why should it take significantly more energy to move a beam of alpha particles than a beam of beta minus particles

An 'alpha particle' is the same thing as the nucleus of a helium atom ... a little bundle made of 2 protons and 2 neutrons.A 'beta' particle is an electron.The mass of an alpha particle is more than 7,000 times the mass of an electron, so it certainly takes more energy to get it moving.