https://www.sciencebuddies.org/mentorin ... ?from=Home has a difficulty rating of 9 to 10 for a reason. It isn't so much that the concept is hard but rather that there is a lot of figuring out how to build a neef vibrator (or interrupter) out of pieces that were NOT designed to be used for that purpose.
The pictoral schematic in figure 1 shows a what I will call a primary circuit that (ignoring the capacitor) consists of the following series: a battery, a push button switch, the Normally Closed (NC) contacts of the relay, several turns of wire wrapped around a bolt, and back to the battery. If you don't understand a "series circuit", try a search engine for more help. The capacitor is in parallel (look up "parallel circuit") with the normally closed relay contacts.
This is effectively an electro-magnet. When you push on the normally open switch, it completes the primary circuit and current flows energizing the electro magnet. Build this first and see if you can get the electro-magnet to pick up something.
At this point, the instructions send you off on your own to figure out how to convert this electro-magnet into something that will open the relay contacts. If you look at the relay, you will see that it has a coil (the one in the project calls for a 110VAC coil; however, a better choice would be the 12 VDC coil version so you could play with using its coil in testing with your battery circuit).
How exactly do you make the neef interupter?
The theory is that you place the carriage bolt or rebar or whatever electro-magnet core you are using so that it will attract the metal piece just like the original relay coil / electro magnet does to open the normally closed contacts. Getting this to happen is left up to you. Last year I bought the relay called for to take a look at it and determined that taking the cover off and placing the end of your electro-magnet up against the end of the core in the U-Frame might work but would probably take more than a few turns which would mean a lot more turns on the secondary.
If you cut apart the frame, then you loose the spring mechanism so you have a lot of playing around to get something that will work.
Which is the spark gap what is it made of.
The spark gap is part of the secondary circuit shown pictorially in red labeled copper terminals. A couple of short pieces of #12 AWG copper wire would work. The main requirement is that you come up with something that will allow you to adjust the gap distance from something the thickness of a business/file/recipe card to an inch or more.
I strongly recommend that you do NOT place the secondary coil over the bolt or rebar until you have the neef vibrator working reliably.
If you bought the 12 VDC relay instead and have another 6 volt battery, you can alter the circuit to make a very reliable relay chatter vibrator / interrupter. Add another circuit as follows: From the negative terminal of the original battery, connect another wire to the positive terminal of the second battery. From the negative terminal of the second battery, connect it to one end of the 12 VDC relay coil. Connect the other end of the relay coil to the side of the normally closed contact that is attached to the "primary" coil. This additional circuitry eliminates the need for the homemade electro-magnet to operate the relay contacts and instead uses the relay coil as the electro-magnet to operate the relay contacts as it was designed to do. Relay current flows through both batteries, the push button switch when closed, the normally closed contact through the relay and back to the batteries. At the same time, current flows through one battery, the push button switch, the normally closed contact, and the "primary" coil back to the "first battery". The relay current opens the relay contacts which shuts off the current in both circuit "loops" which then causes the normally closed contacts to re-close and the process repeats as long as the push button switch is depressed.