Department of Physics and Measurement Technology
Master’s Thesis
Determination of representative spectra for the
characterization of waste from a 450 GeV proton
accelerator (SPS, CERN)
Lisa Bläckberg
LITH-IFM-EX--09/2064--SEDepartment of Physics and Measurement Technology Linköpings universitet
Master’s Thesis LITH-IFM-EX--09/2064--SE
Determination of representative spectra for the
characterization of waste from a 450 GeV proton
accelerator (SPS, CERN)
Lisa Bläckberg
Supervisor: Peter Münger
ifm, Linköpings universitet
Luisa Ulrici
CERN
Examiner: Peter Münger
ifm, Linköpings universitet Linköping, 2 April, 2009
Avdelning, Institution
Division, Department
Department of Physics and Measurement Technology Department of Physics and Measurement Technology Linköpings universitet
SE-581 83 Linköping, Sweden
Datum Date 2009-04-02 Språk Language Svenska/Swedish Engelska/English ⊠ Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport ⊠
URL för elektronisk version
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-19037
ISBN
—
ISRN
LITH-IFM-EX--09/2064--SE
Serietitel och serienummer
Title of series, numbering
ISSN
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Titel
Title Determinering av representativa spektrum för karaktärisering av avfall från en 450GeV protonaccelerator (SPS, CERN) Determination of representative spectra for the characterization of waste from a 450 GeV proton accelerator (SPS, CERN)
Författare
Author
Lisa Bläckberg
Sammanfattning
Abstract
Radioactive waste has been accumulated at CERN as unavoidable consequence of the use of particle accelerators. The elimination of this waste towards the final repositories in France and Switzerland requires the determination of the radionu-clide inventory. In order to calculate the residual induced radioactivity in the waste, it is necessary to determine the spectra of secondary particles which are re-sponsible for the material activation. In complex irradiation environments like in an accelerator tunnel it is expected that the secondary particle spectra vary with the characteristics of the machine components in a given section of tunnel. In order to obtain the production rates of the radionuclides of interest the spectra of secondary particles are to be folded with the appropriate cross sections. Though technically feasible, it would be impractical to calculate the particle spectra in every area of any machine and for all possible beam loss mechanisms. Moreover, a fraction of the waste has unknown radiological history, which makes it impossible to associate an item of waste to a precise area of the machine. Therefore it is use-ful to try to calculate “representative spectra”, which shall apply to a relatively large part of the accelerator complex at CERN. This thesis is dedicated to the cal-culation of representative spectra in the arcs of the 450 GeV proton synchrotron, SPS, at CERN. The calculations have been performed using the Monte Carlo code FLUKA. Extensive simulations have been done to assess the dependence of proton, neutron and pion spectra on beam energy, size of the nearby machine component and position with respect to the beam-loss point. The results obtained suggest that it is possible to define one single set of representative spectra for all the arcs of the SPS accelerator, with a minor error associated with the use of these.
Nyckelord
Abstract
Radioactive waste has been accumulated at CERN as unavoidable consequence of the use of particle accelerators. The elimination of this waste towards the final repositories in France and Switzerland requires the determination of the radionu-clide inventory. In order to calculate the residual induced radioactivity in the waste, it is necessary to determine the spectra of secondary particles which are responsible for the material activation. In complex irradiation environments like in an accelerator tunnel it is expected that the secondary particle spectra vary with the characteristics of the machine components in a given section of tunnel. In order to obtain the production rates of the radionuclides of interest the spectra of secondary particles are to be folded with the appropriate cross sections. Though technically feasible, it would be impractical to calculate the particle spectra in ev-ery area of any machine and for all possible beam loss mechanisms. Moreover, a fraction of the waste has unknown radiological history, which makes it impossible to associate an item of waste to a precise area of the machine. Therefore it is useful to try to calculate “representative spectra”, which shall apply to a relatively large part of the accelerator complex at CERN. This thesis is dedicated to the calcu-lation of representative spectra in the arcs of the 450 GeV proton synchrotron, SPS, at CERN. The calculations have been performed using the Monte Carlo code FLUKA. Extensive simulations have been done to assess the dependence of proton, neutron and pion spectra on beam energy, size of the nearby machine component and position with respect to the beam-loss point. The results obtained suggest that it is possible to define one single set of representative spectra for all the arcs of the SPS accelerator, with a minor error associated with the use of these.
Sammanfattning
Radioaktivt avfall har ackumulerats på CERN som en oundviklig konsekvens av användandet av partikelacceleratorer. Elimineringen av detta avfall mot de slut-giltiga förvaringsplatserna i Frankrike och Schweiz kräver att inventariet av radi-onuklider är känt. För att räkna ut den residuala inducerade radioaktiviteten i avfallet är det nödvändigt att bestämma de spektrum av sekundärpartiklar som är ansvariga för aktiveringen av materialet. I komplexa strålningsmiljöer, såsom en accelerator tunnel, är det väntat att spektrumen av sekundär partiklar varierar med karaktäristiken hos de närvarande maskinkomponenterna i en given tunnel-sektion. För att erhålla produktionshastigheten av de intressanta radionukleiderna ska dessa spektrum av sekundärpartiklar korsas med lämpliga reaktionstvärsnitt. Trots att det är tekniskt möjligt, är det opraktiskt att räkna ut partikelspektrum
i varje del av alla maskiner och för alla möjliga strålförlustmekanismer. Dess-utom har en stor del av de existerande avfallet okänd strålningshistorik, vilket gör det omöjligt att associera ett avfallsföremål till ett precist område i en ma-skin. På grund av detta är det användbart att försöka beräkna “representati-va” spektrum som kan användas för en relativt stor del av acceleratorkomplexet på CERN. Den här rapporten är dedikerad till beräkningen av representativa spektrum i bågarna på proton synchrotronen, SPS, på CERN. Beräkningarna har utförts med Monte Carlo koden FLUKA. Utförliga simuleringar har gjorts för att fastställa hur proton, neutron och pi-meson spektrum beror av energin på den primära protonstrålen, strorleken på närliggande maskinkomponenter, och posi-tion i förhållande till strålförlustpunkten. De erhållna resultaten antyder att det är möjligt att använda representativa spektrum för at karaktärisera avfall som kommer från alla bågarna i SPS-acceleratorn, utan att införa ett för stort fel.
Acknowledgments
First of all I would like to thank my supervisors at CERN, Matteo Magistris and Luisa Ulrici, for all their encouragement and support during the work with this thesis. Especially Matteo’s patience and interest has been invaluable. I would also like to thank my supervisor in Linköping, Peter Münger for his feedback on the report. A thank you also goes to my office mates, flat mates and other friends in Geneva who have been making my year in Switzerland a memorable experience. To my family and my boyfriend Petter, thank you for all your love and support.
Stockholm, april 2009.
Contents
1 Introduction 3
1.1 CERN and its accelerators . . . 3
1.2 Production of waste . . . 4
1.2.1 Beam Dynamics . . . 5
1.2.2 Waste at CERN . . . 6
1.3 The activation process . . . 6
1.3.1 Elastic interactions . . . 7
1.3.2 Inelastic interactions . . . 7
1.3.3 Reaction cross sections . . . 9
1.3.4 Radioactive decay . . . 10
1.3.5 Activity . . . 11
1.3.6 Induced radioactivity in high energy proton accelerators . . 12
1.3.7 The activation formula . . . 13
1.4 The matrix method . . . 14
1.5 FLUKA . . . 17
1.6 Lethargy . . . 18
1.7 Problem description . . . 19
2 Method and Monte Carlo simulations 21 2.1 Characteristics of the SPS . . . 21
2.2 Waste from the SPS . . . 21
2.3 Strategy . . . 22
2.3.1 Naming convention . . . 24
2.3.2 From point losses to uniform losses . . . 25
2.4 The FLUKA input . . . 27
2.4.1 Geometry . . . 27
2.4.2 Beam properties . . . 30
2.4.3 Physics Settings . . . 30
2.4.4 Scoring . . . 30
3 Analysis of the results 33 3.1 Position relative to beam loss point . . . 33
3.1.1 Shape . . . 33
3.1.2 Fractions of LE . . . 35 ix
3.1.3 Intensities . . . 40 3.2 Energy . . . 41 3.2.1 Shape . . . 42 3.2.2 Fractions of LE . . . 44 3.3 Dipole / Quadrupole . . . 44 3.3.1 Neutron moderation . . . 44 3.3.2 Shape . . . 45 3.3.3 Fractions of LE . . . 46
3.4 Propagation of the statistical error . . . 46
3.4.1 Numerical estimation of the statistical error . . . 48
4 Conclusions 51 5 Future work 53 5.1 Characterization of waste from the straight sections of the SPS . . 53
5.2 Characterization of massive objects . . . 53
5.3 Characterization of waste from the Proton Synchrotron . . . 53
Bibliography 55
Nomenclature
Here follows a list of the symbols and abbreviations used in the thesis. Units are given where applicable.
Symbols
α Alpha particle β Beta particle e Electron η Eta meson v Neutrino ¯v Antineutrino n Neutron π Pion γ Photon p Protonσ Reaction cross section [1 b (barn) = 10−24 cm2]
Rb Production rate of nuclide b [s−1 cm−2]
E Particle energy [eV]
λ Decay constant [s−1]
t1/2 Half-life [s]
τ Mean lifetime [s]
A Activity [Bq]
A Specific activity [Bq/kg]
tirr Irradiation time [s]
twait Waiting time [s]
NAv Avogadros number [6.02 × 1023 nuclei/mole]
Lb Reference limit of specific activity of nuclide b [Bq/kg]
u Lethargy
ϕ Normalized particle spectra Φ Particle spectra
d Dipole magnet q Quadrupole magnet
Abbreviations
CERN Conseil Européen pour la Recherche Nuclèaire CNGS CERN Neutrinos to Gran Sasso
COMPASS Common Muon and Proton Apparatus for Structure and Spectroscopy FLUKA FLUktuierende KAskade
LE Limite d’Exemption
LHC Large Hadron Collider
n_TOF neutron Time-Of-Flight facility
PS Proton Synchrotron
PSI Paul Scherrer Institut SPS Super Proton Synchrotron
Chapter 1
Introduction
1.1
CERN and its accelerators
CERN, the European Organization for Nuclear Research, is the world’s largest particle-physics research facility. The laboratory was founded in 1954 and is lo-cated outside Geneva on the Franco-Swiss border. There are currently 20 European member states collaborating to keep the facility running.
At CERN scientists are studying the basic constituents of matter and the forces that act between them. For this purpose the laboratory provides various particle accelerators which are used for a wide range of experiments. With these machines, particles, such as protons, electrons and ions, are accelerated to high energies, using strong electrical and magnetic fields. The accelerated particles can either collide against each other or be sent onto a fixed target, depending on which type of experiment they are used for.
The accelerators at CERN form an extensive accelerator complex (see figure 1.1). Particles are accelerated up to a certain energy in one accelerator and are then injected into the next one in order to achieve an even higher energy. This process continues until the particle beam has attained the energy required by the experiment in which it is to be used. There are both linear and circular accelerators at CERN.
The last part of the accelerator chain is the Large Hadron Collider (LHC). In this circular accelerator with a circumference of 27 km, protons can be accelerated up to an energy of 7 TeV, which corresponds to a velocity very close to the speed of light. The beam has circulated for the first time in the LHC in September 2008. In order to deal with all the experimental data CERN has also been leading in the development of computing tools. In 1990 the World Wide Web was invented at CERN to facilitate the communication between particle physicists from all over the world.
Figure 1.1. Schematic layout of the CERN accelerator complex.
1.2
Production of waste
As a consequence of the operation of the accelerators, machine components can become radioactive. Induced radioactivity in the components of the machine and its surrounding structures is a consequence of the interaction of radiation with matter. This is possible, for example, in case of beam losses. When high-energy particles hit a material this can be activated. Secondary particles are generated in the collisions, which also have the ability of inducing radioactivity.
In this section the mechanisms responsible for the beam losses in the accelera-tors are discussed. A brief description of the radioactive waste generated at CERN is also included.
1.2 Production of waste 5
1.2.1
Beam Dynamics
In any type of accelerator there is an ideal orbit, the so-called design orbit, on which all the particles with zero transversal momentum move. If the design orbit is curved, which is the case in most of CERN’s accelerators, there is a need for bending forces in order to keep the particles on the circular track.
Most of the particles slightly deviate in the transversal plane from the design orbit. To keep these deviations in track there is a need also for focusing forces. Both the bending and the focusing forces are created by means of magnetic fields. The bending forces are created by dipole magnets which constrain the particles to the circular orbit. The focusing forces are created by means of quadrupole magnets. There are two kinds of quadrupole magnets, those which keep the beam focused in the horizontal plane, and those which keep the beam focused in the vertical plane. These are called focusing and defocusing magnets, because when the magnet focuses in one direction it defocuses in the other. A succession of focusing and defocusing magnets is needed in order to keep both kinds of deviations under control. When a particle drifts away from the central orbit, it gets pushed back by the focusing force. The resulting beam will contain particles which oscillate around the central orbit both vertically and horizontally. These oscillations are called the Betatron oscillations and they are described by the so-called Hill’s equation, [1]:
d2x
ds2 + K(s) · x = 0 (1.1)
where x is the displacement from the design orbit, s is the longitudinal position in the accelerator and K is the restoring force which varies with s.
Hills equation is a second order differential equation and its solution is: ( x =√ǫβ cos φ ˙x = −αqβǫcos φ − qǫ βsin φ (1.2) where α and β are parameters which depend on the design of the machine and ǫ is the transverse emittance whose value is determined by the initial beam conditions. A plot of ˙x versus x as φ goes from 0 to 2π results in the so-called phase space ellipse, [1].
The variable β depends on the position s and will change under the influence of the magnets and in turn affect the shape of the ellipse along the machine. However it is only the shape of the ellipse that will change, its area will remain constant throughout the accelerator1. The projection of the ellipse on the x-axis shows how
the transverse beam size will vary along the longitudinal position. A large value of
β(s) will give the phase space ellipse a large projection on the x-axis, and therefore
a large transverse beam size.
There are two concepts regarding the phase space ellipse that need to be men-tioned. The first one is the emittance, which equals the area of the ellipse that contains 95% of the primary particles. The second concept is the acceptance which
1In a real machine, the area of the ellipse actually increases because of non-linearities and errors in the machine components.
is the maximum area of the ellipse which the emittance can attain without loosing any particles. In those parts of the accelerator where the emittance is close to or larger than the acceptance there are significant beam losses. These beam losses lead to the interaction of particles with matter, which in turn leads to induced radioactivity. [1]
Another source of beam losses are beam-gas interactions. These occur when particles from the beam hit residual gas molecules which are present in the beam pipe. This causes the primary particle to scatter away from the design orbit or to undergo nuclear reactions and generate secondary particles.
1.2.2
Waste at CERN
Due to the beam losses the operation of the CERN accelerators unavoidably leads to the activation of materials. Both the components on the beam line (such as magnets) and the objects further out from the beam line (such as cables and other smaller objects) might be activated. The level of activation depends on the energy of the primary beam, the number of primary particles that are accelerated and on the magnitude of the beam losses. The level of activity in a certain item depends on its proximity to the beam losses, its chemical material composition, the time it has been placed in the tunnel and the time which has passed since the activation. During maintenance, dismantling and decommissioning of the accelerators the activated material is removed from the tunnel and is disposed of as radioactive waste, unless it can be reused elsewhere in the machine. In general, smaller objects are removed and exchanged during short-time periods of maintenance while larger objects are removed only during longer periods of shutdown. However, the largest amount of waste comes from the dismantling of the machines.
The radioactive waste is temporarily stored in dedicated buildings or in old accelerator tunnels at CERN. At this moment CERN stores about 200 m3 of
radioactive waste per year. Most of the waste has very low radioactivity (specific activity < 100 Bq/g), but there is also a small fraction which has low to medium level of radioactivity. The interim storage facilities of radioactive waste at CERN are currently close to saturation.
The waste needs to be eliminated towards final repositories in France and Switzerland. According to the CERN safety policy waste which has been activated on the french part of the CERN site has to be disposed of in France, and the waste activated on the swiss part in Switzerland. The treatment of the waste must follow the legislation valid in each of the Host Countries. [5]
1.3
The activation process
When a particle collides with a nucleus in such a way that this nucleus gains internal energy there is a nuclear reaction. The result of the reaction can be a different isotope of the same atom, an isotope of another chemical element or the same atom elevated to an excited state. The resulting nucleus can be found either
1.3 The activation process 7
in a stable or an unstable state. When the nucleus is unstable it can decay and is referred to as radioactive.
These kinds of collisions are called inelastic interactions. Collisions which do not lead to the production of radioactive nuclides are called elastic interactions.
1.3.1
Elastic interactions
A collision in which the sum of the kinetic energy of the incoming particle and the nucleus is conserved is called elastic scattering. In this type of reaction the only things that change are the direction and kinetic energy of the particles. The nucleus is preserved in its original state and no radioactivity is induced. The incident particle however looses some of its energy to the nucleus. Elastic scattering occurs for all incident particle types and for all particle energies.
The forces which are responsible for the change in direction are the electro-static Coulomb force and the nuclear forces. Charged particles are influenced by both forces while uncharged particles are affected only by the nuclear ones. The nuclear forces are the forces which hold the nucleons together in the nucleus, and they do not depend considerably on charge. These forces act on a much shorter distance than the Coulomb force. [4]
1.3.2
Inelastic interactions
In inelastic interactions the composition of the nucleus can be changed, and hence radioactive nuclei can be created. The kind of reaction which takes place when a particle collides with a nucleus depends on the kinetic energy, charge and mass of the incoming particle and of the target nucleus. Here follows a description of the reactions of interest for this study. The reactions are grouped according to the energy of the incident particle.
Low energy inelastic reactions
Low energy interactions take place only when the incident particle is a neutron. This is because the neutron is uncharged and can traverse the Coulomb barrier even with almost zero kinetic energy. Charged particles on the contrary will need a certain amount of energy in order to traverse this barrier and enter the nucleus. When a low energy neutron collides with a nucleus it can get absorbed by the latter. The resulting nucleus will be heavier by one neutron, and thus a new isotope is created. This new nucleus can be either stable or unstable. This kind of reaction is called neutron capture and is written as (n,γ), since the incident particle is a neutron and one or more γ-rays are emitted in the reaction. This reaction is possible with neutrons with down to zero velocity, the so-called thermal neutrons. Thermal neutrons are created when neutrons with higher energies travel through a material and gradually loose their energy by elastic scattering with nuclei which are
at rest (apart from their thermal motions). The energy of such thermal neutrons is typically around 0.025 eV.
Thermal neutrons can also make the nucleus unstable, so that it breaks into two or more fragments. This reaction is called fission and can occur when the nucleus is a heavy element containing a large number of nucleons which are not strongly grouped together.
Also (n,p) and (n,α) reactions, where the neutron enters the nucleus and knocks out a proton or an α-particle, are possible at low energies. [4]
Evaporation
Particles with an intermediate energy can knock out one or more nucleons while colliding with a nucleus. The intermediate energy range needed for this to happen is a few MeV up to about 50 MeV for protons. For neutrons the energy needed is the same or lower, since the neutrons are not affected by the Coulomb barrier and thus need less energy in order to enter the nucleus. The process of an incident particle knocking out one or more nucleons can be described by a sequence of events. Firstly the incoming particle enters, and gets absorbed by, the nucleus. In the physical model which is used to describe this process it is assumed that the energy of the incoming particle is randomly distributed among the nucleons. When the energy of the incoming particle is not too great none of the nucleons achieve enough energy to escape immediately from the nucleus. This process leaves the nucleus in an excited state. The nucleus in this state is called a “compound nucleus”. However, statistical fluctuations in the energy distribution will eventually concentrate enough energy to one nucleon so that it can escape. This can happen more than once, leading to a sequential emission of nucleons. Each emitted particle has a relatively low kinetic energy. This kind of emission of particles from an excited nucleus is called evaporation. [4]
High energy inelastic interactions
When the incoming particles have even higher kinetic energies the situation gets more complicated. When nucleons are struck by high energy particles they can obtain enough energy to move and hit another nucleon within the same nucleus. The hit nucleon can in turn hit another nucleon, and so on. This sequence of events is called an intranuclear cascade of fast nucleons. When a nucleon is hit it can either escape from the nucleus or it can be captured and give away its energy to the nucleus leaving this in an excited state. The excited residual nucleus can de-excite in various ways:
• Evaporation can occur in a way similar to that described in the previous
paragraph. The evaporated nucleons or groups of nucleons are called spalla-tion products.
• The nucleus can be divided into two or more pieces, undergoing so-called
1.3 The activation process 9
The occurrence of intranuclear cascades dramatically increases the number of possible residual nuclei.
Apart from intranuclear cascades there are also other reactions which are pos-sible when a high energy particle collides with a nucleus:
• Fragmentation reactions, which is a direct ejection of high energy ions or
light nuclei from the nucleus.
• Production of new particles which themselves can move within the nucleus
and hit other nucleons. Pions, nucleons and antinucleons are particles which can develop intranuclear cascades of their own. The energy threshold for pion production is of about 290 MeV, [8], and for production of nucleons and antinucleons the threshold is around 4.5 GeV. [4]
Photonuclear reactions
In addition to the previously described interactions, where the incident particle is a hadron (proton, neutron or pion), also photons can interact with the nucleus. These high-energy photons are generally γ-rays, X-rays or synchrotron radiation. The reactions can either be of the type where the nucleus gets excited, or photo-spallation. In photospallation nucleons or fragments of the nucleus can get knocked out by the incident photon. [4]
Particle showers
When the energy of the primary hadron is above a few tens of MeV, the secondary particles emitted in the reactions can themselves trigger further interactions, cre-ating a hadronic shower.
For energies above 290 MeV, pions are produced. These particles can trans-fer energy to the electromagnetic sector. This happens due to the production of mesons (such as π0 and η). The mesons quickly decay into electromagnetic
par-ticles (e± and γ). When the production of pions is significant, it gives rise to electromagnetic showers.
High energy hadronic showers will always create also electromagnetic ones, but the electromagnetic showers however proceed without significant hadronic particle production, due to the small probability of electro and photonuclear interactions. [8]
1.3.3
Reaction cross sections
The reactions described in 1.3.2 will not occur every time a particle hits an atom. The probability of a reaction is proportional to a cross sectional target area pre-sented by the nucleus. The number of reactions that will take place per second and per unit volume of the target can be described by:
N = ϕiσiNT (1.3)
where ϕi is the flux of the particle type i per unit time and per cm2, σi is the
The cross section for a certain reaction depends on the type of incident particle, the type of atom and on the energy of the incident particle. [4]
In the case of a flux of particles of different energy, one has to fold the spectra with the appropriate cross sections in order to calculate the production rate2of a
certain radionuclide from a certain element. This is done according to:
Rb= " X i Z σi,e,b(E)ϕi(E)dE # · NT (1.4)
where σi,e,b(E) is the cross section for the production of the radioisotope b from the element e by the particle type i, and ϕi(E) is the spectrum of the particle type
i.
1.3.4
Radioactive decay
Radioactive nuclei which are created in the nuclear reactions are unstable, and will eventually decay into stable nuclei. The decay occurs by means of the emission of particles from the nucleus. The most common type of decays are α-decay, β-decay,
γ-decay, spontaneous fission and nucleon emission. α-decay
In α-decay an α-particle is emitted from the nucleus. Such particle is composed of 2 protons and 2 neutrons which are bound together like in a helium nucleus. In this process an element which is lighter by 4 nucleons (2p and 2n) is created. The decay process can be written as:
A ZXN →A−4Z−2X ′ N −2+42He2 (1.5) [10] β-decay
In β-decay a proton is converted to a neutron, or a neutron to a proton within the nucleus. There are three possible basic decay processes. In each of them a neutrino (v) or an antineutrino (¯v) is emitted.
• In β−-decay a neutron is converted to a proton, and an electron in addition
to a ¯v is emitted in order to conserve electric charge.
• In β+-decay a proton is converted to a neutron, and a positron in addition
to a v is emitted in order to conserve electric charge.
• In electron capture an atomic electron is swallowed by the nucleus and a
neutron is created from a proton. In addition a v is emitted. [10]
2The production rate is here defined as the number of radioactive nuclei produced per second and per cubic centimeter.
1.3 The activation process 11 γ-decay
γ-decay occurs with the emission of a γ-ray. In this reaction the number of protons
and neutrons in the nucleus (the chemical element) is not changed. Instead it is transformed from one excited state into a lower one, or into the ground state. The energy of the emitted photon equals the energy difference between the nuclear states. γ-decay can occur by itself, or subsequent to α and β decay, since these processes often leave the daughter nucleus in an excited state. [10]
Spontaneous fission
Spontaneous fission is similar to the neutron induced fission described in section 1.3.2, but without a previous neutron capture. In this process a heavy nucleus with an excess of neutrons splits into two lighter nuclei. The types of possible daughter nuclei are statistically distributed over the entire range of medium weight nuclei. [10]
Nucleon emission
This type of decay occurs, as the name suggests, with the emission of a nucleon. The reaction is most common in fission products which have a large neutron excess. [10]
1.3.5
Activity
The rate at which the previously described decays occur and the nuclei return to their stable state is different depending on the kind of radioactive nuclei. The decay process is of statistical nature and it is not possible to predict when a given nucleus will decay. One can only make predictions about the number of radionuclei in the sample as a whole.
If a sample contains only one kind of radioactive nuclei, and at the time t it contains N nuclei and no new radioactive nuclei are created, then the disintegration can be described by:
λ = −(dN/dt)
N (1.6)
where λ is the so-called decay constant and represents the probability per unit time that an atom will decay. This constant is independent of the age of the atom, and specific for every kind of radionuclei.
Integrating equation 1.6 leads to the exponential law of radioactive decay, which describes how the number of radioactive nuclei changes in time:
N (t) = N0e−λt (1.7)
where N0is the number of radioactive nuclei present in the sample at the an initial
time.
Another concept which is often used while talking about radioactivity is the half-life, t1/2. The half-life is the time necessary for half of the radioactive nuclei
in a sample to decay. This constant is also specific for each radionucleus and is related to the decay constant according to:
t1/2= ln(2)
λ (1.8)
This expression is obtained from equation 1.7 by setting N = N0/2.
The inverse of the decay constant is called the mean lifetime, and is the average time that a nucleus is likely to survive before it disintegrates:
τ = 1/λ (1.9)
With equation 1.7 it is possible to predict the number of nuclei present in a sample at a certain time, assuming that the number of nuclei at an initial time is known. However, this is not a very practical way to determine how radioactive a certain item is, since the number of nuclei is a property which is very diffi-cult to measure. Much easier is to measure the number of decays during a certain period of time. This can be done by observing the radiation emitted from the item. Activity, A, is a quantity defined as the number of decays per unit time. The activity can be predicted, by measuring this quantity at an initial time, according to the equation:
A(t) ≡ λN (t) = A0e−λt (1.10)
where A0 is the activity at the initial time. The SI unit for activity is Becquerel
[Bq]. One Bq equals one decay per second. [10]
In practice it is more common to talk about the specific activity, A, of a sample. The specific activity is the activity per unit mass:
A = A
m (1.11)
where m is the mass of the sample. Specific activity is normally expressed in Bq/kg. [9]
1.3.6
Induced radioactivity in high energy proton
accelera-tors
The machines which are of interest in this study are high energy proton accelera-tors. The amount of radioactivity which will be induced in these depends on the beam losses (number of particles lost, their charge and their energy), the compo-sition of the materials in the accelerator, the spectra of secondary particles, and the production cross sections for the radionuclides of interest. The amount of activity present in an item at a certain time will depend on the half lives of the radionuclides, the time during which the item was irradiated, and the time which has passed since the irradiation stopped.
1.3 The activation process 13
Since the incident particles in these kinds of accelerators have very high en-ergies, a large variety of radioisotopes is produced in the reactions. Furthermore the materials present in the machine are of various different compositions, which further increase the number of different isotopes.
The radioisotopes created in high energy accelerators commonly decay by emis-sion of electrons or positrons, or by neutron capture (β-decay). The daughter nuclei also commonly emits γ-rays. [15] The production of α-emitters is not very common in proton accelerators.
Components on the beam line
Items which are placed along the beam line in the accelerator can be activated both by the primary particles lost from the beam as well as by the secondary particles generated in the nuclear reactions. The activation is mostly due to spallation reactions.
Components off the beam line
In hadron accelerators, the activation of items which are placed far from the beam line occurs by means of secondary hadrons (p,n and π±). The activation process is rather complex due to the large energy span of the secondary particles, and their non uniform spectral distribution. The activation mechanisms can be divided into two parts:
• Activation due to spallation reactions by high energy secondary hadrons. • Activation due to evaporation and capture reactions involving low-energy
neutrons.
The activation from low-energy neutrons depends critically on the chemical compositions of the irradiated item, including trace elements. [15]
The ratio between the thermal and high energy particle fluences depends on the layout of the specific machine, and on the beam energy. The ratio can also vary considerably between different positions in the accelerator.
Massive objects
If the objects are massive, regardless of whether they are placed along or far from the beam line, the induced radioactivity will not be homogeneous, since the spectra of activating particles will change with depth inside the material.
1.3.7
The activation formula
The activity of a certain item of waste depends, as already mentioned, on many factors. The irradiation time, the waiting time, the spectra of activating particles, the material composition of the item and the reaction cross sections all affect the amount of activity of the item.
The number of radioactive nuclei of isotope b per gram of target element e produced per unit time by I primary particles per second is:
nb= I NAv Me X i=p,n,π,pho Z Φi(E)σi,e,b(E)dE (1.12)
where NAv is Avogadro’s number (NAv = 6.02 × 1023 nuclei/mole), Me is the atomic weight of the target element e, Φi(E) is the spectrum of particle i (proton, neutron, pion or photon) generated by one primary particle and σi,e,b(E) is the cross section for the production of isotope b from target element e by the particle
i.
This equation can be extended to express the specific activity Ab of the ra-dionuclide b per gram of material after an irradiation time tirrand a waiting time
twait: Ab= I X e NAv Me xe X i=p,n,π,pho Z
Φi(E)σi,e,b(E)dE(1 − e−λbtirr)e−λbtwait (1.13)
where xeis the weight fraction of element e in the item and λbis the decay constant of the isotope b. The term (1 − e−λbtirr)e−λbtwait represents the time build-up.
During the irradiation the number of radioactive nuclei will increase with the production rate Rb. At the same time they will decay due to the radioactive decay. The amount of activity in a sample after an irradiation time tirr can be described by:
Ab= Rb(1 − e−λbtirr) (1.14) When the irradiation stops the activity will only decrease. After a waiting time
twait the activity in the sample will be reduced to (using equation 1.10):
Ab= Rb(1 − e−λbtirr)e−λbtwait (1.15) [10]
1.4
The matrix method
In order to eliminate the radioactive waste generated at CERN it is necessary to determine the radionuclide inventory for all the items. This can be done in a num-ber of ways, of which one is simulations. The Monte Carlo code FLUKA (which will be described in section 1.5) can estimate the residual nuclei in an item, for a given irradiation cycle (tirr and twait). This requires a full implementation of the accelerator components and knowledge of the radiological history of the item. This method also requires one separate set of simulations for each of the items of waste with their exact material composition and geometry. Because of this it is not a very practical way of characterizing all the items of waste since it would require too much time in terms of geometry implementation and CPU time.
1.4 The matrix method 15
In the matrix method, [12], only the spectra of secondary particles in the tunnel are determined by Monte Carlo simulations. The radionuclide inventory is afterwards calculated offline for each of the items of waste.
The method is based on the activation formula, equation 1.13. The idea is to calculate the production rates of all possible residual nuclei, produced from all possible target elements. These production rates are then stored in a matrix and the final activities are calculated a-posteriori using the material composition and irradiation cycle of each item.
The matrix method can be used to characterize waste which fulfills the following criteria:
• The physical processes responsible for the activation are continuous in time. • Known irradiation cycle, with a minimal uncertainty.
• Uniform particle spectra inside the item.
• The particle spectra responsible for the activation should be known and
normalized to one secondary particle per unit surface per unit time, according to: ϕi = Φi(E) P lR Φl(E)dE (1.16) where Φi is the absolute fluence of the particle i obtained by FLUKA and
ϕi is the same fluence normalized to one secondary particle. The sum is performed over all the secondary particles of interest (neutrons, protons, π+
and π−) and the integral is calculated in the energy range from 10−5 eV to
1 TeV for neutrons, and from 1 MeV to 1 TeV for the other particles.
• Uniform material composition of the item. • Known material composition of the item. • Absence of heavy elements.
• No contamination.
• Possibility of measuring a dose rate which is representative for the whole
item.
Apart from the material composition, it is the specific activities which are of interest in order to decide how to eliminate a particular item of waste. Each of the radionuclides emits radiation which can be more or less harmful. Because of this, for a given elimination pathway, there are nuclide-specific limits. The specific activities of the nuclei in the item are to be compared to a set of reference limits in order to choose the appropriate elimination pathway.
The final number of interest is the so-called absolute level of radioactivity:
RAD =X
b
Ab
Lb (1.17)
where Ab is the specific activity of radionuclide b and Lb is its corresponding reference limit.
A combination of equations 1.13 and 1.17 gives:
RAD =X b 1 Lb X e xeI NAv Me X i=p,n,π,pho Z Φi(E)σi,e,b(E)dE(1 − e−λbtirr )e−λbtwait (1.18) This equation can be rewritten in matrix form. The components of the resulting expression are:
• The material composition vector −−−−−−−−−→W (material). This vector contains the
value we=P xeNAv/Mefor each of the elements in the item. weequals the number of atoms of target element e per gram of material.
• The reference limits Lb are coupled with the time dependent function:
gb(tirr, twait) =(1 − e
−λbtirr)e−λbtwait
tirr (1.19)
These components are forming the vector −−−−−−−−→D(irr.cycle) containing one
ele-ment Db(tirr, twait) = gb(tirr, twait)/Lb for each of the radionuclides b.
• For all elements e and all radionuclides b the productions rates are calculated
from the coefficients:
fb,e(ϕ) =X i
Z
σi,e,b(E)ϕi(E)dE (1.20) where σi,e,b(E) is the cross section for production of radionuclide b from target element e by incident particle i and ϕi(E) is the unitary spectrum of particle type i. The sum is calculated over all particle types (p,n, π±
) and the integral over all possible particle energies. These production rates are grouped in a matrix M (spectra). The matrix contains one column for every target element e and one row for every radionuclide b. The appearance of the matrix is:
M (spectra) = fb1,e1(ϕ) · · · · · · · · · fb1,enϕ · · · . .. · · · · · · ... · · · · · · ... · · · fbn,e1(ϕ) · · · · · · · · · fbn,en(ϕ) (1.21)
1.5 FLUKA 17 • A normalization factor is finally defined as:
K = Itot
X
i Z
Φi(E)dE (1.22)
where Itotis the total number of primary particles lost near the item of waste during the time tirr, the sum is calculated over all particle types and the integral over all possible energies. The sum represents the total radiation intensity as a result of one lost primary particle. K can be estimated by measuring the dose rate near the radioactive item.
The resulting expression of the RAD value is:
RAD = Khgb1(tirr,twait)
Lb1 · · · · · · · · · gbn(tirr,twait) Lbn i × × fb1,e1(ϕ) · · · · · · · · · fb1,enϕ · · · . .. · · · · · · ... · · · · · · ... · · · fbn,e1(ϕ) · · · · · · · · · fbn,en(ϕ) we1 .. . .. . .. . wen (1.23)
= K−−−−−−−−→D(irr.cycle)[M (spectra)]−−−−−−−−−→W (material) (1.24)
It should be noted that the calculation of the RAD-value can only be performed if there exists a set of reference limits. Otherwise the matrix method is limited to the estimation of the specific activities of each single radioactive nuclide.
1.5
FLUKA
FLUKA is a Monte Carlo code which is used for calculations involving transport of particles and their interactions with matter.[7], [6] The code is used for a number of applications such as shielding calculations, activation studies, detector design, radiotherapy, etc. The code can simulate the interactions and propagation in matter of about 60 different particles. Photons and electrons can be simulated from 1 keV to thousands of TeV, neutrinos and muons for any energy and hadrons up to 20 TeV. Time evolution and tracking of emitted radiation from unstable nuclei can be performed online.
The code is based on nuclear models, except for interactions of neutrons with energies below 19.6 MeV. The transport and interactions of these particles are treated by a tabulated cross section library. This library contains 72 energy groups.3
3
This is valid for the FLUKA version 2006.3(b) which was the version used for the simulations for this study. However there has recently been a new release, and the FLUKA version 2008.3 has 20 MeV as the transition limit between model and group treatment instead of 19.6 MeV. The new version also contains 260 energy groups instead of 72 for the low energy neutrons.
The code can handle complex geometries implemented with combinatorial ge-ometry. The geometries are formed by a number of regions which are each assigned a material.
To do the simulations one has to provide an input file to the code. The input file is written with special commands, called cards, where the physics settings, problem geometry, beam properties and output options are specified.
1.6
Lethargy
All the particle spectra in this thesis will be presented in fluence per unit lethargy as a function of particle energy. Lethargy, u, is defined as, [16]:
u = lnE0
E (1.25)
where E0 is an arbitrary reference energy. Differentiation of equation 1.25 leads
to:
du = −1
EdE (1.26)
FLUKA generates spectra in the format of differential distribution of fluence in energy:
dΦ(E)
dE (1.27)
The fluence per unit lethargy is obtained using equation 1.26 according to: dΦ du = dΦ dE · dE du = − dΦ dEE (1.28)
In all the graphs of particle spectra presented in this thesis dΦ(E)dE E is plotted¯ against the particle energy E. ¯E is the average energy over the logarithmic energy
bin and calculated as ¯E =pEi· Ei+1, where Ei and Ei+1 are the limits of energy
bin i.
The concept of lethargy was invented for nuclear reactor physics, where the neutron spectra are of great importance. When a neutron collide elastically with an atom in a material it can loose part of its energy to the atom. How much energy it looses depends on the scattering angle and on the weight of the atom. The maximum energy loss (leading to the minimum final energy of the neutron) occurs after a head-on collision. The difference between the initial and the minimum final energy of the neutron can be described by the equation:
Emin
E′ = 1 − 4A
(A + 1)2 = 1 − α (1.29)
where A is the mass number of the struck atom, E′ is the initial energy of the neutron, Emin the minimal final energy and α = 4A
(A+1)2 [16].
This expression is obtained using the momentum and energy conservation prin-ciples for collisions, in a non relativistic frame. Since neutrons are not charged
1.7 Problem description 19
they will not be affected by the electrons of the atom, and the interaction can therefore be described as a simple elastic collision between the neutron and the nucleus. The non relativistic frame can be used since the neutrons participating in these kind of collisions generally have a relatively low energy, and because the nucleus is assumed to be at rest before the collision. After the collision the neutron will have any of the energies between the minimum final energy, Emin, and the initial energy, E′, with equal probability. The average energy loss in a collision is αE′
2 . For hydrogen A = 1, which leads to α = 1 and which in turn gives an
average energy loss of E′
2. The energy loss in elastic collisions is a percentage of
the initial energy E′ rather than a fixed value. This is the result which leads to the convenience of a new scale where the loss is independent of the initial energy: the lethargy scale.
For hydrogen where A = 1 the use of the lethargy scale gives a uniform spec-trum if no neutrons are absorbed by the material, since in hydrogen the neutrons loose on average half of their energy in each collision.
Plotting the fluence per unit lethargy gives the opportunity to deduce physical processes from the shape of the spectrum. It is possible to see for which energies neutrons are absorbed and for wich energies many reactions take place.
The other particles (protons and pions) have a different behavior in the colli-sions since they are charged and will interact with the electrons before reaching the nucleus of the atom. However in this thesis the lethargy scale is adopted also for these particles simply for reasons of readability.
1.7
Problem description
Over the years machine components have become radioactive as a consequence of the operation of the accelerators at CERN. The storage facilities at CERN are close to saturation, and the waste needs to be eliminated towards final repositories in France and Switzerland. In order to do this the radionuclide inventory of the waste needs to be determined.
In the matrix method, described in section 1.4, the radionuclide inventory is explicitly calculated from the spectra of secondary particles responsible for the activation. In complex irradiation environments like an accelerator tunnel it is ex-pected that these spectra vary with the characteristics of the machine components present in a given section of tunnel.
Though technically feasible it would be impractical to calculate the particle spectra for every area of any machine and for all possible beam loss mechanisms. Moreover, a fraction of the waste currently stored at CERN has unknown radio-logical history, which makes it impossible to associate an item of waste to a precise area of the machine.
The number of spectra to be calculated is a compromise between simplicity and accuracy. Only the spectra which differ considerably in terms of production rates shall be implemented in the method.
of representative spectra of secondary particles, which can be applied to all of the arcs of the Super Proton Synchrothron (SPS) at CERN. Such “representative” spectra would be used to calculate the production rates of long lived radionuclides in small accelerator components located in the arcs.
From the material composition of the item of waste, the production rates and the radiological history, and by normalization with a dose-rate measurement one can obtain the activity of the radionuclides in the item. The dependence of the secondary particle spectra on distance from loss point, beam energy and thickness of the nearby accelerator component is investigated.
Chapter 3 concludes with the choice of the representative spectra and the calculation of the associated statistical error.
Chapter 2
Method and Monte Carlo
simulations
2.1
Characteristics of the SPS
The Super Proton Synchrotron (SPS) which was switched on in 1976 is the second largest particle accelerator at CERN, after the LHC. It is a circular accelerator with a circumference of 6.9 km and it is used to accelerate protons and, to a minor extent, ions. The particles are injected to the SPS from the Proton Synchrotron (PS) with an energy of 25 GeV. They are then accelerated to 400 or 450 GeV depending on where they are to be used. The SPS provides proton beams for the LHC, the CNGS experiment and the COMPASS experiment.
The SPS ring is composed of straight sections and arcs. The straight sections are all different from each other and contain an irregular pattern of accelerator components. In the arcs however one can find a repetitive pattern of magnets. Because of these characteristics it was decided to study the arcs first, due to the relatively simple implementation for simulations.
In total the SPS contains 1317 electromagnets, of which 744 are bending dipoles. [13]
2.2
Waste from the SPS
The secondary particles produced during the operation of the SPS may induce radioactivity in items located in the accelerator tunnel. The highest levels of in-duced radioactivity are located around the injection and extraction points as well as beam dump regions. However beam losses can occur anywhere in the acceler-ator which makes it necessary to study the induced radioactivity also in the arcs and straight sections of the ring. Among all radioactive waste from the SPS only a small fraction comes from the so called hot spots (injection, extraction and dump regions). The rest comes from the remaining parts of the tunnel.
The items of waste from the SPS differ considerably both in size and material composition. Typical examples of waste are:
• cables;
• massive magnets; • supports;
• pumps and other elements from the vacuum system.
This variety makes the task of characterizing the waste even more difficult. This thesis is dedicated to the study of the spectra of secondary particles responsible for the activation of small items exposed to secondary radiation in the SPS arcs. Figures 2.1 and 2.2 shows typical waste. Only the induced radioactivity caused by beam losses during acceleration of protons will be considered, since only a limited amount of ions are accelerated, and therefore they do not affect significantly the final activities.
Figure 2.1. Activated magnets. Figure 2.2. Activated cables.
2.3
Strategy
The arcs of the SPS house magnets arranged in a repetitive pattern of one quadrupole followed by four dipoles, which will be referred to as a “magnetic pattern”.
The spectra in one location in the accelerator is assumed not to be affected by beam losses in components more than one magnetic pattern upstream. This assumption is based on a previous study of hadronic cascades in high energy proton accelerators [14]. In this study it is shown that when a 1 TeV proton beam hits a virtually infinite iron target, all the high energy (>50 MeV) interactions take place within the first 2.25 m layer of material. This distance, which is an indicator of the dimension of the geometry to be included in the simulations, is considerably shorter than the length of one magnetic pattern in the SPS, which is 29.5 m.
2.3 Strategy 23
Furthermore such distance decreases with decreasing beam energy, which implies that for a 400 GeV proton beam the length to be considered should be even shorter. The spectra in every magnetic pattern are also expected to be similar in shape. Following these considerations only one magnetic pattern was implemented in the FLUKA simulations.
The beam losses in the SPS can be divided into three main kinds:
• Losses due to beam-gas interactions. • Losses due to betatron oscillations.
• Point losses in the septum magnets located at the injection and extraction
points.
The last kind is beyond the scope of this study since these points are not located in the arcs, and anyway only a very small fraction of the waste from the SPS comes from regions close to these points.
Beam-gas interactions occur when beam particles interact with residual gas molecules in the beam pipe. The intensity of the beam losses depends on the pres-sure in the beam pipe. This prespres-sure changes along the accelerator and therefore so will also the intensities of the losses. In this study they are however assumed to be uniformly distributed along a magnetic pattern.
The losses due to betatron oscillations are not uniform. The longitudinal posi-tion of these losses are determined by the variaposi-tions of the beam pipe aperture A and the β(s)-function, which is described in section 1.2.1. In the position s where the ratio √A
β has a minimum it is expected that the loss is high.
The β(s)-function depends on the operation mode of the accelerator and on the energy of the beam. Depending on these factors the beam losses occur in different longitudinal positions. During its 30 years of operation the SPS has been running with different operation modes and with different beam energies, leading to different losses in various positions. For a particular item of waste it is almost impossible to know where, with respect to the beam losses, it was located dur-ing irradiation. There is therefore no choice but to consider also the losses from betatron oscillations as uniform, rather than point losses, and estimate the error introduced by this approximation.
In order to estimate the error introduced by assuming uniform losses, the losses are simulated as point losses. The spectra in a given point of the tunnel is then obtained by summing the contributions from each loss point. Because of the as-sumption of uniformity, every loss is assigned the same intensity. This strategy allows comparing the spectra from the point losses with the spectra from the uniform losses. By doing this it is possible to calculate the error introduced by assuming that an item has been exposed to uniform losses if in reality it has been
exposed only to secondaries from a point loss.
The study focuses on the losses in the quadrupole and the losses in a dipole, assuming that the irradiation environment produced by a loss in either of the four dipoles would be equivalent. Under this hypothesis two sets of simulations were performed: one with losses in the quadrupole and one with losses in the down-stream dipole. (See figures 2.3 and 2.4. The origin of the arrows corresponds to the beam loss.) The point losses are positioned after 1/3 of the respective magnets length in order to ensure that most of the hadronic cascade is contained inside the magnet. Moreover, it is at the beginning of the magnet that most of the losses due to betatron oscillations actually take place.
The SPS accelerates protons from 25 GeV up to an energy of 450 GeV. In order to study the dependence of the spectra on primary beam energy, three sets of simulations were performed. One with a beam energy of 26 GeV, one with 400 GeV and one with the intermediate energy of 200 GeV. The value of 400 GeV was chosen because a large fraction of the historic waste from the SPS-tunnel was generated during machine operation with such beam energy.
In total, six different scenarios have been studied in order to cover the three beam energies (26, 200 and 400 GeV) and the two loss points (quadrupole and dipole).
It should be noted that the main difference between quadrupoles and dipoles, for the purposes of the present study, lays in the lateral thickness.
Lateral thickness is here defined as the amount of material in the magnet, between the beam pipe and the location of the irradiated item.
This parameter affects both the intensity of the radiation outside the magnet (self-absorption) and the energy dependence of the particle spectra (moderation).
2.3.1
Naming convention
The particle spectra resulting from the FLUKA simulations are normalized to one primary particle. The absolute values of these spectra are of interest only when calculating the spectra from uniform beam losses. Here the absolute values of the contributing spectra reflect how much each of the point losses contributes to the final spectra.
However once these calculations are performed the absolute values of the spec-tra are of minor interest since the intensity of the beam losses, responsible for the activation of a particular item, is not known. The spectra are therefore normalized to one secondary particle, according to the formula 1.16.
In order to distinguish between spectra of different beam energy, type of sec-ondary particle and loss point, subscripts are used, like Φi,j,k,l where
• i is the secondary particle type (ln for low energy neutrons, hn for high
energy neutrons, p for protons, π+ or π− for positive or negative pions).
• j is the beam energy (26, 200 or 400 GeV).
2.3 Strategy 25 • l is the relative position with respect to the loss point.
Φp,200,d,1is for example the absolute fluence of protons resulting from a point loss of a 200 GeV primary beam in the dipole, estimated in the air outside the magnet directly downstream of the loss. The notations are illustrated in figures 2.3 and 2.4.
Figure 2.3. Naming convention for spectra of protons resulting from a beam loss of 200 GeV in a quadrupole.
Figure 2.4. Naming convention for spectra of protons resulting from a beam loss of 200 GeV in a dipole.
2.3.2
From point losses to uniform losses
In order to obtain the spectra in a certain point all the relevant contributing spectra are summed. Two different sums are calculated for each energy, in order to
compare the spectra outside accelerator components of different thicknesses. The first sum concerns the spectra expected outside the quadrupole and the second the spectra outside a dipole.
The contributing spectra in a given point are the ones produced by the losses in the upstream magnets, the losses in the magnet at 90◦ from the point and the backscattering from losses in the first downstream magnet. This is illustrated in figure 2.5.
Figure 2.5. Contributing spectra assuming uniform beam losses.
For items located outside a quadrupole, the resulting spectra of protons for a beam energy of 200 GeV, is given by:
Φp,200,q= Φp,200,d,3+ Φp,200,d,2+ Φp,200,d,1+ Φp,200,q,0+ Φp,200,d,−1 (2.1) Similarly the spectra for items located near a dipole is given by:
Φp,200,d= Φp,200,d,3+ Φp,200,d,2+ Φp,200,q,1+ Φp,200,d,0+ Φp,200,d,−1 (2.2) The third subscript in Φp,200,q and Φp,200,d here indicates that the spectra are either outside the quadrupole (q) or outside the dipole (d).
One should note that the spectra in these sums are numerically the same as the ones illustrated in figure 2.3 and 2.4, but conceptually they have another meaning. For example Φp,200,d,1in equation 2.1 refers to the spectrum near the quadrupole resulting from a point loss in the dipole directly upstream of this. On the other hand in figure 2.4, Φp,200,d,1 refers to the spectrum estimated near the second dipole downstream of the quadrupole, resulting from a loss in the first dipole. In other words, the spectra from a point loss in a given position is assumed to depend on the kind of magnet where the loss has taken place but not on the kind of magnet nearby the position of interest.
2.4 The FLUKA input 27
2.4
The FLUKA input
Each of the six scenarios presented in section 2.3 was addressed with a separate set of simulations. The sets share the same geometry and physics settings. The differences are the energy of the primary beam and the location of the beam loss. The beam energies are 26, 200 and 400 GeV, and the locations of the point losses are inside the quadrupole and inside the first dipole. In the following subsections the common content of the input files is explained.
2.4.1
Geometry
The outer part of the geometry consists of a 36 m long section of the accelerator tunnel. Its walls are 20 cm thick and made of concrete (52.9% O, 33.7% Si, 4.4% Ca, 3.4% Al, 1.6% Al, 1.4% Fe, 1.3% K, 1% H, 0.2% Mg, 0.1% C). The tunnel is represented by a straight cylinder in the geometry, thus neglecting the slight curvature of a real SPS arc.1 The tunnel section houses one magnetic pattern
which has a total length of 29.5 m. The center of the beam line is located 123 cm from the floor of the tunnel, and at 136 and 264 cm distance from each of the tunnel walls. The magnetic pattern is surrounded by air (76% N, 23% O, 1% Ar). Figure 2.6 shows the geometry of the tunnel and the magnetic pattern. Figure 2.7 shows a photography of a section of the SPS arc.
Figure 2.6. Section of the SPS tunnel as implemented in the simulations.
The implemented magnets are one quadrupole and four dipoles, each separated by approximately 40 cm. Both magnets are composed of an iron yoke surrounding copper coils. The density of the iron is 7.874 g/cm3 and the density of the copper
1
The SPS ring has a radius of 1098 m. The piece of the tunnel which is implemented in the geometry has a length of 36 m. Calculations using the laws of sines and cosines show that the end of a straight tunnel is deviated of about 60 cm from the end of a curved tunnel. This deviation can be considered negligible in comparison with the size of the tunnel section.
Figure 2.7. Photography of a section of the SPS arc, containing one magnetic pattern. 8.96 g/cm3. No trace elements are included in these materials since they give a
minor contribution to the characteristics of the spectra of secondary particles. The length of the quadrupole is 305 cm and the total cross section of its yoke is 75.4×75.4 cm2. The details of the cross section can be seen in figure 2.8.
[cm]
[cm]
Cross section of the quadrupole.
-60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60
Figure 2.8. Cross sectional view of the quadrupole
The total length of a dipole is 626 cm. The yoke has a cross section of 84.0×48.1 cm2. The coils of the dipole have a more complicated structure than the ones of the quadrupole, with one part of the coil sticking out of the yoke, as can be seen
2.4 The FLUKA input 29
in figure 2.10. The cross section of a dipole is shown in figure 2.9.
[cm]
[cm]
Cross section of a dipole.
-60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60
Figure 2.9. Cross sectional view of a dipole
[cm]
[cm]
Side wiev of a dipole.
1420 1440 1460 1480 1500 1520 1540 1560 1580 -80 -60 -40 -20 0 20 40 60 80
Figure 2.10. Side view of the beginning of a dipole
The beam pipe connects and goes through all the magnets. It has an elliptical cross section inside the quadrupole and a rectangular one in the rest of the tunnel section. The pipe is made of steel (61.195% Fe, 18.5 % Cr, 14 % Ni, 3% Mo, 2% Mn, 1% Si, 0.2% N, 0.045% P, 0.03% S, 0.03% C) and filled with the FLUKA material “vacuum”. The cross section of the rectangular pipe has an inner size of 14.24×2.97 cm2and an outer one of 14.64×3.85 cm2.
