With this type of subject, the difference between success and failure can be a tiny change to the lighting.
The best approach generally is to have a series of tiny light sources hitting the subject from several different angles, so that some light is refracted through the diamonds and other light is reflected from the surface, and to have a (relatively) large light source overhead, to provide a degree of fill, as necessary. You can easily enhance the colours simply by fitting lighting gels over some of the lights. BTW, a LED torch is pretty massive compared to the size of the subject, you'll need to mask off most of it to turn it into a small light source - or move the torches much further away, but if you do that you'll probably run out of lighting power.
From a physics viewpoint, diamond is a strange material. It's basically just carbon, but with the atoms arranged in straight lines, this causes the light to refract and reflect differently to other materials, and because of this there is no "textbook" way of getting the angles right.