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\begin{document}
J. M. B. Kellogg Phys. Rev. {\bf 56,} 728 \hfill {\large \bf 1939}\\
\vspace{2cm}
\begin{center}
{\bf \Large The Magnetic Moments of the Proton and the Deuteron}
\end{center}
\begin{center}
{\bf The Radiofrequency Spectrum of H$_2$ in Various Magnetic
Fields\footnote{Publication assisted by the Ernest Kempton Adams
Fund for Physical Research of Columbia University.}}
\end{center}
\begin{center}
J.M.B. KELLOGG, I.I. RABI and N.F. RAMSEY, Jr.\\
Columbia University, New York, New York\\
and J.R. ZACHARIAS \\
Hunter College, New York, New York\\
(Received July 31, 1939)
\end{center}
\begin{abstract}
The molecular-beam magnetic-resonance method for
measuring nuclear magnetic moments has been applied to
the proton and the deuteron. In this method the nuclear
moment is obtained by observing the Larmor frequency of
precession ($\nu = \mu H/h I$) in a uniform magnetic field. For
this purpose HD and D$_2$ molecules are most suitable
because they are largely in the state of zero rotational
momentum. Very sharp resonance minima are observed
which makes it possible to show that the observed values of
$\nu/H$ are independent of $H$, and to make a very accurate
determination of the ratio $\mu_P/\mu_D$. With molecules of
ortho-hydrogen in the first rotational state a radiofrequency
spectrum of six resonance minima was obtained. This
spectrum when analyzed yields a set of nine energy levels
from which are obtained (1) the proton moment from its
Larmor precession frequency; (2) the proton moment from
the magnitude of the dipole interaction between the two
proton magnetic moments (the directly measured quantity
is $\mu P/r^3$); and (3) the value of the spin orbit interaction
constant of the proton moment with the rotation of the
molecule or the magnetic field $H'$ produced by the rotation
of the molecule at the position of the nucleus. The
numerical results are $\mu_P = 2.785 \pm 0.02$ nuclear magnetons;
$\mu_D = 0.855 \pm 0.006$ nuclear magneton;
$(\mu_P/\mu_D) = 3.257 \pm 0.001;$ $H' = 27.2 \pm 0.3$
gauss; $\mu_P/r^3 = 34.1 \pm 0.3$ gauss which gives
$\mu_P = 2.785 \pm 0.03$ nuclear magnetons. To within
experimental error there is no disagreement of the results of
these direct measurements with those from atomic beam
measurements of the h.f.s. $\Delta \nu$ of the ground states of H and
D.
\end{abstract}
\section*{Introduction}
~~~In this paper we shall describe experiments in which the
magnetic moments of the proton and deuteron are measured
to a much higher precision than heretofore. In addition
there will be presented some experimental results which
throw light on the inner dynamics of the hydrogen
molecule, such as the magnetic field at the position of the
protons which is produced by the rotation of the molecule
as a whole. The magnetic moments will be deduced from
two independent quantities: (1) the Larmor precession
frequency of the proton or the deuteron in an externally
applied magnetic field, (2) the mutual magnetic energy of
interaction of the two proton moments in the hydrogen
molecule. The first of these two measurements depends on
principles briefly described in the first paragraph of the
section on method and more adequately in a paper by Rabi,
Millman, Kusch and Zacharias\footnote{I.I. Rabi, S. Millman,
P. Kusch and J.R. Zacharias, Phys. Rev. {\bf 55,} 526 (1939).}.
The results are given in the first part of the section
``Evaluation of Experimental Results.''
In a previous paper\footnote{J.M.B. Kellogg, I.I. Rabi and
J.R. Zacharias, Phys. Rev. {\bf 50,} 472 (1936)}
called ``The Gyromagnetic
Properties of the Hydrogens,'' experiments were described
which measured the h.f.s. separation of the normal states of
atomic hydrogen' and deuterium. The magnetic moments of
the proton and the deuteron were evaluated from these
measurements by the application of the Dirac theory of the
hydrogen atom. These calculations depend on the
assumption that the interaction of the nuclear spin with the
external electron is purely electromagnetic in nature. The
values which were obtained from this experiment were 2.85
and 0.85 nuclear magnetons for the proton and deuteron,
respectively, with a precision of about 5 percent.
These values, particularly that of the proton are to be
compared with the earlier value of 2.5 given by Estermann
and Stern\footnote{I. Estermann and 0. Stern, Phys. Zeits. f. Physik
{\bf 85}, 17 (1933).} as a result of a
straightforward molecular beam deflection experiment with
molecular hydrogen. Since this value was not claimed to be
accurate to better than 10 percent the results of the two
experiments, which differ entirely in principle, could be said
to agree within the limits of error. A value of the proton
moment was also given by Lasarew and
Schubnikow\footnote{B.G. Lasarew and L.W. Schubnikow, Physik. Zeits.
Sowj. {\bf 11,} 445 (1937).} who
measured the variation of the diamagnetic susceptibility of
liquid hydrogen with temperature. They interpret the change
in susceptibility as a paramagnetic effect of the nuclear
moment and from measurements at three different
temperatures obtain two values $2.7 \pm 10$ percent and $2.3 \pm 10$
percent. However, later experiments by Estermann,
Simpson, and Stern\footnote{I. Estermann, O.C. Simpson and 0. Stern, Phys. Rev. {\bf 52,} 535 (1937).} yield a value of $2.46 \pm 3$
percent.
They used the same method as in the previous experiments
of Estermann and Stern, but with a more refined technique,
which they applied to both H$_2$ and HD molecules.
The wide divergence between the values 2.85 and 2.46 is
quite outside the sum of the limits of error. As an
explanation the possibility of some sort of spin interaction
between electron and proton not electromagnetic in
character was suggested. Another possibility was a
breakdown of the Dirac equation for the electron in the
hydrogen atom.
These considerations which bring into question the
fundamental entities of nuclear and atomic physics make a
further experimental study of the proton moment a matter
of great interest, particularly if the methods are
independent. Further interest is added to these studies by
recent theoretical investigations\footnote{J. Schwinger, Phys. Rev.
{\bf 55}, 235 (1939)) Abs. {\bf 13;}
H. Fr\"ohlich, W. Heitler and N. Kemmer, Proc. Roy. Soc.
{\bf 166A,} 154 (1938).} of the nature of the proton
and neutron moment and their composition in the deuteron.
\section*{Method}
~~~The method we employ is the molecular-beam
magnetic-resonance method$^1$ modified to apply to gas molecules. As
used in our experiment this method provides a means of
observing the reorientations of molecular and nuclear
moments which may occur with respect to a constant homogeneous
magnetic field when an oscillating or rotating magnetic field
is superposed. In our experiments this oscillating magnetic
field will be weak compared with the constant field and
perpendicular to it in direction. In the absence of any
interaction between the molecular constituents this
reorientation will occur when the Larmor frequency of
precession,
\begin{equation}
\nu = \mu H/h I
\end{equation}
and the frequency of the oscillating field are in resonance.
More generally, when there are interactions between the
molecular constituents, the resonance reorientations occur
when the frequency of the oscillating field is in resonance
with the frequency given by the Bohr relation
\begin{equation}
h \nu_{nm} = \Delta E_{nm} = E_n - E_m,
\end{equation}
where $E_n$ and $E_m$ represent the energies of two states of the
whole molecular system in the homogeneous magnetic
field. There are selection rules which govern these
transitions, and in the cases which we will discuss this
selection rule is $\Delta m = \pm 1$ where $m$ is the magnetic quantum
number of the system. It is to be emphasized that the
method detects not only transitions from state $n$ to $m$, but
also the inverse transition $m$ to $n$. One of these corresponds
to absorption of radiation and the other to stimulated
emission. As Einstein has shown these two processes are
equally probable. The above two descriptions are equivalent
in the absence of interactions within the molecule because
the energy differences between the successive states of spin
quantization are then all equal to $\mu H/I$, whence the Bohr
frequency is the same as the frequency given by Eq. (1).
To detect the reorientations, a beam of molecules is
spread by an inhomogeneous magnetic field and refocused
onto a detector by a subsequent inhomogeneous field. The
transitions between the states of different space quantization
are produced in the homogeneous field of an electromagnet
placed in the region between the two deflecting magnets. In
the gap of this electromagnet is placed a loop of wire which
is connected to a source of current of radio frequency to
produce the oscillating field at right
\begin{figure}[h]
\centerline{\resizebox{14cm}{!}{\includegraphics{fig1.gif}}}
\caption{Schematic diagram of apparatus. Longitudinal dimensions from
source to detector are drawn to scale. Flattened
copper tubes 13.5 cm long to carry radiofrequency current are not shown. They should appear between the pole pieces of magnet $C.$}
\end{figure}
angles to the steady field. If a reorientation of spin occurs in
this field the subsequent conditions in the second deflecting
field are no longer correct for refocusing and the intensity at
the detector goes down. The experimental procedure is to
vary the homogeneous field while maintaining some given
value of the frequency of the oscillating field until a
resonance is observed by a drop in intensity at the detector
and the subsequent recovery when the resonance value is
passed.
\section*{Apparatus}
~~~The apparatus used in these experiments is similar to that
used by {\mbox{R. M. K. Z.,$^1$} the chief modifications being those
which result from the use of gas molecules in the. beam.
The present apparatus is shown schematically in Fig. 1. The
outer $6^{\prime \prime}$ brass tube is divided into three parts
to facilitate
pumping, each part having its own high vacuum system. In
the source chamber, where much gas emerges from the
source, a pressure of $4 \times 10^{-5}$ mm of mercury is maintained
by two brass oil-diffusion umbrella-type pumps with
dry-ice-cooled helical baffles. The pumping speed measured at
the position of the source slit is 100 liters of air per second.
In the separating chamber, which is connected to the source
chamber only by a slit 0.05 mm wide and 3 mm high
through which the molecular beam passes, and to the main
chamber by a slit 0.1 mm wide and 6 mm high, the pressure
is of the order of $10^{-6}$ mm. The purpose of the separating
chamber is to provide vacuum isolation of the main
chamber from the gas in the source chamber. Finally in the
main chamber, where the magnets are located and where
the molecules are for the greatest length of time, a
pressure of better than
$5 \times 10^{-7}$ mm is maintained with another brass oil-diffusion
pump with a speed of 50 liters per second and with a glass
pump of the Zabel design\footnote{R.M. Zabel, Rev. Sci. Inst. {\bf 6,} 54 (1935).}. To diminish pressure fluctuations in this chamber,
an extra section of brass tube 3 ft.
long and $7^{\prime \prime}$ in diameter is added beyond the
Pirani gauge
detector to provide ballast volume. All the diffusion pumps
of the separating and main chambers pump into a glass
pump which in turn pumps into the source chamber which
serves as its fore vacuum. The two brass pumps of the
source chamber are backed by a glass pump which in turn
pumps into a Leybold three stage mercury diffusion pump
capable of pumping against 10 mm of pressure. The gas
from the Leybold pump goes back to the source feed line so
that it is used over again. With this recirculation system as
little as 6 cc of gas at NTP is required for a day's run and, if
desired, even that amount may be recovered at the end of
the day for subsequent use.
The source is mounted at the bottom of an Invar Dewar
to which it is soldered. The molecules being investigated
must pass through a long 2-mm tube in the Dewar before
entering the source so their temperature becomes
approximately that of the refrigerant in the Dewar. The Dewar is
supported at the top by a ground brass joint so mounted that
the trap can be moved both transversely and vertically
under vacuum. The ground joint permits rotation about a
vertical axis. This motion is necessary because of the canal
effect of the slit jaws. In addition to the support at the top of
the trap, two long Invar screws pass through the sides of the
outer brass tube and are used to clamp the source
rigidly in place. This was found to be necessary to
prevent random 0.001 mm lateral motions of the
source. The gas pressure inside the source is usually
of the order of 4 or 5 mm of mercury. The source slit
is 0.015 mm wide, 3 mm high, and 0.25 mm thick.
Although the source slit is made vertical optically
before the apparatus is pumped out, slight changes of
its angle with the vertical can be made under vacuum
by clamping the source box with the Invar screws
and then moving the top of the Invar trap. This
procedure bends the trap a bit and thus changes the
angle of the slit with the vertical.
The collimator consists of two sharp slit jaws
0.015 mm apart mounted eccentrically on a vertical
ground joint. Small rotations of the joint therefore
make possible slight adjustments of the collimator
position while large rotations completely remove the
collimator from the path of the molecular beam.
Although the geometry of the apparatus is such that
the collimator cannot conveniently be put halfway
between the source and the detector, it is placed as
near to this position as possible, since, as pointed out
by Manley\footnote{J.H. Manley, Phys. Rev. {\bf 49,} 921 (1936).},
this is the optimum position. The total
beam length, source to detector, is 77 cm.
The detector is of the Stern-Pirani type and is
shown diagrammatically in Fig. 2. The most
important difference between this detector and the
Pirani gauges previously used' is that use is
\begin{figure}[h]
\centerline{\resizebox{9cm}{!}{\includegraphics{fig2.gif}}}
\caption{ Horizontal cross section of the detector. The block
and tubes are 10 cm long.}
\end{figure}
made of all four arms of the Wheatstone bridge, two
electrically opposite arms being in the detector
chambers and two in the balancing chambers
whereby the sensitivity is increased. Each set of
gauge chambers, which must be of small volume to
diminish the time lag of the detector, is formed by
inlaying in solder two 1.6-mm (inside diameter)
copper tubes in a rectangular trough cut in the surface
of a brass block. A transverse hole is drilled from the
outside through both tubes to form the interconnecting
channel shown in the diagram. The faces of
the block are then surface ground and lapped. The slit
jaws shown in the diagram are carefully lapped brass
blocks screwed to the lapped faces of the gauge
block. The slit itself is formed by forcing these slit
blocks together against 0.015 mm spacers of
aluminum foil to form a channel 0.015 mm wide, 2
mm high, and 4.5 mm long. The gauge wires are
made of nickel ribbon 9 cm long, 0.25 mm wide, and
0.004 mm thick. These ribbons are kept taut by small
spiral springs of 6-mil nickel wire formed on the
point of a needle. Nickel leads are brought out of the
ends of the gauge chambers through small glass tubes
which are ground to fit snugly into the copper tubes.
The ends are then sealed with a drop of clear glyptal
lacquer. Finally, the gauge is baked at a temperature
of $150^{\circ}$ for three days.
To prevent electrical disturbances in the detector
circuit when the oscillator is turned on, the entire
circuit is well shielded and by-pass condensers are
attached between the leads of the hot wire arms of the
bridge and ground.
The portion of the outer brass tube near the
detector is surrounded with rubber tubing through
which is circulated water from a large tank which
stays at the average room temperature. The whole
system is lagged with crumpled Alfol (aluminum
foil) to provide thermal insulation. This is necessary
to diminish the large galvanometer drifts which
accompany even small changes of the detector
temperature. After the apparatus has been opened to
air, it takes about a week for the detector to outgas
sufficiently for accurate measurements to be possible.
Therefore one oil diffusion pump and a mechanical
pump are left running all the time so that ``sticking''
vacuum is maintained even when the apparatus is not
in use.
The inhomogeneous deflecting magnets $A$ and $B$ (Fig. 1)
are of the type described by Millman, Rabi, and
Zacharias\footnote{S. Millman, I.I. Rabi and J.R. Zacharias,
Phys. Rev. {\bf 53,} 384 (1938).}.
They are 19 cm and 24 cm long, respectively. The gaps are
bounded by two cylindrical surfaces, one convex of radius
1.24 mm and the other concave of radius 1.47 mm. The gap
width is 1.06 mm. Each magnet is wound with four turns of
heavy, water-cooled copper. With 300 amperes the field is
about 12,000 gauss and the gradient of the field about
90,000 gauss/cm.
The magnet $C$ producing the homogeneous magnetic field
is a single yoke magnet of Armco iron carefully annealed in
a hydrogen atmosphere. It is wound with 9 turns of heavy,
water-cooled copper. The surfaces of the air gap are 0.6 cm
apart, 3.2 cm high, and 15 cm long. They are surface ground
and lapped, and the gap is spaced with a ground and lapped
spacer so that the air gap should be constant to about
0.05 percent.
Although the gradients of the two inhomogeneous
magnetic fields must be in opposite directions for
refocusing to be possible, it is essential$^1$ that all three
magnetic fields be in the same direction and that the three
magnets be close together to prevent reorientations of the
molecules in the regions of weak rapidly changing field
between the magnets. To further diminish these
reorientations iron slabs are attached to the ends of the
inhomogeneous magnets to increase the value of the
magnetic field in the gaps between the magnets and to make
the changes of field more gradual. With these precautions
the refocused beam intensity in the present apparatus is
about 95 percent of the beam intensity in the absence of the
magnetic fields.
The high frequency magnetic field within the air gap of
the $C$ magnet is produced by a radio-frequency current in a
``hairpin'' of the type described in detail by
Millman\footnote{S. Millman, Phys. Rev. {55}, 628 (1939).}. The
two horizontal wires between which the beam passes are of
flattened $1/8$ in. (outside diameter) copper tubing 13.5 cm long
and 1 mm apart through which water flows for cooling
purposes. This produces a magnetic field of about 3 gauss
per ampere at the position of the beam. As much as
50 amperes of current at several megacycles have been
used. The current comes from a single turn secondary
inductively coupled to the tank coil of a conventional
Hartley oscillator driven by an Eimac 250 TL tube.
Frequencies between 0.5 and 16 megacycles have been
used. The oscillator is sufficiently stable for the frequency
to remain constant to 0.02 percent over the course of a day's
run.
\section*{Calibration}
~~~The measurement of gyromagnetic ratios by the method
of this experiment depends on the absolute measurement of
only two quantities as is shown by Eq. (1): (a) the frequency
of the oscillatory magnetic field and (b) the strength of the
homogeneous magnetic field.
The frequency is measured with the same General Radio
Type 620-A heterodyne frequency meter used by
R. M. K. Z.$^1$ with which frequencies can be measured to 0.01 percent.
In our laboratory this meter was compared with another
meter of the same type and found to agree with it to 0.01
percent.
The homogeneous magnet $C$ is calibrated as a function of
the exciting current in the magnet windings. The current is
measured by the potential drop across a shunt through
which the current passes. This potential drop is measured to
0.01 percent with a type $K$ potentiometer. As the same shunt
is used in the calibration and in the measurement of
moments, its absolute resistance is of no importance.
In the process of calibration it was found that the field
accompanying a current in the windings depended on the
way the magnet was demagnetized. Therefore a
standardized demagnetization procedure was used in both
the calibration and the subsequent measurements.
Calibrations were made for several values of the current in the $A$
and $B$ magnets. When this procedure was followed, the
magnetic fields were found to be reproducible to 0.2
percent.
The calibration of the magnet was performed with flip
coils, ballistic galvanometer and mutual inductance in the
manner described by \mbox{ R. M. K. Z.$^1$} Two Leeds and Northrup
50-millihenry mutual inductances were used in the
calibration. The values of these two mutual
inductances, rated by the manufacturer to $\pm 0.5$ percent, were
found to vary about 0.5 percent depending on humidity.
This uncertainty together with a slight uncertainty in the
area of the flip coil is the chief source of error in the
experiment. The absolute calibration of the field and hence the
absolute values of the magnetic moments should be good to
about 0.5 percent.
Since the moments of a number of nuclei have been
measured$^1$ \footnote{P. Kusch and S. Millman, Bull. Am.
Phys. Soc. {\bf 14,} No. 3, Abs. 35; S. Millman and P. Kusch,
Bull. Am. Phys. Soc. {\bf 14,} No. 3, Abs. 36; P. Kusch, S. Millman
and I.I. Rabi, Phys. Rev. {\bf 55,} 1176 (1939).} on another apparatus
with an independent field
calibration, it is of interest to compare the results of the
measurement of some one nuclear moment on the two
apparatuses. For this purpose we chose fluorine which was
available to us in the form of CCl$_2$F$_2$. We found resonance
minima which may be ascribed to fluorine. While these
minima were not as narrow as those of hydrogen and could
not be located as accurately, it was still possible to measure
the moment of fluorine to an accuracy well within the limits
of error claimed for the calibration of the magnets of either
apparatus. The fluorine moment we
obtain\footnote{J.M.B. Kellogg, I.I. Rabi, N.F. Ramsey, Jr.
and J.R. Zacharias, Phys. Rev. {\bf 55,} 595 (1939), Abs. 24.
} is $2.623 \pm 0.018$
nuclear magnetons. This is to be compared with the value
$2.622 \pm 0.013$ nuclear magnetons previously reported by
R. M. K. Z.
\section*{Procedure}
~~~Although the final values for the nuclear magnetic
moments do not depend on the accuracy of the alignment of
the apparatus or of the parallelism of the source, collimator,
and detector slits, these adjustments must be carefully made
in order that the beam intensity and deflecting power of the
apparatus may be large enough. As in previous molecular
beam experiments$^{2, 9}$ much of the alignment is
accomplished under vacuum with the beam, although part is
done optically in advance. The three magnets are rigidly
attached to a brass base plate which, together with the
separating plate and foreslit between the separating and
detector chambers, can be removed from the outer brass
tube. As the magnets cannot be moved under vacuum, they,
and the foreslit, are optically aligned on the
base plate when it is outside the brass tube by running a
tightly stretched number 40 copper wire down the line of
the beam. The base plate and magnets are then inserted in
the tube and clamped in position. The source slit, foreslits,
collimator, and detector slit are approximately aligned
optically and are carefully made vertical to make them
parallel. The two wires which carry the high frequency
current are set optically so that the beam passes between
them. After these alignments have been made and the
apparatus evacuated, gas is introduced into the source feed
line and the beam is found by trial with the collimator
removed from the beam path. The collimator is then
introduced and the position of the beam relative to the
magnetic field is found by moving the source, collimator,
detector, and narrowest foreslit under vacuum. In this way
the beam may be made to run parallel to the magnets and at
any desired position in the air gap.
The best currents for use in the deflecting and refocusing
magnets $A$ and $B$ are determined by trial. The ratio of the
currents in the two magnets is first selected so that the
refocusing is a maximum. Then with this ratio preserved the
total current is adjusted to a value large enough so that most
of the molecules which have a constituent moment
reoriented in the $C$ field are not refocused, and small enough
so that the deflections of the molecules are not too large to
preclude good refocusing when no reorientations occur. A
careful adjustment of the total current is particularly
important in the measurement of the deuteron moment with
HD where the deuteron magnetic moment being studied
is in the same molecule with a much larger proton moment.
The resonance minima are observed by keeping the
currents in the $A$ and $B$ magnets and the frequency of the
oscillator constant and varying the current in the $C$ magnet
which produces the homogeneous field. A measure of the
number of molecular reorientations produced by the
oscillating field is obtained from the change in beam intensity
which occurs when the oscillator is turned off or on. As
there is always some drift (of the order of $1/2$ cm a minute) of
the detector galvanometer spot, observations of the change
in beam intensity are made by turning the oscillator
on and off at 15-second intervals for two minutes and then
averaging the changes of the galvanometer deflection. In all
of the experiments here described the Dewar to which the
source is attached was filled with liquid nitrogen.
It is very important in these experiments that the correct
high frequency current be used, since, as follows from the
detailed theory of the reorientation probabilities as
developed by Rabi\footnote{I.I. Rabi, Phys. Rev. {\bf 51,} 652 (1937).
}, the half-widths of the resonance
minima are directly dependent on the strength of the
oscillating magnetic field. If too much current is used the widths
of the resonance minima increase proportionally to the
current and if too little is used the depths of the minima
decrease rapidly. The theoretical optimum strength of the
oscillating field and the theoretical half-widths of the curves
when this field is used are discussed in the next section. As
is shown there, the optimum field depends both on the
magnitude of the moment being measured and on the
velocity and hence the mass of the molecule. In practice we
usually determine the best values for the oscillatory current
by trial. Since the positions of the minima are not affected
by the magnitude of the oscillatory current, this
determination is made only approximately.
\section*{Experimental Results}
~~~In this section we shall present our data in the form of
typical curves which represent beam intensity at the detector
as a function of the value of the field in the $C$ magnet while
a fixed frequency and amplitude of the oscillating current is
maintained. These curves we call resonance curves and the
minima of intensity which are observed, resonance minima.
Of physical interest are the numbers, locations, depths,
half-widths, and asymmetries of these resonance minima.
These curves are closely analogous to the absorption and
stimulated emission lines of spectroscopy except for two
distinct differences. First, instead of measuring the
absorption of the radiation we measure the fraction of
molecules which make transitions. Second, we employ a
definite fixed frequency and shift the positions of the energy
levels to correspond to this frequency by varying the magnetic
field rather than scanning
the spectrum through a range of frequencies at a fixed value
of the field. Thus our results are analogous to what would be
obtained if one measured the Zeeman pattern of a spectral
line by measuring the absorption of some narrow spectral
line as a function of the magnetic field in which the absorber
is placed. The Zeeman lines would successively pass across
this frequency interval and one would obtain an unusual set
of absorption lines from which the Zeeman levels could be
deduced. In the present experiment with hydrogen
molecules, as tile strength of the magnetic field is varied the
resonance minima occur in groups, the positions of the
groups being such that the Larmor precession frequency of a
constituent magnetic moment of the molecule is near the
frequency of the oscillating field. Since the regions of field
that are of-interest in the experiment are those in which the
resonance minima occur, only these are included in the
accompanying figures. It is to be emphasized, however, that
between groups of minima there are large regions of the
magnetic field intensity in which no resonance minima
occur.
The gases used in these experiments were H$_2$, D$_2$ and HD,
the latter having been purified for us by Drs. Brickwedde
and Scott of the National Bureau of Standards. All
experiments were performed with the source at liquid
nitrogen temperature. For equilibrium at this temperature,
the distributions of the molecules in the different rotational
states are as given in Table I. The resultant nuclear spins to
be associated with the different rotational states of H$_2$ and
D$_2$ are also included in the table. These are different for
even and odd rotational states since, as is discussed
\newpage
TABLE I. {\it Relative abundance of different rotational
states at 78$^{\circ}$ K. To obtain the abundance of a state
with given $I$ and $J$ ana, $m_I$ and $m_J$ divide the
concentration given by the statistical weight. The
calculations were made assuming an orlho-para ratio
of 3 to 1 for H$_2$ and 1 to 2 for D$_2$. For HD there is no
quantum number of total spin.}
\begin{center}
\begin{tabular}{c|ccc|ccc|cc}
\hline \hline
& \multicolumn{3}{c|}{H$_2$}&\multicolumn{3}{c|}{D$_2$}&\multicolumn{2}{c}{HD}\\
\cline{2-9}
~~J~~&Total&~Stat.~&~Rel.~&Total.&~Stat.~&~Rel.~&~Stat.~&~Rel~\\
&Spin&Wt.&Conc.&Spin&Wt.&Conc.&Wt.&Conc.\\
\hline
0&0&1&0.248&0.2&6&0.559&6&0.628\\
1&1&9&0.745&1&9&0.328&18&0.369\\
2&0&5&0.003&0.2&30&0.105&30&0.003\\
3&1&21&0.000&1&21&0.004&42&0.000\\
\hline \hline
\end{tabular}
\end{center}
\begin{figure}[h]
\centerline{\resizebox{8cm}{!}{\includegraphics{fig3.gif}}}
\caption{Resonance curve for HD molecules chiefly
showing transition of the proton spin (or the zero rotational
state.}
\end{figure}
cussed in the theories of the rotational specific heat of
homonuclear hydrogen molecules, only para-H$_2$ and
ortho-D$_2$ may exist in the even rotational states and vice versa for
odd.
Figure 3 is a resonance curve taken with HD
\begin{figure}[h]
\centerline{\resizebox{10cm}{!}{\includegraphics{fig4.gif}}}
\caption{Resonance curve for HD molecules chiefly showing transition
of the deuteron spin for the zero rotational state.}
\end{figure}
molecules at a frequency of 4.000 million cycles in a region
of magnetic field varied from about 900 to 1000 gauss. In
this region the Larmor precession frequency of the proton
is near the oscillator frequency. The very deep minimum
arises from the reorientation of the proton in this molecule
when the molecule is in a state of zero rotation as will be
made clear in the next section. The small fine structure, on
both sides is due to the same reorientation process, but in
molecules which are in the first rotational state. The
discussion of the results in the first rotational state and of
the similar ones which arise from reorientations of the
deuteron in the first rotational states of HD and D$_2$
molecules will be reserved for another paper. Fig. 4 is a
curve similar to that of Fig. 3 but taken in an entirely
different region of frequency and magnetic field, the region
being such that the Larmor frequency of the deuteron is
near the oscillator frequency. Here again the deep
minimum corresponds to reorientations of the deuteron
when the molecule is in the zero rotational state white the
structure on the two sides arises from similar transitions but
in the first rotational state. Fig. 5 is a curve obtained with
D$_2$. The deep central minimum arises from the reorientation
of the deuteron in molecules in the zero rotational state and
the structure on either side is due to molecules in the first
rotational state.
All of these groups of resonance minima have been
observed at more than one frequency. It is found that the
ratios of $f$, the frequency of the
\begin{figure}[h]
\centerline{\resizebox{10cm}{!}{\includegraphics{fig5.gif}}}
\caption{Resonance curve for D$_2$ molecules showing
transition of the resultant nuclear spin for the zero rotational
state.}
\end{figure}
oscillating field, to $H,$ the magnitude of the homogeneous
field at which the principal minimum of any moment
occurs, are constant. This constancy of $f/H$ would be
expected from Eq. (1) and is strong support for the
assumption that the deepest minimum corresponds to the
interaction of a magnetic moment with the applied magnetic
field, since the constancy of $f/H$ shows that the energy of
interaction is proportional to the strength of the field.
The justification for identifying the resonance minima of
Fig. 3 with proton reorientations and Fig. 4 with deuteron
reorientations is that a group of minima with the same $f/H$
as Fig. 3 is found in H$_2$, and a group with the same $f/H$ as
Fig. 4 in D$_2$ (see Fig. 5).
Each of these curves alone is slightly asymmetrical as
would be expected from the discussion of Millman$^{10}$. As
pointed out by Millman,
\begin{figure}[h]
\centerline{\resizebox{9cm}{!}{\includegraphics{fig6.gif}}}
\caption{Radiofrequency spectrum of ortho-H$_2$ molecules
arising from transitions of the resultant nuclear spin. The
path of the beam in the radiofrequency field is 13.5 cm.}
\end{figure}
the signs of the moments can be deduced from the
asymmetry. The signs of the proton and deuteron moments
found in this way are positive in agreement with the results
of K. R. Z.$^2$
The depth of the minimum of H in HD corresponds to
the occurrence of the reorientation process in about 75
percent of the molecules in the zero rotational state. That
this quantity is not greater is to be expected on the basis of
the theory given by Rabi and by R. M. K. Z. Since the
molecules have a Maxwell distribution of velocity, some of
the molecules are not under the influence of the oscillating
field long enough, because of their high speed, to make the
transition probability unity, while others, because of their
low speed, are in the field too long and are partially returned
to their original state. A discussion of the depths of the
deuteron minima in HD and D$_2$ will be given in a later
paper.
The minimal widths of the resonance minima are given
approximately by the relation $t \Delta \nu \cong 1$
where $t = L/v$ is the
time spent by the molecules in the oscillating field and $\Delta \nu$
is related to the width of the minima in gauss by $\Delta \nu = \mu \Delta
H/hI$.
The minimal width of the resonance minima at half-intensity
(half-depth) is therefore given by $\Delta H \cong hIv/\mu L$. It the
resonance minima are considerably broader than this it is an
indication of lack of resolution or of fine structure in the
minima. If the oscillating field is made very
\begin{figure}[h]
\centerline{\resizebox{10cm}{!}{\includegraphics{fig7.gif}}}
\caption{Radiofrequency spectrum of ortho-H$_2$ molecules
arising from transitions of the resultant nuclear spin. The
path of the beam in the radiofrequency field is only 2.7 cm.
The radiofrequency current was too large.}
\end{figure}
weak, the depth of the resonance minimum decreases, but
its width is approximately unchanged; whereas, if the
oscillating field is made too strong, the half-width of the
minimum increases. The best value of the oscillating field
for maximum depth of the resonance minimum and
narrowest width when we deal with resolved lines is
obtained when the amplitude of the oscillating field is
approximately equal to $hIv/\mu L$. This relation was found to
hold experimentally and is expected from the theory$^1$.
In Fig. 6 we have a set of six resonance minima which
arise from reorientations of the nuclei in Hi molecules in
the first rotational state. Since, as is shown in Table I, the
zero rotational state has zero resultant nuclear spin, this
state has no resultant nuclear magnetic moment and consequently
gives rise to no nuclear resonance minima. The
observed resonance minima can be located to within 0.5
gauss. The depths of the minima are approximately equal.
The sum of the depths of the resonance minima is
approximately equal to the total intensity of ortho-H$_2$ molecules
which reach the detector. This is less than the amount to be
expected under ideal conditions since each minimum is
due to transitions between two states and in our experiment
transitions from $b$ to $a$ are counted as well as those from $a$
to $b$. Since, as will be shown in the next section, there are
nine states all told and since the molecules are equally
distributed amongst the states, the depth of each minimum
should represent $2/9$ of the intensity and the sum of the six
minima $12/9$ of the total intensity. The effect of the velocity
distribution which was discussed above results in a 25
percent reduction of this intensity, so it is reasonable that
the total depth should be approximately equal to
the Intensity of ortho-H$_2$ molecules as found
experimentally.
Figure 7 represents a curve which was obtained with H$_2$
molecules in an earlier form of our apparatus in which the
$C$ magnet as well as the wires which produce the
oscillating field were about a fourth as long as they are
now. This curve was one of the first to be observed and a
value of the oscillating field much greater than the
optimum value discussed above was
\begin{figure}[h]
\centerline{\resizebox{10cm}{!}{\includegraphics{fig8.gif}}}
\caption{ Radiofrequency spectrum for ortho-H$_2$ molecules
arising from transitions of the resultant nuclear spin. The
path of the beam in the radiofrequency field is 2.7 cm.
Optimum r.f. current for this case is 2 amp.}
\end{figure}
used. Fig. 8 is another curve on the same apparatus but with
the use of the optimum oscillating field. A comparison of
these two curves shows how the stronger oscillating field
broadens the resonance minima and causes the curious
over-lappings which are responsible for the peculiar form
of the curve in Fig. 7. By making the $C$ magnet and the
oscillating field four times longer, we were able to reduce
the amplitude of the oscillating field by a factor of four
since the time which the molecules are under its influence
is four times as great. Fig. 6 when compared to Fig. 8
shows the effect of this improvement. The lines in Fig. 6
are much narrower and the resolution is consequently
higher. There are six narrow and well separated minima,
whereas in Fig. 8 there are only five separated minima.
Figure 9 shows how the positions of the H$_2$
\begin{figure}[h]
\centerline{\resizebox{10cm}{!}{\includegraphics{fig9.gif}}}
\caption{Variation of radiofrequency spectrum with magnetic
field to show the progress of the Paschen-Back effect.
The spectra have been shifted to make the positions of $H_0$
coincide. $H_0$ was obtained directly by adding 10 percent of
HD to the sample of H$_2$. In this way one can tell the shift in
field of the individual lines.}
\end{figure}
resonance minima shift when observations are made at
different frequencies. The vertical lines show the observed
positions of the peaks at different frequencies. The positions
of the different groups of minima in this figure are so
adjusted that the field position marked zero is the position on
the different groups at which the Larmor frequency of a
proton is equal to the oscillator frequency. The reason for
the shifts in the peak positions will be discussed in the next
section.
\section*{Evaluation of Experimental Results}
~~~In this section we shall first deduce the magnetic
moments of the proton and the deuteron from the locations
in field and frequency of the deep resonance minima with
HD and D$_2$ molecules as shown in Figs. 3, 4, and 5. Then
we shall discuss in detail the results of the theory of the
energy levels of ortho-H$_2$ in the first rotational state and
apply them to the experimental results obtained from
curves like Fig. 6. Finally we shall present a derivation of
the formulae which we use.
When the molecule of HD or D$_2$ is in the state with
rotational angular momentum, $J$, equal to zero, the
internuclear axis is oriented in every direction with equal
probability. Since these molecules have no resultant
electronic moment ($^1\Sigma$ state) all interactions between each
nucleus and the rest of the molecule become independent of
the orientation of the nucleus in the external magnetic field.
This follows\\
TABLE II. {\it Summary of data taken during the spring of
1938 for the determinations of the proton and deuteron
moments. The magnet used for these measurements
was calibrated with a flip coil and mutual inductance.
The path in the r.f. field was only 2,7 cm.}
\begin{center}
\begin{tabular}{ccccc}
\hline \hline
&Moment&Frequency in&Field in&Nuclear\\
Substance&Measured&Megacycles&Gauss&Moment\\
\hline
H$_2$& H & 5.662 & 1336 & 2.781\\
H$_2$ & H & 7.000&1655&2.775\\
H$_2$&H&8.923&2107&2.779\\
H$_2$& H& 14.007&3299&2.786\\
HD&H&10.420&2456&2.785\\
HD&H&10.420&2461&2.780\\
HD&H&10.430&2457&2.786\\
HD&H&13.535&3180&2.793\\
HD&H&13.190&3106&2.787\\
HD&H&13.212&3106&2.791\\
D$_2$&D&2.103&3232&0.8536\\
D$_2$&D&3.143&4832&0.8537\\
D$_2$&D&2.103&3224&0.8560\\
HD&D&2.021&3106&0.8550\\
\hline \hline
\end{tabular}
\end{center}
because the axis of the molecule has all orientations with
equal probability independent of the nuclear spin
orientation. Further, the magnetic interaction between the
two nuclear magnets averages out to zero, because the
mutual energy of two dipoles averaged over all possible
orientations of the line joining them is zero. Therefore, each
of the two nuclei in the molecule has a definite energy in the
external magnetic field depending on its own orientation but
independent of the orientation of the other nucleus. These
are exactly the circumstances under which the theory of Eq.
(1) may be applied as well as the rigorous
theory$^{13}$ for the
transition probability in the oscillating field.
The experimental results show that these predictions are
indeed true and that the half-widths and depths of the
minima are as to be expected and that the ratio $f/H $ is
constant. From these values of $f/H$, from Eq. (1), and from
the known results that the spin of the proton is $1/2$ and of the
deuteron 1, we compute the value of the magnetic
moments. The results are tabulated in Table II, and show
that those obtained for the deuteron with D$_2$ and HD, and
for the proton with H$_2$ and HD, are in close agreement.
These measurements were made when the homogeneous
field magnet was the small one permitting a length of path
in the radiofrequency field of only 2.7 cm. However, this
magnet is the one that is carefully calibrated so that our
values of the moments depend on this calibration. The
averages, obtained by weighting proportionally to
frequency, are $2.785 \pm 0.02$ nuclear magnetons and
$0.855 \pm 0.006$ nuclear magnetons for the proton and deuteron,
respectively.
The longer magnet, put into the apparatus later, was not
directly calibrated at all. Its field is known only in terms of
the proton moment.
A very careful determination of the ratio $\mu_P/\mu_D$ was
made with the longer magnet. The result is
$\mu_P/\mu_D = 2570 \pm 0.001.$ This agrees with the value
$3.259 \pm 0.007$
obtained with the short magnet. The difference in the
precisions of the two results is chiefly due to the difference
between the lengths of the two radiofrequency fields.
When the rotational quantum number is greater than
zero, the various interactions mentioned in the previous
paragraph are no longer
zero and the energy of a nucleus in the external
magnetic field depends not only on its own
orientation but also on the orientation of the other
nucleus and on the orientation of the rotational
angular momentum of the molecule. We shall
discuss only ortho-H$_2$ in the first rotational state
since, as discussed in the preceding section, this is
the only state which contributes resonance minima
when H$_2$ is observed at $78^{\circ}$K.
In ortho-H$_2$ the two nuclei are in a state where the
spins of the two nuclei are ``parallel,'' that is, the
total nuclear spin angular momentum is 1.
Transitions between this state and the state in which
the total spin is zero cannot occur in our experiment
for two reasons. First, such a change in spin involves
a change from ortho-H$_2$ to para-H$_2$ and hence a
change in rotational quantum number for which the
energy difference is so large that it involves a
frequency thousands of times as great as those we
employ. Second, this transition is highly forbidden
because it is of the triplet to singlet type.
From these considerations we see that in an
externally applied magnetic field, in which the
energy of the protons due to interaction with the
field is much larger than the magnetic energy of
interaction between the two protons and of the
protons with the molecular rotation, we obtain nine
energy levels. These correspond to the three
orientations of the total nuclear spin $I$ with each of
which three orientations of the rotation vector $J$ are
possible. The resonance minima shown in Fig. 6
arise from the six transitions and their inverses
which can occur between these nine magnetic levels
subject to the restriction $\Delta m_I = \pm 1$.
The theory of the location of these levels is very similar
to that of the magnetic levels of a multiplet of the
Paschen-Back effect. If we assume that the orientation dependent
energy of the ortho-H$_2$ is due solely to interactions with the
external field and to the spin-spin magnetic interaction
between the nuclei and the magnetic interaction between
the nuclear spins and the molecular rotation, then the
theoretical expression for the total energy of the ortho-H$_2$
molecule in the magnetic field (apart from constants
independent of the orientation) becomes
$$
E = - \mu_P (\boldsymbol{\sigma}_1 + \boldsymbol{\sigma}_2) \cdot {\bf H} - \mu_R
~{\bf J} \cdot {\bf H} - \mu_P H' (\boldsymbol{\sigma}_1 + \boldsymbol{\sigma}_2)
\cdot {\bf J}\\
$$
\begin{equation}
+ \frac{\mu_P}{r^3} \left\{ \boldsymbol{\sigma}_1 \cdot \boldsymbol{\sigma}_2
- 3 (\boldsymbol{\sigma}_1 \cdot {\bf r}) (\boldsymbol{\sigma}_2 \cdot {\bf r})/r^2
\right\},
\end{equation}
where $\mu_P$ and $\mu_R$ are the proton and rotational magnetic
moments, $\boldsymbol{\sigma}_1$ and $\boldsymbol{\sigma}_2$ the
Pauli matrices for the nuclear
spins, $J$, the rotational angular momentum in units of $h/2 \pi$,
and $r$ the radius vector joining the two nuclei. The first
term is the energy of the two proton moments in the
external magnetic field $H,$ which we take in the $z$ direction.
This term can also be written as $2 \mu_P {\bf I} \cdot {\bf H}$
since, as has
already been stated, the two proton spins, each of value $1/2$,
combine to form a total proton spin $I$ of value 1, and the
two proton moments $\mu_P$ combine to form a total moment
equal to $2 \mu_P$. The second term is the energy arising from
the interaction of the rotational magnetic moment of the
molecule with the external magnetic field. The third term is
the energy of interaction between the proton moment and
the field $H'$ caused by the rotation of the molecule at the
position of the protons.\\
TABLE III. \it Energy levels for the first rotational state of ortho-H$_2$ molecules in a magnetic field, $\alpha= \mu_R/2 \mu_P$.}
{\scriptsize
\begin{center}
\begin{tabular}{rr|ll}
\hline \hline
$m_J$&$m_I$&~~~~~~~Energy\\
\hline
1&1& $- 2 \mu_P \{H + \alpha H - H^{\prime \prime}/5$&$ + H'\}$\\
1&0& $- 2 \mu_P \{ \alpha H + 2 H^{\prime \prime}/5$&$- 3 H^{\prime
\prime}/5 - H')^2/H (1 - \alpha)\}$\\
1&$-1$&$ - 2 \mu_P \{ H + \alpha H - H^{\prime \prime}/5$&$ - H' - [(3
H^{\prime \prime}/5 + H')^2 + 2 (3H^{\prime \prime}/5)^2]/H(1 - \alpha)\}$\\
0&1&$-2 \mu_P \{+2 H^{\prime \prime}/5$&$ +(3H^{\prime \prime}/5
- H')^2/H(1 - \alpha)\}$\\
0&0&$- 2 \mu_P \{ -4H^{\prime \prime}/5\}$&\\
0&$-1$&$- 2 \mu_P \{ -H +2H^{\prime \prime}/5$&$ - (3H^{\prime
\prime}/5 - H')^2/H (1 - \alpha)\}$\\
$-1$&1&$- 2 \mu_P \{ H - \alpha H - H^{\prime \prime}/5$&$ - H'[(3H^{\prime
\prime}/5 + H')^2 + 2 (3H^{\prime \prime}/5)^2]/H(1 - \alpha) \}$\\
$-1$&0&$- 2 \mu_P \{ - \alpha H + 2H^{\prime \prime}/5$&$+ (3H^{\prime
\prime}/5 - H')^2/H(1 - \alpha)\}$\\
$-1$&$-1$&$- 2 \mu_P \{ - H - \alpha H - H^{\prime \prime}/5$&$ + H'\}$\\
\hline \hline
\end{tabular}
\end{center}
}
This field is parallel to the rotational angular momentum $J$.
The last term is the well-known expression for the
interaction between two magnets separated by a distance $r$,
which in this case is the internuclear distance.
From this form of interaction the energy-levels of the
molecule may be obtained by the Schrodinger perturbation
theory. The calculation is given at the end of this section.
The results, good to the second-order perturbation theory,
are given in Table III, where the resultant energy levels are
classified by the quantum numbers $m_I$ and $m_J$, which are
the magnetic quantum numbers of the total nuclear spin and
of the molecular rotation, respectively, where each takes
the values $1,~ 0$ and $-1.$
The terms involving $(H')^2$ and $(H^{\prime \prime})^2$ represent
corrections to the energies which are inversely proportional
to the external field. They arise because the magnetic field
is not sufficiently large for the Paschen-Back effect to be
complete. These corrections though small, are nevertheless
measurable. The quantity $H^{\prime \prime}$ is the value of
$\mu_P/r^2$ and
comes from the term which involves the mutual magnetic
energy of the two protons and is independent of the value
of the external field. $H^{\prime \prime}$ is thus the field at
one proton due to the magnetic moment of the other.
This set of energy levels plus appropriate selection rules
will give the observed spectrum at frequencies equal to the
energy differences divided by $h$, as is shown by Eq. (2). At
high fields where the first three terms in the energy
expression are much larger than the last two, we have
rigorous selection rules, namely:
$$
\Delta m_I = \pm 1,
$$
or
$$
\Delta m_J = \pm 1.
$$
The first corresponds to a change in the orientation
of the total proton spin by one unit while the
component of the rotational angular momentum remains
unchanged; while in the second the orientation of the
nuclear spin remains fixed while the rotational $m_J$ changes.
Since the above mentioned corrections are small compared
with the other terms we obtain two groups of lines, one of
which corresponds to transitions in which $m_I$ changes and
the other when $m_J$ changes. The first set (of six) is grouped
around the region where $f=2 \mu_P H/h$ and the other set of six
around the region $f = \mu_R H/h$.
In this paper only the first group will be discussed. The
second group does not yield any new information except
the magnetic moment of rotation and will be discussed in a
paper devoted to the rotational moments of H$_2$, D$_2$ and HD.
The six lines which we will consider correspond to the
energy differences given in energy units in Table IV.
We must now recall that we use a fixed oscillator
frequency, $f_0$, and vary the magnetic field in the
neighborhood of $H_0$, where $H_0$ is defined by
$H_0 = h f_0/2 \mu_P$.
$H_0$ is thus the field at which the Larmor frequency of the
proton would be equal to the oscillating frequency applied.
Setting $\Delta E$ as given in Table IV equal to $2 \mu_P H_0$ and
dividing through by $2 \mu_P$ it can be seen that resonances will
occur for the values of the magnetic field listed in Table V.
Except for the effect of the small last terms in Table V,
we see that the resonances are symmetrical about the
position $H_0$. This circumstance permits us to deduce a
value of the proton moment from the center of the pattern
formed by the six resonance minima with the results given
in Table II.
A unique identification of the transitions producing the
minima of the curve of Fig. 6 makes possible an evaluation
of the interaction \\
TABLE IV. {\it Changes of energy associated with a change of the orientation of the total nuclear spin by $\pm 1$ for ortho-H$_2$ molecules in the first rotational state.}
{\small
\begin{center}
\begin{tabular}{cr|ll}
\hline \hline
$m_J$&$\Delta m_I$& ~~~~~~$\Delta E$&\\
\hline
1&$0 \rightarrow 1$& $ 2 \mu_P \{ H - (3 H^{\prime \prime}/5 - H')$ &$+
(3 H^{\prime \prime}/5 - H')^2/H(1 - \alpha)\}$\\
1&$- 1 \rightarrow 0$&$2 \mu_P \{ H + (3 H^{\prime \prime}/5 + H')$&$+ 6
H^{\prime \prime}/5 (3H^{\prime \prime}/5 + 2 H')/H(1 - \alpha)\}$\\
0&$0 \rightarrow 1$& $2 \mu_P \{ H+ 6H^{\prime \prime}/5$&$+(3H^{\prime \prime}/5
- H')^2/H (1 - \alpha)\}$\\
0&$-1 \rightarrow 0$&$ 2 \mu_P \{ H - 6H^{\prime \prime}/5$&$+ (3H^{\prime
\prime}/5 - H')^2 /H(1 - \alpha)\}$\\
$-1$& $0 \rightarrow 1$& $2 \mu_P \{ H - (3 H^{\prime \prime} /5 + H')$&$+
6 H^{\prime \prime}/5(3H^{\prime \prime}/5 + 2H')/H(1 - \alpha)\}$\\
$-1$&$- 1 \rightarrow 0$& $2 \mu_P \{ H+ (3H^{\prime \prime}/5 - H')$&$+(3H^{\prime \prime}/5 - H')^2/H(1 - \alpha)\}$\\
\hline \hline
\end{tabular}
\end{center}
}
\newpage
TABLE V. {\it Magnetic fields at which resonances will occur for
a fixed oscillator frequency $f_0 = 2 \mu_P H_0/h$.}
\begin{center}{\small
\begin{tabular}{cr|ll}
\hline \hline
$m_J$&$\Delta m_I$& Magnetic Fields&\\
\hline
1&$0 \rightarrow 1$& $H_0 + (3 H^{\prime \prime}/5 - H')$ &$+
(3 H^{\prime \prime}/5 - H')^2/H(1 - \alpha)$\\
1&$- 1 \rightarrow 0$&$H_0 - (3 H^{\prime \prime}/5 + H')$&$- 6
H^{\prime \prime}/5 (3H^{\prime \prime}/5+ 2H')/H(1 - \alpha)$\\
0&$0 \rightarrow 1$& $H_0 - 6H^{\prime \prime}/5$&$-(3H^{\prime \prime}/5
- H')^2/H (1 - \alpha)$\\
0&$-1 \rightarrow 0$&$ H_0 + 6H^{\prime \prime}/5$&$- (3H^{\prime
\prime}/5 - H')^2 /H(1 - \alpha)$\\
$-1$& $0 \rightarrow 1$& $H_0 + (3 H^{\prime \prime} /5 + H')$&$-
6 H^{\prime \prime}/5(3H^{\prime \prime}/5 + 2H')/H(1 - \alpha)$\\
$-1$&$- 1 \rightarrow 0$& $H_0- (3H^{\prime \prime}/5 - H')$&$-(3H^{\prime \prime}/5 - H')^2/H(1 - \alpha)$\\
\hline \hline
\end{tabular}}q
\end{center}
constants $H'$ and $H^{\prime \prime}$. This identification is
made possible by the
asymmetry of the positions of the two outermost minima
which is especially noticeable in low field. In fact Fig. 9
and Table VI show clearly that the minima labeled
$A_L, B_L, A_R$
and $B_R$ are symmetrically located about $H_0$ at all
frequencies, while $C_L$ and $C_R$ only approach symmetry at
high frequency.
Setting $(1 - \alpha) = 0.843 \approx 1$, reference to Table V reveals
two things. First, if {\it four} minima are symmetrical and two
asymmetrical, the asymmetrically situated minima marked
$C_L$ and $C_R$ on Fig. 6 must be assigned to the transitions
$m_J = 1,~ m_I = -1 \leftrightarrow 0$, and
$m_J = -1,~ m_I = 0 \leftrightarrow1$. Second, the
quantity $(3/5 H^{\prime \prime} - H')^2/H$ must be small.
Reference to Fig. 9
shows that the shifts from symmetry produced by this
second-order term in a field of 1000 gauss must be less
than 1 gauss. Hence $(3/5 H^{\prime \prime} - H') < (1000)^{1/2}$,
and we must identify $A_L$ and $A_R$ with the transition
$m_J = 1,~ m_I = 0 \leftrightarrow1$, and
$m_J = -1,~ m_I = -1 \leftrightarrow0$. This leaves
$B_L$, and $B_R$ as the
remaining minima, and they must be assigned to the other
two transitions.
Once the transitions are identified, the determination of
the separations in field between corresponding minima
gives:
$$
A_R - A_L = 2 \left( \frac{3}{5} H^{\prime \prime} - H' \right) = 13.5 \quad
\mbox{gauss,}
$$
$$
B_R - B_L = (12/5) H^{\prime \prime} = 81.8 \quad
\mbox{gauss,}
$$
$$
C_R - C_L = 2 \left( \frac{3}{5} H^{\prime \prime} - H' \right) = 95.2 \quad
\mbox{gauss,}
$$
This assignment satisfies the obvious requirement that
$(C_R - C_L) = (B_R - B_L) + (A_R - A_L)$, and furthermore fixes the
sign of $H'$ as the same as that of $H^{\prime \prime}$, which
is positive
because the proton moment is positive. The experimental
fact that at low fields the shifts of $C_L$ and $C_R$ are toward
lower field serves only as a check on the correctness of the
applicability of the theory.
The best values deduced from the results in Table VII
are:
$$
\begin{array}{ll}
H' = 27.2 \pm 0.3 \quad \mbox{gauss} \qquad \mbox{and} \qquad &\frac{\displaystyle
3}{\displaystyle 5} H^{\prime \prime} = 20.5 \quad \mbox{gauss}\\
H^{\prime \prime} = 34.1 \pm 0.3 \quad \mbox{gauss}.&
\end{array}
$$
These values can be used to account quantitatively for the
asymmetry. In fact, the term $(6/5) H^{\prime \prime} \left( \frac{3}{5}
H^{\prime \prime} + 2H' \right)/ (1 - \alpha) H = 3660/H$
if we use the value of the rotational moment
given by Rafflsey\footnote{N.F. Ramsey, Jr., Phys. Rev. {\bf 55,} 595
(1939), Abs. 26.}. We compare the expected shift of 3.8
gauss with the experimental value (see Table VI) of 4 gauss
from $(C_R + C_L)/2$ for the 4 Mc data.
The fact that the three intervals and the asymmetry can
be accounted for to such accuracy by only two constants,
strongly supports our theory. Furthermore, these values
agree with similar results to be published by one of us
(N. F. R., Jr.) using the resonances associated with transitions
of the molecular rotation moment of H$_2$. These values for
these same constants are:
$$
H' = 27.0 \quad \mbox{gauss}
$$
$$
H^{\prime \prime} = 34.1 \quad \mbox{gauss}
$$
The quantity $H^{\prime \prime}$ is equal to
$\langle \mu_P /r^3 \rangle_{A_{\mbox{v}}}$.
We are
indebted to Dr. Arnold Nordsieck for supplying. us
with the value of $\langle r^{-3} \rangle_{A_{\mbox{v}}}$,
which he obtained from the
band spectrum analysis of H$_2$ by
C. R. Jeppeson\footnote{C.R. Jeppeson, Phys. Rev. {\bf 44,}
165 (1933).}.
The asymmetry of the potential even\\
TABLE VI. {\it Displacements in gauss of minima for H in H$_2$
from center ($H_0$) obtained from H in HD. These displacements
are considered reliable to $\pm 0.5$ gauss. Note that only
$C_L$ and $C_R$ are unequal; that they approach equality in high
fields.}
\begin{center}
\begin{tabular}{c|ccc}
\hline \hline
&\multicolumn{3}{c}{Displacements in Gauss for Oscillator Frequencies of}\\
&4.000 Mc&7.000 Mc&16.000 Mc\\
\hline
$C_L$&$-51.8$&$-49.3$&$-48.9$\\
$B_L$&$-40.7$&$-40.8$&$-41.0$\\
$A_L$&$-6.6$&$-6.9$&$-6.6$\\
$H_0$&942.3&1649.1&3769.0\\
$A_R$&6.7&6.9&6.6\\
$B_R$&41.0&41.2&41.0\\
$C_R$&43.8&45.5&46.5\\
\hline \hline
\end{tabular}
\end{center}
\vspace{0.3cm}
Table VII.{\it Displacements in gauss between corresponding minima of H in
H$_2$. Note that the separations between corresponding minima are independent
of oscillator frequency and that $(A_R - A_L)+ (B_R - B_L) = (C_R - C_R)$
as expected theoretically.}
\begin{center}
\begin{tabular}{cccccc}
\hline \hline
&\multicolumn{4}{c}{Displacements in Gauss for}&Assigned\\
&\multicolumn{4}{c}{Oscillator Frequencies of}&Theoretical\\
&4 Mc&7 Mc&16 Mc& Average&Expression\\
\hline
$C_R - C_L$& 95.5&94.8&95.4&95.2&$2(H' + 3 H^{\prime \prime}/5)$\\
$B_R - B_L$& 81.6&82.0&81.9&81.8&$12H^{\prime \prime}/5$\\
$A_R - A_L$&13.2&13.8&13.3&13.5&$2(H' - 3 H^{\prime \prime}/5)$\\
\hline \hline
\end{tabular}
\end{center}
for the lowest vibration state and the centrifugal effect
of the first rotational level make the effective
internuclear distance slightly larger than $r_{\epsilon}$ the
equilibrium internuclear distance. This value of
$\langle r^{-3} \rangle_{A_{\mbox{v}}}$
of $2.438 \times 10^{+24}$ cm$^{-3}$ combined with $H^{\prime \prime} = 34.1$
yields a value of $\mu_P = 1.403 \times 10^{-23}$ erg/gauss or
$2.785 \pm 0.03$
nuclear magnetons, in too perfect agreement with the
value obtained from the $\nu/H$ result. It is of interest to
note that this serves as a check on the correctness of
the calibration of the magnet, because if the
calibration constant were changed the two values
would change in opposite directions, and would
disagree if the magnet were incorrectly calibrated.
\section*{Derivation of Table III}
~~~We shall now sketch a derivation of the formulae
which are used to construct Table III, the table of
energy levels obtained from the energy expression of
Eq. (3). Since we use a high value of the external
field, $m_I$ and $m_J$ are ``good'' quantum numbers and we
will use the $m_I$ and $m_J$ representation. The diagonal
elements of the first three terms on the right of Eq. (3)
in this representation are
\begin{equation}
- 2 \mu_P H m_I - \mu_R H m_J - 2 \mu_P H' m_I m_J.
\end{equation}
matrix elements of the fourth term
\begin{equation}
\mu_{P}^2/r^3 (IJm_I m_J |\boldsymbol{\sigma}_1 \cdot
\boldsymbol{\sigma}_2 - 3
(\boldsymbol{\sigma} \cdot {\bf r}) (\boldsymbol{\sigma}_2 \cdot {\bf r}) /r^2
| I J m_I' m_J')
\end{equation}
are evaluated from the nine wave functions of
ortho-H$^2$. To a high approximation these wave functions are
written as a product of the symmetrical spin functions
of the two protons and the coordinate wave functions
of the first rotational state. These wave functions are
given by
\begin{equation}
\Psi_{m_I m_J} = \varphi_{m_I} \psi_{m_J}
\end{equation}
$$
\begin{array}{ll}
\psi_1 = - \frac{\displaystyle 1}{\displaystyle 2} (3/2 \pi)^{1/2} \sin \vartheta
e^{i \varphi},& \psi_0 = \frac{\displaystyle 1}{\displaystyle 2} (3/2 \pi)^{1/2}
\cos \vartheta,\\
& \psi_{-1} = \frac{\displaystyle 1}{\displaystyle 2} (3/2 \pi)^{1/2} \sin
\vartheta e^{- i \varphi},
\end{array}
$$
$$
\begin{array}{lr}
\varphi_1 = \alpha(1) \alpha(2),&\varphi_0 = \sqrt 2 (\alpha(1) \beta(2)
+ \beta(1) \alpha(2))/2\\
& \varphi_{-1} = \beta(1) \beta(2),
\end{array}
$$
where $\alpha$ and $\beta$ are the spin eigenfunctions for the spin
components $1/2$ and $-1/2$, respectively.
The evaluation of these matrix elements is simplified by
use of the following identity which may readily be verified
by expansion:
\begin{eqnarray}
r^2 \boldsymbol{\sigma}_1 \cdot \boldsymbol{\sigma}_2 - 3 (\boldsymbol{\sigma}_1
\cdot {\bf r})(\boldsymbol{\sigma}_2 \cdot {\bf r}) = - (3z^2 - r^2) (3
\sigma_{z_1} \sigma_{z_2} - \boldsymbol{\sigma}_1 \cdot \boldsymbol{\sigma}_2)/2
\nonumber\\
- 3 \{ \frac{\displaystyle 1}{\displaystyle 4} (\sigma_{x_1} + i \sigma_{y_1})(\sigma_{x_2} + i
\sigma_{y_2})(x - iy)^2 \nonumber\\
+ \frac{\displaystyle 1}{\displaystyle 4} (\sigma_{x_1} - i \sigma_{y_1})(\sigma_{x_2}
- i \sigma_{y_2}) (x + iy)^2 \nonumber\\
+ \frac{\displaystyle 1}{\displaystyle 2} [\sigma_{z_2} (\sigma_{x_1} + i \sigma_{y_1}) + \sigma_{z_1}
(\sigma_{x_2} + i \sigma_{y_2})]z(x-iy) \nonumber\\
+ \frac{\displaystyle 1}{\displaystyle 2}[\sigma_{z_2} (\sigma_{x_1} + i \sigma_{y_1}) + \sigma_{z_1}
(\sigma_{x_2} + i \sigma_{y_2})]z(x-iy) \}
\end{eqnarray}
The matrix elements of the first term of Eq. (7) give
all of the diagonal elements of Eq. (5) which are equal
to\footnote{H.B.G. Casimir, ``On the Interaction between Atomic
Nuclei and Electrons,'' Prize Essay published by Teyler's
Tweede Genootschap (1936), page 33.}
$$
(3 m_{J}^2 - 2) (3 \sigma_{x_1} \sigma_{x_2} - \boldsymbol{\sigma}_1 \cdot
\boldsymbol{\sigma}_2) \mu_{P}^2/5r^3.
$$
For ortho-H$_2$, for which the three spin functions are
symmetrical in the proton spin $\boldsymbol{\sigma}_1 \cdot \boldsymbol{\sigma}_2
= 1$ and we then obtain for the diagonal elements of Eq.(5)
$$
(3 m_{J}^2 - 2) (3 m_{I}^2 - 2) 2 \mu_{P}^2/5r^3.
$$
The off diagonal matrix elements are used in our
calculations to evaluate the second-order
perturbations of the energies from complete Paschen-Back
effect and give rise to the quantities listed in
Table III. The off diagonal elements of Eq. (S) are
easily calculated from the three spin and three angular
wave functions. The values of the nonvanishing elements are:
$$
\begin{array}{c}
(m_I, m_J \quad \mbox{to} \quad m_I - 1, m_j + 10
= (2 m_I - 1)(2m_J + 1)\\
\times \{(m_I + 1)(2 - m_I)(1 - m_J)(2 + m_J) \}^{1/2} 3 \mu_{P}^2/5r^3,\\
(m_I, m_J \quad \mbox{to} \quad m_I + 1, m_j - 1) =
= (2 m_I + 1)(2m_J - 1)\\
\times \{(1- m_I)(2 + m_I)(1 + m_J)(2 - m_J) \}^{1/2} 3 \mu_{P}^2/5r^3, \\
(m_I, m_J \quad \mbox{to} \quad m_I - 2, m_j + 2) \\
= \{(1 + m_I)m_I(2 - m_I)(3 - m_I)(1 - m_J)\\
\times (- m_J) (2 + m_J) (3
+ m_J) \}^{1/2} 3 \mu_{P}^2/5r^3,\\
(m_I, m_J \quad \mbox{to} \quad m_I + 2, m_ - 2)
= \{ (1 - m_I)(-m_I)(2+m_I)(3+m_I) \\
\times (1+m_J)(m_J)(2-m_J)(3-m_J)\}^{1/2}
3 \mu_{P}^2/5r^3.
\end{array}
$$
The off diagonal elements of the term $\mu_{P} H' (\boldsymbol{\sigma}_1 +
\boldsymbol{\sigma}_2) \cdot {\bf J}$ are:
$$
\begin{array}{c}
m_I, m_J \quad \mbox{to} \quad m_I + 1, m_J - 1 =\\
\{ (1 - m_I)(2 + m_I)(1 + m_J)(2-m_J)\}^{1/2}\mu_P H',\\
m_I, m_J \quad \mbox{to} \quad m_I - 1, m_J + 1 =\\
\{ (1 + m_I)(2-m_I)(1-m_J)(2+m_J)\}^{1/2}\mu_P H'.
\end{array}
$$
The value of the perturbation energy which is to be
added to the diagonal elements is obtained from the usual
second-order perturbation theory. Since the only
nonvanishing matrix elements are between states with the
same value of $m_I + m_J$ we have
\begin{equation}
W_{m_I, m_J} = \sum_{\delta} \frac{(m_I, m_J |E| m_I + \delta,~ m_J - \delta)
\times (m_I + \delta,~ m_J - \delta |E| m_I, m_J)}{E_{m_I, m_J} - E_{m_I
+ \delta, M_J - \delta}},
\end{equation}
where the sum is over all permissible values of f except
zero. Into this equation are inserted the appropriate
nondiagonal elements listed above and for $E_{m_I, m_J}$ the
diagonal elements of Eq. (4) without the term which
involves $H'$. The second-order terms of Table III are thus
obtained.
\section*{Discussion}
~~The agreement between the measurements of the
magnetic moment of the proton by two theoretically
independent methods, i.e., the Larmor frequency in an
external field and the magnetic interaction of the two
proton moments makes our result quite certain. Our present
result $2.785 \pm 0.02$ nuclear magnetons is in agreement with
the previous atomic beam measurement of the proton
moment through the hyperfine-structure of the normal state
of atomic hydrogen. That result was $2.85 \pm 0.15$ nuclear
magnetons which is well within the limit of error. To this
extent there is no reason for supposing some breakdown of
the Dirac equation for the electron nor the assumption of
any novel interaction between proton and electron. A
similar discussion holds for the deuteron. If we assume that
the neutron moment, is the
difference\footnote{However, see J. Schwinger, reference 6, for
a discussion of the effect of the deuteron quadrupole moment on
these considerations.} between the
deuteron moment and the proton moment then its value is
$-1.93$ nuclear magnetons.
The quantity $H'$ which is the constant of spin-orbit
interaction or the magnetic field at a nucleus produced by
the molecular rotation is 27.2 gauss. Although no exact
theory of this quantity exists in the literature a rough
consideration shows it to be quite reasonable. The magnetic
field produced by the orbital motion of one proton at the
position of the other is $2ev/cr^2$ which can be written as
$(2e/Mc)(Mvr/r^3)$. Setting $Mvr$ equal to $h/2 \pi$ we obtain
$4 \mu_n/r^3$ for this field, where $\mu_n$ is one nuclear magneton.
Inserting numerical values this becomes 49.0 gauss. The
effect of the electrons is in the opposite sense. Since our
results show that $H'$ is positive the magnetic field produced
by the electrons is somewhat less than half that of the
nuclei.
Our results show that it is possible to apply exact
spectroscopic principles and procedures to spectral regions
which correspond to ordinary radio waves. The accuracy to
which the laws of quantum mechanics hold and the
necessity of their application in a region of frequency in
which one is accustomed to think classically we consider a
striking confirmation of the theory.
This research has been aided by grants from the
Research Corporation and the Carnegie Institution of
Washington.
\end{document}
%ENCODED MAY 2003 BY NIS;
%attantion, see Table III, it is misprint, changed \mu_r to \mu_R;