The Photoelectric Effect

Imagine a piece of polished metal. It appears to be made of one smooth, continuous substance. But we know that it’s made of separable parts because we can break it or cut it. The material is divisible, but not infinitely so. The metal possesses a property, mass-energy, whose value must conform to the requirements of a conservation law, so the structure of the metal must conform to the finite-value theorem. Thus we cannot subject the metal to an infinite sequence of cuts: after carrying out a finite number of cuts we must necessarily come to the uncuttable, the minuscule element that, in its myriads, makes up the metal. Although our crude senses see the metal as a smooth, continuous substance, the metal actually consists of a discontinuous agglomeration of atoms.

We conceive electricity as a kind of fluid and imagine it flowing through a metal much as water would flow through a channel filled with cobbles. When there’s no electric current, the electricity pools in the metal just as water pools in the channel when there’s no flow. Because electricity carries a negative charge, which generates an electric force, the atoms of the metal must carry an equal amount of positive charge so that the metal as a whole remains electrically neutral. The electricity is then bound to the metal and removing pieces of it requires the expenditure of energy.

If we make electricity flow through the metal as a current, then the metal becomes hot. The electricity does work, which the metal converts into heat, so we infer that electricity carries energy. That fact necessitates, in accordance with the finite-value theorem, that electricity be only finitely divisible, that it have its own atomos, which we call an electron.

If we blacken the metal and shine a light on it, the metal absorbs the light and becomes hot. Thus we infer that light possesses energy and, therefore, that it has an atomos, which we call a photon. We can characterize atoms and electrons by their masses, but photons have no mass (they couldn’t move at the speed of light if they did), so how can we characterize them?

We first notice of light its color and we take observation as reflecting the most fundamental property of the photon. We can correlate light’s color with a frequency or, alternatively, a wavelength. We might also characterize a photon by where it falls on the visible spectrum: we could pick two bright or dark lines in the solar spectrum, for example, and assign a number to a photon in accordance with how far it falls on the spectrum from one line expressed as a multiple or a fraction of the distance between the defining lines (much as we define and measure temperature). But for this essay let’s use frequency on the assumption that we can, in fact, measure it.

In its ephemeral existence the photon differs from the
electron and the atom in the fact that we can destroy it readily: we need only
cause matter to absorb it. Assume that we have a block of material made of one
specific kind of atom, one that will perfectly absorb any photon of frequency
ν_{0}
and will, of course, emit photons of the same frequency. We generate a photon of
frequency ν_{0}
carrying energy E_{0} and project it at the block. One atom absorbs the
photon and the mass M_{0} of the block increases by
μ=E_{0}/c^{2}.

Imagine either that we have accelerated toward the block or that the block has accelerated toward us until it moves toward us with the relative speed v. In our frame of reference the mass of the block has become

(Eq’n 1)

Now the atom that absorbed our photon emits an identical photon back at us.
In the block’s frame that new photon has frequency
ν_{0},
so in our frame it has frequency

(Eq’n 2)

due to the relativistic Doppler shift.

Calculation of the energy carried by the new photon goes a little beyond simply accounting for the relativistic mass increase. James Clerk Maxwell (in 1873) and Adolfo Bartoli (in 1876) showed that light exerts pressure upon any body that absorbs it, emits it, or reflects it, so our moving block must do work on the new photon in the process of emitting it. Given that the raw energy put into the new photon conforms to

(Eq’n 3)

and that the work done upon the photon by the block conforms to W=E’v/c, we calculate the total energy carried by the new photon as

(Eq’n 4)

Comparing Equations 2 and 4 tells us that in our photon we have a simple linear relation between the photon’s frequency and the energy that it carries. Does that relationship stand true to Reality for all photons? If photons of a given frequency could carry differing amounts of energy, then the photon would not be a true atomos: it would be an indefinite thing that could be divided into more photons of the same frequency. In order for the photon to be a true atomos it must carry the absolute minimum amount of energy possible for a photon of that frequency. Of course, we can split photons, but when we do so the product photons must have frequencies different from and lower than the frequency of the initial photon. In order for the photon to be a true atomos, a definite thing, then, the energy that it carries must be both the absolute minimum and the absolute maximum for the given frequency. That fact necessitates that the proportionality factor between the energy and the frequency be a universal constant (Planck’s constant), so we have

(Eq’n 5)

which encodes Planck’s quantum theorem.

Now let’s go back to our polished metal. We know that it
consists of an array of atoms with a fluid of electrons trapped within it by the
electric attraction between the electrons and the atoms. If we want to extract
an electron from the metal and pull it out into the air, we must do work upon
the electron in order to overcome the electric potential that holds it in the
metal by virtue of the electric force. We call the minimum amount of work that
we must do on the electron to achieve that result the electron’s work function,
W_{e}, and it occurs to us to think that we might be able to supply it
to the electron by shining a ray of light onto the metal.

If we calculate the flux of energy striking the metal through Maxwell’s electromagnetic theory, we find that it’s too low to give any electron enough energy to jump out of the metal. Before it can accumulate enough energy to make the jump the electron shares it out with its surroundings, which turn it into heat. But the electromagnetic calculation gives us an average, one calculated as though the energy carried by the photons were smeared uniformly over the area struck by the ray. If we look instead at the pointillist imagery of the photon picture, we get a different result. When a photon strikes an electron it can deliver all of its energy to the electron in one bump and if that energy exceeds the value of the work function, the electron jumps out of the metal.

We see, then, that the intensity of the light doesn’t matter. If the frequency of the photons is too low, they won’t kick any electrons out of the metal, regardless of how brightly the ray shines. But if the frequency of the photons equals or exceeds a certain threshold value, even a faint glow will have electrons streaming off the metal. In that case each ejected electron carries a net kinetic energy

(Eq’n 6)

That’s the photoelectric effect.

Heinrich Hertz (1857 Feb 22 – 1894 Jan 01) discovered the photoelectric effect in 1887 while he was experimenting with artificially generated electromagnetic waves. He noticed that when light containing an ultraviolet component (such as sunlight) fell into the spark gap in his receiver the maximum lengths of the sparks jumping that gap increased: something seemed to be giving the sparks a boost. Further experiments by Hertz and others produced a full description of the effect, but no one could explain it. The laws of electromagnetism, as they were known at the time, could not provide an accounting for the effect.

In 1905 Albert Einstein explained the photoelectric effect with reference to Max Planck’s quantum hypothesis. In his paper, "On an Heuristic Viewpoint Concerning the Production and Transformation of Light", he pointed out that the energy transfers calculated from Maxwellian electromagnetic theory involve averaged out distributions of the energy while the energy transfers manifested in the photoelectric effect involve the instantaneous distributions of energy within light. Using a thermodynamic approach and some results from the work of Ludwig Boltzmann and Wilhelm Wien, he then deduced Planck’s quantum hypothesis, by showing that light trapped in a cavity behaves like a gas composed of discrete particles, and used it to explain the photoelectric effect as I did above. In so doing he provided an independent verification of the validity of Planck’s quantum theorem and helped launch the quantum revolution of the early Twentieth Century.

Appendix: The Infinitesimal in Physics

Reciprocal of infinity, the infinitesimal seems to play no legitimate role in physics. On occasion we make an error and refer to the differentials of the calculus as infinitesimals, but that’s wrong: properly speaking, the differentials are negligibly small minuscules, things that have definite, if extremely tiny, values. The infinitesimal, crudely described as one divided by infinity, has no definite value and in that fact lies its value to physics.

We know that the number line must be a continuum, that it must be a line without gaps. And yet its very nature, as a set of points, seems to necessitate gaps. Even an infinite set of points, things with zero width, will have gaps. In order to produce a continuum we fill the gaps with the infinitesimals, entities that have the fundamental property of being infinitely small and, therefore, of having an indefinite value. Their fuzziness spreads them out to fill the gaps without creating new gaps.

To gain an understanding of what that last statement means, consider the first infinitesimal, the one that spans the gap between zero and the smallest possible definite number. We understand intuitively that the smallest possible definite number has the form, in our place-value system of Arabic numerals, of a one with a vast number of zeroes between it and the decimal point. How many make a vast number? We don’t know – we can’t know – because the smallest possible definite number lies in the transfinite realm, where the extremely small becomes the infinitely small, where the definite becomes the indefinite. But we do know how we can get there.

In explaining his doctrine of atoms Demokritos of Abdera asked his listeners to imagine taking some small object and cutting it repeatedly. At some stage, he assured his audience, they would come to a piece that they could not cut, the atomos, and the cutting process would come to its end. Let’s try to do something similar. Imagine a small body that in some way is equivalent to the number one and that we have an infinitome, a device that can make an infinite sequence of cuts. Demokritos claims that the device won’t work as advertised, that it cannot make an infinite sequence of cuts, that it must encounter the atomos after making a finite, if large, number of cuts. I disagree: I believe that I can cut our small body into an infinite set of infinitesimal pieces.

Assume that I have done so. The infinitesimal shares one of the fundamental properties of infinity, that of having an indefinite value, so if we reassemble our small body, it will not necessarily come out with a value of one. If that were not the case, then we would know that we didn’t have an infinite set of infinitesimal pieces. In order to gain a clearer understanding of what that statement means consider an example of such a set – space. Comprising an infinite set of infinitesimal points, space does not have a definite extent and, as astronomers have discovered, it does, in fact, expand quite freely.

Now we can conceive the circumstance in which Demokritos was right. If a divisible entity possesses a property that must always have a certain definite value, then we cannot divide it infinitely: we must of necessity come to an atomos. For example, if a body contains a certain amount of energy, which is subject to a conservation law, then any repetitive division of the body must eventually stop when each of the pieces contains a certain minimum of the energy, the atomos that cannot be further divided. Likewise, the action that a particle enacts in traveling between two set points must have a definite value because a closely neighboring potential path must have the same action in order that the subtraction of one path from the other yield a perfect zero, in accordance with the principle of least action; thus, action has its own atomos, an absolute number upon which we have founded the quantum theory.

Given that we can divide an object and its properties into an infinite set of infinitesimals, we can define an infinitesimal of that set as the fundamental unit of the properties. The properties of the reassembled object thus, by that measure, must take on indefinite values. If any one of these properties must take on a definite value, then we cannot divide the object infinitely and the smallest unit of measure of that property must have a finite and definite value. Thus we come back to the finite-value theorem, albeit from the small end of the number line.

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