To determine the longest wavelength that can be analyzed by a NaCl crystal, we utilize Bragg's Law, a fundamental principle describing X-ray diffraction in crystalline solids. This law establishes the condition for constructive interference of X-rays as they interact with the crystal's atomic planes.

Bragg's Law Foundation

Bragg's Law is mathematically expressed as:

$$n\lambda = 2d\sin\theta$$

Where:

• $\lambda$ represents the wavelength of the incident X-rays.
• $d$ is the interplanar spacing between the parallel atomic planes within the crystal.
• $\theta$ is the glancing angle of incidence, which is the angle between the incident X-ray beam and the crystal planes.
• $n$ denotes the order of diffraction (an integer, typically $n=1$ for first order, $n=2$ for second order, and so on).

NaCl Crystal Parameters

The question provides specific details about the NaCl crystal and the diffraction conditions:

• The interplanar spacing ($d$) of the principal planes is given as 0.281 nm. This value represents the distance between consecutive planes of atoms that are diffracting the X-rays.
• The diffraction is considered in the first order, meaning the order of diffraction ($n$) is 1. This is the simplest and often strongest diffraction observed.

Our objective is to find the maximum possible wavelength ($\lambda_{max}$) that can satisfy these conditions for the NaCl crystal.

Wavelength Maximization Principle

To obtain the longest wavelength ($\lambda_{max}$) that can be analyzed using Bragg's Law, the value of $\sin\theta$ must be maximized. The maximum possible value for the sine function is 1, which occurs when $\theta = 90^\circ$. While a physical angle of $90^\circ$ for X-ray incidence is not practically achievable in a typical diffraction setup (as it implies the beam is parallel to the planes), setting $\sin\theta = 1$ provides the theoretical upper limit for the wavelength that can undergo diffraction according to Bragg's Law.

Therefore, for $\lambda_{max}$, we use $\sin\theta = 1$.

Wavelength Calculation Steps

Substitute $n=1$ and $\sin\theta=1$ into the Bragg's Law equation:

$$1 \cdot \lambda_{max} = 2d \cdot (1)$$

This simplifies to:

$$\lambda_{max} = 2d$$

Now, we substitute the given interplanar spacing $d = 0.281 \text{ nm}$:

$$\lambda_{max} = 2 \times 0.281 \text{ nm}$$

$$\lambda_{max} = 0.562 \text{ nm}$$

Wavelength Results Comparison

The calculated longest wavelength that can be analyzed is 0.562 nm. Let's compare this calculated value with the given options:

The calculated value of 0.562 nm is very close to Option 2, which is 0.564 nm. The small difference is likely due to rounding in the provided options. Therefore, 0.564 nm is the most accurate answer among the choices, representing the longest wavelength that can be analyzed by the NaCl crystal under first order diffraction conditions.