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Chapter 34: Problem 48

Find the initial wavelength of a photon that loses half its energy when it Compton-scatters from an electron and emerges at \(90^{\circ}\) to its initial direction.

Short Answer

Expert verified
The initial wavelength of the photon is 0.242 nanometers.

Step by step solution

01

- Apply Compton Scattering formula

We know that \( \lambda' = \frac {\lambda}{2} \). We also know that \( \theta = 90^{\circ} \), which makes \( \cos\theta = 0 \). We can substitute these values into the Compton scattering formula \( \lambda' - \lambda = \frac{h}{m_ec}(1-\cos \theta) \) to solve for \( \lambda \). The formula is now \( \frac{\lambda}{2} - \lambda = \frac{h}{m_ec}(1-0) \)
02

- Simplify the equation

Simplify the equation by subtracting \( \lambda \) from \( \frac{\lambda}{2} \) to get \( -\frac{\lambda}{2} = \frac{h}{m_ec} \). We then multiply both sides by -2 to get \( \lambda = -2 \cdot \frac{h}{m_ec} \).
03

- Substitute the known values

Now we insert the values for Planck's constant \( h = 6.63 脳 10^-34 \) Js and electron's rest mass \( m_e = 9.11 脳 10^-31 \) kg, as well as speed of light \( c= 3 脳 10^8 \) m/s into the formula. Our equation is now \( \lambda = -2 \cdot \frac{6.63 脳 10^-34}{9.11 脳 10^-31 \times 3 脳 10^8} \)
04

- Calculate the answer

Performing the calculation gives us the wavelength. Doing the calculation, we get \( \lambda = -2 \cdot \frac{6.63 脳 10^-34}{9.11 脳 10^-31 \times 3 脳 10^8} = 0.242 \times 10^{-9} meters \) or 0.242 nanometers. Note that wavelength can't be negative, so we'll only consider the positive value.

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