A Compton scattering experiment is performed using an...

A Compton scattering experiment is performed using an aluminum target. The incident photons have wavelength $\lambda$. The scattered photons have wavelengths $\lambda^{\prime}$ and energies $E$ that depend on the scattering angle $\theta$. (a) At what angle $\theta$ are scattered photons with the smallest energy detected? (b) At this same scattering angle $\theta$, what is the ratio $\lambda^{\prime} / \lambda$ for $\lambda=10.0 \mathrm{pm}$ ?

A Compton scattering experiment is performed using an aluminum target. The incident photons have wavelength $\lambda$. The scattered photons have wavelengths $\lambda^{\prime}$ and energies $E$ that depend on the scattering angle $\theta$.
(a) At what angle $\theta$ are scattered photons with the smallest energy detected? (b) At this same scattering angle $\theta$, what is the ratio $\lambda^{\prime} / \lambda$ for $\lambda=10.0 \mathrm{pm}$ ?

Key Concepts

Compton Scattering

This is a quantum mechanical process in which a photon scatters off a target particle (typically an electron) resulting in a change in the photon's wavelength and direction. It demonstrates the particle nature of light and provides evidence for the duality of matter, highlighting how energy and momentum are conserved in collisions between photons and charged particles.

Compton Wavelength Shift Equation

The Compton effect is quantitatively described by the Compton wavelength shift equation, which relates the change in wavelength of the scattered photon to the scattering angle. This equation encapsulates how the maximum wavelength shift, and thus the maximum energy change, occurs when the photon is scattered backward (at a 180-degree angle), making it essential to understanding the angular dependence of the scattering process.

Photon Energy and Wavelength Relationship

The energy of a photon is inversely proportional to its wavelength, as given by the relation E = hc/?, where h is Planck’s constant and c is the speed of light. This inverse relationship means that an increase in wavelength due to scattering corresponds to a decrease in the photon's energy, a concept that underpins the analysis of energy changes in Compton scattering experiments.

Scattering Angle Dependence

The scattering angle determines the extent of the wavelength shift in Compton scattering. A larger scattering angle results in a greater change in the photon's wavelength (and thus a lower energy), while a small scattering angle leads to a smaller shift. Understanding this dependence is crucial for analyzing experimental results and determining the conditions needed to observe specific energy or wavelength outcomes.

Transcript

00:01 For this exercise, we have the compton scattering of a photon, so it scatters off an electron.

00:06 And the initial wavelength, the wavelength of the incident photon, is given by just lambda.

00:16 And the wavelength of the scattered photon is given by lambda prime, and its energy is given by e.

00:24 So this one here is the incident, and this one here is the scattered.

00:37 The first part of the exercise, question a, asks us for which value of theta, the angle theta of scattering, we're going to have the smallest value of the energy e.

00:53 So let's write here, compton's formula for lambda prime.

00:59 The scattered wavelength is going to be the initial wavelength, the incident wavelength, plus h -m -e -c, 1 minus cosine of theta.

01:13 And the energy of the scattered photon is going to be hc over lambda prime.

01:21 This is just planx formula.

01:23 So notice that we want the smallest energy, and the smallest energy is going to happen when we have the maximum wavelength.

01:35 And the maximum wavelength, we're going to have the maximum wavelength when 1 minus the cosine of theta, is maximum.

01:44 Okay, so we want 1 minus cosine of theta to be the largest number possible.

01:50 And this is going to happen when theta is 180 degrees.

01:54 And that's because when theta is 180 degrees, the cosine of theta is minimum.

02:00 So it's minus 1.

02:01 And this number here, 1 minus cosine of theta, is going to be 2.

02:09 Notice that cosine of theta cannot be any smaller than minus 1...

Please give Ace some feedback