Transmuting the nuclear waste needs energy, it does not produce energy.
Unlike the fission from U235 or Pu239, which starts with high energy nuclei and it produces low-energy radioactive nuclei with long lifetimes, so the difference is produced energy, when transmuting long lifetime nuclei into short lifetime nuclei, you end with nuclei having higher energy.
The difference in energy is obtained from the gamma rays. So the gamma ray source must provide a huge amount of energy, of the order of 1 MeV = 1.6E-13 joule per atom, so of the order 1E14 joule = about 28 GWh per kilogram of treated nuclear waste.
This estimation is very approximate, because the required energy per atom depends a lot on the exact composition of the nuclear waste, but even if it would be 10 times lower for certain isotopes, it would still be a huge amount. After being multiplied by the efficiency of the gamma-ray source it would exceed by much the energy output of the nuclear reactor.
A single gamma photon may indeed cause indirectly more than one nuclear reaction, but the mechanism does not matter. Only the initial and final products matter, which determine the energy consumption for waste treatment.
Unlike the fission from U235 or Pu239, which starts with high energy nuclei and it produces low-energy radioactive nuclei with long lifetimes, so the difference is produced energy, when transmuting long lifetime nuclei into short lifetime nuclei, you end with nuclei having higher energy.
The difference in energy is obtained from the gamma rays. So the gamma ray source must provide a huge amount of energy, of the order of 1 MeV = 1.6E-13 joule per atom, so of the order 1E14 joule = about 28 GWh per kilogram of treated nuclear waste. This estimation is very approximate, because the required energy per atom depends a lot on the exact composition of the nuclear waste, but even if it would be 10 times lower for certain isotopes, it would still be a huge amount. After being multiplied by the efficiency of the gamma-ray source it would exceed by much the energy output of the nuclear reactor.
A single gamma photon may indeed cause indirectly more than one nuclear reaction, but the mechanism does not matter. Only the initial and final products matter, which determine the energy consumption for waste treatment.