A photon scatters in the backward direction ($\varphi =180$°) from a free proton that is initially at rest. What must the wavelength of the incident photon be if it is to undergo a $10.0\%$ change in wavelength as a result of the scattering?

Verified step by step guidance

1. Understand the concept of Compton scattering, where a photon scatters off a particle, such as a proton, resulting in a change in the photon's wavelength.
2. Use the Compton wavelength shift formula: $\mathrm{\Delta \lambda }=\frac{h}{mc}(1-\cos \theta )$, where $\mathrm{\Delta \lambda }$ is the change in wavelength, $h$ is Planck's constant, $m$ is the mass of the proton, $c$ is the speed of light, and $\theta$ is the scattering angle.
3. Since the scattering angle $\theta$ is 180°, substitute this into the formula: $\mathrm{\Delta \lambda }=\frac{h}{mc}(1-\cos 180°)$. Note that $\cos 180°$ is -1.
4. Calculate the change in wavelength $\mathrm{\Delta \lambda }$ using the formula: $\mathrm{\Delta \lambda }=\frac{2h}{mc}$. This represents a 10% change in wavelength, so $\mathrm{\Delta \lambda }$ is 0.1 times the initial wavelength $\lambda$₀.
5. Set up the equation $\mathrm{\Delta \lambda }=\frac{2h}{mc}=0.1\lambda$₀ and solve for the initial wavelength $\lambda$₀.

Key Concepts

Compton Scattering

Compton scattering is a phenomenon where a photon scatters off a target particle, such as an electron or proton, resulting in a change in the photon's wavelength. The change in wavelength depends on the angle of scattering and the mass of the target particle. This concept is crucial for understanding how the wavelength of the photon changes during the interaction.

Wavelength Change Calculation

The change in wavelength during Compton scattering can be calculated using the Compton wavelength shift formula: Δλ = λ' - λ = (h/mc)(1 - cos θ), where h is Planck's constant, m is the mass of the target particle, c is the speed of light, and θ is the scattering angle. For a 180° backward scattering, cos θ = -1, which maximizes the wavelength change.

Photon Energy and Momentum

Photons have energy and momentum, which are related to their wavelength by E = hc/λ and p = h/λ, respectively. During scattering, the conservation of energy and momentum principles apply, affecting the photon's wavelength. Understanding these relationships helps in determining the initial wavelength required for a specific percentage change in wavelength.