Before we dive into the calculation, let's first understand some core concepts that we'll use.
1. The energy of an electron can be converted from kilo electron-volts (keV) to Joules through the use of the conversion factor where 1keV equals 1.6 × 10^-16 Joules.
2. The de Broglie wavelength, which is a fundamental concept in quantum mechanics, can be calculated using the Planck's constant (6.62607004 ×10^-34 m^2 kg / s), electron rest mass (9.10938356 ×10^-31 kg), and the energy of the electron in Joules.
3. Bragg's Law is used in X-ray crystallography to determine the angles for coherent and incoherent scattering from a crystal lattice.
Let's proceed with our calculation now.
1. Start by converting the energy of the electron from 2keV to Joules. Using the conversion factor of 1.6 × 10^-16 Joules/keV, the energy becomes 2 * 1.6 × 10^-16 = 3.2 × 10^-16 Joules.
2. With the energy in Joules, we can calculate the de Broglie wavelength. The formula for this is Planck's constant divided by the square root of the product of twice the electron rest mass and the energy of the electron. That comes out to be λ = 6.62607004 ×10^-34 m^2 kg / s / √(2 * 9.10938356 ×10^-31 kg * 3.2 × 10^-16 Joules).
3. Now let's calculate the lattice-plane spacings corresponding to each of the three radii. We do this through Bragg's Law, which can also give us the lattice spacing 'd'. For each radius, we first calculate the diffraction angle using the arctan function: θ = arctan(r/screen_distance), where r is the radius of each ring and screen_distance is 0.35 meters. Afterwards, we find the spacing 'd' by dividing the wavelength by twice sine of the angle 'θ': d = λ / (2 * sin(θ)).
By going through this processo for each radius, we get the following lattice-plane spacings:
For the first ring with radius 2.1 cm:
d = 2.2909764297746585 × 10^-10 meters
For the second ring with radius 2.3 cm:
d = 2.09250958549598 × 10^-10 meters
For the third ring with radius 3.2 cm:
d = 1.507013837871105 × 10^-10 meters
Therefore, the lattice-plane spacings for the rings with radii 2.1 cm, 2.3 cm, and 3.2 cm are 2.2909764297746585 × 10^-10 meters, 2.09250958549598 × 10^-10 meters, and 1.507013837871105 × 10^-10 meters, respectively.