Question

A photon scatters in the backward directionΦ= 180° from a free proton that is initially at rest. What must the wavelength of the incident photon be if it is to undergo a 10.0% change in wavelength as a result of the scattering?

Short answer

The wavelength of the light is$2.65\times {10}^{-14}$ .

Step-by-step solution

Compact Effect

As from compact effect, for particles with mass m, the wavelength of incident photon and the wavelength of the scattered photon scattering angle by the following relation:

$\lambda \text{'}-\lambda =\frac{h}{mc}\left(1-\cos \theta \right)$

Scattering angle and wavelength of light.

The scattering angle of the photon $\varphi =180°$and the mass of proton is$m_p=1.67\times {10}^{-27}kg$ .

For a photon that undergoes a 10.0% change in wavelength as a result of the scattering, we have

$f=\frac{∆\lambda }{\lambda }=\frac{\lambda \text{'}-\lambda }{\lambda }=0.100$

we can get this ratio by dividing equation (1) by ,data-custom-editor="chemistry" $\mathrm{\lambda }$ so we get

$\frac{\lambda \text{'}-\lambda }{\lambda }=\frac{h}{mc\lambda }\left(1-\cos \varphi \right)=f$

Solve for$\lambda$ , we get

$\lambda =\frac{h}{mc\lambda }\left(1-\cos \varphi \right)$

Now, Substitute the values to get the required wavelength of the incident photons

$$\begin{gathered} \lambda =\frac{6.626\times {10}^{-34}J.s}{0.100\left(1.67\times {10}^{-27}kg\right)\left(3.0\times {10}^{8}m/s\right)}\left(1-\cos 180\right) \\ \lambda =2.65\times {10}^{-14} \end{gathered}$$

Hence, the wavelength of the light is$2.65\times {10}^{-14}$ .