Laue lens simulation for gamma radiation in the energy range of 1 MeV

MSc in Photonics PHOTONICSBCN Universitat Politècnica de Catalunya (UPC) Universitat Autònoma de Barcelona (UAB) Universitat de Barcelona (UB) Institut de Ciències Fotòniques (ICFO) http://www.photonicsbcn.eu Master in Photonics MASTER THESIS WORK LAUE LENS SIMULATION FOR GAMMA RADIATION IN THE ENERGY RANGE OF 1 MeV James Perengüez López Supervised by Dr. Jordi Isern Vilaboy and Dr. José Manuel Alvarez Institut de Ciències de l'Espai e Institut d'Estudis Espacials de Catalunya - “ICE (CSIC - IEEC)” Presented on date 20th May 2013 Registered at Laue Lens Simulation for Gamma Radiation in the Energy Range of 1 MeV James Perengüez López Institut de Ciències de l'Espai e Institut d'Estudis Espacials de Catalunya, ICE (CSIC-IEEC), Barcelona, Spain E-mail: [email protected] - [email protected] Abstract. The study of diraction applied to Laue lenses is a very promising tool in the eld of astrophysics of gamma radiation. In the present work we carried out a theoretical study and a computational simulation of a Laue lens that works in the energy range of 1 MeV. First of all, we selected a geometry consisting of a concentric ring structure made of mosaic crystals; the materials used were germanium (Ge), copper (Cu) and silver (Ag). We considered two focal lengths of 10 m and 30 m. Some important parameters such as the arrangement of rings, the crystallographic planes of the crystals, the eective area of the lens and its point spread function, were obtained for these conditions. The results showed that the eective area increases when increases the focal distance, which represents a challenge in the design and construction of a Laue Lens. All calculations were developed using specialized software designed by a French research group, adapted and modied by the author to carry out this work. Keywords: Gamma-ray, Laue lens, Bragg Diraction, Mosaic Crystals, Crystal Diraction, Nuclear Astrophysics. 1. Introduction The soft gamma-ray domain represents a unique window on the non-thermal Universe, allowing violent objects and matter under extreme physical conditions impossible to reproduce in the laboratory. For this reason in astronomy is a crucial window for the study of the most energetic and violent events in the Universe. But still nowadays, observing nuclear gamma-ray lines and continuum emission from celestial sources remains very challenging because of the intense background count rate induced in detectors operated in space by cosmic rays and solar events. After the launch of the several missions (Integral and Swift missions by the European Space Agency (ESA), to keep improving the sensitivity of the future soft gamma-ray telescopes, it seemed mandatory to change radically the principle of the telescopes operating in this energy band [1, 8, 9]. Thanks to eorts on the development of Laue lenses, the focusing Laue Lens Simulation for Gamma Radiation in the Energy Range of 1 MeV 3 of gamma rays with energy of 1MeV is now enabled and ready to be used scientically. In this paper, the simulation of a Laue lens that works in the energy range of 1 MeV was carried out, the simulations were performed for focal lengths of 10 m and 30 m respectively, and the geometric conguration used in this work corresponds to a lens of concentric rings formed of dierent materials (Cu, Ag and Ge) and a gamma ray source positioned on axis. 2. Laue Lens Concept Diraction lenses use the interference between the periodic nature of the electromagnetic radiation and a periodic structure such as the matter in a crystal. In a Laue lens, the photons pass through the full crystal, using its entire volume for interacting coherently. In order to be diracted, an incoming gamma-ray must satisfy the Bragg condition, relating the spacing of lattice planes dhkl with the energy of incident photons E and with the angle of incidence θB with respect to the chosen set of planes ( hkl ) the indices h , k , l , known as Miller indices, are dened as the reciprocals of the fractional intercepts which the lattice plane makes with the crystallographic axes. For example, if the Miller indices of a plane are ( hkl ), written in parentheses, then the plane makes fractional intercepts of 1/h , 1/k , 1/l with the axes, and, if the axial lengths of the unit cell are a , b , c , the plane makes actual intercepts of a/h , b/k , c/l . If a plane is parallel to a given axis, its fractional intercept on that axis is taken as innity and the corresponding Miller index is zero. If the Miller indices [ hkl ] are shown in square brackets, they give the direction of the plane with the same indices [3, 5, 10]. A Laue lens focuses gamma rays by using Bragg diraction in the Laue geometry, which means that the rays go through the crystals. A large number of crystals slabs are arranged in concentric rings and orientated such that they diract radiation coming from innity toward a common point, the focus. In the simplest design each ring is composed of identical crystals and is symmetrical with respect to the line of sight of the lens. θB , the angle of incidence of the rays with respect to the set of atomic planes of Miller index ( hkl ), is related to the diracted energy E at the n-th order, by Bragg's law [1, 2]: 2dhkl sin θB=nhc E (1) where dhkl (in ˚ A ) is the spacing of the lattice planes ( hkl ), n is the diraction order, h Planck constant and c the velocity of light in vacuum, the amount hc = 12.4 keV. ˚ A and E is the energy (in keV) of the gamma-ray photon. An elementary illustration of the Bragg condition, in two dierent congurations (reection and transmission), is given in gure 1, where it can be seen that the incident waves are reected by the parallel planes of the atoms in the crystal. Laue Lens Simulation for Gamma Radiation in the Energy Range of 1 MeV 4 A Laue lens is made of a large number of crystals, in transmission conguration called Laue geometry, that are disposed such that they will concentrate the incident radiation onto a common focal spot. From the Bragg equation, for the rst diraction order ( n= 1 ), it can be seen that the photons incident on a given crystal at distance ri ( rmin ≤ri≤rmax ) from the lens axis can be reected toward the lens focus if their energy E is given by: E=hc 2dhkl sin 1 2arctan f ri≈hcf dhklri ≈12.4f dhkl[˚ A]ri [KeV ] (2) Figure 1. The Bragg condition for constructive interference of a gamma-ray photon beam with the atoms of a given crystalline plane. (a) Bragg diraction in reection conguration (Bragg geometry). (b) Bragg diraction in transmission conguration (Laue geometry).[2] In the expresion (2) considering a focal distance f , the mean energy diracted by a ring is thus given by its radius ri and d spacing of the crystalline planes used in the ring, where the approximated expression is valid for gamma-ray lenses, given the small diration angles involved [1, 2, 7]. Conversely, the lens radius ri at which the photon energy E is reected in the focus is given by: r=ftan[2θB]≈hcf dhklE (3) As we can see from the Bragg's law equation (1), perfect single crystals are not suitable for the realization of a Laue lens because they diract almost only a single wavelength, behaving as a monochromator. In order to obtain an energy bandpass, crystals must present a range of atomic planes orientation [1, 2, 3, 7]. As discussed in the next section, there are two kinds of crystalline structures oering this mandatory feature: either mosaic structures or curved diracting planes In this work Laue Lens Simulation for Gamma Radiation in the Energy Range of 1 MeV 5 we only take into account mosaic crystals to the design of the Laue lens. In both case, the distribution of orientations are characterized by their full width at half maximum (FWHM) that is denoted Ω called mosaicity in the case of mosaic crystal, or angular spread in the general case. The bandpass of a crystal, and therefore of a full ring, is given by the following expresion: ∆E≈Ω Ei θB≈2ΩEi fri (4) It is possible with this principle to cover a wide energy range by overlapping the bandpasses produced in dierent rings. This is obtained by using an identical d-spacing considering same material, same set of diracting planes for several adjacent rings, so that the energy is shifted from one ring to the next of the amount given by Ei−Ei+1 is [1, 7]: Ei−Ei+1 ≈12.4f dhkl[˚ A]ri [KeV ] (5) In this equation, the approximations are valid for small angles, which is the case with the energies of the soft gamma-ray domain. 3. Crystals requirements Selected materials have rst and foremost to exist in a crystalline state and be reproducible with good production yield, and also be mechanically robust enough to allow accurate mounting. Secondly, the crystals have to have an angular spread in their diracting planes, i.e. to have either a mosaic structure or a curvature in their diracting planes. Since we consider entire rings of crystals focusing the same energy band, the lower limit on Ω , the FWHM of the angular distribution of plane orientations, is given by geometrical considerations. In order that the bandpasses of adjacent rings overlap enough to produce a at energy coverage, it is necessary that Ei−Ei+1 ≤∆ E i . By combining equations (3) and (4), one obtains the minimum Ω value for the crystals as a function of the focal distance and the radial size of the crystals slabs [1, 2, 5, 3, 7]: Ω≥ri+1 −ri 2f (6) Both mosaic crystals and curved crystals are suitable to be used for a Laue lens. In the case of mosaic crystals (Gaussian-like energy bandpass), this condition corresponds to a continuous at response at twice the peak eective area of a single ring. For mosaic crystals Ω could in principle be reduced by a factor of two from this value, but this would unduly tighten the constraint on the mounting precision. 3.1. Mosaic Crystals Mosaic crystals are made of many microscopic perfect crystals (crystallites) with their lattice planes slightly misaligned with each other around a mean direction, corresponding to the mean lattice planes ( hkl ) chosen for diraction. In the lens Laue Lens Simulation for Gamma Radiation in the Energy Range of 1 MeV 6 conguration assumed, the mean lattice plane is normal to the surface of the crystals. The distribution function of the crystallite misalignments from the mean direction can be approximated by a Gaussian function. In gure 2, we can apreciate the basic conguration of a Mosaic crystal [1, 5, 3, 7]. Figure 2. Basic structure of a mosaic crystal. A macroscopic crystal is actually an agglomerate of small, perfect crystals. The angular orientation of these small crystals is randomly distributed [3]. 4. The Simulation Performed The only possibility to explore such a variety of design options, and in particular their performance in space, are Laue lens simulations. In order to achieve a realistic estimate of the performance of the telescope, a realistic geometry model, including all passive materials in the instrument itself and the spacecraft, has to be built. 4.1. Code Implemented In order to carry out the computational simulation of a Laue lens, we used an structure composed of mosaic crystals and considering dierent materials. We used a code developed by the French research group whose lider is Dr. Nicolás Barriere. The code was modied and implemented in this work with the express permission of the author. The code was written using the Interactive Data Language (IDL) programming language which allows a variety of test considering various parameters of the lens [11]. 5. Results A large number of simulations, making variations in the type, number of materials and dierent focal distances were achieved. We decided to set a value of energy centered in 847 keV and this value was constant during all simulations, because this is the energy of our interest. The results showed that the Laue lens with focal length f = 10 m and f = 30 m, appear to be the best options to work in this energy value. The materials selected for their best performance were Germanium (Ge), Copper (Cu) and silver (Ag). The simulation allows setting the best location of the rings crystals Laue Lens Simulation for Gamma Radiation in the Energy Range of 1 MeV 7 composed of such materials, the optimum size of each crystal and also the necessary crystallographic conguration. The following sections show the most important results about the performance of the lens. The eective area and the point spread function (PSF) were calculated. 5.1. Laue Lens f = 10 m 5.1.1. Conguration of the lens: gure 3 shows the radius for which each couple [crystal, reection] would need to be placed on a Laue lens to diract at the specied energy. The color of the line marking the radius corresponds to the reectivity of the [crystal, reection] couple. As we can see, is necessary ten systems of crystal rings for this design, each one made for a determined number of crystals. In this case, the lens is 26 cm in diameter and its geometry conguration is as follows: Figure 3. Radius of crystal rings for f = 10 m, and E = 847 Kev. The thickness of the crystals is calculated to maximize the peak reectivit. In this simulation 3.0 mm ≤ T mos ≤ 15.0 mm. The lling of the lens in a histogram, is shown in gure 4 where the y axis indicates the normalized rings lling and the x axis is the radius in which is located each ring. Figure 4. Graphical representation of the population of each ring composing the lens. The number of crystals per ring is normalized to the maximum number that could t in the ring. Laue Lens Simulation for Gamma Radiation in the Energy Range of 1 MeV 8 5.1.2. Eective Area: gure 5 shows the diagram of the eective area of the considered crystals as a function of the energy. The maximum value is around 60 cm2 . Figure 5. Eective area of the ≈ 847 keV lens for a point source on axis. The black curve shows the total eective area while the color curves show the contribution of each crystal material and reection. 5.1.3. Point Spread Function: the expected PSF of the assumed lens (for f = 10 m) is displayed in gure 6 according to the corresponding conguration. Figure 6. Focal spot from the entire lens. We have considered a puntual source, on axis, with a constant spectrum between 700 - 1000 KeV and a ux equal to 1 ph/cm 2 /KeV. Laue Lens Simulation for Gamma Radiation in the Energy Range of 1 MeV 9 5.2. Laue Lens f = 30 m Using the same materials as in the previous case, and the same energy, we simulated a Laue lens but with a focal distance of 30 m . The analysis of the results of the dierent parameters are shown in the following sections. 5.2.1. Conguration of the lens: Like the previous case, gure 7 and gure 8, show the radius for which each couple [crystal, reection] would need to be placed on a Laue lens to diract at the specied energy, and the histogram distribution, but in this case, the focal distance is f= 30 m . In this situation a total of six rings with a specied number of crystals are required for each material. Moreover with these results the lens is 50 cm in diameter. Figure 7. Radius of crystal rings for f = 30 m, and E = 847 Kev. The thickness of the crystals is calculated to maximize the peak reectivit. In this simulation 3.0 mm ≤ T mos ≤ 15.0 mm. Figure 8. Graphical representation of the population of each ring composing the lens. The number of crystals per ring is normalized to the maximum number that could t in the ring. 5.2.2. Eective Area: gure 9 shows the eective area for this conguration. We can observe that all materials diract at the required energy. Compared to the previous case, in the lens designed for a focal length of 30 m , the eective area increases until reaching a maximum peak of ≈ 107 cm 2 .