# A Review of Geometry, Construction and Modelling for Carbon Nanotori
## 2 Geometry and Construction of Carbon Nanotori  
Carbon nanotori are the circular ring structures defined by two parameters describing their geometry, called major and minor radii. The major radius of the carbon nanotori is the radius or the distance between the centre to the centre of the curved tube, while the minor radius is the radius of the closed tube as illustrated in Figure 1. This experimentally determined figure is obtained from [18] and shows a toroidal structure of C\({}_{750}\). The major radii are used to classify carbon nanotori into two groups. If their major radius is greater than 100 nm, they are regarded as large carbon nanotori, and otherwise, they are considered to be small carbon nanotori.  
The larger carbon nanotori were discovered in laser vaporisation [7] and thermal decomposition [19] experiments of single-walled carbon nanotubes as shown in Figure 2. Typically, the major radii of  
Figure 1: Structure of nanotorus showing major and minor radii \(R\) and \(r\)[18]. Reprinted from Journal of Molecular Structure (Theochem), 681, [PERSON] and [PERSON] and [PERSON] and [PERSON], Electronic properties of carbon nanotoroidal structures, 231–234, Copyright (2004), with permission from Elsevier.  
observed carbon nanotori range between 150 and 250 nm, while the minor radii are approximately 2-4 nm. These ring structures might be envisaged as curved carbon nanotubes formed by experimental methods, that are bent in such as manner that their open ends are seamlessly connected together. Such a process is kinetically controlled by the ultrasonic irradiation of nanotubes, and the ring structures obtained are stabilised through van der Waals interactions [20; 21]. The key condition for the stability of these configurations is that their strain energy must not exceed the binding energy [22], so that the small carbon nanotori with major radius less than 100 nm do not arise from this technique according to the elasticity of the carbon nanotubes [23].  
However, smaller carbon nanotori and in particular those with major radii much smaller than 100 nm can at least theoretically be fabricated by introducing a pentagon-heptagon pair into the graphitic honeycomb networks of two carbon nanotubes [24; 25]. The two carbon nanotubes need not be of the same type; for example one might be zigzag and the other tube armchair. In this method, the bend angle between two segments of the two different tubes is approximately 30\({}^{\circ}\), meaning that such 12 segments of carbon nanotubes are needed to construct a closed toroidal molecule. Experimentally, the bend angle is more likely to be 36\({}^{\circ}\) due to strain relaxation around the joints [26], and as a result, only 10 equal segments are required to complete a ring structure. In this event, the condition that each segment of carbon nanotubes be of the same length might be relaxed. In this way, a wide variety of carbon nanotori can be constructed by connecting segments of carbon nanotubes of different lengths, bend angles and types [27; 28]. Figure 3 presents examples of such a construction.  
Carbon nanotori can also be produced by other modelling techniques, such as constructing from fullerenes using the [PERSON]'s prescription [29; 30], sewing the walls of single-walled [31; 32; 33], double-walled [34] or triple-walled [35] carbon nanotubes at both ends and fabricating from only pentagon and heptagon networks [36].  
Figure 2: Experimentally determined carbon nanotorus [7]. Reprinted by permission from Springer Nature: Nature (Fullerene ’crop circles’, [PERSON] and [PERSON] and [PERSON] and [PERSON] and [PERSON], 1997).