= Electric-dipole excitation by a passing ion
{title2=$P_{ji}\simeq[2Ze^2/(\hbar bv)]^2|\langle j|x|i\rangle|^2$}
In the straight-trajectory dipole model, retain the Coulomb-coupling convention of $V(t)=Ze^2(bx+vty)/(b^2+v^2t^2)^{3/2}$. In SI units the coupling denoted $e^2$ here is $e^2/(4\pi\epsilon_0)$. First-order <time-dependent perturbation theory> expresses a transition through the <Fourier transform> of this pulse at the transition angular frequency.
For positive $\omega$ put $\beta=\omega b/v$. The two Cartesian pulse transforms are
$$
I_x=\int_{-\infty}^{\infty}\frac{b e^{i\omega t}}{(b^2+v^2t^2)^{3/2}}dt=\frac{2\beta K_1(\beta)}{bv},\qquad
I_y=\int_{-\infty}^{\infty}\frac{vt e^{i\omega t}}{(b^2+v^2t^2)^{3/2}}dt=\frac{2i\beta K_0(\beta)}{bv},
$$
where $K_\nu$ is a <Modified Bessel function of the second kind>. One may derive these from the Fourier transform of $(1+s^2)^{-1/2}$, differentiating in its scale and integrating its spatial derivative by parts. For $\beta\ll1$, $\beta K_1(\beta)\to1$ while $\beta K_0(\beta)\to0$, so the transverse impulse gives the probability in the title. If the $x$ <matrix element> vanishes, the smaller longitudinal term can be the leading nonzero contribution.
The conditions include $b$ large compared with the atomic extent for the <electric-dipole approximation>, a small total transition <probability> for first-order theory, and a collision time $b/v$ short compared with $1/|\omega|$. Large $\beta$ instead gives exponentially suppressed excitation in this model through the large-argument behavior of $K_\nu$.
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