Question

An incident beam of photons is scattered through \(100.0^{\circ}\) the wavelength of the scattered photons is \(124.65 \mathrm{pm}\). What is the wavelength of the incident photons?

Short answer

The incident photon wavelength is 121.803 pm.

Step-by-step solution

Compton Wavelength Shift Equation

The Compton effect describes how the wavelength of light changes when it scatters off a particle. The equation for the wavelength shift due to this scattering is given by the Compton wavelength shift formula: \[\Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos \theta)\]where \(\lambda'\) is the wavelength of the scattered photons, \(\lambda\) is the wavelength of the incident photons, \(h\) is Planck's constant \((6.626 \times 10^{-34} \, \mathrm{Js})\), \(m_e\) is the electron rest mass \((9.109 \times 10^{-31} \, \mathrm{kg})\), \(c\) is the speed of light \((3.00 \times 10^8 \, \mathrm{m/s})\), and \(\theta\) is the scattering angle.

Calculate Compton Shift

Using the provided scattering angle \( \theta = 100.0^{\circ} \), calculate \(1 - \cos \theta\):\[1 - \cos 100.0^{\circ} = 1 - (-0.1736) = 1.1736\]

Insert Known Values Into the Formula

Now substitute all the known values into the wavelength shift equation:\[\Delta \lambda = \frac{6.626 \times 10^{-34} \, \mathrm{Js}}{9.109 \times 10^{-31} \, \mathrm{kg} \cdot 3.00 \times 10^8 \, \mathrm{m/s}} \times 1.1736\]

Calculate the Wavelength Shift

Calculate the change in wavelength \( \Delta \lambda \):\[\Delta \lambda \approx \frac{2.426 \times 10^{-12} \, \mathrm{m}} \times 1.1736 = 2.847 \times 10^{-12} \, \mathrm{m} = 2.847 \, \mathrm{pm}\]Therefore, the shift in wavelength is approximately \(2.847\, \mathrm{pm}\).

Calculate Incident Wavelength

Use the equation \( \lambda = \lambda' - \Delta \lambda \) to find the incident wavelength:\[\lambda = 124.65 \, \mathrm{pm} - 2.847 \, \mathrm{pm} = 121.803 \, \mathrm{pm}\]

Final Answer

The wavelength of the incident photons is \(121.803 \, \mathrm{pm}\).

Key concepts

Photon Wavelength

The concept of photon wavelength is essential in understanding how light behaves. A photon is a particle representing a quantum of light or electromagnetic radiation. The wavelength of a photon is inversely related to its energy; higher energy photons have shorter wavelengths. Photons travel at the speed of light, and their energy is given by the equation: \[ E = \frac{hc}{\lambda} \]where:

• \(E\) is the energy of the photon,
• \(h\) is Planck's constant,
• \(c\) is the speed of light, and
• \(\lambda\) is the wavelength of the photon.

In the context of the Compton effect, when a photon collides with an electron, its wavelength changes. This shift is a fundamental aspect of understanding photon interactions and energy exchanges.

Scattering Angle

The scattering angle is the angle through which a photon is deflected from its initial path after colliding with a particle, such as an electron. In the analysis of the Compton effect, this deflection leads to a shift in the photon's wavelength. The formula for the Compton wavelength shift includes the scattering angle, \(\theta\). Here’s how the cosine of the scattering angle \(\theta\) influences this shift:

• When \(\theta = 0^{\circ}\), there is no deflection and no wavelength shift occurs.
• As \(\theta\) increases, especially beyond \(90^{\circ}\), the shift becomes more significant.
• At \(\theta = 180^{\circ}\), the shift is at its maximum.

In our example, a scattering angle of \(100.0^{\circ}\) shows that the photon has been significantly deflected, resulting in a noticeable wavelength shift, which is crucial for the calculations in the Compton effect.

Compton Wavelength Shift Equation

The Compton wavelength shift equation is a cornerstone for understanding how photon wavelengths change during scattering. This effect is observable when a photon interacts with a stationary particle like an electron. The equation used to calculate this shift is:\[\Delta \lambda = \frac{h}{m_e c} (1 - \cos \theta)\]where:

• \(\Delta \lambda\) is the change in wavelength,
• \(h\) is Planck's constant,
• \(m_e\) is the electron rest mass,
• \(c\) is the speed of light, and
• \(\theta\) is the scattering angle.

By using this formula, you can predict how much the wavelength of a photon will shift based on the scattering angle. In our exercise, by substituting the values into this formula, we found a wavelength shift of approximately \(2.847\, \mathrm{pm}\). This calculation is essential for determining the information about the incident beam of photons in experiments involving the Compton effect.

Planck's Constant

Planck's constant, denoted as \(h\), is a fundamental constant in physics that plays a vital role in the theory of quantum mechanics. It has a value of approximately \(6.626 \times 10^{-34} \, \mathrm{Js}\). This constant is crucial in the formulas that describe the behavior and properties of photons, especially their energy and wavelength.Here’s how Planck's constant is used:

• In the energy equation: \(E = \frac{hc}{\lambda}\), to link the energy and wavelength of a photon.
• In the Compton wavelength shift equation: \(\Delta \lambda = \frac{h}{m_e c} (1 - \cos \theta)\), where it helps determine the change in wavelength as photons scatter.

Understanding the significance of Planck's constant helps in grasping how energy and wavelength are interconnected in quantum physics. In our specific context, it helps us calculate the precise change in photon wavelength as predicted by the Compton effect.