# A Review of Geometry, Construction and Modelling for Carbon Nanotori

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## Abstract

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## 1. Introduction

## 2. Geometry and Construction of Carbon Nanotori

## 3. Lennard-Jones Potential and Continuum Approximation

## 4. Mathematical and Theoretical Framework for Interaction Energy of Nanotorus with Other Nanostructures

#### 4.1. Interaction Energy of Point and Torus

#### 4.2. Interaction Energy of Sphere Encapsulated Inside Torus

#### 4.3. Interaction Energy of Sphere and Nanotorus

#### 4.4. Interaction Energy of Two Tori

#### 4.5. Interaction Energy of Cylinder and Torus

#### 4.6. Interaction Energy of Offset Molecule and Torus

## 5. Physical Roles, Properties and Applications

#### 5.1. Electronic Properties

#### 5.2. Magnetic Properties

#### 5.3. Some Potential Applications

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Structure of nanotorus showing major and minor radii R and r [18]. Reprinted from Journal of Molecular Structure (Theochem), 681, E. Yazgan and E. Taşci and O.B. Malcioğlu and S. Erkoç, Electronic properties of carbon nanotoroidal structures, 231–234, Copyright (2004), with permission from Elsevier.

**Figure 2.**Experimentally determined carbon nanotorus [7]. Reprinted by permission from Springer Nature: Nature (Fullerene ‘crop circles’, J. Liu and H. Dai and J. H. Hafner and D. T. Colbert and R. E. Smalley, 1997).

**Figure 14.**Charge distributions of carbon nanotori C${}_{170}$, C${}_{250}$, C${}_{360}$, C${}_{520}$ and C${}_{750}$ obtained from [18]. Reprinted from Journal of Molecular Structure (Theochem), 681, E. Yazgan and E. Taşci and O.B. Malcioğlu and S. Erkoç, Electronic properties of carbon nanotoroidal structures, 231–234, Copyright (2004), with permission from Elsevier.

**Figure 15.**Relation between induced magnetic moment and temperature for various carbon nanotori in a perpendicular magnetic field of 0.1 T (solid) and 0.2 T (dashed) obtained from [64]. Reprinted figure with permission from L. Liu and G.Y. Guo and C.S. Jayanthi and S.Y. Wu, Colossal paramagnetic moments in metallic carbon nanotori, 88, 217206 (2002). Copyright (2019) by the American Physical Society.

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**MDPI and ACS Style**

Sarapat, P.; Hill, J.M.; Baowan, D.
A Review of Geometry, Construction and Modelling for Carbon Nanotori. *Appl. Sci.* **2019**, *9*, 2301.
https://doi.org/10.3390/app9112301

**AMA Style**

Sarapat P, Hill JM, Baowan D.
A Review of Geometry, Construction and Modelling for Carbon Nanotori. *Applied Sciences*. 2019; 9(11):2301.
https://doi.org/10.3390/app9112301

**Chicago/Turabian Style**

Sarapat, Pakhapoom, James M. Hill, and Duangkamon Baowan.
2019. "A Review of Geometry, Construction and Modelling for Carbon Nanotori" *Applied Sciences* 9, no. 11: 2301.
https://doi.org/10.3390/app9112301