L E T T E R S TO THE E D I T O R

1197

yield from about 20 g/cm2 of toluene is 0.0030± 0.0005 antineutrons per antiproton with the lead glass, and 0.0028±0.0005 with the liquid scintillator. With the assumption that the interaction cross section for antineutrons is the same as for antiprotons, the inefficiency of the detector due to attenuation in Sh S2, and the lead converter, and to transmission of the detector can be calculated, and is found to be about 50%. From the observed antineutron yield the mean free path for charge exchange of the type detected is about 2300 g/cm2 of toluene (C7H8); or, in other words, the exchange cross section is about 2% of the annihilation cross section for this material. This corresponds to a cross section of approximately 8 millibarns in carbon for this process.
The generous support of many groups, including the Bevatron operating group under Dr. Edward J. Lofgren, is greatly appreciated.
We thank Professor David Frisch of Massachusetts Institute of Technology for the loan of the lead glass used in the Cerenkov counter.
* This work was done under the auspices of the U. S. Atomic Energy Commission.
f On leave of absence from Brookhaven National Laboratory, Upton, New York.
1 Chamberlain, Segre, Wiegand, and Ypsilantis, Phys. Rev. 100, 947(1955).
2 Brabant, Cork, Horwitz, Moyer, Murray, Wallace, and Wenzel, Phys. Rev. 101, 498 (1956).
3 Chamberlain, Chupp, Ekspong, Goldhaber, Goldhaber, Lofgren, Segre, Wiegand, Amaldi, Baroni, Castagnoli, Franzinetti, and Manfredini, Phys. Rev. 102, 921 (1956).
Magnetic Moment of the Proton in Bohr Magnetons*
PETER FRANKENJ AND SIDNEY LIEBES, JR.
Department of Physics, Stanford University, Stanford, California (Received September 18, 1956)

in a fashion that does not require a quantitative knowledge of the electrostatic field distribution.
We make three assumptions which are subjected to experimental verification: (1) The electron orbit radii are small compared to distances in which the electrostatic field varies appreciably. (2) The frequency shift caused by the electrostatic field is small. (3) The electrostatic field is independent of magnetic field in a chosen range of magnetic field variation.

When assumptions (1) and (2) obtain, it follows that

the fractional shift of the observed cyclotron frequency

co/, relative to the unshifted frequency coe, is, in Gaussian

units,

cEr r

/dEz\ i mo2

47Tp-( — )

,

(1)

C0e

v±H

V dz / o\J22eH~2"

wherein Er= (l/2w)Jl2rErd6 represents the average radial electric field at the orbit, vx the magnitude of the component of the electron velocity perpendicular
to the direction of the magnetic field, and p the space
charge density. The electric field derivative is evaluated

at the orbit center and the direction of H is chosen parallel to the z-axis. We take e>0.

The experimentally observed quantity <ae'/o)p can be related to /xo/VP(oii) by the expression

Mir)] we

Mo
1 +

Wp Mp(oil) I I

(2)

The absence of any orbit or velocity parameters in (2) suggests the measurement of coe'/cop as a function of magnetic field. If assumption (3) is satisfied in the range of variation of H, one should observe a linear dependence of coZ/cop with respect to 1/H2. A linear extrapolation to (1/H2) = 0 determines /xo/Mp(oii).

THIS note is a brief report on a recent measurement of the magnetic moment of the proton in units of

the Bohr magneton. The nuclear magnetic resonance

frequency, a)p=2ixP(oii)H/h, of protons in mineral oil and the cyclotron frequency, o)e=eH/mc, of free lowenergy electrons are measured in the same magnetic

field H. The ratio of these two frequencies yields the

proton moment in Bohr magnetons, o)p/coe= Mp(oii)/Mo, uncorrected for environmental shifts due to the mineral oil.

In the previously reported measurements of the proton moment by the electron-cyclotron method,1-2

one of the important limitations on accuracy was

imposed by shifts in the electron-cyclotron frequency

I /hT ( KILOGAUSS"* )

arising from the presence of nonvanishing radial electrostatic fields. These fields can be produced by space charge, externally applied trapping voltages, or stray charges accumulating on the boundaries of the system in which the resonance is studied. The present experiment is specifically designed to correct for these shifts

FIG. 1. aie'/op, for a spherical sample of mineral oil, is plotted versus 1/H2 for each run. The straight lines represent least-squares fits to the data of each run. The vertical dashed lines indicate the interval of magnetic field variation in which the data were taken. The heavy horizontal arrow at 657.476 indicates the terminal value of oe'/oop for zero-energy electrons that should have been obtained in order to yield the theoretical value for the magnetic moment of the free electron.

1198

L E T T E R S TO THE EDITOR

We have studied caj/cap as a function of 1/H2 for magnetic fields ranging from 750 to 1700 gauss. For each run, from three to thirteen points were taken in this interval. Deviations from a least-squares straight line fit to the data of each run were less than three parts per million for over half of all points taken. These deviations were primarily due to random errors in tuning the electron microwave cavity at the individual points. We have found, by analysis of all data without rejection, that any systematic deviations from a straight line are less than one part in one million.
The free electrons were produced by photoelectric emission from a film of a few molecular layers of potassium deposited upon the inner surface of a highly evacuated spherical bulb of Pyrex ~\ cm in diameter. The resonance was observed by the use of typical microwave techniques. The electron line widths in these measurements varied from one part in 2000 to one part in 35 000. The electrostatic fields, and hence the frequency shifts and line widths, were a function of the intensity and distribution of the light over the surface of the bulb; the lighting conditions were varied from run to run.
Figure 1 summarizes all the data taken. Several different electron bulbs, light sources, and cavities were used. The lines represent least-squares fits to the data of each run.
The average of extrapolated intercepts for all these runs, without rejection of any data, is coe/cop= 657.462 ±0.006. The limit of error includes 9 5 % of the runs, and is believed to represent a maximum error.3
A relativistic correction necessitated by the finite velocities of the electrons is taken to be 0.001 ± 0 . 0 0 1 , where the error is again to be regarded as a maximum. Addition of this correction yields

Mo/jMoii)= 657.463-b0.007

(3)

for a spherical sample of mineral oil, where no magnetic corrections have been applied. This result is to be compared with that of Gardner and Purcell :*

Mo/Won) = 657.475±0.008.

(4)

Applying a diamagnetic correction factor4-6 of (2.94 ± 0 . 1 0 ) X 1 0 - 5 to the field at the proton, we obtain for
the final corrected value of the magnetic moment of
the free proton in units of the Bohr magneton:

fip/fio= (657.444i0.007)-1

= (1.521042±0.000016)X10-3.

(5)

The present result (3), uncorrected for the spherical sample of mineral oil, when combined with the data7-9
available for the magnetic moment of the free electron,

/vWoii) = 658.2293±0.0010,

(6)

also referred to a spherical sample of mineral oil, yields for the magnetic moment of the free electron in Bohr

magnetons:

l*e/no= 1.001165±0.000011

= 1 + (<V2TT)+ (0.7±2.0) (a2A2).

(7)

This is to be compared with the current theoretical estimate10,11:

/vVo= 1.0011454

- 1 + (a/2w) - 2.973 (a2/w2).

(8)

A detailed report on this experiment is in preparation.
* This research was supported by Research Corporation and the Office of Naval Research.
f Present address: Department of Physics, University of Michigan, Ann Arbor, Michigan.
I J. H. Gardner and E. M. Purcell, Phys. Rev. 76, 1262 (1949); J. H. Gardner, Phys. Rev. 83, 996 (1951).
2 R . W. Nelson, Ph.D. thesis, Harvard University, 1953 (unpublished).
3 A discussion of errors will be presented in a detailed report on this experiment now in preparation.
4 N. F. Ramsey, Phys. Rev. 78, 699 (1950). 5 H. A. Thomas, Phys. Rev. 80, 901 (1950). 6 H . S. Gutowsky and R. E. McClure, Phys. Rev. 81, 276 (1951). 7 An average of the values given in the papers of reference 8 and reference 9 is used here. 8 Koenig, Prodell, and Kusch, Phys. Rev. 88, 191 (1952). 9 R. Beringer and M. A. Heald, Phys. Rev. 95, 1474 (1954). 10 J. Schwinger, Phys. Rev. 73, 416 (1948). II R. Karplus and N. Kroll, Phys. Rev. 77, 536 (1950).

Angular Distribution of Nuclear Reaction Products
G. R. SATCHLER*
Clarendon Laboratory, Oxford, England (Received September 21, 1956)
IN an earlier paper,1 the author applied the continuum theory of nuclear reactions2,3 to predict the angular distribution of y rays following inelastic neutron scattering. Unfortunately the formulas presented there contain an error and a numerical misprint. We wish now to give the corrected formulas, and to generalize them within the ^-matrix formalism to include interference between two or more compound nucleus levels. Applications of the incorrect formula to two recent experiments have been published4,5; we find that correction of the errors leads to considerably better agreement between experiment and theory.
Let a target nucleus of spin Jo capture particles with total angular momentum j \ to form a compound nucleus with spin Ji. This re-emits particles with total angular momentum j 2 , leaving an excited nucleus with spin 72. Consider now the angular distribution (relative to the incident beam) of radiation with total angular momentum y3, from the decay of J2 to the final nucleus Js. We denote the corresponding orbital angular momentum for the particles by /. When the "particles" are photons, j is the multipole order, and (—)l must be regarded as