29.4

The Momentum of a Photon and the Compton Effect

Although Einstein presented his photon model for the photoelectric effect in 1905, it was not until 1923 that the photon concept began to achieve widespread acceptance. It was then that the American physicist Arthur H. Compton (1892–1962) used the photon model to explain his research on the scattering of X-rays by the electrons in graphite. X-rays are high-frequency electromagnetic waves and, like light, they are composed of photons.
Figure 29-10 illustrates what happens when an X-ray photon strikes an electron in a piece of graphite. Like two billiard balls colliding on a pool table, the X-ray photon scatters in one direction after the collision, and the electron recoils in another direction. Compton observed that the scattered photon has a frequency that is smaller than the frequency f of the incident photon, indicating that the photon loses energy during the collision. In addition, he found that the difference between the two frequencies depends on the angle at which the scattered photon leaves the collision. The phenomenon in which an X-ray photon is scattered from an electron, with the scattered photon having a smaller frequency than the incident photon, is called the Compton effect.

Figure 29-10 In an experiment performed by Arthur H. Compton, an X-ray photon collides with a stationary electron. The scattered photon and the recoil electron depart the collision in different directions.

In Section 7.3 the collision between two objects is analyzed using the fact that the total kinetic energy and the total linear momentum of the objects are the same before and after the collision. Similar analysis can be applied to the collision between a photon and an electron. The electron is assumed to be initially at rest and essentially free— that is, not bound to the atoms of the material. According to the principle of conservation of energy,

| (29.4)

where the relation has been used for the photon energies. It follows, then, that , which shows that the energy and corresponding frequency of the scattered photon are less than the energy and frequency of the incident photon, just as Compton observed. Since (Equation 16.1), the wavelength of the scattered X-rays is larger than that of the incident X-rays.
For an initially stationary electron, conservation of total linear momentum requires that

| (29.5)

To find an expression for the magnitude p of the photon’s momentum, we use Equations 28.3 and 28.4. According to these equations, the momentum p and the total energy E of any particle are

| (28.3)

| (28.4)

Rearranging Equation 28.4 to show that and substituting this result into Equation 28.3 give

A photon travels at the speed of light, so that we have , and the momentum of a photon is

This result only applies to a photon and does not apply to a particle with mass, because such a particle cannot travel at the speed of light. We also know that the energy of a photon is related to its frequency f according to (Equation 29.2) and that the speed c of a photon is related to its frequency and wavelength according to (Equation 16.1). With these substitutions, our expression for the momentum of a photon becomes

| (29.6)

Using Equations 29.4, 29.5, and 29.6, Compton showed that the difference between the wavelength of the scattered photon and the wavelength of the incident photon is related to the scattering angle by

| (29.7)

In this equation m is the mass of the electron. The quantity h/(mc) is called the Compton wavelength of the electron and has the value . Since varies between +1 and −1, the shift in the wavelength can vary between zero and 2h/(mc), depending on the value of , a fact observed by Compton.
The photoelectric effect and the Compton effect provide compelling evidence that light can exhibit particle-like characteristics attributable to energy packets called photons. But what about the interference phenomena discussed in Chapter 27, such as Young’s double-slit experiment and single-slit diffraction, which demonstrate that light behaves as a wave? Does light have two distinct personalities, in which it behaves like a stream of particles in some experiments and like a wave in others? The answer is yes, for physicists now believe that this Wave-Particle duality is an inherent property of light. Light is a far more interesting (and complex) phenomenon than just a stream of particles or an electromagnetic wave.
In the Compton effect the electron recoils because it gains some of the photon’s momentum. In principle, then, the momentum that photons have can be used to make other objects move. Conceptual Example 4 considers a propulsion system for an interstellar spaceship that is based on the momentum of a photon.

| Conceptual Example 4 | | | Solar Sails and the Propulsion of Spaceships

One propulsion method that is currently being studied for interstellar travel uses a large sail. The intent is that sunlight striking the sail creates a force that pushes the ship away from the sun (Figure 29-11), much as the wind propels a sailboat. Does such a design have any hope of working and, if so, should the surface of the sail facing the sun be shiny like a mirror or black, in order to produce the greatest possible force?

Figure 29-11 A solar sail provides the propulsion for this interstellar spaceship.

The physics of solar sails and spaceship propulsion.

Reasoning and Solution There is certainly reason to believe that the design might work. In Conceptual Example 3 in Chapter 7, we found that hailstones striking the roof of a car exert a force on it because they have momentum. Photons also have momentum, so, like the hailstones, they can apply a force to the sail.
As in Chapter 7, we will be guided by the impulse–momentum theorem in assessing the force. This theorem (Equation 7.4) states that when a net force acts on an object, the impulse of the net force is equal to the change in momentum of the object. Greater impulses lead to greater forces for a given time interval. Thus, when a photon collides with the sail, the photon’s momentum changes because of the force applied by the sail to the photon. Newton’s action–reaction law indicates that the photon simultaneously applies a force of equal magnitude, but opposite direction, to the sail. It is this reaction force that propels the spaceship, and it will be greater when the momentum change experienced by the photon is greater. The surface of the sail facing the sun, then, should be such that it causes the largest possible momentum change for the impinging photons.
As we found in Section 13.3, radiation reflects from a shiny mirror-like surface and is absorbed by a black surface. Now, consider a photon that strikes the sail perpendicularly. In reflecting from a mirror-like surface, its momentum changes from its value in the forward direction to a value of the same magnitude in the reverse direction. This is a greater change than what occurs when the photon is absorbed by a black surface. Then, the momentum changes only from its value in the forward direction to a value of zero. Consequently, the surface of the sail facing the sun should be shiny in order to produce the greatest possible propulsion force. A shiny surface causes the photons to bounce like the hailstones on the roof of the car and, in so doing, to apply a greater force to the sail.
Related Homework: Conceptual Question 10, Problem 45

Check Your Understanding 2

In the Compton effect, an incident X-ray photon of wavelength is scattered by an electron, the scattered photon having a wavelength of . Suppose that the incident photon is scattered by a proton instead of an electron. For a given scattering angle , is the change in the wavelength ofthe photon scattered by the proton greater than, less than, or the sam e as that scattered by the electron? (The answer is given at the end of the book.)
Background: The change in the photon’s wavelength is specified by Equation 29.7. How does this change depend on the mass of the particle?
For similar questions (including calculational counterparts), consult Self-Assessment Test 29.2, which is described at the end of Section 29.6.

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