Application of Monte Carlo Method to Solve the Neutron Kinetics Equation for a Subcritical Assembly

Transcription

1 Journal of NUCLEAR SCIENCE and TECHNOLOGY, Vol. 45, No. 11, p (28) ARTICLE Application of Monte Carlo Method to Solve the Neutron Kinetics Equation for a Subcritical Assembly Kohei IWANAGA 1;,y, Hiroshi SEKIMOTO 1 and Takamasa MORI 2 1 Research Laboratory for Nuclear Reactors, Tokyo Institute of Techlogy, O-okayama, Meguro-ku, Tokyo , Japan 2 Japan Atomic Energy Agency, 2-4 Shirane, Shirakata, Tokai-mura, Naka-gun, Ibaraki , Japan (Received February 18, 26 and accepted in revised form March 31, 28) There is a need to understand the space-dependent kinetics of fast or thermal reactor physics and the Monte Carlo method should be implemented in kinetics codes as well. In a transient accident (for example, control rod eection accident or loss of coolant accident), changes in the system are much slower than the prompt neutron lifetime. In the present paper, a method is proposed in which the Monte Carlo method is used to solve the neutron kinetics of reactors, such as the case of a subcritical static reactor with an external neutron source. In the proposed method the time derivative of the neutron equation is set to zero, though the time derivative of the delayed neutron precursor equation is treated without any approximation in the time differentiation of delayed neutron precursor equation. Most kinetics calculation methods originate from the point kinetics method, and they require calculation of excess reactivity at each time step, but the present method does t need calculation of excess reactivity. The neutron counts measured with small fission chambers in the subcritical kinetics experiment using the JAEA fast critical assembly are compared with counts calculated by the proposed method and compared with the measured values. The agreement is good, and the excellent performance of the method for space-dependent neutron kinetics is shown. KEYWORDS: Monte Carlo method, space-dependent kinetics, neutron static approximation, delayed neutron, precursor static approximation, subcritical assembly, accelerator driven system I. Introduction Corresponding author, kiwanaga@mext.go.p y Present address: The Ministry of Education, Culture, Sports, Science and Techlogy, Chiyoda-ku, Tokyo , Japan ÓAtomic Energy Society of Japan In the early history of nuclear reactor theory, the time variation of a neutron population was treated as a whole and its space dependency was t considered. This approach is called the point kinetics method. With the deployment of large power reactors, however, the requirement of spacedependent information has increased and the point kinetics method has been modified to allow evaluation of the space dependency of a neutron population. Many modifications have been investigated, 1 4) but the need for more detailed information about local responses such as the detector response of transient behavior continues to grow. Recently, much development work has been done for an acceleratordriven subcritical system. The space profile of neutron flux in the subcritical assembly is strongly influenced by both the neutron multiplication factor and external neutron source. The kinetics of a subcritical system with an external neutron source has opened a new area in neutron kinetics study. 5) The Monte Carlo method is a powerful method for estimating local values and widely used in nuclear engineering. However, the random walk calculation is difficult in a system where macroscopic cross sections change with time, and development of the time-dependent Monte Carlo method has progressed more slowly than time-dependent deterministic transport calculation methods. In the present paper, a method to solve the neutron kinetics (for example, a subcritical static reactor) employing the Monte Carlo method is proposed. An approximation is introduced such that the time derivative of neutron equation is set to zero, though the time derivative of the delayed the neutron precursor equation is treated. We previously carried out an analytical study on time derivative terms using the point kinetics method. 6) Recently, we have carried out a kinetics experiment using the Fast Critical Assembly (FCA) of the Japan Atomic Energy Agency (JAEA; formerly JAERI) and we measured detector responses and compared them with the calculated values obtained by our newly developed Monte Carlo code. The principle and algorithm of the proposed method are given in Secs. II and III, respectively. The details of the experiment are shown in Sec. IV. The analysis of this experiment using the newly developed computer code is given in Sec. V and comparison between calculated and measured results is given in Sec. VI. 199

2 11 K. IWANAGA et al. II. Principle of Monte Carlo Kinetics Equation 1. Application of Monte Carlo Method to Space-Dependent Kinetics Problems Most reactor cores have complicated structures. It is difficult to simulate these spatial structures in the reactor analysis by using a deterministic neutron transport analysis method. On the other hand, a Monte Carlo method can treat such complicated structures more easily than the deterministic method and it has been widely used in reactor analyses recently. Therefore, in the present paper we use the Monte Carlo method for reactor kinetics analysis. However, the neutron random walk is treated in the steady state in conventional Monte Carlo codes. The difficulty of application of Monte Carlo codes to kinetics problems is shown below. 2. Difficulty of Neutron Random Walk in Changing Medium We consider a neutron that collides with a nucleus after passing through a distance of x freely; the probability distribution function f ðxþ of x can be written by Eq. (1): f ðxþdx ¼ e tx t dx; ð1þ where t is the macroscopic total cross section. The cumulative distribution function is derived from this equation as Eq. (2). FðxÞ Z x f ðx Þdx ¼ 1 e tx Then in the Monte Carlo calculation, 7) we can choose x with the proper probability distribution function in the following way. At first we choose a random number where 1, and substitute it for FðxÞ in Eq. (2). Then we can obtain x by solving this equation as x ¼ 1 t lnð1 Þ: In the derivation of Eq. (3), an important point is that t is t a function of time but a constant. If t is time-dependent, FðxÞ becomes a complicated function of x and generally x cant be written in a simple equation of like Eq. (3). Expected reactor malfunctions and accidents can be described by time constants having units of milliseconds or seconds; the prompt neutron lifetime is typically 1 6 s for assemblies in which fission is caused by fast neutrons. Therefore, almost all random walks in a fast subcritical system will stop within 1 ms. The neutron random walk during its lifetime in this system can be treated with the assumption that the macroscopic cross sections are t changed in this period, and a conventional Monte Carlo code developed for the steady state can be used for the analysis. 3. Neutron Static Approximation The half-lives of delayed neutron precursors are.1 55 s, which are comparable to the time constants of the transient behavior caused by expected reactor malfunctions and accidents mentioned in Sec. II-2. Therefore, in many cases, random walks of delayed neutrons should be treated in a system different from that in which their precursors are born through ð2þ ð3þ fissions. The delayed neutron precursors should be treated time dependently in these cases. In this paper, we call the approximation in which the neutron random walk is treated in a steady state and the time behavior of delayed neutron precursor is treated exactly as a neutron static approximation. It is the same approximation as the prompt ump approximation or zero lifetime approximation. 7) 4. Governing Equations of the Kinetics of Subcritical Assembly At the time t and position r, the flux ðr;;e; tþ of the neutron with the energy E and moving in the direction and the -th delayed neutron precursor density C C ðr; tþ satisfy the following balance equations. þ rþ X ¼ x f x d de E Z þ E Z C þ C ¼ 4 p f d de þ X 4 C þ Q ð4þ f d de ; ð5þ E where ðr; E; tþ: macroscopic total cross section x x ðr; E; tþ: macroscopic cross section of reaction x p f p f ðr; E; tþ: macroscopic fission cross section times average number of prompt fission neutrons f x f x ðr; ; E! ; E; tþ: probability function of neutron transfer from ; E to ; E through reaction x : number of delayed neutrons produced by precursor per fission ¼ X p p ðeþ: prompt fission neutron energy spectrum ðeþ: delayed neutron energy spectrum for precursor : decay constant of delayed neutron precursor Q Qðr; ; E; tþ: external neutron source. Conventional reactor kinetics or dynamics eliminate space and energy dependences from Eqs. (4) and (5) by several methods like weighted integration of the equation or separation of flux into amplitude factor and shape factor; such treatments are called point kinetics. These methods cant treat the transient of a detailed distribution of flux or power density. In the present paper, we try to solve these equations directly by using the Monte Carlo method. We employ the neutron static approximation mentioned in the previous section to solve Eqs. (4) and (5). Then the governing equations of our problem can be written as X ¼ r þ x f x d de Z þ E Z E x6¼f p 4 p f d de þ X 4 C þ Q ð6þ JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

3 Application of Monte Carlo Method to Solve the Neutron Kinetics Equation for a Subcritical Assembly 111 (a) " P (Defined by Eq. (13)) Table 1 Errors caused by neutron and precursor static approximations Inserted reactivity ($) : E E E E E E E E E E E E E E E E E E E 5 k eff E E E 5 1.6E 5 9.9E E 5 3.1E 5 1.2E E E 5 5.1E E E 5 2.9E E E E E 4 1.3E 4 6.7E 5 (b) " D (Defined by Eq. (14)) Inserted reactivity ($) k eff = initial subcriticality ¼ : : E E E E 2 6.E 3 6.4E E E E E E E E 2 3.5E E E E E E 1 k eff E 1 2.8E E 1 1.1E E 2 8.6E 2 2.9E 1 1.4E E E 1 3.2E E E E E E E E E E+2 k eff = initial subcriticality ¼ :7713 For the case of k eff > 1, the steady state solution does t exist, thus the corresponding fields C þ C ¼ f d de : ð7þ E Equation (6) can be solved by the static method for each time step, where all coefficients of the equation including the delayed neutron precursor densities are given. It is thus solved by the Monte Carlo method. On the other hand, Eq. (7) is solved as a time-dependent problem. In the present study, we also investigate the system in which the time derivative of delayed neutron precursor density is assumed to be zero as C ¼ f d de : ð8þ E This treatment is widely used in subcritical kinetics studies. 5) We call this approximation as the precursor static approximation. 5. Investigation of Accuracies of Neutron and Precursor Static Approximations by Point Kinetics Equations We previously examined the subcritical fast reactor system with a source. 6) We explain the approximation with reference to the previous paper. Point kinetics equations are derived from the sets of Eqs. (4) and N ¼ N þ X6 C þ Q C ¼ C þ N ; ð1þ where all the tations are conventional. We te that the symbols and are delayed neutron fractions and different from and used in Sec. II-4. N is the neutron number density. The superscript means the scalar value that does t depend on either space or energy. By introducing the neutron and precursor static approximations, these equations are changed to the following. For the neutron static approximation, ¼ N;P þ X6 ¼1 C ;P þ C;P ¼ C ;P þ N;P : ð12þ For the precursor static approximation, ¼ N;D þ X6 C ;D þ Q ð13þ ¼1 ¼ C ;D N;D : ð14þ The accuracies of these approximations have been studied by the following parameters. " P N ;P N ¼ max t N ð15þ " D N ;D N ¼ max t N ð16þ Some results obtained previously 6) works are shown in Table 1 for a wide range of initial subcriticalities and inserted reactivities, where the reactivity is inserted within.15 s by withdrawing fuel rods from the core. Though the precursor static approximation shows considerable errors for small subcriticalities and large inserted reactivities, the neutron static approximation has very small errors over the whole subcritical region. þ 6. Change of Delayed Neutron Precursor Density The change of delayed neutron precursor density is given by Eq. (7). In the actual calculation, the time variable is divided into time meshes, and the density is calculated along VOL. 45, NO. 11, NOVEMBER 28

4 112 K. IWANAGA et al. f ðr; t Þe ðtiþ1 tþ d de dt : ð17þ Tðt! t iþ1 Þ f ðr; t Þe ðtiþ1 tþ d de dt : ð18þ each successive mesh. The -th precursor density C ðr; t iþ1 Þ at the time t iþ1 is calculated by using C ðr; t i Þ at the previous time mesh t i as C ðr; t iþ1 Þ¼C ðr; t i Þe ðt iþ1 t i Þ þ Z tiþ1 E t i Here, we consider only the production by fission and radioactive decay of a precursor. However, other changes of the system such as movement of a fuel element also causes the change of C ðr; tþ. By deting this change from the time t to t as an operator Tðt! tþ, Eq. (17) is rewritten as Z tiþ1 C ðr; t iþ1 Þ¼Tðt i! t iþ1 ÞC ðr; t i Þe ðt iþ1 t i Þ þ E t i This equation should be employed in the Monte Carlo code. III. Monte Carlo Code 1. Algorithm The neutron static approximation is implemented into the Monte Carlo code MVP, 8) which was developed in JAEA. The algorithm of this code is described here, focusing on the neutron random walk with neutron static approximation. In the present paper, we treat fuel movement in the Monte Carlo analysis. The overall flow chart for the Monte Carlo analysis is shown in Fig. 1(a). In block A, the change of the system is analyzed, and macroscopic cross sections are reevaluated. The movement of a delayed neutron precursor between time meshes i and i-1 is also calculated. In block B, the random walk of a neutron is analyzed at each fixed time mesh i. Before starting the kinetics calculation, an initial state is calculated. The delayed neutron precursor data are transferred to the transient analysis as the delayed neutron source effective for the succeeding time. These data are modified by a system change in block A. The detailed procedure is shown in Fig. 1(b). Adding this delayed neutron source and external source creates the total neutron source, which is used as the neutron source in the neutron random walk performed in block B. The neutron random walk procedure is performed at each time mesh point until the end of the system change. It is divided into 3 blocks, blocks C, D, and E, as shown in Fig. 1(c). The delayed neutron precursors produced in each time mesh are added to the total delayed neutron precursor data, and the procedure goes to the next time mesh. In the present Monte Carlo code the nanalogue method is employed for neutron energies of more than 4.5 ev. For simplicity, the procedure for E < 4:5 ev is omitted in Fig. 1. The neutron random walk procedure is divided into three subprocedures. In each subprocedure, neutron history starts from a different neutron source, as shown in Fig. 1(c); the external source plus the delayed neutron whose precursor is born in a previous time mesh, the fission neutron source produced in this time mesh, and the delayed neuron whose precursor is born in this time mesh. The details of each subprocedure are shown in Figs. 1(d) 1(f). Here, the location of the delayed neutron precursor is selected from the stored collision location. 2. Delayed Neutron Data The role of delayed neutrons becomes more important in kinetics than in statics. Each delayed neutron precursor is treated independently. In the present Monte Carlo code, the delayed neutrons are treated t by the conventional 6- group treatment, but by using the original delayed neutron precursor data shown in Table 2. The data 9 13) in this table are for U-235 fission in a typical fast reactor spectrum, since the system treated in the present paper represents a fast reactor system assembled with highly enrich uranium. For simplicity, we make the approximation that all the precursors are produced directly from fission. The number of delayed neutrons produced by the precursor per fission,, is calculated by ¼ p ; ð19þ where is the cumulative yield of the precursor, and p is the probability of producing a delayed neutron of the precursor. The energy of each neutron from the delayed neutron precursor is selected from the following delayed neutron energy spectrum: N ðeþ ¼E 1=2 exp E ; ð2þ T where T is a parameter that characterizes the spectrum and relates to the average energy E of the neutron from the precursor as T ¼ 2 3 E : ð21þ Necessary data are collected from the literature. 9 13) The total number of employed delayed neutron precursors in the present Monte Carlo calculation is 11, and these data are shown in Tables 2(a) and 2(b). IV. Kinetics Experiment Using the FCA The kinetics experiment is performed using the FCA in JAEA. 14) This core is simulated as an accelerator driven system core and it consists of enriched uranium (93%) and stainless steel. It is surrounded by a soft blanket containing poorly enriched uranium and a blanket containing depleted uranium. The whole system is assembled from drawers, whose size is 5:52 5:52 13 cm 3. This core is 2:8 2:8 2:6 m 3, where 2.6 m is the axial length in the horizontal direction. The core can be separated in the axial direction into two symmetric parts side by side for easy preparation of JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

5 Application of Monte Carlo Method to Solve the Neutron Kinetics Equation for a Subcritical Assembly 113 Initial Core Calculation A in A in System Change Calculation Input System Movement Data DNP External Neutron Source A out Generate Macro. X-sections Create Total Neutron Source DNP System Data at i Select DNP at i-1 Total Neutron Source DNP at i B in Neutron Random walk B out Add DNP at i to DNP Determine Location of DNP at i Calculate Radiative Decay of DNP from i-1 to i Go to i+1 System Change Stop? End (a) DNP All DNP Finish? A out (b) Total Neutron Source C in Neutron Generation B in C in Neutron Random walk from Neutron Source Total Neutron Source Determine Collision Location Add Collision Location to File Reevaluate Weight Collision Locations C out D in Neutron Random walk from Fission in i D out E in Prompt Neutron Source DNP Produced in i Collision Locations w<w L? Russian Roulette kill survive Neutron Random walk from DNP Born in i E out B out DNP Produced in i All Initial Neutrons Finish? Select of Fission Point Set (c) Prompt Neutron Source C out DNP Produced In i (d) Fig. 1 Continued on next page the experiment and easy maintenance. One part is fixed on the floor, and the other is movable. These parts are combined at the time of the experiment. The effective neutron multiplication factor for this core is :9998 :2%. A neutron source of Cf-252 is placed at the core center in the fixed part; its intensity is 4:7 1 8 neutrons/s. The configuration of the separation surface of the fixed part is shown in Fig. 2. We use 4 detectors (M4, M5, M8, and P1) placed inside and outside this core. All the detectors are U-235 fission chambers, and each has a dead time of 1:7 1 6 s. The negative reactivity in this experiment is supplied by withdrawing 16 movable fuel assemblies, whose positions are shown in Fig. 2 as control assemblies. The neutron measurements are started prior to the insertion of the negative reactivity initiated by withdrawing all the control assemblies VOL. 45, NO. 11, NOVEMBER 28

6 114 K. IWANAGA et al. Prompt Neutron Source D in DNP produced in i E in Neutron Generation Neutron Generation Determine Collision Location Determine Collision Location Add Collision Location to File Add Collision Location to File Reevaluate Weight w Collision Locations Reevaluate Weight w w<w L? Collision Locations w <w L Russian Roulette survive Russian Roulette survive kill kill All DNPs Finish? All Prompt Neutrons Finish? Select DNP Produced in i D out (e) DNP Produced in i E out (f) Fig. 1 Flow charts of Monte Carlo code DNP: delayed neutron precursor, i: present time mesh, (a) Overall flow chart (b) System change calculation flow chart (c) Neutron random-walk calculation (d) Neutron random walk from neutron source flow chart (e) Neutron random walk from fission in time-mesh i (f) Neutron random walk from DNP born in time mesh i flow chart at the same time. They are continued after the assemblies are fully withdrawn and until their count levels go down to a certain value. Inserted negative reactivity is about 5.6 $, and the time required for insertion is about.15 s. Within this time period, only one measurement can be made because of the instrumentation and data-handling system performance. V. Monte Carlo Calculation We analyze the kinetics experiment described in the previous section by using the developed Monte Carlo code with the modified MVP code. 8) The geometrical parameters are treated without any approximation except for the neutron source and detectors, which are treated as points. The energy spectrum of the Cf-252 neutron source is calculated using Watt s equation. 15) The time required for the movement of the control assemblies is about.15 s, and this interval is divided into 1 time meshes for the kinetics calculation. The time mesh interval of.15 s is considered fine eugh in consideration of the decay constants of delayed neutron precursors. Each Monte Carlo calculation treats histories. We set one detector, P1, in the core, and three detectors, M4, M5, and M8, on the boundary between the soft blanket and the blanket, as shown in Fig. 2. The point estimator 8) is employed for these detectors. In order to reduce its statistical error, a ncollision region is set around each estimated point; this region is set as 5:52 5:52 5:52 cm 3. These reactive rates are calculated by using the number of densities and the volumes of U-235 based on the specification of the detector. The present calculation is done on the Altix39 (SGI Co.) computer at the Center for Promotion of Computational Science and Engineering in JAEA. The required time for our calculation is from 6 to 1 s per one time step Monte Carlo calculation. VI. Comparison between Calculated and Measured Results 1. Initial Steady State At the beginning the initial state is calculated, which is used as the initial condition of the present kinetics analysis as shown in Sec. II. The Monte Carlo method is also employed in this calculation, but it is performed in a steady state, where the governing equations are Eqs. (6) and (8). Since it is a subcritical system, the analogue Monte Carlo method is employed, where the neutron random walk starts from the external source. However, since subcriticality is very small for the present core, the random walk may t end in a reasonable calculation time. However, after the random walk is continued for a long eugh time, flux ampli- JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

7 Application of Monte Carlo Method to Solve the Neutron Kinetics Equation for a Subcritical Assembly 115 Table 2 Nuclear data of delayed neutron precursors for U-235 fast reactor fission 7 11) (a) Nuclear data of delayed neutron precursors (Br-Cs) Nuclide Half-life (s) p (%) p (%) E (kev) Br E E 2 21 Cs E E 3 78 I E 2 2.E 1 54 Te E E 3 28 Br E E I E E Rb E E Se E E As E 3 4.4E La E E 3 69 Nb E E Rb E E 4 25 Br E 2 2.2E In E E Y E 2 8.6E Te E E Ba E E Ga E 4 2.5E Nb E 3 1.6E Mo E 6 1.9E 5 29 Rb E E Zr E E In 129m E E La E 4 7.8E I E E 2 44 As E E 2 74 Y E 2 6.9E Ge E E Sb E E Br E 3 1.6E Cs E 2 2.1E Ba E E Xe E 3 4.7E 4 36 Cs E 2 5.4E Ga E E Te E E Se E E 3 96 Nb E E 3 29 Ag E E Sn E E 6 29 Y E E Tc E E In 127m E E Kr E E Ga E E 4 37 Xe E E 4 29 Ge E E 3 29 Y 97m E E Xe E E 5 32 Cd E E Sn E E Cs E E Table 2 Nuclear data of delayed neutron precursors for U-235 fast reactor fission 7 11) (b) Nuclear data of delayed neutron precursors (Nb-Rb) Nuclide Half-life (s) p (%) p (%) E (kev) Nb E E In E E Xe E E As E E I E E Sb 134m E E In E E Tc E E Sb E E Y E E 2 5 Kr E E Ba E E Sr E E Y 98m E E Sr E 4 1.8E 3 29 La E E Ga E 3 2.2E 2 31 Br E E 3 51 Sr E E Cs E E 3 45 In E E 4 59 Te E 5 1.3E Se E 4 1.2E Cs E E Sb E E I E E Se E E Sn E E Sr E 2 6.7E Ag E E Rb E E Br E E Kr E E Cs E E Zn E 6 9.6E Ga E 6 1.3E As E E In E E 5 84 Se E E Ge E E Ge E 6 4.5E Kr E E Rb E E I E E Br E E Rb E E 2 53 Rb E E In E 6 2.8E Rb E 5 1.9E p = the probability for producing delayed neutron of the precursor = the cumulative yield of the precursor E = the neutron energy from precursor tude decreases sufficiently and the relative flux distribution is considered to converge to its fundamental mode and its dumping factor (the ratio of the neutron flux amplitudes between successive neutron cycles) converges to a constant which is equal to the neutron multiplication factor of this system without a neutron source. The flux distribution for VOL. 45, NO. 11, NOVEMBER 28

8 116 K. IWANAGA et al. M4 P1 Fig. 2 Core Soft blanket Blanket M8 M5 Control assembly Neutron source Neutron detector Separation surface of fixed side of FCA further generations can be estimated by using the previous distribution. In the present analysis, the neutron random walks are performed with eugh histories for each neutron cycle from the first cycle. We preserve the end point of the random walk of each generation. The method is similar to the conventional critical system Monte Carlo method. The detector counts are estimated analytically by using the assumption of constant amplitude dumping factor after a certain number of random walk stages and by summing up all these contributions. The calculated counts obtained at the steady state before the kinetics experiments are shown in Table 3 with the measured counts for comparison. Since the measurement time is long, the measured errors are very small. (about :8%.) The calculated values are systematically lower than the measured values and statistical error of the former error is about 2%. Both calculated and measured values (counts) agree within 6% or less. We attribute this to neglecting the contribution of the higher modes of neutron flux distribution in the calculation of the contribution of later random walk stages to the neutron counts. However, these discrepancies are small eugh for the initial condition of the present neutron kinetics study, where the measurement is made at a transient state and the calculation is done with some approximations. 2. Transient State The calculated and measured counts at the transient state are shown in Figs. 3(a) 3(d) for each detector, where the origin of the horizontal axis is the starting time for control assemblies to be removed and the starting and stop times of measurement are :1 and.2 s, respectively. The initial counts are also shown at the starting time, but the errors of these counts are t shown, since their values are too small; they are given instead in Table 3. In these figures, the calculated results are shown for both neutron and precursor static approximations. The precursor static approximation gives much lower values than the neutron static approximation. The characteristic of accuracy decrease is consistent with the results shown in Sec. II-5. The horizontal error of the measured value is the gate time width for neutron counting. The calculated values using the neutron static approximation agree well with the measured values except for detector P1 for which the calculated value is lower than the measured value. We attribute this to the treatment of the neutron source and detector in the calculation. For example, in the calculation, both the neutron source and the neutron detector are treated as points, but the actual source and detector have volumes and sizes that are comparable to the distance between the source and the detector. The solid angle of an incidence neutron to the detector increases for the point neutron source. Such approximations usually underestimate the count rate. This effect is smaller for the initial steady state, since the kinetics experiment core is near critical and the neutron spatial distribution becomes flatter. VII. Conclusion When we think about a transient accident from the viewpoint of the time interval, the nuclear system will change with a time constant that is much larger than the prompt neutron lifetime. In the present paper, a method to solve the neutron kinetics by using the Monte Carlo method employing a neutron static approximation was proposed, where the time derivative of the neutron equation was set to zero, though the time derivative of the delayed neutron precursor equation was treated. Most kinetics calculation methods are derived from the point kinetics method, and they require calculation of excess reactivity at each time step. The present method does t need calculation of excess reactivity. The neutron counts of several detectors were calculated using the proposed Monte Carlo method and compared with measured counts of a subcritical kinetics experiment using the JAEA fast critical assembly, and good agreement was Table 3 Calculated and measured neutron count rates at steady state before insertion of negative reactivity Detector Calculated Measured (Cc-Cm)/Cm Count (n/s), Cc Error (%) Count (n/s), Cc Error (%) (%) M M M P JOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY

9 Application of Monte Carlo Method to Solve the Neutron Kinetics Equation for a Subcritical Assembly Calculated counts (neutron static approx.) 16 Calculated counts (neutron static approx.) Count (n/s) Calculated counts (precursor static approx.) Measured counts (average value between -.1s and.2s) Count (n/s) Calculated counts (precursor static approx.) Measured counts (average value between -.1s and.2s) Time after reactivity insertion (s) (a) Time after reactivity insertion (s) (b) Count (n/s) Calculated counts (neutron static approx.) Calculated counts (precursor static approx.) Measured counts (average value between -.1s and.2s) Count (n/s) Calculated counts (neutron static approx.) Calculated counts (precursor static approx.) Measured counts (average value between -.1s and.2s) Time after reactivity insertion (s) (c) Time after reactivity insertion (s) (d) Fig. 3 Calculated and measured counts per.3 s (a) M4 detector (b) M5 detector (c) M8 detector (d) P1 detector obtained. The calculation was also done with the precursor static approximation, where the time derivative of the delayed neutron precursor equation was also set to zero, and the results were compared with the measured data. The discrepancies for this case were too large for practical usage. References 1) G. I. Bell, S. Glasstone, Nuclear Reactor Theory, Van Nostand Reinhold Co., New York (197). 2) K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, La Grange Park, Illiis (1985). 3) D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, La Grange Park, Illiis (1993). 4) W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, Inc., New York (21). 5) A. Gandini, M. Salvatores, The physics of subcritical multiplying systems, J. Nucl. Sci. Techl., 39[6], 673 (22). 6) K. Iwanaga, H. Sekimoto, Study on kinetics of subcritical system; contribution of delayed neutrons to the transition after reactivity insertion, Ann. Nucl. Energy, 32, 1953 (25). 7) S. A. Dupree, S. K. Fraley, A Monte Carlo Primer, Kluwer Academic/Plenum Publishers, New York (22). 8) Y. Nagaya, K. Okumura, T. Mori, M. Nakagawa, MVP/ GMVP II: General Purpose Monte Carlo Codes for Neutron and Photon Transport Calculations based on Continuous Energy and Multigroup Methods, JAERI-Data/Code 98-25, Japan Atomic Energy Agency (JAEA) (24). 9) W. B. Wilson, T. R. England, Delayed neutron study using ENDF/B-VI basic nuclear data, Prog. Nucl. Energy, 41[1], (197). 1) W. B. Wilson, R. E. Schenter, F. M. Mann, Aggregate delayed neutron intensities and spectra using augmented ENDF/B-V precursor data, Nucl. Sci. Eng., 85[2], (1983). 11) T. R. England, B. F. Rider, Fission product chain yields and delayed neutrons: ANS standards 5.2 and 5.8, Trans. Am. Nucl. Soc., 62, (199). 12) W. B. Wilson, T. R. England, Development and status of fission-product-yield data and applications to calculations of decay properties, Trans. Am. Nucl. Soc., 66, (1992). 13) E. Kiefhaber, Influence of delayed neutron spectra on fast reactor criticality, Nucl. Sci. Eng., 111[2], (1992). 14) J. Hirota, Fast Reactor Critical Experiment using FCA and Analysis, JAERI-7753 (1978). 15) G. R. Keepin, Physics of Nuclear Kinetics, Addison Wesley Publishing Co., (1965). VOL. 45, NO. 11, NOVEMBER 28