Phonon Thermal Conductivity — the Umklapp Peak

Material

Conditions

Temperature T 80 K

Grain size L (boundary path) 300 nm

Live Readouts

Conductivity κ(T)

— W/m·K

Mean free path ℓ

— nm

Heat capacity C_v/C_Dulong-Petit

—%

κ(T) — dot = current T, dashed = peak T_peak

How it works

Heat in an insulating crystal is carried by phonons — quantized lattice vibrations that behave like a gas of particles with a kinetic-theory conductivity:

κ = (1/3) · C_v · v · ℓ

C_v(T) is the Debye heat capacity (rises as T³ at low T, saturates at the Dulong–Petit value 3nR):

C_v = 9nR(T/θ_D)³ ∫₀^(θ_D/T) x⁴eˣ/(eˣ−1)² dx

ℓ is the phonon mean free path, set by two competing scattering mechanisms combined via Matthiessen's rule:

1/ℓ = 1/ℓ_boundary + 1/ℓ_Umklapp(T)
ℓ_Umklapp(T) ∝ exp(θ_D/3T) / T

Low T: few phonons exist (C_v tiny) even though ℓ is long — phonons fly ballistically until they hit a grain boundary. κ is small and rises as T³.
High T: C_v has saturated, but phonon-phonon Umklapp collisions (which flip crystal momentum by a reciprocal-lattice vector) multiply fast, collapsing ℓ. κ falls roughly as 1/T.
The crossover between these two regimes produces the characteristic conductivity peak seen in every real crystal's κ(T) curve, typically a few tens of kelvin below θ_D.
The bright sphere in the scene is a single phonon on a schematic random walk — its step length is ℓ(T) mapped onto the visible chain; the glow cloud's brightness/count tracks the live phonon population C_v(T)/C_v,max.

Phonon packet Boundary scatter Umklapp scatter

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