# A Review of Geometry, Construction and Modelling for Carbon Nanotori
## 3 Lennard-Jones Potential and Continuum Approximation
### Interaction Energy of Sphere Encapsulated Inside Torus  
To evaluate the interaction energy of a sphere symmetrically encapsulated inside a torus, it is helpful to first focus only on the interaction energy between a point and a sphere, and then we allow the point to be an arbitrary point on the toroidal surface. Assuming that the centre of the sphere of radius \(a\) is located at the origin and the atomic point is situated along the axis at a distance \(\delta\), we can parameterise any position on the spherical surface as  
\[(x,y,z)=(a\sin\phi\cos\theta,a\sin\phi\sin\theta,a\cos\phi),\]  
Figure 5: Schematic showing atom located on central axis of nanotorus.  
where \(\theta\) and \(\phi\) are polar and azimuthal angles measured counterclockwise from the \(x\)-axis and the \(xy\)-plane, respectively, as shown in Figure 6.  
The energy expression for an atom and a fullerene, denoted by \(I_{n}^{\text{(ps)}}\), can be deduced as follows [41]  
\[I_{n}^{\text{(ps)}} =\int_{0}^{\pi}\int_{-\pi}^{\pi}\frac{a^{2}\sin\phi}{\left[(\delta -a)^{2}+4\delta a\sin^{2}(\phi/2)\right]^{n}}\,\mathrm{d}\theta\mathrm{d}\phi.\] \[=\frac{4\pi a^{2}}{(n-1)(\delta^{2}-a^{2})^{n}}U_{n-2}\left( \frac{\delta^{2}+a^{2}}{\delta^{2}-a^{2}}\right), \tag{6}\]  
where \(U_{n}(x)\) is the Chebyshev polynomial of the second kind, and further details of the calculation can be found in [42]. Subsequently, the arbitrary point in Figure 6 can range over the toroidal surface.  
Assuming that the torus has minor and major radii \(r\) and \(R\), respectively, and translating the centre of the sphere to be situated along the central ring inside the torus as illustrated in Figure 7, we can parameterise any position of the point on the toroidal surface by  
\[(x,y,z)=\big{(}(R+r\cos\phi)\cos\theta,(R+r\cos\phi)\sin\theta,r\sin\phi\big{)}. \tag{7}\]  
In this case, the distance \(\delta\) is no longer a constant, but varies depending on the position of the arbitrary point on the toroidal surface. We now introduce a new distance parameter \(\Delta\) which may be expressed as  
\[\Delta^{2}=r^{2}+4R(R+r\cos\phi)\sin^{2}(\theta/2),\]  
and on substituting \(\delta=\Delta\) into Equation (6) and performing an integration over the toroidal surface, we may deduce  
\[I_{n}^{\text{(st)}} =\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\frac{4\pi a^{2}r(R+r\cos\phi)} {(n-1)\left[r^{2}+4R(R+r\cos\phi)\sin^{2}(\theta/2)-a^{2}\right]^{n}}\] \[\qquad\times U_{n-2}\left(\frac{r^{2}+4R(R+r\cos\phi)\sin^{2}( \theta/2)+a^{2}}{r^{2}+4R(R+r\cos\phi)\sin^{2}(\theta/2)-a^{2}}\right)\, \mathrm{d}\theta\mathrm{d}\phi\] \[=\frac{\pi a^{2}r}{n-1}\sum_{m=0}^{\lfloor n/2-1\rfloor}(-1)^{m} \binom{n-m-2}{m}2^{n-2m+1}\int_{-\pi}^{\pi}(R+r\cos\phi)\bar{I}_{m}^{\text{( st)}}\,\mathrm{d}\phi, \tag{8}\]  
Figure 6: Schematic showing fullerene of radius \(a\) and atom located of a distance \(\delta\) apart.  
where \(U_{n}(x)\) is the Chebyshev polynomial of the second kind and  
\[\begin{split}\bar{I}_{m}^{\rm(st)}&=\frac{\pi(r^{2}+a^ {2})^{n-2m-2}}{(r^{2}-a^{2})^{2n-2m-2}}\sum_{j=0}^{n-2m-1}\frac{(-1)^{j}}{j!} \begin{pmatrix}2j\\ j\end{pmatrix}(-n+2m+2)_{j}\left[\frac{R(R+r\cos\phi)}{r^{2}+a^{2}}\right]^{j }\\ &\quad\times F\left(1/2+j,2n-2m-2;1+j;-\frac{4R(R+r\cos\phi)}{r^{2} -a^{2}}\right),\end{split}\]  
and we refer the reader to the original paper [42] for further details of these calculations. Finally, the total interaction energy for a sphere encapsulated inside a torus, denoted by \(E^{\rm(st)}\), can be obtained as  
\[E^{\rm(st)}=\eta_{\rm s}\eta_{\rm t}(-AI_{3}^{\rm(st)}+BI_{6}^{\rm(st)}), \tag{9}\]  
where \(\eta_{\rm s}\) and \(\eta_{\rm t}\) are the mean atomic surface densities for the sphere and the torus, respectively, and \(I_{n}^{\rm(st)}\) for \(n=3,6\) are defined in (8). Although clearly complicated, these expressions can nevertheless be readily evaluated with the aid of standard algebraic packages, such as MATLAB and MAPLE.