The Cambridge Handbook of Physics Formulas

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The Cambridge Handbook of Physics Formulas 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, the Journal 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.

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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.cambridge.org/9780521573498 © Cambridge University Press 2000 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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Contents

Preface

page vii

How to use this book

1

1

3

Units, constants, and conversions 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 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

105

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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

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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 appreciating 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.

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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 Fr¨ uh, 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 LATEX 2ε using the CUP Times fonts. The software packages used were WinEdt, MiKTEX, Mayura Draw, Gnuplot, Ghostscript, Ghostview, and Maple 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.

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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.

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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 on an electron, are now known to a relative standard uncertainty,2 ur , of less than 10−7 . The least well determined is the Newtonian constant of gravitation, presently standing at a rather lamentable ur of 1.5 × 10−3 , and the best is the Rydberg constant (ur = 7.6 × 10−12 ). 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.

1 The

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. 2 The relative standard uncertainty in x is defined as the estimated standard deviation in x divided by the modulus of x (x = 0).

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4

Units, constants, and conversions

1.2

SI units

SI base units physical quantity length mass time interval electric current thermodynamic temperature amount of substance luminous intensity a Or

name metrea kilogram second ampere kelvin mole candela

symbol m kg s A K mol cd

“meter”.

SI derived units physical quantity catalytic activity electric capacitance electric charge electric conductance electric potential difference electric resistance energy, work, heat force frequency illuminance inductance luminous flux magnetic flux magnetic flux density plane angle power, radiant flux pressure, stress radiation absorbed dose radiation dose equivalenta radioactive activity solid angle temperatureb a To

name katal farad coulomb siemens volt ohm joule newton hertz lux henry lumen weber tesla radian watt pascal gray sievert becquerel steradian degree Celsius

symbol kat F C S V Ω J N Hz lx H lm Wb T rad W Pa Gy Sv Bq sr ◦ C

equivalent units mol s−1 C V −1 As Ω−1 J C−1 V A−1 Nm m kg s−2 s−1 cd sr m−2 V A−1 s cd sr Vs V s m−2 m m−1 J s−1 N m−2 J kg−1 [ J kg−1 ] s−1 m2 m−2 K

distinguish it from the gray, units of J kg−1 should not be used for the sievert in practice. Celsius temperature, TC , is defined from the temperature in kelvin, TK , by TC = TK − 273.15.

b The

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5

1.2 SI units

1 SI prefixesa factor 1024 1021 1018 1015 1012 109 106 103 102 101

prefix yotta zetta exa peta tera giga mega kilo hecto decab

symbol Y Z E P T G M k h da

factor 10−24 10−21 10−18 10−15 10−12 10−9 10−6 10−3 10−2 10−1

prefix yocto zepto atto femto pico nano micro milli centi deci

symbol y z a f p n µ m c d

a The

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. b Or “deka”.

Recognised non-SI units physical quantity area energy length

plane angle

pressure time

mass

volume a These

name barn electron volt ˚ angstr¨ om a fermi microna degree arcminute arcsecond bar minute hour day unified atomic mass unit tonnea,b litrec

are non-SI names for SI quantities. “metric ton.” c Or “liter”. The symbol “l” should be avoided. b Or

symbol b eV ˚ A fm µm

bar min h d

SI value 10−28 m2  1.602 18 × 10−19 J 10−10 m 10−15 m 10−6 m (π/180) rad (π/10 800) rad (π/648 000) rad 105 N m−2 60 s 3 600 s 86 400 s

u t l, L

 1.660 54 × 10−27 kg 103 kg 10−3 m3

◦  

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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 permeability of vacuumb permittivity of vacuum constant of gravitationc Planck constant h/(2π) elementary charge magnetic flux quantum, h/(2e) electron volt electron mass proton mass proton/electron mass ratio unified atomic mass unit fine-structure constant, µ0 ce2 /(2h) inverse Rydberg constant, me cα2 /(2h) Avogadro constant Faraday constant, NA e molar gas constant Boltzmann constant, R/NA Stefan–Boltzmann constant, π 2 k 4 /(60¯h3 c2 ) Bohr magneton, e¯ h/(2me ) a By

2.997 924 58 4π =12.566 370 614 . . . 1/(µ0 c2 ) 0 =8.854 187 817 . . . G 6.673(10) h 6.626 068 76(52) ¯h 1.054 571 596(82) e 1.602 176 462(63) 2.067 833 636(81) Φ0 eV 1.602 176 462(63) 9.109 381 88(72) me 1.672 621 58(13) mp mp /me 1 836.152 667 5(39) u 1.660 538 73(13) α 7.297 352 533(27) 1/α 137.035 999 76(50) 1.097 373 156 854 9(83) R∞ 6.022 141 99(47) NA F 9.648 534 15(39) R 8.314 472(15) k 1.380 650 3(24)

×108 m s−1 ×10−7 H m−1 ×10−7 H m−1 F m−1 ×10−12 F m−1 ×10−11 m3 kg−1 s−2 ×10−34 J s ×10−34 J s ×10−19 C ×10−15 Wb ×10−19 J ×10−31 kg ×10−27 kg

σ

5.670 400(40)

×10−8 W m−2 K−4

µB

9.274 008 99(37)

×10−24 J T−1

c µ0

definition, the speed of light is exact. exact, by definition. Alternative units are N A−2 . c The standard acceleration due to gravity, g, is defined as exactly 9.806 65 m s−2 . b Also

×10−27 kg ×10−3 ×107 m−1 ×1023 mol−1 ×104 C mol−1 J mol−1 K−1 ×10−23 J K−1

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1.3 Physical constants

1 General constants speed of light in vacuum permeability of vacuum permittivity of vacuum impedance of free space constant of gravitation Planck constant in eV s h/(2π) in eV s Planck mass, (¯ hc/G)1/2 Planck length, ¯h/(mPl c) = (¯hG/c3 )1/2 Planck time, lPl /c = (¯ hG/c5 )1/2 elementary charge magnetic flux quantum, h/(2e) Josephson frequency/voltage ratio Bohr magneton, e¯ h/(2me ) −1 in eV T µB /k nuclear magneton, e¯ h/(2mp ) −1 in eV T µN /k Zeeman splitting constant

2.997 924 58 4π =12.566 370 614 . . . 1/(µ0 c2 ) 0 =8.854 187 817 . . . µ0 c Z0 =376.730 313 461 . . . G 6.673(10) h 6.626 068 76(52) 4.135 667 27(16) ¯h 1.054 571 596(82) 6.582 118 89(26) mPl 2.176 7(16) lPl 1.616 0(12) tPl 5.390 6(40) e 1.602 176 462(63) 2.067 833 636(81) Φ0 2e/h 4.835 978 98(19) 9.274 008 99(37) µB 5.788 381 749(43) 0.671 713 1(12) 5.050 783 17(20) µN 3.152 451 238(24) 3.658 263 8(64) µB /(hc) 46.686 452 1(19) c µ0

×108 m s−1 ×10−7 H m−1 ×10−7 H m−1 F m−1 ×10−12 F m−1 Ω Ω ×10−11 m3 kg−1 s−2 ×10−34 J s ×10−15 eV s ×10−34 J s ×10−16 eV s ×10−8 kg ×10−35 m ×10−44 s ×10−19 C ×10−15 Wb ×1014 Hz V −1 ×10−24 J T−1 ×10−5 eV T−1 K T−1 ×10−27 J T−1 ×10−8 eV T−1 ×10−4 K T−1 m−1 T−1

Atomic constantsa fine-structure constant, µ0 ce2 /(2h) inverse Rydberg constant, me cα2 /(2h) R∞ c R∞ hc R∞ hc/e Bohr radiusb , α/(4πR∞ ) a See

also the Bohr model on page 95. b Fixed nucleus.

α 1/α R∞

a0

7.297 352 533(27) 137.035 999 76(50) 1.097 373 156 854 9(83) 3.289 841 960 368(25) 2.179 871 90(17) 13.605 691 72(53) 5.291 772 083(19)

×10−3 ×107 m−1 ×1015 Hz ×10−18 J eV ×10−11 m

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8

Units, constants, and conversions

Electron constants electron mass in MeV electron/proton mass ratio electron charge electron specific charge electron molar mass, NA me Compton wavelength, h/(me c) classical electron radius, α2 a0 Thomson cross section, (8π/3)re2 electron magnetic moment in Bohr magnetons, µe /µB in nuclear magnetons, µe /µN electron gyromagnetic ratio, 2|µe |/¯h electron g-factor, 2µe /µB

9.109 381 88(72) ×10−31 kg 0.510 998 902(21) MeV ×10−4 me /mp 5.446 170 232(12) −e −1.602 176 462(63) ×10−19 C ×1011 C kg−1 −e/me −1.758 820 174(71) Me 5.485 799 110(12) ×10−7 kg mol−1 2.426 310 215(18) ×10−12 m λC re 2.817 940 285(31) ×10−15 m σT 6.652 458 54(15) ×10−29 m2 −9.284 763 62(37) ×10−24 J T−1 µe −1.001 159 652 186 9(41) −1 838.281 966 0(39) 1.760 859 794(71) ×1011 s−1 T−1 γe ge −2.002 319 304 3737(82) me

Proton constants proton mass in MeV proton/electron mass ratio proton charge proton specific charge proton molar mass, NA mp proton Compton wavelength, h/(mp c) proton magnetic moment in Bohr magnetons, µp /µB in nuclear magnetons, µp /µN proton gyromagnetic ratio, 2µp /¯ h

1.672 621 58(13) 938.271 998(38) 1 836.152 667 5(39) 1.602 176 462(63) 9.578 834 08(38) 1.007 276 466 88(13) 1.321 409 847(10) 1.410 606 633(58) 1.521 032 203(15) 2.792 847 337(29) 2.675 222 12(11)

×10−27 kg MeV

1.674 927 16(13) 939.565 330(38) mn /me 1 838.683 655 0(40) mn /mp 1.001 378 418 87(58) Mn 1.008 664 915 78(55) 1.319 590 898(10) λC,n −9.662 364 0(23) µn µn /µB −1.041 875 63(25) µn /µN −1.913 042 72(45) 1.832 471 88(44) γn

×10−27 kg MeV

mp mp /me e e/mp Mp λC,p µp

γp

×10−19 C ×107 C kg−1 ×10−3 kg mol−1 ×10−15 m ×10−26 J T−1 ×10−3 ×108 s−1 T−1

Neutron constants neutron mass in MeV neutron/electron mass ratio neutron/proton mass ratio neutron molar mass, NA mn neutron Compton wavelength, h/(mn c) neutron magnetic moment in Bohr magnetons in nuclear magnetons neutron gyromagnetic ratio, 2|µn |/¯h

mn

×10−3 kg mol−1 ×10−15 m ×10−27 J T−1 ×10−3 ×108 s−1 T−1

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9

1.3 Physical constants

1 Muon and tau constants 1.883 531 09(16) 105.658 356 8(52) 3.167 88(52) mτ 1.777 05(29) mµ /me 206.768 262(30) −e −1.602 176 462(63) −4.490 448 13(22) µµ 4.841 970 85(15) 8.890 597 70(27) −2.002 331 832 0(13) gµ mµ

muon mass in MeV tau mass in MeV muon/electron mass ratio muon charge muon magnetic moment in Bohr magnetons, µµ /µB in nuclear magnetons, µµ /µN muon g-factor

×10−28 kg MeV ×10−27 kg ×103 MeV ×10−19 C ×10−26 J T−1 ×10−3

Bulk physical constants Avogadro constant atomic mass constanta in MeV Faraday constant molar gas constant Boltzmann constant, R/NA in eV K−1 molar volume (ideal gas at stp)b Stefan–Boltzmann constant, π 2 k 4 /(60¯h3 c2 ) Wien’s displacement law constant,c b = λm T

NA mu F R k Vm σ b

6.022 141 99(47) 1.660 538 73(13) 931.494 013(37) 9.648 534 15(39) 8.314 472(15) 1.380 650 3(24) 8.617 342(15) 22.413 996(39) 5.670 400(40) 2.897 768 6(51)

×1023 mol−1 ×10−27 kg MeV ×104 C mol−1 J mol−1 K−1 ×10−23 J K−1 ×10−5 eV K−1 ×10−3 m3 mol−1 ×10−8 W m−2 K−4 ×10−3 m K

mass of 12 C/12. Alternative nomenclature for the unified atomic mass unit, u. temperature and pressure (stp) are T = 273.15 K (0◦ C) and P = 101 325 Pa (1 standard atmosphere). c See also page 121. a=

b Standard

Mathematical constants pi (π) exponential constant (e) Catalan’s constant Euler’s constanta (γ) Feigenbaum’s constant (α) Feigenbaum’s constant (δ) Gibbs constant golden mean Madelung constantb a See

also Equation (2.119). structure.

b NaCl

3.141 2.718 0.915 0.577 2.502 4.669 1.851 1.618 1.747

592 281 965 215 907 201 937 033 564

653 828 594 664 875 609 051 988 594

589 459 177 901 095 102 982 749 633

793 045 219 532 892 990 466 894 182

238 235 015 860 822 671 170 848 190

462 360 054 606 283 853 361 204 636

643 287 603 512 902 203 053 586 212

383 471 514 090 873 820 370 834 035

279 352 932 082 218 466 157 370 544

... ... ... ... ... ... ... ... ...

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10

1.4

Units, constants, and conversions

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 “∗ ” in the 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 abampere abcoulomb abfarad abhenry abmho abohm abvolt acre amagat (at stp) ampere hour ˚ angstr¨ om apostilb arcminute arcsecond are astronomical unit atmosphere (standard) atomic mass unit

value in SI units 10.0∗ A 10.0∗ C 1.0∗ ×109 F 1.0∗ ×10−9 H ∗ 1.0 ×109 S ∗ 1.0 ×10−9 Ω ∗ 10.0 ×10−9 V 4.046 856 ×103 m2 44.614 774 mol m−3 ∗ 3.6 ×103 C ∗ ×10−12 m 100.0 ∗ 1.0 lm m−2 290.888 2 ×10−6 rad 4.848 137 ×10−6 rad ∗ 100.0 m2 149.597 9 ×109 m ∗ 101.325 0 ×103 Pa 1.660 540 ×10−27 kg

bar barn baromil barrel (UK) barrel (US dry) barrel (US liquid) barrel (US oil) baud bayre biot bolt (US) brewster British thermal unit bushel (UK) bushel (US) butt (UK) cable (US) calorie

100.0∗ 100.0∗ 750.1 163.659 2 115.627 1 119.240 5 158.987 3 1.0∗ 100.0∗ 10.0 36.576∗ 1.0∗ 1.055 056 36.36 872 35.23 907 477.339 4 219.456∗ 4.186 8∗

×103 Pa ×10−30 m2 ×10−6 m ×10−3 m3 ×10−3 m3 ×10−3 m3 ×10−3 m3 s−1 ×10−3 Pa A m ×10−12 m2 N−1 ×103 J ×10−3 m3 ×10−3 m3 ×10−3 m3 m J continued on next page . . .

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1.4 Converting between units

1 unit name candle power (spherical) carat (metric) cental centare centimetre of Hg (0 ◦ C) centimetre of H2 O (4 ◦ C) chain (engineers’) chain (US) Chu clusec cord cubit cumec cup (US) curie

value in SI units 4π lm 200.0∗ ×10−6 kg 45.359 237 kg 1.0∗ m2 1.333 222 ×103 Pa 98.060 616 Pa 30.48∗ m 20.116 8∗ m 1.899 101 ×103 J 1.333 224 ×10−6 W 3.624 556 m3 457.2∗ ×10−3 m 1.0∗ m3 s−1 236.588 2 ×10−6 m3 37.0∗ ×109 Bq

darcy day day (sidereal) debye degree (angle) denier digit dioptre Dobson unit dram (avoirdupois) dyne dyne centimetres

986.923 3 86.4∗ 86.164 09 3.335 641 17.453 29 111.111 1 19.05∗ 1.0∗ 10.0∗ 1.771 845 10.0∗ 100.0∗

×10−15 m2 ×103 s ×103 s ×10−30 C m ×10−3 rad ×10−9 kg m−1 ×10−3 m m−1 ×10−6 m ×10−3 kg ×10−6 N ×10−9 J

electron volt ell em emu of capacitance emu of current emu of electric potential emu of inductance emu of resistance E¨ otv¨ os unit esu of capacitance esu of current esu of electric potential esu of inductance esu of resistance erg

160.217 7 1.143∗ 4.233 333 1.0∗ 10.0∗ 10.0∗ 1.0∗ 1.0∗ 1.0∗ 1.112 650 333.564 1 299.792 5 898.755 2 898.755 2 100.0∗

×10−21 J m ×10−3 m ×109 F A ×10−9 V ×10−9 H ×10−9 Ω ×10−9 m s−2 m−1 ×10−12 F ×10−12 A V ×109 H ×109 Ω ×10−9 J

faraday fathom fermi Finsen unit firkin (UK)

96.485 3 1.828 804 1.0∗ 10.0∗ 40.914 81

×103 C m ×10−15 m ×10−6 W m−2 ×10−3 m3 continued on next page . . .

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Units, constants, and conversions

unit name firkin (US) fluid ounce (UK) fluid ounce (US) foot foot (US survey) foot of water (4 ◦ C) footcandle footlambert footpoundal footpounds (force) fresnel funal furlong

value in SI units 34.068 71 ×10−3 m3 28.413 08 ×10−6 m3 29.573 53 ×10−6 m3 ∗ 304.8 ×10−3 m 304.800 6 ×10−3 m 2.988 887 ×103 Pa 10.763 91 lx 3.426 259 cd m−2 42.140 11 ×10−3 J 1.355 818 J 1.0∗ ×1012 Hz 1.0∗ ×103 N 201.168∗ m

g (standard acceleration) gal gallon (UK) gallon (US liquid) gamma gauss gilbert gill (UK) gill (US) gon grade grain gram gram-rad gray

9.806 65∗ 10.0∗ 4.546 09∗ 3.785 412 1.0∗ 100.0∗ 795.774 7 142.065 4 118.294 1 π/200∗ 15.707 96 64.798 91∗ 1.0∗ 100.0∗ 1.0∗

m s−2 ×10−3 m s−2 ×10−3 m3 ×10−3 m3 ×10−9 T ×10−6 T ×10−3 A turn ×10−6 m3 ×10−6 m3 rad ×10−3 rad ×10−6 kg ×10−3 kg J kg−1 J kg−1

hand hartree hectare hefner hogshead horsepower (boiler) horsepower (electric) horsepower (metric) horsepower (UK) hour hour (sidereal) Hubble time Hubble distance hundredweight (UK long) hundredweight (US short)

101.6∗ 4.359 748 10.0∗ 902 238.669 7 9.809 50 746∗ 735.498 8 745.699 9 3.6∗ 3.590 170 440 130 50.802 35 45.359 24

×10−3 m ×10−18 J ×103 m2 ×10−3 cd ×10−3 m3 ×103 W W W W ×103 s ×103 s ×1015 s ×1024 m kg kg

inch inch of mercury (0 ◦ C) inch of water (4 ◦ C)

25.4∗ 3.386 389 249.074 0

×10−3 m ×103 Pa Pa

jansky

10.0∗

×10−27 W m−2 Hz−1 continued on next page . . .

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1.4 Converting between units

1 unit name jar

value in SI units 10/9∗ ×10−9 F

kayser kilocalorie kilogram-force kilowatt hour knot (international)

100.0∗ 4.186 8∗ 9.806 65∗ 3.6∗ 514.444 4

m−1 ×103 J N ×106 J ×10−3 m s−1

lambert langley langmuir league (nautical, int.) league (nautical, UK) league (statute) light year ligne line line (magnetic flux) link (engineers’) link (US) litre lumen (at 555 nm)

10/π ∗ 41.84∗ 133.322 4 5.556∗ 5.559 552 4.828 032 9.460 73∗ 2.256∗ 2.116 667 10.0∗ 304.8∗ 201.168 0 1.0∗ 1.470 588

×103 cd m−2 ×103 J m−2 ×10−6 Pa s ×103 m ×103 m ×103 m ×1015 m ×10−3 m ×10−3 m ×10−9 Wb ×10−3 m ×10−3 m ×10−3 m3 ×10−3 W

maxwell mho micron mil (length) mil (volume) mile (international) mile (nautical, int.) mile (nautical, UK) mile per hour milliard millibar millimetre of Hg (0 ◦ C) minim (UK) minim (US) minute (angle) minute minute (sidereal) month (lunar)

10.0∗ 1.0∗ 1.0∗ 25.4∗ 1.0∗ 1.609 344∗ 1.852∗ 1.853 184∗ 447.04∗ 1.0∗ 100.0∗ 133.322 4 59.193 90 61.611 51 290.888 2 60.0∗ 59.836 17 2.551 444

×10−9 Wb S ×10−6 m ×10−6 m ×10−6 m3 ×103 m ×103 m ×103 m ×10−3 m s−1 ×109 m3 Pa Pa ×10−9 m3 ×10−9 m3 ×10−6 rad s s ×106 s

nit noggin (UK)

1.0∗ 142.065 4

cd m−2 ×10−6 m3

oersted ounce (avoirdupois) ounce (UK fluid) ounce (US fluid)

1000/(4π)∗ 28.349 52 28.413 07 29.573 53

A m−1 ×10−3 kg ×10−6 m3 ×10−6 m3

pace parsec

762.0∗ 30.856 78

×10−3 m ×1015 m continued on next page . . .

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14

Units, constants, and conversions

unit name peck (UK) peck (US) pennyweight (troy) perch phot pica (printers’) pint (UK) pint (US dry) pint (US liquid) point (printers’) poise pole poncelot pottle pound (avoirdupois) poundal pound-force promaxwell psi puncheon (UK)

value in SI units 9.092 18∗ ×10−3 m3 8.809 768 ×10−3 m3 1.555 174 ×10−3 kg ∗ 5.029 2 m 10.0∗ ×103 lx 4.217 518 ×10−3 m 568.261 2 ×10−6 m3 550.610 5 ×10−6 m3 473.176 5 ×10−6 m3 351.459 8∗ ×10−6 m 100.0∗ ×10−3 Pa s 5.029 2∗ m 980.665∗ W 2.273 045 ×10−3 m3 453.592 4 ×10−3 kg 138.255 0 ×10−3 N 4.448 222 N 1.0∗ Wb 6.894 757 ×103 Pa 317.974 6 ×10−3 m3

quad quart (UK) quart (US dry) quart (US liquid) quintal (metric)

1.055 056 1.136 522 1.101 221 946.352 9 100.0∗

×1018 J ×10−3 m3 ×10−3 m3 ×10−6 m3 kg

rad rayleigh rem REN reyn rhe rod roentgen rood (UK) rope (UK) rutherford rydberg

10.0∗ 10/(4π) 10.0∗ 1/4 000∗ 689.5 10.0∗ 5.029 2∗ 258.0 1.011 714 6.096∗ 1.0∗ 2.179 874

×10−3 Gy ×109 s−1 m−2 sr−1 ×10−3 Sv S ×103 Pa s Pa−1 s−1 m ×10−6 C kg−1 ×103 m2 m ×106 Bq ×10−18 J

scruple seam second (angle) second (sidereal) shake shed slug square degree statampere statcoulomb

1.295 978 290.949 8 4.848 137 997.269 6 100.0∗ 100.0∗ 14.593 90 (π/180)2∗ 333.564 1 333.564 1

×10−3 kg ×10−3 m3 ×10−6 rad ×10−3 s ×10−10 s ×10−54 m2 kg sr ×10−12 A ×10−12 C

continued on next page . . .

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1.4 Converting between units

1 unit name statfarad stathenry statmho statohm statvolt stere sth´ene stilb stokes stone

value in SI units 1.112 650 ×10−12 F 898.755 2 ×109 H 1.112 650 ×10−12 S 898.755 2 ×109 Ω 299.792 5 V 1.0∗ m3 1.0∗ ×103 N 10.0∗ ×103 cd m−2 100.0∗ ×10−6 m2 s−1 6.350 293 kg

tablespoon (UK) tablespoon (US) teaspoon (UK) teaspoon (US) tex therm (EEC) therm (US) thermie thou tog ton (of TNT) ton (UK long) ton (US short) tonne (metric ton) torr townsend troy dram troy ounce troy pound tun

14.206 53 14.786 76 4.735 513 4.928 922 1.0∗ 105.506∗ 105.480 4∗ 4.185 407 25.4∗ 100.0∗ 4.184∗ 1.016 047 907.184 7 1.0∗ 133.322 4 1.0∗ 3.887 935 31.103 48 373.241 7 954.678 9

×10−6 m3 ×10−6 m3 ×10−6 m3 ×10−6 m3 ×10−6 kg m−1 ×106 J ×106 J ×106 J ×10−6 m ×10−3 W−1 m2 K ×109 J ×103 kg kg ×103 kg Pa ×10−21 V m2 ×10−3 kg ×10−3 kg ×10−3 kg ×10−3 m3

XU

100.209

×10−15 m

yard year (365 days) year (sidereal) year (tropical)

914.4∗ 31.536∗ 31.558 15 31.556 93

×10−3 m ×106 s ×106 s ×106 s

Temperature conversions TK temperature in kelvin TC temperature in ◦ Celsius

From degrees Celsiusa

TK = TC + 273.15

From degrees Fahrenheit

TK =

TF − 32 + 273.15 1.8

(1.2)

TF

From degrees Rankine

TK =

TR 1.8

(1.3)

TR temperature in ◦ Rankine

a The

(1.1)

temperature in ◦ Fahrenheit

term “centigrade” is not used in SI, to avoid confusion with “10−2 of a degree”.

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1.5

Units, constants, and conversions

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 acceleration action angular momentum angular speed area Avogadro constant bending moment Bohr magneton Boltzmann constant bulk modulus capacitance charge (electric) charge density conductance conductivity couple current current density density electric displacement electric field strength electric polarisability electric polarisation electric potential difference energy energy density entropy Faraday constant force frequency gravitational constant Hall coefficient Hamiltonian heat capacity Hubble constant1 illuminance impedance

symbol a S L, J ω A, S NA Gb µB k, kB K C q ρ G σ G, T I, i J, j ρ D E α P V E, U u S F F ν, f G RH H C H Ev Z

dimensions L T−2 L2 M T−1 L2 M T−1 T−1 L2 N−1 L2 M T−2 L2 I L2 M T−2 Θ−1 L−1 M T−2 L−2 M−1 T4 I2 TI L−3 T I L−2 M−1 T3 I2 L−3 M−1 T3 I2 L2 M T−2 I L−2 I L−3 M L−2 T I L M T−3 I−1 M−1 T4 I2 L−2 T I L2 M T−3 I−1 L2 M T−2 L−1 M T−2 L2 M T−2 Θ−1 T I N−1 L M T−2 T−1 L3 M−1 T−2 L3 T−1 I−1 L2 M T−2 L2 M T−2 Θ−1 T−1 L−2 J L2 M T−3 I−2

SI units m s−2 Js m2 kg s−1 rad s−1 m2 mol−1 Nm J T−1 J K−1 Pa F C C m−3 S S m−1 Nm A A m−2 kg m−3 C m−2 V m−1 C m2 V −1 C m−2 V J J m−3 J K−1 C mol−1 N Hz m3 kg−1 s−2 m3 C−1 J J K−1 s−1 lx Ω

continued on next page . . . Hubble constant is almost universally quoted in units of km s−1 Mpc−1 . There are about 3.1 × 1019 kilometres in a megaparsec.

1 The

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17

1.5 Dimensions

1 physical quantity impulse inductance irradiance Lagrangian length luminous intensity magnetic dipole moment magnetic field strength magnetic flux magnetic flux density magnetic vector potential magnetisation mass mobility molar gas constant moment of inertia momentum number density permeability permittivity Planck constant power Poynting vector pressure radiant intensity resistance Rydberg constant shear modulus specific heat capacity speed Stefan–Boltzmann constant stress surface tension temperature thermal conductivity time velocity viscosity (dynamic) viscosity (kinematic) volume wavevector weight work Young modulus

symbol I L Ee L L, l Iv m, µ H Φ B A M m, M µ R I p n µ  h P S p, P Ie R R∞ µ, G c u, v, c σ σ, τ σ, γ T λ t v, u η, µ ν V, v k W W E

dimensions L M T−1 L2 M T−2 I−2 M T−3 L2 M T−2 L J L2 I L−1 I L2 M T−2 I−1 M T−2 I−1 L M T−2 I−1 L−1 I M M−1 T2 I L2 M T−2 Θ−1 N−1 L2 M L M T−1 L−3 L M T−2 I−2 L−3 M−1 T4 I2 L2 M T−1 L2 M T−3 M T−3 L−1 M T−2 L2 M T−3 L2 M T−3 I−2 L−1 L−1 M T−2 L2 T−2 Θ−1 L T−1 M T−3 Θ−4 L−1 M T−2 M T−2 Θ L M T−3 Θ−1 T L T−1 L−1 M T−1 L2 T−1 L3 L−1 L M T−2 L2 M T−2 L−1 M T−2

SI units Ns H W m−2 J m cd A m2 A m−1 Wb T Wb m−1 A m−1 kg m2 V −1 s−1 J mol−1 K−1 kg m2 kg m s−1 m−3 H m−1 F m−1 Js W W m−2 Pa W sr−1 Ω m−1 Pa J kg−1 K−1 m s−1 W m−2 K−4 Pa N m−1 K W m−1 K−1 s m s−1 Pa s m2 s−1 m3 m−1 N J Pa

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18

1.6

Units, constants, and conversions

Miscellaneous

Greek alphabet A

α

alpha

N

ν

nu

B

β

beta

Ξ

ξ

xi

Γ

γ

gamma

O

o

omicron

∆

δ

delta

Π

π



pi

E



epsilon

P

ρ




rho

Z

ζ

zeta

Σ

σ

ς

sigma

H

η

eta

T

τ

tau

Θ

θ

theta

Υ

υ

upsilon

I

ι

iota

Φ

φ

K

κ

kappa

X

χ

chi

Λ

λ

lambda

Ψ

ψ

psi

M

µ

mu

Ω

ω

omega

ε

ϑ

ϕ

phi

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

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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 is some variation. For example the spherical harmonics, Yl m , 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

Laplacian operator

∇2

derivative

∇ df etc. dx

∂f etc. ∂x n d f dxn
ds

derivative of r with respect to t

partial derivatives nth derivative closed surface integral

˙r


closed loop integral

dl L

matrix

mean value (of x)

x

binomial coefficient

factorial

!

unit imaginary (i2 = −1)

A or aij   n r i

exponential constant

e

modulus (of x)

|x|

natural logarithm

ln

log to base 10

log10

S

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20

2.2

Mathematics

Vectors and matrices

Vector algebra Scalar producta b

Vector product

Product rules

Lagrange’s identity

Scalar triple product

a · b = |a||b|cosθ

(2.1)

  xˆ  a× b = |a||b|sinθ nˆ = ax bx

yˆ ay by

 zˆ  az  bz 

a× b = −b× a a · (b + c) = (a · b) + (a · c) a× (b + c) = (a× b) + (a× c)

(2.4) (2.5) (2.6)

(a× b) · (c× d) = (a · c)(b · d) − (a · d)(b · c)

(2.7)

 ax  (a× b) · c = bx  cx

(2.8)

 az  bz  cz 

= (b× c) · a = (c× a) · b

Vector a with respect to a nonorthogonal basis {e1 ,e2 ,e3 }c a Also

(2.9) (2.10) (2.11)

a× (b× c) = (a · c)b − (a · b)c

(2.12)



b = (c× a)/[(a× b) · c] c = (a× b)/[(a× b) · c]

nˆ (in)

a

(a× b)× c = (a · c)b − (b · c)a

a = (b× c)/[(a× b) · c] Reciprocal vectors

b (2.3)

ay by cy

θ

(2.2)

a·b=b·a

= volume of parallelepiped Vector triple product

a

(2.13) (2.14) (2.15)

(a · a) = (b · b) = (c · c) = 1

(2.16)

a = (e1 · a)e1 + (e2 · a)e2 + (e3 · a)e3

(2.17)

known as the “dot product” or the “inner product.” known as the “cross-product.” nˆ is a unit vector making a right-handed set with a and b. c The prime ( ) denotes a reciprocal vector. b Also

c b

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21

2.2 Vectors and matrices

Common three-dimensional coordinate systems z

2

ρ

point P θ r y x

φ

x = ρcosφ = r sinθ cosφ

(2.18)

y = ρsinφ = r sinθ sinφ

(2.19)

z = r cosθ

(2.20)

coordinate system: coordinates of P : volume element: metric elementsa (h1 ,h2 ,h3 ):

rectangular (x,y,z) dx dy dz (1,1,1)

ρ = (x2 + y 2 )1/2

(2.21)

r = (x2 + y 2 + z 2 )1/2

(2.22)

θ = arccos(z/r)

(2.23)

φ = arctan(y/x)

(2.24)

spherical polar (r,θ,φ) r2 sinθ dr dθ dφ (1,r,r sinθ)

cylindrical polar (ρ,φ,z) ρ dρ dz dφ (1,ρ,1)

a In

an orthogonal coordinate system (parameterised by coordinates q1 ,q2 ,q3 ), the differential line element dl is obtained from (dl)2 = (h1 dq1 )2 + (h2 dq2 )2 + (h3 dq3 )2 .

Gradient Rectangular coordinates

∇f =

∂f ∂f ∂f xˆ + yˆ + zˆ ∂x ∂y ∂z

(2.25)

Cylindrical coordinates

∇f =

1 ∂f ˆ ∂f ∂f ρˆ + φ + zˆ ∂ρ r ∂φ ∂z

(2.26)

Spherical polar coordinates

∇f =

1 ∂f ˆ 1 ∂f ˆ ∂f rˆ + θ+ φ ∂r r ∂θ r sinθ ∂φ

(2.27)

General orthogonal coordinates

∇f =

qˆ 1 ∂f qˆ 2 ∂f qˆ 3 ∂f + + h1 ∂q1 h2 ∂q2 h3 ∂q3

(2.28)

f ˆ

scalar field unit vector

ρ

distance from the z axis

qi hi

basis metric elements

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22

Mathematics

Divergence Rectangular coordinates

∇·A=

∂Ax ∂Ay ∂Az + + ∂x ∂y ∂z

(2.29)

Cylindrical coordinates

∇·A=

1 ∂(ρAρ ) 1 ∂Aφ ∂Az + + ρ ∂ρ ρ ∂φ ∂z

(2.30)

Spherical polar coordinates

∇·A=

General orthogonal coordinates

1 ∂(Aθ sinθ) 1 ∂Aφ 1 ∂(r2 Ar ) + + r2 ∂r r sinθ ∂θ r sinθ ∂φ (2.31)  ∂ ∂ 1 (A1 h2 h3 ) + (A2 h3 h1 ) ∇·A= h1 h2 h3 ∂q1 ∂q2  ∂ (A3 h1 h2 ) (2.32) + ∂q3

A Ai

vector field ith component of A

ρ

distance from the z axis

qi

basis

hi

metric elements

ˆ

unit vector

A

vector field

Ai

ith component of A

ρ

distance from the z axis

qi

basis

hi

metric elements

Curl Rectangular coordinates

Cylindrical coordinates

Spherical polar coordinates

General orthogonal coordinates

  xˆ   ∇× A = ∂/∂x   Ax   ρ/ρ ˆ  ∇× A = ∂/∂ρ   Aρ

 zˆ   ∂/∂z   Az 

yˆ ∂/∂y Ay φˆ ∂/∂φ ρAφ

 rˆ /(r2 sinθ)   ∇× A =  ∂/∂r   Ar

 zˆ /ρ   ∂/∂z   Az 

ˆ sinθ) θ/(r

  qˆ h  1 1 1  ∇× A = ∂/∂q1 h1 h2 h3   h1 A1

(2.33)

∂/∂θ rAθ qˆ 2 h2 ∂/∂q2 h2 A2

(2.34)   ˆ φ/r   ∂/∂φ   rAφ sinθ  qˆ 3 h3   ∂/∂q3   h3 A3 

(2.35)

(2.36)

Radial formsa r ∇r = r ∇·r =3

(2.37) (2.38)

∇r2 = 2r

(2.39)

∇ · (rr) = 4r

(2.40)

a Note

−r r3 1 ∇ · (r/r2 ) = 2 r −2r 2 ∇(1/r ) = 4 r ∇ · (r/r3 ) = 4πδ(r)

∇(1/r) =

that the curl of any purely radial function is zero. δ(r) is the Dirac delta function.

(2.41) (2.42) (2.43) (2.44)

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2.2 Vectors and matrices

Laplacian (scalar) Rectangular ∂2 f ∂2 f ∂2 f 2 f = + + (2.45) ∇ coordinates ∂x2 ∂y 2 ∂z 2   Cylindrical 1 ∂ ∂f 1 ∂2 f ∂2 f 2 f = + (2.46) ∇ ρ + coordinates ρ ∂ρ ∂ρ ρ2 ∂φ2 ∂z 2     Spherical ∂ ∂2 f 1 ∂ 1 ∂f 1 2 2 ∂f f = ∇ r + sinθ + polar 2 2 2 r ∂r ∂r r sinθ ∂θ ∂θ r2 sin θ ∂φ2 coordinates (2.47)      ∂ h2 h3 ∂f h3 h1 ∂f 1 ∂ + ∇2 f = General h1 h2 h3 ∂q1 h1 ∂q1 ∂q2 h2 ∂q2 orthogonal   h1 h2 ∂f ∂ coordinates + (2.48) ∂q3 h3 ∂q3

f scalar field ρ distance from the z axis

qi basis hi metric elements

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)

∇× (A× B) ≡ A(∇ · B) − B(∇ · A) + (B · ∇)A − (A · ∇)B

(2.54)

∇ · (∇f) ≡ ∇ f ≡ f

(2.55)

∇× (∇f) ≡ 0

(2.56)

∇ · (∇× A) ≡ 0

(2.57)

∇× (∇× A) ≡ ∇(∇ · A) − ∇2 A

(2.58)

2

f,g

scalar fields

A,B

vector fields

A dV

vector field volume element

Sc V

closed surface volume enclosed

S ds L

surface surface element loop bounding S

dl

line element

f,g

scalar fields

Vector integral transformations Gauss’s (Divergence) theorem Stokes’s theorem




(∇ · A) dV =


 (∇× A) · ds =

Green’s second theorem

A · dl

S

(2.60)

L

 (f∇g) · ds =

S

V

=


(2.59)

Sc


Green’s first theorem

A · ds

V

∇ · (f∇g) dV

(2.61)

[f∇2 g + (∇f) · (∇g)] dV

(2.62)

V



[f(∇g) − g(∇f)] · ds = S

(f∇2 g − g∇2 f) dV V

(2.63)

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Mathematics

Matrix algebraa



Matrix definition

a11  a21  A= .  ..

a12 a22 .. .

am1

am2

C = A + B if

Matrix addition

C = AB Matrix multiplication

··· ··· ··· ···

 a1n a2n   ..  . 

(2.64)

A m by n matrix aij matrix elements

amn

cij = aij + bij

(2.65)

if cij = aik bkj

(2.66)

(AB)C = A(BC) A(B + C) = AB + AC

(2.67) (2.68)

Transpose matrixb

˜ aij = aji  = N... ˜ B ˜A ˜ (AB...N)

(2.69)

Adjoint matrix (definition 1)c

˜∗ A† = A (AB...N)† = N† ...B† A†

(2.71) (2.72)

Hermitian matrixd

H† = H

(2.73)

(2.70)

˜ aij transpose matrix  (sometimes aT ij , or aij ) ∗ †

complex conjugate (of each component) adjoint (or Hermitian conjugate)

H Hermitian (or self-adjoint) matrix

examples:  a11  A = a21 a31  a11  ˜ A = a12 a13

a12

a13



a22

 a23 

a32

a33

a21

a31

b31 

a22

 a32 

a23

a33

 a11 b11 + a12 b21 + a13 b31  AB = a21 b11 + a22 b21 + a23 b31 a31 b11 + a32 b21 + a33 b31 a Terms

 b11  B = b21

b12

b13



b22

 b23 

b32

b33

 a11 + b11  A + B = a21 + b21 a31 + b31

a12 + b12

a13 + b13



a22 + b22

 a23 + b23 

a32 + b32

a33 + b33

a11 b12 + a12 b22 + a13 b32 a21 b12 + a22 b22 + a23 b32 a31 b12 + a32 b22 + a33 b32

a11 b13 + a12 b23 + a13 b33



 a21 b13 + a22 b23 + a23 b33  a31 b13 + a32 b23 + a33 b33 

are implicitly summed over repeated suffices; hence aik bkj equals k aik bkj . also Equation (2.85). c Or “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)]. d Hermitian matrices must also be square (see next table). b See

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2.2 Vectors and matrices

Square matricesa Trace

Determinantb

Adjoint matrix (definition 2)c

Inverse matrix (detA = 0)

trA = aii tr(AB) = tr(BA)

(2.74) (2.75)

A

square matrix

aij aii

matrix elements  implicitly = i aii

detA = ijk... a1i a2j a3k ... = (−1)i+1 ai1 Mi1 = ai1 Ci1

(2.76) (2.77) (2.78)

tr det Mij

trace determinant (or |A|) minor of element aij

det(AB...N) = detAdetB...detN

(2.79)

Cij

cofactor of the element aij

˜ ij = Cji adjA = C

adj

(2.80) ∼

adjoint (sometimes ˆ written A) transpose

1

unit matrix

δjk

Kronecker delta (= 1 if i = j, = 0 otherwise)

U

unitary matrix Hermitian conjugate

adjA Cji = detA detA AA−1 = 1

a−1 ij =

(AB...N) Orthogonality condition

examples:  a11  A = a21

a12

a31

a32

a22

(2.82) −1

...B A

−1

(2.83) (2.84) (2.85)

˜ A is symmetric A = A, ˜ A is antisymmetric A = −A,

If

Unitary matrix

=N

−1

aij aik = δjk ˜ = A−1 i.e., A If

Symmetry

−1

(2.81)

(2.86) (2.87)

U† = U−1 a13

(2.88)



 a23  a33

trA = a11 + a22 + a33

B=

 b11

b12

b21

b22

†



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 1  −1 A = −a21 a33 + a23 a31 detA a21 a32 − a22 a31   b22 −b12 1 −1 B = detB −b21 b11

−a12 a33 + a13 a32 a11 a33 − a13 a31 −a11 a32 + a12 a31

a12 a23 − a13 a22



 −a11 a23 + a13 a21  a11 a22 − a12 a21

 are implicitly summed over repeated suffices; hence aik bkj equals k aik bkj . b ijk... is defined as the natural extension of Equation (2.443) to n-dimensions (see page 50). Mij is the determinant of the matrix A with the ith row and the jth column deleted. The cofactor Cij = (−1)i+j Mij . c Or “adjugate matrix.” See the footnote to Equation (2.71) for a discussion of the term “adjoint.” a Terms

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Mathematics

Commutators Commutator definition

[A,B] = AB − BA = −[B,A]

(2.89)

[·,·] commutator

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)

(2.94)

σi 1 i

Pauli spin matrices 2 × 2 unit matrix i2 = −1

δij

Kronecker delta

Pauli matrices

  0 1 1 0   1 0 σ3 = 0 −1 σ1 =

Pauli matrices

 0 i  1 1= 0

σ2 =

 −i 0  0 1

Anticommutation

σ i σ j + σ j σ i = 2δij 1

(2.95)

Cyclic permutation

σ i σ j = iσ k

(2.96)

2

(σ i ) = 1

(2.97)

Rotation matricesa

  1 0 0 cosθ sinθ  R1 (θ) = 0 0 −sinθ cosθ   cosθ 0 −sinθ 1 0  R2 (θ) =  0 sinθ 0 cosθ   cosθ sinθ 0 R3 (θ) =  −sinθ cosθ 0 0 0 1

Rotation about x1 Rotation about x2 Rotation about x3 Euler angles



Ri (θ)

matrix for rotation about the ith axis

θ

rotation angle

α β γ

rotation about x3 rotation about x2 rotation about x3

R

rotation matrix

(2.98)

(2.99)

(2.100)

cosγ cosβ cosα − sinγ sinα cosγ cosβ sinα + sinγ cosα R(α,β,γ) = −sinγ cosβ cosα − cosγ sinα −sinγ cosβ sinα + cosγ cosα sinβ cosα sinβ sinα

a Angles

 −cosγ sinβ sinγ sinβ  cosβ (2.101)

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.

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2.3 Series, summations, and progressions

2.3

Series, summations, and progressions

Progressions and summations

Arithmetic progression

Sn = a + (a + d) + (a + 2d) + ··· + [a + (n − 1)d] n = [2a + (n − 1)d] 2 n = (a + l) 2

(2.103)

a d

number of terms sum of n successive terms first term common difference

(2.104)

l

last term

r

common ratio

(2.102)

n Sn

Sn = a + ar + ar2 + ··· + arn−1 1 − rn =a 1−r a S∞ = (|r| < 1) 1−r

(2.106)

Arithmetic mean

1 x a = (x1 + x2 + ··· + xn ) n

(2.108)

. a arithmetic mean

Geometric mean

x g = (x1 x2 x3 ...xn )1/n

(2.109)

. g geometric mean

(2.110)

. h harmonic mean

Geometric progression



Harmonic mean

1 1 1 + + ··· + x h = n x1 x2 xn

Relative mean magnitudes

x a ≥ x g ≥ x h n  i=1 n  i=1 n 

Summation formulas

i=1 n  i=1 ∞  i=1 ∞  i=1 ∞ 

Euler’s constanta a γ  0.577215664...

(2.105)

(2.107)

−1

if xi > 0 for all i

(2.111)

n i = (n + 1) 2

(2.112)

n i2 = (n + 1)(2n + 1) 6

(2.113)

i3 =

n2 (n + 1)2 4

(2.114)

i4 =

n (n + 1)(2n + 1)(3n2 + 3n − 1) 30

(2.115)

1 1 1 (−1)i+1 = 1 − + − + ... = ln2 i 2 3 4

(2.116)

1 1 1 π (−1)i+1 = 1 − + − + ... = 2i − 1 3 5 7 4

(2.117)

π2 1 1 1 1 = 1 + + + + ... = 2 i 4 9 16 6 i=1   1 1 1 γ = lim 1 + + + ··· + − lnn n→∞ 2 3 n

i

dummy integer

γ

Euler’s constant

(2.118) (2.119)

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Mathematics

Power series n(n − 1) 2 n(n − 1)(n − 2) 3 x + x + ··· 2! 3!

Binomial seriesa

(1 + x)n = 1 + nx +

Binomial coefficientb

n

Binomial theorem

(a + b)n =

Taylor series (about a)c

f(a + x) = f(a) + xf (1) (a) +

Taylor series (3-D)

f(a + x) = f(a) + (x · ∇)f|a +

Maclaurin series

f(x) = f(0) + xf (1) (0) +

(2.120)

  n n! Cr ≡ ≡ r r!(n − r)!

(2.121)

n    n n−k k a b k

(2.122)

k=0

x2 (2) xn−1 (n−1) f (a) + ··· + f (a) + ··· 2! (n − 1)!

(2.123)

(x · ∇)2 (x · ∇)3 f|a + f|a + ··· 2! 3!

(2.124)

x2 (2) xn−1 (n−1) f (0) + ··· + f (0) + ··· 2! (n − 1)!

(2.125)

a If n is a positive integer the series terminates and is valid for all x. Otherwise the (infinite) series is convergent for |x| < 1. b The coefficient of xr in the binomial series. c xf (n) (a) is x times the nth derivative of the function f(x) with respect to x evaluated at a, taken as well behaved around a. (x · ∇)n f|a is its extension to three dimensions.

Limits nc xn → 0 as xn /n! → 0

as

n → ∞ if n→∞

|x| < 1 (for any fixed c)

(2.126)

(for any fixed x)

(2.127)

(1 + x/n)n → ex

as

xlnx → 0 as

x→0

(2.129)

sinx → 1 as x

x→0

(2.130)

If

n→∞

f(a) = g(a) = 0 or ∞

then

(2.128)

f(x) f (1) (a) = (1) x→a g(x) g (a) lim

ˆ (l’Hopital’s rule)

(2.131)

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2.3 Series, summations, and progressions

Series expansions exp(x)

1+x+

ln(1 + x)

x−

 ln

1+x 1−x



x2 x3 + + ··· 2! 3!

(2.132)

(for all x)

(2.133)

(−1 < x ≤ 1)

  x3 x5 x7 2 x + + + + ··· 3 5 7

(2.134)

(|x| < 1)

x2 x3 x4 + − + ··· 2 3 4

cos(x)

1−

x2 x4 x6 + − + ··· 2! 4! 6!

(2.135)

(for all x)

sin(x)

x−

x3 x5 x7 + − + ··· 3! 5! 7!

(2.136)

(for all x)

tan(x)

x+

x3 2x5 17x7 + + ··· 3 15 315

(2.137)

(|x| < π/2)

sec(x)

1+

x2 5x4 61x6 + + + ··· 2 24 720

(2.138)

(|x| < π/2)

csc(x)

31x5 1 x 7x3 + + + + ··· x 6 360 15120

(2.139)

(|x| < π)

cot(x)

1 x x3 2x5 − − − − ··· x 3 45 945

(2.140)

(|x| < π)

arcsin(x)a

x+

(2.141)

(|x| < 1)

arctan(x)b

cosh(x)

1 x3 1 · 3 x5 1 · 3 · 5 x7 + + ··· 2 3 2·4 5 2·4·6 7

 x3 x5 x7   x − + − + ···    3 5 7   1 π 1 1 − + 3 − 5 + ···  2 x 3x 5x     1 1 1 π  − − + − + ··· 2 x 3x3 5x5 x2 x4 x6 1 + + + + ··· 2! 4! 6!

(|x| ≤ 1)

(2.142)

(x > 1) (x < −1)

(2.143)

(for all x)

sinh(x)

x+

x3 x5 x7 + + + ··· 3! 5! 7!

(2.144)

(for all x)

tanh(x)

x−

x3 2x5 17x7 + − + ··· 3 15 315

(2.145)

(|x| < π/2)

a arccos(x) = π/2 − arcsin(x).

b arccot(x) = π/2 − arctan(x).

Note that arcsin(x) ≡ sin−1 (x) etc.

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Mathematics

Inequalities Triangle inequality

|a1 | − |a2 | ≤ |a1 + a2 | ≤ |a1 | + |a2 |;  n  n      ≤ a |ai |  i   i=1

Cauchy inequality

2.4

(2.148)

b1 ≥ b2 ≥ b3 ≥ ... ≥ bn  n  n  n    then n ai bi ≥ ai bi and

 n 

i=1



b

i=1

2 ≤

ai bi

i=1

Schwarz inequality

(2.147)

i=1

a1 ≥ a2 ≥ a3 ≥ ... ≥ an

if Chebyshev inequality

(2.146)

n 

a2i

i=1

n 

(2.149) (2.150)

i=1

b2i

(2.151)

i=1

2  b  b f(x)g(x) dx ≤ [f(x)]2 dx [g(x)]2 dx

a

a

(2.152)

a

Complex variables

Complex numbers z

complex variable

Cartesian form

z = x + iy

(2.153)

i x,y

i2 = −1 real variables

Polar form

z = reiθ = r(cosθ + isinθ)

(2.154)

r θ

amplitude (real) phase (real)

Modulusa

|z| = r = (x2 + y 2 )1/2 |z1 · z2 | = |z1 | · |z2 |

(2.155) (2.156)

|z|

modulus of z

argz

argument of z

z∗

complex conjugate of z = reiθ

n

integer

y x arg(z1 z2 ) = argz1 + argz2

θ = argz = arctan Argument

Complex conjugate

Logarithmb a Or

z ∗ = x − iy = re−iθ arg(z ∗ ) = −argz ∗

z · z = |z|

2

lnz = lnr + i(θ + 2πn)

“magnitude.” principal value of lnz is given by n = 0 and −π < θ ≤ π.

b The

(2.157) (2.158) (2.159) (2.160) (2.161) (2.162)

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2.4 Complex variables

Complex analysisa Cauchy– Riemann equationsb Cauchy– Goursat theoremc Cauchy integral formulad

Laurent expansione

if f(z) = u(x,y) + iv(x,y) ∂u ∂v = then ∂x ∂y ∂u ∂v =− ∂y ∂x
f(z) dz = 0

a Closed

(2.164)

complex variable i2 = −1

x,y real variables f(z) function of z u,v real functions

(2.165)

c


f(z) 1 f(z0 ) = dz 2πi c z − z0
f(z) n! dz f (n) (z0 ) = 2πi c (z − z0 )n+1 f(z) =

∞ 

an (z − z0 )n

(2.166) (2.167) (2.168)

(n)

nth derivative

an Laurent coefficients a−1 residue of f(z) at z0 z dummy variable

y

c2

n=−∞


f(z  ) 1 dz   2πi c (z − z0 )n+1
 f(z) dz = 2πi enclosed residues where an =

Residue theorem

(2.163)

z i

c1 (2.169) (2.170)

c

z0 x

c

contour integrals are taken in the counterclockwise sense, once. condition for f(z) to be analytic at a given point. c If 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 z . 0 e Of f(z), (analytic) in the annular region between concentric circles, c and c , centred on z . c is any closed curve 1 2 0 in this region encircling z0 . b Necessary

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2.5

Mathematics

Trigonometric and hyperbolic formulas

Trigonometric relationships sin(A ± B) = sinAcosB ± cosAsinB

(2.171)

cos(A ± B) = cosAcosB ∓ sinAsinB

(2.172)

tanA ± tanB 1 ∓ tanAtanB

1 cosAcosB = [cos(A + B) + cos(A − B)] 2 1 sinAcosB = [sin(A + B) + sin(A − B)] 2 1 sinAsinB = [cos(A − B) − cos(A + B)] 2

(2.173) (2.174) (2.175) 2

cos2 A + sin2 A = 1

(2.177)

1

sec2 A − tan2 A = 1

(2.178)

0

csc A − cot A = 1

(2.179)

sin2A = 2sinAcosA

(2.180)

cos2A = cos A − sin A

(2.181)

2

tan2A =

2

2tanA 1 − tan2 A

(2.183)

cos3A = 4cos3 A − 3cosA

(2.184)

tan x

−6

−4

−2

0

2

4

6

4

6

x

(2.185) (2.186)

4 2

(2.187) (2.188)

tx co

1 cos A = (1 + cos2A) 2 1 2 sin A = (1 − cos2A) 2 1 cos3 A = (3cosA + cos3A) 4 1 sin3 A = (3sinA − sin3A) 4 2

−2

sec x

A−B A+B cos 2 2 A+B A−B sinA − sinB = 2cos sin 2 2 A+B A−B cosA + cosB = 2cos cos 2 2 A+B A−B cosA − cosB = −2sin sin 2 2

−1

(2.182)

sin3A = 3sinA − 4sin3 A

sinA + sinB = 2sin

x

2

sin

2

cos x

(2.176)

csc x

tan(A ± B) =

0 −2 −4

(2.189)

−6

−4

−2

0 x

(2.190) (2.191) (2.192)

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2.5 Trigonometric and hyperbolic formulas

Hyperbolic relationshipsa sinh(x ± y) = sinhxcoshy ± coshxsinhy

(2.193)

cosh(x ± y) = coshxcoshy ± sinhxsinhy

(2.194)

tanh(x ± y) =

tanhx ± tanhy 1 ± tanhxtanhy

1 coshxcoshy = [cosh(x + y) + cosh(x − y)] 2 1 sinhxcoshy = [sinh(x + y) + sinh(x − y)] 2 1 sinhxsinhy = [cosh(x + y) − cosh(x − y)] 2

2

(2.195) (2.196) (2.197) (2.198) 4 co

cosh2 x − sinh2 x = 1

(2.199)

2

sech2 x + tanh2 x = 1

(2.200)

0

coth2 x − csch2 x = 1

(2.201)

−2

sinh2x = 2sinhxcoshx

(2.202)

cosh2x = cosh2 x + sinh2 x

(2.203)

sh

x

tanhx

sin

hx

tanhx

tanh2x =

2tanhx 1 + tanh2 x

(2.205)

cosh3x = 4cosh x − 3coshx

(2.206)

x−y x+y cosh 2 2 x+y x−y sinhx − sinhy = 2cosh sinh 2 2 x+y x−y coshx + coshy = 2cosh cosh 2 2 x+y x−y coshx − coshy = 2sinh sinh 2 2 sinhx + sinhy = 2sinh

1 cosh2 x = (cosh2x + 1) 2 1 sinh2 x = (cosh2x − 1) 2 1 cosh3 x = (3coshx + cosh3x) 4 1 sinh3 x = (sinh3x − 3sinhx) 4 a These

−3

−2

−1

0

1

2

3

x

(2.204)

sinh3x = 3sinhx + 4sinh3 x 3

−4

(2.207) 4

(2.208)

2

cothx

sech x

(2.209)

0

cs

−2

(2.210)

−4 −3

(2.211) (2.212) (2.213) (2.214)

can be derived from trigonometric relationships by using the substitutions cosx → coshx and sinx → isinhx.

ch x

−2

−1

0 x

1

2

3

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34

Mathematics

Trigonometric and hyperbolic definitions de Moivre’s theorem

(cosx + isinx)n = einx = cosnx + isinnx

(2.215)

cosx =

1  ix −ix  e +e 2

(2.216)

coshx =

1  x −x  e +e 2

(2.217)

sinx =

1  ix −ix  e −e 2i

(2.218)

sinhx =

1  x −x  e −e 2

(2.219)

tanx =

sinx cosx

(2.220)

tanhx =

sinhx coshx

(2.221)

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 

in

cs

ar

nx arcta

x

1

0

(2.234)

x

1.6

(2.235) (2.236)

π − arcsinx 2

(2.237)

1

in the angle range 0 ≤ θ ≤ π/2. Note that arcsinx ≡ sin−1 x etc.

0

arccsc x

a Valid

1

(2.233)

  1 x

arccotx = arctan arccosx =

(2.232)

ar cc os x

x cot arc

 x arcsinx = arctan (1 − x2 )1/2   (1 − x2 )1/2 arccosx = arctan x   1 arccscx = arctan (x2 − 1)1/2   arcsecx = arctan (x2 − 1)1/2

1

cx arcse

2

3 x

4

5

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35

2.6 Mensuration

Inverse hyperbolic functions

  arsinhx ≡ sinh−1 x = ln x + (x2 + 1)1/2 (2.238)

2





(2.239)

0

arsi

−1

1+x 1 artanhx ≡ tanh−1 x = ln (2.240) 2 1−x   x+1 1 −1 arcothx ≡ coth x = ln (2.241) 2 x−1   1 (1 − x2 )1/2 −1 + arsechx ≡ sech x = ln x x (2.242)   1 (1 + x2 )1/2 + arcschx ≡ csch−1 x = ln x x (2.243)

|x| < 1

−2

−1

0 x

4 3

arcoth x

0<x≤1

2

x = 0

ar

1

arc

se

Mensuration

Moir´e fringesa −1  1 1 dM =  −  d1 d2

Rotational patternb

d dM = 2|sin(θ/2)|

(2.244)

dM

a From b From

overlapping linear gratings. identical gratings, spacing d, with a relative rotation θ.

Moir´e fringe spacing

d1,2 grating spacings d

(2.245)

1

|x| > 1

sch

x

ch x 1 x

Parallel pattern

2

nh x

0

2.6

x

arc

x≥1

nh

ta

ar

x

  arcoshx ≡ cosh−1 x = ln x + (x2 − 1)1/2

1

osh

for all x

θ

common grating spacing relative rotation angle (|θ| ≤ π/2)

2

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36

Mathematics

Plane triangles Sine formulaa

Cosine formulas

a b c = = sinA sinB sinC

(2.246)

a2 = b2 + c2 − 2bccosA

(2.247)

b +c −a 2bc a = bcosC + ccosB

(2.248)

2

2

cosA =

Tangent formula

C A−B a−b = cot tan 2 a+b 2

Area

1 = absinC 2 a2 sinB sinC = 2 sinA = [s(s − a)(s − b)(s − c)]1/2 1 where s = (a + b + c) 2 area

a The

C

2

(2.249)

b

a A

B c

(2.250) (2.251) (2.252) (2.253) (2.254)

diameter of the circumscribed circle equals a/sinA.

Spherical trianglesa Sine formula Cosine formulas Analogue formula

sinb sinc sina = = sinA sinB sinC

(2.255)

cosa = cosbcosc + sinbsinccosA

(2.256)

cosA = −cosB cosC + sinB sinC cosa

(2.257)

sinacosB = cosbsinc − sinbcosccosA

(2.258)

Four-parts formula

cosacosC = sinacotb − sinC cotB

(2.259)

Areab

E =A+B +C −π

(2.260)

a On

a unit sphere. called the “spherical excess.”

b Also

a

b

C A

B c

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37

2.6 Mensuration

Perimeter, area, and volume Perimeter of circle

P = 2πr

(2.261)

P r

perimeter radius

Area of circle

A = πr2

(2.262)

A

area

Surface area of spherea

A = 4πR 2

(2.263)

R

sphere radius

Volume of sphere

4 V = πR 3 3

(2.264)

V

volume

a

semi-major axis

b E

semi-minor axis elliptic integral of the second kind (p. 45) eccentricity (= 1 − b2 /a2 )

P = 4aE(π/2, e)  2 1/2 a + b2  2π 2

(2.265)

Area of ellipse

A = πab

(2.267)

Volume of ellipsoidc

V = 4π

Surface area of cylinder

b

Perimeter of ellipse

(2.266)

abc 3

e

2

(2.268)

c

third semi-axis

A = 2πr(h + r)

(2.269)

h

height

Volume of cylinder

V = πr2 h

(2.270)

Area of circular coned

A = πrl

(2.271)

l

slant height

Volume of cone or pyramid

V = Ab h/3

(2.272)

Ab

base area

Surface area of torus

A = π 2 (r1 + r2 )(r2 − r1 )

(2.273)

r1

inner radius

r2

outer radius

Volume of torus

V=

Aread of spherical cap, depth d

A = 2πRd

(2.275)

d

cap depth

(2.276)

Ω z α

solid angle distance from centre half-angle subtended

Volume of spherical cap, depth d

π2 2 2 (r − r )(r2 − r1 ) 4 2 1

 2

V = πd



Solid angle of a circle from a point on its axis, z from centre

d R− 3

(2.274)



z Ω = 2π 1 − 2 2 1/2 (z + r ) = 2π(1 − cosα)

 (2.277) (2.278)

defined by x2 + y 2 + z 2 = R 2 . approximation is exact when e = 0 and e  0.91, giving a maximum error of 11% at e = 1. c Ellipsoid defined by x2 /a2 + y 2 /b2 + z 2 /c2 = 1. d Curved surface only. a Sphere b The

α z

r

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38

Mathematics

Conic sections y

y

y

b

equation parametric form foci

x

x

a

a

a

parabola

ellipse

hyperbola

y 2 = 4ax

x2 y 2 + =1 a2 b2

x2 y 2 − =1 a2 b2

x = acost y = bsint √ (± a2 − b2 ,0)

x = ±acosht y = bsinht √ (± a2 + b2 ,0)

x = t2 /(4a) y=t (a,0)

eccentricity

e=1

directrices

x = −a

√

√ a2 − b2 e= a a x=± e

e=

a2 + b2 a

x=±

a e

x

Platonic solidsa solid (faces,edges,vertices)

tetrahedron (4,6,4) cube (6,12,8) octahedron (8,12,6) dodecahedron (12,30,20) icosahedron (20,30,12) a Of

volume

surface area

√ a3 2 12

√ a 3

a3

6a2

√ a3 2 3

√ 2a2 3

√ a3 (15 + 7 5) 4 √ 5a3 (3 + 5) 12

2

 3a2

√ 5(5 + 2 5)

√ 5a2 3

circumradius

inradius

√ a 6 4 √ a 3 2

√ a 6 12

a √ 2

a √ 6  √ a 50 + 22 5 4 5    5 a √ 3+ 4 3

√ a√ 3(1 + 5) 4 a 4



√ 2(5 + 5)

a 2

side a. Both regular and irregular polyhedra follow the Euler relation, faces − edges + vertices = 2.

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39

2.6 Mensuration

Curve measure Length of plane curve Surface of revolution Volume of revolution Radius of curvature

 b

 1+

l= a



dy dx



b

dx 

y 1+

A = 2π a



start point

(2.279)

b y(x) l

end point plane curve length

dy dx

(2.280)

A

surface area

(2.281)

V

volume

ρ

radius of curvature

2 1/2 dx

b

y 2 dx

V =π 

a

2 1/2

a



ρ= 1+

dy dx

2 3/2 

d2 y dx2

−1 (2.282)

Differential geometrya τ r

tangent curve parameterised by r(t)

v

|˙r (t)|

(2.284)

n

principal normal

(2.285)

b

binormal

|˙r× r¨ | |˙r |3

(2.286)

κ

curvature

ρ=

1 κ

(2.287)

ρ

radius of curvature

λ=

... r˙ · (¨r× r ) |˙r× r¨ |2

(2.288)

λ

torsion

Unit tangent

τˆ =

r˙ r˙ = |˙r | v

(2.283)

Unit principal normal

r¨ −˙v τˆ nˆ = |¨r −˙v τˆ |

Unit binormal

bˆ = τˆ × nˆ

Curvature

κ=

Radius of curvature Torsion

nˆ

Frenet’s formulas

τ˙ˆ = κv nˆ

(2.289)

˙ nˆ = −κv τˆ + λv bˆ

(2.290)

˙ˆ b = −λv nˆ

(2.291)

osculating plane normal plane

bˆ

τˆ rectifying plane

r origin

a For

a continuous curve in three dimensions, traced by the position vector r(t).

2

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40

2.7

Mathematics

Differentiation

Derivatives (general) Power Product Quotient Function of a functiona

Leibniz theorem

Differentiation under the integral sign

General integral Logarithm Exponential

Inverse functions

a The

“chain rule.”

d n du (u ) = nun−1 dx dx dv du d (uv) = u +v dx dx dx d u ! 1 du u dv = − dx v v dx v 2 dx d du d [f(u)] = [f(u)] · dx du dx   n   n d u n dv dn−1 u dn [uv] = v + + ··· 0 dxn 1 dx dxn−1 dxn   n   k n−k n d v n d v d u + ··· + u + k n−k n dxn k dx dx   q d f(x) dx = f(q) (p constant) dq p   q d f(x) dx = −f(p) (q constant) dp p   v(x) du d dv − f(u) f(t) dt = f(v) dx u(x) dx dx d (logb |ax|) = (xlnb)−1 dx d ax (e ) = aeax dx  −1 dy dx = dy dx  −3 d2 x d2 y dy = − dy 2 dx2 dx     2 −5 d2 y dy d3 x dy d3 y = 3 − dy 3 dx2 dx dx3 dx

(2.292)

n

power index

(2.293)

u,v

functions of x

(2.294) (2.295)

f(u) function of u(x)

n

(2.296)

k

binomial coefficient

(2.297) (2.298) (2.299) (2.300) (2.301) (2.302) (2.303) (2.304)

b a

log base constant

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41

2.7 Differentiation

Trigonometric derivativesa d (sinax) = acosax dx

(2.305)

d (cosax) = −asinax dx

(2.306)

d (tanax) = asec2 ax dx

(2.307)

d (cscax) = −acscax · cotax dx

(2.308)

d (secax) = asecax · tanax dx

(2.309)

d (cotax) = −acsc2 ax dx

(2.310)

d (arcsinax) = a(1 − a2 x2 )−1/2 dx

(2.311)

d (arccosax) = −a(1 − a2 x2 )−1/2 dx

(2.312)

d (arctanax) = a(1 + a2 x2 )−1 dx

(2.313)

a 2 2 d (arccscax) = − (a x − 1)−1/2 dx |ax|

(2.314)

d (arccotax) = −a(a2 x2 + 1)−1 dx

(2.316)

a 2 2 d (arcsecax) = (a x − 1)−1/2 (2.315) dx |ax| aa

is a constant.

Hyperbolic derivativesa d (sinhax) = acoshax dx

(2.317)

d (coshax) = asinhax dx

(2.318)

d (tanhax) = asech2 ax dx

(2.319)

d (cschax) = −acschax · cothax dx

(2.320)

d (sechax) = −asechax · tanhax dx

(2.321)

d (cothax) = −acsch2 ax dx

(2.322)

d (arsinhax) = a(a2 x2 + 1)−1/2 dx

(2.323)

d (arcoshax) = a(a2 x2 − 1)−1/2 dx

(2.324)

d (artanhax) = a(1 − a2 x2 )−1 dx

(2.325)

a d (arcschax) = − (1 + a2 x2 )−1/2 (2.326) dx |ax|

a d (arsechax) = − (1 − a2 x2 )−1/2 dx |ax| (2.327) aa

is a constant.

d (arcothax) = a(1 − a2 x2 )−1 dx

(2.328)

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42

Mathematics

Partial derivatives Total differential Reciprocity Chain rule

Jacobian

Change of variable

∂f ∂f ∂f dx + dy + dz ∂x ∂y ∂z    ∂g  ∂x  ∂y  = −1 ∂x y ∂y g ∂g x

df =

f

f(x,y,z)

(2.330)

g

g(x,y)

J

Jacobian

u v w

u(x,y,z) v(x,y,z) w(x,y,z)

V V

volume in (x,y,z) volume in (u,v,w) mapped to by V

∂f ∂f ∂x ∂f ∂y ∂f ∂z = + + (2.331) ∂u ∂x ∂u ∂y ∂u ∂z ∂u    ∂x ∂x ∂x     ∂u ∂v ∂w   ∂(x,y,z)  ∂y ∂y ∂y  (2.332) = J=  ∂(u,v,w)  ∂u ∂v ∂w   ∂z ∂z ∂z      ∂u ∂v ∂w   f(x,y,z) dxdy dz = f(u,v,w)J dudv dw V

V

 Euler– Lagrange equation

(2.329)

if

b

I=

(2.333)

F(x,y,y  ) dx

a

then

d ∂F = ∂y dx

δI = 0 when



∂F ∂y 

y dy/dx a,b fixed end points

 (2.334)

Stationary pointsa

saddle point

maximum

minimum

∂f ∂f = = 0 at (x0 ,y0 ) ∂x ∂y Additional sufficient conditions Stationary point if

(necessary condition)

(2.335)

 2 2 ∂ f ∂2 f ∂2 f > 2 2 ∂x ∂y ∂x∂y  2 ∂2 f ∂2 f ∂2 f > ∂x2 ∂y 2 ∂x∂y 2

for maximum

∂2 f < 0, ∂x2

and

for minimum

∂2 f > 0, ∂x2

and

for saddle point

 2 ∂ f ∂2 f ∂2 f < 2 2 ∂x ∂y ∂x∂y

a Of

quartic minimum

a function f(x,y) at the point (x0 ,y0 ). Note that at, for example, a quartic minimum

(2.336) (2.337) (2.338) ∂2 f ∂x2

2

= ∂∂yf2 = 0.

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43

2.7 Differentiation

Differential equations Laplace

∇2 f = 0

(2.339)

f

f(x,y,z)

Diffusiona

∂f = D∇2 f ∂t

(2.340)

D

diffusion coefficient

Helmholtz

∇2 f + α2 f = 0

(2.341)

α

constant

Wave

∇2 f =

(2.342)

c

wave speed

(2.343)

l

integer

(2.344)

m

integer

1 ∂2 f c2 ∂t2

Associated Legendre

  d 2 dy (1 − x ) + l(l + 1)y = 0 dx dx     d m2 2 dy y=0 (1 − x ) + l(l + 1) − dx dx 1 − x2

Bessel

x2

Hermite

d2 y dy + 2αy = 0 − 2x dx2 dx

Laguerre

x

d2 y dy + αy = 0 + (1 − x) dx2 dx

(2.347)

Associated Laguerre

x

dy d2 y + αy = 0 + (1 + k − x) dx2 dx

(2.348)

k

integer

Chebyshev

(1 − x2 )

(2.349)

n

integer

Euler (or Cauchy)

x2

(2.350)

a,b constants

Bernoulli

dy + p(x)y = q(x)y a dx

(2.351)

p,q functions of x

Airy

d2 y = xy dx2

(2.352)

Legendre

d2 y dy + (x2 − m2 )y = 0 +x 2 dx dx

d2 y dy + n2 y = 0 −x dx2 dx

d2 y dy + by = f(x) + ax 2 dx dx

(2.345) (2.346)

a Also 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.

2

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44

Mathematics

2.8

Integration

Standard formsa 



xn+1 x dx = n+1 n





 u dv = [uv] −

v du

 (n = −1)

(2.355)

(2.357)

lnax dx = x(lnax − 1)

(2.359)





 





aa

  x2 1 xlnax dx = lnax − 2 2 1 1 dx = ln(a + bx) a + bx b −1 1 dx = 2 (a + bx) b(a + bx)

1 x dx = ln|x2 ± a2 | x2 ± a2 2

(2.363)  (2.365) 

 (2.369)

x dx = (x2 ± a2 )1/2 (2.373) 2 (x ± a2 )1/2

ax





x 1 − a a2



f  (x) dx = lnf(x) f(x) bax dx =



x! 1 (2.371) dx = arcsin a (a2 − x2 )1/2

and b are non-zero constants.





  1  xn  1 (2.367) dx = ln x(xn + a) an  xn + a 

(2.356)

xe dx = e

(2.361)

 dv dx (2.354) u dx dx

1 dx = ln|x| x ax







u dx −



1 e dx = eax a ax

 

 uv dx = v

(2.353)

bax alnb

(2.358)

(2.360)

(b > 0)

(2.362)

1 1 a + bx dx = − ln x(a + bx) a x

(2.364)

  bx 1 1 dx = arctan 2 2 2 a +b x ab a

(2.366)

  1 1  x − a  ln dx = x2 − a2 2a  x + a 

(2.368)

x −1 dx = (x2 ± a2 )n 2(n − 1)(x2 ± a2 )n−1

(2.370)

1 dx = ln|x + (x2 ± a2 )1/2 | (x2 ± a2 )1/2

(2.372)

x! 1 1 arcsec dx = a a x(x2 − a2 )1/2

(2.374)

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45

2.8 Integration

Trigonometric and hyperbolic integrals 



sinx dx = −cosx

(2.375)





cosx dx = sinx

(2.377)

tanx dx = −ln|cosx|

(2.379)

coshx dx = sinhx

(2.378)

tanhx dx = ln(coshx)

(2.380)

 x   cscx dx = ln tan  2

(2.381)

 x   cschx dx = ln tanh  2

(2.382)

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)

  



  sinmx · sinnx dx = 

sin(m − n)x sin(m + n)x − 2(m − n) 2(m + n)

sinmx · cosnx dx = −  cosmx · cosnx dx =

Error function Complementary error function

sin(m − n)x sin(m + n)x + 2(m − n) 2(m + n)

Gamma function

(2.389)

0

2 π 1/2



(2.390) ∞

exp(−t2 ) dt

(2.391)

x

 x πt2 πt2 dt; S(x) = dt sin 2 2 0 0  1/2  π 1+i erf (1 − i)x C(x) + i S(x) = 2 2  x t e dt (x > 0) Ei(x) = −∞ t  ∞ Γ(x) = tx−1 e−t dt (x > 0) x

cos



φ

F(φ,k) = 

0

1 (1 − k 2 sin2 θ)1/2

0

E(φ,k) = also page 167.

(m2 = n2 )

(2.388)

x

erfc(x) = 1 − erf(x) =

0

Elliptic integrals (trigonometric form)

(m2 = n2 )

(2.387)

exp(−t2 ) dt

π 1/2



Exponential integral



2

erf(x) =

C(x) =

Fresnel integralsa

(m2 = n2 )

cos(m − n)x cos(m + n)x − 2(m − n) 2(m + n)

Named integrals

a See

(2.376)



 

sinhx dx = coshx

φ

dθ

(1 − k 2 sin2 θ)1/2 dθ

(first kind) (second kind)

(2.392) (2.393) (2.394) (2.395) (2.396) (2.397)

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Mathematics

Definite integrals 

∞

e−ax dx = 2

0



∞

1 π !1/2 2 a

xe−ax dx =

1 2a

xn e−ax dx =

n! an+1

2

0



∞

0

(a > 0)

(2.398)

(a > 0)

(2.399)

(a > 0; n = 0,1,2,...)

(2.400)

 2 b π !1/2 exp(2bx − ax ) dx = exp (a > 0) a a −∞ "  ∞ 1 · 3 · 5 · ... · (n − 1)(2a)−(n+1)/2 (π/2)1/2 n > 0 and even n −ax2 x e dx = n > 1 and odd 2 · 4 · 6 · ... · (n − 1)(2a)−(n+1)/2 0  1 p!q! (p,q integers > 0) xp (1 − x)q dx = (p + q + 1)! 0  ∞  ∞ 1 π !1/2 2 cos(ax ) dx = sin(ax2 ) dx = (a > 0) 2 2a 0 0  ∞ 2  ∞ sinx sin x π dx = dx = 2 x x 2 0 0  ∞ 1 π (0 < a < 1) dx = a (1 + x)x sinaπ 0 

2.9

∞

2

(2.401)

(2.402) (2.403) (2.404) (2.405) (2.406)

Special functions and polynomials

Gamma function Definition

 Γ(z) =

∞

tz−1 e−t dt

[(z) > 0]

(2.407)

(n = 0,1,2,...)

(2.408)

0

n! = Γ(n + 1) = nΓ(n) Relations

Stirling’s formulas (for |z|,n  1)

1/2

Γ(1/2) = π   z Γ(z + 1) z! = = w w!(z − w)! Γ(w + 1)Γ(z − w + 1)   1 1 + − ··· Γ(z)  e−z z z−(1/2) (2π)1/2 1 + 12z 288z 2

(2.409) (2.410) (2.411)

n!  nn+(1/2) e−n (2π)1/2

(2.412)

ln(n!)  nlnn − n

(2.413)

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2.9 Special functions and polynomials

Bessel functions x !ν  (−x2 /4)k Jν (x) = 2 k!Γ(ν + k + 1)

Jν (x)

∞

Series expansion

Yν (x) =

Jν (x)cos(πν) − J−ν (x) sin(πν)

Approximations "  x ν 1

π  2   2 1/2 sin x − 12 νπ − π4 πx

(0 ≤ x  ν) (x  ν) (0 < x  ν) (x  ν)

Iν (x) = (−i)ν Jν (ix) π Kν (x) = iν+1 [Jν (ix) + iYν (ix)] 2 π !1/2 jν (x) = Jν+ 1 (x) 2 2x

Modified Bessel functions Spherical Bessel function

(2.415)

order (ν ≥ 0)

ν 1

Jν (x)  Γ(ν+1)  2   2 1/2 cos x − 12 νπ − π4 πx "   −Γ(ν) x −ν Yν (x)  

(2.414)

k=0

Bessel function of the first kind Yν (x) Bessel function of the second kind Γ(ν) Gamma function J0

0.5 J1

(2.416)

0 −0.5

(2.417) (2.418) (2.419)

−1 0 Iν (x)

Y0 Y1 2

4

x

6

8

10

modified Bessel function of the first kind

Kν (x) modified Bessel function of the second kind jν (x)

(2.420)

spherical Bessel function of the first kind [similarly for yν (x)]

Legendre polynomialsa d2 Pl (x) dPl (x) + l(l + 1)Pl (x) = 0 − 2x dx2 dx (2.421)

Legendre equation

(1 − x2 )

Rodrigues’ formula

Pl (x) =

Recurrence relation

(l + 1)Pl+1 (x) = (2l + 1)xPl (x) − lPl−1 (x) 

1

Orthogonality −1

Explicit form

1 dl 2 (x − 1)l 2l l! dxl

Pl (x)Pl  (x) dx =

Pl (x) = 2−l

l/2  m=0

Expansion of plane wave

Kronecker delta

l

binomial coefficients

(2.423) (2.424)

   l 2l − 2m l−2m (2.425) x m l

exp(ikz) = exp(ikr cosθ) ∞  (2l + 1)il jl (kr)Pl (cosθ) = l=0

δll 

(2.422)

2 δll  2l + 1

(−1)m

l

Legendre polynomials order (l ≥ 0)

Pl

m

(2.426)

k z

(2.427)

jl

wavenumber propagation axis z = r cosθ spherical Bessel function of the first kind (order l)

P0 (x) = 1

P2 (x) = (3x2 − 1)/2

P4 (x) = (35x4 − 30x2 + 3)/8

P1 (x) = x

P3 (x) = (5x3 − 3x)/2

P5 (x) = (63x5 − 70x3 + 15x)/8

a Of

the first kind.

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Mathematics

Associated Legendre functionsa Associated Legendre equation

    d dP m (x) m2 P m (x) = 0 (1 − x2 ) l + l(l + 1) − dx dx 1 − x2 l (2.428)

Plm associated Legendre functions

dm Pl (x), 0 ≤ m ≤ l dxm (l − m)! m P (x) Pl−m (x) = (−1)m (l + m)! l

Pl

Legendre polynomials

!!

5!! = 5 · 3 · 1 etc.

δll 

Kronecker delta

Plm (x) = (1 − x2 )m/2

From Legendre polynomials

Recurrence relations

(2.429) (2.430)

m (x) = x(2m + 1)Pmm (x) Pm+1

(2.431)

Pmm (x) = (−1)m (2m − 1)!!(1 − x2 )m/2

(2.432)

m m (l − m + 1)Pl+1 (x) = (2l + 1)xPlm (x) − (l + m)Pl−1 (x)



(2.433) 1

Orthogonality −1

Plm (x)Plm (x) dx =

(l + m)! 2 δll  (l − m)! 2l + 1

(2.434)

P00 (x) = 1

P10 (x) = x

P11 (x) = −(1 − x2 )1/2

P20 (x) = (3x2 − 1)/2

P21 (x) = −3x(1 − x2 )1/2

P22 (x) = 3(1 − x2 )

a Of

the first kind. Plm (x) can be defined with a (−1)m factor in Equation (2.429) as well as Equation (2.430).

Legendre polynomials 1 P3

2

P1

2

P2

0.5

associated Legendre functions

P2

P0

3

P

P4

2

P5

0

1

P00

1

P 0

−0.5

2

0

P1

0

1

P1

−1

−1 −1

−0.5

0 x

0.5

1

−1

0 x

1

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2.9 Special functions and polynomials

Spherical harmonics

   1 ∂ ∂ 1 ∂2 Yl m + l(l + 1)Yl m = 0 sinθ + 2 sinθ ∂θ ∂θ sin θ ∂φ2 (2.435)  1/2 2l + 1 (l − m)! Yl m (θ,φ) = (−1)m Plm (cosθ)eimφ 4π (l + m)! (2.436)



Differential equation

Definitiona



2π



π

φ=0 θ=0

f(θ,φ) =

Plm

associated Legendre functions

Y∗

complex conjugate Kronecker delta

m

Yl (θ,φ)Yl  (θ,φ)sinθ dθ dφ = δmm δll  (2.437)

∞  l 

alm Yl m (θ,φ)

l=0 m=−l  2π

Laplace series

spherical harmonics



m∗

Orthogonality

Yl m



δll 

(2.438) f

continuous function

ψ

continuous function

a,b

constants

π

where alm = φ=0 θ=0

Yl m∗ (θ,φ)f(θ,φ)sinθ dθ dφ (2.439)

if

Solution to Laplace equation

∇2 ψ(r,θ,φ) = 0,

ψ(r,θ,φ) =

∞ 

l 

then $ # Yl m (θ,φ) · alm rl + blm r−(l+1)

l=0 m=−l



1 4π 

Y00 (θ,φ) = Y1±1 (θ,φ) = ∓

 Y2±1 (θ,φ) = ∓ Y30 (θ,φ) =

1 2

Y3±2 (θ,φ) =

1 4

 

3 sinθ e±iφ 8π 15 sinθ cosθ e±iφ 8π

7 (5cos2 θ − 3)cosθ 4π 105 2 sin θ cosθ e±2iφ 2π

(2.440)



3 cosθ 4π    5 3 1 cos2 θ − Y20 (θ,φ) = 4π 2 2  15 sin2 θ e±2iφ Y2±2 (θ,φ) = 32π  1 21 ±1 sinθ(5cos2 θ − 1)e±iφ Y3 (θ,φ) = ∓ 4 4π  1 35 3 ±3iφ ±3 sin θ e Y3 (θ,φ) = ∓ 4 4π Y10 (θ,φ) =

a Defined for −l ≤ m ≤ l, using the sign convention of the Condon–Shortley phase. Other sign conventions are possible.

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Mathematics

Delta functions Kronecker delta

Threedimensional Levi–Civita symbol (permutation tensor)a

" 1 if i = j δij = 0 if i = j δii = 3

(2.442)

123 = 231 = 312 = 1 132 = 213 = 321 = −1

(2.443)

δij Kronecker delta i,j,k,... indices (= 1,2 or 3)

all other ijk = 0 ijk klm = δil δjm − δim δjl δij ijk = 0 ilm jlm = 2δij

(2.444) (2.445) (2.446)

ijk ijk = 6

(2.447)



b

a

 Dirac delta function

(2.441)

b

" 1 if a < 0 < b δ(x) dx = 0 otherwise f(x)δ(x − x0 ) dx = f(x0 )

(2.449)

δ(x − x0 )f(x) = δ(x − x0 )f(x0 ) δ(−x) = δ(x)

(2.450) (2.451)

δ(ax) = |a|−1 δ(x)

(2.452)

δ(x)  nπ

e

(a = 0) (n  1)

Levi–Civita symbol (see also page 25)

δ(x) f(x)

Dirac delta function smooth function of x

a,b

constants

(2.448)

a

−1/2 −n2 x2

ijk

(2.453)

general symbol ijk... is defined to be +1 for even permutations of the suffices, −1 for odd permutations, and 0 if a suffix is repeated. The sequence (1,2,3,... ,n) is taken to be even. Swapping adjacent suffices an odd (or even) number of times gives an odd (or even) permutation. a The

2.10

Roots of quadratic and cubic equations

Quadratic equations (a = 0)

Equation

ax2 + bx + c = 0

Solutions

√ −b ± b2 − 4ac x1,2 = 2a −2c √ = b ± b2 − 4ac

Solution combinations

x1 + x2 = −b/a x1 x2 = c/a

(2.454)

x a,b,c

variable real constants

x1 ,x2

quadratic roots

(2.455) (2.456) (2.457) (2.458)

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2.10 Roots of quadratic and cubic equations

Cubic equations Equation

ax3 + bx2 + cx + d = 0 (a = 0)  3c b2 − 2 a a  3  9bc 27d 1 2b − 2 + q= 27 a3 a a ! ! 3 2 q p + D= 3 2

p= Intermediate definitions

1 3



If D ≥ 0, also define: !1/3 −q + D1/2 u= (2.463) 2 !1/3 −q − D1/2 (2.464) v= 2 y1 = u + v (2.465) u − v 1/2 −(u + v) y2,3 = ±i 3 (2.466) 2 2 1 real, 2 complex roots (if D = 0: 3 real roots, at least 2 equal) Solutionsa

Solution combinations ay

n

(2.459)

xn = yn −

b 3a

2

(2.460) (2.461)

D

discriminant

(2.462) If D < 0, also define:   −3/2  −q |p| φ = arccos 2 3  1/2 |p| φ y1 = 2 cos 3 3  1/2 |p| φ±π y2,3 = −2 cos 3 3

(2.467) (2.468) (2.469)

3 distinct real roots (2.470)

x1 + x2 + x3 = −b/a

(2.471)

x1 x2 + x1 x3 + x2 x3 = c/a x1 x2 x3 = −d/a

(2.472) (2.473)

are solutions to the reduced equation y 3 + py + q = 0.

x variable a,b,c,d real constants

xn

cubic roots (n = 1,2,3)

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Mathematics

2.11

Fourier series and transforms

Fourier series nπx nπx ! a0  an cos + + bn sin 2 L L n=1  L 1 nπx dx an = f(x)cos L −L L  1 L nπx dx bn = f(x)sin L −L L   ∞  inπx cn exp f(x) = L n=−∞    L −inπx 1 f(x)exp cn = dx 2L −L L  L ∞  a2 1   2 1 an + b2n |f(x)|2 dx = 0 + 2L −L 4 2 ∞

(2.474)

f(x) = Real form

Complex form

Parseval’s theorem

(2.475)

=

|cn |

periodic function, period 2L

an ,bn

Fourier coefficients

cn

complex Fourier coefficient

||

modulus

(2.476) (2.477) (2.478) (2.479)

n=1

∞ 

f(x)

2

(2.480)

n=−∞

Fourier transforma



∞

F(s) = −∞ ∞

Definition 1 f(x) =

f(x)e−2πixs dx

(2.481)

2πixs

(2.482)

F(s)e

ds

f(x) F(s)

−∞



∞

f(x)e−ixs dx  1 ∞ f(x) = F(s)eixs ds 2π −∞  ∞ 1 f(x)e−ixs dx F(s) = √ 2π −∞  ∞ 1 f(x) = √ F(s)eixs ds 2π −∞ F(s) =

Definition 2

Definition 3

a All

(2.483)

−∞

(2.484) (2.485) (2.486)

three (and more) definitions are used, but definition 1 is probably the best.

function of x Fourier transform of f(x)

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2.11 Fourier series and transforms

Fourier transform theoremsa 

Convolution

f(x) ∗ g(x) =

∞ −∞

f(u)g(x − u) du

(2.487)

f,g general functions ∗ convolution

Convolution rules

f ∗g =g ∗f f ∗ (g ∗ h) = (f ∗ g) ∗ h

(2.488) (2.489)

f

Convolution theorem

f(x)g(x)   F(s) ∗ G(s)

(2.490)

 

Autocorrelation

f ∗ (x) f(x) =

(2.491)

f∗

complex conjugate of f

Wiener– Khintchine theorem

2 f ∗ (x) f(x)   |F(s)|

Crosscorrelation

f ∗ (x) g(x) =

Correlation theorem

∗ h(x) j(x)   H(s)J (s)

h,j H

real functions H(s)   h(x) J(s)   j(x)

Parseval’s relationb Parseval’s theoremc

Derivatives



∞

−∞



∞

−∞



∞

∞

−∞

 |f(x)| dx = 2

Fourier transform relation correlation

(2.492)

f ∗ (u − x)g(u) du

f(x)g ∗ (x) dx =

−∞



f ∗ (u − x)f(u) du

g

f(x)   F(s)  G(s) g(x) 



(2.493) (2.494)

J ∞

F(s)G∗ (s) ds

(2.495)

−∞ ∞

−∞

|F(s)|2 ds

(2.496)

df(x)  (2.497)  2πisF(s) dx df(x) dg(x) d [f(x) ∗ g(x)] = ∗ g(x) = ∗ f(x) dx dx dx (2.498)

a Defining

the Fourier transform as F(s) = called the “power theorem.” c Also called “Rayleigh’s theorem.” b Also

%∞

−2πixs −∞ f(x)e

dx.

Fourier symmetry relationships f(x) even odd real, even real, odd imaginary, even complex, even complex, odd real, asymmetric imaginary, asymmetric

                   

F(s) even odd real, even imaginary, odd imaginary, even complex, even complex, odd complex, Hermitian complex, anti-Hermitian

definitions real: f(x) = f ∗ (x) imaginary: f(x) = −f ∗ (x) even: f(x) = f(−x) odd: f(x) = −f(−x) Hermitian: f(x) = f ∗ (−x) anti-Hermitian: f(x) = −f ∗ (−x)

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Mathematics

Fourier transform pairsa 

∞

f(x)e−2πisx dx

 

F(s) =

f(ax)

 

1 F(s/a) |a|

f(x − a)

 

e−2πias F(s)

 

(2πis)n F(s)

(2.502)

 

1

(2.503)

δ(x − a)

 

e−2πias

(2.504)

e−a|x|

 

xe−a|x|

 

e−x

/a2

 

sinax

 

cosax

 

δ(x − ma)

 

f(x)

(2.499)

−∞

(a = 0, real) (a real)

(2.500) (2.501)

n

d f(x) dxn δ(x)

∞ 

2

m=−∞

" 0 x<0 f(x) = (“step”) 1 x>0 " 1 |x| ≤ a f(x) = (“top hat”) 0 |x| > a    1 − |x| |x| ≤ a a f(x) = (“triangle”)  0 |x| > a a Equation

2a a2 + 4π 2 s2

(a > 0)

8iπas (a > 0) (a2 + 4π 2 s2 )2 √ 2 2 2 a πe−π a s 1 a ! a ! δ s− −δ s+ 2i 2π 2π  ! a a ! 1 δ s− +δ s+ 2 2π 2π ∞ !  1 n δ s− a n=−∞ a

(2.505) (2.506) (2.507) (2.508) (2.509) (2.510)

 

1 i δ(s) − 2 2πs

(2.511)

 

sin2πas = 2asinc2as πs

(2.512)

 

1 (1 − cos2πas) = asinc2 as (2.513) 2π 2 as2

(2.499) defines the Fourier transform used for these pairs. Note that sincx ≡ (sinπx)/(πx).

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2.12 Laplace transforms

2.12

Laplace transforms

Laplace transform theorems Definitiona

 &

Convolutionb

Inversec

∞

F(s) = L{f(t)} = F(s) · G(s) = L

f(t)e−st dt

0 ∞

L{}

Laplace transform

(2.515)

F(s) G(s)

L{f(t)} L{g(t)}

(2.516)

∗

convolution

γ

constant

n

integer > 0

a

constant

u(t)

unit step function

(2.514) '

f(t − z)g(z) dz

0

= L{f(t) ∗ g(t)}  γ+i∞ 1 f(t) = est F(s) ds 2πi γ−i∞  = residues (for t > 0) &

dn f(t) dtn

'

n−1 

(2.517) (2.518) dr f(t)   dtr t=0 (2.519)

Transform of derivative

L

Derivative of transform

dn F(s) = L{(−t)n f(t)} dsn

(2.520)

Substitution

F(s − a) = L{eat f(t)}

(2.521) (2.522)

Translation

e−as F(s) = L{u(t − a)f(t − a)} " 0 (t < 0) where u(t) = 1 (t > 0)

= sn L{f(t)} −

r=0

sn−r−1

(2.523)

|e−s0 t f(t)| is finite for sufficiently large t, the Laplace transform exists for s > s0 . known as the “faltung (or folding) theorem.” c Also known as the “Bromwich integral.” γ is chosen so that the singularities in F(s) are left of the integral line. a If

b Also

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Mathematics

Laplace transform pairs



f(t) =⇒ F(s) = L{f(t)} =

∞

f(t)e−st dt

(2.524)

0

δ(t) =⇒ 1 1 =⇒ 1/s

(2.525) (s > 0)

n! (s > 0, n > −1) sn+1  π 1/2 t =⇒ 4s3  π −1/2 =⇒ t s 1 eat =⇒ (s > a) s−a 1 teat =⇒ (s > a) (s − a)2 s (1 − at)e−at =⇒ (s + a)2 2 t2 e−at =⇒ (s + a)3 a sinat =⇒ 2 (s > 0) s + a2 s cosat =⇒ 2 (s > 0) s + a2 a sinhat =⇒ 2 (s > a) s − a2 s (s > a) coshat =⇒ 2 s − a2 a e−bt sinat =⇒ (s + b)2 + a2 s+b e−bt cosat =⇒ (s + b)2 + a2 tn =⇒

e−at f(t) =⇒ F(s + a)

(2.526) (2.527) (2.528) (2.529) (2.530) (2.531) (2.532) (2.533) (2.534) (2.535) (2.536) (2.537) (2.538) (2.539) (2.540)

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2.13 Probability and statistics

2.13

Probability and statistics

2

Discrete statistics x =

Mean

1 N

N 

xi

(2.541)

(2.542)

var[·] unbiased variance

(2.543)

σ

i=1

1  (xi − x )2 N −1 N

Variancea

var[x] =

data series series length mean value

xi N ·

i=1

Standard deviation

σ[x] = (var[x])

1/2

standard deviation

3 N   xi − x N skew[x] = (2.544) (N − 1)(N − 2) σ i=1  4  N  1  xi − x kurt[x]  −3 (2.545) N σ

Skewness

Kurtosis

N

i=1

data series to correlate N N r correlation 2 2 i=1 (xi − x ) i=1 (yi − y ) coefficient a If x is derived from the data, {x }, the relation is as shown. If x is known independently, then an unbiased i estimate is obtained by dividing the right-hand side by N rather than N − 1. b Also known as “Pearson’s r.”

Correlation coefficientb

r = 

x,y

i=1 (xi − x )(yi − y )



(2.546)

Discrete probability distributions distribution

pr(x)

Binomial

n

Geometric Poisson

mean

variance

domain

np

np(1 − p)

(x = 0,1,... ,n)

(2.547)

(1 − p)x−1 p

1/p

(1 − p)/p2

(x = 1,2,3,...)

(2.548)

λx exp(−λ)/x!

λ

λ

(x = 0,1,2,...)

(2.549)

x

n

px (1 − p)n−x

x

binomial coefficient

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Mathematics

Continuous probability distributions distribution

pr(x)

mean

variance

domain

Uniform

1 b−a

a+b 2

(b − a)2 12

(a ≤ x ≤ b)

(2.550)

Exponential

λexp(−λx)

1/λ

1/λ2

(x ≥ 0)

(2.551)

Normal/ Gaussian

  −(x − µ)2 1 √ exp 2σ 2 σ 2π

µ

σ2

(−∞ < x < ∞)

(2.552)

r

2r

(x ≥ 0)

(2.553)

(x ≥ 0)

(2.554)

(−∞ < x < ∞)

(2.555)

Chi-squareda Rayleigh Cauchy/ Lorentzian a With

e−x/2 x(r/2)−1 2r/2 Γ(r/2)  2 −x x exp σ2 2σ 2 a π(a2 + x2 )

σ

(

π/2

(none)

2σ 2 1 −

π! 4

(none)

r degrees of freedom. Γ is the gamma function.

Multivariate normal distribution Density function

# $ exp − 12 (x − µ)C−1 (x − µ)T pr(x) = (2π)k/2 [det(C)]1/2 (2.556)

pr k C

probability density number of dimensions covariance matrix

x µ

variable (k dimensional) vector of means

T

Mean

µ = (µ1 ,µ2 ,... ,µk )

(2.557)

transpose det determinant µi mean of ith variable

Covariance

C = σij = xi xj − xi xj

(2.558)

σij

components of C

Correlation coefficient

r=

(2.559)

r

correlation coefficient

(2.560)

xi

normally distributed deviates

yi

deviates distributed uniformly between 0 and 1

Box–Muller transformation

σij σi σj

x1 = (−2lny1 )1/2 cos2πy2 x2 = (−2lny1 )1/2 sin2πy2

(2.561)

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2.13 Probability and statistics

Random walk 

Onedimensional

rms displacement Threedimensional

Mean distance rms distance

pr(x) =

−x 1 exp 2 1/2 2Nl 2 (2πNl )



xrms = N

1/2

l

a !3

(2.563)

exp(−a2 r2 )  1/2 3 where a = 2Nl 2  1/2 8 N 1/2 l r = 3π

(2.564)

rrms = N 1/2 l

(2.566)

pr(r) =

π 1/2



Conditional probability

pr(x) =

Joint probability Bayes’ theorema

pr(y|x) =

pr(x|y  )pr(y  ) dy 

N

displacement after N steps (can be positive or negative) probability density of x %∞ ( −∞ pr(x) dx = 1) number of steps

l

step length (all equal)

xrms

root-mean-squared displacement from start point

r

radial distance from start point probability density of r %∞ ( 0 4πr2 pr(r) dr = 1) (most probable distance)−1

x pr(x)

(2.562)

Bayesian inference

a In

2

pr(r) a

(2.565)

r

mean distance from start point

rrms

root-mean-squared distance from start point

pr(x)

probability (density) of x

(2.567)

pr(x|y  ) conditional probability of x given y 

pr(x,y) = pr(x)pr(y|x)

(2.568)

pr(x,y) joint probability of x and y

pr(x|y) pr(y) pr(x)

(2.569)

this expression, pr(y|x) is known as the posterior probability, pr(x|y) the likelihood, and pr(y) the prior probability.

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Mathematics

2.14

Numerical methods

Straight-line fittinga   {xi },{yi }

Data

n points

(2.570)

{wi }

(2.571)

Model

y = mx + c

(2.572)

Residuals

di = yi − mxi − c

(2.573)

Weighted centre

!   1 wi yi wi xi , (x,y) =  wi

Weights

b

Weighted moment

D=



Intercept

y = mx + c

(0,c) (x,y) x

(2.574) wi (xi − x)2

(2.575)

1 wi (xi − x)yi D  2 1 wi di var[m]  D n−2

m= Gradient

y

c = y − mx   2 1 wi di x2 var[c]   + wi D n−2

(2.576) (2.577) (2.578) (2.579)

a Least-squares b If

fit of data to y = mx + c. Errors on y-values only. the errors on yi are uncorrelated, then wi = 1/var[yi ].

Time series analysisa Discrete convolution

M/2 

(r s)j =

sj−k rk

(2.580)

k=−(M/2)+1

Bartlett (triangular) window Welch (quadratic) window Hanning window Hamming window a The

   j − N/2   wj = 1 −  N/2 

(2.581)



2 j − N/2 N/2    2πj 1 wj = 1 − cos 2 N   2πj wj = 0.54 − 0.46cos N wj = 1 −

ri si

response function time series

M

response function duration

wj

windowing function

N

length of time series 1

(2.582)

w 0.6 (2.583)

Hamming

Welch

0.8 Bartlett

0.4

Hanning

0.2 0

(2.584)

0

0.2

time series runs from j = 0...(N − 1), and the windowing functions peak at j = N/2.

0.4

0.6

j/N

0.8

1

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2.14 Numerical methods

Numerical integration h

2

f(x)

x x0  Trapezoidal rule

x0

 Simpson’s rulea aN

xN

xN

x0

xN h

h f(x) dx  (f0 + 2f1 + 2f2 + ··· 2 + 2fN−1 + fN )

fi

(2.585)

N

= (xN − x0 )/N (subinterval width) fi = f(xi ) number of subintervals

h f(x) dx  (f0 + 4f1 + 2f2 + 4f3 + ··· 3 + 4fN−1 + fN ) (2.586)

must be even. Simpson’s rule is exact for quadratics and cubics.

Numerical differentiationa 1 df [−f(x + 2h) + 8f(x + h) − 8f(x − h) + f(x − 2h)]  dx 12h 1 ∼ [f(x + h) − f(x − h)] 2h

(2.587) (2.588)

d2 f 1 [−f(x + 2h) + 16f(x + h) − 30f(x) + 16f(x − h) − f(x − 2h)]  2 dx 12h2 1 ∼ 2 [f(x + h) − 2f(x) + f(x − h)] h d3 f 1 [f(x + 2h) − 2f(x + h) + 2f(x − h) − f(x − 2h)] ∼ dx3 2h3

(2.589) (2.590) (2.591)

a Derivatives of f(x) at x. h is a small interval in x. Relations containing “” are O(h4 ); those containing “∼” are O(h2 ).

Numerical solutions to f(x) = 0 Secant method

xn+1 = xn −

xn − xn−1 f(xn ) f(xn ) − f(xn−1 )

(2.592)

f xn

function of x f(x∞ ) = 0

Newton–Raphson method

xn+1 = xn −

f(xn ) f  (xn )

(2.593)

f

= df/dx

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Mathematics

Numerical solutions to ordinary differential equationsa if Euler’s method

and then if and

Runge–Kutta method (fourth-order)

then

dy = f(x,y) dx h = xn+1 − xn

(2.594) (2.595) 2

yn+1 = yn + hf(xn ,yn ) + O(h ) dy = f(x,y) dx h = xn+1 − xn k1 = hf(xn ,yn ) k2 = hf(xn + h/2,yn + k1 /2) k3 = hf(xn + h/2,yn + k2 /2) k4 = hf(xn + h,yn + k3 ) k1 k2 k3 k4 yn+1 = yn + + + + + O(h5 ) 6 3 3 6

(2.596) (2.597) (2.598) (2.599) (2.600) (2.601) (2.602) (2.603)

dy 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.

a Ordinary

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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.

1 H.

Goldstein, Classical Mechanics, 2nd ed., 1980, Addison-Wesley.

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3.2

Dynamics and mechanics

Frames of reference

Galilean transformations r = r  + vt t = t

(3.1) (3.2)

Velocity

u = u + v

(3.3)

Momentum

p = p  + mv

(3.4)

Time and positiona

Angular momentum

J = J + mr × v + v× p t

(3.5)

Kinetic energy

1 T = T  + mu · v + mv 2 2

(3.6)

a Frames







r,r  v

position in frames S and S  velocity of S  in S

t,t

time in S and S 

u,u

velocity in frames S and S 

p,p 

particle momentum in frames S and S 

m

particle mass

J ,J 

angular momentum in frames S and S 

T ,T 

kinetic energy in frames S and S 

γ

Lorentz factor

v c

velocity of S  in S speed of light

S

m

S r

r

vt

coincide at t = 0.

Lorentz (spacetime) transformationsa −1/2  v2 γ = 1− 2 c

(3.7)

x = γ(x + vt ); y = y ;

x = γ(x − vt) y = y

(3.8) (3.9)

z = z;

z = z

Lorentz factor Time and position

(3.10)

v ! v ! (3.11) t = γ t + 2 x ; t = γ t − 2 x c c Differential dX = (cdt,−dx,−dy,−dz) four-vectorb (3.12)

 S S

x,x

t,t

X

x-position in frames S and S  (similarly for y and z) time in frames S and S

v  x x

spacetime four-vector

frames S and S  coincident at t = 0 in relative motion along x. See page 141 for the transformations of electromagnetic quantities. b Covariant components, using the (1,−1,−1,−1) signature. a For

Velocity transformationsa Velocity ux + v ux = ; 1 + ux v/c2 uy ; uy = γ(1 + ux v/c2 ) uz uz = ; γ(1 + ux v/c2 ) a For

ux − v 1 − ux v/c2 uy uy = γ(1 − ux v/c2 ) uz uz = γ(1 − ux v/c2 )

ux =

γ

Lorentz factor = [1 − (v/c)2 ]−1/2

v c

velocity of S  in S speed of light

ui ,ui

particle velocity components in frames S and S 

(3.13) (3.14) (3.15)

frames S and S  coincident at t = 0 in relative motion along x.

 S S

u v  x x

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3.2 Frames of reference

Momentum and energy transformationsa Momentum and energy px = γ(px + vE  /c2 ); py = py ;

px = γ(px − vE/c2 ) (3.16) py = py (3.17)

pz = pz ;

pz = pz 

E = γ(E



+ vpx );

E = γ(E − vpx )

2

2 2

E −p c =E −p c 2

2 2

Four-vectorb a For

(3.18) (3.19)

γ

Lorentz factor = [1 − (v/c)2 ]−1/2

v c

velocity of S  in S speed of light

px ,px E,E 

x components of momentum in S and S  (sim. for y and z) energy in S and S  (rest) mass total momentum in S momentum four-vector

= m20 c4

(3.20)

m0 p

P = (E/c,−px ,−py ,−pz )

(3.21)

P

 S S

v  x x

frames S and S  coincident at t = 0 in relative motion along x. components, using the (1,−1,−1,−1) signature.

b Covariant

Propagation of lighta ! v ν = γ 1 + cosα ν c

Doppler effect

(3.22)



cosθ + v/c 1 + (v/c)cosθ cosθ − v/c cosθ = 1 − (v/c)cosθ

cosθ = Aberration

b

Relativistic beamingc

P (θ) =

(3.23) (3.24)

sinθ 2γ 2 [1 − (v/c)cosθ]2

(3.25)

ν ν α

frequency received in S frequency emitted in S  arrival angle in S

γ

Lorentz factor = [1 − (v/c)2 ]−1/2

v velocity of S  in S c speed of light θ,θ emission angle of light in S and S 

S y

c α x

S S  y y

v θ c x x

P (θ) angular distribution of photons in S

frames S and S  coincident at t = 0 in relative motion along x. travelling in the opposite sense has a propagation angle of π + θ radians.% c Angular distribution of photons from a source, isotropic and stationary in S  . π P (θ) dθ = 1. 0 a For

b Light

Four-vectorsa Covariant and contravariant components

x 0 = x0

x1 = −x1

x2 = −x2

x3 = −x3

(3.26)

Scalar product

xi yi = x0 y0 + x1 y1 + x2 y2 + x3 y3

(3.27)

1

0

1

1

0

x = γ[x + (v/c)x ]; 2

2

x =x ;

covariant vector components

xi

contravariant components

xi ,x i four-vector components in frames S and S 

Lorentz transformations x0 = γ[x + (v/c)x ];

xi

x = γ[x0 − (v/c)x1 ] 0

1

x = γ[x − (v/c)x ] 3

x =x

1

3

0

(3.28)

v

Lorentz factor = [1 − (v/c)2 ]−1/2 velocity of S  in S

c

speed of light

γ

(3.29) (3.30)

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). a For

3

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Dynamics and mechanics

Rotating frames Vector transformation



dA dt



 = S

dA dt

 + ω× A

(3.31)

S

Acceleration

˙v = ˙v  + 2ω× v  + ω× (ω× r  )

(3.32)

Coriolis force

F cor = −2mω× v 

(3.33)

F cen = −mω× (ω× r  )

(3.34)

= +mω 2 r ⊥

(3.35)

Centrifugal force

Motion relative to Earth Foucault’s penduluma a The

3.3

A S S

any vector stationary frame rotating frame

ω

angular velocity of S  in S

˙v ,˙v  accelerations in S and S  v r

F cor coriolis force m particle mass

(3.36) ˙ sinλ m¨ y = Fy − 2mωe x ˙ cosλ m¨z = Fz − mg + 2mωe x

(3.37) (3.38)

Ωf = −ωe sinλ

(3.39)

ω

λ

nongravitational force latitude

z y x

local vertical axis northerly axis easterly axis

Ωf

pendulum’s rate of turn

F cen r ⊥

F cen centrifugal force r ⊥ perpendicular to particle from rotation axis Fi

y sinλ − ˙z cosλ) m¨ x = Fx + 2mωe (˙

velocity in S  position in S 

m

r ωe y

z x

λ

ωe Earth’s spin rate sign is such as to make the rotation clockwise in the northern hemisphere.

Gravitation

Newtonian gravitation Newton’s law of gravitation

Newtonian field equationsa

Fields from an isolated uniform sphere, mass M, r from the centre a The

Gm1 m2 rˆ 12 F1= 2 r12

(3.40)

g = −∇φ

(3.41)

∇2 φ = −∇ · g = 4πGρ

(3.42)

 GM  − 2 rˆ (r > a) r g(r) =  − GMr rˆ (r < a) 3  a GM  − (r > a) r φ(r) =   GM (r2 − 3a2 ) (r < a) 2a3

gravitational force on a mass m is mg.

(3.43)

m1,2 masses F 1 force on m1 (= −F 2 ) r 12 ˆ

vector from m1 to m2 unit vector

G g φ

constant of gravitation gravitational field strength gravitational potential

ρ

mass density

r

vector from sphere centre

M a

mass of sphere radius of sphere

M (3.44)

a

r

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3.3 Gravitation

General relativitya Line element

Christoffel symbols and covariant differentiation

ds2 = gµν dxµ dxν = −dτ2

(3.45)

ds dτ gµν

invariant interval proper time interval metric tensor

1 Γαβγ = g αδ (gδβ,γ + gδγ,β − gβγ,δ ) 2 φ;γ = φ,γ ≡ ∂φ/∂xγ Aα;γ = Aα,γ + Γαβγ Aβ

(3.46)

dxµ Γαβγ

differential of xµ Christoffel symbols

(3.47) (3.48)

,α ;α φ

partial diff. w.r.t. xα covariant diff. w.r.t. xα scalar

Bα;γ = Bα,γ − Γβαγ Bβ

(3.49)

Aα Bα

contravariant vector covariant vector

R αβγδ

Riemann tensor

vµ

tangent vector (= dxµ /dλ) affine parameter (e.g., τ for material particles)

R αβγδ = Γαµγ Γµβδ − Γαµδ Γµβγ + Γαβδ,γ − Γαβγ,δ Riemann tensor

Geodesic equation

(3.50)

Bµ;α;β − Bµ;β;α = R γµαβ Bγ

(3.51)

Rαβγδ = −Rαβδγ ; Rβαγδ = −Rαβγδ Rαβγδ + Rαδβγ + Rαγδβ = 0

(3.52) (3.53)

Dv µ =0 Dλ

(3.54)

where

DAµ dAµ ≡ + Γµαβ Aα v β Dλ dλ

(3.55)

λ

Geodesic deviation

D2 ξ µ = −R µαβγ v α ξ β v γ Dλ2

(3.56)

ξµ

geodesic deviation

Ricci tensor

Rαβ ≡ R σασβ = g σδ Rδασβ = Rβα

(3.57)

Rαβ

Ricci tensor

Einstein tensor

Gµν = R µν −

Gµν

Einstein tensor

R

Ricci scalar (= g µν Rµν )

Einstein’s field equations

Gµν = 8πT µν

(3.59)

T µν p

stress-energy tensor pressure (in rest frame)

Perfect fluid

T µν = (p + ρ)uµ uν + pg µν

(3.60)

ρ uν

density (in rest frame) fluid four-velocity

Schwarzschild solution (exterior)

1 µν g R 2

(3.58)

   −1 2M 2M dr2 ds2 = − 1 − dt2 + 1 − r r + r2 (dθ2 + sin2 θ dφ2 )

(3.61)

spherically symmetric mass (see page 183) (r,θ,φ) spherical polar coords. t time M

Kerr solution (outside a spinning black hole) ∆ − a2 sin2 θ 2 2Mr sin2 θ dt − 2a dt dφ
2
2
2 2 (r2 + a2 )2 − a2 ∆sin2 θ 2 2 dr +
2 dθ2 sin θ dφ + +
2 ∆

a ∆

angular momentum (along z) ≡ J/M ≡ r2 − 2Mr + a2


2

≡ r2 + a2 cos2 θ

J

ds2 = −

(3.62)

a General 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.

3

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Dynamics and mechanics

3.4

Particle motion

Dynamics definitionsa F

force

Newtonian force

F = m¨r = p˙

(3.63)

m r

mass of particle particle position vector

Momentum

p = m˙r

(3.64)

p

momentum

Kinetic energy

1 T = mv 2 2

(3.65)

T

kinetic energy

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 (ensemble of N particles)

N mi r i R0 = i=1 N i=1 mi

(3.68)

R0 mi

position vector of centre of mass mass of ith particle

ri

position vector of ith particle

a In

the Newtonian limit, v  c, assuming m is constant.

Relativistic dynamicsa Lorentz factor

 −1/2 v2 γ = 1− 2 c

γ v

Lorentz factor particle velocity

c

speed of light

p

relativistic momentum

m0

particle (rest) mass

(3.71)

F t

force on particle time

(3.69)

Momentum

p = γm0 v

Force

F=

Rest energy

Er = m0 c2

(3.72)

Er

particle rest energy

Kinetic energy

T = m0 c2 (γ − 1)

(3.73)

T

relativistic kinetic energy

E = γm0 c2

(3.74) E

total energy (= Er + T )

Total energy

(3.70)

dp dt

2 2

= (p c

+ m20 c4 )1/2

(3.75)

a It

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 v 2 = u2 + 2as 1 s = ut + at2 2 u+v t s= 2

(3.76) (3.77) (3.78) (3.79)

u

initial velocity

v t s

final velocity time distance travelled

a

acceleration

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3.4 Particle motion

Reduced mass (of two interacting bodies) r m2

m1

centre of mass

r2

r1

m1 m2 m1 + m 2 m2 r r1 = m1 + m 2 −m1 r r2 = m1 + m 2

(3.80)

µ

reduced mass

mi

interacting masses

(3.81)

ri

position vectors from centre of mass

(3.82)

r |r|

r = r1 − r2 distance between masses

Moment of inertia

I = µ|r|2

(3.83)

I

moment of inertia

Total angular momentum

J = µr×˙r

(3.84)

J

angular momentum

Lagrangian

1 L = µ|˙r |2 − U(|r|) 2

(3.85)

L U

Lagrangian potential energy of interaction

µ=

Reduced mass Distances from centre of mass

3

Ballisticsa Velocity

v = v0 cosα xˆ + (v0 sinα − gt) yˆ (3.86)

v0

initial velocity

v α

velocity at t elevation angle

v 2 = v02 − 2gy

(3.87)

g

gravitational acceleration

(3.88)

ˆ t

unit vector time

h

maximum height

l

range

Trajectory

gx2 y = xtanα − 2 2v0 cos2 α

Maximum height

h=

v02 sin2 α 2g

(3.89)

Horizontal range

l=

v02 sin2α g

(3.90)

a Ignoring the curvature and rotation of the Earth and frictional losses. g is assumed constant.

yˆ v0 h

α l

xˆ

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Dynamics and mechanics

Rocketry Escape velocitya Specific impulse Exhaust velocity (into a vacuum)



2GM vesc = r Isp =

vesc G M

escape velocity constant of gravitation mass of central body

r Isp

central body radius specific impulse

u g R

effective exhaust velocity acceleration due to gravity molar gas constant

γ Tc µ ∆v

ratio of heat capacities combustion temperature effective molecular mass of exhaust gas rocket velocity increment

Mi Mf M

pre-burn rocket mass post-burn rocket mass mass ratio

(3.95)

N Mi ui

number of stages mass ratio for ith burn exhaust velocity of ith burn

(3.96)

t θ

burn time rocket zenith angle

∆vah ∆vhb

velocity increment, a to h velocity increment, h to b

ra rb

radius of inner orbit radius of outer orbit

1/2 (3.91)

u g

(3.92)



2γRTc u= (γ − 1)µ 

1/2 (3.93)

Rocket equation (g = 0)

Mi ∆v = uln Mf

Multistage rocket

∆v =

N 

 ≡ ulnM

(3.94)

ui lnMi

i=1

In a constant gravitational field

∆v = ulnM − gtcosθ 

Hohmann cotangential transferb

GM ∆vah = ra 

GM ∆vhb = rb

1/2 

2rb ra + rb

1/2 

 1−

1/2

2ra ra + rb

 −1

(3.97) 1/2 

transfer ellipse, h

a

b

(3.98) a From

the surface of a spherically symmetric, nonrotating body, mass M. between coplanar, circular orbits a and b, via ellipse h with a minimal expenditure of energy.

b Transfer

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3.4 Particle motion

Gravitationally bound orbital motiona α GMm ≡− r r

Potential energy of interaction

U(r) = −

Total energy

J2 α α E =− + =− r 2mr2 2a

(3.100)

Virial theorem (1/r potential)

E = U /2 = −T U = −2T

Orbital equation (Kepler’s 1st law)

r0 = 1 + ecosφ , r a(1 − e2 ) r= 1 + ecosφ

Rate of sweeping area (Kepler’s 2nd law)

J dA = = constant dt 2m

Semi-major axis

a=

r0 α = 2 1−e 2|E|

Semi-minor axis

b=

Eccentricityb Semi-latusrectum Pericentre Apocentre Speed Period (Kepler’s 3rd law)

or

(3.99)

U(r) potential energy G constant of gravitation M central mass m α

orbiting mass ( M) GMm (for gravitation)

E J

total energy (constant) total angular momentum (constant)

(3.101) (3.102)

T

kinetic energy

·

mean value

(3.103)

r0 r

semi-latus-rectum distance of m from M

(3.104)

e φ

eccentricity phase (true anomaly)

(3.105)

A

area swept out by radius vector (total area = πab)

(3.106)

a

semi-major axis

b

semi-minor axis

2a

J r0 = (3.107) (1 − e2 )1/2 (2m|E|)1/2  1/2  1/2 2EJ 2 b2 e= 1+ = 1 − (3.108) mα2 a2

J 2 b2 = = a(1 − e2 ) (3.109) mα a r0 = a(1 − e) (3.110) rmin = 1+e r0 = a(1 + e) (3.111) rmax = 1−e   2 1 2 − v = GM (3.112) r a 1/2  m m !1/2 = 2πa3/2 P = πα 3 2|E| α (3.113) r0 =

m

A

r0

r φ M

ae

2b rmax

rmin

rmin pericentre distance rmax apocentre distance

v

orbital speed

P

orbital period

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. b Note that if the total energy, E, is < 0 then e < 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 > 0 then e > 1 and the orbit becomes a hyperbola (see Rutherford scattering on next page). a For

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Dynamics and mechanics

Rutherford scatteringa y trajectory for α < 0

b

x

scattering centre

a

χ

rmin trajectory for α > 0

Scattering potential energy

Scattering angle

Closest approach

Semi-axis Eccentricity Motion trajectoryb Scattering centrec

Rutherford scattering formulad a Nonrelativistic

rmin

a

(α<0)

(α>0)

α U(r) = − r " < 0 repulsive α > 0 attractive |α| χ tan = 2 2Eb   α |α| χ rmin = csc − 2E 2 |α| = a(e ± 1) |α| 2E 1/2  2 2 4E b χ +1 = csc e= α2 2 a=

4E 2 2 y 2 x − 2 =1 α2 b  2 1/2 α 2 x=± + b 4E 2 dσ 1 dN = dΩ n dΩ α !2 4 χ = csc 4E 2

(3.114) (3.115)

U(r) potential energy r particle separation α constant scattering angle total energy (> 0) impact parameter

(3.116)

χ E b

(3.117)

rmin closest approach

(3.118)

a e

hyperbola semi-axis eccentricity

(3.119)

(3.120) (3.121)

x,y position with respect to hyperbola centre

(3.122) dσ dΩ

(3.123) (3.124)

differential scattering cross section

n beam flux density dN number of particles scattered into dΩ Ω solid angle

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. b The correct branch can be chosen by inspection. c Also the focal points of the hyperbola. d n is the number of particles per second passing through unit area perpendicular to the beam.

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3.4 Particle motion

Inelastic collisionsa m1

m2

v1

m1

v2

Before collision

v2

After collision

Coefficient of restitution

v2 − v1 = (v1 − v2 )  = 1 if perfectly elastic  = 0 if perfectly inelastic

Loss of kinetic energyb

T −T = 1 − 2 T

(3.125) (3.126) (3.127)



coefficient of restitution

vi vi

pre-collision velocities post-collision velocities

T ,T 

total KE in zero momentum frame before and after collision

mi

particle masses

(3.128)

m1 − m2 (1 + )m2 v1 + v2 m1 + m 2 m1 + m 2 m2 − m1 (1 + )m1 v2 = v2 + v1 m1 + m 2 m1 + m 2

v1 = Final velocities

m2

v1

(3.129) (3.130)

the line of centres, v1 ,v2  c. zero momentum frame.

a Along b In

Oblique elastic collisionsa

m1

Directions of motion

Relative separation angle

tanθ1 = θ2 = θ

a Collision

After collision

m1

v2

θ1 v1

v

m2 sin2θ m1 − m2 cos2θ

  > π/2 θ1 + θ2 = π/2   < π/2

θ

angle between centre line and incident velocity

θi mi

final trajectories sphere masses

(3.134)

v

incident velocity of m1

(3.135)

vi

final velocities

(3.131) (3.132)

if m1 < m2 if m1 = m2 if m1 > m2

(m21 + m22 − 2m1 m2 cos2θ)1/2 v m1 + m 2 2m1 v v2 = cosθ m1 + m 2

v1 = Final velocities

m2

θ

Before collision

θ2 m2

(3.133)

between two perfectly elastic spheres: m2 initially at rest, velocities  c.

3

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3.5

Dynamics and mechanics

Rigid body dynamics

Moment of inertia tensor Moment of inertia tensora %



Iij =

(r2 δij − xi xj ) dm

(y 2 + z 2 ) dm %  I =  − xy dm % − xz dm

(3.136)

% − xy dm

%  − xz dm % 2 %  (x + z 2 ) dm − yz dm  % % 2 − yz dm (x + y 2 ) dm

r

r 2 = x2 + y 2 + z 2

δij

Kronecker delta

moment of inertia tensor dm mass element xi position vector of dm Iij components of I

I

(3.137)

tensor with respect to centre of mass ai ,a position vector of centre of mass m mass of body

Iij

I12 = I12 − ma1 a2

(3.138)

I11 = I11 + m(a22 + a23 )

(3.139)

Iij = Iij + m(|a| δij − ai aj )

(3.140)

Angular momentum

J = Iω

(3.141)

J

angular momentum

ω

angular velocity

Rotational kinetic energy

1 1 T = ω · J = Iij ωi ωj 2 2

(3.142)

T

kinetic energy

Parallel axis theorem

2

a I are the moments of inertia of the body. I (i = j) are its products of inertia. The integrals are over the body ii ij volume.

Principal axes

 0 0 I3

Principal moment of inertia tensor

 I1 I =  0 0

Angular momentum

J = (I1 ω1 ,I2 ω2 ,I3 ω3 )

(3.144)

J angular momentum ωi components of ω along principal axes

Rotational kinetic energy

1 T = (I1 ω12 + I2 ω22 + I3 ω32 ) 2

(3.145)

T

Moment of inertia ellipsoida

T = T (ω1 ,ω2 ,ω3 ) ∂T (J is ⊥ ellipsoid surface) Ji = ∂ωi " ≥ I3 generally I1 + I 2 = I3 flat lamina ⊥ to 3-axis

Perpendicular axis theorem

Symmetries

a The

0 I2 0

I1 = I2 = I3

asymmetric top

I1 = I2 = I3 I1 = I2 = I3

symmetric top spherical top

ellipsoid is defined by the surface of constant T .

(3.143)

I

principal moment of inertia tensor

Ii

principal moments of inertia

kinetic energy

(3.146) I3

(3.147) I1

I2

(3.148) lamina (3.149)

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3.5 Rigid body dynamics

Moments of inertiaa Thin rod, length l

Solid sphere, radius r Spherical shell, radius r

Solid cylinder, radius r, length l

Solid cuboid, sides a,b,c

I1 = I2 = I3  0

l

ml 2 12

(3.150)

2 I1 = I2 = I3 = mr2 5

I1 = m(b2 + c2 )/12 I2 = m(c2 + a2 )/12

(3.156) (3.157)

I3 = m(a2 + b2 )/12

(3.158)

I3 =

Elliptical lamina, semi-axes a,b

3

I2

(3.153)

l I1 I2

r

I3

(3.155)

2

a

I3

I2

b

c

 (3.159)

h

I3 I 2 I r

(3.160)

1

2

Solid ellipsoid, semi-axes a,b,c

I3

I1

3 h m r2 + 20 4

3 mr2 10

I1 r

(3.154)

I1 = I2 =

I2

I1

(3.152)

2 I1 = I2 = I3 = mr2 3   m 2 l2 I1 = I2 = r + 4 3 1 I3 = mr2 2



Solid circular cone, base radius r, height hb

I3

(3.151)

2

I1 = m(b + c )/5 I2 = m(c2 + a2 )/5

(3.161) (3.162)

I3 = m(a2 + b2 )/5

(3.163)

I1 = mb2 /4

(3.164)

2

I2 = ma /4 I3 = m(a2 + b2 )/4

I3 a c b I2 I1 I2 b I3 a

(3.165) (3.166)

I1

I2 2

I1 = I2 = mr /4 I3 = mr2 /2

Disk, radius r c

Triangular plate a With

m I3 = (a2 + b2 + c2 ) 36

(3.167) (3.168)

r

I1

I3 a

(3.169)

b I3

c

respect to principal axes for bodies of mass m and uniform density. The radius of gyration is defined as k = (I/m)1/2 . b Origin of axes is at the centre of mass (h/4 above the base). c Around an axis through the centre of mass and perpendicular to the plane of the plate.

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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 2θ

2 sinθ d= r 3 θ

from disk centre

(3.172)

Arc of circle, radius r, angle 2θ

d=r

from circle centre

(3.173)

Arbitrary triangular lamina, height ha

d = h/3

perpendicular from base

(3.174)

Solid cone or pyramid, height h

d = h/4

perpendicular from base

(3.175)

3 (2r − h)2 from sphere centre 4 3r − h shell: d = r − h/2 from sphere centre

(3.176)

solid: d =

Spherical cap, height h, sphere radius r Semi-elliptical lamina, height h ah

sinθ θ

d=

4h 3π

(3.177)

from base

(3.178)

is the perpendicular distance between the base and apex of the triangle.

Pendulums Simple pendulum Conical pendulum Torsional penduluma

   l θ02 1 + + ··· P = 2π g 16

P period g gravitational acceleration

(3.179)

l length θ0 maximum angular displacement

(3.180)

α cone half-angle



1/2 l cosα g  1/2 lI0 P = 2π C P = 2π

Equal double pendulumc a Assuming

P  2π

l √

(2 ± 2)g

α

m I0 moment of inertia of bob

(3.181)

1 (ma2 + I1 cos2 γ1 P  2π mga 1/2 2 2 + I2 cos γ2 + I3 cos γ3 ) (3.182) 

m l



Compound pendulumb

l θ0

C torsional rigidity of wire (see page 81)

l

I0

a distance of rotation axis from centre of mass m mass of body Ii principal moments of inertia γi angles between rotation axis and principal axes

1/2 (3.183)

the bob is supported parallel to a principal rotation axis. b I.e., an arbitrary triaxial rigid body. c For very small oscillations (two eigenmodes).

a

I1

I3 I2

l

m

l m

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3.5 Rigid body dynamics

Tops and gyroscopes J herpolhode

3 ω

space cone

invariable plane

J3 polhode

Ωp body cone

moment of inertia ellipsoid

3

θ support point

a

2 gyroscope

prolate symmetric top

Euler’s equations

a

Free symmetric topb (I3 < I2 = I1 ) Free asymmetric topc

Steady gyroscopic precession

˙ 1 + (I3 − I2 )ω2 ω3 G1 = I 1 ω ˙ 2 + (I1 − I3 )ω3 ω1 G2 = I 2 ω ˙ 3 + (I2 − I1 )ω1 ω2 G3 = I 3 ω I1 − I3 ω3 I1 J Ωs = I1

Ωb =

Ω2b =

(I1 − I3 )(I2 − I3 ) 2 ω3 I1 I2

(3.184) (3.185)

Gi

external couple (= 0 for free rotation)

Ii ωi

principal moments of inertia angular velocity of rotation

(3.187)

Ωb Ωs

body frequency space frequency

(3.188)

J

total angular momentum

Ωp θ J3

precession angular velocity angle from vertical angular momentum around symmetry axis mass

(3.186)

(3.189)

Ω2p I1 cosθ − Ωp J3 + mga = 0

(3.190)

(slow) (fast)

(3.191)

" Mga/J3 Ωp  J3 /(I1 cosθ)

mg

m g a

gravitational acceleration distance of centre of mass from support point moment of inertia about support point

Gyroscopic stability

J32 ≥ 4I1 mgacosθ

(3.192)

Gyroscopic limit (“sleeping top”)

J32  I1 mga

(3.193)

Nutation rate

Ωn = J3 /I1

(3.194)

Ωn

nutation angular velocity

Gyroscope released from rest

Ωp =

(3.195)

t

time

a Components

mga (1 − cosΩn t) J3

I1

are with respect to the principal axes, rotating with the body. 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 J close to 3-axis. If Ω2 < 0, the body tumbles. b b The

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3.6

Dynamics and mechanics

Oscillating systems

Free oscillations x

oscillating variable

t γ ω0

time damping factor (per unit mass) undamped angular frequency

A φ

amplitude constant phase constant

ω

angular eigenfrequency

Ai

amplitude constants

(3.202)

∆ an

logarithmic decrement nth displacement maximum

(3.203)

Q

quality factor

2

Differential equation

dx d x + ω02 x = 0 + 2γ dt2 dt

Underdamped solution (γ < ω0 ) Critically damped solution (γ = ω0 ) Overdamped solution (γ > ω0 )

x = Ae−γt cos(ωt + φ)

(3.197)

where ω = (ω02 − γ 2 )1/2

(3.198)

x = e−γt (A1 + A2 t)

(3.199)

x = e−γt (A1 eqt + A2 e−qt )

(3.200)

− ω02 )1/2

(3.201)

where q = (γ

2

an 2πγ = an+1 ω  π ω0  if Q= 2γ ∆

Logarithmic decrementa

∆ = ln

Quality factor

(3.196)

 Q1

a The

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 Differential equation

Steadystate solutiona

dx d2 x + ω02 x = F0 eiωf t + 2γ dt2 dt x = Aei(ωf t−φ) ,

where

A = F0 [(ω02 − ωf2 )2 + (2γωf )2 ]−1/2 F0 /(2ω0 ) [(ω0 − ωf )2 + γ 2 ]1/2 2γωf tanφ = 2 ω0 − ωf2 

(γ  ωf )

x

oscillating variable

(3.204)

t γ

time damping factor (per unit mass)

(3.205)

ω0

undamped angular frequency

(3.206)

F0

force amplitude (per unit mass)

(3.207)

ωf A φ

forcing angular frequency amplitude phase lag of response behind driving force

(3.208)

Amplitude resonanceb

2 ωar = ω02 − 2γ 2

(3.209)

ωar amplitude resonant forcing angular frequency

Velocity resonancec

ωvr = ω0

(3.210)

ωvr velocity resonant forcing angular frequency

Quality factor

Q=

(3.211)

Q

quality factor

Impedance

Z = 2γ + i

(3.212)

Z

impedance (per unit mass)

a Excluding

ω0 2γ ωf2 − ω02 ωf

the free oscillation terms. frequency for maximum displacement. c Forcing frequency for maximum velocity. Note φ = π/2 at this frequency. b Forcing

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3.7 Generalised dynamics

3.7

Generalised dynamics

Lagrangian dynamics  Action

S q q˙

action (δS = 0 for the motion) generalised coordinates generalised velocities

L t

Lagrangian time

m

mass

v r U

velocity position vector potential energy

T

kinetic energy

m0 γ

(rest) mass Lorentz factor

(3.217)

+e φ A

positive charge electric potential magnetic vector potential

(3.218)

pi

generalised momenta

t2

L(q, q˙,t) dt

S=

(3.213)

t1

d dt

Euler–Lagrange equation



∂L ∂q˙i

 −

∂L =0 ∂qi

(3.214)

1 L = mv 2 − U(r,t) 2 =T −U

Lagrangian of particle in external field Relativistic Lagrangian of a charged particle

L=−

Generalised momenta

pi =

(3.215) (3.216)

2

m0 c − e(φ − A · v) γ

∂L ∂˙ qi

Hamiltonian dynamics Hamiltonian

H=



pi q˙i − L

(3.219)

L pi q˙i

Lagrangian generalised momenta generalised velocities

(3.220)

H qi

Hamiltonian generalised coordinates

(3.221)

v r

particle speed position vector

(3.222)

U T

potential energy kinetic energy

m0 c

(rest) mass speed of light

+e φ

positive charge electric potential

A

vector potential

p t

particle momentum time

i

∂H ; ∂pi

Hamilton’s equations

q˙i =

Hamiltonian of particle in external field

1 H = mv 2 + U(r,t) 2 =T +U

Relativistic Hamiltonian of a charged particle

˙ pi = −

∂H ∂qi

H = (m20 c4 + |p − eA|2 c2 )1/2 + eφ   ∂f ∂g ∂f ∂g  − [f,g] = ∂qi ∂pi ∂pi ∂qi i

Poisson brackets

[qi ,g] =

∂g , ∂pi

∂g = 0, ∂t   ∂S ∂S + H qi , ,t = 0 ∂t ∂qi

[H,g] = 0 if Hamilton– Jacobi equation

∂g ∂qi dg =0 dt

[pi ,g] = −

(3.223)

(3.224) (3.225) (3.226) (3.227)

f,g arbitrary functions [·,·] Poisson bracket (also see Commutators on page 26)

S

action

3

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3.8

Dynamics and mechanics

Elasticity

Elasticity definitions (simple)a Stress

Strain

τ = F/A

e = δl/l

F (3.228)

(3.229)

e

stress applied force cross-sectional area strain

δl l

change in length length

τ F A

Young modulus (Hooke’s law)

E = τ/e = constant

(3.230)

E

Young modulus

Poisson ratiob

σ=−

δw/w δl/l

(3.231)

σ δw

Poisson ratio change in width

A l

w

w width apply to a thin wire under longitudinal stress. b Solids 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. a These

Elasticity definitions (general) Stress tensora Strain tensor

force  i direction area ⊥ j direction   1 ∂uk ∂ul + ekl = 2 ∂xl ∂xk

(3.232)

τij =

τij

stress tensor (τij = τji )

ekl

strain tensor (ekl = elk )

(3.233)

uk xk

displacement  to xk coordinate system

Elastic modulus

τij = λijkl ekl

(3.234)

λijkl

elastic modulus

Elastic energyb

1 U = λijkl eij ekl 2

(3.235)

U

potential energy

Volume strain (dilatation)

δV = e11 + e22 + e33 ev = V

(3.236)

ev δV

volume strain change in volume

V

volume

Shear strain

1 1 ekl = (ekl − ev δkl ) + ev δkl 3 ) *+ , )3 *+ ,

(3.237)

δkl

Kronecker delta

p

hydrostatic pressure

pure shear

Hydrostatic compression

dilatation

τij = −pδij

a τ are normal stresses, τ (i = j) are torsional stresses. ii ij b As usual, products are implicitly summed over repeated

(3.238) indices.

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3.8 Elasticity

Isotropic elastic solids E 2(1 + σ) Eσ λ= (1 + σ)(1 − 2σ)

µ= Lam´e coefficients

Longitudinal modulusa

Ml =

E(1 − σ) = λ + 2µ (1 + σ)(1 − 2σ)

1 [τii − σ(τjj + τkk )] E   σ (ejj + ekk ) τii = Ml eii + 1−σ t = 2µe + λ1tr(e)

Shear modulus (rigidity modulus)

a In

strain in i direction

τii e

stress in i direction strain tensor

(3.246)

9µK µ + 3K

Poisson ratio

eii

(3.245)

E 2(1 + σ) τT = µθsh

3K − 2µ σ= 2(3K + µ)

longitudinal elastic modulus

(3.244)

µ=

E=

Ml

(3.243)

2 E =λ+ µ 3(1 − 2σ) 3 1 1 ∂V  =−  KT V ∂p T −p = Kev

Young modulus

(3.240)

Young modulus Poisson ratio

(3.242)

K= Bulk modulus (compression modulus)

µ,λ Lam´e coefficients E σ

(3.241)

eii = Diagonalised equationsb

(3.239)

t stress tensor 1 unit matrix tr(·) trace K bulk modulus KT isothermal bulk modulus V volume

(3.247)

p T

pressure temperature

(3.248)

ev µ

volume strain shear modulus

(3.249)

τT θsh

transverse stress shear strain

τT

(3.250) θsh (3.251)

an extended medium. aligned along eigenvectors of the stress and strain tensors.

b Axes

Torsion Torsional rigidity (for a homogeneous rod)

G=C

Thin circular cylinder

C = 2πa3 µt

(3.253)

Thick circular cylinder

1 C = µπ(a42 − a41 ) 2

(3.254)

Arbitrary thin-walled tube

4A2 µt C= P

(3.255)

Long flat ribbon

1 C = µwt3 3

(3.256)

φ l

(3.252)

G

twisting couple

C l φ a

torsional rigidity rod length twist angle in length l radius

t µ

wall thickness shear modulus

a1

inner radius

a2

outer radius

A

cross-sectional area perimeter

P w

cross-sectional width

G a

φ

l

A t

w

t

3

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Dynamics and mechanics

Bending beamsa E Gb = Rc EI = Rc

Bending moment

(3.257)

Gb E Rc

bending moment Young modulus radius of curvature

(3.258)

ds ξ

W

area element distance to neutral surface from ds moment of area displacement from horizontal end-weight

l x

beam length distance along beam

w

beam weight per unit length

 2

ξ ds

I y

Light beam, horizontal at x = 0, weight at x = l

x! 2 W l− x y= 2EI 3

(3.259)

4

d y = w(x) dx4  2 2  π EI/l Fc = 4π 2 EI/l 2   2 π EI/(4l 2 )

Heavy beam

Euler strut failure a The

(3.260)

EI

(free ends) (fixed ends) (1 free end) (3.261)

Fc

critical compression force

l

strut length

ds ξ neutral surface (cross section)

x y W free

Fc

Fc fixed

radius of curvature is approximated by 1/Rc  d2 y/dx2 .

Elastic wave velocitiesa In an infinite isotropic solidb

In a fluid

vt vl

speed of transverse wave speed of longitudinal wave

(3.263)

µ ρ

shear modulus density

(3.264)

Ml

longitudinal modulus ! E(1−σ) = (1+σ)(1−2σ)

(3.265)

K

bulk modulus

vl(i)

speed of longitudinal wave (displacement  i)

vt(i) E

speed of transverse wave (displacement  i) Young modulus

σ k t

Poisson ratio wavenumber (= 2π/λ) plate thickness (in z, t  λ)

vφ a

torsional wave velocity rod radius ( λ)

vt = (µ/ρ)1/2

(3.262)

vl = (Ml /ρ)1/2  1/2 2 − 2σ vl = vt 1 − 2σ vl = (K/ρ)1/2

On a thin plate (wave travelling along x, plate thin in z)  vl(x) = k z y

x

In a thin circular rod a Waves

E ρ(1 − σ 2 )

1/2 (3.266)

vt(y) = (µ/ρ)1/2  1/2 Et2 vt(z) = k 12ρ(1 − σ 2 )

(3.267)

vl = (E/ρ)1/2

(3.269)

1/2

(3.270)

vφ = (µ/ρ)  1/2 ka E vt = 2 ρ

(3.268)

(3.271)

that produce “bending” are generally dispersive. Wave (phase) speeds are quoted throughout. waves are also known as shear waves, or S-waves. Longitudinal waves are also known as pressure waves, or P-waves.

b Transverse

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3.8 Elasticity

Waves in strings and springsa In a spring

vl = (κl/ρl )

1/2

(3.272)

vl κ l

speed of longitudinal wave spring constantb spring length

ρl

mass per unit lengthc

On a stretched string

vt = (T /ρl )1/2

(3.273)

vt T

speed of transverse wave tension

On a stretched sheet

vt = (τ/ρA )1/2

(3.274)

τ ρA

tension per unit width mass per unit area

amplitude assumed  wavelength. b In the sense κ = force/extension. c Measured along the axis of the spring. a Wave

Propagation of elastic waves Acoustic impedance

Z=

F force = response velocity ˙u

= (E  ρ)1/2 

E ρ

Wave velocity/ impedance relation

then

Mean energy density (nondispersive waves)

1 U = E  k 2 u20 2 1 = ρω 2 u20 2 P = Uv

Normal coefficientsa

Snell’s lawb

if v =

(3.275)

Z F

impedance stress force

(3.276)

u

strain displacement

(3.277)

E

elastic modulus

(3.278)

ρ v

density wave phase velocity

(3.279)

U

energy density

(3.280)

k ω

wavenumber angular frequency

(3.281)

u0 P

maximum displacement mean energy flux

(3.282)

r

reflection coefficient

(3.283)

t τ

transmission coefficient stress

(3.284)

θi θr

angle of incidence angle of reflection

1/2

Z = (E  ρ)1/2 = ρv

ur τr Z1 − Z2 =− = ui τi Z1 + Z2 2Z1 t= Z1 + Z2

r=

sinθi sinθr sinθt = = vi vr vt

θt angle of refraction 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]. b Angles 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. a For

3

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3.9

Dynamics and mechanics

Fluid dynamics

Ideal fluidsa Continuityb

Kelvin circulation

∂ρ + ∇ · (ρv) = 0 ∂t
Γ = v · dl = constant  = ω · ds

(3.285)

ρ v t

density fluid velocity field time

(3.286)

Γ dl

circulation loop element

ds

element of surface bounded by loop

ω

vorticity (= ∇× v)

(3.287)

S

Euler’s equation

c

∂v ∇p + (v · ∇)v = − +g ∂t ρ ∂ or (∇× v) = ∇× [v× (∇× v)] ∂t

(3.289)

pressure gravitational field strength (v · ∇) advective operator p g

(3.288)

Bernoulli’s equation (incompressible flow)

1 2 ρv + p + ρgz = constant 2

(3.290)

z

altitude

1 2 γ p v + + gz = constant (3.291) 2 γ −1 ρ 1 = v 2 + cp T + gz (3.292) 2

γ

Bernoulli’s equation (compressible adiabatic flow)d

ratio of specific heat capacities (cp /cV ) specific heat capacity at constant pressure temperature

Hydrostatics

∇p = ρg

(3.293)

Adiabatic lapse rate (ideal gas)

g dT =− dz cp

(3.294)

cp T

a No

thermal conductivity or viscosity. generally. c The second form of Euler’s equation applies to incompressible flow only. d Equation (3.292) is true only for an ideal gas. b True

Potential flowa v = ∇φ

(3.295)

v

velocity

∇ φ=0

(3.296)

φ

velocity potential

ω = ∇×v = 0

ω

vorticity

Vorticity condition

(3.297)

F

drag force on moving sphere

Drag force on a sphereb

1 2 F = − πρa3 ˙u = − Md ˙u 3 2

a ˙ u ρ

sphere radius sphere acceleration fluid density

Velocity potential

2

(3.298)

Md displaced fluid mass a For

incompressible fluids. effect of this drag force is to give the sphere an additional effective mass equal to half the mass of fluid displaced.

b The

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3.9 Fluid dynamics

Viscous flow (incompressible)a  Fluid stress

τij = −pδij + η

∂vi ∂vj + ∂xj ∂xi

Navier–Stokes equationb

∇p η ∂v + (v · ∇)v = − − ∇× ω + g ∂t ρ ρ ∇p η 2 + ∇ v +g =− ρ ρ

Kinematic viscosity

ν = η/ρ

a I.e.,

τij p η

fluid stress tensor hydrostatic pressure shear viscosity

vi δij

velocity along i axis Kronecker delta

(3.300)

v ω

fluid velocity field vorticity

(3.301)

g ρ

gravitational acceleration density

(3.302)

ν

kinematic viscosity

 (3.299)

∇ · v = 0, η = 0. bulk (second) viscosity.

b Neglecting

Laminar viscous flow Between parallel plates

vz (y) =

∂p 1 y(h − y) 2η ∂z

Along a circular pipea

1 ∂p vz (r) = (a2 − r2 ) 4η ∂z dV πa4 ∂p = Q= dt 8η ∂z

Circulating between concentric rotating cylindersb

4πηa2 a2 Gz = 2 1 22 (ω2 − ω1 ) a2 − a1

Along an annular pipe

η

flow velocity direction of flow distance from plate shear viscosity

p

pressure

(3.304)

r

(3.305)

a V

distance from pipe axis pipe radius volume

(3.303)

(3.306)

vz z y

Gz

axial couple between cylinders per unit length

ωi

angular velocity of ith cylinder

  a1 π ∂p 4 (a22 − a21 )2 4 a2 Q= a − a1 − 8η ∂z 2 ln(a2 /a1 ) Q (3.307)

inner radius outer radius volume discharge rate

a Poiseuille b Couette

flow. flow.

Draga On a sphere (Stokes’s law)

F = 6πaηv

(3.308)

F

drag force

a

radius

On a disk, broadside to flow

F = 16aηv

(3.309)

v η

velocity shear viscosity

On a disk, edge on to flow

F = 32aηv/3

(3.310)

a For

Reynolds numbers  1.

z

h

y

r

a

a1 ω1 ω2

a2

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Dynamics and mechanics

Characteristic numbers Re

Reynolds number

ρ U

density characteristic velocity

L η

characteristic scale-length shear viscosity

F g

Froude number gravitational acceleration

S

Strouhal number

τ

characteristic timescale

P cp λ

Prandtl number Specific heat capacity at constant pressure thermal conductivity

(3.315)

M c

Mach number sound speed

(3.316)

Ro Ω

Rossby number angular velocity

inertial force ρUL = η viscous force

(3.311)

F=

inertial force U2 = Lg gravitational force

(3.312)

Strouhal numberb

S=

Uτ evolution scale = L physical scale

(3.313)

Prandtl number

ηcp momentum transport = P= λ heat transport

Reynolds number

Re =

Froude numbera

speed U = c sound speed

Mach number

M=

Rossby number

Ro =

a Sometimes b Sometimes

inertial force U = ΩL Coriolis force

(3.314)

the square root of this expression. L is usually the fluid depth. the reciprocal of this expression.

Fluid waves 

Sound waves

K vp = ρ



1/2



=

dp dρ

1/2



In an ideal gas (adiabatic conditions)a

γRT vp = µ

Gravity waves on a liquid surfaceb

ω 2 = gk tanhkh  !  1 g 1/2 vg  2 k  (gh)1/2 ω2 =

Capillary–gravity waves (h  λ)

ω 2 = gk +

b Amplitude

bulk modulus pressure density

γp = ρ

(3.318)

γ R T

ratio of heat capacities molar gas constant (absolute) temperature

(3.319)

µ vg h

mean molecular mass group speed of wave liquid depth

λ k g

wavelength wavenumber gravitational acceleration

ω

angular frequency

σ

surface tension

(3.317) 1/2

(h  λ)

(3.320)

(h  λ) (3.321)

σk 3 ρ

the waves are isothermal rather than adiabatic then vp = (p/ρ)1/2 .  wavelength. c In the limit k 2  gρ/σ. a If

wave (phase) speed

K p ρ

3

Capillary waves (ripples)c

σk ρ

vp

1/2

(3.322)

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3.9 Fluid dynamics

Doppler effecta Source at rest, observer moving at u

|u| ν = 1 − cosθ ν vp

Observer at rest, source moving at u

ν  = ν

a For

1 |u| 1 − cosθ vp

(3.323)

ν  ,ν  observed frequency ν emitted frequency vp wave (phase) speed in fluid

(3.324)

u θ

velocity angle between wavevector, k, and u

k θ u

3

plane waves in a stationary fluid.

Wave speeds Phase speed

vp =

vg = Group speed

ω = νλ k dω dk

= vp − λ

(3.325)

vp

phase speed

ν ω λ

frequency angular frequency (= 2πν) wavelength

k

wavenumber (= 2π/λ)

vg

group speed

(3.326) dvp dλ

(3.327)

Shocks Mach wedgea

sinθw =

Kelvin wedgeb

λK =

Spherical adiabatic shockc

Rankine– Hugoniot shock relationsd

vp vb

4πvb2 3g θw = arcsin(1/3) = 19◦ .5 

r

(3.328)

(3.329) (3.330)

 2 1/5

Et ρ0

p2 2γM21 − (γ − 1) = p1 γ +1 v1 ρ2 γ +1 = = v2 ρ1 (γ − 1) + 2/M21

(3.331)

(3.332) (3.333)

θw

wedge semi-angle

vp vb

wave (phase) speed body speed

λK

characteristic wavelength gravitational acceleration

g r E t

shock radius energy release time

ρ0

density of undisturbed medium

1 2

upstream values downstream values

p v T

pressure velocity temperature

ρ density γ ratio of specific heats M Mach number a Approximating 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 v . b c Sedov–Taylor relation. d Solutions for a steady, normal shock, in the frame moving with the shock front. If γ = 5/3 then v /v ≤ 4. 1 2

T2 [2γM21 − (γ − 1)][2 + (γ − 1)M21 ] = T1 (γ + 1)2 M21

(3.334)

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Dynamics and mechanics

Surface tension surface energy area surface tension = length

σlv = Definition

Laplace’s formulaa

 ∆p = σlv 

Capillary constant

2σlv cc = gρ

Capillary rise (circular tube)

h=

1 1 + R1 R2

(3.335)

surface tension (liquid/vapour interface)

∆p

pressure difference over surface

Ri

principal radii of curvature

cc ρ g

capillary constant liquid density gravitational acceleration rise height

(3.336)

 (3.337)

1/2

2σlv cosθ ρga

σlv

(3.338)

h

surface

R2

R1

h

a

θ

(3.339)

θ contact angle a tube radius σwv wall/vapour surface σwv − σwl tension (3.340) cosθ = Contact angle σwl wall/liquid surface σlv tension a For a spherical bubble in a liquid ∆p = 2σ /R. For a soap bubble (two surfaces) ∆p = 4σ /R. lv lv

σwv σwl θ

σlv

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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 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 , has the operator pˆx = −i¯h∂/∂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.

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4.2

Quantum physics

Quantum definitions

Quantum uncertainty relations De Broglie relation

Planck–Einstein relation

a

Dispersion

h λ p = ¯hk

p=

(4.1) (4.2)

E = hν = ¯hω

(4.3)

(∆a)2 = (a − a )2

(4.4)

p,p h h ¯

particle momentum Planck constant h/(2π)

λ

de Broglie wavelength

k E

de Broglie wavevector energy

ν ω

frequency angular frequency (= 2πν)

a,b ·

observablesb expectation value

(∆a)2

dispersion of a

= a2 − a 2

(4.5)

General uncertainty relation

1 ˆ 2 (∆a)2 (∆b)2 ≥ i[ˆa, b] 4

(4.6)

aˆ [·,·]

operator for observable a commutator (see page 26)

Momentum–position uncertainty relationc

∆p∆x ≥

¯ h 2

(4.7)

x

particle position

Energy–time uncertainty relation

∆E ∆t ≥

¯ h 2

(4.8)

t

time

Number–phase uncertainty relation

∆n∆φ ≥

1 2

(4.9)

n

number of photons

φ

wave phase

a Dispersion

in quantum physics corresponds to variance in statistics. b An observable is a directly measurable parameter of a system. c Also known as the “Heisenberg uncertainty relation.”

Wavefunctions Probability density

Probability density currenta

pr(x,t) dx = |ψ(x,t)|2 dx   ∂ψ ∗ ¯ h ∗ ∂ψ −ψ j(x) = ψ 2im ∂x ∂x # $ h ¯ ψ ∗ (r)∇ψ(r) − ψ(r)∇ψ ∗ (r) j= 2im 1 ˆ = (ψ ∗ pψ) m ∂ (ψψ ∗ ) ∂t

Continuity equation

∇·j =−

Schr¨ odinger equation

∂ψ ˆ = i¯ Hψ h ∂t

Particle stationary statesb

−

a For

¯2 ∂2 ψ(x) h + V (x)ψ(x) = Eψ(x) 2m ∂x2

(4.10) (4.11)

j,j probability density current h (Planck constant)/(2π) ¯

(4.12)

x pˆ

(4.13)

position coordinate momentum operator m particle mass  real part of t time

(4.14) (4.15)

H Hamiltonian

(4.16)

V potential energy E total energy

particles. In three dimensions, suitable units would be particles m−2 s−1 . Schr¨ odinger equation for a particle, in one dimension.

b Time-independent

pr probability density ψ wavefunction

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4.2 Quantum definitions

Operators Hermitian conjugate operator Position operator



Momentum operator

pˆnx =

Kinetic energy operator

h2 ∂ 2 ¯ Tˆ = − 2m ∂x2



∗

(ˆaφ) ψ dx =

aˆ

φ∗ aˆ ψ dx

(4.17)

xˆn = xn

(4.18)

¯n ∂n h in ∂xn

Hermitian conjugate operator ψ,φ normalisable functions ∗

complex conjugate x,y position coordinates

n

arbitrary integer ≥ 1

px

momentum coordinate

T h ¯

kinetic energy (Planck constant)/(2π)

m

particle mass

(4.21)

H V

Hamiltonian potential energy

(4.19)

(4.20)

2

Hamiltonian operator

h ∂2 ˆ =− ¯ + V (x) H 2m ∂x2

Angular momentum operators

Lˆz = xˆ pˆy − yˆ pˆx 2 2 2 Lˆ2 = Lˆx + Lˆy + Lˆz

(4.22)

Lz

(4.23)

L

angular momentum along z axis (sim. x and y) total angular momentum

Parity operator

Pˆ ψ(r) = ψ(−r)

(4.24)

Pˆ

parity operator

r

position vector

a aˆ Ψ x

operator for a (spatial) wavefunction (spatial) coordinate

t h ¯

time (Planck constant)/(2π)

ψn an n

eigenfunctions of aˆ eigenvalues dummy index

cn

probability amplitudes

Expectation value Expectation valuea Time dependence Relation to eigenfunctions

Ehrenfest’s theorem a Equation



a = ˆa =

Ψ∗ aˆ Ψ dx

(4.25)

= Ψ|ˆa|Ψ - . ∂aˆ i ˆ d ˆa = [H, aˆ ] + dt h ¯ ∂t aˆ ψn = an ψn and Ψ =  then a = |cn |2 an

if

(4.26) (4.27) 

cn ψn (4.28)

expectation value of a

d r = p dt

(4.29)

m r

particle mass position vector

d p = −∇V dt

(4.30)

p V

momentum potential energy

m

(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 ˆa are taken as equivalent.

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Quantum physics

Dirac notation  Matrix elementa

n,m eigenvector indices

ψn∗ aˆ ψm

anm =

dx

(4.32)

anm matrix element ψn basis states aˆ operator x spatial coordinate

(4.31)

= n|ˆa|m

Bra vector

bra state vector = n|

(4.33)

·|

bra

Ket vector

ket state vector = |m

(4.34)

|·

ket

Scalar product

n|m =

Ψ cn

wavefunction probability amplitudes



if

Ψ=

ψn∗ ψm dx



then

a =

 m

a The

4.3

(4.36)

cn ψn

n

Expectation

(4.35)

c∗n cm anm

(4.37)

n

Dirac bracket, n|ˆa|m , can also be written ψn |ˆa|ψm .

Wave mechanics

Potential stepa V (x) V0

incident particle i

Potential function Wavenumbers

¯2 k 2 = 2mE h h2 q 2 = 2m(E − V0 ) ¯

Probability currentsb a One-dimensional

x

0

" 0 V (x) = V0

Amplitude reflection coefficient Amplitude transmission coefficient

ii

(x < 0) (x ≥ 0) (x < 0) (x > 0)

(4.38)

V V0

particle potential energy step height

h ¯

(Planck constant)/(2π)

(4.39) (4.40)

k,q particle wavenumbers m particle mass E total particle energy

r=

k−q k+q

(4.41)

r

amplitude reflection coefficient

t=

2k k+q

(4.42)

t

amplitude transmission coefficient

ji

particle flux in zone i

jii

particle flux in zone ii

¯k h (1 − |r|2 ) m hq ¯ jii = |t|2 m ji =

(4.43) (4.44)

interaction with an incident particle of total energy E = KE + V . If E < V0 then q is imaginary and |r|2 = 1. 1/|q| is then a measure of the tunnelling depth. b Particle flux with the sign of increasing x.

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4.3 Wave mechanics

Potential wella V (x) incident particle

i

−a

ii

a iii 0

Potential function

" 0 V (x) = −V0

Wavenumbers

¯2 k 2 = 2mE h h2 q 2 = 2m(E + V0 ) ¯

Amplitude reflection coefficient Amplitude transmission coefficient

(|x| > a) (|x| ≤ a)

(4.45)

(4.46) (4.47) (4.48)

2kqe−2ika 2kq cos2qa − i(q 2 + k 2 )sin2qa

(4.49)

t=

¯k h (1 − |r|2 ) m hk ¯ jiii = |t|2 m

Ramsauer effectc

En = −V0 +

a One-dimensional

−V0

ie−2ika (q 2 − k 2 )sin2qa r= 2kq cos2qa − i(q 2 + k 2 )sin2qa

Probability currentsb

Bound states (V0 < E < 0)d

(|x| > a) (|x| < a)

ji =

(4.50) (4.51)

h2 π 2 n2 ¯ 8ma2

" |k|/q tanqa = −q/|k|

q 2 − |k|2 = 2mV0 /¯h2

(4.52) even parity odd parity

x

V

particle potential energy

V0 h ¯ 2a

well depth (Planck constant)/(2π) well width

k,q particle wavenumbers m E

particle mass total particle energy

r

amplitude reflection coefficient

t

amplitude transmission coefficient

ji jiii

particle flux in zone i particle flux in zone iii

n

integer > 0

En

Ramsauer energy

(4.53) (4.54)

interaction with an incident particle of total energy E = KE + V > 0. flux in the sense of increasing x. c Incident energy for which 2qa = nπ, |r| = 0, and |t| = 1. d When E < 0, k is purely imaginary. |k| and q are obtained by solving these implicit equations. b Particle

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Quantum physics

Barrier tunnellinga V (x) V0

incident particle

i

ii −a

" 0 V (x) = V0

Potential function Wavenumber and tunnelling constant Amplitude reflection coefficient Amplitude transmission coefficient

0

iii a

(|x| > a) (|x| ≤ a)

(4.55)

¯2 k 2 = 2mE h (|x| > a) 2 2 h κ = 2m(V0 − E) (|x| < a) ¯ r=

t=

x

(4.56) (4.57)

−ie−2ika (k 2 + κ2 )sinh2κa 2kκcosh2κa − i(k 2 − κ2 )sinh2κa

V V0 h ¯

particle potential energy well depth (Planck constant)/(2π)

2a k κ

barrier width incident wavenumber tunnelling constant

m E

particle mass total energy (< V0 )

r

amplitude reflection coefficient

t

amplitude transmission coefficient

|t|2

tunnelling probability

ji

particle flux in zone i

jiii

particle flux in zone iii

(4.58)

2kκe−2ika (4.59) 2kκcosh2κa − i(k 2 − κ2 )sinh2κa 4k 2 κ2 (4.60) (k 2 + κ2 )2 sinh2 2κa + 4k 2 κ2 16k 2 κ2 exp(−4κa) (|t|2  1)  2 (k + κ2 )2 (4.61)

|t|2 = Tunnelling probability

Probability currentsb a By

¯k h (1 − |r|2 ) m hk ¯ jiii = |t|2 m ji =

(4.62) (4.63)

a particle of total energy E = KE + V , through a one-dimensional rectangular potential barrier height V0 > E. flux in the sense of increasing x.

b Particle

Particle in a rectangular boxa Eigenfunctions

Energy levels

 Ψlmn =

8 abc

h2 Elmn = 8M

1/2



sin

mπy nπz lπx sin sin a b c (4.64)

l 2 m2 n2 + + a2 b2 c2

 (4.65)

Ψlmn a,b,c l,m,n

eigenfunctions box dimensions integers ≥ 1

Elmn h

energy Planck constant particle mass

M

density of states (per unit volume) a Spinless 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.

Density of states

ρ(E) dE =

4π (2M 3 E)1/2 dE h3

ρ(E)

(4.66)

a b

x z y

c

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4.4 Hydrogenic atoms

Harmonic oscillator Schr¨ odinger equation Energy levelsa

¯2 ∂2 ψn 1 h + mω 2 x2 ψn = En ψn − 2m ∂x2 2 



En = n +

1 hω ¯ 2

(4.68) 2

Eigenfunctions

Hermite polynomials aE 0

4.4

(4.67)

h ¯

(Planck constant)/(2π)

m ψn

mass nth eigenfunction

x n ω

displacement integer ≥ 0 angular frequency

En

total energy in nth state

Hn

Hermite polynomials

y

dummy variable

2

Hn (x/a)exp[−x /(2a )] (n!2n aπ 1/2 )1/2  1/2 h ¯ where a = mω

ψn =

(4.69)

H0 (y) = 1, H1 (y) = 2y, H2 (y) = 4y 2 − 2 Hn+1 (y) = 2yHn (y) − 2nHn−1 (y) (4.70)

is the zero-point energy of the oscillator.

Hydrogenic atoms

Bohr modela Quantisation condition

h µrn2 Ω = n¯

Bohr radius

a0 =

0 h2 α =  52.9pm 2 πme e 4πR∞

Orbit radius

rn =

n2 me a0 Z µ

Total energy

En = −

Fine structure constant

α=

Hartree energy

EH =

Rydberg constant

R∞ =

Rydberg’s formulab

µe4 Z 2 µ Z2 = −R hc ∞ me n2 820 h2 n2

rn Ω

nth orbit radius orbital angular speed

n

principal quantum number (> 0)

(4.72)

a0 µ −e

Bohr radius reduced mass ( me ) electronic charge

(4.73)

Z h h ¯

atomic number Planck constant h/(2π)

(4.74)

En 0 me

total energy of nth orbit permittivity of free space electron mass

α

fine structure constant

µ0

permeability of free space

(4.71)

e2 1 µ0 ce2 =  2h 4π0 ¯hc 137

(4.75)

¯2 h  4.36 × 10−18 J me a20

(4.76)

EH Hartree energy

(4.77)

R∞ Rydberg constant c speed of light

me e4 EH me cα2 = 3 2 = 2h 8h 0 c 2hc   1 1 µ 1 = R∞ Z 2 2 − 2 λmn me n m

(4.78)

λmn photon wavelength m

integer > n

the Bohr model is strictly a two-body problem, the equations use reduced mass, µ = me mnuc /(me +mnuc )  me , where mnuc is the nuclear mass, throughout. The orbit radius is therefore the electron–nucleus distance. b Wavelength of the spectral line corresponding to electron transitions between orbits m and n. a Because

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Quantum physics

Hydrogenlike atoms – Schr¨odinger solutiona Schr¨ odinger equation −

¯2 2 h Ze2 ∇ Ψnlm − Ψnlm = En Ψnlm 2µ 4π0 r

with µ =

me mnuc me + mnuc

(4.79)

Eigenfunctions  1/2  3/2 (n − l − 1)! 2 m xl e−x/2 L2l+1 Ψnlm (r,θ,φ) = n−l−1 (x)Yl (θ,φ) 2n(n + l)! an with a =

me a0 , µ Z

x=

2r , an

and

L2l+1 n−l−1 (x) =

k=0

µe4 Z 2 820 h2 n2

total energy

0

permittivity of free space

h me h ¯

Planck constant mass of electron h/2π

µ mnuc Ψnlm

reduced mass ( me ) mass of nucleus eigenfunctions

(4.85)

Ze −e

charge of nucleus electronic charge

n = 1,2,3,...

(4.86)

Lqp

l = 0,1,2,... ,(n − 1) m = 0,±1,±2,... ,±l ∆n = 0

(4.87) (4.88) (4.89)

a r

associated Laguerre polynomialsc classical orbit radius, n = 1 electron–nucleus separation

Yl m

spherical harmonics

∆l = ±1 ∆m = 0 or

(4.90) (4.91)

a0

0 h Bohr radius = πm e2

En = −

Radial expectation values

a r = [3n2 − l(l + 1)] 2 a2 n2 r2 = [5n2 + 1 − 3l(l + 1)] 2 1 1/r = 2 an 2 2 1/r = (2l + 1)n3 a2

a For

(4.81)

±1

a−3/2 −r/a e π 1/2 a−3/2 r −r/2a e Ψ210 = cosθ 4(2π)1/2 a   r2 −r/3a a−3/2 r + 2 Ψ300 = 27 − 18 e a a2 81(3π)1/2 r ! r −r/3a a−3/2 6− e sinθ e±iφ Ψ31±1 = ∓ 1/2 a a 81π a−3/2 r2 −r/3a Ψ32±1 = ∓ e sinθ cosθ e±iφ 81π 1/2 a2 Ψ100 =

(l + n)!(−x)k (2l + 1 + k)!(n − l − 1 − k)!k! En

Total energy

Allowed quantum numbers and selection rulesb

n−l−1 

(4.80)

(4.82) (4.83) (4.84)

r ! −r/2a a−3/2 2 − e a 4(2π)1/2 a−3/2 r Ψ21±1 = ∓ 1/2 e−r/2a sinθ e±iφ 8π a r ! r −r/3a 21/2 a−3/2 6 − e Ψ310 = cosθ a a 81π 1/2 Ψ200 =

a−3/2 r2 −r/3a e (3cos2 θ − 1) 81(6π)1/2 a2 a−3/2 r2 −r/3a 2 ±2iφ Ψ32±2 = e sin θ e 162π 1/2 a2 Ψ320 =

a single bound electron in a perfect nuclear Coulomb potential (nonrelativistic and spin-free). dipole transitions between orbitals. c The sign and indexing definitions for this function vary. This form is appropriate to Equation (4.80). b For

2

e

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4.4 Hydrogenic atoms

Orbital angular dependence z (s)2 −0.4

(px )2

0.2 −0.4

−0.2

(py )2

−0.2

y

0.2

0.2

x

−0.2

(pz )2

(dx2 −y2 )2

−0.4

(dxz )2

4 (dz 2 )2

(dyz )2

0

0

s orbital (l = 0)

s = Y00 = constant

p orbitals (l = 1)

−1 1 (Y − Y1−1 ) ∝ cosφsinθ 21/2 1 i py = 1/2 (Y11 + Y1−1 ) ∝ sinφsinθ 2 pz = Y10 ∝ cosθ px =

dx2 −y2 =

d orbitals (l = 2)

a See

(dxy )2

1 21/2

(Y22 + Y2−2 ) ∝ sin2 θ cos2φ

−1 dxz = 1/2 (Y21 − Y2−1 ) ∝ sinθ cosθ cosφ 2 dz 2 = Y20 ∝ (3cos2 θ − 1) i dyz = 1/2 (Y21 + Y2−1 ) ∝ sinθ cosθ sinφ 2 −i dxy = 1/2 (Y22 − Y2−2 ) ∝ sin2 θ sin2φ 2

page 49 for the definition of spherical harmonics.

(4.92)

Ylm spherical harmonicsa

(4.93) (4.94)

θ,φ spherical polar coordinates

(4.95) (4.96)

z

(4.97) (4.98) (4.99) (4.100)

θ x

y φ

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4.5

Quantum physics

Angular momentum

Orbital angular momentum

Angular momentum operators

Ladder operators

ˆ = r× pˆ L   ∂ h ¯ ∂ Lˆz = x −y i ∂y ∂x h ∂ ¯ = i ∂φ 2 2 2 ˆ 2 L = Lˆx + Lˆy + Lˆz     1 ∂ ∂ 1 ∂2 2 = −¯ h sinθ + 2 sinθ ∂θ ∂θ sin θ ∂φ2 Lˆ± = Lˆx ± iLˆy   ∂ ∂ ±iφ ± = ¯he icotθ ∂φ ∂θ Lˆ± Y ml = ¯h[l(l + 1) − ml (ml ± 1)]1/2 Y ml ±1 l

l

Eigenfunctions and eigenvalues

(4.101)

L

(4.102)

p r

(4.103)

xyz Cartesian coordinates

(4.104)

rθφ spherical polar coordinates

(4.105) (4.106) (4.107) (4.108)

(l ≥ 0) Lˆ2 Yl ml = l(l + 1)¯h2 Yl ml ml ml ˆ Lz Y = ml ¯ hY (|ml | ≤ l)

(4.110)

Lˆz [Lˆ± Yl ml (θ,φ)] = (ml ± 1)¯hLˆ± Yl ml (θ,φ)

(4.111)

l-multiplicity = (2l + 1)

(4.112)

l

h ¯

angular momentum linear momentum position vector

(Planck constant)/(2π)

Lˆ± ladder operators m Yl l spherical harmonics l,ml integers

(4.109)

l

Angular momentum commutation relationsa Conservation of angular momentumb

[Lˆz ,x] = i¯ hy ˆ [Lz ,y] = −i¯ hx [Lˆz ,z] = 0 hpˆy [Lˆz , pˆx ] = i¯ [Lˆz , pˆy ] = −i¯ hpˆx ˆ [Lz , pˆz ] = 0

ˆ Lˆz ] = 0 [H,

(4.114) (4.115) (4.116) (4.117) (4.118) (4.119) [Lˆ2 , Lˆx ] = [Lˆ2 , Lˆy ] = [Lˆ2 , Lˆz ] = 0

a The b For

(4.113)

L p H Lˆ±

angular momentum momentum Hamiltonian ladder operators

[Lˆx , Lˆy ] = i¯hLˆz [Lˆz , Lˆx ] = i¯hLˆy

(4.120)

[Lˆy , Lˆz ] = i¯hLˆx [Lˆ+ , Lˆz ] = −¯hLˆ+

(4.122)

[Lˆ− , Lˆz ] = ¯hLˆ− [Lˆ+ , Lˆ− ] = 2¯ hLˆz

(4.124)

[Lˆ2 , Lˆ± ] = 0

(4.126)

(4.121) (4.123) (4.125)

(4.127)

commutation of a and b is defined as [a,b] = ab − ba (see page 26). Similar expressions hold for S and J. motion under a central force.

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4.5 Angular momentum

Clebsch–Gordan coefficientsa +1 +3/2 j,−mj |l1 ,−m1 ;l2 ,−m2 = (−1)l1 +l2 −j j,mj |l1 ,m1 ;l2 ,m2 1 1 × 1/2 3/2 +1/2 0 +1/2 +1/2 1 1 +1 +1/2 1 3/2 1/2 0 mj +1/2 −1/2 1/2 1/2 +1 −1/2 1/3 2/3 l1 × l 2 j j ... −1/2 +1/2 1/2 −1/2 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 2 × 1/2 5/2 +3/2 . . . +1 +3/2 +1/2 1 2 +2 +1/2 1 5/2 3/2 1

1/2 × 1/2

+3/2 −1/2 1/4 3/4 0 +1/2 +1/2 3/4 −1/4 2 1 +1/2 −1/2 1/2 1/2 −1/2 +1/2 1/2 −1/2 +2

1×1

2 +1 +1 +1 1 2 1 +1 0 1/2 1/2 0 +1 1/2 −1/2 +1 −1 0 0 −1 +1 +3 2×1 3 +2 +2 +1 1 3 2 +2 0 1/3 2/3 +1 +1 2/3 −1/3

0 2 1 0 1/6 1/2 1/3 2/3 0 −1/3 1/6 −1/2 1/3

+1 3 2 1 +2 −1 1/15 1/3 3/5 +1 0 8/15 1/6 −3/10 0 +1 6/15 −1/2 1/10 +1 −1 0 0 −1 +1

3/2 × 1

+2 −1/2 1/5 4/5 +1/2 +1 +1/2 4/5 −1/5 5/2 3/2 +1 −1/2 2/5 3/5 0 +1/2 3/5 −2/5 +5/2

5/2 +3/2 +3/2 +1 1 5/2 3/2 +3/2 0 2/5 3/5 +1/2 +1/2 +1 3/5 −2/5 5/2 3/2 1/2 +3/2 −1 1/10 2/5 1/2 1/2 0 3/5 1/15 −1/3 +3 −1/2 +1 3/10 −8/15 1/6 3/2 × 3/2 3 +2 +3/2 +3/2 1 3 2 +3/2 +1/2 1/2 1/2 +1 +1/2 +3/2 1/2 −1/2 3 2 1 +3/2 −1/2 1/5 1/2 3/10 +1/2 +1/2 3/5 0 −2/5 0 −1/2 +3/2 1/5 −1/2 3/10 3 2 1 0 0 +3/2 −3/2 1/20 1/4 9/20 1/4 3 2 1 +1/2 −1/2 9/20 1/4 −1/20 −1/4 1/5 1/2 3/10 −1/2 +1/2 9/20 −1/4 −1/20 1/4 3/5 0 −2/5 −3/2 +3/2 1/20 −1/4 9/20 −1/4 1/5 −1/2 3/10

+7/2 7/2 +5/2 +2 +3/2 1 7/2 5/2 +2 +1/2 3/7 4/7 +3/2 +1 +3/2 4/7 −3/7 7/2 5/2 3/2 +2 −1/2 1/7 16/35 2/5 +2 +1 +1/2 4/7 1/35 −2/5 +1/2 4 3 2 0 +3/2 2/7 −18/35 1/5 7/2 5/2 3/2 1/2 3/14 1/2 2/7 +2 −3/2 1/35 6/35 2/5 2/5 4/7 0 −3/7 +1 +1 −1/2 12/35 5/14 0 −3/10 3/14 −1/2 2/7 4 3 2 1 0 +1/2 18/35 −3/35 −1/5 1/5 +2 −1 1/14 3/10 3/7 1/5 −1 +3/2 4/35 −27/70 2/5 −1/10 +1 0 3/7 1/5 −1/14 −3/10 0 +1 3/7 −1/5 −1/14 3/10 0 −1 +2 1/14 −3/10 3/7 −1/5 4 3 2 1 0 +2 −2 1/70 1/10 2/7 2/5 1/5 +1 −1 8/35 2/5 1/14 −1/10 −1/5 0 0 18/35 0 −2/7 0 1/5 −1 +1 8/35 −2/5 1/14 1/10 −1/5 −2 +2 1/70 −1/10 2/7 −2/5 1/5

2 × 3/2

+4 4 +3 +2 +2 1 4 3 +2 +1 1/2 1/2 +1 +2 1/2 −1/2 +2 0 +1 +1 0 +2

2×2

a Or

“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 ≥ 0 are listed here. The coefficients for mj < 0 can be obtained from the symmetry relation j,−mj |l1 ,−m1 ;l2 ,−m2 = (−1)l1 +l2 −j j,mj |l1 ,m1 ;l2 ,m2 .

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Quantum physics

Angular momentum additiona J =L+S Jˆz = Lˆz + Sˆz

(4.128)

·S Jˆ2 = Lˆ2 + Sˆ2 + 2L/ Jˆz ψj,mj = mj ¯ hψj,mj

(4.130)

Jˆ2 ψj,mj = j(j + 1)¯h2 ψj,mj

(4.132)

j-multiplicity = (2l + 1)(2s + 1)

(4.133)

Mutually commuting sets

{L2 ,S 2 ,J 2 ,Jz ,L · S}

(4.134)

{L2 ,S 2 ,Lz ,Sz ,Jz }

(4.135)

Clebsch– Gordan coefficientsb

|j,mj =

Total angular momentum



J ,J total angular momentum L,L orbital angular momentum S,S spin angular momentum ψ eigenfunctions

(4.129)

(4.131)

j,mj |l,ml ;s,ms |l,ml |s,ms

mj

magnetic quantum number |mj | ≤ j

j

(l + s) ≥ j ≥ |l − s|

{}

set of mutually commuting observables

|·

eigenstates

·|· Clebsch–Gordan coefficients

ml ,ms ms +ml =mj

(4.136)

spin and orbital angular momenta as examples, eigenstates |s,ms and |l,ml . “Wigner coefficients.” Assuming no L–S interaction.

a Summing b Or

Magnetic moments Bohr magneton

e¯ h µB = 2me

Gyromagnetic ratioa

γ=

Electron orbital gyromagnetic ratio

Spin magnetic moment of an electronb

Land´e g-factorc

a Or

(4.137)

orbital magnetic moment (4.138) orbital angular momentum

−µB h ¯ −e = 2me

γe =

µB

Bohr magneton

−e h ¯ me

electronic charge (Planck constant)/(2π) electron mass

γ

gyromagnetic ratio

γe

electron gyromagnetic ratio

(4.139) (4.140)

µe,z = −ge µB ms (4.141) h ¯ (4.142) = ±ge γe 2 ge e¯ h =± (4.143) 4me ( µJ = gJ J(J + 1)µB (4.144) µJ,z = −gJ µB mJ (4.145) J(J + 1) + S(S + 1) − L(L + 1) gJ = 1 + 2J(J + 1) (4.146)

µe,z z component of spin magnetic moment ge electron g-factor ( 2.002) ms

spin quantum number (±1/2)

µJ

total magnetic moment

µJ,z z component of µJ mJ magnetic quantum number J,L,S total, orbital, and spin quantum numbers gJ

Land´e g-factor

“magnetogyric ratio.” b The electron g-factor equals exactly 2 in Dirac theory. The modification g = 2 + α/π + ..., where α is the fine e structure constant, comes from quantum electrodynamics. c Relating the spin + orbital angular momenta of an electron to its total magnetic moment, assuming g = 2. e

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4.5 Angular momentum

Quantum paramagnetism 1 0.8 0.6 0.4 0.2 0 −10

−5

B∞ (x) = L(x) B4 (x) B1 (x) B1/2 (x) = tanhx −0.2 −0.4 −0.6 −0.8 −1

5

x

10

  (2J + 1)x x 2J + 1 1 coth coth BJ (x) = − 2J 2J 2J 2J Brillouin function

 J +1x BJ (x)  3J L(x)

BJ (x) Brillouin function

(x  1) (J  1)

B1/2 (x) = tanhx Mean magnetisationa M for isolated spins (J = 1/2) a Of

(4.147)

(4.148) (4.149)

  µB B M = nµB JgJ BJ JgJ (4.150) kT 

µB B M 1/2 = nµB tanh kT

 (4.151)

J

total angular momentum quantum number

L(x)

Langevin function = cothx − 1/x (see page 144)

M n gJ

mean magnetisation number density of atoms Land´e g-factor

µB B k

Bohr magneton magnetic flux density Boltzmann constant

T temperature M 1/2 mean magnetisation for J = 1/2 (and gJ = 2)

an ensemble of atoms in thermal equilibrium at temperature T , each with total angular momentum quantum number J.

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4.6

Quantum physics

Perturbation theory

Time-independent perturbation theory Unperturbed states

ˆ 0 ψn = En ψn H

(4.152)

(ψn nondegenerate)

Perturbed Hamiltonian

ˆ ˆ =H ˆ 0 +H H

Perturbed eigenvaluesa

ˆ  |ψk Ek = Ek + ψk |H  |ψk |H ˆ  |ψn |2 + ... + Ek − En

(4.153)

ˆ0 H ψn En

unperturbed Hamiltonian ˆ0 eigenfunctions of H ˆ0 eigenvalues of H

n

integer ≥ 0

ˆ H ˆ H

perturbed Hamiltonian ˆ 0) perturbation ( H

(4.154)

Ek perturbed eigenvalue ( Ek ) || Dirac bracket

(4.155)

ψk

n=k

Perturbed eigenfunctionsb

ψk = ψk +

 ψk |H ˆ  |ψn n=k

Ek − En

ψn + ...

perturbed eigenfunction ( ψk )

a To

second order. b To first order.

Time-dependent perturbation theory ˆ0 H

Unperturbed stationary states

ˆ 0 ψn = En ψn H

Perturbed Hamiltonian

ˆ  (t) ˆ =H ˆ 0 +H H(t)

Schr¨ odinger equation

(4.156)

(4.157)

t

ˆ  (t)]Ψ(t) = i¯h ˆ 0 +H [H

∂Ψ(t) ∂t

Ψ(t = 0) = ψ0

 Ψ(t) = cn (t)ψn exp(−iEn t/¯h) Perturbed n wavewhere functiona  −i t ˆ  (t )|ψ0 exp[i(En − E0 )t /¯h] dt cn = ψn |H h 0 ¯ Fermi’s golden rule

ψn En n ˆ H ˆ  (t) H

2π ˆ  |ψi |2 ρ(Ef ) Γi→f = |ψf |H h ¯

unperturbed Hamiltonian ˆ0 eigenfunctions of H ˆ0 eigenvalues of H integer ≥ 0 perturbed Hamiltonian ˆ 0) perturbation ( H time

(4.158)

Ψ

wavefunction

(4.159)

ψ0 h ¯

initial state (Planck constant)/(2π)

cn

probability amplitudes

Γi→f

transition probability per unit time from state i to state f

(4.160)

(4.161)

(4.162)

ρ(Ef ) density of final states a To

first order.

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103

4.7 High energy and nuclear physics

4.7

High energy and nuclear physics

Nuclear decay Nuclear decay law

N(t) = N(0)e−λt

Half-life and mean life

T1/2 =

N(t) number of nuclei remaining after time t

(4.163)

ln2 λ T = 1/λ

t

time

(4.164)

λ

decay constant

(4.165)

T1/2 half-life T mean lifetime

Successive decays 1 → 2 → 3 (species 3 stable) N1 (t) = N1 (0)e−λ1 t

(4.166) −λ1 t

−λ2 t

−e N1 (0)λ1 (e N2 (t) = N2 (0)e−λ2 t + λ2 − λ1

N1 N2 N3

)

(4.167)  λ1 e−λ2 t − λ2 e−λ1 t λ1 −λ2 t N3 (t) = N3 (0) + N2 (0)(1 − e ) + N1 (0) 1 + λ2 λ2 − λ1 (4.168) 

population of species 1 population of species 2 population of species 3 decay constant 1 → 2 decay constant 2 → 3

v 3 = a(R − x)

v

velocity of α particle

Geiger’s lawa

(4.169)

Geiger–Nuttall rule

x a R

distance from source constant range

logλ = b + clogR

(4.170)

b, c constants for each series α, β, and γ

N A B

number of neutrons mass number (= N + Z) semi-empirical binding energy

Z av as

number of protons volume term (∼ 15.8MeV) surface term (∼ 18.0MeV)

ac aa ap

Coulomb term (∼ 0.72MeV) asymmetry term (∼ 23.5MeV) pairing term (∼ 33.5MeV)

a For

α particles in air (empirical).

Nuclear binding energy Liquid drop modela B = av A − as A2/3 − ac  −3/4  +ap A δ(A)  −ap A−3/4   0 Semi-empirical mass formula a Coefficient

Z2 (N − Z)2 + δ(A) − aa 1/3 A A Z, N both even Z, N both odd otherwise

M(Z,A) = ZMH + Nmn − B

values are empirical and approximate.

(4.171) (4.172)

(4.173)

M(Z,A) atomic mass MH mass of hydrogen atom mn neutron mass

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Quantum physics

Nuclear collisions Breit–Wigner formulaa

π Γab Γc g k 2 (E − E0 )2 + Γ2 /4 2J + 1 g= (2sa + 1)(2sb + 1)

σ(E) =

Total width

Γ = Γab + Γc

Resonance lifetime

τ=

Born scattering formulab

h ¯ Γ

σ(E) cross-section for a + b → c

(4.174)

k g

incoming wavenumber spin factor

(4.175)

E E0 Γ

total energy (PE + KE) resonant energy width of resonant state R

(4.176)

Γab partial width into a + b Γc partial width into c τ resonance lifetime

(4.177)

J

  2  dσ  2µ ∞ sinKr 2 = 2 V (r)r dr dΩ  ¯ Kr h 0 (4.178)

Mott scattering formulac  α  2χ Acos lntan α !2 dσ χ χ ¯hv 2 = csc4 + sec4 + dΩ 4E 2 2 sin2 2χ cos χ2 (4.179) ! 2 2 dσ 4 − 3sin χ α (A = −1, α  v¯h) (4.180)  dΩ 2E sin4 χ

total angular momentum quantum number of R

sa,b spins of a and b dσ dΩ differential collision cross-section µ reduced mass K = |kin − kout | (see footnote) r radial distance V (r) potential energy of interaction h ¯ (Planck constant)/2π α/r scattering potential energy χ scattering angle v A

closing velocity = 2 for spin-zero particles, = −1 for spin-half particles

the reaction a + b ↔ R → c in the centre of mass frame. 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. c For identical particles undergoing Coulomb scattering in the centre of mass frame. Nonidentical particles obey the Rutherford scattering formula (page 72). a For b For

Relativistic wave equationsa Klein–Gordon equation (massive, spin zero particles) Weyl equations (massless, spin 1/2 particles) Dirac equation (massive, spin 1/2 particles) a Written

(∇2 − m2 )ψ =

∂2 ψ ∂t2

  ∂ψ ∂ψ ∂ψ ∂ψ = ± σx + σy + σz ∂t ∂x ∂y ∂z (iγ µ ∂µ − m)ψ = 0   ∂ ∂ ∂ ∂ , , , where ∂µ = ∂t ∂x ∂y ∂z (γ 0 )2 = 14 ;

in natural units, with c = h¯ = 1.

ψ wavefunction

(4.181)

(4.182) (4.183) (4.184)

(γ 1 )2 = (γ 2 )2 = (γ 3 )2 = −14 (4.185)

m particle mass t time ψ spinor wavefunction σ i Pauli spin matrices (see page 26) i i2 = −1 γ µ Diracmatrices:  1 0 γ0 = 2 0 −12   0 σi γi = −σ i 0 1n n × n unit matrix

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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 = d Q + d W . • 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, equals V /m, where V is the volume of gas and m its mass. Also, the specific heat capacity of a gas at constant pressure is cp = Cp /m, where Cp 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;   is the partial differential of volume with respect to pressure, holding temperature hence ∂V  ∂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.

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5.2

Thermodynamics

Classical thermodynamics

Thermodynamic laws Thermodynamic temperaturea

T ∝ lim(pV )

Kelvin temperature scale

T /K = 273.16

(5.1)

p→0

lim(pV )T

p→0

(5.2)

lim(pV )tr

T V p

thermodynamic temperature volume of a fixed mass of gas gas pressure

K tr

kelvin unit temperature of the triple point of water

p→0

dU change in internal energy

First lawb

dU = d Q + d W

Entropyc

dS =

d Qrev d Q ≥ T T

(5.3)

d W work done on system d Q heat supplied to system S experimental entropy

(5.4)

T

temperature reversible change a As 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. b The d notation represents a differential change in a quantity that is not a function of state of the system. c Associated with the second law of thermodynamics: No process is possible with the sole effect of completely converting heat into work (Kelvin statement). rev

Thermodynamic worka Hydrostatic pressure

d W = −p dV

(5.5)

Surface tension

d W = γ dA

(5.6)

p (hydrostatic) pressure dV volume change d W work done on the system γ surface tension dA change in area

Electric field

d W = E · dp

(5.7)

E

electric field

dp

induced electric dipole moment

Magnetic field

d W = B · dm

(5.8)

B magnetic flux density dm induced magnetic dipole moment

Electric current

d W = ∆φ dq

(5.9)

∆φ potential difference

a The

dq

charge moved

sources of electric and magnetic fields are taken as being outside the thermodynamic system on which they are working.

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5.2 Classical thermodynamics

Cycle efficiencies (thermodynamic)a Heat engine

η=

work extracted Th − Tl ≤ heat input Th

(5.10)

Refrigerator

η=

Tl heat extracted ≤ work done Th − Tl

(5.11)

Heat pump

η=

b

Otto cycle a The

Th heat supplied ≤ work done Th − Tl  γ−1 V2 work extracted =1− η= heat input V1

η Th Tl

efficiency higher temperature lower temperature

V1 V2

compression ratio

γ

ratio of heat capacities (assumed constant)

(5.12) (5.13)

equalities are for reversible cycles, such as Carnot cycles, operating between temperatures Th and Tl . reversible “petrol” (heat) engine.

b Idealised

Heat capacities Constant volume

d Q  ∂U  ∂S  CV =  =  =T  dT V ∂T V ∂T V

Constant pressure

Cp =

d Q  ∂H  ∂S   =  =T  dT p ∂T p ∂T p 

Difference in heat capacities

Cp − CV = =

Ratio of heat capacities

γ=

 ∂U  ∂V   +p  ∂V T ∂T p

V T βp2 κT

Cp κT = CV κS

CV

heat capacity, V constant

Q T

heat temperature

V U S

volume internal energy entropy

Cp p H

heat capacity, p constant pressure enthalpy

βp κT

isobaric expansivity isothermal compressibility

γ κS

ratio of heat capacities adiabatic compressibility

βp V

isobaric expansivity volume

T

temperature

κT

isothermal compressibility

p

pressure

(5.21)

κS

adiabatic compressibility

(5.22)

KT isothermal bulk modulus

(5.23)

KS

(5.14)

(5.15)

(5.16) (5.17) (5.18)

Thermodynamic coefficients Isobaric expansivitya

βp =

1 ∂V   V ∂T p

Adiabatic compressibility

1 ∂V  κT = −  V ∂p T 1 ∂V  κS = −  V ∂p S

Isothermal bulk modulus

KT =

Isothermal compressibility

Adiabatic bulk modulus a Also

1 ∂p  = −V  κT ∂V T 1 ∂p  = −V KS =  κS ∂V S

(5.19) (5.20)

adiabatic bulk modulus

called “cubic expansivity” or “volume expansivity.” The linear expansivity is αp = βp /3.

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Thermodynamics

Expansion processes ∂T  T 2 ∂(p/T )   =−  ∂V U CV ∂T V    1 ∂p  =− T  −p CV ∂T V ∂T  T 2 ∂(V /T )  µ=  =  p ∂p H Cp ∂T    1 ∂V  = T  −V Cp ∂T p η=

Joule expansiona

Joule–Kelvin expansionb a Expansion b Expansion

η T p

Joule coefficient temperature pressure

(5.25)

U CV

internal energy heat capacity, V constant

(5.26)

µ V H

Joule–Kelvin coefficient volume enthalpy

Cp

heat capacity, p constant

(5.24)

(5.27)

with no change in internal energy. with no change in enthalpy. Also known as a “Joule–Thomson expansion” or “throttling” process.

Thermodynamic potentialsa dU = T dS − pdV + µdN

Internal energy

(5.28)

H = U + pV

(5.29)

dH = T dS + V dp + µdN

(5.30)

F =U −TS

(5.31)

dF = −S dT − pdV + µdN

(5.32)

G = U − T S + pV

(5.33)

= F + pV = H − T S dG = −S dT + V dp + µdN

(5.34) (5.35)

Grand potential

Φ = F − µN dΦ = −S dT − pdV − N dµ

(5.36) (5.37)

Gibbs–Duhem relation

−S dT + V dp − N dµ = 0

(5.38)

A = U − T0 S + p0 V

(5.39)

dA = (T − T0 )dS − (p − p0 )dV

(5.40)

Enthalpy

Helmholtz free energyb

Gibbs free energy

Availability

c

U T S

internal energy temperature entropy

µ N

chemical potential number of particles

H p

enthalpy pressure

V

volume

F

Helmholtz free energy

G

Gibbs free energy

Φ

grand potential

A

availability

T0

temperature of surroundings pressure of surroundings

p0 a dN=0

for a closed system. b Sometimes called the “work function.” c Sometimes called the “thermodynamic potential.”

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5.2 Classical thermodynamics

Maxwell’s relations Maxwell 1

∂T  ∂p   =−  ∂V S ∂S V

Maxwell 2

∂V  ∂T   =  ∂p S ∂S p

Maxwell 3 Maxwell 4

 =  =

∂2 U ∂S∂V

∂2 H ∂p∂S



U

internal energy

T V

temperature volume

(5.42)

H S p

enthalpy entropy pressure

(5.43)

F

Helmholtz free energy

(5.44)

G

Gibbs free energy

(5.41)



  ∂p  ∂S  ∂2 F =  =  ∂T V ∂V T ∂T ∂V     ∂V  ∂S  ∂2 G =  =−  ∂T p ∂p T ∂p∂T

Gibbs–Helmholtz equations ∂(F/T )   V ∂T  ∂(F/V )  G = −V 2  T ∂V  2 ∂(G/T )  H = −T  p ∂T

U = −T 2

F U G

Helmholtz free energy internal energy Gibbs free energy

(5.46)

H T

enthalpy temperature

(5.47)

p V

pressure volume

(5.45)

Phase transitions Heat absorbed

Clausius–Clapeyron equationa

Coexistence curveb

L = T (S2 − S1 ) dp S2 − S1 = dT V2 − V1 L = T (V2 − V1 )   −L p(T ) ∝ exp RT

Ehrenfest’s equationc

dp βp2 − βp1 = dT κT 2 − κT 1 1 Cp2 − Cp1 = V T βp2 − βp1

Gibbs’s phase rule

P+F=C+2

L

(latent) heat absorbed (1 → 2)

(5.48)

T S

temperature of phase change entropy

(5.49)

p V

pressure volume

(5.50)

1,2 phase states

(5.51)

R

molar gas constant

(5.52)

βp

isobaric expansivity

(5.53)

κT Cp

isothermal compressibility heat capacity (p constant)

P

number of phases in equilibrium

(5.54)

F number of degrees of freedom C number of components a Phase boundary gradient for a first-order transition. Equation (5.50) is sometimes called the “Clapeyron equation.” b For V  V , e.g., if phase 1 is a liquid and phase 2 a vapour. 2 1 c For a second-order phase transition.

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5.3

Thermodynamics

Gas laws

Ideal gas Joule’s law

U = U(T )

(5.55)

U T

internal energy temperature

Boyle’s law

pV |T = constant

(5.56)

p

pressure

V

volume

Equation of state (Ideal gas law)

pV = nRT

(5.57)

n R

number of moles molar gas constant

pV γ = constant T V (γ−1) = constant

(5.58) (5.59)

γ

ratio of heat capacities (Cp /CV )

Adiabatic equations

γ (1−γ)

= constant 1 (p2 V2 − p1 V1 ) ∆W = γ −1

T p

nRT γ −1

(5.60)

∆W work done on system

(5.61)

Internal energy

U=

Reversible isothermal expansion

∆Q = nRT ln(V2 /V1 )

(5.63)

∆Q heat supplied to system

Joule expansiona

∆S = nR ln(V2 /V1 )

(5.64)

∆S

(5.62)

1,2 initial and final states change in entropy of the system

a Since

∆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 Virial expansion

Boyle temperature

 B2 (T ) pV = RT 1 + V  B3 (T ) + + ··· V2

(5.65)

B2 (TB ) = 0

(5.66)

p

pressure

V R T

volume molar gas constant temperature

Bi

virial coefficients

TB

Boyle temperature

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5.3 Gas laws

Van der Waals gas 

 Equation of state

p+

a (Vm − b) = RT Vm2

(5.67)

T temperature a,b van der Waals’ constants

Tc = 8a/(27Rb)

(5.68)

2

pc = a/(27b ) Vmc = 3b

Critical point



Reduced equation of state

pr +

p pressure Vm molar volume R molar gas constant

 3 (3Vr − 1) = 8Tr Vr2

Tc

critical temperature

(5.69) (5.70)

pc critical pressure Vmc critical molar volume

(5.71)

pr Vr

= p/pc = Vm /Vmc

Tr

= T /Tc

Dieterici gas 

Equation of state

p=



−a RT exp Vm − b RT Vm

 (5.72)

T temperature a ,b Dieterici’s constants

Tc = a /(4Rb ) 

(5.73)

2 2

pc = a /(4b e )

Critical point

(5.74)



Vmc = 2b Reduced equation of state

pr =

(5.75)

  2 Tr exp 2 − 2Vr − 1 Vr Tr

Van der Waals gas 1.4

1.1

1.2 0.8

pr

pr

Tr = 1.2

1.0

1

0.9

0.6 0.4 0.2 0

0.8 0

1

2

3 Vr

4

5

5

p pressure Vm molar volume R molar gas constant

(5.76)

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

Tc critical temperature pc critical pressure Vmc critical molar volume e

= 2.71828...

pr

= p/pc

Vr Tr

= Vm /Vmc = T /Tc

Dieterici gas Tr = 1.2 1.1 1.0 0.9 0.8

0

1

2

3 Vr

4

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5.4

Thermodynamics

Kinetic theory

Monatomic gas 1 p = nmc2 3

Pressure

(5.77)

pV = NkT

Internal energy

N 3 U = NkT = mc2 2 2

(5.79)

3 CV = Nk 2

(5.80)

Entropy (Sackur– Tetrode equation)a a For

(5.78)

5 Cp = CV + Nk = Nk 2 Cp 5 γ= = CV 3  S = Nk ln

the uncondensed gas. The factor

pressure number density = N/V particle mass

c2 mean squared particle velocity

Equation of state of an ideal gas

Heat capacities

p n m

mkT 2π¯ h2

mkT 2π¯h2

(5.81)

volume Boltzmann constant number of particles

T

temperature

U

internal energy

CV Cp γ

heat capacity, constant V heat capacity, constant p ratio of heat capacities

S

entropy

¯ h e

= (Planck constant)/(2π) = 2.71828...

(5.82)

3/2

5/2 V

e

!3/2

V k N



N

(5.83)

is the quantum concentration of the particles, nQ . Their thermal de

−1/3

Broglie wavelength, λT , approximately equals nQ

.

Maxwell–Boltzmann distributiona Particle speed distribution Particle energy distribution

rms speed Most probable speed a Probability



−E 2E exp kT π 1/2 (kT )3/2 1/2 1/2

pr(E) dE = 

Mean speed

pr m

probability density particle mass

k T

Boltzmann constant temperature

c

particle speed

E

particle kinetic energy (= mc2 /2)

(5.86)

c

mean speed

(5.87)

crms root mean squared speed

(5.88)

cˆ

  m !3/2 −mc2 pr(c) dc = exp 4πc2 dc 2πkT 2kT (5.84)

8kT πm  1/2  1/2 3kT 3π = c crms = m 8  1/2 2kT π !1/2 cˆ = = c m 4 c =

density functions normalised so that

%∞ 0

pr(x) dx = 1.

 dE

(5.85)

most probable speed

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5.4 Kinetic theory

Transport properties l d n

mean free path molecular diameter particle number density

pr

probability

x

linear distance

J c

molecular flux mean molecular speed

D

diffusion coefficient

(5.95)

H λ T

heat flux per unit area thermal conductivity temperature

(5.96)

ρ cV

density specific heat capacity, V constant

η

dynamic viscosity

(5.97)

x

displacement of sphere in x direction after time t

(5.98)

k t a

Boltzmann constant time interval sphere radius



dM dt

mass flow rate pipe radius pipe length

Mean free patha

l= √

1 2πd2 n

(5.89)

Survival equationb

pr(x) = exp(−x/l)

(5.90)

Flux through a planec

1 J = nc 4

(5.91)

Self-diffusion (Fick’s law of diffusion)d

J = −D∇n

(5.92)

2 where D  lc 3

(5.93)

Thermal conductivityd

H = −λ∇T 1 ∂T ∇2 T = D ∂t

(5.94)

for monatomic gas Viscosity

d

Brownian motion (of a sphere)

5 λ  ρlc cV 4

1 η  ρlc 2 x2 =

Free molecular flow (Knudsen flow)e

kT t 3πηa 4Rp3

dM = dt 3L



2πm k

1/2 

p1 1/2

T1

−

p2 1/2

T2

Rp L

m particle mass p pressure a For a perfect gas of hard, spherical particles with a Maxwell–Boltzmann speed distribution. b Probability of travelling distance x without a collision. c From the side where the number density is n, assuming an isotropic velocity distribution. Also known as “collision number.” d Simplistic kinetic theory yields numerical coefficients of 1/3 for D, λ and η. e Through a pipe from end 1 to end 2, assuming R  l (i.e., at very low pressure). p

(5.99)

Gas equipartition k T

energy per quadratic degree of freedom Boltzmann constant temperature

CV Cp N

heat capacity, V constant heat capacity, p constant number of molecules

f n R

number of degrees of freedom number of moles molar gas constant

γ

ratio of heat capacities

Eq

Classical equipartitiona

1 Eq = kT 2

Ideal gas heat capacities

1 1 CV = fNk = fnR 2  2 f Cp = Nk 1 + 2 2 Cp =1+ γ= CV f

a System

in thermal equilibrium at temperature T .

(5.100)

(5.101) (5.102) (5.103)

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5.5

Thermodynamics

Statistical thermodynamics

Statistical entropy Boltzmann formulaa Gibbs entropyb

S = k lnW  k lng(E) S = −k



pi lnpi

entropy Boltzmann constant number of accessible microstates

(5.105)

S k W

(5.106)

g(E) density of microstates with energy E  i sum over microstates

(5.104)

i

pi

probability that the system is in microstate i

N two-level systems

W=

N! (N − n)!n!

(5.107)

N n

number of systems number in upper state

N harmonic oscillators

W=

(Q + N − 1)! Q!(N − 1)!

(5.108)

Q

total number of energy quanta available

a Sometimes b Sometimes

called “configurational entropy.” Equation (5.105) is true only for large systems. called “canonical entropy.”

Ensemble probabilities Microcanonical ensemblea

Partition functionb

pi

1 pi = W

Z=

(5.109)



−βEi

e

(5.110)

i

Canonical ensemble (Boltzmann distribution)c

pi =

Grand partition function

Ξ=

Grand canonical ensemble (Gibbs distribution)d

pi =

a Energy

1 −βEi e Z 

−β(Ei −µNi )

e

(5.111)

(5.112)

i

1 −β(Ei −µNi ) e Ξ

(5.113)

fixed. called “sum over states.” c Temperature fixed. d Temperature fixed. Exchange of both heat and particles with a reservoir. b Also

W Z 

probability that the system is in microstate i number of accessible microstates partition function

β Ei

sum over microstates = 1/(kT ) energy of microstate i

k T

Boltzmann constant temperature

Ξ µ

grand partition function chemical potential

Ni

number of particles in microstate i

i

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5.5 Statistical thermodynamics

Macroscopic thermodynamic variables Helmholtz free energy

F = −kT lnZ

(5.114)

Grand potential

Φ = −kT lnΞ

Internal energy

U =F +TS =−

Entropy

S =−

∂ lnZ   ∂β V ,N

∂F  ∂(kT lnZ)   =  V ,N ∂T V ,N ∂T   ∂F  ∂(kT lnZ)  p=− =   T ,N ∂V T ,N ∂V   ∂(kT lnZ)  ∂F  µ=  =−  V ,T ∂N V ,T ∂N

Pressure Chemical potential

F k T

Helmholtz free energy Boltzmann constant temperature

Z

partition function

(5.115)

Φ Ξ

grand potential grand partition function

(5.116)

U β

internal energy = 1/(kT )

S

entropy

N

number of particles

(5.118)

p

pressure

(5.119)

µ

chemical potential

(5.117)

5

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

Bose–Einstein distribution

1.2

(µ = 0)

Fermi–Dirac distribution

1

5

0.6

5

10

β =1

0.4

10

0.2

50 0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2

0 0 0.2 0.4 0.6 0.8

i

Bose–Einstein distributiona Fermi–Dirac distributionb

fi = fi =

1

(5.121)

eβ(i −µ) + 1 2

Bose condensation temperature

Tc =

h ¯ 2m



2

6π n g

2/3

 2/3 n 2π¯ h2 mk gζ(3/2)

bosons. fi ≥ 0. fermions. 0 ≤ fi ≤ 1. c For noninteracting particles. At low temperatures, µ   . F b For

2

fi

mean occupation number of ith state

(5.120)

eβ(i −µ) − 1

F =

1 1.2 1.4 1.6 1.8

i

1

Fermi energyc

a For

(µ = 1)

50

0.8

β =1 fi

fi

Identical particles

(5.122)

β

= 1/(kT )

i µ

energy quantum for ith state chemical potential

F h ¯

Fermi energy (Planck constant)/(2π)

n m g

particle number density particle mass spin degeneracy (= 2s + 1)

ζ

Riemann zeta function ζ(3/2)  2.612

Tc

Bose condensation temperature

(5.123)

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Thermodynamics

Population densitiesa   −(χmj − χlj ) nmj gmj = exp Boltzmann nlj glj kT excitation   −hν g mj lm equation exp = glj kT  −χij ! gij exp Zj (T ) = kT Partition i function nij −χij ! gij exp = Nj Zj (T ) kT Saha equation (general) gij h3 χIj − χij ! (2πme kT )−3/2 exp nij = n0,j+1 ne g0,j+1 2 kT Saha equation (ion populations) Nj Zj (T ) h3 χIj ! (2πme kT )−3/2 exp = ne Nj+1 Zj+1 (T ) 2 kT

nij

(5.124) gij

number density of atoms in excitation level i of ionisation state j (j = 0 if not ionised) level degeneracy

(5.125)

χij

excitation energy relative to the ground state

(5.126)

νij h k

photon transition frequency Planck constant Boltzmann constant

T

temperature

Zj

partition function for ionisation state j

(5.128)

Nj

total number density in ionisation state j

(5.129)

ne me χIj

electron number density electron mass ionisation energy of atom in ionisation state j

(5.127)

a All

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

Pressure fluctuations

pr(x) ∝ exp[S(x)/k]   −A(x) ∝ exp kT −1  2 ∂ A(x) var[x] = kT ∂x2 kT 2 ∂T  var[T ] = kT  = ∂S V CV  ∂V  var[V ] = −kT  = κT V kT ∂p T ∂S  var[S] = kT  = kCp ∂T p KS kT ∂p  var[p] = −kT  = ∂V S V

Density fluctuations

var[n] =

Fluctuation probability General variance Temperature fluctuations Volume fluctuations Entropy fluctuations

a In

n2 n2 κT kT var[V ] = V2 V

(5.130) (5.131)

(5.132)

pr

probability density

x S A

unconstrained variable entropy availability

var[·] mean square deviation k Boltzmann constant T

temperature

V

volume

CV

heat capacity, V constant

(5.134)

p κT

pressure isothermal compressibility

(5.135)

Cp

heat capacity, p constant

(5.136)

KS

adiabatic bulk modulus

(5.137)

n

number density

(5.133)

part of a large system, whose mean temperature is fixed. Quantum effects are assumed negligible.

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5.6 Fluctuations and noise

Noise

Nyquist’s noise theorem

dw = kT · β(eβ − 1)−1 dν = kTN dν  kT dν

(hν  kT )

Johnson (thermal) noise voltagea

vrms = (4kTN R∆ν)1/2

Shot noise (electrical)

Irms = (2eI0 ∆ν)1/2 

Noise figureb

fdB = 10log10 1 + 

Relative power a Thermal b Noise

G = 10log10

P2 P1



TN T0

w

exchangeable noise power

k T

Boltzmann constant temperature

TN β ν

noise temperature = hν/(kT ) frequency

(5.141)

h vrms R

Planck constant rms noise voltage resistance

(5.142)

∆ν Irms −e

bandwidth rms noise current electronic charge

(5.143)

I0 fdB T0

mean current noise figure (decibels) ambient temperature (usually taken as 290 K)

G

decibel gain of P2 over P1

(5.138) (5.139) (5.140)



(5.144)

P1 , P2 power levels

voltage over an open-circuit resistance. figure can also be defined as f = 1 + TN /T0 , when it is also called “noise factor.”

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5.7

Thermodynamics

Radiation processes

Radiometrya

Ω

radiant energy radiance (generally a function of position and direction) angle between dir. of dΩ and normal to dA solid angle

(5.146)

A t

area time

(5.147)

Φe

radiant flux

We radiant energy density dV differential volume of propagation medium

Qe Le

 Radiant energyb

Qe =

Le cosθ dA dΩ dt

J

(5.145) θ

Radiant flux (“radiant power”) Radiant energy densityc

Φe =

∂Qe ∂t 

W

Le cosθ dA dΩ

= We =

∂Qe ∂V

Jm−3

(5.148)

Me =

∂Φe ∂A

Wm−2

(5.149)

Radiant exitanced =

Me radiant exitance

Le cosθ dΩ

(5.150) z

Irradiancee

∂Φe Ee = ∂A =

−2

(5.151)

Le cosθ dΩ

(5.152)

Wm

(normal)

x

θ

dΩ

dA φ

Radiant intensity

∂Φe Ie = ∂Ω =

a Radiometry

(5.153)

Le cosθ dA

(5.154)

1 ∂ 2 Φe cosθ dAdΩ 1 ∂Ie = cosθ ∂A

Le = Radiance

Wsr−1

Wm−2 sr−1

Ee Ie

irradiance radiant intensity

(5.155) (5.156)

is concerned with the treatment of light as energy. called “total energy.” Note that we assume opaque radiant surfaces, so that 0 ≤ θ ≤ π/2. c The 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, M = πL . e e e Power per unit area incident on a surface. b Sometimes

y

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5.7 Radiation processes

Photometrya 

Luminous energy (“total light”)

Qv =

Luminous flux

∂Qv Φv = ∂t 

Lv cosθ dA dΩ dt

= Luminous densityb Luminous exitancec

Illuminance (“illumination”)d

Luminous intensitye

Luminance (“photometric brightness”)

(5.158)

Lv cosθ dA dΩ

(5.159)

∂Qv ∂V

lmsm−3

Mv =

∂Φv ∂A

lx

Ev =

Iv = =

Ω

solid angle

A t Φv

area time luminous flux

Wv luminous density V volume

(5.162)

Lv cosθ dΩ

(5.164)

cd

(5.165)

Lv cosθ dA

(5.166)

1 ∂ 2 Φv cosθ dAdΩ 1 ∂Iv = cosθ ∂A

Lv =

Luminous efficacy

Luminous efficiency

V (λ) =

K(λ) Kmax

z (normal)

(5.163)

Φv Lv Iv K= = = Φe Le Ie

a Photometry

angle between dir. of dΩ and normal to dA

(5.161)

lmm−2

∂Φv ∂Ω

θ

Mv luminous exitance

Lv cosθ dΩ

∂Φv ∂A

=

(5.160)

(lmm−2 )

luminous energy luminance (generally a function of position and direction)

(5.157)

lumen (lm)

Wv =

=

lms

Qv Lv

cdm−2

x

θ

dA φ

y

Ev Iv

illuminance luminous intensity

K Le

luminous efficacy radiance

Φe Ie V

radiant flux radiant intensity luminous efficiency

(5.167) (5.168)

lmW−1

(5.169)

(5.170)

λ wavelength Kmax spectral maximum of K(λ)

is concerned with the treatment of light as seen by the human eye. instantaneous amount of luminous energy contained in a unit volume of propagating medium. c Luminous emitted flux per unit area. d Luminous incident flux per unit area. The derived SI unit is the lux (lx). 1lx = 1lmm−2 . e The SI unit of luminous intensity is the candela (cd). 1cd = 1lmsr−1 . b The

dΩ

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Thermodynamics

Radiative transfera Flux density (through a plane)

z

 Fν =

(normal)

Wm−2 Hz−1

Iν (θ,φ)cosθ dΩ

(5.171)

θ

dΩ

x φ

Mean intensityb

Jν =

1 4π

 Iν (θ,φ) dΩ

Wm−2 Hz−1

(5.172)

Fν Iν Jν

Spectral energy densityc

1 uν = c

Specific emission coefficient

ν jν = ρ

 Iν (θ,φ) dΩ

−3

Jm

−1

Hz

(5.173)

uν Ω θ jν

Wkg−1 Hz−1 sr−1

(5.174)

ν ρ

Gas linear absorption coefficient (αν  1)

αν = nσν =

Opacityd

κν =

Optical depth

τν =

αν ρ 

1 lν

m−1

kg−1 m2 κν ρ ds

Transfer equatione

1 dIν = −κν Iν + jν ρ ds dIν = −αν Iν + ν or ds

Kirchhoff’s lawf

Sν ≡

Emission from a homogeneous medium

Iν = Sν (1 − e−τν )

a The

jν ν = κν αν

y

flux density specific intensity (Wm−2 Hz−1 sr−1 ) mean intensity spectral energy density solid angle angle between normal and direction of Ω specific emission coefficient emission coefficient (Wm−3 Hz−1 sr−1 ) density

αν

linear absorption coefficient

(5.175)

n σν lν

particle number density particle cross section mean free path

(5.176)

κν

opacity

τν ds

optical depth, or optical thickness line element

Sν

source function

(5.177) (5.178) (5.179) (5.180) (5.181)

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 = 4πJ . ν c Assuming a refractive index of 1. d Or “mass absorption coefficient.” e Or “Schwarzschild’s equation.” f Under conditions of local thermal equilibrium (LTE), the source function, S , equals the Planck function, B (T ) ν ν [see Equation (5.182)].

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5.7 Radiation processes

Blackbody radiation 1050

105

1010 K 109 K

νm (T ) = c/λm (T )

108 K

1

107 K 10−5

106 K 105 K

10−10

104 K 103 K

10−15

100K 10−20 106

brightness (Bλ /Wm−2 m−1 sr−1 )

brightness (Bν /Wm−2 Hz−1 sr−1 )

1010

1040 1010 K 1030 1020

107 K 106 K 105 K

1010

104 K 1

103 K 100K

10−10

2.7K

2.7K 108

1010

1012

1014

1016

1018

10−20 10−14 10−12 10−10 10−8 10−6 10−4 10−2

1022

1020

   −1 hν 2hν 3 exp − 1 c2 kT dν Bλ (T ) = Bν (T ) dλ    −1 hc 2hc2 = 5 exp −1 λ λkT

Spectral energy density Rayleigh–Jeans law (hν  kT ) Wien’s law (hν  kT ) Wien’s displacement law Stefan– Boltzmann lawb

4π Bν (T ) c 4π uλ (T ) = Bλ (T ) c

uν (T ) =

Bν

(5.183)

Bλ

(5.184)

h

surface brightness per unit frequency (Wm−2 Hz−1 sr−1 ) surface brightness per unit wavelength (Wm−2 m−1 sr−1 ) Planck constant

Jm−3 Hz−1

(5.185)

c k

speed of light Boltzmann constant

Jm−3 m−1

(5.186)

T temperature uν,λ spectral energy density

2kT 2 2kT ν = 2 c2 λ   −hν 2hν 3 Bν (T ) = 2 exp c kT " 5.1 × 10−3 mK for Bν λm T = 2.9 × 10−3 mK for Bλ  ∞ M =π Bν (T ) dν Bν (T ) =

0 5 4

=

102

(5.182)

Bν (T ) = Planck functiona

1

wavelength (λ/m)

frequency (ν/Hz)

2π k T 4 = σT 4 15c2 h3

Energy density

4 u(T ) = σT 4 c

Greybody

M = σT 4 = (1 − A)σT 4

a With

λm (T )

109 K 108 K

Jm−3

Wm−2

(5.187) (5.188) λm

wavelength of maximum brightness

M σ

exitance Stefan–Boltzmann constant ( 5.67 × 10−8 Wm−2 K−4 )

(5.192)

u

energy density

(5.193)

 A

mean emissivity albedo

(5.189) (5.190) (5.191)

respect to the projected area of the surface. Surface brightness is also known simply as “brightness.” “Specific intensity” is used for reception. b Sometimes “Stefan’s law.” Exitance is the total radiated energy from unit area of the body per unit time.

5

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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.

6

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.

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Solid state physics

6.2

Periodic table 1 name

Hydrogen 1.007 94 1

2

H

1

89 (β) 378 HEX 1.632 13.80 20.28

2

Lithium 6.941

Beryllium 9.012 182

3

Li

4

19

K

Rb

55

Cs

87

300

38

boiling point (K)

56

3

4

5

6

7

8

9

Titanium 47.867

Vanadium 50.941 5

Chromium 51.996 1

Manganese 54.938 049

Iron 55.845

Cobalt 58.933 200

Cr

25 Mn

21

Sc

[Ca]3d1

559 2 992 HEX 1 757 1 813

Sr

39

Y

[Sr]4d1

608 4 475 HEX 1 653 1 798

57 – 71

Fr

502

Zr

[Sr]4d2

72

41

Hf

Nb

[Kr]4d4 5s1

[Ar]3d5 4s1

388 2 943

Molybdenum 95.94

42 Mo [Kr]4d5 5s1

330 10 222 315 BCC 4 973 2 896 4 913

Tantalum 180.947 9

73

24

302 7 194 BCC 3 673 2 180

Niobium 92.906 38

323 8 578 1.593 BCC 4 673 2 750

Hafnium 178.49

V

[Ca]3d3

Ta

[Yb]5d3

Tungsten 183.84

74

W

[Yb]5d4

13 276 319 16 670 330 19 254 316 BCC HEX 1.581 BCC 2 503 4 873 3 293 5 833 3 695 5 823

2 173 Actinides

Rutherfordium [261]

Ra 89 – 103 104

[Rn]7s2

5 000 BCC 923 973

Zirconium 91.224

40

23

295 6 090 1.587 BCC 3 563 2 193

[Yb]5d2

Radium [226]

88

Ti

[Ca]3d2

365 6 507 1.571 HEX 3 613 2 123

Lanthanides

Ba

22

331 4 508 1.592 HEX 3 103 1 943

Yttrium 88.905 85

Barium 137.327

614 3 594 BCC 943.2 1 001

[Rn]7s1

c/a (angle in RHL, c/a b/a in ORC & MCL)

Scandium 44.955 910

[Xe]6s2

Francium [223] 7

Ca

Strontium 87.62

571 2 583 FCC 963.1 1 050

[Xe]6s1

1 900 BCC 301.6

20

[Kr]5s2

Caesium 132.905 45 6

Calcium 40.078

532 1 530 FCC 1 033 1 113

[Kr]5s1

1 533 BCC 312.4

lattice constant, a (fm)

melting point (K)

321 1.624 1 363

[Ar]4s2

Rubidium 85.467 8

37

295 1.587 3 563

[Ne]3s2

429 1 738 HEX 1 153 923

[Ar]4s1

862 BCC 336.5

4 508 HEX 1 943

crystal type

229 1.568 2 745

12 Mg

Potassium 39.098 3

5

density (kgm−3 )

Be

Na

[Ne]3s1

symbol

Ti

22

[Ca]3d2

Magnesium 24.305 0

966 BCC 370.8

4

electron configuration

Sodium 22.989 770

11

Titanium 47.867

[He]2s2

[He]2s1

533 (β) 351 1 846 HEX BCC 453.65 1 613 1 560

3

relative atomic mass (u)

atomic number

1s1

[Ca]3d5

7 473 FCC 1 523

Tc

[Sr]4d5

Fe

[Ca]3d6

891 7 873 BCC 2 333 1 813

Technetium [98]

43

26

Co

[Ca]3d7

287 8 800 () 251 HEX 1.623 3 133 1 768 3 203

Ruthenium 101.07

44

27

Ru

[Kr]4d7 5s1

Rhodium 102.905 50

45

Rh

[Kr]4d8 5s1

11 496 274 12 360 270 12 420 380 HEX 1.604 HEX 1.582 FCC 2 433 4 533 2 603 4 423 2 236 3 973 Rhenium 186.207

75

Re

[Yb]5d5

Osmium 190.23

76

Os

[Yb]5d6

Iridium 192.217

77

Ir

[Yb]5d7

21 023 276 22 580 273 22 550 384 HEX 1.615 HEX 1.606 FCC 3 459 5 873 3 303 5 273 2 720 4 703

Dubnium [262]

Seaborgium [263]

Bohrium [264]

Hassium [265]

Meitnerium [268]

Rf

105 Db

106 Sg

107 Bh

108 Hs

109 Mt

[Ra]5f 14 6d2

[Ra]5f 14 6d3 ?

[Ra]5f 14 6d4 ?

[Ra]5f 14 6d5 ?

[Ra]5f 14 6d6 ?

[Ra]5f 14 6d7 ?

Lanthanum 138.905 5

Cerium 140.116

Promethium [145]

Samarium 150.36

515 1 773

Lanthanides

57

La

[Ba]5d1

6 174 HEX 1 193

89

Ce

[Ba]4f 1 5d1

59

Ac

[Ra]6d1

Thorium 232.038 1

90

Th

[Ra]6d2

Pr

[Ba]4f 3

377 6 711 (γ) 516 6 779 3.23 FCC HEX 3 733 1 073 3 693 1 204

Actinium [227]

Actinides

58

Praseodymium Neodymium 144.24 140.907 65

Pa

[Rn]5f 2 6d1 7s2

Nd

[Ba]4f 4

367 7 000 3.222 HEX 3 783 1 289

Protactinium 231.035 88

91

60

[Ba]4f 5

366 7 220 3.225 HEX 3 343 1 415

Uranium 238.028 9

92

61 Pm

U

[Rn]5f 3 6d1 7s2

Np

[Rn]5f 4 6d1 7s2

Sm

[Ba]4f 6

365 7 536 3.19 HEX 3 573 1 443

Neptunium [237]

93

62

363 7.221 2 063

Plutonium [244]

94

Pu

[Rn]5f 6 7s2

285 20 450 666 19 816 618 10 060 531 11 725 508 15 370 392 19 050 1.736 ORC 0.733 MCL 1.773 FCC TET 0.825 ORC FCC 2.056 0.709 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

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6.2 Periodic table

18 Helium 4.002 602

He

2 BCC CUB DIA FCC HEX MCL ORC RHL TET (t-pt)

1s2

body-centred cubic simple cubic diamond face-centred cubic hexagonal monoclinic orthorhombic rhombohedral tetragonal triple point

13

14

15

16

17

Boron 10.811

Carbon 12.0107

Nitrogen 14.006 74

Oxygen 15.999 4

Fluorine 18.998 403 2

5

B

[Be]2p1

2 466 RHL 2 348

13

Al

[Mg]3p1

11

12

Copper 63.546

Zinc 65.39

Ni

[Ca]3d8

8 907 FCC 1 728

Cu

[Ar]3d10 4s1

30

Pd

[Kr]4d10

Silver 107.868 2

47

Ag

[Pd]5s1

Cadmium 112.411

48

Platinum 195.078

Pt

[Xe]4f 14 5d9 6s1

Gold 196.966 55

79

Au

[Xe]4f 14 5d10 6s1

Cd

[Pd]5s2

11 995 389 10 500 409 8 647 FCC FCC HEX 1 828 3 233 1 235 2 433 594.2

78

Zn

[Ca]3d10

Gallium 69.723

31

49

In

[Cd]5p1

Ge

[Zn]4p2

452 5 323 DIA 2 473 1211

15

Sn

[Cd]5p2

Hg

[Yb]5d10

Thallium 204.383 3

81

Tl

[Hg]6p1

Lead 207.2

82

Pb

[Hg]6p2

P

[Mg]3p3

33

Unununium [272]

Ununbium [285]

63

Eu

[Ba]4f 7

5 248 BCC 1 095

Gadolinium 157.25

64

458 7 870 HEX 1 873 1 587

Americium [243]

95 Am [Ra]5f 7

Gd

[Ba]4f 7 5d1

Terbium 158.925 34

65

363 8 267 1.591 HEX 3 533 1 633

Curium [247]

96 Cm [Rn]5f 7 6d1 7s2

Tb

[Ba]4f 9

97

Bk

[Ra]5f 9

13 670 347 13 510 350 14 780 HEX 3.24 HEX 3.24 HEX 1 449 2 873 1 618 3 383 1 323

114

Dysprosium 162.50

66

Dy

[Ba]4f 10

361 8 531 1.580 HEX 3 493 1 683

Berkelium [247]

F

[Be]2p5

10

S

Chlorine 35.452 7

17

As

Selenium 78.96

34

Cl

[Mg]3p5

331 2 086 1 046 2 030 2.340 ORC ORC 1.229 550 388.47 717.82 172

Se

35

Sb

[Cd]5p3

Tellurium 127.60

52

18

451 6 247 57◦ 7 HEX 1 860 723

Bismuth 208.980 38

Bi

[Hg]6p3

Te

624 1 656 FCC 239.1 83.81

Br

Polonium [209]

84

Po

[Hg]6p4

I

[Cd]5p5

446 4 953 1.33 ORC 1 263 386.7

87.30

Krypton 83.80

36

Kr 581 119.9

Xenon 131.29

54

Xe

[Cd]5p6

727 3 560 FCC 457 161.3

635

1.347 0.659

Astatine [210]

85

532

[Zn]4p6

Iodine 126.904 47

53

Ar

1.324 0.718

[Zn]4p5

[Cd]5p4

27.07

[Mg]3p6

Bromine 79.904

[Zn]4p4

446

Argon 39.948

668 3 000 413 4 808 (γ) 436 3 120 1.308 FCC 54◦ 7 HEX 1.135 ORC 0.672 958 265.90 332.0 115.8 (t-pt) 493

Antimony 121.760

83

16

Ne

[Be]2p6

At

[Hg]6p5

165.0

Radon [222]

86

Rn

[Hg]6p6

337

440

1 233 573

623 202

211

Ununquadium [289]

110 Uun 111 Uuu 112 Uub

Europium 151.964

9

1.320 3.162

[Zn]4p3

51

Sulfur 32.066

21 450 392 19 281 408 13 546 300 11 871 346 11 343 495 9 803 475 9 400 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 Ununnilium [271]

O

[Mg]3p4

Arsenic 74.921 60

566 5 776 RHL 3103 883

Tin 118.710

50

8

[Be]2p4

Phosphorus 30.973 761

543 1 820 ORC 3 533 317.3

Germanium 72.61

1.001 1.695

Indium 114.818

Si

[Mg]3p2

32

N

[Be]2p3

356 1.631 4.22

Neon 20.179 7

550 1 442 357 1 035 (β) 405 1 460 (γ) 683 1 140 1.32 FCC MCL HEX 1.631 CUB 0.61 (t-pt) 63 77.35 54.36 90.19 53.55 85.05 24.56

Silicon 28.085 5

14

7

325 7 285 (β) 583 6 692 298 7 290 1.886 TET 1.521 TET 0.546 RHL 1 043 429.75 2 343 505.08 2 893 903.8

Mercury 200.59

80

Ga

[Zn]4p1

361 7 135 266 5 905 352 8 933 FCC HEX 1.856 ORC 3 263 1 357.8 2 833 692.68 1 183 302.9

Palladium 106.42

46

29

[Be]2p2

2 698 405 2 329 FCC DIA 933.47 2 793 1 683

10 28

C

6

1017 2 266 65◦ 7 DIA 4 273 4 763

Aluminium 26.981 538

Nickel 58.693 4

120 HEX 3-5

98

Cf

[Ra]5f 10

342 15 100 3.24 HEX 1 173

Holmium 164.930 32

67

Ho

[Ba]4f 11

359 8 797 1.573 HEX 2 833 1 743

Californium [251]

Uuq

Es

[Ra]5f 11

338 3.24 HEX 1 133

68

Er

[Ba]4f 12

358 9 044 1.570 HEX 2 973 1 803

Einsteinium [252]

99

Erbium 167.26

Thulium 168.934 21

Ytterbium 173.04

69 Tm

70

[Ba]4f 13

356 9 325 1.570 HEX 3 133 1 823

Yb

[Ba]4f 14

Lutetium 174.967

71

Lu

[Yb]5d1

354 6 966 (β) 549 9 842 1.570 FCC HEX 2 223 1 097 1 473 1 933

351 1.583 3 663

Fermium [257]

Mendelevium [258]

Nobelium [259]

Lawrencium [262]

100 Fm

101 Md

102 No

103 Lr

[Ra]5f 12

[Ra]5f 13

[Ra]5f 14

[Ra]5f 14 7p1

1 803

1 103

1 103

1 903

6

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Solid state physics

6.3

Crystalline structure

Bravais lattices Volume of primitive cell

Reciprocal primitive base vectorsa

V = (a× b) · c

(6.1)

a∗ = 2πb× c/[(a× b) · c] b∗ = 2πc×a/[(a×b) · c]

(6.2) (6.3)

c∗ = 2πa× b/[(a× b) · c]

(6.4)

a · a∗ = b · b∗ = c · c∗ = 2π

(6.5)

∗

Lattice vector Reciprocal lattice vector

∗

a,b,c V

primitive base vectors volume of primitive cell

a∗ ,b∗ ,c∗ reciprocal primitive base vectors

a · b = a · c = 0 (etc.)

(6.6)

Ruvw = ua + vb + wc

(6.7)

Ruvw u,v,w

lattice vector [uvw] integers

G hkl = ha∗ + kb∗ + lc∗

(6.8)

G hkl

reciprocal lattice vector [hkl]

exp(iG hkl · Ruvw ) = 1

(6.9)

i

i2 = −1

Weiss zone equationb

hu + kv + lw = 0

(6.10)

(hkl)

Miller indices of planec

Interplanar spacing (general)

dhkl =

2π Ghkl

(6.11)

dhkl

distance between (hkl) planes

Interplanar spacing (orthogonal basis)

h2 k 2 l 2 1 = 2+ 2+ 2 2 b c dhkl a

(6.12)

a Note

that this is 2π times the usual definition of a “reciprocal vector” (see page 20). for lattice vector [uvw] to be parallel to lattice plane (hkl) in an arbitrary Bravais lattice. c Miller indices are defined so that G hkl is the shortest reciprocal lattice vector normal to the (hkl) planes. b Condition

Weber symbols

Converting [uvw] to [UV T W ]

1 U = (2u − v) 3 1 V = (2v − u) 3 1 T = − (u + v) 3 W =w

(6.13) (6.14) (6.15)

u = (U − T ) v = (V − T )

(6.17) (6.18)

w=W

(6.19)

Zone lawa

hU + kV + iT + lW = 0

(6.20)

trigonal and hexagonal systems.

Weber indices zone axis indices

(hkil)

Miller–Bravais indices

Weber symbol zone axis symbol

(6.16)

Converting [UV T W ] to [uvw]

a For

U,V ,T ,W u,v,w [UV T W ] [uvw]

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6.3 Crystalline structure

Cubic lattices lattice lattice parameter volume of conventional cell lattice points per cell 1st nearest neighboursa 1st n.n. distance 2nd nearest neighbours 2nd n.n. distance packing fractionb reciprocal latticec primitive base vectorsd

primitive (P) a a3 1 6 a 12 √ a 2 π/6 P a1 = axˆ a2 = ayˆ a3 = aˆz

body-centred (I) a a3 2 8 √ a 3/2 6 a √ 3π/8 F ˆ a1 = a2 (yˆ + zˆ − x) a ˆ a2 = 2 (ˆz + xˆ − y) a3 = a2 (xˆ + yˆ − zˆ )

face-centred (F) a a3 4 12 √ a/ 2 6 a √ 2π/6 I a1 = a2 (yˆ + zˆ ) ˆ a2 = a2 (ˆz + x) ˆ a3 = a2 (xˆ + y)

a Or

“coordination number.” √ close-packed spheres. The maximum possible packing fraction for spheres is 2π/6. c The lattice parameters for the reciprocal lattices of P, I, and F are 2π/a, 4π/a, and 4π/a respectively. d x, ˆ y, ˆ and zˆ are unit vectors. b For

Crystal systemsa unit cellb a = b = c; α = β = γ = 90◦

latticesc

system

symmetry

triclinic

none

monoclinic

one diad  [010]

a = b = c; α = γ = 90◦ , β = 90◦

P, C

orthorhombic

three orthogonal diads

a = b = c; α = β = γ = 90◦

P, C, I, F

tetragonal

one tetrad  [001]

a = b = c; α = β = γ = 90◦

P, I

trigonald

one triad  [111]

a = b = c; α = β = γ < 120◦ = 90◦

P, R

hexagonal

one hexad  [001]

a = b = c; α = β = 90◦ , γ = 120◦

P

cubic

four triads  111

a = b = c; α = β = γ = 90◦

P, F, I

a The

P

symbol “=” implies that equality is not required by the symmetry, but neither is it forbidden. cell axes are a, b, and c with α, β, and γ the angles between b : c, c : a, and a : b respectively. c The lattice types are primitive (P), body-centred (I), all face-centred (F), side-centred (C), and rhombohedral primitive (R). d A primitive hexagonal unit cell, with a triad  [001], is generally preferred over this rhombohedral unit cell. b The

6

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Solid state physics

Dislocations and cracks Edge dislocation Screw dislocation Screw dislocation energy per unit lengthb Critical crack lengthc

(6.21)

ˆl · b = b

unit vector  line of dislocation b,b Burgers vectora U

dislocation energy per unit length

(6.23)

µ R r0

shear modulus outer cutoff for r inner cutoff for r

(6.24)

L α

(6.25)

E σ

critical crack length surface energy per unit area Young modulus Poisson ratio

(6.22) 2

R µb ln 4π r0 ∼ µb2

U=

L=

ˆl

ˆl

ˆl · b = 0

4αE π(1 − σ 2 )p20

p0 applied widening stress Burgers vector is a Bravais lattice vector characterising the total relative slip were the dislocation to travel throughout the crystal. b Or “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 p , 0 cracks ≥ L will grow and smaller cracks will shrink.

b

b

ˆl r

L

a The

Crystal diffraction Laue equations

Bragg’s lawa

a(cosα1 − cosα2 ) = hλ b(cosβ1 − cosβ2 ) = kλ

(6.26) (6.27)

c(cosγ1 − cosγ2 ) = lλ

(6.28)

2kin .G + |G|2 = 0 

Atomic form factor

f(G) =

Structure factorb

S(G) =

input wavevector reciprocal lattice vector

(6.30)

f(G) r ρ(r)

atomic form factor position vector atomic electron density

fj (G)e−iG·d j

(6.31)

S(G) n dj

structure factor number of atoms in basis position of jth atom within basis

K

change in wavevector (= kout − kin )

I(K) N

scattered intensity number of lattice points illuminated intensity at temperature T intensity from a lattice with no motion mean-squared thermal displacement of atoms

vol n 

kin G

α2 ,β2 ,γ2

e−iG·r ρ(r) d3 r

j=1

Scattered intensityc

I(K) ∝ N 2 |S(K)|2

Debye– Waller factord

  1 IT = I0 exp − u2 |G|2 3

a Alternatively,

(6.29)

h,k,l λ

lattice parameters angles between lattice base vectors and input wavevector angles between lattice base vectors and output wavevector integers (Laue indices) wavelength

a,b,c α1 ,β1 ,γ1

(6.32)

IT I0

(6.33) u2

see Equation (8.32). 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 |k | = |k out |. in d Effect of thermal vibrations. b The

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6.4 Lattice dynamics

6.4

Lattice dynamics

Phonon dispersion relationsa m

m1

m

m2 (> m1 )

2a

a (2α/µ)1/2

2(α/m)1/2

ω

(2α/m1 )1/2

ω

(2α/m2 )1/2

−

π a

0

k

π a

−

π 2a

monatomic chain   ka α sin2 m 2 ! α 1/2 ω a! sinc vp = = a k m λ  ka α !1/2 ∂ω =a vg = cos ∂k m 2

Diatomic linear chainc

Identical masses, alternating spring constants a Along



ω2 =

1 α 4 ±α 2 − sin2 (ka) µ µ m1 m2

ω α m

phonon angular frequency spring constantb atomic mass

(6.36)

vp vg λ

phase speed (sincx ≡ sinπx πx ) group speed phonon wavelength

(6.37)

k a mi

wavenumber (= 2π/λ) atomic separation atomic masses (m2 > m1 )

µ

reduced mass [= m1 m2 /(m1 + m2 )]

αi

alternating spring constants

(6.34) (6.35)

1/2

α1 + α2 1 2 ± (α1 + α22 + 2α1 α2 coska)1/2 m m (6.38) " 0, 2(α1 + α2 )/m if k = 0 = (6.39) 2α1 /m, 2α2 /m if k = π/a

ω2 =

π 2a

k

diatomic chain

ω2 = 4 Monatomic linear chain

0

m α1 a

m α2

α1

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. b In the sense α = restoring force/relative displacement. c Note 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.

6

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Solid state physics

Debye theory E mean energy in a mode at ω h ¯ (Planck constant)/(2π) ω phonon angular frequency

Mean energy per phonon modea

¯hω 1 ¯ω + E = h 2 exp[¯ hω/(kB T )] − 1

Debye frequency

ωD = vs (6π 2 N/V )1/3 3 1 2 where = + vs3 vl3 vt3

(6.41) (6.42)

vs vl vt

effective sound speed longitudinal phase speed transverse phase speed

Debye temperature

hωD /kB θD = ¯

(6.43)

N V θD

number of atoms in crystal crystal volume Debye temperature

Phonon density of states

g(ω) dω =

3V ω 2 dω 2π 2 vs3 (for 0 < ω < ωD , g = 0 otherwise)

Debye heat capacity

CV = 9NkB

Dulong and Petit’s law

 3NkB

Debye T 3 law



Internal thermal energyb a Or

T3 3 θD



θD /T

0

(T  θD )

T3 12π 4 NkB 3 5 θD

U(T ) = where

x4 ex dx (ex − 1)2



(T  θD )

(6.40)

kB T

(6.44)

(6.45) (6.46) (6.47)

ωD Debye (angular) frequency

g(ω) density of states at ω CV heat capacity, V constant U thermal phonon energy within crystal D(x) Debye function

3NkB CV

0

1

T /θD

2

ωD

h¯ω 3 dω ≡ 3NkB T D(θD /T ) exp[¯ hω/(kB T )] − 1 0  3 x y3 D(x) = 3 dy x 0 ey − 1

9N 3 ωD

any simple harmonic oscillator in thermal equilibrium at temperature T . zero-point energy.

b Neglecting

Boltzmann constant temperature

(6.48) (6.49)

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6.4 Lattice dynamics

Lattice forces (simple) Van der Waals interactiona

hω 3 α2p ¯ φ(r) = − 4 (4π0 )2 r6

(6.50)

B A + r6 r12  σ !12 σ !6 − = 4 r r

φ(r) = − Lennard–Jones 6-12 potential (molecular crystals)

σ = (B/A)1/6 ;

De Boer parameter Coulomb interaction (ionic crystals)

r αp

particle separation particle polarisability

(6.51)

h ¯ 0

(Planck constant)/(2π) permittivity of free space

(6.52)

ω

angular frequency of polarised orbital

(6.53)

A,B constants ,σ Lennard–Jones parameters

 = A2 /(4B) 1/6

at

φmin Λ=

r=

2

σ

h σ(m)1/2

φ(r) two-particle potential energy

(6.54)

Λ h m

de Boer parameter Planck constant particle mass

UC lattice Coulomb energy per ion pair

e2 4π0 r0

Madelung constant electronic charge nearest neighbour separation a London’s formula for fluctuating dipole interactions, neglecting the propagation time between particles.

UC = −αM

(6.55)

αM −e r0

6 Lattice thermal expansion and conduction Gr¨ uneisen parametera

Linear expansivityb Thermal conductivity of a phonon gas Umklapp mean free pathc a Strictly,

γ=−

∂ lnω ∂ lnV

(6.56)

1 ∂p  γCV  = 3KT ∂T V 3KT V

(6.57)

1 CV vs l 3 V

lu ∝ exp(θu /T )

α=

λ=

γ ω V

Gr¨ uneisen parameter normal mode frequency volume

α linear expansivity KT isothermal bulk modulus p pressure T CV λ

temperature lattice heat capacity, constant V thermal conductivity

(6.58)

vs l

effective sound speed phonon mean free path

(6.59)

lu θu

umklapp mean free path umklapp temperature (∼ θD /2)

the Gr¨ uneisen parameter is the mean of γ over all normal modes, weighted by the mode’s contribution to CV . b Or “coefficient of thermal expansion,” for an isotropically expanding crystal. c Mean free path determined solely by “umklapp processes” – the scattering of phonons outside the first Brillouin zone.

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6.5

Solid state physics

Electrons in solids

Free electron transport properties Current density

Mean electron drift velocity

J = −nev d

vd = −

J n −e

current density free electron number density electronic charge

vd τ me

mean electron drift velocity mean time between collisions (relaxation time) electronic mass

(6.62)

E σ0

applied electric field d.c. conductivity (J = σE)

(6.63)

ω a.c. angular frequency σ(ω) a.c. conductivity

(6.60)

eτ E me

(6.61)

2

d.c. electrical conductivity

σ0 =

a.c. electrical conductivitya

σ(ω) =

ne τ me σ0 1 − iωτ

1 CV 2 c τ (6.64) 3 V π 2 nkB2 τT = (T  TF ) 3me (6.65)

λ= Thermal conductivity

Wiedemann– Franz lawb Hall coefficient Hall voltage (rectangular strip)

λ =L= σT c

π 2 kB2 3e2

Ey 1 RH = − = ne Jx Bz VH = RH

Bz Ix w

(6.66)

(6.67)

(6.68)

CV V

total electron heat capacity, V constant volume

c2 mean square electron speed kB Boltzmann constant T temperature TF

Fermi temperature

L λ

Lorenz constant ( 2.45 × 10−8 WΩ K−2 ) Jx thermal conductivity

RH Hall coefficient Ey Hall electric field Jx applied current density Bz magnetic flux density VH Hall voltage Ix w

w

Ey Bz VH

+

applied current (= Jx × cross-sectional area) strip thickness in z

an electric field varying as e−iωt . for an arbitrary band structure. c The charge on an electron is −e, where e 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. a For

b Holds

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133

6.5 Electrons in solids

Fermi gas Electron density of statesa

 3/2 2me V g(E) = 2 E 1/2 2π h2 ¯ 3 nV g(EF ) = 2 EF

E

(6.69) (6.70)

electron energy (> 0)

g(E) density of states V “gas” volume me electronic mass h ¯

(Planck constant)/(2π)

kF

Fermi wavenumber

n

number of electrons per unit volume

Fermi wavenumber

kF = (3π 2 n)1/3

(6.71)

Fermi velocity

vF = ¯hkF /me

(6.72)

vF

Fermi velocity

Fermi energy (T = 0)

EF =

¯2 kF2 h h2 ¯ = (3π 2 n)2/3 2me 2me

(6.73)

EF

Fermi energy

Fermi temperature

TF =

EF kB

(6.74)

TF kB

Fermi temperature Boltzmann constant

π2 g(EF )kB2 T 3 π 2 kB2 = T 2EF

Electron heat capacityb (T  TF )

CV e =

Total kinetic energy (T = 0)

3 U0 = nV EF 5

Pauli paramagnetism

M = χHP H 3n = µ0 µ2B H 2EF

Landau diamagnetism

1 χHL = − χHP 3

(6.75) (6.76) (6.77)

CV e heat capacity per electron T temperature

U0

total kinetic energy

χHP Pauli magnetic susceptibility

a The

(6.78)

H M

magnetic field strength magnetisation

(6.79)

µ0 µB

permeability of free space Bohr magneton

(6.80)

χHL Landau magnetic susceptibility

density of states is often quoted per unit volume in real space (i.e., g(E)/V here). (6.75) holds for any density of states.

b Equation

Thermoelectricity Thermopower

a

Peltier effect Kelvin relation a Or

J E = + ST ∇T σ

(6.81)

H = ΠJ − λ∇T

(6.82)

Π = T ST

(6.83)

E J σ

electrochemical fieldb current density electrical conductivity

ST T

thermopower temperature

H

heat flux per unit area

Π

Peltier coefficient

λ

thermal conductivity

“absolute thermoelectric power.” electrochemical field is the gradient of (µ/e) − φ, where µ is the chemical potential, −e the electronic charge, and φ the electrical potential. b The

6

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Solid state physics

Band theory and semiconductors Bloch’s theorem

Electron velocity Effective mass tensor Scalar effective massa

Ψ(r + R) = exp(ik · R)Ψ(r)

1 v b (k) = ∇k Eb (k) h ¯ 

(6.84)

(6.85)

∂2 Eb (k) ∂ki ∂kj  2 −1 2 ∂ Eb (k) ∗ m = ¯h ∂k 2

(6.87)

m∗ k

scalar effective mass = |k|

µ vd E

particle mobility mean drift velocity applied electric field

−e D T

electronic charge diffusion coefficient temperature

J ne,h µe,h

current density electron, hole, number densities electron, hole, mobilities

kB Eg

Boltzmann constant band gap

m∗e,h

electron, hole, effective masses

(6.88)

Net current density

J = (ne µe + nh µh )eE

(6.89)

Semiconductor equation

ne nh =

(kB T )3 ∗ ∗ 3/2 −Eg /(kB T ) (me mh ) e 2(π¯ h2 )3 (6.90)

Lh = (Dh τh )

1/2

¯ h b

effective mass tensor components of k

eD |v d | = |E| kB T

Le = (De τe )1/2

position vector electron velocity (for wavevector k) (Planck constant)/2π band index

mij ki

µ=

p-n junction

r vb

(6.86)

Mobility

    eV I = I0 exp −1 kB T   De Dh + I0 = en2i A Le Na Lh Nd

electron eigenstate Bloch wavevector lattice vector

Eb (k) energy band

−1

h2 mij = ¯

Ψ k R

(6.91) (6.92) (6.93) (6.94)

I

current

I0 V

saturation current bias voltage (+ for forward)

ni A De,h

intrinsic carrier concentration area of junction electron, hole, diffusion coefficients electron, hole, diffusion lengths

Le,h

electron, hole, recombination times Na,d acceptor, donor, concentrations a Valid for regions of k-space in which E (k) can be taken as independent of the direction of k. b τe,h

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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 0 → 1/(4π)

µ0 → 4π/c2

B → B/c

χE → 4πχE

χH → 4πχH

H → cH/(4π)

M → cM

D → D/(4π)

A → A/c

The quantities ρ, J , E, φ, σ, P, r , and µr are all unchanged.

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7.2

Electromagnetism

Static fields

Electrostatics Electrostatic potential

E = −∇φ

(7.1) 

Potential differencea

φa − φb =

Poisson’s Equation (free space)

∇2 φ = −

b

 E · dl = −

a

a

E · dl

b

(7.2) ρ 0

(7.3)

q (7.4) 4π0 |r − r  | q(r − r  ) E(r) = (7.5) 4π0 |r − r  |3  ρ(r  )(r − r  )  1 dτ (7.6) E(r) = 4π0 |r − r  |3

E φ

electric field electrostatic potential

φa

potential at a

φb dl

potential at b line element

ρ

charge density

0

permittivity of free space

q

point charge

φ(r) =

Point charge at r  Field from a charge distribution (free space) a Between

dτ dτ volume element r position vector of dτ

volume

r

 r

points a and b along a path l.

-

Magnetostaticsa Magnetic scalar potential

B = −µ0 ∇φm

φm in terms of the solid angle of a generating current loop

φm =

Biot–Savart law (the field from a line current)

µ0 I B(r) = 4π

Amp`ere’s law (differential form)

∇× B = µ0 J

Amp`ere’s law (integral form) a In

free space.

(7.7)

IΩ 4π

Ω

loop solid angle

I

current

dl

line element in the direction of the current

r

position vector of dl

(7.10)

J µ0

current density permeability of free space

(7.11)

Itot total current through loop

(7.8)  line

dl× (r − r  ) |r − r  |3

(7.9)


B · dl = µ0 Itot

φm magnetic scalar potential B magnetic flux density

dl

s  W

r

I

r

-

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7.2 Static fields

Capacitancea Of sphere, radius a

C = 4π0 r a

(7.12)

Of circular disk, radius a

C = 80 r a

(7.13)

Of two spheres, radius a, in contact

C = 8π0 r aln2

(7.14)

Of circular solid cylinder, radius a, length l

C  [8 + 4.1(l/a)0.76 ]0 r a

(7.15)

Of nearly spherical surface, area S

C  3.139 × 10−11 r S 1/2

(7.16)

Of cube, side a

C  7.283 × 10−11 r a

(7.17)

Between concentric spheres, radii a < b

C = 4π0 r ab(b − a)−1

(7.18)

Between coaxial cylinders, radii a < b

2π0 r per unit length ln(b/a) π0 r per unit length C= arcosh(d/a) π0 r (d  a)  ln(2d/a)

Between parallel cylinders, separation 2d, radii a Between parallel, coaxial circular disks, separation d, radii a a For

C=

C

0 r πa2 + 0 r a[ln(16πa/d) − 1] d

(7.19) (7.20) (7.21) (7.22)

conductors, in an embedding medium of relative permittivity r .

7 Inductancea Of N-turn solenoid (straight or toroidal), length l, area A ( l 2 )

L = µ0 N 2 A/l

Of coaxial cylindrical tubes, radii a, b (a < b)

L=

Of parallel wires, radii a, separation 2d Of wire of radius a bent in a loop of radius b  a a For

µ0 b ln 2π a

(7.23)

per unit length

µ0 2d ln per unit length, (2d  a) π a   8b L  µ0 b ln − 2 a L

currents confined to the surfaces of perfect conductors in free space.

(7.24) (7.25) (7.26)

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Electromagnetism

Electric fieldsa Uniformly charged sphere, radius a, charge q Uniformly charged disk, radius a, charge q (on axis, z) Line charge, charge density λ per unit length Electric dipole, moment p (spherical polar coordinates, θ angle between p and r) Charge sheet, surface density σ a For

 

q r (r < a) 3 E(r) = 4πq0 a (7.27)  r (r ≥ a) 4π0 r3   1 1 q −√ E(z) = z 2π0 a2 |z| z 2 + a2 (7.28) E(r) =

λ r 2π0 r2



r

pcosθ Er = 2π0 r3 psinθ Eθ = 4π0 r3 σ E= 20

(7.30) −

(7.31)

θ

- K + p

(7.32)

r = 1 in the surrounding medium.

Magnetic fieldsa Uniform infinite solenoid, current I, n turns per unit length Uniform cylinder of current I, radius a

" µ0 nI inside (axial) B= 0 outside " µ0 Ir/(2πa2 ) r < a B(r) = r≥a µ0 I/(2πr)

Magnetic dipole, moment m (θ angle between m and r)

mcosθ Br = µ0 2πr3 µ0 msinθ Bθ = 4πr3

Circular current loop of N turns, radius a, along axis, z

B(z) =

The axis, z, of a straight solenoid, n turns per unit length, current I a For

(7.29)

Baxis =

(7.33)

(7.34) 

(7.35) θ

(7.36)

a2 µ0 NI 2 (a2 + z 2 )3/2

(7.38)

+ Yα1 -

z

⊗

Image charges image point

image charge

b from a conducting plane

−b

−q

b from a conducting sphere, radius a

2

a /b

−qa/b

−b

−q(r − 1)/(r + 1)

b

+2q/(r + 1)

b from a plane dielectric boundary: seen from the dielectric

 α2

µr = 1 in the surrounding medium.

seen from free space

K- m

(7.37)

µ0 nI (cosα1 − cosα2 ) 2

Real charge, +q, at a distance:

r

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7.3 Electromagnetic fields (general)

7.3

Electromagnetic fields (general)

Field relationships Conservation of charge

∇·J =−

Magnetic vector potential

B = ∇× A

Electric field from potentials

E =−

Coulomb gauge condition

∇·A=0

Lorenz gauge condition

∇·A+

Potential field equationsa

Expression for φ in terms of ρa Expression for A in terms of J a

∂ρ ∂t

∂A − ∇φ ∂t

(7.39)

J ρ t

current density charge density time

(7.40)

A

vector potential

(7.41)

φ

electrical potential

c

speed of light

(7.42)

1 ∂φ =0 c2 ∂t

1 ∂2 φ ρ − ∇2 φ = c2 ∂t2 0 1 ∂2 A − ∇2 A = µ0 J c2 ∂t2  ρ(r  ,t − |r − r  |/c)  1 dτ φ(r,t) = 4π0 |r − r  |

(7.43)

dτ

(7.44) r

r

(7.45)

µ0 4π



J (r  ,t − |r − r  |/c)  dτ |r − r  |

-

dτ volume element

(7.46)

r

position vector of dτ

(7.47)

µ0

permeability of free space

volume

A(r,t) =



volume a Assumes

7

the Lorenz gauge.

Li´enard–Wiechert potentialsa Electrical potential of a moving point charge

φ=

q 4π0 (|r| − v · r/c)

(7.48)

q

charge

r

vector from charge to point of observation particle velocity q : v r j

v

Magnetic vector potential of a moving point charge

A=

µ0 qv 4π(|r| − v · r/c)

(7.49)

free space. The right-hand sides of these equations are evaluated at retarded times, i.e., at t = t − |r  |/c, where r  is the vector from the charge to the observation point at time t . a In

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Electromagnetism

Maxwell’s equations Differential form: ∇·E =

ρ 0

Integral form:
 1 E · ds = ρ dτ 0

(7.50)

closed surface

volume




∇·B =0

(7.51)

B · ds = 0

(7.52)

(7.53)

closed surface

∂B ∇× E = − ∂t ∂E ∇× B = µ0 J + µ0 0 ∂t




E · dl = −

(7.54)

dΦ dt

(7.55)

loop




 B · dl = µ0 I + µ0 0

(7.56) loop

E

Equation (7.51) is “Gauss’s law” Equation (7.55) is “Faraday’s law” electric field

B J ρ

magnetic flux density current density charge density

surface

ds

surface element

dτ dl Φ

volume element line element % linked magnetic flux (= B · ds) % linked current (= J · ds) time

I t

∂E · ds (7.57) ∂t

Maxwell’s equations (using D and H) Differential form: ∇ · D = ρfree

Integral form: 
D · ds = ρfree dτ

(7.58)

closed surface




∇·B =0

volume

B · ds = 0

(7.60)

(7.59)

(7.61)

closed surface

∂B ∇× E = − ∂t ∂D ∇× H = J free + ∂t D




E · dl = −

(7.62)

dΦ dt

loop






H · dl = Ifree +

(7.64)

displacement field

ρfree free charge density (in the sense of ρ = ρinduced + ρfree ) B magnetic flux density H magnetic field strength J free free current density (in the sense of J = J induced + J free )

loop

E ds dτ

(7.63) ∂D · ds ∂t

surface

electric field surface element volume element

dl line element % Φ linked magnetic flux (= B · ds) % Ifree linked free current (= J free · ds) t

time

(7.65)

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7.3 Electromagnetic fields (general)

Relativistic electrodynamics Lorentz transformation of electric and magnetic fields

Lorentz transformation of current and charge densities Lorentz transformation of potential fields

Four-vector fields

a

⊥

perpendicular to v

J ρ

current density charge density

φ

electric potential

A

magnetic vector potential

J

current density four-vector

(7.66)

E B

E ⊥ = γ(E + v× B)⊥ B  = B 

(7.67) (7.68)

γ

B ⊥ = γ(B − v× E/c2 )⊥

(7.69)

ρ = γ(ρ − vJ /c2 ) J⊥ = J⊥

(7.70) (7.71)

J = γ(J − vρ)

(7.72)

φ = γ(φ − vA )

(7.73)

A⊥ = A⊥ A = γ(A − vφ/c2 )

(7.74) (7.75)

J = (ρc,J ) ∼   φ A= ,A ∼ c   1 ∂2 2 = 2 2 ,−∇2 c ∂t

(7.76)

2

A = µ0 J ∼

a Other



electric field magnetic flux density measured in frame moving at relative velocity v Lorentz factor = [1 − (v/c)2 ]−1/2 parallel to v

E  = E 

∼

(7.77)



∼

potential four-vector

A ∼

(7.78)

2

D’Alembertian operator

(7.79)

sign conventions are common here. See page 65 for a general definition of four-vectors.

7

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7.4

Electromagnetism

Fields associated with media

Polarisation Definition of electric dipole moment Generalised electric dipole moment

Electric dipole potential Dipole moment per unit volume (polarisation)a Induced volume charge density

p = qa

(7.80)

 p=





r ρ dτ

(7.81)

volume

φ(r) =

p ·r 4π0 r3

(7.82)

±q a p

end charges charge separation vector (from − to +) dipole moment

ρ dτ

charge density volume element

r φ r

vector to dτ dipole potential vector from dipole

0

permittivity of free space

P = np

(7.83)

P n

polarisation number of dipoles per unit volume

∇ · P = −ρ ind

(7.84)

ρ ind

volume charge density

Induced surface charge density

σind = P · sˆ

(7.85)

σind sˆ

unit normal to surface

Definition of electric displacement

D = 0 E + P

(7.86)

D E

electric displacement electric field

Definition of electric susceptibility

P = 0 χE E

(7.87)

χE

electrical susceptibility (may be a tensor)

Definition of relative permittivityb Atomic polarisabilityc

Depolarising fields

Clausius–Mossotti equationd a Assuming

r = 1 + χE

(7.88) (7.89) (7.90)

r

relative permittivity



permittivity

p = αE loc

(7.91)

α

polarisability

E loc

local electric field

Ed

depolarising field

Nd

depolarising factor =1/3 (sphere) =1 (thin slab ⊥ to P)

Nd P 0

(7.92)

nα r − 1 = 30 r + 2

(7.93)

p

-

+

surface charge density

D =  0 r E = E

Ed = −

−

=0 (thin slab  to P) =1/2 (long circular cylinder, axis ⊥ to P)

dipoles are parallel. The equivalent of Equation (7.112) holds for a hot gas of electric dipoles. permittivity as defined here is for a linear isotropic medium. c The polarisability of a conducting sphere radius a is α = 4π a3 . The definition p = α E 0 0 loc is also used. d With the substitution η 2 =  [cf. Equation (7.195) with µ = 1] this is also known as the “Lorentz–Lorenz formula.” r r b Relative

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7.4 Fields associated with media

Magnetisation Definition of magnetic dipole moment

dm = I ds

Generalised magnetic dipole moment

1 m= 2

Magnetic dipole (scalar) potential

φm (r) =

Dipole moment per unit volume (magnetisation)a Induced volume current density



(7.94)





r × J dτ

(7.95)

volume

µ0 m · r 4πr3

(7.96)

dτ r φm r µ0

permeability of free space

M n

magnetisation number of dipoles per unit volume

J ind = ∇× M

(7.98)

J ind

volume current density (i.e., A m−2 )

j ind

surface current density (i.e., A m−1 ) unit normal to surface

Definition of magnetic field strength, H

B = µ0 (H + M )

(7.100)

M = χH H χB B = µ0 χH χB = 1 + χH

(7.101)

Definition of magnetic susceptibility

(7.99)

sˆ B H

magnetic flux density magnetic field strength

χH

magnetic susceptibility. χB is also used (both may be tensors)

µr µ

relative permeability permeability

(7.102) (7.103)

B = µ0 µr H = µH

(7.104) (7.105)

µr = 1 + χH 1 = 1 − χB

(7.106)

dm, ds  out

6

⊗ in

volume element vector to dτ magnetic scalar potential vector from dipole

(7.97)

j ind = M ×sˆ

a Assuming

m J

dipole moment loop current loop area (right-hand sense with respect to loop current) dipole moment current density

M = nm

Induced surface current density

Definition of relative permeabilityb

dm I ds

(7.107)

all the dipoles are parallel. See Equation (7.112) for a classical paramagnetic gas and page 101 for the quantum generalisation. b Relative permeability as defined here is for a linear isotropic medium.

7

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Electromagnetism

Paramagnetism and diamagnetism

Z B

magnetic moment mean squared orbital radius (of all electrons) atomic number magnetic flux density

me −e J

electron mass electronic charge total angular momentum

g

Land´e g-factor (=2 for spin, =1 for orbital momentum)

m r2

Diamagnetic moment of an atom

Intrinsic electron magnetic momenta

Langevin function

m=−

m−

e2 Zr2 B 6me

(7.108)

e gJ 2me

(7.109)

L(x) = cothx −  x/3

1 x < 1) (x ∼



(7.110) (7.111)



Classical gas paramagnetism (|J |  ¯ h)

m0 B M = nm0 L kT

Curie’s law

χH =

µ0 nm20 3kT

(7.113)

Curie–Weiss law

χH =

µ0 nm20 3k(T − Tc )

(7.114)

a See

(7.112)

L(x) Langevin function

M m0 n T

apparent magnetisation magnitude of magnetic dipole moment dipole number density temperature

k χH

Boltzmann constant magnetic susceptibility

µ0 Tc

permeability of free space Curie temperature

also page 100.

Boundary conditions for E, D, B, and H a Parallel component of the electric field Perpendicular component of the magnetic flux density Electric displacementb

E

continuous

(7.115)

B⊥

continuous

(7.116)

sˆ · (D 2 − D 1 ) = σ

(7.117)



component parallel to interface

⊥

component perpendicular to interface

D 1,2

electrical displacements in media 1 & 2 unit normal to surface, directed 1 → 2 surface density of free charge magnetic field strengths in media 1 & 2 surface current per unit width

sˆ σ

Magnetic field strengthc a At

H 1,2

sˆ× (H 2 − H 1 ) = j s

the plane surface between two uniform media. σ = 0, then D⊥ is continuous. c If j = 0 then H is continuous.  s b If

(7.118)

js

2 1

sˆ 6

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7.5 Force, torque, and energy

7.5

Force, torque, and energy

Electromagnetic force and torque Force between two static charges: Coulomb’s law

Force between two current-carrying elements Force on a current-carrying element in a magnetic field Force on a charge (Lorentz force)

F2=

q1 q2 rˆ 2 12 4π0 r12

(7.119)

µ0 I1 I2 [dl 2× (dl 1× rˆ 12 )] dF 2 = 2 4πr12 (7.120)

dF = I dl× B

(7.121)

F2 q1,2 r 12

force on q2 charges vector from 1 to 2

ˆ 0

unit vector permittivity of free space line elements currents flowing along dl 1 and dl 2 force on dl 2

dl 1,2 I1,2 dF 2 µ0

permeability of free space

dl F I

line element force current flowing along dl

B

magnetic flux density

F = q(E + v× B)

(7.122)

E v

electric field charge velocity

Force on an electric dipolea

F = (p · ∇)E

(7.123)

p

electric dipole moment

Force on a magnetic dipoleb

F = (m · ∇)B

(7.124)

m

magnetic dipole moment

Torque on an electric dipole

G = p× E

(7.125)

G

torque

Torque on a magnetic dipole

G = m× B

(7.126)

Torque on a current loop

G = IL

dl L r

line-element (of loop) position vector of dl L

IL

current around loop


loop

r× (dl L× B) (7.127)

simplifies to ∇(p · E) if p is intrinsic, ∇(pE/2) if p is induced by E and the medium is isotropic. b F simplifies to ∇(m · B) if m is intrinsic, ∇(mB/2) if m is induced by B and the medium is isotropic. aF

F2

-

q1

r 12

dl 1

q2

* r 12

W

dlj 2

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Electromagnetism

Electromagnetic energy Electromagnetic field energy density (in free space)

1 B2 1 u = 0 E 2 + 2 2 µ0

Energy density in media

1 u = (D · E + B · H) 2

(7.129)

Energy flow (Poynting) vector

N = E×H

(7.130)

Mean flux density at a distance r from a short oscillating dipole

N =

Total mean power from oscillating dipolea

W=

ω 4 p20 sin2 θ r 32π 2 0 c3 r3

ω 4 p20 /2 6π0 c3 1 2

u E B

energy density electric field magnetic flux density

0 µ0

permittivity of free space permeability of free space

D H c

electric displacement magnetic field strength speed of light

N

energy flow rate per unit area ⊥ to the flow direction

p0 r

amplitude of dipole moment vector from dipole (wavelength)

θ ω

angle between p and r oscillation frequency

(7.132)

W

total mean radiated power

(7.133)

Utot total energy dτ volume element r position vector of dτ

(7.128)

(7.131)



Self-energy of a charge distribution

Utot =

Energy of an assembly of capacitorsb

Utot =

1  Cij Vi Vj 2 i j

Energy of an assembly of inductorsc

Utot =

1  Lij Ii Ij 2 i j

Intrinsic dipole in an electric field

Udip = −p · E

(7.136)

Intrinsic dipole in a magnetic field

Udip = −m · B

(7.137)

Hamiltonian of a charged particle in an EM fieldd a Sometimes

φ(r)ρ(r) dτ

φ ρ

electrical potential charge density

(7.134)

Vi Cij

potential of ith capacitor mutual capacitance between capacitors i and j

(7.135)

Lij

mutual inductance between inductors i and j

volume

H=

|pm − qA|2 + qφ 2m

called “Larmor’s formula.”

b C is the self-capacitance of the ith body. Note that C = C . ii ij ji c L is the self-inductance of the ith body. Note that L = L . ii ij ji d Newtonian limit, i.e., velocity  c.

(7.138)

Udip energy of dipole p

electric dipole moment

m

magnetic dipole moment

H pm

Hamiltonian particle momentum

q m A

particle charge particle mass magnetic vector potential

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7.6 LCR circuits

7.6

LCR circuits

LCR definitions Current

Ohm’s law

I=

dQ dt

V = IR

Ohm’s law (field form)

J = σE

Resistivity

ρ=

1 RA = σ l Q V

C=

Current through capacitor

I =C

Self-inductance

L=

Voltage across inductor

V = −L

Mutual inductance

Φ1 = L21 L12 = I2

Linked magnetic flux through a coil

R V I J

resistance potential difference over R current through R current density

(7.141)

E σ ρ

electric field conductivity resistivity

(7.142)

A

area of face (I is normal to face)

l C V

length capacitance potential difference across C

I t

current through C time

Φ I

total linked flux current through inductor

V

potential difference over L

Φ1 L12

total flux from loop 2 linked by loop 1 mutual inductance

I2

current through loop 2

k

coupling coefficient between L1 and L2 (≤ 1)

Φ N

linked flux number of turns around φ flux through area of turns

(7.144)

Φ I

Φ = Nφ

charge

(7.143)

dV dt

|L12 | = k

current

Q

(7.140)

Capacitance

Coefficient of coupling

I

(7.139)

(7.145) dI dt

(

(7.146)

L1 L2

(7.147)

(7.148)

(7.149)

φ



7 /l A

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Electromagnetism

Resonant LCR circuits Phase resonant frequencya

ω02 =

series

" 1/LC 1/LC − R 2 /L2

(series) (parallel) (7.150)

Tuningb

δω 1 R = = ω0 Q ω0 L

Quality factor

Q = 2π

a At

ω0

(7.151)

stored energy energy lost per cycle

L C

resonant angular frequency inductance capacitance

R

L

C

parallel

R resistance δω half-power bandwidth Q quality factor

(7.152)

which the impedance is purely real. the capacitor is purely reactive. If L and R are parallel, then 1/Q = ω0 L/R.

b Assuming

Energy in capacitors, inductors, and resistors U C

stored energy capacitance

Q V L

charge potential difference inductance

Φ I

linked magnetic flux current

(7.155)

W R

power dissipated resistance

(7.156)

τ r σ

relaxation time relative permittivity conductivity

Energy stored in a capacitor

1 Q2 1 1 U = CV 2 = QV = 2 2 2 C

(7.153)

Energy stored in an inductor

1 Φ2 1 1 U = LI 2 = ΦI = 2 2 2 L

(7.154)

Power dissipated in a resistora (Joule’s law)

W = IV = I 2 R =

Relaxation time

τ=

a This

V2 R

0  r σ

is d.c., or instantaneous a.c., power.

Electrical impedance Impedances in series

Z tot =



Impedances in parallel

Z tot =

(7.157)

Zn

n

 

−1 Z −1 n

(7.158)

n

i ωC

Impedance of capacitance

ZC = −

Impedance of inductance

Z L = iωL

Impedance: Z Inductance: L Conductance: G = 1/R Admittance: Y = 1/Z

Capacitance: C Resistance: R = Re[Z] Reactance: X = Im[Z] Susceptance: S = 1/X

(7.159) (7.160)

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7.6 LCR circuits

Kirchhoff’s laws



Current law

Ii = 0

(7.161)

Ii

currents impinging on node

Vi = 0

(7.162)

Vi

potential differences around loop

node

 Voltage law loop

Transformersa I2

Z1

y

: V1

 I1

V2

z

Z2

9

N1

N2

n

turns ratio

N1 N2 V1

number of primary turns number of secondary turns primary voltage

V2 I1

secondary voltage primary current

I2 Z out Z in

secondary current output impedance input impedance

Z1 Z2

source impedance load impedance

Turns ratio

n = N2 /N1

(7.163)

Transformation of voltage and current

V2 = nV1 I2 = I1 /n

(7.164) (7.165)

Output impedance (seen by Z 2 )

Z out = n2 Z 1

(7.166)

Input impedance (seen by Z 1 )

Z in = Z 2 /n2

(7.167)

a Ideal,

with a coupling constant of 1 between loss-free windings.

Star–delta transformation 1

1

‘Star’ Z1

2 Star impedances Delta impedances

Z2

Z3

Z12 3

2

Z13 Z23

Zij Zik Zij + Zik + Zjk   1 1 1 Zij = Zi Zj + + Zi Zj Zk Zi =

‘Delta’

3 (7.168) (7.169)

i,j,k node indices (1,2, or 3) Zi impedance on node i Zij impedance connecting nodes i and j

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7.7

Electromagnetism

Transmission lines and waveguides

Transmission line relations Loss-free transmission line equations Wave equation for a lossless transmission line Characteristic impedance of lossless line

∂I ∂V = −L ∂x ∂t ∂I ∂V = −C ∂x ∂t 1 ∂2 V ∂2 V = 2 LC ∂x2 ∂t 1 ∂2 I ∂2 I = LC ∂x2 ∂t2  L Zc = C  Zc =

Wave speed along a lossless line

1 vp = vg = √ LC

Input impedance of a terminated lossless line

Z in = Zc

R + iωL G + iωC

Z t coskl − iZc sinkl Zc coskl − iZ t sinkl 2 = Zc /Z t if l = λ/4

r=

Line voltage standing wave ratio

vswr =

(7.171)

L C

inductance per unit length capacitance per unit length

(7.172)

x t

distance along line time

(7.173)

Zc characteristic impedance

(7.174)

ω

resistance per unit length of conductor conductance per unit length of insulator angular frequency

vp vg

phase speed group speed

R

Characteristic impedance of lossy line

Reflection coefficient from a terminated line

I

potential difference across line current in line

V

(7.170)

Zt −Zc Zt +Zc 1 + |r| 1 − |r|

G

(7.175)

(7.176)

Z in (complex) input impedance Z t (complex) terminating impedance k wavenumber (= 2π/λ)

(7.177) (7.178) (7.179)

l

distance from termination

r

(complex) voltage reflection coefficient

(7.180)

Transmission line impedancesa  Coaxial line

Zc = 

Open wire feeder

Zc = 

b 60 µ b ln  √ ln r a 4π 2  a

(7.181)

µ l 120 l ln  √ ln 2 π  r r r

(7.182)

Paired strip

Zc =

µ d 377 d √ w r w

(7.183)

Microstrip line

377 Zc  √ r [(w/h) + 2]

(7.184)

a For

lossless lines.

Zc a

characteristic impedance (Ω) radius of inner conductor

b  µ

radius of outer conductor permittivity (= 0 r ) permeability (= µ0 µr )

r l

radius of wires distance between wires ( r)

d

strip separation

w

strip width ( d)

h

height above earth plane ( w)

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7.7 Transmission lines and waveguides

Waveguidesa

Waveguide equation

kg2 =

ω 2 m2 π 2 n2 π 2 − 2 − 2 c2 a b 

Guide cutoff frequency

νc = c

Phase velocity above cutoff

vp = (

Group velocity above cutoff

m !2 2a c

kg ω a

wavenumber in guide angular frequency guide height

b m,n c

guide width mode indices with respect to a and b (integers) speed of light

(7.186)

νc ωc

cutoff frequency 2πνc

(7.187)

vp ν

phase velocity frequency

(7.188)

vg

group velocity

ZTM

wave impedance for transverse magnetic modes wave impedance for transverse electric modes impedance of free space ( (= µ0 /0 )

(7.185)

n !2 + 2b

1 − (νc /ν)2  vg = c2 /vp = c 1 − (νc /ν)2 

Wave impedancesb

1 − (νc  ZTE = Z0 / 1 − (νc /ν)2

ZTM = Z0

/ν)2

(7.189) (7.190)

ZTE Z0

Field solutions for TEmn modesc ikg c2 ∂Bz ωc2 ∂x ikg c2 ∂Bz By = 2 ωc ∂y nπy mπx cos Bz =B0 cos a b

Bx =

iωc2 ∂Bz ωc2 ∂y −iωc2 ∂Bz Ey = ωc2 ∂x Ez =0

Ex =

Field solutions for TMmn modes ikg c2 ∂Ez ωc2 ∂x ikg c2 ∂Ez Ey = 2 ωc ∂y nπy mπx sin Ez =E0 sin a b

Ex =

a Equations

(7.191)

c

−iω ∂Ez ωc2 ∂y iω ∂Ez By = 2 ωc ∂x Bz =0

Bx =

7

b

x

a

z y (7.192)

are for lossless waveguides with rectangular cross sections and no dielectric. 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(k z − ωt)] on all components. g B0 and E0 are the amplitudes of the z components of magnetic flux density and electric field respectively. b The

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7.8

Electromagnetism

Waves in and out of media

Waves in lossless media ∂2 E ∇ E = µ 2 ∂t

(7.193)

∂2 B ∂t2

(7.194)

2

Electric field Magnetic field

∇2 B = µ

Refractive index

√ η = r µr

Impedance of free space Wave impedance

electric field permeability (= µ0 µr )



permittivity (= 0 r )

B t

magnetic flux density time

v η c

wave phase speed refractive index speed of light

Z0

impedance of free space

Z

wave impedance

H

magnetic field strength

(7.195)

c 1 v= √ = µ η  µ0 Z0 =  376.7 Ω 0  E µr Z = = Z0 H r

Wave speed

E µ

(7.196)

(7.197) (7.198)

Radiation pressurea Radiation momentum density

G=

Isotropic radiation

1 pn = u(1 + R) 3

(7.200)

Specular reflection

pn = u(1 + R)cos2 θi pt = u(1 − R)sinθi cosθi

(7.201) (7.202)

From an extended sourceb

1+R pn = c

From a point source,c luminosity L a On

N c2

(7.199)

 Iν (θ,φ)cos2 θ dΩ dν (7.203)

pn =

L(1 + R) 4πr2 c

G

momentum density

N c pn

Poynting vector speed of light normal pressure

u

incident radiation energy density

R

(power) reflectance coefficient

pt

tangential pressure

θi

angle of incidence

Iν ν Ω

specific intensity frequency solid angle

θ

angle between dΩ and normal to plane

L

source luminosity (i.e., radiant power) distance from source

(7.204) r

an opaque surface. spherical polar coordinates. See page 120 for the meaning of specific intensity. c Normal to the plane. b In

u

θi

z θ

dΩ

(normal)

x φ

y

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7.8 Waves in and out of media

Antennas z

Spherical polar geometry:

Field from a short (l  λ) electric dipole in free spacea Radiation resistance of a short electric dipole in free space

  [˙p] [p] 1 + Er = cosθ 2π0 r2 c r3   [¨p] [˙p] [p] 1 + + sinθ Eθ = 4π0 rc2 r2 c r3   p] [˙p] µ0 [¨ + 2 sinθ Bφ = 4π rc r  2 2πZ0 l ω2 l 2 = R= 6π0 c3 3 λ  2 l  789 ohm λ

r 6  θ U p 6 * φ /x r

distance from dipole

θ

angle between r and p

[p]

c

retarded dipole moment [p] = p(t − r/c) speed of light

(7.208)

l ω λ

dipole length ( λ) angular frequency wavelength

(7.209)

Z0

impedance of free space

ΩA Pn

beam solid angle normalised antenna power pattern Pn (0,0) = 1

dΩ

differential solid angle

(7.211)

G

antenna gain

Ae

effective area

ΩM

main lobe solid angle

(7.205) (7.206) (7.207)

 Beam solid angle

ΩA =

Pn (θ,φ) dΩ

y -

(7.210)

4π

4π ΩA

Forward power gain

G(0) =

Antenna effective area

Ae =

λ2 ΩA

(7.212)

Power gain of a short dipole

3 G(θ) = sin2 θ 2

(7.213)

Beam efficiency

efficiency = 

ΩM ΩA

(7.214)

antenna temperature Tb (θ,φ)Pn (θ,φ) dΩ (7.215) T sky brightness b 4π temperature a All field components propagate with a further phase factor equal to expi(kr − ωt), where k = 2π/λ. b The brightness temperature of a source of specific intensity I is T = λ2 I /(2k ). ν ν B b

Antenna temperatureb

1 TA = ΩA

TA

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Electromagnetism

Reflection, refraction, and transmissiona parallel incidence

perpendicular incidence

Ei

Er



K

Bi

θi /

θr w

Br

> θt

Ei

ηi

Er θi /

 Bi

θr w

ηt

Et *

U Br

> θt Bt =

Bt θi = θr

(7.216)

Snell’s lawb

ηi sinθi = ηt sinθt

(7.217)

Brewster’s law

tanθB = ηt /ηi

(7.218)

(ηi − ηt )2 (ηi + ηt )2 4ηi ηt T= (ηi + ηt )2 R +T =1

refractive index on incident side

ηt

refractive index on transmitted side

θi θr θt

angle of incidence angle of reflection angle of refraction

θB

Brewster’s angle of incidence for plane-polarised reflection (r = 0)

sin(θi − θt ) sin(θi + θt ) 2cosθi sinθt t⊥ = sin(θi + θt ) 2 R⊥ = r⊥ ηt cosθt 2 t T⊥ = ηi cosθi ⊥ ηi − ηt ηi + ηt 2ηi t= ηi + ηt t−r =1

r=

(7.227) (7.228) (7.229) ⊥

R

electric field parallel to the plane of incidence (power) reflectance coefficient

r

electric field perpendicular to the plane of incidence amplitude reflection coefficient

T

(power) transmittance coefficient

t

amplitude transmission coefficient



a For

ηi

r⊥ = −

ηt cosθt 2 t (7.222) T = ηi cosθi  Coefficients for normal incidencec R=

electric field magnetic flux density

Et

Law of reflection

Fresnel equations of reflection and refraction sin2θi − sin2θt r = (7.219) sin2θi + sin2θt 4cosθi sinθt t = (7.220) sin2θi + sin2θt (7.221) R = r2

E B

(7.223) (7.224) (7.225) (7.226)

(7.230) (7.231) (7.232)

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. b The incident wave suffers total internal reflection if ηi sinθ > 1. i ηt c I.e., θ = 0. Use the diagram labelled “perpendicular incidence” for correct phases. i

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7.8 Waves in and out of media

Propagation in conducting mediaa Electrical conductivity (B = 0)

σ ne τc

electrical conductivity electron number density electron relaxation time

µ B

electron mobility magnetic flux density

(7.234)

me −e η

electron mass electronic charge refractive index

(7.235)

0 ν δ

permittivity of free space frequency skin depth

2

σ = ne eµ =

ne e τc me



Refractive index of an ohmic conductorb

σ η = (1 + i) 4πν0

Skin depth in an ohmic conductor

δ = (µ0 σπν)−1/2

(7.233)

1/2

µ0 permeability of free space a relative permeability, µr , of 1. b Taking the wave to have an e−iωt time dependence, and the low-frequency limit (σ  2πν ). 0 a Assuming

Electron scattering processesa σR ω

Rayleigh cross section radiation angular frequency particle polarisability permittivity of free space

Thomson scattering cross sectionc

2  e2 8π σT = (7.237) 3 4π0 me c2 8π = re2  6.652 × 10−29 m2 3 (7.238)

α 0 σT me re

Thomson cross section electron (rest) mass classical electron radius

c

speed of light

Inverse Compton scatteringd

 2 v 4 Ptot = σT curad γ 2 2 3 c

Ptot electron energy loss rate urad radiation energy density γ Lorentz factor = [1 − (v/c)2 ]−1/2

Rayleigh scattering cross sectionb

σR =

Compton scatteringe λ me

4 2

ω α 6π0 c4

(7.236)

(7.239)

v

h (1 − cosθ) me c λ me c2 hν  = 1 − cosθ + (1/ε) θ θ φ cotφ = (1 + ε)tan 2 λ − λ =

(7.240) (7.241) (7.242)

electron speed

λ,λ incident & scattered wavelengths ν,ν  θ h me c ε

incident & scattered frequencies photon scattering angle electron Compton wavelength = hν/(me c2 )

σKN Klein–Nishina cross section

 ' & 1 1 4 2(ε + 1) ln(2ε + 1) + + − 1− σKN = ε ε2 2 ε 2(2ε + 1)2  σT (ε  1)   πr2 1  e ln2ε + (ε  1) ε 2 πre2

Klein–Nishina cross section (for a free electron) a For

Rutherford scattering see page 72. by bound electrons. c Scattering from free electrons, ε  1. d Electron energy loss rate due to photon scattering in the Thomson limit (γhν  m c2 ). e e From an electron at rest. b Scattering

(7.243) (7.244) (7.245)

7

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Electromagnetism

Cherenkov radiation cone semi-angle

θ

Cherenkov cone angle

c sinθ = ηv

c (vacuum) speed of light η(ω) refractive index

(7.246)

particle velocity

v

Radiated powera

e2 µ0 v 4π

Ptot =

ωc

7.9

c2 ω dω v 2 η 2 (ω)

1−

Ptot total radiated power −e electronic charge

(7.247)

µ0 ω ωc

0

where η(ω) ≥ a From



c v

for

0 < ω < ωc

free space permeability angular frequency cutoff frequency

a point charge, e, travelling at speed v through a medium of refractive index η(ω).

Plasma physics

Warm plasmas Landau length

e2 4π0 kB Te  1.67 × 10−5 Te−1

lL =



Electron Debye length

0 kB Te λDe = ne e2

m

m

Debye number

4 NDe = πne λ3De 3

(7.249)

0 kB Te

permittivity of free space Boltzmann constant electron temperature (K)

(7.250)

λDe electron Debye length

φ

effective potential

q r

point charge distance from q

(7.253)

NDe electron Debye number τe

5

Relaxation times (B = 0)b

Characteristic electron thermal speedc a Effective

3/2

Ti τi = 2.09 × 10 ne lnΛ 7



2kB Te vte = me

s 

mi mp

(7.254) 1/2 s (7.255)

1/2

1/2  5.51 × 103 Te

electron number density (m−3 )

(7.252)

3/2

Te τe = 3.44 × 10 ne lnΛ

ne

(7.251)

q exp(−21/2 r/λDe ) 4π0 r

φ(r) =

Landau length electronic charge

1/2

 69(Te /ne )1/2

Debye screeninga

lL −e

(7.248)

(7.256) ms−1

(7.257)

electron relaxation time

τi ion relaxation time Ti ion temperature (K) lnΛ Coulomb logarithm (typically 10 to 20) B magnetic flux density

vte

electron thermal speed

me

electron mass

(Yukawa) potential from a point charge q immersed in a plasma. < T . The Spitzer times for electrons and singly ionised ions with Maxwellian speed distributions, Ti ∼ e conductivity can be calculated from Equation (7.233). c Defined so that the Maxwellian velocity distribution ∝ exp(−v 2 /v 2 ). There are other definitions (see Maxwell– te Boltzmann distribution on page 112). b Collision

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7.9 Plasma physics

Electromagnetic propagation in cold plasmasa (2πνp )2 = Plasma frequency

ne e2 = ωp2 0 m e 1/2

νp  8.98ne

Hz

(7.258)

νp ωp

plasma frequency plasma angular frequency

(7.259)

ne me

electron number density (m−3 ) electron mass

−e 0

electronic charge permittivity of free space

η ν k

refractive index frequency wavenumber (= 2π/λ)

$1/2 # η = 1 − (νp /ν)2

(7.260)

Plasma dispersion relation (B = 0)

c2 k 2 = ω 2 − ωp2

(7.261)

ω c

angular frequency (= 2π/ν) speed of light

Plasma phase velocity (B = 0)

vφ = c/η

(7.262)

vφ

phase velocity

Plasma group velocity (B = 0)

vg = cη vφ vg = c2

(7.263) (7.264)

vg

group velocity

(7.265)

νC cyclotron frequency ωC cyclotron angular frequency νCe electron νC

Plasma refractive index (B = 0)

Cyclotron (Larmor, or gyro-) frequency

Larmor (cyclotron, or gyro-) radius

qB = ωC m νCe  28 × 109 B Hz νCp  15.2 × 106 B Hz

2πνC =

m v⊥ = v⊥ rL = ωC qB rLe = 5.69 × 10−12 rLp = 10.4 × 10−9

v⊥ ! m B! v⊥ m B

Mixed propagation modesb X(1 − X) , η2 = 1 − (1 − X) − 12 Y 2 sin2 θB ± S

(7.266) (7.267)

νCp proton νC q particle charge B

magnetic flux density (T)

(7.268)

m

particle mass (γm if relativistic)

(7.269)

rL rLe

Larmor radius electron rL

(7.270)

rLp proton rL v⊥ speed ⊥ to B (ms−1 ) θB

angle between wavefront ˆ and B normal (k)

(7.271)

where X = (ωp /ω)2 , Y = ωCe /ω, 1 and S 2 = Y 4 sin4 θB + Y 2 (1 − X)2 cos2 θB 4

Faraday rotationc

 µ0 e3 2 ∆ψ = 2 2 λ ne B · dl 8π m c ) *+ e , line 2.63×10−13

= Rλ2 a I.e.,

(7.272)

(7.273)

∆ψ rotation angle λ dl R

wavelength (= 2π/k) line element in direction of wave propagation rotation measure

plasmas in which electromagnetic force terms dominate over thermal pressure terms. Also taking µr = 1. 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. c In a tenuous plasma, SI units throughout. ∆ψ is taken positive if B is directed towards the observer. b In

7

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158

Electromagnetism

Magnetohydrodynamicsa 

Sound speed

γp vs = ρ

1/2



2γkB T = mp

1/2 (7.274)

 166T 1/2 ms−1

vA = Alfv´en speed

B (µ0 ρ)1/2

(7.276)

 2.18 × 10

16

−1/2 Bne

ms

−1

2µ0 p 4µ0 ne kB T 2vs2 = = 2 B2 B2 γvA

Plasma beta

β=

Direct electrical conductivity

σd =

Hall electrical conductivity

σH =

Generalised Ohm’s law

(7.275)

(7.277)

sound (wave) speed ratio of heat capacities hydrostatic pressure

ρ kB

plasma mass density Boltzmann constant

T mp vA

temperature (K) proton mass Alfv´en speed

B µ0 ne

magnetic flux density (T) permeability of free space electron number density (m−3 )

β

plasma beta (ratio of hydrostatic to magnetic pressure)

−e σd

electronic charge direct conductivity

σ

conductivity (B = 0)

σH

Hall conductivity

J E v Bˆ

current density electric field plasma velocity field

µ0 η

(7.278)

n2e e2 σ 2 ne e2 + σ 2 B 2

(7.279)

σB σd ne e

(7.280)

J = σd (E + v× B) + σH Bˆ × (E + v× B) (7.281)

Resistive MHD equations (single-fluid model)b ∂B = ∇× (v× B) + η∇2 B ∂t ∇p 1 ∂v + (v · ∇)v = − + (∇× B)× B + ν∇2 v ∂t ρ µ0 ρ 1 + ν∇(∇ · v) + g 3 Shear Alfv´enic ω = kvA cosθB dispersion relationc Magnetosonic 2 2 4 ω 2 k 2 (vs2 + vA ) − ω 4 = vs2 vA k cos2 θB dispersion d relation

vs γ p

(7.282)

= B/|B|

(7.283)

ν g

permeability of free space magnetic diffusivity [= 1/(µ0 σ)] kinematic viscosity gravitational field strength

(7.284)

ω k

angular frequency (= 2πν) wavevector (k = 2π/λ)

θB

angle between k and B

(7.285)

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. b Neglecting bulk (second) viscosity. c Nonresistive, inviscid flow. d Nonresistive, inviscid flow. The greater and lesser solutions for ω 2 are the fast and slow magnetosonic waves respectively. a For

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7.9 Plasma physics

Synchrotron radiation Power radiated by a single electrona

... averaged over pitch angles

v !2 2 sin θ c v !2 2  1.59 × 10−14 B 2 γ 2 sin θ c

Ptot = 2σT cumag γ 2

v !2 4 Ptot = σT cumag γ 2 3 c  1.06 × 10−14 B 2 γ 2

c

W

Single electron emission spectrumb

Characteristic frequency

eB 3 sinθ νch = γ 2 2 2πme  4.2 × 1010 γ 2 B sinθ

Spectral function

W (7.287) (7.288)

v !2

31/2 e3 B sinθ P (ν) = F(ν/νch ) 4π0 cme  2.34 × 10−25 B sinθF(ν/νch )



(7.286)

Hz

(7.289)

(7.290)

expression also holds for cyclotron radiation (v  c). total radiated power per unit frequency interval.

umag magnetic energy density = B 2 /(2µ0 ) v

electron velocity (∼ c)

γ

Lorentz factor = [1 − (v/c)2 ]−1/2 pitch angle (angle between v and B) magnetic flux density speed of light

θ B c

P (ν) emission spectrum ν frequency νch characteristic frequency −e 0 me

electronic charge free space permittivity electronic (rest) mass

(7.292)

F

spectral function

(7.293)

K5/3 modified Bessel fn. of the 2nd kind, order 5/3

WHz−1 (7.291)

1

∞

F(x) = x K5/3 (y)dy "x 2.15x1/3 (x  1)  1/2 −x (x  1) 1.25x e

Ptot total radiated power σT Thomson cross section

(7.294) (7.295)

F(x) 0.5

0

1

x 2

3

4

a This b I.e.,

7

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Electromagnetism

Bremsstrahlunga Single electron and ionb

     dW ω 2 1 2 ωb Z 2 e6 2 ωb = K + K1 dω γv 24π 4 30 c3 m2e γ 2 v 4 γ 2 0 γv 

Z 2 e6 24π 4 30 c3 m2e b2 v 2

(7.296)

(ωb  γv)

(7.297)

Thermal bremsstrahlung radiation (v  c; Maxwellian distribution)   −hν dP −51 2 −1/2 = 6.8 × 10 Z T ni ne g(ν,T )exp Wm−3 Hz−1 dV dν 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 T ν ) (hν  10 kZ ∼ kT )   10 −1 −1/2 (hν  kT ) (2.1 × 10 T ν ) dP  1.7 × 10−40 Z 2 T 1/2 ni ne dV ω

angular frequency (= 2πν)

Ze −e 0

ionic charge electronic charge permittivity of free space

c me

speed of light electronic mass

b

collision parameterc

Wm−3

P

electron velocity modified Bessel functions of order i (see page 47) Lorentz factor = [1 − (v/c)2 ]−1/2 power radiated

V ν

volume frequency (Hz)

v Ki γ

(7.299)

(7.300) W T ni

energy radiated electron temperature (K) ion number density (m−3 )

ne

electron number density (m−3 )

k h g

Boltzmann constant Planck constant Gaunt factor

a Classical

treatment. The ions are at rest, and all frequencies are above the plasma frequency. spectrum is approximately flat at low frequencies and drops exponentially at frequencies c Distance of closest approach. b The

(7.298)

> ∼ γv/b.

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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

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Optics

8.2

Interference

Newton’s ringsa nth dark ring

rn2 = nRλ0

nth bright ring

rn2 =

a Viewed

(8.1)

  1 n+ Rλ0 (8.2) 2

rn n R

radius of nth ring integer (≥ 0) lens radius of curvature

λ0

wavelength in external medium

R

rn

in reflection.

Dielectric layersa

1

R

η1

η1 N × { ηηab

RN 1

a

single layer η2

multilayer

η3 1−R

Quarter-wave condition

Single-layer reflectanceb

Dependence of R on layer thickness, m

η3

m λ0 a= η2 4

1 − RN

(8.3)

 2 η1 η3 − η22     η η + η2 1 3 2 R=   η − η 2  3   1 η1 + η3

(m odd) (8.4) (m even)

(−1)m (η1 − η2 )(η2 − η3 ) > 0

(8.5)

min if

(−1) (η1 − η2 )(η2 − η3 ) < 0

(8.6)

R = 0 if η2 = (η1 η3 )

1/2

and m odd

film thickness

m

thickness integer (m ≥ 0)

η2 λ0 R

film refractive index free-space wavelength power reflectance coefficient entry-side refractive index exit-side refractive index

η1 η3

max if

m

a

(8.7)

RN multilayer reflectance N number of layer pairs Multilayer η1 − η3 (ηa /ηb ) ηa refractive index of top RN = (8.8) layer reflectancec η1 + η3 (ηa /ηb )2N ηb refractive index of bottom layer a For normal incidence, assuming the quarter-wave condition. The media are also assumed lossless, with µ = 1. r b See page 154 for the definition of R. c For a stack of N layer pairs, giving an overall refractive index sequence η η ,η η ...η η η (see right-hand diagram). a b 3 1 a b a Each layer in the stack meets the quarter-wave condition with m = 1.



 2N 2

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8.2 Interference

Fabry-Perot etalona ∝1 θ

eiφ

e2iφ

η

e3iφ

θ

h

η η

incident rays

Incremental phase differenceb

φ = 2k0 hη  cosθ  

η sinθ = 2k0 hη 1 − η 

2 1/2

= 2πn for a maximum

Coefficient of finesse

Finesse

4R (1 − R)2 π F = F 1/2 2 λ0 =  Q ηh F=

I0 (1 − R)2 1 + R 2 − 2R cosφ I0 = 1 + F sin2 (φ/2) = I0 A(θ)

I(θ) = Transmitted intensity

Fringe intensity profile

incremental phase difference free-space wavenumber (= 2π/λ0 )

h θ θ

cavity width fringe inclination (usually  1) internal angle of refraction

η η n

cavity refractive index external refractive index fringe order (integer)

F R

coefficient of finesse interface power reflectance

(8.13)

F λ0

finesse free-space wavelength

(8.14)

Q

cavity quality factor

I

transmitted intensity

I0 A

incident intensity Airy function

(8.10) (8.11)

(8.12)

(8.15) (8.16) (8.17)

∆φ = 2arcsin(F −1/2 )

(8.18)

−1/2

(8.19)

 2F

Chromatic resolving power

λ0 R 1/2 πn  = nF δλ 1−R 2Fhη   (θ  1) λ0

Free spectral rangec

δλf = Fδλ c δνf =  2η h

a Neglecting

φ k0

(8.9)

(8.20)

∆φ phase difference at half intensity point

δλ

minimum resolvable wavelength difference

(8.21) (8.22) (8.23)

δλf wavelength free spectral range δνf frequency free spectral range

any effects due to surface coatings on the etalon. See also Lasers on page 174. 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 < δλ will not overlap. f b Between

8

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8.3

Optics

Fraunhofer diffraction

Gratingsa

coherent plane waves

Young’s double slitsb

I(s) = I0 cos2

N equally spaced narrow slits

I(s) = I0



kDs 2

(8.24)

sin(Nkds/2) N sin(kds/2)

∞ 



2 (8.25) 

I(s) diffracted intensity I0 peak intensity θ diffraction angle s D λ

= sinθ slit separation wavelength

N k d

number of slits wavenumber (= 2π/λ) slit spacing

n

diffraction order

(8.26)

δ

Dirac delta function

(8.27)

θn

angle of diffracted maximum

nλ d

(8.28)

θi

angle of incident illumination

Reflection grating

nλ sinθn − sinθi = d

(8.29)

Chromatic resolving power

λ = Nn δλ

(8.30)

Grating dispersion

n ∂θ = ∂λ dcosθ

(8.31)

Bragg’s lawc

2asinθn = nλ

(8.32)

Infinite grating

I(s) = I0

Normal incidence

sinθn =

Oblique incidence

sinθn + sinθi =

a Unless

δ s−

n=−∞

nλ d

nλ d

stated otherwise, the illumination is normal to the grating. narrow slits separated by D. c The condition is for Bragg reflection, with θ = θ . n i b Two

D

' N

d

θi θn θi θn

δλ

diffraction peak width

a

atomic plane spacing

θn

a

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8.3 Fraunhofer diffraction

Aperture diffraction y f(

coherent plane-wave illumination, normal to the xy plane

x,

y)

x

sy sx z

 General 1-D aperturea

ψ(s) ∝

∞

f(x)e−iksx dx

(8.33)

ψ I

diffracted wavefunction diffracted intensity

(8.34)

θ s

diffraction angle = sinθ

−∞

I(s) ∝ ψψ ∗ (s)

aperture amplitude transmission function x,y distance across aperture k wavenumber (= 2π/λ) f

General 2-D aperture in (x,y) plane (small angles)

Broad 1-D slitb

 ψ(sx ,sy ) ∝

f(x,y)e−ik(sx x+sy y) dxdy (8.35)

∞

sx sy

deflection  xz plane deflection ⊥ xz plane peak intensity slit width (in x) wavelength

sin2 (kas/2) (kas/2)2

(8.36)

≡ I0 sinc2 (as/λ)

(8.37)

I0 a λ

(8.38)

In

nth sidelobe intensity

(8.39)

a b

aperture width in x aperture width in y

(8.40)

J1 D

first-order Bessel function aperture diameter

λ

wavelength

I(s) = I0

Sidelobe intensity

 2 2 1 In = I0 π (2n + 1)2

Rectangular aperture (small angles)

I(sx ,sy ) = I0 sinc2

Circular aperturec

I(s) = I0

First minimumd

s = 1.22

λ D

(8.41)

First subsid. maximum

s = 1.64

λ D

(8.42)

Weak 1-D phase object

f(x) = exp[iφ(x)]  1 + iφ(x)

Fraunhofer limite

L



asx bsy sinc2 λ λ

2J1 (kDs/2) kDs/2

(∆x)2 λ

(n > 0)

2

(8.43)

φ(x) phase distribution i i2 = −1 L

(8.44)

distance of aperture from observation point aperture size

∆x Fraunhofer integral. b Note that sincx = (sinπx)/(πx). c The central maximum is known as the “Airy disk.” d The “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. e Plane-wave illumination. a The

8

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8.4

Optics

Fresnel diffraction

Kirchhoff’s diffraction formulaa y dS

ρ

dA

x

sˆ

source

θ

r

ψ0

S

P r P

(source at infinity)

Source at infinity

i ψP = − ψ0 λ

 K(θ)

eikr dA r

(8.45)

plane

where: Obliquity factor (cardioid) Source at finite distanceb

1 K(θ) = (1 + cosθ) 2

ψP = −

iE0 λ




z

(8.46)

eik(ρ+r) ˆ dS [cos(ˆs · rˆ ) − cos(ˆs · ρ)] 2ρr

ψP λ k

complex amplitude at P wavelength wavenumber (= 2π/λ)

ψ0 θ r

incident amplitude obliquity angle distance of dA from P ( λ)

dA area element on incident wavefront K dS ˆ

obliquity factor element of closed surface unit vector

s r

vector normal to dS vector from P to dS

closed surface

ρ vector from source to dS E0 amplitude (see footnote) a Also 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 The source amplitude at ρ is ψ(ρ) = E eikρ /ρ. The integral is taken over a surface enclosing the point P . 0

(8.47)

Fresnel zones y source

z1

z2

Effective aperture distancea

1 1 1 = + z z1 z2

(8.48)

Half-period zone radius

yn = (nλz)1/2

(8.49)

Axial zeros (circular aperture)

zm =

a I.e.,

R2 2mλ

(8.50)

observer z z1 z2

effective distance source–aperture distance aperture–observer distance

n λ

half-period zone number wavelength

yn zm

nth half-period zone radius distance of mth zero from aperture

R aperture radius the aperture–observer distance to be employed when the source is not at infinity.

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8.4 Fresnel diffraction

Cornu spiral √

0.8

0.6

√

Cornu Spiral

3

3

2

√

∞

Edge diffraction 2.5

5

S(w)

0.2

2

2

1 2

w

0

1.5 − 21

−0.2

C(w)

−2 −0.4

−1 √ − 5

1

√ − 3

−∞

0.5

intensity   CS(w) + 1 (1 + i)2 2

1

0.4

−0.6 −0.8 −0.8

√ − 2 −0.6

−0.4

−0.2

0 0

0.2

0.4

0.6

0.8

−4

−2

0

2

4

w

Fresnel integralsa

 w πt2 dt C(w) = cos 2 0  w πt2 S(w) = dt sin 2 0

Cornu spiral

CS(w) = C(w) + iS(w) 1 CS(±∞) = ± (1 + i) 2 ψ0 1 [CS(w) + (1 + i)] 2 21/2  1/2 2 where w = y λz

ψP = Edge diffraction

Diffraction from a long slitb

Diffraction from a rectangular aperture

a See b Slit

ψ0 ψP = 1/2 [CS(w2 ) − CS(w1 )] 2  1/2 2 where wi = yi λz ψ0 ψP = [CS(v2 ) − CS(v1 )] × 2 [CS(w2 ) − CS(w1 )]  1/2 2 where vi = xi λz  1/2 2 and wi = yi λz

also Equation (2.393) on page 45. long in x.

(8.51)

C S

Fresnel cosine integral Fresnel sine integral

(8.52) (8.53) (8.54)

(8.55) (8.56)

CS Cornu spiral v,w length along spiral ψP

complex amplitude at P

ψ0 λ z

unobstructed amplitude wavelength distance of P from aperture plane [see (8.48)] position of edge

y

(8.57)

coherent plane waves

8

(8.58) y1

(8.59)

z

(8.60) (8.61) (8.62)

y2

xi yi

positions of slit sides positions of slit top/bottom

P

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8.5

Optics

Geometrical optics

Lenses and mirrorsa object

v

r2

x1

f

f

object

u x2

f

image

r1 lens

r u v f MT

v

R u mirror

sign convention + − centred to right centred to left real object virtual object real image virtual image converging lens/ diverging lens/ concave mirror convex mirror erect image inverted image

 (8.63)

L η

optical path length refractive index

1 1 1 + = u v f

(8.64)

dl u v

ray path element object distance image distance

f

focal length

Newton’s lens formula

x1 x2 = f 2

(8.65)

x1

=v −f

x2

=u−f

Lensmaker’s formula

1 1 1 1 + = (η − 1) − u v r1 r2

ri

radii of curvature of lens surfaces

Mirror formulac

2 1 1 1 + =− = u v R f

(8.67)

R

mirror radius of curvature

Dioptre number

D=

1 f

(8.68)

D

dioptre number (f in metres)

Focal ratiod

n=

f d

(8.69)

n d

focal ratio lens or mirror diameter

Transverse linear magnification

MT = −

(8.70)

MT transverse magnification

Longitudinal linear magnification

ML = −MT2

(8.71)

ML longitudinal magnification

Fermat’s principleb

Gauss’s lens formula

a Formulas

L=

η dl

is stationary



m−1

v u

 (8.66)

assume “Gaussian optics,” i.e., all lenses are thin and all angles small. Light enters from the left. stationary optical path length has, to first order, a length identical to that of adjacent paths. c The mirror is concave if R < 0, convex if R > 0. d Or “f-number,” written f/2 if n = 2 etc. bA

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8.5 Geometrical optics

Prisms (dispersing) α

δ

θi

θt prism

Transmission angle

sinθt =(η 2 − sin2 θi )1/2 sinα − sinθi cosα

(8.72)

θi θt α

angle of incidence angle of transmission apex angle

η

refractive index

δ

angle of deviation

Deviation

δ = θi + θt − α

Minimum deviation condition

sinθi = sinθt = η sin

Refractive index

η=

sin[(δm + α)/2] sin(α/2)

(8.75)

δm

minimum deviation

Angular dispersiona

D=

2sin(α/2) dη dδ = dλ cos[(δm + α)/2] dλ

(8.76)

D λ

dispersion wavelength

a At

(8.73) α 2

(8.74)

minimum deviation.

Optical fibres L

θm cladding, ηc < ηf Acceptance angle Numerical aperture

sinθm =

1 2 (η − ηc2 )1/2 η0 f

N = η0 sinθm 

 ηf −1 ηc

fibre, ηf (8.77)

(8.78)

θm η0 ηf

maximum angle of incidence exterior refractive index fibre refractive index

ηc

cladding refractive index

N

numerical aperture

∆t temporal dispersion L fibre length c speed of light a Of a pulse with a given wavelength, caused by the range of incident angles up to θ . Sometimes called “intermodal m dispersion” or “modal dispersion.”

Multimode dispersiona

∆t ηf = L c

(8.79)

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8.6

Optics

Polarisation

Elliptical polarisationa Elliptical polarisation

y iδ

i(kz−ωt)

E = (E0x ,E0y e )e

(8.80)

Polarisation angleb

2E0x E0y tan2α = 2 cosδ 2 E0x − E0y (8.81)

Ellipticityc

e=

a−b a

(8.82)

E k z

electric field wavevector propagation axis

ωt

angular frequency × time

E0y

E0x

E0x x amplitude of E E0y y amplitude of E δ relative phase of Ey with respect to Ex α polarisation angle e a b

x

α

b

ellipticity semi-major axis semi-minor axis

a

θ

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

d

Jones vectors and matrices Normalised electric fielda

Example vectors:

Jones matrix



 Ex ; |E| = 1 Ey     1 1 1 Ex = E45 = √ 0 2 1     1 1 1 1 Er = √ El = √ 2 −i 2 i E=

E t = AE i

(8.84)

E

electric field

Ex Ey

x component of E y component of E

E45 45◦ to x axis Er right-hand circular

(8.85)

El

left-hand circular

Et Ei A

transmitted vector incident vector Jones matrix

Example matrices:  Linear polariser  x Linear polariser at 45◦ Right circular polariser λ/4 plate (fast  x) a Known

 1 0 0 0   1 1 1 2 1 1   1 1 i 2 −i 1   1 0 eiπ/4 0 i

as the “normalised Jones vector.”



Linear polariser  y Linear polariser at −45◦ Left circular polariser λ/4 plate (fast ⊥ x)

 0 0 0 1   1 1 −1 2 −1 1   1 1 −i 2 i 1   1 0 eiπ/4 0 −i

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8.6 Polarisation

Stokes parametersa E0y y

V χ

pI x

α

Q

E0x

2α

2χ U

2b

Poincar´e sphere

2a

Electric fields

Ex = E0x ei(kz−ωt) Ey = E0y ei(kz−ωt+δ)

Axial ratiob

tanχ = ±r = ±

Stokes parameters

Degree of polarisation

b a

(8.88)

I = Ex2 + Ey2

(8.89)

Q = Ex2 − Ey2 = pI cos2χcos2α

(8.90) (8.91)

U = 2Ex Ey cosδ = pI cos2χsin2α

(8.92) (8.93)

V = 2Ex Ey sinδ = pI sin2χ

(8.94) (8.95)

p=

left circular linear  x linear 45◦ to x unpolarised a Using

(8.86) (8.87)

(Q2 + U 2 + V 2 )1/2 ≤1 I Q/I 0 1 0 0

U/I V /I 0 −1 0 0 1 0 0 0

k

wavevector

ωt δ

angular frequency × time relative phase of Ey with respect to Ex

χ

(see diagram)

r

axial ratio

Ex Ey

electric field component  x electric field component  y

E0x field amplitude in x direction E0y field amplitude in y direction α polarisation angle p ·

degree of polarisation mean over time

(8.96) Q/I U/I right circular 0 0 linear  y −1 0 linear −45◦ to x 0 −1

V /I 1 0 0

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 power flux through a plane ⊥ the beam, i.e., I = (Ex2 + Ey2 )/Z0 etc., where Z0 is the impedance of free space. b The axial ratio is positive for right-handed polarisation and negative for left-handed polarisation using our definitions.

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Optics

8.7

Coherence (scalar theory)

Mutual coherence function Complex degree of coherence

Γ12 (τ) = ψ 1 (t)ψ ∗2 (t + τ) ψ 1 (t)ψ ∗2 (t + τ) [|ψ 1 |2 |ψ 2 |2 ]1/2 Γ12 (τ) = [Γ11 (0)Γ22 (0)]1/2

γ 12 (τ) =

Combined intensitya

Itot = I1 + I2 + 2(I1 I2 )1/2 [γ 12 (τ)]

Fringe visibility

V (τ) =

if |γ 12 (τ)| is a constant: if I1 = I2 : Complex degree of temporal coherenceb Coherence time and length Complex degree of spatial coherencec

(8.97)

(8.98) (8.99)

(8.100) 2(I1 I2 )1/2 |γ 12 (τ)| I1 + I2 Imax − Imin V= Imax + Imin

V (τ) = |γ 12 (τ)| ψ 1 (t)ψ ∗1 (t + τ) |ψ 1 (t)2 | % I(ω)e−iωτ dω = % I(ω) dω

γ(τ) =

∆τc =

1 ∆lc ∼ c ∆ν

ψ 1 ψ ∗2 [|ψ 1 |2 |ψ 2 |2 ]1/2 % I(ˆs)eikD·ˆs dΩ = % I(ˆs) dΩ

γ(D) =

mutual coherence function temporal interval (complex) wave disturbance at spatial point i

t

time

· γ ij

mean over time complex degree of coherence complex conjugate

∗

Itot combined intensity Ii intensity of disturbance at point i  real part of

(8.101) (8.102)

Imax max. combined intensity Imin min. combined intensity

(8.103) (8.104) (8.105)

γ(τ) degree of temporal coherence I(ω) specific intensity ω radiation angular frequency c speed of light ∆τc coherence time

(8.106)

∆lc coherence length ∆ν spectral bandwidth γ(D) degree of spatial coherence

(8.107)

D

(8.108)

I(ˆs) specific intensity of distant extended source in direction sˆ

Intensity correlationd

I1 I2 = 1 + γ 2 (D) [I1 2 I2 2 ]1/2

(8.109)

Speckle intensity distributione

pr(I) =

1 −I/I e I

(8.110)

Speckle size (coherence width)

λ ∆wc  α

a From

Γij τ ψi

spatial separation of points 1 and 2

dΩ differential solid angle sˆ unit vector in the direction of dΩ k wavenumber pr

probability density

∆wc characteristic speckle size

(8.111)

λ α

wavelength source angular size as seen from the screen

interfering the disturbances at points 1 and 2 with a relative delay τ. “autocorrelation function.” c Between two points on a wavefront, separated by D. The integral is over the entire extended source. d For wave disturbances that have a Gaussian probability distribution in amplitude. This is “Gaussian light” such as from a thermal source. e Also for Gaussian light. b Or

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8.8 Line radiation

8.8

Line radiation

Spectral line broadening Natural broadeninga

(2πτ)−1 I(ω) = (2τ)−2 + (ω − ω0 )2

Natural half-width

∆ω =

Collision broadening

I(ω) =

Collision and pressure half-widthc

 −1/2 1 2 πmkT ∆ω = = pπd τc 16

1 2τ

(8.112)

I(ω) normalised intensityb τ lifetime of excited state ω angular frequency (= 2πν)

(8.113)

∆ω half-width at half-power ω0 centre frequency

p d

mean time between collisions pressure effective atomic diameter

m k T

gas particle mass Boltzmann constant temperature

c

speed of light

τc

(πτc )−1 (τc )−2 + (ω − ω0 )2

(8.114)

(8.115)

 1/2  mc2 mc2 (ω − ω0 )2 exp − I(ω) = 2kT 2kT ω02 π ω02 (8.116)  1/2 2kT ln2 ∆ω = ω0 (8.117) mc2 

Doppler broadening Doppler half-width

I(ω) ∆ω ω0

a The

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/ω  1. 0 c The 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 Rij

Absorption

R12 = B12 Iν n1

(8.118)

Spontaneous emission

R21 = A21 n2

(8.119)

Stimulated emission

 = B21 Iν n2 R21

(8.120)

Coefficient ratios

A21 2hν 3 g1 = 2 B12 c g2 B21 g1 = B12 g2

a Note

case

(8.121) (8.122)

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

Bij Einstein B coefficients Iν specific intensity of radiation field A21 Einstein A coefficient ni

number density of atoms in quantum level i (m−3 )

h

Planck constant

ν c gi

frequency speed of light degeneracy of ith level

that the coefficients can also be defined in terms of spectral energy density, uν = 4πIν /c rather than Iν . In this 3 g 1 . See also Population densities on page 116. = 8πhν c3 g

A21 B12

2

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Optics

Lasersa R1

R2

r1

L 

Cavity stability condition

L 0≤ 1− r1

Longitudinal cavity modesb

νn =



L 1− r2

 ≤ 1 (8.123)

c n 2L

(8.124)

2πL(R1 R2 )1/4 λ[1 − (R1 R2 )1/2 ] 4πL  λ(1 − R1 R2 ) νn ∆νc = = 1/(2πτc ) Q Q=

Cavity Q

Cavity line width



∆ν 2πh(∆νc )2 = νn P

Threshold lasing condition

R1 R2 exp[2(α − β)L] > 1

a Also

r1,2 radii of curvature of end-mirrors L distance between mirror centres νn n c

mode frequency integer speed of light

(8.125)

Q quality factor R1,2 mirror (power) reflectances

(8.126)

λ

(8.127)

∆νc cavity line width (FWHP) τc cavity photon lifetime

 gl Nu gl Nu − gu Nl (8.128)

Schawlow– Townes line width

light out

r2

(8.129)

∆ν P

wavelength

line width (FWHP) laser power

gu,l degeneracy of upper/lower levels Nu,l number density of upper/lower levels α β

gain per unit length of medium loss per unit length of medium

see the Fabry-Perot etalon on page 163. Note that “cavity” refers to the empty cavity, with no lasing medium present. b The mode spacing equals the cavity free spectral range.

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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 (1M = 1.989 × 1030 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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9.2

Astrophysics

Solar system data

Solar data equatorial radius mass polar moment of inertia bolometric luminosity effective surface temperature solar constanta absolute magnitude apparent magnitude a Bolometric

R M I L T

= = = = =

MV mV

= =

6.960 × 108 m 1.9891 × 1030 kg 5.7 × 1046 kgm2 3.826 × 1026 W 5770K 1.368 × 103 Wm−2 +4.83; Mbol −26.74; mbol

= = =

109.1R⊕ 3.32946 × 105 M⊕ 7.09 × 108 I⊕

= =

+4.75 −26.82

flux at a distance of 1 astronomical unit (AU).

Earth data equatorial radius flatteninga mass polar moment of inertia orbital semi-major axisb mean orbital velocity equatorial surface gravity polar surface gravity rotational angular velocity af

R⊕ f M⊕ I⊕ 1AU

= = = = =

ge gp ωe

= = =

= 9.166 × 10−3 R 6.37814 × 106 m 0.00335364 = 1/298.183 = 3.0035 × 10−6 M 5.9742 × 1024 kg = 1.41 × 10−9 I 8.037 × 1037 kgm2 11 = 214.9R 1.495979 × 10 m 2.979 × 104 ms−1 (includes rotation) 9.780327ms−2 9.832186ms−2 7.292115 × 10−5 rads−1

equals (R⊕ − Rpolar )/R⊕ . The mean radius of the Earth is 6.3710 × 106 m. the Sun.

b About

Moon data equatorial radius mass mean orbital radiusa mean orbital velocity orbital period (sidereal) equatorial surface gravity a About

Rm Mm am

= = =

1.7374 × 106 m 7.3483 × 1022 kg 3.84400 × 108 m 1.03 × 103 ms−1 27.32166d 1.62ms−2

= = =

0.27240R⊕ 1.230 × 10−2 M⊕ 60.27R⊕

=

0.166ge

the Earth.

Planetary dataa Mercury Venusb Earth Mars Jupiter Saturn Uranusb Neptune Plutob a Using

M/M⊕ 0.055 274 0.815 00 1 0.107 45 317.85 95.159 14.500 17.204 0.00251

R/R⊕ 0.382 51 0.948 83 1 0.532 60 11.209 9.449 1 4.007 3 3.882 6 0.187 36

T (d) 58.646 243.018 0.997 27 1.025 96 0.413 54 0.444 01 0.718 33 0.671 25 6.387 2

P (yr) 0.240 85 0.615 228 1.000 04 1.880 93 11.861 3 29.628 2 84.746 6 166.344 248.348

a(AU) 0.387 10 0.723 35 1.000 00 1.523 71 5.202 53 9.575 60 19.293 4 30.245 9 39.509 0

M R

mass equatorial radius

T

rotational period

P a M⊕

orbital period mean distance 5.9742 × 1024 kg

R⊕

6.37814 × 106 m

1d 1yr

86400s 3.15569 × 107 s

1AU 1.495979 × 1011 m

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. b Retrograde rotation.

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9.3 Coordinate transformations (astronomical)

9.3

Coordinate transformations (astronomical)

Time in astronomy Julian day numbera JD =D − 32075 + 1461 ∗ (Y + 4800 + (M − 14)/12)/4

JD D Y

Julian day number day of month number calendar year, e.g., 1963

(9.1)

M ∗

calendar month (Jan=1) integer multiply

MJD = JD − 2400000.5

(9.2)

/ MJD

integer divide modified Julian day number

W = (JD + 1)

(9.3)

W

day of week (0=Sunday, 1=Monday, ... )

LCT UTC

local civil time coordinated universal time

TZC DSC T

time zone correction daylight saving correction Julian centuries between 12h UTC 1 Jan 2000 and 0h UTC D/M/Y

+ 367 ∗ (M − 2 − (M − 14)/12 ∗ 12)/12 − 3 ∗ ((Y + 4900 + (M − 14)/12)/100)/4 Modified Julian day number Day of week

mod 7

Local civil time

LCT = UTC + TZC + DSC

Julian centuries

T=

Greenwich sidereal time Local sidereal time

JD − 2451545.5 36525

(9.4)

(9.5)

GMST =6h 41m 50s .54841 + 8640184s .812866T + 0s .093104T 2 − 0s .0000062T 3 LST = GMST +

λ◦ 15◦

(9.6)

GMST Greenwich mean sidereal time at 0h UTC D/M/Y (for later times use 1s = 1.002738 sidereal seconds)

(9.7)

LST λ◦

local sidereal time geographic longitude, degrees east of Greenwich

a For

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 Hour angle

H = LST − α

Equatorial to horizon

sina = sinδ sinφ + cosδ cosφcosH −cosδ sinH tanA ≡ sinδ cosφ − sinφcosδ cosH

Horizon to equatorial

sinδ = sinasinφ + cosacosφcosA −cosasinA tanH ≡ sinacosφ − sinφcosacosA

a Conversions

(9.8) (9.9) (9.10) (9.11) (9.12)

LST

local sidereal time

H

(local) hour angle

α δ a

right ascension declination altitude

A φ

azimuth (E from N) observer’s latitude +

+

−

+ A, H

−

−

−

+

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).

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Astrophysics

Ecliptic coordinatesa ε = 23◦ 26 21 .45 − 46 .815T Obliquity of the ecliptic

ε

mean ecliptic obliquity

T

Julian centuries since J2000.0b

(9.14)

α δ

right ascension declination

(9.15)

λ β

ecliptic longitude ecliptic latitude

− 0 .0006T 2 

+ 0 .00181T

3

Equatorial to ecliptic

sinβ = sinδ cosε − cosδ sinεsinα sinαcosε + tanδ sinε tanλ ≡ cosα

Ecliptic to equatorial

sinδ = sinβ cosε + cosβ sinεsinλ sinλcosε − tanβ sinε tanα ≡ cosλ

(9.13)

+

+

(9.16)

−

+

(9.17)

−

−

−

+

λ, α

a Conversions 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). b See Equation (9.5).

Galactic coordinatesa αg = 192◦ 15 δg = 27◦ 24

(9.18) (9.19)

lg = 33◦

(9.20)

Equatorial to galactic

sinb = cosδ cosδg cos(α − αg ) + sinδ sinδg tanδ cosδg − cos(α − αg )sinδg tan(l − lg ) ≡ sin(α − αg )

(9.21)

Galactic to equatorial

sinδ = cosbcosδg sin(l − lg ) + sinbsinδg cos(l − lg ) tan(α − αg ) ≡ tanbcosδg − sinδg sin(l − lg )

Galactic frame

αg δg

lg

ascending node of galactic plane on equator

δ α b

declination right ascension galactic latitude

(9.22) (9.23) (9.24)

right ascension of north galactic pole declination of north galactic pole

l galactic longitude 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).

a Conversions

Precession of equinoxesa In right ascension

α  α0 + (3 .075 + 1 .336sinα0 tanδ0 )N

In declination

δ  δ0 + (20 .043cosα0 )N

a Right

s

s

α

right ascension of date

(9.25)

α0 N

right ascension at J2000.0 number of years since J2000.0

(9.26)

δ δ0

declination of date declination at J2000.0

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.

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9.4 Observational astrophysics

9.4

Observational astrophysics

Astronomical magnitudes Apparent magnitude

m1 − m2 = −2.5log10

F1 F2

(9.27)

Distance modulusa

m − M = 5log10 D − 5 = −5log10 p − 5

(9.28) (9.29)

Luminosity– magnitude relation

Mbol = 4.75 − 2.5log10

mi Fi

apparent magnitude of object i energy flux from object i

M m−M

absolute magnitude distance modulus

D p

distance to object (parsec) annual parallax (arcsec)

(9.30)

L  3.04 × 10(28−0.4Mbol )

Mbol L

bolometric absolute magnitude luminosity (W)

(9.31)

L

solar luminosity (3.826 × 1026 W)

Fbol  2.559 × 10−(8+0.4mbol )

(9.32)

Fbol mbol

bolometric flux (Wm−2 ) bolometric apparent magnitude

Bolometric correction

BC = mbol − mV

(9.33)

= Mbol − MV

(9.34)

BC mV

bolometric correction V -band apparent magnitude

MV

V -band absolute magnitude

Colour indexb

B − V = mB − m V

(9.35)

U − B = mU − m B

(9.36)

B −V U −B

observed B − V colour index observed U − B colour index

Colour excessc

E = (B − V ) − (B − V )0

(9.37)

Flux– magnitude relation

L L

E B − V colour excess (B − V )0 intrinsic B − V colour index

a Neglecting

extinction. the UBV magnitude system. The bands are centred around 365 nm (U), 440 nm (B), and 550 nm (V ). c The U − B colour excess is defined similarly. b Using

Photometric wavelengths Mean wavelength Isophotal wavelength Effective wavelength

% λR(λ) dλ λ0 = % R(λ) dλ % F(λ)R(λ) dλ F(λi ) = % R(λ) dλ % λF(λ)R(λ) dλ λeff = % F(λ)R(λ) dλ

mean wavelength wavelength

(9.38)

λ0 λ

(9.39)

R system spectral response F(λ) flux density of source (in terms of wavelength) λi

(9.40)

isophotal wavelength

λeff effective wavelength

9

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Astrophysics

Planetary bodies Bode’s law

a

4 + 3 × 2n DAU = 10

(9.41)



Roche limit

1/3 100M 9πρ > 2.46R (if densities equal) 0 ∼

> R∼

  1  1  1 = − S P P⊕ 

(9.42) (9.43)

DAU n

planetary orbital radius (AU) index: Mercury = −∞, Venus = 0, Earth = 1, Mars = 2, Ceres = 3, Jupiter= 4, ...

R

satellite orbital radius

M ρ

central mass satellite density

R0

central body radius

S synodic period P planetary orbital period P⊕ Earth’s orbital period a Also 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. b Of a planet.

Synodic periodb

(9.44)

Distance indicators Hubble law

v = H0 d

(9.45)

Annual parallax

Dpc = p−1

Cepheid variablesa

L  1.15log10 Pd + 2.47 (9.47) L MV  −2.76log10 Pd − 1.40 (9.48)

(9.46)

log10

 Tully–Fisher relationb

MI  −7.68log10

Einstein rings

θ2 =

2vrot sini

 − 2.58 (9.49)

4GM c2





ds − dl ds dl



Sunyaev– Zel’dovich effectc

∆T = −2 T

... for a homogeneous sphere

4Rne kTe σT ∆T =− T me c2

ne kTe σT dl me c2

(9.50)

(9.51)

(9.52)

v H0

cosmological recession velocity Hubble parameter (present epoch)

d (proper) distance Dpc distance (parsec) p annual parallax (±p arcsec from mean) L mean cepheid luminosity L Solar luminosity Pd pulsation period (days) MV absolute visual magnitude MI I-band absolute magnitude vrot observed maximum rotation velocity (kms−1 ) i galactic inclination (90◦ when edge-on) θ ring angular radius M ds dl

lens mass distance from observer to source distance from observer to lens

T dl

apparent CMBR temperature path element through cloud

R ne k

cloud radius electron number density Boltzmann constant

Te σT me

electron temperature Thomson cross section electron mass

c speed of light relation for classical Cepheids. Uncertainty in MV is ±0.27 (Madore & Freedman, 1991, Publications of the Astronomical Society of the Pacific, 103, 933). b Galaxy 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). c Scattering 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). a Period–luminosity

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9.5 Stellar evolution

9.5

Stellar evolution

Evolutionary timescales Free-fall timescalea

Kelvin–Helmholtz timescale a For



3π τff = 32Gρ0

1/2 (9.53)

−Ug L GM 2  R0 L

(9.54)

τKH =

(9.55)

τff G ρ0

free-fall timescale constant of gravitation initial mass density

τKH Kelvin–Helmholtz timescale Ug gravitational potential energy M R0 L

body’s mass body’s initial radius body’s luminosity

the gravitational collapse of a uniform sphere.

Star formation 

Jeans lengtha

π dp λJ = Gρ dρ

Jeans mass

π MJ = ρλ3J 6

Eddington limiting luminosityb

1/2 (9.56)

(9.57)

4πGMmp c LE = σT M  1.26 × 1031 M

(9.58) W

(9.59)

λJ G

Jeans length constant of gravitation

ρ p

cloud mass density pressure

MJ (spherical) Jeans mass LE M

Eddington luminosity stellar mass

M solar mass mp proton mass c speed of light σT

Thomson cross section

that (dp/dρ)1/2 is the sound speed in the cloud. b Assuming the opacity is mostly from Thomson scattering. a Note

Stellar theorya r

radial distance

Mr ρ

mass interior to r mass density

(9.61)

p G

pressure constant of gravitation

(9.62)

Lr 

luminosity interior to r power generated per unit mass

T

temperature

Conservation of mass

dMr = 4πρr2 dr

(9.60)

Hydrostatic equilibrium

dp −GρMr = dr r2

Energy release

dLr = 4πρr2  dr

Radiative transport

−3 κ ρ Lr dT = dr 16σ T 3 4πr2

(9.63)

σ Stefan–Boltzmann constant κ mean opacity

Convective transport

dT γ − 1 T dp = dr γ p dr

(9.64)

γ

a For

ratio of heat capacities, cp /cV

stars in static equilibrium with adiabatic convection. Note that ρ is a function of r. κ and  are functions of temperature and composition.

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Astrophysics

Stellar fusion processesa PP i chain

PP ii chain

p + p → 21 H + e+ + νe 2 + 3 1 H + p → 2 He + γ 3 3 4 + 2 He + 2 He → 2 He + 2p +

PP iii chain

p + p → 12 H + e+ + νe 2 + 3 1 H + p → 2 He + γ 3 4 7 2 He + 2 He → 4 Be + γ 7 − 7 4 Be + e → 3 Li + νe

+

+

p + p+ → 12 H + e+ + νe

+

+

2 + 3 1 H + p → 2 He + γ 3 4 7 2 He + 2 He → 4 Be + γ 7 + 8 4 Be + p → 5 B + γ 8 8 + 5 B → 4 Be + e + νe 8 4 4 Be → 2 2 He

7 + 4 3 Li + p → 2 2 He

CNO cycle

triple-α process

12 + 13 6C + p → 7N + γ 13 13 + 7 N → 6 C + e + νe 13 + 14 6C + p → 7N + γ 14 + 15 7N + p → 8O + γ 15 15 + 8 O → 7 N + e + νe

4 4 8  4 Be + γ 2 He + 2 He  8 4  126 C∗ 4 Be + 2 He  12 ∗ 12 6C → 6C + γ

γ

photon

+

p e+

proton positron

e− νe

electron electron neutrino

15 + 12 4 7 N + p → 6 C + 2 He a All

species are taken as fully ionised.

Pulsars ˙ ∝ −ω n ω

Braking index

P P¨ n=2− ˙2 P

Characteristic agea

1 P T= n − 1 P˙

Magnetic dipole radiation

L=

¨ 2 sin2 θ µ0 |m| 6πc3 2πR 6 Bp2 ω 4 sin2 θ = 3c3 µ0

Dispersion measure

(9.65)

ω

rotational angular velocity

(9.66)

P n

rotational period (= 2π/ω) braking index

T L

characteristic age luminosity

µ0 c

permeability of free space speed of light

(9.67)

(9.68) (9.69)

D ne dl

DM =

D dl ne

path length to pulsar path element electron number density

(9.71)

τ ∆τ

pulse arrival time difference in pulse arrival time

(9.72)

νi me

observing frequencies electron mass

(9.70)

0

b

Dispersion

−e2 dτ = 2 DM dν 4π 0 me cν 3   1 1 e2 − ∆τ = 2 DM 8π 0 me c ν12 ν22

pulsar magnetic dipole moment pulsar radius magnetic flux density at magnetic pole θ angle between magnetic and rotational axes DM dispersion measure m R Bp

n = 1 and that the pulsar has already slowed significantly. Usually n is assumed to be 3 (magnetic dipole radiation), giving T = P /(2P˙ ). b The pulse arrives first at the higher observing frequency. a Assuming

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183

9.5 Stellar evolution

Compact objects and black holes rs G M

Schwarzschild radius constant of gravitation mass of body

Schwarzschild radius

rs =

2GM M 3 km c2 M

(9.73)

Gravitational redshift

 1/2 ν∞ 2GM = 1− νr rc2

(9.74)

r ν∞ νr

distance from mass centre frequency at infinity frequency at r

Gravitational wave radiationa

Lg =

(9.75)

mi a Lg

orbiting masses mass separation gravitational luminosity

Rate of change of orbital period

G5/3 m1 m2 P −5/3 96 P˙ = − (4π 2 )4/3 5 5 c (m1 + m2 )1/3

P

orbital period

Neutron star degeneracy pressure (nonrelativistic)

h2 (3π 2 )2/3 ¯ p= 5 mn

p h ¯

pressure (Planck constant)/(2π)

mn ρ

neutron mass density

Relativisticb

p=

(9.78)

u

energy density

Chandrasekhar massc Maximum black hole angular momentum

MCh  1.46M

(9.79)

MCh Chandrasekhar mass

GM 2 c

(9.80)

c speed of light M solar mass

32 G4 m21 m22 (m1 + m2 ) 5 c5 a5

hc(3π 2 )1/3 ¯ 4

Jm =





ρ mn

ρ mn

5/3

4/3

Black hole evaporation time

τe ∼

M3 × 1066 M3

Black hole temperature

T=

M ¯c3 h  10−7 8πGMk M

2 = u 3

1 = u 3

yr

(9.76) (9.77)

(9.81) K

(9.82)

Jm

maximum angular momentum

τe

evaporation time

T

temperature

k

Boltzmann constant

a From two bodies, m and m , in circular orbits about their centre of mass. Note that the frequency of the radiation 1 2 is twice the orbital frequency. b Particle velocities ∼ c. c Upper limit to mass of a white dwarf.

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9.6

Astrophysics

Cosmology

Cosmological model parameters Hubble law

vr = Hd

(9.83)

Hubble parametera

˙ R(t) H(t) = R(t) H(z) =H0 [Ωm0 (1 + z)3 + ΩΛ0

(9.84)

+ (1 − Ωm0 − ΩΛ0 )(1 + z)2 ]1/2 (9.85) Redshift

Robertson– Walker metricb

λobs − λem R0 −1 = λem R(tem )  dr2 ds2 =c2 dt2 − R 2 (t) 1 − kr2 2

2



2

+ r (dθ + sin θ dφ )

Friedmann equationsc

! ¨ = − 4π GR ρ + 3 p + ΛR R 2 3 c 3 2 8π ΛR ˙ 2 = GρR 2 − kc2 + R 3 3

Critical density

ρcrit =

3H 2 8πG

ρ 8πGρ = ρcrit 3H 2 Λ ΩΛ = 3H 2 kc2 Ωk = − 2 2 R H Ωm + ΩΛ + Ωk = 1

Ωm = Density parameters

Deceleration parameter

¨ 0 Ωm0 R0 R − ΩΛ0 q0 = − ˙ 2 = 2 R0

radial velocity Hubble parameter

d

proper distance

0

R t

present epoch cosmic scale factor cosmic time

z

redshift

λobs observed wavelength

(9.86)

z=

2

vr H

λem tem

emitted wavelength epoch of emission

ds c

interval speed of light

(9.87)

r,θ,φ comoving spherical polar coordinates

(9.88)

k G

curvature parameter constant of gravitation

(9.89)

p Λ

pressure cosmological constant

(9.90)

ρ (mass) density ρcrit critical density

(9.91) (9.92) (9.93)

Ωm

matter density parameter

ΩΛ Ωk

lambda density parameter curvature density parameter

q0

deceleration parameter

(9.94) (9.95)

< H < 80kms−1 Mpc−1 ≡ 100hkms−1 Mpc−1 , where called the Hubble “constant.” At the present epoch, 60 ∼ 0∼ h is a dimensionless scaling parameter. The Hubble time is tH = 1/H0 . Equation (9.85) assumes a matter dominated universe and mass conservation. b For 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. c Λ = 0 in a Friedmann universe. Note that the cosmological constant is sometimes defined as equalling the value used here divided by c2 . a Often

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9.6 Cosmology

Cosmological distance measures Look-back time

Proper distance Luminosity distancea Flux density– redshift relation Angular diameter distanced

tlb (z) = t0 − t(z) 

r

dp = R0 0

(9.96)

dr = cR0 (1 − kr2 )1/2 

z

dL = dp (1 + z) = c(1 + z) 0

F(ν) =

L(ν  ) 4πd2L (z)



t0

t

dt R(t)

dz H(z)

where ν  = (1 + z)ν

da = dL (1 + z)−2

tlb (z)light travel time from an object at redshift z t0 present cosmic time t(z) cosmic time at z dp R c

proper distance cosmic scale factor speed of light

0

dL

present epoch luminosity distance

(9.98)

z H F

redshift Hubble parameterb spectral flux density

(9.99)

ν frequency L(ν) spectral luminosityc

(9.97)

(9.100)

da

angular diameter distance

k

curvature parameter

a flat universe (k = 0). The apparent flux density of a source varies as d−2 L . b See Equation (9.85). c Defined as the output power of the body per unit frequency interval. d True for all k. The angular diameter of a source varies as d−1 . a a Assuming

Cosmological modelsa 2c [1 − (1 + z)−1/2 ] H0 H(z) = H0 (1 + z)3/2 q0 = 1/2 2 t(z) = 3H(z) ρ = (6πGt2 )−1 dp =

Einstein – de Sitter model (Ωk = 0, Λ = 0, p = 0 and Ωm0 = 1)

R(t) = R0 (t/t0 ) Concordance model (Ωk = 0, Λ = 3(1 − Ωm0 )H02 , p = 0 and Ωm0 < 1) a Currently

popular.

c dp = H0

 0

z

2/3

(9.101) (9.102) (9.103)

(9.105)

−1/2

H(z) = H0 [Ωm0 (1 + z)3 + (1 − Ωm0 )] q0 = 3Ωm0 /2 − 1   (1 − Ωm0 )1/2 2 −1/2 t(z) = (1 − Ωm0 ) arsinh 3H0 (1 + z)3/2

H 0

z

(9.104)

(9.106)

Ωm0 dz  1/2 [(1 + z  )3 − 1 + Ω−1 m0 ]

dp

(9.107)

speed of light deceleration parameter t(z) time at redshift z c q

R Ωm0

(9.108) (9.109) (9.110)

proper distance Hubble parameter present epoch redshift

cosmic scale factor present mass density parameter

G

constant of gravitation

ρ

mass density

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Index

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

A aberration (relativistic) [3.24], 65 absolute magnitude [9.29], 179 absorption (Einstein coefficient) [8.118], 173 absorption coefficient (linear) [5.175], 120 accelerated point charge bremsstrahlung, 160 Li´enard–Wiechert potentials, 139 oscillating [7.132], 146 synchrotron, 159 acceleration constant, 68 dimensions, 16 due to gravity (value on Earth), 176 in a rotating frame [3.32], 66 acceptance angle (optical fibre) [8.77], 169 acoustic branch (phonon) [6.37], 129 acoustic impedance [3.276], 83 action (definition) [3.213], 79 action (dimensions), 16 addition of velocities Galilean [3.3], 64 relativistic [3.15], 64 adiabatic bulk modulus [5.23], 107 compressibility [5.21], 107 expansion (ideal gas) [5.58], 110 lapse rate [3.294], 84 adjoint matrix definition 1 [2.71], 24 definition 2 [2.80], 25 adjugate matrix [2.80], 25 admittance (definition), 148 advective operator [3.289], 84

Airy disk [8.40], 165 function [8.17], 163 resolution criterion [8.41], 165 Airy’s differential equation [2.352], 43 albedo [5.193], 121 Alfv´en speed [7.277], 158 Alfv´en waves [7.284], 158 alt-azimuth coordinates, 177 alternating tensor (ijk ) [2.443], 50 altitude coordinate [9.9], 177 Amp`ere’s law [7.10], 136 ampere (SI definition), 3 ampere (unit), 4 analogue formula [2.258], 36 angle aberration [3.24], 65 acceptance [8.77], 169 beam solid [7.210], 153 Brewster’s [7.218], 154 Compton scattering [7.240], 155 contact (surface tension) [3.340], 88 deviation [8.73], 169 Euler [2.101], 26 Faraday rotation [7.273], 157 hour (coordinate) [9.8], 177 Kelvin wedge [3.330], 87 Mach wedge [3.328], 87 polarisation [8.81], 170 principal range (inverse trig.), 34 refraction, 154 rotation, 26 Rutherford scattering [3.116], 72 separation [3.133], 73 spherical excess [2.260], 36 units, 4, 5 ˚ angstr¨ om (unit), 5

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188 angular diameter distance [9.100], 185 Angular momentum, 98 angular momentum conservation [4.113], 98 definition [3.66], 68 dimensions, 16 eigenvalues [4.109] [4.109], 98 ladder operators [4.108], 98 operators and other operators [4.23], 91 definitions [4.105], 98 rigid body [3.141], 74 Angular momentum addition, 100 Angular momentum commutation relations, 98 angular speed (dimensions), 16 anomaly (true) [3.104], 71 antenna beam efficiency [7.214], 153 effective area [7.212], 153 power gain [7.211], 153 temperature [7.215], 153 Antennas, 153 anticommutation [2.95], 26 antihermitian symmetry, 53 antisymmetric matrix [2.87], 25 Aperture diffraction, 165 aperture function [8.34], 165 apocentre (of an orbit) [3.111], 71 apparent magnitude [9.27], 179 Appleton-Hartree formula [7.271], 157 arc length [2.279], 39 arccosx from arctan [2.233], 34 series expansion [2.141], 29 arcoshx (definition) [2.239], 35 arccotx (from arctan) [2.236], 34 arcothx (definition) [2.241], 35 arccscx (from arctan) [2.234], 34 arcschx (definition) [2.243], 35 arcminute (unit), 5 arcsecx (from arctan) [2.235], 34 arsechx (definition) [2.242], 35 arcsecond (unit), 5 arcsinx from arctan [2.232], 34 series expansion [2.141], 29 arsinhx (definition) [2.238], 35 arctanx (series expansion) [2.142], 29

Index artanhx (definition) [2.240], 35 area of circle [2.262], 37 of cone [2.271], 37 of cylinder [2.269], 37 of ellipse [2.267], 37 of plane triangle [2.254], 36 of sphere [2.263], 37 of spherical cap [2.275], 37 of torus [2.273], 37 area (dimensions), 16 argument (of a complex number) [2.157], 30 arithmetic mean [2.108], 27 arithmetic progression [2.104], 27 associated Laguerre equation [2.348], 43 associated Laguerre polynomials, 96 associated Legendre equation and polynomial solutions [2.428], 48 differential equation [2.344], 43 Associated Legendre functions, 48 astronomical constants, 176 Astronomical magnitudes, 179 Astrophysics, 175–185 asymmetric top [3.189], 77 atomic form factor [6.30], 128 mass unit, 6, 9 numbers of elements, 124 polarisability [7.91], 142 weights of elements, 124 Atomic constants, 7 atto, 5 autocorrelation (Fourier) [2.491], 53 autocorrelation function [8.104], 172 availability and fluctuation probability [5.131], 116 definition [5.40], 108 Avogadro constant, 6, 9 Avogadro constant (dimensions), 16 azimuth coordinate [9.10], 177

B Ballistics, 69 band index [6.85], 134 Band theory and semiconductors, 134 bandwidth and coherence time [8.106], 172

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Index and Johnson noise [5.141], 117 Doppler [8.117], 173 natural [8.113], 173 of a diffraction grating [8.30], 164 of an LCR circuit [7.151], 148 of laser cavity [8.127], 174 Schawlow-Townes [8.128], 174 bar (unit), 5 barn (unit), 5 Barrier tunnelling, 94 Bartlett window [2.581], 60 base vectors (crystallographic), 126 basis vectors [2.17], 20 Bayes’ theorem [2.569], 59 Bayesian inference, 59 bcc structure, 127 beam bowing under its own weight [3.260], 82 beam efficiency [7.214], 153 beam solid angle [7.210], 153 beam with end-weight [3.259], 82 beaming (relativistic) [3.25], 65 becquerel (unit), 4 Bending beams, 82 bending moment (dimensions), 16 bending moment [3.258], 82 bending waves [3.268], 82 Bernoulli’s differential equation [2.351], 43 Bernoulli’s equation compressible flow [3.292], 84 incompressible flow [3.290], 84 Bessel equation [2.345], 43 Bessel functions, 47 beta (in plasmas) [7.278], 158 binomial coefficient [2.121], 28 distribution [2.547], 57 series [2.120], 28 theorem [2.122], 28 binormal [2.285], 39 Biot–Savart law [7.9], 136 Biot-Fourier equation [5.95], 113 black hole evaporation time [9.81], 183 Kerr solution [3.62], 67 maximum angular momentum [9.80], 183 Schwarzschild radius [9.73], 183

189 Schwarzschild solution [3.61], 67 temperature [9.82], 183 blackbody energy density [5.192], 121 spectral energy density [5.186], 121 spectrum [5.184], 121 Blackbody radiation, 121 Bloch’s theorem [6.84], 134 Bode’s law [9.41], 180 body cone, 77 body frequency [3.187], 77 body-centred cubic structure, 127 Bohr energy [4.74], 95 magneton (equation) [4.137], 100 magneton (value), 6, 7 quantisation [4.71], 95 radius (equation) [4.72], 95 radius (value), 7 Bohr magneton (dimensions), 16 Bohr model, 95 boiling points of elements, 124 bolometric correction [9.34], 179 Boltzmann constant, 6, 9 constant (dimensions), 16 distribution [5.111], 114 entropy [5.105], 114 excitation equation [5.125], 116 Born collision formula [4.178], 104 Bose condensation [5.123], 115 Bose–Einstein distribution [5.120], 115 boson statistics [5.120], 115 Boundary conditions for E, D, B, and H, 144 box (particle in a) [4.64], 94 Box Muller transformation [2.561], 58 Boyle temperature [5.66], 110 Boyle’s law [5.56], 110 bra vector [4.33], 92 bra-ket notation, 91, 92 Bragg’s reflection law in crystals [6.29], 128 in optics [8.32], 164 braking index (pulsar) [9.66], 182 Bravais lattices, 126 Breit-Wigner formula [4.174], 104 Bremsstrahlung, 160 bremsstrahlung

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190 single electron and ion [7.297], 160 thermal [7.300], 160 Brewster’s law [7.218], 154 brightness (blackbody) [5.184], 121 Brillouin function [4.147], 101 Bromwich integral [2.518], 55 Brownian motion [5.98], 113 bubbles [3.337], 88 bulk modulus adiabatic [5.23], 107 general [3.245], 81 isothermal [5.22], 107 bulk modulus (dimensions), 16 Bulk physical constants, 9 Burgers vector [6.21], 128

C calculus of variations [2.334], 42 candela, 119 candela (SI definition), 3 candela (unit), 4 canonical ensemble [5.111], 114 entropy [5.106], 114 equations [3.220], 79 momenta [3.218], 79 cap, see spherical cap Capacitance, 137 capacitance current through [7.144], 147 definition [7.143], 147 dimensions, 16 energy [7.153], 148 energy of an assembly [7.134], 146 impedance [7.159], 148 mutual [7.134], 146 capacitance of cube [7.17], 137 cylinder [7.15], 137 cylinders (adjacent) [7.21], 137 cylinders (coaxial) [7.19], 137 disk [7.13], 137 disks (coaxial) [7.22], 137 nearly spherical surface [7.16], 137 sphere [7.12], 137 spheres (adjacent) [7.14], 137 spheres (concentric) [7.18], 137 capacitor, see capacitance capillary

Index constant [3.338], 88 contact angle [3.340], 88 rise [3.339], 88 waves [3.321], 86 capillary-gravity waves [3.322], 86 cardioid [8.46], 166 Carnot cycles, 107 Cartesian coordinates, 21 Catalan’s constant (value), 9 Cauchy differential equation [2.350], 43 distribution [2.555], 58 inequality [2.151], 30 integral formula [2.167], 31 Cauchy-Goursat theorem [2.165], 31 Cauchy-Riemann conditions [2.164], 31 cavity modes (laser) [8.124], 174 Celsius (unit), 4 Celsius conversion [1.1], 15 centi, 5 centigrade (avoidance of), 15 centre of mass circular arc [3.173], 76 cone [3.175], 76 definition [3.68], 68 disk sector [3.172], 76 hemisphere [3.170], 76 hemispherical shell [3.171], 76 pyramid [3.175], 76 semi-ellipse [3.178], 76 spherical cap [3.177], 76 triangular lamina [3.174], 76 Centres of mass, 76 centrifugal force [3.35], 66 centripetal acceleration [3.32], 66 cepheid variables [9.48], 180 Cerenkov, see Cherenkov chain rule function of a function [2.295], 40 partial derivatives [2.331], 42 Chandrasekhar mass [9.79], 183 change of variable [2.333], 42 Characteristic numbers, 86 charge conservation [7.39], 139 dimensions, 16 elementary, 6, 7 force between two [7.119], 145 Hamiltonian [7.138], 146

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Index to mass ratio of electron, 8 charge density dimensions, 16 free [7.57], 140 induced [7.84], 142 Lorentz transformation, 141 charge distribution electric field from [7.6], 136 energy of [7.133], 146 charge-sheet (electric field) [7.32], 138 Chebyshev equation [2.349], 43 Chebyshev inequality [2.150], 30 chemical potential definition [5.28], 108 from partition function [5.119], 115 Cherenkov cone angle [7.246], 156 Cherenkov radiation, 156 χE (electric susceptibility) [7.87], 142 χH , χB (magnetic susceptibility) [7.103], 143 chi-squared (χ2 ) distribution [2.553], 58 Christoffel symbols [3.49], 67 circle (arc of) centre of mass [3.173], 76 area [2.262], 37 perimeter [2.261], 37 circular aperture Fraunhofer diffraction [8.40], 165 Fresnel diffraction [8.50], 166 circular polarisation, 170 circulation [3.287], 84 civil time [9.4], 177 Clapeyron equation [5.50], 109 classical electron radius, 8 Classical thermodynamics, 106 Clausius–Mossotti equation [7.93], 142 Clausius-Clapeyron equation [5.49], 109 Clebsch–Gordan coefficients, 99 Clebsch–Gordan coefficients (spin-orbit) [4.136], 100 close-packed spheres, 127 closure density (of the universe) [9.90], 184 CNO cycle, 182 coaxial cable capacitance [7.19], 137 inductance [7.24], 137 coaxial transmission line [7.181], 150 coefficient of

191 coupling [7.148], 147 finesse [8.12], 163 reflectance [7.227], 154 reflection [7.230], 154 restitution [3.127], 73 transmission [7.232], 154 transmittance [7.229], 154 coexistence curve [5.51], 109 coherence length [8.106], 172 mutual [8.97], 172 temporal [8.105], 172 time [8.106], 172 width [8.111], 172 Coherence (scalar theory), 172 cold plasmas, 157 collision broadening [8.114], 173 elastic, 73 inelastic, 73 number [5.91], 113 time (electron drift) [6.61], 132 colour excess [9.37], 179 colour index [9.36], 179 Common three-dimensional coordinate systems, 21 commutator (in uncertainty relation) [4.6], 90 Commutators, 26 Compact objects and black holes, 183 complementary error function [2.391], 45 Complex analysis, 31 complex conjugate [2.159], 30 Complex numbers, 30 complex numbers argument [2.157], 30 cartesian form [2.153], 30 conjugate [2.159], 30 logarithm [2.162], 30 modulus [2.155], 30 polar form [2.154], 30 Complex variables, 30 compound pendulum [3.182], 76 compressibility adiabatic [5.21], 107 isothermal [5.20], 107 compression modulus, see bulk modulus compression ratio [5.13], 107 Compton

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192 scattering [7.240], 155 wavelength (value), 8 wavelength [7.240], 155 Concordance model, 185 conditional probability [2.567], 59 conductance (definition), 148 conductance (dimensions), 16 conduction equation (and transport) [5.96], 113 conduction equation [2.340], 43 conductivity and resistivity [7.142], 147 dimensions, 16 direct [7.279], 158 electrical, of a plasma [7.233], 155 free electron a.c. [6.63], 132 free electron d.c. [6.62], 132 Hall [7.280], 158 conductor refractive index [7.234], 155 cone centre of mass [3.175], 76 moment of inertia [3.160], 75 surface area [2.271], 37 volume [2.272], 37 configurational entropy [5.105], 114 Conic sections, 38 conical pendulum [3.180], 76 conservation of angular momentum [4.113], 98 charge [7.39], 139 mass [3.285], 84 Constant acceleration, 68 constant of gravitation, 7 contact angle (surface tension) [3.340], 88 continuity equation (quantum physics) [4.14], 90 continuity in fluids [3.285], 84 Continuous probability distributions, 58 contravariant components in general relativity, 67 in special relativity [3.26], 65 convection (in a star) [9.64], 181 convergence and limits, 28 Conversion factors, 10 Converting between units, 10 convolution definition [2.487], 53

Index derivative [2.498], 53 discrete [2.580], 60 Laplace transform [2.516], 55 rules [2.489], 53 theorem [2.490], 53 coordinate systems, 21 coordinate transformations astronomical, 177 Galilean, 64 relativistic, 64 rotating frames [3.31], 66 Coordinate transformations (astronomical), 177 coordinates (generalised ) [3.213], 79 coordination number (cubic lattices), 127 Coriolis force [3.33], 66 Cornu spiral, 167 Cornu spiral and Fresnel integrals [8.54], 167 correlation coefficient multinormal [2.559], 58 Pearson’s r [2.546], 57 correlation intensity [8.109], 172 correlation theorem [2.494], 53 cosx and Euler’s formula [2.216], 34 series expansion [2.135], 29 cosec, see csc cschx [2.231], 34 coshx definition [2.217], 34 series expansion [2.143], 29 cosine formula planar triangles [2.249], 36 spherical triangles [2.257], 36 cosmic scale factor [9.87], 184 cosmological constant [9.89], 184 Cosmological distance measures, 185 Cosmological model parameters, 184 Cosmological models, 185 Cosmology, 184 cos−1 x, see arccosx cotx definition [2.226], 34 series expansion [2.140], 29 cothx [2.227], 34 Couette flow [3.306], 85 coulomb (unit), 4 Coulomb gauge condition [7.42], 139

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Index Coulomb logarithm [7.254], 156 Coulomb’s law [7.119], 145 couple definition [3.67], 68 dimensions, 16 electromagnetic, 145 for Couette flow [3.306], 85 on a current-loop [7.127], 145 on a magnetic dipole [7.126], 145 on a rigid body, 77 on an electric dipole [7.125], 145 twisting [3.252], 81 coupling coefficient [7.148], 147 covariance [2.558], 58 covariant components [3.26], 65 cracks (critical length) [6.25], 128 critical damping [3.199], 78 critical density (of the universe) [9.90], 184 critical frequency (synchrotron) [7.293], 159 critical point Dieterici gas [5.75], 111 van der Waals gas [5.70], 111 cross section absorption [5.175], 120 cross-correlation [2.493], 53 cross-product [2.2], 20 cross-section Breit-Wigner [4.174], 104 Mott scattering [4.180], 104 Rayleigh scattering [7.236], 155 Rutherford scattering [3.124], 72 Thomson scattering [7.238], 155 Crystal diffraction, 128 Crystal systems, 127 Crystalline structure, 126 cscx definition [2.230], 34 series expansion [2.139], 29 cschx [2.231], 34 cube electrical capacitance [7.17], 137 mensuration, 38 Cubic equations, 51 cubic expansivity [5.19], 107 Cubic lattices, 127 cubic system (crystallographic), 127 Curie temperature [7.114], 144

193 Curie’s law [7.113], 144 Curie–Weiss law [7.114], 144 Curl, 22 curl cylindrical coordinates [2.34], 22 general coordinates [2.36], 22 of curl [2.57], 23 rectangular coordinates [2.33], 22 spherical coordinates [2.35], 22 current dimensions, 16 electric [7.139], 147 law (Kirchhoff’s) [7.161], 149 magnetic flux density from [7.11], 136 probability density [4.13], 90 thermodynamic work [5.9], 106 transformation [7.165], 149 current density dimensions, 16 four-vector [7.76], 141 free [7.63], 140 free electron [6.60], 132 hole [6.89], 134 Lorentz transformation, 141 magnetic flux density [7.10], 136 curvature in differential geomtry [2.286], 39 parameter (cosmic) [9.87], 184 radius of and curvature [2.287], 39 plane curve [2.282], 39 curve length (plane curve) [2.279], 39 Curve measure, 39 Cycle efficiencies (thermodynamic), 107 cyclic permutation [2.97], 26 cyclotron frequency [7.265], 157 cylinder area [2.269], 37 capacitance [7.15], 137 moment of inertia [3.155], 75 torsional rigidity [3.253], 81 volume [2.270], 37 cylinders (adjacent) capacitance [7.21], 137 inductance [7.25], 137 cylinders (coaxial) capacitance [7.19], 137 inductance [7.24], 137

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194 cylindrical polar coordinates, 21

D d orbitals [4.100], 97 D’Alembertian [7.78], 141 damped harmonic oscillator [3.196], 78 damping profile [8.112], 173 day (unit), 5 day of week [9.3], 177 daylight saving time [9.4], 177 de Boer parameter [6.54], 131 de Broglie relation [4.2], 90 de Broglie wavelength (thermal) [5.83], 112 de Moivre’s theorem [2.214], 34 Debye T 3 law [6.47], 130 frequency [6.41], 130 function [6.49], 130 heat capacity [6.45], 130 length [7.251], 156 number [7.253], 156 screening [7.252], 156 temperature [6.43], 130 Debye theory, 130 Debye-Waller factor [6.33], 128 deca, 5 decay constant [4.163], 103 decay law [4.163], 103 deceleration parameter [9.95], 184 deci, 5 decibel [5.144], 117 declination coordinate [9.11], 177 decrement (oscillating systems) [3.202], 78 Definite integrals, 46 degeneracy pressure [9.77], 183 degree (unit), 5 degree Celsius (unit), 4 degree kelvin [5.2], 106 degree of freedom (and equipartition), 113 degree of mutual coherence [8.99], 172 degree of polarisation [8.96], 171 degree of temporal coherence, 172 deka, 5 del operator, 21 del-squared operator, 23 del-squared operator [2.55], 23 Delta functions, 50

Index delta–star transformation, 149 densities of elements, 124 density (dimensions), 16 density of states electron [6.70], 133 particle [4.66], 94 phonon [6.44], 130 density parameters [9.94], 184 depolarising factors [7.92], 142 Derivatives (general), 40 determinant [2.79], 25 deviation (of a prism) [8.73], 169 diamagnetic moment (electron) [7.108], 144 diamagnetic susceptibility (Landau) [6.80], 133 Diamagnetism, 144 Dielectric layers, 162 Dieterici gas, 111 Dieterici gas law [5.72], 111 Differential equations, 43 differential equations (numerical solutions), 62 Differential geometry, 39 Differential operator identities, 23 differential scattering cross-section [3.124], 72 Differentiation, 40 differentiation hyperbolic functions, 41 numerical, 61 of a function of a function [2.295], 40 of a log [2.300], 40 of a power [2.292], 40 of a product [2.293], 40 of a quotient [2.294], 40 of exponential [2.301], 40 of integral [2.299], 40 of inverse functions [2.304], 40 trigonometric functions, 41 under integral sign [2.298], 40 diffraction from N slits [8.25], 164 1 slit [8.37], 165 2 slits [8.24], 164 circular aperture [8.40], 165 crystals, 128 infinite grating [8.26], 164

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Index rectangular aperture [8.39], 165 diffraction grating finite [8.25], 164 general, 164 infinite [8.26], 164 diffusion coefficient (semiconductor) [6.88], 134 diffusion equation differential equation [2.340], 43 Fick’s first law [5.93], 113 diffusion length (semiconductor) [6.94], 134 diffusivity (magnetic) [7.282], 158 dilatation (volume strain) [3.236], 80 Dimensions, 16 diode (semiconductor) [6.92], 134 dioptre number [8.68], 168 dipole antenna power flux [7.131], 146 gain [7.213], 153 total [7.132], 146 electric field [7.31], 138 energy of electric [7.136], 146 magnetic [7.137], 146 field from magnetic [7.36], 138 moment (dimensions), 17 moment of electric [7.80], 142 magnetic [7.94], 143 potential electric [7.82], 142 magnetic [7.95], 143 radiation field [7.207], 153 magnetic [9.69], 182 radiation resistance [7.209], 153 dipole moment per unit volume electric [7.83], 142 magnetic [7.97], 143 Dirac bracket, 92 Dirac delta function [2.448], 50 Dirac equation [4.183], 104 Dirac matrices [4.185], 104 Dirac notation, 92 direct conductivity [7.279], 158 directrix (of conic section), 38

195 disc, see disk discrete convolution, 60 Discrete probability distributions, 57 Discrete statistics, 57 disk Airy [8.40], 165 capacitance [7.13], 137 centre of mass of sector [3.172], 76 coaxial capacitance [7.22], 137 drag in a fluid, 85 electric field [7.28], 138 moment of inertia [3.168], 75 Dislocations and cracks, 128 dispersion diffraction grating [8.31], 164 in a plasma [7.261], 157 in fluid waves, 86 in quantum physics [4.5], 90 in waveguides [7.188], 151 intermodal (optical fibre) [8.79], 169 measure [9.70], 182 of a prism [8.76], 169 phonon (alternating springs) [6.39], 129 phonon (diatomic chain) [6.37], 129 phonon (monatomic chain) [6.34], 129 pulsar [9.72], 182 displacement, D [7.86], 142 Distance indicators, 180 Divergence, 22 divergence cylindrical coordinates [2.30], 22 general coordinates [2.32], 22 rectangular coordinates [2.29], 22 spherical coordinates [2.31], 22 theorem [2.59], 23 dodecahedron, 38 Doppler beaming [3.25], 65 effect (non-relativistic), 87 effect (relativistic) [3.22], 65 line broadening [8.116], 173 width [8.117], 173 Doppler effect, 87 dot product [2.1], 20 double factorial, 48 double pendulum [3.183], 76 Drag, 85

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196 drag on a disk  to flow [3.310], 85 on a disk ⊥ to flow [3.309], 85 on a sphere [3.308], 85 drift velocity (electron) [6.61], 132 Dulong and Petit’s law [6.46], 130 Dynamics and Mechanics, 63–88 Dynamics definitions, 68

E e (exponential constant), 9 e to 1 000 decimal places, 18 Earth (motion relative to) [3.38], 66 Earth data, 176 eccentricity of conic section, 38 of orbit [3.108], 71 of scattering hyperbola [3.120], 72 Ecliptic coordinates, 178 ecliptic latitude [9.14], 178 ecliptic longitude [9.15], 178 Eddington limit [9.59], 181 edge dislocation [6.21], 128 effective area (antenna) [7.212], 153 distance (Fresnel diffraction) [8.48], 166 mass (in solids) [6.86], 134 wavelength [9.40], 179 efficiency heat engine [5.10], 107 heat pump [5.12], 107 Otto cycle [5.13], 107 refrigerator [5.11], 107 Ehrenfest’s equations [5.53], 109 Ehrenfest’s theorem [4.30], 91 eigenfunctions (quantum) [4.28], 91 Einstein A coefficient [8.119], 173 B coefficients [8.118], 173 diffusion equation [5.98], 113 field equation [3.59], 67 lens (rings) [9.50], 180 tensor [3.58], 67 Einstein - de Sitter model, 185 Einstein coefficients, 173 elastic collisions, 73 media (isotropic), 81

Index modulus (longitudinal) [3.241], 81 modulus [3.234], 80 potential energy [3.235], 80 elastic scattering, 72 Elastic wave velocities, 82 Elasticity, 80 Elasticity definitions (general), 80 Elasticity definitions (simple), 80 electric current [7.139], 147 electric dipole, see dipole electric displacement (dimensions), 16 electric displacement, D [7.86], 142 electric field around objects, 138 energy density [7.128], 146 static, 136 thermodynamic work [5.7], 106 wave equation [7.193], 152 electric field from A and φ [7.41], 139 charge distribution [7.6], 136 charge-sheet [7.32], 138 dipole [7.31], 138 disk [7.28], 138 line charge [7.29], 138 point charge [7.5], 136 sphere [7.27], 138 waveguide [7.190], 151 wire [7.29], 138 electric field strength (dimensions), 16 Electric fields, 138 electric polarisability (dimensions), 16 electric polarisation (dimensions), 16 electric potential from a charge density [7.46], 139 Lorentz transformation [7.75], 141 of a moving charge [7.48], 139 short dipole [7.82], 142 electric potential difference (dimensions), 16 electric susceptibility, χE [7.87], 142 electrical conductivity, see conductivity Electrical impedance, 148 electrical permittivity, , r [7.90], 142 electromagnet (magnetic flux density) [7.38], 138 electromagnetic boundary conditions, 144 constants, 7

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Index fields, 139 wave speed [7.196], 152 waves in media, 152 electromagnetic coupling constant, see fine structure constant Electromagnetic energy, 146 Electromagnetic fields (general), 139 Electromagnetic force and torque, 145 Electromagnetic propagation in cold plasmas, 157 Electromagnetism, 135–160 electron charge, 6, 7 density of states [6.70], 133 diamagnetic moment [7.108], 144 drift velocity [6.61], 132 g-factor [4.143], 100 gyromagnetic ratio (value), 8 gyromagnetic ratio [4.140], 100 heat capacity [6.76], 133 intrinsic magnetic moment [7.109], 144 mass, 6 radius (equation) [7.238], 155 radius (value), 8 scattering cross-section [7.238], 155 spin magnetic moment [4.143], 100 thermal velocity [7.257], 156 velocity in conductors [6.85], 134 Electron constants, 8 Electron scattering processes, 155 electron volt (unit), 5 electron volt (value), 6 Electrons in solids, 132 electrostatic potential [7.1], 136 Electrostatics, 136 elementary charge, 6, 7 elements (periodic table of), 124 ellipse, 38 (semi) centre of mass [3.178], 76 area [2.267], 37 moment of inertia [3.166], 75 perimeter [2.266], 37 semi-latus-rectum [3.109], 71 semi-major axis [3.106], 71 semi-minor axis [3.107], 71 ellipsoid moment of inertia of solid [3.163], 75

197 the moment of inertia [3.147], 74 volume [2.268], 37 elliptic integrals [2.397], 45 elliptical orbit [3.104], 71 Elliptical polarisation, 170 elliptical polarisation [8.80], 170 ellipticity [8.82], 170 E = mc2 [3.72], 68 emission coefficient [5.174], 120 emission spectrum [7.291], 159 emissivity [5.193], 121 energy density blackbody [5.192], 121 dimensions, 16 elastic wave [3.281], 83 electromagnetic [7.128], 146 radiant [5.148], 118 spectral [5.173], 120 dimensions, 16 dissipated in resistor [7.155], 148 distribution (Maxwellian) [5.85], 112 elastic [3.235], 80 electromagnetic, 146 equipartition [5.100], 113 Fermi [5.122], 115 first law of thermodynamics [5.3], 106 Galilean transformation [3.6], 64 kinetic , see kinetic energy Lorentz transformation [3.19], 65 loss after collision [3.128], 73 mass relation [3.20], 65 of capacitive assembly [7.134], 146 of capacitor [7.153], 148 of charge distribution [7.133], 146 of electric dipole [7.136], 146 of inductive assembly [7.135], 146 of inductor [7.154], 148 of magnetic dipole [7.137], 146 of orbit [3.100], 71 potential , see potential energy relativistic rest [3.72], 68 rotational kinetic rigid body [3.142], 74 w.r.t. principal axes [3.145], 74 thermodynamic work, 106 Energy in capacitors, inductors, and resistors, 148

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198 energy-time uncertainty relation [4.8], 90 Ensemble probabilities, 114 enthalpy definition [5.30], 108 Joule-Kelvin expansion [5.27], 108 entropy Boltzmann formula [5.105], 114 change in Joule expansion [5.64], 110 experimental [5.4], 106 fluctuations [5.135], 116 from partition function [5.117], 115 Gibbs formula [5.106], 114 of a monatomic gas [5.83], 112 entropy (dimensions), 16 , r (electrical permittivity) [7.90], 142 Equation conversion: SI to Gaussian units, 135 equation of state Dieterici gas [5.72], 111 ideal gas [5.57], 110 monatomic gas [5.78], 112 van der Waals gas [5.67], 111 equipartition theorem [5.100], 113 error function [2.390], 45 errors, 60 escape velocity [3.91], 70 estimator kurtosis [2.545], 57 mean [2.541], 57 skewness [2.544], 57 standard deviation [2.543], 57 variance [2.542], 57 Euler angles [2.101], 26 constant expression [2.119], 27 value, 9 differential equation [2.350], 43 formula [2.216], 34 relation, 38 strut [3.261], 82 Euler’s equation (fluids) [3.289], 84 Euler’s equations (rigid bodies) [3.186], 77 Euler’s method (for ordinary differential equations) [2.596], 62 Euler-Lagrange equation and Lagrangians [3.214], 79

Index calculus of variations [2.334], 42 even functions, 53 Evolutionary timescales, 181 exa, 5 exhaust velocity (of a rocket) [3.93], 70 exitance blackbody [5.191], 121 luminous [5.162], 119 radiant [5.150], 118 exp(x) [2.132], 29 expansion coefficient [5.19], 107 Expansion processes, 108 expansivity [5.19], 107 Expectation value, 91 expectation value Dirac notation [4.37], 92 from a wavefunction [4.25], 91 explosions [3.331], 87 exponential distribution [2.551], 58 integral [2.394], 45 series expansion [2.132], 29 exponential constant (e), 9 extraordinary modes [7.271], 157 extrema [2.335], 42

F f-number [8.69], 168 Fabry-Perot etalon chromatic resolving power [8.21], 163 free spectral range [8.23], 163 fringe width [8.19], 163 transmitted intensity [8.17], 163 Fabry-Perot etalon, 163 face-centred cubic structure, 127 factorial [2.409], 46 factorial (double), 48 Fahrenheit conversion [1.2], 15 faltung theorem [2.516], 55 farad (unit), 4 Faraday constant, 6, 9 Faraday constant (dimensions), 16 Faraday rotation [7.273], 157 Faraday’s law [7.55], 140 fcc structure, 127 Feigenbaum’s constants, 9 femto, 5 Fermat’s principle [8.63], 168 Fermi

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Index energy [6.73], 133 temperature [6.74], 133 velocity [6.72], 133 wavenumber [6.71], 133 fermi (unit), 5 Fermi energy [5.122], 115 Fermi gas, 133 Fermi’s golden rule [4.162], 102 Fermi–Dirac distribution [5.121], 115 fermion statistics [5.121], 115 fibre optic acceptance angle [8.77], 169 dispersion [8.79], 169 numerical aperture [8.78], 169 Fick’s first law [5.92], 113 Fick’s second law [5.95], 113 field equations (gravitational) [3.42], 66 Field relationships, 139 fields depolarising [7.92], 142 electrochemical [6.81], 133 electromagnetic, 139 gravitational, 66 static E and B, 136 velocity [3.285], 84 Fields associated with media, 142 film reflectance [8.4], 162 fine-structure constant expression [4.75], 95 value, 6, 7 finesse (coefficient of) [8.12], 163 finesse (Fabry-Perot etalon) [8.14], 163 first law of thermodynamics [5.3], 106 fitting straight-lines, 60 fluctuating dipole interaction [6.50], 131 fluctuation of density [5.137], 116 of entropy [5.135], 116 of pressure [5.136], 116 of temperature [5.133], 116 of volume [5.134], 116 probability (thermodynamic) [5.131], 116 variance (general) [5.132], 116 Fluctuations and noise, 116 Fluid dynamics, 84 fluid stress [3.299], 85 Fluid waves, 86 flux density [5.171], 120

199 flux density–redshift relation [9.99], 185 flux linked [7.149], 147 flux of molecules through a plane [5.91], 113 flux–magnitude relation [9.32], 179 focal length [8.64], 168 focus (of conic section), 38 force and acoustic impedance [3.276], 83 and stress [3.228], 80 between two charges [7.119], 145 between two currents [7.120], 145 between two masses [3.40], 66 central [4.113], 98 centrifugal [3.35], 66 Coriolis [3.33], 66 critical compression [3.261], 82 definition [3.63], 68 dimensions, 16 electromagnetic, 145 Newtonian [3.63], 68 on charge in a field [7.122], 145 current in a field [7.121], 145 electric dipole [7.123], 145 magnetic dipole [7.124], 145 sphere (potential flow) [3.298], 84 sphere (viscous drag) [3.308], 85 relativistic [3.71], 68 unit, 4 Force, torque, and energy, 145 Forced oscillations, 78 form factor [6.30], 128 formula (the) [2.455], 50 Foucault’s pendulum [3.39], 66 four-parts formula [2.259], 36 four-scalar product [3.27], 65 four-vector electromagnetic [7.79], 141 momentum [3.21], 65 spacetime [3.12], 64 Four-vectors, 65 Fourier series complex form [2.478], 52 real form [2.476], 52 Fourier series, 52 Fourier series and transforms, 52 Fourier symmetry relationships, 53 Fourier transform

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200 cosine [2.509], 54 definition [2.482], 52 derivatives and inverse [2.502], 54 general [2.498], 53 Gaussian [2.507], 54 Lorentzian [2.505], 54 shah function [2.510], 54 shift theorem [2.501], 54 similarity theorem [2.500], 54 sine [2.508], 54 step [2.511], 54 top hat [2.512], 54 triangle function [2.513], 54 Fourier transform, 52 Fourier transform pairs, 54 Fourier transform theorems, 53 Fourier’s law [5.94], 113 Frames of reference, 64 Fraunhofer diffraction, 164 Fraunhofer integral [8.34], 165 Fraunhofer limit [8.44], 165 free charge density [7.57], 140 free current density [7.63], 140 Free electron transport properties, 132 free energy [5.32], 108 free molecular flow [5.99], 113 Free oscillations, 78 free space impedance [7.197], 152 free spectral range Fabry Perot etalon [8.23], 163 laser cavity [8.124], 174 free-fall timescale [9.53], 181 Frenet’s formulas [2.291], 39 frequency (dimensions), 16 Fresnel diffraction Cornu spiral [8.54], 167 edge [8.56], 167 long slit [8.58], 167 rectangular aperture [8.62], 167 Fresnel diffraction, 166 Fresnel Equations, 154 Fresnel half-period zones [8.49], 166 Fresnel integrals and the Cornu spiral [8.52], 167 definition [2.392], 45 in diffraction [8.54], 167 Fresnel zones, 166 Fresnel-Kirchhoff formula

Index plane waves [8.45], 166 spherical waves [8.47], 166 Friedmann equations [9.89], 184 fringe visibility [8.101], 172 fringes (Moir´e), 35 Froude number [3.312], 86

G g-factor electron, 8 Land´e [4.146], 100 muon, 9 gain in decibels [5.144], 117 galactic coordinates [9.20], 178 latitude [9.21], 178 longitude [9.22], 178 Galactic coordinates, 178 Galilean transformation of angular momentum [3.5], 64 of kinetic energy [3.6], 64 of momentum [3.4], 64 of time and position [3.2], 64 of velocity [3.3], 64 Galilean transformations, 64 Gamma function, 46 gamma function and other integrals [2.395], 45 definition [2.407], 46 gas adiabatic expansion [5.58], 110 adiabatic lapse rate [3.294], 84 constant, 6, 9, 86, 110 Dieterici, 111 Doppler broadened [8.116], 173 flow [3.292], 84 giant (astronomical data), 176 ideal equation of state [5.57], 110 ideal heat capacities, 113 ideal, or perfect, 110 internal energy (ideal) [5.62], 110 isothermal expansion [5.63], 110 linear absorption coefficient [5.175], 120 molecular flow [5.99], 113 monatomic, 112 paramagnetism [7.112], 144 pressure broadened [8.115], 173 speed of sound [3.318], 86

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Index temperature scale [5.1], 106 Van der Waals, 111 Gas equipartition, 113 Gas laws, 110 gauge condition Coulomb [7.42], 139 Lorenz [7.43], 139 Gaunt factor [7.299], 160 Gauss’s law [7.51], 140 lens formula [8.64], 168 theorem [2.59], 23 Gaussian electromagnetism, 135 Fourier transform of [2.507], 54 integral [2.398], 46 light [8.110], 172 optics, 168 probability distribution k-dimensional [2.556], 58 1-dimensional [2.552], 58 Geiger’s law [4.169], 103 Geiger-Nuttall rule [4.170], 103 General constants, 7 General relativity, 67 generalised coordinates [3.213], 79 Generalised dynamics, 79 generalised momentum [3.218], 79 geodesic deviation [3.56], 67 geodesic equation [3.54], 67 geometric distribution [2.548], 57 mean [2.109], 27 progression [2.107], 27 Geometrical optics, 168 Gibbs constant (value), 9 distribution [5.113], 114 entropy [5.106], 114 free energy [5.35], 108 Gibbs’s phase rule [5.54], 109 Gibbs–Helmholtz equations, 109 Gibbs-Duhem relation [5.38], 108 giga, 5 golden mean (value), 9 golden rule (Fermi’s) [4.162], 102 Gradient, 21 gradient cylindrical coordinates [2.26], 21

201 general coordinates [2.28], 21 rectangular coordinates [2.25], 21 spherical coordinates [2.27], 21 gram (use in SI), 5 grand canonical ensemble [5.113], 114 grand partition function [5.112], 114 grand potential definition [5.37], 108 from grand partition function [5.115], 115 grating dispersion [8.31], 164 formula [8.27], 164 resolving power [8.30], 164 Gratings, 164 Gravitation, 66 gravitation field from a sphere [3.44], 66 general relativity, 67 Newton’s law [3.40], 66 Newtonian, 71 Newtonian field equations [3.42], 66 gravitational collapse [9.53], 181 constant, 6, 7, 16 lens [9.50], 180 potential [3.42], 66 redshift [9.74], 183 wave radiation [9.75], 183 Gravitationally bound orbital motion, 71 gravity and motion on Earth [3.38], 66 waves (on a fluid surface) [3.320], 86 gray (unit), 4 Greek alphabet, 18 Green’s first theorem [2.62], 23 Green’s second theorem [2.63], 23 Greenwich sidereal time [9.6], 177 Gregory’s series [2.141], 29 greybody [5.193], 121 group speed (wave) [3.327], 87 Gr¨ uneisen parameter [6.56], 131 gyro-frequency [7.265], 157 gyro-radius [7.268], 157 gyromagnetic ratio definition [4.138], 100 electron [4.140], 100

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202 proton (value), 8 gyroscopes, 77 gyroscopic limit [3.193], 77 nutation [3.194], 77 precession [3.191], 77 stability [3.192], 77

H H (magnetic field strength) [7.100], 143 half-life (nuclear decay) [4.164], 103 half-period zones (Fresnel) [8.49], 166 Hall coefficient (dimensions), 16 conductivity [7.280], 158 effect and coefficient [6.67], 132 voltage [6.68], 132 Hamilton’s equations [3.220], 79 Hamilton’s principal function [3.213], 79 Hamilton-Jacobi equation [3.227], 79 Hamiltonian charged particle (Newtonian) [7.138], 146 charged particle [3.223], 79 definition [3.219], 79 of a particle [3.222], 79 quantum mechanical [4.21], 91 Hamiltonian (dimensions), 16 Hamiltonian dynamics, 79 Hamming window [2.584], 60 Hanbury Brown and Twiss interferometry, 172 Hanning window [2.583], 60 harmonic mean [2.110], 27 Harmonic oscillator, 95 harmonic oscillator damped [3.196], 78 energy levels [4.68], 95 entropy [5.108], 114 forced [3.204], 78 mean energy [6.40], 130 Hartree energy [4.76], 95 Heat capacities, 107 heat capacity (dimensions), 16 heat capacity in solids Debye [6.45], 130 free electron [6.76], 133 heat capacity of a gas Cp − CV [5.17], 107

Index constant pressure [5.15], 107 constant volume [5.14], 107 for f degrees of freedom, 113 ratio (γ) [5.18], 107 heat conduction/diffusion equation differential equation [2.340], 43 Fick’s second law [5.96], 113 heat engine efficiency [5.10], 107 heat pump efficiency [5.12], 107 heavy beam [3.260], 82 hectare, 12 hecto, 5 Heisenberg uncertainty relation [4.7], 90 Helmholtz equation [2.341], 43 Helmholtz free energy definition [5.32], 108 from partition function [5.114], 115 hemisphere (centre of mass) [3.170], 76 hemispherical shell (centre of mass) [3.171], 76 henry (unit), 4 Hermite equation [2.346], 43 Hermite polynomials [4.70], 95 Hermitian conjugate operator [4.17], 91 matrix [2.73], 24 symmetry, 53 Heron’s formula [2.253], 36 herpolhode, 63, 77 hertz (unit), 4 Hertzian dipole [7.207], 153 hexagonal system (crystallographic), 127 High energy and nuclear physics, 103 Hohmann cotangential transfer [3.98], 70 hole current density [6.89], 134 Hooke’s law [3.230], 80 ˆ l’Hopital’s rule [2.131], 28 Horizon coordinates, 177 hour (unit), 5 hour angle [9.8], 177 Hubble constant (dimensions), 16 Hubble constant [9.85], 184 Hubble law as a distance indicator [9.45], 180 in cosmology [9.83], 184 hydrogen atom eigenfunctions [4.80], 96 energy [4.81], 96

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Index Schr¨ odinger equation [4.79], 96 Hydrogenic atoms, 95 ¨dinger soHydrogenlike atoms – Schro lution, 96 hydrostatic compression [3.238], 80 condition [3.293], 84 equilibrium (of a star) [9.61], 181 hyperbola, 38 Hyperbolic derivatives, 41 hyperbolic motion, 72 Hyperbolic relationships, 33

I I (Stokes parameter) [8.89], 171 icosahedron, 38 Ideal fluids, 84 Ideal gas, 110 ideal gas adiabatic equations [5.58], 110 internal energy [5.62], 110 isothermal reversible expansion [5.63], 110 law [5.57], 110 speed of sound [3.318], 86 Identical particles, 115 illuminance (definition) [5.164], 119 illuminance (dimensions), 16 Image charges, 138 impedance acoustic [3.276], 83 dimensions, 17 electrical, 148 transformation [7.166], 149 impedance of capacitor [7.159], 148 coaxial transmission line [7.181], 150 electromagnetic wave [7.198], 152 forced harmonic oscillator [3.212], 78 free space definition [7.197], 152 value, 7 inductor [7.160], 148 lossless transmission line [7.174], 150 lossy transmission line [7.175], 150 microstrip line [7.184], 150 open-wire transmission line [7.182], 150

203 paired strip transmission line [7.183], 150 terminated transmission line [7.178], 150 waveguide TE modes [7.189], 151 TM modes [7.188], 151 impedances in parallel [7.158], 148 in series [7.157], 148 impulse (dimensions), 17 impulse (specific) [3.92], 70 incompressible flow, 84, 85 indefinite integrals, 44 induced charge density [7.84], 142 Inductance, 137 inductance dimensions, 17 energy [7.154], 148 energy of an assembly [7.135], 146 impedance [7.160], 148 mutual definition [7.147], 147 energy [7.135], 146 self [7.145], 147 voltage across [7.146], 147 inductance of cylinders (coaxial) [7.24], 137 solenoid [7.23], 137 wire loop [7.26], 137 wires (parallel) [7.25], 137 induction equation (MHD) [7.282], 158 inductor, see inductance Inelastic collisions, 73 Inequalities, 30 inertia tensor [3.136], 74 inner product [2.1], 20 Integration, 44 integration (numerical), 61 integration by parts [2.354], 44 intensity correlation [8.109], 172 luminous [5.166], 119 of interfering beams [8.100], 172 radiant [5.154], 118 specific [5.171], 120 Interference, 162 interference and coherence [8.100], 172 intermodal dispersion (optical fibre) [8.79],

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204 169 internal energy definition [5.28], 108 from partition function [5.116], 115 ideal gas [5.62], 110 Joule’s law [5.55], 110 monatomic gas [5.79], 112 interval (in general relativity) [3.45], 67 invariable plane, 63, 77 inverse Compton scattering [7.239], 155 Inverse hyperbolic functions, 35 inverse Laplace transform [2.518], 55 inverse matrix [2.83], 25 inverse square law [3.99], 71 Inverse trigonometric functions, 34 ionic bonding [6.55], 131 irradiance (definition) [5.152], 118 irradiance (dimensions), 17 isobaric expansivity [5.19], 107 isophotal wavelength [9.39], 179 isothermal bulk modulus [5.22], 107 isothermal compressibility [5.20], 107 Isotropic elastic solids, 81

J Jacobi identity [2.93], 26 Jacobian definition [2.332], 42 in change of variable [2.333], 42 Jeans length [9.56], 181 Jeans mass [9.57], 181 Johnson noise [5.141], 117 joint probability [2.568], 59 Jones matrix [8.85], 170 Jones vectors definition [8.84], 170 examples [8.84], 170 Jones vectors and matrices, 170 Josephson frequency-voltage ratio, 7 joule (unit), 4 Joule expansion (and Joule coefficient) [5.25], 108 Joule expansion (entropy change) [5.64], 110 Joule’s law (of internal energy) [5.55], 110 Joule’s law (of power dissipation) [7.155], 148 Joule-Kelvin coefficient [5.27], 108

Index Julian centuries [9.5], 177 Julian day number [9.1], 177 Jupiter data, 176

K katal (unit), 4 Kelvin circulation theorem [3.287], 84 relation [6.83], 133 temperature conversion, 15 temperature scale [5.2], 106 wedge [3.330], 87 kelvin (SI definition), 3 kelvin (unit), 4 Kelvin-Helmholtz timescale [9.55], 181 Kepler’s laws, 71 Kepler’s problem, 71 Kerr solution (in general relativity) [3.62], 67 ket vector [4.34], 92 kilo, 5 kilogram (SI definition), 3 kilogram (unit), 4 kinematic viscosity [3.302], 85 kinematics, 63 kinetic energy definition [3.65], 68 for a rotating body [3.142], 74 Galilean transformation [3.6], 64 in the virial theorem [3.102], 71 loss after collision [3.128], 73 of a particle [3.216], 79 of monatomic gas [5.79], 112 operator (quantum) [4.20], 91 relativistic [3.73], 68 w.r.t. principal axes [3.145], 74 Kinetic theory, 112 Kirchhoff’s (radiation) law [5.180], 120 Kirchhoff’s diffraction formula, 166 Kirchhoff’s laws, 149 Klein–Nishina cross section [7.243], 155 Klein-Gordon equation [4.181], 104 Knudsen flow [5.99], 113 Kronecker delta [2.442], 50 kurtosis estimator [2.545], 57

L ladder operators (angular momentum) [4.108], 98

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Index Lagrange’s identity [2.7], 20 Lagrangian (dimensions), 17 Lagrangian dynamics, 79 Lagrangian of charged particle [3.217], 79 particle [3.216], 79 two mutually attracting bodies [3.85], 69 Laguerre equation [2.347], 43 Laguerre polynomials (associated), 96 Lam´e coefficients [3.240], 81 Laminar viscous flow, 85 Land´e g-factor [4.146], 100 Landau diamagnetic susceptibility [6.80], 133 Landau length [7.249], 156 Langevin function (from Brillouin fn) [4.147], 101 Langevin function [7.111], 144 Laplace equation definition [2.339], 43 solution in spherical harmonics [2.440], 49 Laplace series [2.439], 49 Laplace transform convolution [2.516], 55 definition [2.514], 55 derivative of transform [2.520], 55 inverse [2.518], 55 of derivative [2.519], 55 substitution [2.521], 55 translation [2.523], 55 Laplace transform pairs, 56 Laplace transform theorems, 55 Laplace transforms, 55 Laplace’s formula (surface tension) [3.337], 88 Laplacian cylindrical coordinates [2.46], 23 general coordinates [2.48], 23 rectangular coordinates [2.45], 23 spherical coordinates [2.47], 23 Laplacian (scalar), 23 lapse rate (adiabatic) [3.294], 84 Larmor frequency [7.265], 157 Larmor radius [7.268], 157 Larmor’s formula [7.132], 146 laser cavity Q [8.126], 174

205 cavity line width [8.127], 174 cavity modes [8.124], 174 cavity stability [8.123], 174 threshold condition [8.129], 174 Lasers, 174 latent heat [5.48], 109 lattice constants of elements, 124 Lattice dynamics, 129 Lattice forces (simple), 131 lattice plane spacing [6.11], 126 Lattice thermal expansion and conduction, 131 lattice vector [6.7], 126 latus-rectum [3.109], 71 Laue equations [6.28], 128 Laurent series [2.168], 31 LCR circuits, 147 LCR definitions, 147 least-squares fitting, 60 Legendre equation and polynomials [2.421], 47 definition [2.343], 43 Legendre polynomials, 47 Leibniz theorem [2.296], 40 length (dimensions), 17 Lennard-Jones 6-12 potential [6.52], 131 lens blooming [8.7], 162 Lenses and mirrors, 168 lensmaker’s formula [8.66], 168 Levi-Civita symbol (3-D) [2.443], 50 ˆ l’Hopital’s rule [2.131], 28 Lie´ nard–Wiechert potentials, 139 light (speed of), 6, 7 Limits, 28 line charge (electric field from) [7.29], 138 line fitting, 60 Line radiation, 173 line shape collisional [8.114], 173 Doppler [8.116], 173 natural [8.112], 173 line width collisional/pressure [8.115], 173 Doppler broadened [8.117], 173 laser cavity [8.127], 174 natural [8.113], 173 Schawlow-Townes [8.128], 174 linear absorption coefficient [5.175], 120 linear expansivity (definition) [5.19], 107

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206 linear expansivity (of a crystal) [6.57], 131 linear regression, 60 linked flux [7.149], 147 liquid drop model [4.172], 103 litre (unit), 5 local civil time [9.4], 177 local sidereal time [9.7], 177 local thermodynamic equilibrium (LTE), 116, 120 ln(1 + x) (series expansion) [2.133], 29 logarithm of complex numbers [2.162], 30 logarithmic decrement [3.202], 78 London’s formula (interacting dipoles) [6.50], 131 longitudinal elastic modulus [3.241], 81 look-back time [9.96], 185 Lorentz broadening [8.112], 173 contraction [3.8], 64 factor (γ) [3.7], 64 force [7.122], 145 Lorentz (spacetime) transformations, 64 Lorentz factor (dynamical) [3.69], 68 Lorentz transformation in electrodynamics, 141 of four-vectors, 65 of momentum and energy, 65 of time and position, 64 of velocity, 64 Lorentz-Lorenz formula [7.93], 142 Lorentzian distribution [2.555], 58 Lorentzian (Fourier transform of) [2.505], 54 Lorenz constant [6.66], 132 gauge condition [7.43], 139 lumen (unit), 4 luminance [5.168], 119 luminosity distance [9.98], 185 luminosity–magnitude relation [9.31], 179 luminous density [5.160], 119 efficacy [5.169], 119 efficiency [5.170], 119 energy [5.157], 119 exitance [5.162], 119 flux [5.159], 119

Index intensity (dimensions), 17 intensity [5.166], 119 lux (unit), 4

M Mach number [3.315], 86 Mach wedge [3.328], 87 Maclaurin series [2.125], 28 Macroscopic thermodynamic variables, 115 Madelung constant (value), 9 Madelung constant [6.55], 131 magnetic diffusivity [7.282], 158 flux quantum, 6, 7 monopoles (none) [7.52], 140 permeability, µ, µr [7.107], 143 quantum number [4.131], 100 scalar potential [7.7], 136 susceptibility, χH , χB [7.103], 143 vector potential definition [7.40], 139 from J [7.47], 139 of a moving charge [7.49], 139 magnetic dipole, see dipole magnetic field around objects, 138 dimensions, 17 energy density [7.128], 146 Lorentz transformation, 141 static, 136 strength (H) [7.100], 143 thermodynamic work [5.8], 106 wave equation [7.194], 152 Magnetic fields, 138 magnetic flux (dimensions), 17 magnetic flux density (dimensions), 17 magnetic flux density from current [7.11], 136 current density [7.10], 136 dipole [7.36], 138 electromagnet [7.38], 138 line current (Biot–Savart law) [7.9], 136 solenoid (finite) [7.38], 138 solenoid (infinite) [7.33], 138 uniform cylindrical current [7.34], 138 waveguide [7.190], 151

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Index wire [7.34], 138 wire loop [7.37], 138 Magnetic moments, 100 magnetic vector potential (dimensions), 17 Magnetisation, 143 magnetisation definition [7.97], 143 dimensions, 17 isolated spins [4.151], 101 quantum paramagnetic [4.150], 101 magnetogyric ratio [4.138], 100 Magnetohydrodynamics, 158 magnetosonic waves [7.285], 158 Magnetostatics, 136 magnification (longitudinal) [8.71], 168 magnification (transverse) [8.70], 168 magnitude (astronomical) –flux relation [9.32], 179 –luminosity relation [9.31], 179 absolute [9.29], 179 apparent [9.27], 179 major axis [3.106], 71 Malus’s law [8.83], 170 Mars data, 176 mass (dimensions), 17 mass absorption coefficient [5.176], 120 mass ratio (of a rocket) [3.94], 70 Mathematical constants, 9 Mathematics, 19–62 matrices (square), 25 Matrix algebra, 24 matrix element (quantum) [4.32], 92 maxima [2.336], 42 Maxwell’s equations, 140 Maxwell’s equations (using D and H), 140 Maxwell’s relations, 109 Maxwell–Boltzmann distribution, 112 Maxwell-Boltzmann distribution mean speed [5.86], 112 most probable speed [5.88], 112 rms speed [5.87], 112 speed distribution [5.84], 112 mean arithmetic [2.108], 27 geometric [2.109], 27 harmonic [2.110], 27 mean estimator [2.541], 57 mean free path

207 and absorption coefficient [5.175], 120 Maxwell-Boltzmann [5.89], 113 mean intensity [5.172], 120 mean-life (nuclear decay) [4.165], 103 mega, 5 melting points of elements, 124 meniscus [3.339], 88 Mensuration, 35 Mercury data, 176 method of images, 138 metre (SI definition), 3 metre (unit), 4 metric elements and coordinate systems, 21 MHD equations [7.283], 158 micro, 5 microcanonical ensemble [5.109], 114 micron (unit), 5 microstrip line (impedance) [7.184], 150 Miller-Bravais indices [6.20], 126 milli, 5 minima [2.337], 42 minimum deviation (of a prism) [8.74], 169 minor axis [3.107], 71 minute (unit), 5 mirror formula [8.67], 168 Miscellaneous, 18 mobility (dimensions), 17 mobility (in conductors) [6.88], 134 modal dispersion (optical fibre) [8.79], 169 modified Bessel functions [2.419], 47 modified Julian day number [9.2], 177 modulus (of a complex number) [2.155], 30 Moire´ fringes, 35 molar gas constant (dimensions), 17 molar volume, 9 mole (SI definition), 3 mole (unit), 4 molecular flow [5.99], 113 moment electric dipole [7.81], 142 magnetic dipole [7.94], 143 magnetic dipole [7.95], 143 moment of area [3.258], 82 moment of inertia

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208 cone [3.160], 75 cylinder [3.155], 75 dimensions, 17 disk [3.168], 75 ellipsoid [3.163], 75 elliptical lamina [3.166], 75 rectangular cuboid [3.158], 75 sphere [3.152], 75 spherical shell [3.153], 75 thin rod [3.150], 75 triangular plate [3.169], 75 two-body system [3.83], 69 moment of inertia ellipsoid [3.147], 74 Moment of inertia tensor, 74 moment of inertia tensor [3.136], 74 Moments of inertia, 75 momentum definition [3.64], 68 dimensions, 17 generalised [3.218], 79 relativistic [3.70], 68 Momentum and energy transformations, 65 Monatomic gas, 112 monatomic gas entropy [5.83], 112 equation of state [5.78], 112 heat capacity [5.82], 112 internal energy [5.79], 112 pressure [5.77], 112 monoclinic system (crystallographic), 127 Moon data, 176 motif [6.31], 128 motion under constant acceleration, 68 Mott scattering formula [4.180], 104 µ, µr (magnetic permeability) [7.107], 143 multilayer films (in optics) [8.8], 162 multimode dispersion (optical fibre) [8.79], 169 multiplicity (quantum) j [4.133], 100 l [4.112], 98 multistage rocket [3.95], 70 Multivariate normal distribution, 58 Muon and tau constants, 9 muon physical constants, 9 mutual capacitance [7.134], 146 inductance (definition) [7.147], 147

Index inductance (energy) [7.135], 146 mutual coherence function [8.97], 172

N nabla, 21 Named integrals, 45 nano, 5 natural broadening profile [8.112], 173 natural line width [8.113], 173 Navier-Stokes equation [3.301], 85 nearest neighbour distances, 127 Neptune data, 176 neutron Compton wavelength, 8 gyromagnetic ratio, 8 magnetic moment, 8 mass, 8 molar mass, 8 Neutron constants, 8 neutron star degeneracy pressure [9.77], 183 newton (unit), 4 Newton’s law of Gravitation [3.40], 66 Newton’s lens formula [8.65], 168 Newton’s rings, 162 Newton’s rings [8.1], 162 Newton-Raphson method [2.593], 61 Newtonian gravitation, 66 noggin, 13 Noise, 117 noise figure [5.143], 117 Johnson [5.141], 117 Nyquist’s theorem [5.140], 117 shot [5.142], 117 temperature [5.140], 117 normal (unit principal) [2.284], 39 normal distribution [2.552], 58 normal plane, 39 Nuclear binding energy, 103 Nuclear collisions, 104 Nuclear decay, 103 nuclear decay law [4.163], 103 nuclear magneton, 7 number density (dimensions), 17 numerical aperture (optical fibre) [8.78], 169 Numerical differentiation, 61 Numerical integration, 61

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209

Index Numerical methods, 60 Numerical solutions to f(x) = 0, 61 Numerical solutions to ordinary differential equations, 62 nutation [3.194], 77 Nyquist’s theorem [5.140], 117

orthorhombic system (crystallographic), 127 Oscillating systems, 78 osculating plane, 39 Otto cycle efficiency [5.13], 107 overdamping [3.201], 78

O

p orbitals [4.95], 97 P-waves [3.263], 82 packing fraction (of spheres), 127 paired strip (impedance of) [7.183], 150 parabola, 38 parabolic motion [3.88], 69 parallax (astronomical) [9.46], 180 parallel axis theorem [3.140], 74 parallel impedances [7.158], 148 parallel wire feeder (inductance) [7.25], 137 paramagnetic susceptibility (Pauli) [6.79], 133 paramagnetism (quantum), 101 Paramagnetism and diamagnetism, 144 parity operator [4.24], 91 Parseval’s relation [2.495], 53 Parseval’s theorem integral form [2.496], 53 series form [2.480], 52 Partial derivatives, 42 partial widths (and total width) [4.176], 104 Particle in a rectangular box, 94 Particle motion, 68 partition function atomic [5.126], 116 definition [5.110], 114 macroscopic variables from, 115 pascal (unit), 4 Pauli matrices, 26 Pauli matrices [2.94], 26 Pauli paramagnetic susceptibility [6.79], 133 Pauli spin matrices (and Weyl eqn.) [4.182], 104 Pearson’s r [2.546], 57 Peltier effect [6.82], 133 pendulum compound [3.182], 76 conical [3.180], 76 double [3.183], 76

Oblique elastic collisions, 73 obliquity factor (diffraction) [8.46], 166 obliquity of the ecliptic [9.13], 178 observable (quantum physics) [4.5], 90 Observational astrophysics, 179 octahedron, 38 odd functions, 53 ODEs (numerical solutions), 62 ohm (unit), 4 Ohm’s law (in MHD) [7.281], 158 Ohm’s law [7.140], 147 opacity [5.176], 120 open-wire transmission line [7.182], 150 operator angular momentum and other operators [4.23], 91 definitions [4.105], 98 Hamiltonian [4.21], 91 kinetic energy [4.20], 91 momentum [4.19], 91 parity [4.24], 91 position [4.18], 91 time dependence [4.27], 91 Operators, 91 optic branch (phonon) [6.37], 129 optical coating [8.8], 162 optical depth [5.177], 120 Optical fibres, 169 optical path length [8.63], 168 Optics, 161–174 Orbital angular dependence, 97 Orbital angular momentum, 98 orbital motion, 71 orbital radius (Bohr atom) [4.73], 95 order (in diffraction) [8.26], 164 ordinary modes [7.271], 157 orthogonal matrix [2.85], 25 orthogonality associated Legendre functions [2.434], 48 Legendre polynomials [2.424], 47

P

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210 simple [3.179], 76 torsional [3.181], 76 Pendulums, 76 perfect gas, 110 pericentre (of an orbit) [3.110], 71 perimeter of circle [2.261], 37 of ellipse [2.266], 37 Perimeter, area, and volume, 37 period (of an orbit) [3.113], 71 Periodic table, 124 permeability dimensions, 17 magnetic [7.107], 143 of vacuum, 6, 7 permittivity dimensions, 17 electrical [7.90], 142 of vacuum, 6, 7 permutation tensor (ijk ) [2.443], 50 perpendicular axis theorem [3.148], 74 Perturbation theory, 102 peta, 5 petrol engine efficiency [5.13], 107 phase object (diffraction by weak) [8.43], 165 phase rule (Gibbs’s) [5.54], 109 phase speed (wave) [3.325], 87 Phase transitions, 109 Phonon dispersion relations, 129 phonon modes (mean energy) [6.40], 130 Photometric wavelengths, 179 Photometry, 119 photon energy [4.3], 90 Physical constants, 6 Pi (π) to 1 000 decimal places, 18 Pi (π), 9 pico, 5 pipe (flow of fluid along) [3.305], 85 pipe (twisting of) [3.255], 81 pitch angle, 159 Planck constant, 6, 7 constant (dimensions), 17 function [5.184], 121 length, 7 mass, 7 time, 7 Planck-Einstein relation [4.3], 90

Index plane polarisation, 170 Plane triangles, 36 plane wave expansion [2.427], 47 Planetary bodies, 180 Planetary data, 176 plasma beta [7.278], 158 dispersion relation [7.261], 157 frequency [7.259], 157 group velocity [7.264], 157 phase velocity [7.262], 157 refractive index [7.260], 157 Plasma physics, 156 Platonic solids, 38 Pluto data, 176 p-n junction [6.92], 134 Poincar´e sphere, 171 point charge (electric field from) [7.5], 136 Poiseuille flow [3.305], 85 Poisson brackets [3.224], 79 Poisson distribution [2.549], 57 Poisson ratio and elastic constants [3.251], 81 simple definition [3.231], 80 Poisson’s equation [7.3], 136 polarisability [7.91], 142 Polarisation, 170 Polarisation, 142 polarisation (electrical, per unit volume) [7.83], 142 polarisation (of radiation) angle [8.81], 170 axial ratio [8.88], 171 degree of [8.96], 171 elliptical [8.80], 170 ellipticity [8.82], 170 reflection law [7.218], 154 polarisers [8.85], 170 polhode, 63, 77 Population densities, 116 potential chemical [5.28], 108 difference (and work) [5.9], 106 difference (between points) [7.2], 136 electrical [7.46], 139 electrostatic [7.1], 136 energy (elastic) [3.235], 80 energy in Hamiltonian [3.222], 79

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211

Index energy in Lagrangian [3.216], 79 field equations [7.45], 139 four-vector [7.77], 141 grand [5.37], 108 Li´enard–Wiechert, 139 Lorentz transformation [7.75], 141 magnetic scalar [7.7], 136 magnetic vector [7.40], 139 Rutherford scattering [3.114], 72 thermodynamic [5.35], 108 velocity [3.296], 84 Potential flow, 84 Potential step, 92 Potential well, 93 power (dimensions), 17 power gain antenna [7.211], 153 short dipole [7.213], 153 Power series, 28 Power theorem [2.495], 53 Poynting vector (dimensions), 17 Poynting vector [7.130], 146 pp (proton-proton) chain, 182 Prandtl number [3.314], 86 precession (gyroscopic) [3.191], 77 Precession of equinoxes, 178 pressure broadening [8.115], 173 critical [5.75], 111 degeneracy [9.77], 183 dimensions, 17 fluctuations [5.136], 116 from partition function [5.118], 115 hydrostatic [3.238], 80 in a monatomic gas [5.77], 112 radiation, 152 thermodynamic work [5.5], 106 waves [3.263], 82 primitive cell [6.1], 126 primitive vectors (and lattice vectors) [6.7], 126 primitive vectors (of cubic lattices), 127 Principal axes, 74 principal moments of inertia [3.143], 74 principal quantum number [4.71], 95 principle of least action [3.213], 79 prism determining refractive index [8.75], 169

deviation [8.73], 169 dispersion [8.76], 169 minimum deviation [8.74], 169 transmission angle [8.72], 169 Prisms (dispersing), 169 probability conditional [2.567], 59 density current [4.13], 90 distributions continuous, 58 discrete, 57 joint [2.568], 59 Probability and statistics, 57 product (derivative of) [2.293], 40 product (integral of) [2.354], 44 product of inertia [3.136], 74 progression (arithmetic) [2.104], 27 progression (geometric) [2.107], 27 Progressions and summations, 27 projectiles, 69 propagation in cold plasmas, 157 Propagation in conducting media, 155 Propagation of elastic waves, 83 Propagation of light, 65 proper distance [9.97], 185 Proton constants, 8 proton mass, 6 proton-proton chain, 182 pulsar braking index [9.66], 182 characteristic age [9.67], 182 dispersion [9.72], 182 magnetic dipole radiation [9.69], 182 Pulsars, 182 pyramid (centre of mass) [3.175], 76 pyramid (volume) [2.272], 37

Q Q, see quality factor Q (Stokes parameter) [8.90], 171 Quadratic equations, 50 quadrature, 61 quadrature (integration), 44 quality factor Fabry-Perot etalon [8.14], 163 forced harmonic oscillator [3.211], 78 free harmonic oscillator [3.203], 78 laser cavity [8.126], 174

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212 LCR circuits [7.152], 148 quantum concentration [5.83], 112 Quantum definitions, 90 Quantum paramagnetism, 101 Quantum physics, 89–104 Quantum uncertainty relations, 90 quarter-wave condition [8.3], 162 quarter-wave plate [8.85], 170 quartic minimum, 42

R Radial forms, 22 radian (unit), 4 radiance [5.156], 118 radiant energy [5.145], 118 energy density [5.148], 118 exitance [5.150], 118 flux [5.147], 118 intensity (dimensions), 17 intensity [5.154], 118 radiation blackbody [5.184], 121 bremsstrahlung [7.297], 160 Cherenkov [7.247], 156 field of a dipole [7.207], 153 flux from dipole [7.131], 146 resistance [7.209], 153 synchrotron [7.287], 159 Radiation pressure, 152 radiation pressure extended source [7.203], 152 isotropic [7.200], 152 momentum density [7.199], 152 point source [7.204], 152 specular reflection [7.202], 152 Radiation processes, 118 Radiative transfer, 120 radiative transfer equation [5.179], 120 radiative transport (in stars) [9.63], 181 radioactivity, 103 Radiometry, 118 radius of curvature definition [2.282], 39 in bending [3.258], 82 relation to curvature [2.287], 39 radius of gyration (see footnote), 75 Ramsauer effect [4.52], 93 Random walk, 59

Index random walk Brownian motion [5.98], 113 one-dimensional [2.562], 59 three-dimensional [2.564], 59 range (of projectile) [3.90], 69 Rankine conversion [1.3], 15 Rankine-Hugoniot shock relations [3.334], 87 Rayleigh distribution [2.554], 58 resolution criterion [8.41], 165 scattering [7.236], 155 theorem [2.496], 53 Rayleigh-Jeans law [5.187], 121 reactance (definition), 148 reciprocal lattice vector [6.8], 126 matrix [2.83], 25 vectors [2.16], 20 reciprocity [2.330], 42 Recognised non-SI units, 5 rectangular aperture diffraction [8.39], 165 rectangular coordinates, 21 rectangular cuboid moment of inertia [3.158], 75 rectifying plane, 39 recurrence relation associated Legendre functions [2.433], 48 Legendre polynomials [2.423], 47 redshift –flux density relation [9.99], 185 cosmological [9.86], 184 gravitational [9.74], 183 Reduced mass (of two interacting bodies), 69 reduced units (thermodynamics) [5.71], 111 reflectance coefficient and Fresnel equations [7.227], 154 dielectric film [8.4], 162 dielectric multilayer [8.8], 162 reflection coefficient acoustic [3.283], 83 dielectric boundary [7.230], 154 potential barrier [4.58], 94 potential step [4.41], 92 potential well [4.48], 93

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213

Index transmission line [7.179], 150 reflection grating [8.29], 164 reflection law [7.216], 154 Reflection, refraction, and transmission, 154 refraction law (Snell’s) [7.217], 154 refractive index of dielectric medium [7.195], 152 ohmic conductor [7.234], 155 plasma [7.260], 157 refrigerator efficiency [5.11], 107 regression (linear), 60 relativistic beaming [3.25], 65 relativistic doppler effect [3.22], 65 Relativistic dynamics, 68 Relativistic electrodynamics, 141 Relativistic wave equations, 104 relativity (general), 67 relativity (special), 64 relaxation time and electron drift [6.61], 132 in a conductor [7.156], 148 in plasmas, 156 residuals [2.572], 60 Residue theorem [2.170], 31 residues (in complex analysis), 31 resistance and impedance, 148 dimensions, 17 energy dissipated in [7.155], 148 radiation [7.209], 153 resistivity [7.142], 147 resistor, see resistance resolving power chromatic (of an etalon) [8.21], 163 of a diffraction grating [8.30], 164 Rayleigh resolution criterion [8.41], 165 resonance forced oscillator [3.209], 78 resonance lifetime [4.177], 104 resonant frequency (LCR) [7.150], 148 Resonant LCR circuits, 148 restitution (coefficient of) [3.127], 73 retarded time, 139 revolution (volume and surface of), 39 Reynolds number [3.311], 86 ribbon (twisting of) [3.256], 81 Ricci tensor [3.57], 67

Riemann tensor [3.50], 67 right ascension [9.8], 177 rigid body angular momentum [3.141], 74 kinetic energy [3.142], 74 Rigid body dynamics, 74 rigidity modulus [3.249], 81 ripples [3.321], 86 rms (standard deviation) [2.543], 57 Robertson-Walker metric [9.87], 184 Roche limit [9.43], 180 rocket equation [3.94], 70 Rocketry, 70 rod bending, 82 moment of inertia [3.150], 75 stretching [3.230], 80 waves in [3.271], 82 Rodrigues’ formula [2.422], 47 Roots of quadratic and cubic equations, 50 Rossby number [3.316], 86 rot (curl), 22 Rotating frames, 66 Rotation matrices, 26 rotation measure [7.273], 157 Runge Kutta method [2.603], 62 Rutherford scattering, 72 Rutherford scattering formula [3.124], 72 Rydberg constant, 6, 7 and Bohr atom [4.77], 95 dimensions, 17 Rydberg’s formula [4.78], 95

S s orbitals [4.92], 97 S-waves [3.262], 82 Sackur-Tetrode equation [5.83], 112 saddle point [2.338], 42 Saha equation (general) [5.128], 116 Saha equation (ionisation) [5.129], 116 Saturn data, 176 scalar effective mass [6.87], 134 scalar product [2.1], 20 scalar triple product [2.10], 20 scale factor (cosmic) [9.87], 184 scattering angle (Rutherford) [3.116], 72 Born approximation [4.178], 104 Compton [7.240], 155

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214 crystal [6.32], 128 inverse Compton [7.239], 155 Klein-Nishina [7.243], 155 Mott (identical particles) [4.180], 104 potential (Rutherford) [3.114], 72 processes (electron), 155 Rayleigh [7.236], 155 Rutherford [3.124], 72 Thomson [7.238], 155 scattering cross-section, see cross-section Schawlow-Townes line width [8.128], 174 Schr¨ odinger equation [4.15], 90 Schwarz inequality [2.152], 30 Schwarzschild geometry (in GR) [3.61], 67 Schwarzschild radius [9.73], 183 Schwarzschild’s equation [5.179], 120 screw dislocation [6.22], 128 secx definition [2.228], 34 series expansion [2.138], 29 secant method (of root-finding) [2.592], 61 sechx [2.229], 34 second (SI definition), 3 second (time interval), 4 second moment of area [3.258], 82 Sedov-Taylor shock relation [3.331], 87 selection rules (dipole transition) [4.91], 96 self-diffusion [5.93], 113 self-inductance [7.145], 147 semi-ellipse (centre of mass) [3.178], 76 semi-empirical mass formula [4.173], 103 semi-latus-rectum [3.109], 71 semi-major axis [3.106], 71 semi-minor axis [3.107], 71 semiconductor diode [6.92], 134 semiconductor equation [6.90], 134 Series expansions, 29 series impedances [7.157], 148 Series, summations, and progressions, 27 shah function (Fourier transform of) [2.510], 54 shear modulus [3.249], 81 strain [3.237], 80 viscosity [3.299], 85 waves [3.262], 82

Index shear modulus (dimensions), 17 sheet of charge (electric field) [7.32], 138 shift theorem (Fourier transform) [2.501], 54 shock Rankine-Hugoniot conditions [3.334], 87 spherical [3.331], 87 Shocks, 87 shot noise [5.142], 117 SI base unit definitions, 3 SI base units, 4 SI derived units, 4 SI prefixes, 5 SI units, 4 sidelobes (diffraction by 1-D slit) [8.38], 165 sidereal time [9.7], 177 siemens (unit), 4 sievert (unit), 4 similarity theorem (Fourier transform) [2.500], 54 simple cubic structure, 127 simple harmonic oscillator, see harmonic oscillator simple pendulum [3.179], 76 Simpson’s rule [2.586], 61 sinx and Euler’s formula [2.218], 34 series expansion [2.136], 29 sinc function [2.512], 54 sine formula planar triangles [2.246], 36 spherical triangles [2.255], 36 sinhx definition [2.219], 34 series expansion [2.144], 29 sin−1 x, see arccosx skew-symmetric matrix [2.87], 25 skewness estimator [2.544], 57 skin depth [7.235], 155 slit diffraction (broad slit) [8.37], 165 slit diffraction (Young’s) [8.24], 164 Snell’s law (acoustics) [3.284], 83 Snell’s law (electromagnetism) [7.217], 154 soap bubbles [3.337], 88 solar constant, 176 Solar data, 176 Solar system data, 176

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Index solenoid finite [7.38], 138 infinite [7.33], 138 self inductance [7.23], 137 solid angle (subtended by a circle) [2.278], 37 Solid state physics, 123–134 sound speed (in a plasma) [7.275], 158 sound, speed of [3.317], 86 space cone, 77 space frequency [3.188], 77 space impedance [7.197], 152 spatial coherence [8.108], 172 Special functions and polynomials, 46 special relativity, 64 specific charge on electron, 8 emission coefficient [5.174], 120 heat capacity, see heat capacity definition, 105 dimensions, 17 intensity (blackbody) [5.184], 121 intensity [5.171], 120 specific impulse [3.92], 70 speckle intensity distribution [8.110], 172 speckle size [8.111], 172 spectral energy density blackbody [5.186], 121 definition [5.173], 120 spectral function (synchrotron) [7.295], 159 Spectral line broadening, 173 speed (dimensions), 17 speed distribution (Maxwell-Boltzmann) [5.84], 112 speed of light (equation) [7.196], 152 speed of light (value), 6 speed of sound [3.317], 86 sphere area [2.263], 37 Brownian motion [5.98], 113 capacitance [7.12], 137 capacitance of adjacent [7.14], 137 capacitance of concentric [7.18], 137 close-packed, 127 collisions of, 73 electric field [7.27], 138 geometry on a, 36 gravitation field from a [3.44], 66

215 in a viscous fluid [3.308], 85 in potential flow [3.298], 84 moment of inertia [3.152], 75 Poincar´e, 171 polarisability, 142 volume [2.264], 37 spherical Bessel function [2.420], 47 spherical cap area [2.275], 37 centre of mass [3.177], 76 volume [2.276], 37 spherical excess [2.260], 36 Spherical harmonics, 49 spherical harmonics definition [2.436], 49 Laplace equation [2.440], 49 orthogonality [2.437], 49 spherical polar coordinates, 21 spherical shell (moment of inertia) [3.153], 75 spherical surface (capacitance of near) [7.16], 137 Spherical triangles, 36 spin and total angular momentum [4.128], 100 degeneracy, 115 electron magnetic moment [4.141], 100 Pauli matrices, 26 spinning bodies, 77 spinors [4.182], 104 Spitzer conductivity [7.254], 156 spontaneous emission [8.119], 173 spring constant and wave velocity [3.272], 83 Square matrices, 25 standard deviation estimator [2.543], 57 Standard forms, 44 Star formation, 181 Star–delta transformation, 149 Static fields, 136 statics, 63 Stationary points, 42 Statistical entropy, 114 Statistical thermodynamics, 114 Stefan–Boltzmann constant, 9 Stefan–Boltzmann constant (dimensions), 17

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216 Stefan-Boltzmann constant, 121 Stefan-Boltzmann law [5.191], 121 stellar aberration [3.24], 65 Stellar evolution, 181 Stellar fusion processes, 182 Stellar theory, 181 step function (Fourier transform of) [2.511], 54 steradian (unit), 4 stimulated emission [8.120], 173 Stirling’s formula [2.411], 46 Stokes parameters, 171 Stokes parameters [8.95], 171 Stokes’s law [3.308], 85 Stokes’s theorem [2.60], 23 Straight-line fitting, 60 strain simple [3.229], 80 tensor [3.233], 80 volume [3.236], 80 stress dimensions, 17 in fluids [3.299], 85 simple [3.228], 80 tensor [3.232], 80 stress-energy tensor and field equations [3.59], 67 perfect fluid [3.60], 67 string (waves along a stretched) [3.273], 83 Strouhal number [3.313], 86 structure factor [6.31], 128 sum over states [5.110], 114 Summary of physical constants, 6 summation formulas [2.118], 27 Sun data, 176 Sunyaev-Zel’dovich effect [9.51], 180 surface brightness (blackbody) [5.184], 121 surface of revolution [2.280], 39 Surface tension, 88 surface tension Laplace’s formula [3.337], 88 work done [5.6], 106 surface tension (dimensions), 17 surface waves [3.320], 86 survival equation (for mean free path) [5.90], 113 susceptance (definition), 148

Index susceptibility electric [7.87], 142 Landau diamagnetic [6.80], 133 magnetic [7.103], 143 Pauli paramagnetic [6.79], 133 symmetric matrix [2.86], 25 symmetric top [3.188], 77 Synchrotron radiation, 159 synodic period [9.44], 180

T tanx definition [2.220], 34 series expansion [2.137], 29 tangent [2.283], 39 tangent formula [2.250], 36 tanhx definition [2.221], 34 series expansion [2.145], 29 tan−1 x, see arctanx tau physical constants, 9 Taylor series one-dimensional [2.123], 28 three-dimensional [2.124], 28 telegraphist’s equations [7.171], 150 temperature antenna [7.215], 153 Celsius, 4 dimensions, 17 Kelvin scale [5.2], 106 thermodynamic [5.1], 106 Temperature conversions, 15 temporal coherence [8.105], 172 tensor Einstein [3.58], 67 electric susceptibility [7.87], 142 ijk [2.443], 50 fluid stress [3.299], 85 magnetic susceptibility [7.103], 143 moment of inertia [3.136], 74 Ricci [3.57], 67 Riemann [3.50], 67 strain [3.233], 80 stress [3.232], 80 tera, 5 tesla (unit), 4 tetragonal system (crystallographic), 127 tetrahedron, 38 thermal conductivity

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Index diffusion equation [2.340], 43 dimensions, 17 free electron [6.65], 132 phonon gas [6.58], 131 transport property [5.96], 113 thermal de Broglie wavelength [5.83], 112 thermal diffusion [5.93], 113 thermal diffusivity [2.340], 43 thermal noise [5.141], 117 thermal velocity (electron) [7.257], 156 Thermodynamic coefficients, 107 Thermodynamic fluctuations, 116 Thermodynamic laws, 106 Thermodynamic potentials, 108 thermodynamic temperature [5.1], 106 Thermodynamic work, 106 Thermodynamics, 105–121 Thermoelectricity, 133 thermopower [6.81], 133 Thomson cross section, 8 Thomson scattering [7.238], 155 throttling process [5.27], 108 time (dimensions), 17 time dilation [3.11], 64 Time in astronomy, 177 Time series analysis, 60 Time-dependent perturbation theory, 102 Time-independent perturbation theory, 102 timescale free-fall [9.53], 181 Kelvin-Helmholtz [9.55], 181 Titius-Bode rule [9.41], 180 tonne (unit), 5 top asymmetric [3.189], 77 symmetric [3.188], 77 symmetries [3.149], 74 top hat function (Fourier transform of) [2.512], 54 Tops and gyroscopes, 77 torque, see couple Torsion, 81 torsion in a thick cylinder [3.254], 81 in a thin cylinder [3.253], 81 in an arbitrary ribbon [3.256], 81 in an arbitrary tube [3.255], 81 in differential geometry [2.288], 39

217 torsional pendulum [3.181], 76 torsional rigidity [3.252], 81 torus (surface area) [2.273], 37 torus (volume) [2.274], 37 total differential [2.329], 42 total internal reflection [7.217], 154 total width (and partial widths) [4.176], 104 trace [2.75], 25 trajectory (of projectile) [3.88], 69 transfer equation [5.179], 120 Transformers, 149 transmission coefficient Fresnel [7.232], 154 potential barrier [4.59], 94 potential step [4.42], 92 potential well [4.49], 93 transmission grating [8.27], 164 transmission line, 150 coaxial [7.181], 150 equations [7.171], 150 impedance lossless [7.174], 150 lossy [7.175], 150 input impedance [7.178], 150 open-wire [7.182], 150 paired strip [7.183], 150 reflection coefficient [7.179], 150 vswr [7.180], 150 wave speed [7.176], 150 waves [7.173], 150 Transmission line impedances, 150 Transmission line relations, 150 Transmission lines and waveguides, 150 transmittance coefficient [7.229], 154 Transport properties, 113 transpose matrix [2.70], 24 trapezoidal rule [2.585], 61 triangle area [2.254], 36 centre of mass [3.174], 76 inequality [2.147], 30 plane, 36 spherical, 36 triangle function (Fourier transform of) [2.513], 54 triclinic system (crystallographic), 127 trigonal system (crystallographic), 127 Trigonometric and hyperbolic defini-

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218 tions, 34 Trigonometric and hyperbolic formulas, 32 Trigonometric and hyperbolic integrals, 45 Trigonometric derivatives, 41 Trigonometric relationships, 32 triple-α process, 182 true anomaly [3.104], 71 tube, see pipe Tully-Fisher relation [9.49], 180 tunnelling (quantum mechanical), 94 tunnelling probability [4.61], 94 turns ratio (of transformer) [7.163], 149 two-level system (microstates of) [5.107], 114

U U (Stokes parameter) [8.92], 171 UBV magnitude system [9.36], 179 umklapp processes [6.59], 131 uncertainty relation energy-time [4.8], 90 general [4.6], 90 momentum-position [4.7], 90 number-phase [4.9], 90 underdamping [3.198], 78 unified atomic mass unit, 5, 6 uniform distribution [2.550], 58 uniform to normal distribution transformation, 58 unitary matrix [2.88], 25 units (conversion of SI to Gaussian), 135 Units, constants and conversions, 3–18 universal time [9.4], 177 Uranus data, 176 UTC [9.4], 177

V V (Stokes parameter) [8.94], 171 van der Waals equation [5.67], 111 Van der Waals gas, 111 van der Waals interaction [6.50], 131 Van-Cittert Zernicke theorem [8.108], 172 variance estimator [2.542], 57 variations, calculus of [2.334], 42 Vector algebra, 20 Vector integral transformations, 23 vector product [2.2], 20 vector triple product [2.12], 20 Vectors and matrices, 20

Index velocity (dimensions), 17 velocity distribution (Maxwell-Boltzmann) [5.84], 112 velocity potential [3.296], 84 Velocity transformations, 64 Venus data, 176 virial coefficients [5.65], 110 Virial expansion, 110 virial theorem [3.102], 71 vis-viva equation [3.112], 71 viscosity dimensions, 17 from kinetic theory [5.97], 113 kinematic [3.302], 85 shear [3.299], 85 viscous flow between cylinders [3.306], 85 between plates [3.303], 85 through a circular pipe [3.305], 85 through an annular pipe [3.307], 85 Viscous flow (incompressible), 85 volt (unit), 4 voltage across an inductor [7.146], 147 bias [6.92], 134 Hall [6.68], 132 law (Kirchhoff’s) [7.162], 149 standing wave ratio [7.180], 150 thermal noise [5.141], 117 transformation [7.164], 149 volume dimensions, 17 of cone [2.272], 37 of cube, 38 of cylinder [2.270], 37 of dodecahedron, 38 of ellipsoid [2.268], 37 of icosahedron, 38 of octahedron, 38 of parallelepiped [2.10], 20 of pyramid [2.272], 37 of revolution [2.281], 39 of sphere [2.264], 37 of spherical cap [2.276], 37 of tetrahedron, 38 of torus [2.274], 37 volume expansivity [5.19], 107 volume strain [3.236], 80 vorticity and Kelvin circulation [3.287],

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219

Index 84 vorticity and potential flow [3.297], 84 vswr [7.180], 150

W wakes [3.330], 87 Warm plasmas, 156 watt (unit), 4 wave equation [2.342], 43 wave impedance acoustic [3.276], 83 electromagnetic [7.198], 152 in a waveguide [7.189], 151 Wave mechanics, 92 Wave speeds, 87 wavefunction and expectation value [4.25], 91 and probability density [4.10], 90 diffracted in 1-D [8.34], 165 hydrogenic atom [4.91], 96 perturbed [4.160], 102 Wavefunctions, 90 waveguide cut-off frequency [7.186], 151 equation [7.185], 151 impedance TE modes [7.189], 151 TM modes [7.188], 151 TEmn modes [7.190], 151 TMmn modes [7.192], 151 velocity group [7.188], 151 phase [7.187], 151 Waveguides, 151 wavelength Compton [7.240], 155 de Broglie [4.2], 90 photometric, 179 redshift [9.86], 184 thermal de Broglie [5.83], 112 waves capillary [3.321], 86 electromagnetic, 152 in a spring [3.272], 83 in a thin rod [3.271], 82 in bulk fluids [3.265], 82 in fluids, 86 in infinite isotropic solids [3.264], 82 magnetosonic [7.285], 158

on a stretched sheet [3.274], 83 on a stretched string [3.273], 83 on a thin plate [3.268], 82 sound [3.317], 86 surface (gravity) [3.320], 86 transverse (shear) Alfv´en [7.284], 158 Waves in and out of media, 152 Waves in lossless media, 152 Waves in strings and springs, 83 wavevector (dimensions), 17 weber (unit), 4 Weber symbols, 126 weight (dimensions), 17 Weiss constant [7.114], 144 Weiss zone equation [6.10], 126 Welch window [2.582], 60 Weyl equation [4.182], 104 Wiedemann-Franz law [6.66], 132 Wien’s displacement law [5.189], 121 Wien’s displacement law constant, 9 Wien’s radiation law [5.188], 121 Wiener-Khintchine theorem in Fourier transforms [2.492], 53 in temporal coherence [8.105], 172 Wigner coefficients (spin-orbit) [4.136], 100 Wigner coefficients (table of), 99 windowing Bartlett [2.581], 60 Hamming [2.584], 60 Hanning [2.583], 60 Welch [2.582], 60 wire electric field [7.29], 138 magnetic flux density [7.34], 138 wire loop (inductance) [7.26], 137 wire loop (magnetic flux density) [7.37], 138 wires (inductance of parallel) [7.25], 137 work (dimensions), 17

X X-ray diffraction, 128

Y yocto, 5 yotta, 5 Young modulus and Lam´e coefficients [3.240], 81

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220 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

Index