To calculate the maximum possible Bragg angle for X-ray diffraction, we use Bragg's Law:

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

Where:

• $n$ is the order of diffraction (we typically take $n = 1$ for the first order),
• $\lambda$ is the wavelength of the X-rays (given as $0.15 \, \text{nm}$),
• $d$ is the interplanar spacing, which for a cubic crystal is related to the lattice constant $a$ by $d = \frac{a}{2}$,
• $\theta$ is the Bragg angle.

We are tasked with finding the maximum Bragg angle, which occurs when $\sin \theta = 1$, or $\theta = 90^\circ$.

Step-by-step calculation:

1. The lattice constant is $a = 0.5 \, \text{nm}$.
2. The interplanar spacing $d = \frac{a}{2} = \frac{0.5}{2} = 0.25 \, \text{nm}$.
3. Bragg's Law simplifies to:

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

For $n = 1$ (first-order diffraction), and solving for $\theta$:

$\sin \theta = \frac{n \lambda}{2d}$

$\sin \theta = \frac{0.15}{2 \times 0.25} = 0.3$

$\theta = \arcsin(0.3)$.

Now let's compute $\theta$.The maximum possible Bragg angle for X-ray diffraction is approximately $17.46^\circ$.

Would you like more details on the steps, or do you have any other questions?

Here are five related questions to consider:

1. How would the Bragg angle change if the wavelength of the X-rays were increased?
2. What effect does increasing the lattice constant have on the Bragg angle?
3. How does the order of diffraction ($n$) influence the Bragg angle?
4. Can the Bragg angle exceed 90 degrees, and if not, why?
5. What happens if the interplanar spacing is much smaller than the wavelength?

Tip: Always ensure the value of $\sin \theta$ is between -1 and 1 when applying Bragg’s law, as this ensures a valid physical solution for the angle.