Compton effect and momentum of recoil electrons

Problem Statement

When an X-ray photon with wavelength $\lambda$λ collides with a stationary free electron of mass $m$m, the X-ray photon is scattered at an angle $\theta$θ relative to its original direction of propagation, and its wavelength changes to $\lambda'$λ′. This phenomenon is called the Compton effect.

The wavelength $\lambda'$λ′ of the scattered X-ray photon satisfies the following relationship, using Planck's constant $h$h and the speed of light in a vacuum $c$c:

$$\lambda' - \lambda = \frac{h}{mc}(1 - \cos\theta)$$λ′−λ=mch​(1−cosθ)

During the collision, the electron receives energy and momentum from the X-ray photon and is ejected as a recoil electron. By considering the conservation of momentum for the entire system before and after a collision, if we know the wavelength of the scattered X-ray photon $\lambda'$λ′, the wavelength of the X-ray photon before the collision $\lambda$λ, and the scattering angle $\theta$θ, we can directly determine the magnitude of the momentum of the ejected recoil electron $p_e$pe​.

Now, when an X-ray photon with wavelength $\lambda$λ collides with a stationary electron, the wavelength of the X-ray photon scattered in the direction of scattering angle $\theta = 90°$θ=90° is $\frac{10¹}{100}$100101​ times the original wavelength. In this collision, express the magnitude of the momentum of the ejected recoil electron $p_e$pe​ using the magnitude of the momentum of the X-ray photon before the collision $p$p.

Constraints

• $m = 9.1 \times 10^{-31} \ \text{kg}$m=9.1×10−31 kg
• $c = 3.0 \times 10^8 \ \text{m/s}$c=3.0×108 m/s
• $h = 6.6 \times 10^{-34} \ \text{J}\cdot\text{s}$h=6.6×10−34 J⋅s

Input Format

When the dimensionless ratio $\frac{p_e}{p}$ppe​​ of the momentum of the recoiling electron $p_e$pe​ and the momentum of the X-ray photon before collision $p$p is found, this value can be expressed as $\frac{\sqrt{N}}{10¹}$101N​​. Enter the value of the natural number $N$N inside the square root.