Another powerful tool for solving particular reactor problems is given by the Monte Carlo method, which replaces a deterministic problem by an analogous one consisting of a game to be played many times. These numerical calculations which make use of statistical variables can be used to simulate the physical phenomena actually happening in a reactor where the neutrons have a stochastic behaviour. A high number of neutron

histories are simulated on the computer, every interesting event is recorded and statistical averages are calculated. Random numbers are used to decide at which place collisions occur and of which type they are (absorption, scattering, etc.). Although only a series of numbers can be defined as random, one speaks usually of random numbers. They are usually generated by arithmetic procedures in the range between 0 and 1.

Apart from the actual simulation of the physical phenomena the Monte Carlo method can be used to obtain numerical solutions of integral or differential equations in many space dimensions. Seen from this second point of view the Monte Carlo methods can be thought of as a numerical tool for solving the Boltzmann transport equation.

An example of the stochastic methods for a non-stochastic problem can be given by the evaluation of the area under the curve of Fig. 4.3. N random points can be chosen within the dotted rectangle and if M is the number of points falling below the curve the area can be obtained after having calculated the ratio M/N. The accuracy of the calculation will of course depend on number of points chosen. Equally well known is the example of the dice. The problem consists of determining whether a dice presents any irregularity in its geometry or mass distribution. The normal solution consists in performing a number of physical measurements on the dice, the Monte Carlo solution consist of playing many times with it in order to determine whether there is any irregularity in the distribution of the results of the game.

The result of a Monte Carlo calculation is, of course, a statistical variable and its accuracy is roughly inversely proportional to the square root of the number of events contributing to it (e. g. number of neutrons undergoing a collision in a certain region). This means that the number of variables which can be calculated by this method should be relatively small in order not to increase enormously the computational effort required (e. g. it is relatively easy to calculate the total neutron flux in a region, but it becomes very difficult to have its accurate space and energy distribution within this region). The square root law means also that in order to increase the accuracy of a calculation by a factor of 10 one must increase the number of samples by a factor of 100 so that it can be very expansive to increase the accuracy beyond a certain limit.

On the other hand, with Monte Carlo methods the transition to a higher number of dimensions implies only a linear increase in computing effort. In the above-mentioned example of the calculation of an area under a curve a numerical integration would have been much more efficient, but for a similar problem in a five — or six-dimensional system the Monte Carlo method would have been more advantageous.

This means that it is relatively easy to calculate with Monte Carlo methods complicated three-dimensional geometries for which S„ codes could prove impossible for most computers now available. Usually the particles which are followed by these computer calculations represent more than one neutron. The number of neutrons represented is called weight of the particle. In this way if, for example, a collision has a 20% absorption and 80% scattering probability the weight after collision is reduced by the factor 0.8, while the rest is recorded as absorbed.

Monte Carlo methods can be very inefficient in some cases and it is then necessary to modify the game in order to reduce the variance or error. For example, in many cases there are neutrons that because of their position, energy and direction can contribute more than others to the final answers. A typical example is given by the study of neutron attenuation through a shield in which, without special variance reduction techniques, an enormous number of neutrons should be followed across the shield in order to have a few neutrons at the end of it. One of the methods of variance reduction consists of changing the weight of the particles, so that regions of high interest are studied by means of many particles having little weight while regions of lower interest are studied by few particles having high weight. Weights can be changed as particles cross given space or energy boundaries. An increase of weight is performed by a game called “Russian Roulette” in which the particle has a probability p of surviving and 1 — p of being killed. The surviving particles are followed with a weight increased by the factor 1 Ip as imposed by the conservation of the number of neutrons. The opposite game, consisting in a decrease in weight and subsequent increase of the number of particles, is called splitting. The Russian Roulette can also be used in cases in which it is not of interest to follow particles with very low weight. Monte Carlo methods are also used successfully to calculate the effect of localized perturbations in a reactor. In that case the unperturbed case is first calculated and then weight standards are chosen so as to enhance greatly the diffusion of particles into the perturbing region. In the field of high-temperature-reactor design Monte Carlo methods, while unsuitable for complete reactor calculations, can be used to treat a number of detailed problems. Among these we can mention resonance absorption calculations, neutron streaming through big holes, shielding. In this last field the Monte Carlo method is very often the only possibility of obtaining sufficiently accurate results. The criticism which is often made to Monte Carlo calculations is that they provide results, but give very little information on the physical phenomena determining them. (For details see ref. 22.)