A photon of wavelength `lamda` is scattered from an electron, which was at rest. The wavelength shift `Delta lamda` is three times of `lamda` and the angle of scattering `theta` is `60^@` . The angle at which the electron recoiled is `phi` . The value of tan `phi` is: (electron speed is much smaller than the speed of light)

Options

A. 0.16
B. 0.22
C. 0.25
D. 0.28

Explanation

To solve the problem, we will use the principles of conservation of momentum and the Compton scattering formula. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a photon of wavelength \( \lambda \) scattering off a stationary electron. The wavelength shift \( \Delta \lambda \) is given as \( 3\lambda \), and the scattering angle \( \theta \) is \( 60^\circ \). We need to find \( \tan \phi \), where \( \phi \) is the angle at which the electron recoils.

### Step 2: Use the Compton Wavelength Shift Equation
The change in wavelength due to Compton scattering is given by:
\[ \Delta \lambda = \lambda_f - \lambda_i = \frac{h}{m_e c}(1 - \cos \theta) \]
Given that \( \Delta \lambda = 3\lambda \), we can express the final wavelength as:
\[ \lambda_f = \lambda_i + 3\lambda_i = 4\lambda_i \]

### Step 3: Write the Conservation of Momentum Equations
For the conservation of momentum in the x-direction:
\[ \frac{h}{\lambda_i} = p_f \cos \phi + \frac{h}{\lambda_f} \cos \theta \]
For the conservation of momentum in the y-direction:
\[ p_f \sin \phi = \frac{h}{\lambda_f} \sin \theta \]

### Step 4: Substitute Known Values
Substituting \( \lambda_f = 4\lambda_i \) and \( \theta = 60^\circ \) into the equations:
1. For the x-direction:
\[ \frac{h}{\lambda_i} = p_f \cos \phi + \frac{h}{4\lambda_i} \cdot \frac{1}{2} \]
Simplifying gives:
\[ \frac{h}{\lambda_i} = p_f \cos \phi + \frac{h}{8\lambda_i} \]
Rearranging gives:
\[ p_f \cos \phi = \frac{h}{\lambda_i} - \frac{h}{8\lambda_i} = \frac{7h}{8\lambda_i} \]

2. For the y-direction:
\[ p_f \sin \phi = \frac{h}{4\lambda_i} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}h}{8\lambda_i} \]

### Step 5: Divide the Two Equations
Now we can find \( \tan \phi \) by dividing the y-direction equation by the x-direction equation:
\[ \tan \phi = \frac{p_f \sin \phi}{p_f \cos \phi} = \frac{\frac{\sqrt{3}h}{8\lambda_i}}{\frac{7h}{8\lambda_i}} = \frac{\sqrt{3}}{7} \]

### Step 6: Calculate \( \tan \phi \)
Using the value of \( \sqrt{3} \approx 1.732 \):
\[ \tan \phi = \frac{1.732}{7} \approx 0.246 \]

### Step 7: Final Answer
The closest option to \( 0.246 \) is \( 0.25 \). Therefore, the value of \( \tan \phi \) is:
\[ \boxed{0.25} \]