Answer
The wavelength of the scattered gamma rays is $~~4.86\times 10^{-12}~m$
Work Step by Step
We can find the original wavelength:
$\lambda = \frac{hc}{E}$
$\lambda = \frac{(6.63\times 10^{-34}~J~s)(3.0\times 10^8~m/s)}{(0.511\times 10^6~eV)(1.6\times 10^{-19}~J/eV)}$
$\lambda = 2.43\times 10^{-12}~m$
We can find the wavelength shift $\Delta \lambda$:
$\Delta \lambda = \frac{h}{mc}(1-cos~\phi)$
$\Delta \lambda = \frac{h}{mc}(1-cos~90.0^{\circ})$
$\Delta \lambda = \frac{h}{mc}$
$\Delta \lambda = \frac{6.63\times 10^{-34}~J~s}{(9.109\times 10^{-31}~kg)(3.0\times 10^8~m/s)}$
$\Delta \lambda = 2.43\times 10^{-12}~m$
We can find the wavelength of the scattered gamma rays:
$\lambda_f = \lambda+\Delta \lambda$
$\lambda_f = 2.43\times 10^{-12}~m+2.43\times 10^{-12}~m$
$\lambda_f = 4.86\times 10^{-12}~m$
The wavelength of the scattered gamma rays is $~~4.86\times 10^{-12}~m$
