Vitalii Tsybulnyk, Single Electron Scattering on = the Nucleus

From: "Saved by Windows Internet Explorer 8" Subject: Vitalii Tsybulnyk, Single Electron Scattering on the Nucleus Date: Thu, 31 Dec 2009 12:33:10 -0800 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_0000_01CA8A15.69765730" X-MimeOLE: Produced By Microsoft MimeOLE V6.0.6000.16669 This is a multi-part message in MIME format. ------=_NextPart_000_0000_01CA8A15.69765730 Content-Type: text/html; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable Content-Location: file://C:\!Personal\Programming\tsybulnyk.com\root\Files\lasopt1.htm Vitalii Tsybulnyk, Single Electron Scattering on = the Nucleus

Laser Optics and Problems of One (Many) Atom(s) (Nuclei) in = Streaming EM=20 Fields

Enhancement and = Absorption=20 of Two Electromagnetic Waves During Single Electron Scattering on the = Nucleus=20

Vitalii=20 Tsybulnyk

(translation by N.N. = Bolotina)=20

It is well known that light going through the matter (glass, water, = air and=20 etc.) is weakening. Most of us don't know and don't even think why it is = so, but=20 this is an experimental fact, which is difficult to have any doubt = about.=20 However, it turned out that under certain circumstances there is a = possibility=20 of amplification of the electromagnetic waves = when they=20 are going through the matter due to the energy of this matter! To be = more=20 specific, laser radiation, going through the plasma, where the processes = of the=20 dispersion of the electrons on the nuclei are going, may intensify!=20

The theory of strong electromagnetic radiation is dealing with this = problem.=20 This theory has been widely developed in some recent years due to the = creation=20 of the lasers and their quick progress in power increase. In last few = years it=20 became possible to check a lots of theories and formulae by = experimenting as a=20 result of creation of compact high-power lasers.=20

It should be mentioned that quantum-electrodynamic processes in the = field of=20 the plane monochromatic electromagnetic wave, which presents the laser=20 radiation, have been studied by such famous scientists as Volkov, = Delone,=20 Krainov, Fedorov and etc. for quite a long time. As early as in 70-80s = Borisov,=20 Zhykovskiy, Karapetyan, Fedorov, Krainov, Roshchupkin and oher = scientists=20 studied the processes of spontaneous and enforced (stimulated) collision = radiation during the dispersion of the electron on the nucleous in the = field of=20 the plane electromagnetic wave in different particular cases of energies = of the=20 electron and intensities of the wave. However, lately scientists are = more=20 interested in studying elementary quantum processes in superposition of = several=20 laser fields, for example forced collision radiation and absorption = during=20 dispersion of the electron on the nucleous with the two plane = electromagnetic=20 waves. While studying this process Roshchupkin S.P. found out a new = interference=20 effect connected to the correlated radiation and absorption of the = electron for=20 the equal number of photons of both waves. More solid research of this = effect=20 shows that light waves when going through the matter may be intensified! =

In a strong field of radiation effects connected with absorption or = releasing=20 into a wave of several quantum at once that play a vital role. Using the = so-called Volkov's solution for wave functions, describing the state of = real=20 particles, it's possible to calculate the probabilities of different = processes=20 in the field of the wave. The mentioned method of calculation of = interaction=20 with the intensive field of the wave is used in many researches, as it = allows to=20 get general solution's formulae correctly, when there are arbitrary = intensions=20 of the wave. This mehod of examination of processes in the intensive=20 electromagnetic field is a half-classic - the falling wave is considered = as a=20 classic field, and all the rest of the particles (electron, absorbed and = radiated quantum) are considered as via the quantum-mechanical notions. = Such=20 description is justified as the radiation of the really acting sources = is=20 practically always corresponding to a large number of photons.

=20

While studying the process of enforced (stimulated) collision = radiation in=20 the field of the external light waves, it's essential to differentiate = the=20 spontaneous and forced radiation. In spite of the commonness of the = nature, the=20 spontaneous radiation may occur on different frequencies and in = different=20 directions, whereas the forced radiation (absorption) occurs only at the = frequency of the external electromagnetic wave. The differential = cross-section=20 of the process for the forced radiation of the photons to the external = field=20 while occurs we will take as the 3D""=20 (emission) and the differential = cross-section=20 for which the force absorption of the photons out of the external wave = occurs -=20 as the 3D"" absorption. It makes sense to take = such a=20 concept as the total differential cross-section 3D"" (total) if we are interested only = in increase=20 or decrease of the number of photons:

=20

The total differential cross-section is connected to the change of = the number=20 of photons of the external wave, i.e. with its amplification or = attenuation.=20

------------=20

We will write down the Dirac's equation in the field of the plane = wave for=20 finding differential cross-sections of radiation and absorption: 3DMATH=20 =20 where 3D"" , 3D""=20 .=20

This equation was first solved by D. Volkov in 1937 and was widely = adopted at=20 the beginning of 60s due to the creation of the laser technology, as = Volkov's=20 solution models the field of the laser. At present time there is a whole = part in=20 physics - the laser physics, which studying the interaction of the laser = radiation with a matter using both experimental and theoretical options. =

Let's write down this solution briefly. We can multiply the Dirac's = equation=20 by 3D"" 3DMATH=20 : =20

And should consider that 3DMATH=20 =20 then after the transformations =20

Thus, the Dirac's equation transforms to the differential equation of = the=20 second order. We should mention that this equation has an excess = solution. The=20 solution we search in the form of 3DMATH=20 =20 We can substitute this expression into the differential equation and = after=20 transformations we get: =20

In this solution the 4-vector 3D""=20 3D$p$=20 is polysemantic and we can assign = an=20 additional condition for it, for example 3DMATH=20 =20

From this it follows that while turning off the external field = 4-vector has a=20 physical sense of the 4-impulse of a free electron =20

where 3DMATH=20 =20 We take the constant as =20 All the degrees of 3D"" those higher than one are equal to = zero as=20 soos as =20 According to that we can replace 3DMATH =20 The final Volkov's solution is: =20 Note, that while turning off the external field the Volkov's function = transforms=20 into the solution of Dirac's equation for a free electron. From this it = follows=20 that 3D"" 3D$u$=20 - is the byspin of the free = electron and we=20 can put the condition of normalization on it 3DMATH=20 =20 =20

which allows to avoid the excess solution.

In this way we've got an expression for Volkov's function which is = the=20 solution of Dirac's equation in the field of the plane electromagnetic = field.=20 This expression will allow us to find the differential radiations of the = processes of forced (stimulated) radiation and absorption of the = photons.

---------------------

At first we examine the passing of the process in the field of one=20 elliptically polarized wave for which 4-potential of the electromagnetic = wave=20 is:

3DMATH=20 =20

where 3D"" 3DMATH=20 , 3D""=20 -- the 4-vectors of wave = polarization, 3D"" -- the parameter of wave = ellipticity, and=20 3D"" 3D$F$=20 and -- are the voltage and the = frequency of a=20 wave; the argument 3D"" has an expresion 3D""

Expanding the solution of Dirac's equation into the Fourier series we = get:=20

3DMATH=20 =20

where partial wave functions 3D""=20 , responding to various processes = of forced=20 radiation and absorption of the electron wave 3D"" 3D$n$=20 photons look like:

3DMATH=20 =20

Here 3D"" -- the phase, which is independant = of=20 summation index, 3D"" -- the 4-vector, 3D"" 3DMATH=20 -- the 4-impulse of the external = field photon,=20 3D"" $3D$u_{p}$=20$ -- the Dirac's byspin, where the = expression=20 with roofs mean scalar products of Dirac's matrixes 3D"" ( 3D""=20 ) with corresponding 4-vectors = ( 3D"" ), 4-vector 3D"" and the 4-quasi-impulse 3D"" are accordingly equal

3DMATH=20 =20

where the relativist-invariant parameter 3D"" 3DMATH=20 is numerically equal to the ratio = of the=20 field's work on the wave length to the electron's ground energy and=20 characterizes the wave intensity.

The function 3D"" $3D$L_{n}$=20$ depends on three = relativist-invariant=20 parameters 3D"" $3D"$\chi$ , 3D""=20 , 3D""=20 $3D"$\beta$ and has the following appearance: =

3DMATH=20 =20

where 3DMATH=20 =20

Here 3D"" $3D"$\phi$ -- is the angle between 3D"" and parallel to the plane surface = 3D"" 3D$(xy)$=20 component 3D"" 3DMATH=20 of the vector 3D"" , and the 4-vector 3D"" and the relativist-invariant = function 3D"" 3D$f$=20 have the following appearance: =

We should mention, that these functions $3D$L_{n}$=20$ may be developed using Bessel's = functions of=20 integer-valued order 3D"" $3D$J_{n}$=20$ 3DMATH=20 =20

Out of this integral presentation $3D$L_{n}$=20$ we see that this function is the = Fourier 3D"" 3D$n$=20 -coefficient for

3DMATH=20 =20

It's essential to stress out that these functions 3D"" $3D$L_{n}$=20$ determine the probability of = multiphoton=20 processes in the field of one plane electromagnetic wave.

---------------

And now we complicate the task and choose the 4-potential of the = external=20 electromagnetic field as a sum of two plane waves propagating along the = axis=20 3D"" 3D$z$=20 :

3DMATH=20 =20

We expand Volkov's solution in this field in a double Fourier series: =

3DMATH=20 =20

where partial wave functions 3D""=20 3DMATH=20 , corresponding to various = processes of=20 enforced radiation and absorption by the electron the 3D"" 3D$n$=20 photons of the first wave and = 3D"" photons of the second wave have = the apperance=20 of:

3DMATH=20 =20 where =20

3DMATH=20 =20 =20 =20 =20

Here 3D"" -- the phase independant of = summation index,=20 3D"" -- the single vector along the = direction of=20 both waves movements.

The function 3D"" depends on the ten = relativist-invariant=20 parameters:

3DMATH=20 =20

where 3D"" 3DMATH=20

Here the parameters of two waves 3D""=20 3D$j=3D1,2$=20 were taken in the following way: =

3DMATH=20 =20

Here 3D"" -- - the angle between 3D"" and , 3D""=20 -- the angle between 3D"" and the parallel to the plane = 3D"" 3D$(xy)$=20 of vector component 3D"" 3DMATH=20 of the vector 3D"" . The 4-vector 3D"" and relativist-invariant = functions 3D"" $3D$f_{j}$=20$ и 3D""=20 have the appearance =20

Out of this integral presentation we see that the functions 3D"" are coefficients of Fourier = function

3DMATH=20 =20

It's essential to mention that these functions 3D"" determine the probability of = multiphoton=20 processes in the field of two plane electromagnetic waves.

-----------------

Now with the help of the obtained dependencies we can find the = differential=20 cross-section of the enforced collision radiation while the dispersion = of the=20 electron on the nucleous in the field of these two electromagnetic waves = in the=20 following way:

3DMATH=20 =20

where the differential Mott section 3DMATH=20 , and determined that 3D"" , 3D""=20 .

And as follow we have the opportunity to find numerically the total=20 differential cross-section normalized by the Mott cross-section for = various=20 intensities 3D"" $3D"$\eta$ , frequencies 3D"" and polarizations 3D"" of the waves.

We come back to finding the total differential cross-sections of = radiation=20 and absorption:

3DMATH=20 =20

We've studied the average cross-section over the angles of incoming = electron:=20

3DMATH=20 =20

The so-called interference and non-interference areas were studied=20 separately. Interference area is considered as an area where the = dispersion=20 occurs in the plane formed by the impulse of the initial electron and = the wave=20 vector. At the same time azimuth direction angles of the electron = movement in=20 the initial and final states are equal and polar - are in the ratio: =

3DMATH=20 =20

where 3D"" $3D$v_{i}$=20$ - the initial speed of the = electron.

Some results of calculations we can show in the form of the following = basic=20 diagrams:

1) the dependence of the average total differential cross-section = 3D"" 3DMATH=20 on the initial speed of the = electron in the=20 non-interference area:

=20

and 2) the dependence of the average total differential cross-section = on the=20 initial speed of the electron in the interference area:

=20

We see that both in the interference and non-interference areas = absorption in=20 general predominates over radiation as the radiation weakens, however = there=20 exist the certain areas of the initial speeds of the electron where = radiation=20 predominates over absorption and as follows the light intensifies. = Comparison of=20 absolute quantities of the total cross-section in the interference and=20 non-interference areas shows that amplification of the electromagnetic = waves=20 usually occurs in the interference area and probability of that event = almost 3=20 times higher than the corresponding probability in the non-interference = area.=20

Thus, our aim is achieved - we have shown that for certain conditions = the=20 laser rays going through the matter can amplify! This is a completely = new result=20 opening a new direction of research in physics of interaction of = intensive=20 electromagnetic radiation with matter. And undoubtedly, this result will = open=20 new horizons of application the examined effects in technology and = engineering.=20

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