arXiv:physics/0506166v1 physics.data-an 21 Jun 2005The pulsing CPSD method for subcriticalassemblies with pulsed sourcesDaniel Ballester andDepartment of Applied Mathematics,Polytechnic University of Valencia, 46022 Valencia, SpainJose L. Munoz-CoboDepartment of Chemical and Nuclear Engineering,Polytechnic University of Valencia, 46022 Valencia, SpainAbstractStochastic neutron transport theory is applied to the derivation of the two-neutron-detectors cross power spectral density for subcritical assemblies when external pulsedsources are used. A general relationship between the two-detector probability gene-rating functions of the kernel and the source is obtained considering the contributionto detectors statistics of both the pulsed source and the intrinsic neutron source. Anexpansion in -eigenvalues is derived for the final solution, which permits to takeinto account the effect of higher harmonics in subcritical systems. Further, expres-sions corresponding to the fundamental mode approximation are compared withrecent results from experiments performed under the MUSE-4 European researchproject.1 IntroductionIn last years, researchers have shown an increasing interest on the conceptualdevelopment of accelerator-driven systems (ADS) for nuclear waste transmuta-tion and energy production purposes. An important issue regarding its futureindustrial applicability is the development of a periodically subcriticality levelmeasurement and monitoring technique, since both its operation safety and Corresponding author: Fax +34 963 877 669Email addresses: dabalbermat.upv.es (Daniel Ballester),jlcobosiqn.upv.es (Jose L. Munoz-Cobo).Preprint submitted to Elsevier Science 2 February 2008its performance as a part of the nuclear fuel cycle shall be seriously affectedby this variable.Following the study of the neutron fluctuations in a multiplying medium,Courant and Wallace (1947), several static and dynamic methods have beenproposed and studied for years concerning their applicability for the determi-nation of some nuclear reactor physics parameters (see Uhrig (1970); Williams(1974); Lewins (1978); Carta and DAngelo (1999). Within the group of dy-namic techniques, apparently the utilisation of the neutron-fission chain fluctu-ations for nuclear assemblies subcriticality determination was firstly suggestedby Bruno Rossi (the Rossi- method). In these methods neutron detectorcounting rates related to individual fission-chain events must be discernedfrom the total counting rate, therefore these methods are applicable to sub-critical systems near delayed critical conditions, Carta and DAngelo (1999).Further, dynamic methods based on a time-dependent external neutron sourcewere proposed by Perez et al. (1964). Recently, the use of these methods hasincreased during the MUSE European experimental studies carried out at theMASURCA facility (Cadarache, France) due to their applicability in order toinvestigate ADS kinetic parameters. In these experiments D-D and D-T neu-tron sources running in pulsed mode have been used, although the utilisationof an external spallation proton source has also been thought. Anyway, mea-surements can be done in two different ways (Valentine et al. (2000); Degweker(2003); Ceder and Pazsit (2003): in the first one, the neutron detector timegate is synchronized with the external neutron pulse injection, thus this me-thod is referred to as deterministic pulsing method, whereas, in the secondcase, the relative delay between the neutron pulse injection and the begin-ning of the neutron counting time is uniformly sampled between zero and thepulsed source period, which is known as stochastic pulsing method. For thelatter case, the neutron source can be assumed to have the formS (t) = ksummationdisplaym=(t( + mT), (1)where k is the number of protons injected per proton pulse, T is the pulsedsource period, and is uniformly sampled within the time interval 0,T.Obviously, for the deterministic pulsing method we will put = 0.On the other hand, Pal (1958) developed a general theory for the study ofthe stochastic neutron field. This model, complemented afterwards by Bell(1965), completely describe the stochastic neutron field in a fissile assembly.Later, Munoz-Cobo et al. (1987) derived expressions for the variance of thenumber of counts in a detector and the CPSD for a Poissonian source withoutdelayed neutrons from the general neutron stochastic transport theory. Theseapproaches permit to go beyond classical point kinetic approximations, taking2into consideration general problems with spatial, spectral, and angular depen-dence. They have been extensively applied to nuclear subcriticality safety andnon-destructive nuclear fuel assay problems.Classical reactor noise methods are not correct when pulsed or correlatedsources are used (Matthes et al. (1988); Behringer and Wydler (1999). Forthese kind of sources, forward Kolmogorovs approach is not valid becauseof the non-Markovi