Bragg's Law and X-ray Diffraction Conditions

Bragg's reflection is a fundamental principle in X-ray diffraction, which is used to study the atomic and molecular structure of crystals. This phenomenon occurs when X-rays, with a specific wavelength, interact with the regular arrangement of atoms within a crystal lattice, leading to constructive interference of the scattered waves. The condition for Bragg's reflection to occur is described by Bragg's Law.

Understanding Bragg's Law for Crystal Reflection

Bragg's Law is mathematically expressed as:

\[ 2d \sin\theta = n\lambda \]

Where:

• \( d \) is the interplanar spacing (the distance between adjacent parallel planes of atoms in the crystal).
• \( \theta \) is the Bragg angle (the angle between the incident X-ray beam and the crystal planes).
• \( n \) is an integer representing the order of diffraction (e.g., \( n=1 \) for first-order reflection, \( n=2 \) for second-order, and so on).
• \( \lambda \) is the wavelength of the incident X-ray beam.

Condition for X-ray Diffraction Not to Occur

To determine when Bragg's reflection CANNOT occur, we need to analyze the variables in Bragg's Law, specifically the sine function. The value of \( \sin\theta \) must be physically possible. We know that for any real angle \( \theta \), the sine function has a range:

\[ 0 \le \sin\theta \le 1 \]

From Bragg's Law, we can rearrange the equation to solve for \( \sin\theta \):

\[ \sin\theta = \frac{n\lambda}{2d} \]

For Bragg's reflection to occur, the condition \( \sin\theta \le 1 \) must be satisfied. Therefore:

\[ \frac{n\lambda}{2d} \le 1 \]

This implies:

\[ n\lambda \le 2d \]

Or, for any given order \( n \):

\[ \lambda \le \frac{2d}{n} \]

The maximum possible wavelength for which Bragg's reflection can occur corresponds to the first order of diffraction, where \( n=1 \). In this case, the condition becomes:

\[ \lambda \le \frac{2d}{1} \]

\[ \lambda \le 2d \]

If the wavelength \( \lambda \) is greater than \( 2d \), then \( \sin\theta \) would need to be greater than 1 for even the first order of diffraction (\( n=1 \)). For example, if \( \lambda > 2d \), then \( \frac{\lambda}{2d} > 1 \). Since \( \sin\theta = \frac{n\lambda}{2d} \), for \( n=1 \), \( \sin\theta = \frac{\lambda}{2d} \). If \( \frac{\lambda}{2d} > 1 \), then \( \sin\theta > 1 \), which is physically impossible.

Therefore, Bragg's reflection CANNOT occur if \( \lambda > 2d \).

Summary of Bragg Reflection Conditions

• For reflection to occur: The wavelength \( \lambda \) must be less than or equal to \( 2d \), i.e., \( \lambda \le 2d \).
• For reflection NOT to occur: The wavelength \( \lambda \) must be greater than \( 2d \), i.e., \( \lambda > 2d \).

Considering the given options, the condition under which Bragg's reflection by a crystal in an X-ray beam CANNOT occur is when \( \lambda > 2d \).