Can Temperature Be Negative? Negative Absolute Temperature, Explained

Absolute zero, 0 K, is the bottom of the temperature scale. Nothing can be cooled below it. And yet physicists have reported negative absolute temperatures since 1951, in nuclear spins and, more recently, in clouds of ultracold atoms. Both statements are true at once, because a negative temperature is not a temperature below zero. It is, in a precise sense, hotter than infinity. Our sticker Invertia is that idea turned into a character.

What does temperature actually measure?

In everyday life, temperature tracks how much energy is jiggling around. In statistical physics it is defined more carefully, through entropy S, the logarithm of how many microscopic arrangements match what you see. Temperature is how entropy responds when you add energy E:

1/T = ∂S/∂E

For ordinary matter, adding energy always opens up more arrangements, so S rises with E and T is positive. Our explainer Why Does Time Only Go Forward? explains why entropy matters so much.

How can temperature be negative?

Now imagine a system whose energy has a ceiling: each particle can sit only in a few levels, and there is a highest one. Nuclear spins in a magnetic field are the classic example; each spin is either aligned with the field (low energy) or against it (high energy).

• All spins in the low state: one arrangement, low entropy.
• Half up, half down: the most arrangements, maximum entropy.
• All spins in the high state: one arrangement again, low entropy.

Past the half-way point, adding energy reduces the number of arrangements. ∂S/∂E becomes negative, so by the definition above T is negative. More particles sit in high-energy states than in low ones: a population inversion. The Braun group put it simply: at negative temperatures, “high-energy states are more occupied than low-energy states” (Braun et al. 2013^[1]).

Is negative temperature colder than absolute zero?

No. This is the most common misunderstanding. If a negative-temperature system touches a positive-temperature one, energy flows from the negative one to the positive one. That is the definition of being hotter. The scale is easiest to read with 1/T (sometimes written β): it runs smoothly from +∞ (near absolute zero), down through 0 (infinitely hot), into negative values. Negative temperatures sit beyond infinite temperature, not below zero. Norman Ramsey worked out the thermodynamics of such systems, and the conditions they need, in 1956 (Ramsey 1956^[2]).

Who first made a negative temperature?

1951: nuclear spins in lithium fluoride

Edward Purcell and Robert Pound reversed the magnetic field on the nuclear spins of a lithium fluoride crystal so fast that the spins could not follow, leaving the spin system with more spins in the high-energy state than the low. Their paper is titled “A Nuclear Spin System at Negative Temperature” (Purcell & Pound 1951^[3]). The inverted state lasted minutes, decaying with the crystal’s spin-lattice time constant (Pound, NAS memoir^[4]). It worked because the spins were, for a while, almost isolated from the vibrations of the crystal lattice, which can never be at negative temperature.

2013: atoms that move

Spins have a bounded energy by nature. Ordinary motion does not, since kinetic energy has no ceiling. In 2013 a team in Munich led by Ulrich Schneider and Immanuel Bloch got around this with ultracold bosonic atoms in an optical lattice, tailoring the system so its energy was bounded above. They reported a negative-temperature state for motional degrees of freedom, stable against collapse, and noted that negative temperatures imply negative pressures (Braun et al. 2013^[1]). Headlines at the time spoke of a gas going “below absolute zero”, which is exactly the phrasing that confuses people. The gas was not colder than 0 K.

Is negative temperature even real? (open question)

Here the honest answer is that physicists still argue, and the argument is about definitions of entropy, not about the experiments.

• Jörn Dunkel and Stefan Hilbert argued that the entropy definition usually used (Boltzmann’s, counting states at a given energy) is the wrong one, and that with the Gibbs definition (counting all states up to that energy) “consistent thermostatistics forbids negative absolute temperatures” (Dunkel & Hilbert 2014^[5]).
• Daan Frenkel and Patrick Warren replied that the Boltzmann definition is the physically sensible one, and that negative temperatures are legitimate (Frenkel & Warren 2015^[6]). Robert Swendsen and Jian-Sheng Wang made a similar case (Swendsen & Wang 2016^[7]).
• Eitan Abraham and Oliver Penrose analysed the physics of negative temperatures and also defended them (Abraham & Penrose 2017^[8]).

Nobody disputes the population inversions; lasers depend on them. The live question is whether “temperature” is the right word for them. We label that point open.

Where does this leave absolute zero?

Untouched. You still cannot cool anything to 0 K, and negative temperatures are not a route there. If anything they live at the other end of the scale. For what happens as matter really does approach absolute zero (superfluids, condensates, zero-point energy), read our explainer What Happens at Absolute Zero?

Wear the inversion

Invertia is also on the Energy Accounts sticker sheet, next to Sortex, Maxwell’s demon. Browse more in the Thermal Realm.

FAQ

Can anything be colder than absolute zero?

No. Negative absolute temperatures are hotter than all positive temperatures, not colder than 0 K.

What is needed for a negative temperature?

A system whose energy is bounded above, well isolated from anything that is not, and prepared with more particles in high-energy states than low.

Who made the first negative temperature?

Purcell and Pound, with nuclear spins in lithium fluoride, in 1951.

Is the concept settled?

The experiments are; the interpretation is debated, because it depends on which definition of entropy you use.

References

1. Braun et al. (2013) Negative Absolute Temperature for Motional Degrees of Freedom, Science 339, 52. https://doi.org/10.1126/science.1227831
2. Ramsey (1956) Thermodynamics and Statistical Mechanics at Negative Absolute Temperatures, Phys. Rev. 103, 20. https://doi.org/10.1103/PhysRev.103.20
3. Purcell & Pound (1951) A Nuclear Spin System at Negative Temperature, Phys. Rev. 81, 279 (reversed state decayed with the 5-min time constant). https://doi.org/10.1103/PhysRev.81.279
4. Pound (2000) Edward Mills Purcell 1912–1997, Biographical Memoirs of the National Academy of Sciences 78 (inverted LiF state decayed with the several-minute spin-lattice time constant). https://www.nationalacademies.org/read/9977/chapter/12
5. Dunkel & Hilbert (2014) Consistent thermostatistics forbids negative absolute temperatures, Nature Phys. 10, 67. https://doi.org/10.1038/nphys2815
6. Frenkel & Warren (2015) Gibbs, Boltzmann, and negative temperatures, Am. J. Phys. 83, 163. https://doi.org/10.1119/1.4895828
7. Swendsen & Wang (2016) Negative temperatures and the definition of entropy, Physica A 453, 24. https://doi.org/10.1016/j.physa.2016.01.068
8. Abraham & Penrose (2017) Physics of negative absolute temperatures, Phys. Rev. E 95, 012125. https://doi.org/10.1103/PhysRevE.95.012125

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.