Formulas, Facts and Constants for Students and Professionals in Engineering, Chemistry, and Physics

The second edition of "Formulas, Facts and Constants" has been enlarged without undue increase in volume. The constants have been revised to 1986 values. Prom~ted by comments of many readers a section on error analysis has been added. host impor­ tantly, the new edition now has a subject index in addition to the detailed Table of Contents. The basic format has not been changed and the aim of the book remains to be a handy source of formulas, conversion factors and constants for everyday use. Fundamental tools of mathematics needed in all areas of the physical sciences are given in Section 1. Conversion factors and fundamental constants are 1 isted in Section 2. Section 3 gives information on terms and facts of spectroscopy and atomic structure. Section 4 reviews basic elements of wave mechanics. Section 5 is a resource for work in the laboratory and is intended to be of assistance in the use of frequently encountered equipment such as electronic devices, detectors, vacuum systems, etc. Material constants and other data frequently needed for esti­ mates and problem solving are provided. This new edition is dedicated to the memory of my father, Kurt Fischbeck, who was instrumental that this book was written in the first place. I have benefitted from his experience as an editor and author. He died shortly before the first edition appeared in print. I wish to thank my colleagues Jack Cohn and Stewart Ryan for their useful comments.

1 Basic mathematical facts and figures.- 1.1 Signs and symbols.- 1.1.1 Arithmetic, algebra and analysis of real scalar quantities.- 1.1.2 Arithmetic, algebra and analysis of complex scalar quantities.- 1.1.3 Matrices.- 1.1.4 Algebra and analysis of vectors and other tensors.- 1.1.5 Greek alphabet.- 1.1.6 Roman numerals.- 1.2 Numerical constants.- 1.3 Exponentials and roots.- 1.4 Logarithm.- 1.5 Compi ex numbers.- 1.6 Factorial.- 1.7 Mensuration formulas.- 1.8 Solid angle.- 1.8.1 Solid angle subtended by apertures.- 1.9 Trigonometric functions.- 1.9.1 Range and sign of trigonometric functions.- 1.9.2 Values of trigonometric functions for multiples of 30° (?/6) and 45° (?/4).- 1.9.2.1 Values of sine and cosine for multiples of ?.- 1.9.3 Relations between trigonometric functions.- 1.9.3.1 Relations involving one angle.- 1.9.3.2 Sums and differences of angles.- 1.9.3.3 Multiples of one angle.- 1.9.3.4 Functions of half angles.- 1.9.3.5 Sums and differences of trigonometric functions.- 1.9.3.6 Products and powers of trigonometric functions.- 1.9.4 Exponential definition of trigonometric functions.- 1.9.5 Inverse trigonometric functions.- 1.9.5.1 Fundamental relations for principal values of inverse trigonometric functions.- 1.9.6 Series expansion of trigonometric functions.- 1.9.6.1 Series expansion of inverse trigonometric functions.- 1.9.7 Small angle approximations.- 1.9.8 Values of trigonometric functions.- 1.10 Relations between sides and angles of a triangle.- 1.11 Hyperbolic functions.- 1.11.1 Relations between hyperbolic functions.- 1.11.1.1 Relations involving one angle.- 1.11.1.2 Sums and differences of two angles.- 1.11.1.3 Multiples of one angle.- 1.11.1.4 Functions of half angles.- 1.11.1.5 Sums and differences of hyperbolic functions.- 1.11.1.6 Products and powers of hyperbolic functions.- 1.11.2 Relations between hyperbolic and trigonometric functions.- 1.11.3 Hyperbolic functions of complex argument.- 1.11.4 The inverse hyperbolic functions.- 1.11.4.1 Fundamental relations for inverse hyperbolic functions.- 1.11.4.2 Relations between inverse hyperbolic and inverse trigonometric functions.- 1.11.5 Series expansion of hyperbolic functions.- 1.11.5.1 Series expansion of inverse hyperbolic functions.- 1.11.6 Values of hyperbolic functions.- 1.12 Algebraic equations and solutions.- 1.12.1 Linear: ax + b = 0.- 1.12.2 Quadratic: ax2 + bx + c = 0.- 1.12.3 Cubic: ax3 + bx2 + cx + d = 0.- 1.12.4 System of linear equations.- 1.12.4.1 Homogeneous system of linear equations.- 1.12.4.2 Inhomogeneous system of linear equations.- 1.13 Calculus.- 1.13.1 Derivatives.- 1.13.1.1 Derivatives of elementary functions.- 1.13.1.2 Higher derivatives of elementary functions.- 1.13.2 Elementary rules of integration.- 1.13.3 Integrals.- 1.13.3.1 Definite integrals.- 1.13.3.2 Elliptic integrals.- 1.13.3.3 Complete elliptical integrals.- 1.14 Series.- 1.14.1 Finite series.- 1.14.2 Infinite series.- 1.14.2.1 Sums of reciprocal powers.- 1.14.3 Binomial series.- 1.14.3.1 Binomial coefficients.- 1.14.4 Fourier series.- 1.14.5 Taylor series.- 1.14.6 MacLaurin series.- 1.15 Polynomials.- 1.15.1 Bernoulli’s polynomials and Bernoulli’s numbers.- 1.15.2 Legendre’s polynomials (First kind).- 1.15.2.1 Associated Legendre’s polynomials (First kind).- 1.15.3 Hermite polynomials.- 1.16 Special functions.- 1.16.1 Error function.- 1.16.1.1 Error function for complex argument.- 1.16.2 Bessel functions.- 1.16.3 Gamma and Beta function.- 1.17 Fourier transformation.- 1.17.1 Fourier cosine transforms.- 1.17.2 Fourier sine transforms.- 1.17.3 Exponential Fourier transforms.- 1.18 Vectors and vector analysis.- 1.18.1 Unit vectors defining right handed orthogonal systems.- 1.18.1.1 Rectangular coordinates (x,y,z).- 1.18.1.2 Cylindrical coordinates (r,?,z).- 1.18.1.3 Spherical coordinates (R,?,?).- 1.18.1.4 Transformation between unit vector systems.- 1.18.1.5 Derivatives of unit vectors in polar coordinates.- 1.18.1.6 Time derivatives of unit vectors in polar coordinates.- 1.18.2 Line elements.- 1.18.3 Area el ements.- 1.18.4 Position vector.- 1.18.5 Vector algebra.- 1.18.5.1 Vector sum.- 1.18.5.2 Scalar or “dot” product.- 1.18.5.3 Vector or “cross” product.- 1.18.6 Derivatives of vectors.- 1.18.6.1 Time derivative of vectors.- 1.18.7 Vector differential operator $$\nabla$$ (del).- 1.18.7.1 Gradient.- 1.18.7.2 Divergence.- 1.18.7.3 Rotation or curl.- 1.18.7.4 Identities involving the vector operator $$\nabla$$.- 1.18.7.5 Laplacian operator $$\overrightarrow {{{\overrightarrow \nabla }^2}}$$.- 1.18.7.6 Identities involving line and surface integrals.- 1.18.7.7 Identities involving surface and volume integrals.- 1.19 Solution of important differential equations.- 1.19.1 Harmonic oscillator.- 1.19.1.1 Oscillator with damping.- 1.19.2 Laplace equation.- 1.19.2.1 Laplace equation in rectangular coordinates.- 1.19.2.2 Laplace equation in polar coordinates.- 1.19.2.3 Example for the solution of the Laplace equation.- 1.19.3 The wave equation.- 1.19.3.1 Telegraph equation (Wave equation in a conducting medium).- 1.19.4 Heat conduction or diffusion equation (linear).- 2 Units, conversion factors and constants.- 2.1 The international system of units (SI).- 2.1.1 The basic SI units.- 2.1.2 Supplementary SI units.- 2.1.3 Derived SI units with special names.- 2.1.4 Commonly used derived SI units without special names.- 2.1.5 Prefixes for multiples and fractions of SI units.- 2.1.6 Multiples and fractions of SI units with special names but not part of the system.- 2.1.7 Other units exactly defined in terms of SI units.- 2.1.8 Units used in the international system whose values are obtained experimentally.- 2.1.9 Symbols for SI units in alphabetical order.- 2.1.10 Derived SI units admitted in connection with human health.- 2.2 Conversion factors listed by physical quantity.- 2.2.1 Length ― macroscopic.- 2.2.1.1 Length ― microscopic.- 2.2.2 Area.- 2.2.3 Volume.- 2.2.4 Mass.- 2.2.5 Speed.- 2.2.6 Force.- 2.2.7 Pressure.- 2.2.8 Power.- 2.2.9 Energy.- 2.2.9.1 Energy ― atomic units.- 2.2.9.2 Energy equivalents.- 2.2.9.3 Energy equivalents in terms of the Rydberg constant.- 2.2.9.4 Energy or voltage to wavelength conversion.- 2.2.10 Temperature.- 2.2.11 Electromagnetic quantities.- 2.2.11.1 Fundamental electromagnetic relations in Gaussian and SI units.- 2.2.12 Factors for converting from half-life to decay constant.- 2.2.13 Time unit conversion.- 2.3 Conversion factors in alphabetical order.- 2.4 Atomic units.- 2.4.1 Values of frequently occurring quantities in atomic units.- 2.5 X-ray units.- 2.6 Electrical units maintained by the Bureau International des Poids et Mesures (BIPM).- 2.7 Relations between fundamental constants.- 2.8 Values of fundamental constants.- 2.8.1 Rydberg constant.- 2.8.1.1 Rydberg constant for some elements.- 2.8.1.2 Rydberg frequency for Hydrogen.- 2.8.2 Planck’s constant in various units.- 2.8.2.1 Planck’s constant times the speed of light.- 2.8.2.2 Planck’s constant times the square of the speed of light.- 2.8.3 Gas constant for various energy units.- 2.8.3.1 Gas constant for various units of pressure and volume.- 2.9 Index of refraction of air.- 3 Spectroscopy and atomic structure.- 3.1 The electromagnetic spectrum.- 3.1.1 The visible spectrum.- 3.2 Energy-level diagram.- 3.2.1 Term value.- 3.2.1.1 Terms for hydrogen and’ hydrogen-like ions.- 3.2.1.2 Terms for alkali atoms.- 3.2.1.3 Term diagrams for H, He, and Na.- 3.3 Spectral lines for transitions between Rydberg terms.- 3.3.1 Hydrogen lines.- 3.3.1.1 Hydrogen lines from transitions between high Rydberg states (hydrogen radio lines).- 3.3.1.2 Hydrogen hyperfine-structure line.- 3.4 Quantum numbers specifying states of an atomic electron.- 3.4.1 Electron orbitals.- 3.4.1.1 Equivalent orbitals.- 3.4.2 Equivalent electrons.- 3.4.3 Shells.- 3.4.3.1 Subshells.- 3.4.4 Spectroscopic notation for orbitals and shells.- 3.4.5 Electron configuration.- 3.4.5.1 Electron configuration of atoms in their ground state.- 3.5 Anguiar momentum.- 3.5.1 Angular momentum addition.- 3.5.1.1 Clebsch-Gordan coefficients.- 3.5.1.2 Clebsch-Gordan coefficients for J2 = 1 and J2 = 1/2.- 3.5.2 Spin-orbit coupling.- 3.6 Term designation.- 3.6.1 LS-coupling terms.- 3.6.2 Multiplicity.- 3.7 Selection rules for electric dipole transitions.- 3.8 Schematic diagram of X-ray lines and terms.- 3.9 Wavelength of prominent spectral lines.- 3.10 Resonance lines of noble gas atoms.- 3.10.1 Resonance lines of some noble gas ions.- 3.11 Atomic data.- 3.11.1 Nuclidic data.- 3.11.2 List of elements and atomic mass in alphabetical order.- 3.11.3 K x-ray energies, relative intensities and fluorescence yield.- 3.11.4 Periodic table.- 4 Basic wave mechanics.- 4.1 Schrödinger equation.- 4.1.1 Solution of the time independent Schrödinger equation for basic potentials.- 4.1.1.1 One dimensional box potential.- 4.1.1.2 One dimensional harmonic oscillator potential.- 4.1.1.3 Coulomb potential (hydrogen atom).- 4.2 Hydrogenic state functions.- 4.3 Expectation values for the radial distance.- 4.4 Expectation value for the root mean square velocity.- 4.5 Dipole moments.- 4.6 Values of hydrogenic radial transition integrals, $$\left\langle {n\ell \left| r \right|n'\ell '} \right\rangle = \int {{R_{n\ell }}{R_{n'\ell '}}{r^3}dr}$$, in units of the Bohr radius.- 5 Facts, figures and data useful in the laboratory.- 5.1 Fundamental relations of vacuum practice.- 5.1.1 Pressure of a homogeneous gas in thermal equilibrium.- 5.1.1.1 Units of pressure used in vacuum work.- 5.1.2 Mass of gas in a vacuum system.- 5.1.3 Amount of gas.- 5.1.4 Mean free path.- 5.1.5 Throughput.- 5.1.6 Impedance and conductance.- 5.1.7 Pumping speed.- 5.1.8 Fundamental vacuum system equation.- 5.1.9 Formulas to estimate the conductance of apertures and pipes for air at 20° C.- 5.1.10 Viscous flow (Streamline flow).- 5.1.11 Molecular flow.- 5.1.12 Pump down time.- 5.1.13 Vacuum properties.- 5.1.14 Composition of standard air at sea-level.- 5.1.14.1 Physical properties of standard air (15° C, 1 atm).- 5.1.15 Physical properties of common gases (15° C).- 5.2 Electronics.- 5.2.1 Electronic circuit symbols.- 5.2.2 Resistors.- 5.2.2.1 Color code.- 5.2.2.2 Resistor characteristics.- 5.2.2.3 Power rating.- 5.2.3 Resistance of standard copper wire.- 5.2.4 Relative electrical resistivity of metals and temperature coefficient.- 5.2.5 Guide to current-carrying capacity of copper wire.- 5.2.6 Basic passive RC networks.- 5.2.6.1 Differentiation (High-pass filter).- 5.2.6.2 Integration (Low-pass filter).- 5.2.6.3 Filter characteristics as a function of frequency.- 5.3 Digital electronics.- 5.3.1 Binary system of numbers.- 5.3.1.1 Decimal to binary conversion.- 5.3.2 Binary terminology.- 5.3.3 Binary to octal and binary to hexadecimal conversion.- 5.3.3.1 Binary equivalents of decimal, octal and hexadecimal numbers.- 5.3.4 Boolean algebra (Two-state algebra).- 5.3.4.1 Basic operations of Boolean algebra.- 5.3.4.2 Theorems and properties of Boolean algebra.- 5.3.5 Digital circuit symbols and functions.- 5.4 Electrons.- 5.4.1 Energy-momentum reiations.- 5.4.2 Magnetic rigidity.- 5.4.2.1 Values for magnetic rigidity, velocity and momentum of electrons as a function of kinetic energy.- 5.4.2.2 Values for energy, magnetic rigidity and momentum of electrons as a function of velocity.- 5.4.3 Space charge limit for electron beam current.- 5.4.4 Range of electrons in aluminum and lucite.- 5.4.4.1 Measured range of electrons in aluminum.- 5.5 Radiation.- 5.5.1 Radiation and dose units.- 5.5.2 Relative Biological Effectiveness factor (RBE).- 5.5.2.1 Neutron biological dose.- 5.5.3 Recommended dose limits.- 5.5.4 Average human radiation exposure.- 5.5.5 Inverse square diminution of radiation intensity.- 5.5.6 Half thickness.- 5.5.6.1 Half thickness for photons in Al and Pb.- 5.5.6.2 Half thickness for neutrons in concrete.- 5.5.7 Range of alpha particles from some alpha emitters in air.- 5.5.8 Gamma ray absorption.- 5.5.8.1 Linear absorption coefficient.- 5.5.8.2 Mass absorption coefficient.- 5.5.8.3 Crossection.- 5.5.8.4 Total photon mass absorption coefficients $${m^\mu }[\frac{{c{m^2}}}{g}$$.- 5.5.9 Absorption of light.- 5.5.9.1 Properties of some window materials.- 5.5.10 Absorbance and transmittance.- 5.6 Calibration sources frequently used in nuclear spectroscopy.- 5.6.1 Gamma-ray standards.- 5.6.2 Average values of Co56 ?-ray energies and intensities.- 5.6.3 Gamma rays from the electron capture decay 13356Ba ? 13355Cs.- 5.6.4 Gamma rays from thermal neutron capture.- 5.6.5 Energy calibration sources for x-ray spectrometers.- 5.6.6 Radioactive sources for x-ray detector efficiency calibration.- 5.6.7 Radioactive sources with known conversion coefficients, suitable for semiconductor electron spectrometer energy and efficiency calibration.- 5.7 Count loss correction.- 5.8 Material constants and other data.- 5.8.1 Normal melting and boiling point for various substances.- 5.8.2 Approximate values for thermal expansion and conductivity for various materials.- 5.8.3 Approximate values for dielectric constant and dielectric strength of various materials.- 5.8.4 Approximate values for the work function of metals in eV.- 5.8.5 Characteristics of various thermocouples.- 5.8.6 Composition and density of common alloys.- 5.8.7 Terrestrial data.- 5.9 Error analysis.