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A novel approach to x-ray mirror bending stability and control

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Title:
A novel approach to x-ray mirror bending stability and control
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Book
Language:
English
Creator:
Weinbaum, Michael
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla
Publication Date:

Subjects

Subjects / Keywords:
Thin Films
Thermal Mismatch
Slope Error
Free Electron Laser
Uneven Heating
Dissertations, Academic -- Mechanical Engineering -- Masters -- USF   ( lcsh )
Genre:
non-fiction   ( marcgt )

Notes

Abstract:
ABSTRACT: A novel, no-contact approach to X-ray mirror bending control is presented here, proposed for use on the beamlines of the European X-ray Free Electron Laser (XFEL) project. A set of mirrors with tunable bending radii are desired, that will maintain their optical properties even as the beam incidence causes local heating. Various mechanical bending mechanisms have been proposed and used on other beamlines, which can take up a lot of physical space, demanding more vacuum power, while using expensive high precision servomotors. Rather than bend the mirror by mechanical means, it is proposed to heat the mirror to produce the desired bending. This could work two ways. One scenario calls for a finely tunable heat lamp to irradiate the back surface of the mirror while the X-ray laser heats the front side. With appropriate tuning, simulations show that this approach can keep the mirror flat, and perhaps produce a circular profile. The second scenario is similar to the first, but a thin film of tungsten is added to the back of the silicon mirror. This scenario calls for the temperature of the mirror to change homogenously to affect the desired bending, and in this case the profile should be cylindrical. In both scenarios the uneven nature of the incident radiation causes distortions that may be undesirable. Both scenarios are simulated and it is shown that the stress produced by a metal film may minimize this distortion. The response time of the mirror and configuration of both the heating and cooling mechanism are also considered.
Thesis:
Thesis (MSME)--University of South Florida, 2010.
Bibliography:
Includes bibliographical references.
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Mode of access: World Wide Web.
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System requirements: World Wide Web browser and PDF reader.
Statement of Responsibility:
by Michael Weinbaum.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains X pages.

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ABSTRACT: A novel, no-contact approach to X-ray mirror bending control is presented here, proposed for use on the beamlines of the European X-ray Free Electron Laser (XFEL) project. A set of mirrors with tunable bending radii are desired, that will maintain their optical properties even as the beam incidence causes local heating. Various mechanical bending mechanisms have been proposed and used on other beamlines, which can take up a lot of physical space, demanding more vacuum power, while using expensive high precision servomotors. Rather than bend the mirror by mechanical means, it is proposed to heat the mirror to produce the desired bending. This could work two ways. One scenario calls for a finely tunable heat lamp to irradiate the back surface of the mirror while the X-ray laser heats the front side. With appropriate tuning, simulations show that this approach can keep the mirror flat, and perhaps produce a circular profile. The second scenario is similar to the first, but a thin film of tungsten is added to the back of the silicon mirror. This scenario calls for the temperature of the mirror to change homogenously to affect the desired bending, and in this case the profile should be cylindrical. In both scenarios the uneven nature of the incident radiation causes distortions that may be undesirable. Both scenarios are simulated and it is shown that the stress produced by a metal film may minimize this distortion. The response time of the mirror and configuration of both the heating and cooling mechanism are also considered.
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Uneven Heating
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A Novel Approach to X-ray Mirror Bending Stability and Control by Michael Weinbaum A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science of Mechanical Engineering Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Alex Volinsky, Ph.D. Nathan Crane, Ph.D. Delcie Durham, Ph.D. Date of Approval: October 22, 2010 Keywords: Thin Films, Thermal Mismatch, Slope Erro r, Free Electron Laser, Uneven Heating Copyright 2010, Michael Weinbaum

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Dedication To my wife

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Acknowledgements The author would like to thank Harald Sinn, Germano Gallasso, Antje Trapp, and Fan Yang and the rest of Workgroup 73 at Europ ean XFEL. Vielen Dank fr euer Hilfe.

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i Table of Contents List of Tables ......................................................................................................... ivList of Figures ....................................................................................................... viiList of Equations ..................................................................................................... xAbstract........ ........................................................................................................... xiCh. 1 Introduction to European XFEL .................................................................... 1Synchrotrons and linear accelerators ...........................................................1X-ray free electron lasers (XFEL) ...............................................................4Applications of XFEL sources .....................................................................7Silicon as a mirro r material ..........................................................................8CVD coating process for silicon mirrors ...................................................10Deformation due to localized heating ........................................................12Curvature and bending ...............................................................................13Warping......................................................................................................15Stress, strain and bending with thin films ..................................................16Cooling and support of ex isting silicon mirrors ........................................18Three spheres support .................................................................18Cylinder bender ..............................................................................18

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ii Epoxy leaf spring bender ...............................................................19Indium-Gallium bath ......................................................................19Liquid metal channels ....................................................................20Cooled copper plate .......................................................................21Ch. 2 Design constraints of mirro rs in the European XFEL ................................. 22Length, height, and flatness .......................................................................22Heat load ....................................................................................................24Bending requirements ................................................................................27Ch. 3 Description, simulation, and analysis of proposed designs ......................... 29Remote cooling ..........................................................................................29Liquid metal cooling on a single surface ...................................................30Liquid metal cooling on a single surface with heat lamp ..........................34Cooling on a second surface ......................................................................38Increasing lamp power to bend mirror ...........................................42Adding a metal film ...................................................................................4320 km bending with 100 micron tungsten film and one cooling surface ......................................................................................4420 km bending with 100 micron tungsten film and two cooled surfaces ....................................................................................47Considering other materials ...........................................................48

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iii 20 km bending with 100 micron nickel film and two cooling surfaces ....................................................................................4920 km bending with 300 micron tungsten film ..............................50Keeping the mirror flat with a metal film ......................................53Buoyant cooling bath .................................................................................53Ch. 4 Recommendations ....................................................................................... 55Response time considerations ....................................................................55Most effective design, conclusions ............................................................57Future work ................................................................................................59References ............................................................................................................. 60Appendices ............................................................................................................ 62Appendix AANSYS Inputs .....................................................................63Units ...............................................................................................64Model (B4, C4, D4) .......................................................................64Steady-State Thermal (B5) ............................................................69Transient Thermal (C5) ..................................................................74Static Structural (D5) .....................................................................78Solution (D6) .................................................................................79Material Data .................................................................................81

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iv List of Tables Table 1Prandtl numbers of selected liquids ............................................................ 20 Table 2Relevant properties of materials discussed here ......................................... 49 Table 3 Temperature change that will keep the mirror flattest in presence of concentrated FEL heating ........................................................................ 53 Table 4 Interventions to bend the mirror to a 20 km radius ................................... 57 Table 5 Interventions to keep the front of the mirror flat. ...................................... 58 Table 6 Simulation units ......................................................................................... 64 Table 7 Model (B4, C4, D4) > geometry ................................................................ 64 Table 8 Model (B4, C4, D4) > geometry > parts .................................................... 65 Table 9 Model (B4, C4, D4) > coordinate system .................................................. 66 Table 10 Model (B4, C4, D4) > connections .......................................................... 66 Table 11 Model (B4, C4, D4) > c onnections > contact region ............................... 66 Table 12 Model (B4, C4, D4) > mesh ..................................................................... 67 Table 13Model (B4, C4, D4) > named selections > named selections ................... 68 Table 14 Model (B4, C4, D4) > analysis ................................................................ 69 Table 15 Model (B4, C4, D4) > steady-stat e thermal (B5) > initial condition ....... 69 Table 16 Model (B4, C4, D4) > stea dy-state thermal (B5) > analysis settings ..................................................................................................... 69 Table 17 Model (B4, C4, D4) > stea dy-state thermal (B5) > loads ........................ 70 Table 18 Model (B4, C4, D4) > stea dy-state thermal (B5) > commands (ANSYS) ................................................................................................. 71 Table 19 Model (B4, C4, D4) > steadystate thermal (B5) > solution ................... 72

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v Table 20 Model (B4, C4, D4) > steady-st ate thermal (B5) > solution (B6) > solution information ................................................................................ 72 Table 21 Model (B4, C4, D4) > steady-st ate thermal (B5) > solution (B6) > results ....................................................................................................... 72 Table 22 Model (B4, C4, D4) > steady-st ate thermal (B5) > solution (B6) > probes ...................................................................................................... 73 Table 23 Model (B4, C4, D4) > analysis ................................................................ 74 Table 24 Model (B4, C4, D4 ) > transient thermal (C5) > initial condition ............ 74 Table 25 Model (B4, C4, D4) > transient thermal (C5) > analysis settings ........... 74 Table 26 Model (B4, C4, D4) > tran sient thermal (C5) > loads ............................. 75 Table 27 Model (B4, C4, D4) > tran sient thermal (C5) > solution ......................... 75 Table 28 Model (B4, C4, D4) > transi ent thermal (C5) > solution (C6) > solution information ................................................................................ 75 Table 29 Model (B4, C4, D4) > transi ent thermal (C5) > solution (C6) > solution information > result charts ......................................................... 76 Table 30 Model (B4, C4, D4) > transi ent thermal (C5) > solution (C6) > results ....................................................................................................... 76 Table 31 Model (B4, C4, D4) > transi ent thermal (C5) > solution (C6) > probes ...................................................................................................... 77 Table 32 Model (B4, C4, D4) > analysis ................................................................ 78 Table 33 Model (B4, C4, D4) > static structural (D5) > an alysis settings.............. 78 Table 34 Model (B4, C4, D4) > static structural (D5) > imported load (setup) ...................................................................................................... 79

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vi Table 35 Model (B4, C4, D4) > static structural (D5) > imported load (setup) > imported body temperature ...................................................... 79 Table 36 Model (B4, C4, D4) > static structural (D5) > solution ........................... 79 Table 37 Model (B4, C4, D4) > static structural (D5) > solution (D6) > solution information ................................................................................ 79 Table 38 Model (B4, C4, D4) > static structural (D5) > solution (D6) > results ....................................................................................................... 80 Table 39 Si > constants ........................................................................................... 81 Table 40 Si > isotropic elasticity ............................................................................. 81 Table 41 W > constants ........................................................................................... 81 Table 42 W > isotr opic elasticity ............................................................................ 81

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vii List of Figures Figure 1 Fleming's rule for direction of induced current. ....................................... 2 Figure 2 View along the beam pipe be tween the magnetic structure of an undulator of the storage ring PETRA III. ................................................... 3 Figure 3 Showing path of electrons and photons in an undulator. .......................... 4 Figure 4 Draft of European XFEL op tics from 2007, presented at Hasylab conference that year. ................................................................................. 6 Figure 5 Photo explaining a ngle of reflection. ........................................................ 8 Figure 6 Sample Reflectivity curves of (a) Silicon and (b) Platinum. .................... 9 Figure 7 Explaining Anticlastic Bending ................................................................ 14 Figure 8 Showing the Shear Center of a typical channel section. ......................... 15 Figure 9 Differences in elastic modulus between a film and a substrate may create warping, if the load and supports are not along the line of symmetry shown. .................................................................................... 16 Figure 10 Taken from the notes of Pr of. W. Nix, used with permission ............... 17 Figure 11 Comparing the X-ray transm ission of Aluminum and Beryllium, angle of incidence = 90. ..................................................................... 24 Figure 12 Describing one fixed-te mperature surface and no other intervention. ........................................................................................ 31 Figure 13 Showing typical temperatur e distribution with cooling on top only and no backlighting. ..................................................................... 32 Figure 14 Z-deflection of weightless mi rror with one cooled surface on top. ................................................................................................................ 33

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viii Figure 15 Describing one cooled surface w ith added heat from a lamp. ............. 34 Figure 16 Temperature di stribution with 43 W back side heat, one cooling surface. ................................................................................................. 35 Figure 17 Z-deflection of mirror with one cooling surface and 43 W heat on back surface. ........................................................................................ 36 Figure 18 Bump created by FEL beam is olated from large-radius circular deflection. ............................................................................................. 37 Figure 19 Variation in vertical slope of the front face due to uneven heating. Single cooled surface. ............................................................................ 38 Figure 20 Showing one possible confi guration to achieve cooling on top and bottom of mirror. ............................................................................. 39 Figure 21 Temperature di stribution with two coo ling surfaces and 43 W backlighting. ......................................................................................... 40 Figure 22 Z-deflection with 43 W backlig ht and two cooling surfaces. ............... 41 Figure 23 Deflection in Z direction al ong center line of mirror face. ................... 41 Figure 24 Results of using back light @ 172 W to bend mirror. ............................. 42 Figure 25 20.3 km circle subtracted fr om deformation in z-direction. ................... 45 Figure 26 Vertical Slop e dx/dy. Tungsten, tf= 100 m, T=36 C, top cooling only ........................................................................................... 45 Figure 27 Temperature distribution with two cooling surfaces and no backlighting. ......................................................................................... 46

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ix Figure 28 Using a 100 micron Tungsten film with a 36 C temperature change to induce bending, graph sh ows deviation from circle due to FEL radiation. .................................................................................. 47 Figure 29 Vertical slopes along mi rror with tungsten film. tf= 100 m, T=36 C, cooling on top and bottom ................................................... 48 Figure 30 The effect of a 100 m Ni film, deposited at 22 C, on mirror behavior.................................................................................................. 50 Figure 31 Adapted from Ashby Materi al Selection Charts, used with permission ............................................................................................ 52 Figure 32 Guide for film select ion using Stoneys Equation. ................................. 59 Figure 33 Simulated mirror, green, with thin film in orange. ................................. 63 Figure 34 Showing mesh ......................................................................................... 68

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x List of Equations Equation 1Stoneys Equation solved for bending radius due to thermal mismatch ................................................................................................ 16 Equation 2Tensile stress in the thin film due to thermal mismatch. ....................... 17 Equation 3Shear stress between film as a function of film tensile stress ............... 17 Equation 4 Image from Wikipedia .......................................................................... 20 Equation 5Exponential decay of X-ray intensity as photons are absorbed by atoms ...................................................................................................... 25 Equation 6Two Dimensional Gaussian Di stribution of photons in FEL beam ....... 26 Equation 7Beam spread from the perspective of the slanted mirror ....................... 26 Equation 8Intensity of FEL and spontan eous photons absorbed as a function of position .............................................................................................. 27 Equation 9 Stoneys Equation solved for film stress with a known radius of bending. .............................................................................................. 51

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xi Abstract A novel, no-contact approach to X-ray mi rror bending control is presented here, proposed for use on the beamlines of the Eu ropean X-ray Free Electron Laser (XFEL) project. A set of mirrors with tunable bendi ng radii are desired, that will maintain their optical properties even as the beam incidence causes local heating. Various mechanical bending mechanisms have been proposed and used on other beamlines, which can take up a lot of physical space, demanding more vacuum power, while using expensive high precision servomotors. Rather than bend th e mirror by mechanical means, it is proposed to heat the mirror to produce the desired be nding. This could work two ways. One scenario calls for a finely tunable heat lamp to irradiate the back surface of the mirror while the X-ray laser heats the front side. W ith appropriate tuning, simulations show that this approach can keep the mirror flat, a nd perhaps produce a circular profile. The second scenario is similar to th e first, but a thin f ilm of tungsten is added to the back of the silicon mirror. This scenario calls fo r the temperature of the mirror to change homogenously to affect the desired bending, and in this case the profile should be cylindrical. In both scenarios the uneven nature of the incident radiation causes distortions that may be undesirable. Both s cenarios are simulated a nd it is shown that the stress produced by a metal film may minimize this distortion. The response time of the mirror and configuration of both the heating and cooling mechanism are also considered.

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1 Ch. 1 Introduction to European XFEL The European XFEL project in many ways can be summed up with two words: bigger and brighter. This study will ask, what are the design constraints on mirrors used in this project, and propose a design. The surface of an X-ray mirror must be almost perfectly smooth while heat orig inating from the X-rays themselves can cause distortions and even damage the mirror surface. This is true whether the surface is designed to be flat, curved, or toroidal. For the European XFEL project, the anticip ated heat loads are orders of magnitude greater than what th e previous generation of beamline components dealt with. Synchrotrons and linear accelerators X-ray light sources, broadly, fall under tw o categories. The first type is a synchrotron and the second type is a linear acce lerator. Both begin by injecting electrons with very high voltage into a vacuum using a klystron or similar device. The synchrotron is a circular path where magnets guide the electrons to keep them travelling in a circle and not run into walls, becoming grounded. Th e linear accelerator simply directs the electrons to a ground; each elec tron travels the path only onc e. Photon emission occurs as these electrons change path due to interactions with magnetic fields.

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2 Figure 1 Fleming's rule for direction of induced current. Extend the thumb, forefinger and middle finger of the right hand [as shown]. Place the hand [so that] the thumb will point in the direction in which the conductor moves, the forefinger in the direction the lines of force (N to S) then will the middle finger point in the direction in which the induced current flows.[1] A magnetic field will deflect any moving charged particle; the effect is similar to gravity except that the the charged particles and the field must be moving relative to each other (see Figure 1). The various magnets in a synchrotron in dividually steer the electrons on a hyperbolic path; multiple magnets are finely tuned along with the initial velocity of the electrons so that th e electrons complete the circuit. In the case of a linear accelerator, the elec trons are typically directed to travel inbetween two sets of magnets, an arrangeme nt called an undulator, shown in Figure 2.

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3 Figure 2 -View along the beam pipe between the magnetic structure of an undulator of the storage ring PETRA III. Retrieved from DESY website and used with permission. These undulators force the electrons to fo llow a sinusoidal path along their length, with many more changes of direction in a shorter space than a typical bending magnet device. The reason the electrons are turned and twisted so much is that, whenever the magnetic field around the electron changes, not only does the acceler ation on the electron change, but a photon is typically emitted. This photons path is usually parallel to the change in acceleration, or along the radius of its curved path. In the case of linear accelerators and undulators, however, the electr ons are already travelling at nearly the speed of light. This means that the emitted radiation is along the same axis as the zero of the sinusoidal path of the electron. More photons are emitted with each successive peak and trough in the electrons pa th, all parallel to the undul ator axis, producing great intensity of radiation in a single dir ection. This is shown in Figure 3.

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4 Figure 3 Showing path of electrons and photons in an undulator. Graphic retrieved from XFEL website and used with permission. The wavelength of the photons emitted, whether by a bending magnet or by an undulator, is a complex function of the kine tic energy of the electron and the magnetic field gradient; the greater either of these is typically, the smaller the wavelength. It is unusual for such a photon to be in the visible light range, typically they range from high ultraviolet, to hard X-ray, meaning that the range of photon energies is 10-30,000 eV. The corresponding range of wavelengths would be 10 nm down to 0.04 nm. X-ray free electron lasers (XFEL) An X-ray free electron laser (XFEL) is a special, new type of undulatoraccelerator assembly where the electrons are entrained into bunches by quantum mechanical effects. Existing XFEL lasers ar e at SPring-8 in Japan, LCLS in California, and FLASH in Hamburg. Large numbers of very high-energy electr ons exit an electron gun and enter a series of cavitie s that each has an alternati ng voltage. The timing of the gunshot is synchronized to the voltage phases in the cavities so that each cavity adds to the electrons electric potential as it passes. This synchronized alternating voltage

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5 accelerates some electrons more than others, depending on their original position and velocity, and the end effect is that the electrons leave the la st cavity in a bunch. These bunches then travel through the undulator at nearly the speed of light, or high relativistic speeds. The magnets in the undul ators are tuned so that each crest and trough in the electrons path causes photons to be em itted that have nearly the same wavelength and direction. At the end of the undulator (or series of undulator s), the electrons are finally diverted by a bending magnet (thi s is a source of wideband radiation or spontaneous radiation) and grounded. Each el ectron burst now corresponds to a flash of X-ray photons. The photons produced in the undulators are an FEL pulse. The pulse itself is already highly monochromatic and spat ially coherent, along the original path of the electrons. When the European XFEL proj ect is complete, it is hoped that these flashes will have brilliance up to 533 (photons / s / mm2 / mrad2 / 0,1% bandwidth) lasting up to 100 femtoseconds. This translat es to a 20 GW peak intensity (integrating the full-width half-maximum) of FEL radiation, concentrated in a small enough spot to recrystallize silicon. The time-averaged intensity is 1.625 (photons / s / mm2 / mrad2 / 0,1% bandwidth) and the corresponding inte nsity is 65 W, according to project documents[2]. Workgroup 73 has already shown that a protective diamond CVD layer, a few nanometers thick, is needed to dissipate this concentrated heat load outwards and save the single crystal mirror from re-c rystallizing on the femtosecond timescale. The pauses in between pulses are so long, however, that on the millisecond timescale only the time-averaged intensity matters. This is th e timescale that our stead y-state and transient simulations will deal with. There is also a great amount of s pontaneous radiation coinciding with these pulses. The spontaneous radiation has much less coherence, both

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6 in terms of spatial distribution and bandw idth, and must be minimized by downstream optics, for the sake of the experiments th at will be done. Some of the spontaneous radiation with a lower photon energy than the FEL beam will be absorbed by solid attenuators ahead of the mirror. The mirrors will be set at an angle just shallow enough to reflect at least 90% of the photons in the FEL beam. This angle will be too shallow for the higher energy spontaneous photons and it is hoped that many of them will be absorbed by the mirror. The optical compone nts are ultimately task ed not only with redirecting the beam away from the undulator axis for safety, but also with absorbing much of the heat loads associated with the undesire d spontaneous radiation. This is especially true of the first mirror, whose position is noted in Figure 4. Figure 4 Draft of European XFEL op tics from 2007, presented at Hasylab conference that year. Mirrors (circled ) shown roughly 500 m away. The current design iteration for SASE 2 calls for mirror 1 to be at 260 m where heat loads will be greater[3].

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7 Applications of XFEL sources Most of the applications of such an intense, collimated, and briefly flashing light are in the fields of biology a nd medicine. The amount of intensity needed to characterize a ceramic powder or metal via X-ray diffraction (XRD) is comparatively low. Repetitions in the crystal pattern of these ma terials selectively diffract different X-rays, and the total intensity of the diffraction pattern is typically similar to the original intensity of the source X-rays. Proteins, on the ot her hand, are much larger molecules, and therefore have larger gaps between the repetitions in their crystal patterns, if they even crystallize at all. It is hoped that an XFEL beam will be able to capture useful data about the structure of such a protein from a single pulse interacting with a single complex molecule, giving off diffractions many orders of magnitude smaller than the original beam intensity. The same logic suggests th at an XFEL beam may be useful to see the steps of a catalyzed biological reaction; again on a molecu le-by-molecule, pulse-by-pulse basis[4]. The properties of the FEL radiation (its intensity and wavelength) will be controlled primarily by the undulator magnets spacing. This spacing could be as small as 6 mm and each setting corresponds to unique values of intensity and wavelength for the FEL radiation, as well as unique values for the spontaneous radiation. The other controlled parameter in this vicinity will be the shutters; they will be opened the minimum amount so that the FEL beam is transmitted while as much as possible of the spontaneous radiation, which is not spatially coherent, is absorbed by the shutter blades.

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8 Silicon as a mirror material Reflectivity is a complex property. It is the fraction of the light incident on a surface that is reflected back by the surface at the reflected a ngle, as shown in Figure 5. Figure 5 Photo explaining angle of reflectio n. By Ztonyi Sndor (ifj.), posted on Wikimedia under a free public license. The fraction cannot be greater than on e without violating the First Law of Thermodynamics. Reflectivity is highly de pendent on surface conditions; the smoother a surface is, the more it reflects, up to some ma ximum that depends on the material of the mirror itself, the incident angle, and the wave length of the photons. Most materials are actually quite poor reflectors in the low wavelength high ultr aviolet to hard X-ray range; though as with visible light metals are still better reflec tors than non-metals. Most materials, metal or non-metal, will only reflect X-rays at very low angles of incidence, on the order of tens of milliradians (mrad). The theoretical reflectivity of any material at a 0 angle is 1; because it can be said that the photons are not interacti ng with the surface at all. As the angle of incidence is increased, the X-ray reflectivity of most materials trends to zero; the photons are absorbed or transmitte d rather than reflected. However at small angles the reflection can be nearly complete Mirrors designed for low-angle reflection

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9 are also sometimes called total reflection mi rrors. The greater the energy of the photon, or smaller its wavelength, the smaller the range of angles that will reflect that photon for a given material will be. For each material and incident wavelength, there is also a critical angle below which (if 0 is taken as parallel to the reflecting surface) no transmission occurs, only reflection and absorption. Figure 6 Sample Reflectivity curves of (a ) Silicon and (b) Platinum. Higher energy X-rays are reflected by a small er range of angles. Data from http://henke.lbl.gov/optical_constants/ Even though metals are better reflectors th an non-metals, silicon is the dominant substrate for X-ray mirrors at these types of facilities. This is because of the surface quality required. Random surface roughness much greater than 2 nm can greatly decrease the surface reflectivity in the hard X -ray regime[5]. This type of perfection is very difficult to achieve in a nything but a single crystal mate rial. In single crystal silicon, roughness less than 0.5 nm is feasible[6]. Pi eces of metal the size of an X-ray mirror cannot match single crystal silicon in term s of dimensional stability. Machining a 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 051015Reflectivity of smooth SiAngle of incidence, mrad(a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 051015Reflectivity of smooth PtAngle of incidence, mrad at 3 keV at 20 keV(b)

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10 polycrystalline piece of metal often reveals vo ids and relieves complex stress patterns so that no matter how many times the perfect t ool passes, roughness is still there. Single crystal metals are not as available as si ngle crystal silicon. This is because of semiconductor industry developments. The front surface of silicon mirrors is often coated w ith platinum by chemical vapor deposition or a similar process, creating a smooth and stress fr ee surface that would not have been possible out of solid, polycrysta lline platinum. A pla tinum coated silicon mirror is more versatile, as shown in Figure 6; it maintains reflection at higher incidence angles. Platinum is one example; tungsten and nickel have very comparable reflectivity and lower cost, though obviously both have a greater tendency to oxidize, potentially compromising their optical properties. Pl atinum and palladium coatings are often primed with chromium to improve their bond strength, other metals need no priming. These coatings are made for optical qualitie s only and are typically less than a micron thick; too small to change the elastic or thermal conduction properties of the mirror significantly. This paper will explore adding a metal film thick enough to bend the mirror but will not be interested in the possi ble small changes to heat conduction caused by the film. CVD coating process for silicon mirrors A metal film is typically added to a silicon mirror by chemical vapor deposition, or CVD. The competing process is PVD, physical vapor deposition. The CVD process involves comparatively little h eat. It typically involves th e mixing of a powder and a liquid or gas that will create a chemical re action on the surface. The chemical reaction creates a free metal ion that will tend to at tach to the surface, gaining electrons in the

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11 process. The other products of the reaction, which are gases or liquids, are carried away by fans or pumps. Though noxious gases ar e often produced, newer methods allow the reaction to take place under a hood, not necessarily under high vacuum. However, earlier methods did require high vacuum because they were very sensitive to water vapor[7]. PVD processes more frequently require a vacuum and, depending on the metal being deposited, are generally more costly. Some metals such as aluminum and copper have chemistries that make CVD difficult to achieve so in this case PVD is preferred. A CVD process typically takes place near room temp erature, and can be finished surprisingly fast; in the case of nickel, a metal film may be deposited on a surface at a rate of 250 microns per hour[8]. Nickel was the first metal to be deposited via CVD and Nickel CVD is still one of the least costly CVD processes. Its low heat and electrical conductivity make it a p oor choice for semiconductors bu t it may be a good choice here. The main reason CVD of metal is used on silicon is to create a small layer of electrically conductive material on top of the semiconductor which is then used to create an integrated circuit. A concern when a ttempting metal CVD on silicon is the formation of metal silicides (analogous to an intermet allic phase; their prope rties are more like a ceramic) at the interface boundary. The presence of metal silicides is problematic for two reasons; one is that the change in crys tal structure may induce high stress, some deformation, and even fracture/delamination as they form. For instance, the large stresses created by molybdenum silicide formation under a molybdenum film were studied recently by Volinsky et al[9]. Anot her problem, for most users, is that these silicides are electrical insulato rs. Though silicides are stable at room temperature, their formation only becomes thermodynamically favorab le at elevated temp eratures. Silicide

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12 formation typically has a starting temperature and an ending temperature. If the piece is held above the starting temperat ure, silicides form (with a sl ow rate and faster as the temperature increases) and they typically remain stable after the temperature is lowered. The ending temperature for a cer tain silicide chemistry, say M2Si, often corresponds to the begining temperature for the formation of a different silicide, pe rhaps MSi. Silicide formation is not a concern, ther efore, as long as the designer knows that the part will not be subjected to temperatures at or above the lowest possible reaction temperature. The lowest reaction temperature for nickel-silicon, for instance, is 300 C[10], while for tungsten this value is higher, about 650 C[1 1]. Avoiding th at threshold should not be a problem in this case. Deformation due to localized heating Most materials, when heated evenly, will expand isotropically and this is called positive thermal strain. However, if only a small part of the solid is heated while the rest remains the same temperature, the heated portion will be under comp ressive stress while the neighboring portions at the lower temper ature will be under tens ile stress. This means that a temperature gradient whose sign never changes w ill typically produce a stress gradient (and the attendant shear st ress) whose sign does change. The heated portion will grow less than would be dict ated by its thermal expansion, while the unheated portion will grow mo re. The thermal conduction of the material and the resulting temperature distribu tion must be understood in detail before the deformation can be predicted; it will be mu ch more difficult to get a steep temperature gradient (and therefore a steep stress gradient) on a materi al with high thermal conductivity and a large

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13 characteristic thickness. Understanding deformation due to uneven heating rapidly becomes a question for either empirical study or finite elements analysis. Recently, the heating of X-ray mirrors and its effect on mirror shape was studied by Yuan et al at Berkeley Labs[12], who attempted to minimize these effects with a Peltier cooling device attached not to the mirro r but to its support. The problem as they describe it is that the mirror, when heated evenly, will also heat the aluminum support below it. The aluminum will expand more than the silicon, creating some extra stress in the mirror and unacceptable slope errors. Th e aluminum support has a ten times greater thermal expansion coefficient than the silic on. So the approach was to add a cooler which would keep the aluminum support at a constant temperature over a range of possible mirror temperatures. Their setup wa s successful in that changes to the mirror curvature no as a function of mirror temperat ure were greatly reduced. However, their tests were conducted in a speci al thermally insulated box without uneven heating from an X-ray beam, and they acknowledge that such in-situ heating woul d be a source of additional slope error wi thout quantifying this. Curvature and bending Bending is a specific type of elastic deformation typical to beams. A beam is any object that is much longer th an it is wide or deep, a nd is typically supported only intermittently along its length. The interacti on between the loads, including the beams own weight, and the supports creates a deflect ed shape. These deflected shapes can be described by singularity equatio ns or other methods, but this is beyond the scope of this paper. Whether a deflected shape follows a 4th order polynomial, or is sinusoidal, or something else entirely, at every point this deflected shape has a derivative and therefore

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14 an instantaneous radius of curvature. An interesting consequence of the curvature along the length of the beam is that all four surfaces of the beam ar e also distortedeven if the beam was square and isotropic to begin with. Figure 7 Explaining Anticlastic Bending When a beam bends, the inner face of the beam is compressed, its length is reduced, while the outer surface expands. The Poisson effect dictates that an infinitesimal volume under uniaxial stress will act ually deflect in all three directions. If the uniaxial stress is compressive, the volume will compress in that direction but bulge or expand in the other two directions. The net effect is that the overall volume, the distorted length width height is nearly preserved (and exac tly preserved in the case of an ideal material whose Poisson ratio is 0.5). The inner face of the beam is in compression, so this face will bulge while the ou ter face will shrink or be sucked in. The faces on the side will be slanted as shown in Figure 7.

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15 Warping The previous discussion of bending dealt with a hypothetical beam that had a square cross-section. While all four faces of the beam were distorted in different directions, the deflected shape had a plane of symmetry. This is because the loads and supports were all in that plane, and the undist orted beam was symmetric about it as well. The original square cross section of the beam has four lines of symmetry. On the other hand, the channel section shown in Figure 8 has only one line of symmetry, the horizontal line. For this reason it will twist out of plane as it bends, unless a) the load is through the shear center shown in the picture, or b) all loads vectors are in to the plane of symmetry. Figure 8 Showing the Shear Center of a typi cal channel section. If the line of the load is not through the shear center, warping is expected. This is counter-intuitive because the shear ce nter is not even on the part itself, so it is difficult to load the beam there. Channel sections are often paired when they are expected to handle a bending load in thei r strong axisthe load center ideally being between the two channels, corre sponding to the shear center of each. If the sections are not paired and the warping is instead c onstrained by a redundant member, this may introduce secondary stresses that the designer must account for.

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16 Figure 9 Differences in elas tic modulus between a film and a substrate may create warping, if the load and supports are not along the line of symme try shown. This representation is simplistic because si ngle crystal silicon is anisotropic. Like the channel section, a mirror with a film also only has one line of symmetry, even though its rectangula r shape still seems to have two lin es of symmetry. Instead it is the mismatch between the Youngs moduli whic h can create some warping, whether the original bending deformation is from internal or external forces. Stress, strain and bending with thin films Stresses in a thin film are known to cause bending deflection in the substrate, though the substrate is much larger than the film. In 1909 Stoney quantified the relationship between the film st ress and the bending in a recta ngular beam that this stress may cause. His equations take many forms, but in our case the source of the film stress is a temperature change working with a mism atch in thermal expansion coefficients between the film and the substrate (thermal mi smatch for short). Equation 1 is Stoneys equation derived for this case and solved for the resulting bending radius. Equation 1Stoneys Equation solved for bending radius due to thermal mismatch

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17 It is taken from lecture notes presented by William Nix at Stanford University and available online[13]. This equa tion relates film thickness (tf), substrate thickness (ts), along with the elastic properties of both (Ef, f; Es, s ) and the thermal mismatch ( f s) and temperature change to find the radius of curvature of the bent shape. Figure 10 taken from the notes of Prof. W. Nix, used with permission Another use of Stoneys equation is to pred ict the tensile stress in the film itself given the same inputs: Equation 2Stress in the thin fi lm due to thermal mismatch. The shear stress between the film and the substrate is related to this value, it is Equation 3Shear stress between film and substrate as a function of film tensile stress These three equations will be used in th e following section in conjunction with ANSYS simulation to validate design proposals incorpora ting a thin film.

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18 Cooling and support of existing silicon mirrors Three spheres support Some recommend supporting the mirror on top of three pins with spherical heads. These pins ideally are only able to return force directly up, and would balance the mirror as a tripod does. This support scheme woul d be non-redundant. The disadvantage is that no part of the support can fail without the mi rror falling. The advantage is that stress caused by misfit parts is eliminated, though bending stress and deflection are often greater in a non-redundant suppor t scheme than a redundant one. The stress state is predictable using a simpler set of equations without needing to take the deflected shape into account. The spheres should also have less friction than a flat or cylindrical support, reducing (but not eliminating) the likelihood of axial load on the mirror due to a change in temperature. If we imagine a mirror that, in the absence of gravity, is completely flat, we know that this mirror will bend when placed on the three spheres support; it will sag. This may create optical distortions. For this reason, wh en the mirror surface is polished with elastic emission machining, the mirror is already sitting on its support, if it was intended to be supported this way. If the s upports are moved, the mirror may sag in a different way and need to be polished again. Cylinder bender The most common way to compensate for the sag that results from non-redundant support is to add precisely c ontrolled bending actuators to the mirror support scheme. This takes the form of three or four cylinders in contact with the flat surface of the mirror,

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19 two on one side and the remaining one or two on the other. The cylinders are only allowed to move in one direction and the position of the cylinde rs is changed with nanometer or even Angstrom precision. Th is precision is typically achieved using stepper motors and stiff levers. The actuators are typically separated from the ultra high vacuum chamber by bellows. The line of contact between the mirror and the cylinder may have a large stress concentration wh ich must be understood and monitored. Epoxy leaf spring bender There are at least two X-ray mirror set-ups one in Berkeley and one in Stanford, which support the mirror vertically on solid metal while allowing it to be bent by attaching leaf springs to the mirrors ends with epoxy. The load on the leaf spring, and therefore the bending radius of the mirror, can be changed by much less precise stepper motors than were required for the cyli nder bender while using epoxy rather than compressive contact produces less stress concen trations. The large so lid contact area can also be useful for cooling. Such an arrangement is also called a u-bender. Indium-Gallium bath Indium-Gallium amalgams are sometimes liquid at room temperature (the Eutectic temperature being lower than bot h pure metal melting points) and much less toxic than mercury. They are already used in some applications as a low temperature solder and are finding use as a thermal conducto r in the liquid state. The X28C beamline at Case Western Reserve University employs a silicon mirror with no rigid support; it is placed in a stainless steel bathtub and liquid Indalloy 51 metal is carefully poured around it, eventually causing the mirror to fl oat. The density of Indalloy 51 is roughly

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20 three times greater than the density of silicon ; for this reason the lower third of the mirror becomes unusable, unless the bobber mechan ism the authors mention is employed[14]. In spite of these difficulties, the advant ages of a liquid support are great. No stress concentration is imposed and there is no need to worry about the variable properties of cured adhesive. Also, the Prandtl number of a liquid metal is exceptionally low compared to other fluids. Equation 4 The Prandtl number is a uni tless ratio. Image from Wikipedia. Table 1Prandtl numbers of selected liquids Liquid Prandtl number Mercury ~0.015 R-12 ~4.5 Water ~7 Whether the mirror actually floats in liquid metal or is simply coated with it, because of the low Prandtl number we can assume that the liquid bath has a negligible temperature gradient and no need to force flow This means that the surfaces wetted by liquid metal can be assumed to have constant temperature, which simplifies analysis. Liquid metal channels While a liquid metal bath would cool the bottom surface of a silicon mirror, for a mirror aligned vertically it may be importa nt to cool the top surface as well. One approach is to paint the top surface with liquid metal and place a cooled copper plate

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21 (typically with internal channels for chilled wa ter) on top of that. Another approach is to machine or otherwise fabricate one or more tr enches into the top surface, which a copper fin will fit into, and fill the gap with liquid me tal. The fin in turn would be brazed or soldered to a copper pipe carrying cold wate r. Such a setup was proposed by Ferm at Socit Europenne de Systmes Optiques[15] It may be significantly less expensive and challenging than a liquid metal bath Cooled copper plate Many silicon mirrors end up be ing attached to a copper pl ate with channels inside for water or another cooling fl uid. A leading manufacturer of such plates is SESO in France. These plates can be manufactured to nearly the flatness requirements of the mirrors they support. They are typically quite thic k and it is commonly practiced to place only a small layer of single crystal silicon on top of them, perhaps 1 cm thick, with an epoxy adhesive. 20 kW cooling power is claimed for a typical 1 m long X-ray mirror[16]. In appli cations where thermal mismatch may be a concern, the mirror may simply rest on a cooled copper plate with a liquid contact, perhaps oil or even liquid metal, carefully spread.

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22 Ch. 2 Design constraints of mirrors in the European XFEL The purpose of the primary mirrors in the European XFEL project is to allow the experiment site to be off-axis from the path the electrons would ta ke if the final bending magnet failed. Though each mirror can only chan ge the path of th e X-ray radiation slightly due to the low angle of reflecti on, with enough distance between the mirrors a 5 m beam displacement is created. The heat load on the second mirror in the pair will be less than that on the first mirror in the pair, so design will focus on the first mirror. There are two beamlines under consideration, SASE 1 and SASE 2. SASE stands for SelfAmplified Spontaneous Emission The goal is for one type of mirror to perform sufficiently in both beamlines. Length, height, and flatness The following relies heavily on the April 14th draft version of Conceptual Design of X-ray Beam Lines by Work Group Packag e 73 members. The mirror will reflect in its vertical plane, displacing the beam horizont ally. A second mirror will return the beam back to its original angle. The distributi on of photons in the FEL beam is Gaussian along two planes. If a significant number of photons does not hit the mirror surface, and instead hits the edge because the mirror is undersized, an inte rference pattern will develop. The further downstream the mirrors are, the wider the Gaussian distribution becomes; this is a linear relationship defi ned by the angle of dive rgence. With the

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23 photons more spread out, the concentrated h eat load associated with absorption becomes easier to deal with. However, the mirror must be bigger. In particular it must be longer due to the very low angles of incidence of less than 8 mrad, depending on the wavelength setting. The low angle stretches the beam f ootprint. Working group 73 has settled on a set of parameters that will require the face of the mirror to be 80 cm long, 5 cm tall. The size requirement derives from the configur ation of SASE 1, where the mirror is 435 m away from the undulators and the beam spread, for this reason, is wider. However, the heat load will be more intense in the SASE 2 beamline than it is in the SASE 1. The first mirror is 260 m away in the cas e of SASE 2. The more concentrated heat load of SASE 2 will therefore be used for our simulations. The total distance from the end of the undulator to the experiment site in both cases is over 900 meters. Kazuto Yamauchi et al from Japan Sync hrotron Radiation Research Institute recently quantified the way small bumps on a silicon mirror will distort an FEL beam[17]. The bumps were im agined as randomly placed bell-curves (Gaussian bumps) of various heights and spreads along the le ngth, and a raytracing finite simulator, integrating the Fresnel-Kresnel integral, was employed. It was found that the spread of a bump is not very important, but the height of the bump is important. With X-ray frequencies, the researchers f ound that 2 nm height is an im portant cutoff point in terms of beam quality. Unacceptable distortions, in the form of diffraction peaks, become likely beyond this point. For this reason, 2 nm flatness seems to be an adequate design goal for this study. This is achievable fr om a manufacturing sta ndpoint. Using elastic emission machining, Mimura et al at SPri ng-8 report that surface roughness down to 0.2 nm RMS is achievable over a length of 96 mm[18]. Therefore, 2 nm RMS roughness

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24 over 80 cm seems to be an easier task. Howe ver, the bumps studied here originate not from machining problems but due to heating. Because the bumps created by FEL radiation may not be ideal, Gaussian bumps, the deflected shape found via simulation should be examined by numerically integra ting the Fresnel-Kirchhoff integral of the deflected shape after this st udy is complete, as long as th e result is bumps within the same order of magnitude as 2 nm. Heat load The latest draft of the Workgroup 73 doc ument proposes that the mirrors will be in front of the double crystal monochromator and behind tungsten slits and a solid attenuator. The attenuator is not specified at this point, but it is likely to be an aluminum or beryllium window. An aluminum or beryllium window will absorb all of the spontaneous radiation below a certain photon energy, as shown in Figure 11, allowing only hard X-rays to pass. Figure 11Comparing the X-ray transmission of Aluminum and Beryllium, angle of incidence = 90. Data from http://henke.lbl.gov/optical_constants/ 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20Transmissivity of 100 micron foilPhoton energy (keV) Aluminum Beryllium

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25 At least 90% of the X-ray photons incident on the first mirror will be reflected, that is, the mirror angle will be set so that the reflectivity is at least 0.9 for the FEL radiation. The other 10% or le ss will be absorbed by the silicon atoms and converted into heat. A larger heat load will come from the spontaneous radiation above the FEL bandwidth that was not affected by the attenuator, most of it will be absorbed by the mirror. The heat reaction is a matter of individual photons being absorbed by individual atoms. Microscopically, the pr ocess at work is heat genera tion varying with respect to position. It is not like absorption of therma l radiation which is a surface phenomenon. The rate of photon absorption is proportional to the percentage of unabsorbed photons remaining, so that the slice closest to the surface absorbs more photons, and has a greater heat load than all the slices below it. This is a case of exponential decay, shown below. Equation 5Exponential decay of X-ray in tensity as photons are absorbed by atoms G0 is the rate of photon absorption/h eat generation at the surface and l is the absorption length which is a function of the wavelength and the size of the atoms absorbing the radiation. In th e case of smaller nuclei or shorter wavelengths, the average X-ray photon may pass many layers of atoms before finally being absorbed and the absorption length is longer. In the case of larger nuclei or longer wavelengths, 99% of the photons may be absorbed in the first fe w nanometers, and the absorption length is shorter. The absorption length can vary but for our purposes, since we are doing a Finite Elements simulation of the entire mirror, th e length will always be small enough that over 99% of the absorbed energy will be absorbed by the nodes along the face of the mirror.

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26 The incident radiation and the heat load will also vary along the length and width of the mirror. While the spont aneous radiation will be nearly evenly distributed over the mirror, the paths of the photons in the laser radiation are all very close together and very close to being parallel. This property of laser light is called spatial coherence. However, all the photons do not have the exact same path; it is more corr ect to say that the paths are very tightly distributed around a mean path. The distribution is normal, and the standard deviation is small. This is called a Gaussi an distribution along th e two axes x and y. Equation 6Two Dimensional Gaussian Distribution of photons in FEL beam H is the peak intensity of the FEL beam in W/m2. If the radiation were hitting the mirror head on at a 90o angle of incidence, x and y would be equal. is the standard deviation, and it is in units of length. At away from the y-axis, the local intensity will be reduced by 63%, at 2 away the reduction is 98%, etc. The y-axis of the mirror and the path of the beam make a plane that is pe rpendicular to the flat face of the mirror, so the beams footprint on the mirror is not stretc hed or compressed in this direction. On the other hand, the x-axis of the mirror and the m ean path of the beam make a plane that intersects the mirror face at the x-axis itself, at a very shallo w angle. For this reason, the beam spread in the x-direction will be much wider, Equation 7Beam spread from the perspective of the slanted mirror is the incident angle on the mirror. will be less than 0.5 or 8 milliradians, so the sine of this angle is equal to the angle itself, in radians. is a property that depends

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27 on the undulator gap setting, and the shutter ga p setting. Its value also grows linearly with the distance between the undulator a nd the mirror. Workgroup 73 has suggested a value of 5 mm be used for in all simulations. We now have enough information to know what the heat generated in a finite volume will be with respec t to position in the mirror. This function is below, where S is the intensity of the spontaneous radiation in W/m2. Equation 8Intensity of FEL and spontane ous photons absorbed as a function of position The total heat load on the mi rror is the volumetric integral of this function. These functions accurately describe the real heat lo ad in terms of space, but not in time. The actual X-ray laser will have many intense puls es lasting less than a picosecond each, with microseconds in between of no photon flux. Howe ver, for the purposes of this study it is sufficient to consider the stea dy state condition of the mirror wh ile the laser is active, and therefore only the average heat load on the mirror is interesting. The total spontaneous heat load due to the S term is 31.5 W, while the total laser heat load with a Gau ssian distribution is 6.5 W. Bending requirements A mirror may be bent in the interest of focusing or de focusing the FEL radiation, to spread the photons out over a larger area, or to compress the photons so that the beam size at the source is the same as the beam size at the experiment site, in spite of divergence. These are both theoretical concerns. In real applications, mirror imperfections, even in costly silicon mirro rs, can destroy desired optical features.

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28 Precisely bending the mirror can compensate fo r these imperfections, so that the bent mirror is straightened. While systems to do this with many actuators along the mirror length are feasible and have been proposed, it is more common to see three and four point bending mechanisms being used. When bendi ng a mirror, slope error becomes an important figure of merit. To find this figure, first the desired, circular slope is defined as a function along the length of the mirror. Then the actual slope of the mirror surface is measured, again as a function along the length of the mirror, and the two are subtracted from each other, giving an error function that al so varies with respect to the length. It is generally acceptable for slope er ror to be 3 microradians ( rad) or less[19]. This is not expected to be different for FEL beams. A mechanical bending mechanism is often a source, rather than a remedy, of slope error. Only the center section of a beam in 4-point bending has a constant radius; no part of a beam in 3-point bending has a constant radius. However, because the radii of focus are often so large that the deflection is on the order of microns, these mechanical benders often gi ve sufficient performance. The European XFEL Work Group Package 73 has requested th at any bending mechanism be able to precisely go from flat to a radius of 20 km for the purposes of focusing. An ability to curve even beyond 20 km, down to 10 km would be desired but not required. The bending only needs to take place in a single axis giving a cylindrical, rather than toroidal, profile.

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29 Ch. 3 Description, simulation, and analysis of proposed designs The following chapter lists five possible designs for the mirror, each with an added bit of complexity compared to the one that came before. The first design was proposed by others, and the next four ar e proposed improvements. Each of these proposals are considered in te rms of the deflected mirror shape resulting from them, and these shapes are simulated w ithout considering gravity or the support arrangement, as if the mirror were floating in space. Deflecti ons caused by gravity and the support scheme will add to the deflection caused by a thermal gradient, and both can be minimized or controlled separately. The last section of th is chapter proposes how to support the mirror. Remote cooling The first instinct of the engineers de signing the European XFEL optics was to cool the mirrors remotely, that is by radiati on only. Because the presence of air or other gas will tend to attenuate and scatter X-rays external convection was out of question for this component. The hope was that by bringing a cold (100 K via a pulse tube refrigerator), black plate near to the front su rface, the mirror would be adequately cooled by near-blackbody thermal radiation exchange. The plate could be moved away when not needed so that the mean temperatur e of the mirror would remain constant. The good thing about this approach is th at the hot spot of the front of the mirror becomes sandwiched between two cold spots crea ted by the cold plate. Since the average temperature of the front and back surfaces of the mirror are the same, this minimizes

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30 bending along the axis, though bending (outwards, towards the beam source) still occurs. The magnitude of the bending is small, with the high point about 30 nm above the low point, and the radius of curvat ure is 3000 kmflat for most practical purposes. However it was found that cooling the remote plate dow n to 100 K would not be cost-effective for the rate of heat removal that could have achieve d. A deficiency of this analysis was that the heat generation was idealized as o ccurring homogenously along a strip going down the front of the beam; in reality the genera tion will be greatest in the middle and decay with a Gaussian relation along the length and height of the face of the mirror. We will show that a heat concentrati on in the center can have an outsized effect on the bending radius. So, while the beginning design constrai nt was dont touch the mirror, this was loosened so that the mirror was imagined as having a liquid metal contact, which will not be affected by the vacuum nor put stress on the mirror, on at least one surface. Not only does this allow the maximum heat removal rate to increase, this de cision also simplifies design analysis because the surface with liquid metal can be idealized as having a homogenous temperature. Liquid metal cooling on a single surface Since the beam is intended to be reflected horizontally, the plane of the mirror is vertical. The mirror will either rest on its cooling surface, or the cooler rest on the mirror. For this reason the main heat flow, and theref ore the likely alignment of the bending, is at 45 to the face of the mirror.

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e i i s u p t h c o A f r a x d a n s u t h m t h T A Figure 12 Cool e This m i ther up or d s not likely t o p or down w h is setup is u o mparison; a To co A NSYS Wo r r om the bea m x es are sho w epth is chos e n d down (in u rface is als o h e coordinat e m esh maxim u h e computin g T he mirror is A NSYS AP D Describin e d surface m m eans that t h d own, away f o be useful, w ill tend to s l u nlikely to g i a control gr o mplete the s r kbench. Th m source. T w n on Figur e e n so the mi the x axis b o held at 22 e system in A um edge len g g power av a divided int o D L is used t o g one fixed m a y also be h e mirror w i f rom the co o because it w l ant the refl e i ve a useful o up. s imulation, a e x-axis is u T he y-axis i s e 13 and the rror will m o b y this simul a C. The ce n A NSYS, w h g th is initial a ilable. The o elements t h o describe th e 3 temperatu r on the bott o i ll want to b e o ler, as well w ill spread, r e cting surfa c mirror, we w a 4 5 80 c u p and dow n s along the l other figure o re easily be n a tion). The n ter of the o p h ich is also t h ly set at 5 m larger mes h h at are each e I(x,y,z) fu n 1 r e surface a o m, mirror e nd both to w The bendi n r ather than f o c e due to an t w ill simulat e cm 3 solid sil n while the l ength of th e s from ANS n d in the de s initial temp e p tical face o h e center of m m b ut this h h size had a m rectangular n ction for th e a nd no othe r assumed w w ards the be a n g towards t o cus the bea m t iclastic effe c e it for the s a icon bar is d z-axis is to w e mirror. T h YS Workb e s ired z-axis r e rature is 2 2 f the mirror the Gaussi a h ad to be rel a m inimal eff e prisms like t e heat load, r interventi o w ei g htless. a m source a n t he beam so u m The ben c ts. Even t h a ke of d escribed in w ards or a w h e x, y, and e nch. The 4 r ather than u 2 C and the is the origi n a n heat loa d a xed to 1 c m e ct on the re s t he mirror i t and this fun c o n. n d u rce ding h ough w ay z cm u p top n of The m for s ults. t self. c tion

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32 is brought into Workbench as a command. First, the steady-state thermal application runs, and the I(x,y,z) function is checked by verifying th at the correct amount of heat, about 38 W, leaves the cooled surface. The temperature distribution is also examined, to make sure that a Gaussian hot spot appears, Figure 13. Next, the steady-state structural application runs with no loads or supports to find the flo ating in space deflected shape in response to the temperature change. Th e results, showing bending in two axes, are Figure 14. Finally, a log of th e steady-state structural resu lts may be made and brought back into ANSYS APDL so the deflected sh ape can be examined in more detail. Figure 13 Showing typical temperature dist ribution with cooling on top only and no backlighting. Distribution remains the same when a film is added.

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33 The result of this first simulation with a single cooling surface is that the mirror bends outwards, opposite to the direction that we want it to bend, and downwards. The radius of the outward bending is 152 km, which is within one order of magnitude of the 20 km inward bending that we hope to achieve This suggests that a backlight with double the power of the FEL beam and spontane ous radiation will stil l be well short of the design goal, giving a ~150 km inward bendi ng, so at that stag e the initial guess for backlight power to intentionally bend th e mirror will be four times beam power. Figure 14 Z-deflection of weightless mirror with one cooled surface on top. The deflected shape bends in z and x.

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f o w L c r s i d t h O t a o m F H t h r a Whe n o r the X-ray w hich is not c L iquid metal Since r eate undesi r i mulate is t o istribution m h e interest o f O ne benefit o a lking about f its own, w e m irror as a h o F i g ure 15D surfac e We e x H owever, if t h e other sid e a diation as i t n only one c o load, the h o c ooled. Thi s coolin g on the first si m r ed bending o use extra h e m ore homog e f not touchi n o f this idea i s X-ray optic e will ideali z o mogenous h D escribin g o e ma y also b x pect the be n t he resistanc e e ANSYS s h t did before, o oling surfa c o ttest spot o n s spot is 0.2 9 a sin g le su rf m ulation con f in the mirr o e at, coming e nous; hope f n g the mirro r s that a halo g s. While a h z e the effec t h eat flow to o ne cooled s b e on the b o n ding in thi s e input is 3 8 h ows that th e only not as 3 4 c e is used, o n n the mirror i 9 C hotter t rf ace with h e f irms that t h o r, the first i d from the ot h f ully the mi r r the heat w g en lamp is h eat lamp c a t of a heat la m the back su r s urface wit h o ttom, and t s case to be u 8 W to matc h e mirror wil l much. Wh y 4 n the top, a n i s on the fro n t han the rest e at lamp h e energy ab s d ea that this h er side, to m r ror will re m w ould come f on the low e a n come wit h m p placed n r face. h added he a t he mirror i u p, towards t h the power o l also bulge t y is this? W h n d no heat c o n t face near of the mirr o s orbed from paper will e m ake the te m m ain straight f rom a lamp e nd in terms h a variety f o n ear the bac k a t from a la m i s assumed w t he cooling p o f the absor b t owards the h ile the hea t o mes in exc e the bottom, o r. the beam w e xplore and m perature in this case. (see Figure of costs wh e o cusing mir r k side of the m p. The co w ei g htless. p late, only. b ed X-rays o incoming t lamp input e pt w ill In 15). e n r ors oled o n is

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d l e t e s e i m w F iffuse, the F e ngth, it has e r m It turn s e t to S dA + 1 m pact on be n w as set to 43 F i g ure 16 T EL radiatio n an greater e s out that be n 1 .7* H(x,y) d n ding as the W. This re l T emperatu r Max t n is not. Si n ffect on ben n ding in the d xdy; in oth e spontaneou l ationship w r e distribu t t emperatur e 3 5 n ce it has a p ding than t h z-plane is m e r words, th e s radiation. w as found by t ion with 43 e is +0.5C i 5 eak intensit y h e more ho m m inimized w h e FEL radia t So in Figur e trial-and-er r W backsi d i n this conf i y centered a l m ogenous he a h en the diff u t ion has alm e s 16 and 1 7 r or. d e heat, one ig uration. l ong the mir a ting due to u se heat la m ost twice th e 7 the heat la coolin g sur ror the S m p is e mp face.

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36 Figure 17 Z-deflection of mirror with one cooling surfac e and 43 W heat on back surface. This is the straightest configuration acheivable while in service using only a heat lamp, no second cooling surface or metal film. Even with this fine tuning, the z-deformation of mirrors front face has an increasingly strong gradient towards the e nds, shown in Figure 17. Rather than being perfectly flat, the deformed shape in this ca se is best modelled by an inward circle of 2600 km radius, which should be near enough to infinity or flat for European XFELs purposes. If the heat lamp used in practice is not diffuse relative to the back surface of the mirror, this relationship will change and the best approach again will be trial and error with the actual lamp. The deviation from th e large circle, shown in Figure 18, is like a cosine wave with an amplitude of 4 nm. Our goal is to reduce this amplitude to 2 nm or

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37 below. The plot is noisy because the large (>106 m) radius compared to the small error pattern (~10-9 m) pushes the graphing software to the limit of its precision. Figure 18 Bump created by FEL beam is olated from large-radius circular deflection. The large radius puts us at the limit of machine precision. Single cooled surface. The bending in the x direction (up) is not without consequence, however. As discussed before, there are anticlastic effect s. In pure bending due to a single applied moment, the anticlastic bending would make the originally vertical planar face of the mirror slope slightly downwards. However, simulation shows that the average slope is actually 0.57 microradia ns upwards. Why is this? This is because isotropic thermal expansion plays a bigger role. The averag e temperature difference between top and bottom, looking back at Figure 16, is about 0.45 C. Both the X-rays on the front and the heat lamp in the back cause this temperature difference. The thermal expansion of silicon is 2.6 m/(m*C), meaning that the expected slope from this effect in isolation is 0.29*2.6 = 0.75 microrad upwards. So the simulated average slope is equal to the effect of isotropic thermal expansion plus the unknown effect of anticla stic bending. The 6.0 4.0 2.0 0.0 2.0 4.0 6.0 00.10.20.30.40.50.60.70.8deviation, nmLength along mirror, m deviation from 2600 km circle

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38 magnitude of anticlastic bending is theref ore about 0.6 microrad, constant along the mirror length. The average vertical slope of 0.57 micror ad is not a big deal; over the ~750 m remaining beam line, it amounts to a vertical beam displacement of 0.4 mm. This will not cause the beam to hit any barriers and is easily accommodated at the experiment site. What may be a greater concern is the variation in the vertical slope against this average, shown in Figure 19. We know that such variation is due to the uneven, Gaussian heat load only, not due to warping, so finding it at this stage will help us isolate the warping effect later. Figure 19 Variation in vertical slope of the front face due to uneven heating. Single cooled surface. Cooling on a second surface There are a couple of ways to achieve cooling on the second surface. The mirror could be sandwiched between two cooled copper plates, wi th a liquid metal interface on both sides. There could possibl y be small channels cut in th e top surface of the mirror as 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 00.20.40.60.8Slope, microradLength along mirror, m Deviation from average vertical slope top half bottom half

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d i n d 5 c h i n T t h w c o a t o iscussed pr e n liquid met a Figure 2 0 If ch a istance fro m cm height. W h annels defi n a pool of l i T he mirror w h e same ass u w ould other w o ol is not es p benefit in t h The r e o bend in th e e viously. F o a l instead o f 0 Showin g a nnels are us m the bottom W e can ass u nes an isoth i quid metal, w ould first b e u mption cou l w ise be the b p ecially im p h e first plac e e sults of A N e x-directio n o r cooling o n f resting on a g one possi b b o ed on top, t h of the chan n u me that a h o ermal surfa c some of the e designed s o l d be made a ottom of th e p ortant at thi e before goi n N SYS simul a n is eliminat e 3 9 n the bottom a cooled plat b le confi g u r o ttom of m i h e overall h e n el to the b o o rizontal pl a c e nearly en o mirror wou o that 5 cm w a bout a con s e mirror. Fo r s stage; it is n g much fur t a tion show t h e d, along wi t 9 surface, the e. This als o r ation to ac h i rror. e ight of the m o ttom of the m a ne intersect i o ugh. Simil a ld be subm e w ould still b s tant-temper a r these reas o more impo r t her. h at with sy m t h its attend a mirror coul o was discus s h ieve coolin m irror will b m irror woul i ng the bott o a rly, if the m e rged and th e e above the a ture horizo n o ns, the choi r tant to und e m metric cool i a nt anticlast i d actually f l s ed previou s g on top an d b e greater bu d be the ori g o m of these m irro r is flo a e refore unus waterline a n tal plane w ce of how t o e rstand if th e i ng, the ten d i c effects in t l oat s ly. d t the g inal a ting able. a nd w hich o e re is d ency t he

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40 Figure 21 Temperature distribution with two cooling surfaces and 43 W backlighting. Max temperature is +0.18 C. Back surface has wide band of higher temperature like front surface, wi thout the ellipse in the middle. z-direction. However, there is still some bending in the z-direction. As before, the flattest shape possible corresponds to a back lighting input of 43 W, and the mirror seems to snake in the z-direction. One bulge is visible for the front si de view of Figure 22, and on the back there are two more. However, these bulges represent deflecti ons that are very small, less than three nanometers in amplitude, shown in Figure 23. While this is an improvement over the 4 nm seen in the simulation with one cooling surface, the tolerable error according to Yamauchi et al was 2 nm. Because there are only three of them, the slope error is much less than the threshold of 3 mi croradians they also proposed.

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41 Figure 22 Z-deflection with 43 W backli ght and two cooling surfaces. Main feature is central bump wi th amplitude of 3 nm. Figure 23 Deflection in Z direction along cen ter line of mirror face. Attempting to keep the mirror flat. -0.1 -0.05 0 0.05 0.1 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 00.20.40.60.8slope, microrad Displacement, nmLength along mirror, m UZ dUZ/dy

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42 In the previous case with a single cooler we closely examined the vertical slope of the mirror, but in this case, as should be e xpected, the deflection profile along the top is identical to that along the bottom; both are the same shape seen in Figure 23 but with smaller amplitude. The vertical slope along th e center is zero for the entire length. Increasing lamp power to bend mirror So far we have only considered how to k eep the mirror flat, not how to use extra heat to bend it with a 20 km radius. Next, simulations were also run where, with two cooled surfaces, the power of the heat lamp is increased beyond optimal to create a surface that may not only reflect the beam but also focus it. Figure 24 Results of using backlight @ 172 W to bend mirror. Amplitude of deviation increases beyond 4 nm. Two cooled surfaces. It turns out that, whether one cooling surf ace or two is used, extra heat does not seem to be the most profitable way to bend th e mirror. With 172 W, thats four times the backlighting required to keep the mirror stra ight, simulation predicts the mirror bending radius as 135 km, still an order of magnitude away from the desired 20 km. The 0.15 0.10 0.05 0.00 0.05 0.10 0.15 7.0 5.0 3.0 1.0 1.0 3.0 5.0 7.0 00.10.20.30.40.50.60.70.8Slope, microrad Deviation, nmLength along mirror, m Deviation from 135 km circle slope error

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43 deviation from true circular de flection due to the concentrated heat of the XFEL beam is slightly greater than it was in the previous case, where flatness was the goal. It seems that using a heat lamp to bend the mirror is not a viable way because as the heat lamp requirement becomes greater, the load on the cooling system also increases. Removing a few hundred watts from a surface of this size is doable; the problem is more doing so in a way that creates a uniform te mperature across the top and bottom. At greater heat loads the cooling fluid going th rough the copper plate or stainless bathtub will be appreciably hotter at near the outle t than near the inlet, creating another temperature gradient whose effect on bending must be considered. For this reason we conclude here that a heat lamp alone may be a viable option to keep a flat mirror flat, or a mirror machined with a curved surface curved, but it is probably not a good way to actively change the focus length of the mirror. Adding a metal film While the heat lamp keeps the mirror flatte r than it would be with no intervention at all, a greater level of fl atness and bending control is desi red. To actively change the bending radius and therefore the focus length of the mirror, while minimizing the appearance of heat bumps, one final solution will be considered which is a metallic film. The film would be de posited by a CVD process at a certain temperature, and when used at temperatures other than the deposit temperature it will tend to form a curved shape, as discussed previously. Simulati ng the metal film is straightforward. The computer is told the bulk properties of th e film and its thickne ss. A starting value 100 microns thickness was chosen after running some numbers through Stoneys equation. ANSYS has a variety of ways that it can model adhesion; we have chosen perfect

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44 adhesion and we will monitor the shear stress to make sure this is always an appropriate choice. Another thing the ANSYS user must consider is the thermal interface between the film and the substrate. We chose the defa ult setting which averages the two values of thermal conductivity for heat flow across the boundary. ANSYS will allow the user to program a unique value of interfacial conductivity in case there are small multilayer structures designed to insulate or conduct. A metal silicide layer, if allowed to form, would be a case like this. This will not be simulated here. 20 km bending with 100 micron tungsten film and one cooling surface For the first iteration of th is design, tungsten was chosen for the film material. The incoming X-ray heat load was kept the sa me, the initial temperat ure set to 22 C, and the temperature of the top cooling surface se t to 58 C, as if the temperature of the cooling water was allowed to change by 36 C The final temperature was selected by examination of Stoneys Equation (Equation 1). With these parameters, Stoneys Equation predicts a bending radius of 20 km as specified by Workgroup 73. The temperature distribution in this case is identical to that shown in Figure 13, which also had one cooling surface and no back light. The difference of course is that now the coldest point is 58 C. This means th at the film does not affect heat conduction. The deformation diagram is completely diffe rent however; the heat bump does not even show up in the display. Instead, the deforma tion at each node must be compared to the nearest circular shape to isolate the effect of uneven heating from the effect of thermal mismatch and also to find if warping is taking place.

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45 Figure 2520.3 km circle subtracted fr om deformation in z-direction. Tungsten, tf= 100 m, T=36 C, top cooling only Figure 25 shows that the shape of the deviation from circular profile is similar to the previous deviation from flat in Figure 24. There is a central bump in a sinusoidal pattern whose amplitude is about 3.5 nm. This suggests that the effect of the film and the effect of the uneven heating have little intera ction; the principle of superposition seems to work here though not perfectly so. Figure 26Vertical Slope dx/dy. Tungsten, tf= 100 m, T=36 C, top cooling only -0.150 -0.100 -0.050 0.000 0.050 0.100 0.150 -5.0 -2.5 0.0 2.5 5.0 00.20.40.60.8slope, microrad displacement, nmlength along mirror, m deviation from 20.3 km circle slope error -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 00.10.20.30.40.50.60.70.8slope, microradlength along mirror, m overall vertical warping slope in top half slope in bottom half

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46 Figure 26 shows that the slopes in the vertical plane have the same inclination as they did in the prior no film single coole r simulation of Figure 19. The magnitude is much smaller, however. This suggests that warping due to stiffness mismatch in fact plays very little role. Instead the increased stiffness seems to greatly reduce the effect that uneven heating has on the vertical slope, even slight ly reducing the amplitude of deviation from 4 nm (Figure 18) to 3.5 nm in Figure 25. Figure 27 Temperature distribution with two cooling surfaces and no backlighting. The distribution remains the sa me when a film is added. Max temp is +0.16 C. Back surface has near constant temperature.

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47 20 km bending with 100 micron tungsten film and two c ooled surfaces The next simulation is the same as the prev ious one, except that this time both the top and bottom are cooled. The nearest circular shape, in th is case, has a radius of 19.5 km, very close to the previous value of 20.3 km and both are sufficiently close to the design value of 20 km. The reason for the vari ance from the 20 km spec is the Gaussian heat generation function in the simulation. We would expect that the uneven heating would create a deviation from the circular shape roughly the same size as the deviation seen in each previous simulation. However, this is not the case. The second cooled surface and removal of the backlight toge ther reduce the overall anomaly in the temperature distribution. With two cooled su rfaces, the deviation from circular with a tungsten film is down to an amplitude of 2 nm compared to 3.5 nm with a film and one cooled surface and 3 nm with no film and a 43 W backlight. Figure 28 Using a 100 micron Tungsten film with a 36 C temperature change to induce bending, graph shows deviation fr om circle due to FEL radiation. Amplitude of deviation less than 2 nm. If the vertical slopes examined previously were a concern, using two coolers eliminates them to a great extent. This is shown in figure 29. Since the temperature gradient shares a plane of symmetry with the undeformed mirror a nd film, the vertical 0.10 0.05 0.00 0.05 0.10 4.0 2.0 0.0 2.0 4.0 00.10.20.30.40.50.60.70.8Slope, microrad Deviation, nmLength along mirror, m deviation from 19.5 km circle slope of deviation

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48 slope in the top half is a mirro r image of the vertical slope in the bottom half and the sum is zero. The magnitudes are also very sma ll, the 0.25 microrad ma ximum is found at the edges only. It was feared that significant vert ical slopes might appear as the film created warping. This did not occur; the fe ar about warping seems to be unfounded. Figure 29 Vertical slopes alon g mirror with tungsten film. tf= 100 m, T=36 C, cooling on top and bottom Considering other materials After examining this first set of results, one concern is that the 36 C temperature difference requirement is too st eep; it may be difficult to find a water delivery system with that kind of range as well as precisi on. Plus, the greater the temperature difference between the flat infinite radius state and the 20 km minimum radius, the more prohibitive changing the mirror st ate will be for researchers, in terms of time required to set up an experiment. One course of action would be to increase the thickness of the tungsten film, but this requires more time and money to fabricate. The obvious step is then to examine other materials with greater thermal mismatch and differing stiffness. Almost every metal has a greater expansion coefficient ( ) than silicon, but as it turns 0.40 0.20 0.00 0.20 0.40 00.10.20.30.40.50.60.70.8Slope, microradLength along mirror, m overall vertical warping slope in top half slope in bottom half

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49 out, tungstens is among the lowest of the metals Metals have a wide range of stiffness values, (E). Table 1 compares tungsten to other candidate metals and Figure 31 graphically shows the selection process. Table 2Relevant properties of materials discussed here Pure Material E (GPa) ( K-1) Deposit Method Notes Silicon 185 2.6 N/A Tungsten 400 4.5 CVD Excellent adhesion Nickel 200 13.4 CVD Good adhesion, no reaction below 300C Copper 110-128 16.5 PVD More reac tive to silicon than others Beryllium 287 11.3 ? Toxic, Reactive in air, may need protective Ni coating electroplated after deposit Nickel and copper immediately stand out as perhaps better choi ces, if maintaining film thickness near 100 m is the goal. Both, however, have lower stiffness than tungsten. At this point it is believed that the greater stiffness of the tungsten reduced the size of the bump created by the concentrated FEL beam. Stoneys Equation suggests that if nickel is used instead of tungsten, only a 12 C change is necessary for a 100 m film to induce a 20 km bent radius, however nickels lower stiffne ss may give inferior results. 20 km bending with 100 micron nickel film and two cooling surfaces So the previous simulation with two cooled surfaces and a metal film was changed from tungsten to nickel, and T was changed from 36 C to 12 C, and the simulation was re-run. Figure 30 shows that th e deviation from flat or circular has an amplitude of 2.5 nm with a 100 micron nickel fi lm, reducing as film stress increases. The

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50 reduced stiffness of nickel doe s not seem to play a role in minimizing the deviation. Interestingly, the deviation also gets smalle r as the mirror bends more due to film action. Figure 30 The effect of a 100 m Ni film, deposited at 22 C, on mirror behavior 20 km bending with 300 mi cron tungsten film If film stiffness is the key to keeping the bent profile as circular as possible, minimizing deviation, taking the stiffest mate rial under considerati on and increasing the thickness seems like a good play. Stoneys Equation says that if a tungsten film is used at 300 microns, the temperature change needed to produce 20 km bending is 12 C, which makes sense because increasing the film thickness by a factor of three should reduce the temperature change by the same factor, ever ything else being constant. However, simulation predicts that increasing the film thickness with tungsten actually makes matters worse. When the film was 100 mi crons tungsten, with two cooling surfaces, the deviation from the nearest circle was 2 nm in amplitude. With increased tungsten thickness it is back up to three. The only in trinsic part of the system that changed when film thickness was increased was the film stress. This suggests that high film stress, and not the thickness of the film alone or its stiffness alone, is what acts to minimize the FEL 6.0 4.0 2.0 0.0 2.0 4.0 6.0 00.10.20.30.40.50.60.70.8Deviation, nmLength along miror, m deviation from 208 km outward circle with mirror at 22C deviation from flat at 23.5C deviation from 22.1 km circle with mirror at 34 C

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51 bump size. The recommendation that will follow then, will likely bring the film stress near to the edge of the tensile or delamination level, whichever is less. Looking at Figure 31, we can now make an informed decision about the best film material. Stoneys equation, as previously derived for the film shear stress, is complicated but it can be shown that the st ress will increase linearly with the films expansion coefficient. However, when there is a fixed goal for a bending radius, the only result of increasing the expans ion coefficient is reducing the needed temperature change. The shear stress is a function of the film thickness, but not the stiffness or thermal expansion of the film, when the radius of cu rvature is fixed. There are stiffness terms, Youngs Modulus E and Poissons Ratio nu, but they belong to the substrate. f Ests 2 6tf1 s r 1 4tfts Equation 9 Stoneys Equation solved for film stress with a known radius of bending. The equation becomes simpler when the radius of curvature, r, is known. Thus the decision of which material to us e is not driven by a desire to minimize the deviation due to concentrat ed heating. Instead it is driven by the power of the cooling system, that is, what magnitude of change in temperature it is capable of over a short period of time. It is also driven by the adhesion strength and yield strength of the material. While a nickel film will achieve desired performance with a smaller change in temperature, there is less documentation available about the adhesion strength of such films, compared to tungsten. For both films, it is presumed that the adhesion strength is the controlling factor; that is that some type of delamination is likely to occur before the film material yields intrinsically.

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52 Figure 31 Adapted from Ashby Materi al Selection Charts, used with permission[20]. The ideal film material, in addition to go od adhesion to silicon and low reactivity, has an elasiticity similar to Silicon with a large coefficient of Thermal Expansion. 10 1 100 Silicon Tungsten Nickel Beryllium

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53 Keeping the mirror flat with a metal film As discussed previously, the XFEL beam alone is already bending the mirror the wrong way before we start to talk about heat lamps and metal films to bend it the right way. No matter what metal film is used, or what the thickness, if the mirror begins to handle X-rays while the cooling system is set to the film deposit temperature, the mirror will bend the wrong way. The magnitude of this bending is reduced by the effect of the film stiffness, but the direction is not changed. For this reason simulations were run for the purpose of knowing what change in temperature gives the flattest mirro r for each film configuration. Table 3 Temperature change that will ke ep the mirror flattest in presence of concentrated FEL heating Film material Thickness Flat temperature Nickel 100 m +1.5K Tungsten 300 m +1.1K Tungsten 100 m +4.6K Buoyant cooling bath A buoyant support has an obvious advantage and an obvious disadvantage. The advantage is that we know that a dens e liquid will support the mirror homogenously, without any risk of creating extra stresses. The disadvantage is that a fraction of the mirror would be submerged, and the mirror ma y tilt in this arrangement. The liquid proposed, as stated before, is Indalloy 51, whic h has a specific gravity of 6.5. The silicon that would float in this liquid has a specific gravity of 2.33, or 35% of the density of the liquid Indalloy 51, therefore 35% of the silicon would be submerged. This means that to have a usable area of 5 cm, the original manufactured mirror height must be 6.7 cm.

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54 A second, related problem to a buoyant suppor t appears if a metal film is used. The solid metal film will be denser than its silicon substrate, unle ss the film is beryllium which is slightly less dense. This difference in density will mean that the center of mass is away from the center of volume. These tw o centers must be vertically aligned for the mirrors floating orientation to be vertical meaning if nothing were done the mirror would tilt backwards as it floats, the side with the film sinking and the optical side rising. This problem could be solved with somethi ng as simple as a well-placed blob of dense putty, or perhaps the optical surface could be machined at the exact tilt needed to counteract this effect. Either way, the tende ncy of the mirror to tilt backwards must be considered if both a mirror with a film is used with a buoyant support.

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55 Ch. 4 Recommendations Response time considerations It is not enough for the mirror to merely exist and hold a ce rtain shape at two different temperatures. We would like to know that it doesnt take 48 hours, or even an hour, for the mirror configuration to change. While the second cooling surface turned out to have less than the expected impact in minimizing displacement along the mirror face, obviously having double the cooling area will ha ve a large impact on the amount of heat stored in the mirror and how much time is requi red to dissipate it and change the mirror configuration. In none of the previous simulations were bumps due to the XFEL beam eliminated. Obviously, however, once the beam is turned off, th ey go away, after a certain period of time. Once the mirror is pe rfectly flat again, it takes the same amount of time for the steady-state deflection patterns pr eviously discussed to reappear after the beam is turned on. The consequence of the transition time betw een beam off and beam on is that experiments carried out within the transition ti me will be exposed to a slightly different beam than those that wait until steady state. If there is an important difference between the two, the experimenter will have to keep the final set of shutters closed during one phase or the other.

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56 So, it will be helpful to make a small in troduction to transient thermal problems, apart from just simulating them. Fouriers law of heat transfer stat es that the rate of change in temperature of any point is proportional to the su m of all thermal gradients at that point. This gives the differential e quation which applies uniquely at each point in space. Equation 10 Simplified Fourier Heat Equation is the heat diffusivity of the material. This typical first-order partial differential equation has solutions for T(x,y,z,t) that are dependent on the initial and boundary conditions but always include a term et for exponential decay. This is typically the only term that involves time, unless one of the bounda ry conditions also varies with time. So, when the question is asked, How long does it take to go from the initia l state to the final state, formally, the answer is Forever. The value of an ex ponential decay function approaches a final value as a limit but theoretically always comes up short. This leaves us dealing with terms such as half life whic h means, the time afte r which the system is halfway between its initial and final states. To describe the time required for the mirror to reach a certain steady state, it seems best to think about % time meaning the time at which the initial difference between th e maximum and minimum temperature of the mirror has reduced to 1% of its original value. This time depends on the material properties and dimensions and alignment of boundary conditions only, and has nothing to do with the initial or final state. For a silicon mirror cooled on the top only, the 99% time is 65 seconds. For a mirror cooled on the top and bottom, the 99% time is only onequarter of that value. This is because th ere is twice as much surface area through which

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57 heat can leave, and the maximum length fr om any point in the mirror to a cooled boundary has decreased by half. These figures were taken from a brief simulation with no film. The thin metal film did not change these values significantly. Most effective design, conclusions The most effective design is one that mi nimizes deformation due to the FEL beam while having a good response time. This study has shown that while an extra heat lamp can reduce this deformation, using a metal film was more effective. The most effective metal film is one that creates the most ther mal stress. Finite Elements simulation shows that thermal stress in the film on the back si de minimizes thermal strain on the front side. The effect is probably analogous to what ha ppens when a bolt is pre-tensioned. The following table lists each confi guration that was considered in Chapter 3 and lists the thermal stress in the film as well as the amplitu de of the deviation from circular shape in the mirror; they seem to be inversely proportional. Table 4 Interventions to bend the mirror to a 20 km radius Film material Film Thickness Temp change for r= 20 km Shear stress Amplitude of deviation Top cooling Sym cooling Tungsten 100 m 33 K 34.2 MPa 3.5 nm 2 nm Tungsten 300 m 11.5 K 11.5 MPa 3.8 nm 3 nm Nickel 100 m 11 K 34.2 MPa -2nm No film, 172 W lamp power causing 135 km bending -3.5 nm

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58 Table 5 Interventions to keep the front of the mirror flat. Film material Film Thickness T Cooling Shear stress Amplitude of deviation Tungsten 100 m 4.5 K Top only 4.7 MPa 2.5 nm 3.3 K Top and bottom 3.4 MPa 3.4 nm Tungsten 300 m 1.1 K Top and bottom 1.5 MPa 5 nm Nickel 100 m 1.5 K Top only 4.7 MPa 3.4 nm No film, 43 W backlight -Top only -~4 nm -Top and bottom -3 nm For the design process to continue from here, more information will be needed about the CVD process and the adhesion stre ngth expected, and the cooling system and how quickly the temperature of the cooling water may be changed. Cooling both on the top and the bottom are recommended for the 75% shorter response time and the reduced deviation both from fl at and from circular. For film selection, the rule of thumb will be that thinner metal films working with larger temperature changes will produce the best results both in the flat state and in the curved state. A second rule of thumb is that for a given thickness of tungsten film on a 4 cm thick silicon mirror, the temperature change, in degrees Kelvin, required to create a 20 km bend is roughly equal to the shear stress generated, in MPa.

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59 Figure 32 Guide for film selection using Stoneys Equation. Future work Future work could go in two directions fr om this juncture. For those working at European XFEL, future work would center on practical considerations such as the water/coolant delivery system, whose characteristics will drive the desired film properties. Commercially available metal co ating processes must al so be investigated and compared to the ideal room-temperatu re CVD envisioned here; again the three requirements for the film metal are 1) able to be strongly and ine xpensively deposited to silicon (maximizing film stress without failu re), 2) minimally reactive to air and liquid metals, and 3) the higher the coefficient of th ermal expansion, the better. If the deposit temperature selected is much warmer or co lder than room temperature, the cooling system much account for this. The second direction that future work could go in is to simulate multilayer systems that might more effectively create the kind of constraining stress documented here, over a more practical range of temperatures. 0 100 200 300 400 0 100 200 300 400 050100150200250300, MPa T KFilm thickness, microns Temperature change for tungsten film, K Temperature change for nickel film, K Shear stress between film and substrate, MPa

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60 References [1] Original image and caption from Hawkin s and Staff Hawkins Electrical Guide Number One (New York: Theo. Aude l and Company, 1917) 133, reworked, Copyright: 2009, Florida Center for Instructional Technology. [2] Proprietary claim made online, http:// www.xfel.eu/overview/ facts_and_figures/ Retrieved 10/4/2010 [3] Harald Sinn, Heat load estimates for XFEL beamline optics. Internal XFEL development document. [4] Cameron Kewish, et al, The potential for two-dimensional crystallography of membrane proteins at future X-ray fr ee-electron laser sources, New Journal of Physics, 12, 035005 (2010). [5] Kazuto Yamuchi et al, Wave-optical evaluation of interf erence fringes and wavefront phase in a hard-X-ray beam totally reflected by mirror optics, Applied Optics, 44, 32, 6927 (2005) [6] Michael Sullivan, et al, White light X-ray focusing mirror, Rev. Sci Instrum. 79, 025101 (2008) [7] Michael Hitchman, et al, Some considerations of the thermodynamics and kinetics of the Chemical Vapor Deposition of Tungsten, Applied Surface Science, 38, 312 (1989) [8] Marketing claim made by manufacturer online, http://www.cvmr.ca/nvd/proces s.html Retrieved 10/4/2010 [9] D.C. Meyer, T. Leisegang, A.A. Levin, P. Paufler, A.A. Volinsky, Tensile Crack Patterns in Mo/Si Multilaye rs on Si Substrates Under Hightemperature Bending, Appl. Phys. A 78, pp. 303-305, 2004 [10] George Vander Voort, Atlas of time-temperature digrams for nonferrous alloys, A S M International, 1991 p. 462 [11] K.N. Tu et al, Thin FilmsInterdiffusion and Reactions, Interscience, New York, 1978, p. 359 [12] Sheng Yuan et al, Elliptically Bent X-ray Mirrors with Active Temperature Stabilization, X-ray optics and Instrumentation, 2010 784732.

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61 [13] W.D. Nix, Mechanical Properties of Thin Films, Lecture notes prepared for Stanford University in 2 005 and available online at http://imechanica.org/no de/530. Retrieved 9/16/2010. [14] Michael R. Sullivan et al, Installation and testing of a focusing mirror at beamline X28C for high flux x-ray radiolysis of biological macromolecules, Rev. Sci. Instrum. 79, 025101 (2008) [15] Jean-Jacques Ferm, New Improvements in Bendable Mirrors, Proc. SPIE 3152 (1997) [16] Marketing claim made by manufacturer online, http://seso.com/pub/Glid cop%20cooled%20mirror.ppt Retrieved 8/16/2010. [17] Kazuto Yamuchi et al, Wave-optical evaluation of interf erence fringes and wavefront phase in a hard-X-ray beam totally reflected by mirror optics, Applied Optics, 44, 32, 6927 (2005) [18] Hidekazu Mimura et al, Direct determ ination of the wave field of n X-ray nanobeam, Physical Review A, 77, 015812 (2008) [19] See 16 [20] Lecture notes available online at http://mielsvr1.ecs.umass.edu/mie 497a/Ashby%20Materials%20Selection% 20Charts.PDF. Reprinted here with permission from Dr. Mike Ashby.

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62 Appendices

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63 Appendix AANSYS Inputs This Appendix section shows the report generated by a typical run of ANSYS 12 Workbench. This particular case is with a 100 micron tungsten film, one cooling surface, and a 4.5 K temperature change meant to keep the mirror flat while the beam is on. All other cases will be similar. This report should answer any detailed question about how the model was set up. Figure 33 Simulated mirror, gree n, with thin film in orange.

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Appendix A (Continued) 64 Units Table 6 Simulation units Unit SystemMetric (m, kg, N, s, V, A) Degrees rad/s Celsius Angle Degrees Rotational Velocity rad/s Temperature Celsius Model (B4, C4, D4) Geometry Table 7 Model (B4, C4, D4) > geometry Object Name Geometry State Fully Defined Definition Source E:\THESIS\Ansyssimulation\091710_files\dp0\Geom-1\DM\Geom1.agdb Type DesignModeler Length Unit Millimeters Element Control Program Controlled Display Style Part Color Bounding Box Length X 5.e-002 m Length Y 0.8 m Length Z 4.01e-002 m Properties Volume 1.604e-003 m Mass 3.805 kg Scale Factor Value 1. Statistics Bodies 2 Active Bodies 2 Nodes 12027 Elements 2000 Mesh Metric None Preferences Import Solid Bodies Yes Import Surface Bodies Yes Import Line Bodies No Parameter Processing Yes Personal Parameter Key DS CAD Attribute Transfer No Named Selection Processing No

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Appendix A (Continued) 65 Material Properties Transfer No CAD Associativity Yes Import Coordinate Systems No Reader Save Part File No Import Using Instances Yes Do Smart Update No Attach File Via Temp File Yes Temporary Directory C:\Documents and Settings\EN B229.FOREST.005\Application Data\Ansys\v120 Analysis Type 3-D Mixed Import Resolution None Enclosure and Symmetry Processing Yes Table 8 Model (B4, C4, D4) > geometry > parts Object Name substrate film State Meshed Graphics Properties Visible Yes Transparency 1 Definition Suppressed No Stiffness Behavior Flexible Coordinate SystemDefault Coordinate System Reference Temperature By Environment Material AssignmentSi W Nonlinear Effects Yes Thermal Strain Effects Yes Bounding Box Length X 5.e-002 m Length Y 0.8 m Length Z4.e-002 m 9.9998e-005 m Properties Volume1.6e-003 m 4.e-006 m Mass3.728 kg 7.7e-002 kg Centroid X 0. m Centroid Y 0. m Centroid Z-2.e-002 m -4.005e-002 m Moment of Inertia Ip10.19932 kgm 4.1068e-003 kgm Moment of Inertia Ip21.2737e-003 kgm1.6042e-005 kgm Moment of Inertia Ip30.1996 kgm 4.1228e-003 kgm Statistics Nodes8799 3228 Elements1600 400 Mesh Metric None

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Appendix A (Continued) 66 Coordinate Systems Table 9 Model (B4, C4, D4) > coordinate system Object Name Global Coordinate System StateFully Defined Definition TypeCartesian Ansys System Number 0. Origin Origin X 0. m Origin Y 0. m Origin Z 0. m Directional Vectors X Axis Data[ 1. 0. 0. ] Y Axis Data[ 0. 1. 0. ] Z Axis Data[ 0. 0. 1. ] Connections Table 10 Model (B4, C4, D4) > connections Object Name Connections StateFully Defined Auto Detection Generate Contact On UpdateYes Tolerance TypeSlider Tolerance Slider0. Tolerance Value2.0064e-003 m Face/FaceYes Face/EdgeNo Edge/EdgeNo PriorityInclude All Group ByBodies Search AcrossBodies Revolute JointsYes Fixed JointsYes Transparency EnabledYes Table 11 Model (B4, C4, D4) > connections > contact region Object Name Bonded substrate To film StateFully Defined Scope Scoping MethodGeometry Selection Contact1 Face Target1 Face

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Appendix A (Continued) 67 Contact Bodiessubstrate Target Bodies film Definition TypeBonded Scope ModeManual BehaviorSymmetric Suppressed No Advanced FormulationPure Penalty Normal StiffnessProgram Controlled Update Stiffness Never Thermal ConductanceProgram Controlled Pinball RegionProgram Controlled Mesh Table 12 Model (B4, C4, D4) > mesh Object Name Mesh StateSolved Defaults Physics PreferenceMechanical Relevance 0 Sizing Use Advanced Size FunctionOn: Fixed Relevance CenterCoarse Initial Size SeedActive Assembly SmoothingMedium TransitionFast Min SizeDefault (4.0003e-004 m) Max Face Size1.e-002 m Max Tet SizeDefault (8.0006e-002 m) Growth RateDefault (1.850 ) Minimum Edge Length1.e-004 m Inflation Use Automatic Tet InflationNone Inflation OptionSmooth Transition Transition Ratio0.272 Maximum Layers 5 Growth Rate 1.2 Inflation Algorithm Pre View Advanced Options No Advanced Shape CheckingStandard Mechanical Element Midside NodesProgram Controlled Straight Sided Elements No Number of Retries 0 Rigid Body BehaviorDimensionally Reduced Mesh MorphingDisabled

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Appendix A (Continued) 68 Pinch Pinch ToleranceDefault (3.6002e-004 m) Generate on Refresh No Statistics Nodes12027 Elements2000 Mesh MetricNone Figure 34 Showing mesh Named Selections Table 13Model (B4, C4, D4) > named selections > named selections Object Name all StateFully Defined Definition Send to SolverYes VisibleYes Scope Geometry2 Bodies

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Appendix A (Continued) 69 Statistics TypeManual Total Selection2 Bodies Suppressed0 Hidden0 Steady-State Thermal (B5) Table 14 Model (B4, C4, D4) > analysis Object Name Steady-State Thermal (B5) StateSolved Definition Physics TypeThermal Analysis TypeSteady-State Solver TargetANSYS Mechanical Options Generate Input Only No Table 15 Model (B4, C4, D4) > steady-sta te thermal (B5) > initial condition Object Name Initial Temperature StateFully Defined Definition Initial TemperatureUniform Temperature Initial Temperature Value22. C Table 16 Model (B4, C4, D4) > steady-sta te thermal (B5) > analysis settings Object Name Analysis Settings State Fully Defined Step Controls Number Of Steps 1. Current Step Number 1. Step End Time 1. s Auto Time Stepping Program Controlled Solver Controls Solver Type Program Controlled Nonlinear Controls Heat Convergence Program Controlled Temperature Convergence Program Controlled Line Search Program Controlled Output Controls Calculate Thermal Flux Yes Calculate Results At All Time Points

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Appendix A (Continued) 70 Analysis Data Management Solver Files DirectoryE:\THESIS\Ansyssimulation\ 091710_files\dp0\SYS\MECH\ Future Analysis None Scratch Solver Files Directory Save ANSYS db No Delete Unneeded Files Yes Nonlinear Solution No Solver Units Active System Solver Unit System mks Table 17 Model (B4, C4, D4) > steady-state thermal (B5) > loads Object Name Temperature Heat Flow StateFully Defined Suppressed Scope Scoping MethodGeometry Selection Geometry 1 Face Definition TypeTemperature Heat Flow Magnitude26.5 C (ramped)43. W (ramped) SuppressedNo Yes Define As Heat Flow

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Appendix A (Continued) 71 Table 18 Model (B4, C4, D4) > steady-state thermal (B5) > commands (ANSYS) Commands inserted into this file will be executed just prior to the Ansys SOLVE command. These commands may supersede command settings set by Workbench. Active UNIT system in Workbench when this object was created: Metric (m, kg, N, s, V, A) *SET,_FNCNAME,'x091310' *SET,_FNCCSYS,0 /INPUT,F:\THESIS\Ansyssimulation\091310.func,,,1 *DIM,%_FNCNAME%,TABLE,6,23,1,,,,%_FNCCSYS% ! Begin of equation: (472400+400000*EXP((125000*{X}^2+10.33*{Y}^2)))*EXP (2000*{Z}) *SET,%_FNCNAME%(0,0,1), 0.0, -999 *SET,%_FNCNAME%(2,0,1), 0.0 *SET,%_FNCNAME%(3,0,1), 0.0 *SET,%_FNCNAME%(4,0,1), 0.0 *SET,%_FNCNAME%(5,0,1), 0.0 *SET,%_FNCNAME%(6,0,1), 0.0 *SET,%_FNCNAME%(0,1,1), 1.0, -1, 0, 0, 0, 0, 0 *SET,%_FNCNAME%(0,2,1), 0.0, -2, 0, 1, 0, 0, -1 *SET,%_FNCNAME%(0,3,1), 0, -3, 0, 1, -1, 2, -2 *SET,%_FNCNAME%(0,4,1), 0.0, -1, 0, 2, 0, 0, 2 *SET,%_FNCNAME%(0,5,1), 0.0, -2, 0, 1, 2, 17, -1 *SET,%_FNCNAME%(0,6,1), 0.0, -1, 0, 125000, 0, 0, -2 *SET,%_FNCNAME%(0,7,1), 0.0, -4, 0, 1, -1, 3, -2 *SET,%_FNCNAME%(0,8,1), 0.0, -1, 0, 2, 0, 0, 3 *SET,%_FNCNAME%(0,9,1), 0.0, -2, 0, 1, 3, 17, -1 *SET,%_FNCNAME%(0,10,1), 0.0, -1, 0, 10.33, 0, 0, -2 *SET,%_FNCNAME%(0,11,1), 0.0, -5, 0, 1, -1, 3, -2 *SET,%_FNCNAME%(0,12,1), 0.0, -1, 0, 1, -4, 1, -5 *SET,%_FNCNAME%(0,13,1), 0.0, -2, 0, 1, -3, 3, -1 *SET,%_FNCNAME%(0,14,1), 0.0, -1, 7, 1, -2, 0, 0 *SET,%_FNCNAME%(0,15,1), 0.0, -2, 0, 400000, 0, 0, -1 *SET,%_FNCNAME%(0,16,1), 0.0, -3, 0, 1, -2, 3, -1 *SET,%_FNCNAME%(0,17,1), 0.0, -1, 0, 472400, 0, 0, -3 *SET,%_FNCNAME%(0,18,1), 0.0, -2, 0, 1, -1, 1, -3 *SET,%_FNCNAME%(0,19,1), 0.0, -1, 0, 2000, 0, 0, 4 *SET,%_FNCNAME%(0,20,1), 0.0, -3, 0, 1, -1, 3, 4 *SET,%_FNCNAME%(0,21,1), 0.0, -1, 7, 1, -3, 0, 0 *SET,%_FNCNAME%(0,22,1), 0.0, -3, 0, 1, -2, 3, -1 *SET,%_FNCNAME%(0,23,1), 0.0, 99, 0, 1, -3, 0, 0 End of equation: (900000+50000000*EXP((125000*{X}^2+10.33*{Y}^2)))*EXP(2000* {Z}) !--> LGWRITE,'091310','lgw','F:\thesis\Ansyssimulation\',COMMENT bf,all,hgen,%x091310%

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Appendix A (Continued) 72 Solution (B6) Table 19 Model (B4, C4, D4) > steady-state thermal (B5) > solution Object Name Solution (B6) StateSolved Adaptive Mesh Refinement Max Refinement Loops1. Refinement Depth2. Table 20 Model (B4, C4, D4) > steady-state thermal (B5) > solution (B6) > solution information Object Name Solution Information StateSolved Solution Information Solution OutputSolver Output Update Interval2.5 s Display PointsAll Table 21 Model (B4, C4, D4) > steady-state thermal (B5) > solution (B6) > results Object Name Temperature StateSolved Scope Scoping MethodGeometry Selection GeometryAll Bodies Definition TypeTemperature ByTime Display TimeLast Calculate Time HistoryYes Identifier Results Minimum26.5 C Maximum26.793 C Minimum Occurs Onsubstrate Maximum Occurs Onsubstrate Information Time1. s Load Step1 Substep1 Iteration Number1

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Appendix A (Continued) 73 Table 22 Model (B4, C4, D4) > steady-state thermal (B5) > solution (B6) > probes Object Name Reaction Probe StateSolved Definition TypeReaction Location MethodBoundary Condition Boundary ConditionTemperature Options Display TimeEnd Time Results Heat-38.043 W Maximum Value Over Time Heat-38.043 W Minimum Value Over Time Heat-38.043 W Information Time1. s Load Step1 Substep1 Iteration Number1

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Appendix A (Continued) 74 Transient Thermal (C5) Table 23 Model (B4, C4, D4) > analysis Object Name Transient Thermal (C5) StateSolved Definition Physics TypeThermal Analysis TypeTransient Solver TargetANSYS Mechanical Options Generate Input Only No Table 24 Model (B4, C4, D4) > transien t thermal (C5) > initial condition Object Name Initial Temperature StateFully Defined Definition Initial TemperatureNon-Uniform Temperature Initial Temperature EnvironmentSteady-State Thermal TimeEnd Time Table 25 Model (B4, C4, D4) > transien t thermal (C5) > analysis settings Object Name Analysis Settings State Fully Defined Step Controls Number Of Steps 1. Current Step Number 1. Step End Time 110. s Auto Time Stepping Program Controlled Initial Time Step 1.1 s Minimum Time Step 0.11 s Maximum Time Step 11. s Time Integration On Solver Controls Solver Type Program Controlled Nonlinear Controls Heat Convergence Program Controlled Temperature Convergence Program Controlled Line Search Program Controlled Nonlinear Formulation Program Controlled Output Controls Calculate Thermal Flux Yes Calculate Results At All Time Points

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Appendix A (Continued) 75 Analysis Data Management Solver Files DirectoryE:\THESIS\An syssimulation\0917 10_files\dp0\SYS-1\MECH\ Future Analysis None Scratch Solver Files Directory Save ANSYS db No Delete Unneeded Files Yes Nonlinear Solution No Solver Units Active System Solver Unit System mks Table 26 Model (B4, C4, D4) > tr ansient thermal (C5) > loads Object Name Temperature 2 StateFully Defined Scope Scoping MethodGeometry Selection Geometry1 Face Definition TypeTemperature Magnitude34. C (step applied) SuppressedNo Solution (C6) Table 27 Model (B4, C4, D4) > tran sient thermal (C5) > solution Object Name Solution (C6) StateSolved Adaptive Mesh Refinement Max Refinement Loops1. Refinement Depth2. Table 28 Model (B4, C4, D4) > transient th ermal (C5) > solution (C6) > solution information Object Name Solution Information StateSolved Solution Information Solution OutputSolver Output Update Interval2.5 s Display PointsAll

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Appendix A (Continued) 76 Table 29 Model (B4, C4, D4) > transient th ermal (C5) > solution (C6) > solution information > result charts Object Name Temperature Global Maximum Temperature Global Minimum State Solved Scope Scoping Method Global Maximum Global Minimum Definition Type Temperature Results Minimum 34. C 26.622 C Maximum 34. C 33.995 C Table 30 Model (B4, C4, D4) > transient th ermal (C5) > solution (C6) > results Object Name Temperature StateSolved Scope Scoping MethodGeometry Selection GeometryAll Bodies Definition TypeTemperature ByTime Display Time76.653 s Calculate Time HistoryYes Identifier Results Minimum33.961 C Maximum34. C Minimum Occurs Onfilm Maximum Occurs Onsubstrate Minimum Value Over Time Minimum26.622 C Maximum33.995 C Maximum Value Over Time Minimum34. C Maximum34. C Information Time76.653 s Load Step1 Substep18 Iteration Number18

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Appendix A (Continued) 77 Table 31 Model (B4, C4, D4) > transient thermal (C5) > solution (C6) > probes Object Name Heat Flux Probe StateSolved Definition TypeHeat Flux Location MethodGeometry Selection Geometry1 Face OrientationGlobal Coordinate System Options Result SelectionZ Axis Display Time6.0237 s Spatial ResolutionUse Maximum Results Z Axis0.64973 W/m Maximum Value Over Time Z Axis2.2654 W/m Minimum Value Over Time Z Axis3.5898e-003 W/m Information Time 1. s Load Step 1 Substep 1 Iteration Number 1

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Appendix A (Continued) 78 Static Structural (D5) Table 32 Model (B4, C4, D4) > analysis Object Name Static Structural (D5) StateSolved Definition Physics TypeStructural Analysis TypeStatic Structural Solver TargetANSYS Mechanical Options Environment Temperature22. C Generate Input OnlyNo Table 33 Model (B4, C4, D4) > static structural (D5) > analysis settings Object Name Analysis Settings State Fully Defined Step Controls Number Of Steps 1. Current Step Number 1. Step End Time 1. s Auto Time Stepping Program Controlled Solver Controls Solver Type Program Controlled Weak Springs Program Controlled Large Deflection Off Inertia Relief Off Nonlinear Controls Force Convergence Program Controlled Moment Convergence Program Controlled Displacement Convergence Program Controlled Rotation Convergence Program Controlled Line Search Program Controlled Output Controls Calculate Stress Yes Calculate Strain Yes Calculate Results At All Time Points Analysis Data Management Solver Files DirectoryE:\THESIS\An syssimulation\0917 10_files\dp0\SYS-2\MECH\ Future Analysis None Scratch Solver Files Directory Save ANSYS db No Delete Unneeded Files Yes Nonlinear Solution No Solver Units Active System Solver Unit System mks

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Appendix A (Continued) 79 Table 34 Model (B4, C4, D4) > static stru ctural (D5) > imported load (setup) Object Name Imported Load (Setup) StateFully Defined Definition TypeImported Data Interpolation TypeMechanical Results Transfer Suppressed No Table 35 Model (B4, C4, D4) > static stru ctural (D5) > imported load (setup) > imported body temperature Object Name Imported Body Temperature State Solved Scope Scoping MethodGeometry Selection Geometry2 Bodies Definition TypeImported Body Temperature Suppressed No Source EnvironmentSteady-State Thermal (B5) Solution (D6) Table 36 Model (B4, C4, D4) > static structural (D5) > solution Object Name Solution (D6) StateSolved Adaptive Mesh Refinement Max Refinement Loops1. Refinement Depth2. Table 37 Model (B4, C4, D4) > static struct ural (D5) > solution (D6) > solution information Object Name Solution Information StateSolved Solution Information Solution OutputSolver Output Newton-Raphson Residuals0 Update Interval2.5 s Display PointsAll

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Appendix A (Continued) 80 Table 38 Model (B4, C4, D4) > static stru ctural (D5) > solution (D6) > results Object Name Directional Deformation StateSolved Scope Scoping MethodGeometry Selection GeometryAll Bodies Definition TypeDirectional Deformation OrientationZ Axis By Time Display Time Last Coordinate SystemGlobal Coordinate System Calculate Time History Yes Identifier Results Minimum-2.1714e-007 m Maximum3.0483e-007 m Minimum Occurs On film Maximum Occurs Onsubstrate Information Time 1. s Load Step 1 Substep 1 Iteration Number 1

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Appendix A (Continued) 81 Material Data Si Table 39 Si > constants Density2330 kg m^-3 Coefficient of Thermal Expansion2.6e-006 C^-1 Thermal Conductivity149 W m^-1 C^-1 Specific Heat710 J kg^-1 C^-1 Table 40 Si > isotropic elasticity Temperature C Young's Modulus Pa Poisson's Ratio 1.85e+011 0.31 W Table 41 W > constants Density19250 kg m^-3 Coefficient of Thermal Expansion4.5e-006 C^-1 Thermal Conductivity173 W m^-1 C^-1 Specific Heat131 J kg^-1 C^-1 Table 42 W > isotropic elasticity Temperature C Young's Modulus Pa Poisson's Ratio 4.e+011 0.28