**Abstract**

In the introduction and the second part of the chapter, we discuss the Compton effect in general, and the modern viewpoint on the 2-dimensional carbon crystals called graphene, where graphene unique properties arise from he collective behavior of electrons governed by the Dirac equation. The Dirac equation in graphene physics is used for so called pseudospin of pseudoelectron formed by the hexagonal lattice composed of the systems of two equilateral triangles with the corresponding particular wave functions. The total wave function of an electron moving in the hexagonal system is superposition of the particular wave functions. The crucial step in the graphene physics is the definition of the new spinor function where spinor function is solution of the Pauli equation in the nonrelativistic situation and Dirac equation of the generalized case. The corresponding mass of such effective electron is proved to be approximately zero.

In the third part of the chapter, we deal with the Dirac equation and its Volkov solution and in the fourth part of the chapter, we discuss the Volkov solution in a dielectric medium.

The fifth part of the chapter, deals with the Compton effect derived from Volkov solution of the Dirac equation while the sixth part of the chapter, deals with the calculation of the Compton effect with ultrashort laser pulse, where the pulse is of the Dirac delta-function form.

The seven part of the chapter, deals with Compton effect initiated by two orthogonal plane waves. We solve the Dirac equation for two different four-potentials of the plane electromagnetic waves and we specify the solutions of the Dirac equation for two orthogonal plane waves. The modified Compton formula for the scattering of two photons on an electron is determined.

The conclusion eighth part involves possible perspectives of the Compton effect with regard to the scientific and technological meaning of the results derived in our contribution.

**Keywords**: Compton effect, Dirac equation, Volkov solution, dielectric, laser pulse, graphene, silicene.

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