The Theory of Alpha Decay

One of the relatively simple physical processes that can be described successfully in the framework of introductory quantum theory is alpha decay. In this process, two protons and two neutrons out of a relatively large nucleus dissociate from the rest of the nucleus, form a separate subsystem and subsequently depart. In its full complexity, this process is not be describable, however a reasonable simplified version of it is readily tractable and will be studied in this note. Several alternative models will also be described, for the interested reader to pursue in greater detail. 1 The Physical Description of the Problem The process of α-decay starts with a nucleus, say 234 92U (Uranium). This nucleus has 92 protons and 142 neutrons, and is obviously too complicated for a many-body type of analysis. That is, one would need to set up 3×234 = 702 equations of motion for the three components of the radius vector of each of the 234 particles, and these would involve (234 2 ) = 27, 261 2-particle potential terms to describe the pair-wise interactions. At some point, among these 234 particles, somehow two protons and two neutrons end up forming a little subsystem within which the pair-wise binding forces are stronger than the binding forces to any of the other 230 nucleons. Gradually1 , the little subsystem becomes an entity which may be thought of as a separate particle trapped within the confines of the ‘rest’ of the nucleus. Eventually, the little subsystem will tunnel outside the reach of the strong nuclear forces, at which point the repulsive Coulomb force expels it as the α-particle which is detected well outside the nucleus. The remaining 230-nucleon nucleus is identified as 230 90Th (Thorium). The 234 92U nucleus is called the parent nucleus, 230 90Th the daughter nucleus, and the general formula for the α-decay would be A+4 Z+2X → 4 2He ++ + Z Y −−, or A+4 Z+2X α −−→ Z Y. (1) In the first formula, we have used that the α-particle is in fact the nucleus of Helium atom, which would have two electrons in its stable state. The double positive charge then indicates that only the nucleus appears on the right, and so leaves the daughter atom twice negatively charged. In nuclear physics, the same symbols would be used to denote merely the nuclei (with no concern about the electrons), and so no charges are indicated in the second form of the formula, and the decay process (arrow) is labeled by ‘α’. Exceptional cases occur when the number of protons, Z, and/or the number of neutrons, (A−Z), equal 2, 8, 20, 28, 50, 82 or 126 (the so-called magic numbers). If, say Z = 82 (as in 208 82Pb (lead), which in fact is doubly magical, since also (A−Z) = 126), the nuclear shell model predicts that the 82 protons (and here also the 126 neutrons) form a strongly bound closed ‘shell’. Thus, the nearby Polonium nucleus 212 84Po is rather accurately described as a system of a doubly closed shell of 82 protons and 126 neutrons, plus two extra protons and two extra neutrons. In this exceptional case, the four extra nucleons are easily identifiable and it does make sense to think of the 212 84Po nucleus as an α-particle bound to a 208 82Pb ‘shell’. Then, the α-decay is simply a dissociation of this bond and the subsequent escape of the α-particle. It should also be clear that this situation is extremely rare: most of α-radioactive nuclei will not be this simple. Since the physical process of the α-decay is rather involved, we are forced to make an approximation. This will consist of two separate assumptions. The above discussion notwithstanding: 1. the α-particle will be considered as if a well-defined separate entity and simply trapped within the confines of the nucleus. 1 The description of the process here is purely conceptual and qualitative. The α-decay process is ultimately governed by the so-called ‘nuclear strong interaction’, the characteristic time for which is 10−23s; therefore, “gradually” here still happens incredibly swiftly when compared to typical macroscopic processes, such as a cell division or the burst of an automobile tire.