Temperature?
Is a negative Kelvin temperature possible?

Thermodynamics is older than the atomistic view of matter, hence thermodynamic quantities such as temperature are not defined atomistically. The definition of temperature allows negative absolute temperatures. But do such temperature have any empirical meaning? While this might be questionable, it is certainly interesting to discuss it because it helps understand the concept of temperature. As follows, a simplified statistical concept of temperature is discussed and used to explain the state of matter in lasers as well as the technique to attain very low temperature (adiabatic demagnetization)

Temperature

The most often given answer to this question is something like "no, because at zero K, molecules are not moving anymore and you cannot have less than zero motion."

This answer, while not completely wrong, ignores that temperature is not defined as motion of molecules. It is, for certain systems, equal to the mean kinetic energy of molecules, but this is an equality, not a definition.

Temperature is usually taken as simple, because we all can feel it. But have you ever thought of explaining it to an alien who does not feel it? It would be very difficult. Entropy, for instance, is much simpler, yet seems difficult, probably because we cannot feel it.

What is correct in the above-cited answer is that negative Kelvin temperature cannot be achieved by cooling; moreover, not even zero K can be achieved by cooling, as you probably know. Let's have a closer look at that equality between temperature and mean kinetic energy of molecules, and see what we can get out of this.

Statistical model

In a gas made of tiny hard billard balls temperature is proportional to mean kinetic energy of the particles. But certainly, nobody expects all molecules to have the same speed. The image (Maxwellian distribution) shows the distribution of molecular speeds for three different temperatures.

Maxwellian distribution of particle speeds
The Maxwellian distribution of molecular speeds for three different temperatures. Even at low temperature, there is a small fraction of molecules having high speed. This fraction increases with temperature, while the fracition of molecules with low speed becomes smaller but does not vanish

Obviously, the distribution of molecular speeds depends on temperature. Conversely, temperature is determined by the ratio of the number of molecules at high speed to the number of molecules at low speed. A model can help us understand this relationship.

To keep things simple, I will assume a system that can have but two levels of energy, E(hi) and E(lo): the particles either have energy or they have not.

When the system is heated, energy is transferred to it; the particles must somehow accomodate this energy. In this model, this can only happen if some change from a low to a high energy state.

What is the number of particles that will be on the i-th energy level? The answer is given by the Boltzmann distribution:


N(i) = C*exp (-E(i)/kT),

where

C: a constant (at a given temperature)
N(i): the number of particles with energy E(i)
E(i): is the energy portion, according to our simplifying assumption it is either nought or it is something
k: the Boltzmann constant
T: absolute temperature

For only two levels of energy the ratio of the population of these levels is:

N(hi)/N(lo) = exp (-DeltaE/kT)

where

DeltaE = E(hi)-E(lo)

Solving this equation for T

- DeltaE/k T = ------------------- ln [N(hi)/N(lo)]

Behaviour of the model

Temperature of a body

Does this temperature, as defined here, behave as we expect it to? For bodies at thermal equilibrium


N(lo) > N(hi),

it is not possible to populate the high energy level more by adding heat to a body. Thus for usual systems the logarithm is negative and temperature is positive.

Two level model
In this example there are eight atoms at the low level and four at the high level; this system has some usual temperature. Assuming that the difference of the energy of the two levels is 10k (k is the Boltzmann constant, 1.38*10^-23, what would the temperature of this system be?

Heating and cooling

If heat is added, more and more particles change from the low energy level to the high energy level (note that the levels themselves are unchanged); the number of particle with high energy, N(hi), grows and N(lo) decreases so the logarithm becomes less negative and temperature rises. The energy of the whole ensemble rises because now there are more particles on the high energy level.

Try calculating the temperature for the system shown in picture (two level model), but one particle changed his place from low to high level.

On the other hand, if energy is withdrawn from a body, E(lo), the low energy level, is populated at the cost of E(hi). The value of the logarithm becomes more and more negative

and temperature goes towards zero. Also, from the expression can be concluded that it must be difficult to reach absolute zero, as a diagram of the logarithm versus the ratio shows.

Logarithm function
The logarithm changes much when argument is small.

At low ratio of N(hi)/N(lo) the logarithm is very steep; hence if only a few particles change from low to high, temperature rises appreciably. As can easily be imagined, it is not a simple task to keep some 10²³ particles on the low state; keep in mind that it takes very little energy to rise but a few atoms to the high level.

As can be seen from these examples, the formula behaves as we would expect temperature to behave.

Infinite temperature

System at infitine temperature
At equally populated levels, temperature, but not the energy, of a system is infinite

I would like to elaborate another nice conclusion from the formula. If both levels are equally populated, N(hi) = N(lo) and the logarithm vanishes, so that

The value of the temperature becomes infinite (note that DeltaE does not change). Since number of particles are always finite, only an finite amount of energy is needed to get infinite temperature in this model. Infinite temperature cannot be reached anyway by heating, although not because of the energy needed, but because heat flows from high temperature to low temperature. To heat a body, a hotter body is required.

Negative absolute temperature?

System at negative absolute temperature
If more particles are at the upper level than at the lower one, absolute temperature of a system is negative

As soon as the high energy level is populated more that the low energy one, we have negative absolute temperature. Can such a state be realised?

Yes, it can, but not by cooling. The formula, as well as the picture, show that a state of matter to which negative absolute temperature can be attributed has more energy than the states at usual temperatures, because more particles are at high energy level than at low energy level. Thus one has to add energy to get negative absolute temperature. It has been emphasized that such states cannot be reached by adding heat to a body.

Systems at negative absolute temperature

Nevertheless, both systems at infinite temperature and at negative absolute temperature can be prepared. Let us look first at negative temperature. In most lasers atoms are excited electronically to a high energy level. Prior to laser light emission of the whole system, more atoms have to be excited than are not, and a negative absolute temperature can be ascribed to the system. Note that, as expected, atoms are not excited by heating.

Systems at infinite temperature

Systems at infinite temperature are used to attain the lowest temperatures possible. The technique is that of adiabatic demagnetization. To get a rough idea of the subject, let us look at magnetism due to electron spin. Each electron can be seen as a tiny magnet, except that it can have only two orientations in an external magnetic field, namely parallel to the field and antiparallel to it. Without a field, these "electron magnets" are randomly oriented. Energy of both states is equal, so it is equally probable to find each orientation. If an external field is applied, the energy of the parallel state is lowered and energy of the antiparallel state goes up.

This phenomenon can be used to cool samples of suitable materials (specific magnetic properties are required). A sample is cooled down to the temperature of liquid helium. Still submerged in liquid helium, a magnetic field is applied to it. At the very first moment, the number of atoms in the parallel equals that in the antiparallel state. But these states are no more energetically equivalent in the presence of an external magnetic field. Our formula says that if two states differ in energy but are equally populated, then the temperature is infinite. This is an example for the previously stated assertion that one does not need infinite energy to get infinite temperature. In fact, the energy of the sample does not rise, on the contrary! Because it is at infinite temperature, the system will now lose energy as antiparallel states change to parallel. The energy is transferred as heat from the sample to the helium bath. Its temperature changes from infinity to that of the helium bath.

Now the sample is insulated from the helium bath and the external magnetic field is reduced to zero. The energy that was transferred to the helium bath cannot go back to the sample. Hence in this step, while the populations of the antiparallel and the parallel states are gradually equalized, the sample cools down. This is how some of the lowest available temperatures are attained.

What about systems with more than two energy levels?

Usual systems have more than two excited states and hence more than two energy levels. Can these temperatures be attributed to such systems, too?

Looking at the limits of this two level model may help to elucidate further the concept of temperature. First, each level must be populated enough to avoid random fluctuations of particle number. Keep in mind that the systems are dynamic, particles exchanging energy continually. So the model fails when there are only a few particles on each level.

The above given Boltzmann relation holds true for multi level systems, of course. Now we can clarify what it means when it is said that a system must be in thermal equilibrium:

A system is in thermal equilibrium if particle number on each energy level follows the Boltzmann distribution.

Random fluctuations cause deviations from this distribution, but the larger the fluctuation, the less it is probable, so most of the time the system will be close to that distribution.

For systems with an infinite number of states infinite temperature would also mean infinite energy. Now the relation

E_kin = ³/₂kT

shows that, in contradiction to our conclusions above, infinite temperature means infinite energy. In fact, the number of translational states is very large, near infinite. For a system with a Volume of about 1 cm³ there are more translational states than molecules.

Translational states thus cannot be described by the above model. Free particles that store energy in their translational motion cannot have negative Kelvin temperature.

If you put negative temperature into the Boltzmann formula, the function value goes up exponetially. Each energy level must be populated more than its lower neighbour. Of course it is not possible to keep populating levels like this. It follows that in system that have inversely populated states, these states cannot be in thermal equilibrium with all other states of the same system. Negative absolute temperature is not an equilibrium quantity.

In fact, one may find the notion a little bit far fetched. But see "What Does Negative Temperature Mean?" by Scott I. Chase.

© Copyright on text and pictures 1998 ... 2009 Gian Vasta, except for the reading daemon (thank you very much, Marijke!)
You may use this text freely only for non-commercial and (logical and! <g>) educational purpose. If you find it useful, please drop a note to me.

Last update Mittwoch, 1. April 2009 gVa

Temperature? Is a negative Kelvin temperature possible?