The silicon wafer is coated with a layer of metallic aluminum which acts as an electrical contact. The x-ray diffraction pattern of aluminum is measured in a diffractometer with Fe Kα radiation. At what angle, θ, do you expect to observe the first reflection of aluminum, i.e., the reflection at the lowest angle? Express your answer in degrees.

Question

The silicon wafer is coated with a layer of metallic aluminum which acts as an electrical contact. The x-ray diffraction pattern of aluminum is measured in a diffractometer with Fe Kα radiation. At what angle, θ, do you expect to observe the first reflection of aluminum, i.e., the reflection at the lowest angle? Express your answer in degrees.

Answer 1

To determine the angle at which you would observe the first reflection of aluminum in the x-ray diffraction pattern, we need to use the Bragg Equation. The Bragg Equation relates the wavelength of a diffracted X-ray, the angle of incidence, and the spacing between atomic planes in a crystal lattice.

The Bragg Equation is given by:

nλ = 2dsinθ

where:
- n is the order of the reflection (in this case, n = 1 for the first reflection),
- λ is the wavelength of the incident X-ray radiation (in this case, Fe Kα radiation),
- d is the spacing between atomic planes in the crystal lattice of aluminum, and
- θ is the angle at which the reflection occurs.

We are given that Fe Kα radiation is used, so we need to find the wavelength of that radiation. The wavelength of Fe Kα radiation is approximately 1.937 Å (angstroms).

Next, we need to find the spacing between atomic planes in the crystal lattice of aluminum (d). The spacing between atomic planes is determined by the Miller indices of the planes involved in the reflection. The Miller indices for the aluminum reflection can be determined from the aluminum crystal structure.

For aluminum, we have the face-centered cubic (FCC) crystal structure, which has Miller indices of (111) for the planes involved in the first reflection.

Using crystallographic tables or software, we can find that the spacing between (111) planes in aluminum is approximately 2.090 Å.

Now, we can plug the values into the Bragg Equation to find the angle θ:

(1)(1.937 Å) = 2(2.090 Å) sinθ

1.937 Å = 4.180 Å sinθ

sinθ = 1.937 Å / 4.180 Å

sinθ ≈ 0.4639

To find the angle θ, we can take the inverse sine (sin^(-1)) of 0.4639:

θ = sin^(-1)(0.4639)

Using a calculator or trigonometric table, you will find:

θ ≈ 27.9 degrees.

Therefore, the first reflection of aluminum in the x-ray diffraction pattern is expected to occur at an angle of approximately 27.9 degrees.