calculate the lowest values of the energies of an electron, a neutron and a photon which allows diffraction from the (1,0,1) planes of silicon, using Bragg diffraction.

lattice parameter 0.542nm

To calculate the lowest values of the energies for diffraction from the (1,0,1) planes of silicon using Bragg diffraction, we need to use the Bragg's law equation:

nλ = 2dsinθ

In this equation, n is the order of diffraction, λ is the wavelength of the particle/wave, d is the spacing between the crystal planes, and θ is the angle of incidence relative to the plane.

First, we need to determine the spacing between the (1,0,1) planes of silicon. The spacing can be calculated using the Miller indices of the plane and the lattice parameter.

To calculate the spacing, we use the formula:

d = a / sqrt(h^2 + k^2 + l^2)

Here, a is the lattice parameter, h, k, and l are the Miller indices of the plane.

Given that the lattice parameter of silicon is 0.542 nm and the Miller indices of the (1,0,1) plane are (1,0,1), we can substitute these values into the equation to find the plane spacing.

d = 0.542 nm / sqrt(1^2 + 0^2 + 1^2)
d = 0.542 nm / sqrt(2)
d ≈ 0.383 nm

Now that we have the plane spacing, we can proceed to calculate the lowest value of energy for diffraction for each particle/wave.

1. Electron:
The energy of an electron can be calculated using its de Broglie wavelength:

λ = h / p

Where λ is the wavelength, h is the Planck's constant (6.626 x 10^-34 J·s), and p is the momentum of the electron.

Since we are looking for the lowest energy, we assume n = 1 in Bragg's law. Therefore, we can rearrange Bragg's law to solve for λ:

λ = 2d / sinθ

The electron's momentum can be determined using the equation:

p = mv

Where m is the mass of the electron (me = 9.109 x 10^-31 kg) and v is the velocity of the electron.

To calculate the velocity, we use the relativistic equation for kinetic energy:

KE = (γ - 1)mc^2

Where KE is the kinetic energy, γ is the Lorentz factor (√(1 - v^2/c^2)), m is the mass, and c is the speed of light (3.00 x 10^8 m/s).

Now, let's calculate the energy of an electron that allows diffraction from the (1,0,1) planes of silicon:

1. Calculate the momentum:
v = √(c^2 - (mc^2 / KE)^2)
p = mv

2. Calculate the wavelength using Bragg's law:
λ = 2d / sinθ
λ = 2 * 0.383 nm / sinθ

3. Calculate the energy using the de Broglie wavelength equation:
λ = h / p

By substituting the values into the equations and solving, you can find the lowest energy of an electron that allows diffraction from the (1,0,1) planes of silicon.

Repeat the same process for the neutron and photon, using their respective formulas for momentum and energy.