PHYSICAL REVIEW ACCELERATORS AND BEAMS 19, 112801 (2016)
Formation region effects in transition radiation, bremsstrahlung, and ionization loss of ultrarelativistic electrons
† S. V. Trofymenko* and N. F. Shul’ga Akhiezer Institute for Theoretical Physics of National Science Center ‘Kharkov Institute of Physics and Technology’, Akademicheskaya st. 1, 61108 Kharkov, Ukraine and Karazin Kharkov National University, Svoboda sq. 4, 61022 Kharkov, Ukraine (Received 10 April 2016; published 3 November 2016) The processes of transition radiation and bremsstrahlung by an ultrarelativistic electron as well as the effect of transition radiation influence upon the electron ionization loss in thin layer of substance are theoretically investigated in the case when radiation formation region has macroscopically large size. Special attention is drawn to transition radiation (TR) generated during the traversal of thin metallic plate by the electron previously deflected from its initial direction of motion. In this case TR characteristics are calculated for realistic (circular) shape of the electron deflection trajectory. The difference of such characteristics under certain conditions from the ones obtained previously with the use of approximation of anglelike shape of the electron trajectory (instant deflection) is shown. The problem of measurement of bremsstrahlung characteristics in the prewave zone is investigated. The expressions defining the measured radiation distribution for arbitrary values of the size and the position of the detector used for radiation registration are derived. The problem of TR influence upon the electron ionization loss in thin plate and in a system of two plates is discussed. The proposal for experimental investigation of such effect is formulated.
DOI: 10.1103/PhysRevAccelBeams.19.112801
I. INTRODUCTION of the experimental facility as well. Therefore within the formation length the particle, being half-bare, can undergo At high energies the electromagnetic processes, which further interactions with matter (for example, detectors). take place during a charged particle interactions with The character of such interactions should significantly matter, develop within large macroscopic distances along differ from the character of interaction with matter of a the direction of the particle motion. These distances are particle which has equilibrium Coulomb field. known as formation or coherence lengths of the electro- The manifestations of large size of the formation length magnetic processes [1–3]. Within these distances from the and the half-bare state of a particle were previously interaction region the particle exists in the state with investigated for the process of bremsstrahlung by relativ- significantly nonequilibrium electromagnetic field around istic electron during its motion through substance. It led to it. This field lacks part of Fourier-components (virtual the prediction of the Landau-Pomeranchuk-Migdal effect photons) comparing to equilibrium Coulomb field of a [7,8] of Bethe-Heitler spectrum suppression at low frequen- moving particle. The possibility of manifestation of such cies as well as the Ternovsky-Shul’ga-Fomin effect [9,10] state of the particle in different processes involving both of bremsstrahlung suppression in thin layers of substance. electromagnetic and strong interactions was firstly pointed The experimental investigations of these effects, which out in [4] in the framework of quantum consideration. In confirmed the theoretical predictions, were recently carried [5,6] the examples of such manifestation were qualitatively out at SLAC [11,12] and CERN [13–15]. considered with the use of classical electrodynamics. In In the present work we consider a series of processes these papers such particle with incomplete or suppressed involving low-energy relativistic electrons (several tens of electromagnetic field was named “half-bare” particle. The MeV) in which the half-bare state of electron or the size of the coherence length can sometimes substantially nonequilibrium state of free electromagnetic field (radia- exceed not only the size of the region of the particle tion) produced by electron can manifest itself in measured interaction with matter or external field but the whole size characteristics of these processes. This involves millimeter wavelength TR by the half-bare electron and the prewave *[email protected] zone effect in the electron bremsstrahlung in the same † [email protected] wavelength region. The situation in which the nonequili- brium state of the electron’s field manifests itself in the Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- particle ionization loss in thin layers of substance is bution of this work must maintain attribution to the author(s) and considered as well for the case of higher particle energies the published article’s title, journal citation, and DOI. (several hundred MeV).
2469-9888=16=19(11)=112801(16) 112801-1 Published by the American Physical Society S. V. TROFYMENKO and N. F. SHUL’GA PHYS. REV. ACCEL. BEAMS 19, 112801 (2016)
In the usual statement of the problem about TR by a both half-bare and overdressed states of the electron upon charged particle the incident particle is supposed to move its ionization loss is formulated. Such study may be of uniformly and rectilinearly during long period of time interest in connection with the development of noninvasive before impinging upon the target and have equilibrium techniques of particle identification on the basis of ultrathin Coulomb field around it. The change of the motion state of detectors [23,24]. such particle (e.g., deflection) before hitting the target significantly disturbs its electromagnetic field. It results in II. TRANSITION RADIATION BY AN ELECTRON modification of characteristics of TR generated by this WITH NONEQUILIBRIUM FIELD particle. Such situation takes place, for instance, in the problems about TR produced by the beams consisting of TR is one of the effective tools for analysis of various particles extracted from the initial linear beam by deflection characteristics of relativistic particle beams. Different (e.g., with the use of bent crystal) or the beams extracted techniques are based on registration of TR generated by from accelerating storage rings. charged particle bunches during their traversal of thin plates Previously the characteristics of such TR were inves- (conducting or dielectric) in different frequency domains: tigated with the use of the simplifying approximation of X-ray, optical, millimeter wavelength etc. Under certain anglelike shape for the trajectory of the deflecting electron conditions such radiation characteristics significantly (instant deflection) [16,17]. In the present work, along with depend on the size of the bunch, particle energy and spatial the further analysis of such approximation, the calculation distribution within the bunch, beam divergence etc. It with the use of realistic (circular) shape of the electron allows defining such parameters of the beam on the basis trajectory is made and the applicability of the considered of the observed radiation characteristics. approximation is discussed. In this respect naturally arises the problem of the At sufficiently high energies of the particles or in rather applicability of such techniques for extracted beams con- long wavelength region of the considered radiation its sisting of particles which deflect from the initial direction formation length may exceed the size of the whole labo- of motion before the moment of diagnostics through TR ratory. In this case the necessity of making measurements production (i.e., before impinging on the plates). The within the formation region (in the prewave zone) arises. For perturbation of electromagnetic field around the particles the case of TR such situation was first theoretically con- caused by such deflection can be rather long lasting. It sidered in [18,19]. The experimental investigations of TR results in modification of characteristics of TR generated by such particles even for macroscopically large separations characteristics in the prewave zone are reported in [20–22]. 1 In these papers it was shown that radiation spectral-angular between the radiating plate and the deflection area. These make the investigation of characteristics of TR generated distribution registered by a pointlike detector in this situation by particles with perturbed electromagnetic field required. significantly differs from the one registered in the wave zone Let us consider a process in which the electron moves (outside the formation region). along the z-axis and in the point z ¼ 0 deflects to a large In the present work the analogous effects of the prewave angle (Fig. 1). Such simplified model (instant deflection) of zone are investigated for the case of bremsstrahlung which the deflection process is usually used for consideration of is emitted in the process of the electron deflection to a large electromagnetic fields produced in this process with the angle. Both the spatial distribution of radiation energy and wavelengths having rather large radiation formation dis- its distribution over wave vector directions in the prewave tance. Namely, such distance should significantly exceed zone are studied. Special attention is drawn to investigation the size of the region in which the deflection occurs (the of the dependence of the results on the size and the position spatial size of the deflecting field). The dependence of the of the detector. The expression for the radiation spectral- formation length in the direction of the electron velocity angular density, which can be obtained with the use of (both initial and final) on the radiated wavelength λ is detector of arbitrary transversal size, is derived. 2 defined by the expression: lC ∼ γ λ, where γ is the The third problem, which the present paper deals with, is ’ the influence of the nonequilibrium state of the electron’s electron s Lorentz-factor. For example, for 50 MeV elec- field upon its ionization loss in thin films. Such state of the trons and the measured wavelength of 1 mm such formation field around the electron in the considered cases is caused distance reaches 10 m. by the interference of the electron’s own Coulomb field After the deflection the total electromagnetic field in space with the field of TR produced either during the particle splits into two parts, the first of which as if tears away from traversal of the thin film itself or during the electron interactions with the other media before impinging upon 1For instance, the modification of characteristics of millimeter the film. In this case both the effects of reduction (when the wavelength TR produced by deflected several-tens-MeV elec- “ ” trons should be observed even for 10 m of such separation, while electron is half-bare) and enhancement ( overdressed for several TeV protons in the same wavelength region such electron) of the ionization loss in thin film are possible. distance reaches several kilometers (in the optical frequency The proposal to investigate experimentally the influence of region—several tens of meters).
112801-2 FORMATION REGION EFFECTS IN TRANSITION … PHYS. REV. ACCEL. BEAMS 19, 112801 (2016)
Let us analyze in detail the influence of the electron deflection upon the spectral-angular characteristics of its TR. The expression (1) has the structure of a product of the formula for distribution of TR by the electron having equilibrium Coulomb field and the interference term in braces with coefficient two in front of it. It shows that, unlike the TR by the electron with equilibrium field, the characteristics of radiation by the half-bare electron depend on the frequency ω ¼ 2πc=λ of the radiated waves. The value of the interference factor also depends on the distance between the plate and the deflection point l and on the angle θ between the observation direction and the plate normal. Figures 2(a),2(b) show the difference between the distributions of TR by the half-bare electron (red lines) FIG. 1. The traversal of metallic plate by deflected half-bare and by the electron having equilibrium Coulomb field electron. (green lines) for γ ¼ 100. Here we use a denomination 2 2 2 Fω θ ¼ðcπ =e Þd W=dωdo. the electron and, moving in the initial direction, gradually ; Namely, Fig. 2(a) shows that the angular distribution of transforms into spherically diverging waves of forward TR by the half-bare electron is wider than the one by the bremsstrahlung. The second part is the nonequilibrium field electron with equilibrium field. In this case the angular of the deflected half-bare electron which gradually recon- position of the main radiationp maximumffiffiffiffiffiffiffi is approximately structs into equilibrium Coulomb field. The deflection angle 2 defined by the relation θ ∼ λ=l.Forl<γ λ it is larger χ is supposed here to have the value which allows neglecting max than the value 1=γ for the electron having equilibrium field. the interference between forward bremsstrahlung and the Moreover, additional maxima in the angular distribution field of half-bare electron and consider the evolution of these appear in this case. fields separately. This imposes the following condition upon From Fig. 2(b) we see that for small values of distance l this angle: χ ≫ 1=γ. For instance, for 50 MeV electrons this between the deflection point and the plate the TR intensity gives χ ≫ 0.01 (χ ≫ 0.60). Such deflection should be ≪ is suppressed while for larger l it may significantly exceed reached within the distance L lC. the intensity of TR by the electron with equilibrium field. Since the distance from the deflection area within which Previously we considered characteristics of TR by the the electron exists in the half-bare state is very large (it is ∼ half-bare electron with the use of the approximation of the lC) it is possible to place a conducting plate in the final anglelike shape for the real (circular, for instance) trajectory direction of the particle motion and observe the backward of the electron (approximation of instant deflection). Such TR generated when the half-bare electron traverses the approximation can be violated in the case when the distance plate. The spectral-angular density of such radiation is between the plate and the deflection region is comparable defined by the expression derived in [16] (for more detailed 2 with the size of this region. In this case it is necessary to use consideration see [17]) : the real shape of the electron trajectory. Let us obtain the expression defining TR characteristics in this case. d2W e2 θ2 ωl ¼ 2 1 − ðθ2 þ γ−2Þ ; Let the electron move along the x-axis with the velocity v ω π2 ðθ2 þ γ−2Þ2 cos 2 d do c c and in the origin of the coordinate system at the moment of ð1Þ time t ¼ 0 begin deflecting along the circular trajectory with the curvature radius R (without changing the absolute where l is the distance between the plate and the deflection value of the velocity) (Fig. 3). Let the total deflection angle point. Let us note that in the considered case the electron be χ. After the deflection the electron moves again becomes half-bare due to destructive interference of its rectilinearly along the final direction during some period Coulomb field with the bremsstrahlung emitted in the final of time and covers distance l before impinging on the plate direction of the electron motion during its deflection. The situated in the plane perpendicular to the electron velocity. discussed modification of TR characteristics is nothing else In the millimeter range of radiated wavelengths the than the result of interference of TR produced by electron metallic plate can be considered ideally conducting. This having a Coulomb field with such bremsstrahlung reflected allows using the method of images [3,26] for calculation of from the plate. the spectral-angular distribution of the radiation outburst which takes place at the electron impinging upon the plate. 2The similar expression was obtained in [25] for distribution of In this method the backward TR produced by the electron is TR generated by an electron traversing a system of two metallic considered as radiation (bremsstrahlung) produced by the foils. electron’s image, which moves the mirror symmetrically to
112801-3 S. V. TROFYMENKO and N. F. SHUL’GA PHYS. REV. ACCEL. BEAMS 19, 112801 (2016)
FIG. 2. (a) TR angular distribution for l ¼ 2 m, λ ¼ 2 mm; (b) dependence of TR intensity on l for θ ¼ 1=γ, λ ¼ 2 mm. the electron with respect to the plane in which the plate is where n~ is a unit vector in the radiation direction, while situated and abruptly stops at the point of its “collision” ~r0ðtÞ defines here the image position at the moment of with the electron on the plate surface.3 The calculation of time t. such radiation spectral-angular density can be made with The integration region in (2) splits into three parts. They the use of the well-known expression for the distribution of respectively correspond to rectilinear motion of the image radiation by a particle with a known law of motion [29,30]: before the deflection along the circular trajectory (t<0), deflection (0 Zχ 2 2 iw sinðθþχÞ d W e β sinðθ þ χÞe α ¼ i þ w dα sinðα − χ − θÞ exp iw − sinðα − χ − θÞ dωdo 4π2c 1 − β cosðθ þ χÞ β 0 β θ 1 1 2 sin vT ð θþχ θÞ − i exp iwχ − cos θ − exp iw − cos θ eiw sin cos ; ð3Þ 1 − β cos θ β R β where β ¼ v=c, T ¼ðRχ þ lÞ=v is the moment of time Figures 4(a) and 4(b) show the difference between the at which the electron (and its image) impinges upon the TR angular distribution given by the expression (3) and the plate, w ¼ ωR=c ¼ 2πR=λ and θ is the angle between the corresponding distribution (1) obtained with the use of the radiation direction and the direction normal to the plate. θ approximation of anglelike trajectory. is positive to the left of this normal and negative to the Here we see that the TR distribution given by (3) right (see Fig. 3). For simplicity we consider the radiation significantly differs from the one defined by (1) in the in the plane of the electron trajectory where in ultra- relativistic case the major part of radiation is concentrated. The first term in (3) describes the bremsstrahlung which is emitted in the direction close to the electron initial velocity during its deflection and reflected by the plate. Provided the deflection angle is rather large (χ ≫ 1=γ), if we consider TR at rather small angles to the plate normal (θ ∼ 1=γ), this term can be neglected. 3Let us note that radiation produced by a particle moving along the arc of a circle (without traversal of a conducting plate afterwards) was considered in a series of papers. See, for FIG. 3. Normal incidence of the deflected electron upon thin example, [27] as well as [2,28] and references therein. metallic plate. 112801-4 FORMATION REGION EFFECTS IN TRANSITION … PHYS. REV. ACCEL. BEAMS 19, 112801 (2016) FIG. 4. Half-bare electron TR angular distribution for γ ¼ 100, λ ¼ 3 mm, χ ¼ π=4, R ¼ 50 cm (electron path within the deflection region δ ¼ Rχ ≈ 40 cm) and (a) l ¼ 40 cm, (b) l ¼ 2 m. Red lines—calculation with the use of (2), black lines—with the use of (1), green line—TR by electron with equilibrium field. case when the distance l between the plate and the For instance, in the case of 3 mm wavelength this leads deflection region is close to the size δ of this region itself. to condition l<3 m. For the case of smaller wavelengths With the increase of l the distribution (3) gradually the distances l defined by (5) can become close to the size δ approaches (1). of the deflection region. In this case there appears inter- 2 Let us note that the formula (3) was obtained for the mediate region of distances lC=π 112801-5 S. V. TROFYMENKO and N. F. SHUL’GA PHYS. REV. ACCEL. BEAMS 19, 112801 (2016) ΔW ∼ 100 MeV: It is necessary to note that 1 m separation between the plate and the detector means the measurement in the prewave zone. In this case additional equipment as, for example, a parabolic mirror focusing the radiation on the detector [21] could be used for eliminating the prewave zone effect (see the next section). Otherwise expressions (1) and (3) should be modified to take into account the considered effect. III. PREWAVE ZONE EFFECT FOR FIG. 5. Measurement of forward bremsstrahlung by a point BREMSSTRAHLUNG detector in the prewave zone. A. Spatial distribution of radiated energy A series of other interesting effects associated with the that the characteristic values of the transverse components manifestation of the nonequilibrium state of electromag- q of the wave vectors in the considered packet are much netic field produced by electron concern the influence of smaller than the absolute value k ¼ ω=c of the correspond- the size of the transverse radiation formation length upon ing wave vectors (which stipulates small values of radiation radiation characteristics measured in the prewave zone. Let angles θ). The magnetic field of the considered packet is us consider such effect for the case of forward brems- orthogonal to the electric one and equals it by the absolute strahlung emitted by a relativistic electron being deflected value. Thus, from (6) the expression for the spectral- to a large angle in an external field (the process considered angular density of the energy flux associated with this in the previous section) (Fig. 5). In this case by the field can be directly derived (see [16,17]): transversal (with respect to the direction of electron initial ∞ 2 velocity) formation length lT we mean the effective linear 2 2 2 Z 2 ð ρÞ 2 d W e z q J1 q − q cz transversal size of the region of space in which the Fourier- ¼ dq e i 2ω ; ð7Þ component with the wavelength λ of the electron’s dωdo π2c q2 þ ω2=c2γ2 0 Coulomb field is concentrated: lT ∼ γλ=2π. During the ’ electron s deflection this whole region is responsible for ð Þ radiation of the waves with the wavelength λ. where J1 x is the Bessel function which appeared in the For the case of 50 MeV electrons in the millimeter result of integration in (6) with respect to the angles ρ wavelength region of the radiated waves the longitudinal between q~ and ~ (which lie in the same plane). The expression (7) describes the value of the bremsstrahlung formation length lC is macroscopically large and the detector used for radiation registration may be situated spectral-angular density which can be obtained if the within this distance from the deflection point (in the measurement is made by a small (pointlike) detector ρ prewave zone). The electromagnetic field (packet of free situated in the point with the coordinates and z waves) which forms the forward bremsstrahlung pulse is (Fig. 5). Such detector registers the total radiation energy ω the part of the field around the electron outside the sphere (in the vicinity of the considered frequency ) which falls (as well as on its surface) of radius R ¼ ct with the center on its surface. As could be seen from (6) and (7), the ðρ Þ situated in the deflection point. After the particle deflection electric field in the point ;z consists of the components this field continues moving in the initial direction along the with different directions of the wave vector k~ (different q). z-axis. In the ultrarelativistic case the angles θ at which the The total electromagnetic energy in the considered point is most part of radiation is concentrated are much smaller than defined by the sum of the energies associated with each unit. At such angles the electric field of the discussed component with the given value of q as well as by packet can be considered transverse (having only compo- interference of the components with different q. It is such nent orthogonal to z-axis). A single-frequency component interference pattern which the point detector registers in the of this field can be presented in the form of the following prewave zone. In this case the measured radiation charac- Fourier integral [17]: teristics will significantly differ from the corresponding Z characteristics obtained by the measurements in the wave P ie iωz=c 2 q~ iq~ ρ~ −iq2cz=2ω zone. E~ ωðρ~;zÞ¼− e d q e : πc q2 þ ω2=c2γ2 Namely, in order to describe the results of the brems- strahlung spectral-angular distribution measurement by a ð6Þ point detector in the wave zone (z ≫ lC) the stationary Here the expansion of the expression in the exponent phase method can be used for calculation of the integral with respect to qc=ω is made. It is possible due to the fact in (7). As a result we obtain the well-known expression: 112801-6 FORMATION REGION EFFECTS IN TRANSITION … PHYS. REV. ACCEL. BEAMS 19, 112801 (2016) radiation field is a spherical wave diverging from the point in which the electron deflection occurred. The direction of the wave vector k~ in each point of the spherical wave front coincides with the direction of the radius-vector ~r of this point, which originates from the point of the electron deflection. Hence, in the considered case the angles α ¼ q=k and θ ¼ ρ=z are the same and the distribution of the radiated energy over α coincides with the one over θ. This is however not the case if the measurement is made in the prewave zone. Here the radiation field consists of both the spherically diverging wave (the field on the surface of the sphere) and a part of a Coulomb field outside the sphere. FIG. 6. Radiation angular distributions for different distances z The latter one is a superposition of the waves with different between the deflection point and the detector. Green line wave vector directions in arbitrary point of space. Therefore —z>γ2λ ¼ 30 m, red line—z ¼ 3 m, blue line—z ¼ 1 m. in the prewave zone there is no unique correspondence between the observation point position and the radiation dW e2 θ2 wave vector direction in this point. Each point contains a ¼ : ð8Þ “ ” dωdo π2c ðθ2 þ γ−2Þ2 whole bunch of wave vectors with different directions. This leads to difference between the results obtained with In [16,17] the analytical expression for the radiation the use of different approaches to radiation distribution spectral-angular density on very small distance from the measurement in the prewave zone considered here. deflection point was derived from (7). Here, on Fig. 6,we In order to measure the radiation energy distribution with present the results of numerical calculations with the use of respect to α each set of waves with the given value α ¼ (7) for larger values of this distance (however, still much less qc=ω should be collected together and separated from the than γ2λ). It illustrates the dependence of the radiation waves with the different values of α. As was shown in [21] angular distribution on the detector position along the for the case of TR, such problem can be solved by using a z-axis for γ¼100 and λ¼3mm calculated with the use of large parabolic mirror (Fig. 7). It collects all the waves with the expressions (7) (red and blue lines) and (8) (green line). wave vector directions parallel to its axis in its focal point Figure 6 shows that in the prewave zone the point (in which a pointlike detector can be placed). By rotating detector registers wider radiation angular distribution than such mirror (by changing the direction of its axis) it is in the wave zone. The angle of the radiationpffiffiffiffiffiffiffi distribution possible to collect the waves with different directions of θ ∼ λ maximum can be estimated as max =z, which in the wave vectors and thus measure the considered radiation prewave zone (z ≪ γ2λ) exceeds the corresponding value energy distribution. Therefore further we will use the 1=γ which can be obtained by measurements in the wave example of measurement with the use of a parabolic mirror zone (z>γ2λ). Let us note that the bremsstrahlung dis- in our investigation of the problem of the dependence of the tributions in the prewave zone presented on Fig. 6 resemble results obtained in the prewave zone on the size of the the corresponding distributions for TR presented in detector used for radiation registration.5 [18–22]. This fact is associated with the structural analogy The approach considered here provides certain advan- (see [16,17]) of the electromagnetic field generated in the tages for the measurement of radiation distribution in the considered bremsstrahlung process with the one produced case when the radiation formation length (prewave zone) is in the TR process discussed in these papers. rather large. In this case the measurements of radiation characteristics in the wave zone6 require large separations B. Radiation energy distribution over between the radiation source and the detector. It often wave vector directions Let us now consider a somewhat different approach to the 5In the prewave zone a pointlike detector registers all the waves problem of bremsstrahlung spectral-angular distribution with different wave vector directions arriving at the point of the measurement in the prewave zone. The idea of this approach detector location and, thus, measures radiation spatial distribu- presupposes the measurement of the radiation energy tion. By a detector of arbitrary finite size we mean a detector which collects the radiated waves with a certain wave vector distribution over components with different directions of direction, falling on its surface, and allows measuring radiation the wave vector (which is the distribution over α ¼ q=k, energy distribution over the wave vector directions (e.g., where α is the angle between the wave vector k~ and the z- detecting system including lens or collecting parabolic mirror). 6which is convenient (e.g., for beam diagnostics purposes) axis). Such approach coincides with the one considered since the radiation distribution (8) in the wave zone does not previously if the measurement is made in the wave zone of depend on frequency and the distance between the detector and the radiation process (z>γ2λ). Indeed, in the wave zone the the deflection point. 112801-7 S. V. TROFYMENKO and N. F. SHUL’GA PHYS. REV. ACCEL. BEAMS 19, 112801 (2016) equals the radiation angular (over θ) distribution (8) registered by a pointlike detector in the wave zone. Let us note that for the distribution (11) to be registered in the prewave zone the mirror does not need to be infinitely large. It should just significantly exceed the effective transversal size lT ∼ γλ=2π of the region responsible for radiation of the considered wavelength. In this case all the energy radiated at certain frequency is collected by the mirror and the distribution (11) is registered. The transversal radiation formation length lT is propor- tional both to the incident electron energy and the radiated wavelength. With the increase of the particle energy or at larger values of the considered radiation wavelengths lT FIG. 7. Parabolic mirror with the edge of radius ρ0 lying in the may reach and exceed the transversal radius ρ0 of the mirror ¼ plane z z0. used for the measurement of the radiation distribution in the prewave zone. In this case the problem of the radiation makes such investigations impossible because of the angular distribution measurement with the use of the mirror limited size of the laboratory, huge radiation background of arbitrary radius (which cannot be considered infinitely from the accelerator collected by the detector in this case large) arises. Let us consider this case more properly and etc. As was pointed out in [21], the use of a parabolic mirror derive the expression for radiation spectral-angular distri- in the prewave zone allows obtaining here the result (8) for bution registered in this case. the radiation angular distribution which a pointlike detector It might be thought that the considered angular distri- would register in the wave zone. Such effect is caused by bution in this situation is defined by the integrand in (9) (the the fact that the detector situated in the focal point of the expression in braces) as in the case ρ0 → ∞. However, this mirror collects all the waves with a certain wave vector is not the case. The point is that for arbitrary values of ρ0 direction α just like the detector situated in the wave zone in this integrand is complex and both its real and imaginary the direction θ ¼ α. Indeed, the expression for radiation parts are not positively defined (being integrated with distribution over α measured with the use of the mirror in respect to dΩ it becomes real and positive as the quantity the prewave zone can be obtained in the following way. dW=dω should be). This fact does not allow considering Integrating (7) with respect to do ¼ 2πθdθ (where such expression as the angular distribution of radiation θ ¼ ρ=z) we can calculate the spectral density of the total spectral density. The considered integrand with a certain radiation energy flux which falls on the mirror (the total value of α describes not only the amount of energy flux which traverses the circular region in the plane z ¼ z0 associated with the waves having such α. It also contains restricted by the edge of the mirror): the result of interference of such waves with the waves having different wave vector directions α0 in the plane Z Z 02 2 2 02 iz0ωα =2c dW e ω α 2 α e z ¼ z0. Such interference pattern disappears after averaging ¼ dΩ e−iz0ωα =2c dα0 dω π2c3 α2 þ γ−2 α02 þ γ−2 over the whole considered plane (which includes the 0 ρ0 ∞ “ ” Zρ0 integration with respect to < < ). Only diagonal α ¼ α0 ρρ ðωαρ Þ ðωα0ρ Þ ð Þ terms (with ), which define the energy density at × d J1 =c J1 =c ; 9 certain value of α, survive in this case and the expression 0 (11) for radiation energy distribution is obtained. For arbitrary values of ρ0 a somewhat different approach Ω ¼ 2πα α where d d . If the mirror has infinite transverse should be used for calculation of the bremsstrahlung ρ → ∞ ρ size ( 0 ) the integral with respect to gives spectral-angular distribution with respect to wave vector δðα − α0Þ 2 ω2α δð Þ δ c = , where x is the -function and the directions registered by a pointlike detector situated in the expression (9) can be presented in the following form: focal point of a collecting mirror. Such approach consists in Z 2 α2 calculation of the part of radiated energy with a certain dW ¼ e Ω ð Þ λ 2 d 2 −2 2 ; 10 wavelength , which is reflected by the mirror of finite size. dω π c ðα þ γ Þ It should be presented in the form of an integral with respect to wave vector directions (with respect to dΩ). The which shows that the distribution of the radiated energy integrand in this case will be the considered distribution. α over the directions of the wave vectors Let us note that the amount of energy carried by the ~ 2 α2 electromagnetic waves with the certain wave vector k dW ¼ e ð Þ 2 2 −2 2 11 collected by the mirror in the focal point can be calculated dωdΩ π c ðα þ γ Þ in the following simplified way. Namely, it can be 112801-8 FORMATION REGION EFFECTS IN TRANSITION … PHYS. REV. ACCEL. BEAMS 19, 112801 (2016) considered as the amount of energy carried by the waves Fourier components. Making the Fourier decomposition of with the wave vector k~, which are reflected from an the relation (14) with respect to ρ~, we can finally obtain the ¼ R imaginary circular plane mirror in the plane z z0 with quantity E~ q;ω in the following form: the radius ρ0. Such mirror is the projection of the parabolic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2 2 02 ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 02 iz0 ω =c −q surface of the real mirror on the plane z z0. Such 8π ie q~ 2 2 2 q e R iz0 ω =c −q 0 E~ ω ¼ − e dq imaginary substitution of the parabolic mirror by the plane q; c q q02 þ ω2=c2γ2 one simplifies the calculations of the radiation field Zρ0 reflected from the mirror. The plane mirror, however, does ρ0ρ0 ð ρ0Þ ð 0ρ0Þ ð Þ not collect all the waves with a definite wave vector × d J1 q J1 q : 15 direction in a singular point on finite distance from itself. 0 Therefore in order to calculate the distribution of the As was pointed out before, in the ultrarelativistic case it reflected radiation energy over the wave vector directions 0 is possible to make a decomposition in such expression it is necessary to consider the angular (over θ , see Fig. 7) for radiation field with respect to small radiation angle distribution of such radiation in the wave zone (in the α ¼ qc=ω. Substituting (15) into (13) and integrating it region z<−γ2λ). As was pointed out, such angular with respect to the angle between q~ and ρ~ we obtain the distribution coincides with the distribution of the reflected following expression for the field of the reflected radiation: part of the bremsstrahlung energy over the wave vector directions α. R e ρ~ Let us consider the problem of impinging of the wave ~ ðρ Þ¼2 −iωðz−2z0Þ=c Eω ;z e ρ packet (6), radiated by the electron at its deflection, upon cZ ρ ¼ 2 the circular mirror of radius 0 situated in the plane z z0 iq cðz−z0Þ=2ω × dqqJ1ðqρÞFðq; ρ0Þe ; ð16Þ with the center lying on the z-axis. The total field in space P in this case consists of the field of the considered packet E~ ω R where and the field E~ ω reflected from the mirror. The latter one ∞ ρ can be presented in the following form [1,31]: Z 2 Z0 η2 −iz0η c=2ω Z ð ρ Þ¼ η e ð Þ ðη Þ ð Þ 1 F q; 0 d 2 2 2 2 dxxJ1 qx J1 x : 17 R 3 R 2 2 2 ik~~ r η þ ω =c γ E~ ωðρ~;zÞ¼ d kE~ ωδðω =c − k Þe ; ð Þ ð2πÞ3 k; 12 0 0 The field (16) is a packet of free waves with different where δ-function restricts the integration region by the directions α ¼ qc=ω of the wave vector k~. In order to obtain values of k satisfying the dispersion relation for electro- the distribution of the reflected radiation energy over α we magnetic waves in vacuum. The integration of (12) with have to calculate the radiation angular distribution with respect to kz leads to the following representation of the 0 respect to θ ¼ ρ=jz − z0j (see Fig. 7) in the wave zone for field of the packet of the reflected waves: 2 the reflected radiation field (for z<−γ λ). Such distribu- Z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tion will coincide with the required one since in the wave R 2 R i~q~ρ −iz ω2=c2−q2 0 E~ ωðρ~;zÞ¼ d qE~ ωe ; ð Þ α ¼ θ ð2πÞ3 q; 13 zone it is only the wave with the direction of the vector k~ that arrives at the observation point in the direction θ0 ~ R ¼ ~ R j2 jj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi defined by . where we denoted Eq;ω Ek;ω= kz ¼ ω2 2− 2 . kz =c q The reflected radiation angular distribution in the wave The total field should satisfy the boundary condition on zone can be obtained with the use of the asymptotic R the surface of the mirror, where it should equal zero. expression for E~ ω at large negative values of z. It can be Outside this surface (for z ¼ z0 and ρ > ρ0) the total field derived from (16) with the use of the stationary phase in the considered ultrarelativistic case can be approximately method. As follows from (9) and (10), such distribution can set to equal the field of the incident packet (6). This leads to be also calculated in a simpler way by integration of the the following form of the boundary condition: expression for reflected radiation spectral-angular density7 R P 2 2 E~ ωðρ~;z0Þ¼−hðρ0 − ρÞE~ ωðρ~;z0Þ; ð14Þ d W cðz − z0Þ R 2 ¼ jE~ ωðρ;zÞj ð18Þ dωdo0 4π2 where hðxÞ is the Heaviside step function, which equals 0 0 0 2 unit for x>0 and zero for x<0. with respect to do ¼ 2πθ dθ ¼ 2πρdρ=ðz − z0Þ in the In order to derive the expression for a single Fourier limits 0 < ρ < ∞. The result of such integration should be ~ R component Eq;ω in the decomposition (13) of the reflected ¼ ωα 7 0 0 0 field over wave vector directions (since q =c)we For θ ≪ 1 we use the approximation r ≈ jz − z0j, where r is should rewrite the condition (14) in the terms of such the distance from the mirror center to the observation point. 112801-9 S. V. TROFYMENKO and N. F. SHUL’GA PHYS. REV. ACCEL. BEAMS 19, 112801 (2016) presented in the form of an integral with respect to Coulomb field of the electron moving along the z-axis. dΩ ¼ 2παdα. In the considered case this gives: Thus the distribution (21) coincides with the spectral- angular density (measured in the wave zone) of backward 2 2 Z dW e ω 2 TR produced by the electron during its normal traversal of a ¼ dΩjFðq; ρ0Þj : ð19Þ dω π2c3 circular conducting mirror of radius ρ0 [31]. Let us now simplify the expression (20) taking into From (19) we finally obtain the required distribution for account the dispersion of the field (6) (its difference from the reflected radiation energy: the Coulomb field) more accurately (1-approximation). The integral with respect to η can be identically presented as Zρ0 Z∞ 2 2 2 4 2 Z∞ d W e w η J1ðwηρÞ − η2 2 2 ¼ ρρ ð αρÞ η iwz0 = η J1ðwηρÞ 2 2 d J1 w d 2 −2 e ; η −iwz0η =2 ¼ − ð Þ dωdΩ π c η þ γ d η2 þ γ−2 e P1 P2; 22 0 0 0 ð20Þ where ¼ ω ¼ 2π λ ∞ where w =c = . The result (20) is calculated as the Z 2 2 2i wρ 2 energy distribution of radiation reflected from a circular −iwz0η =2 iwρ =4z0 P1 ¼ dηJ1ðwηρÞe ¼ − sin e plane mirror of radius ρ0. It describes the bremsstrahlung wρ 4z0 energy distribution with respect to wave vector directions 0 measured with the use of the parabolic mirror of arbitrary and transversal radius ρ0 situated on arbitrary distance z0 from Z∞ 1 J1ðwηρÞ 2 the point of electron deflection to a large angle. −iwz0η =2 P2 ¼ dη e : In the case ρ0 → ∞, the expression (20) coincides with γ2 η2 þ γ−2 the result (11) for the radiation distribution obtained with 0 the use of the mirror of infinite transverse size. The values of η which mainly contribute to the integral For arbitrary values of ρ0 the integrals in (20) can be P2 do not exceed 1=γ. For such values of η in the prewave calculated numerically. Nevertheless it is possible to zone the expression in the exponent is much less than unit ρ simplify the expression (20) if 0 is not much smaller than (for arbitrary ρ) and the exponent can be set to equal unit. l ∼ γλ=2π. Indeed, the integral with respect to η converges T The integral P2 in this case is defined by the following both due to the presence of the exponent and the corre- expression: sponding Bessel function depending on η in the integrand. The latter one defines the values of η which make the main P2 ¼ 1=wρ − K1ðwρ=γÞ=γ: η η ∼ 1 ρ contribution to the considered integral as < max =w . At such values of η the expression in the exponent is of the Substituting (22) into (20) we can finally present the 2 2 order of z0=2wρ . In the prewave zone z0 ≪ γ λ and the expression for the radiation distribution in the following considered expression is much less than ðγλ=2πρÞ2. Hence form: for the values of ρ of the order of γλ=2π the exponent in d2Wð1Þ e2 α2 (20) can be set to equal unit. In this case all the integrals in ¼ j ð0Þ þ ð1Þj2 ð Þ ω Ω 2 2 −2 2 S S ; 23 (20) can be calculated analytically and for the radiation d d π c ðα þ γ Þ distribution we obtain: where 2 ð0Þ 2 2 d W e α ð0Þ 2 ¼ j j ð Þ Zρ0 2 2 −2 2 S ; 21 dωdΩ π c ðα þ γ Þ w 2 ð1Þðα ω ρ Þ¼− ðα2 þ γ−2Þ ρ ð αρÞ iwρ =2z0 S ; ; 0 α d J1 w e : where 0 ρ ð0Þðα ω ρ Þ¼1 þ w 0 ð αρ Þ ð ρ γÞ S ; ; 0 γ J2 w 0 K1 w 0= Figures 8 and 9 represent the distributions of the bremsstrahlung spectral density with respect to the wave wρ0 − ð αρ Þ ð ρ γÞ vector directions, which can be measured with the use of a γ2α J1 w 0 K2 w 0= parabolic mirror of a certain transverse radius ρ0, for and KðxÞ is the Macdonald function. different values of ρ0 and the mirror position z0 along The approximation which is used here (we will call it the z-axis. The distributions are calculated for the case of 0-approximation), in which the exponent in (20) is set to 3 mm radiation wavelength and the energy of the incident equal unit, corresponds to the situation when the dispersion electrons equal to 50 MeV (γ ¼ 100). In this case the size of the wave packet (6) ‘torn away’ from the electron at its of the longitudinal formation length can be estimated as 2 scattering is neglected. In such case this field equals the lC ∼ γ λ ∼ 30 m, while the size of the transversal one 112801-10 FORMATION REGION EFFECTS IN TRANSITION … PHYS. REV. ACCEL. BEAMS 19, 112801 (2016) FIG. 8. Radiation distribution with respect to α for γ ¼ 100, λ ¼ 3 mm, z0 ¼ 20 cm and (a) ρ0 ¼ 3 cm, (b) ρ0 ¼ 10 cm. Red lines— numerical calculation with the use of expression (20), blue lines—0-approximation (21), dashed lines—1-approximation (23), green line—asymptotic distribution (11) for ρ0 ≫ γλ=2π. lT ∼ γλ=2π ∼ 5 cm. The results of precise numerical cal- this case is still nearly twice as large as the corresponding culations on the basis of the expression (20) as well as the position of the asymptotic distribution maximum. approximate results obtained from formulas (21) and (23) The presented figures show that for the considered are presented here. values of z0 ≪ lC the formula (23) approximates very well The figures show that for the values ρ0 FIG. 9. The same as in Fig. 8 but for z0 ¼ 100 cm. 112801-11 S. V. TROFYMENKO and N. F. SHUL’GA PHYS. REV. ACCEL. BEAMS 19, 112801 (2016) electromagnetic processes during the interaction of such emission from solid into gas. In this case at the very electron with substance. Previously (Sect. II) we considered beginning of its motion through gas the particle still has such modification with the use of the example of TR significantly screened electromagnetic field (as it had in generated by the electron which was partly “undressed” in solid) which lacks part of Fourier components comparing to the result of deflection. The electron’s field may also the equilibrium field of the particle in gas. The overdressed become significantly perturbed if the electron moves case corresponds to the opposite situation of the particle through substance with inhomogeneous dielectric proper- entrance to solid from gas or vacuum. In such case the ties. The simplest situations corresponding to such case particle still has vacuumlike field around it in the boundary include the electron entrance into a substance from vacuum layer of the solid. and its emission to vacuum from substance. In this case the reconstruction of the electromagnetic field around the A. Overdressed case electron occurs which leads to generation of TR. Such According to theoretical predictions by Garibian [32], reconstruction, however, can influence upon the electron the boundary effects may cause the total absence of the ionization loss in the vicinity of the boundary of the density effect for particle ionization loss in sufficiently thin substance as well. Such influence is mostly pronounced plates at arbitrary energies of the particle. So far only a in sufficiently thin layers of substance, in which the single experimental investigation [33,34] confirming this boundary effects are significant within the whole volume effect was made and it has never been repeated in other of the layer. laboratories.8 A series of similar experiments was made in The value of the low-energy relativistic particle ioniza- connection with investigation of characteristic X-ray emis- tion loss during its motion through the substance is defined sion during electrons traversal of thin plates (see references by the Bethe-Bloch formula, according to which the in [36] dedicated to theoretical consideration of this ionization loss grows logarithmically with the increase problem), however not with investigation of the restricted of the particle energy. This happens due to relativistic ionization loss. increase of the maximum impact parameters ρ ∼ γ c=I max Thorough experimental investigation of Garibian effect (I is the mean ionization potential of the atoms of the (suppression of the density effect in thin plates) is desirable substance) at which the particle still effectively interacts since thereby the process of the particle electromagnetic with the atoms. However, at higher energies partial screen- field reconstruction during its traversal of the boundary ing of the particle’s own field by the polarization of the between different media can be visualized and the influence substance (comparing to the vacuumlike field which is used of this reconstruction upon ionization loss could be studied. for derivation of the Bethe-Bloch formula) leads to satu- ρ ρ ∼ ω ω Such information might be valuable for elaboration of ration of max to the value max c= p ( p is the plasma ultrathin detectors for noninvasive particle identification frequency of the substance) and further independence of purposes [23,24]. this quantity and of ionization loss on the particle energy. Among one of the controversial questions in this respect Such effect is known as the density effect. The particle is the question about the distance a from the border (or, ionization loss in this case is defined by the Fermi formula analogously, the thickness of thin plate) for which the (or the Bethe-Bloch formula with the additional term taking boundary effect is significant for ionization loss. On the one into account the density effect). hand, this distance can be thought of as the distance within Let us note that we consider here the so-called restricted which the transition radiation contribution to ionization of ionization loss. It is defined by the particle collisions with the substance is significant. In this case a is defined by the atoms at which the momentum transfer does not exceed absorption length of TR photons of frequencies which some maximum value q0. For relativistic electrons in make the main contribution to ionization and should not solids, for instance, the density effect becomes significant depend on the particle energy. On the other hand, according at the energies of the order of 10 MeV. Bethe-Bloch and to Garibian’s estimations, this distance should grow log- Fermi formulas are valid in homogeneous media in which arithmically with the increase of the particle energy. This the effect of finiteness of the medium (the influence of the may occur, for instance, if, along with absorption, TR boundary conditions upon the ionization loss) is negligible. formation effects play their role in this process. This is however not the case for sufficiently thin layers of The absorption length of about 100 eV photons (those substance. Namely, if a particle traverses the border with frequencies of the order of mean ionization potential) between two different media the reconstruction of the in solids is estimated as ∼10−5 cm. Therefore in order to ’ particle s electromagnetic field takes place. Such field measure the distance a a series of plates of different reconstruction leads to TR generation which changes the value of the particle ionization loss in the vicinity of the 8Let us note that recently the investigation of such kind at much border. In this case we deal with ionization loss of either higher particle energies was also reported in [35].However,very half-bare or overdressed particle. The half-bare situation thick plate (535 μm thick silicon detector) was used here and the may take place, for example, in the case of the particle observed effect was very small. 112801-12 FORMATION REGION EFFECTS IN TRANSITION … PHYS. REV. ACCEL. BEAMS 19, 112801 (2016) thicknesses in the range L ∼ 10−6–10−4 cm is needed (in B. Half-bare case the experiments [33,34] luminescent polystyrene foils of An interesting case concerning ionization loss of an ≪ 20 μ ≫ thickness 10 nm ( a) and m( a) were used). The electron with nonequilibrium field can take place during dependence of ionization loss on the plate thickness is the the electron normal traversal of a system consisting of necessary data on the basis of which the considered two plates of different thickness (Fig. 10) situated in distance could be directly determined. vacuum.9 Since the predicted energy dependence of a is rather Let the plate thicknesses satisfy the conditions a1 ≫ a weak, in order to investigate it, it is necessary to have the and a2 However, from the known value of the most probable ð2Þ 2 dε ω e q0vγ 1 ω γ ionization loss in thin layer of silicon dε=dz ≈ ¼ p ln − þ ln p −1 30 keV=100 μm (which grows faster than linearly with dz v I 2 I the increase of the layer thickness) and the fact that the − cos zpCiðzp þ zγÞ − sin zpSiðzp þ zγÞ average loss in thin layer exceeds the most probable one we may assume that the considered value should be higher. þ CiðzγÞþzγSiðzγÞþcos zγ ; ð25Þ For the thinnest plates proposed to be used in the experiment (L ∼ 10−6 cm) the total signal from a bunch 8 of n ¼ 10 electrons (provided the signals from single 9This process is analogous to the one studied in [37,38] where particles can be integrated) should be ∼200 MeV. the thicker plate was considered as a semi-infinite medium. 112801-13 S. V. TROFYMENKO and N. F. SHUL’GA PHYS. REV. ACCEL. BEAMS 19, 112801 (2016) FIG. 11. Dependence of the electron ionization loss in the second (thin) plate on the distance between the plates for the electron energy 500 MeV. FIG. 10. Schematic picture of evolution of the electron’s field during its traversal of a system of two plates. 2 At high energies the distance lC ∼ γ c=I, within which the considered change of the ionization loss takes place, has ð1Þ2 2 where zp ¼ z0ωp =2cI, zγ ¼ Iz0=2γ c, I—is the mean large macroscopic size (for 100 GeVelectrons, for instance, ionization potential of the atoms of the second plate, v is the it can reach 10 m). The minimal electron energies at which ð1;2Þ such change could still be observed may be considered as electron velocity, ωp are the plasma frequencies of the plates and SiðxÞ and CiðxÞ–are Sine and Cosine integrals. several hundreds MeV. For example, for 500 MeVelectrons Here we see that the value of electron ionization loss in thin the distance lC is about 1 mm and it is still possible to make measurements with the use of thin plate of submicrometer plate significantly depends on the distance z0 between the plates. thickness situated on different distances within lC. From (25) it follows that in the case when the separation Figure 11 shows the dependence of the electron ioniza- ð1Þ tion loss in the second plate on the distance z0 between the between the plates satisfies the condition z0 112801-14 FORMATION REGION EFFECTS IN TRANSITION … PHYS. REV. ACCEL. BEAMS 19, 112801 (2016) produced by an electron which had undergone deflection principle, using the same detecting facility for such study as from the initial direction of motion and became half-bare. in the case of the considered experiments. The special attention was drawn to the consideration with the use of realistic (namely, circular) trajectory of the ACKNOWLEDGMENTS electron motion in the deflection process. It was shown that the TR spectral-angular distribution noticeably differs from The research is conducted in the scope of the IDEATE the one obtained with the use of previously considered International Associated Laboratory (LIA). The work was [16,17] approximation of the anglelike trajectory (instant partially supported by the project No. 0115U000473 of the deflection) in the case when the size of the deflection area is Ministry of Education and Science of Ukraine, by the comparable to the distance between this area and the project CO1-8/2016 and the project for young researchers radiating foil. The numerical estimations of the radiation (contract M63/56-2016) of the National Academy of yield are made for the beam parameters which approx- Sciences of Ukraine and by the project F64/17-2016 of imately correspond to the ones planned to be achieved at State Fund for Fundamental Research of Ukraine. The ThomX injector at LAL Orsay. The obtained results might authors are grateful to S. Barsuk, O. Bezshyyko, L. be important in connection with the problems of TR-based Burmistrov and N. Delerue for useful discussions. diagnostics of the extracted beams. In the next section the theoretical aspects of the problem of measurement of bremsstrahlung characteristics in the prewave zone is considered. Following [19,21] we distin- guish two different approaches to measurement of the [1] M. L. Ter-Mikaelyan, High-Energy Electromagnetic Processes in Media (Wiley, New York, 1972). radiation spectral-angular distribution in the prewave zone. [2] A. I. Akhiezer and N. F. 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