Thursday, January 17, 2013

Degenerate Matter: Introduction

Degenerate matter, simply speaking, is matter at very high densities. Environments where such matter exists are relatively rare throughout the Universe, and they most commonly occur in the cores of stars, or stellar remnants.

Formally, ordinary matter assumes a "degenerate" state when atoms are packed in such close proximity that the very structure of atoms, and eventually, their constituent particles, breaks down. This happens in stages, and at each stage, there is a different type of degenerate matter.

Two major quantum mechanical principles come into play in determining the nature of degenerate matter. The first is the Heisenberg Uncertainty Principle. It states that there are certain pairs of properties of a particle has that can only be simultaneously known to a certain accuracy. In the context of degenerate matter, the pair of position and momentum is the most important. Therefore, as the position of a particle is known with greater and greater accuracy, the momentum is more and more uncertain, and vice versa. The principle is represented by the equation below.

In this equation, ∆χ is the error in the position, and ∆ρ is the error in momentum. The right side of the equation is simply a constant, specifically the reduced Planck Constant divided by 2. The product of the two errors is always greater than this value. It is clear that, if ∆χ is reduced, ∆ρ will increase, and vice versa.

The second principle of quantum mechanics necessary for an analysis of degenerate matter is the Pauli Exclusion Principle. This principle establishes that no two identical particles can be in the same quantum state simultaneously, i.e. have the same quantum numbers. Since these numbers can take only specific values, one particle literally "excludes" another from having the same properties. The most well-known example of this is the principle's application to elections, which defines the structure of the atom.

The above is a visualization of the electron structure of the element Xenon. For each electron shell, i.e. each concentric ring in the above image, the Pauli Exclusion Principle precludes "stuffing in" any more electrons beyond full capacity. The first three shells here have a capacity of 2, 8, and 18 electrons respectively, all of which are filled; no more electrons could fit within them, and the rest therefore had to next occupy the fourth and fifth shells. The key idea is that electron states are discrete; only a finite number of electrons can fill a certain level. There is also a minimum energy level, below which no electron can exist.

Armed with these ideas, one can now understand how degeneracy comes about. Consider a gas under normal conditions stored in a container (below).

The individual particles (molecules or atoms) have varying momenta, with some moving faster, and some slower, but the average speed of all the particles in the container can easily be found. It is commonly known as the temperature of the gas. Also, the gas produces a pressure pushing on the container, created when particles bounce off all sides. This pressure is practically uniform, due to the minuteness of the particles involved, and is called the thermal pressure. This pressure is what keeps a balloon inflated, for example.

Clearly, this thermal pressure varies with temperature. When the temperature is increased, the particles involved are more energetic, move faster, and therefore exert a higher pressure on the sides of the container. Similarly, if one shrinks the container, more particles will bounce off the sides of the container per unit time, and consequently, the pressure on the container will increase. Therefore, pressure on the container is proportional to the temperature of the gas and inversely proportional to the space which it occupies. These phenomena are part of the kinetic theory of gases, and are summarized in the equation below:

p = kt/s

where p is pressure on the container, t is temperature, s is size of the container, and k is the associated proportionality constant, which varies from gas to gas.

By the above theory, when the temperature of a gas is lowered, the size would have be lowered correspondingly to keep the pressure constant. If the temperature were lowered to absolute zero, however, the particles of the gas would no longer be in motion, and no pressure would be exerted on the sides of the container no matter how small it was. The gas could then be compressed into an arbitrarily small volume with no resistance.

However, this does not occur, as the kinetic theory of gases in an approximation, and does not take quantum phenomena into account. When the electrons associated with particles of gas are put into a very small space, their positions are easily measured, and almost precisely pinpointed. Returning to the Heisenburg Uncertainty Principle, this implies that to compensate for the very low uncertainty in position, there must be a high uncertainty in momentum, meaning that the electrons must have very high velocities despite a temperature of near absolute zero.

Note that this principle does not dictate that a particular electron is suddenly imparted with a higher momentum, but merely states that, on average, the uncertainty must increase, and there therefore are some electrons traveling faster. At the same time, many electrons are being pushed into the same space as the container is compressed, and, by the Pauli Exclusion Principle, cannot occupy the same quantum state. They are then pushed into higher energy states, generally producing an outward force.

The combination of these two effects produces an intense pressure against further compression not predicted by the kinetic theory. This is known as electron degeneracy pressure. When more of the counter-force pushing outward on the container is supplied by electron degeneracy pressure than by thermal pressure, then the matter involved is known as electron-degenerate.

To find a naturally occurring example of electron-degenerate matter, one must look no further than the night sky, see the next post.


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