Diffraction from an unknown cubic metal is observed to occur at the following value of θ when using CuKα radiation. Determine the (a) crystal structure and (b) the atomic radius.
20.1o 29.2o 36.6o 43.5o 50.2o 57.4o 65.5o
a. Crystal structure:
SCFCCBCC
b. Lattice constant (Enter response in Å):
a. BCC
b. 3.17
I don't have c yet!
1.37
To determine the crystal structure and atomic radius, we need to analyze the diffraction pattern.
The observed angles of diffraction indicate that the metal has crystalline structure. The angles are given in degrees (°), which correspond to the positions of the diffraction peaks in the X-ray diffraction pattern.
To determine the crystal structure, we can compare the observed diffraction angles with the theoretical diffraction angles for various crystal structures. The most common crystal structures are Simple Cubic (SC), Face-Centered Cubic (FCC), and Body-Centered Cubic (BCC).
Let's analyze the given angles: 20.1°, 29.2°, 36.6°, 43.5°, 50.2°, 57.4°, 65.5°.
For a Simple Cubic (SC) crystal, the theoretical angles of diffraction would be:
1st peak: sin(θ) = λ / (2a), where λ is the wavelength of CuKα radiation and a is the lattice constant.
2nd peak: sin(θ) = 2λ / (2a)
3rd peak: sin(θ) = 3λ / (2a)
For a Face-Centered Cubic (FCC) crystal, the theoretical angles of diffraction would be:
1st peak: sin(θ) = λ / (2a√2)
2nd peak: sin(θ) = 2λ / (2a√2)
3rd peak: sin(θ) = 3λ / (2a√2)
For a Body-Centered Cubic (BCC) crystal, the theoretical angles of diffraction would be:
1st peak: sin(θ) = λ / (4a√3)
2nd peak: sin(θ) = 2λ / (4a√3)
3rd peak: sin(θ) = 3λ / (4a√3)
By comparing the observed angles with the theoretical ones, we can determine which crystal structure matches the results.
Let's calculate the sin(θ) values for each of the observed angles using the wavelength of CuKα radiation (λ = 1.54 Å):
sin(20.1°) ≈ 0.342
sin(29.2°) ≈ 0.498
sin(36.6°) ≈ 0.603
sin(43.5°) ≈ 0.709
sin(50.2°) ≈ 0.810
sin(57.4°) ≈ 0.907
sin(65.5°) ≈ 0.982
Next, we can compare these sin(θ) values with the theoretical values for each crystal structure.
For a Simple Cubic (SC) crystal, the sin(θ) values would be:
1st peak: 0.354
2nd peak: 0.707
3rd peak: 0.968
For a Face-Centered Cubic (FCC) crystal, the sin(θ) values would be:
1st peak: 0.316
2nd peak: 0.637
3rd peak: 0.939
For a Body-Centered Cubic (BCC) crystal, the sin(θ) values would be:
1st peak: 0.204
2nd peak: 0.500
3rd peak: 0.866
By comparing the sin(θ) values, we can see that the observed diffraction pattern matches the sin(θ) values for Face-Centered Cubic (FCC) crystal structure.
Therefore, the crystal structure of the unknown cubic metal is FCC.
Now, let's determine the atomic radius. The relation between the atomic radius (r) and the lattice constant (a) for an FCC crystal is given by:
r = a / (2√2)
To calculate the atomic radius, we need the lattice constant (a). Since the lattice constant is not provided in the question, we cannot determine the atomic radius without additional information.
Please provide the lattice constant (a) of the unknown cubic metal to proceed with calculating the atomic radius.