Question

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

Short answer

The initial wavelength is approximately \(1.32 \times 10^{-14} \text{ m}\).

Step-by-step solution

Understand the Problem

We need to calculate the initial wavelength of a photon that undergoes Compton scattering, resulting in a 10% increase in wavelength when scattering backward from a proton at rest.

Compton Wavelength Change Formula

The change in wavelength \(\Delta \lambda\) for Compton scattering is given by the formula:\[\Delta \lambda = \frac{h}{{m_pc}}(1 - \cos \phi)\]where \(h\) is Planck's constant, \(m_p\) is the proton mass, \(c\) is the speed of light, and \(\phi\) is the scattering angle.

Substitute Known Values

Since the photon scatters backward, \(\phi = 180^\circ\), hence \(\cos 180^\circ = -1\). Substitute this into the equation:\[\Delta \lambda = \frac{h}{{m_pc}}(1 - (-1)) = \frac{2h}{{m_pc}}\]

Express Wavelength Change as Percentage

We know that the change in wavelength is 10% of the initial wavelength, \(\lambda\). So, \(\Delta \lambda = 0.1 \lambda\). Thus,\[0.1 \lambda = \frac{2h}{{m_pc}}\]

Solve for Initial Wavelength

Rearrange the equation to solve for \(\lambda\):\[\lambda = \frac{20h}{{m_pc}}\]

Substitute Constants and Compute

Substitute numerical values \(h = 6.626 \times 10^{-34} \text{J} \cdot \text{s}\), \(m_p = 1.672 \times 10^{-27} \text{kg}\), and \(c = 3.00 \times 10^8 \text{m/s}\) into the equation and calculate \(\lambda\):\[\lambda = \frac{20(6.626 \times 10^{-34})}{{1.672 \times 10^{-27} \times 3.00 \times 10^8}} \approx 1.32 \times 10^{-14} \text{m}\]

Key concepts

Photon Wavelength

The wavelength of a photon is a crucial factor in the phenomenon of Compton scattering. In this process, a photon collides with a particle, like a proton, leading to a change in its wavelength. This change is a direct result of the transfer of energy and momentum between the photon and the particle it interacts with.

A photon's initial wavelength before scattering is denoted as \( \lambda \), and after the scattering event, the change in wavelength is represented by \( \Delta \lambda \). The incident wavelength can be crucial in determining properties such as energy and frequency of a photon. In our exercise, understanding how much the wavelength changes during scattering helps us find the initial wavelength. Knowing that the wavelength increases by 10% aids in solving for the original wavelength using the Compton wavelength change formula.

Overall, the wavelength is an essential parameter that connects directly with the energy level of the photon and is a telling factor in interactions involving photons.

Scattering Angle

The scattering angle, denoted as \( \phi \), plays a significant role in Compton scattering. This angle determines the direction in which the photon moves after interacting with a particle. It is a pivotal parameter in calculating the change in the photon's properties, especially the change in wavelength.

When a photon is scattered at an angle \( \phi \), the greater the angle (up to 180 degrees), the more significant the change in wavelength. In our exercise, the scattering angle is specified to be \( 180^\circ \), implying a full reversal of direction in the scattered photon. This specific angle leads to the maximum possible wavelength change in Compton scattering because \( \cos 180^\circ = -1 \). As such, the equation for wavelength change, \( \Delta \lambda = \frac{h}{{m_pc}}(1 - \cos \phi) \), shows a doubling of Planck's constant over the particle's rest mass and light speed product, resulting in maximum wavelength elongation.

Understanding the concept of scattering angles is crucial in physics problems because it influences the calculation and outcome of photon interactions, directly affecting the energy and direction of scattered photons.

Planck's Constant

Planck's constant, denoted by \( h \), is a fundamental constant in physics with the value \( 6.626 \times 10^{-34} \text{J} \cdot \text{s} \). It is a cornerstone of quantum mechanics, relating to the quantization of various physical properties. It appears prominently in Planck's law, Einstein's photoelectric equation, and, relevant to our topic, the Compton scattering formula.

In Compton scattering, Planck's constant helps determine the amount of wavelength change a photon experiences. The equation \( \Delta \lambda = \frac{h}{{m_pc}}(1 - \cos \phi) \) encapsulates the relationship between the scattering angle, the mass of the scattering particle, and the speed of light. Planck's constant is pivotal, demonstrating how tiny units of energy, or quanta, lead to measurable changes in a particle's wavelength when scattering occurs.

The presence of Planck's constant in these calculations emphasizes the deep connections between wave and particle characteristics, forming the basis of quantum mechanical principles that govern behavior at microscopic scales.