By definition, directivity is the ability of an antenna to focus the radiation to a particular direction, whereas, the gain is the ability of the antenna to convert the input power into radio waves in a particular direction. Gain is the combination of the directivity and the electrical efficiency of the antenna where, electrical efficiency of an antenna takes into account the matching between the feed line and antenna as well as the internal losses in the antenna. Hence gain is always less than the directivity because most of the antennas have some internal losses. Also, note that gain is practical and measurable term.
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This is accomplished by the Half Power Beam Width (HPBM) specification which indicates the broadness or the narrowness of the antenna radiation pattern. Further, this HPBM is divided into horizontal and vertical beamwidth to cover azimuthal and elevation planes, respectively.
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