In condensed matter physics, the Slater–Pauling rule states that adding an element to a metal alloy will reduce the alloy's saturation magnetization by an amount proportional to the number of valence electrons outside of the added element's d shell.^[1] Conversely, elements with a partially filled d shell will increase the magnetic moment by an amount proportional to number of missing electrons. Investigated by the physicists John C. Slater^[2] and Linus Pauling^[3] in the 1930s, the rule is a useful approximation for the magnetic properties of many transition metals.

The use of the rule depends on carefully defining what it means for an electron to lie outside of the d shell. The electrons outside a d shell are the electrons which have higher energy than the electrons within the d shell. The Madelung rule (incorrectly) suggests that the s shell is filled before the d shell. For example, it predicts Zinc has a configuration of [Ar] 4s² 3d¹⁰. However, Zinc's 4s electrons actually have more energy than the 3d electrons, putting them outside the d shell. Ordered in terms of energy, the electron configuration of Zinc is [Ar] 3d¹⁰ 4s². (see: the n+ℓ energy ordering rule)

Slater–Pauling rule (nearest integer)

Element	Electron configuration	Magnetic valence	Predicted moment per atom
Tin	[Kr] 4d10 5s2 5p2	-4	-4 ${\displaystyle \mu _{\mathrm {B} }}$
Aluminum	[Ne] 3s2 3p1	-3	-3 ${\displaystyle \mu _{\mathrm {B} }}$
Zinc	[Ar] 3d10 4s2	-2	-2 ${\displaystyle \mu _{\mathrm {B} }}$
Copper	[Ar] 3d10 4s1	-1	-1 ${\displaystyle \mu _{\mathrm {B} }}$
Palladium	[Kr] 4d10	0	0 ${\displaystyle \mu _{\mathrm {B} }}$
Cobalt	[Ar] 3d7 4s2	+1	+1 ${\displaystyle \mu _{\mathrm {B} }}$
Iron	[Ar] 3d6 4s2	+2	+2 ${\displaystyle \mu _{\mathrm {B} }}$
Manganese	[Ar] 3d5 4s2	+3	+3 ${\displaystyle \mu _{\mathrm {B} }}$

The basic rule given above makes several approximations. One simplification is rounding to the nearest integer. Because we are describing the number of electrons in a band using an average value, the s and d shells can be filled to non-integer numbers of electrons, allowing the Slater–Pauling rule to give more accurate predictions. While the Slater–Pauling rule has many exceptions, it is often a useful as an approximation to more accurate, but more complicated physical models.

Building on further theoretical developments done by physicists such as Jacques Friedel,^[4] a more widely applicable version of the rule, known as the generalized Slater–Pauling rule was developed.^[5][6]

• Spin states (d electrons)
• Ferromagnetism
• Metallic bonding

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