I'm aware that, like all quantum objects (I think), an electron can act as both a wave and a particle. Electron diffraction is a good example of how an electron can act as a wave, but I'm struggling to come up with an example of how it acts as a particle.
Any help would be much appreciated!
In an old-school TV picture tube, electrons were shot into one end of it, accelerated, steered into specific directions, and then collided with a thin phosphor coating on the inside of the picture end of the tube. Each collision created a burst of light, building up a visible picture for the TV watchers.
This process is well-modeled by envisioning the electrons as particles.
The original SLAC particle accelerator can be thought of as a machine for adding huge amounts of energy to electrons by shooting them down an evacuated tube two miles long, in which the electrons were separated into individual bunches and then made to surf on the crests of an intense microwave beam traveling down the same tube.
This process is also well-modeled by envisioning the electrons as particles.
Some examples of electrons behaving like a particle:
The photoelectric effect : As beautifully described by Einstein, electromagnetic radiation hits a material which leads to the emission of electrons. This example is perfectly based on the particle-only view and, amazingly, disagrees with the results of classical electromagnetism (The experimental results proved this claim). This effect was the heart of the technology of the early TV systems in which the electrons are always found to be absorbed at the screen at discrete points, as individual particles. I think there is no need for any reference.
Confinement of single electron : A number of studies have been confirmed that individual electrons can be confined in ultra small transistors, seemingly indicating the validity of particle-only view. For example, see Refs. [1-3]. These studies are primarily based on the discovery of the single-electron transistor in which requires only the (classical) quantization of charge (for this one, see Ref. 4). Note that these observations can be interpreted classically as well as quantum mechanically. Quantum mechanically, the wave packet of a single electron can be technically confined/localized in a very small region.
Synchrotron radiation : Assuming the particle nature for electrons, radiation from electrons in a Synchrotron were announced in Ref. 5, proving that electrons radiates radially when like charged particles are accelerated. This phenomenon is completely explained according to the particle-only view without considering any wave-only view.
Other particle-only view proofs are scatterings including Muller (electron-electron) scattering, Compton (electron-photon) scattering etc (the relevant references can be found easily in modern physics textbooks).
I know these categories for the particle-only view of electron. Others may suggest different proposals such as the experiments of J. J. Thomson (the discoverer of electron with the famous measurement of the charge-to-mass ratio of the electron) or Robert Andrews Millikan's famous oil-drop experiments, which are almost good evidence in support of particle-only view, but, I think these experiments reveal the quantized nature of electric charge more than the particle-only view. In my opinion, photoelectric effect, is the best example for particle-only view. However, I think the answer by @niels nielsen meets your question as well. In addition, one can cleverly deduce the particle-only view of electron from the Feynman's double-slit experiment, but this one needs principles of quantum physics, and its discussion here is controversial as well.
References:
1 M.A. Kastner, The single-electron transistor, Reviews of modern physics 64, 849 (1992)
2 E. Prati et al, Few electron limit of n-type metal oxide semiconductor single electron transistors, Nanotechnology 23, 215204 (2012)
3 E. Prati, Single electron effects in silicon quantum devices, Journal of nanoparticle research 15, 1 (2013)
4 T. Fulton et al, Observation of single-electron charging effects in small tunnel junctions. Physical review letters 59, 109 (1987)
5 F. Elder et al, Radiation from electrons in a synchrotron, Physical Review 71, 829 (1947)
Since the idea of the electron as a particle predates the idea of it as a wave by a quarter century, why not ask why electrons were thought of as particles at first? And for that, why not go to their discoverer? In 1897 J. J. Thomson performed experiments on "cathode rays", the rays emitted when a voltage is applied between metal plates in vacuum. At this point, the idea of matter being made of atoms was almost a century old, if we count from Dalton. If we count from the Greeks, millennia. Boltzmann had published his statistical theory of gases 25 years previously, although it would not be universally accepted for another decade or so (Einstein's Brownian motion paper). Measurements of the charge-to-mass ratios of ions had been performed, Becquerel and Curie were working under atomism, etc. Thomson argued
The experiments* discussed in this paper were undertaken in the hope of gaining some information as to the nature of the Cathode Rays. The most diverse opinions are held as to these rays; according to the almost unanimous opinion of German physicists they are due to some process in the aether to which--inasmuch as in a uniform magnetic field their course is circular and not rectilinear--no phenomenon hitherto observed is analogous: another view of these rays is that, so far from being wholly aetherial, they are in fact wholly material, and that they mark the paths of particles of matter charged with negative electricity. It would seem at first sight that it ought not to be difficult to discriminate between views so different, yet experience shows that this is not the case, as amongst the physicists who have most deeply studied the subject can be found supporters of either theory.
The electrified-particle theory has for purposes of research a great advantage over the aetherial theory, since it is definite and its consequences can be predicted; with the aetherial theory it is impossible to predict what will happen under any given circumstances, as on this theory we are dealing with hitherto unobserved phenomena in the aether, of whose laws we are ignorant.
The following experiments were made to test some of the consequences of the electrified-particle theory.
By measuring the charge-to-mass ratio, he could rule out that cathode rays were atoms or molecules, for the particles are far too small or light.
Thus for the carriers of the electricity in the cathode rays $m/e$ is very small compared with its value in electrolysis. The smallness of $m/e$ may be due to the smallness of $m$ or the largeness of $e$, or to a combination of these two. That the carriers of the charges in the cathode rays are small compared with ordinary molecules is shown, I think, by Lenard's results as to the rate at which the brightness of the phosphorescence produced by these rays diminishes with the length of path travelled by the ray. If we regard this phosphorescence as due to the impact of the charged particles, the distance through which the rays must travel before the phosphorescence fades to a given fraction (say $1/e$, where $e = 2.71$) of its original intensity, will be some moderate multiple of the mean free path. Now Lenard found that this distance depends solely upon the density of the medium, and not upon its chemical nature or physical state. In air at atmospheric pressure the distance was about half a centimetre, and this must be comparable with the mean free path of the carriers through air at atmospheric pressure. But the mean free path of the molecules of air is a quantity of quite a different order. The carrier, then, must be small compared with ordinary molecules.
Being conceptually simple, in agreement with experiments, and in line with the general paradigm of atomism, it is not difficult to find plausible the idea that electrons are particles. (Perhaps it is a twist of irony that de Broglie and Schrödinger then some 20-30 years later showed that the Germans were -- in a very loose sense! -- also right.)
Reference
Thomson, J.J. (1897). "Cathode Rays". Philosophical Magazine. 44 (269): 293–316
My absolute favorite example of the electron acting as just a particle is the bubble chamber.
Is it possible to produce images of pair production in home-made cloud chamber?
As you can see on the left, the electron you are asking about is just a "scratch", leaving nothing but a track (of microscopic bubbles) behind as it spirals inwards. I believe this is the best example of the electron being a point particle that is able to leave this linear track. Just a (ionization) track, no waves.
Some experiments use some kind of medium analogous to ballistics jell. The electron travels through it and it's trajectory is traced out or left in it, so that it is easily seen. When there are electric fields the trajectory will spiral in accordance with lorents force and classical laws of electrodynamics. These are individual electron trajectories, and for intents and purposes, completely adheres to electrodynamics and Maxwell equations.
There are similar experiments that involve particle collisions or decay, were one particle splits into two and the two trajectories can be seen. I think this is common with early particle accelerators. Not sure if cern still uses such things, because here the scales are very large probably in correspondence with q.m. approaching classical mechanics. Cern however is completely dealing with q.m. phenomena I would presume. Sorry no sources as of yet only my vague recollection.
There are no experiments or main stream theories in which the electron is not a point particle. That being said, there are no experiments or main stream theories in which the electron's behavior is not predicted by a wave function.
A simple model for the cascade theory of electronic showers can be formulated as a set of integro-partial differential equations. Let $\Pi (E,x) dE$ and $\Gamma (E,x) dE$ be the number of particles and photons with energy between $E$ and $E+dE$ respectively (here $x$ is the distance along the material). Similarly let $\gamma (E,E')dE'$ be the probability per unit path length for a photon of energy $E$ to produce an electron with energy between $E'$ and $E'+dE'$. Finally let $\pi (E,E')dE'$ be the probability per unit path length for an electron of energy $E$ to emit a photon with energy between $E'$ and $E'+dE'$. The set of integro-differential equations which govern $\Pi$ and $\Gamma$ are given by...etc.
I think that this example (from Wikipedia, see comment below) shows clearly that the electrons (and photons) are treated in a particle-like way.
