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

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Lecture 11: The Grand Canonical Ensemble

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Lecture 11: The Grand Canonical Ensemble

Schroeder Ch. 7.1-7.3

Gould and Tobochnik 6.3-6.5, 6.8


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


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.


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


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


Derivation of Gibbs Distribution

  • Expanding in a Taylor series about gives

  • Writing the above expression in terms of temperature and chemical potential gives


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


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


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?


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


Example

  • If we treat the electrons as a monatomic ideal gas, then the chemical potential for electrons is given by

  • Therefore, we have


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


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


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.


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.


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


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.


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


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


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.


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.


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


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


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.


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).


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


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


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


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


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


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.


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


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.


Sommerfeld Expansion

  • Performing integration by parts gives

  • The boundary term vanishes at both limits and using a suitable change in variables, we have


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


Sommerfeld Expansion

  • Solving for gives

  • Performing the same expansion for the total energy gives

  • From this result, we can easily calculate the heat capacity


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.


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.


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.


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.


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


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.


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