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Layer 18 / Materials
From molecules to collective matter

One atom has properties.
A material has phases.

A material is not just a larger molecule. Once enormous numbers of atoms interact, collective modes, electronic bands, magnetic order and phase transitions become possible. This lab makes that change of scale visible while keeping each model explicit.

crystal orderdefectsphononselectronic bandsmagnetismphase transitions
01 ORDERRepeated local structure builds a lattice.
02 VIBRATECoupled atoms move in collective normal modes.
03 DELOCALIZEMany atomic states broaden into energy bands.
04 ALIGNLocal interactions can create bulk magnetization.
05 TRANSFORMAn order parameter reorganizes at a phase transition.
01 / ORDER

Periodic order creates a crystal; defects make real materials interesting.

Switch between square and triangular order, then insert a vacancy or a schematic edge dislocation. Local coordination changes even though most of the lattice remains ordered.

lattice sitelocally disturbed site
02 / VIBRATE

Coupled atoms do not vibrate independently.

Excite one normal mode of a one-dimensional monoatomic chain. The same spring coupling that binds neighboring masses produces a dispersion relation, so different wavelengths move and oscillate differently.

atomsselected wavevector on dispersion
03 / DELOCALIZE

Many atomic states can broaden into a band.

Use a one-dimensional tight-binding model. Increasing orbital overlap widens the band; changing the filling moves the Fermi edge. A partly filled band has nearby unoccupied states available in this idealized picture.

04 / ALIGN

Local spin interactions can become bulk magnetization.

A finite two-dimensional Ising model provides a stripped-down demonstration: thermal agitation competes with neighbor alignment and an external field biases the collective state.

05 / TRANSFORM

A phase transition reorganizes the preferred macroscopic state.

Landau theory compresses complex microscopic behavior into an order parameter. Above a critical point one minimum is preferred; below it, symmetry-related minima can appear.

free energy Forder parameter m