The reason behind the lower atomic radius of \(\mathrm{Ga}\) as compared to \(\mathrm{Al}\) is (a) poor screening effect of \(d\)-electrons for the outer electrons from increased nuclear charge (b) increased force of attraction of increased nuclear charge on electrons (c) increased ionisation enthalpy of \(\mathrm{Ga}\) as compared to \(\mathrm{Al}\) (d) anomalous behaviour of Ga.

In the world of atomic physics, the screening effect, or shielding effect, plays a critical role in understanding the forces at play within an atom. Fundamentally, it describes how the inner electrons of an atom provide a shield for outer electrons against the pull of the nucleus. This results in the outer electrons experiencing a reduced effective nuclear charge–in other words, the net positive charge they 'feel' is less than the actual charge of the nucleus.

Imagine the nucleus of an atom as a strong magnet and the inner electrons as layers of material wrapped around it; these layers weaken the magnet's attraction on objects further away. In elements with a poor screening effect, the 'magnet's pull' reaches further, effectively reducing the atomic radius as the outer electrons are drawn more closely to the nucleus.

However, not all electrons contribute equally to this shielding. Electrons situated in inner shells, especially those in the s and p orbitals, are more effective at blocking the nuclear pull. When it comes to the exercise at hand, understanding the screening effect is crucial for comprehending why gallium (Ga) has a smaller atomic radius compared to aluminium (Al), despite both being in the same group of the periodic table.

Transitioning to the concept of d-electrons, these are electrons that occupy the d subshell of an atom, which begins to fill from the fourth period onwards in the periodic table. Gallium (Ga) falls into this category as it lies in the fourth period and therefore possesses d-electrons. These electrons are particularly important when analyzing atomic size and the shielding effect.

What sets d-electrons apart is their lower ability to shield compared to s or p electrons. This is due to their higher energy and less penetration towards the nucleus which allows them to be less effective in blocking the nuclear charge. Therefore, as Ga has d-electrons and Al does not, Ga experiences a greater effective nuclear charge on its valence electrons, leading to a decreased atomic radius—a concept highlighted in the original exercise.

This can lead to counterintuitive trends across a period, where one might expect the atomic radius to increase with more electrons. However, due to the poor shielding of d-electrons, the addition of these electrons can often lead to a reduction in size, as is the case when comparing Ga with Al.

Finally, exploring the grander scheme of things within the periodic table brings us to periodic trends. These are patterns that emerge when we examine the chemical and physical characteristics of elements as we move across a period or down a group. These trends are incredibly useful for predicting properties of elements and understanding their behaviors.

As you move from left to right across a period, the atomic number increases, which typically results in a greater pull from the nucleus as there are more protons. Additionally, down a group, each element has an additional shell of electrons, generally leading to an increase in atomic radius. However, anomalies arise, such as the one observed between Al and Ga, due to the screening effect and presence of d-electrons.

Understanding these trends aids students in pinpointing the reasons behind the varying atomic radii in elements like Ga and Al. It empowers them to predict and rationalize the physical and chemical nature of other elements they encounter in their studies, particularly when engaging with challenging concepts such as the poor screening effect by d-electrons.