Sec. 1.1] PROPERTIES OF NUCLEI 5 



lower quantum levels as indicated under Rule 2. Nevertheless, such nuclei 

 are stable against K capture or beta emission when the odd particle is added 

 to the next lowest level. The type of stable nucleus that can be formed by 

 the addition of a particle to an (E, E) nucleus of given atomic weight then 

 depends upon the next lowest level to be filled. If this is a proton level but a 

 neutron is added instead, the nucleus is unstable against beta decay and the 

 neutron is transformed to a proton. 



4. (0, 0) nuclei, with the exception of H 2 , Li 6 , B 10 , and N 14 , are unstable. 

 The existence of only four such stable isotopes indicates immediately the 

 relative instability of this type of nucleus in all but the lightest elements. If 

 each proton-neutron pair occupies the same quantum state, the resulting 

 nucleus is stable and the numbers of protons and neutrons are equal. If, 

 however, the number of neutrons is greater by two or more, they will lie in 

 successively higher levels above the last filled proton level and will transform 

 by beta emission to the lower lying proton level to form nuclei of the type 

 (0, 0). Thus, nuclei of this type are stable only if the numbers of protons 

 and neutrons are equal. This condition carv be found, however, only for the 

 lightest elements where the electrostatic forces are still small. In heavier 

 elements, stable nuclei containing equal numbers of protons and neutrons 

 cannot exist since the electrostatic repulsion then diminishes the nuclear 

 binding energy as compared with nuclei with the same total number of 

 particles but with a greater proportion of neutrons. 



5. For any even Z, there exists only one or at the most two stable isotopes of odd 

 A; if two, they differ by two mass units. Many isotopes of even A may exist. 



6. For any odd Z, there exists only one or, at most, two isotopes; if two, they 

 differ by two mass units. 



7. For any even A, only two stable isobars may exist and they differ in charge 

 by two units and are even in Z. 



8. For any odd A , only one stable nucleus exists (no isobars) and its Z may be 

 even or odd. 



A qualitative proof of rules 5, 6, 7, and 8 follows from energy considerations 

 for the various possible combinations of protons and neutrons. A nucleus of 

 mass A is stable only for that combination of protons and neutrons which 

 provides the maximum binding energy or, alternatively, the minimum exact 

 mass. This can be estimated from the principal terms in the binding-energy 

 formula 



„ . . (A - 2Z) 2 Z 2 



E = a A — b - 



A A* 



£ is a maximum when the last two terms are equal in magnitude and opposite 

 in sign. This requires that {A — 2Z)/Z ~ A'& or that the excess of neutrons 

 over protons increase as A^ for stable nuclei. Further, the binding energy of 



