If a 6.0keV photon scatters from a fee proton at rest, what is the change in the photon's wavelength if the photon recoils at 90 degrees?
To find the change in the photon's wavelength after scattering from a free proton at rest, we can use the conservation of energy and momentum. The formula for Compton scattering is given by:
λ' - λ = (h / m_e c) * (1 - cosθ)
Where:
- λ' is the wavelength of the scattered photon,
- λ is the initial wavelength of the photon,
- h is Planck's constant (6.626 x 10^-34 J⋅s),
- m_e is the mass of an electron (9.10938356 x 10^-31 kg),
- c is the speed of light (2.998 x 10^8 m/s),
- θ is the scattering angle.
Given that the photon has an initial wavelength of λ = hc / E, where E is its energy, and the energy E = 6.0 keV, we can calculate the initial wavelength:
λ = hc / E = (6.626 x 10^-34 J⋅s * 2.998 x 10^8 m/s) / (6.0 x 10^3 eV * 1.602 x 10^-19 J/eV)
λ ≈ 1.2398 x 10^-6 m
Now, let's substitute the values into the Compton scattering formula. Since the photon recoils at 90 degrees (θ = 90°), the cosine term becomes 0:
λ' - λ = (h / m_e c) * (1 - cos90°)
λ' - λ = (h / m_e c) * (1 - 0)
λ' - λ = (h / m_e c)
Substituting the known values:
λ' - λ = (6.626 x 10^-34 J⋅s / 9.10938356 x 10^-31 kg * 2.998 x 10^8 m/s)
Calculating the change in wavelength:
λ' - λ ≈ 0.229 x 10^-12 m
Therefore, the change in the photon's wavelength if it recoils at 90 degrees is approximately 0.229 x 10^-12 m.
To determine the change in the photon's wavelength when it scatters from a free proton at rest, we can make use of the Compton scattering formula:
Δλ = λ' - λ = (h / (m_e * c)) * (1 - cos(θ)),
where:
- Δλ is the change in wavelength of the photon,
- λ' is the final wavelength of the scattered photon,
- λ is the initial wavelength of the photon,
- h is the Planck's constant (6.62607015 × 10^-34 J·s),
- m_e is the electron mass (9.10938356 × 10^-31 kg),
- c is the speed of light (299,792,458 m/s),
- θ is the scattering angle.
In this case, the photon scatters at 90 degrees (θ = 90°). Now we can calculate:
Δλ = (h / (m_e * c)) * (1 - cos(θ))
= (6.62607015 × 10^-34 J·s / (9.10938356 × 10^-31 kg * 299,792,458 m/s)) * (1 - cos(90°))
= (6.62607015 × 10^-34 J·s / (9.10938356 × 10^-31 kg * 299,792,458 m/s)) * (1 - 0)
= (6.62607015 × 10^-34 J·s / (9.10938356 × 10^-31 kg * 299,792,458 m/s)) * 1
= 2.42659184 × 10^-12 m (approximately)
Therefore, the change in the photon's wavelength when it scatters from a free proton at rest is approximately 2.42659184 × 10^-12 meters.