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J. Schwinger Phys. Rev. {\bf 52,} 1250 \hfill {\large \bf 1937}\\
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\begin{center}
{\bf LETTERS TO THE EDITOR}
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~~\\
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{\small \it ~~~~Prompt publication of brief reports of important discoveries in physics may be secured by addressing them to
this department. Closing dales for this department are, for the first issue of the month, the eighteenth of the
preceding month, for the second issue, the third of the month. Because of the late closing dates for the section
no proof can be shown to authors. The Board of Editors does not hold itself responsible for the opinions
expressed by the correspondents,
Communications should not in general exceed 600 words in length.}
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~~\\
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{\small \bf Communications should not in general exceed 600 words in length.}
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\begin{center}
Julian Schwinger\\
University of Wisconsin\\
Madison, Wisconsin\\
November 17, 1937
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\begin{center}
{\bf On the Spin of the Neutron}
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The intrinsic angular momenta of the proton and the
deuteron imply a neutron spin (in units of $\hbar$) of either
$\frac{1}{2}$ or $\frac{3}{2}$.
The usual assumption of $\frac{1}{2}$ for the neutron spin is based
entirely upon arguments of simplicity, since either of these
two possible values is consistent with data on nuclear spins.
In view of the importance of the neutron spin in nuclear
theory, it would be desirable to determine this quantity by
direct experiment. It has recently been
shown\footnote{J. Schwinger and E. Teller, Phys. Rev. {\bf 52},
286 (1937).} that
experiments on the scattering of neutrons by ortho- and
parahydrogen would enable one to obtain information about
the spin dependence and the range of the neutron-proton
interaction. It is the purpose of this note to point out that
such experiments also permit the determination of the
neutron spin.
A system composed of a proton and a neutron with $S_n$
units of spin may have a resultant spin angular momentum
of either $S_n + 1/2$ or $S_n - 1/2$. If the neutron spin
is $3/2$ the excited
state of the deuteron is a quintet state, as compared with a
singlet excited state for $S_n = 1/2$. In either case, the position of
the excited level is determined by the requirement that $\sigma_0$,
the cross section for the scattering of slow neutrons by free
protons, equal the experimental value of $14 \times 10^{-24}$ cm$^2$.
For both possible values of the neutron spin, we may write
\begin{equation}
\sigma_0 = 4 \pi \left( \frac{S_n + 1}{2 S_n + 1} a^2_{S_n + 1/2} +
\frac{S_n}{2 S_n + 1} a^2_{S_n - 1/2} \right),
\end{equation}
where the $a$'s denote the amplitudes of the waves scattered
in the states of corresponding resultant spin angular
momenta. The triplet state amplitude was determined by an
integral equation method from the interaction potential
$- Be^{-(r/b)^2,}$
with $B=36.8$ Mev and $b = 2.25 \times 10^{-13}$ cm. The resultant
value of $a_1 = - 5.73 \times 10^{-13}$ cm corresponds to a triplet
scattering cross section of $4.13 \times 10^{-24}$ cm$^2$.
The values of $a_0$
and $a_2$ for the respective spins of $S_n = 1/2$ and $S_n = 3/2$
may then be
calculated, with the results $a_0/a_1 = \pm 3.31$
and $a_2/a_1 = \pm 2.22$.
For both spins, a plus sign indicates a real excited state, and
a minus sign a virtual excited state.
The appropriate extension of Fermi's
theorem\footnote{E. Fermi, Ricerca
Scient. {\bf 7,} 13 (1936).} to the,
situation under consideration states that
\begin{equation}
- \frac{4 \pi \hbar^2}{M} \left( \frac{S_n + 1}{2 S_n + 1} a_{S_n + 1/2}
+ \frac{S_n}{2S_n + 1} a_{S_n - 1/2} \right.$$
$$
\left. + \frac{1}{2S_n + 1} (a_{S_n + 1/2}
- a_{S_n - 1/2}) \boldsymbol{\sigma}_p \cdot {\bf S}_n \right)
\delta({\bf r}_n -{\bf r_p}),
\end{equation}
where ${\bf S}_n$ and $\frac{1}{2} {\bf \sigma}_p$ are the spin
operators of the neutron and
the proton, is the effective neutron-proton interaction
to be inserted in the Born approximation formula. By
utilizing methods almost identical with those employed in
reference 1, we may calculate the cross sections for the
various transitions excited in molecular hydrogen by neutron
impact. The cross sections thus obtained for {\it para $\rightarrow$
para, para $\rightarrow$ ortho, ortho $\rightarrow$ para,} and
{\it ortho $\rightarrow$ ortho} transitions
are, respectively, proportional to
$$
\left(\frac{S_n + 1}{2 S_n + 1} a_{S_n + 1/2} + \frac{S_n}{2S_n + 1} a_{S_n
- 1/2} \right)^2,
$$
$$
3 \frac{S_n(S_n + 1)}{(2 S_n + 1)^2} (a_{S_n + 1/2} - a_{S_n - 1/2})^2,
$$
$$
\frac{S_n(S_n + 1)}{(2S_n + 1)^2}(a_{S_n + 1/2} - a_{S_n - 1/2})^2,
$$
and
$$
\left( \frac{S_n + 1}{2S_n + 1} a_{S_n + 1/2} + \frac{S_n}{2S_n + 1} a_{S_n
- 1/2} \right)^2 + \frac{2}{3} \frac{S_n (S_n + 1}{(2S_n + 1)^2} (a_{S_n
+ 1/2} - a_{S_n - 1/2})^2.
$$
The actual cross sections are these quantities multiplied by
functions of the neutron energy which involve only
properties of the hydrogen molecule.
The experiments of both Dunning\footnote{J.R. Dunning,
F.G. Brickwedde, J.H. Manley and H.J. Hoge, to be
published shortly.} and Stern\footnote{J. Halpern, I. Estermann,
O.C. Simpson and 0. Stern, Phys. Rev. {\bf 52,} 142
(1937).} and their
collaborators show that the scattering cross section of
{\it ortho}--H$_2$ at liquid-air neutron temperatures
($T = 100^{\circ}$K) is much
larger than the corresponding ра{\it para}--H$_2$ cross section. It has
already been pointed out$^1$ that this result is in agreement with
the theoretical expectations for a virtual singlet state.
Assuming a neutron spin of $3/2$ the theoretical value of the
ratio $\sigma_{\it ortho}/\sigma_{\it para}$, at an energy
of $3kT/2 = 0.012$ ev, is 3.11
for a virtual quintet state and 1.09 for a real quintet state. In
either case, the two cross sections are quite comparable in
magnitude, in contradiction with experiment. On the basis of
these experiments the conclusion must be drawn that the
intrinsic angular momentum of the neutron is,
in reality, $\frac{1}{2} \hbar$.
The author wishes to express his deep gratitude to
Professors Breit and Wigner for the benefit of stimulating
conversations on this and other subjects.\\\\
\hfill Julian Schwinger\footnote{Tyndall Fellow of Columbia University.}
University of Wisconsin,\\
\hbox to 1cm{} Madison, Wisconsin, \\
\hbox to 1cm{} November 17, 1937.
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