The Physical Chemist's Toolbox

By Robert M. Metzger

The working tools of the physical sciences, expertly organized into one volume

Covering the basic concepts and working tools in the physical sciences, this reference is a unique, indispensable guide for students and researchers in chemistry, physics, and related disciplines. Everyone from novices to experienced researchers can turn to this book to find the essential equations, theories, and working tools needed to conduct and interpret contemporary research. Expertly organized, the book.

• Summarizes the core theories common to chemistry and physics

• Introduces topics and techniques that lay the foundations of instrumentation

• Discusses basic as well as advanced instrumentation and experimental methods

• Guides readers from crystals to nanoparticles to single molecules

Readers gain access to not only the core concepts of the physical sciences, but also the underlying mathematics. Among the topics addressed are mechanics, special relativity, electricity and magnetism, quantum chemistry, thermodynamics, electrochemistry, symmetry, solid state physics, and electronics. The book also addresses energy and electrical sources, detectors, and algorithms. Moreover, it presents state-of-the-technology instrumentation and techniques needed to conduct successful experiments.

Each chapter includes problems and exercises ranging from easy to difficult to help readers master core concepts and put them into practice. References lead to more specialized texts so that readers can explore individual topics in greater depth.

The Physical Chemist's Toolbox is recommended not only as a general reference, but also as a textbook for two-semester graduate courses in physical and analytical chemistry.

• Language: English
• Publisher: Wiley
• Release date: Apr 30, 2012
• ISBN: 9781118195574

Book preview

Chapter 1

Introduction: A Physical Chemists's Toolbox

Indocti discant, amentque meminisse periti.Charles Jean Hénault (1685–1770) in Nouvel Abrégé Chronologique de l'Histoire de France jusqu'à la Mort de Louis XIV

Jack Sherman: Dr. Pauling, how does one get good ideas?

Linus Pauling: Well, I guess one must have many ideas, and throw away the bad ones.

Linus Carl Pauling (1901–1994)

Never give in. Never give in. Never, never, never, never, never give in.

Sir Winston Churchill (1874–1965) at Harrow School, 29 October 1941

This compendium, vademecum, or toolbox is an abbreviated introduction to, or review of, theory and experiment in physics and chemistry. The term vade mecum or go with me was the first tentative title for this book; it was associated with the learned and boring Baedeker® guidebooks for travel in the early 1900s: These Baedekers have been replaced with heavily illustrated and less boring Dorling–Kindersley® guides. Most students in 2011 who know some Latin would ask vade mecum? go with me? where? why?

The intended audience for this toolbox is the beginning researcher, who often has difficulty in reconciling recent or past classroom knowledge in the undergraduate or first-year graduate curriculum with the topics and research problems current in research laboratories in the twenty-first century. While several excellent and specialized monographs exist for all the topics discussed in this book, to my knowledge there is no single compact book that covers adequately the disparate techniques needed for scientific advances in the twenty-first century. In particular, there is a need to find What will this or that technique do for my research problem? The aim of this toolbox is thus fourfold:

1. Summarize the theory common to chemistry and physics (Chapters 2–6).

2. Introduce topics and techniques that lead to instrumentation (Chapters 7–9).

3. Discuss the advanced instrumentation available in research (Chapters 10 and 11).

4. Travel a path from crystals to nanoparticles to single molecules (Chapter 12).

The book is interspersed with problems to do: some trivial, some difficult. This expedient can keep the volume more compact, and it becomes a useful pedagogical tool. This book tries to be a mathematically deep, yet brief and useful compendium of several topics, which can and should be covered by more specialized books, courses, and review articles.

Throughout, the aim is to bring the novice up to speed. The teaching of chemistry leaned rather heavily toward mathematical and physical rigor in the 1960s, but this fervor was lost, as chemical, physical, and biochemical complexity eluded simple mathematical precision. Alas, chemical and biological phenomena are usually determined by small but significant differences between two very large quantities, whose accurate calculation is often difficult!

Lamentably, the recent educational trend has been to train what could be called one-dimensional scientists, very good in one subfield but blissfully unaware of the rest. It is sad that we no longer produce those broadly trained scientists of past generations, who were willing to delve into new problems far from their original interest: I am thinking of Hans Bethe, Peter Debye, Enrico Fermi, Linus Pauling, or Edward Teller. This toolbox tries to adhere to this older and broader tradition, redress the temporary malady, and help restore the universality of scientific inquiry.

To the instructor: This toolbox could form the basis of a one-year graduate course in physical chemistry and/or analytical chemistry, perhaps team-taught; it should be taught with mandatory problem sets (students will connect the dots by doing the suggested problems) and with recourse to traditional texts that cover, for example, quantum mechanics or statistical mechanics in much greater detail. I am reminded of the very successful one-year team-taught courses such as Western Civilization at Stanford University in the 1960s! I have taught the toolbox several times at the University of Alabama as a one-semester course, but found the pace exhausting.

To the many students who took my course: Thanks for being so patient.

To Chemistry and Physics departments: The toolbox could become a valuable resource for all entering graduate students, so maybe students, even in areas far from physical chemistry, should be encouraged to buy it and work at it on their own.

To the student: (1) Do the problems; (2) read around in specialized reference texts that may be suggested either in this toolbox or by your instructor(s); (3) discover whether the toolbox could be developed in new directions.

To myself: To adapt Tom Lehrer's (1928–) famous quip, I am embarrassed to realize that at my present age Mozart had been dead for 36 years.

Alan MacDiarmid (1927–2007) once said Chemistry is about people: In this spirit, full names and birth and death dates are given to all the scientists quoted in this book; such brief historical data may help illuminate how and when science was done. I have resisted mentioning who was a Nobel prize winner: too many to list, and some worthy scientists—for example, Mendeleyeff, Eyring, Edison, Slater, and Tesla—were not honored. I owe a deep debt of gratitude to many people who have educated me over several decades, as live teachers and silent authors. In particular, I am indebted to Professor Willard Frank Libby (1908–1980), who taught us undergraduates at UCLA to love current research problems and led us into quite a few wild-goose chases; Professor Harden Marsden McConnell (1927–), who led us at Caltech and Stanford by example to see what are the interesting problems and what are trivial problems; Professor Linus Carl Pauling (1901–1994), who taught me electrical and magnetic susceptibilities with his incomparable photographic recall of data and dates, and with his insight and humanity about current events; Dr. Richard Edward Marsh (1922–) and Professor Paul Gravis Simpson (1937–1978), who taught me crystallography; Professor Michel Boudart (1925–), who introduced me to heterogeneous catalysis; Mr. William D. Good (1937–1978), who taught me combustion calorimetry; Professor Sukant Kishore Tripathy (1952–2000), who introduced me to Langmuir–Blodgett films; and finally, Professor Richard Phillips Feynman (1918–1983), who taught me about the Schwartzschild singularity and event horizons and who was a source of deep inspiration, pleasant conversations, and mischievous fun. Thanks are also due to two persons who helped me greatly in my academic career and taught me a thing or two about what good science really means: Professor Andrew Peter Stefani (1927–) of the University of Mississippi and Professor Michael Patrick Cava (1926–2010) of the University of Alabama. Professor Carolyn J. Cassady (University of Alabama) kindly allowed me to use an experiment she had devised for students of mass spectrometry.

The following books have inspired me: (1) Principles of Modern Physics by Robert B. Leighton, (2) Theoretical Physics by Georg Joos, (3) The Feynman Lectures on Physics by Richard P. Feynman, and (4) Principles of Instrumental Analysis by Doug Skoog, James Holler, and Stanley Crouch. In this twenty-first century, much help was obtained on-line from Wikipedia, but "caveat emptor"!

Writing is teaching but also learning; Marcus Porcius Cato (234–149 bc), who was echoing Solon (630–560 bc), said "I dare to say again: ‘senesco discens plurima.’"

Thanks are due to several friends and colleagues, who corrected errors and oversights in the early drafts: Professor Massimo Carbucicchio (University of Parma, Italy), Professor Michael Bowman, Dan Goebbert, Shanlin Pan, and Richard Tipping (University of Alabama), Professor Harris J. Silverstone (Johns Hopkins University), Professor Zoltán G. Soos (Princeton University), Dr. Ralph H. Young (Eastman Kodak Co.), and Adam Csoeke-Peck (Brentwood, California). The errors that remain are all mine; errare humanum est, sed perseverare diabolicum [Lucius Anneus Seneca (ca. 4 bc–ad 66)]. To the reader who finds errors, my apologies: I will try to correct the errors for the next edition; echoing what Akira Kurosawa (1910–1998) said in 1989, when he received an honorary Oscar for lifetime achievements in cinematography: So sorry, [I] hope to do better next time.

Chapter 2

Particles, Forces, and Mathematical Methods

Viribus Unitis

[with united forces]

Emperor Franz Josef the First (and the Last) (1830–1916)

It is difficult to make predictions, especially about the future.

Yogi Berra (1925–)

This chapter summarizes the fundamental forces in nature, reviews some mathematical methods, and discusses electricity, magnetism, special relativity, optics, and statistics.

Sideline

The name physics derives from the Greek word ϕυσις (= nature, essence): Early physicists like Newton were called natural philosophers. The word chemistry, through its Arabic precursor alchimya, derives from the Greek word χημι (= black earth), a tribute to the Egyptians' embalming arts. Mathematics comes from the Greek μαθημα (= learning, study). Algebra comes from the Arabic al-jabr (= transposition [to the other side of an equation]). Calculus (as in infinitesimal calculus) is the Latin word for a small pebble.

2.1 Fundamental Forces, Elementary Particles, Nuclei and Atoms

The four fundamental forces, their governing equations, the mediating particles, their relative magnitudes, and their ranges are listed in Table 2.1.

Table 2.1 The Fundamental Forces.

The first (and weakest) force is Newton's¹ force of universal gravitation (1687) [2]:

(2.1.1) equation

which describes the attractive force F12 between two bodies of masses m1 and M2 placed a distance r12 apart, where G is the constant of gravitation. The largest visible objects in the universe (galaxies, stars, quasars, planets, satellites, comets) are held together by this weakest force, which may be transmitted by a presumed but hitherto unobserved mediating particle called the graviton. Its range extends to the whole universe. Masses are always positive.

The second force is the electrical force, which obeys Coulomb's² law (1785) [3]:

(2.1.2) equation

which describes the attractive (or repulsive) force F12 between two electrical charges q1 and q2 (positive or negative) placed r12 apart, where ε0 is the electrical permittivity of vacuum. The fundamental electrical monopole (electron) is probably infinitely stable; the mediating particle for the electrical force (photon) is observed and well understood. Magnetism is usually due to moving electrical charges, but its monopole has never been seen, so magnetism is not really an independent force; atoms have magnetic properties, and in wires the gegenions of electrical currents are stationary, yet the overall charge is zero: Hence magnetism is a special relativistic effect. As explained below, electricity and magnetism are well described by Maxwell's³ four field equations [4].

The third force is the weak nuclear or Fermi⁴ force (1934), which stabilizes many radioactive particles and the free neutron; it explains beta decay and positron emission (e.g., the free neutron decays within a half-life of 13 minutes into a proton, an electron, and an electron antineutrino). The weak force has a very narrow range.

The fourth and strongest force in the universe is the strong nuclear force, which binds together the nuclei and the constituents of atomic nuclei, but has an extremely narrow range. Indirect experimental evidence exists for a mediating particle (gluon). Nucleons (neutrons, protons) and maybe nuclei consist of elementary particles called quarks, which have never been seen free, although proton–proton scattering experiments show that protons consist of lumps, which may be the best experimental evidence for quarks.

Between 1900 and 1960 a zoo of 100-odd stable and unstable elementary particles were discovered; the shortest-lived among them were called resonances; quarks were proposed in 1964 by Zweig⁵ and Gell-Mann⁶ to help order this zoo. Within the nucleus, the inter-nucleon strong force was traditionally thought of as being mediated by pions (themselves combinations of two quarks). The nuclear shell model assigns quantum numbers to the protons and neutrons that have merged to form a certain nucleon. Certain magic values of these nuclear quantum numbers explain why certain nuclei are more stable (have longer lives) than others.

Sideline

The name quark comes from a sentence in Joyce's⁷ Finnegan Wake; a free quark has never been isolated, but physicists have not looked in German grocery stores, where Quark is a well-known special soft cheese!

In 1960 electrical and weak forces were merged by Glashow⁸ into electroweak theory. Evolving in the 1960s and 1970s from the quark hypothesis, the Standard Model of Glashow, Weinberg⁹, and Salam¹⁰ explains nucleons and other particles (hadrons, baryons, and mesons) as unions of either three or two quarks each, with a new set of ad hoc quantum numbers. This Standard Model has a symmetry basis in the finite special unitary group SU(3), along with a mathematical expression in quantum chromodynamics, but does not yield a force field. These seemingly provisional ex post facto arguments and quantum numbers are reminiscent of the chemical arguments used by Mendeleyeff¹¹ in 1869 to construct the Periodic Table of chemical elements (whose explanation had to wait for quantum mechanics in the 1920s).

Sideline

Mendeleyeff divorced his wife in 1882 and married a student: By the rules of the Russian Orthodox Church, he became a bigamist, and according to an Edict of the Russian Czar, only members of the Church in good standing could teach in Russian Universities. When apprised of the dilemma, Czar Alexander III¹² said: Mendeleyeff may have two wives, but I only have one Mendeleyeff: Professor. Mendeleyeff kept his job!

Table 2.2 lists the presently known fundamental particles (unobserved quarks and some neutrinos), the elementary particles, and the observed (photon, vector bosons) and unobserved (gluons, gravitons, Higgs¹³ bosons) particles that mediate the interactions (strong, electromagnetic, weak, and gravitational) between them. Here fundamental is used for the presumed building blocks, while elementary is used for the experimentally observed smallest constituents of matter.

Table 2.2 Fundamental (Quark, Gluon, Graviton, Neutrino) and Elementary (= Fundamental Plus 2-Quark and 3-Quark Combinations) Particlesa

The lifetimes t, or half-lives τ can measured directly when or so (τ is defined as the time elapsed from the initial formation of a number N of these particles to the time when their population has decreased to N/2). Shorter half-lives ( ) are inferred from a measured natural or Breit¹⁴–Wigner¹⁵ or Lorentzian¹⁶ linewidth ΔE of their energy E and the Heisenberg¹⁷ uncertainty principle condition (discussed further in Section 3.1) (h/4π ≡ η/2), where is Planck's¹⁸ constant of action (one can argue whether τ is a lifetime or a half-life in the Heisenberg sense). In practice, one estimates the half-life τ from the width of the resonance ΔE12 (= width at half-maximum height; see Fig. 2.1)

(2.1.3) equation

Figure 2.1 Natural or Breit–Wigner or Lorentzian linewidth for ΔE = 1.5 MeV at half-maximum centered around .

Searches for individual quarks using high-energy accelerators have failed, up to rest-mass energies in vast excess of the masses of the stable known leptons and hadrons. Searches for quarks in minerals and seawater, potentially left over when hadrons and leptons first formed, and focused on their putative fractional electrical charge, have also failed. Therefore an explanation of quark confinement has emerged: Quarks are confined by twos and threes in a very deep potential well (Fig. 2.2), and are held together by forces so strong that only maybe future high-energy accelerator experiments may (if ever) detect an individual quark. The sum of the probable rest masses of one u, one , and one d quark, namely 4.6 + 4.6 + 16 = 25.2 me is far short of the rest mass of the proton (1836 me).

Figure. 2.2 This crude model tries to show the confinement of three quarks [up (u), conjugate up ( ), and down (d), with electrical charges +2/3, +2/3, and −1/3] inside a proton. The springs depict the mutual interactions, meditated by virtual gluons, which are (somehow) limited by the inter-quark potential to remain within the inside of the proton.

Efforts to unify all four forces into a single grand unified theory have failed. The very elegant string theory has provided no measurable predictions.

The rest masses of the particles cannot yet be predicted; the proposed Higgs boson, which has not yet been detected, may explain why particles have mass. Experimental searches are ongoing for free quarks, gluons, and the Higgs boson. After 15 years of construction, in March 2010 the 27-km radius Large Hadron Collider at the Centre Européen de Recherches Nucléaires near Geneva, Switzerland has reached an energy of 7 TeV (1.12 µJ per particle), so there is some hope for future discoveries.

Given the dearth of new results from high-energy scattering studies, high-energy physics has turned to the universe and to astrophysics for clues. The existence of dark matter (75% of all matter in the universe?) has been postulated, to account for the stability of galaxies; similarly, the existence of dark energy is also guessed at. But we must always remember Newton's stern warning: Physica, cave metaphysicam and Occam's¹⁹ razor: Entia non sunt multiplicanda praeter necessitatem.

Nuclei of atoms can be thought of as super-dense combinations of Z protons and N neutrons, that is, A = Z + N nucleons, and their mass is M(Z, A); there is a mass loss (defect) or nuclear binding energy ΔE when nuclei are formed from Z protons and N neutrons; ΔE (in atomic mass units: the mass of one 6C¹² nucleus is 12.000 atomic mass units) is

(2.1.4)

equation

The quantity ΔE/A rises from He to Fe, and it declines thereafter (Fig. 2.3); this explains why successive gravitational collapses of dying stars form first He stars, then C stars, then Ne stars, then finally Fe stars. When all nuclear fuel is depleted in an Fe star, an ultimate gravitational collapse into black holes must occur, if the star mass exceeds 8 solar masses, that is, >1.6 × 10³¹ kg; otherwise the star will decay into a white dwarf.

Figure. 2.3 Average binding energy per nucleon ΔE/N for stable nuclei, as a function of atomic number A [5].

Several models were adopted to explain the structure of stable and radioactive nuclei. The liquid drop model assumes that protons and neutrons coalesce to form a liquid drop of high density (spherical, or prolate spheroidal, or oblate spheroidal); Weizsäcker's²⁰ semiempirical mass formula of 1935 accounts fairly well for the masses M(Z, A) (in atomic mass units) for stable nuclei:

(2.1.5)

equation

However, the liquid-drop model does not account for the relative stability of certain nuclei called islands of (relative) nuclear stability (Z and/or N = 2, 8, 20, 28, 50, 82, 126, 184).

The shell model of Göppert-Mayer²¹ and Jensen²² posits populating nuclear states as if the nucleons occupied the lowest possible quantum states for a three-dimensional harmonic oscillator, but with an energy correction due to a nuclear spin–orbit interaction: The nuclear spin quantum numbers I and orbital quantum numbers M, couple strongly as I•M; this nuclear spin–orbit interaction (invented in analogy to the electron spin-orbit interaction) is, however, due to an unknown potential function; nevertheless, this model does account nicely for magic number stability and nuclear excited states. There is an acrostic "spuds if pug dish of pig" that serves as a mnemonic for the ordering 1s, 1p, 1d, 2s, 1f, 2p, 1g, 2d, 3s, 1h, 2f, 3p, 1i, 2g (before nuclear spin–orbit splittings).

The final model that accounts for nuclear stabilities must, of course, be the strong force, or rather the residual component of the strong force that works outside of quark confinement. Natural or artificial radioactive nuclei can exhibit several decay modes: α decay (

with emission of a 2He⁴ nucleus), which is dominant for elements of atomic number greater than Pb; β–decay or electron emission ( this involves the weak force and the extra emission of a neutrino); positron or β+ decay ( emission of a positron and an antineutrino; this also involves the weak force); γ decay: no changes in N or Z, and electron capture ( emission of electron; this involves the weak force). There is also internal conversion, from a metastable nucleus to a more stable nucleus with no particle emission. Very useful is a wall chart of all nuclides developed by the Knolls Atomic Power Laboratory of the General Electric Company in the 1950s and subsequently updated often (Appendix, Table A) [6].

The Periodic Table of The Chemical Elements (Table 2.3) was first organized by Mendeleyeff in 1869 [7] well before quantum mechanics and the modern theory of atomic structure, by using group analogies in chemical and physical properties; Mendeleyeff even predicted two as yet undiscovered elements (Ga, Ge) and left spaces for them in his table.

Table 2.3 The Periodic Table of the Known Chemical Elementsa

aThe 18 groups (Grp n) are the modern ones; the older grouping is given inside slashes.

Sideline

In the 1780s Lavoisier first pinpointed the irreducibility of chemical elements (like hydrogen and oxygen) and their combination in chemical compounds (like water). In the early 1800s Dalton²³ revived the ancient idea of indivisible fundamental atoms proposed by Leucippus²⁴ and his pupil Democritus.²⁵ Dalton also demonstrated two laws (of definite and multiple proportions); as a result, relative empirical formulas and tables of relative atomic weights were established for a growing list of chemical compounds. But for several decades the molecular structure of water was erroneously assumed to be HO. Avogadro's²⁶ 1811 principle that at constant pressure and temperature equal volumes of gases contained an equal number of molecules was ignored until Cannizzaro's 1858 work,²⁷ circulated at the Karlsruhe conference of 1860, convinced the German chemists to finally take Avogadro's principle seriously: Shazam! The molecular formula for water became H2O, all relative scales rolled into one, and Mendeleyeff could then build his periodic table!

Table 2.4 Fundamental Constantsa

Source: P.J. Mohr, B.N. Taylor, and D.B. Newell, The 2010 CODATA Recommended Values of the Fundamental Physical Constants (National Institute of Standards and Technology, Gaithersburg, MD 20899, 2011).

Table 2.5 Other Constants.

2.2 Gravitation

The gravitational force is the weakest force in nature, but it binds together the most massive bodies in the universe. The force is in newtons (N) in the SI system, but in dynes in the cgs system (see Appendix, Table A). This force can be rewritten in terms of a vector gravitational fieldΦ1(r2) experienced by particle 2 at position r2, due to the existence of a particle 1 of mass m1 at r1, and mediated by a continuous, if virtual, flow of gravitons emanating from particle 1:

(2.2.1) equation

where r12 = r2 − r1. This gravitational field can be integrated once to yield the scalar gravitational potential U1 (an energy):

(2.2.2) equation

where the field is the negative gradient of the potential U1 evaluated at any field point r, for example, r = r2 (except at the singular position r = r1):

(2.2.3) equation

where er is the unit vector in the radial direction. This potential energy is measured in joules, J (1 J ≡ 1 N m) in the SI system, or in erg (1 erg ≡ 1 dyne cm) in the cgs system. We can also define the gravitational potential energy U12 as the potential energy of the two-body system:

(2.2.4) equation

2.3 Review of Mathematical Concepts

When a function y = f(x) is specified (e.g, y = x⁴ + 3 sin x + tanh x), then x is the independent variable and y is the dependent variable, whose value is computed once numbers are assigned to the (one or more) independent variables for f. In other words, a function is a recipe for going from a variable (x) to a number (f(x)).

An equation is when the function is restricted by a definite value it must obtain after evaluation, for example, x³ + 3 sin x + tanh x = 55 means that we must solve the equation for x (i.e., compute x) such that that value of x will satisfy the given equation.

A functional F[g] = 33 means that the explicit functional form of g is not known, or not knowable, but its use must yield the definite value of 33. In other words, a functional is a recipe for going from a function (g) to a number (F[g]).

Algebraic equations with one variable λ of order n = 1 through 4 can be solved explicitly.

Ifn = 2, the quadratic equation

(2.3.1) equation

has two solutions:

(2.3.2) equation

If the discriminant D ≡ a² − 4b < 0, then both roots are complex; if D = 0, then the roots are real and degenerate (equal to each other). The solution for n = 2 was known to the Egyptians in the Middle Kingdom (ca. 2160–1700 bc), the Hindus (Brahmagupta²⁸ in 628 ad), and in its geometrical form to the ancient Greeks (Euclid²⁹ and Diophantus³⁰).

If n = 3, for the cubic equation:

(2.3.3) equation

the solution was found by del Ferro³¹ and Tartaglia,³² published by Cardano³³ in 1545, and confirmed by Ferrari³⁴ as

(2.3.4)

equation

Two distinct roots are possible, for the two alternatives for ±. The three cube roots of 1, namely, (i) exp(2πi/3) = cos(120°) + i sin(120°) = −0.500000 + i0.866025 = −(1/2) + i(3¹/²/2), (ii) exp(4πi/3) = cos(240°) + isin(240°) = −0.500000 − i0.866025 = −(1/2) − i(3¹/²/2), and (iii) exp(6πi/3) = 1, provide three roots; this times two is six roots, which do reduce to only three. To get the three correct roots λ1, λ2, and λ3, it is essential that u (and not λ) be premultiplied by factors of exp(6πi/3), exp(2πi/3), and exp(4πi/3), or else wrong results will be obtained. The discriminant Δ ≡ 18abc −4a³c + a²b² − 4b³ − 27c² determines the nature of the three roots: If Δ > 0, then there are three distinct real roots; if Δ = 0, then all three roots are real (but some are degenerate); if Δ < 0, then there are one real root and two complex and mutually conjugate roots. An umbrella or monic formula, which is foolproof, is

(2.3.5)

equation

Equivalently, one can define:

(2.3.6) equation

The three solutions are then

(2.3.7) equation

Consider δ ≡ (1/4)q² + p³/27; within overall multiplicative factors, this δ is equivalent to, but opposite in sign to, the discriminant Δ ≡ 18abc − 4a³c + a²b² − 4b³ − 27c² defined above.

If δ > 0, there will be 1 real root and 2 conjugate imaginary roots.

If δ = 0, there will be 3 real roots, of which at least 2 are equal.

If δ < 0, there will be 3 real & unequal roots; if δ < 0, then define the following:

(2.3.8)

equation

If n = 4, the general quartic equation

(2.3.9) equation

has a solution:

(2.3.10)

equation

where

(2.3.11) equation

where all the upper signs travel together. This solution was found by Ferrari in 1545.

If n > 4, Abel³⁶ showed in 1824 that there can be no general closed-form solution [8]. Thus, numerical methods must be used when n > 4.

Plane Trigonometric Functions

In a right plane triangle with right angle γ = 90°, sin α ≡ A/C, where A is the segment opposite to the angle α, and C is the hypotenuse; cos α ≡ B/C; tan α ≡ A/B = sin α/cos α; sin β ≡ B/C; cos β ≡ A/C; α + β = 90°; sec α ≡ 1/cos α; cosec α ≡ 1/sin α; cotan α ≡ 1/tan α; sin² α + cos² α = 1; sin (−x) = −sin x; cos (−x) = cos x; tan (−x) = −tan (x); sin (x ± y) = sin x cos y ± cos x sin y; cos(x ± y) = cos x cos y sin x sin y; 2 sin x cos y = cos(x + y) + sin(x − y); 2 cos x cos y = cos(x + y) + cos(x − y); 2 sin x sin y = cos(x − y) − cos(x + y). Thus, sin x and tan x are odd functions of x, while cos x is an even function of x. In some countries, tan x is written as tg x, and cotan x is written cotg x.

Inverse functions: If x = cos y, then y = cos−1 x. Be careful: cos−1 x ≠ 1/cos x!!!

Hyperbolic Functions

(2.3.12) equation

(2.3.13) equation

(2.3.14) equation

(2.3.15)

equation

(2.3.16)

equation

Differential calculus was developed independently by Newton and Leibniz.³⁷

Derivatives

(2.3.17) equation

(2.3.18) equation

(2.3.19) equation

(2.3.20) equation

(2.3.21) equation

(2.3.22) equation

(2.3.23) equation

(2.3.24) equation

(2.3.25) equation

(2.3.26) equation

(2.3.27) equation

(2.3.28) equation

(2.3.29) equation

Differential operators

Chain Rule

(2.3.30) equation

Integrals

(2.3.31) equation

where C is a constant;

(2.3.32) equation

(2.3.33) equation

(2.3.34) equation

(2.3.35) equation

Integration by Parts

(2.3.36) equation

Taylor³⁸ Series

(2.3.37)

equation

Maclaurin³⁹ Series

(= Taylor series for a = 0):

(2.3.38)

equation

(2.3.39)

equation

(2.3.40) equation

(2.3.41) equation

(2.3.42)

equation

(2.3.43)

equation

(2.3.44)

equation

Euler⁴⁰ Formula

(2.3.45) equation

where i ≡ (−1)¹/², which gives the funny-looking but nevertheless true result:

(2.3.46) equation

Problem 2.3.1. Prove Eq. (2.3.21) by using the chain rule.

Sums

(2.3.47) equation

The Einstein⁴¹ summation convention is that a sum is understood for any repeated indices over their full range:

(2.3.48) equation

Arithmetic Series

If ai ≡ a + di, then

(2.3.49) equation

Geometric Series

If ai ≡ adi, then

(2.3.50)

equation

If n is infinite and ai ≡ adi, then

(2.3.51) equation

Problem 2.3.2. Verify Eq. (2.3.50).

Problem 2.3.3. Verify Eq. (2.3.51).

Partial Fractions

It is often convenient or desirable (e.g., in some difficult integrations) to break up a complicated factored polynomial expression in the denominator into partial fractions involving new denominators of order no higher than 2. For instance, it can be shown that the fraction on the left can be decomposed into a sum of the simpler fractions on the right:

where the coefficients A, B, C, D, and especially E and F are nonzero. These coefficients are found by brute force. The two rules for how to set up the partial fractions are as follows (i) If a linear factor ax + b occurs n times in the denominator, then to this factor will correspond a sum of n partial fractions:

with ; (ii) if a quadratic factor ax² + bx + c occurs n times as factors in the denominator, then to this factor will correspond a sum of n partial fractions:

Problem 2.3.4. Evaluate the coefficients A, B, C, D, E, and F in the equation

equation

The Lagrange Method of Undetermined Multipliers

To prove important statistical mechanical results in Chapter 5, we need the method of undetermined multipliers, due to Lagrange.⁴² This method can be enunciated as follows: Assume that a function f(x1, x2,. . ., xn) of n variables x1, x2,. . ., xn is subject to two auxiliary conditions:

(2.3.52) equation

(2.3.53) equation

We seek extrema (maxima, minima, or saddle points) of f, subject to these two conditions. We shall show that there exist two constants, defined as α and β (these two are known as the Lagrange multipliers), such that the system of n + 2 equations

(2.3.54)

equation

(2.3.55) equation

(2.3.56) equation

when solved, will provide the desired extremum for f.

We want to find the conditions for which df = 0. Remember that, while the constraints g = 0 and h = 0, in general f ≠ 0.

Define the differential:

(2.3.57) equation

We want to find when df = 0. Since g = 0 and h = 0, therefore also, a fortiori

(2.3.58) equation

and also

(2.3.59) equation

Rewriting Eq. (2.3.54) now yields

(2.3.60)

equation

so that finally

(2.3.61)

equation

that is, we found the condition of Eq. (2.3.54), that df = 0, as desired, as conditions for α and β.

Proving whether this extremum in f(x1, x2, . . ., xn) is a maximum, a minimum, or zero is usually not done analytically (e.g., by further differentiation to make sure that d²f > 0 for a minimum, d²f < 0 for a maximum, etc.), but instead by recourse to physical arguments. Indeed, the values of the Lagrange multipliers α and β can often be found from physical arguments.

2.4 Mechanics, Vectors, Tensors, and Determinants

Force is defined by Newton's second law:

(2.4.1) equation

where F is the force, t is the time, p is the particle's linear momentum, and m is its mass (this equation is relativistically correct). When the momentum is given by the product of mass time velocity v:

(2.4.2) equation

(this is not valid at relativistic speeds) and if a = dv/dt, then

(2.4.3) equation

The rest mass m of any particle or celestial body can be considered in three ways:

i. as a proportionality constant between force and acceleration;

ii. as a curvature of the space–time continuum around a massive body (the effects of Einstein's theory of general relativity were relabeled by Wheeler⁴³ as geometrodynamics [9]);

iii. as a fundamental property, of dimension [M], defined by Eq. (2.1.2) or by Eq. (2.4.3) as inertial mass in outer space, or as amount of material.

Interpretation (ii) has triumphed, but one may still argue about what m really is. If m is a fundamental essence, of dimension [M], then force and field have dimensions [M] [L] [T]−2, while energy has units [M] [L]² [T]−2. What rest mass an elementary particle should have may be predictable if the Higgs boson is ever found.

Five unit systems should be summarized here:

A. The SI (Système International) units use kilograms, meters, seconds, ampères, kelvin, mole (6.022 × 10²³ molecules per gram-mole, and not per kg-mole), and candela for [M], [L], [T], current, absolute temperature, mole, and luminous intensity, respectively. It started from an MKS (m-kg-s) system and included an electrical unit as part of the definition, as first suggested by Giorgi⁴⁴ in 1904. There is a very slight modification of SI, used in nonlinear optics, confusingly dubbed MKS by its users, but called SI′ here.

B. The older cgs units started from the French Academy work of 1793 defining the gram and the meter, and we use grams, centimeters, and seconds for [M], [L], and [T], respectively. To define electrical and magnetic quantitites, cgs comes in two flavors: cgs-esu, or simply esu (where statCoulombs are the units for electrical charge), and cgs-emu, or simply emu, where the Oersted⁴⁵ is the unit of magnetic field. There are other variants of the cgs units: Gaussian⁴⁶ and Heaviside⁴⁷-Lorentz.

C. In addition to the SI and cgs systems, we can define a system of Hartree⁴⁸atomic units (a.u.). Alas, a slightly different set of Rydberg⁴⁹ a.u. also exists, but will not be discussed here. The Hartree atomic units are defined so that (i) the unit of length [L] is a0 = 1 bohr = 5.29177 × 10−11 m = radius of the Bohr⁵⁰ orbit for hydrogen, (ii) the unit of mass [M] is 1 electron mass = me = 9.109 × 10−31 kg, (iii) the unit of [action] is [M] [L]² [T]−1 = (h/2π) ≡ = reduced Planck constant of action = 1.055 × 10−34 J s (iv) the unit of electrical charge is the proton charge e = 1.602 × 10−19 C. This is equivalent to putting = 1, me = 1, e = 1 in all formulas. As a consequence, (v) the unit of time [T] = time for 1 electron to travel 1 Bohr radius = 2.419 × 10−17 s, (vi) the unit of energy [M] [L]² [T]−2 = 1 hartree = twice the ionization energy of the hydrogen atom = 4.360 × 10−18 J.

D. In Planck units, of interest to quantum gravity and to early cosmology, = 1, as in the atomic units, but c = speed of light in vacuo = 1, and G = gravitational constant = 1.

E. In Astronomical units (unfortunately, also called a.u.) the unit of mass is the solar mass (1.98892 × 10¹¹ kg), and the unit of length is the mean distance from earth to sun (1.49597871464 × 10¹¹ m).

For a single particle of mass m and momentum p, or velocity v = (dr/dt), the kinetic energyT (originally dubbed vis viva, or live energy!) is defined as

(2.4.4) equation

T is positive definite. The potential energy U introduced above depends on an arbitrary choice of its zero, which depends on tradition,—that is, on the whim of the first or loudest experimenter: U becomes positive or negative, relative to that zero, depending on that tradition.

A concept helpful for solving simple celestial mechanics problems is the centrifugal accelerationa of any body moving with a speed v in a circular orbit of radius r:

(2.4.5) equation

which, expressed as a vector, is a = d²r/dt² = −v²rr−2. Its opposite is the centripetal acceleration, which will keep a body on its circular path. In general, the acceleration a will have a radial component, the centripetal acceleration (v²/r) along an unit (inward) vector en, and a tangential component, along the unit tangent vector et: a = d²r/dt² = (dv/dt) et + (v²/r) en.

Problem 2.4.1. Given that the average earth–moon distance is Rem = 3.844 × 10⁸ m and that the moon's revolution around the earth is 27.3 days (from which its tangential orbital velocity is vm = 1.0186 × 10³ m s−1), compute the mass of the earth.

Problem 2.4.2. Given the mass of the earth Me = 5.977 × 10²⁴ kg and its mean radius Re = 6371 km, verify that the acceleration due to mean gravity at sea level at the equator is

(2.4.6) equation

Problem 2.4.3. Show that the relative gravitational potential energy at a height h above the earth's surface is

(2.4.7) equation

where Urel = 0 at the earth's surface. Use the Maclaurin series:

(2.4.8) equation

Problem 2.4.4. Show that the escape velocity vesc from the earth's gravitational field is 1.1 × 10⁴ m s−1. Given the necessary escape kinetic energy where kB = 1.3807 × 10−23 J K−1 atom−1 is Boltzmann's⁵¹ constant, which molecules, at an effective temperature of 30,000 K, can leak out from the earth's atmosphere into space? Is this temperature reasonable?

Problem 2.4.5. Show that the gravitational potential energy of an object of mass m at the earth's surface is only 7% due to the earth, and 93% due to the sun (the earth–sun distance is 149,600,000 km; the mass of the sun is 1.985 × 10³⁰ kg). So why do we not fall off the earth and tumble toward the sun?

Problem 2.4.6. What is the center of gravity in the earth–sun trajectory (sun–earth distance = 1.496 × 10¹¹ m; earth mass = 5.977 × 10²⁴ kg; sun mass = 1.985 × 10³⁰ kg)?

Problem 2.4.7. If a satellite is to reach an orbit 100 km above the surface of the earth, what tangential velocity must it have as it enters the orbit? How long will it take to make one revolution around the earth (earth mass = 5.977 × 10²⁴ kg; earth radius = 6.371 × 10⁶ m)?

Since forces are represented by vectors, we next review some properties of vectors, with particular applications to crystals. The position vector r is usually given in a Cartesian (orthogonal) system, but crystals are defined as symmetric objects with translational symmetry, with a fundamental or unit cell, which is the basic nonrepeating unit that is not necessarily orthogonal (see Sections 7.1 and 7.10)]. The unit cell has axes a, b, c (measured in nm or in Å); the axes do form a right-handed system, and are, in general, the corners of an oblique parallelopiped (in the lowest symmetry system, the triclinic system, see Fig. 2.4); the angle between a and b is called γ; the angle between b and c is called α, and the angle between c and a is called β.

Figure 2.4 The general unit cell in a triclinic (lowest symmetry) crystal. The unit cell has sides a, b, c, angles α, β, and γ and volume V.

The Cartesian system is named after Descartes.⁵²

Sideline

Descartes, a late riser, died in Sweden, maybe of pneumonia, because Queen Christina, who had hired him, insisted that he teach her philosophy at dawn. The French broadsheet La Gazette d'Anvers announced Descartes' death by En Suède un sot vient de mourir qui disait qu'il pouvait vivre aussi longtemps qu'il voulait.

The dot or scalar inner product between vectors a and b is a scalar quantity, defined by

(2.4.11) equation

where the angle between the a and b axes is γ; similarly:

(2.4.12) equation

(2.4.13) equation

One can also define the a, b, c system in terms of any arbitrarily oriented Cartesian system:

(2.4.14) equation

but in crystallography it is customary to align either the b axis with ey or the c axis with ez. The inner product can be defined in a space of any dimensions from two to infinity, and it obeys the associative and commutative laws. Of course, if a and b are orthogonal, then the dot product a · b is zero; in the Cartesian system used above we have

(2.4.14) equation

and the length, or norm, of the unit vectors is, of course, unity:

(2.4.15) equation

The vector product or cross product (term coined by Gibbs⁵³) is defined only in three-dimensional space: The vector product, or cross product, of vectors a and b is a vector v, whose magnitude is |a| |b| sin γ, where γ is the angle between a and b, and whose direction is perpendicular to both a and b, and whose orientation is such that a, b, and v form a right-handed system:

(2.4.16) equation

In a Cartesian coordinate system, by applying the definition of a cross product in this orthogonal system, the unit vectors ex, ey, ez are related as follows:

(2.4.17)

equation

while the cross product of any vector with itself vanishes:

(2.4.18) equation

By using the distributive property of the vector product (Problem 2.4.8) and using Eq. (2.4.17):

(2.4.19)

equation

and then remembering the properties of a 3 × 3 determinants, one sees

(2.4.20) equation

Problem 2.4.8. Verify the distributive property of the vector product:

(2.4.21)

equation

Problem 2.4.9. Prove

(2.4.22) equation

Problem 2.4.10. Prove

(2.4.23)

equation

Problem 2.4.11. Prove

(2.4.24)

equation

The cross product is anticommutative, that is, it changes sign when the factors are reversed. Indeed, the cross product is really a "pseudovector, or polar vector, which has all the desirable properties of vectors, plus one undesirable one: A pseudovector is married to a right-handed system (in a left-handed system its magnitude is the same, but its sign changes). To differentiate them from pseuodovectors, the normal vectors are also called sometimes axial vectors." The anticommutation is also implicit in the properties of a determinant. Geometrically, the magnitude of the vector product of a and b is the area A of the parallelogram with a and b as the sides:

(2.4.25) equation

We next calculate the volume V of the parallelopiped of Fig. 2.4: We need the projection of c onto the direction normal to both a and b; this demands the dot product of c times the cross product of a and b; indeed (in a cyclic fashion) it can seen that

(2.4.26) equation

and using the commutative properties of the scalar product and the anticommutative properties of the vector product one sees the cyclic permutation of vectors as giving a positive value for the volume, while any twofold permutation yields a negative volume (left-handed system!):

(2.4.27)

equation

V is also married to a right-handed system, so V should be called a pseudoscalar. To evaluate V, we need the difficult projection of c onto the direction normal to both a and b; this is not easy (see Problem 2.4.12). The result is

(2.4.28)

equation

An easier result to see from Eqs. (2.4.21) and (2.4.22) is

(2.4.29) equation

A detailed discussion of reciprocal space is given in Section 7.10.

Problem 2.4.12. Prove Eq. (2.4.28).

Ski Slopes, Hernias, and Curls

In Cartesian space the del (or nabla ≡ Assyrian harp or atled ≡ backwards delta) operator or function is given by

(2.4.30) equation

The gradg ≡ g could also be called the ski-slope function: It shows the vector sum of the partial derivatives along three Cartesian axes for a scalar function g(x, y, z) and has the value of a vector:

(2.4.31)

equation

When applied to a mountain, g gives the direction of steepest descent for the most ardent and risk-averse skier!

We could call the div u = u function the hernia function: It shows how a vector function u(x, y, z) herniates, that is, sprouts in the x, y, and z directions at the point (x, y, z); in Cartesian space the div or del dot operator can be represented by

(2.4.32)

equation

The curl u, or × u, or rot u, or curly function describes how curly a vector field (array) of vectors u is. Verify that in a Cartesian system the pseudovector:

(2.4.33)

equation

As defined above, × u is a pseudovector. Incidentally, consider the face of a hairy monkey, or the face of an ape-man: if it is taken to be spherical, it can be covered by facial hair everywhere, but the topology of putting hair all over a sphere requires that there be at least two whorls, or points on the sphere where the hair density is zero, and the curl of the hair is maximized.

The Laplacian⁵⁴, or del-squared, operator is · = ² (sometimes shown as Δ to further confuse the poor reader); · is given in Cartesian space as

(2.4.34)

equation

Problem 2.4.13. Verify the following identity, valid for any vector v:

(2.4.35)

equation

or, in a different notation:

(2.4.36) equation

Problem 2.4.14. Calculate the volume of a tetrahedron of sides a, b, c (where |a| = |b| = |c|).

Problem 2.4.15. Prove that · × U = 0 for any vector U.

Tensors

While a scalar (a pure number, real or complex) p merely multiplies the length of a vector V:

(2.4.37) equation

(i.e., multiplies all three components of V by p), there are 3 × 3 tensors α that can rotate them: they are known as tensors of rank two:

(2.4.37) equation

These tensors are square matrices. If we are dealing with a (column) vector V (or tensor of rank one) in three-dimensional space, then the tensor α of rank 2 will have, in general, 3 × 3 = 9 independent elements:

(2.4.38) equation

where each tensor element αij is a scalar (real or complex).

If a tensor β is symmetric, that is, if βij = βji, then this tensor has only six independent elements:

(2.4.39) equation

A tensor γ is Hermitian⁵⁵ if its off-diagonal elements are the complex conjugates of the corresponding elements across the main diagonal γij = γji* (this Hermitian condition is written in matrix form as γ = γ†); a tensor δ is unitary if the inverse of the matrix is equal to its Hermitian conjugate: δ = (δ†)−1.

The trace Tr of a square tensor is the scalar sum of its diagonal elements:

(2.4.40)

equation

Of course, n × n tensors of rank 2 can be defined for dimensions n > 3: They occur frequently in four-dimensional special and general relativity theories.

Determinants

To each square matrix α of dimension n we can associate a determinant det α as follows:

(2.4.41)

equation

where the indices (j1, j2,. . ., jn) are permutations of the natural or perfectly sequential order of the first n integers (1, 2, 3, . . ., n) and h is the number of twofold exchanges of any two elements needed to recreate this natural order; the cofactoror minorAik of any element αik is defined as the subdeterminant created by eliminating the ith row and k-column of the original determinant, multiplied by (−1)i + k. A determinant can be computed as a sum of the minors of any one given row (or column) (choose one):

(2.4.42)

equation

where the operator also means determinant. Det α = 0 if (1) all elements of any row are zero, or (2) all elements of any column are all zero, or (3) any two rows have equal elements in the same order, or (4) any two columns have equal elements in the same order. The evaluation of determinant with n ≥ 4 rows and columns is laborious. For 3 × 3 determinants, Sarrus'⁵⁶ rule is simple: Repeat and append the first two columns at the end, and multiply diagonally down three times with a +1 prefactor (a11a22a33, then a12a23a31, then a13a21a32), then multiply diagonally up three times with a −1 prefactor (−a31a22a13, then −a32a23a11, then −a33a21a12). For example:

A matrix A is singular if det A = 0; conversely, A is nonsingular if det A ≠ 0.

If the matrix A is symmetric, Hermitian, or unitary, then there is a system of 3 × 3 rotation matrices R (and their inverses R−1) which will rotate the matrix elements Aij so that the only nonzero elements will appear on the diagonal; this is known as a similarity transformation or as a principal-axis transformation or diagonalization:

(2.4.43) equation

This transformation is crucial when some measured 3 × 3 matrix has nine experimental values, and a coordinate system is sought which will highlight the physically significant components of this matrix, which may be as few as three after the appropriate similarity transformation into the right coordinate system. Written out in full, the relationship between a matrix and its inverse is

where E is the unit matrix:

(2.4.44) equation

The problem of finding the rotation matix that will diagonalize some symmetric, Hermitian, or unitary matrix A can be recast as an eigenvalue–eigenvector problem: We seek the characteristic solutions to the problem

(2.4.45) equation

where λ is one of n scalar (real or complex) characteristic values or eigenvalues, and K is the characteristic matrix of A (a matrix constructed by adjoining all column vectors to each other). In particular the scalar nth-degree equation:

(2.4.46) equation

has n solutions (the n eigenvalues λ1, λ2,. . ., λn), so that K(λ) can rewritten as

The determinant may factor naturally: this is rare. For n = 2, 3, or 4 one seeks the roots of a quadratic, cubic, or quartic polynomial equation in λ, or one must resort to numerical methods. In the case n = 7, the eigenvalue matrix Λ ≡ R−1AR is the diagonal matrix:

(2.4.47) equation

The problem of finding the n eigenvalues (λi, i = 1, 2, . . ., n) and the n corresponding eigenvectors x:

(2.4.48) equation

is thus equivalent to diagonalizing the matrix A. The n × neigenvalue–eigenvector equation or secular equation (so-called because astronomers used it to describe motions of distant planets tracked over the course of several centuries, or saecula) is given by

(2.4.49) equation

where E is the n × n unit matrix and O is the n × n null matrix. Its solution x has the eigenvector components

(2.4.50)

equation

where means determinant; and the Aij are the (n − 1) × (n − 1) minors of the original n × n determinant A − λE , obtained by eliminating the ith row and jth column of the original determinant. The eigenvectors coefficients are redefined by normalization. If the eigenvalues are not degenerate, then the eigenvectors are automatically mutually orthogonal; if the eigenvalues are degenerate, then the eigenvectors can be made to be orthogonal. The n × n rotation matrix R is constructed by accosting to each other the n × 1 eigenvectors. The inverse rotation matrix R−1 is the transpose of R. The transformation of A into Λ is known as a similarity transformation:

(2.4.51) equation

It is important to realize that the trace of the matrix A (the sum of the diagonal terms) is invariant under a similarity transformation.

Problem 2.4.16. Invert the symmetric matrix:

Problem 2.4.17. Diagonalize the 3 × 3 symmetric real matrix:

by first solving the determinantal equation:

Finally, check that the trace of A is equal to the trace of Λ.

Problem 2.4.18. Find the eigenvectors for the matrix:

which in Problem 2.4.17 was transformed into the diagonal eigenvalue matrix:

Problem 2.4.19. What is the inverse of the rotation matrix R

which was computed in Problem 2.4.17?

Problem 2.4.20. We need to learn how to rotate coordinate systems. In the xy plane a counterclockwise rotation by an angle ϕ causes a rotation from vectors x, y to new vectors x′, y′:

(2.4.52) equation

(2.4.53) equation

which can be represented as a column vector X = (x y) rotated to a new column vector X′ = (x′ y′) by premultiplication by the square rotation matrix A = (aij):

(2.4.54) equation

(2.4.55) equation

Since the coordinate system is simply rotated, not shrunk or expanded in any way, the matrix A must be orthonormal:

(2.4.56) equation

To do a full rotation in three-dimensional space, Eulerian rotation angles can be defined, as three successive rotations, each in two dimensions.

The first rotation (rotation matrix A) is a counterclockwise rotation by α degrees about the z axis: It leaves the z axis unchanged (z′ = z), but it rotates the axis x to x′ by α degrees and rotates the axis y to y' by α degrees.

(2.4.57) equation

The second rotation (rotation matrix B) rotates by a counterclockwise rotation by β degrees about the axis x′: It leaves the x′ axis unchanged (x″ = x′), but it rotates y′ to y″ by β degrees and rotates z′ to z″ by β degrees:

(2.4.58) equation

The third rotation (rotation matrix C) rotates by a counterclockwise rotation by γ degrees about the axis z′: it leaves the z″ axis unchanged: z = z″, but it rotates x″ to x by γ degrees and rotates y′ to y by γ degrees:

X = CX″. Show that the rotation matrices are

(2.4.59)

equation

Show also that the overall Eulerian rotation matrix E = ABC is given by

(2.4.60)

equation

Show also that the inverse matrix E−1 is equal to the transpose matrix ET (where rows and columns are interchanged):

(2.4.61)

equation

Problem 2.4.21. If the Eulerian matrix E given by Eq. (2.4.60) must equal the rotation matrix R determined in Problem 2.4.20 above:

compute the relevant Eulerian angles α, β, and γ. This is useful, for example when the principal axes of a physically measured tensor in a crystal must be oriented relative to laboratory Cartesian axes.

When we rotate a contravariant n × 1 column vector (for position, velocity, momentum, electric field, etc.) we premultiply it by an n × n rotation tensor R. When, instead, we transform the coordinate system in which such vectors are defined, then the coordinate system and, for example, the operator are covariant 1 × n row vectors, which are transformed by the tensor R−1 that is the reciprocal of R. A dot product or inner product a·b must be the multiplication of a row vector a by a column vector b, to give a single number (scalar) as the result. This will be expanded further in the discussion of special relativity (Section 2.13) and of crystal symmetry (Section 7.10).

Spherical Trigonometry

Spherical trigonometry is essential for terrestrial and celestial navigation (but not really for the physics and chemistry discussed in this book). Nevertheless, its brief presentation here should help scientific travelers estimate their travel distance and appreciate how the angles we all learned in plane trigonometry are quite different on a spherical surface. In a spherical triangle (a triangle localized on the surface of a sphere), angles A, B, C as well as sides a, b, c are measured in radians (Fig. 2.5) [10].

Figure. 2.5 (Left) Spherical triangle with sides a, b, c and opposite angles A, B, and C. (Right) Computation of orthodrome between San Francisco (SFO) and Tokyo (TYO), and plane triangle ABD.

Here is a summary of the properties of spherical triangles: (1) Any angle A, B, or C must be less than 180°; (2) a + b + c < 360°; (3) any side is less than the sum of the other two sides; (4) 180° < A + B + C < 540°; (5) the sine law is

(2.4.62) equation

(6) the cosine law for the sides is

(2.4.63) equation

(7) the cosine law for the angles is

(2.4.64) equation

(8) the half-angle formulas are

(2.4.65)

equation

and

(2.4.66)

equation

(9) The four analogies of Napier:

(2.4.67)

equation

and cyclically:

(2.4.68)

equation

(10) Gauss's formula:

(2.4.69)

equation

Note that one, two, or three right angles may coexist in the same right spherical triangle!

The orthodrome between points A and B on the surface of a geode or earth is the shortest distance between A and B on this surface; the orthodrome is a segment of a great circle (e.g., a meridian) passing through both A and B.

A loxodrome (from the Greek loxos for slanted and dromos for course) is a path on the earth's surface that is followed when a compass is kept pointing in the same direction: It is a straight line on a Mercator⁵⁷ projection of the globe, precisely because such a projection is designed to have the property that all paths along the earth's surface that preserve the same directional bearing appear as straight lines. Nunes⁵⁸ thought that the loxodrome was the shortest distance between two points on a sphere (he was wrong). Many centuries ago, it was difficult for a ship's navigator to follow a great circle, because this required constant changes of compass heading. The solution was to follow a loxodrome, also known as a rhumb line, by navigating along a constant direction. In middle latitudes, at least, this didn't lengthen the journey unduly. If a loxodrome is continued indefinitely around a sphere, it will produce a spherical spiral, or a logarithmic spiral on a polar projection. The distance between two points, measured along a loxodrome, is simply the absolute value of the secant of the bearing (azimuth) times the north–south distance (except for circles of latitude). The airports of Helsinki, Finland and Anchorage, Alaska are almost on the same parallel, so one can fly due west from Helsinki and get comfortably close to Anchorage, but it surely is not the shortest path (the loxodrome distance would be 9493 km, while the orthodrome, the shortest distance, is 6520 km). The loxodrome spirals from one pole to the other, with an angle setting equal to the compass setting. Close to the poles, the loxodromes resemble closely logarithmic spirals. The total length of the loxodrome from the N pole to the S pole is, assuming a perfect sphere, the length of the meridian divided by the cosine of the bearing away from true north. On a sphere that has coordinates ϕ (latitude), λ (longitude), and α (azimuth), the equation of a loxodrome is

(2.4.70)

equation

(2.4.71) equation

where arcgd(ϕ) ≡ gd−1(ϕ) is the inverse Gudermannian⁵⁹ function, and λ0 is the longitude where the loxodrome passes the equator. Here the Gudermannian function is gd z ≡