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1/y/V factor 4-vectors analogous to equation appendix G bodies correct phase space Cucinotta d3Pi d3Pl Dalitz plot Deutchman dLips equa equation 9 equation Dl equations 14 evaluate extra 1/y/V Formalism for Phase function of dE\22 Galactic cosmic rays Green's function Idaho identical to E2 ij\t\kl incident particles integral iV-body final Jacobian Langley Research Center Lorentz-invariant phase space mass of particles matrix elements metric tensor momentum conservation NASA noninvariant phase space nonrelativistic limit Norbury and Philip Nuclear cross sections nucleon number of incident obtain Old Dominion University one-body wave function overall CM frame particular reaction Performing Organization Phase Space Calculations phase space element phase space factors pion Recurrence Relations reduced mass reference 12 Research Center Hampton Space The two-body T-matrix three-body density three-body final three-body phase space three-body problem transition rate two-body density units of dNips units of Energy University of Idaho volume factors volume terms cancel