# Compton Scattering and Recoiling Electron Momentum

• CoffeeCrow
In summary, an x-ray which is scattered from a valence electron at an angle of 180 degrees has a wavelength of 50.14 pm which is different from the wavelength of the x-ray before the collision.
CoffeeCrow

## Homework Statement

X-rays with wavelength 55pm are scattered from a graphite target. Consider an x-ray which is scattered from a valence electron at an angle of 180 degrees (back towards the x-ray source).

a. Is the wavelength of the X-ray greater or less than before the collision?
b. What is the momentum of the recoiling electrons?

## Homework Equations

##\Delta\lambda=\frac {h} {mc}(1-cos(\theta))##
##p=\frac {h} {\lambda}##

## The Attempt at a Solution

Part a. seems obvious. The X-ray collides with an electron, transfers some kinetic energy to it, losing some energy in the process. Less energy, longer wavelength, right?

Part b however, is giving me a few issues.
I start by finding the Compton shift for 180 degrees, which gives $\Delta\lambda=4.85pm$.
This means that the scattered X-ray should have a wavelength of 55+4.85=59.85pm, correct?
This is where things get a bit hairy. The solution for this problem however, states that the recoiling X-ray has a wavelength of 50.14 pm. This doesn't seem to make sense, as the energy of the X-ray has increased, despite transferring energy to the electron (these solutions have had issues in the past). Anyway,they then simply use $p=\frac{h}{\lambda}$ to calculate the momentum of the recoiling electron, yielding (with their wavelength), 1.32E-23 Ns.
I however have tried the following:
##p_{i}=p_{f}##
##\frac{h}{\lambda_{i}}=p_{electron}-\frac{h}{\lambda_{f}}##
##p_{electron}=h(\frac{1}{\lambda_{i}}+\frac{1}{\lambda_{f}})##
##p_{electron}=2.31E-23Ns## (using my values for wavelength, 55pm and 59.85pm)
OR:
##p_{electron}=2.53E-23Ns##(using their values)
Which of course, doesn't agree with the solutions.
So, I was wondering, am I missing something fundamental and conceptual in regards to energy and wavelength shift and, is my momentum approach valid?

CoffeeCrow said:
Part a. seems obvious. The X-ray collides with an electron, transfers some kinetic energy to it, losing some energy in the process. Less energy, longer wavelength, right?
Right.
CoffeeCrow said:
This means that the scattered X-ray should have a wavelength of 55+4.85=59.85pm, correct?
I didn't check the number but the approach is right.
50.14 pm is clearly wrong.

CoffeeCrow
Alright, thanks, I was just worrying I'd managed to miss something fundamental!

## 1. What is the Compton Shift phenomenon?

The Compton Shift is a phenomenon in which the wavelength of a photon increases after it collides with an electron. This is due to the transfer of momentum from the electron to the photon.

## 2. How does the Compton Shift relate to the momentum of the particles involved?

The Compton Shift is directly related to the momentum of the particles involved. The change in wavelength of the photon is equal to the change in momentum of the electron, as described by the Compton formula: Δλ = h/mc (1-cosθ), where h is Planck's constant, m is the mass of the electron, c is the speed of light, and θ is the angle of deflection of the photon.

## 3. What is the significance of the Compton Shift in the field of physics?

The Compton Shift is significant because it provides evidence for the particle nature of light and the wave-particle duality of matter. It also has practical applications in fields such as X-ray imaging and spectroscopy.

## 4. How does the Compton Shift affect the energy of the photon?

The Compton Shift results in a decrease in the energy of the photon, as the increase in wavelength corresponds to a decrease in frequency according to the equation E = hc/λ. This energy loss is equal to the energy gained by the electron during the collision.

## 5. Can the Compton Shift be observed in everyday life?

The Compton Shift can be observed in everyday life through the phenomenon of X-ray diffraction. X-rays are emitted from a source and scattered by electrons in a crystal, resulting in a shift in their wavelength. This shift can be measured and used to determine the structure of the crystal.

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