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The Fowler-Nordheim equation is derived in terms of total electron energy so as to obtain the total-energy distribution of field-emitted electrons. At 0^ the new distribution width is less than 13 of that obtained from the previous "normal-energy" theory. A surprising mirror-image symmetry is observed between the zero-temperature total-energy field emission and the zero-field normal-energy thermionic emission distributions. The total-energy distribution is also derived for the zero-field thermionic case and this is found to be the mirror image of the normal-energy zero-temperature field emission distribution. The new distribution applies to problems involving the total energy of electrons before and after emission.
Russell D. Young (1959) studied this question.