Confocal X-ray technology based on capillary X-ray optics

Basics of confocal XRF

XRF is the characteristic “secondary” (or fluorescent) X-rays from a material that has been excited by bombarding with high-energy X-rays, protons, electrons, and so on (Zhang et al. 2014). According to the way of obtaining characteristic XRF spectrum, the XRF analysis method can be divided into a wavelength dispersive mode and an energy dispersive mode. Confocal XRF is most the energy dispersive mode (Woll et al. 2006, Samber et al. 2008, Sun et al. 2011, Kanngießer et al. 2012) and can also be designed into the wavelength dispersive mode (Hoszowska et al. 2011). For all the available confocal XRF facilities based on CXRO, the focusing X-ray optics in the detection channel is PXRO or MXRO and the focusing X-ray optics in the excitation channel may be MXRO, PXRO, compound refractive lenses (CRLs), Kirkpatrick-Baez (KB) mirrors, Fresnel zone plates, and so on. The performances of CXRO for confocal technology have been studied by many researchers (Janssens et al. 2004, Vekemans et al. 2004, Sun et al. 2009c). The order of magnitude of the gain of the PFXRL in the excitation channel is 10³, and the transmission efficiency of the PPXRL is about 30%, which is a function of the energy of X-rays. The order of magnitude of the size of the output focal spot of the PFXRL and the input focal spot of the PPXRL is 10 μm, which is also dependent on the energy of X-rays. In general, the size of the foci of PXRO decreases with the increasing energy of the X-rays. The reason for this is as follows. For the PFXRL and PPXRL, their focal spot size ϕ can be roughly written as (MacDonald 2010)

where f is the output focal distance and input focal distance of the PFXRL and PPXRL, respectively, and θ_c is the critical angle of total reflection. The change in the focal distance f with increasing energies is very small. But the decrease in critical angle θ_c with increasing energies is larger because, for the borosilicate glass capillary, the critical angle of the total reflection is determined by the X-ray energy through

where E is photon energy in keV. Therefore, the focal spot size decreases with increasing energies. The energy dependence of the focal spot size of PXRO results in the profile size of the confocal volume overlapped by the output focal spot of the PFXRL and the input focal spot of the PPXRL being different at various energies of the X-rays. An adjustment method for the confocal configuration based on CXRO with a liquid secondary target was designed. Compared with the theoretical value of volume size, the relative error of the adjustment with a liquid secondary target is 9.8%, and this value is more accurate than that observed with a metal secondary target. Moreover, the developed adjustment method is more efficient than the usual method (Peng et al. 2013).

Experimental design for confocal XRF

CXRO is versatile so that it can be used to design different confocal XRF facilities based on various X-ray sources.

There are many confocal XRF facilities based on the synchrotron radiation source. Such confocal XRF facilities can base on a PHFXRL in the excitation channel and a PPXRL in the detection channel. For such confocal 3D micro-XRF facility, the absolute and relative detection abilities for the elements Fe-Zr are 0.3–1.0 fg and 0.1–0.3 ppm, respectively. The spatial resolution for the sensitive trace analysis of the transition elements Fe-Zr is in the range 15–30 μm at a bending magnet beam line of a second-generation synchrotron when operated in pink beam mode. To demonstrate the 3D visualization capabilities of major, minor, and trace constituents in materials of geological and biological origin, maps of the elements Fe and Sr for a polished granite section and maps of the element Br for the specimen of Calanus helgolandicus have been provided, respectively (Janssens et al. 2004). The confocal XRF facilities based on synchrotron radiation can also use MXRO in the excitation channel and a PPXRL in the detection channel. The spatial resolution of such confocal XRF facilities with MXRO in the excitation channel is smaller than that of those with a PHFXRL in the excitation channel. This results from the smaller focal spot size of MXRO than that of PHFXRL. Such borosilicate MXRO working on a single bounce has working distances up to several centimeters and reflection efficiencies upward 90%. It has been demonstrated to function well up to 40 keV. A semiempirical model for simple and layered structures was designed to demonstrate its use in a four-layer paint structure on a glass slide (Woll et al. 2006). Because CRLs are often used to focus the synchrotron radiation, the confocal 3D micro-XRF facility can base on the CRL to focus the X-rays from a third-generation synchrotron source in the excitation channel and a PPXRL in the detection channel. Compared with PXRO, the focal spot size of CRL is smaller. This results in a smaller spatial resolution of the confocal 3D micro-XRF facility with the CRL in the excitation channel than that of those with PXRO in the excitation channel. However, CRL is large and expensive, and moreover, it is not convenient to be used to focus the X-rays from the conventional X-ray source. This confocal setup with the CRL in the excitation channel provides the possibility of sample depth scans with an energy-dependent resolution of 10–35 μm in the energy range of 3–23 keV. The absolute and relative detection limits of this confocal facility are subfemtogram and about 0.6–10 ppm, respectively. The sensitivity of the confocal XRF setup is sufficient to perform a fully 3D imaging based on the trace elements within microscopic inclusions in geological samples down to concentration levels of 50 ppm and buried liquid inclusions in quartz (Vincze et al. 2004). Like CRL, KB mirrors are also often used to focus the synchrotron radiation, and the confocal 3D micro-XRF facility can accordingly use KB mirrors in the excitation channel and a PPXRL in the detection channel. The depth resolution of this confocal facility is 77.1 μm. The distribution of Fe and Zn in the twig of Buxus microphyllaat various depths was obtained by using this confocal facility. The faux bamboo paint in Emperor Qianlong’s Lodge of Retirement in Forbidden City and airborne particles were analyzed with this confocal XRF (Sun et al. 2008, Wei et al. 2008). The focal spot size of KB mirrors can be down to submicron. This is helpful for optimizing the spatial resolution of the confocal 3D micro-XRF facility based on KB mirrors in the excitation channel. However, the collecting angle of KB mirrors is so small that they are almost used to focus the synchrotron radiation. Confocal XRF facilities based on the synchrotron radiation source mentioned above are all energy dispersive. In fact, they can be designed into a wavelength dispersive model. For example, a wavelength dispersive confocal spectrometer can base on a Fresnel zone plate or a KB mirror in the excitation channel and PPXRL in the detection channel. Its energy resolution is compared with the results of Monte Carlo (MC) simulations. Further improvement in energy resolution, down to the eV range, by employing a double-crystal geometry has been examined. The double K-shell vacancy states in solid Mg, Al, and Si were produced by means of monochromatized synchrotron radiation, and the two-electron one-photon radiative transitions were observed by using this wavelength dispersive confocal facility. The two-electron one-photon transition energies and the branching ratios of the radiative one-electron to two-electron transitions were determined and compared to available perturbation theory predictions and configuration interaction calculations (Szlachetko et al. 2010, Hoszowska et al. 2011).

There are many confocal XRF spectrometers using a laboratory conventional X-ray source. For example, confocal XRF spectrometers with a PFXRL in the excitation channel and a PPXRL in the detection channel base on the conventional X-ray source can be designed under air condition (Sun et al. 2010, Cordes et al. 2014). There are certain limitations for such confocal XRF spectrometers under air condition, namely, it cannot detect XRF of low-atomic-number elements due to absorption of low-energy X-ray photons by ambient air. This limitation can be overcome with piezo positioners for optimum alignment of both PFXRL and PPXRL installed inside the vacuum chamber. Such confocal XRF spectrometer working at vacuum condition was designed for light element analysis (6≤Z≤14). Lower limits of its detection have been determined to be in the ppm region for the confocal geometry. Depth resolution varies between 100 μm at 1 keV (Na-Kα) and 30 μm at 17.5 keV (Mo-Kα) (Smolek et al. 2012, Nakazawa and Tsuji 2013a,b). In order to increase exciting X-ray incidents on a sample with low-power tabletop X-ray sources, a confocal micro-XRF instrument using two independent micro-X-ray tubes was used. One PFXRL is attached to each X-ray tube. Another PPXRL is attached to the detector. The focal spots of these three lenses are adjusted in a confocal configuration. Using the excitation of two X-ray beams increased the incident X-rays on the sample and accordingly increased the XRF intensity from the sample. The elemental depth distributions for Ca, K, and Fe of wheat grain were measured. Moreover, the merits and disadvantages of both the micrograzing exit XRF and confocal micro-XRF methods have been discussed in detail (Tsuji et al. 2007, Nakano and Tsuji 2009). In order to optimize the detection limits of the confocal XRF spectrometers based on conventional X-ray sources, a total reflection confocal XRF based on X-ray tube was designed to analyze a single particle on a flat Si substrate (Nakano et al. 2006). Besides these confocal XRF instruments, a confocal micro-XRF spectrometer based on a PFXRL in the excitation channel and a PPXRL in the detection channel was designed to obtain not only elemental mapping of the sample but also simultaneously its own X-ray absorption imaging. The spatial resolutions for fluorescence and for absorption imaging of this confocal spectrometer are <100 μm and <10 μm, respectively (Hampai et al. 2008).

Besides the synchrotron radiation source and conventional X-ray source, a proton microbeam and a micro electron beam can also be used in confocal X-ray analysis. For example, a confocal micro proton-induced X-ray emission (PIXE) facility was developed. The size of a proton microbeam with an energy of 3 MeV was 1.6×2.4 μm², and the focal spot size of the PPXRL in the detection channel was 26 μm at 8 keV. A 3D element-specific distribution of aerosol microparticles captured in a thick quartz filter was reconstructed by using this confocal micro-PIXE facility. The depth intensity profiles of the major elements that compose the patina layer of a quaternary bronze alloy were also measured (Kanngießer et al. 2007, Žitnik et al. 2008, Sokaras et al. 2009, Grlj et al. 2011). There is similar confocal technology based on a micro electron beam in the excitation channel and a PPXRL or a PFXRL in the detection channel for energy dispersive or wavelength dispersive X-ray analysis (Schields et al. 2002, Tanaka et al. 2008).

Quantitative methods for confocal XRF

For the quantitative analysis of a sample using confocal XRF based on PXRO, a few methods have so far been developed (Vekemans et al. 2004, Schmitz et al. 2009, Sun et al. 2009b, Sun et al. 2010, Fittschen and Falkenberg 2011, Mazel et al. 2011, Mantouvalou et al. 2012, 2014, Schoonjans et al. 2012). The fundamental parameter approach assuming a spherical probing volume has been proposed for the analysis of paint layers. The procedure of evaluation of concentrations is based on the independent parameter method and included absorption of radiation in the outer layers and secondary fluorescence enhancement induced by hard X-rays of the same and neighboring layers. Although the measurements are not calibrated absolutely, this model yields correct concentration ratios (Šmit et al. 2004). Such quantification for confocal XRF can be carried out with a fundamental parameter scheme based on calibration with a glass reference material (Vekemans et al. 2004). Moreover, a more detailed model of confocal volume has been proposed, and a general equation for the depth-dependent intensity of XRF radiation in confocal geometry as well as a calibration procedure has been derived. Fundamental parameter expressions for the primary intensity of various types of 3D micro-XRF experiments have been presented. Fundamental parameter equations for calibrating a 3D micro-XRF spectrometer with thick reference materials have been developed and describe the fluorescence intensity profile measured rather accurately (Malzer and Kanngießer 2005). This method is extended to reconstruct the composition of stratified materials. A reconstruction algorithm for stratified materials was developed and its validation was demonstrated with thick glass standards and stratified polymer materials, which shows the capabilities of confocal micro-XRF to determine the thickness and composition of stratified materials. This model treats only primary fluorescence (Mantouvalou et al. 2008). A global theoretical model that accounts for the secondary fluorescence enhancement with either particle (3D micro-PIXE) or photon (3D micro-XRF) microbeams used in the excitation channel has been presented based on the study on the influence of secondary fluorescence enhancement in confocal X-ray microscopy analysis with stratified materials. The contribution of the secondary fluorescence effect to the confocal X-ray intensity profiles was calculated for some typical representative cases, and the influence of several experimental parameters in terms of their influence on the absolute intensity and shape of the secondary fluorescence intensity profile was also studied (Sokaras and Karydas 2009). These models are based on multidimensional integrals, which must be computed with numerical methods. A quantification approach based on a MC simulation code has been presented, which is devoted to multilayered materials. In this work, a new MC code for simulation of XRF from stratified materials, which is recorded in the 3D confocal configuration, was developed. The experimental profiles of X-ray peak intensities collected during sample scan in depth were compared against the results of the MC simulation and the analytical expression. Agreement between these profiles was noticed for monoelement thin foils. A disadvantage of the MC method is the long time simulation. Since deconvolution is executed by the iterative procedure, a convergence of the sequence of the successive steps could be rather slow in case of very complex structures. Therefore, it is necessary to increase simulation efficiency with more advanced variation reduction techniques (Czyzycki et al. 2011). In order to use confocal XRF to determine the element concentration profiles in stratified materials, a direct deconvolution of the measured depth-dependent XRF intensity signal with the established response function of the spectrometer was developed. Since this method neglects the absorption of primary and secondary radiation within the probing volume, it is applicable only to low absorbing samples and small probing volumes (Wrobel and Czyzycki 2013). Moreover, a generalized mathematical model to describe the intensity of primary XRF radiation collected in the tilted confocal geometry mode has been provided, where the collimating optics in the detection channel is rotated over an angle relative to a horizontal plane. The model was verified with a multilayer test sample scanned in depth. It was proven that for low-Z matrices, the rotation of the detection channel does not induce any significant differences in reconstruction of the thickness and chemical composition of layers (Czyzycki et al. 2014). A method with confocal micro-XRF to measure the thickness of multi-ply films was developed. This method is suitable for relatively thick multi-ply films. It is convenient in in situ and elementally resolved analysis of the thickness of multi-ply films without a cumbersome theoretical correction model. However, there is a limitation of this method that because it depends on measuring the change in the size of confocal volume overlapped by foci of CXRO, the errors of this method are large if the thickness of the film is small, for example, a thickness of several microns (Peng et al. 2014b). Moreover, confocal micro-XRF based on a rotating anode X-ray generator with a PFXRL in the excitation channel and a PPXRL in the detection channel was designed to carry out element-resolved and in situ quantitative analysis of ion distribution near the surface of the electrode in a steady-state diffusion in an electrolytic tank. The standard curve of the Cu-Kα fluorescence intensity corresponding to the concentration of CuCl₂ was measured to quantitatively determine the ion distribution near the surface of the electrode in a steady-state diffusion. The distribution of the electrolytic ions around the surface of the electrode in the electrolytic tank was measured in situ, and the effects of the concentration of the electrolyte and the bath voltage on the shape of the layer with a nonuniform distribution of the Cu²⁺ ions near the cathode surface in a steady state were analyzed with confocal XRF, which shows that confocal XRF has potential applications in spatially resolved analysis of the liquid mass transfer in electrolytic tanks in situ. This method is suitable for analyzing liquid samples (Peng et al. 2014a).

All in all, the existing quantitative methods of confocal XRF have advantages and disadvantages. There are many opportunities for researchers to design advanced relative methods for confocal XRF.

Applications of confocal XRF

Confocal XRF can be used in many fields, such as chemical analysis, materials science, environmental science, geology, life science, cultural heritage, food science, medicine science, electrochemistry, and forensic science (Perez et al. 2010, Sun et al. 2011, Liu et al. 2013b, Cordes et al. 2014, 2015, Peng et al. 2014a, Choudhury et al. 2015). As an example of the application of confocal XRF in cultural heritage, an archaeometric sample was analyzed by confocal XRF. The capabilities of this confocal XRF method were evaluated and illustrated with depth-sensitive investigations of paint layers in ancient Indian Mughal miniatures. Successive paint layers can be distinguished nondestructively with a depth resolution of about 10 μm. Major and minor elements are detectable and can be discriminated in different layers. The results show that new light can be shed on ancient painting techniques and materials with this confocal micro-XRF setup (Kanngießer et al. 2003). As an example of application in forensic science, several forensic samples of multilayered automotive paint fragments and leather samples were analyzed with confocal micro-XRF system. Elemental depth profiles and mapping images of forensic samples were obtained, which show that multilayered structures can be distinguished in forensic samples by their elemental depth profiles (Nakano et al. 2011). For application in food science, for instance, elemental distributions in rice grains were studied with confocal micro-XRF and inductively coupled plasma sector field mass spectrometry. Brown rice was analyzed to find that a surface layer having a thickness of about 80 μm is the richest region in elements. The element of Ti is detectable only in this so-called skin region. Thus, in regions affected by heavy metal and other toxic element contamination, those rice dishes would be preferred whose preparation should need abundant amounts of water for washing and cooking (Mihucz et al. 2010). As for application in medicine science, for example, pharmaceutical tablets were nondestructively analyzed using confocal XRF. It was shown that it is possible to use confocal XRF to measure the distribution of several inorganic elements (Zn, Fe, Ti, Mn, Cu) from the surface to a depth of several hundred microns under the surface. A model to correct the absorption from the matrix was designed in order to accurately measure the true elemental distribution. These chemical imaging studies of pharmaceutical tablets are currently an important emerging field in the pharmaceutical industry since they are an important issue for production quality control but also for counterfeit detection (Mazel et al. 2011). As an example of application in life science, confocal micro-XRF was used to unravel the tissue-specific 3D distribution of metals down to trace concentration levels in a nondestructive manner within the crustacean Daphnia magna, an ecotoxicological model organism. Specific areas of metal accumulation in different cross-sections of interest within the organism were investigated. Full 3D element-to-tissue correlation can be derived coupled with the obtained elemental information with microscopic morphological data obtained by laboratory absorption microtomography. This allows a more detailed interpretation of the obtained results with respect to metal accumulation within this model organism (Samber et al. 2010). For application of confocal XRF in materials science, confocal XRF was used to determine surface topography. The surface topography based on such confocal XRF has potential applications in materials sciences (Zhao et al. 2013). There are so many applications of confocal XRF that they are not mentioned here one by one. With the development of the confocal XRF technology, there will be more new applications of this versatile method.

Trends of confocal XRF

The trend of confocal XRF is the development of the confocal XRF method with an optimized spatial resolution (Dehlinger et al. 2013a). For example, the confocal XRF technology can base on a PFXRL in the excitation channel and a cylindrical monocapillary with radii ranging from 50 μm down to 5 μm in the detection channel to improve spatial resolution (Dehlinger et al. 2013b). The extrapolated value for a 0.5-μm radius capillary suggests that sub-1-μm resolution XRF should be possible with a laboratory source. Of course, increasing the source brightness, i.e. working with liquid-metal or synchrotron sources, could probably lead to reaching 100-nm resolution (Dehlinger et al. 2013b). The confocal XRF facility can base on Fresnel zone plates in the excitation channel and a PPXRL in the detection channel. The spatial resolution of such facility can be down to 300×300 nm², which allows the detailed nondestructive in situ study of impacted grains at the sub-micrometer level (Schmitz et al. 2009). There is a method for the quantification for confocal nano-XRF based on the fundamental parameter method. A thorough error estimation algorithm based on the MC method is applied to produce a detailed analysis of the uncertainties of the quantification results (Schoonjans et al. 2012).

Confocal XRF is often used coupled with confocal XAFS to analyze the elemental composition and the structures of the sample.

Basics of confocal XAFS

XAFS is the modulation of the X-ray absorption coefficient at energies near and above an X-ray absorption edge. XAFS includes both extended XAFS (EXAFS) and X-ray absorption near edge structure (XANES). In XAFS, the measurement of the X-ray absorption coefficient of a material as a function of energy can be performed with such methods as a fluorescence mode and a transmission detection. Confocal XAFS is based on the fluorescence mode (Lühl et al. 2013).

It is well known that in XAFS experiments, an energy scan across the absorption edge is performed. For EXAFS and XANES, the scanning energy range is about 1 keV and 0.5 keV, respectively. Therefore, the energy dependence of the performances of CXRO is important for XAFS experiments. Experimental results show that a change in the focal distance of PXRO with a change in energy can be ignored at a scanning range of about 1 keV. XAFS experiments are generally carried out with synchrotron radiation, and the position of the beam of the synchrotron radiation might change at different energies. However, when PXRO is used to focus such synchrotron radiation beam, the change in the position of the focal spot of PXRO is very slight because of PXRO not being an imaging optics and can be accordingly ignored at a scanning energy range of about 1 keV (Sun et al. 2007c).

Experimental design of confocal XAFS

Confocal XAFS is based on confocal XRF, and therefore, its experimental designs are similar to those of confocal XRF. Some specially designed confocal XAFS facilities are introduced as follows. A confocal XANES based on a PHFXRL in the excitation channel and a PPXRL in the detection channel was designed. Because these two PXROs have a focal distance shorter than 4 mm, the profile size of the confocal volume of such confocal setup can be 18.5×12.0×10.0 μm³. This improves the ability of confocal XANES for high spatially resolved analysis of samples. A confocal 3D resolved study of mineral inclusions in rare natural diamonds at the Fe-K edge was performed (Silversmit et al. 2010). In order to improve micro-XRF and XAFS experiments with high-pressure diamond-anvil cells, a confocal XAFS and XRF setup based on a single-bounce MXRO in the excitation channel and a PPXRL in the detection channel was designed. The output focal distance of such single-bounce MXRO is longer than that of the PHFXRL, and therefore, it is convenient for performing the in situ analysis with high-pressure diamond-anvil cells. This confocal setup enhances the quality of the fluorescence and XAFS spectra and, thus, the sensitivity for detecting elements at low concentrations. It efficiently suppresses signal from outside the sample chamber, which stems from elastic and inelastic scattering of the incoming beam by the diamond anvils as well as from excitation of fluorescence from the body of the diamond-anvil cell (Wilke et al. 2010).

An approach for chemical speciation in stratified systems using confocal micro-XAFS spectroscopy and conventional XAFS was designed. A reliable reconstruction algorithm for obtaining undistorted spectra for superficial and in-depth layers was developed to calculate the attenuation coefficients of the analyte for successive layers, which facilitates a new spectroscopic tool for 3D resolved nondestructive chemical speciation (Lühl et al. 2012, 2013).

Applications of confocal XAFS

The XAFS technology is a powerful and versatile technique for studying the local atomic coordination and oxidation state of samples in chemistry, physics, biology, and other fields (Bressler et al. 2009, Katz et al. 2012). One advantage of confocal XAFS is that it can be used to perform in situ analysis for samples. For example, the cathode material LiNi_0.5Mn_1.5O₄ for lithium-ion batteries was studied with confocal XANES at the Mn-K edge and the Ni-K edge combined with confocal micro-XRF. The results show that the degradation of Mn³⁺ to Mn⁴⁺ was observable only at the surface of the electrode. The spatially resolved nondestructive analysis provides knowledge helpful for further understanding of deterioration and the development of high-voltage battery materials, and because of its nondestructive nature, it will be also suitable to monitor processes during battery cycling (Menzel et al. 2013). Another advantage of confocal XAFS is that it can be used to analyze samples with micrometer-scale spatial resolution. As an example, a bore core section of a uranium-rich tertiary sediment was investigated with micrometer-scale confocal XAFS and XRF in order to assess the mechanisms leading to immobilization of the uranium during diagenesis. The results show that uranium is present as a tetravalent phosphate and U (IV) is associated with As (V) and that the arsenopyrite might act as reductant of groundwater-dissolved U (VI), leading to precipitation of less soluble U (IV) and thereby forming As (V) (Denecke et al. 2005). The correlation of the iron oxidation state in the black glaze layer with the manufacturing process was studied by means of conventional and confocal XANES. The enhanced surface sensitivity of confocal XANES was combined with conventional XANES, resulting in higher counting rates to reliably evaluate the iron oxidation state (Fe³⁺/ΣFe) of the surface layer. The three-stage firing process was retraced by correlating selected attic black-glazed specimens from different periods (Archaic, Classical, Hellenistic) with laboratory reproductions. The modern black-glazed specimens serving as reference samples were produced by following the three-stage firing process at different top temperatures, using clay suspensions of different particle size produced with treatment of raw illitic clays from Attica (Lühl et al. 2014).

As an application of confocal XAFS in analyzing the local atomic structure, stream mineral inclusions having the negative crystal shape of its host within an “ultra-deep” diamond from Rio Soriso (Juina area, Mato Grosso State, Brazil) were studied with confocal XANES at the Fe-K and Mn-K edges. The observed Fe-rich inclusions were identified to be ferropericlase (Fe, Mg) O, hematite, and a mixture of these two minerals, and confocal XRF further showed that Ca-rich inclusions are present as well, which are spatially separated from or in close contact with the Fe-rich inclusions. The results imply that an imposed negative diamond shape of an inclusion alone does not exclude epigenetic formation or intense late-stage overprint (Silversmit et al. 2011). There will be many new applications of confocal XAFS technology with its increasing development.

In the future, time-resolved confocal XAFS might be designed. This time-resolved confocal XAFS is actually a 4D technology. In other words, this time-resolved confocal XAFS could be used to obtain the information of 3D of space and 1D of time.

Basics of confocal XRD

XRD is a versatile and nondestructive analytical technique (Mankowsky et al. 2014), and it can be divided into two types: angle-dispersive XRD (ADXRD) and energy-dispersive XRD (EDXRD). The resolution of lattice spacing of ADXRD is better than that of EDXRD. However, EDXRD is able to collect full diffraction patterns quickly by using polychromatic X-rays as the source and with no need for a goniometer. Moreover, because the whole spectrum of diffracted radiation is obtained simultaneously, EDXRD enables studies where structural changes can be determined over time. Now, the available confocal XRD setup is EDXRD (Sun et al. 2007d). As mentioned above, a change in the output focal distance of PXRO with a change in the X-ray energy can be ignored. In other words, PXRO can focus simultaneously the X-rays with different energies into the same focal spot. This is the reason that PXRO can be used in confocal EDXRD, which works on polychromatic X-rays.

The diffraction peak position should satisfy the Bragg condition, as follows:

Here, d is the lattice spacing, θ is the Bragg angle, hc is a constant as 12.398, and E is X-ray energy. According to Eq. (3), the resolution of lattice spacing Δd/d of this confocal EDXRD diffractometer can be written as follows:

Here, ΔE/E is the relative energy resolution of the detector system, and the angular resolution Δθ can be calculated with Δθ_p and Δθ_s as follows:

Here, Δθ_p is the convergence angle of the focused beam from PFXRL in the excitation channel and Δθ_s is the divergence angle of the beam, which can be collected by the PPXRL. Such Δθ_p and Δθ_s are energy dependent, and the resolution of lattice spacing is accordingly energy dependent (Sun et al. 2014b).

Experimental design and applications of confocal XRD

Confocal XRD can be based on the synchrotron radiation and conventional laboratory X-ray sources. For example, confocal EDXRD based on the synchrotron radiation source with a PHFXRL in the excitation channel and a PPXRL in the detection channel was designed to perform a structural and multielemental X-ray microanalysis. The performance of the confocal spectrometer was illustrated with the analysis of two specific artificial samples. The setup successfully combined the spatial resolution with short acquisition times for elemental and structural microanalysis (Sosa et al. 2014). As examples of confocal XRD based on tabletop source, one confocal EDXRD spectrometer based on conventional X-ray source and a combination of a polycapillary slightly focusing X-ray lens in the excitation channel and a PPXRL with a long input focal distance in the detection channel was designed. Because such polycapillary slightly focusing X-ray lens and PPXRL with a long input focal distance have smaller convergence angle and divergence angle mentioned above, respectively, such confocal EDXRD setup has improved the resolution of lattice spacing. The confocal EDXRD setup was used to analyze common plastics to show that such confocal EDXRD has potential applications in the identification of plastic evidences (Liu et al. 2013a). Another confocal micro-EDXRD spectrometer based on a conventional X-ray source and a combination of a PFXRL in the excitation channel and a PPXRL in the detection channel was designed to measure large grain size by using point-to-point XRD signals. A metallographic specimen of a nickel alloy was analyzed by this confocal method. The limitation of this confocal method was that its error was large when the grain size and the absolute error for determining the profile size of the confocal micro volume were at the same order of magnitude. To some extent, this limitation might be decreased by using X-ray focusing optics with smaller focal spot size in confocal EDXRD. Confocal EDXRD has potential applications in measuring large grain size (Sun et al. 2014c). This confocal micro-EDXRD spectrometer can also be used for the phase and structure formation investigation of metal electrocrystallization nucleation and growth. The experimental results demonstrate the possibilities of the in situ and time-resolved analysis of the electrocrystalization process of metal electrodeposition by laboratory confocal EDXRD (Li et al. 2015b).

The main limitation of the existing confocal EDXRD spectrometer is its high detection limit and modest resolution for analysis. Taking these into account, further works need to be done to optimize the confocal EDXRD technique for its potential applications in such fields as chemicals, pharmaceuticals, polymers, semiconductors, thin films, and minerals.

In the future, such confocal EDXRD has potential applications in the higher energy XRD (Tschauner et al. 2014) and fast XRD (Anzellini et al. 2013).