To solve this question, we need to understand the basic properties of atomic structure, particularly the components of an atom and their roles:
Protons: These are positively charged particles found in the nucleus of an atom. The number of protons in the nucleus determines the atomic number, which defines the element. Changing the number of protons in an atom will change the element itself, altering its chemical properties. Hence, adding protons will change the atom’s chemical properties.
Electrons: These are negatively charged particles that orbit the nucleus. While electrons do not alter the identity of the element, they do influence chemical properties such as reactivity and bonding due to their involvement in forming chemical bonds. Therefore, adding electrons can change chemical properties.
Neutrons: These are neutral particles found within the nucleus alongside protons. Neutrons do not affect the chemical properties of an atom directly because they do not contribute to the charge. They mainly contribute to the isotope nature of the element without altering the chemical properties, such as reactions and bonds. Therefore, adding neutrons is the correct answer as they can be added without changing the chemical properties of the atom.
$\alpha$-particles: These consist of 2 protons and 2 neutrons, effectively making them helium nuclei. Introducing $\alpha$-particles to an atom could significantly alter its nuclear structure and potentially change the element, thus affecting chemical properties.
Given this analysis, the only particle that can be added to the nucleus of an atom without changing its chemical properties is neutrons.