= Solution
A parent initially at rest emits an <alpha particle> and a recoiling heavy daughter. <momentum conservation> gives equal and opposite momenta, so their nonrelativistic <kinetic energies> satisfy
$$
\frac{T_d}{T_\alpha}=\frac{m_\alpha}{M_d},\qquad
T_\alpha=Q\frac{M_d}{M_d+m_\alpha},\qquad
T_d=Q\frac{m_\alpha}{M_d+m_\alpha}.
$$
Here $Q$ is the <kinetic energy> available to this branch; excitation of the daughter reduces it and may be followed by photon emission. For an illustrative $Q=5$ MeV, $m_\alpha=4\,\mathrm u$ and $M_d=236\,\mathrm u$, the energies are \b[$4.917$ MeV in the <alpha particle> and $83.3$ keV in the daughter]. The heavy daughter receives much less energy, but that does not imply much less atomic damage.
The fast <alpha particle> loses most of its energy through <electronic stopping>: <ionization> and <electronic excitation> along a relatively long path. It also undergoes nuclear collisions, particularly towards the end of its range, and can produce <displacement damage>. The slower, heavy, highly charged recoil couples strongly to surrounding nuclei and generates a compact <displacement cascade>. Successive knock-on atoms create <vacancies>, <interstitial defects> and clusters. Some <Frenkel pairs> recombine as the cascade relaxes; the initial collision count is therefore not the number of defects that remain.
A quantitative illustration comes from <brannerite> rather than an assumed universal <ceramic>. Simulations in the primary natural-mineral study used a $4.5$ MeV <alpha particle> and an $80$ keV daughter recoil. The corresponding ranges were about \b[$13$ micrometres and $23$ nanometres]. At the study's assumed $50$ eV <threshold displacement energy>, its tabulated results give roughly $560$–$571$ recoil <atomic displacements> and $91$–$100$ <alpha particle> <atomic displacements> per <radioactive decay>. The numbers depend strongly on the displacement threshold and material, but the shorter recoil path contains the dominant and much denser <ballistic radiation damage>.<Chemistry, microstructure, and alpha decay damage of natural brannerite by G. R. Lumpkin and others>[Chemical Geology 291, section 2.3 and Table 5; simulated ranges and threshold-dependent displacement counts]{ref}
In a susceptible <ceramic>, cascade cores can accumulate into an <amorphous fraction>; defects and associated relaxation can also change volume, <thermal conductivity> and mechanical response. <molecular dynamics> simulations of <zirconolite> found a strongly disordered core surrounded by more isolated <point defects>, supporting this distinction between local cascade damage and an entirely transformed specimen.<Molecular dynamics simulation of the α-recoil nucleus displacement cascade in zirconolite by L. Veiller and others>[Journal of Nuclear Materials 306, 61–72, abstract and cascade-core/point-defect results]{ref}
The <alpha particle> eventually becomes a <helium> atom after slowing and <charge exchange>. <helium> accumulation can have later microstructural effects distinct from the initial knock-on cascade. Material-dependent recombination and <annealing (materials science)> limit surviving disorder; a change in <diffraction> or <radiation-induced swelling> is not automatically <radiation-induced amorphization>, and <radiation-induced amorphization> is not automatically <dissolution> or release of the contained radionuclide.
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