Visualizing the electron’s quantization with a ruler

More than 100 years ago, Robert Millikan demonstrated the quantization of the electron using charged, falling droplets, but the statistical analysis on many falling droplets did not allow a direct visualization of the quantization of charge. Instead of letting the droplets fall, we have used optical levitation to create a single droplet version of Millikan’s experiment where the effects of a single electron removal can be observed by the naked eye and measured with a ruler. As we added charges to the levitated droplet, we observed that its equilibrium position jumped vertically in quantized steps. The discrete nature of the droplet’s jumps is a direct consequence of the single-electron changes in the charge on the droplet, and therefore clearly demonstrates the quantization of charge. The steps were optically magnified onto a wall and filmed. We anticipate that the video of these single electron additions can become a straightforward demonstration of the quantization of charge for a general audience.

www.nature.com/scientificreports/ We produced the electric field by applying a voltage difference of 666 ±0.5 V across a pair of horizontal electrodes with a vertical separation of 1.0 mm. The laser passes through each electrode via a circular hole with a radius of 1.0 mm. The small distance between the electrodes and the large voltage difference were chosen to produce a strong electric field.
We used a numerical simulation to determine the magnitude and homogeneity of the electric field. We defined a working volume between the electrodes inside of which we know the experiment took place. In this volume, the electric field was found to have a vertical direction and a magnitude of 360 ± 45 kV/m. In a smaller volume sufficient for the droplet to make a couple of jumps, the homogeneity of the field is even greater, which results in equispaced jumps (see Supplementary Information).
We added individual electrons to a previously neutralized droplet (see Supplementary Information) using a 241 Am alpha radiation source. The emitted alpha particles produced free electrons in the trap either directly by striking the droplet or indirectly by striking the electrodes or ionizing the air in the trap. The free electrons were subsequently deposited on the surface of the droplet. In this manner, we were able to change the charge on the droplet in randomly distributed steps of either single or small multiples of the elementary charge.
We projected an image of the droplet onto a wall in the laboratory using an aspheric planoconvex lens (f = 50 mm, Thorlabs AL2550G) with a magnification of 73 ± 1.4 . With this magnification, we were able to observe micro-metric movements of the droplet with the naked eye. The light scattered from the droplet comes mostly from the bottom where the laser beam hits the droplet and the top where the beam leaves the droplet. Hence, the image of the droplet is observed as two separate images on the screen, as seen in Fig. 1.
When we set the alpha radiation source at the appropriate distance from the droplet, it gains charges randomly. This causes it to jump discontinuously from one equilibrium point to another. We filmed a video of a series of electron additions (see online). Some selected frames are shown in Fig. 2a. Between each of the frames, the droplet gained an additional amount of charge causing it to jump from one equilibrium position to another. Equally spaced horizontal reference lines are added to Fig. 2a. The separation between the equally spaced horizontal lines corresponds to the distance the droplet moves when absorbing a single elementary charge. The quantization of the charge immediately stands out. The position of the droplet always falls on one of the horizontal lines and, by simply counting the number of lines between the steps, one can determine the number of electrons the droplet has gained. Figure 2b presents the position as function of time for the droplet shown in the video. The magenta equally spaced lines represent the displacement, y , caused by a single electron addition and all the steps are multiples of this distance. In this graph, one can clearly see the full series of 8 steps that fall on a horizontal line, where the droplet gains 2, 3, 1, 1, 4, 6, 3, and 9 electrons. Once again, one can observe the quantization of the electronic charge arising from the discrete nature of the individual steps.
We determined the displacement y by fitting the positions of the droplet to a step function, which is shown in Fig. 3a. The fit produced a value of y of 10.36 ± 0.26 µ m, which is almost three times smaller than the droplet's diameter. The two fit variables were a global single electron displacement y , and an individual number of electrons added for each point in time. The uncertainty in y stems mostly from the droplet's oscillation around the stability positions.  www.nature.com/scientificreports/ The residue between the fit and the data is plotted in magenta at the bottom of Fig. 3a and a histogram of this residue is plotted in Fig. 3b. The histogram follows a normal distribution around zero and its FWHM is less than half an electron step, providing further evidence that we are observing quantized steps.
To confirm that these steps were indeed caused by electron additions, we used them to calculate the charge of the electron. Balancing the electrostatic ( F e = qE ) and optical restoring ( F r = k y ) forces results in where q is the charge of the electron, y the displacement caused by a single electron addition, k the trap stiffness and E the magnitude of the electric field. The method to determine the trap stiffness is described in the methods section. Using Eq. (1), we calculated the charge of the electron to be of 1.44 ± 0.25 × 10 −19 C , which agrees within the statistical uncertainty with the known value of 1.602 × 10 −19 C 12 . The uncertainty was calculated through error propagation in Eq. (1) where the biggest contributors were the uncertainties of the electric field and the trap stiffness.
The series of electron jumps serve as straight forward evidence of the quantization of the electric charge. We have magnified the effect to a level where the step caused by adding a single electron can be seen with the naked eye and measured with a simple ruler. The discrete steps are the result of the charge on the droplet changing by a single electron. In contrast, other methods of observing the effects of quantization such as the photoelectric effect or atomic emission lines are indirect in the sense that they involve the use of many photons or electrons. Our experiment allows one to directly visualize charge quantization, a quantum phenomenon, in the macroscopic world.

Data availibility
All relevant data generated or analysed for this study are available within the article and the associated Supplementary Information. Any other data are are available from J.T.M (javier.marmolejo@physics.gu.se) upon reasonable request.