They were using a so called accelerator mass spectrometer (AMS). The main challenge, as I understood it from talking with them, was in the chemistry required to prepare the samples for injection into the AMS. Specifically ensuring that there's no background signal created from the preparation itself. So there was a lot of work done just to get the process right (the less steps to prepare the sample, the less steps that might contaminate it). Then a lot of verification, calibration on the AMS and optimizing the beam transport to reduce losses as far as possible.

EDIT: The AMS was built and used in a lot of experiments before that. Among other things done with that instrument was measuring the neutron flux of the nuclear blast at various locations in Hiroshima, by determining the isotopic composition of the copper on building facade elements. Neutron capture produces some radioactive as well as "unnatural" stable isotopes and by determining their ratio, one can determine the neutron flux and energies.

https://www.researchgate.net/publication/6012967_Neutron-ind...

It saddens me, that the accelerator they (and I for that matter) used for their research, has been decommissioned as of last year and is being dismantled right now :( – I could start a major rant about it, that would touch the subjects of funding, some people's personal pride and good science vs. fashionable science.