Re: [Phys-l] temperature : definition and fundamental properties

[John Denker <jsd@av8n.com>]

On 01/25/2012 05:51 AM, chuck britton wrote:
> Just to shift the topic maybe a little (or maybe not?) could you expound a bit
>   on the practical problem of thermal equilibrium of a substance that 
> is cooled by nuclear demagnetization?
> Temperatures down into the microKelvin.

Depending on how you keep score, the world record might be
quite a bit lower than that ... more like 0.1 nanokelvin, 
i.e. 100 picokelvin.  Reference:
  http://ltl.tkk.fi/wiki/LT/%C2%B5KI_Group/Cryogenics

Note that the thermodynamics of a demagnetization fridge is
profoundly analogous to the thermodynamics of the rubber band 
that was discussed yesterday.  You pull on the rubber band.
It gets hot.  You suck away the heat.  You relax the rubber
band.  It gets cold.  At this level of detail, all refrigerators
are the same.  As RPF was fond of saying:  the same equations
have the same solutions.

> Are the electrons at the same temperature?

Well, that depends on how you keep score.  As mentioned in the 
aforementioned article from Otaniemi, there are two scenarios: 
 a) the refrigerator refrigerates something OTHER than itself, 
versus
 b) the coolant cools only itself.

Concept-wise, everybody in the business recognizes this distinction.

Terminology-wise, some folks call (a) refrigeration and (b) cooling,
but not everybody uses this terminology, and nobody I know uses it
consistently.  I am *not* going to use this terminology.  I will use
heating and cooling to refer to any change in temperature.  If/when
I need to say _what_ is being cooled, I will spell it out.

As the temperature decreases, the spin-lattice coupling gets weaker
and weaker, so it becomes harder and harder to use the nuclei to
cool anything other than the nuclei themselves.  
 a) The record low _lattice_ temperature is on the order of 1 microkelvin
 b) The picokelvin experiments involve the nuclei only, quite decoupled 
  from the host lattice.

In both cases, the state of the art in the demagnetization world
(i.e. excluding laser trapping and cooling) is /double/ demag.
That is, an ordinary dilution refrigerator is used to pre-cool
the first demag stage, which is then demagnetized to pre-cool
the second demag stage, which is then demagnetized to reach the
lowest temperature.

In cases where you want the spins to couple to the lattice, it
takes time on the order of days for the two subsystems to come 
into some semblance of equilibrium.  This makes for some painfully 
slow experiments.

> What are the thermal wavelengths of the various constituents?

For a classical demag stage, made of copper or the like, I'm not
so sure the thermal de Broglie length is relevant, or even well 
defined.  AFAICT the thermal de Broglie length is only relevant 
in the context of exchange, and in a solid the nuclei aren't 
undergoing any exchange.  (I've never really thought about this, 
but there's a reason why not.)

As for the conduction electrons in such a system, they are a
degenerate Fermi gas.  They behave as either a normal metal, or
a ferromagnet, or a superconductor.  I don't think there are any
other possibilities.

Meanwhile, there is such a thing as
  Medley et al.
  "Spin gradient demagnetization cooling of ultracold atoms"
  http://arxiv.org/abs/1006.4674

That involves the entropy of mixing -- on top of laser trapping and
cooling -- rather than classical nuclear demagnetization.  So if you 
are looking for interesting topic-drift and/or looking for seriously 
low temperatures, that's a good place to look.

In general, independent of what mechanism is used to cool things,
all of the following are related to exchange and degeneracy, all of
which is related to the thermal de Broglie length-scale:
 -- conduction electrons are already degenerate at room temperature
 -- exchange can lead to ferromagnetic ordering of conduction 
  electrons, with a transition temperature possibly above room 
  temperature, or possibly much lower.
 -- electron Cooper pairs can go degenerate resulting in super-
  conductivity at temperatures on the order of a few kelvin, 
  sometimes higher, sometimes much lower.
 -- liquid 4He goes superfluid at a few kelvin.
 -- liquid 3He goes superfluid (Cooper pairs) at a few millikelvin.
 -- gas-phase atoms can go fully degenerate at low enough temperatures:
  http://www.nobelprize.org/nobel_prizes/physics/laureates/2001/

Degeneracy just means that we have more than one particle in a 
box λ on a side, where λ is the thermal de Broglie length. 

Even when things are not degenerate enough to be superconducting
or superfluid, you get to see the effect of the degeneracy in the
transport properties.  The electrical conductivity of an ordinary
metal /at room temperature/ is profoundly affected by the fact that
the electrons are degenerate.   Similarly, the viscosity of 3He at
moderately low temperatures is enormous, because it is a degenerate
Fermi liquid.  Also you get to see exotic transport phenomena such 
as spin waves in ferromagnets, in 3He, in spin-polarized atomic
hydrogen gas, et cetera.

Also FWIW neutron stars are degenerate and presumably superfluid.
The density is such that a few thousand kelvin is a very low
temperature for them.  Similarly all of nuclear physics starts
with accounting for the fact that the nucleons in the nucleus
are degenerate ... and any first-principles understanding of
chemistry (aka atomic physics) starts with accounting for the 
fact that the valence electrons in the atom are degenerate.

If you want some additional topic-drift, you can ask about degeneracy
in reduced dimensions, such as quantum Hall effect, quantum wires,
et cetera.

Note that I call it the thermal de Broglie length or length-scale
(not wavelength) because it has precious little to do with wavelength.  
It has dimensions of length, but that doesn't make it a wavelength.
It has more to do with the size of the /envelope/ of a wave packet, 
rather than the wavelength inside the envelope.

===========

Enough babbling for now.

If that didn't answer the question, please clarify the question.