Compton Scattering Homework: Wavelength & Angle Calculation

In summary, the conversation discusses a Compton scattering experiment where an x-ray photon with an incident wavelength is scattered at an angle of 17.40 degrees from a free electron at rest. The electron recoils with a speed of 2180 km/s. The problem asks to calculate the wavelength of the incident photon and the angle through which the electron scatters. Using the equations for energy and kinetic energy, the final solution for the wavelength is 1nm.
  • #1
romsoy
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Homework Statement


In a Compton scattering experiment, an x-ray photon scatters through an angle of 17.40 from a free electron that is initially at rest. The electron recoils with a speed of 2180 km/s. Calculate (a) the wavelength of the incident photon and (b) the angle through which the electron scatters.

Homework Equations


$$\Delta \lambda = \lambda^{‘} − \lambda_{0} = \frac{h}{mc} (1−cos\theta)$$
$$K = \frac{1}{2}mv^{2}$$
$$E = hc(\frac{1}{\lambda^{‘}} − \frac{1}{\lambda_{0}})$$

The Attempt at a Solution


For (a), I set the kinetic energy gained by the electron equal to the energy lost by the scattered photon - so I basically equated my second and third equations above. Then I managed to get it into a quadratic form where I solve for $\lambda _{0}$, which I got by eliminating $$\lambda'$$ with $$\lambda' = \Delta \lambda + \lambda_{0}$$. I ended up getting $$\frac{−A\Delta \lambda \pm \sqrt{(A\Delta \lambda)^2 − 4A\Delta \lambda}}{2A}$$ where $$A = \frac{mv^{2}}{2hc}$$
But turns out the term under the square root is negative :/. Can anyone help me out please?
 
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  • #2
Hello and welcome to PF!
romsoy said:
$$E = hc(\frac{1}{\lambda^{‘}} − \frac{1}{\lambda_{0}})$$
Does E represent a positive number? Does the right hand side of the equation yield a positive number?
 
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  • #3
TSny said:
Hello and welcome to PF!
Does E represent a positive number? Does the right hand side of the equation yield a positive number?
Ah...E would be the energy lost by the photon so it'd be negative... so the A term will be negative. Turns out that results nicely in 1nm, thanks!
 
  • #4
Yes. Good work.
 

FAQ: Compton Scattering Homework: Wavelength & Angle Calculation

1. What is Compton scattering and how does it relate to wavelength and angle calculations?

Compton scattering is a phenomenon that occurs when a photon (light particle) collides with a free electron. The photon transfers some of its energy to the electron, causing it to recoil and change direction. This results in a change in the wavelength and angle of the scattered photon.

2. How is the wavelength of the scattered photon calculated in Compton scattering?

The wavelength of the scattered photon can be calculated using the Compton scattering formula: λ' = λ + h/mc (1 - cosθ), where λ is the initial wavelength, λ' is the final wavelength, h is Planck's constant, m is the mass of the electron, c is the speed of light, and θ is the angle of scattering.

3. What is the significance of the angle of scattering in Compton scattering?

The angle of scattering is important because it determines the amount of energy transferred between the photon and the electron, and therefore affects the wavelength of the scattered photon. It also provides information about the direction in which the photon was scattered.

4. Can Compton scattering be used to determine the properties of a material?

Yes, Compton scattering can be used to analyze the atomic structure of materials, as the change in wavelength and angle can provide information about the electron density and atomic number of the material.

5. Are there any factors that can affect the accuracy of Compton scattering calculations?

Yes, there are several factors that can affect the accuracy of Compton scattering calculations, such as the energy of the incident photon, the angle of scattering, and the density and composition of the material being analyzed. Additionally, factors like experimental error and uncertainties in measurement can also impact the accuracy of the calculations.

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