Understanding Bragg's Diffraction and Interplanar Spacing

Bragg's diffraction is a phenomenon observed when X-rays interact with the atomic planes of a crystal. It occurs when waves scattered by atoms in adjacent planes interfere constructively. This constructive interference happens only at specific angles, which are determined by the wavelength of the X-rays and the spacing between the atomic planes.

The condition for Bragg's diffraction is described by Bragg's Law.

Bragg's Law and the Diffraction Condition

Bragg's Law is given by the equation:

\(n\lambda = 2d \sin\theta\)

Let's break down the terms in this equation:

• \(\lambda\) is the wavelength of the X-rays.
• \(d\) is the distance between the atomic planes (the interplanar spacing).
• \(\theta\) is the glancing angle or Bragg angle, which is the angle between the incident X-ray beam and the crystal plane.
• \(n\) is an integer representing the order of the diffraction (n = 1, 2, 3, ...). For first-order diffraction, n=1.

Bragg's Law describes the condition for constructive interference, which leads to the observed diffraction peaks in an X-ray diffraction pattern. Constructive interference occurs when the path difference between waves scattered by successive planes is an integer multiple of the wavelength.

Finding the Minimum Interplanar Spacing

We are looking for the minimum interplanar spacing (\(d\)) required for Bragg's diffraction to occur. From Bragg's Law, \(n\lambda = 2d \sin\theta\), we can rearrange it to solve for \(d\):

\(d = \frac{n\lambda}{2 \sin\theta}\)

For a given wavelength \(\lambda\) and diffraction order \(n\), the value of \(d\) depends on the angle \(\sin\theta\).

To find the minimum possible value of \(d\), we need to find the maximum possible value of \(\sin\theta\). The sine function, \(\sin\theta\), has a maximum value of 1. This occurs when \(\theta = 90^\circ\). However, physically, \(\theta\) is the angle between the incident beam and the plane, which is usually considered between 0 and 90 degrees. For diffraction to be possible, there must be some angle \(\theta\) between 0 and 90 degrees for which Bragg's Law is satisfied.

The maximum value of \(\sin\theta\) is 1 (theoretically at \(\theta = 90^\circ\)). If we consider the lowest order of diffraction, which is \(n=1\), this gives the fundamental condition.

Setting \(n=1\) and \(\sin\theta = 1\) in the equation for \(d\):

\(d_{min} = \frac{1 \cdot \lambda}{2 \cdot 1}\)

\(d_{min} = \frac{\lambda}{2}\)

This calculation shows that the minimum interplanar spacing (\(d\)) for which Bragg's diffraction is possible for a given wavelength \(\lambda\) is \(\lambda/2\). If the interplanar spacing is less than \(\lambda/2\), then \(2d\) will be less than \(\lambda\). Even with the maximum possible value of \(\sin\theta = 1\), \(2d \sin\theta\) would be less than \(\lambda\), meaning Bragg's Law ($n\lambda = 2d \sin\theta$) cannot be satisfied for \(n=1\) or any higher integer \(n\). Therefore, no Bragg's diffraction peak would be observed.

Conclusion on Minimum Interplanar Spacing

The minimum interplanar spacing required for Bragg's diffraction is \(\lambda/2\). This is derived directly from Bragg's law by considering the first order of diffraction (\(n=1\)) and the maximum possible value of \(\sin\theta\).

This concept of minimum spacing is crucial in understanding X-ray diffraction patterns and determining crystal structures. For a given wavelength of X-rays, only crystal planes with an interplanar spacing equal to or greater than \(\lambda/2\) will contribute to the Bragg's diffraction pattern.

Looking at the options, the value \(\lambda/2\) matches our derived minimum interplanar spacing required for Bragg's diffraction.