OCTOBER 27, 1928] NATURE 649 The Magnetic Moment of the Electron. Rayleigh's address to the Royal Society in 1907: ''. In looking into the more recen~ progress of Geomet1;'1cal THE hypothesis of the spinning electron assigns to Optics, I have been astomshed to fin~ ho\,: htt~e the electron an angular momentum h/41r and a mag­ 1 correlation there has been. In this subJect 1t netic moment eh/41rmc. Dirac showed that the would appear that a man cannot succeed in making spectroscopic duplexity poi1_1ted o~t by Heise~b~rg even his own countrymen attend to him." Lord can be explained more sat1sfactor1ly by mod1fymg Rayleigh himself appears to have fared no bettor than the wave equation. In his theory . the angular his great predecessors. More than forty years ago he momentum is still h/41r. The magnetic moment µ, discussed, in terms of the wave theory, the accuracy however, is only approximately the Bohr magneton 2 necessary in focussing, and verified his theory by µ 0• Using Darwin's explicit expressions for the experiment. Nevertheless, so far as I can recall, ever_y four y;' s, the magnetic moment of the electron in th_e book on optics or photography that deal~ with this :field of a charge Ze is easily calculated. The result IS question bases the discussion on the geometrical theory. 2 µ = µ 0 (1 + 2,v'l - a Z2)/3, The tables for finding depth of field published each year in the British Journal Photographic Almanac, and where a is the fine structure constant 7·3 x 10-3 • a recent contribution to NATURE, rest on the same Substituting Z = 92 for uranium, we obtainµ =_0·83µ 0• assumptions. Since in the nucleus there may be very mtense Unfortunately, the geometrical method has a habit lields, the conditions in it are approximated by using of returning the wrong answer to questions affecting large values of Z. The highest value that may be the use of optical instruments. It leads us to expect used is 137, because higher values make the solution an improvement in the definition given by a lens when inapplicable. For this Z the momen~ decrease~ to the aperture is diminished, but the wave theory leads (1/3)µ0 • Performing the same calculat10n for excited to the opposite conclusion. It declares that the depth states with radial quantum number= 0, we find, of field should vary inversely as the first power of the using Darwin's notation and his first type of solution, diameter of the aperture, but the wave theory sub­ 2 µ= (1 + 2y'(k + 1) - a 2Z 2 )(2k + 3J- 1µH, stitutes the second power. These two examples, out of many that might be cited, show the importance of where µH is the ordinary magnetic moment of the considering optical questions in accordance with the state in question calculated neglecting relativity. concepts of the wave theory. The geometrical The theoretically possible minimum of the above method, if employed at all, should only be used to find expression is /2k + 3)-1µH. Thus in inten_se fields such relations between loci where perfect ray convergence as may exist 1n the nucleus, the magnetic moment of may be assumed. \1/hen tolerances are discusse~ ?Y the electron may be less than a Bohr magneton. the wave method, it turns out that the quantities Dr. R. S. Mulliken pointed out to me that in several involved are invariants as found by this restricted instances isotopes supposedly containing odd and geometrical method. The reason for this corre­ even numbers of nuclear electrons have practically spondence becomes clear when the wave theory is identical spectra. If the magnetic moment of the employed throughout ; it will be sufficient here to give electron were always µ one of the two isotopes 0 examples. would have a resultant nuclear magnetic moment If y is a small length perpendicular to the axis, which would modify its spectrum very noticeably. meeting it in the same point as a ray inclined at an Therefore he concluded that for the nuclear electrons angle 0 with the axis, µy sin 0 is invariant on refraction µ may be much smalle~ than µ0 • Dir3:c's th~ory is if this elementary length is imaged without aberration. qualitatively at least m agreement with this coi:-­ The parallel theorem states that two points in the same clusion of Mulliken. It shows that the electron spm transverse plane will not appear distinct in the image can be modified by the presence of intense electric if their separation y is less than the value which fields, and that in the cases mentioned above the satisfies µy sin 0 = KA, where K is a constant and 0 is maanetic moment is smaller than a Bohr magneton. the inclination to the axis of the extreme ray trans­ 3 4 Barnett's and Emil Beck's measurements of the mitted by the lens. If x represents a small length gyromagnetic ratio gave in contradiction to those of measured along the axis of the lens, the invariance of 5 Chattock, Sucksmith, and Bates a value which µx( 1 - cos 0) corresponds to a range of focus :r, deter­ indicates a somewhat smaller µ than would correspond mined by the condition µx(l - cos 0)=K' "/\, within to a Bohr magneton. There may be other reasons which the minimum standard of definition depends on for this than the dependence of µ on the type and the constant K'. For sensibly perfect imagery K' strength of the field. This may be, however, one of should not exceed 1/4. In most photographic work the reasons. A central field of the order of about 50e values as high as 4, or even 8, are generally accepted as corresponds to the observed deviation. the equivalent of good definition. Except with high G. BREIT. power microscope lenses, sin 0 and 1 - cos 0 may be Department of Terrestrial Magnetism, replaced by a/2u and a 2 /8u2 resrectively, where a is the Carnegie Institution of Washington. diameter of the aperture and u 1s the distance from the (Temporarily in Zurich.) lens of the point from which the elementary displace­ ments x and y are made. A third law of geometrical optics, which is in effect a The Depth of Field and Resolving Power of combination of the two just mentioned, states that the Optical Instruments. longitudinal magnification is proportional to the THE belief of some physicists that the significance of square of the transverse magnification ; i~ symbols, Airy's work of a hundred year~ ago on the characte! of x/µy 2 is invariant. The related theorem 1s that the optical images is rarely appreciated by users of optical depth of field associated with the distinct rendering of 2 instruments, will be strengthened by a recent letter points distant y apart is x, where "/\x=Kµy , K being a in NATURE. It calls to mind a statement in Lord constant depending on the quality of definition con­ sidered satisfactory. Evidently K = 2K'/K 2 , and 1 P. A. M. Dirac, Proc. Roy. Soc., A, vol. 117, p. 610; vol. 118, sensibly perfect definition, with the central diffraction p. ;3~~- G. Darwin, idem, vol. 119. discs of distinctly rendered points in contact, is • s. J. Barnett, Proc. Amer. Acad., vol. 60, p. 127. attained with K = 1/3. • Emil Beck, Ann. der Phys., vol. 60, p. 109. • Sucksmith and Bates, Proc. Roy. Soc., vol. 104, p. 499; vol. 108, In all these expressions "I\ is the wave-length of the p. 638. light forming the image, measured in the medium for No. 3078, VoL. 122] R2 © 1928 Nature Publishing Group.