- Email: [email protected]
Mosaic crystals as X-ray phase plates
m-_
Nuclear Instruments and Methods in Physics Research A 361 (199.5)354-357
__ ki!iB
NUCLEAR INSTRUMENTS 8 METHODS IN PHYStCS RESEARCH
ELSEVIER
Mosaic crystals as X-ray phase plates Carlos Giles a-b,Ckile Malgrange b3*, Fraqois de Bergevin aye,JosC Goulon a, Fransois Baudelet d, Alain Fontaine d, Christian Vettier a, Andreas Freund a a ESRF(European Synchrotron Radiation Facility), B.P. 220. F-38 043 Grenoble Cedex, France b Laboratoire de Min.&alogie-Cristallographie, assock! au CNRS, lJniuersitt% Paris VI et Paris VII, 4 place Jussieu, F-75252, Paris Cedex 05, France ’ Laboratoire de Cristallographie du CNRS associC d I’lJniversiti J. Fourier de Grenoble, B.P. 066 X, F-38043 Grenoble Cedex, France ’ LURE, laboratoire mixte du CEA-CNRS-MEN* associt! & I’lrniuersitt? de Paris-Sud (Orsay), Batiment 209d. F-91 405 Orsay Cedex, France Received 10 November 1994; revised form received 19 January 1995 Abstract It is shown that a beryllium crystal whose mosaicity is of the order of 80 arcsec can be used as an efficient X-ray quarter-wave plate. The experiment is realized on the energy dispersive absorption spectrometer at LURE (Orsay, France), where X-ray circular magnetic dichroic spectra, proportional to the circular polarization rate, are recorded.
Circularly polarized X-rays are a very useful probe for magnetic materials since they interact with angular momenta. Various experiments have been carried out using generally the elliptically polarized photons delivered by bending magnets below or above the synchrotron orbit plane (see for example Refs. [l-5]). Another technique consists in using linearly polarized photons transformed into circularly polarized ones by X-ray phase plates. Up to now most experiments that have been performed to show the efficiency of X-ray phase plates have used a very well collimated monochromatic wave and perfect silicon crystal phase plates [6-91. Conversely, some recent experiments have proven [lo-121 that diamond crystals act as very efficient quarter-wave plates in an energy-dispersive absorption spectrometer where the beam is polychromatic and divergent. Furthermore, the experimental conditions were such that the monochromatic divergence was of the order of one minute of arc. These results prompted us to measure the efficiency of imperfect crystals, with a mosaic spread comparable with the above mentioned beam divergence, as phase plates. We report here the results which have been obtained with a beryllium crystal, grown at the Max Planck Institut fiir Metallforschung (Stuttgart), whose mosaic spread was about 80 arcsec. The efficiency of the Be phase plate is here determined
by transforming a linearly polarized beam into a circularly polarized one and measuring the circular magnetic dichroic spectrum (CMXD) of HoFe, at the Ho L,,, absorption edge on an energy dispersive absorption spectrometer (Dll station at LURE Orsay-France). The spectrum thus obtained is compared to the one measured without quarterwave plate but using the elliptically polarized beam delivered by a bending magnet below the orbit plane of the positrons in the DC1 synchrotron ring at LURE. The principle of the phase plate used here, first suggested by Dmitrienko and Belyakov [13], and experimentally tested by Hirano et al. [8] is described in Ref. [lo]. The phase plate is adjusted near the Bragg peak but outside the diffraction profile at an angular distance of its center A6 called the offset and the transmitted beam is used. The two components u and n of the electric field perpendicular and parallel to the plane of diffraction respectively, are transmitted with different indices of refraction n, and n,,,. When the offset is big compared to the width of the rocking-curve their difference is r,A2 ’ F,Fi; sin2Ba n -_n ZZ D ?I A8 i 2lTv i and the resulting phase-shift nents is 4=
* Corresponding author. Tel. + 33 44 27 52 21, fax + 33 44 27 37 85, e-mail [email protected]. Elsevier Science B.V. SSDI 0168-9002(95)00132-8
E+,-,,,t=
_
4 between
(1) the two compo-
t t=Az,
1
(2)
C. Giles et al. /Nucl.
Instr. and Me& in Phys. Res. A 361 (1995) 354-357
where t is the beam-path inside the crystal, F, and FL are the structure factors for h and h reflections respectively, A the wavelength, V the volume of the unit cell, 6, the Bragg angle and re the classical electron radius. Since Eq. (1) is an asymptotic form of the exact solution given by the dynamical theory of X-ray diffraction, a more straightforward derivation is expected to come through an asymptotic approximation to that theory. Such an approximation actually relies on a multiple scattering expansion written up to the second order term. As it is known, the normal index of refraction is obtained by adding the wave scattered once in the forward direction by all the electrons in the material to the incident wave. This normal index does not depend on polarization. The next term in the expansion comes from the waves scattered twice before leaving the material in the forward direction. They add incoherently except near the Bragg condition, where they produce some contribution to the indices no and n=, whose difference can be shown to be given by Eq. (1) [ 141. These doubly scattered waves add coherently, provided that the crystalline coherence is maintained between all pairs of lattice planes whose distance is smaller than some range. This minimum coherence range is equal to a few times the extinction length divided by the ratio of the angular offset to the Darwin width. Consequently, the bigger is the offset, the smaller is the thickness along which the crystal has to be perfect, which explains that some mosaicity does not destroy the birefringence at large offsets. Let us consider a linearly polarized X-ray beam and a phase plate whose plane of diffraction makes an angle + with the polarization vector. If the plate is adjusted in order to get a given phase-shift do between u and 1~ components, the wave after the phase plate is partially circularly polarized with a circular polarization rate T (defined as the ratio of the difference between the right and left-handed circularly polarized intensity (I, - 1,) and the total intensity)
I, - IL
7= = sin2*sin+. IR + II_ If $ = n/4 and 4 = f n/2, then the beam transmitted by the plate is circularly polarized. The energy dispersive absorption spectrometer has been described elsewhere [15,16]. Let us only recall that the synchrotron beam is focused by a curved silicon crystal with a curvature such that the source is far outside the Rowland circle. The reflected beam is then polychromatic and focused to a point inside the Rowland circle where the sample is placed. Rays of each energy converge on the Rowland circle where a position sensitive detector made of a photodiode array is placed. The detector collects simultaneously the full spectrum diffracted by the curved crystal after its partial absorption by the sample. The circular magnetic X-ray dichroic spectrum (CMXD) is the differ-
355
ence between the absorption spectra of right and left-handed circularly polarized X-rays but it is usually easier to use an either right or left-handed polarized beam and to record the difference between two spectra with the magnetic field successively parallel and antiparallel to the photon wave vector. This approach is valid as long as the magnetization is saturated. The energy and the direction of the wave-vector of the photons in the beam issued from the curved polychromator are correlated. In order to achieve a constant offset with respect to the phase plate for all the energies, the bent polychromator and the flat phase plate should form a non-dispersive setup. This nondispersive condition links the interplanar distances of the reflecting planes used in the bent crystal (d,) and in the phase plate (d,,) through the relationship [lo]
$! =
[sine:+ E$it]“‘,
(4)
where /3 is proportional to cos+ and, in a symmetric geometry for the bent crystal, equal to
(W. - SinWnJ P=coS~(1/R,-sin8,/p,)’
(5)
where Br is the mean Bragg angle on the bent crystal, R, its radius of curvature and p. the source to bent crystal distance. Eq. (5) applied to a bent Si(ll1) crystal in symmetric Bragg geometry and a beryllium phase plate, at a photon energy near the HoL,,, absorption edge (8071 eV), is best satisfied by the 100 reflection on the Be crystal. The t,f~ angle is then equal to 39.4”. In order to reduce the sensitivity to the crystal mosaicity and to the divergence of each monochromatic beam, the working offset has to be of the order of 100 arcsec. In this condition, a a/2 phase-shift would be obtained for a beam-path in beryllium equal to 18.7 mm resulting in an extremely weak transmission of 0.006. To reduce the effective beampath for a given offset we have to consider a stronger reflection. The 002 reflection is the strongest Be reflection and gives a n/2 phaseshift between u and rr polarizations at an angular offset of 100 arcsec for a beampath limited to 4.6 mm and a transmission factor of about 0.29, which is really good. On the other hand the nondispersive condition is satisfied for an angle $ equal to 29.6”, resulting in a still acceptable degradation of the circular polarization rate (by a factor 0.86 as given by sin2JI in Eq. (3)). The beryllium crystal which has been used is a 2.9 mm thick (100) slice with a diameter around 12 mm. The (001) lattice planes are normal to the surface and the geometry is a symmetric Laue case. The Bragg angle for the 002 reflection at the Ho L,,, absorption edge is 25.4”. The beam-path far from Bragg reflection is 3.2 mm resulting in a transmission of 0.41 and an offset equal to 70.5 arcsec
356
C. Giles et al./Nucl.
Instr. and Meth. in Phys. Res. A 361 (1995) 354-357
-0.02
-0.04
e -50
0
50
100
150 200
E - E, (eV) Fig. 1. CMXD spectra obtained at HoL,,, absorption edge (I$, = 8071 eV) on a HoFez sample. (a) with a diamond phase plate, (b) with a Be phase plate, (c) superposition of spectrum (a) and spectrum (b) multiplied by a 1.12 factor.
0.08 2 0.06 E 0.04
-50
0
50
100 150 200
E - E, (eV) Fig. 2. CMXD spectra obtained for various offsets. From the bottom to the top the offsets are respectively equal to -501.4, -213.4, -87.4, -47.8, 6.2, 63.8, 117.8, 200.6, 254.6, 506.6, 1010.6 arcsec. The different spectra are translated by 0.01 one with respect to the other.
-0.8 t..,..i.,,.,I,....I.....i -1200 -600 0 600 A0 (arcsec)
1200
Fig. 3. Amplitudes of the CMXD signal as a function of the offset. The experimental points have been multiplied by an arbitrary factor in order to fit a theoretical curve obtained form the convolution of 7 by a Gaussian with a 95 arcsec FWHM.
for a 1r/2 phase-shift. It will be shown below that the optimum offset is significantly larger than this calculated value because of the monochromatic divergence. Fig. 1 shows the CMXD spectrum obtained with the beryllium quarter-wave plate set at an offset equal to 117.8 arcsec compared to what has been obtained with a diamond quarter-wave plate (QWP) [12]. The amplitude of the CMXD signal is only slightly lower with the beryllium QWP. The energy range of the collected spectrum is much larger due to the larger lateral size of the beryllium crystal. Different CMXD spectra have been recorded at different offsets of the beryllium QWP (Fig. 2). The amplitude of the CMXD signal, proportional to the circular polarization rate and defined here as the peak-to-peak difference between the maximum and minimum on each side of the absorption edge is plotted as a function of the offset in Fig. 3. It has been multiplied by an arbitrary factor in order to fit the theoretical T values deduced from Eqs. (3) and (2) and convoluted by a Gaussian profile in order to take into account the monochromatic divergence of the beam. The best fit is obtained for a FWHM of the Gaussian equal to 95 arcsec. One can then deduce the value of the polarization rate for every offset. For example, for the offset at point A in Fig. 3 the absolute value of the polarization rate is found equal to 0.63. The same series of CMXD data (same sample and absorption edge) had been performed with a diamond phase plate [12] and fitted by the same procedure which allows to determine the polarization rate. For a given offset the polarization rate thus determined was 0.69. The ratio of these two polarization rates, which have been independently determined, should be equal to the ratio of the CMXD signals. The agreement is good since one gets 0.69/0.63 = 1.10 while the ratio of the measured CMXD signals is equal to 1.12. Note that the here mentioned polarization rate values are obtained under the assumption of a complete linear polarization in the incident beam. In fact, a preliminary measurement of the linear polarization in this experiment [17] has given 0.90 leading to an expected polarization rate of 0.57. The orbital plane of the synchrotron was experimentally determined by measuring CMXD spectra without the phase plate for different vertical positions of the horizontal slit and noting the position where the CMXD signal is zero. At such a position the slit selects a beam composed of linearly poiarized and unpolarized X-rays. Since the helicity can be reversed by reversing the offset one can think of getting a CMXD spectrum by switching the offset instead of changing the direction of the magnetic field. This helicity flip has been carried out successfully with a diamond phase-plate [12]. However it was necessary to take the half-difference of two such CMXD spectra recorded successively with two opposite magnetic fields in order to eliminate the small background which appeared on each spectrum. This background comes probably from a difference of transmission by the diamond plate on each side of the diffraction profile but its com-
C. Giles et al. /iiucl.
Ins@. and
Meth. in Phys. Res. A 361 (1995) 354-357
357
be 0.73. The possibility to use mosaic crystals enlarges the choice of suitable crystals for phase plates.
would
Acknowledgements One of us (C.G.) acknowledges a Ph.D. grant from the Brazilian government through the RI-L4E/CNPq.
References
-0.011 -40
,‘I
’ ’ ’
0
40
’ ’ .* ’ 80
120
1 160
E - E, (eV)
Fig. 4. Upper CUNCS:each curve represents the difference between normalized absorption spectra obtained with a fixed magnetic field (+ B and - B respectively) and a Be phase plate at two opposite offsets. The signals have been divided by 10 for representation. Lower curve: CMXD signal obtained as the half-difference between the two upper curves.
analysis requires deeper investigation. Such a procedure is absolutely essential here as shown on Fig. 4. Each CMXD spectrum obtained by switching the offset of the beryllium phase plate and keeping fixed the magnetic field does not present any feature of a Ho Ln, CMXD spectrum but their difference gives a surprisingly satisfactory spectrum even if not perfect. In conclusion, it has been shown that imperfect crystals can be used as X-ray phase plates using the transmitted beam far from Bragg angle. The beryllium crystal with a 80 arcsec mosaicity used at a 100 arc set offset gives a circular polarization rate evaluated to 0.63 assuming an incident linear polarization equal to 100%. The nondispersive condition required to incline the QWP diffraction plane with respect to the horizontal plane by a I) angle equal to 30”. If the I& angle were equal to 45” and the incident beam fully linearly polarized, the polarization rate
plete
[l] G. Schiitz, R. Frahm, P. Mautner, R. Wienke, W. Wagner, W. Wilhelm and P. Kienle, Phys. Rev. Lett. 62 (1989) 2620. [2] C.T. Chen, F. Sette, Y. Ma, S. Modesti and N.V. Smith, Phys. Rev. B 42 (1990) 7262. [3] F. Baudelet, E. Dartyge, G. Krill, J.P. Klapper, C. Brouder, M. Piecuch and A. Fontaine, Phys. Rev. B 43 (1991) 5857. [4] S.P Collins, D. Laundy and A.J. Rollason, Phil. Mag. B 65 (1992) 37. [5] M.J. Cooper, D. Laundy, D.A. Cardwell, D.N. Timms, R.S. Holt and G. Clark, Phys. Rev. B 34 (1986) 5984. [6] J.A. Golovchenko, B.M. Kincaid, R.A. Levesque, A.E. Meixner and D.R. Kaplan, Phys. Rev. Lett. 57 (1986) 202. [7] D.M. Mills, Nucl. Instr. and Meth. A 266 (1988) 531 [8] K. Hirano, K. Izumi, T. Ishikawa, S. Annaka and S. Kikuta, Jpn. J. Appl. Phys. 30 (1991) L407. [9] B.W. Batterman, Phys. Rev. B 45 (1992) 12677. [IO] C. Giles, C. Malgrange, J. Goulon, F. de Bergevin, C. Vettier, E. Dartyge, A. Fontaine, C. Giorgetti and S. Pizzini, J. Appl. Cryst. 27 (1994) 232. [ll] C. Giles, C. Malgrange, J. Goulon, C. Vettier, F. de Bergevin, A. Freund, P. Elleaume, E. Dartyge, A. Fontaine, C. Giorgetti and S. Pizzini SPIE vol. 2010, X-ray and Ultraviolet Polarimetry (1993) pp. 136-149. [12] C. Giles, C. Malgrange, J. Goulon, F. de Bergevin, C. Vettier, A. Fontaine, E. Dartyge and S. Pizzini, Nucl. Instr. and Meth. A 349 (1994) 622. [13] V.E. Dmitrienko and V.A. Belyakov, Sov. Techn. Phys. Lett. 6 (1980) 621. [14] F. de Bergevin, in preparation. [15] H. Tolentino, E. Dartyge, A. Fontaine and G. Tourillon, J. Appl. Cryst. 21 (1988) 15. 1161 H. Tolentino, F. Baudelet, E. Dartyge, A. Fontaine and G. Tourillon, Nucl. Instr. and Meth. A 289 (1990) 307. [17] C. Malgrange, C. Giles, F. de Bergevin, C. Vettier, A. Fontaine, E. Dartyge and F. Baudelet, in preparation.
Nuclear Instruments and Methods in Physics Research A 361 (199.5)354-357
__ ki!iB
NUCLEAR INSTRUMENTS 8 METHODS IN PHYStCS RESEARCH
ELSEVIER
Mosaic crystals as X-ray phase plates Carlos Giles a-b,Ckile Malgrange b3*, Fraqois de Bergevin aye,JosC Goulon a, Fransois Baudelet d, Alain Fontaine d, Christian Vettier a, Andreas Freund a a ESRF(European Synchrotron Radiation Facility), B.P. 220. F-38 043 Grenoble Cedex, France b Laboratoire de Min.&alogie-Cristallographie, assock! au CNRS, lJniuersitt% Paris VI et Paris VII, 4 place Jussieu, F-75252, Paris Cedex 05, France ’ Laboratoire de Cristallographie du CNRS associC d I’lJniversiti J. Fourier de Grenoble, B.P. 066 X, F-38043 Grenoble Cedex, France ’ LURE, laboratoire mixte du CEA-CNRS-MEN* associt! & I’lrniuersitt? de Paris-Sud (Orsay), Batiment 209d. F-91 405 Orsay Cedex, France Received 10 November 1994; revised form received 19 January 1995 Abstract It is shown that a beryllium crystal whose mosaicity is of the order of 80 arcsec can be used as an efficient X-ray quarter-wave plate. The experiment is realized on the energy dispersive absorption spectrometer at LURE (Orsay, France), where X-ray circular magnetic dichroic spectra, proportional to the circular polarization rate, are recorded.
Circularly polarized X-rays are a very useful probe for magnetic materials since they interact with angular momenta. Various experiments have been carried out using generally the elliptically polarized photons delivered by bending magnets below or above the synchrotron orbit plane (see for example Refs. [l-5]). Another technique consists in using linearly polarized photons transformed into circularly polarized ones by X-ray phase plates. Up to now most experiments that have been performed to show the efficiency of X-ray phase plates have used a very well collimated monochromatic wave and perfect silicon crystal phase plates [6-91. Conversely, some recent experiments have proven [lo-121 that diamond crystals act as very efficient quarter-wave plates in an energy-dispersive absorption spectrometer where the beam is polychromatic and divergent. Furthermore, the experimental conditions were such that the monochromatic divergence was of the order of one minute of arc. These results prompted us to measure the efficiency of imperfect crystals, with a mosaic spread comparable with the above mentioned beam divergence, as phase plates. We report here the results which have been obtained with a beryllium crystal, grown at the Max Planck Institut fiir Metallforschung (Stuttgart), whose mosaic spread was about 80 arcsec. The efficiency of the Be phase plate is here determined
by transforming a linearly polarized beam into a circularly polarized one and measuring the circular magnetic dichroic spectrum (CMXD) of HoFe, at the Ho L,,, absorption edge on an energy dispersive absorption spectrometer (Dll station at LURE Orsay-France). The spectrum thus obtained is compared to the one measured without quarterwave plate but using the elliptically polarized beam delivered by a bending magnet below the orbit plane of the positrons in the DC1 synchrotron ring at LURE. The principle of the phase plate used here, first suggested by Dmitrienko and Belyakov [13], and experimentally tested by Hirano et al. [8] is described in Ref. [lo]. The phase plate is adjusted near the Bragg peak but outside the diffraction profile at an angular distance of its center A6 called the offset and the transmitted beam is used. The two components u and n of the electric field perpendicular and parallel to the plane of diffraction respectively, are transmitted with different indices of refraction n, and n,,,. When the offset is big compared to the width of the rocking-curve their difference is r,A2 ’ F,Fi; sin2Ba n -_n ZZ D ?I A8 i 2lTv i and the resulting phase-shift nents is 4=
* Corresponding author. Tel. + 33 44 27 52 21, fax + 33 44 27 37 85, e-mail [email protected]. Elsevier Science B.V. SSDI 0168-9002(95)00132-8
E+,-,,,t=
_
4 between
(1) the two compo-
t t=Az,
1
(2)
C. Giles et al. /Nucl.
Instr. and Me& in Phys. Res. A 361 (1995) 354-357
where t is the beam-path inside the crystal, F, and FL are the structure factors for h and h reflections respectively, A the wavelength, V the volume of the unit cell, 6, the Bragg angle and re the classical electron radius. Since Eq. (1) is an asymptotic form of the exact solution given by the dynamical theory of X-ray diffraction, a more straightforward derivation is expected to come through an asymptotic approximation to that theory. Such an approximation actually relies on a multiple scattering expansion written up to the second order term. As it is known, the normal index of refraction is obtained by adding the wave scattered once in the forward direction by all the electrons in the material to the incident wave. This normal index does not depend on polarization. The next term in the expansion comes from the waves scattered twice before leaving the material in the forward direction. They add incoherently except near the Bragg condition, where they produce some contribution to the indices no and n=, whose difference can be shown to be given by Eq. (1) [ 141. These doubly scattered waves add coherently, provided that the crystalline coherence is maintained between all pairs of lattice planes whose distance is smaller than some range. This minimum coherence range is equal to a few times the extinction length divided by the ratio of the angular offset to the Darwin width. Consequently, the bigger is the offset, the smaller is the thickness along which the crystal has to be perfect, which explains that some mosaicity does not destroy the birefringence at large offsets. Let us consider a linearly polarized X-ray beam and a phase plate whose plane of diffraction makes an angle + with the polarization vector. If the plate is adjusted in order to get a given phase-shift do between u and 1~ components, the wave after the phase plate is partially circularly polarized with a circular polarization rate T (defined as the ratio of the difference between the right and left-handed circularly polarized intensity (I, - 1,) and the total intensity)
I, - IL
7= = sin2*sin+. IR + II_ If $ = n/4 and 4 = f n/2, then the beam transmitted by the plate is circularly polarized. The energy dispersive absorption spectrometer has been described elsewhere [15,16]. Let us only recall that the synchrotron beam is focused by a curved silicon crystal with a curvature such that the source is far outside the Rowland circle. The reflected beam is then polychromatic and focused to a point inside the Rowland circle where the sample is placed. Rays of each energy converge on the Rowland circle where a position sensitive detector made of a photodiode array is placed. The detector collects simultaneously the full spectrum diffracted by the curved crystal after its partial absorption by the sample. The circular magnetic X-ray dichroic spectrum (CMXD) is the differ-
355
ence between the absorption spectra of right and left-handed circularly polarized X-rays but it is usually easier to use an either right or left-handed polarized beam and to record the difference between two spectra with the magnetic field successively parallel and antiparallel to the photon wave vector. This approach is valid as long as the magnetization is saturated. The energy and the direction of the wave-vector of the photons in the beam issued from the curved polychromator are correlated. In order to achieve a constant offset with respect to the phase plate for all the energies, the bent polychromator and the flat phase plate should form a non-dispersive setup. This nondispersive condition links the interplanar distances of the reflecting planes used in the bent crystal (d,) and in the phase plate (d,,) through the relationship [lo]
$! =
[sine:+ E$it]“‘,
(4)
where /3 is proportional to cos+ and, in a symmetric geometry for the bent crystal, equal to
(W. - SinWnJ P=coS~(1/R,-sin8,/p,)’
(5)
where Br is the mean Bragg angle on the bent crystal, R, its radius of curvature and p. the source to bent crystal distance. Eq. (5) applied to a bent Si(ll1) crystal in symmetric Bragg geometry and a beryllium phase plate, at a photon energy near the HoL,,, absorption edge (8071 eV), is best satisfied by the 100 reflection on the Be crystal. The t,f~ angle is then equal to 39.4”. In order to reduce the sensitivity to the crystal mosaicity and to the divergence of each monochromatic beam, the working offset has to be of the order of 100 arcsec. In this condition, a a/2 phase-shift would be obtained for a beam-path in beryllium equal to 18.7 mm resulting in an extremely weak transmission of 0.006. To reduce the effective beampath for a given offset we have to consider a stronger reflection. The 002 reflection is the strongest Be reflection and gives a n/2 phaseshift between u and rr polarizations at an angular offset of 100 arcsec for a beampath limited to 4.6 mm and a transmission factor of about 0.29, which is really good. On the other hand the nondispersive condition is satisfied for an angle $ equal to 29.6”, resulting in a still acceptable degradation of the circular polarization rate (by a factor 0.86 as given by sin2JI in Eq. (3)). The beryllium crystal which has been used is a 2.9 mm thick (100) slice with a diameter around 12 mm. The (001) lattice planes are normal to the surface and the geometry is a symmetric Laue case. The Bragg angle for the 002 reflection at the Ho L,,, absorption edge is 25.4”. The beam-path far from Bragg reflection is 3.2 mm resulting in a transmission of 0.41 and an offset equal to 70.5 arcsec
356
C. Giles et al./Nucl.
Instr. and Meth. in Phys. Res. A 361 (1995) 354-357
-0.02
-0.04
e -50
0
50
100
150 200
E - E, (eV) Fig. 1. CMXD spectra obtained at HoL,,, absorption edge (I$, = 8071 eV) on a HoFez sample. (a) with a diamond phase plate, (b) with a Be phase plate, (c) superposition of spectrum (a) and spectrum (b) multiplied by a 1.12 factor.
0.08 2 0.06 E 0.04
-50
0
50
100 150 200
E - E, (eV) Fig. 2. CMXD spectra obtained for various offsets. From the bottom to the top the offsets are respectively equal to -501.4, -213.4, -87.4, -47.8, 6.2, 63.8, 117.8, 200.6, 254.6, 506.6, 1010.6 arcsec. The different spectra are translated by 0.01 one with respect to the other.
-0.8 t..,..i.,,.,I,....I.....i -1200 -600 0 600 A0 (arcsec)
1200
Fig. 3. Amplitudes of the CMXD signal as a function of the offset. The experimental points have been multiplied by an arbitrary factor in order to fit a theoretical curve obtained form the convolution of 7 by a Gaussian with a 95 arcsec FWHM.
for a 1r/2 phase-shift. It will be shown below that the optimum offset is significantly larger than this calculated value because of the monochromatic divergence. Fig. 1 shows the CMXD spectrum obtained with the beryllium quarter-wave plate set at an offset equal to 117.8 arcsec compared to what has been obtained with a diamond quarter-wave plate (QWP) [12]. The amplitude of the CMXD signal is only slightly lower with the beryllium QWP. The energy range of the collected spectrum is much larger due to the larger lateral size of the beryllium crystal. Different CMXD spectra have been recorded at different offsets of the beryllium QWP (Fig. 2). The amplitude of the CMXD signal, proportional to the circular polarization rate and defined here as the peak-to-peak difference between the maximum and minimum on each side of the absorption edge is plotted as a function of the offset in Fig. 3. It has been multiplied by an arbitrary factor in order to fit the theoretical T values deduced from Eqs. (3) and (2) and convoluted by a Gaussian profile in order to take into account the monochromatic divergence of the beam. The best fit is obtained for a FWHM of the Gaussian equal to 95 arcsec. One can then deduce the value of the polarization rate for every offset. For example, for the offset at point A in Fig. 3 the absolute value of the polarization rate is found equal to 0.63. The same series of CMXD data (same sample and absorption edge) had been performed with a diamond phase plate [12] and fitted by the same procedure which allows to determine the polarization rate. For a given offset the polarization rate thus determined was 0.69. The ratio of these two polarization rates, which have been independently determined, should be equal to the ratio of the CMXD signals. The agreement is good since one gets 0.69/0.63 = 1.10 while the ratio of the measured CMXD signals is equal to 1.12. Note that the here mentioned polarization rate values are obtained under the assumption of a complete linear polarization in the incident beam. In fact, a preliminary measurement of the linear polarization in this experiment [17] has given 0.90 leading to an expected polarization rate of 0.57. The orbital plane of the synchrotron was experimentally determined by measuring CMXD spectra without the phase plate for different vertical positions of the horizontal slit and noting the position where the CMXD signal is zero. At such a position the slit selects a beam composed of linearly poiarized and unpolarized X-rays. Since the helicity can be reversed by reversing the offset one can think of getting a CMXD spectrum by switching the offset instead of changing the direction of the magnetic field. This helicity flip has been carried out successfully with a diamond phase-plate [12]. However it was necessary to take the half-difference of two such CMXD spectra recorded successively with two opposite magnetic fields in order to eliminate the small background which appeared on each spectrum. This background comes probably from a difference of transmission by the diamond plate on each side of the diffraction profile but its com-
C. Giles et al. /iiucl.
Ins@. and
Meth. in Phys. Res. A 361 (1995) 354-357
357
be 0.73. The possibility to use mosaic crystals enlarges the choice of suitable crystals for phase plates.
would
Acknowledgements One of us (C.G.) acknowledges a Ph.D. grant from the Brazilian government through the RI-L4E/CNPq.
References
-0.011 -40
,‘I
’ ’ ’
0
40
’ ’ .* ’ 80
120
1 160
E - E, (eV)
Fig. 4. Upper CUNCS:each curve represents the difference between normalized absorption spectra obtained with a fixed magnetic field (+ B and - B respectively) and a Be phase plate at two opposite offsets. The signals have been divided by 10 for representation. Lower curve: CMXD signal obtained as the half-difference between the two upper curves.
analysis requires deeper investigation. Such a procedure is absolutely essential here as shown on Fig. 4. Each CMXD spectrum obtained by switching the offset of the beryllium phase plate and keeping fixed the magnetic field does not present any feature of a Ho Ln, CMXD spectrum but their difference gives a surprisingly satisfactory spectrum even if not perfect. In conclusion, it has been shown that imperfect crystals can be used as X-ray phase plates using the transmitted beam far from Bragg angle. The beryllium crystal with a 80 arcsec mosaicity used at a 100 arc set offset gives a circular polarization rate evaluated to 0.63 assuming an incident linear polarization equal to 100%. The nondispersive condition required to incline the QWP diffraction plane with respect to the horizontal plane by a I) angle equal to 30”. If the I& angle were equal to 45” and the incident beam fully linearly polarized, the polarization rate
plete
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