1-a) The time difference measured between the detection moments of γ1 and γ2 stems from the nuclei’s intermediate state lifetime. Given that not every nucleus in an intermediate state “lives” for an equal amount of time, an exponential decrease in the intermediate states will be observed as part of the nuclear decay process, thereby progressively increasing the time difference between the detection of γ1 and γ2.
1-b) It is important to remember that this experiment involves a situation without hyperfine interaction, so the nuclear spin will not evolve in time and it’s orientation will remain the same for the duration of the experiment. When using the 1-3 detector combination, the detector that measures the stop signal is looking along the spin direction and captures the high intensity part of the radiation distribution, while detector 2 is monitoring its least intensive part. This means that while the shape of the exponential curve will remain the same, the consequence of this particular detector arrangement means that there will be less events registered with the 1-2 detector combination than with the 1-3 detector combination.