The number of displacements $D\left(E\right)\mathrm{}$ and the number of vacancies $V\left(E\right)\mathrm{}$ produced in a monatomic solid as a result of collisions due to an incident ion of initial energy $E$, are obtained as solutions of the equation $f\left(E\right)=\int 0^{E}dyK(E, y)\left\{p\right(y\left)\right[f(y-\alpha )+1-\theta q(E-y)]+[1-p(E-y)q\left(y\right)\left]f\right(y\left)\right\},$ where $f\left(E\right)=D\left(E\right)\mathrm{} or V\left(E\right), p\left(y\right)\mathrm{}$ denotes the probability that a struck atom is displaced when it has received energy $y$, $q(E-y)\mathrm{}$ is the probability that the striking atom replaces it if displacement has occurred, $K(E, y)$ is the scattering kernel, and $\alpha$ is the minimum amount of energy that is assumed to be necessary to displace an atom (it is assumed that the struck atom loses energy $\alpha$ in breaking away from its lattice site). In the equation, $f\left(E\right)=0\mathrm{}$ for $E<\mathrm{}\alpha$, with $\theta =0\mathrm{}$ for displacements and $\theta =1\mathrm{}$ for vacancies.

This equation is solved for some representative cases of $p\left(y\right)\mathrm{}$ and $q\left(y\right)\mathrm{}$. The functions $p\left(y\right)\mathrm{}$ and $q\left(y\right)\mathrm{}$ can be chosen to fit experimental estimates of either $D\left(E\right)\mathrm{}$ or $V\left(E\right)\mathrm{}$ singly but indicate a fundamental discrepancy of the joint estimates. The discrepancy, if not due to inaccuracy in the interpretation of experimental results, suggests that a mathematical model based on individual collisions is inadequate.

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