MAT-VET-F 20009
Examensarbete 15 hp Juni 2020
Simulation of fission fragments in VERDI
A study of the Plasma Delay Time phenomenon
Lovisa Rygaard
Andreas Ström
Teknisk- naturvetenskaplig fakultet UTH-enheten
Besöksadress:
Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0
Postadress:
Box 536 751 21 Uppsala
Telefon:
018 – 471 30 03
Telefax:
018 – 471 30 00
Hemsida:
http://www.teknat.uu.se/student
Abstract
Simulation of fission fragments in VERDI
Lovisa Rygaard & Andreas Ström
The purpose of this project is to study the plasma delay time phenomenon in preparation for the construction of the VERDI spectrometer. To accomplish this, simulations of the spontaneous fissioning process of Cf-252 were created using the fission code GEF, as well as MATLAB. GEF has produced one million fission events, from which the time of flight and kinetic energy of each fission fragment have been calculated with classical mechanics, to replicate the experiment. To imitate the plasma delay time phenomenon, three different models, found in the literature, have been compared. Accounting for other realistic resolution effects and using the first model as the plasma delay time phenomenon, the absolute errors of the mass-yields reaches up to 4 u, whereas the second and third models display absolute errors up to 3 u.
Furthermore, it is found that, despite the significant differences in the models' dependencies, the resulting effects are quite similar. All models are found to have a narrowing influence on the pre-neutron emission mass- yield distributions, resulting in an increased peak-to-valley ratio. In the detection of fission fragments, a higher peak-to-valley resolution is often associated with a better mass resolution. This study shows that the plasma delay time could have a misleading influence in regards to estimating an experimental mass resolution.
Ämnesgranskare: Stephan Pomp Handledare: Ali Al-Adili & Diego Tarrío
Popul¨ arvetenskaplig sammanfattning
F¨orm˚agan att unders¨oka olika atomprocesser har lagt grunden f¨or den moderna forskningen inom fysik. Fr˚an att Lise Meitner och Otto Hahn uppt¨ackte det revolutionerande fenomenet att klyva atomk¨arnor, kallat fission, i Tyskland ˚ar 1938 har det utvecklats och anv¨ants till b˚ade gott och ont. Fission har sedan dess pr¨aglat v˚art moderna samh¨alle, och ¨aven om skr¨ammande vapen och katastrofer har haft en central roll i diskussionen om fission, ¨ar det dess potential och m¨ojligheter inom k¨arnkraft och forskning som kommer leda oss in i framtiden. Idag har forskningen n˚att nya h¨ojder med utf¨orlig kunskap om atomers uppbyggnad och processer, och n¨asta steg ¨ar att utveckla samt skapa f¨orb¨attrade experimentella anl¨aggningar.
I Belgien byggs f¨or tillf¨allet den nya spektrometern kallad VERDI (VElocity foR Direct particle Identifi- cation) vars m˚al ¨ar att bli v¨arldsledande genom precisa m¨atningar av fissionsfragment, med h¨ogre uppl¨osning
¨
an vad som finns tillg¨anglig idag. F¨or att uppn˚a dessa m˚al beh¨ovs kunskap om anl¨aggningens detektorer och deras p˚averkan p˚a uppl¨osningen. Detta projekt fokuserar p˚a en av VERDIs st¨orsta utmaningar; Plasma Delay Time (PDT), vilket f¨ordr¨ojer registreringen av m¨atningar och d¨armed ger missvisande resultatv¨arden. F¨or att kunna kompensera f¨or dessa m¨atfel har olika teoretiska modeller av PDT anv¨ants, i detta projekt unders¨oks modellernas beroende p˚a uppm¨atta massor och energier. Eftersom VERDI ¨annu ¨ar under konstruktion och inga experimentella m¨atningar har utf¨ardats har ist¨allet simulerade v¨arden av fissionsevent skapats med pro- grammet GEF (A General description of Fission observables). Dessa simuleringar ˚aterskapar ideala m¨atdata, varp˚a givna v¨arden p˚a VERDIs tids- och energiuppl¨osningsfel, tillsammans med de olika PDT-modellerna, har applicerats f¨or att skapa realistika m¨atdata. D¨arefter har f¨ordelningen av fissionsfragmentens massa ber¨aknats och analyserats. Projektets resultat best˚ar av grafer av de olika PDT-modellernas beroende p˚a massa och energi samt massf¨ordelningar, p˚averkade av olika PDT-modeller. ¨Aven om modellerna p˚avisade tydliga skillnader i mass- och energiberoenden, visar projektet att dessa har snarlik inverkan p˚a massf¨ordel- ningarna.
Contents
1 Introduction 1
2 Theory 2
2.1 Experimental Setup . . . 2
2.2 Fission . . . 2
2.3 Semiconductors . . . 5
2.3.1 Doping and Leakage Current . . . 5
2.3.2 Junctions, Depletion Region and Reverse Bias . . . 5
2.3.3 Neutron Transmutation-Doping . . . 6
2.3.4 Surface Barrier Detectors . . . 7
2.4 Plasma Delay Time . . . 7
2.4.1 Bias Dependency . . . 7
2.4.2 Model 1 . . . 8
2.4.3 Model 2 . . . 8
2.4.4 Model 3 . . . 9
2.5 Other Time Resolution Effects . . . 9
2.6 Energy Resolution Effects . . . 9
2.7 Simulations with GEF code . . . 10
2.8 Kinematics . . . 10
3 Method 12 4 Results 15 4.1 Resolution Effects, excluding PDT . . . 15
4.2 Plasma Delay Time . . . 17
4.2.1 Model 1 . . . 17
4.2.2 Model 2 . . . 18
4.2.3 Model 3 . . . 20
4.2.4 Uncertainties . . . 21
4.3 Effects on Individual Masses . . . 22
5 Discussion 24 5.1 The Impact of Time and Energy Resolution Effects . . . 24
5.2 The Impact of Plasma Delay Time . . . 24
5.3 Sources of Error . . . 26
6 Summary and Conclusions 26
Refrences 27
Appendix 28
1 Introduction
In the last several decades, great progress has been made towards achieving new, profound knowledge of the elements shaping our reality. The recent advancements in nuclear physics have been among some of the most important contributions to research in modern physics. Today, the next step in nuclear research is to improve experimental facilities and their detectors to achieve more precise and reliable results. As part of this progress, the VERDI (VElocity foR Direct particle Identification) spectrometer is being constructed to measure fission fragments with better performance than what is possible for the spectrometers operating today. The design of VERDI was created to achieve high geometrical efficiency. It follows the main ideas of the already existing spectrometer COSI FAN TUTTE but with about 200 times higher efficiency. With this design, the goal is to attain mass resolutions greater than 2 atomic mass units.
One of the main challenges to reach the goals of the VERDI spectrometer is the Plasma Delay Time (PDT) phenomenon in the silicon detectors. The PDT delays the registration of measurements in the detec- tors, extending the fission fragments’ perceived Time Of Flight (TOF) and thereafter affecting the further computation of fragment properties. The PDT can be compensated for in calibrations of the experimental setup, but as the precise nature of the PDT effects is unknown, a better understanding of the PDT’s behavior is required to attain successful adjustments.
As a first step towards a solution to the described problem above, this project has been designed to simulate fission events with the purpose to study the influence of resolution effects, including PDT, on the fission mass-yields. Three theoretical models defining possible behaviors of the PDT are used and reconstructed to emphasize the mass and energy dependency. By applying the PDT models to the data simulated by GEF [14], along with other time and energy resolution effects, the change in mass-yields can be compared to the ideal case. The work process of the conducted study is illustrated in the flow chart in Figure 1. It is expected that different areas of the restriction can be identified by examining the effects on the mass-yields. This would predict what resolution of the TOF is needed to avoid loss of accuracy, as well as the effect the PDT would have on the results. The results will hopefully be used in a proposed experiment in France, conducted by the nuclear reactions group within the division of applied nuclear physics at Uppsala University, and with the simulations of this project the analysis of the experimental data can be streamlined.
Figure 1: A flow chart to illustrate the order in which the various parts of this work are conducted.
2 Theory
This project will focus on two types of detectors in the VERDI spectrometer and study the possible in- fluences the detectors have on simulated measurements. Therefore, the preparatory theory below includes an introduction to the VERDI spectrometer and the properties of its detectors. Furthermore, fundamental knowledge of the fissioning processes and semiconductors is presented, as well as necessary kinematics. These fundamentals provide sufficient knowledge to understand the following theory of the important aspect of PDT as well as other time and energy resolution effects.
2.1 Experimental Setup
The VERDI spectrometer is under construction at the European Commission’s Joint Research Centre (JRC) in Geel (Belgium) to measure fission fragments. The setup consists of silicon detectors, Micro Channel Plates (MCP), and electrostatic mirrors, illustrated in Figure 2. There are in total 16 silicon detectors, which are spherically placed about 0.5 meters away on both sides of the sample. The specific detectors used can vary, and an example of a typical detector chosen is explained in section 2.3. This placement makes up a two-armed spectrometer detecting the properties of two fission fragments, with antiparallel velocities, simultaneously.
The MCPs are placed close to the target nuclei to measure the time of the fission event, these are implemented in a FIETS (FIssion Electron Time-of-Flight Start) detector designed to record the time of electron emissions with minimal energy losses. To measure the electron emissions, the electrostatic mirror is used to redirect electrons to the FIETS detector. The measurements in VERDI will be able to determine the distribution of mass and total kinetic energy for the fission fragments, both before and after neutron emission. The mass differences will thereafter give the neutron multiplicity in each event.[2]
Figure 2: The experimental setup showing the silicon detectors, MCPs, and electrostatic mirrors (green).
252Cf is an example of a possible target to use in VERDI, fissioning into two fission fragments (red). [1]
2.2 Fission
The fission phenomenon means that one nucleus splits into, for most events, two smaller isotopes, called fission fragments. In this section the fundamentals behind this process are discussed. The strong force is the primary binding agent between the different nucleons of a nucleus, but as this force has a very short range (around 2 fm) it is not always adequate to keep the heavier nuclei together. Nucleon Pairing, Odd-Even effect, and Coulomb repulsion effect are a few phenomena that alter this binding and as the Coulomb force is inversely proportional to nucleons’ distance apart, these effects help put an upper limit on the mass of
nuclei. Since these effects have different dependencies on the mass and atomic numbers, heavier nuclei can not be assumed to be spherically symmetric, which further encourages these nuclei to split into smaller, and more stable, nuclei.
In the liquid drop model (LDM) a nucleus in the fissioning process is seen as having the shape of a liquid drop affected by surface tension (Figure 3). This model describes a nucleus’s change in shape from the start of a fission event to the scission point, the point where the original nucleus is split. The corresponding potential barrier attributed to a nucleus by this model is seen in Figure 4 (blue, dashed curve). However, this model only treats average nuclear properties and even if this model has had a great impact on nuclear physics, corrections to it had to be made [9]. These corrections alter the LDM based on nuclear pairing and the shell model, changing the shape of the fission barrier as seen in Figure 4 (black, solid curve). This mainly affects heavier nuclei that, in the LDM, only would exist for short periods of time. Figure 4 is however a simplified version as the fission barrier, in reality, depends on a plethora of quantities, making it a complex multi-dimensional characteristic. An example of this is seen in Figure 5, showing a complex fission barrier with various deformation parameters, here projected on the two dependencies elongation and mass asymmetry.
Figure 3: The model of a fissioning nucleus in the LDM. [9]
Figure 4: A schematic representation of the fission barrier according to the LDM (blue, dashed line), with shell corrections (black, solid line) and an example of spontaneous fission via tunneling. Different energy levels the nuclei can occupy are also marked as black lines.
Figure 5: A more in-depth visualisation of the fission barrier, rendered by Karpov et. al. [7]
To get a basic understanding of what enables fission, the energies of the nuclei can be examined. As an example, the heavy nucleus 25298 Cf, which has a binding energy of 1881 MeV, undergoes fission into the two equally heavy nuclei12649 In with binding energies of about 1060 MeV each [12]. The resulting energy difference in the decay would be 2 ∗ 1060 − 1881 = 239 MeV, which is given to the fission fragments as kinetic energy or excitation energy. The amount of excess energy determines the fission fragments’ energy level (Figure 4) and affects the probability of spontaneous fission. These larger excess energies are achieved for heavier nuclei, meaning that spontaneous fission primarily is probable for nuclei with mass numbers above 250. Although, the possibility of fissioning is significantly reduced by the Coulomb barrier of the nucleus. For the given example, the value of the Coulomb barrier is 276 MeV, according to equation (1) with Z1 = Z2 = 46 and R1= R2= 1.25(126)1/3fm for the fission fragments. The fact that the energy difference of the reaction does not surpass the energy needed to transverse the Fission barrier would make it impossible for this reaction to occur in classical mechanics. However, with the introduction of quantum tunneling the probability of splitting reactions, with energies lower than the energy of the fission barrier, becomes a non-zero entity.
This is the effect demonstrated in Figure 4, where the fission event occurs from an energy state below the fission barrier height. The height of the fission barrier will decrease for fission fragments with larger mass differences, to an extent, which is a reason behind the shape of the mass-yields of Figure 9.
V = 1 4πε0
Z1Z2e2 R1+ R2
(1) Fission fragments are very neutron-rich nuclei, meaning that the proportion of neutrons to protons is quite high, creating unstable isotopes. The excess of neutrons leads to emission of neutrons, both instantly after fissioning (in about 10−16 s) as prompt neutrons and after (normally) a few seconds as delayed neutrons.
Because the TOFs in this project are of the magnitude of 10−8 s, the neutrons studied at VERDI will be prompt neutrons. The number of these emitted neutrons is dependent on the fission fragments and the type of fission, such as neutron-induced or spontaneous fission. [9]
2.3 Semiconductors
A semiconductor is a material that has a specific conductivity, determined by its bandgap of about 1 eV. In the following sections semiconductor properties and applications, such as doping, leakage current, junctions and particle detection will be discussed.
Semiconducting materials are signified by their atoms mainly having four valence electrons, which can create a diamond lattice structure through covalent bonds between these atoms. This results in a material that has a uniformly filled valence band and an empty conduction band. As previously stated these semiconductors have a bandgap of about 1 eV, meaning that the energy difference between the semiconductors valence and conduction band is around 1 eV. Theoretically this would be an insulator (given an absence of thermally created electron-hole pairs), since all electrons are tightly bound. However, once external forces act upon the material these valence electrons can be excited to the conduction band, and, given the presence of an electrical field, these electrons act as charge carriers since they now freely move around the conduction band.
The necessity for a significant electric field arises as each excited electron leaves a hole behind, which the electron is free to occupy again unless moved away. The hole in itself also acts as a positive charge relative to its surrounding. These electron-hole pairs comprise the charge carriers of the semiconductors. [8]
2.3.1 Doping and Leakage Current
When a semiconductor is doped it means that it has a small designed impurity-concentration, usually made up of elements with either three or five valence electrons. This leads to another crystalline lattice of atoms, only now with these impurities spread around. Each of these impurities contributes with an extra electron or hole, depending on the choice of dopant, and are called acceptor or donor sites. The benefit of doping a semiconductor is that it has a certain number of extra charge carriers, originating from the aforementioned impurities. These charge carriers are extra susceptible to excitation owing to the fact that they are more loosely bound in relation to the rest of the electrons.
As a substantial electrical field usually is necessary to collect charge carriers effectively after an ionizing event, there is always a resulting leakage current in semiconductors (especially doped ones), since even small fluctuations in thermal conditions or purity lead to a production of charge carriers. This leakage current can be a significant source of noise when collecting charge carriers originating from ionizing radiation, the different ways this can be combated are brought up in the next section. [8]
2.3.2 Junctions, Depletion Region and Reverse Bias
Once a positively doped (p-doped) and a negatively doped (n-doped) area are created next to one another, there is an immediate diffusion of charge carriers as the electrons of the n-doped area combine with the holes of the p-doped area. This diffusion leads to a non-zero net charge distribution (ρ(x)) within what is named the depletion region (Figure 6), where all, or a fraction, of the donor-/acceptor-sites, are neutralized by the opposite charge carriers. This charge distribution is what constitutes an n-p junction, as seen from Figure 6 there is a difference in electric potential (called ”contact potential”), which gives rise to an inherent electrical field. The good thing about this field is that it opposes further diffusion and becomes somewhat of a steady-state electrical current of newly created charge carriers since these are accelerated to either side by the field. This allows for a decently efficient collection of said charge carriers, as they are sorted and should not interfere more than necessary with one another, through e.g. recombination effects. Applying an external electrical field (a bias) leads to the junction obtaining some useful properties.
Figure 6: The charge density with highlighted depletion region over an n-p junction. [8]
A forward bias is where the n- and p-doped sides get a negative respectively positive voltage applied. The applied voltage leads to a counteraction to the aforementioned charge distribution (Figure 6), which tech- nically would work as a detector since the charge carriers would be sorted. However, this would be highly inefficient as a lot of the majority charge carriers would recombine as they pass through each other. Instead, a reverse bias is applied, enhancing the junction’s intrinsic properties. Since the resistivity of the depletion region is huge compared to the n- and p-doped areas, virtually all of the reverse bias voltage is applied to this area. This leads to an efficient sorting of electrons and holes, where instead of moving the majority charge carriers through each other (e.g. the doped electrons moved through the p-doped area and vice versa), the minority charge carriers are sorted to the corresponding area. This means that e.g. the electrons are moved to (and through) the n-doped area instead of having to transverse the p-doped area, facing the sea of holes existing there.
The depletion region is created by the electrical field and the field strength correlates to the size of this region. As it is the charge carriers created within the depletion region that are accelerated by the bias, a larger depletion region means that a larger area can be used as a detector. Depending on if the entire semiconductor is depleted it is named fully depleted or partially depleted. However, if the reverse bias becomes too large the covalent bindings will start to break down, leading to permanent damage to the junction, meaning that there is an upper limit to this applied voltage. [8]
2.3.3 Neutron Transmutation-Doping
Neutron Transmutation-Doping (NTD) is a specific method used to dope semiconductors, further explained below. There exists a variety of methods for doping, such as electrochemical, magnetic, modulation, and transmutation doping. Transmutation effects usually involve introducing high energy charged particles, photons, or, as in this case, neutrons to a nucleus. If the nucleus absorbs the neutron, it might produce a new unstable isotope, leading to a deterioration to a stable state. NTD is superior to other methods due to the resulting, extremely uniform, dopant concentration.
NTD is an efficient method for doping silicon. This comes from the fact that natural silicon is composed of three main isotopes, 28,29,30Si, where30Si compose about 3% of the material. Adding an extra neutron to the other two isotopes (28,29Si) only transform them into other stable isotopes. However, when the 31Si isotope is created it deteriorates via β−-decay into31P (2).
30Si + n →31Si∗→ e−+ ¯ν +31P (2)
As phosphorus has five valence electrons this creates a n-doped silicon semiconductor with an extremely uniform dopant concentration. [5]
2.3.4 Surface Barrier Detectors
A Surface Barrier Detectors (SBD) is a form of detector where the junction is created by starting with an n- or p-doped semiconductor, and re-doping one end of it, to obtain a p-n junction. This is a quite common form of detector and can be achieved by etching the surface of an n-doped semiconductor and then evaporating a golden foil, to create a p-doped area with a good electrical connection to the n-doped area. The process would be similar if it begins with an n-type semiconductor, only now there would be a thin aluminum foil to create an n-doped area. In these detectors the resulting dead layer is quite thin, which is preferable as the distance traveled through this layer correlates to the energy loss of the ion, further discussed in section 2.6.
[8]
2.4 Plasma Delay Time
Plasma Delay Time (PDT) is a time resolution effect arising from the properties of the detectors, which will be of great importance to this project. Therefore, the origin of PDT is presented, along with different properties that affect its value. The three models below are examples of experimentally tested equations to calculate the PDT.
The PDT originates from a plasma-like cloud shielding the inner charge carriers in the material. The cloud is created from high densities of charge carriers, which is common when detecting heavy ions such as fission fragments. As the exterior charge carriers drift away, the interior of the plasma cloud will gradually be exposed to the electrical field and the plasma decays. This results in the PDT phenomenon, which is the added time for the plasma cloud to dissolve. The delay will, therefore, be the difference in time from the moment the heavy ions impact the detector to when the electrical signal reaches the electrodes. The high density of charge carriers also interferes with the applied electrical field by slowing down the movement of charges. The PDT does most likely depend on the bias voltage, as well as the mass and energy of the fission fragments. [8]
2.4.1 Bias Dependency
Practically every model for PDT (τd) in the current literature treat the PDT as having a correlation to the bias (F ) [4, 10, 11, 13, 15], discussed below. The dependency of the bias for the PDT has been experimentally examined in several experiments, such as by Neidel and Henschel in [10] for fission fragments of252Cf. The experimental results are shown (together with results for alpha decays) in Figure 7 together with a fitted curve. The results show an inverse linear dependency of the bias (Electric Field in the figure) for values above 16 kV/cm, which can be expressed as τd = 27.75/F (ns cm/kV) for 16 . F . 25 (kV/cm). If the electric field can be assumed to be below 16 (kV/cm) (meaning that (Electric Field)−1 is higher than 0.0625 (cm/kV)), the value of the PDT can be approximated to 2.5 ns according to Figure 7.
Figure 7: The dependency of the (Electric Field)−1 [cm/kV] for the PDT, for fission fragments of252Cf and alpha particles. [10]
2.4.2 Model 1
According to the article [15] written by Velovska and McGrath, the PDT can be calculated using equation (3).
τd= Z¯2 2Aexp
−E 3.75A
Cρ
d
2/5 1
F (3)
The parameters thickness d, capacity C and resistivity ρ are properties of the detector and ¯Z is the effective charge of the incoming ion. [15] This model’s PDT also has an inverse proportionality to the bias F , which has been quite common throughout the literature.
2.4.3 Model 2
H.-O. Neidel and H. Henschel in [10] has also made an approximation of the PDT depending on the mass (A), energy (E) and bias (F ) according to equation (4).
τd= 1.33A1/6E1/2
F (4)
This equation has been tested in different experiments for alpha particles, 252Cf [10] and later on also for
238U [11]. The latter experiment was able to confirm that the PDT will increase with the energy of 238U, which follows the predicted relation of equation (4). This also contradicts Model 1.
2.4.4 Model 3
M. Ogihara et al. describes another model depending on the energy in [13], as shown below.
τd= 3.25 ∗ 10−12ρ0.14Z0.36(7.5 ∗ 1014SEd)1/3F−0.85 (5) The energy in the detector Ed is given as the energy of the fission fragments subtracted with the window defect. Similarly to Model 1, this equation have detector dependent properties such as the resistivity ρ and stopping power S. This model differs from the others by not being explicitly dependent on the mass number A, but includes the atomic charge Z which is roughly proportional to the mass.
2.5 Other Time Resolution Effects
Apart from the PDT, various other phenomena affect the measured time, to account for this in the simulations, a Monte-Carlo type approach will be used. For each calculated TOF a randomized value will be extracted from a Gaussian distribution and added to the TOF. This section describes these phenomena originating from time resolution effects other than PDT.
The uncertainties of the time resolutions are given by two categories. For input pulses with constant amplitudes, the size and shape of the signal are affected by random fluctuations such as noise of the detector’s electronic components. This category is called time jitter. Other examples of this are statistical fluctuations originating from a decrease of charge carriers or the leakage current, which affects the discrete signal of the detector. This will also result in a changed size and shape of the signal. The other category of time resolution effects is time slewing, which occurs when the input pulses have varying amplitudes. For ideal cases, the amplitudes should, therefore, be constant, so that the resolution only depends on jitter, but since the amplitudes of fission fragments’ energies vary, the time slewing also has to be accounted for. This effect is more important for detectors with fewer charge carriers that are more sensitive to fluctuations. [8]
The silicon detectors and the MCPs in this experiment will both have fluctuations affecting the timing resolution. These effects will originate from different aspects since the silicon detectors receive heavier particles with higher energies than the electrons detected by the MCPs. In 2015 the timing resolution was said to be 140 ps for the FIETS detectors (including the MCPs) and 150 ps for the NTD silicon SBDs. This is given by M. O. Fr´egeau and S. Oberstedt in [3]. These values result in a combined timing resolution of 205 ps for the VERDI spectrometer.
2.6 Energy Resolution Effects
Similarly to the section above, there will also be effects that affect the energy measurement. Analogously these phenomena will be simulated as randomized values extracted from a Gaussian distribution, and are described below.
One factor that affects the resolution is the dead layer of the detector, mentioned in section 2.3.4, this is based on the fact that ionizing radiation deposits energy along its path in matter, as in Figure 8. For particle detectors, this phenomenon plays a role in increasing the uncertainty of the measured energy, as a thicker dead layer means a larger possibility for energy loss of the charged particle.
Figure 8: A typical Bragg curve, in this case with a shape similar to that created by α-particles.
The PDT is another factor that affects the energy measurement. This arises as the plasma cloud (explained in depth in section 2.4) gives the charge carriers a slight window to recombine, leading to a varying pulse height defect (Ephd). Said pulse height defect (PHD) is calculated as equation 6 by Velkovska et. al. [15]
this is originating from an empirical formula derived in an earlier paper, ”Systematic Measurements of Pulse Height Defect for Heavy Ions in Surface-Barrier Detectors” [13], where it is stated that this effect is not as prominent as other factors regarding the PHD. Ogihara et. al. describes these three effects as the window, nuclear stopping, and residual defect. Where the residual defect is the one correlating to the aforementioned plasma cloud.
Ephd= 2.33 × 10−4(AZ)65 rE
A (6)
However, none of these factors specifically tackle the variance of these phenomena σ2E, which is the more interesting value for these simulations. This quantity is difficult to pinpoint for a large spectrum of fission fragments and, originating mainly from the energy loss in the target of252Cf, the standard deviation σE will have a magnitude in the order of a few tens of keVs.
2.7 Simulations with GEF code
For this project fission events are simulated with the code GEF (a GEneral description of Fission observables).
This code can generate simulated data of spontaneous fission, neutron-induced fission, and fission of other nuclei with a given energy and angular momentum. The data is based on the physical and mathematical properties of the fundamentals of fission and are, in this work, assumed to be the ”true” values of the fission, based on the description of the code [14].
2.8 Kinematics
The calculations in the project are based on the non-relativistic kinematics presented below. Because of the two-armed setup of VERDI with detectors on both sides, the 2E-2v method can be used advantageously for calculations. The fission fragments before neutron emission will also have the same amplitude of momentum but in opposite directions (7). By using the classic mechanical equation for the kinetic energy, the conservation
of momentum can be rewritten to depend on the kinetic energy (8). The mass m∗ and velocity v∗represent fission fragment properties before neutron emission, while m and v represent their properties after neutron emission.
m∗1v1∗= m∗2v∗2 (7)
m∗1E1∗= m∗2E2∗ (8)
Because the kinetic energy of each individual fission fragment is not given by the event-by-event generator in GEF, the energies can instead be calculated using the given relations above and the total kinetic energy T KE∗ = E1∗+ E2∗ (9). Rewriting of the equation yields both the individual kinetic energy of the second fission fragment (10), and also of the first fission fragment by changing the nominator to m∗2.
m∗1
m∗2 = E2∗
T KE∗− E∗2 (9)
E∗2= T KE∗ m∗1
m∗1+ m∗2 (10)
Even if equation (11) is an approximation for an individual fission fragment, the velocities of the fragment, once averaged over multiple events, will be the same before and after the neutron emissions. This is an effect that emerges as the neutron emission is a process which can be assumed to be isotropic, in the frame of the individual fragment. Consequently, the energies post-neutron emission can be calculated using the approximation shown in equation (12). Using the described relations, the velocities, post-neutron emission, are attained (13), and, with the distance d to the detectors, also the ideal TOF ti (14).
vi∗≈ vi (11)
Ei≈ mi
m∗iEi∗ (12)
vi=r 2Ei
mi (13)
ti= d
vi (14)
The calculations above are used to attain a representation of values similar to what is detected by VERDI.
Therefore, a PDT τdi and other time resolution effects ∆ti, described in the subsections 2.4 and 2.5, are applied to the TOF for each event (15). Similarly, the calculated energies are applied with energy resolution effects (16), see section 2.6. The values of each individual resolution effect (∆ti, ∆Ei) are random values from Gaussian distributions (N (µ, σ2)) with standard deviations (σTOF, σE) obtained from the literature, as shown in equation (17). The resulting values will thereby resemble the measured data, including resolution effects originating from the detectors.
TOFi= ti+ ∆ti+ τdi (15)
∆ti ∼ N (0, σTOF2 )
∆Ei∼ N (0, σ2E)
(17)
To recalculate the masses of the fission fragments, both before and after neutron emission, now with resolution effects, the conservation of momentum is used again. With the mass of the fissioning nucleus mCN = m∗1+ m∗2 the mass of fission fragment pre-neutron emission m∗1is achieved (18), as well as m∗2with v1∗in the nominator.
The velocities v∗1and v2∗are given by equations (13) and (11). The mass of the fission fragments post-neutron emission are given by the calculated kinetic energies and velocities (19).
m∗1= mCN v∗2
v1∗+ v2∗ (18)
mi=2Eicalc
vi2 (19)
3 Method
The practicalities that went into performing this project mainly consisted of extracting fission data from GEF, processing the resulting data to mimic measured data from VERDI, and adding resolution effects according to literature. An explanation of these parts is what follows.
The first step of the project was to obtain knowledge of the different components of VERDI, mainly of the operating detectors and their impact on the time and energy resolutions. The three presented PDT models in section 2.4, as well as the current values of the detectors’ energy and other time resolution effects (sections 2.5 and 2.6) in VERDI were derived from the existing literature.
The simulations have been created using data from GEF. The target nucleus that has been studied is
252
98 Cf, which undergoes spontaneous fission with a decay probably of 3.1%. Therefore, the fissioning nucleus in GEF has been specified as25298 Cf, the fissioning process as spontaneous, and the enhancement factor set to 10. The enhancement factor directs GEF to simulate one million events, otherwise set to 100000 by default.
Under output options the desired file type is given as “.lmd”, to receive an ASCII file of raw data, here the
“list-mode” values are specified to Z1, Z2, A1pre, A2pre, A1post, A2post, TKEpre, and TKEpost. This data has been considered to be ideal for upcoming comparisons, and an example can be seen below in Figure 9.
60 80 100 120 140 160 180 200 Mass (u)
0 1 2 3 4 5 6 7
Yield (%)
Comparison masses Pre-neutron emission
mean d = 0 ns TOF = 0 ns
GEF
Calculated values
Figure 9: An ideal mass-yield distribution for252Cf, created by the code GEF (red) and recalculated without PDT τd or other time resolution effects σT OF (blue).
The raw data from GEF was used to calculate values representing the measurements in VERDI. The nuclear masses (A1pre, A2pre, A1post, A2post) and the total kinetic energies (TKEpre, TKEpost), both post- and pre-neutron emission, have been used in equations (7)-(14) to calculate the fragment specific TOF and kinetic energy for each simulated event.
After a fragment’s TOF and kinetic energy were calculated, the energy and time resolution effects were added using typical values derived from existing work in the literature. The effects were modeled as random values from Gaussian distributions, created using MATLAB’s “randn” [6] function. The standard deviations of these distributions have been varied to show both the realistic effects and exaggerated effects, which better demonstrate these effects’ impact on the simulations. The time and energy resolution effects were then added to the calculated TOF and kinetic energy, respectively.
As this project has not taken the specific detector used into account, various approximations regarding the PDTs have been made. The three PDT models from literature have been modified to only depend on mass (m) and kinetic energy (E) of the fission fragments, shown in equations (20), (21), and (22). These simplifications were made to compensate for the unknown detector specific parameters needed to calculate the correct PDT in VERDI. To calculate each individual event’s PDT the three constants c1, c2 and c3
have been calculated using the mean values for each fission fragment’s mass and kinetic energy (both in MeV), together with the approximation of 2.5 ns for the PDT (τd) described in section 2.4.1. As this project is mainly interested in the general effect PDT has on the detection of fission fragments and because the detector specific properties are unknown, the PDT’s value τd mainly acts as a “normalization” constant in this project, which gives some leeway when choosing its specific value. Once the constants were determined each event’s individual PDT was calculated and added to the corresponding TOF. Figure 10 displays a schematic representation of the three different PDT models’ mass and energy dependency, which shows a similarity between Models 2 and 3, and their difference from Model 1. For all three models, the calculated PDT for each event was plotted against the individual fragment’s mass and kinetic energy, in a bivariate histogram. Mean values of the PDT for each mass respectively kinetic energy were also included in these histograms. In equation (5) Model 3 is shown, it has a dependence on the fragment’s charge number Z, which is proportionally dependent on a fragment’s mass m. In this project this has been omitted, and the model is seen as lacking a mass dependence (equation (22)). Moreover, the stopping power S in Model 3
τd1= c1
1
mexp −E m
(20)
τd2= c2m1/6E1/2 (21)
τd3= c3E1/3 (22)
2 75 2.5
95 128
3
PDT (ns)
115 110
3.5
PDT Model 1 schematic mesh
Mass (u)
93 Energy (MeV) 4
134154 59 76
174 41
2 2.5 3 3.5 4
PDT (ns)
(a) Model 1.
1.5 75 2
95 128
2.5
PDT (ns)
115 110
PDT Model 2 schematic mesh
3
Mass (u)
93 Energy (MeV) 134
3.5
154 59 76
174 41
1.6 1.8 2 2.2 2.4 2.6 2.8 3
PDT (ns)
(b) Model 2.
1.8 75 2
95 2.2
128
PDT (ns)
2.4
115 110
PDT Model 3 schematic mesh
Mass (u) 2.6
93 Energy (MeV)
134154 59 76
174 41
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7
PDT (ns)
(c) Model 3.
Figure 10: Schematic representation of the different PDT models’ mass and energy dependency.
The various resolution effects described above were used to re-calculate the fission fragment masses. Once the TOFs were adjusted, each event’s (pre-neutron emission) mass was calculated, using the non-relativistic kinematics in equations (15)-(19). The resulting mass-yields and corresponding error plots were made in both linear and logarithmic scales (although only presented in linear scale), together with a comparison to the ideal values obtained from GEF. All calculations and plots were made using MATLAB and the main script is shown in the Appendix.
4 Results
This section begins by presenting the inherent relationship between a fission fragment’s kinetic energy and its mass. Thereafter results regarding the time and energy resolution effects (excluding PDT), the various PDT models, errors of these models, and how they affect singular masses follow.
Figure 11 shows how the simulation given by GEF has a wide spread of values of kinetic energy for each mass of the fission fragments. A colour representation of the data is also displayed, showing the high concentration of energy values around the mass values 110 u and 145 u. The mean values of the energies given for each mass (red stars) appear to be almost constant for lower masses, while there is a linear energy decrease for heavier masses.
80 100 120 140 160
Mass (u) 50
60 70 80 90 100 110 120
Kinetic energy (MeV)
Mass-Energy plot, pre-neutron emission
0 1000 2000 3000 4000 5000 6000
Counts (#)
Figure 11: The relationship between mass and kinetic energy for each pre-neutron emission fragment, without any added effects. Mean values of the kinetic energy for each mass are represented in red.
4.1 Resolution Effects, excluding PDT
Below are the calculated mass-yields with time (excluding PDT) and energy resolution effects. The resolution effects are given by random numbers attained from Gaussian distributions with realistic standard deviations (σTOF, σE) obtained in the literature, and also exaggerated values of the standard deviations (σTOFex, σEex) to illustrate the resulting shapes of the mass-yields.
In Figure 12 the time resolutions are 0.14 ns for the MCP and 0.15 ns for the silicon detectors. As shown in the figure, the resulting realistic resolution effect is 0.205 ns while the exaggerated value is 1 ns. The results are displayed for both pre- and post-neutron emission masses and although the realistic effects barely show any change in mass-yields, it is clear that the post-neutron masses are more affected by the resolutions effects.
60 80 100 120 140 160 180 200 Mass (u)
0 1 2 3 4 5 6 7
Yield (%)
Comparison masses Pre-neutron emission
mean d = 0 ns
TOF = 0.205 ns TOFex = 1 ns
GEF Realistic effects Exaggerated effects
(a) Mass-yield of pre-neutron emission masses
60 80 100 120 140 160 180 200
Mass (u) 0
1 2 3 4 5 6 7
Yield (%)
Comparison masses Post-neutron emission
mean d = 0 ns
TOF = 0.205 ns TOFex = 1 ns E = 0 keV
GEF Realistic effects Exaggerated effects
(b) Mass-yield of post-neutron emission masses
Figure 12: Time resolution effects, excluding PDT, with a realistic standard deviation of the time resolution effects σTOF (blue) and an exaggerated standard deviation σTOFex(green).
Figure 13 shows the calculated mass-yield with energy resolutions for only post-neutron emission masses, since the energy resolution does not affect the mass-yield before neutron emission. The realistic value of the energy resolution is 50 keV and the exaggerated value is 5000 keV. Similarly to the time resolution, the realistic energy resolution values show almost no difference from GEF’s simulated values.
60 80 100 120 140 160 180 200
Mass (u) 0
1 2 3 4 5 6 7
Yield (%)
Comparison masses Post-neutron emission
mean d = 0 ns
TOF = 0.205 ns E = 50 keV Eex = 5000 keV
GEF Realistic effects Exaggerated effects
Figure 13: Energy resolution effects, with a realistic standard deviation of the energy resolution effects σE (blue) and an exaggerated standard deviation σEex(green).
4.2 Plasma Delay Time
The various PDT models’ dependencies on mass and energy, as well as the mass-yields obtained after applying these effects, with accompanying errors, are presented below. As described in section 3 the constants c1, c2 and c3 are calculated using an average value τd of 2.5 ns and the mean values of the fragments’ mass and energy in natural units, resulting in the different constants’ values shown in Table 1. These constants together with a fragment’s mass and kinetic energy then yield the PDT for each fragment of each event.
c1 290000 (ns MeV)
c2 0.037 (ns MeV−1/12)
c3 0.55 (ns MeV−1/3)
Table 1: The resulting of the constants values used for the three different PDT models.
4.2.1 Model 1
Model 1 displays a dominant dependency on mass (Figure 14a) and in comparison an almost non-existent dependency on energy (Figure 14b), which can be seen from the lack of, respectively, the large spread of counts in the histograms (Figure 14). It can seem like the model has a distinct energy dependency, however, this is mainly a consequence of the strong correlation between a fission fragment’s mass and energy, arising from the Kinematics. This correlation explains how the general shape in Figure 14b bears resemblance to Figure 11, with a rotation due to the change in axes. This model displays a total spread barely over 2 ns.
In Figure 15 Model 1’s raw effect on the mass-yields are shown in green. Making the simple alteration of shifting the position of these peaks to the theoretically expected position of the peaks (in this case given by GEF), the more realistic (blue) data set is obtained. In the following subsections, 4.2.2 and 4.2.3, only these more realistic data sets are shown. Figure 15 also shows a clear increase in the peak-to-valley ratio, both for the pre- and post-neutron emission mass-yields, which will be discussed in section 5.
80 100 120 140 160 Mass (u)
2 2.5 3 3.5 4
PDT (ns)
PDT Model 1
0 1 2 3 4 5 6
Counts (#)
104
(a) The PDT’s mass dependency.
50 60 70 80 90 100 110 120
Energy (MeV) 2
2.5 3 3.5 4
PDT (ns)
PDT Model 1
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Counts (#)
(b) The PDT’s energy dependency.
Figure 14: PDT Model 1, with mean values of the PDT for mass respectively energy displayed in red.
60 80 100 120 140 160 180 200
Mass (u) 0
1 2 3 4 5 6 7 8
Yield (%)
Comparison masses Pre-neutron emission
PDT Model 1
mean d = 2.561 ns TOF = 0.205 ns
GEF
Calculated values Unshifted values
(a) Mass-yield of pre-neutron emission masses.
60 80 100 120 140 160 180 200
Mass (u) 0
1 2 3 4 5 6 7 8
Yield (%)
Comparison masses Post-neutron emission
PDT Model 1
mean d = 2.561 ns TOF = 0.205 ns E = 50 keV
GEF
Calculated values Unshifted values
(b) Mass-yield of post-neutron emission masses.
Figure 15: PDT Model 1, displaying both the effect a PDT has on the mass-yields in green, with the same calculated mass-yields in blue, now also containing a compensation of the shift the PDT produces. Realistic energy resolution effects and other time resolution effects also accounted for as σEand σTOF.
4.2.2 Model 2
As can be seen from the spread of the counts in Figures 16a and 16b, Model 2 has a dominant energy dependence, but is still significantly affected by a fragment’s mass. In Figure 16a it appears that the PDT has a strong correlation to the fragment’s mass, but the close resemblance to Figure 11 is a telltale sign that
this mainly is a consequence of a fission fragment’s inherent mass-energy correlation. The slope of the mean PDT values for lower masses in Figure 16a signifies that Model 2 does have a mass dependency, although this is not as strong as if one neglects the aforementioned mass-energy dependency. The spread of the PDT reaches about 1 ns for this model. The pre-neutron emission mass-yields (Figure 17a) obtained using this model have a shape akin to that of Model 1 (Figure 15a). However, the post-neutron emission mass-yields in Figure 17b are , for lighter masses, distinctly different than those obtained by Model 1 (Figure 15b).
80 100 120 140 160
Mass (u) 1.8
2 2.2 2.4 2.6 2.8
PDT (ns)
PDT Model 2
0 500 1000 1500 2000 2500 3000 3500
Counts (#)
(a) The PDT’s mass dependency.
50 60 70 80 90 100 110 120
Energy (MeV) 1.8
2 2.2 2.4 2.6 2.8
PDT (ns)
PDT Model 2
0 2000 4000 6000 8000 10000 12000 14000 16000
Counts (#)
(b) The PDT’s energy dependency.
Figure 16: PDT Model 2, with mean values of the PDT for mass respectively energy displayed in red.
60 80 100 120 140 160 180 200
Mass (u) 0
1 2 3 4 5 6 7
Yield (%)
Comparison masses Pre-neutron emission
PDT Model 2
mean d = 2.483 ns TOF = 0.205 ns
GEF
Calculated values
(a) Mass-yield of pre-neutron emission masses.
60 80 100 120 140 160 180 200
Mass (u) 0
1 2 3 4 5 6 7 8
Yield (%)
Comparison masses Post-neutron emission
PDT Model 2
mean d = 2.483 ns TOF = 0.205 ns E = 50 keV
GEF
Calculated values
(b) Mass-yield of post-neutron emission masses.
Figure 17: PDT Model 2, with realistic energy and other time resolution effects also accounted for as standard
4.2.3 Model 3
PDT Model 3 only exhibits an energy dependence, as per equation (22). This results in a model quite similar to Model 2, which can seen through a comparison of the models’ dependencies (Figures 18 and 16). Precisely as in Model 2 this model appears to have a mass dependency in Figure 18a, but for Model 3 this is an incorrect conclusion as it once again only is a consequence of a fragment’s mass-energy correlation. The total spread of the PDT almost reaches 1 ns for this model, which is the lowest total spread between the three models. The mass-yields of Model 2 and 3 are also almost identical (Figures 19 and 17). The iconic increase in peak-to-valley ratio is seen in the pre-neutron emission mass-yields (Figure 19a) whilst the post-neutron emission mass-yields’ peak for lower masses is broadened and the peak for higher masses see an converse alteration (Figure 19b), precisely as in Model 2.
80 100 120 140 160
Mass (u) 2
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8
PDT (ns)
PDT Model 3
0 1000 2000 3000 4000 5000
Counts (#)
(a) The PDT’s mass dependency.
50 60 70 80 90 100 110 120
Energy (MeV) 2
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8
PDT (ns)
PDT Model 3
0 1 2 3 4 5 6
Counts (#)
104
(b) The PDT’s energy dependency.
Figure 18: PDT Model 3, with mean values of the PDT for mass respectively energy displayed in red.
60 80 100 120 140 160 180 200 Mass (u)
0 1 2 3 4 5 6 7
Yield (%)
Comparison masses Pre-neutron emission
PDT Model 3
mean d = 2.492 ns TOF = 0.205 ns
GEF
Calculated values
(a) Mass-yield of pre-neutron emission masses.
60 80 100 120 140 160 180 200
Mass (u) 0
1 2 3 4 5 6 7
Yield (%)
Comparison masses Post-neutron emission
PDT Model 3
mean d = 2.492 ns TOF = 0.205 ns E = 50 keV
GEF
Calculated values
(b) Mass-yield of post-neutron emission masses.
Figure 19: PDT Model 3, with realistic energy and other time resolution effects also accounted for as standard deviations σEand σTOF.
4.2.4 Uncertainties
Below (in Figure 20) are the uncertainties displayed as calculated mass errors for each PDT model, determined by the absolute value of the difference between the ideal mass values produced by GEF and the calculated values with applied resolution effects. Common for all models is that there are lower errors around the mean values of the fission fragments’ masses (108 and 144 u), and all models produce mass errors of 2 u or more for each mass. However, the effect is more dominant for PDT model one (20a), which also reaches higher error values up to 4 u, while the other models only have errors of 3 u at most.
(a) Model 1. (b) Model 2. (c) Model 3.
Figure 20: The absolute error of each event caused by aforementioned effects and it’s dependency on fragment mass.
4.3 Effects on Individual Masses
Previously the PDTs’ and other resolution effects’ influence on the entire mass-yield spectra have been explored, what follows is an examination of these effects’ impact on single masses of these spectra. In Figure 21 this is shown for the mean mass of the lighter and heavier fission fragments, and Figure 22 is an example of this impact on masses located beside the peaks. Gaussian curves have been fitted to the resulting distributions of observed fission fragments, for each PDT model. The mean and standard deviation of these Gaussian curves are displayed in the figures. Figure 23 displays how the properties of these Gaussian curves depend on the fragment mass, both pre- and post-neutron emission. As can be seen from a fission events mass-yield (Figure 9) particular fission fragment masses are more prominent than others, meaning that they occur more often. This also entails that other masses are less prominent, and in this project certain mass numbers need to be omitted from these graphs, to ensure an adequate number of counts are used for the fitting of aforementioned Gaussian curves.
105.5 106 106.5 107 107.5 108 108.5 109 109.5 110 110.5 Mass (u)
0 1000 2000 3000 4000 5000
Counts (#)
Mass 108 isolated, pre-neutron emission 95% Confidence Interval
of mass (u)
Model 1: [0.4089, 0.41351]
Model 2: [0.41609, 0.42078]
Model 3: [0.41239, 0.41705]
= 108.0031 = 108.0239
= 108.0216
Ideal value, 61006 counts PDT model 1 PDT model 2 PDT model 3
(a) The three PDT models’ and other resolution effects’ impact on the distribution of fragments with mass 108 (u), which is the mean mass of the lighter fragments.
141.5 142 142.5 143 143.5 144 144.5 145 145.5 146 146.5 Mass (u)
0 1000 2000 3000 4000 5000
Counts (#)
Mass 144 isolated, pre-neutron emission 95% Confidence Interval
of mass (u)
Model 1: [0.4089, 0.41351]
Model 2: [0.41609, 0.42078]
Model 3: [0.41239, 0.41705]
= 143.9969 = 143.9761
= 143.9784
Ideal value, 61006 counts PDT model 1 PDT model 2 PDT model 3
(b) The three PDT models’ and other resolution effects’ impact on the distribution of fragments with mass 144 (u), which is the mean mass of the heavier fragments.
Figure 21: Showcases of how the effects affect isolated masses, pre-neutron emission, with Gaussian curves fitted to the resulting distributions.
