Simulation of stray radiation and demagnetization of permanent magnets in EXFEL undulators

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DEPARTMENT OF PHYSICS Master’s Thesis

Simulation of stray radiation and demagnetization in the EXFEL Undulators

Author: Supervisors:

Sadia Fazil Fredrik Hellberg Anders Hedqvist

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Abstract

The European X-ray Free Electron Laser (EXFEL) generates ultrashort X-ray flashes that are used by researchers for the analysis of atomic and molecular structures. The facility consists of various components, amongst them one of the main components is undulator system that consists of several undulator cells containing permanent magnets. The permanent magnets are getting demagnetized due to the high dosage absorption of radiation, and it is believed that if the dosage remains sufficiently high it will degrade the magnetic properties of the permanent magnet.

The goal of this research is to highlight this problem by studying different sources of radiation in the undulator systems that are creating the stray radiation. To find out the reasons for stray radiation many experiments and simulations were performed and it is believed that the stray radiation is created because of the misalignment of the electron beam and beam halo particles hitting the beam pipe. In this study Monte Carlo simulations were performed with Geant4 in order to compare the measured results with the simulations to investigate where the beam hit the vacuum pipe at several positions in XFEL undulators and also investigate the beam properties of photons.

Two different methods were used to investigate to distributions of absorbed dosage. In the first method several points were selected manually with particular angle and position for the beam to collide with the vacuum pipe in order to reproduce the measured dosage absorption pattern taken from the XFEL.

The second method was done using least square fitting technique by taking surface energy distribution from several different cells. Those two methods were used in simulations for many different cells and some interesting results were observed from both methods. It has also been observed from simulations that measurement data from different consecutive cells has shown a pattern of dosage which indicates beam misalignment near the quadrupole magnet. The results also indicate that in the last cell of undulator radiation were originated from photons that are also contributing in the actual dosage of XFEL undulator.

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Table of content

1 Introduction ... 3

2 Background ... 5

2.1 European XFEL Facility ... 5

2.2 Undulators in European XFEL ... 6

2.3 Demagnetization of Permanent Magnets ... 7

2.4 RADFET sensors and Gafchromic Films ... 7

3 Theory ... 9

3.1 Radiation Matter Interaction ... 9

3.1.1 Energy transfer of Electrons in matter ... 9

3.1.2 Energy Transfer of Photons in matter ... 11

4 Simulations and Methods ... 14

4.1 Monte Carlo Simulations ... 14

4.2 Geant4 code ... 14

4.3 Methods ... 15

4.3.1 Input parameters in simulation environment ... 15

4.3.2 Geometry Setup ... 15

4.3.3 Beam Input and calculations ... 17

4.3.4 Method 1... 19

4.3.5 Method 2... 20

5 Results and Discussion ... 22

5.1 Method 1 ... 22

5.1.1 Photons Absorption in Cell 31 ... 26

5.2 Method 2 ... 29

6 Conclusions ... 40

7 References ... 43

8 Appendix ... 45

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1 Introduction

From the beginning of 21^st century the European free electron laser (FEL) facility has opened completely new research opportunities for scientists. FELs are used to generate ultra-short, high- intensity x-rays flashes that can unravel the mysteries of the chemical reactions on atomic level.

Scientists can now use these flashes to map the atomic details of viruses, to interpret the molecular composition of cells, take 3-D images of nanoworld and can even study the processes that happen deep inside the planets [1].

The European X-ray Free Electron Laser (EU-XFEL) is an X-ray research laser facility in Hamburg, Germany and can be seen in figure 1 [2]. FELs generate high intensity electromagnetic radiation by accelerating bunches of electrons to relativistic speed and direct them through special magnetic structures (undulators) to produce extremely brilliant and short pulses synchrotron radiation. The European XFEL is world’s largest X-ray laser that generates ultrashort X-ray flashes 27,000 times per second with a brilliance that is billion times higher than that of best conventional X-ray radiation sources [1].

The facility consists of injectors, linac, beam distribution system, undulators, photon beam lines and instruments in experimental stations. In the injector, electron bunches are extracted from a cathode by laser beam. The electrons are then directed towards linear accelerator with energy of 120 MeV. The electrons can be accelerated up to energies of 17.5 GeV. At the end of linear accelerator individual electron bunches are channeled down to one or two electron beam lines by beam distribution system.

The electron beam then passes through the undulators, which consist of periodic structure of dipole magnets with alternating polarity. The magnets produce a magnetic field which forces the beam to move in a zig-zag pattern, and with every turn electrons emit X-rays. After going through the

Figure 1: Part of the Undulator system at the XFEL facility. (Image taken from XFEL.eu)

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undulators, electrons in the beam are departed towards the beam dump. The photons generated by undulators are then transported towards experimental stations for research purposes.

During EXFEL operations the permanent magnetic material in undulators are exposed to stray radiation. The permanent magnets can lose their magnetic properties if the radiation absorbed by the magnets is sufficiently high and this will affect the performance of the machine. Thus stray radiation should be reduced in order to avoid the demagnetization of the permanent magnets. To measure the spatial intensity distribution of radiation, Gafchromic films are used inside the undulator segments.

To investigate the electron beam properties that are creating stray radiation, for simulation purposes a Monte Carlo code has been generated using Geant4. The simulated results were then compared with the measurement data taken from the XFEL undulators. Photon beam properties were also investigated to get better understanding of radiation absorbed in the undulators.

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2 Background

This section contains a short description of European XFEL as well as general discussion about radiation in undulator section and demagnetization of permanent magnets.

2.1 European XFEL Facility

The construction of European Free Electron Laser (EXFEL) was started in 2005 and user operations started in September 2017. It is a fourth generation light source in Hamburg, Germany. The European XFEL consists of a superconducting linear accelerator that is 1.7 km long and can deliver 17.5 GeV electron beam with an average beam power of 600 kW. The superconducting technology was adopted because it allows the production of x-rays up to 27000 pulse/s, in comparison with the typical normal conduction Linac FELs that produces approximately 100 pulse/s [1]. To generate X-ray flashes bunches of electrons are accelerated to high energies at nearly the speed of light. Those electrons are accelerated in resonators in which the oscillating microwave transfers its energy to the electrons. The resonators are superconducting and made of the metal niobium; they lose their electrical resistance when cooled to a temperature -271 degree Celsius. When electrical current flows through those resonators nearly the entire electrical power is transferred to particles. The resonators also deliver a very fine and even electron beam.

At the end of Linac, electron bunches can be focused in either of two electron beamlines, and pass through long undulators. The undulators consist of more than a few undulator cells. Each cell contains two undulator segments and each segment comprises of permanent magnets and the vacuum pipe passes in between these two segments through which the electron beam travels. In between two undulator segments a quadrupole magnet and phase shifter is placed to steer and focus the beam. A schematic view of undulator system can be seen in figure 2.

Undulator Cell Permanent Magnet

Undulator Segment

Quadrupole Magnet Phase Shifter

Vacuum Pipe

Figure 2: Schematic view of the undulator system.

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In the undulators, the electron bunches produce photon pulses by the Self Amplification Spontaneous Emission (SASE) process [1]. The first beamline comprises a hard X-ray undulator SASE1 that generates photons that are tunable between 3 - 25 keV (at 17.5 GeV) and a soft X-ray undulator SASE3 that generate photons that are tunable between 0.26 - 3 keV range (at 17.5 GeV). The second beamline contains a second hard X-ray undulator SASE2, identical to SASE1.

The hard X-ray undulators consist of 35 segments and the soft X-ray undulator SASE3 consists of 21 segments and each segment is 5m long. SASE1 and SASE2 are 200 m long, very similar in their construction and produce extremely short-wavelength X-ray light. SASE3 is behind SASE1 and it is 120 m long, a little shorter then SASE1 and produces longer wavelength X-ray light [13].

In the initial phase of the facility, each of the three installed SASE undulators works with six experimental stations. It might be expanded to more undulators and experimental stations. The X-ray flashes of the European XFEL enable a large variety of very different experiments at several different experiment stations.

After leaving the electron bunches at the end of the undulators, the photons are still very far away from the experimental stations. To cover this distance they have to pass through tunnels with a diameter 4.5 m and depth 6-13 m underground, which lead the photons to the experimental hall.

There are six scientific instruments in the experiment hall of the XFEL. All those instruments are optimized for different purposes and they have immensely different uses. Every experiment requires light with special properties, such that experiments are permanently assigned to different beamlines [13].

In the European XFEL basic setup is similar for different experiments. X-ray flashes can be extended, focused or filtered by using optical materials such as mirrors, gratings, slits, or crystals that depends on the experimental requirements. The samples are then given to the experiment station, where they interact with the X-ray flashes and then measured the interactions using detectors [1]. That data is then treated for analysis.

The facility has successfully started its experiments from September 2017. Using the X-ray flashes of European XFEL, scientists will be able to read the atomic details of viruses, to make out the molecular composition of cells and study those processes that are occurring deep inside planets [1].

The purpose of the facility is to generate extremely brilliant ultra-short (<100 fs) pulses of spatially coherent x-rays with wavelengths 0.05 - 5 nm [11], and to exploit them for revolutionary scientific experiments in a variety of disciplines including physics, chemistry, materials science and biology.

2.2 Undulators in European XFEL

Undulators at XFEL consist of permanent magnets that are placed in an alternating pattern. The magnets are made from the alloy of Neodymium, iron and boron to form Nd2Fe14B crystalline structure which is particularly the strongest magnet [8]. When the accelerated electrons enter in an undulator, the alternating field forces the electrons to move in sinus curve. With every turn, the electrons wiggle and emit light characteristic of the undulator strength but within a certain energy bandwidth. The emitted photons travel somewhat faster than electrons and with each undulator period they interact with them. The electrons gain and lose energy depending on the phase with each other i.e.

the faster electron catch up to the slower electrons [12]. Due to this the electron bunch density is periodically adjusted by the radiation resulting in microbunching. For microbunching to occur the

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electron beam intensity should be high enough. At XFEL one can observe microbunching at the level of intensity gain in excess of 10⁷ – 10⁸ [13]. When microbunching occur almost all the electrons radiates in same phase and the emitted beam is quasi-coherent. This is called SASE process, in which laser like beam is created from high energy electrons.

2.3 Demagnetization of Permanent Magnets

The undulators at the facility consist of several permanent magnetic materials and the magnets are supposed to produce a stable magnetic field that gives the electrons their desired slalom path.

However, the magnets can lose their magnetic strength and become weaker when they are exposed to radiation. Different types of radiation e.g. neutron, electron, gamma rays may alter the properties of permanent magnets causing demagnetization [4].

Demagnetization of permanent magnet material induced by radiation is a great concern in the manufacturing of insertion devices (undulators). The number of high energy electrons scattered from the electron beam absorbed in the permanent magnets results in radiation dose and demagnetization of the permanent magnets. With increasing time the radiation that is absorbed by the permanent magnets degrades the performance of the insertion device [9]. The magnetic field will be changed if the permanent magnet material becomes demagnetized because of radiation. If the magnetic field generated by the undulators varies a lot from the required strength then XFEL will not be able to function properly as expected, then that magnet will need to be replaced. The process of replacing the magnet is very expensive and during this time the machine would not be operational.

To see which type of radiation have been involved in demagnetization of the permanent magnets various simulations have been performed. From these simulations it is found that the radiation is mostly formed by electrons, photons and neutrons [3]. A theoretical reason for this was that a few halo particles that were away from the electron beam that hit the vacuum wall, results in shower of radiation that can be absorbed by the permanent magnets. The halo is very large at the point between the undulators near the quadrupole magnets. There magnets are placed between the cell to focus the beam vertically and horizontally, and focusing is done once in one direction in between two cells.

2.4 RADFET sensors and Gafchromic Films

The radiation damage on magnetic structures of XFEL is a great concern during the operations. The RADFET sensors and Gafchromic films are used to measure the dose distribution of ionizing radiation [26]. In the facility the RADFET sensors were designed for radiation dose measurements for all SASE lines. RADFET systems have a chip containing metal-oxide semiconductor field-effect transistors (MOSFETs). When ionizing radiation go through the metal oxide layer, electron-hole pairs will be generated there, which lead to a permanent change of the layer with positive charges trapped inside. In result the threshold voltage of the transistor is being modified. The voltage shift of these transistors varies with the amount of radiation which can then provide a reliable measurement of absorbed radiation [28]. To measure the amount of radiation absorbed dosage, threshold voltage shift has to be monitored.

In contrast with the measurements performed by the RADFET, Gafchromic films were placed in front of the RADFET dosimeter and on top of the magnets inside some of the chosen cells to get better picture of the dose absorption cell to cell. The size of each film is 8x10 inches and can be cut to any desired size. The property of the Gafchromic film is that the amount of dosage absorbed changes the color of the film. When radiation exposed on the films, depending upon their composition, the

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radiochromic films become polymerized and turn red or blue [27]. During this process where the molecules are affected on the film, it can be dark in color. With the increasing irradiation the colored chain increases, and the colored polymer chain is a result of polymerization [27]. And the equipment which is required to measure the amount of dosage is scanner and software.

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3 Theory

The permanent magnets are getting demagnetized in undulators because of the showers of ionizing radiation. It is assumed that the ionizing radiation is mainly caused by the misalignment of the beam and electrons in the beam halo hitting the beam pipe. To investigate these radiation Gafchromic films were used to find the deposited energy caused by stray radiation in the magnetic structures. These two dimensional Gafchromic film dosimeters were placed at different locations in the XFEL undulator to get multiple samples of radiation absorption.

The purpose of this work is to investigate the distribution of absorbed dosage both spatially in the undulator geometry as well as with respect to the type of the particles and energy range of different particles in matter. For that purpose electron beam properties were investigated, also the measured data from XFEL (using Gafchromic films) was compared with the simulated data to see where the beam strikes at different positions. To get better understanding of radiation contribution in actual dosage absorption, photon beam properties were also investigated at different energies.

In the undulator system as the electron beam moves their Gaussian spread will increase and there is more chance of electrons to move closer to vacuum pipe. In order to overcome this problem there are quadrupole magnets placed between each undulator cell. The magnet applies a force on electrons, horizontally or vertically to keep the beam close to the center of the vacuum pipe [16].The particles from the beam which stray far off from the center are called beam halo particles.

The beam halo particles that hit the vacuum vessel wall can interact with both the nucleus of an atom and with the electrons bound to the atoms in the material. A code has been developed using the Geant4 software to find out how the halo particles interact with the vacuum wall and which kind of radiation they create.

3.1 Radiation Matter Interaction

When charge particle interact with atomic electrons and nuclei of the matter, they can change their direction of motion and lose energy. In result the atoms of the material become excited or ionized. A charge particles when pass through the material, it interacts elastically with almost all the atoms along its path, losing their energy and changing direction. An electron interacts with all the atoms in its surrounding transferring its kinetic energy gradually to the electrons of the collided atoms. Therefore an electron loses its energy with a friction like process called Continuous slowing down approximation [20].

Photons can interact with the matter through Photoelectric effect, Compton scattering, Pair Production, Rayleigh scattering and photonuclear interaction depending on the energy of the photon and atomic number of the material. This has been further explained below in detail.

3.1.1 Energy transfer of Electrons in matter

Electrons can interact with matter both by elastic scattering and inelastic scattering. Elastic scattering occurs when incident electron collides and changes its direction without kinetic energy loss. And inelastic scattering occurs when after interaction the incident particle’s energy is converted to excitation energy. Electrons also interact with the nucleus of atom both elastically and inelastically.

The inelastic scattering of particles results in Bremsstrahlung x-rays emitted from the electron.

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Electrons of the beam also interact with electrons of the matter and can transfer its kinetic energy to these electrons [24].

Electrons interact with matter through several mechanisms. Electron loses its energy through different processes including ionization, Moller scattering (electron-electron scattering), Bremsstrahlung and pair production. Electron scattering is considered as ionization when energy loss per collision is below 0.25 MeV, above this energy Moller scattering dominates and at high energies (few tens of MeV) Bremsstrahlung dominates.

One of the basic interactions of electrons with matter is Bremsstrahlung process which can convert a large fraction of the electron beam energy into X-ray energy. The change in kinetic energy of electrons results in X-rays photons being produce which have equal energy to the amount of kinetic energy lost.

Those x-rays emitted into various angles and energies.

Bremsstrahlung radiation can also produce large number of fast moving neutrons through photodisintegration process. In photodisintegration process the atomic nucleus absorbs a high energy gamma ray, which gets excited the neutron and immediately decays by emitting subatomic particle.

Those fast moving neutrons can pass through the material and can also be absorbed in permanent magnets.

The main source for the production of neutrons from electron beam is Bremsstrahlung photons produced by the electron beam. Electron can directly interact with nucleus and produce a neutron but the probability of that is 100 times less than gamma ray interaction to produce neutron [25]. It is necessary for a neutron to be produced that the photon absorbed by the nucleus must have energy greater than the binding energy of neutron to nucleus. In most of the cases the neutron binding energies lies between 7 and 15 MeV but there are small exceptional cases in which binding energy is less than 7 MeV and greater than 15 MeV as well.

The emission of neutrons from nucleus results in the production of an isotope, which is unstable and ready for positron emission. For each positron emitted two 0.51 MeV of photons will be produced in annihilation process.

The magnitude of radioactivity produced as a result of neutron absorption depends on the energy of the neutrons in a process called neutron capture. Neutron capture is a type of nuclear reaction in which a target nucleus absorbs a neutron that emits a discrete quantity of electromagnetic energy (gamma-ray photon). Some of the photons then helped another process to happen called pair production. For pair production to occur, the electromagnetic energy called a photon must be at least equivalent to the 0.51 MeV of energy. To produce an electron-positron pair the energy of photon must be equal to 1.02 MeV.

When this process occurs, photon energy in excess of this amount is converted into electron-positron motion. The positrons then help in the emission of gamma rays from the nucleus.

Therefore, when electron beam interact with matter the radiation are mostly formed from electrons, Bremsstrahlung photons and fast moving neutrons. Below figure 3 represents the radiation matter interaction (Reproduced from “Shielding for high energy electron accelerator installation” from the introduction section of the book [25]). When electron beam interacts with matter Bremsstrahlung photons are produced. These photons through photodisintegration process produce fast moving neutrons and during that process each positron emits two photons through annihilation. The photons through pair production create electron positron pair near a nucleus. Pair production is a creation of subatomic particle and its anti-particle and it is a high energy phenomena.

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11 3.1.2 Energy Transfer of Photons in matter

When x-rays or gamma rays are directed into a material, some of the photons interact with the particles of the matter and their energy will be absorbed and scattered. This absorption and scattering is called attenuation. Other photons will pass through the material without interacting with any of the object’s particles. The number of photons transmitting through the material depends on the thickness, density, energy of the photon and atomic number of the material.

When photons interact with matter there are five different types of interactions, these are Photoelectric effect, Compton scattering, Pair production, Rayleigh scattering and photonuclear interaction. The significance of interaction depends on both the photon energy and the atomic number of the absorbing material.

The Photoelectric effect, Compton scattering and Pair Production are important, for example the Photoelectric effect is considered more important at lower photon energies ranging up to 0.5 MeV.

The Compton scattering takes over at medium energies up to 5 MeV and Pair production at higher energies [20]. Also there are some points where two interactions are equally significant depending on the atomic number of the absorbing material. The Rayleigh scattering is elastic, which only changes the direction of photon with no energy loss. On the other hand the photonuclear interactions are significant for high energy photons, where they can create radiation protection problems.

Intensity of photon can tell us about the quantity of incoming photons that can pass through the material. It can be explained as;

IPh = Ioe-µt

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Where µ is the linear attenuation coefficient which describes how easily a volume of material can be penetrated by a beam of particles and t is the thickness of the material. The linear attenuation

Figure 3: Interaction of radiation with matter. “Shielding for high energy electron accelerator installation”. U.S Department of Commerce. A.V Astin Director Secretary National Bureau of Standards. Reproduced with permission.

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coefficient of a material can be calculated from the mass attenuation coefficient and density of the material. Mass attenuation coefficient describes how easily the particles penetrate through the material and its unit is cm²/g. To convert the mass attenuation coefficient of a material into linear attenuation coefficient, we can multiple it by the density of the material. The table presenting the mass attenuation coefficients for many different materials at different energies can be found in reference [19].

Equation 1 is used to estimate the transmission of photon through the aluminum wall, the attenuation coefficient of the material needs to be calculated. The vacuum pipe through which the electron beam is passing is made of aluminum which has a density of 2.7 g/cm³ and attenuation coefficient of the aluminum for 10 keV photon energy is 26.21 cm²/g [19]. The thickness of the material is 0.55cm through which the particles pass. Therefore, after calculating with mass attenuation coefficient and density we get µ as 70.76 cm^-1. In the EXFEL the energy of photons for SASE1 and SASE2 is between 3-25 keV. To calculate the transmission of photons through matter with fundamental frequency the photon’s energy is taken as 10 keV [1]. For 10 keV energy the transmission of photons through matter is approximately 1.25x10^-17. As only odd multiple harmonics contributes in a photon’s energy that is why only 1st, 3rd and 5th harmonics are considered. For the first harmonics the energy of photon is approximately 30keV and similarly transmission of photon is calculated to be approximately 0.19 and for third harmonics the calculated transmission is approximately 0.37.

Low energy photons mostly interact with matter through Photoelectric effect. It has been observed that when photons with fundamental harmonic frequency hit the Aluminum wall (atomic number 13) the photoelectric effect dominates more than Compton scattering. Below in figure 4 the relative importance of these Gamma ray interactions with matter is shown [30]. During photoelectric interaction the photon is totally absorbed by the bounded electrons in matter, so photons with fundamental frequency are difficult to pass through the matter.

Figure 4: Relative importance of three major types of gamma ray interactions.

“Introduction to radiological physics and radiation dosimetry”. Frank Herbert Attix.

Page(125). Copyright Wiley-VCH GmbH. Reproduced with permission

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With photon’s energy as 30 keV both Photoelectric and Compton scattering are important. In Compton scattering the incoming photon interacts with the stationary and unbound electrons of the material. After interaction the electron departs at an angle with some energy and momentum, and photon scatter at an angle with lower quantum energy and momentum. As these photons are not absorbed by the bounded electrons so few of them will pass through the material.

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4 Simulations and Methods

4.1 Monte Carlo Simulations

The Monte Carlo (MC) method is a statistical simulation method based on random sampling for modeling uncertainty of the system. It’s a stochastic technique that is based on the use of random numbers and probability statistic to determine problems. By using suitable numerical computations different simulations are run for generating paths to achieve desired results. When a complex system or a model having uncertain parameter needs to be analyzed, MC method is the most plausible method that has been used to solve those kinds of problems [22].

A wide variety of MC models have been used in studies aimed at describing various aspects of the interaction of electron beams with solid specimens including backscattering, x-ray emission and bremsstrahlung. One of the ways to use Monte Carlo method is simulating interactions of particles in matter. In the simulations it is possible to create an environment similar to facility (e.g. like in XFEL) for the particle with different properties to transport through any medium and geometry. For this in the simulations one can decide about the interaction type, beam particle energy, number of the particles etc. Through this method user can determine e.g. output of the deposited energy and energy distribution of particles passing through some media.

In this research Monte Carlo code has been used in connection with Geant4, to simulate the interaction of different particles in materials. The code was already generated by Anders Hedqvist, in this research it is a continuation of the previous work [3]. The beam interact with the vacuum pipe produced different particles, those simulated particles energies between 1 keV to 1000 MeV have been used. The simulated particle filter used to pass radiation from particles like neutron, proton, gamma, electron, positron and photon. A simulated electron beam has been used with energy as 17.5 GeV. For photons the energy were chosen as 0.01 MeV same as according to the fundamental frequency of photons in XFEL facility.

4.2 Geant4 code

Geant4 (Geometry and tracking) is a platform used for the simulation of the passage of particles from matter using Monte Carlo methods. The Geant4 toolkit is widely used in various radiation physics researches, from high energy physics to accelerator physics. The toolkit allows the user to create detector geometry with a given set of physics acting inside it. The toolkit includes a wide range of functionality including the physics models of geometry and tracking.

In Geant4 for describing the software components, all aspects of simulations has been included for example the geometry of the system, kinds of the material involved, the tracking of particles through material, particles of interest, and all the physics processes involved in particles interaction [23].

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4.3 Methods

In this research project one of the objectives was to do simulation that in combination with measurements to measure the integrated dosage in each undulator segment. And find the position for the beam that contributes to demagnetize the permanent magnets in XFEL undulator. The simulations were made to investigate the level and spatial distribution of stray radiation (dosage) absorbed in the machine. For this purpose two different kind of simulation methods were made (those will be explained below in later subsections) to compare them with measured dosage from different cells in XFEL.

To investigate the spatial distribution of the absorbed dosage in the XFEL undulator a similar geometry setup in Geant4 has been prepared. Below in the table are the input parameters and geometry setup for the simulations.

4.3.1 Input parameters in simulation environment

Input parameters Settings for Simulation

Number of Particles 1000

Electrons Gun Energy 17.5 GeV

Gamma particles Gun Energy 0.01-0.03 MeV

4.3.2 Geometry Setup

To do the simulations a similar geometry like XFEL undulators was created in Geant4 code and it is an approximation of actual undulators. As this work is the continuation of the previous one therefore this setup is also made to replicate the work which has been already done [3].The undulator consists of 5m long module, separated by 1.1 m long intersection. At the intersection quadrupole magnet and phase shifter have been placed. The undulator segment consists of an upper and a lower aluminum block. Each block of the segment has series of permanent magnet material (Nd2Fe14B). The permanent magnet structure includes a series of one cobalt iron block and other NdFeB. In Figure 5 a rough sketch of the undulator structure is shown. The undulator beam pipe has rectangular shaped outside (70x12 mm wide and high) and a circular interior (9mm diameter). There is 4mm distance between beam pipe and permanent magnetic structure on each side of the pipe. The actual beam pipe has an elliptical cross-section and it was difficult to consider all the directions for the beam to pass. Therefore it was observed that the shortest dimension is in the vertical direction and which is closest to the permanent magnet so vertical direction was considered in the simulations for the beam particles to pass.

The undulator beam pipe is 250 m long. In the simulation it has been designed in such way so that the electron beam can start from any point in the undulator beam pipe i.e. by changing the starting position of the electron beam different positions can be investigated for the undulator segment.

It was difficult to make the actual geometry of the quadrupole magnet in simulations. Therefore it was convenient to use a rectangular shape quadrupole magnet for approximation because most importantly

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the setup was made to see the behavior of simulated particles. In the horizontal direction in between the undulators about 1 m long intersection contains a quadrupole magnet and a phase shifter. The dimension of the quadrupole magnet is 200x200 mm block, 100 mm long with rectangular central opening, where 20x78 mm opening for the vacuum pipe leaving 4 mm on either side. The phase shifter consists of two blocks in the beam direction those are 100x240x200 mm wide high and long with rectangular central opening. The Gafchromic films are placed on top of the magnets inside some of the cell. The films made it possible to see the behavior of radiation by changing distance. This is shown below in Figure 5. This simplified geometry has been used in all the simulations.

Figure 5: Undulator geometry setup for simulation.

1000 mm 5000 mm

650 mm

20 mm

70 mm

12 mm 9 mm 200 mm

200 mm 78 mm

20 mm

100 mm

20 mm

240 mm

100 mm

200 mm Quadrupole Magnet Phase Shifter

Undulator # 1 Undulator # 2

Vacuum pipe

Quadrupole magnet

Phase Shifter y

z x

Center axis Point (0,0,0)

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17 4.3.3 Beam Input and calculations

In order to simulate beam halo electrons intersecting the vacuum vessel wall, all simulations were made by creating Gaussian electron beam, directed from the center of the vacuum pipe before the first undulator segment to hit the wall at certain point in the vacuum chamber. Electrons of 17.5 GeV hit the wall at quadrupole magnet which is positioned before the first undulator segment under angle of 200 µrad. This was simulated several times just by changing the starting point of the beam position, which gives beam a different interaction point with the wall.

The distance from the beam to vacuum pipe is 4.5 mm and adjusting the position where the beam strike with the angle taken as 200 µrad gives different interaction points. This was done with simulations to study where the energy deposited at different points.

4.3.3.1 Trigonometric Relation:

Position Calculation

From the starting position in undulator: (X, Y, Z) = (0, 0, -6500) From Figure

Z = L + Zo

For the case Zo = -6500 Z = 100 mm L = 6600 mm

Zo Z

Y

h Δy

µrad L

Figure 6 : Geometric relationships between angle and distances. Zo is the starting position of the beam, Y is the position where the beam strike, L is the horizontal distance between the starting position and the point where the beam strike, h is the distance between Zo and Y.

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18 Δy = 4.5 mm (Radius of the vacuum pipe)

Tan (0.0002) = Δy/6600;

Δy = 1.32;

Angle Calculation

For angle calculation starting position of the beam have to be changed Z = 100 mm;

Zo = -6400;

L = 6500 mm

Tan(α) = 4.5 / 6500;

In the first simulation the starting point in the axis was -6500 from the center point and the first point which was taken for the beam to hit was -100 before the center point of the axis. The reason this position was chosen as the first point was that before this point there was no significant radiation absorption found in the undulator cell. Hence the distance 6400mm was taken as the first point and then with the interval of 200 mm to 300 mm distance the beam were hit on the whole undulator cell.

In figure 5 the point between two undulator cells were taken as center axis point (Cartesian coordinate). At center axis point all the axis (x, y, z) are (0, 0, 0). Starting from that point the beam hit the wall at different positions with the distance of 200 mm or 300 mm. The above mentioned trigonometric relations are used to calculate the positions of the beam where the beam strikes in the simulation. At the end of the report Table 1 has shown all the interaction points with different starting positions keeping the angle 200 µrad. The results were calculated as deposited energy over two planes, which is relatable to actual dosage (J/Kg) in the matter. All showed results were calculated along the length of the pole magnet as z-y plane, with x thickness of 10 cm and another at the front cross section of the pole magnet as x-y plane with z thickness of 2 cm as demonstrated by the Figure 6.

yo = 4.5 – Δy = 3.18 mm

α = 0.00069 rad

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19 4.3.3.2 Illustrating deposited energy

The deposited energy was plotted using Matlab, from each simulation separately and as a sum of the deposited energy from the simulation of all interaction points.

To see where the energy has been deposited in the undulator, a color graph was plotted to see the amount of energy deposited along the undulator segment. This has been done to observe the contribution of absorbed dosage at different positions. This way it’s easy to compare the measured dosage with the simulated dosage and generate a result about if the beam strike at a certain position has how much contribution in actual dosage absorption.

4.3.3.3 Radiation Origin

To study how the generated radiation and the dosage were delivered to the permanent magnet at XFEL, a similar model like undulators in XFEL was created in Geant4. An electron beam was directed towards different interaction points in the vacuum pipe. From the resulting distributions of deposited energy the surface distribution were obtained by calculating the sum of the deposited energy. The surface distribution was fitted to the measurement data of previous studies using the following methods. The simulation results were then compared with Gafchromic films results.

4.3.4 Method 1

Changing Position

In the first method one of the approaches was the interaction of beam with vacuum pipe along the z axis of the magnet segment over a certain point in simulations which had the largest contribution to the deposited energy. By looking at the measured plot from XFEL (using Gafchromic films) it was visually considered that beam hit the vacuum wall at a certain position. Using the trigonometric relation the position and distance has been calculated. That certain position was then used in simulations as indication of the location of main contribution in the distribution of deposited energy.

Changing Angle

Another approach which has been considered was by changing the angle of the electron beam in the vacuum pipe. Looking at the measured graphs one can see that there are different patterns of radiation absorption peaks. Using one specific angle would give same kind of peak at different positions by changing the position of the electron beam. By changing the angle it was assumed that it will have different effect on beam distribution energy. When the beam with sharp angle hit the vacuum wall then in simulations it gives a narrow peak, and that has been used below.

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20 4.3.5 Method 2

The second method was to make a least square fit, in Matlab using the measured surface energy distribution from many different points as dependent variables and the simulated data as independent variables. The first method was like a hit and trial method, it was visual observation of the measured dosage. In this method it was assumed that by looking at the measured graphs where there is high absorption of dosage in the simulations visually the beam hit at those particular points only. On the other hand the second method was more scientific in a way that various striking positions were selected with a certain distance in between them. The beam then hit at those selected points in the simulation and fitted plot was made from them.

In simulations first an environment was created in Geant4. Then an output file was generated from the simulations, which contain dosage absorption at every point in the magnet along with xyz axis. That output file was then used in Matlab to generate a plot from the data. The dimension of the undulator block was 500x5x7 (z, y, x) in the generated output. In the simulations a cross-section of the block was taken by fixing x=3 and y=1 along the z-axis shown below in the figure.

Those points are chosen because there are more chances to find the best dosage absorption points. As the x,y dimensions of the undulator block was 5x7, so x=3 (in the simulation x start from 0 to 6) was selected to be in the middle and y=1 (in the simulation y start from 0 to 4) was selected to be near the bottom of the undulator where most of the dosage should be absorbed. Also the gafchromic films used to obtain the measured values from the XFEL were also placed on the bottom of the undulator.

4.3.5.1 Least Square Fitting Technique

Least square method is a statistical procedure to find the best fit for a set of data points by minimizing the sum of the points from the curve. In the second method least square fitting method was used to find the values of unknown. Here in this case as various points were considered at once where the beam hit. Linear least square fit equation is:

x x y

z

x =0 x =6 x =3 y =0

y =4

y =1

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There are various beams that have been considered at the same time. So instead of linear least square fitting, multiple linear regression (MLR) has been used, which is a process of fitting model to the data.

MLR is used on several explanatory variables to predict outcome of response variable. It also helps to understand how much the dependent variable changes when independent variable has been changed.

This gives the coefficient according to regress equation.[17]

Negative values at MRF shows this model may not fit into measured data.

R² shows how accurate is the model depends how close it is to 1 like 0.9 – 1 accurate.

The drawback of this solution was that this method gives negative coefficients as well as positive ones. In this case negative coefficients do not suggest anything about the absorbed dosage in the undulators. As normally negative coefficients means that increasing the corresponding variable decreases the effect on the output, as increasing particle collisions will always increase the absorbed dosage the negative coefficients doesn’t make sense in this case. So in order to fix this problem another method was taken into account that only provides positive correlations.

4.3.5.2 Non-negative least squares (nonneglsq)

To solve the problem of negative coefficients the non-negative least squares has been introduced here.

It is a type of constrained least squares problem where the coefficients are not allowed to become negative. That is, given a matrix C and a (column) vector of response variables d, the goal is to find the correlation as shown in equation below. The following equation is used in Matlab to find the non- neg least square fit. More detail regarding the non-negative least squares fitting method can be found in [18].

x = lsqnonneg(C, d)

Here x ≥ 0 means that each component of the vector x should be non-negative.

Using the input arguments the real valued matrix C of dimension (m x n) as the normalized simulated values which are set of 20 simulated dosage values. And the other input argument which is a vector d of dimension m as the measured dosage from the actual measurements. It returns the vector which minimizes norm(C.x - d) subject to x ≥ 0 and additionally some other output arguments, e.g. lambda and exitflag that satisfies the answer to be true [14].

The output argument x describes the contribution of each position in the actual measurement (as in measured graph). In addition with x another output argument was lambda which in correspondence with x describes their relation with coefficients. If the value of lambda in correspondence with x is approximately equal to 0 which means, that certain position has a contribution in actual measurement.

And the others with values less than 0 have no such contributions.

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5 Results and Discussion 5.1 Method 1

In order to determine the amount of dosage, in the beginning only few undulator cells were chosen to place the Gafchromic films on them. The chosen cells were from the beginning, middle and end of the undulator section. Those cells were cell 4, 12 and 31. The data from different films were analyzed in the report [5] [32]. Looking at the measured dosage plots from the Gafchromic films it was decided that in the simulation beam should hit the vacuum pipe manually at selected points. Below in figure 7 the results from Gafchromic films have been shown with measured dosage.

Figure 7: Measurement of absorbed dosage from cell 4, 12 and 31 at XFEL from previous studies [15].

(Retrieved: Cell 4: 07-12-2017, Cell 12: 14-03-2018, Cell 31: 21-11-2017 {inserted for about 3 months})

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By observing these plots it was considered that in the simulations the beam must hit the vacuum wall at more than one point with a certain angle. In cell 4 the dosage starts from 4.5 [Gy] and goes up till 5 [Gy] and then steadily goes down till the end of the cell. So it was decided that in the simulations the electron beam must hit the vacuum pipe in the beginning of the cell around the quadrupole magnet to have maximum dosage at the start of the cell. But at the same time if we look at cell 31 both of them look alike in a way that both cells have high dosage in the beginning and then then gradually decreases till the end of cell. The electron beam must hit the vacuum pipe somewhere around the quadrupole magnet in the beginning of cell. Comparing both of them one can see that the plot of cell 4 has broad spectrum of dosage whereas in cell 31 the beam strike at one position but with a very sharp angle.

Various simulations were performed using different beam striking points with different angles to generate a similar dosage pattern.

Starting with cell 4 after hitting the beam at different positions one suitable position was selected.

With an angle of 200 µrad the electron beam hit the vacuum wall at distance 1800 mm starting from the middle of two undulator cells (center axis point as shown in figure 5 ) striking the vacuum wall at the start of undulator. The generate dosage plot is shown below in Figure 8.

Figure 8: Measurement of the surface dosage in cell 4 using the simulation surface energy deposit.

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Similarly by analyzing dosage plot of cell 12 in Figure 7, it can be observed that the beam is hitting the wall at different positions. In the beginning of the cell the dosage is high which then drops steadily and then gradually increases giving some peaks till the end of undulator. So in the simulations three points were selected manually to get the similar plot like measurements. First the beam hit the vacuum wall near the quadrupole magnet at distance 100 mm after the center with angle of 200µrad and the second one was approximately in the middle of the undulator cell at 3300 mm with angle of 100µrad and then the beam hit at 3900 mm also with 100µrad angle.

The absorbed dosage pattern of cell 31 is little different from cell 4 and 12. It can be observed that in the beginning of the cell there was a sharp peak and then the radiation go down to minimum. When the beam strike at the beginning of the cell with angle 200 µrad the dosage peak pattern was broader as compared to the measured dosage is shown in Figure 10.

Figure 9: Measurement of the surface dosage in cell # 12 using the simulation surface energy deposit from different interaction point.

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Most of the absorbed dosage is concentrated in one region and the rest of the cell has low dosage.

Because the angle 200 µrad was resulting in a relatively broad dosage peak which doesn’t match with the measured absorbed pattern so it was decided to use a different angle that can result in a sharper peak so various simulations were performed using different positions and angles. The purpose of doing this is to find out where and how actually the beam strikes in vacuum pipe giving this sharp peak. To find out this with a very sharp angle electron beam was then hit at the vacuum wall by changing the starting position of the beam. In the simulation by changing the starting point the electron beam was then hit at an angle of 10 mrad to give a relatively sharp peak as shown in Figure 11.

Figure 10: Measurement of the surface dosage in cell # 31 using the simulation surface energy deposit.

Figure 11: Measurement of the surface dosage in cell # 31 using the simulation surface energy deposit.

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26 5.1.1 Photons Absorption in Cell 31

In Self Amplified Spontaneous Emission (SASE) process a group of electrons pass through an undulator with nearly the speed of light which results in photon emission. These emitted photons interact with electrons which are then influenced by the electric field of x-ray light. Fast electrons catch up with slower ones, depending on the phase with photons the electrons gain or lose their energy [29]. The electron bunch density is modulated by radiation called microbunching. The arranged electron beam amplifies only certain photon energies using kinetic energy until the system goes into saturation. This process continue till the last cell of undulator but it has been observed that in cell 31 when saturation point has reached photon field is still growing at the end of undulator cell. Photons field was still growing and there some photons could pass through the vacuum pipe and were absorbed by the permanent magnets resulting in dosage absorption and demagnetization.

To observe the behavior of photons in matter, a photon beam is used in the simulation with an angle of 200 µrad to hit the vacuum pipe near the quadrupole magnet at a distance of 7000 mm (from figure 5).

To see the absorbed dosage pattern in simulation, photon beam was directed at the vacuum pipe with fundamental frequency of 10 keV energy. It has been observed through simulations that with this energy there were no particles that can pass through the wall to produce radiation. Therefore, it was concluded (from section 3.1.2) that for particles with 10 keV energy it is very difficult for them to pass through the wall because most of the particles have been absorbed by the vacuum wall (Aluminum) due to Photoelectric effect.

The photon energy was then increased to 30 keV energy (Photon’s first harmonic energy is 30 keV).

Below in figure 12 one can see the absorbed dosage of photon beam with 30 keV energy and with angle of 200 µrad at distance 7000 mm. By looking at the absorbed dosage pattern it was observed that when photon beam hit the vacuum wall it gives the dosage in the form of very sharp and narrow peak.

Figure 12: Photons in Cell 31 with energy 30 keV

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It is believed that the radiation field is growing continuously through the undulator and it is maximum at the end of the undulator cell. Therefore to see if there is radiation as we go downstream the undulators, the amplitude of the radiation is increases as well. With the increasing radiation field the photons with different energies can pass through the vacuum pipe. To examine this if there are radiation from photons in the last undulator cells, one can observe the measured absorbed dosage of Radfet detector from XFEL. Below in the Figure 13 the measured absorbed dosage from Radfet detectors with and without lead shield from the beginning and last undulator cells [31].

The 4mm lead shield was chosen to remove x-rays. And one can observe in the figure 13 that in the beginning undulator there is not much difference of absorbed dosage with and without lead shield. But later in cell 31 the difference with and without shield is observable also the overall absorbed dosage is very low in comparison with cell 3. It is assumed that this could be because in the beginning undulators the electron beam is not well aligned and is observed that most of them are stray radiation.

But later in the last cells the beam is mostly aligned but the radiation field is increasing that there are photons that are contributing in the dosage absorption. Figure 14 shows that the downstream undulators mainly absorb radiation lower than 100-200 keV [31].

Here from the theory also it can be concluded that, in the last cell of XFEL where amplitude of the photon radiation has exceeded the saturation point, at that point around 99% of photons with fundamental frequency will transmit through the Aluminum wall with 1.25x10^-17 transmittance. Most of the photons in this case are absorbed by the matter due to photoelectric effect. These photons do not contribute much to the actual dosage absorption and the little dosage absorption observed is due to synchrotron radiation. Remaining approx. 1% of the photons from third harmonics will pass through the wall with the transmittance of 0.19. Around 0.03% of photons from fifth harmonics will pass through the wall with the transmittance of 0.37.

Figure 13: Measured absorbed dosage with and without lead shield from cell 3 and cell 31.

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The difference between intensity ratio of fundamental frequency and third harmonic frequency is approximately seventeen orders of magnitude. And the intensity ratio in fifth harmonics is two times higher than third harmonics. The transmission of photons increases in higher harmonics but the number of photons decreases. It has been observed that most of the photons that pass through the Aluminum wall and contribute in the dosage are from third harmonics with 1% of total number of photons with 0.19 intensity ratio. In higher harmonics the number of photons that can contribute in actual dosage is very low. For the European XFEL, simulations predict that relative contribution to the total power, around 99% of radiation with fundamental frequency are contributing to the total power, 1% from third and 0.03% from fifth harmonics [13].

Therefore it can be assumed that the photon field is still growing at the end of the undulator cell, at that point some photons have reached the vacuum pipe. Figure 12 shows the absorbed dosage at the beginning of undulator cell; however it can be assumed that the similar dosage absorption peak will be present throughout the whole undulator cell because the radiation field is increasing all the way till the end of the cell. This can result in a low but uniform radiation pattern throughout the cell. After comparing with the measured data from cell 31 in figure 7 it can be explained in a way that the uniform absorbed dosage from the tail of the spectra can be created due to photons absorption.

Figure 14: Absorbed radiation from downstream undulators with lead shield.

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5.2 Method 2

Another measurement was taken for six different cells to see if previously the measurement was taken have given same dosage pattern or it changes with time. The significance of taking these measurements was that the data were taken from several consecutive cells. The reason of choosing consecutive cells was to have a better understanding of dosage absorption pattern. The idea was to investigate the dosage absorption pattern at the end of one undulator cell and the pattern after the quadrupole magnet at the beginning of next cell. The quadrupole magnet focuses the beam at one direction and defocuses on other therefore cell 10, 11 and 12 were chosen to observe the magnets effect on the beam pattern. This may provide us a clear picture of the beam path pattern throughout the whole undulator segment. This can enable us to find out the position at which the beam is misaligned at its path and how the dosage is absorbed at different points at the permanent magnet. One of chosen cells was from the beginning segment of the undulator which was cell 4. Other consecutive cells were 10, 11 and 12. And the last dosage measurements were performed on two consecutive cells number 30 and 31. The dosage measurements plots from these cells are shown in Figure 15 and Figure 16.

Figure 15: Measurement of absorbed dosage from XFEL through Gafchromic films of cell 4, 10,11and 12. (Retrieved: Cell 4: 14-03-2018, Cell 10: 25-09-2018, Cell 11: 25-09-2018,

Cell 12: 21-06-2018)

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In the second method a library of output files were created by hitting the beam at various positions by fixing the angle at 200 µrad. The selected positions are mentioned in table 1 in Appendix at the end of the report. Figure 17 represent individual surface dosage spectra from selected interaction points and the corresponding points are mentioned in the table 1. Once the library was created it was then used for every measured case for each cell to find the best fit for simulated plots. The purpose of doing this process is to find out the approximate positions where high amount of dosage has been absorbed in undulators.

By solving the equation of least square, the coefficients were calculated from the equation as described in 4.3.5.2. The bar graph was then plotted from the coefficients which indicate the positions of maximum dosage absorption in the undulator cell. Using method 2 for the measurement of the dosage distribution along the surface of the undulator segments in cell 4, 10, 11, 12, 30 and 31 at EXFEL are discussed below.

Figure 16: Measurement of absorbed dosage from EXFEL through Gafchromic films of cell 30 and 31. (Retrieved: Cell 30: 20-12-2018, Cell 31: 14-03-2018)

Figure 17: Deposited energy deposition from six interaction points out of total fourteen used.

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Figure 18: Measurement of surface dosage in cell 4 with curve fitting using the simulation surface energy deposit from various interaction points.

Figure 19: Contribution of absorbed dosage in cell 4 using simulation surface energy deposit from different interaction points

Length of undulator (5000 mm)

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Figure 18 is a comparison of measured dosage and simulated data using least square fitting method.

Using the library of the beam interaction points with nonneglsq method provided positive coefficients, which will indicate the dosage absorption at various points in the undulator cells. Looking at Figure 10 one can observe that in the beginning of cell 4 the dosage starts from 4.5 Gy and goes up to 5 Gy and then decreases steadily till the end of the cell, where it increases a little again.

In simulations the beam interaction points in the vacuum pipe starts from near the quadrupole magnet.

It was observed that the interaction points at the beginning of cell 4 have more contribution in the absorbed dosage. The beam halo particles which left defocused from quadrupole magnet have more probability to interact with the vacuum pipe and create stray radiation.

Nonneglsq function in Matlab will provide many useful output parameters, one of them is the list of coefficients which represents the contribution of surface absorb dosage at different positions. To show the contribution of dosage absorption bar graphs were created.

Looking at the bar graphs in Figure 19 one can observe where the beam strikes at different points. The points where beam strikes give the dosage absorption points and their value represent the amount of dosage in the undulator. Most of them are from the beginning points of undulator cell, but few other points from the cell are also contributing in dosage absorption.

Below in the figures there are plots from cell 10, 11 and 12. Which are cells in a series and it would be interesting to see their behavior.

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Figure 21: Contribution of absorbed dosage in cell 10 using simulation surface energy deposit from different interaction points.

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Figure 15 contains the measurements from cell 10, 11 and 12 which shows absorbed dosage using Gafchromic films. By observing these cells in the simulations one can see that there is a pattern of dosage in these consecutive cells. In all these cells the beginning and ending pattern is different and it can be significant for us to draw some conclusion about them.

In cell 10 high amount of radiation were received throughout the entire cell, which is quite unique. At some points in the middle of the cell where radiation peaks were found alternately, the least square method used was unable to provide a very accurate representation. Looking at the bar graph (Figure 19) one can see that the radiation were absorbed here at the entire area but the non-negative least square function (nonneglsq) was unable to account for the high dosage peaks. One reason for this could be that there are so many interaction points through the entire area that it was difficult for the method to find correlation for many consecutive peaks without giving some distance between them. At the end of the cell 10 high amount of energy was deposited there.

In cell 11 in the beginning there is low amount of radiation coming from all the interaction points. At the end of cell 10 there was high dosage but at start of the cell 11 radiation drop to minimum. Then from different interaction points low amount of radiation was observed till the end of the cell. The least square fit and bar graph suggests that in both cell 10 and 11 relatively dosage absorption is high and it was absorbed throughout the undulator cell. This means the electron beam hit many interaction points over the whole cell.

In cell 12 the dosage is very high at the start of the undulator and then it drops gradually. Observing the bar graph (Figure 25) one can see that the interaction points in the beginning is results in most of the radiation. The beam is then aligned in a direction that there are only few halo particles that are interacting with the vacuum wall resulting in small amount of dosage.

After observing these three consecutive cells it can be assumed that when the particle beam was passing through the vacuum pipe there were few beam halo particles that intersect the vacuum vessel wall. Those halo particles entered the vacuum pipe with electron beam from the beginning of the undulators. The quadrupole magnet focuses the beam in one direction, the other direction where the beam is unfocused the size of the beam would increase and chances are halo particles can hit the vacuum pipe at different positions. In cell 10 and 11 few dose absorption peaks were found almost throughout the cell. This suggests that in between these two cells the probability is higher for electrons to interact with the pipe approximately after every 0.5 - 1 m distance.

Comparing the cell 10, 11 and 12 it is to be believed that the adjacent cell 10 and 12 has almost similar kind of pattern and cell 11 has opposite pattern to those two cells. It was observed in both cell 10 and 12 that the dosage absorption is high in the beginning and is low at the end of cells excluding some sharp peaks in cell 10. Whereas cell 11 has opposite pattern, the dosage absorption is low in the beginning and high at the end of the cell. It suggests that the difference between the results of these three cells could be due to the quadrupole magnets which are focusing and defocusing in alternating horizontal and vertical directions (FODO lattice). The quadrupole magnets are placed between every two cells which are focusing and defocusing in horizontal and vertical directions alternatively, so it is possible to have high dosage in the beginning of the alternative cells. Likewise we see high dosage at the end of the next cell.

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