**8. Perspectives**

with the standard conditions for *ϕ*, *χ*, *k*, *κ*, and *δ* being the phase shift.

*<sup>q</sup><sup>µ</sup>* <sup>=</sup> *<sup>p</sup><sup>µ</sup>* <sup>−</sup> *<sup>e</sup>*

*m*2 <sup>∗</sup> <sup>=</sup> *<sup>m</sup>*<sup>2</sup>

The matrix element involves the extended law of conservation. Namely:

generalized equation of the double Compton process for *s* = *t* = 1:

which is for the massless graphene limit, the following one *m*<sup>2</sup>

We can write eq. (170) in the equivalent form:

1 *<sup>ω</sup>* <sup>−</sup> <sup>1</sup>

*<sup>ω</sup>* <sup>=</sup> <sup>1</sup> *m*∗


by Lyulka [22].

and

and *κ* .

the *κ*

It was shown in [22] that

140 Graphene - New Trends and Developments

The two-wave Volkov solution is given by eq. (166), and the matrix elements with corresponding calculation ingredients are given by the standard method as it was shown

> <sup>2</sup> *<sup>a</sup>*<sup>2</sup> 1 2(*kp*)

<sup>1</sup> <sup>−</sup> *<sup>e</sup>*2*a*<sup>2</sup> 1 *<sup>m</sup>*<sup>2</sup> <sup>−</sup> *<sup>e</sup>*2*a*<sup>2</sup>

*sk* + *tκ* + *q* = *q* + *k* + *κ*

where the *s* and *t* are natural numbers. The last equation has natural interpretations. The photon object with momenta *sk* and *tκ* interacts with electron with momentum *q*. After interaction, the electron has a momentum *q* and two photons are emitted with momenta *k*

From the squared form of the last equation and after some modification, we get the

where Ξ is the angle between the 3-momentum of the *κ*-photon and the 3-momentum of

Let us remark that if the frequencies of the photons of the first wave substantially differ from the frequencies of the photons of the second wave, then eq. (172) can be experimentally verified by the same method as the original Compton formula. To our knowledge, formula

*ωω* <sup>−</sup> ΩΩ

*ωωm*<sup>∗</sup>

, respectively. In the situation of graphene with the

(<sup>1</sup> <sup>−</sup> cos <sup>Θ</sup>) + <sup>Ω</sup> <sup>−</sup> <sup>Ω</sup>

massless limit, we replace the renormalized mass in eq. (172) by *m*∗(*m* = 0) = *e*

(172) is not involved in the standard textbooks on quantum electrodynamics.

*<sup>k</sup><sup>µ</sup>* <sup>−</sup> *<sup>e</sup>*

<sup>2</sup> *<sup>a</sup>*<sup>2</sup> 2 2(*κp*)

> 2 *m*<sup>2</sup>

*sk* + *q* − *k* = *q* + *κ* − *tκ*. (171)

*κ<sup>µ</sup>* (168)

, (169)

, (170)

(1 − cos Ξ), (172)

 <sup>−</sup>*a*<sup>2</sup> <sup>1</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup> 2.

<sup>1</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup> 2).

<sup>∗</sup>(*<sup>m</sup>* <sup>=</sup> <sup>0</sup>) = <sup>−</sup>*e*2(*a*<sup>2</sup>

We have considered the Compton effect in the framework of the Volkov solution of the Dirac equation assuming that the process occurred in vacuum and in medium with the index of refraction *n*. The determination of the index of refraction follows from the Compton effect. Mass renormalization of electron is involved in the Volkov solution.

The harmonic oscillator with frequency *ω*<sup>0</sup> and the dispersion theory leads to the known formula for the index of refraction of matter [26]. The index of refraction derived in the dispersion theory based on the damped oscillator is given by the formula [27]

$$m = 1 + 2\pi N \frac{e^2}{m} \frac{\omega\_0^2 - \omega^2}{(\omega\_0^2 - \omega^2)^2 + \gamma^2 \omega^2},\tag{173}$$

where *N* is number of electrons in the unit of volume. The modern aspects can be found in the Crenshaw article [28]. So, to consider the Compton effect in dielectric medium is the perspective problem.

The interesting result of our article is the derivation that for some scattering angles given by eq. (70), there exists the so-called anomalous Compton effect, where the wavelengths of scattered photons are shorter than the wavelengths of the original photons. To our knowledge, information of this effect was not published in the physical journals.

The Compton scattering is, at the present time, the elementary laboratory problem because for the monochromatic X-rays for *λ* = 1, the shift of wavelength is several percent. This is quantity which can be easily measured. On the other hand, the Compton wavelength shift for the visible light is only 0,01 percent. It means that the measurement of the Compton effect for the visible light in the dielectric medium involves the subtle approach.

We have also discussed the problem of the Dirac equation with the two-wave potentials of the electromagnetic fields. While the Volkov solution for one potential is well known for a long time, the Compton process with two beams was not investigated experimentally by any laboratory.

It is possible to consider the situation with the sum of N waves, or

$$V = \sum\_{i=1}^{N} A\_i(\varphi\_i) \quad \varphi\_i = k\_i \ge \tag{174}$$

The problem of the laser compression of target by many beams, involving the Compton effect, is one of the actual problems of the contemporary laser physics. The goal of the experiments is to generate the physical implosion in the spherical target. The light energy is absorbed by the target and generates a high-temperature plasma with high pressure of a few hundred megabars. For the process sufficiently spherically symmetric, the central area is heated up to 5–10 keV and fusion reaction starts [29] . The solution of that problem in the general form is a difficult one, and it can be solved only by the special laser institutions such as the Lebedev Institute of Physics, the Lawrence Livermore National Laboratory, and so on. New experiments can be realized and new measurements performed by means of the laser pulses, giving new results and discoveries.
