Crystal-based intensive gamma-ray light sources
aa r X i v : . [ phy s i c s . acc - ph ] A ug Crystal-based intensive gamma-ray light sources
Andrei V. Korol and Andrey V. Solov’yov ∗ ‡ MBN Research Center, Altenh¨oferallee 3, 60438 Frankfurt am Main, GermanyE-mail: [email protected]
Abstract.
We discuss design and practical realization of novel gamma-ray Crystal-based Light Sources (CLS) that can be constructed through exposure of orientedcrystals (linear, bent, periodically bent) to beams of ultrarelativistic charged particles.In an exemplary case study, we estimate brilliance of radiation emitted in a CrystallineUndulator (CU) LS by available positron beams. Intensity of CU radiation in thephoton energy range 10 − MeV, which is inaccessible to conventional synchrotrons,undulators and XFELs, greatly exceeds that of laser-Compton scattering LSs andcan be higher than predicted in the Gamma Factory proposal to CERN. Brillianceof CU-LSs can be boosted by up to 8 orders of magnitude through the process ofsuperradiance by a pre-bunched beam. Construction of novel CLSs is a challenging taskwhich constitutes a highly interdisciplinary field entangling a broad range of correlatedactivities. CLSs provide a low-cost alternative to conventional LSs and have enormousnumber of applications.
1. Introduction
The development of light sources (LS) for wavelengths λ well below 1 angstrom(corresponding photon energies E ph ≫
10 keV) is a challenging goal of modern physics.Sub-angstrom wavelength powerful spontaneous and, especially, coherent radiation willhave many applications in the basic sciences, technology and medicine. They may havea revolutionary impact on nuclear and solid-state physics, as well as on the life sciences.At present, several X-ray Free-Electron-Laser (XFEL) sources are operating (EuropeanXFEL, FERMI, LCLS, SACLA, PAL-XFEL) or planned (SwissFEL) for X-rays downto λ ∼ λ decreases.Therefore, to create a powerful LS in the range well below 1 ˚A, i.e. in the hard Xand gamma ray band, one has consider new approaches and technologies.In this article we discuss possibilities and perspectives for designing andpractical realization of novel gamma-ray Crystal-based LSs (CLS) operating at ‡ On leave from A.F. Ioffe Physical-Technical Institute, St. Petersburg, Russia rystal-Based Light Sources E ph & keV and above that can be constructed throughexposure of oriented crystals (linear, bent and periodically bent crystals) to beams ofultrarelativistic charged particles. CLSs include Channeling Radiation (ChR) emitters,crystalline synchrotron radiation emitters, crystalline Bremsstrahlung radiationemitters, Crystalline Undulators (CU) and stacks of CUs. This interdisciplinaryresearch field combines theory, computational modeling, beam manipulation, design,manufacture and experimental verification of high-quality crystalline samples andsubsequent characterization of their emitted radiation as novel LSs. In an exemplarycase study, we estimate the characteristics (brilliance, intensity) of radiation emitted inCU-LS by positron beams available at present. It is demonstrated that peak brillianceof the CU Radiation (CUR) at E ph = 10 − − MeV is comparable to or even higherthan that achievable in conventional synchrotrons but for much lower photon energies.Intensity of radiation from CU-LSs greatly exceeds that available in the laser-Comptonscattering LSs and can be made higher than predicted in the Gamma Factory proposal toCERN [9,10]. The brilliance can be boosted by orders of magnitude through the processof superradiance by a pre-bunched beam. We show that brilliance of superradiant CURcan be comparable with the values achievable at the current XFEL facilities whichoperate in much lower photon energy range.CLSs can generate radiation in the photon energy range where the technologiesbased on the charged particles motion in the fields of permanent magnets becomeinefficient or incapable. The limitations of conventional LS is overcome by exploitingvery strong crystalline fields that can be as high ∼ V/cm, which is equivalentto a magnetic field of 3000 Tesla whilst modern superconducting magnets provide 1-10 Tesla [11]. The orientation of a crystal along the beam enhances significantly thestrength of the particles interaction with the crystal due to strongly correlated scatteringfrom lattice atoms. This allows for the guided motion of particles through crystals ofdifferent geometry and for the enhancement of radiation.Examples of CLSs are shown in Figure 1. The synchrotron radiation is emittedby ultra-relativistic projectiles propagating in the channeling regime through a bentcrystal, panel a). A CU, panel b), contains a periodically bent crystal and a beam ofchanneling particles which emit CUR following the periodicity of the bending [12–14].A CU-based LS can generate photons of E ph = 10 keV - 10 GeV range (correspondingto λ from 0.1 to 10 − ˚A). Under certain conditions, CU can become a source of thecoherent light within the range λ = 10 − − − ˚A [14–16]. An LS based on a stack ofCUs is shown in panel c) [17].Practical realization of CLSs often relies on the channeling effect. The basic phenomenon of channeling is in a large distance which a projectile particle penetratesmoving along a crystallographic plane or axis and experiencing collective action ofthe electrostatic fields of the lattice atoms [18]. A typical distance covered by aparticle before it leaves the channeling mode due to uncorrelated collisions is calledthe dechanneling length , L d . It depends on the type of a crystal and its orientation,on the type of channeling motion, planar or axial, and on the projectile energy and rystal-Based Light Sources Figure 1.
Selected examples of the novel CLSs: (a) bent crystal, (b) periodically bentcrystal, (c) a stack of periodically bent crystals. Black circles and lines mark atomsof crystallographic planes, wavy curves show trajectories of the channeling particles,shadowed areas refer to the emitted radiation. charge. In the planar regime, positrons channel in between two adjacent planes whereaselectrons propagate in the vicinity of a plane thus experiencing more frequent collisions.As a result, L d for electrons is much less than for positrons. To ensure enhancementof the emitted radiation due to the dechanneling effect, the crystal length L must bechosen as L ∼ L d [12–14].The motion of a projectile and the radiation emission in bent and periodically bentcrystals are similar to those in magnet-based synchrotrons and undulators. The maindifference is that in the latter the particles and photons move in vacuum whereas incrystals they propagate in medium, thus leading to a number of limitations for thecrystal length, bending curvature, and beam energy. However, the crystalline fieldsare so strong that they steer ultra-relativistic particles more effectively than the mostadvanced magnets. Strong fields bring bending radius in bent crystals down to the cmrange and bending period λ u in periodically bent crystals to the hundred or even tenmicrons range. These values are orders of magnitude smaller than those achievable withmagnets [4]. As a result, the radiators can be miniaturized thus lowering dramaticallythe cost of CLSs as compared to that of conventional LSs. Figure 2 matches the magneticundulator for the European XFEL with the CU manufactured in University of Aarhusand used in recent experiments [19].Modern accelerator facilities make available intensive electron and positron beamsof high energies, from the sub-GeV up to hundreds of GeV. These energies combinedwith large bending curvature achievable in crystals provide a possibility to consider novelCLSs of the synchrotron type (continuous spectrum radiation) and of the undulator type(monochromatic radiation) of the energy range up to tens of GeV. Manufacture of highquality bent and periodically bent crystals is at the edge of current technologies.
2. Exemplary crystal-based LSs
A number of theoretical and experimental studies of the channeling phenomenon inoriented linear crystals have been carried out (see, e.g., a review [21]). A channelingparticle emits intensive ChR, which was predicted theoretically [22] and shortly afterobserved experimentally [23]. Since then there has been extensive theoretical andexperimental investigation of ChR. The energy of emitted photons E ph scales with thebeam energy as ε / and thus can be varied by changing the latter. For example, by rystal-Based Light Sources Figure 2.
Left:
Magnetic undulator for the European XFEL.
Right:
A Si − x Ge x superlattice CU (the upper layer) build atop the silicon substrate (the lower layer)with the face normal to the h i crystallographic direction. In the superlattice, the(110) planes are bent periodically. The picture courtesy of J.L. Hansen, A. Nylandstedand U. Uggerhøj (University of Aarhus). The whole figure is adapted from Ref. [20]. propagating electrons of moderate energies, ε = 10 −
40 MeV, through a linear crystalit is possible to generate ChR with photon energy E ph = 10 −
80 keV [24, 25]. Thisrange can be achieved in magnetic undulators but with much higher beam energy. High-quality electron beams of (tunable) energies within the tens of MeV range are availableat many facilities. Hence, it has become possible to consider ChR from linear crystalsas a new powerful LS in the X-ray range [24].In the gamma-range, ChR can be emitted by higher energy ε & MeV beams.However, modern accelerator facilities operate at a fixed value of ε (or, at several fixedvalues) [26–28]. This narrows the options for tuning the ChR parameters, in particular,the wavelength. Hence, the corresponding CLS lack the tunability option. From thisviewpoint, the use of bent and, especially, periodically bent crystals can become analternative as they provide tunable emission in the hard X- and gamma-ray range.Strong crystalline fields give rise to channeling in a bent crystal. Since its prediction[29] and experimental support [30], the idea to deflect high-energy beams of chargedparticles by means of bent crystals has attracted a lot of attention [21, 31]. Theexperiments have been carried with ultra-relativistic protons, ions, positrons, electrons, π − -mesons [19, 32–35]. Steering of highly energetic electrons and positrons in bentcrystals with small bending radius R gives rise to intensive synchrotron radiation with E ph & MeV. The parameters of radiation can be tuned by varying R within therange 10 − cm [36, 37].Even more tunable is a CU-LS. In this system CUR and ChR are emitted indistinctly different photon energy ranges so that CUR is not affected by ChR. Theintensity, photon energy and line-width of CUR can be varied and tuned by changing ε ,bending amplitude a and period λ u , type of crystal, its length and detector aperture [16].Since introducing the concept of CU, major theoretical studies have been devoted tothe large-amplitude large-period bending λ u ≫ a > d [12–14]. In this regime, a projectilefollows the shape of periodically bent planes. CUR is emitted at the frequencies ω u wellbelow those of ChR, ω ch . By varying a , λ u , ε and the crystal length one can tunethe CUR peaks positions and intensities. Small-amplitude small-period regime, whichimplies a ≪ d and λ u less than period of channeling oscillations [38–42]. This schemeallows emission of photons of the higher energies, ω u > ω ch , makes feasible construction rystal-Based Light Sources
3. Practical realization of CU
In the current paper a case study of a tunable
CU-based
LS is presented. Thereforebelow we focus on the methods which allow one to produce undulating crystal samples.The feasibility of the CU concept was verified theoretically in Refs. [12–14,49] whereessential conditions and limitations which must be met were formulated. These papersboosted theoretical and experimental investigation of the CU and CUR phenomenaworldwide, so that nowadays these topics represent a new and very rich field of research.Theoretical and experimental studies of the CU and CUR phenomena hasascertained the importance of the high quality of the undulator material needed toachieve strong effects in the emission spectra. Up to now, several methods to createperiodically bent crystalline structures have been proposed and/or realized.Figure 3 provides schematic illustration of the ranges of a and λ u within whichthe emission of intensive CUR is feasible. Shadowed areas mark the ranges currentlyachievable by different technologies.Several approaches have been applied to produce static bending. The greenisharea marks the area achievable by means of technologies based on surface deformations.These include mechanical scratching [50], laser ablation technique [51], grooving method[52, 53], tensile/compressive strips deposition [52, 54, 55], ion implantation [56]. Themost recent techniques proposed is based on sandblasting one of the major sides ofa crystal to produce an amorphized layer capable of keeping the sample bent [57].Another technique, which is under consideration in for manufacturing periodicallybent silicon and germanium crystals, is pulsed laser melting processing that produceslocalized and high-quality stressing alloys on the crystal surface. This technology isused in semiconductor processing to introduce foreign atoms in crystalline lattices [58].Currently, by means of the surface deformation methods the periodically bent crystalswith large period, λ u & microns, can be produced. rystal-Based Light Sources λ u one can rely on production of graded composition strainedlayers in an epitaxially grown Si − x Ge x superlattice [59,60]. Both silicon and germaniumcrystals have the diamond structure with close lattice constants. Replacement of afraction of Si atoms with Ge atoms leads to bending crystalline directions. By means ofthis method sets of periodically bent crystals have been produced and used in channelingexperiments [61]. A similar effect can be achieved by graded doping during synthesisto produce diamond superlattice [62]. Both boron and nitrogen are soluble in diamond,however, higher concentrations of boron can be achieved before extended defects appear[62,63]. The advantage of a diamond crystal is radiation hardness allowing it to maintainthe lattice integrity in the environment of very intensive beams [21]. The grey area in3 marks the ranges of parameters achievable by means of crystal growing.The bluish area indicates the range of parameters achievable by means of anothermethod, realization of which is although still pending, based on propagation of atransverse acoustic wave in a crystal [16]. Figure 3.
Shadowing indicates the ranges accessible by means of modern technologies:superlattices (grey), surface deformations (green), acoustic waves (blue). Slopingdashed lines indicate the boundaries of the stable channeling for ε = 0 . λ u ≫ a ) lies to the right from the line. Thehorizontal line a/d = 1 ( d is the interplanar spacing) separates the Large-Amplitude(LA) and Small-Amplitude (SA) bending. The boundaries of the most favourable CUregime, LALP, are marked by thick red lines. In a Crystalline Undulator (CU), a projectile’s trajectory follows the profile ofperiodic bending. This is possible when the electrostatic crystalline field exceeds thecentrifugal force acting on the projectile. This condition, which entangles bendingamplitude and period, the projectile’s energy and the crystal field strength, impliesthat the bending parameter C is less than one, see Eq. (B.4). Two sloping dashed linesin Fig. 3 show the dependences a = a ( λ u ) corresponding to the extreme value C = 1 rystal-Based Light Sources ε = 0 . λ u ≫ a , and (ii) λ u greatly exceeds the periodof channeling oscillations. The horizontal line a/d = 1 ( d stands for the interplanardistance) divides the CU domain into two parts: the Large-Amplitude (LA), a > d , andthe Small-Amplitude (SA), a < d , regions. Larger amplitudes are more favourable fromthe viewpoint of achieving higher intensities of CUR. The red lines delineate the domainwhere the LALP periodic bending can be considered.Another regime of periodic bending, Small-Amplitude Short-Period (SASP), canbe realized in the domain a < d and λ u < λ u are much smallerthat channeling periods of projectiles with ε & dynamic bending can be achieved by propagating a transverseacoustic wave along a particular crystallographic direction [12, 13, 64–68]. This can beachieved, for example, by placing a piezo sample atop the crystal and generating radiofrequencies to excite the oscillations. The advantage of this method is in its flexibility:the bending amplitude and period can be changed by varying the wave intensity andfrequency [12, 13]. Although the applicability of this method has not yet been checkedexperimentally, we note that a number of experiments has been carried out on thestimulation of ChR by acoustic waves excited in piezoelectric crystals [69]. bending period λ u ( µ m) bend i ng a m p li t ude a ( Å ) N d =5N d =10N d =20N d =30 AW frequency (MHz) a/d=1
C=1 C=0.1C=0.01C=0.2 ε =0.5 GeV bending period λ u ( µ m) bend i ng a m p li t ude a ( Å ) N d =5N d =10N d =20N d =50N d =100N d =150 AW frequency (MHz) a/d=1
C=1 C=0.1C=0.01C=0.2 ε =50 GeV Figure 4.
Ranges of acoustic wave frequency ν (upper horizontal axis), of bendingperiod equal to the wave wavelength λ u = λ AW (lower horizontal axis) and of amplitude a (vertical axis) to be probed to construct a silicon(110)-based CU in the LALP regime.The data refer to ε = 0 . ε = 50 GeV positrons. Figure 4 allows one to estimate the acoustic wave frequencies ν needed to achieve theLALP periodic bending of the silicon(110) planes. The diagonal dashed lines correspondto the dependences a = a ( λ u ) obtained for several values (as indicated) of the bendingparameter C . The CU cannot be realized in the domain lying to the left from the line rystal-Based Light Sources C = 1. The solid curves present the dependences a = a ( λ u ) calculated for the fixedvalues (indicated in the legends) of the number of undulator periods N d = L d ( C ) /λ u within the dechanneling length L d ( C ), Eq. (B.5). It is seen that the values λ u ∼ . . . microns correspond to the frequencies ν = v s /λ u ∼ . . . MHz, which are achievableexperimentally ( v s = 4 . × cm/s is the speed of sound) [69–71].
4. CU-LS versus state-of-the-art LS
In this section, we present quantitative estimates for the CUR brilliance using theparameters of high-energy positron beams either available at present or planned tobe commissioned in near future (see Table A1 in Appendix A). We demonstrate thatby means of CU-LS, which operates in the LALP regime, one can achieve much higherphoton yield as compared to the values achievable in modern LS facilities operating inthe gamma-ray range, E ph & keV.The relevant modern facilities are synchrotrons and undulators based on the actionof magnetic field. § Another type of modern LS, which does not utilize magnets, is basedon the Compton scattering process [72]. In this process, a low-energy (eV) laser photonbackscatters from an ultra-relativistic electron thus acquiring increase in the energyproportional to the squared Lorentz factor γ = ε/mc . This method has been used forproducing gamma-rays in a broad, 10 keV – 10 MeV, energy range [73, 74].The Compton scattering also occurs if the scatterer is an atomic (ionic) electronwhich moves being bound to a nucleus. This phenomenon is behind the Gamma Factoryproposal for CERN [9, 10] that implies using a beam of ultra-relativistic ions in thebackscattering process. In this case, an ionic electron is resonantly excited by absorbinga laser photon. The subsequent radiative de-excitation produces a gamma-photon.
The radiometric unit frequently used to compare different LS is brilliance , B . It isdefined in terms of the number of photons ∆ N ω of frequency ω within the interval[ ω − ∆ ω/ , ω + ∆ ω/
2] emitted in the cone ∆Ω per unit time interval, unit sourcearea, unit solid angle and per a bandwidth (BW) ∆ ω/ω [75–77]. To calculatethis quantity is it necessary to know the beam electric current I , transverse sizes σ x,y and angular divergences φ x,y as well as the divergence angle φ of the radiationand the ’size’ σ of the photon beam. Explicit expression for B measured in h photons/s/mrad /mm / . i reads [78] B = ∆ N ω (∆ ω/ω ) (2 π ) ε x ε y Ie , (1)where e is the elementary charge. The quantities ǫ x,y = p σ + σ x,y p φ + φ x,y are thetotal emittance of the photon source in the transverse directions with φ = p ∆Ω / π § Fore the sake of comparison we also match our data to the brilliance available at the XFEL facilitiesfor much lower energy of the emitted radiation. rystal-Based Light Sources σ = λ/ πφ being the ‘apparent’ source size calculated in the diffraction limit [79].In (1) σ, σ x,y are measured in millimeters, φ, φ x,y – in milliradians.The product ∆ N ω I/e on the right-hand side of Eq. (1), represents the numberof photons per second (intensity) emitted in the cone ∆Ω and frequency interval ∆ ω (see Eq. (B.1) in Section Appendix B). Using the peak value of the current, I max , onecalculates the peak brilliance , B peak .Let us compare the brilliance of CUR with that available at modern synchrotronfacilities. Figure 5 presents the peak brilliance calculated for positron-baseddiamond(110) and Si(110) CUs and that for several synchrotrons. The CUR curvesrefer to the optimal parameters of CU (see Appendix B), i.e. those which ensure thehighest values of B peak ( ω ) of CUR for each positron beam indicated. -2 -1 photon energy (MeV) pea k b r illi an c e ( pho t on s / s m r ad mm . % B W ) C(110)Si(110)
PETRAESRFSPring8APS SuperBSuperKEKBCEPC FACETDA Φ NE VEPP4BEPC2
Figure 5.
Comparison of the peak brilliance available at several synchrotron radiationfacilities (APS, ESRF, PETRA, SPring8) with that for CUR from diamond(110)- andSi(110)-based CUs for several positron beams listed in Table A1 in Appendix A. TheCUR data refer to the emission in the fundamental harmonic. The data on APS (USA),ESRF (France), PETRA (DESY, Germany), SPring8 (Japan) are from [8, 75].
To be noted is that for the well-collimated intensive beams with small transversesizes (SuperB, FACET, SuperKEK, CEPC) the peak brilliance of CUR in the photonenergy range from 10 keV to 10 MeV (the corresponding wavelengths vary from 10 − down to 10 − ˚A) is comparable to (the case of SuperB, FACET and SuperKEK beams)or even higher (CEPC beam) than that achievable in conventional LS for much lowerphoton energies.We stress that the values of bending amplitude and periods, which maximize theCUR brilliance over broad range of photon energies, are accessible by means of moderntechnologies (compare Figs. B1 & B2 with Figs. 3 & 4 in Appendix A).The Gamma Factory proposal for CERN discusses a concept of the LS basedon the resonant absorption of laser photons by the ultra-relativistic ions [9, 10]. It rystal-Based Light Sources -2 -1 photon energy (MeV) I n t en s i t y ( pho t on s / s ) Figure 6.
Peak intensity (number of photons per second, ∆ N ω I max /e ) ofdiamond(110)-based CUs calculated for positron beams at different facilities: 1 -DAΦNE, 2 - VEPP4M, 3 - BEPC-II, 4 - SuperB, 5 - SuperKEK, 6 - FACET-II, 7 - CEPC. Solid and dashed lines correspond to the emission in the first andthird harmonics, respectively. Open circles indicate the data on the laser-Comptonbackscattering [73]. The horizontal dash-dotted line marks the intensity 10 photon/sindicated in the Gamma Factory (GF) proposal for CERN [9]. is expected that the intensity of the LS will be orders of magnitude higher that thepresently operating LS aiming at the values of 10 photons/s in the gamma-ray domain1 ≤ E ph ≤
400 MeV. To this end, it is instructive to compare the intensity of CUR withthe quoted value as well as with the intensities currently achievable by means of the LSbased on laser-Compton scattering from electron beam [73].Figure 6 presents the peak intensities, ∆ N ω I max /e , of the first (solid lines) and third(dashed lines) harmonics of CUR from diamond(110)-based CU with the optimizedparameters (see Fig. B2 in Appendix B). Different curves correspond to differentpositron beams as specified in the caption. Most of the curves presented show orders ofmagnitude higher intensities in the photon energy range one to tens of MeV than thatfrom the laser-Compton scattering LS (open circles). Within the same photon energyinterval the CUR intensity can be comparable with or even higher (see the curves forthe SuperB, SuperKEK and FACET-II beams) than the value predicted in the GammaFactory proposal (marked with the horizontal dash-dotted line).Figures 5 and 6 demonstrate also the tunability of a CU-LS. For any positron beamwith specified parameters the photon yield can be maximized (more generally, varied)over broad range of photon energies by properly choosing parameters of the CU (bendingamplitude and period, crystal, plane). rystal-Based Light Sources The radiation emitted in an undulator is coherent (at the harmonics frequencies) withrespect to the number of periods, N u , but not with respect to the emitters since thepositions of the beam particles are not correlated. As a result, the intensity of ofradiation emitted in a certain direction is proportional to N and to the number ofparticles, I inc ∝ N p N (the subscript ‘inc’ stands for ‘incoherent’). In conventionalundulators, N u is on the level of 10 . . . [75], therefore, the enhancement due tothe factor N is large making undulators a powerful source of spontaneous radiation.However, the incoherence with respect to the number of the radiating particles causesa moderate (linear) increase in the radiated energy with the beam density.More powerful and coherent radiation will be emitted by a beam in which position ofthe particles is modulated in the longitudinal direction with the period equal to integermultiple to the radiation wavelength λ . In this case, the electromagnetic waves emittedby different particles have approximately the same phase. Therefore, the amplitude ofthe emitted radiation is a coherent sum of the individual waves, so that the intensitybecomes proportional to the number of particles squared, I coh ∝ N N [80]. Thus,the increase in the photon yield due to the beam pre-bunching (other terms usedare ‘bunching’ [5] or ‘microbunching’ [6]) can reach orders of magnitudes relative toradiation by a non-modulated beam of the same density (see the data on N p in TableA1 in Appendix A). Following Ref. [81] we use the term ’superradiant’ to designate thecoherent emission by a pre-bunched beam of particles.In what follows we assume that the beam is fully modulated at the crystalentrance. The description on the methods of preparation of a pre-bunched beam withthe parameters needed to amplify CUR one finds in [15] and in Section 8.5 in Ref. [16].For a pre-bunched beam, the intensity is sensitive not only to the shape of thetrajectory but also to the relative positions of the particles along the undulator axis.In the course of beam propagation through the crystal these positions become randomdue to both the collisions with crystal atoms and the non-similarity of the channelingtrajectories for different particles [82]. This leads to the beam demodulation and, as aresult, to the loss of the superradiance effect.For an unmodulated beam, the CU length L is limited mainly by the dechannelingprocess. For a pre-bunched the demodulation becomes the phenomenon which imposesmost restrictions on the parameters of a CU. In Ref. [82] the demodulation length , L dm , was introduced to quantify the spatial scale at which a modulated beam becomesdemodulated. To preserve the modulation and to maintain the coherence of radiationthe crystal length must be less than L dm (see Appendix C where essential details aresummarized).Quantitative analysis and numerical data on the parameters of a CU whichmaximize the brilliance of CUR in presence of the demodulation process is presented inSection Appendix D. These data have been used to calculate the peak brilliance of thesuperradiant CUR. rystal-Based Light Sources -3 -2 -1 photon energy (MeV) pea k b r illi an c e ( pho t on s / s m r ad mm . % B W ) PETRAESRFSPring8APS SuperBSuperKEKB CEPCFACETTESLA spontaneous undulator10 GeV 20 GeVLCLS TESLA SASE FEL
Figure 7.
Peak brilliance of superradiant CUR (thick solid curves) and spontaneousCUR (dashed lines) from diamond(110) CUs calculated for the SuperKEKB, SuperB,FACET-II and CEPC positron beams versus modern synchrotrons, undulators andXFELs. The data on the latter are taken from Ref. [8].
Figure 7 illustrates a boost in peak brilliance due to the beam modulation. Thickcurves correspond to superradiant CUR calculated for fully modulated positron beams(as indicated) propagating in the channeling mode through diamond(110)-based CU. Inthe photon energy range 10 − . . . MeV the brilliance of superradiant CUR by ordersof magnitudes (up to 8 orders in the case of CEPC) exceeds that of the spontaneousCUR (dash-dotted curves) emitted by the random beams. Remarkable feature is thatthe superradiant CUR brilliance can not only be much higher that the spontaneousemission from the state-of-the-art magnetic undulator (see the curves for the TESLAundulator) but also be comparable with the values achievable at the XFEL facilities(LCLC (Stanford) and TESLA SASE FEL) which operate in much lower photon energyrange.
5. Discussion
Construction of novel CLSs is an extremely challenging task which constitutes a highlyinterdisciplinary field. To accomplish this task, a broad collaboration is needed ofresearch groups with different but mutually complementary expertise, such as materialscience, nanotechnology, particle beam and accelerator physics, radiation physics, X-raydiffraction imaging, acoustics, solid state physics, structure determination, advancedcomputational modeling methods and algorithms, high-performance computing as wellas industries specializing in manufacturing of crystalline structures and in design andconstruction of complete accelerator systems.As a first step towards achieving the major breakthrough in the field, one can rystal-Based Light Sources ε = 10 − − GeV, including anauthoritative study of the structure sustainability with respect to beam intensity, aswell as explicit experimental characterization of the emission spectra; (iv) Advancein computational and numerical methods for multiscale modeling of nanostructuredmaterials with extremely high, reliable levels of prediction (from atomistic to mesoscopicscale), of particle propagation, of irradiation-induced solid state effects, and forcalculation of spectral-angular distribution of emitted radiation and for modeling[83]. Ultimately, this will enable better experimental planning and minimisationof experimental costs. The knowledge gained the studies (i)-(v) will provide CLSsprototypes and a roadmap for practical implementation by CLS system manufacturersand accelerator laboratories/users worldwide.Sub-angstrom wavelength powerful and tunable CLSs will have a broad range ofexciting potential applications.A micron-sized narrow photon beam may be used in cancer therapy . This wouldgreatly improve the precision and effectiveness of the therapy for the destruction oftumour by collimated radiation. Furthermore, it would allow delicate operations to beperformed in close vicinity of vital organs. Taking into account the experience gainedto date in the field of radio-therapy, one can expect that practical manipulations withmicro-sized beams will become active soon after the novel LSs become available.Gamma-rays induce nuclear reactions by photo-transmutation . For instance, inthe experiment of Ref. [84] a long-lived isotope was converted into a short-lived oneby irradiation with a gamma-ray bremsstrahlung pulse. However, the intensity ofbremsstrahlung is orders of magnitudes less than of CUR. Moreover, to increase theeffectiveness of the photo-transmutation process is it desirable to use photons whoseenergy is in resonance with the transition energies in the irradiated nucleus [73]. Byvarying the CU parameters one can tune the energy of CUR to values needed to inducethe transmutation process in various isotopes. This opens the possibility for a noveltechnology for disposing of nuclear waste . Photo-transmutation can also be used toproduce medical isotopes . Another possible application of the CU-LSs concerns photo-induced nuclear fission when a heavy nucleus is split into two or more fragments due rystal-Based Light Sources γ, n ) reaction in theregion of the giant dipole resonance (typically 20-40 MeV). The PET isotopes can beused directly for medial PET and for Positron Emission Particle Tracking experiments.Powerful monochromatic radiation within the MeV range can be used as an alternativesource for producing beams of MeV protons by focusing a photon pulse on to a solidtarget [84, 85]. Such protons can induce nuclear reactions in materials producing, inparticular, light isotopes which serve as positron emitters to be used in PET. Irradiationby hard X-ray strongly decreases the effects of natural surface tension of water [86]. Thepossibility to tune the surface tension by the irradiation can be exploited to study themany phenomena affected by this parameter in physics, chemistry, and biology such as,for example, the tendency of oil and water to segregate.
6. Conclusion
The exemplary case study of a tunable CU-based LS considered in the paper hasdemonstrated that peak brilliance of CUR emitted in the photon energy range 10 keV up to 10 MeV by currently available (or planned to be available in near future)positron beams channeling in periodically bent crystals is comparable to or even higherthan that achievable in conventional synchrotrons in the much lower photon energyrange. Intensity of CUR greatly exceeds the values provided by LSs based on Comptonscattering and can be made higher than the values predicted in the Gamma Factoryproposal in CERN. By propagating a pre-bunched beam the brilliance in the energyrange 10 keV up to 10 MeV can be boosted by orders of magnitude reaching thevalues of spontaneous emission from the state-of-the-art magnetic undulators and beingcomparable with the values achievable at the XFEL facilities which operate in muchlower photon energy range. Important is that by tuning the bending amplitude andperiod one can maximize brilliance for given parameters of a positron beam and/orchosen type of a crystalline medium. Last but not least, it is worth to mention that thesize and the cost of CLSs are orders of magnitude less than that of modern LSs basedon the permanent magnets. This opens many practical possibilities for the efficientgeneration of gamma-rays with various intensities and in various ranges of wavelengthby means of the CLSs on the existing and newly constructed beam-lines.Though we expect that, as a rule, the highest values of brilliance can be reachedin CU-based LSs (or, in those based on stacks of CUs) the analysis similar to the onepresented can be carried out for other types of CLSs based on linear and bent crystals.This will allow one to make an optimal choice of the crystalline target and the CLS typeto be used in a particular experimental environment or/and to tune the parameters ofthe emitted radiation matching them to the needs of a particular application.The case study presented has been focused on the positron beams, which have rystal-Based Light Sources
15a clear advantage since the dechanneling length of positrons is order of magnitudelarger than that of electrons of the same energy. This allows one to use thickercrystals in channeling experiments with positrons thus enhancing the photon yield.Nevertheless, experimental studies of CLSs with electron beams are worth to be carriedout. Indeed, high-quality electron beams of energies starting from sub-GeV rangeand onward are more available than their positron counterparts. Therefore, theselaboratories provide more options for the design, assembly and practical implementationof a full suite of correlated experimental facilities needed for operational realizationand exploitation of the novel CLSs. In this connection we note that in the course ofchanneling experiments at the Mainz Microtron facility with ε = 190 −
855 MeV electronspropagating in various CUs, which have been carried out over the last decade within theframeworks of several EU-supported collaborative projects (FP6-PECU, FP7-CUTE,H2020-PEARL), a unique experience has been gained. This experience has ascertainedthat the fundamental importance of the quality of periodically bent crystals, which,in turn is based on the cutting-edge technologies used to manufacture the crystallinestructures, of modern techniques for non-destructive characterization of the samples,of the necessity of using advanced computational methods for numerical modeling of avariety of phenomena involved. On the basis of this experience the bottlenecks on theway to practical realization of the CLSs concept have been established.To quantify the scale of the impact within Europe and worldwide which thedevelopment of radically novel CLSs might have, we can draw historical parallelswith synchrotrons, optical lasers and FELs. In each of these technologies there wasa significant time lag between the formulation of a pioneering idea, its practicalrealization and follow-up industrial exploitation. However, each of these inventionshas subsequently launched multi-billion dollar industries. The implementation of CLS,operating in the photon energy range up to hundreds of MeV, is expected to lead toa similar advance and CLSs have the potential to become the new synchrotrons andlasers of the mid to late 21st century, stimulating many applications in basic sciences,technology and medicine. The development of CLS will therefore herald a new age inphysics, chemistry and biology.
Acknowledgments
We acknowledge support by the European Commission through the N-LIGHT Projectwithin the H2020-MSCA-RISE-2019 call (GA 872196). The work was also supportedby Deutsche Forschungsgemeinschaft (Project No. 413220201).
Appendix A. Beam Parameters
The data on positron and electron beams energy ε , bunch length L b , number of particlesper bunch N , beam sizes σ x,y and divergences φ x,y (the subscripts x, y refer to thehorizontal and vertical dimensions, respectively), volume density n , and peak current rystal-Based Light Sources I max are summarized in Table A1. The table compiles the data for the following facilities:VEPP4M (Russia), BEPCII (China), DAΦNE (Italy), SuperKEKB (Japan) [26], SLAC(the FACET-II beams, Ref. [87]), SuperB (Italy) [88], and CEPC (China) [28]. Notethat the SuperB data are absent in the latest review by Particle Data Group [26] sinceits construction was canceled [89]. Table A1.
Parameters of positron (’p’) and electron (’e’) beams: beam energy, ε ,bunch length, L b , number of particles per bunch, N , beam size, σ x,y , beam divergence φ x,y , volume density n = N / ( πσ x σ y L b ) of particles in the bunch, peak current I max = e N c/L b . In the cells with no explicit reference to either ’e’ or ’p’ the data referto both modalities.Facility VEPP4M BEPCII DAΦNE SuperKEKB SuperB FACET-II CEPCRef. [26] [26] [26] [26] [88, 89] [87] [28] ε (GeV) 6 1.9-2.3 0.51 p: 4 p: 6.7 10 45.5e: 7 e: 4.2 N
15 3.8 p: 2.1 p: 9.04 p: 6.5 p: 0.375 8(units 10 ) e: 3.2 e: 6.53 e: 5.1 e: 0.438 L b (cm) 5 1.2 1-2 p: 0.6 0.5 p: 0.00076 0.85e: 0.5 e: 0.00011 σ x ( µ m) 1000 347 260 p: 10 8 p: 10.1 6e: 11 8 e: 5.5 σ y ( µ m) 30 4.5 4.8 p: 0.048 0.04 p: 7.3 0.04e: 0.062 e: 5.9 φ x (mrad) 0.2 0.35 1 p: 0.32 p: 0.250 p: 0.178 0.03e: 0.42 e: 0.313 e: 0.073 φ y (mrad) 0.67 0.35 0.54 p: 0.18 p: 0.125 p: 0.044 0.04e: 0.21 e: 0.150 e: 0.044 I max (A) 144 152 p: 50-100 p: 723 p: 624 p: 12.1 × × n (10 cm − ) 3.2 65 p: 54 p:1 . × p:1 . × p: 2 × . × e: 82 e:0 . × e:1 . × e: 3 . × Appendix B. Optimal Length of a CU
With account for the dechanneling and the photon attenuation, the number of photons∆ N ω n of the frequency within the interval h ω n − ∆ ω n / , ω n + ∆ ω n / i emitted in theforward direction within the cone ∆Ω n by a projectile in a CU is given by the followingexpression (see Refs. [16, 90] for the details):∆ N ω n = A ( C ) 4 πα nζ h J n − ( nζ ) − J n +12 ( nζ ) i N eff ( N d ; x, κ d ) ∆ ω n ω n , (B.1)where ζ = K / (4 + 2 K ), J ν ( nζ ) is the Bessel function and K = 2 πγa/λ u is theundulator parameter. The subscript n enumerates the harmonics of CUR. The frequency rystal-Based Light Sources ω n = nω of the n -th harmonic is expressed in terms of the fundamental harmonic givenby ω = 2 γ K / πcλ u . (B.2)The quantity A stands for the channel acceptance, which is defined as a fraction of theincident particles captured into the channeling mode at the crystal entrance (anotherterm used is surface transmission, see e.g. Ref. [91]).Apart from the factor A , the difference between (B.1) and the formula for an idealundulator (see, e.g., [78]) is that the number of undulator periods N u , which enters thelatter, is substituted with the effective number of periods , N eff ( N d , x, κ d ) ≡ N eff , whichdepends on the number of periods within the dechanneling length, N d = L d /λ u , and onthe ratios x = L d /L a and κ d = L/L d where L d denotes the dechanneling length and L a is the photon attenuation length. The effective number of periods is given by [16, 90]: N eff = 4 N d xκ d (cid:20) x e − xκ d (1 − x )(2 − x ) − e − κ d − x + 2e − (2+ x ) κ d / − x (cid:21) r κ ( x − + 14 π . (B.3)In the limit L d , L a → ∞ , i.e. when the dechanneling and the attenuation are neglected, N eff → N u = L/λ u , as it must be in the case of an ideal undulator. In this case one can,in principle, increase infinitely the number of periods by considering larger values of theundulator length L . This will lead to the increase of the number of photons and thebrilliance since these quantities are proportional to N u . The limitations on the valuesof L and N u are mainly of a technological nature.The situation is different for a CU, where the number of channeling particles andthe number of photons, which can emerge from the crystal, decrease with the growth of L . It is seen from (B.3), that in the limit L → ∞ the parameters κ d and xκ d = L/L a alsobecome infinitely large leading to N eff →
0. This result is quite clear, since in this limit L ≫ L a so that all emitted photons are absorbed inside the crystal. Another formal (andphysically trivial) fact is that N eff = 0 also for a zero-length undulator L = 0. Vanishingof a positively-defined function N eff ( N d , x, κ d ) at two extreme boundaries suggests thatthere is a length L ( x ) which corresponds to the maximum value of the function.To define the value of L ( x ) or, what is equivalent, of the quantity κ d ( x ) = L ( x ) /L d ,one carries out the derivative of f ( x, κ d ) with respect to κ d and equalizes it to zero. Theanalysis of the resulting equation shows that for each value of x = L d /L a ≥ κ d . Hence, the equation defines, in an inexplicit form, a single-valuedfunction κ d ( x ) = L ( x ) /L d which ensures the maximum of N eff ( x, κ d ) for given L a , L d and λ u .Note that the crystal length enters Eq. (B.1) only via the ratio κ d . It was shown[16,90] that the quantity L ( x ) ensures the highest values of the number of photons ∆ N ω n and of the brilliance B n of the CUR. Therefore, L ( x ) can be called the optimal length that corresponds to a given value of the ratio x = L d /L a .The following multi-step procedure has been adopted to calculate the highestbrilliance of CUR. rystal-Based Light Sources • Fix crystal and crystallographic direction.
In the current paper we have focusedon the (110) planar channels in diamond and silicon crystals, which are commonlyused in channeling experiments. We note that other crystals/channels, availableor/and studied experimentally, can also be considered [13, 92, 93]. • Fix parameters of the positron beam: energy ε , sizes σ x,y and divergence φ x,y , peakbeam current I max . • Scan over photon energy ω . For each ω value:(i) Determine the attenuation length L a ( ω ) (for the photon energies above 1 keVthe data are compiled in Ref. [94]).(ii) Scan over a and λ u consistent with the stable channeling condition [12, 13]: C = 4 π aλ εU ′ max < . (B.4)The bending parameter C is defined as the ratio F cf /U ′ max where F cf ≈ ε/R is the centrifugal force in a channel bent with curvature radius R and U ′ max isthe maximum force due to the interplanar potential. Channeling motion in thebent crystal is possible if C <
1. In a periodically bent crystal, the bendingradius in the points of maximum curvature equals to λ / π a which explainsthe right-hand side of (B.4).(iii) Determine dechanneling length L d ( C ).The data on the dechanneling length can be extracted (when available) fromthe experiments [95, 96] or obtained by means of highly accurate numericalsimulation of the channeling process [16, 20, 97]. For positrons, a very goodestimation for L d ( C ) can be obtained by means of the following formulae[16, 31]: L d ( C ) = (1 − C ) L d (0) ,L d (0) = 2569 π a TF dr m e c ε Λ (B.5)Here L d (0) is the dechanneling length in the straight channel, r cm is theclassical electron radius, Z and a TF are, respectively, the atomic number andthe Thomas-Fermi radius of the constituent atom, Λ = 13 . . ε [GeV]) − . Z ) is the Coulomb logarithm.(iv) Determine the maximum value of N eff and the optimal length L .(v) Determine the channel acceptance.The acceptance A ( C ) of a bent channel can be estimated as follows [31]: A ( C ) = (1 − C ) A . (B.6)Here A = 1 − u T /d ( u T is the amplitude of thermal vibrations of the crystalatoms) is the acceptance of the straight channel.(vi) Substituting the quantities obtained into Eq. (B.1) and Eq.(1) in the maintext one calculates the highest available peak brilliance B peak ( ω ).As formulated, the items (iii)-(vi) listed above are applicable for a fully collimatedpositron beam with zero divergence. In reality, the beams have non-zero divergence rystal-Based Light Sources φ , see Table A1, so that only a fraction ξ of the beam particles gets accepted intothe critical angle Θ L for channeling. To estimate this fraction we assume the normaldistribution of the beam particles with respect to the incident angle and calculate ξ asfollows: ξ = (2 πφ ) − / Z Θ L − Θ L exp (cid:18) − θ φ (cid:19) d θ. (B.7)The values of ξ calculated for the beams listed in Table A1 are presented in Table B1.For each beam, Lindhard’s critical angle Θ L = p U /ε is estimated using the value U = 20 eV (which corresponds, approximately, to the interplanar potential depth inSi(110) and diamond(110)) and the indicated values of the beam divergence is calculatedas φ = min[ φ x , φ y ].To account for the non-zero divergence one multiplies the value B peak ( ω ), calculatedas described above, by the factor ξ . Table B1.
Fraction ξ of the beams particles with incident angle less than Lindhard’scritical angle Θ L (in mrad). For each beam indicated the parameter φ (in mrad) standsfor the minimum of two divergences φ x and φ y , see Table A1.Facility VEPP4M BEPCII DAΦNE SuperKEKB SuperB FACET-II CEPC φ L ξ Figures B1 and B2 show the results of calculations performed for silicon(110)- anddiamond(110)-based CU using and for the positron beams specified in Table A1. Thedependences presented were obtained by maximizing the brilliance of CUR emitted inthe fundamental harmonic. It is seen, that within the range of moderate values of thebending amplitude ( a/d varies from several units up to several tens, graphs (e); d = 1 . N eff ≈ . . . λ u ≈ . . . µ m(graphs (d)) which is achievable by different methods of preparation of periodically-bent crystalline structures, see Section 3. It is seen from Figs. B1 and B2 that out of allcalculated quantities the peak brilliance B peak ( ω ), graphs (f), is the most sensitive to theparameters of the positron beam. The variation in the magnitude of B peak ( ω ) is oversix orders of magnitude, from ∼ up to ∼ photons/s/mrad / mm / . Appendix C. Beam Demodulation in CU
In a CU, a channeling particle, while moving along the channel centerline, undergoestwo other types of motion in the transverse directions with respect to the CU axis rystal-Based Light Sources C SuperKEKBFACET-IICEPC 10100 λ u ( µ m ) L / L d DA Φ NEVEPP4MBEPC-IISuperB 10 a / d -1 h _ ω (MeV) N e ff -1 h _ ω (MeV) P ea k B r illi an c e (a)(b)(c) (d)(e)(f) Figure B1.
Parameters of the silicon(110)-based CU, - C , λ u , a (measured in theinterplanar distances d ), N eff , L (measured in the units of L d ( C )), that ensure thehighest peak brilliance B peak ( ω ), graph (f). Different curves correspond to severalcurrently achievable positron beams as indicated in the legend (see also Table A1). z . First, there are channeling oscillations along the y direction perpendicular to thecrystallographic planes. Second, the particle moves along the planes (the x direction).To be noted is that different particles have different (i) amplitudes a ch of the channelingoscillations, and (ii) momenta p x in the ( xz ) plane due to the distribution in thetransverse energy of the beam particles as well as the result of multiple scattering fromcrystal atoms. Therefore, even if the speed of all particles along their trajectories isthe same, the difference in a ch or/and in p x leads to different values of the velocitieswith which the particles move along the undulator axis. As a result, the beam loses itsmodulation while propagating through the crystal.For an unmodulated beam, the CU length L is limited mainly by the dechannelingprocess. A dechanneled particle does not follow the periodic shape of the channel, and,thus, does not contribute to the CUR spectrum. Hence, it is reasonable to estimate L on the level of several dechanneling lengths L d (see panels (b) in Figs. B1 and B2).Longer crystals would attenuate rather then produce the radiation. Since the intensityof CUR is proportional to the undulator length squared, the dechanneling length andthe attenuation length are the main restricting factors (see Section Appendix B) whichmust be accounted for.For a modulated beam, the intensity is sensitive not only to the shape of thetrajectory but also to the relative positions of the particles along the undulator axis. rystal-Based Light Sources C SuperKEKBFACET-IICEPC 10100 λ u ( µ m ) L / L d DA Φ NEVEPP4MBEPC-IISuperB 10 a / d -1 h _ ω (MeV) N e ff -1 h _ ω (MeV) P ea k B r illi an c e (a)(b)(c) (d)(e)(f) Figure B2.
Same as in Fig. B1 but for the diamond(110)-based CU.
If these positions become random because of the beam demodulation, the coherence ofCUR is lost even for the channeled particles. Hence, the demodulation becomes thephenomenon which imposes most restrictions on the parameters of a CU.In Ref. [82] an important quantity, – the demodulation length , was introduced. Itrepresents the characteristic scale of the penetration depth at which a modulated beamof channeling particles becomes demodulated. Within the framework of the approachdeveloped in the cited papers the demodulation length L dm is related to the dechannelinglength L d ( C ) in a bent channel: L dm = L d ( C ) α ( ξ ) + √ ξ/j , . (C.1)Here j , = 2 . . . . is the first root of the Bessel function J ( x ). The dimensionlessparameter ξ is expressed in terms of the emitted radiation frequency ω , the dechannelinglength L d ( C ) and Lindhard’s critical angle Θ L ( C ) in the bent channel: ξ = ωL d ( C )Θ ( C ) / c (see [92] for the details). The function α ( ξ ) is related to the realand imaginary parts of the first root (with respect to ν ) of the equation [92] F ( − ν, , z ) (cid:12)(cid:12)(cid:12) z =(1+i) j , √ ξ/ = 0 (C.2)where F ( . ) stands for Kummer’s confluent hypergeometric function (see, e.g., [98]).Eqs. (C.1) and (C.2) can be analyzed numerically to derive the dependence ofthe demodulation length on the radiation energy ~ ω for a particular crystal channel.The result of such analysis is illustrated by Fig. C1 where the dependences of the ratio L dm /L d ( C ) on the photon energy are presented for the (110) planar channels in diamond rystal-Based Light Sources C as indicated. To be noted,is that for all values of the bending parameter C and over broad energy range of theemitted radiation, the demodulation length is noticeably less than the dechanneling one. -1 h _ ω (MeV) L d m / L d ( C ) diamond(110) -1 h _ ω (MeV) L d m / L d ( C ) C=0C=0.1C=0.2C=0.3C=0.4C=0.5
Si(110)
Figure C1.
The ratio L dm /L d ( C ) vs. photon energy for diamond(110) (left panel) andsilicon(110) (right panel) channels calculated for various values of bending parameter C . To preserve the beam modulation during its channeling in a crystal and, as a result,to maintain the coherence of the radiation the crystal length L must be less than thedemodulation length: L . L dm < L d ( C ) . (C.3)It follows from (C.1) that in a periodically bent crystal L dm depends on the crystal type,on the parameters of the channel (its width, strength of the interplanar field), on thebending amplitude and period, on the projectile energy and its type (these are ”hidden”in L d ( C ), C , and ξ ) as well as on the emitted photon energy (enters the parameter ξ ).Therefore, Eq. (C.3) imposes addition restriction on the CU length as compared to thecase of the CUR emission by the unmodulated beam.In Ref. [92] it is also shown that the phase velocity of the modulated beam alongthe CU channel is modified as compared to the unmodulated one. The modificationchanges the resonance condition which links the parameters of the undulator and theradiated wavelength (energy). The expression for the fundamental harmonic frequency ω ≡ ω acquires the following form (compare with Eq. (B.2)): ω = 2 γ K / β / πcλ u (C.4)where the additional term in the denominator is given by∆ β = 4 γ Θ L ( C ) (cid:18) β ( ξ ) + 12 j , √ ξ (cid:19) (C.5)with β ( ξ ) being another function related to the real and imaginary parts of the first rootof Eq. (C.2) (details can be found in Refs. [16, 92]). The quantity ξ = ωL d ( C )Θ ( C ) / c depends on ω . Therefore, Eq. (C.4) represents a transcendent equation which relates ω to the projectile energy and to the bending amplitude and period. rystal-Based Light Sources ω and ε all other quantities which characterize the CU and the demodulation process can beexpressed in terms of a single independent variable, for example, the bending amplitude a . Then, scanning over the a values it is possible to determine the whole set of theparameters (these include a , λ u , C , L dm ( C )) which maximize the peak brilliance of thesuperradiant emission (see section Appendix D). λ u ( µ m ) a / d N d m C -1 h _ ω (MeV) -2 -1 L d m ( c m ) -1 h _ ω (MeV) K (a)(b)(c) (d)(e)(f) Figure C2.
Parameters of the diamond(110)-based CU, - λ u , a (measured in units ofthe interplanar distance d = 1 .
26 ˚A), C , the undulator parameter K = 2 πγa/λ u ,the demodulation length L dm ( C ) ≡ L dm and the number of periods within L dm , N dm = L dm /λ u that ensure the highest peak brilliance of the radiation emitted bythe fully modulated FACET-II positron beam. Figure C2 shows the results of calculations of the parameters of the diamond(110)-based CU which maximize the peak brilliance of the radiation of energy ~ ω emitted bythe FACET-II positron beam, see Table A1. The dependences presented correspondto the emission in the fundamental harmonic. The crystal thickness was set to thedemodulation length L = L dm ( C ), graph (e). The quantity N dm stands for the numberof undulator periods within the demodulation length, N dm = L dm ( C ) /λ u . Only the datacorresponding to N dm ≥
10 are shown in the panels. The dependences presented refer tothe Large-Amplitude regime of the periodic bending, which implies that the amplitude a exceeds the interplanar distance d .Noticing that the factor 2 π/λ u can be written in terms of the undulator parameter K = 2 πγa/λ u , one writes Eq. (C.4) as a quadratic equation with respect to K .Resolving it one finds that K is a two-valued function of ω , which is reflected in graph(f). As a result, all dependences presented contain two branches related to the smaller rystal-Based Light Sources K . Appendix D. Brilliance of the Superradiant Emission in CU
Powerful superradiant emission by ultra-relativistic particles channeled can be achievedif the probability density of the particles in the beam is (uniformly) modulated in thelongitudinal direction with the period equal to integer multiple to the wavelength λ ofthe emitted radiation [81].To prevent the demodulation of the beam as it propagates through the crystal, thecrystal length L must satisfy condition (C.3). In a wide range of photon energies, startingwith ~ ω ∼ keV, the demodulation length is noticeably less than the dechannelinglength L d . In addition to this, in this energy range the photon attenuation length L a insilicon and diamond greatly exceeds the dechanneling length of positrons with energiesup to several tens of GeV [16]. Therefore, on the spatial scale of L dm the dechannelingand the photon attenuation effects can be disregarded.In what follows, we carry out quantitative estimates of the characteristics ofthe superradiant CU radiation (CUR) emitted by a fully modulated positron beamchanneled in periodically bent diamond and silicon (110) oriented crystals in the absenceof the dechanneling and the photon attenuation. The beam represents a train of buncheseach of the length L b containing N particles. The crystal length (along the beamdirection) is set to the demodulation length, L = L dm . The transverse sizes of a crystalare assumed to be larger than those of the beam, i.e. than σ x,y .For the sake of clarity, below we consider the emission in the first harmonics ofCUR, see Eq. (C.4)Final width ∆ ω of the CUR peak introduces a time interval τ coh = 1 / ∆ ω withinwhich two particles separated in space can emit coherent waves. Hence, one canintroduce a coherence length [75] L coh = cτ coh = λ π ω ∆ ω (D.1)where λ is the radiation wavelength, and the band-width (BW) ∆ ω/ω ≈ /N dm with N dm = L dm /λ u standing for the number of periods within the demodulation length.The number of the particles from the bunch which emit coherently is calculated as N coh = L coh L b N . (D.2)Their radiated energy is proportional to N . The number of such sub-bunchesis L b /L coh , therefore, the energy emitted by the whole bunch contains the factor( L b /L coh ) N = N N coh .Another important quantity to be estimated is the solid angle ∆Ω coh within whichthe waves emitted by the particles of the sub-bunch are coherent. This angle can bechosen as the minimum value from the natural emission cone of the first harmonics∆Ω = 2 πλ u /L dm and the angle ∆Ω ⊥ which ensures transverse coherence of the emission rystal-Based Light Sources σ x,y of the bunch. Assuming the elliptic form for the bunch crosssection one derives ∆Ω ⊥ ≤ λ / πσ x σ y . Therefore, the solid angle ∆Ω coh is found from∆Ω coh = min h ∆Ω ⊥ , ∆Ω i . (D.3)The number of photons ∆ N ω emitted by the bunch particles one obtains multiplyingthe spectral-angular distribution of the energy emitted by a single particle by the factor N N coh ∆Ω coh (∆ ω/ω ). The result reads:∆ N ω = 4 πα N N coh ζ [ J ( ζ ) − J ( ζ )] N dm ∆Ω coh ∆Ω ∆ ωω . (D.4)where ζ = ( K + ∆ β ) / K + ∆ β ) with ∆ β defined in (C.5).The number of photons emitted by the particles of the unmodulated beam in a CUof the same length and number of periods one calculates from Eq. (B.1) written for n = 1 by setting N eff = N dm , substituting K → K + ∆ β and multiplying the right-hand side by N . Comparing the result with Eq. (D.4) one notices that the enhancementfactor due to the coherence effect is N coh ∆Ω coh / ∆Ω.Another quantity of interest is the flux F ω of photons. Measured in the units of (cid:16) photons / s / . (cid:17) , it is related to ∆ N ω as follows: F ω = 110 (∆ ω/ω ) ∆ N ω ∆ t b (D.5)where ∆ t b = L b /c = e N /I max is the time flight of the bunch and I max stands for thepeak current.The peak brilliance , B peak , of the superradiant CUR one obtains substituting ∆ N ω from (D.4) into Eq. (1) in the main text and using there peak current I max instead of I .Figure D1 shows peak brilliance of radiation formed in the diamond(110)-based CUas functions of the first harmonic energy. Four graphs correspond to the positron beams(as indicated) the parameters of which are listed Table A1. In each graph, the dashedline refers to the the emission of the spontaneous CUR formed in the undulator withoptimal parameters, see Fig. B2. The thick curves present the peak brilliance of thesuperradiant CUR maximized by the proper choice of the bending amplitude and period(as described in Section Appendix C). Two branches of this dependence, seen in graphs(a)-(c), are due to the two-valued character of the dependence of undulator parameter K on the radiation frequency ω . For the CEPC beam, graph (d), this peculiarity manifestsitself in the frequency domain beyond 40 MeV, therefore it is not seen in the graph. References [1] E. A. Seddon, J. A. Clarke, D. J. Dunning, C. Masciovecchio, C. J. Milne, F. Parmigiani, D. Rugg,J. C. H. Spence, N. R. Thompson, K. Ueda, S. M. Vinko, J. S. Wark, E. Wurth. Short-wavelengthfree-electron laser sources and science: a review.
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