How s, p, d, and f Orbitals Get Their Shapes
On this page · Compare the shapes
Start by comparing shapes
Choose an orbital family, then choose one of its orientations. Rotate the model to see the full three-dimensional pattern.
Rotate by dragging or with the arrow keys; zoom with pinch, scroll, or +/−. Denser points show where the electron is more likely to be found. Color shows the wave's sign, not a different charge. The shaded envelope is a guide to the shape, not a hard edge.
Model limits and physical scale
These are hydrogen-like models, not measured orbitals for neutral cerium. The scale bar uses Bohr radii (); 90% of this orbital's radial probability lies within of the nucleus. The camera fits each selection, so compare scale bars instead of the on-screen diameters.
Try subshells occupied in cerium
These buttons select representative hydrogen-like shapes for occupied subshells. They do not assign cerium's d or f electron to one unique orientation.
Notice where the pattern has lobes and where it has gaps. The gaps are places where the wave is zero; those surfaces help determine each orbital's shape.
Angular nodes make the shapes
The angular pattern describes how the wave changes with direction. Its zero-amplitude surfaces, called angular nodes, divide the surrounding space into lobes. Changing the angular quantum number changes this pattern.
This cut shows how one angular pattern changes with direction. It is not the full orbital or a hard boundary. gives 1 angular node and 3 real-basis shapes.
The sequence is , and . Each family has orientations: 1, 3, 5, and 7. The magnetic quantum number distinguishes angular patterns within a family; it does not change the family itself.
An orbital is a probability pattern
The wavefunction can have positive or negative values. Its sign is not electric charge. Squaring its magnitude, , gives the probability of finding the electron in each small region of space.
Blue and orange mean opposite wave signs, not different charges.
The colored lobes are not solid walls. They summarize where the electron is more likely to be found; the color shows the wave's sign, not a different kind of charge.
Same shape, different shell
The shell number changes the radial pattern: how far from the nucleus the electron is likely to be found and how many spherical nodes appear. Changing can add radial structure while keeping the same angular family.
For a hydrogen-like orbital, the radial node count is . These spherical surfaces are separate from the angular nodes that shape the lobes.
has 1 radial node. The probability graph is , not just : larger spherical shells contain more space. Curves are rescaled separately for comparison.
Follow the math, if you want
Start with hydrogen: one electron attracted to one proton. Its time-independent Schrödinger equation asks which wave patterns can have a definite energy.
The first term describes how the wave bends through space; the second is the attractive electric potential.
Try moving away from the nucleus
Only distance matters here. That spherical symmetry lets us split the equation into radial and angular parts.
Hydrogen's attraction depends only on distance from the nucleus. That spherical symmetry lets us separate a wavefunction into a radial part and an angular part:
The radial part changes with distance. The angular part changes with direction. Probe the real example below; is measured from the vertical axis used by the 3D viewer.
Probe a wave at one position
These values omit a shared normalization constant. Crossing flips the wave’s sign; it does not make the electron’s charge change.
The quantum numbers obey , , and . The real orbitals shown here are combinations of the complex states when is nonzero.
Cerium (Ce, atomic number 58) is the first neutral ground-state atom with occupied s, p, d, and f subshells. Its configuration is ; the core includes filled p subshells. Lanthanum (57) has no occupied f subshell in its ground state.
The interactive shapes above are hydrogen-like models. They isolate the shape families; they are not exact solutions for the many-electron cerium atom.
To see how electrons fill these subshells, continue to the electron-configuration experiment.
Sources and next steps
- OpenStax: The Hydrogen Atom — equation, separation, and probability.
- OpenStax: Development of Quantum Theory — orbital families and quantum numbers.
- NIST: Cerium and NIST: Lanthanum — measured ground-state configurations.