This list is by no means exhaustive, but has what I think are some things you can do for a nice research project. Depending on the type of project, just about any working crystal set will do.
I. Crystal set design for sensitivity and selectivity:
a. Compare the performance of a variety of manufactured diodes. Both loudness and/or number of stations heard can be the basis for comparison. For more objective comparison criteria, look at my page on testing with a meter. A call for help on Rap 'n Tap should give you a nice selection of germanium diodes to use. I myself probably have in my junk box at least a half dozen varieties of this old standard. Basic.
b. Compare the detection ability of mineral detectors, such as galena, iron pyrite, and others. This is a more advanced aspect of detector research.
c. Compare the efficiency of different size wire used in tuning coils. This is a bit more complex than it might appear. The coils should all have the same value of inductance, and probably the same geometry, which means that the ratio of length to diameter should be held constant. The value of the coil doesn't necessarily have to give you full coverage of the band, but needs to enable you to pick up stations you expect to hear; this can save you some wire. Basic to Advanced
d. Compare
the efficiency of different coil designs, such as close wound vs space wound,
basket wound, and "spider web".
Make each coil of the same type and size
wire, and also of about the same value. Basic to very Advanced
e. Compare different configurations of antenna tuning. This can be pretty basic, such as comparing a tapped or "slider" tuned coil alone to a coil and capacitor combination. Opportunities exist for exploring more advanced schemes as well.
f. Compare different methods of getting the signal from the antenna to the set. Advanced to Basic, depending on methods and test criteria.
g. Which works better, listening with two ears or one? Use a double headset or pair of crystal earplugs for this. You can also compare left vs right ear. Put a hearing protection earplug into the ear not being tested. Basic.
h. Optimizing the crystal set for selectivity and sensitivity. Very advanced and long term project.
II. Antenna design: (some climbing required, as well as suitable supports for the antenna)
a. For a set length of wire "out the window", compare the effect of making the antenna higher versus longer in the horizontal section versus a gradual slope from one height to another. Compare center or off-center fed versus end fed. Try the same antenna pointed in different directions for the horizontal section. Advanced climbers.
b. Compare the effect of making the horizontal section longer for the same vertical height, or try using the same horizontal length but at different heights. Basic to advanced.
c. Try the same antenna configuration but with different size wire, or solid versus stranded wire.
d. Try antennas with single wires in the horizontal section versus two or more parallel wires.
III. Propagation:
a. Picking the best time of day to hear distant stations. Basic
b. Gray line propagation, variations in reception around sunset and sunrise. Advanced