Shape of a core for use in a solenoid
BAHAHAHAHAHAHHAHA!  This is a project I've thought about since I started learning about electricity.  I was fascinated with coil guns but don't really want to make a weapon/useless weapon.


  I am currently designing an electromagnetic accelerator meant to accelerate a spherical nickel-plated NdFeB (neodymium) projectile along a circular tube, in a similar manner to a particle accelerator. The projectile is to be accelerated by a series of eight solenoids positioned along the tube.


So you need a preferably transparent non-ferrous tube!  Might I recommend heat bending acrylic?  See Marble Machine X/Wintergatan Wednesdays videos on youtube for a method of using a silicon rod to preserve inner diameter when you're bending it.  Problem with using a magnet is that the only thin it allows you to do that a regular ball bearing won't is use a push/pull arrangement, which I think you will find is far far above beginner skill level, so consider just using a regular iron ball bearing.


  The dimensions of the solenoids and the predicted dimensions of the tube (the details of the tubing have yet to be finalized) suggest that there is likely to be a gap of 2-3 millimeters between the solenoids and the tube.


2-3 millimeters!  2-3 millemeters?  That's like 2 or three thousandths of a meter!  Yeah, if you use the tube bending method mentioned above and order correctly sized bearings you should be able to keep it tighter than that.  


  I know that placing an iron, or any material with µ>1, core within a solenoid increases the strength of the magnetic field produced by the solenoid. 


Yes.  In this case, the iron will be the projectile, or the ball bearing.  


  My question is if I place a ring of iron, approximately 2-3mm in thickness, 32-33mm in diameter, in this gap between the solenoid and the tube, will it increase the strength of the magnetic field generated by the solenoid as in an iron core?


Nope nope nope, don't want to do this.  You can put iron on the outside of the solenoid, but if you put it on the inside, it will give the magnetic field a circuit to bypass the projectile instead of increasing the force on it.  Make sure you consider solenoid length as additional layers of turns become less effective and the solenoid can only pull the ball bearing up to the point the bearing reaches its center, after which any current/field left in the solenoid will slow the projectile down instead of speeding it up.


  Additionally, should such a design increase the strength of the solenoid's magnetic field, are there any other materials, which are generally not unreasonably difficult to acquire, that would be more effective in such a role?


For repeatedly changing fields(Like if you wanted to attempt a push/pull arrangement) you might want to consider specifically silicon steel, but if you can't find a way to salvage it, it will be hard or impossible to find or expensive.  I'd get your solenoid working in a basic manner first, and then do a few tests with simple iron pipe fitted tightly on the outside of the coil.  Keep the gap between the inside of the solenoid and the bearing as tight as possible, so thin acrylic tube and a tightly fitting bearing(the bearing should still roll freely through the whole tube, just use close tolerances.

The other thing I think would make this project amusing would be to use sensors to sense the ball's position/velocity and use that to time the firing of the coils.  If you use a microcontroller for the control, you could extensively tweak your program to provide the best and most adaptive timing possible, maybe get that bearing going really fast.  

Bear in mind that generally speaking coil guns operate at significant voltages for beginners and even weak ones are often capable of discharging lethal current, so think everything through and don't use anything you don't understand.  Just messing around willy nilly with voltage multiplier circuits could see you badly hurt.

Finally, experiment with magnetic projectiles if you wish, but bear in mind that forbids use of a sphere, which will increase your gap, and the output velocity will be a function of the input power, the weight of the projectile, friction and efficiency, so if you don't have specific reason to believe you can achieve higher efficiency and or lower friction with a rare earth magnet, don't use one.