Compton effect and momentum of recoil electrons
Problem Statement
Compton effect and momentum of recoil electrons
Problem Statement
When an X-ray photon with wavelength λ collides with a stationary free electron of mass m, the X-ray photon is scattered at an angle θ relative to its original direction of propagation, and its wavelength changes to λ′. This phenomenon is called the Compton effect.
The wavelength λ′ of the scattered X-ray photon satisfies the following relationship, using Planck's constant h and the speed of light in a vacuum c:
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 λ′, the wavelength of the X-ray photon before the collision λ, and the scattering angle θ, we can directly determine the magnitude of the momentum of the ejected recoil electron pe.
Now, when an X-ray photon with wavelength λ collides with a stationary electron, the wavelength of the X-ray photon scattered in the direction of scattering angle θ=90° is 100101 times the original wavelength. In this collision, express the magnitude of the momentum of the ejected recoil electron pe using the magnitude of the momentum of the X-ray photon before the collision p.
Constraints
- m=9.1×10−31 kg
- c=3.0×108 m/s
- h=6.6×10−34 J⋅s
Input Format
When the dimensionless ratio ppe of the momentum of the recoiling electron pe and the momentum of the X-ray photon before collision p is found, this value can be expressed as 101N. Enter the value of the natural number N inside the square root.
Solution
Explanation
Vector composition of momentum conservation law
Let p be the momentum vector of the X-ray photon before the collision, p′ be the momentum vector of the X-ray photon after scattering, and pe be the momentum vector of the recoil electron. Since the electron was at rest before the collision, the law of conservation of momentum before and after the collision can be expressed as a vector equation as follows.
p=p′+peLet's organize this in terms of the momentum of recoil electrons.
pe=p−p′In order to obtain the relationship between both sizes and the scattering angle θ, we calculate the self-inner product (square) of both sides.
∣pe∣2=(p−p′)⋅(p−p′) pe2=p2+(p′)2−2pp′cosθAccording to the problem condition, the scattering angle is θ=90∘, so cos90∘=0, and the above momentum relation can be reduced to the form of the simple Pythagorean theorem as follows.
pe2=p2+(p′)2Relationship between photon momentum and wavelength
Since the magnitude of the photon's momentum is inversely proportional to its wavelength, the magnitudes of the photon's momentum before and after the collision, p and p′, are as follows.
p=λh,p′=λ′hAccording to the conditions of the problem statement, the wavelength after scattering is given as λ′=100101λ, so the momentum p′ can be expressed using p as follows.
p′=100101λh=101100λh=101100pCalculating the momentum of recoil electrons
Substitute the obtained relational expression for p′ into the equation for the law of conservation of momentum from earlier.
pe2=p2+(101100p)2 pe2=p2(1+101210000) pe2=p2(10121012+10000)Here, 1012=(100+1)2=10000+200+1=10201. Substitute this into the numerator and organize it.
pe2=p2(101210201+10000)=p2(101220201)By taking the square root of both sides, we can find the momentum pe of the recoil electron.
pe=p10120201Therefore, the required momentum ratio ppe is as follows.
ppe=10120201Determining the number of inputs
Comparing with the expression 101N defined in the input format specification, the natural number N in the radical is determined to be 20201.
The natural number you should enter is 20201.