Foundations — Lanthanides — electronic configuration, lanthanide contraction, oxidation states (mostly +3)

Before you can enjoy that story, you need to read a dozen little symbols the parent note throws around: $[\text{Xe}]$, $4f^n$, $Z$, $\ell$, $m_{\ell }$, $Z_{\text{eff}}$, "pm", ${\text{Ln}}^{3+}$. This page builds each one from nothing, in an order where every symbol is earned before it is used.

1. $Z$ — the atomic number (the counter of protons)

The picture: think of the nucleus as a tiny ball of positive charge in the middle, and $Z$ counts the little "+" marks inside it.

Why the topic needs it: the whole lanthanide story is "walk from $Z=57$ (La) to $Z=71$ (Lu), adding one proton each step." Every trend — contraction, filling of 4f — is indexed by $Z$. If you can't read $Z$, you can't read the row.

2. Electron shells and subshells — where electrons live

Electrons don't sit anywhere; they live in shells (labelled by $n$) and within each shell in subshells labelled $s,p,d,f$.

The picture: $s$ is a smooth ball hugging the nucleus; $f$ is a many-petalled flower that reaches out in odd directions and has thin, wispy regions. This shape is the whole reason 4f shields poorly — hold that thought.

Why the topic needs it: the parent note writes $4f$, $5d$, $6s$. That is (shell number)(subshell letter). $4f$ = 4th shell, f-shape. You cannot decode a single configuration without this.

3. How many electrons fit — and where "14" comes from

Each subshell holds a fixed number of electrons, and here is exactly why.

Let's list the values the number $m_{\ell }$ can take — $m_{\ell }$ just labels the different orientations of the same shape (which way the petal points):

$\ell =3\Rightarrow m_{\ell }=-3,-2,-1,0,+1,+2,+3(\text{that’s }7\text{ orbitals})$

$\Rightarrow 7\times 2=\boxed{14}\text{electrons in the }f\text{ subshell}$

Why the topic needs it: "there are exactly 14 lanthanides" is not a random fact — it is $2(2\cdot 3+1)=14$. See Half-filled and Fully-filled Stability for why the middle ($4f^7$) and full ($4f^{14}$) are special.

4. Superscript notation $4f^n$ — reading a configuration

Check the electron count: $54+7+2=63=Z$ of Europium. It must add up to $Z$ — that's your always-available sanity check.

Why the topic needs it: every configuration in the parent note is written this way. ${\text{Eu}}^{3+}=[\text{Xe}]4f^6$ is meaningless until $[\text{Xe}]$, $4f$, and the superscript all make sense — which now they do.

5. The filling order — the $(n+\ell )$ rule

Why does $6s$ fill before $4f$, even though $6$ is bigger than $4$? Because electrons fill lowest-energy first, and energy tracks $n+\ell$.

So $6s$ (sum 6) fills first; then $4f$ and $5d$ tie at 7, and 4f (smaller $n$) usually wins. Because they tie so closely, small stability effects can flip which one gets an electron — that's the source of the La/Ce/Gd/Lu exceptions.

Why the topic needs it: this is the machinery behind $[\text{Xe}]4f^{1\text{–}14}5d^{0\text{–}1}6s^2$. Full detail lives in Aufbau Principle and (n+l) Rule.

6. Ions and the charge symbol ${\text{Ln}}^{3+}$

The picture: a neutral atom is balanced (protons = electrons). Pluck out 3 electrons and the leftover positive nucleus "shows through" as a $+3$ charge.

Why the topic needs it: the headline result "lanthanides are mostly +3" is written ${\text{Ln}}^{3+}:[\text{Xe}]4f^n$. You now know it means: take a neutral lanthanide, remove 3 electrons (the two $6s$ + one $5d/4f$), and read off what's left.

7. Effective nuclear charge $Z_{\text{eff}}$ and "shielding"

This is the engine of the lanthanide contraction, so we build it carefully.

The picture and the punchline: because the $4f$ cloud is a wispy, spread-out flower (Section 2), an added 4f electron is a lousy screen — it contributes little to $S$. So each step across the row adds one full proton to $Z$ but only a tiny amount to $S$. Net: $Z_{\text{eff}}$ on the outer electrons rises, pulling the whole atom inward. That inward pull, repeated 14 times, is the lanthanide contraction.

$\text{step right}\Rightarrow \underset{\text{big}}{\underbrace{Z\uparrow \text{ by }1}}-\underset{\text{small}}{\underbrace{S\uparrow \text{ tiny (4f screens poorly)}}}\Rightarrow Z_{\text{eff}}\uparrow \Rightarrow \text{radius}\downarrow$

Why the topic needs it: "poor shielding raises $Z_{\text{eff}}$" is the root cause line the parent note keeps repeating. Quantitative shielding rules are in Effective Nuclear Charge & Slater's Rules.

8. Radius and the unit "pm"

The picture: draw the atom as a fuzzy ball; the radius is the ball's radius. Contraction = this ball shrinking, e.g. $r({\text{La}}^{3+})\approx 103\text{pm}\to r({\text{Lu}}^{3+})\approx 86\text{pm}$.

Sense of scale: the shrink per step is only about $1\text{pm}$ — invisible in one hop, but $103-86=17\text{pm}$ over the full row, which is enough to make Zr and Hf near-twins.

Why the topic needs it: every "contraction" claim is a statement about radius in pm. Without the unit and picture, "$103\to 86$" is just numbers. Downstream, this tiny-difference-in-size is why separation is hard — see Separation of Lanthanides — Ion Exchange.

9. How it all connects

Read it as: the counter $Z$, the shape-letter $\ell$, and the shorthand pieces all feed reading a configuration; that feeds the +3 ion. Separately, poor 4f shielding + rising $Z$ feed $Z_{\text{eff}}$, which drives the shrinking radius. Both streams pour into the lanthanide story.

Equipment checklist

Test yourself — you should be able to answer each before moving to the main topic.