The Diffraction of Microparticles on Single-layer and Multi-layer Statistically Uneven Surfaces
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ The Diffraction of Microparticles on Single-layer and Multi-layer Statistically Uneven Surfaces
Mikhail Batanov-Gaukhman Ph.D., Associate Professor, Institute No. 2 “Aircraft, rocket engines and power plants”, Federal State Budgetary Educational Institution of Higher Education “Moscow Aviation Institute (National Research University)”, Volokolamsk highway 4, Moscow, Russian Federation
Abstract:
In this article: a ) a method is developed for calculating volumetric diagrams of elastic scattering of microparticles (in particular, electrons and photons) on single-layer and multi-layer statistically uneven surfaces; b ) the diffraction of elementary particles on crystals is explained without involving de Broglie's idea of the wave properties of matter; c ) the probability density functions of the derivative of various stationary random processes are obtained; d ) volumetric dia-grams of the scattering of particles and photons on homogeneous and isotropic uneven surfaces with Gaussian, uniform, Laplace, sinusoidal, and other distributions of unevenness are obtained. Keywords: electron diffraction on a crystal, particle scattering diagram, wave scattering diagram, volumetric scattering diagram, statistically uneven surface, derivative of a stationary random pro-cess, Kirchhoff approximation 03.65.−w (Quantum mechanics) 05.30.−d (Quantum statistical mechanics) [email protected] M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
In this article: a ) a method is developed for calculating volumetric diagrams of elastic scattering of microparticles (in particular, electrons and photons) on single-layer and multi-layer statistically uneven surfaces; b ) the diffraction of elementary particles on crystals is explained without involving de Broglie's idea of the wave properties of matter; c ) the probability density functions of the derivative of various stationary random processes are obtained; d ) volumetric dia-grams of the scattering of particles and photons on homogeneous and isotropic uneven surfaces with Gaussian, uniform, Laplace, sinusoidal, and other distributions of unevenness are obtained. Keywords: electron diffraction on a crystal, particle scattering diagram, wave scattering diagram, volumetric scattering diagram, statistically uneven surface, derivative of a stationary random pro-cess, Kirchhoff approximation 03.65.−w (Quantum mechanics) 05.30.−d (Quantum statistical mechanics) [email protected] M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
In 1924, Louis de Broglie suggested that a uniformly and rectilinearly moving particle with mass m and velocity v can be associated with a plane wave = exp i ( Et – pr ) /h , (1.1) where E is the kinetic energy of the particle; p = m v is its momentum; h is Planck's constant. The length of such a monochromatic wave is determined by the de Broglie formula λ b = h/m v. (1.2) This idea served as the basis for the development of wave-particle duality and, in particular, made it possible to explain a number of experiments on the diffraction of electrons, neutrons, and atoms by crystals and thin films [1, 2]. Since then, it has been assumed that the diffraction maxima in the Dewisson and Germer experiment appear in directions that meet the Wolfe - Bragg condi-tion ebs nd = sin2 , or taking into account the refraction of “electron waves” in a crystal [1]: ( ) ebse nnd =− cos2 , (1.3) where d is the interplanar distance of the crystal lattice, θ s is the Bragg’s glancing angle (Figure 1), n = 1, 2, 3 ... is the order of interference (or reflection), λ eb is the de Broglie electron wavelength, n e is the refractive index of the de Broglie electron wave. Fig. 1
Wulff - Bragg’s condition for diffraction of microparticles (in particular, electrons or photons) on the surface of a crystal. is the direction of motion of the falling microparticles; is the direction of movement of the reflected microparticles
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ However, over the past 95 years, de Broglie waves have not been detected experimentally. They remained an auxiliary mental construction, which allows one to describe the phenomenon mathematically, without revealing the essence of the events occurring in this case. This article shows that the diffraction of microparticles on a crystal can be described without involving de Broglie's idea of the wave properties of matter. Based on the laws of reflection in geometric optics and probability theory, at the end of this article, the formula (3.9) [or (3.10)] is obtained for calculating the diagram of elastic scattering of microparticles (DESM) on a multilayer crystal surface. The results of calculations using this for-mula are consistent with experimentally obtained electron diffraction patterns (EDP) (Figure 1a). а ) b ) Fig. 1a a ) The volumetric diagram of elastic scattering of microparticles on a multilayer crystal surface, obtained as a result of calculations by the formula (3.9); b In addition, this article develops a method for calculating volumetric diagrams of elastic scattering of microparticles (DESM) on uneven surfaces with various statistics of the heights of the irregularities. By “microparticles” in this paper we mean any particles (fermions and bosons) whose sizes (or wavelength) are much smaller than the characteristic sizes of the irregularities of the reflecting 4
M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________ surface (Kirchhoff approximation), and whose reflection occurs according to the laws of geometric optics. For example, an electron can be called a "microparticle" with an effective size of about 10 –13 cm, which is reflected from the surface of a crystal with characteristic sizes of irregularities greater than 10 –11 cm. Also, a football can be considered a "microparticle" with a diameter of about 22.3 cm, reflected from an uneven solid surface, the average radius of curvature of which is more than 20 m. "Microparticles" also include photons and phonons with a wavelength λ two or-ders of magnitude smaller than the autocorrelation radius of the heights of the reflecting surface irregularities (see Appendix 1). Extensive literature is devoted to the scattering of particles and waves on the uneven (rough) boundary of two media, for example, [3 – 27]. However, the formulas for calculating volumetric diagrams of scattering of particles or waves on surfaces with different statistics of roughness (ir-regularities) heights in the case of the Kirchhoff approximation are practically absent in the litera-ture. Extensive literature is devoted to the scattering of particles and waves on the uneven (rough) boundary of two media, for example, [3 – 27]. However, the formulas for calculating volumetric diagrams of scattering of particles or waves on surfaces with different statistics of roughness (ir-regularities) -- in the case of the Kirchhoff approximation -- are practically absent in the literature. This article presents for the first time volumetric diagram of elastic scattering of microparti-cles on homogeneous and isotropic uneven surfaces with Gaussian, uniform, Laplace, sinusoidal, and other distributions of height of irregularities. Data from the scattering diagram refer to any of the above microparticles (fermions and bosons). The purpose of this section of the article is to develop a method for calculating volumetric dia-grams of elastic scattering of microparticles (DESM) (in particular, electrons, photons or phonons) on statistically uneven surfaces under the conditions of the Kirchhoff approximation (i.e., when the averaged radius of curvature or the radius of autocorrelation of irregularities reflecting surface is much larger than the size or wavelength of the microparticles).
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ Consider the incidence of microparticles on the surface of a solid (or liquid) body (Figure 2) at the angles ϑ and γ (Figure 3), and their reflection from this surface at the angles ν and ω . Fig. 2
Scattering of microparticles (in particular, electrons or photons, that is, a ray of light) on a reflective surface, where: 1 is a microparticle generator; 2 is microparticle detector; 3 is solid or liquid body (in particular, a metal crystal or a volume of water) Let's imagine the upper layer of a body as a two-dimensional statistically uneven surface ( x , y ), which repeats the structure of its atomic lattice (Figure 3), or the excitement of a liquid. It is known that elastic particles (or waves) moving at the speed of v are reflected from the smooth surface of a solid (or liquid) body according to the laws of geometric optics (specular re-flection): 1) the incident particle (or light ray), reflected particle (or light ray) and the perpendicu-lar (normal) to the two media border in the point of particle (or light ray) incidence lie in one plane; 2) the angle of incidence Q is equal to angle of reflection Q . This phenomenon is called “specular reflection” or “elastic scattering” of microparticles. Under the condition of elastic scattering, the fact that the two-dimensional probability densi-ty function (TPDF) ρ ( ν , ω ) of the microparticle is reflected from the uneven reflecting surface at 6 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Scattering of microparticles (in particular, electrons or photons, that is, a ray of light) on a reflective surface, where: 1 is a microparticle generator; 2 is microparticle detector; 3 is solid or liquid body (in particular, a metal crystal or a volume of water) Let's imagine the upper layer of a body as a two-dimensional statistically uneven surface ( x , y ), which repeats the structure of its atomic lattice (Figure 3), or the excitement of a liquid. It is known that elastic particles (or waves) moving at the speed of v are reflected from the smooth surface of a solid (or liquid) body according to the laws of geometric optics (specular re-flection): 1) the incident particle (or light ray), reflected particle (or light ray) and the perpendicu-lar (normal) to the two media border in the point of particle (or light ray) incidence lie in one plane; 2) the angle of incidence Q is equal to angle of reflection Q . This phenomenon is called “specular reflection” or “elastic scattering” of microparticles. Under the condition of elastic scattering, the fact that the two-dimensional probability densi-ty function (TPDF) ρ ( ν , ω ) of the microparticle is reflected from the uneven reflecting surface at 6 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________ angles ν , ω and corresponds to the fact that the TPDF ρ ( , φ ) of the unit vector normal to the sur-face n , at the point where the microparticle falls, will be directed at angles , φ (Figures 3,4,5). Therefore, from the TPDF ρ ( , φ ) it is possible to obtain the TPDF ρ (ν,ω/ ϑ , γ ) using the trans-formation of variables: ρ ( , φ ) = ρ { =f ( ν , ω / ϑ , γ ); φ=f ( ν , ω / ϑ , γ )} | G νω | = ρ ( ν , ω / ϑ , γ )| G νω |, (2.1) where = f ( ν , ω / ϑ , γ ) (2.2) is the functional relationship between the angle and the angles ν , ω, for given angles ϑ, γ ; φ = f ( ν , ω / ϑ , γ ) (2.3) is the functional relationship between the angle φ and the angles ν , ω, for given angles ϑ, γ ; ρ { ν , ω / ϑ , γ }| G νω | is the TPDF of the microparticle which is reflected from an uneven surface in the direction given by the angles ν , ω , if angles ϑ , γ are known; | G νω | is the Jacobian of the transformation of the variables , φ into the variables ν , ω . In this case, the probability that a particle whose initial direction of motion is given by the angles ϑ and γ will be reflected from the surface in a direction limited by the angular ranges dν and dω is P ( ν,ω ) = ρ ( ν,ω/ϑ,γ ) |G νω |dνdω . This formula essentially shows what portion of the total number of microparticles (or the to-tal wave energy) that fall on the reflective surface is scattered in the direction given by the angles ν and ω within the element of the solid angle dΩ=dνdω . If the generator and the detector of microparticles are located at a large distance from the considered part of the reflecting surface (Figure 2), then the TPDF ρ ( ν , ω / ϑ , γ )| G νω | (2.1) determines the volumetric diagram of elastic scattering of these particles on this surface D ( ν , ω / ϑ , γ ) = ρ ( ν , ω / ϑ , γ ) |G νω | . (2.4) Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ Fig. 3
Area of uneven surface, reflecting microparticles, where: ϑ , γ are the angles defining the direction of microparticle incidence on a reflecting surface; ν , ω are the angles defining the direction of reflection of the microparticle from this surface; a f is a unit vector indicating the direction to the microparticle generator; n is the unit normal vector to the surface at the point where the microparticle incidence; a r is a unit vector indicating the direction of motion of the microparticle after an elastic colli-sion with a reflective surface Fig. 4
Specular reflection of a microparticle from a portion of an uneven surface according to the laws of geometric optics: 1) elastic reflection of a particle (or light ray) occurs in the plane of its incidence; 2) the angle of reflection of the particle (or ray of light) Q is equal to the angle of its incidence Q (i.e., the condition Q = Q is satisfied) 8 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Specular reflection of a microparticle from a portion of an uneven surface according to the laws of geometric optics: 1) elastic reflection of a particle (or light ray) occurs in the plane of its incidence; 2) the angle of reflection of the particle (or ray of light) Q is equal to the angle of its incidence Q (i.e., the condition Q = Q is satisfied) 8 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Fig. 5
Illustration for determining the functional relationship between angles , φ and angles ν, ω , if angles ϑ , γ are known: where Q = Q and vectors naa rf ,, lie in the same plane AOB In the case where the microparticle detector is located at a short distance from the consid-ered part of the reflecting surface, then in order to find the volumetric DESM D ( ν , ω / ϑ , γ ), the right part of Expression (2.4) should be integrated over all angles ν and ω , along which reflected micro-particles can enter the detector aperture (or at one point on the plate of the electron diffraction pat-tern or radiographs). ( ) ( ) .,/,,/,
21 21 ddGD = (2.5) This case is not considered in this article. That is, in the future we will assume that the gen-erator and the detector of microparticle are so far from the reflecting surface area that it is permis-sible to use the simplified formula (2.4). Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ , φ and ν , ω / ϑ , γ Let's find the functional relationships (2.2) and (2.3). Figure 5 shows the unit vectors ,,, naa rf whose tails coincides with the origin of the local reference system XYZ (located at the point of collision of the microparticle with the surface), and their heads are given by the following coordi-nates: sin,coscos,sincos,, == fzfyfxf aaaa (2.6) –- a unit vector, indicating the direction on the microparticle generator (Figure 2 and 3); sin,coscos,sincos,, == rzryrxr aaaa (2.7) – a unit vector, indicating the direction of movement of a microparticles after an elastic collision with a reflecting surface. sin,coscos,sincos,, == rzryx nnnn – a unit normal vector to the surface at the point of incidence of the microparticle; Figure 5 shows that when the laws of geometric optics are satisfied (i.e., when Q = Q ), the normal vector n determines the direction of the bisector of the isosceles triangle AOB whose sides are the unit vectors a f and a r , Obviously by setting the coordinates of the point N that divides the segment AB in half, we get the coordinates of the head of the vector b , the direction of which coincides with that of the normal vector n . Using the coordinates of the head of the vector a r (2.6) and the head of the vector a r (2.7), and based on the methods of analytical geometry [29 - 31], we obtain .2 sinsin,2 coscoscoscos,2 sincossincos,, +++== zyx bbbb From the scalar product of the vectors b and k = {0,0,1} (where k indicates the direction of the OZ axis, see Figure 5) ( ) sin)2/cos( kbkbkb =−= , we define the angle ( ) ( ) ( ) ( ) ,sinsincoscoscoscossincossincos sinsinarcsinarcsin +++++ += = kb kb (2.8) which is the desired functional relationships (2.2). 10 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
21 21 ddGD = (2.5) This case is not considered in this article. That is, in the future we will assume that the gen-erator and the detector of microparticle are so far from the reflecting surface area that it is permis-sible to use the simplified formula (2.4). Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ , φ and ν , ω / ϑ , γ Let's find the functional relationships (2.2) and (2.3). Figure 5 shows the unit vectors ,,, naa rf whose tails coincides with the origin of the local reference system XYZ (located at the point of collision of the microparticle with the surface), and their heads are given by the following coordi-nates: sin,coscos,sincos,, == fzfyfxf aaaa (2.6) –- a unit vector, indicating the direction on the microparticle generator (Figure 2 and 3); sin,coscos,sincos,, == rzryrxr aaaa (2.7) – a unit vector, indicating the direction of movement of a microparticles after an elastic collision with a reflecting surface. sin,coscos,sincos,, == rzryx nnnn – a unit normal vector to the surface at the point of incidence of the microparticle; Figure 5 shows that when the laws of geometric optics are satisfied (i.e., when Q = Q ), the normal vector n determines the direction of the bisector of the isosceles triangle AOB whose sides are the unit vectors a f and a r , Obviously by setting the coordinates of the point N that divides the segment AB in half, we get the coordinates of the head of the vector b , the direction of which coincides with that of the normal vector n . Using the coordinates of the head of the vector a r (2.6) and the head of the vector a r (2.7), and based on the methods of analytical geometry [29 - 31], we obtain .2 sinsin,2 coscoscoscos,2 sincossincos,, +++== zyx bbbb From the scalar product of the vectors b and k = {0,0,1} (where k indicates the direction of the OZ axis, see Figure 5) ( ) sin)2/cos( kbkbkb =−= , we define the angle ( ) ( ) ( ) ( ) ,sinsincoscoscoscossincossincos sinsinarcsinarcsin +++++ += = kb kb (2.8) which is the desired functional relationships (2.2). 10 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Figure 5 shows that φ is the angle between vectors j = {0,1,0} and c , where j defines the di-rection of the axis OY , and vector c is the projection of vector b onto the XOY plane. .0,2 coscoscoscos,2 sincossincos0,, ++== yx ссc (2.9) From the scalar product of the vectors ( ) cos jcjc = , we define the angle φ ( ) ( ) ( ) ,coscoscoscossincossincos coscoscoscosarccosarccos +++ += = jc jc (2.10) which is the second desired functional relationship (2.3). , φ into the variables ν , ω for given ϑ , γ Let's introduce the notations a = cos ν cos ω + cos ϑ cos γ ; b = cos ν sin ω + cos ϑ sin γ ; d = sin ν + sin ϑ ; a ν = – sin ν cos ω ; b ν = – sin ν sin ω ; c ν = cos ν ; a ω = – cos ν sin ω ; b ω = cos ν cos ω . (2.11) In this case, expressions (2.8) and (2.10), taking into account the polysemy of inverse trigonomet-ric functions, take the form ( ) ,arcsin1 ++−+= dba dm m (2.12) ,arccos2 += ba am (2.13) where m = 0, 1, 2, 3, … From Figures 2, 3, 5 it can be seen that the angles , φ can take the values [0, π/2], φ [0, π]. Let's also take into account that the principal branches of inverse trigonometric functions are enclosed within: arcsin( x ) [– π/2, π/2], arccos( x ) [0, π]. Therefore, in expressions (2.12) and (2.13) we assume m = 0 and choose (+), and as a result we obtain unambiguous functional rela-tionships Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ ( ) ,arcsin,/, ++== dba df (2.14) ( ) .arccos,/, +== ba af (2.15) Let's find the Jacobian of the transformation | G νω | variables , φ into variables ν , ω . To do this, we calculate the determinant of the matrix [32, 33] . −= = ffffff ffG (2.16) Substituting functions (2.14) and (2.15) into the determinant (2.16), and taking into account the notation (2.11), we obtain the desired Jacobian of the transformation ( ) ( ) ( ) . dbaba baabcbabadG +++ −+−= (2.17) This result was obtained by the author together with Dr. S.V. Kostin. ρ ( , φ ) Let’s obtain the ТPDF ρ ( , φ ) (2.1). To do this, represent a homogeneous and isotropic uneven reflecting surface as a two-dimensional stationary random process ( x , y ) of changing the height of the irregularities (Figure 3). Any azimuthal cross-section (for example, along the Y axis) of the process ( x , y ) is a one-dimensional stationary random process ( y ). Suppose we know the one-dimensional probability density function (OPDF) ρ [ ( y )] of the heights of irregularities ( y ). It will be shown below that on the basis of the OPDF ρ [ ( y )], it is possible to obtain the OPDF ρ [ ( y )] of the derivative of this stationary random process ( y ). Taking into account that ( y ) = tg , where is the angle between the tangent to the process ( y ) and the Y axis (see Figure 4), we make in ρ [ ( y )] the change of the variable to . As a result, we obtain the OPDF of angles in the azimuthal section ( y ) under study 12 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Figure 5 shows that φ is the angle between vectors j = {0,1,0} and c , where j defines the di-rection of the axis OY , and vector c is the projection of vector b onto the XOY plane. .0,2 coscoscoscos,2 sincossincos0,, ++== yx ссc (2.9) From the scalar product of the vectors ( ) cos jcjc = , we define the angle φ ( ) ( ) ( ) ,coscoscoscossincossincos coscoscoscosarccosarccos +++ += = jc jc (2.10) which is the second desired functional relationship (2.3). , φ into the variables ν , ω for given ϑ , γ Let's introduce the notations a = cos ν cos ω + cos ϑ cos γ ; b = cos ν sin ω + cos ϑ sin γ ; d = sin ν + sin ϑ ; a ν = – sin ν cos ω ; b ν = – sin ν sin ω ; c ν = cos ν ; a ω = – cos ν sin ω ; b ω = cos ν cos ω . (2.11) In this case, expressions (2.8) and (2.10), taking into account the polysemy of inverse trigonomet-ric functions, take the form ( ) ,arcsin1 ++−+= dba dm m (2.12) ,arccos2 += ba am (2.13) where m = 0, 1, 2, 3, … From Figures 2, 3, 5 it can be seen that the angles , φ can take the values [0, π/2], φ [0, π]. Let's also take into account that the principal branches of inverse trigonometric functions are enclosed within: arcsin( x ) [– π/2, π/2], arccos( x ) [0, π]. Therefore, in expressions (2.12) and (2.13) we assume m = 0 and choose (+), and as a result we obtain unambiguous functional rela-tionships Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ ( ) ,arcsin,/, ++== dba df (2.14) ( ) .arccos,/, +== ba af (2.15) Let's find the Jacobian of the transformation | G νω | variables , φ into variables ν , ω . To do this, we calculate the determinant of the matrix [32, 33] . −= = ffffff ffG (2.16) Substituting functions (2.14) and (2.15) into the determinant (2.16), and taking into account the notation (2.11), we obtain the desired Jacobian of the transformation ( ) ( ) ( ) . dbaba baabcbabadG +++ −+−= (2.17) This result was obtained by the author together with Dr. S.V. Kostin. ρ ( , φ ) Let’s obtain the ТPDF ρ ( , φ ) (2.1). To do this, represent a homogeneous and isotropic uneven reflecting surface as a two-dimensional stationary random process ( x , y ) of changing the height of the irregularities (Figure 3). Any azimuthal cross-section (for example, along the Y axis) of the process ( x , y ) is a one-dimensional stationary random process ( y ). Suppose we know the one-dimensional probability density function (OPDF) ρ [ ( y )] of the heights of irregularities ( y ). It will be shown below that on the basis of the OPDF ρ [ ( y )], it is possible to obtain the OPDF ρ [ ( y )] of the derivative of this stationary random process ( y ). Taking into account that ( y ) = tg , where is the angle between the tangent to the process ( y ) and the Y axis (see Figure 4), we make in ρ [ ( y )] the change of the variable to . As a result, we obtain the OPDF of angles in the azimuthal section ( y ) under study 12 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________ ρ ( ) = ρ (tg ) cos1 , (2.18) where cos1 = G is the Jacobian of the transformation. Figure 4 shows that between the angles and there is a unambiguous unique functional dependence + + π /2 = π , whence it follows = π /2 – . (2.18) In view of this expression, we make in OPDF ρ ( ) (2.18) the change of the variable to ρ ( ) = ρ [tg(π/2 – )] ) /2(cos 1 − , with the Jacobian of the transformation .1 = G Take into account that tg( /2 – ) = ctg , cos( /2– ) = sin . As a result, we obtain the OPDF of angles ( ) ( ) ,sin1 ctg = (2.19) In the case of statistical independence of the angles and φ (which is typical for many une-ven surfaces), the joint ТPDF ρ ( , φ ) (2.1) can be represented as ( ) ( ) ( ) ( ) ( ) .sin1, ctg == (2.20) For homogeneous and isotropic statistically uneven surfaces, the angle φ, which determines the azimuthal direction of the projection of the normal to the XOY plane (see Figure 5), can be uniformly distributed in the interval from 0 to 2 π , and the OPDF ρ ( φ ) can be given by the expres-sion ( ) .21 = (2.21) Substituting (2.21) into (2.20), we obtain the TPDF (2.1) ( ) ( ) .sin121, ctg == (2.22) Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ If the reflecting surface is non-isotropic, then the OPDF ρ ( φ ) can be specified by another function, for example, ( ) ,sin2 = (2.23) or ( ) .cossin4 = (2.23a) In this case, the TPDF (2.1) will have the form ( ) ( ) ( ) .sin1)cos1(2sin1sin2, ctgctg −== (2.24) or ( ) ( ) ( ) .sin1cos)cos1(4sin1cossin4, ctgctg −== (2.24a) Note once again that the ТPDF ρ ( , φ ) (2.22), (2.24) and (2.24a) are obtained for the case where an uneven surface can be represented as a two-dimensional homogeneous random process of changing the heights of the irregularities ( x , y ) (see Figure 3). At each point with x , y coordi-nates, the random variable has the same averaged characteristics: OPDF, expected value, vari-ance, and other moments and central moments. The overall form of volumetric DESM is expressed by (2.4) D ( ν , ω / ϑ , γ ) = ρ ( ν , ω / ϑ , γ ) |G νω | = ρ { =f ( ν , ω / ϑ , γ ); φ=f ( ν , ω / ϑ , γ )}| G νω |. (2.25) In view of (2.17) and (2.22), expression (2.25) takes the form ( ) ( ) ( ) ( ) ( ) ( ) .,/,,/,sin 121,/, dbaba baabcbabadfctgfD +++ −+−==== We take into account that sin11 =+ ctg , whence follows −= ctg , therefore, this expression can be represented in the form ( ) ( ) ( ) ( ) ( ) ( ) .1,/,sin 1,/,sin 121,/, dbaba baabcbabadffD +++ −+− −=== (2.26) 14 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Figure 5 shows that φ is the angle between vectors j = {0,1,0} and c , where j defines the di-rection of the axis OY , and vector c is the projection of vector b onto the XOY plane. .0,2 coscoscoscos,2 sincossincos0,, ++== yx ссc (2.9) From the scalar product of the vectors ( ) cos jcjc = , we define the angle φ ( ) ( ) ( ) ,coscoscoscossincossincos coscoscoscosarccosarccos +++ += = jc jc (2.10) which is the second desired functional relationship (2.3). , φ into the variables ν , ω for given ϑ , γ Let's introduce the notations a = cos ν cos ω + cos ϑ cos γ ; b = cos ν sin ω + cos ϑ sin γ ; d = sin ν + sin ϑ ; a ν = – sin ν cos ω ; b ν = – sin ν sin ω ; c ν = cos ν ; a ω = – cos ν sin ω ; b ω = cos ν cos ω . (2.11) In this case, expressions (2.8) and (2.10), taking into account the polysemy of inverse trigonomet-ric functions, take the form ( ) ,arcsin1 ++−+= dba dm m (2.12) ,arccos2 += ba am (2.13) where m = 0, 1, 2, 3, … From Figures 2, 3, 5 it can be seen that the angles , φ can take the values [0, π/2], φ [0, π]. Let's also take into account that the principal branches of inverse trigonometric functions are enclosed within: arcsin( x ) [– π/2, π/2], arccos( x ) [0, π]. Therefore, in expressions (2.12) and (2.13) we assume m = 0 and choose (+), and as a result we obtain unambiguous functional rela-tionships Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ ( ) ,arcsin,/, ++== dba df (2.14) ( ) .arccos,/, +== ba af (2.15) Let's find the Jacobian of the transformation | G νω | variables , φ into variables ν , ω . To do this, we calculate the determinant of the matrix [32, 33] . −= = ffffff ffG (2.16) Substituting functions (2.14) and (2.15) into the determinant (2.16), and taking into account the notation (2.11), we obtain the desired Jacobian of the transformation ( ) ( ) ( ) . dbaba baabcbabadG +++ −+−= (2.17) This result was obtained by the author together with Dr. S.V. Kostin. ρ ( , φ ) Let’s obtain the ТPDF ρ ( , φ ) (2.1). To do this, represent a homogeneous and isotropic uneven reflecting surface as a two-dimensional stationary random process ( x , y ) of changing the height of the irregularities (Figure 3). Any azimuthal cross-section (for example, along the Y axis) of the process ( x , y ) is a one-dimensional stationary random process ( y ). Suppose we know the one-dimensional probability density function (OPDF) ρ [ ( y )] of the heights of irregularities ( y ). It will be shown below that on the basis of the OPDF ρ [ ( y )], it is possible to obtain the OPDF ρ [ ( y )] of the derivative of this stationary random process ( y ). Taking into account that ( y ) = tg , where is the angle between the tangent to the process ( y ) and the Y axis (see Figure 4), we make in ρ [ ( y )] the change of the variable to . As a result, we obtain the OPDF of angles in the azimuthal section ( y ) under study 12 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________ ρ ( ) = ρ (tg ) cos1 , (2.18) where cos1 = G is the Jacobian of the transformation. Figure 4 shows that between the angles and there is a unambiguous unique functional dependence + + π /2 = π , whence it follows = π /2 – . (2.18) In view of this expression, we make in OPDF ρ ( ) (2.18) the change of the variable to ρ ( ) = ρ [tg(π/2 – )] ) /2(cos 1 − , with the Jacobian of the transformation .1 = G Take into account that tg( /2 – ) = ctg , cos( /2– ) = sin . As a result, we obtain the OPDF of angles ( ) ( ) ,sin1 ctg = (2.19) In the case of statistical independence of the angles and φ (which is typical for many une-ven surfaces), the joint ТPDF ρ ( , φ ) (2.1) can be represented as ( ) ( ) ( ) ( ) ( ) .sin1, ctg == (2.20) For homogeneous and isotropic statistically uneven surfaces, the angle φ, which determines the azimuthal direction of the projection of the normal to the XOY plane (see Figure 5), can be uniformly distributed in the interval from 0 to 2 π , and the OPDF ρ ( φ ) can be given by the expres-sion ( ) .21 = (2.21) Substituting (2.21) into (2.20), we obtain the TPDF (2.1) ( ) ( ) .sin121, ctg == (2.22) Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ If the reflecting surface is non-isotropic, then the OPDF ρ ( φ ) can be specified by another function, for example, ( ) ,sin2 = (2.23) or ( ) .cossin4 = (2.23a) In this case, the TPDF (2.1) will have the form ( ) ( ) ( ) .sin1)cos1(2sin1sin2, ctgctg −== (2.24) or ( ) ( ) ( ) .sin1cos)cos1(4sin1cossin4, ctgctg −== (2.24a) Note once again that the ТPDF ρ ( , φ ) (2.22), (2.24) and (2.24a) are obtained for the case where an uneven surface can be represented as a two-dimensional homogeneous random process of changing the heights of the irregularities ( x , y ) (see Figure 3). At each point with x , y coordi-nates, the random variable has the same averaged characteristics: OPDF, expected value, vari-ance, and other moments and central moments. The overall form of volumetric DESM is expressed by (2.4) D ( ν , ω / ϑ , γ ) = ρ ( ν , ω / ϑ , γ ) |G νω | = ρ { =f ( ν , ω / ϑ , γ ); φ=f ( ν , ω / ϑ , γ )}| G νω |. (2.25) In view of (2.17) and (2.22), expression (2.25) takes the form ( ) ( ) ( ) ( ) ( ) ( ) .,/,,/,sin 121,/, dbaba baabcbabadfctgfD +++ −+−==== We take into account that sin11 =+ ctg , whence follows −= ctg , therefore, this expression can be represented in the form ( ) ( ) ( ) ( ) ( ) ( ) .1,/,sin 1,/,sin 121,/, dbaba baabcbabadffD +++ −+− −=== (2.26) 14 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Substituting the functional dependence (2.14) ,arcsin ++= dba d in (2.26), we obtain ( ) ( ) ( ) ( ) .21,/, dbaba baabcbabadd bad dbaD +++ −+− +++= Simplifying this expression, we find the overall form of volumetric DESM (2.25) ( ) ( ) ( ) ,21,/, ++ −+−== d babad baabcbabadD (2.27) where a, b, d, a ν , b ν , c ν , a ω , b ω are given by expressions (2.11). Note that formally in expression (2.27), the derivative was replaced by a quantity d ba + with the Jacobian of the transformation ( ) ( ) bad baabcbabadG + −+−= . In the case where the ТPDF ρ ( , φ ) has the form (2.24) or (2.24a) (i.e., when the reflecting surface is non-isotopic), then instead of the ТPDF (2.27), taking into account (2.15), we obtain ( ) ( ) ( ) ,2,/, ++ −+− += d babad baabcbabadba bD (2.28) or ( ) ( ) ( ) .)(4,/, ++ −+−+== d babad baabcbabadba baD (2.28a) Formulas (2.27), (2.28) and (2.28a) can be considered as DESM on a statistically uneven surface under the following conditions (see §2.1): - the uneven surface is statistically homogeneous; - the irregularities of this surface are quite smooth and large-scale in comparison with the size of the microparticles; the reflection of microparticles from all local sections of the uneven surface occurs according to the laws of geometric optics (see Figures 4 and 5); Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ - the portion of the uneven surface involved in the reflection of microparticles is located at a large distance from the generator and detector of microparticles (Figureу 2). ρ [ ( r )] derivative of the stationary random process ( r ) In § 2.4 it was shown that in order to determine the DESM on a statistically uneven reflecting sur-face ( x , y ), any cross-section of which is described by a stationary random process ( r ) (Figure 3), it is necessary to find the OPDF ρ [ ( r )] derivative of this process. To search for ρ [ ( r )], let's use the method proposed in [34, 35]. If the OPDF ρ ( ) of the one-dimensional stationary random process (SRP) ( r ) = is known, then the OPDF ρ ( ) of the derivative of this process can be obtained on the basis of the following formal procedure [34, 35]: a ) The given OPDF ρ ( ξ ) is represented as the product of two probability amplitudes ψ ( ξ ): ( ) ( ) = )( . (2.29) b ) Two Fourier transforms are performed [34, 35] − = di }/exp{)(21)( , (2.30) − −= di }/exp{)(21)( . (2.31) where cor r = , (2.32) ξ is the standard deviation of the stationary random process ξ ( r ) = ξ ; r cor is the radius of autocorrelation of this process. c ) The desired OPDF of the derivative of the SRP ξ ( r ) = ξ is [34, 35]: ( ) ( ) ( ) .)( == (2.33) Let’s apply the procedure (2.29) through (2.33) to find the OPDF ρ [ ( r )] of the derivative of stationary random process with different statistics of heights of irregularities. 16 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Substituting the functional dependence (2.14) ,arcsin ++= dba d in (2.26), we obtain ( ) ( ) ( ) ( ) .21,/, dbaba baabcbabadd bad dbaD +++ −+− +++= Simplifying this expression, we find the overall form of volumetric DESM (2.25) ( ) ( ) ( ) ,21,/, ++ −+−== d babad baabcbabadD (2.27) where a, b, d, a ν , b ν , c ν , a ω , b ω are given by expressions (2.11). Note that formally in expression (2.27), the derivative was replaced by a quantity d ba + with the Jacobian of the transformation ( ) ( ) bad baabcbabadG + −+−= . In the case where the ТPDF ρ ( , φ ) has the form (2.24) or (2.24a) (i.e., when the reflecting surface is non-isotopic), then instead of the ТPDF (2.27), taking into account (2.15), we obtain ( ) ( ) ( ) ,2,/, ++ −+− += d babad baabcbabadba bD (2.28) or ( ) ( ) ( ) .)(4,/, ++ −+−+== d babad baabcbabadba baD (2.28a) Formulas (2.27), (2.28) and (2.28a) can be considered as DESM on a statistically uneven surface under the following conditions (see §2.1): - the uneven surface is statistically homogeneous; - the irregularities of this surface are quite smooth and large-scale in comparison with the size of the microparticles; the reflection of microparticles from all local sections of the uneven surface occurs according to the laws of geometric optics (see Figures 4 and 5); Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ - the portion of the uneven surface involved in the reflection of microparticles is located at a large distance from the generator and detector of microparticles (Figureу 2). ρ [ ( r )] derivative of the stationary random process ( r ) In § 2.4 it was shown that in order to determine the DESM on a statistically uneven reflecting sur-face ( x , y ), any cross-section of which is described by a stationary random process ( r ) (Figure 3), it is necessary to find the OPDF ρ [ ( r )] derivative of this process. To search for ρ [ ( r )], let's use the method proposed in [34, 35]. If the OPDF ρ ( ) of the one-dimensional stationary random process (SRP) ( r ) = is known, then the OPDF ρ ( ) of the derivative of this process can be obtained on the basis of the following formal procedure [34, 35]: a ) The given OPDF ρ ( ξ ) is represented as the product of two probability amplitudes ψ ( ξ ): ( ) ( ) = )( . (2.29) b ) Two Fourier transforms are performed [34, 35] − = di }/exp{)(21)( , (2.30) − −= di }/exp{)(21)( . (2.31) where cor r = , (2.32) ξ is the standard deviation of the stationary random process ξ ( r ) = ξ ; r cor is the radius of autocorrelation of this process. c ) The desired OPDF of the derivative of the SRP ξ ( r ) = ξ is [34, 35]: ( ) ( ) ( ) .)( == (2.33) Let’s apply the procedure (2.29) through (2.33) to find the OPDF ρ [ ( r )] of the derivative of stationary random process with different statistics of heights of irregularities. 16 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
1] OPDF of the derivative of a Gaussian stationary random process
Suppose that at each point r of the SRP ξ ( r ), the random variable ξ (in particular, the height of irregularities) is distributed according to the Gaussian law a −−= , (2.34) where ξ and а ξ are the variance and expected value of the given process ξ ( r ). According to (2.29), we represent the OPDF (2.34) as the product of two probability ampli-tudes ( ) ( ) ,)( = where .4/)(exp2 1)( a −−= (2.35) Let's insert (2.35) into (2.30) and (2.31) − −−= dia }/exp{4/)(exp2 121)( , (2.36) − −−−= dia }/exp{4/)(exp2 121)( . (2.37) Let's perform the integration },/exp{)]2/(2/[exp)2/(2 1)( −= ia (2.38) }./exp{)]2/(2/[exp)2/(2 1)(* −−= ia (2.39) In accordance with (2.33), we multiply (2.38) and (2.39), as a result, we obtain ,2/exp21)( −= (2.40) where according to (2.32) = ξ /r cor1 (2.41) is the standard deviation of the differentiated stationary random process ( r ) = ; Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ r cor1 is the autocorrelation radius of the initial SRP ξ ( r ) = ξ with the Gaussian distribution of the height of the irregularities. 2] OPDF derivative of the SRP with a uniform distribution of the heights of the irregularities
Suppose that at each point r of the SRP ξ ( r ), the random variable ξ is distributed according to the uniform law in the interval ξ < ξ < ξ −= . (2.42) According to (2.29), we represent the OPDF (2.42) as the product of two probability ampli-tudes ( ) ( ) ,)( = (2.43) where .1)( −= (2.44) Let's insert (2.43) into (2.30) and (2.31) −= }/exp{121)( di , (2.45) −−= }/exp{121)( di . (2.46) As a result of calculation by the formula (2.45), we obtain ./)(2 }/exp{}/exp{}/exp{211)(
12 1212 − −=−= i iidi (2.47) Take into account that ( ξ – ξ )/2 = а ξ is the expected value, ξ – ξ = l is the base of the con-sidered SRP ξ ( r ). Now we can write ξ = а ξ – l /2 and ξ = а ξ + l /2, while the expression (2.47) takes the form .}/exp{/2 )}2/(exp{)}2/(exp{/2 }/)2/(exp{}/)2/(exp{)( aili lilili lailai −−= −−+= (2.48) Using the expression ieex ixix − −= , we represent (2.48) in the form 18 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
12 1212 − −=−= i iidi (2.47) Take into account that ( ξ – ξ )/2 = а ξ is the expected value, ξ – ξ = l is the base of the con-sidered SRP ξ ( r ). Now we can write ξ = а ξ – l /2 and ξ = а ξ + l /2, while the expression (2.47) takes the form .}/exp{/2 )}2/(exp{)}2/(exp{/2 }/)2/(exp{}/)2/(exp{)( aili lilili lailai −−= −−+= (2.48) Using the expression ieex ixix − −= , we represent (2.48) in the form 18 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________ .}/exp{/2 )}2/(sin{2)( aill = (2.49) As a result of similar calculations by the formula (2.46), we obtain .}/exp{/2 )}2/(sin{2)(* aill − = (2.50) Substituting (2.49) and (2.50) in (2.33), we finally find ,}{sin)(
22 22 kk = (2.51) where ,332
12 222 −=== corcor rlrlk (2.52) r cor 2 is the autocorrelation radius of the initial SRP ξ ( r ) = ξ with a uniform distribution of the height of the irregularities. It is taken into account that according to (2.32) ,62 corcor rlr == (2.53) where
12 )(12 −== l is the dispersion of the SRP ξ ( r ) = ξ with a uniform distribution of the heights of the irregularities (2.42). Thus, for an SRP ξ ( r ) = ξ with a uniform distribution of the heights of the irregularities, OPDF ρ ( ) its derivative is a distribution of the type sin / (2.51) with the scale parameter k (2.52).
3] OPDF derivative of the SRP with the Laplace distribution of the heights of irregularities
Let at each point r of the SRP ξ ( r ) = ξ a random variable ξ is distributed according to the Laplace law ,}/exp{21)( LL a −−= (2.54) where 1/ L is the scale parameter of this process ξ ( r ) = ξ ; а ξ is the shift parameter (expected value). According to (2.29), we represent the OPDF (2.54) as the product of two probability ampli-tudes Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ ( ) ( ) ,)( = (2.55) where .}2/exp{21)( LL a −−= (2.56) Let's insert (2.56) into (2.30) and (2.31) − −−= dia LL }/exp{}2/exp{2121)( , (2.57) − −−−= dia LL }/exp{}2/exp{2121)( . (2.58) We rewrite these expressions in the form ,}/2/)(exp{42)( −−−= a LL dia (2.59) .}/2/)(exp{42)(* − +−−= a LL dia (2.60) Let's perform the integration ,2/ )2/exp()( LLL i aa + −= (2.61) .2/ )2/exp()(* LLL i aa − +−= (2.62) Substituting (2.61) and (2.62) in (2.33), we find .)4/()( LL += (2.63) The dispersion of the Laplace distribution (2.54) is L = , therefore, according to (2.32), in this case ,4 corL r = (2.64) where r cor3 is the autocorrelation radius of the initial SRP ξ ( r ) = ξ with the Laplace distribution of the height of irregularities. Substituting (2.64) in (2.63), we obtain 20 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Let at each point r of the SRP ξ ( r ) = ξ a random variable ξ is distributed according to the Laplace law ,}/exp{21)( LL a −−= (2.54) where 1/ L is the scale parameter of this process ξ ( r ) = ξ ; а ξ is the shift parameter (expected value). According to (2.29), we represent the OPDF (2.54) as the product of two probability ampli-tudes Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ ( ) ( ) ,)( = (2.55) where .}2/exp{21)( LL a −−= (2.56) Let's insert (2.56) into (2.30) and (2.31) − −−= dia LL }/exp{}2/exp{2121)( , (2.57) − −−−= dia LL }/exp{}2/exp{2121)( . (2.58) We rewrite these expressions in the form ,}/2/)(exp{42)( −−−= a LL dia (2.59) .}/2/)(exp{42)(* − +−−= a LL dia (2.60) Let's perform the integration ,2/ )2/exp()( LLL i aa + −= (2.61) .2/ )2/exp()(* LLL i aa − +−= (2.62) Substituting (2.61) and (2.62) in (2.33), we find .)4/()( LL += (2.63) The dispersion of the Laplace distribution (2.54) is L = , therefore, according to (2.32), in this case ,4 corL r = (2.64) where r cor3 is the autocorrelation radius of the initial SRP ξ ( r ) = ξ with the Laplace distribution of the height of irregularities. Substituting (2.64) in (2.63), we obtain 20 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________ ,)()(
223 3 += k k (2.65) where corL rk = is the scale parameter. Thus, for the Laplace SRP ξ ( r ) = ξ of the OPDF ρ ( ) its derivative is the Cauchy distri-bution (2.65) with the expected value (i.e., the shift parameter) equal to zero.
4] OPDF derivative of the SRP with the distribution of the height of the irregularities ac-cording to the Cauchy law
Let at each point r of the SRP ξ ( r ) = ξ a random variable ξ be distributed according to the Cauchy law ,])([)( a K K −+= (2.66) where K is the scale parameter of this process ξ ( r ) = ξ ; а ξ is the shift parameter (expected value). Performing actions (2.29) through (2.33), the reverse of transformations (2.57) through (2.63), we obtain .}/exp{21)( kk −= (2.67) To find the scale parameter k , we note that the dispersion (variance) of the Cauchy distribu-tion is not defined, i.e., tends to infinity, but the heights of the real surface irregularities can be distributed only by a truncated Cauchy law with effective dispersion ξ ⁓ 25 K . Therefore, in this case, according to (2.32), we can write ,50 cor K r (2.68) where r cor4 is the autocorrelation radius of the initial SRP ξ ( r ) = ξ with the distribution of the heights of irregularities ξ according to the Cauchy law (2.66). In this case, we obtain the following estimate of the scale parameter Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ .25/2 KcorK rk (2.69) Thus, for an SRP with a distribution of roughness heights ξ ( r ) = ξ according to the Cauchy law (2.66), the OPDF ρ ( ) of its derivative is the Laplace distribution (2.67) with the scale parameter (2.69) and the shift parameter (expected value) equal to zero.
5] OPDF derivative of the SRP with the distribution of the heights of the irregularities ac-cording to the multilayer sinusoidal law
Consider the scattering of microparticles (in particular, electrons or high-frequency photons) on a single crystal. Falling microparticles can be reflected from different atomic layers of the crys-tal lattice of a single crystal (Figure 6 a ). This case is equivalent to the scattering of microparticles on a multilayer reflecting surface, each layer of which can be defined by a two-dimensional ho-mogeneous SRP ( x , y ), repeating on average the structure of a planar atomic lattice (Figure 6 b ). а ) b ) c ) F ig. 6 a ) Reflection of falling microparticles from various atomic layers of the crystal lattice of a single crystal; b ) Scattering of microparticles on the multilayer surface of the crystal, with each layer being considered as a separate uneven surface of the sinusoidal type; с ) Multi-humped sinusoidal OPDF of the height of the irregularities of the multilayer surface of the crystal (2.71) If only the upper layer of the crystal is involved in the scattering of microparticles, then it can be assumed that in each azimuthal cross-section r of such an SRP ξ ( r ) = ξ , a random value ξ (i.e., the height of the unevenness of the upper layer of the crystal surface) is distributed according to the sinusoidal law 22 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Consider the scattering of microparticles (in particular, electrons or high-frequency photons) on a single crystal. Falling microparticles can be reflected from different atomic layers of the crys-tal lattice of a single crystal (Figure 6 a ). This case is equivalent to the scattering of microparticles on a multilayer reflecting surface, each layer of which can be defined by a two-dimensional ho-mogeneous SRP ( x , y ), repeating on average the structure of a planar atomic lattice (Figure 6 b ). а ) b ) c ) F ig. 6 a ) Reflection of falling microparticles from various atomic layers of the crystal lattice of a single crystal; b ) Scattering of microparticles on the multilayer surface of the crystal, with each layer being considered as a separate uneven surface of the sinusoidal type; с ) Multi-humped sinusoidal OPDF of the height of the irregularities of the multilayer surface of the crystal (2.71) If only the upper layer of the crystal is involved in the scattering of microparticles, then it can be assumed that in each azimuthal cross-section r of such an SRP ξ ( r ) = ξ , a random value ξ (i.e., the height of the unevenness of the upper layer of the crystal surface) is distributed according to the sinusoidal law 22 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________ = ,,00 ;,0)/(sin2)(
111 12 lпри lприl l (2.70) where l is the thickness of one (i.e., the first) reflecting layer of a single crystal (Figure 6 c ). In the case where several identical crystal layers are effectively involved in the scattering of microparticles (Figure 6 b ), then the multi-humped sinusoidal OPDF of the heights of irregularities (Figure 6 c ) should be used. = ,,00 ;,0)/(sin2)(
222 212 lпри lприl ln (2.71) where n is the number of identical uneven sinusoidal layers lying in the interval [0, l ], here l = n l is the depth of the multilayer crystal surface of the effectively scattering microparticles. According to (2.29), we represent the OPDF (2.71) as the product of two probability ampli-tudes ( ) ( ) ,)( = where .)/sin(2)( lnl = Let's insert (2.72) into (2.30) and (2.31) = }/exp{)/sin(221)( l dilnl , (2.73) −= }/exp{)/sin(221)( l dilnl . Performing integration [see Appendix 2 expressions (A.2.12) and (A.2.13)], we obtain ,)//( 1)//( 14 1)(
21 )/(21 )/(2 − −++ −−= −−+ lnelnel lnilni (2.74) .)//( 1)//( 14 1)(*
21 )/(21 )/(2 + −+− −−= +−− lnelnel lnilni (2.75) Substituting (2.74) and (2.75) into (2.33) [see Appendix 3 in archive], we find
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ ( ) .// 1)/cos()/cos()cos()(cos1)(*)()(
221 212221 21122 + −+− − −== ln lnln lnnlp (2.76) The dispersion (variance) of the multi-humped sinusoidal distribution (2.71) is equal to .12 )6(12 )6(12 )6( −=−=−= nlnnnlnnl (2.77) Therefore, in this case, according to (2.32), we have the scale parameter ,6 )6(2
52 212215241 corcor rnlr −== (2.78) where r cor5 is the autocorrelation radius of one uneven layer of a sinusoidal crystal. OPDF (2.76) can be represented in the form ( ) .// 1)/cos()/cos()cos()cos(1)(
221 212221 2112 + −+− − −= ln lnln lnnlp (2.79) Taking into account the trigonometric formula − +=− xyxyyx from expression (2.79), we obtain another form of the desired OPDF ( ) .// 1)/cos(2/sin2/sin)cos(21)(
221 212221 121212 + −+− − − += ln lnln nlnlnlp (2.80) Thus, for an SRP with a multi-humped sinusoidal OPDF of the heights of irregularities (2.71) of the OPDF ρ ( ) its derivative is distribution (2.76) [or in another form (2.80)] with scale parameter (2.78). Using the formal procedure (2.29) through (2.33), we can obtain the OPDF ρ ( ) of the de-rivative for many other stationary random processes with different statistics of the height of irregularities . 24 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
221 212221 121212 + −+− − − += ln lnln nlnlnlp (2.80) Thus, for an SRP with a multi-humped sinusoidal OPDF of the heights of irregularities (2.71) of the OPDF ρ ( ) its derivative is distribution (2.76) [or in another form (2.80)] with scale parameter (2.78). Using the formal procedure (2.29) through (2.33), we can obtain the OPDF ρ ( ) of the de-rivative for many other stationary random processes with different statistics of the height of irregularities . 24 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
OPDF ρ ( ) (2.40), (2.51), (2.65), (2.67), (2 of irregularities.76) and others can be used in many problems of static physics. Based on the method proposed in Section 2, in this part of the paper we obtain formulas for calcu-lating volumetric diagrams of elastic scattering of microparticles (DESM) on statistically uneven surfaces when the conditions of the Kirchhoff approximation are met.
1] Volumetric diagram of elastic scattering of microparticles on a reflecting surface with a Gauss-ian distribution of the heights of irregularities
As an example, we consider the procedure for obtaining a volumetric DESM D ( ν,ω/ϑ,γ ) (2.27) for the case where the homogeneous and isotropic irregularities ( x , y ) of the reflecting surface (see Figure 3) at each point with coordinates x , y are distributed according to the law of Gauss (2.34). In this case, the OPDF ρ ( ) of the derivative of this stationary random process is also Gaussian (2.40) ( ) ,2/exp21)( −= (3.1) where = ξ /r cor1. In accordance with the algorithm described at the end of § 2.5, instead of in (3.1), we substitute d ba + , as a result, we obtain .2exp21
22 2222 22 +−= + d bad ba (3.2) Substituting (3.2) into (2.27), we obtain the explicit form of the desired DESM in the case of a Gaussian distribution of the heights of the irregularities of the reflecting surface
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ ( ) ( ) ( ) ,2exp8 1,/, bad baabcbabadd baD + −+− +−= (3.3) where = ξ /r cor1 ; the values of a, b, d, a ν , b ν , c ν , a ω , b ω are given by expressions (2.11). Under the conditions specified at the end of §2.5, the expression (3.3) is a formula for calcu-lating volumetric DESM on a large-scale (as compared to the size of microparticles) uneven re-flecting surface with a Gaussian distribution of the heights of the irregularities ( x , y ). The scattering diagrams calculated by formula (3.3) for various values of ϑ , γ , ξ and r cor1 are shown in Figure 7 (see Appendix 4 in archive). a ) ϑ = 65 , γ = 0 , ξ = 3, r cor1 = 5 b ) ϑ = 43 , γ = 0 , ξ = 3, r cor1 = 5 c ) ϑ = 60 , γ = 0 , ξ = 5, r cor1 = 5 d ) ϑ = 23 , γ = 0 , ξ = 17, r cor1 = 5 Fig. 7
Volumetric DESM on a homogeneous and isotropic uneven surface with a Gaussian distri-bution of the heights of irregularities. The calculations were performed according to the formula (3.3) for various values of the parameters ϑ , γ , ξ and r cor1 . Here and further, DESM are calculated using MathCad software 26 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Volumetric DESM on a homogeneous and isotropic uneven surface with a Gaussian distri-bution of the heights of irregularities. The calculations were performed according to the formula (3.3) for various values of the parameters ϑ , γ , ξ and r cor1 . Here and further, DESM are calculated using MathCad software 26 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
2] Volumetric DESM on a reflecting surface with a uniform distribution of the heights of irregu-larities
Let the homogeneous and isotropic irregularities ( x , y ) of the reflecting surface, at each point with coordinates x , y be distributed according to the uniform law (2.42). In this case, we use the OPDF ρ ( ) (2.51). Performing actions similar to (3.1) through (3.3), we obtain a formula for calculating volumetric DESM on a given surface ( ) ( ) ( ) ( ) ,sin2 1,/, ba baabcbabadd bakkD + −+− += (3.4) where
12 222 −== corcor rlrk is the scale parameter. The results of calculations using the formula (3.4) for different values of the parameters ϑ , γ , l and r cor2 are shown in Fig. 8 (see Appendix 5 in archive). a ) ϑ = 65 , γ = 0 , l = 3, r cor2 = 5 b ) ϑ = 43 , γ = 0 , l = 3, r cor2 = 5 c ) ϑ = 60 , γ = 0 , l = 5, r cor2 = 5 d ) ϑ = 23 , γ = 0 , l = 17, r cor2 = 5 Fig. 8
Volumetric DESM on a homogeneous and isotropic uneven reflecting surface with a uniform distribution of the heights of irregularities. The calculations are performed according to the formula (3.4)
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________
2] Volumetric DESM on a reflecting surface with a Laplace distribution of the heights of irregu-larities
Let the homogeneous and isotropic irregularities ( x , y ) of the reflecting surface, at each point with coordinates x , y be distributed according to the Laplace law (2.54). In this case, we use the OPDF ρ ( ) (2.65). Performing actions similar to (3.1) through (3.3), we obtain a formula for calculating volumetric DESM on a given surface ( ) ( ) ( ) ,)(2,/, ba baabcbabadbadk kD + −+−++= (3.5) where corL rk = is the scale parameter. The results of calculations using the formula (3.5) for different values of the parameters ϑ , γ , L and r cor3 are shown in Figure 9 (see Appendix 6 in archive). a ) ϑ = 65 , γ = 0 , L = 3, r cor3 = 5 b ) ϑ = 43 , γ = 0 , L = 3, r cor3 = 5 c ) ϑ = 60 , γ = 0 , L = 5, r cor3 = 5 d ) ϑ = 23 , γ = 0 , L = 17, r cor3 = 5 Fig. 9
Volumetric DESM on a homogeneous and isotropic uneven surface with a Laplace distribu-tion of the heights of irregularities. The calculations are performed according to the formula (3.5) 28
M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
4] Volumetric DESM on a reflecting surface with a distribution of the heights of irregularities according to the Cauchy law
Let the homogeneous and isotropic irregularities ( x , y ) of the reflecting surface, at each point with coordinates x , y be distributed according to the Cauchy law (2.66). In this case, we use the OPDF ρ ( ) (2.67). Performing actions similar to (3.1) through (3.3), we obtain a formula for calculating volumetric DESM on a given surface ( ) ( ) ( ) ,exp21,/, bad baabcbabadkd bakD + −+− +−= (3.6) where Kcor rk is the scale parameter. The results of calculations using the formula (3.6) for different values of the parameters ϑ , γ , K and r cor4 are shown in Figure 9 (see Appendix 7 in archive). a ) ϑ = 75 , γ = 0 , K = 3, r cor4 = 8 b ) ϑ = 38 , γ = 0 , K = 9, r cor4 = 8 c ) ϑ = 16 , γ = 0 , K =12, r cor4 = 8 d ) ϑ = 87 , γ = 0 , K = 5, r cor4 = 6 Fig. 10
Volumetric DESM on a homogeneous and isotropic uneven surface with distribution of the heights of irregularities according to the truncated Cauchy law. The calculations are performed according to the formula (3.6)
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ Analysis of the scattering diagrams shown in Figures 7 through 10, as well as of other DESMs calculated by the formulas (3.3) through (3.6), shows that the elastic scattering of micro-particles on one upper layer of a statistically uneven reflecting surface weakly depends on the sta-tistics of the heights of its irregularities. Single-layer surfaces with a distribution of the heights of irregularities according to the Gauss law, the uniform law, the Laplace law, and the Cauchy law scatter microparticles almost equally. Some differences between DESMs shown in Figures 7 through 10 are observed at small slip angles of incident microparticles ϑ and large values of the ratio ξm / r cor . Let’s consider the scattering of microparticles on an n -layer surface of a crystal, each layer of which is a homogeneous and isotropic two-dimensional uneven surface ( x , y ) with sinusoidal ir-regularities (Figure 6 b ). In this case, we use the multi-humped sinusoidal distribution of the height of the irregularities ξ (2.71), while the OPDF of ρ ( ) its derivative is expression (2.76) with the scale parameter η (2.78) ( ) ,// 1)/cos()/cos()cos()(cos1)(
221 212221 21122 + −+− − −= ln lnln lnnlp (3.7) or the same function in the form (2.80) ( ) .// 1)/cos(22sin22sin)cos(21)(
221 212221 121212 + −+− − − += ln lnln nlnlnlp (3.8) To obtain a volumetric DESM on a multilayer crystal surface, we use the method described in § 2.5 [similar to (3.1) through (3.3)]. Replace the derivative in (3.7) [or in (3.8)] by the value 30 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
221 212221 121212 + −+− − − += ln lnln nlnlnlp (3.8) To obtain a volumetric DESM on a multilayer crystal surface, we use the method described in § 2.5 [similar to (3.1) through (3.3)]. Replace the derivative in (3.7) [or in (3.8)] by the value 30 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________ d ba + , and substitute the resulting expression in (2.27). As a result, we obtain the formula for calculating the DESM on a multilayer crystal surface ( ) ( ) ( ) , / 1/cos/ /cos)cos()(cos2 1,/,
222 22 2221 22 221222 22221 22 2211222 bad baabcbabad d baln ld banld baln ld bannlD + −+− ++ − ++− +− +−= (3.9) or the same formula in another form ( ) ( ) ( ) , / 1/cos/ 22sin22sin)cos(22 1,/,
222 22 2221 22 221222 22221 122 22122 22122 bad baabcbabad d baln ld banld baln nld banld banlD + −+− ++ − ++− +− −+ ++= (3.10) where according to (2.11) a = cos ν cos ω + cos ϑ cos γ ; b = cos ν sin ω + cos ϑ sin γ ; d = sin ν + sin ϑ ; a ν = – sin ν cos ω ; b ν = – sin ν sin ω ; c ν = cos ν ; a ω = – cos ν sin ω ; b ω = cos ν cos ω ; according to (2.78) ,6 )6(
52 21221 cor rnl −= (3.11) where l is the thickness of one reflecting layer (i.e., the horizontal atomic plane) of the crystal (Fig. 6 b ); l = l n is the depth of the multilayer surface of the single crystal, effectively involved in the elastic scattering of microparticles (s); Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ n is the number of uneven layers of a single-crystal (sinusoidal type) that fit in the interval [0, l ]; r cor5 is the autocorrelation radius of one uneven layer of a sinusoidal type. This autocorrelation radius is approximately equal to the average radius of curvature of the sinusoidal irregularities of a single crystal layer; ϑ, γ are the angles that specify the direction of motion of the microparticle beam incident on the crystal surface (Figures 2, 3, 5); ν, ω are the angles that specify the direction of movement of microparticles reflected from the surface of the crystal toward the detector (Figures 2, 3, 5). Expressions (3.9) and (3.10) are the same formula for calculating DESM on a multilayer crystal surface, only written in different trigonometric forms. The formula (3.9) will be called the cosine-version of the DESM on the multilayer resulting surface, and the formula (3.10) is the si-nus-version of the same DESM. The DESM calculated by the formula (3.9) for various values of the five parameters ϑ , γ , l , n and r cor5 are shown in Figure 11 (see Appendix 8 in archive). a ) ϑ = 45 , γ = 0 , n = 64, b ) ϑ = 45 , γ = 0 , n = 65, l =10 – cm, r cor5 = 6×10 – cm l =10 – cm, r cor5 = 6×10 – cm 32 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
52 21221 cor rnl −= (3.11) where l is the thickness of one reflecting layer (i.e., the horizontal atomic plane) of the crystal (Fig. 6 b ); l = l n is the depth of the multilayer surface of the single crystal, effectively involved in the elastic scattering of microparticles (s); Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ n is the number of uneven layers of a single-crystal (sinusoidal type) that fit in the interval [0, l ]; r cor5 is the autocorrelation radius of one uneven layer of a sinusoidal type. This autocorrelation radius is approximately equal to the average radius of curvature of the sinusoidal irregularities of a single crystal layer; ϑ, γ are the angles that specify the direction of motion of the microparticle beam incident on the crystal surface (Figures 2, 3, 5); ν, ω are the angles that specify the direction of movement of microparticles reflected from the surface of the crystal toward the detector (Figures 2, 3, 5). Expressions (3.9) and (3.10) are the same formula for calculating DESM on a multilayer crystal surface, only written in different trigonometric forms. The formula (3.9) will be called the cosine-version of the DESM on the multilayer resulting surface, and the formula (3.10) is the si-nus-version of the same DESM. The DESM calculated by the formula (3.9) for various values of the five parameters ϑ , γ , l , n and r cor5 are shown in Figure 11 (see Appendix 8 in archive). a ) ϑ = 45 , γ = 0 , n = 64, b ) ϑ = 45 , γ = 0 , n = 65, l =10 – cm, r cor5 = 6×10 – cm l =10 – cm, r cor5 = 6×10 – cm 32 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________ c ) ϑ = 45 , γ = 0 , n = 126, d ) ϑ = 45 , γ = 0 , n = 127, l =10 – cm, r cor5 = 6×10 – cm l =10 – cm, r cor5 = 6×10 – cm e ) ϑ = 45 , γ = 0 , n = 46, f ) ϑ = 45 , γ = 0 , n = 47, l =10 – cm, r cor5 = 1,4×10 – cm l =10 – cm, r cor5 = 1,4×10 – cm g ) ϑ = 45 , γ = 0 , n = 24, h ) ϑ = 45 , γ = 0 , n = 23, l =10 – cm, r cor5 = 4×10 – cm l =10 – cm, r cor5 = 4×10 – cm Fig. 11
Volumetric diagrams of elastic scattering of microparticles on a multilayer crystal surface, calculated by the formula (3.9) for various values of the parameters ϑ , l , n and r cor5 Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ If each crystal layer has the same anisotropy, for example, of type (2.23), then, taking into account (2.24) and (2.28), we obtain the following formula for calculating the volumetric DESM for this case ( ) ( ) ( ) . / 1/cos/ /cos)cos()(cos2,/,
222 22 2221 22 221222 22221 22 2211222 222 bad baabcbabad d baln ld banld baln ld bannba blD + −+− ++ − ++− +− +− += (3.12) The results of calculations by the formula (3.12) are shown below in Figure 11a (see Appen-dix 9 in archive). a ) b ) Fig. 11a
Volumetric DESM on a multilayer non-isotropic crystal surface calculated by the formula (3.12) at ϑ = 45 , γ = 0 , l =10 – cm, r cor5 = 4×10 – cm, a ) n = 48 and b ) n = 47 If each crystal layer has the same anisotropy, for example, of type (2.23a), then, taking into account (2.24a) and (2.28a), we obtain the following formula for calculating volumetric DESM for this case 34 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Volumetric DESM on a multilayer non-isotropic crystal surface calculated by the formula (3.12) at ϑ = 45 , γ = 0 , l =10 – cm, r cor5 = 4×10 – cm, a ) n = 48 and b ) n = 47 If each crystal layer has the same anisotropy, for example, of type (2.23a), then, taking into account (2.24a) and (2.28a), we obtain the following formula for calculating volumetric DESM for this case 34 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________ ( ) ( ) ( ) . / 1/cos/ /cos)cos()(cos)(4,/,
222 22 2221 22 221222 22221 22 22112222 2222 bad baabcbabad d baln ld banld baln ld bannba balD + −+− ++ − ++− +− +− += (3.12а) The results of calculations by the formula (3.12a) are shown in Fig. 11 b (see Appendix 10 in archive). a ) b ) Fig. 11a
Volumetric DESM on a multilayer non-isotropic crystal surface calculated by the formula (3.12a) at ϑ = 45 , γ = 0 , l =10 – cm, r cor5 = 4×10 – cm, a ) n = 42 and b ) n = 37 Let’s analyze the volumetric diagrams of elastic scattering of microparticles shown in Fig-ures 11, 11 a and 11 b . 1] Agreement with experiment
A separate study should be devoted to comparing the calculations according to the formulas presented in this article with experimental data. But, already at this stage, it can be noted that vol-umetric DESMs calculated by the formula (3.9) (Figure 11) correspond to the results of experi-ments on the diffraction of particles and high-frequency electromagnetic waves on a crystal (see Figures 12, 12 a ). Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ a ) b ) c ) Fig. 12 a ) Electron diffraction pattern of the Ti Ni Cu alloy (http://dream-journal. org/issues/2018-6/2018-6_233.html); b ) Electron diffraction on gold. The thickness of the gold plate was about 250Å = 2.5×10 – cm. The size of the gold atom is approximately equal to 0,28 nm = 2.8×10 – cm. Thus, in the gold plate there were approximately 100 layers (i.e., atomic planes); b ) Illustration of an X -ray diffraction pattern obtained by diffraction of photons on a crys-tal. Photos and drawing are taken from the World Wide Web in the public domain a ) b ) c ) Fig. 12a a ) Electron diffraction pattern of the NaCl standard; b ) Electron diffraction pattern of a polycrystal of hexagonal nickel hydride NiH (http://ignorik.ru/docs/lekciya-13-eksperimentalenie -metodi-kristallofiziki.html); c M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
A separate study should be devoted to comparing the calculations according to the formulas presented in this article with experimental data. But, already at this stage, it can be noted that vol-umetric DESMs calculated by the formula (3.9) (Figure 11) correspond to the results of experi-ments on the diffraction of particles and high-frequency electromagnetic waves on a crystal (see Figures 12, 12 a ). Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ a ) b ) c ) Fig. 12 a ) Electron diffraction pattern of the Ti Ni Cu alloy (http://dream-journal. org/issues/2018-6/2018-6_233.html); b ) Electron diffraction on gold. The thickness of the gold plate was about 250Å = 2.5×10 – cm. The size of the gold atom is approximately equal to 0,28 nm = 2.8×10 – cm. Thus, in the gold plate there were approximately 100 layers (i.e., atomic planes); b ) Illustration of an X -ray diffraction pattern obtained by diffraction of photons on a crys-tal. Photos and drawing are taken from the World Wide Web in the public domain a ) b ) c ) Fig. 12a a ) Electron diffraction pattern of the NaCl standard; b ) Electron diffraction pattern of a polycrystal of hexagonal nickel hydride NiH (http://ignorik.ru/docs/lekciya-13-eksperimentalenie -metodi-kristallofiziki.html); c M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Formula (3.9) has a significant advantage in that it allows a more subtle analysis of the pro-cess of scattering of microparticles on a crystal than methods based on the idea of the existence of de Broglie waves. A selection of the parameters ϑ , l , n and r cor5 can lead to similarity with exper-imentally obtained electron diffraction patterns or X -ray diffraction patterns, and more detailed information on the structure of the crystal or other multilayer reflecting surface is disclosed. 2] "Adjustment" of the scale parameter η To take into account the various features of the crystal lattice, a scale parameter (2.78)
52 21221 cor rnl −= can be "adjusted" to the results of experiments. For example, can change the values of numeric constants and / or enter functional dependencies on parameters l , n and r cor5 :
52 21221
12 )3( cor rnl −= or
16 )127( cor rnl −= or
52 21221
12 )334( cor rnl −= and etc. (3.13)
52 112221 cor r Nnl −= or
52 112221 cor r Nntgl −= and etc. Perhaps such an "adjustment" η will lead to greater similarity of the results of calculations by the formula (3.9) with real electron diffraction patterns or radiographs. At the same time, the "adjustment" of the scale parameter (2.78) can make it possible to evaluate additional features of the structure and / or defects of the crystal lattice. When "adjusting" η , however, it is necessary to consider that equation (3.9) must satisfy the condition ( ) ,1,/,
20 20 = dd (3.14) where the angle ν varies from 0 to π /2; the angle ω varies from 0 to 2 π . Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ Even and odd number of crystal layers
From the diagrams shown in Figure 11, it can be seen that if the even number of layers n ef-fectively involved in the reflection of microparticles, then a minimum (dip) is observed in the very center of the diagram; and if the number of reflecting layers is odd, then a maximum (peak) is ob-served in the very center of the diagram. The same effect is found in experiments (Figure13). It should be noted, however, that for n = 4 (i.e., for an even number of layers) in the center of the diagram, there is not a minimum, but a maximum (peak) (Figure 20). а ) b ) Fig. 13 a ) In a number of experiments on the diffraction of microparticles, a dark spot is observed in the center of the electron diffraction pattern or X-ray diffraction pattern. b ) In a number of other similar experiments, a bright spot is observed in the center of the electron diffraction pattern or radiograph 4] The falling velocity v of microparticles and the number of reflective layers n It should be expected that the number of layers n , which are penetrated by microparticles incident to the surface of the crystal, mainly depends on their energy E [i.e. n = f ( Е )]. More pre-cisely, for incident fermions (in particular, electrons), the immersion depth in the thickness of the reflecting surface (i.e., the number of layers n ) mainly depends on their speed (the momentum or kinetic energy), and for incident bosons (in particular, photons) from their frequency. More gener-ally, it is possible to find the dependence n = f ( Е , l , r cor5 , ϑ , γ ). (3.15) 38 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
From the diagrams shown in Figure 11, it can be seen that if the even number of layers n ef-fectively involved in the reflection of microparticles, then a minimum (dip) is observed in the very center of the diagram; and if the number of reflecting layers is odd, then a maximum (peak) is ob-served in the very center of the diagram. The same effect is found in experiments (Figure13). It should be noted, however, that for n = 4 (i.e., for an even number of layers) in the center of the diagram, there is not a minimum, but a maximum (peak) (Figure 20). а ) b ) Fig. 13 a ) In a number of experiments on the diffraction of microparticles, a dark spot is observed in the center of the electron diffraction pattern or X-ray diffraction pattern. b ) In a number of other similar experiments, a bright spot is observed in the center of the electron diffraction pattern or radiograph 4] The falling velocity v of microparticles and the number of reflective layers n It should be expected that the number of layers n , which are penetrated by microparticles incident to the surface of the crystal, mainly depends on their energy E [i.e. n = f ( Е )]. More pre-cisely, for incident fermions (in particular, electrons), the immersion depth in the thickness of the reflecting surface (i.e., the number of layers n ) mainly depends on their speed (the momentum or kinetic energy), and for incident bosons (in particular, photons) from their frequency. More gener-ally, it is possible to find the dependence n = f ( Е , l , r cor5 , ϑ , γ ). (3.15) 38 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
The expression (3.15) can also take into account the effects of shading part of the deep sec-tions of the reflecting surface at small angles ϑ , etc. The determination of the functional dependence (3.15) will make it possible to more accu-rately match the results of calculations by the formula (3.9) with experimental data on the diffrac-tion of microparticles on periodic structures such as crystals and to obtain additional information about the structure of the reflecting surface. In particular, we consider the DESM (3.9) as a function of the number of layers n of the re-flecting surface of the single crystal D ( n ) with the six fixed parameters ϑ , γ , ν, ω, l , r cor5 . The results of calculations by the formula (3.9) D ( n ) in this case are shown in Fig. 14a (see Appendix 11 in archive ) a ) b ) Fig. 14 a ) Dependence of the DESM (3.9) on the number of layers n of the reflecting surface of the single crystal, which, in turn, depends on the velocity v (more precisely, energy E) of the mi-croparticles incident on this surface (3.15). The calculations were performed according to the for-mula (3.9) D ( n ) as a function of the number n , which varies in the range from 40 to 50 layers, with the following constant parameters: ϑ = 45 , γ = 0 , ν = 45 , ω = 0 , l =10 – cm, r cor5 = 9×10 – cm; b ) The intensity of an electron beam I scattered on a nickel single crystal at a constant reflection angle, depending on the square root of the voltage U , accelerating particles in an electron gun (electron generator). This experimental dependence was first obtained in 1927 by Clinton Davisson and Lester Germer [1] n D ( n ) I Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ Considering that the number of crystal layers penetrated by incident microparticles depends on their speed n = f (v), these calculations using the formula (3.9) D ( n ) are in good agreement with the results of K. Davisson and L. Germer's experiment (1927) on electron diffraction on a nickel crystal [1] (Figure14 b ). Formula (3.9) D ( n ) allows one to perform calculations in a much wider range of n values (Figure 15). At the same time, Figure 15 a shows that the in range n from 0 to 40 layers D ( n ) can take negative values. Since formula (3.9) D ( n ) is an OPDF, then at first glance this looks like an absurd result. a ) b ) Fig. 15
Results of calculations by the formula (3.9) D ( n ) as a function of the number of layers n , varying in the range from a ) 0 to 100 layers; b ) 40 to 100 layers, with the following constant pa-rameters ϑ = 45 , γ = 0 , ν = 45 , ω = 0 , l =10 – cm, r cor5 = 9 × – cm. Calculations performed using MathCad software n D ( n ) D ( n ) D ( n ) n M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Results of calculations by the formula (3.9) D ( n ) as a function of the number of layers n , varying in the range from a ) 0 to 100 layers; b ) 40 to 100 layers, with the following constant pa-rameters ϑ = 45 , γ = 0 , ν = 45 , ω = 0 , l =10 – cm, r cor5 = 9 × – cm. Calculations performed using MathCad software n D ( n ) D ( n ) D ( n ) n M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Negative values in the calculations by formula (3.9) can be “eliminated” by assuming that in the case under consideration (that is, with l =10 – cm и r cor5 = 9×10 – cm), the depth of the reflect-ing layer cannot be less than l = l n = 40×10 – = 4×10 – cm. In another case, when l =10 – cm and r cor5 = 2×10 – cm - corresponds to the size of an at-om that effectively reflects electrons or high-frequency photons, a calculation by the formula (3.9) D ( n ) leads to the result shown in Figure 16. In this case, the prohibition applies only to l in the 3 through 4 first layers. Fig. 16
The result of the calculation by the formula (3.9) D ( n ) for the following unchanged pa-rameters ϑ = 45 , γ = 0 , ν = 45 , ω = 0 , l =10 – cm, r cor5 = 2×10 – cm, and the experimental dependence obtained by K. Davisson and L. Germer in the study of electron diffraction on a nickel crystal [1] On the other hand, as will be shown below, the negative results of calculations by the formu-la (3.9) D ( n ) can mean that, when scattering microparticles on the thin films (i.e., for n <12), part of the microparticles pass through the atomic lattice. In this article, the problem D ( n ) < 0 has no final solution. A separate theoretical and exper-imental study should be devoted to this issue. n D ( n ) Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________
5] Scattering of microparticles on a single crystal layer
When scattering microparticles on one layer of the crystal (i.e., for n = 1), the calculation by the formula (3.9) leads to the result shown in Figure 17 a , b . a ) n = 1, ϑ = 45 , γ = 0 , b ) n = 1, ϑ = 45 , γ = 0 , l =10 – cm, r cor5 = 6·10 – cm l =10 – cm, r cor5 = 6×10 – cm Fig. 17
Diagrams of elastic scattering of microparticles (DESM) on one layer of the crystal ( n = 1), calculated by the formula (3.9) for various l If the thickness of the first layer is l =10 – cm, then the calculation result by the formula (3.9) is negative (Figure 17 a ). This can be explained by the fact that microparticles do not reflect from this layer, but pass through it. If the first layer is thicker, for example, l =10 – cm, then the reflection from such a layer (Figure 17 b ) is similar to the reflection from the top layer of an une-ven surface with other statistics of the height of irregularities (Figures 7 through 10). An interesting calculation results using the formula (3.9) is observed for n = 1 and l = 4×10 – cm (Figure 18). This case can be interpreted as a prediction that part of the microparti-cles will reflect from one layer of the crystal, and the other part of the microparticles will pass through it. 42 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Diagrams of elastic scattering of microparticles (DESM) on one layer of the crystal ( n = 1), calculated by the formula (3.9) for various l If the thickness of the first layer is l =10 – cm, then the calculation result by the formula (3.9) is negative (Figure 17 a ). This can be explained by the fact that microparticles do not reflect from this layer, but pass through it. If the first layer is thicker, for example, l =10 – cm, then the reflection from such a layer (Figure 17 b ) is similar to the reflection from the top layer of an une-ven surface with other statistics of the height of irregularities (Figures 7 through 10). An interesting calculation results using the formula (3.9) is observed for n = 1 and l = 4×10 – cm (Figure 18). This case can be interpreted as a prediction that part of the microparti-cles will reflect from one layer of the crystal, and the other part of the microparticles will pass through it. 42 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________ а ) n = 1, ϑ = 45 , γ = 0 , b ) n = 1, ϑ = 45 , γ = 0 , l = 4×10 – cm, r cor5 = 6·10 – cm l = 3.4×10 – cm, r cor5 = 6×10 – cm Fig. 18
Diagram of elastic scattering of microparticles on one layer of the crystal, calculated by the formula (9.3) for n = 1, а ) l = 4×10 – cm and b ) l = 3.4×10 – cm
6] Scattering of microparticles on two, three and four layers of the crystal
Diagrams of elastic scattering of microparticles on two, three and four layers of the crystal, calcu-lated by the formula (9.3), are shown in Figures 19 and 20. a ) n = 2, ϑ = 45 , γ = 0 , b ) n = 3, ϑ = 45 , γ = 0 , l =3×10 – cm, r cor5 = 6×10 – cm l = 1,1×10 – cm, r cor5 = 6×10 – cm Fig. 19
DESM on two ( a ) and three ( b ) layers of the crystal, calculated by the formula (3.9) Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ Fig. 20
Two angles of the DESM on four layers of the crystal, calculated by the formula (3.9) for n = 4, ϑ = 45 , γ = 0 , l =1,2×10 – cm, r cor5 = 9×10 – cm
7] The fifth parameter γ
As shown above, by selecting four parameters: H’ V H I ϑ , l , n , r cor5 it is possible to achieve that the calculations by the formula (3.9) correspond to different diffrac-tion patterns of microparticles on a multilayer statistically uneven surface of the crystal. The fifth parameter (quintessence from lat. quīnta essentia “fifth essence”) is the angle γ (see Figures 3 and 5), in all the previously considered cases it remained equal to zero (γ = 0 o ). a ) n = 66, ϑ = 45 , γ = 35 , b ) n = 66, ϑ = 45 , γ = 55 , l =10 – cm, r cor5 = 6×10 – cm l = 10 – cm, r cor5 = 6×10 – cm 44 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
As shown above, by selecting four parameters: H’ V H I ϑ , l , n , r cor5 it is possible to achieve that the calculations by the formula (3.9) correspond to different diffrac-tion patterns of microparticles on a multilayer statistically uneven surface of the crystal. The fifth parameter (quintessence from lat. quīnta essentia “fifth essence”) is the angle γ (see Figures 3 and 5), in all the previously considered cases it remained equal to zero (γ = 0 o ). a ) n = 66, ϑ = 45 , γ = 35 , b ) n = 66, ϑ = 45 , γ = 55 , l =10 – cm, r cor5 = 6×10 – cm l = 10 – cm, r cor5 = 6×10 – cm 44 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________ c ) n = 66, ϑ = 45 , γ = 80 , d ) n = 66, ϑ = 45 , γ = 155 , l =10 – cm, r cor5 = 6×10 – cm l = 10 – cm, r cor5 = 6×10 – cm Fig. 21
DESM on a crystal, calculated by the formula (3.9), for the identical ϑ, n , l , r cor5 and different angles γ When deriving the formula (3.9), it was taken into account that all azimuthal cross-sections in different directions of a homogeneous and isotropic uneven surface of the crystal are the same. Therefore, it was expected that when the azimuthal angle γ changes, the scattering diagram should remain unchanged, and only its azimuthal direction should change. From the diagrams shown in Figure 21 a , b , it can be seen that for small angles γ equal to 35 and 55 , only the azimuthal direc-tion of the whole diagram shifts. But with a further increase in the angle γ , the scattering diagram changes significantly with the remaining four parameters ϑ , l , n , r cor5 unchanged (Figure 21 c , d ). At this stage of the study, it is difficult to es-tablish whether this change is a drawback of the formula (3.9), or is it a reflection of reality that can be experimentally confirmed. It can be assumed that the DESM depends on the angle α between the projection of the azimuthal direction of motion of the incident microparticles on the XOY-plane and the direction of the rows of at-oms in the crystal lattice (Figure 22). From Figure 22 can be seen that rotation of the plane of incidence of the microparticles at the angle Fig. 22
Angle α between the projection of the azimuthal direction of motion of incident mi-croparticles on the XOY -plane and the direction of the rows of atoms in the crystal lattice
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ α is accompanied by the effect of increasing the distance between the atoms of the crystal lattice, which are effectively involved in their scattering. This effect can be taken into account by increas-ing the correlation radius of the heights of the surface irregularities r cor5 . The scattering diagrams at γ = 75 and enlarged in comparison with the previous case of r cor5 and l are shown in Figure 23. a ) n = 66, ϑ = 45 , γ = 75 , b ) n = 66, ϑ = 45 , γ = 75 , l =10 – cm, r cor5 = 6×10 – cm l = 2,5×10 – cm, r cor5 = 8×10 – cm Fig. 23
DESM calculated by the formula (3.9), at γ = 75 and increased r cor5 and l These calculation results by the formula (3.9) are subject to experimental verification. If the distortions of the DESM due to a change in the angle γ are not experimentally confirmed, then this disadvantage can be compensated for by a change in the orientation of the reference frame. In many cases, the coordinate axis from which the angle γ is measured can be initially combined with the azimuthal direction of motion of the microparticles incident on the crystal surface. That is, in a number of experiments, taking advantage of the arbitrariness in choosing a reference frame, it is possible from the very beginning to achieve that γ = 0 . 8] Diffraction of microparticles on thin films
The DESM calculation procedure presented in §2.1 and §2.5 was developed on the basis that microparticles, after a collision with a solid surface, are reflected from it according to the laws of geometric optics, and do not pass through this body. But it turned out that the formula (3.9) makes it possible to calculate the scattering diagram when microparticles pass through thin films. 46
M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Figure 24 shows the scattering diagrams of microparticles on thin films consisting of 14 and 15 layers of the crystal lattice. Diffraction maxima are obtained when microparticles fall on thin films at angles ϑ from 25 to 65 . In this case, some of the microparticles are reflected from the uneven layers (i.e., atoms) of the thin film, and the other part passes through them. a ) n = 14, ϑ = 45 , γ = 0, b ) n = 15, ϑ = 45 , γ = 0, l =10 – cm, r cor5 = 9×10 – cm l = 2×10 – cm, r cor5 = 9×10 – cm Fig. 24
Diffraction maxima of microparticles passing through thin films calculated by the formula (3.9) When microparticles fall vertically on the surface (i.e., at ϑ = 90 ), calculations using the formula (3.9) lead to absurd results. In other words, the method of calculating DESM proposed in this article does not apply to this case. It should be noted that diffraction maxima are obtained when microparticles fall on thin films at angles ϑ from 25 to 65 . In this case, some of the micro-particles are reflected from the uneven layers (i.e., atoms) of the thin film, and the other part pass-es through them. 9] Overall remarks
Summarizing this section, we note that the formula (3.9) [or in another form (3.10)] opens up wide opportunities for studying the properties of solid materials by analyzing the results of scattering of microparticles on them.
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ By selecting five parameters ϑ , l , n , r cor5 and γ , which are associated with some properties of the atomic or molecular structure of a solid, one can achieve a similarity of the scattering dia-gram calculated by the formula (3.9) with an electron diffraction or X -ray, and thereby obtain in-formation about the structure of this body. In general, the formula (9.3) with the five parameters ϑ , l , n , r cor5 and γ generates an infi-nite set of two-dimensional surfaces in which individual forms can exist that reflect the outlines or essence of processes in the surrounding reality. However, all these surfaces have the following common property. Since the formula (3.9) is the one-dimensional probability density function (OPDF) ( ) ( ) GD ,/,,/, = , the total area of all these surfaces is equal to one (3.14). The formula (3.9) is suitable for describing elastic diffraction not only of elementary parti-cles, atoms, and photons, but also for scattering macroscopic elastic bodies (such as soccer balls or tennis balls) on large multi-layer periodic structures. Let, for example, a three-dimensional grid with the edge length of one cubic cell of 3400 cm = 34 m be assembled from metal pipes with a diameter of 30 through 50 cm, whereby metal balls with a diameter of 50 to 80 cm are placed in the nodes of this grid (see Figure 25). Fig. 25
Cubic lattice consisting of metal pipes and balls of different diameters If a stream of soccer balls with a diameter of 22.3 cm is directed at an angle ϑ = 45 at such a cubic lattice, then their scattering is also described by the formula (3.9). Indeed, if instead of l = 10 – cm, r cor5 = 6×10 – cm and n = 66, substitute l = 50 см, r cor5 = 3400 cm and n = 18 in the 48 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Cubic lattice consisting of metal pipes and balls of different diameters If a stream of soccer balls with a diameter of 22.3 cm is directed at an angle ϑ = 45 at such a cubic lattice, then their scattering is also described by the formula (3.9). Indeed, if instead of l = 10 – cm, r cor5 = 6×10 – cm and n = 66, substitute l = 50 см, r cor5 = 3400 cm and n = 18 in the 48 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________ scale parameter η (3.11), then the diagram of elastic scattering of soccer balls on such a cubic lat-tice, calculated by the formula (3.9), will be approximately the same as shown in Figure 11 a . If the case of diffraction of soccer balls is confirmed experimentally, then we can argue that the formula (3.9) turned out to be universal with respect to the different scales of the events stud-ied, and the phenomena of the microworld are indistinguishable from the phenomena of the mac-rocosm (under similar conditions). It is possible to pose the inverse problem of simulating processes occurring in the micro-world using similar processes of the macrocosm. This will allow a more detailed understanding of the essence of microscopic phenomena. The following results are obtained in this article.
In §§2.1 through 2.5, a method has been developed for calculating volumetric DESM on statisti-cally uneven surfaces with various statistics of the height of irregularities. This method is applica-ble for describing the scattering of elastic particles and waves (photons and phonons), under the conditions of the Kirchhoff approximation: - irregularities of the reflecting surface are statistically uniform, smooth and large-scale in comparison with the sizes of microparticles (their radius or wavelength); - reflection of microparticles from all local sections of an uneven surface occurs according to the laws of geometric optics. For brevity, such a reflection of microparticles in the article is called "elastic"; - the section of the uneven reflecting surface is at a great distance from the generator and de-tector of the microparticles (see Figure 2). In this work, attention is focused on the scattering of elementary particles (in particular, electrons and high-frequency photons). However, the article suggests that the proposed method is suitable for describing elastic scattering of large-scale bodies as well (for example, soccer or ten-nis balls), if the above conditions are met. In other words, it is assumed that there are no funda-
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ mental differences between the diffraction of particles of the microworld and compact elastic bod-ies of the macroworld under similar conditions. Based on the procedure (2.29) through (2.32) given in [34, 35], in this article, we obtained: 1) an OPDF of the derivative of a Gaussian stationary random process (SRP) (2.40); 2) an OPDF derivative of the SRP with a uniform distribution of the heights of irregularities (2.51); 3) an OPDF derivative of the SRP with the Laplace distribution of the height of irregularities (2.65); 4) an OPDF derivative of the SRP with the distribution of the height of irregularities according to the Cauchy law (2.67); 5) an OPDF derivative of the SRP with the distribution of the heights of the irregularities ac-cording to the multilayer sinusoidal law (2.76). The obtained OPDF ρ [ ( r )] of derivatives of various SRP can be of interest for many branches of statistical physics. For example, since the momentum of a particle moving in the di-rection of the x-axis is related to the derivative of its coordinate by the ratio p x = mv x = mdx/dt = mx , the procedure (2.29) through (2.32) essentially means a transition from the coordinate a real repre-sentation of the statistical system, to its impulsive representation, with all the many consequences arising from this. Based on the method described in §§ 2.1 through 2.5 and the OPDF of derivatives of stationary random processes obtained in §2.6, the following formulas are derived for calculating the elastic scattering diagrams of microparticles on single-layer, large-scale uneven surfaces: 1) a formula for calculating DESM D ( ν,ω/ϑ,γ ) on a single-layer surface with Gaussian dis-tribution of the heights of irregularities (3.3); 50 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ mental differences between the diffraction of particles of the microworld and compact elastic bod-ies of the macroworld under similar conditions. Based on the procedure (2.29) through (2.32) given in [34, 35], in this article, we obtained: 1) an OPDF of the derivative of a Gaussian stationary random process (SRP) (2.40); 2) an OPDF derivative of the SRP with a uniform distribution of the heights of irregularities (2.51); 3) an OPDF derivative of the SRP with the Laplace distribution of the height of irregularities (2.65); 4) an OPDF derivative of the SRP with the distribution of the height of irregularities according to the Cauchy law (2.67); 5) an OPDF derivative of the SRP with the distribution of the heights of the irregularities ac-cording to the multilayer sinusoidal law (2.76). The obtained OPDF ρ [ ( r )] of derivatives of various SRP can be of interest for many branches of statistical physics. For example, since the momentum of a particle moving in the di-rection of the x-axis is related to the derivative of its coordinate by the ratio p x = mv x = mdx/dt = mx , the procedure (2.29) through (2.32) essentially means a transition from the coordinate a real repre-sentation of the statistical system, to its impulsive representation, with all the many consequences arising from this. Based on the method described in §§ 2.1 through 2.5 and the OPDF of derivatives of stationary random processes obtained in §2.6, the following formulas are derived for calculating the elastic scattering diagrams of microparticles on single-layer, large-scale uneven surfaces: 1) a formula for calculating DESM D ( ν,ω/ϑ,γ ) on a single-layer surface with Gaussian dis-tribution of the heights of irregularities (3.3); 50 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
2) a formula for calculating DESM D ( ν,ω/ϑ,γ ) on a single-layer surface with a uniform dis-tribution of the heights of irregularities (3.4); 3) a formula for calculating DESM D ( ν,ω/ϑ,γ ) on a single-layer surface with a Laplace dis-tribution of the heights of irregularities (3.5); 4) a formula for calculating DESM D ( ν,ω/ϑ,γ ) on a single-layer surface with the distribu-tion of the heights of irregularities according to Cauchy's law (3.6). Due to limitations imposed on the length of the paper, there is no detailed comparison of volumetric DESM calculated by the formulas (3.3) through (3.6) with experimental data. Howev-er, we note that in some cases the obtained DESMs are in good agreement with experiments (un-der the conditions of the Kirchhoff approximation) described in the extensive literature on the scattering of waves and particles on statistically uneven surfaces [17 through 27]. For example, Figure 26 compares the DESM calculated by formula (3.5) with the experimentally obtained dia-gram of neutron scattering on a single crystal CsHSeO [36]. a ) b ) Fig. 26 a ) The diffraction maximum of the neutron intensity reflected from the single crystal CsHSeO [36]; b ) Volumetric DESM, calculated by formula (3.5) for the case of the Laplace dis-tribution of the heights of irregularities of reflecting surface, for ϑ = 60 , γ = 0 , L = 7, r cor3 = 5 On the basis of the method described in §§ 2.1 through 2.6, and the OPDF of the derivative of a multilayer sinusoidal stationary random process (2.76), in this paper we obtained the formula (3.9)
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ for calculating the DESM on large-scale (compared to microparticles) irregularities of the multi-layer surface of crystal. By selecting the five parameters ϑ , l , n , r cor5 and γ included in the equation (3.9), it is pos-sible to achieve similarity of the scattering diagram of microparticles on the multilayer surface of the crystal calculated using this formula with experimentally obtained electron diffraction patterns (Figures 23, 24) or radiographs. a ) b ) Fig. 27 a ) The volumetric diagram of the elastic scattering of microparticles on the multilayer surface of a crystal, calculated by the formula (3.9), for ϑ = 45 , γ = 0 , n = 64, l =10 – cm, r cor5 = 6×10 – cm; b ) Experimentally obtained electron diffraction pattern with a dark spot in the middle. Photo taken from a source that is freely available on the Internet a ) b ) Fig. 28 a ) The volumetric diagram of the elastic scattering of microparticles on the multilayer surface of a crystal, calculated by the formula (3.9), for ϑ = 45 , γ = 0 , n = 46 l =10 – cm, r cor5 = 1.4×10 – cm; b ) Experimentally obtained electron diffraction pattern with a dark spot in the middle. Photo taken from a source that is freely available on the Internet. 52 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ for calculating the DESM on large-scale (compared to microparticles) irregularities of the multi-layer surface of crystal. By selecting the five parameters ϑ , l , n , r cor5 and γ included in the equation (3.9), it is pos-sible to achieve similarity of the scattering diagram of microparticles on the multilayer surface of the crystal calculated using this formula with experimentally obtained electron diffraction patterns (Figures 23, 24) or radiographs. a ) b ) Fig. 27 a ) The volumetric diagram of the elastic scattering of microparticles on the multilayer surface of a crystal, calculated by the formula (3.9), for ϑ = 45 , γ = 0 , n = 64, l =10 – cm, r cor5 = 6×10 – cm; b ) Experimentally obtained electron diffraction pattern with a dark spot in the middle. Photo taken from a source that is freely available on the Internet a ) b ) Fig. 28 a ) The volumetric diagram of the elastic scattering of microparticles on the multilayer surface of a crystal, calculated by the formula (3.9), for ϑ = 45 , γ = 0 , n = 46 l =10 – cm, r cor5 = 1.4×10 – cm; b ) Experimentally obtained electron diffraction pattern with a dark spot in the middle. Photo taken from a source that is freely available on the Internet. 52 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Once again, we note that these results were obtained without using the idea of Louis de Broglie on the wave properties of elementary particles.
The article presents formulas for calculating the volumetric diagrams of elastic scattering of mi-croparticles (DESM) (fermions and bosons) on uneven single-layer and multilayer surfaces with different statistics of the height of irregularities, when the conditions of the Kirchhoff approxima-tion are met. At the same time, the one-dimensional probability density functions (OPDF) of the derivative of various stationary random processes are obtained, which can be used in a number of other problems of statistical physics. In addition to solving the above practical problems, this article is aimed at introducing ra-tional clarity into the conceptual problem associated with discussing the idea of the possible "existence" of de Broglie waves. The laws of geometric optics and the probabilistic methods of statistical physics applied here, according to the author, have allowed an explanation of the dif-fraction of elementary particles and atoms by crystals without using this hypothesis of Louis de Broglie about matter-waves. Moreover, this paper suggests that the phenomenon of particle dif-fraction on solid periodic structures can occur not only in the microcosm, but also in the macro-cosm under similar conditions.
I thank my mentors, Dr. A.A. Kuznetsov and Dr. A.I. Kozlov for the formulation and discussion the tasks outlined in this article. When performing calculations, Dr. S.V. Kostin provided invalua-ble assistance. During the preparation of the manuscript, valuable comments were made by D. Reid, Dr. G.I. Shipov, Dr. V.A. Lukyanov and Dr. Gubarev.
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ Appendix 1
The reflection of a plane electromagnetic wave from a square surface area
Let the length of a plane monochromatic electromagnetic wave λ be much less than the characteristic dimensions of the surface irregularities of a solid or liquid substance conducting electric current (i.e., λ « r cor , where r cor is the autocorrelation radius of the heights of the bumps in the reflecting surface). In this case, the uneven surface can be divided into many flat square sec-tions (facets). Consider the reflection of the rays of the electromagnetic wave from each facet sep-arately (Figure A.1.1 a , b ). a ) b ) Fig. A.1.1
Scattering of an electromagnetic wave on a surface approximated by smooth square sections (facets). a ) The maximum of the main lobe of the scattering diagram of each facet is di-rected according to the laws of geometric optics: lies in the plane of incidence and the angle of reflection is equal to the angle of incidence; b ) Only the main lobes of the scattering diagrams whose facets are oriented accordingly are directed towards the receiver antenna Transmitter Receiver M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Scattering of an electromagnetic wave on a surface approximated by smooth square sections (facets). a ) The maximum of the main lobe of the scattering diagram of each facet is di-rected according to the laws of geometric optics: lies in the plane of incidence and the angle of reflection is equal to the angle of incidence; b ) Only the main lobes of the scattering diagrams whose facets are oriented accordingly are directed towards the receiver antenna Transmitter Receiver M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
The beam of an electromagnetic wave here refers to a cylinder whose axis connects the source of the electromagnetic wave to the center of the reflecting facet, and the diameter of the base of this cylinder approximately coincides with the size of one of the sides of the b n square fac-et. We define the scattering diagram of a flat monochromatic electromagnetic wave (EMW) on a single facet that perfectly conducts an electric current. Let's assume that the radiation point (emitter, Fig. A.1.1 b ) and the observation point (receiver) are located at a great distance from the facet (i.e. b n « r and b n « r ), so that the EMW rays incident on the facet and reflected from the facet can be considered almost parallel. In this case, the signal sent from any point on the square facet to the receiver antenna has the form ,)coscoscos(cos )sincossin(cos2exp),( −++ +++= ryxtirEyxE mi (A.1.1) where x and y determine the coordinates of each point on the square facet; E m is the amplitude of the monochromatic electromagnetic field near the emitter; r is the distance from the source of EMW to the center of the facet (Figure A.1.1 b ); r is the distance from the center of the facet to the antenna of the receiver (Figure A.1.1 b ); ω is the oscillation frequency of a monochromatic electromagnetic wave; ϑ , γ are angles that specify the direction of the EMW ray incident on the facet (Figure A.1.2); ν , ω are the angles that specify the direction of the EMW ray reflected from the facet. Fig. A.1.2
Angles ϑ , γ determine the direction of the EMW ray incident on the facet; angles ν , ω determine the direction of the EMW ray reflected from the facet Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ ( )( )( )( ) .2expcoscoscoscos2 coscoscoscos2sin sincossincos2 sincossincos2sin),( −+ + + +== rtirEb b b bdxdyyxEE mn nb n ni n (А.1.2) The first and second multipliers in the expression (A.1.2), squared, is the desired power scat-tering diagram of a flat, monochromatic EMW from a perfectly conducting square sections of the surface (facets) ( ) ( )( ) ( )( ) .coscoscoscos2 coscoscoscos2sinsincossincos2 sincossincos2sin,/, + + + += n nn nr b bb bD (A.1.3) The scattering diagrams calculated using the formula (A.1.3) are shown in Figure A.1.3 (see Appendix 12 in archive) Fig. A.1.3.
Power scattering diagrams of a flat, monochromatic EMW from a perfectly conducting square sections of the surface (facets). The calculations are performed according to the formula (A.1.3) using the MathCad software a ) ϑ = 45 , γ = 0 , b n / λ = 3 b ) ϑ = 45 , γ = 0 , b n / λ = 50 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Power scattering diagrams of a flat, monochromatic EMW from a perfectly conducting square sections of the surface (facets). The calculations are performed according to the formula (A.1.3) using the MathCad software a ) ϑ = 45 , γ = 0 , b n / λ = 3 b ) ϑ = 45 , γ = 0 , b n / λ = 50 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
The cross section of the scattering diagram (A.1.3) in the plane of incidence and reflection of the EMW beam shown in Figure A.1.4.
Fig. A.1.4
Cross section of the scattering diagram of a beam flat electromagnetic wave from a flat square surface area (facet) conducting an electric current From the scattering diagrams (DR) shown in Fig. A.1.3, it is seen that with an increase in the ratio b n / λ , the main lobe of the DR becomes thinner and elongates, and the side lobes disap-pear. For large b n with respect to λ (i.e., when b n / λ → ∞), the scattering diagram (A.1.3) degener-ates into a delta function, i.e. the EMW beam reflected by the large facet becomes infinitely thin. In this case, the laws of reflection of a light ray from a facet (i.e., the laws of geometric optics) completely coincide with the laws of elastic reflection of particles from a solid surface under simi-lar conditions (i.e., when the particles are much smaller than the dimensions of a solid surface). In other words, in this case, the behavior of the light beam completely corresponds to the behavior of the particle (which can be called a photon). A photon is reflected almost lossless from a "mirror" surface according to the laws of geometric optics, just as elastic electrons or protons are reflected from a solid surface. Energy losses due to heating of the reflecting surface during colli-sions with particles and other secondary effects are not taken into account in the model under con-sideration. Therefore, in this article, microparticles are any particles: fermions (e.g., electrons) and bos-ons (e.g., photons), whose sizes are much smaller than the characteristic irregularities of the re-flecting surface (Kirchhoff approximation), and reflected from this surface according to the laws of geometric optics. Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ All conclusions made in this article relate to both elastic particles and electromagnetic radia-tion (light) rays, under the above conditions. In connection with the foregoing, all conclusions made in this article relate to both elastic particles and EMW (light) rays, if the above conditions are met. Appendix 2
The Calculation of Integrals
Calculate the integrals (2.73) = }/exp{)/sin(221)( l dilnl , (A.2.1) −= }/exp{)/sin(221)( l dilnl (A.2.2) We start with the integral (A.2.1), and use the formula ieex ixix − −= and represent (A.2.1) in the form − −= l ilnilni deieel (A.2.3) Let's perform the following transformations − −= l ilniilni di eeeel − +−+ −= dieel ilniilni ////2 −−+ −= l lnilni dieel −−+ −=
222 1)( l lnilni deeli
As a result of these transformations, we obtain 58
M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________ −= −−+
222 1)( l lnil lni dedeli (A.2.4) Let's calculate the first integral in (A.2.4) )//( )]//([
21 210 )//()//( + += +− + lni lnidede l lnilni
021 )//(21 210 )//( )//()//( )//( llnil lni lni elni lndie +=+ + ++ )//()//()//(
21 )//(021 )//(021 )//( +−+=+ +++ lni elni elni e lnilnilllni )//( 1)//()//( +−+=+ ++ lnilni elni e lnilllni
As a result of these calculations, we obtain )//( 1)//(
21 )//(021 )//( + −=+ ++ lnielni e lnilllni (A.2.5) Let's calculate the second integral in (A.2.4) )//( )]//([
21 210 )//(0 )//( −− −−= −−−− lni lnidede l lnil lni
021 )//(21 210 )//( )//()//( )//( llnil lni lnielni lnide −−=−− −− −−−− )//( 1)//()//( −−−=−− −−−− lnilni elnie lnilllni As a result of these calculations, we obtain )//( 1)//(
21 )//(021 )//( −− −=−− −−−− lnielnie lnilllni (A.2.6) Substituting (A.2.5) and (A.2.6) and (A.2.4), we have
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ −− −−+ −= −−+ )//( 1)//( 1222 1)(
21 )//(21 )//(2 lnielnieli lnillnil
Let's do the transformations − −++ −= −−+ )//( 1)//( 1222 1)(
21 )/(21 )/(2 lnelnelii lnilni − −++ −−= −−+ )//( 1)//( 1222 1)(
21 )/(21 )/(2 lnelnel lnilni
Finally we get the result of integration (A.2.1) − −++ −−= −−+ )//( 1)//( 14 1)(
21 )/(21 )/(2 lnelnel lnilni (A.2.7) Similarly, we take the integral (A.2.2) −= }/exp{)/sin(221)( l dilnl Let's represent (A.2.2) in the form −− −= l ilnilni deieel −−− −= l ilniilni di eeeel − −−− −= dieel ilniilni ////2 +−− −= l lnilni dieel +−− −=
222 1)(* l lnilni deeli
As a result of these transformations, we obtain −= +−−
222 1)(* l lnil lni dedeli (A.2.8) We calculate the first integral in (A.2.8) 60
M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________ )//( )]//([
21 210 )//()//( − −= −− + lni lnidede l lnilni
021 )//(21 210 )//( )//()//( )//( llnil lni lni elni lndie −=− − −− )//()//()//(
21 )//(021 )//(021 )//( −−−=− −−− lni elni elni e lnilnilllni )//( 1)//()//( −−−=− −− lnilni elni e lnilllni )//( 1)//(
21 )//(021 )//( − −=− −− lnielni e lnilllni (A.2.9) We calculate the second integral in (A.2.8) )//( )]//([
21 210 )//(0 )//( +− +−= +−+− lni lnidede l lnil lni
021 )//(21 210 )//( )//()//( )//( llnil lni lnielni lnide +−=+− +− +−+− )//( 1)//(
21 )//(021 )//( +− −=+− +−+− lnielnie lnilllni (A.2.10) Substituting (A.2.5) and (A.2.6) and (A.2.4), we have +− −−− −= +−− )//( 1)//( 1222 1)(*
21 )//(21 )//(2 lnielnieli lnillnil
Let's do the transformations + −+− −= +−− )//( 1)//( 1222 1)(*
21 )/(21 )/(2 lnelnelii lnilni + −+− −−= +−− )//( 1)//( 1222 1)(*
21 )/(21 )/(2 lnelnel lnilni
Finally, we obtain the result of integration (A.2.2)
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ + −+− −−= +−− )//( 1)//( 14 1)(*
21 )/(21 )/(2 lnelnel lnilni (A.2.11) So, the results of taking the integrals (A.2.1) and (A.2.2) are the expressions (A.2.7) and (A.2.11): − −++ −−= −−+ )//( 1)//( 14 1)(
21 )/(21 )/(2 lnelnel lnilni (A.2.12) + −+− −−= +−− )//( 1)//( 14 1)(*
21 )/(21 )/(2 lnelnel lnilni (A.2.13)
Appendix 3
The product of factors
Product of expressions (2.74) and (2.75) {or (A.2.12) and (A.2.13)} − −++ −−= −−+ )//( 1)//( 14 1)(
21 )/(21 )/(2 lnelnel lnilni (A.3.1) + −+− −−= +−− )//( 1)//( 14 1)(*
21 )/(21 )/(2 lnelnel lnilni (A.3.2) equally + −+− − − −++ −== +−−−−+ )//( 1)//( 1)//( 1)//( 14 1)(*)()(
21 )/(21 )/(21 )/(21 )/(2 lnelnelnelnelp lnilnilnilni (A.3.3) Opening large brackets, we multiply the terms in pairs ( )( ) − −−=− −+ − −+−+ ln eelnelne lnilnilnilni M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
21 )/(21 )/(21 )/(21 )/(2 lnelnelnelnelp lnilnilnilni (A.3.3) Opening large brackets, we multiply the terms in pairs ( )( ) − −−=− −+ − −+−+ ln eelnelne lnilnilnilni M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
Add the resulting expressions ( )( ) ( )( ) ( ) ( )( ) − −−++ −−+ − −− +−−−+−+−+ ln eeln eeln ee lnilnilnilnilnilni
Rearranging terms and summing up them, we get ( )( ) ( )( ) ( )( ) ( )
221 )/()/(2221 )/()/()/()/( // 1121111 + −−+ − −−+−− +−++−−−−+ ln eeln eeee lnilnilnilnilnilni (A.3.4) Performing calculations 1. ( )( ) )/()/(2)/()/( +−−=−− −+−+ lnilninilnilni eeeee ( )( ) )/()/(2)/()/( +−−=−− +−−−−+−−− lnilninilnilni eeeee ( )( ) =+−−=−− +−++−+ )/()/(0)/()/( lnilnilnilni eeeee ( )( ) ( )
221 )/()/(21 )/(21 )/( // 11)//( 1)//( 1 + −−=+ −+ − +−++−+ ln eelnelne lnilnilnilni ( )( ) ( )
221 )/()/(21 )/(21 )/( // 11)//( 1)//( 1 + −−=+ −+ − +−++−+ ln eelnelne lnilnilnilni ( )( ) − −−=+ −− − +−−−+−−− ln eelnelne lnilnilnilni
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ =++−= +−+ )/()/( lnilni ee =++− +−+ )/()/( lnilni ee ]12/)[(2 )/()/( −+− +−+ lnilni ee = ]1)/[cos(2 −+− ln (A.3.5) where the expression iхix eex − += is taken into account. Add 1 and 2 )/()/(2)/()/(2 +−−+−− +−−−−−+ lnilninilnilnini eeeeee Let’s regroup the terms )/()/()/()/(22 ++−+−+ −−−+−+− lnilnilnilninini eeeeee or ]12/)(2/)(2/)[(2 )/()/()/()/(22 ++−+−+ −−−+−+− lnilnilnilninini eeeeee = = ]1)/cos()/cos(2[cos2 +−−+− lnlnn (A.3.6) Let’s substitute the terms (A.3.5) and (A.3.6) into (A.3.4), we obtain ( )
221 212221 21211 // ]1)/[cos(4]1)/cos()/cos(2[(cos2 + −+− − +−−+− ln lnln lnlnn (A.3.7) Now insert (A.3.7) into (A.3.3) ( ) + −+− − +−−+−=
221 212221 212112 // ]1)/[cos(4]1)/cos()/cos(2[(cos24 1)( ln lnln lnlnnlp (A.3.8) We use two trigonometric formulas 64
M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________ xx += and )]cos()[cos(21coscos yxyxyx ++−= (A.3.9) Where should cos212cos nn =+ (A.3.10) )/cos()cos(2)/cos()/cos( lnlnln =++− (A.3.11) In view of (A.3.10) and (A.3.11), the expression (A.3.8) takes the form ( ) + −+− − −=
221 212221 21122 // ]1)/[cos(4)]/cos()cos(2cos2[24 1)( ln lnln lnnlp
Performing simplifications ( ) + −+− − −=
221 212221 21122 // ]1)/[cos(4)]/cos()cos([cos44 1)( ln lnln lnnlp finally get ( ) + −+− − −=
221 212221 21122 // ]1)/[cos()]/cos()cos([cos1)( ln lnln lnnlp (A.3.12)
DESM is diagram of elastic scattering of microparticles; EDP is electron diffraction pattern; OPDF is one-dimensional probability density function; SD is standard deviation; SRP is stationary random process; TPDF is two-dimensional (or joint) probability density function.
Microparticle is a solid elastic compact body or a ray of light (i.e., a photon) whose size or wave-length is much smaller than the characteristic size of the irregularities of the reflecting surface, upon collision with which they are reflected according to the laws of geometric optics (see §1).
Diffraction of microparticles on uneven surfaces _______________________________________________________________________________________________________________ Elastic scattering is the reflection of a particle from a surface according to the laws of geometric optics (see Figures 3, 4): 1) The reflection of an elastic particle from a solid surface occurs in the plane of its incidence; 2) The angle of reflection Q is equal to the angle of incidence Q . The author states that there are no competing interests.
There is permission from the Moscow Aviation Institute for publication.
10 Ethics approval and consent to participate
Not applicable.
11 Funding
There is no funding for this work.
12 Data availability
Data confirming the results of this study can be obtained from the author of this article upon re-quest at: [email protected] and at http://metraphysics.ru/.
13 References [1] C.J. Davisson, L.H. Germer, Reflection of Electrons by a Crystal of Nickel, Proceedings of the National Academy of Sciences of the United States of America. vol. (4), (1928), pp. 317–322. doi:10.1073/pnas.14.4.317, PMC 1085484, PMID 16587341. [2] W.L. Bragg, The Diffraction of Short Electromagnetic Waves by a Crystal. Proceeding of the Cambridge Philosophical Society, , 43 (1914). [3] K.R. Gehrenbeck, "Electron diffraction: fifty years ago". Physics Today. 31 (1): 34–41(1978). doi:10.1063/1.3001830. [4] B.K. Vainshtein, Structure Analysis by Electron Diffraction, Pergamon Press, Oxford (1964). 66 M. S. Batanov-Gaukhman ____________________________________________________________________________________________________________________________________________________________________________________________
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