Univerza v Ljubljani Fakulteta za matematiko in fiziko. X-ray monochromators. Author: Agata Müllner Mentor: prof. dr. Iztok Arčon

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1 Univerza v Ljubljani Fakulteta za matematiko in fiziko X-ray monochromators Author: Agata Müllner Mentor: prof. dr. Iztok Arčon April 005

2 Abstract Due to the divergent character of the synchrotron radiation and large distance from the source to the experiment focusing optical elements are needed to collect as much of the beam intensity as possible and to monochromatize the beam. Monochromatizing can be done with x-ray mirrors, crystals or with multilayers. In this work different methods of focusing with crystal monochromators are presented. The basis of the multilayers performance is described. Contents 1. Introduction...3. Crystal diffraction Crystal optics Sagittal focusing Bending devices Multilayers Preparation of multilayers Conclusions...1 References...13

3 1. Introduction For the x-ray absorption spectroscopy monochromatic photon beam of high intensity is required [1]. The synchrotron radiation is divergent which together with the large distance from the source to the experiment causes the increase of the beam cross section and the decrease of the flux. Therefore focusing optical elements must be used. In the x-ray range such optical elements are mirrors, crystals and multilayers []. The schematic view of the experimental set-up is shown in Fig. 1. Figure 1: Schematic view of an experimental station for x-ray absorption spectroscopy with x-ray synchrotron radiation [3]. There are two main goals in the designing of x-ray optics: to collect as much radiation as possible (this means collecting as much horizontal divergence as possible) and to monochromatize the collected radiation as efficiently as possible. In this work the crystal optics needed to achieve these goals will be discussed.. Crystal diffraction X-ray diffraction in a crystal occurs if the Bragg condition [4] is fulfilled: dsinθ B = nλ, (1) where d is the spacing of crystal lattice planes, θ B is the Bragg angle measured from the diffracting planes, λ is the wavelength of the incident x-rays and n is an integer (the order of diffraction). The incidence angle θ i does not need to precisely equal the Bragg angle θ B but must lie within the range of angles given by the Darwin width (or rocking curve width) ω of the crystal (typically few arc seconds). The Darwin width is the full width at half-maximum (FWHM) of the total reflective profile of the monochromator crystal and is given by [1]: λ NF h ωn =.1re n 1 + πsin( θ ), () B for single Bragg reflection of polarized x-rays, where re is the electron radius, λ is the incident photon wave length, N is the atomic density, Fh is the structural factor: F = F(sin λ / d) /( h + k + l ), where h, k, l are Miller indices of the h reflection plane, θ B is the Bragg angle and n is the order of harmonic present in the beam. The rocking curve for 3

4 Si(111) is shown in Fig. and the energy dependence of the Darwin width (or rocking curve width) in Fig. 3. The crystal energy resolution is given by [1]: E.1 d = re E π ( n + 1). (3) Figure : Single crystal reflectivity curve for Si(111) at 10 kev [1]. Figure 3: Comparison of the synchrotron radiation opening angle α with some typical silicon Bragg reflections [1]. Fig. 3 shows the opening angle of an x-ray storage ring which is wider than the acceptance angles of the monochromator crystals. If the full opening angle is allowed to strike the crystal the energy resolution is degraded. If slits are used to collimate the beam the intensity is decreased. The x-ray spectroscopy requires the suppression of higher harmonics in the incident beam. This can be achieved if two parallel monochromator crystals are used [1]. The order sorting is achieved by adjusting the pair to be slightly nonparallel. Thus the higher-order reflections are extinguished leaving only the fundamental energy (Fig. 4). The fundamental intensity, however, is decreased by detuning the crystals. Figure 4: Reduction of the higher-order reflection by detuning for the Si(111) reflection. When the angle between the two crystals is 3.5 arcsec the fundamental (111) reflection at 10 kev has about 50% of its original intensity while the (333) reflection is reduced by about 10 3 [1]. 4

5 3. Crystal optics Although crystals are dispersive elements they can be used as focusing optical elements. But one condition must be fulfilled: all of the incoming x-rays must make the same angle with respect to the lattice plane normal in order to maintain the energy resolution. Most commonly curved crystals are used to obtain focusing. Focusing can take place in two directions: in the scattering plane (meridional focusing) or in the plane perpendicular to the scattering plane (sagittal focusing). For the meridional or dispersive focusing the most common focusing geometry is the Rowland circle geometry [1,, 5, 6, 7]. In this geometry the curved crystal with lattice planes of radius R is arranged tangentially to the focusing (Rowland) circle of radius R/. The source and the focus lie on the circle (Fig 5). In this geometry rays from an extended source can be brought to focus if one assumes a virtual point source [5]. Figure 5: Geometry of focusing monochromator crystals: a) Johann, b) Johansson. Rays diverging from source S are focused at image I. R and C are the radius and center of the focal circle respectively. C is the center of curvature of the c rystal planes. θ i and θ r are the incidence and reflection angles [6]. Changing energy of the monochromatized beam requires changing the radius of the crystal. The crystal bending techniques will be discussed in section Sagittal focusing The focusing in non-dispersive direction or sagittal focusing is most commonly used for the x-ray absorption spectroscopy. The ideal geometry of the crystal for this kind of focusing is ellipsoid of revolution but a good approximation to this geometry is given by a cylindrical surface. Fig. 6 shows the use of a cylindrically bent crystal for sagittal focusing. Figure 6: Sagittal focusing with a first flat and a second cylindrically bent crystal [1]. 5

6 Fig. 7 shows the optical path of rays leaving a point source S with an arbitrary horizontal deviation from the central ray α, and which are focused in an image I through Bragg diffraction from the crystal. The relation between Bragg angle θ, cylinder radius R S, source-to-crystal distance p and crystal-to-image distance q is given by [7]: 1/ p + 1/ q = (sin θ )/ RS. (4) Figure 7: Schematic view of sagittal focusing using Bragg diffraction from a cylindrically bent crystal of radius R S. a) Optical system composed of one cylindrically bent crystal. b) Optical system composed of one flat crystal and one cylindrically bent crystal [7]. In the general case, the Bragg angle θ is a function of α: θ ( ) = θ α. We can see from Fig. 7 that RS + z sin( θ ) =. (5) p A s can be seen from equations 4 and 5 the magnification M = q/ p is related to the relative position of the source S a long the z axis [7]: M = 1/ 1 + ( z/ R S ). (6) As the source S is moved from the axis of the cylinder ( z = 0 ) to the surface of the cylinder ( z = R S ), the magnification M changes from a value of M = 1 to M = 1/3. The (monochromatic beam) horizontal acceptance of this system is limited to a value α max given by solving the following equation [7]: θ ( α) θ (0) = ω, (7) where ω is the Darwin width of the crystal. 6

7 Let us now discuss the optical system shown in Fig. 6 and 7b. It is composed of a first flat and of a second cylindrically bent crystal. Its horizontal acceptance is limited to a value α max given by the solution of the equation [7]: θ = θ ( α) θ = ω, (8) 1 where θ 1 is the Bragg angle on the flat crystal. A rocking curve analogue is obtained by rocking one of the crystals throug h the parallelism condition θ = θ (0) 1. However, the shape and width of this curve are, in the general case, different from those relative to a flat-crystal s rocking curve. A method to illustrate the optics of sagittal focusing with a flat and a cylindrically bent crystal for M = 1 is summarized in Fig. 7b. The circle C represents intersection points on the plane of the first crystal of rays from the source S which are reflected from the first crystal with constant Bragg angle. The hyperbola H represents the intersection points on the first crystal of rays from the virtual source S which strike the second crystal with a constant Bragg angle θ.the circle C and the hyperbola H have been replaced by corresponding footprints of width equal to the Darwin width, ω (inset of Fig. 7b). Rays are transmitted by both crystals only if they belong to the intersection of the two footprints. Figure 8: Rocking-curve scans in a) the M = 1 and b) the M = 1/3 geometry. The footprint curves have been calculated for Si(311) crystals at 5 kev. On the left side of each panel the result of calculations is shown, while o n the right side corresponding distributions of the reflection in a vertical plane between the second crystal and the image is shown. δ is the relative angle between the first and the second crystal: δ = θ θ (0) 1. It can be shown that in 1:3 geometry any ray from the virtual source S (which now lies at z = R ) forms exactly the same angle with the flat and with the cylindrical crystal [7]. S 7

8 In Fig. 8 the situation which occurs during a rocking-curve scan in the two geometries, the 1:1 and the 1:3 is illustrated. Fig. 9 shows the photographs of the beam in the geometries discuss above. Figure 9: Sequence of photographs of the monochromatic reflection at three different alignments of the two crystals: a) in the M = 1 geometry, and b) in the M = l/3 geometry. The inset shows the corresponding points on the rocking curve at which the photographs were taken. In the 1:1 geometry the reflection appears at the centre of the image and gradually separates into two spots which drift further and further apart, while in the 1:3 geometry the whole width of the image appears and disappears simultaneously [7]. 3.. Bending devices Changing energy of the monochromatized beam requires changing the radius of the sagittally focusing crystal. Several different bending techniques can be used. Fig. 10 shows the four cylinder mechanism for bending rectangular crystals. Bending to a cylinder is achieved by applying forces of the same size to the outer bars. By using different torques asymmetrical shapes can be obtained. Figure 10: Schematic view of the four cylinder bender in plane and in face view []. The inner cylinders are segmented into two pieces so that the crystal accepts x- rays at grazing incidence. Figure 11: Triangular crystal designed for cylindrical bending [1]. The stiffening ribs prevent anticlastic bending. 8

9 Fig. 11 shows a triangular crystal which is particularly suitable since a cylindrical bend is achieved by simply clamping the base of the triangle and pushing on the apex. A diamond shaped crystal with the central part held fixed and equal forces applied on the apices can also be used. S. Pascarelli et al. [7] used a 8cm long Si(311) diamond shaped crystal with the values of the curvature radius ranging be tween 1m and 13 m. When bending the crystal one problem arises: bending in the perpendicular direction or anticlastic bending. It can be reduced by adding ribs to the back of the crystal (Fig. 11). Rib height h, width w, and spacing d, and crystal thickness t are very important in the optimization of a sagittal-focusing crystal for these reasons [7]: 1. the ratio R S /R a of the sagittal/anticlastic radii of curvature is a third-power function of t/h, with a linear dependence on the ratio d/w;. the ultimate limit for the horizontal spot size is a function of w; 3. the ratio t/w strongly influences beam horizontal intensity homogeneity. Because the crystal surface under the rib does not bend, the presence of the ribs induces inhomogeneities in the reflected beam related to periodic oscillations of the sagittal radius of curvature along the surface of the crystal. An appropriate design of the ribs can reduce this effect considerably. 4. Multilayers A multilayer is a thin film coating consisting of (usually two) alternating thin layers (A, B) of high-z and lowmaterials deposited N times on each other [8, 9]. The structure has a repetition period Λ = t A + t B where t i is the single Z layer thickness (Fig. 1). Layer A usually consists of a strongly absorbing material (metal). Layer B is a spacer made of a low-density material. Some materials used for multilayers are Ni/C, Ni/B 4 C, Mo/ B 4 C, W/ B 4 C [10], Cr/SC [11]. Figure 1: A schematic multilayer structure and a typical measured reflectivity spectrum [8]. A multilayer diffracts x-rays in a fashion analogous to a crystal. Alternating layers of high-z and low-z materials create a periodic structure of differing electron densities, like the atomic planes in crystals. Thus the x-ray diffraction from a multilayer can be described with the modified Bragg equation [8]: 9

10 mλ = Λ n cos θ, (9) taking into account refraction effects. Since the multilayer period Λ ranges typically between.0 nm and 10.0 nm, the Bragg angle θ for hard X-rays (E = 5 to 100 kev) is rather small (0. to.0 ) [8]. A very important parameter of the multilayer is the thickness ratio Γ = t A / Λ (the equivalent of the structure factor of single crystals). By variation of Γ the undesired harmonics in the reflected spectrum can be reduced. To achieve this Γ value should be equal to 1/m, where m is the order of the reflection [9]. As an example, attenuating the 3rd order reflection requires a gamma value of 1/3. Figure 13: Applying lateral variation of the layer thickness and/or curvature of the layers allows to influence the geometrical properties of the reflected beam [1]. Multilayers are often used as focusing elements on synchrotron beamlines. For this purpose, lateral gradient of the layer thickness (Fig. 13) and curvature o f the layers have to be applied. Different focusing and collimating geometries are shown in Fig Figure 14: The multilayer must have an elliptical figure of curvature to focus a divergent incident beam. The source and the focus are located in the foci of the ellipse [13]. Figure 15: Parabollically bent multilayers are called "Goebel mirrors". Adjusting the x-ray source to the focus of the parabola results in a parallel reflected beam [13]. Figure 16: Schematic view of convex curved parallel beam multilayers [13]. 10

11 Fig. 16 shows the parabolically bent Göbel mirror. To convert a divergent beam (emitted from a point source) into a parallel beam of very low divergence, the partial beam intensity within a defined acceptance angle must be reflected at the surface of a parabolic mirror segment. This guar antees that in each point of the segment, Bragg s law is fulfilled for the angle of incidence. For optimum conditions, it is important that the period width of the multilayer follows Bragg s law too, so that for lateral direction a continuo usly changing angle of incidence is compensated by a graded period width Preparation of multilayers The x-ray optical performance of the multilayers is determined by the period number N, the period thickness Λ, and by the interface roughness σ R, the variation of period thickness across the total layer stack σ D and the x-ray optical constants of the alternating deposited spacer and absorbing material. Typical values of these characteristic parameters of a nanometer-multilayer for x-ray optical applications are shown in Fig. 17. Figure 17: Typical parameters of nanometermultilayers for x-ray optical applications [14] Extremely high qualified deposition techniques are demanded to meet these requirements across macroscopic substrate dimensions. Some of these techniques are electron beam evaporation, sputter deposition [11] or pulsed laser deposition (PLD) [14, 15]. The PLD will be discussed here. With the right equipment, PLD is a useful technique for mirror synthesis. The use of a precisely controllable system can create uniform and precisely adjusted average layer thicknesses in the sub-nm range. The coating process involves pulsed laser ablation (i.e. atomization) of the coating material target with simultaneous ignition of a plasma plume, because of the high power density in the laser beam cross section at the target surface. Subsequent condensation of the atomic flux emitted by this plasma causes the formation of a thin solid film at the surface of a high quality substrate. The striking feature of this nm-film is an intrinsic surface roughness of infinitesimal amplitude (e.g. σ has an order of magnitude of typically 0.1 nm). Fig. 18 shows the basic principle of the PLD target/substrate handling. Curved multilayers are produced by applying the layers on the pre-curved substrate. 11

12 Figure 18: Schematic diagram of target/substrate handling in large area PLD [15]. 5. Conclusions The x-ray optical elements must bear high thermal and radiation loads, therefore they must be very robust. Certain common monochromator crystals have only limited lifetime in the beam. Temperature changes can cause large expansion or contraction in a crystal - effectively changing the crystal's layer spacing and making analysis more difficult. Complex temperature control systems are therefore required. Multilayers, however, are thermally much more stable and do not require additional temperature controls. Despite the benefits that the multilayers offer, the monochromator crystals are more widely used in the synchrotron beamlines, especially because the manufacturing is easier. 1

13 References [1] X-ray Absorption Principles, Applications, Techniques of EXAFS, SEXAFS and XANES, ed. by D. C. Koningsberger and R. Prins, John Wiley (1988) [] M. Krisch, Focusing crystal optics, ESRF internal report (1988) [3] [4] N. W. Ashcroft, N. D. Mermin, Solid State Physics, Harcourt Brace (1996) [5] U. Boltin, Spektralni analizator za rentgensko svetlobo s sferično ukrivljenim kristalom, diplomsko delo, mentor A. Kodre, FMF (1988) [6] D. B. Wittry, N. C. Barbi, Microsc. Microanal. 7, 14 (001) [7] S. Pascarelli, F. Boscherini, F. D Acapito, J. Hrdy, C. Meneghini, S. Mobilio, J. Synchrotron Rad. 3, 147 (1996) [8] [9] [10] [11] F. Eriksson, G. A. Johansson, H. M. Hertz, J. Birch, Opt. Eng. 41, 903 (00) [1] [13] [14] R. Dietsch, T. Holz, D. Weißbach, R. Scholz, Appl. Surf. Sci , 169 (00) [15] 13