For Bragg's diffraction by a crystal to occur, then the X-ray of wavelength `lambda` and interatomic distance d must be

Options

A. `lambda` is greater than 2 d
B. `lambda` equals 2 d
C. `lambda` is smaller than or equal to 2d
D. `lambda` is smaller than 2 d

Answer

To solve the problem regarding Bragg's diffraction by a crystal, we need to analyze the relationship between the wavelength of X-rays (λ) and the interatomic distance (d).

### Step-by-Step Solution:

1. **Understanding Bragg's Law**:
Bragg's law states that for constructive interference to occur when X-rays are diffracted by a crystal, the following condition must be satisfied:
\[ 2d \sin \theta = n \lambda \]
where:
- \(d\) = interatomic distance
- \(\theta\) = angle of incidence
- \(n\) = order of diffraction (an integer)
- \(\lambda\) = wavelength of the X-rays

2. **Rearranging Bragg's Law**:
To express the wavelength \(\lambda\) in terms of \(d\) and \(\theta\), we rearrange the equation:
\[ \lambda = \frac{2d \sin \theta}{n} \]

3. **Considering Maximum Wavelength**:
For the maximum wavelength, we take \(n = 1\):
\[ \lambda = 2d \sin \theta \]

4. **Analyzing the Sine Function**:
The sine function, \(\sin \theta\), has a maximum value of 1 (when \(\theta = 90^\circ\)). Therefore, the maximum possible value of \(\lambda\) occurs when \(\sin \theta = 1\):
\[ \lambda_{\text{max}} = 2d \]

5. **Conclusion**:
From the above analysis, we conclude that for Bragg's diffraction to occur, the wavelength \(\lambda\) must satisfy:
\[ \lambda \leq 2d \]
This means that the wavelength of the X-rays must be less than or equal to twice the interatomic distance.

### Final Answer:
The correct relationship is:
\[ \lambda \leq 2d \]