Microstates, Entropy and Quanta Don Koks

Microstates, Entropy and Quanta An Introduction to Statistical Mechanics

123 ISBN 978-3-030-02428-4 ISBN 978-3-030-02429-1 (eBook) https://doi.org/10.1007/978-3-030-02429-1

Library of Congress Control Number: 2018960736

© Springer Nature Switzerland AG 2018, corrected publication 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Cover art: Central to statistical mechanics is the idea of counting the states accessible to a system. When these states exist within some continuous space, they cannot be counted. Instead, we “tile” the space into cells, with each cell defining a state, and then we count those cells. The ball on the front cover is a schematic of this tiling of the velocity space of a free particle that moves in three spatial dimensions. For real particles, the cells are so much smaller than the size of the ball that, to all intents and purposes, the ball is a smooth sphere. The number of cells can then easily be found from the sphere’s volume.

This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland For my ancestors, and all those who have gone before. Preface

Another book on introductory statistical mechanics? You might think that a century-old subject would have nothing left unsaid; but that is perhaps not the case. Unlike most other fields of physics, one can compare a dozen books on statistical mechanics and find a dozen different approaches to the discipline. At one extreme are authors who revel in arcane abstraction, but whose books go mostly unread. At the other extreme are very readable books that lack the mathematics to carry the reader very far beyond a set of physical assumptions. Most readers are looking for something in between; but that space is vast and subjective, with plenty of room for another book to aim for the “Goldilocks Zone” of being just right. That’s why I wrote this book: I think that the field of introductory statistical mechanics still has plenty of scope for an author to try a different mix of mathematical exposition and physical reasoning. The physics part of this mix is a build of statistical mechanics from the ground up, anchored to a bedrock of physical concepts. With this approach, I hope to have revealed the necessity and importance of the subject’s core ideas, such as entropy and temperature. The mathematics part of the mix has been an emphasis on a strong logical reasoning that has a clean outline, yet avoids the notational clutter and obscure discussions that are so often associated with statistical mechanics, and which can make it so hard to learn. Thus, beside the calculations of representative physical quantities, you will find here various mathematical analyses that I believe are important to physicists. Much of this mathematical foundation is given in the first chapter, such as details of integrating the gaussian function, and the correct use of infinitesimals, partial derivatives, and units of measurement. By the time you reach that chapter’s end, you might be wondering whether you are really reading a book on statistical mechanics after all. And yet, you will encounter those topics time and again as you work through the rest of the book. The choice of how and where to begin describing a subject is always highly author dependent. The concepts that I introduce methodically, as needed,

vii viii Preface are sometimes merely postulated with a breezy stroke of the pen in books that announce themselves as introductions. Postulatory approaches to other subjects can certainly work well; for instance, I admire Feynman’s approach to electromagnetism in his Lectures on Physics, since, although he postulates Maxwell’s equations at the very start, we never lose sight of the physics in his discussions. In contrast, I struggle to see any physics at all in some postulatory approaches to statistical mechanics, which can so easily ignore the difficult questions that interest physicists. I commence the subject of statistical mechanics with an archetypal obser- vation: why does a drop of ink placed in a bathtub disperse? Once dispersed, might it ever re-assemble into a drop? This question showcases the impor- tance of counting the number of ways in which a system’s constituents can be arranged, and leads to statistical mechanics proper via its fundamental postulate. That discussion demands knowledge of the concept of energy, a concept that was useful and intriguing to early astronomers studying plane- tary orbits, but whose wider application was not well understood in the early days of thermodynamics, 150 years ago. With a more modern understanding of energy (or perhaps “acceptance” is a better word, since we still don’t know what it is—if, indeed, asking what it is has any meaning), we are in a good position to write down the laws of thermodynamics. Then, we can explore heat engines, chemical processes and equilibria, and heat flow. The flow of heat is a stepping stone to appreciating diverse related areas, such as parti- cle diffusion and, in fact, the signal processing performed in a modern radar receiver. But no system is ever truly isolated; and the question of how to analyse a system in contact with the wider world brings us to the Boltzmann dis- tribution, with examples in paramagnetism, atomic energy levels, molecular and crystal heat capacities, and data-transmission theory. The Boltzmann distribution also sheds light on the motion of gas particles. I use that theory to explore an atmosphere, as well as the molecular details of viscosity and thermal conductivity. Quantum ideas then emerge, via Einstein’s and Debye’s theories of heat capacity. The notion of fermions and bosons forms a springboard to the study of electronic heat capacity, electrical conduction, thermal noise in electric circuits, the spectra of light produced by hot bodies, some cosmology, the greenhouse effect, and the modern technologies of light-emitting diodes and the laser. I have sprinkled the text with occasional short digressions, discussing top- ics such as the factorial function in number theory, the energy–momentum tensor in relativity, a little bit of signal processing, and decrying the short- comings of modern analytical astronomy. Hopefully, these asides will only enrich your interest without being a distraction. Unlike some books on statistical mechanics, I have chosen to discuss a lot of material before introducing the Boltzmann distribution. Thus, in those pre- Boltzmann chapters, I invoke the equipartition theorem to approximate the Preface ix

particles in an ideal gas of temperature T as each having, say, translational energy 3/2 kT . Later, when studying the Boltzmann distribution, we learn that only their average translational energy is 3/2 kT . Some authors will avoid this initial simplification by introducing the Boltzmann distribution very early on. But I think that using the simple approximation initially, and leaving Boltzmann for later, is useful pedagogically. A subject as old as statistical mechanics is bound to carry baggage picked up along the way, created as a normal part of its development, when physicists and chemists were searching for the best path through the new forest they had discovered. The choice of what might best be discarded is a little subjective. I have tended to minimise the use of phrases and topics that appear to be generally confusing, unattractive, or not useful. For example, I cannot imagine that the conventional vernacular that de- scribes various flavours of ensemble, along with free energies, partition func- tions, and Maxwell relations, does anything to attract new adherents to sta- tistical mechanics. Pseudo wisdom that you can find in books on the subject, such as “The trick to solving this problem is to use the grand canonical en- semble”, is apt to give the impression that statistical mechanics is all about finding the right trick using the right ensemble to get the right answer. The language of ensembles is not especially deep, and after explaining what it means and how it’s used, I tend to avoid it, because the “correct ensemble to use” should be clear from the context being studied; it is not some arbitrary choice that we make. Free energies have a range of uses in thermodynamics (and I certainly use them in this book), but they are probably more relevant to the history of the subject, when early physicists and chemists worked hard to ascertain the nature of what was then a cutting-edge new quantity called energy. Nowadays, we view free energies as useful combinations of more fun- damental parameters such as energy, temperature, and entropy. I think that the use of partition functions can be minimised in a book on introductory sta- tistical mechanics: they are sufficient but not necessary to the field; and yet, all too often, books seem to suggest that the partition function is the answer to every problem. Lastly, Maxwell relations are a useful but straightforward application of basic partial derivatives to thermodynamics, and that they have a name at all is probably just historical. More generally, long parades of partial derivatives and an endless swapping and tabulation of independent variables appear in so many books on statistical mechanics. These relics of history are best left to museums. No one really uses them! One deliberate nuance in some of my numerical calculations should be ex- plained: I tend not to choose nicely rounded versions of some parameters that turn up again and again. For example, I use 298 kelvins for room tempera- ture and 9.8 m/s2 for Earth’s gravity, instead of the simpler-looking rounded values of 300 kelvins and 10 m/s2. The reason here is that, when you see “298” and “9.8” in a calculation, you will perhaps find it easier to digest the various parameters quickly through recognising those quirky numbers at a glance, as opposed to seeing the more generic-looking numbers 300 and 10. x Preface

Also, whenever I have a quantity written in both non-bold and bold font in one context—such as “a” and “a”—then a should be understood to be the length of the vector a. This book has benefitted from the contributions of my family, friends, and colleagues—although, of course, I claim full ownership of my often-strong opinions about physics in general. My undergraduate lecturers at Auckland University, Graeme Putt and Paul Barker, provided my first instruction in thermodynamics and statistical mechanics in the mid-1980s, and so laid out the backbone for a later set of lectures of my own, which became this book. All manner of details were donated by others. Brad Alexander gave me a computer scientist’s view of entropy. Colin Andrew discussed scuba diving and ocean pressure. Shayne Bennetts listened to my views on the principle of detailed balance. Encouragement and some discussion of grammar came from Ine Brummans. The modern puzzle that is liquid helium was spelled out for me by Peter McClintock. I discussed some ideas of presentation with Steven Cohen. Occasional technical discussions took place with Scott Foster. Roland Keir contributed his knowledge of physical chemistry. Harry Koks informed me of some evolved wording in combinatorics, and Rudolf Koks explained osmosis in humans. Mark Krieg improved my grammar. Hans Laue discussed atmospheric temperature. Nadine Pesor helped me settle on the use of some jargon. Robert Purvinskis was a sounding board on occasion. Andy Rawlinson gave feedback on many ideas. Keith Stowe helped untangle some knotty problems in the subject. Vivienne Wheaton prompted some early deliberation on the Boltzmann distribution. The feedback of two anonymous early referees certainly helped make a better final product. Springer’s proof reader, Marc Beschler, gave a final and detailed burnish to my words. And the entire text was much improved by the careful reading and many thoughtful suggestions of Alice von Trojan. Beyond that, I thank Springer’s Tom Spicer for having the confidence to allow the project to go ahead, and Cindy Zitter for the details of making it happen.

Adelaide, Australia Don Koks August 2018 Contents

Preface ...... vii

1 Preliminary Ideas of Counting, and Some Useful Mathematics ...... 1 1.1 The Spreading of an Ink Drop...... 2 1.1.1 Identical-Classical Particles...... 6 1.2 Wandering Gas Particles...... 7 1.3 Fluctuations in the Binomial Distribution...... 17 1.3.1 Expected Value and Standard Deviation of a Random Variable...... 18 1.3.2 The Random Walk...... 23 1.4 Gaussian Approximation of the Binomial Distribution...... 25 1.5 Integrals of the Gaussian Function...... 30 1.5.1 Calculating the Error Function Numerically...... 36 1.5.2 The 3-Dimensional Gaussian...... 37 1.6 Increases and Infinitesimals...... 39 1.6.1 Basis Vectors...... 48 1.6.2 The Probability Density...... 50 1.7 Exercising Care with Partial Derivatives...... 53 1.8 Exact and Inexact Differentials...... 60 1.9 Numerical Notation, Units, and Dimensions...... 64 1.9.1 Units versus Dimensions...... 67 1.9.2 Function Arguments Must Be Dimensionless...... 75 1.9.3 Distinguishing Between an Entity and its Representation 80

2 Accessible States and the Fundamental Postulate of Statistical Mechanics ...... 83 2.1 States and Microstates...... 83

xi xii Contents

2.2 Energy Spacing of States...... 87 2.3 Position–Momentum and Phase Space...... 91 2.4 Microstates Are Cells of Phase Space...... 95 2.4.1 A System’s Quadratic Energy Terms...... 112 2.4.2 When Particles are Identical Classical...... 114 2.5 The Density of States...... 116

2.6 Ωtot for Massless Particles...... 119

3 The Laws of Thermodynamics ...... 125 3.1 The Concept of Energy for a Central Force...... 125 3.2 Force and Potential Energy...... 132 3.3 Interaction Types and the Zeroth Law of Thermodynamics.. 136 3.4 The First Law of Thermodynamics...... 138 3.4.1 Expressions for Quasi-Static Mechanical Work...... 140 3.4.2 The dC Term and Chemical Potential...... 146 3.5 The Definition of Temperature...... 148 3.5.1 Accessible Microstates for Thermally Interacting Systems...... 149 3.5.2 Temperature and the Equipartition Theorem...... 152 3.6 The Ideal Gas and Temperature Measurement...... 154 3.6.1 Measuring Temperature: the Constant-Volume Gas Thermometer...... 159 3.6.2 Temperature of Our Upper Atmosphere...... 161 3.7 The Non-Ideal Gas and van der Waals’ Equation...... 162 3.8 Entropy and the Second Law of Thermodynamics...... 167 3.8.1 Entropy of an Ideal Gas of Point Particles...... 170 3.8.2 The Canonical Example of Entropy Growth...... 171 3.8.3 Reversible and Cyclic Processes...... 174 3.8.4 The Use of Planck’s Constant for Quantifying Entropy 176 3.9 Can Temperature Be Negative?...... 177 3.10 Intensive and Extensive Variables, and the First Law...... 181 3.11 A Non-Quasi-static Process...... 183 3.12 The Ideal-Gas Law from Entropy...... 185 3.13 Relation of Entropy Increase to Interaction Direction...... 187 3.14 Integrating the Total Energy...... 191 3.14.1 Swapping the Roles of Conjugate Variables...... 193 3.14.2 Maxwell Relations...... 198 3.15 Excursus: Pressure and Temperature of a Star’s Interior..... 200 Contents xiii

4 The First Law in Detail...... 207 4.1 The First Term: Thermal Interaction...... 207 4.1.1 The Third Law of Thermodynamics...... 216 4.1.2 Heat Flow and the Thermal Current Density...... 220 4.1.3 The Continuity Equation...... 224 4.1.4 The Heat Equation, or Diffusion Equation...... 226 4.2 The Second Term: Mechanical Interaction...... 233 4.2.1 Heat Engines and Reversibility...... 233 4.2.2 The Joule–Thomson Process...... 238 4.3 The Third Term: Diffusive Interaction...... 247 4.3.1 Pressure and Density of the Atmosphere...... 247 4.3.2 Pressure and Density of the Ocean...... 251 4.3.3 Pressure and Density from the Chemical Potential.... 255 4.3.4 Phase Transitions and the Clausius–Clapeyron Equation...... 257 4.3.5 Chemical Equilibrium...... 270

5 The Non-Isolated System: the Boltzmann Distribution . . . 275 5.1 The Boltzmann Distribution...... 277 5.1.1 The Exponential Atmosphere Again...... 278 5.2 Paramagnetism...... 279 5.3 Energy Levels, States, and Bands...... 283 5.4 Hydrogen Energy Levels...... 287 5.5 Excitation Temperature...... 290 5.6 Diatomic Gases and Heat Capacity...... 291 5.6.1 Quantised Rotation...... 292 5.6.2 Quantised Vibration...... 298 5.7 Another Look at the Hydrogen Atom...... 300 5.8 Equipartition for a System Contacting a Thermal Bath..... 303 5.8.1 Fluctuation of the System’s Energy...... 306 5.9 The Partition Function in Detail...... 307 5.10 Entropy of a System Contacting a Thermal Bath...... 312 5.11 The Brandeis Dice...... 319 5.12 Entropy and Data Transmission...... 324

6 The Motion of Gas Particles, and Transport Processes . . . . 333 6.1 The Maxwell Velocity Distribution...... 338 6.1.1 Alternative Derivation of the Velocity Distribution... 342 6.2 The Maxwell Speed Distribution...... 344 6.2.1 Alternative Derivation of the Speed Distribution..... 345 xiv Contents

6.3 Representative Speeds of Gas Particles...... 346 6.4 Doppler Broadening of a Spectral Line...... 350 6.5 Temperature Gradient in a Weatherless Atmosphere...... 353 6.6 Gaseous Makeup of Planetary Atmospheres...... 358 6.7 Mean Free Path of Gas Particles...... 365 6.7.1 Excursus: The Proof of (6.123)...... 368 6.8 Viscosity and Mean Free Path...... 371 6.9 Thermal Conductivity and Mean Free Path...... 376 6.10 Excursus: The Energy–Momentum Tensor...... 379

7 Introductory Quantum Statistics ...... 385 7.1 Einstein’s Model of Heat Capacity...... 385 7.2 A Refinement of Einstein’s Model of Heat Capacity...... 389 7.3 Debye’s Model of Heat Capacity...... 394 7.4 Gibbs’ Paradox and Its Resolution...... 403 7.5 The Extent of a System’s Quantum Nature...... 405 7.5.1 Average de Broglie Wavelength...... 407 7.6 Fermions and Bosons...... 409 7.7 Occupation Numbers of Fermion and Boson Gases...... 418 7.7.1 Calculating µ(T ) and n(E,T ) for Fermions...... 421 7.7.2 Calculating µ(T ) and n(E,T ) for Bosons...... 424 7.8 Low-Temperature Bosons and Liquid Helium...... 426 7.9 Excursus: Particle Statistics from Counting Configurations.. 432 7.9.1 Fermi–Dirac and Bose–Einstein from Configurations.. 442

8 Fermion Statistics in Metals ...... 445 8.1 Conduction Electrons’ Contribution to Heat Capacity...... 445 8.1.1 A More Accurate Approximation of n(E,T )...... 454 8.2 Electrical Conductivity of Metals...... 458 8.3 Thermal Conductivity of Metals...... 465 8.3.1 The Lorenz Number...... 467 8.4 Insulators and Semiconductors...... 468 8.5 Diodes...... 473

9 Boson Statistics in Blackbody Radiation ...... 481 9.1 Spectrum of Radiation Inside an Oven...... 482 9.1.1 Mean “Extractable” Energy of an Oscillator, ε(f)..... 484 9.2 The One-Dimensional Oven: an Electrical Resistor...... 488 9.2.1 Calculating the Density of Wave States, g(f)...... 489 Contents xv

9.2.2 Excursus: Thermal Noise in a Resistor, and Some Communications Theory...... 496 9.3 The Three-Dimensional Oven...... 502 9.4 The End Product: Planck’s Law...... 506 9.4.1 Planck’s Law Expressed Using Wavelength...... 508 9.5 Total Energy of Radiation in the Oven...... 510 9.6 Letting the Radiation Escape the Oven...... 511 9.7 Blackbody Radiation...... 514 9.7.1 The Large-Scale Universe Is a Very Cold Oven...... 517 9.7.2 Total Power Emitted by a Black Body...... 520 9.8 The Greenhouse Effect...... 523 9.9 Photon Absorption and Emission: the Laser...... 525

Correction to: Microstates, Entropy and Quanta...... C1

Index ...... 535 List of Common Symbols

Chapter1 : Preliminary Ideas of Counting, and Some Useful Mathe- matics

NA Avogadro’s number. n Cx Number of combinations (selections) of x objects taken from a total of n objects. N (x; µ, σ2) Normal distribution of x, with mean µ and variance σ2. N (x; µ,P ) Multi-dimensional normal distribution of x, with mean µ and covariance matrix P . eq Basis vector for coordinate q. uq Unit-length basis vector for coordinate q. dx Infinitesimal, an “exact differential” of state variable x. dA Infinitesimal, an “inexact differential” of quantity A that is not a state variable. λ(x) Linear mass density as a function of position x.

Mmol Molar mass. [L]S Representation of L in system S.

Chapter2 : Accessible States and the Fundamental Postulate of Sta- tistical Mechanics D Number of internal variables in which a particle can store its energy. Ω(E) Number of microstates that each have energy E.

Ωtot(E) Total number of microstates that each have energy somewhere in the range 0 to E. ν Number of quadratic energy terms of a particle, meaning the number of quadratic coordinates that describe the particle’s energy. (In other texts, this is called the number of degrees of

xvii xviii List of Common Symbols

freedom of the particle.) The particle need not be an atom; it could be a molecule. ic Ωtot(E) Ωtot(E) for identical-classical particles. g(E), g(f) Density of states as functions of energy E and frequency f.

Chapter3 : The Laws of Thermodynamics ur Unit-length radial vector. b(r) A function describing the central force as a function of radial distance r. U Potential energy of a particle. dQ Thermal energy put into a system. dW Mechanical work performed on a system. dC Energy brought into a system by incoming particles or envi- ronmental changes. E, p Electric field and electric dipole moment. B, µ Magnetic field and magnetic dipole moment. µ Chemical potential.

γi νiNi/2, where νi is the number of quadratic energy terms (“de- grees of freedom” in other texts) per particle in system i, and Ni is the number of particles in system i. N Number of particles. n Number of moles.

Ebi Value of energy Ei at which Ω(Ei) peaks. dist Ωtot Ωtot for distinguishable particles. Sdist Entropy of distinguishable particles. Sic Entropy of identical-classical particles. F Helmholtz energy. G Gibbs energy. H Enthalpy. κ Coefficient of isothermal compressibility. β Coefficient of thermal expansion.

Chapter4 : The Three Interactions of the First Law

CP ,CV Heat capacities at constant pressure and volume. Csp Specific heat capacity. Cmol Molar heat capacity. List of Common Symbols xix

sp sp mol mol γC P /CV , which also equals CP /CV and CP /CV . µJT Joule–Thomson coefficient. a, b Van der Waals parameters. amol, bmol Van der Waals parameters for the molar form of van der Waals’ equation. J Current density, also known as flux density. κ Thermal conductivity. (a, b) Angle between vectors a and b (between 0 and π). I Heat current. R Thermal resistance. % Thermal resistivity.

%E Energy content per unit volume. %m Mass per unit volume. K Diffusion constant. ∗ Convolution operator. % Mass per unit volume. ν Number of particles per unit volume. B Bulk modulus. φ Ratio of salt particles to total number of salt and water par- ticles. mol mol Lvap ,Lfusion Molar latent heats of vaporisation and fusion.

Chapter5 : The Non-Isolated System: the Boltzmann Distribution plevel n Probability that a system occupies any state at energy level n. pstate n Probability that a system occupies a specific state n. β 1/(kT ).

En,Vn Energy and volume of a hydrogen atom at energy level n. Z Partition function.

Te Excitation temperature of a system. TR,TV Characteristic temperatures of the onsets of rotation and vi- bration.

E Abbreviated version of Es, the mean energy of the system.

Chapter6 : The Motion of Gas Particles and Transport Processes v, v Speed and velocity of a particle. 3 d v dvx dvy dvz. xx List of Common Symbols

3 Nvel(v) d v Infinitesimal number of particles with velocities in the range v to v + dv.

Nx(vx) dvx Infinitesimal number of particles with x velocities in the range vx to vx + dvx. Nsp(v) dv Infinitesimal number of particles with speeds in the range v to v + dv.

Ntot Total number of particles. dΩtot Number of microstates in the energy range E to E + dE.

Nz(z, vz) dz dvz Infinitesimal number of particles with heights in z to z + dz, and z velocities in vz to vz + dvz. Nsp(z, v) dz dv Infinitesimal number of particles with heights in z to z + dz, and speeds in v to v + dv. λ Mean free path. ν Number of particles per unit volume. σ Collision cross section. η Coefficient of viscosity. κ Thermal conductivity.

Chapter7 : Introductory Quantum Statistics n The energy level of a one-dimensional oscillator, and also the number of quantum particles per state, in which each state denotes one dimension of oscillation of a single molecule in a crystal. E Mean energy of a crystal molecule (a quantised oscillator that can oscillate in three dimensions). Eventually redefined to ex- clude zero-point energy.

E1D Mean energy of a one-dimensional quantised oscillator.

TE,TD Einstein and Debye temperatures. n Occupation number of a crystal, the arithmetic mean of n: the mean number of quantum particles present per 1D-oscillator in the crystal. A function of temperature.

Etot Total energy of all oscillators in the crystal. n(E,T ) Occupation number treated as a function of energy and tem- perature: the mean number of quantum particles per state. N Number of quantum particles with energies up to E. λ De Broglie wavelength. E Energy of a state, which that state “bestows” on each particle occupying it. pn Probability of n quantum particles being present in a state. List of Common Symbols xxi

N Total number of quantum particles of all energies (in a later section to N immediately above). C A constant for a gas of massive particles, encoding spin, vol- ume, and particle mass.

EF Fermi energy.

Tc Critical temperature of liquid helium. N Total number of balls to be placed on shelves. ni Number of balls on shelf i.

Chapter8 : Fermion Statistics in Metals

mol CV (electrons) Valence-electron contribution to a crystal’s molar heat ca- pacity. E Mean energy of one valence electron. E Energy of one valence electron. N Total number of valence electrons. n(E,T ) Occupation number of valence electrons.

TF Fermi temperature of valence electrons. vF Fermi speed of valence electrons. α A number in the region of 1 or 2, modelling the characteristic width of the fall-off of the Fermi–Dirac occupation number with energy. % Electrical resistivity. κ Thermal conductivity.

Ne Number of electrons in a conduction band. Various parameters are also defined in (8.65).

Chapter9 : Boson Statistics in Blackbody Radiation %(f) Spectral energy density as a function of frequency f. ε(f) Mean energy of a single oven-wall oscillator of frequency f.

λ0 Wavelength corresponding to the peak radiated power from a black body. σ Stefan–Boltzmann constant. N Number of frequency-f photons produced by any process in a laser. List of Common Constants

23 Avogadro’s number NA 6.022 ×10 (or 6.022 ×1023 mol−1) Boltzmann’s constant k 1.381×10−23 J/K Gas constant R = NAk 8.314 J/K (or J K−1mol−1) Planck’s constant h 6.626 ×10−34 J s ~ = h/(2π) 1.0546 ×10−34 J s Speed of light in vacuum c 2.998 ×108 m/s

Proton mass 1.67 ×10−27 kg Electron mass 9.11 ×10−31 kg Electron charge −e −1.602 ×10−19 C Electron volt 1 eV = 1.602 ×10−19 J

Room temperature 298 K (25◦C) Ground temperature 288 K (15◦C) Air temperature 253 K (−20◦C) (mean atmospheric)

Mass of a generic air molecule 4.8 ×10−26 kg Molar mass of air 29.0 g Molar mass of sodium 23.0 g

Earth gravity 9.8 m/s2 Atmospheric pressure at sea level 101,325 Pa

Copper’s molar mass 63.5 g Copper’s Fermi temperature 81,000 K Number density of copper’s 8.47 ×1028 m−3 valence electrons

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