Not quite sure how we got onto the subject of tabletop neutron sources, but if it were me, I make up a little H2 plasma bottle, strip off electrons to get protons, accelerate the protons across whatever potential I choose, and then slam them into paraffin. Neutrons aplenty. Looking at that 'zetatron' schematic, I'd guess that the "target" is a paraffin block. As I learned form my visit yesterday, if you want gobs of neutrons but don't want a reactor, then you'd use the technique at the SNS in Oak Ridge- liquid mercury getting slammed with protons from a simple linac. A cuter and more compact technology is the DWA (dielectric wall accelerator) now under development for medical applications at Livermore in collaboration with Tomotherapy. That proton therapy device will be in every cancer hospital within 2-3 decades, mark my words.
I am skeptical that one would want all that zetatron complication in a weapon... it's easy to generate a shower of neutrons from fission. OTOH maybe the zetatron finds application in the baby nukes that contain a net sub-critical mass. But I digress