The beam or power gain of a radio antenna is nothing but directional gain, the gain that varies toward different direction written as $G(\theta,\phi)$. It is normalized to be $1$ at the center of the field of view of an antenna.

So beam can be defined as the direction-dependent gain of a directional antenna with respect to the gain of an isotropic antenna:

$$ G(\theta,\phi) = 10 \log_{10} \frac{G}{G_i} $$

where $G_i$ is the gain of an isotropic antenna. This value is given in units of decibel (dB). By definition, the gain integrated over the surface of a sphere has to be unity:

$$ \int_{sphere} G \ d\Omega = 1 $$

which gives rise to the definition of beam solid angle:

$$ \Omega_A = \frac{4\pi}{G_0} $$

where $G_0$ is the peak gain. For example, the gain pattern or radiation pattern of a short dipole

$$ G = \frac{3}{2} \sin^2\theta = G_0 \sin^2\theta $$

which can be derived from the formula of power for a dipole antenna. Here $G_0=3/2 = 1.5 = 1.76$ dB is the peak gain because this would be the gain at $\theta=90^\circ$.

For a receiving antenna, we use effective area instead of power gain for defining the beam. It is defined by the fact that flux is nothing but power per unit area. So the collecting area of a radio telescope (also called effective area)

$$ A_e = \frac{2P}{S} $$

where the factor $2$ is there because an antenna detects only half of the incident light from an unpolarized source. This can be derived using the following thought experiment.

Two cavities are in thermodynamic equilibrium, one has a resistor, the other an antenna. The antenna is connected to the resistor via a wire, so current of a specific frequency range can pass, but em wave cannot. The power received by the antenna

$$ P_\nu = \frac{1}{2} \int_{4\pi} A_e(\theta,\phi) B_\nu d\Omega $$

which can be calculated using Planck's law and Nyquist formula. Planck’s law gives the radiation from a blackbody, a random mess of particles in thermal equilibrium. And the Nyquist formula gives the radiation from a warm a resistor, one that has been heated. All heated resistors radiate. The power of a warm resistor

$$ P_\nu = kT \frac{h\nu/kT}{e^{h\nu/kT} - 1} $$

and the intensity of a blackbody

$$ B_\nu = \frac{2kT}{\lambda^2} \frac{h\nu/kT}{e^{h\nu/kT} - 1} $$

which leads to

$$ \langle A_e \rangle = \frac{\lambda^2}{4\pi} $$

which means all isotropic lossless antennas have the same collecting area irrespective of their shape. This is the reason why GPS antennas, FM radio and dipole radio telescopes all work at long wavelengths, to increase the collecting area which is proportional to $\lambda^2$.

And the beam solid angle of a lossless isotropic antenna

$$ \Omega_A = \int_{4\pi} \frac{A_e}{A_0} d\Omega \Rightarrow A_0 \Omega_A = \lambda^2 $$

where $A_0$ is the maximum or peak collecting area.