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Spectral-angular and polarization properties of near-axis channeling radiation of 31 MeV electrons in silicon
Nuclear Instruments and Methods in Physics Research B 145 (1998) 113±119
Spectral-angular and polarization properties of near-axis channeling radiation of 31 MeV electrons in silicon P.M. Weinmann a,*, M. Rzepka a, G. Buschhorn a, R. Kotthaus a, K.H. Schmidt a, J. Freudenberger b, H. Genz b, V.V. Morokhovskii b, A. Richter b a
Max-Planck-Institut f ur Physik (Werner-Heisenberg-Institut), D-80805 M unchen, Germany Institut f ur Kernphysik, Technische Universit at Darmstadt, D-64289 Darmstadt, Germany
b
Received 8 December 1997; received in revised form 5 March 1998
Abstract The spectral-angular and the linear polarization properties of channeling radiation emitted by 30.8 MeV electrons incident on a 13 lm thick silicon crystal at small angles with respect to various major crystal axes have been investigated. The transition from axial to planar channeling radiation properties is well described by a two-dimensional manybeam calculation based on a periodic string potential. At energies above the strongest planar channeling radiation lines coherent bremsstrahlung linearly polarized perpendicular to the polarization of the corresponding planar channeling radiation is observed and found to be well reproduced by the model calculations. Axial channeling radiation of the h1 1 0i axis shows a small amount of linear polarization due to the `double-well' potential in agreement with the calculations. Ó 1998 Elsevier Science B.V. All rights reserved. PACS: 61.85.+p; 78.70.En; 78.90.+t; 41.60.-m Keywords: Channeling radiation; Coherent bremsstrahlung; Potential calculation; Polarization
1. Introduction While a relativistic electron traverses a crystal it interacts with the electric ®eld of the crystal atoms leading to the emission of radiation. In addition to ordinary bremsstrahlung characteristic forms of radiation occur due to the coherent interaction with the periodically distributed atoms of the
* Corresponding author. Tel.: 49 89 323 54 402; fax: 49 89 32 26 704; e-mail: [email protected].
crystal lattice, i.e. channeling radiation [1,2] and coherent bremsstrahlung [3]. These types of radiation have been studied intensively over the last decades as is documented in numerous articles and monographs [4]. In a qualitative picture, channeling radiation is emitted as a consequence of the oscillatory path of an electron along a crystal axis (axial channeling) or plane (planar channeling), while coherent bremsstrahlung originates from the periodic crossing of crystal planes. The main parameter which determines the radiation characteristics is the
0168-583X/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 8 ) 0 0 3 2 0 - 6
114
P.M. Weinmann et al. / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 113±119
electron beam direction with respect to the crystal lattice. If the electron direction is nearly parallel to a crystal axis, in addition to axial channeling radiation coherent bremsstrahlung is emitted due to the crossing of planes perpendicular to this axis. As the distance between such planes is short, i.e. of the order of the lattice constant d, this type of radiation ± in Ref. [5] called coherent bremsstrahlung `type B' ± is restricted to photon energies well above the axial channeling radiation energies and thus is of no concern for this study. Enlarging the angle # between the electron and the crystal axis changes the radiation characteristics drastically. If the electron direction is chosen parallel to a crystal plane, in addition to planar channeling radiation coherent bremsstrahlung is emitted due to the crossing of planes normal to the channeling plane and containing the axis. Since the distance between consecutive crossings in this case is of the order d= sin # lower energy radiation (`type A' according to [5]) results which can contribute within the energy range of planar channeling radiation. The interplay of channeling radiation and coherent bremsstrahlung has been studied at low electron energies both theoretically and experimentally by Andersen et al. [6,7]. In Ref. [6] the authors investigate the modulation of energy spectra of coherent bremsstrahlung emitted by 3.5 MeV electrons in a thin silicon crystal under planar channeling conditions by varying the beam direction across the (1 1 0) crystal plane at a constant angle of about 2° to the h1 1 0i direction, thus keeping the coherence condition for bremsstrahlung ®xed. The main objective of the present study is the investigation of channeling radiation properties when changing from axial to planar channeling. We therefore varied the electron beam direction inside crystallographic planes in the vicinity of the h1 0 0i and h1 1 0i axes of a silicon crystal within angular ranges of several times the respective critical channeling angle of typically a few milliradians. The electron energy of 30.8 MeV was chosen such that signi®cant contributions of coherent bremsstrahlung are limited to photon energies well above the main spectral lines of planar channeling radiation.
We compare intensity and polarization spectra with many-beam calculations following the general formalism ®rst developed by [6,7]. The linear polarization allows us to probe the crystal potential in a particularly sensitive way. So far, in the energy range of interest for this investigation, there exist detailed measurements of the linear polarization only for pure axial and planar channeling radiations in silicon and diamond crystals [8]. These measurements show that axial channeling radiation produced along the azimuthally symmetric h1 0 0i axis is not linearly polarized, while planar channeling radiation is completely polarized perpendicular to the channeling plane. Unpolarized axial radiation will thus have to change into completely linearly polarized planar radiation within a narrow angular range in the vicinity of the axis which had not been explored in detail in ref. [8]. The linear polarization analysis is also a very sensitive probe for radiation components polarized dierently than planar channeling radiation such as coherent bremsstrahlung of type A which is linearly polarized perpendicular to the planar channeling radiation [3]. In Section 2 we describe the method used to calculate spectral and polarization properties of axial and planar channeling radiations. In Section 3 the numerical results of these calculations are compared with the measurements of intensity and linear polarization spectra at small angles with respect to the h1 0 0i and h1 1 0i axes of silicon.
2. Theoretical concepts An electron moving at a small angle # to a given crystal axis (z-direction) is subject to a rapid ¯uctuation of the longitudinal electric ®eld component (responsible for coherent bremsstrahlung type B) and to a slow variation of the transverse force (responsible for channeling radiation and coherent bremsstrahlung type A). We have calculated the spectral radiation properties using the socalled two-dimensional continuum potential [9] due to strings of charge along the z-axis. This transversely periodic potential is the longitudinal average of the real three-dimensional crystal
P.M. Weinmann et al. / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 113±119
potential thus neglecting high energy coherent bremsstrahlung. Such a string potential was ®rst utilized by Andersen et al. [10] for many-beam calculations of axial channeling radiation. The results of such calculations are generally in good agreement with experiments [10±12]. For the calculation of planar channeling radiation properties usually a one-dimensional continuum potential is used which is the average of the two-dimensional potential along the given plane. This reduces the numerical eort, but does not allow us to calculate the transition of axial to planar channeling. In order to calculate coherent radiation properties in this transition region we use a two-dimensional periodic string potential. In the following we summarize the main steps of the many-beam calculations performed with such a potential following the general formalism ®rst developed in Refs. [6,7]. The two-dimensional continuum potential can be expanded in the Fourier series X r U~gn : exp i~ gn~ 1 U ~ r U x; y n
The sum is over all reciprocal lattice vectors ~ gn in the plane normal to the z-axis (x±y plane). The Fourier coecients of the potential are given by U~g
2ph2 X exp ÿi~ g~ rj Aj ~ g meVE j 1 2 2 exp ÿ g q : 2
2
The sum is over all atoms with relative positions ~ rj within the primitive cell of volume VE ( d 3 for the cubic silicon crystal). For the scattering amg we have used tabulated values [13] of plitude Aj ~ a spherically symmetric Hartree±Fock potential of an isolated atom. The last term is the Debye± Waller factor and takes into account thermal vibrations with mean square amplitude in one dimension of q2 . As the potential is constant in z-direction, the wave function of the electron factorizes in a plane wave corresponding to the free motion in the zdirection and a Bloch function W for the motion in the transverse plane
X
W ~ r exp ÿi~ k?~ r
n
exp i~ gn~ r C~gn :
115
3
Introducing this wave function into the Schr odinger equation for the nonrelativistic transverse motion h2 2 r U ~ r W ~ r E? W ~ r; 4 ÿ 2cm one obtains the eigenwert equation in the reciprocal space for the in®nite matrix H u H~ u E?~
5
with elements Hmn U~gm ÿ~gn
h2 ~ 2 k? ÿ ~ gm dmn ; 2cm
~ u . . . ; C~gÿ2 ; C~gÿ1 ; C~g0 ; C~g1 ; C~g2 ; . . .:
6 7
The occupation density of the channeling states is given by the projection of the incoming plane wave ~r onto W ~ exp ÿiK~ r X ! ~ri C~gn d K? ÿ ~ gn k? ~ hW ~ rj exp ÿiK~ n
! k? ~ g0 : C~g0 d K? ÿ ~
8
(The last proportionality holds because the dfunction singles out a particular vector ~ g. The transformation properties of Eqs. (5) and (6) cause k? ÿ ~ gn to be identical the solutions for ~ k? and ~ except for a shift in the indici of the eigenvectors: g C~g ! C~gÿ~gn :) We have numbered the vectors ~ in such a way, that ~ g0 0. Thus only chan~? K ~ sin # cos u; neling states with ~ k? K sin # sin u; 0 are populated at the crystal surface with a probability C~g20 . The redistribution of this inital population along the path through the crystal is considered by an average over a distribution of incident angles. The width of this distribution is mainly given by the mean de¯ection due to multiple scattering in the crystal. The intensity of the transition between two states is given by the product of the population of the initial state and the square of the transition amplitude ~ S given in dipole approximation by X ~ g: 9 C~gi C~gf ~ S ~ g
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P.M. Weinmann et al. / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 113±119
Each single transition is completely linearly polarized along the direction of ~ S. To compare the solution of Eq. (5) with experimental results one has to consider the line shape of the radiation spectrum due to transverse bound state transitions and the smearing of the incident electron direction due to beam divergence and multiple scattering. The line width C is mainly given by incoherent scattering of the incoming electron on thermal atomic vibrations or atomic electrons. As the observed line shape is also aected by various experimental conditions we have chosen to ®t an empirical function to the measured width rather than to include C into the model calculations. A common value was taken for the fractional line width for all crystal orientations determined by ®tting the calculated to the experimental spectra yielding C 0:1 keV 0:14 Ephoton . The shape of a single transition line is then given by a skew Lorentzian (Lorentzian times the photon phase space factor Ephoton ) with line width C. Whereas Eq. (5) gives the coherent radiation spectrum for a given angle of incidence in the experiment the electron direction is distributed about a mean value. At the crystal surface, the width of this distribution is given by the electron beam divergence and increases due to multiple scattering inside the crystal. We have taken the electron angular distribution into account by averaging over about 7000 theoretical spectra for directions distributed around the mean angle in steps of 0.01° in the transverse angular coordinates. Each spectrum is weighted by a gaussian of width r 2.3 mrad determined by ®tting the results of the calculation for the width of an angular scan normal to the (1 0 0) plane at a large angle to the h1 0 0i axis to the experimental data. This angular width is consistent with the beam divergence of about 0.6 mrad and a mean angle of multiple scattering in a 13 lm thick silicon crystal of 2.5 mrad [14]. In order to keep the numerical eort manageable we have carefully investigated the convergence of the solution of Eq. (5) with increasing matrix size N. Whereas for large angles of incidence to the h1 0 0i axis a matrix size of N 593 is sucient for an accuracy of about 1% for pure
axial channeling a value of N 1389 leads to the same accuracy. 3. Experiment and results The experiment was carried out at the superconducting electron linear accelerator S-DALINAC [15]. An uncollimated continuous-wave beam of (30:8 0:3) MeV electrons has been steered onto a 13 lm thick silicon crystal mounted on a three-axes goniometer. For a detailed description of the experimental setup we refer to [8]. For the new experiment described here we have improved the Compton polarimeter [16] now employing two identical germanium detectors to measure simultaneously 90° Compton scattering at two orthogonal azimuths. The pair of detectors can be rotated about the beam axis to measure the complete azimuthal angular distribution of Compton scattering yields. The azimuthal symmetry of the polarimeter was adjusted and controlled carefully. In particular, the scattering target, a 40 mm long polyethylene cylinder of 14 mm diameter, was centered relative to the photon beam with high precision in order to minimize photon absorption asymmetries. The analyzing power of the polarimeter was determined by a Monte Carlo simulation [17] based on experimental channeling spectra measured in the forward direction and taking into account absorption and multiple Compton scattering in the target. In Fig. 1(a) we show intensity spectra for channeling radiation after subtraction of ordinary bremsstrahlung which was measured independently by rotating the crystal to an arbitrary orientation. Given here is the transition of axial channeling at the h1 0 0i axis (0.00°) to planar channeling at the (1 1 0) plane for selected angles between the electron beam and the h1 0 0i axis. The spectrum for 0.50° is very similar to the pure planar channeling spectrum far away from the axis which is not shown in Fig. 1. The spectra are normalized to each other and show a sharp decrease in the integral intensity while the planar channeling radiation lines emerge. Fig. 1(b) shows the corresponding results of our calculation which
P.M. Weinmann et al. / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 113±119
Fig. 1. Intensity spectra of channeling radiation produced by 30.8 MeV electrons in silicon: (a) measurements after subtraction of ordinary bremsstrahlung, (b) model calculation results based on a two-dimensional periodic string potential, shown is the 1 1 0 ! h1 0 0i transition for four angles of incidence relative to the h1 0 0i axis.
are generally in good agreement with the measured spectra. Fig. 2 gives the degree of linear polarization as measured (a) and as calculated (b) for the same experimental conditions as in Fig. 1. Axial channeling radiation (0.00°) along the h1 0 0i axis shows no linear polarization in agreement with the expectation for the rotationally symmetric string potential of this axis. Enlarging the incident angle causes the degree of linear polarization to increase mainly for planar bound state transition energies. Pure planar channeling at large angles of incidence (2.56°) is 100% polarized for all photon energies. The calculated degree of linear polarization shown in Fig. 2(b) is again in good agreement with the measurements. Deviations, as e.g. observed at 0.50°, are accounted for by temporary beam instabilities.
117
Fig. 2. Polarization spectra of channeling radiation produced by 30.8 MeV electrons in silicon, shown is the 1 1 0 ! h1 0 0i transition for ®ve angles of incidence relative to the h1 0 0i axis.
Figs. 3 and 4 show the high energy parts of the intensity and polarization spectra beyond the range of Figs. 1 and 2 which is expected to contain coherent bremsstrahlung corresponding to the crossing of (1 1 0) planes orthogonal to the channeling planes. While the smooth intensity spectra of Fig. 3 do not give clear evidence of coherent bremsstrahlung the polarization spectra of Fig. 4 demonstrate its presence by the observed change of sign of the linear polarization. As expected, the linear polarization of coherent bremsstrahlung type A is negative, i.e. orthogonal to the linear polarization direction of planar channeling radiation. The observed polarization minima in Fig. 4(a) coincide with the energies of the expected strong contributions of the (2 2 0) re¯ex at hx 200 and 400 keV for # 0.25° and 0.50°, respectively. In Fig. 5 we show the intensity and polarization spectra for pure axial channeling along the h1 1 0i axis. This axis consists of two parallel strings of charges forming a `double-well' potential which is not rotationally symmetric. Therefore, one expects
118
P.M. Weinmann et al. / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 113±119
Fig. 3. Intensity spectra at energies beyond the energy range of Fig. 1 for two selected angles of incidence relative to the h1 0 0i axis. Bremsstrahlung contributions have been subtracted, negative ¯uctuations in the dierence spectra have been truncated.
axial channeling radiation along the h1 1 0i axis to be partially linearly polarized. This is indeed observed (Fig. 5(a)) and reproduced in the model calculation (Fig. 5(b)) of polarization spectra. 4. Summary The comparison of experimental results on spectral intensities and linear polarization properties of radiation emitted by 30.8 MeV electrons traversing a 13 lm thick silicon crystal close to major crystal axes with those of two-dimensional many-beam calculations based on a periodic string potential shows good overall agreement. In particular, detailed measurements of the linear polarization in the transition region between axial and planar channeling are well reproduced. The two-dimensional many-beam calculation therefore describes all features of planar and axial channeling radiations in a uni®ed manner. Moreover, the linear polarization proves to be a very sensitive
Fig. 4. Polarization spectra at energies beyond the energy range of Fig. 2 for two selected angles of incidence relative to the h1 0 0i axis. Coherent bremsstrahlung type A is polarized perpendicular to the polarization direction of the planar channeling radiation and thus causes negative values of the degree of polarization. The range of negative linear polarization is shifted towards larger energies with increasing angle, as expected for coherent bremsstrahlung contributions.
observable to distinguish between channeling radiation and coherent bremsstrahlung at energies where intensity spectra are not distinctly dierent. We have shown for the ®rst time that axial channeling radiation for the h1 1 0i axis of silicon is partially linearly polarized in contrast to the radiation along the rotationally symmetric h1 0 0i axis which is not linearly polarized. Acknowledgements This work was supported by the German Federal Minister for Research and Technology (BMBF) under contract No. 06DA820. The authors wish to thank the technical sta of the MaxPlanck-Institut f ur Physik, in particular W. Erbe, for their contributions to the construction and
P.M. Weinmann et al. / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 113±119
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References
Fig. 5. Intensity (a) and linear polarization (b) spectra of channeling radiation produced along the h1 1 0i axis of silicon. Due to the double-well potential of this axis the radiation is partially linearly polarized in agreement with the result of the calculation.
installation of the experiment and H.-D. Gr af and the S-DALINAC group for the excellent electron beams. The MPI authors gratefully acknowledge the hospitality extended to them at the Institut f ur Kernphysik at Darmstadt.
[1] M.A. Kumakhov, Phys. Lett. A 57 (1976) 17. [2] M.J. Alguard, R.L. Swent, R.H. Pantell, B.L. Berman, S.D. Bloom, Phys. Rev. Lett. 42 (1979) 1148. [3] G.D. Palazzi, Rev. Mod. Phys. 40 (1968) 611. [4] A.W. Saenz, H. Uberall (Eds.), Coherent Radiation Sources, Springer, Berlin, 1985; N.P. Kalashnikov, Coherent Interactions of Charged Particles in Single Crystals, Harwood academic publishers, Chur, 1988; M.A. Kumakhov, R. Wedell, Radiation of Relativistic Light Particles during Interaction with Single Crystals, Spektrum Akademischer Verlag, Heidelberg, 1991. [5] A.W. Saenz, H. Uberall, Phys. Rev. B 25 (1982) 4418. [6] J.U. Andersen, E. Lñgsgaard, Nucl. Instr. and Meth. B 33 (1988) 11. [7] J.U. Andersen, K.R. Eriksen, E. Lñgsgaard, Physica Scripta 24 (1981) 588. [8] M. Rzepka et al., Phys. Rev. B 52 (1995) 771. [9] J. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 34 (14) (1965). [10] J.U. Andersen, E. Bonderup, E. Lñgsgaard, B.B. Marsh, A.H. Sùrensen, Nucl. Instr. and Meth. 194 (1982) 209. [11] R.K. Klein, J.O. Kephart, R.H. Pantell, H. Park, B.L. Berman, R.L. Swent, S. Datz, R.W. Fearick, Phys. Rev. B 31 (1985) 68. [12] W. Lotz, H. Genz, P. Homann, U. Nething, A. Richter, A. Weickenmeier, H. Kohl, W. Kn upfer, J.P.F. Sellschop, Nucl. Instr. and Meth. B 48 (1990) 256. [13] P.A. Doyle, P.S. Turner, Acta Cryst. A 24 (1968) 390. [14] V.L. Highland, Nucl. Instr. and Meth. 129 (1975) 497. [15] J. Auerhammer et al., Nucl. Phys. A 553 (1993) 841c. [16] P.M. Weinmann, Ph.D. thesis, Technische Universit at M unchen, in preparation. [17] M. Rzepka, Ph.D. thesis, Technische Universit at M unchen, 1994; Report MPI-PhE/94-19, 1994.
Spectral-angular and polarization properties of near-axis channeling radiation of 31 MeV electrons in silicon P.M. Weinmann a,*, M. Rzepka a, G. Buschhorn a, R. Kotthaus a, K.H. Schmidt a, J. Freudenberger b, H. Genz b, V.V. Morokhovskii b, A. Richter b a
Max-Planck-Institut f ur Physik (Werner-Heisenberg-Institut), D-80805 M unchen, Germany Institut f ur Kernphysik, Technische Universit at Darmstadt, D-64289 Darmstadt, Germany
b
Received 8 December 1997; received in revised form 5 March 1998
Abstract The spectral-angular and the linear polarization properties of channeling radiation emitted by 30.8 MeV electrons incident on a 13 lm thick silicon crystal at small angles with respect to various major crystal axes have been investigated. The transition from axial to planar channeling radiation properties is well described by a two-dimensional manybeam calculation based on a periodic string potential. At energies above the strongest planar channeling radiation lines coherent bremsstrahlung linearly polarized perpendicular to the polarization of the corresponding planar channeling radiation is observed and found to be well reproduced by the model calculations. Axial channeling radiation of the h1 1 0i axis shows a small amount of linear polarization due to the `double-well' potential in agreement with the calculations. Ó 1998 Elsevier Science B.V. All rights reserved. PACS: 61.85.+p; 78.70.En; 78.90.+t; 41.60.-m Keywords: Channeling radiation; Coherent bremsstrahlung; Potential calculation; Polarization
1. Introduction While a relativistic electron traverses a crystal it interacts with the electric ®eld of the crystal atoms leading to the emission of radiation. In addition to ordinary bremsstrahlung characteristic forms of radiation occur due to the coherent interaction with the periodically distributed atoms of the
* Corresponding author. Tel.: 49 89 323 54 402; fax: 49 89 32 26 704; e-mail: [email protected].
crystal lattice, i.e. channeling radiation [1,2] and coherent bremsstrahlung [3]. These types of radiation have been studied intensively over the last decades as is documented in numerous articles and monographs [4]. In a qualitative picture, channeling radiation is emitted as a consequence of the oscillatory path of an electron along a crystal axis (axial channeling) or plane (planar channeling), while coherent bremsstrahlung originates from the periodic crossing of crystal planes. The main parameter which determines the radiation characteristics is the
0168-583X/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 8 ) 0 0 3 2 0 - 6
114
P.M. Weinmann et al. / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 113±119
electron beam direction with respect to the crystal lattice. If the electron direction is nearly parallel to a crystal axis, in addition to axial channeling radiation coherent bremsstrahlung is emitted due to the crossing of planes perpendicular to this axis. As the distance between such planes is short, i.e. of the order of the lattice constant d, this type of radiation ± in Ref. [5] called coherent bremsstrahlung `type B' ± is restricted to photon energies well above the axial channeling radiation energies and thus is of no concern for this study. Enlarging the angle # between the electron and the crystal axis changes the radiation characteristics drastically. If the electron direction is chosen parallel to a crystal plane, in addition to planar channeling radiation coherent bremsstrahlung is emitted due to the crossing of planes normal to the channeling plane and containing the axis. Since the distance between consecutive crossings in this case is of the order d= sin # lower energy radiation (`type A' according to [5]) results which can contribute within the energy range of planar channeling radiation. The interplay of channeling radiation and coherent bremsstrahlung has been studied at low electron energies both theoretically and experimentally by Andersen et al. [6,7]. In Ref. [6] the authors investigate the modulation of energy spectra of coherent bremsstrahlung emitted by 3.5 MeV electrons in a thin silicon crystal under planar channeling conditions by varying the beam direction across the (1 1 0) crystal plane at a constant angle of about 2° to the h1 1 0i direction, thus keeping the coherence condition for bremsstrahlung ®xed. The main objective of the present study is the investigation of channeling radiation properties when changing from axial to planar channeling. We therefore varied the electron beam direction inside crystallographic planes in the vicinity of the h1 0 0i and h1 1 0i axes of a silicon crystal within angular ranges of several times the respective critical channeling angle of typically a few milliradians. The electron energy of 30.8 MeV was chosen such that signi®cant contributions of coherent bremsstrahlung are limited to photon energies well above the main spectral lines of planar channeling radiation.
We compare intensity and polarization spectra with many-beam calculations following the general formalism ®rst developed by [6,7]. The linear polarization allows us to probe the crystal potential in a particularly sensitive way. So far, in the energy range of interest for this investigation, there exist detailed measurements of the linear polarization only for pure axial and planar channeling radiations in silicon and diamond crystals [8]. These measurements show that axial channeling radiation produced along the azimuthally symmetric h1 0 0i axis is not linearly polarized, while planar channeling radiation is completely polarized perpendicular to the channeling plane. Unpolarized axial radiation will thus have to change into completely linearly polarized planar radiation within a narrow angular range in the vicinity of the axis which had not been explored in detail in ref. [8]. The linear polarization analysis is also a very sensitive probe for radiation components polarized dierently than planar channeling radiation such as coherent bremsstrahlung of type A which is linearly polarized perpendicular to the planar channeling radiation [3]. In Section 2 we describe the method used to calculate spectral and polarization properties of axial and planar channeling radiations. In Section 3 the numerical results of these calculations are compared with the measurements of intensity and linear polarization spectra at small angles with respect to the h1 0 0i and h1 1 0i axes of silicon.
2. Theoretical concepts An electron moving at a small angle # to a given crystal axis (z-direction) is subject to a rapid ¯uctuation of the longitudinal electric ®eld component (responsible for coherent bremsstrahlung type B) and to a slow variation of the transverse force (responsible for channeling radiation and coherent bremsstrahlung type A). We have calculated the spectral radiation properties using the socalled two-dimensional continuum potential [9] due to strings of charge along the z-axis. This transversely periodic potential is the longitudinal average of the real three-dimensional crystal
P.M. Weinmann et al. / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 113±119
potential thus neglecting high energy coherent bremsstrahlung. Such a string potential was ®rst utilized by Andersen et al. [10] for many-beam calculations of axial channeling radiation. The results of such calculations are generally in good agreement with experiments [10±12]. For the calculation of planar channeling radiation properties usually a one-dimensional continuum potential is used which is the average of the two-dimensional potential along the given plane. This reduces the numerical eort, but does not allow us to calculate the transition of axial to planar channeling. In order to calculate coherent radiation properties in this transition region we use a two-dimensional periodic string potential. In the following we summarize the main steps of the many-beam calculations performed with such a potential following the general formalism ®rst developed in Refs. [6,7]. The two-dimensional continuum potential can be expanded in the Fourier series X r U~gn : exp i~ gn~ 1 U ~ r U x; y n
The sum is over all reciprocal lattice vectors ~ gn in the plane normal to the z-axis (x±y plane). The Fourier coecients of the potential are given by U~g
2ph2 X exp ÿi~ g~ rj Aj ~ g meVE j 1 2 2 exp ÿ g q : 2
2
The sum is over all atoms with relative positions ~ rj within the primitive cell of volume VE ( d 3 for the cubic silicon crystal). For the scattering amg we have used tabulated values [13] of plitude Aj ~ a spherically symmetric Hartree±Fock potential of an isolated atom. The last term is the Debye± Waller factor and takes into account thermal vibrations with mean square amplitude in one dimension of q2 . As the potential is constant in z-direction, the wave function of the electron factorizes in a plane wave corresponding to the free motion in the zdirection and a Bloch function W for the motion in the transverse plane
X
W ~ r exp ÿi~ k?~ r
n
exp i~ gn~ r C~gn :
115
3
Introducing this wave function into the Schr odinger equation for the nonrelativistic transverse motion h2 2 r U ~ r W ~ r E? W ~ r; 4 ÿ 2cm one obtains the eigenwert equation in the reciprocal space for the in®nite matrix H u H~ u E?~
5
with elements Hmn U~gm ÿ~gn
h2 ~ 2 k? ÿ ~ gm dmn ; 2cm
~ u . . . ; C~gÿ2 ; C~gÿ1 ; C~g0 ; C~g1 ; C~g2 ; . . .:
6 7
The occupation density of the channeling states is given by the projection of the incoming plane wave ~r onto W ~ exp ÿiK~ r X ! ~ri C~gn d K? ÿ ~ gn k? ~ hW ~ rj exp ÿiK~ n
! k? ~ g0 : C~g0 d K? ÿ ~
8
(The last proportionality holds because the dfunction singles out a particular vector ~ g. The transformation properties of Eqs. (5) and (6) cause k? ÿ ~ gn to be identical the solutions for ~ k? and ~ except for a shift in the indici of the eigenvectors: g C~g ! C~gÿ~gn :) We have numbered the vectors ~ in such a way, that ~ g0 0. Thus only chan~? K ~ sin # cos u; neling states with ~ k? K sin # sin u; 0 are populated at the crystal surface with a probability C~g20 . The redistribution of this inital population along the path through the crystal is considered by an average over a distribution of incident angles. The width of this distribution is mainly given by the mean de¯ection due to multiple scattering in the crystal. The intensity of the transition between two states is given by the product of the population of the initial state and the square of the transition amplitude ~ S given in dipole approximation by X ~ g: 9 C~gi C~gf ~ S ~ g
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Each single transition is completely linearly polarized along the direction of ~ S. To compare the solution of Eq. (5) with experimental results one has to consider the line shape of the radiation spectrum due to transverse bound state transitions and the smearing of the incident electron direction due to beam divergence and multiple scattering. The line width C is mainly given by incoherent scattering of the incoming electron on thermal atomic vibrations or atomic electrons. As the observed line shape is also aected by various experimental conditions we have chosen to ®t an empirical function to the measured width rather than to include C into the model calculations. A common value was taken for the fractional line width for all crystal orientations determined by ®tting the calculated to the experimental spectra yielding C 0:1 keV 0:14 Ephoton . The shape of a single transition line is then given by a skew Lorentzian (Lorentzian times the photon phase space factor Ephoton ) with line width C. Whereas Eq. (5) gives the coherent radiation spectrum for a given angle of incidence in the experiment the electron direction is distributed about a mean value. At the crystal surface, the width of this distribution is given by the electron beam divergence and increases due to multiple scattering inside the crystal. We have taken the electron angular distribution into account by averaging over about 7000 theoretical spectra for directions distributed around the mean angle in steps of 0.01° in the transverse angular coordinates. Each spectrum is weighted by a gaussian of width r 2.3 mrad determined by ®tting the results of the calculation for the width of an angular scan normal to the (1 0 0) plane at a large angle to the h1 0 0i axis to the experimental data. This angular width is consistent with the beam divergence of about 0.6 mrad and a mean angle of multiple scattering in a 13 lm thick silicon crystal of 2.5 mrad [14]. In order to keep the numerical eort manageable we have carefully investigated the convergence of the solution of Eq. (5) with increasing matrix size N. Whereas for large angles of incidence to the h1 0 0i axis a matrix size of N 593 is sucient for an accuracy of about 1% for pure
axial channeling a value of N 1389 leads to the same accuracy. 3. Experiment and results The experiment was carried out at the superconducting electron linear accelerator S-DALINAC [15]. An uncollimated continuous-wave beam of (30:8 0:3) MeV electrons has been steered onto a 13 lm thick silicon crystal mounted on a three-axes goniometer. For a detailed description of the experimental setup we refer to [8]. For the new experiment described here we have improved the Compton polarimeter [16] now employing two identical germanium detectors to measure simultaneously 90° Compton scattering at two orthogonal azimuths. The pair of detectors can be rotated about the beam axis to measure the complete azimuthal angular distribution of Compton scattering yields. The azimuthal symmetry of the polarimeter was adjusted and controlled carefully. In particular, the scattering target, a 40 mm long polyethylene cylinder of 14 mm diameter, was centered relative to the photon beam with high precision in order to minimize photon absorption asymmetries. The analyzing power of the polarimeter was determined by a Monte Carlo simulation [17] based on experimental channeling spectra measured in the forward direction and taking into account absorption and multiple Compton scattering in the target. In Fig. 1(a) we show intensity spectra for channeling radiation after subtraction of ordinary bremsstrahlung which was measured independently by rotating the crystal to an arbitrary orientation. Given here is the transition of axial channeling at the h1 0 0i axis (0.00°) to planar channeling at the (1 1 0) plane for selected angles between the electron beam and the h1 0 0i axis. The spectrum for 0.50° is very similar to the pure planar channeling spectrum far away from the axis which is not shown in Fig. 1. The spectra are normalized to each other and show a sharp decrease in the integral intensity while the planar channeling radiation lines emerge. Fig. 1(b) shows the corresponding results of our calculation which
P.M. Weinmann et al. / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 113±119
Fig. 1. Intensity spectra of channeling radiation produced by 30.8 MeV electrons in silicon: (a) measurements after subtraction of ordinary bremsstrahlung, (b) model calculation results based on a two-dimensional periodic string potential, shown is the 1 1 0 ! h1 0 0i transition for four angles of incidence relative to the h1 0 0i axis.
are generally in good agreement with the measured spectra. Fig. 2 gives the degree of linear polarization as measured (a) and as calculated (b) for the same experimental conditions as in Fig. 1. Axial channeling radiation (0.00°) along the h1 0 0i axis shows no linear polarization in agreement with the expectation for the rotationally symmetric string potential of this axis. Enlarging the incident angle causes the degree of linear polarization to increase mainly for planar bound state transition energies. Pure planar channeling at large angles of incidence (2.56°) is 100% polarized for all photon energies. The calculated degree of linear polarization shown in Fig. 2(b) is again in good agreement with the measurements. Deviations, as e.g. observed at 0.50°, are accounted for by temporary beam instabilities.
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Fig. 2. Polarization spectra of channeling radiation produced by 30.8 MeV electrons in silicon, shown is the 1 1 0 ! h1 0 0i transition for ®ve angles of incidence relative to the h1 0 0i axis.
Figs. 3 and 4 show the high energy parts of the intensity and polarization spectra beyond the range of Figs. 1 and 2 which is expected to contain coherent bremsstrahlung corresponding to the crossing of (1 1 0) planes orthogonal to the channeling planes. While the smooth intensity spectra of Fig. 3 do not give clear evidence of coherent bremsstrahlung the polarization spectra of Fig. 4 demonstrate its presence by the observed change of sign of the linear polarization. As expected, the linear polarization of coherent bremsstrahlung type A is negative, i.e. orthogonal to the linear polarization direction of planar channeling radiation. The observed polarization minima in Fig. 4(a) coincide with the energies of the expected strong contributions of the (2 2 0) re¯ex at hx 200 and 400 keV for # 0.25° and 0.50°, respectively. In Fig. 5 we show the intensity and polarization spectra for pure axial channeling along the h1 1 0i axis. This axis consists of two parallel strings of charges forming a `double-well' potential which is not rotationally symmetric. Therefore, one expects
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Fig. 3. Intensity spectra at energies beyond the energy range of Fig. 1 for two selected angles of incidence relative to the h1 0 0i axis. Bremsstrahlung contributions have been subtracted, negative ¯uctuations in the dierence spectra have been truncated.
axial channeling radiation along the h1 1 0i axis to be partially linearly polarized. This is indeed observed (Fig. 5(a)) and reproduced in the model calculation (Fig. 5(b)) of polarization spectra. 4. Summary The comparison of experimental results on spectral intensities and linear polarization properties of radiation emitted by 30.8 MeV electrons traversing a 13 lm thick silicon crystal close to major crystal axes with those of two-dimensional many-beam calculations based on a periodic string potential shows good overall agreement. In particular, detailed measurements of the linear polarization in the transition region between axial and planar channeling are well reproduced. The two-dimensional many-beam calculation therefore describes all features of planar and axial channeling radiations in a uni®ed manner. Moreover, the linear polarization proves to be a very sensitive
Fig. 4. Polarization spectra at energies beyond the energy range of Fig. 2 for two selected angles of incidence relative to the h1 0 0i axis. Coherent bremsstrahlung type A is polarized perpendicular to the polarization direction of the planar channeling radiation and thus causes negative values of the degree of polarization. The range of negative linear polarization is shifted towards larger energies with increasing angle, as expected for coherent bremsstrahlung contributions.
observable to distinguish between channeling radiation and coherent bremsstrahlung at energies where intensity spectra are not distinctly dierent. We have shown for the ®rst time that axial channeling radiation for the h1 1 0i axis of silicon is partially linearly polarized in contrast to the radiation along the rotationally symmetric h1 0 0i axis which is not linearly polarized. Acknowledgements This work was supported by the German Federal Minister for Research and Technology (BMBF) under contract No. 06DA820. The authors wish to thank the technical sta of the MaxPlanck-Institut f ur Physik, in particular W. Erbe, for their contributions to the construction and
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References
Fig. 5. Intensity (a) and linear polarization (b) spectra of channeling radiation produced along the h1 1 0i axis of silicon. Due to the double-well potential of this axis the radiation is partially linearly polarized in agreement with the result of the calculation.
installation of the experiment and H.-D. Gr af and the S-DALINAC group for the excellent electron beams. The MPI authors gratefully acknowledge the hospitality extended to them at the Institut f ur Kernphysik at Darmstadt.
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