The second order Bragg diffraction of X-rays with `lambda=1.0 Å` from a set of parallel planes in a metal occurs at an angle `60^(@)`. The distance between the scattering planes in the crystals is

Options

A. `0.575Å`
B. `1.00Å`
C. `2.00Å`
D. `1.17Å`

Explanation

To solve the problem of finding the distance between the scattering planes in a crystal using Bragg's law, we will follow these steps:

### Step 1: Write down Bragg's Law
Bragg's law is given by the equation:
\[ n\lambda = 2d \sin \theta \]
where:
- \( n \) is the order of diffraction,
- \( \lambda \) is the wavelength of the X-rays,
- \( d \) is the distance between the scattering planes,
- \( \theta \) is the angle of diffraction.

### Step 2: Identify the given values
From the problem statement, we have:
- Wavelength (\( \lambda \)) = 1.0 Å (or \( 1 \times 10^{-10} \) m),
- Angle (\( \theta \)) = 60°,
- Order of diffraction (\( n \)) = 2.

### Step 3: Substitute the values into Bragg's Law
Substituting the known values into the equation:
\[ 2 \cdot 1.0 = 2d \sin(60°) \]

### Step 4: Calculate \( \sin(60°) \)
We know that:
\[ \sin(60°) = \frac{\sqrt{3}}{2} \]

### Step 5: Substitute \( \sin(60°) \) into the equation
Now substituting \( \sin(60°) \):
\[ 2 \cdot 1.0 = 2d \cdot \frac{\sqrt{3}}{2} \]

### Step 6: Simplify the equation
The equation simplifies to:
\[ 2 = d \cdot \sqrt{3} \]

### Step 7: Solve for \( d \)
Now, we can solve for \( d \):
\[ d = \frac{2}{\sqrt{3}} \]

### Step 8: Calculate the numerical value of \( d \)
To find the numerical value, we can approximate \( \sqrt{3} \approx 1.732 \):
\[ d \approx \frac{2}{1.732} \approx 1.1547 \text{ Å} \]

### Step 9: Round the answer
Rounding to two decimal places, we get:
\[ d \approx 1.15 \text{ Å} \]

### Conclusion
The distance between the scattering planes in the crystal is approximately \( 1.15 \text{ Å} \).

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