Question

X rays of wavelength 2.63 A were used to analyze a crystal.The angle of first- order diffraction \((n=1 \text { in the Bragg }\) 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 \(4.98Å\), and the angle for the second-order diffraction is \(31.62°\).

Step-by-step solution

Write down the given information

Given information: - Wavelength of X-rays: \(\lambda = 2.63 Å\) - Angle of first-order diffraction: \(\theta_1 = 15.55°\) - First-order diffraction: \(n_1 = 1\)

Apply Bragg's Law to find spacing between crystal planes

Bragg's Law states that \(n \lambda = 2d\sin{\theta}\), where \(n\) is the order of diffraction, \(\lambda\) is the wavelength of X-rays, \(d\) is the spacing between crystal planes, and \(\theta\) is the angle of diffraction. From the given information, we have: \(n_1 = 1\), \(\lambda = 2.63 Å\), and \(\theta_1 = 15.55°\) Substitute the given values in Bragg's Law: \(1 \cdot 2.63 = 2d \cdot \sin{15.55°}\) Now, we will solve for \(d\), the spacing between crystal planes.

Calculate the spacing between crystal planes (d)

To find the spacing between crystal planes, \(d\), rearrange the equation and solve for \(d\): \(d = \frac{1 \cdot 2.63}{2 \cdot \sin{15.55°}}\) After calculating, we get the spacing between the crystal planes: \(d = 4.98Å\)

Determine the angle of second-order diffraction

Now, we need to find the angle for the second-order diffraction (\(\theta_2\)), with \(n_2 = 2\). We will use Bragg's Law for this purpose: \(2 \lambda = 2d\sin{\theta_2}\) To find the angle \(\theta_2\), we will rearrange the equation and substitute the known values.

Calculate the angle for second-order diffraction (\(\theta_2\))

Rearrange the equation to solve for \(\theta_2\): \(\theta_2 = \sin^{-1}{\frac{2 \lambda}{2d}}\) Now, substitute the given values of \(\lambda = 2.63Å\) and \(d = 4.98Å\): \(\theta_2 = \sin^{-1}{\frac{2 \cdot 2.63}{2 \cdot 4.98}}\) After calculating, we get the angle for the second-order diffraction: \(\theta_2 = 31.62°\)

Final result

The spacing between the crystal planes is \(4.98Å\), and the angle for the second-order diffraction is \(31.62°\).

Key concepts

X-ray Diffraction

X-ray diffraction is a powerful tool used to study the structure of crystalline materials. It allows scientists to understand how atoms are arranged in a crystal by observing how X-rays are scattered when they encounter the crystal's surface. This scattering results from the interaction between the X-rays and the electrons in the crystal.

When X-rays hit a crystal, they are reflected in many directions due to the regular arrangement of atoms. By analysing the patterns of these reflections, called diffraction patterns, it is possible to determine the distances between the planes of atoms in the crystal. This is extremely helpful for identifying chemical structures and understanding material properties.

• X-rays are a type of electromagnetic radiation, with wavelengths in the range of 0.01 to 10 nanometers.
• The study of diffraction patterns helps in the determination of unknown substances.

X-ray diffraction is widely used in fields like chemistry, physics, and materials science for crystal analysis.

Wavelength

The wavelength of X-rays is a crucial characteristic that influences their interaction with matter. In Bragg's Law, the wavelength, denoted as \( \lambda \), is a fundamental parameter, representing the distance between successive peaks of the X-ray wave. It determines how X-rays will diffract when they encounter a crystal structure.

Wavelengths for X-rays typically range from 0.01 nm to 10 nm. In the context of X-ray diffraction, having the right wavelength is essential for achieving effective diffraction and obtaining clear diffraction patterns.

• In the case of the exercise, the wavelength used is 2.63 Å.
• The choice of wavelength can impact the resolution and sharpness of the diffraction pattern.

A smaller wavelength can result in more detailed diffraction patterns and is particularly important in the study of atomic and molecular structures.

Crystal Plane Spacing

Crystal plane spacing, represented by \( d \), refers to the distance between parallel planes of atoms within a crystal. It is a key feature in determining the crystal structure and is closely involved in the application of Bragg's Law for X-ray diffraction.

When X-rays are directed at a crystal, the specific angle at which they are diffracted depends on the spacing between these atomic planes. By measuring these diffraction angles and using Bragg's Law, scientists can calculate the crystal plane spacing.

In the exercise, the calculated spacing, \( d \), is 4.98 Å, derived from the equation \( d = \frac{n\lambda}{2\sin{\theta}} \). This parameter provides insight into the physical structure of the crystal and its underlying symmetrical properties.

• The accurate determination of \( d \) is essential for the identification of crystalline substances.
• Crystals with smaller plane spacing will diffract X-rays at larger angles.

Understanding crystal plane spacing is pivotal in material science and engineering for analyzing solid materials.

Diffraction Order

Diffraction order, denoted by \( n \), describes the multiple reflections or peaks observed in diffraction experiments. It relates to the number of wavelengths covering the distance between successive maxima in the diffraction pattern.

In Bragg’s Law, the diffraction order plays a significant role: \( n\lambda = 2d\sin{\theta} \), where \( n \) is the number of wavelengths fitting into the crystal spacing. Each increment in \( n \) indicates a higher order of diffraction, leading to a different angle of diffraction.

In the exercise, two orders of diffraction were considered:

• First-order diffraction with \( n=1 \) resulted in an angle of 15.55°.
• Second-order diffraction with \( n=2 \) resulted in a calculated angle of 31.62°.

The concept of diffraction order helps in understanding the periodicity and range of wave interference within a crystal. This information is fundamental for precise structural analysis of crystalline materials and can influence the design and fabrication of various technological applications.