1.1. Distribution of Electromagnetic Field Momentum in dielectrics in stipulation of self-induced transparency
Interaction of quantums of electromagnetic radiation with substance can be investigated both from a wave position, and from a quantum position. From a wave position under action of an electromagnetic wave there are compelled fluctuations of an electronic orbit and nucleus of atoms. The energy of electromagnetic radiation going on oscillation of nucleus passes in heat. Energy of fluctuations of an electronic orbit causes repeated electromagnetic radiation with energy, smaller, than initial radiation.
From a quantum position character of interaction is more various. Interaction without absorption of quantums is possible: resonant absorption, coherent dispersion. The part of quantums is completely absorbed. Quantums can be absorbed without occurrence secondary electrons. Thus all energy of quantums is transferred fonons - to mechanical waves in a crystal lattice, and the impulse is transferred all crystal lattice of substance. At absorption of quantums can arise secondary electrons, for example, at an internal photoeffect. Absorption of quantums with radiation of secondary quantums of smaller energy and frequency is possible, for example, at effect of Compton or at combinational dispersion.
All these processes define as formation of impulses of electromagnetic radiation in substance, and absorption of radiation by substance.
1.2. Coordination of the electromagnetic impulse with the substance
Firstly, consider the one-dimensional task the electric part of electromagnetic field momentum with the dielectric substance, which posses a certain numerical concentration
We accept that there takes place the interaction of quantum of electromagnetic radiation with nuclear electrons, thus quantum are absorbed by the electrons. By gaining the energy of quantum the electrons shift to the advanced power levels. Further, by means of resonate shift of electrons back, appears the quantum radiation forward. The considered medium lacks non-radiating shift of electrons, i.d. the power of quantum is not transfered to the atom.
Thus, the absorption of electromagnetic radiation in the case of its power dissipation in the substance, owing to SIT, is disregarded. There appears the atomic sypraradiation of quantum. Thus, the forefront of momentum passes the power on to the atomic electrons of the medium, forming its back front.
The probabilities of quantum's absorption and radiation by the electrons in the unity of time, with a large quantity of quantum in the impulse, according to Einstein, can be referred to as the approximately identical . For the separate interaction of the with the electron this very probability is the same and is proportional to the cube of the fine-structure constant ~ (1/137)3
. Consider a random quantity – the number of interactions of quantum with atomic electrons in the momentum. In accordance with the Poisson law of distribution, the probability of that will not be swallowed up any quantum atomic’s electrons (will not take place any interaction), at rather low probability of separate interaction, is equal an exponent from the mathematical expectation of a random variable − an average quantity of interactions
The index of interaction is
Taking into account that the wave intensity is we shall have
− the amplitudes of electric and magnetic fields' strength of the impulse on longitudinal coordinate
In the formula (1.3) and further the upper variables in parentheses are referred to electric field, and lower – to the magnetic field of impulse.
By the ratio (1.2) it is possible to find
The formula (1.4) demands some further consideration. If
We consider the dependence of the average of filling on the time
Secondly, the period of variable
while the dependence
1.3. Non-linear Schrödinger equation
One-dimensional wave equation for electric and magnetic aspects of electromagnetic field for the considered problem is 
We introduce the transformation of electric field intensity be formula
We estimate the relative variable of first and second items in the parenthesis of the left side (1.8). for this purpose we would introduce the scales of variables time
where the asterisk designates dimensionless parameters. For the time scale the duration (period) of impulse
that dimensionless second derivative
and the dimensionless function
Similarly, it can be presented that the second item in the round brackets (1.8) is far more that the first one.
Hence, by disregarding the small item in (1.8), we observe
By accepting vector of polarization
We consider the strength of electric and magnetic fields of impulse as
By transforming (1.11) we have
The similar ratio can be also referred to the function |
For the variable
, where χ – relative magnetic permittivity of substance, we get the ratio, similar to (1.14), except that the right part lacks
In the equation (1.15) the variable χ is meaningful to dielectric permittivity for electric and magnetic permittivity for the magnetic aspects of electromagnetic field.
The non-linear Schrödinger equation with complicated type of linearity is received. We introduce the signs: , , where – relative permittivities of the substance. Hence, the equation (1.15) will be
If to permit that as there should not be any imaginary items in (1.18), this equation is transformed to
We consider the solution of the equation (1.19) by
The formulas (1.21) associate the frequency and the wave number of oscillations of function
The most simple ratios between the parameters are gained, when . In this case , . From the equations in (1.21), and concerning there is
By concerning that
, we have
. This inequality is true, as for the rarefied gas
It should be stressed, that though, the ratios for the electric aspect of impulse in  and (1.23) are similar to each other and feature the same phases of oscillations, that is possible on some distance from the over-radiating atom, the non-linear Schrödinger equations are differ in type of non-linearity. The reason of this lies in the fact that in  the impulse was considered with regard to low intensity, the one that does not lead to the energetic saturation of medium, in which it is disseminated.
For the estimation, like in  we have , , ω =6,28.1012 s—1, δ = 6,28.1013 s—1.
Taking in to account the reciprocal orthogonality of planes of vectors' envelopes of electric and magnetic fields impulse, we could gain the type of electromagnetic soliton, fig. 3.
Figure 4 shows the envelopes of electric field impulse in the SIT, based on formula (1.23), curve 1, and by formula (1.24), being the consequence of Maxwell-Bloch theory, curve 2. the impulse envelope of electric field strength in this theory is expressed as the first derivative of the Sin-Gordon equation solving and is
Evidently, the first derivative of Sin-Gordon equation solving is similar to the soliton envelope in the non-linear Schrödinger equation with cube non-linearity solving (27). Curves 1and 2 in fig. 4 are designed for the same parameters as the function in fig. 2. We can infer from fig. 4 that impulse, referred to formula (1.23), curve 1, is broader in its central part, but asymptotically shorter than impulse, inferred by the Maxwell-Bloch theory, curve 2. Evidently, its is bound with the energetic permittivity of medium in the central part of impulse.
2. Angular distribution of photoelectrons during irradiation of metal surface by electromagnetic waves
There is the problem of achieving of the maximum photoelectric flow during irradiation of the metal by flow of electromagnetic waves while designing of photoelectrons. The depth of radiation penetration into metal during irradiation of its surface is defined by the Bouguer low :
where I0 − is the intensity of the incident wave, I − is the intensity on z-coordinate,
directioned depthward the metal, λ − is the wavelength of radiation, − is the product of refractive index by extinction coefficient.
Let's estimate the thickness of the metal at which intensity of light decreases in е = 2,718 times:
Average wavelength of a visible light for gold λ=550 nm, , therefore z = 15,5 nm. Considering  that lattice constant for gold а = 0,408 nm, it is possible to deduce that electromagnetic radiation penetrates into the metal on 40 atomic layers.
Therefore radiation interaction occurs basically of the top layers of atoms and angular distribution of electron escape from separate atoms, i.e. during the inner photoemissive effect, it will appreciably have an impact on distribution of electron escape from the metal surface.
As a result it is interesting to consider angular distribution of photoelectrons during the inner photoemissive effect.
2.1. Nonrelativistic case
Although Einstein has explained the photoeffect nature in the early 20th century, various aspects of this phenomenon draw attention, till nowadays for example, the role of tunnel effect is investigated during the photoeffect .
In the description of angular distribution of the photoelectrons which are beaten out by photons from atoms, there are also considerable disagreements. For example it is possible to deduce that the departure of photoelectrons forward of movement of the photon and back in approach of the main order during the unitary photoeffect is absent, using the computational method of Feynman diagrams . I is marked that photoelectrons don't take off in the direction of distribution of quantum . This conclusion is made on the basis of positions which in the simplified variant are represented by the following.
The momentum of the taken off electron is defined basically by action produced by the electric vector of quantum of light on electron. If electron takes off in the direction of an electric vector of quantum it gets the momentum. On a plane set at an angle to a plane of polarization of quantum of light, (fig. 5) electron momentum value will be .
Besides, if the electron momentum is set at an angle θ to the direction of quantum of light its value will be:
Therefore, photoelectron energy is equal to:
where m1 – is the electronic mass.If then photoelectron energy . Photoelectrons take off readies its maximum in the direction of a light vector or a polarization vector, i.e. an electric field vector of quantum of light. The same dependence is offered in the work . The formula (2.2) has the simplified nature in comparison with [7, 14], but convey correctly the basic dependence of distribution energy of a photoelectrons escape from the corners φ and θ.
The lack of dependence (2.2) is that at its conclusion the law of conservation of momentum, wasn't used and therefore there is no electron movement to the direction . Usage of the momentum conservation equation in [7, 14] can't be considered satisfactory since in the analysis made by the authors it has an auxiliary character. At the heart of the analysis [7, 14] is the passage of electron from a discrete energy spectrum to a condition of a continuous spectrum under the influence of harmonious indignation, i.e. the matrix element of the perturbation operator is harmonious function of time. In other words, the emphasis is on the wave nature of the quantum cooperating with electron. Angular distribution of electron energy in the relative units, made according the formula (2.2) is shown on fig. 6, a curve 1. Let's illustrate the correction to the formula (2.2) connected with presence of photon momentum, following .
Fig.7 demonstrates change of photoelectron momentum in the presence of a photon momentum. The conclusion made on the basis of the is . Let's find . Considering that δ is too small we find . The law of sines for a triangle on fig. 7 is used.
Further consideration , where – is the relation of photoelectron speed to a speed of light in vacuum, W − is the work function of electrons from atom, we have . Taking for granted that β′ is small we will transform (2.2) into . Angular distribution of electron energy for , made according to the (2.2) taking into account the correction is shown on fig. 7, a curve 2.
Thus scattering indicatrix of photoelectrons has received some slope forward, but to the direction of quantum momentum, i.e. at electrons don't take off as before.
The formula (2.2) is accounted as a basis of the wave nature of light. For the proof of this position we will consider interaction of an electromagnetic wave with orbital electron. The description of orbital movement electron is done on the basis of Bohr semiclassical theory since interacting process of electron with an electromagnetic wave is investigated from the positions of classical physics, fig. 8.
By the sine law from a triangle of speeds we find:
where Vt – is the speed of electron movement round the nucleus, V1 − is the total speed of electron considering the influence on it of an electromagnetic wave.
By the law of cosines we have:
where Vn – is the component of the general speed of electron movement after its detachment from a nucleus which arises under the influence of dielectric field intensity in the electromagnetic wave.
Solving (2.4) rather V1, we find:
The condition of detachment electron from atom at any position of electron .
In case of equality of speeds we have:
Distribution of speeds (2.6) corresponds to (2.2) and fig. 6, a curve 1. Thus, the parity (2.6) arises if to consider only the wave nature of the electromagnetic wave cooperating with orbital electron.
In  distribution of an angle of the electron escape is investigated only for a relativistic case. It is thus received that electrons are emanated mainly to a direction of photon distribution. However the done conclusion is also actually based on the formula (2.1). Therefore the drawback of the conclusion  is in absence in definitive formulas of angular distribution of electrons of nuclear mass m2. And after all the nuclear mass defines a share of the photon momentum which can incur a nuclear.
Let's consider the phenomenon of the inner photoemissive effect from positions of corpuscular representation of quantum of light, fig. 5. The quantum of light by momentum and energy Е beats out electron from atom, making A a getting out. Thus both laws of conservation of energy should be observed:
Where Е1 – is the kinetic energy of taken off electron, Е2 – is the kinetic energy of nucleus as well as the law of conservation of momentum:
Where − is the momentum of taken off electron, − is the momentum transferred to a nucleus.
The formula (2.7) differs from Einstein's standard formula . The point is that Einstein's formula means the absence of angular distribution of photoelectrons speed. Really, if energy of photon Е is set and work function A for the given chemical element is determined certain speed of the electron escape from atom is thereby set. It means that speeds of electrons, taking off to every possible directions are identical, and the problem of finding out their angular distribution is becoming incorrect.
The value of the momentum transferred to a nucleus can be found using the formula, following (2.8):
The system of equations (2.7) and (2.9) to obtain a combined solution and the equation (2.9) are convenient to express through energy. Taking into account , where c − is the speed of light in vacuum, and , we find:
where m1 – is the electronic mass, m2 – is the nuclear mass.
Let us introduce the following notation , , , . Then the equation (2.11) will be transformed into:
Solving quadratic equation (2.12) provided (electronic mass is much less that nuclear mass), we find:
Substituting in (2.13) accepted notation we have:
Considering that , where V1 – speed of photoelectrons provided , we find:
Provided that nuclear mass is aiming to infinity the formula (2.15) is transformed into Einstein's standard law for the photoeffect. Besides, this, as if it has been specified earlier, angular distribution of speed of photoelectrons disappears.
The condition is fair in outer photoemissive effect when the photon momentum is transferred to the whole metal through single atoms. Therefore for an outer photoemissive effect, i.e. for interaction of the solid and the photon, Einstein's formula is applicable absolutely.
For the inner photoemissive effect in the formula (2.15) it is necessary to use effective nuclear mass , considering attractive powers between atoms in substance.
Transforming the formula (2.15), we get:
Let us nominate . Distribution of photoelectrons will arise at . In the right part of the received inequality there is a very small value, therefore distribution of photoelectrons will arise practically at .
Let us nominate , where characterizes the value of exceedance of photon energy over work function in relative units. Thus the formula (2.16) takes the form:
The analysis of the formula (2.17) shows that the root must to taking a plus since otherwise electron scattering basically goes aside, contrary to the direction of a falling photon. Angular distribution of the electron escape during the inner photoemissive effect in the relative units is shown on fig. 9, made according to the formula (2.17) with several values η for copper.
The figure makes it evident that speeds of photoelectrons become almost identical in all directions already at . Then Einstein's formula becomes fair and for the inner photoemissive effect. Considering that, for example, for copper the relation equivalent as far as the order of value is concerned in the field of red photoelectric threshold (λr = 250 nm), it is possible to draw the conclusion that the evident difference of distribution of photoelectrons speeds from spherical, i.e. actually formula is violated , can be observed only in very short wave part of spectrum γ-radiations.
The observed data of angular distribution of the photoelectrons which have been beaten out from a monolayer of atoms of copper by covering the nickel surface are shown in fig. 9 by black small squares . The wavelength of quanta allowed observing the photoeffect with 2р-atom shell of copper, but the photoeffect on nickel thus was absent. Experimental distribution of photoelectrons contradicts calculated distribution in fig. 6. Moreover, in distinction in fig. 6, small maxima of indicatrix of the distributions directed to an opposite direction of flight of light quanta at an angle of approximately 45º to the direction of light flux are observed. In  these maxima are explained by focusing properties of all population of atoms of the surface. The amplitude of maxima ascends with the increase of quantity of the monolayers of copper atoms on nickel.
Thus, angular distribution of photoelectrons will be absolutely various depending on whether what properties, wave or corpuscular are reveal by the light quantum in interaction with orbital electron. Only experiment can give the answer to the question what distribution it is true, fig. 6 or fig. 9. However existence of electron flux from an illuminated surface at normal light incidence , in the direction opposite to intensity of light, shows at the prevalence of corpuscular properties of light in its interaction with atoms.
2.2. Relativistic case
Dealing with relativistic case of the inner photoemissive effect, the law of conservation of energy needs to be written down as:
Where – is the kinetic energy of photoelectron.
The law of conservation of momentum remains in the form (2.9). Using relativistic relation between the energy and the momentum for electron:
where Е1 – is the total energy of electron, m1 – is the electron rest mass, we will express the momentum of electron from (2.19)and we will substitute in (2.9). For convenience of the further transformations we will write down (2.19) into:
Formulating (2.20) the relation has been used:
The equation (2.9) will be transformed into:
Because of that the nucleus that has a big mass and a relatively low speed after interaction with the photon, expression for relation of the momentum of the nucleus with its kinetic energy Е2 is used in the nonrelativistic form.
Let us nominate:
As a result (2.23) will be transformed into:
The notation corresponds to item 1 section.
It is thus accounted for that .
Solving the equation (2.25), we get:
Substituting notations, we find:
which includes also kinetic energy . But dependence of value α on not strong as the total energy structure includes rather big rest energy of electron .
Considering that , where is the relative speed of the photoelectron, we find:
Considering that , at , we find:
The formula (2.30) allows to consider relativistic effects at the photoeffect, in case of rather big speeds of photoelectrons. Thus, in contrast to (2.17), relativistic coefficient μ is introduced under the root. The calculation of dependence μ(β) shows on fig. 10, relativistic effects while calculating distribution of photoelectrons escape, can be neglected and (2.17) can be used while the photoelectron speeds read approximately half the value of the light speed in the vacuum.
The laws of formation of the impulse of electromagnetic radiation in dielectric environment for conditions self-induced transparency are considered. The insufficiency of the description of such impulse with the help of the equations Maxwell - Bloch are shown. The impulse of electromagnetic radiation in conditions of a self-induced transparency submits to nonlinear equation of Schrödinger with logarithmic nonlinearity. The way of connection of an average number filling and energy of the impulse taking into account energy saturation of environment are offered. The calculation of a electrical component of the impulse is submitted.
Angular distribution of photoelectrons is investigated during the inner photoemissive effect for two variants: quantum of light basically reveals wave and basically corpuscular properties interacting with orbital electron. Distinction in angular distribution of photoelectrons for these variants is demonstrated. If electromagnetic radiation shows basically quantum properties during a photoeffect there is an emission of photoelectrons on a direction of movement of quantums. It corresponds Einstein's to formula. In Einstein's formula there is no corner of a start of photoelectrons. Angular distribution in the second variant is investigated for the nonrelativistic and relativistic cases.
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