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

Options

A. `0.575Å`
B. `1.00Å`
C. `2.00Å`
D. `1.15Å`

Explanation

To find the distance between the scattering planes in the crystal using Bragg's law, we can 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 values
From the problem statement, we have:
- \( \lambda = 1 \) Å (1 Angstrom = \( 1 \times 10^{-10} \) m),
- \( \theta = 60^\circ \),
- \( n = 2 \) (since it is second order diffraction).

### Step 3: Substitute the values into Bragg's Law
Substituting the known values into the equation:
\[ 2 \times 1 \text{ Å} = 2d \sin(60^\circ) \]

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

### Step 5: Substitute \( \sin(60^\circ) \) into the equation
Now substituting \( \sin(60^\circ) \) into the equation:
\[ 2 \text{ Å} = 2d \left(\frac{\sqrt{3}}{2}\right) \]

### Step 6: Simplify the equation
We can simplify the equation:
\[ 2 \text{ Å} = d \sqrt{3} \]

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

### Step 8: Calculate the numerical value of \( d \)
Calculating \( d \):
\[ d \approx \frac{2}{1.732} \approx 1.1547 \text{ Å} \]

### Step 9: Round off the result
Rounding off the result, we get:
\[ d \approx 1.15 \text{ Å} \]

### Final Answer
The distance between the scattering planes in the crystal is approximately \( 1.15 \) Å.

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