Carole, when I started making pens I had a "drill chuck" mandrel from PSI. That is because I did not have the Jet Mini yet, and the lathe I was using does not have a Morse taper in it. The PSI mandrels have a 1/4"-20 thread and nut. Since I bought the Jet I have been using Morse taper mandrels from Woodcraft.
Drill chucks for lathe use usually have a Morse taper. They will work ok if they run true. You can easily test by putting a mandrel in the chuck, with the live center about 1/8" from the end of the mandrel. Rotate the spindle by hand, and you will see whether it runs true.
Mandrels are a little under 1/4", something like .246 (depending on the manufacturer). Size "D" drill rod is the same size. So is "allthread" 1/4" rod from the hardware store. I have found at Lowe's that they have allthread in a rack, and also in the nuts and bolts drawers, where it is sold in 12" lengths in plastic bags. The ones in the drawers seem to be better quality, they are straight and much stiffer. I have used them as mandrels, and they worked ok for me. What I did was to epoxy a "threaded rod connector" which is like a long nut, near one end, leaving enough rod to go in the chuck. I used a nylon wing nut on the other end.
However, since Woodcraft sells replacement rods for $3. and complete mandrels for $10, I no longer use the threaded rod.
Of course, no matter where you get the rod, if you are going to use multiple mandrels you still need more bushings for them. Since they cost money, and the mandrels include some bushings, I find buying complete mandrels the best way to go.
Hope this helps.