These are separates from a altered detrital sample. Barite is obviously heavy and non-magnetic, so we are dealing with the final non-magnetic heavy fraction. Obviously the goal here is the zircon, not the barite.
Zircon is non-conducting, slightly radioactive and flourescent, with Specific Gravity of 4.85 gm/cc, with hardness of 7.5, which is greater than that of baryte, i.e., 3-3.5.
The specific gravity of baryte is also less than zircon, i.e., 4.48gm/cc with careful use of heavy media and other criteria such as it is not radioactive and not flourescent,
I think you should be able to separate.
You can selectively leach out baryte mixed with zircon by using hot sulfuric acid (approx. 38 wt%).
At the Arizona Laserchron Center we use a dentistry tool called the Wig-L-Bug to pulverize soft minerals (commonly barite) that make it through density and magnetic separation. This process is detailed on the ALC web site (laserchron.org) and in this PDF
https://drive.google.com/file/d/0B9ezu34P5h8eYzRmZGYwMGItMGI1MC00MGQ3LWE4YzEtNWZiZmFiNDQ5OGRh/view?pli=1
One issue here is the removal of barite, but the retention of all zircon. For typical zircon, we are fine, but with high-damage zircon there is a concern with chemical approaches.
I also think you can use gravity separation, flotation and magnetic separation.
you can maybe find some helpfull process in the following pdf.
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