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Degenerate matter

In an atom, the negative electrons are bound to the positive nucleus by forces of electrical attraction and constantly orbit around it. Just as the repeated shocks of gas molecules hitting the walls of a container generate a pressure, so the electrons bound to a nucleus are responsible for a pressure which prevents matter from contracting beyond a certain limit. This limit is determined by the Exclusion Principle discovered by Wolfgang Pauli in 1925.

In pictorial terms this fundamental principle of physics establishes the existence of elementary cells which can contain a maximum of two inhabitants. In ‘ordinary’ matter (whose density is similar to that of water), most of these cells are unoccupied. It is for this reason that we can say that there is a lot of vacuum in matter: each atom consists of a core which contains most of the mass, surrounded by electrons moving on such distant orbits that if the nucleus was the size of a marble, the atom would measure 2 kilometres across.

However, at the same time as explaining a property of matter which has long been observed, quantum mechanics predicts the possible existence of so-called degenerate states of matter, charac­terised by the fact that all the elementary cells are occupied by particles.

Not all types of matter can become degenerate. Elementary particles are divided into two categories with different collective behaviour at high density or very low temperature: fermions (named after the Italian physicist, Enrico Fermi) and bosons (named after the Indian physicist, Satyendra Bose, who collaborated with Einstein on the subject). The important characteristic which differ­entiates between these two large classes of elementary particles is their spin. Spin is an intrinsic property of an elementary particle associated with its angular momentum. One of the important things that quantum mechanics revealed is that spin is quantised, that is, it can take only certain discrete values, integer or half­integer multiples of a fundamental constant called the ‘normalised Planck’s constant’, h (read h bar). In our daily life, the discrete values of spin pass completely unnoticed because h is so tiny that macroscopic objects have a gigantic spi . The spin of a simple child’s top is as great as 1030h. Thus it is only on the atomic scale that the discontinuity of spin becomes noticeable, along with that of the other quantised physical variables such as energy.

The difference between fermions and bosons is that fermions have half-integer spins (1/2h,3/2 h and so on), while bosons have inte­ger spins (0h , 1h, 2h and so on). The fundamental components of atoms, protons, neutrons and electrons, are fermions with 1/2 h  spin.

The photon (a light particle) is a boson with 1 h spin. Pauli demon­strated a fundamental principle: two identical fermions cannot be found in the same quantum state (this rule does not apply to bosons). This very important law rules out very tightly packed groups of fermions. Let us see in more detail how it works.

In an atom, the quantum state of an electron is defined by its energy (a function of the orbit in which the electron is found) and by the orientation of its spin. This can have one of two directions, either ‘up’ or ‘down’, depending on whether it spins in the same or in the opposite sense as its orbit. From Pauli’s Exclusion Principle one can deduce that an orbit of given energy can be occupied by two electrons which have two opposite spin orientations. The presence of any further electron in the same orbit is forbidden by nature.

Let us now consider an electron gas in a box. An electron’s quantum state is defined by its energy, its linear momentum  and its spin. According to quantum mechanics, energy and momentum are also ‘quantised’ parameters and can take only discrete values. Therefore if electrons are placed in a smaller and smaller volume there will come a point when all the energy and momentum levels are occupied by electrons having all the possible spin orientations. The Exclusion Principle then comes into play and prevents the volume from being populated any more. Consequently the electrons resist any further attempts at decreasing the volume by exerting a colossal internal ‘quantum' pressure, called degeneracy pressure. The characteristic property of this pressure is its indepen­dence of temperature, unlike ordinary gas pressure which increases in proportion to the gas temperature.

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