Compton Effect

1

Even by the early 1920's, the particle-like nature of light, i.e. the photon, as implied by the photoelectric effect in 1905, was still being debated. In this context, an experiment carried out by Arthur Compton, in 1922, would be forwarded as further independent evidence in support of light’s particle-like behaviour. The Compton Effect, also referred to as Compton Scattering, is analogous in set-up to the photoelectric effect in that it involves the collision between high-energy photons and electrons in some given target. However, in this case, the focus of the observations is on the fact that the scattered radiation experiences a wavelength shift, which could not be explained in terms of classical wave theory. During the experiment, Compton observed the scattering of X-rays by electrons in a carbon target and found that the scattered X-rays had a longer wavelength than the X-ray originally incident upon the target. The change in the wavelength increased with scattering angle according to the Compton formula:

[1]     1

Compton then explained the results of [1] by assuming the light had a particle-like nature in which kinetic energy was conserved in the collision between the photon and the electron. In this context, the scattered photon would have lower energy, after the collision, and therefore have longer wavelength in accordance to the Planck relationship:

[2]      2

As the diagram above indicates, Compton used X-ray photons, which exist in the frequency range of 3*1016 to 3*10 19 hertz, simply because they have enough energy [E=hf] to recoil an electron. However, there is another implication that follows on from the particle-like nature, which allowed Compton to also equate [2] to Einstein’s famous energy equation

[3]      3

The numeric values shown are based on the assumption that the scattering occurs in the rest frame of the electron. In the Bohr model of hydrogen, the orbital velocity of the electron in the ground state of hydrogen was estimated to only be around 1% of the speed of light [c], as such, we might assume negligible relativistic effects on the rest mass of the electron.

It should be noted that the idea of electrons physically orbiting the nucleus will later be questioned by quantum theory.

However, the result in [3] defined what subsequently became known as Compton’s wavelength [λ] for the electron, because the mass of the electron was substituted into [3]. Whether this wavelength really has anything to do with the electron might be questioned further in the context of a particle-like collision between a photon and an electron. For example, the original assumption in [3] can be transformed such that it yields an equivalent kinetic mass for a photon:

[4]      4

At this point, the subscription of [mi] reflects the initial effective kinetic mass of the incident X-ray photon in Compton’s experiment, i.e. prior to the collision, which we want to interpret as having a particle-like mass. If we make the assumption that this collision conforms to an elastic collision, we can describe the conservation of kinetic energy as follows:

[5]      5

What we see in [5] is that some portion of the kinetic energy of the incident photon is transferred to the ‘stationary’ electron, whose rest mass [me] remains unchanged, but which then acquires a non-relativistic velocity [v2] due to the collision. The only way to maintain the conservation of energy in this case is to assume that the initial kinetic mass [mi] of the photon falls to [mf], which actually reflects a decrease in the frequency [f] or alternatively an increase in the wavelength [λ] of the photon.

[6]      6

In the context of [6], Compton’s wavelength is simply the wavelength of the post-collision photon. As such, there is no physical inference that this wavelength is an attribute of the particle; as in the context of an elastic collision, it always remains an attribute of either the incident or scattered photon. As such, Compton was only pointing towards the fact that light had a particle-like nature. The inference that particles, e.g. electrons, had a wave-like natue would have to wait for de Broglie's hypothesis in 1924.

Footnotes

Although Einstein had won the Nobel Prize in 1921 for his work on the photoelectric effect, this prize was given for the discovery of the law associated with the effect and did not endorse the idea of light quanta. However, Einstein did not give his Nobel lecture until July 1923 at which Bohr also spoke and made a comment along the following lines:

In spite of its heuristic value, the hypothesis of light quanta is irreconcilable with inference phenomena and is not able to throw light on the nature of radiation.”

In the context of the historical timeline being followed, this comment was made after Bohr had learnt about the Compton Effect. However, at this point, Bohr was still convinced about the wave-like nature of light such that he even suggested that the conservation of energy, on which Compton had predicated his analysis, might not be totally applicable on the sub-atomic scale. While Compton was able to eventually prove Bohr wrong on this point in 1925, Einstein had already hinted at the prospect of wave-particle duality in 1924 along the following lines:

"There are now two theories of light, both indispensable and, as one must admit today despite 20 years  of effort, without any logic connection”

However, the debate surrounding the wave-particle nature of light would quickly be extended to matter particles,  in 1924, when Louis de Broglie submitted his PhD thesis entitled “Researches on the quantum theory” in which he outlined the idea that particles, like electrons, might also have wave-like attributes.