Question

Under what condition, the diffraction from the lattice will not occur :

• A. $\frac{n\lambda}{2d} > 1$
• B. $\frac{n\lambda}{2d} < 1$
• C. $\frac{n\lambda}{2d} = 1$
• D. $\frac{n\lambda}{d} > 3$

Hint:

Because $\sin\theta \le 1$, Bragg diffraction is impossible if $\lambda > 2d$. This is exactly why visible light ($\lambda \sim 500\text{ nm}$) cannot diffract through crystal lattices ($d \sim 0.1\text{ nm}$); you must use X-rays or electrons with much smaller wavelengths!

Answer 1

The Correct Option is A

Step 1: Concept:
This problem assesses the fundamental mathematical limits of X-ray diffraction in crystal lattices, governed by Bragg's Law.

Step 2: Key Formula or Approach:
Bragg's Law establishes the condition for constructive interference (diffraction) from crystalline lattice planes:
\[ n\lambda = 2d \sin \theta \]
Where:
- $n$ is the order of diffraction (integer)
- $\lambda$ is the wavelength of the incident wave
- $d$ is the interplanar spacing of the lattice
- $\theta$ is the scattering angle

Step 3: Step-by-step Explanation:

• To determine when diffraction is physically impossible, we rearrange Bragg's Law to solve for the trigonometric component:
\[ \sin \theta = \frac{n\lambda}{2d} \]

• From basic trigonometry, the value of the sine function for any real angle $\theta$ is strictly bounded between -1 and +1. Therefore, for a physical solution (a valid diffraction angle) to exist, the absolute value must satisfy:
\[ |\sin \theta| \le 1 \]
\[ \frac{n\lambda}{2d} \le 1 \]

• If the term $\frac{n\lambda}{2d}$ exceeds 1, there is no real angle $\theta$ that can satisfy the Bragg condition.

• Consequently, constructive interference cannot happen, and no diffraction peak will be observed.

• This physically implies that if the wavelength $\lambda$ is more than twice the lattice spacing ($2d$) for first-order diffraction, the wave is simply too large to resolve the crystal planes.

Step 4: Final Answer:
Diffraction will not occur when $\frac{n\lambda}{2d} > 1$, which aligns with option (A).