Temperature gradient → Electrical work

Matrix context

D = disequilibrium row · C = coupling family column. A cell reads the state of the atlas for that pair.

D.01 × C.03 · Temperature gradient × Thermoacousticfamily: ThermoacousticD.01 × C.35 · Temperature gradient × Acoustoelectricfamily: Acoustoelectric

Typed bounds on this route

Every limit the atlas holds for this route, as constraint records: reached through bounded_by claims on the route's own entities. Hard bounds (an upper bound or a formula) decide the thermodynamic-bound check above; benchmarks, constitutive relations and resource limits are listed and never decisive. No prose limit exists anywhere (pass 42).

• Carnot limit · formula-bound · hard, applies to this route
  η ≤ 1 − T_c/T_h for any heat engine between reservoirs at T_h and T_c

Steps

1. 01
   Temperature gradient —drives→ Thermoacoustic effectestablishedclaim record

   thermal → acoustic · loss to thermal

   |∇T_crit| = β T_m ω |p₁| / (ρ_m c_p |u₁|); prime-mover onset requires |∇T| > |∇T_crit| · critical-gradient factor β T_m ω/(ρ_m c_p u₁) [K/(Pa·m)]

   p₁ and u₁ are the local first-order acoustic pressure and particle-velocity amplitudes; β is the volumetric thermal expansion coefficient (β T_m ≈ 1 for an ideal gas); this is the linear short-stack / standing-wave criterion of Swift 1988 — above the critical gradient the stack amplifies the wave, below it the same interaction pumps heat — not a closed-form performance law for every travelling-wave regenerator.

   • temperature gradient along the stack steeper than the critical gradient set by the acoustic pressure and displacement amplitudes
   • a resonant gas column

   gas · temperature gradient required

   refs [1]
2. 02
   Thermoacoustic effect —produces→ Acoustic waveestablishedclaim record

   acoustic → acoustic

   gas

   refs [1]
3. 03
   Acoustic wave —drives→ Acoustoelectric effectdemonstratedclaim record

   acoustic → electrical · loss to thermal

   |j_AE| = μ |Γ| I_s / v_s · Weinreich acoustoelectric responsivity μ Γ / v_s [A/W]

   I_s is positive along the acoustic wavevector and Γ > 0 is the intensity attenuation coefficient due to the carriers; momentum transfer is along propagation, so electron drift gives conventional current opposite to propagation while hole current is along it — the magnitude form keeps the carrier sign out of μ.

   • a travelling longitudinal bulk acoustic wave propagates through a semiconductor, or a travelling surface acoustic wave is coupled to mobile carriers
   • net DC Weinreich drag requires directed acoustic momentum; a pure standing wave does not by itself supply the required net momentum
   • a gas-borne acoustic wave may couple mechanically into the solid at a gas–solid boundary; an active piezoelectric launcher is an optional implementation, not part of the acoustoelectric effect

   solid · semiconductor required

   refs [2] [3] [4] [5] [6]
4. 04
   Acoustoelectric effect —produces→ Charge carriersdemonstratedclaim record

   electrical → electrical

   refs [2] [3]
5. 05
   Charge carriers —converts into→ Electrical workestablishedclaim record

   electrical → electrical · loss to thermal

   • carriers driven through an external load at the potential difference the source maintains

   refs [7]

Evidence for the complete composition · 0

No source on record for this exact composition. The sources below support the individual steps, not the route as a whole.

Evidence for the constituent relations · 7

1. [1]
   G. W. Swift. Thermoacoustic engines (1988)

   Journal of the Acoustical Society of America 84, 1145–1180 · doi:10.1121/1.396617 · crossref ✓ 2026-09-19 · record
2. [2]
   G. Weinreich, H. G. White. Observation of the Acoustoelectric Effect (1957)

   Physical Review 106, 1104-1106 · doi:10.1103/PhysRev.106.1104 · crossref ✓ 2026-09-20 · record
3. [3]
   R. H. Parmenter. The Acousto-Electric Effect (1953)

   Physical Review 89, 990-998 · doi:10.1103/PhysRev.89.990 · crossref ✓ 2026-09-20 · record
4. [4]
   G. Weinreich. Ultrasonic Attenuation by Free Carriers in Germanium (1957)

   Physical Review 107, 317–318 · doi:10.1103/PhysRev.107.317.2 · record
5. [5]
   G. Weinreich, T. M. Sanders, H. G. White. Acoustoelectric Effect in n-Type Germanium (1959)

   Physical Review 114, 33–44 · doi:10.1103/PhysRev.114.33 · record
6. [6]
   M. Atashbar et al.. Observation of giant acoustoelectric currents in graphene on lead magnesium niobate–lead titanate (2026)

   Nature Communications 17, 8932 · doi:10.1038/s41467-026-74813-3 · record
7. [7]
   N. W. Ashcroft, N. D. Mermin. Solid State Physics (1976)

   Holt, Rinehart and Winston · book, no DOI · record