Final answer:
The angle for the first-order diffraction of a crystal with spacings of 0.394 nm, using X-rays with a wavelength of 0.147 nm, is calculated using Bragg's law. The resulting diffraction angle is approximately 10.7°.
Step-by-step explanation:
The subject of the question concerns the principles of diffraction and Bragg's law in Physics. More specifically, this question is related to first-order diffraction angles when X-rays are incident upon a crystal lattice. Bragg's law is fundamental in determining the angles where diffracted rays will constructively interfere and create diffraction patterns based on the spacing between lattice planes and the wavelength of the incident rays.
To find the angle for the first-order diffraction, we use the formula nλ = 2d sin(θ), where n is the order of the reflection, λ is the wavelength, d is the spacing between planes in the crystal, and θ is the diffraction angle. For first-order diffraction (n=1), rearranging the formula gives θ = arcsin(λ / (2d)). Plugging in the given values, 0.394 nm for d and 0.147 nm for λ, we can calculate the first-order diffraction angle.
To solve this problem: θ = arcsin(0.147 nm / (2 × 0.394 nm)) = arcsin(0.147 nm / 0.788 nm) = arcsin(0.1866) ≈ 10.7 degrees. Thus, the angle for the first order diffraction is approximately 10.7°.