Dual-energy crystal-analyzer scheme for spectral tomography

The principles of using the Laue analyzer as an X-ray optical element for separating two characteristic lines of an X-ray tube are presented.

Abstract

In the present work, a method for adjusting a crystal analyzer to separate two characteristic lines from the spectrum of a conventional X-ray tube for simultaneous registration of tomographic projections is proposed. The experimental implementation of this method using radiation of a molybdenum anode (Kα₁, Kβ lines) and a silicon Si(111) crystal analyzer in Laue geometry is presented. Projection images at different wavelengths are separated in space and can be recorded independently for further processing. Potential uses of this scheme are briefly discussed.

1. Introduction  

Currently, laboratory tomographs usually use continuous-energy-spectrum X-ray tubes and detectors without either spatial or energy resolution. Therefore, the main efforts in this area are focused on the search for methods that allow taking into account or using the polychromatic nature of X-ray sources.

In the incident polychromatic radiation, low-energy quanta are mainly absorbed more strongly by the sample under study. Thus, with increasing sample thickness, the high-energy component of the radiation in the beam recorded at the output begins to prevail: ‘beam hardening’ occurs. In this case, the corresponding mathematical model of signal formation is not linear. However, the conventional methods of tomographic reconstruction can only process linearized projection data. Attempts to linearize the problem if the linearity conditions are not met at the physical level generate additional reconstruction artifacts, such as the ‘cupping effect’ (Fox et al., 2018 ▸; Cao et al., 2018 ▸). To reduce inaccuracies in the reconstructed structure of the object, several essentially different approaches are used.

The approach that does not affect the hardware of the tomographic setup involves pre-processing projection data (Cao et al., 2018 ▸), developing special reconstruction algorithms, in which the polychromaticity is incorporated into the mathematical model used in the back projection (Blumensath & Boardman, 2015 ▸; Jin et al., 2015 ▸), and/or post-processing reconstructed images (Krumm et al., 2008 ▸). None of the methods of this approach will provide reliable information about the volume being restored, even if the visual representation improves (increased contrast, reduced noise). These methods cannot be used in areas where computed tomography results have to be interpreted quantitatively. For example, this is required in cases where the subject matter of the study is not the spatial structure of the object but rather its chemical composition. Besides that, for objects with high porosity and/or granularity, the artifacts of beam hardening often significantly distort the result of the reconstruction of their internal structure, although they are not noticeable at the boundaries of large objects.

Other approaches include modifying the hardware of the projection image registration scheme. Usually, energy-dispersive detectors are used to resolve radiation by energy. However, this type of detector is not spatially sensitive, i.e. to acquire a complete projection image, point-to-point scanning of a sample is required (and, accordingly, using X-ray optical elements, such as lenses, to form a probe). These methods are implemented at synchrotrons (Mino et al., 2018 ▸).

‘Sandwich detectors’, consisting of two (or more) registering layers, were introduced by Brooks & Chiro (1978 ▸), Barnes et al. (1985 ▸), Ishigaki et al. (1986 ▸), Chakraborty & Barnes (1989 ▸) and Boone et al. (1990 ▸). In this case, the upper layer registers the image formed by low-energy photons, while the subsequent layer or layers receive the already filtered high-energy radiation. The filter can be either the top detector layer itself or a thin metal foil between the detector layers. Such an approach allows acquiring images in different energy spectra, but they remain polychromatic, which requires data pre-processing and special reconstruction algorithms.

Prospective studies to develop detectors with both spatial and energy resolution are also underway (Firsching et al., 2009 ▸; Taguchi & Iwanczyk, 2013 ▸). However, their characteristics do not yet satisfy the requirements of modern research.

Another possible way to obtain energy resolution is the use of dispersive X-ray optical elements. For example, a crystal is placed in front of an object to monochromatize the radiation (Buzmakov et al., 2018 ▸). In this case, the use of a crystal monochromator leads to physical change in the spectrum of the probe radiation and, hence, reconstruction artifacts caused by the polychromaticity do not occur.

Thus, consecutive changes of wavelength in repeated experiments will allow one to obtain sets of projection data with different energy resolution. However, this will lead to both a significant increase in the total time of the experiment and difficulties in comparing reconstructed data arising from the inability to exactly match elements of the geometric setup in time-separated experiments.

One development of such an approach is a scheme using a Laue–Bragg monolithic crystalline L-shaped splitter to form a ‘stereo image’ at different wavelengths (Oberta & Mokso, 2013 ▸; Mokso & Oberta, 2015 ▸). The primary X-ray radiation is divided into polychromatic and diffracted monochromatic beams. One more monochromatic part is separated by the second crystal in the Bragg geometry from the passed polychromatic beam. The object under investigation is placed at the point of intersection of the diffracted beams, and the image is recorded on two detectors. However, the comparison of the reconstructed data is difficult owing to the different projection angles.

On the other hand, if the crystal is placed not before but after the object, as a crystal analyzer, mostly similar results can be obtained. In the same way as for the crystal monochromator, the radiation on the detector will be monochromated. However, it is possible to adjust the crystal analyzer in such a way that the diffraction conditions for several different wavelengths are simultaneously met (for a laboratory setup, two characteristic lines of the X-ray source). In this case, the projection images are separated in space and can be registered independently for further processing (Villanueva-Perez et al., 2018 ▸). At the same time, the problem of comparing the reconstructed data does not occur, since the change of the object position is synchronous for all wavelengths.

In this paper, the technique of adjusting a crystal analyzer is proposed as a means to separate two previously specified spectral lines from the incident radiation. Section 2.1 describes the general principle of selecting pairwise reciprocal lattice (RL) points to satisfy the diffraction conditions for two wavelengths at the same time. In Section 2.2, the mutual arrangements of the Si(111) crystal-analyzer surface and pre-selected vectors of the reciprocal lattice are defined, and we calculate the rotation angles of the crystal relative to the primary beam that are required to adjust the experimental scheme. Section 3 demonstrates the experimental implementation of this technique for separating the Kα and Kβ lines from the polychromatic spectrum of an X-ray tube with a molybdenum anode. An Si(111) crystal in Laue geometry was used as a crystal analyzer for two different configurations of reflective planes {111} and {220}.

2. Main idea  

2.1. Preliminaries  

Let a collimated polychromatic beam from the X-ray tube fall on the crystal. The fulfillment of the simultaneous diffraction condition for two wavelengths, say, the Kα and Kβ lines of the used anode, means that two different crystallographic planes are under the Bragg reflection conditions. At the same time, the corresponding RL vectors and are the bases of isosceles triangles with diffraction vectors , and , , respectively. How can this be achieved?

First, the reflecting plane for the Kα line, and therefore the RL vector which is designated as , is selected. In most cases, this choice is related to the convenience of adjusting the crystal under experimental conditions. For the given RL vector and the given wavelength of the incident Kα radiation, we can construct the Ewald sphere (James, 1962 ▸) so that the Bragg diffraction conditions are satisfied. Such a construction is usually performed relative to the diffraction plane, i.e. the plane in which both the chosen RL vector and the wavevector of the incident radiation lie. Since all spectral components in the incident collimated polychromatic beam have the same angle of incidence, i.e. the wavevectors of all components are collinear , we can continue the vector and also build the Ewald sphere for the Kβ line.

Now, if this construction of vectors , and the related Ewald spheres are rotated around the RL vector , then the starting points of the vectors will trace circles that determine all possible directions of the incident radiation, while maintaining the diffraction conditions for the Kα line [Fig. 1 ▸(a)], and the set of Ewald spheres will give us a torus [Fig. 1 ▸(b)].

Figure 1.

Schematic construction to find the conditions for the simultaneous diffraction of the Kα and Kβ lines of an X-ray tube for a selected pair of points in the RL crystal space: (a) intersection of the Ewald spheres for Kα and Kβ lines by the diffraction plane for the Kα line; (b) a torus formed by the rotation of the Ewald spheres around an axis passing through the RL point ; (c) rotation of the Kβ Ewald sphere around to get diffraction from the RL point ; (d) same as (c), but top view.

Fig. 1 ▸(c) shows the construction of the Ewald spheres for the Kβ line in the case of two different (azimuthal relative to the RL vector) incident angles. Fig. 1 ▸(d) shows the top view of the same structure. To emphasize the quasi-3D character of the drawing, parallels on the Ewald sphere are shown as thin lines.

It is seen from this graphical representation that activation of the reflection for the Kβ line from any RL node within the toroidal surface can be achieved by selecting the azimuthal angle of incidence. This is possible because the surface of the Ewald sphere intersects the RL points one by one during rotation, the intersections corresponding to the Bragg diffraction conditions for this RL point. At the same time, the diffraction conditions for the Kα line are strictly fixed by the fact that the rotation occurs around the axis passing through the corresponding RL point.

Hence, the main idea of the presented paper: having fixed the conditions for diffraction of one wavelength, we use the remaining degree of freedom to fulfill the diffraction of another one. That is, by changing the azimuthal angle, which corresponds to the rotation of the crystal around the selected axis, we achieve the result that the diffraction conditions are also fulfilled for another family of reflective planes. It is clear that, in general, this scheme corresponds to non-coplanar diffraction, since the diffraction vectors and the RL vectors lie in different planes , , and , , . But often, this is exactly what is required because images at different wavelengths are separated in space. Note also that this construction and the algorithm based on it should be valid for both Laue and Bragg diffraction geometries.

2.2. Implementation  

The previous considerations were related to the infinite crystal, and, consequently, no restrictions on the rotation of the crystals around an arbitrary axis were implied. However, real crystals have specific directions in space (orientation) owing to the crystal cut along a certain crystallographic plane.

In the case of the symmetrical Bragg geometry, when the reciprocal lattice vector for this crystallographic plane is directed along the normal to the surface, the required reflection for another spectral line can be found by a simple turn of the crystal (around the normal). In all other cases, when the rotation should be performed around a direction that does not coincide with the normal to the surface, it is necessary to determine the mutual orientation of the crystal and the pre-selected diffraction vectors and calculate its rotation angles relative to the primary beam. This is the purpose of this section.

When it is said that a family of reflective planes has the Miller indices (hkl), this means that they belong to the coordinate system in which the OX axis corresponds to the direction [100], the OY axis to [010] and the OZ axis to [001]. This is known as the crystallographic coordinate system (CS). But under experimental conditions, the CS often changes. For example, in our case, when an Si(111) crystal is mounted on a goniometer head, this corresponds to the fact that the crystallographic direction [111] is located along the OZ axis in the laboratory CS (Fig. 2 ▸), the crystallographic direction along the OX axis and the crystallographic direction along the OY axis. The coordinates of the RL point selected in the crystallographic CS change in this CS. Thus, first the coordinates of the selected RL points should be calculated for the reflections of the Kα and Kβ lines in the new CS associated with the crystal surface. It should be remembered that the RL point for the Kβ line must be inside the torus formed by the rotation of the corresponding Ewald sphere around the axis passing through the point for the Kα line, as described in the previous section.

Figure 2.

The scheme of experimental setup of DITOM-M.

Also, since we are only interested in the mutual orientation of the RL vectors relative to the primary beam, we operate with unit-length-normalized vectors. In this case, their coordinates are the direction cosines in the respective CS. The following equation holds for unit vectors of the CS associated with an Si(111) crystal analyzer (initially they coincide with the laboratory CS):

For the first experiment, a plane corresponding to the RL vector ( for brevity) in the laboratory CS is chosen as the reflection plane for Kα, i.e.

For the Kβ line, () is chosen:

So, we have the coordinates (direction cosines) of two unit vectors oriented along the RL vectors, as well as the values of the angles that they form with the incident beam vector, thus fixing the beam orientation relative to the crystal. These angles are determined by the Bragg relations for the corresponding reflections and give us two equations to find the unknown direction cosines of the vector of the incident beam. But, more precisely, since we are in the CS (and look at the radiation source from the CS center) this vector associated with the crystal is directed in the opposite direction with respect to the incident beam vector:

where d is the lattice spacing. One more equation missing if we are to close the equation system. Since, as mentioned above, we are only interested in the directions and the ends of all vectors lie in the unit sphere, then, in principle, the third equation could be

But this equation makes the system nonlinear and thus makes it harder to solve. Therefore, another way is chosen to find the third, linearly independent equation for the system [equation (3)]. The result is

The derivation of and notations for this equation are presented in Appendix A .

Solving this system of equations provides the angles (relative to the coordinate axes of the crystal) at which the radiation should fall on the crystal. The crystal should be rotated with the motors so it occupies a reflecting position relative to the incident beam directed along the OZ axis of the laboratory CS. In principle, two consecutive rotations around any coordinate axis are sufficient to adjust any direction to the required direction. But since the rotations are non-commutative operations, the result depends on the order of their execution. And it, in turn, depends on the specific mechanics of the motors: which of them turns the goniometer head with the crystal first, and which one after.

The best way to find the required rotation matrix is through the rotation of the CS associated with the crystal. In this case, passive rotation matrices that rotate the CS axis and not the vector itself are used. Let the connection between the old and new coordinates after the first turn of the CS be described as

and that after the second rotation as

Then the final rotation matrix will be described as

And if M are the matrices of rotation around the axes, it follows from equation (5) that the second rotation is made around the axes of the previously rotated coordinate system. Thus, by double rotating the CS related to the crystal, we make sure that, for the vector with the coordinates from the previous section, the coordinates become in the stationary laboratory CS, i.e. along the OZ axis of the incident beam.

Writing these matrices explicitly for our case – rotations around the OX (OY) axis, and then OZ (OX) – a system of trigonometric equations is obtained. For the XZ rotations,

and for the YX rotations,

This system having been solved, the required rotation angles α and γ (or α and β) for the motors can be obtained, and the angles can then be used in an experiment (as described in the next section).

The main advantage of our technique is that the calculation of the required motor rotation angles and the subsequent fine adjustment of the crystal analyzer take only a few minutes. After that the experimental scheme is ready to go. This saves time, which is so essential for tomographic experiments (data acquisition).

This method can also be used to set up multi-beam diffraction schemes.

The adjustment of such schemes using existing methods requires a special technique (such as a Renninger scan) (Chang, 1984 ▸) and takes quite a long time. Using our technique, the adjustment of the scheme, at least for three-beam diffraction, becomes more simple.

Our technique also allows selection of reflective planes from different families (h ₁, k ₁, l ₁) and (h ₂, k ₂, l ₂). To do this, we just need to substitute λ_Kβ = λ_Kα into the above equations for the RL point with the indices (h ₂, k ₂, l ₂), or vice versa λ_Kα = λ_Kβ for (h ₁, k ₁, l ₁). For example, the angles for three-beam diffraction between the families {111} and {022} are found in the next section. We are not specialists in multibeam diffraction, but such an opportunity – the creation of different systems of X-ray standing waves inside the crystal – can be useful in some circumstances (Greiser & Materlik, 1987 ▸; Zegenhagen, 2019 ▸).

For other interesting references see Liu et al. (2016 ▸) and Tsusaka et al. (2016 ▸). From the point of view of X-ray optics, the multi-beam diffraction mode can also be used as a kind of beam splitter: one beam falls on the crystal and two come out. This can be used for development of X-ray interferometer schemes based on crystals (Graeff & Bonse, 1977 ▸; Yoneyama et al., 2002 ▸).

How this technique is implemented in an experiment can be seen in the next section.

3. Results and discussion  

Currently, silicon crystals are most widely used as monochromators for laboratory and synchrotron radiation sources owing to their accessibility and low cost of production (Caciuffo et al., 1987 ▸). When using a crystal in Laue geometry, it is necessary to take into account the spectrum of the incident radiation and the thickness of the crystal to minimize the influence of absorption. From previous experience (Zolotov et al., 2017 ▸; Asadchikov et al., 2018 ▸), the attenuation value μt should be less than 1 (where μ is the linear absorption coefficient and t is the crystal thickness). To satisfy this condition, it was decided to use an X-ray tube with a molybdenum anode with the thickness of the silicon monochromator crystal not exceeding 700 µm.

When choosing the corresponding X-ray reflections from the diverse possibilities in the case of Laue diffraction, the following criteria were used. Firstly, it is known that the Kα_{1, 2} and Kβ characteristic lines of the anode used in the spectrum of polychromatic radiation are the most intense. Secondly, the fact that reflections with smaller Miller indices (i.e. crystallographic planes of the type {111}, {220}, {331} etc. in order of their brightness reduction) are the most intense for silicon is taken into account.

The LauePT (Huang, 2010 ▸) software tool is a good support for experimental crystal adjustment in Laue geometry. With its help, one can perform the modulation of experimental Laue-grams using a polychromatic beam, which allows subsequent direct comparison of the location of the calculated reflections with experimental ones (Danilewsky et al., 2008 ▸; Asadchikov et al., 2016 ▸). The calculation takes into account the crystal orientation, rotation angles, the probe polychromatic beam size, the detector (or film) size, the distance to the detector and the spectrum of the X-ray source.

In principle, using LauePT, it is possible to solve the crystal adjustment problem for simultaneous diffraction of two spectral lines by selecting angles. This corresponds to solving the inverse problem by a fitting method. The method described in Section 2 directly solves the inverse problem by calculating the required angles. Thus, in this case, LauePT is considered as a supplementary tool to check the final result.

The experiments were carried out on the DITOM-M X-ray diffractometer (Zolotov et al., 2019 ▸), the scheme of which is presented in Fig. 2 ▸.

A silicon crystal analyzer with a thickness of 520 µm was mounted on a goniometer perpendicular to the X-ray beam (Fig. 2 ▸). The source of X-ray radiation was an X-ray tube with a molybdenum anode with a focus size of 0.4 × 12 mm. Taking into account the size of the focal spot of the source (∼1 mm), the size of the probe crystal of the polychromatic X-ray beam was regulated by two mutually perpendicular slits and was 2 mm both vertically and horizontally. Initially, the crystal was set so that its (111) plane was perpendicular to the beam. By rotation around the X, Y and Z axes (see Fig. 2 ▸), the analyzer was adjusted to the maximum reflections for the Kα₁ and Kβ lines in the Laue geometry. At this position of the crystal, X-ray images of two characteristic lines, Kα₁ and Kβ, were acquired on the two-dimensional CCD camera (Ximea XiRay11). The exposure time of one frame was 100 s. The size of the sensitive element (pixel size) of the detector was 9 µm. The source–crystal distance was 1000 mm and the crystal–detector distance 20 mm.

For the first experimental implementation of the proposed approach, a pair of reflections from the family of reflective planes {111} was chosen: for Mo Kα₁ and for Mo Kβ. Using the rotation angles described in the previous section, the crystal was set to the working position: angle α = −24.59° (rotation around the X axis), angle γ = −1.035° (rotation around the Z axis).

The results are shown in Fig. 3 ▸(a). Two reflections corresponding to the Kα₁ and Kβ lines of the molybdenum anode are clearly visible, which is also confirmed by the calculated Laue-gram. Here, the rotation angles of the crystal used in modeling the Laue-gram [Fig. 3 ▸(b)] completely coincided with the previously calculated angles. The input radiation spectrum for the LauePT software package was calculated using the algorithm described by Ingacheva (2019 ▸) with the following parameters of the X-ray tube operating mode: accelerating voltage 40 kV and current 40 mA.

Figure 3.

Experimental results of separation of two characteristic lines from the incident polychromatic beam of the molybdenum anode (a), (c), (e) in comparison with the Laue-grams calculated by LauePT (b), (d), (f): (a), (b) reflection for Mo Kα₁ and for Mo Kβ; (c), (d) reflection for Mo Kα₁ and for Mo Kβ; (e), (f) reflection for Mo Kα₁ and for Mo Kβ.

As a further example, the reflections from the {022} family were chosen: for Mo Kα₁ and for Mo Kβ. The results are shown in Fig. 3 ▸(c). To adjust the crystal to the desired position, consecutive YX rotations with the following rotation angles were used: β = −1.149° (rotation around the Y axis), α = 11.639° (rotation around the X axis). Again, the rotation angles of the crystal used in modeling of the Laue-gram completely coincided with the angles calculated by our technique. In addition, the reflection corresponding to plane can be seen on the simulated Laue-gram [Fig. 3 ▸(d)] as well as on the experimental topograph. At these rotation angles, the crystal reflects some of the Bremsstrahlung of the X-ray tube spectrum, but the reflection is clearly visible. Therefore, for the final example of the use of reflective planes from different families, we chose for Mo Kα₁ and for Mo Kβ. The result is shown in Figs. 3 ▸(e) and 3 ▸(f). The crystal adjustment angles are as follows: β = −3.362° (Y-axis rotation), α = 12.965° (X-axis rotation). Surprisingly, the rotation angles of the crystal for modeling the Laue-gram again turned out to be the same as those calculated according the technique in Section 2. For this case, we also calculated the angles that were necessary to fulfill the three-ray diffraction scheme with Mo Kα₁: β = −1.113° (Y-axis rotation), α = 12.975° (X-axis rotation).

Thus, on the basis of the results obtained, it can be concluded that the algorithm presented in this paper allows the determination of the location and rotation angles of the crystal analyzer relative to the laboratory coordinate system for subsequent experiments.

Finally, one tomographic projection of the test object, obtained simultaneously in two characteristic lines, corresponding to Fig. 3 ▸(a) ( reflection for Mo Kα₁ and for Mo Kβ), is shown in Fig. 4 ▸. A video (formed from several projections) obtained by the rotation of this object is provided as supporting information.

Figure 4.

Tomographic projections as a result of separation of two characteristic lines for the case of {111} reflections: (a) Mo Kα₁ line; (b) Mo Kβ line.

4. Conclusions  

The technique of adjustment of the crystal analyzer presented in this paper to separate two previously specified spectrum lines from the incident radiation is very simple to understand and implement: simple geometry and nothing more. At the same time, it allows us to expand our capabilities both in terms of experimental research and in the development of spectral tomography methods.

In particular, three sets of tomographic data (projections) will be obtained: two monochromatic (Kα ₁ + Kβ) and one polychromatic projection, which passes through the crystal analyzer. In this paper, the polychromatic beam was beam-stopped. However, in subsequent experiments for tomography, it will be taken into account (as beam attenuated). It should be noted that monochromatic images do not coincide with ordinary projection images, because their spatial Fourier filtration occurs after the crystal diffraction. The interpretation of such images is a separate task.

Now there is a possibility to directly compare the result of reconstruction algorithms for polychromatic radiation with experimental data for a specific wavelength. In addition, it is possible by varying the anode material to select a characteristic line of radiation which is located near the absorption edge of one of the elements that make up the object under study. In this case, the scheme is configured in such a way that projections before and after the absorption edge of a particular element are added to the standard tomographic projection. For example, the distribution of Nb and Zr in the case of application of an Mo anode, or Ni for a Cu anode, can be obtained. It is expected that the number of different options for the use of this technique will increase as it is applied.

Appendix A. The linearly independent equation closing the system (3)

This appendix provides supplementary information to find the linearly independent equation mentioned in Section 2.2.

A spherical triangle on the unit sphere is considered in detail, based on the triangle of three vectors (Fig. 5 ▸). Note that the three vectors , , do not make any physical sense, i.e. there are no relations of the type , between the RL vectors and the incident and diffracted waves. These are just unit vectors whose direction coincides with the corresponding vectors.

Figure 5.

A spherical triangle formed by three unit vectors , , .

The perpendicular is dropped from the C vertex to the plane formed by the vectors and as well as along the edges of COB and COA. On one hand, the length of the perpendicular d can be calculated as

where and are the angles at the vertices of the spherical triangle and are equal to the dihedral angles of the corresponding edges. Their values can be calculated with the help of the law of cosines from spherical geometry (Stepanov, 1948 ▸):

The equivalent of calculating d in equation (8) through different edges follows from the law of sines for spherical triangles:

On the other hand, d is a projection of the vector onto the normal to the plane that can be calculated as . (This, by the way, guarantees the linear independence of the third equation.) Therefore the next expression is true:

Since the desired equation is obtained:

where d is calculated from equations (8)–(10). When writing this equation it is necessary to make sure that the normal is directed inside the trihedral angle formed by the three vectors , , since the normal is defined through a vector product. Equation (13) together with equations (3a) and (3b) in the body of the article makes the system of linear equations closed and solvable by standard methods.

Funding Statement

This work was funded by Russian Foundation for Basic Research grant 18-29-26036. Ministry of Science and Higher Education of the Russian Federation grant .