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This is definitely not a new problem. Studies of such lattices, theoretical as well as experimental, were pioneered by Enrico Fermi, and then continued by many others (among whom were many scientists of well-known names, e.g., Edward Teller, or Stanislaw Ulam). The results of their theoretical investigations can be found now in numerous textbooks. However, those “classical” results were obtained primarily by analytical methods, using quite advanced tools such as, e.g., the Boltzmann Equation and other differential equations of a similar level of complexity. For students who are entering this field, understanding all the “meanders” of the analytical theory and “digesting it” in order to clearly see the “physical picture” emerging from that sophisticated math may take months of really hard studying.
An alternative approach to the problem is numerical simulation – and the Monte Carlo technique is very well suited for investigating the scattering of neutrons and related phenomena in physical systems because neutrons do not interact with each other. No many-body effects of any kind are involved; the basic Monte Carlo algorithm in such modeling is a one that simulates a “random trek” of an individual neutron in the system investigated. This basic algorithm is repeated for a large number of neutrons, and the effects of interest are obtained by averaging all the single-neutron results.
The underlying physics in modeling the “Fermi lattice” is surprisingly simple, and the program can be written in a transparent form. I will argue that such a program may be of considerable help for students at the early stage of learning because it will enable them to quickly find out how essential physical characteristics of the system depend on a few crucial parameters that can be readily modified in the program. After “playing” with such numerical model, the understanding of “what and why” is done in the analytical approach to the problem may become much easier.
I will demonstrate how such program works, and I will run some simulations for a model closely resembling the set-up that Fermi and co-workers used in their historic experiments nearly seventy years ago. A large part of the success of those experiments was owed to Leo Szilard who realized that too high a contamination of industrial-grade graphite with Boron could doom the experiment. He proposed changes in the manufacturing technology, which lowered the neutron absorption cross section from 0.0075 millibarn in industrial graphite to 0.0045 mb. The program that I will demonstrate indeed shows that the experiment might have not worked if 0.0075 mb graphite were used.