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

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.