LETTERS TO THE EDITOR eo l' $ I I I t I t t } 05 I.O FRACTION OF RANGE Fia. 2. Feather plot for Ca&, 12,000 counts per minute, and the contribution due to gamma-rays and other unabsorbed contaminants was less than one part in 3000 with the strongest source, thus indicating the absence of any appreciable amount of gamma-radiation. The absorption curve obtained with the strongest source is shown in Fig. 1. The Feather plot, shown in Fig. 2, gives a range of 64+1 mg/cm'. Glendenin' has shown that a reliable range-energy curve for the low energy region can be derived from the data of Marshall and %'ard' for monoenergetic electrons and beta- ray spectrograph data on low energy beta-emitters. Glen- denin's curve is identical with that of Marshall and Nard below 0.5 Mev. Using this range-energy curve, we have found that the Ca~ beta-radiation has a maximum energy of 260+5 kev. %'e have found no evidence of any harder — beta-radiation, course of this or of any investigation. g'amma-radiation at all in the Ackeomledgesents. This work has been supported with funds from the Office of Naval Research. The authors wish to express their appreciation to Miss Jacqueline Becker for her assistance in making the counts. ~%'alke, Thompson, and Holt, Phys. Rev. 5V, 171 (1940). ~ Solomon. Gould, and Anlnsen, Phys. Rev. 2'2, 1097 (1947). ~ Feather, Proc. Camb. PhiL Soc. 35, 599 (1938). ' Glendenin, Nucleonics, in press for January, 1948. J. ~ Marshall and Ward, Can. Research 15, 29 (i939). ~ This result Redioiwtopes, CisekianbggooaIdsdagIr'rekeemeJnitsfwNitoh. a value of Z, revised 250 kev. given in September, 1947, distributed, by Isotopes Branch, United States Atomic Energy Commis- sion. Uafoftunately, the Atomic Energy Commission's result is not supported by any published experimental evidence. On Quan&~m-Electredyne~Ics and the Mgnetlc Moment of the Electron Jm.net ScawtNGER Heroerd Ueisersigy, Cnmbridgc, Massachusetts December 30, 1947 TTEMPTS to evaluate radiative corrections to elec- tron phenomena have heretofore been beset by divergence difficulties, attributable to self-energy and vacuum polarization effects. Electrodynamics unquestion- ably requires revision at ultra-relativistic energies, but is presumably accurate at moderate relativistic energies. It would be desirable, therefore, to isolate those aspects of the current theory that essentially involve high energies, and are subject to modification by a more satisfactory theory, from aspects that involve only moderate energies and are thus relatively trustworthy. This goal has been achieved by transforming the Hamiltonian of current hole theory electr&ynamics to exhibit explicitly the logarithmically divergent self-energy oF a free electron, which arises from the virtual emission and absorption of light quanta. fhe electromagnetic self-energy of a free electron can be ascribed to an electromagnetic mass, which must be added to the mechanical mass of the electron. Indeed, the only meaningful statements of the theory involve this combina- tion of masses, which is the experimental mass of a free electron. It might appear, from this point of view, that the divergence of the electromagnetic mass is unobjection- able, since the individual contributions to the experimental mass are unobservable. However, the transformation of the Hamiltonian is based on the assumption of a weak inter- action between matter and radiation, which requires that the electromagnetic mass be a small correction ( (8/Ac)mo) to the mechanical mass mo. The new Hamiltonian is superior to the original one in essentially three ways: it involves the experimental elec- tron mass, rather than the unobservable mechanical mass; an electron now interacts with the radiation 6eld only in the presence of an external field, that is, only an accelerated electron can emit or absorb a light quantum;~ the inter- action energy of an electron with ah external field is now subject to a fixate radiative correction. In connection with the last point, it is important to note that the inclusion of the electromagnetic mass with the mechanical mass does not avoid all divergences; the polarization of' the vacuum produces a logarithmically divergent term proportiona1 to the interaction energy of the electron in an external field. However, it has long been recognized that such a term is equivalent to altering the value of the electron charge by a constant factor, only the final value being properly identi- fied with the experimental charge. Thus the interaction between matter and radiation produces a renormalization of the electron charge and mass, all divergences being contained in the renormalization factors. The simplest example of a radiative correction is that for the energy of an electron in an external magnetic field. The detailed application of the theory shows that the radiative correction to the magnetic interaction energy corresponds to an additional magnetic moment associated with the electron spin, of magnitude bp/p, =($x)H/hc =0.001162, It is indeed gratifying that recently acquired expen'mental data con6rm this prediction. Measurements on the hyperfine splitting of the ground states of atomic hydrogen and deuterium' have yielded values that are definitely larger than those to be expected from the directly measured nuclear moments and an electron moment of one Bohr magneton. These by a smaB additional discrepancies electron spin camnagbneetaicccoumnotemdentf.or' Recalling that the nuclear moments have been calibrated in terms of the electron moment, we find the additional moment necessary to account for the measured hydrogen and deuterium hyper6ne structures to be bp/p, =0.00126 %0.00019and Spa/p, ~ 0.00131&00. 0025, respectively. These values are not in disagreement with the theoretical predic- tion. More precise conformation is provided by measure- ' ment of the g values for the sSy, sP~, and sPIg~ states of' sodium and gallium. To account for these results, it is necessary to ascribe the following additional spin magnetic moment to the electron, 8g/p, =0.00118~0.00003. LETTERS TO THE EDITOR The radiative correction to the energy of an electron in a Coulomb 6eld wiB produce a shift in the energy levels of hydrogen-like atoms, and modify the scattering of elec- trons in a Coulomb field. Such energy level displacements ' have recently been observed in the 6ne structures of hydro- gen, deuterium, and ionized helium. s The values yielded by our theory differ only slightly from those conjectured by Bethe' on the basis of a non-relativistic calculation, and are, thus, in good accord with experiment. Finally, the 6nite radiative correction to the elastic s'cattering of electrons by a Coulomb 6eld provides a satisfactory termination to a subject that has been beset with much confusion. A paper dealing with the details of this theory and its applications is in course of preparation. +A classical non-relativistic theory of this type was discussed by H. A. Kramers at the Shelter Island Conference, held in June 1947 J. under the auspices t D. E. Nafe, E. Nagel, E. R. SBo..fJNutehlilesaonnN,,ataainondndalJI..AIR.ca.RdZaebamic,yhaPrhoiayfss,S, cRiPeehnvyc.es.s7.1R, e9v1.4 (1947); VZ, 971 (1947). s G. Breit, Phys. Rev. 71, 984 (1947).However, Sreit has not correctly drawn the consequences of his empiriml hypothesis. The e8ects of a nuclear magnetic field and a constant magnetic 6eld do not involve different combinations of p, and bla. ~ P. Kusch and H. M. Foley, Phys. Rev. 'N, 1256 (1947), and further unpublished work. 4 s JW..EE. .MLaacmkb,anJdr.Na.ndAuRst.erCn,. Retherford, Phys. Rev. Phys. Rev. V2, 241 VZ, 972 (1947). (1947). & H. A. Bethe, Phys, Rev. 72, 339 (1947). Excitation Curves of (e, n); (I, 2n); (e, 3n) Reactions on Silver 8. N. Gmosm. r. Dcpertnccnt of Physks, Veiscrsity of California, Bcrkcky, California January 5, 1948 ILVER bombarded with e-particles from the 60-in. cyclotron produces radioactive substances with the following three half-lifes: 65 min. , 5.2 hr. , and 2.7 d. All of these activities have been chemically attributed to indium and have been assigned by mass-spectrograph separation to In"' In'o', and In», respectively. Tendam and Bradt' recently announced similar activities. Their assignment of 65-min. and 2./-d activities agrees with ours. The 23-min. activity found by them was not looked for in the present experiment. The excitation curves for the isotopes reported above have been determined for a-energies up to 37 Mev and are reproduced in Fig. 1. The abscissae give the energy in Mev, the ordinates the cross sections in arbitrary units. The ordinate units are, ho~ever, the same for reactions leading to the formation of the same isotope. Evaluation of absolute cross sections has not yet been possible due to lack of knowledge regarding the efficiencies of the diferent radiations for the ionization chamber used. From the 6gure it is seen that the 65-min. activity belonging to In»s (emitting positrons of 1.7 Mev), a product of Ag'"{ot, e} reaction, has a threshold of 11 Mev. ~ The yield after attaining a peak at 17.5 Mev drops rapidly to low values when the (a, 2n) process appears as a competing process. After attaining a minimum, the 65-min. activity again increases and does not reach saturation even at 37 Mev. Apparently this part of the curve is due to Ag"'(a, 3e}In»s. The sharpness of the peak at 1i'.5 Mev is also interesting. The di8erence of 4 Mev between (e, n) and (a, 2e) thresholds is much smaller than that between (a, 2e) and {a,3e) thresholds (~8 Mev}. This difference seems to be due to the Coulomb barrier which cuts off the production of any alpha-reaction below 11 Mev. The 2.7-d activity belonging to In"' has a threshoM of about 15 Mev, which is in agreement with that found by Tendam and Bradt. ' This activity is produced by the Agi (a, 2n) process, and emits a p-ray of about 0.2 Mev (no positrons). After attaining a peak around 27 Mev, the yield begins to drop and reaches about 16 percent of maximum at 37 Mev. The 5.2-hr. period is produced by Ag"'{a, 2e)In'0' reac- tion. The excitation cur ve is similar to the excitation curve of In"', as is expected, since both are products of (n, 2n) reactions. The threshold of In'so is about 13.5 Mev, slightly lower than that of In'". At higher energies, how- ever, the two curves dier widely. Instead of decreasing, the 5.2-hr. curve goes on increasing even beyond 30 Mev, after which it drops slightly, the yield at 37 Mev being 80 percent of the maximum. This suggests the production of a different isotope at higher e-energies having a very similar half-life. A comparison with the Ag'os{a, 3n)In»s curve and with the Agis {a,2g}In'» curve suggests that this new activity is probably due to Ag's7{a, 3n}In«'. The possibility of its being due to Ag'oo(a, 3n}In»o (an isomer of 65-min. period} is ruled out by the fact that the threshold and low energy part of the curve is similar to the other (a, 2n} curve and not to the (a, e) curve. To verify this conclusion, two foils were bombarded, one with 37-Mev alphas (foil 1) and the other with 20-Mev alphas (foil 2). The latter is not likely to have any In'Os in it, while the former should mostly contain In'" with little In' '. The absorption curves for the radiations from the two foils, corrected for In»', showed marked differences. Foil 1 showed a p-ray of about 0.65 Mev, whi1e foil 2 showed a y-ray of about 0.5 Mev. No positrons were detected. These conclusions were also corroborated by 0 ln so tn (+In ?) 0' I lO ZO 30 Mcv 10 PEG. 1. The abscissa represents energy of the bombarding a-particles in Mev. The ordinate represents cross section in arbitrary units. The curve with open circles represents the cross section for the formation of In~+. The one with crosses represents the cross section for the formation of In»o, while the curve with solid circles represents the cross section for the formation of In~os. and at the higher energies probably of Intos also.