10 ISOTOPIC TRACERS AND NUCLEAR RADIATIONS [Chap. 1 



exhibit complete saturation of their nuclear fields as do those lying deeper. 

 The particle is bound to the nucleus by only a portion of its available energy, 

 and the remainder cannot contribute to the total binding energy but gives 

 rise, instead, to a surface tension analogous to that of a liquid drop. The total 

 amount of the energy associated with the surface tension or unsaturated fields 

 must be proportional to the nuclear surface and hence to A& if it is assumed 

 that nuclear matter is incompressible. Referred to as surface energy, this 

 term is deducted from the .binding energy first estimated from the nuclear 

 volume. Since the volume-to-surface ratio increases with the radius as A 1 ^, 

 the relative magnitude of the surface effect in heavy nuclei is smaller than in 

 light nuclei where it causes an appreciable reduction in the binding energy. 



The long-range electrostatic forces due to a uniform volume distribution of 

 Z protons are taken into account by the third term. These fields cannot be 

 saturated, and since their range is very much greater than nuclear dimensions, 

 each proton is influenced by the repulsive electrostatic fields of the remaining 

 Z — 1 protons. From electrostatics, this term is found to be proportional to 

 Z(Z — \)/R ~ Z(Z — 1)/A^. The term is relatively unimportant in light 

 nuclei where it amounts to but a small fraction of the energy but increases 

 rapidly with atomic number and becomes, in the heaviest nuclei, a dominant 

 factor in reducing the binding energy and, particularly, the stability against 

 fission (see Fission, Chap. 6). 



The symmetry or isotopic-spin [7] energy term is based on the observation 

 that the most stable nuclei are those for which Z = A/2. From the statisti- 

 cal model of the nucleus [1,2,7] this effect is found to be proportional to 

 P/A, where I = A — 2Z = N — Z is the isotopic number. 



The complete expression for the total binding energy may now be written 



E = aA + bA& + cZ(Z - 1)A~X + dPA~ l + AE 

 where 



[ O, for A odd 



Z even | 



AE = 



+ eA~ 3/ *, for A even 



Z odd _ 



The coefficients a, volume energy, b, surface energy, c, electrostatic energy, 

 and d, symmetry energy, have been determined by fitting the equation to the 

 packing-fraction curve of isotopes for which the masses have been measured 

 with accuracy (see Table 2 and Fig. 2). Table 1 gives three sets of values 

 based on calculations of coefficients in similar formulas for the packing 

 fraction and exact atomic mass. The form of AE used here is based 

 on that suggested by Deutsch [4]. For a detailed discussion of AE see 

 reference 7. 



The semiempirical binding-energy formula above represents an energy 

 surface in the form of a trough. Stable nuclei lie at points along the bottom 



