POSSIBILITIES OF DELAYED NEUTRON FRACTION (βeff) CALCULATION AND MEASUREMENT Svetozár Michálek – Ján Haščík – Gabriel Farkas Slovak University of Technology, Faculty of Electrical Engineering and Information Technology, Department of Nuclear Physics and Technology, Ilkovičova 3, 81219 Bratislava, Slovakia, [email protected] 1 ABSTRACT The influence of the delayed neutrons on the reactor dynamics can be understood through their impact on the reactor power change rate. In spite of the fact that delayed neutrons constitute only a very small fraction of the total number of neutrons generated from fission, they play a dominant role in the fission chain reaction control. If only the prompt neutrons existed, the reactor operation would become impossible due to the fast reactor power changes. The exact determination of delayed neutrons main parameter, the delayed neutron fraction (βeff), is very important in the field of reactor physics. The interest in the delayed neutron data accuracy improvement started to increase at the end of 80-ties and the beginning of 90-ties, after discrepancies among the results of calculations and experiments. In consequence of difficulties in βeff experimental measurement, this value in exact state use to be determined by calculations. Subsequently, its reliability depends on the calculation method and the delayed neutron data used. Determination of βeff requires criticality calculations. In the past, keff used to be traditionally calculated by taking the ratio of the adjoint- and spectrum-weighted delayed neutron production rate to the adjoint- and spectrum- weighted total neutron production rate [1]. An alternative method has also been used in which βeff is calculated from simple k-eigenvalue solutions. In this work, a summary of possible βeff calculation methods can be found and a calculation of βeff for VR-1 training reactor in one operation state is made using the prompt method, by MCNP5 code. Also a method of βeff kinetic measurement on VR-1 training reactor at CTU in Prague using in-pile kinetic technique is outlined. 2 INTRODUCTION Determination of the value of effective delayed neutron fraction is not an easy task. Delayed neutrons constitute only a small (∼0.7%) fraction of all neutrons generated from fission, therefore the value of βeff is a very small number. To determine this value by calculation using stochastic codes, one needs to have a very good statistic. This fact requires a very precise calculation model, a lot of active cycles and number of neutrons per cycle. For our calculation a stochastic code MCNP5 was chosen. As a verification of the MCNP5 calculation, a method for determining the βeff value based on measuring power oscillations in zero-power reactor is being developed. This method is unique in that sense, that this value is derived from the dynamic behavior of the reactor, more specifically from the reactor’s transfer function. 3 POSSIBLE METHODS OF βeff CALCULATION In general, one writes for the total neutron production rate by fission for a reactor near criticality, and without external source [2]: P = ν (E)Σ (r, E)ϕ(r, E,Ω)dEdrdΩ ∫ f where E, Ω, and r - the energy, solid angle, and position of the neutrons, ϕ - neutron flux, r Σf - macroscopic fission cross section of the material at position r , ν - average neutron multiplicity per fission. For the production Pd of delayed neutrons, one the factor ν(E) is replaced by νd(E), which is the average delayed neutron multiplicity per fission. If the difference between the several delayed neutron time groups, which each have their own energy spectrum, is not considered, the fundamental delayed neutron fraction β0 is defined as β 0 = Pd P . But this definition is not effective in terms of reactor dynamics, because the effect that neutrons have on reactor behaviour is through their ability to generate power, i.e. to induce the next fission. It means that one should compute the number of fissions that are induced by delayed neutrons, as well as by all neutrons. In transport theory, the effectiveness in generating fission is calculated by multiplying by the energy spectrum of the generated neutrons, χ(E'), and by the adjoint function ψ ( r ,E′,Ω′). The adjoint function is defined as the fundamental mode eigenfunction of the equation adjoint to the time independent transport equation. The r , E', and Ω’ are the position, energy, and solid angle of neutrons generated by the fissions that were induced by the incident neutrons characterized by r , E, Ω. The position r is obviously the same for both. The effectiveness of neutrons with initial energy E' in inducing fission is characterized by the factor χ(E'). The adjoint function ψ ( r ,E′,Ω′) is used because it is an importance function, which represents the significance of a neutron with properties r ,E′,Ω’ for producing fission and is proportional to the asymptotic power level resulting from the introduction of a neutron in a critical system at zero power [1]. This leads to the so-called spectrum and adjoint weighted neutron production: P = ψ (r, E',Ω')χ(E')ν (E)Σ (r, E)ϕ(r, E,Ω)dEdΩdE'dΩ'dr (1) eff ∫ f The same quantity for delayed neutrons only (Pd,eff) can be calculated by replacing χ(E‘) by χd(E‘) and ν(E) by νd(E). The ratio Pd ,eff Peff is the Keepin definition of βeff: ψχ mν mΣ 'mϕ'dΩ'dE'dΩdEdr ∑∑∫ di di f Pd ,eff im β eff = = m m 'm , (2) Peff ∑ψχ t ν t Σ f ϕ'dΩ'dE'dΩdEdr m where m – mth-isotope, i – ith-delayed neutron group, other symbols mentioned earlier. In general, P means the neutron source (the number of neutrons produced per unit of time), and Peff as the number of fissions produced by this source per unit of time. In Monte Carlo calculations the physical processes are simulated as realistically as possible. Consider the introduction of a neutron with properties r ,E′,Ω’ in a critical system with zero power. This neutron will produce other neutrons by inducing fission and these neutrons will in turn cause fission and thereby lead to further neutrons, etc. The number of fissions produced in this way will approach a limit, which is given by the iterated fission probability, which is proportional to the adjoint function. Using the number of fissions counted and the knowledge of whether the neutron was prompt or delayed at its start, one can easily calculate βeff [2]. 3.1 Prompt method Denoting the integral in Eq. (1) as χν , one can rewrite the expression for βeff (making use of the fact that the integrals are linear), as follows: χ ν χν − χ ν χν p − (χ d − χ )ν d χ ν β = d d = 1− d d = 1− ≅ 1− p p , eff χν χν χν χν where: ν p =ν −ν d . The approximation in the last step is based on the following arguments. The term (χd - χ)νd is two orders of magnitude smaller than the one with χνp, because νd is two orders of magnitude smaller than νp. For the same reason, the shape of χ is almost equal to that of χp. But, often it is simply stated that: χ pν p k k = p → β ≅ 1− p . (3) χν k eff k In fact, this is an approximation. The difficulty lies in the definition of kp. Since this parameter is supposed to be calculated by means of a transport theory code, it should be defined as the eigenvalue pertaining to a reactor with χ = χp, and ν = νp. The difference is that in such a calculation, the shapes of ϕ and ψ will not be the same as for the original system with χ and ν, for which the eigenvalue is k. This subtlety is generally ignored in papers dealing with the prompt method of calculating βeff based on Eq. (3) [3], where is aforementioned that, since the neutron multiplication factor for a reactor is proportional to the total fission yield (ν), the multiplication factor for prompt neutrons only is proportional to (ν − Eν d ) =ν (1− β eff ). Here νd is the total delayed fission neutron yield and E is the average effectiveness of these delayed neutrons. The total effective delayed neutron fraction is therefore: k − k p β eff = = ∑ β eff ,i , k i where k is the eigenvalue for all neutrons and kp is the eigenvalue for prompt neutrons only. 3.2 Spriggs method The delayed neutron fraction is traditionally counted from: ψχ mν mΣ'mΦ'dΩ'dE'dΩdEdr ∑∑∫ di di f im β eff = m m 'm ∑ψχ t ν t Σ f Φ'dΩ'dE'dΩdEdr m Alternatively, Spriggs et al. [4] rewrite βeff as: χ dν d χ dν d χ dν ' χ dν β eff = = 1− ⋅ = β 0 ⋅ , χν χ dν χν χν ' where, after Spriggs et al., yet another delayed neutron fraction β0 is introduced. To ' ' calculate β0 , adjoint weighting has to be performed. Approximating β 0 ≅ β 0 , the calculation can be simplified to something that can easily be implemented in a Monte Carlo code. As remarked by Spriggs et al., this approximation works well for homogeneous cases. Also with the introduction of a ratio of k-values: χ ν k k d = d → β = β ' ⋅ d χν k eff 0 k As in the case of the prompt method, this is also an approximation. Here the problem lies in the definition of kd. Since this parameter is supposed to be calculated by means of a transport theory code, it should be defined as the eigenvalue pertaining to a reactor with χ = χd and ν = νd. Again, the shapes of ϕ and ψ will not be the same as for the original system with χ and ν, for which the eigenvalue is k.