Catching/harvesting neutrons This question has several parts. 


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*How does modern science catch free neutrons that are just floating around?

*Is there anyway to "pull" a neutron off any atom that does not involve our current nuclear reactors? 


They're not very controlled with tons of other forms of radiation so, I was wondering if there is a clean, and more controlled way to get our hands on neutrons alone.
 A: Neutron sources
You can buy a commercial off-the shelf "neutron generator", or you can use a radioactive source. 
Neutron generators are accelerator-based fusion reactors1 and have the advantage of being able to simply turn the neutron supply on and off.
The most common source is AmBe (Americium-241/Beryllium), though Californium-252 and tritium both have their uses. I also built a Po-210/C-13 source once as part of my work on KamLAND. Sources require no external power supply and no maintenance (though they do decay away if you use a short half-life progenitor like Po-210), but they run continually.
Free neutrons
You basically can't "catch" free neutrons in the sense of collecting a bunch of them in one place. 
On the other hand, thermal neutrons (those whose kinetic energy in on scale of that of room temperature gasses) can "capture" on a variety of atoms. Hydrogen-1 and Carbon-12 are the most common nuclei that can capture a free neutron. Chlorine, boron and especially Gadolinium are useful because they have high thermal neutron capture cross-sections.

1 Yes, fusion reactors. They've been around for many decades. The fusion "problem" is in getting ones that generate more power than you put in.
A: This is a bit pedantic, but when you say catch free neutrons -- I assume you mean those that exist from non-discrete sources undergoing reactions (transmutation, decay processes, etc.) rather than neutrons that are produced in the unbound state from a specific source, be it "fusors" or a radioactive source coupled with a target material or a source that has sufficient spontaneous decay events (Cf-252) which result in neutrons being freed.  The answer, to either source of neutrons, is yes we can, and it involves electromagnetic confinement.  The problem is, they have short lives, averaging around 15 minutes, so there isn't a tremendous amount you can do with them.  The real uses for neutrons come not from storing them, but rather their passage through materials, starting as particles whose lack of charge reduces deflections and thus allows materials it travels through to be transparent.  When they become thermalized, several things can happen, but the one most exploited is an activation, that is the neutron is absorbed by a nucleus.  The other way activation products are formed through this neutron bombardment is by induced fission.  It's essentially only when they interact with they environment that useful measurements and thus useful information can be obtained.
Fusors are not capable of producing neutrons of sufficient energies to pull off most threshold reactions, which why radiological materials sources are very useful.  They are safe and very easy to build, the problem tends to lie in acquiring sufficient radioactive sources, something that an occasional clever physics student will figure out ;)
To the second question, this may not be what you mean, but when neutron sources employing radioactive isotopes are designed, the initial consideration has to be whether you want an alpha particle bombardment of a low Z material, or a gamma ray bombardment of the target material AKA photonuclear source.  In the either example, we have nowhere near the particulate radiations flux or the photonic flux as that of a commercial nuclear reactor, but we are pulling off neutrons at very high KE's...maybe I misunderstood the query.
