Radiation damage in a crystal Take a single crystal as close to the ideal as possible, say, Si, GaAs, or SiGe, etc. Immerse the crystal in some fixed rate ionizing radiation and measure the radiation damage it has caused to the crystal per unit time. How does the resulting radiation damage depend on the temperature of the crystal, especially, as it is cooled to cryogenic temperatures?
 A: The colder the crystal gets, the more difficult become diffusion processes in the crystal. Since radiation damage to crystalline materials often involves atoms getting knocked out of their positions in the lattice, and since healing that damage requires diffusion to be enabled, exposing the crystal to radiation at cryogenic temperatures should lock in the damage and allow it to accumulate.
A: To first order, my guess is that room temperature vs cryogenic temperature doesn't matter. The energy needed to knock an atom out of place is significantly higher than the thermal energy of the atoms at either temperature, so the temperature of the crystal shouldn't play a direct role. (If you want to heal Si in a reasonable timeframe, you've got to heat it up above room temperature.)
I can imagine some second order effects. All things being equal, colder materials are more dense, so I'd expect that cooling a material down will increase the rate of collisions and thus increase the rate of damage. However, I expect that to be a small effect.
Radiation damage to semiconductors is an important topic for designing radiation-hardened electronics (which are primarily used in space applications), so there's a fair amount of research in that area. However, since it's mostly of commercial and defense interest, less information is public than in many research areas. I've never studied it myself, but I've spent time with people who have, and I've never heard of them discuss this type of temperature dependence. The distinction I'm familiar with is basically: is the crystal hot enough to heal the damage (i.e. above room temperature) or is it not hot enough to heal the damage (which includes both room and cryogenic temperatures --- at least for Si).

EDIT:
Two things to start:

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*If you're asking a math question (specifically, a geometric probability question), may I suggest using math stack exchange? Use the "geometric-probability" tag. (You could probably use the "billiards" tag too.)

*I see where this toy model is coming from, but I'm not convinced that it's useful for modeling actual radiation damage. It has too much of a spherical cow feel to it. For example, if the ionizing radiation is fast-moving particles (which happens in cosmic rays), then particles could scatter and cause multiple defects. In fact, if a relativistic particle hits an atom in the crystal, that atom will probably go flying and cause even more defects. None of this is captured in a hard sphere model.

That said, here's my attempt to model radiation damage using hard sphere scattering --- and why I still think that temperature doesn't matter.
First, the relevant comparison isn't a perfect vs an imperfect lattice. The lattices will be imperfect in both cases --- just more imperfect at higher temperatures.
Second, assuming the incoming radiation can be thought of as rays or point particles, you have three important length scales: the lattice spacing, the rms value of the atoms' displacement from the ideal location, and the radius of the atoms. The first is always going to be much greater than the other two. The interesting bit is the relative size of the second two length scales. Suppose that the RMS displacement >> atomic radius. The top half of the below figure sketches out a column in the lattice for two different RMS displacements. Because RMS displacement >> atomic radius, the area that's "shaded" by the atoms (i.e. how much of the incoming radiation flux is blocked) is the same in both cases because none of the spheres "shade" (overlap with) each other.

Now, say that it's no longer true that RMS displacement >> atomic radius. In the bottom half of the figure, we see that no spheres shade each other in the high RMS displacement case, but some do shade each other in the low RMS displacement case. This happens because we're no longer in the case where RMS displacement >> atomic radius.
Now, you may object that I only showed a few layers. If I had enough layers, even with RMS displacement >> atomic radius, then atoms would start to shade each other, and this would be more pronounced with smaller RMS displacement. This is true, and the effect starts to become important when you have roughly (RMS displacement / atomic radius) layers. However, in "real life", atoms can be displaced in both x and y, so the shading effect becomes important when you have about (RMS displacement / atomic radius)^2 layers. Say that you have a 1 mm thick Si wafer. The lattice constant for Si is about 0.5 nm, so that would be order 1 million layers. Say that the RMS displacement is 5% of the lattice constant -- about 25 pm --- and that the sphere has the same radius as the atomic nucleus --- something like 2 fm. Then (25 pm/2 fm)^2 = 156 million. 1 million << 156 million, so shading would not be important; the RMS displacement wouldn't matter much, and by extension, the temperature wouldn't matter much.
Of course, if you plug in different numbers, you could get different results. I'm not sure exactly what effective size to use for the atom. Still, remember that other effects work in the other direction. Making Si colder will make it more dense, which will increase the scattering rate. Plus, I don't think this is a useful model in the first place!
In any case, I'm sticking by my original reaction, which is that temperature doesn't matter much one way or another.
A: I may depends on what exactly you mean by ionizing radiation, which can be very high energy or just high enough (comparable to bond energy) to pop out a local particle.
I'd like to think of it in a low-energy, single particle quantum mechanical point of view. This may not be applicable to very high energy ionizing radiation but may provide some insight for low energy radiation.
Quantum mechanically, damaging a crystal, inducing a displacement or a dangling bond means the kinetic energy scale of that atom or electron exceeds the potential of its environment that tends to localize it. In the extreme case of super fluid Helium, all He atoms are always randomly displaced which makes it a liquid that never orders. In a coarse picture this can be attributed to the fact that He atoms are very small in size, so that according to De broglie's theorem $p \sim h/\lambda$, for small $\lambda$ the zero-point kinetic energy is so large that the potential between particles are not enough to localize $He$ atoms. In this respect the $He$ "crystal" would spontaneously destroy itself by the large ratio between kinetic energy and binding potential, without any external energy input.
In the case of ordinary crystals where the zero-point energy is low so that lattice order becomes stable, popping out a particle from its bound state requires an energy input from, say, electron beam or X/$\gamma$-rays, that is larger than (but not too large), or close to the potential well (or chemical bond) from it's neighbors, so that the probability amplitude to tunnel out of its localized position is high enough for a damage to occur. Therefore at higher temperature, I'll expect a higher probability for a damage to occur since there's a higher probability that the atom stays in the excited level of its bound state thus requires less energy to escape from the potential.
