Electron diffraction experiment

Figure 1: Schematic of electron diffraction experiment performed in lecture

In the spirit of observing and characterizing the behavior of elementary particles without prejudice, we performed an experiment in lecture on electrons where we observed their trajectories in vacuum after being sent through a series of slits formed by atoms of aluminum.

Figure 1 shows the experiment. The experiment takes place in a cathode ray tube (crt) similar to almost all computer monitors before the advent of the lcd flat panel display. Such a tube is filled with vacuum and has on one end a source of electrons and on the other end an observation screen coated with a chemical which gives a little flash of light at the location where each electron hits it. In general, the source emits electrons at such a high rate that we do not perceive the individual flashes; we just see a glow with brightness proportional to the rate or, equivalently, probability at which electrons arrive at each point.

After emerging from the source, the electrons pass through a parallel plate capacitor, made from two metal screens rather than solid plates so that the electrons can pass through. A knob controls the voltage $V$ across the capacitor so that the electrons can be accelerated to different speeds, ultimately picking up a kinetic energy equal to the electron charge $e$ times the voltage $V$,

$$KE = \frac{1}{2} mv^2 = eV.$$ (1)

Some additional equipment (not shown) focuses the accelerated electrons into a narrow beam which impinges on a target of aluminum metal (Al). Finally, after interacting with the aluminum metal, the electrons travel off to the observation screen at a distance $R=0.20$ m from the target.

Figure 2: Small section of target from electron beam's head-on viewpoint

Aluminum metal is generally polycrystalline, consisting of many tiny crystallites of aluminum stuck together at all different random angles, where each little crystal is a nearly perfect periodic array of atoms of aluminum. Figure 2 illustrates a tiny portion of such a polycrystal, showing five crystallites stuck together. We will not be going into three-dimensional crystal structure in any detail in this course other than to say that aluminum forms a so-called face centered cubic (fcc) crystal in which the atoms are arranged tightly into planes of spacing $d=2.34$ Å.²

Upon sending the electrons through this series of slits, we observed something truly remarkable (Figure 3). The probability of finding electrons arranges itself into thin circles of narrowly defined radii. The geometry of the experiment (Figure 1), means that the narrowly defined radius of each of these circles implies that electrons emerging from each crystallite come out at certain specific highly preferred angles, with the arrangement into circles coming from the fact that the crystallites occur at all possible angles. The radii which we see, in addition to be very narrowly defined, also always seem to occur in multiples: if there is a circle of radius $r$, we also find circles of radii $2r$, $3r$, .... This is extremely reminiscent of the sharply defined principle maxima of the intensity pattern for $N$-slits and leads to the hypothesis that the probability of finding an electron emerging at an angle $\theta$ from one of the crystallites is exactly what we would compute for the intensity pattern of interfering waves passing through the slits formed by the planes of atoms in the crystal. We are not saying that the electrons are waves, only that we can compute the probabilities using the same methods which we have already developed for computing intensities of waves.

Figure 3: Image on surface of cathode ray tube

The above hypothesis has been verified in detail many times. In fact, the radii of the different sets of rings correspond precisely to the spacings of the various planes of atoms in a truly three-dimensional treatment of the fcc crystal of aluminum. Moreover, many different types of barrier (other than crystals) have been tried, and each and every time the above hypothesis has held true. No one has yet to see a violation of it, for electrons or any other particle. It has even been verified for composite particles like helium atoms.

To turn the hypothesis into something useful for the central task of quantum mechanics, calculating probabilities, the final thing needed is to know the wave vector $k$ to use in the phase factors in the sum over histories prescription for evaluating intensities. To understand what affects $k$, we observed how the radius of the innermost circle, which corresponds to the first-order principle maximum of the interference from the planes of spacing $d=2.34$ Å, depends on the applied voltage. To relate this to the radius, we first note that from the N-slit interference formula, the n$^{th}$-order principle maximum occurs at angle $\theta$, where

$$k d \sin\theta = 2 \pi n.$$ (2)

Then, to convert the observed radius $r$ to the angle $\theta$, we use the geometry in Figure 1 to find

$$\tan \theta = \frac{r}{R}.$$ (3)

The first two columns of Table 1 summarize the raw results from one lecture for the radius of the smallest ring $r$ for two different values of the applied voltage $V$. Then, to produce the third column we use $\theta=\tan^{-1}(r/R)$ from Eq. (3). To produce the forth column, we solve Eq. (2) with $n=1$ (for the first-order principle maximum) to find $k=2 \pi/(d \sin\theta)$. The results for $k$ show that it does depend on the voltage $V$.

To determine the functional dependence, Figure 4 shows the value of $k$ as a function of $V$ where we have added the additional data point (found by continuing to decrease $V$) that indeed $k \rightarrow 0$ as $V \rightarrow 0$. The data in the figure approach the origin with increasing slope, something characteristic of the square-root function (curve sketched in the figure), suggesting that the dependence is as the square-root of the voltage,

$$k \propto \sqrt{V}.$$ (4)

Table 1: Data on smallest ring from electron diffraction experiment

$V$ (volts)	$r$ (m)	$\theta$ (rad)	$k$ (m$^{-1}$)	$p$ (kg$\cdot$m/s)	$\hbar$ (J$\cdot$s)
10,000	$0.93\times 10^{-2}$	$4.6 \times 10^{-2}$	$58 \times 10^{10}$	$5.4 \times 10^{-23}$	$0.93 \times 10^{-34}$
5,000	$1.36\times 10^{-2}$	$6.7 \times 10^{-2}$	$40 \times 10^{10}$	$3.8\times 10^{-23}$	$0.95 \times 10^{-34}$

Figure 4: Plot of results from Table 1 (crosses) with sketch of square-root function for comparison (curve)

The difficulty with Eq. (4) as a new general physical law is that the quantity $V$ is quite specific to our experiment and that there are many other ways to accelerate electrons. As a reflection of this, the proportionality constant in Eq. (4) turns out to be different for nearly every particle. To find a truly general law of physics, it is better to relate the intrinsic property $k$ to some intrinsic property which the voltage controls. One possible choice would be to use the electron's velocity, but it turns out that using the momentum, which is even more fundamental as it is a conserved quantity, leads to a much more general relation that applies to all known particles. To relate $k$ to the momentum $p$ of the electrons, we first relate the kinetic energy of the electrons directly to the momentum,

$$KE=\frac{1}{2} m v^2 = \frac{1}{2} m \left(\frac{p}{m}\right)^2 = \frac{p^2}{2m}.$$ (5)

This form for the kinetic energy turns out to be very useful in advanced applications because it actually relates two conserved quantities, energy and momentum. It is in fact the form for kinetic energy always used in quantum mechanics and even more advanced courses on mechanics. It is worth memorizing.

The final relation between $V$ and $p$ we find by using $KE=eV$ from Eq. (1) and solving Eq. (5) for $p$,

$$p =\sqrt{2meV}.$$ (6)

Thus, we have that $p \propto \sqrt{V}$ from basic considerations of energy and momentum, and we also have from our experimental observations (Figure 4) that $k \propto \sqrt{V}$. Therefore, we conclude that $k \propto p$ in our experiment. In fact, this relation has been observed in many different experiments for electrons and every other known elementary particle, with the same constant of proportionality for all particles!.³ The standard form for writing the proportionality is

$$k=\frac{p}{\hbar},$$ (7)

where the proportionality constant $\hbar$ is a universal constant of nature known as Planck's constant. The symbol for this constant is an `$h$' with a bar through the top and is pronounced ```h'-bar''. This is to distinguish it from another constant unfortunately also known as Planck's constant but written as a plain `$h$'. This other constant, which you may find in tables of physical constants, is directly related to the constant appearing in Eq. (7) through $h=2 \pi \hbar$. The reasons for this are purely historical. In this course, we shall always use the version with the bar.

Now that we have the general relation (7), the last thing we need is the value of $\hbar$. We could look this up in a table, but we can also determine an approximate value right from the experiment which we did in lecture. To do so, the fifth column of Table 1 lists the values of $p$ determined from Eq. (6) using the voltage from the first column and the known values for the mass $m=9.11 \times 10^{-31}$ kg and charge $e=1.602\times 10^{-19}$ C of the electron. Finally, the sixth column gives the value for $\hbar$ determined by solving Eq. (7) to give $\hbar=p/k$. The values are not quite the same because of errors in our experiment and the difference between them gives some idea of our experimental errors. We find a value of about $1 \times 10^{-34}$ J$\cdot$s, which is quite close to the official value of $\hbar = 1.055\ldots \times 10^{-34}$ J$\cdot$s, especially considering how roughly we did all of the measurements!

The following statement summarizes the lessons learned from our experiment in a form that has been verified in many experiments and never yet observed to be violated.

de Broglie hypothesis (generalized): The probability ${\mathcal P}(x)$ of finding a particle at point $x$ is proportional to the intensity $I(x)$ we would compute for waves of wave vector $k=p/\hbar$ (and frequency $\omega = E/\hbar$) at $x$, where $p$ and $E$ are the momentum and energy of the particle, respectively.

We call this form of the hypothesis ``generalized'' because de Broglie did not spell out the connection between probabilities and intensities quite so clearly. He was mostly responsible for the connections between momentum and wave vector and energy and frequency. The frequency-energy connection is not something that we will deal with in this course. It comes from other experiments (such as the photo-electric effect) which we will not cover; we include it here only for completeness.