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. This subtlety is explained by Spriggs et al. for their method of calculating βeff.

3.3 The Holland MC Method

This method was proposed by van der Mark and K. Meulekamp from Petten NRG, Holland. It is based on the Spriggs method. In MCNP code, the neutron production and the spectrum weighting is taken into account in the correct way. The only missing link to βeff is the adjoint weighting [2]. Here, adjoint weighting is implemented into MCNP by some minor bookkeeping, which will give a result for βeff in the same run with which one calculates keff. The implementation is based on the application of the generation definition of history to the βeff calculation. This introduces an approximation, because history now only runs until the next fission. Therefore the counting of the number of induced fissions also counts only until the next fission, i.e., not the iterated fission probability, but the next fission probability is calculated. This is no longer exactly proportional to the adjoint function, but it is a useful approximation. For the calculation of βeff it is not required to know the adjoint function very precisely. The value for βeff is largely determined by the value of β0, the fraction of delayed neutrons produced. For most critical systems, the value for βeff differs <20% from β0. Since the effectiveness of neutrons, i.e., the adjoint function only influences this small difference between βeff and β0, one only needs to know the adjoint function with 10% accuracy to get 2% accurate results for βeff [2].

3.4 βeff calculation

The effective delayed neutron fraction (βeff) in training reactor VR-1 is calculated by MCNP5 transport code using prompt method, which requires two calculations. βeff is defined, according to [3] as: β eff ≅ 1− k p k .

The required value of the effective multiplication factor keff taking both prompt and delayed neutrons into account was acquired in the straight calculation mode of MCNP5 calculation, using the data card KCODE. In the KCODE mode the mean values of both prompt and delayed neutrons (if these are included in the used cross-section libraries) are used in criticality calculations. To prevent the influence of the delayed neutrons, TOTNU data card with entry NO had to be used, to obtain the value of effective multiplication factor for prompt neutrons (kp). A TOTNU card with NO as the entry causes νp to be used, and consequently kp to be calculated, for all fissionable nuclides for which prompt values are available. If the TOTNU card is used and has no entry after it, the total average number of neutrons from fission (ν) using both prompt and delayed neutrons is used and the total effective multiplication factor (k) is calculated. The calculation was realized for the complex MCNP5 model of training reactor VR-1, composed and provided by Mr. Rataj and Mr. Sklenka, from the Department of Nuclear Reactors of CTU in Prague, under an international cooperation. In the model, several simplifications were considered:

a) The tool for dynamic studies is not included – model of fuel assembly is used instead b) The fission chambers RJ 1300 for power measurement during operation are not considered in the model c) The SNM corona boron counters are not considered in the model

The positions of the rods considered in the model were as follows:

a) Safety rod: B1, B2, B3 in top level, which corresponds to a height of 680 mm b) Experimental rods E1 at 350 mm and E2 at 340 mm c) Control rods R1 in top level and R2 at the bottom level of the core

The spatial distribution of source neutrons was created using the KCODE data card. The source neutrons in each cycle are determined by the points generated in the previous cycle. For the initial cycle, the point neutron sources are necessary to specify. These are specified explicitly by KSRC data card. In this case 8 source neutron points were defined. The fuel rods are divided into 8 layers, and in each layer there is one source neutron point specified. 1000 active cycles and 30000 neutrons per cycle were used in the calculations.

3.4.1 Calculation results

The first calculation in KCODE mode with TOTNU card with NO, brought following value of effective multiplication factor for prompt neutrons (kp):

k p = 0.99734 ± 0.00015

The second calculation result was the value of total effective multiplication factor (k):

k = 1.00419 ± 0.00016

The final effective delayed neutron fraction for the model of VR-1 training reactor, calculated using prompt method (4):

k 0.99734 ± 0.00015 β = 1− p = 1− = 0.0068214 ± 0.00031 eff k 1.00419 ± 0.00016

4 βeff MEASUREMENT

Results of more than a hundred measurements were published to determine delayed neutron group parameters. In general, these measurements can be divided into two groups: out-of-pile and in-pile measurements. Most of them were out-of-pile measurements. The method for our measurement of βeff on VR-1 training reactor at CTU in Prague is in- pile kinetic method, which uses the reactor itself as a measurement system. This method is based on sinusoidal (and trapezoidal) reactivity of ±0.05$ (and ±0.11$) insertion to a critical reactor by periodical movement of the control rod and following power oscillation measurement. The reactor’s initial power was 1W. Nuclear reactor with power on this level, where temperature changes have no influence on physical properties of the core is called zero- power reactor. The power oscillations were measured by 4 fission chambers (power measurement during operation) and also by two 3He detectors. 36 measurements were performed with oscillation period 20.25 ÷ 300 s for ±0.05$ (40.5 ÷ 300 s for ±0.11$) reactivity insertion.

4.1 Theoretical analysis

When reactivity feedbacks are negligible (e.g. when reactor power is very low), and reactivity oscillations are small compared to the average reactor power (∆P << P0 ), integral form of the point kinetic equation changes to [5]:

t ∆P(t) = P G (t − t′)ρ(t′)dt′ 0 ∫ 0 0 where ρ(t) and ∆P(t)=P(t)-P0 are reactivity and power oscillations, respectively, and G0(t) is the zero-power transfer function. Applying Laplace transform on previous equation, the transfer of linearized model of zero-power reactor can be expressed as follows:

−1 ~ m 1 ∆P(s) 1 ⎡ β i ⎤ ~ = G0 (s) = ⎢Λ + ∑ ⎥ , P0 ρ(s) s ⎣ i=1 s + λi ⎦ where G0(s) is linearized zero-power reactor transfer function, Λ - prompt neutron generation th time, βi, λi – delayed neutron fraction, decay constant of i -delayed neutron group, s – L- operator. According to [5], the frequency response of zero-power reactor has three areas:

1 1. Slow reactivity changes λ 〉〉 s ⇒ G (s) = i min 0 sβλ−1 m β i 1 2. Intermediate reactivity changes λ i max 〈〈 s ∧ Λ 〈〈 ∑ ⇒ G 0 ( s ) = i =1 s + λ i β m β i 1 3. Fast reactivity changes λi max 〈〈 s ∧ Λ 〉〉 ∑ ⇒ G0 (s) = i=1 s + λi sl

Minimal oscillation period Tmin reached by sinusoidal oscillation of control rod R2 was 20.25 s (ω = 0.31rad / s ). This period is not suitable for determination of βeff by none of -1 previous areas, because ω = 0.31 rad/s is not much lesser than λi min = 0.0127 s and not much -1 greater than λi max = 3.87 s . Measurements with oscillation period of 1s and less are needed nd to reach the 2 area, whereG0 (s) ≈ 1/ β . Delayed neutron data were taken from Table 1.2 in Ref. [5].

Figure 1: Reactor response to periodical reactivity insertion

4.2 Experimental results and data analysis

By fitting the reactor response function (reactor power as function of time) is possible to get the zero-power reactor transfer function in polynomial form. Response and transfer function of reactor VR-1 for oscillation period 22.5 s is depicted on Fig. 2. Reactor power and R2 position values were set to the axes origin [0,0], therefore the power oscillates to negative values.

Figure 2: VR-1 reactor response (blue) and transfer function (red) for 22.5 s oscillation period

Transfer function was obtained using ARMAX model in numerical computing program MATLAB®. In order to get more precise fitting of response function and subsequently more precise transfer function, the oscillation period has to be significantly lower. Its value should be 0.5 ÷ 1 s. Since the mechanical constraints of common regulating rods do not enable such short oscillation period, use of a device with a piston containing neutron absorbing material as the oscillation generator is required. This is essential to in order to resolve the shortest-living delayed neutrons precursors [6].

5 CONCLUSION

The final value of the effective delayed neutron fraction 0.00682 with the standard deviation of 0.031% is 4.7% lesser than the theoretical value estimated and used by the VR-1 operators. Currently used value of βeff is 0.00714 [7]. The discrepancies between these two values were caused by uncertainties of selected calculation method, using the simplifications in the MCNP model of the reactor, so as the assumptions summarized above. The results of βeff in-pile kinetic measurements show, that the oscillation frequency has to be decreased to approximately 1s to obtain response function whose average power level rises very slowly. Fitting of this type of function is much more precise and it also increases the precision of the final βeff value.

6 LIST OF NOMENCLATURE

ARMAX - Auto Regressive Moving Average with eXternal input model estimator βeff - Effective delayed neutron fraction CTU - Czech Technical University in Prague KCODE - Monte Carlo eigenvalue algorithm to determine kfor nuclear criticality KSRC - K-code SouRce Card MATLAB - MATrix LABoratory (numerical computing program) MCNP5 - Monte Carlo N-Particle Transport Code,version 5 NRG - Nuclear Research and consultancy Group TOTNU - TOTal NUbar data card (mean value of neutrons keff from fission) VR-1 - Research Reactor at CTU in Prague

7 REFERENCES

[1] KEEPIN, G. R.: “Physics of Nuclear Kinetics”, Addison Wesley, Reading, Massachusetts (1965) [2] MARCK, S.C. and KLEIN MEULEKAMP, R., Calculation of the effective delayed neutron fraction by Monte Carlo, subm. to Nucl. Sci. Eng. (2005) [3] BRETSCHER, M.M., “Evaluation of reactor kinetic parameters without the need for perturbation codes“, Proc. Int. Meeting on Reduced Enrichment for Research and Test Reactors, Oct. 1997, Wyoming, USA [4] SPRIGGS, R.D. et.al., “Calculation of the delayed neutron effectiveness factor using ratios of k-eigenvalues“, Annals of Nuclear Energy 28 (2001), Pages 477-478 [5] HEŘMANSKÝ, B.: “Dynamika jaderných reaktorů”, MŠ ČSR (1987) [6] YEDVAB, Y. et al., “Determination of Delayed Neutrons Source in he Frequency Domain based on in-pile Oscillation Measurements”, Proceedings PHYSOR 2006 [7] KROPIK, M. et al., “Eugene Wiegner” Training Course at VR-1 Reactor (May 2003).Textbook for practical Course on Reactor Physics in Framework of European Nuclear Engineering Network, DNR FNSPE CTU Prague (2003), Page 11