Pseudo-spherical stepped diffractor constructed under constant step width conditions (multi-stepped monochromator)

Abstract

A diffractor for electromagnetic radiation is based on a pseudo-spherical stepped geometry designed under the constant step width conditions. The diffractor consists of a few small pseudo-spherical curved dispersive elements (oriented crystal surfaces or gratings) that are located on a focal circle. The location on the focal circle of each element is made to guarantee the same Bragg angle for the incident radiation. Thus a diffractor is an array of diffracting elements (“steps”), each one contributing to the total solid angle of the diffractor, that increase the spectral output of the device without decreasing the resolution.

Description

FIELD OF THE INVENTION

[0002] The present invention relates to the field of electromagnetic radiation diffraction such as x-rays or neutrons. More particularly, it consists essentially of a diffractor for electromagnetic radiation based on a pseudo-spherical stepped geometry designed under the constant step width conditions.

BACKGROUND OF THE INVENTION

[0003] In many spectral devices characteristic radiation is generated by a small area of the sample surface. In such a case the source can be considered point like. In this case the conventional energy dispersive focusing technique is made either by cylindrical curved crystal (Advances in X-Ray Spectroscopy, Eds. C. Bonnelle, C. Mande, (Oxford, U.K., 1982)) or by doubly curved, on spherical or toroidal surfaces, crystal monochromators (U.S. Pat. No. 4,882,780; U.S. Pat. No. 4,807,268). These diffractors focus the monochromatic radiation onto the entrance detector slit. According to the Bragg's equation, the spectral resolution Δλ/λ depends both on θ and Δθ:

Δλ/λ=Δθ/tan θ  (1)

[0004] The intensity of the monochromatic radiation is proportional to the area of the diffractor surface, that reflects x-rays under the given Bragg's angle θ within the range ±Δθ. However, increasing the reflecting area, a widening of the aperture ratio of the diffractor occurs associated to a simultaneous decrease of the spectral resolution.

[0005] In the last decade analytical investigations of the shape and size of the reflecting area of a crystal-monochromator surface, employing different focusing methods have been carried out (D. B. Wittry and S. Sun, J. Appl. Phys. 67, 1633 (1990); D. B. Wittry and S. Sun, J. Appl. Phys. 68, 387 (1990); D. B. Wittry and S. Sun, J. Appl. Phys. 69, 3886 (1991); D. B. Wittry and S. Sun, J. Appl. Phys. 71, 564 (1992); W. Z. Chand and D. B. Wittry, J. Appl. Phys. 74, 2999 (1993)). Indeed, x-ray diffractors with double curved crystal provide significantly greater aperture ratio compared to that based on the cylindrical Johann or Johannson geometries. For such devices assuming an incidence angle θ>20° and a crystal height of L<0.1R, the reflecting surface projection on the XZ plane is rectangular and the projection on the focal circle plane (XY plane) is an arc of radius R=2r, where r is the focal circle radius. The knowledge of the shape of the reflecting surface allows an estimation of the parameters of a spherical diffractor designed with a stepped surface (D. B. Wittry and S. Sun, J. Appl. Phys. 69, 3886 (1991)) and in the case of constant step height, the aperture of this diffractor is larger than a spherical curved crystal.

SUMMARY OF THE INVENTION

[0006] An aim of the invention is to provide a diffractor specially dedicated to the x-rays range, based on a pseudo-spherical geometry directed to replace the plane or curved crystals in various apparatus (e.g., x-ray microanalyzer, x-ray photoelectron, spectrometer for chemical analysis, etc.).

[0007] For this purpose, the diffractor according to the present invention consists of a few small spherical curved dispersive elements (oriented crystal surfaces or gratings) that are located on a focal circle. The location on the focal circle of each element is made to guarantee the same Bragg angle for the incident radiation. Thus a diffractor according to the present invention is an array of diffracting elements (“steps”), each one contributing to the total solid angle of the diffractor, that increases the spectral output of the device without decreasing the resolution.

[0008] In the present invention a stepped diffractor is based on the physical condition that the single element of the diffractor subtends a constant angle width. With such design the efficiency of the diffractor is maximized and it is greater than both a spherical diffractor (U.S. Pat. No. 4,882,780; U.S. Pat. No. 4,807,268) and a stepped diffractor constructed under the constant step height condition (D. B. Wittry and S. Sun, J. Appl. Phys. 67, 1633 (1990)).

BRIEF DESCRIPTION OF THE DRAWINGS

[0009] The accompanying drawings, which are incorporated in and form a part of this application, illustrate the embodiments of the present invention and, together with the following detailed description, serve to explain the principles of the invention.

[0010] In the drawings:

[0011] FIG. 1 is a pictorial view of a three step (because of the symmetry there are really five steps) diffractor in accordance with the present invention;

[0012] FIG. 2 is a detailed view of the upper part of a single step of the diffractor (the bottom part of a step is the same according to the symmetry with respect to the focal plane);

[0013] FIG. 3 is the topography of the reflecting area of the radiation focused by the diffractor surface under different Bragg angle deviation (see text);

[0014] FIG. 4 is the scheme of a stepped surface diffractor under the φ=constant condition (φ is the half-width angle of the i-th step) in the focusing circle plane;

[0015] FIG. 5 is a diagram of the gain obtained by replacing a spherical diffracting crystal as that reported in the publication: “W. A. Caliebe, S. Bajt and C-C. Kao, Rev. Sci. Instrum. 67(1996) 1” with a multi-stepped diffractor using a germanium (Ge) crystal with different reflecting planes at the energy of Cr-Kβ; and

[0016] FIG. 6 is similar to the diagram of FIG. 5 replacing the Ge crystal with a silicon (Si) crystal that has different reflecting planes.

[0017] For purposes of this application, no distinction should be made between upper and lower case letters in the Greek alphabet, i.e., upper and lower case Greek letters refer to and mean the same thing.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

[0018] In describing a preferred embodiment of the invention, specific terminology will be selected for the sake of clarity. However, the invention is not intended to be limited to the specific terms so selected, and it is understood that each specific term includes all technical equivalents that operate in a similar manner to accomplish a similar purpose.

[0019] FIG. 1 shows a scheme of a three step device (actually the array is formed by five elements because of the symmetry). The shadowed areas show the reflecting surface of each step of the device placed on a Rowland circle of radius R.

[0020] FIG. 2 shows in more detail the upper part of a single diffracting element (step) of the diffractor (the bottom part of the step is omitted according to the fact that there is a symmetry with respect to the focal plane). In addition, FIG. 3 shows the calculated topography of the radiation focused on the diffractor surface under different Bragg angle deviations. As the deviations from the Bragg angle lead out of the Bragg's conditions, such deviations result in a lower resolution. Three areas are reported in FIG. 3 that produce different spectral resolutions (the darker area corresponds to the higher resolution). The results of FIG. 3 are very important to the understanding of the gain in the spectral output of the present device when compared with a spherical bent diffracting monocrystal like those in U.S. Pat. No. 4,882,780 and U.S. Pat. No. 4,807,268. Indeed, this figure shows that only a small region (the darkest region somewhat resembling the form of a butterfly) of the entire surface contributes to the high resolution reflection of the electromagnetic radiation. Therefore, it is necessary, in an ordinary device, to limit the reflecting area of the crystal to achieve a high spectral resolution, clearly in contrast with the need to use a large area to increase the throughput.

[0021] In the present multi-stepped geometry, the special geometric design allows one to enhance the contribution of the central and best part of the topography (the dark butterfly) increasing the number of the diffracting elements. Thus we can obtain both a high resolution condition for the entire surface of the diffractor, and an increase of the total solid angle.

[0022] FIG. 4 presents the scheme of the stepped surface diffractor in the focal plane with all parameters used of the calculation of the theoretical parameters of the device. In the XZ plane each step is a curved strip along the z-axis of length L (total crystal height). It lies on a spherical surface of radius Ri where Ri=OAi; O′ is the center of the focal circle; and hi=BiDi is the height of the i-th step. L is the sum of all of the hi; E is the step height in the XZ plane where L=NE. Here φ is the half-width angle of the i-th step in the focal circle plane. In particular, OA0=R0=2r; where r is the focusing circle radius; Ai is the point of intersection between the i-th step surface and the focal circle; OAi is the curvature radius of the i-th step; S is the focusing point; EiBi is the part of the i-th step shadowed by the (i+1)-th step. In the plane of FIG. 4 the steps are shown by the arcs 2φi=2(AiBi), where i=0, 1, . . . , N.

[0023] Ai is also the point of intersection between the i-th step in the focal circle plane. The point O is clearly geometrically defined in both FIGS. 2 and 4. It is the point lying on the Rowland circle that define this geometry, indeed OA0=RO0=2r, where r is the focusing circle radius. As a consequence of the design, all the segments OAi define the radius of the generic i-th step.

[0024] The extension of the arc may be expressed by angle (2φi). It is clear the strategy of the device, each step has the same arc in term of angle, but not in length because the radius of curvature (R) of each step, has to change. Moreover, Ai and Bi represent the center of the curved surface of the generic i-th step and the point on the outer edge (as viewed in FIG. 4, Bi is the left edge) of the same step, respectively. Both points lie on the ideal curved crystal surface of the generic i-th step; similarly the point Di also lies on the ideal curved crystal surface of the generic i-th step and is the symmetric of Bi.

[0025] Herein below, the equations to calculate this diffractor scheme are discussed. The value of the Bragg angle deviation Δθ is calculated for given magnitudes of: λ, Δλ and θ. The initial value φ0=Y0 was determined by using the equation (see D. B. Wittry and S. Sun, J. Appl. Phys. 67, 1633 (1990))

2Δθ=X2 cot θ+X3 cot2 θ−XZ2 cot2 θ  (2)

[0026] Eq. (2) determines the coordinates of the boundaries of the reflection area on the spherical crystal surface. The radius R0=2r establishes the curvature of the central step. One time fixed the focal circle radius r, the parameters of the remaining steps are evaluated with the following formulae:

R i =R i−1 sin(Yi−1)/sin(Δi)  (3a)

tan(αi)={square root}{square root over ( )}(R02−R12)/R1  (3b)

γi=αi+φi  (3c)

φi=α1+Δi  (3d)

[0027] where αi=A0OA0; Δi=A1ODi; γi=AiOBi and φi=AiOBi (i≧1). In the present work Δi is evaluated by using an iterative method according to the condition φ=constant. The source is placed on the focal circle and the half-width angle of the diffractor on the focal plane, excluding the areas where shadow effect occurs, is given by:

Ω=2(Ω0+ΣΩi)  (4)

[0028] where Ω0=A0SE0 and Ωi=Di−1SEi. Taking into account the increase of the angle width of the reflecting surface, the diffractor aperture ratio is given by the central step aperture times the factor Ω/Ω0. The ratio Ω/Ωsp (where Ωsp is the solid angle of a spherical monocrystal diffractor and Ω is the solid angle of the multi-stepped diffractor here described) is an actual estimation of the gain of this device when compared with an ideal spherical crystal device with the same focal circle and size. U.S. Pat. No. 4,882,780 and U.S. Pat. No. 4,807,268 are incorporated by reference as if both were set forth in their entirety herein.

[0029] For a better understanding of the advantages of the diffractor according to the present invention, the performances of the present device with previous stepped diffractor constructed under the constant step height conditions (D. B. Wittry and S. Sun, J. Appl. Phys. 69, 3886 (1991)) are hereinafter evaluated. In order to compare these two designs we performed calculations with the same set of parameters of D. B. Wittry and S. Sun, J. Appl. Phys. 69, 3886 (1991), i.e., focal circle radius r=50 mm, Bragg's angle θ=22.76°, h=0.15 mm and central step half-width A0B0=3.87 mm. For the h=constant condition, when the step number increases its size decreases. As a consequence, a diffractor aperture based on the φ=constant condition is slightly bigger. Although the number of steps are the same in both cases, for i=7, Bx increases from 15.05 mm to 38.60 mm, moreover, the solid angle of the present diffractor is about 5 times larger than a stepped diffractor under constant step height conditions.

[0030] For practical applications, as already described before, a fundamental property of this device is that the increase of the aperture ratio does not determine a reduction of the resolution. Indeed, the latter combines both high aperture ration and high resolution at the same time. This really unique condition is fulfilled for different Bragg's angles and focal circle radii. In the resolution range of 5·10−5<Δλ/λ<10−4 that corresponds to the range of the more recent high resolution spectrometers (V. Stojanoff, K. Hämäläinen, D. P. Siddons, J. B. Hastings, L. E. Berman, S. Cramer, G. Smith, Rev. Sci. Instrum. 63, 1125 (1992)), that also coincide with the typical silicon rocking curve width (5″) in the angular range 25°<θ<45°, (T. Matsushita and H. -O Hashizume, in: Handbook on Synchrotron Radiation, ed. E. E. Koch (North-Holland, Amsterdam, 1983) p. 261.) the average value of the aperture for this device is in all the range greater than 0.058 sr with fluctuations of about 15%.

[0031] Another example of x-ray monochromator based on the present type of diffractor is illustrated at the energy of the cerium (Ce) L2-edge (λ=2.012 Å). With a resolution of ≈9*10−5 and a focal circle radius of R=400 mm, a device constructed with thin silicon (Si) (220) crystals collects a solid angle fifty times greater than a spherically curved crystal (U.S. Pat. No. 4,882,780; U.S. Pat. No. 4,807,268).

[0032] Other kind of x-ray instruments are those used as x-ray microanalyzers. The latter need a high aperture ratio but such instruments are not so demanding in resolution (Δλ/λ>10−3). Actually, in these devices the diffractor surface could use plates of muscovite (mica), which typically provides high plasticity.

[0033] A diffractor for a microanalyzer of the CAMEBAX-micro type with 160 mm focal circle radius, in the x-ray range suitable to analyze iron (Fe) (Z=26) and titanium (Ti) (Z=22), could use the muscovite crystallographic planes (311) and (201) under the Bragg's angles of 34.83° and of 32.82° respectively. With a diffractor height of L=16 mm a wide aperture (0.066 sr) associated to a moderate resolution of Δλ/λ=10−3 is obtained for the first case, while in the second case an aperture of 0.039 sr and a resolution of Δλ/λ=7*10−4 are calculated. The estimated gain is about 15-30 times that of the spherical diffractor of U.S. Pat. No. 4,882,780 and U.S. Pat. No. 4,807,268.

[0034] A further example of experimental apparatus which could implement the embodiment shown in FIG. 1 is an ESCA spectrometer. The characteristic radiations of the x-ray photons (Ai, Kα or Mg Kα) of an x-ray photoelectron spectrometer, when absorbed by the sample, are known to induce the appearance of free photoelectrons. The intensity and the contrast of x-ray photoelectron spectra are remarkably magnified by intense incident quantum flux.

[0035] The properties of the stepped surface permit to design new kind of spectrometers that allow to obtain high contrast spectra in micro-samples. Here we report the parameters for an x-ray monochromator at the Mg Kα energy (λ=9.89 Å). An acceptance of 0.17 sr is obtained with the following parameters: a focal circle radius of r=100 mm, a diffractor height of L=20 mm and a resolution Δλ/λ=3*10−3. The x-ray beams should be diffracted by a device made again by muscovite planes (001) (2d=19.93 Å) working at θ=29.90°. The diffractor should consist of only five steps, but the intensity advantage over a conventional spherical curved crystal should be larger than 30.

[0036] More recently a diffraction based device using spherical crystal elements was designed for high resolution x-ray fluorescence measurements in the Cr Kβ energy (W. A. Caliebe, S. Bajt and C-C. Kao, Rev. Sci. Instrum. 67(1996)-1.). In FIGS. 5 and 6 we present the comparison of the device of W. A. Caliebe, S. Bajt and C-C. Kao, Rev. Sci. Instrum. 67(1996)-1 when the spherical diffracting crystal is replaced by the multi-stepped diffractor of similar size according to the present invention. The gain here is very high over the entire range of the resolution both with Ge (FIG. 5) and Si (FIG. 6) crystals. Moreover it is very important to underline, looking at the results of these two figures, in order to achieve the larger gain one only has to use the Si(111) surface; there is no need to use higher order reflections, or other crystals and orientations with different spacing.

[0037] Another useful application of the present multi-stepped diffractor should be the replacement of the conventional curved monocrystal in laboratory EXAFS spectrometer based on x-ray diffractometer coupled to a rotating anode, like the RIGAKU one (K. Tohji, Y. Udagava, T. Kawasaki, K. Massuda, Rev. Sci. Instrum.), with a proper multi-stepped device.

[0038] Finally, it should be noted that the device according to the invention, realized locating some small spherically curved dispersive elements (monocrystals or gratings) on a focal circle and disposing said elements in a way to fulfill the same Bragg reflection condition for the incident radiation, allows advantageously to increase the throughput with respect to a spherical bent crystal device without affecting the resolution, where usually mechanical stresses disturb the performance of these devices.

[0039] Although this invention has been described and illustrated by reference to specific embodiments, it will be apparent to those skilled in the art that various changes and modifications may be made which clearly fall within the scope of this invention. The present invention is intended to be protected broadly within the spirit and scope of the appended claims.

Claims

1. A diffractor for electromagnetic radiation, comprising: a crystal having a plurality of stepped, curved surfaces, the stepped surfaces of the crystal being defined by the arcs 2φi=2(AiBi); wherein the diffractor has N steps, i=0, 1, . . . , N; the arc of each curved surface is determined in an X-Y-Z coordinate system by placing the crystal on the perimeter of a focal circle in the XY plane having a radius “r”; wherein the crystal has a total height of “L” in the XZ plane; φ is the half-width angle of the i-th step in the focal circle plane; Ai, is the center of the curved surface of the i-th step and also the point of intersection between the i-th step surface and the focal circle plane; Bi is a point on the outer edge of the i-th step; Di is the inner edge of the i-th step; OAi is the curvature radius of the i-th step; S is the focusing point; hi=BiDi is the height of the i-th step; L is the sum of all of the hi terms; OA0=R0=2r; EiBi is defined as the part of the i-th step shadowed by the (i+1)-th step; and the crystal lies on a spherical surface of radius Ri where Ri=OAi.

2. A diffractor for electromagnetic radiation, comprising: a crystal having a plurality of stepped, curved surfaces, the stepped surfaces of the crystal being defined by the arcs 2φi=2(AiBi); wherein the diffractor has N steps, i=0, 1, . . . , N; the arc of each curved surface is determined in an X-Y-Z coordinate system by placing the crystal on the perimeter of a focal circle in the XY plane having a radius “r”; wherein the crystal has a total height of “L” in the XZ plane; E is the step height in the XZ plane; φ is the half-width angle of the i-th step in the focal circle plane; Ai is the center of the curved surface of the i-th step and also the point of intersection between the i-th step surface and the focal circle plane; Bi; is a point on the outer edge of the i-th step; Di is the inner edge of the i-th step; OAi; is the curvature radius of the i-th step; S is the focusing point; hi=BiDi is the height of the i-th step; L is the sum of all of the hi terms and NE=L; OA0=R0=2r; EiBi is defined as the part of the i-th step shadowed by the (i+1)-th step; and the crystal lies on a spherical surface of radius Ri where Ri=OAi.