It has bee ages since I looked at that formula; you may need to remind me of some of the definitions! To my recollection, there may be two answers. First, at present I evaluate the GDH model (in code alice) using a monte-carlo approach. In this mode, a result is calculated for two particle energies/charge plus hole energy depending on impact parameter/density of nucleus. For one of these two particles nu,e, the emission to spreading+emission ratio is applied; if the nucleon is emitted, it is added to the emisson spectrum, and the second particle is treated similarly. No depletion coefficient need be

computed. If that nucleon rescatters rather than being emitted, it is treated as if it were a new beam particle starting its own cascade but of course with the energy e' it has from the exciton energy choice made. It and any excitons it creates are similarly followed until they are below the binding energy, at which point it is assumed that all the excitons it creates have become part of the equilibrated residue to be treated by W-E evaporation model. Then the second particle from the initial two-particle,one hole distribution is put under the microscope, and we monte-carlo decide using its emission to spreading rate, if it will be emitted or rescatter.  This repeats what was done for the

first exciton selected ( with a conservation of energy applied between the 2 particles and one hole). When all excitons from these processes have become particle bound/equilibrated, a new random number is selected to begin a new projectile-

target interaction beginning with a 2 part-1hole state, etc. For a serious calculation

I run 10**5 cascade events- which goes pretty quickly on my old desktop computer.

In this monte-carlo approach, depletion takes care of itself thru the exciton energy

dependent emission to spreading+emission ratio of the formula.

  Let now return to your question, which I have un-artfully begged! The question of the closed form formula. I should re-read the paper- it has been nearly 40 years?

It appears to me that it is e.g. looking at the distribution of exciton energies 

beginning with the first 3 exciton configuration, multiplying the relative population in each interval times the emission  probability at that energy ( the depletion factor

would I think be 1/2 the emission sum over all energies, since two particles could be emitted).

Then the next term would be multiplied for each emitted exciton by 1.-D1, but in retrospect it also seems to me that a 3 exciton state would be followed by a 7 exciton state as both initial particle  excitons would have undergone a 2p1h process.

  You can see why I prefer the monte-carlo approach. I shall try to see if I still have an early non-monte carlo version of alice to see how I treated the depletion question- I hope that this is helpful. Contact me at [email protected] if further

discussion would be helpful  regards,marshall blann

After thinking further, let me modify my answer slightly. The original formula was for

precompound emissions at lower-( tens of MeV) excitations. So there are factors which

calculate for each energy interval e to e+de, the number of particles of type nu=n or p.

This is multiplied by the emission fraction for those nucleons given by the emission

rate divided by sum of emission+re-scattering rate. For the first event wherein a projectile has interacted with a nucleus to make a 2p1h exciton distribution, the 'depletion' factor (maybe better called  population factor?) is 1.00, because all

the excitons of the 2p1h are there. Next we apply the formula to see what fraction

of the exciton we are looking at is emitted at each energy. So we are integrating the fraction of the exciton at e to e+de times emission ratio (emission rate / emission+re-scatter)

for this first exciton configuration.For calculation of contribution from the next 5 exciton configuration formed by the fraction of those excitons which rescatter rather than havingg been emitted, one minus that integral is the 'depletion factor', which again is the fraction of population surviving to the next exciton number.

    This works o.k. at tens of MeV, where single precompound emission is a good

approximation, but at higher energies,e.g. 60-200 MeV incident nucleons, or incident heavy ions, the question of multiple precompound decay from a single

exciton number becomes significant. I think that the Monte-Carlo method of making and decaying 2p1h configurations and following each cascade is a powerful approach which

is able to address the multiple precompound decay which becomes increasingly important as excitation/nucleon increases..

  marshall blann

Im glad to read from you Mr marshall blan and I would like to thank you for the explicit explanation for the depletion factor or as you said the population factor. at first I'll try to evaluate the GDH model at low energies and if I do it well then i'll try the higher energies. YES passed many years when you work on this but it still powerful and I always like your work. I attach one paper of your work on GDH model just to remind you and thank you so much