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\begin{document}
G. Breit, Phys. Rev., {\bf Vol. 49,} 519 \hfill {\large \bf 1936}\\
\vspace{2cm}
\begin{center}
{\Large \bf Capture of Slow Neutrons}
\end{center}
\vspace{0.5cm}
\begin{center}
G. BREIT and E. WIGNER, \\
Institute of Advanced Study and Princeton University \\
(Received February 15, 1936)
\end{center}
\begin{abstract}
Current theories of the large cross sections of slow
neutrons are contradicted by frequent absence of strong
scattering in good absorbers as well as the existence of
resonance bands. These facts can be accounted for by
supposing that in addition to the usual effect there exist
transitions to virtual excitation states of the nucleus in which
not only the captured neutron but, in addition to this, one of
the particles of the original nucleus is in an excited state.
Radiation damping due to the emission of $\gamma$-rays broadens
the resonance and reduces scattering in comparison with
absorption by a large factor. Interaction
with the nucleus is most probable through the $s$ part of the
incident wave. The higher the resonance region, the smaller
will be the absorption. For a resonance region at 50 volts the
cross section at resonance may be as high as l0$^{-19}$ cm$^2$ and
$0.5 \times 10^{20}$ cm$^2$ at thermal energy. The estimated
probability of having a nuclear level in the low energy region is
sufficiently high to make the explanation reasonable.
Temperature effects and absorption of filtered radiation point
to the existence of bands which fit in with the present theory.
\end{abstract}
\section{Introduction}
~~~~Bethe,\footnote{H. A. Bethe, Phys. Rev. {\bf 47,} 747 (1935).
We refer to this paper as H. B. in the text.}
Fermi,\footnote{E. Amaldi, 0. d'Agostino, E. Fermi, B. Pontecorvo,
F. Rasetti, E. Segre, Proc. Roy. Soc. {\bf A149,} 522 (1935).} Perrin
and Elsasser,\footnote{Perrin and Elsasser, Comptes rendus {\bf 200,}
450 (1935).} Beck and
Horsley\footnote{Beck and Horsley, Phys. Rev. {\bf 47,} 510 (1935).}
gave theories of the anomalously large cross
sections of nuclei for the capture of slow neutrons.
These theories are essentially alike and explain the
anomalously large capture cross sections as a sort of
resonance of the $s$ states of the incident particle.
Resonance is usually helpful in causing a large
scattering as well as a large probability of capture and
it has been shown [H. B. Eq. (35)] that large
scattering is to be expected by nuclei showing
anomalously large capture at thermal energies. This
consequence of the current theories is apparently in
contradiction with experiment, there being no
evidence of a large scattering in good absorbers. If
also follows from current theories that with very few
exceptions the capture cross section should vary
inversely as the velocity of the slow neutrons.
Experiments on selective absorption recently
performed\footnote{Moon and Tillman,
Nature {\bf 135,} 904 (1935); Bjerge and Westcott, Proc. Roy. Soc.
{\bf A150,} 709 (1935); Arsimovitch, Kourtschatow, Miccovskii
and Palibin, Comptes rendus {\bf 200,} 2159 (1935); Ridenour and
Yost, Phys. Rev. {\bf 48,} 383 (1935); Pontecorvo, Ricerca
scientifica {\bf 6-7,} 145 (1935).} indicate that there are
absorption bands characteristic of different nuclei and it appears from
the experiments of Szilard\footnote{L. Szilard, Nature {\bf 136,} 950
(1935).} that these bands have fairly
well-defined edges. It has been pointed out by Van
Vleck\footnote{J. H. Van Vleck, Phys. Rev. {\bf 48,} 367 (1935).}
that it is hard and probably impossible to
reconcile the difference in internal phase required by
the Bethe-Fermi theory with reasonable pictures of the
structure of the nucleus. The combined evidence of
experimental results and theoretical expectation is thus
against a literal acceptance of the current theories and
it is our purpose to outline an extension which is
capable of explaining the above facts by a mechanism
similar to that used for the inverse of the Auger effect
by Polanyi and Wigner.\footnote{O. K. Rice,
Phys. Rev. {\bf 33,} 748 (1929); {\bf 35,} 1551 (1930);
{\bf 38,} 1943 (1931); J. Chem. Phys. {\bf 1,} 375 (1933). A
similar process was used by M. Polanyi and E. Wigner, Zeits. f.
Physik {\bf 33,} 429 (1925).}
It will be supposed that there exist quasi-stationary
(virtual) energy levels of the system nucleus$+$neutron
which happen to fall in the region of thermal energies
as well as somewhat above that region. The incident
neutron will be supposed to pass from its incident state
into the quasi-stationary level. The excited system
formed by the nucleus and neutron will then jump into
a lower level through the emission of
$\gamma$-radiation or perhaps at times in some other fashion.
The presence of the quasi-stationary level, $Q,$ will also
affect scattering because the neutron can be returned
to its free condition during the mean life of $Q.$ If the probability
of $\gamma$-ray emission from Q were negligible there would
be in fact strong scattering at the resonance, the
scattering cross section being then
of the order of the square of the wave-length.
Estimates of order of magnitude show that it is
reasonable to assign 12 volts to the ``half-value
breadth'' of $Q$ due to radiation damping and that the
``half-value breadth'' due to passing back into the free
state is about one-fortieth of the above amount. This
means that when the system passes into the state $Q$ it
radiates practically immediately and the neutron has
no time to be rescattered. It will, in fact, be seen from
the calculations that follow that the ratio of scattering
to absorption is essentially the ratio of the
corresponding half value breadths. The hardness of the
emitted $\gamma$-rays is of primary importance for the small
ratio of scattering to absorption because it makes the
probability of $\gamma$-ray emission sufficiently high.
Inasmuch as the interesting phenomena occur for low
energies we may suppose that in most cases the
coupling of the incident state occurs through its $s$ state,
i.e., in virtue of head on collisions. It will be seen,
however, that the possibility of obtaining observable
effects by means of $p$ states is not excluded even
though it is less probable and leads to smaller cross
sections. Calculation shows that with resonances of the
type considered here one may obtain appreciable
probability of capture at energies of the order of 1000
volts. It is possible to have at such energies cross
sections of roughly 10$^{-22}$ cm$^2$ with a half-value breadth
of about 20 volts. It is therefore not necessary to
ascribe all large cross sections to neutrons of thermal
velocities and the probability of finding a quasi-stationary
level in a suitable region is not so small as
to make the process improbable.
We are presenting below the theory of capture on
this basis in some detail not because we believe it to be
a final theory but because further development may be
helped by having the preparatory structure well
cemented.
\section{Theory of Damping}
~~~~The process of absorption from the continuum into a
quasi-stationary level and a subsequent reemission of a
photon is related to the phenomena of predissociation
discussed by O. K. Rice$^8$ who made the first
application of quantum mechanics to this type of
process since Dirac's
first approach.\footnote{P. A. M. Dirac, Zeits. f. Physik
{\bf 44,} 594 (1927)} It is essential for us to consider two
continua and in this respect the present problem is
more general. It resembles closely the problem of
absorption of light from a level $a$ to a level $c$ which is
strongly damped by radiation in jumps to a third level
$b.$ The absorption from $a$ to $c$ corresponds to the
transition of the neutron into the quasi-stationary level
and the jumps from $c$ to $b$ correspond to the emission
of $\gamma$-rays in a transition to a more stable level of the
nucleus. The absorption probabilities can be obtained
by using the principle of detailed balance from the
solution which represents
emission\footnote{V. Weisskopf and E. Wigner, Zeits. f. Physik
{\bf 63,} 54 (1930).} from the level $c$
to the levels $a,~b$ or else by a direct application of the
theory of absorption.\footnote{V. Weisakopf, Ann. d. Physik
{\bf 9,} 23 (1931).} The usual theory as developed
for either process is not accurate enough to represent
the effect of the variation of matrix elements with
velocity which is essential for our purpose, inasmuch
as it is responsible for the existence of two regions of
large absorption. The usual type of calculation will
now be generalized so as to take the variation into
account.
\subsection*{
(a) Calculation of the absorption and scattering
process}
~~~~Let $a_s$ denote the probability amplitudes of states in
which the neutron is free and in a state $s.$ Similarly let
$b_r$ stand for the probability amplitude of a state in
which the neutron is captured and there is a photon $r$
emitted and let $c$ be the probability amplitude of the
quasi-stationary state having energy $h \nu$. The states
$r,~s$ are here considered to be discrete but very closely
spaced in energy. The average spacing of the levels $r,~s$
are written $\Delta E_r = h \Delta \nu_r$, $E_s = h \Delta \nu_s$
so that the number
of levels $s$ per unit energy range is $1/ \Delta E_s$.
The matrix element of the interaction energy responsible for
transitions from $a_s$ to $c,$ $c$ to $b_r$ will be written,
respectively,
\begin{equation}
M_s = h A_s, \quad M_r = h B_r.
\end{equation}
The damping constants for $c$ due, respectively, to the
possibility of emitting $a_s$ or $b_r$ are then\footnote{G. Breit,
Rev. Mod. Phys. {\bf 5,} 91, 104, 117 (1933).}
\begin{equation}
(4 \pi \tau_a)^{-1} =\Gamma_s = [ \pi \overline{|A_s|^2}/\Delta \nu_s]_{[\nu_s
= \nu_0};
\end{equation}
$$
(4 \pi \tau_b)^{-1} = \Gamma_r = [\pi \overline{|B_r|^2}/ \Delta \nu_r]_{\nu_r
= \nu_0}; \quad \Gamma = \Gamma_s + \Gamma_r,
$$
where $\tau_a~, ~\tau_b,$ are respective mean lives of
$c$ due to emission of $a_s$, and $b_r$. The quantities $\Gamma$
represent one-half of the ``half-value breadth'' measured in
frequency. In discussing line
emission and absorption${10,~11}$ the directional averages of
$|A_s|^2$ and $|B_r|^2$ can be taken for any energy within the
breadth of the
line because the line can be usually considered to be sharp. In
the present case it will be necessary to distinguish among
directional averages of $|A_s|^2$ for different energies.
The states $s$ will be thought of as plane waves modified by
a central field due to the nucleus and satisfying boundary
conditions at the surface of a fundamental cube of volume $V.$
The equations satisfied by $a_s,~ b_r,~ c$ are
\begin{equation}
\left( \frac{d}{2 \pi idt} + \nu_s \right) a_s = A_s c; \quad \left( \frac{d}{2
\pi idt} + \nu_r \right) b_r = B_r c,
\end{equation}
$$
\left( \frac{d}{2 \pi idt} + \nu \right) c = \Sigma A_s^{\ast} c_s + \Sigma
B_r^{\ast} b_r.
$$
In these equations the influence of only one quasi-stationary
level is taken into account and for this reason they are not
quite accurate. They are sufficiently good for the present
purpose because it will be supposed that different
quasi-stationary levels do not fall closely together.
At $t=0$ it will be supposed that
\begin{equation}
a_s = \delta_{s s_0}, \quad b_r = 0, \quad c=0, \quad (t=0).
\end{equation}
An approximate solution of (3) satisfying this initial
condition can be obtained by forming a linear combination of
\begin{equation}
\begin{array}{l}
c = e^{- 2 \pi i(\nu - i\Gamma')t};\\\\
a_s = A_s[e^{- 2 \pi i(\nu - i \Gamma')t} - e^{- 2 \pi i \nu_s t}]/(\nu_s - \nu + i \Gamma'),\\\\
b_r = B_r[e^{- 2 \pi i (\nu - i \Gamma')t} - e^{- 2 \pi i \nu_s t}]/(\nu_r
- \nu + i \Gamma'),
\end{array}
\end{equation}
with
$$
\Gamma' = [ \pi \overline{|A_s|^2}/\Delta \nu_s + \pi \overline{|B_r|^2}/
\Delta \nu_r]~ \mbox{resonance region,}
\eqno(5^{\prime})
$$
and
\begin{equation}
\begin{array}{l}
a_{s_0} = e^{-2 \pi i(\nu_0 - i \gamma)t}; \quad c = A_{}{s_0}^{\ast}{}
e^{- 2 \pi i (\nu_0 - i \gamma)t}/(\nu - \nu_0 - i \Gamma),\\\\
a_s = A_s A_{}{s_0}^{\ast}{}[e^{- 2 \pi i (\nu_0 - i \gamma)t} - e^{- 2 \pi
i \nu_s t}]/ (\nu_s - \nu_0 + i \gamma)(\nu - \nu_0 - i \Gamma),\\\\
b_r = B_r A_{}{s_0}^{\ast}{}[e^{-2 \pi i(\nu_0 - i \gamma)t} - e^{- 2 \pi
i \nu_r t}]/(\nu_r - \nu_0 + i \gamma)(\nu - \nu_0 - i \Gamma),\\\\
\Gamma = [\pi \overline{|A_s|^2}/ \Delta \nu_s + \pi \overline{|B_r|^2}/
\Delta \nu_r]_{\nu_s = \nu_0}.
\end{array}
\end{equation}
In Eq, (6) $s \ne s_0$. The quantities $\gamma$ and $\nu_0 - \nu_{s_0}$, are small compared with $\Gamma$; they will go to zero with increasing
volume. From (3), one finds for them the equation:
\begin{equation}
(\nu_{s_0} - \nu_0 + i \gamma)(\nu - \nu_0 - i \Gamma) = |A_{s_0}|^2,
\end{equation}
so that
\begin{equation}
\gamma = |A_{s_0}|^2 \Gamma /[(\nu - \nu_0)^2 + \Gamma^2]; \quad \nu_0 =
\nu_{s_0} + (\nu_0 - \nu)(\gamma/\Gamma).
\end{equation}
In obtaining Eq. (7) the approximations
\begin{equation}
\Sigma'_s |A_s|^2 \frac{1 - e^{2 \pi i(\nu_0 - \nu_s - i \gamma)t}}{\nu_s
- \nu_0 + i \gamma} = \pi i \overline{|A_s|^2}/ \Delta \nu_s
\end{equation}
are made. These correspond to replacing the sums by integrals and
extending the range of integration from
$\nu_s = - \infty$ to $\nu_s + \infty$ and similarly for $\nu_r$.
In addition it is
supposed that $\overline{|A_s|^2}$, $\overline{|B_r|^2}$ vary so
slowly through the region in
which the integrand is large that they may be taken outside
the integral sign. These approximations are, therefore, valid
only if the contributions to the sums (9a), (9b) are localized in
a sharp maximum. Such a maximum exists for $\nu_s \cong \nu_0$
because: (1) $\gamma$ vanishes as the fundamental volume is
increased and therefore one may consider $\gamma t \ll 1$ and (2)
for any $\nu_s - \nu_0$ it is possible to choose $t$ sufficiently
large to make
$|\nu_s - \nu_0| t \gg 1$. For such times the most important part
of the integrand oscillates rapidly with $\nu_s$. However for
$|\nu_s - \nu_0| \sim \gamma$, the values of $t$ which satisfy $\gamma t
\ll 1$
are always such that $|\nu_s - \nu_0|t \ll 1$. The integrand is
thus not
oscillatory for $\nu_s = \nu_0 \pm \gamma$ and the values of
$\overline{|A_s|^2}$, $\overline{|B_r|^2}$ on the
right side of (9) are to be understood as corresponding to
$\nu_s = \nu_0$ with an uncertainty of the order $\gamma$. It can
be verified by calculation that the contribution to (9) due
to a finite region at
a distance $|\nu_s - \nu_0| \gg \gamma$ contributes imaginary quantities decreasing exponentially with $2 \pi |\nu_s - \nu_0|t$ and real
quantities which contribute to a frequency shift$^9$ of $\nu$. For the
present this shift will be neglected. Eqs. (6) are thus
approximate solutions which become increasingly better as $t$
increases, provided $\gamma t \ll 1$. In our application $\Gamma$ is
mostly due to
the radiation damping $\Gamma_r$. The directional averages of $|B_r|^2$
vary smoothly since the energy of the $\gamma$-ray is of the order of
several million volts and is large compared to $\Gamma$.
The quantity $\Gamma'$ which enters (5) is not determined
accurately by the present method because $\overline{|A_s|^2}$ which enters in this case is some sort of average over the resonance width.
This complication causes no trouble because: (a) for times
$t \gg 1/4 \pi \Gamma'$ the rates of emission of states $a_s, b_r$
are, respectively,
$4 \pi \gamma \Gamma_s/\Gamma,~ 4 \pi \gamma \Gamma_r/\Gamma$ and
depend\footnote{Appendix I.} only on $\Gamma$ and not on $\Gamma'$;
(b) the
largeness of $\Gamma_r$ in comparison with $\Gamma_s$ makes
$|\Gamma' - \Gamma| \ll \Gamma$. Thus $\Gamma'$
is of importance only in determining the initial transients but
not the steady rate of absorption. This can be expected from
the fact that the solutions (6) represent a condition in which
$s_0$ is absorbed at the rate $4 \pi \gamma$. The addition of the
``emission solution'' (5) is only needed to enforce the condition
$c=0$ at
$t=0$; it modifies the emission of states $b_r, a_s$ during times
comparable with the mean life of the nucleus but leaves them
unchanged over longer periods very similarly to the way in
which analogous transient conditions are of no importance in
the absorption of monochromatic radiation by classical
vibrating systems.
The total cross section $\sigma$ which corresponds to the
disappearance of the incident states $s_0$ is given by
\begin{equation}
\sigma=4 \pi \gamma V/v,
\end{equation}
where $v$ is the neutron velocity because the modified
plane waves denoted by $s$ were normalized in the volume
$V$ and thus represent states of density $1/V$.
The number of possible plane waves in $V$ per unit
frequency range is
\begin{equation}
1/ \Delta \nu_s = 4 \pi V/v \Lambda^2,
\end{equation}
where $\Lambda$ is the de Broglic wave-length. From (2), (10), (11)
we have
\begin{equation}
\sigma = \gamma \Lambda^2/ \Delta \nu_s = \frac{\Lambda^2}{\pi} S \frac{\Gamma_s
\Gamma}{(\nu - \nu_0)^2 + \Gamma^2}.
\end{equation}
Here the statistical factor $S$ takes account of the fact that the
state $s_0$ may be more or less effective in its coupling to the
quasi-stationary level than the average modified plane wave
in the same energy region. If the quasi-stationary level has an
orbital angular momentum $L \hbar$ and if there is no spin orbit
interaction then $|A_{s_0}|^2 = (2L+1) \overline{|A_s|^2}$ because
coupling to c can
take place only through $1/(2L+1)$ of the total number of
states. Thus.
\begin{equation}
S = 2L+1
\end{equation}
in these special circumstances. For $s$ terms $S=1$. The total
cross section
$$
\sigma = \sigma_c + \sigma_s,
$$
where $\sigma_s$ is the cross section due to scattering and $\sigma_c$ is the cross section due to capture. We have
\begin{equation}
\sigma_c = \frac{\Lambda^2}{\pi} S \frac{\Gamma_s \Gamma_r}{(\nu - \nu_0)^2
+ \Gamma^2}; \quad \sigma_s = \frac{\Lambda^2}{\pi} S \frac{\Gamma_s^2}{(\nu
- \nu_0)^2 + \Gamma^2}.
\end{equation}
The above value of $\sigma_s$ corresponds to the value $\Sigma |a_s|^2$
and does
not take into account the fact that there is scattering in the
abscence of the quasi-stationary level. If this is strong one
must correct $\sigma_s$ for interference of the states $s$ with the
spherical wave present in $s_0$. In the applications made below
the scattering effect due to either cause will be small and the
correction need not be considered. According to (14) the
extra scattering can be expected to be of the order
$\Gamma_s / \Gamma_r$ times
the capture and is quite small for small $\Gamma_s$.
It should be noted that the order of magnitude of $\sigma_c$ at
resonance is changed by taking into account the radiation
damping. If this were
neglected and if one were to calculate simply by using
Einstein's emission probability for the stationary states
of matter then one would obtain an incorrect value,
$$
\sigma_c^{\prime} = \frac{\Lambda^2}{\pi} S \frac{\Gamma_s \Gamma_r}{(\nu
- \nu_0)^2 + \Gamma_s^2}.
\eqno(14')
$$
For resonance $\sigma_c/ \sigma_c^{\prime} = \Gamma_s^2/\Gamma^2$ and
approximately
$\int \sigma_c dE / \int \sigma^{\prime}_c dE$ is $\Gamma_s/\Gamma_r$.
No paradox is involved here
because it is not legitimate to apply Einstein's emission
probability formula to levels separated by less than
their breadth due to radiation damping. Eq. (14$'$) gives
too high values to the cross section. If $\nu - \nu_0 \gg \Gamma$
there is no difference between $\sigma^{\prime}_c$ and $\sigma_c$.
For sufficiently
large values of $\nu - \nu_0$ the discussion which led to Eqs.
(13), (14) will break down because Dirac's frequency
shift$^9$ is neglected in these formulas. A more complete
formal discussion including the frequency shift is
given in Appendix I. The calculation shows that one
should change the frequency of the quasi-stationary
level $\nu$ by
\begin{equation}
\nu \rightarrow \nu - \int \frac{|A_s|^2}{\Delta \nu_s}\frac{d \nu_s}{\nu_s -\nu_0} - \int \frac{|B_r|^2}{\Delta \nu_r} \frac{d \nu_r}{\nu_r - \nu_0}
\end{equation}
where the integrations are extended over the complete
range of states $s,r$ and where the principal values of
the integrals are to be taken. The last part of (15)
represents the frequency shift due to electromagnetic
radiation and can be incorporated in $\nu$ as a constant
because $\nu_0$ need be varied only in a range small in
comparison with the frequency of the $\gamma$-ray. It is
dangerous to take this shift into account on account of
the well known inconsistency of quantum electrodynamics.
The second term on the right side of Eq.
(15) is due to interactions between free neutron states
and the quasi-stationary state. It is physically correct
and it is necessary in order to bring about agreement
between (14) and calculations away from resonance by
means of the Einstein emission probabilities. The shift
is large in the applications. Nevertheless changes in it
are small in the relatively small range of values which
need be considered and its effect is therefore primarily
that of displacing the resonance frequency by a constant amount.
\subsection*{
(b) Resonance of one-body systems}
~~~~{\small The above discussion cannot be applied directly to cases
in which resonance consists simply in a sharp increase of the
wave function of one neutron to a maximum inside the
nucleus because there is no intermediate state $c$ under such
conditions in the same sense as in the previous section. For
low velocity neutrons such resonance can be sharp for states
with $L \geq 1$. Formally one could try to apply the discussion
already given by starting with wave functions which are
solutions of the wave equation for an infinitely high barrier
somewhat outside the nucleus. The difference between the
actual height of the barrier and $\infty$ can be then treated as a
perturbation essentially responsible for the matrix elements
$h A_s$. Such a procedure leads apparently to correct results
which can be verified by other methods. It is troublesome to
justify it completely because the region where the infinite
barrier must be erected should be such that the wave functions
within are small for all energies. It is preferable to use a more
direct calculation for such a case. We consider a plane wave
of neutrons incident on the nucleus. Resonance takes place to
the wave functions of angular momentum $L \hbar$. We surround
the nucleus by a large perfectly reflecting sphere of radius $R$
and we calculate the rate at which states of angular
momentum $L \hbar$ disappear by radiation. There is no essential
restriction on the possibility of forming wave packets out of
the plane waves if We admit only those states $L$ which satisfy
the boundary conditions on the sphere. The radius will be
made finally infinitely large and the spacing between the
levels infinitely small. This provides the necessary flexibility
for the formation of the wave packets.
The spacing between successive possible neutron levels is
given by
\begin{equation}
\Delta \nu = v/2R.
\end{equation}
The radial function will be expressed as $F/r$ where $F$ will be by
definition a sine wave with unit amplitude at a large distance
from the nucleus. The normalized wave function is then
$Y_L(F/r)(2/R)^{1/2}$ where $Y_L$ is a spherical harmonic normalized
so as to have $\int |Y_L|^2 d \Omega = 1$. The wave function for the bound state will be written
\begin{equation}
Y_{L \pm 1} f/r; \quad \int \limits^{\infty}_0 f^2 dr = 1.
\end{equation}
The damping constant which corresponds to the emission of
radiation from the state $F$ is obtained by using the formula for
Einstein's emission probability and is
\begin{equation}
\gamma E = (C/R)|\int \limits^{\infty}_0 F frdr|^2,
\end{equation}
where
$$
C = \frac{32 \pi^3 e^{'2} \nu^3}{3hc^2} \frac{L + 1/2 \pm 1/2}{2L + 1}, \eqno(18')
$$
the upper sign applying to jumps $L \rightarrow L+1$ and
$e' \sim e/2$ is the
effective charge of the neutron nucleus system. As $R \rightarrow \infty$,
both $\Delta \nu$ and $\gamma E$ decrease towards zero but their
ratio remains constant. The cross section due to capture computed directly from the emission probability is
\begin{equation}
\sigma_{C'} = (2L + 1) \Lambda^2 \gamma E/\Delta \nu.
\end{equation}
If this expression approaches $\Lambda^2$ then $\gamma E/ \Delta \nu$
becomes comparable with unity and the levels are close enough together to make Eq. (19) meaningless. It is then necessary to take into
account the mutual influence of neighboring levels. This can be
done by means of the damping matrix.\footnote{G. Breit, Rev. Mod. Phys. {\bf
5}, 117 (1933); G. Breit and I. S. Lowen, Phys. Rev. {\bf 46}, 590 (1934).} The successive states of angular momentum $L \hbar$ will be denoted by
indices $j,l$ and their probability amplitudes by $a_j$. These satisfy
\begin{equation}
\left( \frac{d}{2 \pi i dt} + \nu_j \right) a_j = i \Sigma \gamma^{jl}a_l,
\end{equation}
where
$$
\gamma^{jl} = CJ_{}j^{\ast}{} J_l/R; \quad J_j = \int F_j fr dr.
\eqno(20')
$$
In our case only states with the same magnetic quantum
number can interact so that a complete specification of the
states is obtained through their energy. Solutions of (20) in
which all quantities vary as $\mbox{exp} \{ - 2 \pi i (\nu_0 - i \gamma)t\}$ correspond as closely as possible to the notion of a stationary
state decaying under influence of radiation damping. From
(20) one obtains
$$
(\nu_j - \nu_0 + i \gamma)a_j = i \Sigma \gamma^{jl} a_l. \eqno(20'')
$$
These equations with the complex eigenwert $\nu_0 - i \gamma$ can be
reduced making use o( the fact that $\gamma^{jl}$ is a matrix of
rank 1. Thus eliminating the $a_l$ one finds
\begin{equation}
1 = \frac{iC}{R} \Sigma \frac{|J_j|^2}{\nu_j - \nu_0 + i \gamma}
\end{equation}
for the secular equation which determines $\nu_0$ and $\gamma$. This
equation will be solved approximately for the case of sharp
resonance. The resonance will be supposed to take place at an
energy $h \nu_F$ and to have a ``half-value breadth'' $2 h \Gamma_F$.
Close to resonance
\begin{equation}
|J_j|^2 = \frac{\Gamma_{F^2}|I|^2}{(\nu_j - \nu_F)^3 + \Gamma_{F^2}},
\end{equation}
where $|I^2|$ is the maximum value of $|J|^2$. This approximation
will usually apply only in a region of a few $\Gamma_F$. The value of
$\Gamma_F$ can be estimated using\footnote{G. Breit and F. L. Yost, Phys.
Rev. {\bf 48}, 203 (1935). See also Eq. (32$'$).}
\begin{equation}
4 \pi \Gamma_F = v_r/ \int \overline{G}^2 dr,
\end{equation}
where $\overline{G}$ is $F$ for resonance, $v_r$ is the velocity
at resonance,
and the integration is to be carried through the range of large
values of $\overline{G}$. The quantity $\Gamma_F$ is analogous to
$\Gamma_s$, of section (a).
The state represented by $\overline{G}$ is analogous to the quasi-
stationary state of section (a). In order to bring out the
analogy we introduce a damping constant similar to the
previous $\Gamma_r$
$$
\Gamma_R = C|I|^2/2 \int \overline{G}^2 dr = 2 \pi C |I|^2 \Gamma_F/v_r,
\eqno(23')
$$
which is the damping constant of the state represented by $\overline{G}$
when that state is normalized within the nucleus and its
immediate vicinity. Substituting (23$'$) into (21), replacing the
sum by an integral everywhere except in the vicinity of $\nu_0$
and performing the summation in that region on the
assumption that the $\Delta \nu$ can be considered as equal to each
other in that region gives
\begin{equation}
1 = i \{ a \cot \left[ \frac{\pi (\nu_{j0} - \nu_0 + i \gamma)}{\Delta \nu_{j0}}
\right] + ia + \int \limits^{\infty}_0 \frac{v_r|J_j|^2 \Gamma_R d \nu_j}{\pi
v_j|I^2|\Gamma_F (\nu_j - \nu_0 + i \gamma)}
\end{equation}
$$
a = \left[ \frac{v_r|J_j|^2 \Gamma_R}{v_j|I|^2 \Gamma_F} \right]_{\nu_0},
$$
where the integral must be extended over all $\nu_j$ and the region
around $\nu_0$ is integrated over the real axis. The quantity
$\nu_{j0}$ is
any one of the $\nu_j$ located so close to $\nu_0$ that the variation
in $\Delta \nu$
in between can be neglected. In the approximation of Eq.
(22) the integration over the resonance region $\Gamma_F$ leads to an
equation which to within a sufficient approximation reduces to
\begin{equation}
1 + it T h = (Th + it)(a + ib); \quad b = \frac{v_{j0}|I|^2 q}{v_r|J_{j0}|^2
(1+q_2)}
\end{equation}
with
$$
\nu_0 - \nu_F = q \Gamma_F; \quad Th = \tan h \frac{\pi \gamma}{\Delta \nu_{j0}};
\quad t = \tan \frac{\pi (\nu_0 - \nu_{j0})}{\Delta \nu_{j0}}. \eqno(25')
$$
By eliminating $t$
$$
Th + 1/Th = a + 1/a+b^2/a, \eqno(25'')
$$
which has the approximate solution
\begin{equation}
1/Th = a + 1/a+b^2/a.
\end{equation}
For values of $\nu_0$ which lie in the region where Eq.. (22)
applies and where $\Delta \nu_{j0} \sim \Delta \nu_r$ we have approximately
$$
\frac{\pi \gamma}{\Delta \nu} = \frac{\Gamma_R \Gamma_F}{\Gamma_{R^2} + \Gamma_{F^2}(1
+ q^2)}, \eqno(26')
$$
where it is supposed that $\Gamma_r \gg \Gamma_F$. If, however,
$\Gamma_F \gg \Gamma_R$ then
$$
\frac{\pi \gamma}{\Delta \nu} = \frac{\Gamma_R}{\Gamma_F (1 + q^2)}, \eqno(26'')
$$
which is equivalent to using the $\gamma E$ of Eq. (18), (19); in this
case one may compute using emission probabilities. If one is
so far away from resonance that $b^2/a < a,~ 1/a$ Eq. (25") gives
$$
\gamma = \gamma^{j0j0}, \eqno(26''')
$$
provided the right side is $\ll 1.$ Here again the simple emission
point of view applies. For $\Gamma_R \gg \Gamma_F$ all regions are
approximated by
\begin{equation}
\frac{\pi \gamma}{\Delta \nu_{j0}} = \frac{v_r|J_{j0}|^2 \Gamma_R
\Gamma_F(1+q^2)}{v_{j0}|I|^2[\Gamma_{R^2} + \Gamma_{F^2}(1+ q^2)]}
\end{equation}
The treatment of scattering by means of the damping matrix
is somewhat involved and will not be reproduced here. The
phase shift due to $\nu_0 - \nu_{j0}$ when added to the phase shift
already present in $F_{j0}$ gives the phase shift required. The
scattering is diminished by $\Gamma_R$ in much the same way as it
was diminished by it in section (a). By comparing (27) with (19)
\begin{equation}
\sigma_C = (2L + 1) \frac{\Lambda^2 v_r|J_{j0}|^2 \Gamma_R \Gamma_F(1+q^2)}{\pi
v_{j0}|I|^2[\Gamma_{R^2} + \Gamma_{F^2}(1+q^2)]},
\end{equation}
which is similar to Eq. (14), close to resonance. The factors
$|J|^2/|I|^2$ and $v_r/v_{j0}$ take into account the deviations
from the dependence of $|J|^2$ on $\nu$ given by (22). In (14) this is
analogous to the dependence of $\Gamma_s$ on $\nu_{j0}$ combined with
Dirac's frequency shift.
\subsection*{
(c) Sharpness of resonance for single-body problem}
~~~The upper limit of integration in Eq. (23) has been left
indefinite. By Green's theorem
$$
\frac{d}{dr} \left[ F_1 \frac{dF_2}{dr} - F_2 \frac{dF_1}{dr} \right] + \frac{2
\mu}{\hbar^2} (E_2 - E_1) F_1 F_2 = 0,
$$
where $F_1, F_2$ correspond to energies $E_1,E_2$ and need not be
regular at $r=0$. Hence\footnote{J. A. Wheeler. We are indebted to
Dr. Wheeler for communicating to us other applications of this relation.}
\begin{equation}
\frac{\partial}{\partial r} \left[ F^2 \frac{\partial}{\partial E} \frac{\partial
F}{F \partial r} \right] + \frac{2 \mu}{\hbar^2} F^2 = 0.
\end{equation}
In this section let $F_i$ be the function inside the nucleus, and
let $F$ stand for the regular solution of the wave equation for
$r \times ~{\rm radial}$ function on the absence of the nuclear
field. The
normalization of $F$ is such as to make it a sine wave
$\sin (kr + \varphi)$ of unit amplitude at $\infty$. Similarly $G$
is defined as
satisfying the same differential equation as $F$ but it is to be
90$^{circ}$ out of phase with $F$ at $\infty$ i.e.
$\cos(kr + \varphi)$. The regular
solution of the differential equation in the presence of the
nuclear field, normalized in the same way as $F$ and $G$, will be
called $\bar F.$ At the nuclear radius $r_0$
\begin{equation}
F^2 = G^2 / \left\{ \left[ G^2 \left( \frac{G'}{G} - \frac{F_1'}{F_i} \right)
\right]^2 + \left[ FG \left( \frac{F'}{F} - \frac{F_i'}{F_i} \right) \right]^2
\right\}.
\end{equation}
Here the accent stands for differentiation with respect to $kr$.
At resonance $F_i'/F_i=G'/G$ and the second term in the curly
bracket is then 1, while the first term is zero. As $E$ changes to
either side of the resonance value $E_r$ the first term may
become 1 for $E=E_r \pm \Delta E$ where $\Delta E$ is properly
chosen. The half-value breadth is then $2 \Delta E$ and
$\Delta E = h \Gamma_F$. The value of
$\Delta E$ can be estimated by
\begin{equation}
\Delta E \frac{\partial}{\partial E} \left[ G^2 \left( \frac{G'}{G} - \frac{F'_i}{F_i}
\right) \right]_{r_0} = 1.
\end{equation}
Using Eq. (29) and calculation the $\partial/\partial E$ for $E=E_r$,
one obtains
a result which can be expressed in terms of integrals up to $R$
where $R$ is any value of $r$ which is greater than $r_0$. The
function which is $G$ for $r > r_0$ and $F_i(G/F_i)_{r_0}$, for
$r < r_0$ is
continuous at $r_0$ and at resonance its derivative with respect to
$r$ is also continuous. The function will be called $\bar G$ for
$0 < r < \infty$. We have then\footnote{Cf. Eq. (22) reference 15.
In calculations with Coulombian fields it is sometimes convenient
to transform Eq. (32) of the text into
$$
\frac{E}{\Delta E} = G^2 \left[ \frac{k}{F_j^2} \int \limits^{r_0}_0 F_i^2
dr - \frac{k}{F^2} \int \limits^{r_0} F^2 dr - \frac{EG^2 \partial}{kr \partial
E} \left( \frac{kr}{FG} \right) \right]
$$
all quantities outside the integrals being taken for $r=r_0$.}
\begin{equation}
\frac{E}{\Delta E} = k \int \limits^R_0 \bar G^2 dr + \left[ \frac{G^2 E
\partial}{k \partial E} \frac{\partial G}{G \partial r} \right]_{r=k}; \quad
\Delta E = h \Gamma_F.
\end{equation}
The right side of this result is independent of $R$ and is finite.
The term outside the integral should be included in Eq. (23) changing
$$
\int G^2 dr \rightarrow \int \limits^R_0 \bar{G}^2 dr + \left[ \frac{G^2 E \partial}{k^2
\partial E} \frac{\partial G}{G \partial r} \right]_{r=k}. \eqno(32')
$$
Eq. (32) has a well-defined meaning only if resonance is
sharp. Otherwise the $\partial/\partial E$ entering in Eq. (31)
cannot be
supposed to be sufficiently constant through the half-breadth
$2h \Gamma_F$. It cannot be expected to hold for the broad $S$
resonance discussed by Bethe.}
\subsection*{(d) Capture by $p$ states}
~~~~For a potential well of constant depth
$$
F_i = \sin z/z - \cos z; \quad F = \sin \rho/\rho - \cos \rho;
$$
\begin{equation}
G = \cos \rho/\rho + \sin \rho,
\end{equation}
where
$$
z =Kr; \quad \rho = kr; \quad K = \mu v_i / \hbar; \quad k = \mu v/\hbar
\eqno(33')
$$
$v_i, v$ being, respectively, the velocities inside and
outside the nucleus. The resonance condition is
$$
z \sin z/[\sin z/z - \cos z]=\rho \cos \rho/\rho[\cos \rho/\rho + \sin \rho].
$$
For slow neutrons $\rho \ll 1$ the right side is $\ll \rho^2$
and therefore very small. The first resonance point is
obtained for $z= \pi - \epsilon,~ \epsilon \sim \rho^2/ \pi$.
It will suffice to take
$z = \pi$. By substituting into Eq. (32) it follows that
$$
\Delta E/E = 2 \rho/3 = 4 \pi r_0/3 \Lambda. \eqno(33'')
$$
For $E=(1/40)$ volt, $\Lambda = 1.8 \times 10^{-8}$ cm,
$h \Gamma_f = 5.8 \times 10^{-6}$ volt.
For 3-MEV $\gamma$-rays a reasonable value of $h \Gamma_R$
is 5.8 volts. The cross section at resonance is by Eq. (28)
$3 \Lambda \Gamma_F/ \pi \Gamma_R = 300 \times 10^{-24}$ cm$^2$.
Since scattering is of
the order $\Gamma_F/\Gamma_R$ times capture the scattering cross
section is small. According to Eq. (33$''$) the cross
section at resonance for $p$ terms with $\Gamma_R \gg \Gamma_F$ can be
expected to vary as $v$ and $h \Gamma_F$ as $v^2$. The range in which
$p$ terms can be expected to give large capture cross
sections and small scattering is therefore roughly from
$1/40$ volt to 1 volt. At higher velocities $h \Gamma_F$ is likely to
be higher than $h \Gamma_R$. In the absence of an apparent
reason for nuclear $p$ levels to fall in this narrow
velocity range, an explanation in terms of $p$ terms
although possible is improbable on account of the
small range of neutron velocities required.
\section{Capture Through $s$ Wave}
\subsection*{(a)}
~~~~This section will contain the calculation of the $A_s$
used in 2a. It is supposed that the system
``nucleus+neutron'' can be treated in first approximation
by means of an effective central field
acting on the neutron. The difference between the
Hamiltonian of the system and the Hamiltonian
corresponding to the central field will be called $H'$. On
account of this difference there exist transitions from
the $s$ wave of the incident state to quasi-stationary
excited states of the ``nucleus +neutron'' system.
Normalizing the $s$ waves within a sphere of radius $R$
the wave function inside the nucleus is
$$
C \sin Kr/r; \quad C^{-2}[1 + (U/E) \cos^2 Kr_0]2 \pi R,
$$
where
$$
K^2/k^2 = (U+E)/E
$$
and $U$ is the depth of the potential hole. The interaction
energy $H'$ involves besides $r$ also internal coordinates
$x.$ The wave function of the whole system in the
incident state may be written $C \psi_0(x) \sin Kr/r$ and in the
quasi-stationary state $\psi_q(r,x)$. The matrix element $M_s$
of Eq. (1) is then
\begin{equation}
M_s = \int \psi_Q(r,x)H'C \sin Kr \psi_0(x) dv/r,
\end{equation}
where $dv$ is the volume element of the whole system.
The state $Q$ is by definition such that the integral of
$|\psi_Q|^2$ through nuclear dimensions is unity. The order
of magnitude of $M_s$ is therefore
$$
M_s = C \bar H r_0^{1/2}, \eqno(34')
$$
where $\bar H$ is an average of $H'$ through the nucleus and
may have reasonably a value of 0.5 MEV. It cannot be
specified further without detailed calculation which
would probably be unsatisfactory in the present state of
nuclear theory. Since $\Delta E = hv/2R$,
$$
h \Gamma_s \cong \frac{\bar H^2 r_0}{2 \Lambda U \cos^2 Kr_0}. \eqno(34'')
$$
According to Eq. (14)
\begin{equation}
\sigma_c = \frac{\Lambda r_0}{2 \pi} \frac{\bar H}{U \cos^2 Kr_0} \frac{\bar
Hh \Gamma_r}{h^2 (\Gamma_r + \Gamma_s)^2 + (E - E_r)^2,}
\end{equation}
where $E_r$ is the value of $E$ for resonance; According to
this formula there are two maxima\\
\newpage
TABLE I. {\it Calculated cross sections for neutron capture.}
\begin{center}
\begin{tabular}{rrrcrc}
\hline \hline
Position of&&&$h \Gamma_s$&\multicolumn{2}{c}{$10^{24} \sigma$ (cm$^2$)}\\
Resonance&$\bar H$& $h \Gamma_r$&at resonance&&Thermal\\
(volts)&(MEV)&(volts)&(volts)&Resonance&Energies\\
\hline
$1/40$&0.1&10&0.01&90000&\\
&.1&1&.01&900000&\\
1&.1&10&.05&14000&\\
&.1&1&.05&140000&\\
50&.1&10&.37&2000&3500\\
&.5&10&9&13400&80000\\
&.1&1&0.37&11000&350\\
&.5&1&9&4800&9000\\
1000&.1&10&1.6&320&9\\
&.5&10&40&420&200\\
10000&.1&10&5&60&0.09\\
&.5&10&125&18&2\\
\hline\hline
\end{tabular}
\end{center}
for $\sigma_c$, one for $E=E_r$ and one for $E=0.$ The expected
cross sections are given in Table I to about ten percent
accuracy. The numbers correspond to
$\Lambda(kT) = 1.8 \times 10^{-8}$
cm; $\Lambda(1~ \mbox{volt}) = 2.9 \times 10^{-9}$ cm;
$r_0 = 3 \times 10^{-13}$ cm; $U \cos^2 Kr_0 = 10^7$
volts. For $E_r = 140$ 1 volt the table shows
large cross sections at thermal energies and above. The
condition is similar to Bethe's except for a relatively
sharper resonance determined by $h \Gamma_r$. For 50 volts one
sees the development of two maxima one at resonance
and one at thermal energies. For $E=1000$ and 10,000
volts the maximum at thermal energies decreases as
$E_r^{-2}$ and the maximum at resonance roughly as $E_r^{-1/2}.$
For such high values of $E_r$ scattering has a chance of
becoming comparable with absorption or even greater
than the absorption at resonance. In the thermal energy
region the $1/v$ law is obeyed for high values of $E_r$; for
low $E_r$r the maximum at $E_r$ interferes with the $1/v$ law
and the region of its validity is displaced below
thermal energies.
In Table I only the effect of a quasi-stationary level
at $E_r$ is considered. In addition there may be effects of
other levels as well as radiation jumps of the kind
considered by Bethe and Fermi which do not depend
on the existence of virtual levels. It is thus probable
that in most cases there is a region with a $1/v$
dependence although it may be at times masked by a
resonance region.
\subsection*{(b) Dirac's frequency shift}
~~~~In the above estimates the effect of Dirac's frequency
shift was neglected. This is given by
$$
(h \Delta \nu)_D = \int \limits^{\infty}_0 \frac{h \Gamma_s}{\pi} \frac{dE}{E-E_0}
= \frac{\bar H^2 r_0}{\pi \Lambda_0 x_0} \int \limits^{\infty}_0 \frac{x^2
dx}{(x^2 - x_0^2)(x^2 + a^2)},
$$
where $x = E^{1/2},~ a^2 = U \cos^2 Kr_0$ and the value of $h\Gamma_s$
was substituted by means of Eq. (34$''$). Here the subscript 0
refers to the neutron energy $E_0$ and the principal value
of the integral is understood. Evaluating the expression
\begin{equation}
(h \Delta \nu)_D = \frac{\bar H^2 r_0 U^{1/2} \cos Kr_0}{2 \Lambda_0(E_0
+ U \cos^2 Kr_0)E_0^{1/2}}.
\end{equation}
The shift is seen to be of the order of 3000 times $h\Gamma_r$
for $E_0=l$ volt. The shift is nearly independent of the
velocity. In the approximation of Eq. (36)
$$
\frac{d(h \Delta \nu)_D}{d E_0} = \frac{h \Gamma_s}{E_0} \frac{E_0^{1/2}}{U^{1/2}|\cos
Kr_0|},
\eqno(36')
$$
which shows that the variation in the shift is small and
of the order of $2 \times 10^{-5} (E - E_r)$ for $\bar H = 0.1$ MEV.
\section{Discussion}
\subsection*{(a)
Absence of scattering}
~~~~According to Dunning, Pegram, Fink and
Mitchell\footnote{J. R. Dunning, G. B. Pegram, G. A. Fink
and D. P. Mitchell, Phys. Rev. {\bf 48,} 265 (1935).}
the elastic scattering of slow neutrons by Cd is less
than one percent of the number captured. According to
A. C. G. Mitchell and E. J. Murphy\footnote{A. C. G. Mitchell and
E. J. Murphy, Phys. Rev. {\bf 48,} 653
(1935). Cf. also Bull. Am. Phys. Soc. {\bf 11,} paper 27, Feb.
4, 1936.} scattering as
detected by silver is about the same as absorption for
Fe, Pb, Cu, Zn, Sn while for Hg scattering is about
$1/80$ of the absorption. In the later communication of
Mitchell and Murphy$^{19}$ it is also found that Ag, Hg, Cd
are poor scatterers of slow neutrons detected by silver.
It is interesting that Ag shows small scattering in these
experiments because the detection took place by
means of silver and that Hg and Cd show small
scattering because they have large absorption cross
sections.$^{18}$ The observation of scattering by a material
having large absorption is difficult because the
neutrons entering the material are absorbed before they
can be scattered and it is possible that to some extent
the failure to observe scattering in good absorbers is
due to this cause. The absence of observed scattering
in the region of strong absorption is therefore not a
surprise, particularly in view of the relatively small
numbers of neutrons available for experimentation. It
seems more significant, however, that strong absorbers
do not show, so far, strong scattering in any velocity
region because, according to the Fermi-Bethe theory,
the scattering cross section should be large in a wide
range of energies. The experimental evidence says
little about the ratio of scattering to absorption near
resonance. It indicates that this ratio is less than $1/10$
in most cases. It is impossible, therefore, to ascertain
definitely the ratio $\Gamma_s/\Gamma_r$ until more detailed
experimental data are available. According to Table I
the condition $\Gamma_s/\Gamma_r < 1/10$ can be satisfied in many ways
up to velocities of over 1000 ev.
\subsection*{
(b) Magnitude of interaction with internal states and
probability of internal state in required region}
~~~In Table I arbitrary assignments of values of $\Gamma_s, \Gamma_r$
were made. It will be noted that at low neutron
velocities the desired large capture cross sections are
easily obtained through relatively wide bands having a
half-value breadth $2 \Gamma_r$. Keeping $\Gamma_r$ fixed one can
decrease the interaction energy $\bar H$ to 10,000 ev for
$h \Gamma_r = 1$ volt, $E_r = 1$ volt and still have a cross section of
$1000 \times 10^{-24}$ cm$^2$ in an energy range up to 2 volts. In
some cases relatively weak radiative transitions will
come into consideration leading to smaller $\Gamma_r$. For such
transitions $\bar H$ need not be as large as 10,000 ev in order
to have cross sections of $1000 \times 10^{-24}$ cm$^2$ in the
resonance region. For the large energies involved in
nuclear structure it is reasonable to expect interaction
energies of the order of 10,000 volts between
practically any pair of levels not isolated by a selection
rule and interaction energies of the order 100,000 volts
between a great many levels.
There are about ten elements among 72 observed
that show cross sections of more than
$500 \times 10^{-24}$ cm$^2$. Allowing for the fact that there are
more isotopes than elements it appears fair to say that
the chance of such an anomalous cross section is about
$1/20.$ One can try to account for these solely by the
low velocity regions which exist for any resonance
level, thus probably overestimating the necessary
number of levels. In order that $\sigma_c > 500 \times 10^{-24}$ cm$^2$
at $1/40$ volt for a nucleus having $r_0 = 10^{-12}$ cm and
$h \Gamma_r = 10$ volts the resonance region must be not farther than
at $|E_r| = \bar H/420$ from thermal energies by Eq. (35). We
do not wish $h\Gamma_s$. at thermal energies to be greater than
0.1 volt so as not to have too much scattering and
therefore $\bar H$ should be below $2 \times 10^5$ ev at the higher
$E_r.$ Thus $E_r$ should be kept below about 460 volts in
order to give the large capture cross sections for
$E=1/40$ volt together with small scattering. A level
below ionization will also be effective in producing an
increased absorption. The observed number of large
absorptions corresponds in this way to one level in 900
volts for $1/20$ of nuclei or one level every 18,000 volts
for a single nucleus. In addition some cross sections
will be caused by direct resonance. Just how many is
uncertain but it is clear that such effects exist in Cd,
Ag, Au, Rh, In.
The average spacing between the $\gamma$ ray levels of
Th C$''$ as given in Gamow's book is about 100,000 volts
and this is apparently the order of magnitude usual for
$\gamma$-ray levels of radioactive nuclei. There appears to be
no reason why the energy levels found through the
analysis of $\gamma$-ray spectra should include all the
nuclear levels and there may be as many as one level
in 20,000 of a land that may be responsible for
coupling to incident neutrons. It should be
remembered here that some of the levels may be active
even though the coupling is weak so that more
possibilities are likely to matter than for the $\gamma$-rays of
radioactive nuclei.
For a complicated configuration of particles it seems
reasonable to consider a total number of 100 possible
levels per configuration because protons and neutrons
can be combined separately to give different states. On
this basis we deal with an average spacing between
configurations of about 2 MEV which is not
excessively small. It is, of course, impossible to prove
anything definitely without calculating the levels; this
appears to be premature at present on account of
uncertainties in nuclear theories.
\subsection*{
(c) Existence of two maxima}
~~~~According to the calculation given above it is
expected that there will be in general two maxima one
of which should be at resonance and another at $v = 0.$
According to the experiments of Rasetti, Segre, Fink,
Dunning and Pegram\footnote{F. Rasetti, E. Segre, G. Fink,
J. R. Dunning and G. B.
Pegram, Phys. Rev. {\bf 49,} 103 (1936).} the $1/v$ law is not
obeyed by Cd
but is obeyed by Ag. Cadmium has therefore a
resonance region close to thermal velocities. In the
classification of Fermi and Amaldi\footnote{E. Amaldi and E. Fermi,
Ricerca scientifica {\bf 2,} 9 (1936);
E. Fermi and E. Amaldi, Recerca scientifica {\bf 2,} 1 (1936).}
this region must
be affected by the $C$ group since absorption measurements
by the Li ionization chamber which was used in
these experiments agree for most elements with the
measurements of Fermi and Amaldi on the $C$
group.\footnote{Unpublished results of F. Rasetti. We are very
grateful to
Professor Rasetti for informing us of these results.}
The verification of the $1/v$ law for Ag by the rotating
wheel indicates that in Ag the resonance band is
located above thermal energies. This conclusion is in
agreement with the smallness of the temperature effect
for the $A$ neutrons detected by silver which was
recently established by Rasetti and Fink.\footnote{F. Rasetti and
George A. Fink, Bull. Am. Phys. Soc. {\bf 11,}
Paper 28, Feb. 4, 1936.} Since Rh
behaves similarly to Ag in these temperature
experiments Rh also has a resonance region above
thermal velocities. Fermi and Amaldi have evidence
that $D$ neutrons, which affect Rh, are different from $A$
neutrons which affect Ag. It is very probable that both
of these groups lie above the thermal region and they
may reasonably cover a range of 30 volts inasmuch as
the $B$ group overlaps weakly with both $A$ and $D.$
According to Szilard$^6$ In shows strong selective
effects outside the $C$ group and according to Fermi and
Amaldi$^{21}$ the same period of In (54 min.) detects the $D$
group. The number of neutrons in the groups is
presumably in the ratios $C/80=B/20=D/15=A/1$. One
could try to conclude that the order of increasing
energies is $C, B, D, A$ on the assumption that the
number of neutrons increases towards low energies.
Such a conclusion is dangerous because little is known
about the velocity distribution, because within
each group there may be several bands at different
velocities, and also because the number of expected
neutrons in a group should depend on its width.
Temperature effects show that practically all captures
increase as the energy is lowered. The effects are
strongest\footnote{P. B. Moon and R. R. Tillman, Proc. Roy. Soc.
{\bf A153,} 476 (1936).} for Cu, V are smaller for Ag, Dy weaker
for Rh and weakest for I. The absorption coefficient
for $C$ neutrons is, however, larger for Rh than for Ag
indicating that the smaller temperature effect in Rh is
due to a relatively greater importance in it of a band
above thermal energies. All temperature effects agree
in indicating the presence of a region in which the $l/v$
law is followed approximately but again no definite
conclusion about the order of bands is possible. The
low temperature effect in I would tend to indicate that
its absorption region is high and detection-absorption
experiments on I and Br tend to indicate that their
bands are isolated from the others discussed here;
perhaps these isotopes have resonance bands at higher
energies. A new band was recently discovered in Au
by Frisch, Hevesy and McKay\footnote{0. R. Frisch, G. Hevesy
and H. A. C. McKay, Nature {\bf 137,}
149 (1936).} which represents
strong absorption on a weaker background. The large
number of selective effects observed makes the present
explanation reasonable and the existence of a region of
low energies in which the absorption decreases with
energy is seen to fit in well with expectation.
\subsection*{
(d) Other possibilities}
~~~~One may consider weak long range forces as a
possible explanation of the same phenomenon.
Potentials of the order of neutron energies in a region
comparable with the neutron wave-length would
produce strong effects on absorption and scattering.
For thermal energies the wave-length is of atomic
dimensions and one would therefore expect the
binding energy of deuterium compounds to be
different from that of hydrogen compounds by an
amount comparable to $1/40$ volt if such potentials were
present. Such energy differences do not exist. It would
be possible to devise potentials which fall off
sufficiently rapidly with distance to make the
interaction potential negligible for chemical binding
and which would
cover a total region appreciably larger than the
nucleus. Such hypotheses seem improbable without
additional argument. Besides special relations between
the phase integrals through the nuclear interior and the
part of the range of force outside the nucleus would
have to be set up in order to make absorption large and
scattering small. It is improbable that the large number
of bands could be accounted for by any single particle
picture.
Forces between electrons and neutrons even though
they may exist are not likely to have much to do with
the bands. Thus it has been shown by Condon\footnote{E. U. Condon, in
press. We are indebted to Professor
Condon for showing us his manuscript before publication.} that
electron neutron interactions would give rise to
scattering cross sections varying roughly as the square
of the atomic number $Z$ on the assumption that the
electron-neutron forces alone are responsible for the
scattering. Forces inside the nucleus must also be
supposed to contribute to the phase shifts responsible
for scattering. Since these forces also vary with $Z$ one
could obtain a more complicated dependence of the
scattering cross section by suitably adjusting the
nucleus-neutron and electron-neutron potentials. On
such a picture one could try to account for sharp
resonances by making the electron neutron interaction
repulsive. However, Condon's calculation shows that
isotope shifts would be also produced by these
interactions. It is improbable that the isotope shift is
due solely to neutron-electron interaction because the
deviation from the inverse square law inside the
nucleus due to smearing out of protons produces a
considerably larger effect than the observed shift. But
it would also be unreasonable to try to combine the
proton and neutron effects in the nucleus so as to have
each large but their difference small. It is therefore
probable that the electron-neutron interaction is not
much larger than that which corresponds to the
observed isotope shift. Since the density of the
Fermi-Thomas distribution varies for small $r$ as $r^{-3/2}$ the
effective potential acting on the neutron will become
high for small $r.$ However, calculation shows that it is
not high in a wide enough region to account for sharp
resonances if the limitation due to the isotope shift is
considered.
Bombardment of light nuclei with charged particles
has also shown the existence of resonances. Thus there
are resonances\footnote{L. R. Hafstad and M. A. Tuve, Phys. Rev.
{\bf 48,} 306 (1935); P. Savel, Comptes rendus {\bf 198,} 1404 (1934),
Ann. de physique {\bf 4,} 88 (1935).} for the emission of
$\gamma$-rays in proton
bombardment of Li, C, F and similarly there are the
well known resonances in disintegrations produced by
$\alpha$ particles. Experimental methods have not been very
suitable so far for the detection of resonance regions
on account of the scarcity of monochromatic sources
and the necessity of using thin films. In Li protons are
apparently able to produce $\gamma$-rays in two ways; by
resonance at 450 kv and by another process at higher
energies. In fluorine there are several peaks. In carbon
there was an indication of the main resonance peak
being double. It appears possible that many more
levels will be detected inasmuch as neutron
experiments indicate a high density of levels.
Calculations on the radiative capture of carbon under
proton bombardment$^{15}$ lead to a higher yield than is
observed by a factor of several thousand. In these
calculations the capture was supposed to occur by a
jump from the $p$ state of the incident wave to an $s$
state of the N$^{13}$ nucleus. The calculated half-value
breadth due to proton escape from the quasi-stationary $p$ level was
of the order $h \Gamma_F \sim 10,000$ ev and
thus much larger than the width due to radiation
damping. The yield in thick targets under these
conditions is nearly independent of the special value of
$h \Gamma_F$. It is clear from the formulas given here for
neutron capture that one can decrease the theoretically
expected yield either by ascribing the capture to a
transition having a small probability of radiation
(small $\Gamma_r$) such as would correspond to quadruple or
other forbidden transitions or else by using an
intermediate state
of excitation of the nucleus with a small transition
probability to the incident state of the proton (small $\Gamma_s$).
In the latter case this transition probability would have
to be made so small as to have $\Gamma_s < \Gamma_r$ and the observed
width of resonance would have to be ascribed to
experimental effects. If $\Gamma_s < \Gamma_r$ the thick target yield
depends on $\Gamma_s$ and is proportional to it for small $\Gamma_s$.
The apparent disagreement between theory and experiment
previously found for carbon is thus not alarming from
the many-body point of view and supports the belief
that excitation states of the nucleus have often to do
with the simultaneous excitation of more than one
particle.
The excitation states responsible for the neutron
absorption bands make it possible for a fast neutron to
lose energy by inelastic impact with the nucleus.
Estimates show that the cross sections for such
processes are likely to be small when energy losses are
high. The cross section is estimated to be
$$
\frac{\Lambda_1}{4 \pi \Lambda_2} \frac{\bar H^2 r_0^2}{U^2 \cos^4 Kr_0},
$$
where $\Lambda_1, \Lambda_2$ are, respectively, neutron wavelengths in
the incident and final states. For large energy losses
$\Lambda_2 \gg \Lambda_1$ and only a small effect need be expected. The
excitation levels responsible for neutron capture will
give small values $\Lambda_1/\Lambda_2$. Excitation levels located
lower are more favorable and probably the excitation
of Pb to about 1.5 MEV has to do with such a
possibility.\footnote{J. Chadwick and M. Goldhaber, Proc. Roy. Soc.
{\bf A151,} 479 (1935).}
We are very grateful to Professors R. Ladenburg
and F. Rasetti for interesting discussions of the
experimental material.
\section*{Appendix I}
\subsection*{
Variation of damping constant with energy and Dirac's
frequency shift}
{\small
~~~Eq. (6) of the text lead to [cf. Eqs. (126$'$) to (129') of
reference in footnote (12)]
$$
(\nu_{s0} - \nu_0 + i \gamma)(\nu - \nu_0 + i \gamma) = |A_{s0}|^2 + (\nu_{s0}
- \nu_0 + i\gamma)[\Sigma'|A_s|^2 \frac{1 - e^{2 \pi (\nu_0 - \nu_s - i \gamma
t}}{\nu_s
- \nu_0 + i \gamma}
$$
\setcounter{equation}{27}
\begin{equation}
\left. + \Sigma|B_r|^2 \frac{1 - e^{2 \pi (\nu_0 - \nu_r - i \gamma)t}}
{\nu_r -\nu_0 + i \gamma} \right]
\end{equation}
which determines $\Gamma$ by comparison with (7) $[\Gamma \gg \gamma]$.
It is by no means natural that this equation can be satisfied
because the right side depends on $t$. If the $A_s$ as well as
the $B_r$
were all essentially equal and if $\gamma$ were great in comparison
with the frequency differences of consecutive levels the sums
could be transformed, into integrals in the well-known
way$^{10-12}$ so that (9) as well as (6) would follow. We shall
attempt here a more exact procedure.
Consider the $\Sigma'_s$ in the square brackets. It is natural to
divide the range of $\nu_s$ into two parts: one for which
$|\nu_s - \nu_0| > a \gg \gamma$ and one for which $|\nu_s - \nu_0| \le
a$. Since
$|A_s|^2/ \Delta \nu_s$. changes slowly this quantity will be
replaced by a
constant in $|\nu_s - \nu_0| \le a$ and its value may be taken to be
that at $\nu_0$ for the evaluation of the contribution of this region.
We have then to consider
$$
(\overline{|A_s|^2})_{r0} \sum \limits^{\nu_0 + a}_{\nu_0 - a} \frac{1 - e^{2
\pi i(\nu_0 - \nu_s - i \gamma)t}}{\nu_s - \nu_0 + i \gamma}. \eqno(39a)
$$
An exact evaluation of this sum is not simple because $\gamma$ and
$\Delta \nu$ are of the same order of magnitude and the replacement of
(39a) by an integral is somewhat objectionable. This point has
never been completely cleared up and we have only
qualitative arguments in favor of the correctness of the
replacement of (39a) by an integral. For $\nu_s - \nu_0$ of the
order of
a few $\gamma$ such a replacement is indeed meaningless but
fortunately this region is not vital for $t \ll 1/\gamma$ since the
numerator of (39a) is then small. For larger $|\nu_s - \nu_0|$
the terms
of (39a) vary more smoothly and finally they become rapidly
oscillating for $|\nu_s - \nu_0| t \gg 1$ which can be satisfied
simultaneously with $t \ll 1/\gamma$ provided $a \gg \gamma$.
The smallness of
$\gamma$ is thus not as serious as might appear from the fact that
$\gamma/ \Delta \nu \sim 1$. It should also be observed that
the treatment of
Rice$^8$ using real eigenwerte for a single one-dimensional
continuum is in agreement with replacing (39a) by an integral.
The result of doing so is given by (9).
In addition one has the contribution of $|\nu_s - \nu_0| > a$.
This integral can be treated neglecting $\gamma$ because it is
of interest to
evaluate $\gamma$ only to quantities of order $\gamma/(\nu - \nu_0)$
and because
the discussion is supposed to apply only to $\gamma t \ll 1$. This
integration gives
$$
\left( \int \limits^{\nu_0 - a}_0 + \int \limits^{\infty}_{\nu_0 + a} \right)
\frac{\overline{|A_s|}^2}{\Delta \nu_s} \frac{d \nu_s}{\nu_s - \nu_0}
\eqno(39b)
$$
which means that the principal value of the $\int$ is understood.
Similarly one obtains a contribution due to $|B-r|^2$. These two
integrals give the Dirac frequency shift which is included in
Eq. (15).
As stated in the text the difference between $\Gamma'$ and $\Gamma$ does not
affect the absorption for $t \gg 1/\Gamma$. Thus for the initial
condition given by Eq. (4)
$$
|b_r|^2 =\frac{|B_r|^2 |A_{s0}|^2}{(\nu - \nu_0)^2 + \Gamma^2} \left\{ \frac{1
+ e^{- 4 \pi \gamma t} - 2e^{- 2 \pi \gamma t} \cos 2 \pi (\nu_r - \nu_0)t}{(\nu_r
- \nu_0)^2 + \gamma^2} \right.
$$
$$
\left. + \frac{1 + e^{- 4 \pi \Gamma' t} - 2e^{- 2 \pi \Gamma' t} \cos 2
\pi (\nu_r - \nu_0)t}{(\nu_r - \nu) + \Gamma^{'2}} + \mbox{cross product
term} \right\}.
$$
Only the first fraction in the curly brackets contributes to the
steady increase of $\Sigma|b_r|^2$ in times $\gg 1/\Gamma$. Its
contribution is
$$
\frac{\pi |A_{s0}|^2 \overline{|B_r|^2}}{\gamma[(\nu - \nu_0)^2 + \Gamma^2]
\Delta \nu_r} (1 - e^{- 4 \pi \gamma t}).
$$
The last factor is for practical purposes $4 \pi \gamma t$. The
second and
third terms in the curly bracket give terms $\mbox{exp} (- 4 \pi \Gamma'
t)$,
$\mbox{exp} (- 2 \pi \Gamma' t)$ and constants. The first two
kinds die off and the
last kind represents the effect of transients which do not
matter in the long run, so that for times not too large as
compared with $1/\gamma$ and yet great as compared with $1/\Gamma$ one
may consider the rates of change of $\Sigma|b_r|^2$ and of
$\Sigma'|a_s|^2$ to be
$4 \pi \gamma \Gamma_r/\Gamma$ and $4 \pi \gamma \Gamma_s/\Gamma$.
These are the results used in the text.
}
\end{document}
%ENCODED JUNE 2003 BY Natalia Sukhikh;