SPAGINS: semiempirical parameterization for fragments in gamma-induced nuclear spallation

Abstract

From the empirical phenomena of fragment distributions in nuclear spallation reactions, semiempirical formulas named SPAGINS were constructed to predict fragment cross-sections in high-energy $\gamma$-induced nuclear spallation reactions (PNSR). In constructing the SPAGINS formulas, theoretical models, including the TALYS toolkit, SPACS, and Rudstam formulas, were employed to study the general phenomenon of fragment distributions in PNSR with incident energies ranging from 100 to 1000 MeV. Considering the primary characteristics of PNSR, the SPAGINS formulas modify the EPAX and SPACS formulas and efficiently reproduce the measured data. The SPAGINS formulas provide a new and effective tool for predicting fragment production in PNSR.

Appendices

Appendix I: SPAGINS formulas

Many SPACS and EPAX formulas are incorporated into the SPAGINS formulas. Owing to the definition of the individual parameters in the formula, SPAGINS adopts gamma quanta as the target. To present the SPAGINS formula more clearly, detailed formulas are provided in this section.

For a fragment with mass and charge numbers (A, Z), the production cross-section in PNSR is

$$\begin{aligned} \sigma (A,Z)=\sigma _\text{R} Y(A) Y(Z_\text {prob}-Z)|_A. \end{aligned}$$

(11)

For the first term, the reaction cross-section is

$$\begin{aligned} \sigma _\text {R}=10 \pi r_0^2\left(A_\text {proj}^{\frac{1}{3}}+A_\text {tar}^{\frac{1}{3}}+\delta _\textrm{E}\right)^2(1-B/E_\text {cm})\chi _\text {m}, \end{aligned}$$

(12)

where $r_0=$ 1.1 fm, and

$$\begin{aligned}{} & {} B=1.44 Z_\text {proj}Z_\text {tar}/H, \end{aligned}$$

(13)

$$\begin{aligned}{} & {} E_\text {cm}=E_\text {proj}A_\text {proj}/(A_\text {proj}+A_\text {tar}), \end{aligned}$$

(14)

$$\begin{aligned}{} & {} H=r_\text {proj}+r_\text {tar}+1.2\left( A_\text {proj}^{\frac{1}{3}}+A_\text {tar}^{\frac{1}{3}}\right) /E_\text {cm}^{\frac{1}{3}}. \end{aligned}$$

(15)

One can determine $r_\text {proj}=1.29r_\text {proj}^\text {rms}$, and $r_\text {proj}^\text {rms}=0.891A_\text {proj}^{\frac{1}{3}} \left( 1+5.565A_\text {proj}^{-\frac{2}{3}}-1.04A_\text {proj}^{-\frac{4}{3}}\right)$. $r_\text {tar}^\text {rms}=$ 0.85.

$$\begin{aligned} \delta _\text {E}=1.85S+\left( 0.16S/E_\text {cm}^{\frac{1}{3}}\right) -C_\text {E}, \end{aligned}$$

(16)

where

$$\begin{aligned} S=A_\text {tar}^{\frac{1}{3}}A_\text {proj}^{\frac{1}{3}}/\left( A_\text {tar}^{\frac{1}{3}}+A_\text {proj}^{\frac{1}{3}}\right) , \end{aligned}$$

(17)

and

$$\begin{aligned}{} & {} C_\text {E}=2.05[1-\exp (-E_\text {proj}/40)-0.292\exp (-E_\text {proj}/792) \nonumber \\{} & {} \cos (0.229E_\text {proj}^{0.453})] \end{aligned}$$

(18)

For $A_\text {proj}<$ 200,

$$\begin{aligned} \chi _\text{m}=1-\chi _\text{l}\exp [-E_\text {proj}/(\chi _\text{l} s_\text{l})], \end{aligned}$$

(19)

in which,

$$\begin{aligned} \chi _\text{l}=2.83-3.1\times 10^{-2}A_\text {proj}+1.7\times 10^{-4}A_\text {proj}^2, \end{aligned}$$

(20)

and

$$\begin{aligned}{} & {} s_\text{l}=0.6, \quad \text {for} \quad A_\text {proj}< 12, \end{aligned}$$

(21)

$$\begin{aligned}{} & {} s_\text{l}=1.6, \quad \text{for} \quad A_\text {proj}= 12, \end{aligned}$$

(22)

$$\begin{aligned}{} & {} s_\text{l}=1.0, \quad \text{for} \quad A_\text {proj}> 12. \end{aligned}$$

(23)

For the second term, the mass distribution considers the contributions from the central ($Y(A)_\text {cent}$) and peripheral collisions ($Y(A)_\text {prph}$),

$$\begin{aligned} Y(A)=Y(A)_\text {cent}+Y(A)_\text {prph}. \end{aligned}$$

(24)

When considering the incident energy dependence of the mass yield,

$$\begin{aligned} Y'(A)=G \cdot Y(A) \end{aligned}$$

(25)

where G is sorted into the following three regions:

$$\begin{aligned} G=\left\{ \begin{aligned}&10^{(E_\gamma -100)/100},&100 \le E_\gamma< 220; \\&10^{(E_\gamma -220)/300},&220 \le E_\gamma < 500; \\&10^{(E_\gamma -500)/2000},&500 \le E_\gamma \le 1000. \end{aligned} \right. \end{aligned}$$

(26)

The contribution of central collisions to a specific fragment cross-section is

$$\begin{aligned} Y(A)_\text {cent}= & {} (A_\text {length}/\{1+\exp [(A_\text {proj} -A_\text {cent}^{'} \nonumber \\{} & {} -A)/A_\text {cent-fluct}]\})/A_\text {cent}^{'}, \end{aligned}$$

(27)

where

$$\begin{aligned}{} & {} A_\text {cent}=\alpha _\text {cent}/\exp (A_\text {proj}/\beta _\text {cent}), \nonumber \\{} & {} A_\text {cent}^{'}=A_\text {proj}\{1-\exp [-\ln (A_\text {cent})(E_\text {proj}/1000)^{\epsilon _\text {cent}}]\}, \nonumber \\{} & {} A_\text {length}=1, \nonumber \\{} & {} A_\text {cent-fluct}=a_\text {fluct}A_\text {cent}^{\delta _\text {fluct}}(E_\text {proj}/1000)^{\epsilon _\text {cent}}, \nonumber \\{} & {} a_\text {fluct}=\alpha _\text {fluct}+\beta _\text {fluct}A_\text {proj}. \nonumber \end{aligned}$$

The contribution of peripheral collisions to a specific fragment cross-section is

$$\begin{aligned} Y(A)_\text {periph}= & {} A_\text {periph}\exp \{[A-(A_{\text {proj}-A_\text {diff}})]/A_\text {diff}\} \nonumber \\{} & {} \times (E_\text {proj}/1000)^{B_\text {periph}}/A_\text {cent}^{'}, \end{aligned}$$

(28)

where

$$\begin{aligned}{} & {} A_\text {periph}=\alpha _\text {periph}+\beta _\text {periph}A_\text {proj} \nonumber \\{} & {} A_\text {diff}=\alpha _\text {diff}+\beta _\text {diff}A_\text {proj} \nonumber \end{aligned}$$

The third term $\sigma (A,Z)$ is parameterized by the number of neutron- and proton-rich fragments. For neutron-rich ones,

$$\begin{aligned} Y(Z_\text {prob}-Z)|_{A}=nf_\text{n}\exp (-R|{Z_\text {prob}}-Z|^{u_\text {n}}), \end{aligned}$$

(29)

and proton-rich

$$\begin{aligned} Y(Z_\text {prob}-Z)|_{A}=nf_\text{p}\exp (-R|{Z_\text {prob}}-Z|^{u_\text{p}}). \end{aligned}$$

(30)

$Z_\text {prob}$ and $Z_{\beta }$ take the form of

$$\begin{aligned}{} & {} Z_\text {prob}=Z_{\beta }+\Delta _\text {m}^\text {n,p}+0.002A+\Delta ^{'}, \end{aligned}$$

(31)

$$\begin{aligned}{} & {} Z_{\beta }=\frac{A}{1.98+0.0155A^{2/3}}. \end{aligned}$$

(32)

For a neutron-rich projectile,

$$\begin{aligned} \Delta _\text {m}^\text {n}= & {} \{n_1(A/A_\text {proj})^6+n_2[(A_\text {proj}-A)/A_\text {proj}]\}^2 \nonumber \\{} & {} \times (Z_\text {proj}-Z_{\beta \text {p}}), \end{aligned}$$

(33)

For proton-rich projectiles,

$$\begin{aligned} \Delta _\text {m}^\text {p}=[\exp (p_1+p_2A/A_\text {proj})](Z_\text {proj}-Z_{\beta \text {p}}), \end{aligned}$$

(34)

in which $\Delta =\Delta _5A^2$, $\Delta ^{'}=\Delta [1+d_1(A/A_\text {proj}-d_2)^2]$, $n=\sqrt{R/\pi }$, and $R=R_{0}^\text {n,p}R^\text {phy}$.

For neutron-rich spallation targets,

$$\begin{aligned} R_{0}^\text {n}=r_0\exp [r_3(Z_\text {proj}-Z_{\beta \text {p}})], \end{aligned}$$

(35)

For a proton-rich spallation target,

$$\begin{aligned} R_{0}^\text {p}=r_0\exp [r_4(Z_\text {proj}-Z_{\beta \text {p}})]. \end{aligned}$$

(36)

One also has,

$$\begin{aligned} R^\text {phy}=\exp \{-\ln [R_1(A/t_2)]\}. \end{aligned}$$

(37)

For neutron-rich fragments,

$$\begin{aligned} U^\text {n}=U_\text {n1}+U_\text {n2}A/A_\text {proj}, \end{aligned}$$

(38)

and proton-rich fragments.

$$\begin{aligned} U^\text {p}=U_\text {p1}+U_\text {p2}A_\text {proj}. \end{aligned}$$

(39)

The brute force factors $f_\text {n}$ presented in Sect. 6.

Appendix II: brute force factor

The “brute force factor” in Eqs. (9) and (10) was parameterized according to the incident energy of the reaction and different ranges of fragment charge numbers. For the energy range of 100 MeV $\le E_\gamma<$ 220 MeV, when $Z\le 26$,

$$\begin{aligned} \begin{aligned}&f_\text {p}=\frac{1}{10^{(\textrm{d}F/\textrm{d}Z)(Z-Z_\text {exp})}}\times 10^{[-12+0.5(Z-1)]}\\&\qquad \times 10^{[0.5(29-Z_\text {proj})][5.575-0.175Z]}, \\&f_\text {n}=10^4\times 10^{[-23+(Z-1)]} \\&\qquad \times 10^{[0.5(29-Z_\text {proj})][4.7-0.1Z]} \\ \end{aligned} \end{aligned}$$

(40)

where

$$\begin{aligned} \begin{aligned}&\textrm{d}F/\textrm{d}Z=1.2+0.647(A/2)^{0.3}, \\&Z_\text {exp} =Z_\text {prob}+\textrm{d}F/\textrm{d}Z \ln (10)/(2R); \end{aligned} \end{aligned}$$

(41)

when $Z>26$,

$$\begin{aligned} \begin{aligned}&f_\text {p}=\frac{1}{10^{(\textrm{d}F/\textrm{d}Z)(Z-Z_\text {exp})}}\times 10^{[-24+(Z-1)]},\\&f_\text {n}=10^4\times 10^{[-22.5+(Z-1)]}. \end{aligned} \end{aligned}$$

(42)

For the incident energy range of 220 MeV$\le E_\gamma<$ 500 MeV, when the charge of the fragments $Z\le 21$,

$$\begin{aligned} \begin{aligned}&f_\text {p}=\frac{1}{10^{(\textrm{d}F/\textrm{d}Z)(Z-Z_\text {exp})}}\times 10^{[5.5-0.3(Z-1)]},\\&f_\text {n}=10^4\times 10^{[-15.5+0.75(Z-1)]}; \end{aligned} \end{aligned}$$

(43)

when $21<Z\le 26$,

$$\begin{aligned} \begin{aligned}&f_\text {p}=\frac{1}{10^{(\textrm{d}F/\textrm{d}Z)(Z-Z_\text {exp})}}\times 10^{[-8.2+0.4(Z-1)]},\\&f_\text {n}=10^4\times 10^{[-12.6+0.6(Z-1)]}; \end{aligned} \end{aligned}$$

(44)

when $Z>26$,

$$\begin{aligned} \begin{aligned}&f_\text {p}=\frac{1}{10^{(\textrm{d}F/\textrm{d}Z)(Z-Z_\text {exp})}} \times 10^{[-10+0.5(Z-1)]},\\&f_\text {n}=10^4\times 10^{[-27.7+1.2(Z-1)]}; \end{aligned} \end{aligned}$$

(45)

For the incident energy range of 500 MeV $\le E_\gamma \le$ 1000 MeV, when $Z\le 21$,

$$\begin{aligned} \begin{aligned}&f_\text {p}=\frac{1}{10^{(\textrm{d}F/\textrm{d}Z)(Z-Z_\text {exp})}}\times 10^{[5.7-0.3(Z-1)]}\\&\qquad \times 10^{[0.5(29-Z_\text {proj})][2.2-0.1Z]}, \\&f_\text {n}=10^4\times 10^{[-19.6+0.6(Z-1)]} \\&\qquad \times 10^{[0.5(29-Z_\text {proj})][1.15-0.05Z]}; \\ \end{aligned} \end{aligned}$$

(46)

when $21<Z\le 26$,

$$\begin{aligned} \begin{aligned}&f_\text {p}=\frac{1}{10^{(\textrm{d}F/\textrm{d}Z)(Z- Z_\text {exp})}}\times 10^{[-7.4+0.4(Z-1)]} \\&\qquad \times 10^{[0.5(29-Z_\text{proj})][-2+0.1Z]}, \\&f_\text {n}=10^4\times 10^{[-9.5+0.5(Z-1)]} \\&\qquad \times 10^{[0.5(29-Z_\text{proj})][-1.9+0.1Z]}, \\ \end{aligned} \end{aligned}$$

(47)

and when $Z>26$,

$$\begin{aligned} \begin{aligned}&f_\text {p}=\frac{1}{10^{(\textrm{d}F/\textrm{d}Z)(Z-Z_\text {exp})}}\times 10^{[-9.8+0.5(Z-1)]},\\&f_\text {n}=10^4\times 10^{[-14.2+0.7(Z-1)]}. \end{aligned} \end{aligned}$$

(48)

where parameters from Eqs. (3) to (48) can be found in Refs. [46, 47].