Classical and Quantum Gases Fundamental Ideas Density of States Internal Energy Fermi-Dirac and Bose-Einstein Statistics Chemical potential Quantum concentration

Density of States Derived by considering the gas particles as wave-like and confined in a certain volume, V. Density of states as a function of momentum, g(p), between p and p + dp: gs = number of polarisations 2 for protons, neutrons, electrons and photons

Internal Energy The energy of a particle with momentum p is given by: Hence the total energy is: No. of quantum states in p to p +dp Average no. of particles in state with energy Ep

Total Number of Particles No. of quantum states in p to p +dp Average no. of particles in state with energy Ep

Fermi-Dirac Statistics For fermions, no more than one particle can occupy a given quantum state Pauli exclusion principle Hence:

Bose-Einstein Statistics For Bosons, any number of particles can occupy a given quantum state Hence:

F-D vs. B-E Statistics

The Maxwellian Limit Note that Fermi-Dirac and Bose-Einstein statistics coincide for large E/kT and small occupancy Maxwellian limit

Ideal Classical Gases Classical Þ occupancy of any one quantum state is small I.e., Maxwellian Equation of State: Valid for both non- and ultra-relativistic gases

Ideal Classical Gases Recall: Non-relativistic: Ultra-relativistic Pressure = 2/3 kinetic energy density Hence average KE = 2/3 kT Ultra-relativistic Pressure = 1/3 kinetic energy density Hence average KE = 1/3 kT

Ideal Classical Gases Total number of particles N in a volume V is given by:

Ideal Classical Gases Rearranging, we obtain an expression for m, the chemical potential

Ideal Classical Gases Interpretation of m From statistical mechanics, the change of energy of a system brought about by a change in the number of particles is:

Ideal Classical Gases Interpretation of nQ (non-relativistic) Consider the de Broglie Wavelength Hence, since the average separation of particles in a gas of density n is ~n-1/3 If n << nQ , the average separation is greater than l and the gas is classical rather than quantum

Ideal Classical Gases A similar calculation is possible for a gas of ultra-relativistic particles:

Quantum Gases Low concentration/high temperature electron gases behave classically Quantum effects large for high electron concentration/”low” temperature Electrons obey Fermi-Dirac statistics All states occupied up to an energy Ef , the Fermi Energy with a momentum pf Described as a degenerate gas

Quantum Gases Equations of State: (See Physics of Stars secn 2.2) Non-relativistic: Ultra-relativistic:

Quantum Gases Note: Pressure rises more slowly with density for an ultra-relativistic degenerate gas compared to non-relativistic Consequences for the upper mass of degenerate stellar cores and white dwarfs

The Saha Equation Atoms within a star are ionised via interaction with photons We have a dynamic equilibrium between photons and atoms on one hand and electrons and ions on the other Considering the case of hydrogen: H + g « e- + p

The Saha Equation Thermodynamic equilibrium is reached when the chemical potentials on both sides of the equation are equal I.e, changes in numbers of particles doesn’t affect the energy, hence: m(H) + m(g) = m(e-) + m(p)

The Saha Equation Chemical potential of a photon: m(g) = 0 Also have to allow the hydrogen atom to be in any electronic quantum state, q, with energy: Eq = -13.6/q2 eV Then: m(Hq) = m(e-) + m(p) (1)

The Saha Equation Assuming the density is low and energies are non-relativistic: See Workshop 3, Question 1 Evaluate the chemical potentials in terms of the quantum concentrations using functions derived in Lecture 5:

The Saha Equation For protons: For atoms: For electrons: (Note nQ depends on mass and is almost identical for protons and hydrogen atoms)

The Saha Equation Also, gse = gsp = 2 and gsH = gq gse gsp with gq =q2 Note that the total energy of a hydrogen atom in state q is: Also, gse = gsp = 2 and gsH = gq gse gsp with gq =q2

The Saha Equation Combining these relationships with the condition for equilibrium (equation (1)), we obtain:

Consequences Degree of ionisation Sum over all q levels to obtain the ratio of protons to all neutral states of H

Degree of Ionisation We may re-write this in the form: The summation is truncated at a value of q such that the spatial extent of the atom is comparable to the separation of the atoms In practice, the summation ~1

Degree of Ionisation The ratio of ionised to neutral hydrogen (or indeed, any atom) can now be written as:

Degree of Ionisation Degree of ionisation in the sun: Average density, r~1.4x103 kg/m-3 Typical temperature T~ 6x106 K nQe ~ 1021T3/2 Assume electron density ~ proton density

Degree of Ionisation Denote the fraction of hydrogen ionised as x(H+). Then, we can write: ne = n(H+) = x(H+).r/mH n(H) = (1-x(H+)).r/mH We can now re-write (2) as:

Degree of Ionisation Substituting the values for the sun and for hydrogen, we obtain x(H+) ~ 95% I.e., the interior of the sun is almost completely ionised For further discussion, see Phillips, ch. 2 secn 2.5

Balmer Absorption To find the degree of Balmer absorption in a stellar atmosphere, we require: n(H(2))/(n(H)+n(H+)) (3) Saha gives us n(H+)/n(H) Boltzmann gives us n(H(2))/n(H(1)) Assume n(H(1))~ n(H)

Balmer Absorption From Boltzmann We can rewrite (3) as: From Saha

Balmer Absorption Hence: Typically, ne ~ 1019 m-3

Balmer Absorption A B F O G K M

Reminder Assignment 1 available today on module website

Next Week Private Study Week - Suggestions Assessment Worksheet Review Lectures 1-3 Photons in Stars (Phillips ch. 2 secn 2.3) The Photon Gas Radiation Pressure Reactions at High Temperatures (Phillips ch. 2 secn 2.6) Pair Production Photodisintegration of Nuclei