Here is a tip. One type of carbon rod that is readily available and rather cheap is 10 mm square tubes with an 8 mm hole. They are used for building drones and model airplanes (I think), not that hard to find on eBay. I use them instead of a truss rod to stiffen parlor necks.
Besides being cheap and available, they are surprisingly stiff for its weight. On the net I found this:
"In a round rod with a 1 unit thick wall and a 3 unit diameter hole: The hollow shape maintains 87% of its strength, with only 63% of its mass (or to put in another way, you only lose 13% of the strength, but lose 37% of the weight)."
The drawback is that a rod with a hole will split faster than a solid one when bending. Not a problem in bedded in a guitar neck...
Another great thing is that you have two alternatives, leaving the hole empty or filling it up with a 8 mm round rod. I use birch rods as a standard and glue it with thin epoxy. Sometimes when I need more strength I use a solid carbon rod making it a solid 10 mm square rod. I love the idea to have more wood in the mix, but I can't say that I can hear it in the sound of the guitar!
Here is the carbon tube with a birch rod in the hole mounted in a parlor neck. A strip of birch is glued on top (epoxy glue) for the hide glue to bond with
Roger, thanks for the great post. One question, what do you use to thin the epoxy to glue the 8mm rods onto the square tubes? Also, it looks like the birch cap is quite thin, almost veneer. Is it's only function to create a bonding surface for the fingerboard?
I don't use epoxy glue but the resin used to bind carbon fibers into rods. That stuff is thin, not as strong as the glue though. For the square rod itself I use epoxy glue. I pour resin into the hole from both sides and move the round rod to make sure the resin gets everywhere between the rod and the inside of the hole.
The birch cap is thin and creates a bonding surface for hide glue as you said. It is thicker to begin with, after the glue is set I sand it flush to the neck surface.
This is messy business. I wrap the whole neck/instrument in common kitchen wrapping film and tape the top side of the neck close to the routed channel to protect the wood from resin and epoxy glue. VERY important to wear plastic gloves, epoxy is dangerous stuff!