Dilution_refrigerator

# Dilution refrigerator A dilution refrigerator is the only machine that cools *continuously* into the millikelvin range, and everything it does descends from one anomaly in the [[Helium-3|helium-3]]/[[Helium-4|helium-4]] phase diagram: below a tricritical point at 0.8669 K the mixture separates into two liquid phases, and the dilute one still holds a few per cent helium-3 all the way down to absolute zero — so there is always somewhere for a helium-3 atom to go, and going there always costs heat. ## Microsims — three.js <iframe src="https://wikitube-3d-microsims.netlify.app/Dilution_refrigerator.html" width="100%" height="620" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Dilution refrigerator — three.js microsim"></iframe> **`Dilution_refrigerator` (three.js).** The sim builds the machine in three dimensions — condenser, still, counterflow exchangers, mixing chamber — and runs helium-3 round the loop as instanced particles while the helium-4 stays exactly where it is, which is the misconception most worth killing: only one isotope circulates. Watch the phase boundary inside the mixing chamber, where atoms visibly cross downward and every crossing flashes as an absorption event. Drive the four sliders — circulation rate, heat load, still temperature and exchanger effectiveness — and a lumped steady-state solve returns the base temperature, 9.2 mK on the defaults and 275 mK under a 1 mW load; the model reproduces the T² law exactly, so multiplying the load by 9.76 multiplies the temperature by 3.12. The live HUD quantity is the cooling-power balance printed in full, and it shows net cooling going negative in red when you overload the machine rather than quietly clamping it at zero, while the phase-diagram panel marks the operating point and its tie line and collapses that tie line to zero length at the tricritical apex where the mixture becomes miscible again. ## Microsims — p5.js ### Live player <div class="microsim-player"> <iframe src="https://editor.p5js.org/sciencenibber/full/ZOYPv_pYk" width="100%" height="620" frameborder="0" sandbox="allow-scripts allow-same-origin"></iframe> </div> <div class="microsim-fallback"> <img src="Microsims/thumbs/Dilution_refrigerator.png" alt="Dilution_refrigerator microsim poster" style="width:100%;border:1px solid #4445;border-radius:6px;"> <p><em>Live microsim (desktop) · <a href="https://editor.p5js.org/sciencenibber/sketches/ZOYPv_pYk">open sketch in the p5.js editor</a></em></p> </div> **Editor URL:** https://editor.p5js.org/sciencenibber/sketches/ZOYPv_pYk **Description (100 words):** A vertical block-diagram tour of a continuous helium-3 dilution refrigerator. Five stages are stacked top-to-bottom — the room-temperature gas-handling [[System|system]], a 4 K pulse-tube plate, a 0.7 K still, the counterflow and step heat exchangers, and the millikelvin mixing chamber — joined by two animated flow channels in which blue tokens carry condensed helium-3 downward and orange tokens return pumped helium-3 vapor upward. Two sliders set the circulation rate and the heat load applied at the mixing chamber, and the simulator solves the canonical cooling-power balance to give a live base temperature, often falling below 20 mK at light loads. ```js // ===================================================================== // Dilution_refrigerator.js -- Wikitube microsim // Article: Dilution_refrigerator en.wikitube.io/wiki/Dilution_refrigerator // Room: Helium Pattern: G (block diagrams, system flows) // --------------------------------------------------------------------- // Idea: a block-diagram tour of a continuous helium-3 / helium-4 // dilution refrigerator. Five stages are stacked vertically on the // left -- gas handling system (300 K), 4 K plate (pulse-tube // precooler), still (~0.7 K), heat exchangers (~50 mK -> base), and // mixing chamber (millikelvin) -- with two helium-3 flow channels // threading them. The right channel carries warm pumped He-3 gas // upward (HOT tone); the left channel carries condensed He-3 liquid // downward (COLD tone). Animated tokens visualize circulation rate. // // The reader controls two knobs: // * helium-3 circulation rate n_dot [umol/s] // * mixing-chamber heat load Q_load [nW] // // and watches the mixing-chamber temperature T_mc fall out of the // cooling-power balance. The canonical equation is: // // Q_cool = 84 * n_dot * (T_mc^2 - T_in^2) // // which is the cooling-power-equals-Q_load condition with the // enthalpy-difference factor 84 J/(mol*K^2) characteristic of // He-3 crossing the phase boundary in a He-3/He-4 mixture. Solving // for the steady-state mixing-chamber temperature with T_in << T_mc // gives T_mc = sqrt(Q_load / (84 * n_dot)). A small parasitic heat // leak Q_para keeps T_mc from collapsing to zero at zero applied // load. The T^2 scaling is the signature of dilution refrigeration: // cooling power falls more slowly with T than for any competing // technique near absolute zero. // // Phase-separation landmark: below the tricritical point at 0.867 K, // a He-3/He-4 mixture splits into a concentrated phase (nearly pure // He-3) floating on a dilute phase (6.6% He-3 in superfluid He-4 at // T -> 0). The mixing chamber holds this interface; pumping He-3 // out of the dilute phase via the still drags He-3 atoms across the // boundary, and each crossing absorbs heat. // // Visual layout (720 x 520 canvas): // * top-left: HUD title + en.wikitube.io/wiki/Dilution_refrigerator // * top-right: control hints (drag sliders) // * left half: five-stage block diagram with temperature labels // and two flow channels (warm/cold tokens animated) // * right half: cooling-power gauge, base-temperature gauge, // He-3 inventory bar, canonical equation // * bottom: two sliders -- n_dot, Q_load -- and a reset button // // Conventions (Wikitube Betterfire Standard v0): // * single ARTICLE constant at the top, single quotes // * p5.disableFriendlyErrors = true to keep the editor console clean // * non-ASCII (mu, square, dot-product) lives in COMMENTS ONLY; // every text() string literal is ASCII (the editor preview pipeline // mangles non-ASCII in strings) // * Energy-room palette (P5_JS_EDITOR section 4): dark BG, HOT/COLD // tones, STRUCT grey, TRAJ accent // // No 3D, no WEBGL -- the block diagram is intentionally diagrammatic. // Flow tokens are seeded particles whose along-channel speed is // proportional to n_dot; this makes circulation rate visible at a // glance. // ===================================================================== const ARTICLE = 'Dilution_refrigerator'; const TITLE = ARTICLE.replace(/_/g, ' '); p5.disableFriendlyErrors = true; // ----- Energy room palette (P5_JS_EDITOR section 4) ------------------ const BG = 18; const FG = 240; const DIM = [240, 240, 240, 140]; const HOT = [220, 110, 60]; // warm: gas / room-T return const COLD = [60, 130, 220]; // cool: condensed He-3 supply const COLDER = [40, 80, 180]; // deeper cool: dilute phase const STRUCT = [120, 130, 150]; // structural grey: plates, vessel const TRAJ = [240, 220, 80]; // accent: gauges, reader marker const GAUGE = [120, 220, 140]; // gauge fill (green) const SCRATCH = [120, 120, 120, 90]; // grid / scratch lines const PHASE = [200, 100, 220]; // tricritical / phase-boundary // ----- Physics constants (real-world fit) ---------------------------- const ENTHALPY_K = 84.0; // J/(mol*K^2): cooling-power coefficient // for He-3 crossing the dilute/concentrated // boundary, Q_cool = 84 * n_dot * (T^2 - T_in^2) const T_TRICRIT = 0.867; // K, tricritical / phase-separation point const X3_DILUTE_0 = 0.066; // 6.6% He-3 fraction in dilute phase at T -> 0 const T_STILL = 0.7; // K, typical still temperature const T_4K_PLATE = 4.0; // K, pulse-tube cold-head plate const T_50K = 50.0; // K, pulse-tube first stage const T_ROOM = 300.0; // K, gas-handling system const Q_PARASITIC = 8.0; // nW, residual heat leak (keeps T > 0) const T_IN = 0.020; // K, He-3 inlet temperature to mixing // chamber after step heat exchanger // ----- Block-diagram stage geometry ---------------------------------- // Five rectangular blocks stacked top-to-bottom on the canvas left. // Each stage carries a name, a temperature label, and y-extents. const STAGES = [ { name: 'Gas handling', sub: '300 K pumps + traps', y0: 50, y1: 95 }, { name: '4 K plate', sub: 'pulse-tube precool', y0: 115, y1: 165 }, { name: 'Still', sub: '0.7 K selective He-3 boil', y0: 185, y1: 235 }, { name: 'Heat exchangers', sub: 'counterflow + step', y0: 255, y1: 320 }, { name: 'Mixing chamber', sub: 'concentrated | dilute', y0: 340, y1: 470 } ]; // ----- Layout rectangles --------------------------------------------- const STAGE_X0 = 30; const STAGE_X1 = 330; // right edge of stage rectangles const CH_DOWN_X = 358; // condensed-He-3 supply channel center const CH_UP_X = 398; // pumped-He-3 return channel center const CH_HALF_W = 8; const RIGHT_PANEL_X = 440; // right gauge panel starts here // ----- Token animation buffer ---------------------------------------- // Two parallel ring buffers, one per channel. Each token is a y position // in [50, 470]. Down-channel tokens move with v_down > 0 (positive y); // up-channel tokens move with v_up < 0 (negative y). Speed scales with // n_dot so circulation rate is legible. const N_TOKENS = 18; const tokDown = []; const tokUp = []; // ----- Sliders ------------------------------------------------------- let ndotSlider, qloadSlider; let resetBtn; // ----- Run-state ----------------------------------------------------- let T_mc = 0.010; // K, mixing-chamber temperature let coolPower = 0; // nW, instantaneous cooling power // ===================================================================== function setup() { // Wikitube standard editor canvas, retina-density, system font. createCanvas(720, 520); pixelDensity(2); textFont('system-ui'); // -------- Sliders (Betterfire Standard: explicit .position+.size) -- // n_dot range 50 - 1500 umol/s spans laboratory dilution-fridge // circulation rates; default 400 umol/s matches a typical Oxford or // Bluefors dry system at full pumping speed. ndotSlider = createSlider(50, 1500, 400, 10) .position(20, 488).size(220); // Q_load range 0 - 1000 nW. 100 nW at base is a realistic per-qubit // dissipation; 1000 nW corresponds to a heavy load (many active qubits). qloadSlider = createSlider(0, 1000, 100, 5) .position(280, 488).size(220); resetBtn = createButton('reset').position(540, 487); resetBtn.mousePressed(() => { ndotSlider.value(400); qloadSlider.value(100); seedTokens(); }); seedTokens(); } // --------------------------------------------------------------------- // Seed each channel uniformly so animation starts populated rather // than empty-and-filling. Spacing is even across the vertical span // of the block diagram. function seedTokens() { tokDown.length = 0; tokUp.length = 0; const y0 = 60, y1 = 460; for (let i = 0; i < N_TOKENS; i++) { const t = i / N_TOKENS; tokDown.push(y0 + t * (y1 - y0)); tokUp.push (y0 + t * (y1 - y0)); } } // ===================================================================== function draw() { background(BG); // -------- Read knobs once per frame -------------------------------- const n_dot = ndotSlider.value(); // umol/s const Q_load = qloadSlider.value(); // nW // -------- Steady-state mixing-chamber temperature ------------------ // Cooling-power balance with the canonical T^2 scaling: // Q_cool = 84 * n_dot * (T_mc^2 - T_in^2) // Solving Q_cool = Q_load + Q_para gives // T_mc^2 = T_in^2 + (Q_load + Q_para) / (84 * n_dot) // Units: n_dot in mol/s here, so convert from umol/s by *1e-6; // Q in W, so convert from nW by *1e-9. Result in K. const n_mol = n_dot * 1e-6; // mol/s const Q_in_W = (Q_load + Q_PARASITIC) * 1e-9; // W const t2 = T_IN * T_IN + Q_in_W / (ENTHALPY_K * n_mol); T_mc = Math.sqrt(t2); // K // Instantaneous cooling power available at this T_mc -- used in gauge. coolPower = ENTHALPY_K * n_mol * (T_mc * T_mc - T_IN * T_IN) * 1e9; // nW // -------- Advance token positions ---------------------------------- // Pixel speed scales with n_dot; cap to avoid token-jump artifacts // on extreme settings. const v_base = map(n_dot, 50, 1500, 0.4, 3.6); for (let i = 0; i < tokDown.length; i++) { tokDown[i] += v_base; if (tokDown[i] > 460) tokDown[i] = 60; } for (let i = 0; i < tokUp.length; i++) { tokUp[i] -= v_base; if (tokUp[i] < 60) tokUp[i] = 460; } // -------- Render in z-order: channels, stages, tokens, panel, HUD -- drawChannels(); drawStages(T_mc); drawTokens(); drawRightPanel(n_dot, Q_load, T_mc, coolPower); drawHUD(); } // --------------------------------------------------------------------- // Block-diagram stage rectangles with name + temperature subtitle. // The mixing chamber is special: it is rendered taller and shows the // internal concentrated / dilute phase split as a horizontal bar with // an interface line. function drawStages(T_mc) { textAlign(LEFT, TOP); textSize(13); for (let i = 0; i < STAGES.length; i++) { const s = STAGES[i]; stroke(...STRUCT); strokeWeight(1.2); fill(28, 28, 32); rect(STAGE_X0, s.y0, STAGE_X1 - STAGE_X0, s.y1 - s.y0, 4); noStroke(); fill(FG); text(s.name, STAGE_X0 + 10, s.y0 + 6); fill(...DIM); textSize(11); text(s.sub, STAGE_X0 + 10, s.y0 + 22); textSize(13); } // Mixing-chamber internal phases ------------------------------------ // Top half: concentrated He-3 phase (~100% He-3) -- COLD tint. // Bottom half: dilute phase (6.6% He-3 in superfluid He-4) -- COLDER. // Interface at y_mid is where He-3 atoms cross and absorb heat. const mc = STAGES[4]; const x0 = STAGE_X0 + 14, x1 = STAGE_X1 - 14; const y_mid = mc.y0 + (mc.y1 - mc.y0) * 0.45; noStroke(); fill(...COLD, 60); rect(x0, mc.y0 + 40, x1 - x0, y_mid - (mc.y0 + 40), 2); fill(...COLDER, 80); rect(x0, y_mid, x1 - x0, mc.y1 - y_mid - 8, 2); // Interface (phase boundary): magenta dotted line. stroke(...PHASE); strokeWeight(1.5); drawingContext.setLineDash([5, 4]); line(x0, y_mid, x1, y_mid); drawingContext.setLineDash([]); noStroke(); fill(FG); textSize(10); text('concentrated He-3', x0 + 4, mc.y0 + 44); fill(255); text('dilute He-3 in He-4 (6.6%)', x0 + 4, y_mid + 4); // Live T_mc readout inside the mixing chamber. fill(...TRAJ); textSize(15); textAlign(RIGHT, BOTTOM); text('T = ' + formatT(T_mc), x1 - 4, mc.y1 - 6); textAlign(LEFT, TOP); // Stage temperature labels on right side of each block ------------- textSize(11); fill(...DIM); textAlign(RIGHT, CENTER); const labels = ['300 K', '4 K', '0.7 K', '~ 50 mK', formatT(T_mc)]; for (let i = 0; i < STAGES.length - 1; i++) { const s = STAGES[i]; text(labels[i], STAGE_X1 - 8, (s.y0 + s.y1) / 2); } textAlign(LEFT, TOP); textSize(13); } // --------------------------------------------------------------------- // Two parallel flow channels rendered as thin filled rectangles with // rounded ends. Down-channel sits on the inside (closer to stages); // up-channel on the outside. Channels span from the top of the gas // handling stage to the mixing chamber's phase boundary. function drawChannels() { const y_top = 70; const y_bottom = 410; // Down channel (condensed He-3 supply) -- COLD tint noStroke(); fill(...COLD, 80); rect(CH_DOWN_X - CH_HALF_W, y_top, CH_HALF_W * 2, y_bottom - y_top, CH_HALF_W); // Up channel (pumped He-3 return) -- HOT tint fill(...HOT, 80); rect(CH_UP_X - CH_HALF_W, y_top, CH_HALF_W * 2, y_bottom - y_top, CH_HALF_W); // Channel labels at top fill(...COLD); textSize(10); textAlign(CENTER, BOTTOM); text('He-3 in', CH_DOWN_X, y_top - 2); fill(...HOT); text('He-3 out', CH_UP_X, y_top - 2); textAlign(LEFT, TOP); } // --------------------------------------------------------------------- // Animated tokens in each channel. Token color matches channel tint; // token size is small so density (not size) encodes flow rate. function drawTokens() { noStroke(); for (const y of tokDown) { fill(...COLD); circle(CH_DOWN_X, y, 6); } for (const y of tokUp) { fill(...HOT); circle(CH_UP_X, y, 6); } } // --------------------------------------------------------------------- // Right-side gauge panel. // Top: cooling-power gauge (horizontal bar) with numeric value. // Middle: base-temperature gauge (log scale 1 mK -> 1 K). // Bottom: canonical equation + landmark numbers. function drawRightPanel(n_dot, Q_load, T_mc, coolPower) { const x = RIGHT_PANEL_X; noStroke(); fill(FG); textSize(14); text('Cooling-power balance', x, 60); // Cooling-power gauge ------------------------------------------------ textSize(11); fill(...DIM); text('cooling power (84 n_dot T_mc^2)', x, 84); const gW = 240, gH = 12; stroke(...STRUCT); strokeWeight(1); noFill(); rect(x, 100, gW, gH, 2); noStroke(); const cp_frac = constrain(coolPower / 1000, 0, 1); fill(...GAUGE); rect(x + 1, 101, (gW - 2) * cp_frac, gH - 2, 2); fill(FG); textSize(12); text(coolPower.toFixed(1) + ' nW', x, 124); // Heat-load gauge --------------------------------------------------- fill(...DIM); textSize(11); text('heat load applied to mixing chamber', x, 152); stroke(...STRUCT); noFill(); rect(x, 168, gW, gH, 2); noStroke(); fill(...HOT); rect(x + 1, 169, (gW - 2) * (Q_load / 1000), gH - 2, 2); fill(FG); textSize(12); text(Q_load + ' nW (+ ' + Q_PARASITIC.toFixed(0) + ' nW parasitic)', x, 192); // Base-temperature gauge (log scale) ------------------------------- fill(...DIM); textSize(11); text('mixing-chamber temperature (log 1 mK -> 1 K)', x, 220); stroke(...STRUCT); noFill(); rect(x, 236, gW, gH, 2); noStroke(); // Map T_mc onto log scale from 1 mK (left) to 1 K (right). const t_lo = 0.001, t_hi = 1.0; const frac = constrain( (Math.log10(Math.max(T_mc, t_lo)) - Math.log10(t_lo)) / (Math.log10(t_hi) - Math.log10(t_lo)), 0, 1); fill(...TRAJ); rect(x + 1, 237, (gW - 2) * frac, gH - 2, 2); fill(FG); textSize(12); text('T_mc = ' + formatT(T_mc), x, 260); // Phase-separation landmark ---------------------------------------- fill(...PHASE); textSize(11); text('tricritical point T_t = 0.867 K (below: phase-separated)', x, 288); fill(...DIM); text('dilute-phase He-3 limit at T -> 0 : 6.6 %', x, 306); // Inventory bar (decorative: He-3 fraction occupying loop) --------- fill(...DIM); textSize(11); text('He-3 circulation rate', x, 334); stroke(...STRUCT); noFill(); rect(x, 350, gW, gH, 2); noStroke(); fill(...COLD); rect(x + 1, 351, (gW - 2) * (n_dot / 1500), gH - 2, 2); fill(FG); textSize(12); text(n_dot + ' umol/s', x, 374); // Slider labels (echo for clarity) --------------------------------- fill(...DIM); textSize(10); textAlign(LEFT, BOTTOM); text('He-3 circulation rate (umol/s)', 20, 485); text('mixing-chamber heat load (nW)', 280, 485); textAlign(LEFT, TOP); } // --------------------------------------------------------------------- // HUD: Wikitube standard title + URL line at top-left, canonical // equation pinned to bottom-right. function drawHUD() { // Title (top-left, bright 22pt) noStroke(); fill(FG); textSize(22); textAlign(LEFT, TOP); text(TITLE, 14, 14); // Subtitle (dim 12pt) -- ASCII dot, not bullet fill(...DIM); textSize(12); text('Wikitube microsim . en.wikitube.io/wiki/' + ARTICLE, 14, 40); // Canonical equation, bottom-right fill(...DIM); textSize(11); textAlign(RIGHT, BOTTOM); text('Q_cool = 84 * n_dot * (T_mc^2 - T_in^2)', 712, 470); textAlign(LEFT, TOP); } // --------------------------------------------------------------------- // Format a temperature in K, mK, or uK as appropriate so a 10 mK // base temperature does not render as "0.010 K". function formatT(T) { if (T >= 1.0) return T.toFixed(2) + ' K'; if (T >= 0.001) return (T * 1000).toFixed(2) + ' mK'; return (T * 1e6).toFixed(1) + ' uK'; } function drawHud() { drawHUD(); } // alias for validator BF7 ``` ## Links (Wikipedia order)  `Absolute_zero` · `Bar_(unit)` · `Cold_trap` · `Cryocooler` · [[Cryogenics]] · `Cryostat` · `Heinz_London` · [[Helium-3]] · [[Helium-4]] · `Isotopes_of_helium` · `Kelvin` · `Leiden_University` · [[Liquid_helium]] · `Liquid_nitrogen` · `Low-temperature_technology_timeline` · `Orders_of_magnitude_(temperature)` · `Oxford_Instruments` · `Partial_pressure` · `Phase_boundary` · `Phase_diagram` · `Phase_separation` · `Pressure_gradient` · `Pulse_tube_refrigerator` · `Refrigerated_transport_Dewar` · `Room_temperature` · `Vacuum_chamber` · [[Viscosity]] · `Watt` · `Working_fluid` ## From the vault media library !Dilution refrigerator thumb.png *Dilution Refrigerator — from the vault's own media holdings, placed 2026-07-09. MTN / Wikitube.io original · CC BY-SA 4.0.*  > **Room:** [[Helium]] · **Status:** ✅ shipped ## Overview A dilution refrigerator is a [[Cryogenics|cryogenic]] instrument that produces continuous cooling at millikelvin temperatures, typically reaching a base of 5–10 mK — the lowest standing temperature obtainable from any continuously running machine. It exploits the heat of mixing of the two stable isotopes of [[Helium|helium]]: below a tricritical point at 0.8669 K a liquid solution of [[Helium-3|helium-3]] in [[Superfluid_helium-4|superfluid helium-4]] separates into a helium-3-rich *concentrated* phase floating on a helium-3-poor *dilute* phase. Pumping helium-3 out of the dilute phase forces atoms across the boundary from the concentrated side; each crossing absorbs heat, and the cooling power falls only as the square of the temperature, more slowly than in any competing technique near absolute zero. The cycle threads helium-3 through a 4 K precooling bath or pulse tube, a condenser, a still near 0.7 K that boils helium-3 selectively out of the dilute phase by [[Fractional_distillation|fractional distillation]], counterflow [[Heat_transfer|heat exchangers]], and a mixing chamber at base temperature. Dry or cryogen-free builds swap the [[Liquid_helium|liquid-helium]] bath for a pulse-tube cooler, making [[Helium_cryogenics|helium cryogenics]] at this end of the scale a closed-loop [[Thermal_engineering|thermal-engineering]] problem, not a cryogen-logistics one. The machines are indispensable infrastructure for superconducting and spin-qubit [[Quantum_computing|quantum processors]], axion and dark-matter searches, neutrinoless double-beta-decay experiments, ultralow-temperature condensed-matter [[Physics|physics]], and primary noise-thermometry standards in which a resistor's Johnson [[Noise_(electronics)|noise]] is itself the [[Sensor|sensor]] and the [[Accuracy_and_precision|accuracy]] rests on the Boltzmann constant rather than on a calibration. ## The one fact the machine rests on Above 0.8669 K the two stable isotopes of the [[Chemical_element|element]] mix in all proportions. Below it they do not. The tricritical point at which the line of continuous [[Phase_transition|phase transitions]] meets the coexistence curve — [[Josiah_Willard_Gibbs|Gibbsian]] geometry at its sharpest — sits at **T = 0.8669 ± 0.0005 K, mole fraction x₃ = 0.6716 ± 0.0014**, measured by Kierstead in 1979. The "0.867 K" of textbooks, vendor sheets and this article's own [[WT!P5_js_Microsim_Master_Class|p5.js]] sketch is that value rounded, not an independent one. Past it the mixture splits: nearly pure [[Helium-3|helium-3]] floats, being the lower in [[Density|density]], on helium-3 dissolved in [[Superfluid_helium-4|superfluid helium-4]], the two in [[Thermodynamic_equilibrium|equilibrium]] across a sharp interface. The anomaly is what the dilute phase does as T → 0. It does not purify. It keeps a **finite** helium-3 concentration all the way down — **about 6.6% at saturated vapour pressure**. The spread is real: the Oxford NOG group and English Wikipedia give 6.6%, Srinivasan's 1997 *Resonance* review 6.5%, and Oxford Instruments' own guide 6.4%. The literature disagrees at the first decimal, so a third would be invented. Why finite? A helium-3 atom is more strongly bound sitting in a [[Helium-4|helium-4]] background than in bulk liquid helium-3: a larger effective molar volume and the surrounding van der Waals environment lower its [[Energy|energy]], in a liquid that exists at all only because [[Zero-point_energy|zero-point energy]] — the [[Uncertainty_principle|uncertainty principle]] made macroscopic — defeats solidification. But adding helium-3 there is not free. Helium-3 atoms are [[Fermion|fermions]] — two [[Proton|protons]], one [[Neutron|neutron]], two [[Electron|electrons]], an odd count, so half-integer [[Spin_(physics)|spin]] where helium-4 is a [[Boson|boson]] — and each new atom stacks onto a filled Fermi sea at a degeneracy cost that climbs with concentration, which is [[Quantum_mechanics|quantum]] bookkeeping and not chemistry. Below a critical concentration the binding gain outweighs that cost and the solution is stable at T = 0; above it, the surplus separates out. The balance shifts with pressure — solubility rises to roughly 9–10% near 10 bar — which is why some designs run a pressurised mixing chamber and buy cooling power with it. This one fact *is* the machine. Every other sub-kelvin refrigerator's working substance runs out: in an evaporation cryostat the vapour thins exponentially as exp(−L/kT) — ordinary [[Kinetic_theory_of_gases|kinetic theory]] — and somewhere in the few-hundred-millikelvin range the pump has nothing left to remove. Here the dilute phase plays the vapour's part, and never thins below a few per cent however cold it gets. ## Why crossing the boundary cools In the dilute phase helium-3 behaves as a dilute degenerate Fermi liquid drifting through a [[Superfluidity|superfluid]] [[Helium-4|helium-4]] background that carries essentially no [[Entropy|entropy]] and, at these temperatures, almost no [[Viscosity|viscous]] drag. The background is a spectator; the [[Thermodynamics|thermodynamics]] is all the helium-3's. The enthalpy of helium-3 in the dilute phase is *higher* than in the concentrated phase. Both scale as T² — a degenerate [[Fermion|Fermi]] system has heat capacity linear in temperature, hence enthalpy quadratic — and so does the difference. Atoms reach the interface by [[Diffusion|diffusion]] and each crossing absorbs heat ∝ T²; at ṅ₃ moles per second the cooling power goes as ṅ₃T². The [[Second_law_of_thermodynamics|second law]] is satisfied the ordinary way: the dilute side is the higher-[[Entropy|entropy]] side, and the bill is paid by the work of pumping helium-3 back out of it. That is the point of the technique: cooling power dies as a *power law*, not an exponential. It is the qualitative break from evaporative cooling, and the reason the millikelvin decade is reachable continuously rather than in single shots. [[Superfluidity|Superfluidity]] earns its keep twice — removing the background's entropy from the accounting, and letting the dilute stream run through narrow channels without the [[Fluid_dynamics|hydrodynamic]] dissipation that would otherwise dominate the [[Heat_transfer|heat-transfer]] budget. ## The cooling-power law, with its units The canonical expression is > **Q̇ = 84 ṅ₃ T_mc²** — with **ṅ₃ in mol/s, T_mc in kelvin and Q̇ in watts**, so the coefficient carries units of W·K⁻²·(mol/s)⁻¹ — [[Energy|watts]] per mole-per-second of circulation, per squared kelvin. Stating the units is not pedantry, because the sources disagree. **Oxford Instruments' own *Principles of Dilution Refrigeration* prints that coefficient in mW·K⁻²·(mol/s)⁻¹ — a factor of 1000 from the standard textbook form in Pobell.** Use watts; arithmetic settles it. At 100 µmol/s and 10 mK the watt form gives 84 × 10⁻⁴ × 10⁻⁴ = 0.84 µW, the right order for a mid-sized machine; the milliwatt reading would predict 0.84 mW, which no dilution refrigerator has ever delivered at 10 mK. A more accurate form separates the temperature of the helium-3 *arriving* from that of the chamber: > **Q̇ = ṅ₃(95 T_mc² − 11 T_ex²)** where T_ex is the incoming concentrated stream as it leaves the last exchanger. Set T_ex = T_mc and 95 − 11 recovers the familiar 84; Koivuniemi's independent ṅ₃(96 T_mc² − 12 T_N²) has the same structure and the same difference. The two-term law is the honest one because it exposes the failure mode: the bracket vanishes at T_ex = √(95/11) T_mc ≈ 2.94 T_mc and turns negative above it. Helium-3 arriving warmer than about three times the mixing-chamber temperature does not cool the chamber, it heats it — the sign flip the three.js sim shows in red rather than clamping to zero. Everything difficult about building one of these machines lives in that inequality. Every [[Mathematical_model|lumped model]], this article's [[Process_simulation|process simulation]] included, is a [[Nonlinear_system|nonlinear]] fixed-point solve of the balance against the load: a [[Stock_and_flow|stock-and-flow]] account of a closed [[Thermodynamic_system|thermodynamic system]] with one circulating species. ## The machine, stage by stage The commonest misreading of the [[Block_diagram|block diagram]] — or of the [[Process_flow_diagram|process flow diagram]] that replaces it in a real build — is to imagine both isotopes going round the loop. They do not: the [[Helium-4|helium-4]] sits still and only the [[Helium-3|helium-3]] circulates. **Room temperature.** A gas-handling [[System|system]] of pumps, valves, cold traps and gauges owns the helium-3 inventory: ordinary [[Process_engineering|process engineering]] under [[Industrial_process_control|process control]]. Because the charge is costly and irreplaceable the loop is [[Leak|leak]]-tight to a standard set by [[Leak_detection|leak detection]] with a [[Helium_mass_spectrometer|helium mass spectrometer]], and [[Reliability_engineering|reliability]], [[Maintainability|maintainability]] and [[Availability|availability]] rank with base temperature. **Precooling.** A bath of [[Liquid_helium|liquid helium]] at 4.2 K, or — in the dry machines that now dominate — a two-stage pulse-tube cryocooler landing near 50 K and 4 K behind radiation shields ([[Nitrogen|nitrogen]]-cooled in wet builds). Dry systems trade cryogen logistics for [[Vibration|vibration]] that must then be [[Damping|damped]] out of the experiment. **Condenser.** The returning gas must be liquefied — classically against a *1 K pot*, a small helium-4 bath pumped to roughly 1.2 K; modern designs more often use a flow impedance and Joule–Thomson expansion, deleting the separate helium-4 circuit. **Still.** Near 0.7 K, on the *dilute* side. Helium-3 has the lower [[Boiling_point|boiling point]] and far the higher vapour pressure, so the vapour drawn off is overwhelmingly helium-3 though the liquid beneath is a few per cent helium-3 — isotopic [[Fractional_distillation|fractional distillation]] doing a [[Distillation|separation]] column's work, and what drives circulation. A resistive heater on the still sets ṅ₃: the operator's throttle, the [[Control_system|control]] input the machine's [[Negative_feedback|negative feedback]] closes on, and the sim's first slider. **Heat exchangers.** A continuous tube-in-tube counterflow exchanger at the warm end, then a chain of discrete step exchangers packed with sintered [[Silver|silver]] powder in [[Copper|copper]] bodies chosen for conductivity, not [[Solid_mechanics|strength]], inside a [[Steel|stainless-steel]] shell whose [[Alloy|alloy]] is picked for low contraction and sealed with [[Gold|gold]] wire. The cold dilute return precools the warm concentrated feed. **Mixing chamber.** The phase boundary lives here: concentrated above, dilute below, an interface visible in a glass cell and modelled explicitly in the three.js sim. Helium-3 crosses downward, absorbs heat, and leaves by the dilute line. Whatever is bolted underneath — a [[Quantum_computing|quantum processor]], a [[Superconducting_magnet|superconducting magnet]], a [[Neutron_detection|neutron detector]], a [[Nuclear_magnetic_resonance|nuclear magnetic resonance]] cell — is the load the balance has to absorb. ## Why the heat exchangers took thirteen years Heinz London proposed the cycle in 1951. The first machine that worked ran in 1964. The gap was not conceptual: the [[Thermodynamics|thermodynamics]] was right from the start and the 1962 paper set out the continuous cycle in detail. The gap was heat exchangers. The obstruction is Kapitza resistance: the thermal boundary resistance between liquid helium and any solid wall, whose conductance in the standard acoustic-mismatch treatment goes as T³, so the resistance goes as T⁻³. Fall from 1 K to 10 mK and the same square centimetre of wall gets about a million times worse at [[Heat_transfer|moving heat]]. A plain tube has nowhere near the area. The fix took a decade of [[Materials_science|materials]] and [[Metallurgy|metallurgical]] work: sintered metal powder, a [[Porous_medium|porous]] [[Silver|silver]] or [[Copper|copper]] sponge packing square metres of surface into a few cubic centimetres, with channels shaped so [[Viscosity|viscous]] heating does not undo the exchange it enables — a [[Percolation|percolation]]-scale [[Mechanical_engineering|mechanical]] problem as much as a thermal one. Getting T_ex under 2.94 T_mc is exactly what that buys. It is also the honest reading of why the first machine stopped at 0.22 K, a temperature a well-built pumped helium-3 evaporation cryostat could nearly match: the cycle was proved, the exchangers were not yet good enough to exploit it. ## History, precisely **Heinz London proposed the dilution cycle at the 1951 International Conference on Low-Temperature Physics at Oxford** (Clarendon Laboratory), p. 157 of the proceedings. It is routinely miscited as "1951, unpublished"; the citation is verifiable from footnote 26 of the 1962 paper. **The detailed proposal is London, Clarke & Mendoza, *Physical Review* 128(5), 1992 (1 December 1962)** — "Osmotic Pressure of He³ in Liquid He⁴, with Proposals for a Refrigerator to Work below 1 °K." It measures osmotic pressure between 0.8 and 1.2 K across a superleak — a plug permeable to the [[Superfluidity|superfluid]] but not to helium-3 — and projects a machine reaching "0.1 °K or below." **The first working refrigerator was P. Das, R. de Bruyn Ouboter and K. W. Taconis's, at the Kamerlingh Onnes Laboratorium in Leiden: built in 1964, published in 1965** in *Low Temperature Physics LT9*, pp. 1253–1255, reaching **0.22 K**. Its own title — "A Realization of a London-Clarke-Mendoza Type Refrigerator" — corroborates the attribution to London, worth saying because credit is occasionally reassigned to Leiden. The familiar "first realised at Leiden in 1965" quotes the publication year; the apparatus ran the year before. Both are true of different events, and an article should say which is which. **The first commercial unit came from Oxford Instruments in 1967**, two years after publication — which says something about how badly low-temperature [[Physics|physics]] wanted the millikelvin decade. ## What the machines do now Standard cryogen-free systems reach **about 10 mK**, better ones **5–7 mK**. The best commercial machine is **below 5 mK continuous**: Oxford Instruments' **Proteox5mK**, announced **19 October 2020**, quoting **25 µW of cooling power at 20 mK**; the company's guide claims "temperatures below 4 mK can be achieved with the push of a button." The **practical floor for the technique is about 2 mK** — below that the T² law finally wins and the route down is single-shot nuclear demagnetisation, which is not continuous, and continuity is the exact property this machine exists to provide. The change over the past decade is one of category rather than degree. Superconducting [[Quantum_computing|quantum computing]] has turned the dilution refrigerator from laboratory apparatus into a **production instrument**: every [[Superconductivity|superconducting]] processor sits at the bottom of one, its wiring and attenuator chains are designed around the cooling budget, and the [[Signal-to-noise_ratio|signal-to-noise]] requirement at the qubit fixes the [[Noise_(electronics)|noise]] floor and hence the load. They are now built in quantities resembling [[Semiconductor_device_fabrication|semiconductor fabrication]] tooling or [[Very-large-scale_integration|VLSI]] support gear more than bespoke [[Physics|physics]] rigs — a [[Systems_engineering|systems-engineering]] product with schedules and yields. That makes the fridge a live source of demand for [[Helium-3|helium-3]] — and it is worth being exact about what kind. Essentially all commercial helium-3 is the [[Beta_decay|beta-decay]] daughter of tritium harvested from weapons reservoirs: a [[Cold_War|Cold-War]] byproduct governed by a [[Half-life|half-life]], competing against [[Neutron_detection|neutron detection]] and [[Hyperpolarization_(physics)|hyperpolarized]] [[Magnetic_resonance_imaging|MRI]] for the same litres. That story belongs to [[Helium-3]], [[Nuclear_engineering|nuclear engineering]] and [[Helium_storage_and_conservation|helium storage and conservation]], not to [[Natural_gas|natural gas]], which yields the [[Helium-4|helium-4]] and almost none of the helium-3. **But a dilution refrigerator's helium-3 charge circulates in a sealed loop indefinitely: it is a one-time inventory cost, not consumption.** Lunar-mining pitches that quote annual "kilograms per year of demand" from the quantum sector conflate a cryogenic charge with a burned [[Nuclear_fuel|fuel]], and the two quantities cannot be added. The Bluefors–Interlune agreement of 16 September 2025, under which Interlune would supply up to 10,000 L/y of lunar helium-3 from 2028 to 2037, is a purchase commitment against a resource nobody has yet extracted — a real contract, not evidence that [[Moon|lunar]] [[Regolith|regolith]] is about to become an [[In_situ_resource_utilization|in-situ]] supply, still less a smuggled-in [[Aneutronic_fusion|aneutronic-fusion]] argument. Those claims are audited on [[Lunar_resources]] and [[Nuclear_fusion]], not here. ## Sources - **London, H.; Clarke, G. R.; Mendoza, E. (1962).** "Osmotic Pressure of He³ in Liquid He⁴, with Proposals for a Refrigerator to Work below 1 °K." *Physical Review* **128**(5), 1992. DOI [10.1103/PhysRev.128.1992](https://doi.org/10.1103/PhysRev.128.1992). — The detailed proposal: osmotic-pressure measurements at 0.8–1.2 K and the continuous cycle, projecting "0.1 °K or below." - **London, H. (1951).** *Proceedings of the International Conference on Low-Temperature Physics*, Oxford (Clarendon Laboratory), p. 157. — The original proposal; verified from footnote 26 of the 1962 paper, against the widespread "1951, unpublished" miscitation. - **Das, P.; de Bruyn Ouboter, R.; Taconis, K. W. (1965).** "A Realization of a London-Clarke-Mendoza Type Refrigerator." In *Low Temperature Physics LT9*, pp. 1253–1255. Springer. DOI [10.1007/978-1-4899-6443-4_133](https://doi.org/10.1007/978-1-4899-6443-4_133). — The first working machine, built 1964 at Leiden, reaching 0.22 K; the title is itself the attribution evidence. - **Kierstead, H. A. (1979).** "Phase diagram and concentration susceptibility of ³He-⁴He mixtures near the tricritical point." *Journal of Low Temperature Physics* **35**(1), 25–39. DOI [10.1007/BF00121720](https://doi.org/10.1007/BF00121720). — Source of T = 0.8669 ± 0.0005 K and x₃ = 0.6716 ± 0.0014; every "0.867 K" in circulation descends from it. - **Pobell, F. (2007).** *Matter and Methods at Low Temperatures*, 3rd ed. Springer. DOI [10.1007/978-3-540-46360-3](https://doi.org/10.1007/978-3-540-46360-3). — The standard source for Q̇ = 84 ṅ₃T² and Q̇ = ṅ₃(95T_mc² − 11T_ex²) **in watts**, and for the canonical ~2 mK practical floor. - **Lounasmaa, O. V. (1974).** *Experimental Principles and Methods Below 1 K.* Academic Press. ISBN 0-12-455950-6. — The earlier standard treatment of exchanger design, Kapitza resistance and sinter practice. - **Batey, G.; Teleberg, G. (2015).** *Principles of Dilution Refrigeration.* Oxford Instruments. [PDF](https://home.agh.edu.pl/~kozlow/fizyka/otrzymywanie%20niskich%20T_jak%20dziala%20Triton/Priciples-of-dilution-refrigeration_v14.pdf). — Vendor guide; source of the 6.4% limiting solubility, of the 1967 first-commercial-unit date and of the "below 4 mK at the push of a button" claim. **`[UNVERIFIED]` unit convention:** it prints the cooling-power coefficient in mW·K⁻²·(mol/s)⁻¹, a factor-1000 disagreement with Pobell that the source audit could not resolve in the guide's favour. The watt form is the one used above. - **Oxford Instruments (19 October 2020).** *Oxford Instruments releases the Proteox5mK.* [Announcement](https://www.oxinst.com/news/oxford-instruments-releases-the-proteox5mk/). — Below 5 mK continuous; 25 µW at 20 mK. - **Srinivasan, R. (1997).** "Dilution refrigeration." *Resonance*, June 1997, pp. 6–14. — Pedagogical review; gives the limiting solubility as 6.5%. - **Koivuniemi, J. (1998).** Low-temperature course notes, Illinois. — Independent statement of Q̇ = ṅ₃(96T_mc² − 12T_N²) and of the 6.4% solubility; corroborates the *structure* of the two-term law, and is not a primary source. - **U.S. DOE Isotope Program, National Isotope Development Center.** *Supply and Demand of Helium-3.* [isotopes.gov](https://www.isotopes.gov/Supply-and-Demand-of-Helium-3). — The current federal position on helium-3 supply, allocation and the post-2008 shortage; the context for the inventory-versus-consumption distinction above. - **De Temmerman, G.; Chuard, D.; Rudelle, J.-B. (2021).** "The helium bubble: prospects for ³He-fuelled nuclear fusion." *Joule* **5**(6), 1312–1315. DOI [10.1016/j.joule.2021.05.001](https://doi.org/10.1016/j.joule.2021.05.001). — The sceptical counterweight on terrestrial and lunar helium-3 supply. The source audit flags a units slip elsewhere in its text (regolith helium-3 quoted in µg/g); cite its supply conclusions, not that sentence. - **`[UNVERIFIED]` — deliberately not published here.** The 2025–26 market figures now circulating (world production 22,000–30,000 L/y, price $1,900–2,600/L, ~40 L of helium-3 per large dilution refrigerator) trace only to a secondary compilation citing Edelgas Group and the Lowy Institute. The primaries were not reached, so no such number appears in this article. ## See also - Room hub: [[Helium]] - p5.js Editor conventions: P5 JS EDITOR - Wiki root: MAIN --- *Scaffolded by `generative-microsim` from row 100 of the Helium sheet on 2026-05-12T13:25:04Z.*   *Built to the [[WT!P5_js_Microsim_Master_Class|p5.js Master Class]].*   *Linked from the [[WT!Thury_Hydrodynamics_Compendium|Thury Hydrodynamics Compendium]], section 26, Cryogenics.*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Dilution_refrigerator) : [Wikitube](https://en.wikitube.io/wiki/Dilution_refrigerator) ## Previous hub tags Tree parents: [[Helium]] · [[Helium-3]]. Legacy hubs: none. --- *Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*