1. Introduction

Thin crystals of highly oriented pyrolytic graphite (HOPG) are of particular interest for the use as dispersive x-ray optics because their unique structure enables them to be highly efficient in x-ray diffraction in an energy range between 2 keV up to several 10 keV [1]. HOPG is a mosaic crystal, which consists of a large number of small crystallites (Fig.1). The angular distribution of the crystallites, with plane orientations off the normal axis to the surface, is called mosaic spread. Mosaicity makes it possible that even for a fixed angle of incidence to the crystal surface, an energetic distribution of photons can be reflected, because each photon of this energetic distribution can find a crystallite plane at the right Bragg angle. Therefore the mosaicity is responsible for the dramatic increase of integral reflectivity for mosaic crystals in comparison to perfect crystals. The width of the reflected energetic distribution depends on the mosaic spread. The mosaicity also gives rise to mosaic focusing (parafocusing) [2], which further enhances the intensity in the image plane. In addition thin HOPG films give the opportunity to realize crystal optics with arbitrary geometry by mounting it with adhesive on a polished mould of any shape. This enables the design of crystal optics with high collecting efficiencies. In contrast to ideal crystals, as Ge, which can be bent only slightly, thin films of HOPG can be bent much stronger, without reducing the energy resolution. Latter can be understood if considering the crystal structure of HOPG. The crystal structure of HOPG can be described as an agglomeration of a large number of small perfect crystallites with sub-mm size. Consequently, the strain due to bending is significantly reduced in comparison to bent ideal crystals, even for small bending radii. Potential fields of applications for such optics are plasma diagnostics or x-ray spectroscopy and x-ray diffraction with low averaged power x-ray sources, such as ultrashort pulses emitting laser plasma sources [3–7].

Applications in x-ray spectroscopy and diffractometry require dispersive optics with high spectral resolution. Up to now it is commonly assumed that the energy resolution delivered by HOPG crystals is not sufficient for applications where a broad energy range should be scanned at once, even if its high integral reflectivity would be useful. Previous investigations [8] have shown a dependency of the energy resolution on the crystal thickness and the distances between source, crystal and detector, which both can be explained by the mosaicity of the crystal. Reported values were E/ΔE=300–1000 for the first order reflection [8–11] and up to 1200 for the second order reflection [8]. In contrast to ideal crystals, an incident photon must penetrate more deeply into the mosaic crystal, before it finds a well aligned crystallite, from which it can be reflected. That means that the effective depth, from which diffraction in mosaic crystals occurs, is much larger as compared to ideal crystals. This diffraction out of the depth reduces the energy resolution, as demonstrated recently [12], and can only be overcome by using thinner HOPG films or by increasing the distances between source, crystal and detector, as we have done so far [8]. Both are possible, because the thickness of HOPG crystals can be easily adjusted. To prove in the experiment, whether the spectral resolution of a HOPG crystal can be significantly enhanced by reducing its thickness, measurements with low foil thickness were performed and are presented in this article.

2. General considerations

While the mosaicity of a HOPG crystal determines the energetic band width which can be reflected at once and the peak reflectivity for a given crystal thickness, an upper limit for the energy resolution of a mosaic crystal is given by the intrinsic width of Bragg reflection. This intrinsic width of the Bragg reflection refers to the diffraction properties of the small crystallites and is called the Darwin width for nearly perfect crystals. It results from particle size and/or strain broadening and can vary for different crystal thickness as well as the mosaicity depending on the production process.

 

Fig. 1. Diffraction properties of HOPG. The mosaic focusing is illustrated for a monochromatic beam (thick lines). Rays emitted by a point source are focused into a point in the image plane if the crystallites are lying on a Rowland circle. This parafocusing occurs in 1:1 magnification geometry, for which the distance F between source and crystal and crystal and image plane are equal. In this geometry the best energy resolution is expected. Also shown is the focusing error arising from reflection of crystallites out of the depth. For the experiment a CCD was placed in the image plane and a microfocus x-ray tube was used as source.

Download Full Size | PDF

An upper limit for the energy resolution is given by the intrinsic width of Bragg reflection Δθ according to following equation:

Here θ_B is the Bragg angle. With a reported intrinsic width of 26 arcs for the HOPG (002)- reflection [9] (7 arcs for Si(111)), an energy resolution up to E/ΔE=1870 is theoretically possible in this reflection order. The energy smearing in the image plane resulting from this intrinsic width of reflection is 2 eV for all distances. This energy resolution can hardly be achieved in practice, because of an extended source, aberrations of the spectrometer geometry and other focusing errors causing further broadening. The influence of the crystal thickness on the energy resolution can be estimated from geometrical considerations by the equation (cp. Fig. 1):

where D is the thickness of the crystal and Δs is the spatial smearing in the detector plane. Because the spatial smearing due to reflection out of the depth is independent of the distance F its contribution to the energy resolution can be neglected for very large distances. Considering absorption, attenuation of the incident beam by diffraction inside each crystallite (primary extinction) and screening of the lower lying crystallites due to Bragg reflection by crystallites near the crystal surface (secondary extinction), the thickness D can be replaced by a beam penetration depth. The secondary extinction depends on mosaicity, because screening is stronger for smaller mosaicity. Photons reflected out of the depth give rise to an asymmetric reflection profile. Making HOPG films thinner as the penetration depth can enhance the energy resolution, if the source size is small enough, however at the cost of intensity.

3. Experiment

The experimental arrangement of our test setup consisted of flat HOPG crystals (thickness 15 µm and 150 µm, from Optigraph), an x-ray tube as source and a CCD camera placed in the image plane (cp. Fig. 1). The HOPG crystals were mounted on a polished plane glass plate. The x-ray source, which was used for the measurements, is a low power microfocus x-ray tube (IfG [13]) with a source diameter of about 50 µm. Measurements were performed with the Cu K_α emission of the Cu anode at 8 keV. The spectra were collected with a 16-bit deep depletion CCD camera (Roper Scientific model PI-LCX 1300) with a quantum efficiency of about 50% at 8 keV. A thin (250 µm) Be window in front of the deep depletion CCD was used for vacuum sealing of the camera, so that a deep cooling (down to -50° C) of the CCD was possible. Measurements were performed for different distances in two reflection orders, (002)- reflection and (004)-reflection, whereby the distances between source and crystal and between crystal and detector were equal in each measurement. In this configuration parafocusing can take place. All measurements were carried out with the detector plane oriented perpendicularly to the reflected x-ray beam.

The images obtained for both crystals are summarized in Fig. 2. As shown in Fig. 2 for the 15 µm thick crystal the Cu K_α1 and K_α2 lines, which are separated by 20 eV, are clearly resolved for all distances, whereas for the 150 µm crystal only in the second order reflection the lines are clearly separated. In Fig. 3 the cross-sections of the images of the 15 µm thick crystal (averaged over a stripe of 200 pixels) are shown.

The energy resolution of the HOPG films at different distances was obtained from the recorded spectra by a convolution procedure (see [8]). In this procedure, by convolution with the natural Cu emission line profile [14] the spectrometer function was determined, which reproduces the shape of the measured data. The width (FWHM) of the spectrometer function then gives the energy resolution of the crystal. The best spectral resolution of E/ΔE=2900 was obtained for the 15 µm at the largest distance of F=260 mm (E/ΔE=2200 for F=210 mm) in (004)-reflection. For the largest distance of 310 mm in (002)-reflection an energy resolution of E/ΔE=1800 (E/ΔE=900 for F=210 mm) was found for the 15 µm. This value is very close to that, which is predicted by the measured intrinsic width of reflection. These measured resolutions are by far the highest spectral resolutions reported for HOPG crystals.

 

Fig. 2. The images of the recorded Cu K_α1 and K_α2 lines for different source-crystal and crystal-detector distances F in (002)- reflection (upper picture) and in (004)-reflection (lower picture) obtained with 15 µm and 150 µm thick HOPG films. The acquisition time to record the spectra of the 15 µm was 5 sec and 1 sec for the 150 µm crystal.

Download Full Size | PDF

The results demonstrate that the energy resolution of the HOPG crystals can be drastically improved by reducing the thickness of the crystal. On the other hand, by decreasing the thickness of the HOPG films the integral reflectivity is also reduced. The integral reflectivity can be determined from the images in Fig. 2, because the geometry of the used setup allows to collect all photons reflected from the crystal film by the CCD. In order to determine the integral reflectivity from the images it should be ensured, that the acceptance angle of the crystal film given by the mosaicity is smaller than the geometrical acceptance angle given by the distance and the crystal size. Latter is surely fulfilled for the smallest investigated distance F=210 mm, assuming a mosaicity smaller than 1°. Consequently all possible orientations of crystallites in the crystal can participate in the reflection process of the Cu K_α photons emitted by the source. In that case the integral over the number of photons collected by the CCD corresponds to the integral value which one would obtain by integrating over a rocking curve measurement. Dividing the photons collected on the CCD (calibrated with a Fe55 radioactive source) by the number of emitted Cu K_α photons in an solid angle given by (unit solid angle [sr]/crystal height [rad]), the integral reflectivity can be estimated. The number of photons emitted by the x-ray tube was determined with an energy dispersive SDD. This procedure gives an integral reflectivity of 4.6×10^-3 rad for the 150 µm thick HOPG in (002)-reflection and 0.9×10^-3 rad in (004)-reflection. In a previous work for a 150 µm thick HOPG in (004)-reflection a corresponding integral reflectivity of 1.2×10^-3 [rad] at 8 keV was measured [8]. For the 15 µm thick crystal the integral reflectivity is 0.7×10^-3 rad in (002)-reflection and 0.08×10^-3 rad in (004)-reflection. In Table 1 the crystal parameter for the 15 µm thick crystal are summerized and compared with the values of Ge(111) crystals.

 

Fig. 3. Reflection profiles of the 15 µm for the (002)-reflection (grey curves) and the (004)- reflection (dark curves) for different distances.

Download Full Size | PDF

Even if the integral reflectivity of the 15 µm crystal is about ten times lower than for the 150

Table 1. Comparison of crystal properties at 8 keV.

View Table

µm crystal, this integral reflectivity of the 15 µm thick HOPG crystal is still comparable with the values obtained for Ge(111) crystals [16]. In contrast the peak reflectivity could be lower by a factor two to three even if a very low mosaicity of 0.1° is considered, as can be estimated from rocking curve calculations [1]. On the other hand, bright reflection in a wide spectral range as a consequence of mosaicity is favorable in x-ray spectroscopy, because it reduces the recording time and enables single shot spectroscopy. Due to the above mentioned possibility to realize crystal optics in arbitrary geometry, high collection efficiency can be reached by optics design. That is e.g. interesting for point sources, emitting in a large solid angle, as e.g. laser plasma x-ray sources. That means, in contrast to bent ideal crystals, such as Ge(111), the design of x-ray optics using HOPG is much more flexible. Ideal crystals can only be bent very slightly and the fabrication process of these crystals is much more complicated. Consequently, the price for HOPG optics is much lower. In addition, because HOPG films are relative insensitive to temperature fluctuations and mechanical effort also the handling of HOPG optics is much easier. Critical for the performance of HOPG optics is the accuracy of the shape and for the bonding process the surface roughness of the mould. However, the above mentioned flexibility makes HOPG crystals to an interesting option for a broad field of application.

4. Conclusion

In conclusion, the highest measured spectral resolution obtained with HOPG films was presented. For thin HOPG films an integral reflectivity and energy resolution comparable to that obtained for Ge(111) could be demonstrated. The main advantage of these thin HOPG films is the possibility, to realize optics in any geometry, by mounting these films on a mould of any shape. In contrast to ideal crystals, such as e.g. Ge(111), which can be bent only slightly, any desired shape can be designed with HOPG films. That means the geometry of these crystals can be optimized for an application, e.g. to collect the light emitted from a point source in a large solid angle. Latter can be done e.g. by using cylindrically shaped optics in the von Hamos geometry [8] or ellipsoidally shaped optics [17] depending on the desired field of application. Examples of such applications are the x-ray diagnostics (e.g. plasma diagnostics), x-ray diffractometry and x-ray spectroscopy.