Conduction in copper and magnetism in iron

The partially filled d-shell has two fates. A filled 3d¹⁰ band beneath one 4s electron makes copper a free-electron conductor; an exchange-split d-band leaves iron a net moment ≈ 2.2 μ_B and ferromagnetism. Both follow from d-band occupation, not from adjustable parameters, with a stated falsification criterion.

The partially filled d-shell splits into two fates. In copper a filled 3d¹⁰ band beneath a single 4s electron gives free-electron conduction; in iron an exchange-split d-band leaves a net moment ≈ 2.2 μ_B and ferromagnetism. Both follow from d-band occupation, not from adjustable parameters.

(chemistry). Status grades as in Chapter EM. Numbers reproduced by vp_copper_conduction.py and vp_iron_magnetism.py (deterministic, standard-library, self-checking).

Copper conducts and iron magnetizes for the same root reason, read two ways. Copper’s inner d-shell is full, freeing its outer electron into a conducting sea; iron’s d-shell is partly empty, leaving unpaired rotations that align into a magnet. A filled shell makes a conductor; an unfilled shell makes a magnet.

CM.0 The two questions

This chapter answers two concrete questions about metals using the electromagnetism of Chapter EM:

1. Electricity is an electromagnetic wave — how does it pass through the inside of copper, when copper (a metal) reflects ordinary light?
2. What is the principle of iron’s magnetism — and in particular, why iron (and cobalt, nickel) but not copper, and not even manganese, which has more unpaired electrons than iron? (The physics whitepaper took the “why Fe and not Cu” criterion as an empirical input, §14; here it is derived.)

CM.1 The paradox

Light is a transverse electromagnetic wave, and metals reflect it — that is why copper is shiny and opaque. Yet electricity is also electromagnetic, and it travels straight through the body of a copper wire. How can the same medium reflect one electromagnetic disturbance and transmit another?

The resolution is the propagation angle of Chapter EM (§EM.6). Electricity is the extreme longitudinal limit (χ → 0) of the electromagnetic wave, while visible light is the transverse limit (χ → 90^∘). The same free-electron sea reflects the transverse wave and transmits the longitudinal one. Same medium, opposite fates, set by angle.

CM.2 Copper’s free-electron sea [F]/[CAL]

Copper is [Ar] 3d¹⁰4s¹: the inner 3d shell is full, and the single 4s electron is delocalized into a conducting sea. The free-electron density is

(one free electron per atom; ρ, M measured [CAL]). This sea is the “internal quanta” through which conduction propagates.

CM.3 Why transverse light is reflected — the plasma frequency [F]/[CAL]

The free-electron sea has a natural oscillation frequency, the plasma frequency

Below ωₚ — radio, infrared, all of visible light — an incident transverse wave drives the free electrons, which re-radiate and reflect it: this is exactly why metals are shiny and opaque (vp_copper_conduction.py, §1). Above ωₚ (X-rays) the electrons cannot follow and the metal becomes transparent. The transverse electromagnetic wave is turned back by the very sea that carries the current.

CM.4 Why longitudinal electricity is transmitted — conduction as χ → 0 [F]

The longitudinal mode is different. A longitudinal (χ → 0) electromagnetic disturbance is a density/pressure wave of the electron sea that threads straight along the wire — straighter even than a γ-ray. It is not reflected because it is not a transverse oscillation of the surface electrons; it propagates through the internal quanta of the copper atoms, not through vacuum or air (physics §14). This longitudinal disturbance is the current. The angle that makes light reflect (transverse) is the angle that makes electricity transmit (longitudinal). Two physical facts fix conduction at χ → 0: small amplitude (a low-amplitude disturbance threads straight; if the amplitude grows the path tilts transverse and the signal leaves the copper as radiative loss) and medium (it runs through the atomic-internal quanta).

CM.5 The skin effect [F]/[CAL]

The transverse and longitudinal pictures are quantitatively distinct. A transverse wave that does enter the metal decays within the skin depth

High-frequency transverse fields are confined to a micron-thin surface layer, while the direct/low-frequency longitudinal current fills the whole cross-section (vp_copper_conduction.py, §3).

CM.6 Three velocities — drift, signal, Fermi [F]/[CAL]

A common confusion is dissolved by separating three speeds (vp_copper_conduction.py, §4):

• Drift velocity v_d = J/(ne): for J = 10⁶ A/m², only 0.074 mm/s — the electrons themselves crawl.
• Signal velocity ∼ c: the longitudinal field disturbance travels near light speed, so a lamp lights essentially instantly even though the electrons barely move.
• Fermi velocity v_F = 1.57× 10⁶ m/s: the electrons at the Fermi surface move fast, but their net drift is the slow v_d.

The predicted Fermi energy E_F = ℏ²(3π² n)⁽2/3)/(2mₑ) = 7.04 eV matches the measured ≈ 7.0 eV.

CM.7 Why copper conducts so well [F]/[CAL]

The Drude relation σ = ne²τ/mₑ with σ₍Cu) = 5.96× 10⁷ S/m gives a relaxation time τ = 2.5× 10⁻¹⁴ s and a mean free path ℓ = v_Fτ = 39 nm. Copper’s filled 3d¹⁰ shell hands its 4s¹ electron to a low-scattering sea — exactly the property that, in iron, is absent. This is the hinge of the whole chapter: a filled inner shell yields conduction; an unfilled one yields magnetism.

CM.8 The magnetic field of a current — and Joule loss [F]

A drifting charge tilts its rotation axis by ∼ v/c (§EM.4), shedding a small transverse twist perpendicular to the drift; propagating sideways it wraps the wire azimuthally as the magnetic field B = μ₀ I/(2π r), with circulation sense set by the rotation handedness (the right-hand rule). The transverse twist has two fates: the returned part is the reactive B field itself (present even at zero loss, as in a superconductor), and the un-returned fraction is dissipation (Joule loss). Thus B and resistive loss share one origin — the transverse twist of moving charge — as its returned versus un-returned parts; B itself is not the loss.

CM.9 Iron’s magnetism — the principle [F]/[CAL]/[O]

Magnetism is aligned rotation. A magnetic moment is the rotation of an unpaired electron — the same rotation that is electric charge (§EM.1) and helicity (§EM.5). For a material to be a permanent (ferro)magnet, two conditions must both hold, and the pair derives exactly which elements magnetize.

CM.10 Criterion 1 — unpaired d-electrons (this is why not copper) [F]

A localized moment requires an unfilled d-shell with unpaired electrons (Hund’s rule). For 3d⁽n_d) the unpaired count is n_d if n_dle 5 else 10-n_d, and the spin-only moment is μ = √(n(n+2)) μ_B (vp_iron_magnetism.py, condition 1):

Copper’s 3d¹⁰ shell is full: zero unpaired d-electrons, no localized moment. Its 4s¹ is a free conduction electron, not a localized moment. Copper fails condition 1 and cannot be magnetic — the same filled shell that makes it a superb conductor.

CM.11 Criterion 2 — positive exchange (this is why not manganese) [F]/[CAL]

Having moments is not enough; neighbouring d-rotations must align parallel (ferromagnetic), not antiparallel (antiferromagnetic). The Bethe–Slater criterion uses the ratio D/d = interatomic distance / 3d-orbital diameter: above a critical value (≈ 1.5) the exchange integral is positive (parallel, ferromagnetic); below it the exchange is negative (antiparallel, antiferromagnetic) (vp_iron_magnetism.py, condition 2):

Manganese has five unpaired electrons — more than iron — yet D/d = 1.47 < 1.5 gives J<0, so its moments align antiparallel and cancel: manganese fails condition 2 and is antiferromagnetic. Cobalt (D/d = 1.82) sits near the exchange maximum.

CM.12 The derivation — 7/7 classification [F]

Applying both conditions reproduces the observed magnetic class of the late 3d series with no fitted magnetic parameter (vp_iron_magnetism.py, summary):

7/7. Copper is non-magnetic by condition 1, manganese is antiferromagnetic by condition 2, and only iron, cobalt, and nickel satisfy both — the “why Fe and not Cu” the physics whitepaper left as input is here derived.

CM.13 Curie temperature, domains, and ∇ ·mathbf B = 0 [F]/[CAL]

Thermal rotation destroys alignment above the Curie temperature, where k_BT_C is of order the exchange energy: Fe 1043 K, Co 1388 K (highest, matching the strongest exchange), Ni 627 K (vp_iron_magnetism.py). Below T_C, exchange aligns moments into domains; an external field moves the domain walls. Because mathbf B is a rotation axis (§EM.4), it is divergence-free, ∇ ·mathbf B = 0: the poles are the two ends of one axis and cannot be separated — no monopoles, and cutting a magnet yields two fresh N/S pairs. Hysteresis is the part of the alignment that does not return — the permanent magnet — and heating above T_C erases it. The ∇ ·mathbf B = 0 seen abstractly in Chapter EM becomes, in iron, the macroscopic fact of domains and permanent magnets. (The metallic moments are non-integer — Fe 2.22, Co 1.72, Ni 0.62 μ_B — an itinerant/band effect marked [O]; the localized Hund count gives the correct ordering and the magnetic class.)

CM.14 The unifying picture

Copper and iron are the two fates of the d-shell:

• Copper — filled 3d¹⁰ → free 4s electron → conduction. Electricity passes through copper as the longitudinal (χ→ 0) electromagnetic mode of that free sea, while light (transverse) is reflected by it.
• Iron — unfilled 3d⁶ → unpaired moments → magnetism. With positive exchange (D/d>1.5) the moments align into a ferromagnet; the aligned rotation is the magnetic field.

A filled inner shell makes a conductor; an unfilled one makes a magnet. Both are the same rotation — the rotation that is charge, light, and helicity — expressed as a delocalized current in copper and as an aligned moment in iron.

CM.15 Falsification and reproducibility [F]

• If a metal’s plasma frequency did not predict its reflectivity edge (shiny below ωₚ, transparent above), the plasma-reflection account of conduction-vs-light fails.
• If manganese (more unpaired electrons than iron) were ferromagnetic, or copper (3d¹⁰) were ferromagnetic, the two-condition criterion fails.
• If the derived magnetic class disagreed with any late-3d metal, the derivation fails (it is 7/7).

python3 code/foundation/vp_copper_conduction.py   # plasma freq, skin depth, drift/signal/Fermi
python3 code/foundation/vp_iron_magnetism.py      # two-condition ferromagnetic criterion (7/7)

Both modules are deterministic (stdout sha256 identical on re-run) and standard-library only.

Copper and iron close the loop opened in Chapter EM: charge is rotation, and that one rotation appears as a longitudinal current threading copper’s filled-shell electron sea, and as an aligned moment in iron’s unfilled d-shell. The conductor and the magnet are the same physics, divided by whether the inner shell is full.