X-rays of wavelength 0.10 nm fall on a microcrystalline powder sample. The sample is located 15 cm from a photographic sensor. The crystal structure of the sample has an atomic spacing of 0.22 nm. Calculate the radii of the diffraction rings corresponding to first- and second-order scattering. Note in Fig. 35–28 that the X-ray beam is deflected through an angle 2Φ.

Verified step by step guidance

1. Understand the problem: The X-rays are diffracted by the crystal structure, and we are tasked with finding the radii of the diffraction rings for the first- and second-order scattering. This involves using Bragg's law and some geometry to relate the diffraction angle to the radius of the rings on the photographic sensor.
2. Apply Bragg's law to determine the diffraction angle (Φ). Bragg's law is given by: $n\lambda =2dsin\Phi$, where $n$ is the order of diffraction, $\lambda$ is the wavelength of the X-rays, $d$ is the atomic spacing, and $\Phi$ is the diffraction angle. Solve for $\Phi$ for both $n=1$ and $n=2$.
3. Once the diffraction angle $\Phi$ is determined, note that the X-ray beam is deflected through an angle $2\Phi$. Use this deflection angle to calculate the radius of the diffraction ring on the photographic sensor. The radius is related to the angle by the formula: $r=Ltan\left(2\Phi \right)$, where $L$ is the distance from the sample to the sensor (15 cm in this case).
4. Substitute the values for $\lambda$, $d$, and $L$ into the equations. For the first-order diffraction ($n=1$), calculate $\Phi$ using Bragg's law, then find the radius $r$. Repeat the process for the second-order diffraction ($n=2$).
5. Verify the results: Ensure that the calculated radii for the first- and second-order diffraction rings are reasonable and consistent with the given parameters. This involves checking the units and ensuring that the angles and radii are physically meaningful.

Key Concepts

Bragg's Law

Bragg's Law relates the wavelength of X-rays to the angles at which they are diffracted by a crystal lattice. It is expressed as nλ = 2d sin(θ), where n is the order of diffraction, λ is the wavelength, d is the distance between atomic planes, and θ is the angle of incidence. This principle is fundamental for analyzing diffraction patterns and determining crystal structures.

Diffraction Rings

Diffraction rings are circular patterns formed when X-rays are scattered by the atomic planes in a crystal. The radius of these rings is related to the angle of diffraction and the distance from the sample to the detector. The first- and second-order rings correspond to different values of n in Bragg's Law, indicating different angles of scattering.

Geometric Relationships in Diffraction

In diffraction experiments, geometric relationships help determine the radii of diffraction rings based on the distance from the sample to the detector and the angles of scattering. The radius of a diffraction ring can be calculated using the formula r = L tan(θ), where r is the radius, L is the distance to the detector, and θ is the angle of diffraction. Understanding these relationships is crucial for accurately interpreting diffraction data.