Linear Separation Between Minima in Diffraction Pattern

To find the linear separation between the $1^{\text{st}}$ and $3^{\text{rd}}$ minima in a single-slit diffraction pattern, we use the condition for diffraction minima.

In a single-slit diffraction of slit width $a$ illuminated by light of wavelength $\lambda$, the angular position $\theta_n$ of the $n^{\text{th}}$ minimum is given by: $a \sin \theta_n = n \lambda \quad \text{for } n = 1, 2, 3, \dots$

For small angular deviations ($\sin \theta_n \approx \tan \theta_n \approx \frac{y_n}{D}$), where $y_n$ is the linear distance of the $n^{\text{th}}$ minimum from the central maximum on a screen placed at distance $D$, the position is expressed as: $y_n = \frac{n D \lambda}{a}$

1. The linear distance of the $1^{\text{st}}$ minimum ($n = 1$) from the central maximum is: $y_1 = \frac{D \lambda}{a}$

2. The linear distance of the $3^{\text{rd}}$ minimum ($n = 3$) from the central maximum is: $y_3 = \frac{3 D \lambda}{a}$

3. The linear separation $\Delta y$ between the $1^{\text{st}}$ and $3^{\text{rd}}$ minima on the same side of the central maximum is: $\Delta y = y_3 - y_1 = \frac{3 D \lambda}{a} - \frac{D \lambda}{a} = 2 \frac{D \lambda}{a}$

Hence, the correct option is C.