In our recent work [

20], we reported TTY for

^{43}Ca(p,n)

^{43}Sc on

^{43}CaCO

_{3} (90%

^{43}Ca) targets,

^{44}Ca(p,n)

^{44g}Sc and

^{44}Ca(p,n)

^{44m}Sc on

^{44}CaCO

_{3} (94.8%

^{44}Ca) as well as

^{48}Ca(p,2n)

^{47}Sc and

^{48}Ca(p,n)

^{48}Sc on

^{48}CaCO

_{3} (97.1%

^{48}Ca). Those TTY values are directly related to cross-sections by the following formula [

22,

23]:

where

H is target enrichment,

N_{A} is Avogadro’s number,

τ is the mean lifetime of a radioisotope,

Z is the ionization number of the projectile,

e is the elementary charge,

m is the atomic mass of the target,

E_{max} and

E_{min} are the maximal and minimal energy of the projectile penetrating the target (in case of TTY,

E_{min} <= reaction threshold), respectively,

σ is the cross-section for the nuclear reaction, and d

E/d

x is the stopping-power of the projectile according to the aerial density of the target. Here, we describe the attempt to obtain the energy dependence of the cross-section (the excitation function) based on the experimental

TTY_{exp}(

E) [MBq/µAh] values for different projectile energies

E. These data are supplemented by an assumption

TTY_{exp}(

E_{thr}) = 0, where

E_{thr} denotes the energy threshold for this reaction.

The crucial factor is the choice of the function used to describe the TTY energy dependence. The number of parameters of the function used to fit the data should be restricted, as the number of the experimental data points is usually limited. Therefore, we propose a simple shape,

which fulfils several important criteria. This function is monotonically increasing, as

TTY(

E) should be. Most importantly, its derivative is a modified q-Weibull distribution [

24],

which reflects the global shape of the (p,n) and (p,2n) excitation functions, commonly used in the field of the production of medical radioisotopes. The request

TTY_{exp}(

E_{thr}) = 0 provides the condition

and limits the number of

TTY_{fit} parameters to 3:

a,

b and

c. Once those parameters are obtained, the cross-section values can be estimated as

In our case, TTY measurements were obtained on CaCO

_{3} targets instead of metallic Ca. Therefore, we used d

E/d

x(

E) values corresponding to the energy loss in calcium carbonate (provided by SRIM software [

25]),

m = 100 u to address the mass of CaCO

_{3}, and

H as the level of enrichment of employed material. We have also calculated the 95% confidence band for

TTY_{fit}(

E) fit and reconstructed the cross-section. Details of our calculations are shown and explained in the Python code attached to this paper.

Alternatively, in [

21], the cross-section was reconstructed after fitting the TTY curve by calculating target yields (TY) for thicknesses corresponding to 0.1 MeV projectile energy loss each 1 MeV and multiplied by projectile range. This method assumes the constant stopping-power in each layer. In our approach, this simplification was not necessary.