Antenna directivity ($D(\Omega )=4\pi U(\Omega )/P_{\mathrm{rad}}$)

The antenna directivity compares angular radiated power $U=d{P}_{\mathrm{rad}}/d\Omega$ with its average over directions. Its integral is $4\pi$. Unlike antenna gain, it uses radiated power rather than accepted input power, so it omits dissipative losses. A narrow beam can have high directivity without a larger total radiated power.