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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. Angular distribution in the second variant is investigated for the nonrelativistic and relativistic cases.

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 [

There is the problem of achieving of the maximum photoelectric flow during irradiation of the metal by flow of electromagnetic waves while designing of photocells. The depth of radiation penetration into metal during irradiation of its surface is defined by the Bouguer low [

,

where I_{0}—is the intensity of the incident wave, I—is the intensity on z-coordinate, directioned depthward the metal, l—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 [

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.

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 [

Therefore, photoelectron energy is equal to:

,(1.2)

where m_{1}—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 lack of dependence (1.2) is that at its conclusion the law of conservation of impulse, wasn’t used and therefore there is no electron movement to the direction. Usage of the impulse conservation equation in [6,7] can’t be considered satisfactory since in the analysis made by the authors it has an auxiliary character. At the heart of the analysis [6,7] 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 (1.2) is shown on

Let’s illustrate the correction to the formula (1.2) connected with presence of photon impulse, following [

.

Considering that δ is too small we find

.

The law of sines for a triangle on

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 is small, we will transform (1.2) into

Angular distribution of electron energy for, made according to the (1.2), taking into account the correction is shown on

Thus scattering indicatrix of photoelectrons has received some slope forward, but to the direction of quantum impulse, i.e., at electrons don’t take off as before.

The formula (1.2) 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,

,(1.3)

where V_{t}—is the speed of electron movement round the nucleus, V_{1}—is the total speed of electron considering the influence on it of an electromagnetic wave.

By the law of cosines we have:

where V_{n}—is the component of the general speed of electron movement after its detachment from a nucleus which arises under the influence of electric field in the electromagnetic wave.

Solving (1.4) rather V_{1}, we find:

.(1.5)

The condition of detachment electron from atom at any position of electron.

In case of equality of speeds we have:

.(1.6)

Distribution of speeds (1.6) corresponds to (1.2) and

In [_{2}. And after all the nuclear mass defines a share of the photon impulse which can incur a nuclear.

Let’s consider the phenomenon of the inner photoemissive effect from positions of corpuscular representation of quantum of light,

,(1.7)

where Е_{1}—is the kinetic energy of taken off electron, Е_{2}—is the kinetic energy of nucleus as well as the law of conservation of impulse:

,(1.8)

where—is the impulse of taken off electron,—is the impulse transferred to a nucleus.

The formula (1.7) 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 impulse transferred to a nucleus can be found using the formula, following (1.8):

.(1.9)

The system of Equations (1.7) and (1.9) to obtain a combined solution and the Equation (1.9) is convenient to express through energy. Taking into account, where c—is the speed of light in vacuum, and, we find:

where m_{1}—is the electronic mass, m_{2}—is the nuclear mass.

Substituting in (1.10) kinetic energy of nuclear Е_{2} by (1.7), we have:

Let us introduce the following notation

, , ,.

Then the Equation (1.11) will be transformed into:

. (1.12)

Solving quadratic Equation (1.12) provided (electronic mass is much less that nuclear mass), we find:

Substituting in (1.13) accepted notation we have:

(1.14)

Considering that

where V_{1}—speed of photoelectrons provided

we find:

.(1.15)

Provided that nuclear mass is aiming to infinity the formula (1.15) 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 impulse 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 (1.15) it is necessary to use effective nuclear mass, considering attractive powers between atoms in substance.

Transforming the formula (1.15), we get:

.(1.16)

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 (1.16) takes the form:

.(1.17)

The analysis of the formula (1.17) 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 in

The

equivalent as far as the order of value is concerned

in the field of red photoelectric threshold (l_{r} = 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 g-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

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,

Dealing with relativistic case of the inner photoemissive effect, the law of conservation of energy needs to be written down as:

,(2.1)

where—is the kinetic energy of photoelectron.

The law of conservation of impulse remains in the form (1.9). Using relativistic relation between the energy and the impulse for electron:

where Е_{1}—is the total energy of electron, m_{1}—is the electron rest mass, we will express the impulse of electron from (2.2) and we will substitute in (1.9). For convenience of the further transformations we will write down (2.2) into:

.(2.3)

Formulating (2.3) the relation has been used:

.(2.4)

The Equation (1.9) will be transformed into:

.(2.5)

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 impulse of the nucleus with its kinetic energy Е_{2} is used in the nonrelativistic form.

Substituting value Е_{2} in (2.5) from the Equation (2.1), we get:

Let us nominate:

As a result (2.6) will be transformed into:

. (2.8)

The notation

corresponds to item 1 section.

The value

.

It is thus accounted for that.

Solving the Equation (2.8), we get:

.(2.9)

Substituting notations, we find:

.(2.10)

In contrast to the nonrelativistic case, the formula (1.17), formula (2.10) possesses in its right part value

which depends on the total energy of electron Е_{1} the structure of which includes also kinetic energy. But dependence of value a 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:

.(2.11)

Substituting the Equation (2.11) in the Equation (2.10) and considering that, we get:

where.

Considering that

, atwe find:

.(2.13)

The formula (2.13) allows to consider relativistic effects at the photoeffect, in case of rather big speeds of photoelectrons. Thus, in contrast to (1.17), relativistic coefficient μ is introduced under the root.

The calculation of dependence m(b) shows on

be used while the photoelectron speeds read approximately half the value of the light speed in the vacuum.

The analysis has shown that the start of electrons from atom at a photoeffect is almost spherical symmetric. It corresponds Einstein’s to formula. In Einstein’s formula there is no corner of a start of photoelectrons.The assumption, that in a direction of movement of a photon at a photoeffect of an electron does not take off unfairly. At designing photocells it is necessary to take into account presence of a stream electrons in a direction of electromagnetic radiation and in an opposite direction.