Sec. 1.5] 



PROPERTIES OF NUCLEI 



11 



of the trough, while radioactive nuclei are found at points on the sides and 

 move toward the bottom by emission of radiations. 



The packing fraction of an atom of atomic weight A and charge Z is found 

 from the energy formula by dividing through by 9314. Similarly an esti- 

 mate of the exact mass of the atom is obtained by dividing the formula by 931 

 and adding the terms 1.00893(4 - Z) + 1.00812Z. 



The detailed structure of the observed packing-fraction curve for stable 

 isotopes is not accounted for by the semiempirical formula since it is evaluated 



by an averaging process and no terms of short period are included. A closer 

 fit to the fluctuations in the actual curve must await more detailed informa- 

 tion on nuclear structure. 



The total binding energy is an important criterion of nuclear stability. At 

 least the lightest nuclei are stable against spontaneous radioactive decay 

 only when their binding energy is greater, i.e., their mass is less, than that 

 for any combination of lighter nuclei containing the same total number of 

 protons and neutrons. 



An important application of binding energy is found in the calculation of 

 the exact atomic mass of radioactive nuclei. In the case of beta decay, the 

 energy released (rest mass plus kinetic energy) is exactly equivalent to the 

 difference in the atomic masses of the initial and final atoms, i.e., 

 E = c 2 (M z — Mz+i). When a positron is emitted, however, the residual 

 atom is lighter by the equivalent energy carried off by the particle together 

 with its rest energy plus the rest energy of an orbital electron which 

 is also lost. E = c 2 (M z — M Z -i — 2m), where m is the mass of the 

 positron. Similarly, when an alpha particle is emitted, the energy release is 

 E = c 2 {M A , z — Af (4 _4, z _ 2 ) — 2m). It must be kept in mind, however, 

 that when gamma radiation is emitted its energy must also be taken into 

 account. 



1.5. Nuclear Spin. The resultant angular momentum or spin of a nucleus 

 is observed in all instances to be either half-integer or integer multiples of 

 h/2ir. In particular, the spins of all nuclei with even atomic weight are 

 integral multiples of h/2ir, while for nuclei of odd atomic number, the spins are 



