Libby calculated the half-life of c14 as 5568 ± 30 years.
However, to avoid confusion all radiocarbon laboratories continue to use the half-life calculated by Libby, sometimes rounding it to 5570 years.
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.
For example, it was once standard practice to simply burn whole bones, but the results were eventually seen to be unreliable.
Chemical methods for separating the organic (collagen) from the inorganic (apatite) components of bone created the opportunity to date both components and compare the results.