Positive and negative picokelvin temperatures

Nuclear magnetic ordering has been investigated at T>0 in copper and more recently at T>0 and at T<0 in silver and rhodium. In copper and silver, competition between the dipolar force and the conduction-electron mediated exchange interaction leads, at T>0, to antiferromagnetic structures. At negative spin temperatures, ferromagnetic order has been found in silver. In this metal, the phase transitions occur at 560 pK and at -1.9 nK, respectively. In rhodium, 280 pK and -750 pK have been reached; these are the current world records of low temperatures.

Fig. 8. Schematic illustration, on a temperature vs. entropy diagram, of the procedure for cooling an assembly of silver or rhodium nuclei to negative nanokelvin temperatures. At left, the energy level diagram of the nuclear spins is shown.
(H -> A) Both nuclear stages, are cooled to Ti = 15 mK by the dilution refrigerator and (A -> B) polarized in a magnetic field slowly increased to Bi = 8 T.
(B -> C) The nuclei of the first stage, made of 20 mol of copper, are adiabatically demagnetized to Bf = 100 mT, producing a low temperature Tf = Ti(Bf /Bi) = 200 µK.
(B -> D) The 2-gram silver or rhodium specimen of thin polycrystalline foils, acting as the second nuclear stage, cools under the field Bi = 8 T, by thermal conduction, to Tf = 200 µK.
(D -> E) The specimen is adiabatically demagnetized from 8 T to 400 µT, whereby the spin assembly cools into the nanokelvin range, thermally isolated by the slow spin-lattice relaxation ($\tau$1 = 14 h) from the conduction electrons which are anchored to 200 µK by the first nuclear stage at (C). By continuing demagnetization of the specimen to zero field (not shown in the diagram), the record temperature of 280 pK was reached in rhodium.
(E -> F) Finally, production of the negative temperature is achieved in the system of silver or rhodium nuclei by flipping the 400-µT magnetic field in about 1 ms. The rapid inversion results in some loss of polarization. By continuing demagnetization to zero field, the record temperature of -750 pK was reached in rhodium (not shown in the diagram).
(F -> G -> H -> A) The system starts to lose its negative polarization, crossing in a few hours, via infinity, from negative to positive temperatures.
(C -> A) The first nuclear stage warms slowly, under the 100 mT field, from 200 µK to 15 mK. A new experimental sequence can then be started.

The energy level diagram for an assembly of silver or rhodium nuclei is illustrated schematically in the left part of Fig. 8; the spin I = 1/2, so there are just two levels, corresponding to the nuclear magnetic moment m parallel and antiparallel to the external magnetic field B. The distribution of the nuclei among the Zeeman energy levels is determined by the Boltzmann factor, exp(m·B/kBT). At positive temperatures the number of nuclei in the upper level, with m antiparallel to B, is always smaller than in the lower. At the absolute zero, all nuclei are in the ground level with m parallel to B.

As the temperature is increased from T= +0, nuclei flip into the upper energy band and, at T = +$\infty$, there is an equal number of spins in both levels. When the energy is increased further, the distribution of the nuclear spins can still be described by the Boltzmann factor but now with T<0. Finally, when approaching zero from the negative side, T-> -0, eventually only the highest energy level is populated. Since heat is transferred from the warmer to the colder part when two systems are brought into thermal contact, negative temperatures are actually hotter than positive ones. The crystalline lattice and conduction electrons cannot be cooled to a negative temperature because in this case the energy spectrum has no upper bound: at T<0 the energy of the system would be infinite.

At T = +0, an isolated nuclear spin assembly thus has the lowest and, at T = -0, the highest possible Helmholtz free energy. The tendency to maximize energy, instead of minimizing it, is the basic difference between negative and positive temperatures. In silver, the nearest-neighbor antiferromagnetic Ruderman-Kittel exchange interaction, three times stronger than the dipolar force, favors antiparallel alignment of the nuclear magnetic moments and thus leads to antiferromagnetism when T>0. At T<0, since the Helmholtz free energy now must be maximized, the very same interactions produce ferromagnetic nuclear order.

Experimentally, population inversion from T>0 to T<0 is possible in silver and rhodium (see Fig. 8). This can be accomplished at ultralow temperatures by reversing the magnetic field B quickly, in a time t << $\tau$2 = 10 ms, where $\tau$2 is the spin-spin relaxation time, so that the nuclei have no chance to rearrange themselves among the energy levels. In a certain sense, the system passes from positive to negative temperatures via T = +$\infty$ = -$\infty$, without crossing the absolute zero. Therefore, the third law of thermodynamics is not invalidated. In copper $\tau$2 is 100 times shorter than in silver. Producing spontaneous nuclear ordering at T<0 is thus more difficult in copper.

To obtain nuclear temperatures in the nano- and picokelvin range, an elaborate "brute force" cooling apparatus, with a dilution refrigerator and two nuclear refrigeration stages in cascade, was employed. The cooling procedure is described in detail with the aid of Fig. 8.

One of the difficult problems in these experiments is to measure the absolute temperature of the nuclei. The second law of thermodynamics, T=$\Delta$Q/$\Delta$S, was used directly, i.e., the nuclear spin system was supplied with a small amount of heat $\Delta$Q and the ensuing entropy increase $\Delta$S was calculated from the measured decrease of polarization.

NMR measurements showed that, instead of absorption as at T>0, the spin system in silver was emitting energy when T<0. In rhodium antiferromagnetic susceptibility was found at T>0 while, at T<0, the ferromagnetic tendency changed at |T| = 5 nK into a tendency towards antiferromagnetism; phase transitions were not seen.

Fig. 9. Phase diagram of nuclear spins in silver for positive (blue) and negative (red) absolute temperatures in the magnetic field (B) versus reduced entropy (S/R ln2) plane. The ferromagnetic spin arrangement at T<0 is illustrated in the insert: across a domain boundary the tangential component of magnetization is continuous while the perpendicular component changes sign.

It has been argued that negative temperatures are fictitious quantities because they do not represent true thermal equilibrium in the sample consisting of nuclei, conduction electrons, and the lattice. However, the experiments on silver, in particular, show conclusively that this is not the case. The same interactions produce ferro- or antiferromagnetic order in the spin assembly, just depending on whether the nuclear temperature T<0 or T>0 (see Fig. 9).

Metals offer good models to investigate magnetism in general. The nuclei are well localized, they are isolated from the electronic and lattice degrees of freedom at low temperatures, and the interactions between nuclei can be calculated from first principles. Therefore, these systems are particularly suitable for testing theory against experiments. The realm of negative spin temperatures offers new possibilities for studies of magnetism.

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