Effective Nuclear Charge Calculator

Enter Calculation

Atomic Number (Z)

Shielding Constant (S)

Results

Effective Nuclear Charge

2.2

Z_eff (units of elementary charge e)

Atomic Number (Z)	11
Shielding Constant (S)	8.8
Equation	Z_eff = Z − S

What Is Effective Nuclear Charge?

The effective nuclear charge (Z_eff) is the net positive charge actually experienced by a particular electron in a multi-electron atom. While the nucleus carries a full charge equal to the atomic number Z, inner electrons partially "screen" or shield the outer electrons from that pull. This calculator computes Z_eff using the simple relationship \(Z_{\text{eff}} = Z - S\), where S is the shielding (screening) constant.

Inner electrons shield the outer electron, reducing the nuclear charge it actually feels.

How to Use the Calculator

Enter the atomic number Z of the element (for example, 11 for sodium) and the shielding constant S for the electron of interest. S is often estimated with Slater's rules. The calculator instantly returns Z_eff, the effective charge felt by that electron. A higher Z_eff means the electron is held more tightly, which correlates with smaller atomic radius and higher ionization energy.

The Formula Explained

The governing equation is $$Z_{\text{eff}} = Z - S.$$ Z is a whole number equal to the number of protons. S accounts for repulsion from other electrons: core electrons shield strongly, electrons in the same shell shield partially, and electrons in outer shells contribute essentially nothing. Slater's rules provide systematic values for S, so the precision of Z_eff depends on the chosen shielding estimate.

Z_eff = Z − S: the full nuclear charge minus the shielding from inner electrons.

Worked Example

Consider a 3s valence electron in sodium (Z = 11). Using Slater's rules, the shielding constant is approximately S = 8.8. Then $$Z_{\text{eff}} = 11 - 8.8 = 2.2.$$ So the valence electron in sodium feels an effective charge of about +2.2, far less than the full nuclear charge of +11.

FAQ

Why is Z_eff less than Z? Because inner electrons partially block the attraction between the nucleus and an outer electron, reducing the charge that electron actually feels.

Where do I get the shielding constant S? Most commonly from Slater's rules, which assign screening contributions based on the electron's shell and subshell relative to the others.

Can Z_eff be negative? In practice no — S never exceeds Z for a real electron, so Z_eff remains positive, though crude estimates can occasionally produce small values.