**2. Binomial and Poisson Distribution Functions for the Analysis of the Ion Track Overlapping Effect**

The binomial distribution function,

$$f(n,k,p) = \frac{n!}{k!(n-k)!}p^k(1-p)^{n-k},\tag{1}$$

generally presents the discrete probability distribution of the number of successes, *k*, in a sequence of *n* independent trials, where *p* is the probability of the success and 1 − *p*, that of the failure for each trial. The paper of Ishikawa et al. [23] and our previous report [24] have shown that the probability of the ion-track overlapping can be described by the binomial distribution function. When the fluence of ions is Φ and the total area of the target is *S*0, the number of irradiating ions, *n* (=Φ*S*0) corresponds to the trial number. The ratio of the cross section of each ion track, *S*, to *S*<sup>0</sup> (*s* = *S*/*S*0) and the number of track impacts, *r*, correspond to *p* and *k* in Equation (1), respectively. The probability of *r* times track impacts is given as

$$a(n,r,s) = \frac{n!}{r!(n-r)!}s^r(1-s)^{n-r} \tag{2}$$

If the ion track overlapping is discussed for the unit irradiation area (*S*<sup>0</sup> = 1 cm<sup>2</sup> ), *n* is replaced by the ion fluence, Φ, and *s* is replaced by the track cross section itself, *S*. Then, the fraction of the area for the *r* times track impacts in the unit area of the target, *A*(Φ,*r*) is,

$$A(\Phi, r) = \frac{\Phi!}{r!(\Phi - r)!} S^r (1 - S)^{\Phi - r} \text{ for } r = 1, 2, 3, \dots \text{-} \Phi \tag{3}$$

Although the dimensions of Φ and *S* are cm−<sup>2</sup> and cm<sup>2</sup> as experimental parameters, these parameters can be treated in Equation (3) and other equations in this paper as dimensionless.

For the usual irradiation experiments of the ion tracks in materials, as a value of Φ is very large and a value of *S* is very small, the binomial distribution function can be approximated by the following Poisson distribution function [25],

$$A(\Phi, r) = \frac{(\Phi S)^r \exp(-\Phi S)}{r!} \tag{4}$$

We have confirmed that the result calculated by the Poisson distribution function (Equation (4)) is completely the same as that calculated by the binomial distribution function (Equation (3)) in the case of the irradiation studies referred in the present report [16,21,24]. Equation (4) will be used later for the comparison of the analytical result with that by the Monte Carlo simulation.
