Question

The first-order Bragg reflection \((n=1)\) from a \(\mathrm{NaCl}\) crystal with a spacing of \(282 \mathrm{pm}\) is seen at \(23.0^{\circ} .\) Calculate the wavelength of the \(\mathrm{X}\) -ray radiation used.

Short answer

The wavelength of the X-ray radiation is approximately 220.3 pm.

Step-by-step solution

Identify the Known Values

We have the following known values from the problem:- Order of reflection, \( n = 1 \)- Crystal spacing (lattice spacing), \( d = 282 \text{ pm} \)- Angle of reflection, \( \theta = 23.0^{\circ} \)

Understand the Bragg's Law Formula

Bragg's Law relates the wavelength of X-ray radiation to the angle of reflection and the crystal spacing. The formula is given by:\[ n\lambda = 2d \sin \theta \]where \( \lambda \) is the wavelength, \( n \) is the order of reflection, \( d \) is the lattice spacing, and \( \theta \) is the angle of incidence/reflection.

Plug Values into Bragg's Law

Substitute the known values into Bragg's Law formula:\[ 1 \times \lambda = 2 \times 282 \times 10^{-12} \text{ m} \times \sin(23.0^{\circ}) \]

Calculate Sin of the Angle

Use a calculator to find the sine of \( 23.0^{\circ} \):\[ \sin(23.0^{\circ}) \approx 0.3907 \]

Solve for Wavelength \( \lambda \)

Substitute the sine value back into the equation:\[ \lambda = 2 \times 282 \times 10^{-12} \times 0.3907 \]Calculate the result:\[ \lambda \approx 220.3 \times 10^{-12} \text{ meters} = 220.3 \text{ pm} \]

Confirm Unit Consistency

Ensure that the final unit of the wavelength is in picometers (pm), as the original lattice spacing was given in pm.

Key concepts

X-ray diffraction

X-ray diffraction is a fascinating phenomenon that occurs when X-rays are directed at a crystal structure. The X-rays interact with the crystal lattice, causing the rays to be scattered in various directions. This scattering is not random; instead, it is highly specific and governed by the arrangement of atoms within the crystal. The way X-rays bend around the crystal lattice essentially creates a pattern, similar to a fingerprint, which can be analyzed.

Understanding X-ray diffraction is key in fields such as physics, chemistry, and material science. This process allows scientists to explore the arrangement of atoms in a crystal, thus providing crucial information about the material's internal structure. For example:

• X-ray diffraction helps in determining the physical properties of crystals.
• It aids in the identification of unknown crystalline substances.
• It is instrumental in figuring out how atoms or molecules pack within the crystal structure.

By studying the diffraction pattern, researchers can infer the spacing between planes of atoms in the crystal, which is where Bragg's Law comes into play.

crystal lattice spacing

Crystal lattice spacing refers to the distance between adjacent layers or planes of atoms in a crystal. This spacing is crucial because it influences how the X-rays will be reflected by the crystal. In the given example of NaCl, the lattice spacing is 282 pm.

Think of a crystal lattice like a three-dimensional grid where atoms are positioned at intersections. The spacing between these planes affects numerous properties of materials, such as:

• The mechanical strength of the crystal.
• The electrical conductivity.
• Thermal properties.

The lattice spacing is fundamental to calculating exact values using Bragg's Law, which we touched on earlier. By knowing the crystal lattice spacing, scientists can determine other unknown values such as wavelength of X-rays, as shown in the exercise. This interplay between lattice spacing and diffraction forms the basis of several analytical techniques in material science and crystallography.

first-order reflection

First-order reflection, denoted often as \( n = 1 \), is a specific term in Bragg's Law. It refers to the simplest case where the path difference between successive rays is exactly one wavelength. This is the most straightforward reflection scenario and is frequently used to interpret X-ray diffraction patterns.

In Bragg's Law, the order of reflection \( n \) represents the number of wavelengths by which the path difference exceeds a whole number multiple of the wavelength. Here's what this means for our calculation:

• First-order reflection is typically the strongest and clearest to observe, hence often the most useful.
• For higher orders, the reflection angle becomes greater, which can weaken the reflection intensity.
• In the given exercise, \( n = 1 \) simplifies our task of calculating the wavelength of X-rays.

This concept helps in prioritizing reflections when working with X-ray diffraction data, focusing on the clearest and most precise signals to analyze the crystal structure.