Z*≤*L*, we find(3.11)which implies that the buckyball travels inside the carbon nanotube at the constant speed . Alternatively, equation (3.11) can also be formally obtained using the Lorentzian limit as shown in appendix B.

On using equation (3.7) and the constants given in table 1, we obtain the velocity *v*=932 m s^{−1} for the case when the C_{60} molecule is initially at rest outside the carbon nanotube (10, 10) and the molecule gets sucked into the tube owing to the attractive force. This gives rise to the frequency *f*=*v*/(4*L*)=36.13 GHz. In figure 9, we plot the oscillatory frequency with respect to the length of the nanotube. The result obtained agrees with the molecular dynamics study of Liu *et al*. (2005). It confirms their finding that the shorter the nanotube, the higher the oscillatory frequency. Further, we consider the case where the C_{60} molecule is fired on the tube axis towards the open end of the carbon nanotube of radius *a*<6.338 Å, which does not accept a C_{60} molecule by suction forces alone owing to the strong repulsive force of the carbon nanotube (see part I of this paper). For the carbon nanotube (9, 9) (*a*=6.106 Å), the initial velocity *v*_{0} needs to be approximately 1268 m s^{−1} for the C_{60} molecule to penetrate into the tube. A result similar to that given by Qian *et al*. (2001) is also found here, where a C_{60} molecule cannot penetrate into either of (8, 8), (7, 7), (6, 6), (5, 5) even though it is fired into the tube with an initial velocity as high as 1600 m s^{−1}. In addition, we find for (8, 8) with *a*=5.428 Å from our model that the minimum initial velocity must be approximately 8210 m s^{−1} for the C_{60} molecule to penetrate into the tube.

## 4. Frictional force

Although an ultra low-frictional effect in nano-oscillators has been observed through experiments and molecular dynamics studies, we still need to properly understand the frictional behaviour at the molecular level. There are a number of studies on the frictional effect of the sliding of the inner shell inside the outer shell of double- and multi-walled carbon nanotubes. Rivera *et al*. (2003) and Servantie & Gaspard (2006) assume that the friction is dependent on the sliding velocity. The latter authors also consider friction to depend on the position of the inner shell with respect to the outer shell, and they find that this dependence gives rise to a relatively small effect. While Zheng & Jiang (2002) neglect the friction effect between the inner and outer shells in their calculation of the oscillatory frequency, they introduce an inter-atomic locking force which acts against the intershell sliding. This force is also incorporated in the model of Zheng *et al*. (2002). Further, Zheng & Jiang (2002) state that the frictional force resulting from inter-atomic locking is a potential force and therefore is non-dissipative. This approach is different from Zhao *et al*. (2003) and Ma *et al*. (2005), where the frictional effect is assumed to be a result of a dissipative energy during an off-axial rocking motion of the inner tube and which in consequence gives rise to a wavy deformation on the outer shell.

In contrast to the double- and multi-walled carbon nanotube oscillators, there are no proposed frictional force models for a C_{60} molecule oscillating inside a single-walled carbon nanotube. For a C_{60} spherical fullerene, the tendency for a rocking motion is considerably reduced. Accordingly, we propose here a model for the frictional term for the spherical C_{60} as that arising from a ring of contact of a certain prescribed length. Following Zheng *et al*. (2002), we assume a periodic inter-atomic locking force(4.1)where is the spatial period of the inter-atomic locking, typically for an armchair or (*n*, *n*) carbon nanotube, where *n* is a positive integer, it is the distance between opposite bonds of the carbon ring and *κ*_{0}=*τ*_{s}*α*, where *τ*_{s} denotes the resistance strength and *α* denotes the area of a ring of contact of a certain prescribed length of the sphere, and which is given by , for a certain angle *θ*_{0}. For multi-walled carbon nanotube oscillators, Zheng *et al*. (2002) state that the spatial period is dependent on the helicities of both tubes in the intershell sliding, and that the resistance force increases with the degree of commensurability of the two shells. Here, we assume high commensurability between a C_{60} molecule and a single-walled carbon nanotube, which then leads to assuming (Zheng *et al*. 2002), where *σ* is the bond length. By introducing equation (4.1) into equation (3.6), we obtain(4.2)Again, by multiplying d*Z*/d*t* on both sides of equation (4.2), we deduce(4.3)which upon integrating with respect to *t*, applying the zero initial condition, and assuming that there is no frictional force at *t*=0 since the C_{60} molecule is assumed outside the nanotube, we find(4.4)which for the buckyball inside the carbon nanotube (−*L*≤*Z*≤*L*), equation (4.4) gives rise to(4.5)or(4.6)Upon integrating equation (4.6), we have the time *T* where the buckyball travels from −*L* to *L* as(4.7)From equation (4.7), on making the substitution , we obtain(4.8)where the modulus *k*^{2} is defined by(4.9)

The integral appearing in equation (4.8) is the normal elliptic integral of the first kind, usually denoted by *F*(*ϕ*, *k*), thus(4.10)and equation (4.8) becomes(4.11)

Generally, the position of the buckyball is determined from(4.12)which on non-dimensionalizing by *T* we obtain(4.13)which is shown graphically in figure 10, noting that it is almost a straight line owing to the small value of *k*^{2}.

## 5. Conclusions

In part I of this paper, the condition for a C_{60} fullerene initially outside the carbon nanotube to be sucked into the tube is given. Here in part II, we assume that the C_{60} molecule is sucked in and oscillates inside the carbon nanotube of a finite length. For simplicity, we first consider the ideal situation of an oscillating single atom and a similar but more involved method can be employed in the case of C_{60}–nanotube oscillator.

For both the atom and the C_{60} molecule, using the Lennard-Jones potential, we determine its most stable positions with reference to the cross-section of the carbon nanotube. The preferred location is where the potential interaction energy between the particle and the nanotube is a minimum. Generally, we find that inside the carbon nanotube, the atom or the C_{60} molecule is at an inter-atomic distance from the tube wall. However, in particular, inside the carbon nanotube (6, 6), the atom is most probably at the centre of the cross-section of the carbon nanotube and inside the carbon nanotube (10, 10), the centre of C_{60} molecule is most probably at the centre of the cross-section of the tube.

For both an oscillating single atom and an oscillating buckyball C_{60} within the interior of a carbon nanotube, we use the Lennard-Jones potential to calculate the van der Waals restoring force. Owing to the symmetry of the problem, only the force in the axial direction needs to be considered. In both cases, we demonstrate that the resultant van der Waals axial force can be approximated by two equal and opposite Dirac delta functions operating at the two extremities of the carbon nanotubes. Assuming zero friction, this model implies that the atom or buckyball C_{60} oscillates at constant velocity and the instantaneous forces at the extremities serve only to change direction. Our model also predicts the oscillating frequency of the C_{60} molecule to be in the gigahertz range, which is in agreement with molecular dynamical simulations.

## Acknowledgments

The authors are grateful to the Australian Research Council for their support through the Discovery Project Scheme and the provision of an Australian Professorial Fellowship for J.M.H. The authors also wish to acknowledge Professor Julian Gale of Curtin University of Technology for his many helpful comments and discussions on this and other related work, and for suggesting the calculations given in §2.

## Footnotes

- Received March 29, 2006.
- Accepted August 17, 2006.

- © 2006 The Royal Society