---
title: "Dilution Refrigerator Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/dilution-refrigerator/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/dilution-refrigerator/"
last_modified: "2026-11-12"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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 chamber Sample

$t/\mathrm{s}$ · $T/\mathrm{mK}$

### Recorded heater steps and conditional mixing-chamber cooling

Measured heater power Enthalpy 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.

Physics tutorial

## Follow heat into the coldest stage

**Background** Helium-3 absorbs heat as it crosses from the upper concentrated liquid into the lower dilute liquid. Circulation removes helium-3 from the dilute side and returns it to the concentrated side.

**Why it matters** The sample still needs wiring and a thermometer. Each connection adds a possible route for 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 misconception** A cold mixing chamber guarantees an equally cold sample.

**Better mental model** A 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 misconception** One temperature determines every heat leak.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

Physical conclusion

Read the sample temperature together with the mixing-chamber temperature, the wire route and the thermometer self-heating check.

## Related experiments and articles

How physics instruments measure · Reading path

### How physics instruments measure

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How a dilution refrigerator keeps carrying heat away**](https://sandboxphysics.com/stories/instruments/dilution-refrigerator/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)