Foundations — Diagonal relationship — Li - Mg, Be - Al, B - Si

Before you can read the parent note, you need to own every piece of vocabulary and every symbol it fires at you. Below, each term is built from nothing: plain words → the picture → why the topic needs it. Read top to bottom; each rung stands on the one before.

1. Atom, nucleus, electron cloud

Picture it (figure below): a dark dot in the middle (nucleus), a soft blue haze around it (the cloud). The haze is not solid — it is a probability fog showing where electrons are likely to be.

Why the topic needs it: the whole story is about one atom's cloud getting pulled and stretched by a neighbour. You cannot talk about "distorting the electron cloud" until you can see the cloud.

2. Charge and the symbols $z$, $+$, $-$

• The little superscript in ${\text{Li}}^{+}$ means "this particle carries one extra unit of positive charge" — it lost one electron.
• ${\text{Mg}}^{2+}$ lost two electrons → charge $+2$.
• ${\text{Cl}}^{-}$ gained one electron → charge $-1$.

Why the topic needs it: the pulling strength of a cation grows with how much positive charge it carries. We need a plain number, $z$, to plug into a formula later.

3. Ion, cation, anion

Picture it: neutral atom = balanced scale. Remove an electron → the positive nucleus "wins" → the ball shrinks a little and becomes a cation. Add an electron → negative "wins" → it puffs up into an anion.

Why the topic needs it: the parent note constantly writes ${\text{Li}}^{+}$, ${\text{Mg}}^{2+}$, ${\text{Cl}}^{-}$. The diagonal story is a cation (small positive tugger) pulling on an anion (the cloud that gets stretched).

4. Radius and the symbol $r$ — with a size ruler

Picture it (figure below): the same ball drawn twice — a big one and a small one — with a double-headed arrow labelled $r$ from centre to edge. Bigger $r$ = bigger, "softer" ion.

Two facts you must carry into the topic:

1. Across a period (left → right), $r$ shrinks. More protons pull the same shell in tighter.
2. Down a group (top → bottom), $r$ grows. New, bigger shells get added.

These two facts are the engine of the whole diagonal effect — see Periodic Trends — Atomic and Ionic Radii for the full build.

Why the topic needs it: the closeness of the tug depends on $r$. A closer (smaller-$r$) cation tugs harder. And for Li/Mg the punchline is that their radii come out nearly equal (76 pm vs 72 pm).

5. Coulomb's pull — why closeness matters

Picture it: a magnet and a paperclip. Use a stronger magnet (bigger $z$) → stronger pull. Bring it closer (smaller $r$) → stronger pull. Same idea, electric version.

Why the topic needs it: it justifies the shape of the ionic-potential formula: charge on top, size on the bottom.

6. Polarising power and the symbol $\varphi$

Picture it (figure below): a round anion cloud next to a small hard cation. The cation's positive charge drags the cloud toward it, denting the round shape — the shared, dented region looks like a covalent bond forming.

Why the topic needs it: $\varphi$ is the single number the whole diagonal relationship is judged by. Two elements are "diagonal twins" when one of the inputs of $\varphi$ (either $r$, or $\varphi$ itself) comes out matched. This is Fajans' Rules in one symbol.

7. Reading a fraction as "trends"

The parent note writes things like $\varphi =\frac{z\uparrow }{r\downarrow }$. Decode the arrows:

• $z\uparrow$ = "$z$ is going up". Bigger top → bigger fraction.
• $r\downarrow$ = "$r$ is going down". Smaller bottom → bigger fraction (dividing by a smaller number gives more).

So $\frac{z\uparrow }{r\downarrow }$ = both effects push $\varphi$ up strongly. That is the "across a period" case. And $\frac{z}{r\uparrow }$ (bottom grows) = $\varphi$ goes down — the "down a group" case.

Why the topic needs it: the "diagonal step = do both, they partly cancel" argument is literally about the top and bottom of this fraction moving in opposite directions.

8. Covalent vs ionic bonding (the outcome of polarising)

Picture it: low polarising power → the anion cloud stays round → ionic. High polarising power → the cation dents the cloud so much that electron density piles up between them → sharing → covalent. So high $\varphi$ ⇒ more covalent character.

Why the topic needs it: the parent explains why ${\text{BeCl}}_2$ and ${\text{AlCl}}_3$ are covalent — both cations have high, matched $\varphi$, so they polarise ${\text{Cl}}^{-}$ heavily. This is the direct payoff of §6.

9. The two supporting energies (named only)

The Li/Mg solubility argument leans on two energies you will fully build in Lattice Energy vs Hydration Energy. For now, just the plain idea:

Why the topic needs it: these two words are the mechanism behind "Li resembles Mg in solubility". You don't need the numbers here — just know which is which.

How the foundations feed the topic

Read it as: charge and radius build $\varphi$; the periodic trends decide how $\varphi$ moves; $\varphi$ decides bonding and energetics; matched $\varphi$-inputs diagonally = the diagonal relationship.

Equipment checklist

Cover the right side and answer aloud. If any line is fuzzy, reread its section.

Ready? Now open the parent: the main note and every symbol will already feel like an old friend.