Atoms are composed of a heavy nucleus surrounded by light electrons. The behaviour of the electrons is governed by the rules of quantum mechanics. Those rules allow electrons to occupy specific regions called orbitals. The interactions of atoms are almost exclusively through their outermost electrons, so the shape of those orbitals becomes very important. For example, when atoms are brought next to each other, if their outermost orbitals overlap then they create a strong chemical bond; so some knowledge of the shape of the orbitals is important for understanding atomic interactions.
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Quantum Numbers and Orbitals
Physicists have found it convenient to use shorthand to describe the characteristics of electrons in an atom. The shorthand is in terms of quantum numbers; these numbers can only be whole numbers, not fractions. The principal quantum number, n, is related to the energy of the electron; then there's the orbital quantum number, l, and the angular momentum quantum number, m. There are other quantum numbers, but they aren't directly related to the shape of the orbitals. Orbitals are not orbits, in the sense of being paths around the nucleus; instead, they represent the positions where the electron is most likely to be found.
Lowest Order Orbital
For each value of n, there is one orbital where both l and m are equal to zero. Those orbitals are spheres. The higher the value of n, the larger the sphere --- that is, the more likely it is that the electron will be found farther from the nucleus. The spheres are not equally dense throughout; they are more like nested shells. For historical reasons, this is called an s orbital. Because of the rules of quantum mechanics, the lowest energy electrons, with n=1, must have both l and m equal to zero, so the only orbital that exists for n=1 is the s orbital. The s orbital also exists for every other value of n.
When n is larger than one, more possibilities open up. L, the orbital quantum number, can have any value up to n-1. When l equals one, the orbital is called a p orbital. P orbitals look kind of like dumbbells. For each l, m goes from positive to negative l in steps of one. So, for n=2, l=1, m can equal 1, 0, or -1. That means there are three versions of the p orbital: one with the dumbbell up and down, another with the dumbbell left-to-right, and another with the dumbbell at right angles to both of the others. P orbitals exist for all principal quantum numbers greater than one, although they have additional structure as n gets higher.
When n=3, then l can equal 2, and when l=2, m can equal 2, 1, 0, -1, and -2. The l=2 orbitals are called d orbitals, and there are five different ones corresponding to the different values of m. The n=3, l=2, m=0 orbital also looks like a dumbbell, but with a doughnut around the middle. The other four d orbitals look like four eggs stacked on end in a square pattern. The different versions just have the eggs pointing in different directions.
The n=4, l=3 orbitals are called f orbitals, and they're difficult to describe. They have multiple complex features. For example, the n=4, l=3, m=0; m=1; and m=-1 orbitals are shaped like dumbbells again, but now with two doughnuts between the ends of the barbell. The other m values look kind of like a bundle of eight balloons, with all their knots tied together in the centre.
The mathematics governing the electron orbitals is pretty complex, but there are many online resources that provide graphical realisations of the different orbitals. Those tools are very helpful in visualising the behaviour of electrons around atoms.
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