Subject: Re: Misused Units From: Kent Budge <"kgbudge,"@,sandia,.gov> Date: Feb 26 1997 Newsgroups: rec.arts.sf.science,sci.astro,sci.physics,alt.folklore.science, rec.arts.sf.tv Mikko Juhani Vilenius wrote: ... > I saw a document for quite a while ago about this. > I believe it had something to do with the spin of electrons. > > There was some material (hydrogen?) that was frozen very near to 0K > and then put into powerful magneticfield, so that something happened > to the spin of the electrons, which still dropped the energy level making > the material "colder". > > The scientists said that infact the energy level was increased radically, > thus making the material "very hot".. This is where I drop off. I'm not > too familiar with quantum mechanics and this was about four years ago. > > Could somebody clarify this? Or did I get it totally wrong the first time? ... What you are saying mostly makes sense, but *could* use some clarifying. I'll give it a shot (takes deep breath and rubs sleepiness out of his eyes): The hydrogen nucleus (which is a single proton) has a quantum mechanical spin of 1/2. This gives it magnetic "north" and "south" poles, like a bar magnet. Unlike a bar magnet, the proton can only have two orientations with respect to a magnetic field; it either has its north pole pointing towards the north direction of the field, or towards the south. Orientations in between simply don't exist, because the orientation is "quantized" (that's where quantum mechanics gets its name.) Weird, but experimentally verified. (Hope I haven't lost you already *OR* insulted your intelligence -- I have no idea how extensive your knowledge is.) When the north pole of the proton is pointed in the north direction, the energy of the proton is greater than when the north pole of the proton is pointed in the south direction. (Like poles repel each other, so it takes energy to force the north pole of the proton to point north.) The energy difference depends on the strength of the magnetic field, but it is quite small for ordinary field strengths. Now let's look at a frozen hydrogen crystal. The electrons are essentially all frozen into their lowest energy state, so we'll just consider the protons. Turn on the magnetic field. Each proton will be either aligned with or against the field. If most are aligned with the field, the total energy is high. If most are aligned against the field, the total energy is low. Now let's talk about temperature and statistical mechanics. (very DEEP breath.) You can't possibly keep track of the sagans of protons in even a very tiny hydrogen crystal. You have to examine the problem statistically, which is where "statistical mechanics" comes in. The fundamental principle of statistical mechanics is: All accessible states of a system are equally probable. In other words, for a given total energy within a system, all states of the system that have that same total energy are equally probable. This principle is based on the assumption that the system is constantly moving between states more or less at random, so a snapshot at any given moment is as likely to catch it in one state as another. Suppose we have just enough energy in the hydrogen crystal to allow half the protons to be aligned with the magnetic field. There are sagans of protons involved. Let's suppose the crystal is very small, so that there are just 2e20 protons involved. (That's less than a milligram of hydrogen.) Think of all the ways you can divide up 2e20 objects into two groups of 1e20 objects. (Note to gurus: yes, I am aware that I am neglecting certain technical issues regarding the distinguishability of like particles. It doesn't matter for this discussion.) There are about (2e20)!/(1e20!)/(1e20!) = 1e(1.386e20) ways to do it. That last quantity is meant to represent 1 followed by 1.386e20 zeros. Carl Sagan would be proud. In other words, there are 1e(1.386e20) states accessible to the crystal, each equally probable. Now consider what happens if there is enough energy for all the protons to be aligned with the field. There's only one way to align all the protons with the field; it's the state in which all the protons are aligned with the field. ;-) So there is exactly one accessible state, with probability 1.000... The same is true when you have the minimum possible energy, forcing all the protons to be aligned against the field. There's only one such state. If you plot the number of accessible states versus the energy, you get a function that is strongly peaked at the energy necessary to align half the protons with the field. (I mean *STRONGLY* peaked.) The "entropy" of this system, incidentally, is proportional to the logarithm of the number of accessible states. So the entropy is highest for the half-aligned system, and is zero for either the fully-aligned or fully-unaligned state. Suppose this system comes into contact with another system, so that energy can flow between them. Suppose it's another hydrogen crystal. Suppose one is fully unaligned and the other is half aligned. If no energy flows, the one crystal has one accessible state and the other has 1e(1.386e20) accessible states. The number of accessible states for the combination of the two crystals is the product of these two numbers, or 1e(1.386e20). If, on the other hand, energy flows until the two crystals each have an equal energy -- enough to align 25% of the protons -- each crystal has (2e20)!/(5e19)!/(1.5e20)! = 1e(1.125e20) accessible states. The number of accessible states for the combination of the two crystals is then 1e(2.249e20), way more than for the case where no energy flows. So its *extremely* probable that energy will flow from the half-aligned crystal to the fully unaligned crystal until the energy is equally shared between them. Let U be the energy in a crystal. Let S(U) be the log of the number of accessible states of the crystal for an energy U (the "entropy.") Then the log of the total number of accessible states for the combination of two crystals is S(U1) + S(U2). (Note to gurus: This is why entropy is the *log* of the number of accessible states -- it makes it an extensive quantity.) Suppose we increase U2 at the expense of U1 (that is, energy flows to crystal 2 from crystal 1.) If this increases S(U2) more than it diminishes S(U1), the total number of accessible states increases. So such an energy exchange is likely to take place. If S(U2) is increased less than S(U1) is diminished, the total number of accessible states is decreased, and the energy exchange is unlikely to take place. We therefore define the temperature of the system to be 1/T = dS/dU You can see that a crystal with a higher temperature will tend to give its energy to a crystal with a lower temperature, because the higher temperature means a smaller reduction in entropy when energy is given up, and entropy is the measure of the number of states accessible to the system. For most systems, dS/dU is always positive. But for our hydrogen crystal, dS/dU is negative if the crystal has enough energy to align more than half the protons. So negative temperatures actually correspond to more energy than do positive temperatures. If you put a negative temperature crystal in contact with a positive temperature crystal, the entropy of both crystals increases as energy flows from the negative temperature crystal to the positive temperature crystal. So, odd as it seems, energy flows from negative temperature systems to positive temperature systems. The negative temperature systems are "hotter." Stranger still, the negative temperature system becomes more negative in temperature as energy flows, and the rate of change in temperature accelerates as more energy flows until the system passes from infinitely negative temperature to infinitely positive temperature in the blink of an eye. This is purely an artifact of the weird reciprocal definition of temperature; the *energies* involved certainly aren't infinite. I hope this helps, though I fear it may be more than anyone can digest in a single reading. -- -Kent (usual disclaimer) Return address hacked to foil junk mail; edit before replying.

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