Lecture 11: The Grand Canonical Ensemble

1. Lecture 11: The Grand Canonical Ensemble Schroeder Ch. 7.1-7.3 Gould and Tobochnik 6.3-6.5, 6.8

2. Outline • Derivation of the Gibbs distribution • Grand partition function • Bosons and fermions • Degenerate Fermi gases • White dwarfs and neutron stars • Density of states • Sommerfeld expansion • Semiconductors

3. Introduction • We have already described the canonical ensemble, which is defined as a collection of closed systems. • If we relax the condition that no matter is exchanged between the system and its reservoir, we obtain the grand canonical ensemble. • In this ensemble, the systems are all in thermal equilibrium with a reservoir at some fixed temperature, but they are also able to exchange particles with this reservoir.

4. Derivation of Gibbs Distribution • What is the probability of finding a member of the ensemble in a given state with energy and containing particles? • Consider a system in thermal and diffusive contact with a reservoir, , whose temperature and chemical potential are effectively constant. • If has energy and particles, then the total number of states available to the combined system is

5. Derivation of Gibbs Distribution • Since the combined system belongs to a microcanonical ensemble, the probability of finding our system with energy and particles is given by • Since the reservoir is very large compared with our system, then and

6. Derivation of Gibbs Distribution • Expanding in a Taylor series about gives • Writing the above expression in terms of temperature and chemical potential gives

7. Derivation of the Gibbs Distribution • Since the combined system is in the microcanonical ensemble, then the probability of finding in any one state of energy and particles is given by • Each of the exponential factors is called a Gibbs factor • If we define the constant as , then for any state , the probability is given by • This is the Gibbs distribution and it characterizes the grand canonical ensemble

8. Grand Partition Function • The quantity is called the grand partition function. • By requiring that the sum of the probabilities of all states to equal 1, it can be shown that

9. Example • Consider a system consisting of a single hydrogen atom, which has two possible states: • Unoccupied (i.e. no electron present) • Occupied (one electron present in the ground state) • Q: What is the ratio of the probabilities of these two states?

10. Example • If we neglect the spin states of the electron and the excited states of the hydrogen atom, this system has just two states • Unoccupied: • Occupied: • The ratio of the probabilities of these two states s given by

11. Example • If we treat the electrons as a monatomic ideal gas, then the chemical potential for electrons is given by • Therefore, we have

12. Example • Taking electron spin into account, the hydrogen atom now has two occupied states, each with the same energy, so the ratio of unoccupied to occupied atoms is • Now, a free electron has two degenerate states so the chemical potential of the electron gas is • Therefore, we have

13. Thermodynamics in the Grand Canonical Ensemble • It can be shown that all the thermodynamic functions can be expressed in terms of the grand partition function and its derivatives. • The average internal energy is • The average number of particles is • The generalized forces are • The entropy is given by

14. Grand Potential • Recall that in the canonical ensemble, there is a relationship between the Helmholtz free energy and the partition function: . • Using an analogous argument, we can derive the grand potential: • The grand potential is the maximum amount of energy available to do external work for a system in contact with both a heat and a particle reservoir.

15. Bosons and Fermions • The most important application of Gibbs factors is to quantum statistics: the study of dense systems in which 2+ identical particles have a probability of occupying the same single-particle state. • Particles that can share a state with another of the same species are called bosons. • Examples include photons, helium-4 atoms, etc. • Particles that cannot share a state with another of the same species are called fermions. • Examples include protons, neutrons, neutrinos, etc.

16. Bosons and Fermions • Because individual particles have wave-like properties, the state of a particle can be described by a wavefunction • The indistinguishability of quantum-mechanical particles implies that • Observations show that both signs are possible for quantum-mechanical particles For bosons For fermions

17. Bosons and Fermions • Suppose that the two-particle wavefunction can be decomposed into a product of single-particle wavefunctions • For bosons, this equation guarantees that will be symmetric under interchange of and . • For fermions, this equation guarantees that . • The rule that two identical fermions cannot occupy the same state is called the Pauli exclusion principle • Pauli also demonstrated that all particles with integer spins are bosons, whereas all particles with half-integer spin are fermions.

18. Bosons and Fermions • When , the chance of any two particles occupying the same state is negligible. • For an ideal gas, the chance of any two particles occupying the same state is negligible only if . • There are a number of systems that violate this condition • Neutron star • Liquid helium • Electrons in metals • Photons

19. Fermi-Dirac Distribution • Consider a single-particle state of a system whose energy when occupied by a single particle is . The probability of the state being occupied by particles is • If the particles in question are fermions, then can only be 0 or 1, which implies that • The average number of particles in the state (also called the occupancy of the state) is given by • This distribution is called the Fermi-Dirac distribution

20. Fermi-Dirac Distribution • The Fermi-Dirac distribution goes to zero when and goes to 1 when . • At very low temperature, fermions will distribute themselves in the energy levels below the chemical potential and all the levels above are empty. • As the temperature rises, energy levels above the chemical potential begin to be occupied.

21. Degenerate Fermi Gases • As an application of the Fermi-Dirac distribution, let’s examine degenerate Fermi gases. • Examples of degenerate Fermi gases are • Conduction electrons in a metal • Electrons in a white dwarf star • Neutrons in a neutron star • Let’s first consider the properties of an low-temperature electron gas.

22. Low-Temperature Electron Gas • At , the Fermi-Dirac distribution becomes a step function in which all single-particle states with energy less than are occupied, while all states with energy greater than are unoccupied. • In this context, is also called the Fermi energy • When a gas of fermions is so cold that nearly all states below are occupied, it is said to be degenerate. • It can be shown that the Fermi energy is given by

23. Low-Temperature Electron Gas • Thus, the Fermi energy is the highest energy of all the electrons. • To calculate the total energy of all the electrons, we can add up the energies of the electrons in all occupied states. • It can be shown that the total energy is

24. Low-Temperature Electron Gas • Therefore, the average energy of the electrons is 3/5 the Fermi energy. • The temperature that a Fermi gas would have to have in order for is called the Fermi temperature. • The pressure of a degenerate electron gas (also called the degeneracy pressure) is given by • The degeneracy pressure is what keeps matter from collapsing under the electrostatic forces that attract electrons and protons.

25. White Dwarfs and Neutron Stars • A star that has consumed all its nuclear fuel will undergo gravitational collapse, but may end up in a stable state as a white dwarf or a neutron star. • This occurs when the mass that remains in the core after the outer layers are blown away doesnot exceed a particular limit, called the Chandrashekar limit. • Stars that succeed in forming such stable remnants owe their existence to the high degeneracy pressure exerted by electrons (for white dwarfs) and neutrons (for neutron stars).

26. White Dwarf Stars • A white dwarf star can be considered as a degenerate electron gas. • The nuclei present within the white dwarf balances the charge and provides the gravitational attraction that holds the star together. • The total kinetic energy of the degenerate electrons is given by the Fermi energy

27. White Dwarf Stars • If we assume that the star contains one proton and one neutron for each electron, then and thus we have • The Fermi energy and Fermi temperature for the white dwarf star is given by • Here, is the equilibrium radius of a white dwarf star, which can be determined by finding the minimum of the total energy

28. White Dwarf Stars • It can be shown that the gravitational potential energy of a white dwarf is given by • The total energy of a white dwarf can be given by • The equilibrium radius of a white dwarf is determined by the minimum of the total energy • For a one solar mass white dwarf, and thus, the Fermi energy and the Fermi temperature for the white dwarf star is given by

29. Neutron Stars • A neutron star is made entirely of neutrons and is supported against gravitational collapse by degenerate neutron pressure. • The total kinetic energy of the degenerate electrons is given by the Fermi energy • Thus, the kinetic energy comes from the neutrons, and the number of these is simply . Therefore, we have • The Fermi energy and the Fermi temperature for the neutron star is given by

30. Neutron Stars • The total energy of a neutron star can be given by • The equilibrium radius of a white dwarf is determined by the minimum of the total energy • For a one solar mass white dwarf, and thus, the Fermi energy and the Fermi temperature for the neutron star is given by

31. Density of States • One property of a Fermi gas that we cannot calculate using the approximation is the heat capacity. • Therefore, we must examine the Fermi gas at small nonzero temperatures • To better visualize – and quantify – the behavior of a Fermi gas at small temperatures, we will introduce a new concept called the density of states.

32. Density of States • Using a suitable change of variables, it can be shown that the energy integral for a Fermi gas at zero temperature becomes • The quantity in square brackets is the number of single particle states per unit energy, also known as the density of states. • Using the density of states, we can obtain the total number of electrons at zero temperature

33. Density of States • For nonzero temperature, the total number and energy of electrons can be determined as follows • Note that the chemical potential is the point where the probability of being occupied is exactly ½. • In the limit , we can use the Sommerfeld expansion to evaluate the above integrals.

34. Sommerfeld Expansion • Performing integration by parts gives • The boundary term vanishes at both limits and using a suitable change in variables, we have

35. Sommerfeld Expansion • We can make two approximations • Extend the lower limit to • Expand in a Taylor series about the point • With these approximations, we have • This integral can be performed giving

36. Sommerfeld Expansion • Solving for gives • Performing the same expansion for the total energy gives • From this result, we can easily calculate the heat capacity

37. Electrons in Metals • Atoms in a metal are closely packed, which causes their outer shell electrons to break away from the parent atoms and move freely through the solid. • The set of electron energy levels for which they are more or less free to move in the solid is called the conduction band. • Energy levels below the conduction band form the valence band and electrons at energies below that are strongly bound to the atoms. • The work function, , is the energy that an electron must acquire to escape the metal.

38. Conductors and Insulators • In a conductor, the Fermi energy lies within one of the bands, whereas in an insulator, the Fermi energy lies within a gap. • Therefore, at , the band below the gap is completely occupied while the band above the gap is unoccupied. • Because there are no empty states close in energy to those that are occupied, the electrons are “stuck in place” and the material does not conduct electricity.

39. Semiconductors • A semiconductor is an insulator in which the gap is narrow enough for a few electrons to jump across it at room temperature. • The figure below shows the density of states in the vicinity of the Fermi energy for an idealized semiconductor.

40. Semiconductors • As an approximation, let’s model the density of states near the bottom of the conduction band using the same function as for a free Fermi gas with an appropriate zero point. • Let’s model the density of states near the top of the valence band as a mirror image of this model. • In this approximation, the chemical potential must always lie precisely in the middle of the gap, regardless of the temperature.

41. Semiconductors • The number of electrons in the conduction band is given by • If the width of the gap is much greater than , then we can approximate the above integral as • After a suitable change in variables, this integral can be evaluated to give

42. Semiconductors • The above result indicates that a pure semiconductor will conduct much better at higher temperature because there are much more electrons in the conduction band. • Moreover, in order to produce an insulator, the gap would have to become significantly wider.