After a .800-nm x-ray photon scatters from a free electron, the electron recoils with a speed equal to 1.20 x 10^6 m/s. (a) What was the Compton Shift in the photon's wavelength? (b) Through what angle was the photon scattered?
I got an answer for (a), but when i use it to find (b) the solution does not exist. I clearly did something wrong, and I can't figure it out.
a) follows from conservation of energy:
Photon energy + m = photon energy after collision plus gamma m
(in c = 1 units)
In b) you have to take into account that both the photon after the collision and the electron after the collison are not moving in the same direction the incident photon was moving.
The simplest way to solve this problem is by writing conservation of energy and momentum as a single four-momentum equation:
pf + pe = qf + qe (1)
where
pf = four-mometum of photon before collision
pe = four-momentum of electron before collision
qf = four-mometum of photon after collision
qe = four-momentum of electron after collision
We are not interested in qe, so we want to eliminate it from the equation. What we do is we write (1) as:
qe = pf + pe - qf
We then square both sides:
qe^2 = (pf + pe - qf)^2 =
pf^2 + pe^2 + qf^2 + 2 pf dot pe +
- 2 pf dot qf -2 pe dot qf
Next substitute the energy momentum relation p^2 = m^2, so we have:
qe^2 = m^2
pe^2 = m^2
pf^2 = 0
qf^2 = 0
and the equation simplifies to:
2 pf dot pe - 2 pf dot qf -2 pe dot qf = 0
Pf dot qf contains the angle you want to know. Pe = (m,0,0) because the electron before the collision was at rest, so the Lorentz inner product of the two other terms are just the electron mass times the photon energies.
How much force will be needed to give a 10kg body resting on the ground an upward acceleration of 5m/s if acceleration due to gravity at the place is 10.02m/s?
To determine the Compton shift in the photon's wavelength, we can use the Compton scattering formula:
Δλ = λ' - λ = λ - λ₀,
where:
Δλ is the change in wavelength (Compton shift),
λ' is the final wavelength (after scattering),
λ is the initial wavelength,
λ₀ is the wavelength of the scattered photon.
For part (a), since the initial and final wavelengths are given, we can calculate the Compton shift:
Δλ = λ' - λ = 0.800 nm - λ = λ - λ₀.
However, the problem does not provide any information about the wavelength of the scattered photon (λ₀). Therefore, we cannot determine the Compton shift accurately. It seems there is missing information in the problem statement.
For part (b), to find the angle at which the photon is scattered (θ), we can use the Compton scattering formula again:
λ' - λ = λ - λ₀ = (h / mₑc) * (1 - cos θ),
where:
h is Planck's constant (6.626 x 10^-34 J s),
mₑ is the mass of the electron (9.109 x 10^-31 kg),
c is the speed of light (3.00 x 10^8 m/s).
However, without the Compton shift (Δλ), we cannot compute the angle (θ). Hence, we are unable to determine the angle at which the photon is scattered.
To resolve the issue and find a solution, it is essential to have all the required information, such as the initial and final wavelengths or the Compton shift. Please verify the problem statement or provide any additional information available.
To find the Compton Shift in the photon's wavelength and the angle through which the photon is scattered, we can use the laws of energy and momentum conservation in Compton scattering.
(a) Let's start by finding the Compton Shift in the photon's wavelength. The Compton Shift, denoted by Δλ, is defined as the difference between the initial and final wavelengths of the photon. It can be calculated using the equation:
Δλ = λ' - λ
where λ' is the wavelength of the scattered photon and λ is the wavelength of the incident photon.
To determine λ' and λ, we can use the equation relating the change in wavelength to the momentum of the photon:
Δλ = h / (m_e * c) * (1 - cosθ)
where h is the Planck's constant (approximately 6.626 x 10^(-34) J·s), m_e is the mass of the electron (approximately 9.11 x 10^(-31) kg), c is the speed of light (approximately 3.00 x 10^8 m/s), and θ is the scattering angle.
Given that λ = 800 nm (8.00 x 10^(-7) m), we need to find λ'. Rearranging the equation, we get:
λ' = λ + Δλ
Substituting the values into the equation, we have:
λ' = (8.00 x 10^(-7) m) + Δλ
Now, we can substitute this expression for λ' into the equation relating the change in wavelength to the scattered angle:
1.20 x 10^6 m/s = h / (m_e * c) * (1 - cosθ) * (8.00 x 10^(-7) m + Δλ) - λ
We can rearrange this equation to solve for Δλ:
Δλ = (1.20 x 10^6 m/s - h / (m_e * c) * (1 - cosθ) * λ) / (h / (m_e * c) * (1 - cosθ) + 1)
Now, substitute the known values into the equation (h, m_e, c, λ) and solve for Δλ numerically using a calculator or computational software.
(b) After finding the value of Δλ, you can then use it to find the scattering angle θ. The formula to calculate the scattering angle is:
cosθ = 1 - (Δλ * (m_e * c) / (h * λ))
Simply rearrange this equation to solve for θ. Plug in the values of Δλ, m_e, c, h, and λ, and solve for θ numerically.
If, after following these steps, you are still not getting a solution for (b), it is possible that there was an error in your calculations or you made a mistake somewhere along the way. Double-check your calculations, your input values, and any algebraic steps you performed to ensure accuracy.