an interesting problem of probability as applied in nuclear chemistry

Today my roommate asked me for the explanation of a crude estimation of a quantity in an applied nuclear chemistry problem. The set up is as follows: one shoots a neutron into a slab of uranium atoms, and tries to determine the thickness of the slab by considering the chance of the neutron being absorbed by one of the atoms it passes through. Assuming the absorptions are independent events when they occur in sequence, let p be the probability of being absorbed by a single atom. One can determine the number of uranium in a vertical slice of the slab using the following formula

   (1-p)^n = 1- q    (1)
 where q is the proportion of the neutrons that are absorbed by the slab. So solving for n one gets n = log q / log(1-p)
 But one could also approximate n by the following formula
  np = q              (2)
  The mathematical reasoning behind the approximation is as follows: 1-p is roughly equal to e^{-p}, and therefore e^{-np} = e^{-q),  hence np = q follows by taking log of both sides, at least when q and p are both very small.
  The physical justification is that one could consider the absorptions of the neutron by each atom as either serial events or parallel events. In the former case, one would assume in addition that the events are independent, hence we get the product formula (1), but one could also treat the absorptions as if the neutron first chooses an atom, and then decide whether to get absorbed or not by that atom, thus the total probability of absorption is the sum of the probability of individual absorption, since they are disjoint events, from which (2) follows.

   Now at a deeper level, one could ask whether the physical explanation can be related to the mathematical one, because the physical reasoning given above is based on two different underlying assumptions of the physical process, but maybe the two processes can be thought of as approximation of each other as well. And indeed this can be done. If p is rather small, then the chance of any particular atom absorbing the neutron is negligible. Hence if we consider the kth atom, we might as well ignore the first k-1 atoms and think of the absorption at the kth atom as disjoint from the first k-1 events. Thus the law of addition naturally comes to mind and the parallel model is indeed a mathematical approximation of the serial model. There is nothing to be ashamed of when we impose such approximation schemes, as the independence assumption of the serial model itself is a simplifying assumption; to be quantum mechanically correct, one would have to treat the entire slab of uranium as a huge entangled quantum potential well and figure out the wave function of the neutron under this potential. Thus the various absorption events are necessarily correlated with each other at the most basic level.

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About aquazorcarson

math PhD at Stanford, studying probability
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