Question

What is the wavelength of the X-rays if the first-order Bragg diffraction is observed at \(23.0^{\circ}\) related to the crystal surface, with inter atomic distance of \(0.256 \mathrm{nm} ?\)

Short answer

Answer: The wavelength of the X-rays is approximately \(0.200 \mathrm{nm}\).

Step-by-step solution

Write down the given information and the Bragg's Law equation.

First, let's write down the information given in the problem statement: Order of diffraction (n) = 1 (first-order), Diffraction angle (\(\theta\)) = \(23.0^{\circ}\), Interatomic distance (d) = \(0.256 \mathrm{nm}\). Now, write down the Bragg's Law formula: \(n\lambda = 2d\sin{\theta}\)

Plug the given values into the Bragg's Law equation.

Next, we'll plug the given values into the Bragg's Law formula: \(1\cdot\lambda = 2(0.256\,\mathrm{nm})\sin{23.0^{\circ}}\)

Calculate the sin of the angle.

Now, calculate the sin value for \(23.0^{\circ}\): \(\sin{23.0^{\circ}} \approx 0.391\)

Multiply the interatomic distance by 2 and the sin value.

Multiply the interatomic distance (0.256 nm) by 2 and the sin value (0.391) to find the wavelength: \(\lambda = 2(0.256\,\mathrm{nm})(0.391)\)

Calculate the wavelength.

Calculate the wavelength: \(\lambda \approx 0.200\,\mathrm{nm}\) The wavelength of the X-rays for the first-order Bragg diffraction at \(23.0^{\circ}\) related to the crystal surface with an interatomic distance of \(0.256 \mathrm{nm}\) is approximately \(0.200 \mathrm{nm}\).

Key concepts

X-ray diffraction

X-ray diffraction (XRD) is a powerful non-destructive technique used to analyze the structure of crystalline materials. By directing X-rays at a crystal and then detecting the angles and intensities of the reflected beams, scientists can infer information about the crystal's structure. This is based on the principle that when X-rays hit a crystal, their waves will scatter in various directions. If these scattered waves interfere constructively (their peaks match up), they will amplify each other and lead to a detectable signal. On the other hand, if they interfere destructively (their peaks and troughs cancel out), the signal will be minimized or absent.

These patterns of interference can create a 'diffraction pattern' unique to the atomic structure of the crystal. Analysis of the diffraction pattern enables determination of the crystal structure, including interatomic distances and the arrangement of atoms within the crystal. The core formula used in XRD is known as Bragg's Law, which relates the wavelength of X-rays to the diffraction angle and the distances between atomic planes within the crystal.

Interatomic distance

The interatomic distance refers to the distance between the centers of adjacent atoms within a material. In the context of XRD, the interatomic distance is among the critical parameters. It is often calculated from the spacing between planes of atoms in the crystal lattice, which diffract the incoming X-rays. This spacing is denoted by the variable 'd' in Bragg's Law and is typically measured in nanometers (nm), which is one billionth of a meter.

Importance in XRD

Understanding the interatomic distance is vital because the arrangement and distance between atoms affects the diffraction pattern that XRD detects. Larger distances will result in a different set of allowable angles for constructive interference than smaller distances. Therefore, knowledge about these distances is essential in determining the crystal's structure and is directly related to the physical properties of the material.

Diffraction angle

In XRD, the diffraction angle, often represented as \(\theta\), is the angle at which constructive interference occurs, resulting in a diffraction peak in the pattern. It is the angle between the incident X-ray beam and the diffracted beam that exits the crystal at which specific conditions for constructive interference are met, satisfying Bragg's Law.

Role in XRD Analysis

The diffraction angle is crucial for characterizing materials because it is directly related to the crystallographic planes' orientations and distances. By measuring the angles at which these peaks occur, scientists can resolve the structural details of the crystal, which is helpful in fields ranging from materials science to biology. Each mineral or compound has a characteristic set of diffraction angles that can be used as a 'fingerprint' to identify unknown substances or to learn more about their crystal structure and purity.

Wavelength calculation

The calculation of the wavelength of X-rays (\(\lambda\)) in XRD analysis is a fundamental component of understanding and using Bragg's Law. Bragg's Law is given by the equation: \[n\lambda = 2d\sin{\theta}\], where \(n\) is the order of diffraction, \(\lambda\) is the wavelength, \(d\) is the interatomic distance, and \(\theta\) is the Bragg angle.

Application in Exercise

To calculate the wavelength, as seen in the exercise, each term except for \(\lambda\) is known, allowing for the isolation and calculation of the wavelength. This calculation is pivotal for identifying the energy of the X-rays and for the thorough characterization of the crystal structure. The accuracy of this calculation has direct implications for the outcome of an XRD analysis and subsequent interpretations. Moreover, knowing the X-ray wavelength is essential in various applications like determining bond lengths and angles, and it serves as a basis for further calculations in materials science and chemistry.