The Statistics of the Number of Neutron Collisions Prior to Absorption

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1 NUCLEAR SCIENCE AND ENGINEERING: 153, ~2006! The Statistics of the Number of Neutron Collisions Prior to Absorption Sara A. Pozzi* Oak Ridge National Laboratory, P.O. Box 2008, MS 6010, Oak Ridge, Tennessee and Imre Pázsit Chalmers University of Technology, Department of Reactor Physics, SE Göteborg, Sweden Received May 19, 2005 Accepted July 25, 2005 Abstract We propose a simple analytical model to describe the statistics of the number of collisions undergone by fast neutrons during slowing down until they are absorbed. We assume that the moderator is homogeneous and account for scattering and absorption, but we do not consider thermalization. Although the problem cannot be solved in a compact form, a simple recurrent formula provides the solution in a very transparent way. The model can be readily evaluated numerically, and the results are in excellent agreement with the corresponding Monte Carlo simulations. Both the mean number and the variance of the number of collisions are calculated. The results are discussed and compared with the classical case of neutron slowing down to or past a given energy in a moderating medium without absorption. I. INTRODUCTION * pozzisa@ornl.gov The process of neutron slowing down via elastic collisions on nuclei is a random process with peculiar and intriguing properties. For example, a well-known and counterintuitive fact 1 4 is that the average number of collisions undergone by a neutron to slow down to, or below, a certain energy E from a starting energy E 0 cannot be calculated from the mean energy loss ratio per collision in the form of a geometric series, nor from the average logarithmic energy loss as an arithmetic series, although the latter yields a good approximation for large energy losses. Moreover, higher moments of the number of collisions do not seem to have been discussed even for the classical case of elastic scattering without absorption; we will address this question in this paper. The number of collisions undergone by a neutron during slowing down in a scattering and absorbing medium was the object of recent Monte Carlo investigations. 5 The motivation arises from the need for enhanced information extraction from neutron detection methods for nuclear material management and accounting purposes, in connection with nuclear safeguards, nonproliferation, and homeland security. One application considers detectors that use organic scintillators doped with an absorbing medium, such as boron or lithium. The neutron detection occurs in two steps: First, the neutron generates a scintillation pulse by multiple scatterings in the scintillating material ~hydrogen and carbon!; then, the neutron is captured in the absorbing medium. This capture occurs predominantly when the neutron has lost most of its energy in the previous scatterings. In the instrumentation, the scintillation pulse is gated by the subsequent neutron capture, which produces a pulse having fixed amplitude. Because of the nonlinear relationship between energy transfer and light output, the latter is determined not only by the total energy transferred to the medium by the neutron but also by the entire energy transfer statistics. Moreover, the energy transfer and the collision number are related quantities, and therefore, the statistics of 60

2 NUMBER OF NEUTRON COLLISIONS PRIOR TO ABSORPTION 61 the number of collisions is of prime interest to understand the process of unfolding the detector light output into the incoming neutron energy spectrum. This heretofore disregarded fact constitutes much of the motivation behind the present work. In particular, we shall discuss the mean and the variance of the collision number leading to absorption, but higher moments can be computed as well. Regarding terminology, we have to distinguish between the number of scatterings ~excluding the capture event! and the total number of collisions ~including the capture event! that occur in the life history of the neutron. The scintillation light generation is related to the number of scatterings; however, when making a comparison with the classical case of slowing down without absorption, it is more natural to consider the total number of collisions. The first two moments of these random numbers are very simply related; the average total number of collisions exceeds by one the average number of scatterings, whereas their variances are equal. It follows that there will be some difference in the relative variances. The interest in the statistics of the collision number comes also from another direction. Namely, the quantitative treatment of particle slowing down and spatial transport is simulated very accurately by Monte Carlo methods. Originally, Monte Carlo codes were developed to calculate first-order moments of particle transport based on probabilistic modeling of the stochastic transport, such as in the well-known MCNP package 6 for calculating neutron fluxes, leakage currents, and reaction rates. However, recently, it has become clear that higher-order moments of the particle distribution are needed in a number of problems of interest. Examples are the variances of various quantities, as well as two-point ~in time! distributions of the neutron detection process. 7 Hence, a new generation of particle transport Monte Carlo codes was developed that makes it possible to calculate entire probability distributions of various transport quantities, as well as temporal correlations among detections of various particles ~neutrons, gammas, etc.!, by implementing the correct statistics of individual reactions on an eventresolved basis. To carry out such simulations, the original MCNP code was modified to a full-statistics code for neutron and gamma-ray transport: MCNP-PoliMi ~Ref. 8!. Using MCNP-PoliMi, various higher-order moments of the distribution of the collision number, or indeed any statistical quantity related to the neutron and photon transport, can be easily calculated. However, as is often the case with inverse tasks such as unfolding a measured light output into an incoming energy spectrum of source neutrons, a basic understanding of the process in simple terms is highly desirable. The objective of this paper is to provide a simple and accurate analytical model for the statistics of the neutron collisions upon slowing down in a homogeneous and absorbing medium. The present work started out by performing Monte Carlo simulations to treat the detection of fast neutrons in a capture-gated organic scintillator. It was noticed that the average number of collisions before a neutron is captured depends very weakly on the initial energy of the neutron. Although it soon turned out that the reason for this weak dependence is the finite size of the detector and hence the leakage effect, the problem turned our attention to the question of the average number of collisions prior to capture. Surprisingly, this problem does not seem to have been discussed in the literature. Much work has been done on the calculation of the energy spectrum during slowing down in an absorbing medium ~see, e.g., Refs. 9 and 10! but not in a collision-numberresolved way. The present paper completes this lack of treatment in the literature. A discussion of the variance of the collision number is also included in the paper. We have not found any discussion of the variance of the number of collisions in the classical problem of slowing down in the absence of absorption, although the formalism is available in the literature for the calculation of any moments of the classical collision number. II. ANALYTICAL MODEL The calculation of the number of collisions required to slow down a neutron in hydrogen from energy E 0 to energy E E 0 has been treated extensively. 1 4 The neutron energy distribution following the n th collision is f n ~E! 1 E 0 ~n 1!! lnn 1 E 0 E, n 1. ~1! For n 0, i.e., for the spectrum of the uncollided neutrons, one can write f 0 ~E! d~e E 0!. ~2! It is interesting to note that among the earliest derivations of Eq. ~1!, there is a reference to work in Russia; according to Mukhin, 11 this formula was derived by Kurchatov, Artsimovich, and others in 1935, although he only gives a reference to a publication in the Russian Journal of Theoretical and Experimental Physics by the same authors in 1965 ~Ref. 12!. This derivation is of course not restricted to hydrogen, and it can be extended to other moderating materials. For the case of A. 1, no simple closed form for the distribution functions f n ~E!, such as Eq. ~1!, can be given. Various forms of calculation are found in Refs. 1, 3, 4, and 10. One way to determine f n ~E! is to use a recursion formula of the type E0a f n 1 ~E '! f n ~E! E ~1 a!e de ', ~3! ' where, as usual, a ~A 1! 2 0~A 1! 2, with f 0 ~E! given by Eq. ~2!. As is known, the f n ~E! of Eq. ~3! is nonvanishing only for a n E 0 E E 0.

3 62 POZZI and PÁZSIT where q ln a. A representation of Eq. ~5! is given in Fig. 1 with dotted lines for n 1to8. Equations ~1! through ~6! have been used in the past to calculate the average number of collisions m~e! that a neutron suffers to slow down to a certain energy interval E de, or the average number n * ~E! to slow down below a certain energy E. This notation was introduced by Cohen. 3 As is well known, these two numbers are the same for slowing down in hydrogen ~and only in hydrogen!, which is a fact that is surprising in itself: m~e! n * ~E! u 1, ~7! Fig. 1. Neutron energy spectra as a function of neutron lethargy for 1-MeV neutrons after 1,2,...,8 scatterings. Solid lines show spectra with absorption, and dotted lines show spectra without ~5!#. In calculations, in general it is practical to turn to the lethargy variable u ln E 0 E. ~4! Then, a simple substitution of Eq. ~4! into Eq. ~1! yields the energy distributions for scattering on hydrogen, as functions of lethargy per unit lethargy interval, as f n ~u! e u u n 1 ~n 1!! ; f 0 ~u! d~u!. ~5! The latter form is normalized both in u and n, so f n ~u! is a probability density in u for a given n, and a discrete probability distribution in n for a given u. For the case A. 1, one has q f n 1 ~u u '!e u' f n ~u! du 0 ', ~6! 1 a where u is the neutron lethargy, defined in Eq. ~4!. Table I shows the number of collisions needed, on average, to slow down a neutron from E 0 1to15MeVtoE 1eV in hydrogen. To our knowledge, no attempt has been made to analyze higher-order moments of the distribution f n ~E! in the collision number, of which Table I reports the average. Our derivation consists of developing a simple analytical model to describe the neutron slowing-down process in a moderating medium that contains an absorber. In line with the calculations of the mean number of collisions to slow down a neutron to a given energy or past a given energy E, starting from energy E 0, the central quantity to be calculated is the probability P n that the neutron will be absorbed exactly at the n th collision. This quantity can be calculated by knowledge of the relevant cross sections and of the neutron energy spectra at the ~n 1! th collision. In our simple model consisting of only hydrogen and boron, capture occurs predominantly on 10 B and scattering on hydrogen. The total cross section is then S T ~E! S c ~E! S s ~E!, and the probability of absorption in a collision with energy E is given as W~E! [ S c~e! 1. ~8! S T ~E! We can now calculate recursively both the energy distributions f n ~a! ~E! of n-times collided neutrons in the presence of absorption and the probability P n that the neutron TABLE I Average Number of Collisions for Neutron Slowing Down in Hydrogen Incident Neutron Energy, E 0 ~MeV! Average number of collisions n * ~E!# to slow down to ~or past! E 1 ev

4 S NUMBER OF NEUTRON COLLISIONS PRIOR TO ABSORPTION 63 will be captured at the n th collision. For slowing down on hydrogen, the latter reads 0 P n 0E f ~a! n 1 ~E!W~E! de 0 ` f ~a! n 1 ~u!w~u! du, ~9! whereas the energy spectrum after the n th collision, i.e., the energy spectrum of n-times collided neutrons, is E 0 fn 1 ~E '! f ~a! n W~E E!# de ' E ', ~10! or in the lethargy domain, f n ~a! ~u! 0 u f ~a! n 1 ~u u '!@1 W~u u '!#e u' du '. ~11! In particular, P 1 W~E 0! W~u 0! ~12! and or f 1 ~a! ~E! 1 W~E 0! E 0 ~13! f 1 ~a! ~u! 1 W~0!. ~14! For the case of A. 1, the formulas are modified as follows: and f n ~a! ~E! E or f n ~a! ~u! 0 E 0 P n a n 1 E 0 0 E0a ~n 1!q f ~a! n 1 ~E!W~E! de f ~a! n 1 ~u!w~u! du ~15! f ~a! n 1 ~E '! ~1 W~E '!# de ' ~16! ' q f ~a! n 1 ~u u '!e u' 1 W~u u '!# du '. ~17! From Eqs. ~9!, ~10!, and ~11!, or Eqs. ~15!, ~16!, and ~17!, all the quantities P n and f n can be calculated recursively. The average number of collisions to capture ns is calculated as ` n ( np n, ~18! n 1 and all higher-order moments can be calculated in a similar way. It should be noted here that ns is a tally of the number of neutron collisions that includes the capture event. The number of elastic scatterings prior to capture is therefore ns 1. III. NUMERICAL WORK The numerical work that follows below will be restricted to the case of the scattering on hydrogen. Before calculating the moments of the collision number, we consider the energy spectra of n-times collided neutrons in the presence of absorption. As the above formulas show, these can be calculated recursively, without calculating the absorption probabilities. In Fig. 1 we show the energy distributions per unit lethargy for the few first collisions. For the sake of comparison, the classical spectra in an absorption-free medium, Eq. ~5!, are also plotted. The spectra in Fig. 1 show the effect of neutron absorption, which is evident at low energies and with increasing n. The spectra with absorption decay faster with increasing lethargy than the classical spectra. III.A. Average Values The average number of collisions was calculated for neutrons with initial energies from 1 to 15 MeV in a hypothetical detector consisting of a mixture of hydrogen and 10 B, with approximately 36 atoms of hydrogen for each atom of 10 B. Scattering on boron and absorption on hydrogen were neglected. The scattering and absorption cross sections were taken from the evaluated ENDF0 B-VI data and are shown in Fig. 2 as a function of neutron energy. Because the calculations were made in the lethargy domain, the same cross sections are shown as functions of the lethargy, with E 0 15 MeV in Fig. 3. Figures 2 and 3 also show the weighting function W~E!@or W~u!#, defined in Eq. ~8!. The calculated values of the P n for various values of E 0 from 1 to 15 MeV, in 1-MeV steps, are shown in Fig. 4. The values of n, S calculated with Eq. ~18!, are given in Table II. The results obtained with the analytical model were compared with Monte Carlo simulations, performed with the MCNP-PoliMi code 8 and a specifically designed postprocessing code. Monoenergetic neutrons having energy 1 to 15 MeV were simulated impinging on a detector comprising hydrogen and boron. In order to make a proper comparison, a large detector with dimensions cm was simulated, so leakage did not occur: All the neutrons were slowed down and captured inside the detector. To perform a meaningful comparison with

5 64 POZZI and PÁZSIT Fig. 2. Hydrogen and boron ENDF0B-VI cross sections and weighting function W~E! as a function of neutron energy. Fig. 3. Hydrogen and boron ENDF0B-VI cross sections and weighting function W~u! as a function of neutron lethargy, with E 0 15 MeV.

6 NUMBER OF NEUTRON COLLISIONS PRIOR TO ABSORPTION 65 Fig. 4. P ~9!# as a function of the number of neutron collisions n. the numerical model described previously, the Monte Carlo histories that included neutron scattering on boron or absorption by hydrogen were not considered in the analysis performed with the postprocessor. These histories were few compared with the number of histories comprising scatterings on hydrogen and final capture on boron. The results are shown in Fig. 5, and the data are reported in Table II. Figure 5 shows excellent agreement between the Monte Carlo simulations and the simple model calculations. The relative difference between the model and the simulations is within 0.5% on the entire energy range. The question might be raised whether these results can be converted into simpler formulas, based on the classical result of the number of collisions n * ~E!# required to slow down to ~or past! a certain energy E, which was discussed earlier. In particular, one may ask what energy level must be chosen for the average number of collisions required to reach that energy, or to slow past that energy, to equal Fig. 5. Average number of neutron collisions to capture as a function of incident neutron energy ~MeV! for a large detector comprising hydrogen and boron. the average number of collisions to absorption calculated by Eq. ~18!. From Eq. ~6!, one obtains that this energy can be calculated by setting from which one obtains the energy as ns u 1, ~19! E e ~ n 1! S E 0. ~20! For E 0 1 MeV, Table II gives ns 13. This number includes the capture event, so we must subtract 1 from it, and from Eq. ~17! we obtain E 16.7 ev. Similarly, for E 0 10 MeV, one obtains E 22.6 ev. This shows that it is not possible to use one single energy as the slowingdown limit and apply the classical formula to calculate from it the average number of collisions to absorption; however, as a rough estimate, one can take the neighborhood of 18 ev as the equivalent slowing-down limit. This energy is significantly larger than the most likely TABLE II Average Number of Collisions to Neutron Capture Incident Neutron Energy, E 0 ~MeV! Average number of collisions ~n! obtained with analytical model Average number of collisions ~n! simulated with MCNP-PoliMi and, shown in parenthesis, s n ~0.01! ~0.01! ~0.01! ~0.01! ~0.01! ~0.01! ~0.01! ~0.01! ~0.01! ~0.01! ~0.01! ~0.01! ~0.01! ~0.01! ~0.01!

7 66 POZZI and PÁZSIT energy of the thermal distribution and is also higher than the ~obviously arbitrary! upper limit of the thermal region, which is usually set as 1 ev. The above threshold energy, ;18 ev, can, on the other hand, be compared with the average energy of the neutrons prior to absorption, which can also be calculated using Monte Carlo. In the present model it varies between 400 and 600 ev with neutron initial energies of 1 to 15 MeV ~see Fig. 6!, which is actually significantly higher than the hypothetical limit of slowing down to give the same average number of collisions. Finally, we mention that the classical formula can be used as a rule of thumb for estimating the difference ~as opposed to the absolute value! between the number of collisions needed for absorption with source neutrons of two different energies. To slow down to any given lower energy E from initial energies E 1 and E 2 E 1 requires u 1 1 and u 2 1 collisions, respectively. Hence, the difference in the number of collisions is u 1 u 2. For E 1 10 MeV and E 2 1 MeV, this gives a difference of ln~10! 2.3 collisions. From the model ~and also from the more accurate Monte Carlo simulations!, this difference is approximately 2 collisions. Hence, the classical formula remains suitable to give a rough estimate of the differences between the number of collisions to absorption from different starting energies, at least in this particular case. III.B. Second Moments Fig. 6. Average neutron energy at capture as a function of incident neutron energy ~MeV! for a large detector comprising hydrogen and boron. When calculating the second moments, it is again worth making a comparison with the classical case of slowing down in a purely scattering medium. First, a general expression will be derived for the second moments because such a formula is not available in the literature. Then, the variance will be calculated and compared with the variance for a medium with absorption. To this end, it is practical to introduce the generating function g~z, u! of the distribution function f n ~u!, given in Eq. ~5!, for n 1, as follows: ` g~z, u! ( z n f n ~u!. ~21! n 1 Using Eqs. ~5! and ~21!, we obtain g~z, u! ze u~z 1!. ~22! Then, the factorial moments can be calculated in the usual way, by taking the derivatives with respect to z as follows. For the first moment, one obtains ]g~z, u! n~u! S ]z u 1 ~23! z 1 as expected. For the second factorial moment, one has ^n~n 1!& ] 2 g~z, u! 2u u 2. ~24! ]z 2 z 1 Hence, from Eqs. ~23! and ~24!, one readily obtains the variance as s 2 n ~u! u. ~25! It might be more interesting to discuss the behavior of the relative variance s 2 n ~u! u n~u! S u 1. ~26! Equation ~26! shows that for small u values, the relative variance starts out from zero in a linear way. This is just an expression of the fact that for vanishing lethargy values, the probability of the event of one collision is very close to unity and all the other probabilities are approximately zero; i.e., the distribution is deterministic. With increasing lethargies, the variance starts to increase, and it approaches asymptotically the value one, which is similar to the relative variance of a Poisson distribution. Physically, this means that just as with the case of the Poisson distribution, the individual collisions become independent if the average number is large enough. There are also similarities between Eq. ~26! and the so-called Fano factor of atomic collision cascades. This factor is also the relative variance of certain quantities, such as the number of charges induced by an incoming energetic particle in a detector, or the number of Frenkel pairs ~vacancy-interstitial! in materials The Fano factor is usually less or much less than unity because the total number of events is usually bounded by some conservation relation ~e.g., conservation of energy! that drives the process toward deterministic behavior by inducing

8 NUMBER OF NEUTRON COLLISIONS PRIOR TO ABSORPTION 67 TABLE III Relative Variance of the Number of Neutron Collisions Incident Neutron Energy, E 0 ~MeV! Relative variance ~0.07! ~0.07! ~0.07! ~0.07! ~0.07! ~0.07! ~0.07! ~0.06! ~0.06! ~0.06! ~0.06! ~0.06! ~0.06! ~0.06! ~0.06! negative correlations ~transfer of a large amount of energy in one collision makes it more likely to transfer a small amount of energy in the next!. This is in contrast to the case when branching is present, as in a fission chain, where there is no particle conservation; moreover, the simultaneous generation of several particles in the fission event leads to positive correlations and, hence, larger than Poisson variance. This is the case of the Feynmanalpha measurements. In the present case, although the relative variance is less than unity for small lethargies, in the limit of energetic particles slowing down to thermal energies, the relative variance approaches unity, i.e., the result for a Poisson distribution ~no correlations!. The relative variance of the number of collisions ~including capture! in the presence of absorption can also be easily calculated from the collision distribution P n. The results are shown in Table III. It is seen that the relative variance exceeds unity; i.e., the variance is larger than that of a corresponding Poisson process. This is because the absorption, which can happen randomly at any time ~i.e., after any collision number in the slowingdown process!, introduces an additional randomness in the history of the process. This effect is expected to be even more pronounced in finite detectors, where the neutron leakage influences both the mean value and the variance of the number of collisions. The relative variance of the number of scatterings behaves in a similar manner, but it will be slightly larger. This is because the expected number of scatterings is one less than the total number of collisions, whereas the variance remains the same. At any rate, the relative variance of both the scatterings and the collisions is over- Poisson; this is clearly different from the relative variance of the number of scatterings to slow down in a nonabsorbing medium, which is slightly ~23!#. IV. CONCLUSIONS The simple model introduced in this paper yields the complete statistics of the number of collisions leading to capture in a homogeneous mixture of a scattering and absorbing material. We obtained excellent agreement between the model results and corresponding Monte Carlo simulations. A deeper understanding of the collisionnumber-resolved statistics of the slowing-down process was gained, which will be used in the future to interpret measurements in the field of fissile material accounting. The result is also useful for the basic understanding of the neutron slowing-down process. The model can be extended to include neutron multiplication and transport in an inhomogeneous medium. ACKNOWLEDGMENTS We thank V. Protopopescu for his careful revision of this manuscript and for many valuable comments and discussions. The Oak Ridge National Laboratory is managed and operated for the U.S. Department of Energy by UT-Battelle, LLC, under contract DE-AC05-00OR REFERENCES 1. L. RUBY, Dancoff s Solution for the Number of Collisions Necessary to Slow Down, Nucl. Sci. Eng., 86, 110 ~1984!. 2. H. VAN DAM, A Difference of at Least Three Collisions, Nucl. Sci. Eng., 89, 365 ~1985!. 3. E. R. COHEN, How Many Collisions to Slow Down a Neutron, Nucl. Sci. Eng., 89, 99~1985!. 4. P. REUSS, Additional Remarks on the Number of Collisions Required to Slow Down a Neutron, Nucl. Sci. Eng., 93, 105 ~1986!. 5. S. A. POZZI, R. B. OBERER, and J. S. NEAL, Analysis of the Response of Capture-Gated Organic Scintillators, Proc. IEEE Nuclear Science Symp., Rome, Italy, October 16 22, 2004, Institute of Electrical and Electronics Engineers. 6. MCNP A General Monte Carlo N-Particle Transport Code, LA M, J. F. BRIESMEISTER, Ed., Los Alamos National Laboratory ~2000!. 7. J. T. MIHALCZO, J. A. MULLENS, J. K. MATTINGLY, and T. E. VALENTINE, Physical Description of Nuclear Materials Identification System ~NMIS! Signatures, Nucl. Instrum. Methods Phys. Res. A, 450, 2 3, 531 ~2000!.

9 68 POZZI and PÁZSIT 8. S. A. POZZI, E. PADOVANI, and M. MARSEGUERRA, MCNP-PoliMi: A Monte Carlo Code for Correlation Measurements, Nucl. Instrum. Methods Phys. Res. A, 513, 3, 550 ~2003!. 9. M. M. R. WILLIAMS, The Slowing Down and Thermalisation of Neutrons, North-Holland Publishing Company, Amsterdam ~1966!. 10. R. E. MARSHAK, Theory of Slowing Down of Neutrons by Elastic Collision with Atomic Nuclei, Rev. Mod. Phys., 19, 185 ~1947!. 11. K. N. MUKHIN, Experimental Nuclear Physics ~1985! ~Hungarian translation!. 12. I. V. KURCHATOV et al., Sov. Phys. JETP, 5, 659 ~1965! ~in Russian!. 13. U. FANO, Phys. Rev., 72, 26~1947!. 14. U. FANO, Phys. Rev., 2, 328 ~1953!. 15. G. F. KNOLL, Radiation Detection and Measurement, John Wiley and Sons, New York ~1979!.

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