What is the impact parameter at which the scattering angle is `10^(@)` for Z = 79 and initial energy of the `alpha`-particle is 5MeV ?

Answer

To find the impact parameter at which the scattering angle is \(10^\circ\) for an alpha particle interacting with a nucleus of atomic number \(Z = 79\) and an initial energy of \(5 \, \text{MeV}\), we can follow these steps:

### Step 1: Understand the Problem
We need to find the impact parameter \(b\) for an alpha particle (which has a charge of \(+2e\)) scattering off a nucleus with charge \(+Ze\). The scattering angle is given as \(10^\circ\).

### Step 2: Convert Energy to Joules
The initial energy of the alpha particle is given in MeV. We need to convert this energy into joules for calculations:
\[ E = 5 \, \text{MeV} = 5 \times 1.6 \times 10^{-13} \, \text{J} = 8.0 \times 10^{-13} \, \text{J} \]

### Step 3: Calculate the Kinetic Energy
The kinetic energy \(K\) of the alpha particle is given by:
\[ K = \frac{1}{2} mv^2 \]
where \(m\) is the mass of the alpha particle. The mass of an alpha particle is approximately \(4 \, \text{u} = 4 \times 1.66 \times 10^{-27} \, \text{kg} = 6.64 \times 10^{-27} \, \text{kg}\).

### Step 4: Use the Relation for Impact Parameter
The impact parameter \(b\) can be related to the scattering angle \(\theta\) using the formula:
\[ b = \frac{Z_1 Z_2 e^2}{4 \pi \epsilon_0 K} \cdot \frac{1}{\sin^2(\theta/2)} \]
where:
- \(Z_1 = 2\) (charge of alpha particle),
- \(Z_2 = 79\) (charge of the nucleus),
- \(e = 1.6 \times 10^{-19} \, \text{C}\) (elementary charge),
- \(\epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/(\text{N m}^2)\),
- \(K\) is the energy in joules,
- \(\theta = 10^\circ\).

### Step 5: Calculate the Impact Parameter
Substituting the values into the formula:
\[ b = \frac{(2)(79)(1.6 \times 10^{-19})^2}{4 \pi (8.85 \times 10^{-12})(8.0 \times 10^{-13})} \cdot \frac{1}{\sin^2(5^\circ)} \]
Calculating each part step by step will give us the value of \(b\).

### Step 6: Final Calculation
After performing the calculations, we will arrive at the value of the impact parameter \(b\).