Problem 22 - University Physics with Modern Physics

Question

A photon scatters in the backward direction \(\left(\phi=180^{\circ}\right)\) 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?

Step-by-Step Solution

Answer

The initial wavelength of the photon is approximately \(7.920 \times 10^{-14}\, \text{m}\).

Step 1 - Understand the Problem

We need to find the initial wavelength of a photon, which undergoes a 10% increase in wavelength after scattering off a stationary proton at an angle \(\phi = 180^\circ\). This involves using the Compton scattering equation to find the initial wavelength that leads to a specified percentage change.

Step 2 - Review Compton Scattering Formula

The Compton wavelength shift formula is: \[\Delta \lambda = \lambda' - \lambda = \frac{h}{m c} (1 - \cos \phi)\]where \(h\) is Planck's constant, \(m\) is the mass of the proton, \(c\) is the speed of light, and \(\phi\) is the scattering angle. Since \(\phi = 180^\circ\), \(1 - \cos \phi = 2\).

Step 3 - Calculate the Wavelength Change

The change in wavelength \(\Delta \lambda\) is given as 10% of the initial wavelength \(\lambda\). Thus,\[\Delta \lambda = 0.1 \lambda\]Using the Compton equation:\[\Delta \lambda = \frac{h \cdot 2}{m c}\]

Step 4 - Equate and Solve for Initial Wavelength

Equating the expressions for \(\Delta \lambda\), \[0.1 \lambda = \frac{2h}{m c}\]Solving for \(\lambda\),\[\lambda = \frac{20h}{m c}\]Substituting known values: \(h = 6.626 \times 10^{-34}\, \text{Js}\), \(m = 1.672 \times 10^{-27}\, \text{kg}\), and \(c = 3.00 \times 10^8\, \text{m/s}\), we calculate \(\lambda\).

Step 5 - Perform the Calculation

Substituting the constants into the formula:\[\lambda = \frac{20 \times 6.626 \times 10^{-34}}{1.672 \times 10^{-27} \times 3.00 \times 10^8}\]This yields:\[\lambda \approx 7.920 \times 10^{-14} \text{m}\]Thus, the initial wavelength of the photon is approximately \(7.920 \times 10^{-14}\, \text{m}\).

Key Concepts

Photon Wavelength

When we talk about the 'photon wavelength', we refer to the distance between consecutive peaks of the electromagnetic wave that the photon travels along. A photon, being a quantum of light, has a wavelength that directly affects both its energy and color. The formula connecting wavelength \(\lambda\), frequency \(f\), and the speed of light \(c\) is \(c = \lambda f\). Since the speed of light is constant in a vacuum, any change in the wavelength must result in a change in the frequency as well. In the context of Compton scattering, understanding the initial photon wavelength is crucial because the interaction changes this wavelength. This change provides insights into the energy transferred during the event. With a known wavelength, one can deduce aspects like the energy and momentum of the photon before and after interacting with a particle such as a proton.

Scattering Angle

The 'scattering angle', denoted as \(\phi\), in the Compton scattering context, plays a crucial role in determining the change in a photon's wavelength. It is the angle at which the photon is deflected as a result of the interaction with another particle. In formula terms, the scattering angle appears in the cosine term of the Compton formula: \(\Delta \lambda = (h/mc)(1 - \cos \phi)\).

• When \(\phi = 0^\circ\), the photon continues in its initial direction, and there is no change in wavelength (\(\Delta \lambda = 0\)).
• When \(\phi = 180^\circ\), as in the exercise, the photon is deflected backwards, and the change in wavelength is maximized, equal to twice the Compton wavelength of the proton involved.

Thus, the scattering angle is integral in calculating how much the photon's wavelength will shift during the scattering event, making it a pivotal factor when predicting photon behavior post-interaction.

Wavelength Shift

The 'wavelength shift', \(\Delta \lambda\), in Compton scattering specifically describes the difference in the wavelength of a photon before and after it scatters off a target particle. This shift arises due to the transfer of energy and momentum between the photon and the other particle, such as a proton in our problem.The shift can be calculated using Compton's wavelength shift formula: \[\Delta \lambda = \lambda' - \lambda = \frac{h}{mc}(1 - \cos \phi) \]This formula tells us how photon properties change with scattering. Key steps include:

• Identifying that the scattering angle \(\phi\) influences the amount of shift. A \(180^\circ\) angle results in maximum possible shift.
• Understanding that the shift as 10% of the initial wavelength in this exercise implies \(0.1\lambda\).
• Equating the calculated shift to 10% of the initial wavelength helps in solving exercises like determining initial photon wavelengths given post-scattering conditions.

The significance of wavelength shift lies in informing us about the energy dynamics during the photon-material interaction, which are fundamental in understanding quantum mechanical processes.