Final answer:
The value of θ for the first diffraction line in bcc iron when the X-ray wavelength is given can be found using the Bragg's Law equation. Plugging in the values and solving the equation gives a value of θ ≈ 60.43°.
Step-by-step explanation:
The Bragg's Law equation is used to solve this type of problem. The equation is given by:
nλ = 2dsin(θ)
Where:
- n = order of diffraction (in this case, n = 1)
- λ = wavelength of the X-ray (58.0 pm = 0.0580 nm)
- d = spacing between the closest set of lattice planes
- θ = angle of diffraction (we need to find this)
Plugging in the values, we have:
1 * (0.0580 nm) = 2d * sin(θ)
We can rearrange the equation to solve for θ:
θ = arcsin[(1 * (0.0580 nm)) / (2d)]
The formula for the atomic radius of the bcc lattice is:
d = (4/√3) * r
Plugging in the values, we have:
d = (4/√3) * 126 pm = 290.07 pm = 0.0290 nm
Finally, we can substitute the value of d into the equation for θ:
θ = arcsin[(1 * (0.0580 nm)) / (2 * 0.0290 nm)]
Solving this equation gives:
θ ≈ 60.43°