# What is a galaxy distribution function

## X-ray astronomy. Christoph Braig

Transcript

1 High-resolution image in the X-ray astronomy Christoph Braig Munich 2005

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3 High-resolution image in X-ray astronomy Christoph Braig dissertation at the Faculty of Physics at the Ludwig Maximilians University in Munich, presented by Christoph Braig from Kösching Garching, August 24, 2005

4 first reviewers: Prof. Dr. Joachim Trümper Second reviewer: Prof. Dr. Ralf Bender Oral exam day: January 12, 2006

5 Contents Summary xiii 1 Introduction 1 2 Fundamentals of X-ray optics Radiation-matter interaction The paraxial diffraction integral Refractive and diffractive lenses Properties of the refractive X-ray lens The generalized Fresnel lens The zone plate in various designs Correction of the dispersion The achromatic hybrid lens Reduction of the refractive profile component Dialytic configurations Paths for practical implementation Principles and construction of parallel systems Basics of transmissive lens arrays X-ray lenses based on hydrogen and helium Development of exemplary model systems The segmented aperture Diffractive-segmented monoband lenses Hybrid-segmented monoband lenses Development of various multiband lenses Construction of dialytic telescopes Detection and formation flight Detectors and detection sensitivity Comments for formation flight

6 vi Contents 7 Applications in astronomy Micro-gravitational lenses Stellar coronae Supernovae and SNR X-ray binary stars, AGN and GRB First experimental steps Diffractive and refractive miniature lenses Segmentation in the visual spectral range Aspects of numerical simulation Permeation at low temperatures A Geometric aberration theory 235 Acknowledgments 245

7 List of figures 2.1 Path and angle tolerances of reflective and transmissive optics Course of the critical zone number N 0 with the nuclear charge Z Reflection and refraction of X-rays at an interface Influence of absorption on the refraction angle Reflectivity and polarization rotation on material surfaces Diffraction at the circular aperture PSF of the diffraction at the Covered aperture Performance integral of the PSF variants of concave lens profiles PSF and MTF of solid lenses Lateral resolution of the solid lens Peak efficiency and axial intensity distribution of X-ray lenses Seidel aberrations of the parabolic lens profile Focal length and max. Lens energy spectrum of the general Fresnel lens transition from Gaussian to diffractive focus Angular resolution of the generalized FL for EE 3.12 transmission profiles of different diffractive lenses Effiz ience of multi-level zone plates Efficiency of different diffractive lenses Use of Fresnel lenses in higher diffraction orders Seidel aberrations of the zone plate Capability of the diffractive lens Focal dispersion of the hybrid achromatic dispersion correction with diffractive-refractive lens combinations Bandpass of the absorption-free hybrid achromatic absorption function of the hybrid achromatic transfer function absorbing achromatic lenses. Apodization of the transmission profile using gray filters

8 viii LIST OF FIGURES 4.7 Suitability of various elements for the manufacture of refractive correction lenses Suitability of multi-element materials for the manufacture of refractive correction lenses Gain factor of hybrid achromats Usability of solid hybrid achromats Focusing by means of Fresnel achromats in higher orders Seidel aberrations of the parabolic hybrid profile Hybrid profile beam path in the aplanatic Seidel aberrations of the aplanatic hybrid profile Light intensity of hybrid lenses Bandpass of reduced hybrid achromats Local dispersion of the reduced hybrid profile Comb structure with single and multiple reductions Increase in light intensity through profile reduction Dimensions of a dialytic doublet Length scales of dialytic configurations Focal position and singularity of the dialyte PSF radius and dispersion of the dialyte in 2nd and 3rd

9 LIST OF FIGURES ix 5th order Beam geometry on the Bragg monochromator Refocusing using capillary optics Spectral resolution of CCD-based detectors Spectral resolution of the XEUS micro-calorimeter The diffuse X-ray background radiation Simulated image of a distant star Orbits of extraterrestrial telescopes Influence on the lateral uncertainty of the detector unit PSF quality Permissible torsion angle of the detector module Apparent angular distance Overview of X-ray sources on small angular scales X-ray scattering spectra of the Fe-K α radiation Geometry of the gravitational lens effect X-ray brightness of quasars Galaxy distribution functions and microlens statistics Spectral type and angle size of the brightest stars

10 x List of Figures 7.8 Observation-relevant parameters of stellar coronae Structures of X-ray emission from pre-main sequence stars Temporal evolution of supernova remnants Observed X-ray flux from supernova remnants Properties of the supernova 1987A Unified model of active black holes Dynamics of merging super-) massive black holes Dimensions of the PANTER test facility construction of a simple multiband lens Test setup for optics and permeation of cool gases A.1 Notation for calculating the geometric image errors A.2 Off to new shores in high-energy astrophysics

11 List of tables 2.1 On the real part of the refractive index between 1 and 10 kev Optimization of the magnification of refractive lenses Dispersion factor of the general Fresnel lens Angular resolution of the absorbing hybrid lens Relative gain of hybrid lenses Maximum gain of hybrid lenses Pitch angle of biconvex hybrid lenses Permissible aperture ratio for parabolic hybrid profiles Efficiency of profile reduction Optimized profile reduction of Hybrid lenses Ratio of the 3rd and 2nd order bandwidths Examples of compact achromatic lenses Aperture radii of the FL matrix for ɛ = 0.1 mas Zone numbers N of the FL matrix for F = 15 km Number of segments in the kth ring Optical parameters of diffractive objectives Fresnel diffraction efficiency first order for Si and Ti Fresnel diffraction efficiency first order for Be and polycarbonate Total transmission of the filter-free hybrid segment Data on absorption-related reduction in resolution Increase in performance through filter-free dispersion Correction Optimized number of zones of segmented Li-achromats Examples of performance-optimized Li-Hybrid-Achromats Optimized number of zones of segmented Be-Achromats Examples of performance-optimized Be-Hybrid-Achromats Optimized numbers of zones of segmented B 10 H 14 -Achromats Examples of performance-optimized B 10 H 14 -Hybrid-achromats Performance data of diffractive multiband Objective performance data of conventional multiband achromatic lenses. Possible values for k and n for N = zones

12 xii List of Tables 5.19 Absorption-related spatial resolution in the reduced hybrid segment Examples of reduced Li achromats Number of maxima in the comb structure of the reduced achromatic examples of reduced Be achromatic examples Examples of reduced achromats made of polycarbonate Performance data of partially reduced dual band achromats Theoretical spatial resolution of the tunable dialyte Optimized number of zones of segmented Li Dialytes Optimized number of zones of segmented Be dialytes Examples of segmented dialytes in 3rd order Optimized number of zones of coherent Li dialytes Optimized number of zones of coherent Be dialytes Examples of coherent Li and Be dialytes in 3rd order Bragg crystals for spectrally selective detection in different energy bands counting rates of the diffuse X-ray background with diffractive optics Axial tolerance depending on the aperture ratio known X-ray binary stars and active galaxy cores Transmission of Fresnel lenses made of polycarbonate at 6 kev absorption nslengths of Li and Be between 1 kev E 10 kev

13 Summary The subject of the present work was the development and analysis of a new type of imaging optics with the aim of improving the spatial resolution in the X-ray band between 1 kev E 20 kev to at least 10 3 arcsec. Due to their high tolerance to manufacturing defects, transmissive lenses have the potential for diffraction-limited imaging. Depending on the design, there may be deviations of several nm compared to the ideal shape. In contrast to the solid version, which is subject to absorption, the diffractive, profile-optimized Fresnel lens has a diffraction efficiency of between 40% and 100% even in higher orders. The contamination of the image plane by scattered radiation from neighboring orders may have to be countered by a sufficient central obstruction, the radius of which corresponds to twice the detector radius. Radiation-optical calculations show diffractive lenses to be comparatively tolerant of aberrations of spherical and angle-dependent origin. Typical aperture ratios f) allow tilts of 1. The light intensity, defined as the product of the effective collecting area and bandpass, for Fresnel lenses only scales linearly with the focal length, but remains limited to a few cm 2 kev even for focal distances of 10 2 km. With the segmented aperture, however, the light intensity can be increased without giving up the principle of diffraction-limited imaging and the classic single focus. With a spatial resolution of 10 3 mm, such incoherently operating lenses achieve a light intensity of 10 3 cm 2 kev. Using a crystal spectrograph that is adequate for the diffractive bandpass requires large radii of 10 m and typical focal lengths in the range of a few 10 2 km. Furthermore, in the course of this work, it is believed that for the first time multiband lenses for the scientifically advantageous simultaneous focusing of up to three energy bands were implemented. Consisting of partial lenses with different grid frequencies, they prove to be in principle equal to the monoband version in terms of resolution, focal length and light intensity. The dispersion correction by means of an additive refractive lens profile extends the spectral bandpass on the detector directly accessible 10 2 ev or more. The absorption is accompanied by a reduced collecting area for compact hybrid lenses. Nevertheless, while maintaining the angular resolution for materials such as Li or Be beyond less kev, the result is an improved light intensity compared to the diffractive analogue. Optimized with regard to material and energy, such achromatic lenses increase the detection sensitivity

14 xiv Summary up to 40 times that corresponds to a light intensity of 10 2 cm 2 kev with focal lengths of 10 2 km. Once again segmented, the absorption has a comparatively minor effect on the angular resolution; the sensitivity of the dispersion-corrected lens increases by up to two orders of magnitude compared to the diffractive version under otherwise identical conditions. With a given spatial resolution of m, there are optimized light intensities between 10 3 and 10 4 cm 2 kev for Li above 6 kev and Be beyond 8 kev, comparable to those of the currently active observatories Chandra and XMM-Newton. The angular sharpness scales inversely with the focal distance, for 10 3 arcsec there are focal lengths of) km. Plano-convex profiles generally do not do justice to the mostly small radii of curvature of the refractive component with regard to their third-degree aberrations. In contrast, the aplanatic, almost biconvex profile reduces both spherical and angle-dependent image errors to their diffractive contributions and therefore suggests the construction of symmetrical, prism-like building blocks in the segmented hybrid achromatic lens. The interference associated with the coherent profile reduction requires the use of imaging spectrographs with a resolution close to 1 possibly. Above all, optically weak materials such as e.g. Polycarbonate C 16 H 14 O 3) benefit from the increased transparency with constant spatial and angular resolution; Example configurations optimized in the energy interval 9 kev E 12 kev provide a light intensity of at least cm 2 kev. Models made of Li and Be achieve a similar performance above 4 kev and 7 kev, respectively. Unlike diffractive, simultaneous focussing objectives, multiband hybrid systems allow detection using conventional CCDs. The configurations consisting of Li and Be each map two energy bands at the same time and, with a spatial resolution in the sub-mm range and focal lengths of a few 10 2 km, prove to be a competitive alternative to the monoband telescope in terms of their overall light intensity: Ideally, one obtains 4 7) 10 3 cm 2 kev. Dialytic model telescopes, the refractive component of which is spatially separated from the diffractive one, offer the option of dispersion-corrected optics that can be tuned over several kevs. A usable energy interval of 6 kev E 14 kev is obtained by varying the lens spacing. The light intensity increases from cm 2 kev in the second to cm 2 kev in the third order of dispersion. Compact dialytes with a diameter of 1 m have the potential for an angular resolution of a few 10 5 arcsec and a light intensity of several 10 3 cm 2 kev. The spectral bandpass of such models is 1 kev. Estimates of the signal-to-noise ratio show that if the detector is adequately shielded and the source flow is moderate, a signal or photon-limited observation situation can be expected. In view of the discrete X-ray background, this also applies to a large extent when several telescopes are connected in parallel. Supplementary considerations on potential astronomical observation objects show that coronaes of neighboring stars, jets of X-ray double stars and active galaxy nuclei, supernova remnants with regard to their extension have a resolution of 10 3 arcsec. Considerations of merging, supermassive black holes should also be of great interest with a view to future gravitational wave experiments.

15 Chapter 1 Introduction The knowledge gained over the last few decades about high-energy cosmic radiation sources literally made the universe appear in a new light relative to its familiar visual appearance. While the instrument-based observation of the starry sky was already possible with Galileo in the 17th century AD. began, various analyzes of the ionosphere [1, 2] first indicated the existence of atomic, extreme UV and X-rays of solar origin in the 1930s. After initially vague measurements [3], their undoubted proof was finally achieved in 1949 with the help of Geiger counters [4], which, mounted on a rocket, could briefly operate outside the Earth's atmosphere absorbing the X-ray light. Optical imaging in the sense of a two-dimensional representation of the intensity distribution of the object in an image plane turned out to be difficult, however, since the extraordinarily high absorption in the X-ray range is the imaginary part of the complex refractive index n = 1 δ iβ for all materials usually. several orders of magnitude above the corresponding value for visible light does not allow the construction of conventional reflection or refraction telescopes. Nevertheless, several researchers subsequently succeeded in photographing the sun in the said spectral range, initially with the help of simple pinhole cameras, and later using Fresnel zone plates [5, 6]. As a focusing transmission grating, these lenses, first described by Soret [7], are based on the diffraction of electromagnetic radiation and thus avoid the problem of absorption. However, in their simple form, zone plates show a rather low focus efficiency of less than 10% as well as a strong chromatic aberration on circumstances which led to the development of various alternatives to the zone plate from the sixties. On the one hand, based on the above-mentioned pioneering work [3], the largest possible counter module, e.g. upstream honeycomb-like collimators. In this way, X-ray sources with an angular resolution of up to about 1 can be detected with a spectrally broadband and with maximum utilization of the collecting surface [8]. In fact, the combination of sufficient sensitivity and resolution achieved with such collimator tubes in various rocket- and balloon-supported experiments made it possible for the first time from the early 1970s onwards for UHURU to analyze extrasolar X-ray sources, especially Cen X-3 and Cyg X-1 [9 ]. The X-ray sources of these binary star systems turned out to be

16 2 1. Introduction to pulsar or stellar black hole. Although the scheme of the collimator-reinforced proportional counter led to the discovery of numerous new emitters and in some cases allowed their rough analysis, the method was not satisfactory from the optical point of view. The reasons for this are not only in their angular resolution, which is limited by the collimator dimensions, but also in the fact that this coincides with the usable field of view (FOV). Theoretical work by H. Wolter, dated to the year 1952, suggested the double external total reflection on a coaxial arrangement of a paraboloid and hyperboloid mirror surface for bundling incident X-rays in microscopy [10].The possibility of focusing using a simple parabolic mirror and the phenomenon of total reflection, which is external due to Ren) 1, were already known, but the construction of such an optical system has so far failed due to the serious aberrations in the case of rays that are not axially parallel. As Wolter showed, the aberrations can be drastically reduced by a second reflection on a hyperboloid mirror shell. A useful side effect of the configuration transferred to telescopic applications by Giacconi and Rossi [11] is that the focal length is reduced to a few meters. In the mid-sixties, the technical difficulties associated with the macroscopic shape and the surface finish on the scale of 10 9 m had been overcome to such an extent that the first focusing X-ray telescopes could initially be put into operation with the help of rockets, shortly afterwards on board Skylab [ 9]. These mirror telescopes, dedicated to the observation of the solar corona, were followed in 1978 with Einstein, the first Wolter1) telescope 1 with sufficient collecting surface to be able to examine extrasolar targets. Mention should be made of the discovery of the shock wave emission from supernovae or the thermal emission of the hot gas in galaxy clusters along with several thousand new point sources. The ROSAT mission, launched in 1990, added two orders of magnitude to the catalog of known X-ray sources. The success of this and other missions helped the WolterI) mirror to achieve a breakthrough over competing approaches, so that almost all X-ray satellites up to about kev are based on its scheme to this day. If the focusing total reflection is ruled out at higher energies, it is advisable to only arrange a two-dimensional mask that absorbs in places according to a certain pattern in front of the detector. It creates a geometric shadow from the X-ray source; Since the resulting intensity distribution in the detector plane depends on the angular position of the point source, the superposition of the individual shadow images results in a clear counting rate distribution in the detector for extended objects, from which the real image can then be reconstructed. In the absence of better alternatives, this so-called Coded Aperture Imaging is still used today in the hard X-ray and gamma range, i.e. above around kev. The compact and energy-independent design is, however, in addition to the geometric design, typically around a minute of arc. arcmin) limited angular resolution a bad signal-to-noise 1 H. Wolter also investigated modifications of this so-called type I) configuration, e.g. on SOHO or in the form of EUV collimators, but should be left out here.

17 3 ratio SNR) compared to. This property of every indirect imaging method is based on the fact that the noise of all detected photons is superimposed on the signal of a point source distributed over the entire detector surface and can be eliminated by using direct, focusing optics. In 1999, XMM-Newton and Chandra, the two most powerful X-ray telescopes to date, went into operation. With regard to their main scientific objective, the observatories initiated by ESA and NASA are complementary to each other: XMM-Newton was optimized with regard to the sensitivity, i.e. the lowest still detectable radiation flux, so that the temporally and spectrally high-resolution spectra of even weak X-ray sources such as e.g. accretive binary stars in neighboring galaxies. For this purpose, XMM-Newton has 3 Wolter telescopes, which in turn consist of 58 mirror shells pushed into one another and have a total effective area of up to 4500 cm 2 beyond around 0.5 kev. The large effective telescope aperture is at the expense of the angular resolution, since the mirror shells, which are necessarily thin-walled for reasons of weight and space, cannot be configured with the required precision. This leads to a broadening of the point spread. point spread function, PSF) corresponding to a resolution element of about 6 arc seconds engl. arcsec). With Chandra, on the other hand, the main focus was placed on mirrors that were shaped and adjusted as precisely as possible. In this way, the angular resolution reaches 0.5 arcsec, the best value ever achieved with an X-ray telescope. The small number and greater thickness of the nested mirrors now result in a collecting area of up to 800 cm 2 beyond approx. 1 kev. After several decades of continuous improvement in optical quality, however, serious technical difficulties are now evident, which requires further optimization the mirror quality and thus the achievable angular resolution. The problems lie in both high and low frequency deviations from the ideal parabolic-hyperbolic mirror profile in terms of surface roughness and form defects. In the meantime, a value of around 0.1 arcsec is regarded as the actual minimum for telescopic applications. However, in order to define the starting point of the present work here, the theoretically achievable diffraction-limited image sharpness is missed by 2 3 orders of magnitude. The blatant deviation between the real and the possible image sharpness must be viewed from a physical point of view as all the more unsatisfactory as numerous astronomical objects in the X-ray range appear punctiform under the current observation-technical circumstances and, on closer inspection, suggest previously unknown structures. In addition, there is the relatively high surface density, i.e. the ratio of mass to effective collecting surface, as it is typical and unavoidable for reflector telescopes at grazing incidence, typical in the sense of values of hardly less than 10 1 kg cm 2 for missions optimized in this regard, such as the one in XEUS is currently in the project phase, but is also unavoidable in view of the growing thickness of the shell with increasing demands on image quality. As the example just mentioned of the two complementary observatories XMM-Newton and Chandra shows, the superior angular resolution of the latter is at the expense of lower efficiency.

18 4 1. Introduction The following is an attempt to develop the basic features of future X-ray optics for astronomical applications that will overcome these obstacles. The already mentioned zone plate serves as a basis as a paradigm of a transmitting imaging optics based on diffraction. Chapter 2 accordingly leads from an overview to abstract principles of X-ray and diffraction optics through the properties of general X-ray lenses to detailed analyzes of massive refractive optical elements, zone plates and their derivatives, i.e. generally diffractive lenses. According to their structure, the latter can be differentiated into amplitude- and phase-modulated circular gratings. Although the desired diffraction-limited resolution can already be achieved in this way with lenses with a large collecting surface and low mass, the correction of the intrinsic chromatic aberration remains the central challenge. The physically most effective approach to achieve this by means of refractive divergent lenses according to previous knowledge is the subject of the next chapter 4. While the work has so far mainly examined elementary theoretical models, in the further course of Sect. 5 aspects of their application in large-scale telescopes come to the fore. These include, in particular, questions of the segmentation of large-area lenses, the possible implementations of which are presented taking into account the previously developed concepts. A distinction is made between primarily diffractive and dispersion-corrected versions. Furthermore, in this part of the thesis possibilities are to be addressed to use optics connected in parallel and based in the soft X-ray band H 2. Inevitably long focal lengths of) km make the spatial separation of lens and detector inevitable, which is why an analysis of the scattered light caused by the diffuse background is of particular importance. Furthermore, concepts for the design of the detector are outlined. In order not to impair the image quality through temporal fluctuations in the relative position, the two modules must be mechanically stabilized axially and laterally against each other. In the context of a brief presentation, based on considerations of possible orbits on different scales, Chap. 6 is also dedicated to gravitational gradient forces and other influences with regard to their effects on the positioning of the two spacecraft. The performance parameters on which the telescope is based specify its astronomical field of application. The extremely small field of view compared to previous missions and the long integration times required because of the rather modest light intensity make the instrument appear unsuitable for large-scale surveys of the sky. However, its high spatial resolution makes it ideal for analyzing individual objects such as the already mentioned stellar coronae, supernovae or the accretion disks of active galaxy nuclei. In chap. 7, therefore, potential observation objects are compiled with concrete examples and corresponding astrophysical questions. The last chap. 8 finally goes into various possibilities to experimentally check the knowledge gained and to further develop it technically. In this context, special attention should be paid to the scale invariance, which allows the geometric dimensions to be reduced to practically accessible lengths.

19 Chapter 2 Fundamentals of X-ray Optics Since a few years ago, in the course of the studies preparing the planned XEUS mission, the option was considered of not using massive mirror shells, but precisely polished, appropriately milled and oriented Si wafers This gives rise to the hope that, given the total effective area of up to 30 m 2 at 1 kev), the still critical surface density of kg cm 2 can be reduced many times over, thus making transport and calibration much easier. At the same time, the angular resolution remains limited to a few arc seconds, since this construct is a double cone as an approximation of a shortened Wolter optic, stabilized in the form of numerous superimposed wafers. The example makes it clear that turning away from the concept of massive mirror modules and turning to thin, lens-like geometries is not enough to significantly improve both the collecting surface and the resolution. Rather, it is principally new methods such as diffraction and refraction to focus X-rays that promise success. Compared to reflective optics, the latter allow significantly greater tolerances in terms of optical path difference (OPD) and angular scattering, both of which are semi-quantitative standards for the imaging errors that limit the resolution. Rough estimates of the respective design-related errors can be found in Fig. 2.1. OPD and angular spread δφ result from small displacements δx and tilts δθ according to the inserted sketches and otherwise follow elementary laws of geometrical optics. 2.1 Radiation-matter interaction However, the refractive index ne) as the expression describing the interaction WW) with media as a function of the energy E in the X-ray range above less than 100 ev shows a fundamentally different behavior than in the visual. Because of the no longer negligible absorption i.a. as n = 1 δ iβ with δ, β ɛ R + 2.1)

20 6 2. Fundamentals of X-ray optics Figure 2.1: Path and angle tolerances of reflective and transmissive optics. Estimates for the scattering angle (solid lines) depend on the tilt relative to the optical axis from left). Typical error limits of the optical path difference OPD) with regard to the Rayleigh's λ / 4 criterion are plotted as a function of the energy on the right). The tolerances of the refractive optics with δ = 10 3 are to be regarded as lower limits, while the other values represent approximate orders of magnitude. Expressed, the increment δ 1 stands for a very small deviation from the vacuum value n vac 1. The likewise small quantity β nevertheless exceeds the corresponding value in the visible spectral range by orders of magnitude and, depending on the material and energy, leads to a 1 / e attenuation on a scale of mostly) m. In order to gain a semi-quantitative understanding 1 of the radiation-matter interaction in the X-ray range, we consider the semiclassical model of a medium, consisting of atomic nuclei and more or less harmoniously bound e. Accordingly, the e accelerated by the incident transverse) field E in r, t) emit dipole waves. The field E out r, t) resulting from the superposition of all elementary contributions according to Huygens' principle is essentially summed up over all N atoms and their Z n electrons, complex-valued scattering amplitude f = NZ nn = 1 s = 1 ˆf n, s) k, ω 2.2). It in turn depends on the momentum transfer kk out k in with k = 2k in sin θ and the scattering angle θ, which is included in the scattering amplitude in the form of separable exponential factors and refers to the positions rn, s of the nuclei or their e: ˆf n, s) k, ω = Q n, s ω) eikrn, s) 2.3) The spatial position of the nuclei and also of the e are now of decisive importance: 1 A strictly analytical treatment is dispensed with here for reasons of space, a more detailed presentation can be found e.g. in [12].

21 2.1 Radiation-matter-interaction 7 Solid body with periodic structure (crystal). The atoms, which are regularly arranged at intervals d as scattering centers, enable fixed phase relationships between the partial waves. With bond lengths d λ, sharp interference maxima result in the X-ray band according to the Bragg / Laue condition even with large scattering angles θ. Metals such as Li and Be in particular also fall into this category. Amorphous solids, liquids and gases. Irregular and partly temporally variable arrangement of the cores and e leads i. a. to statistically distributed phase factors in Eq. Apparently, however, for θ 1 ˆf n, s k, ω) = Qn, s ω), so that with the forward scattering in the zeroth order, the trivial case of interference can be observed. Incidentally, diffraction in the crystalline medium is also limited to this if d λ applies there. The borderline case eikrn, s) 1 finally motivates the notation) 2 c ne) = 1 2π renaf 0 1 E) + if2 0 E)), 2.4) E where na for the atomic number density, re for the electron radius and the superscript 0 des total scatter factor f 0 1 E) + if 0 2 E) stands for the limit krn, s) 0 considered here. A further differentiation applies to the energy or frequency ω of the radiation, in particular in relation to the energy levels or natural frequencies of the atom or the chemical compound. Strong absorption basically suffers from soft X-rays of up to a few kev by ionizing the atoms by knocking out internal e. Consequently, there is a strong energy dependency, which in the range between around 1 and 20 kev away from absorption edges mostly runs like E γ, with 3 γ 4. The functional relationship with the nuclear charge Z, which is caused by numerous jumps corresponding to the position of the Element in the periodic table. Elastic and coherent scattering occurs mainly on strongly bound e, which guarantee fixed phase differences between incident and reflected or refracted wave in the sense of geometrical optics. In the image of the harmonic oscillator, the e with natural frequencies ω s follow the excitation with the frequency ω ω s only weakly and out of phase. This is the cause of the low refractive power δ 1). As a frequency-preserving scattering, it dominates with low energy over the inelastic or Compton scattering, which only comes into play with negligible binding forces and E E K with E K as the binding energy of the e in the K shell). It takes place with partial energy transfer to only weakly bound e and ergo undefined phase relationships. In the context of the forward scatter, which is important in this work, however, it does not play a role.

22 8 2. Fundamentals of X-ray optics Because of the complex WW processes, f 0 E) must be determined semi-empirically for each medium, usually by measuring the absorption length determining the imaginary part with subsequent calculation of f 0 1 E) from the Kramers-Kronig relations . While the βλ) determined within the scope of this method is only roughly approximated by a function λ 4 and errors of several 10% can occur, δ follows a λ 2 curve very precisely, at least in the energy range under consideration. It is therefore δλ) = δ c λ λ c) 2) 2 or δe) Ec = δ c, 2.5) E if δ c = δ λ c) and for E analogous terms are used. Table 2.1 gives an overview of the fit parameters and their ± 1σ error limits in the energy interval 1 kev E 10 kev. In the context of diffraction theory, only the substance α ± σ α in kev) 2 substance α ± σ α in kev) 2 H ± He ± Li ± Be ± B ± C ± N ± O ± Table 2.1: The real part of the refractive index between 1 and 10 kev. Far from absorption edges, δ = 1 Ren) decreases with the energy E to a good approximation according to δe) = α E 2. The slope parameters α are listed with their standard errors. Proportionality 2.5 referred to, however, absolute values of the real refractive index play a role in the context of the theory of image errors.With the critical number of zones N 0 δ according to [16], we now introduce the number of 2πβ phase inversions per absorption length, the parameter characteristic of the optical quality of the medium. In contrast to δ and β itself, it shows a relatively smooth course with regard to the nuclear charge Z, as illustrated in Fig. 2.2. As can be seen from the graphic, elements with low Z and relatively hard X-rays are to be preferred in terms of low absorption. Where this is not possible, a material choice that is favorable with regard to the K electrons can improve N 0 by a factor of 10, e.g. via 15 Ph instead of 13 Al at 2.0 kev. In addition, N 0 runs roughly proportional to λ 2, at least in the soft X-ray band of a few kev, and results as the quotient of the functions of δ and 2πβ. So far, only elementary substances have been considered. The optical properties of compounds such as B. Hybrid compositions of the form X m H n or plastics are made up of those of the components additively:) 2 c δ + iβ = 2π r e n mol E f 0 1, k E) + if2, ke)) 0 2.6) k

23 2.1 Radiation-matter interaction 9 Figure 2.2: Course of the critical zone number N 0 with the nuclear charge Z. Away from absorption edges, the measurement data show points) for N 0 for Z 2 in good approximation a Z 3 -dependence solid curves). The discontinuity in N 0 E) indicated here by arrows is due to the K 1s binding energy of the respective element. Index k) of the molecule is added up over all atoms, the n mol of which can be accommodated in the unit volume. The fact that the molecular structure, i.e. the geometric arrangement of the atoms, does not normally play a role in the case of forward scattering according to the above argumentation is also relevant with regard to the choice of material. The only exception are the narrow spectral bands in the area of the absorption edges, which are characterized by anomalous dispersion and correlate not only with the usual intra- but also with the structure-dependent interatomic bonds. According to Eq. 2.6 follows for the critical number of zones of a multi-element material k N 0 = f 1, k 0 E) 2π kf 2, k 0 E) = jm jf1, je) 0 2π jm 2.7) jf2, j 0 E), if mj is the number of the atoms of type j per molecule. This relation plays a role especially in connection with H 2 admixtures to increase the optical quality of a substance; on the other hand, it results in the problematic effect of carbon due to the absorption, for example in the polycarbonate C 16 H 14 O 3). The microscopic peculiarities of ne) now affect the macroscopic physical and formal laws of X-ray optics. In the further will

24 10 2. Fundamentals of X-ray optics basically assumed a refractive index n r, ω) that only varies on very large scales relative to the wavelength. This assumption includes discontinuous changes with rnr, ω) b ± δ rrb) at adequately dimensioned orifices as well as continuous variations of the form rnr, ω) 0, which are to be considered as local constants in the differentiation of Maxwell's equations leading to the wave equation an analogous relationship applies to B r, t)): 2 t 2) 2 c) E r, t) = 0 2.8) nr, ω) Accordingly, the general solution consists of a linear superposition of the form) A r, t) Ã keikr ωk) t) d 3 k, 2.9) where A r, t) stands for any component of the fields E r, t) and B r, t). The requirement of sufficient homogeneity of the medium limits both to wave packets with E B) and E, B k defined by the) k spectrum Ã k. In the simplest case, plane waves E r, t) and B r, t) result, which only depend on time in the form of a separable factor e iωt. The simplified stationary variant of Eq. 2.8 thus reads + k 2) E r) = 0, 2.10) with an analogous expression for B r). Continuity conditions for the fields are now provided by the well-known laws of reflection and refraction, see Fig. 2.3), which we list here because of their quantitative deviations from their visual counterparts. To this end, we consider the components of the electric and magnetic fields that are perpendicular and parallel to the plane of incidence at the interface between two media with refractive indices n 1 and n 2. Analogous to visual optics, continuity requirements for E p and H p as well as D s and B s initially provide the trivial law of reflection φ e = φ r. The law of refraction can be derived in a similar way, n 1 sin φ e = n 2 sin φ g with n 1,2 ɛ C. 2.11) At this point, the slight but fundamental modification of that from Re n 1,2) is possible following direction of deflection mentioned by Im n 1,2). At first glance, the influence of complex refractive indices on the refractive angle β seems quite unclear. This problem has been discussed controversially by various authors [20]. Various models are in circulation that have so far been awaiting experimental verification or falsification. Nevertheless, there seem to be at most marginal quantitative differences between them. The graphs in Fig. 2.4 are based on the theory favored by the authors of [20]. As in particular in Chap. 5, due to the large focal lengths, the deflection angles are at most a few arc seconds arcsec). Furthermore, according to [20], relative angular deviations 1 φ g, β 0 φ g, β = are to be expected for φ e 90 and maximum refractive index parameters δ and β.

25 2.1 Radiation-matter interaction 11 Figure 2.3: Reflection and refraction of X-rays at an interface between two media, which are assumed to be linear, homogeneous and isotropic. Incident index e), reflected and refracted waves consist of field components E s and E p perpendicular) and parallel) to the plane of incidence. In the extreme case of almost grazing incidence, an absolute angular error of 1 mas, that is to say of the order of magnitude of the desired resolution, would therefore have to be expected. The assumed deviations from the absorption-free angle of refraction therefore turn out to be negligibly small as a rule, as long as the angle of incidence does not exceed moderate values φ e. A significant influence on the imaging quality of the optical systems discussed in this work is therefore generally not to be expected and is therefore not taken into account for the time being. With δ, β 1 the practical formula sin φ g n sin φ e results after successive series expansion, with n 1 δ 1 δ 2) i β 1 β 2). The glancing angle θ c π φ 2 c of total reflection, which is especially important for mirror optics, follows under the assumption β i 0 to θ c = 2 δ2 δ 1). 2.12) In practice, again assuming maximum refractive indices δ, glancing angles of a maximum of 2, but mostly of significantly less than 1, occur. We now come to the Fresnel formulas for the reflection and transmission coefficients, which in turn are calculated from the continuity requirements for the tangential, stationary components of the electric and magnetic fields E and H, as in the visual: Er E e) s = n 1 n 2 cos φ en 1 n 2 cos φ en 1 n 2) 2 sin 2 φ en 1 n 2) 2 sin 2 φ e and Eg E e) s = n 1 n 2 cos φ e + 2 n 1 n 2 cos φ e 1 n 1 n 2) 2 sin 2 φ e

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