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So how would we explain why the atomic radius of a phosphorus atom be smaller than the atomic radius of a magnesium atom?

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So how would we explain why the atomic radius of a sodium atom smaller than the atomic radius of a cesium atom?

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You might ask why doesn't the radius get smaller as we go from sodium to potassium? Potassium has a greater nuclear charge compared to sodium. You're right potassium does have a greater nuclear charge compared to sodium. The difference is the valence electron in potassium is in a different and HIGHER level compared to the valence electron in sodium. So the valence electron in potassium is further away and can not feel the higher nuclear charge. When going across a period each added electron goes into the SAME level. Since the nuclear charge is also increasing the valence electron feel a greater attraction to the nucleus and the atom's size gets smaller.