Lanthanoid contraction is caused due to: (a) The appreciable shielding of outer electrons by 4felectrons from the nuclear charge (b) The appreciable shielding on outer electrons by 5 delectrons from the nuclear charge (c) The same effective nuclear charge from \(\mathrm{Ce}\) to \(\mathrm{Lu}\) (d) The imperfect shielding of outer electrons by \(4 \mathrm{f}\) electrons from the nuclear charge

The shielding effect is crucial in understanding why atomic and ionic sizes tend to vary across different elements. At its core, the shielding effect describes how inner-shell electrons act as a 'shield' for outer-shell electrons against the attractive force of the nucleus. More specifically, these inner electrons repel the ones in the outer shell, reducing the effective nuclear charge that the outer electrons experience. In the context of the lanthanoids, the 4f electrons are relatively poor at shielding.

This inefficiency arises because the 4f orbitals are deeply buried within the atom and have a more complex shape, leading to a weaker shielding impact on the outer electrons. Consequently, as the proton number increases across the lanthanoid series, adding more positive charge in the nucleus, the outer electrons do not experience a correspondingly increased repulsion to balance this. This results in the outer electrons being drawn more tightly towards the nucleus, decreasing the size of the atom or ion. The imperfect shielding by 4f electrons is why lanthanoid atoms become smaller, even as more electrons are added to the 4f subshell with each successive element.

The atomic radius is the distance from the nucleus of an atom to the outermost stable electron orbital in a neutrally charged atom, while the ionic radius is the distance in a charged ion. For instance, as you move through the lanthanoid series from cerium (Ce) to lutetium (Lu), the atomic and ionic radii decrease.

This decrement, known as lanthanoid contraction, primarily results from the increased nuclear charge without adequately compensated shielding by inner electrons. Because the new electrons are entering the 4f orbitals, which are inefficient at shielding, the outermost electrons are pulled closer to the nucleus by the increasing nuclear charge. This results in a smaller radius for both atoms and ions incrementally. As a tangible example, the ionic radius of a lanthanum ion is larger than that of a gadolinium ion, even though both have a similar electron structure - the difference in their ionic radii being a direct result of lanthanoid contraction.

The 4f orbitals are the inner orbitals of the lanthanoids, starting from cerium (Ce) to lutetium (Lu). These orbitals have a more complex shape compared to the s or p orbitals, which means that they have a larger number of lobes and nodes. The electrons in these orbitals are closer to the nucleus and, as such, are impacted significantly by the positive charge of the nucleus.

However, they are not particularly effective at providing shielding for the valence electrons from the nuclear charge. This is because the spatial orientation and characteristics of 4f orbitals limit their ability to overlap with the outer orbits, preventing them from exerting a stronger repulsive force on the valence electrons. Their inability to offer effective shielding is directly related to the aforementioned lanthanoid contraction. Since each added electron enters a 4f orbital, the increasing number of protons in the nucleus does not lead to a proportional increase in size, due to the limited shielding these 4f electrons provide.