Demodulation of a positron beam in a bent crystal channel
aa r X i v : . [ phy s i c s . acc - ph ] J a n Demodulation of a positron beam in a bent crystalchannel
A. Kostyuk a , A.V. Korol a,b , A.V. Solov’yov a , Walter Greiner a a Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universit¨at,Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany b Department of Physics, St Petersburg State Maritime Technical University, StPetersburg, Russia
Abstract
The evolution of a modulated positron beam in a planar crystal channelis investigated within the diffusion approach. A detailed description of theformalism is given. A new parameter, the demodulation length, is introduced,representing the quantitative measure of the depth at which the channellingbeam preserves its modulation in the crystal. It is demonstrated that thereexist crystal channels with the demodulation length sufficiently large for usingthe crystalline undulator as a coherent source of hard X rays. This finding isa crucial milestone in developing a new type of lasers radiating in the hardX ray and gamma ray range.
Keywords:PACS:
1. Introduction
In this article we study the evolution of a modulated positron beam instraight and bent planar crystal channels. Some key ideas of this researchwere briefly communicated in [1] and [2]. In this paper we present a system-atic and detailed description of the formalism and the obtained results. Theoutcome of the research is of crucial importance for the theory of the crystal
Email addresses: [email protected] (A. Kostyuk), [email protected] (A.V. Korol), [email protected] (A.V. Solov’yov)
Preprint submitted to Elsevier November 11, 2018 ndulator based laser (CUL) [3, 4, 5] — a new electromagnetic radiationsource in hard x- and gamma-ray range.Channelling takes place if charged particles enter a single crystal at smallangle with respect to crystallographic planes or axes [6]. The particles getconfined by the interplanar or axial potential and follow the shape of thecorresponding planes and axes. This suggested the idea [7] of using bentcrystals to steer the particles beams. Since its first experimental verification[8] the idea to deflect or extract high-energy charged particle beams by meansof tiny bent crystals replacing huge dipole magnets has been attracting alot of interest worldwide. Bent crystal have been routinely used for beamextraction in the Institute for High Energy Physics, Russia [9]. A series ofexperiments on the bent crystal deflection of proton and heavy ion beams wasperformed at different accelerators [10, 11, 12, 13, 14] throughout the world.The bent crystal method has been proposed to extract particles from thebeam halo at CERN Large Hadron Collider [15] The possibility of deflectingpositron [16] and electron [14, 17] beams has been studied as well. zyx da Λ u Periodically bentcrystallographicplaneTrajectory ofa channelingparticle Radiation
Figure 1: Schematic representation of the crystalline undulator.
A single crystal with periodically bent crystallographic planes can forcechannelling particles to move along nearly sinusoidal trajectories and radiatein the hard x- and gamma-ray frequency range (see figure 1). The feasibility2f such a device, known as the ’crystalline undulator‘, was demonstratedtheoretically a decade ago [3] (further developments as well as historicalreferences are reviewed in [18]). More recently, an electron based crystallineundulator has been proposed [19].It was initially suggested to obtain sinusoidal bending by the propagationof an acoustic wave along the crystal [3, 4]. The advantage of this approachis its flexibility: the period of deformation can be chosen by tuning the fre-quency of the ultrasound. However, this approach is rather challenging tech-nologically and yet to be tested experimentally. Several other technologiesfor the manufacturing of periodically bent crystals have been developed andtested. These include making regularly spaced grooves on the crystal surfaceeither by a diamond blade [20, 21] or by means of laser-ablation [22], depo-sition of periodic Si N layers onto the surface of a Si crystal [21], growingof Si − x Ge x crystals[23] with a periodically varying Ge content x [24, 25].Experimental studies of the crystalline undulator are currently in progress.The first results are reported in [26] and [27].The advantage of the crystalline undulator is in extremely strong electro-static fields inside a crystal which are able to steer the particles much moreeffectively than even the most advanced superconductive magnets. This factallows to make the period λ u of the crystalline undulator in the range of hun-dreds or tens micron which is two to three orders of magnitude smaller thanthat of conventional undulator. Therefore the wavelength of the producedradiation λ ∼ λ u / (2 γ ) ( γ ∼ –10 being the Lorentz factor of the par-ticle) can reach the (sub)picometer range, where conventional sources withcomparable intensity are unavailable [28].Even more powerful and coherent radiation will be emitted if the prob-ability density of the particles in the beam is modulated in the longitudinaldirection with the period λ , equal to the wavelength of the emitted radia-tion (see figure 2). In this case, the electromagnetic waves emitted in theforward direction by different particles have approximately the same phase[29]. Therefore, the intensity of the radiation becomes proportional to thebeam density squared (in contrast to the linear proportionality for an un-modulated beam). This increases the photon flux by orders of magnitude relative to the radiation of unmodulated beam of the same density. The ra-diation of a modulated beam in an undulator is a keystone of the physics offree-electron lasers (FEL) [30, 31]. It can be considered as a classical coun-terpart of the stimulated emission in quantum physics. Therefore, if similarphenomenon takes place in a crystalline undulator, it can be referred to as3
1 2 3 4 5 6 7 8 9 10 x o r y z/ λ
1 2 3 4 5 6 7 8 9 10 x o r y z/ λ Figure 2: In an unmodulated beam (the upper panel) the particles are randomly dis-tributed. In a completely modulated beam (the lower panel) the distance between anytwo particles along the beam direction is an integer multiple of the modulation period λ . the lasing regime of the crystalline undulator .The feasibility of CUL radiating in the hard x-ray and gamma-ray rangewas considered for the fist time in [3, 4]. Recently, a two-crystal scheme, thegamma klystron, has been proposed [5].A simplified model used in the cited papers assumed that all particle tra-jectories follow exactly the shape of the bent channel. In reality, however, theparticle moving along the channel also oscillates in the transverse directionwith respect to the channel axis (see the shape of the trajectory in figure1). Different particles have different amplitudes of the oscillations inside thechannel (figure 3, upper panel). Similarly, the directions of particle momentain ( xz ) plane are slightly different (figure 3, lower panel). Even if the speedof the particles along their trajectories is the same, the particles oscillat-4 / a
1 2 3 4 5 6 7 8 9 10 x z/ λ Figure 3: Due to different amplitudes of channelling oscillation (upper panel) and differentmomentum directions in the ( xz ) plane (lower panel), the initially modulated beam getsdemodulated. The open and filled circles denote the same particles at the crystal entranceand after travelling some distance in the crystal channel, respectively. ing with different amplitudes or the particles with different trajectory slopeswith respect to z axis have slightly different components of their velocitiesalong the channel. As a result, the beam gets demodulated. An additionalcontribution to the beam demodulation comes from incoherent collisions ofthe channelling particles with the crystal constituents.In the case of an unmodulated beam, the length of the crystalline undu-lator and, consequently, the maximum accessible intensity of the radiationare limited by the dechannelling process. The channelling particle gradu-ally gains the energy of transverse oscillation due to collisions with crystalconstituents. At some point this energy exceeds the maximum value of theinterplanar potential and the particle leaves the channel. The average pen-5tration length at which this happens is known as the dechannelling length .The dechannelled particle no longer follows the sinusoidal shape of the chan-nel and, therefore, does not contribute to the undulator radiation. Hence,the reasonable length of the crystalline undulator is limited to a few dechan-nelling lengths. A longer crystal would attenuate rather then produce theradiation. Since the intensity of the undulator radiation is proportional tothe undulator length squared, the dechannelling length and the attenuationlength are the main restricting factors that have to be taken into accountwhen the radiation output is calculated.In contrast, not only the shape of the trajectory but also the particlespositions with respect to each other along z axis are important for the lasingregime. If these positions become random because of the beam demodulation,the intensity of the radiation drops even if the particles are still in the chan-nelling mode. Hence, it is the beam demodulation rather than dechannellingthat restricts the intensity of the radiation of CUL. Understanding this pro-cess and estimating the characteristic length at which this phenomenon takesplace is, therefore, a cornerstone of the theory of this new radiation source.
2. Diffusion Equation
We adopt the following model of the planar crystal channel (see Fig. 2.1):- the interplanar potential is approximated with a parabola U ( ρ ) = U max (cid:18) ρρ max (cid:19) (1)( ρ is the distance from the potential minimum) so that the channeling oscil-lations are assumed to be harmonic;- the electron density within the distance of one Thomas-Fermi radius of thecrystal atoms from the crystallographic plane is assumed to be so high thatthe particle gets quickly scattered out of the channel. Therefore, the particleis considered dechanneled just after it enters this region. So that the effec-tive channel width is 2 ρ max = d − a TF , where d and a TF are respectively theinterplanar distance and is the Thomas-Fermi radius.As is seen from the figure, the parabolic approximation is quite reason-able. The real potential differs from the parabola mostly in the region ofhigh electron density, where the particle assumes to be dechanneled.6 U ( ρ ) / U ( d / ) ρ /d a TF U max ρ max Molier potentialParabolic approximation
Figure 4: The model of the plane crystal channel. The interplanar potential is approxi-mated by a parabola. It is assumed that the particle dechannels if it enters the vicinity ofthe crystallographic plane within the Thomas-Fermi radius, a TF . Let us consider the distribution f ( t, s ; ξ, E y ) of the beam particles withrespect to the angle between the particle trajectory and axis z in the ( xz )plane ξ = arcsin p x /p ≈ p x /p and the energy of the channeling oscillation E y = p y / E + U ( y ) . Here p , p x and p y are, respectively, the particlemomentum and its x and y components, and E is the particle energy (wewill consider only ultrarelativistic particles, therefore E ≈ p ). We chose the system of units in such a way that the speed of light is equal to unity.Therefore, mass, energy and momentum have the same dimensionality. This is also truefor length and time. .3. Kinetic equation In absence of random scattering, the distribution function f ( t, z ; ξ, E y )would satisfy the differential equation ∂f∂t + ∂f∂z v z = 0, where v z ≡ ∂z∂t . Inreality, however, the right-hand-side of the equation is not zero. It containsthe collision integral. After averaging over the period of the channeling os-cillation, the kinetic equation takes the form ∂f∂t + ∂f∂z h v z i = (cid:28)ZZ dξdE ′ y (cid:2) f ( t, z ; ξ ′ , E ′ y ) w ( ξ ′ , E ′ y ; ξ, E y ) − f ( t, z ; ξ, E y ) w ( ξ, E y ; ξ ′ , E ′ y ) (cid:3)(cid:11) (2)where w ( ξ, E y ; ξ ′ , E ′ y ) dz is the probability that the particle changes its angle ξ and transverse energy from ξ and E y to, respectively, ξ ′ and E ′ y whiletravelling the distance dz . The angular brackets stand for averaging over theperiod of the channeling oscillations.Due to the detailed equilibrium w ( ξ, E y ; ξ ′ , E ′ y ) = w ( ξ ′ , E ′ y ; ξ, E y ) (3) We assume that soft scattering dominates, i.e. the function w ( ξ ′ , E ′ y ; ξ, E y )is not negligible only if | ξ ′ − ξ | and | E ′ y − E y | are small so that f ( t, z ; ξ ′ , E ′ y ) ≡ f ( t, z ; ξ + ϑ x , E y + q y ) can be expanded into the Taylor series with respect to ϑ x and q y . Then, up to the second order in ϑ x and q y , one obtains ∂f∂t + ∂f∂z h v z i = D ξ ∂f∂ξ + D y ∂f∂E y + D ξξ ∂ f∂ξ + D ξy ∂ f∂ξ∂E y + D yy ∂ f∂E y (4)where D ξ = (cid:28)Z dϑ x ϑ x Z dq y w ( ξ, E y ; ξ + ϑ x , E y + q y ) (cid:29) (5) D y = (cid:28)Z dϑ x Z dq y q y w ( E ξ , E y ; ξ + ϑ x , E y + q y ) (cid:29) (6) D ξξ = 12 (cid:28)Z dϑ x ϑ x Z dq y w ( ξ, E y ; ξ + ϑ x , E y + q y ) (cid:29) (7) D ξy = (cid:28)Z dϑ x ϑ x Z dq y q y w ( ξ, E y ; ξ + ϑ x , E y + q y ) (cid:29) (8) D yy = 12 (cid:28)Z dϑ x Z dq y q y w ( ξ, E y ; ξ + ϑ x , E y + q y ) (cid:29) (9)8 . Diffusion Coefficient Let us consider a channeling positron colliding with a target electron. If θ is the scattering angle in the lab frame and ϕ is the angle between thescattering plane and the ( xz )-plane then the transverse components of theparticle momentum are changed by δp x = p sin θ cos ϕ, (10) δp y = p sin θ sin ϕ, (11)As far as θ ≪
1, we can use the approximation sin θ ≈ θ . Then ϑ x = p x + δp x p − p x p = δp x p = θ cos ϕ. (12)and q y = (cid:18) ( p y + δp y ) E + U ( y ) (cid:19) − (cid:18) p y E + U ( y ) (cid:19) (13)= p y θ sin ϕ + p θ sin ϕ. The probability for the particle to be scattered by an electron from thestate ( ξ, E y ) to the state ( ξ + ϑ x , E y + q y ) while travelling the distance dz canbe related to the differential cross section of positron-electron scattering: w ( ξ, E y ; ξ + ϑ x , E y + q y ) dz = n e dz Z dθ Z π dϕ d σdθdϕ (14) δ ( θ cos ϕ − ϑ x ) δ (cid:16) p y θ sin ϕ + p θ sin ϕ − q y (cid:17) Because both target and projectile are not polarized, the cross section doesnot depend on ϕ : d σdθdϕ = 12 π dσdθ . (15)9ubstituting (14) into (5)–(9) and integrating over ϑ x and q y one obtains D ξ = 12 π (cid:28) n e Z dθ Z π dϕ dσdθ θ cos ϕ (cid:29) (16) D y = 12 π (cid:28) n e Z dθ Z π dϕ dσdθ (cid:16) p y θ sin ϕ + p θ sin ϕ (cid:17)(cid:29) (17) D ξξ = 14 π (cid:28) n e Z dθ Z π dϕ dσdθ θ cos ϕ (cid:29) (18) D ξy = 12 π (cid:28) n e Z dθ dσdθ Z π dϕθ cos ϕ (cid:16) p y θ sin ϕ + p θ sin ϕ (cid:17)(cid:29) (19) D yy = 14 π (cid:28) n e Z dθ dσdθ Z π dϕθ sin ϕ (cid:16) p y + p θ sin ϕ (cid:17) (cid:29) (20)Then integration over ϕ and neglecting higher order terms with respectto θ yields D ξ = 0 (21) D y = p h n e i Z dθ dσdθ θ (22) D ξξ = 14 h n e i Z dθ dσdθ θ (23) D ξy = 0 (24) D yy = 14 h n e p y i Z dθ dσdθ θ (25)Here h n e i is the electron density along the particle trajectory averagedover the period of the channeling oscillations. Generally speaking, h n e i de-pends on the transverse energy E y We assume, however, that the electrondensity does not change essentially within the channel. Therefore, h n e i canbe treated as a constant. For the same reason, we can make the approxi-mation h n e p y i ≈ h n e ih p y i Then h p y i = 2 E D p y E E = EE y . due to the virialtheorem for the harmonic potential: p y / (2 E ) = E y / D y ≡ D (26) D ξξ = 1 E D (27) D yy = E y D (28)10he diffusion equation takes the form ∂f∂t + ∂f∂z h v z i = D (cid:20) ∂∂E y (cid:18) E y ∂f∂E y (cid:19) + 1 E ∂ f∂ξ (cid:21) . (29)Equation (29) is akin to the equation describing dechanneling process (seee.g. [32]). The novel feature of it is the presence of time variable, whichallows to describe time dependent (modulated) beams. Additionally, it takesinto account scattering in the ( x, z ) plane.
4. Solving the diffusion equation
The particle velocity along z axes averaged over the period of channelingoscillations can be represented as h v z i = q − γ cos ξ k c π R π/k c q bk c sin( k c z )] dz . (30)Here p − /γ is the particle speed along the trajectory, cos ξ ≈ (1 − ξ / ξ ≪ z axis in ( xz ) plane,and the denominator is due to the sinusoidal channeling oscillations in ( xy )plane with the amplitude b and the period λ c = 2 π/k c . Taking into accountthat the amplitude of the channeling oscillations is much smaller than theirperiod, bk c ≪
1, the denominator can be approximated by 1 + ( bk c ) /
4. Forthe harmonic potential (1) (see Fig. 2.1) the amplitude b is related to thetransverse energy E y by b = ρ max r E y U max . (31)Using the formula for the frequency of the harmonic oscillator one finds k c = s E d Udρ = 1 ρ max r U max E (32)So that bk c = q E y E . Finally, neglecting higher order terms h v z i ≈ (cid:18) − γ − ξ − E y E (cid:19) (33)11 .2. Excluding the time variable If the beam is periodically modulated (bunched) the distribution f ( t, z ; ξ, E y )can be represented as a Fourier series: f ( t, z ; ξ, E y ) = ∞ X j = −∞ g j ( z ; ξ, E y ) exp( ijωt ) . (34)with g ∗ j ( z ; ξ, E y ) = g − j ( z ; ξ, E y ) to ensure the real value of the particle distri-bution. Since Eq. (29) is linear, it is sufficient to consider only one harmonic.Substituting f ( t, z ; ξ, E y ) = g ( z ; ξ, E y ) exp( iωt ) into (29) one obtains iωg ( z ; ξ, E y ) + ∂g∂z h v z i = D (cid:20) ∂∂E y (cid:18) E y ∂g∂E y (cid:19) + 1 E ∂ g∂ξ (cid:21) . (35) To simplify this equation, we make the substitution g ( z ; ξ, E y ) = exp ( − iωz ) ˜ g ( z ; ξ, E y ) , (36)where ˜ g ( s ; ξ, E y ) varies slowly comparing to exp ( − iωz ): ∂ ˜ g/∂z ≪ ω ˜ g ( z ; ξ, E y ) . (37)Equation (35) takes the form ∂ ˜ g∂z h v z i + iω ˜ g ( z ; ξ, E y )(1 − h v z i ) = D (cid:20) ∂∂E y (cid:18) E y ∂ ˜ g∂E y (cid:19) + 1 E ∂ ˜ g∂ξ (cid:21) . (38)In the first term, the velocity can be approximated by unity: h v z i ≈
1, i.e. theterm ∂ ˜ g/∂z (1 − h v z i ) can be neglected. However the term iω ˜ g ( z ; ξ, E y )(1 −h v z i ) has to be kept because of (37). Using the expression (33) for h v z i , oneobtains from (38) the following partial differential equation for ˜ g ( z ; ξ, E y ) ∂ ˜ g ( z ; ξ, E y ) ∂z + iω γ ˜ g ( z ; ξ, E y ) = D ∂∂E y (cid:18) E y ∂ ˜ g ( z ; ξ, E y ) ∂E y (cid:19) (39) − iω E y E ˜ g ( z ; ξ, E y ) + D E ∂ ˜ g ( z ; ξ, E y ) ∂ξ − iω ξ g ( z ; ξ, E y )12his equation can be solved by the method of separation of variables. Putting˜ g ( z ; ξ, E y ) = Z ( z )Ξ( ξ ) E ( E y ), after substitution into (39) we obtain a set ofordinary differential equations: D E ξ ) d Ξ( ξ ) dξ − iω ξ C ξ , (40) D E ( E y ) ddE y (cid:18) E y d E ( E y ) dE y (cid:19) − iω E y E = C y , (41)1 Z ( z ) d Z ( z ) dz + iω γ = C z , (42)where C z , C ξ and C y do not depend on any of the variables z , ξ and E y andsatisfy the condition C z = C ξ + C y . (43) Ξ( ξ )Equation (40) can be rewritten as d Ξ( ξ ) dξ − i ωE D ξ Ξ( ξ ) = ED C ξ Ξ( ξ ) . (44)We change the variable χ = e iπ/ r ωE D ξ (45)and introduce the notationΩ = − e − iπ/ r EωD C ξ . (46)This results into d Ξ( χ ) dχ − χ Ξ( χ ) = − Ω Ξ( χ ) . (47)This equation has the form of the Schr¨odinger equation for the harmonicoscillator. Its eigenvalues and integrable eigenfunctions are well known:Ω n = 2 n + 1 (48)Ξ n ( χ ) = H n ( χ ) exp( − χ / , (49)13here n = 0 , , , . . . and H n ( χ ) = e χ (cid:16) − ddχ (cid:17) n e − χ are Hermite Polynomialssatisfying the orthogonality condition Z + ∞−∞ dχ e − χ H n ( χ ) H n ′ ( χ ) = δ nn ′ n n ! √ π (50)which is equivalent to Z + ∞−∞ dχ Ξ n ( χ ) Ξ n ′ ( χ ) = δ nn ′ n n ! √ π. (51)Returning back to the variable ξ one obtainsΞ n ( ξ ) = H n e iπ/ r ωE D ξ ! exp − i r ωED ξ ! , (52)Any integrable function F ( ξ ) can be represented as series F ( ξ ) = F e − iπ/ r D ωE χ ! = ∞ X n =0 b n Ξ n ( χ ) (53)Let us multiply the above expression by Ξ n ′ ( χ ) and integrate over χ Z + ∞−∞ dχF e − iπ/ r D ωE χ ! Ξ n ′ ( χ ) = ∞ X n =0 b n Z + ∞−∞ dχ Ξ n ( ξ )Ξ n ′ ( ξ ) (54)Using (50) one finds b n = 12 n n ! √ π Z + ∞−∞ dχ e − χ / F e − iπ/ r D ωE χ ! H n ( χ ) (55)From (46) and (48) one finds C ξ,n = − (1 + i ) r ωD E (cid:18) n + 12 (cid:19) , n = 0 , , , . . . (56)14 .5. Solving the equation for E ( E y )Equation (41) can be rewritten as ddE y (cid:18) E y d E ( E y ) dE y (cid:19) − (cid:18) iω D E E y + C y D (cid:19) E ( E y ) = 0 , (57)By the substitution E y = 1 − i r D Eω ε (58)equation (57) can be reduced to ε d E dε + d E dε − (cid:18) ε − ν + 12 (cid:19) E = 0 (59)with 2 ν + 1 = − (1 − i ) r EωD C y . (60)Further substitution E ( ε ) = exp( − ε/ L ( ε ) results into the Laguerreequation: ε d Ldε + (1 − ε ) dLdε + νL = 0 (61)One of two linearly independent solutions of this equation is logarithmi-cally divergent at ε → L ν ( ε ), is finite at ε = 0 and is known as the Laguerre function. Returning back to the variable E y , the solution of equation (57) can berepresented as E ( E y ) = exp (cid:18) − i r ωD E E y (cid:19) L ν (cid:18) (1 + i ) r ωD E E y (cid:19) (62)The eigenvalues can be found by imposing the boundary conditions. Ifthe energy of the channeling oscillations exceeds the value U max (see Fig. 2.1)the particle enters the region of high electron density, get scattered by crystalconstituents and becomes dechanneled. Therefore, the distribution function At nonnegative integer values of ν , the Laguerre function is reduced to the well knownLaguerre polynomials. In the general case that is relevant to our consideration, it can berepresented by an infinite series (Appendix A.12).
15f channeling particles has to be zero at E y = U max . This results into thefollowing boundary condition L ν (cid:18) (1 + i ) r ωD E U max (cid:19) = 0 . (63)Equation (63) has to be solved for ν . Then, according to (60), the eigenvalue C y,k can be found from C y = − (1 + i )2 r D ωE (2 ν + 1) . (64)The subscript k = 1 , , , . . . enumerates different roots of equation (63).We introduce a dimensionless parameter κ = 4 j , ωD E U (65)( j ,k is k -th zero of the 0-th order Bessel function: J ( j ,k ) = 0). Thenequation (63) can be rewritten as L ν (cid:18) i j , √ κ (cid:19) = 0 . (66)This equation has infinite number of complex roots (see Appendix) which wedenote as ν k ( κ ), k = 1 , , , . . . . The equation does not have any analyticalsolution and therefore has to be solved numerically.Instead of the complex function ν k ( κ ), it is more convenient to introducetwo real functions: α k ( κ ) = √ κj , [1 + 2 ( ℜ [ ν k ( κ )] − ℑ [ ν k ( κ )])] (67) β k ( κ ) = 12 j , √ κ [1 + 2 ( ℜ [ ν k ( κ )] + ℑ [ ν k ( κ )])] . (68)The eigenvalues (64) can be represented in the form C y,k = − α k ( κ ) L d − iωθ β k ( κ ) . (69)Here L d = 4 U max / ( j , D ) and (70) θ L = p U max /E (71)16re, respectively, the dechanneling length [32] and Lindhard’s angle. Theparameter κ (65) can be rewritten in terms of L d and θ L κ = π L d λ θ , (72)where λ = 2 π/ω is the spatial period of the modulation. Z ( z )Equation (42) has the solution Z ( z ) = exp (cid:18) C z − i ω γ (cid:19) (73)The value of C z can be found using (43), (56) and (64). Then the solution(73) takes the form Z n,k ( z ) = exp (cid:26) − zL d (cid:20) α k ( κ ) + (2 n + 1) √ κj , (cid:21) − (74) iωz (cid:20) γ + θ β k ( κ ) + θ (2 n + 1)2 j , √ κ (cid:21)(cid:27) . Hence, the solution of Eq. (35) is represented as g ( z ; ξ, E y ) = exp ( − iωz ) ∞ X n =0 ∞ X k =1 a n,k Ξ n ( ξ ) E k ( E y ) Z n,k ( z ) , (75)where the coefficients a n,k are found from the particle distribution at theentrance of the crystal channel: a n,k = R + ∞−∞ dξ R U max dE y g (0; ξ, E y )Ξ n ( ξ ) E k ( E y )2 n n ! √ π R U max dE y [ E k ( E y )] . (76)
5. The demodulation length
Due to the exponential decrease of Z n,k ( z ) with z (see (74)), the asymp-totic behaviour of ˜ g ( z ; ξ, E y ) at large z is dominated by the term with n = 0and k = 1 having the smallest value of the factor [ α k ( κ ) + (2 n + 1) √ κ/j , ]17n the exponential. Therefore, at sufficiently large penetration depths, theparticle distribution depends on z as g ( z ; ξ, E y ) ∝ exp ( − z/L dm − iω/u z z ) (77)where L dm is the newly introduced parameter — the demodulation length : L dm = L d α ( κ ) + √ κ/j , (78)and u z is the phase velocity of the modulated beam along the crystal channel u z = (cid:20) γ + θ (cid:18) β k ( κ ) + 12 j , √ κ (cid:19)(cid:21) − . (79)This parameter is important for establishing the resonance conditions be-tween the undulator parameters and the radiation wavelength.In this article we concentrate our attention on the demodulation length.This parameter represents the characteristic scale of the penetration depthat which a beam of channeling particles looses its modulation.Fig. 5.1 presents the dependence of the ratio L dm /L d on the parameter κ .It is seen that the demodulation length approaches the dechanneling lengthat κ .
1. On the contrary, the ratio noticeably drops for κ & x and y direction on the demodulation length separately. Replacing α ( κ ) in (78)with unity means neglecting the motion in the y direction, while omitting thesecond term in the denominator ignores the motion in x direction. One seesfrom Fig. 5.1 that it is mostly the motion in x direction that diminishes thedemodulation length at κ .
10, while the influence of channeling oscillationsis negligible. This suggests the idea that for the axial channeling, i.e. whenmotion in both x and y directions has the nature of channeling oscillations,the demodulation length L dm may practically coincide with the dechannelinglength L d at higher frequencies of the beam modulation than in the case ofplanar channeling. So far, beam demodulation in a straight channel has been considered.The channels of a crystalline undulator, however, have to be periodicallybent. Therefore the above formalism has to be modified for the case of abent channel. 18 L d m / L d κ Motion in x-direction onlyMotion in y-direction onlyBoth contributions are included
Figure 5: The ratio of the demodulation length L dm (78) to the dechanneling length L d versus the parameter κ (72). See text for details. Let us consider a crystal that is bent in the (yz) plane so that the crystalchannel has a constant curvature with the radius R . An ultrarelativisticparticle with energy E moving in such a channel experiences the action thecentrifugal force F c . f . = ER . (80)It is convenient to characterise the channel curvature by the dimensionlessparameter C defined as C = (cid:12)(cid:12)(cid:12)(cid:12) F c . f . U ′ max (cid:12)(cid:12)(cid:12)(cid:12) , (81)where U ′ max is the maximum value of the derivative of the particle potentialenergy in the channel, i.e. the maximum transverse force acting on theparticle in the interplanar potential. Channeling is possible at 0 ≤ C < C = 0 corresponds to a straight channel. The critical radius R c (known also as Tsyganov radius) at which the interplanar potential becomesunable to overcome the centrifugal force corresponds to C = 1.In the case of potential energy (1), U ′ max = U ′ ( ρ max ) = 2 U max ρ max . (82)19o that C = ρ max F c . f . U max . (83) U ( ρ ) / U ( d / ) ρ /dU max (1-C) ρ max (1-C) C=0C=0.2 Figure 6: The potential energy of a particle in the planar crystal channel for a straight, C = 0, and for a bent, C = 0, crystal. The effective width of the bent channel is2 ρ max (1 − C ) and the depth of the potential well is U max (1 − C ) , where 2 ρ max and U max are, respectively, the effective width and the depth of the straight channel (cf. Fig. 2.1). The potential energy is modified by the centrifugal force in the followingway U C ( ρ ) = U ( ρ ) − ρF c . f . . (84)For the parabolic potential energy (1) the modified potential can be conve-20iently rewritten in terms of the parameter C : U C ( ρ ) = U max "(cid:18) ρρ max − C (cid:19) − C (85)The potential energy U C ( ρ ) reaches its minimum at ρ = Cρ max . The effectivewidth of the channel becomes (see Fig. 5.2) ρ max − ρ = ρ max (1 − C ) . (86)The depth of the potential energy well is U C ( ρ max ) − U C ( ρ ) = U max (1 − C ) . (87)So to obtain the solution of the diffusion equation for the bent crystal we canuse the results of Sec. 4 with the substitution U max → U max (1 − C ) . (88) Substitution (88) modifies the demodulation length and the Lindhard’sangle the parameter κ in the following way: L d → L d (1 − C ) (89) θ L → θ L (1 − C ) (90)Consequently, the the modification of parameter κ is κ → κ (1 − C ) . (91)It is convenient to introduce modified functions α k ( κ, C ) and β k ( κ, C ): α k ( κ, C ) = α k ( κ (1 − C ) )(1 − C ) (92) β k ( κ, C ) = (1 − C ) β k (cid:0) κ (1 − C ) (cid:1) . (93)In terms of these functions, the eigenvalue C y,k has the form C y,k = − α k ( κ, C ) L d − iωθ β k ( κ, C ) . (94)21his exactly coincides with (69) up to replacing α k ( κ ) and β k ( κ ) with α k ( κ, C )and β k ( κ, C ), respectively. Note that L d and θ L in (94) have the same mean-ing as in (69): they are related to the straight channel.Similarly, the demodulation length in the bent channel is given by L dm = L d α ( κ, C ) + √ κ/j , (95) L d m / L d κ C=0C=0.1C=0.2C=0.3
Figure 7: The ratio of the demodulation length L dm (95) to the dechannelling length in thestraight channel L d versus the parameter κ (72) for different values of curvature parameter C . The corresponding asymptotic values at κ → Fig. 5.1 presents the dependence of the ratio L dm /L d on the parameter κ . At κ →
0, the demodulation length approaches (1 − C ) L d which isthe dechannelling length in a bent crystal. It is seen that the demodulationlength is smaller than dechannelling length by only 20–30% at κ . C ranging from 0 to 0 .
3. It noticeably drops, however, at κ & C varying between0 and 0 . κ . ∼
100 keV (softer photons are strongly absorbed in the crystal). Itwill be shown in the next section that such crystal channels do exist. κ To evaluate the parameter κ (72) we shall use the approximate formulafor the dechannelling length [32]: L d = 2569 π Em e a TF r d Λ . (96)Here m e and r are, respectively, the electron mass and the classical radius, d is the distance between the crystal planes, and the Coulomb logarithm Λfor positron projectiles is defined as [18]:Λ = log √ Em e I − , (97)with I ≈ Z . eV (98)being the ionization potential of the crystal atom with the atomic number Z . The Thomas-Fermi radius of this atom is related to the Bohr radius a B by the formula a TF = a B . √ Z . (99)Substituting (96) into (72) and taking into account (71), one obtains κ = 5129 π a TF Λ r U max m e dλ . (100)As is seen from the above formula the value of κ is determined by the potentialdepth U max , by the distance between the planes d and the modulation period λ . It also depends on the atomic number of the crystal atoms Z via (98) and(99). These parameters are listed in table 5.4 for several crystal channels.23 able 1: The parameters of the crystalchannels used in the calculations (see text). For(111) plane of diamond, only the larger of two channels is presented. Crystal
Z I (eV) a TF (˚A) Plane d (˚A) U max (eV)Diamond 6 80 0.26 (100) 0.9 2.2(110) 1.3 7.3(111)L 1.5 10.8Graphite 6 80 0.26 (0002) 3.4 37.9Silicon 14 172 0.19 (100) 1.4 6.6(110) 1.9 13.5Germanium 32 362 0.15 (100) 1.4 14.9Tungsten 74 770 0.11 (100) 1.6 56.3The dependence on the particle energy is cancelled out, except the weakdependence due to the logarithmic expression (97).The dependence of the parameter κ on the energy of the emitted photons, ~ ω = 2 π ~ /λ , is shown in Fig. 5.4. The calculation was done for 1 GeVpositrons. Due to the weak (logarithmic) dependence of κ on the particleenergy, changing this energy by an order of magnitude would leave Fig. 5.4practically unaltered.As one sees from the figure, κ ∼ ~ ω = 100 −
300 keVfor (100) and (110) planes in Diamond and (100) plane in Silicon. So thesechannels are the most suitable candidates for using in CUL. This is, however,not the case for a number of other crystals e.g. for graphite and tungstenhaving κ &
10 in the same photon energy range.At ~ ω ∼
10 MeV, κ becomes larger than 10 for all crystal channels. Thisputs the upper limit on the energies of the photons that can be generated byCUL. It is expected to be most successful in the hundred keV range, whilegenerating MeV photons looks more challenging.
6. Discussion and Conclusion
One may expect that the demodulation is not limited to the processesillustrated in figure 3. An additional contribution can come from the energyspread of the channelling particles, as it usually happens in ordinary FELs.In fact, the contribution of the energy spread to the beam demodulation24 κ − h ω (MeV) C (diam.) (100)C (diam.) (110)C (diam.) (111)C (graph.) (0002)Si (100)Si (110)Ge (100)W (100)
Figure 8: The parameter κ (72) versus the photon energy ~ ω for the crystal channels listedin Table 5.4. on the distance of a few dechannelling lengths is negligible. It would besubstantial if the relative spread δE/E of particle energies would be compa-rable to or larger than the ratio λ u /L d . The latter ratio, however, can notbe made smaller than 10 − [18], while modern accelerators usually have amuch smaller relative energy spread. The same is true for the energy spreadinduced by the stochastic energy losses of the channelling particles due to theinteraction with the crystal constituents and the radiation of photon. It wasshown in Ref.[33] that at initial energies of ∼ GeV or smaller, the averagerelative energy losses of a positron in the crystalline undulator ∆
E/E aresmaller than 10 − . Clearly, the induced energy spread δE/E ≪ ∆ E/E issafely below the ratio λ u /L d . From these reasons, we ignored energy spreadof the particles in our calculations.In conclusion, we have studied the propagation of a modulated positronbeam in straight and bent planar crystal channel within the diffusion ap- Note that the corresponding quantity in ordinary ultraviolet and soft x-ray FELs, theinverse number of undulator periods 1 /N u = λ u /L , is usually of the order of 10 − –10 − [31]. That is why these FELs are so demanding to the small energy spread of the electronbeam. Appendix A. Appendix: Solving equation (66).
Appendix A.1. A series expansion at κ ≪ κ , the solution of equation (66) can be found in theform of power series.The Laguerre function L ν ( z ) (which is a special case of the Kummerfunction, L ν ( z ) ≡ M ( − ν, , z ) can be represented as L ν ( z ) = exp( z / ∞ X n =0 A n (cid:20) z ν ) (cid:21) n/ J n (cid:16)p ν ) z (cid:17) , (Appendix A.1)where J n ( . . . ) are Bessel functions and the coefficients A n are defined by thefollowing recurrence relation: A = 1 (Appendix A.2) A = 0 (Appendix A.3) A = 12 (Appendix A.4) A n +1 = 1 n + 1 [ nA n − − (1 + 2 ν ) A n − ] . (Appendix A.5)Keeping only the leading term in (Appendix A.1) (this approximation isvalid if z ≪ J (cid:18)q (1 + 2 ν (0) )(1 + i ) j , √ κ (cid:19) = 0 , (Appendix A.6)where ν (0) is the zero-order approximation to the root of equation (66).26quation (Appendix A.6) is satisfied if q (1 + 2 ν (0) k )(1 + i ) j , √ κ = j ,k , (Appendix A.7) j ,k is a root of the Bessel function: J ( j ,k ) = 0. Here the subscript k =1 , , , . . . enumerates the roots of the Bessel function and the correspondingapproximate solutions of equation (66). Solving (Appendix A.7) for ν (0) k results into ν (0) k ( κ ) = 1 − i j ,k j , √ κ − . (Appendix A.8)Keeping higher order terms in (Appendix A.1) and expanding the Besselfunctions around the the zero-order approximation (Appendix A.8), one ob-tains a series expansion of ν k ( κ ): ν k ( κ ) = 1 − i j ,k j , √ κ −
12 + 1 + i j , j ,k − j ,k √ κ − − i j , j ,k − j ,k + 54 j ,k (cid:0) √ κ (cid:1) + . . . (Appendix A.9)(Dots stand for higher order terms with respect to κ ). For the functions (67)and (68) expansion (Appendix A.9) takes the form α k ( κ ) = j ,k j , − j , (cid:0) j ,k − j ,k + 54 (cid:1) j ,k κ + . . . (Appendix A.10) β k ( κ ) = j ,k − j ,k + . . . (Appendix A.11) Appendix A.2. Numerical solution
Expansions (Appendix A.9), (Appendix A.10) and (Appendix A.11)fail at κ &
1. Therefore, equation (66) has to be solved numerically. Inour numerical procedure, we use the series representation for L ν ( z ): L ν ( z ) = ∞ X j =0 Q j − m =0 ( m − ν )( j !) z j (Appendix A.12)Equation was solved by Newton’s method. At small κ the value foundfrom the series expansion (Appendix A.9) was used as initial approxima-tion. Then κ was gradually increasing. At each step, the equation was27olved and the solution was used as initial approximation for the next step.During this procedure, the roots ν k ( κ ) were slowly moving in the complexplane along the trajectories shown in figure Appendix A.2. The functions -2-1.5-1-0.5 0 0.5 1 1.5 2 0 1 2 3 I m ( ν ) Re( ν )k = 1k = 2k = 3k = 4k = 5k = 6 Figure A.9: The trajectories drawn by the roots ν k ( κ ) of (66) in the complex plain atvarying κ . The arrows show the direction of motion of the roots when κ increases. α k and β k are plotted in Figures Appendix A.2 and Appendix A.2. Appendix A.3. Asymptotic behaviour at κ ≫ ν k ( κ ) approaches integerreal numbers as κ → ∞ . This the case for k = 1 , , ,
6. For these solutions,the asymptotic behaviour can be found.Let us represent ν k ( κ ) in the form ν k ( κ ) = n k + δ k ( κ ) , (Appendix A.13)28 α ( κ ) κ k = 1k = 2k = 3k = 4k = 5k = 6 Figure A.10: The function α k ( κ ) obtained by numerical analysis. where n k = ν k ( ∞ ) is an integer number and the function δ k ( κ ) goes to zeroat κ → ∞ .Substituting (Appendix A.13) into (Appendix A.12) and expanding around δ k = 0 one obtains L ν ( z ) = n k ! " n k X j =0 ( − j ( j !) [ n k − j ]! z j (Appendix A.14) − δ k P n k ( z ) + ( − n k z n k +1 ∞ X j =0 j ![( j + n k + 1)!] z j ! Here P n k ( z ) is a polynomial of the order n k whose explicit form will not be29 β ( κ ) κ k = 1k = 2k = 3k = 4k = 5k = 6 Figure A.11: The function β k ( κ ) obtained by numerical analysis. needed in the following.At | z | ≪
1, the infinite sum in (Appendix A.14) can be approximated byan integral and evaluated by Laplace’s method: ∞ X j =0 j ![( j + n k + 1)!] z j ≍ z − n k +1) e z (Appendix A.15)The polynomial P n k ( z ) in (Appendix A.14) becomes negligible with respectto the exponential at large z . Similarly, the leading order term dominates thefirst sum in (Appendix A.14). The asymptotic expression for L ν ( z ) takes,30herefore, the following form L ν ( z ) ≍ ( − n k n k ! (cid:20) z n k ( n k !) − δ k z − ( n k +1) e z (cid:21) (Appendix A.16)Using (Appendix A.16) and taking into account (Appendix A.13) oneobtains the asymptotic expression for the root of equation (66): ν k ( κ ) ≍ n k + 1( n k !) (cid:18) i j , √ κ (cid:19) n k +1 exp (cid:18) − i j , √ κ (cid:19) . (Appendix A.17)This equivalent to the following asymptotic behaviour of the functions (67)and (68) α k ( κ ) ≍ n k + 1 j , √ κ (Appendix A.18)+ ( j , ) n k κ n k +1 n k − ( n k !) exp (cid:18) − j , √ κ (cid:19) sin (cid:18) j , √ κ − π n k (cid:19) β k ( κ ) ≍ n k + 12 j , √ κ (Appendix A.19)+ ( j , ) n k κ n k n k ( n k !) exp (cid:18) − j , √ κ (cid:19) cos (cid:18) j , √ κ − π n k (cid:19) It has to be stressed once more, that not all solutions of equation (66) havethe above asymptotic behaviour. Among the solutions represented in figuresAppendix A.2-Appendix A.2, (Appendix A.17)–(Appendix A.19) is validonly for k = 1 , , , n k = 0 , , ,
3, respectively.
Acknowledgement
This work has been supported in part by the European Commission (thePECU project, Contract No. 4916 (NEST)) and by Deutsche Forschungsge-meinschaft. 31 eferences [1] A. Kostyuk, A.V. Korol, A.V. Solov’yov and W. Greiner, J. Phys. B:At. Mol. Opt. Phys. et al. , Phys. Lett. B88 (1979) 387.[9] A.G. Afonin et al. , Nucl. Instrum. Meth. B234 (2005) 14.[10] G. Arduini et al. , Phys. Lett. B422 (1998) 325.[11] W. Scandale et al. , Phys. Rev. ST Accel. Beams 11 (2008) 063501.[12] R.A. Carrigan et al. , Phys. Rev. ST Accel. Beams 5 (2002) 043501.[13] R.P. Fliller et al. , Phys. Rev. ST Accel. Beams 9 (2006) 013501.[14] S. Strokov et al. , J. Phys. Soc. Jap. 76 (2007) 064007.[15] E. Uggerhøj and U.I. Uggerhøj, Nucl. Instrum. Meth. B234 (2005) 31.[16] S. Bellucci et al. , Nucl. Instrum. Meth. B252 (2006) 3.3217] S. Strokov et al. , Nucl. Instrum. Meth. B252 (2006) 16.[18] A.V. Korol, A. V. Solov’yov and W. Greiner, Int. J. Mod. Phys. E13(2004) 867–916.[19] M. Tabrizi, A.V. Korol, A. V. Solov’yov and W. Greiner, Phys. Rev.Lett. 98 (2007) 164801.[20] S. Bellucci et al. , Phys. Rev. Lett. 90 (2003) 034801.[21] V. Guidi et al. , Nucl. Inst. and Meth. B234, (2005) 40.[22] P. Balling et al. , Nucl. Inst. and Meth. B267 (2009) 2952.[23] M. B. H. Breese, Nucl. Inst. and Meth. B132 (1997) 540.[24] U. Mikkelsen and E. Uggerhøj, Nucl. Inst. and Meth. B160 (2000) 435.[25] A.V. Korol, W. Krause, A. V. Solov’yov, and W. Greiner Nucl. Inst.and Meth. A483 (2002) 455.[26] V. T. Baranov et al.et al.