How this instrument works
Every electron in a multi-electron atom is attracted to the nucleus, but it's also pushed away from the nucleus by every other electron sitting between it and the center. The net attraction it actually experiences — the effective nuclear charge, Zeff — is always less than the full atomic number Z, because the other electrons partially cancel, or 'shield,' the nuclear pull. Zeff = Z − S, where S is the shielding constant: the total canceling effect of every other electron in the atom.
Slater's rules, published by physicist John C. Slater in 1930, are a set of bookkeeping rules for estimating S without solving the full quantum mechanics. Electrons are sorted into groups by shell and subshell, and each group contributes a fixed shielding amount depending on where it sits relative to the electron you're calculating for: other electrons in the same (n, l) group contribute 0.35 each (0.30 if the electron of interest is itself a 1s electron); electrons one shell closer to the nucleus (n−1) contribute 0.85 each; electrons two or more shells closer contribute a full 1.00 each. Electrons farther from the nucleus than the one you're studying contribute nothing at all — they're outside the electron of interest and don't get in the way of the nuclear pull.
Zeff explains periodic trends that a bare atomic number can't: it's why atomic radius shrinks and ionization energy climbs moving left to right across a period, even though every added electron in that direction is, in principle, adding negative charge that should push back against the nucleus. Because same-shell shielding (0.35) is much weaker than the +1 added to the nuclear charge with each new proton, Zeff creeps steadily upward across a period, pulling the outer electrons in tighter as you go.
- Enter the atomic number, Z, of the atom.
- Enter the number of other electrons sharing the same (n, l) group as the electron of interest, not counting the electron itself.
- Enter the number of electrons one shell closer to the nucleus (n−1).
- Enter the number of electrons two or more shells closer to the nucleus (n−2 or lower).
- Indicate whether the electron of interest is itself a 1s electron, since Slater's shielding constant for that case is 0.30 rather than the general 0.35.
- Read the total shielding constant, σ, and the resulting effective nuclear charge, Zeff.
Worked example — selenium's outermost 3p electron
Selenium (Z=34) has the ground-state configuration 1s² 2s² 2p⁶ 3s² 3p⁶ 3d¹⁰ 4s² 4p⁴. Take one of its six 3p electrons as the electron of interest: the same-group count (the rest of the 3s²3p⁶ electrons, minus the one under study) is 2+6−1=7; the n−1 shell (n=2: the 2s²2p⁶ electrons) holds 8; the n−2-or-lower shell (n=1: 1s²) holds 2. The 3d, 4s, and 4p electrons don't count at all here, because under Slater's rules electrons farther from the nucleus than the one being studied contribute zero shielding. Shielding: σ = 0.35×7 + 0.85×8 + 1.00×2 = 2.45 + 6.8 + 2 = 11.25. Effective nuclear charge: Zeff = 34 − 11.25 = 22.75.
That 22.75 is a meaningfully smaller number than selenium's full atomic number of 34 — nearly a third of the nuclear charge is shielded away by selenium's own inner-shell electrons before it ever reaches that 3p electron, which is exactly the physical reason inner-shell (core) electrons are held so much more tightly than an atom's outermost valence electrons.
Questions
What is effective nuclear charge?
It's the net positive charge an electron actually experiences from the nucleus, after accounting for the shielding, or blocking, effect of every other electron between it and the nucleus. It's always smaller than the atom's full atomic number (total proton count), because inner and same-shell electrons partially cancel the nuclear pull rather than letting the outer electron feel the full, unshielded charge.
Why is the shielding constant 0.30 instead of 0.35 for 1s electrons?
Slater's rules treat the 1s subshell as a special case because two 1s electrons sit unusually close to each other and to the nucleus, with no inner shell at all to compare against. Slater found that using 0.30 for mutual 1s-1s shielding, rather than the general same-group value of 0.35, matched experimental and calculated results better for this innermost-shell case, so it's kept as a documented exception to the general rule rather than folded into it.
How accurate are Slater's rules compared to real calculations?
They're a hand-calculable approximation, not an exact result — real effective nuclear charges come from far more computationally intensive self-consistent-field (SCF) quantum calculations. Slater's rules were designed to be quick enough to do by hand while still capturing the right qualitative trends (shielding increases with proximity to the nucleus, valence electrons are shielded more than core electrons), and they remain a standard teaching tool for exactly that reason, even though the numbers they produce are approximate.
Why does effective nuclear charge increase across a period?
Because each additional proton adds a full +1 to the nuclear charge, but each additional electron added in the same period sits in the same shell and only contributes about 0.35 to the shielding constant under Slater's rules. Since the increase in nuclear charge (+1 per element) consistently outpaces the increase in same-shell shielding (+0.35 per element), Zeff climbs steadily left to right across a period — the underlying reason atomic radius shrinks and ionization energy rises across the same span.
Do inner-shell electrons shield more than outer electrons of the same atom?
Yes, substantially more. Under Slater's rules, electrons one full shell closer to the nucleus than the electron of interest shield 0.85 each, and electrons two or more shells closer shield a full 1.00 each — both far higher than the 0.35 (or 0.30) contributed by electrons in the same shell. This reflects a real physical difference: an inner-shell electron sits mostly between the outer electron and the nucleus and blocks the nuclear pull almost completely, while a same-shell electron sits roughly alongside the electron of interest and blocks it only partially.