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?
To answer these questions, we can use the formulas and concepts from neutron scattering and crystallography. Let's go through each question step-by-step:
a) To determine whether phonons are destroyed or created in the crystal, we need to compare the initial and final wavelengths of the neutrons. If the final wavelength (λ') is shorter than the initial wavelength (λ), phonons are created. Conversely, if the final wavelength is longer than the initial wavelength, phonons are destroyed.
In this case, we have λ = 1.8 Å and λ' = 1.15 Å. Since λ' < λ, phonons are created in the crystal.
b) The wavevector (k) of a neutron is related to its wavelength (λ) by the equation: k = 2π/λ. Similarly, the wavevector (k') of the scattered neutron is related to the final wavelength (λ') by the same equation: k' = 2π/λ'.
Using the given values, 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: ω = ck, where ω is the angular frequency and c is the speed of light. We know that the speed of light is approximately 3 × 10^8 m/s.
ω = ck = (3 × 10^8 m/s)(3.49 Å⁻¹) ≈ 1.05 × 10^9 s⁻¹ ≈ 1.05 THz
Therefore, the frequency of the involved phonon is approximately 1.05 THz.
c) To determine the involved reflection, we need to consider the direction of the incident neutron beam (along [100]) and the scattering plane (within (001)). We can use the Bragg's law equation: λ = 2d_hkl sinθ, where λ is the wavelength, d_hkl is the net plane distance, and θ is the scattering angle.
Given that the scattering angle is 30°, we can plug in the values to find the net plane distance: λ = 2d_hkl sin(30°).
Let's evaluate using both initial and final wavelengths:
For λ = 1.8 Å: 1.8 Å = 2d_hkl sin(30°) => d_hkl ≈ 1.04 Å
For λ' = 1.15 Å: 1.15 Å = 2d_hkl sin(30°) => d_hkl ≈ 0.664 Å
Comparing the net plane distances, the value of d_hkl ≈ 1.04 Å is more realistic and plausible. Therefore, the involved reflection is likely one that corresponds to the net plane distance of 1.04 Å.
d) To draw a sketch of the scattering process, we need to visualize the incident neutron beam, the scattering plane, and the scattered neutron beam. Unfortunately, text-based communication limits my ability to create visual sketches. However, you can imagine a neutron beam coming from the direction [100] and being scattered within the (001) plane by 30°. The scattered neutron beam will have a different direction, but its exact position can vary depending on the crystal structure and scattering geometry.
e) The wavevector of a phonon can be determined using the formula: g_hkl = 2π/d_hkl, where g_hkl is the magnitude of the corresponding reciprocal lattice vector and d_hkl is the net plane distance.
From the previous calculations, we found that the net plane distance for the involved reflection is approximately 1.04 Å. Using this value, we can calculate the magnitude of the reciprocal lattice vector:
g_hkl = 2π/d_hkl = 2π/1.04 Å ≈ 6.02 Å⁻¹
Therefore, the magnitude of the wavevector of the phonon is approximately 6.02 Å⁻¹.
I hope this explanation helps! Let me know if you need any further clarification.