Magnetic properties — paramagnetism via spin-only formula μ = √(n(n+2)) BM: Foundations

This page assumes you know nothing about the symbols on the parent page. We will earn each one — plain words first, then a picture, then the reason the topic can't live without it. Read top to bottom; every item leans on the one above it.

1. What is an electron doing? (the moving-charge picture)

Why do we start here? Because magnetism in this whole topic comes from one fact of physics: a moving electric charge creates a magnetic field. No moving charge → no magnetism.

Figure — Magnetic properties — paramagnetism via spin-only formula μ = √(n(n+2)) BM

Hold that image: charge in motion = magnet. Everything else is bookkeeping about how the electron moves.

2. The two ways an electron moves → two kinds of magnetism

An electron has two separate motions, and each one makes its own magnetism:

Here $L$ and $S$ are just labels for amounts of "twirl." Bigger $L$ = more orbital twirl; bigger $S$ = more spin twirl. We will make $S$ into an actual number in Section 7.

3. Spin has only two settings: up or down

Why does this matter so much? Because two electrons sharing a room point opposite ways, and that is the entire reason paired magnets cancel.

4. Orbitals — the "rooms" electrons live in

5. Hund's rule — why electrons prefer to be lonely first

Why does the topic need this? Because it decides $n$ , the count of lonely electrons — and $n$ is the only input to the formula. Get Hund wrong and every $μ$ you compute is wrong.

Recall Why aligned, not opposite?

Electrons repel each other. Sitting in separate rooms keeps them apart, which costs less energy than crowding two into one room. Nature is lazy, so it spreads them out first. ::: correct

6. $n$ — the number of unpaired electrons

This is the single most important symbol on the whole parent page. Everything the formula does is: "give me $n$ , I'll give you the magnet strength $μ$ ."

7. From counting to the spin number $S$

Each lonely electron carries spin projection $m_{s} = + \frac{1}{2}$ (Section 3), and Hund aligns them all the same way (all $+ \frac{1}{2}$ , Section 5). When spins all point the same direction, adding them up (this is called vector addition of spins, and for aligned spins it is just ordinary addition) gives the total spin quantum number:

$S = n times \frac{1}{2} + \frac{1}{2} + \dots + \frac{1}{2} = \frac{n}{2} .$

8. Why $μ = n (n + 2)$ and not the full formula

The complete magnetic moment (both motions) is written: $μ_{S + L} = 4 S (S + 1) + L (L + 1) μ_{B} .$

Two pieces of this need explaining before we may use it:

Set $L = 0$ : the $L (L + 1)$ term vanishes, leaving only spin: $μ = 4 S (S + 1) μ_{B} .$ Now substitute the bridge $S = \frac{n}{2}$ : $μ = 4 \cdot \frac{n}{2} (\frac{n}{2} + 1) = 4 \cdot \frac{n}{2} \cdot \frac{n + 2}{2} = \frac{4 n ( n + 2 )}{4} = n (n + 2) μ_{B} .$

Why the square-root and not just $n$ itself? Because quantum mechanics measures the length of the total-spin arrow, and the length of an arrow always comes from a square-root (like the diagonal of a box). You don't need the full proof — just know the root is where the "" comes from, not a mistake.

9. Bohr Magneton — the unit BM

So when the boxed formula outputs " $35$ ," the full reading is " $35 μ_{B}$ ," i.e. $5.92$ Bohr Magnetons — and chemists write that unit as BM. That is the unit hiding behind every $μ$ value on the parent page.

Why invent a special unit? The magnet of one electron is unimaginably tiny in everyday units (joules per tesla). Rather than write ugly powers of ten every time, chemists say "this ion is $5.92$ BM strong" — clean and comparable. When the formula outputs " $35$ ," that answer is already in BM; you never square it or add units by hand.

μ

stand for?

The magnetic moment — a single number for the atom's overall magnet strength.

What makes an electron magnetic at all?

It is a charge in motion; a moving charge creates a magnetic field.

What are the electron's two magnetism-producing motions?

Orbital (circling the nucleus,

L

) and spin (spinning in place,

S

What does the spin projection

m_{s}

label, and what values can it take?

The direction of one electron's spin;

+ \frac{1}{2}

(up) or

- \frac{1}{2}

(down).

How is the total spin

S

different from

m_{s}

m_{s}

is one electron's direction label;

S

is the sum of all the aligned

m_{s}

values for the whole atom.

How many electrons fit in one orbital, and with what spins?

Two, and they must be opposite (

↑↓

How many electrons can the five d-orbitals hold in total?

Ten (

5 \times 2

What does Hund's rule tell you to do first?

Put one electron in each equal-energy room (all spins aligned) before pairing any.

What exactly does the symbol

n

count?

Only the unpaired (lonely) electrons — not the total.

What is the bridge from

n

to total spin

S

S = \frac{n}{2}

Why is there a "

4

" in front of

S (S + 1)

It is

g^{2}

; the electron-spin Landé

g

-factor is

\approx 2

, and

2^{2} = 4

Why does an angular-momentum number

X

enter as

X (X + 1)

That combination is its quantum-mechanical eigenvalue ("size") — true for both

S (S + 1)

and

L (L + 1)

Why does the orbital term

L (L + 1)

vanish in the spin-only model?

Ligands lock the d-orbitals, "quenching" orbital motion so

L \to 0

What unit is the formula's answer in, and its abbreviation?

Bohr Magnetons,

μ_{B}

, abbreviated BM.

n = 0

means the substance is…?

Diamagnetic (

μ = 0

, weakly repelled).

Connections

Yeh note Hinglish mein
Hund's Rule & Electron Configuration — fixes how many electrons stay lonely.
Electronic Configuration of Ions — decides the d-electron count to fill the boxes.
Crystal Field Theory — why ligands quench $L$ and set high/low spin.
Bohr Magneton — the unit $μ_{B}$ .
Colour in Transition Metal Complexes — same d-electrons, different observable.
Lanthanide Magnetism — where $L$ is not quenched, so the full formula returns.

Foundations — Magnetic properties — paramagnetism via spin-only formula μ = √(n(n+2)) BM