# Symmetry and the Nanoscale: Advances in Analytical Modeling in the Perspective of Holistic Unification

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. A Recently Appeared Drude–Lorentz-Type Model

## 3. Analytical Expression of the Mean Square Deviation of Position in the Quantum-Relativistic Case

- -
- ${\omega}_{i}=({E}_{i}-{E}_{0})/\hslash ;$
- -
- ${E}_{i}$, ${E}_{0}$ energies of the excited and the ground states, respectively;
- -
- ${\Gamma}_{i}=1/{\tau}_{i}$ inverse of the decay time of every mode;
- -
- N density of carriers.

_{1}) Case ${\Delta}_{{R}_{quant}}>0$

_{2}) Case ${\Delta}_{{I}_{quant}}<0$

_{0}is the rest mass, m* is the effective mass, K is the Boltzmann’s constant, T is the system’s temperature, ${\omega}_{i}$ and ${\tau}_{i}$ are the frequencies and decaying times of each mode.

_{1}) Case ${\Delta}_{{R}_{rel}}>0$

_{2}) Case ${\Delta}_{{I}_{rel}}<0$

_{1}) Case ${\Delta}_{i{R}_{Q-R}}>0$

_{2}) Case ${\Delta}_{{iI}_{Q-R}}<0$

## 4. Results and Applications

_{2}nanoparticles, ZnO films, InP nanoparticles, semiconducting polymer molecules and carbon NTs [21,22,23].

- (1)
- Photon-Induced Near-Field Electron Microscopy: this inspection technique connects the spatial resolution at the nanoscale of the electron microscopy with the femto-second temporal resolution of extreme fast light impulses; it can be used to check very fast occurrences present at very small length scales. A way for raising the electron-light interactions in very short intervals consists of enlarging the light field through two synchronized femto-second light impulses. Variations of the time delay among the exciting light impulses and the electronic imaging ones allow one to obtain snapshots of the evanescent field as it evolves on femto-second intervals. The application of still shorter pulses can allow us to keep trace of the extreme fast processes happening in photonic and plasmonic devices [28,29].
- (2)
- Graphene based plasmonics: the non-linear optical properties of a plasma expected in the relativistic movement of electrons subjected to a high laser field are of central significance in the present research. Recently herein showed fast progress in the sector of graphene plasmonics, especially considering graphene’s special global properties. The application of graphene plasmonics will give stimulating results in the little-exploited terahertz to mid-infrared regime.

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**${R}^{2}(t)$ vs. t with ${\alpha}_{{R}_{Q-R}}^{}=10$, $T=300K$; $v={10}^{7}cm/s$ (blue solid line), $v={10}^{10}cm/s$ (red dashed line) and $v=2.5\cdot {10}^{10}cm/s$ (green dot-dashed line).

**Figure 2.**${R}^{2}(t)$ vs. t with ${\alpha}_{{I}_{Q-R}}^{}=0.5$, $T=300K$; $v={10}^{7}cm/s$ (blue solid line), $v={10}^{10}cm/s$ (red dashed line) and $v=2.5\cdot {10}^{10}cm/s$ (green dot-dashed line).

**Figure 7.**${R}^{2}(t)$ vs. t for a state given by the sum of the previous three states (data from Table 2), considering three different velocities ($v={10}^{7}cm/s$ (blue solid line), $v={10}^{10}cm/s$ (red dashed line) and $v=2.5\cdot {10}^{10}cm/s$ (green dot-dashed line)).

v (cm/s) | β^{2} | 1/ρ | γ |
---|---|---|---|

10^{7} | 0.11 × 10^{−6} | 0.998 | 1.001 |

10^{10} | 0.11 | 0.888 | 1.061 |

2.5 × 10^{10} | 0.69 | 0.31 | 1.796 |

States | ω_{i} (×10^{−12} Hz) | τ_{i} (×10^{12} Hz) | f_{i} |
---|---|---|---|

1 | 6.59 | 0.0042 | 0.312 |

2 | 1166.01 | 0.0037 | 0.176 |

3 | 2000.05 | 0.0014 | 0.512 |

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**MDPI and ACS Style**

Di Sia, P.
Symmetry and the Nanoscale: Advances in Analytical Modeling in the Perspective of Holistic Unification. *Symmetry* **2023**, *15*, 1611.
https://doi.org/10.3390/sym15081611

**AMA Style**

Di Sia P.
Symmetry and the Nanoscale: Advances in Analytical Modeling in the Perspective of Holistic Unification. *Symmetry*. 2023; 15(8):1611.
https://doi.org/10.3390/sym15081611

**Chicago/Turabian Style**

Di Sia, Paolo.
2023. "Symmetry and the Nanoscale: Advances in Analytical Modeling in the Perspective of Holistic Unification" *Symmetry* 15, no. 8: 1611.
https://doi.org/10.3390/sym15081611