X rays of wavelength 0.0166 nm are directed in the positive direction of an x axis onto a target containing loosely bound electrons. For Compton scattering from one of those electrons, at an angle of 169°, what are (a) the Compton shift, (b) the corresponding change in photon energy, (c) the kinetic energy of the recoiling electron, and (d) the angle between the positive direction of the x axis and the electron's direction of motion? The electron Compton wavelength is 2.43 × 10e-12 m.
I am struggling with part d.
My answers:
a) 4.82pm
b) -16.8keV
c) 16.8keV
To solve this problem, we will need to use the principles of Compton scattering and the formulas associated with it. Compton scattering describes the scattering of X-rays or gamma rays by charged particles, such as electrons.
Let's go through each part of the problem one by one.
(a) The Compton shift refers to the change in wavelength of the incident photon after scattering. It can be calculated using the formula:
Δλ = λ' - λ = λc(1 - cosθ)
Where:
- Δλ is the Compton shift,
- λ' is the wavelength of the scattered photon,
- λ is the initial wavelength (in this case, given as 0.0166 nm),
- λc is the Compton wavelength of the electron,
- θ is the angle between the incident and scattered photon.
Plugging in the given values, we have:
Δλ = 0.0166 nm * (1 - cos(169°))
(b) The corresponding change in photon energy can be calculated using the formula:
ΔE = h / λ = hc / λ^2
Where:
- ΔE is the change in energy,
- h is Planck's constant (6.626 x 10^-34 J.s),
- c is the speed of light (3.0 x 10^8 m/s).
Substituting the given values, we have:
ΔE = (6.626 x 10^-34 J.s * 3.0 x 10^8 m/s) / (0.0166 nm)^2
(c) The kinetic energy of the recoiling electron can be calculated using the formula:
K.E. = ΔE
Since the photon loses energy and the electron gains an equal amount of energy, the magnitude of the photon's energy change is equal to the kinetic energy of the recoiling electron.
(d) Finally, the angle between the positive direction of the x-axis and the electron's direction of motion can be calculated using the relationship:
sin(Θ) = λ / λc
Where:
- Θ is the angle between the x-axis and the electron's direction of motion.
Substituting the given values, we have:
sin(Θ) = (0.0166 nm) / (2.43 x 10^-12 m)
Once you calculate the above expressions, you will have the answers to all the subparts of the problem.