SPACS: A semi-empirical parameterization for isotopic spallation cross sections

A new semi-empirical parameterization for residue cross sections in spallation reactions is presented. The prescription named SPACS, for spallation cross sections, permits calculating the fragment production in proton- and neutron-induced collisions with light up to heavy non-fissile partners from the Fermi regime to ultra-relativistic energies. The model is fully analytical, based on a new parameterization of the mass yields, accounting for the dependence on bombarding energy. The formalism for the isobaric distribution consists of a commonly used functional form, borrowed from the empirical parameterization of fragmentation cross sections EPAX, with the observed suited adjustments for spallation, and extended to the charge-pickup channel. Structural and even-odd staggering related to the last stage of the primary-residue deexcitation process is additionally explicitly introduced with a new prescription. Calculations are benchmarked with recent data collected at GSI, Darmstadt as well as with previous measurements employing various techniques. The dependences observed experimentally on collision energy, reaction-partner mass, and proton-neutron asymmetry are well described. A fast analytical parameterization, such as SPACS, can be relevant to be implemented in complex simulations as used for practical issues at nuclear facilities and plants. Its predictive power also makes it useful for cross-section estimates in astrophysics and biophysics.

Figure 1

Schematic behind the functional form used to parameterize the mass yield $Y$($A$) in SPACS. See the text for the signification of the indicated quantities.

Figure 2

Mass distributions of the residues produced in the reactions induced by ${}^{208}\mathrm{Pb}$ projectiles on a hydrogen target at 500-MeV/nucleon [74] (circles and stars) and 1000-MeV/nucleon [70] (squares) beam energy. The 500-MeV/nucleon data with $A$ < 176 (stars) are incomplete and represent lower limits only. The measurements are compared to the predictions by the SPACS code (dashed blue and full red lines at 500 and 1000 MeV/nucleon, respectively). Not to overload the figure, experimental error bars (below 10% in most cases) are not shown.

Figure 3

Sample of experimental isotopic distributions of the residues produced in the reactions induced by ${}^{208}\mathrm{Pb}$ projectiles on a hydrogen target at 500-MeV/nucleon [74] (circles) and 1000-MeV/nucleon [50, 70] (squares) beam energy. For clarity, the data at 1000 MeV/nucleon have been scaled by a factor of 10. The isotopic distributions of elements below $Z$ = 69 and in the charge-pickup channel $(Z$ = 83) are not available yet for the 500-MeV/nucleon run. Wherever not visible, experimental error bars are smaller than the symbols. The measurements are compared to the predictions by SPACS (dashed blue and full red lines at 500 and 1000 MeV/nucleon, respectively).

Figure 4

Sample of experimental isotopic distributions of the residues produced in the reactions induced by ${}^{197}\mathrm{Au}$ projectiles on a hydrogen target at 800-MeV/nucleon beam energy [48] (circles). The measurement is compared to the predictions by SPACS (full red lines). Wherever not visible, experimental error bars are smaller than the symbols.

Figure 5

Sample of experimental isotopic distributions of the residues produced in the reactions induced by ${}^{136}\mathrm{Xe}$ projectiles on a hydrogen target at 200-MeV/nucleon [76] (circles), 500-MeV/nucleon [60] (triangles), and 1000-MeV/nucleon [77] (squares) beam energy. Experimental error bars are, in general, smaller than the symbols and are not shown. For clarity, the data at 200 and 1000 MeV/nucleon have been scaled by factors of ${10}^{-2}$ and ${10}^2$, respectively. The measurements are compared to the predictions by SPACS (dashed blue, dashed-dotted green, and full red lines at 200, 500, and 1000 MeV/nucleon, respectively). Experimental distributions are not available yet below $Z$ = 51 for the 200-MeV/nucleon run.

Figure 6

Experimental element distributions of the residues produced in the reactions induced by ${}^{56}\mathrm{Fe}$ projectiles on a hydrogen target at 300-MeV/nucleon (downwards triangles), 500-MeV/nucleon (diamonds), 750-MeV/nucleon (upwards triangles), 1000-MeV/nucleon (circles), and 1500-MeV/nucleon (squares) beam energy [29, 78]. Error bars are smaller than the symbols. For clarity, the data have been scaled by factors of ${10}^2,{10}^4,{10}^5$, and ${10}^6$ at 500, 750, 1000, and 1500 MeV/nucleon, respectively. Lines correspond to model predictions with SPACS.

Figure 7

Sample of experimental isotopic distributions of the residues produced in the reactions induced by ${}^{56}\mathrm{Fe}$ projectiles on a hydrogen target at 300-MeV/nucleon (circles), 500-MeV/nucleon (triangles), and 1000-MeV/nucleon (squares) beam energy [29, 78]. Error bars are smaller than the symbols. For clarity, the data at 300 MeV/nucleon have been scaled by a factor of ${10}^{-2}$, and those at 1000 MeV/nucleon have been scaled by a factor of ${10}^2$. The measurements are compared to the predictions by SPACS (dashed blue, dashed-dotted green, and full red lines at 300, 500, and 1000 MeV/nucleon, respectively).

Figure 8

Experimental and calculated $Z$ distributions of the residues produced in the reaction ${}^{56}\mathrm{Fe}$ (1000 MeV/nucleon) + $p$ for various selections in $(N$ − $Z)$ from (a) $(N$ − $Z)$ = −1 to (g) $(N$ − $Z)$ = +5. The data are shown with black dots joined by full lines, whereas SPACS calculations are represented by red open circles joined by dashed lines. Model calculations without shell-structure and even-odd effects are also shown for comparison (thin blue lines). For clarity, experimental error bars are omitted.

Figure 9

Isotopic distribution of element $Z$ = 74 for the system ${}^{208}\mathrm{Pb}$ (1000 MeV/nucleon) + $p$. The data are shown with black dots joined by a full line, together with three calculations: accounting for both $\gamma$-decay competition and angular momentum effects (red open circles joined by a dashed line), accounting for $\gamma$-decay competition but neglecting angular momentum effects (green open triangles joined by a dotted line), neglecting both $\gamma$-decay competition and angular momentum effects (blue open squares joined by a dashed-dotted line).

Figure 10

Sample of experimental isotopic distributions of the residues produced in the reactions induced by ${}^{40}\mathrm{Ca}$ projectiles on a hydrogen target at 763 MeV/nucleon [80] (black dots joined by full lines). The measurement is compared to the predictions by SPACS (red open circles joined by dashed lines).

Figure 11

Experimental $Z$ distribution of the residues produced in the reaction induced by ${}^{22}\mathrm{Ne}$ projectiles on a hydrogen target at 894 MeV/nucleon [31]. The data (black dots joined by a full line) are compared to the predictions by SPACS with (red open circles joined by a dashed line) and without structural staggering (blue open squares joined by a dashed-dotted line).

Figure 12

Excitation function of ${}^{206}\mathrm{Bi},{}^{196}\mathrm{Au},{}^{188}\mathrm{Pt},{}^{183}\mathrm{Re},{}^{181}\mathrm{Re},{}^{175}\mathrm{Hf},{}^{167}\mathrm{Tm}$, and ${}^{149}\mathrm{Tm}$ produced in direct-kinematics experiments with the indicated reactions. The experimental data [87, 88, 89, 90, 91] are shown by black dots; error bars are smaller than the symbols wherever not visible. The predictions by SPACS are displayed by open red squares.

Figure 13

Element distributions of the residues produced in reactions induced by ${}^{197}\mathrm{Au}$ at 10.6 GeV/nucleon (left panel) and ${}^{208}\mathrm{Pb}$ at 158 GeV/nucleon on hydrogen. Experimental cross sections [23, 92] (black dots) are compared to predictions by SPACS (red circles joined by lines). Error bars are not shown for clarity; they are smaller than the symbols in most cases.

Figure 14

First row: Chart of the nuclides produced in the interaction of ${}^{56}\mathrm{Fe}$ on hydrogen at 300 MeV/nucleon: Experimental data from Ref. [29] (left), predictions by SPACS (middle) and predictions by the parameterization of Silberberg and Tsao [14] and Barghouty [93] (right). Second row: Identical to the first row for ${}^{136}\mathrm{Xe}$ on hydrogen at 1000 MeV/nucleon. The data are from Ref. [77]. Third row: Identical to the first row for ${}^{208}\mathrm{Pb}$ on hydrogen at 1000 MeV/nucleon. The data are from Ref. [70]. Fourth row: Isotopic distributions of element $Z$ = 61 (left), $Z$ = 72 (middle), and $Z$ = 80 (right) produced in ${}^{208}\mathrm{Pb}$ (1000 MeV/nucleon) + $p$. The data (open squares) are compared to predictions by SPACS (full red line) and predictions by Silberberg and Tsao (dashed-dotted blue line).

Figure 15

Sample of experimental isotopic distributions of the residues produced in the reactions induced by ${}^{197}\mathrm{Au}$ projectiles on a hydrogen target at 800-MeV/nucleon beam energy [48] (circles). The measurement is compared to the predictions by incl4.6-abla07 (dotted blue lines) [97] and to those by SPACS (full red lines). Wherever not visible, the experimental error bars are smaller than the symbols. The error bars of the Monte Carlo simulation are of statistical nature.