Adiabatic Demagnetization — Cooling Toward Absolute Zero

Magnetic Field

Magnetizing field B_high 5.0 T

Residual field B_low 0.10 T

Cycle Control

Ready. Sample thermalized at bath temperature, field at B_low.

Live Readouts

Temperature T

4.000 K

Field B

0.10 T

Entropy S / Nk

0.692

Cycle n

Spins aligned with field

51.7%

Entropy–Temperature Staircase

How it works

Each spin in the lattice is a two-level paramagnet (up/down) in field B at temperature T. Its dimensionless coupling and per-spin entropy are:

x = (μ_B / k_B) · B / T  ≈ 0.671 K/T · B/T
S(T,B)/Nk = ln(2cosh x) − x·tanh x
P(aligned) = 1 / (1 + e^(−2x))

A cycle has two strokes:

Isothermal magnetization — the sample stays in contact with the bath at temperature T while B rises B_low→B_high. Spins align, entropy drops, and the expelled heat flows into the bath.
Adiabatic demagnetization — the sample is thermally isolated and B falls back to B_low at constant entropy. Since S depends only on B/T in this model, constant S means B/T is unchanged, so cooling the field by a factor r = B_low/B_high cools the sample by the same factor: T_new = T · r.

Running the cycle repeatedly multiplies T by r every time — an infinite sequence of finite steps, each one getting you closer to 0 K but never landing on it. That is exactly the Nernst / Third Law statement: absolute zero is unattainable in any finite number of operations, because it would require an infinite number of magnetization–demagnetization strokes. Real dilution and nuclear-demagnetization refrigerators use this same staircase to reach the millikelvin–microkelvin range.

Aligned with field Thermally disordered

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