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Chapter 6: Problem 40

Speculate as to how the diffraction patterns of a typical crystal would be affected if \(\gamma\) -rays were used instead of \(\mathrm{X}\) - rays.

Short Answer

Expert verified
In conclusion, using gamma-rays instead of X-rays for crystal diffraction patterns would lead to higher scattering angles and/or higher orders of diffraction due to their shorter wavelengths. This could potentially provide more detailed information about the crystal lattice structure but may require higher precision in experimental setups to observe and measure the diffraction patterns accurately.

Step by step solution

01

Understand diffraction

Diffraction is a phenomenon that occurs when electromagnetic waves, such as X-rays or gamma-rays, bend around obstacles or interact with periodic structures such as crystal lattices. The phenomenon leads to a distinct pattern that can be observed and analyzed to understand the properties of the crystal lattice structure.
02

Recall Bragg's Law

Bragg's Law is used to describe the condition required for constructive interference to occur between waves scattered by different planes of atoms in a crystal lattice. Bragg's Law is given by: \[ n\lambda = 2d\sin\theta \] Where: - \(n\) is an integer known as the order of diffraction, - \(\lambda\) is the wavelength of the incident electromagnetic wave, - \(d\) is the distance between planes in the crystal lattice, and - \(\theta\) is the angle of incidence or scattering.
03

Compare the wavelengths of X-rays and gamma-rays

Gamma-rays have shorter wavelengths and higher energies compared to X-rays. The wavelength of gamma-rays can be as short as \(10^{-12}\) m, while the wavelength of X-rays usually falls within the range of \(10^{-11}\) m to \(10^{-9}\) m.
04

Analyze the effect of shorter wavelengths on diffraction patterns

Since the wavelength of gamma-rays is shorter than that of X-rays, using gamma-rays instead of X-rays would cause a shift in the diffraction pattern. According to Bragg's Law, as the wavelength \(\lambda\) decreases, either the angle \(\theta\) or the order of diffraction \(n\) must increase to maintain the equation for constructive interference. This means that when gamma-rays are used, the diffraction patterns will usually have higher angles of scattering or higher orders of diffraction.
05

Summarize the difference in diffraction patterns

In conclusion, when gamma-rays are used instead of X-rays to study diffraction patterns in crystals, the resulting patterns will have higher scattering angles and/or higher orders of diffraction due to the shorter wavelengths of gamma-rays. This could potentially provide more detailed information about the crystal lattice structure, but may also demand higher precision in experimental setups to observe and measure the diffraction patterns accurately.

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