The Spallation Neutron Source (all details taken from this link) is described as firing a 1 GeV proton beam into a mercury target. As this makes the proton beam relativistic, a cyclotron cannot be used (not for all of it, anyway), and therefore a very large linac is used instead.
Could the setup be made smaller by accelerating mercury to 1 GeV and firing it into a hydrogen target? As mercury atoms have a mass of 186.85 GeV/c², I make the speed and γ at 1 GeV to be:
$$ \small {\begin{alignat}{7} && 1 \, \mathrm{GeV} &~=~ (\gamma -1) \cdot 186.85 \, \frac{\mathrm{GeV}}{c^2} c^2, \qquad γ = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \\[2.5px] \therefore~~ && 1 \, \mathrm{GeV} &~=~ \left(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1\right) \cdot 186.85 \, \mathrm{GeV} \\[2.5px] \therefore~~ && \frac{1}{186.85} &~=~ \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1 \\[2.5px] \therefore~~ && 1 + \frac{1}{186.85} &~=~ \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \\[2.5px] \therefore~~ && \sqrt{1-\frac{v^2}{c^2}} &~=~ 1/(1 + \frac{1}{186.85}) \\[2.5px] \therefore~~ && 1-\frac{v^2}{c^2} &~=~ \frac{1}{\left(1 + \frac{1}{186.85}\right)^2} \\[2.5px] \therefore~~ && -\frac{v^2}{c^2} &~=~ \frac{1}{\left(1 + \frac{1}{186.85}\right)^2}-1 \\[2.5px] \therefore~~ && \frac{v^2}{c^2} &~=~ -\frac{1}{\left(1 + \frac{1}{186.85}\right)^2}+1 \\ && &~=~ 1-\frac{1}{\left(1 + \frac{1}{186.85}\right)^2} \\[2.5px] \therefore~~ && v^2 &~=~ c^2 \left(1-\frac{1}{\left(1 + \frac{1}{186.85}\right)^2}\right) \\[2.5px] \therefore~~ && v &~=~ \sqrt{c^2 \left(1-\frac{1}{\left(1 + \frac{1}{186.85}\right)^2}\right)} \\[2.5px] \therefore~~ && v &~≅~ 0.103 c, \qquad γ ≅ 1.0053471 \end{alignat}} $$
which is close to what I've been told is the relativistic limit for a cyclotron (I think on the lower side of the limit, but can't remember).
However, even if it's on the wrong side of that limit: a heavy element like mercury can be multiply-ionised, and even fully ionising mercury requires far less energy than accelerating it to that speed, so even a linac with that design could be 20 times shorter. If spallation works like that.
I'm mainly thinking of this in terms of a spacecraft powered by an accelerator-driven subcritical reactor, and reducing the linac length from 335 m to 16.75 m seems like a significant improvement for such a task.
[My physics level is {UK: AS-level equivalent, USA: probably highschool, I'm not sure}; my maths level is {UK: double-A2-level, USA: probably between freshman and somophore year at university, but I'm not sure}, please target answers at that sort of level]