X-rays of wavelength 22 pm are scattered from a carbon target at an angle of ` 85^(@)` to the incident beam The Compton shift for X-rays is ` (cos 85^(@) = 0.088)`

Options

A. 2.2 pm
B. 1.1 pm
C. 0.55 pm
D. 4.4 pm

Answer

To solve the problem of calculating the Compton shift for X-rays scattered from a carbon target at an angle of 85 degrees, we will follow these steps:

### Step 1: Understand the Compton Shift Formula
The Compton shift (Δλ) is given by the formula:
\[ \Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos \theta) \]
Where:
- \( \Delta \lambda \) is the change in wavelength,
- \( h \) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)),
- \( m_e \) is the mass of the electron (\(9.11 \times 10^{-31} \, \text{kg}\)),
- \( c \) is the speed of light (\(3.00 \times 10^{8} \, \text{m/s}\)),
- \( \theta \) is the scattering angle.

### Step 2: Calculate the Compton Wavelength Shift Constant
First, we calculate the constant \( \frac{h}{m_e c} \):
\[ \frac{h}{m_e c} = \frac{6.626 \times 10^{-34}}{9.11 \times 10^{-31} \times 3.00 \times 10^{8}} \approx 2.43 \times 10^{-12} \, \text{m} \, \text{or} \, 2.43 \, \text{pm} \]

### Step 3: Convert the Angle to Cosine Value
The angle given is \( 85^\circ \). We need to find \( \cos(85^\circ) \):
\[ \cos(85^\circ) \approx 0.088 \]

### Step 4: Substitute Values into the Compton Shift Formula
Now we can substitute the values into the Compton shift formula:
\[ \Delta \lambda = 2.43 \, \text{pm} \times (1 - 0.088) \]
\[ \Delta \lambda = 2.43 \, \text{pm} \times 0.912 \approx 2.216 \, \text{pm} \]

### Step 5: Conclusion
The Compton shift for the X-rays scattered from the carbon target at an angle of \( 85^\circ \) is approximately \( 2.216 \, \text{pm} \).

### Final Answer
\[ \Delta \lambda \approx 2.216 \, \text{pm} \]

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