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Electron degeneracy pressure

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Title: Electron degeneracy pressure  
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Electron degeneracy pressure

Electron degeneracy pressure is a particular manifestation of the more general phenomenon of quantum degeneracy pressure. The Pauli exclusion principle disallows two identical half-integer spin particles (electrons and all other fermions) from simultaneously occupying the same quantum state. The result is an emergent pressure against compression of matter into smaller volumes of space. Electron degeneracy pressure results from the same underlying mechanism that defines the electron orbital structure of elemental matter. Freeman Dyson showed that the imperviousness of solid matter is due to quantum degeneracy pressure rather than electrostatic repulsion as had been previously assumed.[1][2][3] Furthermore, electron degeneracy creates a barrier to the gravitational collapse of dying stars and is responsible for the formation of white dwarfs.

When electrons are squeezed together too closely, the exclusion principle requires them to have different energy levels. To add another electron to a given volume requires raising an electron's energy level to make room, and this requirement for energy to compress the material manifests as a pressure.

Electron degeneracy pressure in a material can be computed as[4]

P= \frac{2}{3}\frac{E_{tot}}{V}=\frac{2}{3}\frac{\hbar^2 k_{\rm{F}}^5}{10 \pi^2 m_{\rm{e}}}=\frac{(3 \pi^2)^{2/3} \hbar^2}{5 m_{\rm{e}}}\rho_N^{5/3} ,

where \hbar is the reduced Planck constant, m_{\rm e} is the mass of the electron, and \rho_N is the free electron density (the number of free electrons per unit volume). When particle energies reach relativistic levels, a modified formula is required.

This is derived from the energy of each electron with wave number k = \frac{2 \pi}{\lambda}, having E = \frac{p^2}{2m} = \frac{\hbar^2 k^2}{2 m}, and every possible momentum state of an electron within this volume up to the Fermi energy being occupied.

This degeneracy pressure is omnipresent and is in addition to the normal gas pressure P = NkT/V. At commonly encountered densities, this pressure is so low that it can be neglected. Matter is electron degenerate when the density (proportional to n/V) is high enough, and the temperature low enough, that the sum is dominated by the degeneracy pressure.

Perhaps useful in appreciating electron degeneracy pressure is the Heisenberg uncertainty principle, which states that

\Delta x \Delta p \ge \frac{\hbar}{2}

where Δx is the uncertainty of the position measurements and Δp is the uncertainty (standard deviation) of the momentum measurements. A material subjected to ever-increasing pressure will compact more, and, for electrons within it, their delocalization, Δx, will decrease. Thus, as dictated by the uncertainty principle, the spread in the momenta of the electrons, Δp, will grow. Thus, no matter how low the temperature drops, the electrons must be traveling at this "Heisenberg speed", contributing to the pressure. When the pressure due to this "Heisenberg motion" exceeds that of the pressure from the thermal motions of the electrons, the electrons are referred to as degenerate, and the material is termed degenerate matter.

Electron degeneracy pressure will halt the gravitational collapse of a star if its mass is below the Chandrasekhar limit (1.44 solar masses[5]). This is the pressure that prevents a white dwarf star from collapsing. A star exceeding this limit and without significant thermally generated pressure will continue to collapse to form either a neutron star or black hole, because the degeneracy pressure provided by the electrons is weaker than the inward pull of gravity.

References

  1. ^ Dyson, F. J.; Lenard, A. (March 1967). "Stability of Matter I". J. Math. Phys. 8 (3): 423–434.  
  2. ^ Lenard, A; Dyson, F. J. (May 1968). "Stability of Matter II". J. Math. Phys. 9 (5): 698–711.  
  3. ^ Dyson, F. J. (August 1967). "Ground‐State Energy of a Finite System of Charged Particles". J. Math. Phys. 8 (8): 1538–1545.  
  4. ^ Griffiths (1994). Introduction to Quantum Mechanics. London, UK: Prentice Hall.  Equation 5.46
  5. ^ Mazzali, P. A.; K. Röpke, F. K.; Benetti, S.; Hillebrandt, W. (2007). "A Common Explosion Mechanism for Type Ia Supernovae". Science 315 (5813): 825–828.  
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