= Solution
The assumption is a \b[leading approximation for concentrated <ballistic radiation damage>], not an exact statement that the <alpha particle> displaces no atoms. The daughter has only about $m_\alpha/M_d$ times the <alpha particle>'s <kinetic energy>, but it is much slower and loses a much larger fraction of that energy in collisions with target nuclei. Those collisions initiate closely packed recoil cascades. The <alpha particle> deposits most of its much larger energy into the electronic system over a path hundreds of times longer, with a less concentrated displacement contribution.
Thus the dense region susceptible to one-event loss of crystalline order can reasonably be associated chiefly with the daughter recoil. This explains why the effective damaged volume in the <direct-impact amorphization model> can be estimated or fitted from recoil-dominated cascades. It is the \b[spatial density and stopping channel of the damage], not simply which particle carries more energy, that matters.
The approximation has a measurable limit. Turcotte's comparison of actinide-dioxide self-damage and external alpha irradiation found a nonzero alpha contribution, reporting about $19\%$ of total damage in the studied case. Comparison of diffraction-derived lattice changes with internal-reference measurements helps separate real <radiation-induced swelling> from instrument shifts, while the distinction between self-decay and external-alpha irradiation probes the omitted recoil channel.<Alpha radiation damage in the actinide dioxides by R. P. Turcotte>[BNWL-SA-5333, summary and experimental/results discussion; nonzero alpha damage and diffraction measurements]{ref}
Consequently “all <ballistic radiation damage>” is useful as the stated model assumption but should not be promoted to an exact material-independent law. <electronic excitation> can also couple to structural change, and recovery alters what survives. The derivation in part (b) remains valid for an effective one-event volume fitted consistently to the relevant regime; it does not establish that all <ceramics> or spent fuels share the same $V_\alpha$ or the same eventual <amorphous fraction>.
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