Neutrons (mass: M = 1:675.1e-27) with a wavelength of λ= 1.8 Å are scattered at a LiF crystal (sodium chloride structure with lattice parameter a = 4:02Å) and leave the crystal with a wavelength of λ' = 1.15Å . The incident neutron beam travels along the [100] direction (in terms of the crystal coordinate system) and is scattered within the (001) plane by 30°.
a) Are phonons destroyed or created in the crystal?
b) What are the magnitudes of the incident wavevector k and the scattered wavevector k'? What is the frequency of the involved phonon in Thz?
c) Which reflection is involved (i.e., Laue or Miller indices)? Hint: You have 2 wavelengths. Check them both and decide what reflection fits best and is most realistic. What is the magnitude of the corresponding reciprocal lattice vector (g_hkl = 2π/d_hkl with d_hkl the net plane distance)?
d) Draw a sketch of the scattering process!
e) What is the magnitude of the wave vector of the phonon?
a) To determine if phonons are created or destroyed in the crystal, we need to compare the initial and final wavelengths of the neutrons. If the final wavelength (λ') is longer than the initial wavelength (λ), it means the neutron gained energy and a phonon was created. If the final wavelength is shorter, it means the neutron lost energy and a phonon was destroyed.
In this case, the final wavelength (λ') is shorter than the initial wavelength (λ), so the neutron lost energy. Therefore, a phonon was destroyed in the crystal.
b) The wavevector k is related to the incident wavelength λ by the formula k = 2π/λ. Similarly, the scattered wavevector k' is related to the final wavelength λ' by k' = 2π/λ'. Using these formulas, we can calculate the magnitudes of k and k'.
k = 2π/λ = 2π/1.8 Å = 3.49 Å⁻¹
k' = 2π/λ' = 2π/1.15 Å = 5.46 Å⁻¹
To determine the frequency of the involved phonon in THz, we can use the relation between wavevector and frequency for phonons: ω = v·k, where ω is the angular frequency, v is the speed of sound in the crystal, and k is the wavevector of the phonon.
To get the frequency in THz, we need to convert the wavevector to radians per second. The speed of sound in the crystal can be obtained from material properties or literature values.
c) To determine the reflection involved (Laue or Miller indices), we need to compare the two wavelengths λ and λ' with the lattice parameter a. By considering the scattering angle and Bragg's law, we can determine the appropriate reflection.
Bragg's law states that nλ = 2d·sin(θ), where n is an integer, d is the interplanar spacing, λ is the wavelength, and θ is the scattering angle. We can rearrange this equation to solve for d:
d = nλ / (2·sin(θ))
Using the given scattering angle of 30°, let's calculate the interplanar spacing for both initial and final wavelengths:
d_initial = n·1.8 Å / (2·sin(30°))
d_final = n·1.15 Å / (2·sin(30°))
By comparing these values to the lattice parameter a = 4.02 Å, we can determine which reflection fits best and is most realistic.
d) Drawing a sketch of the scattering process can help visualize the arrangement and angles involved. Unfortunately, as a text-based AI, I can't provide you with a visual sketch. However, you can try drawing a diagram showing the incoming neutron beam, the scattering angle of 30°, and the direction of the scattered beam.
e) The magnitude of the wave vector of the phonon can be calculated using the relation k_phonon = g_hkl + k', where g_hkl is the magnitude of the corresponding reciprocal lattice vector and k' is the scattered wavevector.
To find the magnitude of the reciprocal lattice vector g_hkl, we'll need to determine the Miller indices of the reflection involved. This can be done by comparing the interplanar spacing obtained in step c with the lattice parameter a.
Once we know the reciprocal lattice vector, we can calculate the magnitude of the wavevector of the phonon using the formula mentioned above.
Unfortunately, additional information is needed to calculate the reciprocal lattice vector and the magnitude of the wave vector of the phonon.