Answer
Wavelength of scattered radiation will be $\lambda=0.005211nm$
Work Step by Step
Wavelength of incident X-ray $\lambda_{0}=0.0045nm$
suppose wavelength of X-ray after scattering is $\lambda$
Angle of scattering is given as $\theta= 45^{0}$
for scattering by electron $\lambda_{C}=\frac{h}{m_{e}c}=2.43\times10^{-3}nm$
Wavelength shift or Compton shift is given by
$\Delta \lambda= \lambda- \lambda_{0}=\lambda_{C} (1-cos\theta)$
putting the values in above equation we will get
$\lambda- \lambda_{0}=2.43\times10^{-3}nm(1-cos45^{0})$
$\lambda- \lambda_{0}=2.43\times10^{-3}nm(1-0.7071)=0.7117\times10^{-3}nm$
$\lambda- \lambda_{0}=0.0007117nm$
$\lambda=0.0007117nm+\lambda_{0}=0.0007117nm+0.0045nm$
$\lambda=0.005211nm$
