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An Xray diffractometer using mirage diffraction
^{a}Saitama Institute of Technology, Fukaya, Saitama 3690293, Japan, ^{b}Institute of Material Structure Science, KEKPF, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 3050801, Japan, and ^{c}University of Yamanashi, Kofu, 4008510, Japan
^{*}Correspondence email: tomoef@wonder.ocn.ne.jp
Some characteristics are reported of a triplecrystal diffractometer with a (+, −, +) setting of Si(220) using mirage diffraction. The first crystal is flat, while the second and third crystals are bent. Basically, the first crystal is used as a collimator, the second as a monochromator and the third as the sample. The third crystal also works as an analyzer. The advantages of this diffractometer are that its setup is easy, its structure is simple, the divergence angle from the second crystal is small and the energy resolution of the third crystal is high, of the order of submeV.
Keywords: mirage diffraction; mirage fringes; interference fringes; Xray difractometers; monochromators; dynamical theory of Xray diffraction.
1. Introduction
The refracted beam of an Xray in a bent perfect crystal propagates along a hyperbolic trajectory and comes back to the incident surface in the Bragg geometry. The refracted beam is referred to a beam representing the Poynting vector of the Xray in this paper. Authier (2001, p. 355) pointed out that the behavior of the refracted beam in a bent crystal resembled a mirage in optics. We call the diffracted beam coming out of the crystal a mirage diffraction beam. Under conditions, the divergence angle of the refracted beam is quite large compared with that of the incident beam. The refracted beam can be regarded as a quasispherical wave, even when the divergence angle of the incident beam is smaller than 1′′ (Authier, 2001, p. 313). When mirage diffraction beams interfere with each other, this results in interference fringes, which are called mirage interference fringes (Fukamachi et al., 2010). Such mirage interference fringes were first observed by Zaumseil (1978). Mirage interference fringes have been used for evaluation of the strain gradient in a bent crystal (Jongsukswat et al., 2012).
In this paper, we will describe a triplecrystal diffractometer using mirage interference fringes of Si(220) and some characteristics of the diffractometer.
2. Theoretical basis
According to Gronkowski & Malgrange (1984), the trajectory of the refracted beam in a bent crystal is given for the symmetric Bragg geometry as
for , where s(W_{s}) = −1 for W_{s} < −1 and s(W_{s}) = 1 for W_{s} > 1. is the and x and z are the coordinates parallel and normal to the crystal surface in the incident plane, respectively, with the origin at the incident point of the Xray on the surface. The parameter W_{s} is the initial value of the deviation from the Bragg condition W, which is defined by
with being the incident glancing angle and C the polarization factor. is the hth Fourier coefficient of the Xray polarizability and is a parameter corresponding to the strain gradient in the crystal, defined by
Here, is the Xray wavelength, h the reciprocal vector corresponding to the hth reflection and the displacement vector; x_{0} and x_{h} are the coordinates in the directions of the transmitted and diffracted beams, respectively. Under conditions (), the trajectory of the refracted beam has a hyperbolic form, as shown in Fig. 1 for < 0. The eccentricity of the hyperbola is related to . By taking derivatives of equation (2), the divergence angle () of the incident beam is related to the change of W as
when is fixed. In Fig. 1(a), the angle between the refracted beam and the surface is given by
Here the reflectivity (r) is defined by with and being the vectors of the incident and diffracted beams, respectively. The superscript on represents the branch index. For a nonabsorbing crystal, r is expressed by
For W = −1, = 1 and = 0, and for W = −2, = 0.07 and is approximately equal to . The angle amplification rate A between the changes in and in reflection geometry is given by Authier (2001, p. 313) as
Since A is between 10^{4} and 10^{6}, even when is less than 1′′ and the incident beam can be regarded as a quasiplane wave, is nearly equal to the and the refracted beam can be regarded as a quasispherical wave if is close to 1. This means that this single crystal works as a lens. This angle amplification can also be applied to a monochromator reflecting Xrays with very small divergence angle as well as to an analyzer with high energy resolution. Authier (1960) obtained a highly collimated incident beam by using this angle amplification in the transmission geometry so as to verify the double refraction.
In Fig. 1(b) is shown a schematic illustration of interference fringes between two mirage diffraction beams. The refracted beams and correspond to the incident beams for the values of W_{1} and W_{2}, respectively, satisfying the relation W_{2} =  (W_{1}^{2} + 3)^{1/2}/2. In Fig. 1(c), trajectories are shown of the refracted beams when W_{s} changes from −1.1 to −1.5 in the case of the Si(220) reflection. The Xray energy is 11 100 eV and is 1 mm^{−1}.
The intensities of the mirage diffraction beams are measured as a function of the distance x from the incident point of the beam to the emission point of the mirage diffraction beam. By using equation (1), the deviation () of the parameter of W_{s} from −1 is given by
It is possible to get the value of by measuring the position of mirage interference fringes x. Fig. 2 shows the diffraction geometry for an Xray with energy E and glancing angle . The thick solid line shows the dispersion surface in the crystal for the in the Bragg arrangement. L_{o} is the Lorentz point, L_{a} is the Laue point and c_{1} on the X axis is the point corresponding to W_{s} = −1. The nomenclature of the points L_{o} and L_{a} is adopted according to the books by Pinsker (1977) and Authier (1960, pp. 68–71). The line T_{0}′ represents the dispersion surface in a vacuum and is parallel to the asymptote T_{0} of the hyperbola of the dispersion surface in the crystal. is the wavevector of the incident beam whose glancing angle is and energy is E. We assume that the perpendicular line v_{1} passing the tie point corresponding to crosses the dispersion surface at c_{1}. The refracted beam excited at the point c_{1} runs parallel to the crystal surface. The distance X_{0} from L_{o} to c_{1} is given by
where , C = 1 for polarization and . When the glancing angle changes from to , the corresponding perpendicular line changes from v_{1} to v_{2} and the parameter W_{s} changes from −1 to . The tie point in a vacuum moves from a_{1} to a_{2} and that in the crystal moves from c_{1} to c_{2}. From the tie point c_{2}, the refracted beam is excited. In Fig. 2, the relation
is obtained.
3. Experimental
The experiment was carried out by using Xrays from synchrotron radiation at BL15C, KEKPF, Tsukuba, Japan. The optical system is shown in Fig. 3(d). The Xrays were polarized and the energy was tuned to 11 100 ± 0.5 eV by using a doublecrystal Si(111) monochromator. After Slit 1, the first crystal was basically used as a collimator, the second as a monochromator and the third as the sample, as shown in Fig. 3(d). The first crystal was flat. The second and third crystals were bent by applying force in the backward and forward directions of gravity, as shown in Figs. 3(b) and 3(c), respectively. The three plane parallel crystals were prepared by nondisturbance polishing at Sharan Instrument Corporation. The crystals were 50 mm long, 15 mm wide and 0.28 mm thick. The usual Bragg diffracted beam (^{1}P_{h}) passed through the second slit (Slit 2), and the first peak of the mirage interference fringes () passed through the third slit (Slit 3). Here the left superscript on P represents either the first (1), second (2) or third (3) crystal. The right superscript on P_{m} represents a serial peak number of the mirage interference fringes.
The mirage diffraction intensities of from the second crystal are shown as a function of the distance x in Fig. 4(a), where the glancing angle was fixed and β was 0.73 mm^{−1}. The intensities were measured by moving Slit 3 in front of the SC2 in Fig. 3(d) after removing the third crystal. The value of β was determined by measuring the position of the third peak as reported by Jongsukswat et al. (2012). The mirage diffraction intensities of from the third crystal are shown as a function of x in Fig. 4(b), where the glancing angle was fixed and β was 0.63 mm^{−1}. The intensities were measured by moving Slit 4 in front of the SC1. The vertical width of Slit 2 was equal to 0.02 mm and those of Slit 3 and Slit 4 were 0.04 mm.
Rocking curves of ^{1}P_{h} from the first crystal and ^{2}P_{h} from the second crystal are shown in Fig. 5(a). The FWHM of the curve ^{1}P_{h} is 9′′, which is twice as large as that of the curve ^{2}P_{h} of 4.5′′. The rocking curve of the first peak () of the mirage interference fringes, with an FWHM of 3.8′′, is shown in Fig. 5(b) (dots). It has an asymmetric form characteristic of the rocking curve from a weakly absorbing crystal. The slopes of both shoulders of the peak are steeper than those of ^{2}P_{h}. The rocking curve of ^{3}P_{h} from the third crystal is shown in Fig. 5(c) (dots). It also has an asymmetric form and is in good agreement with the curve (solid line) calculated by taking the absorption effect into account. The FWHM of the peak is 4′′. The rocking curves of ^{3}P_{h}, and ^{3}P_{t} +^{3}P_{lt} from the third crystal are shown in Fig. 6. The ordinate is the intensities and the abscissa is the incident angle. The origin of the angle (0) is taken at the center of ^{3}P_{h}, which corresponds to the Lorentz point. The mirage fringe intensity is measured by the SC1 after setting Slit 4 at the peak position of the curve of . ^{3}P_{t} is the intensity of the transmitted beam and ^{3}P_{lt} is the intensity of the emitted beam in the direction of the transmitted beam from the lateral surface of the crystal. These intensities are measured by the SC2. The sharp peak of , with an FWHM of 0.6′′, appears between the peaks of ^{3}P_{h} and ^{3}P_{t} +^{3}P_{lt}. The peaks of and ^{3}P_{t} + ^{3}P_{lt} appear on the negative angle side where the occurs.
In Fig. 7 are shown the four rocking curves for n = 1, 2, 3 and 4 from the third crystal, together with the rocking curves of ^{3}P_{t} + ^{3}P_{lt}. The origin of the angle is taken at the center of the peak of . Each curve of was measured by the SC1 after setting Slit 4 at the peak position of the mirage interference fringes, and the curve of ^{3}P_{t} + ^{3}P_{lt} was measured simultaneously by the SC2, as shown in Fig. 3(d). When n increases, the peak position of moves to the lower incident angle side, close to the peak of the curve ^{3}P_{t} + ^{3}P_{lt}. The angle difference between the two peaks and is 0.3′′. The average angle difference between two adjacent peaks is 0.1′′. The curve of ^{3}P_{t} + ^{3}P_{lt} shows two peaks: one, corresponding to the peak of ^{3}P_{lt}, appears around the origin, and the other, corresponding to the peak of the transmitted beam ^{3}P_{t}, appears around the angle of −1.3′′. When the rocking curves of the nth (n = 1, 2, 3 and 4) peak of the interference fringes are measured after setting Slit 4 at its peak position, the measured peak of ^{3}P_{t} + ^{3}P_{lt} stays at the same position as shown in Fig. 7. The peak of ^{3}P_{t} + ^{3}P_{lt} is a good reference point for the incident angle. It is possible to determine a very small angle difference between two peaks of and by measuring the curves of and ^{3}P_{t} + ^{3}P_{lt} simultaneously.
4. Discussion
Equation (1) derived by Gronkowski & Malgrange (1984) is applicable only to a monochromatic Xray of a plane wave. In the present experiment, however, since the Xrays from synchrotron radiation are emitted from a source of finite size and are monochromated by a crystal monochromator, they have a small energy bandwidth as well as a small divergence angle. It is noted that denotes the divergence angle for monochromatic Xrays and denotes the angle shift corresponding to the energy shift , which can be estimated from the observed range of mirage interference fringes. is equal to the maximum value of . It is necessary to have the relation between the diffraction of monochromatic Xrays and that of Xrays with a finite energy bandwidth and a finite divergence angle. If the Xray energy changes from E to and the amplitude of the wavevector changes from to while the glancing angle is fixed, the dispersion surface T_{0}′ moves to T_{0}′′ and the changes from to as shown in Fig. 2. Here is the amplitude of the wavevector , which is related to the energy deviation by
In Fig. 2, we have the relation between and as
Using equations (9), (11) and (12), is expressed by
When the dispersion angle changes from to , the relation holds in Fig. 2. is related to as
by using the relation or .
By inserting equation (13) into equation (8), the energy shift is related to the distance x by
It is possible to estimate by measuring x. In the present experiment, since the measured maximum value of x is 3 mm, as shown in Fig. 4(b), the maximum value of is obtained as 11 meV by using equation (15) and the maximum value of is obtained as 0.65′′ by using equation (14). By differentiating equation (15), we have the energy resolution dE obtained from the position of the mirage interference fringes with the position resolution dx as
When dx is 0.1 mm and = 0.63 mm^{−1} in the case of Si(220), is approximately 0.4 meV at x = 1.5 mm. This means that the mirage interference fringes from the third crystal can be used for energy analysis of the beams with an energy resolution of submeV. The value of 0.1 mm for dx is nearly the same as the projected width of Slit 4 on the sample surface ( mm).
In the case of angle dispersive diffractometry, an asymmetric reflection is usually used as a monochromator or an analyzer with high energy resolution from meV to submeV. For example, two asymmetrically cut crystals with (+n, +m, −m, −n) setting were used as a monochromator with the energy bandwidth of meV (Ishikawa et al., 1992; Yabashi & Ishikawa, 2000). According to the relation , it is also possible to have Xrays with a small energy bandwidth by using back reflection. Xrays with an energy resolution of 0.45 meV were obtained by using Si(13 13 13) reflection with = 89.98° for Xrays of 25.70 keV (Verbeni et al., 1996). Another monochromator with an energy resolution of submeV was designed by combining asymmetric reflection with back reflection (Baron et al., 2001; Stoupin et al., 2013). Stoupin et al. designed a monochromator with high spectral efficiency by combining asymmetric reflection from an asymmetrically cut diamond with back reflection from a silicon crystal. In these angle dispersive monochromators, it is necessary to choose an appropriate crystal and its Bragg reflection after setting the Xray energy. In contrast, it is possible to adjust the energy resolution for any energy of Xrays just by choosing a Bragg reflection and changing the strain gradient parameter for the current monochromator using mirage interference fringes.
As the FWHM of the rocking curve of in Fig. 6 is 0.6′′, which is much smaller than that of ^{1}P_{h} (9′′) in Fig. 5(a), the rocking curve of is regarded to be the intrinsic curve of the primary diffraction beam of ^{1}P_{h}. The value of 0.6′′ actually corresponds to the FWHM of . However, according to the estimation by using the beam width passing through Slit 2 with a width of 0.02 mm, the FWHM of the rocking curve of should be 0.2′′. The measured FWHM is three times larger than this estimated value. The cause of this difference is probably the source size (0.06 mm) of the synchrotron radiation Xrays from the bending magnet. If the source size is assumed to be 0.06 mm, the FWHM of the rocking curves of becomes 0.6′′, which agrees with the observed maximum value of ≃ 0.65′′. If we use an undulator instead of the bending magnet, we can obtain a beam with a much smaller divergence angle by using this mirage diffractometer, because the source size of the undulator is approximately ten times smaller than that of the bending magnet.
Fig. 8 shows a DuMond diagram to demonstrate the angle and energy resolutions of the current triplecrystal difftactometer with (+, −, +) setting. The reflection indices of 220 are the same for the three Si crystals. The glancing angle of the incident beam on each crystal is fixed when the mirage interference fringes are observed by moving Slit 4 in Fig. 3(d). The divergence of the (= 0.07′′) and the bandwidth of the wavelength (= 1.1 × 10^{  7} nm) of the second crystal used as a monochromator are determined according to the source size of the synchrotron radiation Xrays as described above. The divergence angle d of the beam from the first peak of the mirage interference fringes in Fig. 4(a) is found to be 0.003′′ by using equations (4), (14) and (16), after passing Slit 3 of 40 µm width. The value of d corresponds to the angular width of the − curve. As the mirage interference fringes from the third crystal are observed by moving Slit 4 with a vertical width of 40 µm, the angular width d of the beam from Slit 4 is 0.003′′ and the bandwidth of the wavelength d is 4 × 10^{  9} nm, which is much smaller than , as shown in Fig. 8. Slit 4 is located at when x = 0 mm and at when x = x_{max}. The energy width dE is approximately 0.4 meV by using the relation .
5. Summary
In the present experiment, the divergence angle of the beam from the first Si crystal is 4′′. When this beam is incident on the second Si crystal, the divergence angle of the mirage interference fringes is reduced to 0.6′′. The second crystal works as a monochromator to obtain a small divergence angle and thus high energy resolution. Table 1 shows the peak positions x and the corresponding values of W_{s1} (W_{s2}), and of mirage interference fringes from the third crystal. W_{s1} and W_{s2} are the values of W_{s} obtained by using equation (8) for the refracted beams and in Fig. 1(b). The intensity of the diffracted beam corresponding to is approximately five times larger than that corresponding to , as shown in Fig. 7. This is because beam directly reaches A_{2}, while beam reaches point A_{2} after being once reflected from the top surface. Then we use the value of W_{s1} for estimating and . The average value of the difference between two adjacent values of given by is 0.1′′, which agrees with the angular difference between two adjacent peaks of in Fig. 7. The value of energy width obtained in Table 1 shows that the third crystal works as an analyzer with high energy resolution.

In order to have Xrays with a very small divergence angle, an asymmetric crystal monochromator is widely used. When we use an Si(220) asymmetric monochromator for an Xray energy of 11 100 eV, for example, the b [= with a = 15.9°] is 31. The divergence angle from the asymmetric crystal monochromator is 1/5.6 (= 1/b^{1/2}) of the divergence angle from a symmetric crystal, and the is reduced by 1/31. If we use the mirage interference fringes from Si(220) as a monochromator, on the other hand, the divergence angle is 1/8 of the incident Xray and the intensity is reduced by 1/50, because the number of photons of the incident beams through the second crystal is approximately 25 000 s^{−1} and that of the mirage interference fringes from the third crystal is 500 s^{−1} according to the present experiment, as shown in Fig. 4. The intensity is the same as or slightly weaker than that from an asymmetric crystal. It is an advantage of this monochromator using the mirage interference fringes that the setup is quite easy and its structure is simple, although the divergence angle and the intensity are the same orders of magnitude as those of an angle dispersive monochromator using an asymmetric crystal.
is 16.9° and the asymmetric factorFukamachi et al. (2011) reported a monochromator with a very small divergence angle using the multipleBragg Laue mode diffraction from a lateral surface. The characteristics and usability of this instrument are nearly the same as those of the present monochromator using mirage interference fringes, but the divergence angle from the monochromator using the multipleBragg Laue mode is about twice that using mirage diffraction.
In this experiment, we have measured the intensities of mirage diffraction with a x is regarded as a spectrum of the incident Xrays projected onto the crystal according to equation (15). If we use an Xray CCD camera or a position sensitive detector instead of a the energy resolution of the spectrum should be improved to a large extent and the measuring time should be reduced. As we have very large angle amplification in the mirage diffraction, in the near future we can expect to achieve high energy resolution (less than meV) by using this diffractometer if it is combined with Xrays from an undulator beamline and a CCD camera.
by moving a slit of very small width in front of it. The intensity distribution of the mirage interference fringes as a function ofAcknowledgements
This work was carried out under the approval of the Program Advisory Committee of PF (proposal No. 2012G606). This work was partially supported by the Nano Technology Project for Private Universities (from 2011), with matching fund subsidy from the Ministry of Education, Culture, Sports, Science and Technology.
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