by Richard Keesing

My interest in this subject was awakened as an undergraduate when I found that I could not get a simple experiment to measure Planck's constant using the visible photoelectric effect to work. As you probably know this experiment measures the photo-current versus voltage in a vacuum photo-diode which is illuminated by monochromatic light. The object of the exercise is to find the retarding voltage at which the photo-current 'cut off' occurs as a function of the frequency of the light. It is well known that Einstein predicted that the energy E of emission of electrons from a metal illuminated with monochromatic light of frequency n is governed by:

E = hn - f

where f is the work function of the metal.

I noticed that a reverse photo-current existed, (due to photo-electrons being emitted from the anode) and try as I might I could not get rid of it. As a result the apparent 'cut off' simply occurred when the forward and reverse currents were equal and opposite. Needless to say I failed to get a value of Planck's constant. Several years later I was demonstrating in our first year lab here and found that the apparatus we had for measuring Planck's constant had similar problems. As a consequence I decided to build a photo-diode which I hoped would not suffer from this problem, had it built and after a few modifications produced a tube which had a forward to reverse current ratio of about 106 in the red and ~104 in the near UV1. And this is where my troubles really started!

My first disquieting observation with the new tube was that the I/V curves had high energy tails on them and always approached the voltage axis asymptotically. I had been brought up to believe that the current would show a well defined cut off, however my curves just refused to do so. What on earth was wrong with my apparatus; why were my results so different from those in all the text books? After considerable soul searching it suddenly occurred on me that there was something wrong with the theory of the photoelectric effect which one finds in elementary treatments. The problem with this theory is that it conflicts with some fundamental aspects of physics. The most important of these, at least to me, is the condition that mathematical functions which can be applied to the physical world are generally continuous in amplitude and gradient. A discrete cut off implies a discontinuity in gradient and immediately makes one very suspicious about the function.

The second problem is that the conduction electrons in a metal (the source of the photoelectrons) have an energy distribution which possesses a 'thermal tail'. This causes the (Fermi-Dirac) energy distribution function to approach the energy axis asymptotically: as it should because it is a well behaved physical function. The third problem is that in any real apparatus, even if the temperature were absolute zero and there were consequently no thermal tail, the photoelectrons must leave the photo-cathode over a finite range of angles. Unless the retarding device is a point source at the centre of a sphere, this will cause a finite energy resolution and this in turn causes the current-voltage function to approach the energy axis asymptotically (as it should because physical systems obey the first condition). For these reasons one would not expect an observable cut off. This being the case how on earth did Robert Millikan2 obtain the results he published in 1916 for they appear highly unphysical?

Before starting this investigation I had read Millikan's 1916 papers in a rather cursory way but had not read Einstein's 1905 paper3 at all. I now returned to these papers with an earnest intent.

Einstein's actual statements on the photoelectric effect.

It is worthwhile reviewing exactly what Einstein wrote in his 1905 paper. Quoting verbatim from the English translation:

'...According to the concept that the light consists of energy quanta of magnitude Rbn /N (i.e. hn) however one can conceive of the ejection of electrons by light in the following way. Energy quanta penetrate into the surface layer of the body, and their energy is transformed, at least in part, into kinetic energy of electrons. The simplest way to imagine this is that a light quantum delivers its entire energy to a single electron; we shall assume that this is what happens. The possibility should not be excluded, however, that electrons might receive their energy only in part from the light quanta.' (I have highlighted this last sentence, Einstein did not)

Einstein then goes on to derive the photoelectric equation by considering the emission of photo-electrons from a body having a work function f which is at a positive potential P surrounded by earthed conductors. He argues that we can make the potential of the photo-emitter sufficiently positive just to stop current reaching the earthed conductors. Under this condition:

eP = hn - f

where e is the electronic charge.

In this Einstein seems to be assuming, that there are no other sources of energy for the electrons. Thus he actually makes the prediction that there is a maximum energy of photo-emission although he covers himself against all eventualities with the condition I have highlighted above.

Millikan's Investigation of the photoelectric effect

In his day Millikan was determined to perform experiments which would confront the most modern theories and have a real impact on the course of physics. This he did. To this end he spent about ten years investigating Einstein's predictions for the photoelectric effect which culminated in his several papers of around 1916. It is important to realise that although Millikan was using state of the art; vacuum equipment, monochromators, light sources and electrical measuring instruments these were somewhat rudamentry in comparison with today.

His investigations of photo-emission from billets of vacuum cast sodium, potassium and lithium having freshly cut surfaces give one the impression that the surfaces he examined were the alkali metals from which the billets were cast. However his vacuua, although good for 1905, were only 10-5 torr and would allow complete contamination of a surface in the order of one second. He noted that after freshly cutting his surfaces in vacuum the photo-currents and contact potentials were very unstable and it was his practice to increase the pressure in his systems to 10-2 torr to allow these to stabilise before experiments could be performed.

Fig 1. Millikan's graphs of photocurrent vs retarding potential for Lithium with and without his aesculin UV filter showing the asymptotic tail and the tail removed

The work on Lithium: Using a mercury arc lamp and Hilger monochromator to isolate the spectral line at 433.9 nm Millikan measured the photo-current collected in a flared oxidised copper cylinder as a function of retarding potential. The I/V curve he obtained is reproduced from his paper as I in Fig 1. It will be noticed that it has an ~ 1eV tail which approaches the energy axis asymptotically. After a deal of experimentation Millikan convinced himself that this asymptotic tail was entirely due to the presence of short wavelength light scattered in his monochromator which he removed by 'interposing a filter of aesculin in a glass trough' . He reported that the asymptotic tail completely disappeared leaving curve II of fig 1. As this filter reduced the saturation photo-current by about a factor four from 2,595 to 591 units he removed it and so subsequently collected I/V curves with these long asymptotic tails in situ.

Quoting Millikan; 'Hence after it has been established that the flat 'feet' of the curves are due to stray light, it is perhaps just as well to reduce these feet as much as possible, without the use of filters, that is, by having all transmitting and reflecting surfaces as clean and as perfect as possible and all absorbing surfaces as black as possible and then simply to cut the feet off'

Millikan repeated on many occasions that the effect of the optical filter was to cause the asymptotic tail to disappear completely and allow him to observe the current plunging into the axis. On page 369 Millikan makes the following statement:

'These curves and a great many similar which I have taken seem to me to establish beyond question the contention that there is a definite, exactly determinable maximum velocity of emission of corpuscles from a metal under the influence of light of a given frequency'.

Millikan was aware that the presence of these distinct maximum energies implied that the source of the photo-electrons could not originate in a thermal distribution and so concluded that the photo-electrons had to come from the atoms themselves.

Let us for the moment accept that Millikan's photoelectrons were monoenergetic or at least had a distinct maximum energy. In order to observe a discontinuity in the I/V curve Millikan's retarding field analyser would have to have had an infinite energy resolution. The sole geometry which has this property is a point source at the centre of a perfect sphere the collection efficiency of which is unity for all energies. Millikan's geometry was clearly not this.

Reading Millikan's papers leaves one somewhat perplexed. Having performed experiments on photo-emission and modelled the process in planar and spherical diodes and found that one can understand the phenomenon quantitatively one is faced with Millikan's results. For his results to be physically meaningful his experiment would have to have suppressed the energy spread of his photoelectrons, suppressed the finite energy resolution of his energy analyser, and in doing so violated the requirement that physical functions should be continuous in amplitude and gradient.

Millikan fortunately included tables of data with his photocurrent vs retarding potential curves and I have re-plotted his curves again in the figure 2 below. Curves 1 and 2 are plots of the actual data from the tables through which I have fitted simple splines. It is clear that there is no discernable cut off in curve 1 or 2 although Millikan claims otherwise. Interestingly though Millikan's curve II is not a plot of his actual data. It has in fact been scaled by multiplying by a factor of 93/24 so that it passes through the point at 1.00 volts.

Fig 2. Curves 1 and 2 show Millikan's data taken from the tables of Fig 1, replotted on the same current scale. Also included is his scaled curve II for reference. The data points on all three curves have been connected with simple splines.

I have shown this as curve 3 the points being joined by a simple spline. Millikan claimed that his curve II plunged into the energy axis at an exactly definable point. My curve 2, which is a plot of his actual data clearly does no such thing and is manifestly asymptotic to the energy axis. The numbers he quotes in his tables represent actual physical measurements and these cannot violate the continuity of functions and clearly do not no matter how often Millikan claims to the contrary. Why then you might ask does Millikan's curve II appear to plunge into the energy axis for all he appears to have done is scale all the points by a constant which would seem to be an entirely legitimate procedure. And here lies the rub: all the points up to a retarding voltage of 0.7 are recorded as being exactly zero current. Thus if one multiplies them by any finite constant they remain zero and the zero current point at V = 0.7 effectively pins the curve to the energy axis. Millikan then projected his curve II into the energy axis from the currents at 0.9 and 0.8 volts. Had Millikan measured the current at 0.7 volts more precisely one feels sure that he would have found it to be finite and not zero and this really would have put the cat amongst his pigeons.

Fig 3. Millikan's curves of photo-current vs retarding potential in sodium with the asymptotic tails cut off: the data often reproduced in text books.

Once Millikan had convinced himself that curve II plunged into the energy axis he felt justified in cutting off the tails from all his other curves and published the results shown in fig 3. This is the figure which is often mistakenly published in textbooks as undoctored experimental observations.

The moral of this story? Don't fiddle your data!

References:

1 R G Keesing, Eur. J Phys. 2 (1981) 139-149.

2 A Einstein: (A B Arons & M B Pepperd) Am. J. Phys.(1965) 33,5,367-374.

3 R A Millikan: Phys Rev (1916) VII, 3, 355-388.

Figs 1 & 3 Copyright (1915) by The American Physical Society.