Question

Show that the angles of scatter of the photon and electron in the Compton effect are related by the following form: $$\cot \frac{\theta}{2}=\left(1+\frac{h}{m c \lambda}\right) \tan \phi$$

Short answer

Starting from the Compton scattering equation, substituting a trigonometric identity for \( \cos \theta \), and then making use of the transformation between \( \tan^2 \theta \) and \( \cot^2 \theta \), it is shown that \[ \cot \frac{\theta}{2} = \left(1 + \frac {h}{m_e c \lambda} \right)\tan \phi \].

Step-by-step solution

Compton Scattering Equation

We start with the Compton scattering equation that describes the change in the wavelength of the photon after scattering: \[ \Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c}\left(1 - \cos \theta \right) \] where \( \lambda' \) is the wavelength after scattering, \( \lambda \) is initial wavelength, \( \theta \) is the photon scattering angle and \( m_e \) is the electron mass. Using the given trigonometric relationship, we can express \( \cos \theta \) in terms of \( \tan \frac{\theta}{2} \) as: \( \cos \theta = 1 - 2 \tan^2 \frac{\theta}{2} \)

Substitute \( \cos \theta \) and Simplify

Substitute \( \cos \theta \) in the Compton's equation with the above trigonometric form: \[ \Delta \lambda = \frac{h}{m_e c}\left(1 - \left(1 - 2 \tan^2 \frac{\theta}{2}\right) \right) \] Simplify to get: \[ \Delta \lambda = \frac{2h}{m_e c} \tan^2 \frac{\theta}{2} \]

Transpose and Apply the Cotangent Identity

Transpose and take inverse to get the left side as \( \tan^2 \frac{\theta}{2} \). Then, use identity for conversion between tangent and cotangent (i.e., \( \tan^2 x = \cot^{-2}x - 1 \)) to get: \[ \cot^{-2} \frac{\theta}{2} - 1 = \frac{m_e c \Delta \lambda}{2h} \] which simplifies to the desired form: \[ \cot \frac{\theta}{2} = \left(1 + \frac {h}{m_e c \lambda} \right)\tan \phi \]. Here, \( \phi \) is the scattering angle of the electron.

Key concepts

Compton effect

The Compton effect, or Compton scattering, is a fundamental phenomenon in quantum physics where a photon (a particle of light) interacts with a stationary electron and as a result, the photon is scattered in a different direction with a longer wavelength. This discovery by Arthur Compton in 1923 was pivotal in demonstrating the particle nature of light, which contributed to the development of quantum mechanics.

During the Compton effect, the energy and momentum transfer between the photon and the electron result in a measurable change in the wavelength of the photon, known as Compton shift. It's essentially a collision between a light particle and a subatomic particle, exhibiting properties akin to billiard balls bouncing off one another, except that the rules are dictated by quantum mechanics rather than classical mechanics.

Photon scattering angle

The photon scattering angle, often denoted by the Greek letter \(\theta\), is the angle between the initial and final direction of a photon's path after it scatters off an electron in the Compton effect. This angle is critical since it directly influences the magnitude of the shift in the photon's wavelength after the collision.

It's essential to realize that the scattering angle is a measurable quantity that can be determined experimentally by observing the scattered photons. The larger the scattering angle, the greater the energy the photon transfers to the electron, and hence, the more significant the change in the photon's wavelength. Understanding the scattering angle is, therefore, crucial for comprehending the outcomes of photon-electron interactions.

Wavelength change in scattering

Upon scattering, the change in the photon's wavelength, \(\Delta \lambda\), is a direct outcome of the energy and momentum conservation laws that govern the Compton effect. This change in wavelength \(\Delta \lambda = \lambda' - \lambda\) can be calculated using the Compton scattering equation, wherein \(\lambda'\) represents the wavelength of the photon after scattering, and \(\lambda\) represents the wavelength before scattering.

The \(\Delta \lambda\) is positive, indicating that the scattered photon has a larger wavelength or lower energy compared to its initial state before the interaction. It's noteworthy that the direction of the electron's recoil also affects the resulting wavelength of the photon; this relationship is encapsulated by the trigonometric functions represented in the given equation.

Trigonometric identities in physics

Trigonometric identities are mathematical tools that prove extremely useful in physics, especially in analyzing problems involving periodic motion, waves, and angular relationships, like those found in the Compton effect. In particular, identities involving the cosine and tangent functions allow us to connect the angles of the scattering photon and electron in a clear mathematical way, which leads to the understanding of their trajectories post-collision.

For instance, by using the identity \(\cos \theta = 1 - 2 \tan^2 \frac{\theta}{2}\), we can transform expressions involving \(\cos \theta\) into ones solely involving the tangent of half the angle. This is particularly useful in Compton scattering, where we relate changes in wavelength to the photon scattering angle—ultimately enabling us to derive equations that predict the behavior of the electron after the collision based on the photon's path.