NF,ACRP-,$-L&3 l THE REACTIVITY SCALE FOR FAST-SPECTRUMCRITICALS E. F. Bennett ArSome National Laboratory l September 1981 . THE REACTIVITY SCALE FOR FAST-SPECTRDMCRITICALS INTRODUCTION The effect of a perturbation of materials in a reactor induces a time- dependent flux change, the nature of which depends upon reactor prompt-neutron lifetime and delayed-neutron parameters. For an asympototic positive period, at powers far in excess of source levels and neglecting the prompt-neutron lifetime (which is always justified for fast reactors), the relationship between reactivity Ak/k and period T is given by the In hour equation Ak -= k i=l 1 + 1iT ai and ,4i in Eqn. 1 are the relative delayed-neutron precursor yields and decay constants respectively and S is the effective beta. The number of delayed-neutron families, N, is large but experimental evidence has repeatedly confirmed that N=6 is sufficient for use in the point-reactor kinetics model. Small-sample worths, which ara a basic type of measurement in mockups, are seldom measured by asymptotic period. To minimise sensitivity to drift while retaining statistical precision, a periodic modulation of the reactor by a sample is usually made. The induced flux modulation (or the modulation in position of a balance control rod) is analysed using the point kinetics model to deduce the magnitude of the reactivity effect involved. For small sample worths, corresponding to large values of period in Eqn. 1, QT is much greater than unity for all ii and Eqn. 1 may be written as where 8d is a characteristic delayed-neutron time scale . , 0f msgnituae 10 t0 14 ssc aspenaing upon the typs 0f reactor. T in Eqn. 2 is observable ana if B9.a is known, Ak/k csn be inferrea. B!J.a is the “Reactivity Scale Factor” ana is the only quantity in the point kinetics moael requirea to reauce s small-sample worth measurement to Ak/k if the conaitions of the ex- periment involve s long measurement time scale, such ss the measurement of the sverage position of s balance roa over an interval of time of several minutes, 9 or the aetermfnation of s slow asymptotic perioa. Since the reactivity scale factor, B&a, proviaes the proportional link between worth measurements ana calculations of worth, it is not surprising that it has been subjectea to consiaerable scrutiny wherever aisagreement in C/E for worths hss been fauna. Estimates of the range of uncertainty in the computea scale factor inaucea by computational uncertainties relating to flux, aajoint ana chi spectra in aaaition to uncertainties in the basic ENDF aata files for aelayea-neutrons hsve been maae. The conclusions sre that errors of 4 to 5% sre likely to exist in the calculatea scale factor. An error in scale factor of this msgnituae is s major part of sny systematic aiscrepancy between measurea sna ddstsa worths. Recognition of this sizeable error in reactivity scale calculation hss lea to numerous integral measurements from which the scale factor (or one of its parts) csn be inferrea. A. Characteristic Delayea-Neutron Lifetime Different experimental methoas have been usea to measure .ta on fast mockups. One of these, aescribea briefly below, is similar to methoas usea for si, Ai aeterminations on pure fissile specimens. Typically, a fissile specimen is exposed to an accelerator neutron source for a time long compared to the largest value of l/ii, at which time all of the precursor concentrations are Ln equilibrium. The neutron source is then switched off and the decay is assumed to follow the time-dependent shape function Six delayed neutron families are found to be adequate to describe the decay although many more precursors than six ara known to exist. The in-reactor technique requires that the reactor be brought to a steady powar (usually fairly high) and operated for a time sufficient to assure equilibrium. All of the fast-acting safety rods ara scrammed simultaneously and the decay of the neutron flux is observed. If the reactor ceased to multiply at all, the decay of the neutron flux would be the same as in Eqn. 4. A fitting procedure would then provide directly the values of ai and Ai. The major difference between the idealizad accelerator experiment and the reactor experiment is that ai values for the reactor ara a space-and-energy weighting of adjoint with delayed-neutron emission spectra and some localised changes in adjoint and real flux spectrum will occur. The integrated effects will be small however. Since only relative values for the ai ara required, no absolute measurements are needed and detectors of arbitrary (but presumably high) neutron detection efficiency suffice. However, since the real reactor will retain the capacity to multiply after the safety rod scram, the effects of this residual multiplication must be taken into account. Equation 4 will not be valid but instead a die-away profile of the form ,.. !.. , 4 ; Qie-v (5) i=l will occur where the number of families is the same as the number of delayed neutron fSmiliSS but the Qi and Si ~II Eqn. 5 are dependent upon the ai, Ai in such a way that Qi •, ai and % + Ai in the limit as subcriticality increases. The relationship between the ai, ).i, and the Qi, Si as observed from the flux decay following a rod drop leading to a configuration subcritical by an amount $ dollars can readily be developed from the point kinetics model. It may be of considerable interest to know the full detail of the ai, ii set (for instance, full detail is necessary to analyse rod-drop subcriticality measurements by inverse kinetics) but only the characteristic delayed-neutron lifetime, id, enters the reactivity scale factor needed for interpreting small-sample worth measurements. To determine the dependence of id upon the maasured Qi, Si of the decay, one only needs to integrate the point-kinetics expression directly from a time 0+ to infinity where 0+ is sufficiently greater than 0 to insure that prompt-neutron and rod-motion effects have become negligible. This integral, in terms of the Qi, Si of Eqn. 4, is defined to be the quantity and from the point kinetics equations it is readily shown that 5 Results from measurements on two uranium-fueled mockups are given in Table I. Table I. Summary of Results of Characteristic Delayed-Neutron Lifetime Measurements for ZPR-9 Mockups Subcriticality Mockup ~8s Csecl CS) &j (='d (EN&) C/E STFl/l 15.57 (iO.6%) 5.10 (k5%) 13.02 (*I.,?%) 12.77 0.981 u9 13.51 (*0.6%) 3.74 (?4%) 10.66 (Al.3%) 10.40 0.976 B. Measurements of Effective Beta The californium source method has been used for effective beta measurements beginning with mockups on ZPR-III. The method consists of a determination of the equivalent reactivity effect at a known absolute reactor fission power of a neutron source of known absolute intensity. l Another widely used kinetic technique for the measurement of effective beta is by the use of pile noise. The noise method is unlike the other methods and the errors are different although noise measurements and the Cf z5z source measurements both require that the absolute total fission rate be known. Noise measurements consist of observations of the time-correlation of events occurring in neutron-sensitive detectors. Rossi alpha measurements are noise measurements performed a short time (high frequency) domain empha- sizing prompt neutron behavior. Effective beta measurements are, by contrast, . , 6 noise measurements performed in the long time (low frequency) domain empha- sizing delayed-neutron behavior. The unadorned point reactor kinetics model is e very good model for low frequency phenomena; its application to the interpretation of asymptotic period measurements is fundamental to reactor kinetics. The point model fails, period measurements is fundamental to reactor kinetics. The point model fails, however, at the very high frequencies (-t3/f,o) involving prompt-neutron lifetime to for large LMFBR-prototypical reactor cores using existing detector systeras. For measurements performed on reactors operated at stationary flux levels, noise fluctuations are proportional to the square of the reactor transfer function per unit total fission rate. By limiting the frequency response to the region between (but well removed from) f3/Zo and the largest delayed-neutron decay constant, there will be no dependence of noise on either prompt-neutron effects, which occur on e frequency scale S/.&o, or upon delayed-neutron effects, which occur on .a frequency scale of Al. The relationship between noise, as measured, and reactor parameters of the point kinetics model takes the form F * VJ~ =JD/@'(l+$)Z 2 for a reactor subcritical by $ dollars. wr2 in Eqn. 7 is an experimentally- determined measure of neutron noise. The precise definition of ?a2 is dependent upon the experimental technique. F in Eqn. 7 is the absolute total fission rate for all fissionable isotopes at the stationary flux level adhered to during the measurement. D is a spatially-weighted dispersion in neutron emission per fission. It is the product of two dimensionless quantities : Dv or Ds defined by Dv = Core Average of V(V-l)/~2 F&d:! 7 ZZ F(;)12(:)d: . % = a y is the average neutron emission/fission and V(V-1)/y 2 is the Diven facto?. I(:) is the fission spectrum-weighted adjoint and F(;) is the total fission rate at position ;. F(T) and I(T) are available as output from reactor computations codes. Ds, the spatial dispersion factor, can be calculated with negligible error introduced by nonina modeling and cross section uncertainties. Its value ranges from 1.1 to 1.2, depending upon the details of the core, for LMFBR- prototypical mockups.