Editing Negative temperature (section)

=== Nuclear spins ===
The previous example is approximately realized by a system of nuclear spins in an external magnetic field.<ref name="PuPo" /><ref>{{cite journal|doi=10.1103/PhysRevE.57.6487|title=Minimax games, spin glasses, and the polynomial-time hierarchy of complexity classes|year=1998|last1=Varga|first1=Peter|journal=Physical Review E|volume=57|issue=6|pages=6487–6492|arxiv=cond-mat/9604030|bibcode = 1998PhRvE..57.6487V |citeseerx=10.1.1.306.470|s2cid=10964509}}</ref> This allows the experiment to be run as a variation of [[nuclear magnetic resonance spectroscopy]]. In the case of electronic and nuclear spin systems, there are only a finite number of modes available, often just two, corresponding to [[Spin (physics)#Properties of spin|spin up and spin down]]. In the absence of a [[magnetic field]], these spin states are ''degenerate'', meaning that they correspond to the same energy. When an external magnetic field is applied, the energy levels are split, since those spin states that are aligned with the magnetic field will have a different energy from those that are anti-parallel to it.

In the absence of a magnetic field, such a two-spin system would have maximum entropy when half the atoms are in the spin-up state and half  are in the spin-down state, and so one would expect to find the system with close to an equal distribution of spins. Upon application of a magnetic field, some of the atoms will tend to align so as to minimize the energy of the system, thus slightly more atoms should be in the lower-energy state (for the purposes of this example we will assume the spin-down state is the lower-energy state).  It is possible to add energy to the spin system using [[radio frequency]] techniques.<ref>{{cite book|last=Ramsey|first=Norman F.|author-link=Norman Ramsey|title=Spectroscopy with coherent radiation: selected papers of Norman F. Ramsey with commentary|year=1998|publisher=World Scientific|location=Singapore; River Edge, N.J.|isbn= 9789810232504 |pages=417|oclc=38753008|series=World Scientific series in 20th century physics, v. 21}}</ref> This causes atoms to ''flip'' from spin-down to spin-up.

Since we started with over half the atoms in the spin-down state, this initially drives the system towards a 50/50 mixture, so the entropy is increasing, corresponding to a positive temperature. However, at some point, more than half of the spins are in the spin-up position.<ref>{{cite book|last=Levitt|first=Malcolm H.|title=Spin Dynamics: Basics of Nuclear Magnetic Resonance|year=2008|publisher=John Wiley & Sons Ltd|location=West Sussex, England|isbn= 978-0-470-51117-6|pages=273}}</ref> In this case, adding additional energy reduces the entropy, since it moves the system further from a 50/50 mixture. This reduction in entropy with the addition of energy corresponds to a negative temperature.<ref name="kylma">{{cite web|url=http://ltl.tkk.fi/triennial/positive.html |title=Positive and negative picokelvin temperatures}}</ref>

In [[NMR spectroscopy]], such spin flips correspond to pulses with pulse widths over 180° (for a given spin). While relaxation is fast in solids, it can take several seconds in solutions and even longer in gases and in ultracold systems; several hours were reported for silver and rhodium at picokelvin temperatures.<ref name="kylma" /> It is still important to understand that the temperature is negative only with respect to nuclear spins. Other degrees of freedom, such as molecular vibrational, electronic and electron spin levels are at a positive temperature, so the object still has positive sensible heat. Relaxation actually happens by exchange of energy between the nuclear spin states and other states (e.g. through the [[nuclear Overhauser effect]] with other spins).