Simulation of Channeling and Radiation of 855 MeV Electrons and Positrons in a Small-Amplitude Short-Period Bent Crystal
Andrei V. Korol, Victor G. Bezchastnov, Gennady B. Sushko, Andrey V. Solov'yov
aa r X i v : . [ phy s i c s . acc - ph ] J un Simulation of Channeling and Radiation of 855 MeV Electronsand Positrons in a Small-Amplitude Short-Period Bent Crystal
Andrei V. Korol a , Victor G. Bezchastnov,
2, 3
Gennady B. Sushko, and Andrey V. Solov’yov MBN Research Center, Altenh¨oferallee 3, 60438 Frankfurt am Main, Germany A.F. Ioffe Physical-Technical Institute,Politechnicheskaya Str. 26, 194021 St. Petersburg, Russia Peter the Great St. Petersburg Polytechnic University,Politechnicheskaya 29, 195251 St. Petersburg, Russia.
Channeling and radiation are studied for the relativistic electrons and positronspassing through a Si crystal periodically bent with a small amplitude and a shortperiod. Comprehensive analysis of the channeling process for various bending am-plitudes is presented on the grounds of numerical simulations. The features of thechanneling are highlighted and elucidated within an analytically developed continu-ous potential approximation. The radiation spectra are computed and discussed.
PACS numbers: 02.70.Ns, 41.60.-m, 61.85.+p, 83.10.Rs a E-mail: [email protected]
I. INTRODUCTION
Channeling of the charged projectiles in crystals establishes a field of research importantboth with respect to the fundamental theoretical studies as well as to the ongoing exper-iments (see, e.g., the monograph [1] and the references therein). Starting from the firsttheoretical predictions by Lindhard [2], the channeling is known to occur when the projec-tiles move in the crystals preferably along the crystalline planes or axes. This phenomenonopens a possibility to manipulate the beams of projectiles, in particular by deflecting themin bent crystals.Of particular interest for the theory and applications is the radiation produced by thechanneling projectiles. The radiation spectra display distinct lines related to the inter-planaroscillations or axial rotations of the particles involved into the planar or axial channeling,respectively, in contrast to the bremsstrahlung-type spectra produced by the non-channelingprojectiles. In periodically bent crystals, the channeling radiation exhibit additional undu-lator lines as a result of modulation of the transverse velocity of the moving particles by thebent crystalline structure. The positions of the undulator lines depend on the bending pe-riod and the projectile energies and thereby can be tuned in an experiment. Theoretically, acrystalline undulator has been suggested [3, 4] as a source of the monochromatic radiation ofsub-MeV to MeV energies, and different experiments have been being performed to produceand detect the undulator radiation.The original concept of the crystalline undulator assumes the projectiles to move throughthe crystal following the periodically bent crystalline planes or axes. For such motions,the undulator modulation frequencies are smaller than the frequencies of the channelingoscillations or rotations providing the undulator spectral lines to arise at the energies belowthe energies of the channeling lines. Recently, channeling has been studied for periodicallybent crystals with the bending period shorter than the period of channeling oscillations inthe straight crystal [5, 6]. The emergent radiation was shown to display spectral lines atthe energies exceeding the energies of the channeling peaks. The corresponding theoreticalsimulations have been performed for the electron and positron channeling in the siliconcrystals with the (110)-planes bent according to the shapes δ ( z ) = a cos(2 πz/λ u ) , (1)where z is a coordinate along the planes in the straight crystal, whereas a and λ u arethe bending amplitude and period, respectively. The simulations have also assumed theamplitude a to be substantially smaller than the inter-planner distance d in the straightcrystal. In contrast to the channeling in conventional crystalline undulator, for the newlyproposed crystalline structures the channeling projectiles do not follow the short-period bentplanes. The positrons move bouncing between the bent planes, whereas the electrons movepreferable through the small-amplitude “waves” of the bent structures. Yet the motion of theprojectiles acquires a short-period modulation resulting from the bending and explicitly seenin the theoretically simulated trajectories [6]. These modulations are of a regular jitter-typeand responsible for producing the high-energy monochromatic radiation. Interestingly, asimilar radiative mechanism has recently been studied with respect to the radiation producedby relativistic particles in interstellar environments with turbulent small-scale fluctuationsof the magnetic field [7, 8].It requires, in our view, some future clarifications on whether the jitter-type channelingmotions in crystals provide observable properties (in particular for the radiation detected)inherent to the conventional undulator. Within the current studies, we adopt the term small-amplitude short-period (SASP) to designate the bent crystalline structures suggestedin Ref. [5]. Important is that, the SASP bent crystals can be regarded as possible efficientsources of tunable monochromatic hard-energy radiation and therefore are of immediateinterest for applications and further theoretical studies.Since introducing the SASP bent crystals [5], several theoretical and experimental stud-ies have already been performed on the new regime of channeling and the radiation pro-duced [6, 9]. However, further efforts are certainly required for more detailed investigationson statistical properties of channeling as well as on computing the radiation spectra for avariety of conditions including different bending parameters, lengths of the crystalline sam-ples, beam energies and angular apertures for detecting the radiation. The focus of thepresent paper is on the theoretical simulations for the electrons and positrons with the en-ergies of 855 MeV that are the beam energies achievable in experiments at Mainz MicrotronFacility [9, 24].As in a number of our recent studies, three-dimensional simulations of the propagationof ultra-relativistic projectiles through the crystal are performed by using the MBN Ex-plorer package [10, 11]. This package was originally developed as a universal numerical toolto study structure and dynamics on the spatial scales from nanometers and above for a widerange of complex atomic and molecular systems. In order to address the channeling phe-nomena, an additional module has been incorporated into the
MBN Explorer to computethe motion for relativistic projectiles along with dynamical simulations of the propagationenvironments, including the crystalline structures, in the course of projectile’s motion [12].These computations advance to account for the interaction of the projectiles with the sep-arate atoms of the environments, whereas a variety of interatomic potentials implementedin
MBN Explorer support rigorous simulations of various media. The developed softwarepackage can be regarded as a powerful numerical tool to uncover the dynamics of relativisticprojectiles in crystals, amorphous bodies, as well as in biological environments. Its efficiencyand reliability has already been benchmarked for the channeling in crystals [6, 12–16]. Thecalculated with the
MBN Explorer relativistic motion, represented by the projectile’scoordinates and velocities at the instances of the propagation time, is used as the input datato compute the spectral and/or the spectral-angular distributions of the emitted radiation.A module for calculating the radiation emergent from the channeling is included into thelatest version of
MBN Explorer [11].
II. THEORETICAL FRAMEWORK
We consider the motion of the projectiles through the crystal as being governed by thelaws of classical relativistic dynamics: ∂ r /∂t = v , ∂ p /∂t = − q ∂U/∂ r , (2)where r ( t ) is the coordinate, v ( t ) is the velocity and p ( t ) = mγ v ( t ) is the momentum of theparticle at the propagation time t , γ = (1 − v /c ) − / = ε/mc is the relativistic Lorentz-factor, ε and m are the energy and the rest mass of the particle, respectively, and c is thespeed of light. The driving force, sensitive to the charge q of the projectile, stems froman electrostatic potential U = U ( r ) for the particle-crystal interaction. MBN Explorer allows for computing the latter potential from the interactions of the projectile with theindividual atoms, U ( r ) = X j U at ( | r − R j | ) . (3)The spacial locations R j of the atoms are selected according to the crystalline structuresof the interest (including the effect of thermal fluctuations). The atomic potentials U at inour simulations are evaluated within the Moliere approximation [17]. A rapid decrease ofthese potentials with increasing the distances from the atoms allows to truncate the sum(3) in practical calculations. MBN Explorer provides an option to restrict the atomscontributing to the interaction U ( r ) to these located inside a cut-off sphere around thecoordinate r , as well as invokes an efficient linked-cell algorithm to search for the atomsinside this sphere.A particular feature of MBN Explorer is simulating the environment “on the fly” i.e. inthe course of computing the motion of the projectiles. For channeling, the crystalline latticeis simulated inside a box surrounding the position of the projectile and the simulation boxshifts accordingly to the motion of the projectile. The coordinate frame for the simulationshas the z -axis along the incoming beam and parallel to the crystalline planes responsible forthe channeling, whereas the y axis is set perpendicular to these planes. In order to excludean accidental axial channeling, the z -axis should avoid major crystallographic directions.For channeling along the (110) planes, we opted to define the z -axis by a direction h− nnm i with n ≫ m ∼ R j = R (0) j + ∆ j ,selected with account for the thermal displacements ∆ j of the atomic nuclei with respect tothe equilibrium positions R (0) j . The equilibrium nodes correspond to the Bravais cells for thecrystal, whereas the Cartesian components ∆ jk ( k = x, y, z ) of the thermal displacementsare selected randomly according to the normal distribution w (∆ jk ) = 1 p πu T exp (cid:18) − ∆ jk u T (cid:19) . (4)The values of the amplitude u T of the thermal vibrations are well-known for various crystalsand can be found in Ref. [18]. For the silicon crystals at room temperature we use the value u T = 0 .
075 ˚A.The equations (2) are numerically integrated forth from t = 0 when the particle entersthe crystal at z = 0. The initial values x and y for the transverse coordinates are selectedrandomly from an entrance domain where the beam can be guided by the crystalline planesto get into the inside of the crystal. The size of this domain is taken between 2 d and 5 d in the x -direction and between d and 3 d in the y -direction, where d is the inter-plannerseparation for the crystal. The initial velocity v = ( v x , v y , v z ) has the value determinedby the beam energy and is predominantly oriented in the z -direction, i.e. the values v x and v y are small compared to the value v z ≈ c . The non-vanishing v x and v y can be adjustedto account for the beam emittance. The data presented below in Sects. III A and III B areobtained for the zero emittance, v x = v y = 0.When computing the motion of the projectile through the crystal with MBN Explorer ,an efficient algorithm of “dynamic simulation box” [12] is used as follows. Inside a simulationbox, the particle interacts with the atoms of the cutoff sphere. As the particle moves, thesphere shifts and at some point meets an edge of the box. Once this happens, a newsimulation box is introduced being centered at the current position of the projectile. Thenew box is then filled with the crystalline lattice such that the nodes inside the intersection ofthe old and the new simulation boxes remain unchanged i.e. not being simulated anew withincluding the effect of the thermal vibrations. This allows to avoid a spurious change in thedriving force as well as reduces numerical efforts in simulating the crystalline environmentin the course of the projectile’s motion.The above described numerical propagation procedure terminates when the z -coordinateof the projectile approaches the thickness L of the crystal under study. To simulate a peri-odically bent crystal, the y -coordinates for each lattice node R j = ( X j , Y j , Z j ) are obtainedfrom these for the straight crystal by the transformation Y j → Y j + δ ( Z j ) with δ ( Z j ) beingdetermined by the bending profile (1).Employing the Monte-Carlo technique for sampling the incoming projectiles as well as foraccounting for the thermal fluctuations of the crystalline lattice yields a statistical ensembleof trajectories simulated with MBN Explorer . The trajectories then can be used forcomputing the radiation from the projectiles passing through the crystal. The energy emittedper unit frequency ω within the cone θ ≤ θ max along the z axis is computed as followsd E ( θ ≤ θ max )d ω = 1 N N X n =1 2 π Z d φ θ max Z θ d θ d E n d ω dΩ , (5)where the sum is carried over the simulated trajectories of the total number N , Ω is the solidangle corresponding to the emission angles θ and φ , and d E n / d ω dΩ is the energy per unitfrequency and unit solid angle emitted by the projectile moving along the n th trajectory. Thegeneral equation (5) accounts for contributions of all the segments of simulated trajectoriesi.e. the segments of the channeling motion as well as the segments of motion out of thechanneling regime.The radiation emitted by the individual projectiles is computed within the quasi-classicalapproach developed by Baier and Katkov. For the details of the formalism as well asvarious applications to radiative processes we refer to the monograph [19] (we also mentionAppendix A in Ref. [20] where a number of intermediate steps of the formalism are evaluatedexplicitly). It is to be pointed out that, along with the classical description of the motion ofprojectiles, the formalism includes the effect of quantum radiative recoil i.e. it accounts forthe change of the projectile energy due to the photon emission. The impact of the recoil onthe radiation is governed by the ratio ~ ω/ε , and the limit ~ ω/ε → r = r ( t ) in the crystal of the thickness L , we use the equation [1, 12]:d E ~ d ω dΩ = αq ω (1 + u )(1 + w )4 π " w | I z | γ (1 + w ) + | sin φI x − cos φI y | + | θI z − cos φI x − sin φI y | , (6)where α is the fine structure constant, q is the charge of the projectile in units of theelementary charge, u = ~ ω/ ( ε − ~ ω ) and w = u / (1 + u ). The quantities I x,y,z involve theintegrals with the phase functions, I z = τ Z d t e i ω ′ Φ( t ) − i ω ′ (cid:18) e i ω ′ Φ(0) D (0) − e i ω ′ Φ( τ ) D ( τ ) (cid:19) ,I x,y = τ Z d t v x,y ( t ) c e i ω ′ Φ( t ) − i ω ′ (cid:18) v x,y (0) c e i ω ′ Φ(0) D (0) − v x,y ( τ ) c e i ω ′ Φ( τ ) D ( τ ) (cid:19) , (7)where τ = L/c is the time of flight through the crystal, D ( t ) = 1 − n · v ( t ) /c , Φ( t ) = t − n · r ( t ) /c , n is the unit vector in the emission direction, and ω ′ = (1 + u ) ω . The quantumradiative recoil is accounted for by the parameter u which vanishes in the classical limit. III. CHANNELING AND RADIATION FOR
MEV ELECTRONS ANDPOSITRONS
Within the above described theoretical framework and by exploiting the
MBN Ex-plorer package, the trajectories and radiation spectra have been simulated for the ε = 855MeV electrons and positrons. The projectiles were selected as incoming along the (110)crystallographic planes in a straight silicon crystal, and the SASP bent crystalline profileswere introduced according to Eq. (1).The amplitude and period of bending were varied within the intervals a = 0 . . . . . a to be smaller than the distance d = 1 .
92 ˚A between the (110) planes inthe straight Si crystal) and λ u = 100 . . . . θ was carried out for two particular detector apertures determined by the values θ max = 0 . γ − ≈ . γ − providing the emission cone with θ ≤ θ max to collect almost all the radiationfrom the relativistic projectiles. The latter situation corresponds to the conditions at theexperimental setup at SLAC [25]. The discussion of the calculated emission spectra is givenin Section III B. A. Statistical Properties of Channeling
For each type of the projectiles and different sets of the bending amplitudes and periods(including the case of the straight crystal with a = 0), the numbers N of the simulatedtrajectories were sufficiently large (between 4000 and 7000) thus enabling a reliable statis-tical quantification of the channeling process. Below we define and describe the quantitiesobtained.A randomization of the “entrance conditions” for the projectiles (in particular samplingthe entrance locations as explained in Sect. II) makes the different projectiles to encounterdifferently scattering with the crystalline atoms upon entering the crystal. As a result,not all the simulated projectiles start moving through the crystal in a channeling mode.A commonly used parameter to quantify the latter property is acceptance defined as theratio A = N acc /N of the number N acc of particles captured into the channeling mode onceentering the crystal (the accepted particles) to the number N of the incident particles.The non-accepted particles experience over-barrier motion unrestricted in the inter-planardirections.It is order to notice that different theoretical approaches to the interaction of the projec-tiles with the crystalline environments are also different in criteria distinguishing between thechanneling and the over-barrier motions. For example, the continuous potential approxima-tion [2] decouples the transverse (inter-planar) and longitudinal motions of the projectiles.As a result, it is straightforward to distinguish the channeling projectiles as those withtransverse energies not exceeding the height of the inter-planar potential barrier. In oursimulations, the projectiles interact, as in reality, with the individual atoms of the crystal.The inter-planar potential experienced by the projectiles vary rapidly in the course of theirmotion and couples the transverse and longitudinal degrees of freedom. Therefore, anothercriteria are required to select the channeling episodes in the projectile’s motion. We assumethe channeling to occur when a projectile, while moving in the same channel, changes thesign of the transverse velocity v y at least two times [26]. The latter criteria have also tobe supplemented by geometrical definitions of the crystalline channels. Here, we refer tothe straight crystals where the positron channels are the volume areas restricted by theneighboring (110) planes, whereas the electron channels are the areas between the corre-sponding neighboring mid-planes. For SASP bent crystals, the simulations show that, atsmall values of the bending amplitude, the channeling occur in the above defined channelsfor the straight crystal. We will refer to such situation as to the “conventional” channeling.With the amplitudes increasing above some values, the positrons tend to channel in theelectron channels for the straight crystal, and vice versa, the electrons become channeling0in the positron channels. The latter situation will be regarded as the “complementary”channeling. We will provide below a quantification for both channeling regimes.For an accepted projectile, the channeling episode lasts until an event of de-channelingwhen the projectile leaves the channel. The de-channeling events occur mostly as cumulativeoutcomes of multiple collisions of the projectiles with the crystalline constituents. Also, arare large-angle scattering collision can lead to de-channeling. De-channeling of the acceptedparticles is conveniently quantified by the penetration depth L p1 evaluated as the meanlongitudinal extension of the primary channeling segments of the trajectories. The lattersegments are those that start at the crystalline entrance and end at the first de-channelingevent experienced by the particle inside the crystal [12]. The penetration depth can berelated to a commonly used de-channeling length L d . Within the framework of the diffusiontheory of de-channeling (see, e.g., Ref. [27]), the fraction of the channeling particles at largedistances z from the entrance decays exponentially, ∝ exp( − z/L d ), with increasing z [28].Assuming the decay law to be applicable for all z , the penetration depth can be evaluated asthe integral L p1 = R L ( z/L d ) exp( − z/L d )d z and appears to be smaller than L d for the finitelengths L of the crystalline samples. For sufficiently long crystals, L ≫ L d , the penetrationdepth approaches the de-channeling length, L p1 → L d .Random scattering of the projectiles on the crystalline constituents can also result ina re-channeling process of capturing the over-barrier particles into the channeling mode ofmotion. This process is quantified by the re-channeling length , L rech , defined as an averagedistance along the crystal between the trajectory points corresponding to the successiveevents of de-channeling and re-channeling. We notice that, the re-channeling events aremore frequent (and, correspondingly, the re-channeling lengths are shorter) for the negativelycharged projectiles than these for the positively charged ones. For a qualitative explanationof this feature we refer to Ref. [26].In a sufficiently long crystal, the projectiles can experience de-channeling and re-channeling several times in the course of propagation. These multiple events are accountedfor by an additional pair of lengths, the penetration length L p2 and the total channelinglength L ch that characterize the channeling process in the whole crystal. The depth L p2 isthe average extension calculated with respect to all channeling segments in the trajectories,i.e. not only with respect to the segments that start from the crystalline entrance and areused to evaluate L p1 but also including the segments that appear inside the crystal due1to the re-channeling events followed by de-channeling. The total channeling length L ch iscomputed as the average length with respect to all channeling segments per trajectory.As a result of re-channeling, the projectiles become captured into the channels whenpossess, statistically, the incident angles of the values smaller than the Lindhard’s criticalangle values Θ L [2]. Therefore, for sufficiently thick crystals, the lengths L p2 provide theestimates of the de-channeling lengths L d for the beams with the emittance values ≈ Θ L .The above described statistical quantities have been calculated from the simulations onthe channeling for the 855 MeV electrons and positrons in the straight and SASP bentSi(110) crystals of the thickness L = 150 micron. The value of the bending period hasbeen fixed at λ u = 400 nm, and different values of the bending amplitude a have beenselected. The results on the acceptance and characteristic lengths are presented in Table I.All the data refer to the zero emittance beams. Statistical uncertainties due to the finite(but sufficiently big) numbers of the simulated trajectories correspond to the confidenceprobability value of 0 . a ≤ . L p1 , L p2 and L ch for the electrons withthese for the positrons reveals that, all three lengths for the positrons noticeably exceed,up to the order of magnitude, the corresponding lengths for the electrons. The latter isnot surprising given that fact that the channels optimally guiding the particles through thecrystal are different for the positrons and electrons due to the different character of the theparticle-crystal interactions. As the interactions with the crystalline atoms are repulsive forthe positrons and attractive for the electrons, the channels guide these particles to move inthe domains with low and high content of the crystalline atoms, respectively. In course ofthe projectile’s motion, the electrons experience scattering on the atoms more frequent asthe positrons. As a result, for the electrons the de-channeling process develops faster andresults in shorter penetration and channeling lengths than for the positrons. The penetrationlengths L p1 and L p1 for the electrons appear to be close to each other within the statisticaluncertainties, for almost all the bending profiles studied in Table I, as well as to be muchshorter than the thickness L = 150 micron for the simulated crystalline sample. Therefore,either of these lengths can be regarded as the de-channeling length for the electrons. Forthe positrons, two penetration scales, L p1 and L p2 , appear to be different, with the values L p2 being systematically smaller than the values L p1 . For small bending amplitudes, the2 TABLE I. Acceptance A , penetration depths L p1 , , total channeling length L ch and re-channelinglength L rech for 855 MeV electrons and positrons channeling in L = 150 µ m thick straight andSASP bent Si(110). The bending period is λ u = 400 nm, and the different values of the bendingamplitude a are examined. Two sets of statistical quantities given in the upper and bottom linesfor a = 0 . , . electron channeling a (˚A) A L p1 ( µ m) L p2 ( µ m) L ch ( µ m) L rech ( µ m)0 .
00 0 .
64 11 . ± .
49 11 . ± .
19 42 . ± .
21 24 . ± . .
05 0 .
61 9 . ± .
43 8 . ± .
17 39 . ± .
49 21 . ± . .
10 0 .
54 6 . ± .
32 6 . ± .
10 34 . ± .
40 19 . ± . .
15 0 .
48 5 . ± .
30 5 . ± .
09 30 . ± .
39 19 . ± . .
20 0 .
48 5 . ± .
33 5 . ± .
10 26 . ± .
16 23 . ± . .
25 0 .
50 6 . ± .
43 6 . ± .
14 25 . ± .
17 26 . ± . .
30 0 .
56 8 . ± .
43 7 . ± .
17 26 . ± .
02 30 . ± . .
40 0 .
70 10 . ± .
57 10 . ± .
26 30 . ± .
28 32 . ± . .
50 0 .
73 11 . ± .
57 10 . ± .
26 31 . ± .
43 30 . ± . .
60 0 .
78 10 . ± .
55 10 . ± .
27 32 . ± .
61 29 . ± . .
70 0 .
81 8 . ± .
47 9 . ± .
23 27 . ± .
43 31 . ± . .
64 5 . ± .
25 5 . ± .
11 20 . ± .
10 29 . ± . .
80 0 .
71 4 . ± .
23 5 . ± .
10 21 . ± .
08 26 . ± . .
79 7 . ± .
33 7 . ± .
16 26 . ± .
05 29 . ± . .
90 0 .
36 2 . ± .
10 2 . ± .
03 13 . ± .
67 22 . ± . .
72 8 . ± .
17 7 . ± .
17 27 . ± .
34 28 . ± . a (˚A) A L p1 ( µ m) L p2 ( µ m) L ch ( µ m) L rech ( µ m)0 .
00 0 .
95 131 . ± .
11 100 . ± .
11 131 . ± .
02 29 . ± . .
10 0 .
92 124 . ± .
78 84 . ± .
09 122 . ± .
66 29 . ± . .
20 0 .
89 115 . ± .
99 65 . ± .
62 111 . ± .
81 27 . ± . .
30 0 .
88 99 . ± .
44 48 . ± .
25 100 . ± .
04 25 . ± . .
40 0 .
86 84 . ± .
05 36 . ± .
12 87 . ± .
53 26 . ± . .
50 0 .
84 63 . ± .
94 23 . ± .
47 70 . ± .
33 24 . ± . .
60 0 .
80 41 . ± .
23 13 . ± .
84 52 . ± .
64 22 . ± . .
70 0 .
74 13 . ± .
56 6 . ± .
29 31 . ± .
58 20 . ± . .
50 6 . ± .
34 3 . ± .
09 22 . ± .
03 16 . ± . .
80 0 .
58 3 . ± .
19 3 . ± .
04 20 . ± .
77 19 . ± . .
66 9 . ± .
43 6 . ± .
15 27 . ± .
01 21 . ± . .
90 0 .
29 2 . ± .
09 2 . ± .
02 17 . ± .
77 18 . ± . .
80 15 . ± .
76 12 . ± .
34 40 . ± .
56 27 . ± . L d ≈
570 micronessentially longer than the crystalline thickness. Therefore, the values for the positronpenetration scales L p1 presented in Table I can be considered only as the lower bounds forthe positron de-channeling length. penetration distance, z ( µ m) -1-0.500.511.52 d i s t an c e f r o m s t r a i gh t p l ane ( i n un i t s o f d ) penetration distance, z ( µ m) -1-0.500.511.52 a =0 a =0.9 Åe + e-e + e- FIG. 1. Channeling trajectories for the electrons( e − ) and positrons ( e + ) in the straight (left plot)and periodically bent (right plot) Si(110). Thick dashed lines correspond to the straight (110)-planes, chained lines mark the centerlines between the planes. Thin dashed lines in the right plotshow the profiles of the planes bent with a = 0 . λ u = 400 nm. The trajectories shown in theleft and right plot can be regarded as “conventional” and “complementary”, respectively. Noticethe shift d/ d is the inter-planar distance) between the equilibrium positions for the transverseoscillations of the projectiles in the left- and right-plot trajectories. A close look at the trajectories simulated for various bending amplitudes reveals thechanneling segments in the trajectories to be of two different kinds that can be called “con-ventional” and “complementary”. The fraction of “complementary” segments in the simu-lated trajectories is small and the corresponding channeling lengths are very short for thebending amplitudes a ≤ . a the latter fraction extends and dominates4over the “conventional” fraction as the amplitude approaches the value d/ d beingthe inter-planar distance in straight Si(110). The two kinds of the channeling trajectorysegments yield essentially different values for the statistical quantities. In Table I, the quan-tities deduced from the “complementary” segments are given in the lower lines for a = 0 . . . a we refer to Fig. 1. The left plot in thefigure shows the channeling segments of the trajectories in the straight crystal. There, theprojectiles exhibit the “conventional” channeling where the electrons oscillate around thecrystalline planes whereas the positrons oscillate around the centerlines between the planes.In the bent crystal studied in the right plot for a = 0 . d/ a ≤ . a ≈ d/ L p1 , and L ch change with varying the bending period a for the electrons and for the positrons in essen-tially different manner. To illustrate the difference, Fig. 2 shows the primary penetrationlengths L p1 as functions of a for the electrons (left plot) and the positrons (right plot). Thelength computed from the “conventional” trajectory segments (solid curves in the figure)vary monotonously with a for the positrons and exhibits pronounced local minimum andmaximum in the range a ≤ . a (dashed curves in the figure) be-ing smaller than the “conventional” lengths unless a approaches the values ≈ .
75 ˚A forboth types of the projectiles. For larger bending amplitudes the “complementary” channel-ing segments become on average longer than the “conventional” segments that display thevalues for the penetration lengths close to each other for the electrons and positrons.5
Bending amplitude (Å) P ene t r a t i on l eng t h L p1 ( µ m ) Bending amplitude (Å) "normal""complementary" electron positron
FIG. 2. Penetration length L p1 versus bending amplitude for the 855 MeV electrons (left plot) andpositrons (right plot) in SASP bent Si(110). The bending period is fixed at 400 nm. The solid anddashed curves represent the dependencies calculated from the “conventional” and “complementary”channeling trajectory segments, respectively. See also explanations in the text. The origin of the peculiarities described above are the changes in the particle-crystalinteractions with changing the bending amplitudes. As already discussed, the positronsand electrons tend to channel in the domains with low and high content of the crystallineconstituents, respectively. In order to highlight how these domains change with changing thebending amplitude, a continuous inter-planar potential experienced by the moving projectilescan be considered. To calculate this potential for a periodically bent crystal we used theapproach described in A. The non-periodic part of the emerging potential, Eq. (A1), isevaluated by using the Moli`ere atomic potentials. This part is presented in Fig. 3 for theelectrons (left plot) and positrons (right plot), for different values of the bending amplitudes(given in ˚A near the curves). Fig. 4 shows the corresponding volume densities of thecrystalline electrons and nuclei.In Fig. 3, the potential curves labeled with “0” correspond to the straight crystal. Forsmall amplitude values, a ≤ . a aredecreasing potential barriers and, for the electrons, a broadening potential well. As the a values approach 0 . . . . . -1.5 -1 -0.5 0 0.5 transverse coordinate (in units of d ) -50510152025 e l e c t r on i n t e r p l ana r po t en t i a l ( e V ) -1 -0.5 0 0.5 1 transverse coordinate (in units of d ) -50510152025 po s i t r on i n t e r p l ana r po t en t i a l ( e V ) FIG. 3. Continuous inter-planar potentials for the electrons (left plot) and positrons (right plot)for different values of the bending amplitude indicated in ˚A near the curves ( a = 0 corresponds tothe straight crystal). The potentials shown in the figure are evaluated for the temperature 300 Kby using the Moli`ere atomic potentials and averaging the individual particle-atom interactionsover the bending period λ u = 400 nm. The unit for the transverse coordinate in the plots is theinter-planar distance d = 1 .
92 ˚A. The vertical lines mark the adjacent (110)-planes in the straightcrystal. -1.5 -1 -0.5 0 0.5 transverse coordinate (in units of d ) e l e c t r on den s i t y , n S i / n S i a m -1.5 -1 -0.5 0 0.5 transverse coordinate (in units of d ) nu c l ea r den s i t y , n S i / n S i a m FIG. 4. Distributions of the electronic (left graph) and nuclear (right graph) densities along thetransverse coordinate. The densities are normalized with respect to the values for amorphoussilicon and correspond to the potentials shown in Fig. 3. more dramatic way as the additional potential wells appear. These wells force the projectileto channel in spatial regions different from the channels for the straight and small-amplitudebent crystal. The latter regions are the “complementary” channels that appear in vicinities7of the centerlines for the periodically bent planes for the positrons, and are shifted awayfrom the centerlines for the electrons. For the largest bending amplitude studied, a = 0 . a both theelectronic and nuclear crystalline densities gradually increase in the region between the twocrystalline planes. As a result, motion of the positrons in the “conventional” channels isaccompanied by increasing probability of collisions with the nuclei. This effect, togetherwith lowering potential well, leads to a monotonous decrease of the channeling lengths.In contrast, in the “complementary” positron channels the densities decrease gradually, sothat the corresponding channeling lengths increase. The non-monotonous variation with a of the lengths for “conventional” channeling of the electrons can be interpreted as follows.For small values of a , the decrease in the potential barrier out-powers the decrease in thedensities in the central part of the channel. As a result, the channeling lengths initiallydecrease with increasing a . For 0 . ≤ a . . . . . . L p1 , and L ch with a . Forthe amplitudes larger than 0 . a ≥ . fraction ofthe incident particles captured into the channeling mode at the entrance. The artifact valuesfor the sum of “conventional” and “complementary” acceptances arise from double-countingof some particles as being accepted by both kinds of channels. Channeling in a SASP bentcrystal develops with the trajectory oscillations of distinctly different periods, the long period λ ch of channeling oscillations and the short period λ u ≪ λ ch of oscillations due to the bending.At the entrance, a particle experiencing the short-period oscillations is distinguished in thesimulations as channeling regardless of the type, “conventional” or “complementary”. Forillustration, we refer to the electron trajectory marked with the open circles in the right plotof Fig. 1. Near the crystalline entrance, this trajectory exhibits several oscillations with theperiod λ u becoming thereby accepted by both the “conventional”, − . < y/d < − .
5, andthe “complementary”, − < y/d <
0, electron channels.
B. Radiation Spectra
The above described statistical studies on the channeling properties show the penetrationlengths for the 855 MeV electrons to not exceed a value of about 12 µ m. We have thereforeopted to study first the radiation for the 12 µ m thick silicon crystal. The radiation spectraproduced by the particles incoming along the (110) crystallographic planes are presented inFigs. 5 and 6. In Fig. 5, the spectra are studied for the fixed value of the bending period, λ u = 400 nm, and different values of the bending amplitudes in the range a = 0 . . . . a = 0 . λ u = 200 . . . θ max = 0 .
21 mrad and θ max = 4 mrad (top andbottom, respectively). The smaller aperture value refers to a nearly forward emission from855 MeV projectiles ( θ max ≈ (3 γ ) − ) wheres the second value corresponds to the emissioncone collecting almost all the radiation from the relativistic particles ( θ max ≫ γ − ).The spectra computed for the straight crystal and various bending amplitudes display avariety of features seen in Fig. 5. To be noticed are the pronounced peaks of the channelingradiation in the spectra for the straight crystal (the black solid-line curves). Nearly perfectly9 d E / d ( h _ ω ) ( × ) straight a = 0.1 Å a = 0.2 Å a = 0.4 Å a = 0.5 Å a = 0.6 Å a = 0.8 Å a = 0.9 ÅB-H positron d E / d ( h _ ω ) ( × ) straight a = 0.1 Å a = 0.2 Å a = 0.4 Å a = 0.5 Å a = 0.6 Å a = 0.8 Å a = 0.9 ÅB-H electron d E / d ( h _ ω ) ( × ) straight a =0.1 Å a =0.2 Å a =0.4 Å a =0.6 Å a =0.8 Å a =0.9 ÅB-H positron d E / d ( h _ ω ) ( × ) straight a = 0.1 Å a = 0.2 Å a = 0.4 Å a = 0.5 Å a = 0.6 Å a = 0.8 Å a = 0.9 ÅB-H electron FIG. 5. Spectral distribution of radiation emitted by 855 MeV positrons (left) and electrons (right)in straight (thick solid lines) and SASP bent Si(110) with the period λ u = 400 nm and variousamplitudes a as indicated. Dotted lines mark the Bethe-Heitler spectra for amorphous silicon. Theupper and lower plots correspond to the aperture values θ max = 0 .
21 mrad and θ max = 4 mrad,respectively. All spectra refer to the crystal thickness L = 12 micron. harmonic channeling oscillations in the positron trajectories (the examples of the simulatedtrajectories can be found in [1, 12, 14]) lead to the undulator-type spectra of radiation withsmall values of the undulator parameter, K < ≈ . K . In particular, for the smalleraperture value, θ max = 0 .
21 mrad, the maximal spectral spectral intensity in the fundamentalpeak is an order of magnitude larger than that for the second harmonics displayed by a smallpeak at about 5 MeV, and only a tiny hump of the third harmonics of channeling radiationcan be recognized at about 7.5 MeV (see the top left plot in the figure). For the electrons0passing through the straight crystal, the channeling radiation peaks are less intensive andmuch broader than these for the positrons, as a result of stronger anharmonicity of thechanneling oscillations in the trajectories. For the larger aperture value, θ max = 4 mrad,a sizable part of the energy is radiated at the angles γ − < θ < θ max . For relatively largeemission angles, the emission energy for the first channeling spectral harmonics decreaseswith increasing angle. As a result, the fundamental peaks of channeling radiation broadenand shift towards softer radiation energies.The radiation spectra produced by the projectiles passing the SASP bent crystals displayadditional peaks, more pronounced for the smaller aperture value, which emerge from theshort-period modulations of the projectile trajectories by the bent crystalline structure. Themajor novel feature of the radiation is that the peaks due the bending appear at the emissionenergies larger than the energies of the channeling peaks. For both types of the projectiles,the fundamental spectral peaks in the radiation emergent from SASP bending correspondto the emission energy about 16 MeV significantly exceeding the energy 2 . a , the spectral peaks disappear because the positrons experience mainly “conventional”channeling staying away from the crystalline atoms and being therefore less affected by theSASP bent planes. In contrast, the electrons experience the impact of the bent crystallinestructure at lower bending amplitudes. As seen in the right upper plot for the fundamentalspectral peaks emergent from the bending, the peak for a = 0 . a = 0 . a exceeding 0 . d E / d ( h _ ω ) ( × ) straight λ u =2500nm λ u =1000nm λ u =600nm λ u =400nm λ u =300nm λ u =200nmB-H positron up to1.2 up to0.6 d E / d ( h _ ω ) ( × ) straight λ u =2500 nm λ u =1000 nm λ u =600 nm λ u =400 nm λ u =300 nm λ u =200 nmB-H electron d E / d ( h _ ω ) ( × ) straight λ u =600nm λ u =400nm λ u =300nm λ u =200nm electron up to2.2 FIG. 6. Spectral distribution of radiation emitted by 855 MeV positrons (left) and electrons (right)in straight (thick solid curves) and SASP bent Si(110) with the bending amplitude a = 0 . λ u as indicated. Dotted lines mark the Bethe-Heitler spectra for amorphous silicon.The spectra are computed for the crystal thickness L = 12 µ m, and two values for the radiationaperture, θ max = 0 .
21 mrad (upper plots) and θ max = 4 mrad (lower plots). Fig. 6. The positions of the lines due to the bending are clearly seen to be in reciprocal rela-tion with the bending period and shift towards hard-range radiation energies with decreasing λ u . In contrast, the positions of channeling spectral lines, especially for the positrons, donot noticeably change with varying λ u . Yet the shapes of the channeling lines are differentfor different bending amplitudes: the lines decrease in height as well as slightly shift towardsthe softer radiation energies with increasing bending period. The changes in the channelingradiation develop when the channeling lines and the lines resulting from the bending appearclose to each other in the spectra and therefore interfere. The latter effects are more promi-nent for the larger aperture value (bottom plots in the figure), and in the spectra for theelectrons (right plots) then in the spectra for the positrons (left plots).2To complete the studies on the radiation from SASP bent Si(110), we show in Fig. 7 thespectra simulated for four different thicknesses of the crystal, from 12 up to 150 micron.We can conclude that, the spectral shapes remain fairly the same with increasing crystallinethickness, with naturally increasing spectral intensities. The spectra in the figure also displaythe effects of varying bending amplitude. For both types of the projectiles, increasing theamplitude from 0 . . d E / d ( h _ ω ) ( × ) µ m25 µ m75 µ m150 µ m a =0.4 Å, λ u =400nm positron up to 5.2 d E / d ( h _ ω ) ( × ) µ m25 µ m75 µ m150 µ m a =0.4 Å, λ u =400nm electron up to 0.95 d E / d ( h _ ω ) ( × ) µ m25 µ m75 µ m150 µ m a =0.6 Å, λ u =400nm positron d E / d ( h _ ω ) ( × ) µ m25 µ m75 µ m150 µ m a =0.6 Å, λ u =400nm electron FIG. 7. Spectral distributions of radiation emitted by the 855 MeV electrons (right plots) andpositrons (left plots) for SASP bending with the period λ u = 400 nm and the amplitudes a =0 . . θ max = 0 .
21 mrad and different crystal thicknesses indicated in the plots. IV. CONCLUSIONS
In this paper, we have provided a systematic analysis of the channeling and radiation inshort-period small-amplitude bent silicon crystals.The statistical properties of channeling were described in terms of the lengths quantifyingthe spatial scales of de-channeling and re-channeling processes experienced by the projec-tiles. We have particularly focused on behavior of the lengths with varying the bendingamplitude of the crystalline planes. With the amplitude increasing above already somemoderate values, we have encountered a drastic change in the channeling process for theboth types of projectiles. To elucidate the underlying physics, an analytical model has beendeveloped in terms of the charge densities for the bent crystalline media which influence theparticles moving through the crystal. The model reveals the channels optimal for “binding”the transverse motion of the projectiles to shift in the inter-planar direction with increasingthe bending amplitude. The varying properties of channeling can be clarified in terms oftwo groups of the trajectories, the “conventional” and “complementary” ones. The supple-mentary analytical model of the continuous potential turns out to be helpful proving thesimulations performed with
MBN Explorer to be reliable, in particular, with respect toadvance account for the interaction of the projectiles with individual atoms of the crystallinemedia.
ACKNOWLEDGMENTS
The work was supported by the European Commission (the PEARL Project within theH2020-MSCA-RISE-2015 call, GA 690991).
Appendix A: Continuous Potential in a SASP Bent Crystal
In this supplementary section we develop an approximation of continuous inter-planarpotential. This potential and the corresponding distributions of the crystalline charge den-sities help in qualitative explanations of the results of numerical simulations on the motionand radiation for the electrons and positrons in SASP bent crystal, discussed above in Sects.III A and III B.4
1. Continuous Potential of a Periodically Bent Plane
To derive the approximation of continuous potential in a crystal with periodically bentplanes we use a general approach developed to study the radiation by fast projectiles inacoustically excited crystals (see, for example, Ref. [32]).For the cosine periodic bending (1), the continuous potential U pl of a single plane can bepresented in the form of a series: U pl ( a ; y, z ) = U ( a ; y ) + ∞ X m =1 cos(2 πmz/λ u ) U m ( a ; y ) . (A1)The expansion potentials U m ( a ; y ) can be related to the atomic potentials according to theexpressions given in Ref. [32]. In the limit a → n > U m (0; y ) = 0, and the zero-order (non-periodic in z ) term yields the continuous potentialof the straight plane, U pl (0; y, z ) = U (0; y ) ≡ U pl ( y ). The later planar potential dependsonly on the transverse coordinate y . In the following, we assume the bending amplitude andperiod to satisfy the SASP bending condition a < d ≪ λ u , (A2)and relate the non-periodic term U ( a ; y ) to the planar potential U pl ( y ).For a straight plane, the continuous potential is obtained by summing up the atomicpotentials as exerted by the atoms distributed uniformly along the plane [2]. The surfacedensity N of the atoms is related to the volume density n as N = nd , where d is theinter-planar spacing.For the periodically bent plane we notice, that the atoms located withing the interval[ y ′ , y ′ + d y ′ ] with respect to the centerline y ′ = 0 are distributed along the planar surfacewith the density 2d l N /λ u . By virtue of the strong inequality a ≪ λ u , Eq. (A2) above, wehave d l ≈ d z and can approximate the planar atomic density as 2d z N /λ u (see Fig. 8). Thecontinuous potential then can be calculated as follows U ( a ; y ) = Z λ u / − λ u / zλ u U pl (cid:16) | y − a cos(2 πz/λ u ) | (cid:17) = 1 π Z π/ − π/ U pl (cid:16) | y − a cos ξ | (cid:17) d ξ , (A3)where ξ = 2 πz/λ u , and appears to not depend on the bending period λ u but be affected bythe amplitude a . For a = 0 the potential reduces to potential U pl ( y ) for the straight plane.5 a y y’ dy’ λ u FIG. 8. Illustrative figure to the method of calculating the continuous potential for a periodicallybent crystallographic plane (thick solid curve represents the bending profile (1)). See explanationsin the text.
Similar procedure can be used to derive a non-periodic part of the distribution n (+)0 ( a ; y )of the nuclei in the periodically bent plane. One obtains n (+)0 ( a ; y ) = 1 π Z π/ − π/ n (+)pl (cid:16) | y − a cos ξ | (cid:17) d ξ , (A4)where n (+)pl is the nuclear volume density for the straight plane. With account for the thermalvibrations, the latter density is given by the formula n (+)pl ( y ) = n p πu T exp (cid:18) − y u T (cid:19) , (A5)where u T is the root-mean-square amplitude of the vibrations. The corresponding non-periodic part of distribution n ( − )0 ( a ; y ) of the crystalline electrons can be calculated from thePoisson equation as n ( − )0 ( a ; y ) = 14 πe d U ( a ; y )d y + Zn (+)0 ( a ; y ) , (A6)where Z is the nucleus charge.
2. Continuous Inter-planar Potential
The non-periodic part U ( a ; y ) of the inter-planar potential is obtained by summing upthe potentials (A3) of the separate planes. For the electrons it can be presented in the form U ( a ; y ) = U ( a ; y ) + N max X n =1 [ U ( a ; y + nd ) + U ( a ; y − nd )] + C , (A7)where y is the transverse coordinate with respect to an arbitrary selected reference plane,and the sum describes a balanced contribution from the neighboring planes. Since the planar6potential (A3) falls off rapidly with increasing distance from the plane, Eq. (A7) providesa good approximation for the inter-planar potential at already moderate numbers of theterms included in the sum. In our calculations we use N max = 2. The constant term C can be varied to adjust U ( a ; y = 0) = 0. For the positrons, the inter-planar potential canbe obtained from Eq. (A7) by reversing the signs of the planar potentials and selecting theconstant C to adjust U ( a ; y = ± d/
2) = 0. Similar summation schemes allow to calculatethe charge densities, nuclear and electronic, across the periodically bent channels.
3. The Moli`ere Approximation
The integral in the right-hand side of Eq. (A3) for the continuous potential for a separateperiodically bent plane can be evaluated using various model atomic potentials to quantifythe potential U pl ( y ) for a straight crystalline plane. A variety of model potentials can befound in, e.g., Refs. [2, 18, 19, 27, 33]. In our studies we use the Moli`ere approximation forthe atomic potentials [17] and evaluate the planar potential with accounting for the thermalvibrations of the atoms (cf. Eq. (A5)). This approach yields the planar potential in a closedanalytical form [34, 35] U pl ( y ) = 2 πn am d Ze a TF 3 X i =1 (cid:16) F i ( y ) + F i ( − y ) (cid:17) (A8)with F i ( ± y ) = α i β i exp (cid:18) β i u T a ± β i ya TF (cid:19) erfc (cid:20) √ (cid:18) β i u T a TF ± yu T (cid:19)(cid:21) . (A9)In the above equations, n am is the mean nuclear density in the amorphous medium, Z ischarge number of the crystalline atoms, a TF = 0 . a B Z − / is the Thomas-Fermi radius( a B = 0 .
529 ˚A is the Bohr radius), α , , = (0 . , . , .
35) and β , , = (6 . , . , .
3) arethe parameters of the Moli`ere approximation for the atomic potential. The complementaryerror functions, erfc( ζ ) = 2 π − / R ∞ ζ exp( − t ) d t , in Eq. (A9) result from averaging over thethermal vibrations of atomic nuclei.The inter-planar potentials, presented in Fig. 3, were calculated from Eqs. (A3), (A7),(A8) and (A9). These potentials are also helpful in understanding the evolution of thelines of channeling radiation with varying bending period. To clarify this evolution wehave investigated the periods λ ch of the channeling oscillations in the trajectories and the7 distance from midplane (Å) pe r i od o f c hanne li ng o sc ill a t i on s , λ c h ( µ m ) distance from midplane (Å) a =0 Å a =0.2 Å a =0.3 Å a =0.4 Å a =0.5 Å a =0.6 Å positronelectron0 0.5 1 distance from midplane (Å) h _ ω c h ( M e V ) distance from midplane (Å) a =0 Å a =0.2 Å a =0.3 Å a =0.4 Å a =0.5 Å a =0.6 Å positronelectron FIG. 9. Periods (upper plots) and energies (lower plots) of channeling oscillations for 855 MeVelectrons (left panels) and positrons (right panels). The periods and energies are studied as func-tions of the distance from the channel centerlines for the continuous potentials shown in Fig. 3.The curves correspond to different bending amplitudes a as indicated ( a = 0 corresponds to thestraight crystal). See also explanations in the text. corresponding radiation energies ~ ω ch . These quantities have been evaluated as the functionsof the amplitude of channeling trajectory oscillations. The amplitudes were fixed by themaximal transverse displacements of the projectiles from the channel centerlines, and theperiods have been evaluated from the corresponding classical turning points and the shapesfor the continuous inter-planar potential (see, e.g., Sec. C.2 in Ref. [1]). The channelingenergies we deduced from the periods according to the relation ω ch = 2 γ (2 πc/λ ch ). Theresults for both types of the projectiles in the straight and bent with different amplitudeschannels are shown in Fig. 9. For the straight crystals, the channeling radiation energies varywith the amplitude of the trajectory oscillations in a narrow energy range for the positrons8and in a prominently broader range for the electrons (see the lower plots in the figure). Thelatter properties correspond to the narrow channeling lines in the radiation spectra for thepositrons, and the the broader lines in the spectra for the electrons, as seen in Fig. 5. Withchanging bending amplitude, the variation ranges of the radiation energy increase for thepositrons and decrease for the electrons, and correspond to the evolution of the channelinglines in the simulated spectra. REFERENCES [1] A.V. Korol, A.V. Solov’yov, Walter Greiner,
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