Cross section calculation

Greetings! We are working on the production of Ac-225 via proton beam hitting Th-232 target. And recently we want to extract the cross section of Ac-225 production from the fluka results. First, we choose the beam intensity as 1e+12 p/s with 1 sec radition time, so that the decay of Ac-225 and other elements can be mostly avoided. The target is set as a cylinder with 3mm thickness. And the material density is set as 11.899. Then at the cooling time 0sec, we found the activity density for Ac-225 is 1.98E+01 Bq/cm^3 (the target volume is set as 1.0 cm^3 in the input file). Since not all the beam protons really involved the inelastic scattering, we count the protons running out from the target, and obtain the ratio 0.98137175136. Also we obtain the total cross section of proton scattering Th-232 as 4713 mb. And the Ac-225 lambda for converting from activity to element number is 8.087234689552827e-07. Therefore, we calculate the cross section for the Ac-225 production as following

sigma = 1.98E+01 / 8.087234689552827e-07 / ( 1e+12 * (1-0.98137175136) ) * 4713 = 6.194276073460916 mb

The problem appears when we change the target density into 118.99. We obtain the changed activity density as 2.66E+01 Bq/cm^3 and the ratio of proton running out from target is 0.00013069056, which seems reasonable with much higher density. Then we calculate the cross section for the Ac-225 production

sigma’ = 2.66E+01 / 8.087234689552827e-07 / ( 1e+12 * (1-0.00013069056) ) * 4713 = 0.155037155568 mb

The obvious change on the cross section surely does not make sense. So we were wondering if we have missed something in our calculation. Thank you.

Dear @Ivan_Martin,

Sorry for the delay in response. Could you please share your input files and information about FLUKA version?

Best regards,
Volodymyr

Actually, using the escaping proton fraction (with 118.99 target density) to estimate the number of interactions assumes that most protons stopping in the target contribute only to inelastic reactions, but this is a coarse approximation. Not all stopped protons undergo single inelastic interactions; some may interact multiple times, while others may lose energy through non-reactive processes. As a result, the effective cross sections are modified indirectly via changes in the energy spectrum. FLUKA handles all of this internally.

The better solution would be to use a thin target of Th-232 where you will bias the inelastic cross-section via LAM-BIAS to collect more statistics. In this way, you will ensure that only a single interaction happens inside the target, closer to the first-principles definition of cross-section. Also, you should use the number of proton stars from .out, which represents a number of inelastic interactions.