IN37 · Intercept heat along the wires

Dilution Refrigerator

Why can a bundle of wires warm the coldest sample stage? Wires conduct heat from warmer stages to the sample, adding to the heat the refrigerator must remove. A clamp on an intermediate cold plate intercepts much of that heat. Poor thermal contact can also leave the sample warmer than the mixing chamber.

Watch helium-3 cross a liquid phase boundary and carry heat from the mixing chamber. Drag the wire clamp to compare heat interception, weak exchange and poor sample contact.

Where does the wire heat go?

Drag the gold wire clamp left to loosen it. The plots retain three heater steps for each setting.

Sweep-end mixing chamber $23.0\,\mathrm{mK}$

Sweep-end sample $26.3\,\mathrm{mK}$

Wire heat into mixing chamber $8.88\,\mathrm{nW}$

Conditional cooling $13.0\,\mathrm{\mu W}$

Heater-step slope $24.6\,\mathrm{mW\,K^{-2}}$

Helium-3 flow

\dot n_3/(\mathrm{mmol\,s^{-1}})

0.3

First heater step

P_h/\mathrm{\mu W}

2

Wire clamp contact

a

1

Final heat exchange

\epsilon

1

Wire count

N

96

Sample contact and thermometry

Sample contact

K_4/(\mathrm{W\,K^{-4}})

5

Thermometer excitation

I/\mathrm{nA}

0.1

Noise seed

s

370210

Model assumptions

Starts with established liquid phases in an already precooled 20–40 millikelvin operating window. Uses the 1971 NBS low-temperature 94/12 enthalpy approximation, its equal-temperature coefficient 82, and a dilute heat-exchanger stream heat capacity of 107.2 times temperature. Exchanger effectiveness, wire conductivity, node heat capacities, sample contact and thermometer curves are declared synthetic teaching parameters. Mixing chamber and sample store energy; the fixed four-kelvin boundary and 80-millikelvin cold plate receive wire heat through a finite, storage-free junction. Upper stages are controlled boundaries and upper pipe routes are schematic. Room-temperature precooling, pumping hydrodynamics, liquid levels, viscous heating and commercial performance are omitted. Integration stops at the temperature boundary. Reversed four-wire observations, independent bath standards, held-out checks and a self-heating check support thermometry. Three actually settled heater steps test differential cooling; absolute cooling remains conditional on the stated enthalpy law and flow calibration. Geometry is defined in metres; wire thickness and flow markers are enlarged. Replay runs at four times physical time.

Already precooled. The apparatus replays the first heater step at four times physical time. Surface tint follows model temperature; pipes and markers are enlarged.

Display run

Import raw run

Temperature reconstructed from reversed sensor observations

Mixing chamberSample

$t/\mathrm{s}$ · $T/\mathrm{mK}$

Recorded heater steps and conditional mixing-chamber cooling

Measured heater powerEnthalpy model with measured flow and temperature

$t/\mathrm{s}$ · $P/\mathrm{\mu W}$

Raw observations, standards and uncertainty

Each sensor has three fitted reference baths, two withheld temperatures, a second excitation and an actual bath-step response check. Flow, current and voltage each carry separate fit and withheld standards. Reference baths and uncertainties are declared synthetic inputs.

Scroll the table sideways to see every column.

First eight raw observations; full channels and references are exported
$t/\mathrm{s}$	$R_m/\Omega$	$R_s/\Omega$	$V_h/\mathrm{V}$	$I_h/\mathrm{\mu A}$	Flow signal $V/\mathrm{V}$

Scroll the table sideways to see every column.

Eight independently calibrated thermometer channels
Sensor	$T/\mathrm{K}$	Withheld error $\Delta T/\mathrm{K}$	Excitation check $\Delta T/\mathrm{K}$

Thermal assumptions and source range

The 1971 NBS cooling coefficients are 94 and 12; the equal-temperature difference is 82. The dilute exchanger stream uses heat capacity 107.2 times temperature. The finite junction sends wire heat into the fixed cold plate, whose external cooler removes that heat. Wire conductivity, thermal masses and contact coefficients are synthetic. Quantitative integration covers the two lowest nodes from 20 to 40 millikelvin.

Three settled heater steps test a differential cooling slope. The absolute cooling curve uses the stated enthalpy law with calibrated flow and temperature. A single temperature does not determine the unknown parasitic heat.

Start with the essentials

Focus question: Why can a bundle of wires warm the coldest sample stage?
One-sentence intuition: A thermal anchor diverts wire heat to an intermediate plate; the sample also needs good contact with the mixing chamber.

Core mathematical model

Heat absorbed in the mixing chamber

\dot Q_m=\dot n_3(94T_m^2-12T_i^2)

This 1971 NBS approximation is used only in the selected 20–40 millikelvin window. Temperatures are kelvin and flow is moles per second. The coefficients have units of joules per mole per kelvin squared. Warmer incoming liquid reduces cooling.

Heat conducted along a wire

\dot Q_{u\to v}=\frac{A}{L}\int_{T_v}^{T_u}\kappa(T)\,dT,\qquad\kappa(T)=k_1T

Larger area and shorter length carry more heat. The synthetic effective alloy is used from 20 millikelvin to four kelvin. A finite anchor junction must send away exactly the heat that arrives.

Energy stored in two nodes

U_i=\tfrac12\gamma_iT_i^2,\qquad\dot U_m+\dot U_s=P_{\mathrm{in}}+\dot Q_{\mathrm{wire}}-\dot Q_m

Sample-to-mixing heat cancels when the nodes are added. Electrical heat, parasitic heat and thermometer dissipation enter; helium circulation removes energy.

Resistance from reversed excitation

R=\frac{V_+-V_-}{I_+-I_-}

Current reversal removes a stable thermoelectric offset. Each sensor has its own reference baths. Held-out temperatures and a second excitation check must pass before resistance is reported as a platform temperature.

Common difficulties

The sample temperature

Typical misconceptionA cold mixing chamber guarantees an equally cold sample.

Better mental modelA heated sample passes heat back through a finite contact. That heat needs a temperature difference, so the sample can remain warmer.

What heater steps determine

Typical misconceptionOne temperature determines every heat leak.

Better mental modelSettled heater steps show how an added heat load changes temperature. An absolute heat leak also needs independent background and conductance measurements.

Run the experiment

01
Loosen the wire clamp
Hold the gold clamp and drag left, or focus the canvas and press the left arrow.
What to observe: Weaker anchor contact warms the junction and sends more wire heat to the mixing chamber. The thermal graph and newly acquired curve change before release.
02
Stop circulation
Choose Circulation stops and follow the end of the temperature trace.
What to observe: Positive heat input keeps warming the nodes. The run stops at the 40-millikelvin boundary and withholds a cooling inference.
03
Compare the sample and mixing chamber
Choose Weak sample contact and inspect the sample mount.
What to observe: The same sample heater needs a larger temperature difference to pass its heat through weaker contact.
04
Check thermometer excitation
Choose Thermometer self-heating and open the raw observations and standards.
What to observe: The sensor warms itself and fails the independent high-excitation bath check. Raw resistance remains available; platform thermometry is withheld.
05
Keep a cooling scan
Keep the run, change the anchor, select the old record and export CSV or JSON.
What to observe: The three heater steps, unsettled observations, reversed currents and references remain fixed and support independent replay.

Dilution Refrigerator

Temperature reconstructed from reversed sensor observations

Recorded heater steps and conditional mixing-chamber cooling

Follow heat into the coldest stage

Core mathematical model

Heat absorbed in the mixing chamber

Heat conducted along a wire

Energy stored in two nodes

Resistance from reversed excitation

Common difficulties

The sample temperature

What heater steps determine

Run the experiment

Loosen the wire clamp

Stop circulation

Compare the sample and mixing chamber

Check thermometer excitation

Keep a cooling scan

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