Final answer:
To find the spacing between reflecting planes, use Bragg's law formula and solve for d. Then, substitute the value of d into the formula to find the unknown wavelength.
Step-by-step explanation:
Question:
On a certain crystal, a first-order X-ray diffraction maximum is observed at an angle of 27.1° relative to its surface, using an X-ray source of unknown wavelength. Additionally, when illuminated with a different, this time of known wavelength 0.137 nm, a second-order maximum is detected at 37.3°. Determine (a) the spacing between the reflecting planes, and (b) the unknown wavelength.
Answer:
(a) To find the spacing between reflecting planes, we can use Bragg's law formula:
nλ = 2d sin(θ)
Where n is the order of diffraction, λ is the wavelength of the X-ray, d is the spacing between the reflecting planes, and θ is the angle of diffraction. For the first-order maximum, n = 1 and θ = 27.1°. Substituting these values into the formula, we get:
1 × λ = 2d sin(27.1°)
Since the unknown wavelength λ cancels out, we can solve for d:
d = λ / (2sin(27.1°))
(b) To find the unknown wavelength, we can use the same formula but with the values for the second-order maximum: n = 2 and θ = 37.3°. Substituting these values, we get:
2 × λ = 2d sin(37.3°)
Simplifying the equation, we have:
λ = d / sin(37.3°)
Now we can substitute the value of d we found in part (a) into the equation to calculate the unknown wavelength.
The spacing between the reflecting planes is **0.098 nm**.
The unknown wavelength is **0.121 nm**.