## Dxu

Fig 10.10.

The Ring Nebula, shown in Fig. 10.8(a), is shown here in four different colors of light, highlighting gas with different physical conditions.The wavelengths are indicated below each frame. [NOAO/AURA/NSF]

### 10.4 I White dwarfs 10.4.1 Electron degeneracy

The material left behind after the planetary nebula is ejected is the remnant of the core of the star. It is mostly carbon or oxygen, and its temperature is not high enough for further nuclear fusion to take place. The gas pressure is not high enough to support the star against gravitational collapse. This collapse would continue forever if not for an additional source of pressure when a high enough density is reached. This pressure arises from electron degeneracy.

Electron degeneracy arises from the Pauli exclusion principle, which states that no two electrons can be in the same state. For two electrons to be in the same state, all of the quantum numbers describing that state must be the same. For example, in an atom, there is a quantum number describing which orbit the electron is in, and another describing how that orbit is oriented (by giving the component of the angular momentum along some axis). In addition, we must take into account the fact that the electron has intrinsic angular momentum, called "spin". The spin can have two opposite orientations. For convenience, we call them "up" and "down" (depending on the direction of the angular momentum vector). An up electron and a down electron in the same energy level are considered to be in different states. However, two is the limit. We can only put two electrons into each energy level.

We can see how this affects the properties of atoms with many electrons. Suppose we build the atom by adding electrons one at a time. The first electron goes into the lowest energy level. The second also goes into the lowest level, but with the opposite spin orientation. The first level is now full. The third electron must go into the next level. After we have added all of the electrons, we can add up the excitation energies of all of the electrons. We will find that the average excitation energy of the electrons in the atom is much greater than kT. This means that electrons are in higher levels than we would guess by just considering the thermal energy available. The problem is that the electron cannot jump into the filled lower states.

We can apply the same idea to a solid, which we can think of as a structure with many energy levels. The electrons fill the lowest energy levels first, but as they fill the electrons end up in higher and higher levels, as shown in Fig. 10.11. The average energy of the electrons is, again, much greater than kT. In fact, the distribution of energies of electrons in a solid at room temperature is negligibly different from that in a solid at absolute zero. We call an electron gas in which all of the electrons are in the lowest energy states allowed by the exclusion principle a degenerate gas.

In a degenerate gas, most of the electrons will have energies much greater than they would in an ordinary gas. These high energy electrons also have high momenta. They can therefore exert a pressure considerably in excess of the pressure exerted by an ideal gas at the same temperature. The higher pressure is called degeneracy pressure. We have everyday examples of this pressure. For example, it is responsible for the hardness of metals. (Metals consist of a regular arrangement of positive ions, held together by many shared electrons. The exclusion principle results in those shared electrons

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