An X-ray scattering from an electron with negligible initial kinetic energy is observed to undergo a change in wavelength by 2.561 pm. i.e. The scattered X-ray photon has a wavelength 2.561 pm larger than the incident X-ray photon.
Find the direction of propagation of the scattered electron relative to the direction of the incident X-ray, given that the incident X-ray has a wavelength of 0.3276 nm. (i.e. Find the electron scattering angle in degrees.)
(Take the Compton wavelength of the electron to be 2.424 pm. Use the Compton formula, and both components of momentum conservation. You should work algebraically as far as possible, and hence should only need the Compton wavelength of the electron in terms of physical constants. In order: find the final photon wavelength and the photon scattering angle. Store those two values to high precision. I then suggest algebraically eliminating the unknown electron momentum magnitude from the two momentum conservation equations in favor of the electron scattering angle. Solve for the electron scattering angle in terms of the photon scattering angle and the initial and final photon wavelengths. The final algebraic equation is not very difficult.)
To find the direction of propagation of the scattered electron relative to the direction of the incident X-ray, we will use the Compton formula and both components of momentum conservation.
Let's go step by step:
1. Start by converting all the given values to standard SI units. The incident X-ray wavelength is 0.3276 nm, which is equivalent to 3.276 × 10^-10 m. The observed change in wavelength is 2.561 pm, which is equivalent to 2.561 × 10^-12 m. The Compton wavelength of the electron is given as 2.424 pm, which is 2.424 × 10^-12 m.
2. Use the Compton formula to relate the change in wavelength to the scattering angle. The Compton formula is given by: Δλ = (h / m_e * c) * (1 - cosθ), where Δλ is the change in wavelength, h is the Planck constant, m_e is the mass of the electron, c is the speed of light, and θ is the scattering angle.
Rearranging the formula, we get: cosθ = 1 - (Δλ * m_e * c) / h
Substituting the values, we have: cosθ = 1 - ((2.561 × 10^-12 m) * (9.10938356 × 10^-31 kg) * (3 × 10^8 m/s)) / (6.62607015 × 10^-34 J·s)
Calculating this expression will give us the value of cosθ.
3. To solve for the electron scattering angle, we will use momentum conservation. Since the electron initially has negligible kinetic energy, its initial momentum is given by P_initial = hf / c, where f is the frequency of the incident X-ray and h is the Planck constant.
The final momentum of the electron can be given by P_final = mv, where m is the mass of the electron and v is the velocity of the scattered electron.
Using the principle of conservation of momentum, we can equate the initial momentum with the final momentum: P_initial = P_final.
Substituting the values, we have: (hf / c) = (m * v)
4. Now, let's algebraically eliminate the unknown electron momentum magnitude from the two momentum conservation equations in favor of the electron scattering angle.
Using the de Broglie wavelength equation: λ = h / (mv), we can rewrite the equation as: (hf / c) = (h / (λ * c)) * v
Simplifying this equation, we obtain: v = (hf) / (λ * c)
Substituting this value of v into the momentum conservation equation, we get: (hf) / (λ * c) = mv
Canceling 'm' from both sides and rearranging the equation, we have: λ = (h / p) = h / ((hf) / (λ * c))
Simplifying, we get: λ = c / f
We know that Δλ = 2.561 × 10^-12 m, and the initial wavelength λ = 3.276 × 10^-10 m.
Substituting these values into the equation, we can solve for the final photon wavelength.
5. Finally, we can solve for the photon scattering angle using the equation: sin(θ/2) = (λ_f - λ_i) / (2d), where θ is the scattering angle, λ_f is the final photon wavelength, λ_i is the initial photon wavelength, and d is the distance between successive crystal planes.
Using the given change in wavelength, we can calculate λ_f. Substituting the values, we get: sin(θ/2) = ((3.276 × 10^-10 m) + (2.561 × 10^-12 m)) / (2 * d)
Simplifying this equation will help us find the value of sin(θ/2). From this, we can calculate the scattering angle θ.
Keep in mind that the final algebraic equation might be quite complex, but solving it will give you the electron scattering angle in terms of the photon scattering angle and the initial and final photon wavelengths.