Statistical Physics Quantum Distributions In quantum systems only

Statistical Physics Quantum Distributions In quantum systems only certain energy values are allowed. In quantum theory particles are described by wave functions. Identical particles cannot be distinguished from one another if there is a significant overlap of their wave functions. Pauli principle has a significant impact on how energy states can be occupied and therefore on the corresponding energy distribution.

Statistical Physics Quantum Distributions Fermions: particles with half-integer spins The Fermi-Dirac distribution, which is valid for fermions:

Statistical Physics Quantum Distributions Bosons: particles with zero or integer spins The Bose-Einstein distribution, which is valid for bosons:

Statistical Physics Fermi-Dirac Statistics It provides the basis for our understanding of the behavior of a collection of fermions. Applying it to the problem of electrical conduction.

Statistical Physics Fermi-Dirac Statistics From Fermi-Dirac distribution: The factor B 1 is computed by integrating n(E)d. E over allo energies. The parameter β(=1/k. T) is contained in FFD, so that B 1 should be temperature dependent. B 1 = exp(-βEF)

Statistical Physics Fermi-Dirac Statistics EF : Fermi energy FFD = ½ for E = EF FFD = 1 for E < EF FFD = 0 for E > EF

Statistical Physics Fermi-Dirac Statistics At T=0 fermions occupy the lowest energy levels available to them. They cannot all be in the lowest level, since that would viola the Pauli principle. Rather, fermions will fill all the available energy levels up to a particular energy ( the Fermi energy ). At T=0 there is no chance that thermal agitation will kick a fermion to an energy greater than EF.

Statistical Physics Fermi-Dirac Statistics The Fermi-Dirac factor FFD at various temperatures

Statistical Physics Fermi-Dirac Statistics As the temperature increases from T=0, more fermions ma be in higher energy levels. The Fermi-Dirac factor “smears from the sharp step function [ Figure (a) ] to a smoother cu [ Figure (b) ]. A Fermi temperature’s defined as TF= EF /k which shown in Figure (c).

Statistical Physics Fermi-Dirac Statistics When T << TF the step function approximation for FFD is reasonably accurate. When T>> TF , FFD approaches a simple decaying exponential [ Figure (d) ]. As the temperature increases, the step is gradually rounded Finally, at very high temperatures, the distribution approach the simple decaying exponential of Maxwell-Boltzmann distribution.

Statistical Physics Fermi-Dirac Statistics Classical Theory of Electrical Conducti In 1900 Paul Drude developed a theory of electrical conduction in an effort to explain the observed conductivity of metals. Drude model assumed that the electrons in a metal existed as a gas of free particles. Paul Drude (1863 -1906)

Statistical Physics Fermi-Dirac Statistics Classical Theory of Electrical Conducti In this model the metal is thought of as a lattice of positive ions with a gas of electrons free to flow through it. The electron have a thermal kinetic energy proportional to temperature. The mean speed of an electron at room temperature can be calculated to be about 105 The velocities of the particles in a gas are directed random m/s. Therefore, there will be no flow of electrons unless an elec field is applied to the conductor.

Statistical Physics Fermi-Dirac Statistics Classical Theory of Electrical Conducti When an electric field is applied, the negatively charged electrons flow in the opposite direction to the field. Drude was able to show that the current in conductor shou be linearly proportional to the applied electric field.

Statistical Physics Fermi-Dirac Statistics Classical Theory of Electrical Conducti The principal success of Drude’s theory was that it did predict Ohm’s law. Unfortunately, the numerical predictions of theory wer not so successful. One important prediction was that the electrical conductivit could be expressed as

Statistical Physics Fermi-Dirac Statistics Classical Theory of Electrical Conducti n is the number density of conduction electrons e is the electronic charge is the average time between electron-ion collisions m is the electronic mass When combined with the other parameters in above Equat produced a value of σ that is about one order of magnitude too small for most conductors. The Drude theory is therefo incorrect in this prediction.

Statistical Physics Fermi-Dirac Statistics Classical Theory of Electrical Conducti Drude model, the conductivity should be proportional to T-1/2. But for most conductors the conductivity is nearly proportional to T-1 except at very low temperatures, where it no longer follows a simple relation. Clearly the classical model of Drude has failed to predict this important experimental fact.

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conducti How electron energies are distributed in a conductor? The real problem we face is to find g(E), the number of allowed states per unit energy. What energy values should we use? From assumption of the Drude model about “free electron” and use the results obtained in Quantum theory for a three-dimensional infinite square well potential.

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conducti The allowed energies are Where L is the length of side of the cube and ni are the integer quantum numbers. Above Equation can be rewritten as E = r 2 E 1 is just a constant, not the ground-state energy.

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conducti The number of allowed states up to “radius” r will be directly related to The spherical “volume”(4/3)πr 3. The exact number of states up to r is The extra factor of 2 is due to spin degeneracy

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conducti The factor 1/8 is necessary because we are restricted to positive quantum numbers and, therefore, to one octant of the three-dimensional number space. Nr as a function of E :

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conducti At T=0 the Fermi energy is the energy of the highest occup energy level. If there a total of N electrons, then

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conducti The density of states can be calculated by differentiating Equation of N with respect to energy :

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conducti At T = 0 we have n(E) = g(E) for E < EF , n(E) =0 for E > EF. With the distribution function n(E) the mean electronic energy can be calculated easily :

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conducti Therefore the internal energy (U) of the system is: The fraction of electrons capable of participating in this thermal process is on the order of k. T/EF. The exact number of electrons depends on temperature, because the shape of the curve n(E) changes with temperature.

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conducti T = 300 K T= 0 K

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conducti The heat capacity

Statistical Physics Fermi-Dirac Statistics Quantum Theory of Electrical Conducti The electrical conductivity varies inversely with temperatu This is another striking success for the quantum theory.

Statistical Physics Bose-Einstein Statistics Blackbody Radiation The intensity of the emitted radiation as a function of temperature and wavelength as:

Statistical Physics Bose-Einstein Statistics Blackbody Radiation The electromagnetic radiation is really a collection of photo of energy hc/λ. Use the Bose-Einstein distribution to find how the photons distributed by energy, and then use the relationship E=hc/λ turn the energy distribution into a wavelength distribution. The desired temperature dependence should already be included in the Bose-Einstein factor.

Statistical Physics Bose-Einstein Statistics Blackbody Radiation The energy of a photon is pc, so

Statistical Physics Bose-Einstein Statistics Blackbody Radiation The density of states g (E) is

Statistical Physics Bose-Einstein Statistics Blackbody Radiation The next step is to convert from a number distribution to an energy density distribution u(E). To do this it is necessary to multiply by the factor E/L 3 ( that is, energy per unit volume):

Statistical Physics Bose-Einstein Statistics Blackbody Radiation Using E=hc/λ and |d. E|=(hc/λ 2)dλ, we find In the SI system multiplying by a constant factor c/4 is requi to change the energy density to a spectral intensity:

Statistical Physics Bose-Einstein Statistics Blackbody Radiation u(λ, T) is energy per unit volume per unit wavelength inside the cavity. U(λ, T) is power per unit area per unit wavelength for radiation emitted from the cavity.

Quantum Statistics Summary Fermi-Dirac distribution Bose-Einstein distribution Quantum Particles Fermions e. g. , electrons, neutrons, protons and quarks Bosons e. g. , photons, Cooper pairs and cold Rb Spins 1/2 integer Properties At temperature of 0 K, each energy level is occupied by two Fermi particles with opposite spins. Pauli exclusion principle At very low temperature, large numbers of Bosons fall into lowest energy state. Bose-Einstein condensation Function Energy Dependence