91Ó°ÊÓ

(a) Derive an expression for the total shift in photon wave- length 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 wave-length 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 \(\theta=0\) , for which the total shifts are the same? (c) Use the result of part (a) to calculate the total wave- length 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

01

Understanding Compton Scattering

Compton scattering describes the phenomenon where a photon scatters off an electron, resulting in a change in the photon's wavelength. The wavelength shift \( \Delta \lambda \) after scattering can be determined by the formula: \[ \Delta \lambda = \frac{h}{mc} (1 - \cos \theta), \]where \( h \) is Planck's constant, \( m \) is the electron rest mass, \( c \) is the speed of light, and \( \theta \) is the scattering angle.

02

Deriving Total Wavelength Shift For Two Scatterings

For two successive scatterings, the total shift in wavelength is the sum of the shifts from each scattering. The expression for the total shift \( \Delta \lambda_{total} \) is:\[ \Delta \lambda_{total} = \frac{h}{mc} (1 - \cos \theta_1) + \frac{h}{mc} (1 - \cos \theta_2), \]which simplifies to:\[ \Delta \lambda_{total} = \frac{h}{mc} [(1 - \cos \theta_1) + (1 - \cos \theta_2)]. \]

03

Comparing Two Successive Scatterings at \( \theta/2 \) with One at \( \theta \)

For two scatterings each at an angle \( \theta/2 \), the total shift becomes:\[ \Delta \lambda_{two} = 2 \times \frac{h}{mc} (1 - \cos(\theta/2)). \]The shift for a single scattering at \( \theta \) is:\[ \Delta \lambda_{one} = \frac{h}{mc} (1 - \cos \theta). \]Generally, \( \Delta \lambda_{two} eq \Delta \lambda_{one} \) except when \( \cos(\theta/2) = \pm \cos(\theta) \). This condition holds for \( \theta = 0 \) and \( \theta = \pi \), but not for other angles.

04

Calculating Total Shift for Two Scatterings of \(30^\circ\) Each

Using the derived formula for two successive scatterings, substitute \( \theta_1 = 30^\circ \) and \( \theta_2 = 30^\circ \):\[ \Delta \lambda_{total} = \frac{h}{mc} [2 \times (1 - \cos 30^\circ)]. \]Since \( \cos 30^\circ = \sqrt{3}/2 \), calculate:\[ \Delta \lambda_{total} = \frac{h}{mc} [2 \times (1 - \sqrt{3}/2)] = \frac{h}{mc} [2 - \sqrt{3}]. \]

05

Calculating Single Scattering at \(60^\circ\)

For a single 60-degree scattering, calculate \( \Delta \lambda \) using the standard Compton formula:\[ \Delta \lambda = \frac{h}{mc} (1 - \cos 60^\circ). \]Given that \( \cos 60^\circ = 0.5 \),\[ \Delta \lambda = \frac{h}{mc} (1 - 0.5) = \frac{h}{2mc}. \]

06

Comparing the Results

Compare the wavelength shifts from two successive 30-degree scatterings and a single 60-degree scattering: - Two 30-degree scatterings: \( \Delta \lambda_{total} = \frac{h}{mc} [2 - \sqrt{3}] \).- One 60-degree scattering: \( \Delta \lambda = \frac{h}{2mc} \).These two results are not equal, showing that the shifts are different.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Photon Wavelength Shift

When a photon collides with an electron, its wavelength changes, a process known as Compton scattering. The change in wavelength, denoted as \( \Delta \lambda \), is given by the formula: \[ \Delta \lambda = \frac{h}{mc} (1 - \cos \theta) \]where \( h \) is Planck's constant, \( m \) is the electron's rest mass, \( c \) is the speed of light, and \( \theta \) is the scattering angle. This equation shows how the scattering angle affects the wavelength shift of a photon.
Understanding this concept is crucial because the wavelength shift reflects how energy and momentum are transferred between the photon and the electron.

• Larger scattering angles lead to more significant wavelength changes.
• The process conserves energy and momentum, which are fundamental principles in physics.

Successive Scatterings

Successive Compton scatterings occur when a photon scatters off electrons multiple times in succession. Each scattering results in an additional wavelength shift. The total shift after two such scatterings can be determined by adding the individual shifts. Following the formula for wavelength change from a single scattering, we derive the total shift for two scatterings:\[ \Delta \lambda_{total} = \frac{h}{mc} [(1 - \cos \theta_1) + (1 - \cos \theta_2)] \]This formula demonstrates that the total wavelength shift is additive for successive events.
Understanding successive scatterings is essential in studies of photon interactions in complex environments, such as astrophysical phenomena or medical imaging technologies.
Key points to remember:

• The sequence of scatterings impacts the final wavelength but the principle of linear addition remains.
• Each scattering event follows the same fundamental principles as single scattering but with compound effects.

Scattering Angle

The scattering angle \( \theta \) is vital when analyzing Compton scattering because it significantly impacts the resultant wavelength shift of the photon.
For single scattering, the formula is:\[ \Delta \lambda = \frac{h}{mc} (1 - \cos \theta) \]For two scatterings, if each is at an angle \( \theta/2 \), we compare it to a single scattering at \( \theta \):

• The wavelength shift from two scatterings at angles \( \theta/2 \) is \( 2 \times \frac{h}{mc} (1 - \cos(\theta/2)) \).
• Generally, \( \Delta \lambda_{two} eq \Delta \lambda_{one} \) except in specific cases like \( \theta = 0 \) or \( \theta = \pi \).

This comparison highlights how different scattering scenarios impact energy transfer between photons and electrons, proving crucial for applied physics, such as quantum mechanics and materials science.

Planck's Constant

Planck's constant (\( h \)) is a fundamental constant in physics that plays a crucial role in quantifying the energy and momentum of particles like photons in the Compton scattering formula:\[ \Delta \lambda = \frac{h}{mc} (1 - \cos \theta) \]In this context, it links the photon's energy with its wavelength shift during scattering.
Planck's constant is valued at approximately \( 6.626 \times 10^{-34} \text{ m}^2\text{kg/s} \), a pivotal figure that supports the concept of quantized electromagnetic radiation.
Importance of Planck's Constant:

• Enables calculations involving the quantum nature of light.
• Integral in understanding energy transfer in quantum mechanics.
• Essential for bridging classical and quantum physics.

Exploring Planck's constant allows for deeper insight into fundamental physics, assisting in both theoretical study and practical applications.