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The problem isn't in finding the so-called "rare earths" ~ it's been known for decades that they are quite commonly found EVERYWHERE.

At the same time they are found in mineable quantities in a limited number of spots ~ a good dozen are in New Mexico alone (just for example).

Because of the concentrations of the rare earths mixed together in the ore bodies the resulting processing for extraction is quite complex. Then there's this problem with THORIUM ~ which is mildly radioactive ~ and as the processing proceeds it gets "concentrated" and next thing you know it's a "radioactive waste" product.

Just busting up these ore bodies at the mechanical crushing stage releases radioactive materials ~ so most mines treat it all initially as posing some degree of radioactive danger to people ~ EXCEPT IN CHINA.

You might want to read this piece to get an idea of the problems faced in mining, processing, extracting and working with the rare earths, et al. http://geology.com/usgs/ree-geology/