In Rutherford `alpha`-sattering experiment , the ratio of number of particles scattered at an angle of `180^@` to the number of particles scattered at an angle of `90^@` is `alpha:4`. What is the value of `alpha` ?

Explanation

To solve the problem, we need to find the value of α given the ratio of the number of particles scattered at angles of 180° and 90° in Rutherford's alpha scattering experiment. Let's break it down step by step.

### Step 1: Understand the Given Information
We know that the ratio of the number of particles scattered at an angle of 180° to those scattered at an angle of 90° is given as α:4. This can be expressed mathematically as:
\[ \frac{N(180^\circ)}{N(90^\circ)} = \frac{\alpha}{4} \]

### Step 2: Use the Scattering Formula
In Rutherford's alpha scattering experiment, the number of particles scattered at an angle θ is inversely proportional to \( \sin^4(\theta/2) \). Therefore, we can write:
\[ N(\theta) \propto \frac{1}{\sin^4(\theta/2)} \]
This means:
\[ N(180^\circ) \propto \frac{1}{\sin^4(90^\circ)} \quad \text{and} \quad N(90^\circ) \propto \frac{1}{\sin^4(45^\circ)} \]

### Step 3: Calculate the Values of Sine
Now we need to calculate the sine values:
- \( \sin(90^\circ) = 1 \)
- \( \sin(45^\circ) = \frac{1}{\sqrt{2}} \)

### Step 4: Substitute the Sine Values into the Formula
Substituting these values into our expressions for \( N(180^\circ) \) and \( N(90^\circ) \):
\[ N(180^\circ) \propto \frac{1}{1^4} = 1 \]
\[ N(90^\circ) \propto \frac{1}{\left(\frac{1}{\sqrt{2}}\right)^4} = \frac{1}{\frac{1}{4}} = 4 \]

### Step 5: Set Up the Ratio
Now we can set up the ratio:
\[ \frac{N(180^\circ)}{N(90^\circ)} = \frac{1}{4} \]

### Step 6: Relate to the Given Ratio
From the problem statement, we have:
\[ \frac{N(180^\circ)}{N(90^\circ)} = \frac{\alpha}{4} \]
Setting the two expressions for the ratio equal to each other gives:
\[ \frac{1}{4} = \frac{\alpha}{4} \]

### Step 7: Solve for α
To find α, we can multiply both sides by 4:
\[ 1 = \alpha \]

Thus, the value of α is:
\[ \alpha = 1 \]

### Final Answer
The value of α is **1**.

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