# Case Study: Photoelectric Effect

The photoelectric effect is observed when electromagnetic radiation strikes the surface of a metal and the resulting energy transfer causes the metal to emit electrons. This phenomenon played a major role in the rejection of classical physics and the development of quantum mechanics.

### Introduction

The photoelectric effect was first documented in 1887 by the German physicist Heinrich Hertz and is therefore sometimes referred to as the Hertz effect. While working with a spark-gap transmitter (a primitive radio-broadcasting device), Hertz discovered that upon absorption of certain frequencies of light, substances would give off a visible spark. In 1899, this spark was identified as light-excited electrons (also called photoelectrons) leaving the metal's surface by J.J. Thomson. One of Hertz’' former assistants named Philipp Lenard went on to study this effect and was awarded the Nobel Prize in physics for his efforts. In 1905, Albert Einstein explained the photoelectric effect mathematically by proposing the concept of light quanta, or photons. This conclusion runs counter to the classic understanding of physics and is better understood in the context of wave-particle duality.

### Classical Explanation

According to the classical understanding of physics, when light shines on a surface, it slowly transfers energy into the substance. This increases the kinetic energy of the particles until finally, they give off excited electrons. This process is called thermal emission and it was considered the most likely explanation for the photoelectric effect. Given this justification, it was expected that increasing light intensity, regardless of frequency, would result in photoelectrons with higher kinetic energies. In addition, since the substance must first reach a critical temperature before it can begin ejecting electrons, it was expected that the photoelectric effect would not be observed immediately.

### Lenard's Experiment

To test the theories proposed by classical mechanics, Lenard built the experimental device shown below.

Figure 1: Lenard's experimental setup

When light reached the cathode, electrons were emitted and traveled down the vacuum tube until they reached the anode. Lenard could then determine the amount of electrons reaching the anode by measuring the current through the wire using a set potential voltage (battery). Using this device, Lenard ran a series of experiments in which he varied the frequency and intensity of the light. Surprisingly, Lenard found that below a certain threshold frequency, no matter how intense the light was, there was no emission of electrons. Above the threshold frequency, the current (i.e. the # of electrons reaching the anode) was directly proportional to the light intensity.

Figure 2: Current Intensity and Threshold frequency

Moreover, the current appeared almost instantaneously after the light was turned on (Lenard measured this to within 0.1 s, but today it has been observed to occur within 1 ns). Finally, by varying the potential and observing the change in current, Lenard was able to determine the kinetic energy of the ejected electrons. Interestingly, he found that higher frequency light increased the kinetic energy of the electrons, while changing the light intensity had no effect on the kinetic energy. Clearly, these findings could not be explained by classical physics and there must be some other explanation for the photoelectric effect.

### Quantum Explanation

Based on Lenard's experiment, the young physicist Albert Einstein set about explaining the photoelectric effect using the concept of photons (i.e. distinct "packets" of light). This controversial theory states that light, while it may have wave-like properties, can also be described by small, massless particles of energy. This complex understanding of electromagnetic radiation is referred to as wave-particle duality. With this theory, Einstein proposed that in the photoelectric effect, each photon was striking a single electron and causing it to break its association with the atom.

Figure 3: Absorption of a photon by an electron

However, each electron will only absorb the energy and be ejected if the frequency of the light is of sufficient energy according to the equation

$E = h\nu \tag{1}$

where $$\nu$$ is the frequency of the incident light and $$h$$ is Planck's constant = $$h=6.626*10^{-34} Js$$. The required energy to free an electron from an atom is called the work function and is designated by the symbol $$\phi$$. The threshold frequency is the lowest energy light particle needed to satisfy this work function (i.e. overcome the electron's affinity for the atom). Higher frequency light increases the kinetic energy ($$KE$$) of the ejected electron according to the equation:

$KE = h \nu - \Phi \tag{2}$

The following table summarizes the work functions for several elements:

Table 1: Work Functions of Select Elements
Element  Work Function $$\phi$$ (eV) Ionization Energy (kJ/mole)
Potassium 2.30 418.8
Sodium 2.75 495.8
Aluminum  4.28 577.5
Tungsten 4.55 770
Copper 4.65 745.5
Iron  4.70 762.5
Gold 5.10 890.1

Since every photon of sufficient energy excites only one electron, increasing the light's intensity (i.e. the number of photons/sec) only increases the number of released electrons and not their kinetic energy. In addition, no time is necessary for the atom to be heated to a critical temperature and therefore the release of the electron is nearly instantaneous upon absorption of the light. Finally, because the photons must be above a certain energy to satisfy the work function, a threshold frequency exists below which no photoelectrons are observed. This frequency is measured in Hertz (1/second) in honor of the discoverer of the photoelectric effect.

Thus in summary, Einstein's simple explanation completely accounted for the observed phenomenon in Lenard's experiment and began an investigation into the field we now call quantum mechanics. This new field seeks to provide a quantum explanation for classical mechanics and create a more unified theory of physics and thermodynamics. The study of the photoelectric effect has also lead to the creation of new photoelectron spectroscopy theory and applications.

### References

1. Knight, Randall D. Physics for Scientists and Engineers. 1st ed. Pearson Education Inc.; San Francisco, CA. 2004. 1220-1230.
2. McQuarrie, Donald A. Quantum Chemistry. 2nd ed. United States Of America: University Science Books, 2008. 321-24.