More Fermi-Dirac Statistics: Protons, Neutrons & Neutron Stars

1. More Fermi-Dirac Statistics: Protons, Neutrons & Neutron Stars

2. The Fermi Gas of Nucleons in a Nucleus Let’s consider the system of nucleons in a large nucleus (both protons and neutrons are fermions). In heavy elements, the number of nucleons in the nucleus is large and statistical treatment is a reasonable approximation. We need to estimate the density of protons/neutrons in the nucleus. The radius of the nucleus that contains A nucleons: Thus, the density of nucleons is: For simplicity, assume that the # of protons = the # of neutrons, so their density is The Fermi energy EF >>> kBT – at room temperature, the system is strongly degenerate (all nucleons are in their ground state) The average kinetic energy in a degenerate Fermi gas = 0.6 of the Fermi energy - the nucleons are non-relativistic

3. Fermi Velocity and Fermi Momentum (non-relativistic gas) In metals, the electrons at the Fermi level are moving at a very high speed (at T=0!): - vF is the Fermi velocity When an electron is confined in a very small space, it "flies about its tiny cell at high speed, kicking with great force against adjacent electrons in their cells. This degenerate motion ... cannot be stopped by cooling the matter. Nothing can stop it; it is forced on the electron by the laws of quantum mechanics, even when the matter is at absolute zero temperature" (Thorne 1994). This velocity is of the same order of magnitude as the orbital velocity of the outer electrons in an atom, and ~10 times the mean thermal velocity that a non-degenerate electron gas would have at room temperature. Still, since vF<<c, we can treat the mobile electrons as non-relativistic particles. The corresponding momentum: the Fermi momentum

4. Pressure of a Degenerate (non-relativistic) Fermi Gas The pressure of a degenerate Fermi gas (Fermi or degeneracy pressure) at T=0 can be very high (in stark contrast to the classical ideal gas described by the equation of state PV=RT) density n 1m2 vF 1s assuming all Fermions have the same v and 1/6 of them move towards the wall exact result obtained by integration over all angles and velocities The degeneracy pressure depends on n (as n5/3) and does not depend on T at T << EF. Let’s estimate the electron pressure for a typical metal: ! In metals, this enormous pressure is counteracted by the Coulomb attraction of the electrons to the positive ions.

5. Fermi Pressure and Stellar Evolution How stars can support themselves against gravity: • gas and radiation pressure supports stars in which thermonuclear reactions occur • pressure of a degenerate electron gas at high densities supports the objects with no fusion: dead stars (white dwarfs) and the cores of giant planets (Jupiter, Saturn) • pressure of a degenerate neutron gas at high densities support neutron stars 3Msun 1.4Msun

6. Fermi Pressure of a Relativistic Fermi Gas - scales with the density as n5/3 provided that the electrons remain non-relativistic (speeds v << c). In a white dwarf or a neutron star, this approximation breaks down. For the relativistic degenerate matter, the equation of state is “softer” The number of fermions per unit volume with k<kF: At T=0, all the states up to kFare occupied: In the ultra-relativistic case The relativistic and non-relativistic expressions for electron degeneracy pressure are equal at ne=1036 m-3, about that of the core of a 0.3 Mwhite dwarf. As long as the star is not too massive, the Fermi pressure prevents it from collapsing under gravity and becoming a black hole.

7. White Dwarfs, Chandrasekhar Limit By combining the ideas of relativity and quantum mechanics, Chandrasekhar made important contributions to our understanding of the star evolution. The Fermi pressure of a degenerate electron gas prevents the gravitational collapse of the star if the star is not too massive (the white dwarf). Nobel 1983 An estimate the upper limit of the white dwarf’s mass (the Chandrasekhar mass): Total # neutrons in the star  the number of protons: proton mass Total potential (gravitational) energy of the star: Total (kinetic) energy of the Fermi gas of electrons (relativistic case): -Ug Et energy If M<MCh, the Fermi pressure is sufficient to prevent the collapse. MCh M (more accurately – 1.4 M)

8. Neutron Stars [M=(1.4-3)M ] For a dead star more massive than 1.4 MSun, the electron degeneracy pressure cannot prevent the gravitational collapse. During the collapse an extra energy liberated is sufficient to drive a nuclear reaction: If the mass is not too high (<3 MSun), the further contraction is stopped by the degeneracy pressure of neutrons (otherwise – black hole). By knowing the parameters of neutron stars, we can explain why these stars are made of neutrons. A typical neutron star has a mass like the Sun (MSun = 2 × 1033 g), but a much smaller radius R ~10 km. The neutron density: n ~ 1044 m-3 The Fermi energy of the neutrons: EF 30MeV. comparable to the nuclear density Consider a hypothetical star made of protons and electrons, each with the concentration n ~ 1044 m-3 (the # of protons = the # of electrons, the star is electrically neutral). The Fermi energy for protons would be nearly the same as that for the neutrons. On the other hand, the electron EFwould be greater: by the ratio Mn/me ~1838 in the non-relativistic case, but we need to consider ultra-relativistic electrons. For ultra-relativistic electrons with this density EF~280 MeV! Obviously, the energy difference EF(electrons)-EF(neutrons) is sufficient to drive the nuclear reaction Note that in a white dwarf [n ~ 1036 m-3EF(electrons)  0.5MeV], this reaction is still energetically unfavorable:

9. Thermonuclear Runaway in Supernova Thus, on one hand, the Fermi pressure helps to stabilize dead stars against gravitational collapse. On the other hand, reaching the state of degeneracy in a still-alive star may have dramatic consequences. The structure of young stars (low densities, non-degenerate core) includes a built-in thermostat (negative feedback) that automatically adjusts the temperature to just the value needed to make the reaction go at the correct rate. This negative feedback fails when the core reaches the state of degeneracy: the pressure of degenerate electron gas does not depend on temperature, therefore a slight rise in T is not compensated by expansion. burning H shell Thermonuclear runaway: degenerate He core • rise temperature • accelerate nuclear reactions • increase energy production He core grows by H-shell burning until He-burning sets in

10. The outer layers start moving inward, crushing the interior into a dense core. The neutron core is formed at such a terrific speed, its surface rebounds, meeting the matter from outer layers slowly falling inwards. neutron star The shock wave travels outwards. When it meets the outer layers, compresses them and heats them up, the thermo-nuclear reactions are triggered. The Crab Nebula, which was formed by a supernova explosion recorded in 1054. An explosion then results and a there is a bright supernova flash.

11. Applications to White Dwarf Stars The temperature inside the core of a typical star is at the order of 107K. The atoms are completely ionized at such a high T, which creates a hugh electron gas The loss of gravitational energy balances with an increase in the kinetic energy of the electrons and ions, which prevent the collapse of star!

12. Example: The pressure of the electron gas in Sirius B can be calculated with the formula Using the following numbers Mass M = 2.09 × 1030 kg Radius R= 5.57 × 106 m Volume V= 7.23 × 1020 m3

13. Assuming that nuclear fusion has ceased after all the core hydrogen has been converted to helium! The number nucleons = Since the ratio of nucleons and electrons is 2:1 there are electrons

14. Therefore, T(=107 K) is much smaller then TF . i.e. is a valid assumption ! Thus: P can be calculated as

15.  A white dwarf is stable when its total energy is minimum For Since can be expressed as Where

16. For gravitational energy of a solid With In summary To find the minimum U with respect to R

17. Consider the collapse of the sun into a white dwarf. For the sun, M= 2 x 1030 kg, R = 7 x 108 m, V= 1.4 x 1027 m3. Calculate the Fermi energy of the Sun’s electrons.