The Casimir Torque Bound Between Anisotropic Plates: Calcite at 50 nm to 1 µm, Its Use for Micro-Mechanical Devices, and the Effect of Surface Roughness

The Casimir torque between two birefringent plates twisted relative to each other is derived from the Lifshitz framework in the weak-anisotropy approximation and evaluated for two calcite plates across a vacuum gap at 300 K, using a stated two-oscillator model dielectric function for each principal axis of calcite (static principal values 8.50 and 8.00; optical values 2.75 and 2.21). The maximum torque per unit area, at a twist of 45 degrees, is 8.58e-9 N/m at 50 nm, 1.06e-9 N/m at 100 nm, 1.18e-10 N/m at 200 nm and 4.14e-13 N/m at 1 micrometre, with a local power-law exponent steepening from 3.0 to 3.8. The torque follows -sin(2 theta) and restores the plates toward aligned optic axes. For a one-millimetre square plate the total maximum torque is 8.6e-15 N m at 50 nm, deflecting a soft torsional suspension (1e-12 N m/rad) by 8.6e-3 rad but a stiff micro-mechanical element (1e-9 N m/rad) by only 8.6e-6 rad: the torque is usable only as an alignment or measurement torque for a large, softly suspended plate at or below about 100 nm, and not as an actuator for ordinary stiff micro-mechanical devices. Surface roughness slightly increases the height-averaged torque at fixed mean gap, but derates the usable torque to a factor 0.37 at 50 nm with 10 nm rms roughness once the closest asperities must be kept clear of contact. All derived numbers are produced by the deposited compute.py. This is a calculation, not a measurement: simulated, not measured. AI use is disclosed: this paper was produced by Dave, an AI system, which carried out the derivation, computation and drafting under the direction of the named author. The author, William Alexander Aldred, is sole author of record.