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**CHAPTER 9 Statistical Physics**

underpins thermodynamics, ideal gas (a classical physics model), ensembles of molecules, solids, liquids … the universe 9.1 Justification for its need ! 9.2 Classical distribution functions as examples of distributions of velocity and velocity2 in ideal gas 9.3 Equipartition Theorem 9.4 Maxwell Speed Distribution 9.5 Classical and Quantum Statistics 9.6 Black body radiation, Liquid Helium, Bose-Einstein condensates, Bose-Einstein statistics, 9.7 Fermi-Dirac Statistics … Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906 by his own hand. Paul Ehrenfest, carrying on his work, died similarly in Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously. - David L. Goldstein (States of Matter, Mineola, New York: Dover, 1985)

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**First there was classical physics with a cause (or causes)**

Newton’s three force laws, first unification in physics Lagrange around 1790 and Hamilton around 1840 added significantly to the computational power of Newtonian mechanics. Pierre-Simon de Laplace ( ) Made major contributions to the theory of probability and well known clockwork universe statement: It should be possible in principle to have perfect knowledge of the universe. Such knowledge would come from measuring at one time the position and velocities of every particle of matter and then applying Newton’s law. As they are cause and effect relations that work forwards and backwards in time, perfect knowledge can be extended all the way back to the beginning of the universe and all the way forward to its end. So no uncertainty principle allowed …

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then there was the realization that one does not always need to know the cause (causes), can do statistical analyses instead Typical problem, flipping of 100 coins, One can try to identify all physical condition before the toss, model the toss itself, and then predict how the coin will fall down if all done correctly, one will be able to make a prediction on how many heads or tails one will obtain in a series of experiments Statistics and probabilities would just predict 50 % heads 50% tails by ignoring all of that physics, The more experimental trials, 100,000 coin tosses, the better this prediction will be borne out

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**<KE>= <p2>/2m**

Speed distribution of particles in an ideal gas in equilibrium, instead of analyzing what each individual particle is going to do, one derives a distribution function, determines the density of states, and then calculates the physical properties of the system (always by the same procedures) <KE>= <p2>/2m There is one characteristic kinetic energy (or speed) distribution for each value of T, so we would like to have a function that gives these distribution for all temperatures !!!

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**Path to statistical physics from classical to quantum for bosons and fermions**

Benjamin Thompson (Count Rumford) 1753 – 1814 Put forward the idea of heat as merely the kinetic energy of individual particles in an ideal gas, speculation for other substances. James Prescott Joule 1818 – 1889 Demonstrated the mechanical equivalent of heat, so central concept of thermodynamics becomes internal energy of systems (many many particles at once)

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**Beyond first or second year college physics**

James Clark Maxwell 1831 – 1879, Josiah Willard Gibbs 1839 – 1903, Ludwig Boltzmann 1844 – 1906 (all believing in reality of atoms, tiny minority at the time) Brought the mathematical theories of probability and statistics to bear on the physical thermodynamics problems of their time. Showed that statistical distributions of physical properties of an ideal gas (in equilibrium – a stationary state) can be used to explain the observed classical macroscopic phenomena (i.e. gas laws) Gibbs invents notation for vector calculus, the form in which we use Maxwell’s equations today Maxwell’s electromagnetic theory succeeded his work on statistical foundation of thermodynamics – so he was a genius twice over.

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**and then there came modern physics …**

Einstein 1905 PhD thesis, the correct theory of Brownian motion, a theory that required atoms to be real, because there are measurable consequences to their motion, (also start of quantitative nanoscience as size of a common sugar molecule (1 nm) was determined correctly) Bose (with Einstein’s generalization) 1924 Statistics of indistinguishable particles that are bosons (photons: Bose, all other bosons: Einstein’s generalization) Fermi and Dirac independently 1926 Statistics of indistinguishable particles that are fermions

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**9.2: Maxwell Velocity and Velocity2 Distribution**

internal energy in an ideal gas depends only on the movements of the entities that make up that gas. Define a velocity distribution function . = the probability of finding a particle with velocity between . where is similar to the product of a wavefunction with its complex conjugate (in 3D), from it we can calculate expectation values (what is measured on average) by the same integration procedure as in previous chapters !!

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**Maxwell Velocity Distribution**

Maxwell proved that the velocity probability distribution function is proportional to exp(−½ mv2 / kT), special form of exp(-E/kT) – the Maxwell-Boltzmann statistics distribution function. Therefore where C is a proportionality factor and β ≡ (kT)−1. k: Boltzmann constant, which we find everywhere in this field Because v2 = vx2 + vy2 + vz2 then Rewrite this as the product of three factors. Is the product of the three functions gx, gy gz which are just for one variable (1D) each

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**Maxwell Velocity Distribution**

g(vx) dvx is the probability that the x component of a gas molecule’s velocity lies between vx and vx + dvx. if we integrate g(vx) dvx over all of vx and set it equal to 1, we get the normalization factor The mean value (expectation value) of vx Full Widths at Half Maximum e-0.5 = g(0) That is similar to the expectation value of momentum in the square wells

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**Maxwell Velocity2 Distribution**

The mean value of vx2, also an expectation value that is a simple function of x This is not zero because it is related to kinetic energy, remember the expectation value of p2 was also not zero It relates the human invented energy scale (at the individual particle level) to the absolute temperature scale (a physical thing) (13)×10−23 J K−1 (78)×10−5 eV K−1 gas constant R divided by Avogadro’s number NA

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**Maxwell Velocity2 Distribution**

The results for the x, y, and z velocity2 components are identical. The mean translational kinetic energy of a molecule: Equipartion of the kinetic energy in each of 3 dimension a particle may travel, in each degree of freedom of its linear movement this result can be generalized to the equipartition theorem

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**9.3: Equipartition Theorem**

For a system of particles (e.g. atoms or molecules) in equilibrium a mean energy of ½ kT per system member is associated with each independent quadratic term in the energy of the system member. That can be movement in a direction, rotation about an axis, vibration about an equilibrium position, …, 3D vibrations in a harmonic oscillator Each independent phase space coordinate: degree of freedom

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**Equipartition Theorem**

In a monatomic ideal gas, each molecule has There are three degrees of freedom. Mean kinetic energy is 3(1/2 kT) = 3/2 kT In a gas of N helium atoms, the total internal energy is CV = 3/2 N k For the heat capacity for 1 mole The ideal gas constant R = 8.31 J/K

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**As predicted, only 3 translational degrees of freedom**

2 more (rotational) degrees of freedom 2 more (vibrational) degrees of freedom plus vibration, which also adds two times 1/2 kBT discrepancies due to quantized vibrations, not due to high particle density We get excellent agreement for the noble gasses, they are just single particles and well isolated from other particles

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**Molar Heat Capacity Example of H2**

The heat capacities of diatomic gases are temperature dependent, indicating that the different degrees of freedom are “turned on” at different temperatures. Example of H2

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**The Rigid Rotator Model**

For diatomic gases, consider the rigid rotator model. The molecule rotates about either the x or y axis. The corresponding rotational energies are ½ Ixωx2 and ½ Iyωy2. There are five degrees of freedom, three translational and two rotational. (I is rotational moment of inertia)

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**Two more degrees of freedom, ½ Ixωx2**

and ½ Ixωx2 Two more degrees of freedom, because there are kinetic and potential energy, both are “quadratic” (both have variables that appear squared in a formula of energy is a degree of freedom, ½ m (dr/dt)2 and ½ κ r2

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**Using the Equipartition Theorem**

In the quantum theory of the rigid rotator the allowed energy levels are From previous chapters, the mass of an atom is largely confined to its nucleus Iz is much smaller than Ix and Iy. Only rotations about x and y are allowed at reasonable temperatures. Model of diatomic molecule, two atoms connected to each other by a massless spring. The vibrational kinetic energy is ½ m(dr/dt)2, there is kinetic and potential energy ½ κ r2 in a harmonic vibration, so two extra degrees of freedom There are seven degrees of freedom (three translational, two rotational, and two vibrational for a two-atom molecule in a gas).

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**not that simple six degrees of freedom**

according to classical physics, Cv should be 3 R = 6/2 kBT NA for solids and independent of the temperature We will revisit this problem when we have learned of quantum distributions, concept of phonons, which are quasi-particle that are not restricted by the Pauli exclusion principle

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**Maxwell’s speed (v) distribution**

Slits have small widths, size of it defines dv (a small speed segment of the speed distribution)

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**9.4: Maxwell Speed Distribution ΙvΙ**

Maxwell velocity distribution: Where let’s turn this into a speed distribution. F(v) dv = the probability of finding a particle with speed between v and v + dv. One cannot derive F(v) dv (i.e. a distribution of a scalar entity) simply from f(v) d3v (the velocity distribution function, i.e. a distribution of vectors and their components), we need idea of phase space for this derivation

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**Maxwell Speed Distribution**

Idea of phase space, to count how many states there are Suppose some distribution of particles f(x, y, z) exists in normal three-dimensional (x, y, z) space. The distance of the particles at the point (x, y, z) to the origin is the probability of finding a particle between

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**Maxwell Speed Distribution**

Radial distribution function F(r), of finding a particle between r and r + dr {sure not equal to f(x,y,z) as we want to go from coordinates to length of the vector, a scalar} The volume of any spherical shell is 4πr2 dr. now replace the 3D space coordinates x, y, and z with the velocity space coordinates vx, vy, and vz Maxwell speed distribution: It is only going to be valid in the classical limit, as a few particles would have speeds in excess of the speed of light. note speed distribution function is different to velocity distribution function, but both have the same Maxwell-Boltzmann statistical factor

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**Maxwell Speed Distribution**

The most probable speed v*, the mean speed , and the root-mean-square speed vrms are all different.

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**Maxwell Speed Distribution**

Most probable speed (at the peak of the speed distribution), simply plot the function, take the derivative and set it zero, derive the consequences: Average (mean) speed, will be an expectation value that we calculate from on an integral on the basis of the speed distribution function

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average (mean) of the square of the speed, will be an expectation value that we calculate from another integral on the basis of the speed distribution function We define root mean square speed on its basis which is of course associated with the mean kinetic energy We can also calculate the spread (standard deviation) of the speed distribution function in analogy to quantum mechanical spreads Note that σv in proportional to So now we understand the whole function, can make calculations for all T

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Straightforward: turn speed distribution into kinetic energy (internal energy of ideal gas) distribution

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**So we recover the equipartition theorem for a mono-atomic gas**

Number of particles with energy in interval E and E + dE

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**9.5: Needs for Quantum Statistics**

If molecules, atoms, or subatomic particles are fermions, i.e. most of matter, in the liquid or solid state, the Pauli exclusion principle prevents two particles with identical wave functions from sharing the same space. The spatial part of the wavefunction can be identical for two particles in the same state, but the spin part f the wavefunction has to be different to fulfill the Pauli exclusion principle. If the particles under consideration are indistinguishable and Bosons, they are not subject to the Pauli exclusion principle, i.e. behave differently There are only certain energy values allowed for bound systems in quantum mechanics. There is no restriction on particle energies in classical physics.

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**Classical physics Distributions**

Boltzmann showed that the statistical factor exp(−βE) is a characteristic of any classical system in equilibrium (in agreement with Maxwell’s speed distribution) {quantities other than molecular speeds may affect the energy of a given state (as we have already seen for rotations, vibrations)} Maxwell-Boltzmann statistics for classical system: β ≡ (kBT)−1 The energy distribution for classical system: n(E) dE = the number of particles with energies between E and E + dE. g(E) = the density of states, is the number of states available per unit energy range. FMB gives the relative probability that an energy state is occupied at a given temperature. A is a normalization factor, problem specific

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**Classical / quantum distributions**

Characteristic of indistinguishability that makes quantum statistics different from classical statistics. The possible configurations for distinguishable particles in either of two (energy or anything else) states: There are four possible states the system can be in. State 1 State 2 AB A B

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**Quantum Distributions**

If the two particles are indistinguishable: There are only three possible states of the system. Because there are two types of quantum mechanical particles, two kinds of quantum distributions are needed. Fermions: Particles with half-integer spins, obey the Pauli principle. Bosons: Particles with zero or integer spins, do not obey the Pauli principle. State 1 State 2 XX X

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Multiply each state with its number of microstates for distinguishable particles – sum it all up and you get the distribution of classical physics particles Ignore all microstates for indistinguishable particles – sum it all up, that would be the distribution for bosons Ignore all microstates and states that have more than one particle at the same energy level, - sum it all up, that would be the distribution of fermions Serway et al, chapter 10 for details Realize, there must be three different distribution functions !!

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**Quantum Distributions**

Fermi-Dirac distribution: where Bose-Einstein distribution: Where In each case Bi (i = 1 or 2) is a normalized factor which depends on the problem. Both distributions reduce to the classical Maxwell-Boltzmann distribution when Bi exp(βE) is much greater than 1, this happens at low densities (i.e. in a dilute gas at moderately high temperatures, i.e. room temperature

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**Classical and Quantum Distributions**

For photons in cavity, Planck, A = 1, α = 0 E is quantized in units of h if part of a bound system

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**Quantum Distributions**

If all three normalization factors = 1, just for comparison has to do with specific normalization factor The exact forms of normalization factors for the distributions depend on the physical problem being considered. Because bosons do not obey the Pauli exclusion principle, more bosons can fill lower energy states (are actually attracted to do so) All three graphs coincide at high energies – the classical limit. Maxwell-Boltzmann statistics may be used in the classical limit when particles are so far apart that they are distinguishable, can be tracked by their paths

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**When there are so many states that there is a very low probability of occupation**

also if the particles are heavy (macroscopic), i.e. a bunch of classical physics particle, Bohr’s correspondence principle again

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Anything to do with solids, when high probability of occupancy of energy states, e.g. electrons in a metal, which are fermions

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**anything to do with liquids, when high probability of occupancy of energy states**

Bose-Einstein condensate for at 2.17 K superfluidity (explained later on)

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**degeneracy of the first exited state in H atom**

l ml ms up ms down 2 +1/2 -1/2 1 -1 g functions, density of states, how many states there are per unit energy value, in other words: the degeneracy if we talk about a hydrogen atom g functions are problem specific !!

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revisited Einstein’s assumptions in 1907, atoms vibrate independently of each other he used Maxwell-Boltzmann statistics because there are so many possibly vibration states that only a few of the available states will be occupied, (and the other distribution functions were not known at the time) (starting from zero point energy, due to uncertainty principle) A. Einstein, "Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme", Annalen der Physik 22 (1907) 180–190 i.e. at high temperatures is approaches the classical value of 2 degrees of freedom with ½ kT each times 3 vibration direction (Bohr’s correspondence principle once more)

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**To account for different bond strength, different spring constants**

hetero-polar bond in diamond much stronger than metallic bond in lead and aluminum, so much larger Einstein Temperature for diamond (1,320 K) >> K for typical metals

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Peter Debye lifted the assumption that atoms vibrate independably, similar statistics, Debye temperature TD even better modeling with phonons, which are pseudo-particle of the boson type

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**Blackbody Radiation Blackbody Radiation**

Intensity of the emitted radiation is Use Bose-Einstein distribution because photons are bosons with spin 1 (they have two polarization states) For a free particle in terms of momentum in a 3D infinitely deep well: E = pc = hf so we need the equivalent of this formulae in terms of momentum (KE = p2 / 2m) now our particles are measles

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Phase space again Density of states in cavity, we can assume the cavity is a sphere, we could alternatively assume it is any kind of shape that can be filled with cubes …

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**Bose-Einstein Statistics**

The number of allowed energy states within “radius” r of a sphere is Where 1/8 comes from the restriction to positive values of ni and 2 comes from the fact that there are two possible photon polarizations. Resolve Energy equation for r, and substitute into the above equation for Nr Then differentiate to get the density of states g(E) is Multiply the Bose-Einstein factor in: For photons, the normalization factor is 1, they are created and destroyed as needed

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**Bose-Einstein Statistics**

Convert from a number distribution to an energy density distribution u(E). For all photons in the range E to E + dE Using E = hf and |dE| = (hc/λ2) dλ In the SI system, multiplying by c/4 is required. and world wide fame for Satyendra Nath Bose 1894 – 1974 !

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Liquid Helium Has the lowest boiling point of any element (4.2 K at 1 atmosphere pressure) and has no solid phase at normal pressure. The density of liquid helium as a function of temperature.

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Liquid Helium The specific heat of liquid helium as a function of temperature The temperature at about 2.17 K is referred to as the critical temperature (Tc), transition temperature, or lambda point. As the temperature is reduced from 4.2 K toward the lambda point, the liquid boils vigorously. At 2.17 K the boiling suddenly stops. What happens at 2.17 K is a transition from the normal phase to the superfluid phase. Thermal conductivity goes to infinity at lambda point, so no hot bubbles can form while the liquid is boiling,

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Liquid Helium The rate of flow increases dramatically as the temperature is reduced because the superfluid has an extremely low viscosity. Creeping film – formed when the viscosity is very low and some helium condenses from the gas phase to the glass of some beaker.

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Liquid Helium Liquid helium below the lambda point is part superfluid and part normal. As the temperature approaches absolute zero, the superfluid approaches 100% superfluid. The fraction of helium atoms in the superfluid state: Superfluid liquid helium is referred to as a Bose-Einstein condensation. not subject to the Pauli exclusion principle because (the most common helium atoms are bosons all particles are in the same quantum state

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**9.7: Fermi-Dirac Statistics**

EF is called the Fermi energy. When E = EF, the exponential term is 1. FFD = ½ In the limit as T → 0, At T = 0, fermions occupy the lowest energy levels. Near T = 0, there is no chance that thermal agitation will kick a fermion to an energy greater than EF.

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**Fermi-Dirac Statistics**

As the temperature increases from T = 0, the Fermi-Dirac factor “smears out”. Fermi temperature, defined as TF ≡ EF / k T = TF T >> TF When T >> TF, FFD approaches a decaying exponential of the Maxwell Boltzmann statistics. At room temperature, only tiny amount of fermions are in the region around EF,i.e. can contribute to elecric current, …

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**Classical Theory of Electrical Conduction**

Paul Drude (1900) showed on the basis of the idea of a free electron gas inside a metal that the current in a conductor should be linearly proportional to the applied electric field, that would be consistent with Ohm’s law. His prediction for electrical conductivity: Mean free path is . Drude electrical conductivity:

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**Classical Theory of Electrical Conduction**

From Maxwell’s speed distribution According to the Drude model, the conductivity should be proportional to T−1/2. But for most metals is very nearly proportional to T−1 !! This is not consistent with experimental results. l and τ make only sense for a realistic microscopic model, so whole approach abandoned, but free electron gas idea kept, just a different kind of gas

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**All condensed matter (liquids and solids) problems are statistical quantum mechanics problems !!**

Quantum condensed matter physics problems are typically low temperature problems Ideal gasses can be modeled classically, because they have very low matter densities

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**Quantum Theory of Electrical Conduction**

The allowed energies for electrons are The parameter r is the “radius” of a sphere in phase space. The volume is (4/3)πr 3. The exact number of states up to radius r is

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**Quantum Theory of Electrical Conduction**

Rewrite as a function of E: At T = 0, the Fermi energy is the energy of the highest occupied level. Total of electrons Solve for EF: The density of states with respect to energy in terms of EF:

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**Quantum Theory of Electrical Conduction**

At T = 0, The mean electronic energy: Internal energy of the system: Only those electrons within about kT of EF will be able to absorb thermal energy and jump to a higher state. Therefore the fraction of electrons capable of participating in this thermal process is on the order of kT / EF.

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**Quantum Theory of Electrical Conduction**

In general, Where α is a constant > 1. The exact number of electrons depends on temperature. Heat capacity is Molar heat capacity is

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**Quantum Theory of Electrical Conduction**

Arnold Sommerfeld used correct distribution n(E) at room temperature and found a value for α of π2 / 4. With the value TF = 80,000 K for copper, we obtain cV ≈ 0.02R, which is consistent with the experimental value! Quantum theory has proved to be a success. Replace mean speed in Eq (9,37) by Fermi speed uF defined from EF = ½ uF2. Conducting electrons are loosely bound to their atoms. these electrons must be at the high energy level. at room temperature the highest energy level is close to the Fermi energy. We should use

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**Quantum Theory of Electrical Conduction**

Drude thought that the mean free path could be no more than several tenths of a nanometer, but it was longer than his estimation. Einstein calculated the value of ℓ to be on the order of 40 nm in copper at room temperature. The conductivity is Sequence of proportions.

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**Rewrite Maxwell speed distribution in terms of energy.**

For a monatomic gas the energy is all translational kinetic energy. where

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