The motion of charged particles in a crystal in the axial channeling regime can be both regular and chaotic. The chaos in quantum case manifests itself in the statistical properties of the energy levels set. These properties have been studied previously for the electrons channeling along direction of the silicon crystal, in the case when the classical motion was completely chaotic. It is demonstrated that the level spacing distribution for both electrons and positrons can be better described by Berry-Robnik distribution than by both Wigner (completely chaotic case) or Poisson (completely regular case) distribution

Dronik, V. I.

Isupovd, A. Yu.

Shul’ga, N. F.

Syshchenko, V. V.

Tarnovsky, A. I.

arXiv

DSpace at Belgorod State University

Prepared for submission to JINSTXIIIth International Symposium «Radiation from Relativistic Electrons in PeriodicStructures»September 16-20, 2019Belgorod, Russian FederationRegular and Chaotic Motion Domains in the ChannelingElectron’s Phase Space and Mean Level Density for ItsTransverse Motion EnergyN.F. Shul’ga,a,b V.V. Syshchenko,c,1 A.I. Tarnovsky,c V.I. Dronikc and A.Yu. IsupovdaAkhiezer Institute for Theoretical Physics of the NSC “KIPT”,Akademicheskaya Street, 1, Kharkov 61108, UkrainebV.N. Karazin National University,Svobody Square, 4, Kharkov 61022, UkrainecBelgorod State University,Pobedy Street, 85, Belgorod 308015, Russian FederationdLaboratory of High Energy Physics (LHEP), Joint Institute for Nuclear Research (JINR),Dubna 141980, Russian FederationE-mail: syshch@yandex.ruAbstract: The motion of charged particles in a crystal in the axial channeling regime can be bothregular and chaotic. The chaos in quantum case manifests itself in the statistical properties ofthe energy levels set. These properties have been studied previously for the electrons channelingalong [110] direction of the silicon crystal, in the case when the classical motion was completelychaotic. The case of channeling along [100] direction is of special interest because the classicalmotion here can be both regular and chaotic for the same energy depending on the initial conditions.The semiclassical energy level density (as well as its part that corresponds to the regular motiondomains in the phase space) is computed for the 10 GeV channeling electrons and positrons. Itis demonstrated that the level spacing distribution for both electrons and positrons can be betterdescribed by Berry–Robnik distribution than by both Wigner (completely chaotic case) or Poisson(completely regular case) distributions.Keywords: Interaction of radiation with matterArXiv ePrint: 1909.132381Corresponding author.Contents1 Introduction 12 Method and potential wells 23 Results and discussion 44 Conclusion 61 IntroductionWhen a fast charged particle is incident on a crystal at a small angle to any crystallographic axisdensely packed with atoms, it can perform the finite motion in the transverse plane. This motionis known as the axial channeling [1–3]. The particle motion in this case could be described witha good accuracy as the one in the continuous potential of the atomic string. During motion inthis potential the longitudinal particle momentum p‖ is conserved, so the motion description isreduced to two-dimensional problem of motion in the transversal plane. From the viewpoint of thedynamical systems theory, the channeling particle’s motion could be either regular or chaotic. Thequantum chaos theory [4–6] predicts qualitative differences for these alternatives.Themanifestations of chaos in quantum systems are found, first of all, in the statistical propertiesof their energy spectra. The quantum chaos theory predicts (see, e.g., [4–6]) that the energy levelsnearest-neighbor distribution of the chaotic system obeys Wigner distributionp(s) = (πρ2s/2) exp(−πρ2s2/4) (1.1)(where s is the distances between consequent energy levels, ρ is the mean level density on the energyrange under consideration) while the regular system — the exponential one (frequently referred asPoisson distribution)p(s) = ρ exp(−ρs) . (1.2)The level statistics for the electrons channeling near [110] direction in silicon crystal has beenstudied in [7, 8]. In that case each pair of the closest parallel atomic strings forms the two-wellpotential (see, e.g., [2]), in which the motion above the saddle point is chaotic for the major partof the initial conditions. As a result, the level spacing statistics in that case is well described byWigner distribution (1.1).The aim of the present paper is to study the level spacing statistics in the case of co-existenceof regular and chaotic motion domains. Such situation takes the place when the electron channelsnear [100] direction of silicon crystal. The level spacing statistics is described in this case byBerry–Robnik distribution [9]p(s) =1ρexp (−ρ1s){ρ21erfc(√π2ρ2s)+(2ρ1ρ2 +π2ρ32s)exp(−π4ρ22s2)}, (1.3)– 1 –whereerfc (x) =2√π∫ ∞xe−t2dt = 1 − erf (x) . (1.4)It is presupposed that the regular motion domains and (single) chaotic motion domain generateindependent level sequences with the densities ρ1 and ρ2 (ρ = ρ1 + ρ2) respectively. Our goal isto calculate the relative contribution of the regular and chaotic motion domains into the mean leveldensity.2 Method and potential wellsThe electron transversal motion in the atomic string continuous potential is described by the two-dimensional Schrödinger equation with HamiltonianĤ = −(c2~2/2E‖)[(∂2/∂x2) + (∂2/∂y2)]+U(x, y) , (2.1)where the value E‖/c2 (here E‖ = (m2c4 + p2‖c2)1/2) plays the role of the particle mass [1]. TheHamiltonian eigenvalues E⊥ are found numerically using the so-called spectral method [10–12].Here we consider the particle’s motion near direction of the atomic string [100] of the Si crystal.The continuous potential could be represented by the modified Lindhard potential [1]U(1)(x, y) = −U0 ln[1 + βR2/(x2 + y2 + αR2)], (2.2)where U0 = 66.6 eV, α = 0.48, β = 1.5, R = 0.194 Å (Thomas–Fermi radius). These stringsform in the plane (100) the square lattice with the period a ≈ 1.92 Å. The additional contributionsfrom the eight closest neighboring strings lead to the following potential energy of the channelingelectron (figure 1, left):U(−)(x, y) =1∑i=−11∑j=−1U(1)(x − ia, y − ja) . (2.3)Axial channeling of positrons near [100] direction is possible in the small potential pit near thecenter of the square cell with repulsive potentials −U(1) in the corners of the square (figure 1, right):U(+)(x, y) = −U(1)(x − a/2, y − a/2) −U(1)(x − a/2, y + a/2)− (2.4)−U(1)(x + a/2, y − a/2) −U(1)(x + a/2, y + a/2) − 7.9589 eV ,where the constant is chosen to achieve zero potential in the center of the cell. The spectral methodfor the channeling electrons and positrons near [100] direction had been tested in [13] for small E‖values, when the total number of energy levels in the potential well is small. Here we put E‖ = 10GeV to achieve the semiclassical domain, where the energy level density is high, as it is needed forquantum chaos investigations.The semiclassical mean level density is described by the integral [9] in the 4-dimensional phasespace with the classical counterpart H(x, y, px, py) of the quantum Hamiltonian (2.1):ρ(E⊥) = (2π~)−2∫dxdydpxdpy δ(E⊥ − H(x, y, px, py))= (2.5)– 2 –Figure 1. Potentials (2.3) (left) and (2.4) (right).=2(2π~)2∫dxdydpx|vy(x, y, px)|, (2.6)where vy is the y-component of the electron’s velocity, and the integration in (2.6) is performedover the domainc2p2x2E‖+U(x, y) ≤ E⊥ . (2.7)The integral (2.6) is computed using Monte-Carlo method. Under that, if the random point fallsinto a regular motion domain (see figure 2), its contribution is accounted both in the total densityof states and in the density of states related to the regular motion domain. The boundaries of theregular motion domains are found using Poincaré sections method (see, e.g., [1]).Figure 2. Left: the surface E = H(x, y, vx, vy = 0) that bounds the phase space domain (2.7) permitted formotion under the laws of classical mechanics. The random points in this domain are plotted in red. Right:the phase trajectories started from these points are traced up to their intersection with the plane y = 0. Ifthe final point falls into a regular motion domain on the Poincaré section, the initial point also belongs to theregular motion domain in the phase space.– 3 –3 Results and discussionSo, the electrons channeling along [100] direction move in a weakly disturbed, almost axiallysymmetric potential (2.3). It leads to the approximate conservation of the electron’s angularmomentum (and hence to the regular dynamics) in the range far from the edges of the potentialwell. As a consequence, there exists (among other regular domains) the regular motion domainmarked 1 on the Poincaré sections in the figure 3 (b) and (c); the electron’s transverse motion thereis similar to ones in a central field (the orbit is presented on the bottom of the figure 3 (a)). Suchdomain exists in the whole range of transverse motion energies under consideration from E⊥ ≈ −14eV (completely regular motion) up to E⊥ = −12, 0883 eV (the upper edge of the potential well).E||= 10 GeV,  E = -13.8924 eV11222222(b)-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 x,  Angstrom-1.5-1-0.500.511.5 vx /  c10-4E||= 10 GeV,  E = -12.0883 eV11222222(c)-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 x,  Angstrom-1.5-1-0.500.511.5 vx /  c10-4Figure 3. Middle: Poincaré sections for the lowest and highest transverse energy levels of the electron withthe energy of longitudinal motion 10 GeV in the range under consideration. Left and right: projections ofthe phase space domains of types 1 (left) and 2 (right) onto the 3-dimensional space (x, y, vx) as well as thecorresponding orbits and Poincaré sections.It turns out that the relative contribution of this domain to the mean level density ρ1/ρ isapproximately constant in the energy range under consideration, −13.19 ≤ E⊥ ≤ −12.0883 eV; it isabout 34%. The average contribution from all regular motion domains in that range is about 42%(dashed line in the figure 4 (a)).Themean level density ρ (2.6) as well as the contributions to it from the regular motion domainsare presented in figure 4 (a). Note that ρ varies in the range under consideration. In this case thegeneric set of energy levels has to be subjected the so-called unfolding procedure [6], which leads todimensionless values of s with the mean levels density ρ = 1 for the E⊥ range under consideration.Note also that the channeling particle’s eigenstates of the transverse motion can be classifiedusing the group theory. The potential (2.3) possesses the symmetry of the square thus isomorphicto dihedral group D4 (or C4v). This group has four one-dimensional irreducible representationsand one two-dimensional one, denoted A1, A2, B1, B2, E [13, 14]. So, the eigenstates of thefirst four types are non-degenerated while the eigenstates of the last type are twice degenerated.The histogram in the figure 5 represents the levels spacings distribution for that four types ofnon-degenerated levels.The levels spacings distribution is compared to the predictions (1.1), (1.2) and (1.3), the lastcase with the minimal and the average estimations of the regular motion domains contribution,– 4 –ρ1/ρ ≈ 34% and ρ1/ρ ≈ 42% (left and middle panels in the figure 5, respectively). We seethat Berry–Robnik distribution (1.3) describes the level spacing distribution much better than pureWigner or Poisson ones (that is confirmed by the χ2 values calculated for all three hypotheses).In contrast to the case of electron, the potential well for the channeling positrons (2.4) couldnot be considered as a slightly perturbed axially symmetric well. Hence the structure of regularmotion domains stays rather complex yet near the upper edge of the potential pit (see figures 6 and7).-13.6 -13.4 -13.2 -13 -12.8 -12.6 -12.4 -12.2 -12 E  ,  eV01002003004005006007008009001000 ,  eV-1(a)0 0.5 1 1.5 E  ,  eV050100150200250300350400450500 ,  eV-1(b)Figure 4. (a) Semiclassical mean energy level density (2.6) (solid line), the contribution to it from thedomain of the type 1 (circles), and the total contribution from all domains of regular motion (dots; errorsare due to difficulty of precise determination of the domains’ boundaries). The dashed line corresponds tothe average value of the regular motion contribution in the interval under consideration that is equal to 42%.(b) The same for the channeling positrons, while the circles are referred to the contribution from the regularmotion domains of the types 1, 2 and 3 in figure 6 (or the domain enveloping them for the lower energylevels).0 1 2 3 4s020406080100Number of level spacings 2W = 1632.45122P = 34.40142BR = 12.98360 1 2 3 4s020406080100Number of level spacings 2W = 1632.45122P = 34.40142BR = 8.5450 1 2 3 4s020406080100Number of level spacings 2W = 10.63822P = 34.07972BR = 6.3349Figure 5. Left: nearest-neighbor spacing distribution for the E‖ = 10 GeV channeling electrons in the−13.19 ≤ E⊥ ≤ −12.0885 eV range in comparison with Wigner (1.1), Poisson (1.2), and Berry–Robnik(1.3) distributions (dashed, dotted, and solid lines, respectively); ρ1 = 0.34. Middle: the same for ρ1 = 0.42.Right: the same for the channeling positrons in the 0.9 ≤ E⊥ ≤ 1.4886 eV range with ρ1 = 0.26.There is no the energy range with approximately constant relative contribution of the regularmotion domains to the mean level density ρ1/ρ. However we consider the energy range 0.9 ≤E⊥ ≤ 1.4886 eV with the value ρ1/ρ about 26% in it (dashed line in the figure 4 (b)). We see thatin this case Berry–Robnik function also describes the level spacing distribution better than Wigneror Poisson ones (figure 5, right panel).– 5 –Figure 6. Main types of regular orbits and corresponding Poincaré sections for the channeling E‖ = 10GeV positrons with E⊥ = 1.4857 eV.Figure 7. The positron’s regular motion domains of various sorts from the figure 6 in the (x, y, vx) space.The panel 1 presents the orbit 1 from the figure 6 (dark yellow) and its counterpart rotated on 90◦ (brightyellow) making the same contribution to the total level density, the panel 2 presents the orbit 2 (blue)and its counterpart (red); in these cases both counterparts make the same trace on the Poincaré sectionplane. The panel 9 presents the orbit 9 (blue) and its counterpart (red); in this case the Poincaré sectionsof these counterparts are essentially different. The panel 4, 5, 6 presents the orbit 4 (dark green) but not itscounterpart and also the orbits 5 and 6 that differ from each other only by the direction of motion, clockwiseor counterclockwise, their Poincaré sections are symmetric to each other (purple and bright green).4 ConclusionThe energy levels of the transverse motion of E‖ = 10GeV electrons and positrons channeling in the[100] direction of a silicon crystal are found using the spectral method of numerical integration ofthe time-dependent Schrödinger equation. The case of channeling in this direction is interesting forthe quantum chaos investigations due to the co-existence of the regular and chaotic motion domainswith given transverse motion energy value. The Berry-Robnik theory predicts the formula for thelevel spacing distribution for such cases.– 6 –We have estimated the relative contribution of the regular motion domains in the phase spacein the semiclassical mean level density. This value is needed as a parameter in Berry-Robnikdistribution.There exists the electron’s transverse energy interval where the contribution of the regularmotion domains is approximately constant, satisfying the condition under which Berry–Robniktheory is built; it comprises approximately 42%. The level spacing distribution in this intervalis much better described by Berry-Robnik distribution rather than pure Wigner or Poisson ones(χ2 = 8.545 for 11 degrees of freedom that corresponds to p = 0.66; for Wigner and Poissondistributions we have p values 0 and 3.1 · 10−4, respectively).There is no wide enough interval with approximately constant regular contribution in the caseof channeling positrons. However, even in this case Berry-Robnik distribution, with the meanregular contribution as the parameter, describes the level spacing distribution better than Wignerand Poisson ones (χ2 = 6.3349 for 11 degrees of freedom that corresponds to p = 0.85; for Wignerand Poisson distributions we have p values 0.47 and 3.5 · 10−4, respectively).References[1] A.I. Akhiezer, N.F. Shul’ga, High-Energy Electrodynamics in Matter, Gordon and Breach (1996).[2] A.I. Akhiezer, N.F. Shul’ga, V.I. Truten’, A.A. Grinenko, V.V. Syshchenko, Physics-Uspekhi 38(1995) 1119.[3] U.I. Uggerhøj, The interaction of relativistic particles with strong crystalline fields, Rev. Mod. Phys.77 (2005) 1131.[4] M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer-Verlag (1990).[5] H.-J. Stöckmann, Quantum Chaos. An Introduction, Cambridge University Press (2000).[6] L.E. Reichl, The Transition to Chaos, Springer (2004).[7] N.F. Shul’ga, V.V. Syshchenko, A.I. Tarnovsky and A. Yu. Isupov, Quantum Chaos Effects in theAxial Channeling of Electrons, Journal of Surface Investigation 9, No. 4 (2015) 731.[8] N.F. Shul’ga, V.V. Syshchenko, A.I. Tarnovsky and A. Yu. Isupov, Structure of the channelingelectrons wave functions under dynamical chaos conditions, Nucl. Instrum. Methods in Phys. Res. B370 (2016) 1.[9] M.V. Berry, M. Robnik, Semiclassical level spacings when regular and chaotic orbits coexist, J. Phys.A 17 (1984) 2413.[10] M.D. Feit, J.A. Fleck, Jr., F. Steiger, Solution of the Schrödinger equation by a spectral method, J.Comput. Phys. 47 (1982) 412.[11] S. Dabagov, L.I. Ognev, Passage of MeV-energy electrons through monocrystals, Nucl. Instrum. andMethods in Phys. Res. B 30 (1988) 185.[12] N.F. Shul’ga, V.V. Syshchenko, V.S. Neryabova, On spectral method in the axial channeling theory,Nucl. Instrum. Methods B 309 (2013) 153.[13] N.F. Shul’ga, V.V. Syshchenko, A.I. Tarnovsky, I.I. Solovyev and A.Yu. Isupov, Positrons vs electronschanneling in silicon crystal: energy levels, wave functions and quantum chaos manifestations, JINST13 (2018) C01017.[14] L.D. Landau, E.M. Lifshitz, Quantum Mechanics. Non-Relativistic Theory, Pergamon Press (1977).– 7 –

Regular and chaotic motion domains in the channeling electron's phase space and mean level density for its transverse motion energy

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Regular and chaotic motion domains in the channeling electron's phase space and mean level density for its transverse motion energy

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