A Photon of energy 240 KeV is scattered by a free electron. If the recoil electron has a Kinetic energy of 190 KeV, what is the wavelength of the scattered photon?
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200kev
To find the wavelength of the scattered photon, we can use the energy-momentum conservation equation. According to this equation, the initial energy of the photon must be equal to the final sum of the energies of the scattered photon and the recoil electron.
Given:
Initial photon energy (E_i) = 240 KeV
Recoil electron kinetic energy (KE_e) = 190 KeV
To convert KeV to Joules, we can use the conversion factor:
1 KeV = 1.602 x 10^-16 J
Converting the energies to Joules:
E_i = 240 KeV * (1.602 x 10^-16 J/KeV) = 3.85 x 10^-14 J
KE_e = 190 KeV * (1.602 x 10^-16 J/KeV) = 3.05 x 10^-14 J
Using the conservation equation:
E_i = E_f
E_f = E_photon + E_recoil_electron
Substituting the values:
3.85 x 10^-14 J = E_photon + 3.05 x 10^-14 J
Solving for E_photon:
E_photon = 3.85 x 10^-14 J - 3.05 x 10^-14 J = 8 x 10^-15 J
To find the wavelength of the scattered photon, we can use the equation:
E_photon = hc/λ
Where:
h = Planck's constant = 6.626 x 10^-34 J·s
c = speed of light = 3 x 10^8 m/s
λ = wavelength of the scattered photon
Substituting the values:
8 x 10^-15 J = (6.626 x 10^-34 J·s)(3 x 10^8 m/s) / λ
Solving for λ:
λ = ((6.626 x 10^-34 J·s)(3 x 10^8 m/s)) / (8 x 10^-15 J)
λ ≈ 24.8 nm
Therefore, the wavelength of the scattered photon is approximately 24.8 nm.
To find the wavelength of the scattered photon, we can use the Compton scattering equation:
λ' - λ = (h / mc) * (1 - cosθ)
where:
- λ' is the wavelength of the scattered photon
- λ is the wavelength of the incident photon
- h is Planck's constant (6.626 x 10^-34 Js)
- m is the mass of the electron (9.109 x 10^-31 kg)
- c is the speed of light (3 x 10^8 m/s)
- θ is the scattering angle between the incident and scattered photons
First, we need to calculate the change in wavelength (Δλ = λ' - λ). We can do this by rearranging the equation:
Δλ = (h / mc) * (1 - cosθ)
Next, we need to calculate the scattering angle (θ) using the kinetic energy of the recoil electron. The kinetic energy of the recoil electron can be related to the scattering angle by the equation:
K.E. = (2 * h * c / λ) * (1 - cosθ)
Rearranging this equation, we can solve for cosθ:
cosθ = 1 - (K.E. * λ) / (2 * h * c)
Given the recoil electron's kinetic energy (190 KeV) and the energy of the incident photon (240 KeV), we can calculate the change in wavelength and the scattering angle:
K.E. = 190 KeV = 190 x 10^3 eV
λ = c / f = (3 x 10^8 m/s) / (240 x 10^3 eV)
cosθ = 1 - (K.E. * λ) / (2 * h * c)
Finally, we can use these values to find the change in wavelength (∆λ) and the wavelength of the scattered photon (λ'):
∆λ = (h / mc) * (1 - cosθ)
λ' = λ + ∆λ
Plug in the values into the equations and calculate λ'.