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Ratio of the contributions real and virtual photons diffraction in thin perfect crystals. Comparison of calculation and experiment
Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx
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Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Ratio of the contributions real and virtual photons diffraction in thin perfect crystals. Comparison of calculation and experiment Yu.A. Goponov, S.A. Laktionova, M.A. Sidnin, I.E. Vnukov ⇑ Belgorod National Research University, Belgorod, Russia
a r t i c l e
i n f o
Article history: Received 9 December 2016 Received in revised form 7 February 2017 Accepted 24 February 2017 Available online xxxx Keywords: Thin crystal Fast electron Parametric X-ray radiation Diffracted transition radiation Diffracted bremsstrahlung
a b s t r a c t To evaluate and improve the previously proposed method of calculating diffracted photon yields in thin perfect crystals, a comparison between calculated and experimental results in wide range of photons and electrons energy was carried out. It is shown that the proposed method describes all investigated experimental results for bremsstrahlung diffraction and transition radiation one with an error less than tenfifteen percent. Consequently, the method may be used for calculation of the electron beam divergence influence on the diffracted transition radiation angular distribution. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction Beam size and angular divergence are one of the most important parameters in the field of accelerators, and many beam diagnostic methods have been developed. For electron accelerators, one conventional method is to use optical radiation such as fluorescent light, optical transition radiation (OTR) [1], optical diffraction radiation (ODR) [2], and Smith-Purcell radiation [3]. However, it was recently found that radiation in optical region cannot be used for profile measurements of an electron beam at modern accelerators and the projected electron–positron linear colliders such as International Linear Collider (ILC) [4] and Compact Linear Collider (CLIC) [5] because it becomes coherent when the bunch size of the beam is sufficiently small [6] in comparison with the measured photons wavelength. In order to avoid the coherence, photons with shorter wavelength are required. Some years ago, the use of radiation in the X-ray region, the so-called parametric X-ray radiation (PXR) were proposed [7,8]. To a first approximation, PXR can be considered as coherent scattering of the charged particle’s electromagnetic field on the electron shells of periodically arranged atoms in a target, see e.g. [9] and reference therein. It is emitted in the Bragg direction when a relativistic charged particle moves across a crystalline target. Recently, proof-of-principle experiments on beam
⇑ Corresponding author. E-mail address: [email protected] (I.E. Vnukov).
profile measurements using PXR have been demonstrated at SAGA Light Source (SAGA-LS) [8,10] and Mainzer Microtron (MAMI) [11]. For fast electrons PXR is always accompanied by radiation diffracted in the crystal which is born directly inside the target or on its surface [12,13]. In the first case we can talk about diffracted bremsstrahlung (DB) and in the second one about diffracted transition radiation (DTR). The first is dominated by under the condition x cxp , where x is photon’s energy, c-Lorentz factor, and xp -plasma frequency of the medium, [14,15] and the second under opposite conditions [13]. If condition x cxp is true, the contributions of both mechanisms of radiation are observed [12]. Method for calculation of real photons diffraction contribution into measured photon yield was suggested in Ref. [16]. In Ref. [17] using the method [16] we have shown that for high electron energy the DTR angular density becomes essentially larger than PXR one, therefore the DTR angular distribution measurements are more preferable for estimation of high energy electron beam parameters in comparison with PXR one. Advantage of a beam diagnostic method using diffracted transition radiation for future electron–positron colliders [4,5] is connected with the fact that the beam divergences in both planes are not so small (of the order of tens lrad) and is by far greater than the transition radiation characteristic angle c1 . Therefore measurements of DTR angular distributions in both planes provides estimation of electron beam divergences in these planes and the electron beam spatial sizes if we know the beam emittance which may be calculated or measured on the previous stage of acceleration [18].
http://dx.doi.org/10.1016/j.nimb.2017.02.068 0168-583X/Ó 2017 Elsevier B.V. All rights reserved.
Please cite this article in press as: Y.A. Goponov et al., Ratio of the contributions real and virtual photons diffraction in thin perfect crystals. Comparison of calculation and experiment, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.02.068
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The transition radiation arises when a fast charged particle crosses an interface of media with different dielectric constants. If one medium is a single crystal and another is vacuum, transition radiation is produced at the entrance surface of the crystal and then a part of its satisfying Bragg’s law is diffracted by the crystal target and emitted in the Bragg direction. PXR is generated in the crystal volume, therefore the DTR contribution in comparison with PXR one is comparatively large for thin crystals only. As it was shown in Ref. [19] the DB contribution may be also clearly observed for thin crystals only. Contribution of real photons diffraction was observed not only in the above mentioned experimental works, however quantitative comparison between calculations and experiments up to date was not made jet. Consequently, it is clear that a comparison of the results of total emission angular distribution measurements in thin crystals with calculations taking account all emission mechanisms and the experiments features is important and relevant. 2. Calculation technique As it was remarked above the main goal of the paper is to compare known results of the experiments devoted to investigation of X-ray emission from the thin crystal irradiated by fast electrons with calculation in accordance with the method proposed in Ref. [16]. It takes into account such types of the electron emission in crystals as PXR, DB and DTR. The special features of the method are following: (a) The kinematic PXR theory [20] is used; (b) The bremsstrahlung suppression due to the density effect [21] is taken into account; (c) The so-called Garibian formula for the transition radiation (TR) spectral-angular distribution [9] is used. It is supposed that the TR is generated directly at the inlet into crystal and then it is diffracted therein. TR photons polarization isn’t taken into account; (d) The electron beam divergence and multiple scattering in the crystal, the emission collimation angle and other experimental conditions are taking into account in accordance with the methodology of Ref. [22]. The main peculiarity of the method [16] is the definition of the X-ray reflectivity in a perfect crystal. As it is well known the angular distribution of diffracted radiation relative to the center of reflex along axis x(see, for example, [12]) can be represented as:
Z
Z Y DR ðx; hx Þ ¼
dx
d I Rðx; ~ n; ~ g ; HD ÞSðx; ~ nÞdX; dxdX 2
because of the refraction of waves in a crystal, d ¼ ðxp =xÞ2 =2 is a difference between the refractive index from 1, and g ¼ f ð~ g Þð1 þ cosð2HB ÞÞ=2f ð0Þ. f ð~ g Þ is the Fourier component of the spatial distribution of electrons in a crystal atom. The characteristic parameter of the model is the primary extinction length, which is dependent on the photon energy, reflection order and the parameters described above, and may be written as [23]:
lex ¼ d=ð2n sin HB Þ;
ð2Þ
where dis an inter planar distance, and expð2 nÞ is an impairment of the intensity of the primary wave as it passes through a plane with the reciprocal lattice vector ~ g. For the part of the crystal with a thickness much smaller than lex , the probability of reflection of photons with an energy of x and direction ~ n, for which the Bragg condition is satisfied, is proportional to the number of planes crossed by them [23]. Therefore, the dependence of the number of photons, which have not undergone the reflection, on the way length in the crystal t can be written as N c ðtÞ ¼ N c ð0Þ expðt=lex Þ [23], where N c ð0Þ is the number of photons at a starting point. Taking into account the Bragg reflection, photon absorption and scattering on atoms the dependence of the number of photons on passable way can be rewritten as:
Nc ðx; ~ n; tÞ ¼ Nc ð0Þ expðltot ðx; ~ g; ~ nÞtÞ;
ð3Þ
where ltot ðx; ~ g; ~ nÞ ¼ lðxÞ þ ldif ðx; ~ g; ~ nÞ is a total coefficient of linear absorption of radiation with energy x, for the direction of the crystal reflecting plane ~ g and the photon velocity direction ~ n. Here lðxÞ is a linear absorption coefficient of photons due to the all processes on separate atoms and ldif ðx; ~ g; ~ nÞ ¼ 1=lex ðx; ~ g; ~ nÞ is due to diffraction. The possibility of such notation allows us to use a well-known in experimental physics method of statistical simulation of photon transmission through a matter for perfect crystals also, see [16] for details. We may ignore the influence of the radiation polarisation on the diffraction process for small Bragg angles only when cosð2HB Þ is about one. In this case photons with r and p polarization are diffracted with close efficiency and have the same extinction length value. In the contrary case the extinction length for photons with different polarization differs. As the transition radiation is polarized in outlet plane [9] the simulation process was made for both polarization components separately.
ð1Þ
d2 I dxdX
where is a spectral-angular distribution of the radiation, taking into account the divergence of the primary electron beam, multiple electron scattering an so on. Rðx; ~ n; ~ g ; HD Þ is reflectivity for these directions of the vectors ~ n and ~ g , defined the crystal orientation n is the individual angle H and the location of the detector HD . Here, ~ vectors corresponding to the initial photon (with the energy x), ~ g is the reciprocal lattice vector. Sðx; ~ nÞ is a function taking into account the photon absorption in the crystal and the geometry of the experiment. To determine the output of the diffracted radiation we need information about Rðx; ~ n; ~ g ; HD Þ. In accordance with [23] for a fixed photon direction ~ n from the beam with the spectral-angular distribution of d I =dxdX satisfying Bragg’s condition for photons with energies of x only photons in the energy range Dx ¼ x cosðHB Þ= sinðHB ÞDH will be reflected. For unpolarized 2
radiation and lack of absorption DH ¼ 2 gDh0 , where Dh0 ¼ 2 d= sin 2HB is an amendment to the Bragg angle HB
3. Comparison of calculation and experiment results The main aim of this study is to compare results of the experiments where contribution of real photon diffraction was clearly observed with the calculation in accordance with techniques reported in Ref. [16] and briefly described in the previous section. These are experiments with thin crystals, strong emission collimation and known values of photon yield [12,14,15] and measurements for large observation angles, where PXR contribution and diffracted real photons one are separated by the emission angle relatively Bragg direction [13]. The experiment [14] was performed at Kharkov linear accelerator. Electron beam energy of 900 MeV with divergence #e 0:2 mrad hits on a silicon crystal thickness of 30 lm. Reflection (2 2 0) in Laue geometry was being investigated. The system of detecting with the round aperture diameter of 5 mm was located at a distance of 8.2 m from the crystal at an angle
Please cite this article in press as: Y.A. Goponov et al., Ratio of the contributions real and virtual photons diffraction in thin perfect crystals. Comparison of calculation and experiment, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.02.068
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HD ¼ 2HB ¼ 17:8 mrad. Only the spectral-angular density of emission was measured in the experiment. Its value was about 0.3 photon/(MeV*electron*sr.). Calculated value is equal to 0.314 photon/ (MeV*electron*sr.). For this experimental condition x 350 keV cxp ¼ 54:2 keV, therefore only PXR and DB contributions was observed. DB contribution was about 15% from the total radiation intensity maximum value. This result is a rather interesting because of the crystal thickness was by far less than the extinction length for this crystal orientation and the photon energy, therefore simple method suggested and used in Ref. [13,19] is invalid. The experiment [15] was performed at electron linac LUE-40 of Kharkov Institute of Physics and Technology for particles energy of 15.7 and 25.7 MeV. The beam divergence was about 1 mrad. Silicon crystal thickness of 17 lm installed in Laue geometry was used. Reflection (1 1 1) in Laue geometry was being investigated. The semiconductor Silicon detector with round collimator Ø = 5 mm was placed at the observation angle HD ¼ 305:9 mrad at 4520 mm from the target. Measured theta-scans for both energy of electrons and photons of the first reflection order are presented on the Fig. 1– points. Calculated dependencies of the parametric X-ray radiation and the diffracted bremsstrahlung and total radiation yield calculated by the technique [16], so these are the curves 1–3, respectively. For the experimental conditions [15] the condition x 13 keV cxp 1 keV for the both electrons energy is satisfied, so we can observe PXR and DB contribution only.
3
In accordance with the PXR theory a virtual photons angular distribution in a medium has a minimum along an electron movement direction, see. e.g. [9], and we should observe two peaks and a deep between them (see curve 1). The deep position corresponds strong execution of Bragg’s law H ¼ HD =2, where H is a plane misalignment angle accordingly electron beam direction. DB yield has maximum value for this crystal orientation (see curve 2), therefore the experimental yield in this orientation angle is greater than the PXR theory prediction and agrees well with the calculation result including both mechanism of photon emission (see curve 3) for electron energy of 15.7 MeV (see Fig. 1a). Difference on the right part of the theta-scan may be connected with interference between PXR and coherent bremsstrahlung [24] which was not taken into account. Agreement for another energy is worse (see Fig. 1b). For execution of Bragg’s condition measured and calculated results agree well, but for PXR region, outside the deep in the theta-scan, experimental photon yield is greater than the calculation results. One of the possible reason may be contribution of the so-called diffracted channelling radiation (DCR) [25]. This type of emission may arise when a fast particle moves simultaneously along a crystallographic plane in the planar channelling regime and emits parametric X-ray on another plane. If the photons energy for both mechanisms are close we may observe DCR as a result of interference between PXR and channelling radiation. It was discussed in some theoretical papers [26,27], but aren’t observed up to date, see [28] for explanation.
Fig. 1. Theta-scan of the of the X-ray yield for [15] condition and the first reflection order for electrons energy of 15.7 MeV (a) and 25.7 MeV (b).
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The theta-scans in the experiment [15] were measured for the electron motion along the (1 1 2) plane. It is likely that for electron energy of 25.7 MeV the (1 1 2) planar channelling radiation energy was about the photon energy of the first order reflection and DCR was observed. For another energy of electrons the (1 1 2) planar channelling radiation energy was less then the PXR one, therefore the measured and calculated results was close. The experiment in the photon energy region x cxp ¼ 51:5 keV, where the DB and DTR contributions into measured emission yield was comparable and clearly observed, was performed at MAMI [12] for electron energy of 855 MeV with a silicon crystalline target thickness of 124 lm and a Si step shaped target with 6 steps of 1 mm width and 100lm height. The crystals surface for both targets was perpendicular to the h1 0 0i axis. The measurements were done with a semiconductor detector which was placed at the angle HD ¼ 45 at distance 1001 mm. The round collimator aperture was equal to 1 mm. Horizontal angular distributions of total emission for the (1 1 1) and (3 3 3) reflection orders and total emission photon flux in the maximum of angular distributions for three investigated crystal orientations were presented in the analysed paper [12]. Results of the photon flux measurements for the investigated reflection orders and our calculation ones are presented in the table. Photon absorption in the entrance window of the detector was not taken into account under calculation procedure.
Order (1 1 1) (2 2 0) (2 2 4)
Measured flux [12] (4.5 ± 0.5)10
3
(6.5 ± 0.8)10
3
(5.4 ± 0.5)10
3
Calculated flux
5.1103 phot./(e.*sr.)
6.7103 phot./(e.*sr.)
5.7103 phot./(e.*sr.)
phot./(e .*sr.) phot./(e .*sr.) phot./(e .*sr.)
As can be seen from the table the measured and calculated values agree a rather well. Difference for the (1 1 1) reflection order with x 5:16 keV may be connected with absorption of the photons in the detector entrance window. Calculation shows that intensity of diffracted real photons for (2 2 0) and (2 2 4) reflection are about a quarter of PXR yield maximum value. For the (1 1 1) reflection this contribution is almost negligible because of small value of the absorption length ła 20 lm in comparison with the targets thickness. Transition radiation which was born on the entrance surface of the crystal is almost absorbed. Fig. 2 shows the calculated horizontal angular distributions for the (3 3 3) reflection order with x 15:7 keV and all emission mechanisms (Fig. 2a) and comparison between calculated and measured flux angular dependencies for the experiment [12] condition (Fig. 2b). From the Fig. 2a it is seen that PXR (curve 1) possesses bigger intensity than the DTR and DB (curves 2 and 3), and its angular distribution is broader. The multiple scattering effect of electrons
Fig. 2. Horizontal distribution of the X-ray yield for the [12] condition. a- Calculated dependences. Curve 1- PXR; curve 2 – DTR; curve 3 – DB and curve 4 -total emission yield. b – Comparison between measured angular distribution [12] – points and calculated one – line.
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leads to some swelling of the PXR distribution minimum. The absorption length la 450 lm is greater than the target thickness, therefore the DTR contribution is about a half of the PXR maximum value. Because of the density effect the DB contribution is rather small, but isn’t negligible. As a result the typical deep in the total emission angular distribution (see curve 4) is nearly absent. Comparison between the experiment and calculation (see Fig. 2b) shows satisfactory agreement especially in absolute value and confirms the above mentioned statement about significant contribution of real photons diffraction. It is necessary to remark that the agreement was obtained without any fitting procedure as it was done in the Ref. [12]. Next experiment, where the real photon diffraction contribution was clearly observed, was made at Tomsk synchrotron for electron energy of 900 MeV and observation angle HD ¼ 90 . The condition x 6:96 keV cxp ¼ 67 keV is executed. Preliminary results for Bragg geometry measurements were partially published in Ref. [13,29]. Measurements were performed for a diamond crystal size of 6 10 0:35 mm3 with proportional counter. The large surface of the target was perpendicular the h1 1 0i axis. The (001) crystallographic plane coincides with horizontal one. The detector with round collimator Ø4 mm was placed at the distance 1.06 m from the target. One of most interesting features of the experiment was a possibility to use both Bragg and Laue geometry for the same experimental equipment. For obtaining of needful alignment the
5
crystal was turned around vertical axis on ±45° from the h1 1 0i axial alignment. More detailed description of the experimental set-up and method is presented in Ref. [13,29]. In the Tomsk synchrotron, the electrons are focused on a target due to the losses on the synchrotron radiation. After the desired energy of particles is achieved, the increase in the magnetic field stops and for 20 ms the field is kept constant. To increase the duration of the radiation pulse, the accelerating high frequency field smoothly decreases. The electrons leave synchronism with the accelerating field and are thrown at the target for 10–15 ms. The maximum range of horizontal oscillations is from 1.5 to 2.5 mm, depending on the energy of electrons. The distribution of electrons on the coordinate of hitting the target in the horizontal plane is determined by the oscillation and described well by the function f ðxÞ expðx=sÞ, where x is the coordinate counted from the target edge, and s is the characteristic parameter, see for detail [29,30] and reference therein. Measurements of the diamond target darkness under the electron beam influence have shown that for the particles energy of 900 MeV the s value is about 0.7 mm [30]. These measurements were done when multiple crossing of the electron through the target wasn’t limited [31], therefore for condition of Ref. [13,29] true value of this parameter may be less. Results of the vertical angular distribution measurements are presented on Figs. 3a and 3b for Bragg geometry and Laue one,
Fig. 3. Vertical angular distribution of the X-ray yield for the experiment [13] condition. a – Bragg geometry; b – Laue geometry. Points – experiment. Line – calculation. Curve 1- PXR; curve 2 – DTR; curve 3 – total emission yield.
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accordingly – points. The angular distribution was measured with a differential discriminator tuned to the energy of photons of 5:0 < x < 10:0 keV. When measuring the angular distribution of the background, a thin titanium absorber 50 lm thick was mounted on the path of the radiation beam from the target to the detector, which provided absorption of practically all photons for the (2 2 0) reflection without changing other experimental conditions. The emission angular distribution for the first reflection order was obtained by subtracting the angular distributions measured without and with the absorber. The counter efficiency and photon absorption on the path from the crystal to the detector were taken into account. The calculation results are shown by curves. Curves 1–3 are PXR, DTR and total emission angular distributions, respectively. Due to the normalization error 15% and the uncertainty of the average number of electrons passages through the thin target [13,29] the measurements results are normalized to the calculation ones in the range of the angles hy > 5 mrad, where contribution of diffracted real photons was negligible. Due to the crystal turning and horizontal distribution of electrons on the crystal hitting position some part of electrons moves in the target less distance than the whole target thickness t 0:5 mm. It leads to some decreasing of the registered PXR yield. Moreover the PXR and DTR photons may leave the crystal via lateral edge. This effect decreases the photon absorption in the target and increases the measured emission yield especially for DTR and Laue geometry case. The calculations were done taking into account this effect with s ¼ 0:7 mm. From the figure one can see that for both geometries the ‘‘pure” PXR yield in the angular distribution centre is essentially less than measured one. The DTR contribution enlarges the total emission yield, therefore the calculation and experimental results become close. For Laue geometry the path of the DTR photons in the target is greater, therefore the DTR yield for Laue case is less than for Bragg one. On both figures the measured yield is about 15% greater than the calculated one. The difference may be connected with s value which was used in the calculation. For less value of this parameter contribution of the DTR yield will be greater and the difference will be less. Another reason may be assumption that TR is born on the crystal surface and diffracted in its volume made in [16]. 4. Summary and conclusions Results of the study may be briefly stated as follows: (1) The contribution of diffraction of real photons is maximum for thin crystals and strong collimation, where it is manifested in the ‘‘swelling” of the minimum in the theta-scan or angular distribution and may be compared with PXR yield.
(2) Method [16] describes the measurement results of total emission yield in thin crystals in wide range of electrons and photons energies with accuracy about 10–15% and may be used for calculation of X-ray emission in thin crystal and estimation of high energy electron beam parameters using diffracted transition radiation as it was proposed in Ref. [18].
Acknowledgement This work was supported by a grant from the Russian Science Foundation (Project N 15–12-10019). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
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Please cite this article in press as: Y.A. Goponov et al., Ratio of the contributions real and virtual photons diffraction in thin perfect crystals. Comparison of calculation and experiment, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.02.068
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Ratio of the contributions real and virtual photons diffraction in thin perfect crystals. Comparison of calculation and experiment Yu.A. Goponov, S.A. Laktionova, M.A. Sidnin, I.E. Vnukov ⇑ Belgorod National Research University, Belgorod, Russia
a r t i c l e
i n f o
Article history: Received 9 December 2016 Received in revised form 7 February 2017 Accepted 24 February 2017 Available online xxxx Keywords: Thin crystal Fast electron Parametric X-ray radiation Diffracted transition radiation Diffracted bremsstrahlung
a b s t r a c t To evaluate and improve the previously proposed method of calculating diffracted photon yields in thin perfect crystals, a comparison between calculated and experimental results in wide range of photons and electrons energy was carried out. It is shown that the proposed method describes all investigated experimental results for bremsstrahlung diffraction and transition radiation one with an error less than tenfifteen percent. Consequently, the method may be used for calculation of the electron beam divergence influence on the diffracted transition radiation angular distribution. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction Beam size and angular divergence are one of the most important parameters in the field of accelerators, and many beam diagnostic methods have been developed. For electron accelerators, one conventional method is to use optical radiation such as fluorescent light, optical transition radiation (OTR) [1], optical diffraction radiation (ODR) [2], and Smith-Purcell radiation [3]. However, it was recently found that radiation in optical region cannot be used for profile measurements of an electron beam at modern accelerators and the projected electron–positron linear colliders such as International Linear Collider (ILC) [4] and Compact Linear Collider (CLIC) [5] because it becomes coherent when the bunch size of the beam is sufficiently small [6] in comparison with the measured photons wavelength. In order to avoid the coherence, photons with shorter wavelength are required. Some years ago, the use of radiation in the X-ray region, the so-called parametric X-ray radiation (PXR) were proposed [7,8]. To a first approximation, PXR can be considered as coherent scattering of the charged particle’s electromagnetic field on the electron shells of periodically arranged atoms in a target, see e.g. [9] and reference therein. It is emitted in the Bragg direction when a relativistic charged particle moves across a crystalline target. Recently, proof-of-principle experiments on beam
⇑ Corresponding author. E-mail address: [email protected] (I.E. Vnukov).
profile measurements using PXR have been demonstrated at SAGA Light Source (SAGA-LS) [8,10] and Mainzer Microtron (MAMI) [11]. For fast electrons PXR is always accompanied by radiation diffracted in the crystal which is born directly inside the target or on its surface [12,13]. In the first case we can talk about diffracted bremsstrahlung (DB) and in the second one about diffracted transition radiation (DTR). The first is dominated by under the condition x cxp , where x is photon’s energy, c-Lorentz factor, and xp -plasma frequency of the medium, [14,15] and the second under opposite conditions [13]. If condition x cxp is true, the contributions of both mechanisms of radiation are observed [12]. Method for calculation of real photons diffraction contribution into measured photon yield was suggested in Ref. [16]. In Ref. [17] using the method [16] we have shown that for high electron energy the DTR angular density becomes essentially larger than PXR one, therefore the DTR angular distribution measurements are more preferable for estimation of high energy electron beam parameters in comparison with PXR one. Advantage of a beam diagnostic method using diffracted transition radiation for future electron–positron colliders [4,5] is connected with the fact that the beam divergences in both planes are not so small (of the order of tens lrad) and is by far greater than the transition radiation characteristic angle c1 . Therefore measurements of DTR angular distributions in both planes provides estimation of electron beam divergences in these planes and the electron beam spatial sizes if we know the beam emittance which may be calculated or measured on the previous stage of acceleration [18].
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The transition radiation arises when a fast charged particle crosses an interface of media with different dielectric constants. If one medium is a single crystal and another is vacuum, transition radiation is produced at the entrance surface of the crystal and then a part of its satisfying Bragg’s law is diffracted by the crystal target and emitted in the Bragg direction. PXR is generated in the crystal volume, therefore the DTR contribution in comparison with PXR one is comparatively large for thin crystals only. As it was shown in Ref. [19] the DB contribution may be also clearly observed for thin crystals only. Contribution of real photons diffraction was observed not only in the above mentioned experimental works, however quantitative comparison between calculations and experiments up to date was not made jet. Consequently, it is clear that a comparison of the results of total emission angular distribution measurements in thin crystals with calculations taking account all emission mechanisms and the experiments features is important and relevant. 2. Calculation technique As it was remarked above the main goal of the paper is to compare known results of the experiments devoted to investigation of X-ray emission from the thin crystal irradiated by fast electrons with calculation in accordance with the method proposed in Ref. [16]. It takes into account such types of the electron emission in crystals as PXR, DB and DTR. The special features of the method are following: (a) The kinematic PXR theory [20] is used; (b) The bremsstrahlung suppression due to the density effect [21] is taken into account; (c) The so-called Garibian formula for the transition radiation (TR) spectral-angular distribution [9] is used. It is supposed that the TR is generated directly at the inlet into crystal and then it is diffracted therein. TR photons polarization isn’t taken into account; (d) The electron beam divergence and multiple scattering in the crystal, the emission collimation angle and other experimental conditions are taking into account in accordance with the methodology of Ref. [22]. The main peculiarity of the method [16] is the definition of the X-ray reflectivity in a perfect crystal. As it is well known the angular distribution of diffracted radiation relative to the center of reflex along axis x(see, for example, [12]) can be represented as:
Z
Z Y DR ðx; hx Þ ¼
dx
d I Rðx; ~ n; ~ g ; HD ÞSðx; ~ nÞdX; dxdX 2
because of the refraction of waves in a crystal, d ¼ ðxp =xÞ2 =2 is a difference between the refractive index from 1, and g ¼ f ð~ g Þð1 þ cosð2HB ÞÞ=2f ð0Þ. f ð~ g Þ is the Fourier component of the spatial distribution of electrons in a crystal atom. The characteristic parameter of the model is the primary extinction length, which is dependent on the photon energy, reflection order and the parameters described above, and may be written as [23]:
lex ¼ d=ð2n sin HB Þ;
ð2Þ
where dis an inter planar distance, and expð2 nÞ is an impairment of the intensity of the primary wave as it passes through a plane with the reciprocal lattice vector ~ g. For the part of the crystal with a thickness much smaller than lex , the probability of reflection of photons with an energy of x and direction ~ n, for which the Bragg condition is satisfied, is proportional to the number of planes crossed by them [23]. Therefore, the dependence of the number of photons, which have not undergone the reflection, on the way length in the crystal t can be written as N c ðtÞ ¼ N c ð0Þ expðt=lex Þ [23], where N c ð0Þ is the number of photons at a starting point. Taking into account the Bragg reflection, photon absorption and scattering on atoms the dependence of the number of photons on passable way can be rewritten as:
Nc ðx; ~ n; tÞ ¼ Nc ð0Þ expðltot ðx; ~ g; ~ nÞtÞ;
ð3Þ
where ltot ðx; ~ g; ~ nÞ ¼ lðxÞ þ ldif ðx; ~ g; ~ nÞ is a total coefficient of linear absorption of radiation with energy x, for the direction of the crystal reflecting plane ~ g and the photon velocity direction ~ n. Here lðxÞ is a linear absorption coefficient of photons due to the all processes on separate atoms and ldif ðx; ~ g; ~ nÞ ¼ 1=lex ðx; ~ g; ~ nÞ is due to diffraction. The possibility of such notation allows us to use a well-known in experimental physics method of statistical simulation of photon transmission through a matter for perfect crystals also, see [16] for details. We may ignore the influence of the radiation polarisation on the diffraction process for small Bragg angles only when cosð2HB Þ is about one. In this case photons with r and p polarization are diffracted with close efficiency and have the same extinction length value. In the contrary case the extinction length for photons with different polarization differs. As the transition radiation is polarized in outlet plane [9] the simulation process was made for both polarization components separately.
ð1Þ
d2 I dxdX
where is a spectral-angular distribution of the radiation, taking into account the divergence of the primary electron beam, multiple electron scattering an so on. Rðx; ~ n; ~ g ; HD Þ is reflectivity for these directions of the vectors ~ n and ~ g , defined the crystal orientation n is the individual angle H and the location of the detector HD . Here, ~ vectors corresponding to the initial photon (with the energy x), ~ g is the reciprocal lattice vector. Sðx; ~ nÞ is a function taking into account the photon absorption in the crystal and the geometry of the experiment. To determine the output of the diffracted radiation we need information about Rðx; ~ n; ~ g ; HD Þ. In accordance with [23] for a fixed photon direction ~ n from the beam with the spectral-angular distribution of d I =dxdX satisfying Bragg’s condition for photons with energies of x only photons in the energy range Dx ¼ x cosðHB Þ= sinðHB ÞDH will be reflected. For unpolarized 2
radiation and lack of absorption DH ¼ 2 gDh0 , where Dh0 ¼ 2 d= sin 2HB is an amendment to the Bragg angle HB
3. Comparison of calculation and experiment results The main aim of this study is to compare results of the experiments where contribution of real photon diffraction was clearly observed with the calculation in accordance with techniques reported in Ref. [16] and briefly described in the previous section. These are experiments with thin crystals, strong emission collimation and known values of photon yield [12,14,15] and measurements for large observation angles, where PXR contribution and diffracted real photons one are separated by the emission angle relatively Bragg direction [13]. The experiment [14] was performed at Kharkov linear accelerator. Electron beam energy of 900 MeV with divergence #e 0:2 mrad hits on a silicon crystal thickness of 30 lm. Reflection (2 2 0) in Laue geometry was being investigated. The system of detecting with the round aperture diameter of 5 mm was located at a distance of 8.2 m from the crystal at an angle
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HD ¼ 2HB ¼ 17:8 mrad. Only the spectral-angular density of emission was measured in the experiment. Its value was about 0.3 photon/(MeV*electron*sr.). Calculated value is equal to 0.314 photon/ (MeV*electron*sr.). For this experimental condition x 350 keV cxp ¼ 54:2 keV, therefore only PXR and DB contributions was observed. DB contribution was about 15% from the total radiation intensity maximum value. This result is a rather interesting because of the crystal thickness was by far less than the extinction length for this crystal orientation and the photon energy, therefore simple method suggested and used in Ref. [13,19] is invalid. The experiment [15] was performed at electron linac LUE-40 of Kharkov Institute of Physics and Technology for particles energy of 15.7 and 25.7 MeV. The beam divergence was about 1 mrad. Silicon crystal thickness of 17 lm installed in Laue geometry was used. Reflection (1 1 1) in Laue geometry was being investigated. The semiconductor Silicon detector with round collimator Ø = 5 mm was placed at the observation angle HD ¼ 305:9 mrad at 4520 mm from the target. Measured theta-scans for both energy of electrons and photons of the first reflection order are presented on the Fig. 1– points. Calculated dependencies of the parametric X-ray radiation and the diffracted bremsstrahlung and total radiation yield calculated by the technique [16], so these are the curves 1–3, respectively. For the experimental conditions [15] the condition x 13 keV cxp 1 keV for the both electrons energy is satisfied, so we can observe PXR and DB contribution only.
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In accordance with the PXR theory a virtual photons angular distribution in a medium has a minimum along an electron movement direction, see. e.g. [9], and we should observe two peaks and a deep between them (see curve 1). The deep position corresponds strong execution of Bragg’s law H ¼ HD =2, where H is a plane misalignment angle accordingly electron beam direction. DB yield has maximum value for this crystal orientation (see curve 2), therefore the experimental yield in this orientation angle is greater than the PXR theory prediction and agrees well with the calculation result including both mechanism of photon emission (see curve 3) for electron energy of 15.7 MeV (see Fig. 1a). Difference on the right part of the theta-scan may be connected with interference between PXR and coherent bremsstrahlung [24] which was not taken into account. Agreement for another energy is worse (see Fig. 1b). For execution of Bragg’s condition measured and calculated results agree well, but for PXR region, outside the deep in the theta-scan, experimental photon yield is greater than the calculation results. One of the possible reason may be contribution of the so-called diffracted channelling radiation (DCR) [25]. This type of emission may arise when a fast particle moves simultaneously along a crystallographic plane in the planar channelling regime and emits parametric X-ray on another plane. If the photons energy for both mechanisms are close we may observe DCR as a result of interference between PXR and channelling radiation. It was discussed in some theoretical papers [26,27], but aren’t observed up to date, see [28] for explanation.
Fig. 1. Theta-scan of the of the X-ray yield for [15] condition and the first reflection order for electrons energy of 15.7 MeV (a) and 25.7 MeV (b).
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The theta-scans in the experiment [15] were measured for the electron motion along the (1 1 2) plane. It is likely that for electron energy of 25.7 MeV the (1 1 2) planar channelling radiation energy was about the photon energy of the first order reflection and DCR was observed. For another energy of electrons the (1 1 2) planar channelling radiation energy was less then the PXR one, therefore the measured and calculated results was close. The experiment in the photon energy region x cxp ¼ 51:5 keV, where the DB and DTR contributions into measured emission yield was comparable and clearly observed, was performed at MAMI [12] for electron energy of 855 MeV with a silicon crystalline target thickness of 124 lm and a Si step shaped target with 6 steps of 1 mm width and 100lm height. The crystals surface for both targets was perpendicular to the h1 0 0i axis. The measurements were done with a semiconductor detector which was placed at the angle HD ¼ 45 at distance 1001 mm. The round collimator aperture was equal to 1 mm. Horizontal angular distributions of total emission for the (1 1 1) and (3 3 3) reflection orders and total emission photon flux in the maximum of angular distributions for three investigated crystal orientations were presented in the analysed paper [12]. Results of the photon flux measurements for the investigated reflection orders and our calculation ones are presented in the table. Photon absorption in the entrance window of the detector was not taken into account under calculation procedure.
Order (1 1 1) (2 2 0) (2 2 4)
Measured flux [12] (4.5 ± 0.5)10
3
(6.5 ± 0.8)10
3
(5.4 ± 0.5)10
3
Calculated flux
5.1103 phot./(e.*sr.)
6.7103 phot./(e.*sr.)
5.7103 phot./(e.*sr.)
phot./(e .*sr.) phot./(e .*sr.) phot./(e .*sr.)
As can be seen from the table the measured and calculated values agree a rather well. Difference for the (1 1 1) reflection order with x 5:16 keV may be connected with absorption of the photons in the detector entrance window. Calculation shows that intensity of diffracted real photons for (2 2 0) and (2 2 4) reflection are about a quarter of PXR yield maximum value. For the (1 1 1) reflection this contribution is almost negligible because of small value of the absorption length ła 20 lm in comparison with the targets thickness. Transition radiation which was born on the entrance surface of the crystal is almost absorbed. Fig. 2 shows the calculated horizontal angular distributions for the (3 3 3) reflection order with x 15:7 keV and all emission mechanisms (Fig. 2a) and comparison between calculated and measured flux angular dependencies for the experiment [12] condition (Fig. 2b). From the Fig. 2a it is seen that PXR (curve 1) possesses bigger intensity than the DTR and DB (curves 2 and 3), and its angular distribution is broader. The multiple scattering effect of electrons
Fig. 2. Horizontal distribution of the X-ray yield for the [12] condition. a- Calculated dependences. Curve 1- PXR; curve 2 – DTR; curve 3 – DB and curve 4 -total emission yield. b – Comparison between measured angular distribution [12] – points and calculated one – line.
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leads to some swelling of the PXR distribution minimum. The absorption length la 450 lm is greater than the target thickness, therefore the DTR contribution is about a half of the PXR maximum value. Because of the density effect the DB contribution is rather small, but isn’t negligible. As a result the typical deep in the total emission angular distribution (see curve 4) is nearly absent. Comparison between the experiment and calculation (see Fig. 2b) shows satisfactory agreement especially in absolute value and confirms the above mentioned statement about significant contribution of real photons diffraction. It is necessary to remark that the agreement was obtained without any fitting procedure as it was done in the Ref. [12]. Next experiment, where the real photon diffraction contribution was clearly observed, was made at Tomsk synchrotron for electron energy of 900 MeV and observation angle HD ¼ 90 . The condition x 6:96 keV cxp ¼ 67 keV is executed. Preliminary results for Bragg geometry measurements were partially published in Ref. [13,29]. Measurements were performed for a diamond crystal size of 6 10 0:35 mm3 with proportional counter. The large surface of the target was perpendicular the h1 1 0i axis. The (001) crystallographic plane coincides with horizontal one. The detector with round collimator Ø4 mm was placed at the distance 1.06 m from the target. One of most interesting features of the experiment was a possibility to use both Bragg and Laue geometry for the same experimental equipment. For obtaining of needful alignment the
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crystal was turned around vertical axis on ±45° from the h1 1 0i axial alignment. More detailed description of the experimental set-up and method is presented in Ref. [13,29]. In the Tomsk synchrotron, the electrons are focused on a target due to the losses on the synchrotron radiation. After the desired energy of particles is achieved, the increase in the magnetic field stops and for 20 ms the field is kept constant. To increase the duration of the radiation pulse, the accelerating high frequency field smoothly decreases. The electrons leave synchronism with the accelerating field and are thrown at the target for 10–15 ms. The maximum range of horizontal oscillations is from 1.5 to 2.5 mm, depending on the energy of electrons. The distribution of electrons on the coordinate of hitting the target in the horizontal plane is determined by the oscillation and described well by the function f ðxÞ expðx=sÞ, where x is the coordinate counted from the target edge, and s is the characteristic parameter, see for detail [29,30] and reference therein. Measurements of the diamond target darkness under the electron beam influence have shown that for the particles energy of 900 MeV the s value is about 0.7 mm [30]. These measurements were done when multiple crossing of the electron through the target wasn’t limited [31], therefore for condition of Ref. [13,29] true value of this parameter may be less. Results of the vertical angular distribution measurements are presented on Figs. 3a and 3b for Bragg geometry and Laue one,
Fig. 3. Vertical angular distribution of the X-ray yield for the experiment [13] condition. a – Bragg geometry; b – Laue geometry. Points – experiment. Line – calculation. Curve 1- PXR; curve 2 – DTR; curve 3 – total emission yield.
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accordingly – points. The angular distribution was measured with a differential discriminator tuned to the energy of photons of 5:0 < x < 10:0 keV. When measuring the angular distribution of the background, a thin titanium absorber 50 lm thick was mounted on the path of the radiation beam from the target to the detector, which provided absorption of practically all photons for the (2 2 0) reflection without changing other experimental conditions. The emission angular distribution for the first reflection order was obtained by subtracting the angular distributions measured without and with the absorber. The counter efficiency and photon absorption on the path from the crystal to the detector were taken into account. The calculation results are shown by curves. Curves 1–3 are PXR, DTR and total emission angular distributions, respectively. Due to the normalization error 15% and the uncertainty of the average number of electrons passages through the thin target [13,29] the measurements results are normalized to the calculation ones in the range of the angles hy > 5 mrad, where contribution of diffracted real photons was negligible. Due to the crystal turning and horizontal distribution of electrons on the crystal hitting position some part of electrons moves in the target less distance than the whole target thickness t 0:5 mm. It leads to some decreasing of the registered PXR yield. Moreover the PXR and DTR photons may leave the crystal via lateral edge. This effect decreases the photon absorption in the target and increases the measured emission yield especially for DTR and Laue geometry case. The calculations were done taking into account this effect with s ¼ 0:7 mm. From the figure one can see that for both geometries the ‘‘pure” PXR yield in the angular distribution centre is essentially less than measured one. The DTR contribution enlarges the total emission yield, therefore the calculation and experimental results become close. For Laue geometry the path of the DTR photons in the target is greater, therefore the DTR yield for Laue case is less than for Bragg one. On both figures the measured yield is about 15% greater than the calculated one. The difference may be connected with s value which was used in the calculation. For less value of this parameter contribution of the DTR yield will be greater and the difference will be less. Another reason may be assumption that TR is born on the crystal surface and diffracted in its volume made in [16]. 4. Summary and conclusions Results of the study may be briefly stated as follows: (1) The contribution of diffraction of real photons is maximum for thin crystals and strong collimation, where it is manifested in the ‘‘swelling” of the minimum in the theta-scan or angular distribution and may be compared with PXR yield.
(2) Method [16] describes the measurement results of total emission yield in thin crystals in wide range of electrons and photons energies with accuracy about 10–15% and may be used for calculation of X-ray emission in thin crystal and estimation of high energy electron beam parameters using diffracted transition radiation as it was proposed in Ref. [18].
Acknowledgement This work was supported by a grant from the Russian Science Foundation (Project N 15–12-10019). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
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