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Subba Tata",coverURL:"https://cdn.intechopen.com/books/images_new/7999.jpg",editedByType:"Edited by",editors:[{id:"187859",title:"Prof.",name:"Kusal",surname:"Das",slug:"kusal-das",fullName:"Kusal Das"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"16405",title:"Interaction of Electromagnetic Radiation with Substance",doi:"10.5772/10586",slug:"interaction-of-electromagnetic-radiation-with-substance",body:'\n\t\t
\n\t\t\t
1. Introduction
\n\t\t\t
\n\t\t\t\t
1.1. Distribution of Electromagnetic Field Momentum in dielectrics in stipulation of self-induced transparency
\n\t\t\t\t
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.
\n\t\t\t\t
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.
\n\t\t\t\t
All these processes define as formation of impulses of electromagnetic radiation in substance, and absorption of radiation by substance.
\n\t\t\t
\n\t\t\t
\n\t\t\t\t
1.2. Coordination of the electromagnetic impulse with the substance
\n\t\t\t\t
Firstly, consider the one-dimensional task the electric part of electromagnetic field momentum with the dielectric substance, which posses a certain numerical concentration n of centrosymmetrical atoms – oscillators. For the certainty of the analysis we suggest the atom to be one-electronic. It is also agreed, that no micro current or free charge are present in the medium. The peculiarities of interaction between magnetic aspect of momentum and the atoms will be considered later.
\n\t\t\t\t
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.
\n\t\t\t\t
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.
\n\t\t\t\t
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 [6]. 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\n\t\t\t\t\t[7]. 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 λ of quantums and electrons in impulse, taken with the minus\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Therefore, as it will be explained further, it is possible that the intensity of non-absorbed power of impulse by the atomic electrons of the medium in it forefront is determined by the exponential Bouguer law [3] (in German tradition - Beer law)
where α – index of electromagnetic wave and substance interaction, l – length of interaction layer, I\n\t\t\t\t\t0 - intensity of incident wave. Thus, the intensity of atomic electron\'s power recoil into impulse on its back front could be described with the help of the Bouguer law with the negative index of absorption [8].
\n\t\t\t\t
The index of interaction is\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, where α − effective section of atom-oscillator interaction with the wave. Hence,
where V\n\t\t\t\t\teff – the effective volume of interaction. In defying (1.2) the right part of the formula is multiplied and divided by the geometric volume V, in which there is M of particles interacting with the radiation. The ratio\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. The ratio of effective volume of interaction to the geometric volume characterizes the medium possibility of electromagnetic radiation\'s interaction with the atom. Hence, by exponential function in the Bouguer law (1.1) the mathematical expectation of random variable is supposed, which subdues to the Poisson law distribution – average variable of atoms interacting with the electromagnetic radiation in the area of impulse influence\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
\n\t\t\t\t
Taking into account that the wave intensity is \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t~\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t we shall have
where\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t− the amplitudes of electric and magnetic fields\' strength of the impulse on longitudinal coordinate X = 0.
\n\t\t\t\t
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.
The formula (1.4) demands some further consideration. If E<E\n\t\t\t\t\t0\n\t\t\t\t\t, that reflects the process of wave absorption by atomic electrons N>0 and classical consideration of electromagnetic wave interaction with the atom is quite admissible. The case when E>E\n\t\t\t\t\t0 reflects the process of wave over-radiation. Thus, N<0 and variable N can not be considered as the probability of electromagnetic wave interaction with the atom. In this case we speak about the quantum-mechanical character of the process of interaction between the quantum and the bi-level power system of the atom, provided that the power transition\'s radiation is reversed. Variable N in this case possess the notion of united average of filling by atom (-1<N<1). Due to the use of the average of filling to raise the atom and bend of its magnetic moment in the magnetic field of the impulse, the existence of bi-level quantum system by magnetic quantum numbers. Thus, the variable N provides with the measure of inversion of the system of atom-radiators by the raised atoms [2] as well as the measure of inversion of the magnetic moment of the atom\'s system by magnetic quantum numbers. If N=-1 all the atoms occur in the basic condition [3].\n\t\t\t\t
\n\t\t\t\t
Figure 1.
Dependence of volumetric density of energy of electromagnetic radiation impulse w (curve 1) and average on atoms of number of filling N (curve 2) from time; 3 and 4 - points of an excess of function w(t)\n\t\t\t\t\t\t
\n\t\t\t\t
We consider the dependence of the average of filling on the time N(t). If to accept the proportion of polarization of separate bi-level atom to the intensity of electric field in the impulse, then, in accordance with the Maxwell-Bloch equations, the average by atoms of considered volume, the filling number is proportional to the volumetric density of electromagnetic wave power N~w\n\t\t\t\t\t[3]. However such a monotonous dependence between these variables can not remain on the whole extent of the impulse. Firstly, by the high volumetric density of impulse power w, typical of SIT, when the central part of impulse power is higher than any variable w, there exists energetic saturation of the medium. The average filling number thus N=1, all the atoms are raised, fig. 1 (curve 1 - the dependence w of time, thicker curve 2 – the considered dependence N of time). The violation of proportion N~w in the central part of impulse is the basic drawback of frequently used system of Maxwell-Bloch equations for the SIT description.
\n\t\t\t\t
Secondly, the period of variable N relaxation is not less than 1 ns\n\t\t\t\t\t[2] that is why the dependence N(t) can not repeat high-frequently oscillations on both fronts of the impulse. The dependence N~w could characterize the proportion of average filling number and envelope w (curve 1) in the impulse. However, in two points of the fold (3 and 4 fig. 1) on the sites of increase and decrease if the envelope w the variable \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tw\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t hence, also\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Besides, the dependence N(t) has the symmetrical character as at the SIT impulse becomes the conservative system (there is no reverse dispersion and dissipation of power) [2]. Therefore, it could be thoroughly concerned that on the whole extent of impulse, except the points of curve\'s N(t) fold, the condition remains
while the dependence N(t) has the character as shown on the fig. 1, curve 2. it could be also highlighted the high generality of formula (1.5), which is possible for any piecewise linear function N(t). Thus, the points of function break are excluded, as the derivates undergo the break.
\n\t\t\t
\n\t\t\t
\n\t\t\t\t
1.3. Non-linear Schrödinger equation
\n\t\t\t\t
One-dimensional wave equation for electric and magnetic aspects of electromagnetic field for the considered problem is [2]\n\t\t\t\t
where\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t≡\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tor\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t≡\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\tZ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t,\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t≡\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tor\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t≡\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\tZ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, X and t – accordingly the coordinate alongside of which the impulse and the time are distributed, P − polarization of substance, J – its magnetization, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t − electrical and magnetic constant, ε − relative static permittivity of substance, μ – relative magnetic permittivity,\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t– speed of light in vacuum.
\n\t\t\t\t
We introduce the transformation of electric field intensity be formula
The function Ф(X, t) is less rapidly changing one in time then E(X,t) or H(X,t),\n\t\t\t\t\tω\n\t\t\t\t\t0 – aspect of cyclic frequency of high-frequent oscillations of the field.
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 t and Ф\n\t\t\t\t
where the asterisk designates dimensionless parameters. For the time scale the duration (period) of impulse T should be logically chosen. The scale Ф\n\t\t\t\t\t0 is chosen from a condition
\n\t\t\t\t
that dimensionless second derivative \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tФ\n\t\t\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and the dimensionless function Ф* are in the same order. Hence, the the first item in round brackets (1.8) is\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tФ\n\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tФ\n\t\t\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, and the last one\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tФ\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tФ\n\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Instead of impulse T period we introduce cyclic frequency of impulse\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. By comparing these items, it is realized, that \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tФ\n\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t4\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tФ\n\t\t\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tФ\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tФ\n\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t as the cyclic frequency of impulse is far less than infrequences of field\'s oscillations, especially when\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
\n\t\t\t\t
Similarly, it can be presented that the second item in the round brackets (1.8) is far more that the first one.
\n\t\t\t\t
Hence, by disregarding the small item in (1.8), we observe
By accepting vector of polarization P or magnetizing J to be directly proportional, accordingly, to the electric and magnetic fields strength, we could derive the wave equation from (1.6), which is possible to any form of the wave. However, there exists a physical mechanism, which restrict the wave form. This mechanism is connected with the way of over-radiating of electromagnetic impulse with the atomic electrons. This process is precisely considered further.
\n\t\t\t\t
We consider the strength of electric and magnetic fields of impulse as
The similar ratio can be also referred to the function |H|. These ratios should not be regarded as the equations to define the module of electric and magnetic aspect of impulse. It is the approximate expression of the second derivative \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t or \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t for the considered physical model and reflects several non-linear effects of interaction between electromagnetic radiation and substance. The approximate ratio (1.12) defines the connection of medium polarization P with the strength of impulse electric field (similarly to the magnetization J with the magnetic field strength), that would be considered further. The electromagnetic field impulse strengths should be estimated from the equation (1.6) taking into account the ratio (1.12).
\n\t\t\t\t
In accordance with (1.10), \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, hence, from (1.12) we estimate equation for the electromagnetic field impulse
The same ratio exists for the magnetic field also. Passing over to (1.13) to the function Ф(X,t) by formula (1.7) and by concerning\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tχ\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, where χ – relative dielectric permittivity of substance, we have
For the variable \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t by using\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tχ\n\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, where χ – relative magnetic permittivity of substance, we get the ratio, similar to (1.14), except that the right part lacks ε\n\t\t\t\t\t0.
\n\t\t\t\t
The variables\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tФ\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\texp\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. By comparing (1.7) and (1.10) we state\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\tФ\n\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tФ\n\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\to\n\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
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.
\n\t\t\t\t
The non-linear Schrödinger equation with complicated type of linearity is received. We introduce the signs:\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\tχ\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\tχ\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\tχ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tχ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\tχ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t – relative permittivities of the substance. Hence, the equation (1.15) will be
where the type of the function \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is still unknown. The variables k, ω and δ\n\t\t\t\t\t* – constants. By marking\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tζ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, and substituting (1.17) in (1.16) and concerning \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\tФ\n\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tФ\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tζ\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t we get
If to permit that \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tс\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t as there should not be any imaginary items in (1.18), this equation is transformed to
where C\n\t\t\t\t\t1 and C\n\t\t\t\t\t2 – constants. By substituting (1.20) into (1.19) we get that the constant C\n\t\t\t\t\t1 could be the arbitrary variable,\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tχ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tс\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
\n\t\t\t\t
The constant C\n\t\t\t\t\t2 could not depend upon the parameters of equation. It is accepted that C\n\t\t\t\t\t2\n\t\t\t\t\t=-1. Then the frequency and the wave number in (1.17), accordingly, are
The formulas (1.21) associate the frequency and the wave number of oscillations of function Ф(X,t) with the parameters of substance and electromagnetic field impulse.
\n\t\t\t\t
The most simple ratios between the parameters are gained, when\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. In this case\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t,\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. From the equations in (1.21), and concerning \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tχ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tс\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t there is
By concerning that\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tχ\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, we have\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\tχ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. This inequality is true, as for the rarefied gas (n < 1018\n\t\t\t\t\tatoms/cm\n\t\t\t\t\t3) \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t>> χ and the frequency of wave filling of impulse δ is far more than frequency of impulse envelope ω.
\n\t\t\t\t
Taking into account (1.10), (1.20) and the\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\tФ\n\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, we can find the laws of electromagnetic field strengths shifting by
It should be stressed, that though, the ratios for the electric aspect of impulse in [1] 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 [1] 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.
\n\t\t\t\t
For the estimation, like in [1] we have\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t10\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t4\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t10\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t5\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, ω =6,28.1012 s—1, δ = 6,28.1013 s—1.
\n\t\t\t\t
For instance, the result of strength estimation of the electric filed impulse by the coordinate X, calculated with the MathCAD system by formula (1.23), is shown in fig.2.
\n\t\t\t\t
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.
\n\t\t\t\t
\n\t\t\t\t\tFigure 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
Calculation of the electric component of electromagnetic radiation impulse in dielectric
\n\t\t\t\t
Figure 3.
Intensity of electric and magnetic fields electromagnetic solitone in dielectric in conditions of the self-induced transparency
\n\t\t\t\t
Figure 4.
Comparison bending around of the electromagnetic field impulse, received on the basis of the offered theory, a curve 1, and the equations the Maxwell - Bloch, curve 2
\n\t\t\t\t
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.
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\n\t\t
\n\t\t
\n\t\t\t
2. Angular distribution of photoelectrons during irradiation of metal surface by electromagnetic waves
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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 [10]:\n\t\t\t
where I0 − is the intensity of the incident wave, I − is the intensity on z-coordinate,
\n\t\t\t
directioned depthward the metal, λ − is the wavelength of radiation, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\tχ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t− is the product of refractive index by extinction coefficient.
\n\t\t\t
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, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\tχ\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t83\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, therefore z = 15,5 nm. Considering [11] 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.
\n\t\t\t
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.
\n\t\t\t
As a result it is interesting to consider angular distribution of photoelectrons during the inner photoemissive effect.
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\n\t\t\t\t
2.1. Nonrelativistic case
\n\t\t\t\t
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 [12].
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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 [13]. I is marked that photoelectrons don\'t take off in the direction of distribution of quantum [14]. This conclusion is made on the basis of positions which in the simplified variant are represented by the following.
\n\t\t\t\t
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\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
\n\t\t\t\t
Figure 5.
Direction of vectors of particle momentum at the inner photoemissive effect
\n\t\t\t\t
Besides, if the electron momentum is set at an angle θ to the direction of quantum of light its value will be:
If \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tthen photoelectron energy\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. 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 [7]. 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.
Figure 6.
Angular distribution of photoelectrons during interaction of orbital electron with the electromagnetic wave
\n\t\t\t\t
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\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. 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 [15].\n\t\t\t\t
\n\t\t\t\t
\n\t\t\t\t\tFig.7 demonstrates change of photoelectron momentum in the presence of a photon momentum. The conclusion made on the basis of the is\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Let\'s find\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tsin\n\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tsin\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\tsin\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Considering that δ is too small we find\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tsin\n\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tsin\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tsin\n\t\t\t\t\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tsin\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tsin\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. The law of sines for a triangle on fig. 7 is used.
\n\t\t\t\t
Further consideration\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\tν\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tW\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t – is the relation of photoelectron speed to a speed of light in vacuum, W − is the work function of electrons from atom, we have\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tsin\n\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tsin\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Taking for granted that β′ is small we will transform (2.2) into\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tsin\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Angular distribution of electron energy for\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t15\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, made according to the (2.2) taking into account the correction is shown on fig. 7, a curve 2.
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Figure 7.
The account of the momentum of quantum \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t using wave approach of electron interaction with the electromagnetic wave
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Thus scattering indicatrix of photoelectrons has received some slope forward, but to the direction of quantum momentum, i.e. at \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t electrons don\'t take off as before.
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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.
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Figure 8.
Attitude of components velocity of orbital electron during its interaction with the electromagnetic wave
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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.
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 \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t in the electromagnetic wave.
The condition of detachment electron from atom at any position of electron\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
\n\t\t\t\t
In case of equality of speeds \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t 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.
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In [6] 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 [6] 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.
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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 \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and energy Е beats out electron from atom, making A a getting out. Thus both laws of conservation of energy should be observed:
Where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t− is the momentum of taken off electron, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t− is the momentum transferred to a nucleus.
\n\t\t\t\t
The formula (2.7) differs from Einstein\'s standard formula\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. 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.
\n\t\t\t\t
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\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, where c − is the speed of light in vacuum, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, we find:
Let us introduce the following notation\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t,\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Then the equation (2.11) will be transformed into:
Solving quadratic equation (2.12) provided \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t (electronic mass is much less that nuclear mass), we find:
Considering that\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, where V1 – speed of photoelectrons provided\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, we find:
Provided that nuclear mass is aiming to infinity \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t 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.
\n\t\t\t\t
The condition \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t 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 \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is applicable absolutely.
\n\t\t\t\t
For the inner photoemissive effect in the formula (2.15) it is necessary to use effective nuclear mass\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, considering attractive powers between atoms in substance.
Let us nominate\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Distribution of photoelectrons will arise at\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. In the right part of the received inequality there is a very small value, therefore distribution of photoelectrons will arise practically at\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
\n\t\t\t\t
Let us nominate\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tη\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tη\n\t\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t 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 \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is shown on fig. 9, made according to the formula (2.17) with several values η for copper.
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Figure 9.
Angular distribution of photoelectrons during the inner photoemissive effect depending on parameter η during interaction of orbital electrons with light quantum. The results of experiments [16] are shown by black small squares
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The figure makes it evident that speeds of photoelectrons become almost identical in all directions already at\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tη\n\t\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t01\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Then Einstein\'s formula \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t becomes fair and for the inner photoemissive effect. Considering that, for example, for copper the relation \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t4\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t10\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t11\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t equivalent as far as the order of value is concerned \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\tη\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t 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\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, can be observed only in very short wave part of spectrum γ-radiations.
\n\t\t\t\t
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 [16]. 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 [16] 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.
\n\t\t\t\t
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 [16], in the direction opposite to intensity of light, shows at the prevalence of corpuscular properties of light in its interaction with atoms.
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\n\t\t\t
\n\t\t\t\t
2.2. Relativistic case
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Dealing with relativistic case of the inner photoemissive effect, the law of conservation of energy needs to be written down as:
Where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t – is the kinetic energy of photoelectron.
\n\t\t\t\t
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:
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.
\n\t\t\t\t
Substituting value Е2 in (2.22) from the equation (2.18), we get:
The notation \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\tη\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t corresponds to item 1 section.
It is thus accounted for that\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
In contrast to the nonrelativistic case, the formula (2.17), formula (2.27) possesses in its right part value \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t which depends on the total energy of electron Е1 the structure of
\n\t\t\t\t
which includes also kinetic energy\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. But dependence of value α on \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t not strong as the total energy structure includes rather big rest energy of electron\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
\n\t\t\t\t
Considering that\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the relative speed of the photoelectron, we find:
Substituting the equation (2.28) in the equation (2.27) and considering that\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, we get:
Considering that\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, at\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, 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.
\n\t\t\t\t
Figure 10.
Dependence of relativistic coefficient β on relative speed of photoelectrons \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t
\n\t\t\t
\n\t\t
\n\t\t
\n\t\t\t
3. Conclusion
\n\t\t\t
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.
\n\t\t\t
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.
\n\t\t
\n\t\n',keywords:null,chapterPDFUrl:"https://cdn.intechopen.com/pdfs/16405.pdf",chapterXML:"https://mts.intechopen.com/source/xml/16405.xml",downloadPdfUrl:"/chapter/pdf-download/16405",previewPdfUrl:"/chapter/pdf-preview/16405",totalDownloads:1855,totalViews:256,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,dateSubmitted:null,dateReviewed:null,datePrePublished:null,datePublished:"July 5th 2011",dateFinished:null,readingETA:"0",abstract:null,reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/16405",risUrl:"/chapter/ris/16405",book:{slug:"behavior-of-electromagnetic-waves-in-different-media-and-structures"},signatures:"Andrey N. Volobuev",authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_1_2",title:"1.1. Distribution of Electromagnetic Field Momentum in dielectrics in stipulation of self-induced transparency",level:"2"},{id:"sec_2_2",title:"1.2. Coordination of the electromagnetic impulse with the substance",level:"2"},{id:"sec_3_2",title:"1.3. Non-linear Schrödinger equation",level:"2"},{id:"sec_5",title:"2. Angular distribution of photoelectrons during irradiation of metal surface by electromagnetic waves ",level:"1"},{id:"sec_5_2",title:"2.1. Nonrelativistic case",level:"2"},{id:"sec_6_2",title:"2.2. Relativistic case",level:"2"},{id:"sec_8",title:"3. Conclusion ",level:"1"}],chapterReferences:[{id:"B1",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tVolobuev\n\t\t\t\t\t\t\tA. N.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNeganov\n\t\t\t\t\t\t\tV. A.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2002\n\t\t\t\t\tThe Electromagnetic Envelope Soliton Propagating in Dielectric.\n\t\t\t\t\tTechnical Physics Letters. S-Petersburg. 28\n\t\t\t\t\t2\n\t\t\t\t\t15\n\t\t\t\t\t20 .\n\t\t\t'},{id:"B2",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tDodd\n\t\t\t\t\t\t\tR. K.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tEilbeck\n\t\t\t\t\t\t\tJ. C.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tGibbon\n\t\t\t\t\t\t\tJ. D.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMorris\n\t\t\t\t\t\t\tH. C.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1982\n\t\t\t\t\tSolitons and Nonlinear Wave Equations. Harcourt Brace Jovanovich, Publishers. London, New York, Toronto. 545 588, 601.\n\t\t\t'},{id:"B3",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAblowitz\n\t\t\t\t\t\t\tM. J.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tSegur\n\t\t\t\t\t\t\tH.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1981\n\t\t\t\t\tSolitons and the Inverse Scattering Transform. SIAM. Philadelphia. 374\n\t\t\t\t\t378 .\n\t\t\t'},{id:"B4",body:'\n\t\t\t\t\n\t\t\t\t\tVolobuev A.N. Modeling of Physical Processes, to Describe by Nonlinear Schrodinger Equation. Mathematical modelling. 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A. \n\t\t\t\t\t\t\n\t\t\t\t\tA doubl enu clear photoeffectin relativistic region. Angular and energy distributions of photoelectrons. Moscow, JETP. 1998\n\t\t\t\t\t114 №. 5. 1537 1554.\n\t\t\t'},{id:"B14",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tLevich\n\t\t\t\t\t\t\tV. G.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1962\n\t\t\t\t\tThe Course of theoretical physics. 2 Мoscow, PhysMathGiz. 658\n\t\t\t\t\n\t\t\t'},{id:"B15",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBlohin\n\t\t\t\t\t\t\tM. A.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1957 Physics of Roentgenrays Мoscow, State Publishing House. 261\n\t\t\t\t\n\t\t\t'},{id:"B16",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tSteigerwald\n\t\t\t\t\t\t\tD. A.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tEgelhoff\n\t\t\t\t\t\t\tW. F. jr\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1988Phys Rev. Lett 60\n\t\t\t\t\t2558\n\t\t\t\t\n\t\t\t'}],footnotes:[],contributors:[{corresp:null,contributorFullName:"N. 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Samara State University, Russia
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Nickelson, S. Asmontas and T. Gric",authors:[{id:"19853",title:"Prof.",name:"Liudmila",middleName:null,surname:"Nickelson",fullName:"Liudmila Nickelson",slug:"liudmila-nickelson"},{id:"24827",title:"Prof.",name:"Steponas",middleName:null,surname:"Asmontas",fullName:"Steponas Asmontas",slug:"steponas-asmontas"},{id:"24828",title:"Mr.",name:"Tatjana",middleName:null,surname:"Gric",fullName:"Tatjana Gric",slug:"tatjana-gric"}]},{id:"15089",title:"Silicon Carbide Based Transit Time Devices: The New Frontier in High-power THz Electronics",slug:"silicon-carbide-based-transit-time-devices-the-new-frontier-in-high-power-thz-electronics",signatures:"Moumita Mukherjee",authors:[{id:"21168",title:"Dr.",name:"Moumita",middleName:null,surname:"Mukherjee",fullName:"Moumita Mukherjee",slug:"moumita-mukherjee"}]},{id:"15090",title:"Contact Formation on Silicon Carbide by Use of Nickel and Tantalum in a Materials Science Point of View",slug:"contact-formation-on-silicon-carbide-by-use-of-nickel-and-tantalum-in-a-materials-science-point-of-v",signatures:"Yu Cao and Lars Nyborg",authors:[{id:"18297",title:"Dr.",name:"Yu",middleName:null,surname:"Cao",fullName:"Yu Cao",slug:"yu-cao"},{id:"21819",title:"Prof.",name:"Lars",middleName:null,surname:"Nyborg",fullName:"Lars Nyborg",slug:"lars-nyborg"}]},{id:"15091",title:"Properties and Applications of Ceramic Composites Containing Silicon Carbide Whiskers",slug:"properties-and-applications-of-ceramic-composites-containing-silicon-carbide-whiskers",signatures:"Brian Bertram and Rosario Gerhardt",authors:[{id:"19005",title:"Prof.",name:"Rosario",middleName:null,surname:"Gerhardt",fullName:"Rosario Gerhardt",slug:"rosario-gerhardt"},{id:"21440",title:"Mr.",name:"Brian D.",middleName:null,surname:"Bertram",fullName:"Brian D. Bertram",slug:"brian-d.-bertram"}]},{id:"15092",title:"Spectroscopic Properties of Carbon Fibre Reinforced Silicon Carbide Composites for Aerospace Applications",slug:"spectroscopic-properties-of-carbon-fibre-reinforced-silicon-carbide-composites-for-aerospace-applica",signatures:"Davide Alfano",authors:[{id:"19679",title:"Dr.",name:"Davide",middleName:null,surname:"Alfano",fullName:"Davide Alfano",slug:"davide-alfano"}]},{id:"15093",title:"Effect of Self-Healing on Fatigue Behavior of Structural Ceramics and Influence Factors on Fatigue Strength of Healed Ceramics",slug:"effect-of-self-healing-on-fatigue-behavior-of-structural-ceramics-and-influence-factors-on-fatigue-s",signatures:"Wataru Nakao",authors:[{id:"21462",title:"Dr.",name:"Wataru",middleName:null,surname:"Nakao",fullName:"Wataru Nakao",slug:"wataru-nakao"}]},{id:"15094",title:"Contribution to the Evaluation of Silicon Carbide Surge Arresters",slug:"contribution-to-the-evaluation-of-silicon-carbide-surge-arresters",signatures:"Arnaldo Gakiya Kanashiro and Milton Zanotti Jr.",authors:[{id:"18365",title:"Prof.",name:"Arnaldo",middleName:"Gakiya",surname:"Kanashiro",fullName:"Arnaldo Kanashiro",slug:"arnaldo-kanashiro"},{id:"21617",title:"MSc.",name:"Milton",middleName:null,surname:"Zanotti, Jr.",fullName:"Milton Zanotti, Jr.",slug:"milton-zanotti-jr."}]},{id:"15095",title:"Silicon Carbide Neutron Detectors",slug:"silicon-carbide-neutron-detectors",signatures:"Fausto Franceschini and Frank H. Ruddy",authors:[{id:"21548",title:"Dr.",name:"Fausto",middleName:null,surname:"Franceschini",fullName:"Fausto Franceschini",slug:"fausto-franceschini"},{id:"21551",title:"Dr.",name:"Frank",middleName:null,surname:"Ruddy",fullName:"Frank Ruddy",slug:"frank-ruddy"}]},{id:"15096",title:"Fundamentals of Biomedical Applications of Biomorphic SiC",slug:"fundamentals-of-biomedical-applications-of-biomorphic-sic",signatures:"Mahboobeh Mahmoodi and Lida Ghazanfari",authors:[{id:"17813",title:"Dr.",name:"Mahboobeh",middleName:null,surname:"Mahmoodi",fullName:"Mahboobeh Mahmoodi",slug:"mahboobeh-mahmoodi"},{id:"21336",title:"Prof.",name:"Lida",middleName:null,surname:"Ghazanfari",fullName:"Lida Ghazanfari",slug:"lida-ghazanfari"}]},{id:"15097",title:"Silicon Carbide Whisker-mediated Plant Transformation",slug:"silicon-carbide-whisker-mediated-plant-transformation",signatures:"Shaheen Asad and Muhammad Arshad",authors:[{id:"18190",title:"Dr.",name:"Shaheen",middleName:null,surname:"Asad",fullName:"Shaheen Asad",slug:"shaheen-asad"},{id:"21743",title:"Dr.",name:"Muhammad",middleName:null,surname:"Arshad",fullName:"Muhammad Arshad",slug:"muhammad-arshad"}]},{id:"15098",title:"Silicon Carbide: Synthesis and Properties",slug:"silicon-carbide-synthesis-and-properties",signatures:"Houyem Abderrazak and Emna Selmane Bel Hadj Hmida",authors:[{id:"18643",title:"Dr.",name:"Houyem",middleName:null,surname:"Abderrazak",fullName:"Houyem Abderrazak",slug:"houyem-abderrazak"},{id:"23082",title:"Dr.",name:"Emna Selmane",middleName:null,surname:"Bel Hadj Hmida",fullName:"Emna Selmane Bel Hadj Hmida",slug:"emna-selmane-bel-hadj-hmida"}]},{id:"15099",title:"Combustion Synthesis of Silicon Carbide",slug:"combustion-synthesis-of-silicon-carbide",signatures:"Alexander S. Mukasyan",authors:[{id:"21434",title:"Dr.",name:"Alexander S.",middleName:null,surname:"Mukasyan",fullName:"Alexander S. Mukasyan",slug:"alexander-s.-mukasyan"}]},{id:"15100",title:"In Situ Synthesis of Silicon-Silicon Carbide Composites from SiO2-C-Mg System via Self-Propagating High-Temperature Synthesis",slug:"in-situ-synthesis-of-silicon-silicon-carbide-composites-from-sio2-c-mg-system-via-self-propagating-h",signatures:"Sutham Niyomwas",authors:[{id:"17615",title:"Prof.",name:"Sutham",middleName:null,surname:"Niyomwas",fullName:"Sutham Niyomwas",slug:"sutham-niyomwas"}]},{id:"15101",title:"High Reliability Alumina-Silicon Carbide Laminated Composites by Spark Plasma Sintering",slug:"high-reliability-alumina-silicon-carbide-laminated-composites-by-spark-plasma-sintering",signatures:"Vincenzo M. Sglavo and Francesca De Genua",authors:[{id:"17425",title:"Dr.",name:"Francesca",middleName:null,surname:"De Genua",fullName:"Francesca De Genua",slug:"francesca-de-genua"},{id:"17426",title:"Prof.",name:"Vincenzo Maria",middleName:null,surname:"Sglavo",fullName:"Vincenzo Maria Sglavo",slug:"vincenzo-maria-sglavo"}]},{id:"15102",title:"High Temperature Phase Equilibrium of SiC-Based Materials",slug:"high-temperature-phase-equilibrium-of-sic-based-materials",signatures:"Yuhong Chen, Laner Wu ,Wenzhou Sun, Youjun Lu and Zhenkun Huang",authors:[{id:"21059",title:"Dr.",name:"Laner",middleName:null,surname:"Wu",fullName:"Laner Wu",slug:"laner-wu"}]},{id:"15103",title:"Liquid Phase Sintering of Silicon Carbide with AlN-Re2O3 Additives",slug:"liquid-phase-sintering-of-silicon-carbide-with-aln-re2o3-additives",signatures:"Laner Wu, Yuhong Chen ,Yong Jiang, Youjun Lu and Zhenkun Huang",authors:[{id:"21059",title:"Dr.",name:"Laner",middleName:null,surname:"Wu",fullName:"Laner Wu",slug:"laner-wu"},{id:"135946",title:"Prof.",name:"Yuhong",middleName:null,surname:"Chen",fullName:"Yuhong Chen",slug:"yuhong-chen"},{id:"135947",title:"Prof.",name:"Yong",middleName:null,surname:"Jiang",fullName:"Yong Jiang",slug:"yong-jiang"},{id:"135948",title:"Prof.",name:"Youjun",middleName:null,surname:"Lu",fullName:"Youjun Lu",slug:"youjun-lu"},{id:"135949",title:"Prof.",name:"Zhenkun",middleName:null,surname:"Huang",fullName:"Zhenkun Huang",slug:"zhenkun-huang"}]},{id:"15104",title:"Investigations on Jet Footprint Geometry and its Characteristics for Complex Shape Machining With Abrasive Waterjets in Silicon Carbide Ceramic Material",slug:"investigations-on-jet-footprint-geometry-and-its-characteristics-for-complex-shape-machining-with-ab",signatures:"S. Srinivasu D. and A. Axinte D.",authors:[{id:"23402",title:"Dr.",name:"Srinivas",middleName:null,surname:"Devadula",fullName:"Srinivas Devadula",slug:"srinivas-devadula"},{id:"23403",title:"Dr.",name:"Dragos",middleName:null,surname:"Axinte",fullName:"Dragos Axinte",slug:"dragos-axinte"}]},{id:"15105",title:"Ductile Mode Micro Laser Assisted Machining of Silicon Carbide",slug:"ductile-mode-micro-laser-assisted-machining-of-silicon-carbide",signatures:"Deepak Ravindra, Saurabh Virkar and John Patten",authors:[{id:"21605",title:"Dr.",name:"Deepak",middleName:null,surname:"Ravindra",fullName:"Deepak Ravindra",slug:"deepak-ravindra"},{id:"21773",title:"Dr.",name:"John",middleName:null,surname:"Patten",fullName:"John Patten",slug:"john-patten"},{id:"21774",title:"Mr.",name:"Saurabh",middleName:null,surname:"Virkar",fullName:"Saurabh Virkar",slug:"saurabh-virkar"},{id:"22753",title:"Mr",name:"Bogac",middleName:null,surname:"Poyraz",fullName:"Bogac Poyraz",slug:"bogac-poyraz"}]}]}]},onlineFirst:{chapter:{type:"chapter",id:"67431",title:"Incorporating Sustainable Practices in Asphalt Industry",doi:"10.5772/intechopen.86702",slug:"incorporating-sustainable-practices-in-asphalt-industry",body:'\n
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1. Introduction
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The massive demand for asphalt pavement is a response to the growing length of highways and the number of vehicles worldwide. In the USA, for example, highway networks are approximately 8 million miles and facilitate over 3 million vehicle miles yearly [1]. According to Green Car Reports, there are nearly 1.2 billion vehicles in operation today, and this number is constantly growing to reach the estimated number of 2.5 billion by 2050. The massive number of vehicles has expanded the need for more streets and highways most of which require asphalt paving. Additionally, the pavements that are in use require periodic repairs and replacements of asphalt. This explains an enormous increasing demand for asphalt.
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This chapter discusses a comprehensive approach to maximize the sustainability of the asphalt industry. This approach is presented in the two following major parts. (1) Assessment of asphalt’s environmental impacts in its life cycle. In this section, the sustainable practices in the four phases of asphalt’s life cycle (material procurement, processing of raw materials, transportation of products, and end of a product’s life) are discussed. (2) Sustainable approaches in the asphalt industry. This section is divided into two parts that discuss two main directions in pushing the asphalt industry to improve its contribution to sustainability. The first direction is enhancing the technology of warm mix asphalts, which explains the benefits and the existing shortcomings of these mixes that require further research. The second part discusses the opportunities for reducing the transfer of pollutants through runoffs by using permeable asphalt pavements. The barriers to more widespread use of permeable asphalt pavements that need to be considered by the asphalt industry are also presented.
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The chapter is expected to provide an understanding of the adverse impacts of the asphalt industry on the environment, how these impacts can be mitigated, and what additional steps must be taken to enhance the existing sustainable methods in different stages of asphalt’s life cycle.
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2. Is asphalt an unsustainable product by itself?
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Despite the impression from the detrimental impacts of asphalt on the environment, in compassion with other pavements, asphalt can be addressed as a sustainable material. According to Gambetes, producing asphalt pavements requires almost 20% less energy than other pavements. Additionally, the construction and rehabilitation of these pavements are relatively quick as no curing is needed before the repaired road segment is ready to use. This is beneficial in reducing the exposure to construction process pollutions, in saving taxpayers’ money, and in reducing the time of road closure, which contributes to the societal bottom line of suitability.
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Recycling in the asphalt industry is not a goal for the future. It is already happening in massive amounts. For example, the USA, as the leading recycler of asphalt, reclaims approximately 65 million tons of its products annually and reused almost 99% of it [2]. The asphalt industry also contributes to the reuse of other materials such as tires, glass, blast furnace slag, and asphalt roofing shingles.
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In terms of maintenance and rehabilitation, asphalt pavements are also less expensive than concrete pavements. Although their service life is relatively shorter compared with concrete pavements, with proper design, construction, and maintenance, they can continue to serve for decades without wearing out. Their maintenance is faster and less expensive than concrete pavements. This is why rubbishing or covering worn out concrete pavements with an asphalt surface is a common practice for repairing concrete pavements.
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Research findings support that asphalt pavements provide superior contact between the vehicle tires and the road surface, which enhances vehicle safety. When designed with open-graded, asphalt pavements can effectively reduce the rainwater splash, which can potentially improve visibility and reduce accidents.
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Asphalt pavements are relatively low noise pavements and are beneficial to be used in urban areas where reducing noise pollution is essential. According to the research [2], asphalt pavements can lower the noise between 3 and 10 dB, which affects similar to doubling the distance from the road to the people who can hear the road noise (Figure 1).
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Figure 1.
The sustainability benefits of asphalt pavements compared with other pavements.
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To answer the question if asphalt pavements contribute to sustainability or are detrimental to it, certain facts must be considered as follows:
The sustainability must be compared with other alternatives of pavements. If a pavement is more sustainable than other alternatives, even if, it has a burden on the sustainability triple bottom lines, it is considered as the sustainable option. Based on this comparison, asphalt pavements are one of the most sustainable options for pavements.
The sustainability must be compared with the optimum potential sustainability that can be achieved in the procurement, design, production, shipping, construction, rehabilitation, replacement, and reuse of the product. Based on this approach, there is high potential in modifying the technology of asphalt pavements to enhance their sustainability.
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3. How does the asphalt industry affect the environment?
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The asphalt industry negatively impacts the environment in three ways. This is shown in Figure 2. As a petroleum-based product, asphalt mixes carry a significant amount of chemicals that can impact the environment during extraction, transportation of raw materials, processing, and using.
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Figure 2.
The primary adverse impacts of the asphalt industry on the planet.
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Another primary adverse environmental impact of asphalt is on water. Hazardous chemicals, dust, and other particles can penetrate to asphalt paving surfaces and then be transferred through stormwater runoffs. The polluted runoffs can eventually affect water bodies and endanger aquatic life, reduce water quality, and degrade the views to name a few. This explains the asphalt industry must focus vigorously on the ways to mitigate the accumulation and transfer of pollutants. The third major adverse impact of the asphalt industry is generating atmospheric emissions. A wide range of hazardous emissions are released particularly in the process of heating the mix. These emissions contribute to global warming and lead to various impacts on the environment as well as on the human health.
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4. What are the main approaches in improving the asphalt industry’s sustainability?
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Once the detrimental impacts of asphalt on the environment are identified, practices to minimize these impacts must be established and followed. Figure 3 shows some of the indicators of asphalt sustainability as it relates to material, construction, maintenance, and use [3]. The figure indicates why the sustainability of asphalt industry has significant potentials for improvement in multiple aspects. The challenge is the tradeoff of compromising some of the mechanical properties of asphalt to make it more sustainable. For instance, a wide range of emissions can be reduced by lowering the temperature of heating the asphalt mixes, but there are some uncertainties if the reduced temperature can maintain an asphalt pavements resistance against fatigue or heavy loads. This will be discussed in more details in Section 5.2.2. Therefore, the sustainable approaches must keep a balance between lowering the adverse environmental impacts and the functionality, mechanical property, and durability of asphalt pavements.
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Figure 3.
Potential approaches to improve environmental sustainability in the asphalt industry.
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5. Mitigating asphalt’s negative environmental impacts in its life cycle
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The life cycle assessment or cradle-to-grave is an integral element of sustainability evaluation. This approach must be applied for asphalt analysis to assess its impacts on global warming and climate change, depletion of fossil fuels, and human well-being to name a few [4]. Life cycle assessment encompasses a wide range of decisions and activities in different phases of asphalt production, transportation, use, and reuse. In addition to extraction, significant amounts of materials such as bitumen, fuel oil, and gasoline are transferred and stored in asphalt plants. Opportunities to modify the traditional practices that are commonly used in asphalt plants must be studied for each plant. There exist common best practices that can be followed by the managers of asphalt plants. Table 1 shows the four primary steps in life assessment of asphalt products (data from [5]). These four stages and some best practices to apply in each of them are discussed in the following.
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5.1 Material procurement (phase 1)
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The practices used in the procurement of virgin materials can significantly identify the level of sustainable production of asphalt mixes. While main asphalt ingredients are primarily byproducts of the petroleum industry; considering the magnitude of their production, there are independent practices for enhancing the sustainable procurement of virgin materials in the asphalt industry. Based on the discussed sustainability concerns about the asphalt industry, the following target can be defined to push this industry toward more sustainable practices:
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5.1.1 Extraction of materials
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The enormous demand for nonrenewable raw materials to produce asphalt pavements is a threat to sustainability. For instance, 83% of all street pavements in the USA are made of asphalt [6]. The primary step in contributing to sustainability in the procurement of asphalt materials is reducing the demand for the extraction of nonrenewable petroleum resources and other virgin materials. One of the greatest advantages of asphalt is its recyclability for almost unlimited times. The fact that asphalt pavements are 100% recyclable can be interpreted that there is no certain end life for them [7]. Figure 4 shows the sustainable procurement process of materials in an asphalt plant. As it is seen in each of the six following steps, there are potentials to maintain the sustainability of operations. The asphalt plant’s determination and leadership in establishing these practices is pivotal in the successful application of these practices and policies.
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Figure 4.
Sustainable procurement process of materials in an asphalt plant.
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5.2 Processing of raw materials (phase 2)
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The most energy-intensive phase of asphalt production is the processing phase as it requires a high temperature for burning and processing of raw materials. This phase is also the most polluting phase due to the emissions that are normally generated in different heating processes. Therefore, energy and pollution reduction are the primary sustainability goals of this phase. This is explained in the following.
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5.2.1 Energy consumption reduction
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To have a holistic energy consumption reduction approach, the consumers of energy in the typical production phase should first be identified. In the following, some of the main energy consumers in the production process are identified. (1) Fuel or electricity needed for construction equipment that is applied in the production process, (2) energy to operate burners that dry aggregates, and (3) fuel for heating liquid asphalt binder and other mixing processes (natural gas, propane, diesel fuel, recycled fuel, biofuels, etc.)
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The comprehensive study of the system to identify the potentials for energy consumption reduction is required. Some of the best practices in this regard are the following:
Supplying energy from more sustainable sources (solar and wind power)
Constant maintenance of the tools and equipment to maintain their efficiency
Minimizing wastes of energy in heating processes
Replacing low-efficiency tools with new efficient ones
Optimizing the layout, production queuing, and supply of raw materials to minimize energy consumption
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5.2.2 Pollution and emission reduction
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The second main goal in sustainable processing of asphalt is pollution and emission reduction. To enhance the environmental sustainability of the asphalt industry, controlling certain chemicals must be the target. Table 2 shows the primary hazardous chemicals, how they are generated in the asphalt industry, and why they must be controlled.
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Cradle to grave process
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Phases
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How the assessment must be performed
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Cradle to gate
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Phase 1. Material procurement
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Extraction of materials
Reuse and recycle
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Phase 2. Processing of raw materials
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Energy consumption reduction
Pollution and emission reduction
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Phase 3. Transportation of product
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Vehicles’ air pollution reduction
Preventing the spill of polluting materials during transportation
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Gate to grave
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Phase 4. End of a product’s life
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Connecting end use point to reuse and recycle
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Table 1.
The primary steps of a life-cycle assessment of asphalt products.
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Hazardous chemicals
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Why they must be controlled (selected hazards and the chemical causes)
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Primary generation area
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Carbon dioxide (CO2)
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Global warming, climate change, human toxicity
\n
Construction
\n
\n
\n
Methane (CH4)
\n
Global warming (21 times more than carbon dioxide), photo-oxidant formation
\n
Construction
\n
\n
\n
Nitrous oxide (N2O)
\n
Global warming
\n
Construction
\n
\n
\n
Sulfur dioxide (SO2)
\n
Acidification, photo-oxidant formation, human toxicity
\n
Construction
\n
\n
\n
Nitrogen dioxide (NO2)
\n
Acidification, eutrophication, human toxicity, photo-oxidant formation
\n
Construction + End of life
\n
\n
\n
Ammonia (NH3)
\n
Acidification, eutrophication, human toxicity
\n
Construction + end of life
\n
\n
\n
Carbon monoxide (CO)
\n
Human toxicity, photo-oxidant formation
\n
Construction
\n
\n
\n
Nonmethane volatile organic compound (NMVOC)
\n
Ecotoxicity, human toxicity, photo-oxidant formation
\n
Construction + maintenance
\n
\n
\n
Hydrocarbon (HC)
\n
Ecotoxicity, human toxicity
\n
Construction + maintenance + use
\n
\n
\n
Particulate matter (PM)
\n
Human toxicity
\n
Construction
\n
\n
\n
Heavy metals
\n
Human toxicity
\n
Construction
\n
\n
\n
Arsenic (As)
\n
Ecotoxicity, human toxicity
\n
Maintenance + use
\n
\n
\n
Cadmium (Cd)
\n
Ecotoxicity, human toxicity
\n
Maintenance + use
\n
\n
\n
Lead (Pb)
\n
Ecotoxicity, human toxicity
\n
Maintenance + use
\n
\n
\n
Mercury (Hg)
\n
Ecotoxicity, human toxicity
\n
Maintenance + use
\n
\n
\n
Chemical oxygen demand (COD)
\n
Eutrophication
\n
End of life
\n
\n
\n
Phosphate (PO4)
\n
Eutrophication
\n
End of life
\n
\n
\n
Nitrate
\n
Eutrophication
\n
End of life
\n
\n
\n
Phosphorus (P)
\n
Eutrophication
\n
End of life
\n
\n
\n
Nitrogen (N)
\n
Eutrophication
\n
End of life
\n
\n\n
Table 2.
Hazardous chemicals that must be controlled to enhance the environmental sustainability of the asphalt industry.
\n
It should be noted that lowering the mixing temperature can substantially contribute to both energy consumption and emission reduction. Therefore, it is crucial to invest in modifying asphalt technology in this direction. Warm mix asphalt is the product of this approach, which is one of the most impactful modifications in asphalt technology to make it more sustainable. Warm mixes will be discussed in Section 6.1.
\n
Research shows that the reduced energy consumption required in the manufacture of warm mix asphalts can lower the carbon dioxide emission between 30 to 40% [8] and 30 to 50% reduction on the fumes at bituminous plants, which is significantly beneficial for workers on a paving team [9]. Additionally, reduced emissions enable to establish asphalt plants closer to urban areas [10]. This, in turn, can reduce transportation pollution, cost, and increase the accessibility of asphalt materials for urban construction. Table 3 compares the emission generated by hot and warm mix asphalts ([11] as cited by [8]).
\n
\n
\n
\n
\n
\n\n
\n
Type of emission
\n
Hot mix asphalt
\n
Warm mix asphalt
\n
Reduction
\n
\n\n\n
\n
Temperatures (°C)
\n
155–165
\n
110–120
\n
45
\n
\n
\n
Gases temperature (°C)
\n
65.6
\n
50.3
\n
15
\n
\n
\n
CO2 (%)
\n
2.12
\n
1.59
\n
25
\n
\n
\n
CO (ppm)
\n
217
\n
151.6
\n
30
\n
\n
\n
NOx (mg/m3, e.g., NO2)
\n
26.8
\n
21.5
\n
25
\n
\n
\n
Environmental dust (mg/m3)
\n
168
\n
21
\n
88
\n
\n\n
Table 3.
Comparing the emission generated by hot and warm mix asphalt.
\n
As it is seen in Table 3, the amount of emission reductions for replacing hot mix with warm mix asphalts is significant. These reductions contribute to reducing the global warming effects at a large scale, pollution for neighboring urban areas at a middle scale, and a healthier environment for the people working at the asphalt plant in a small scale.
\n
\n
\n
\n
5.3 Transportation of asphalt raw materials and products (phase 3)
\n
Asphalt industry is involved with a massive demand for transportation to (1) supply the raw materials for the production and (2) deliver the products to consumers in an almost nonstop basis. Additionally, being a petroleum product, asphalt materials and products carry high loads of chemicals that can potentially be hazardous to the environment (as it was listed in the Table 3). Table 4 shows some of the indicators of sustainable transportation of asphalt raw materials and products.
\n
\n
\n
\n\n
\n
Sustainable transportation approach
\n
How to implement
\n
\n\n\n
\n
Reducing the number of trips
\n
\n
Increase the capacity of delivery in each trip
Optimize trips to deliver more in fewer trips
\n
\n
\n
\n
Enhancing the safety of trips
\n
\n
Building private roads for main access to the asphalt plant
Putting signs around the heavy vehicle
\n
\n
\n
\n
Reducing the distance of trips
\n
\n
Locating the asphalt plants with optimal distance to multiple consumers
Creating temporary asphalt plants near road projects
\n
\n
\n
\n
Protecting health and well-being
\n
\n
Reduce the vehicles’ pollution or use hybrid vehicles
Reduce transportation noise
\n
\n
\n
\n
Protecting the hosting roads environment
\n
\n
Minimize spill of chemicals form the vehicle
Be responsible for fixing possible damages to the environment
Ongoing maintenance and smog check of vehicles
Replacing polluting and old vehicles with efficient and green vehicles
\n
\n
\n
\n
Promote local consumption
\n
\n
Give priority to local projects to avoid long trips
\n
\n
\n\n
Table 4.
Principles of sustainable transportation of asphalt raw materials and products.
\n
\n
\n
5.4 End of a product’s life (phase 4)
\n
Considering the huge amount of asphalt-paved streets and highways, an enormous amount of asphalt pavements can reach their end-of-life point in any selected interval. The sustainable approach of cradle to cradle attempts to connect the end of life to reuse of asphalt pavements. The primary solution to reduce the demand for more raw materials for asphalt production is the reuse of asphalt and mixing it with virgin materials. Asphalt mixes contain a considerable amount of recyclable materials such as coarse and fine aggregates. To maximize the asphalt reuse, production of certain types of asphalts must be encouraged.
\n
\n
5.4.1 Reclaimed asphalt pavements (RAP)
\n
Using reclaimed asphalt pavement (RAP) in producing new bituminous mixes has multiple sustainability advantages including significant cost and energy savings and environmental benefits due to a reduction in the waste generated in road maintenance and rehabilitation processes [12, 13].
\n
According to the National Asphalt Pavement Association of the USA, reclaimed asphalt pavements are removed or reprocessed pavement materials containing asphalt and aggregates. These materials are the debris of construction activities such as reconstruction, resurfacing, and diggings to gain access to buried utilities [14]. Based on the estimates, 41 million metric tons of RAP is produced yearly in the United States [14]. According to EAPA [15], in Europe, approximately 50 million tons of RAP is produced annually. This enormous volume of RAP can be reused to produce new bituminous mixtures [8].
\n
The environmental and economic advantages of applying RAP in hot mix asphalts can be maximized by producing totally recycled hot mixed produced with 100% RAP. The challenge to do this is improving the mechanical properties of these mixes such as workability, durability, binder aging, and fatigue cracking resistance [16].
\n
\n
\n
5.4.2 Recycled asphalt shingles (RAS)
\n
\n
5.4.2.1 Waste plastic
\n
Waste plastic is another massive source of material reuse to procure the asphalt ingredients. Only in the USA, 42.6 1-l bottles are purchased every year. It is estimated that on a global scale, 200 billion plastic water bottles are consumed by people [17]. The abundant amount of plastic waste and its suitability to be used in the asphalt industry creates a massive opportunity to reduce the need for virgin materials. The research has identified and encouraged this opportunity [18, 19].
\n
\n
\n
\n
\n
\n
6. Sustainable approaches in the asphalt industry
\n
This section discusses the next steps in improving the sustainability of asphalt industry. The two main approaches discussed are (1) warm mix asphalts and (2) permeable asphalt pavements. In each section, the sustainability advantages are explained. Additionally, the shortcomings of the existing technologies and potentials for improvement are discussed. Table 5 provides a summary of this section.
\n
\n
6.1 Warm mix asphalts
\n
Reducing the temperature of mixes is a fundamental sustainability approach in asphalt industry as it contributes to both energy reduction and emission mitigation [9, 10, 20, 21]. Warm mix asphalt is the result of the temperature reduction approach. Research findings support that using them contributes to reducing energy consumption and air emissions [22]. Regular hot mixes need a temperature of 150°C or more for production and compaction [23]. Warm mixes can lower the demand for energy by reducing the heat needed between 16 and 55°C (Figure 5) [24].
\n
Figure 5.
Low temperature of production for warm mix asphalt (courtesy of Federal Highway administration).
\n
The decline of required temperature for processing asphalt mixes is proportional with requiring less fuel/ton to heat the mix. For instance, hot mix asphalt may require a temperature of 180°C and more than 7 l of fuel per ton and half warm asphalt may require to be heated between 60 and 80°C and less than 4 l of fuel per ton [9]. Reduction of fuel, in turn, leads to a reduction in fuel costs between 25 and 35% [11]. Figure 6 shows some of the sustainability advantages of these mixes (data from [25]).
\n
Figure 6.
How warm mixes contribute to sustainability.
\n
Another advantage of warm mixes is the possibility of using reclaimed asphalt pavements in them [8]. In addition, sustainability advantages (see Figure 7) and research findings indicate that the mechanical properties of these mixes are not compromised to make them more environmentally friendly. These mixes have stiffness very close to hot mixes, and their water sensitivity and fatigue resistance are comparable with hot mixes.
\n
Figure 7.
Sustainability advantages of warm mix asphalt.
\n
\n
6.1.1 Research status of warm mix asphalt mixtures
\n
An impactful approach in increasing the sustainability of warm mix asphalts is the use of rubberized asphalt concrete (RAC). This novel technology, which is addressed as WarmRAC, is both economically and environmentally sound [26]. Although these mixes have been used since the 1960s in the US [27], their technology is still evolving. Research has shown a satisfactory functionality of these pavements for improving aging, oxidation, fatigue, skid, reflection cracking, and rutting resistance as well as mitigating noise generation [26], improved durability and lower maintenance costs [28, 29]. To enhance various aspects of warm mix pavements, the research findings can be categorized into design, construction, performance, and recycling [26] as shown in Table 6 (data from [26]).
\n
\n
\n
\n
\n\n
\n
Technology type
\n
How it contributes to sustainability
\n
Shortcomings that need to be resolved
\n
\n\n\n
\n
Warm mix asphalts
\n
\n
Reduces energy demand for heating the mix
Reduces emissions
\n
\n
\n
Low resistance to fatigue
Moisture damage due to incomplete drying of aggregates
\n
\n
\n
\n
Permeable asphalt pavements
\n
\n
Controls urban runoffs
Prevents transfer of pollution to water bodies
\n
\n
\n
Low resistance to high compressive loads
Clogging
Freeze-thaw issues
\n
\n
\n\n
Table 5.
Sustainable technologies in asphalt industry.
\n
\n
\n
\n
\n\n
\n
Main category
\n
Subcategories
\n
Potentials for improvement
\n
\n\n\n
\n
Design
\n
Material selection
\n
Responsiveness to climate Responsiveness to high loads of traffic
\n
\n
\n
Mix gradation
\n
Gap and open gradation Using in dense graded mixtures
\n
\n
\n
Optimum bitumen content selection
\n
The relationship between bitumen content and durability, permeability, water susceptibility, and compaction
\n
\n
\n
Laboratory performance evaluation
\n
Enhancing lab methods for conditioning/curing of test samples, workability, etc.
\n
\n
\n
Construction
\n
Temperature issues
\n
Solving temperature issues during construction (through reducing the viscosity of binder)
\n
\n
\n
Production rate
\n
Increasing the production rate to increase the construction speed
\n
\n
\n
Compaction
\n
Achieving the target compaction density of WMA
\n
\n
\n
Performance
\n
Comparing different types of WarmRAC
\n
Warm rubberized asphalt binder, Warm mix rubberized asphalt concrete, etc.
\n
\n
\n
Recycling
\n
Recycling methods for WarmRAC
\n
The differences between recycling WarmRAC and conventional mixes Increasing the reusability
\n
\n
\n
Environmental analysis
\n
Potential environmental effects of RAC pavements
\n
Air quality and occupational exposure Water quality
\n
\n\n
Table 6.
Research approaches in enhancing the warm mix asphalt technology.
\n
\n
\n
6.1.2 Potentials for improvement in warm mix asphalt technology
\n
Warm mix asphalt technology has been evolving since they were introduced [30, 31, 32, 33]. The prerequirement of reducing the energy needed for processing and compacting the mat is reducing the shear resistance of the mixture. Research shows that the primary approach in extending asphalt pavement’s durability is the prevention of fatigue in them resulting from repeated loading and unloading by the traffic [33, 34]. The findings [11] indicate that warm mix asphalts have quite satisfactory performance in resisting cracking caused by fatigue [8].
Further reduction of required temperature for processing. As it was explained above, the reduction of the temperature in proceeding warm mix asphalts is proportional to energy consumption reduction, fewer fuel costs, lowered emission, and mitigated pollutions. Therefore, research must invest in further reduction of the processing temperature of these mixes. This reduction must maintain the expected mechanical properties of mixes. Optimizing the trade-off between lower temperature and the mechanical performance required further research.
Incorporating RAP in warm mix asphalt. Both warm mix asphalts and RAP are the results of sustainable approaches in asphalt production. The combination of both approaches is expected to maximize the sustainability of asphalt production. These innovative warm mixes are recycled asphalt with bitumen emulsion. Applying them is expected to contribute to both the economy and the environment. The lowered temperature allows using a larger amount of materials for recycling. This can reduce material use and increase energy savings and reduce costs [35]. A variety of lab tests have been conducted to evaluate the mechanical properties of these mixes including fatigue, bending, rutting, Marshall immersion, aging, and freeze-thaw-cycle splitting. The findings show that application of RAP can negatively impact the moisture and low-temperature cracking resistance, rutting resistance, aging, tensile strength ratio (TSR), and fatigue resistance [36]. On the other hand, there are findings that indicated an improved performance of mixes that use RAP [37]. Further research is required to resolve the possible poor performance of these mixes considering their significant contributions to sustainability.
\n
\n
\n
\n
6.2 Reducing the transfer of pollutants through runoffs by using permeable asphalt pavements
\n
One of the primary negative impacts of the expansion of asphalt pavements is modifying the natural conditions of runoff paths by extending impervious surfaces that cannot effectively capture runoffs [38]. Stormwater runoffs contribute to various environmental issues including pollution of water bodies, reduction of groundwater recharge, and increasing the rivers’ water temperature [39]. According to USEPA, urban runoffs are the leading source of water quality impairment to surveyed estuaries and the third-largest source of impairments to surveyed lakes [40]. Considering these significant detrimental impacts on the environment, the asphalt industry has an indispensable role in developing asphalt design and production modifications that contribute to stormwater runoff control. A primary approach in doing so is increasing the runoff capturing capacity of pavements (Figure 8).
\n
Figure 8.
Typical porous asphalt pavement with stone reservoir cross section (courtesy of Federal Highway Administration).
\n
\n
6.2.1 The sustainability advantages of using permeable asphalt pavements
\n
Permeable, pervious, or porous asphalt pavements have been identified as one of the dominant solutions and best management practices (BMPs) in controlling urban runoffs. They assist the transfer of runoffs to drain through the layers of the pavement into the ground below. The technology of permeable asphalt is still evolving, and significant ongoing research is currently followed to enhance the functionality of these pavements in different aspects such as durability, runoff capturing capacity, and ease of maintenance. Some of the sustainability advantages of using permeable pavements are shown in Figure 9 [41].
\n
Figure 9.
The advantages of using permeable pavements for the triple sustainability bottom lines.
\n
As it can be seen (from Figure 9), the advantages of pervious pavements are significant, and a vigorous attempt must be made to encourage the asphalt industry to replace impervious pavements with permeable pavements wherever they can be applied. While this transition may be costly, the savings that can be made from eliminating the need to make more expensive stormwater runoff control devices such as retention ponds and swales, specifically in urban areas where land is very expensive, compensate the additional costs [42].
\n
Additionally, the application of permeable asphalt pavements is more critical in urban areas for two reasons. One is the rapid growth of cities that is accompanied by replacing the soil and natural runoff paths with concrete and asphalt pavements [43]. According to the United Nations’ report, in 2010, more than half of the earth’s population lived in cities [44]. According to the research, between 60 and 100% of the total urban areas have impermeable surfaces [45]. This is exacerbated by the multiple negative impacts of construction projects with long and often extended project durations [46, 47]. The second reason is the high presence of chemical pollutants and particles in urban runoffs [48]. This can be attributed to housing, vehicles, and industry activities in cities [49, 50, 51].
\n
\n
\n
6.2.2 Research status of permeable asphalt pavements and filling the knowledge gap
\n
Although there are adequate reasons for increasing the application of permeable pavements, there are certain barriers that limit their application. The research in the asphalt industry should focus on minimizing these barriers. The predominant barrier with this regard is the reduced resistance of permeable pavements against high loads of traffic. Considering that parking lots are one of the sensitive urban areas to collect polluted runoffs, the low compressive strength of permeable pavements limit their application for the parking lots where heavy equipment tools are parked. Also, the resistance of these pavements must be improved to withstand high-speed traffic. Another potential for mechanical property improvement is the permeable pavement’s resistance against freeze-thaw as these pavements can capture and store water particularly when they are poorly constructed, and the infiltration process in them takes longer than it should normally take. Also, the clogging of these pavements over time and as a result of exposure to fine particles reduced their functionality [41]. Figure 10 shows four potentials for improvement in the permeable pavements that require further research.
\n
Figure 10.
Potentials for pervious asphalt pavements’ mechanical property advancement.
\n
\n
\n
\n
\n
7. Summary and discussion
\n
This chapter discusses the instrumental role of expanding asphalt pavements in nations’ growth and development and the enormous and ever-rising demand to asphalt. It was stated that although asphalt production, in comparison, with other types of pavements is not considered an unsustainable industry, due to massive volumes of its production, it can have significant impacts on the sustainability triple bottom line and, more particularly, on the environment. The main sustainability in this regard was consumption of virgin materials, energy, and water as well as emission of pollutants. A key solution for pushing asphalt industry was suggested to be a comprehensive life-cycle analysis to identify all the possible potentials to implement sustainable practices in the four phases of material procurement, processing, transportation, and end of life point. Some policy recommendations were provided in each category. This chapter also discusses the opportunities for further advancement of the current sustainable approaches in the asphalt industry. The two main approaches were warm mix asphalts and pervious asphalt pavements. It was explained that although these technologies have significant sustainability advantages, there are some drawbacks in their application that need to be resolved through a collaborative effort of the industry and research. Some of the potentials of mechanical improvements of the two technologies were mentioned. It is essential that the individuals involved in the asphalt industry understand the essentiality of applying sustainable practices, gain enough knowledge and expertise of how to implement them, and then pursue the application of sustainable practices throughout the production process. Researchers, on the other hand, are responsible to target mitigating the existing environmental risks of asphalt pavements.
\n
\n\n',keywords:"asphalt pavement sustainability, warm mix, permeability, emission",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/67431.pdf",chapterXML:"https://mts.intechopen.com/source/xml/67431.xml",downloadPdfUrl:"/chapter/pdf-download/67431",previewPdfUrl:"/chapter/pdf-preview/67431",totalDownloads:436,totalViews:0,totalCrossrefCites:0,dateSubmitted:"January 15th 2019",dateReviewed:"May 6th 2019",datePrePublished:"July 13th 2019",datePublished:"December 11th 2019",dateFinished:"May 30th 2019",readingETA:"0",abstract:"Shrink of nonrenewable natural resources and the pollution generated by many manufacturing industries have initiated a global determination for pushing the industry toward more sustainable products. Asphalt as the principal element of almost any street and highway pavement is integral in transportation development, which, in turn, is pivotal for sustainable development. On the other hand, the material consumption and pollution generated in the asphalt industry pose significant threats to the environment and, therefore, to sustainability. This chapter reflects some of the sustainability concerns of the asphalt industry and discusses some of the possible solutions to mitigate them. The sustainability considerations are categorized into four phases of asphalt life cycle namely: (1) extraction of materials, (2) processing of raw materials, (3) transportation of asphalt raw materials and products, and (4) reusing and recycling at the end of a pavement’s life. In each phase, best practices to improve and maintain the sustainability of asphalt pavements are discussed. This chapter also discusses sustainable approaches and technologies in the asphalt industry such as warm mixes, permeable asphalt pavements as well as the potentials for improving the mechanical properties of asphalt pavements particularly in terms of resisting heavy load traffic, clogging, and freeze-thaw.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/67431",risUrl:"/chapter/ris/67431",signatures:"Mohammadsoroush Tafazzoli",book:{id:"8747",title:"Asphalt and Asphalt Mixtures",subtitle:null,fullTitle:"Asphalt and Asphalt Mixtures",slug:"asphalt-and-asphalt-mixtures",publishedDate:"December 11th 2019",bookSignature:"Haitao Zhang",coverURL:"https://cdn.intechopen.com/books/images_new/8747.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",isbn:"978-1-78984-769-7",printIsbn:"978-1-78984-768-0",pdfIsbn:"978-1-83968-109-7",editors:[{id:"260604",title:"Prof.",name:"Haitao",middleName:null,surname:"Zhang",slug:"haitao-zhang",fullName:"Haitao Zhang"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:[{id:"283849",title:"Ph.D.",name:"Mohammadsoroush",middleName:null,surname:"Tafazzoli",fullName:"Mohammadsoroush Tafazzoli",slug:"mohammadsoroush-tafazzoli",email:"tafazzoli@wsu.edu",position:null,institution:null}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Is asphalt an unsustainable product by itself?",level:"1"},{id:"sec_3",title:"3. How does the asphalt industry affect the environment?",level:"1"},{id:"sec_4",title:"4. What are the main approaches in improving the asphalt industry’s sustainability?",level:"1"},{id:"sec_5",title:"5. Mitigating asphalt’s negative environmental impacts in its life cycle",level:"1"},{id:"sec_5_2",title:"5.1 Material procurement (phase 1)",level:"2"},{id:"sec_5_3",title:"5.1.1 Extraction of materials",level:"3"},{id:"sec_7_2",title:"5.2 Processing of raw materials (phase 2)",level:"2"},{id:"sec_7_3",title:"5.2.1 Energy consumption reduction",level:"3"},{id:"sec_8_3",title:"Table 1.",level:"3"},{id:"sec_10_2",title:"5.3 Transportation of asphalt raw materials and products (phase 3)",level:"2"},{id:"sec_11_2",title:"5.4 End of a product’s life (phase 4)",level:"2"},{id:"sec_11_3",title:"5.4.1 Reclaimed asphalt pavements (RAP)",level:"3"},{id:"sec_12_3",title:"5.4.2 Recycled asphalt shingles (RAS)",level:"3"},{id:"sec_12_4",title:"5.4.2.1 Waste plastic",level:"4"},{id:"sec_16",title:"6. Sustainable approaches in the asphalt industry",level:"1"},{id:"sec_16_2",title:"6.1 Warm mix asphalts",level:"2"},{id:"sec_16_3",title:"Table 5.",level:"3"},{id:"sec_17_3",title:"6.1.2 Potentials for improvement in warm mix asphalt technology",level:"3"},{id:"sec_19_2",title:"6.2 Reducing the transfer of pollutants through runoffs by using permeable asphalt pavements",level:"2"},{id:"sec_19_3",title:"6.2.1 The sustainability advantages of using permeable asphalt pavements",level:"3"},{id:"sec_20_3",title:"6.2.2 Research status of permeable asphalt pavements and filling the knowledge gap",level:"3"},{id:"sec_23",title:"7. Summary and discussion",level:"1"}],chapterReferences:[{id:"B1",body:'Santero NJ, Horvath A. Global warming potential of pavements. Environmental Research Letters. 2009;4(3):034011\n'},{id:"B2",body:'Asphalt Pavement Alliance. Asphalt is the Sustainable Material for Constructing Pavements. 2019. Available from: http://www.asphaltroads.org/why-asphalt/environment/ [Accessed: April 14, 2019]\n'},{id:"B3",body:'Mazumder M, Sriraman V, Kim HH, Lee SJ. Quantifying the environmental burdens of the hot mix asphalt (HMA) pavements and the production of warm mix asphalt (WMA). International Journal of Pavement Research and Technology. 2016;9(3):190-201\n'},{id:"B4",body:'Rebitzer G, Ekvall T, Frischknecht R, Hunkler D, Norris T, Rydberg W, et al. Life cycle assessment, part 1: Framework, goal and scope definition, inventory analysis, and applications. Environment International. 2004;30:701-720\n'},{id:"B5",body:'Mukherjee A. Life cycle assessment of asphalt mixtures in support of an environmental product declaration. In: National Asphalt Pavement Institution. Maryland: Lanham; 2016\n'},{id:"B6",body:'Zapata P, Gambatese JA. Energy consumption of asphalt and reinforced concrete pavement materials and construction. Journal of Infrastructure Systems. 2005;11(1):9-20\n'},{id:"B7",body:'Butt AA, Mirzadeh I, Toller S, Birgisson B. Life cycle assessment framework for asphalt pavements: Methods to calculate and allocate energy of binder and additives. International Journal of Pavement Engineering. 2014;15(4):290-302\n'},{id:"B8",body:'Dinis-Almeida M, Afonso ML. Warm mix recycled asphalt—A sustainable solution. Journal of Cleaner Production. 2015;107:310-316\n'},{id:"B9",body:'D’Angelo J, Harm E, Bartoszek J, Baumgardner G, Corrigan M, Cowsert J, et al. Warm-Mix Asphalt: Europe Practice. Federal Highway Administration. U.S. DoT, AASHTO, NCHRP, Report No. FHWA PL-08-007; Alexandria; 2008\n'},{id:"B10",body:'Capitão SD, Picado-Santos LG, Martinho F. Pavement engineering materials: Review on the use of warm-mix asphalt. Construction and Building Materials. 2012;36:1016-1024\n'},{id:"B11",body:'Carvalho T, Barreno IP. Ligantes betuminosos temperados. In: VII. Congresso Rodovi-Ario Portugues e Novos Desafios Para a Atividade Rodovi-Aria; 10-12 April 2013; Lisboa, Portugal (in Portuguese). 2013\n'},{id:"B12",body:'Miró R, Valdés G, Martínez A, Segura P, Rodríguez C. Evaluation of high modulus mixture behaviour with high reclaimed asphalt pavement (RAP) percentages for sustainable road construction. Construction and Building Materials. 2011;25(10):3854-3862\n'},{id:"B13",body:'Croteau JM. Performance and cost benefits of asphalt pavement recycling. In: Semi-Annual Meeting: Asphalt Recycling and Reclaiming Association (ARRA). 2005\n'},{id:"B14",body:'Chesner WH, Collins RJ, MacKay MH, Emery J. User Guidelines for Waste and by-Product Materials in Pavement Construction, No. FHWA-RD-97-148, Guideline Manual, Rept No. 480017. Recycled Materials Resource Center; 2002\n'},{id:"B15",body:'EAPA. Asphalt in Figures 2013. Brussels, Belgium: European Asphalt Pavement Association; 2014\n'},{id:"B16",body:'Silva HM, Oliveira JR, Jesus CM. Are totally recycled hot mix asphalts a sustainable alternative for road paving? Resources, Conservation and Recycling. 2012;60:38-48\n'},{id:"B17",body:'Lake R. Bottled Water Statistics: 23 Outrageous Facts. 2015. Available from: https://www.creditdonkey.com/bottled-water-statistics.html [Accessed: April 14, 2019]\n'},{id:"B18",body:'Ahmadinia E, Zargar M, Karim MR, Abdelaziz M, Ahmadinia E. Performance evaluation of utilization of waste polyethylene terephthalate (PET) in stone mastic asphalt. Construction and Building Materials. 2012;36:984-989\n'},{id:"B19",body:'Rahman WMNWA, Wahab AFA. Green pavement using recycled polyethylene terephthalate (PET) as partial fine aggregates replacement in modified asphalt. Procedia Engineering. 2013;53:124-128\n'},{id:"B20",body:'EAPA. The Use ofWarm Mix Asphalt. Brussels, Belgium: European Asphalt Pavement Association; 2010\n'},{id:"B21",body:'Oliveira JRM, Silva HMRD, Abreu LPF, Fernades SRM. Use of a warm mix asphalt additive to reduce the production temperatures and to improve the performance of asphalt rubber mixtures. Journal of Cleaner Production. 2013;41:15-22\n'},{id:"B22",body:'Hassan M. Evaluation of the environmental and economic impacts of warm-mix asphalt using life-cycle assessment. International Journal of Construction Education and Research. 2010;6(3):238-250\n'},{id:"B23",body:'Johnston A, Da Silva M, Soeymani H, Yeung CK. An early evaluation of the initial Canadian experience with warm asphalt mix. In: Annual General Conference of the Canadian Society for Civil Engineering; Calgary, Alberta, Canada. 2006\n'},{id:"B24",body:'Newcomb D. Warm mix: The wave of the future. In: HMAT: Hot Mix Asphalt Technology. Vol. 10. 2005. p. 4\n'},{id:"B25",body:'Hurley GC, Prowell BD. Evaluation of potential processes for use in warm mix asphalt. Journal of the Association of Asphalt Paving Technologists. 2006;75:41-90\n'},{id:"B26",body:'Wang H, Liu X, Apostolidis P, Scarpas T. Review of warm mix rubberized asphalt concrete: Towards a sustainable paving technology. Journal of Cleaner Production. 2018;177:302-314\n'},{id:"B27",body:'Epps LA. Thermal Behavior of Crumb-Rubber Modified Asphalt Concrete Mixtures. Berkeley, California, USA: Civil Engineering, Univeristy of California; 1997\n'},{id:"B28",body:'Lo Presti D. Recycled tyre rubber modified bitumens for road asphalt mixtures: A literature review. Construction and Building Materials. 2013;49:863-881\n'},{id:"B29",body:'Shu X, Huang BS. Recycling of waste tire rubber in asphalt and Portland cement concrete: An overview. Construction and Building Materials. 2014;67:217-224\n'},{id:"B30",body:'Kheradmand B, Muniandy R, Hua LT, Yunus R, Solouki A. An overview of the emerging warm mix asphalt technology. International Journal of Pavement Engineering. 2014;15(1):79-94\n'},{id:"B31",body:'Prowell B, Hurley G, Frank B. Warm-Mix Asphalt: Best Practices. 2nd ed. National Asphalt Pavement Association: Quality Improvement Publication; 2008. p. 125\n'},{id:"B32",body:'Rubio MC, Martínez G, Baena L, Moreno F. Warm mix asphalt: An overview. Journal of Cleaner Production. 2012;24:76-84\n'},{id:"B33",body:'Zaumanis M. Warm mix asphalt investigation [Master of Science thesis]. Kgs. Lyngby: Technical University of Denmark in cooperation with the Danish Road Institute, Department of Civil Engineering; 2010\n'},{id:"B34",body:'Teixeira A. Caracterização da resistência a fadiga de misturas betuminosas em equipamento servopneum atico [MSc thesis] (in Portuguese). Porto, Portugal: Faculdade de Engenharia da Universidade do Porto; 2000\n'},{id:"B35",body:'Dinis-Almeida M, Castro-Gomes J, de Lurdes Antunes M. Mix design considerations for warm mix recycled asphalt with bitumen emulsion. Construction and Building Materials. 2012;28(1):687-693\n'},{id:"B36",body:'Guo N, You Z, Zhao Y, Tan Y, Diab A. Laboratory performance of warm mix asphalt containing recycled asphalt mixtures. Construction and Building Materials. 2014;64:141-149\n'},{id:"B37",body:'Sengoz B, Oylumluoglu J. Utilization of recycled asphalt concrete with different warm mix asphalt additives prepared with different penetration grades bitumen. Construction and Building Materials. 2013;45:173-183\n'},{id:"B38",body:'Tafazzoli M. Investigating the Impacts of Green Roofs’ Vegetation Properties on their Function in Controlling Urban Runoffs. International Low Impact Development Conference2018. pp. 176-183\n'},{id:"B39",body:'Driscoll E, Shelley PE, Strecker EW. Pollutant Loadings and Impacts from Highway Stormwater Runoff, Vol. I–IV, FHWA/RD-88-006-9. Oakland, Calif: Federal Highway Administration, Woodward-Clyde Consultants; 1990\n'},{id:"B40",body:'USEPA. Protecting Water Quality from Urban Runoff. EPA 841- F-03-003. Washington, DC: USEPA; 2003\n'},{id:"B41",body:'Tafazzoli M. Enhancing the Functionality of Pervious Concrete Pavements through Design and Maintenance. International Low Impact Development Conference2018. pp. 184-192\n'},{id:"B42",body:'Tennis PD, Leming ML, Akers DJ. Pervious Concrete Pavements. No. PCA, Serial No. 2828. Skokie, IL: Portland Cement Association; 2004\n'},{id:"B43",body:'Bates B, Kundzewicz Z, Wu S. Climate Change and Water. Intergovernmental Panel on Climate Change Secretariat. Geneva: IPCC Secretariat; 2008. p. 210\n'},{id:"B44",body:'United Nations. Population Distribution, Urbanization, Internal Migration and Development: An International Perspective. Department of Economic and Social Affairs Population Division; 2011\n'},{id:"B45",body:'Lu D, Weng Q. Use of impervious surface in urban land-use classification. Remote Sensing of Environment. 2006;102(1–2):146-160\n'},{id:"B46",body:'Tafazzoli M. 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However, several infectious diseases caused by many phytopathogens are deteriorating mango quality and quantity. Mango tree and fruit have been affected by about 83 diseases reported worldwide, and in Pakistan, 27 diseases are recognized as more important. Disease control always remains a challenge for the farmers to get optimum production especially due to pesticide resistance. Resistance to fungicide in current days is a major threat to plant disease management. In many cases, plant pathogen resistance could develop naturally; thus, several newly developed chemistries of fungicides remain at high risk. However, research toward an increase of resistance and delay in disease development has been undertaken. Existing fungicide chemistry, sometimes, renamed with new trade name does not satisfy the farmer to apply such fungicides for disease management. However, chemical fungicides are believed to be a significant way to control fungal pathogens or sometimes to inhibit and prevent the development of pathogens. However, due to pathogen resistance development, it is hard to manage plant diseases. 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