In the Auger process as atom makes a transition to a lower state without emitting a photon. The excess energy is transferred to an outer electron which may be ejected by the atom (this is called an Auger electron). Assuming the nucleus to be massive, calculate the kinetic energy of an `n=4` Auger electron emitted by chromium by absorbing the energy from a `n=2` to `n=1` transition .

Options

A. 4.6
B. 7.5
C. 5.38
D. 3.36

Explanation

To calculate the kinetic energy of an Auger electron emitted by chromium during a transition from \( n=2 \) to \( n=1 \), we will follow these steps:

### Step 1: Understand the Energy Levels
The energy of an electron in a hydrogen-like atom is given by the formula:
\[ E_n = -\frac{Z^2 R}{n^2} \]
where:
- \( Z \) is the atomic number (for chromium, \( Z = 24 \)),
- \( R \) is the Rydberg constant (\( R \approx 13.6 \, \text{eV} \)),
- \( n \) is the principal quantum number.

### Step 2: Calculate the Energy for \( n=1 \) and \( n=2 \)
First, we calculate the energy for \( n=1 \):
\[ E_1 = -\frac{Z^2 R}{1^2} = -\frac{24^2 \cdot 13.6}{1} = -576 \, \text{eV} \]

Next, we calculate the energy for \( n=2 \):
\[ E_2 = -\frac{Z^2 R}{2^2} = -\frac{24^2 \cdot 13.6}{4} = -144 \, \text{eV} \]

### Step 3: Calculate the Energy Difference
The energy difference between the two states (transition from \( n=2 \) to \( n=1 \)) is:
\[ \Delta E = E_1 - E_2 = (-576) - (-144) = -576 + 144 = -432 \, \text{eV} \]

### Step 4: Calculate the Energy for \( n=4 \)
Now, we calculate the energy for \( n=4 \):
\[ E_4 = -\frac{Z^2 R}{4^2} = -\frac{24^2 \cdot 13.6}{16} = -36 \, \text{eV} \]

### Step 5: Calculate the Kinetic Energy of the Auger Electron
The kinetic energy \( KE \) of the Auger electron is given by the energy difference between the transition energy and the energy of the electron in the \( n=4 \) state:
\[ KE = \Delta E - E_4 = (-432) - (-36) = -432 + 36 = -396 \, \text{eV} \]

### Step 6: Final Calculation
Thus, the kinetic energy of the emitted Auger electron is:
\[ KE = 396 \, \text{eV} \]

### Summary
The kinetic energy of the Auger electron emitted by chromium during the transition from \( n=2 \) to \( n=1 \) is \( 396 \, \text{eV} \).

---