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Polarimetry of coherent polarization radiation
Nuclear Instruments and Methods in Physics Research B 201 (2003) 55–66 www.elsevier.com/locate/nimb
Polarimetry of coherent polarization radiation D. Pugachov a, J. They a, G. Buschhorn a, R. Kotthaus a,*, V.L. Morokhovskii 1, H. Genz b, A. Richter b, A. Ushakov b a b
Max-Planck-Institut f€ur Physik (Werner-Heisenberg-Institut), F€ohringer Ring 6, D-80805 M€unchen, Germany Institut f€ ur Kernphysik, Technische Universit€at Darmstadt, Schlossgartenstr. 9, D-64289 Darmstadt, Germany Received 30 January 2002
Abstract Polarization properties of coherent polarization X-radiation produced by a low-emittance electron beam of 72 MeV interacting with a silicon monocrystal have been investigated experimentally at the superconducting Darmstadt linear accelerator S-DALINAC. Spatially and energetically resolved measurements of the direction and the degree of linear polarization show good agreement with expectations based on the kinematical theory of coherent polarization radiation erenkov nature of the radiation. and rule out a quasi-C Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 41.60.-m; 07.85.-m; 47.75.Ht erenkov radiation; Compton polarimeter; Linear Keywords: Parametric X-radiation; Coherent polarization radiation; Quasi-C polarization
1. Introduction The linear polarization properties of coherent polarization radiation (CPR), sometimes called ‘‘Parametric X-radiation’’, produced by electrons of 72 MeV in a silicon monocrystal and observed at an angle of 21° have been studied with an energy resolving Compton scatter polarimeter of high analyzing power. In contrast to the well-known forms of coherent radiation in the near-forward direction, i.e. chan* Corresponding author. Tel.: +49-89-32354265; fax: +49-893226704. E-mail address: [email protected] (R. Kotthaus). 1 Guest Scientist from Kharkov Institute of Physics and Technology, Kharkov, Ukraine.
neling radiation and coherent bremsstrahlung (CBS), due to the periodic perturbation of the electron trajectory inside a crystal CPR occurs as the result of coherent superposition of the electromagnetic waves emitted by the polarized crystal atoms along the particle trajectory in close analogy to ordinary Bragg diffraction of a beam of real photons. Since the elementary radiation process according to this model is polarization bremsstrahlung of crystal atoms the name ‘‘coherent polarization radiation’’ (CPR) [1] is most appropriate for this form of X-radiation. For highly relativistic charged particles (Lorentz factor c ¼ 1 þ E=mc2 1, E is the kinetic energy) CPR is strongly collimated and the intensity maximum is along a narrow cone of opening angle c1 about the center of the reflex which is near the reflection of the
0168-583X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 0 9 5 0 - 3
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incident particle trajectory at the diffracting crystallographic plane (Bragg direction). The spectral–angular properties of CPR produced by electrons mostly in silicon and diamond crystals at observation angles H of typically a few tens of degrees off the incident beam direction have been studied extensively over a broad range of electron energies from a few MeV up to several GeV. For a recent review we refer to [2] and references therein. In general, the experimental results are well described by classical and quantum theories in ‘‘kinematical’’, i.e. lowest order, approximation. Recently, it was shown [3] that small higher order (‘‘dynamical’’) corrections to the kinematical theory vanish for relatively low electron 1=2 energies (c < v0 ). v0 is the average susceptibility 1=2 of the crystal material (for Si: v0 ¼ 270). Fig. 1 shows calculated spectral properties of CPR produced by 72 MeV (c ¼ 142) electrons at the (1 1 1) plane of silicon and observed at an angle H ¼ 21° which are the conditions of this experiment. Parts
(a) and (b) give the flux distribution about the minimum near the Bragg direction (Hx ¼ Hy ¼ 0). The energy-angular correlation is displayed in (c). Parallel to the diffraction plane (x-direction) the energy is dispersed to a narrow band about 11 keV whereas perpendicular (y-direction) it is almost constant. These well-known spectral properties of CPR are exploited in this experiment to probe the polarization vector locally with a small aperture polarimeter. Up to now the polarization properties of CPR have not yet been studied systematically. Two pioneering experiments [4,5] analyzed the linear polarization of the (2 2 0) reflex of silicon at H 20° corresponding to photon energies of about 18 keV. Away from the center of the reflex in both experiments the expected high degree of linear polarization is observed. But the measurements disagree on the orientation of the polarization plane, i.e. the plane containing the electric field vector. Adishchev and coworkers [4,6] using 900 MeV elec-
Fig. 1. Spectral properties of CPR: (1 1 1) reflex of silicon produced by 72 MeV electrons at an observation angle H ¼ 21°. (a) Flux distribution, (b) central part of flux distribution, (c) energy-angular correlation.
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trons and conventional 90°-Compton scattering for polarization analysis find the polarization plane to be oriented radially with respect to the axis of the radiation cone in analogy to the po erenkov radiation. The larization behaviour of C analysis by our group [5] of CPR produced by 80.5 MeV electrons exploiting directional information of the photoeffect in a finely segmented CCD [7] revealed a different pattern of polarization directions. Within one azimuthal quadrant of the reflex the polarization plane rotates oppositely to the azimuth. Such a behaviour is expected within the kinematical theory of CPR [5,8,9]. For small angular distances to the Bragg direction (Hx ; Hy 1) the polarization angle W is given to a very good approximation by tan W ¼
1 Hy : cos H Hx
ð1Þ
W is the angle between the polarization and the observation planes. In the forward hemisphere (cos H P 0) the polarization plane should thus rotate against the azimuth (‘‘hyperbolic’’ behav-
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iour) whereas for backward diffraction (cos H < 0) the polarization angle should follow the azimuth (‘‘radial’’ behaviour). Fig. 2 shows the distribution of linear polarization directions of CPR in the vicinity of the Bragg direction for three different observation angles H. The local linear polarization is tangential to the curves shown. The objective of the present study is to perform a substantially improved spatially resolved measurement of the polarization vector and thereby to clarify the experimental controversy and the nature of the radiation. The experiment combines the respective merits of the two previous analyses [4,5] by utilizing a Compton polarimeter of high analyzing power as in [4,6] and by choice of a moderately low electron energy similar as in [5,7] such that the angular spread of the radiation reflex is sufficiently large for a spatially resolved analysis. The crucial signal-to-background ratio is enhanced by selecting the (1 1 1) reflex which is over four times more intense than the (2 2 0) reflex used in the two previous experiments. Some preliminary results of this experiment have already been published [10].
Fig. 2. Orientation of linear polarization of CPR in the vicinity of the Bragg direction (Hx ¼ Hy ¼ 0) for three different observation angles H. (a) H ¼ 21°, (b) H ¼ 135°, (c) H ¼ 180°. Local linear polarization directions are tangential to the curves shown.
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2. Experiment 2.1. Setup at S-DALINAC The experiment [11] was carried out at the Darmstadt superconducting linear accelerator SDALINAC [12] with a low-emittance electron beam of 72 MeV. The accelerator is operated in continuous wave (cw) mode which maximizes the macroscopic duty fraction and thus allows for high average beam currents (of the order of 1 lA) at tolerable instantaneous CPR rates incident to the polarimeter. The electron beam spot radius at the Si crystal is typically 0.5 mm (rms) at a divergence of about 0.5 mrad. Both, the cw mode of operation of the accelerator and the low-emittance electron beam are indispensable for a sensitive spatially resolved polarization analysis of CPR. Fig. 3 gives a schematic view of the setup of the experiment. The electron beam traverses the CPR producing Si crystal and is deflected into a Faraday cup, FC to measure the beam current and the integrated charge with an accuracy of 2–3% [13]. The crystal is mounted on a three-axes goniometer allowing rotations about the electron beam axis (a-rotation with minimal step width of 0.001°) and two mutually perpendicular axes (U- and W-rotations, both with minimal step width of 0.01°) which
Fig. 3. Schematic view of the setup of the experiment at SDALINAC (see text).
change their direction in space on a-rotations. The Si crystal has been prepared by etching a standard 2-in. wafer cut along the (1 0 0) crystallographic plane from 300 lm thickness to about 20 lm such that the multiple scattering angle (1r 2 mrad) is reasonably matched to the low electron beam divergence. The CPR beam generated in the Si crystal passes through a circular collimator C1 (/ ¼ 15 mm) and leaves the target chamber through a 50 lm thick Kapton window K. The Compton polarimeter is positioned at a fixed observation direction (H ¼ 21°) 700 mm downstream of the Si crystal inside a hermetic Pb shield. A pinhole C2 (/ ¼ 3 mm) defines the aperture and limits the angular acceptance of the polarimeter to 4.1 mrad. Compared to the opening angle c1 ¼ 7:2 mrad of the CPR cone this acceptance results in a reasonable compromise between the conflicting needs of a sufficiently high data rate and a good spatial resolution. 2.2. Compton polarimeter The Compton scattering polarimeter (Fig. 4) employs five thermoelectrically cooled commercial Si PIN photodiode detectors (models XR-100T and XR-100CR produced by AMPTEC Inc.) of 7 mm2 sensitive area each. Four diodes (D1 –D4 in Fig. 3) act as scattering detectors viewing a 2 mm thick planar Be scatterer intercepting the incident CPR beam at right angle. The detector azimuths / are spaced by 45°. The scattering angle is 110° which according to Monte Carlo simulation results optimizes the polarization sensitivity for the planar Be scatterer in the energy range of interest [14]. The scattering probability at an energy around 11 keV is of the order of 104 . The measurement of scattering yields at four azimuths / spaced by 45° fully constrains the cos2 /-modulation due to the linear polarization of the incident CPR beam. The polarimeter thus determines the polarization vector, i.e. the orientation of the polarization plane and the degree of linear polarization, in one single measurement not requiring to rotate the polarimeter. Ka fluorescence radiation from a 10 lm thick Fe foil attached to the backside of the Be scatterer is used to monitor the time stability of the polarimeter. Detector D5 (dismounted for the photo in
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Fig. 4. 110°-Compton scattering polarimeter mounted at the e -beam line before installation of shielding materials (view against the incident CPR beam). The forward detector used for alignment has been removed for better visibility of the square Be scatterer and the arrangement of scattering detectors.
Fig. 4) is positioned on the polarimeter axis behind the scatterer to facilitate the alignment of the CPR cone with respect to this axis by measuring energy and angular distributions (see Fig. 1) of photons penetrating the Be scatterer and the Fe foil. Within the CPR energy band of the (1 1 1) reflex (Fig. 1(c)) the transmission through all the materials in front of detector D5 varies between 16% and 26% [11]. The absorption losses as well as the detection probability in detector D5 (ranging from 80% to 90%) have been corrected for in order to arrive at absolute yields of unscattered CPR (see Section 2.4). 2.3. Polarimeter calibration and performance The compact design of the 110°-Compton polarimeter utilizing small volume scattering detectors (Fig. 4) is vulnerable to various instrumental asymmetries affecting the analyzing power and mimicking false polarization signals. Therefore, a careful calibration of the polarization sensitivity and a measurement of instrumental asymmetries of the polarimeter was carried out at HASYLAB/ DESY utilizing an unfocussed monochromatized beam of 11 keV synchrotron radiation of known linear polarization [11]. Fig. 5 shows measurements of scattering yields obtained for 14 different azimuthal orientations of the polarimeter about the synchrotron radiation beam spanning a range
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Fig. 5. Azimuthal angular distribution of Compton scattered synchrotron radiation of 11 keV for different polarimeter rotation angles about the photon beam. Fractional yields obtained in the four scattering detectors are displayed in different symbols.
of about 50°. Different symbols give the yields of each of the four scattering detectors normalized to the total number of counts registered in all four detectors. The analyzing power a is determined by the measured asymmetry A of yields at the ‘‘allowed’’ (/ ¼ 90°) and ‘‘forbidden’’ (/ ¼ 0°) azimuths and the calculated degree of linear polarization P ¼ 97%: a ¼ A=P ¼ ð72 2Þ%. The error is obtained from comparisons of individual cos2 -fits to the four scattering yields at each of the polarimeter settings. Within the /-range covered by the measurements the analyzing power a does not depend on the orientation of the polarimeter with respect to the polarization plane. The cos2 -fits also determine the azimuth /0 of the polarization plane (forbidden azimuth) with a statistical uncertainty of r/0 ’ 0:3° which is small compared to the statistical and systematical errors of the CPR polarization measurements (see Section 4) and thus does not enter the final result. From measurements with pairs of detectors oriented symmetrically with respect to the polarization plane the instrumental asymmetry of the polarimeter is determined. Fig. 6 shows the corresponding modulation curve which would be flat for a perfectly symmetric polarimeter. The CPR
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with the calculated effective degree of linear polarization out to large dip angles. 2.4. Experimental procedures The local polarization properties of CPR are probed by sweeping the reflex over the polarimeter aperture through crystal rotations exploiting the well-established angular distribution and energyangular correlation shown in Fig. 1. The proce-
Fig. 6. Azimuthal modulation of Compton scattering yields introduced by instrumental asymmetries.
measurements are corrected for the instrumental asymmetry of Ainstr ¼ 0:058. As an overall test of the polarimeter performance the vertical polarization profile of the synchrotron radiation beam has been measured. The result is given in Fig. 7 and shows good agreement
Fig. 7. Vertical polarization profile of 11 keV synchrotron radiation. The dip angle W is given in units of c1 ¼ 0:11 mrad. The curve is the calculated effective degree of linear polarization normalized to the measurements (triangles) at the synchrotron plane (cW ¼ 0).
Fig. 8. Calculated contours of constant CPR flux as a function of the goniometer rotation angles U and a. Dots mark probing locations for two sets of measurements at slightly different observation angles: (a) H ¼ 20:86°, (b) 20.34°.
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dure and the theoretical distributions used to fit the measurements are presented in detail elsewhere [10,11]. The majority of measurements were done for the (1 1 1) reflex which is of highest intensity and therefore leads to the best signalto-background ratio. Fig. 8 shows calculated contours of constant flux of the (1 1 1) reflex along the polarimeter axis as a function of the goniometer rotation angles U and a for two different sequences of measurements between which a small change of the observation angle was noticed (see below). The dots (labelled A; B; C; . . . ; H) mark angular settings at which measurements were done. The probing locations are reached by angular movements from the center of the reflex at the Bragg direction after alignment with the polarimeter axis by measuring energy spectra of unscattered CPR with the forward detector D5 for different crystal orientations a and U. The spectrum measured at location A is given in Fig. 9. The dominant (1 1 1) reflex is measured with very little background. The peak energy of 10.66 keV and the line width (310 eV FWHM) correspond to the expected energy-angular correlation and the angular acceptance of the polarimeter [11]. In addition, the higher harmonics ((3 3 3) and (4 4 4)) show at their proper energies and strengths. The
Fig. 9. Energy spectrum measured by forward detector D5 at probing location A of Fig. 8(a) (see text).
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broad enhancement at the upper end of the spectrum due to minimum ionizing particles is used for online normalization of the CPR flux during the alignment procedure. Fig. 10 shows an example of a pair of scans in the angles a and U of the CPR flux and energy extracted from energy spectra measured by detector D5 . The curves correspond to fits of the theoretical distributions [10,11] to the measurements not carrying out the integration over the detector acceptance. The minima of the intensity distributions mark the alignment of the Bragg
Fig. 10. Angular scans for CPR cone alignment with polarimeter axis. Shown are CPR flux and energy measured by forward detector D5 for crystal rotations about goniometer axes a (a) and U (b). Curves correspond to fits of the theoretical flux distribution and energy-angular relation to the measurements. Sketch (c) shows where the angular cross sections were taken.
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direction with the polarimeter axis. The energy measurements show the expected dispersion in the diffraction plane (angle U) and constant energy perpendicular (angle a) (see Fig. 1). The measured energy dependence on U allows to determine the absolute value of the observation angle H ¼ 2UBragg with very high accuracy. For the first measurements (at positions A and B in Fig. 8(a)) we obtain H ¼ ð20:86 0:01Þ°. The second sequence (C to H in Fig. 8(b)) was done at H ¼ ð20:34 0:01Þ°. The shift is due to a horizontal beam displacement. The systematical error of the crystal alignment is given by beam instabilities. The very precise energy measurement in the forward detector D5 (see Fig. 10) is a sensitive monitor of the horizontal beam position (affecting angle U). From observed energy shifts of 50–100 eV uncertainties of the angle U of 1–2 mrad are inferred. Vertical beam instabilities (changing the angle a) cannot be observed by energy measurements but are generally expected to be less pronounced as all beam deflections occur in the horizontal plane. Unnoticed beam movements and corresponding uncertainties of probing locations are a major contribution to the systematic error of the local CPR polarization vector (see Section 3).
3. Polarimetry results The CPR polarization vector was first measured at two positions (A and B in Fig. 8(a)) close to the
centers of adjacent quadrants of the radiation cone where according to (1) the polarization directions should differ by 90° and where the distinction between the radial and the hyperbolic behaviour is most pronounced. The energy spectra measured by the scattering detectors D1 to D4 at points A and B are shown in Fig. 11. The strongly different yields of Compton scattered CPR (peaks at 10.37 keV (A) and 10.78 keV (B)) in detectors D1 to D4 reflect the high degree of linear polarization. At position A the smallest yield, marking the forbidden azimuth along the polarization plane, is measured in detector D4 whereas at position B the minimum occurs close to detector D2 . The azimuthal angular distributions extracted from the energy spectra are shown in Fig. 12. The measured yields are corrected for the relative efficiencies of detectors D1 to D4 (see Fig. 6) and are normalized to the total number of counts in all four scattering detectors. From cos2 -fits (solid and dashed lines in Fig. 12) the degree and the orientation of the linear polarization are obtained. The phase difference of the modulation curves of approximately 90° signals a corresponding change of the polarization direction from A to B. The phase values are as expected for the polarization angle W given by (1) (hyperbolic behaviour). The rotation of the local polarization vector within one quadrant of the radiation cone is demonstrated by a series of measurements at five different azimuths (labels C to H in Fig. 8(b)). The measured Compton scattering distributions and
Fig. 11. Energy spectra measured by polarimeter detectors D1 to D4 at positions A and B of Fig. 8(a). Scattered CPR peaks at 10.37 keV (A) and at 10.78 keV (B) of different strengths reflect a high degree of linear polarization. The line at 6.4 keV is Ka fluorescence from the Fe monitor foil at the back of the Be scatterer.
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Fig. 12. Azimuthal angular distributions of Compton scattered CPR near the centers of two adjacent quadrants of the radiation cone (locations A, B of Fig. 8(a)) The lines are cos2 -fits to the measurements.
the corresponding cos2 -fits (Fig. 13) show a phase progression due to the rotation of the polarization plane oppositely to the azimuth. The average local polarization angle W within the aperture of the polarimeter extracted from the Compton scatter-
Fig. 13. Azimuthal angular distributions of Compton scattered CPR for a series of measurements at different azimuths in one quadrant of the radiation cone (locations C to G of Fig. 8(b)). The lines are cos2 -fits to the measurements.
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ing distributions is displayed in Fig. 14. The measurements follow the expectation (1) for CPR (solid line) and rule out a center symmetric radial polarization pattern (dotted line) as would be ex erenkov radiation. pected for quasi-C The numerical results for the local polarization angle W and the effective degree of linear polarization P of the Si(1 1 1) reflex measured within the aperture of the polarimeter are given in Table 1. In all cases the error due to the statistical uncertainty of the number of registered Compton scattered photons is smaller than the systematical errors. The major contribution to the systematical error of the polarization angle given in Table 1 is due to the uncertainty of the probing location (dHx ¼ dHy ¼ 1:4 mrad) estimated from observed shifts of the energy of the incident CPR measured in the forward detector D5 [11]. A minor contribution is due to the uncertainty of the polarimeter asymmetry correction (see Section 2.3). The systematical error of the effective degree of polarization P (not shown in Table 1) receives contributions of comparable size from various sources (analyzing power, polarimeter asymmetry, background
Fig. 14. Measurements of the polarization angle at five different azimuths in one quadrant of the Si(1 1 1) radiation cone (locations C to H of Fig. 8(b)). The error bars represent the statistical and systematical measurement errors added quadratically.
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Table 1 Comparison of measured and expected polarization results for Si(1 1 1) plane Orientation
A
U (°) a (°) Reflex quadrant
10.18 0.902 1
Polarization angle (°) (experiment) W Statistical error Systematical error (theory) W (Monte Carlo) W
45.8 2.0 8.1 34.4 34.3
Degree of polarization P (experiment) 0.86 Statistical error 0.05 P (theory) 0.97 P (Monte Carlo) 0.92
B 10.60 0.902 2 52.4 2.3 11.2 45.7 46.4 0.66 0.05 0.94 0.86
C 10.10 1.085 1 73.5 1.2 10.5 66.2 70.0 0.87 0.03 0.93 0.84
subtraction, electron beam instabilities) and ranges between 0.074 and 0.135 depending on the correlation between the individual contributions [11]. The measurements are compared to theoretical expectations and to Monte Carlo simulation results also listed in Table 1. Using the CPR angular distribution (Fig. 1(a) and (b)) and the polarization angle calculated in kinematical approximation the average polarization direction W inside the polarimeter aperture and the effective degree of polarization P ¼ cos 2ðW WÞ are computed. Smearing effects due to the e -beam emittance, energy spread and multiple scattering in the Si crystal reduce the local degree of polarization to less than one. In order to take these instrumental effects into account Monte Carlo simulations [13] have been carried out. Comparing the simulation results with the analytical integrations (‘‘theory’’ in Table 1) one notices that the effect of experimental smearings on the polarization angle W is minimal whereas the effective degree of polarization P is noticeably reduced, as expected.
4. Conclusions and outlook The measured polarization directions are in good agreement with expectations within the kinematical theory of CPR. The polarization angle
D
E
F
G
H
10.02 0.999 1
9.96 0.752 1
9.91 0.384 1
58.6 1.6 9.7 50.8 50.5
43.7 2.1 9.6 33.4 33.2
12.3 1.3 9.9 15.2 15.1
5.7 1.4 10. 2 1.95 1.9
0.79 0.04 0.94 0.87
0.75 0.05 0.95 0.88
0.85 0.04 0.95 0.89
0.84 0.04 0.95 0.89
9.90 0.05 4
10.10 1.085 1 68.0 1.7 10.5 66.2 70.0 0.79 0.05 0.93 0.84
W follows the hyperbolic pattern (see Figs. 2 and 14) as was first measured less accurately for the Si(2 2 0) reflex in [5]. The measurements definitely rule out the center symmetric radial polarization pattern suggested by measurements of [4,6] and as erenkov radiation. would be expected for quasi-C The results obtained for the effective degree of polarization inside the narrow polarimeter aperture vary between P ¼ 0:66 and 0.87 depending on the location inside the radiation reflex (Table 1). Within combined systematical and statistical errors of 9% to 15% the measurements agree marginally with expectations for CPR but are systematically lower by about 10%. Possible reasons for this slight discrepancy have been investigated [11]. A small contamination (of the order of 10%) of CPR by X-rays of a different polarization behaviour would not spoil the agreement of the measured polarization angle W with expectations but could reduce the effective degree of linear polarization P as is observed. Such a background would not be noticed if it had spectral–angular characteristics similar to CPR (see Fig. 1). CBS and diffracted forward going radiation, like transition radiation or bremsstrahlung accompanying the e -beam, are potential sources. The interference of CPR and CBS (being linearly polarized in the diffraction plane) has been investigated in [13,15]. The effect is strong only at observation angles comparable to c1 ( ¼ 7.2 mrad in this ex-
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periment). At observation angles H ’ 21° of this experiment CBS is not a significant contribution. Diffracted transition radiation (DTR) has similar polarization characteristics as CPR and therefore would not reduce the effective degree of polarization. Moreover, at the relatively low e -energy of 72 MeV the CPR energy of 11 keV is far beyond the cutoff energy cxp ¼ 4:37 keV (plasma frequency xp ¼ 31 eV for Si) for transition radiation. Therefore, DTR should be negligible. Diffracted bremsstrahlung (DBS) can in principle be discriminated from CPR by the angular distribution (peaking at the Bragg direction) and by a slightly different energy-angular correlation (Fig. 15). In practice, however, a contribution of DBS of the order of 10% to the measured CPR yield cannot be excluded. Another possible cause of the small difference between the measured and the expected degree of linear polarization is a slight underestimation of experimental smearing effects assumed for the Monte Carlo simulations. Somewhat larger values of the e -beam divergence and transverse dimensions or slightly enhanced multiple scattering in the Si crystal would still be compatible with the measured CPR energy line width of about 150 eV and at the same time would more strongly reduce the local polarization.
Fig. 15. Ratio of energies of Bragg scattered photons and of CPR as a function of crystal angle /. Within the range covered by the scan shown in Fig. 10(b) (9:8° 6 / 6 11:0°) the two energies differ by less than 0.4% or 40 eV.
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For a complete verification of CPR polarization properties measurements at larger observation angles H need to be carried out. In particular, the radial orientation of polarization planes expected according to (1) for backward CPR (see Fig. 2) should be experimentally confirmed. Polarimetry of backward CPR will be a very difficult task since the CPR yield (Fig. 16) as well as the energy (Fig. 17) fall with increasing observation angle. In view of the very low energy of the (1 1 1) reflex of Si (E < 3 keV in the entire backward hemisphere) and correspondingly large absorption effects it appears more promising to study the less intense (by a factor of 4.4) but more energetic (2 2 0) reflex. An interesting special case is CPR emitted at H ¼ 90°. Here the p-polarization component vanishes (Fig. 16) such that the entire CPR reflex is expected to be linearly polarized perpendicular to the diffraction plane (r-polarization) to a very good approximation [5,8,9]. Polarimetry of large-angle CPR requires a polarimeter of high analyzing power and energy resolution for low energy X-rays. A 90°-Compton polarimeter utilizing Si drift detectors has been built for this purpose and is now being calibrated
Fig. 16. Relative CPR yield as a function of observation angle H. The total yield and the p- and r-polarization components are shown separately.
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BMBF (contract no. 06DA9151) and by the German Research Society DFG (contract no. FK410). References [1] [2] [3] [4] [5]
[6]
[7]
Fig. 17. CPR energy at center of (1 1 1) and (2 2 0) reflex of Si as a function of observation angle H.
[8]
[16,17]. Measurements at H ¼ 90° are presently being prepared.
[9] [10]
[11]
Acknowledgements The authors wish to thank Dr. H.-D. Gr€ af and his crew for the expert operation of the accelerator and Dr. V.V. Morokhovskii for his kind support of the Monte Carlo simulations. The MPI authors gratefully acknowledge the hospitality extended to them at S-DALINAC and the generous support of polarimeter calibrations given by HASYLAB at the Deutsches Elektronen-Synchrotron DESY. The work has been supported by the German Federal Minister for Education and Research
[12]
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Polarimetry of coherent polarization radiation D. Pugachov a, J. They a, G. Buschhorn a, R. Kotthaus a,*, V.L. Morokhovskii 1, H. Genz b, A. Richter b, A. Ushakov b a b
Max-Planck-Institut f€ur Physik (Werner-Heisenberg-Institut), F€ohringer Ring 6, D-80805 M€unchen, Germany Institut f€ ur Kernphysik, Technische Universit€at Darmstadt, Schlossgartenstr. 9, D-64289 Darmstadt, Germany Received 30 January 2002
Abstract Polarization properties of coherent polarization X-radiation produced by a low-emittance electron beam of 72 MeV interacting with a silicon monocrystal have been investigated experimentally at the superconducting Darmstadt linear accelerator S-DALINAC. Spatially and energetically resolved measurements of the direction and the degree of linear polarization show good agreement with expectations based on the kinematical theory of coherent polarization radiation erenkov nature of the radiation. and rule out a quasi-C Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 41.60.-m; 07.85.-m; 47.75.Ht erenkov radiation; Compton polarimeter; Linear Keywords: Parametric X-radiation; Coherent polarization radiation; Quasi-C polarization
1. Introduction The linear polarization properties of coherent polarization radiation (CPR), sometimes called ‘‘Parametric X-radiation’’, produced by electrons of 72 MeV in a silicon monocrystal and observed at an angle of 21° have been studied with an energy resolving Compton scatter polarimeter of high analyzing power. In contrast to the well-known forms of coherent radiation in the near-forward direction, i.e. chan* Corresponding author. Tel.: +49-89-32354265; fax: +49-893226704. E-mail address: [email protected] (R. Kotthaus). 1 Guest Scientist from Kharkov Institute of Physics and Technology, Kharkov, Ukraine.
neling radiation and coherent bremsstrahlung (CBS), due to the periodic perturbation of the electron trajectory inside a crystal CPR occurs as the result of coherent superposition of the electromagnetic waves emitted by the polarized crystal atoms along the particle trajectory in close analogy to ordinary Bragg diffraction of a beam of real photons. Since the elementary radiation process according to this model is polarization bremsstrahlung of crystal atoms the name ‘‘coherent polarization radiation’’ (CPR) [1] is most appropriate for this form of X-radiation. For highly relativistic charged particles (Lorentz factor c ¼ 1 þ E=mc2 1, E is the kinetic energy) CPR is strongly collimated and the intensity maximum is along a narrow cone of opening angle c1 about the center of the reflex which is near the reflection of the
0168-583X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 0 9 5 0 - 3
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incident particle trajectory at the diffracting crystallographic plane (Bragg direction). The spectral–angular properties of CPR produced by electrons mostly in silicon and diamond crystals at observation angles H of typically a few tens of degrees off the incident beam direction have been studied extensively over a broad range of electron energies from a few MeV up to several GeV. For a recent review we refer to [2] and references therein. In general, the experimental results are well described by classical and quantum theories in ‘‘kinematical’’, i.e. lowest order, approximation. Recently, it was shown [3] that small higher order (‘‘dynamical’’) corrections to the kinematical theory vanish for relatively low electron 1=2 energies (c < v0 ). v0 is the average susceptibility 1=2 of the crystal material (for Si: v0 ¼ 270). Fig. 1 shows calculated spectral properties of CPR produced by 72 MeV (c ¼ 142) electrons at the (1 1 1) plane of silicon and observed at an angle H ¼ 21° which are the conditions of this experiment. Parts
(a) and (b) give the flux distribution about the minimum near the Bragg direction (Hx ¼ Hy ¼ 0). The energy-angular correlation is displayed in (c). Parallel to the diffraction plane (x-direction) the energy is dispersed to a narrow band about 11 keV whereas perpendicular (y-direction) it is almost constant. These well-known spectral properties of CPR are exploited in this experiment to probe the polarization vector locally with a small aperture polarimeter. Up to now the polarization properties of CPR have not yet been studied systematically. Two pioneering experiments [4,5] analyzed the linear polarization of the (2 2 0) reflex of silicon at H 20° corresponding to photon energies of about 18 keV. Away from the center of the reflex in both experiments the expected high degree of linear polarization is observed. But the measurements disagree on the orientation of the polarization plane, i.e. the plane containing the electric field vector. Adishchev and coworkers [4,6] using 900 MeV elec-
Fig. 1. Spectral properties of CPR: (1 1 1) reflex of silicon produced by 72 MeV electrons at an observation angle H ¼ 21°. (a) Flux distribution, (b) central part of flux distribution, (c) energy-angular correlation.
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trons and conventional 90°-Compton scattering for polarization analysis find the polarization plane to be oriented radially with respect to the axis of the radiation cone in analogy to the po erenkov radiation. The larization behaviour of C analysis by our group [5] of CPR produced by 80.5 MeV electrons exploiting directional information of the photoeffect in a finely segmented CCD [7] revealed a different pattern of polarization directions. Within one azimuthal quadrant of the reflex the polarization plane rotates oppositely to the azimuth. Such a behaviour is expected within the kinematical theory of CPR [5,8,9]. For small angular distances to the Bragg direction (Hx ; Hy 1) the polarization angle W is given to a very good approximation by tan W ¼
1 Hy : cos H Hx
ð1Þ
W is the angle between the polarization and the observation planes. In the forward hemisphere (cos H P 0) the polarization plane should thus rotate against the azimuth (‘‘hyperbolic’’ behav-
57
iour) whereas for backward diffraction (cos H < 0) the polarization angle should follow the azimuth (‘‘radial’’ behaviour). Fig. 2 shows the distribution of linear polarization directions of CPR in the vicinity of the Bragg direction for three different observation angles H. The local linear polarization is tangential to the curves shown. The objective of the present study is to perform a substantially improved spatially resolved measurement of the polarization vector and thereby to clarify the experimental controversy and the nature of the radiation. The experiment combines the respective merits of the two previous analyses [4,5] by utilizing a Compton polarimeter of high analyzing power as in [4,6] and by choice of a moderately low electron energy similar as in [5,7] such that the angular spread of the radiation reflex is sufficiently large for a spatially resolved analysis. The crucial signal-to-background ratio is enhanced by selecting the (1 1 1) reflex which is over four times more intense than the (2 2 0) reflex used in the two previous experiments. Some preliminary results of this experiment have already been published [10].
Fig. 2. Orientation of linear polarization of CPR in the vicinity of the Bragg direction (Hx ¼ Hy ¼ 0) for three different observation angles H. (a) H ¼ 21°, (b) H ¼ 135°, (c) H ¼ 180°. Local linear polarization directions are tangential to the curves shown.
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2. Experiment 2.1. Setup at S-DALINAC The experiment [11] was carried out at the Darmstadt superconducting linear accelerator SDALINAC [12] with a low-emittance electron beam of 72 MeV. The accelerator is operated in continuous wave (cw) mode which maximizes the macroscopic duty fraction and thus allows for high average beam currents (of the order of 1 lA) at tolerable instantaneous CPR rates incident to the polarimeter. The electron beam spot radius at the Si crystal is typically 0.5 mm (rms) at a divergence of about 0.5 mrad. Both, the cw mode of operation of the accelerator and the low-emittance electron beam are indispensable for a sensitive spatially resolved polarization analysis of CPR. Fig. 3 gives a schematic view of the setup of the experiment. The electron beam traverses the CPR producing Si crystal and is deflected into a Faraday cup, FC to measure the beam current and the integrated charge with an accuracy of 2–3% [13]. The crystal is mounted on a three-axes goniometer allowing rotations about the electron beam axis (a-rotation with minimal step width of 0.001°) and two mutually perpendicular axes (U- and W-rotations, both with minimal step width of 0.01°) which
Fig. 3. Schematic view of the setup of the experiment at SDALINAC (see text).
change their direction in space on a-rotations. The Si crystal has been prepared by etching a standard 2-in. wafer cut along the (1 0 0) crystallographic plane from 300 lm thickness to about 20 lm such that the multiple scattering angle (1r 2 mrad) is reasonably matched to the low electron beam divergence. The CPR beam generated in the Si crystal passes through a circular collimator C1 (/ ¼ 15 mm) and leaves the target chamber through a 50 lm thick Kapton window K. The Compton polarimeter is positioned at a fixed observation direction (H ¼ 21°) 700 mm downstream of the Si crystal inside a hermetic Pb shield. A pinhole C2 (/ ¼ 3 mm) defines the aperture and limits the angular acceptance of the polarimeter to 4.1 mrad. Compared to the opening angle c1 ¼ 7:2 mrad of the CPR cone this acceptance results in a reasonable compromise between the conflicting needs of a sufficiently high data rate and a good spatial resolution. 2.2. Compton polarimeter The Compton scattering polarimeter (Fig. 4) employs five thermoelectrically cooled commercial Si PIN photodiode detectors (models XR-100T and XR-100CR produced by AMPTEC Inc.) of 7 mm2 sensitive area each. Four diodes (D1 –D4 in Fig. 3) act as scattering detectors viewing a 2 mm thick planar Be scatterer intercepting the incident CPR beam at right angle. The detector azimuths / are spaced by 45°. The scattering angle is 110° which according to Monte Carlo simulation results optimizes the polarization sensitivity for the planar Be scatterer in the energy range of interest [14]. The scattering probability at an energy around 11 keV is of the order of 104 . The measurement of scattering yields at four azimuths / spaced by 45° fully constrains the cos2 /-modulation due to the linear polarization of the incident CPR beam. The polarimeter thus determines the polarization vector, i.e. the orientation of the polarization plane and the degree of linear polarization, in one single measurement not requiring to rotate the polarimeter. Ka fluorescence radiation from a 10 lm thick Fe foil attached to the backside of the Be scatterer is used to monitor the time stability of the polarimeter. Detector D5 (dismounted for the photo in
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Fig. 4. 110°-Compton scattering polarimeter mounted at the e -beam line before installation of shielding materials (view against the incident CPR beam). The forward detector used for alignment has been removed for better visibility of the square Be scatterer and the arrangement of scattering detectors.
Fig. 4) is positioned on the polarimeter axis behind the scatterer to facilitate the alignment of the CPR cone with respect to this axis by measuring energy and angular distributions (see Fig. 1) of photons penetrating the Be scatterer and the Fe foil. Within the CPR energy band of the (1 1 1) reflex (Fig. 1(c)) the transmission through all the materials in front of detector D5 varies between 16% and 26% [11]. The absorption losses as well as the detection probability in detector D5 (ranging from 80% to 90%) have been corrected for in order to arrive at absolute yields of unscattered CPR (see Section 2.4). 2.3. Polarimeter calibration and performance The compact design of the 110°-Compton polarimeter utilizing small volume scattering detectors (Fig. 4) is vulnerable to various instrumental asymmetries affecting the analyzing power and mimicking false polarization signals. Therefore, a careful calibration of the polarization sensitivity and a measurement of instrumental asymmetries of the polarimeter was carried out at HASYLAB/ DESY utilizing an unfocussed monochromatized beam of 11 keV synchrotron radiation of known linear polarization [11]. Fig. 5 shows measurements of scattering yields obtained for 14 different azimuthal orientations of the polarimeter about the synchrotron radiation beam spanning a range
59
Fig. 5. Azimuthal angular distribution of Compton scattered synchrotron radiation of 11 keV for different polarimeter rotation angles about the photon beam. Fractional yields obtained in the four scattering detectors are displayed in different symbols.
of about 50°. Different symbols give the yields of each of the four scattering detectors normalized to the total number of counts registered in all four detectors. The analyzing power a is determined by the measured asymmetry A of yields at the ‘‘allowed’’ (/ ¼ 90°) and ‘‘forbidden’’ (/ ¼ 0°) azimuths and the calculated degree of linear polarization P ¼ 97%: a ¼ A=P ¼ ð72 2Þ%. The error is obtained from comparisons of individual cos2 -fits to the four scattering yields at each of the polarimeter settings. Within the /-range covered by the measurements the analyzing power a does not depend on the orientation of the polarimeter with respect to the polarization plane. The cos2 -fits also determine the azimuth /0 of the polarization plane (forbidden azimuth) with a statistical uncertainty of r/0 ’ 0:3° which is small compared to the statistical and systematical errors of the CPR polarization measurements (see Section 4) and thus does not enter the final result. From measurements with pairs of detectors oriented symmetrically with respect to the polarization plane the instrumental asymmetry of the polarimeter is determined. Fig. 6 shows the corresponding modulation curve which would be flat for a perfectly symmetric polarimeter. The CPR
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with the calculated effective degree of linear polarization out to large dip angles. 2.4. Experimental procedures The local polarization properties of CPR are probed by sweeping the reflex over the polarimeter aperture through crystal rotations exploiting the well-established angular distribution and energyangular correlation shown in Fig. 1. The proce-
Fig. 6. Azimuthal modulation of Compton scattering yields introduced by instrumental asymmetries.
measurements are corrected for the instrumental asymmetry of Ainstr ¼ 0:058. As an overall test of the polarimeter performance the vertical polarization profile of the synchrotron radiation beam has been measured. The result is given in Fig. 7 and shows good agreement
Fig. 7. Vertical polarization profile of 11 keV synchrotron radiation. The dip angle W is given in units of c1 ¼ 0:11 mrad. The curve is the calculated effective degree of linear polarization normalized to the measurements (triangles) at the synchrotron plane (cW ¼ 0).
Fig. 8. Calculated contours of constant CPR flux as a function of the goniometer rotation angles U and a. Dots mark probing locations for two sets of measurements at slightly different observation angles: (a) H ¼ 20:86°, (b) 20.34°.
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dure and the theoretical distributions used to fit the measurements are presented in detail elsewhere [10,11]. The majority of measurements were done for the (1 1 1) reflex which is of highest intensity and therefore leads to the best signalto-background ratio. Fig. 8 shows calculated contours of constant flux of the (1 1 1) reflex along the polarimeter axis as a function of the goniometer rotation angles U and a for two different sequences of measurements between which a small change of the observation angle was noticed (see below). The dots (labelled A; B; C; . . . ; H) mark angular settings at which measurements were done. The probing locations are reached by angular movements from the center of the reflex at the Bragg direction after alignment with the polarimeter axis by measuring energy spectra of unscattered CPR with the forward detector D5 for different crystal orientations a and U. The spectrum measured at location A is given in Fig. 9. The dominant (1 1 1) reflex is measured with very little background. The peak energy of 10.66 keV and the line width (310 eV FWHM) correspond to the expected energy-angular correlation and the angular acceptance of the polarimeter [11]. In addition, the higher harmonics ((3 3 3) and (4 4 4)) show at their proper energies and strengths. The
Fig. 9. Energy spectrum measured by forward detector D5 at probing location A of Fig. 8(a) (see text).
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broad enhancement at the upper end of the spectrum due to minimum ionizing particles is used for online normalization of the CPR flux during the alignment procedure. Fig. 10 shows an example of a pair of scans in the angles a and U of the CPR flux and energy extracted from energy spectra measured by detector D5 . The curves correspond to fits of the theoretical distributions [10,11] to the measurements not carrying out the integration over the detector acceptance. The minima of the intensity distributions mark the alignment of the Bragg
Fig. 10. Angular scans for CPR cone alignment with polarimeter axis. Shown are CPR flux and energy measured by forward detector D5 for crystal rotations about goniometer axes a (a) and U (b). Curves correspond to fits of the theoretical flux distribution and energy-angular relation to the measurements. Sketch (c) shows where the angular cross sections were taken.
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direction with the polarimeter axis. The energy measurements show the expected dispersion in the diffraction plane (angle U) and constant energy perpendicular (angle a) (see Fig. 1). The measured energy dependence on U allows to determine the absolute value of the observation angle H ¼ 2UBragg with very high accuracy. For the first measurements (at positions A and B in Fig. 8(a)) we obtain H ¼ ð20:86 0:01Þ°. The second sequence (C to H in Fig. 8(b)) was done at H ¼ ð20:34 0:01Þ°. The shift is due to a horizontal beam displacement. The systematical error of the crystal alignment is given by beam instabilities. The very precise energy measurement in the forward detector D5 (see Fig. 10) is a sensitive monitor of the horizontal beam position (affecting angle U). From observed energy shifts of 50–100 eV uncertainties of the angle U of 1–2 mrad are inferred. Vertical beam instabilities (changing the angle a) cannot be observed by energy measurements but are generally expected to be less pronounced as all beam deflections occur in the horizontal plane. Unnoticed beam movements and corresponding uncertainties of probing locations are a major contribution to the systematic error of the local CPR polarization vector (see Section 3).
3. Polarimetry results The CPR polarization vector was first measured at two positions (A and B in Fig. 8(a)) close to the
centers of adjacent quadrants of the radiation cone where according to (1) the polarization directions should differ by 90° and where the distinction between the radial and the hyperbolic behaviour is most pronounced. The energy spectra measured by the scattering detectors D1 to D4 at points A and B are shown in Fig. 11. The strongly different yields of Compton scattered CPR (peaks at 10.37 keV (A) and 10.78 keV (B)) in detectors D1 to D4 reflect the high degree of linear polarization. At position A the smallest yield, marking the forbidden azimuth along the polarization plane, is measured in detector D4 whereas at position B the minimum occurs close to detector D2 . The azimuthal angular distributions extracted from the energy spectra are shown in Fig. 12. The measured yields are corrected for the relative efficiencies of detectors D1 to D4 (see Fig. 6) and are normalized to the total number of counts in all four scattering detectors. From cos2 -fits (solid and dashed lines in Fig. 12) the degree and the orientation of the linear polarization are obtained. The phase difference of the modulation curves of approximately 90° signals a corresponding change of the polarization direction from A to B. The phase values are as expected for the polarization angle W given by (1) (hyperbolic behaviour). The rotation of the local polarization vector within one quadrant of the radiation cone is demonstrated by a series of measurements at five different azimuths (labels C to H in Fig. 8(b)). The measured Compton scattering distributions and
Fig. 11. Energy spectra measured by polarimeter detectors D1 to D4 at positions A and B of Fig. 8(a). Scattered CPR peaks at 10.37 keV (A) and at 10.78 keV (B) of different strengths reflect a high degree of linear polarization. The line at 6.4 keV is Ka fluorescence from the Fe monitor foil at the back of the Be scatterer.
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Fig. 12. Azimuthal angular distributions of Compton scattered CPR near the centers of two adjacent quadrants of the radiation cone (locations A, B of Fig. 8(a)) The lines are cos2 -fits to the measurements.
the corresponding cos2 -fits (Fig. 13) show a phase progression due to the rotation of the polarization plane oppositely to the azimuth. The average local polarization angle W within the aperture of the polarimeter extracted from the Compton scatter-
Fig. 13. Azimuthal angular distributions of Compton scattered CPR for a series of measurements at different azimuths in one quadrant of the radiation cone (locations C to G of Fig. 8(b)). The lines are cos2 -fits to the measurements.
63
ing distributions is displayed in Fig. 14. The measurements follow the expectation (1) for CPR (solid line) and rule out a center symmetric radial polarization pattern (dotted line) as would be ex erenkov radiation. pected for quasi-C The numerical results for the local polarization angle W and the effective degree of linear polarization P of the Si(1 1 1) reflex measured within the aperture of the polarimeter are given in Table 1. In all cases the error due to the statistical uncertainty of the number of registered Compton scattered photons is smaller than the systematical errors. The major contribution to the systematical error of the polarization angle given in Table 1 is due to the uncertainty of the probing location (dHx ¼ dHy ¼ 1:4 mrad) estimated from observed shifts of the energy of the incident CPR measured in the forward detector D5 [11]. A minor contribution is due to the uncertainty of the polarimeter asymmetry correction (see Section 2.3). The systematical error of the effective degree of polarization P (not shown in Table 1) receives contributions of comparable size from various sources (analyzing power, polarimeter asymmetry, background
Fig. 14. Measurements of the polarization angle at five different azimuths in one quadrant of the Si(1 1 1) radiation cone (locations C to H of Fig. 8(b)). The error bars represent the statistical and systematical measurement errors added quadratically.
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Table 1 Comparison of measured and expected polarization results for Si(1 1 1) plane Orientation
A
U (°) a (°) Reflex quadrant
10.18 0.902 1
Polarization angle (°) (experiment) W Statistical error Systematical error (theory) W (Monte Carlo) W
45.8 2.0 8.1 34.4 34.3
Degree of polarization P (experiment) 0.86 Statistical error 0.05 P (theory) 0.97 P (Monte Carlo) 0.92
B 10.60 0.902 2 52.4 2.3 11.2 45.7 46.4 0.66 0.05 0.94 0.86
C 10.10 1.085 1 73.5 1.2 10.5 66.2 70.0 0.87 0.03 0.93 0.84
subtraction, electron beam instabilities) and ranges between 0.074 and 0.135 depending on the correlation between the individual contributions [11]. The measurements are compared to theoretical expectations and to Monte Carlo simulation results also listed in Table 1. Using the CPR angular distribution (Fig. 1(a) and (b)) and the polarization angle calculated in kinematical approximation the average polarization direction W inside the polarimeter aperture and the effective degree of polarization P ¼ cos 2ðW WÞ are computed. Smearing effects due to the e -beam emittance, energy spread and multiple scattering in the Si crystal reduce the local degree of polarization to less than one. In order to take these instrumental effects into account Monte Carlo simulations [13] have been carried out. Comparing the simulation results with the analytical integrations (‘‘theory’’ in Table 1) one notices that the effect of experimental smearings on the polarization angle W is minimal whereas the effective degree of polarization P is noticeably reduced, as expected.
4. Conclusions and outlook The measured polarization directions are in good agreement with expectations within the kinematical theory of CPR. The polarization angle
D
E
F
G
H
10.02 0.999 1
9.96 0.752 1
9.91 0.384 1
58.6 1.6 9.7 50.8 50.5
43.7 2.1 9.6 33.4 33.2
12.3 1.3 9.9 15.2 15.1
5.7 1.4 10. 2 1.95 1.9
0.79 0.04 0.94 0.87
0.75 0.05 0.95 0.88
0.85 0.04 0.95 0.89
0.84 0.04 0.95 0.89
9.90 0.05 4
10.10 1.085 1 68.0 1.7 10.5 66.2 70.0 0.79 0.05 0.93 0.84
W follows the hyperbolic pattern (see Figs. 2 and 14) as was first measured less accurately for the Si(2 2 0) reflex in [5]. The measurements definitely rule out the center symmetric radial polarization pattern suggested by measurements of [4,6] and as erenkov radiation. would be expected for quasi-C The results obtained for the effective degree of polarization inside the narrow polarimeter aperture vary between P ¼ 0:66 and 0.87 depending on the location inside the radiation reflex (Table 1). Within combined systematical and statistical errors of 9% to 15% the measurements agree marginally with expectations for CPR but are systematically lower by about 10%. Possible reasons for this slight discrepancy have been investigated [11]. A small contamination (of the order of 10%) of CPR by X-rays of a different polarization behaviour would not spoil the agreement of the measured polarization angle W with expectations but could reduce the effective degree of linear polarization P as is observed. Such a background would not be noticed if it had spectral–angular characteristics similar to CPR (see Fig. 1). CBS and diffracted forward going radiation, like transition radiation or bremsstrahlung accompanying the e -beam, are potential sources. The interference of CPR and CBS (being linearly polarized in the diffraction plane) has been investigated in [13,15]. The effect is strong only at observation angles comparable to c1 ( ¼ 7.2 mrad in this ex-
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periment). At observation angles H ’ 21° of this experiment CBS is not a significant contribution. Diffracted transition radiation (DTR) has similar polarization characteristics as CPR and therefore would not reduce the effective degree of polarization. Moreover, at the relatively low e -energy of 72 MeV the CPR energy of 11 keV is far beyond the cutoff energy cxp ¼ 4:37 keV (plasma frequency xp ¼ 31 eV for Si) for transition radiation. Therefore, DTR should be negligible. Diffracted bremsstrahlung (DBS) can in principle be discriminated from CPR by the angular distribution (peaking at the Bragg direction) and by a slightly different energy-angular correlation (Fig. 15). In practice, however, a contribution of DBS of the order of 10% to the measured CPR yield cannot be excluded. Another possible cause of the small difference between the measured and the expected degree of linear polarization is a slight underestimation of experimental smearing effects assumed for the Monte Carlo simulations. Somewhat larger values of the e -beam divergence and transverse dimensions or slightly enhanced multiple scattering in the Si crystal would still be compatible with the measured CPR energy line width of about 150 eV and at the same time would more strongly reduce the local polarization.
Fig. 15. Ratio of energies of Bragg scattered photons and of CPR as a function of crystal angle /. Within the range covered by the scan shown in Fig. 10(b) (9:8° 6 / 6 11:0°) the two energies differ by less than 0.4% or 40 eV.
65
For a complete verification of CPR polarization properties measurements at larger observation angles H need to be carried out. In particular, the radial orientation of polarization planes expected according to (1) for backward CPR (see Fig. 2) should be experimentally confirmed. Polarimetry of backward CPR will be a very difficult task since the CPR yield (Fig. 16) as well as the energy (Fig. 17) fall with increasing observation angle. In view of the very low energy of the (1 1 1) reflex of Si (E < 3 keV in the entire backward hemisphere) and correspondingly large absorption effects it appears more promising to study the less intense (by a factor of 4.4) but more energetic (2 2 0) reflex. An interesting special case is CPR emitted at H ¼ 90°. Here the p-polarization component vanishes (Fig. 16) such that the entire CPR reflex is expected to be linearly polarized perpendicular to the diffraction plane (r-polarization) to a very good approximation [5,8,9]. Polarimetry of large-angle CPR requires a polarimeter of high analyzing power and energy resolution for low energy X-rays. A 90°-Compton polarimeter utilizing Si drift detectors has been built for this purpose and is now being calibrated
Fig. 16. Relative CPR yield as a function of observation angle H. The total yield and the p- and r-polarization components are shown separately.
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BMBF (contract no. 06DA9151) and by the German Research Society DFG (contract no. FK410). References [1] [2] [3] [4] [5]
[6]
[7]
Fig. 17. CPR energy at center of (1 1 1) and (2 2 0) reflex of Si as a function of observation angle H.
[8]
[16,17]. Measurements at H ¼ 90° are presently being prepared.
[9] [10]
[11]
Acknowledgements The authors wish to thank Dr. H.-D. Gr€ af and his crew for the expert operation of the accelerator and Dr. V.V. Morokhovskii for his kind support of the Monte Carlo simulations. The MPI authors gratefully acknowledge the hospitality extended to them at S-DALINAC and the generous support of polarimeter calibrations given by HASYLAB at the Deutsches Elektronen-Synchrotron DESY. The work has been supported by the German Federal Minister for Education and Research
[12]
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