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R.A.~ Millikan, Phys. Rev. {\bf 7,} 18 \hfill {\large \bf 1916}\\
\vspace{2cm}
\begin{center}
{\large \bf Einstein's Photoelectric Equation and Contact Electromotive Force}\\
\end{center}
\begin{center}
R.A.~Millikan,\\
{\it University of Chicago}\\
(Received September 15, 1915)\\
\end{center}
\section*{
{\bf Introductory}}
\vspace{0.5cm}
EINSTEIN's photoelectric equation for the maximum energy of emission
of a negative electron under the influence of
ultra--violet light, namely,
\begin{equation}
^1/_2~ m v^2 = V e = h \nu - p
\end{equation}
cannot in my judgment be looked upon at present as resting upon any
sort of a satisfactory theoretical foundation. Its credentials are
thus far purely empirical, but it is an equation which, if correct,
is certainly destined to play a scarcely less important r\^o1e in
the future development of the relations between radiant
electromagnetic energy and thermal energy than Maxwell's equations
have played in the past.
I have in recent years been subjecting this equation to some searching
experimental tests from a variety of viewpoints and have been led to
the conclusion that, whatever its origin, it actually represents very
accurately the behavior, as to both photoelectric and contact E.M.F.
relations, of all the substances with which I have worked. The
precision which I have been able to attain in these tests has been
due to the following precautions.
\begin{itemize}
\item[1.]
I have made simultaneous measurements in extreme vacua of
photo-currents and contact E.M.F.'s and have thus been able to
eliminate the considerable influence which these latter have on
photo-potentials.
\item[2.]
I have worked with surfaces newly formed in extreme vacua and with
very large photo--currents of saturation value of the order 20,000
scale divisions in 30 secs. so that I have thus been able to locate
the intercept of the photo--current curve on the $PD$ axis with much
precision.
\item[3.]
I have used substances which are photo-sensitive practically
throughout the whole length of the visible spectrum and have thus been
able to use a large range of wave-lengths all of which were above the
long wave-length limit of the receiving Faraday cylinder - a matter of
no little importance.
\item[4.]
I have, with the aid of filters, carefully chosen for the principal
lines of the mercury spectrum, eliminated from the photo--current
potential curves corresponding to the longer wave--lengths, the
effects of the scattered short wave--length light, which not
infrequently falsifies entirely the shape of these curves in the
neighborhood of the intercept on the potential axis.
\end{itemize}
My conclusions, however, reported briefly last year\footnote{Phys.
Rev., IV., p. 73, '14.} and this\footnote{Phys. Rev., VI., p. 55,
'15.} are directly at variance with results recently reported in a
very notable paper by Ramsauer\footnote{Ann. der Phys., 45., p. 1120,
1914. Also 45, p. 961.} who finds that there is no definite maximum
velocity of emission of corpuscles from metals under the influence of
ultra violet light. Before considering, then, any of the theoretical
consequences of Einstein's equation it is necessary to first present
such evidence as exists for believing that, in spite of Ramsauer's
results, there is in fact a definite and accurately determinable maximum
velocity of emission for each exciting wave--length.
\vspace{0.5cm}
\section*{$\S~2$
{\bf Proof of the Existence of a Maximum energy of Emission of
Photoelectrons and Discussion of Ramsauer's Experiments}}
\vspace{0.5cm}
Ramsauer's method is notable in that he makes the first direct
measurement by a magnetic deflection, of the velocity of emission of
photo--corpuscles. By this method he is able to choose those
corpuscles which emerge in one particular direction only, for example,
the direction of the normal to the surface, and he finds that these
have a certain distribution about a most frequent value. This
distribution he finds the same for all
wave--lengths, of the incident
light, for all substances, and for all angles of emission. His source,
like Kadesch's, and like my own in much of my former work, is a
powerful condensed spark between zinc electrodes. This source I
discarded in my most recent tests on Einstein's equation because the
mercury arc was found to give greater reliability in the settings and
to have greater monochromatism in its lines.
The substances which Ramsauer studies are gold, brass, zinc and
carbon. His range of wave--lengths, obtained with a quartz
spectrometer, is quite narrow, being the same as that used by Hughes,
namely, from 186 $\mu \mu$ to 256 $\mu \mu$ for gold, and 186 $\mu
\mu$ to 334 $\mu \mu$ for zinc. As is well known lines of wave--length
below 220 $\mu \mu$ pass with great difficulty through a quartz
spectrometer. Ramsauer further works quite largely with waves which
are shorter than the long wave--length limit of his receiving surface,
and in fact his corrections for "falsches licht" (this term actually
covers several different effects such as the emission of corpuscles
from the illuminated surface itself by stray light, and the emission
of corpuscles from the surrounding walls by light reflected to them
from the illuminated surface) are very large, amounting to as much as
$1/10$ of the maximum photo--currents. This seems to me to rob the
lower parts of his velocity distribution curves of practically all
significance. His procedure differs from mine most vitally in that
while I measure very precisely, as I think, the maximum velocity of
emission he measures instead the most frequent velocity of emission.
This was the quantity from which Richardson and
Compton\footnote{Phil. Mag., 24, 572, 1912.} drew most of their
conclusions and upon which they placed their chief reliance, though
their method of obtaining it differed from Rumsauer's.
In his results Ramsauer agrees with Richardson and Compton in finding
this most frequent velocity (expressed in energy units) a linear
function of the frequency and in finding that the long wave--length
limit is given by the intercept of this line on the frequency axis.
Question may be raised regarding the certainty with which Richardson
and Compton could determine this intercept, since its location
involves the contact E.M.F. and they made no contact E.M.F.
measurements in vacuo. Ramsauer eliminates contact E.M.F. entirely by
surrounding the emitting surface by walls of the same metal as the
emitter itself. He agrees with Richardson and Compton\footnote{Phil.
Mag., 24, p. 572, 1912.} also in finding the slopes of the
volt--frequency lines differing among themselves by 20 or more per
cent. and like Richardson and Compton he finds these slopes all lower
than $h/e$ by large per cents., which vary in his case, from 35 to 50
per cent. {\it None of these results so far are at variance either
with my work or with Einstein's equation, for measurements on the most
probable velocity of emission are not capable of furnishing a test
of Einstein's equation.}
But Ramsauer's results are at variance with mine and with Einstein's
equation in that he finds no definite maximum velocity of emission at
all for when he plots energies of emission as ordinates and deflecting
magnetic field strengths as absciss\ae he finds these curves run off
asymptotically to the axis of absciss\ae. In my judgment this is
because the "falsche's licht" errors mask entirely the phenomenon
under investigation in the region corresponding to the lower parts of
his velocity distribution curves. My own experiments seem to me
approximately 1,000 times better adapted to the testing of this point
than are Ramsauer's, since my maximum currents are about 1,000 times
larger than his, as measured in scale divisions of deflection, and if
his distribution curve is the correct one, I should obtain very large
currents at potentials at which, in fact, I get none at all. Thus in
the case of the mercury lines 2535 \AA, if the potential applied to my
lithium surface was 0.02 volt to the left of the intercept shown in
Fig. 1, there was not a trace of deflection in 30 seconds.
But with lines 5461 and 4337 in the case of sodium, and line 4339 in
the case of lithium, although I used a Hilgar monochromator and a
narrow, slit (or inch) I did obtain definite indications of
deflections due to stray short--wave--length light {\it which,
however, disappeared entirely as soon as I used filters which
cut out all lines of shorter wave--length than that under
examination.} Fig 1 furnishes a very good illustration of this
effect. Without a filter the curves corresponding to line 4339 seemed
to approach the axis asymptotically, as in Ramsauer's experiments
(note the curve marked I) but with a filter of \ae sculin in a glass
trough which cut out entirely all lines below 4339 including the
strong adjacent line 4047, the asymptotic character disappeared
completely and the curve shot suddenly into the axis and gave no
indications what
ever of deflection either at $- 0.6$ or at $- 0.7$ with volts (see
Fig. 1) The curves shown in the figure and a great many other similar
ones which I have taken seem to me to establish beyond question the
contention that {\it there is a definite maximum velocity of emission
of corpuscles from a metal under the influence of ultra--violet
light,} or in other words that the curves due to a particular spectral
line do plunge sharply into the potential axis and do not approach it
asymptotically.
\begin{figure}[h]
\centerline{\epsfig{file=fig1e.eps,width=12cm}}
\caption{}
\end{figure}
The work on the photoelectric determination of $h$ will be reported
more fully in another paper but the data furnished in Fig 1 suffices
to determine $h$ from lithium with no little precision. Thus, since
frequencies of 2535 and 4339 are $118.2 \times 10^{13}$ and $69.1
\times 10^{13}$ respectively, we see from the intercepts of the figure
that the slope
\begin{equation}
\frac{dV}{dv} = \frac{h}{e} = \frac{129 + 0.74}{(118.2 - 69.1) \times
10^{13}} = 4.13 \times 10^{-15} ~ \frac{\mbox{Volt}}{\mbox{
Frequency}};
\end{equation}
$$
\therefore h = \frac{4.13 - 10^{-15} \times 4.774 \times 10^{-10}}{300}
= 6.58 \times 10^{-27} ~ \frac{\mbox{erg}}{\mbox{Frequency}}.
$$
The error here can scarcely exceed 0,02 volts in 2 volts or 1 per
cent.
The second conclusion of Ramsauer's which seems at variance with Einstein's
equation is that each corpuscle liberated by a
given wave--length does not
leave the atom with a constant energy but that a given
wave--length may
liberate corpuscles from a given kind of atom with a large range of energies.
But if the $p$ in Einstein's equation is indeed a characteristic constant of
the material as he assumed it to be, then the corpuscles are all expelled from
the atom with a constant speed and any differences which may be shown by the
velocities of the corpuscles which have escaped from the
surface of the metal
at a given angle are due to differences
in the retardations which they have
encountered in getting out from different depths beneath the surface.
Ramsauer,
however, concludes from the fact that he apparently gets the same curve of
distribution of velocities for all wave--lengths and for all angles of
emission that his observed external distribution of velocities is the same as
the "internal distribution," that is that the corpuscles are emitted from the
atoms themselves with precisely the same distribution of speeds as that which
he measures outside the metal.
Now I am not at all convinced that Ramsauer's results actually do show
that the distribution of velocities is the same for different
wave--lengths, for the range of wave--lengths (185 $\mu \mu$ to 256
$\mu \mu$ for gold) seems to me too small and the experimental
uncertainties too large to permit of such a conclusion. According to
his own statement the curve corresponding to 256 $\mu \mu$, is badly
falsified by stray short--wave--length light (the maximum deflections
obtained with this wave--length were but 17 mm. as against the 20,000
mm. which I have used in my work with sodium). The same is true of all
his curves corresponding to the longer wave--lengths. The point in
question could be convincingly tested only by using widely different
wave--lengths like those corresponding to the lines 2535 and 5461 as I
have done in the work with sodium.
Secondly, even if the large experimental uncertainties should be
reduced 10 times, and the distribution of velocities for different
wave--lengths and different angles then shown to be the same, I should
still consider Ramsauer's argument for the identity of the external
and internal distribution of velocities to be quite unconvincing. For
even though the corpuscles make perfectly elastic impact with the
atoms, as Ramsauer assumes, according to the Maxwell--Boltzmann law
their energy of agitation must decrease continually with successive
impacts until they are in temperature equilibrium with the atoms. In
other words the corpuscles which have made many impacts before
emerging in a given direction must have a smaller velocity than those
which have made few. And as a matter of fact Ramsauer's observed
velocity distribution curve, ignoring the asymptotic portion, is one
which differs from all his suggested energy distribution curves in
being too steep on the high velocity side just as would be the case if
all the corpuscles had started with a common velocity and only those
which came from appreciable depths beneath the surface had fallen
below this velocity.
Thirdly Ramsauer in identifying the internal and external distribution
of velocities appears to me to overlook the fact that the mere
phenomenon of a free charge remaining on a charged conductor
necessitates the existence of a surface force which prevents its
escape. This is the force which Helmholtz conceived of as arising from
"the specific attraction of matter for electricity." It is not a force
which in any way impedes the free movement of electricity over or
through the conductor, else the body would not act like a conductor,
and it is with conductors alone that we are here concerned. The force
considered is then one which acts on the conduction electrons, that
is, on the so-called free electrons as distinguished from those which
are permanent constituents of the atoms. Hence, even after an electron
has escaped from the interior of an atom it cannot escape from the
metal until this force is overcome. It is this force which is
responsible for about 999 thousandths of the contact E.M.F. which we
measure between metals. The other thousandth, measured by the Peltier
effect, has a kinetic, instead of a static, origin. These relations
have been the occasion of much confusion among writers on contact
effects, though they have been stated with admirable clearness by
Kelvin, Helmholtz and others.\footnote{A remarkably lucid presentation
is found in W\"ullner's Experimental Physic, Vol III., pp. 736-755. See
also kelvin, Phil. Mag., 46, p. 82, 1898.}
I am inclined to think then that neither Ramsauer's of new
conclusions, (1) that there is no definite maximum energy of emission
and (2) that the external and internal distribution of velocities are
the same can, possibly stand. At any rate the correctness of the
second has in no way been demonstrated, while the incorrectness of the
first seems to me to have been established.
\vspace{0.5cm}
\section*{$\S~3$
{\bf The relation of Contact E.M.F. and Einstein's Equation}}
\vspace{0.5cm}
The precise tests which I have reported of Einstein's equation consist
in showing (1) that there is a very exact linear relation between the
maximum P.D. and the frequency, (2) that the slope of this line
yields very accurately Planck's $h$, and (3) that the intercept of
this line on the frequency axis is the frequency at which the metal
first becomes photosensitive. In order to test this last point it was
necessary to displace in the direction of positive potentials the
observed P.D. $\nu$. line by the exact amount of the measured contact
E.M.F. When this was done the observed long wave--length limit, as
directly determined, agreed quite accurately with the intercept (see
Fig. 2). This means that in Einstein's
equation $p$ represents not the among of work necessary of remove the
corpuscle entirely from the influence of the metal, that is, to carry
it out beyond the influence of the latter's contact field, but rather
the work necessary to just free it from the surface so that a
relatively large accelerating field can then remove it, for it is in
just this way that we actually make the test. The quantity $p$, then,
is the work necessary to just detach a corpuscle from the surface of
the metal and we have, by putting in (1) $v = o$
\begin{figure}[h]
\centerline{\epsfig{file=fig2e.eps,width=12cm}}
\caption{}
\end{figure}
\begin{equation}
p = h v_0
\end{equation}
Now since both the independence of photo--emission upon temperature,
and also the fact that gases show the photo--effect, indicate that
the electrons which are ejected by light from metals are not the free
electrons of the metal, but rather electrons which are constituents
of the atoms, we would naturally consider $p$ as made up of two
parts, (1) the work $p_1$ necessary to detach the electron from its
parent atom and make it a free electron of the metal, and (2) the
work $p_2$ necessary to detach this free, or conduction, electron
from the surface of the metal.
If we consider two opposed metal surfaces, for example one of pure
zinc and one of pure copper separated only by the ether, and imagine
that they have been put initially into the same electrical condition,
so that no electrical field exists between them, then if a wire of
copper be run from the copper plate to the zinc plate, we find by
experiment that upon making contact an electric field is established
between the plates. We say that the P.D. which now exists has arisen
because of a contact. E.M.F. at the junction of copper and zinc
which causes an electrical flow from copper to zinc until equilibrium
is set up, and we measure this contact E.M.F. by the observed P.D.
which it creates, taken of course with the opposite sign. By
definition then the contact E.M.F. is the amount of work which,
before any electrical field exists, would be required to transfer one
unit of free positive electricity from the zinc over to the copper
against the superior attraction of zinc for this unit. Alter the
contact has been made and equilibrium set up, it, of course, requires
no work to carry electricity across the zinc--copper junction other
than that represented by the Peltier effect, which has another cause
and is of an altogether different order of magnitude. Writing then
the above definition in symbols we have
\begin{equation}
\mbox{Contact E.M.F.} = \frac{p_2 - p_2'}{e}.
\end{equation}
in which $p_2$ relates to the zinc and $p'_2$ to the copper.
If, as in the case we are considering, e is negative, Then this
contact E.M.F. is negative. Now if we write for each of any two
opposed metals
$$
h \nu_0 = p = (p_1 + p_2)
$$
and
$$
h \nu_0' = p' = (p_1' + p_2')
$$
and subtract we obtain
\begin{equation}
h \nu_0 - h \nu_0' = (p_1 - p_1') + (p_2 - p_2')
\end{equation}
which in view of (4) becomes
\begin{equation}
\mbox{Contact E.M.F.} = \frac{h \nu_0 - h \nu_0' - (p_1 - p_1')}{e},
\end{equation}
an expression which shows that Einstein's equation does not at all
demand that the contact E.M.F., even between two pure metals, be
equal to the difference between the frequencies corresponding to long
wave--length limits of the two metals multiplied by $h/e$. If this
latter relation is found by experiment to hold for any two metals it
is an exceedingly interesting and important fact which, however, has
no bearing on the validity or invalidity of Einstein's equation. If
p$(p_1 - p'_1)$ which in turn might mean that the energy of {\it
escape} of the corpuscle from the atom is always equal to $h \nu$,
the absorbed energy being, contrary to the physical theory which
guided Einstein greater than $h \nu$ or it might mean that, though
the absorbed energy it but $h \nu$, the work necessary to detach
corpuscles from the atoms of the two metals are the same, or that
these works are both so small in comparison with the works necessary
to detach conduction electrons from metallic surfaces that $(p_1 -
p'_1)$ is in any case negligible in comparison with
$(p_2 - p'_2)$.
Now, the experimental situation is as follows: Richardson and
Compton,\footnote{Phil. Mag., 24, p. 592, 1912.} although they
made their contact E.M.F. and their
photoelectric measurements under
different conditions, namely the former in air and the latter in
vacuo, yet found that for any two metals $h/e (\nu_0 - \nu'_0)$ was
at least of the same order of magnitude as the contact E.M.F. between
these same two metals, and last year I also found that in the case of
sodium and copper oxide, while the measured contact P.D. between
them was 2.51 volt, $h/e$ times the difference in the frequencies
corresponding to the long wave--length limits was 2.79 volts, a
result which seemed almost near enough to be in accord with
Richardson and Compton's conclusion. Furthermore if Einstein's
equation is correct, this conclusion can be tested very accurately
without any contact potential measurement at all by a method which
has already been tried a number of times, though the results have
been quite discordant, and so far as, I know the relation of the
result to contact E.M.F. has not been clearly pointed out. This
relation appears at once as soon as we determine just what are the
demands which Einstein's equation imposes on contact E.M.F.
Consider a corpuscle ejected from any conducting surface by light of
frequency $\nu$ into a Faraday cylinder made for example of a metal
more electronegative than the emitter. Then, if the corpuscle is to
be brought to rest just as it reaches the wall of the Faraday
cylinder the energy of ejection must just equal the work done
against the applied positive potential and the contact potential and
this by Einstein's equation is equal to $h \nu - p$. Thus denoting by
$V_0$ the {\it observed} maximum positive potential, and by $K$ the
contact E.M.F. between the Faraday cylinder and the emittor, and
remembering that $p = h \nu_0$ in which, if the emitting surface is
inhomogeneous $\nu_0$ is the long wave--length limit of the most
electropositive element in the surface, that is, the element which
loses negative electrons most easily, we have
$$
^1/_2 ~m v^2 = \left( V_0 + K \right) \cdot e = h \nu - h \nu_0.
$$
Writing a similar equation for an electron ejected by light of the
same frequency from the surface of any other conducting material into
the same Faraday cylinder we obtain
$$
^1/_2~ m v^2 = \left( V'_0 + K' \right) \cdot e = h \nu - h \nu'_0.
$$
By subtraction we find since the contact E.M.F. between the two
metals used as emitters is $K' - K$.
\begin{equation}
\mbox{Contact E.M.F.} = \frac{h}{e} \cdot (\nu_0 - \nu'_0)
- \left( V_0 - V'_0 \right).
\end{equation}
If then Einstein's equation is correct this should furnish a
perfectly general way of measuring contact E.M.F. between any two
conducting surfaces, pure or impure, and if it can be experimentally
verified, then we have one more proof of the correctness of
Einstein's equation. This is the equation which I have been
submitting to careful experimental test in the case of the alkali
metals, and I have thus far found it in perfect agreement with
experiment (see below).
If (7) may be regarded as established by the work which follows, then
it will be obvious that we can test whether or not, with any
particular metals,
\begin{equation}
\mbox{Contact E.M.F.} = \frac{h}{e} \cdot (\nu_0 - \nu'_o)
\end{equation}
by simply observing whether these metals, when placed before the same
Faraday cylinder and stimulated by a given wave--length, show the
same value of the observed maximum positive potential, i. e., whether
$V_0 - V'_0$.
What I wish to point out then is that, though Einstein's equation
does demand that certain relations exist between contact E.M.F.'s and
photo--potentials, namely those stated in equation (7) it does not
demand the relation (8) suggested by the experimental work of
Richardson and Compton, and with about the same sort of roughness, by
my results on sodium. If however (7) is a correct and perfectly
general relation, as it must he if Einstein's equation is a rigorous
one, then (8) can be tested for any particular substances by seeing
whether, for these substances, the last term of (7) vanishes. This
point was tested carefully in 1906 by Millikan and
Winchester\footnote{Phil. Mag., 14, p. 20, 1907.} who found marked
differences in the $V_0$'s for eleven different metals and whose
results therefore seen to conflict with (8). It was again tested by
Page in 1913\footnote{Am. Jr. Sci., 36, p. 501, 1913.} who found that
with freshly scraped surfaces of copper, aluminum and zinc the
$V_0$'s were all alike. Kadesch\footnote{Phys. Rev., III., p. 367,
May, 1914.} corresponding to freshly cut sodium and potassium in front
of the same receiving chamber, and although he did not discuss the
question here under consideration, a glance at his curves shows that
for a given wave--length these $V_0$'s differ by much as 85 volt. Last
winter I found that the directly measured contact E.M.F. in vacuo
between lithium and copper oxide was more than a volt less than the
observed value of $h/e~(\nu_0 - \nu'_0)$ in which $\nu_0$ was
the observed long wave--length limit of lithium freshly cut in vacuo
and $\nu'_0$ the observed long wave--length limit of CuO in the same
vessel. (It is to be remembered that $e$ is here negative.) These
results appear to show conclusively that (8) does not in general hold,
and in a paper presented at the April meeting of the American Physical
Society\footnote{Phys. Rev., VI., p. 55, '15.} I suggested that the
failure of (8) in some cases and its apparent validity in others might
be explained by the influence of surface inhomogeneities. For
obviously, in measuring $(h/e) ~(\nu_0 - \nu'_0)$ one is always making
his long wave--length limit tests on the most photosensitive, i.e.,
the most electropositive constituent of a given surface, while in
measuring contact E.M.F.'s one is testing the {\it mean} effect
outside the surface of all the surface constituents, so that in the
case of a lithium surface which is discharging electrons into a
copper oxide cylinder, if the lithium surface were a mixture of
substances, some of which were much more electropositive
than others, the measurement of $(h/e)~(\nu_o - \nu'_0)$
would correspond, if (8) were correct for homogeneous surfaces, to a
determination of the contact E.M.F. of the most electropositive
element in the lithium surface. Hence the measured contact E.M.P.
would he expected to fall below the value of
$(h/e)~(\nu_0 - \nu_0')$ as results showed that it did.
Nevertheless this was not a necessary cause of the failure of (8),
for equation (6) shows that there is no reason other than an
experimental one for supposing that (8) ever holds. Accordingly,
shortly after the above--mentioned meeting, I suggested to Doctors
Kadesch and Hennings that they re\"examine the point tested first by
Millikan and Winchester and last by Page, using as nearly as possible
the latter's experimental conditions in order to find out whether the
difference between the two sets of results was due to the fact that we
worked in these early experiments with old surfaces while Page had
tested newly scraped metals, or whether new surfaces of the ordinary
metals do actually show differences in the $V_0$'s which escaped
Page's detection, as equation (6) indicates that they might well do.
They found that they could use with some modification the apparatus
on which Dr. Hennings had worked with contact E.M.F.'s in this
laboratory some years previously. Their results are given in papers
which follow and seem to support Page's conclusion for the ordinary
metals when newly scraped. It is useless, however, to attempt to put
any interpretation upon these results until Einstein's equation is
shown to be a reliable tool with which to work, that is, until
equation (7) is shown to be able to predict accurately and invariably
observed contact E.M.F.'s. The following results show that in all the
cases thus far examined in which (8) breaks down completely (7)
nevertheless yields the most beautiful agreement.
Thus in the experiments reported in April to the Physical
Society\footnote{Phys. Rev.,VI., p. 55, '15.} the measured contact
E.M.F. between lithium and copper oxide was found to be 1.52 volts.
The long wave--length limit of the lithium was found to correspond
accurately to $\nu_0 = 57.0 \times 10^{13}$. This was determined most
reliably by displacing the P.D., $\nu$ line toward positive potentials
by the amount of the measured contact E.M.F. and then taking the
intercept of this line on the $\nu$ axis. Direct observation checked
closely however the value thus obtained. The long wave--length limit
of the receiving copper oxide cylinder was directly determined at line
2535 with an uncertainty of perhaps 50 \AA. This
corresponds to
$\nu'_0 = 118.2 \times 10^{13}$. These figures give (see equation (2))
$$
\frac{h}{e} \cdot (\nu_0 - \nu'_0) = 4.13 \times 10^{-15} ~ (118.2 -
57.0) \times 10^{13} = 2.53 \quad \mbox{volts.}
$$
Now line 2535 was just at the long wave--length limit of the
copper oxide, so that for this line $V'_0$ between CuO and Faraday
cylinder of CuO was zero. On the other hand, for line 2535 the $V_0$
between the lithium and the CuO Faraday cylinder was found to
be just $+ 1.00$ volts, so that (7) becomes
$$
1.52 = 2.53 - 1.00 = 1.53.
$$
These measurements were made on a newly cut lithium surface. Several
months later the measurements were repeated and the contact E.M.F.
between the then old lithium surface and the Faraday cylinder had
changed to 1.11 volts. The $V_0$ between the lithium and cylinder tor
line 2535 had changed to 1.29 volts (these are the measurements shown
in Fig. 1) and the long wave--length limit $\nu_0$ was now measured
at $59.7 \times 10^{13}$. The $\nu'_0$ had not changed. These figures
give
$$
\frac{h}{e} \cdot (\nu_0 - \nu'_0) = 4.13 \times 10^{-11}~ (118.2 -
59.6) \times 10^{13} = 2.42 \quad \mbox{volt}
$$
and equation (7) now becomes
$$
1.11 = 2.42 - 129 = 1.13
$$
Although the agreements in both these cases are exceedingly dose the
uncertainties in the long wave--length limit of the CuO amount to
possibly 50 \AA. so that $\nu_0 - \nu'_0$ is uncertain
by as much as 3 per cent., or possibly a trifle more.
Again in the case of the sodium I stated above that the results of the
measurements gave contact E.M.F. = 2.51 volts, while $h/e \cdot
(\nu_0 - \nu'_0)$ came out 2.79 volts, which looked at first like fair
agreement with equation (8), but the agreement is well nigh perfect
when the second term of equation (7) is taken into account, for the
$\nu'_ 0$ for this Faraday cylinder (it was a different one from that
used with the lithium.) corresponded to $\lambda_0 = 26 85$ \AA.
instead $\lambda_0 = 2535$ \AA. This gives $\nu'_0 = 111.8 \times
10^{13}$. For the sodium $\nu_0$ came out accurately 43,.ly $43.9
\times 10^{13}$. From the relation $(d \quad \mbox{volt})/d \nu =
4.13 \times 10^{-15}$ we can compute the maximum energy of emission
$V_0$ of corpuscles under the influence of line 2535 from a surface
for which this energy is zero at $\lambda = 2685$. It is
$$
4.13 \times 10^{-15} ~ (118.2 - 111.8) \times 10^{13} =
0.26 \quad \mbox{volt.}
$$
This is $V'_0$. With line 2535 the observed $V_0$ for the sodium was
0.52 volt, so that
$$
\left( V_0 - V'_0 \right) = 0.52 - 0.26 = 0.26 \quad \mbox{volt.}
$$
Then equation (7) becomes
$$
2.51 = 2.79 - 0.26 = 2.53
$$
which is but one per cent. in error. All of these results then, are in
perfect agreement with the demands of equation (7) although they are
definitely opposed to the generality of the conclusion that contact
$\mbox{E.M.F.} = h/e \cdot (\nu_0 - \nu'_0)$ which is identical
with the conclusion that the observed photo--potentials of different
metals are all the same when light of a given wave--length eject
corpuscles from metals into the same Faraday cylinder. Whether the
differences between the alkali metals and ordinary metals in regard to
equation (8) are due to surface inhomo-geneities, as I stated last
April, or that it might be to intrinsic properties of the metals must
be determined by further experiments. Some data has already been
accumulated, but its presentation will be deferred to another
occasion.
Since we are here concerned primarily with the testing of Einstein's pt
equation, the important result already attained and so far as I am
aware not hitherto shown is that Einstein's equation {\it appears to
predict accurately and generally the relations between Contact
E.M.F.'s and photo--potentials.}
\vspace{0.5cm}
\section*{$\S~4$
{\bf Contact E.M.F.'s and Temperature}}
\vspace{0.5cm}
There are some interesting relations between Einstein's photoelectric
equation and the effect of temperature on contact E.M.F.'s which, so
far as I am aware, have not hitherto been pointed out. In 1906
Millikan Winchester\footnote{Phys. Rev., 24, p.16, and Phil. Mag. (6),
14, 188.} and Lienhop\footnote{Ann. d. Phys. (4), 21, 284.}
independently established be lack of dependence upon temperature of
the $v$ of equation (l). Since $\nu$ is not dependent on temperature
it follows from (1) that $p \cdot (= h \nu_0)$ also is not a function
of temperature. It follows then from equation (7), since all the terms
on the right were definitely shown in Millikan and Winchester's
experiments to be independent of temperature, that contact E.M.F. must
also be independent of temperature.
Now W. Schottky\footnote{Ann. d. Phys., 44, p. 1011, 1914. Phys.
Zeit., p. 624, June 15, 1914.} has recently made measurements on
contact E.M.F.'s at incandescent temperatures and obtained results
which are the same, within the rather wide limits of uncertainty, as
those commonly obtained at ordinary temperatures, that they are the
same as at ordinary temperatures, so that, so far as
experiment has now gone, Einstein's photoelectric equation, whatever
may be said of its origin, seems to stand up accurately under all of
the tests to which it has been subjected.
\vspace{0.5cm}
\section*{$\S~5$
{\bf Summary}}
\vspace{0.5cm}
The tests of Einstein's photoelectric equation which I have considered
and, save in the case of the last, subjected to accurate experimental
verification are:
\begin{itemize}
\item[1.]
The existence of a definite and exactly determinable maximum energy of
emission of corpuscles under the influence of a given wave--length.
\item[2.]
The existence of a linear relationship between photo--potentials and
the frequency of the incident light. (This has not been shown in the
present paper.)
\item[3.]
The exact appearance of Planck's $h$ in the slope of the
potential--frequency line. The photoelectric method is one of the most
accurate available methods for fixing this constant.
\item[4.]
The agreement of the long wave--length limit with the intercept of the
P.D., $\nu$ line, when the latter has been displaced by the amount of
the contact E.M.F.
\item[5.]
Contact E.M.F.'s are accurately given by
$$
\frac{h}{e} \cdot (\nu_0 - \nu'_0) - \left( V_0 - V'_0 \right).
$$
\item[6.]
Contact E.M.F.'s are independent of temperature. This last result
follows from Einstein's equation taken in conjunction with the
experimentally well established fact of the independence of
photo--potentials on temperature. If the surface changes in the heating
so as to change the photoelectric currents, the contact E.M.F.
should change also, otherwise not.
\end{itemize}
\begin{minipage}[t]{8cm}
\begin{center}
Ryerson Physical Laboratory,\\
University of Chicago\\
{\small September 15, 1915}
\end{center}
\end{minipage}
% SB. = ENCODED BY NIS 1 of March 2000
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