Submit manuscript...
eISSN: 2576-4543

Physics & Astronomy International Journal

Review Article Volume 2 Issue 6

A Gamma-ray in a uniform compton scatterer

Vladimir Uchaikin

Professor, Uljanovsk State University, Russia

Correspondence: Prof Vladimir Uchaikin, Uljanovsk State University, L.Tolstoi str. 42, 432000 Russia, Tel 79876875971

Received: February 01, 2018 | Published: December 6, 2018

Citation: Uchaikin V. A Gamma-ray in a uniform compton scatterer. Phys Astron Int J. 2018;2(6):573-576. DOI: 10.15406/paij.2018.02.00144

Download PDF

Abstract

The structure of a narrow ray of gamma radiation propagating through a medium filled with free electrons (called Compton scatterer) is investigated. The dual representation is used including primary (basic) and adjoint forms of kinetic equations. In the ray propagation problem primary form possesses cylindrical symmetry whereas the adjoint form (in case of isotropic detector in an unbounded homogeneous medium) has the spherical symmetry. The analysis performed on the base of adjoint function singularities shows, in particular, that in a vicinity of the narrow gamma-ray single-scattered radiation predominates over all other components. Analytical representation of the field in the vicinity of a cylindrical primary ray has been found. The result can be important in gamma astronomy processing.

Keywords: Compton Scatterer, neumann series, boltzmann equation

Introduction

The concept of adjoint function (or importance function) can be unlocked as follows. Let the problem under consideration requires finding some linear functional
J[f()]=(P,f) P(x)f(x)dx MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGkbGaaG4wai aadAgacaaIOaGaeyyXICTaaGykaiaai2facaaI9aGaaGikaiaadcfa caaISaGaamOzaiaaiMcacqGHHjIUdaWdbaqabSqabeqaniabgUIiYd GccaWGqbGaaGikaiaahIhacaaIPaGaamOzaiaaiIcacaWH4bGaaGyk aiaadsgacaWH4baaaa@4FD3@ (1)
from a solution  of the equation

Lf(x)=Q(x), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGmbGaamOzai aaiIcacaWH4bGaeyykaKIaeyypa0JaamyuaiabgIcaOiaahIhacqGH PaqkcaaISaaaaa@4151@  (2)

Describing the transport of particles in a medium from a source with phase density Q(x)  immersed in a medium, interaction of which with the particles is described by the operator L. In the problem under consideration,Q(x)  may denote the flux of photons, or their concentration, or some other local characteristics of the photon field at a pointx=(r,p) of the phase space. For the sake of convenience, we will represent the photon momentum p through the pair of variables Ω=p/p(the direction unit vector) and E=cp (photon’s energy)

The direct way of finding the value of the functional (1), expressing the reading of a photon detector, lies through solving of the kinetic equation (2) and computing integral J=f+(x)Q(x)dx over the photon field with a weighting function P(x) expressing the contribution of a single photon placed at point X into resulting reading of the detector. But this is not a unique way to reach this result. Another way is based on the other equivalent representation of the detector reading,1−3 using so-called adjoint (or importance) function f+(x) being a solution of the adjoint (in Lagrange’s sense) equation
L+f+(x)=P(x).  (3)

By definition, the primary L and adjoint L+ operators are connected via relation

( f + , Lf ) = ( L + f + , f ) .

Inserting now Eq.(2) into the LHS of the latter equation and Eq.(3) into the RHS of it, and using Eq.(1), we understand that numerically f+(x0)is the detector reading under condition that the source is localized at the point x0 and has a unite power: Q(x)=δ(xx0). A more deep exposition of the concept of the importance function is given in books.13

Integral equations and neumann series

The mathematical description of penetration of gamma radiation (X-rays) through a scattering matter is given in the well-known monograph.4,5 Primary transport equation (2) is written as a linearized kinetic Boltzmann equation, for time-independent problem having the form:

ΩI(r,Ω,E)+μ(r,E)I(r,Ω,E)=Q(r,Ω,E)+dE'μs(r,E')I(r,ΩE')W(ΩE'Ω,E). (4)

Here I=I(r,Ω,E) denotes the differential intensity (so, IdSdΩdE is a mean number of particles, crossing over a unit elementary area dSΩ and belonging to intervals ×dE), μ and μs are linear attenuation (total and scattering) coefficients Q(r,Ω,E) is the source phase density, W the scattering indicatrix, normalized to 1:

4π0E'dEW(ΩE'Ω,E)=1.

The correspondent adjoint function is of the form:

Ω I + (r,Ω,E)+μ(r,E) I + (r,Ω,E)=P(r,Ω,E)+ dΩ dE μ s (r,E)W(Ω,EΩ', E ) I + (r,Ω', E ). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiabgkHiTi abfM6axjabgwSixlabgEGirlaadMeadaahaaWcbeqaaiabgUcaRaaa kiaaiIcacaWHYbGaaGilaiabfM6axjaaiYcacaWGfbGaaGykaiabgU caRiabeY7aTjaaiIcacaWHYbGaaGilaiaadweacaaIPaGaamysamaa CaaaleqabaGaey4kaScaaOGaaGikaiaahkhacaaISaGaeuyQdCLaaG ilaiaadweacaaIPaGaaGypaiaadcfacaaIOaGaaCOCaiaaiYcacqqH PoWvcaaISaGaamyraiaaiMcacqGHRaWkaeaadaWdbaqabSqabeqani abgUIiYdGccaWGKbGaeuyQdC1aa8qaaeqaleqabeqdcqGHRiI8aOGa amizaiaadweacqaH8oqBdaWgaaWcbaGaam4CaaqabaGccaaIOaGaaC OCaiaaiYcacaWGfbGaaGykaiaadEfacaaIOaGaeuyQdCLaaGilaiaa dweacqGHsgIRcqqHPoWvcaaINaGaaGilaiqadweagaqbaiaaiMcaca WGjbWaaWbaaSqabeaacqGHRaWkaaGccaaIOaGaaCOCaiaaiYcacqqH PoWvcaaINaGaaGilaiqadweagaqbaiabgMcaPiabg6caUaaaaa@8115@ (5)

Both the equation can be transformed to the equivalent integral forms:67

I(r,Ω,E)= 0 e τ(r,rΩξ,E) Q ˜ (rξΩ,Ω,E)dξ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbGaaGikai aahkhacaaISaGaeuyQdCLaaGilaiaadweacaaIPaGaaGypamaapeha beWcbaGaaGimaaqaaiabg6HiLcqdcqGHRiI8aOGaamyzamaaCaaale qabaGaeyOeI0IaeqiXdqNaaGikaiaahkhacaaISaGaaCOCaiabgkHi TiabfM6axjabe67a4jaaiYcacaWGfbGaaGykaaaakmaaxacabaGaam yuaaqabeaacWaGOVibUdlaaiaaiIcacaWHYbGaeyOeI0IaeqOVdGNa euyQdCLaaGilaiabfM6axjaaiYcacaWGfbGaaGykaiaadsgacqaH+o aEaaa@6376@  (6)

with

Q ˜ (r,Ω,E)=Q(r,Ω,E)+ 4π dΩ' E E d E I(r,Ω', E ) μ s (r, E )W(Ω', E Ω,E), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfGaqaaiaadg faaeqabaGamai6lsG7WcaacaaIOaGaaCOCaiaaiYcacqqHPoWvcaaI SaGaamyraiaaiMcacaaI9aGaamyuaiaaiIcacaWHYbGaaGilaiabfM 6axjaaiYcacaWGfbGaaGykaiabgUcaRmaapefabeWcbaGaaGinaiab ec8aWbqab0Gaey4kIipakiaadsgacqqHPoWvcaaINaWaa8qCaeqale aacaWGfbaabaGabmyrayaafaaaniabgUIiYdGccaWGKbGabmyrayaa faGaamysaiaaiIcacaWHYbGaaGilaiabfM6axjaaiEcacaaISaGabm yrayaafaGaaGykaiabeY7aTnaaBaaaleaacaWGZbaabeaakiaaiIca caWHYbGaaGilaiqadweagaqbaiaaiMcacaWGxbGaaGikaiabfM6axj aaiEcacaaISaGabmyrayaafaGaeyOKH4QaeuyQdCLaaGilaiaadwea cqGHPaqkcqGHSaalaaa@7395@  (7)

and

I + (r,Ω,E)= 0 e τ(r,r+Ωξ;E) P ˜ (r+Ωξ,Ω;E)dξ, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaaWbaaS qabeaacqGHRaWkaaGccaaIOaGaaCOCaiaaiYcacqqHPoWvcaaISaGa amyraiaaiMcacaaI9aWaa8qCaeqaleaacaaIWaaabaGaeyOhIukani abgUIiYdGccaWGLbWaaWbaaSqabeaacqGHsislcqaHepaDcaaIOaGa aCOCaiaaiYcacaWHYbGaey4kaSIaeuyQdCLaeqOVdGNaaG4oaiaadw eacaaIPaaaaOWaaCbiaeaacaWGqbaabeqaaiadaI+Ie4oSaaGaaGik aiaahkhacqGHRaWkcqqHPoWvcqaH+oaEcaaISaGaeuyQdCLaaG4oai aadweacaaIPaGaamizaiabe67a4jaaiYcaaaa@654C@  (8)

with

P ˜ (r,Ω,E)=P(r,Ω,E)+ μ s (r,E) 4π dΩ' 0 E d E I + (r,Ω,E)W(Ω,EΩ', E ). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfGaqaaiaadc faaeqabaGamai6lsG7WcaacaaIOaGaaCOCaiaaiYcacqqHPoWvcaaI SaGaamyraiaaiMcacaaI9aGaamiuaiaaiIcacaWHYbGaaGilaiabfM 6axjaaiYcacaWGfbGaaGykaiabgUcaRiabeY7aTnaaBaaaleaacaWG ZbaabeaakiaaiIcacaWHYbGaaGilaiaadweacaaIPaWaa8quaeqale aacaaI0aGaeqiWdahabeqdcqGHRiI8aOGaamizaiabfM6axjaaiEca daWdXbqabSqaaiaaicdaaeaacaWGfbaaniabgUIiYdGccaWGKbGabm yrayaafaGaamysamaaCaaaleqabaGaey4kaScaaOGaaGikaiaahkha caaISaGaeuyQdCLaaGilaiaadweacaaIPaGaam4vaiaaiIcacqqHPo WvcaaISaGaamyraiabgkziUkabfM6axjaaiEcacaaISaGabmyrayaa faGaeyykaKIaaGOlaaaa@739B@  (9)

In these equations, τ(r,r±ΩξE) denotes the optical distance between the points for γ-quanta with energy E indicated in the argument:

τ(r,r±Ωξ;E)= 0 ξ μ(r±Ω ξ )d ξ . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDcaaIOa GaaCOCaiaaiYcacaWHYbGaeyySaeRaeuyQdCLaeqOVdGNaaG4oaiaa dweacaaIPaGaaGypamaapehabeWcbaGaaGimaaqaaiabe67a4bqdcq GHRiI8aOGaeqiVd0MaaGikaiaahkhacqGHXcqScqqHPoWvcuaH+oaE gaqbaiaaiMcacaWGKbGafqOVdGNbauaacaaIUaaaaa@56E4@  (10)

Inserting (7) into (6) and (9) into (8), we will see initial terms of the Neumann series representing solution of the integral equations. The terms with n=0 relate to non-scattered radiation, whereas the other terms describe contribution of scattered quanta (the integer n=1,2,3,...  indicate scattering multiplicity. In particular,

I 0 (r,Ω,E)= 0 e τ(r,rΩξ;E) Q(rΩξ,Ω,E)dξ, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaaSbaaS qaaiaaicdaaeqaaOGaaGikaiaahkhacaaISaGaeuyQdCLaaGilaiaa dweacaaIPaGaaGypamaapehabeWcbaGaaGimaaqaaiabg6HiLcqdcq GHRiI8aOGaamyzamaaCaaaleqabaGaeyOeI0IaeqiXdqNaaGikaiaa hkhacaaISaGaaCOCaiabgkHiTiabfM6axjabe67a4jaaiUdacaWGfb GaaGykaaaakiaadgfacaaIOaGaaCOCaiabgkHiTiabfM6axjabe67a 4jaaiYcacqqHPoWvcaaISaGaamyraiaaiMcacaWGKbGaeqOVdGNaaG ilaaaa@614A@  (11)

I 0 + (r,Ω,E)= 0 e τ(r,r+Ωξ;E) P(r+Ωξ,Ω,E)dξ, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaa0baaS qaaiaaicdaaeaacqGHRaWkaaGccaaIOaGaaCOCaiaaiYcacqqHPoWv caaISaGaamyraiaaiMcacaaI9aWaa8qCaeqaleaacaaIWaaabaGaey OhIukaniabgUIiYdGccaWGLbWaaWbaaSqabeaacqGHsislcqaHepaD caaIOaGaaCOCaiaaiYcacaWHYbGaey4kaSIaeuyQdCLaeqOVdGNaaG 4oaiaadweacaaIPaaaaOGaamiuaiaaiIcacaWHYbGaey4kaSIaeuyQ dCLaeqOVdGNaaGilaiabfM6axjaaiYcacaWGfbGaaGykaiaadsgacq aH+oaEcaaISaaaaa@6216@  (12)

I(r,Ω,E)= n=0 I n (r,Ω,E), I + (r,Ω,E)= n=0 I n + (r,Ω,E), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbGaeyikaG IaaCOCaiaaiYcacqqHPoWvcaWHSaGaaCyraiabgMcaPiabg2da9maa qahabeWcbaGaamOBaiaai2dacaaIWaaabaGaeyOhIukaniabggHiLd GccaWGjbWaaSbaaSqaaiaad6gaaeqaaOGaeyikaGIaaCOCaiaaiYca cqqHPoWvcaaISaGaamyraiabgMcaPiaaiYcacaaMf8UaaGzbVlaadM eadaahaaWcbeqaaiabgUcaRaaakiabgIcaOiaahkhacaaISaGaeuyQ dCLaaCilaiaahweacqGHPaqkcqGH9aqpdaaeWbqabSqaaiaad6gaca aI9aGaaGimaaqaaiabg6HiLcqdcqGHris5aOGaamysamaaDaaaleaa caWGUbaabaGaey4kaScaaOGaeyikaGIaaCOCaiaaiYcacqqHPoWvca aISaGaamyraiabgMcaPiaaiYcaaaa@6C1F@

and next terms are computed by induction:

I n (r,Ω,E)= 0 dξ e τ(r,rΩξ;E) μ s (rΩξ,E)× 4π dΩ' E d E W(Ω', E Ω,E) I n1 (rΩξ,Ω, E ) ,n=1,2, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiaadMeada WgaaWcbaGaamOBaaqabaGccaaIOaGaaCOCaiaaiYcacqqHPoWvcaaI SaGaamyraiaaiMcacaaI9aWaa8qCaeqaleaacaaIWaaabaGaeyOhIu kaniabgUIiYdGccaWGKbGaeqOVdGNaamyzamaaCaaaleqabaGaeyOe I0IaeqiXdqNaaGikaiaahkhacaaISaGaaCOCaiabgkHiTiabfM6axj abe67a4jaaiUdacaWGfbGaaGykaaaakiabeY7aTnaaBaaaleaacaWG ZbaabeaakiaaiIcacaWHYbGaeyOeI0IaeuyQdCLaeqOVdGNaaGilai aadweacaaIPaGaey41aqlabaWaamWaaeaadaWdrbqabSqaaiaaisda cqaHapaCaeqaniabgUIiYdGccaWGKbGaeuyQdCLaaG4jamaapehabe WcbaGaamyraaqaaiabg6HiLcqdcqGHRiI8aOGaamizaiqadweagaqb aiaadEfacaaIOaGaeuyQdCLaaG4jaiaaiYcaceWGfbGbauaacqGHsg IRcqqHPoWvcaaISaGaamyraiaaiMcacaWGjbWaaSbaaSqaaiaad6ga cqGHsislcaaIXaaabeaakiaaiIcacaWHYbGaeyOeI0IaeuyQdCLaeq OVdGNaaGilaiabfM6axjaahYcaceWHfbGbauaacaaIPaaacaGLBbGa ayzxaaGaaGilaiaaywW7caWGUbGaeyypa0JaaGymaiaaiYcacaaIYa GaaGilaiablAcilbaaaa@9466@  (13)

I n + (r,Ω,E)= 0 dξ e τ(r,r+Ωξ; E ) μ s (r+Ωξ,E)× 4π dΩ' 0 E d E W(Ω,EΩ', E ) I n1 + (r+Ωξ,Ω', E ) ,n=1,2, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiaadMeada qhaaWcbaGaamOBaaqaaiabgUcaRaaakiaaiIcacaWHYbGaaGilaiab fM6axjaaiYcacaWGfbGaaGykaiaai2dadaWdXbqabSqaaiaaicdaae aacqGHEisPa0Gaey4kIipakiaadsgacqaH+oaEcaWGLbWaaWbaaSqa beaacqGHsislcqaHepaDcaaIOaGaaCOCaiaaiYcacaWHYbGaey4kaS IaeuyQdCLaeqOVdGNaaG4oaiqadweagaqbaiaaiMcaaaGccqaH8oqB daWgaaWcbaGaam4CaaqabaGccaaIOaGaaCOCaiabgUcaRiabfM6axj abe67a4jaaiYcacaWGfbGaaGykaiabgEna0cqaamaadmaabaWaa8qu aeqaleaacaaI0aGaeqiWdahabeqdcqGHRiI8aOGaamizaiabfM6axj aaiEcadaWdXbqabSqaaiaaicdaaeaacaWGfbaaniabgUIiYdGccaWG KbGabmyrayaafaGaam4vaiaaiIcacqqHPoWvcaaISaGaamyraiabgk ziUkabfM6axjaaiEcacaaISaGabmyrayaafaGaaGykaiaadMeadaqh aaWcbaGaamOBaiabgkHiTiaaigdaaeaacqGHRaWkaaGccaaIOaGaaC OCaiabgUcaRiabfM6axjabe67a4jaaiYcacqqHPoWvcaaINaGaaGil aiqadweagaqbaiaaiMcaaiaawUfacaGLDbaacaaISaGaamOBaiabg2 da9iabggdaXiabgYcaSiabgkdaYiabgYcaSiabgAci8caaaa@95AF@  (14)

A point mono-directional source

Now we consider the field created by a point source of gamma-quanta placed at the origin O of the Cartesian coordinate and directed along z-axis, and let the field be measured by a point isotropic detector showing the integral

J= 4π dΩ 0 dEP(E)I( r 1 ,Ω,E) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGkbGaaGypam aapefabeWcbaGaaGinaiabec8aWbqab0Gaey4kIipakiaadsgacqqH PoWvdaWdXbqabSqaaiaaicdaaeaacqGHEisPa0Gaey4kIipakiaads gacaWGfbGaamiuaiaaiIcacaWGfbGaaGykaiaadMeacaaIOaGaaCOC amaaBaaaleaacaaIXaaabeaakiaaiYcacqqHPoWvcaaISaGaamyrai aaiMcaaaa@51CD@

with  standing for the detector response to a single photon with energy E. Thus, the free term in the adjoint equation (5) takes the form

P(r,Ω,E)=P(E)δ(r r 1 ), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaaGikai aahkhacaaISaGaeuyQdCLaaGilaiaadweacaaIPaGaaGypaiaadcfa caaIOaGaamyraiaaiMcacqaH0oazcaaIOaGaaCOCaiabgkHiTiaahk hadaWgaaWcbaGaaGymaaqabaGccqGHPaqkcaaISaaaaa@4A3A@  (15)

Where as the primary equation (4) has it in the form

Q(r,Ω,E)=δ(r)δ(Ω Ω 0 ) Q 0 (E) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGrbGaaGikai aahkhacaaISaGaeuyQdCLaaGilaiaadweacaaIPaGaaGypaiabes7a KjaaiIcacaWHYbGaaGykaiabes7aKjaaiIcacqqHPoWvcqGHsislcq qHPoWvdaWgaaWcbaGaaGimaaqabaGccaaIPaGaamyuamaaBaaaleaa caaIWaaabeaakiaaiIcacaWGfbGaaGykaaaa@4F75@  (16)

Q0(E)  denotes the source energy spectrum). The medium is supposed to be homogeneous and isotropic, so a suitable choice of coordinate system is dictated by specifics of the source-detector arrangement. An important role in this is given to symmetries, allowing for the reduction of the number of independent variables in the problem and thereby facilitating its solution (Figure 1). As we saw above, the same physical problem is expressed in two different forms differing in the degree of symmetry: Eq.(15) exhibits spherical symmetry whereas Eq.(16) reveals cylindrical one. So, it is naturally to choose the spherical symmetry version, possessing higher degree of symmetry and requiring only one spatial variable|rr1|. In what follows, we put r1=0, i.e. combine the origin of coordinates with the detector position, use the vector r for the initial position of the quantum, and introduce the angle θ via relation:

cosθ= Ωr r . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGGJbGaai4Bai aacohacqaH4oqCcaaI9aGaeyOeI0YaaSaaaeaacqqHPoWvcaWHYbaa baGaamOCaaaacaaIUaaaaa@423A@

On substitution (15) into Eq.(12), we obtain non-scattered term in adjoint function decomposition:

I 0 + (r,Ω,E)= 1 2π r 2 e τ(r,r';E) P(E)δ(1cosθ). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaa0baaS qaaiaaicdaaeaacqGHRaWkaaGccaaIOaGaaCOCaiaaiYcacqqHPoWv caWHSaGaaCyraiaaiMcacaaI9aWaaSaaaeaacaaIXaaabaGaaGOmai abec8aWjaadkhadaahaaWcbeqaaiaaikdaaaaaaOGaamyzamaaCaaa leqabaGaeyOeI0IaeqiXdqNaaGikaiaahkhacaaISaGaaCOCaiaaiE cacaaI7aGaamyraiaaiMcaaaGccaWGqbGaaGikaiaadweacaaIPaGa eqiTdqMaeyikaGIaeyymaeJaeyOeI0Iaai4yaiaac+gacaGGZbGaeq iUdeNaeyykaKIaaGOlaaaa@5E15@  (18)

For a homogeneous medium τ(r,r+Ωξ;E)=μ(E)ξ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDcaaIOa GaaCOCaiaaiYcacaWHYbGaey4kaSIaeuyQdCLaeqOVdGNaaG4oaiaa dweacaaIPaGaaGypaiabeY7aTjaaiIcacaWGfbGaaGykaiabe67a4b aa@49BE@ , so

I 0 + (r,Ω,E)= 1 2π r 2 e μ(E)ξ P(E)δ(1cosθ). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaa0baaS qaaiaaicdaaeaacqGHRaWkaaGccaaIOaGaaCOCaiaaiYcacqqHPoWv caWHSaGaaCyraiaaiMcacaaI9aWaaSaaaeaacaaIXaaabaGaaGOmai abec8aWjaadkhadaahaaWcbeqaaiaaikdaaaaaaOGaamyzamaaCaaa leqabaGaeyOeI0IaeqiVd0MaaGikaiaadweacaaIPaGaeqOVdGhaaO GaamiuaiaaiIcacaWGfbGaaGykaiabes7aKjabgIcaOiabggdaXiab gkHiTiaacogacaGGVbGaai4CaiabeI7aXjabgMcaPiabg6caUaaa@5BD7@  (19)

Figure 1Geometry of narrow ray. Q is the source place, P is a counter place.

Computing single scattered intensity, we take into account that the scattering indicatrix includes the delta-function expressing the Compton interrelation between initial quantum energy E, scattering angle θ'  and scattered quantum energy

E 1 = E 1+(1cos θ )E/m c 2 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaSbaaS qaaiaaigdaaeqaaOGaaGypamaalaaabaGaamyraaqaaiabggdaXiab gUcaRiabgIcaOiaaigdacqGHsislcaGGJbGaai4BaiaacohacuaH4o qCgaqbaiaaiMcacaWGfbGaaG4laiaad2gacaWGJbWaaWbaaSqabeaa caaIYaaaaaaakiaai6caaaa@49B5@

Denoting r+Ωξ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHYbGaey4kaS IaeuyQdCLaeqOVdGhaaa@3CE5@ by r' and the regular multiplier in the indicatrix by V(cos θ ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGwbGaaGikai aacogacaGGVbGaai4CaiqbeI7aXzaafaGaaGykaaaa@3E8A@ , we represent the latter as

W(ΩΩ')=V(cos θ )δ( E E 1 (cos θ ,E)), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbGaaGikai abfM6axjabgkziUkabfM6axjaaiEcacaaIPaGaaGypaiaadAfacaaI OaGaai4yaiaac+gacaGGZbGafqiUdeNbauaacaaIPaGaeqiTdqMaaG ikaiqadweagaqbaiabgkHiTiaadweadaWgaaWcbaGaaGymaaqabaGc caaIOaGaai4yaiaac+gacaGGZbGafqiUdeNbauaacaaISaGaamyrai abgMcaPiabgMcaPiaaiYcaaaa@5658@

where

cos θ = Ωr' r MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGGJbGaai4Bai aacohacuaH4oqCgaqbaiaai2dadaWcaaqaaiabfM6axjaahkhacaaI NaaabaGabmOCayaafaaaaaaa@415E@  (20)

and

V(cos θ )= 1 σ dσ dΩ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGwbGaaGikai aacogacaGGVbGaai4CaiqbeI7aXzaafaGaaGykaiaai2dadaWcaaqa aiaaigdaaeaacqaHdpWCaaWaaSaaaeaacaWGKbGaeq4WdmhabaGaam izaiabfM6axbaaaaa@4712@

is the normalized differential cross-section of the Compton scattering. In an explicit form,

dσ dΩ = r 0 2 2 E 1 E 2 E E 1 + E 1 E sin 2 θ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiaads gacqaHdpWCaeaacaWGKbGaeuyQdCfaaiaai2dadaWcaaqaaiaadkha daqhaaWcbaGaaGimaaqaaiaaikdaaaaakeaacaaIYaaaamaabmaaba WaaSaaaeaacaWGfbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamyraaaa aiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGcdaqadaqaamaala aabaGaamyraaqaaiaadweadaWgaaWcbaGaaGymaaqabaaaaOGaey4k aSYaaSaaaeaacaWGfbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamyraa aacqGHsisldaqfGaqabSqabeaacaaIYaaakeaacaGGZbGaaiyAaiaa c6gaaaGafqiUdeNbauaaaiaawIcacaGLPaaaaaa@545B@

(Klein-Nishina-Tamm formula) and

σ ε = 3 4 σ 0 1+ε ε 3 2ε(1+ε) 1+2ε ln(1+2ε) + 3 4 σ 0 1 2ε ln(1+2ε) 1+3ε (1+2ε) 2 ,ε= E m e c 2 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiabeo8aZn aaBaaaleaacqaH1oqzaeqaaOGaaGypamaalaaabaGaaG4maaqaaiaa isdaaaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaOWaaiWaaeaadaWcaa qaaiaaigdacqGHRaWkcqaH1oqzaeaacqaH1oqzdaahaaWcbeqaaiaa iodaaaaaaOWaamWaaeaadaWcaaqaaiaaikdacqaH1oqzcqGHOaakca aIXaGaey4kaSIaeqyTduMaeyykaKcabaGaaGymaiabgUcaRiaaikda cqaH1oqzaaGaeyOeI0IaaiiBaiaac6gacqGHOaakcqGHXaqmcqGHRa WkcaaIYaGaeqyTduMaaGykaaGaay5waiaaw2faaaGaay5Eaiaaw2ha aiabgUcaRaqaamaalaaabaGaaG4maaqaaiaaisdaaaGaeq4Wdm3aaS baaSqaaiaaicdaaeqaaOWaaiWaaeaadaWcaaqaaiaaigdaaeaacaaI YaGaeqyTdugaaiaacYgacaGGUbGaeyikaGIaaGymaiabgUcaRiaaik dacqaH1oqzcqGHPaqkcqGHsisldaWcaaqaaiaaigdacqGHRaWkcaaI ZaGaeqyTdugabaGaeyikaGIaaGymaiabgUcaRiaaikdacqaH1oqzcq GHPaqkdaahaaWcbeqaaiaaikdaaaaaaaGccaGL7bGaayzFaaGaaGil aiaaywW7cqaH1oqzcaaI9aWaaSaaaeaacaWGfbaabaGaamyBamaaBa aaleaacaWGLbaabeaakiaadogadaahaaWcbeqaaiaaikdaaaaaaOGa aGOlaaaaaa@866E@  (21)

Inserting n=1 into Eq.(14), using Eq.(19) and property of delta-function, we arrive at the following expression for single scattering contribution into the sought quantity

J 1 = I + (r,Ω,E)= 0 e μ(E)ξμ( E 1 ) r μ s (E)V(cos θ )P( E 1 ) dξ r 2 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGkbWaaSbaaS qaaiaaigdaaeqaaOGaaGypaiaadMeadaahaaWcbeqaaiabgUcaRaaa kiaaiIcacaWHYbGaaGilaiabfM6axjaaiYcacaWGfbGaaGykaiaai2 dadaWdXbqabSqaaiaaicdaaeaacqGHEisPa0Gaey4kIipakiaadwga daahaaWcbeqaaiabgkHiTiabeY7aTjaaiIcacaWGfbGaaGykaiabe6 7a4jabgkHiTiabeY7aTjaaiIcacaWGfbWaaSbaaeaacaaIXaaabeaa caaIPaGabmOCayaafaaaaOGaeqiVd02aaSbaaSqaaiaadohaaeqaaO GaaGikaiaadweacaaIPaGaamOvaiaaiIcacaGGJbGaai4Baiaacoha cuaH4oqCgaqbaiaaiMcacaWGqbGaaGikaiaadweadaWgaaWcbaGaaG ymaaqabaGccaaIPaWaaSaaaeaacaWGKbGaeqOVdGhabaGabmOCayaa faWaaWbaaSqabeaacaaIYaaaaaaakiaai6caaaa@6B5B@

With fixed variables r and Ω, the variable ξ is uniquely linked with the scattering angle θ'(Figure 1). Applying the sinus-theorem yields interrelations

r =r sinθ sin θ ξ=r sin( θ θ) sin θ ,dξ=r sinθ sin 2 θ d θ . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGYbGbauaaca aI9aGaamOCamaalaaabaGaai4CaiaacMgacaGGUbGaeqiUdehabaGa ai4CaiaacMgacaGGUbGafqiUdeNbauaaaaGaeqOVdGNaaGypaiaadk hadaWcaaqaaiaacohacaGGPbGaaiOBaiaaiIcacuaH4oqCgaqbaiab gkHiTiabeI7aXjaaiMcaaeaacaGGZbGaaiyAaiaac6gacuaH4oqCga qbaaaacaaISaGaaGzbVlaadsgacqaH+oaEcaaI9aGaamOCamaalaaa baGaai4CaiaacMgacaGGUbGaeqiUdehabaWaaubiaeqaleqabaGaaG OmaaGcbaGaai4CaiaacMgacaGGUbaaaiqbeI7aXzaafaaaaiaadsga cuaH4oqCgaqbaiaai6caaaa@68C9@

Using these formulae for passage to other variables leads us to expression

J= μ s (E) rsinθ θ θ s exp r sin θ [μ(E)sin( θ θ)+μ( E 1 )sinθ] V(cos θ )P( E 1 )d θ . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGkbGaaGypam aalaaabaGaeqiVd02aaSbaaSqaaiaadohaaeqaaOGaaGikaiaadwea caaIPaaabaGaamOCaiaacohacaGGPbGaaiOBaiabeI7aXbaadaWdXb qabSqaaiabeI7aXbqaaiabeI7aXnaaBaaabaGaam4Caaqabaaaniab gUIiYdGccaGGLbGaaiiEaiaacchadaGadaqaaiabgkHiTmaalaaaba GaamOCaaqaaiaacohacaGGPbGaaiOBaiqbeI7aXzaafaaaaiaaiUfa cqaH8oqBcaaIOaGaamyraiaaiMcacaGGZbGaaiyAaiaac6gacaaIOa GafqiUdeNbauaacqGHsislcqaH4oqCcaaIPaGaey4kaSIaeqiVd0Ma aGikaiaadweadaWgaaWcbaGaaGymaaqabaGccaaIPaGaai4CaiaacM gacaGGUbGaeqiUdeNaaGyxaaGaay5Eaiaaw2haaiaadAfacaaIOaGa ai4yaiaac+gacaGGZbGafqiUdeNbauaacaaIPaGaamiuaiaaiIcaca WGfbWaaSbaaSqaaiaaigdaaeqaaOGaaGykaiaadsgacuaH4oqCgaqb aiaai6caaaa@7D42@  (22)

In case of an infinite homogeneous medium (coefficients μ and μs are independent of spatial variables), the upper limit of integration θs=, but this formula can be used and on the plain boundary with an absorbing medium, when the source is in a homogeneous semi-space and the point of measurement is on a plane surface perpendicular to direction Ω MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHPoWvaaa@3945@ . In this case the lower limit of integrating is zero upper limit θ s MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH4oqCdaWgaa WcbaGaam4Caaqabaaaaa@3A91@ should be taken π/2

The formulae for single-scattered quanta obtained above, includes multiplier [rsinθ]1, which grows unboundedly with decreasing 0. Consequently, at small θ or r the rest terms (with n>1) may be ignored.

Singularity of the adjoint function at great depths

Let us pass to the cylindrical coordinate system each point of which is characterized by longitudinal (z=rcosθ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGGOaGaamOEai aai2dacaWGYbGaai4yaiaac+gacaGGZbGaeqiUdeNaaiykaaaa@4054@  coordinate (depth) and transverse one (ρ=rsinθ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGGOaGaeqyWdi NaaGypaiaadkhacaGGZbGaaiyAaiaac6gacqaH4oqCcaGGPaaaaa@411A@ (radius). Then

I 1 + (r,Ω,E)= μ s (E) ρ θ θ s exp μ(E)ρctg θ μ(E)ρcosec θ V(cos θ )P( E 1 )d θ . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaa0baaS qaaiaaigdaaeaacqGHRaWkaaGccaaIOaGaaCOCaiaaiYcacqqHPoWv caaISaGaamyraiaaiMcacaaI9aWaaSaaaeaacqaH8oqBdaWgaaWcba Gaam4CaaqabaGccaaIOaGaamyraiaaiMcaaeaacqaHbpGCaaWaa8qC aeqaleaacqaH4oqCaeaacqaH4oqCdaWgaaqaaiaadohaaeqaaaqdcq GHRiI8aOGaaiyzaiaacIhacaGGWbWaaiWaaeaacqaH8oqBcaaIOaGa amyraiaaiMcacqaHbpGCcaaMi8Uaae4yaiaabshacaqGNbGaaGjcVl qbeI7aXzaafaGaeyOeI0IaeqiVd0MaaGikaiaadweacaaIPaGaeqyW diNaaGjcVlaabogacaqGVbGaae4CaiaabwgacaqGJbGaaGjcVlqbeI 7aXzaafaaacaGL7bGaayzFaaGaamOvaiaaiIcacaGGJbGaai4Baiaa cohacuaH4oqCgaqbaiaaiMcacaWGqbGaaGikaiaadweadaWgaaWcba GaaGymaaqabaGccaaIPaGaamizaiqbeI7aXzaafaGaaGOlaaaa@7FB7@  (23)

Expand the exponential function in series and denote ith term of this series by φi:i=0,1,2,... :

φ 0 = μ s (E) ρ e μ(E)z θ θ s V(cos θ )P( E 1 )d θ , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHgpGAdaWgaa WcbaGaaGimaaqabaGccaaI9aWaaSaaaeaacqaH8oqBdaWgaaWcbaGa am4CaaqabaGccaaIOaGaamyraiaaiMcaaeaacqaHbpGCaaGaaGjcVl aadwgadaahaaWcbeqaaiabgkHiTiabeY7aTjaaiIcacaWGfbGaaGyk aiaadQhaaaGcdaWdXbqabSqaaiabeI7aXbqaaiabeI7aXnaaBaaaba Gaam4CaaqabaaaniabgUIiYdGccaWGwbGaaGikaiaacogacaGGVbGa ai4CaiqbeI7aXzaafaGaaGykaiaadcfacaaIOaGaamyramaaBaaale aacaaIXaaabeaakiaaiMcacaWGKbGafqiUdeNbauaacaaISaaaaa@5FB3@  (24)

φ 1 = μ s (E) e μ(E)z θ θ s [μ(E)ctg θ μ( E 1 )cosec θ ]V(cos θ )P( E 1 )d θ , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHgpGAdaWgaa WcbaGaaGymaaqabaGccaaI9aGaeqiVd02aaSbaaSqaaiaadohaaeqa aOGaaGikaiaadweacaaIPaGaamyzamaaCaaaleqabaGaeyOeI0Iaeq iVd0MaaGikaiaadweacaaIPaGaamOEaaaakmaapehabeWcbaGaeqiU dehabaGaeqiUde3aaSbaaeaacaWGZbaabeaaa0Gaey4kIipakiaaiU facqaH8oqBcaaIOaGaamyraiaaiMcacaaMi8Uaae4yaiaabshacaqG NbGaaGjcVlqbeI7aXzaafaGaeyOeI0IaeqiVd0MaaGikaiaadweada WgaaWcbaGaaGymaaqabaGccaaIPaGaaGjcVlaabogacaqGVbGaae4C aiaabwgacaqGJbGaaGjcVlqbeI7aXzaafaGaaGyxaiaadAfacaaIOa Gaai4yaiaac+gacaGGZbGafqiUdeNbauaacaaIPaGaamiuaiaaiIca caWGfbWaaSbaaSqaaiaaigdaaeqaaOGaaGykaiaadsgacuaH4oqCga qbaiaaiYcaaaa@78F2@  (25)

φ 2 = 1 2 μ s (E)ρ e μ(E)z θ θ s [μ(E)ctg θ μ( E 1 )cosec θ ] 2 V(cos θ )P( E 1 )d θ , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHgpGAdaWgaa WcbaGaaGOmaaqabaGccaaI9aWaaSaaaeaacaaIXaaabaGaaGOmaaaa cqaH8oqBdaWgaaWcbaGaam4CaaqabaGccaaIOaGaamyraiaaiMcacq aHbpGCcaWGLbWaaWbaaSqabeaacqGHsislcqaH8oqBcaaIOaGaamyr aiaaiMcacaWG6baaaOWaa8qCaeqaleaacqaH4oqCaeaacqaH4oqCda WgaaqaaiaadohaaeqaaaqdcqGHRiI8aOGaaG4waiabeY7aTjaaiIca caWGfbGaaGykaiaayIW7caqGJbGaaeiDaiaabEgacaaMi8UafqiUde NbauaacqGHsislcqaH8oqBcaaIOaGaamyramaaBaaaleaacaaIXaaa beaakiaaiMcacaaMi8Uaae4yaiaab+gacaqGZbGaaeyzaiaabogaca aMi8UafqiUdeNbauaacaaIDbWaaWbaaSqabeaacaaIYaaaaOGaamOv aiaaiIcacaGGJbGaai4BaiaacohacuaH4oqCgaqbaiaaiMcacaWGqb GaaGikaiaadweadaWgaaWcbaGaaGymaaqabaGccaaIPaGaamizaiqb eI7aXzaafaGaaGilaaaa@7D2D@  (26)

and so on.

Let the counter be far enough from the source and from the boundary (i.e. z>ρ and h<ρ). In this case, θ ρ z 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH4oqCcqGHij YUdaWcaaqaaiabeg8aYbqaaiaadQhaaaGaeyOKH4QaaGimaaaa@4094@ and θ s π ρ h π MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH4oqCdaWgaa WcbaGaam4CaaqabaGccqGHijYUcqaHapaCcqGHsisldaWcaaqaaiab eg8aYbqaaiaadIgaaaGaeyOKH4QaeqiWdahaaa@455D@ , therefore,

φ 0 = μ s (E) ρ e μ(E)z 0 π V(cos θ )P( E 1 )d θ . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHgpGAdaWgaa WcbaGaaGimaaqabaGccaaI9aWaaSaaaeaacqaH8oqBdaWgaaWcbaGa am4CaaqabaGccaaIOaGaamyraiaaiMcaaeaacqaHbpGCaaGaaGjcVl aadwgadaahaaWcbeqaaiabgkHiTiabeY7aTjaaiIcacaWGfbGaaGyk aiaadQhaaaGcdaWdXbqabSqaaiaaicdaaeaacqaHapaCa0Gaey4kIi pakiaadAfacaaIOaGaai4yaiaac+gacaGGZbGafqiUdeNbauaacaaI PaGaamiuaiaaiIcacaWGfbWaaSbaaSqaaiaaigdaaeqaaOGaaGykai aadsgacuaH4oqCgaqbaiaai6caaaa@5DA7@  (27)

The integrand in Eq.(25) has a singularity at θ=0 and θ=π. Let us break the integral into three pieces θ θ s = ρ/z ε + ε πε + πε πρ/h MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWdXbqabSqaai abeI7aXbqaaiabeI7aXnaaBaaabaGaam4CaaqabaaaniabgUIiYdGc caaI9aWaa8qCaeqaleaacqaHbpGCcaaIVaGaamOEaaqaaiabew7aLb qdcqGHRiI8aOGaey4kaSYaa8qCaeqaleaacqaH1oqzaeaacqaHapaC cqGHsislcqaH1oqza0Gaey4kIipakiabgUcaRmaapehabeWcbaGaeq iWdaNaeyOeI0IaeqyTdugabaGaeqiWdaNaeyOeI0IaeqyWdiNaaG4l aiaadIgaa0Gaey4kIipaaaa@5E05@ , where ε MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzaaa@395E@ is a small positive number choosing in such a way that all terms in this sum are positive. Neglecting the change of μ,K and P in the ε-interval and leaving only singular terms, we obtain

φ 1 = μ s (E) e μ(E)z [μ(E)+μ( E π )]V(cosπ)P( E z )lnρ, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHgpGAdaWgaa WcbaGaaGymaaqabaGccaaI9aGaeqiVd02aaSbaaSqaaiaadohaaeqa aOGaaGikaiaadweacaaIPaGaamyzamaaCaaaleqabaGaeyOeI0Iaeq iVd0MaaGikaiaadweacaaIPaGaamOEaaaakiaaiUfacqaH8oqBcaaI OaGaamyraiaaiMcacqGHRaWkcqaH8oqBcaaIOaGaamyramaaBaaale aacqaHapaCaeqaaOGaaGykaiaai2facaWGwbGaaGikaiaacogacaGG VbGaai4Caiabec8aWjaaiMcacaWGqbGaaGikaiaadweadaWgaaWcba GaamOEaaqabaGccaaIPaGaaGjcVlaabYgacaqGUbGaaGjcVlabeg8a YjaaiYcaaaa@661B@  (28)

where

EπE1(cosπ,E).

Acting in a similar way, one can show that  and next terms in this sum at ρ0 do not have singularity.
If the counter is placed on the very boundary, then singularity characterizes only the first term (24) in the expansion of (23)

φ 0 = μ s (E) ρ e μ(E)z 0 π/2 V(cos θ )P( E 1 )d θ . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHgpGAdaWgaa WcbaGaaGimaaqabaGccaaI9aWaaSaaaeaacqaH8oqBdaWgaaWcbaGa am4CaaqabaGccaaIOaGaamyraiaaiMcaaeaacqaHbpGCaaGaaGjcVl aadwgadaahaaWcbeqaaiabgkHiTiabeY7aTjaaiIcacaWGfbGaaGyk aiaadQhaaaGcdaWdXbqabSqaaiaaicdaaeaacqaHapaCcaaIVaGaaG OmaaqdcqGHRiI8aOGaamOvaiaaiIcacaGGJbGaai4BaiaacohacuaH 4oqCgaqbaiaaiMcacaWGqbGaaGikaiaadweadaWgaaWcbaGaaGymaa qabaGccaaIPaGaamizaiqbeI7aXzaafaGaaGOlaaaa@5F1C@  (29)

Let us introduce the notations

Θ(ϕ,ψ;E)= 2π P(E) ϕ ψ V(cos θ )P( E 1 )d θ , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHyoqucaaIOa Gaeqy1dyMaaGilaiabeI8a5jaaiUdacaWGfbGaaGykaiaai2dadaWc aaqaaiaaikdacqaHapaCaeaacaWGqbGaaGikaiaadweacaaIPaaaam aapehabeWcbaGaeqy1dygabaGaeqiYdKhaniabgUIiYdGccaWGwbGa aGikaiaacogacaGGVbGaai4CaiqbeI7aXzaafaGaaGykaiaadcfaca aIOaGaamyramaaBaaaleaacaaIXaaabeaakiaaiMcacaWGKbGafqiU deNbauaacaaISaaaaa@5AF5@

and Θ=Θ(0,ψ;E) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHyoqucqGH9a qpcqqHyoqucqGHOaakcaaIWaGaaGilaiabeI8a5jaaiUdacaWGfbGa eyykaKcaaa@4232@ , then for both above mentioned positions of the counter the expression

I + (r,Ω,E) I 0 + (r,Ω,E)+ I 1 + (r,Ω,E) 1 2πρ [δ(ρ)+ μ s (E)Θ]P(E) e μ(E)z ,ρ0, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiaadMeada ahaaWcbeqaaiabgUcaRaaakiaaiIcacaWHYbGaaGilaiabfM6axjaa iYcacaWGfbGaaGykaiablYJi6iaadMeadaqhaaWcbaGaaGimaaqaai abgUcaRaaakiaaiIcacaWHYbGaaGilaiabfM6axjaaiYcacaWGfbGa aGykaiabgUcaRiaadMeadaqhaaWcbaGaaGymaaqaaiabgUcaRaaaki aaiIcacaWHYbGaaGilaiabfM6axjaaiYcacaWGfbGaaGykaiabgIKi 7cqaamaalaaabaGaaGymaaqaaiaaikdacqaHapaCcqaHbpGCaaGaaG 4waiabes7aKjaaiIcacqaHbpGCcaaIPaGaey4kaSIaeqiVd02aaSba aSqaaiaadohaaeqaaOGaaGikaiaadweacaaIPaGaeuiMdeLaaGyxai aadcfacaaIOaGaamyraiaaiMcacaWGLbWaaWbaaSqabeaacqGHsisl cqaH8oqBcaaIOaGaamyraiaaiMcacaWG6baaaOGaaGilaiaaywW7cq aHbpGCcqGHsgIRcaaIWaGaaGilaaaaaa@796D@  (30)

stay be valid, if put θs=π/2  in the boundary case, and π otherwise (more weak logarithmic singularity are omitted). As a result, we get:

I + (r,Ω,E;)= 1 2πρ δ(ρ)+ μ s Θ P(E) e μz . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaaWbaaS qabeaacqGHRaWkaaGccaaIOaGaaCOCaiaaiYcacqqHPoWvcaaISaGa amyraiaaiUdacaaIPaGaaGypamaalaaabaGaaGymaaqaaiaaikdacq aHapaCcqaHbpGCaaWaamWaaeaacqaH0oazcaaIOaGaeqyWdiNaaGyk aiabgUcaRiabeY7aTnaaBaaaleaacaWGZbaabeaakiabfI5arbGaay 5waiaaw2faaiaadcfacaaIOaGaamyraiaaiMcacaWGLbWaaWbaaSqa beaacqGHsislcqaH8oqBcaWG6baaaOGaaGOlaaaa@5ACA@

Let I 0 (x,y) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaaSbaaS qaaiaaicdaaeqaaOGaaGikaiaadIhacaaISaGaamyEaiaaiMcaaaa@3D8B@  be the initial profile of the ray at z=0 and I(x,y,z) the ray profile at depth . According to the above result, the latter is composed from two terms: unscattered part repeating the shape of the incident ray, I nsc (x,y,z)= I 0 (x,y) e μz , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaaWbaaS qabeaacaqGUbGaae4CaiaabogaaaGccaaIOaGaamiEaiaaiYcacaWG 5bGaaGilaiaadQhacaaIPaGaaGypaiaadMeadaWgaaWcbaGaaGimaa qabaGccaaIOaGaamiEaiaaiYcacaWG5bGaaGykaiaadwgadaahaaWc beqaaiabgkHiTiabeY7aTjaadQhaaaGccaaISaaaaa@4D68@ and scattered component

I sc (x,y,z)= μ s Θ e μz Σ I 0 ( x , y )d x d y (x x ) 2 + (y y ) 2 , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaaWbaaS qabeaacaqGZbGaae4yaaaakiaaiIcacaWG4bGaaGilaiaadMhacaaI SaGaamOEaiaaiMcacaaI9aGaeqiVd02aaSbaaSqaaiaadohaaeqaaO GaeuiMdeLaamyzamaaCaaaleqabaGaeyOeI0IaeqiVd0MaamOEaaaa kmaapefabeWcbaGaeu4OdmfabeqdcqGHRiI8aOWaaSaaaeaacaWGjb WaaSbaaSqaaiaaicdaaeqaaOGaaGikaiqadIhagaqbaiaaiYcaceWG 5bGbauaacaaIPaGaamizaiqadIhagaqbaiaadsgaceWG5bGbauaaae aadaGcaaqaaiaaiIcacaWG4bGaeyOeI0IabmiEayaafaGaaGykamaa CaaaleqabaGaaGOmaaaakiabgUcaRiaaiIcacaWG5bGaeyOeI0Iabm yEayaafaGaaGykamaaCaaaleqabaGaaGOmaaaaaeqaaaaakiaaiYca aaa@6442@  (31)

where Σ  stands for a cross section of the incident ray. The formula establishes the connection between the clear image of the ray given by the unscattered component, and the diffusion "halo," responsible for which is the scattering process. This connection is reminiscent of the relationship known in electrostatics between the charge distribution σ( x , y )= μ s Θ I 0 ( x , y ) e μz MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCcaaIOa GabmiEayaafaGaaGilaiqadMhagaqbaiaaiMcacaaI9aGaeqiVd02a aSbaaSqaaiaadohaaeqaaOGaeuiMdeLaamysamaaBaaaleaacaaIWa aabeaakiaaiIcaceWG4bGbauaacaaISaGabmyEayaafaGaaGykaiaa dwgadaahaaWcbeqaaiabgkHiTiabeY7aTjaadQhaaaaaaa@4D6F@  on the plane z=const MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG6bGaaGypai aabogacaqGVbGaaeOBaiaabohacaqG0baaaa@3E33@ and the potential ϕ(x,y)= I sc (x,y) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHvpGzcaaIOa GaamiEaiaaiYcacaWG5bGaaGykaiaai2dacaWGjbWaaWbaaSqabeaa caqGZbGaam4yaaaakiaaiIcacaWG4bGaaGilaiaadMhacaaIPaaaaa@4555@ created by it in this plane.

In case of a uniform distribution of the initial flux over cross-section having the circular form with radius a,

σ(r)= σ 0 , r<a; 0, r>0, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCcaaIOa GaamOCaiaaiMcacaaI9aWaaiqaaeaafaqaaeWacaaabaGaeq4Wdm3a aSbaaSqaaiaaicdaaeqaaOGaaGilaaqaaiaadkhacaaI8aGaamyyai aaiUdaaeaacaaIWaGaaGilaaqaaiaadkhacqGH+aGpcqGHWaamcaaI SaaabaaabaaaaaGaay5Eaaaaaa@49AB@

and

ϕ(r)= 0 a 0 2π σ 0 r d r dϑ r 2 + r 2 2r r cosϑ . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHvpGzcaaIOa GaamOCaiaaiMcacaaI9aWaa8qCaeqaleaacaaIWaaabaGaamyyaaqd cqGHRiI8aOWaa8qCaeqaleaacaaIWaaabaGaaGOmaiabec8aWbqdcq GHRiI8aOWaaSaaaeaacqaHdpWCdaWgaaWcbaGaaGimaaqabaGcceWG YbGbauaacaWGKbGabmOCayaafaGaamizaiabeg9akbqaamaakaaaba GaamOCamaaCaaaleqabaGaaGOmaaaakiabgUcaRiqadkhagaqbamaa CaaaleqabaGaaGOmaaaakiabgkHiTiaaikdacaWGYbGabmOCayaafa Gaai4yaiaac+gacaGGZbGaeqy0dOealeqaaaaakiaai6caaaa@5C5B@

Integration with respect to r' yields

ϕ(r)=2 σ 0 0 π ( r 2 + a 2 2racosϑ r+ +rcosϑln arcosϑ+ r 2 + a 2 2racosϑ r(1cosϑ) )dϑ. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiabew9aMj aaiIcacaWGYbGaeyykaKIaeyypa0JaaGOmaiabeo8aZnaaBaaaleaa caaIWaaabeaakmaapehabeWcbaGaaGimaaqaaiabec8aWbqdcqGHRi I8aOGaaGikamaakaaabaGaamOCamaaCaaaleqabaGaaGOmaaaakiab gUcaRiaadggadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaIYaGaam OCaiaadggacaGGJbGaai4BaiaacohacqaHrpGsaSqabaGccqGHsisl caWGYbGaey4kaScabaGaey4kaSIaamOCaiaacogacaGGVbGaai4Cai abeg9akjaayIW7caqGSbGaaeOBaiaayIW7daWcaaqaaiaadggacqGH sislcaWGYbGaai4yaiaac+gacaGGZbGaeqy0dOKaey4kaSYaaOaaae aacaWGYbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamyyamaaCaaa leqabaGaaGOmaaaakiabgkHiTiaaikdacaWGYbGaamyyaiaacogaca GGVbGaai4Caiabeg9akbWcbeaaaOqaaiaadkhacaaIOaGaaGymaiab gkHiTiaacogacaGGVbGaai4Caiabeg9akjaaiMcaaaGaaGykaiaads gacqaHrpGscaaIUaaaaaa@8279@

Further, introducing

χ= πϑ 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHhpWycqGH9a qpdaWcaaqaaiabec8aWjabgkHiTiabeg9akbqaaiaaikdaaaaaaa@3F92@

and integrating the term with logarithm by parts, we obtain

ϕ(r)=2 σ 0 (a+r)E(k)+(ar)K(k) ,k= 2 ra r+a , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHvpGzcaaIOa GaamOCaiabgMcaPiabg2da9iabgkdaYiabeo8aZnaaBaaaleaacaaI WaaabeaakmaadmaabaGaaGikaiaadggacqGHRaWkcaWGYbGaaGykai aadweacaaIOaGaam4AaiaaiMcacqGHRaWkcaaIOaGaamyyaiabgkHi TiaadkhacaaIPaGaam4saiaaiIcacaWGRbGaaGykaaGaay5waiaaw2 faaiaaiYcacaaMf8Uaam4Aaiaai2dadaWcaaqaaiaaikdadaGcaaqa aiaadkhacaWGHbaaleqaaaGcbaGaamOCaiabgUcaRiaadggaaaGaaG ilaaaa@5C58@  (32)

where K MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbaaaa@3887@  and E stand for total elliptic integrals of the first and second kind respectively. At r=0, E=K= π 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbGaaGypai aadUeacaaI9aWaaSaaaeaacqaHapaCaeaacaaIYaaaaaaa@3D68@ and Eq. (32) becomes ϕ(0)=2πaσ0.

In the general case of an axially symmetrical initial profile, the haloϕ(r) Is expressed via integral

ϕ(r)= 0 0 2π σ( r ) r d r dϑ r 2 r 2 2r r cosϑ . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHvpGzcaaIOa GaamOCaiaaiMcacaaI9aWaa8qCaeqaleaacaaIWaaabaGaeyOhIuka niabgUIiYdGcdaWdXbqabSqaaiaaicdaaeaacaaIYaGaeqiWdahani abgUIiYdGcdaWcaaqaaiabeo8aZjaaiIcaceWGYbGbauaacaaIPaGa bmOCayaafaGaamizaiqadkhagaqbaiaadsgacqaHrpGsaeaadaGcaa qaaiaadkhadaahaaWcbeqaaiaaikdaaaGccqGHsislceWGYbGbauaa daahaaWcbeqaaiaaikdaaaGccqGHsislcaaIYaGaamOCaiqadkhaga qbaiaacogacaGGVbGaai4Caiabeg9akbWcbeaaaaGccaaIUaaaaa@5E69@  (33)

Expanding inverse distance with respect to Legendre polynomials,

1 r 2 + r 2 2r r cosϑ = 1 r > n=0 r < r > n P n (cosϑ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiaaig daaeaadaGcaaqaaiaadkhadaahaaWcbeqaaiaaikdaaaGccqGHRaWk ceWGYbGbauaadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaIYaGaam OCaiqadkhagaqbaiaacogacaGGVbGaai4Caiabeg9akbWcbeaaaaGc caaI9aWaaSaaaeaacaaIXaaabaGaamOCamaaBaaaleaacaaI+aaabe aaaaGcdaaeWbqabSqaaiaad6gacaaI9aGaaGimaaqaaiabg6HiLcqd cqGHris5aOWaaeWaaeaadaWcaaqaaiaadkhadaWgaaWcbaGaaGipaa qabaaakeaacaWGYbWaaSbaaSqaaiaai6daaeqaaaaaaOGaayjkaiaa wMcaamaaCaaaleqabaGaamOBaaaakiaadcfadaWgaaWcbaGaamOBaa qabaGccaaIOaGaai4yaiaac+gacaGGZbGaeqy0dOKaaGykaaaa@5DC5@

and inserting this into (33) yields:

ϕ(r)= n=0 C n 0 r r r n+1 σ( r )d r + r r r n σ( r )d r , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHvpGzcaaIOa GaamOCaiaaiMcacaaI9aWaaabCaeqaleaacaWGUbGaaGypaiaaicda aeaacqGHEisPa0GaeyyeIuoakiaadoeadaWgaaWcbaGaamOBaaqaba GcdaWadaqaamaapehabeWcbaGaaGimaaqaaiaadkhaa0Gaey4kIipa kmaabmaabaWaaSaaaeaaceWGYbGbauaaaeaacaWGYbaaaaGaayjkai aawMcaamaaCaaaleqabaGaamOBaiabgUcaRiaaigdaaaGccqaHdpWC caaIOaGabmOCayaafaGaaGykaiaadsgaceWGYbGbauaacqGHRaWkda WdXbqabSqaaiaadkhaaeaacqGHEisPa0Gaey4kIipakmaabmaabaWa aSaaaeaacaWGYbaabaGabmOCayaafaaaaaGaayjkaiaawMcaamaaCa aaleqabaGaamOBaaaakiabeo8aZjaaiIcaceWGYbGbauaacaaIPaGa amizaiqadkhagaqbaaGaay5waiaaw2faaiaaiYcaaaa@6895@

where coefficients

Cn=02πPn(cosϑ)=2πPn2(0)

are equal to 0 for odd n, and to 2π 1 2 n n n/2 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIYaGaeqiWda 3aamWaaeaadaWcaaqaaiaaigdaaeaacaaIYaWaaWbaaSqabeaacaWG Ubaaaaaakmaabmaabaqbaeqabmqaaaqaaiaad6gaaeaacaWGUbGaey 4la8IaaGOmaaqaaaaaaiaawIcacaGLPaaaaiaawUfacaGLDbaadaah aaWcbeqaaiaaikdaaaaaaa@44E0@ for even. Expression (33) can be integrated over the angle directly.

ϕ(r)=4 0 K( k ) σ( r ) r d r r+ r , k = 2 r r r+ r . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=wjYlH8qqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHvpGzcaaIOa GaamOCaiabgMcaPiabg2da9iaaisdadaWdXbqabSqaaiaaicdaaeaa cqGHEisPa0Gaey4kIipakiaadUeacaaIOaGabm4AayaafaGaaGykam aalaaabaGaeq4WdmNaaGikaiqadkhagaqbaiaaiMcaceWGYbGbauaa caWGKbGabmOCayaafaaabaGaamOCaiabgUcaRiqadkhagaqbaaaaca aISaGaaGzbVlqadUgagaqbaiaai2dadaWcaaqaaiaaikdadaGcaaqa aiaadkhaceWGYbGbauaaaSqabaaakeaacaWGYbGaey4kaSIabmOCay aafaaaaiaai6caaaa@5A26@

In case of absence of axial symmetry the problem can be solved by numerical integration in expression (31).

Acknowledgments

This work is partially supported by the Russian Foundation of Basic Research (Projects 16-01-00556 and 18-51-53018).

Conflict of interest

Authors declare there is no conflicts of interest.

References

Creative Commons Attribution License

©2018 Uchaikin. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.