Diffraction from an unknown cubic metal is observed to occur at the following values of θ when using CuKα radiation. Determine the crystal structure to find the lattice constant.
20.1⁰, 29.2⁰, 36.6⁰, 43.5⁰, 50.2⁰, 57.4⁰, 65.5⁰
a) What is the crystal structure?
SC, BCC or FCC
b)Express the value of the lattice constant in m
a. BCC
i think b only ask for the expression, not the actual value????...beside, in order to get the actual value of lattice constant, we need the radius...is the radius given or can it be determined???...
i tried expressing the lattice constant for the BCC as 4r/sqrt3 but can't get the correct format.
The value of the radius is not given
For CuKα-radiation λ=1.541•10⁻¹⁰ m.
From equation
sin²θ₁/(h²+k²+l²)₁=sin²θ₂/(h²+k²+l²)₂=…. const,
sin²θ₁/1= sin²θ₂/2= sin²θ₃/3=… sin²θ ₇/7.
For the given data
0.118/1 = 0.238/2=0.3555/3=0.4738/4=0.59/5=
=0.7097/6=0.828/7=0.118 =>
The crystal structure is simple cubic –SC.
sin²θ/(h²+k²+l²)=const=λ²/4a²
a= sqrt{λ²(h²+k²+l²)/2sin²θ }= >
a= sqrt{λ²(1²+0²+0²)₁/2sin²θ₁}=
= sqrt{(1.541•10⁻¹⁰)²(1²+0²+0²)/2sin²20.1⁰}=3.17•10⁻¹⁰m.
3.17*10^-10
To determine the crystal structure and lattice constant, we can use the Bragg's Law formula:
n * λ = 2 * d * sin(θ)
Where:
- n is the order of the diffraction peak (usually n = 1 for first-order diffraction)
- λ is the wavelength of the X-ray radiation (in this case CuKα radiation, which has a known wavelength)
- d is the interplanar spacing (lattice constant) of the crystal
- θ is the angle of diffraction
First, let's calculate the wavelength of CuKα radiation. The wavelength of CuKα radiation is approximately 1.54 Å (angstroms), which is equivalent to 1.54 x 10^-10 meters.
Now we can calculate the interplanar spacing (d) for each observed diffraction angle (θ) using Bragg's Law. We can rearrange the formula to solve for d:
d = (n * λ) / (2 * sin(θ))
For each given value of θ, we can substitute the values and solve for d. The order (n) of the diffraction peak is 1 for all calculations.
For the given values of θ:
θ1 = 20.1⁰
d1 = (1 * 1.54 x 10^-10 m) / (2 * sin(20.1⁰))
Similarly, calculate d for the remaining values of θ.
Once we have the values of interplanar spacing (d), we need to identify the crystal structure based on these values.
For a simple cubic (SC) crystal structure:
The interplanar spacing (d) is given by:
d_SC = a
For a body-centered cubic (BCC) crystal structure:
The interplanar spacing (d) is given by:
d_BCC = a * sqrt(2) / 2
For a face-centered cubic (FCC) crystal structure:
The interplanar spacing (d) is given by:
d_FCC = a * sqrt(2) / 2
Now, we need to investigate if the calculated interplanar spacing (d) values match any of these crystal structures. If the d values match only one of the structures, then that is the crystal structure of the unknown cubic metal.
Finally, to express the value of the lattice constant in meters, we can take the average of the interplanar spacing (d) values obtained for the identified crystal structure (SN, BCC, or FCC) and express it in meters.