The presence in the universe of an apparently uniform ``sea'' of blackbody radiation at a temperature of roughly 3K gives one mechanism for an upper limit on the energies of photons that have traveled an appreciable distance since their creation. Photon-photon collisions can result in the creation of a charged particle and its antiparticle (``pair creation'') if there is sufficient energy in the center of ``mass'' of the two photons. The lowest threshold and also the largest cross section occurs for an electron-positron pair.
(a) Taking the energy of a typical 3K photon to be E=2.5×10−4 eV, calculate the energy for an incident photon such that there is energy just sufficient to make an electron-positron pair. For photons with energies larger than this threshold value, the cross section increases to a maximum of the order of (e2/mc2)2 and then decreases slowly at higher energies. This interaction is one mechanism for the disappearance of such photons as they travel cosmological distances.
(b) There is some evidence for a diffuse x-ray background with photons having energies of several hundred electron volts or more. Beyond 1 keV the spectrum falls as E−n with n≈1.5. Repeat the calculation of the threshold incident energy, assuming that the energy of the photon in the ``sea'' is 500 eV.