The time required for the detector output to go from the initial value to a percentage (e.g. 99%) of the final value. In the case of an exponential behaviour of the detector

can be related to the
time constant

.
The rise time

is the time required for the detector output to vary between given percentages (e.g. from 10% to 90%) of the final value. Similarly,
the
fall time

is the time required for the detector output to vary between given percentages (e.g. from 90% to 10%) of the initial value.
Source:
PAC, 1995, 67, 1745
(Nomenclature, symbols, units and their usage in spectrochemical analysis-XI. Detection of radiation (IUPAC Recommendations 1995))
on page 1751