A gamma ray with 700KeV energy is Compton-Scattered from an electron. Find the energy of the photon scattered at 110 degrees, the energy of the scattered electron, and the recoil angle of the electron.
To solve this problem, we can use the Compton scattering formula, which relates the initial and final photon energies to the scattering angle and the electron mass. The Compton formula is:
λ' - λ = (h / (m_e c)) * (1 - cos(theta))
Where:
- λ' is the wavelength of the scattered photon
- λ is the wavelength of the incident photon
- h is the Planck's constant (6.62607015 × 10^-34 J·s)
- m_e is the mass of the electron (9.10938356 × 10^-31 kg)
- c is the speed of light (299,792,458 m/s)
- theta is the scattering angle
First, we need to convert the energy of the gamma ray to its corresponding wavelength using the equation:
E = h * c / λ
We know that E = 700 KeV = 700 × 10^3 × 1.602176634 × 10^-19 J. Plugging this into the equation above, we can solve for λ.
λ = h * c / E
Next, we substitute the given values into the Compton scattering formula to find the wavelength shift, Δλ = λ' - λ:
Δλ = (h / (m_e * c)) * (1 - cos(theta))
Since wavelength and energy are inversely related (λ = c / ν), we can find the energy shift, ΔE, using the relationship:
ΔE = E - E'
where E is the incident photon energy, and E' is the final photon energy.
Now, we can use the energy shift to find the energy of the scattered photon:
E' = E - ΔE
Finally, we can determine the energy and recoil angle of the scattered electron using the conservation of energy and momentum equations:
E_e = E - E'
and
sin(theta_e) = p' / (m_e * c)
where E_e is the energy of the scattered electron, p' is its momentum, and theta_e is the recoil angle.
Note that we are assuming non-relativistic scattering, as the problem does not indicate high photon energies or relativistic electron momenta. If relativistic effects are significant, additional calculations are required to account for relativistic corrections.
To find the energy of the photon scattered at 110 degrees, we can use the Compton scattering formula:
λ' - λ = (h / me c) * (1 - cosθ)
Where:
λ' = Compton wavelength of the scattered photon
λ = Compton wavelength of the initial photon
h = Planck's constant (6.626 x 10^-34 J·s)
me = mass of the electron (9.10938356 x 10^-31 kg)
c = speed of light (2.998 x 10^8 m/s)
θ = scattering angle in radians
First, let's convert the scattering angle from degrees to radians:
θ = 110 degrees * (π / 180 degrees)
θ = 1.91986 radians
Now, let's calculate the energy of the scattered photon:
E' = (hc / λ')
E = (hc / λ)
The energy of the scattered photon (E') and the initial photon (E) are related to their respective wavelengths (λ' and λ) by the following equations:
E' = (1 / λ') * hc
E = (1 / λ) * hc
Substituting the values of h and c into the equations, we have:
E' = (1 / λ') * (6.626 x 10^-34 J·s) * (2.998 x 10^8 m/s)
E = (1 / λ) * (6.626 x 10^-34 J·s) * (2.998 x 10^8 m/s)
Next, we need to calculate the Compton wavelengths of the initial and scattered photons:
λ' = λ + Δλ
Δλ = (h / me c) * (1 - cosθ)
Substituting the values of h, me, c, and θ, we can solve for Δλ:
Δλ = (6.626 x 10^-34 J·s) / [(9.10938356 x 10^-31 kg) * (2.998 x 10^8 m/s)] * (1 - cos(1.91986 radians))
Now, substitute the values of Δλ and λ into the equations for E' and E:
E' = (1 / (λ + Δλ)) * (6.626 x 10^-34 J·s) * (2.998 x 10^8 m/s)
E = (1 / λ) * (6.626 x 10^-34 J·s) * (2.998 x 10^8 m/s)
Finally, calculate the energy of the scattered electron using the energy-momentum conservation principle:
Ee' = E - E'
To find the recoil angle of the electron, we can use the relationship between the scattering angle of the photon and the recoil angle of the electron:
tan(ϑe) = (me * c * sinθ) / (E - me * c^2 * (1 - cosθ))
Where:
ϑe = recoil angle of the electron
Now, substitute the values of me, c, θ, and the calculated values of E and E' into the equation to solve for ϑe.
Please note that the above calculations involve several intermediate steps and may require a scientific calculator or software to obtain accurate results.