content_image/372071.jpg content_image/372068.jpg content_image/372070.jpg content_image/372069.jpg content_image/372067.jpg content_image/372066.jpg content_image/372065.jpg content_image/372064.jpg
After rewriting the densities of fragment states in terms of the internal Helmholtz free energies, defined for fragment j by[25] 

<img src='content_image/372071.jpg'>

all but one of the integrals in Eq. (9) can be performed analytically, as is shown in the Appendix. We can then write the density of final states ω$_{n}$ as 

where 

and the total Helmholtz free energy F$_{n}$ ( β ) has been defined as 

with the sum over fragments replaced by a sum over distinct fragments times their multi- plicities. The translational Helmholtz free energies are given by 

<img src='content_image/372068.jpg'>

<img src='content_image/372070.jpg'>

<img src='content_image/372069.jpg'>

<img src='content_image/372067.jpg'>

where we write the mas of fragment l as m$_{l}$ = m$_{N}$A$_{l}$ and the mass of the decaying nucleus as m$_{0}$ = m$_{N}$A$_{0}$ , with m$_{N}$ the nucleon mass. We emphasize that we have made no approxi- mations up to this point. The expression given in Eq. (11) is exactly equivalent to that of Eq. (9). 

To approximate the final integral, we use the method of steepest descent. We look for a value β$_{0}$ for which 

Using the relations of the Helmholtz free energy to the entropy and energy, 

<img src='content_image/372066.jpg'>

<img src='content_image/372065.jpg'>

respectively, we find the saddle point condition to be equivalent to the requirement that energy is conserved, 

<img src='content_image/372064.jpg'>

