Eddie, a 21 year old male from Toronto asks on June 22, 2003,I know the basic details of how an ionization smoke detector works, where particles enter the chamber and block the alpha particles so that the current generated is affected. But the question I have trouble with is the equation and formula relating to the inner effect inside the ionization chamber. How do you relate the amount and the size of the smoke particles with the current that it generated?
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As I understand it, the alpha particles are NOT blocked by the smoke particles as the questioner states. The alpha particles from the radioactive source (almost always Am-241 today) ionize the air between two electrodes. The negative and positive ions travel to the anode and cathode respectively. The resulting current is monitored. Some of the smoke particles entering the chamber combine with the ions. Since the smoke particles are now charged, they move towards the electrodes. Being relatively large, these charged smoke particles are not as mobile as the ionized air molecules and therefore move more slowly. The slower movement of these particles allows some of them to be swept out of the chamber by air currents. The effect is to reduce the number of charged particles/ions reaching the electrodes. The drop in current sets off the alarm. The best description I have run across is in NUREG 1717. The document is fairly big--about 3 megs. See page 2-217 for the description.
As far as relating the size of the smoke particles with the current that it generated, you don't try to relate the amount and size of the smoke particles to the current. The change in current would be affected by many things: velocity of the air currents, the chemical nature of the smoke particles, the concentration of the smoke particles, and their size. A purely mathematical description as to how these things work would be very complex and nobody to my knowledge has decided to devote the time to generating the equations. For what its worth, the ionization type of smoke detector that we are talking about is best for smaller smoke particles e.g., 0.01 to 1 micrometer.
My short answer is that I suspect you would need to solve the transport equation, accounting for energy degradation. Cross sections and energy deposition rates are very energy dependent. Messy. There is a code called MCNP that can solve that using Momte Carlo techniques.
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