Bachelor thesis. Helium ray-tracing. Simulating the neutral helium atom microscope NEMI using McStas. by Anders Komár Ravn (sdz360) - PDF

Bachelor thesis Helium ray-tracing Simulating the neutral helium atom microscope NEMI using McStas by Anders Komár Ravn (sdz360) Submitted to the Faculty of Science at the University of Copenhagen in partial

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Bachelor thesis Helium ray-tracing Simulating the neutral helium atom microscope NEMI using McStas by Anders Komár Ravn (sdz360) Submitted to the Faculty of Science at the University of Copenhagen in partial fulfillment of the degree of Bachelor of Science in the subject of Physics. June 10, 2015 Supervisors: Kim Lefmann, Niels Bohr Institute, University of Copenhagen, Denmark Bodil Holst, Department for Physics and Technology, University of Bergen, Norway Erik B. Knudsen, DTU Physics, Technical University of Denmark, Denmark Abstract This thesis investigates the validity of making simulations of neutral matter-wave microscope instrumentation, by modifying the well established McStas neutron ray tracing software. Specifically this thesis models the neutral helium atom microscope, NEMI, located at the University of Bergen. Two new instrument components were written in order to simulate NEMI. The new components were a source with a Lorentzian wavelength distribution and Gaussian spatial distribution, and a Fresnel zone plate. The simulations comply with the physics intended to be simulated, and show a subtle, but important, dependency on wavelength that was not present in previous geometrical calculations done by the NEMI group in Bergen. Dansk resumé Dette bachelorprojekt undersøger gyldigheden af at lave simulationer af neutral stof-bølgemikroskopinstrumentering, ved at modificere det veletablerede McStas neutron ray-tracing software. Konkret handler denne afhandling om at modellere det neutrale helium atom mikroskop, NEMI, der står ved Universitetet i Bergen. To nye instrument komponenter blev skrevet for at simulere NEMI. De nye komponenter var en kilde med en Lorentzfordeling i bølgelængder og en Gaussisk rumlig fordeling, og en Fresnel zone plade. Simuleringerne overholder den kendte fysik bag komponenterne, og viser en subtil, men dog vigtig, afhængighed af bølgelængde, der ikke var til stede i tidligere geometriske beregninger udført af NEMI-gruppen i Bergen. Acknowledgements I would like to thank my supervisor for his patience with me and for recommending me this quirky project. I would also like to thank Erik B. Knudsen for accompanying me to Bergen, to see NEMI and the group surrounding the project, and sparring early in the process. And finally I would also like to thank Bodil Holst, Sabrina Eder and Thomas Kaltenbacher for their hospitality in Bergen and for introducing me to the world of microscopy instrumentation. It s been a pleasure. i Contents 1 Introduction 1 2 NEMI 2 3 Monte Carlo simulation using McStas Ray-tracing, McStas and the Monte Carlo method The McStas weight factor Building NEMI in McStas The source The aperture The Fresnel zone plate The detectors Results and discussion Sanity check Comparison between the simulation and real data A note on the zone plate component Conclusion 18 A Detailed description of the code 21 A.1 The Fresnel zone plate (FZP) A.2 The source B The Code 24 B.1 nemi.instr B.2 Source_Lorentz_He.comp B.3 FZP_simple.comp ii 1. Introduction Microscopy has for long been a key method for scientists to study and observe features smaller than the naked eye can see. In the 16th century the first microscopes were constructed[17]. They were optical in design where light would be scattered by the object being studied and focused through an optical lens system to the eye of the observer. To this day microscopes have been a key tool in answering fundamental questions set by investigators of the physical world. We have come a long way since the earliest microscopes, however, and there have been many landmark achievements in the field of microscopy in recent years. The prime objective of most modern research in microscopy techniques is to find ways to overcome or circumvent the Abbe resolution limit. Abbe discovered back in 1837 that light with wavelength λ travelling in a medium with refractive index n converging on a spot with the angle θ will make a spot with radius d = λ /2n sin θ for optical microscopes[14]. In 2014 the Nobel prize in chemistry was given to William Moerner, Erik Betzig and Stefan Hell [15] for their work and contribution to the field of super resolution microscopy 1, especially the development of several kinds of stimulated emission depletion microscopy (STED) where, with a clever use of so-called excitation and depletion lasers, fluorophores in the sample of interest can be observed in high detail. Examples of non-optical microscopes is scanning probe microscopes (SPM) such as the atomic force microscope (AFM) and the scanning tunnelling microscope (STM)[10]. These SPM microscopes works by recording the interaction between a sharp tip and a sample, either by measuring the change in current at a constant height or vice versa in the case of STM; or by measuring the inter-atomic forces between the tip and surface of the sample in the case of ATM. In either case only samples of a certain smoothness and stiffness are easy to image in a quickly and reliable manner. If the sample is too soft or rough, or has a high aspect ratio, then a slow scanning speed is required to get an image and not damage the sample or tip. Another type of microscope is the matter-wave microscopes such as the scanning electron microscope (SEM/ESEM) and the transmission electron microscope (TEM)[6], and the very new helium-ion microscope (HIM)[11]. In TEM images are taken of electrons transmitted through very thin samples. In SEM and ESEM the image is recorded from the backscattered and secondary electrons exited from the sample. Normally samples used in these microscopes have to be conducting, 1 A common denomination for all optical techniques that in some way or other bypasses the Abbe limit. 1 otherwise the sample may get charged and that distorts the image, and the sample may also get damaged by heating or ionization. All in all these microscopes are very useful and powerful techniques to overcome the problem with the Abbe limit for optical microscopes, however, most of these require the samples in question to be either very smooth, sturdy, conducting or a combination of these requirements. In the interest of coming up with a technique with the ability to fast and reliably image fragile, insulating samples with high aspect ratios, researchers came up with the idea of microscopes based on neutral atom or molecule beams scattering. Even though this notion is not new, the technical realization of such an instrument has proven to be a challenge. The very nature of neutral matter-waves dictate for them to be focused requires they are manipulated through their de Broglie wavelength. This has to be done analogous to classical optics since neutral atoms do not interact with the electro- and magneto-optical lenses used in electron and ion microscopes. Since low-energy molecules do not penetrate solid matter, lenses are out, and the only two focusing elements left are Fresnel zone plates and mirrors. The first images obtained using a neutral helium beam was in 2007 by Koch et al.[12]. The focusing element in their set-up was a Fresnel zone plate. The Neutral Microscope, NEMI 2, is one of these neutral matter-wave microscopes. NEMI is located at the Department for Physics and Technology at the University of Bergen. The aim of this work is to simulate the Neutral Microscope, using the neutron raytracing tool McStas by a modification of the program to simulate helium physics, and verify the simulation with experimental data taken with the NEMI instrument. The NEMI team is particularly interested in finding out what the smallest achievable focus is at a certain speed distribution when using a zone plate as the focusing element. Simple geometrical models are not accurate enough. It should be mentioned that this project is the first steps toward a planned expansion of Mc- Stas into McHe (working title), a ray-tracing software package to simulate helium beam experiments. 2. NEMI The Neutral Microscope, NEMI for short, is a reflecting neutral matter-wave scanning microscope that uses neutral ground-state helium-4 as the probing beam. 2 Although the acronym never was in question, what it s an acronym of is a matter of taste. Examples are: neutral helium atom microscope, neutral matter-wave microscope and neutral helium scattering microscope, and the list goes on. 2 It works in principle similarly to an SEM, creating an image by scanning the focused beam across a sample. In this case this is achieved by having movable sample stage which can rotate and translate for the beam to scan. The image is a record of the backscattered atoms by a detector placed at an angle from the optical axis. Figure 2.1 is a conceptual drawing of this. Figure 2.1: Schematic drawing of the working principle of NEMI. The source is on a typical nozzle mounted on a cooling block through which a high-pressure tube runs. A skimmer selects the central part of the beam to be imaged onto the sample by the zone plate. Between the nozzle and skimmer is low pressure, and beyond the skimmer everything is in vacuum. Image taken from [5] by Eder. The main components on NEMI is the beam source, the skimmer, the optical element and the detection system. The beam source is in principle no different from the sources found in other modern helium atom scattering devices. It produces a beam by expanding a highpressure helium gas through a small nozzle aperture into a low-pressure ambient chamber[5]. By choosing the right nozzle diameter, gas pressure and temperature a so called free-jet expansion can be achieved, resulting in an almost monochromatic beam of neutral helium atoms[5, 16]. The distribution of wavelengths is Lorentzian, 3 and is characterized by the speed ratio[5] S = 2 ln 2 v v, where v is the mean velocity of the atoms, and v is the FWHM of the speed distribution[4, 5, 12]. It is shown in [5] that the FWHM of the Lorentzian wavelength distribution can be expressed as λ = 2 ln 2 λ 0 S 1, (1) where λ 0 is the median and mode of the distribution. The skimmer is the beam shaping element. It has a conical shape with a small orifice at the source-facing side. Figure 2.2 is a photograph of the skimmer used in NEMI. The skimmer has several functions in the microscope: (i) It separates the source chamber and the pumping stage vacuum chamber thereby working as a differential pumping stage. (ii) It confines the helium beam it restricts the spatial intensity distribution of the free-jet expansion. And finally (iii) the skimmer diameter defines the object size which is being imaged by the zone plate[5]. The optical elements used to focus the beam is a Fresnel zone plate, as there is not a whole lot of alternatives as outlined in the introduction. The choice of zone plate Figure 2.2: Photograph of the skimmer used in the NEMI instrument taken form [5]. The source-facing side (tip) has an opening with radius 5 µm. instead of a mirror is simply because there are more technical limitations to using mirrors 3 and there have been several previous successful attempts at using a zone plate to focus a helium beam [3, 4, 12]. A Fresnel zone plate is a circular diffraction grating. Figure 2.3 is a sketch of a zone plate. The radius of the boundary between the n th successive opaque and transparent rings on the zone plate is given by the formula r 2 n = nλf + nλ 4, (2) 3 For a detailed technical description of this and more related to the construction of NEMI the author refers the reader to [5]. 4 Figure 2.3: A sketch of the principle idea behind a Fresnel zone plate. Marked on the drawing is the radius R, the width of the outermost transmission zone r, the source at point S radiating waves of wavelength λ, which diffract on the grating to converge again in an image at point P. The distance to the image point is a function of all the marked parameters P = P (S, R, r, λ). The image is taken from [1]. Figure 2.4: Sketch of the first three orders of focus in both convergent and divergent. Marked are the different focal and path lengths of the rays. Also drawn is a so-called order-sorting aperture (OSA), an aperture designed to sort out most rays of the other foci as to only let the first-order focused rays pass. Image taken from [1]. 5 where λ is the wavelength of the incoming wave and f is the focal length of the plate. The fractional term nλ /4 represents spherical aberration and is usually omitted since f nλ for most practical applications[1] including NEMI[5]. Equation (2) also imply that f is proportional to λ 1 i.e. a function of wavelength. Then, when the zone plate is exposed to waves with a distribution of wavelengths, chromatic aberration occurs, since each different wavelength will have a different focal length and therefore will be imaged at different positions[9]. As previously mentioned a Fresnel zone plate is a circular diffraction grating that can focus waves to a point. But unlike a lens, the transmission function allows for multiple foci. The transmission function can be expressed as a cosine series[1] T (u) = m= c m cos mu, where u = πr2 fλ, m is the order and c m is the Fourier component c m = 1 π π 2 0 cos mu du. Now calculating the value of T for the different orders, m, shows that 50 % of the oncoming wave is reflected, 25 % pass through unchanged (the so-called 0- order beam), 10 % goes to the first-order focus, the same amount goes to the divergent first-order (m = 1) focus and 1 % to the third-order focus and so on[1]. Figure 2.4 is a drawing of the first orders of focus. Figure 2.5: Sketch of the different pumping stages, from left to right: SC the source chamber, PST the pumping stage chamber, ZPC the zone plate chamber and SDC the sample and detector chamber. The black lines represent the dividing apertures. The image is taken from [5]. The detection system is a so called Pitot tube. The Pitot tube has a pinhole aperture. When the entering beam flow is equal to the effusive flow back out of the aperture, an equilibrium pressure can be measured. The change in pressure is 6 then taken as the measure of beam intensity[5]. In NEMI these are placed in different chambers separated by differential pumping stages. Figure 2.5 is an illustration of this. The first separation is, as previously mentioned, the skimmer between the source chamber and the pumping stage vacuum chamber. The zone plate chamber is isolated on either side by a 5 mm aperture separating it from the pumping stage vacuum chamber and the sample and the detector chamber. 3. Monte Carlo simulation using McStas McStas is a software package developed specifically with the purpose of simulating neutron scattering experiments. Besides being a software package also a so called domain specific language (DLS) built on ANSI-C[19]. There are three levels of coding in McStas. The bottom level is the McStas kernel. This is where all low level particle transportation routines, geometry engines and such lay. It is written entirely in C, and also provides the basis for the MsStas DSL and compiler used on the other levels. The middle level is the component files. These files are the building blocks of the McSats simulated instruments. It is here on this level the Monte Carlo choices are taken. These files are for example beam sources, optical elements or detectors used in neutron scattering experiments. It is also on this level the samples are. The top level is the instrument files. The instrument files consist of a number of calls to different components, usually a source, then different optical components to manipulate the beam and finally a sample and detectors. This project works on the middle and top level. 3.1 Ray-tracing, McStas and the Monte Carlo method Traditional ray-tracing methods, as the ones used to generate computer graphics, trace several rays from the eye through each pixel on the image frame to their sources and assign a value to the pixel based on sophisticated recursive algorithms taking into account material properties as well as reflections, refractions and shading[2, 7]. The McStas method is different. It utilises what is known as backwards raytracing 4 in the CG-business, tracing rays from the source to the image. The difference is that each ray s properties is determined by one or more Monte Carlo choices. 4 According to [2]. [7] disagrees on the term for direction. The point is: McStas does it the opposite way of traditional ray-tracing for graphics rendering. 7 There is not just one Monte Carlo method. Looking up the encyclopedic definition 5 one finds that the Monte Carlo methods span a broad class of computer algorithms that rely on repeated random sampling to obtain numerical results. Monte Carlo methods typically follow a particular pattern: 1. Define a domain of possible inputs. 2. Generate inputs randomly from a probability distribution over the domain. 3. Perform a determistic computation on the input. 4. Aggregate the result. At first glance this pattern is very clear in McStas: The end user defines the domain by setting the parameter values for the source. McStas then generates the actual simulation inputs with specific probability distribution in a number of parameters set in the source component and propagates the rays through the virtual instrument performing deterministic calculations at each component. Then finally it aggregates the result at a monitor which records either energies, wavelengths, position or some other parameter or parameters at the time the ray passed the monitor area. As an addendum to point number 3, it should be mentioned that McStas is also equipped to perform stochastic choices in a component or sample[13]. The Monte Carlo method used in McStas is a variation on the Monte Carlo integration method[19]. The Monte Carlo integration method[8] is a method to solve a multi-dimensional integral of the kind I = Ω f(x) dx numerically using random numbers, where Ω is a subset of R m, Ω dx = 1 and f is square integrable on Ω. The Monte Carlo way to integrate such a function f is to generate N uniform samples x 1,..., x N Ω and approximate I by E N 1 N N f(x i ). i=1 The law of large numbers is applicable, and shows lim E N = I. N 5 8 This, in turn, indicates that E N is an unbiased estimator[8] of I, and the variance is σ 2 E N = 1 N Ω (f(x) I) 2 dx = σ2 I N. In practice the true variance is rarely known and is therefore estimated by s 2 = 1 N 1 N (f(x i ) E N ) 2 (3) i=1 3.2 The McStas weight factor One of the fundamental concepts to understand when dealing with McStas is the so-called weight factors. In the real world a neutron (or in this case a helium atom) is either present or lost, but, since only a fraction of the initial particles make it to the place where they are detected, McStas assigns each ray a weight factor. If for example the reflectivity of a certain component is 10 % and only the reflected rays are considered later in the simulation, the weight of every ray will be multiplied by 0.10 when passing the component and then reflected. This approach works around the problem of making a realistic simulation where only one in ten neutrons are reflected in the component. Another way of expressing this, is by saying, if the real life probability of an outcome ϵ of an interaction between a ray and a component is P ϵ, and this particular outcome is the only one of interest later in the simulation, the Monte Carlo probability of this outcome is set to one, f ϵ = 1, but the weight of the ray is multiplied by a weight multiplier π ϵ, such that equation π ϵ f ϵ = P ϵ (4) is always met, while keeping in mind P ϵ = ϵ ϵ f ϵ = 1. Equation (4) is called the fundamental multiplier law[13], and programmers should always bear this in mind when coding components for a McStas instrument. The weight factor of a ray at any point is given by w = w 0 n i=1 π i, where w 0 is the initial weight of the ray and this is multiplied by all weight multipliers of the previous components. McStas uses these weigh
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