## Range in the Range-Energy Calculator

The range of an arbitrary ion in a specified stopping material is given by the relationship:

$\displaystyle{ R = \frac{M}{Z^2} [ \lambda(\beta) + B_Z(\beta) ] \text{,} }$ | (1) |

where $R$ is the range, $\lambda(\beta)$ is the range of an ideal proton with velocity $\beta/c$ in the same material, $B_Z(\beta)$ is the range extension, $M$ is the mass of the ion compared to a proton, $Z$ is the charge of the ion, and $E$ is the kinetic energy of the ion [E.V. Benton and R.P. Henke, "Heavy particle range-energy relations for dielectric nuclear track detectors," Nucl. Instr. Meth. 67(1), 87 (1969)]. The linear energy transfer $\mathrm{d} E/\mathrm{d} x$ is obtained by first differentiating Equation (1):

$\displaystyle{ \frac{\mathrm{d} R}{\mathrm{d} E} = \frac{M}{Z^2} [ \lambda'(\beta) + B'_Z(\beta) ] \frac{\mathrm{d} \beta}{\mathrm{d} E} \text{,} }$ | (2) |

where $B'_Z(\beta) \ne 0$. In practice, $\lambda$ is a function of the proton kinetic energy $\tau$, so Equation (2) becomes

$\displaystyle{ \frac{\mathrm{d} R}{\mathrm{d} E} = \frac{M}{Z^2} [ \lambda'(\tau) \frac{\mathrm{d} \tau}{\mathrm{d} E} + B'_Z(\beta) ] \frac{\mathrm{d} \beta}{\mathrm{d} E} \text{.} }$ | (3) |

In Equation (3), $\tau$ is the kinetic energy of a proton with the same $\beta$ as the ion for which we would like to know the range. The kinetic energy of the ion is expressed as

$\displaystyle{ E = \tau M \text{.} }$ | (4) |

The differential $\mathrm{d} \tau/\mathrm{d} E$ in Equation (3) is therefore

$\displaystyle{ \frac{\mathrm{d} \tau}{\mathrm{d} E} = \frac{1}{M} \text{.} }$ | (5) |

The contribution of $B_Z$ in Equation (1) is only significant when $137 \beta / Z \le 2$, so $\mathrm{d} \beta/\mathrm{d} E$ in Equation (3) can be replaced by its non-relativistic form

$\displaystyle{ E = \frac{1}{2} M m_\text{p} c^2 \beta^2 \text{,} }$ | (6) |

where $m_\text{p}$ is the mass of the proton. Differentiating Equation (6) with respect to $\beta$ gives

$\displaystyle{ \frac{\mathrm{d} E}{\mathrm{d} \beta} = M m_\text{p} c^2 \beta \text{.} }$ | (7) |

Plugging Equations (5) and (7) into Equation (3) and inverting the result gives the linear energy transfer:

$\displaystyle{ \frac{\mathrm{d} E}{\mathrm{d} x} = \left( \frac{\mathrm{d} R}{\mathrm{d} E} \right)^{-1} = \frac{Z^2}{\lambda'(\tau) + B'_Z(\beta) / m_{\text{p}} c^2 \beta} \text{,} }$ | (8) |

where $\beta = \sqrt{E_0 (2 + E_0)} / (1 + E_0)$, $E_0 = E / M m_{\text{p}} c^2$, $\tau = E/M$, $M = 1.0008 A$, and $A$ is the atomic mass of the ion.

The proton range $\lambda(\tau)$ is obtained from

$\displaystyle{ \ln{ \lambda } = \ln{ \langle A/Z \rangle } + \sum_{n=0}^N \sum_{m=0}^M a_{m n} \ln{ I^m } \ln{ \tau^n } \text{,} }$ | (9) |

where $\langle A/Z \rangle$ is the average number of nucleons per electron in the stopping material, and $I$ is the adjusted mean ionization potential of the stopping medium. The fraction by weight of the $i$^{th} stopping component is

$\displaystyle{ f_i = \frac{\mu_i A_i}{\sum \mu_i A_i} \text{,} }$ | (10) |

where $\mu_i$ and $A_i$ are the mass fraction and the atomic mass of the $i$^{th} component of the stopping material, respectively. The average number of nucleons per electron in the stopping material is given by

$\displaystyle{ \langle Z/A \rangle = \sum_i f_i \frac{Z_i}{A_i} = \frac{\sum \mu_i Z_i}{\sum \mu_i A_i} \text{,} }$ | (11) |

where $Z_i$ is the atomic number of the $i$^{th} component of the stopping material. The coefficients $a_{m n}$ in Equation (9) are divided into three kinetic energy regimes,

$\displaystyle{ \begin{matrix} \text{Region 1:} \\ \text{Region 2:} \\ \text{Region 3:} \end{matrix} } \displaystyle{ \begin{matrix} \tau \le J_1 \\ J_1 < \tau \le J_2 \\ J_2 < \tau \le 1200 \end{matrix} }$ | (12) |

The energy boundaries $J_1$ and $J_2$ (in units of MeV/n) are empirical functions of $I$ and are given by:

$\displaystyle{ J_1 = 0.94 + 20.19 I_3 - 84.08 I_3^2 + 132.98 I_3^3 - 30.77 I_3^4 - 102.29 I_3^5 + 64.03 I_3^6 \text{;} }$ | (13) |

$\displaystyle{ J_2 = 12.62 - 51.96 I_3 + 199.14 I_3^2 - 367.04 I_3^3 + 327.06 I_3^4 - 108.57 I_3^5 \text{,} }$ | (14) |

where $I_3 = I/1000$ eV. The functions $J_1$ and $J_2$ have been chosen such that Equation (9) is continuous.

To obtain $\lambda'(\tau)$, Equation (9) is differentiated with respect to $\tau$:

$\displaystyle{ \frac{1}{\lambda} \frac{\mathrm{d} \lambda}{\mathrm{d} \tau} = \frac{1}{\tau} \sum_{n=1}^N \sum_{m=0}^M n a_{m n} \ln{I^m} \ln{\tau^{n-1}} \text{,} }$ | (15) |

or

$\displaystyle{ \lambda'(\tau) = \frac{\lambda}{\tau} + \sum_{n=1}^N \sum_{m=0}^M n a_{m n} \ln{I^m} \ln{\tau^{n-1}} \text{.} }$ | (16) |

To make Equation (16) continuous, different functions of $J_1$ and $J_2$ than those given in Equations (13) and (14) must be used:

$\displaystyle{ J_1 = 0.63 + 6.81 I_3 - 7.65 I_3^2 + 42.03 I_3^3 - 131.10 I_3^4 - 146.51 I_3^5 + 57.52 I_3^6 \text{;} }$ | (17) |

$\displaystyle{ J_2 = 8.56 - 50.50 I_3 + 213.59 I_3^2 - 430.10 I_3^3 + 403.41 I_3^4 - 139.40 I_3^5 \text{.} }$ | (18) |

The value of $\lambda$ used in Equation (16) is obtained from Equation (9) using the functions $J_1$ and $J_2$ given by Equations (7) and (18) rather than Equations (13) and (14). Then Equation (16) is a continuous function of $\tau$.

The value of $I$ for a given material is computed from the $I$s of the individual constituents. For the $i$^{th} constituent:

$\displaystyle{ I_i = 12 Z_i + 7 \text{ for } Z \le 13 \text{;} }$ | (19) |

$\displaystyle{ I_i = 9.76 Z_i + 58.8 / Z_i^{0.19} \text{ for } Z \le 13 \text{.} }$ | (20) |

Equations (19) and (20) are approximate for values of $Z \le 13$. The adjusted mean ionization potential of the stopping medium is then given by

$\displaystyle{ \ln I = \frac{\sum \mu_i \ln I_i}{\sum \mu_i Z_i} = \langle A/Z \rangle \sum \frac{f_i Z_i}{A_i} \ln I_i \text{.} }$ | (21) |

Equations (9) and (16) can be expressed in the form

$\displaystyle{ \ln{\lambda} = \ln{ \langle A/Z \rangle } + \sum_{n=0}^N \sum_{m=0}^M a_{m n} (\ln{I})_{10}^m (\ln{\tau})_{10}^n \text{;} }$ | (22) |

$\displaystyle{ \lambda'(\tau) = \frac{\lambda}{10 \tau} + \sum_{n=1}^N \sum_{m=0}^M n a_{m n} (\ln{I})_{10}^m (\ln{\tau})_{10}^{n-1} \text{,} }$ | (23) |

where $(\ln{I})_{10} \equiv \ln{I/10}$ and $(\ln{\tau})_{10} \equiv \ln{\tau/10}$. The values of $a_{m n}$ are given in Table 1, Table 2, and Table 3. The coefficients given in Table 1 were obtained by making a fit to the range data of Whaling for low energy ions. Prior to making the fit, the range extension was subtracted. Thus, $\lambda$ is the range of an ideal proton, as desired. The coefficients given in Table 2 and Table 3 are taken from Barkas and Berger's contribution to the National Research Council Publication.

The range extension is computed using the range extension for emulsion $C_Z(137 \beta / Z)$ scaled to an arbitrary material. The scaling is such that $B_Z(\beta)$ has the same asymptotic value given by Barkas and Berger:

$\displaystyle{ (7.0 + 0.85 I^{5/8}) \langle A/Z \rangle 10^{-6} Z^{8/3} \text{.} }$ | (24) |

Thus

$\displaystyle{ B_Z(\beta) = (31.8 + 3.86 I^{5/8}) \langle A/Z \rangle 10^{-6} Z^{8/3} C_Z (137 \beta / Z) \text{,} }$ | (25) |

where the functions $C_Z(137 \beta / Z = x)$ are given by

$\displaystyle{ C_Z(x) = -0.00006 + 0.05252 x + 0.12847 x^2 \text{ for } x \le 0.2 \text{;} }$ | (26) |

$\displaystyle{ C_Z(x) = -0.00185 + 0.07355 x + 0.07171 x^2 - 0.2723 x^3 \text{ for } 0.2 \lt x \le 2.0 \text{;} }$ | (27) |

$\displaystyle{ C_Z(x) = -0.0793 + 0.3323 x - 0.1234 x^2 + 0.0153 x^3 \text{ for } 2.0 \lt x \le 3.0 \text{;} }$ | (28) |

$\displaystyle{ C_Z(x) = 0.220 \text{ for } x \ge 3.0 \text{.} }$ | (29) |

$C_Z(x)$ for $x \le 0.2$ was obtained from measurements by Henke and Benton [R.P. Henke and E.V. Benton, "Range-Momentum Relation for Heavy Recoil Ions in Emulsion," Phys. Rev. 139(6A), A2017 (1965)]. For $x \gt 0.2$, the fit was made to data of Heckman et al. [Heckman et al., "Ranges and Energy-Loss Processes of Heavy Ions in Emulsion," Phys. Rev. 117(2), 544 (1960)].

$B'_Z(\beta)$ is obtained by taking the derivative of Equation (25) with respect to $\beta$:

$\displaystyle{ B'_Z(\beta) = (4.357 + 0.5288 I^{5/8}) \langle A/Z \rangle 10^{-3} Z^{5/3} C'_Z (137 \beta / Z) \text{.} }$ | (30) |

It follows that

$\displaystyle{ C'_Z(x) = 0.05252 + 0.25694 x \text{ for } x \le 0.2 \text{;} }$ | (31) |

$\displaystyle{ C'_Z(x) = 0.07355 + 0.14342 x - 0.8169 x^2 \text{ for } 0.2 \lt x \le 2.0 \text{;} }$ | (32) |

$\displaystyle{ C'_Z(x) = 0.3323 - 0.2468 x + 0.0459 x^2 \text{ for } 2.0 \lt x \le 3.0 \text{;} }$ | (33) |

$\displaystyle{ C'_Z(x) = 0.0 \text{ for } x \ge 3.0 \text{.} }$ | (34) |

The scaling of $C_Z(\beta/Z)$ to obtain $B_Z$ and the arguments for this particular method of scaling are shown by the following. Barkas and Berger use the following expression for $R$:

$\displaystyle{ R = \frac{M}{Z^2} [\lambda(\beta) + B_Z(\beta)] \text{,} }$ | (35) |

where

$\displaystyle{ R_{\text{ext}} = \frac{M}{Z^2} B_Z(\beta) }$ | (36) |

is the range extension in units of g/cm^{2}. $B_Z$, also in units of g/cm^{2}, is given by

$\displaystyle{ B_Z(I, \beta) = 48 (1 + 0.121 I^{5/8}) \langle A/Z \rangle 10^{-5} Z^{5/3} \beta }$ | (37) |

for $\beta \lt 2 Z / 137$, and

$\displaystyle{ B_Z(I, \beta) = B_Z(I) = 7 (1 + 0.121 I^{5/8}) \langle A/Z \rangle 10^{-6} Z^{8/3} }$ | (38) |

for $\beta \gt 2 Z / 137$.

Plugging Equations (37) and (38) into Equation (36) gives

$\displaystyle{ R_{\text{ext}} = 4.8 \times 10^{-4} (1 + 0.121 I^{5/8}) \langle A/Z \rangle \frac{M}{Z^{1/3}} \beta }$ | (39) |

for $\beta \lt 2 Z / 137$, and

$\displaystyle{ R_{\text{ext}} = 7.0 \times 10^{-6} (1 + 0.121 I^{5/8}) \langle A/Z \rangle M Z^{2/3} }$ | (40) |

for $\beta \gt 2 Z / 137$. Heckman uses the expression for $R$ in emulsion:

$\displaystyle{ R = \frac{M}{Z^2} \beta + M Z^{2/3} C_Z(\beta /Z) \text{,} }$ | (41) |

where

$\displaystyle{ R_{\text{ext}} = M Z^{2/3} C_Z(\beta /Z) \text{.} }$ | (42) |

Empirically, $C_Z(x)$ is found to be dependent on $Z$ only through its argument $x \equiv \beta / Z$. Thus it is a universal function in $x$ that can be given as an approximation. The asymptotic value of $C_Z(\beta/Z)$ as $\beta \to \infty$ is 0.22 $\mu$m.

$R_{\text{ext}}$ shows an asymptotic dependence on $M$ and $Z$ through the factor $M Z^{2/3}$, as seen from Barkas and Berger, Equation (40), and also from Heckman, Equation (42). It is also seen that the dependence of Equations (39) and (40) on the stopping material is $(1 + 0.121 I^{5/8}) \langle A / Z \rangle$. If instead of using the functions (in units of g/cm^{2})

$\displaystyle{ f' = 4.8 \times 10^{-4} \frac{\beta}{Z^{1/3}} }$ | (43) |

for $\beta \lt 2 Z / 137$, and

$\displaystyle{ f' = 7.0 \times 10^{-6} Z^{2/3} }$ | (44) |

for $\beta \gt 2 Z / 137$, a scaled value of $C_Z(\beta/Z)$ is used, then the function $f$ is given by

$\displaystyle{ f(\beta /Z) = \frac{7.0 \times 10^{-6}}{0.22} C_Z(\beta /Z) = 3.18 \times 10^{-5} C_Z(\beta /Z) \text{.} }$ | (45) |

From Equation (42) we have

$\displaystyle{ R_{\text{ext}} = (1 + 0.121 I^{5/8}) \langle A/Z \rangle M Z^{2/3} f(\beta /Z) \text{,} }$ | (46) |

with the dependence on the stopping material present as a multiplicative factor. The range extension $B_Z(\beta)$ is now given by

$\displaystyle{ B_Z(\beta) = 3.18 \times 10^{-5} (1 + 0.121 I^{5/8}) \langle A/Z \rangle M Z^{2/3} C_Z(\beta /Z) \text{.} }$ | (47) |

Equation (47) preserves both the asymptotic $M$ and $Z$ dependence of Equations (38), (40), and (41) while also containing the stopping material dependence of Equations (37)–(40). The asymptotic value of Equation (47) is equal to the value of the asymptotic Barkas and Berger $B_Z$, Equation (38). The derivative of Equation (47) is continuous, which is the requirement of $\mathrm{d} E/\mathrm{d} x$ calculations. For very small values of $\beta$, the term $B_Z$ is about the same size as the term $\lambda$, the proton range, in Equation (35). It is expected that Equation (47) is a better representation of $B_Z$ than Equations (37) and (38).