**1. Introduction**

The Compton effect is the light-particle interaction where the wavelength of scattered photon is changed. The difference between the Compton effect and the Einstein photoeffect consists in the fact that during the photoeffect, the energy of photon is transmitted to electron totally.

Compton used in his original experiment [1] the energy of the X-ray photon (∼20 keV) which was very much larger than the binding energy of the atomic electron, so the electrons could be treated as being free. Compton scattering usually refers to the interaction involving only the electrons of an atom. However, the nuclear Compton effect was confirmed too. The effect is important because it demonstrates that light cannot be explained purely as a wave phenomenon. The Compton experiment proved that light is composed of particle-like objects with energy *E* = *h*¯ *ω*. The interaction between electrons and high-energy photons is such that the electron takes part of the photon initial energy, and a photon containing the remaining energy is emitted in a different direction from the original. If the scattered photon still has enough energy left, the process may be repeated.

Compton, in his paper [1], derived a simple formula relating the shift of wavelength to the scattering angle of the X-rays by postulating that each scattered X-ray photon interacted with only one electron. His paper involves the information on experiments for verification of the equation:

$$
\lambda' - \lambda = \frac{h}{mc}(1 - \cos\theta),
\tag{1}
$$

where *λ* is wavelength of the scattered X-ray and *θ* is the angle between the incident and scattered X-ray. The scattering was considered in the laboratory frame where electron was at rest. Let us remark immediately that eq. (1) has also a limit for *m* → 0, if and only if *θ* → 0. It corresponds to the situation of graphene sheet, where the mass of the so-called pseudoelectron can be considered zero. This limit can be verified immediately by the visible light and not by the X-rays. In case of using the X-rays in graphene, we get still the original Compton process with the real electrons (ionization process in graphene) and not the process with the so-called pseudoelectrons. The considered process was the so-called one-photon process with the symbolic equation

$$
\gamma + e \to \gamma + e.\tag{2}
$$

The differential cross section corresponding to eq. (2) was derived by Klein and Nishina in the form

$$\frac{d\sigma}{d\Omega} = \frac{1}{2} \frac{\varepsilon^4}{m^2 c^4} \left(\frac{\omega'}{\omega}\right)^2 \left(\frac{\omega'}{\omega} + \frac{\omega}{\omega'} - \sin^2\theta\right). \tag{3}$$

It is the ratio of the number of scattered photons into the unit solid angle Ω over the number of incident photons. At the present time with the high-power lasers, there is a possibility to realize so-called multiphoton scattering according to equation *Nγ* + *e* → *Mγ* + *e*, where N and M are numbers of photons participating in the scattering. *N* photons are absorbed at a single point and, after some time, *M* photons are emitted at the distant point. Let us remark that in case of the Raman effect, the equation describing the Raman process is *γ* + *A* → *γ* + *A*<sup>∗</sup> where *A* denotes atom, or molecule, and *A*<sup>∗</sup> denotes excited atom, or excited molecule. This Raman process involves also the interaction of the fullerene *C*<sup>60</sup> with photons.

Equation (3) has also limit for *m* → 0, if and only if the angle *θ* is solution of eq. (*ω* /*ω* + *<sup>ω</sup>*/*<sup>ω</sup>* <sup>−</sup> sin2 *<sup>θ</sup>*) = 0. Such limit corresponds to the Compton effect in graphene with zero mass of pseudoelectron.

We calculate the Compton process as a result of the Volkov solution [2] of the 3D and 2D Dirac equation. The Volkov solution involves not only the one-photon scattering but also the multiphoton scattering of photons on electron. In the time of the Compton experiment in 1922, the Volkov solution was not known, because the Dirac equation was published in 1928 and the Volkov solution in 1935 [2].

In the next part of the chapter, we discuss the modern viewpoint on the two-dimensional carbon crystals – graphene. Then we discuss the Dirac equation with plane wave and its Volkov solution including dielectric medium. Then we deal with the calculation of the Compton effect with ultrashort laser pulse in graphene and graphene-like sheet as silicene. We consider the case that a charged particle moves in the parallel direction to the silicene sheet. We include also the classical situation of a charged particle accelerated by the *δ*-function form of impulsive force. Then, we discuss the corresponding quantum theory based on the Volkov solution of the Dirac equation. The modified Compton formula for frequency of photons generated by the laser pulse is derived. The discussion is extended to the Dirac equation for two different waves, and the Volkov solution is then determined for the orthogonal two plane waves. The last part of the chapter is the conclusion, where we consider the scientific and technological perspectives of the results derived in our contribution.
