328

LETTERS TO THE EDITOR

B" seems justiGable to assume that the transition to the
ground state is allowed in the usual sense.

The bound

next states

quoefstBion" inisthtehapt -coafptturraensiptirooncsesst. o

excited, The fact

tohfaBt "oinmlyplie1s3%thaotf

all absorptions high excitations

lead to bound states are favored. Appreci-

able formation of excited states would wash out the

orientation in the ground state because of the smearing

over magnetic quantum numbers that occurs in the

process of de-excitation by p-ray emission. Fortunately,

the situation here seems favorable. There are only four

' known excited states below the threshold for particle
emission. While no firm arguments can be made, what is

known of the spins and parities of these states makes

it seem
events

probable that the leading to holed

large states

majonty
of B"

of p,-capture actually go

directly to the ground state.

Another effect which must be considered is possible
B" depolarization of the nucleus due to hyperGne inter-

action with the atomic electrons. Rough estimates

indicate that the atom is probably ionized due to
recoil at the instant of absorption of the p, meson. If the

atom is always ionized and then re-forms again after

it stops, we can calculate the depolarization under the

assumption that the Gne-structure substates are popu-
lated statistically. This gives, for the resultant B"

polarization,

(J)=-;(0.54)(n) =0.36(n).

(2)

3" Thus, if ~(a)~ equals 15%, the final polarization

~(J) ~

of the is probably closer to 5% than to the value of

10% given above.

There is an additional depolarization due to the
environment in which the B"atom 6nds itself. But the

relaxation time for this effect in graphite is presumably
B" longer than the mean life of since metals show

relaxation times of the order of tens of milliseconds.

In any event, such solid-state effects can be essentially

eliminated by a suitable choice of organic material as

target.

*This work was supported, in part, by the Ofhce of Naval Research and the U. S. Atomic Energy Commission.
t Visiting Guggenheim Fellow, on leave of absence from McGill
University, Montreal, Canada.
~T. D. Lee and R. P. Feynman, Proceedings of the Seventh
Annual Rochester Conference on High-Energy Euclear Physics,
'April, 1957 (to be published).
R. L, Garwin, L. Lederman, and co-workers at Columbia have
observed the longitudinal polarization of the electrons in y decay
(L. Lederman, in reference 1).On the basis of a theory of p, decay
the direction of the y meson s polarization can then be inferred. 3 Wu, Ambler, Hayward, Hoppes, and Hudson, Phys. Rev. 105,
1413 (1957). 4 Garwin, Lederman, and Weinrich, Phys. Rev. 105, 1415
(1957). ~ T. N. K. Godfrey, Princeton University thesis, 1954
J J' {u'npubalnisdhed). are the anal and initial nuclear spins respectively,
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
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).

'' T. D. Lee and C. ¹

Yang, Phys. Rev. 104, 254 (1956).

Since the larger fraction of p, mesons bound in carbon decay

before nuclear capture, the directional asymmetry of the prompt

electrons can be used to measure the magnitude of (e) directly,

while the asymmetry of the delayed electrons will determine (J).

9 F. A. Ajzenberg and T. Lauritsen, Revs. Modern Phys. 27,

77 (1955).

Magnetic Dipole Moment of the Electron
CHARLES M. SO~RPIELD*
IIaruard University, Cambridge, Massachusetts (Received May 6, 1957)

~HE fourth-order radiative corrections to the

magnetic dipole calculated by Karplus

moment and Kroll

of in

1th9e49e.'leTchtreoinr

were result

is contained in the complete expression for the moment,

y, ,/pp = 1+(n/2w) —2.973(n /pr ) = 1.0011454, (1)

where po is the Bohr magneton.

The stance

calculation has been using the mass-operator

redone in formalism

the of

Spcrhesweinntgeri.n-'

We consider a single electron moving in a constant

(in space and time) electromagnetic field. The expectation value of the mass operator in the lowest state

represents the self or proper energy of the electron. The magnetic moment is identified from that part of the

self-energy which is linear in the external Geld.
The electron Green's function 6, the photon Green's
function 8, and the interaction operator I', which
appear in the symbolic expression for the mass operator,

+ M =m, te' TryGI'g,

are computed in the presence of (as functions of) the

external held. To do this it is sufficient to replace the

— electron's momentum
by the combination

operator, p,
II=p eA,

where it
provided

occurs, by that full

account is taken of the commutation properties of II.

Units are such that A=c=i. Renormalized quantities

are used throughout the perturbation calculation.

The fourth-order contribution to the moment is

found to be

— + li. '4'

n' (197

~' +-',t'(3) —-'m, '

q
ln2

= —0.328n—' ,

(2)

dup m' (144 12

~

where l'(3) is the Riemann zeta function of 3. Thus
p,/pp = 1.0011596.

The discrepancy between (1) and (2) has been traced to the term iir+ii'r' of Karplus and Kroll. In
other words, terms li"' and iirr'+ii"" appear unchanged
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
in at least two different ways.

LETTERS TO THE E D I TOR

329

The theoretical magnetic moment may be compared

with the experimental moment; it is also used in

determining the 6ne-structure constant 0. , and it contributes to the Lamb shift. The magnetic moment is

measured by determining p,/p~ and p„/p&, where p~ is

the proton been quite

moment. accurate.

'

The On

measurements the other hand,

of p,/p~ have
there are two

confhcting experimental determinations" of p„/p, s,

which result in two diferent values for the magnetic

moment:

References
3and4
3 and 5

It e/Po
1.001146+0.000012 1.001165~0.000011.

The theoretical value' for the hyper6ne splitting in hydrogen is proportional to the quantity

=(—0.53+0.37)os/ss, which is consistent with the

value presented above.

'*RN.atKioanrapllusScaienndceNF. oMun.dKatrioolnl,

Predoctoral Phys. Rev.

Fellow.
77, 536 (1950).

2
'

J. Schwinger, Proc. Natl. Acad. Sci.
Koenig, Prodell, and Kusch, Phys.

37, 452, 455 (1951). Rev. 88, 191 (1952);

R.

Be'riJn.gHer.

and M.
Gardner

A. Heald, Phys R.ev. 95,
and E. M. Purcell, Phys.

1474 Rev.

(1954). 76, 1262

(1949);

J.

'''HAP. ..GFCarr.adnZniememrn,acPha,hnydPs.hSyR.sL.eviRe.be8evs3.,,
Cohen, DuMond, Layton,

996 Jr., 104,
and

(1951). Phys. Rev. 104, 1197 1771 (1956).
Rollet, Revs. Modern

(1956). Phys.

27, 363 (1955}.

"s
'

E. T.

E. Salpeter,
Fulton and

"W. Aron and

Phys. Rev. 89, 92 (1953). P. C. Martin, Phys. Rev. 95, 811 (1954).
A. J. Zuchelli, Phys. Rev. 105, 1681 (1957).

Triebwasser, Dayhoff, and Lamb, Phys. Rev. 89, 98 (1953).

""NHo.vSicuku,ra

Lipworth, and Yergin,
and E. H. Wichmann,

Phys. Rev. 100, 1153 (1955). Phys. Rev. 105, 1930 (1957};

A."PAete.rPmeatenrnm, anPnhys(.priRveavte.

105, 1931 (1957). communication) (to

be

published).

present value of e,v to determine a new value. This turns out to be
1/n = 137.039.

The theoretical Lamb shifts in hydrogen, deuterium,

and singly ionized helium are aRected by the changes

in both n and p, Incorporating these changes into the

' calculations of Salpeter,
recoil corrections of Fulton

along with and Martin,

t'heandprtohteonp-rreoctoonil-

" structure corrections of Aron and Zuchelli, we obtain

the following results in Mc/sec:

Theoretical

Experimental

Reference

SH

1057.99+0.13

1057.77m 0.10

11

SDS—DSH

1059.23~0.13 1.24&0.04

1059.00~ 0.10 11 1.23~ 0.15 11

" SHe

14055.9 &2.1

The experimental values"

14043 w13.0

12

have been listed for

comparison. There remain several uncomputed theoreti-

cal e8ects which are expected to be of the same order

of magnitude as the indicated theoretical uncertainties.

" The magnetic moment of the p, meson, as computed
by Suura and Wichmann, and Petermann, would be

changed to read

— —. — o.

n'q eh

p„= i

1+ 2~

+0.75

x')

i

2ns„c

I J. would like to thank Professor Schwinger, Pro-
fessor P. C. Martin, Professor E. M. Purcell, and
Professor R. J. Glauber, and Dr. K. A. Johnson for
their helpful comments and discussion related to this
— 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

P. F. ZWEIPEL
Knolls Atomic Power Laboratory, * Schenectady,
(Received April 18, 1957)

Sew Fork

' 'N a previous paper, ' tables of allowed X capture-

- - positron branching ratios were presented. However,

it was pointed. out by Wapstra' and Perlman' that

numerical errors existed in the table. These errors
appear in the first, third, and fifth columns of Table II

of reference 1, each entry of which should be multiplied.

by the factors of 0.5018, 1.2244, and 0.6462, respec-

I tively. In Table of this communication, the corrected

E table of allowed

to positron branching ratios is

given. In this work, the eftect of the 6nite nuclear size

' on the bound electron wave functions, which was
ignored in reference 1, was taken into account. This

eGect, which is negligible for low Z, reduces the branch-
ing ratio by about 10% for Z=84 and by about 15% for Z=92. EBects of finite size on the positron wave

' functions was ignored, since it is a considerably smaller
effect.

As in reference 1, the bound electron wave functions
were taken from Reitz's thesis' except for Z=16, for

E TABLE I. Allowed to positron branching ratios.

W0/mC2+g
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

29

46.6

707

1.208X104 4.56X105 8.92X105

8.65

112

1.58X108 4.50X104 8.41X104

2.83

38.6

425

1.08X104 1.84X104

1.24

18.9

164

8.57X108 5.01X108

0.641

6.91

77.6

1.57X108 2.67X108

0.378

8.91

42.8

807

1.86X108

0.190

1.60

16.4

289

479

0.0613

0.597

5.86

96.4

158

0.0169

0.160

1.51

28.6

89.0

7.00X10 8 0.0648

0.608

9.10

15.7

8.56X10 8 0.0828

0.802

4.82

8.05

2.06X10 8 0.0188

0.178

2.82

4.75

1.30X10 8 0.0118

0.109

1.80

8.06

8.85X10 4 V.93X10 8 0.0729

1.28

2.10

6.29X10 4 5.60X10 8 0.0513

0.879

1.52

4.48X10 4 4.09X10 8 0.0377

0.652

1.18

8.37X10 4 3.07X10 8 0.0281

0.498

0.869

2.60X10 4 2.37X10 8 0.0219

0.893

0.685