Question

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 crystal planes is approximately \(6.11 Å\), and the angle for second-order diffraction is approximately \(25.81°\).

Step-by-step solution

Compute the value of d

From the equation above, we use a calculator to compute the value of d: \(d ≈ 6.11 Å\) So, the spacing between crystal planes is approximately 6.11 Å. #Step 2: Find the angle for second-order diffraction (n=2)# In this step, we will use the Bragg's law equation and the spacing value obtained in the previous step to find the angle for the second-order diffraction, where \(n = 2\). The new equation becomes: \(2\cdot 2.63 Å = 2(6.11 Å)\sin{\theta_2}\) Now we will solve for \(\theta_2\): \(\sin{\theta_2} = \frac{2\cdot 2.63 Å}{2(6.11 Å)}\)

Compute the value of \(\theta_2\)

From the equation above, we use a calculator to compute the value of sin(\(\theta_2\)): \(\sin{\theta_2} ≈ 0.4314\) Now, use sin.backwardfunction (inverse sin, arcsin) to find the value of \(\theta_2\): \(\theta_2 ≈ \arcsin(0.4314)\) \(\theta_2 ≈ 25.81°\) So, the angle for second-order diffraction is approximately 25.81 degrees.

Key concepts

X-ray Diffraction

X-ray diffraction is a powerful technique used to study the arrangement of atoms within a crystal. When X-rays, which are a form of electromagnetic radiation with very short wavelengths, hit a crystalline material, they can be scattered by the atoms in the crystal lattice. This scattering can produce a diffraction pattern, which helps scientists understand the structure of the crystal. The diffraction pattern is created because the X-rays are reflected from different crystal planes within the material. As these reflections combine, they create a pattern of bright and dark spots, much like the patterns seen when light passes through a prism.

• This technique is crucial in fields like materials science and chemistry, as it helps in determining the arrangement and spacing of atoms in crystals.
• Bragg's Law is essential in interpreting the diffraction patterns, allowing us to calculate crystal plane spacing and the angles at which different orders of diffraction occur.

Crystal Plane Spacing

Crystal plane spacing refers to the distance between two adjacent parallel planes of atoms in a crystalline structure. This spacing is crucial for understanding the internal structure of a material, as it determines the angles at which diffraction occurs. In Bragg's Law, denoted as \(d\), this spacing can be calculated using the formula:\[ n\lambda = 2d\sin{\theta} \]where:

• \(n\) is the order of diffraction
• \(\lambda\) is the wavelength of the X-rays used
• \(\theta\) is the angle of diffraction

Knowing \(d\) provides insights into the size and shape of the unit cell in the crystal, making it a fundamental parameter in crystallography.

First-order Diffraction

First-order diffraction occurs when the path difference between the reflected X-rays from consecutive crystal planes results in constructive interference for the first time. In Bragg's Law, this is when \(n=1\). At this order, the path difference is equal to one wavelength of the X-rays used:

• This means that the reflection conditions are satisfied for the smallest angle, which is the first angle where a peak in intensity is observed.
• This is the simplest case of diffraction and often the easiest to observe and measure.

For example, with X-rays of a given wavelength, this first-order angle can help calculate the crystal plane spacing, providing a baseline measurement for further investigations.

Second-order Diffraction

Second-order diffraction involves the next level, where \(n=2\), providing another peak of intensity beyond the first-order diffraction. This takes place when the path difference equals two wavelengths.

• The calculation of this angle uses the same Bragg's Law formula, but with \(n=2\), indicating a higher diffraction angle.
• The angle for second-order diffraction is always larger than the first-order, as the wave travels a longer path to meet the diffraction condition.

In crystallographic studies, second-order diffraction is significant because it offers additional information about the crystal structure, such as finer details of atomic spacing and arrangement. It serves as a cross-check to ensure accuracy in measuring the first-order results, confirming the consistency of measurements through multiple orders of diffraction.