X-rays are electro-magnetic radiation

Transcription

1 X-rays are electro-magnetic radiation Just like visible light, X-rays are waves - cos[ 2π ( x / λ ft)] X-rays travel at the speed of light (c) Quantum Mechanics gives the energy of an X-ray photon c = E = fλ hf This allows the energy in electron- Volts and wavelength in Angstroms to be related Or if the energy is in kev λ(a) = hc ee λ(a) ~ ~ E(eV) 12.4 E(keV) Just light like, X-rays show refraction and diffraction but the wavelength of an X-ray photon is 2000 times smaller and the energy is 2000 times larger.

2 Diffraction of X-rays The wavelength of X-rays is of order 1 Å. In order to see diffraction requires a very small aperture. Using an aperture of width 0.5 microns a classic single slit diffraction can be measured. X-ray beam X-ray pinhole photographic film Diffraction is the interference of waves come from different parts of the aperture. Path differences of 1/2 a wavelength cause the waves to interfere destructively. Interference causes the beam to be spread out vertically with the appearance of maxima in intensity and minima in intensity.

3 If X-rays are waves like visible light, can we focus X-rays with lenses? focus source Yes but, the refractive index of glass for light is about the refractive index of glass for X-rays is about Refractive index tells how much the rays are deflected. X-rays need concave lenses refractive lenses source focus To get enough focusing, need to stack a lot of lenses together

4 Can we focus X-rays using a mirror? Yes but, the refractive index (n) of glass for light is about the refractive index of glass for X-rays is about Total internal reflection occurs inside a glass prism for light. For X-rays, we need to go from vacuum into a dense material. A. θ>θc B. θ=θc Aluminium X-ray mirror 12.4keV C. θ<θc reflectivity angle (degrees) A. θ>θc C. θ<θc

5 At angle of 3 milliradians (~0.2º), a one metre long mirror collects only 3mm height of beam. If the mirror is made of aluminium, the critical energy is 11keV. The critical energy depends on density of the material so for higher critical energy, we need a denser material. focus source C. θ<θc reflectivity reflectivity of aluminium energy (kev) With optical photons we can also get reflection from a shiny surface near back reflection.with X-rays, we have to use total external reflection and work at very low angles. This makes focusing mirrors hard to build. 30 milli-radians 3 milli-radians

6 What else? Diffraction gratings exist for optical photons. Diffraction gratings also work for X-rays when the wavelength is long. For X-rays, we can use single crystals. Bragg diffraction - discovered by Bragg - confirmed that X-rays are waves. A single crystal is a periodic array of atoms. The X-rays penetrate into the crystal and Bragg scattering is seen when the wave scattered from each plane of atoms is in phase (constructive interference). If we hit a crystal with a parallel beam of X-rays, the diffraction from a set of crystal planes is monochromatic. Braggs Law - X-ray beam d θ θ crystal planes λ = 2d sinθ B For 111 planes in silicon d=1.135 Å λ(a) ~ 12.4 E(keV) silicon crystal

7 Absorbers for X-rays For light, refraction is important and light often scatters from the surface of materials where there is a boundary between regions of different refractive index. With X-rays, refraction is much less important. X-rays penetrate into matter. Photo-electric absorption occurs when the X-ray is absorbed by an atom and the energy is used in exciting an atomic electron. Often, we need to absorb X-rays to define a beam shape or to cut down scattered radiation. Other times we need a window that will allow X-rays to pass. Absorption increases as atomic number increases. Beryllium makes a good window, lead makes a good absorber. transmission Be Transmission through 0.1mm filters transmission transmission through 0.1mm iron filter energy(kev) Fe energy (kev) Pb

8 X-ray optics. Why do we need X-ray optics? This is a typical beamline Dipole or ID electron beam shutter shield wall optics hutch experimental hutch Optics hutch contains elements for conditioning the X-ray beam. X-rays produced by the electron beam in the dipole or ID. Shutter in front end isolates optics from synchrotron. PSS forces optics hutch and experimental hutches to be locked when the shutter is open. There is not usually enough room in the front end for much optics. Beryllium is usually the first optical element in the beamline. Dipole or ID dipole radiation electron beam SR

9 What is the optics in a beamline for? to condition the X-ray beam 1. Change the energy (wavelength) spread or distribution 2. Cut down the size of the beam 3. Collect a fan of radiation and focus down to a small spot 4. Improve the divergence of the X-rays 5. Control polarisation, coherence Y' Y X X' Source of X-rays in real space Source of X-rays in angle space Properties of the radiation are dependent on the source. Source is characterised by a size and a divergence (vertical and horizontal) φ

10 1mm thick beryllium window transmission energy (kev) A total thickness of 1mm restricts lower energy to about 5keV. The practical limit using thinner windows is about 3keV. Air absorption can be significant. transmission through air (1m) transmission energy (kev) Beryllium window - required to protect the ring. The beryllium window has an important affect on the X-ray spectrum.in order to withstand atmospheric pressure, the window must be 0.25mm or more thick (depending on size). Station 7.6 has two Be windows each 0.5mm thick. Be windows can introduce phase contrast. ph/mm/mm/sec SRS, Station 7.6, 200mA 4.E+08 4.E+08 3.E+08 3.E+08 2.E+08 2.E+08 1.E+08 5.E+07 No beryllium window 1mm Be window 0.E Energy (kev)

11 Diffraction from perfect crystals Bragg rocking curve for silicon 111 at 12.4 kev θ θ reflectivity When a crystal diffracts dynamically, there is a totally reflecting region where the reflectivity is nearly 100%. Outside this region the reflectivity has Lorentzian tails. The reflectivity curve also known as the Darwin curve is very narrow (~10-5 radians). With a monochromatic beam, the divergence is reduced to the Darwin width. For a parallel beam, the crystal acts as a monochromator. 0-1.E-05 0.E+00 1.E-05 2.E-05 3.E-05 4.E-05 5.E-05 K0 θ silicon crystal angle (radians) diffraction vector (H) crystal planes θ KH

12 reflectvity silicon keV energy (ev) The energy in the plot is the deviation from the position predicted by Braggs Law. The shift of about -1.5eV (shown with arrow) is due to refraction of the X-rays. There is a slight difference in the curve for vertical (σ) and horizontal (π) polarisation. K0 diffraction vector (H) KH θ θ silicon crystal crystal planes

13 Large perfect single crystals of silicon and germanium are cheap and easily available. Silicon is most commonly used as it can be obtained with the highest crystal perfection and it is the cheapest and easiest to cut. Germanium can be used where a broader bandpass is required. Diamond has many advantages as a monochromator - low X-ray absorption, high thermal conductivity, high mechanical stiffness - but crystal quality is poor, only small crystals are available and it is expensive. reflectivity Bragg reflectivity at 12.4keV Si 111 Diamond Ge E (ev)

14 silicon keV reflectvity By using higher order crystal planes, narrower reflectivity curves are obtained K0 diffraction vector (H) KH energy (ev) θ silicon crystal crystal planes θ diffracting surface X-ray beam θ single crystal silicon strain cut

15 DuMond Diagram - used to visualise the way a monochromator selects wavelength and angle. Bragg s Law λ = 2d sinθ B source θ This relates λ and θ. We plot λ against θ. The Darwin curve shows that there is a narrow reflecting region for a crystal near the Bragg angle and we mark that region on the graph. λ Bragg rocking curve for silicon 111 at 12.4 kev 1 reflectivity θ -1.E-05 0.E+00 1.E-05 2.E-05 3.E-05 4.E-05 5.E-05 angle (radians) SR source has small divergence but a large wavelength spread λ (Angstroms) θ (degrees)

16 reflectivity reflectivity of germanium 10keV 20keV 5keV 15keV keV Germanium -4 form factor E (ev) 6 The width of the Darwin curve increases as the energy of the X-rays increases. Near the K-absorption edge there is a reduction in reflectivity but in general the width of the curve E is roughly proportional to energy - or E/E is approximately constant. 4 2 electrons energy (kev)

17 reflectivity ph/mm/mm Harmonic contamination E+14 1.E+13 1.E+12 1.E+11 1.E+10 1.E+09 1.E+08 1.E+07 1.E+06 1.E+05 1.E+04 1.E+03 1.E+02 1.E+01 1.E+00 wiggler 16.3 E (ev) harmonic content, Si energy (kev) dipole 7.6 If we attempt to use a monochromator reflection then harmonics may also be present. For example, a monochromator set up for the 111 reflection from silicon will also reflect from the 333, 444, 555 etc crystal planes (the 222 planes are forbidden). The energies are 1, 2, 3 the 111 energy. If there is synchrotron flux at these energies, then these harmonics will be present in the reflected beam.

18 Problems with the single crystal monochromator. 1. The monochromatic beam moves when the energy is changed θ crystal 2 θ 2. High harmonic content 3. Big tails θ crystal 1 θ These problems can be helped by a double crystal design. The simplest design is a channel cut monochromator so called because it is made by cutting a channel for the beam in a silicon block. θ crystal 1 crystal 2 θ

19 Channel cut monochromator on station 16.3 (M Hart design). Station 16.3 is designed for high resolution diffraction so there is no focusing. In the standard configuration there is a single silicon 111 channel cut monochromator. There is water cooling of the first crystal.

20 A double crystal monochromator can also be made using two independent crystals. The two crystals are mounted in a crystal cage which is attached to the θ rotation stage. The second crystal must be adjusted precisely to achieve the same Bragg angle as the first crystal makes with the incoming beam. If the height of the second crystal is adjustable, the height of the monochromatic beam can be made constant as the energy (and Bragg angle) is changed. incident beam mono beam

21 A major advantage of the double crystal design with two independent crystals is that the Bragg angle of the second crystal can be offset and harmonic rejection achieved. reflectivity harmonic reduction with 1µ radian offset of a double crystal mono E (ev) incident beam mono beam

22 reflectivity multiple parallel reflections from silicon keV E (ev) 1 bounce 2 bounces 3 bounces 4 bounces In some applications, it is important to reduce the tails of the Bragg reflection. This can be achieved by using multiple reflections. For example for a multiple bounce channel cut crystal each bounce reduces the tails by the same factor. multiple parallel reflections from silicon keV 1.00E E E-02 crystal 2 θ reflectivity 1.00E E E E-06 θ crystal E E-08 E (ev)

23 rhodium 4 milliradians osmium 4 milliradians reflectivity reflectivity energy (kev) The critical energy depends on density of the mirror. Osmium is the densest element but has awkward absorption edges. Rhodium is often used instead. The mirror can be used to remove monochromator harmonics. The main use of mirrors is to focus the X-rays. electrons energy(kev) Osmium form factor energy (kev)

24 silicon substrate surface coating X-ray mirrors are usually made by polishing a base and vapour depositing a thin layer of dense material (eg rhodium) on top. Due to the low critical angle mirrors may be over 1m in length. The most common base is single crystal silicon because of the surface finish that can be obtained, chemical stability and the mechanical properties but other materials such as quartz (silicon dioxide) have also been used. In situations were there is a high heat load from the synchrotron X-ray beam, there may be cooling holes drilled through the base block and water circulated to regulate the temperature.

25 In order two achieve focusing of the X-ray beam, the reflecting surface must have some curvature. In the simplest case, the mirror is made with a flat surface and curvature is creates by applying bending couples to both ends of the mirror. This gives a cylindrical figure for the surface. A. 1:1 focusing B. collimating The cylindrical mirror focuses in one plane only (the radiation continues to diverge in the other plane). A. the mirror produces an image of the source in the vertical direction with unity magnification. B. the mirror produces a parallel beam to improve energy resolution of a following monochromator. Bending radii can be of order 10 kilometres.

26 To achieve simultaneous focusing in horizontal and vertical planes requires ideally an ellipsoidal figure to the surface. This can be achieved by polishing the mirror to a cylindrical figure along the axis and then bending the mirror along its length. This gives a toroidal figure which is a good approximation to the required ellipsoidal figure. source focus The sagittal radius is a few 10s of millimetres. This limits the horizontal fan that can be collected. There are some aberrations when the horizontal and vertical focusing are combined in a single element.

27 How perfect can an X-ray mirror be? Conventionally, the quality of the mirror surface is measured by two parameters - A slope error and B surface roughness. Slope error is usually specified as the RMS slope variation along the length and perpendicular to the length of the mirror. Roughness is usually specified as the RMS variation of surface height. Slope error is caused by long period ripples on the surface where as surface roughness is caused by atomic scale fluctuations in the surface height. A. slope error B. roughness Slope error degrades the focusing performance of the mirror. The increase in focus spot size is given approximately by Where L is the distance from the mirror to the focus. y = L ψ δψ ~ 1 arc-sec ~ 1 Angstrom Surface roughness affects the reflectivity curve and smoothes out the intensity drop off at the critical energy. Surface roughness also causes diffuse scattering back ground.

28 Monochromators as focusing elements. If a monochromator crystal is bent then the lattice planes that diffract the radiation are also bent. This gives the possibility of simultaneously monochromating and focusing the X-rays. The monochromator requires a mechanism to apply the bend uniformly. F A single crystal, bent into a cylindrical shape is the simplest arrangement to implement. In this case the diffraction plane and the focusing plane are both horizontal. S O source focus

29 source focus sagittal bent monochromator A sagittal bent monochromator deflects the beam in the vertical plane but focuses in the horizontal plane. The crystal has ribs to reduce anticlastic curvature. A sagittal bent crystal is usually combined in a double crystal monochromator with a plane first crystal. Since the bending is perpendicular to the deflection the Bragg angle is unaffected. Sagittal crystals work at higher angles than toroidal mirrors and therefore do not require such small curvature. Typically 1 to 2m bending radius. 1. Plane crystals - no focusing 2. Sagittal mono - horizontal focusing

30 The simplest station is a white beam station. Could be used for energy dispersive diffraction, Laue diffraction or topography. There is no optics except for beryllium windows and slits. On high power beamlines the slits may be water cooled. If a small beam is used, most of the X-ray flux is wasted - but energy dispersive experiments are often detector limited.

31 Monochromatic experiment. There is no focusing, just a channel cut monochromator. Station 9.1 is a powder diffraction station close into a wiggler source. If the sample is small then most of the available X-ray flux is lost. For flat plate powder diffraction geometry, large X-ray beams are used.

32 Monochromatic station with bent crystal (single bounce) monochromator. A bent crystal monochromatic can collect a large horizontal fan of radiation and focus to a line. If the power loading is high, the crystal may be cooled from its low edge by dipping it into a liquid metal bath (Ga In alloy). The focused flux can be further increased by using a cylindrical mirror to do vertical focusing. Due to movement of the focus spot when the monochromator Bragg angle is changed, the energy is not easily tuned.

33 Combined sagittal monochromator and mirror. This arrangement gives tunability, good energy resolution, good focusing but requires the most complicated optical arrangement. The optical elements are - A. collimating mirror - a cylindrically bent mirror that focuses the source to infinity thereby reducing the vertical divergence of the beam. B. monochromator crystal 1. - this is a plane crystal - usually water cooled to take out the heat loading. With the collimated incident beam low bandwidth is obtained. C. monochromator crystal 2. - this is sagittally bent to focus the beam horizontally. Motorised adjustments bend, pitch, yaw, roll allow the focus to be optimised and the Bragg angle to be matched to crystal 1. Adjustments to the height of crystal 2 keep the exit beam at constant height for the final mirror. D. mirror 2 - a cylindrical bent mirror that completes the vertical focusing.

34 MPW10 & MPW6.2 two designs considered A. collimating mirror 2 crystal mono toroidal mirror B. collimating mirror 2 crystal mono with sagittal crystal focusing mirror Ray tracing with SHADOW

35 Sagittal crystal bending mechanism The monochromator vessel Sagittal crystal Crystal cage

36 1. Sagittal mono 2. Toroidal mirror Source-mirror m 13.1m Mirror1-mono 1.7m 1.7m Mono-mirror2 1.93m 1.93m Mirror-2-focus 5.47m 8.365m Source-focus 22.2m m Vertical magnification Horizontal 1/2 1/2 magnification Mirror 1 bending radius 7486m 7486m Monochromator 1.95m - bending radius Mirror 2 bending radius 3126m 4780m (longitudinal), 39.0mm (sagittal) Horizontal fan accepted 2mrads 2mrads from source Vertical focus spot size 0.114mm 0.226mm from ray tracing (sigma) Horizontal focus spot 0.613mm 0.436mm size from ray tracing (sigma) Flux into 50µ circular aperture Flux into 100µ circular aperture Flux into 200µ circular aperture Total flux in focus

37 2 1 y (mm) x(mm) Figure 1. Ray plot sagittal monochromator arrangement 2:1 horizontal focusing

38 Figure 1. Contour plot sagittal monochromator arrangement 2:1 horizontal focusing

39 2 1 y (mm) x(mm) Figure 1. Ray plot toroidal mirror arrangement 2:1 horizontal focusing

40 Figure 1. Contour plot toroidal mirror arrangement 2:1 horizontal focusing

41 sag mono (2:1) tor mirror (2:1) sag mono (3;1) tor mirror (3;1) Counts Aperture dimension (mm) Figure 1. Flux through circular aperture at the focus

42 Other topics - 1. Laue monochromators 2. Cooling of monochromators - liquid nitrogen 3. Multilayers 4. Fresnel zone plates 5. Polarisation - generating linear polarisation using multi bounce monochromators, generating circular polarisation using quarter phase plates 6. Micro focusing 7. Focusing high energy radiation 8. Using diamond monochromators 9. Optics for ultra high energy resolution (e.g. for inelastic scattering)