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The exponential decay is due to the lifetime of the intermediate state, which gives rise to this curve for the gamma 2 decay probability.
The different count rate is due to the orientation of the nuclear spin. Since the first gamma is observed in detector 1, it is most likely parralel with that direction. Since the transitions go fast, its spin will not change much before sending out the second gamma, which will then again be send out most often parallel. Since detector 3 is also parallel to the spin, it will get more gamma 2’s.
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