Experiment to obtain the X-ray diffraction peaks of an unknown metal.
Radiation is K_alpha from Cu.
Through these diffraction peaks you wanted to determine:
(a) whether the cell is SC, BCC, or FCC
(b) the (hkl) value of the peaks
(c) the lattice parameter a of the metal
sin^2(theta)
0.120
0.239
0.480
0.600
0.721
0.841
0.956
--------------------------------
a) I think this is SC
c) a = 2.4 A
what about b) ?
To determine the value of (hkl) for the X-ray diffraction peaks, we can use Bragg's Law:
n * λ = 2 * d * sin(θ)
where:
- n is the order of the diffraction peak
- λ is the wavelength of the X-ray radiation
- d is the interplanar spacing between the crystal lattice planes
- θ is the angle of incidence of the X-ray radiation
Assuming the unknown metal has a simple crystal lattice structure (SC, BCC, or FCC), we can calculate the interplanar spacing using the lattice parameter (a):
1. For Simple Cubic (SC) lattice:
The interplanar spacing (d) can be calculated using the formula:
d = a / √(h^2 + k^2 + l^2)
2. For Body-Centered Cubic (BCC) lattice:
The interplanar spacing (d) can be calculated using the formula:
d = a / √((h^2 + k^2 + l^2) / 4)
3. For Face-Centered Cubic (FCC) lattice:
The interplanar spacing (d) can be calculated using the formula:
d = a / √(h^2 + k^2 + l^2)
Let's calculate the (hkl) values for the given sin^2(theta) values and determine which lattice structure (SC, BCC, or FCC) corresponds to the interplanar spacing calculations:
Given:
sin^2(theta):
0.120
0.239
0.480
0.600
0.721
0.841
0.956
λ (wavelength of Cu K_alpha radiation) = 1.5406 Å (angstrom = 10^-10 meters)
For each sin^2(theta) value, calculate the value of sin(theta) using the identity sin^2(theta) + cos^2(theta) = 1:
sin(theta) = √sin^2(theta)
Using Bragg's Law, rearrange the formula to solve for d:
d = n * λ / (2 * sin(theta))
For each sin(theta) value, calculate d for possible lattice structures (SC, BCC, and FCC) using their respective formulas for d mentioned above. Compare the calculated values of d to the given lattice parameter (a) value to determine the appropriate lattice structure.
Finally, using the calculated d values, determine the (hkl) values by rearranging the formula: h^2 + k^2 + l^2 = (a / d)^2, and substitute the corresponding values for d and a. The (hkl) values will represent the indices of the crystal planes that are responsible for the X-ray diffraction peaks.
With the given parameters, we can now proceed to calculate the (hkl) values and determine the appropriate lattice structure.