Question

\(\mathrm{X}\) rays of wavelength \(2.63\) Å were used to analyze a crystal. The angle of first-order diffraction \((n=1\) in the Bragg equation) was \(15.55\) degrees. What is the spacing between crystal planes, and what would be the angle for second-order diffraction \((n=2)\) ?

Short answer

The spacing between the crystal planes is approximately \(4.97\) Å, and the angle for second-order diffraction is about \(30.24\degree\).

Step-by-step solution

Recall the Bragg's equation

First, let's recall the Bragg's equation which is given as follows: \[n\lambda = 2d\sin{\theta}\] where \(n\) is the order of diffraction, \(\lambda\) is the wavelength of the X-rays, \(d\) is the spacing between crystal planes, and \(\theta\) is the angle of diffraction.

Calculate the spacing between crystal planes

Given the first-order diffraction (\(n = 1\)), we have the wavelength \(\lambda = 2.63\) Å and the angle of diffraction \(\theta = 15.55\) degrees. We can plug these values into the Bragg's equation and solve for the spacing between crystal planes, \(d\): \[d = \frac{1 \times 2.63 Å}{2\sin{15.55°}}\] First, we convert the angle from degrees to radians: \[15.55° = 15.55 \times \frac{\pi}{180} \approx 0.2713 \; \text{radians}\] Now, we can calculate \(d\): \[d \approx \frac{2.63 Å}{2\sin{0.2713}} \approx 4.97 Å\] So, the spacing between crystal planes is approximately \(4.97\) Å.

Calculate the angle for second-order diffraction

Now, we need to find the angle of the second-order diffraction (\(n = 2\)). We can use the Bragg's equation again, this time with \(n = 2\) and the previously calculated spacing \(d = 4.97\) Å: \[2 \lambda = 2(4.97 Å) \sin{\theta_2}\] We can solve for the angle of second-order diffraction, \(\theta_2\): \[\theta_2 = \sin^{-1}\left(\frac{2\lambda}{2(4.97 Å)}\right)\] \[\theta_2 = \sin^{-1}\left(\frac{2(2.63 Å)}{2(4.97 Å)}\right)\] \[\theta_2 \approx 0.5277 \; \text{radians}\] Now, we convert the angle from radians to degrees: \[\theta_2 \approx 0.5277 \times \frac{180}{\pi} \approx 30.24\degree\] So, the angle for second-order diffraction is approximately \(30.24\degree\). In conclusion, the spacing between crystal planes is approximately \(4.97\) Å and the angle for second-order diffraction is about \(30.24\degree\).

Key concepts

Diffraction Angle

When we discuss diffraction angles in the context of X-rays and crystals, we're talking about the specific angle at which X-rays scatter off the evenly spaced layers within a crystal. The significance of the diffraction angle is underscored by its use in Bragg's law, which describes how these angles relate to the crystalline structure.
In our exercise, the first-order diffraction angle was given as 15.55 degrees, and we were tasked with determining what the angle would be for second-order diffraction. It's crucial to note that the diffraction angle will change with the order of diffraction, keeping the wavelength constant, as every order represents a different level of path difference, or interference, between the waves reflecting off different crystal planes. Knowing the specific angles for different orders can help in identifying the crystal structure by matching theoretical and observed diffraction patterns, which is a cornerstone of methods like X-ray crystallography.

Spacing Between Crystal Planes

The spacing between crystal planes, symbolized as 'd' in Bragg's equation, is essentially the distance between layers of atoms within the crystal lattice. This measurement is crucial because it's unique to each material's crystal structure and determines how the X-rays will diffract when they encounter the crystal.

• Essentially, 'd' is a reflection of the crystal's internal symmetry and the arrangement of its constituent atoms or molecules.
• It's not directly observable, but it can be calculated using the diffraction pattern produced when X-rays interact with the crystal.
• In the provided exercise, we calculated the spatial separation between the crystal planes to be approximately 4.97 Å using the given wavelength and first-order diffraction angle.

Understanding this spacing allows us to make predictions about the material's properties and can be applied in fields such as materials science, geology, and chemistry.

X-ray Wavelength

X-ray wavelength, expressed in angstroms (Å) or nanometers (nm), is a crucial factor in studying crystal structures through diffraction methods. X-rays with the appropriate wavelength can interact with the electron clouds surrounding the atomic nuclei in a crystal, causing diffraction, which is the basis for determining the crystal structure.

• The choice of X-ray wavelength depends on the material being studied and the desired resolution.
• In the context of Bragg's law, the wavelength is one of the known variables used to solve for unknown quantities like the crystal spacing or the diffraction angle.
• In our exercise scenario, the wavelength given was 2.63 Å, which we used, along with the first-order diffraction angle, to uncover the crystal spacing and predict the angle for the second-order diffraction.

A thorough understanding of the relationship between the X-ray wavelength and crystal structure can lead to profound insights into the material's properties and behavior.

Order of Diffraction

The order of diffraction, denoted by 'n' in Bragg's equation, indicates whether the scattered X-rays are undergoing first, second, third, (and so on) order diffraction. Each order corresponds to an increment in the path difference of one wavelength, which will affect the angle at which constructive interference occurs and hence the overall diffraction pattern.

• First-order diffraction (n=1) typically involves the least bending of the X-ray and is often the most intense reflection observed.
• With increasing order, the diffraction angle usually increases since the path difference is a multiple of the wavelength.
• In our exercise, knowing the first-order diffraction angle allowed us to calculate the spacing between crystal planes, and then using this information for the second-order (n=2), we found the larger diffraction angle corresponding to it.

The concept of diffraction order is vital in understanding and interpreting X-ray diffraction patterns, which are foundational in materials characterization and the identification of unknown substances.