X-rays with wavelengths of 0.2599 nm are detected at a scattering angle of = 132° in a Compton-scattering experiment, as shown in the drawing. Find (a) the wavelength of an incident photon, (b) the energy of an incident photon (in eV), (c) the energy of a scattered photon (in eV), and (d) the kinetic energy of the recoil electron (in eV). (For accuracy, use h = 6.626 x 10-34 J·s and c = 2.998 x 108 m/s.)

Question

X-rays with wavelengths of 0.2599 nm are detected at a scattering angle of = 132° in a Compton-scattering experiment, as shown in the drawing. Find (a) the wavelength of an incident photon, (b) the energy of an incident photon (in eV), (c) the energy of a scattered photon (in eV), and (d) the kinetic energy of the recoil electron (in eV). (For accuracy, use h = 6.626 x 10-34 J·s and c = 2.998 x 108 m/s.)

Answer 1

To solve this problem, we will use the principles of Compton scattering and the Compton wavelength shift equation. Let's break down the problem step by step:

Step 1: Understand the problem and identify the given values:
We are given:
- Wavelength of scattered X-rays (λ') = 0.2599 nm (nanometers)
- Scattering angle (θ) = 132°
- Planck's constant (h) = 6.626 x 10^-34 J·s
- Speed of light (c) = 2.998 x 10^8 m/s

Step 2: Convert the given values to appropriate units:
Since energy calculations involve the standard SI units (meters and joules), we need to convert the wavelength from nanometers to meters:

λ' = 0.2599 nm = 0.2599 x 10^-9 m

Step 3: Calculate the wavelength of the incident photon (λ):
To find the wavelength of the incident photon, we will use the Compton wavelength shift equation, which relates the change in wavelength (Δλ) to the scattering angle (θ). The equation is given by:

Δλ = λ' - λ = (h / (mec)) * (1 - cos(θ))

Where:
Δλ = Change in wavelength = λ' - λ
λ = Wavelength of the incident photon
mec = Compton wavelength = h / mc
m = Mass of electron = 9.1 x 10^-31 kg

Since Δλ, λ', and θ are known, we can solve for the incident wavelength (λ).

Step 4: Solve for the wavelength of the incident photon (λ):
Rearranging the equation, we have:

λ = λ' - Δλ

Using the given values, we can now calculate λ.

Step 5: Calculate the energy of the incident photon (E):
The energy of a photon can be calculated using the equation:

E = hc / λ

Where:
E = Energy of the photon
h = Planck's constant
c = Speed of light
λ = Wavelength of the incident photon

Using the calculated value of λ, we can calculate the energy (E) of the incident photon.

Step 6: Calculate the energy of the scattered photon (E'):
The energy of the scattered photon can be calculated using the same equation as in Step 5, but with the wavelength of the scattered photon (λ'):

E' = hc / λ'

Step 7: Calculate the kinetic energy of the recoil electron (KE):
The kinetic energy of the recoil electron can be calculated using the equation:

KE = E - E'

Where:
KE = Kinetic energy of the recoil electron
E = Energy of the incident photon
E' = Energy of the scattered photon

Step 8: Substitute the known values and calculate the final answers:
Using Steps 4-7, substitute the known values into the respective equations and calculate the final values for (a), (b), (c), and (d).

Note: Since the question asks for answers in electron volts (eV), you will need to convert the final energies from joules to eV using the conversion factor: 1 eV = 1.6 x 10^-19 J.

Once you have calculated these values, you will have the answers to all four parts of the question: (a) the wavelength of the incident photon, (b) the energy of an incident photon, (c) the energy of a scattered photon, and (d) the kinetic energy of the recoil electron.