The Cambridge Handbook of Physics Formulas

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The Cambridge Handbook of Physics Formulas is a quick-reference aid for students and professionals in the physical sciences and engineering. It contains more than 2 000 of the most useful formulas and equations found in undergraduate physics courses, covering mathematics, dynamics and mechanics, quantum physics, thermodynamics, solid state physics, electromagnetism, optics, and astrophysics. An exhaustive index allows the required formulas to be located swiftly and simply, and the unique tabular format crisply identiﬁes all the variables involved. The Cambridge Handbook of Physics Formulas comprehensively covers the major topics explored in undergraduate physics courses. It is designed to be a compact, portable, reference book suitable for everyday work, problem solving, or exam revision. All students and professionals in physics, applied mathematics, engineering, and other physical sciences will want to have this essential reference book within easy reach.

Graham Woan is a senior lecturer in the Department of Physics and Astronomy at the University of Glasgow. Prior to this he taught physics at the University of Cambridge where he also received his degree in Natural Sciences, specialising in physics, and his PhD, in radio astronomy. His research interests range widely with a special focus on low-frequency radio astronomy. His publications span journals as diverse as Astronomy & Astrophysics, Geophysical Research Letters, Advances in Space Science,theJournal of Navigation and Emergency Prehospital Medicine. He was co-developer of the revolutionary CURSOR radio positioning system, which uses existing broadcast transmitters to determine position, and he is the designer of the Glasgow Millennium Sundial. main April 22, 2003 15:22 main April 22, 2003 15:22

The Cambridge Handbook of Physics Formulas

2003 Edition

GRAHAM WOAN Department of Physics & Astronomy University of Glasgow    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University Press The Edinburgh Building, Cambridge  ,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridg e.org /9780521573498

This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

First published in print format 2000

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Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. main April 22, 2003 15:22

Contents

Preface page vii How to use this book1

1 Units, constants, and conversions 3 1.1 Introduction,3 • 1.2 SI units,4 • 1.3 Physical constants,6 • 1.4 Converting between units, 10 • 1.5 Dimensions, 16 • 1.6 Miscellaneous, 18 2 Mathematics 19 2.1 Notation, 19 • 2.2 Vectors and matrices, 20 • 2.3 Series, summations, and progressions, 27 • 2.4 Complex variables, 30 • 2.5 Trigonometric and hyperbolic formulas, 32 • 2.6 Mensuration, 35 • 2.7 Diﬀerentiation, 40 • 2.8 Integration, 44 • 2.9 Special functions and polynomials, 46 • 2.10 Roots of quadratic and cubic equations, 50 • 2.11 Fourier series and transforms, 52 • 2.12 Laplace transforms, 55 • 2.13 Probability and statistics, 57 • 2.14 Numerical methods, 60 3 Dynamics and mechanics 63 3.1 Introduction, 63 • 3.2 Frames of reference, 64 • 3.3 Gravitation, 66 • 3.4 Particle motion, 68 • 3.5 Rigid body dynamics, 74 • 3.6 Oscillating systems, 78 • 3.7 Generalised dynamics, 79 • 3.8 Elasticity, 80 • 3.9 Fluid dynamics, 84 4 Quantum physics 89 4.1 Introduction, 89 • 4.2 Quantum deﬁnitions, 90 • 4.3 Wave mechanics, 92 • 4.4 Hydrogenic atoms, 95 • 4.5 Angular momentum, 98 • 4.6 Perturbation theory, 102 • 4.7 High energy and nuclear physics, 103 5 Thermodynamics 105 5.1 Introduction, 105 • 5.2 Classical thermodynamics, 106 • 5.3 Gas laws, 110 • 5.4 Kinetic theory, 112 • 5.5 Statistical thermodynamics, 114 • 5.6 Fluctuations and noise, 116 • 5.7 Radiation processes, 118 main April 22, 2003 15:22

6 Solid state physics 123 6.1 Introduction, 123 • 6.2 Periodic table, 124 • 6.3 Crystalline structure, 126 • 6.4 Lattice dynamics, 129 • 6.5 Electrons in solids, 132

7 Electromagnetism 135 7.1 Introduction, 135 • 7.2 Static ﬁelds, 136 • 7.3 Electromagnetic ﬁelds (general), 139 • 7.4 Fields associated with media, 142 • 7.5 Force, torque, and energy, 145 • 7.6 LCR circuits, 147 • 7.7 Transmission lines and waveguides, 150 • 7.8 Waves in and out of media, 152 • 7.9 Plasma physics, 156

8 Optics 161 8.1 Introduction, 161 • 8.2 Interference, 162 • 8.3 Fraunhofer diﬀraction, 164 • 8.4 Fresnel diﬀraction, 166 • 8.5 Geometrical optics, 168 • 8.6 Polarisation, 170 • 8.7 Coherence (scalar theory), 172 • 8.8 Line radiation, 173

9 Astrophysics 175 9.1 Introduction, 175 • 9.2 Solar system data, 176 • 9.3 Coordinate transformations (astronomical), 177 • 9.4 Observational astrophysics, 179 • 9.5 Stellar evolution, 181 • 9.6 Cosmology, 184 Index 187 main April 22, 2003 15:22

Preface

In A Brief History of Time, Stephen Hawking relates that he was warned against including equations in the book because “each equation... would halve the sales.” Despite this dire prediction there is, for a scientific audience, some attraction in doing the exact opposite. The reader should not be misled by this exercise. Although the equations and formulas contained here underpin a good deal of physical science they are useless unless the reader understands them. Learning physics is not about remembering equations, it is about appreci- ating the natural structures they express. Although its format should help make some topics clearer, this book is not designed to teach new physics; there are many excellent textbooks to help with that. It is intended to be useful rather than pedagogically complete, so that students can use it for revision and for structuring their knowledge once they understand the physics. More advanced users will benefit from having a compact, internally consistent, source of equations that can quickly deliver the relationship they require in a format that avoids the need to sift through pages of rubric. Some difficult decisions have had to be made to achieve this. First, to be short the book only includes ideas that can be expressed succinctly in equations, without resorting to lengthy explanation. A small number of important topics are therefore absent. For example, Liouville’s theorem can be algebraically succinct (˙ = 0) but is meaningless unless ˙ is thoroughly (and carefully) explained. Anyone who already understands what ˙ represents will probably not need reminding that it equals zero. Second, empirical equations with numerical coefficients have been largely omitted, as have topics significantly more advanced than are found at undergraduate level. There are simply too many of these to be sensibly and confidently edited into a short handbook. Third, physical data are largely absent, although a periodic table, tables of physical constants, and data on the solar system are all included. Just a sighting of the marvellous (but dimensionally misnamed) CRC Handbook of Chemistry and Physics should be enough to convince the reader that a good science data book is thick. Inevitably there is personal choice in what should or should not be included, and you may feel that an equation that meets the above criteria is missing. If this is the case, I would be delighted to hear from you so it can be considered for a subsequent edition. Contact details are at the end of this preface. Likewise, if you spot an error or an inconsistency then please let me know and I will post an erratum on the web page. main April 22, 2003 15:22

Acknowledgments This venture is founded on the generosity of colleagues in Glasgow and Cambridge whose inputs have strongly influenced the final product. The expertise of Dave Clarke, Declan Diver, Peter Duffett-Smith, Wolf-Gerrit Fruh,¨ Martin Hendry, Rico Ignace, David Ireland, John Simmons, and Harry Ward have been central to its production, as have the linguistic skills of Katie Lowe. I would also like to thank Richard Barrett, Matthew Cartmell, Steve Gull, Martin Hendry, Jim Hough, Darren McDonald, and Ken Riley who all agreed to field-test the book and gave invaluable feedback. My greatest thanks though are to John Shakeshaft who, with remarkable knowledge and skill, worked through the entire manuscript more than once during its production and whose legendary red pen hovered over (or descended upon) every equation in the book. What errors remain are, of course, my own, but I take comfort from the fact that without John they would be much more numerous. Contact information A website containing up-to-date information on this handbook and contact details can be found through the Cambridge University Press web pages at us.cambridge.org (North America) or uk.cambridge.org (United Kingdom), or directly at radio.astro.gla.ac.uk/hbhome.html. Production notes This book was typeset by the author in LATEX2ε using the CUP Times fonts. The software packages used were WinEdt, M EX, Mayura Draw, Gnuplot, Ghostscript, Ghostview,andMaple V.

Comments on the 2002 edition I am grateful to all those who have suggested improvements, in particular Martin Hendry, Wolfgang Jitschin, and Joseph Katz. Although this edition contains only minor revisions to the original its production was also an opportunity to update the physical constants and periodic table entries and to reﬂect recent developments in cosmology. main April 22, 2003 15:22

How to use this book

The format is largely self-explanatory, but a few comments may be helpful. Although it is very tempting to flick through the pages to find what you are looking for, the best starting point is the index. I have tried to make this as extensive as possible, and many equations are indexed more than once. Equations are listed both with their equation number (in square brackets) and the page on which they can be found. The equations themselves are grouped into self-contained and boxed “panels” on the pages. Each panel represents a separate topic, and you will find descriptions of all the variables used at the right-hand side of the panel, usually adjacent to the first equation in which they are used. You should therefore not need to stray outside the panel to understand the notation. Both the panel as a whole and its individual entries may have footnotes, shown below the panel. Be aware of these, as they contain important additional information and conditions relevant to the topic. Although the panels are self-contained they may use concepts defined elsewhere in the handbook. Often these are cross-referenced, but again the index will help you to locate them if necessary. Notations and definitions are uniform over subject areas unless stated otherwise. main April 22, 2003 15:22 main January 23, 2006 16:6

1 Chapter 1 Units, constants, and conversions

1.1 Introduction The determination of physical constants and the definition of the units with which they are measured is a specialised and, to many, hidden branch of science. A quantity with dimensions is one whose value must be expressed relative to one or more standard units. In the spirit of the rest of the book, this section is based around the International System of units (SI). This system uses seven base units1 (the number is somewhat arbitrary), such as the kilogram and the second, and defines their magnitudes in terms of physical laws or, in the case of the kilogram, an object called the “international prototype of the kilogram” kept in Paris. For convenience there are also a number of derived standards, such as the volt, which are defined as set combinations of the basic seven. Most of the physical observables we regard as being in some sense fundamental, such as the charge 2 −7 on an electron, are now known to a relative standard uncertainty, ur, of less than 10 . The least well determined is the Newtonian constant of gravitation, presently standing at a −3 −12 rather lamentable ur of 1.5 × 10 , and the best is the Rydberg constant (ur =7.6 × 10 ). The dimensionless electron g-factor, representing twice the magnetic moment of an electron measured in Bohr magnetons, is now known to a relative uncertainty of only 4.1 × 10−12. No matter which base units are used, physical quantities are expressed as the product of a numerical value and a unit. These two components have more-or-less equal standing and can be manipulated by following the usual rules of algebra. So, if 1 · eV = 160.218 × 10−21 · J then 1 · J=[1/(160.218 × 10−21)] · eV. A measurement of energy, U, with joule as the unit has a numerical value of U/J. The same measurement with electron volt as the unit has a numerical value of U/eV = (U/J) · (J/ eV) and so on.

1The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second. The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 × 10−7 newton per metre of length. The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12; its symbol is “mol.” When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles. The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. 2The relative standard uncertainty in x is defined as the estimated standard deviation in x divided by the modulus of x (x =0). main January 23, 2006 16:6

4 Units, constants, and conversions

1.2 SI units

SI base units physical quantity name symbol length metrea m mass kilogram kg time interval second s electric current ampere A thermodynamic temperature kelvin K amount of substance mole mol luminous intensity candela cd aOr “meter”.

SI derived units physical quantity name symbol equivalent units catalytic activity katal kat mol s−1 electric capacitance farad F C V−1 electric charge coulomb C A s electric conductance siemens S Ω−1 electric potential difference volt V J C−1 electric resistance ohm Ω V A−1 energy, work, heat joule J N m force newton N m kg s−2 frequency hertz Hz s−1 illuminance lux lx cd sr m−2 inductance henry H V A−1 s luminous flux lumen lm cd sr magnetic flux weber Wb V s magnetic flux density tesla T V s m−2 plane angle radian rad m m−1 power, radiant flux watt W J s−1 pressure, stress pascal Pa N m−2 radiation absorbed dose gray Gy J kg−1 radiation dose equivalenta sievert Sv [ J kg−1] radioactive activity becquerel Bq s−1 solid angle steradian sr m2 m−2 temperatureb degree Celsius ◦CK aTo distinguish it from the gray, units of J kg−1 should not be used for the sievert in practice. b The Celsius temperature, TC, is defined from the temperature in kelvin, TK,byTC = TK − 273.15. main January 23, 2006 16:6

1.2 SI units 5 1

SI prefixesa factor prefix symbol factor prefix symbol 1024 yotta Y 10−24 yocto y 1021 zetta Z 10−21 zepto z 1018 exa E 10−18 atto a 1015 peta P 10−15 femto f 1012 tera T 10−12 pico p 109 giga G 10−9 nano n 106 mega M 10−6 micro µ 103 kilo k 10−3 milli m 102 hecto h 10−2 centi c 101 decab da 10−1 deci d aThe kilogram is the only SI unit with a prefix embedded in its name and symbol. For mass, the unit name “gram” and unit symbol “g” should be used with these prefixes, hence 10−6 kg can be written as 1 mg. Otherwise, any prefix can be applied to any SI unit. bOr “deka”.

Recognised non-SI units physical quantity name symbol SI value area barn b 10−28 m2 energy electron volt eV 1.602 18 × 10−19 J length ˚angstrom¨ A10˚ −10 m fermia fm 10−15 m microna µm10−6 m plane angle degree ◦ (π/180) rad arcminute (π/10 800) rad arcsecond (π/648 000) rad pressure bar bar 105 Nm−2 time minute min 60 s hour h 3 600 s day d 86 400 s mass uniﬁed atomic mass unit u 1.660 54 × 10−27 kg tonnea,b t103 kg volume litrec l, L 10−3 m3 aThese are non-SI names for SI quantities. bOr “metric ton.” cOr “liter”. The symbol “l” should be avoided. main January 23, 2006 16:6

6 Units, constants, and conversions

1.3 Physical constants The following 1998 CODATA recommended values for the fundamental physical constants can also be found on the Web at physics.nist.gov/constants. Detailed background information is available in Reviews of Modern Physics, Vol. 72, No. 2, pp. 351–495, April 2000. The digits in parentheses represent the 1σ uncertainty in the previous two quoted digits. For example, G =(6.673±0.010)×10−11 m3 kg−1 s−2. It is important to note that the uncertainties for many of the listed quantities are correlated, so that the uncertainty in any expression using them in combination cannot necessarily be computed from the data presented. Suitable covariance values are available in the above references.

Summary of physical constants speed of light in vacuuma c 2.997 924 58 ×108 ms−1 b −7 −1 permeability of vacuum µ0 4π ×10 Hm =12.566 370 614 ... ×10−7 Hm−1 2 −1 permittivity of vacuum 0 1/(µ0c )Fm =8.854 187 817 ... ×10−12 Fm−1 − constant of gravitationc G 6.673(10) ×10−11 m3 kg 1 s−2 Planck constant h 6.626 068 76(52) ×10−34 Js h/(2π) h¯ 1.054 571 596(82) ×10−34 Js elementary charge e 1.602 176 462(63) ×10−19 C −15 magnetic ﬂux quantum, h/(2e) Φ0 2.067 833 636(81) ×10 Wb electron volt eV 1.602 176 462(63) ×10−19 J −31 electron mass me 9.109 381 88(72) ×10 kg −27 proton mass mp 1.672 621 58(13) ×10 kg

proton/electron mass ratio mp/me 1 836.152 667 5(39) unified atomic mass unit u1.660 538 73(13) ×10−27 kg 2 −3 fine-structure constant, µ0ce /(2h) α 7.297 352 533(27) ×10 inverse 1/α 137.035 999 76(50) 2 7 −1 Rydberg constant, mecα /(2h) R∞ 1.097 373 156 854 9(83) ×10 m 23 −1 Avogadro constant NA 6.022 141 99(47) ×10 mol 4 −1 Faraday constant, NAe F 9.648 534 15(39) ×10 C mol − molar gas constant R 8.314 472(15) J mol 1 K−1 −23 −1 Boltzmann constant, R/NA k 1.380 650 3(24) ×10 JK Stefan–Boltzmann constant, − − − σ 5.670 400(40) ×10 8 Wm 2 K 4 π2k4/(60¯h3c2) −24 −1 Bohr magneton, eh/¯ (2me) µB 9.274 008 99(37) ×10 JT aBy definition, the speed of light is exact. bAlso exact, by definition. Alternative units are N A−2. cThe standard acceleration due to gravity, g, is defined as exactly 9.806 65 m s−2. main January 23, 2006 16:6

1.3 Physical constants 7 1 General constants speed of light in vacuum c 2.997 924 58 ×108 ms−1 −7 −1 permeability of vacuum µ0 4π ×10 Hm =12.566 370 614 ... ×10−7 Hm−1 2 −1 permittivity of vacuum 0 1/(µ0c )Fm =8.854 187 817 ... ×10−12 Fm−1

impedance of free space Z0 µ0c Ω =376.730 313 461 ... Ω − constant of gravitation G 6.673(10) ×10−11 m3 kg 1 s−2 Planck constant h 6.626 068 76(52) ×10−34 Js in eV s 4.135 667 27(16) ×10−15 eV s h/(2π) h¯ 1.054 571 596(82) ×10−34 Js in eV s 6.582 118 89(26) ×10−16 eV s 1/2 −8 Planck mass, (¯hc/G) mPl 2.176 7(16) ×10 kg 3 1/2 −35 Planck length, ¯h/(mPlc)=(¯hG/c ) lPl 1.616 0(12) ×10 m 5 1/2 −44 Planck time, lPl/c =(¯hG/c ) tPl 5.390 6(40) ×10 s elementary charge e 1.602 176 462(63) ×10−19 C −15 magnetic ﬂux quantum, h/(2e) Φ0 2.067 833 636(81) ×10 Wb Josephson frequency/voltage ratio 2e/h 4.835 978 98(19) ×1014 Hz V−1 −24 −1 Bohr magneton, eh/¯ (2me) µB 9.274 008 99(37) ×10 JT in eV T−1 5.788 381 749(43) ×10−5 eV T−1 −1 µB/k 0.671 713 1(12) K T −27 −1 nuclear magneton, eh/¯ (2mp) µN 5.050 783 17(20) ×10 JT in eV T−1 3.152 451 238(24) ×10−8 eV T−1 −4 −1 µN/k 3.658 263 8(64) ×10 KT −1 −1 Zeeman splitting constant µB/(hc)46.686 452 1(19) m T

Atomic constantsa 2 −3 ﬁne-structure constant, µ0ce /(2h) α 7.297 352 533(27) ×10 inverse 1/α 137.035 999 76(50) 2 7 −1 Rydberg constant, mecα /(2h) R∞ 1.097 373 156 854 9(83) ×10 m 15 R∞c 3.289 841 960 368(25) ×10 Hz −18 R∞hc 2.179 871 90(17) ×10 J R∞hc/e 13.605 691 72(53) eV b −11 Bohr radius , α/(4πR∞) a0 5.291 772 083(19) ×10 m aSee also the Bohr model on page 95. bFixed nucleus. main January 23, 2006 16:6

8 Units, constants, and conversions

Electron constants −31 electron mass me 9.109 381 88(72) ×10 kg in MeV 0.510 998 902(21) MeV −4 electron/proton mass ratio me/mp 5.446 170 232(12) ×10 electron charge −e −1.602 176 462(63) ×10−19 C 11 −1 electron speciﬁc charge −e/me −1.758 820 174(71) ×10 Ckg −7 −1 electron molar mass, NAme Me 5.485 799 110(12) ×10 kg mol −12 Compton wavelength, h/(mec) λC 2.426 310 215(18) ×10 m 2 −15 classical electron radius, α a0 re 2.817 940 285(31) ×10 m 2 × −29 2 Thomson cross section, (8π/3)re σT 6.652 458 54(15) 10 m −24 −1 electron magnetic moment µe −9.284 763 62(37) ×10 JT

in Bohr magnetons, µe/µB −1.001 159 652 186 9(41)

in nuclear magnetons, µe/µN −1 838.281 966 0(39) 11 −1 −1 electron gyromagnetic ratio, 2|µe|/h¯ γe 1.760 859 794(71) ×10 s T

electron g-factor, 2µe/µB ge −2.002 319 304 3737(82)

Proton constants −27 proton mass mp 1.672 621 58(13) ×10 kg in MeV 938.271 998(38) MeV

proton/electron mass ratio mp/me 1 836.152 667 5(39) proton charge e 1.602 176 462(63) ×10−19 C 7 −1 proton speciﬁc charge e/mp 9.578 834 08(38) ×10 Ckg −3 −1 proton molar mass, NAmp Mp 1.007 276 466 88(13) ×10 kg mol −15 proton Compton wavelength, h/(mpc) λC,p 1.321 409 847(10) ×10 m −26 −1 proton magnetic moment µp 1.410 606 633(58) ×10 JT −3 in Bohr magnetons, µp/µB 1.521 032 203(15) ×10

in nuclear magnetons, µp/µN 2.792 847 337(29) 8 −1 −1 proton gyromagnetic ratio, 2µp/h¯ γp 2.675 222 12(11) ×10 s T

Neutron constants −27 neutron mass mn 1.674 927 16(13) ×10 kg in MeV 939.565 330(38) MeV

neutron/electron mass ratio mn/me 1 838.683 655 0(40)

neutron/proton mass ratio mn/mp 1.001 378 418 87(58) −3 −1 neutron molar mass, NAmn Mn 1.008 664 915 78(55) ×10 kg mol −15 neutron Compton wavelength, h/(mnc) λC,n 1.319 590 898(10) ×10 m −27 −1 neutron magnetic moment µn −9.662 364 0(23) ×10 JT −3 in Bohr magnetons µn/µB −1.041 875 63(25) ×10

in nuclear magnetons µn/µN −1.913 042 72(45) 8 −1 −1 neutron gyromagnetic ratio, 2|µn|/h¯ γn 1.832 471 88(44) ×10 s T main January 23, 2006 16:6

1.3 Physical constants 9 1 Muon and tau constants −28 muon mass mµ 1.883 531 09(16) ×10 kg in MeV 105.658 356 8(52) MeV −27 tau mass mτ 3.167 88(52) ×10 kg in MeV 1.777 05(29) ×103 MeV

muon/electron mass ratio mµ/me 206.768 262(30) muon charge −e −1.602 176 462(63) ×10−19 C −26 −1 muon magnetic moment µµ −4.490 448 13(22) ×10 JT −3 in Bohr magnetons, µµ/µB 4.841 970 85(15) ×10

in nuclear magnetons, µµ/µN 8.890 597 70(27)

muon g-factor gµ −2.002 331 832 0(13)

Bulk physical constants 23 −1 Avogadro constant NA 6.022 141 99(47) ×10 mol a −27 atomic mass constant mu 1.660 538 73(13) ×10 kg in MeV 931.494 013(37) MeV − Faraday constant F 9.648 534 15(39) ×104 C mol 1 − molar gas constant R 8.314 472(15) J mol 1 K−1 −23 −1 Boltzmann constant, R/NA k 1.380 650 3(24) ×10 JK in eV K−1 8.617 342(15) ×10−5 eV K−1 b −3 3 −1 molar volume (ideal gas at stp) Vm 22.413 996(39) ×10 m mol Stefan–Boltzmann constant, π2k4/(60¯h3c2) σ 5.670 400(40) ×10−8 Wm−2 K−4 c −3 Wien’s displacement law constant, b = λmT b 2.897 768 6(51) ×10 mK a= mass of 12C/12. Alternative nomenclature for the uniﬁed atomic mass unit, u. bStandard temperature and pressure (stp) are T = 273.15 K (0◦C) and P = 101 325 Pa (1 standard atmosphere). cSee also page 121.

Mathematical constants pi (π) 3.141 592 653 589 793 238 462 643 383 279 ... exponential constant (e) 2.718 281 828 459 045 235 360 287 471 352 ... Catalan’s constant 0.915 965 594 177 219 015 054 603 514 932 ... Euler’s constanta (γ) 0.577 215 664 901 532 860 606 512 090 082 ... Feigenbaum’s constant (α) 2.502 907 875 095 892 822 283 902 873 218 ... Feigenbaum’s constant (δ) 4.669 201 609 102 990 671 853 203 820 466 ... Gibbs constant 1.851 937 051 982 466 170 361 053 370 157 ... golden mean 1.618 033 988 749 894 848 204 586 834 370 ... Madelung constantb 1.747 564 594 633 182 190 636 212 035 544 ... aSee also Equation (2.119). bNaCl structure. main January 23, 2006 16:6

10 Units, constants, and conversions

1.4 Converting between units The following table lists common (and not so common) measures of physical quantities. The numerical values given are the SI equivalent of one unit measure of the non-SI unit. Hence 1 astronomical unit equals 149.597 9 × 109 m. Those entries identiﬁed with a “∗”inthe second column represent exact conversions; so 1 abampere equals exactly 10.0 A. Note that individual entries in this list are not recorded in the index, and that values are “international” unless otherwise stated. There is a separate section on temperature conversions after this table.

unit name value in SI units abampere 10.0∗ A abcoulomb 10.0∗ C abfarad 1.0∗ ×109 F abhenry 1.0∗ ×10−9 H abmho 1.0∗ ×109 S abohm 1.0∗ ×10−9 Ω abvolt 10.0∗ ×10−9 V acre 4.046 856 ×103 m2 amagat (at stp) 44.614 774 mol m−3 ampere hour 3.6∗ ×103 C ˚angstrom¨ 100.0∗ ×10−12 m apostilb 1.0∗ lm m−2 arcminute 290.888 2 ×10−6 rad arcsecond 4.848 137 ×10−6 rad are 100.0∗ m2 astronomical unit 149.597 9 ×109 m atmosphere (standard) 101.325 0∗ ×103 Pa atomic mass unit 1.660 540 ×10−27 kg bar 100.0∗ ×103 Pa barn 100.0∗ ×10−30 m2 baromil 750.1 ×10−6 m barrel (UK) 163.659 2 ×10−3 m3 barrel (US dry) 115.627 1 ×10−3 m3 barrel (US liquid) 119.240 5 ×10−3 m3 barrel (US oil) 158.987 3 ×10−3 m3 baud 1.0∗ s−1 bayre 100.0∗ ×10−3 Pa biot 10.0A bolt (US) 36.576∗ m brewster 1.0∗ ×10−12 m2 N−1 British thermal unit 1.055 056 ×103 J bushel (UK) 36.36 872 ×10−3 m3 bushel (US) 35.23 907 ×10−3 m3 butt (UK) 477.339 4 ×10−3 m3 cable (US) 219.456∗ m calorie 4.186 8∗ J continued on next page ... main January 23, 2006 16:6

1.4 Converting between units 11 1 unit name value in SI units candle power (spherical) 4π lm carat (metric) 200.0∗ ×10−6 kg cental 45.359 237 kg centare 1.0∗ m2 centimetre of Hg (0 ◦C) 1.333 222 ×103 Pa ◦ centimetre of H2O(4 C) 98.060 616 Pa chain (engineers’) 30.48∗ m chain (US) 20.116 8∗ m Chu 1.899 101 ×103 J clusec 1.333 224 ×10−6 W cord 3.624 556 m3 cubit 457.2∗ ×10−3 m cumec 1.0∗ m3 s−1 cup (US) 236.588 2 ×10−6 m3 curie 37.0∗ ×109 Bq darcy 986.923 3 ×10−15 m2 day 86.4∗ ×103 s day (sidereal) 86.164 09 ×103 s debye 3.335 641 ×10−30 Cm degree (angle) 17.453 29 ×10−3 rad denier 111.111 1 ×10−9 kg m−1 digit 19.05∗ ×10−3 m dioptre 1.0∗ m−1 Dobson unit 10.0∗ ×10−6 m dram (avoirdupois) 1.771 845 ×10−3 kg dyne 10.0∗ ×10−6 N dyne centimetres 100.0∗ ×10−9 J electron volt 160.217 7 ×10−21 J ell 1.143∗ m em 4.233 333 ×10−3 m emu of capacitance 1.0∗ ×109 F emu of current 10.0∗ A emu of electric potential 10.0∗ ×10−9 V emu of inductance 1.0∗ ×10−9 H emu of resistance 1.0∗ ×10−9 Ω Eotv¨ os¨ unit 1.0∗ ×10−9 ms−2 m−1 esu of capacitance 1.112 650 ×10−12 F esu of current 333.564 1 ×10−12 A esu of electric potential 299.792 5 V esu of inductance 898.755 2 ×109 H esu of resistance 898.755 2 ×109 Ω erg 100.0∗ ×10−9 J faraday 96.485 3 ×103 C fathom 1.828 804 m fermi 1.0∗ ×10−15 m Finsen unit 10.0∗ ×10−6 Wm−2 ﬁrkin (UK) 40.914 81 ×10−3 m3 continued on next page ... main January 23, 2006 16:6

12 Units, constants, and conversions

unit name value in SI units firkin (US) 34.068 71 ×10−3 m3 fluid ounce (UK) 28.413 08 ×10−6 m3 fluid ounce (US) 29.573 53 ×10−6 m3 foot 304.8∗ ×10−3 m foot (US survey) 304.800 6 ×10−3 m foot of water (4 ◦C) 2.988 887 ×103 Pa footcandle 10.763 91 lx footlambert 3.426 259 cd m−2 footpoundal 42.140 11 ×10−3 J footpounds (force) 1.355 818 J fresnel 1.0∗ ×1012 Hz funal 1.0∗ ×103 N furlong 201.168∗ m g (standard acceleration) 9.806 65∗ ms−2 gal 10.0∗ ×10−3 ms−2 gallon (UK) 4.546 09∗ ×10−3 m3 gallon (US liquid) 3.785 412 ×10−3 m3 gamma 1.0∗ ×10−9 T gauss 100.0∗ ×10−6 T gilbert 795.774 7 ×10−3 A turn gill (UK) 142.065 4 ×10−6 m3 gill (US) 118.294 1 ×10−6 m3 gon π/200∗ rad grade 15.707 96 ×10−3 rad grain 64.798 91∗ ×10−6 kg gram 1.0∗ ×10−3 kg gram-rad 100.0∗ Jkg−1 gray 1.0∗ Jkg−1 hand 101.6∗ ×10−3 m hartree 4.359 748 ×10−18 J hectare 10.0∗ ×103 m2 hefner 902 ×10−3 cd hogshead 238.669 7 ×10−3 m3 horsepower (boiler) 9.809 50 ×103 W horsepower (electric) 746∗ W horsepower (metric) 735.498 8 W horsepower (UK) 745.699 9 W hour 3.6∗ ×103 s hour (sidereal) 3.590 170 ×103 s Hubble time 440 ×1015 s Hubble distance 130 ×1024 m hundredweight (UK long) 50.802 35 kg hundredweight (US short) 45.359 24 kg inch 25.4∗ ×10−3 m inch of mercury (0 ◦C) 3.386 389 ×103 Pa inch of water (4 ◦C) 249.074 0 Pa jansky 10.0∗ ×10−27 Wm−2 Hz−1 continued on next page ... main January 23, 2006 16:6

1.4 Converting between units 13 1 unit name value in SI units jar 10/9∗ ×10−9 F kayser 100.0∗ m−1 kilocalorie 4.186 8∗ ×103 J kilogram-force 9.806 65∗ N kilowatt hour 3.6∗ ×106 J knot (international) 514.444 4 ×10−3 ms−1 lambert 10/π∗ ×103 cd m−2 langley 41.84∗ ×103 Jm−2 langmuir 133.322 4 ×10−6 Pa s league (nautical, int.) 5.556∗ ×103 m league (nautical, UK) 5.559 552 ×103 m league (statute) 4.828 032 ×103 m light year 9.460 73∗ ×1015 m ligne 2.256∗ ×10−3 m line 2.116 667 ×10−3 m line (magnetic flux) 10.0∗ ×10−9 Wb link (engineers’) 304.8∗ ×10−3 m link (US) 201.168 0 ×10−3 m litre 1.0∗ ×10−3 m3 lumen (at 555 nm) 1.470 588 ×10−3 W maxwell 10.0∗ ×10−9 Wb mho 1.0∗ S micron 1.0∗ ×10−6 m mil (length) 25.4∗ ×10−6 m mil (volume) 1.0∗ ×10−6 m3 mile (international) 1.609 344∗ ×103 m mile (nautical, int.) 1.852∗ ×103 m mile (nautical, UK) 1.853 184∗ ×103 m mile per hour 447.04∗ ×10−3 ms−1 milliard 1.0∗ ×109 m3 millibar 100.0∗ Pa millimetre of Hg (0 ◦C) 133.322 4 Pa minim (UK) 59.193 90 ×10−9 m3 minim (US) 61.611 51 ×10−9 m3 minute (angle) 290.888 2 ×10−6 rad minute 60.0∗ s minute (sidereal) 59.836 17 s month (lunar) 2.551 444 ×106 s nit 1.0∗ cd m−2 noggin (UK) 142.065 4 ×10−6 m3 oersted 1000/(4π)∗ Am−1 ounce (avoirdupois) 28.349 52 ×10−3 kg ounce (UK fluid) 28.413 07 ×10−6 m3 ounce (US fluid) 29.573 53 ×10−6 m3 pace 762.0∗ ×10−3 m parsec 30.856 78 ×1015 m continued on next page ... main January 23, 2006 16:6

14 Units, constants, and conversions

unit name value in SI units peck (UK) 9.092 18∗ ×10−3 m3 peck (US) 8.809 768 ×10−3 m3 pennyweight (troy) 1.555 174 ×10−3 kg perch 5.029 2∗ m phot 10.0∗ ×103 lx pica (printers’) 4.217 518 ×10−3 m pint (UK) 568.261 2 ×10−6 m3 pint (US dry) 550.610 5 ×10−6 m3 pint (US liquid) 473.176 5 ×10−6 m3 point (printers’) 351.459 8∗ ×10−6 m poise 100.0∗ ×10−3 Pa s pole 5.029 2∗ m poncelot 980.665∗ W pottle 2.273 045 ×10−3 m3 pound (avoirdupois) 453.592 4 ×10−3 kg poundal 138.255 0 ×10−3 N pound-force 4.448 222 N promaxwell 1.0∗ Wb psi 6.894 757 ×103 Pa puncheon (UK) 317.974 6 ×10−3 m3 quad 1.055 056 ×1018 J quart (UK) 1.136 522 ×10−3 m3 quart (US dry) 1.101 221 ×10−3 m3 quart (US liquid) 946.352 9 ×10−6 m3 quintal (metric) 100.0∗ kg rad 10.0∗ ×10−3 Gy rayleigh 10/(4π) ×109 s−1 m−2 sr−1 rem 10.0∗ ×10−3 Sv REN 1/4 000∗ S reyn 689.5 ×103 Pa s rhe 10.0∗ Pa−1 s−1 rod 5.029 2∗ m roentgen 258.0 ×10−6 Ckg−1 rood (UK) 1.011 714 ×103 m2 rope (UK) 6.096∗ m rutherford 1.0∗ ×106 Bq rydberg 2.179 874 ×10−18 J scruple 1.295 978 ×10−3 kg seam 290.949 8 ×10−3 m3 second (angle) 4.848 137 ×10−6 rad second (sidereal) 997.269 6 ×10−3 s shake 100.0∗ ×10−10 s shed 100.0∗ ×10−54 m2 slug 14.593 90 kg square degree (π/180)2∗ sr statampere 333.564 1 ×10−12 A statcoulomb 333.564 1 ×10−12 C continued on next page ... main January 23, 2006 16:6

1.4 Converting between units 15 1 unit name value in SI units statfarad 1.112 650 ×10−12 F stathenry 898.755 2 ×109 H statmho 1.112 650 ×10−12 S statohm 898.755 2 ×109 Ω statvolt 299.792 5 V stere 1.0∗ m3 sthene´ 1.0∗ ×103 N stilb 10.0∗ ×103 cd m−2 stokes 100.0∗ ×10−6 m2 s−1 stone 6.350 293 kg tablespoon (UK) 14.206 53 ×10−6 m3 tablespoon (US) 14.786 76 ×10−6 m3 teaspoon (UK) 4.735 513 ×10−6 m3 teaspoon (US) 4.928 922 ×10−6 m3 tex 1.0∗ ×10−6 kg m−1 therm (EEC) 105.506∗ ×106 J therm (US) 105.480 4∗ ×106 J thermie 4.185 407 ×106 J thou 25.4∗ ×10−6 m tog 100.0∗ ×10−3 W−1 m2 K ton (of TNT) 4.184∗ ×109 J ton (UK long) 1.016 047 ×103 kg ton (US short) 907.184 7 kg tonne (metric ton) 1.0∗ ×103 kg torr 133.322 4 Pa townsend 1.0∗ ×10−21 Vm2 troy dram 3.887 935 ×10−3 kg troy ounce 31.103 48 ×10−3 kg troy pound 373.241 7 ×10−3 kg tun 954.678 9 ×10−3 m3 XU 100.209 ×10−15 m yard 914.4∗ ×10−3 m year (365 days) 31.536∗ ×106 s year (sidereal) 31.558 15 ×106 s year (tropical) 31.556 93 ×106 s

Temperature conversions

TK temperature in From degrees kelvin a TK = TC + 273.15 (1.1) Celsius TC temperature in ◦Celsius − From degrees TF 32 TF temperature in T = + 273.15 (1.2) ◦ Fahrenheit K 1.8 Fahrenheit

From degrees TR TR temperature in T = (1.3) ◦ Rankine K 1.8 Rankine aThe term “centigrade” is not used in SI, to avoid confusion with “10−2 of a degree”. main January 23, 2006 16:6

16 Units, constants, and conversions

1.5 Dimensions The following table lists the dimensions of common physical quantities, together with their conventional symbols and the SI units in which they are usually quoted. The dimensional basis used is length (L), mass (M), time (T), electric current (I), temperature (Θ), amount of substance (N), and luminous intensity (J).

physical quantity symbol dimensions SI units acceleration a LT−2 ms−2 action S L2 MT−1 Js angular momentum L, J L2 MT−1 m2 kg s−1 angular speed ω T−1 rad s−1 area A, S L2 m2 −1 −1 Avogadro constant NA N mol 2 −2 bending moment Gb L MT Nm 2 −1 Bohr magneton µB L I JT 2 −2 −1 −1 Boltzmann constant k, kB L MT Θ JK bulk modulus K L−1 MT−2 Pa capacitance C L−2 M−1 T4 I2 F charge (electric) q TI C charge density ρ L−3 TI Cm−3 conductance G L−2 M−1 T3 I2 S conductivity σ L−3 M−1 T3 I2 Sm−1 couple G, T L2 MT−2 Nm current I, i I A current density J , j L−2 I Am−2 density ρ L−3 M kg m−3 electric displacement D L−2 TI Cm−2 electric field strength E LMT−3 I−1 Vm−1 electric polarisability α M−1 T4 I2 Cm2 V−1 electric polarisation P L−2 TI Cm−2 electric potential difference V L2 MT−3 I−1 V energy E, U L2 MT−2 J energy density u L−1 MT−2 Jm−3 entropy S L2 MT−2 Θ−1 JK−1 Faraday constant F TIN−1 C mol−1 force F LMT−2 N frequency ν, f T−1 Hz gravitational constant G L3 M−1 T−2 m3 kg−1 s−2 3 −1 −1 3 −1 Hall coefficient RH L T I m C Hamiltonian H L2 MT−2 J heat capacity C L2 MT−2 Θ−1 JK−1 Hubble constant1 H T−1 s−1 −2 illuminance Ev L J lx impedance Z L2 MT−3 I−2 Ω continued on next page ... 1The Hubble constant is almost universally quoted in units of km s−1 Mpc−1. There are about 3.1 × 1019 kilometres in a megaparsec. main January 23, 2006 16:6

1.5 Dimensions 17 1 physical quantity symbol dimensions SI units impulse I LMT−1 Ns inductance L L2 MT−2 I−2 H −3 −2 irradiance Ee MT Wm Lagrangian L L2 MT−2 J length L, l L m luminous intensity Iv J cd magnetic dipole moment m, µ L2 I Am2 magnetic field strength H L−1 I Am−1 magnetic flux Φ L2 MT−2 I−1 Wb magnetic flux density B MT−2 I−1 T magnetic vector potential A LMT−2 I−1 Wb m−1 magnetisation M L−1 I Am−1 mass m, M M kg mobility µ M−1 T2 I m2 V−1 s−1 molar gas constant R L2 MT−2 Θ−1 N−1 J mol−1 K−1 moment of inertia I L2 M kg m2 momentum p LMT−1 kg m s−1 number density n L−3 m−3 permeability µ LMT−2 I−2 Hm−1 permittivity L−3 M−1 T4 I2 Fm−1 Planck constant h L2 MT−1 Js power P L2 MT−3 W Poynting vector S MT−3 Wm−2 pressure p, P L−1 MT−2 Pa 2 −3 −1 radiant intensity Ie L MT Wsr resistance R L2 MT−3 I−2 Ω −1 −1 Rydberg constant R∞ L m shear modulus µ, G L−1 MT−2 Pa specific heat capacity c L2 T−2 Θ−1 Jkg−1 K−1 speed u, v, c LT−1 ms−1 Stefan–Boltzmann constant σ MT−3 Θ−4 Wm−2 K−4 stress σ, τ L−1 MT−2 Pa surface tension σ, γ MT−2 Nm−1 temperature T ΘK thermal conductivity λ LMT−3 Θ−1 Wm−1 K−1 time t T s velocity v, u LT−1 ms−1 viscosity (dynamic) η, µ L−1 MT−1 Pa s viscosity (kinematic) ν L2 T−1 m2 s−1 volume V, v L3 m3 wavevector k L−1 m−1 weight W LMT−2 N work W L2 MT−2 J Young modulus E L−1 MT−2 Pa main January 23, 2006 16:6

18 Units, constants, and conversions

1.6 Miscellaneous

Greek alphabet

Aα alpha Nν nu Bβ beta Ξ ξ xi Γ γ gamma Oo omicron ∆ δ delta Π πpi Eεepsilon Pρrho Zζ zeta Σ σςsigma Hη eta Tτ tau Θ θϑtheta Υ υ upsilon Iι iota Φ φϕphi Kκ kappa Xχ chi Λ λ lambda Ψ ψ psi Mµ mu Ω ω omega

Pi (π) to 1 000 decimal places

3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989

e to 1 000 decimal places

2.7182818284 5904523536 0287471352 6624977572 4709369995 9574966967 6277240766 3035354759 4571382178 5251664274 2746639193 2003059921 8174135966 2904357290 0334295260 5956307381 3232862794 3490763233 8298807531 9525101901 1573834187 9307021540 8914993488 4167509244 7614606680 8226480016 8477411853 7423454424 3710753907 7744992069 5517027618 3860626133 1384583000 7520449338 2656029760 6737113200 7093287091 2744374704 7230696977 2093101416 9283681902 5515108657 4637721112 5238978442 5056953696 7707854499 6996794686 4454905987 9316368892 3009879312 7736178215 4249992295 7635148220 8269895193 6680331825 2886939849 6465105820 9392398294 8879332036 2509443117 3012381970 6841614039 7019837679 3206832823 7646480429 5311802328 7825098194 5581530175 6717361332 0698112509 9618188159 3041690351 5988885193 4580727386 6738589422 8792284998 9208680582 5749279610 4841984443 6346324496 8487560233 6248270419 7862320900 2160990235 3043699418 4914631409 3431738143 6405462531 5209618369 0888707016 7683964243 7814059271 4563549061 3031072085 1038375051 0115747704 1718986106 8739696552 1267154688 9570350354 main January 23, 2006 16:6

Chapter 2 Mathematics 2

2.1 Notation Mathematics is, of course, a vast subject, and so here we concentrate on those mathematical methods and relationships that are most often applied in the physical sciences and engineering. Although there is a high degree of consistency in accepted mathematical notation, there m is some variation. For example the spherical harmonics, Yl , can be written Ylm, and there is some freedom with their signs. In general, the conventions chosen here follow common practice as closely as possible, whilst maintaining consistency with the rest of the handbook. In particular:

scalars a general vectors a unit vectors aˆ scalar product a·b vector cross-product a×b gradient operator ∇ df Laplacian operator ∇2 derivative etc. dx ∂f derivative of r with partial derivatives etc. ˙r ∂x respect to t dnf nth derivative n closed loop integral dl dx L

closed surface integral ds matrix A or aij S n mean value (of x) x binomial coeﬃcient r factorial ! unit imaginary (i2 = −1) i exponential constant e modulus (of x) |x|

natural logarithm ln log to base 10 log10 main January 23, 2006 16:6

20 Mathematics

2.2 Vectors and matrices

Vector algebra

Scalar producta a·b = |a||b|cosθ (2.1) a xˆ yˆ zˆ b Vector product a×b = |a||b|sinθnˆ = ax ay az (2.2) θ bx by bz b nˆ (in) a·b = b·a (2.3) a×b = −b×a (2.4) Product rules a·(b+c)=(a·b)+(a·c) (2.5) a×(b+c)=(a×b)+(a×c) (2.6)

Lagrange’s (a×b)·(c×d)=(a·c)(b·d)−(a·d)(b·c) (2.7) identity ax ay az (a×b)·c = bx by bz (2.8) Scalar triple cx cy cz a product =(b×c)·a =(c×a)·b (2.9) c = volume of parallelepiped (2.10) b

Vector triple (a×b)×c =(a·c)b−(b·c)a (2.11) product a×(b×c)=(a·c)b−(a·b)c (2.12)

a =(b×c)/[(a×b)·c] (2.13) b =(c×a)/[(a×b)·c] (2.14) Reciprocal vectors c =(a×b)/[(a×b)·c] (2.15) (a ·a)=(b ·b)=(c ·c) = 1 (2.16)

Vector a with respect to a a =(e ·a)e +(e ·a)e +(e ·a)e (2.17) nonorthogonal 1 1 2 2 3 3 c basis {e1,e2,e3} aAlso known as the “dot product” or the “inner product.” bAlso known as the “cross-product.” nˆ is a unit vector making a right-handed set with a and b. cThe prime () denotes a reciprocal vector. main January 23, 2006 16:6

2.2 Vectors and matrices 21

Common three-dimensional coordinate systems

z ρ 2

point P θ r

y x φ

ρ =(x2 +y2)1/2 (2.21) x = ρcosφ = rsinθcosφ (2.18) r =(x2 +y2 +z2)1/2 (2.22) y = ρsinφ = rsinθsinφ (2.19) θ = arccos(z/r) (2.23) z = rcosθ (2.20) φ =arctan(y/x) (2.24)

coordinate system: rectangular spherical polar cylindrical polar coordinates of P :(x,y,z)(r,θ,φ)(ρ,φ,z) volume element: dx dy dzr2 sinθ dr dθ dφρdρ dz dφ a metric elements (h1,h2,h3): (1,1,1) (1,r,rsinθ)(1,ρ,1) a In an orthogonal coordinate system (parameterised by coordinates q1,q2,q3), the diﬀerential line 2 2 2 2 element dl is obtained from (dl) =(h1 dq1) +(h2 dq2) +(h3 dq3) .

Gradient Rectangular ∂f ∂f ∂f f scalar ﬁeld ∇f = xˆ + yˆ + zˆ (2.25) coordinates ∂x ∂y ∂z ˆ unit vector Cylindrical ∂f 1 ∂f ∂f ∇f = ρˆ + φˆ + zˆ (2.26) ρ distance from the coordinates ∂ρ r ∂φ ∂z z axis Spherical polar ∂f 1 ∂f 1 ∂f ∇f = rˆ + θˆ + φˆ (2.27) coordinates ∂r r ∂θ rsinθ ∂φ General qˆ ∂f qˆ ∂f qˆ ∂f q basis orthogonal ∇f = 1 + 2 + 3 (2.28) i h metric elements coordinates h1 ∂q1 h2 ∂q2 h3 ∂q3 i main January 23, 2006 16:6

22 Mathematics

Divergence Rectangular ∂A ∂A ∂A A vector ﬁeld ∇·A = x + y + z (2.29) coordinates Ai ith component ∂x ∂y ∂z of A Cylindrical 1 ∂(ρA ) 1 ∂A ∂A ∇·A = ρ + φ + z (2.30) ρ distance from coordinates ρ ∂ρ ρ ∂φ ∂z the z axis 1 ( 2 ) 1 ( sin ) 1 Spherical polar ∇· ∂ r Ar ∂ Aθ θ ∂Aφ A = 2 + + coordinates r ∂r rsinθ ∂θ rsinθ ∂φ (2.31) ∇· 1 ∂ ∂ General A = (A1h2h3)+ (A2h3h1) qi basis h1h2h3 ∂q1 ∂q2 orthogonal hi metric coordinates ∂ elements + (A3h1h2) (2.32) ∂q3

Curl ˆ unit vector xˆ yˆ zˆ Rectangular A vector ﬁeld ∇×A = ∂/∂x ∂/∂y ∂/∂z (2.33) coordinates Ai ith component Ax Ay Az of A ρˆ/ρ φˆ zˆ/ρ Cylindrical ρ distance from ∇×A = ∂/∂ρ ∂/∂φ ∂/∂z (2.34) coordinates the z axis Aρ ρAφ Az rˆ/(r2 sinθ) θˆ/(rsinθ) φˆ /r Spherical polar ∇×A = ∂/∂r ∂/∂θ ∂/∂φ (2.35) coordinates Ar rAθ rAφ sinθ qˆ h qˆ h qˆ h General 1 1 2 2 3 3 qi basis 1 orthogonal ∇×A = ∂/∂q ∂/∂q ∂/∂q (2.36) h h h 1 2 3 hi metric coordinates 1 2 3 elements h1A1 h2A2 h3A3

Radial formsa −r r ∇(1/r)= (2.41) ∇r = (2.37) r3 r 1 ∇· ∇·(r/r2)= (2.42) r = 3 (2.38) r2 2 − ∇r =2r (2.39) ∇ 2 2r (1/r )= 4 (2.43) ∇·(rr)=4r (2.40) r ∇·(r/r3)=4πδ(r) (2.44) aNote that the curl of any purely radial function is zero. δ(r) is the Dirac delta function. main January 23, 2006 16:6

2.2 Vectors and matrices 23

Laplacian (scalar) Rectangular ∂2f ∂2f ∂2f ∇2f = + + (2.45) f scalar ﬁeld coordinates ∂x2 ∂y2 ∂z2 Cylindrical 1 ∂ ∂f 1 ∂2f ∂2f ρ distance 2 ∇2 from the f = ρ + 2 2 + 2 (2.46) coordinates ρ ∂ρ ∂ρ ρ ∂φ ∂z z axis Spherical 1 ∂ ∂f 1 ∂ ∂f 1 ∂2f ∇2f = r2 + sinθ + polar r2 ∂r ∂r r2 sinθ ∂θ ∂θ r2 sin2 θ ∂φ2 coordinates (2.47) ∇2 1 ∂ h2h3 ∂f ∂ h3h1 ∂f General f = + qi basis h1h2h3 ∂q1 h1 ∂q1 ∂q2 h2 ∂q2 orthogonal h metric ∂ h h ∂f i coordinates + 1 2 (2.48) elements ∂q3 h3 ∂q3

Diﬀerential operator identities

∇(fg) ≡ f∇g +g∇f (2.49) ∇·(fA) ≡ f∇·A+A·∇f (2.50) ∇×(fA) ≡ f∇×A+(∇f)×A (2.51) ∇(A·B) ≡ A×(∇×B)+(A·∇)B +B×(∇×A)+(B ·∇)A (2.52) ∇·(A×B) ≡ B ·(∇×A)−A·(∇×B) (2.53) f,g scalar ﬁelds ∇×(A×B) ≡ A(∇·B)−B(∇·A)+(B ·∇)A−(A·∇)B (2.54) A,B vector ﬁelds ∇·(∇f) ≡∇2f ≡ f (2.55) ∇×(∇f) ≡ 0 (2.56) ∇·(∇×A) ≡ 0 (2.57) ∇×(∇×A) ≡ ∇(∇·A)−∇2A (2.58)

Vector integral transformations A vector field Gauss’s dV volume element (Divergence) (∇·A)dV = A· ds (2.59) Sc closed surface theorem V Sc V volume enclosed S surface Stokes’s ds surface element (∇×A)· ds = A· dl (2.60) theorem S L L loop bounding S dl line element (f∇g)· ds = ∇·(f∇g)dV (2.61) Green’s first S V f,g scalar fields theorem = [f∇2g +(∇f)·(∇g)] dV (2.62) V Green’s second [f(∇g)−g(∇f)]· ds = (f∇2g −g∇2f)dV theorem S V (2.63) main January 23, 2006 16:6

24 Mathematics

Matrix algebraa   a11 a12 ··· a1n  ···  a21 a22 a2n  A m by n matrix Matrix deﬁnition A =  . . .  (2.64)  . . ··· .  aij matrix elements am1 am2 ··· amn

Matrix addition C = A+B if cij = aij +bij (2.65)

C = AB if cij = aikbkj (2.66) Matrix (AB)C = A(BC) (2.67) multiplication A(B+C)=AB+AC (2.68)

a˜ij = aji (2.69) b a˜ij transpose matrix Transpose matrix T (AB...N)=N˜ ...B˜A˜ (2.70) (sometimes aij ,oraij )

∗ † ˜ ∗ complex conjugate (of Adjoint matrix A = A (2.71) each component) † † † † (deﬁnition 1)c (AB...N) = N ...B A (2.72) † adjoint (or Hermitian conjugate) † Hermitian matrixd H = H (2.73) H Hermitian (or self-adjoint) matrix examples:     a11 a12 a13 b11 b12 b13     A = a21 a22 a23 B = b21 b22 b23

a31 a32 a33 b31 b32 b33     a11 a21 a31 a11 +b11 a12 +b12 a13 +b13     A˜ = a12 a22 a32 A+B = a21 +b21 a22 +b22 a23 +b23

a13 a23 a33 a31 +b31 a32 +b32 a33 +b33

  a11 b11 +a12 b21 +a13 b31 a11 b12 +a12 b22 +a13 b32 a11 b13 +a12 b23 +a13 b33   AB = a21 b11 +a22 b21 +a23 b31 a21 b12 +a22 b22 +a23 b32 a21 b13 +a22 b23 +a23 b33

a31 b11 +a32 b21 +a33 b31 a31 b12 +a32 b22 +a33 b32 a31 b13 +a32 b23 +a33 b33 a Terms are implicitly summed over repeated suﬃces; hence aikbkj equals k aikbkj. bSee also Equation (2.85). cOr “Hermitian conjugate matrix.” The term “adjoint” is used in quantum physics for the transpose conjugate of a matrix and in linear algebra for the transpose matrix of its cofactors. These deﬁnitions are not compatible, but both are widely used [cf. Equation (2.80)]. dHermitian matrices must also be square (see next table). main January 23, 2006 16:6

2.2 Vectors and matrices 25

Square matricesa

A square matrix trA = aii (2.74) Trace a matrix elements tr(AB)=tr(BA) (2.75) ij aii implicitly = i aii 2

detA = ijk...a1ia2j a3k ... (2.76) tr trace i+1 det determinant (or |A|) =(−1) a 1M 1 (2.77) b i i Determinant Mij minor of element aij = ai1Ci1 (2.78) Cij cofactor of the det(AB...N)=detAdetB...detN (2.79) element aij

Adjoint matrix adj adjoint (sometimes ˜ written Aˆ ) c adjA = Cij = Cji (2.80) (deﬁnition 2) ∼ transpose

−1 Cji adjA aij = = (2.81) Inverse matrix detA detA −1 1 unit matrix (detA =0) AA = 1 (2.82) (AB...N)−1 = N−1 ...B−1A−1 (2.83)

aij aik = δjk (2.84) Orthogonality δjk Kronecker delta (= 1 condition i.e., A˜ = A−1 (2.85) if i = j, = 0 otherwise)

If A = A˜, A is symmetric (2.86) Symmetry If A = −A˜, A is antisymmetric (2.87)

† − U unitary matrix Unitary matrix U = U 1 (2.88) † Hermitian conjugate examples:  a11 a12 a13   b11 b12 A = a21 a22 a23 B = b21 b22 a31 a32 a33

trA = a11 +a22 +a33 trB = b11 +b22

detA = a11 a22 a33 −a11 a23 a32 −a21 a12 a33 +a21 a13 a32 +a31 a12 a23 −a31 a13 a22

detB = b11 b22 −b12 b21   a22 a33 −a23 a32 −a12 a33 +a13 a32 a12 a23 −a13 a22 1   A−1 = −a a +a a a a −a a −a a +a a  detA 21 33 23 31 11 33 13 31 11 23 13 21 a21 a32 −a22 a31 −a11 a32 +a12 a31 a11 a22 −a12 a21 1 b22 −b12 B−1 = detB −b21 b11 a Terms are implicitly summed over repeated suﬃces; hence aikbkj equals k aikbkj. b ijk... is deﬁned as the natural extension of Equation (2.443) to n-dimensions (see page 50). Mij is the determinant i+j of the matrix A with the ith row and the jth column deleted. The cofactor Cij =(−1) Mij . cOr “adjugate matrix.” See the footnote to Equation (2.71) for a discussion of the term “adjoint.” main January 23, 2006 16:6

26 Mathematics

Commutators Commutator [A,B]=AB−BA = −[B,A] (2.89) [·,·] commutator deﬁnition

† † † Adjoint [A,B] =[B ,A ] (2.90) † adjoint

Distribution [A+B,C]=[A,C]+[B,C] (2.91)

Association [AB,C]=A[B,C]+[A,C]B (2.92)

Jacobi identity [A,[B,C]] = [B,[A,C]]−[C,[A,B]] (2.93)

Pauli matrices 01 0 −i σ = σ = 1 10 2 i 0 σi Pauli spin matrices Pauli matrices 1 2×2 unit matrix 10 10 2 σ = 1 = (2.94) ii= −1 3 0 −1 01 Anticommuta- σ σ +σ σ =2δ 1 (2.95) δ Kronecker delta tion i j j i ij ij

Cyclic σiσj = iσk (2.96) 2 permutation (σi) = 1 (2.97)

Rotation matricesa   Rotation 10 0 Ri(θ) matrix for rotation   about the ith axis R1(θ)= 0cosθ sinθ (2.98) about x1 0 −sinθ cosθ θ rotation angle   cosθ 0 −sinθ Rotation   R2(θ)= 01 0 (2.99) about x2 sinθ 0cosθ   cosθ sinθ 0 α rotation about x3 Rotation   R3(θ)= −sinθ cosθ 0 (2.100) β rotation about x2 about x3 001 γ rotation about x3 Euler angles R rotation matrix   cosγcosβ cosα−sinγsinα cosγcosβ sinα+sinγcosα −cosγsinβ R(α,β,γ)=−sinγcosβ cosα−cosγsinα −sinγcosβ sinα+cosγcosα sinγsinβ  sinβ cosα sinβ sinα cosβ (2.101) aAngles are in the right-handed sense for rotation of axes, or the left-handed sense for rotation of vectors. i.e., a vector v is given a right-handed rotation of θ about the x3-axis using R3(−θ)v → v . Conventionally, x1 ≡ x, x2 ≡ y, and x3 ≡ z. main January 23, 2006 16:6

2.3 Series, summations, and progressions 27

2.3 Series, summations, and progressions

Progressions and summations

Sn = a+(a+d)+(a+2d)+··· n number of terms 2 +[a+(n−1)d] (2.102) Sn sum of n successive Arithmetic n terms progression = [2a+(n−1)d] (2.103) a ﬁrst term 2 n d common diﬀerence = (a+l) (2.104) 2 l last term

2 n−1 Sn = a+ar+ar +···+ar (2.105) n Geometric 1−r = a (2.106) r common ratio progression 1−r a S∞ = (|r| < 1) (2.107) 1−r Arithmetic 1 x = (x +x +···+x ) (2.108) . a arithmetic mean mean a n 1 2 n

Geometric 1/n x =(x x x ...x ) (2.109) . geometric mean mean g 1 2 3 n g −1 1 1 1 Harmonic mean x h = n + +···+ (2.110) . h harmonic mean x1 x2 xn Relative mean x ≥x ≥x if x > 0foralli (2.111) magnitudes a g h i n n i = (n+1) (2.112) 2 i=1 n n i2 = (n+1)(2n+1) (2.113) 6 i=1 n n2 i3 = (n+1)2 (2.114) 4 i=1 Summation n n i4 = (n+1)(2n+1)(3n2 +3n−1) (2.115) i dummy integer formulas 30 i=1 ∞ (−1)i+1 1 1 1 =1− + − +...= ln2 (2.116) i 2 3 4 i=1 ∞ (−1)i+1 1 1 1 π =1− + − +...= (2.117) 2i−1 3 5 7 4 i=1 ∞ 1 1 1 1 π2 =1+ + + +...= (2.118) i2 4 9 16 6 i=1 Euler’s 1 1 1 a γ = lim 1+ + +···+ −lnn (2.119) γ Euler’s constant constant n→∞ 2 3 n aγ 0.577215664... main January 23, 2006 16:6

28 Mathematics

Power series n(n−1) n(n−1)(n−2) Binomial (1+x)n =1+nx+ x2 + x3 +··· (2.120) seriesa 2! 3! Binomial n n n! Cr ≡ ≡ (2.121) coeﬃcientb r r!(n−r)! n n Binomial (a+b)n = an−kbk (2.122) theorem k k=0

x2 xn−1 Taylor series f(a+x)=f(a)+xf(1)(a)+ f(2)(a)+···+ f(n−1)(a)+··· (2.123) (about a)c 2! (n−1)!

Taylor series (x·∇)2 (x·∇)3 f(a+x)=f(a)+(x·∇)f|a + f|a + f|a +··· (2.124) (3-D) 2! 3!

x2 xn−1 Maclaurin f(x)=f(0)+xf(1)(0)+ f(2)(0)+···+ f(n−1)(0)+··· (2.125) series 2! (n−1)! aIf n is a positive integer the series terminates and is valid for all x. Otherwise the (inﬁnite) series is convergent for |x| < 1. bThe coeﬃcient of xr in the binomial series. cxf(n)(a)isx times the nth derivative of the function f(x) with respect to x evaluated at a, taken as well behaved n around a.(x·∇) f|a is its extension to three dimensions.

Limits ncxn → 0asn →∞ if |x| < 1 (for any ﬁxed c) (2.126)

xn/n! → 0asn →∞ (for any ﬁxed x) (2.127)

(1+x/n)n → ex as n →∞ (2.128)

xlnx → 0asx → 0 (2.129)

sinx → 1asx → 0 (2.130) x f(x) f(1)(a) If f(a)=g(a)=0 or ∞ then lim = (l’Hopital’sˆ rule) (2.131) x→a g(x) g(1)(a) main January 23, 2006 16:6

2.3 Series, summations, and progressions 29

Series expansions x2 x3 exp(x) 1+x+ + +··· (2.132) (for all x) 2! 3! x2 x3 x4 2 ln(1+x) x− + − +··· (2.133) (−1 1)  − + − +··· (2.142)  2 x 3x3 5x5   π 1 1 1 − − + − +··· (x<−1) 2 x 3x3 5x5 x2 x4 x6 cosh(x) 1+ + + +··· (2.143) (for all x) 2! 4! 6! x3 x5 x7 sinh(x) x+ + + +··· (2.144) (for all x) 3! 5! 7! x3 2x5 17x7 tanh(x) x− + − +··· (2.145) (|x| <π/2) 3 15 315 − aarccos(x)=π/2−arcsin(x). Note that arcsin(x) ≡ sin 1(x)etc. barccot(x)=π/2−arctan(x). main January 23, 2006 16:6

30 Mathematics

Inequalities | |−| |≤| |≤| | | | a1 a2 a1 +a2 a1 + a2 ; (2.146) Triangle n n inequality ≤ | | ai ai (2.147) i=1 i=1

if a1 ≥ a2 ≥ a3 ≥ ...≥ an (2.148) and ≥ ≥ ≥ ≥ (2.149) Chebyshev b1 b2 b3 ... bn inequality n n n then n aibi ≥ ai bi (2.150) i=1 i=1 i=1 2 Cauchy n n n ≤ 2 2 inequality aibi ai bi (2.151) i=1 i=1 i=1 Schwarz b 2 b b ≤ 2 2 inequality f(x)g(x)dx [f(x)] dx [g(x)] dx (2.152) a a a

2.4 Complex variables

Complex numbers z complex variable Cartesian form z = x+iy (2.153) ii2 = −1 x,y real variables r amplitude (real) Polar form z = reiθ = r(cosθ +isinθ) (2.154) θ phase (real)

|z| = r =(x2 +y2)1/2 (2.155) Modulusa |z| modulus of z |z1 ·z2| = |z1|·|z2| (2.156)

y θ =argz =arctan (2.157) Argument x argz argument of z arg(z1z2)=argz1 +argz2 (2.158)

z∗ = x−iy = re−iθ (2.159) Complex ∗ ∗ arg(z )=−argz (2.160) z complex conjugate of conjugate z = reiθ z ·z∗ = |z|2 (2.161)

Logarithmb lnz =lnr +i(θ +2πn) (2.162) n integer aOr “magnitude.” bThe principal value of lnz is given by n = 0 and −π<θ≤ π. main January 23, 2006 16:6

2.4 Complex variables 31

Complex analysisa

if f(z)=u(x,y)+iv(x,y) z complex variable Cauchy– ∂u ∂v ii2 = −1 then = (2.163) Riemann ∂x ∂y x,y real variables 2 b equations ∂u ∂v f(z) function of z = − (2.164) ∂y ∂x u,v real functions Cauchy– Goursat f(z)dz = 0 (2.165) theoremc c 1 f(z) (n) nth derivative Cauchy f(z0)= dz (2.166) 2πi z −z a Laurent coeﬃcients integral c 0 n d (n) n! f(z) a−1 residue of f(z)atz0 formula f (z )= dz (2.167) 0 − n+1 z dummy variable 2πi c (z z0) ∞ n y f(z)= an(z −z0) (2.168) c2 Laurent n=−∞ e c1 expansion 1 f(z) where a = dz (2.169) z n − n+1 c 0 2πi c (z z0) Residue f(z)dz =2πi enclosed residues (2.170) x theorem c aClosed contour integrals are taken in the counterclockwise sense, once. bNecessary condition for f(z) to be analytic at a given point. cIf f(z) is analytic within and on a simple closed curve c. Sometimes called “Cauchy’s theorem.” d If f(z) is analytic within and on a simple closed curve c, encircling z0. e Of f(z), (analytic) in the annular region between concentric circles, c1 and c2, centred on z0. c is any closed curve in this region encircling z0. main January 23, 2006 16:6

32 Mathematics

2.5 Trigonometric and hyperbolic formulas

Trigonometric relationships

sin(A±B)=sinAcosB ±cosAsinB (2.171) cos(A±B)=cosAcosB ∓sinAsinB (2.172) tanA±tanB tan(A±B)= (2.173) 1∓tanAtanB 1 cosAcosB = [cos(A+B)+cos(A−B)] (2.174) 2 1 sinAcosB = [sin(A+B)+sin(A−B)] (2.175) 2 1 sinAsinB = [cos(A−B)−cos(A+B)] (2.176) 2 2

2 cos2 A+sin A = 1 (2.177) 1 x sin x 2 − 2 cos sec A tan A = 1 (2.178) 0 csc2 A−cot2 A = 1 (2.179) −1

tan sin2A =2sinAcosA (2.180) −2 −6 −4 −2 0 2 4 6 2 2 cos2A =cos A−sin A (2.181) x 2tanA tan2A = (2.182) 1−tan2 A sin3A =3sinA−4sin3 A (2.183) cos3A =4cos3 A−3cosA (2.184)

A+B A−B sinA+sinB =2sin cos (2.185) 2 2 − − A+B A B 4 sinA sinB =2cos sin (2.186) sec 2 2 x x 2 csc A+B A−B cot cosA+cosB =2cos cos (2.187) x 2 2 0 A+B A−B cosA−cosB = −2sin sin (2.188) −2 2 2 −4 2 1 cos A = (1+cos2A) (2.189) −6 −4 −2 0 2 4 6 2 x 1 sin2 A = (1−cos2A) (2.190) 2 1 cos3 A = (3cosA+cos3A) (2.191) 4 1 sin3 A = (3sinA−sin3A) (2.192) 4 main January 23, 2006 16:6

2.5 Trigonometric and hyperbolic formulas 33

Hyperbolic relationshipsa

sinh(x±y)=sinhxcoshy ±coshxsinhy (2.193) cosh(x±y)=coshxcoshy ±sinhxsinhy (2.194) 2 tanhx±tanhy tanh(x±y)= (2.195) 1±tanhxtanhy 1 coshxcoshy = [cosh(x+y)+cosh(x−y)] (2.196) 2 1 sinhxcoshy = [sinh(x+y)+sinh(x−y)] (2.197) 2 1 sinhxsinhy = [cosh(x+y)−cosh(x−y)] (2.198) 2 4 2 − 2 cosh cosh x sinh x = 1 (2.199) 2 x tanhx 2 2 sech x+tanh x = 1 (2.200) 0 tanhx 2 − 2 coth x csch x = 1 (2.201) −2 x

sinh −4 sinh2x = 2sinhxcoshx (2.202) −3 −2 −1 0 1 2 3 2 2 cosh2x = cosh x+sinh x (2.203) x 2tanhx tanh2x = (2.204) 1+tanh2 x sinh3x = 3sinhx+4sinh3 x (2.205) cosh3x =4cosh3 x−3coshx (2.206)

x+y x−y sinhx+sinhy = 2sinh cosh (2.207) 2 2 4 x+y x−y sinhx−sinhy =2cosh sinh (2.208) 2 2 2 cothx sechx x+y x−y 0 coshx+coshy =2cosh cosh (2.209) csch 2 2 x −2 x+y x−y coshx−coshy = 2sinh sinh (2.210) 2 2 −4 −3 −2 −1 0 1 2 3 2 1 cosh x = (cosh2x+1) (2.211) x 2 1 sinh2 x = (cosh2x−1) (2.212) 2 1 cosh3 x = (3coshx+cosh3x) (2.213) 4 1 sinh3 x = (sinh3x−3sinhx) (2.214) 4 aThese can be derived from trigonometric relationships by using the substitutions cosx → coshx and sinx → isinhx. main January 23, 2006 16:6

34 Mathematics

Trigonometric and hyperbolic deﬁnitions

de Moivre’s theorem (cosx+isinx)n =einx =cosnx+isinnx (2.215)

1 1 cosx = eix +e−ix (2.216) coshx = ex +e−x (2.217) 2 2 1 1 sinx = eix −e−ix (2.218) sinhx = ex −e−x (2.219) 2i 2 sinx sinhx tanx = (2.220) tanhx = (2.221) cosx coshx

cosix = coshx (2.222) coshix =cosx (2.223)

sinix = isinhx (2.224) sinhix = isinx (2.225)

cotx =(tanx)−1 (2.226) cothx =(tanhx)−1 (2.227)

secx =(cosx)−1 (2.228) sechx =(coshx)−1 (2.229)

cscx =(sinx)−1 (2.230) cschx = (sinhx)−1 (2.231)

1.6 Inverse trigonometric functionsa arccos x x arcsinx =arctan (2.232) 1 x (1−x2)1/2 arctan x (1−x2)1/2 arccosx =arctan (2.233) arcsin x 1 0 1 arccscx =arctan (2.234) x (x2 −1)1/2 1.6 2 − 1/2 arccsc arcsecx =arctan (x 1) (2.235) arccot x arcsec

x x 1 1 arccotx =arctan (2.236) x π arccosx = −arcsinx (2.237) 2 − aValid in the angle range 0 ≤ θ ≤ π/2. Note that arcsinx ≡ sin 1 x etc. 0 1 2345 x main January 23, 2006 16:6

2.6 Mensuration 35

Inverse hyperbolic functions 2 −1 2 1/2 arsinhx ≡ sinh x =ln x+(x +1) (2.238) for all x 1 x

artanh x − ≡ 1 2 − 1/2 0 arcoshx cosh x =ln x+(x 1) ≥ 2 x 1 arcosh (2.239) x −1 arsinh − 1 1+x ≡ 1 | | 1 artanhx tanh x = ln − (2.240) x < −2 2 1 x −1 0 1 x − 1 x+1 arcothx ≡ coth 1 x = ln (2.241) |x| > 1 2 x−1 4 − 2 1/2 ≡ −1 1 (1 x ) arsechx sech x =ln + 0 ≤ 1 x x

2 x 2 1/2 arcsch −1 1 (1+x ) arcschx ≡ csch x =ln + arsech x x x x =0 1 (2.243) x 0 1 2 x

2.6 Mensuration

Moire´ fringesa − 1 d Moire´ fringe spacing Parallel pattern 1 − 1 M dM = (2.244) d grating spacings d1 d2 1,2 d common grating Rotational d spacing d = (2.245) patternb M 2|sin(θ/2)| θ relative rotation angle (|θ|≤π/2) aFrom overlapping linear gratings. bFrom identical gratings, spacing d, with a relative rotation θ. main January 23, 2006 16:6

36 Mathematics

Plane triangles a b c Sine formulaa = = (2.246) sinA sinB sinC

a2 = b2 +c2 −2bccosA (2.247) Cosine b2 +c2 −a2 C b cosA = (2.248) a formulas 2bc a = bcosC +ccosB (2.249) A B c Tangent A−B a−b C tan = cot (2.250) formula 2 a+b 2 1 area = absinC (2.251) 2 a2 sinB sinC = (2.252) Area 2 sinA =[s(s−a)(s−b)(s−c)]1/2 (2.253) 1 where s = (a+b+c) (2.254) 2 aThe diameter of the circumscribed circle equals a/sinA.

Spherical trianglesa sina sinb sinc Sine formula = = (2.255) sinA sinB sinC

Cosine cosa =cosbcosc+sinbsinccosA (2.256) formulas cosA = −cosB cosC +sinB sinC cosa (2.257)

Analogue a C b sinacosB =cosbsinc−sinbcosccosA (2.258) formula A Four-parts B cosacosC =sinacotb−sinC cotB (2.259) formula c

Areab E = A+B +C −π (2.260) aOn a unit sphere. bAlso called the “spherical excess.” main January 23, 2006 16:6

2.6 Mensuration 37

Perimeter, area, and volume P perimeter Perimeter of circle P =2πr (2.261) r radius

2 Area of circle A = πr (2.262) A area 2

2 Surface area of spherea A =4πR (2.263) R sphere radius

4 Volume of sphere V = πR3 (2.264) V volume 3 a semi-major axis P =4aE(π/2,e) (2.265) b semi-minor axis b E elliptic integral of the Perimeter of ellipse a2 +b2 1/2 2π (2.266) second kind (p. 45) 2 e eccentricity (= 1−b2/a2)

Area of ellipse A = πab (2.267)

abc Volume of ellipsoidc V =4π (2.268) c third semi-axis 3 Surface area of A =2πr(h+r) (2.269) h height cylinder

Volume of cylinder V = πr2h (2.270)

Area of circular coned A = πrl (2.271) l slant height

Volume of cone or V = A h/3 (2.272) A base area pyramid b b

2 r1 inner radius Surface area of torus A = π (r1 +r2)(r2 −r1) (2.273) r2 outer radius π2 Volume of torus V = (r2 −r2)(r −r ) (2.274) 4 2 1 2 1 Aread of spherical cap, A =2πRd (2.275) d cap depth depth d Ω solid angle Volume of spherical d V = πd2 R − (2.276) z distance from centre cap, depth d 3 α half-angle subtended z Solid angle of a circle Ω=2π 1− (2.277) r 2 2 1/2 α from a point on its (z +r ) axis, z from centre =2π(1−cosα) (2.278) z aSphere deﬁned by x2 +y2 +z2 = R2. bThe approximation is exact when e = 0 and e 0.91, giving a maximum error of 11% at e =1. cEllipsoid deﬁned by x2/a2 +y2/b2 +z2/c2 =1. dCurved surface only. main January 23, 2006 16:6

38 Mathematics

Conic sections

y y y

b x x x a a a

parabola ellipse hyperbola

x2 y2 x2 y2 equation y2 =4ax + =1 − =1 a2 b2 a2 b2 parametric x = t2/(4a) x = acost x = ±acosht form y = t y = bsint y = bsinht √ √ foci (a,0) (± a2 −b2,0) (± a2 +b2,0) √ √ a2 −b2 a2 +b2 eccentricity e =1 e = e = a a a a directrices x = −a x = ± x = ± e e

Platonic solidsa

solid volume surface area circumradius inradius (faces,edges,vertices) √ √ √ tetrahedron a3 2 √ a 6 a 6 a2 3 (4,6,4) 12 4 12 √ cube a 3 a a3 6a2 (6,12,8) 2 2 √ octahedron a3 2 √ a a 2a2 3 √ √ (8,12,6) 3 2 6 √ √ √ √ dodecahedron a3(15+7 5) √ a a 50+22 5 3a2 5(5+2 5) 3(1+ 5) (12,30,20) 4 4 4 5 √ √ √ icosahedron 5a3(3+ 5) √ a a 5 5a2 3 2(5+ 5) 3+ (20,30,12) 12 4 4 3 aOf side a. Both regular and irregular polyhedra follow the Euler relation, faces−edges+vertices = 2. main January 23, 2006 16:6

2.6 Mensuration 39

Curve measure a start point 1/2 Length of plane b dy 2 b end point curve l = 1+ dx (2.279) y(x) plane curve a dx l length 2 1/2 Surface of b 2 dy A surface area revolution A =2π y 1+ dx (2.280) a dx Volume of b 2 V volume revolution V = π y dx (2.281) a 2 3/2 −1 Radius of dy d2y ρ radius of ρ = 1+ (2.282) curvature dx dx2 curvature

Diﬀerential geometrya r˙ r˙ τ tangent Unit tangent τˆ = = (2.283) r curve parameterised by r(t) |r˙| v v |r˙(t)| r¨ −˙vτˆ Unit principal normal nˆ = (2.284) n principal normal |r¨ −˙vτˆ|

Unit binormal bˆ = τˆ×nˆ (2.285) b binormal

|r˙×r¨| Curvature κ = (2.286) κ curvature |r˙|3 1 Radius of curvature ρ = (2.287) ρ radius of curvature κ ... r˙ ·(r¨× r ) Torsion λ = (2.288) λ torsion |r˙×r¨|2

nˆ τ˙ˆ = κvnˆ (2.289) osculating plane ˙ normal plane Frenet’s formulas nˆ = −κvτˆ +λvbˆ (2.290) τˆ rectifying ˙ r bˆ = −λvnˆ (2.291) bˆ plane origin aFor a continuous curve in three dimensions, traced by the position vector r(t). main January 23, 2006 16:6

40 Mathematics

2.7 Diﬀerentiation

Derivatives (general) d du Power (un)=nun−1 (2.292) n power dx dx index d dv du Product (uv)=u +v (2.293) u,v functions dx dx dx of x d u 1 du u dv Quotient = − (2.294) dx v v dx v2 dx Function of a d d du [f(u)] = [f(u)]· (2.295) f(u) function of functiona dx du dx u(x) n n n−1 d n d u n dv d u ··· [uv]= v + − + dxn 0 dxn 1 dx dxn 1 n binomial Leibniz theorem (2.296) k n dkv dn−ku n dnv coeﬃcient + +···+ u k dxk dxn−k n dxn d q Diﬀerentiation f(x)dx = f(q)(p constant) (2.297) dq under the integral p d q sign f(x)dx = −f(p)(q constant) (2.298) dp p d v(x) dv du General integral f(t)dt = f(v) −f(u) (2.299) dx u(x) dx dx d b log base Logarithm (log |ax|)=(xlnb)−1 (2.300) dx b a constant d Exponential (eax)=aeax (2.301) dx − dx dy 1 = (2.302) dy dx − d2x d2y dy 3 Inverse functions = − (2.303) dy2 dx2 dx 2 − d3x d2y dy d3y dy 5 = 3 − (2.304) dy3 dx2 dx dx3 dx aThe “chain rule.” main January 23, 2006 16:6

2.7 Diﬀerentiation 41

Trigonometric derivativesa d d (sinax)=acosax (2.305) (cosax)=−asinax (2.306) dx dx d d 2 (tanax)=asec2 ax (2.307) (cscax)=−acscax·cotax (2.308) dx dx d d (secax)=asecax·tanax (2.309) (cotax)=−acsc2 ax (2.310) dx dx d d (arcsinax)=a(1−a2x2)−1/2 (2.311) (arccosax)=−a(1−a2x2)−1/2 (2.312) dx dx d d a (arctanax)=a(1+a2x2)−1 (2.313) (arccscax)=− (a2x2 −1)−1/2 (2.314) dx dx |ax| d a d (arcsecax)= (a2x2 −1)−1/2 (2.315) (arccotax)=−a(a2x2 +1)−1 (2.316) dx |ax| dx aa is a constant.

Hyperbolic derivativesa d d (sinhax)=acoshax (2.317) (coshax)=asinhax (2.318) dx dx d d (tanhax)=asech2 ax (2.319) (cschax)=−acschax·cothax (2.320) dx dx d d (sechax)=−asechax·tanhax (2.321) (cothax)=−acsch2 ax (2.322) dx dx d d (arsinhax)=a(a2x2 +1)−1/2 (2.323) (arcoshax)=a(a2x2 −1)−1/2 (2.324) dx dx d d a (artanhax)=a(1−a2x2)−1 (2.325) (arcschax)=− (1+a2x2)−1/2 (2.326) dx dx |ax|

d a 2 2 −1/2 (arsechax)=− (1−a x ) d − dx |ax| (arcothax)=a(1−a2x2) 1 (2.328) dx (2.327) aa is a constant. main January 23, 2006 16:6

42 Mathematics

Partial derivatives Total ∂f ∂f ∂f df = dx+ dy + dz (2.329) ff(x,y,z) diﬀerential ∂x ∂y ∂z ∂g ∂x ∂y Reciprocity = −1 (2.330) gg(x,y) ∂x y ∂y g ∂g x ∂f ∂f ∂x ∂f ∂y ∂f ∂z Chain rule = + + (2.331) ∂u ∂x ∂u ∂y ∂u ∂z ∂u ∂x ∂x ∂x J Jacobian ∂u ∂v ∂w ∂(x,y,z) ∂y ∂y ∂y uu(x,y,z) Jacobian J = = (2.332) vv(x,y,z) ∂(u,v,w) ∂u ∂v ∂w ww(x,y,z) ∂z ∂z ∂z ∂u ∂v ∂w V volume in (x,y,z) Change of f(x,y,z)dxdydz = f(u,v,w)J dudvdw V volume in (u,v,w) variable V V (2.333) mapped to by V b Euler– if I = F(x,y,y )dx y dy/dx Lagrange a ∂F d ∂F a,b ﬁxed end points equation then δI =0 when = (2.334) ∂y dx ∂y

Stationary pointsa

saddle pointmaximum minimum quartic minimum

∂f ∂f Stationary point if = =0 at(x ,y ) (necessary condition) (2.335) ∂x ∂y 0 0 Additional suﬃcient conditions ∂2f ∂2f ∂2f ∂2f 2 for maximum < 0, and > (2.336) ∂x2 ∂x2 ∂y2 ∂x∂y ∂2f ∂2f ∂2f ∂2f 2 for minimum > 0, and > (2.337) ∂x2 ∂x2 ∂y2 ∂x∂y ∂2f ∂2f ∂2f 2 for saddle point < (2.338) ∂x2 ∂y2 ∂x∂y 2 2 aOf a function f(x,y) at the point (x ,y ). Note that at, for example, a quartic minimum ∂ f = ∂ f =0. 0 0 ∂x2 ∂y2 main January 23, 2006 16:6

2.7 Diﬀerentiation 43

Diﬀerential equations

2 Laplace ∇ f = 0 (2.339) ff(x,y,z)

∂f Diffusiona = D∇2f (2.340) D diffusion 2 ∂t coefficient

2 2 Helmholtz ∇ f +α f = 0 (2.341) α constant

1 ∂2f Wave ∇2f = (2.342) c wave speed c2 ∂t2 d dy Legendre (1−x2) +l(l +1)y = 0 (2.343) l integer dx dx Associated d dy m2 (1−x2) + l(l +1)− y = 0 (2.344) m integer Legendre dx dx 1−x2 d2y dy Bessel x2 +x +(x2 −m2)y = 0 (2.345) dx2 dx d2y dy Hermite −2x +2αy = 0 (2.346) dx2 dx d2y dy Laguerre x +(1−x) +αy = 0 (2.347) dx2 dx Associated d2y dy x +(1+k −x) +αy = 0 (2.348) k integer Laguerre dx2 dx d2y dy Chebyshev (1−x2) −x +n2y = 0 (2.349) n integer dx2 dx Euler (or d2y dy x2 +ax +by = f(x) (2.350) a,b constants Cauchy) dx2 dx dy Bernoulli +p(x)y = q(x)ya (2.351) p,q functions of x dx d2y Airy = xy (2.352) dx2 aAlso known as the “conduction equation.” For thermal conduction, f ≡ T and D, the thermal diﬀusivity, ≡ κ ≡ λ/(ρcp), where T is the temperature distribution, λ the thermal conductivity, ρ the density, and cp the speciﬁc heat capacity of the material. main January 23, 2006 16:6

44 Mathematics

2.8 Integration

Standard formsa dv u dv =[uv]− v du (2.353) uv dx = v u dx− u dx dx (2.354) dx xn+1 1 xn dx = (n = −1) (2.355) dx =ln|x| (2.356) n+1 x 1 x 1 eax dx = eax (2.357) xeax dx =eax − (2.358) a a a2 f(x) lnax dx = x(lnax−1) (2.359) dx =lnf(x) (2.360) f(x) x2 1 bax xlnax dx = lnax− (2.361) bax dx = (b>0) (2.362) 2 2 alnb 1 1 1 1 a+bx dx = ln(a+bx) (2.363) dx = − ln (2.364) a+bx b x(a+bx) a x 1 −1 1 1 bx dx = (2.365) dx = arctan (2.366) (a+bx)2 b(a+bx) a2 +b2x2 ab a − n 1 1 x a 1 1 x dx = ln (2.368) dx = ln (2.367) x2 −a2 2a x+a x(xn +a) an xn +a x 1 x −1 dx = ln|x2 ±a2| (2.369) dx = (2.370) x2 ±a2 2 (x2 ±a2)n 2(n−1)(x2 ±a2)n−1

1 x 1 dx = arcsin (2.371) dx =ln|x+(x2 ±a2)1/2| (2.372) (a2 −x2)1/2 a (x2 ±a2)1/2 x 1 1 x dx =(x2 ±a2)1/2 (2.373) dx = arcsec (2.374) (x2 ±a2)1/2 x(x2 −a2)1/2 a a aa and b are non-zero constants. main January 23, 2006 16:6

2.8 Integration 45

Trigonometric and hyperbolic integrals sinx dx = −cosx (2.375) sinhx dx = coshx (2.376) 2 cosx dx =sinx (2.377) coshx dx = sinhx (2.378) tanx dx = −ln|cosx| (2.379) tanhx dx = ln(coshx) (2.380) x x cscx dx =lntan (2.381) cschx dx =lntanh (2.382) 2 2 secx dx =ln|secx+tanx| (2.383) sechx dx = 2arctan(ex) (2.384) cotx dx =ln|sinx| (2.385) cothx dx =ln|sinhx| (2.386) sin(m−n)x sin(m+n)x sinmx ·sinnx dx = − (m2 = n2) (2.387) 2(m−n) 2(m+n) cos(m−n)x cos(m+n)x sinmx ·cosnx dx = − − (m2 = n2) (2.388) 2(m−n) 2(m+n) sin(m−n)x sin(m+n)x cosmx ·cosnx dx = + (m2 = n2) (2.389) 2(m−n) 2(m+n)

Named integrals 2 x Error function erf( )= exp(− 2)d (2.390) x 1/2 t t π 0 Complementary error 2 ∞ erfc( )=1−erf( )= exp(− 2)d (2.391) x x 1/2 t t function π x x πt2 x πt2 C(x)= cos dt;S(x)= sin dt (2.392) 2 2 Fresnel integralsa 0 0 1+i π1/2 C(x)+i S(x)= erf (1−i)x (2.393) 2 2 x et Exponential integral Ei(x)= dt (x>0) (2.394) −∞ t ∞ Gamma function Γ(x)= tx−1e−t dt (x>0) (2.395) 0 φ 1 F(φ,k)= dθ (ﬁrst kind) (2.396) Elliptic integrals (1−k2 sin2 θ)1/2 0 (trigonometric form) φ E(φ,k)= (1−k2 sin2 θ)1/2 dθ (second kind) (2.397) 0 aSee also page 167. main January 23, 2006 16:6

46 Mathematics

Deﬁnite integrals ∞ 2 1 π 1/2 e−ax dx = (a>0) (2.398) 0 2 a ∞ 2 1 xe−ax dx = (a>0) (2.399) 0 2a ∞ n −ax n! x e dx = n+1 (a>0; n =0,1,2,...) (2.400) 0 a ∞ π 1/2 b2 exp(2bx −ax2)dx = exp (a>0) (2.401) −∞ a a ∞ · · · · − −(n+1)/2 1/2 n −ax2 1 3 5 ... (n 1)(2a) (π/2) n>0 and even x e dx = −(n+1)/2 (2.402) 0 2·4·6·...·(n−1)(2a) n>1 and odd 1 p!q! xp(1−x)q dx = (p,q integers > 0) (2.403) 0 (p+q +1)! ∞ ∞ 1 π 1/2 cos(ax2)dx = sin(ax2)dx = (a>0) (2.404) 0 0 2 2a ∞ sinx ∞ sin2 x π dx = 2 dx = (2.405) 0 x 0 x 2 ∞ 1 π a dx = (0

2.9 Special functions and polynomials Gamma function ∞ Deﬁnition Γ(z)= tz−1e−t dt [(z) > 0] (2.407) 0 n!=Γ(n+1)=nΓ(n)(n =0,1,2,...) (2.408) Γ(1/2) = π1/2 (2.409) Relations z z! Γ(z +1) = = (2.410) w w!(z −w)! Γ(w +1)Γ(z −w +1) 1 1 Γ(z) e−zzz−(1/2)(2π)1/2 1+ + −··· (2.411) 12z 288z2 Stirling’s formulas − (for |z|,n 1) n! nn+(1/2) e n(2π)1/2 (2.412) ln(n!) nlnn−n (2.413) main January 23, 2006 16:6

2.9 Special functions and polynomials 47

Bessel functions ∞ ( ) Bessel function of the ﬁrst 2 k Jν x x ν (−x /4) kind Jν (x)= (2.414) Series 2 k!Γ(ν +k +1) Y (x) Bessel function of the k=0 ν expansion second kind Jν (x)cos(πν)−J−ν (x) 2 Yν (x)= (2.415) Γ(ν) Gamma function sin(πν) ν order (ν ≥ 0) Approximations 1 J 0 J1 1 x ν (0 ≤ x ν) 0.5 Γ(ν+1) 2 Jν (x) 1/2 (2.416) 2 cos x− 1 νπ− π (x ν) 0 πx 2 4 Y − 0 −Γ(ν) x −ν 0.5 (0

− ν Iν (x) modified Bessel function of Modified Bessel Iν (x)=( i) Jν (ix) (2.418) the first kind π ν+1 functions Kν (x)= i [Jν (ix)+iYν (ix)] (2.419) Kν (x) modified Bessel function of 2 the second kind Spherical Bessel π 1/2 jν (x) spherical Bessel function jν (x)= Jν+ 1 (x) (2.420) of the first kind [similarly function 2 2x for yν (x)]

Legendre polynomialsa

2 P Legendre Legendre 2 d Pl(x) dPl(x) l (1−x ) −2x +l(l +1)Pl(x)=0 polynomials equation dx2 dx (2.421) l order (l ≥ 0) Rodrigues’ 1 dl P (x)= (x2 −1)l (2.422) formula l 2ll! dxl Recurrence (l +1)P (x)=(2l +1)xP (x)−lP − (x) (2.423) relation l+1 l l 1 1 2 Orthogonality Pl(x)Pl (x)dx = δll (2.424) δll Kronecker delta −1 2l +1 l/2 − −l m l 2l 2m l−2m l Explicit form P (x)=2 (−1) x (2.425) m binomial coeﬃcients l m l m=0 k wavenumber exp(ikz)=exp(ikrcosθ) (2.426) z propagation axis Expansion of ∞ z = rcosθ plane wave l j spherical Bessel = (2l +1)i jl(kr)Pl(cosθ) (2.427) l function of the ﬁrst l=0 kind (order l)

2 4 2 P0(x)=1 P2(x)=(3x −1)/2 P4(x)=(35x −30x +3)/8 3 5 3 P1(x)=xP3(x)=(5x −3x)/2 P5(x)=(63x −70x +15x)/8 aOf the ﬁrst kind. main January 23, 2006 16:6

48 Mathematics

Associated Legendre functionsa m 2 Associated d dP (x) m P m associated (1−x2) l + l(l +1)− P m(x)=0 l Legendre dx dx 1−x2 l Legendre equation (2.428) functions dm From m − 2 m/2 ≤ ≤ Pl (x)=(1 x ) m Pl(x), 0 m l (2.429) Legendre dx Pl Legendre (l −m)! polynomials polynomials P −m(x)=(−1)m P m(x) (2.430) l (l +m)! l

m m Pm+1(x)=x(2m+1)Pm (x) (2.431) Recurrence m m 2 m/2 P (x)=(−1) (2m−1)!!(1−x ) (2.432) !! 5!! = 5·3·1etc. relations m − m m − m (l m+1)Pl+1(x)=(2l +1)xPl (x) (l +m)Pl−1(x) (2.433) 1 δ Kronecker m m (l +m)! 2 ll Orthogonality Pl (x)Pl (x)dx = − δll (2.434) delta −1 (l m)! 2l +1

0 0 1 − − 2 1/2 P0 (x)=1 P1 (x)=xP1 (x)= (1 x ) 0 2 − 1 − − 2 1/2 2 − 2 P2 (x)=(3x 1)/2 P2 (x)= 3x(1 x ) P2 (x)=3(1 x ) a m − m Of the ﬁrst kind. Pl (x) can be deﬁned with a ( 1) factor in Equation (2.429) as well as Equation (2.430).

Legendre polynomials associated Legendre functions 3 1 P P 2 0 2 P P 1 2 2 P 0.5 3 P4 P 1 P 2 P 0 5 1 0 0 0 P 0 2 P 1 0 −0.5 1 P 1 − −1 1

−1 −0.5 0 0.5 1 −1 0 1 x x main January 23, 2006 16:6

2.9 Special functions and polynomials 49

Spherical harmonics 1 ∂ ∂ 1 ∂2 Differential m m Y m spherical sinθ + 2 Yl +l(l +1)Yl =0 l equation sinθ ∂θ ∂θ sin θ ∂φ2 harmonics (2.435) 2 1/2 2l +1(l −m)! P m associated a Y m(θ,φ)=(−1)m P m(cosθ)eimφ l Definition l 4π (l +m)! l Legendre (2.436) functions Y ∗ complex 2π π m∗ m conjugate Orthogonality Y (θ,φ)Y (θ,φ)sinθ dθ dφ = δ δ (2.437) l l mm ll δ Kronecker φ=0 θ=0 ll delta ∞ l m f(θ,φ)= almYl (θ,φ) (2.438) l=0 m=−l Laplace series f continuous 2π π function m∗ where alm = Yl (θ,φ)f(θ,φ)sinθ dθ dφ φ=0 θ=0 (2.439) ∇2 Solution to if ψ(r,θ,φ)=0, then ψ continuous Laplace ∞ l function m · l −(l+1) equation ψ(r,θ,φ)= Yl (θ,φ) almr +blmr (2.440) a,b constants − l=0 m= l 1 3 Y 0(θ,φ)= Y 0(θ,φ)= cosθ 0 4π 1 4π ± 3 5 3 1 Y 1(θ,φ)=∓ sinθe±iφ Y 0(θ,φ)= cos2 θ − 1 8π 2 4π 2 2 ± 15 ± 15 Y 1(θ,φ)=∓ sinθcosθe±iφ Y 2(θ,φ)= sin2 θe±2iφ 2 8π 2 32π 1 7 ± 1 21 Y 0(θ,φ)= (5cos2 θ −3)cosθY1(θ,φ)=∓ sinθ(5cos2 θ −1)e±iφ 3 2 4π 3 4 4π ± 1 105 ± 1 35 Y 2(θ,φ)= sin2 θcosθe±2iφ Y 3(θ,φ)=∓ sin3 θe±3iφ 3 4 2π 3 4 4π aDefined for −l ≤ m ≤ l, using the sign convention of the Condon–Shortley phase. Other sign conventions are possible. main January 23, 2006 16:6

50 Mathematics

Delta functions 1ifi = j δij = (2.441) δ Kronecker delta Kronecker delta 0ifi = j ij i,j,k,... indices (= 1,2 or 3) δii = 3 (2.442)

123 = 231 = 312 =1 − Three- 132 = 213 = 321 = 1 (2.443) dimensional all other ijk =0 Levi–Civita − ijk Levi–Civita symbol symbol ijkklm = δilδjm δimδjl (2.444) (see also page 25) (permutation δij ijk = 0 (2.445) a tensor) ilmjlm =2δij (2.446)

ijkijk = 6 (2.447) b 1ifa<0

2.10 Roots of quadratic and cubic equations Quadratic equations x variable Equation ax2 +bx +c =0 (a = 0) (2.454) a,b,c real constants √ −b± b2 −4ac x1,2 = (2.455) Solutions 2a x ,x quadratic roots −2c 1 2 = √ (2.456) b± b2 −4ac − Solution x1 +x2 = b/a (2.457) combinations x1x2 = c/a (2.458) main January 23, 2006 16:6

2.10 Roots of quadratic and cubic equations 51

Cubic equations x variable Equation ax3 +bx2 +cx +d =0 (a = 0) (2.459) a,b,c,d real constants 1 3c b2 p = − (2.460) 2 3 a a2 Intermediate 1 2b3 9bc 27d q = − + (2.461) D discriminant deﬁnitions 27 a3 a2 a p 3 q 2 D = + (2.462) 3 2

If D ≥ 0, also deﬁne: If D<0, also deﬁne: −q 1/3 − | | −3/2 u = +D1/2 (2.463) q p 2 φ = arccos (2.467) 2 3 −q 1/3 v = −D1/2 (2.464) | | 1/2 2 p φ y1 =2 cos (2.468) y = u+v (2.465) 3 3 1 − − 1/2 (u+v) ± u v 1/2 |p| φ±π y2,3 = i 3 (2.466) y = −2 cos (2.469) 2 2 2,3 3 3 1 real, 2 complex roots (if D = 0: 3 real roots, at least 2 equal) 3 distinct real roots

b Solutionsa x = y − (2.470) xn cubic roots n n 3a (n =1,2,3)

x1 +x2 +x3 = −b/a (2.471) Solution x x +x x +x x = c/a (2.472) combinations 1 2 1 3 2 3 x1x2x3 = −d/a (2.473) a 3 yn are solutions to the reduced equation y +py +q =0. main January 23, 2006 16:6

52 Mathematics

2.11 Fourier series and transforms

Fourier series ∞ a nπx nπx f(x)= 0 + a cos +b sin (2.474) 2 n L n L n=1 f(x) periodic function, L Real form 1 nπx period 2L an = f(x)cos dx (2.475) L − L a ,b Fourier L n n 1 L nπx coeﬃcients bn = f(x)sin dx (2.476) L −L L ∞ inπx f(x)= cn exp (2.477) Complex L cn complex n=−∞ Fourier form 1 L −inπx coeﬃcient cn = f(x)exp dx (2.478) 2L −L L ∞ L 2 1 | |2 a0 1 2 2 f(x) dx = + an +bn (2.479) Parseval’s 2L −L 4 2 n=1 || modulus theorem ∞ 2 = |cn| (2.480) n=−∞

Fourier transforma ∞ F(s)= f(x)e−2πixs dx (2.481) −∞ f(x) function of x Definition 1 ∞ F(s) Fourier transform of f(x) f(x)= F(s)e2πixs ds (2.482) −∞ ∞ F(s)= f(x)e−ixs dx (2.483) Definition 2 −∞ 1 ∞ f(x)= F(s)eixs ds (2.484) 2π −∞ 1 ∞ F(s)=√ f(x)e−ixs dx (2.485) Definition 3 2π −∞ 1 ∞ f(x)=√ F(s)eixs ds (2.486) 2π −∞ aAll three (and more) definitions are used, but definition 1 is probably the best. main January 23, 2006 16:6

2.11 Fourier series and transforms 53

Fourier transform theoremsa ∞ ∗ − f,g general functions Convolution f(x) g(x)= f(u)g(x u)du (2.487) ∗ −∞ convolution ∗ ∗ 2 Convolution f g = g f (2.488) ff(x) F(s) rules f ∗(g ∗h)=(f ∗g)∗h (2.489) gg(x) G(s)

Convolution f(x)g(x) F(s)∗G(s) (2.490) Fourier transform theorem relation Autocorrela- ∞ correlation f∗(x) f(x)= f∗(u−x)f(u)du (2.491) ∗ tion f complex −∞ conjugate of f Wiener– ∗ Khintchine f (x) f(x) |F(s)|2 (2.492) theorem Cross- ∞ f∗(x) g(x)= f∗(u−x)g(u)du (2.493) correlation −∞ h,j real functions Correlation ∗ h(x) j(x) H(s)J (s) (2.494) HH(s) h(x) theorem JJ(s) j(x) ∞ ∞ Parseval’s ∗ ∗ b f(x)g (x)dx = F(s)G (s)ds (2.495) relation −∞ −∞ ∞ ∞ Parseval’s 2 2 c |f(x)| dx = |F(s)| ds (2.496) theorem −∞ −∞ df(x) 2πisF(s) (2.497) dx Derivatives d df(x) dg(x) [f(x)∗g(x)] = ∗g(x)= ∗f(x) dx dx dx (2.498) a ∞ −2πixs Deﬁning the Fourier transform as F(s)= −∞ f(x)e dx. bAlso called the “power theorem.” cAlso called “Rayleigh’s theorem.”

Fourier symmetry relationships f(x) F(s) deﬁnitions ∗ even even real: f(x)=f (x) ∗ odd odd imaginary: f(x)=−f (x) real, even real, even even: f(x)=f(−x) real, odd imaginary, odd odd: f(x)=−f(−x) ∗ imaginary, even imaginary, even Hermitian: f(x)=f (−x) ∗ complex, even complex, even anti-Hermitian: f(x)=−f (−x) complex, odd complex, odd real, asymmetric complex, Hermitian imaginary, asymmetric complex, anti-Hermitian main January 23, 2006 16:6

54 Mathematics

Fourier transform pairsa

∞ − f(x) F(s)= f(x)e 2πisx dx (2.499) −∞ 1 f(ax) F(s/a)(a = 0, real) (2.500) |a| − f(x−a) e 2πiasF(s)(a real) (2.501) dn f(x) (2πis)nF(s) (2.502) dxn δ(x) 1 (2.503) − δ(x−a) e 2πias (2.504)

− | | 2a e a x (a>0) (2.505) a2 +4π2s2

− | | 8iπas xe a x (a>0) (2.506) (a2 +4π2s2)2 √ − 2 2 − 2 2 2 e x /a a πe π a s (2.507) 1 a a sinax δ s− −δ s+ (2.508) 2i 2π 2π 1 a a cosax δ s− +δ s+ (2.509) 2 2π 2π ∞ ∞ 1 n δ(x−ma) δ s− (2.510) −∞ a −∞ a m= n= 0 x<0 1 i f(x)= (“step”) δ(s)− (2.511) 1 x>0 2 2πs 1 |x|≤a sin2πas f(x)= (“top hat”) =2asinc2as (2.512) 0 |x| >a πs   |x| 1− |x|≤a 1 f(x)= a (“triangle”) (1−cos2πas)=asinc2 as (2.513)  2π2as2 0 |x| >a aEquation (2.499) deﬁnes the Fourier transform used for these pairs. Note that sincx ≡ (sinπx)/(πx). main January 23, 2006 16:6

2.12 Laplace transforms 55

2.12 Laplace transforms

Laplace transform theorems ∞ L{} Deﬁnitiona L{ } −st Laplace 2 F(s)= f(t) = f(t)e dt (2.514) transform 0 ∞ F(s)·G(s)=L f(t−z)g(z)dz (2.515) F(s) L{f(t)} Convolutionb 0 G(s) L{g(t)} = L{f(t)∗g(t)} (2.516) ∗ convolution 1 γ+i∞ f(t)= estF(s)ds (2.517) c 2πi − ∞ Inverse γ i γ constant = residues (for t>0) (2.518)

−1 dnf(t) n drf(t) Transform of L = snL{f(t)}− sn−r−1 dtn dtr t=0 n integer > 0 derivative r=0 (2.519) Derivative of dnF(s) = L{(−t)nf(t)} (2.520) transform dsn

at Substitution F(s−a)=L{e f(t)} (2.521) a constant

e−asF(s)=L{u(t−a)f(t−a)} (2.522) Translation u(t) unit step 0(t<0) function where u(t)= (2.523) 1(t>0) a −s t If |e 0 f(t)| is ﬁnite for suﬃciently large t, the Laplace transform exists for s>s0. bAlso known as the “faltung (or folding) theorem.” cAlso known as the “Bromwich integral.” γ is chosen so that the singularities in F(s) are left of the integral line. main January 23, 2006 16:6

56 Mathematics

Laplace transform pairs ∞ f(t)=⇒ F(s)=L{f(t)} = f(t)e−st dt (2.524) 0 δ(t)=⇒ 1 (2.525) 1=⇒ 1/s (s>0) (2.526)

n ⇒ n! − t = n+1 (s>0,n> 1) (2.527) s π t1/2 =⇒ (2.528) 4s3 π t−1/2 =⇒ (2.529) s 1 eat =⇒ (s>a) (2.530) s−a 1 teat =⇒ (s>a) (2.531) (s−a)2 s (1−at)e−at =⇒ (2.532) (s+a)2 2 t2e−at =⇒ (2.533) (s+a)3 a sinat =⇒ (s>0) (2.534) s2 +a2 s cosat =⇒ (s>0) (2.535) s2 +a2 a sinhat =⇒ (s>a) (2.536) s2 −a2 s coshat =⇒ (s>a) (2.537) s2 −a2 a e−bt sinat =⇒ (2.538) (s+b)2 +a2 s+b e−bt cosat =⇒ (2.539) (s+b)2 +a2 e−atf(t)=⇒ F(s+a) (2.540) main January 23, 2006 16:6

2.13 Probability and statistics 57

2.13 Probability and statistics

Discrete statistics 2 x data series 1 N i Mean x = x (2.541) N series length N i i=1 · mean value N 1 var[·] unbiased Variancea var[x]= (x −x )2 (2.542) N −1 i variance i=1 Standard σ[x] = (var[x])1/2 (2.543) σ standard deviation deviation N N x −x 3 Skewness skew[x]= i (2.544) (N −1)(N −2) σ i=1 1 N x −x 4 Kurtosis kurt[x] i −3 (2.545) N σ i=1 N − − x,y data series to Correlation i=1(xi x)(yi y ) correlate b r = (2.546) coeﬃcient N (x −x )2 N (y −y )2 r correlation i=1 i i=1 i coeﬃcient a If x is derived from the data, {xi}, the relation is as shown. If x is known independently, then an unbiased estimate is obtained by dividing the right-hand side by N rather than N −1. bAlso known as “Pearson’s r.”

Discrete probability distributions

distribution pr(x) mean variance domain n binomial n x − n−x − x Binomial x p (1 p) np np(1 p)(x =0,1,...,n)(2.547) coeﬃcient

Geometric (1−p)x−1p 1/p (1−p)/p2 (x =1,2,3,...)(2.548)

Poisson λx exp(−λ)/x! λλ (x =0,1,2,...)(2.549) main January 23, 2006 16:6

58 Mathematics

Continuous probability distributions

distribution pr(x) mean variance domain

1 a+b (b−a)2 Uniform (a ≤ x ≤ b)(2.550) b−a 2 12 Exponential λexp(−λx)1/λ 1/λ2 (x ≥ 0) (2.551) 1 −(x−µ)2 Normal/ √ exp µσ2 (−∞

Multivariate normal distribution pr probability density − exp − 1 (x−µ)C 1(x−µ)T k number of dimensions pr(x)= 2 Density function (2π)k/2[det(C)]1/2 C covariance matrix (2.556) x variable (k dimensional) µ vector of means T transpose Mean µ =(µ1,µ2,...,µk) (2.557) det determinant

µi mean of ith variable − Covariance C = σij = xixj xi xj (2.558) σij components of C

Correlation σij r = (2.559) r correlation coeﬃcient coeﬃcient σiσj − 1/2 Box–Muller x1 =( 2lny1) cos2πy2 (2.560) xi normally distributed deviates transformation − 1/2 yi deviates distributed x2 =( 2lny1) sin2πy2 (2.561) uniformly between 0 and 1 main January 23, 2006 16:6

2.13 Probability and statistics 59

Random walk x displacement after N steps (can be positive or negative) 1 −x2 One- pr( )= exp pr(x) probability density of x x 2 1/2 2 ∞ dimensional (2πNl ) 2Nl ( −∞ pr(x)dx =1) 2 (2.562) N number of steps l step length (all equal) rms x = N1/2l (2.563) xrms root-mean-squared displacement rms displacement from start point 3 a 2 2 r radial distance from start pr(r)= exp(−a r ) (2.564) point Three- π1/2 pr(r) probability density of r 1/2 ∞ dimensional 3 ( 4πr2 pr(r)dr =1) where a = 0 2Nl2 a (most probable distance)−1 1/2 8 r mean distance from start Mean distance r = N1/2l (2.565) 3π point

1/2 rrms root-mean-squared distance rms distance rrms = N l (2.566) from start point

Bayesian inference pr(x) probability (density) of x Conditional pr(x)= pr(x|y )pr(y )dy (2.567) pr(x|y) conditional probability of x probability given y Joint pr(x,y) = pr(x)pr(y|x) (2.568) pr(x,y) joint probability of x and y probability pr(x|y)pr(y) Bayes’ theorema pr(y|x)= (2.569) pr(x) aIn this expression, pr(y|x) is known as the posterior probability, pr(x|y) the likelihood, and pr(y) the prior probability. main January 23, 2006 16:6

60 Mathematics

2.14 Numerical methods

Straight-line ﬁttinga Data {xi},{yi} n points (2.570) y y = mx +c b Weights {wi} (2.571) (0,c) Model y = mx +c (2.572) (x,y) Residuals di = yi −mxi −c (2.573) x 1 Weighted (x,y)= wixi , wiyi centre wi (2.574) Weighted D = w (x −x)2 (2.575) moment i i 1 m = w (x −x)y (2.576) D i i i Gradient 1 w d2 var[m] i i (2.577) D n−2

c = y −mx (2.578) Intercept 1 x2 w d2 var[c] + i i (2.579) wi D n−2 aLeast-squares ﬁt of data to y = mx +c.Errorsony-values only. b If the errors on yi are uncorrelated, then wi =1/var[yi].

Time series analysisa

M/2 r response function Discrete i s time series convolution (r s)j = sj−krk (2.580) i k=−(M/2)+1 M response function duration Bartlett j −N/2 wj windowing function (triangular) w =1− (2.581) j N length of time series window N/2 Welch 1 Welch Hamming j −N/2 2 (quadratic) w =1− (2.582) 0.8 j N/2 window w 0.6 Bartlett Hanning 1 2πj 0.4 w = 1−cos (2.583) Hanning window j 2 N 0.2 0 Hamming 2πj 0 0.2 0.4 0.6 0.8 1 wj =0.54−0.46cos (2.584) window N j/N aThe time series runs from j =0...(N −1), and the windowing functions peak at j = N/2. main January 23, 2006 16:6

2.14 Numerical methods 61

Numerical integration

h f(x) 2

x x0 xN

h =(xN −x0)/N xN h (subinterval f(x)dx (f +2f +2f +··· 2 0 1 2 width) Trapezoidal rule x0 fi fi = f(xi) +2f −1 +f ) (2.585) N N N number of subintervals xN h f(x)dx (f0 +4f1 +2f2 +4f3 +··· a 3 Simpson’s rule x0 +4fN−1 +fN ) (2.586) aN must be even. Simpson’s rule is exact for quadratics and cubics.

Numerical diﬀerentiationa df 1 [−f(x+2h)+8f(x+h)−8f(x−h)+f(x−2h)] (2.587) dx 12h 1 ∼ [f(x+h)−f(x−h)] (2.588) 2h d2f 1 [−f(x+2h)+16f(x+h)−30f(x)+16f(x−h)−f(x−2h)] (2.589) dx2 12h2 1 ∼ [f(x+h)−2f(x)+f(x−h)] (2.590) h2 d3f 1 ∼ [f(x+2h)−2f(x+h)+2f(x−h)−f(x−2h)] (2.591) dx3 2h3 aDerivatives of f(x)atx. h is a small interval in x. Relations containing “”areO(h4); those containing “∼”areO(h2).

Numerical solutions to f(x)=0

xn −xn−1 f function of x Secant method xn+1 = xn − f(xn) (2.592) f(xn)−f(xn−1) xn f(x∞)=0 ( ) Newton–Raphson − f xn xn+1 = xn (2.593) f =df/dx method f (xn) main January 23, 2006 16:6

62 Mathematics

Numerical solutions to ordinary diﬀerential equationsa dy if = f(x,y) (2.594) dx Euler’s method and h = xn+1 −xn (2.595) 2 then yn+1 = yn +hf(xn,yn)+O(h ) (2.596)

dy if = f(x,y) (2.597) dx and h = xn+1 −xn (2.598) Runge–Kutta k1 = hf(xn,yn) (2.599) method k2 = hf(xn +h/2,yn +k1/2) (2.600)

(fourth-order) k3 = hf(xn +h/2,yn +k2/2) (2.601)

k4 = hf(xn +h,yn +k3) (2.602) k k k k then y = y + 1 + 2 + 3 + 4 +O(h5) (2.603) n+1 n 6 3 3 6 a dy Ordinary diﬀerential equations (ODEs) of the form dx = f(x,y). Higher order equations should be reduced to a set of coupled ﬁrst-order equations and solved in parallel. main January 23, 2006 16:6

Chapter 3 Dynamics and mechanics

3 3.1 Introduction Unusually in physics, there is no pithy phrase that sums up the study of dynamics (the way in which forces produce motion), kinematics (the motion of matter), mechanics (the study of the forces and the motion they produce), and statics (the way forces combine to produce equilibrium). We will take the phrase dynamics and mechanics to encompass all the above, although it clearly does not! To some extent this is because the equations governing the motion of matter include some of our oldest insights into the physical world and are consequentially steeped in tradition. One of the more delightful, or for some annoying, facets of this is the occasional use of arcane vocabulary in the description of motion. The epitome must be what Goldstein1 calls “the jabberwockian sounding statement” the polhode rolls without slipping on the herpolhode lying in the invariable plane, describing “Poinsot’s construction” – a method of visualising the free motion of a spinning rigid body. Despite this, dynamics and mechanics, including ﬂuid mechanics, is arguably the most practically applicable of all the branches of physics. Moreover, and in common with electromagnetism, the study of dynamics and mechanics has spawned a good deal of mathematical apparatus that has found uses in other ﬁelds. Most notably, the ideas behind the generalised dynamics of Lagrange and Hamilton lie behind much of quantum mechanics.

1H. Goldstein, Classical Mechanics, 2nd ed., 1980, Addison-Wesley. main January 23, 2006 16:6

64 Dynamics and mechanics

3.2 Frames of reference Galilean transformations S r,r position in frames S m Time and r = r +vt (3.1) and S S a r position t = t (3.2) v velocity of S in S r t,t time in S and S vt u,u velocity in frames S Velocity u = u +v (3.3) and S p,p particle momentum Momentum p = p +mv (3.4) in frames S and S m particle mass Angular J = J +mr ×v +v×p t (3.5) J ,J angular momentum momentum in frames S and S

Kinetic 1 2 T,T kinetic energy in T = T +mu ·v + mv (3.6) energy 2 frames S and S aFrames coincide at t =0.

Lorentz (spacetime) transformationsa − γ Lorentz factor 2 1/2 v Lorentz factor − v velocity of S in S γ = 1 2 (3.7) c c speed of light Time and position − S S x = γ(x +vt ); x = γ(x vt) (3.8) x,x x-position in frames y = y ; y = y (3.9) S and S (similarly v for y and z) z = z ; z = z (3.10) x t,t time in frames S and x v v t = γ t + x ; t = γ t− x (3.11) S c2 c2 Diﬀerential dX =(cdt,−dx,−dy,−dz) b X spacetime four-vector four-vector (3.12) aFor frames S and S coincident at t = 0 in relative motion along x. See page 141 for the transformations of electromagnetic quantities. bCovariant components, using the (1,−1,−1,−1) signature.

Velocity transformationsa γ Lorentz factor Velocity =[1−(v/c)2]−1/2 u +v u −v u = x ; u = x (3.13) x 2 x − 2 1+uxv/c 1 uxv/c v velocity of S in S S uy uy c speed of light S uy = ; u = (3.14) u 2 y − 2 ui,u particle velocity γ(1+uxv/c ) γ(1 uxv/c ) i v components in u u u = z ; u = z (3.15) frames S and S x z 2 z − 2 x γ(1+uxv/c ) γ(1 uxv/c ) aFor frames S and S coincident at t = 0 in relative motion along x. main January 23, 2006 16:6

3.2 Frames of reference 65

Momentum and energy transformationsa γ Lorentz factor Momentum and energy =[1−(v/c)2]−1/2

2 − 2 v velocity of S in S px = γ(px +vE /c ); px = γ(px vE/c ) (3.16) p = p ; p = p (3.17) c speed of light y y y y S S px,px x components of pz = pz; pz = pz (3.18) momentum in S and S (sim. for y and z) v E = γ(E +vp ); E = γ(E −vpx) (3.19) x E,E energy in S and S x x 2 2 2 − 2 2 − 2 2 4 m0 (rest) mass 3 E p c = E p c = m0c (3.20) p total momentum in S

b P momentum Four-vector P =(E/c,−px,−py,−pz) (3.21) four-vector aFor frames S and S coincident at t = 0 in relative motion along x. bCovariant components, using the (1,−1,−1,−1) signature.

Propagation of lighta ν frequency received in S Doppler ν v S c = γ 1+ cosα (3.22) ν frequency emitted in S y eﬀect ν c α arrival angle in S α γ Lorentz factor x cosθ +v/c =[1−(v/c)2]−1/2 cosθ = (3.23) 1+(v/c)cosθ v velocity of S in S S S Aberrationb y y cosθ −v/c c speed of light v cosθ = − (3.24) θ,θ emission angle of light c 1 (v/c)cosθ θ in S and S x x Relativistic sinθ P (θ)= (3.25) P (θ) angular distribution of beamingc 2γ2[1−(v/c)cosθ]2 photons in S aFor frames S and S coincident at t = 0 in relative motion along x. bLight travelling in the opposite sense has a propagation angle of π +θ radians. c π Angular distribution of photons from a source, isotropic and stationary in S . 0 P (θ)dθ =1. Four-vectorsa Covariant and 0 1 x0 = x x1 = −x xi covariant vector contravariant components − 2 − 3 components x2 = x x3 = x (3.26) xi contravariant components

i 0 1 2 3 Scalar product x yi = x y0 +x y1 +x y2 +x y3 (3.27)

xi,xi four-vector components in Lorentz transformations frames S and S 0 0 1 0 0 − 1 x = γ[x +(v/c)x ]; x = γ[x (v/c)x ] (3.28) γ Lorentz factor − 2 −1/2 x1 = γ[x 1 +(v/c)x 0]; x 1 = γ[x1 −(v/c)x0] (3.29) =[1 (v/c) ] v velocity of S in S 2 2 3 3 x = x ; x = x (3.30) c speed of light aFor frames S and S , coincident at t = 0 in relative motion along the (1) direction. Note that the (1,−1,−1,−1) signature used here is common in special relativity, whereas (−1,1,1,1) is often used in connection with general relativity (page 67). main January 23, 2006 16:6

66 Dynamics and mechanics

Rotating frames A any vector Vector trans- dA dA S stationary frame = +ω×A (3.31) S rotating frame formation dt dt S S ω angular velocity of S in S ˙v,˙v accelerations in S and S × × × Acceleration ˙v =˙v +2ω×v +ω×(ω×r ) (3.32) v velocity in S r position in S − × F cor coriolis force Coriolis force F cor = 2mω v (3.33) ω m particle mass F cen F centrifugal force r − × × cen ⊥ m F cen = mω (ω r ) (3.34) Centrifugal r⊥ perpendicular to 2 force =+mω r⊥ (3.35) particle from r rotation axis F nongravitational ωe − i z mx¨ = Fx +2mωe(y˙sinλ ˙zcosλ) force y Motion (3.36) λ latitude relative to x my¨ = F −2mω x˙sinλ (3.37) z local vertical axis Earth y e y northerly axis λ m¨z = Fz −mg +2mωex˙cosλ (3.38) x easterly axis Foucault’s Ωf pendulum’s rate Ω = −ω sinλ (3.39) of turn penduluma f e ωe Earth’s spin rate aThe sign is such as to make the rotation clockwise in the northern hemisphere.

3.3 Gravitation

Newtonian gravitation

m1,2 masses Newton’s law of Gm1m2 F 1 force on m1 (= −F 2) F 1 = rˆ12 (3.40) gravitation 2 r vector from m to m r12 12 1 2 ˆ unit vector G constant of gravitation −∇ Newtonian ﬁeld g = φ (3.41) g gravitational ﬁeld strength equationsa ∇2φ = −∇·g =4πGρ (3.42) φ gravitational potential ρ mass density   GM r vector from sphere centre − rˆ (r>a) Fields from an r2 M mass of sphere g(r)= (3.43) isolated −GMr a radius of sphere 3 rˆ (ra) M r the centre φ(r)= r (3.44) GM a  (r2 −3a2)(r

3.3 Gravitation 67

General relativitya ds invariant interval 2 µ ν 2 Line element ds = gµν dx dx = −dτ (3.45) dτ proper time interval

gµν metric tensor µ µ 1 dx differential of x α αδ − α Γ βγ = g (gδβ,γ +gδγ,β gβγ,δ) (3.46) Γ βγ Christoffel symbols Christoffel 2 α γ ,α partial diff. w.r.t. x symbols and φ;γ = φ,γ ≡ ∂φ/∂x (3.47) ;α covariant diff. w.r.t. xα covariant α α α β A;γ = A,γ +Γ βγA (3.48) φ scalar differentiation − β Aα contravariant vector 3 Bα;γ = Bα,γ Γ αγBβ (3.49) Bα covariant vector α α µ − α µ R βγδ =Γ µγΓ βδ Γ µδΓ βγ α − α +Γ βδ,γ Γ βγ,δ (3.50) α − γ R βγδ Riemann tensor Riemann tensor Bµ;α;β Bµ;β;α = R µαβBγ (3.51)

Rαβγδ = −Rαβδγ ; Rβαγδ = −Rαβγδ (3.52)

Rαβγδ +Rαδβγ +Rαγδβ = 0 (3.53)

µ Dv µ = 0 (3.54) v tangent vector Geodesic Dλ (= dxµ/dλ) equation DAµ dAµ λ aﬃne parameter (e.g., τ where ≡ +Γµ Aαvβ (3.55) Dλ dλ αβ for material particles) Geodesic D2ξµ = −Rµ vαξβvγ (3.56) ξµ geodesic deviation deviation Dλ2 αβγ ≡ σ σδ Ricci tensor Rαβ R ασβ = g Rδασβ = Rβα (3.57) Rαβ Ricci tensor

µν µν µν − 1 µν G Einstein tensor Einstein tensor G = R g R (3.58) µν 2 R Ricci scalar (= g Rµν ) Einstein’s field T µν stress-energy tensor Gµν =8πT µν (3.59) equations p pressure (in rest frame) ρ density (in rest frame) Perfect fluid T µν =(p+ρ)uµuν +pgµν (3.60) uν fluid four-velocity −1 Schwarzschild 2 2M 2 2M 2 M spherically symmetric ds =− 1− dt + 1− dr mass (see page 183) solution r r (r,θ,φ) spherical polar coords. (exterior) 2 2 2 2 +r (dθ +sin θ dφ ) (3.61) t time

Kerr solution (outside a spinning black hole) ∆−a2 sin2 θ 2Mrsin2 θ J angular momentum ds2 = − dt2 −2a dt dφ 2 2 (along z) a ≡ J/M 2 2 2 − 2 2 2 (r +a ) a ∆sin θ 2 2 2 2 2 ∆ ≡ r2 −2Mr+a2 + 2 sin θdφ + dr + dθ (3.62) ∆ 2 ≡ r2 +a2 cos2 θ aGeneral relativity conventionally uses the (−1,1,1,1) metric signature and “geometrized units” in which G = 1 and c = 1. Thus, 1kg = 7.425×10−28 m etc. Contravariant indices are written as superscripts and covariant indices as subscripts. Note also that ds2 means (ds)2 etc. main January 23, 2006 16:6

68 Dynamics and mechanics

3.4 Particle motion

Dynamics deﬁnitionsa F force Newtonian force F = m¨r = p˙ (3.63) m mass of particle r particle position vector

Momentum p = m˙r (3.64) p momentum

1 T kinetic energy Kinetic energy T = mv2 (3.65) 2 v particle velocity

Angular momentum J = r×p (3.66) J angular momentum

Couple (or torque) G = r×F (3.67) G couple Centre of mass N R0 position vector of centre of mass i=1 miri (ensemble of N mi mass of ith particle R0 = N (3.68) particles) i=1 mi ri position vector of ith particle aIn the Newtonian limit, v c, assuming m is constant.

Relativistic dynamicsa − γ Lorentz factor v2 1/2 Lorentz factor − v particle velocity γ = 1 2 (3.69) c c speed of light p relativistic momentum Momentum p = γm0v (3.70) m0 particle (rest) mass

dp F force on particle Force F = (3.71) dt t time

2 Rest energy Er = m0c (3.72) Er particle rest energy

2 Kinetic energy T = m0c (γ −1) (3.73) T relativistic kinetic energy

2 E = γm0c (3.74) Total energy 2 2 2 4 1/2 E total energy (= Er +T ) =(p c +m0c ) (3.75) aIt is now common to regard mass as a Lorentz invariant property and to drop the term “rest mass.” The symbol m0 is used here to avoid confusion with the idea of “relativistic mass” (= γm0) used by some authors.

Constant acceleration v = u+at (3.76) u initial velocity 2 2 v = u +2as (3.77) v ﬁnal velocity 1 s = ut+ at2 (3.78) t time 2 s distance travelled u+v s = t (3.79) a acceleration 2 main January 23, 2006 16:6

3.4 Particle motion 69

Reduced mass (of two interacting bodies)

m2 centre m1 of mass r2 r1

m1m2 µ reduced mass Reduced mass µ = (3.80) m1 +m2 mi interacting masses 3

m2 r position vectors from centre of r1 = r (3.81) i Distances from m1 +m2 mass − centre of mass m1 rr= r1 −r2 r2 = r (3.82) | | m1 +m2 r distance between masses

Moment of 2 I = µ|r| (3.83) I moment of inertia inertia Total angular J = µr×˙r (3.84) J angular momentum momentum

1 L Lagrangian Lagrangian L = µ|˙r|2 −U(|r|) (3.85) 2 U potential energy of interaction

Ballisticsa v0 initial velocity yˆ v = v cosαxˆ +(v sinα−gt)yˆ 0 0 v velocity at t v (3.86) 0 Velocity α elevation angle 2 2 − α h xˆ v = v0 2gy (3.87) g gravitational acceleration l gx2 ˆ unit vector Trajectory y = xtanα− (3.88) 2 2 t time 2v0 cos α 2 Maximum v h maximum h = 0 sin2 α (3.89) height 2g height Horizontal v2 l = 0 sin2α (3.90) l range range g aIgnoring the curvature and rotation of the Earth and frictional losses. g is assumed constant. main January 23, 2006 16:6

70 Dynamics and mechanics

Rocketry vesc escape velocity Escape 2GM 1/2 G constant of gravitation a v = (3.91) velocity esc r M mass of central body r central body radius I specific impulse Specific u sp I = (3.92) u effective exhaust velocity impulse sp g g acceleration due to gravity R molar gas constant Exhaust 1/2 γ ratio of heat capacities 2γRTc velocity (into u = (3.93) Tc combustion temperature ( −1) a vacuum) γ µ µ effective molecular mass of exhaust gas ∆v rocket velocity increment Rocket M M pre-burn rocket mass equation ∆v = uln i ≡ ulnM (3.94) i M post-burn rocket mass (g =0) Mf f M mass ratio N number of stages Multistage N M M mass ratio for ith burn rocket ∆v = ui ln i (3.95) i i=1 ui exhaust velocity of ith burn In a constant t burn time gravitational ∆v = ulnM−gtcosθ (3.96) θ rocket zenith angle field ∆v velocity increment, a to h 1/2 1/2 ah GM 2rb ∆vhb velocity increment, h to b ∆vah = −1 r r +r r radius of inner orbit Hohmann a a b a (3.97) rb radius of outer orbit cotangential b 1/2 1/2 transfer ellipse, h transfer GM 2ra ∆vhb = 1− ab rb ra +rb (3.98) aFrom the surface of a spherically symmetric, nonrotating body, mass M. bTransfer between coplanar, circular orbits a and b, via ellipse h with a minimal expenditure of energy. main January 23, 2006 16:6

3.4 Particle motion 71

Gravitationally bound orbital motiona U(r) potential energy G constant of gravitation Potential energy GMm α U(r)=− ≡− (3.99) M central mass of interaction r r m orbiting mass ( M) αGMm(for gravitation) α J2 α E total energy (constant) Total energy − − total angular momentum E = + 2 = (3.100) J r 2mr 2a (constant) − 3 Virial theorem E = U /2= T (3.101) T kinetic energy (1/r potential) U = −2T (3.102) · mean value

r Orbital 0 =1+ecosφ, or (3.103) r0 semi-latus-rectum equation r r distance of m from M 2 (Kepler’s 1st a(1−e ) e eccentricity r = (3.104) law) 1+ecosφ φ phase (true anomaly) Rate of sweeping area dA J = = constant (3.105) A area swept out by radius (Kepler’s 2nd dt 2m vector (total area = πab) law) r0 α a semi-major axis Semi-major axis a = = (3.106) 1−e2 2|E| b semi-minor axis r J 2a Semi-minor axis b = 0 = (3.107) (1−e2)1/2 (2m|E|)1/2 m A r 2 1/2 2 1/2 r0 b 2EJ b Eccentricity e = 1+ = 1− (3.108) φ mα2 a2 M 2 2 ae Semi-latus- J b 2 2b r0 = = = a(1−e ) (3.109) rectum mα a rmax rmin

r0 Pericentre rmin = = a(1−e) (3.110) r pericentre distance 1+e min

r0 Apocentre rmax = = a(1+e) (3.111) r apocentre distance 1−e max 2 1 Speed v2 = GM − (3.112) v orbital speed r a 1/2 1/2 Period (Kepler’s m 3/2 m P = πα =2πa P orbital period 3rd law) 2|E|3 α (3.113) aFor an inverse-square law of attraction between two isolated bodies in the nonrelativistic limit. If m is not M, then the equations are valid with the substitutions m → µ = Mm/(M +m) and M → (M +m) and with r taken as the body separation. The distance of mass m from the centre of mass is then rµ/m (see earlier table on Reduced mass). Other orbital dimensions scale similarly, and the two orbits have the same eccentricity. bNote that if the total energy, E,is< 0thene<1 and the orbit is an ellipse (a circle if e = 0). If E =0,then e =1 and the orbit is a parabola. If E>0thene>1 and the orbit becomes a hyperbola (see Rutherford scattering on next page). main January 23, 2006 16:6

72 Dynamics and mechanics

Rutherford scatteringa

y trajectory b for α<0

scattering x centre χ a a

rmin (α<0) trajectory for 0 α> rmin (α>0) −α U(r)= (3.114) U(r) potential energy Scattering potential r r particle separation energy < 0 repulsive α (3.115) α constant > 0 attractive χ scattering angle χ |α| Scattering angle tan = (3.116) E total energy (> 0) 2 2Eb b impact parameter |α| χ α − r closest approach rmin = csc | | (3.117) min Closest approach 2E 2 α a hyperbola semi-axis = a(e±1) (3.118) e eccentricity

|α| Semi-axis a = (3.119) 2E 4E2b2 1/2 χ Eccentricity e = +1 =csc (3.120) α2 2

2 2 4E y x,y position with respect to Motion trajectoryb x2 − = 1 (3.121) α2 b2 hyperbola centre 2 1/2 c α Scattering centre x = ± +b2 (3.122) 4E2 dσ dΩ differential scattering dσ 1 dN cross section = (3.123) Rutherford n beam flux density dΩ n dΩ scattering formulad α 2 χ dN number of particles = csc4 (3.124) 4E 2 scattered into dΩ Ω solid angle aNonrelativistic treatment for an inverse-square force law and a fixed scattering centre. Similar scattering results from either an attractive or repulsive force. See also Conic sections on page 38. bThe correct branch can be chosen by inspection. cAlso the focal points of the hyperbola. dn is the number of particles per second passing through unit area perpendicular to the beam. main January 23, 2006 16:6

3.4 Particle motion 73

Inelastic collisionsa

m2 m2 m1 v1 v2 m1 v1 v2

Before collision After collision

− − v2 v1 = (v1 v2) (3.125) coeﬃcient of restitution Coeﬃcient of = 1 if perfectly elastic (3.126) vi pre-collision velocities restitution 3 = 0 if perfectly inelastic (3.127) vi post-collision velocities

T,T total KE in zero Loss of kinetic T −T =1−2 (3.128) momentum frame energyb T before and after collision

m1 −m2 (1+)m2 v1 = v1 + v2 (3.129) m1 +m2 m1 +m2 Final velocities mi particle masses m2 −m1 (1+)m1 v2 = v2 + v1 (3.130) m1 +m2 m1 +m2 a Along the line of centres, v1,v2 c. bIn zero momentum frame.

Oblique elastic collisionsa v2 θ2 m2 m2 Before collision θ After collision m1 v m1 θ1 v1

m sin2θ θ angle between 2 centre line and Directions of tanθ1 = (3.131) m1 −m2 cos2θ incident velocity motion θ2 = θ (3.132) θi ﬁnal trajectories m sphere masses  i >π/2ifm1 m2 2 2 − 1/2 (m1 +m2 2m1m2 cos2θ) v1 = v (3.134) v incident velocity m1 +m2 Final velocities of m1 2m1v v ﬁnal velocities v2 = cosθ (3.135) i m1 +m2 a Collision between two perfectly elastic spheres: m2 initially at rest, velocities c. main January 23, 2006 16:6

74 Dynamics and mechanics

3.5 Rigid body dynamics

Moment of inertia tensor 2 2 2 2 Moment of 2 − rr= x +y +z a Iij = (r δij xixj )dm (3.136) inertia tensor δij Kronecker delta   I moment of inertia (y2 +z2)dm − xy dm − xz dm   tensor  − 2 2 −  I = xy dm (x+z )dm yz dm dm mass element − xz dm − yz dm (x2 +y2)dm xi position vector of (3.137) dm Iij components of I tensor with respect I = I −ma a (3.138) Iij 12 12 1 2 to centre of mass Parallel axis 2 2 I = I +m(a +a ) (3.139) a ,a position vector of theorem 11 11 2 3 i 2 centre of mass Iij = I +m(|a| δij −aiaj ) (3.140) ij m mass of body

Angular J angular momentum J = Iω (3.141) momentum ω angular velocity Rotational 1 1 T = ω ·J = I ω ω (3.142) T kinetic energy kinetic energy 2 2 ij i j a Iii are the moments of inertia of the body. Iij (i = j) are its products of inertia. The integrals are over the body volume.

Principal axes   I principal moment of Principal I1 00   inertia tensor moment of I = 0 I2 0 (3.143) Ii principal moments of inertia tensor 00I3 inertia Angular J angular momentum J =(I ω ,I ω ,I ω ) (3.144) momentum 1 1 2 2 3 3 ωi components of ω along principal axes Rotational 1 T = (I ω2 +I ω2 +I ω2) (3.145) T kinetic energy kinetic energy 2 1 1 2 2 3 3

Moment of T = T (ω1,ω2,ω3) (3.146) inertia ∂T ⊥ I3 ellipsoida Ji = (J is ellipsoid surface) (3.147) ∂ωi I1 I2 Perpendicular ≥ I3 generally axis theorem I1 +I2 (3.148) = I3 ﬂat lamina ⊥ to 3-axis lamina

I1 = I2 = I3 asymmetric top

Symmetries I1 = I2 = I3 symmetric top (3.149)

I1 = I2 = I3 spherical top aThe ellipsoid is deﬁned by the surface of constant T . main January 23, 2006 16:6

3.5 Rigid body dynamics 75

Moments of inertiaa ml2 l I1 = I2 = (3.150) Thin rod, length l 12 I3 I2 I3 0 (3.151) I1 2 I Solid sphere, radius r I = I = I = mr2 (3.152) 1 1 2 3 5 r I3 2 2 Spherical shell, radius r I = I = I = mr (3.153) I2 1 2 3 3 3 l 2 m 2 l I1 = I2 = r + (3.154) Solid cylinder, radius r, 4 3 I1 I3 r length l 1 I2 I = mr2 (3.155) 3 2 I1 2 2 I1 = m(b +c )/12 (3.156) 2 2 I Solid cuboid, sides a,b,c I2 = m(c +a )/12 (3.157) 3 a I 2 2 2 I3 = m(a +b )/12 (3.158) b c 3 h2 I = I = m r2 + (3.159) Solid circular cone, base 1 2 20 4 h radius r,heighthb I3 I 3 2 2 I3 = mr (3.160) r 10 I1

2 2 I3 I1 = m(b +c )/5 (3.161) Solid ellipsoid, semi-axes 2 2 c I2 = m(c +a )/5 (3.162) a b a,b,c 2 2 I2 I3 = m(a +b )/5 (3.163) I1

I2 I = mb2/4 (3.164) 1 b I1 Elliptical lamina, I = ma2/4 (3.165) a 2 I3 semi-axes a,b 2 2 I3 = m(a +b )/4 (3.166) I2 2 r I1 I1 = I2 = mr /4 (3.167) I3 Disk, radius r 2 I3 = mr /2 (3.168) a c m 2 2 2 Triangular plate I3 = (a +b +c ) (3.169) b I 36 3 c aWith respect to principal axes for bodies of mass m and uniform density. The radius of gyration is deﬁned as k =(I/m)1/2. bOrigin of axes is at the centre of mass (h/4 above the base). cAround an axis through the centre of mass and perpendicular to the plane of the plate. main January 23, 2006 16:6

76 Dynamics and mechanics

Centres of mass Solid hemisphere, radius r d =3r/8 from sphere centre (3.170)

Hemispherical shell, radius r d = r/2 from sphere centre (3.171)

Sector of disk, radius r, angle 2 sinθ d = r from disk centre (3.172) 2θ 3 θ Arc of circle, radius r, angle sinθ d = r from circle centre (3.173) 2θ θ Arbitrary triangular lamina, d = h/3 perpendicular from base (3.174) height ha Solid cone or pyramid, height d = h/4 perpendicular from base (3.175) h 3 (2r −h)2 solid: d = from sphere centre (3.176) Spherical cap, height h, 4 3r −h sphere radius r shell: d = r −h/2 from sphere centre (3.177)

Semi-elliptical lamina, 4h d = from base (3.178) height h 3π ah is the perpendicular distance between the base and apex of the triangle.

Pendulums P period g gravitational acceleration l θ0 Simple l θ2 P =2π 1+ 0 +··· (3.179) l length pendulum g 16 θ0 maximum angular m displacement l Conical l cosα 1/2 α α cone half-angle pendulum P =2π (3.180) g m 1/2 I0 moment of inertia of bob Torsional lI0 penduluma P =2π (3.181) C torsional rigidity of wire l I C (see page 81) 0 a distance of rotation axis from centre of mass 1 2 2 a P 2π (ma +I1 cos γ1 m mass of body I Compound mga 1 I3 pendulumb 1/2 Ii principal moments of 2 2 inertia I2 +I2 cos γ2 +I3 cos γ3) (3.182) γi angles between rotation axis and principal axes l Equal l 1/2 m double P 2π √ (3.183) c ± l pendulum (2 2)g m aAssuming the bob is supported parallel to a principal rotation axis. bI.e., an arbitrary triaxial rigid body. cFor very small oscillations (two eigenmodes). main January 23, 2006 16:6

3.5 Rigid body dynamics 77

Tops and gyroscopes

J 3 herpolhode ω space J 3 invariable cone polhode plane Ωp body moment cone θ of inertia support point 3 ellipsoid a mg 2

prolate symmetric top gyroscope

G1 = I1ω˙ 1 +(I3 −I2)ω2ω3 (3.184) Gi external couple (= 0 for free rotation) a Euler’s equations G2 = I2ω˙ 2 +(I1 −I3)ω3ω1 (3.185) Ii principal moments of inertia G = I ω˙ +(I −I )ω ω (3.186) 3 3 3 2 1 1 2 ωi angular velocity of rotation

I1 −I3 Ωb = ω3 (3.187) Ωb body frequency Free symmetric I1 b Ωs space frequency top (I3

Ωp precession angular velocity Ω2I cosθ −Ω J +mga = 0 (3.190) θ angle from vertical p 1 p 3 Steady gyroscopic J3 angular momentum around precession Mga/J3 (slow) symmetry axis Ωp (3.191) m mass J3/(I1 cosθ) (fast) g gravitational acceleration a distance of centre of mass Gyroscopic 2 ≥ from support point J3 4I1mgacosθ (3.192) stability I1 moment of inertia about support point Gyroscopic limit J2 I mga (3.193) (“sleeping top”) 3 1

Nutation rate Ωn = J3/I1 (3.194) Ωn nutation angular velocity

Gyroscope mga Ωp = (1−cosΩnt) (3.195) t time released from rest J3 aComponents are with respect to the principal axes, rotating with the body. bThe body frequency is the angular velocity (with respect to principal axes) of ω around the 3-axis. The space frequency is the angular velocity of the 3-axis around J , i.e., the angular velocity at which the body cone moves around the space cone. c 2 J close to 3-axis. If Ωb < 0, the body tumbles. main January 23, 2006 16:6

78 Dynamics and mechanics

3.6 Oscillating systems

Free oscillations x oscillating variable Diﬀerential d2x dx t time +2γ +ω2x = 0 (3.196) γ damping factor (per unit equation 2 0 dt dt mass)

ω0 undamped angular frequency −γt A amplitude constant Underdamped x = Ae cos(ωt+φ) (3.197) 2 2 1/2 φ phase constant solution (γ<ω0) where ω =(ω −γ ) (3.198) 0 ω angular eigenfrequency

Critically damped −γt x =e (A1 +A2t) (3.199) Ai amplitude constants solution (γ = ω0)

−γt qt −qt Overdamped x =e (A1e +A2e ) (3.200) 2 − 2 1/2 solution (γ>ω0) where q =(γ ω0) (3.201)

Logarithmic an 2πγ ∆ logarithmic decrement a ∆=ln = (3.202) decrement an+1 ω an nth displacement maximum ω0 π Quality factor Q = if Q 1 (3.203) Q quality factor 2γ ∆ aThe decrement is usually the ratio of successive displacement maxima but is sometimes taken as the ratio of successive displacement extrema, reducing ∆ by a factor of 2. Logarithms are sometimes taken to base 10, introducing a further factor of log10 e. Forced oscillations x oscillating variable Diﬀerential d2 d x x 2 iωf t t time +2γ +ω x = F0e (3.204) equation dt2 dt 0 γ damping factor (per unit mass)

i(ωf t−φ) x = Ae , where (3.205) ω0 undamped angular frequency A = F [(ω2 −ω2)2 +(2γω )2]−1/2 (3.206) F0 force amplitude (per unit Steady- 0 0 f f mass) F /(2ω ) state 0 0 ( ) (3.207) ωf forcing angular frequency a 2 2 1/2 γ ωf solution [(ω0 −ωf ) +γ ] A amplitude 2γω φ phase lag of response behind tanφ = f (3.208) 2 − 2 driving force ω0 ωf Amplitude ω2 = ω2 −2γ2 (3.209) ωar amplitude resonant forcing resonanceb ar 0 angular frequency Velocity ω = ω (3.210) ωvr velocity resonant forcing resonancec vr 0 angular frequency

Quality ω0 Q = (3.211) Q quality factor factor 2γ ω2 −ω2 Impedance Z =2γ +i f 0 (3.212) Z impedance (per unit mass) ωf aExcluding the free oscillation terms. bForcing frequency for maximum displacement. cForcing frequency for maximum velocity. Note φ = π/2 at this frequency. main January 23, 2006 16:6

3.7 Generalised dynamics 79

3.7 Generalised dynamics

Lagrangian dynamics action ( = 0 for the motion) t2 S δS Action S = L(q,q˙,t)dt (3.213) q generalised coordinates t1 q˙ generalised velocities L Lagrangian Euler–Lagrange d ∂L ∂L − = 0 (3.214) t time equation dt ∂q˙ ∂q i i m mass 3 v velocity Lagrangian of 1 2 − L = mv U(r,t) (3.215) r position vector particle in 2 U potential energy external ﬁeld = T −U (3.216) T kinetic energy

m0 (rest) mass Relativistic 2 γ Lorentz factor m0c Lagrangian of a L = − −e(φ−A·v) (3.217) +e positive charge charged particle γ φ electric potential A magnetic vector potential Generalised ∂L pi = (3.218) pi generalised momenta momenta ∂q˙i

Hamiltonian dynamics L Lagrangian − Hamiltonian H = piq˙i L (3.219) pi generalised momenta i q˙i generalised velocities Hamilton’s ∂H ∂H H Hamiltonian q˙i = ; p˙i = − (3.220) equations ∂pi ∂qi qi generalised coordinates v particle speed Hamiltonian 1 2 H = mv +U(r,t) (3.221) r position vector of particle in 2 U potential energy external ﬁeld = T +U (3.222) T kinetic energy m (rest) mass Relativistic 0 c speed of light Hamiltonian 2 4 2 2 1/2 H =(m c +|p −eA| c ) +eφ (3.223) +e positive charge of a charged 0 φ electric potential particle A vector potential ∂f ∂g ∂f ∂g [f,g]= − (3.224) ∂q ∂p ∂p ∂q p particle momentum i i i i i Poisson t time ∂g − ∂g brackets [qi,g]= , [pi,g]= (3.225) f,g arbitrary functions ∂pi ∂qi [·,·] Poisson bracket (also see ∂g dg Commutators on page 26) [H,g]=0 if =0, = 0 (3.226) ∂t dt Hamilton– ∂S ∂S Jacobi +H qi, ,t = 0 (3.227) S action equation ∂t ∂qi main January 23, 2006 16:6

80 Dynamics and mechanics

3.8 Elasticity

a Elasticity deﬁnitions (simple) F τ stress A Stress τ = F/A (3.228) F applied force A cross-sectional l area e strain Strain e = δl/l (3.229) δl change in length l length w Young modulus E = τ/e = constant (3.230) E Young modulus (Hooke’s law) σ Poisson ratio δw/w Poisson ratiob σ = − (3.231) δw change in width δl/l w width aThese apply to a thin wire under longitudinal stress. bSolids obeying Hooke’s law are restricted by thermodynamics to −1 ≤ σ ≤ 1/2, but none are known with σ<0. Non-Hookean materials can show σ>1/2.

Elasticity deﬁnitions (general) a force i direction Stress tensor τ = (3.232) τij stress tensor (τij = τji) ij area ⊥ j direction ekl strain tensor (ekl = elk) 1 ∂uk ∂ul Strain tensor ekl = + (3.233) uk displacement to xk 2 ∂xl ∂xk xk coordinate system

Elastic modulus τij = λijklekl (3.234) λijkl elastic modulus

1 Elastic energyb U = λ e e (3.235) U potential energy 2 ijkl ij kl e volume strain Volume strain δV v e = = e +e +e (3.236) δV change in volume (dilatation) v V 11 22 33 V volume 1 1 ekl =(ekl − evδkl)+ evδkl (3.237) Shear strain 3 3 δkl Kronecker delta pure shear dilatation Hydrostatic τ = −pδ (3.238) p hydrostatic pressure compression ij ij a τii are normal stresses, τij (i = j) are torsional stresses. bAs usual, products are implicitly summed over repeated indices. main January 23, 2006 16:6

3.8 Elasticity 81

Isotropic elastic solids E µ = (3.239) µ,λ Lame´ coeﬃcients 2(1+σ) Lame´ coeﬃcients E Young modulus Eσ λ = (3.240) σ Poisson ratio (1+σ)(1−2σ) Longitudinal E(1−σ) M = = λ+2µ (3.241) Ml longitudinal elastic modulusa l (1+σ)(1−2σ) modulus 1 eii strain in i direction 3 eii = [τii −σ(τjj +τkk)] (3.242) τ stress in i direction E ii Diagonalised σ e strain tensor b τ = M e + (e +e ) (3.243) equations ii l ii 1−σ jj kk t stress tensor t =2µe+λ1tr(e) (3.244) 1 unit matrix tr(·) trace E 2 K = = λ+ µ (3.245) K bulk modulus 3(1−2σ) 3 Bulk modulus KT isothermal bulk modulus (compression 1 − 1 ∂V = (3.246) V volume modulus) KT V ∂p T p pressure −p = Kev (3.247) T temperature

E ev volume strain Shear modulus = (3.248) µ µ shear modulus (rigidity modulus) 2(1+σ) τT transverse stress τT = µθsh (3.249) θsh shear strain τ 9µK T Young modulus E = (3.250) µ+3K θsh 3K −2µ Poisson ratio σ = (3.251) 2(3K +µ) aIn an extended medium. bAxes aligned along eigenvectors of the stress and strain tensors.

Torsion G Torsional rigidity G twisting couple (for a φ C torsional rigidity a G = C (3.252) l rod length φ homogeneous l rod) φ twist angle in l length l a radius Thin circular 3 C =2πa µt (3.253) t wall thickness cylinder µ shear modulus

Thick circular 1 4 − 4 a1 inner radius C = µπ(a2 a1) (3.254) cylinder 2 a2 outer radius A Arbitrary 4 2 A cross-sectional A µt area thin-walled tube C = (3.255) P P perimeter t 1 Long ﬂat ribbon C = µwt3 (3.256) w cross-sectional 3 width t w main January 23, 2006 16:6

82 Dynamics and mechanics

Bending beamsa G bending moment b ds E Young modulus E 2 Gb = ξ ds (3.257) Rc radius of curvature ξ Bending Rc moment EI ds area element neutral surface = (3.258) ξ distance to neutral R c surface from ds (cross section) I moment of area Light beam, y displacement from horizontal x horizontal at W x y = l − x2 (3.259) W end-weight x =0,weight y 2EI 3 l beam length at x = l x distance along beam W 4 d y w beam weight per Heavy beam EI = w(x) (3.260) dx4 unit length free  π2EI/l2 (free ends) Fc critical compression F Euler strut 2 2 Fc c Fc = 4π EI/l (ﬁxed ends) force failure  π2EI/(4l2) (1 free end) l strut length (3.261) ﬁxed a 2 2 The radius of curvature is approximated by 1/Rc d y/dx .

Elastic wave velocitiesa vt speed of transverse wave 1/2 vt =(µ/ρ) (3.262) vl speed of longitudinal wave 1/2 In an inﬁnite vl =(Ml/ρ) (3.263) µ shear modulus b isotropic solid 1/2 ρ density vl 2−2σ = (3.264) Ml longitudinal modulus vt 1−2σ E(1−σ) = (1+σ)(1−2σ) 1/2 In a ﬂuid vl =(K/ρ) (3.265) K bulk modulus

v(i) speed of longitudinal On a thin plate (wave travelling along x, plate thin in z) l wave (displacement i) 1/2 (i) (x) E vt speed of transverse wave v = (3.266) l ρ(1−σ2) (displacement i) k E Young modulus (y) 1/2 z vt =(µ/ρ) (3.267) σ Poisson ratio 2 1/2 k wavenumber (= 2π/λ) x (z) Et y v = k (3.268) t 12ρ(1−σ2) t plate thickness (in z, t λ)

1/2 vl =(E/ρ) (3.269) 1/2 In a thin circular vφ =(µ/ρ) (3.270) vφ torsional wave velocity rod a rod radius ( λ) ka E 1/2 v = (3.271) t 2 ρ aWaves that produce “bending” are generally dispersive. Wave (phase) speeds are quoted throughout. bTransverse waves are also known as shear waves, or S-waves. Longitudinal waves are also known as pressure waves, or P-waves. main January 23, 2006 16:6

3.8 Elasticity 83

Waves in strings and springsa

vl speed of longitudinal wave b 1/2 κ spring constant In a spring vl =(κl/ρl) (3.272) l spring length c ρl mass per unit length

On a stretched 1/2 vt speed of transverse wave vt =(T/ρl) (3.273) string T tension

On a stretched 1/2 τ tension per unit width vt =(τ/ρA) (3.274) sheet ρA mass per unit area 3 aWave amplitude assumed wavelength. bIn the sense κ =force/extension. cMeasured along the axis of the spring.

Propagation of elastic waves force F Z impedance Acoustic Z = = (3.275) response velocity u˙ F stress force impedance 1/2 =(E ρ) (3.276) u strain displacement 1/2 E Wave velocity/ if v = (3.277) E elastic modulus impedance ρ ρ density relation then Z =(Eρ)1/2 = ρv (3.278) v wave phase velocity

U 1 2 2 U energy density Mean energy = E k u0 (3.279) 2 k wavenumber density 1 2 2 ω angular frequency (nondispersive = ρω u0 (3.280) 2 u maximum displacement waves) U 0 P = v (3.281) P mean energy ﬂux

ur τr Z1 −Z2 r = = − = (3.282) r reflection coefficient Normal ui τi Z1 +Z2 t transmission coefficient coefficientsa 2Z t = 1 (3.283) τ stress Z1 +Z2

θi angle of incidence b sinθi sinθr sinθt Snell’s law = = (3.284) θr angle of reflection vi vr vt θt angle of refraction aFor stress and strain amplitudes. Because these reflection and transmission coefficients are usually defined in terms of displacement, u, rather than stress, there are differences between these coefficients and their equivalents defined in electromagnetism [see Equation (7.179) and page 154]. bAngles defined from the normal to the interface. An incident plane pressure wave will generally excite both shear and pressure waves in reflection and transmission. Use the velocity appropriate for the wave type. main January 23, 2006 16:6

84 Dynamics and mechanics

3.9 Fluid dynamics

Ideal fluidsa ρ density ∂ρ Continuityb +∇·(ρv) = 0 (3.285) v fluid velocity field ∂t t time Γ circulation Γ= v · dl = constant (3.286) dl loop element Kelvin circulation ds element of surface = ω · ds (3.287) bounded by loop S ω vorticity (= ∇×v) ∂v ∇p +(v ·∇)v = − +g (3.288) p pressure ∂t ρ Euler’s equationc g gravitational field ∂ strength or (∇×v)=∇×[v×(∇×v)] (3.289) ∂t (v ·∇) advective operator Bernoulli’s equation 1 ρv2 +p+ρgz = constant (3.290) z altitude (incompressible flow) 2 γ ratio of specific heat 1 2 γ p Bernoulli’s equation v + +gz = constant (3.291) capacities (c /c ) 2 γ −1 ρ p V (compressible cp specific heat capacity d 1 2 at constant pressure adiabatic flow) = v +cpT +gz (3.292) 2 T temperature

Hydrostatics ∇p = ρg (3.293)

Adiabatic lapse rate dT g = − (3.294) (ideal gas) dz cp aNo thermal conductivity or viscosity. bTrue generally. cThe second form of Euler’s equation applies to incompressible ﬂow only. dEquation (3.292) is true only for an ideal gas.

Potential ﬂowa ∇ v = φ (3.295) v velocity Velocity potential ∇2φ = 0 (3.296) φ velocity potential

ω vorticity Vorticity condition ω = ∇×v = 0 (3.297) F drag force on moving sphere a sphere radius Drag force on a 2 3 1 u˙ sphere acceleration F = − πρa u˙ = − Mdu˙ (3.298) sphereb 3 2 ρ fluid density Md displaced fluid mass aFor incompressible fluids. bThe effect of this drag force is to give the sphere an additional effective mass equal to half the mass of fluid displaced. main January 23, 2006 16:6

3.9 Fluid dynamics 85

Viscous ﬂow (incompressible)a

τij fluid stress tensor p hydrostatic pressure ∂vi ∂vj Fluid stress τij = −pδij +η + (3.299) η shear viscosity ∂xj ∂xi vi velocity along i axis δij Kronecker delta ∂v ∇p η v fluid velocity field +(v ·∇)v = − − ∇×ω +g (3.300) Navier–Stokes ∂t ρ ρ ω vorticity equationb ∇p η g gravitational acceleration = − + ∇2v +g (3.301) 3 ρ ρ ρ density Kinematic ν = η/ρ (3.302) ν kinematic viscosity viscosity aI.e., ∇·v =0, η =0. bNeglecting bulk (second) viscosity.

Laminar viscous flow vz flow velocity z direction of flow Between 1 ∂p v (y)= y(h−y) (3.303) y distance from z parallel plates z 2η ∂z plate h η shear viscosity y p pressure

1 2 2 ∂p r distance from vz(r)= (a −r ) (3.304) Along a 4η ∂z pipe axis circular pipea dV πa4 ∂p a pipe radius Q = = (3.305) r a dt 8η ∂z V volume Circulating Gz axial couple between 4πηa2a2 between cylinders G = 1 2 (ω −ω ) concentric z 2 − 2 2 1 per unit length a2 a1 a1 rotating (3.306) ωi angular velocity b of ith cylinder cylinders ω1 a2 a inner radius π ∂p (a2 −a2)2 1 Along an 4 − 4 − 2 1 outer radius Q = a2 a1 a2 ω2 annular pipe 8η ∂z ln(a2/a1) Q volume discharge (3.307) rate aPoiseuille ﬂow. bCouette ﬂow.

Draga F drag force On a sphere (Stokes’s law) F =6πaηv (3.308) a radius v velocity On a disk, broadside to ﬂow F =16aηv (3.309) η shear viscosity

On a disk, edge on to ﬂow F =32aηv/3 (3.310) aFor Reynolds numbers 1. main January 23, 2006 16:6

86 Dynamics and mechanics

Characteristic numbers Re Reynolds number ρ density Reynolds ρUL inertial force Re = = (3.311) U characteristic velocity number η viscous force L characteristic scale-length η shear viscosity Froude U2 inertial force F Froude number F = = (3.312) numbera Lg gravitational force g gravitational acceleration

Strouhal Uτ evolution scale S Strouhal number S = = (3.313) numberb L physical scale τ characteristic timescale P Prandtl number Prandtl ηc momentum transport P = p = (3.314) cp Speciﬁc heat capacity at number λ heat transport constant pressure λ thermal conductivity Mach U speed M Mach number M = = (3.315) number c sound speed c sound speed

Rossby U inertial force Ro Rossby number Ro = = (3.316) number ΩL Coriolis force Ω angular velocity aSometimes the square root of this expression. L is usually the ﬂuid depth. bSometimes the reciprocal of this expression.

Fluid waves vp wave (phase) speed K 1/2 dp 1/2 K bulk modulus Sound waves v = = (3.317) p ρ dρ p pressure ρ density γ ratio of heat capacities In an ideal gas 1/2 1/2 R molar gas constant (adiabatic γRT γp vp = = (3.318) T (absolute) temperature conditions)a µ ρ µ mean molecular mass

vg group speed of wave ω2 = gktanhkh (3.319) h liquid depth  Gravity waves on 1 g 1/2 λ wavelength b (h λ) a liquid surface k wavenumber vg 2 k (3.320) (gh)1/2 (h λ) g gravitational acceleration ω angular frequency Capillary waves σk3 ω2 = (3.321) σ surface tension (ripples)c ρ Capillary–gravity σk3 ω2 = gk+ (3.322) waves (h λ) ρ a 1/2 If the waves are isothermal rather than adiabatic then vp =(p/ρ) . bAmplitude wavelength. cIn the limit k2 gρ/σ. main January 23, 2006 16:6

3.9 Fluid dynamics 87

Doppler effecta Source at rest, ν ,ν observed frequency ν |u| observer =1− cosθ (3.323) ν emitted frequency moving at u ν vp vp wave (phase) speed k in fluid θ Observer at ν 1 = (3.324) u velocity rest, source | | u ν − u θ angle between moving at 1 cosθ wavevector, k, and u u vp aFor plane waves in a stationary fluid. 3 Wave speeds

vp phase speed ω ν frequency Phase speed vp = = νλ (3.325) ω angular frequency (= 2πν) k λ wavelength k wavenumber (= 2π/λ) dω v = (3.326) g dk Group speed v group speed dv g = v −λ p (3.327) p dλ

Shocks θw wedge semi-angle a vp Mach wedge sinθw = (3.328) vp wave (phase) speed vb vb body speed 4πv2 λ characteristic λ = b (3.329) K Kelvin K 3g wavelength wedgeb ◦ g gravitational θw = arcsin(1/3)=19 .5 (3.330) acceleration

r shock radius Spherical E energy release Et2 1/5 adiabatic r (3.331) t time c ρ0 shock ρ0 density of undisturbed medium 1 upstream values p 2γM2 −(γ −1) 2 = 1 (3.332) 2 downstream values Rankine– p1 γ +1 p pressure Hugoniot v1 ρ2 γ +1 v velocity = = (3.333) − 2 temperature shock v2 ρ1 (γ 1)+2/M1 T d relations T [2γM2 −(γ −1)][2+(γ −1)M2] ρ density 2 = 1 1 (3.334) 2 2 γ ratio of speciﬁc heats T1 (γ +1) M 1 M Mach number aApproximating the wake generated by supersonic motion of a body in a nondispersive medium. b For gravity waves, e.g., in the wake of a boat. Note that the wedge semi-angle is independent of vb. cSedov–Taylor relation. d Solutions for a steady, normal shock, in the frame moving with the shock front. If γ =5/3thenv1/v2 ≤ 4. main January 23, 2006 16:6

88 Dynamics and mechanics

Surface tension surface energy σlv = (3.335) surface area σlv surface tension Deﬁnition (liquid/vapour surface tension = (3.336) interface) length R2 ∆p pressure diﬀerence R1 Laplace’s 1 1 over surface a ∆p = σlv + (3.337) formula R1 R2 Ri principal radii of curvature cc capillary constant 1/2 Capillary 2σlv ρ liquid density constant cc = (3.338) gρ g gravitational h θ acceleration a h rise height Capillary rise 2σlv cosθ h = (3.339) θ contact angle (circular tube) ρga a tube radius

σwv wall/vapour surface σwv σ −σ tension Contact angle cosθ = wv wl (3.340) σlv σwl wall/liquid surface tension σwl θ σlv a For a spherical bubble in a liquid ∆p =2σlv/R. For a soap bubble (two surfaces) ∆p =4σlv/R. main January 23, 2006 16:6

Chapter 4 Quantum physics

4.1 Introduction Quantum ideas occupy such a pivotal position in physics that different notations and algebras appropriate to each field have been developed. In the spirit of this book, only those formulas that are commonly present in undergraduate courses and that can be simply presented in 4 tabular form are included here. For example, much of the detail of atomic spectroscopy and of specific perturbation analyses has been omitted, as have ideas from the somewhat specialised field of quantum electrodynamics. Traditionally, quantum physics is understood through standard “toy” problems, such as the potential step and the one-dimensional harmonic oscillator, and these are reproduced here. Operators are distinguished from observables using the “hat” notation, so that the momentum observable, px,hastheoperatorpˆx = −ih∂/∂x¯ . For clarity, many relations that can be generalised to three dimensions in an obvious way have been stated in their one-dimensional form, and wavefunctions are implicitly taken as normalised functions of space and time unless otherwise stated. With the exception of the last panel, all equations should be taken as nonrelativistic, so that “total energy” is the sum of potential and kinetic energies, excluding the rest mass energy. main January 23, 2006 16:6

90 Quantum physics

4.2 Quantum deﬁnitions Quantum uncertainty relations h p,p particle momentum p = (4.1) h Planck constant De Broglie relation λ p =¯hk (4.2) hh/¯ (2π) λ de Broglie wavelength k de Broglie wavevector Planck–Einstein E energy E = hν =¯hω (4.3) relation ν frequency ω angular frequency (= 2πν)

b (∆a)2 = (a−a )2 (4.4) a,b observables Dispersiona · expectation value = a2 −a 2 (4.5) (∆a)2 dispersion of a

General uncertainty 1 aˆ operator for observable a (∆a)2(∆b)2 ≥ i[a,ˆ bˆ] 2 (4.6) relation 4 [·,·] commutator (see page 26) Momentum–position h¯ ∆p∆x ≥ (4.7) x particle position uncertainty relationc 2 Energy–time h¯ ∆E ∆t ≥ (4.8) t time uncertainty relation 2

Number–phase 1 n number of photons ∆n∆φ ≥ (4.9) uncertainty relation 2 φ wave phase aDispersion in quantum physics corresponds to variance in statistics. bAn observable is a directly measurable parameter of a system. cAlso known as the “Heisenberg uncertainty relation.”

Wavefunctions Probability pr probability density pr(x,t)dx = |ψ(x,t)|2 dx (4.10) density ψ wavefunction ∗ j,j probability density current h¯ ∗ ∂ψ ∂ψ j(x)= ψ −ψ (4.11) h¯ (Planck constant)/(2π) 2im ∂x ∂x Probability x position coordinate h¯ density j = ψ∗(r)∇ψ(r)−ψ(r)∇ψ∗(r) (4.12) pˆ momentum operator currenta 2im m particle mass 1 = (ψ∗pˆψ) (4.13) real part of m t time Continuity ∂ ∇·j = − (ψψ∗) (4.14) equation ∂t Schrodinger¨ ∂ψ Hψˆ = ih¯ (4.15) H Hamiltonian equation ∂t Particle h¯2 ∂2ψ(x) V potential energy stationary − +V (x)ψ(x)=Eψ(x) (4.16) 2 E total energy statesb 2m ∂x aFor particles. In three dimensions, suitable units would be particles m−2 s−1. bTime-independent Schrodinger¨ equation for a particle, in one dimension. main January 23, 2006 16:6

4.2 Quantum deﬁnitions 91

Operators Hermitian aˆ Hermitian conjugate conjugate (aφˆ )∗ψ dx = φ∗aψˆ dx (4.17) operator operator ψ,φ normalisable functions Position ∗ complex conjugate xˆn = xn (4.18) operator x,y position coordinates n Momentum h¯ ∂n n arbitrary integer ≥ 1 ˆn px = (4.19) operator in ∂xn px momentum coordinate T kinetic energy Kinetic energy h¯2 ∂2 Tˆ = − (4.20) h¯ (Planck constant)/(2π) operator 2 2m ∂x m particle mass Hamiltonian h¯2 ∂2 H Hamiltonian Hˆ = − +V (x) (4.21) operator 2m ∂x2 V potential energy 4

Angular Lˆz = xˆpˆy −yˆpˆx (4.22) Lz angular momentum along momentum 2 2 2 z axis (sim. x and y) ˆ2 ˆ ˆ ˆ operators L = Lx +Ly +Lz (4.23) L total angular momentum

Pˆ parity operator Parity operator Pψˆ (r)=ψ(−r) (4.24) r position vector

Expectation value a expectation value of a ∗ Expectation a = aˆ = Ψ aˆΨdx (4.25) aˆ operator for a a value = Ψ|aˆ|Ψ (4.26) Ψ (spatial) wavefunction x (spatial) coordinate Time d i ∂aˆ t time aˆ = [H,ˆ aˆ] + (4.27) dependence dt h¯ ∂t h¯ (Planck constant)/(2π) ψn eigenfunctions of aˆ if aψˆ n = anψn and Ψ = cnψn Relation to an eigenvalues eigenfunctions 2 n dummy index then a = |cn| an (4.28) cn probability amplitudes d m particle mass m r = p (4.29) Ehrenfest’s dt r position vector theorem d p momentum p = −∇V (4.30) dt V potential energy aEquation (4.26) uses the Dirac “bra-ket” notation for integrals involving operators. The presence of vertical bars distinguishes this use of angled brackets from that on the left-hand side of the equations. Note that a and aˆ are taken as equivalent. main January 23, 2006 16:6

92 Quantum physics

Dirac notation n,m eigenvector indices ∗ a matrix element anm = ψnaψˆ m dx (4.31) nm a Matrix element ψn basis states = n|aˆ|m (4.32) aˆ operator x spatial coordinate

Bra vector bra state vector = n| (4.33) ·| bra

Ket vector ket state vector = |m (4.34) |· ket | ∗ Scalar product n m = ψnψm dx (4.35) if Ψ = cnψn (4.36) Ψ wavefunction Expectation n ∗ cn probability amplitudes then a = cncmanm (4.37) m n a The Dirac bracket, n|aˆ|m , can also be written ψn|aˆ|ψm .

4.3 Wave mechanics Potential stepa V (x)

incident particle V0 iii

0 x V particle potential energy Potential 0(x<0) V (x)= (4.38) V0 step height function ≥ V0 (x 0) h¯ (Planck constant)/(2π) h¯2k2 =2mE (x<0) (4.39) k,q particle wavenumbers Wavenumbers m particle mass h¯2q2 =2m(E −V )(x>0) (4.40) 0 E total particle energy Amplitude k −q reflection r amplitude reflection r = (4.41) coefficient coefficient k +q Amplitude 2k transmission t amplitude transmission t = (4.42) coefficient coefficient k +q

hk¯ 2 ji = (1−|r| ) (4.43) Probability m ji particle ﬂux in zone i b ii currents hq¯ 2 jii particle ﬂux in zone jii = |t| (4.44) m a One-dimensional interaction with an incident particle of total energy E =KE+V .IfE

4.3 Wave mechanics 93

Potential wella

V (x) incident particle i ii iii −a a x 0 −V0

V particle potential energy Potential 0(|x| >a) V0 well depth V (x)= (4.45) function h¯ (Planck constant)/(2π) −V0 (|x|≤a) 2a well width h¯2k2 =2mE (|x| >a) (4.46) k,q particle wavenumbers Wavenumbers 2 2 m particle mass h¯ q =2m(E +V0)(|x|

Amplitude −2ika 2kqe t amplitude transmission transmission t = (4.49) − 2 2 coeﬃcient coeﬃcient 2kqcos2qa i(q +k )sin2qa

hk¯ 2 ji = (1−|r| ) (4.50) Probability m ji particle ﬂux in zone i b iii currents hk¯ 2 jiii particle ﬂux in zone jiii = |t| (4.51) m

Ramsauer n2h¯2π2 n integer > 0 c En = −V0 + (4.52) eﬀect 8ma2 En Ramsauer energy |k|/q even parity tanqa= (4.53) Bound states −q/|k| odd parity (V 0. bParticle ﬂux in the sense of increasing x. cIncident energy for which 2qa= nπ, |r| = 0, and |t| =1. dWhen E<0, k is purely imaginary. |k| and q are obtained by solving these implicit equations. main January 23, 2006 16:6

94 Quantum physics

Barrier tunnellinga

V (x) V0 incident particle i ii iii

−a 0 a x V particle potential energy Potential 0(|x| >a) V0 well depth V (x)= (4.55) function h¯ (Planck constant)/(2π) V0 (|x|≤a) 2a barrier width k incident wavenumber Wavenumber 2 2 | | h¯ k =2mE ( x >a) (4.56) κ tunnelling constant and tunnelling h¯2κ2 =2m(V −E)(|x|

Amplitude −2ika 2kκe t amplitude transmission transmission t = (4.59) − 2 − 2 coeﬃcient coeﬃcient 2kκcosh2κa i(k κ )sinh2κa 4k2κ2 |t|2 = (4.60) 2 2 2 2 2 2 Tunnelling (k +κ ) sinh 2κa+4k κ |t|2 tunnelling probability probability 16k2κ2 exp(−4κa)(|t|2 1) (k2 +κ2)2 (4.61)

hk¯ 2 ji = (1−|r| ) (4.62) Probability m ji particle flux in zone i b iii currents hk¯ 2 jiii particle flux in zone jiii = |t| (4.63) m a By a particle of total energy E =KE+V , through a one-dimensional rectangular potential barrier height V0 >E. bParticle flux in the sense of increasing x.

Particle in a rectangular boxa 1/2 Eigen- 8 lπx mπy nπz Ψlmn eigenfunctions Ψ = sin sin sin functions lmn abc a b c a,b,c box dimensions l,m,n integers ≥ 1 (4.64) a Elmn energy x b 2 2 2 2 Energy h l m n h Planck z Elmn = + + (4.65) levels 8M a2 b2 c2 constant y M particle mass c Density of 4π ρ(E)densityof 3 1/2 states (per unit ρ(E)dE = 3 (2M E) dE (4.66) states h volume) aSpinless particle in a rectangular box bounded by the planes x =0, y =0, z =0, x = a, y = b, and z = c. The potential is zero inside and inﬁnite outside the box. main January 23, 2006 16:6

4.4 Hydrogenic atoms 95

Harmonic oscillator h¯ (Planck constant)/(2π) 2 2 Schrodinger¨ h¯ ∂ ψn 1 2 2 m mass − + mω x ψn = Enψn (4.67) equation 2m ∂x2 2 ψn nth eigenfunction x displacement n integer ≥ 0 Energy 1 E = n+ hω¯ (4.68) ω angular frequency levelsa n 2 En total energy in nth state 2 2 Hn(x/a)exp[−x /(2a )] ψn = (4.69) Eigen- (n!2naπ1/2)1/2 Hn Hermite polynomials functions h¯ 1/2 where a = mω 2 − Hermite H0(y)=1,H1(y)=2y, H2(y)=4y 2 4 y dummy variable polynomials Hn+1(y)=2yHn(y)−2nHn−1(y) (4.70) a E0 is the zero-point energy of the oscillator.

4.4 Hydrogenic atoms Bohr modela rn nth orbit radius Quantisation 2 Ω orbital angular speed µrnΩ=nh¯ (4.71) condition n principal quantum number (> 0)

2 a0 Bohr radius 0h α Bohr radius µ reduced mass ( me) a0 = 2 = 52.9pm (4.72) πmee 4πR∞ −e electronic charge

2 Z atomic number n me Orbit radius rn = a0 (4.73) h Planck constant Z µ hh/¯ (2π) E total energy of nth orbit µe4Z 2 µ Z 2 n Total energy E = − = −R∞hc (4.74) 0 permittivity of free space n 2 2 2 2 8 h n me n 0 me electron mass Fine structure µ ce2 e2 1 α ﬁne structure constant α = 0 = (4.75) constant 2h 4π0hc¯ 137 µ0 permeability of free space 2 h¯ −18 Hartree energy × EH Hartree energy EH = 2 4.36 10 J (4.76) mea0 2 4 Rydberg mecα mee EH R∞ Rydberg constant R∞ = = = (4.77) constant 2h 8h32c 2hc c speed of light 0 Rydberg’s 1 µ 2 1 1 λmn photon wavelength ∞ − b = R Z 2 2 (4.78) formula λmn me n m m integer >n a Because the Bohr model is strictly a two-body problem, the equations use reduced mass, µ = memnuc/(me+mnuc) me, where mnuc is the nuclear mass, throughout. The orbit radius is therefore the electron–nucleus distance. bWavelength of the spectral line corresponding to electron transitions between orbits m and n. main January 23, 2006 16:6

96 Quantum physics

Hydrogenlike atoms – Schrodinger¨ solutiona Schrodinger¨ equation 2 2 h¯ 2 Ze memnuc − ∇ Ψnlm − Ψnlm = EnΨnlm with µ = (4.79) 2µ 4π0r me +mnuc

Eigenfunctions (n−l −1)! 1/2 2 3/2 Ψ (r,θ,φ)= xle−x/2L2l+1 (x)Y m(θ,φ) (4.80) nlm 2n(n+l)! an n−l−1 l − −1 m a 2r nl (l +n)!(−x)k with a = e 0 ,x= , and L2l+1 (x)= µ Z an n−l−1 (2l +1+k)!(n−l −1−k)!k! k=0

4 2 µe Z En total energy Total energy En = − (4.81) 2 2 2 0 permittivity of free space 80h n a r = [3n2 −l(l +1)] (4.82) h Planck constant 2 me massofelectron 2 2 a n hh/¯ 2π Radial r2 = [5n2 +1−3l(l +1)] (4.83) 2 µ reduced mass ( me) expectation 1 mnuc mass of nucleus values 1/r = 2 (4.84) an Ψnlm eigenfunctions 2 Ze charge of nucleus 1/r2 = (4.85) (2l +1)n3a2 −e electronic charge

q n =1,2,3,... (4.86) Lp associated Laguerre c l =0,1,2,...,(n−1) (4.87) polynomials Allowed classical orbit radius, =1 ± ± ± a n quantum m =0, 1, 2,..., l (4.88) r electron–nucleus separation numbers and m ∆n = 0 (4.89) Yl spherical harmonics b 2 selection rules 0h ∆l = ±1 (4.90) a0 Bohr radius = 2 πmee ∆m =0 or ±1 (4.91)

a−3/2 a−3/2 r Ψ = e−r/a Ψ = 2− e−r/2a 100 π1/2 200 4(2π)1/2 a −3/2 −3/2 a r −r/2a a r −r/2a ±iφ Ψ = e cosθ Ψ ± = ∓ e sinθe 210 4(2π)1/2 a 21 1 8π1/2 a a−3/2 r r2 21/2a−3/2 r r Ψ = 27−18 +2 e−r/3a Ψ = 6− e−r/3a cosθ 300 81(3π)1/2 a a2 310 81π1/2 a a −3/2 −3/2 2 a r r −r/3a ±iφ a r −r/3a 2 Ψ ± = ∓ 6− e sinθe Ψ = e (3cos θ −1) 31 1 81π1/2 a a 320 81(6π)1/2 a2 −3/2 2 −3/2 2 a r −r/3a ±iφ a r −r/3a 2 ±2iφ Ψ ± = ∓ e sinθcosθe Ψ ± = e sin θe 32 1 81π1/2 a2 32 2 162π1/2 a2 aFor a single bound electron in a perfect nuclear Coulomb potential (nonrelativistic and spin-free). bFor dipole transitions between orbitals. cThe sign and indexing deﬁnitions for this function vary. This form is appropriate to Equation (4.80). main January 23, 2006 16:6

4.4 Hydrogenic atoms 97

Orbital angular dependence

2 0.2 2 2 (s) (px) (py) −0.4 −0.4 −0.2 −0.2

0.2 0.2 y x −0.2

2 2 2 (pz) (dx2−y2 ) (dxz) −0.4 4

2 2 2 (dz2 ) (dyz) (dxy )

0 0

s orbital 0 Y m spherical s = Y = constant (4.92) l (l =0) 0 harmonicsa −1 p = (Y 1 −Y −1) ∝ cosφsinθ (4.93) x 21/2 1 1 p orbitals i θ,φ spherical polar p = (Y 1 +Y −1) ∝ sinφsinθ (4.94) (l =1) y 21/2 1 1 coordinates 0 ∝ pz = Y1 cosθ (4.95)

1 2 −2 ∝ 2 dx2−y2 = (Y +Y ) sin θcos2φ (4.96) 21/2 2 2 z −1 d = (Y 1 −Y −1) ∝ sinθcosθcosφ (4.97) θ xz 21/2 2 2 d orbitals 0 2 d 2 = Y ∝ (3cos θ −1) (4.98) y (l =2) z 2 x i φ d = (Y 1 +Y −1) ∝ sinθcosθsinφ (4.99) yz 21/2 2 2 −i d = (Y 2 −Y −2) ∝ sin2 θsin2φ (4.100) xy 21/2 2 2 aSee page 49 for the deﬁnition of spherical harmonics. main January 23, 2006 16:6

98 Quantum physics

4.5 Angular momentum

Orbital angular momentum

Lˆ = r×pˆ (4.101) L angular momentum h¯ ∂ ∂ Lˆ = x −y (4.102) p linear momentum z i ∂y ∂x Angular r position vector h¯ ∂ momentum = (4.103) xyz Cartesian coordinates operators i ∂φ 2 2 2 rθφ spherical polar Lˆ2 = Lˆ +Lˆ +Lˆ (4.104) x y z coordinates 2 2 1 ∂ ∂ 1 ∂ h¯ (Planck = −h¯ sinθ + (4.105) constant)/(2π) sinθ ∂θ ∂θ sin2 θ ∂φ2

Lˆ± = Lˆx ±iLˆy (4.106) Lˆ± ladder operators Ladder ±iφ ∂ ∂ ml =¯he icotθ ± (4.107) Yl spherical operators ∂φ ∂θ harmonics ± ˆ ml − ± 1/2 ml 1 l,ml integers L±Yl =¯h[l(l +1) ml(ml 1)] Yl (4.108)

ˆ2 ml 2 ml ≥ L Yl = l(l +1)¯h Yl (l 0) (4.109) Eigen- ml ml LˆzY = mlhY¯ (|ml|≤l) (4.110) functions and l l ˆ ˆ ml ± ˆ ml eigenvalues Lz[L±Yl (θ,φ)]=(ml 1)¯hL±Yl (θ,φ) (4.111) l-multiplicity = (2l +1) (4.112)

Angular momentum commutation relationsa L angular momentum Conservation of angular p momentum [H,ˆ Lˆz] = 0 (4.113) momentumb H Hamiltonian Lˆ± ladder operators

[Lˆx,Lˆy]=ih¯Lˆz (4.120) [Lˆz,x]=ihy¯ (4.114) [Lˆz,Lˆx]=ih¯Lˆy (4.121) [Lˆz,y]=−ihx¯ (4.115) [Lˆy,Lˆz]=ih¯Lˆx (4.122) [Lˆz,z] = 0 (4.116) [Lˆ+,Lˆz]=−h¯Lˆ+ (4.123) [Lˆz,pˆx]=ih¯pˆy (4.117) [Lˆ−,Lˆz]=¯hLˆ− (4.124) [Lˆz,pˆy]=−ih¯pˆx (4.118) [Lˆ+,Lˆ−]=2¯hLˆz (4.125) [Lˆz,pˆz] = 0 (4.119) [Lˆ2,Lˆ±] = 0 (4.126)

ˆ2 ˆ2 ˆ2 [L ,Lˆx]=[L ,Lˆy]=[L ,Lˆz] = 0 (4.127) aThe commutation of a and b is deﬁned as [a,b]=ab−ba (see page 26). Similar expressions hold for S and J. bFor motion under a central force. main January 23, 2006 16:6

4.5 Angular momentum 99

Clebsch–Gordan coeﬃcientsa

l1+l2−j +1 j,−mj |l1,−m1;l2,−m2 =(−1) j,mj |l1,m1;l2,m2 +3/2 1/2×1/2 1 0 1×1/2 3/2 +1/2 +1/2+1/2 1 10 +1 +1/2 1 3/21/2 m +1/2 −1/2 1/21/2 j +1 −1/2 1/32/3 × j j ... −1/2+1/2 1/2 −1/2 l1 l2 0+1/2 2/3 −1/3 m1 m2 coeﬃcients m1 m2 j,mj |l1,m1;l2,m2 +2 . . . +5/2 . . . 3/2×1/2 2 +1 . . . 2×1/2 5/2 +3/2 +3/2+1/2 1 21 +2 +1/2 1 5/23/2 +3/2 −1/2 1/43/4 0 +2 −1/2 1/54/5 +1/2 +1/2+1/2 3/4 −1/4 21 +1 +1/2 4/5 −1/5 5/23/2 +1/2 −1/2 1/21/2 +1 −1/2 2/53/5 − − − +2 1/2+1/2 1/2 1/2 +5/2 0+1/2 3/5 2/5 1×1 2 +1 3/2×1 5/2 +3/2 +1 +1 1 21 +3/2+1 1 5/23/2 +1 0 1/21/2 0 +3/202/53/5 +1/2 4 0+11/2 −1/2 21 0 +1/2+13/5 −2/5 5/23/21/2 +1 −1 1/61/21/3 +3/2 −1 1/10 2/51/2 − − 002/30 1/3 +3 1/203/51/15 1/3 −1+11/6 −1/21/3 −1/2+13/10 −8/15 1/6 +3 3/2×3/2 3 +2 2×1 3 +2 +3/2+3/2 1 32 +2 +1 1 32 +3/2+1/2 1/21/2 +1 − +2 0 1/32/3 +1 +1/2+3/2 1/2 1/2 32 1 +1 +1 2/3 −1/3 32 1 +3/2 −1/2 1/51/23/10 +2 −1 1/15 1/33/5 +1/2+1/2 3/50−2/5 0 − − +1 0 8/15 1/6 −3/10 0 1/2+3/2 1/5 1/23/10 32 1 0 0+16/15 −1/21/10 32 1 +3/2 −3/2 1/20 1/49/20 1/4 +1 −1 1/51/23/10 +1/2 −1/2 9/20 1/4 −1/20 −1/4 003/50−2/5 −1/2+1/2 9/20 −1/4 −1/20 1/4 −1+11/5 −1/23/10 −3/2+3/2 1/20 −1/49/20 −1/4

+7/2 2×3/2 7/2 +5/2 +2 +3/2 1 7/25/2 +4 × +2 +1/2 3/74/7 +3/2 2 2 4 +3 +1 +3/2 4/7 −3/7 7/25/23/2 +2 +2 1 43 +2 −1/2 1/716/35 2/5 +2 +1 1/21/2 +2 +1 +1/2 4/71/35 −2/5 +1/2 − +1 +2 1/2 1/2 43 2 0+3/2 2/7 −18/35 1/5 7/25/23/21/2 +2 0 3/14 1/22/7 +2 −3/2 1/35 6/35 2/52/5 +1 +1 4/70−3/7 +1 +1 −1/2 12/35 5/14 0 −3/10 0+23/14 −1/22/7 43 2 1 0+1/2 18/35 −3/35 −1/51/5 +2 −1 1/14 3/10 3/71/5 −1+3/2 4/35 −27/70 2/5 −1/10 +1 0 3/71/5 −1/14 −3/10 0+13/7 −1/5 −1/14 3/10 0 −1+21/14 −3/10 3/7 −1/5 43210 +2 −2 1/70 1/10 2/72/51/5 +1 −1 8/35 2/51/14 −1/10 −1/5 0018/35 0 −2/70 1/5 −1+1 8/35 −2/51/14 1/10 −1/5 −2+2 1/70 −1/10 2/7 −2/51/5 aOr “Wigner coefficients,” using the Condon–Shortley sign convention. Note that a square root is assumed over all coefficient digits, so that “−3/10” corresponds to − 3/10. Also for clarity, only values of mj ≥ 0are listed here. The coefficients for mj < 0 can be obtained from the symmetry relation j,−mj |l1,−m1;l2,−m2 = l +l −j (−1) 1 2 j,mj |l1,m1;l2,m2 . main January 23, 2006 16:6

100 Quantum physics

Angular momentum additiona

J = L+S (4.128) J ,J total angular momentum

Jˆz = Lˆz +Sˆz (4.129) L,L orbital angular momentum ˆ2 ˆ2 ˆ2 · Total angular J = L +S +2L S (4.130) S,S spin angular momentum momentum ˆ ψ eigenfunctions Jzψj,mj = mjhψ¯ j,mj (4.131) mj magnetic quantum ˆ2 2 | |≤ J ψj,mj = j(j +1)¯h ψj,mj (4.132) number mj j j-multiplicity = (2l +1)(2s+1) (4.133) j (l +s) ≥ j ≥|l −s|

Mutually { 2 2 2 · } L ,S ,J ,Jz,L S (4.134) {} commuting set of mutually { 2 2 } commuting observables sets L ,S ,Lz,Sz,Jz (4.135) Clebsch– |j,mj = j,mj |l,ml;s,ms |l,ml |s,ms |· eigenstates Gordan ml ,ms ·|· Clebsch–Gordan m +m =m b s l j coefficients coefficients (4.136) a Summing spin and orbital angular momenta as examples, eigenstates |s,ms and |l,ml . bOr “Wigner coefficients.” Assuming no L–S interaction.

Magnetic moments

µB Bohr magneton eh¯ −e electronic charge Bohr magneton µB = (4.137) 2me h¯ (Planck constant)/(2π) me electron mass Gyromagnetic orbital magnetic moment γ = (4.138) γ gyromagnetic ratio ratioa orbital angular momentum −µ Electron orbital γ = B (4.139) e h¯ gyromagnetic −e γe electron gyromagnetic ratio ratio = (4.140) 2me

µe,z = −geµBms (4.141) Spin magnetic µe,z z component of spin ± h¯ magnetic moment moment of an = geγe (4.142) 2 electron -factor ( 2 002) b ge g . electron g eh¯ ± = ± e (4.143) ms spin quantum number ( 1/2) 4me µJ total magnetic moment µJ = gJ J(J +1)µB (4.144) µJ,z z component of µJ µJ,z = −gJ µBmJ (4.145) Lande´ g-factorc mJ magnetic quantum number J(J +1)+S(S +1)−L(L+1) g =1+ J,L,S total, orbital, and spin J 2J(J +1) quantum numbers (4.146) gJ Lande´ g-factor aOr “magnetogyric ratio.” b The electron g-factor equals exactly 2 in Dirac theory. The modiﬁcation ge =2+α/π + ...,whereα is the ﬁne structure constant, comes from quantum electrodynamics. c Relating the spin + orbital angular momenta of an electron to its total magnetic moment, assuming ge =2. main January 23, 2006 16:6

4.5 Angular momentum 101

Quantum paramagnetism 1 0.8 B L 0.6 ∞(x)= (x) B4(x) 0.4 B1(x) 0.2 B1/2(x)=tanhx 0 − − 10 5 −0.2 510x −0.4 −0.6 −0.8 −1 2J +1 (2J +1)x 1 x B (x)= coth − coth (4.147) J 2J 2J 2J 2J B  J (x) Brillouin function 4 J +1 Brillouin ( 1) J total angular momentum B x x quantum number function J (x)  3J (4.148) L(x)(J 1) L(x) Langevin function =cothx−1/x (see page 144) B (x)=tanhx (4.149) 1/2 M mean magnetisation n number density of atoms gJ Lande´ g-factor Mean µBB M = nµ Jg B Jg (4.150) µB Bohr magneton magnetisationa B J J J kT B magnetic ﬂux density k Boltzmann constant M for isolated µ B temperature M = nµ tanh B (4.151) T 1/2 B spins (J =1/2) kT M 1/2 mean magnetisation for J =1/2 (and gJ =2) aOf an ensemble of atoms in thermal equilibrium at temperature T , each with total angular momentum quantum number J. main January 23, 2006 16:6

102 Quantum physics

4.6 Perturbation theory

Time-independent perturbation theory Hˆ unperturbed Hamiltonian ˆ 0 Unperturbed H0ψn = Enψn (4.152) ψn eigenfunctions of Hˆ 0 ˆ states (ψn nondegenerate) En eigenvalues of H0 n integer ≥ 0 Perturbed ˆ ˆ ˆ Hˆ perturbed Hamiltonian H = H0 +H (4.153) Hamiltonian Hˆ perturbation ( Hˆ 0) | ˆ | Ek = Ek + ψk H ψk Perturbed E perturbed eigenvalue ( E ) 2 k k a |ψ |Hˆ |ψ | eigenvalues + k n +... (4.154) || Dirac bracket Ek −En n= k Perturbed | ˆ | ψk H ψn ψ perturbed eigenfunction eigen- ψ = ψ + ψ +... (4.155) k k k n ( ) b Ek −En ψk functions n= k aTo second order. bTo ﬁrst order.

Time-dependent perturbation theory Hˆ unperturbed Hamiltonian Unperturbed 0 ψn eigenfunctions of Hˆ 0 stationary Hˆ 0ψn = Enψn (4.156) E eigenvalues of Hˆ states n 0 n integer ≥ 0 Hˆ perturbed Hamiltonian Perturbed Hˆ (t)=Hˆ +Hˆ (t) (4.157) Hˆ (t) perturbation ( Hˆ ) Hamiltonian 0 0 t time

ˆ ˆ ∂Ψ(t) Ψ wavefunction Schrodinger¨ [H0 +H (t)]Ψ(t)=ih¯ (4.158) ∂t ψ initial state equation 0 Ψ(t =0)=ψ0 (4.159) h¯ (Planck constant)/(2π) Perturbed Ψ(t)= cn(t)ψn exp(−iEnt/h¯) (4.160)

wave- n cn probability amplitudes a where function t −i cn = ψn|Hˆ (t )|ψ0 exp[i(En −E0)t /h¯]dt (4.161) h¯ 0

Γi→f transition probability per Fermi’s 2π 2 unit time from state i to Γ → = |ψ |Hˆ |ψ | ρ(E ) (4.162) golden rule i f h¯ f i f state f ρ(Ef ) density of ﬁnal states aTo ﬁrst order. main January 23, 2006 16:6

4.7 High energy and nuclear physics 103

4.7 High energy and nuclear physics

Nuclear decay N(t) number of nuclei Nuclear decay − N(t)=N(0)e λt (4.163) remaining after time t law t time

ln2 λ decay constant Half-life and T1/2 = (4.164) λ T half-life mean life 1/2 T =1/λ (4.165) T mean lifetime

Successive decays 1 → 2 → 3 (species 3 stable)

−λ1t N1(t)=N1(0)e (4.166) 4 − − N1 population of species 1 (0) (e λ1t −e λ2t) −λ2t N1 λ1 N2(t)=N2(0)e + (4.167) N2 population of species 2 λ2 −λ1 N3 population of species 3 − − e λ2t − e λ1t → −λ2t λ1 λ2 λ1 decay constant 1 2 N3(t)=N3(0)+N2(0)(1−e )+N1(0) 1+ λ2 −λ1 λ2 decay constant 2 → 3 (4.168) v velocity of α particle 3 Geiger’s lawa v = a(R −x) (4.169) x distance from source a constant Geiger–Nuttall R range logλ = b+clogR (4.170) rule b, c constants for each series α, β, and γ aFor α particles in air (empirical).

Nuclear binding energy N number of neutrons Liquid drop modela A mass number (= N +Z) Z 2 (N −Z)2 B semi-empirical binding energy B = a A−a A2/3 −a −a +δ(A) v s c A1/3 a A Z number of protons (4.171) ∼  av volume term ( 15.8MeV) −3/4 as surface term (∼ 18.0MeV) +apA Z, N both even a Coulomb term (∼ 0.72MeV) − −3/4 c δ(A) apA Z, N both odd (4.172) ∼  aa asymmetry term ( 23.5MeV) 0otherwise ap pairing term (∼ 33.5MeV)

M(Z,A) atomic mass Semi-empirical M(Z,A)=ZM +Nm −B (4.173) M mass of hydrogen atom mass formula H n H mn neutron mass aCoeﬃcient values are empirical and approximate. main January 23, 2006 16:6

104 Quantum physics

Nuclear collisions σ(E) cross-section for a+b → c π ΓabΓc σ(E)= 2 g 2 2 (4.174) k incoming wavenumber Breit–Wigner k (E −E0) +Γ /4 g spin factor formulaa 2J +1 g = (4.175) E total energy (PE + KE) (2s +1)(2s +1) a b E0 resonant energy Γ width of resonant state R

Total width Γ=Γab +Γc (4.176) Γab partial width into a+b Γc partial width into c τ resonance lifetime Resonance h¯ τ = (4.177) J total angular momentum lifetime Γ quantum number of R sa,b spins of a and b dσ dΩ diﬀerential collision cross-section ∞ 2 dσ 2µ sinKr 2 µ reduced mass Born scattering = V (r)r dr b 2 | − | formula dΩ h¯ 0 Kr K = kin kout (see footnote) (4.178) r radial distance V (r) potential energy of interaction Mott scattering formulac h¯ (Planck constant)/2π dσ α 2 χ χ Acos α lntan2 χ = csc4 +sec4 + hv¯ 2 α/r scattering potential energy dΩ 4E 2 2 2 χ χ sin 2 cos 2 χ scattering angle (4.179) v closing velocity − dσ α 2 4−3sin2 χ A = 2 for spin-zero particles, = 1 (A = −1,α vh¯) (4.180) for spin-half particles dΩ 2E sin4 χ aFor the reaction a+b ↔ R → c in the centre of mass frame. bFor a central ﬁeld. The Born approximation holds when the potential energy of scattering, V , is much less than the total kinetic energy. K is the magnitude of the change in the particle’s wavevector due to scattering. cFor identical particles undergoing Coulomb scattering in the centre of mass frame. Nonidentical particles obey the Rutherford scattering formula (page 72).

Relativistic wave equationsa Klein–Gordon ψ wavefunction equation ∂2ψ (∇2 −m2)ψ = (4.181) m particle mass (massive, spin 2 ∂t t time zero particles) Weyl equations ψ spinor wavefunction ∂ψ ± ∂ψ ∂ψ ∂ψ (massless, spin = σx +σy +σz (4.182) σi Pauli spin matrices 1/2 particles) ∂t ∂x ∂y ∂z (see page 26) ii2 = −1 (iγµ∂µ−m)ψ = 0 (4.183) γµ Dirac matrices: Dirac equation ∂ ∂ ∂ ∂ 0 12 0 γ = − (massive, spin where ∂µ= , , , (4.184) 0 12 ∂t ∂x ∂y ∂z 0 σ 1/2 particles) γi = i 0 2 1 2 2 2 3 2 − 0 (γ ) = 14 ;(γ ) =(γ ) =(γ ) = −14 (4.185) σi 1n n×n unit matrix aWritten in natural units, with c =¯h =1. main January 23, 2006 16:6

Chapter 5 Thermodynamics

5.1 Introduction The term thermodynamics is used here loosely and includes classical thermodynamics, statistical thermodynamics, thermal physics, and radiation processes. Notation in these subjects can be confusing and the conventions used here are those found in the majority of modern treatments. In particular:

• The internal energy of a system is defined in terms of the heat supplied to the system plus the work done on the system, that is, dU =dQ+dW . 5 • The lowercase symbol p is used for pressure. Probability density functions are denoted by pr(x) and microstate probabilities by pi. • With the exception of specific intensity, quantities are taken as specific if they refer to unit mass and are distinguished from the extensive equivalent by using lowercase. Hence specific volume, v,equalsV/m,whereV is the volume of gas and m its mass. Also, the specific heat capacity of a gas at constant pressure is cp = Cp/m,whereCp is the heat capacity of mass m of gas. Molar values take a subscript “m” (e.g., Vm for molar volume) and remain in upper case.

• The component held constant during a partial diﬀerentiation is shown after a vertical bar; hence ∂V is the partial diﬀerential of volume with respect to pressure, holding temperature ∂p T constant.

The thermal properties of solids are dealt with more explicitly in the section on solid state physics (page 123). Note that in solid state literature speciﬁc heat capacity is often taken to mean heat capacity per unit volume. main January 23, 2006 16:6

106 Thermodynamics

5.2 Classical thermodynamics

Thermodynamic laws T thermodynamic temperature Thermodynamic T ∝ lim(pV ) (5.1) V volume of a ﬁxed mass of gas temperaturea p→0 p gas pressure

lim(pV )T K kelvin unit Kelvin p→0 T/K = 273.16 (5.2) tr temperature of the triple point temperature scale lim(pV )tr p→0 of water dU change in internal energy First lawb dU =dQ+dW (5.3) dW work done on system dQ heat supplied to system S experimental entropy dQrev dQ Entropyc dS = ≥ (5.4) T temperature T T rev reversible change aAs determined with a gas thermometer. The idea of temperature is associated with the zeroth law of thermodynamics: If two systems are in thermal equilibrium with a third, they are also in thermal equilibrium with each other. bThe d notation represents a diﬀerential change in a quantity that is not a function of state of the system. cAssociated with the second law of thermodynamics: No process is possible with the sole eﬀect of completely converting heat into work (Kelvin statement).

Thermodynamic worka Hydrostatic p (hydrostatic) pressure dW = −p dV (5.5) pressure dV volume change dW work done on the system Surface tension dW = γ dA (5.6) γ surface tension dA change in area E electric field Electric field dW = E · dp (5.7) dp induced electric dipole moment B magnetic flux density Magnetic field dW = B · dm (5.8) dm induced magnetic dipole moment

∆φ potential diﬀerence Electric current dW =∆φ dq (5.9) dq charge moved aThe sources of electric and magnetic ﬁelds are taken as being outside the thermodynamic system on which they are working. main January 23, 2006 16:6

5.2 Classical thermodynamics 107

Cycle eﬃciencies (thermodynamic)a η eﬃciency work extracted Th −Tl Heat engine η = ≤ (5.10) Th higher temperature heat input Th Tl lower temperature heat extracted T Refrigerator η = ≤ l (5.11) work done Th −Tl heat supplied T Heat pump η = ≤ h (5.12) work done Th −Tl − V1 γ 1 V compression ratio b work extracted V2 2 Otto cycle η = =1− (5.13) γ ratio of heat capacities heat input V1 (assumed constant) a The equalities are for reversible cycles, such as Carnot cycles, operating between temperatures Th and Tl. bIdealised reversible “petrol” (heat) engine.

Heat capacities

CV heat capacity, V constant Q heat Constant dQ ∂U ∂S CV = = = T (5.14) T temperature volume dT V ∂T V ∂T V V volume 5 U internal energy S entropy Constant dQ ∂H ∂S Cp heat capacity, p constant Cp = = = T (5.15) pressure dT p ∂T p ∂T p p pressure H enthalpy ∂U ∂V Cp −CV = +p (5.16) Diﬀerence in ∂V T ∂T p βp isobaric expansivity 2 heat capacities VTβ κT isothermal compressibility = p (5.17) κT Ratio of heat C κ γ ratio of heat capacities γ = p = T (5.18) capacities CV κS κS adiabatic compressibility

Thermodynamic coeﬃcients β isobaric expansivity Isobaric 1 ∂V p a βp = (5.19) V volume expansivity V ∂T p T temperature Isothermal 1 ∂V κT isothermal compressibility κT = − (5.20) compressibility V ∂p T p pressure Adiabatic 1 ∂V κS = − (5.21) κS adiabatic compressibility compressibility V ∂p S Isothermal bulk 1 ∂p KT = = −V (5.22) KT isothermal bulk modulus modulus κT ∂V T Adiabatic bulk 1 ∂p KS = = −V (5.23) KS adiabatic bulk modulus modulus κS ∂V S a Also called “cubic expansivity” or “volume expansivity.” The linear expansivity is αp = βp/3. main January 23, 2006 16:6

108 Thermodynamics

Expansion processes ∂T T 2 ∂(p/T ) η Joule coeﬃcient η = = − (5.24) T temperature Joule ∂V U C ∂T V V p pressure expansiona 1 ∂p = − T −p (5.25) U internal energy V CV ∂T CV heat capacity, V constant ∂T T 2 ∂(V/T) µ = = (5.26) µ Joule–Kelvin coeﬃcient Joule–Kelvin ∂p H Cp ∂T p V volume b expansion 1 ∂V H enthalpy = T −V (5.27) Cp heat capacity, p constant Cp ∂T p aExpansion with no change in internal energy. bExpansion with no change in enthalpy. Also known as a “Joule–Thomson expansion” or “throttling” process.

Thermodynamic potentialsa U internal energy T temperature Internal energy dU = T dS −pdV +µdN (5.28) S entropy µ chemical potential N number of particles H = U +pV (5.29) H enthalpy Enthalpy p pressure dH = T dS +V dp+µdN (5.30) V volume

Helmholtz free F = U −TS (5.31) F Helmholtz free energy energyb dF = −S dT −pdV +µdN (5.32)

G = U −TS+pV (5.33) Gibbs free energyc = F +pV = H −TS (5.34) G Gibbs free energy dG = −S dT +V dp+µdN (5.35)

Φ=F −µN (5.36) Grand potential Φ grand potential dΦ = −S dT −pdV −N dµ (5.37)

Gibbs–Duhem −S dT +V dp−N dµ = 0 (5.38) relation A availability A = U −T0S +p0V (5.39) Availability T0 temperature of dA =(T −T0)dS −(p−p0)dV (5.40) surroundings p0 pressure of surroundings a dN=0 for a closed system. bSometimes called the “work function.” cSometimes called the “thermodynamic potential.” main January 23, 2006 16:6

5.2 Classical thermodynamics 109

Maxwell’s relations U internal energy ∂T ∂p ∂2U Maxwell 1 = − = (5.41) T temperature ∂V S ∂S V ∂S∂V V volume H enthalpy ∂T ∂V ∂2H Maxwell 2 = = (5.42) S entropy ∂p S ∂S p ∂p∂S p pressure ∂p ∂S ∂2F Maxwell 3 = = (5.43) F Helmholtz free energy ∂T V ∂V T ∂T∂V ∂V ∂S ∂2G Maxwell 4 = − = (5.44) G Gibbs free energy ∂T p ∂p T ∂p∂T

Gibbs–Helmholtz equations F Helmholtz free energy 2 ∂(F/T) U = −T (5.45) U internal energy ∂T V G Gibbs free energy ∂(F/V) 5 G = −V 2 (5.46) H enthalpy ∂V T T temperature ∂(G/T ) H = −T 2 (5.47) p pressure ∂T p V volume

Phase transitions L (latent) heat absorbed (1 → 2) Heat absorbed L = T (S2 −S1) (5.48) T temperature of phase change S entropy

dp S2 −S1 = (5.49) p pressure Clausius–Clapeyron dT V2 −V1 V volume equationa L = (5.50) 1,2 phase states T (V2 −V1) −L Coexistence curveb p(T ) ∝ exp (5.51) R molar gas constant RT

dp βp2 −βp1 = (5.52) βp isobaric expansivity Ehrenfest’s dT κT 2 −κT 1 c κT isothermal compressibility equation 1 Cp2 −Cp1 = (5.53) Cp heat capacity (p constant) VT βp2 −βp1 P number of phases in equilibrium Gibbs’s phase rule P+F = C+2 (5.54) F number of degrees of freedom C number of components aPhase boundary gradient for a ﬁrst-order transition. Equation (5.50) is sometimes called the “Clapeyron equation.” b For V2 V1, e.g., if phase 1 is a liquid and phase 2 a vapour. cFor a second-order phase transition. main January 23, 2006 16:6

110 Thermodynamics

5.3 Gas laws

Ideal gas U internal energy Joule’s law U = U(T ) (5.55) T temperature p pressure Boyle’s law pV |T = constant (5.56) V volume Equation of state n number of moles pV = nRT (5.57) (Ideal gas law) R molar gas constant

pV γ = constant (5.58) (γ−1) TV = constant (5.59) ratio of heat capacities Adiabatic γ γ (1−γ) (C /C ) equations T p = constant (5.60) p V ∆W work done on system 1 ∆W = (p V −p V ) (5.61) γ −1 2 2 1 1 nRT Internal energy U = (5.62) γ −1 Reversible ∆Q heat supplied to system isothermal ∆Q = nRT ln(V2/V1) (5.63) 1,2 initial and ﬁnal states expansion

a ∆S change in entropy of the Joule expansion ∆S = nRln(V2/V1) (5.64) system aSince ∆Q = 0 for a Joule expansion, ∆S is due entirely to irreversibility. Because entropy is a function of state it has the same value as for the reversible isothermal expansion, where ∆S =∆Q/T .

Virial expansion p pressure B2(T ) pV = RT 1+ V volume V Virial expansion (5.65) R molar gas constant B3(T ) ··· T temperature + 2 + V Bi virial coeﬃcients Boyle B (T ) = 0 (5.66) T Boyle temperature temperature 2 B B main January 23, 2006 16:6

5.3 Gas laws 111

Van der Waals gas p pressure V molar volume a m Equation of state − R molar gas constant p+ 2 (Vm b)=RT (5.67) Vm T temperature a,b van der Waals’ constants

Tc =8a/(27Rb) (5.68) Tc critical temperature 2 Critical point pc = a/(27b ) (5.69) pc critical pressure

Vmc =3b (5.70) Vmc critical molar volume

p = p/p Reduced equation 3 r c p + (3V −1) = 8T (5.71) Vr = Vm/Vmc of state r V 2 r r r Tr = T/Tc

Dieterici gas p pressure 5 V molar volume RT −a m Equation of state R molar gas constant p = − exp (5.72) Vm b RTVm T temperature a,b Dieterici’s constants Tc = a /(4Rb ) (5.73) Tc critical temperature 2 2 pc critical pressure Critical point pc = a /(4b e ) (5.74) Vmc critical molar volume Vmc =2b (5.75) e=2.71828... pr = p/pc Reduced equation Tr 2 pr = exp 2− (5.76) Vr = Vm/Vmc of state 2Vr −1 VrTr Tr = T/Tc

Van der Waals gas Dieterici gas 2 1.4 1.1 Tr =1.2 1.8 1.2 1.6 Tr =1.2 1.0 1 1.4 1.1 1.2 1.0 r 0.8 r

p p 1 0.9 0.6 0.8 0.9 0.4 0.6 0.8 0.4 0.2 0.8 0.2 0 0 0 1 2 3 4 5 0 1 2 3 4 5 Vr Vr main January 23, 2006 16:6

112 Thermodynamics

5.4 Kinetic theory Monatomic gas p pressure n number density = N/V 1 2 Pressure p = nmc (5.77) m particle mass 3 c2 mean squared particle velocity V volume Equation of k Boltzmann constant state of an ideal pV = NkT (5.78) N number of particles gas T temperature 3 N Internal energy U = NkT = mc2 (5.79) U internal energy 2 2 3 C = Nk (5.80) V 2 5 CV heat capacity, constant V Heat capacities C = C +Nk= Nk (5.81) C heat capacity, constant p p V 2 p γ ratio of heat capacities C 5 γ = p = (5.82) CV 3 Entropy S entropy (Sackur– mkT 3/2 V S = Nkln e5/2 (5.83) h¯ = (Planck constant)/(2π) Tetrode 2 2πh¯ N e=2.71828... equation)a 3/2 aFor the uncondensed gas. The factor mkT is the quantum concentration of the particles, n . Their thermal de 2πh¯2 Q −1/3 Broglie wavelength, λT , approximately equals nQ .

Maxwell–Boltzmann distributiona pr probability density m 3/2 −mc2 m particle mass Particle speed pr(c)dc = exp 4πc2 dc k Boltzmann constant distribution 2πkT 2kT (5.84) T temperature c particle speed 1/2 Particle energy 2E −E E particle kinetic pr(E)dE = exp dE (5.85) 2 distribution π1/2(kT)3/2 kT energy (= mc /2) 8kT 1/2 Mean speed c = (5.86) c mean speed πm 1/2 1/2 3kT 3π c root mean squared rms speed c = = c (5.87) rms rms m 8 speed Most probable 2kT 1/2 π 1/2 cˆ = = c (5.88) cˆ most probable speed speed m 4 a ∞ Probability density functions normalised so that 0 pr(x)dx =1. main January 23, 2006 16:6

5.4 Kinetic theory 113

Transport properties 1 l mean free path Mean free patha l = √ (5.89) d molecular diameter 2 2πd n n particle number density Survival pr probability pr(x)=exp(−x/l) (5.90) equationb x linear distance

Flux through a 1 J molecular ﬂux J = nc (5.91) planec 4 c mean molecular speed

Self-diffusion J = −D∇n (5.92) (Fick’s law of 2 D diffusion coefficient d where D lc (5.93) diffusion) 3

H = −λ∇T (5.94) H heat flux per unit area λ thermal conductivity 1 ∂T Thermal ∇2T = (5.95) T temperature d conductivity D ∂t ρ density 5 c specific heat capacity, V for monatomic gas λ ρlc cV (5.96) V 4 constant η dynamic viscosity 5 d 1 Viscosity η ρlc (5.97) x displacement of sphere in 2 x direction after time t Brownian k Boltzmann constant kTt motion (of a x2 = (5.98) t time interval 3πηa sphere) a sphere radius dM mass flow rate dt 3 1/2 Free molecular dM 4R 2πm p p Rp pipe radius = p 1 − 2 flow (Knudsen dt 3L k 1/2 1/2 L pipe length e T1 T2 flow) (5.99) m particle mass p pressure aFor a perfect gas of hard, spherical particles with a Maxwell–Boltzmann speed distribution. bProbability of travelling distance x without a collision. cFrom the side where the number density is n, assuming an isotropic velocity distribution. Also known as “collision number.” dSimplistic kinetic theory yields numerical coefficients of 1/3forD, λ and η. e Through a pipe from end 1 to end 2, assuming Rp l (i.e., at very low pressure).

Gas equipartition

Eq energy per quadratic degree of Classical 1 freedom E = kT (5.100) equipartitiona q 2 k Boltzmann constant T temperature 1 1 CV heat capacity, V constant CV = fNk= fnR (5.101) C heat capacity, p constant 2 2 p N number of molecules Ideal gas heat f C = Nk 1+ (5.102) f number of degrees of freedom capacities p 2 n number of moles C 2 γ = p =1+ (5.103) R molar gas constant CV f γ ratio of heat capacities aSystem in thermal equilibrium at temperature T . main January 23, 2006 16:6

114 Thermodynamics

5.5 Statistical thermodynamics

Statistical entropy S entropy Boltzmann S = klnW (5.104) k Boltzmann constant formulaa klng(E) (5.105) W number of accessible microstates g(E) density of microstates with energy E b − i sum over microstates Gibbs entropy S = k pi lnpi (5.106) p probability that the system is in microstate i i i N two-level N! N number of systems W = (5.107) systems (N −n)!n! n number in upper state N harmonic (Q+N −1)! W = (5.108) Q total number of energy quanta available oscillators Q!(N −1)! aSometimes called “conﬁgurational entropy.” Equation (5.105) is true only for large systems. bSometimes called “canonical entropy.”

Ensemble probabilities

pi probability that the system is in Microcanonical 1 microstate i p = (5.109) ensemblea i W W number of accessible microstates Z partition function −βE sum over microstates Partition functionb Z = e i (5.110) i β =1/(kT) i Ei energy of microstate i Canonical ensemble 1 − k Boltzmann constant (Boltzmann p = e βEi (5.111) i T temperature distribution)c Z Ξ grand partition function − − Grand partition Ξ= e β(Ei µNi) (5.112) µ chemical potential function i Ni number of particles in microstate i Grand canonical 1 −β(Ei−µNi) ensemble (Gibbs pi = e (5.113) distribution)d Ξ aEnergy fixed. bAlso called “sum over states.” cTemperature fixed. dTemperature fixed. Exchange of both heat and particles with a reservoir. main January 23, 2006 16:6

5.5 Statistical thermodynamics 115

Macroscopic thermodynamic variables F Helmholtz free energy Helmholtz free k Boltzmann constant F = −kT lnZ (5.114) energy T temperature Z partition function Φ grand potential Grand potential Φ=−kT lnΞ (5.115) Ξ grand partition function ∂lnZ U internal energy Internal energy U = F +TS= − (5.116) ∂β V,N β =1/(kT) ∂F ∂(kT lnZ) S entropy Entropy S = − = (5.117) ∂T V,N ∂T V,N N number of particles ∂F ∂(kT lnZ) Pressure p = − = (5.118) p pressure ∂V T,N ∂V T,N Chemical ∂F ∂(kT lnZ) µ = = − (5.119) µ chemical potential potential ∂N V,T ∂N V,T

Identical particles 5

Bose–Einstein distribution Fermi–Dirac distribution 2 1.2 1.8 (µ =0) (µ =1) 1.6 1 50 1.4 5 0.8 10 1.2 β =1 i i f 1 5 f 0.6 β =1 0.8 0.4 0.6 10 0.4 50 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 i i Bose–Einstein 1 fi mean occupation number of ith state a fi = − (5.120) distribution β(i µ) − e 1 β =1/(kT)

Fermi–Dirac 1 i energy quantum for ith state b fi = − (5.121) distribution eβ(i µ) +1 µ chemical potential F Fermi energy 2 2 2/3 c h¯ 6π n h¯ (Planck constant)/(2π) Fermi energy = (5.122) F 2m g n particle number density m particle mass g spin degeneracy (= 2s+1) Bose 2πh¯2 n 2/3 ζ Riemann zeta function condensation T = (5.123) ζ(3/2) 2.612 c mk gζ(3/2) temperature Tc Bose condensation temperature a For bosons. fi ≥ 0. b For fermions. 0 ≤ fi ≤ 1. c For noninteracting particles. At low temperatures, µ F. main January 23, 2006 16:6

116 Thermodynamics

Population densitiesa nij number density of atoms in nmj gmj −(χmj −χlj) Boltzmann = exp (5.124) excitation level i of ionisation n g kT state j (j = 0 if not ionised) excitation lj lj − gij level degeneracy equation gmj hνlm = exp (5.125) χij excitation energy relative to glj kT the ground state − χij ν photon transition frequency Zj (T )= gij exp (5.126) ij Partition kT h Planck constant i function nij gij −χij k Boltzmann constant = exp (5.127) T temperature Nj Zj (T ) kT

Saha equation (general) Zj partition function for ionisation state j g h3 χ −χ ij −3/2 Ij ij total number density in nij = n0,j+1ne (2πmekT) exp (5.128) Nj g0,j+1 2 kT ionisation state j Saha equation (ion populations) ne electron number density 3 me electron mass Nj Zj (T ) h −3/2 χIj = ne (2πmekT) exp (5.129) χIj ionisation energy of atom in Nj+1 Zj+1(T ) 2 kT ionisation state j aAll equations apply only under conditions of local thermodynamic equilibrium (LTE). In atoms with no magnetic splitting, the degeneracy of a level with total angular momentum quantum number J is gij =2J +1.

5.6 Fluctuations and noise Thermodynamic fluctuationsa pr probability density pr(x) ∝ exp[S(x)/k] (5.130) Fluctuation x unconstrained variable −A(x) probability ∝ exp (5.131) S entropy kT A availability − var[·] meansquaredeviation General ∂2A(x) 1 var[ ]= (5.132) k Boltzmann constant variance x kT 2 ∂x T temperature Temperature ∂T kT 2 V volume var[T ]=kT = (5.133) fluctuations ∂S V CV CV heat capacity, V constant Volume ∂V p pressure var[V ]=−kT = κT VkT (5.134) fluctuations ∂p T κT isothermal compressibility Entropy ∂S var[S]=kT = kCp (5.135) Cp heat capacity, p constant fluctuations ∂T p Pressure ∂p KS kT var[p]=−kT = (5.136) KS adiabatic bulk modulus fluctuations ∂V S V Density n2 n2 var[n]= var[V ]= κ kT (5.137) n number density fluctuations V 2 V T aIn part of a large system, whose mean temperature is fixed. Quantum effects are assumed negligible. main January 23, 2006 16:6

5.6 Fluctuations and noise 117

Noise w exchangeable noise power k Boltzmann constant · β − −1 dw = kT β(e 1) dν (5.138) T temperature Nyquist’s noise = kT dν (5.139) T noise temperature theorem N N kT dν (hν kT) (5.140) β = hν/(kT) ν frequency h Planck constant

Johnson vrms rms noise voltage 1/2 (thermal) noise vrms =(4kTNR∆ν) (5.141) R resistance voltagea ∆ν bandwidth

Irms rms noise current Shot noise 1/2 I =(2eI ∆ν) (5.142) −e electronic charge (electrical) rms 0 I mean current 0 fdB noise figure (decibels) b TN Noise figure fdB =10log10 1+ (5.143) T0 ambient temperature (usually T0 taken as 290 K) P2 G decibel gain of P2 over P1 Relative power G =10log10 (5.144) P1 P1,P2 power levels aThermal voltage over an open-circuit resistance. 5 b Noise figure can also be defined as f =1+TN/T0, when it is also called “noise factor.” main January 23, 2006 16:6

118 Thermodynamics

5.7 Radiation processes

Radiometrya

Qe radiant energy Le radiance (generally a function of position b Radiant energy Qe = Le cosθ dA dΩ dt J (5.145) and direction) θ angle between dir. of dΩ and normal to dA Ω solid angle

∂Qe Φ = W (5.146) A area Radiant ﬂux e ∂t t time (“radiant power”) = Le cosθ dA dΩ (5.147) Φe radiant ﬂux

Radiant energy ∂Q We radiant energy density W = e Jm−3 (5.148) densityc e dV diﬀerential volume of ∂V propagation medium

∂Φe −2 Me = Wm (5.149) d ∂A Radiant exitance Me radiant exitance = Le cosθdΩ (5.150) z (normal) ∂Φe −2 θ dΩ Ee = Wm (5.151) e ∂A Irradiance dA = Le cosθdΩ (5.152) x φ y

∂Φe −1 Ie = Wsr (5.153) Ω Ee irradiance Radiant intensity ∂ Ie radiant intensity = Le cosθ dA (5.154)

2 1 ∂ Φe −2 −1 Le = Wm sr (5.155) Radiance cosθ dAdΩ 1 ∂I = e (5.156) cosθ ∂A aRadiometry is concerned with the treatment of light as energy. bSometimes called “total energy.” Note that we assume opaque radiant surfaces, so that 0 ≤ θ ≤ π/2. cThe instantaneous amount of radiant energy contained in a unit volume of propagation medium. d Power per unit area leaving a surface. For a perfectly diﬀusing surface, Me = πLe. ePower per unit area incident on a surface. main January 23, 2006 16:6

5.7 Radiation processes 119

Photometrya

Qv luminous energy Lv luminance (generally a Luminous energy function of position Q = L cosθ dA dΩ dt lms (5.157) and direction) (“total light”) v v θ angle between dir. of dΩ and normal to dA Ω solid angle

∂Qv Φ = lumen (lm) (5.158) A area v ∂t Luminous ﬂux t time = Lv cosθ dA dΩ (5.159) Φv luminous ﬂux

Luminous ∂Q W luminous density W = v lmsm−3 (5.160) v densityb v ∂V V volume ∂Φ M = v lx (lmm−2) (5.161) Luminous v ∂A M luminous exitance exitancec v = Lv cosθdΩ (5.162) z (normal) θ dΩ 5 ∂Φv −2 Ev = lmm (5.163) Illuminance ∂A d dA (“illumination”) x = Lv cosθdΩ (5.164) φ y

∂Φv Iv = cd (5.165) Luminous ∂Ω Ev illuminance e intensity Iv luminous intensity = Lv cosθ dA (5.166)

2 1 ∂ Φv −2 Luminance Lv = cdm (5.167) (“photometric cosθ dAdΩ 1 ∂I brightness”) = v (5.168) cosθ ∂A K luminous efficacy Φ L I L radiance Luminous efficacy K = v = v = v lmW−1 (5.169) e Φe Le Ie Φe radiant flux Ie radiant intensity V luminous efficiency Luminous K(λ) V (λ)= (5.170) λ wavelength efficiency Kmax Kmax spectral maximum of K(λ) aPhotometry is concerned with the treatment of light as seen by the human eye. bThe instantaneous amount of luminous energy contained in a unit volume of propagating medium. cLuminous emitted flux per unit area. dLuminous incident flux per unit area. The derived SI unit is the lux (lx). 1lx = 1lmm−2. eThe SI unit of luminous intensity is the candela (cd). 1cd = 1lmsr−1. main January 23, 2006 16:6

120 Thermodynamics

Radiative transfera

z (normal) dΩ Flux density −2 −1 θ (through a Fν = Iν (θ,φ)cosθ dΩ Wm Hz plane) (5.171) x φ y Fν ﬂux density b 1 −2 −1 Iν speciﬁc intensity Mean intensity J = I (θ,φ)dΩ Wm Hz (5.172) − − − ν 4π ν (Wm 2 Hz 1 sr 1) Jν mean intensity uν spectral energy density Spectral energy 1 −3 −1 Ω solid angle c uν = Iν (θ,φ)dΩ Jm Hz (5.173) density c θ angle between normal and direction of Ω

jν specific emission Specific coefficient ν −1 −1 −1 emission jν = Wkg Hz sr (5.174) ν emission coefficient ρ −3 −1 −1 coefficient (Wm Hz sr ) ρ density Gas linear αν linear absorption coefficient absorption 1 −1 αν = nσν = m (5.175) n particle number density coefficient lν σν particle cross section (αν 1) lν mean free path α d ν −1 2 Opacity κν = kg m (5.176) κ opacity ρ ν τν optical depth, or Optical depth τν = κν ρ ds (5.177) optical thickness ds line element 1 dI ν = −κ I +j (5.178) Transfer ρ ds ν ν ν equatione dI or ν = −α I + (5.179) ds ν ν ν

f jν ν Kirchhoff’s law Sν ≡ = (5.180) Sν source function κν αν Emission from −τν a homogeneous Iν = Sν (1−e ) (5.181) medium aThe definitions of these quantities vary in the literature. Those presented here are common in meteorology and astrophysics. Note particularly that the ambiguous term specific is taken to mean “per unit frequency interval” in the case of specific intensity and “per unit mass per unit frequency interval” in the case of specific emission coefficient. b In radio astronomy, flux density is usually taken as S =4πJν . cAssuming a refractive index of 1. dOr “mass absorption coefficient.” eOr “Schwarzschild’s equation.” f Under conditions of local thermal equilibrium (LTE), the source function, Sν , equals the Planck function, Bν (T ) [see Equation (5.182)]. main January 23, 2006 16:6

5.7 Radiation processes 121

Blackbody radiation

1050 1010 ) 1010 K ) 40 − 1 10 5 − 1 10 sr 10 10 K 9 sr νm(T )=c/λm(T ) 10 K λm(T ) − 1

− 1 30 9 8 10 10 K

10 K m Hz 1 108 K

7 − 2 − 2 10 K 1020 107 K −5 6 10 10 K 6 Wm

Wm 10 K / /

5 λ ν 10 K 10 105 K −10 10 10 4 10 K 104 K 3 − 10 K 1 3 10 15 10 K 100K 100K −10 −20 10 brightness ( B

brightness ( B 10 2.7K 2.7K 10−20 106 108 1010 1012 1014 1016 1018 1020 1022 10−14 10−12 10−10 10−8 10−6 10−4 10−2 1 102 frequency (ν/Hz) wavelength (λ/m) 3 −1 2hν hν surface brightness per B (T )= exp −1 (5.182) Bν ν c2 kT unit frequency (Wm−2 Hz−1 sr−1) Planck dν 5 a Bλ(T )=Bν (T ) (5.183) Bλ surface brightness per function dλ unit wavelength − −2 −1 −1 2hc2 hc 1 (Wm m sr ) = exp −1 (5.184) λ5 λkT h Planck constant

4π −3 −1 c speed of light uν (T )= Bν (T )JmHz (5.185) Spectral energy c k Boltzmann constant density 4π −3 −1 T temperature uλ(T )= Bλ(T )Jmm (5.186) c uν,λ spectral energy density Rayleigh–Jeans 2kT 2kT B (T )= ν2 = (5.187) law (hν kT) ν c2 λ2 Wien’s law 2hν3 −hν B (T )= exp (5.188) (hν kT) ν c2 kT Wien’s −3 5.1×10 mK for B λ wavelength of displacement λ T = ν (5.189) m m × −3 maximum brightness law 2.9 10 mK for Bλ ∞ M exitance Stefan– M = π Bν (T )dν (5.190) Boltzmann 0 σ Stefan–Boltzmann 5 4 constant ( b 2π k 4 4 −2 law = T = σT Wm (5.191) 5.67×10−8 Wm−2 K−4) 15c2h3

4 − Energy density u(T )= σT 4 Jm 3 (5.192) u energy density c mean emissivity Greybody M = σT 4 =(1−A)σT 4 (5.193) A albedo aWith respect to the projected area of the surface. Surface brightness is also known simply as “brightness.” “Speciﬁc intensity” is used for reception. bSometimes “Stefan’s law.” Exitance is the total radiated energy from unit area of the body per unit time. main January 23, 2006 16:6 main January 23, 2006 16:6

Chapter 6 Solid state physics

6.1 Introduction This section covers a few selected topics in solid state physics. There is no attempt to do more than scratch the surface of this vast ﬁeld, although the basics of many undergraduate texts on the subject are covered. In addition a period table of elements, together with some of their physical properties, is displayed on the next two pages.

Periodic table (overleaf) Data for the periodic table of elements are taken from Pure Appl. Chem., 71, 1593–1607 (1999), from the 16th edition of Kaye and Laby Tables of Physical and Chemical Constants (Longman, 1995) and from the 74th edition of the CRC Handbook of Chemistry and Physics (CRC Press, 1993). Note that melting and boiling points have been converted to kelvins by adding 273.15 to the Celsius values listed in Kaye and Laby. The standard atomic masses reflect the relative isotopic abundances in samples found naturally on Earth, and the number of significant figures reflect the variations between samples. Elements with atomic masses shown in square brackets have no stable nuclides, and the values reflect the mass numbers of the longest-lived isotopes. Crystallographic data are based on the most common forms of the elements (the α-form, unless stated otherwise) stable under standard conditions. Densities are for the solid state. For full details and footnotes for each element, the reader is advised to consult the original texts. Elements 110, 111, 112 and 114 are known to exist but their names are not yet permanent. main January 23, 2006 16:6

124 Solid state physics

6.2 Periodic table 1 Hydrogen name 1.007 94 1 H 1 1s1 atomic number relative atomic mass (u) 89 (β)378 Titanium HEX 1.632 electron conﬁguration 47.867 symbol 13.80 20.28 2 22 Ti Lithium Beryllium [Ca]3d2 6.941 9.012 182 density (kgm−3) 4 508 295 lattice constant, a (fm) 3 Li 4 Be HEX 1.587 2 [He]2s1 [He]2s2 1 943 3 563 c/a (angle in RHL, crystal type 533 (β)351 1 846 229 c/a in ORC & MCL) BCC HEX 1.568 b/a 453.65 1 613 1 560 2 745 melting point (K) boiling point (K) Sodium Magnesium 22.989 770 24.305 0 11 Na 12 Mg 3 [Ne]3s1 [Ne]3s2 966 429 1 738 321 BCC HEX 1.624 370.8 1 153 923 1 363 3456789 Potassium Calcium Scandium Titanium Vanadium Chromium Manganese Iron Cobalt 39.098 3 40.078 44.955 910 47.867 50.941 5 51.996 1 54.938 049 55.845 58.933 200 19 K 20 Ca 21 Sc 22 Ti 23 V 24 Cr 25 Mn 26 Fe 27 Co 4 [Ar]4s1 [Ar]4s2 [Ca]3d1 [Ca]3d2 [Ca]3d3 [Ar]3d54s1 [Ca]3d5 [Ca]3d6 [Ca]3d7 862 532 1 530 559 2 992 331 4 508 295 6 090 302 7 194 388 7 473 891 7 873 287 8 800 () 251 BCC FCC HEX 1.592 HEX 1.587 BCC BCC FCC BCC HEX 1.623 336.5 1 033 1 113 1 757 1813 3103 1 943 3 563 2 193 3 673 2 180 2 943 1 523 2 333 1813 3133 1 768 3 203 Rubidium Strontium Yttrium Zirconium Niobium Molybdenum Technetium Ruthenium Rhodium 85.467 8 87.62 88.905 85 91.224 92.906 38 95.94 [98] 101.07 102.905 50 37 Rb 38 Sr 39 Y 40 Zr 41 Nb 42 Mo 43 Tc 44 Ru 45 Rh 5 [Kr]5s1 [Kr]5s2 [Sr]4d1 [Sr]4d2 [Kr]4d45s1 [Kr]4d55s1 [Sr]4d5 [Kr]4d75s1 [Kr]4d85s1 1 533 571 2 583 608 4 475 365 6 507 323 8 578 330 10 222 315 11 496 274 12 360 270 12 420 380 BCC FCC HEX 1.571 HEX 1.593 BCC BCC HEX 1.604 HEX 1.582 FCC 312.4 963.1 1 050 1 653 1798 3613 2 123 4 673 2 750 4 973 2 896 4 913 2 433 4 533 2 603 4 423 2 236 3 973 Caesium Barium Lanthanides Hafnium Tantalum Tungsten Rhenium Osmium Iridium 132.905 45 137.327 178.49 180.947 9 183.84 186.207 190.23 192.217 55 Cs 56 Ba 57–71 72 Hf 73 Ta 74 W 75 Re 76 Os 77 Ir 6 [Xe]6s1 [Xe]6s2 [Yb]5d2 [Yb]5d3 [Yb]5d4 [Yb]5d5 [Yb]5d6 [Yb]5d7 1 900 614 3 594 502 13 276 319 16 670 330 19 254 316 21 023 276 22 580 273 22 550 384 BCC BCC HEX 1.581 BCC BCC HEX 1.615 HEX 1.606 FCC 301.6 943.2 1 001 2 173 2 503 4 873 3 293 5 833 3 695 5 823 3 459 5 873 3 303 5 273 2 720 4 703 Francium Radium Actinides Rutherfordium Dubnium Seaborgium Bohrium Hassium Meitnerium [223] [226] [261] [262] [263] [264] [265] [268] 87 Fr 88 Ra 89 – 103 104 Rf 105 Db 106 Sg 107 Bh 108 Hs 109 Mt 7 [Rn]7s1 [Rn]7s2 [Ra]5f146d2 [Ra]5f146d3? [Ra]5f146d4? [Ra]5f146d5? [Ra]5f146d6? [Ra]5f146d7? 5 000 515 BCC 300 923 973 1 773

Lanthanum Cerium Praseodymium Neodymium Promethium Samarium 138.905 5 140.116 140.907 65 144.24 [145] 150.36 57 La 58 Ce 59 Pr 60 Nd 61 Pm 62 Sm Lanthanides [Ba]5d1 [Ba]4f15d1 [Ba]4f3 [Ba]4f4 [Ba]4f5 [Ba]4f6 6 174 377 6 711 (γ) 516 6 779 367 7 000 366 7 220 365 7 536 363 HEX 3.23 FCC HEX 3.222 HEX 3.225 HEX 3.19 HEX 7.221 1 193 3 733 1 073 3 693 1 204 3 783 1 289 3 343 1415 3573 1 443 2 063 Actinium Thorium Protactinium Uranium Neptunium Plutonium [227] 232.038 1 231.035 88 238.028 9 [237] [244] 89 Ac 90 Th 91 Pa 92 U 93 Np 94 Pu 3 1 2 4 1 2 6 2 Actinides [Ra]6d1 [Ra]6d2 [Rn]5f26d17s2 [Rn]5f 6d 7s [Rn]5f 6d 7s [Rn]5f 7s 10 060 531 11 725 508 15 370 392 19 050 285 20 450 666 19 816 618 1.736 0.733 1.773 FCC FCC TET 0.825 ORC 2.056 ORC 0.709 MCL 0.780 1 323 3 473 2 023 5 063 1 843 4 273 1 405.3 4 403 913 4 173 913 3 503 main January 23, 2006 16:6

6.2 Periodic table 125

18 Helium 4.002 602 2 He 2 BCC body-centred cubic 1s 120 356 CUB simple cubic HEX 1.631 DIA diamond 13 14 15 16 17 3-5 4.22 FCC face-centred cubic Boron Carbon Nitrogen Oxygen Fluorine Neon HEX hexagonal 10.811 12.0107 14.006 74 15.999 4 18.998 403 2 20.179 7 MCL monoclinic 5 B 6 C 7 N 8 O 9 F 10 Ne 5 ORC orthorhombic [Be]2p1 [Be]2p2 [Be]2p3 [Be]2p4 [Be]2p [Be]2p6 RHL rhombohedral 2 466 1017 2 266 357 1 035 (β)405 1 460 (γ) 683 1 140 550 1 442 446 ◦ 1.32 TET tetragonal RHL 65 7 DIA HEX 1.631 CUB MCL 0.61 FCC (t-pt) triple point 2 348 4 273 4 763 (t-pt) 63 77.35 54.36 90.19 53.55 85.05 24.56 27.07 Aluminium Silicon Phosphorus Sulfur Chlorine Argon 26.981 538 28.085 5 30.973 761 32.066 35.452 7 39.948 13 Al 14 Si 15 P 16 S 17 Cl 18 Ar 3 4 5 [Mg]3p1 [Mg]3p2 [Mg]3p [Mg]3p [Mg]3p [Mg]3p6 2 698 405 2 329 543 1 820 331 2086 1046 2 030 624 1 656 532 1.320 2.340 1.324 FCC DIA ORC 3.162 ORC 1.229 ORC 0.718 FCC 10 11 12 933.47 2 793 1 683 3 533 317.3 550 388.47 717.82 172 239.1 83.81 87.30 Nickel Copper Zinc Gallium Germanium Arsenic Selenium Bromine Krypton 58.693 4 63.546 65.39 69.723 72.61 74.921 60 78.96 79.904 83.80 28 Ni 29 Cu 30 Zn 31 Ga 32 Ge 33 As 34 Se 35 Br 36 Kr 1 5 [Ca]3d8 [Ar]3d104s1 [Ca]3d10 [Zn]4p [Zn]4p2 [Zn]4p3 [Zn]4p4 [Zn]4p [Zn]4p6 8 907 352 8 933 361 7 135 266 5 905 452 5 323 566 5 776 413 4 808 (γ) 436 3 120 668 3 000 581 1.001 ◦ 1.308 FCC FCC HEX 1.856 ORC 1.695 DIA RHL 54 7 HEX 1.135 ORC 0.672 FCC 1 728 3 263 1 357.8 2 833 692.68 1 183 302.9 2 473 1211 3103 883 (t-pt) 493 958 265.90 332.0 115.8 119.9 Palladium Silver Cadmium Indium Tin Antimony Tellurium Iodine Xenon 106.42 107.868 2 112.411 114.818 118.710 121.760 127.60 126.904 47 131.29 46 Pd 47 Ag 48 Cd 49 In 50 Sn 51 Sb 52 Te 53 I 54 Xe 6 5 [Kr]4d10 [Pd]5s1 [Pd]5s2 [Cd]5p1 [Cd]5p2 [Cd]5p3 [Cd]5p4 [Cd]5p [Cd]5p6 11 995 389 10 500 409 8 647 298 7 290 325 7 285 (β)583 6 692 451 6 247 446 4 953 727 3 560 635 ◦ 1.347 FCC FCC HEX 1.886 TET 1.521 TET 0.546 RHL 57 7 HEX 1.33 ORC 0.659 FCC 1 828 3 233 1 235 2 433 594.2 1 043 429.75 2 343 505.08 2 893 903.8 1 860 723 1 263 386.7 457 161.3 165.0 Platinum Gold Mercury Thallium Lead Bismuth Polonium Astatine Radon 195.078 196.966 55 200.59 204.383 3 207.2 208.980 38 [209] [210] [222] 78 Pt 79 Au 80 Hg 81 Tl 82 Pb 83 Bi 84 Po 85 At 86 Rn [Xe]4f145d96s1 [Xe]4f145d106s1 [Yb]5d10 [Hg]6p1 [Hg]6p2 [Hg]6p3 [Hg]6p4 [Hg]6p5 [Hg]6p6 21 450 392 19 281 408 13 546 300 11 871 346 11 343 495 9 803 475 9 400 337 440 FCC FCC RHL 70◦32 HEX 1.598 FCC RHL 57◦14 CUB 2 041 4 093 1 337.3 3 123 234.32 629.9 577 1743 600.7 2 023 544.59 1 833 527 1 233 573 623 202 211 Ununnilium Unununium Ununbium Ununquadium [271] [272] [285] [289] 110 Uun 111 Uuu 112 Uub 114 Uuq

Europium Gadolinium Terbium Dysprosium Holmium Erbium Thulium Ytterbium Lutetium 151.964 157.25 158.925 34 162.50 164.930 32 167.26 168.934 21 173.04 174.967 63 Eu 64 Gd 65 Tb 66 Dy 67 Ho 68 Er 69 Tm 70 Yb 71 Lu [Ba]4f7 [Ba]4f75d1 [Ba]4f9 [Ba]4f10 [Ba]4f11 [Ba]4f12 [Ba]4f13 [Ba]4f14 [Yb]5d1 5 248 458 7 870 363 8 267 361 8 531 359 8 797 358 9 044 356 9 325 354 6 966 (β) 549 9 842 351 BCC HEX 1.591 HEX 1.580 HEX 1.573 HEX 1.570 HEX 1.570 HEX 1.570 FCC HEX 1.583 1 095 1 873 1 587 3 533 1 633 3 493 1 683 2 833 1 743 2 973 1 803 3 133 1 823 2 223 1 097 1 473 1 933 3 663 Americium Curium Berkelium Californium Einsteinium Fermium Mendelevium Nobelium Lawrencium [243] [247] [247] [251] [252] [257] [258] [259] [262] 95 Am 96 Cm 97 Bk 98 Cf 99 Es 100 Fm 101 Md 102 No 103 Lr [Ra]5f7 [Rn]5f76d17s2 [Ra]5f9 [Ra]5f10 [Ra]5f11 [Ra]5f12 [Ra]5f13 [Ra]5f14 [Ra]5f147p1 13 670 347 13 510 350 14 780 342 15 100 338 HEX 3.24 HEX 3.24 HEX 3.24 HEX 3.24 HEX 1 449 2 873 1 618 3 383 1323 1 173 1 133 1 803 1 103 1 103 1 903 main January 23, 2006 16:6

126 Solid state physics

6.3 Crystalline structure Bravais lattices Volume of a,b,c primitive base vectors V =(a×b)·c (6.1) primitive cell V volume of primitive cell

∗ a =2πb×c/[(a×b)·c] (6.2) ∗ b =2πc×a/[(a×b)·c] (6.3) Reciprocal ∗ ∗ ∗ ∗ primitive base c =2πa×b/[(a×b)·c] (6.4) a ,b ,c reciprocal primitive base vectors a vectors a·a∗ = b·b∗ = c·c∗ =2π (6.5) a·b∗ = a·c∗ = 0 (etc.) (6.6)

Ruvw lattice vector [uvw] Lattice vector Ruvw = ua+vb+wc (6.7) u,v,w integers

G = a∗ + b∗ + c∗ (6.8) Reciprocal lattice hkl h k l Ghkl reciprocal lattice vector [hkl] 2 − vector exp(iGhkl ·Ruvw) = 1 (6.9) ii= 1

Weiss zone hu+kv+lw= 0 (6.10) (hkl) Miller indices of planec equationb Interplanar 2π = (6.11) dhkl distance between (hkl) dhkl planes spacing (general) Ghkl Interplanar 1 h2 k2 l2 spacing 2 = 2 + 2 + 2 (6.12) (orthogonal basis) dhkl a b c aNote that this is 2π times the usual deﬁnition of a “reciprocal vector” (see page 20). bCondition for lattice vector [uvw] to be parallel to lattice plane (hkl) in an arbitrary Bravais lattice. c Miller indices are deﬁned so that Ghkl is the shortest reciprocal lattice vector normal to the (hkl) planes.

Weber symbols 1 U = (2u−v) (6.13) 3 U,V,T,W Weber indices Converting 1 − V = (2v u) (6.14) u,v,w zone axis indices [uvw]to 3 1 [UVTW] Weber symbol [UVTW] T = − (u+v) (6.15) 3 [uvw] zone axis symbol W = w (6.16)

− Converting u =(U T ) (6.17) [UVTW]to v =(V −T ) (6.18) [uvw] w = W (6.19)

Zone lawa hU +kV +iT +lW = 0 (6.20) (hkil) Miller–Bravais indices aFor trigonal and hexagonal systems. main January 23, 2006 16:6

6.3 Crystalline structure 127

Cubic lattices lattice primitive (P) body-centred (I) face-centred (F) lattice parameter aa a volume of conventional cell a3 a3 a3 lattice points per cell 1 2 4 a 1st nearest neighbours 68√ 12√ 1st n.n. distance aa3/2 a/ 2 2nd nearest neighbours 12√ 6 6 2nd n.n. distance a 2 √ aa√ packing fractionb π/6 3π/8 2π/6 reciprocal latticec PF I a − a a1 = axaˆ 1 = 2 (yˆ +zˆ xˆ) a1 = 2 (yˆ +zˆ) d a − a primitive base vectors a2 = ayaˆ 2 = 2 (zˆ +xˆ yˆ) a2 = 2 (zˆ +xˆ) a − a a3 = azaˆ 3 = 2 (xˆ +yˆ zˆ) a3 = 2 (xˆ +yˆ) a Or “coordination number.” √ bFor close-packed spheres. The maximum possible packing fraction for spheres is 2π/6. cThe lattice parameters for the reciprocal lattices of P, I, and F are 2π/a,4π/a, and 4π/a respectively. dxˆ, yˆ, and zˆ are unit vectors.

Crystal systemsa system symmetry unit cellb latticesc 6 a = b = c; triclinic none P α = β = γ =90 ◦ a = b = c; monoclinic one diad [010] P, C α = γ =90◦, β =90 ◦ a = b = c; orthorhombic three orthogonal diads P, C, I, F α = β = γ =90◦ a = b = c; tetragonal one tetrad [001] P, I α = β = γ =90◦ a = b = c; trigonald one triad [111] P, R α = β = γ<120◦ =90 ◦ a = b = c; hexagonal one hexad [001] P α = β =90◦, γ = 120◦ a = b = c; cubic four triads 111 P, F, I α = β = γ =90◦ aThe symbol “=” implies that equality is not required by the symmetry, but neither is it forbidden. bThe cell axes are a, b, and c with α, β, and γ the angles between b : c, c : a, and a : b respectively. cThe lattice types are primitive (P), body-centred (I), all face-centred (F), side-centred (C), and rhombohedral primitive (R). dA primitive hexagonal unit cell, with a triad [001], is generally preferred over this rhombohedral unit cell. main January 23, 2006 16:6

128 Solid state physics

Dislocations and cracks ˆ ˆl Edge l unit vector line of ˆl ·b = 0 (6.21) dislocation dislocation b,b Burgers vectora b Screw U dislocation energy per ˆl ·b = b (6.22) unit length dislocation µ shear modulus ˆl R outer cutoﬀ for r Screw µb2 R r inner cutoﬀ for b dislocation U = ln (6.23) r0 r 4π r0 energy per L critical crack length ∼ µb2 (6.24) unit lengthb α surface energy per unit area E Young modulus Critical crack 4αE c L = (6.25) σ Poisson ratio length π(1−σ2)p2 L 0 p0 applied widening stress aThe Burgers vector is a Bravais lattice vector characterising the total relative slip were the dislocation to travel throughout the crystal. bOr “tension.” The energy per unit length of an edge dislocation is also ∼ µb2. c For a crack cavity (long ⊥ L) within an isotropic medium. Under uniform stress p0, cracks ≥ L will grow and smaller cracks will shrink.

Crystal diﬀraction a,b,c lattice parameters − a(cosα1 cosα2)=hλ (6.26) α1,β1,γ1 angles between lattice base Laue vectors and input wavevector b(cosβ1 −cosβ2)=kλ (6.27) equations α2,β2,γ2 angles between lattice base c(cosγ1 −cosγ2)=lλ (6.28) vectors and output wavevector h,k,l integers (Laue indices) λ wavelength a | |2 Bragg’s law 2kin.G + G = 0 (6.29) kin input wavevector G reciprocal lattice vector f(G) atomic form factor Atomic form − · f(G)= e iG rρ(r)d3r (6.30) r position vector factor vol ρ(r) atomic electron density n S(G) structure factor Structure − · S(G)= f (G)e iG dj (6.31) n number of atoms in basis factorb j j=1 dj position of jth atom within basis K change in wavevector (= kout −kin) Scattered 2 2 I(K) ∝ N |S(K)| (6.32) I(K) scattered intensity intensityc N number of lattice points illuminated

IT intensity at temperature T Debye– 1 I0 intensity from a lattice with no Waller I = I exp − u2 |G|2 (6.33) motion T 0 3 factord u2 mean-squared thermal displacement of atoms aAlternatively, see Equation (8.32). bThe summation is over the atoms in the basis, i.e., the atomic motif repeating with the Bravais lattice. c The Bragg condition makes K a reciprocal lattice vector, with |kin| = |kout|. dEﬀect of thermal vibrations. main January 23, 2006 16:6

6.4 Lattice dynamics 129

6.4 Lattice dynamics

Phonon dispersion relationsa

m m m1 m2 (>m1)

a 2a (2α/µ)1/2 2(α/m)1/2 1/2 ω (2α/m1) ω

1/2 (2α/m2)

π π π π − 0 k − 0 k a a 2a 2a monatomic chain diatomic chain α ka ω2 =4 sin2 (6.34) ω phonon angular frequency m 2 α spring constantb Monatomic ω α 1/2 a m atomic mass vp = = a sinc (6.35) ≡ sinπx linear chain k m λ vp phase speed (sincx πx ) ∂ω α 1/2 ka vg group speed 6 v = = a cos (6.36) g ∂k m 2 λ phonon wavelength k wavenumber (= 2π/λ) 1/2 a atomic separation Diatomic 2 α 1 4 2 c ω = ±α − sin (ka) (6.37) mi atomic masses (m2 >m1) linear chain 2 µ µ m1m2 µ reduced mass [= m1m2/(m1 +m2)]

α1 +α2 1 Identical 2 ± 2 2 1/2 αi alternating spring constants ω = (α1 +α2 +2α1α2 coska) masses, m m (6.38) m m alternating α1 α2 α1 spring 0, 2(α1 +α2)/m if k =0 constants = (6.39) a 2α1/m, 2α2/m if k = π/a aAlong infinite linear atomic chains, considering simple harmonic nearest-neighbour interactions only. The shaded region of the dispersion relation is outside the first Brillouin zone of the reciprocal lattice. bIn the sense α = restoring force/relative displacement. cNote that the repeat distance for this chain is 2a, so that the first Brillouin zone extends to |k| <π/(2a). The optic and acoustic branches are the + and − solutions respectively. main January 23, 2006 16:6

130 Solid state physics

Debye theory E mean energy in a mode at ω Mean energy h¯ (Planck constant)/(2π) 1 hω¯ per phonon E = hω¯ + (6.40) ω phonon angular frequency a 2 exp[¯hω/(kBT )]−1 mode kB Boltzmann constant T temperature

2 1/3 ωD Debye (angular) frequency ωD = vs(6π N/V) (6.41) Debye vs eﬀective sound speed 3 1 2 frequency vl longitudinal phase speed where 3 = 3 + 3 (6.42) vs vl vt vt transverse phase speed N number of atoms in crystal Debye θ =¯hω /k (6.43) V crystal volume temperature D D B θD Debye temperature g(ω) density of states at ω 3Vω2 Phonon C heat capacity, V constant g(ω)dω = 2 3 dω (6.44) V density of 2π vs U thermal phonon energy states (for 0 <ω<ωD, g = 0 otherwise) within crystal D(x) Debye function θ /T Debye heat T 3 D x4ex 3NkB C =9Nk dx (6.45) capacity V B 3 x − 2 θD 0 (e 1) Dulong and CV 3Nk (T θ ) (6.46) Petit’s law B D Debye T 3 12π4 T 3 Nk (T θ ) (6.47) law B 3 D 012 5 θD T/θD 9N ωD hω¯ 3 U(T )= dω ≡ 3Nk T D(θ /T ) (6.48) Internal 3 − B D ω 0 exp[¯hω/(kBT )] 1 thermal D 3 x y3 energyb where D(x)= dy (6.49) 3 y − x 0 e 1 aOr any simple harmonic oscillator in thermal equilibrium at temperature T . bNeglecting zero-point energy. main January 23, 2006 16:6

6.4 Lattice dynamics 131

Lattice forces (simple) φ(r) two-particle potential Van der Waals α2hω¯ energy −3 p a φ(r)= 2 6 (6.50) r particle separation interaction 4 (4π0) r αp particle polarisability A B φ(r)=− + (6.51) r6 r12 h¯ (Planck constant)/(2π) Lennard–Jones 12 6 0 permittivity of free space σ − σ 6-12 potential =4 (6.52) ω angular frequency of r r polarised orbital (molecular 1/6 2 A,B constants crystals) σ =(B/A) ; = A /(4B) 21/6 ,σ Lennard–Jones φ at r = (6.53) parameters min σ Λ de Boer parameter De Boer h Λ= (6.54) h Planck constant parameter σ(m)1/2 m particle mass

UC lattice Coulomb energy per ion pair Coulomb e2 α Madelung constant interaction U = −α (6.55) M C M −e electronic charge (ionic crystals) 4π0r0 r0 nearest neighbour separation aLondon’s formula for ﬂuctuating dipole interactions, neglecting the propagation time between particles. 6

Lattice thermal expansion and conduction γ Gruneisen¨ parameter Gruneisen¨ ∂lnω γ = − (6.56) ω normal mode frequency parametera ∂lnV V volume α linear expansivity

KT isothermal bulk modulus Linear 1 ∂p γCV b α = = (6.57) p pressure expansivity 3K ∂T V 3K V T T T temperature

CV lattice heat capacity, constant V Thermal λ thermal conductivity 1 CV conductivity of λ = v l (6.58) vs effective sound speed 3 V s a phonon gas l phonon mean free path Umklapp mean ∝ lu umklapp mean free path c lu exp(θu/T ) (6.59) free path θu umklapp temperature (∼ θD/2) aStrictly, the Gruneisen¨ parameter is the mean of γ over all normal modes, weighted by the mode’s contribution to CV . bOr “coefficient of thermal expansion,” for an isotropically expanding crystal. cMean free path determined solely by “umklapp processes” – the scattering of phonons outside the first Brillouin zone. main January 23, 2006 16:6

132 Solid state physics

6.5 Electrons in solids

Free electron transport properties J current density n free electron number density Current density J = −nevd (6.60) −e electronic charge

vd mean electron drift velocity Mean electron eτ τ mean time between collisions (relaxation vd = − E (6.61) time) drift velocity me me electronic mass d.c. electrical ne2τ E applied electric ﬁeld σ0 = (6.62) conductivity me σ0 d.c. conductivity (J = σE) a.c. electrical σ0 ω a.c. angular frequency σ(ω)= (6.63) conductivitya 1−iωτ σ(ω) a.c. conductivity C total electron heat capacity, V constant 1 C V λ = V c2 τ (6.64) V volume Thermal 3 V c2 mean square electron speed 2 2 conductivity π nkBτT k Boltzmann constant = (T TF) B 3me T temperature (6.65) TF Fermi temperature 2 2 × −8 −2 Wiedemann– λ π kB L Lorenz constant ( 2.45 10 WΩK ) b = L = (6.66) Franz law σT 3e2 λ thermal conductivity Jx

RH Hall coefficient w Ey c 1 Ey Ey Hall electric field Bz Hall coefficient RH = − = (6.67) ne JxBz Jx applied current density Bz magnetic flux density + Hall voltage VH Hall voltage VH BzIx (rectangular V = R (6.68) I applied current (= J × cross-sectional area) H H w x x strip) w strip thickness in z aFor an electric field varying as e−iωt. bHolds for an arbitrary band structure. cThe charge on an electron is −e,wheree is the elementary charge (approximately +1.6 × 10−19 C). The Hall coefficient is therefore a negative number when the dominant charge carriers are electrons. main January 23, 2006 16:6

6.5 Electrons in solids 133

Fermi gas 3/2 E electron energy (> 0) V 2me g(E)= E1/2 (6.69) g(E) density of states Electron density 2π2 h¯2 a V “gas” volume of states 3 nV me electronic mass g(EF)= (6.70) 2 EF h¯ (Planck constant)/(2π) Fermi kF Fermi wavenumber k =(3π2n)1/3 (6.71) wavenumber F n number of electrons per unit volume

Fermi velocity vF =¯hkF/me (6.72) vF Fermi velocity

2 2 2 Fermi energy h¯ kF h¯ 2 2/3 EF = = (3π n) (6.73) EF Fermi energy (T =0) 2me 2me

Fermi EF TF Fermi temperature TF = (6.74) temperature kB kB Boltzmann constant 2 π 2 Electron heat CV e = g(EF)kBT (6.75) 3 CV e heat capacity per electron capacityb 2 2 T temperature π kB (T TF) = T (6.76) 2EF Total kinetic 3 U = nV E (6.77) U0 total kinetic energy energy (T =0) 0 5 F

χHP Pauli magnetic susceptibility 6 M = χ H (6.78) H magnetic ﬁeld strength Pauli HP M magnetisation paramagnetism 3n 2 = µ0µBH (6.79) µ0 permeability of free space 2EF µB Bohr magneton Landau 1 χ = − χ (6.80) χHL Landau magnetic diamagnetism HL 3 HP susceptibility aThe density of states is often quoted per unit volume in real space (i.e., g(E)/V here). bEquation (6.75) holds for any density of states.

Thermoelectricity E b J electrochemical ﬁeld Thermopowera E = +S ∇T (6.81) J current density σ T σ electrical conductivity

ST thermopower Peltier effect H =ΠJ −λ∇T (6.82) T temperature H heat flux per unit area Π Peltier coefficient Kelvin relation Π=TST (6.83) λ thermal conductivity aOr “absolute thermoelectric power.” bThe electrochemical field is the gradient of (µ/e)−φ,whereµ is the chemical potential, −e the electronic charge, and φ the electrical potential. main January 23, 2006 16:6

134 Solid state physics

Band theory and semiconductors Ψ electron eigenstate k Bloch wavevector Bloch’s theorem Ψ(r +R)=exp(ik ·R)Ψ(r) (6.84) R lattice vector r position vector

vb electron velocity (for wavevector k) Electron 1 v (k)= ∇ E (k) (6.85) h¯ (Planck constant)/2π velocity b k b h¯ b band index

Eb(k) energy band −1 2 m effective mass tensor Effective mass 2 ∂ Eb(k) ij tensor mij =¯h (6.86) k components of k ∂ki∂kj i −1 ∗ 2 m scalar effective mass Scalar effective ∗ 2 ∂ Eb(k) a m =¯h (6.87) mass ∂k2 k = |k| µ particle mobility

vd mean drift velocity |vd| eD E applied electric field Mobility µ = = (6.88) − |E| kBT e electronic charge D diffusion coefficient T temperature J current density Net current J =(n µ +n µ )eE (6.89) n electron, hole, number densities density e e h h e,h µe,h electron, hole, mobilities 3 ( ) kB Boltzmann constant kBT ∗ ∗ 3/2 −Eg/(kBT ) Semiconductor nenh = (m m ) e 2 3 e h Eg band gap equation 2(πh¯ ) ∗ (6.90) me,h electron, hole, effective masses I current I saturation current eV 0 I = I0 exp −1 (6.91) V bias voltage (+ for forward) kBT ni intrinsic carrier concentration 2 De Dh A area of junction p-n junction I0 = eni A + (6.92) LeNa LhNd De,h electron, hole, diffusion 1/2 coefficients Le =(Deτe) (6.93) Le,h electron, hole, diffusion lengths 1/2 Lh =(Dhτh) (6.94) τe,h electron, hole, recombination times

Na,d acceptor, donor, concentrations a Valid for regions of k-space in which Eb(k) can be taken as independent of the direction of k. main January 23, 2006 16:6

Chapter 7 Electromagnetism

7.1 Introduction The electromagnetic force is central to nearly every physical process around us and is a major component of classical physics. In fact, the development of electromagnetic theory in the nineteenth century gave us much mathematical machinery that we now apply quite generally in other ﬁelds, including potential theory, vector calculus, and the ideas of divergence and curl. It is therefore not surprising that this section deals with a large array of physical quantities and their relationships. As usual, SI units are assumed throughout. In the past electromagnetism has suﬀered from the use of a variety of systems of units, including the cgs system in both its electrostatic (esu) and electromagnetic (emu) forms. The fog has now all but cleared, but some specialised areas of research still cling to these historical measures. Readers are advised to consult the section on unit conversion if they come across such exotica in the literature. Equations cast in the rationalised units of SI can be readily converted to the once common Gaussian (unrationalised) units by using the following symbol transformations:

7 Equation conversion: SI to Gaussian units 2 0 → 1/(4π) µ0 → 4π/c B → B/c

χE → 4πχE χH → 4πχH H → cH/(4π) A → A/c M → cMD→ D/(4π)

The quantities ρ, J, E, φ, σ, P, r,andµr are all unchanged. main January 23, 2006 16:6

136 Electromagnetism

7.2 Static ﬁelds

Electrostatics Electrostatic E electric ﬁeld E = −∇φ (7.1) potential φ electrostatic potential b a φ potential at a Potential − · − · a φa φb = E dl = E dl φ potential at b diﬀerencea a b b (7.2) dl line element Poisson’s Equation ρ ρ charge density ∇2φ = − (7.3) (free space) 0 permittivity of 0 free space q φ(r)= | − | (7.4) 4π0 r r Point charge at r q point charge q(r −r) E(r)= 3 (7.5) 4π0|r −r | dτ Fieldfroma 1 ρ(r )(r −r ) dτ volume element charge distribution E(r)= 3 dτ (7.6) r position vector r 4π0 |r −r | (free space) volume of dτ r - aBetween points a and b along a path l.

Magnetostaticsa

φm magnetic scalar Magnetic scalar potential B = −µ0∇φm (7.7) potential B magnetic flux density φm in terms of the solid angle of a IΩ Ω loop solid angle φm = (7.8) generating current 4π I current loop dl line element in dl s Biot–Savart law (the µ I dl×(r −r ) the direction of 0 W field from a line B(r)= 3 (7.9) the current r 4π |r −r | I current) line r position vector of r - dl Ampere’s` law J current density ∇×B = µ J (7.10) (differential form) 0 µ0 permeability of free space Ampere’s` law (integral B · dl = µ I (7.11) Itot total current form) 0 tot through loop aIn free space. main January 23, 2006 16:6

7.2 Static ﬁelds 137

Capacitancea

Of sphere, radius a C =4π0ra (7.12)

Of circular disk, radius a C =80ra (7.13)

Of two spheres, radius a,in C =8π aln2 (7.14) contact 0 r Of circular solid cylinder, C [8+4.1(l/a)0.76] a (7.15) radius a,lengthl 0 r Of nearly spherical surface, − C 3.139×10 11 S 1/2 (7.16) area S r

−11 Of cube, side a C 7.283×10 ra (7.17)

Between concentric spheres, − C =4π ab(b−a) 1 (7.18) radii a

Between parallel, coaxial 2 0rπa circular disks, separation d, C +0ra[ln(16πa/d)−1] (7.22) radii a d a For conductors, in an embedding medium of relative permittivity r. 7

Inductancea Of N-turn solenoid 2 (straight or toroidal), L = µ0N A/l (7.23) length l,areaA ( l2) Of coaxial cylindrical µ b L = 0 ln per unit length (7.24) tubes, radii a, b (a

138 Electromagnetism

Electric ﬁeldsa  q  r (r

seen from free space −b −q(r −1)/(r +1)

seen from the dielectric b +2q/(r +1) main January 23, 2006 16:6

7.3 Electromagnetic ﬁelds (general) 139

7.3 Electromagnetic ﬁelds (general)

Field relationships J current density Conservation of ∂ρ ∇·J = − (7.39) ρ charge density charge ∂t t time Magnetic vector B = ∇×A (7.40) A vector potential potential Electric ﬁeld from ∂A E = − −∇φ (7.41) φ electrical potentials ∂t potential Coulomb gauge ∇·A = 0 (7.42) condition Lorenz gauge 1 ∂φ ∇·A+ = 0 (7.43) c speed of light condition c2 ∂t

2 1 ∂ φ −∇2 ρ dτ 2 2 φ = (7.44) Potential ﬁeld c ∂t 0 r equationsa 1 ∂2A −∇2A = µ J (7.45) r - c2 ∂t2 0 −| − | dτ volume element Expression for φ 1 ρ(r ,t r r /c) φ(r,t)= dτ (7.46) a | − | r position vector of in terms of ρ 4π0 r r volume dτ µ0 J (r ,t−|r −r |/c) Expression for A A(r,t)= dτ (7.47) µ0 permeability of in terms of J a 4π |r −r| free space volume aAssumes the Lorenz gauge. 7

Lienard–Wiechert´ potentialsa q charge Electrical potential of q r vector from charge φ = | |− · (7.48) to point of a moving point charge 4π0( r v r/c) observation v particle velocity Magnetic vector q : µ0qv potential of a moving A = (7.49) v 4π(|r|−v ·r/c) r j point charge aIn free space. The right-hand sides of these equations are evaluated at retarded times, i.e., at t = t−|r|/c,wherer is the vector from the charge to the observation point at time t. main January 23, 2006 16:6

140 Electromagnetism

Maxwell’s equations Differential form: Integral form: ρ 1 ∇·E = (7.50) E · ds = ρ dτ (7.51) 0 0 closed surface volume ∇·B = 0 (7.52) B · ds = 0 (7.53) closed surface ∂B · − dΦ ∇×E = − (7.54) E dl = (7.55) ∂t dt loop ∂E · ∂E · ∇×B = µ J +µ (7.56) B dl = µ0I +µ00 ds (7.57) 0 0 0 ∂t ∂t loop surface Equation (7.51) is “Gauss’s law” ds surface element Equation (7.55) is “Faraday’s law” dτ volume element E electric field dl line element B magnetic flux density · Φ linked magnetic flux (= B ds) J current density I linked current (= J · ds) ρ charge density t time

Maxwell’s equations (using D and H) Diﬀerential form: Integral form: · ∇·D = ρfree (7.58) D ds = ρfree dτ (7.59) closed surface volume ∇·B = 0 (7.60) B · ds = 0 (7.61) closed surface ∂B · − dΦ ∇×E = − (7.62) E dl = (7.63) ∂t dt loop ∂D · ∂D · ∇×H = J + (7.64) H dl = Ifree + ds (7.65) free ∂t ∂t loop surface

D displacement ﬁeld E electric ﬁeld

ρfree free charge density (in the sense of ds surface element ρ = ρinduced +ρfree) dτ volume element B magnetic flux density dl line element H magnetic field strength · Φ linked magnetic flux (= B ds) J free free current density (in the sense of Ifree linked free current (= J free · ds) J = J +J ) induced free t time main January 23, 2006 16:6

7.3 Electromagnetic ﬁelds (general) 141

Relativistic electrodynamics E electric field E = E (7.66) B magnetic flux density Lorentz measured in frame moving × transformation of E⊥ = γ(E +v×B)⊥ (7.67) at relative velocity v electric and B = B (7.68) γ Lorentz factor − 2 −1/2 =[1 (v/c) ] magnetic fields − × 2 B⊥ = γ(B v E/c )⊥ (7.69) parallel to v ⊥ perpendicular to v

2 Lorentz ρ = γ(ρ−vJ/c ) (7.70) transformation of J current density J⊥ = J⊥ (7.71) current and charge ρ charge density − densities J = γ(J vρ) (7.72)

− Lorentz φ = γ(φ vA) (7.73) φ electric potential transformation of A⊥ = A⊥ (7.74) A magnetic vector potential 2 potential ﬁelds A = γ(A −vφ/c ) (7.75)

J =(ρc,J ) (7.76) ∼ φ A = ,A (7.77) J current density four-vector ∼ c ∼ Four-vector ﬁeldsa A potential four-vector 1 ∂2 ∼ 2 = ,−∇2 (7.78) 2 D’Alembertian operator c2 ∂t2 2A = µ J (7.79) ∼ 0∼ aOther sign conventions are common here. See page 65 for a general deﬁnition of four-vectors. 7 main January 23, 2006 16:6

142 Electromagnetism

7.4 Fields associated with media

Polarisation ± Deﬁnition of electric q end charges − p + p = qa (7.80) - dipole moment a charge separation vector (from − to +) p dipole moment Generalised electric p = rρ dτ (7.81) ρ charge density dipole moment dτ volume element volume r vector to dτ φ dipole potential · Electric dipole p r r vector from dipole φ(r)= 3 (7.82) potential 4π0r 0 permittivity of free space Dipole moment per P polarisation unit volume P = np (7.83) n number of dipoles per (polarisation)a unit volume Induced volume ∇·P = −ρ (7.84) ρ volume charge density charge density ind ind

Induced surface σind surface charge density σind = P ·sˆ (7.85) charge density sˆ unit normal to surface Definition of electric D electric displacement D = 0E +P (7.86) displacement E electric field Definition of electric P = χ E (7.87) χE electrical susceptibility susceptibility 0 E (may be a tensor)

r =1+χE (7.88) Deﬁnition of relative r relative permittivity D = 0rE (7.89) permittivityb permittivity = E (7.90)

Atomic α polarisability c p = αEloc (7.91) polarisability Eloc local electric ﬁeld

Ed depolarising ﬁeld

Nd depolarising factor =1/3 (sphere) NdP Depolarising ﬁelds Ed = − (7.92) =1 (thin slab ⊥ to P) 0 =0 (thin slab to P) =1/2 (long circular cylinder, axis ⊥ to P)

Clausius–Mossotti nα r −1 d = (7.93) equation 30 r +2 aAssuming dipoles are parallel. The equivalent of Equation (7.112) holds for a hot gas of electric dipoles. bRelative permittivity as deﬁned here is for a linear isotropic medium. c 3 The polarisability of a conducting sphere radius a is α =4π0a . The deﬁnition p = α0Eloc is also used. d 2 With the substitution η = r [cf. Equation (7.195) with µr = 1] this is also known as the “Lorentz–Lorenz formula.” main January 23, 2006 16:6

7.4 Fields associated with media 143

Magnetisation dm dipole moment Deﬁnition of dm,ds I loop current 6 magnetic dipole dm = I ds (7.94) ds loop area (right-hand ⊗ moment sense with respect to out in loop current) m dipole moment Generalised 1 J current density magnetic dipole m = r ×J dτ (7.95) 2 dτ volume element moment volume r vector to dτ

φm magnetic scalar potential Magnetic dipole µ0m·r φ (r)= (7.96) r vector from dipole (scalar) potential m 4πr3 µ0 permeability of free space Dipole moment per M magnetisation unit volume M = nm (7.97) n number of dipoles (magnetisation)a per unit volume Induced volume J = ∇×M (7.98) J ind volume current current density ind density (i.e., A m−2)

jind surface current −1 Induced surface × density (i.e., A m ) jind = M sˆ (7.99) current density sˆ unit normal to surface Definition of B magnetic flux density magnetic field B = µ0(H +M) (7.100) H magnetic field strength, H strength

M = χH H (7.101) Deﬁnition of χH magnetic χBB magnetic = (7.102) susceptibility. χB is µ0 also used (both may 7 susceptibility χH be tensors) χB = (7.103) 1+χH

B = µ0µrH (7.104) = µH (7.105) Deﬁnition of relative µr relative permeability permeabilityb µr =1+χH (7.106) µ permeability 1 = (7.107) 1−χB aAssuming all the dipoles are parallel. See Equation (7.112) for a classical paramagnetic gas and page 101 for the quantum generalisation. bRelative permeability as deﬁned here is for a linear isotropic medium. main January 23, 2006 16:6

144 Electromagnetism

Paramagnetism and diamagnetism m magnetic moment r2 mean squared orbital radius (of all electrons) Diamagnetic e2 − 2 Z atomic number moment of an atom m = Z r B (7.108) 6me B magnetic ﬂux density

me electron mass −e electronic charge Intrinsic electron e J total angular momentum m − gJ (7.109) magnetic momenta 2m g Lande´ g-factor (=2 for spin, e =1 for orbital momentum) 1 L(x)=cothx− (7.110) Langevin function x L(x) Langevin function x/3(x ∼< 1) (7.111)

 Classical gas M apparent magnetisation m B paramagnetism L 0 m0 magnitude of magnetic dipole M = nm0 (7.112) moment (|J|h¯) kT n dipole number density

2 T temperature µ0nm Curie’s law χ = 0 (7.113) k Boltzmann constant H 3kT χH magnetic susceptibility 2 µ0nm0 µ0 permeability of free space Curie–Weiss law χH = (7.114) 3k(T −Tc) Tc Curie temperature aSee also page 100.

Boundary conditions for E, D, B, and Ha Parallel component parallel to component of the E continuous (7.115) interface electric ﬁeld Perpendicular component of the ⊥ component B⊥ continuous (7.116) perpendicular to magnetic ﬂux interface density

D1,2 electrical displacements in media 1 & 2 Electric 6sˆ sˆ ·(D −D )=σ (7.117) sˆ unit normal to surface, 2 displacementb 2 1 directed 1 → 2 σ surface density of free 1 charge

H1,2 magnetic ﬁeld strengths Magnetic ﬁeld × − in media 1 & 2 c sˆ (H2 H1)=js (7.118) strength js surface current per unit width aAt the plane surface between two uniform media. bIf σ =0,then D⊥ is continuous. c If js = 0 then H is continuous. main January 23, 2006 16:6

7.5 Force, torque, and energy 145

7.5 Force, torque, and energy

Electromagnetic force and torque

F 2 force on q2

q1,2 charges Force between two q q F-2 1 2 r12 vector from 1 to 2 - static charges: F 2 = 2 rˆ12 (7.119) 4π0r ˆ unit vector q1 q2 Coulomb’s law 12 r12 0 permittivity of free space dl line elements 1,2 dl1* I currents flowing along Force between two µ0I1I2 1,2 dF = [dl ×(dl ×rˆ )] dl and dl r current-carrying 2 2 2 1 12 1 2 12 4πr12 dF 2 force on dl2 W elements (7.120) j µ0 permeability of free dl2 space Force on a dl line element current-carrying F force dF = I dl×B (7.121) element in a I current flowing along dl magnetic field B magnetic flux density Force on a charge E electric field F = q(E +v×B) (7.122) (Lorentz force) v charge velocity Forceonanelectric F =(p ·∇)E (7.123) p electric dipole moment dipolea Force on a magnetic F =(m·∇)B (7.124) m magnetic dipole moment dipoleb Torque on an G = p×E (7.125) G torque electric dipole 7 Torque on a G = m×B (7.126) magnetic dipole dlL line-element (of loop) Torque on a × × G = IL r (dlL B) (7.127) r position vector of dl current loop L loop IL current around loop aF simplifies to ∇(p ·E)ifp is intrinsic, ∇(pE/2) if p is induced by E and the medium is isotropic. bF simplifies to ∇(m·B)ifm is intrinsic, ∇(mB/2) if m is induced by B and the medium is isotropic. main January 23, 2006 16:6

146 Electromagnetism

Electromagnetic energy

Electromagnetic field 2 u energy density 1 2 1 B energy density (in free u = 0E + (7.128) E electric field space) 2 2 µ0 B magnetic flux density

0 permittivity of free space Energy density in 1 µ permeability of free space u = (D ·E +B ·H) (7.129) 0 media 2 D electric displacement H magnetic field strength Energy flow (Poynting) c speed of light N = E×H (7.130) vector N energy flow rate per unit area ⊥ to the flow direction

p0 amplitude of dipole moment Mean ﬂux density at a ω4p2 sin2 θ r vector from dipole distance r from a short 0 (wavelength) N = 2 3 3 r (7.131) oscillating dipole 32π 0c r θ angle between p and r ω oscillation frequency Total mean power ω4p2/2 from oscillating 0 W total mean radiated power W = 3 (7.132) dipolea 6π0c U total energy tot dτ volume element Self-energy of a 1 Utot = φ(r)ρ(r)dτ (7.133) r position vector of dτ charge distribution 2 volume φ electrical potential ρ charge density 1 Vi potential of ith capacitor Energy of an assembly U = C V V (7.134) of capacitorsb tot 2 ij i j Cij mutual capacitance between i j capacitors i and j Energy of an assembly 1 L mutual inductance between Utot = Lij IiIj (7.135) ij of inductorsc 2 inductors i and j i j

Intrinsic dipole in an Udip energy of dipole Udip = −p ·E (7.136) electric field p electric dipole moment Intrinsic dipole in a U = −m·B (7.137) m magnetic dipole moment magnetic field dip H Hamiltonian Hamiltonian of a p particle momentum |p −qA|2 m charged particle in an H = m +qφ (7.138) q particle charge EM fieldd 2m m particle mass A magnetic vector potential aSometimes called “Larmor’s formula.” b Cii is the self-capacitance of the ith body. Note that Cij = Cji. c Lii is the self-inductance of the ith body. Note that Lij = Lji. dNewtonian limit, i.e., velocity c. main January 23, 2006 16:6

7.6 LCR circuits 147

7.6 LCR circuits

LCR definitions dQ I current Current I = (7.139) dt Q charge R resistance Ohm’s law V = IR (7.140) V potential difference over R I current through R J current density Ohm’s law (field J = σE (7.141) E electric field form) σ conductivity ρ resistivity 1 RA Resistivity = = (7.142) A area of face (I is ρ normal to face) 7 σ l / l l length A Q C capacitance Capacitance C = (7.143) V potential difference V across C Current through dV I current through C I = C (7.144) capacitor dt t time Φ Φ total linked flux Self-inductance L = (7.145) I current through I inductor Voltage across dI V = −L (7.146) V potential difference inductor dt over L Φ1 total flux from loop 2 7 Mutual Φ linkedbyloop1 = 1 = (7.147) L12 L21 L mutual inductance inductance I2 12 I2 current through loop 2 k coupling coefficient Coefficient of |L12| = k L1L2 (7.148) between L1 and L2 coupling (≤ 1)

Φ linked flux Linked magnetic N number of turns Φ=Nφ (7.149) around φ flux through a coil φ flux through area of turns main January 23, 2006 16:6

148 Electromagnetism

Resonant LCR circuits series ω0 resonant angular Phase 2 1/LC (series) frequency R L C ω0 = resonant 1/LC −R2/L2 (parallel) L inductance a parallel frequency (7.150) C capacitance R resistance δω half-power δω 1 R bandwidth Tuningb = = (7.151) ω0 Q ω0L Q quality factor Quality stored energy Q =2π (7.152) factor energy lost per cycle aAt which the impedance is purely real. b Assuming the capacitor is purely reactive. If L and R are parallel, then 1/Q = ω0L/R.

Energy in capacitors, inductors, and resistors U stored energy Energy stored in a 1 1 1 Q2 C capacitance U = CV2 = QV = (7.153) capacitor 2 2 2 C Q charge V potential diﬀerence L inductance Energy stored in an 1 1 1 Φ2 2 Φ linked magnetic ﬂux inductor U = LI = ΦI = (7.154) 2 2 2 L I current Power dissipated in V 2 W power dissipated a resistora (Joule’s W = IV = I2R = (7.155) R resistance law) R τ relaxation time 0r Relaxation time τ = (7.156) relative permittivity σ r σ conductivity aThis is d.c., or instantaneous a.c., power.

Electrical impedance Impedances in series Ztot = Zn (7.157) n −1 Impedances in parallel −1 Ztot = Zn (7.158) n i Impedance of capacitance Z = − (7.159) C ωC

Impedance of inductance ZL = iωL (7.160)

Impedance: Z Capacitance: C Inductance: L Resistance: R =Re[Z] Conductance: G =1/R Reactance: X =Im[Z] Admittance: Y =1/Z Susceptance: S =1/X main January 23, 2006 16:6

7.6 LCR circuits 149

Kirchhoﬀ’s laws I currents impinging Current law Ii = 0 (7.161) i on node node V potential diﬀerences Voltage law Vi = 0 (7.162) i around loop loop

Transformersa n turns ratio

N1 number of primary turns

I2 N2 number of secondary turns Z - 1 : y V1 primary voltage V2 secondary voltage

V1 V2 Z2 I1 primary current I secondary current z 9 2 Zout output impedance

I1 N1 N2 Zin input impedance Z1 source impedance Z2 load impedance

Turns ratio n = N2/N1 (7.163)

V2 = nV1 (7.164) Transformation of voltage and current I2 = I1/n (7.165)

2 Output impedance (seen by Z2) Zout = n Z1 (7.166) 7 2 Input impedance (seen by Z1) Zin = Z2/n (7.167) aIdeal, with a coupling constant of 1 between loss-free windings.

Star–delta transformation 1 i,j,k node indices (1,2, or 3) 1 ‘Star’ ‘Delta’ Zi impedance on node i Z1 Zij impedance connecting Z12 Z13 nodes i and j

2 3 2 Z2 Z3 3 Z23

Star Zij Zik Zi = (7.168) impedances Zij +Zik +Zjk Delta 1 1 1 Zij = ZiZj + + (7.169) impedances Zi Zj Zk main January 23, 2006 16:6

150 Electromagnetism

7.7 Transmission lines and waveguides Transmission line relations ∂V ∂I V potential diﬀerence across Loss-free = −L (7.170) line ∂x ∂t transmission line I current in line ∂I ∂V equations = −C (7.171) L inductance per unit length ∂x ∂t C capacitance per unit length 1 ∂2V ∂2V Wave equation for a = (7.172) 2 2 x distance along line lossless transmission LC ∂x ∂t 1 ∂2I ∂2I t time line = (7.173) LC ∂x2 ∂t2 Characteristic L impedance of Zc = (7.174) Zc characteristic impedance lossless line C R resistance per unit length Characteristic of conductor R +iωL impedance of lossy Zc = (7.175) G conductance per unit line G+iωC length of insulator ω angular frequency

Wave speed along a 1 vp phase speed vp = vg = √ (7.176) lossless line LC vg group speed Z cos − sin Z (complex) input impedance Input impedance of t kl iZc kl in Zin = Zc − (7.177) a terminated lossless Zc coskl iZt sinkl Zt (complex) terminating 2 impedance line = Z /Zt if l = λ/4 (7.178) c k wavenumber (= 2π/λ)

Reflection coefficient l distance from termination Zt −Zc from a terminated r = (7.179) r (complex) voltage + line Zt Zc reflection coefficient Line voltage 1+|r| vswr = (7.180) standing wave ratio 1−|r|

Transmission line impedancesa Zc characteristic impedance (Ω) µ b 60 b a radius of inner conductor Coaxial line √ Zc = 2 ln ln (7.181) 4π a r a b radius of outer conductor permittivity (= 0r) µ permeability (= µ µ ) µ l 120 l 0 r Open wire feeder √ r radius of wires Zc = 2 ln ln (7.182) π r r r l distance between wires ( r) 377 d strip separation µ d √ d Paired strip Zc = (7.183) w r w w strip width ( d) 377 Microstrip line Z √ (7.184) h height above earth plane c ( w) r[(w/h)+2] aFor lossless lines. main January 23, 2006 16:6

7.7 Transmission lines and waveguides 151

Waveguidesa

kg wavenumber in guide ω angular frequency 2 2 2 2 2 a guide height Waveguide 2 ω m π n π k = − − (7.185) b guide width equation g c2 a2 b2 m,n mode indices with respect to a and b (integers) c speed of light Guide cutoﬀ m 2 n 2 νc cutoﬀ frequency νc = c + (7.186) frequency 2a 2b ωc 2πνc

Phase velocity c vp phase velocity vp = (7.187) above cutoﬀ 1−(ν /ν)2 ν frequency c Group velocity v = c2/v = c 1−(ν /ν)2 (7.188) v group velocity above cutoﬀ g p c g

ZTM wave impedance for 2 transverse magnetic modes ZTM = Z0 1−(νc/ν) (7.189) Wave ZTE wave impedance for b impedances 2 transverse electric modes ZTE = Z0/ 1−(νc/ν) (7.190) Z0 impedance of free space (= µ0/0)

c Field solutions for TEmn modes

2 ikgc ∂Bz 2 Bx = iωc ∂Bz 2 Ex = ωc ∂x 2 ωc ∂y 2 ikgc ∂Bz − 2 (7.191) By = iωc ∂Bz 2 Ey = ωc ∂y ω2 ∂x mπx nπy c B =B cos cos E =0 b z 0 a b z 7 c a Field solutions for TMmn modes z x 2 ikgc ∂Ez − Ex = iω ∂Ez y 2 Bx = ωc ∂x 2 ωc ∂y ik c2 ∂E E = g z iω ∂Ez (7.192) y 2 By = ωc ∂y ω2 ∂x mπx nπy c E =E sin sin Bz =0 z 0 a b aEquations are for lossless waveguides with rectangular cross sections and no dielectric. bThe ratio of the electric field to the magnetic field strength in the xy plane. c Both TE and TM modes propagate in the z direction with a further factor of exp[i(kgz −ωt)] on all components. B0 and E0 are the amplitudes of the z components of magnetic flux density and electric field respectively. main January 23, 2006 16:6

152 Electromagnetism

7.8 Waves in and out of media

Waves in lossless media 2 E electric field 2 ∂ E Electric field ∇ E = µ (7.193) µ permeability (= µ0µr) ∂t2 permittivity (= 0r) ∂2B B magnetic flux density Magnetic field ∇2B = µ (7.194) ∂t2 t time √ Refractive index η = rµr (7.195)

wave phase speed 1 c v Wave speed v = √ = (7.196) η refractive index µ η c speed of light µ Impedance of free space 0 Z0 impedance of free Z0 = 376.7 Ω (7.197) space 0 E µr Z wave impedance Wave impedance Z = = Z0 (7.198) H r H magnetic ﬁeld strength

Radiation pressurea Radiation G momentum density N momentum G = (7.199) N Poynting vector c2 density c speed of light

pn normal pressure Isotropic 1 u incident radiation p = u(1+R) (7.200) energy density radiation n 3 R (power) reflectance coefficient u θ 2 i Specular pn = u(1+R)cos θi (7.201) pt tangential pressure reflection pt = u(1−R)sinθi cosθi (7.202) θi angle of incidence

Iν speciﬁc intensity z From an 1+R ν frequency θ dΩ p = I (θ,φ)cos2 θ dΩ dν (normal) extended n c ν Ω solid angle b source (7.203) θ angle between dΩ x and normal to plane φ y From a point L(1+R) L source luminosity source,c p = (7.204) (i.e., radiant power) n 4 2 luminosity L πr c r distance from source aOn an opaque surface. bIn spherical polar coordinates. See page 120 for the meaning of speciﬁc intensity. cNormal to the plane. main January 23, 2006 16:6

7.8 Waves in and out of media 153

Antennas z 6 r θ U Spherical polar geometry: y p 6 - * φ / x 1 [p˙] [p] r distance from E = + cosθ (7.205) dipole r 2π r2c r3 Fieldfromashort 0 θ angle between r and (l λ) electric 1 [p¨] [p˙] [p] p E = + + sinθ (7.206) dipole in free θ 2 2 3 [p] retarded dipole 4π0 rc rc r spacea moment µ0 [p¨] [p˙] [p]=p(t−r/c) Bφ = + 2 sinθ (7.207) 4π rc r c speed of light Radiation 2 2 2 l dipole length ( λ) ω l 2πZ0 l resistance of a R = 3 = (7.208) ω angular frequency 6π0c 3 λ short electric wavelength 2 λ dipole in free l 789 ohm (7.209) Z0 impedance of free space λ space

ΩA beam solid angle Pn normalised antenna Beam solid angle Ω = ( ) dΩ (7.210) power pattern A Pn θ,φ P (0,0) = 1 4π n dΩ differential solid angle Forward power 4π G(0) = (7.211) G antenna gain gain ΩA Antenna effective λ2 7 Ae = (7.212) Ae effective area area ΩA Power gain of a 3 G(θ)= sin2 θ (7.213) short dipole 2 Ω Beam efficiency efficiency = M (7.214) ΩM main lobe solid angle ΩA

TA antenna Antenna 1 temperature b TA = Tb(θ,φ)Pn(θ,φ) dΩ (7.215) temperature ΩA 4π Tb sky brightness temperature aAll ﬁeld components propagate with a further phase factor equal to expi(kr−ωt), where k =2π/λ. b 2 The brightness temperature of a source of speciﬁc intensity Iν is Tb = λ Iν /(2kB). main January 23, 2006 16:6

154 Electromagnetism

Reﬂection, refraction, and transmissiona

parallel incidence perpendicular incidence E electric ﬁeld

Ei Er B magnetic ﬂux density ηi K Ei Er ηi refractive index on θ θ θ θ incident side i r i r U Bi / w Br / w ηt refractive index on Bi Br transmitted side θi angle of incidence

ηt θr angle of reﬂection > Et > θt angle of refraction θt * θt E = t Bt Bt

Law of reﬂection θi = θr (7.216)

b Snell’s law ηi sinθi = ηt sinθt (7.217)

θB Brewster’s angle of incidence for Brewster’s law tanθB = ηt/ηi (7.218) plane-polarised reﬂection (r =0)

Fresnel equations of reﬂection and refraction

sin2θi −sin2θt sin(θi −θt) r = (7.219) r⊥ = − (7.223) sin2θi +sin2θt sin(θi +θt) 4cosθi sinθt 2cosθi sinθt t = (7.220) t⊥ = (7.224) sin2θi +sin2θt sin(θi +θt) 2 2 R = r (7.221) R⊥ = r⊥ (7.225)

ηt cosθt 2 ηt cosθt 2 T = t (7.222) T⊥ = t⊥ (7.226) ηi cosθi ηi cosθi Coeﬃcients for normal incidencec 2 (η −η ) ηi −ηt i t r = (7.230) R = 2 (7.227) (ηi +ηt) ηi +ηt 4η η 2ηi i t t = (7.231) T = 2 (7.228) (ηi +ηt) ηi +ηt R +T = 1 (7.229) t−r = 1 (7.232)

electric field parallel to the plane of ⊥ electric field perpendicular to the incidence plane of incidence R (power) reflectance coefficient r amplitude reflection coefficient T (power) transmittance coefficient t amplitude transmission coefficient aFor the plane boundary between lossless dielectric media. All coefficients refer to the electric field component and whether it is parallel or perpendicular to the plane of incidence. Perpendicular components are out of the paper. bThe incident wave suffers total internal reflection if ηi sinθ > 1. ηt i c I.e., θi = 0. Use the diagram labelled “perpendicular incidence” for correct phases. main January 23, 2006 16:6

7.8 Waves in and out of media 155

Propagation in conducting mediaa σ electrical conductivity

Electrical 2 ne electron number density nee conductivity σ = neeµ = τc (7.233) τc electron relaxation time (B =0) me µ electron mobility B magnetic ﬂux density m electron mass Refractive index e 1/2 −e electronic charge of an ohmic σ η =(1+i) (7.234) η refractive index conductorb 4πν0 0 permittivity of free space ν frequency Skin depth in an −1/2 δ =(µ σπν) (7.235) δ skin depth ohmic conductor 0 µ0 permeability of free space a Assuming a relative permeability, µr,of1. b −iωt Taking the wave to have an e time dependence, and the low-frequency limit (σ 2πν0).

Electron scattering processesa σ Rayleigh cross section Rayleigh R ω4α2 ω radiation angular frequency scattering σ = (7.236) R 4 α particle polarisability cross sectionb 6π0c 0 permittivity of free space 8π e2 2 σT Thomson cross section Thomson σT = 2 (7.237) 3 4π0mec m electron (rest) mass scattering e 8π r classical electron radius cross sectionc = r2 6.652×10−29 m2 e 3 e c speed of light (7.238) Ptot electron energy loss rate Inverse 2 4 2 v urad radiation energy density Compton P = σ cu γ (7.239) − tot T rad 2 γ Lorentz factor = [1−(v/c)2] 1/2 scatteringd 3 c 7 v electron speed Compton h − − scatteringe λ λ = (1 cosθ) (7.240) λ,λ incident & scattered wavelengths mec ν,ν incident & scattered frequencies λ m c2 hν = e (7.241) θ photon scattering angle λ me − h θ 1 cosθ +(1/ε) electron Compton wavelength mec φ θ 2 cotφ =(1+ε)tan (7.242) ε = hν/(mec ) 2

σKN Klein–Nishina cross section πr2 2(ε+1) 1 4 1 σ = e 1− ln(2ε+1)+ + − (7.243) Klein–Nishina KN ε ε2 2 ε 2(2ε+1)2 cross section σT (ε 1) (7.244) (for a free πr2 1 electron) e ln2ε+ (ε 1) (7.245) ε 2 aFor Rutherford scattering see page 72. bScattering by bound electrons. cScattering from free electrons, ε 1. d 2 Electron energy loss rate due to photon scattering in the Thomson limit (γhν mec ). eFrom an electron at rest. main January 23, 2006 16:6

156 Electromagnetism

Cherenkov radiation θ cone semi-angle Cherenkov c c (vacuum) speed of light sinθ = (7.246) cone angle ηv η(ω) refractive index v particle velocity

ωc 2 2 Ptot total radiated power e µ0 c Ptot = v 1− ω dω (7.247) −e electronic charge Radiated 4π v2η2(ω) µ free space permeability powera 0 0 c ω angular frequency where η(ω) ≥ for 0 <ω<ωc v ωc cutoﬀ frequency aFrom a point charge, e, travelling at speed v through a medium of refractive index η(ω).

7.9 Plasma physics

Warm plasmas

lL Landau length e2 l = (7.248) −e electronic charge Landau L 4π k T 0 B e 0 permittivity of free space length × −5 −1 1.67 10 Te m (7.249) kB Boltzmann constant Te electron temperature (K) 1/2 0kBTe λDe electron Debye length Electron λDe = 2 (7.250) nee ne electron number density Debye length − 1/2 (m 3) 69(Te/ne) m (7.251)

φ eﬀective potential Debye qexp(−21/2r/λ ) De q point charge screeninga φ(r)= (7.252) 4π0r r distance from q

Debye 4 3 N = πn λ (7.253) NDe electron Debye number number De 3 e De

τe electron relaxation time T 3/2 5 e τi ion relaxation time τe =3.44×10 s (7.254) Relaxation ne lnΛ Ti ion temperature (K) b times (B =0) 3/2 1/2 lnΛ Coulomb logarithm T mi τ =2.09×107 i s (7.255) (typically 10 to 20) i n lnΛ m e p B magnetic flux density 1/2 Characteristic 2kBTe electron vte = (7.256) v electron thermal speed me te thermal m electron mass × 3 1/2 −1 e speedc 5.51 10 Te ms (7.257) aEffective (Yukawa) potential from a point charge q immersed in a plasma. b < Collision times for electrons and singly ionised ions with Maxwellian speed distributions, Ti ∼ Te. The Spitzer conductivity can be calculated from Equation (7.233). c ∝ − 2 2 Defined so that the Maxwellian velocity distribution exp( v /vte). There are other definitions (see Maxwell– Boltzmann distribution on page 112). main January 23, 2006 16:6

7.9 Plasma physics 157

Electromagnetic propagation in cold plasmasa 2 n e νp plasma frequency (2 )2 = e = 2 (7.258) πνp ωp ω plasma angular frequency 0me p Plasma frequency −3 1/2 ne electron number density (m ) νp 8.98ne Hz (7.259) me electron mass −e electronic charge Plasma refractive 2 1/2 0 permittivity of free space η = 1−(νp/ν) (7.260) index (B =0) η refractive index ν frequency k wavenumber (= 2π/λ) Plasma dispersion 2 2 2 2 c k = ω −ω (7.261) ω angular frequency (= 2π/ν) relation (B =0) p c speed of light Plasma phase v = c/η (7.262) v phase velocity velocity (B =0) φ φ

Plasma group vg = cη (7.263) 2 vg group velocity velocity (B =0) vφvg = c (7.264)

ν cyclotron frequency qB C 2πν = = ω (7.265) ωC cyclotron angular frequency Cyclotron C m C νCe electron νC (Larmor, or gyro-) ν 28×109B Hz (7.266) Ce νCp proton νC frequency 6 νCp 15.2×10 B Hz (7.267) q particle charge B magnetic ﬂux density (T) v⊥ m rL = = v⊥ (7.268) m particle mass (γm if relativistic) ωC qB Larmor rL Larmor radius −12 v⊥ (cyclotron, or rLe =5.69×10 m (7.269) rLe electron rL B gyro-) radius rLp proton rL −9 v⊥ r =10.4×10 m (7.270) ⊥ −1 7 Lp B v⊥ speed to B (ms ) θ angle between wavefront b B Mixed propagation modes normal (kˆ) and B X(1−X) η2 =1− , (7.271) − − 1 2 2 ± (1 X) 2 Y sin θB S 2 where X =(ωp/ω) ,Y= ωCe/ω, 1 and S 2 = Y 4 sin4 θ +Y 2(1−X)2 cos2 θ 4 B B 3 µ0e 2 · ∆ψ rotation angle ∆ψ = 2 2 λ neB dl (7.272) 8π me c λ wavelength (= 2π/k) c Faraday rotation line dl line element in direction of 2.63×10−13 wave propagation 2 = Rλ (7.273) R rotation measure a I.e., plasmas in which electromagnetic force terms dominate over thermal pressure terms. Also taking µr =1. bIn a collisionless electron plasma. The ordinary and extraordinary modes are the + and − roots of S 2 when θB = π/2. When θB = 0, these roots are the right and left circularly polarised modes respectively, using the optical convention for handedness. cIn a tenuous plasma, SI units throughout. ∆ψ is taken positive if B is directed towards the observer. main January 23, 2006 16:6

158 Electromagnetism

Magnetohydrodynamicsa vs sound (wave) speed 1/2 1/2 γp 2γkBT γ ratio of heat capacities vs = = (7.274) p hydrostatic pressure Sound speed ρ mp ρ plasma mass density 166T 1/2 ms−1 (7.275) kB Boltzmann constant T temperature (K)

mp proton mass B ´ = (7.276) vA Alfven speed vA 1/2 Alfven´ speed (µ0ρ) B magnetic flux density (T) − 16 1/2 −1 µ0 permeability of free space 2.18×10 Bne ms (7.277) ne electron number density (m−3) 2µ p 4µ n k T 2v2 β plasma beta (ratio of Plasma beta 0 0 e B s hydrostatic to magnetic β = 2 = 2 = 2 (7.278) B B γvA pressure) − 2 2 e electronic charge Direct electrical ne e σ σ = (7.279) σd direct conductivity conductivity d 2 2 2 2 ne e +σ B σ conductivity (B =0) Hall electrical σB σH = σd (7.280) σH Hall conductivity conductivity nee J current density Generalised E electric field J = σd(E +v×B)+σHBˆ ×(E +v×B) (7.281) Ohm’s law v plasma velocity field Bˆ = B/|B| Resistive MHD equations (single-fluid model)b ∂B = ∇×(v×B)+η∇2B (7.282) ∂t µ0 permeability of free space ∂v ∇p 1 η magnetic diffusivity ·∇ − ∇× × ∇2 +(v )v = + ( B) B +ν v [= 1/(µ0σ)] ∂t ρ µ0ρ ν kinematic viscosity 1 + ν∇(∇·v)+g (7.283) g gravitational field strength 3 Shear Alfvenic´ ω angular frequency (= 2πν) dispersion ω = kvA cosθB (7.284) k wavevector (k =2π/λ) c relation θB angle between k and B Magnetosonic 2 2 2 2 − 4 2 2 4 2 dispersion ω k (vs +vA) ω = vs vAk cos θB (7.285) relationd a + − For a warm, fully ionised, electrically neutral p /e plasma, µr = 1. Relativistic and displacement current effects are assumed to be negligible and all oscillations are taken as being well below all resonance frequencies. bNeglecting bulk (second) viscosity. cNonresistive, inviscid flow. dNonresistive, inviscid flow. The greater and lesser solutions for ω2 are the fast and slow magnetosonic waves respectively. main January 23, 2006 16:6

7.9 Plasma physics 159

Synchrotron radiation v 2 Ptot total radiated power P =2σ cu γ2 sin2 θ (7.286) Power radiated tot T mag σ Thomson cross section c T by a single 2 u magnetic energy −14 2 2 v 2 mag a 1.59×10 B γ sin θ W 2 electron c density = B /(2µ0) (7.287) v electron velocity (∼ c) γ Lorentz factor 2 2 −1/2 4 2 v =[1−(v/c) ] ... averaged Ptot = σTcumagγ (7.288) 3 θ pitch angle (angle over pitch c v 2 between v and B) angles 1.06×10−14B2γ2 W (7.289) c B magnetic ﬂux density c speed of light P (ν) emission spectrum 31/2e3B sinθ ν frequency Single electron P (ν)= F(ν/ν ) (7.290) ch ν characteristic frequency emission 4π0cme ch − − − b × 25 1 e electronic charge spectrum 2.34 10 B sinθF(ν/νch)WHz (7.291) 0 free space permittivity me electronic (rest) mass

3 2 eB νch = γ sinθ (7.292) F spectral function Characteristic 2 2πme frequency K5/3 modiﬁed Bessel fn. of 4.2×1010γ2B sinθ Hz (7.293) the 2nd kind, order 5/3 1 ∞

F(x)=x K5/3(y)dy (7.294) F(x) Spectral x 0.5 function 1/3 2.15x (x 1) 1/2 −x (7.295) 1.25x e (x 1) 0 1234x aThis expression also holds for cyclotron radiation (v c). bI.e., total radiated power per unit frequency interval. 7 main January 23, 2006 16:6

160 Electromagnetism

Bremsstrahlunga

Single electron and ionb dW Z 2e6 ω2 1 ωb ωb = 2 + 2 (7.296) 4 3 3 2 2 4 2 K0 K1 dω 24π 0c me γ v γ γv γv Z 2e6 ( ) (7.297) 4 3 3 2 2 2 ωb γv 24π 0c me b v

Thermal bremsstrahlung radiation (v c; Maxwellian distribution) dP −hν =6.8×10−51Z 2T −1/2n n g(ν,T)exp Wm−3 Hz−1 (7.298) d d i e V ν  kT 16 3 −2 −2 5 2 0.28[ln(4.4×10 T ν Z )−0.76] (hν kT ∼< 10 kZ ) 10 −1 5 2 where g(ν,T) 0.55ln(2.1×10 Tν )(hν 10 kZ ∼< kT) (7.299)  (2.1×1010Tν−1)−1/2 (hν kT)

dP 1.7×10−40Z 2T 1/2n n Wm−3 (7.300) dV i e W energy radiated ω angular frequency (= 2πν) v electron velocity T electron temperature (K) Ze ionic charge Ki modiﬁed Bessel functions of order i (see page 47) n ion number density (m−3) −e electronic charge i γ Lorentz factor n electron number density permittivity of free space − e 0 =[1−(v/c)2] 1/2 (m−3) c speed of light P power radiated k Boltzmann constant m electronic mass e V volume h Planck constant b collision parameterc ν frequency (Hz) g Gaunt factor aClassical treatment. The ions are at rest, and all frequencies are above the plasma frequency. bThe spectrum is approximately ﬂat at low frequencies and drops exponentially at frequencies ∼> γv/b. cDistance of closest approach. main January 23, 2006 16:6

Chapter 8 Optics

8.1 Introduction Any attempt to unify the notations and terminology of optics is doomed to failure. This is partly due to the long and illustrious history of the subject (a pedigree shared only with mechanics), which has allowed a variety of approaches to develop, and partly due to the disparate fields of physics to which its basic principles have been applied. Optical ideas find their way into most wave-based branches of physics, from quantum mechanics to radio propagation. Nowhere is the lack of convention more apparent than in the study of polarisation, and so a cautionary note follows. The conventions used here can be taken largely from context, but the reader should be aware that alternative sign and handedness conventions do exist and are widely used. In particular we will take a circularly polarised wave as being right-handed if, for an observer looking towards the source, the electric field vector in a plane perpendicular to the line of sight rotates clockwise. This convention is often used in optics textbooks and has the conceptual advantage that the electric field orientation describes a right-hand corkscrew in space, with the direction of energy flow defining the screw direction. It is however opposite to the system widely used in radio engineering, where the handedness of a helical antenna generating or receiving the wave defines the handedness and is also in the opposite sense to the wave’s own angular momentum vector.

8 main January 23, 2006 16:6

162 Optics

8.2 Interference

Newton’s ringsa

rn radius of nth ring 2 ≥ nth dark ring rn = nRλ0 (8.1) n integer ( 0) R R lens radius of curvature 1 wavelength in external nth bright ring r2 = n+ Rλ (8.2) λ0 n 2 0 medium r aViewed in reﬂection. n

Dielectric layersa

η1 RN 1 R 1 N ×{ ηa η1 ηb

single layerη2 a multilayer

η3

1−R η3 1−RN

a film thickness Quarter-wave m λ = 0 (8.3) m thickness integer a (m ≥ 0) condition η2 4 η2 film refractive index  λ0 free-space wavelength 2  − 2 R power reflectance  η1η3 η2  2 (m odd) coefficient Single-layer η1η3 +η2 b R = (8.4) η1 entry-side refractive reflectance  2  η −η index  1 3 ( even) m η exit-side refractive η1 +η3 3 index − m − − Dependence of max if ( 1) (η1 η2)(η2 η3) > 0 (8.5) − m − − R on layer min if ( 1) (η1 η2)(η2 η3) < 0 (8.6) thickness, m 1/2 R =0 if η2 =(η1η3) and m odd (8.7)

RN multilayer reflectance N number of layer pairs − 2N 2 Multilayer η1 η3(ηa/ηb) ηa refractive index of top c R = (8.8) reflectance N 2N layer η1 +η3(ηa/ηb) ηb refractive index of bottom layer a For normal incidence, assuming the quarter-wave condition. The media are also assumed lossless, with µr =1. bSee page 154 for the definition of R. c For a stack of N layer pairs, giving an overall refractive index sequence η1ηa,ηbηa ...ηaηbη3 (see right-hand diagram). Each layer in the stack meets the quarter-wave condition with m =1. main January 23, 2006 16:6

8.2 Interference 163

Fabry-Perot etalona

∝ 1 eiφ θ 2iφ e η e3iφ θ h η

incident rays η

φ incremental phase difference k free-space wavenumber (= 2π/λ ) φ =2k0hη cosθ (8.9) 0 0 h cavity width Incremental 2 1/2 − ηsinθ θ fringe inclination (usually 1) phase =2k0hη 1 (8.10) b η θ internal angle of refraction difference =2πn for a maximum (8.11) η cavity refractive index η external refractive index n fringe order (integer) Coefficient of 4R F coefficient of finesse F = (8.12) finesse (1−R)2 R interface power reflectance π F = F1/2 (8.13) F 2 finesse Finesse λ0 free-space wavelength λ0 = Q (8.14) Q cavity quality factor ηh I (1−R)2 I(θ)= 0 (8.15) 2 − 1+R 2Rcosφ I transmitted intensity Transmitted I 0 I0 incident intensity intensity = 2 (8.16) 1+F sin (φ/2) A Airy function

= I0A(θ) (8.17)

Fringe ∆φ = 2arcsin(F−1/2) (8.18) intensity ∆φ phase diﬀerence at half intensity −1/2 point proﬁle 2F (8.19) 8

λ R1/2πn Chromatic 0 F − = n (8.20) resolving δλ 1 R δλ minimum resolvable wavelength 2Fhη diﬀerence power (θ 1) (8.21) λ0

δλf = Fδλ (8.22) Free spectral δλf wavelength free spectral range c c range δν = (8.23) δνf frequency free spectral range f 2ηh aNeglecting any eﬀects due to surface coatings on the etalon. See also Lasers on page 174. bBetween adjacent rays. Highest order fringes are near the centre of the pattern. c At near-normal incidence (θ 0), the orders of two spectral components separated by <δλf will not overlap. main January 23, 2006 16:6

164 Optics

8.3 Fraunhofer diﬀraction

Gratingsa

coherent plane waves

I(s) diffracted intensity Young’s I peak intensity kDs 0 double I(s)=I cos2 (8.24) θ diffraction angle D 0 2 slitsb s =sinθ D slit separation λ wavelength N equally 2 N number of slits spaced sin(Nkds/2) d I(s)=I0 (8.25) k wavenumber N narrow slits N sin(kds/2) (= 2π/λ) d slit spacing ∞ Infinite nλ n diffraction order I(s)=I δ s− (8.26) δ Dirac delta grating 0 d n=−∞ function Normal nλ sinθ = (8.27) θn angle of diffracted incidence n d maximum Oblique nλ θ sinθ +sinθ = (8.28) θi angle of incident i incidence n i d illumination θn Reflection nλ sinθn −sinθi = (8.29) θ θ grating d i n Chromatic λ resolving δλ diffraction peak = Nn (8.30) width power δλ Grating ∂θ n = (8.31) dispersion ∂λ dcosθ Bragg’s 2asinθ = nλ (8.32) a atomic plane lawc n spacing a Unless stated otherwise, the illumination is normal to the grating. θ a bTwo narrow slits separated by D. n c The condition is for Bragg reflection, with θn = θi. main January 23, 2006 16:6

8.3 Fraunhofer diﬀraction 165

Aperture diﬀraction

y ) x x,y f( coherent plane-wave sy illumination, normal to the xy plane sx

z ∞ ψ diffracted wavefunction ψ(s) ∝ f(x)e−iksx dx (8.33) General 1-D −∞ I diffracted intensity aperturea ∗ θ diffraction angle I(s) ∝ ψψ (s) (8.34) s =sinθ f aperture amplitude General 2-D transmission function

aperture in −ik(sxx+sy y) x,y distance across aperture ψ(sx,sy) ∝ f(x,y)e dxdy (8.35) (x,y) plane k wavenumber (= 2π/λ) ∞ (small angles) sx deflection xz plane sy deflection ⊥ xz plane sin2(kas/2) I(s)=I (8.36) I0 peak intensity Broad 1-D 0 (kas/2)2 b a slit width (in x) slit 2 ≡ I0 sinc (as/λ) (8.37) λ wavelength 2 Sidelobe In 2 1 = (n>0) (8.38) In nth sidelobe intensity intensity 2 I0 π (2n+1) Rectangular as bs a aperture width in x aperture I(s ,s )=I sinc2 x sinc2 y (8.39) x y 0 b aperture width in y (small angles) λ λ 2 Circular 2J1(kDs/2) J1 first-order Bessel function c I(s)=I (8.40) aperture 0 kDs/2 D aperture diameter First λ 8 s =1.22 (8.41) λ wavelength minimumd D First subsid. λ s =1.64 (8.42) maximum D Weak 1-D φ(x) phase distribution f(x)=exp[iφ(x)] 1+iφ(x) (8.43) phase object ii2 = −1 Fraunhofer (∆x)2 L distance of aperture from observation point limite L (8.44) λ ∆x aperture size aThe Fraunhofer integral. bNote that sincx = (sinπx)/(πx). cThe central maximum is known as the “Airy disk.” dThe “Rayleigh resolution criterion” states that two point sources of equal intensity can just be resolved with diffraction-limited optics if separated in angle by 1.22λ/D. ePlane-wave illumination. main January 23, 2006 16:6

166 Optics

8.4 Fresnel diffraction Kirchhoff’s diffraction formulaa y S x ρ dS dA source sˆ r θ ψ0 P

r z

(source at inﬁnity) P

ψP complex amplitude at P λ wavelength ikr Source at i e k wavenumber (= 2π/λ) ψP = − ψ0 K(θ) dA (8.45) infinity λ r ψ0 incident amplitude plane θ obliquity angle where: r distance of dA from P ( λ) Obliquity dA area element on incident 1 wavefront factor K(θ)= (1+cosθ) (8.46) K obliquity factor (cardioid) 2 dS element of closed surface ˆ unit vector ik(ρ+r) Source at iE0 e s vector normal to dS ψP = − [cos(sˆ ·rˆ)−cos(sˆ ·ρˆ)] dS finite λ 2ρr r vector from P to dS closed surface distanceb ρ vector from source to dS (8.47) E0 amplitude (see footnote) aAlso known as the “Fresnel–Kirchhoff formula.” Diffraction by an obstacle coincident with the integration surface can be approximated by omitting that part of the surface from the integral. b ikρ The source amplitude at ρ is ψ(ρ)=E0e /ρ. The integral is taken over a surface enclosing the point P .

Fresnel zones

sourcez1 z2 observer

z effective distance Effective aperture 1 1 1 a = + (8.48) z1 source–aperture distance distance z z1 z2 z2 aperture–observer distance n half-period zone number Half-period zone 1/2 y =(nλz) (8.49) λ wavelength radius n yn nth half-period zone radius Axial zeros (circular 2 zm distance of mth zero from R aperture aperture) zm = (8.50) 2mλ R aperture radius aI.e., the aperture–observer distance to be employed when the source is not at infinity. main January 23, 2006 16:6

8.4 Fresnel diﬀraction 167

Cornu spiral

0.8 √ 3 2

0.6 √ √ Edge diﬀraction 2.5 Cornu Spiral 3 ∞ 5 0.4 1 )

w 2 2 2 0.2 S( 1 w 2 i )

0 1.5 (1+

− 1 1 2 2 −0.2 C(w) intensity −2 1 )+

−0.4 −1 √ −∞ √ − 5 − 3 CS( w

0.5 −0.6

√ − 2 −0.8 0 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 −4 −240 2 w w πt2 C(w)= cos dt (8.51) Fresnel 0 2 C Fresnel cosine integral a integrals w πt2 S Fresnel sine integral S(w)= sin dt (8.52) 0 2 CS(w)=C(w)+iS(w) (8.53) CS Cornu spiral Cornu spiral 1 CS(±∞)=± (1+i) (8.54) v,w length along spiral 2

ψP complex amplitude at P ψ0 1 ψ = [CS(w)+ (1+i)] (8.55) ψ0 unobstructed amplitude P 1/2 2 2 λ wavelength Edge diffraction 1/2 2 z distance of P from where w = y (8.56) λz aperture plane [see (8.48)] y position of edge ψ0 coherent ψP = [CS(w2)−CS(w1)] (8.57) Diffraction 21/2 plane waves from a long 2 1/2 8 slitb where w = y (8.58) i i λz y2 ψ0 y1 P ψP = [CS(v2)−CS(v1)]× (8.59) 2 z Diffraction [CS(w2)−CS(w1)] (8.60) x positions of slit sides from a 1/2 i 2 y positions of slit rectangular where vi = xi (8.61) i λz top/bottom aperture 2 1/2 and w = y (8.62) i i λz aSee also Equation (2.393) on page 45. bSlit long in x. main January 23, 2006 16:6

168 Optics

8.5 Geometrical optics

Lenses and mirrorsa

r2 v object f x1 object f u v image R x2 f r1 u lens mirror sign convention + − r centred to right centred to left u real object virtual object v real image virtual image f converging lens/ diverging lens/ concave mirror convex mirror

MT erect image inverted image L optical path length Fermat’s principleb L = η dl is stationary (8.63) η refractive index dl ray path element 1 1 1 u object distance Gauss’s lens formula + = (8.64) v image distance u v f f focal length

Newton’s lens 2 x1 = v −f x1x2 = f (8.65) formula x2 = u−f Lensmaker’s 1 1 1 1 − − ri radii of curvature of + =(η 1) (8.66) lens surfaces formula u v r1 r2 1 1 2 1 Mirror formulac + = − = (8.67) R mirror radius of u v R f curvature 1 Dioptre number D = m−1 (8.68) D dioptre number (f in f metres)

f n focal ratio Focal ratiod n = (8.69) d d lens or mirror diameter Transverse linear v M = − (8.70) MT transverse magnification T u magnification Longitudinal linear M = −M2 (8.71) ML longitudinal magnification L T magnification aFormulas assume “Gaussian optics,” i.e., all lenses are thin and all angles small. Light enters from the left. bA stationary optical path length has, to first order, a length identical to that of adjacent paths. cThe mirror is concave if R<0, convex if R>0. dOr “f-number,” written f/2ifn = 2 etc. main January 23, 2006 16:6

8.5 Geometrical optics 169

Prisms (dispersing)

α δ

θi θt

prism

θi angle of incidence 2 − 2 1/2 Transmission sinθt =(η sin θi) sinα θt angle of transmission angle −sinθi cosα (8.72) α apex angle η refractive index

Deviation δ = θi +θt −α (8.73) δ angle of deviation

Minimum α deviation sinθi =sinθt = ηsin (8.74) 2 condition

Refractive sin[(δm +α)/2] η = (8.75) δm minimum deviation index sin(α/2)

Angular dδ 2sin(α/2) dη D dispersion a D = = (8.76) dispersion dλ cos[(δm +α)/2] dλ λ wavelength aAt minimum deviation.

Optical ﬁbres L

θm

cladding, ηc <ηf ﬁbre, ηf

θm maximum angle of incidence 8 1 2 − 2 1/2 η0 exterior refractive index Acceptance angle sinθm = (ηf ηc ) (8.77) η0 ηf ﬁbre refractive index ηc cladding refractive index Numerical N = η sinθ (8.78) N numerical aperture aperture 0 m ∆t temporal dispersion Multimode ∆t ηf ηf = −1 (8.79) L ﬁbre length dispersiona L c ηc c speed of light a Of a pulse with a given wavelength, caused by the range of incident angles up to θm. Sometimes called “intermodal dispersion” or “modal dispersion.” main January 23, 2006 16:6

170 Optics

8.6 Polarisation

Elliptical polarisationa y E electric ﬁeld E0y k wavevector Elliptical E =(E ,E eiδ)ei(kz−ωt) 0x 0y z propagation axis α x polarisation (8.80) ωt angular frequency × time E0x

E0x x amplitude of E 2E E amplitude of E Polarisation tan2α = 0x 0y cosδ E0y y b 2 − 2 δ relative phase of E a angleb E0x E0y y (8.81) with respect to Ex α polarisation angle e ellipticity a−b Ellipticityc e = (8.82) a semi-major axis θ a b semi-minor axis I(θ) transmitted intensity

d 2 I0 incident intensity Malus’s law I(θ)=I0 cos θ (8.83) θ polariser–analyser angle aSee the introduction (page 161) for a discussion of sign and handedness conventions. bAngle between ellipse major axis and x axis. Sometimes the polarisation angle is deﬁned as π/2−α. cThis is one of several deﬁnitions for ellipticity. dTransmission through skewed polarisers for unpolarised incident light.

Jones vectors and matrices E electric ﬁeld Normalised Ex E = ; |E| = 1 (8.84) Ex x component of E electric ﬁelda E y E y component of E y 1 1 1 ◦ Ex = E45 = √ E 45 to x axis Example 0 2 1 45 E right-hand circular vectors: r 1 1 1 1 E left-hand circular Er = √ El = √ l 2 −i 2 i

Et transmitted vector

Jones matrix Et = AEi (8.85) Ei incident vector A Jones matrix

Example matrices: 10 00 Linear polariser x Linear polariser y 00 01 1 11 1 1 −1 Linear polariser at 45◦ Linear polariser at −45◦ 2 11 2 −11 1 1 i 1 1 −i Right circular polariser Left circular polariser 2 −i 1 2 i 1 10 10 λ/4 plate (fast x)eiπ/4 λ/4 plate (fast ⊥ x)eiπ/4 0 i 0 −i aKnown as the “normalised Jones vector.” main January 23, 2006 16:6

8.6 Polarisation 171

Stokes parametersa

y V E0y

χ pI α x Q 2χ E0x 2α U 2b 2a Poincare´ sphere

k wavevector i(kz−ωt) Ex = E0xe (8.86) ωt angular frequency × time Electric fields i(kz−ωt+δ) Ey = E0ye (8.87) δ relative phase of Ey with respect to Ex b χ (see diagram) Axial ratiob tanχ = ±r = ± (8.88) a r axial ratio 2 2 I = Ex + Ey (8.89) 2 − 2 Ex electric field component x Q = Ex Ey (8.90) Ey electric field component y = pI cos2χcos2α (8.91) E field amplitude in x direction Stokes 0x U =2E E cosδ (8.92) E field amplitude in y direction parameters x y 0y = pI cos2χsin2α (8.93) α polarisation angle p degree of polarisation V =2ExEy sinδ (8.94) · mean over time = pI sin2χ (8.95)

Degree of (Q2 +U2 +V 2)1/2 p = ≤ 1 (8.96) polarisation I Q/I U/I V /I Q/I U/I V /I 8 left circular 0 0 −1 right circular 0 0 1 linear x 100linear y −100 linear 45◦ to x 010linear −45◦ to x 0 −10 unpolarised 0 0 0 aUsing the convention that right-handed circular polarisation corresponds to a clockwise rotation of the electric field in a given plane when looking towards the source. The propagation direction in the diagram is out of the plane. The parameters I, Q, U, and V are sometimes denoted s0, s1, s2, and s3, and other nomenclatures exist. There is no generally accepted definition – often the parameters are scaled to be dimensionless, with s0 = 1, or to represent ⊥ 2 2 power flux through a plane the beam, i.e., I =( Ex + Ey )/Z0 etc., where Z0 is the impedance of free space. bThe axial ratio is positive for right-handed polarisation and negative for left-handed polarisation using our definitions. main January 23, 2006 16:6

172 Optics

8.7 Coherence (scalar theory)

Mutual Γij mutual coherence function ∗ τ temporal interval coherence Γ12(τ)= ψ1(t)ψ2(t+τ) (8.97) function ψi (complex) wave disturbance at spatial point i ψ (t)ψ∗(t+τ) γ (τ)= 1 2 (8.98) t time 12 | |2 | |2 1/2 · Complex degree [ ψ1 ψ2 ] mean over time of coherence Γ12(τ) γij complex degree of coherence = (8.99) ∗ 1/2 complex conjugate [Γ11(0)Γ22(0)]

Itot combined intensity Combined 1/2 Itot = I1 +I2 +2(I1I2) [γ (τ)] Ii intensity of disturbance at a 12 intensity (8.100) point i real part of 1/2 2(I1I2) | | Fringe visibility V (τ)= γ12(τ) (8.101) I1 +I2 if |γ (τ)| is a I −I I max. combined intensity 12 V = max min (8.102) max constant: Imax +Imin Imin min. combined intensity | | if I1 = I2: V (τ)= γ12(τ) (8.103)

∗ ψ1(t)ψ1(t+τ) γ(τ) degree of temporal coherence Complex degree γ(τ)= (8.104) |ψ (t)2| I(ω) speciﬁc intensity of temporal 1 −iωτ b I(ω)e dω ω radiation angular frequency coherence = (8.105) I(ω)dω c speed of light

∆τc coherence time Coherence time ∆lc 1 ∆τ = ∼ (8.106) ∆lc coherence length and length c c ∆ν ∆ν spectral bandwidth ∗ γ(D) degree of spatial coherence ψ1ψ2 Complex degree γ(D)= (8.107) D spatial separation of points 1 [|ψ |2 |ψ |2 ]1/2 and 2 of spatial 1 2 ikD·sˆ c I(sˆ)e dΩ I(sˆ) speciﬁc intensity of distant coherence = (8.108) extended source in direction sˆ I(sˆ)dΩ dΩ diﬀerential solid angle Intensity I I sˆ unit vector in the direction of 1 2 =1+γ2(D) (8.109) dΩ correlationd 2 2 1/2 [ I1 I2 ] k wavenumber Speckle 1 − intensity I/ I pr probability density pr(I)= e (8.110) distributione I Speckle size ∆wc characteristic speckle size λ λ wavelength (coherence ∆wc (8.111) width) α α source angular size as seen from the screen aFrom interfering the disturbances at points 1 and 2 with a relative delay τ. bOr “autocorrelation function.” cBetween two points on a wavefront, separated by D. The integral is over the entire extended source. dFor wave disturbances that have a Gaussian probability distribution in amplitude. This is “Gaussian light” such as from a thermal source. eAlso for Gaussian light. main January 23, 2006 16:6

8.8 Line radiation 173

8.8 Line radiation

Spectral line broadening I(ω) normalised intensityb Natural (2πτ)−1 I(ω)= (8.112) τ lifetime of excited state broadeninga −2 − 2 (2τ) +(ω ω0) ω angular frequency (= 2πν) Natural 1 ∆ω half-width at half-power ∆ω = (8.113) half-width 2τ ω0 centre frequency Collision (πτ )−1 τc mean time between I(ω)= c (8.114) collisions broadening −2 − 2 (τc) +(ω ω0) p pressure d effective atomic diameter Collision and − m gas particle mass 1 πmkT 1/2 pressure ∆ω = = pπd2 (8.115) k Boltzmann constant half-widthc τc 16 T temperature c speed of light mc2 1/2 mc2 (ω −ω )2 Doppler I(ω)= exp − 0 2 2 I(ω) broadening 2kTω0π 2kT ω0 (8.116) ∆ω Doppler 2kT ln2 1/2 ∆ω = ω (8.117) half-width 0 2 mc ω0 aThe transition probability per unit time for the state is = 1/τ. In the classical limit of a damped oscillator, the e-folding time of the electric field is 2τ. Both the natural and collision profiles described here are Lorentzian. b The intensity spectra are normalised so that I(ω)dω = 1, assuming ∆ω/ω0 1. cThe pressure-broadening relation combines Equations (5.78), (5.86) and (5.89) and assumes an otherwise perfect gas of finite-sized atoms. More accurate expressions are considerably more complicated.

Einstein coeﬃcientsa −3 −1 Rij transition rate, level i → j (m s )

Absorption R12 = B12Iν n1 (8.118) Bij Einstein B coefficients Iν specific intensity of radiation field 8 Spontaneous A21 Einstein A coefficient R21 = A21n2 (8.119) ni number density of atoms in quantum emission − level i (m 3) Stimulated R = B I n (8.120) emission 21 21 ν 2

3 A21 2hν g1 h Planck constant = 2 (8.121) Coefficient B12 c g2 ν frequency ratios B21 g1 c speed of light = (8.122) g degeneracy of ith level B12 g2 i a Note that the coefficients can also be defined in terms of spectral energy density, uν =4πIν /c rather than Iν .Inthis 3 case A21 = 8πhν g1 .SeealsoPopulation densities on page 116. B12 c3 g2 main January 23, 2006 16:6

174 Optics

Lasersa

R1 R2 r1

light out r L 2 Cavity stability L L r radii of curvature of end-mirrors 0 ≤ 1− 1− ≤ 1 (8.123) 1,2 condition r1 r2 L distance between mirror centres ν mode frequency Longitudinal c n ν = n (8.124) n integer cavity modesb n 2L c speed of light 1/4 2πL(R1R2) = (8.125) Q quality factor Q 1/2 λ[1−(R1R2) ] Cavity Q R1,2 mirror (power) reﬂectances 4πL (8.126) λ wavelength λ(1−R1R2)

Cavity line νn ∆νc cavity line width (FWHP) ∆νc = =1/(2πτc) (8.127) width Q τc cavity photon lifetime ∆ν line width (FWHP) 2 Schawlow– ∆ν 2πh(∆νc) glNu P laser power Townes line = νn P glNu −guNl gu,l degeneracy of upper/lower levels width (8.128) Nu,l number density of upper/lower levels Threshold α gain per unit length of medium R1R2 exp[2(α−β)L] > 1 (8.129) lasing condition β loss per unit length of medium aAlso see the Fabry-Perot etalon on page 163. Note that “cavity” refers to the empty cavity, with no lasing medium present. bThe mode spacing equals the cavity free spectral range. main January 23, 2006 16:6

Chapter 9 Astrophysics

9.1 Introduction Many of the formulas associated with astronomy and astrophysics are either too specialised for a general work such as this or are common to other fields and can therefore be found elsewhere in this book. The following section includes many of the relationships that fall into neither of these categories, including equations to convert between various astronomical coordinate systems and some basic formulas associated with cosmology. Exceptionally, this section also includes data on the Sun, Earth, Moon, and planets. Observational astrophysics remains a largely inexact science, and parameters of these (and other) bodies are often used as approximate base units in measurements. For example, the masses of stars and galaxies are frequently quoted as multiples of the mass of the Sun 30 (1M =1.989×10 kg), extra-solar system planets in terms of the mass of Jupiter, and so on. Astronomers seem to find it particularly difficult to drop arcane units and conventions, resulting in a profusion of measures and nomenclatures throughout the subject. However, the convention of using suitable astronomical objects in this way is both useful and widely accepted.

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176 Astrophysics

9.2 Solar system data Solar data 8 equatorial radius R = 6.960×10 m = 109.1R⊕ 30 5 mass M = 1.9891×10 kg = 3.32946×10 M⊕ 46 2 8 polar moment of inertia I = 5.7×10 kgm = 7.09×10 I⊕ 26 bolometric luminosity L = 3.826×10 W effective surface temperature T = 5770K solar constanta 1.368×103 Wm−2 absolute magnitude MV =+4.83; Mbol =+4.75 apparent magnitude mV = −26.74; mbol = −26.82 aBolometric flux at a distance of 1 astronomical unit (AU). Earth data 6 −3 equatorial radius R⊕ = 6.37814×10 m = 9.166×10 R flatteninga f =0.00335364 = 1/298.183 24 −6 mass M⊕ = 5.9742×10 kg = 3.0035×10 M 37 2 −9 polar moment of inertia I⊕ = 8.037×10 kgm = 1.41×10 I b 11 orbital semi-major axis 1AU = 1.495979×10 m = 214.9R mean orbital velocity 2.979×104 ms−1 −2 equatorial surface gravity ge = 9.780327ms (includes rotation) −2 polar surface gravity gp = 9.832186ms −5 −1 rotational angular velocity ωe = 7.292115×10 rads a 6 f equals (R⊕ −Rpolar)/R⊕. The mean radius of the Earth is 6.3710×10 m. bAbout the Sun. Moon data 6 equatorial radius Rm = 1.7374×10 m =0.27240R⊕ 22 −2 mass Mm = 7.3483×10 kg = 1.230×10 M⊕ a 8 mean orbital radius am = 3.84400×10 m =60.27R⊕ mean orbital velocity 1.03×103 ms−1 orbital period (sidereal) 27.32166d −2 equatorial surface gravity 1.62ms =0.166ge aAbout the Earth. Planetary dataa M/M⊕ R/R⊕ T (d) P (yr) a(AU) M mass Mercury 0.055 274 0.382 51 58.646 0.240 85 0.387 10 R equatorial radius Venusb 0.815 00 0.948 83 243.018 0.615 228 0.723 35 T rotational period Earth 1 1 0.997 27 1.000 04 1.000 00 P orbital period Mars 0.107 45 0.532 60 1.025 96 1.880 93 1.523 71 a mean distance Jupiter 317.85 11.209 0.413 54 11.861 3 5.202 53 M⊕ 5.9742×1024 kg Saturn 95.159 9.449 1 0.444 01 29.628 2 9.575 60 R⊕ 6.37814×106 m Uranusb 14.500 4.007 3 0.718 33 84.746 6 19.293 4 1d 86400s Neptune 17.204 3.882 6 0.671 25 166.344 30.245 9 1yr 3.15569×107 s Plutob 0.00251 0.187 36 6.387 2 248.348 39.509 0 1AU 1.495979×1011 m aUsing the osculating orbital elements for 1998. Note that P is the instantaneous orbital period, calculated from the planet’s daily motion. The radii of gas giants are taken at 1 atmosphere pressure. bRetrograde rotation. main January 23, 2006 16:6

9.3 Coordinate transformations (astronomical) 177

9.3 Coordinate transformations (astronomical) Time in astronomy a Julian day number JD Julian day number JD =D −32075+1461∗(Y +4800+(M −14)/12)/4 D day of month number +367∗(M −2−(M −14)/12∗12)/12 Y calendar year, e.g., 1963 M calendar month (Jan=1) −3∗((Y +4900+(M −14)/12)/100)/4 (9.1) ∗ integer multiply Modiﬁed / integer divide − Julian day MJD = JD 2400000.5 (9.2) MJD modiﬁed Julian day number number Day of W =(JD +1) mod 7 (9.3) W day of week (0=Sunday, week 1=Monday, ... ) LCT local civil time Local civil UTC coordinated universal time LCT = UTC+TZC+DSC (9.4) time TZC time zone correction DSC daylight saving correction Julian JD −2451545.5 T Julian centuries between T = (9.5) 12h UTC 1 Jan 2000 and centuries 36525 0h UTC D/M/Y GMST =6h41m50s 54841 . GMST Greenwich mean sidereal Greenwich +8640184s.812866T time at 0h UTC D/M/Y sidereal (for later times use +0s.093104T 2 time 1s=1.002738 sidereal −0s.0000062T 3 (9.6) seconds)

Local ◦ LST local sidereal time λ ◦ sidereal LST = GMST+ ◦ (9.7) λ geographic longitude, time 15 degrees east of Greenwich aFor the Julian day starting at noon on the calendar day in question. The routine is designed around integer arithmetic with “truncation towards zero” (so that −5/3=−1) and is valid for dates from the onset of the Gregorian calendar, 15 October 1582. JD represents the number of days since Greenwich mean noon 1 Jan 4713 BC. For reference, noon, 1 Jan 2000 = JD2451545 and was a Saturday (W =6).

Horizon coordinatesa LST local sidereal time Hour angle H =LST−α (9.8) H (local) hour angle α right ascension sina =sinδ sinφ+cosδ cosφcosH (9.9) δ declination Equatorial −cosδ sinH a altitude 9 to horizon tanA ≡ (9.10) sinδ cosφ−sinφcosδ cosH A azimuth (E from N) φ observer’s latitude + + sinδ =sinasinφ+cosacosφcosA (9.11) − + Horizon to A, H −cosasinA equatorial tanH ≡ (9.12) − − sinacosφ−sinφcosacosA − + aConversions between horizon or alt–azimuth coordinates, (a,A), and celestial equatorial coordinates, (δ,α). There are a number of conventions for deﬁning azimuth. For example, it is sometimes taken as the angle west from south rather than east from north. The quadrants for A and H can be obtained from the signs of the numerators and denominators in Equations (9.10) and (9.12) (see diagram). main January 23, 2006 16:6

178 Astrophysics

Ecliptic coordinatesa

ε =23◦2621.45−46.815T ε mean ecliptic obliquity Obliquity of − 2 0 .0006T T Julian centuries since the ecliptic b +0.00181T 3 (9.13) J2000.0

α right ascension sinβ =sinδ cosε−cosδ sinεsinα (9.14) Equatorial to δ declination sinαcosε+tanδ sinε ecliptic tanλ ≡ (9.15) λ ecliptic longitude cosα β ecliptic latitude + + sinδ =sinβ cosε+cosβ sinεsinλ (9.16) − + Ecliptic to λ, α sinλcosε−tanβ sinε equatorial tanα ≡ (9.17) − − cosλ − + aConversions between ecliptic, (β,λ), and celestial equatorial, (δ,α), coordinates. β is positive above the ecliptic and λ increases eastwards. The quadrants for λ and α can be obtained from the signs of the numerators and denominators in Equations (9.15) and (9.17) (see diagram). bSee Equation (9.5).

Galactic coordinatesa ◦ αg = 192 15 (9.18) αg right ascension of Galactic ◦ north galactic pole δg =27 24 (9.19) frame ◦ δg declination of north lg =33 (9.20) galactic pole

sinb =cosδ cosδ cos(α−α )+sinδ sinδ (9.21) Equatorial g g g lg ascending node of − − galactic plane on to galactic tanδ cosδg cos(α αg)sinδg tan(l −lg) ≡ (9.22) equator sin(α−αg) δ declination sinδ =cosbcosδ sin(l −l )+sinbsinδ (9.23) Galactic to g g g α right ascension − equatorial cos(l lg) b galactic latitude tan(α−αg) ≡ (9.24) tanbcosδg −sinδg sin(l −lg) l galactic longitude aConversions between galactic, (b,l), and celestial equatorial, (δ,α), coordinates. The galactic frame is deﬁned at epoch B1950.0. The quadrants of l and α can be obtained from the signs of the numerators and denominators in Equations (9.22) and (9.24).

Precession of equinoxesa α right ascension of date

In right s s α0 right ascension at J2000.0 α α0 +(3 .075+1 .336sinα0 tanδ0)N (9.25) ascension N number of years since J2000.0 In δ declination of date δ δ0 +(20 .043cosα0)N (9.26) declination δ0 declination at J2000.0 aRight ascension in hours, minutes, and seconds; declination in degrees, arcminutes, and arcseconds. These equations are valid for several hundred years each side of J2000.0. main January 23, 2006 16:6

9.4 Observational astrophysics 179

9.4 Observational astrophysics

Astronomical magnitudes

Apparent − − F1 mi apparent magnitude of object i m1 m2 = 2.5log10 (9.27) magnitude F2 Fi energy ﬂux from object i M absolute magnitude − − Distance m M = 5log10 D 5 (9.28) m−M distance modulus a − − modulus = 5log10 p 5 (9.29) D distance to object (parsec) p annual parallax (arcsec) − L Luminosity– Mbol =4.75 2.5log10 (9.30) Mbol bolometric absolute magnitude magnitude L L luminosity (W) (28−0.4Mbol) relation L 3.04×10 (9.31) L solar luminosity (3.826×1026 W)

Flux– −2 −(8+0.4mbol) Fbol bolometric ﬂux (Wm ) magnitude Fbol 2.559×10 (9.32) m bolometric apparent magnitude relation bol − BC bolometric correction Bolometric BC = mbol mV (9.33) mV V -band apparent magnitude correction = Mbol −MV (9.34) MV V -band absolute magnitude − − Colour B V = mB mV (9.35) B −V observed B −V colour index b index U −B = mU −mB (9.36) U −B observed U −B colour index

Colour − − − EB−V colour excess c E =(B V ) (B V )0 (9.37) excess (B −V )0 intrinsic B −V colour index aNeglecting extinction. bUsing the UBV magnitude system. The bands are centred around 365 nm (U), 440 nm (B), and 550 nm (V ). cThe U −B colour excess is deﬁned similarly.

Photometric wavelengths λ mean wavelength Mean λR(λ)dλ 0 λ wavelength wavelength λ0 = (9.38) R(λ)dλ R system spectral response Isophotal F(λ)R(λ)dλ F(λ) flux density of source (in 9 terms of wavelength) wavelength F(λi)= (9.39) R(λ)dλ λ isophotal wavelength i Effective λF(λ)R(λ)dλ λ = (9.40) λeff effective wavelength wavelength eff F(λ)R(λ)dλ main January 23, 2006 16:6

180 Astrophysics

Planetary bodies

DAU planetary orbital radius (AU) n 4+3×2 −∞ Bode’s lawa D = (9.41) n index: Mercury = , Venus AU 10 = 0, Earth = 1, Mars = 2, Ceres = 3, Jupiter= 4, ... 1/3 R satellite orbital radius > 100M R ∼ (9.42) M central mass Roche limit 9πρ > ρ satellite density ∼ 2.46R0 (if densities equal) (9.43) R0 central body radius S synodic period Synodic 1 1 1 b = − (9.44) P planetary orbital period period S P P⊕ P⊕ Earth’s orbital period aAlso known as the “Titius–Bode rule.” Note that the asteroid Ceres is counted as a planet in this scheme. The relationship breaks down for Neptune and Pluto. bOf a planet.

Distance indicators v cosmological recession velocity

Hubble law v = H0d (9.45) H0 Hubble parameter (present epoch) d (proper) distance

Dpc distance (parsec) Annual −1 D = p (9.46) ± parallax pc p annual parallax ( p arcsec from mean) L mean cepheid luminosity L log10 1.15log10 Pd +2.47 (9.47) L Solar luminosity Cepheid L a variables − − Pd pulsation period (days) MV 2.76log10 Pd 1.40 (9.48) MV absolute visual magnitude

MI I-band absolute magnitude 2 Tully–Fisher − vrot − vrot observed maximum rotation MI 7.68log10 2.58 velocity (kms−1) relationb sini (9.49) i galactic inclination (90◦ when edge-on) θ ring angular radius 4GM d −d M lens mass Einstein rings 2 s l θ = 2 (9.50) c dsdl ds distance from observer to source dl distance from observer to lens T apparent CMBR temperature Sunyaev– dl path element through cloud ∆T nekTeσT Zel’dovich − R cloud radius = 2 2 dl (9.51) c T mec effect ne electron number density k Boltzmann constant T electron temperature ... for a e ∆T 4Rn kT σ σ Thomson cross section homogeneous = − e e T (9.52) T 2 m electron mass sphere T mec e c speed of light a Period–luminosity relation for classical Cepheids. Uncertainty in MV is ±0.27 (Madore & Freedman, 1991, Publications of the Astronomical Society of the Pacific, 103, 933). bGalaxy rotation velocity–magnitude relation in the infrared I waveband, centred at 0.90µm. The coefficients depend on waveband and galaxy type (see Giovanelli et al., 1997, The Astronomical Journal, 113,1). cScattering of the cosmic microwave background radiation (CMBR) by a cloud of electrons, seen as a temperature decrement, ∆T , in the Rayleigh–Jeans limit (λ 1mm). main January 23, 2006 16:6

9.5 Stellar evolution 181

9.5 Stellar evolution

Evolutionary timescales τ free-fall timescale Free-fall 3π 1/2 ﬀ a τ = (9.53) G constant of gravitation timescale ﬀ 32Gρ 0 ρ0 initial mass density

τKH Kelvin–Helmholtz timescale −Ug τ = (9.54) U gravitational potential energy Kelvin–Helmholtz KH L g M body’s mass timescale GM2 (9.55) R0 body’s initial radius R0L L body’s luminosity aFor the gravitational collapse of a uniform sphere.

Star formation λJ Jeans length π dp 1/2 G constant of gravitation Jeans lengtha λ = (9.56) J Gρ dρ ρ cloud mass density p pressure

π 3 Jeans mass MJ = ρλ (9.57) M (spherical) Jeans mass 6 J J

LE Eddington luminosity 4πGMmpc M stellar mass Eddington LE = (9.58) σ M solar mass limiting T M mp proton mass luminosityb 1.26×1031 W (9.59) M c speed of light σT Thomson cross section aNote that (dp/dρ)1/2 is the sound speed in the cloud. bAssuming the opacity is mostly from Thomson scattering.

Stellar theorya r radial distance Conservation of dMr 2 =4πρr (9.60) Mr mass interior to r mass dr ρ mass density Hydrostatic dp −GρM p pressure = r (9.61) equilibrium dr r2 G constant of gravitation

dL L luminosity interior to r Energy release r =4πρr2 (9.62) r dr power generated per unit mass T temperature 9 Radiative dT −3 κ ρ Lr = (9.63) σ Stefan–Boltzmann constant transport dr 16σ T 3 4πr2 κ mean opacity Convective dT γ −1 T dp = (9.64) γ ratio of heat capacities, c /c transport dr γ p dr p V aFor stars in static equilibrium with adiabatic convection. Note that ρ is a function of r. κ and are functions of temperature and composition. main January 23, 2006 16:6

182 Astrophysics

Stellar fusion processesa PP i chain PP ii chain PP iii chain + + → 2 + + + → 2 + + + → 2 + p +p 1H+e +νe p +p 1H+e +νe p +p 1H+e +νe 2 + → 3 2 + → 3 2 + → 3 1H+p 2He+γ 1H+p 2He+γ 1H+p 2He+γ 3 3 → 4 + 3 4 → 7 3 4 → 7 2He+ 2He 2He+2p 2He+ 2He 4Be+γ 2He+ 2He 4Be+γ 7 − → 7 7 + → 8 4Be+e 3Li+νe 4Be+p 5B+γ 7 + → 4 8 → 8 + 3Li+p 22He 5B 4Be+e +νe 8 → 4 4Be 22He CNO cycle triple-α process 12 + → 13 4 4 8 6C+p 7N+γ 2He+ 2He 4Be+γ γ photon 13 → 13 + 8 4 12 ∗ + 7N 6C+e +νe 4Be+ 2He 6C p proton 13 + → 14 12 ∗ → 12 e+ positron 6C+p 7N+γ 6C 6C+γ − 14 + → 15 e electron 7N+p 8O+γ 15 → 15 + νe electron neutrino 8O 7N+e +νe 15 + → 12 4 7N+p 6C+2He aAll species are taken as fully ionised.

Pulsars n ω˙ ∝−ω (9.65) ω rotational angular velocity Braking P rotational period (= 2π/ω) index P P¨ n =2− (9.66) n braking index P˙ 2 T characteristic age Characteristic 1 P L luminosity a T = (9.67) age n−1 P˙ µ0 permeability of free space c speed of light m pulsar magnetic dipole moment µ |m¨|2 sin2 θ Magnetic L = 0 (9.68) R pulsar radius 6πc3 dipole Bp magnetic ﬂux density at 2πR6B2ω4 sin2 θ magnetic pole radiation = p (9.69) 3c3µ θ angle between magnetic and 0 rotational axes DM dispersion measure D Dispersion D path length to pulsar DM = n dl (9.70) measure e dl path element

0 ne electron number density dτ −e2 = DM (9.71) τ pulse arrival time dν 4π2 m cν3 ∆τ difference in pulse arrival time Dispersionb 0 e 2 e 1 1 νi observing frequencies ∆τ = − DM (9.72) 2 2 2 me electron mass 8π 0mec ν1 ν2 aAssuming n = 1 and that the pulsar has already slowed significantly. Usually n is assumed to be 3 (magnetic dipole radiation), giving T = P/(2P˙). bThe pulse arrives first at the higher observing frequency. main January 23, 2006 16:6

9.5 Stellar evolution 183

Compact objects and black holes

rs Schwarzschild radius G constant of gravitation Schwarzschild 2GM M rs = 3 km (9.73) M mass of body radius c2 M c speed of light M solar mass 1/2 r distance from mass centre Gravitational ν∞ 2GM = 1− (9.74) ν∞ frequency at inﬁnity redshift ν rc2 r νr frequency at r m orbiting masses Gravitational 32 G4 m2m2(m +m ) i L = 1 2 1 2 (9.75) a mass separation wave radiationa g 5 c5 a5 Lg gravitational luminosity 96 G5/3 m m P −5/3 Rate of change of ˙ = − (4 2)4/3 1 2 P π 5 1/3 P orbital period orbital period 5 c (m1 +m2) (9.76) Neutron star p pressure degeneracy (3π2)2/3 h¯2 ρ 5/3 2 h¯ (Planck constant)/(2π) pressure p = = u (9.77) m neutron mass 5 mn mn 3 n (nonrelativistic) ρ density hc¯ (3π2)1/3 ρ 4/3 1 Relativisticb p = = u (9.78) u energy density 4 mn 3 Chandrasekhar M 1.46M (9.79) M Chandrasekhar mass massc Ch Ch Maximum black GM2 hole angular Jm maximum angular Jm = (9.80) momentum momentum c Black hole M3 ∼ × 66 τ evaporation time τe 3 10 yr (9.81) e evaporation time M 3 Black hole hc¯ M T temperature T = 10−7 K (9.82) temperature 8πGMk M k Boltzmann constant a From two bodies, m1 and m2, in circular orbits about their centre of mass. Note that the frequency of the radiation is twice the orbital frequency. bParticle velocities ∼ c. cUpper limit to mass of a white dwarf.

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184 Astrophysics

9.6 Cosmology

Cosmological model parameters

vr radial velocity Hubble law vr = Hd (9.83) H Hubble parameter d proper distance R˙(t) H(t)= (9.84) 0 present epoch ( ) R cosmic scale factor Hubble R t 3 t cosmic time parametera H(z)=H0[Ωm0(1+z) +ΩΛ0 2 1/2 z redshift +(1−Ωm0 −ΩΛ0)(1+z) ] (9.85)

λobs observed wavelength λobs −λem R0 Redshift z = = −1 (9.86) λem emitted wavelength λem R(tem) tem epoch of emission 2 2 2 2 2 dr Robertson– ds =c dt −R (t) ds interval 1−kr2 Walker c speed of light b r,θ,φ comoving spherical polar metric 2 2 2 2 +r (dθ +sin θ dφ ) (9.87) coordinates 4π p ΛR R¨ = − GR ρ+3 + (9.88) k curvature parameter Friedmann 3 c2 3 G constant of gravitation equationsc 8π ΛR2 p pressure R˙ 2 = GρR2 −kc2 + (9.89) 3 3 Λ cosmological constant

Critical 3H2 ρ (mass) density ρcrit = (9.90) density 8πG ρcrit critical density ρ 8πGρ Ωm = = 2 (9.91) ρcrit 3H Λ Ωm matter density parameter Density ΩΛ = 2 (9.92) 3H ΩΛ lambda density parameter parameters 2 kc Ω curvature density parameter Ω = − (9.93) k k R2H2 Ωm +ΩΛ +Ωk = 1 (9.94)

Deceleration R0R¨0 Ωm0 q = − = −Ω (9.95) q0 deceleration parameter parameter 0 ˙ 2 Λ0 R0 2 a < < −1 −1 −1 −1 Often called the Hubble “constant.” At the present epoch, 60 ∼ H0 ∼ 80kms Mpc ≡ 100hkms Mpc ,where h is a dimensionless scaling parameter. The Hubble time is tH =1/H0. Equation (9.85) assumes a matter dominated universe and mass conservation. bFor a homogeneous, isotropic universe, using the (−1,1,1,1) metric signature. r is scaled so that k =0,±1. Note that ds2 ≡ (ds)2 etc. cΛ = 0 in a Friedmann universe. Note that the cosmological constant is sometimes deﬁned as equalling the value used here divided by c2. main January 23, 2006 16:6

9.6 Cosmology 185

Cosmological distance measures

tlb(z)light travel time from Look-back an object at redshift z tlb(z)=t0 −t(z) (9.96) time t0 present cosmic time t(z) cosmic time at z dp proper distance Proper r dr t0 dt R cosmic scale factor d = R = cR (9.97) p 0 2 1/2 0 c speed of light distance 0 (1−kr ) t R(t) 0 present epoch d luminosity distance Luminosity z dz L d = d (1+z)=c(1+z) (9.98) z redshift distancea L p H(z) 0 H Hubble parameterb Flux density– F spectral ﬂux density L(ν ) redshift F(ν)= where ν =(1+z)ν (9.99) ν frequency 4 2 ( ) relation πdL z L(ν) spectral luminosityc

Angular da angular diameter −2 diameter da = dL(1+z) (9.100) distance distanced k curvature parameter a −2 Assuming a flat universe (k = 0). The apparent flux density of a source varies as dL . bSee Equation (9.85). cDefined as the output power of the body per unit frequency interval. d −1 True for all k. The angular diameter of a source varies as da .

Cosmological modelsa

2c − −1/2 dp proper dp = [1 (1+z) ] (9.101) distance H0 3/2 H Hubble Einstein – de H(z)=H0(1+z) (9.102) parameter Sitter model q0 =1/2 (9.103) 0 present epoch (Ωk =0, 2 z redshift t(z)= (9.104) Λ=0, p =0 3H(z) c speed of light and Ωm0 =1) 2 −1 q deceleration ρ =(6πGt ) (9.105) parameter 2/3 t(z) time at R(t)=R0(t/t0) (9.106) redshift z

− z 1/2 R cosmic scale 9 c Ωm0 dz Concordance d = (9.107) factor p 3 − −1 1/2 H0 0 [(1+z ) 1+Ωm0] model Ωm0 present mass 3 (Ωk =0,Λ= H(z)=H0[Ωm0(1+z) +(1−Ωm0)] (9.108) density − 2 parameter 3(1 Ωm0)H0 , q =3Ω /2−1 (9.109) 0 m0 G constant of p =0 and 1/2 2 − (1−Ωm0) gravitation Ωm0 < 1) t(z)= (1−Ω ) 1/2 arsinh (9.110) m0 3/2 mass density 3H0 (1+z) ρ aCurrently popular. main January 23, 2006 16:6 main January 23, 2006 16:6

Index

Section headings are shown in boldface and panel labels in small caps. Equation numbers are contained within square brackets.

Airy A disk [8.40], 165 aberration (relativistic) [3.24],65 function [8.17], 163 absolute magnitude [9.29], 179 resolution criterion [8.41], 165 absorption (Einstein coefficient) [8.118], Airy’s differential equation [2.352],43 173 albedo [5.193], 121 absorption coefficient (linear) [5.175], 120 Alfven´ speed [7.277], 158 accelerated point charge Alfven´ waves [7.284], 158 bremsstrahlung, 160 alt-azimuth coordinates, 177 Lienard–Wiechert´ potentials, 139 alternating tensor (ijk) [2.443],50 oscillating [7.132], 146 altitude coordinate [9.9], 177 synchrotron, 159 Ampere’s` law [7.10], 136 acceleration ampere (SI definition), 3 constant, 68 ampere (unit), 4 dimensions, 16 analogue formula [2.258],36 due to gravity (value on Earth), 176 angle in a rotating frame [3.32],66 aberration [3.24],65 acceptance angle (optical fibre) [8.77], acceptance [8.77], 169 169 beam solid [7.210], 153 acoustic branch (phonon) [6.37], 129 Brewster’s [7.218], 154 acoustic impedance [3.276],83 Compton scattering [7.240], 155 action (definition) [3.213],79 contact (surface tension) [3.340],88 action (dimensions), 16 deviation [8.73], 169 addition of velocities Euler [2.101],26 Galilean [3.3],64 Faraday rotation [7.273], 157 relativistic [3.15],64 hour (coordinate) [9.8], 177 adiabatic Kelvin wedge [3.330],87 bulk modulus [5.23], 107 Mach wedge [3.328],87 compressibility [5.21], 107 polarisation [8.81], 170 expansion (ideal gas) [5.58], 110 principal range (inverse trig.), 34 lapse rate [3.294],84 refraction, 154 adjoint matrix rotation, 26 I definition 1 [2.71],24 Rutherford scattering [3.116],72 definition 2 [2.80],25 separation [3.133],73 adjugate matrix [2.80],25 spherical excess [2.260],36 admittance (definition), 148 units, 4, 5 advective operator [3.289],84 ˚angstrom¨ (unit), 5 main January 23, 2006 16:6

188 Index angular diameter distance [9.100], 185 artanhx (definition) [2.240],35 Angular momentum,98 area angular momentum of circle [2.262],37 conservation [4.113],98 of cone [2.271],37 definition [3.66],68 of cylinder [2.269],37 dimensions, 16 of ellipse [2.267],37 eigenvalues [4.109] [4.109],98 of plane triangle [2.254],36 ladder operators [4.108],98 of sphere [2.263],37 operators of spherical cap [2.275],37 and other operators [4.23],91 of torus [2.273],37 definitions [4.105],98 area (dimensions), 16 rigid body [3.141],74 argument (of a complex number) [2.157], Angular momentum addition, 100 30 Angular momentum commutation rela- arithmetic mean [2.108],27 tions,98 arithmetic progression [2.104],27 angular speed (dimensions), 16 associated Laguerre equation [2.348],43 anomaly (true) [3.104],71 associated Laguerre polynomials, 96 antenna associated Legendre equation beam efficiency [7.214], 153 and polynomial solutions [2.428],48 effective area [7.212], 153 differential equation [2.344],43 power gain [7.211], 153 Associated Legendre functions,48 temperature [7.215], 153 astronomical constants, 176 Antennas, 153 Astronomical magnitudes, 179 anticommutation [2.95],26 Astrophysics, 175–185 antihermitian symmetry, 53 asymmetric top [3.189],77 antisymmetric matrix [2.87],25 atomic Aperture diffraction, 165 form factor [6.30], 128 aperture function [8.34], 165 mass unit, 6, 9 apocentre (of an orbit) [3.111],71 numbers of elements, 124 apparent magnitude [9.27], 179 polarisability [7.91], 142 Appleton-Hartree formula [7.271], 157 weights of elements, 124 arc length [2.279],39 Atomic constants,7 arccosx atto, 5 from arctan [2.233],34 autocorrelation (Fourier) [2.491],53 series expansion [2.141],29 autocorrelation function [8.104], 172 arcoshx (definition) [2.239],35 availability arccotx (from arctan) [2.236],34 and fluctuation probability [5.131], arcothx (definition) [2.241],35 116 arccscx (from arctan) [2.234],34 definition [5.40], 108 arcschx (definition) [2.243],35 Avogadro constant, 6, 9 arcminute (unit), 5 Avogadro constant (dimensions), 16 arcsecx (from arctan) [2.235],34 azimuth coordinate [9.10], 177 arsechx (definition) [2.242],35 arcsecond (unit), 5 B arcsinx Ballistics,69 from arctan [2.232],34 band index [6.85], 134 series expansion [2.141],29 Band theory and semiconductors, 134 arsinhx (definition) [2.238],35 bandwidth arctanx (series expansion) [2.142],29 and coherence time [8.106], 172 main January 23, 2006 16:6

Index 189

and Johnson noise [5.141], 117 Schwarzschild solution [3.61],67 Doppler [8.117], 173 temperature [9.82], 183 natural [8.113], 173 blackbody of a diffraction grating [8.30], 164 energy density [5.192], 121 of an LCR circuit [7.151], 148 spectral energy density [5.186], 121 of laser cavity [8.127], 174 spectrum [5.184], 121 Schawlow-Townes [8.128], 174 Blackbody radiation, 121 bar (unit), 5 Bloch’s theorem [6.84], 134 barn (unit), 5 Bode’s law [9.41], 180 Barrier tunnelling,94 body cone, 77 Bartlett window [2.581],60 body frequency [3.187],77 base vectors (crystallographic), 126 body-centred cubic structure, 127 basis vectors [2.17],20 Bohr Bayes’ theorem [2.569],59 energy [4.74],95 Bayesian inference,59 magneton (equation) [4.137], 100 bcc structure, 127 magneton (value), 6, 7 beam bowing under its own weight [3.260], quantisation [4.71],95 82 radius (equation) [4.72],95 beam efficiency [7.214], 153 radius (value), 7 beam solid angle [7.210], 153 Bohr magneton (dimensions), 16 beam with end-weight [3.259],82 Bohr model,95 beaming (relativistic) [3.25],65 boiling points of elements, 124 becquerel (unit), 4 bolometric correction [9.34], 179 Bending beams,82 Boltzmann bending moment (dimensions), 16 constant, 6, 9 bending moment [3.258],82 constant (dimensions), 16 bending waves [3.268],82 distribution [5.111], 114 Bernoulli’s differential equation [2.351], entropy [5.105], 114 43 excitation equation [5.125], 116 Bernoulli’s equation Born collision formula [4.178], 104 compressible flow [3.292],84 Bose condensation [5.123], 115 incompressible flow [3.290],84 Bose–Einstein distribution [5.120], 115 Bessel equation [2.345],43 boson statistics [5.120], 115 Bessel functions,47 Boundary conditions for E, D, B,and beta (in plasmas) [7.278], 158 H, 144 binomial box (particle in a) [4.64],94 coefficient [2.121],28 Box Muller transformation [2.561],58 distribution [2.547],57 Boyle temperature [5.66], 110 series [2.120],28 Boyle’s law [5.56], 110 theorem [2.122],28 bra vector [4.33],92 binormal [2.285],39 bra-ket notation, 91, 92 Biot–Savart law [7.9], 136 Bragg’s reflection law Biot-Fourier equation [5.95], 113 in crystals [6.29], 128 black hole in optics [8.32], 164 evaporation time [9.81], 183 braking index (pulsar) [9.66], 182 I Kerr solution [3.62],67 Bravais lattices, 126 maximum angular momentum [9.80], Breit-Wigner formula [4.174], 104 183 Bremsstrahlung, 160 Schwarzschild radius [9.73], 183 bremsstrahlung main January 23, 2006 16:6

190 Index

single electron and ion [7.297], 160 constant [3.338],88 thermal [7.300], 160 contact angle [3.340],88 Brewster’s law [7.218], 154 rise [3.339],88 brightness (blackbody) [5.184], 121 waves [3.321],86 Brillouin function [4.147], 101 capillary-gravity waves [3.322],86 Bromwich integral [2.518],55 cardioid [8.46], 166 Brownian motion [5.98], 113 Carnot cycles, 107 bubbles [3.337],88 Cartesian coordinates, 21 bulk modulus Catalan’s constant (value), 9 adiabatic [5.23], 107 Cauchy general [3.245],81 differential equation [2.350],43 isothermal [5.22], 107 distribution [2.555],58 bulk modulus (dimensions), 16 inequality [2.151],30 Bulk physical constants,9 integral formula [2.167],31 Burgers vector [6.21], 128 Cauchy-Goursat theorem [2.165],31 Cauchy-Riemann conditions [2.164],31 C cavity modes (laser) [8.124], 174 calculus of variations [2.334],42 Celsius (unit), 4 candela, 119 Celsius conversion [1.1],15 candela (SI definition), 3 centi, 5 candela (unit), 4 centigrade (avoidance of), 15 canonical centre of mass ensemble [5.111], 114 circular arc [3.173],76 entropy [5.106], 114 cone [3.175],76 equations [3.220],79 definition [3.68],68 momenta [3.218],79 disk sector [3.172],76 cap, see spherical cap hemisphere [3.170],76 Capacitance, 137 hemispherical shell [3.171],76 capacitance pyramid [3.175],76 current through [7.144], 147 semi-ellipse [3.178],76 definition [7.143], 147 spherical cap [3.177],76 dimensions, 16 triangular lamina [3.174],76 energy [7.153], 148 Centres of mass,76 energy of an assembly [7.134], 146 centrifugal force [3.35],66 impedance [7.159], 148 centripetal acceleration [3.32],66 mutual [7.134], 146 cepheid variables [9.48], 180 capacitance of Cerenkov, see Cherenkov cube [7.17], 137 chain rule cylinder [7.15], 137 function of a function [2.295],40 cylinders (adjacent) [7.21], 137 partial derivatives [2.331],42 cylinders (coaxial) [7.19], 137 Chandrasekhar mass [9.79], 183 disk [7.13], 137 change of variable [2.333],42 disks (coaxial) [7.22], 137 Characteristic numbers,86 nearly spherical surface [7.16], 137 charge sphere [7.12], 137 conservation [7.39], 139 spheres (adjacent) [7.14], 137 dimensions, 16 spheres (concentric) [7.18], 137 elementary, 6, 7 capacitor, see capacitance force between two [7.119], 145 capillary Hamiltonian [7.138], 146 main January 23, 2006 16:6

Index 191

to mass ratio of electron, 8 coupling [7.148], 147 charge density finesse [8.12], 163 dimensions, 16 reflectance [7.227], 154 free [7.57], 140 reflection [7.230], 154 induced [7.84], 142 restitution [3.127],73 Lorentz transformation, 141 transmission [7.232], 154 charge distribution transmittance [7.229], 154 electric field from [7.6], 136 coexistence curve [5.51], 109 energy of [7.133], 146 coherence charge-sheet (electric field) [7.32], 138 length [8.106], 172 Chebyshev equation [2.349],43 mutual [8.97], 172 Chebyshev inequality [2.150],30 temporal [8.105], 172 chemical potential time [8.106], 172 definition [5.28], 108 width [8.111], 172 from partition function [5.119], 115 Coherence (scalar theory), 172 Cherenkov cone angle [7.246], 156 cold plasmas, 157 Cherenkov radiation, 156 collision χE (electric susceptibility) [7.87], 142 broadening [8.114], 173 χH , χB (magnetic susceptibility) [7.103], elastic, 73 143 inelastic, 73 chi-squared (χ2) distribution [2.553],58 number [5.91], 113 Christoffel symbols [3.49],67 time (electron drift) [6.61], 132 circle colour excess [9.37], 179 (arc of) centre of mass [3.173],76 colour index [9.36], 179 area [2.262],37 Common three-dimensional coordinate perimeter [2.261],37 systems,21 circular aperture commutator (in uncertainty relation) [4.6], Fraunhofer diffraction [8.40], 165 90 Fresnel diffraction [8.50], 166 Commutators,26 circular polarisation, 170 Compact objects and black holes, 183 circulation [3.287],84 complementary error function [2.391],45 civil time [9.4], 177 Complex analysis,31 Clapeyron equation [5.50], 109 complex conjugate [2.159],30 classical electron radius, 8 Complex numbers,30 Classical thermodynamics, 106 complex numbers Clausius–Mossotti equation [7.93], 142 argument [2.157],30 Clausius-Clapeyron equation [5.49], 109 cartesian form [2.153],30 Clebsch–Gordan coefficients,99 conjugate [2.159],30 Clebsch–Gordan coefficients (spin-orbit) [4.136], logarithm [2.162],30 100 modulus [2.155],30 close-packed spheres, 127 polar form [2.154],30 closure density (of the universe) [9.90], Complex variables,30 184 compound pendulum [3.182],76 CNO cycle, 182 compressibility coaxial cable adiabatic [5.21], 107 I capacitance [7.19], 137 isothermal [5.20], 107 inductance [7.24], 137 compression modulus, see bulk modulus coaxial transmission line [7.181], 150 compression ratio [5.13], 107 coefficient of Compton main January 23, 2006 16:6

192 Index

scattering [7.240], 155 derivative [2.498],53 wavelength (value), 8 discrete [2.580],60 wavelength [7.240], 155 Laplace transform [2.516],55 Concordance model, 185 rules [2.489],53 conditional probability [2.567],59 theorem [2.490],53 conductance (definition), 148 coordinate systems, 21 conductance (dimensions), 16 coordinate transformations conduction equation (and transport) [5.96], astronomical, 177 113 Galilean, 64 conduction equation [2.340],43 relativistic, 64 conductivity rotating frames [3.31],66 and resistivity [7.142], 147 Coordinate transformations (astronomical), dimensions, 16 177 direct [7.279], 158 coordinates (generalised ) [3.213],79 electrical, of a plasma [7.233], 155 coordination number (cubic lattices), 127 free electron a.c. [6.63], 132 Coriolis force [3.33],66 free electron d.c. [6.62], 132 Cornu spiral, 167 Hall [7.280], 158 Cornu spiral and Fresnel integrals [8.54], conductor refractive index [7.234], 155 167 cone correlation coefficient centre of mass [3.175],76 multinormal [2.559],58 moment of inertia [3.160],75 Pearson’s r [2.546],57 surface area [2.271],37 correlation intensity [8.109], 172 volume [2.272],37 correlation theorem [2.494],53 configurational entropy [5.105], 114 cosx Conic sections,38 and Euler’s formula [2.216],34 conical pendulum [3.180],76 series expansion [2.135],29 conservation of cosec, see csc angular momentum [4.113],98 cschx [2.231],34 charge [7.39], 139 coshx mass [3.285],84 definition [2.217],34 Constant acceleration,68 series expansion [2.143],29 constant of gravitation, 7 cosine formula contact angle (surface tension) [3.340], planar triangles [2.249],36 88 spherical triangles [2.257],36 continuity equation (quantum physics) [4.14], cosmic scale factor [9.87], 184 90 cosmological constant [9.89], 184 continuity in fluids [3.285],84 Cosmological distance measures, 185 Continuous probability distributions, Cosmological model parameters, 184 58 Cosmological models, 185 contravariant components Cosmology, 184 in general relativity, 67 cos−1 x, see arccosx in special relativity [3.26],65 cotx convection (in a star) [9.64], 181 definition [2.226],34 convergence and limits, 28 series expansion [2.140],29 Conversion factors,10 cothx [2.227],34 Converting between units,10 Couette flow [3.306],85 convolution coulomb (unit), 4 definition [2.487],53 Coulomb gauge condition [7.42], 139 main January 23, 2006 16:6

Index 193

Coulomb logarithm [7.254], 156 Curie’s law [7.113], 144 Coulomb’s law [7.119], 145 Curie–Weiss law [7.114], 144 couple Curl,22 definition [3.67],68 curl dimensions, 16 cylindrical coordinates [2.34],22 electromagnetic, 145 general coordinates [2.36],22 for Couette flow [3.306],85 of curl [2.57],23 on a current-loop [7.127], 145 rectangular coordinates [2.33],22 on a magnetic dipole [7.126], 145 spherical coordinates [2.35],22 on a rigid body, 77 current on an electric dipole [7.125], 145 dimensions, 16 twisting [3.252],81 electric [7.139], 147 coupling coefficient [7.148], 147 law (Kirchhoff’s) [7.161], 149 covariance [2.558],58 magnetic flux density from [7.11], covariant components [3.26],65 136 cracks (critical length) [6.25], 128 probability density [4.13],90 critical damping [3.199],78 thermodynamic work [5.9], 106 critical density (of the universe) [9.90], transformation [7.165], 149 184 current density critical frequency (synchrotron) [7.293], dimensions, 16 159 four-vector [7.76], 141 critical point free [7.63], 140 Dieterici gas [5.75], 111 free electron [6.60], 132 van der Waals gas [5.70], 111 hole [6.89], 134 cross section Lorentz transformation, 141 absorption [5.175], 120 magnetic flux density [7.10], 136 cross-correlation [2.493],53 curvature cross-product [2.2],20 in differential geomtry [2.286],39 cross-section parameter (cosmic) [9.87], 184 Breit-Wigner [4.174], 104 radius of Mott scattering [4.180], 104 and curvature [2.287],39 Rayleigh scattering [7.236], 155 plane curve [2.282],39 Rutherford scattering [3.124],72 curve length (plane curve) [2.279],39 Thomson scattering [7.238], 155 Curve measure,39 Crystal diffraction, 128 Cycle efficiencies (thermodynamic), 107 Crystal systems, 127 cyclic permutation [2.97],26 Crystalline structure, 126 cyclotron frequency [7.265], 157 cscx cylinder definition [2.230],34 area [2.269],37 series expansion [2.139],29 capacitance [7.15], 137 cschx [2.231],34 moment of inertia [3.155],75 cube torsional rigidity [3.253],81 electrical capacitance [7.17], 137 volume [2.270],37 mensuration, 38 cylinders (adjacent) Cubic equations,51 capacitance [7.21], 137 I cubic expansivity [5.19], 107 inductance [7.25], 137 Cubic lattices, 127 cylinders (coaxial) cubic system (crystallographic), 127 capacitance [7.19], 137 Curie temperature [7.114], 144 inductance [7.24], 137 main January 23, 2006 16:6

194 Index cylindrical polar coordinates, 21 delta–star transformation, 149 densities of elements, 124 D density (dimensions), 16 d orbitals [4.100],97 density of states D’Alembertian [7.78], 141 electron [6.70], 133 damped harmonic oscillator [3.196],78 particle [4.66],94 damping profile [8.112], 173 phonon [6.44], 130 day (unit), 5 density parameters [9.94], 184 day of week [9.3], 177 depolarising factors [7.92], 142 daylight saving time [9.4], 177 Derivatives (general),40 de Boer parameter [6.54], 131 determinant [2.79],25 de Broglie relation [4.2],90 deviation (of a prism) [8.73], 169 de Broglie wavelength (thermal) [5.83], diamagnetic moment (electron) [7.108], 112 144 de Moivre’s theorem [2.214],34 diamagnetic susceptibility (Landau) [6.80], Debye 133 T 3 law [6.47], 130 Diamagnetism, 144 frequency [6.41], 130 Dielectric layers, 162 function [6.49], 130 Dieterici gas, 111 heat capacity [6.45], 130 Dieterici gas law [5.72], 111 length [7.251], 156 Differential equations,43 number [7.253], 156 differential equations (numerical solutions), screening [7.252], 156 62 temperature [6.43], 130 Differential geometry,39 Debye theory, 130 Differential operator identities,23 Debye-Waller factor [6.33], 128 differential scattering cross-section [3.124], deca, 5 72 decay constant [4.163], 103 Differentiation,40 decay law [4.163], 103 differentiation deceleration parameter [9.95], 184 hyperbolic functions, 41 deci, 5 numerical, 61 decibel [5.144], 117 of a function of a function [2.295], declination coordinate [9.11], 177 40 decrement (oscillating systems) [3.202], of a log [2.300],40 78 of a power [2.292],40 Definite integrals,46 of a product [2.293],40 degeneracy pressure [9.77], 183 of a quotient [2.294],40 degree (unit), 5 of exponential [2.301],40 degree Celsius (unit), 4 of integral [2.299],40 degree kelvin [5.2], 106 of inverse functions [2.304],40 degree of freedom (and equipartition), 113 trigonometric functions, 41 degree of mutual coherence [8.99], 172 under integral sign [2.298],40 degree of polarisation [8.96], 171 diffraction from degree of temporal coherence, 172 N slits [8.25], 164 deka, 5 1 slit [8.37], 165 del operator, 21 2 slits [8.24], 164 del-squared operator, 23 circular aperture [8.40], 165 del-squared operator [2.55],23 crystals, 128 Delta functions,50 infinite grating [8.26], 164 main January 23, 2006 16:6

Index 195

rectangular aperture [8.39], 165 disc, see disk diffraction grating discrete convolution, 60 finite [8.25], 164 Discrete probability distributions,57 general, 164 Discrete statistics,57 infinite [8.26], 164 disk diffusion coefficient (semiconductor) [6.88], Airy [8.40], 165 134 capacitance [7.13], 137 diffusion equation centre of mass of sector [3.172],76 differential equation [2.340],43 coaxial capacitance [7.22], 137 Fick’s first law [5.93], 113 drag in a fluid, 85 diffusion length (semiconductor) [6.94], electric field [7.28], 138 134 moment of inertia [3.168],75 diffusivity (magnetic) [7.282], 158 Dislocations and cracks, 128 dilatation (volume strain) [3.236],80 dispersion Dimensions,16 diffraction grating [8.31], 164 diode (semiconductor) [6.92], 134 in a plasma [7.261], 157 dioptre number [8.68], 168 in fluid waves, 86 dipole in quantum physics [4.5],90 antenna power in waveguides [7.188], 151 flux [7.131], 146 intermodal (optical fibre) [8.79], 169 gain [7.213], 153 measure [9.70], 182 total [7.132], 146 of a prism [8.76], 169 electric field [7.31], 138 phonon (alternating springs) [6.39], energy of 129 electric [7.136], 146 phonon (diatomic chain) [6.37], 129 magnetic [7.137], 146 phonon (monatomic chain) [6.34], field from 129 magnetic [7.36], 138 pulsar [9.72], 182 moment (dimensions), 17 displacement, D [7.86], 142 moment of Distance indicators, 180 electric [7.80], 142 Divergence,22 magnetic [7.94], 143 divergence potential cylindrical coordinates [2.30],22 electric [7.82], 142 general coordinates [2.32],22 magnetic [7.95], 143 rectangular coordinates [2.29],22 radiation spherical coordinates [2.31],22 field [7.207], 153 theorem [2.59],23 magnetic [9.69], 182 dodecahedron, 38 radiation resistance [7.209], 153 Doppler dipole moment per unit volume beaming [3.25],65 electric [7.83], 142 effect (non-relativistic), 87 magnetic [7.97], 143 effect (relativistic) [3.22],65 Dirac bracket, 92 line broadening [8.116], 173 Dirac delta function [2.448],50 width [8.117], 173 Dirac equation [4.183], 104 Doppler effect,87 I Dirac matrices [4.185], 104 dot product [2.1],20 Dirac notation,92 double factorial, 48 direct conductivity [7.279], 158 double pendulum [3.183],76 directrix (of conic section), 38 Drag,85 main January 23, 2006 16:6

196 Index drag modulus (longitudinal) [3.241],81 on a disk to flow [3.310],85 modulus [3.234],80 on a disk ⊥ to flow [3.309],85 potential energy [3.235],80 on a sphere [3.308],85 elastic scattering, 72 drift velocity (electron) [6.61], 132 Elastic wave velocities,82 Dulong and Petit’s law [6.46], 130 Elasticity,80 Dynamics and Mechanics, 63–88 Elasticity definitions (general),80 Dynamics definitions,68 Elasticity definitions (simple),80 electric current [7.139], 147 E electric dipole, see dipole e (exponential constant), 9 electric displacement (dimensions), 16 e to 1 000 decimal places,18 electric displacement, D [7.86], 142 Earth (motion relative to) [3.38],66 electric field Earth data, 176 around objects, 138 eccentricity energy density [7.128], 146 of conic section, 38 static, 136 of orbit [3.108],71 thermodynamic work [5.7], 106 of scattering hyperbola [3.120],72 wave equation [7.193], 152 Ecliptic coordinates, 178 electric field from ecliptic latitude [9.14], 178 A and φ [7.41], 139 ecliptic longitude [9.15], 178 charge distribution [7.6], 136 Eddington limit [9.59], 181 charge-sheet [7.32], 138 edge dislocation [6.21], 128 dipole [7.31], 138 effective disk [7.28], 138 area (antenna) [7.212], 153 line charge [7.29], 138 distance (Fresnel diffraction) [8.48], point charge [7.5], 136 166 sphere [7.27], 138 mass (in solids) [6.86], 134 waveguide [7.190], 151 wavelength [9.40], 179 wire [7.29], 138 efficiency electric field strength (dimensions), 16 heat engine [5.10], 107 Electric fields, 138 heat pump [5.12], 107 electric polarisability (dimensions), 16 Otto cycle [5.13], 107 electric polarisation (dimensions), 16 refrigerator [5.11], 107 electric potential Ehrenfest’s equations [5.53], 109 from a charge density [7.46], 139 Ehrenfest’s theorem [4.30],91 Lorentz transformation [7.75], 141 eigenfunctions (quantum) [4.28],91 of a moving charge [7.48], 139 Einstein short dipole [7.82], 142 A coefficient [8.119], 173 electric potential difference (dimensions), B coefficients [8.118], 173 16 diffusion equation [5.98], 113 electric susceptibility, χE [7.87], 142 field equation [3.59],67 electrical conductivity, see conductivity lens (rings) [9.50], 180 Electrical impedance, 148 tensor [3.58],67 electrical permittivity, , r [7.90], 142 Einstein - de Sitter model, 185 electromagnet (magnetic flux density) [7.38], Einstein coefficients, 173 138 elastic electromagnetic collisions, 73 boundary conditions, 144 media (isotropic), 81 constants, 7 main January 23, 2006 16:6

Index 197

fields, 139 the moment of inertia [3.147],74 wave speed [7.196], 152 volume [2.268],37 waves in media, 152 elliptic integrals [2.397],45 electromagnetic coupling constant, see fine elliptical orbit [3.104],71 structure constant Elliptical polarisation, 170 Electromagnetic energy, 146 elliptical polarisation [8.80], 170 Electromagnetic fields (general), 139 ellipticity [8.82], 170 Electromagnetic force and torque, 145 E = mc2 [3.72],68 Electromagnetic propagation in cold emission coefficient [5.174], 120 plasmas, 157 emission spectrum [7.291], 159 Electromagnetism, 135–160 emissivity [5.193], 121 electron energy charge, 6, 7 density density of states [6.70], 133 blackbody [5.192], 121 diamagnetic moment [7.108], 144 dimensions, 16 drift velocity [6.61], 132 elastic wave [3.281],83 g-factor [4.143], 100 electromagnetic [7.128], 146 gyromagnetic ratio (value), 8 radiant [5.148], 118 gyromagnetic ratio [4.140], 100 spectral [5.173], 120 heat capacity [6.76], 133 dimensions, 16 intrinsic magnetic moment [7.109], dissipated in resistor [7.155], 148 144 distribution (Maxwellian) [5.85], 112 mass, 6 elastic [3.235],80 radius (equation) [7.238], 155 electromagnetic, 146 radius (value), 8 equipartition [5.100], 113 scattering cross-section [7.238], 155 Fermi [5.122], 115 spin magnetic moment [4.143], 100 first law of thermodynamics [5.3], thermal velocity [7.257], 156 106 velocity in conductors [6.85], 134 Galilean transformation [3.6],64 Electron constants,8 kinetic , see kinetic energy Electron scattering processes, 155 Lorentz transformation [3.19],65 electron volt (unit), 5 loss after collision [3.128],73 electron volt (value), 6 mass relation [3.20],65 Electrons in solids, 132 of capacitive assembly [7.134], 146 electrostatic potential [7.1], 136 of capacitor [7.153], 148 Electrostatics, 136 of charge distribution [7.133], 146 elementary charge, 6, 7 of electric dipole [7.136], 146 elements (periodic table of), 124 of inductive assembly [7.135], 146 ellipse, 38 of inductor [7.154], 148 (semi) centre of mass [3.178],76 of magnetic dipole [7.137], 146 area [2.267],37 of orbit [3.100],71 moment of inertia [3.166],75 potential , see potential energy perimeter [2.266],37 relativistic rest [3.72],68 semi-latus-rectum [3.109],71 rotational kinetic semi-major axis [3.106],71 rigid body [3.142],74 I semi-minor axis [3.107],71 w.r.t. principal axes [3.145],74 ellipsoid thermodynamic work, 106 moment of inertia of solid [3.163], Energy in capacitors, inductors, and 75 resistors, 148 main January 23, 2006 16:6

198 Index energy-time uncertainty relation [4.8],90 calculus of variations [2.334],42 Ensemble probabilities, 114 even functions, 53 enthalpy Evolutionary timescales, 181 definition [5.30], 108 exa, 5 Joule-Kelvin expansion [5.27], 108 exhaust velocity (of a rocket) [3.93],70 entropy exitance Boltzmann formula [5.105], 114 blackbody [5.191], 121 change in Joule expansion [5.64], luminous [5.162], 119 110 radiant [5.150], 118 experimental [5.4], 106 exp(x) [2.132],29 fluctuations [5.135], 116 expansion coefficient [5.19], 107 from partition function [5.117], 115 Expansion processes, 108 Gibbs formula [5.106], 114 expansivity [5.19], 107 of a monatomic gas [5.83], 112 Expectation value,91 entropy (dimensions), 16 expectation value , r (electrical permittivity) [7.90], 142 Dirac notation [4.37],92 Equation conversion: SI to Gaussian from a wavefunction [4.25],91 units, 135 explosions [3.331],87 equation of state exponential Dieterici gas [5.72], 111 distribution [2.551],58 ideal gas [5.57], 110 integral [2.394],45 monatomic gas [5.78], 112 series expansion [2.132],29 van der Waals gas [5.67], 111 exponential constant (e), 9 equipartition theorem [5.100], 113 extraordinary modes [7.271], 157 error function [2.390],45 extrema [2.335],42 errors, 60 escape velocity [3.91],70 F estimator f-number [8.69], 168 kurtosis [2.545],57 Fabry-Perot etalon mean [2.541],57 chromatic resolving power [8.21], 163 skewness [2.544],57 free spectral range [8.23], 163 standard deviation [2.543],57 fringe width [8.19], 163 variance [2.542],57 transmitted intensity [8.17], 163 Euler Fabry-Perot etalon, 163 angles [2.101],26 face-centred cubic structure, 127 constant factorial [2.409],46 expression [2.119],27 factorial (double), 48 value, 9 Fahrenheit conversion [1.2],15 differential equation [2.350],43 faltung theorem [2.516],55 formula [2.216],34 farad (unit), 4 relation, 38 Faraday constant, 6, 9 strut [3.261],82 Faraday constant (dimensions), 16 Euler’s equation (fluids) [3.289],84 Faraday rotation [7.273], 157 Euler’s equations (rigid bodies) [3.186], Faraday’s law [7.55], 140 77 fcc structure, 127 Euler’s method (for ordinary differential Feigenbaum’s constants, 9 equations) [2.596],62 femto, 5 Euler-Lagrange equation Fermat’s principle [8.63], 168 and Lagrangians [3.214],79 Fermi main January 23, 2006 16:6

Index 199

energy [6.73], 133 flux density–redshift relation [9.99], 185 temperature [6.74], 133 flux linked [7.149], 147 velocity [6.72], 133 flux of molecules through a plane [5.91], wavenumber [6.71], 133 113 fermi (unit), 5 flux–magnitude relation [9.32], 179 Fermi energy [5.122], 115 focal length [8.64], 168 Fermi gas, 133 focus (of conic section), 38 Fermi’s golden rule [4.162], 102 force Fermi–Dirac distribution [5.121], 115 and acoustic impedance [3.276],83 fermion statistics [5.121], 115 and stress [3.228],80 fibre optic between two charges [7.119], 145 acceptance angle [8.77], 169 between two currents [7.120], 145 dispersion [8.79], 169 between two masses [3.40],66 numerical aperture [8.78], 169 central [4.113],98 Fick’s first law [5.92], 113 centrifugal [3.35],66 Fick’s second law [5.95], 113 Coriolis [3.33],66 field equations (gravitational) [3.42],66 critical compression [3.261],82 Field relationships, 139 definition [3.63],68 fields dimensions, 16 depolarising [7.92], 142 electromagnetic, 145 electrochemical [6.81], 133 Newtonian [3.63],68 electromagnetic, 139 on gravitational, 66 charge in a field [7.122], 145 static E and B, 136 current in a field [7.121], 145 velocity [3.285],84 electric dipole [7.123], 145 Fields associated with media, 142 magnetic dipole [7.124], 145 film reflectance [8.4], 162 sphere (potential flow) [3.298],84 fine-structure constant sphere (viscous drag) [3.308],85 expression [4.75],95 relativistic [3.71],68 value, 6, 7 unit, 4 finesse (coefficient of) [8.12], 163 Force, torque, and energy, 145 finesse (Fabry-Perot etalon) [8.14], 163 Forced oscillations,78 first law of thermodynamics [5.3], 106 form factor [6.30], 128 fitting straight-lines, 60 formula (the) [2.455],50 fluctuating dipole interaction [6.50], 131 Foucault’s pendulum [3.39],66 fluctuation four-parts formula [2.259],36 of density [5.137], 116 four-scalar product [3.27],65 of entropy [5.135], 116 four-vector of pressure [5.136], 116 electromagnetic [7.79], 141 of temperature [5.133], 116 momentum [3.21],65 of volume [5.134], 116 spacetime [3.12],64 probability (thermodynamic) [5.131], Four-vectors,65 116 Fourier series variance (general) [5.132], 116 complex form [2.478],52 Fluctuations and noise, 116 real form [2.476],52 I Fluid dynamics,84 Fourier series,52 fluid stress [3.299],85 Fourier series and transforms,52 Fluid waves,86 Fourier symmetry relationships,53 flux density [5.171], 120 Fourier transform main January 23, 2006 16:6

200 Index

cosine [2.509],54 plane waves [8.45], 166 definition [2.482],52 spherical waves [8.47], 166 derivatives Friedmann equations [9.89], 184 and inverse [2.502],54 fringe visibility [8.101], 172 general [2.498],53 fringes (Moire),´ 35 Gaussian [2.507],54 Froude number [3.312],86 Lorentzian [2.505],54 shah function [2.510],54 G shift theorem [2.501],54 g-factor similarity theorem [2.500],54 electron, 8 sine [2.508],54 Lande´ [4.146], 100 step [2.511],54 muon, 9 top hat [2.512],54 gain in decibels [5.144], 117 triangle function [2.513],54 galactic Fourier transform,52 coordinates [9.20], 178 Fourier transform pairs,54 latitude [9.21], 178 Fourier transform theorems,53 longitude [9.22], 178 Fourier’s law [5.94], 113 Galactic coordinates, 178 Frames of reference,64 Galilean transformation Fraunhofer diffraction, 164 of angular momentum [3.5],64 Fraunhofer integral [8.34], 165 of kinetic energy [3.6],64 Fraunhofer limit [8.44], 165 of momentum [3.4],64 free charge density [7.57], 140 of time and position [3.2],64 free current density [7.63], 140 of velocity [3.3],64 Free electron transport properties, 132 Galilean transformations,64 free energy [5.32], 108 Gamma function,46 free molecular flow [5.99], 113 gamma function Free oscillations,78 and other integrals [2.395],45 free space impedance [7.197], 152 definition [2.407],46 free spectral range gas Fabry Perot etalon [8.23], 163 adiabatic expansion [5.58], 110 laser cavity [8.124], 174 adiabatic lapse rate [3.294],84 free-fall timescale [9.53], 181 constant, 6, 9, 86, 110 Frenet’s formulas [2.291],39 Dieterici, 111 frequency (dimensions), 16 Doppler broadened [8.116], 173 Fresnel diffraction flow [3.292],84 Cornu spiral [8.54], 167 giant (astronomical data), 176 edge [8.56], 167 ideal equation of state [5.57], 110 long slit [8.58], 167 ideal heat capacities, 113 rectangular aperture [8.62], 167 ideal, or perfect, 110 Fresnel diffraction, 166 internal energy (ideal) [5.62], 110 Fresnel Equations, 154 isothermal expansion [5.63], 110 Fresnel half-period zones [8.49], 166 linear absorption coefficient [5.175], Fresnel integrals 120 and the Cornu spiral [8.52], 167 molecular flow [5.99], 113 definition [2.392],45 monatomic, 112 in diffraction [8.54], 167 paramagnetism [7.112], 144 Fresnel zones, 166 pressure broadened [8.115], 173 Fresnel-Kirchhoff formula speed of sound [3.318],86 main January 23, 2006 16:6

Index 201

temperature scale [5.1], 106 general coordinates [2.28],21 Van der Waals, 111 rectangular coordinates [2.25],21 Gas equipartition, 113 spherical coordinates [2.27],21 Gas laws, 110 gram (use in SI), 5 gauge condition grand canonical ensemble [5.113], 114 Coulomb [7.42], 139 grand partition function [5.112], 114 Lorenz [7.43], 139 grand potential Gaunt factor [7.299], 160 definition [5.37], 108 Gauss’s from grand partition function [5.115], law [7.51], 140 115 lens formula [8.64], 168 grating theorem [2.59],23 dispersion [8.31], 164 Gaussian formula [8.27], 164 electromagnetism, 135 resolving power [8.30], 164 Fourier transform of [2.507],54 Gratings, 164 integral [2.398],46 Gravitation,66 light [8.110], 172 gravitation optics, 168 field from a sphere [3.44],66 probability distribution general relativity, 67 k-dimensional [2.556],58 Newton’s law [3.40],66 1-dimensional [2.552],58 Newtonian, 71 Geiger’s law [4.169], 103 Newtonian field equations [3.42],66 Geiger-Nuttall rule [4.170], 103 gravitational General constants,7 collapse [9.53], 181 General relativity,67 constant, 6, 7, 16 generalised coordinates [3.213],79 lens [9.50], 180 Generalised dynamics,79 potential [3.42],66 generalised momentum [3.218],79 redshift [9.74], 183 geodesic deviation [3.56],67 wave radiation [9.75], 183 geodesic equation [3.54],67 Gravitationally bound orbital motion, geometric 71 distribution [2.548],57 gravity mean [2.109],27 and motion on Earth [3.38],66 progression [2.107],27 waves (on a fluid surface) [3.320], Geometrical optics, 168 86 Gibbs gray (unit), 4 constant (value), 9 Greek alphabet,18 distribution [5.113], 114 Green’s first theorem [2.62],23 entropy [5.106], 114 Green’s second theorem [2.63],23 free energy [5.35], 108 Greenwich sidereal time [9.6], 177 Gibbs’s phase rule [5.54], 109 Gregory’s series [2.141],29 Gibbs–Helmholtz equations, 109 greybody [5.193], 121 Gibbs-Duhem relation [5.38], 108 group speed (wave) [3.327],87 giga, 5 Gruneisen¨ parameter [6.56], 131 golden mean (value), 9 gyro-frequency [7.265], 157 I golden rule (Fermi’s) [4.162], 102 gyro-radius [7.268], 157 Gradient,21 gyromagnetic ratio gradient definition [4.138], 100 cylindrical coordinates [2.26],21 electron [4.140], 100 main January 23, 2006 16:6

202 Index

proton (value), 8 constant pressure [5.15], 107 gyroscopes, 77 constant volume [5.14], 107 gyroscopic for f degrees of freedom, 113 limit [3.193],77 ratio (γ) [5.18], 107 nutation [3.194],77 heat conduction/diffusion equation precession [3.191],77 differential equation [2.340],43 stability [3.192],77 Fick’s second law [5.96], 113 heat engine efficiency [5.10], 107 H heat pump efficiency [5.12], 107 H (magnetic field strength) [7.100], 143 heavy beam [3.260],82 half-life (nuclear decay) [4.164], 103 hectare, 12 half-period zones (Fresnel) [8.49], 166 hecto, 5 Hall Heisenberg uncertainty relation [4.7],90 coefficient (dimensions), 16 Helmholtz equation [2.341],43 conductivity [7.280], 158 Helmholtz free energy effect and coefficient [6.67], 132 definition [5.32], 108 voltage [6.68], 132 from partition function [5.114], 115 Hamilton’s equations [3.220],79 hemisphere (centre of mass) [3.170],76 Hamilton’s principal function [3.213],79 hemispherical shell (centre of mass) [3.171], Hamilton-Jacobi equation [3.227],79 76 Hamiltonian henry (unit), 4 charged particle (Newtonian) [7.138], Hermite equation [2.346],43 146 Hermite polynomials [4.70],95 charged particle [3.223],79 Hermitian definition [3.219],79 conjugate operator [4.17],91 of a particle [3.222],79 matrix [2.73],24 quantum mechanical [4.21],91 symmetry, 53 Hamiltonian (dimensions), 16 Heron’s formula [2.253],36 Hamiltonian dynamics,79 herpolhode, 63, 77 Hamming window [2.584],60 hertz (unit), 4 Hanbury Brown and Twiss interferometry, Hertzian dipole [7.207], 153 172 hexagonal system (crystallographic), 127 Hanning window [2.583],60 High energy and nuclear physics, 103 harmonic mean [2.110],27 Hohmann cotangential transfer [3.98], Harmonic oscillator,95 70 harmonic oscillator hole current density [6.89], 134 damped [3.196],78 Hooke’s law [3.230],80 energy levels [4.68],95 l’Hopital’sˆ rule [2.131],28 entropy [5.108], 114 Horizon coordinates, 177 forced [3.204],78 hour (unit), 5 mean energy [6.40], 130 hour angle [9.8], 177 Hartree energy [4.76],95 Hubble constant (dimensions), 16 Heat capacities, 107 Hubble constant [9.85], 184 heat capacity (dimensions), 16 Hubble law heat capacity in solids as a distance indicator [9.45], 180 Debye [6.45], 130 in cosmology [9.83], 184 free electron [6.76], 133 hydrogen atom heat capacity of a gas eigenfunctions [4.80],96 Cp −CV [5.17], 107 energy [4.81],96 main January 23, 2006 16:6

Index 203

Schrodinger¨ equation [4.79],96 paired strip transmission line [7.183], Hydrogenic atoms,95 150 Hydrogenlike atoms – Schrodinger¨ so- terminated transmission line [7.178], lution,96 150 hydrostatic waveguide compression [3.238],80 TE modes [7.189], 151 condition [3.293],84 TM modes [7.188], 151 equilibrium (of a star) [9.61], 181 impedances hyperbola, 38 in parallel [7.158], 148 Hyperbolic derivatives,41 in series [7.157], 148 hyperbolic motion, 72 impulse (dimensions), 17 Hyperbolic relationships,33 impulse (specific) [3.92],70 incompressible flow, 84, 85 I indefinite integrals, 44 I (Stokes parameter) [8.89], 171 induced charge density [7.84], 142 icosahedron, 38 Inductance, 137 Ideal fluids,84 inductance Ideal gas, 110 dimensions, 17 ideal gas energy [7.154], 148 adiabatic equations [5.58], 110 energy of an assembly [7.135], 146 internal energy [5.62], 110 impedance [7.160], 148 isothermal reversible expansion [5.63], mutual 110 definition [7.147], 147 law [5.57], 110 energy [7.135], 146 speed of sound [3.318],86 self [7.145], 147 Identical particles, 115 voltage across [7.146], 147 illuminance (definition) [5.164], 119 inductance of illuminance (dimensions), 16 cylinders (coaxial) [7.24], 137 Image charges, 138 solenoid [7.23], 137 impedance wire loop [7.26], 137 acoustic [3.276],83 wires (parallel) [7.25], 137 dimensions, 17 induction equation (MHD) [7.282], 158 electrical, 148 inductor, see inductance transformation [7.166], 149 Inelastic collisions,73 impedance of Inequalities,30 capacitor [7.159], 148 inertia tensor [3.136],74 coaxial transmission line [7.181], 150 inner product [2.1],20 electromagnetic wave [7.198], 152 Integration,44 forced harmonic oscillator [3.212], integration (numerical), 61 78 integration by parts [2.354],44 free space intensity definition [7.197], 152 correlation [8.109], 172 value, 7 luminous [5.166], 119 inductor [7.160], 148 of interfering beams [8.100], 172 lossless transmission line [7.174], 150 radiant [5.154], 118 I lossy transmission line [7.175], 150 specific [5.171], 120 microstrip line [7.184], 150 Interference, 162 open-wire transmission line [7.182], interference and coherence [8.100], 172 150 intermodal dispersion (optical fibre) [8.79], main January 23, 2006 16:6

204 Index

169 Julian centuries [9.5], 177 internal energy Julian day number [9.1], 177 definition [5.28], 108 Jupiter data, 176 from partition function [5.116], 115 ideal gas [5.62], 110 K Joule’s law [5.55], 110 katal (unit), 4 monatomic gas [5.79], 112 Kelvin interval (in general relativity) [3.45],67 circulation theorem [3.287],84 invariable plane, 63, 77 relation [6.83], 133 inverse Compton scattering [7.239], 155 temperature conversion, 15 Inverse hyperbolic functions,35 temperature scale [5.2], 106 inverse Laplace transform [2.518],55 wedge [3.330],87 inverse matrix [2.83],25 kelvin (SI definition), 3 inverse square law [3.99],71 kelvin (unit), 4 Inverse trigonometric functions,34 Kelvin-Helmholtz timescale [9.55], 181 ionic bonding [6.55], 131 Kepler’s laws, 71 irradiance (definition) [5.152], 118 Kepler’s problem, 71 irradiance (dimensions), 17 Kerr solution (in general relativity) [3.62], isobaric expansivity [5.19], 107 67 isophotal wavelength [9.39], 179 ket vector [4.34],92 isothermal bulk modulus [5.22], 107 kilo, 5 isothermal compressibility [5.20], 107 kilogram (SI definition), 3 Isotropic elastic solids,81 kilogram (unit), 4 kinematic viscosity [3.302],85 J kinematics, 63 Jacobi identity [2.93],26 kinetic energy Jacobian definition [3.65],68 definition [2.332],42 for a rotating body [3.142],74 in change of variable [2.333],42 Galilean transformation [3.6],64 Jeans length [9.56], 181 in the virial theorem [3.102],71 Jeans mass [9.57], 181 loss after collision [3.128],73 Johnson noise [5.141], 117 of a particle [3.216],79 joint probability [2.568],59 of monatomic gas [5.79], 112 Jones matrix [8.85], 170 operator (quantum) [4.20],91 Jones vectors relativistic [3.73],68 definition [8.84], 170 w.r.t. principal axes [3.145],74 examples [8.84], 170 Kinetic theory, 112 Jones vectors and matrices, 170 Kirchhoff’s (radiation) law [5.180], 120 Josephson frequency-voltage ratio, 7 Kirchhoff’s diffraction formula, 166 joule (unit), 4 Kirchhoff’s laws, 149 Joule expansion (and Joule coefficient) [5.25], Klein–Nishina cross section [7.243], 155 108 Klein-Gordon equation [4.181], 104 Joule expansion (entropy change) [5.64], Knudsen flow [5.99], 113 110 Kronecker delta [2.442],50 Joule’s law (of internal energy) [5.55], kurtosis estimator [2.545],57 110 Joule’s law (of power dissipation) [7.155], L 148 ladder operators (angular momentum) [4.108], Joule-Kelvin coefficient [5.27], 108 98 main January 23, 2006 16:6

Index 205

Lagrange’s identity [2.7],20 cavity line width [8.127], 174 Lagrangian (dimensions), 17 cavity modes [8.124], 174 Lagrangian dynamics,79 cavity stability [8.123], 174 Lagrangian of threshold condition [8.129], 174 charged particle [3.217],79 Lasers, 174 particle [3.216],79 latent heat [5.48], 109 two mutually attracting bodies [3.85], lattice constants of elements, 124 69 Lattice dynamics, 129 Laguerre equation [2.347],43 Lattice forces (simple), 131 Laguerre polynomials (associated), 96 lattice plane spacing [6.11], 126 Lame´ coefficients [3.240],81 Lattice thermal expansion and conduc- Laminar viscous flow,85 tion, 131 Lande´ g-factor [4.146], 100 lattice vector [6.7], 126 Landau diamagnetic susceptibility [6.80], latus-rectum [3.109],71 133 Laue equations [6.28], 128 Landau length [7.249], 156 Laurent series [2.168],31 Langevin function (from Brillouin fn) [4.147], LCR circuits, 147 101 LCR definitions, 147 Langevin function [7.111], 144 least-squares fitting, 60 Laplace equation Legendre equation definition [2.339],43 and polynomials [2.421],47 solution in spherical harmonics [2.440], definition [2.343],43 49 Legendre polynomials,47 Laplace series [2.439],49 Leibniz theorem [2.296],40 Laplace transform length (dimensions), 17 convolution [2.516],55 Lennard-Jones 6-12 potential [6.52], 131 definition [2.514],55 lens blooming [8.7], 162 derivative of transform [2.520],55 Lenses and mirrors, 168 inverse [2.518],55 lensmaker’s formula [8.66], 168 of derivative [2.519],55 Levi-Civita symbol (3-D) [2.443],50 substitution [2.521],55 l’Hopital’sˆ rule [2.131],28 translation [2.523],55 Lienard–Wiechert´ potentials, 139 Laplace transform pairs,56 light (speed of), 6, 7 Laplace transform theorems,55 Limits,28 Laplace transforms,55 line charge (electric field from) [7.29], 138 Laplace’s formula (surface tension) [3.337], line fitting, 60 88 Line radiation, 173 Laplacian line shape cylindrical coordinates [2.46],23 collisional [8.114], 173 general coordinates [2.48],23 Doppler [8.116], 173 rectangular coordinates [2.45],23 natural [8.112], 173 spherical coordinates [2.47],23 line width Laplacian (scalar),23 collisional/pressure [8.115], 173 lapse rate (adiabatic) [3.294],84 Doppler broadened [8.117], 173 Larmor frequency [7.265], 157 laser cavity [8.127], 174 I Larmor radius [7.268], 157 natural [8.113], 173 Larmor’s formula [7.132], 146 Schawlow-Townes [8.128], 174 laser linear absorption coefficient [5.175], 120 cavity Q [8.126], 174 linear expansivity (definition) [5.19], 107 main January 23, 2006 16:6

206 Index linear expansivity (of a crystal) [6.57], intensity (dimensions), 17 131 intensity [5.166], 119 linear regression, 60 lux (unit), 4 linked flux [7.149], 147 liquid drop model [4.172], 103 M litre (unit), 5 Mach number [3.315],86 local civil time [9.4], 177 Mach wedge [3.328],87 local sidereal time [9.7], 177 Maclaurin series [2.125],28 local thermodynamic equilibrium (LTE), Macroscopic thermodynamic variables, 116, 120 115 ln(1+x) (series expansion) [2.133],29 Madelung constant (value), 9 logarithm of complex numbers [2.162], Madelung constant [6.55], 131 30 magnetic logarithmic decrement [3.202],78 diffusivity [7.282], 158 London’s formula (interacting dipoles) [6.50], flux quantum, 6, 7 131 monopoles (none) [7.52], 140 longitudinal elastic modulus [3.241],81 permeability, µ, µr [7.107], 143 look-back time [9.96], 185 quantum number [4.131], 100 Lorentz scalar potential [7.7], 136 broadening [8.112], 173 susceptibility, χH , χB [7.103], 143 contraction [3.8],64 vector potential factor (γ) [3.7],64 definition [7.40], 139 force [7.122], 145 from J [7.47], 139 Lorentz (spacetime) transformations,64 of a moving charge [7.49], 139 Lorentz factor (dynamical) [3.69],68 magnetic dipole, see dipole Lorentz transformation magnetic field in electrodynamics, 141 around objects, 138 of four-vectors, 65 dimensions, 17 of momentum and energy, 65 energy density [7.128], 146 of time and position, 64 Lorentz transformation, 141 of velocity, 64 static, 136 Lorentz-Lorenz formula [7.93], 142 strength (H) [7.100], 143 Lorentzian distribution [2.555],58 thermodynamic work [5.8], 106 Lorentzian (Fourier transform of) [2.505], wave equation [7.194], 152 54 Magnetic fields, 138 Lorenz magnetic flux (dimensions), 17 constant [6.66], 132 magnetic flux density (dimensions), 17 gauge condition [7.43], 139 magnetic flux density from lumen (unit), 4 current [7.11], 136 luminance [5.168], 119 current density [7.10], 136 luminosity distance [9.98], 185 dipole [7.36], 138 luminosity–magnitude relation [9.31], 179 electromagnet [7.38], 138 luminous line current (Biot–Savart law) [7.9], density [5.160], 119 136 efficacy [5.169], 119 solenoid (finite) [7.38], 138 efficiency [5.170], 119 solenoid (infinite) [7.33], 138 energy [5.157], 119 uniform cylindrical current [7.34], exitance [5.162], 119 138 flux [5.159], 119 waveguide [7.190], 151 main January 23, 2006 16:6

Index 207

wire [7.34], 138 and absorption coefficient [5.175], wire loop [7.37], 138 120 Magnetic moments, 100 Maxwell-Boltzmann [5.89], 113 magnetic vector potential (dimensions), 17 mean intensity [5.172], 120 Magnetisation, 143 mean-life (nuclear decay) [4.165], 103 magnetisation mega, 5 definition [7.97], 143 melting points of elements, 124 dimensions, 17 meniscus [3.339],88 isolated spins [4.151], 101 Mensuration,35 quantum paramagnetic [4.150], 101 Mercury data, 176 magnetogyric ratio [4.138], 100 method of images, 138 Magnetohydrodynamics, 158 metre (SI definition), 3 magnetosonic waves [7.285], 158 metre (unit), 4 Magnetostatics, 136 metric elements and coordinate systems, magnification (longitudinal) [8.71], 168 21 magnification (transverse) [8.70], 168 MHD equations [7.283], 158 magnitude (astronomical) micro, 5 –flux relation [9.32], 179 microcanonical ensemble [5.109], 114 –luminosity relation [9.31], 179 micron (unit), 5 absolute [9.29], 179 microstrip line (impedance) [7.184], 150 apparent [9.27], 179 Miller-Bravais indices [6.20], 126 major axis [3.106],71 milli, 5 Malus’s law [8.83], 170 minima [2.337],42 Mars data, 176 minimum deviation (of a prism) [8.74], mass (dimensions), 17 169 mass absorption coefficient [5.176], 120 minor axis [3.107],71 mass ratio (of a rocket) [3.94],70 minute (unit), 5 Mathematical constants,9 mirror formula [8.67], 168 Mathematics, 19–62 Miscellaneous,18 matrices (square), 25 mobility (dimensions), 17 Matrix algebra,24 mobility (in conductors) [6.88], 134 matrix element (quantum) [4.32],92 modal dispersion (optical fibre) [8.79], maxima [2.336],42 169 Maxwell’s equations, 140 modified Bessel functions [2.419],47 Maxwell’s equations (using D and H), modified Julian day number [9.2], 177 140 modulus (of a complex number) [2.155], Maxwell’s relations, 109 30 Maxwell–Boltzmann distribution, 112 Moirefringes´ ,35 Maxwell-Boltzmann distribution molar gas constant (dimensions), 17 mean speed [5.86], 112 molar volume, 9 most probable speed [5.88], 112 mole (SI definition), 3 rms speed [5.87], 112 mole (unit), 4 speed distribution [5.84], 112 molecular flow [5.99], 113 mean moment arithmetic [2.108],27 electric dipole [7.81], 142 I geometric [2.109],27 magnetic dipole [7.94], 143 harmonic [2.110],27 magnetic dipole [7.95], 143 mean estimator [2.541],57 moment of area [3.258],82 mean free path moment of inertia main January 23, 2006 16:6

208 Index

cone [3.160],75 inductance (energy) [7.135], 146 cylinder [3.155],75 mutual coherence function [8.97], 172 dimensions, 17 disk [3.168],75 N ellipsoid [3.163],75 nabla, 21 elliptical lamina [3.166],75 Named integrals,45 rectangular cuboid [3.158],75 nano, 5 sphere [3.152],75 natural broadening profile [8.112], 173 spherical shell [3.153],75 natural line width [8.113], 173 thin rod [3.150],75 Navier-Stokes equation [3.301],85 triangular plate [3.169],75 nearest neighbour distances, 127 two-body system [3.83],69 Neptune data, 176 moment of inertia ellipsoid [3.147],74 neutron Moment of inertia tensor,74 Compton wavelength, 8 moment of inertia tensor [3.136],74 gyromagnetic ratio, 8 Moments of inertia,75 magnetic moment, 8 momentum mass, 8 definition [3.64],68 molar mass, 8 dimensions, 17 Neutron constants,8 generalised [3.218],79 neutron star degeneracy pressure [9.77], relativistic [3.70],68 183 Momentum and energy transformations, newton (unit), 4 65 Newton’s law of Gravitation [3.40],66 Monatomic gas, 112 Newton’s lens formula [8.65], 168 monatomic gas Newton’s rings, 162 entropy [5.83], 112 Newton’s rings [8.1], 162 equation of state [5.78], 112 Newton-Raphson method [2.593],61 heat capacity [5.82], 112 Newtonian gravitation,66 internal energy [5.79], 112 noggin, 13 pressure [5.77], 112 Noise, 117 monoclinic system (crystallographic), 127 noise Moon data, 176 figure [5.143], 117 motif [6.31], 128 Johnson [5.141], 117 motion under constant acceleration, 68 Nyquist’s theorem [5.140], 117 Mott scattering formula [4.180], 104 shot [5.142], 117 µ, µr (magnetic permeability) [7.107], 143 temperature [5.140], 117 multilayer films (in optics) [8.8], 162 normal (unit principal) [2.284],39 multimode dispersion (optical fibre) [8.79], normal distribution [2.552],58 169 normal plane, 39 multiplicity (quantum) Nuclear binding energy, 103 j [4.133], 100 Nuclear collisions, 104 l [4.112],98 Nuclear decay, 103 multistage rocket [3.95],70 nuclear decay law [4.163], 103 Multivariate normal distribution,58 nuclear magneton, 7 Muon and tau constants,9 number density (dimensions), 17 muon physical constants, 9 numerical aperture (optical fibre) [8.78], mutual 169 capacitance [7.134], 146 Numerical differentiation,61 inductance (definition) [7.147], 147 Numerical integration,61 main January 23, 2006 16:6

Index 209

Numerical methods,60 orthorhombic system (crystallographic), 127 Numerical solutions to f(x)=0,61 Oscillating systems,78 Numerical solutions to ordinary dif- osculating plane, 39 ferential equations,62 Otto cycle efficiency [5.13], 107 nutation [3.194],77 overdamping [3.201],78 Nyquist’s theorem [5.140], 117 P O p orbitals [4.95],97 Oblique elastic collisions,73 P-waves [3.263],82 obliquity factor (diffraction) [8.46], 166 packing fraction (of spheres), 127 obliquity of the ecliptic [9.13], 178 paired strip (impedance of) [7.183], 150 observable (quantum physics) [4.5],90 parabola, 38 Observational astrophysics, 179 parabolic motion [3.88],69 octahedron, 38 parallax (astronomical) [9.46], 180 odd functions, 53 parallel axis theorem [3.140],74 ODEs (numerical solutions), 62 parallel impedances [7.158], 148 ohm (unit), 4 parallel wire feeder (inductance) [7.25], Ohm’s law (in MHD) [7.281], 158 137 Ohm’s law [7.140], 147 paramagnetic susceptibility (Pauli) [6.79], opacity [5.176], 120 133 open-wire transmission line [7.182], 150 paramagnetism (quantum), 101 operator Paramagnetism and diamagnetism, 144 angular momentum parity operator [4.24],91 and other operators [4.23],91 Parseval’s relation [2.495],53 definitions [4.105],98 Parseval’s theorem Hamiltonian [4.21],91 integral form [2.496],53 kinetic energy [4.20],91 series form [2.480],52 momentum [4.19],91 Partial derivatives,42 parity [4.24],91 partial widths (and total width) [4.176], position [4.18],91 104 time dependence [4.27],91 Particle in a rectangular box,94 Operators,91 Particle motion,68 optic branch (phonon) [6.37], 129 partition function optical coating [8.8], 162 atomic [5.126], 116 optical depth [5.177], 120 definition [5.110], 114 Optical fibres, 169 macroscopic variables from, 115 optical path length [8.63], 168 pascal (unit), 4 Optics, 161–174 Pauli matrices,26 Orbital angular dependence,97 Pauli matrices [2.94],26 Orbital angular momentum,98 Pauli paramagnetic susceptibility [6.79], orbital motion, 71 133 orbital radius (Bohr atom) [4.73],95 Pauli spin matrices (and Weyl eqn.) [4.182], order (in diffraction) [8.26], 164 104 ordinary modes [7.271], 157 Pearson’s r [2.546],57 orthogonal matrix [2.85],25 Peltier effect [6.82], 133 I orthogonality pendulum associated Legendre functions [2.434], compound [3.182],76 48 conical [3.180],76 Legendre polynomials [2.424],47 double [3.183],76 main January 23, 2006 16:6

210 Index

simple [3.179],76 plane polarisation, 170 torsional [3.181],76 Plane triangles,36 Pendulums,76 plane wave expansion [2.427],47 perfect gas, 110 Planetary bodies, 180 pericentre (of an orbit) [3.110],71 Planetary data, 176 perimeter plasma of circle [2.261],37 beta [7.278], 158 of ellipse [2.266],37 dispersion relation [7.261], 157 Perimeter, area, and volume,37 frequency [7.259], 157 period (of an orbit) [3.113],71 group velocity [7.264], 157 Periodic table, 124 phase velocity [7.262], 157 permeability refractive index [7.260], 157 dimensions, 17 Plasma physics, 156 magnetic [7.107], 143 Platonic solids,38 of vacuum, 6, 7 Pluto data, 176 permittivity p-n junction [6.92], 134 dimensions, 17 Poincare´ sphere, 171 electrical [7.90], 142 point charge (electric field from) [7.5], of vacuum, 6, 7 136 permutation tensor (ijk) [2.443],50 Poiseuille flow [3.305],85 perpendicular axis theorem [3.148],74 Poisson brackets [3.224],79 Perturbation theory, 102 Poisson distribution [2.549],57 peta, 5 Poisson ratio petrol engine efficiency [5.13], 107 and elastic constants [3.251],81 phase object (diffraction by weak) [8.43], simple definition [3.231],80 165 Poisson’s equation [7.3], 136 phase rule (Gibbs’s) [5.54], 109 polarisability [7.91], 142 phase speed (wave) [3.325],87 Polarisation, 170 Phase transitions, 109 Polarisation, 142 Phonon dispersion relations, 129 polarisation (electrical, per unit volume) phonon modes (mean energy) [6.40], 130 [7.83], 142 Photometric wavelengths, 179 polarisation (of radiation) Photometry, 119 angle [8.81], 170 photon energy [4.3],90 axial ratio [8.88], 171 Physical constants,6 degree of [8.96], 171 Pi (π) to 1 000 decimal places,18 elliptical [8.80], 170 Pi (π), 9 ellipticity [8.82], 170 pico, 5 reflection law [7.218], 154 pipe (flow of fluid along) [3.305],85 polarisers [8.85], 170 pipe (twisting of) [3.255],81 polhode, 63, 77 pitch angle, 159 Population densities, 116 Planck potential constant, 6, 7 chemical [5.28], 108 constant (dimensions), 17 difference (and work) [5.9], 106 function [5.184], 121 difference (between points) [7.2], 136 length, 7 electrical [7.46], 139 mass, 7 electrostatic [7.1], 136 time, 7 energy (elastic) [3.235],80 Planck-Einstein relation [4.3],90 energy in Hamiltonian [3.222],79 main January 23, 2006 16:6

Index 211

energy in Lagrangian [3.216],79 deviation [8.73], 169 ﬁeld equations [7.45], 139 dispersion [8.76], 169 four-vector [7.77], 141 minimum deviation [8.74], 169 grand [5.37], 108 transmission angle [8.72], 169 Lienard–Wiechert,´ 139 Prisms (dispersing), 169 Lorentz transformation [7.75], 141 probability magnetic scalar [7.7], 136 conditional [2.567],59 magnetic vector [7.40], 139 density current [4.13],90 Rutherford scattering [3.114],72 distributions thermodynamic [5.35], 108 continuous, 58 velocity [3.296],84 discrete, 57 Potential flow,84 joint [2.568],59 Potential step,92 Probability and statistics,57 Potential well,93 product (derivative of) [2.293],40 power (dimensions), 17 product (integral of) [2.354],44 power gain product of inertia [3.136],74 antenna [7.211], 153 progression (arithmetic) [2.104],27 short dipole [7.213], 153 progression (geometric) [2.107],27 Power series,28 Progressions and summations,27 Power theorem [2.495],53 projectiles, 69 Poynting vector (dimensions), 17 propagation in cold plasmas, 157 Poynting vector [7.130], 146 Propagation in conducting media, 155 pp (proton-proton) chain, 182 Propagation of elastic waves,83 Prandtl number [3.314],86 Propagation of light,65 precession (gyroscopic) [3.191],77 proper distance [9.97], 185 Precession of equinoxes, 178 Proton constants,8 pressure proton mass, 6 broadening [8.115], 173 proton-proton chain, 182 critical [5.75], 111 pulsar degeneracy [9.77], 183 braking index [9.66], 182 dimensions, 17 characteristic age [9.67], 182 ﬂuctuations [5.136], 116 dispersion [9.72], 182 from partition function [5.118], 115 magnetic dipole radiation [9.69], 182 hydrostatic [3.238],80 Pulsars, 182 in a monatomic gas [5.77], 112 pyramid (centre of mass) [3.175],76 radiation, 152 pyramid (volume) [2.272],37 thermodynamic work [5.5], 106 waves [3.263],82 Q primitive cell [6.1], 126 Q, see quality factor primitive vectors (and lattice vectors) [6.7], Q (Stokes parameter) [8.90], 171 126 Quadratic equations,50 primitive vectors (of cubic lattices), 127 quadrature, 61 Principal axes,74 quadrature (integration), 44 principal moments of inertia [3.143],74 quality factor principal quantum number [4.71],95 Fabry-Perot etalon [8.14], 163 I principle of least action [3.213],79 forced harmonic oscillator [3.211], prism 78 determining refractive index [8.75], free harmonic oscillator [3.203],78 169 laser cavity [8.126], 174 main January 23, 2006 16:6

212 Index

LCR circuits [7.152], 148 random walk quantum concentration [5.83], 112 Brownian motion [5.98], 113 Quantum definitions,90 one-dimensional [2.562],59 Quantum paramagnetism, 101 three-dimensional [2.564],59 Quantum physics, 89–104 range (of projectile) [3.90],69 Quantum uncertainty relations,90 Rankine conversion [1.3],15 quarter-wave condition [8.3], 162 Rankine-Hugoniot shock relations [3.334], quarter-wave plate [8.85], 170 87 quartic minimum, 42 Rayleigh distribution [2.554],58 R resolution criterion [8.41], 165 Radial forms,22 scattering [7.236], 155 radian (unit), 4 theorem [2.496],53 radiance [5.156], 118 Rayleigh-Jeans law [5.187], 121 radiant reactance (definition), 148 energy [5.145], 118 reciprocal energy density [5.148], 118 lattice vector [6.8], 126 exitance [5.150], 118 matrix [2.83],25 flux [5.147], 118 vectors [2.16],20 intensity (dimensions), 17 reciprocity [2.330],42 intensity [5.154], 118 Recognised non-SI units,5 radiation rectangular aperture diffraction [8.39], blackbody [5.184], 121 165 bremsstrahlung [7.297], 160 rectangular coordinates, 21 Cherenkov [7.247], 156 rectangular cuboid moment of inertia [3.158], field of a dipole [7.207], 153 75 flux from dipole [7.131], 146 rectifying plane, 39 resistance [7.209], 153 recurrence relation synchrotron [7.287], 159 associated Legendre functions [2.433], Radiation pressure, 152 48 radiation pressure Legendre polynomials [2.423],47 extended source [7.203], 152 redshift isotropic [7.200], 152 –flux density relation [9.99], 185 momentum density [7.199], 152 cosmological [9.86], 184 point source [7.204], 152 gravitational [9.74], 183 specular reflection [7.202], 152 Reduced mass (of two interacting bod- Radiation processes, 118 ies),69 Radiative transfer, 120 reduced units (thermodynamics) [5.71], radiative transfer equation [5.179], 120 111 radiative transport (in stars) [9.63], 181 reflectance coefficient radioactivity, 103 and Fresnel equations [7.227], 154 Radiometry, 118 dielectric film [8.4], 162 radius of curvature dielectric multilayer [8.8], 162 definition [2.282],39 reflection coefficient in bending [3.258],82 acoustic [3.283],83 relation to curvature [2.287],39 dielectric boundary [7.230], 154 radius of gyration (see footnote), 75 potential barrier [4.58],94 Ramsauer effect [4.52],93 potential step [4.41],92 Random walk,59 potential well [4.48],93 main January 23, 2006 16:6

Index 213

transmission line [7.179], 150 Riemann tensor [3.50],67 reflection grating [8.29], 164 right ascension [9.8], 177 reflection law [7.216], 154 rigid body Reflection, refraction, and transmis- angular momentum [3.141],74 sion, 154 kinetic energy [3.142],74 refraction law (Snell’s) [7.217], 154 Rigid body dynamics,74 refractive index of rigidity modulus [3.249],81 dielectric medium [7.195], 152 ripples [3.321],86 ohmic conductor [7.234], 155 rms (standard deviation) [2.543],57 plasma [7.260], 157 Robertson-Walker metric [9.87], 184 refrigerator efficiency [5.11], 107 Roche limit [9.43], 180 regression (linear), 60 rocket equation [3.94],70 relativistic beaming [3.25],65 Rocketry,70 relativistic doppler effect [3.22],65 rod Relativistic dynamics,68 bending, 82 Relativistic electrodynamics, 141 moment of inertia [3.150],75 Relativistic wave equations, 104 stretching [3.230],80 relativity (general), 67 waves in [3.271],82 relativity (special), 64 Rodrigues’ formula [2.422],47 relaxation time Roots of quadratic and cubic equations,50 andelectrondrift[6.61], 132 Rossby number [3.316],86 in a conductor [7.156], 148 rot (curl), 22 in plasmas, 156 Rotating frames,66 residuals [2.572],60 Rotation matrices,26 Residue theorem [2.170],31 rotation measure [7.273], 157 residues (in complex analysis), 31 Runge Kutta method [2.603],62 resistance Rutherford scattering,72 and impedance, 148 Rutherford scattering formula [3.124],72 dimensions, 17 Rydberg constant, 6, 7 energy dissipated in [7.155], 148 and Bohr atom [4.77],95 radiation [7.209], 153 dimensions, 17 resistivity [7.142], 147 Rydberg’s formula [4.78],95 resistor, see resistance resolving power S chromatic (of an etalon) [8.21], 163 s orbitals [4.92],97 of a diffraction grating [8.30], 164 S-waves [3.262],82 Rayleigh resolution criterion [8.41], Sackur-Tetrode equation [5.83], 112 165 saddle point [2.338],42 resonance Saha equation (general) [5.128], 116 forced oscillator [3.209],78 Saha equation (ionisation) [5.129], 116 resonance lifetime [4.177], 104 Saturn data, 176 resonant frequency (LCR) [7.150], 148 scalar effective mass [6.87], 134 Resonant LCR circuits, 148 scalar product [2.1],20 restitution (coefficient of) [3.127],73 scalar triple product [2.10],20 retarded time, 139 scale factor (cosmic) [9.87], 184 I revolution (volume and surface of), 39 scattering Reynolds number [3.311],86 angle (Rutherford) [3.116],72 ribbon (twisting of) [3.256],81 Born approximation [4.178], 104 Ricci tensor [3.57],67 Compton [7.240], 155 main January 23, 2006 16:6

214 Index

crystal [6.32], 128 shear modulus (dimensions), 17 inverse Compton [7.239], 155 sheet of charge (electric field) [7.32], 138 Klein-Nishina [7.243], 155 shift theorem (Fourier transform) [2.501], Mott (identical particles) [4.180], 104 54 potential (Rutherford) [3.114],72 shock processes (electron), 155 Rankine-Hugoniot conditions [3.334], Rayleigh [7.236], 155 87 Rutherford [3.124],72 spherical [3.331],87 Thomson [7.238], 155 Shocks,87 scattering cross-section, see cross-section shot noise [5.142], 117 Schawlow-Townes line width [8.128], 174 SI base unit definitions, 3 Schrodinger¨ equation [4.15],90 SI base units,4 Schwarz inequality [2.152],30 SI derived units,4 Schwarzschild geometry (in GR) [3.61], SI prefixes,5 67 SI units,4 Schwarzschild radius [9.73], 183 sidelobes (diffraction by 1-D slit) [8.38], Schwarzschild’s equation [5.179], 120 165 screw dislocation [6.22], 128 sidereal time [9.7], 177 secx siemens (unit), 4 definition [2.228],34 sievert (unit), 4 series expansion [2.138],29 similarity theorem (Fourier transform) [2.500], secant method (of root-finding) [2.592], 54 61 simple cubic structure, 127 sechx [2.229],34 simple harmonic oscillator, see harmonic second (SI definition), 3 oscillator second (time interval), 4 simple pendulum [3.179],76 second moment of area [3.258],82 Simpson’s rule [2.586],61 Sedov-Taylor shock relation [3.331],87 sinx selection rules (dipole transition) [4.91], and Euler’s formula [2.218],34 96 series expansion [2.136],29 self-diffusion [5.93], 113 sinc function [2.512],54 self-inductance [7.145], 147 sine formula semi-ellipse (centre of mass) [3.178],76 planar triangles [2.246],36 semi-empirical mass formula [4.173], 103 spherical triangles [2.255],36 semi-latus-rectum [3.109],71 sinhx semi-major axis [3.106],71 definition [2.219],34 semi-minor axis [3.107],71 series expansion [2.144],29 − semiconductor diode [6.92], 134 sin 1 x, see arccosx semiconductor equation [6.90], 134 skew-symmetric matrix [2.87],25 Series expansions,29 skewness estimator [2.544],57 series impedances [7.157], 148 skin depth [7.235], 155 Series, summations, and progressions,27 slit diffraction (broad slit) [8.37], 165 shah function (Fourier transform of) [2.510], slit diffraction (Young’s) [8.24], 164 54 Snell’s law (acoustics) [3.284],83 shear Snell’s law (electromagnetism) [7.217], 154 modulus [3.249],81 soap bubbles [3.337],88 strain [3.237],80 solar constant, 176 viscosity [3.299],85 Solar data, 176 waves [3.262],82 Solar system data, 176 main January 23, 2006 16:6

Index 215 solenoid in a viscous fluid [3.308],85 finite [7.38], 138 in potential flow [3.298],84 infinite [7.33], 138 moment of inertia [3.152],75 self inductance [7.23], 137 Poincare,´ 171 solid angle (subtended by a circle) [2.278], polarisability, 142 37 volume [2.264],37 Solid state physics, 123–134 spherical Bessel function [2.420],47 sound speed (in a plasma) [7.275], 158 spherical cap sound, speed of [3.317],86 area [2.275],37 space cone, 77 centre of mass [3.177],76 space frequency [3.188],77 volume [2.276],37 space impedance [7.197], 152 spherical excess [2.260],36 spatial coherence [8.108], 172 Spherical harmonics,49 Special functions and polynomials,46 spherical harmonics special relativity, 64 definition [2.436],49 specific Laplace equation [2.440],49 charge on electron, 8 orthogonality [2.437],49 emission coefficient [5.174], 120 spherical polar coordinates, 21 heat capacity, see heat capacity spherical shell (moment of inertia) [3.153], definition, 105 75 dimensions, 17 spherical surface (capacitance of near) [7.16], intensity (blackbody) [5.184], 121 137 intensity [5.171], 120 Spherical triangles,36 specific impulse [3.92],70 spin speckle intensity distribution [8.110], 172 and total angular momentum [4.128], speckle size [8.111], 172 100 spectral energy density degeneracy, 115 blackbody [5.186], 121 electron magnetic moment [4.141], definition [5.173], 120 100 spectral function (synchrotron) [7.295], Pauli matrices, 26 159 spinning bodies, 77 Spectral line broadening, 173 spinors [4.182], 104 speed (dimensions), 17 Spitzer conductivity [7.254], 156 speed distribution (Maxwell-Boltzmann) [5.84], spontaneous emission [8.119], 173 112 spring constant and wave velocity [3.272], speed of light (equation) [7.196], 152 83 speed of light (value), 6 Square matrices,25 speed of sound [3.317],86 standard deviation estimator [2.543],57 sphere Standard forms,44 area [2.263],37 Star formation, 181 Brownian motion [5.98], 113 Star–delta transformation, 149 capacitance [7.12], 137 Static fields, 136 capacitance of adjacent [7.14], 137 statics, 63 capacitance of concentric [7.18], 137 Stationary points,42 close-packed, 127 Statistical entropy, 114 I collisions of, 73 Statistical thermodynamics, 114 electric field [7.27], 138 Stefan–Boltzmann constant, 9 geometry on a, 36 Stefan–Boltzmann constant (dimensions), gravitation field from a [3.44],66 17 main January 23, 2006 16:6

216 Index

Stefan-Boltzmann constant, 121 susceptibility Stefan-Boltzmann law [5.191], 121 electric [7.87], 142 stellar aberration [3.24],65 Landau diamagnetic [6.80], 133 Stellar evolution, 181 magnetic [7.103], 143 Stellar fusion processes, 182 Pauli paramagnetic [6.79], 133 Stellar theory, 181 symmetric matrix [2.86],25 step function (Fourier transform of) [2.511], symmetric top [3.188],77 54 Synchrotron radiation, 159 steradian (unit), 4 synodic period [9.44], 180 stimulated emission [8.120], 173 Stirling’s formula [2.411],46 T Stokes parameters, 171 tanx Stokes parameters [8.95], 171 definition [2.220],34 Stokes’s law [3.308],85 series expansion [2.137],29 Stokes’s theorem [2.60],23 tangent [2.283],39 Straight-line fitting,60 tangent formula [2.250],36 strain tanhx simple [3.229],80 definition [2.221],34 tensor [3.233],80 series expansion [2.145],29 volume [3.236],80 tan−1 x, see arctanx stress tau physical constants, 9 dimensions, 17 Taylor series in fluids [3.299],85 one-dimensional [2.123],28 simple [3.228],80 three-dimensional [2.124],28 tensor [3.232],80 telegraphist’s equations [7.171], 150 stress-energy tensor temperature and field equations [3.59],67 antenna [7.215], 153 perfect fluid [3.60],67 Celsius, 4 string (waves along a stretched) [3.273], dimensions, 17 83 Kelvin scale [5.2], 106 Strouhal number [3.313],86 thermodynamic [5.1], 106 structure factor [6.31], 128 Temperature conversions,15 sum over states [5.110], 114 temporal coherence [8.105], 172 Summary of physical constants,6 tensor summation formulas [2.118],27 Einstein [3.58],67 Sun data, 176 electric susceptibility [7.87], 142 Sunyaev-Zel’dovich effect [9.51], 180 ijk [2.443],50 surface brightness (blackbody) [5.184], fluid stress [3.299],85 121 magnetic susceptibility [7.103], 143 surface of revolution [2.280],39 moment of inertia [3.136],74 Surface tension,88 Ricci [3.57],67 surface tension Riemann [3.50],67 Laplace’s formula [3.337],88 strain [3.233],80 work done [5.6], 106 stress [3.232],80 surface tension (dimensions), 17 tera, 5 surface waves [3.320],86 tesla (unit), 4 survival equation (for mean free path) [5.90], tetragonal system (crystallographic), 127 113 tetrahedron, 38 susceptance (definition), 148 thermal conductivity main January 23, 2006 16:6

Index 217

diffusion equation [2.340],43 torsional pendulum [3.181],76 dimensions, 17 torsional rigidity [3.252],81 free electron [6.65], 132 torus (surface area) [2.273],37 phonon gas [6.58], 131 torus (volume) [2.274],37 transport property [5.96], 113 total differential [2.329],42 thermal de Broglie wavelength [5.83], 112 total internal reflection [7.217], 154 thermal diffusion [5.93], 113 total width (and partial widths) [4.176], thermal diffusivity [2.340],43 104 thermal noise [5.141], 117 trace [2.75],25 thermal velocity (electron) [7.257], 156 trajectory (of projectile) [3.88],69 Thermodynamic coefficients, 107 transfer equation [5.179], 120 Thermodynamic fluctuations, 116 Transformers, 149 Thermodynamic laws, 106 transmission coefficient Thermodynamic potentials, 108 Fresnel [7.232], 154 thermodynamic temperature [5.1], 106 potential barrier [4.59],94 Thermodynamic work, 106 potential step [4.42],92 Thermodynamics, 105–121 potential well [4.49],93 Thermoelectricity, 133 transmission grating [8.27], 164 thermopower [6.81], 133 transmission line, 150 Thomson cross section, 8 coaxial [7.181], 150 Thomson scattering [7.238], 155 equations [7.171], 150 throttling process [5.27], 108 impedance time (dimensions), 17 lossless [7.174], 150 time dilation [3.11],64 lossy [7.175], 150 Time in astronomy, 177 input impedance [7.178], 150 Time series analysis,60 open-wire [7.182], 150 Time-dependent perturbation theory, 102 paired strip [7.183], 150 Time-independent perturbation theory, reflection coefficient [7.179], 150 102 vswr [7.180], 150 timescale wave speed [7.176], 150 free-fall [9.53], 181 waves [7.173], 150 Kelvin-Helmholtz [9.55], 181 Transmission line impedances, 150 Titius-Bode rule [9.41], 180 Transmission line relations, 150 tonne (unit), 5 Transmission lines and waveguides, 150 top transmittance coefficient [7.229], 154 asymmetric [3.189],77 Transport properties, 113 symmetric [3.188],77 transpose matrix [2.70],24 symmetries [3.149],74 trapezoidal rule [2.585],61 top hat function (Fourier transform of) triangle [2.512],54 area [2.254],36 Tops and gyroscopes,77 centre of mass [3.174],76 torque, see couple inequality [2.147],30 Torsion,81 plane, 36 torsion spherical, 36 in a thick cylinder [3.254],81 triangle function (Fourier transform of) I in a thin cylinder [3.253],81 [2.513],54 in an arbitrary ribbon [3.256],81 triclinic system (crystallographic), 127 in an arbitrary tube [3.255],81 trigonal system (crystallographic), 127 in differential geometry [2.288],39 Trigonometric and hyperbolic defini- main January 23, 2006 16:6

218 Index

tions,34 velocity (dimensions), 17 Trigonometric and hyperbolic formulas,32 velocity distribution (Maxwell-Boltzmann) Trigonometric and hyperbolic integrals, [5.84], 112 45 velocity potential [3.296],84 Trigonometric derivatives,41 Velocity transformations,64 Trigonometric relationships,32 Venus data, 176 triple-α process, 182 virial coefficients [5.65], 110 true anomaly [3.104],71 Virial expansion, 110 tube, see pipe virial theorem [3.102],71 Tully-Fisher relation [9.49], 180 vis-viva equation [3.112],71 tunnelling (quantum mechanical), 94 viscosity tunnelling probability [4.61],94 dimensions, 17 turns ratio (of transformer) [7.163], 149 from kinetic theory [5.97], 113 two-level system (microstates of) [5.107], kinematic [3.302],85 114 shear [3.299],85 viscous flow U between cylinders [3.306],85 U (Stokes parameter) [8.92], 171 between plates [3.303],85 UBV magnitude system [9.36], 179 through a circular pipe [3.305],85 umklapp processes [6.59], 131 through an annular pipe [3.307],85 uncertainty relation Viscous flow (incompressible),85 energy-time [4.8],90 volt (unit), 4 general [4.6],90 voltage momentum-position [4.7],90 across an inductor [7.146], 147 number-phase [4.9],90 bias [6.92], 134 underdamping [3.198],78 Hall [6.68], 132 unified atomic mass unit, 5, 6 law (Kirchhoff’s) [7.162], 149 uniform distribution [2.550],58 standing wave ratio [7.180], 150 uniform to normal distribution transfor- thermal noise [5.141], 117 mation, 58 transformation [7.164], 149 unitary matrix [2.88],25 volume units (conversion of SI to Gaussian), 135 dimensions, 17 Units, constants and conversions, 3–18 of cone [2.272],37 universal time [9.4], 177 of cube, 38 Uranus data, 176 of cylinder [2.270],37 UTC [9.4], 177 of dodecahedron, 38 of ellipsoid [2.268],37 V of icosahedron, 38 V (Stokes parameter) [8.94], 171 of octahedron, 38 van der Waals equation [5.67], 111 of parallelepiped [2.10],20 Van der Waals gas, 111 of pyramid [2.272],37 van der Waals interaction [6.50], 131 of revolution [2.281],39 Van-Cittert Zernicke theorem [8.108], 172 of sphere [2.264],37 variance estimator [2.542],57 of spherical cap [2.276],37 variations, calculus of [2.334],42 of tetrahedron, 38 Vector algebra,20 of torus [2.274],37 Vector integral transformations,23 volume expansivity [5.19], 107 vector product [2.2],20 volume strain [3.236],80 vector triple product [2.12],20 vorticity and Kelvin circulation [3.287], Vectors and matrices,20 main January 23, 2006 16:6

Index 219

84 on a stretched sheet [3.274],83 vorticity and potential flow [3.297],84 on a stretched string [3.273],83 vswr [7.180], 150 on a thin plate [3.268],82 sound [3.317],86 W surface (gravity) [3.320],86 wakes [3.330],87 transverse (shear) Alfven´ [7.284], 158 Warm plasmas, 156 Waves in and out of media, 152 watt (unit), 4 Waves in lossless media, 152 wave equation [2.342],43 Waves in strings and springs,83 wave impedance wavevector (dimensions), 17 acoustic [3.276],83 weber (unit), 4 electromagnetic [7.198], 152 Weber symbols, 126 in a waveguide [7.189], 151 weight (dimensions), 17 Wave mechanics,92 Weiss constant [7.114], 144 Wave speeds,87 Weiss zone equation [6.10], 126 wavefunction Welch window [2.582],60 and expectation value [4.25],91 Weyl equation [4.182], 104 and probability density [4.10],90 Wiedemann-Franz law [6.66], 132 diffracted in 1-D [8.34], 165 Wien’s displacement law [5.189], 121 hydrogenic atom [4.91],96 Wien’s displacement law constant, 9 perturbed [4.160], 102 Wien’s radiation law [5.188], 121 Wavefunctions,90 Wiener-Khintchine theorem waveguide in Fourier transforms [2.492],53 cut-off frequency [7.186], 151 in temporal coherence [8.105], 172 equation [7.185], 151 Wigner coefficients (spin-orbit) [4.136], impedance 100 TE modes [7.189], 151 Wigner coefficients (table of), 99 TM modes [7.188], 151 windowing TEmn modes [7.190], 151 Bartlett [2.581],60 TMmn modes [7.192], 151 Hamming [2.584],60 velocity Hanning [2.583],60 group [7.188], 151 Welch [2.582],60 phase [7.187], 151 wire Waveguides, 151 electric field [7.29], 138 wavelength magnetic flux density [7.34], 138 Compton [7.240], 155 wire loop (inductance) [7.26], 137 de Broglie [4.2],90 wire loop (magnetic flux density) [7.37], photometric, 179 138 redshift [9.86], 184 wires (inductance of parallel) [7.25], 137 thermal de Broglie [5.83], 112 work (dimensions), 17 waves capillary [3.321],86 X electromagnetic, 152 X-ray diffraction, 128 in a spring [3.272],83 in a thin rod [3.271],82 Y I in bulk fluids [3.265],82 yocto, 5 in fluids, 86 yotta, 5 in infinite isotropic solids [3.264],82 Young modulus magnetosonic [7.285], 158 and Lame´ coefficients [3.240],81 main January 23, 2006 16:6

220 Index

and other elastic constants [3.250], 81 Hooke’s law [3.230],80 Young modulus (dimensions), 17 Young’s slits [8.24], 164 Yukawa potential [7.252], 156 Z Zeeman splitting constant, 7 zepto, 5 zero-point energy [4.68],95 zetta, 5 zone law [6.20], 126