In low energy nuclear reactions or radioactive decays the re

In low energy nuclear reactions or radioactive decays, the reaction or decay energy is typically on the order of 1 MeV. Show why we have to treat electrons relativistically while we can treat nucleons non-relativistically.

Solution

we consider relativistic treatment for the particles moving close to the velocity of light .

in the case of electron decay with energy of 1MeV which is in the form of kinetic energy

                             mv^2 / 2 = E = 1 Mev or 1.602* 10^ -13 J

                                 m = mass of electron = 9.109 * 10^31 kg

                                     from the above relation we can measure the velocity of the electron

                                       v =( 2E / m ) ^1/2

                                          v = 5.930 * 10^ 8 which is > c, velocity of light

                              this velocity of the electron exceeding light velocity is not possible according to relativity

hence we have to consider relativistic effects on electron for 1MeV kinetic energy

where as in the case of protons and neutrons which are simillar in mass

                  the velocity        v =( 2E / m ) ^1/2

                                              m = mass of proton = 1.672* 10^-27 kg

                                              v = 1.3842 * 10 ^7 < c .

                   since the velocity is with in the limit of relativistic velocity for the case of protons and neutron non relativistic approach is valid .