The nucleus of a Uranium atom has 92 protons. (This can be seen from the periodic table, as the atomic number of the element Uranium is 92.) The Uranium-238 isotope has 146 neutrons in the nucleus as well: 238 - 92 = 146.
An alpha-particle, by definition, is a nucleus of a Helium atom. It consists of two neutrons and two protons. Therefore, when Uranium-238 is bombarded with an alpha-particle, a Uranium nucleus might capture the two protons, thus turning a Uranium nucleus into a Plutonium nucleus, because the atomic number of Plutonium is 94. The mass number of the resultant nucleus is 94 + 146 = 240. For Plutonium-239 to be produced, one of the neutrons must separate from the nucleus. This neutron, together with two additional neutrons from the alpha-particle, will become the by-product.
So, three neutrons will be produced as a by-product.
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