610 lines
11 KiB
Plaintext
610 lines
11 KiB
Plaintext
328
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LETTERS TO THE EDITOR
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B" seems justiGable to assume that the transition to the
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ground state is allowed in the usual sense.
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The bound
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next states
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quoefstBion" inisthtehapt -coafptturraensiptirooncsesst. o
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excited, The fact
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tohfaBt "oinmlyplie1s3%thaotf
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all absorptions high excitations
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lead to bound states are favored. Appreci-
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able formation of excited states would wash out the
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orientation in the ground state because of the smearing
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over magnetic quantum numbers that occurs in the
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process of de-excitation by p-ray emission. Fortunately,
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the situation here seems favorable. There are only four
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' known excited states below the threshold for particle
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emission. While no firm arguments can be made, what is
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known of the spins and parities of these states makes
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it seem
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events
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probable that the leading to holed
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large states
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majonty
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of B"
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of p,-capture actually go
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directly to the ground state.
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Another effect which must be considered is possible
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B" depolarization of the nucleus due to hyperGne inter-
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action with the atomic electrons. Rough estimates
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indicate that the atom is probably ionized due to
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recoil at the instant of absorption of the p, meson. If the
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atom is always ionized and then re-forms again after
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it stops, we can calculate the depolarization under the
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assumption that the Gne-structure substates are popu-
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lated statistically. This gives, for the resultant B"
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polarization,
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(J)=-;(0.54)(n) =0.36(n).
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(2)
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3" Thus, if ~(a)~ equals 15%, the final polarization
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~(J) ~
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of the is probably closer to 5% than to the value of
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10% given above.
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There is an additional depolarization due to the
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environment in which the B"atom 6nds itself. But the
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relaxation time for this effect in graphite is presumably
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B" longer than the mean life of since metals show
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relaxation times of the order of tens of milliseconds.
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In any event, such solid-state effects can be essentially
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eliminated by a suitable choice of organic material as
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target.
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*This work was supported, in part, by the Ofhce of Naval Research and the U. S. Atomic Energy Commission.
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t Visiting Guggenheim Fellow, on leave of absence from McGill
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University, Montreal, Canada.
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~T. D. Lee and R. P. Feynman, Proceedings of the Seventh
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Annual Rochester Conference on High-Energy Euclear Physics,
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'April, 1957 (to be published).
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R. L, Garwin, L. Lederman, and co-workers at Columbia have
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observed the longitudinal polarization of the electrons in y decay
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(L. Lederman, in reference 1).On the basis of a theory of p, decay
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the direction of the y meson s polarization can then be inferred. 3 Wu, Ambler, Hayward, Hoppes, and Hudson, Phys. Rev. 105,
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1413 (1957). 4 Garwin, Lederman, and Weinrich, Phys. Rev. 105, 1415
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(1957). ~ T. N. K. Godfrey, Princeton University thesis, 1954
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J J' {u'npubalnisdhed). are the anal and initial nuclear spins respectively,
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while 'Ag g is a numerical factor de6ned in the appendix of Jackson, Treiman, and Wyld, Phys. Rev. 106, 517 (1957). For a transition with AJ=O, the polarization of the daughter nucleus is of the
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form of Eq. (1) with the factor )p.J replaced by E/(1+5) the coefficients X and b being given in the above reference (with E,=m, and the sign appropriate for electrons).
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'' T. D. Lee and C. ¹
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Yang, Phys. Rev. 104, 254 (1956).
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Since the larger fraction of p, mesons bound in carbon decay
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before nuclear capture, the directional asymmetry of the prompt
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electrons can be used to measure the magnitude of (e) directly,
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while the asymmetry of the delayed electrons will determine (J).
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9 F. A. Ajzenberg and T. Lauritsen, Revs. Modern Phys. 27,
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77 (1955).
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Magnetic Dipole Moment of the Electron
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CHARLES M. SO~RPIELD*
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IIaruard University, Cambridge, Massachusetts (Received May 6, 1957)
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~HE fourth-order radiative corrections to the
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magnetic dipole calculated by Karplus
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moment and Kroll
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of in
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1th9e49e.'leTchtreoinr
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were result
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is contained in the complete expression for the moment,
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y, ,/pp = 1+(n/2w) —2.973(n /pr ) = 1.0011454, (1)
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where po is the Bohr magneton.
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The stance
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calculation has been using the mass-operator
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redone in formalism
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the of
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Spcrhesweinntgeri.n-'
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We consider a single electron moving in a constant
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(in space and time) electromagnetic field. The expectation value of the mass operator in the lowest state
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represents the self or proper energy of the electron. The magnetic moment is identified from that part of the
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self-energy which is linear in the external Geld.
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The electron Green's function 6, the photon Green's
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function 8, and the interaction operator I', which
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appear in the symbolic expression for the mass operator,
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+ M =m, te' TryGI'g,
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are computed in the presence of (as functions of) the
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external held. To do this it is sufficient to replace the
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— electron's momentum
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by the combination
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operator, p,
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II=p eA,
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where it
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provided
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occurs, by that full
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account is taken of the commutation properties of II.
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Units are such that A=c=i. Renormalized quantities
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are used throughout the perturbation calculation.
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The fourth-order contribution to the moment is
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found to be
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— + li. '4'
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n' (197
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~' +-',t'(3) —-'m, '
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q
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ln2
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= —0.328n—' ,
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(2)
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dup m' (144 12
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~
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where l'(3) is the Riemann zeta function of 3. Thus
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p,/pp = 1.0011596.
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The discrepancy between (1) and (2) has been traced to the term iir+ii'r' of Karplus and Kroll. In
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other words, terms li"' and iirr'+ii"" appear unchanged
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in the new result. A further point-by-point comparison of the two answers is not readily accomplished because the grouping of the terms divers markedly in the two cases. The present calculation has been checked several times and all of the auxiliary integrals have been done
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in at least two different ways.
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LETTERS TO THE E D I TOR
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329
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The theoretical magnetic moment may be compared
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with the experimental moment; it is also used in
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determining the 6ne-structure constant 0. , and it contributes to the Lamb shift. The magnetic moment is
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measured by determining p,/p~ and p„/p&, where p~ is
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the proton been quite
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moment. accurate.
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'
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The On
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measurements the other hand,
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of p,/p~ have
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there are two
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confhcting experimental determinations" of p„/p, s,
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which result in two diferent values for the magnetic
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moment:
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References
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3and4
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3 and 5
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It e/Po
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1.001146+0.000012 1.001165~0.000011.
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The theoretical value' for the hyper6ne splitting in hydrogen is proportional to the quantity
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=(—0.53+0.37)os/ss, which is consistent with the
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value presented above.
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'*RN.atKioanrapllusScaienndceNF. oMun.dKatrioolnl,
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Predoctoral Phys. Rev.
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Fellow.
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77, 536 (1950).
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2
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'
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J. Schwinger, Proc. Natl. Acad. Sci.
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Koenig, Prodell, and Kusch, Phys.
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37, 452, 455 (1951). Rev. 88, 191 (1952);
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R.
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Be'riJn.gHer.
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and M.
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Gardner
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A. Heald, Phys R.ev. 95,
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and E. M. Purcell, Phys.
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1474 Rev.
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(1954). 76, 1262
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(1949);
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J.
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'''HAP. ..GFCarr.adnZniememrn,acPha,hnydPs.hSyR.sL.eviRe.be8evs3.,,
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Cohen, DuMond, Layton,
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996 Jr., 104,
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and
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(1951). Phys. Rev. 104, 1197 1771 (1956).
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Rollet, Revs. Modern
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(1956). Phys.
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27, 363 (1955}.
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"s
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'
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E. T.
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E. Salpeter,
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Fulton and
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"W. Aron and
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Phys. Rev. 89, 92 (1953). P. C. Martin, Phys. Rev. 95, 811 (1954).
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A. J. Zuchelli, Phys. Rev. 105, 1681 (1957).
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Triebwasser, Dayhoff, and Lamb, Phys. Rev. 89, 98 (1953).
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""NHo.vSicuku,ra
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Lipworth, and Yergin,
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and E. H. Wichmann,
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Phys. Rev. 100, 1153 (1955). Phys. Rev. 105, 1930 (1957};
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A."PAete.rPmeatenrnm, anPnhys(.priRveavte.
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105, 1931 (1957). communication) (to
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be
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published).
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present value of e,v to determine a new value. This turns out to be
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1/n = 137.039.
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The theoretical Lamb shifts in hydrogen, deuterium,
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and singly ionized helium are aRected by the changes
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in both n and p, Incorporating these changes into the
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' calculations of Salpeter,
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recoil corrections of Fulton
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along with and Martin,
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t'heandprtohteonp-rreoctoonil-
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" structure corrections of Aron and Zuchelli, we obtain
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the following results in Mc/sec:
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Theoretical
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Experimental
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Reference
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SH
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1057.99+0.13
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1057.77m 0.10
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11
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SDS—DSH
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1059.23~0.13 1.24&0.04
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1059.00~ 0.10 11 1.23~ 0.15 11
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" SHe
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14055.9 &2.1
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The experimental values"
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14043 w13.0
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12
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have been listed for
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comparison. There remain several uncomputed theoreti-
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cal e8ects which are expected to be of the same order
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of magnitude as the indicated theoretical uncertainties.
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" The magnetic moment of the p, meson, as computed
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by Suura and Wichmann, and Petermann, would be
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changed to read
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— —. — o.
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n'q eh
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p„= i
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1+ 2~
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+0.75
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x')
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i
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2ns„c
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I J. would like to thank Professor Schwinger, Pro-
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fessor P. C. Martin, Professor E. M. Purcell, and
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Professor R. J. Glauber, and Dr. K. A. Johnson for
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their helpful comments and discussion related to this
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— work. Note added its proof. Petermann" has placed upper and lower bounds on the separate terms of Karplus and Kroll. He 6nds that their value for pIIc does not lie within the appropriate bounds. Assuming the other terms to be correct, he concludes that p'/ps
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P. F. ZWEIPEL
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Knolls Atomic Power Laboratory, * Schenectady,
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(Received April 18, 1957)
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Sew Fork
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' 'N a previous paper, ' tables of allowed X capture-
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- - positron branching ratios were presented. However,
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it was pointed. out by Wapstra' and Perlman' that
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numerical errors existed in the table. These errors
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appear in the first, third, and fifth columns of Table II
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of reference 1, each entry of which should be multiplied.
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by the factors of 0.5018, 1.2244, and 0.6462, respec-
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I tively. In Table of this communication, the corrected
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E table of allowed
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to positron branching ratios is
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given. In this work, the eftect of the 6nite nuclear size
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' on the bound electron wave functions, which was
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ignored in reference 1, was taken into account. This
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eGect, which is negligible for low Z, reduces the branch-
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ing ratio by about 10% for Z=84 and by about 15% for Z=92. EBects of finite size on the positron wave
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' functions was ignored, since it is a considerably smaller
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effect.
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As in reference 1, the bound electron wave functions
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were taken from Reitz's thesis' except for Z=16, for
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E TABLE I. Allowed to positron branching ratios.
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W0/mC2+g
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1.28 1.44 1.60 1.76 1.92 2.08 2.40 2.88 8.84 4.80 5.76 6.72 7.68 8.64 9.60 10.56 11.52 12.48
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29
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46.6
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707
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1.208X104 4.56X105 8.92X105
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8.65
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112
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1.58X108 4.50X104 8.41X104
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2.83
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38.6
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425
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1.08X104 1.84X104
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1.24
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18.9
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164
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8.57X108 5.01X108
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0.641
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6.91
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77.6
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1.57X108 2.67X108
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0.378
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8.91
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42.8
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807
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1.86X108
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0.190
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1.60
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16.4
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289
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479
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0.0613
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0.597
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5.86
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96.4
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158
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0.0169
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0.160
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1.51
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28.6
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89.0
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7.00X10 8 0.0648
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0.608
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9.10
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15.7
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8.56X10 8 0.0828
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0.802
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4.82
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8.05
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2.06X10 8 0.0188
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0.178
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2.82
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4.75
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1.30X10 8 0.0118
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0.109
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1.80
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8.06
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8.85X10 4 V.93X10 8 0.0729
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1.28
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2.10
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6.29X10 4 5.60X10 8 0.0513
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0.879
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1.52
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4.48X10 4 4.09X10 8 0.0377
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0.652
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1.18
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8.37X10 4 3.07X10 8 0.0281
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0.498
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0.869
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2.60X10 4 2.37X10 8 0.0219
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0.893
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0.685
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