Channeling of relativistic electrons and positrons

The quanta-mechanical problem of relativistic electron (positron) movement through the electro-static crystal field for the case of small angle between initial particle momentum and the atomic plane is considered for the case of electron energy larger than several MeV. The analytical expressions f...

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Datum:	2009
1. Verfasser:	Morokhovskii, V.L.
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Sprache:	English
Veröffentlicht:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2009
Schriftenreihe:	Вопросы атомной науки и техники
Schlagworte:	Электродинамика
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spelling	nasplib_isofts_kiev_ua-123456789-965192025-02-09T13:15:47Z Channeling of relativistic electrons and positrons Каналювання релятивiстських електронiв та позитронiв Каналирование релятивистских электронов и позитронов Morokhovskii, V.L. Электродинамика The quanta-mechanical problem of relativistic electron (positron) movement through the electro-static crystal field for the case of small angle between initial particle momentum and the atomic plane is considered for the case of electron energy larger than several MeV. The analytical expressions for the wave functions for electrons and positrons in the model potentials, which are similar to the state of the ”one-dimensional relativistic atom”, are presented. Розглянуто квантово-механiчна проблема руху релятивiстського електрона (позитрона) в електричному полi кристалу у випадку, коли початковий iмпульс електрона направлений пiд малим кутом вiдносно атомної площини кристалу, для енергiй електронiв бiльших кiлькох МеВ. Приведено аналiтичнi вирази для хвильових функцiй релятивiстського електрона (позитрона) в модельному усередненому потенцiалi суцiльної площини, якi описують зв’язанi стани релятивiстського електрона з атомною площиною, якi подiбнi станам "одновимiрного релятивiстського атома". Рассмотрена квантово-механическая задача движения релятивистского электрона (позитрона) в электрическом поле кристалла в случае, когда начальный импульс электрона направлен под малым углом к атомной плоскости кристалла, для энергий электронов больших нескольких МэВ. Приведены аналитические выражения для волновых функций релятивистского электрона (позитрона) в модельном усредненном потенциале непрерывной плоскости, которые описывают связанные состояния релятивистского электрона с атомной плоскостью, подобные состояниям "одномерного релятивиcтского атома". 2009 Article Channeling of relativistic electrons and positrons / V.L. Morokhovskii // Вопросы атомной науки и техники. — 2009. — № 5. — С. 122-129. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS:29.20.Dh. https://nasplib.isofts.kiev.ua/handle/123456789/96519 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Электродинамика Электродинамика
spellingShingle	Электродинамика Электродинамика Morokhovskii, V.L. Channeling of relativistic electrons and positrons Вопросы атомной науки и техники
description	The quanta-mechanical problem of relativistic electron (positron) movement through the electro-static crystal field for the case of small angle between initial particle momentum and the atomic plane is considered for the case of electron energy larger than several MeV. The analytical expressions for the wave functions for electrons and positrons in the model potentials, which are similar to the state of the ”one-dimensional relativistic atom”, are presented.
format	Article
author	Morokhovskii, V.L.
author_facet	Morokhovskii, V.L.
author_sort	Morokhovskii, V.L.
title	Channeling of relativistic electrons and positrons
title_short	Channeling of relativistic electrons and positrons
title_full	Channeling of relativistic electrons and positrons
title_fullStr	Channeling of relativistic electrons and positrons
title_full_unstemmed	Channeling of relativistic electrons and positrons
title_sort	channeling of relativistic electrons and positrons
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2009
topic_facet	Электродинамика
url	https://nasplib.isofts.kiev.ua/handle/123456789/96519
citation_txt	Channeling of relativistic electrons and positrons / V.L. Morokhovskii // Вопросы атомной науки и техники. — 2009. — № 5. — С. 122-129. — Бібліогр.: 10 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT morokhovskiivl channelingofrelativisticelectronsandpositrons AT morokhovskiivl kanalûvannârelâtivistsʹkihelektronivtapozitroniv AT morokhovskiivl kanalirovanierelâtivistskihélektronovipozitronov
first_indexed	2025-11-26T02:50:02Z
last_indexed	2025-11-26T02:50:02Z
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fulltext	CHANNELING OF RELATIVISTIC ELECTRONS AND POSITRONS V.L. Morokhovskii ∗ National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine (Received July 29, 2009.) The quanta-mechanical problem of relativistic electron (positron) movement through the electro-static crystal field for the case of small angle between initial particle momentum and the atomic plane is considered for the case of electron energy larger than several MeV. The analytical expressions for the wave functions for electrons and positrons in the model potentials, which are similar to the state of the ”one-dimensional relativistic atom”, are presented. PACS: PACS:29.20.Dh. 1. INTRODUCTION Directional effects of positively charged heavy parti- cles, which are penetrating through the single crystal targets, can be investigated in the frame of the classi- cal theory of Lindhard [1], where the applicability of classical theory is justified. Such a way the finite motion of positively charged heavy particles along the crystal atomic plane (or atomic string) can be described. This directional effect, which is a result of correlated small angular scattering on the multi- tude of crystal atoms, which belongs to the plane (or string) named ”channeling”. Plane channeling can be described with good exactness using crystal potential averaged over the atomic plane. Investigation of the directional effects of electrons (positrons), which are penetrating through the sin- gle crystal targets, requires different theoretical de- scriptions for different particle energies. It is well- known that for electrons with kinetic energy in the range from tens eV up to a few hundreds keV the angular distribution of scattered electrons in the thin mono-crystal targets demonstrate diffraction pattern [2]. For electrons in such energy range a classical approach fails, and the directional effects can be de- scribed in the terms of a few beam dynamical dif- fraction theory. For particle energies from hundreds keV up to the several Mev the wave packet is com- posed of a large number of partial waves and thus requires a many-beam description. If electrons with energies less than several MeV move along the atomic plane, they can not create the bound states with sep- arate crystal plane. Only if electron energy is larger than several MeV, it can create the bound state with separate crystal plane, which can be considered as ”one dimensional atom”. Such result for electrons in quantum-mechanical description contrasts with re- sult for heavy particles in classical description, where finite motion is possible for arbitrary small kinetic energies. For electron with energies are larger than several MeV, the wave packet localized well enough, particularly in the transverse plane. With increasing the electron energy the number of partial plane waves increasing drastically, and therefore we need to search another type of basis functions, which are not similar to the plane waves, and which are suitable for pre- sentation of electron bound state by superposition of small number of waves. The convenient basis can be created from solutions of Dirac equation with model continued plane potentials, which have simple ana- lytical form and small difference from the real crystal potential. The task of the present work is to search such solutions. It should be noted that such quan- tum description becomes inconvenient in the case, when electron creates a large number of bound states with small difference between adjacent energy levels, which are realized at electron energies order of GeV and larger. In this case it’s more convenient to apply the quasi-classical and classical descriptions [3]. 2. PLANAR CHANNELING We are interested in the investigation of the rela- tivistic electron motion in the crystal potential in the case, when initial electron momentum ~p creates small angle respectively one of the main crystal planes (for example plane Y Z in the Cartesian coordinate sys- tem XY Z). We assume that the projection of the electron momentum ~p on the X-axes satisfies the in- equality p2 x/(2mγ) < V ′, where V ′ is the value of the potential energy of the interaction between relativis- tic particle and the atomic plane (V ′ = \| ± eUp(x)\|, where Up(x) is the plane potential). In our task the action of the real plane potential can be replaced by action of the averaged potential over the square of the crystal plane. The plane potential Up(x) creates the ∗E-mail: victor@kipt.kharkov.ua 122 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2009, N5. Series: Nuclear Physics Investigations (52), p.122-129. potential prison for negative charged particle. The potential of the couple of planes creates the potential prison for positive charged particle. So the electron (positron) motion is characterized by its localization near the potential deep minimum and may be con- sidered as analogy with ”relativistic one-dimensional atom”, that is moving along the crystal plane in the direction ~n = ~ey sin(θ) + ~ez cos(θ). So we will search solution of the Dirac equation ih̄ ∂ ∂t ψ = Ĥ (1) with Hamiltonian Ĥ = c~̂α~̂p + β̂mc2 ± eUp(x), where one-dimensional potential Up(x) is the poten- tial of the separated crystal plane (or channel) aver- aged over the square in the Y Z plane. One easily veri- fies the commutation relations and the corresponding constants of motion [ Ĥ, ih̄ ∂ ∂t ] = 0, with eigenvalue E (2) [ Ĥ, p̂y ] = 0, with eigenvalue py (3) [ Ĥ, p̂z ] = 0, with eigenvalue pz (4) [ Ĥ, p̂x ] 6= 0. (5) Using integrals of motion E, py, pz one can look for solution of Dirac equation in the following form ψ(~r, t) = ψ(x)ei(pyy+pzz−Et)/h̄. (6) So as the second order differential equations may be solved only for some simple potentials, and taking into account that coordinates of localization centers for particles with different sign do not coincide, we will use different model functions V (x) = ±eUp(x) for description the potential energies of electron and positron in the plane potential. Therefore solutions for electron and positron will be considered sepa- rately, and Dirac equation in the considered case has the form: [ α̂1∇x − iV h̄c ] ψ(x) = (7) = i h̄c [ E −mc2β̂ − p2cα̂2 − p3cα̂3 ] ψ(x). Let us designate the momentum component, which is parallel to the atomic plane by h̄~k and angle between ~k and axis ~z by θ. Then p2 = h̄k sin θ and p3 = h̄k cos θ. If we multiply equation (7) from the right side by α̂1 and take into account that α̂1α̂2 ≡ iŜ3 and α̂1α̂3 ≡ −iŜ2, we can represent Dirac equation (7) in the following form: [ ∇x − k(sin θŜ3 − cos θŜ2) ] ψ(x) = (8) = i h̄c [ (E + V (x)) α̂1 −mc2α̂1β̂ ] ψ(x). Expression sin θŜ3 − cos θŜ2 is equal the x projection of the vector product of the spin operator ~̂S on the direction ~n = ~k/k i.e. [ ~̂S, ~n]x. The eigenvalues of op- erator sin θŜ3 − cos θŜ2 are: λ1 = +1, λ2 = −1, λ3 = +1, λ4 = −1, (9) and corresponding eigenvectors are: ū1 =   1 −iξ 0 0   , ū2 =   −iξ 1 0 0   , ū3 =   0 0 1 −iξ   , ū4 =   0 0 −iξ 1   , (10) where ξ(θ) = (cos(θ/2)− sin(θ/2))/(cos(θ/2) + sin(θ/2)). Eigenvectors (10) satisfies the following condition: ūiū + j = ū+ i ūj = (1 + ξ2)δij . (11) It’s easy to show, that: α̂1ū1 = ū4, α̂1ū2 = ū3, α̂1ū3 = ū2, α̂4ū1 = ū1; α̂1β̂ū1 = ū4, α̂1β̂ū2 = ū3, α̂1β̂ū3 = −ū2, α̂1β̂ū4 = −ū1. (12) If we search solution of the Dirac equation (8) in the form ψ(x) = 4∑ j=1 ūjfj(x), (13) where fj(x) are arbitrary functions, and use the con- ditions (11) and (12), we obtain the following system of equations: ∇xf1(x)− kf1(x) = i h̄c [ E + mc2 + V (x) ] f4(x), ∇xf2(x) + kf2(x) = i h̄c [ E + mc2 + V (x) ] f3(x), ∇xf3(x)− kf3(x) = i h̄c [ E −mc2 + V (x) ] f2(x), ∇xf4(x) + kf4(x) = i h̄c [ E −mc2 + V (x) ] f1(x). (14) The system of four equations (14) is shared into two pairs of independent and identical systems of equa- tions. Using the first equation we can write: f4(x) = − ih̄c (∇x − k) E + mc2 + V (x) f1(x). (15) Substituting this expression into the fourth equation of the system (14) we obtain the second order differ- ential equation: [ h̄2c2 2E ∇2 xx + E2 − h̄2k2c2 −m2c4 2E + V (x)+ (16) + V 2(x) 2E − h̄2c2 2E V ′(x)x (∇x − k) E + mc2 + V (x) ] f1(x) = 0. 123 The last two terms in (16) are small, and in the first approximation we can neglect them. Then for obtaining basis functions we can use model poten- tial V (x), which has sufficient analytical form and small difference from the real potential. In the first approximation function f1(x) does not differ from f2(x), because equation for f2(x) differs from (16) only by the sign near the k in the small term. Gen- eral solution of the Dirac equation for electrons can be represented in the form: ψ(−e)(x) = A ( ū1 − ih̄c ∇x − k E + mc2 + V (x) ū4 ) f1(x) + B ( ū2 − ih̄c ∇x + k E + mc2 + V (x) ū3 ) f2(x), (17) and for positrons in the form: ψ(+e)(x) = C ( ū3 − ih̄c ∇x − k E −mc2 + V (x) ū2 ) f3(x) + D ( ū4 − ih̄c ∇x + k E −mc2 + V (x) ū1 ) f4(x), (18) where arbitrary constants A, B, C and D become definite after using the boundary conditions. 2.1. ELECTRON WAVE FUNCTIONS 2.1.1. Model potential The averaged crystal plane potential can be repre- sented (in units mec 2) in the form of the sum of po- tentials of the separate planes Up(x): U(x) = +∞∑ n=−∞ Up(x + d · n), (19) Up(x) = 2Zα νd vc 3∑ i=1 aiUi(x), where : Ui(x) = π βi exp(−βix) exp(u2β2 i )× × { 1 2 exp(2βix)erfc( x 2u + uβi) + 1− − 1 2 erfc( x 2u − uβi) } , (20) Z is the atomic number, α = 1/137, ν is the number of atoms in elementary cell, vc is the volume of the elementary cell, d is the distance between planes, u2 is the root mean atomic thermal displacement. Here we used the Moliere-type of atomic potential, where constants ai and βi defined as: ai = {0.1; 0.55; 0.35} (21) bi = {6.0; 1.2; 0.3} βi = ( Z1/3/121 ) bi. At a distance \|x\| ≥ d from the plane the potential Ui(x) has the following form Ui(x) ≈ π βi exp(−βi\|x\|). We get the sum of potentials of all planes (19) by adding to Ui(x) the sum of asymptotical plane po- tentials from (20), which is a geometrical progression π βi ∞∑ n=1 {exp[−βi(dn− x)] + exp[−βi(dn + x)]} = = 2π βi exp(−βid) 1− exp(−βid) ch(βix). (22) In order to investigate equation (16) let’s suppose that potential energy of interaction between relativis- tic electron with plane potential can be presented by model function V (x) = V0/ ch2(x/x0) + C, where constants V0, x0 and C can be obtained by fitting the model potential to the corresponding exact crys- tal plane potential [4]. For the case of relativis- tic electron motion with Lorentz factor γ ∼ 10 near (100)-plane of the Si crystal the orders of these con- stants are: \|V0\| ∼ 10−4mc2 and x0 ∼ 1.3× 102h̄/mc (look at (20)). Fig.1. Potential energy of electron interacting with continuum potential of (110) plane of dia- mond crystal. Solid line represents the model potential V (x) = V0/ch2(x/x0) + C, where: C = −47.3085 eV , V0 = −21.0048 eV , x0 = 0.2215 Å. Dashed line represents potential calculated using for- mulas 19 and 20 2.1.2. Solution of the Schrödinger wave equ- ation with potential V (x) = V0/ ch2(x/x0) + C Representing 2E = 2mc2γ, we can see, that equation (16) is a Schrödinger wave equation for the particle with mass M = mγ. The last two terms in the rec- tangular brackets of (16) are small corrections to the potential energy V (x). Total expression of the po- tential energy for our case looks like V (x)(1 + O(x)), where O(x) ∼ 10−3. Let’s neglect small terms in Schrödinger wave equation (16), replace V (x) by model potential V (x) = V0/ ch2(x/x0) + C and [ − h̄2 2M ∇2 xx + V0 ch2(x/x0) ] f1(x) = E⊥f1(x), (23) 124 where E⊥ = (E2 − h̄2k2c2 −m2c4 − 2EC)/2E. As it was first shown in [5] and [6], the Schrödinger equation with such potential (23) can be solved analytically. We will go on to the new variable y = ch2(x/x0), and then we obtain equation: y(1−y)f ′′1yy + ( 1 2 − y ) f ′1y− ( u y − κ2 ) f1 = 0, (24) where −κ2 = [E2 −m2c4 − h̄2k2c2]x2 0/(4h̄2c2); u = 1 2 ( V0 mc2 )( 2πx0 λc )2 γ; (25) Then with the help of substitutions f = yν · w(y), where while ν-is some arbitrary constant, we’ll ob- tain the equation y(1− y)w′′yy + [ (2ν + 1 2 )− (2ν + 1)y ] w′y + + [ (ν2 − ν 2 − u)y−1 − (ν2 − κ2) ] w = 0, (26) which turns to hyper geometric equation of Gauss if a coefficient equates to zero with y−1. Setting this coefficient to be zero, we get the necessary value of ν ν = ( 1±√1 + 16u ) /4. (27) For reducing the hyper geometric equation to the standard form, let’s introduce some new constants a = ν − κ, b = ν + κ, c = 2ν + 1 2 , (28) where κ was defined in (25) (κ > 0). We are interested in solution of the equation (26) in the area of changing the argument 0 ≤ \|x\| < ∞, which corresponds to the sphere of changing the function 1 ≤ \|y\| < ∞. The standard sphere of changing the argument of hyper geometrical function is a half line −∞ < z ≤ 0. So going on to the variable quantity z = 1− y we’ll satisfy this requirement and obtain the following equation: z(1− z)w′′zz +[c∗− (a+ b+1)z] ·w′z −abw = 0, (29) where c∗ = a + b + 1− c ≡ 1/2. Solution of the hy- pergeometric Gauss equation (29) may be presented by series w(z) = zδ ∞∑ k=0 Ckzk, where coefficients Ck satisfy the following request: Ck+1 = Ck (a + k + δ)(b + k + δ) (c∗ + k + δ)(1 + k + δ) , and δ will be determined from the equation δ(δ + c∗ − 1) = 0. So as c∗ is not integer number and therefore the hy- pergeometric Gauss equation (29) has two linear in- dependent solutions: w1(z) = F (a, b, c∗; z) = = 1 + ab c∗ z + a(a + 1)b(b + 1) c∗(c∗ + 1) z2 2! + + a(a + 1)(a + 2)b(b + 1)(b + 2) c∗(c∗ + 1)(c∗ + 2) z3 3! + ..., (30) and w2(z) = = z1−c∗ ( 1 + (a− c∗ + 1)(b− c∗ + 1) 2− c∗ z + ... ) = = z1−c∗F (a− c∗ + 1, b− c∗ + 1, 2− c∗; z). (31) So the general solution of hyper geometric equation (29) has the following form: w(z) = A ·F (a, b, 1 2 ; z)+B · z1/2F (a+ 1 2 , b+ 1 2 , 3 2 ; z), (32) where we denote the hyper geometric Gauss functions by F (a, b, c; z) and the arbitrary constants by A and B. While A = 1 and B = 0 we’ll obtain the standard even solution of the equation (24) f (+) 1 (x) = ch2ν(x/x0)F (a, b, 1 2 ;−sh2(x/x0)). (33) While A = 0 and B = i - we’ll obtain the odd solution of (24) f (−) 1 (x) = ch2ν(x/x0)sh(x/x0) · F (a + 1 2 , b + 1 2 , 3 2 ;−sh2(x/x0)). (34) Wave functions f+ 1 (x) and f−1 (x) should be normal- ized therefore it is necessary to satisfy the condition lim\|x\|→∞ f±1 (x/x0) = 0. For investigation of solu- tions of (33) and (34) with large values of the ar- gument let’s use asymptotic equations: ch(x/x0) ≈ 2−1 exp(\|x/x0\|), sh(x/x0) ≈ ±2−1 exp(\|x/x0\|), −sh2(x/x0) ≈ −2−2 exp(2\|x/x0\|), (35) and let’s use the identity F (a, b, c; z) = Γ(c)Γ(b− a) Γ(b)Γ(c− a) (−z)a · ·F (a, 1− c + a, 1− b + a; 1 z ) + + Γ(c)Γ(a− b) Γ(a)Γ(c− b) (−z)−b · ·F (b, 1− c + b, 1− a + b; 1 z ). (36) While argument z is increasing infinitely, the hyper geometric functions in the right part of the equation 125 (36) run to the unity. Thus while the values of the ar- gument are big, the asymptotic equations take place: f (+) 1 (x) ≈ 2−2ν exp ( 2ν ∣∣∣∣ x x0 ∣∣∣∣ ) Γ ( 1 2 ) × [ Γ(b− a) Γ(b)Γ( 1 2 − a) 22a exp ( −2a ∣∣∣∣ x x0 ∣∣∣∣ ) + Γ(a− b) Γ(a) · Γ( 1 2 − b) 22b exp ( −2b ∣∣∣∣ x x0 ∣∣∣∣ )] , (37) and f (−) 1 (x) ≈ ±2−(2ν+1) exp ( (2ν + 1) ∣∣∣∣ x x0 ∣∣∣∣ ) Γ ( 3 2 ) × [ Γ(b− a) Γ(b + 1 2 )Γ(1− a) 22a+1 exp ( −(2a + 1) ∣∣∣∣ x x0 ∣∣∣∣ ) + Γ(a− b) Γ(a + 1 2 ) · Γ(1− b) 22b+1 exp ( −(2b + 1) ∣∣∣∣ x x0 ∣∣∣∣ )] . In the last expression sign ”+” corresponds to x > 0, and ” − ” corresponds to x < 0. The first and the second items in brackets of the expressions (37) and (38) behave themselves as exp(κ\|x\|) and exp(−κ\|x\|). Since for the bound states conditions κ is real and is larger than zero therefore in order to obtain normal- ized solution it is necessary to go to zero the coeffi- cients near exp(κ\|x\|). This is possible when functions Γ(ζ)−1 turn to zero. As a result we obtain arguments ζ are equal integer numbers −n, (n = 0; 1; 2; ...). This condition is necessary, but not enough. It should be added by one more. In all cases of physical applica- tion of Gauss hyper geometric functions the condi- tion c− a > 0 or c− b > 0 should be satisfied, there- fore in expression (27) before the square root we choose the sign ” − ”, and it guarantees the values to be ν < 0. Let’s demand, that b = −n for the even function ψ+(x) and b + 1/2 = −n, for the odd ψ−(x) function. Then to determine the energy levels of the bound states we’ll obtain the conditions: κ = −ν − n (38) for the even function, and κ = −ν − n− 1/2 (39) for the odd function. Then taking into account (16) and our definition (25), we see that the energy of the crosswise motion E⊥ respectively the crystal plane we can present in the form E⊥ = h̄2c2k⊥ 2/2E. so finally we obtain E (n) ⊥ = − h̄2c2 2E ( 2ν + n x0 )2 , (40) where even n corresponds to the even function, and the odd n corresponds the odd function. Such de- pendence is shown on Fig.2. Analogous result was obtained in [7] and [8]. Fig.2. Transverse electron energy as a function of γ = E/mc2 in the plane (110) of the diamond crystal for quantum numbers n = 0, ..., 15 While the condition that was discussed above is ful- filled, then the hyper geometric rows in the expres- sions (37) and (38) are broken on the term with number n and thus they are degenerated into Jacobi polynomials. Therefore the physical solutions of the equation (23) have the following forms: f (+) 1 (n, x) = Nnch2ν(x/x0)× (41) Gn(2ν + n/2, 1/2;−sh2(x/x0)), f (−) 1 (n, x) = Nnch2ν(x/x0)sh(x/x0)× Gn(2ν + (n− 1)/2, 3/2;−sh2(x/x0)), where Gn(p, q; z) ≡ 1 + n∑ k=1 (−1)k ( n k ) × (42) (p)(p + 1)...(p + k − 1) q(q + 1)...(q + k − 1) zk, and ( n k ) are binomial coefficients. 2.1.3. Normalization Functions f (+) 1 (n, x) and f (−) 1 (m,x) for m 6= n are orthogonal, because they are eigenfunctions of Hermitian operator. Besides they must be normalized, i.e. must satisfy the condition〈 f (±) m (x) \| f (±) n (x) 〉 = δmn. Let us designate in (42) ck - the coefficient with zk. Then we can write down the condition of normalizing of wave functions f±(x) in the following form: 〈 f (±) m (x) \| f (±) n (x) 〉 = N2 n n∑ k=0 n∑ j=0 ckcjx0 × ∫ +∞ −∞ ch4ν(x/x0)sh2(k+j)(x/x0)dx = 1. (43) Integral in (43) can be transformed to the new vari- able t = th(x/x0) and can be represented in the form 126 of: x0 ∫ 1 0 (1− t2)ptqdt = x0B(p, q) , B(p, q) = Γ(p)Γ(q)/Γ(p− q) , (44) where designations p = k + j + 1/2 and q = −(ν + 2 + k + j) are used. Whence we find out that for normalizing of functions which are presented by ex- pression (43) it is necessary to multiply them by the normalizing factor: Nn =  x0 n∑ k=0 n∑ j=0 ckcjB(p, q)   −1/2 . (45) Samples of bound state electron wave functions are shown on Fig.3 and in [4]. Fig.3. Bound state wave functions (n = 0; 1; 2; 3) for electrons with Lorentz factor γ = 500, which moving along the plane (110) of the diamond crystal 2.2. POSITRON WAVE FUNCTIONS 2.2.1. Solution of the Schrödinger wave equation for channeled positrons Potential energy of positron interaction with planar continuous crystal potential near the center of iso- lated plane channel has its minimum, and in this re- gion it can be approximated by parabolic function V (x) = Kx2/2 + C, (Fig.4), [9]. We can see that such simple model potential is a good enough ap- proximation for the planar channel potential. In this case from the main equation (16) we obtain one- dimensional equation for function f3(x) , which is well-known equation of harmonic oscillator: [ − h̄2 2M ∇2 xx + Kx2 2 ] f3(x) = E⊥f3(x), (46) where designation E⊥ = ((E + C)2 − h̄2k2c2 − m2c4)/2E is used. Introducing new variable z = αx and chose α to be α = (KM/h̄2)1/4, we obtain equa- tion: (∇2 z + µ− z2 ) f(z) = 0, (47) where µ = 2E⊥/h̄ωc and ωc = (K/M)1/2 is a clas- sical oscillator frequency. Such representation is con- venient because it operates by dimensionless values. Fig.4. Potential energy of positron interacting with continuum potential of (110) plane of diamond crystal. Solid line represents the model potential V (x) = C + (x/x0)2, where: C = 47.04 ± 0.04 eV , x0 = 570.2215 Å. Dashed line represents potential calculated using formulas 19 and 20 Asymptotic solution of the equation (47) for z →∞ must satisfy the equation (∇2 z − z2)f(z) = 0 . Gen- eral solution of this equation is C1e −z2/2 + C2e z2/2 . Normalized solution must satisfy the asymptotic con- dition limz→∞ f(z) = 0 i.e. we must put C2 = 0 . So we must look for the solution of (47) in the form f(z) = e−z2/2H(z) . Then we obtain equation for function H(z) H ′′(z)− 2zH ′(z) + (µ− 1)H(z) = 0, (48) that is the Hermite’s equation. Let us look for the solution of (48) in the form of series H(z) = zs ∑∞ k=0 akzk . Substituting such series into equation (48) one can obtain the requirements (s+k+2)(s+k+1)ak+2−(2s+2k+1−µ)ak = 0. (49) From the first equation for index k = 0 s(s− 1)a0 = 0 . If a0 6= 0 , then s = 0 or s = 1 . The second equation (s + 1)sa1 = 0 also satisfies by s = 0 . Then it may be a0 6= 0 or a0 = 0 . But if s = 1 then must be a1 = 0 . General solution of (48) looks like H(z) = e−z2/2 ×{ a0 [ 1 + ∞∑ p=1 (−1)p (µ− 1)...(µ− 4p + 3) (2p)! z2p ] + a1z [ 1 + ∞∑ p=1 (−1)p (µ− 3)...(µ− 4p + 1) (2p + 1)! z2p ]} , (50) where a0 and a1 are arbitrary constants. The first term is a sum of even powers of z and the second term is a sum of odd powers of z. In order to obtain the normalized solutions we must choose µ = 4p + 1 and 127 a1 = 0 , or µ = 4p + 3 and a0 = 0 . Under this re- quirement our series degenerate into the finite sums. In the first case we obtain the even solution and in the second case the odd one. Let us introduce new princi- ple integer number n = 2p if n even and n = 2p + 1 if n odd. Then we can join both solutions into the general expression. Each principle quantum number n corresponds to the positron transverse energy E (n) ⊥ E (n) ⊥ = C + h̄ωcγ −1/2(n + 1/2). (51) Such dependence is shown on the Fig.5. Fig.5. Transverse positron energy as a function of γ = E/mc2 in the plane (110) of the diamond crystal for quantum numbers n = 0, ..., 25 Corresponding wave functions are represented by formula: f (±) n (x) = Nne−α2x2/2Hn (αx) , (52) where Hn(αx) are the Hermite’s polynomials, Nn are normalization factors. 2.2.2. Normalization Functions f (+) 1 (n, x) and f (−) 1 (m, x) for m 6= n are orthogonal, because they are eigenfunctions of Hermitian operator. Besides they must be normalized, i.e. must satisfy the condition〈 f (±) m (αx) \| f (±) n (αx) 〉 = δmn. Such property of wave functions (52) can be proved by using Hermitian pro- ducing function1. We can write: ∫ ∞ −∞ e−t2+2txe−s2+2sxe−x2 dx = ∞∑ n,m=0 tnsm n!m! ∫ ∞ −∞ Hn(x)Hm(x)e−x2 dx . (54) After integration on the left we obtain the following result: π1/2e2ts = ∞∑ n=0 (2ts)n n! . (55) Equating the coefficients near the equal powers of t and s in the right part of (57) and in the expression (55), we obtain: ∫ ∞ −∞ Hn(αx)Hm(αx)e−α2x2 dx = { π1/22nn!/α, n = m, 0, n 6= m. (56) So the orthogonality of the wave function is proved, and the normalization factors Nn are: Nn = ( α 2nn! √ π )1/2 . (57) Samples of bound state electron wave functions with principle quantum numbers n = 0, 1, 2, 3, 4 are shown on Fig.6. Fig.6. Bound state wave functions (n = 0; 1; 2; 3, 4) for positrons with γ = 500, which are moving along the plane (110) of the diamond crystal 2.2.3. Population probabilities The distribution in the transverse energy varies sig- nificantly when the particle progresses through a crystal due to transitions between the transverse states. Here we shall not consider this problem. Now we shall consider only the initial distribution in the positron transverse energy at the crystal entrance. We propose that the particle enters the crystal at the definite angle θ0 respectively to the plane. The probability Pn(q) of capture to the state \|n〉, where q = Eθ0/h̄c, can be expressed in the terms of the plane wave expansion in the transverse wave func- tions: Pn(q) = \|An(q)\|2 , An(q) = ∫ ∞ −∞ f (±) n (αx)eiqxdx . (58) 1Hermitian producing function is: e−t2+2tx = ∞X n=0 Hn(x) n! tn . (53) 128 Using the integral representation of the Hermitian functions: Hn(x) = C ∫ +0 −∞ e−t2+2xtt−(n+1)dt and changing the order of the priority of integration, we obtain: An(q) = Nn(−i)−n(2π)1/2α−1e−(q/α)2/2Hn(q/α) . (59) 3. OUTLOOK The wave functions, which was obtained above for the model plane potentials, can be combined with or- thogonalized plane waves and used as basis functions for representation the wave functions of relativistic particles in the crystal potential. Using these basis functions, we can modify the method of numerical calculations, which was developed by [10] for calcula- tion the wave functions of the crystal electrons, and apply them for obtaining wave functions of the rela- tivistic electrons and positrons in the close to reality crystal potential. References 1. J. Lindhard. Influence of crystal lattice on mo- tion of energetic charged particles // K. Dansk. Vid. Selsk. Mat. Phys. Medd. 1965, 34, p.1-64. 2. C.J.Davisson, L.H.Germer. Diffraction of elec- trons by a crystal of nickel //Phys. Rev. 1927, v.30, p.705. 3. A.I. Akhiezer, N.F. Shul’ga. High-Energy Electro- dynamics in Matter. Amsterdam: ”Gordon and Breach”, 1996, 388 p. 4. V.L.Morokhovskii. Wave functions of the bound states of relativistic electrons in crystal for elec- tron movement along the crystal plane: Preprint, M: ”CNIIatominform”. 1987, 6p. (in Russian). 5. N. Rosen, P.M. Morse. On the vibrations of po- liatomic moleculs // Phys. Rev. 1932, v.42, p.210. 6. G. Pöschl and E. Teller. Bemerkungen zur quan- tenmechanik des anharmonischen oszillators // Zs. Phys. 1933, v.83, p.143 (in German). 7. V.A.Bazylev, V.I. Glebov, N.K. Zhevago. Depen- dence of the radiation spectra from channeled electrons on their energy // Radiation Effects. 1981, v.56, p.99-104. 8. K. Komaki, F. Fujimoto. Energy levels of chan- neled electrons and channeling radiation // Ra- diation Effects. 1981, v.56, p.13-16. 9. M.A. Kumakhov, R. Wedell. Theory of radiation of relativistic channelled particles // Phys. stat. sol. 1977, v.84(b), p.581. 10. C.Herring. A new method of calculating wave functions in crystals // Phys. Rev. 1940, v.15, p.1169. КАНАЛИРОВАНИЕ РЕЛЯТИВИСТСКИХ ЭЛЕКТРОНОВ И ПОЗИТРОНОВ В.Л. Мороховский Рассмотрена квантово-механическая задача движения релятивистского электрона (позитрона) в электрическом поле кристалла в случае, когда начальный импульс электрона направлен под малым углом к атомной плоскости кристалла, для энергий электронов больших нескольких МэВ. Приведены аналитические выражения для волновых функций релятивистского электрона (позитрона) в модель- ном усредненном потенциале непрерывной плоскости, которые описывают связанные состояния ре- лятивистского электрона с атомной плоскостью, подобные состояниям "одномерного релятивиcтского атома". КАНАЛЮВАННЯ РЕЛЯТИВIСТСЬКИХ ЕЛЕКТРОНIВ ТА ПОЗИТРОНIВ В.Л. Мороховський Розглянуто квантово-механiчна проблема руху релятивiстського електрона (позитрона) в електрич- ному полi кристалу у випадку, коли початковий iмпульс електрона направлений пiд малим кутом вiд- носно атомної площини кристалу, для енергiй електронiв бiльших кiлькох МеВ. Приведено аналiтичнi вирази для хвильових функцiй релятивiстського електрона (позитрона) в модельному усередненому потенцiалi суцiльної площини, якi описують зв’язанi стани релятивiстського електрона з атомною пло- щиною, якi подiбнi станам "одновимiрного релятивiстського атома". 129

Channeling of relativistic electrons and positrons

Institution

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