Understanding Bragg's Law for X-ray Diffraction

Understanding Bragg's Law for X-ray Diffraction

X-ray diffraction is a powerful tool used to study the atomic structure of materials. Central to this technique is Bragg's Law, an equation that relates the wavelength of incident X-rays, the distance between crystal planes, and the angle of incidence to produce constructive interference.

Introduction to Bragg's Law

In 1913, William Lawrence Bragg and his father, William Henry Bragg, formulated Bragg's Law to explain the diffraction patterns observed when X-rays interact with crystalline substances. The law is given by the equation:

nλ = 2d sin θ

where:

• n is the order of reflection (an integer).
• λ (lambda) is the wavelength of the X-rays (measured in meters).
• d is the distance between the planes in the crystal lattice (measured in meters).
• θ (theta) is the angle of incidence (measured in degrees).

How Bragg's Law Works

When X-rays hit a crystal, they are scattered by the atoms within the crystal. If scattered X-rays from different layers of atoms are in phase, they will constructively interfere, leading to a detectable reflected beam. The angle at which this constructive interference occurs is what Bragg’s Law helps to determine.

Consider a real-world example: Imagine holding a finely woven net under a lamp such that you see bright reflective spots. These spots form because light waves reflect off the net's threads and interfere constructively at specific angles. Similarly, in a crystal, the layers of atoms act like the threads of the net, reflecting X-rays at precise angles.

Importance of Each Parameter

• Wavelength (λ): The wavelength of the X-rays determines how they interact with the atomic planes. X-rays typically have wavelengths in the range of 0.01 to 10 nanometers (1 nm = 10^-9 meters).
• Distance Between Planes (d): The distance between the planes in the crystal lattice influences the angle at which constructive interference occurs. This distance, known as the interplanar spacing, is usually on the order of a few angstroms (1 Å = 10^-10 meters).
• Angle of Incidence (θ): The angle at which the X-ray hits the crystal plane, measured relative to the plane. Reflected beams will only be observed at specific angles where Bragg's Law is satisfied.

Visualizing Bragg's Law

Let’s say we have X-rays with a wavelength (λ) of 0.154 nanometers, and we’re examining a crystal with planes spaced 0.5 nanometers apart. We want to find the angle θ for the first-order reflection (n = 1).

Plugging in the values into Bragg's Law:

1 * 0.154 nm = 2 * 0.5 nm * sin θ

Simplifying, we get:

sin θ = 0.154 / (2 * 0.5)

sin θ = 0.154 / 1

sin θ = 0.154

Thus:

θ = sin-1(0.154)

θ ≈ 8.84°

Applying Bragg’s Law

Bragg's Law is pivotal in X-ray crystallography, a technique used to determine the atomic and molecular structure of a crystal. By measuring the angles and intensities of the diffracted beams, scientists can produce a three-dimensional picture of the electron density within the crystal. This method has been fundamental in discovering the structures of many biological molecules, including DNA.

FAQs

Is Bragg's Law only applicable to X-rays?

No, Bragg's Law can be applied to any wave-based phenomena where diffraction occurs, including neutron and electron diffraction. However, it is most commonly associated with X-rays due to their suitable wavelength for studying atomic-scale structures.

Can Bragg's Law be used for all types of crystals?

Yes, Bragg's Law is universally applicable to any crystalline material. However, the clarity and precision of the diffraction pattern may vary depending on the quality and type of the crystal.

Why is the order of reflection (n) important?

The order of reflection indicates that there can be multiple angles that satisfy Bragg's condition for different integers of n. Higher-order reflections occur at larger angles and correspond to the constructive interference of waves that have traveled different path lengths within the crystal.

Example Description

Imagine you are working in a lab with a high-quality NaCl (table salt) crystal. Using an X-ray machine, you measure a first-order reflection angle (θ) of 20° using X-rays with a wavelength (λ) of 0.154 nm. To find the interplanar spacing (d), you can rearrange Bragg’s Law:

d = nλ / (2 sin θ)

Given n = 1, λ = 0.154 nm, and θ = 20°:

d = 0.154 / (2 * sin 20°)

d ≈ 0.154 / 0.684

d ≈ 0.225 nm

Thus, the distance between the planes in the NaCl crystal is approximately 0.225 nanometers.

Conclusion

Bragg's Law holds a fundamental place in the field of materials science and crystallography. By understanding the interplay between X-ray wavelength, crystal plane spacing, and incidence angles, we unravel the complex atomic structures hidden within substances. Whether uncovering the double helix of DNA or exploring new materials, Bragg's Law continues to shed light on the microscopic world in profound ways.

Tags: Physics, X-ray Diffraction, Bragg's Law