Monte Carlo Methods in Reactor Physics

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1 UNIVERSITY OF LJUBLJANA Faculty of Mathematics and Physics Department of Physics Seminar on Monte Carlo Methods in Reactor Physics Author: Andrej Kavčič Mentor: prof. dr. Matjaž Ravnik Ljubljana, January 2008 Abstract The Monte Carlo method is being widely used to solve neutron transport problems in nuclear reactor cores along with the advancements in computer technology. The method is originally as old as neutron is, however, it was not until the last decade that it became popular due to growing computer capacities. Today almost all neutron parameters and their effects on reactor behaviour can be simulated. The next pages represent a step by step introduction, from the primary Monte Carlo idea to the calculation of three most important parametres in nuclear reactor physics: multiplication factor, delayed neutrons factor and prompt neutron lifetime.

2 Contents Introduction Nuclear power Introduction to Nuclear Reactor Technology Reactor kinetic equations Principles of Monte Carlo Neutron multiplication factor Delayed neutrons Prompt neutron lifetime Prompt jump Summary References Introduction The Monte Carlo method provides approximate solutions to a variety of mathematical problems by performing statistical sampling experiments on a computer. The method applies to problems with no probabilistic content as well as to those with inherent probabilistic structure. Among all numerical methods that rely on N-point evaluations in M-dimensional space to produce an approximate solution, the Monte Carlo method has absolute error that decreases as N -/2 whereas, in the absence of exploitable special methods all others have errors that decrease as N -/M at best. The method was born in 930s and began to be studied in depth during Second World War in Los Alamos National Laboratory. It is called after the city in the Monaco principality, famous by its roulette, a simple random number generator. The name and the systematic development of Monte Carlo methods dates from about 944. At that time basic steps were made by physics researchers Ulam, Fermi, Von Neumann and Metropolis while analysing neutron behaviour. At that time, there was no typical realization because of poor computational technology, but nowadays the method corresponds perfectly to digital processors, that are 2

3 getting faster every day, and is widely used in all fields of science, especially in nuclear physics. 2. Principles of Fission Chain Reaction Nuclear reactors use a chain reaction to induce a controlled rate of nuclear fission in fissile material, releasing both energy and free neutrons. The splitting of the nucleus is possible if it is energetically feasible and the potential barrier is passed. Since the probability for spontaneous fission of naturally occuring elements (uranium) is very small and cannot be controlled externally, nuclear reactors are driven by induced fission with neutrons. Neutrons are used because every fission of a heavy nucleus apart from two smaller nuclei and a lot of kinetic energy produces 2 or 3 additional neutrons. The newly produced neutrons can either trigger new fission to produce new generation of neutrons or be absorbed by other material. The other reason why neutrons are convenient is that in addition to the kinetic energy they carry also binding energy when they enter nuclei. Because of the spin-coupling effect the difference in energy released by neutron capture of two nuclei of similar but different parity mass and atomic number is about 2MeV. Therefore nuclei (around uranium) with even number of protons and odd number of neutrons (for example 235 U) can be split by thermal neutron, whereas odd-odd nuclei (for example 238 U) can be split only by neutron with kinetic energy over MeV. Fissioned nucleus usually splits into two lighter nuclei and neutrons. The former are called the fission products. They are mostly radioactive because of the overplus of neutrons. Due to the ''shell'' structure of the target nucleus, asymmetric decay is energetically more favourable that the symmetric one []. Therefore, if the incident neutron energy is low, the probability for asymmetric decay is much bigger compared to the symmetric decay. On the other hand, if most of the neutrons causing fission have high kinetic energies, the symmetric decay predominates []. The neutron population at any instant is a function of the rate of neutron production (due to fission processes) and the rate of neutron losses (via non-fission absorption mechanisms and leakage from the system). When a reactor s neutron population remains steady from one generation to the next, the fission chain reaction is self-sustaining and the reactor s condition 3

4 is referred to as "critical". In order to be steady state, the multiplication factor for the chain reaction, defined as: k n n i+ = () i where n i is the number of fissions in i-th generation, has to be exactly. This means that the power of the reactor is constant. However, because the time between successive reactions, which is per definition equal to the average neutron lifetime in reactor, is of order 0-5 s, even small change of k would cause explosion or shut down before the operator could interfere. Luckily, aproximately % of the neutrons (called delayed neutrons) arise from fission products with lifetimes of up to around 0s. Although their relative number is small, they increase the average neutron lifetime to about s and the chain reaction becomes controlable. Fissile isotopes are capable of sustaining fission chain reaction in collisions with low-energy (or thermal) neutrons [2]. Examples are 235 U, 239 Pu and 233 U but only 235 U is present in nature, so others have to be produced artificially. In general fissile materials have large cross sections ( > 0 2 barn) for slow (E < 0.eV) neutrons. The probability for inducing fission is higher for slow neutrons. To reduce the critical mass and required fissile fuel concentration (enrichment) most reactors use materials to slow down neutrons, called moderators. A good moderator has a high cross section for neutron (mostly elastic) scattering and on the other hand the cross section of neutron absorption has to be as low as possible. The moderator should be dense (to decrease the mean free path of the neutron) and, most importantly, contain low mass nuclei in order to increase neutron energy loss per collision [3]. Examples of good moderators are: hydrogen, deuterium, graphite or beryllium. 3. Reactor Kinetic Equations All the equations written down are based on independence of space and time distribution of neutron flux, which for thermal neutrons can be explain as follows: Φ (, ret,) = vnret (,,) (2) 4

5 where Φ( ret,, ) represents the space, energy and time distribution of neutron flux, is distribution of neutron density and v is velocity of thermal neutrons. nret (,,) During a transient time distribution of neutron density mostly depends on multiplicator factor k of fission assembly, prompt neutron lifetime and delayed neutron parameters. In one neutron lifetime, l, the number of neutrons changes as Δ n= kn n= ( k ) n (3) We can rewrite upper equation in the infinitesimal form (4), count out the delayed neutrons ( - β ) and add the factor that actually describes them ( λ C ) in order to get the reactor kinetic equation: dn k = n (4) dt l where k dn k( β ) = n+ λc, C is calculated as dc β = λc+ n (5) dt l dt l is multiplication factor, l - prompt neutron lifetime in fission assembly, β is delayed neutron fraction,c precursors concentration of delayed neutrons and λ decay constant of precursors of delayed neutrons. The prompt neuron lifetime neutron reproduction number l is related to prompt neutron generation time k. Relation between bought prompt neutron times is: Λ through the l Λ= (6) k Using term for reactivity r [4] and prompt neutron generation time the kinetic equations can be edited by following: dn ( ρ β ) = n+ λc dt Λ (7) It is clear that the the most important factors how the reactor behaves are: reactivity, effect of delayed neutrons and neutron generation time. In the next chapter there will be shown how to calculate these crucial parametres. There are some deterministic principles, but it is almost impossible to get the final result without approximations in comparison to Monte Carlo method. 5

6 4. Principles of Monte Carlo Basic idea of Monte Carlo method is to calculate the travelling distance of each neutron. We need to propagate the neutron trajectory in the reactor like it really behaves (the best way is 'to think like a neutron'). The probability of a first collision for particle between l and l + dl along its line of flight is given by: Σl p() ldl=σ e dl (8) where S is the macroscopic total cross section of the medium and is interpreted as the probability per unit length of a collision. Setting ξ the random number on [0,) it follows that and l Σs ξ = Σ e ds= e 0 l = ln( ξ ) Σ Σl (9) (0) But, because - ξ is distributed in the same manner as ξ, and hence may be replaced by ξ, we obtain the well-known expression for the distance to collision: l = ln( ξ ) () Σ and after i collisions in one dimmension it can be written: l = i l + i ln( ξ ) Σ (2) In this chapter there will not be discussion about random numbers, let just say that they could be generated with quite many modern mathematical algorhythms. Untill now there has not been anybody who could proove unrandomness in these generators. In the beginning there were actually some problems, but nowadays modern mathematical software corporations (like Wolfram Research Inc.) have officially published invitation for those who will prove unrandomness in their generators. In equation (2) there is random walk of neutron in one dimension. The next logical decision is to write it down in three dimensions. What we need are two more additional random 6

7 numbers to specify the direction of flight of the neutron. One way is to write them with the azimuthal and polar angle, but because of practical reasons the more common expression is to write as j and Cos q. For the homogenous distribution it can be written: dp dp dp = =, =, dω dϕd( Cosϑ) 4π dϕ 2π dp = (3) dcosϑ ( ) 2 If we define another two random numbers x i, x i+, any direction of the neutron flight could be defined as: ϕi = 2πξi and Cosϑ i = 2ξ i + (4) Figure : randomly sampled 500 directions from the same starting point (Eq. 3). They are actually distributed on the surface of a sphere. In the next step the length has to be included in each directon. For the coordinates we write x = l sin q cosj, y = l sin q sin j and z = l cos q, where l is random lenght of neutron flight, and a propagation of elastic scattering looks like: 2 xj+ = xj ln( ξi) (2ξi+ ) Cos(2 πξ i + 3) Σ 2 yj+ = yj ln( ξi) (2ξi+ ) Sin(2 πξ i + 3) (5) Σ zj+ = zj ln( ξi)(2ξi+ 3 ) Σ 7

8 Figure 2: random neutron walk in three dimensions. The next logical step is to correct our model for the in-elastic scattering. When a neutron collides with the hydrogen nucleus (proton) it looses for about half of its energy. That is also the reason why the water is so good moderator. At this point we need exact information about matherials (nuclear fuel, control rods, moderator...) for defining the loss of energy for each collision. Here it is usefull to include all the neutron interactions with matter, especially capture and fission. When a particle collides with a nucleus, the following sequence should occur: - the collision nuclide is identified, - either elastic or inelastic scattering is selected, - check probability for fission and determine the new energy and direction of the outgoing neutrons, - check probability for capture and kill the neutron. All the probability informations are given in the evaluated nuclear data files [5] containing evaluated (recommended) cross sections, spectra, angular distributions, fission product yields and other data, with emphasis on neutron-induced reactions. Figure 3: Fission cross sections s for interaction with neutrons for 235 U as a function of incident neutron kinetic energy E in log-log scale [7]. 8

9 At this point we have achieved the first reliable model of neutron transport. On the bottom picture (Figure 4) there is the starting neutron (black line from the orange square) that scatters and absorbs in the uranium. After fission we get two new neutrons, the green one and the blue one with their own trajectories. The green one is captured in the absorber (for example B, In, Cd, Ag ) and the blue one produces another fission with two new neutrons (light blue and red one). Figure 4: model of neutron fission (black or blue) and capture (green one). The basic idea of such transport via Monte Carlo method is actually as old as neutron is. Enrico Fermi used Monte Carlo (at that time the method had no special name) in the calculation of neutron diffusion in 930s but until now it was almost impossible to make an exact calculation of a neutron transport. As soon as we have Monte Carlo neutron transport code, we can start calculating major reactor paramethers. 4.. Neutron multiplication factor The neutron multiplication factor k, is the average number of neutrons from one fission that cause another fission. The remaining neutrons either are absorbed in non-fission reactions or leave the system without being absorbed. The value of k determines how a nuclear chain reaction proceeds: 9

10 - k < (sub-critical mass): The system cannot sustain a critical reaction, and any beginning of a chain reaction dies out over time. For every fission that is induced in the system, an average total of /( k) fissions occur. - k = (critical mass): Every fission causes an average of one more fission, leading to a fission (and power) level that is constant. Nuclear power plants operate with k =. - k > (super-critical mass): For every fission in the material, it is likely that there will be k fissions after the next mean generation time. The result is that the number of fission reactions increases exponentially, according to the equation (7). After fission 2 neutrons are created (blue and green line) Red one produces another fission (k=) After fission 3 neutrons are created (red, brue and green line) Two neutrons (green and red line) produce another fission (k=2) Figure 5: production of fission neutrons and calculation of multiplication factor k. After a lot of fission neutrons during each generation the effective multiplication factor can be defined (or the reactivity). In practice reactivity is written to five decimal places and the unit is pcm (''percent milli''). 0

11 4.2. Delayed neutrons: As it was already mentioned nearly all of the fission neutrons more than 99% of them are emitted just after the fission process, without considerable delay. The fission fragments that usually have much higher excitation energy than the neutron separation energy emit these prompt neutrons. The half-life of neutron-emission of these highly excited states is in the order of 0-5 s or even smaller. However, not all fission fragments emit neutrons, some of them decay by b - emission. After the emission of the prompt neutrons there is usually no more neutron emissions, the fission products undergo several successive b - decays to reduce their neutron excess. However, in some cases a daughter nucleus is formed after a b - decay, where the excitation energy is higher than the neutron separation energy. This nucleus will emit again a neutron, nearly promptly after its formation. These are the delayed neutrons. For example the decay chain for 87 Br is summarised as follows: Br Kr Kr+ n (6) 87 β 87 * 86 The 87 Br nucleus is called delayed-neutron precursor, the 87 Kr nucleus is called delayed-neutron emitter. Obviously, for these neutrons the delay time is determined by the half-life of the precursor nucleus, which can be quite large, since the b decay is governed by the weak interaction. Another interesting consequence of this decay chain is that the excitation energy of the delayed-neutron emitter nucleus is usually much lower than the excitation energies of the direct fission fragments. Therefore, the average energy of the delayed neutrons is also smaller ( kev) than that of the prompt neutrons (2 MeV in average). 235 U Delayed neutron is born. 46 La fission 87 Br Radioactive decay 87 Kr Figure 6: birth of delayed neutrons.

12 The total yield of the fission neutrons (average number of neutrons, ν) is the sum of the yield of the prompt and the delayed neutrons (ν p, and ν d ): v = v p + v d And delayed neutron fraction is defined as: (6) v d β = v. (7) Figure 7. shows the dependence of the delayed-neutron yield for 235 U and 239 Pu on the energy of the fission-inducing neutron. Delayed neutron yield (delayed neutrons / fission) 235 U 0,05 0,0 0, Pu 0, 0,3,0 3,0 0,0 (MeV) Energy of the neutron, which induced the fission Figure 7: Delayed neutron yield in function of the energy of the fission inducing neutron[6]. The curves show that the delayed neutron yield is practically independent of the neutron energy, at least in the energy interval 0 < E < 4 MeV. However, the total delayed neutron yields are strongly dependent on the composition of the fissile nucleus. The values are shown in Table. Fissile nucleus ν d (neutron/00 fissions) 233 U 0,667+0, U,62+0, U* 4,39+0,0 239 Pu 0,628+0, Pu* 0,95+0,08 24 Pu,52+0, 242 Pu* 2,2+0,26 *Data for fast-neutron induced fission. Table. Total delayed neutron yields (number of delayed neutrons on 00 fission events) for thermal neutron induced fission of different isotopes [6]. 2

13 The protocol of calculating the effectiveness of delayed neutrons (example for TRIGA type reactor) i generation: 000 At the begining there are N fission neutrons (000 in the left example). Fission (average n = 2.439) Prompt neutrons Delayed neutrons b 0 = 0.69% χhel Energijski spekter takojš njih nevtronov χhel Energijski spekter zakasnelih nevtronov E HMeV L 0.2 E HMeV L Prompt and delayed neutron energies are sampled from different sprectrums. After picking time, energy and angle parametres the journey starts. 4% survival i+ generation: % survival b eff = 0.8% Because of different starting energies, delayed neutrons appear to be more effective (this statement is valid only for small reactors - like TRIGA) [7]. The results of kinetic parametres for the upper example would be: ni k = = =, β 0 = = 0.69%, n i 8 β eff = = 0.80% (8) 000 3

14 4.3. Prompt neutron lifetime The prompt neutron lifetime is the average time between the emission of neutrons and either their absorption in the system or their escape from the system [8]. The term lifetime is used because the emission of a neutron is often considered its birth, and the subsequent absorption is considered its death. For thermal (slow-neutron) fission reactors, the typical prompt neutron lifetime is on the order of 0-4 seconds, and for fast fission reactors, the prompt neutron lifetime is on the order of 0-7 seconds [9]. In the beginning when the neutron was born its energy has been defined from the neutron fission spectrum. Along its journey we have the information about scattering and a loos of energy which gives us exact information about velocity. And together with the travelling distance we receive the neutron lifetime. The problem is that all the neutrons are not important for the reactor core. In other words, neutrons that escape out of the reactor core can either return or not. Only the lifetime of the returned neutrons is important but it is hard to separate between them. Because of that we will rather choose another method, simulation of a neutron prompt jump Prompt jump (or drop): The prompt jump factor is the factor by which power undergoes immediate (but not instantaneous) change in response to a step change in reactivity. This power change occurs over several hundred prompt neutron lifetimes, through alteration of the prompt neutron population because of change in the source multiplication factor. The change is so rapid that delayed neutrons can be neglected in the point kinetic equation (7), thus it follows: dn ( ρ β ) = n dt Λ (9) ρ β eff t Λ Which solution is: nt () = ne 0 (20) In earlier chapters we have already calculated the multiplication factor and delayed neutrons factor. What we need now is the change of the neutrons density during time and we can get the mean generation time from the equation (20). 4

15 The prompt jump beings in the order of 0-4 seconds and involves only a small fraction of one second (exact time depends on kinetic parametres). At t = 0 ms, the neutron source is put in the reactor and after 5ms we get the homogenous neutron flux across the whole reactor (Fig. 8). After 35ms the source is put outside the core and the flux decreases with tipical period which is defined from the kinetic parametres. φhtl thms L Figure 8: the curve of neutron flux when we put the neutron source into the reactor for 35ms In the last step the exponential function has to be fitted and the time we are looking for can be easily calculated by following: ρ β eff t Λ t0 nt () = ne 0 or nt () = ne, where Λ 0 t = 0 ρ β eff t (2) The final result of a prompt neutron lifetime for TRIGA type reactor is: Λ = 38μs φhtl Prompt drop L=38.8 ms Prompt jump L=37.6ms Figure 9: the promt jump and drop. thms L 5. Summary The professional Monte Carlo neutron codes are rapidly developing from year to year, for instance, delayed neutrons were regulary implemented in the last two years. However, there are still some problems with the prompt neutron lifetime calculations. In general, the method 5

16 (especially General Monte Carlo N-particle Code [0]) is gaining a major role in kinetic calculations and stands as highly perspective. Exact calculations of kinetic parametres are cruical for the reactor safety and Monte Carlo method is highly respected when talking about precise evaluations. References: [] M. Rosina, Jedrska fizika, FMF (2005). [2] J. R. Lamarsh and A. J. Baratta, Introduction to Nuclear Engineering, Third Edition, Prentice Hall, Inc. (200). [3] M. Ravnik, Reaktorska in radiacijska fizika, FMF (2005). [4] Criticality is another expression for multiplication factor (especially usefull for operators in nuclear power plants) and is defined as: k ρ =. k [5] Available at Janis 2.2 Java-based Nuclear Data Display Program: Accessed: December, [6] G. R. Keepin: Physics of Nuclear Reactors, Addison-Wesley Publishing Co., Massachusetts (965). [7] Nuclear Power Fundamentals, Effective Delayed Neutron Fraction, , [8] Lamarsh, John; Baratta, Anthony. Introduction to Nuclear Engineering. Prentice Hall (200). [9] Duderstadt, James; Hamilton, Louis. Nuclear Reactor Analysis. John Wiley & Sons, Inc. (976). [0] MCNP5 - A General Monte Carlo N-Particle Transport Code, Version 5 Los Alamos Diagnostics Applications Group (2007). 6

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