Problem 40 - University Physics with Modern Physics

Question

(a) Derive an expression for the total shift in photon wavelength after two successive Compton scatterings from electrons at rest. The photon is scattered by an angle \(\theta_{1}\) in the first scattering and by \(\theta_{2}\) in the second. (b) In general, is the total shift in wavelength produced by two successive scatterings of an angle \(\theta / 2\) the same as by a single scattering of \(\theta\) ? If not, are there any specific values of \(\theta\) , other than \(\theta=0^{\circ}\) , for which the total shifts are the same? (c) Use the result of part (a) to calculate the total wavelength shift produced by two successive Compton scatterings of \(30.0^{\circ}\) each. Express your answer in terms of \(h / m c .\) (d) What is the wavelength shift produced by a single Compton scattering of \(60.0^{\circ}\) . Compare to the answer in part (c).

Step-by-Step Solution

Answer

The shifts are not generally equal; only specific angles like 180° match. For 30° each, total shift is \(\frac{h}{m c}(2 - \sqrt{3})\); a 60° shift is equivalent.

Step 1: Compton Wavelength Shift Formula

The Compton scattering formula gives the change in wavelength \( \Delta \lambda \) of a photon when it scatters off a stationary electron: \[ \Delta \lambda = \frac{h}{m c} (1 - \cos \theta) \] where \( h \) is Planck's constant, \( m \) is the mass of the electron, \( c \) is the speed of light, and \( \theta \) is the scattering angle.

Step 2: Total Shift for Two Successive Compton Scatterings

For two successive scatterings, use the formula twice to find the total wavelength shift \( \Delta \lambda_{\text{total}} \): \[ \Delta \lambda_{\text{total}} = \Delta \lambda_1 + \Delta \lambda_2 = \frac{h}{m c} (1 - \cos \theta_1) + \frac{h}{m c} (1 - \cos \theta_2) \] \[ \Delta \lambda_{\text{total}} = \frac{h}{m c} ((1 - \cos \theta_1) + (1 - \cos \theta_2)) \]

Step 3: Comparison for Different Angles

Assess if \( \theta/2 \) twice yields a total shift equivalent to a single shift of \( \theta \): \[ 2 \cdot \frac{h}{m c} (1 - \cos \frac{\theta}{2}) = \frac{h}{m c} (1 - \cos \theta) \] Simplifying gives \(1 - \cos \theta = 2 (1 - \cos \frac{\theta}{2})\). This is only true for specific angles such as \( \theta = 180^\circ \) or \(0^\circ\).

Step 4: Specific Calculation for Two 30 Degree Scatterings

Use the expressions derived to find the total shift for \( \theta_1 = \theta_2 = 30^\circ \): \[ \Delta \lambda_{\text{total}} = \frac{h}{m c} (1 - \cos 30^\circ + 1 - \cos 30^\circ) \] The cosine of 30 degrees is \( \frac{\sqrt{3}}{2} \), so \( 1 - \cos 30^\circ = 1 - \frac{\sqrt{3}}{2} \). \[ \Delta \lambda_{\text{total}} = 2 \cdot \frac{h}{m c} \left(1 - \frac{\sqrt{3}}{2}\right) \]

Step 5: Calculation for Single 60 Degree Scattering

For a single 60 degree scattering, calculate \( \Delta \lambda \) using: \[ \Delta \lambda = \frac{h}{m c} (1 - \cos 60^\circ) \] Since \( \cos 60^\circ = \frac{1}{2} \), this becomes: \[ \Delta \lambda = \frac{h}{m c} \left(1 - \frac{1}{2}\right) = \frac{h}{2 m c} \]

Step 6: Compare Wavelength Shifts

From part (c), \( \Delta \lambda_{\text{total}} = \frac{h}{m c} (2 \times \frac{1}{2}) = \frac{h}{m c} \cdot \frac{1}{2} = \frac{h}{m c} (2 - \sqrt{3}) \). This matches with the calculated shift for a single 60-degree scattering.

Key Concepts

Photon Wavelength Shift

When a photon collides with an electron, its direction changes, and its wavelength also shifts. This phenomenon is known as Compton scattering. The change or shift in the wavelength, represented as \( \Delta \lambda \), can be calculated using the Compton scattering formula. This formula helps to understand how energy and momentum are transferred during the collision. In this case, the wavelength shift articulates the difference between the initial and final wavelengths of the photon after the scattering event.

The focus on the wavelength shift allows scientists and students to measure the exact impact of the scattering process on the photon's properties. Understanding this shift is crucial for various applications and further studies in quantum physics and energy dynamics.

Compton Wavelength Formula

The Compton wavelength formula is a key tool in quantifying the wavelength shift experienced by a photon during scattering. The formula is given by: \[ \Delta \lambda = \frac{h}{m c} (1 - \cos \theta) \] Here, \( h \) represents Planck's constant, \( m \) is the mass of the electron, and \( c \) is the speed of light. The variable \( \theta \) represents the scattering angle. This formula demonstrates how the shift depends on the cosine of the scattering angle, indicating that different angles cause different shifts in wavelength.

This formula provides insight into how energy levels are conserved during the scattering process. By altering the angle, one can see significant changes in the photon's wavelength. Such calculations are immensely valuable in experimental physics, where precise measurements of subatomic particles are critical.

Successive Scatterings

Successive scatterings involve a photon undergoing multiple scatter events one after another. Each scattering event shifts the photon's wavelength by a certain amount. The total shift in wavelength after multiple, or in this case, two successive scatterings can be determined by applying the Compton wavelength formula sequentially.

For two scatterings at angles \( \theta_1 \) and \( \theta_2 \), the total shift is calculated by summing the individual shifts: \[ \Delta \lambda_{\text{total}} = \frac{h}{m c} ((1 - \cos \theta_1) + (1 - \cos \theta_2)) \] By analyzing how the photon energy is altered through each interaction, students can gain insights into how energy and momentum conservation are maintained across multiple scattering events.

Scattering Angles

The scattering angle, denoted as \( \theta \), is integral to determining the extent of the wavelength shift in Compton scattering. It refers to the angle at which the photon is deflected as a result of colliding with an electron. This angle directly influences the energy transfer and, consequently, the wavelength change. The formula \( \Delta \lambda = \frac{h}{m c} (1 - \cos \theta) \) shows that the shift relies significantly on \( 1 - \cos \theta \).

When assessing successive scatterings, comparing the effects of two angles (e.g., \( \theta \) and \( \theta / 2 \)) helps to evaluate under what conditions the wavelength shift might be equal. For certain specific angles like 0° and 180°, the outcomes align due to the properties of cosine values. Understanding how scattering angles impact outcomes deepens comprehension of comparative results in analytical physics contexts.