Can impact offset of an electron on an ion affect the resulting particle velocities?

Question

A free electron, of mass $m_e$ and velocity $v_e$ collides with an ion of mass $m_i$ and velocity $v_i$. When they recombine, a photon of exactly the ionization energy $E_i$ will be emitted, moving in direction $\theta_p$, and travelling at the speed of light $c$. The electron and ion will form a single neutral atom, moving at speed $|v_a|$ and angle $\theta_a$.

Consider only 2 axes for simplicity, $x$ and $y$. Conservation of energy give one equation of scalars, and conservation of momentum gives two equations of scalars: one for each axis.

Thus we have three simultaneous equations. Meanwhile we have three unknowns: $|v_a|$, $\theta_a$, and $\theta_p$.

Thus, we can solve for the three unknowns using the three equations.

However, we did not consider the impact location of the electron on the photon. The electron could strike dead center, or off to one side. Think of a billiard ball striking another billiard ball: the offset of the collision affects the resulting ball angles and speeds, even if the initial billiard ball velocities are identical in each case.

Given that the laws of conservation of energy and momentum have already fully defined the speed and direction of the resulting particles, there seems no obvious way for these to be affected by the position of impact of the electron. Does this mean that the position of impact of the electron does not affect the resulting speed and directions of the resulting particles?

Edit1: by comparison, when two billiard balls collide, the resulting balls each have two unknowns, i.e. the speed and the angle (assuming 2 dimensions); or alternatively the x and y velocities (again, assuming 2 dimensions). The laws of conservation of energy and linear momentum again comprise three equations (given 2 dimensions), so now we have 1 more unknown than we have equations. And the additional unknown gives us the possibility to take into account the impact position of one billiard ball on the other.

Edit2: note that some people might observe that I've ignored conservation of angular momentum, but that would only add additional equations? Conservation of angular momentum does not provide any "loopholes" around conservation of linear momentum. Oh, hmmm, I suppose we could have rotational kinetic energy, which would thus reduce the linear kinetic energy, and thus the velocity of the resulting atom?

Edit 3: In the case of perfectly smooth billiard balls, the impact of the balls does not affect the rotation of the balls, by virtue of there being no friction, and thus no angular torque. Maybe this is not the case for the impact of an electron on an ion?

Edit 4: yes, I suppose the capture of the electron by the ion is kind of the opposite of the frictionless billiard balls. It's in essence infinite friction, and the impacting electron will in fact impart a torque on the ion, and thus be converted partially into rotational kinetic energy. Oh, maybe this is the answer?

Answer 1

In the case of two perfectly smooth, perfectly elastic billiard balls colliding, the impact of the perfectly smooth billiard balls does not impart a torque of one ball on the other. So, we can ignore rotational kinetic energy, and conservation of angular momentum.

However, when the electron is completely absorbed by the ion, an off-center impact will impart a rotation to the ion. We cannot ignore rotational dynamics. Off-center impacts will lead to rotational kinetic energy, which will decrease the linear kinetic energy of the resulting atom, and lead to a different speed after collision.

Thus, the offset of collision will affect the resulting atom velocity, and it can do this because the linear kinetic energy doesn't need to be conserved: only the total energy, which is a sum of linear kinetic energy, rotational kinetic energy, and ionization energy.

The linear momentum will still need to be conserved. However, if the resulting atom is not moving along the direction of the momentum - which is quite likely unless the photon moves along that direction too - then changing the speed of the resulting atom will also change its direction, without violating either the laws of conservation of momentum or the laws of conservation of energy.