Monochromatic x-rays (λ = 0.166 nm) from a nickel target are incident on a potassium chloride (KCl) crystal surface. The spacing between planes of atoms in KCl is 0.314 nm. At what angle (relative to the surface) should the beam be directed for a second-order maximum to be observed?
To find the angle (θ) at which the second-order maximum is observed, we can use Bragg's Law. Bragg's Law states that the condition for constructive interference occurs when 2d sin(θ) = nλ, where d is the spacing between planes of atoms, θ is the angle of incidence, n is the order of the maximum, and λ is the wavelength of the incident wave.
In this case, we're looking for the second-order maximum, so n = 2. We know the wavelength of the incident x-rays (λ = 0.166 nm) and the spacing between planes of atoms in KCl (d = 0.314 nm). We can substitute these values into Bragg's Law and solve for θ.
2d sin(θ) = nλ
2 * 0.314 nm * sin(θ) = 2 * 0.166 nm
0.628 nm * sin(θ) = 0.332 nm
sin(θ) = 0.332 nm / 0.628 nm
sin(θ) = 0.528
Now, to find θ, we can take the inverse sine (or arcsine) of 0.528.
θ = arcsin(0.528)
Using a calculator, we find that θ ≈ 31.21 degrees.
Therefore, the angle at which the beam should be directed for the second-order maximum to be observed is approximately 31.21 degrees relative to the surface.