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Libby calculated the half-life of c14 as 5568 ± 30 years.
This means that half of the c14 has decayed by the time an organism has been dead for 5568 years, and half of the remainder has decayed by 11,136 years after death, etc.
A more recent innovation is the direct counting of c14 atoms by accelerator mass spectrometers (AMS).
The sample is converted to graphite and mounted in an ion source from which it is sputtered and accelerated through a magnetic field.
Bases may be used to remove contaminating humic acids.
Some types of samples require more extensive pre-treatment than others, and these methods have evolved over the first 50 years of radiocarbon dating.
the field deflects atoms of different masses differently (heavier atoms deflect less).