The lattice constant of a crystal is 0.5. Calculate the maximum possible Bragg angle (degrees) for X-ray diffraction with a wavelength of 0.15 nm

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.

The lattice constant of a crystal is 0.5. Calculate the maximum possible Bragg angle (degrees) for X-ray diffraction with a wavelength of 0.15 nm.

Learn how to calculate the maximum possible Bragg angle for X-ray diffraction given a lattice constant of 0.5 nm and an X-ray wavelength of 0.15 nm using Bragg's Law. This problem explores concepts in crystallography, trigonometry, and materials science.

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