ATOMIC RADIATIVE PROCESSES
CONTENTS GENERAL LAWS OF THE ELECTROMAGNETIC FIELD MAXWELL’S EQUATIONS ENERGY TRANSPORTED BY THE ELECTRIC FIELD MOMENTUM TRANSPORTED BY THE ELECTROMAGNETIC FIELD ELECTROMAGNETIC POTENTIALS GAUGE INVARIANCE SOLVING MAXWELL’S EQUATIONS IN VACUUM RADIATION PRESSURE SINUSOIDAL PLANE WAVES SPECTRUM AND POLARISATION SPECTRUM OF THE RADIATION SPECTRA OF SOME PARTICULAR PULSE SHAPES SPECTRA OF STOCHASTIC AND PERIODIC SIGNALS DIFFRACTION GRATING SPECTROSCOPE POLARISATION OF A MONOCHROMATIC WAVE SPECTROPOLARIMETRIC MEASUREMENTS PROPERTIES OF THE STOKES PARAMETERS RADIATION FROM MOVING CHARGES ELECTROMAGNETIC POTENTIALS DUE TO CHARGES AND CURRENTS THE LIÉNARD AND WIECHART POTENTIALS THE ELECTROMAGNETIC FIELD OF A MOVING CHARGE RADIATION FROM A MOVING CHARGE THOMSON SCATTERING RAYLEIGH SCATTERING BREMSSTRAHLUNG CYCLOTRON RADIATION SYNCHROTRON RADIATION MULTIPOLAR EXPANSION IN THE RADIATION ZONE RADIATION DIAGRAM FOR THE MULTIPOLAR COMPONENTS QUANTISATION OF THE ELECTROMAGNETIC FIELD HARMONIC OSCILLATOR, OPERATORS OF CREATION AND ANNIHILATION EXPANSION OF THE ELECTROMAGNETIC FIELD IN FOURIER SERIES THE QUANTUM ANALOG INTENSITY AND PHOTONS RELATIVISTIC WAVE EQUATIONS THE DIRAC EQUATION FOR A FREE PARTICLE THE DIRAC EQUATION FOR THE ELECTRON IN AN ELECTROMAGNETIC FIELD NONRELATIVISTIC LIMIT OF THE DIRAC EQUATION ZERO ORDER LIMIT, PAULI EQUATION FIRST ORDER LIMIT THE DIRAC EQUATION DESCRIBES A PARTICLE OF SPIN 1/2 SOLUTION OF THE DIRAC EQUATION IN A MAGNETIC FIELD ATOMS WITH A SINGLE VALENCE ELECTRON HYDROGEN ATOM, BOHR THEORY SCHRÖDINGER’S EQUATION IN SPHERICAL COORDINATES HYDROGEN ATOM, QUANTUM THEORY HYDROGEN ATOM, RELATIVISTIC CORRECTIONS SPECTRA OF ALKALINE METALS ATOMS WITH MULTIPLE VALENCE ELECTRONS THE PAULI EXCLUSION PRINCIPLE THE NONRELATIVISTIC HAMILTONIAN: GOOD QUANTUM NUMBERS THE CENTRAL FIELD APPROXIMATION THE THOMAS-FERMI METHOD THE VARIATIONAL METHOD AND THE HARTREE-FOCK METHOD CON?GURATIONS THE PRINCIPLE OF FORMATION OF THE PERIODIC TABLE CON?GURATIONS OF EXCITED ELECTRONS A SUMMARY OF ANGULAR MOMENTUM THEORY TERMS ORIGINATING FROM GIVEN CON?GURATIONS THE EIGENFUNCTIONS OF THE NON-RELATIVISTIC HAMILTONIAN THE HELIUM ATOM THE CARBON ATOM TERM ENERGIES THE DIAGONAL SUM RULE CALCULATION OF DIAGONAL MATRIX ELEMENTS SINGLE PARTICLE MATRIX ELEMENTS MATRIX ELEMENTS OF THE COULOMB INTERACTION SUMS OVER CLOSED SUBSHELLS TERMS STRUCTURE MORE DETAILS ON ATOMIC SPECTRA THE SPIN-ORBIT INTERACTION THE WIGNER-ECKART THEOREM AND THE PROJECTION THEOREM THE LANDÉ INTERVAL RULE THE J-J COUPLING AND THE INTERMEDIATE COUPLING THE ZEEMAN EFFECT (CLASSICAL APPROACH) THE ZEEMAN EFFECT (QUANTUM APPROACH) THE PASCHEN-BACK EFFECT HYPER?NE STRUCTURE, ISOTOPE EFFECT HYPER?NE STRUCTURE, EFFECT DUE TO THE NUCLEAR SPIN LAWS OF THERMODYNAMIC EQUILIBRIUM THE PRINCIPLES OF STATISTICAL EQUILIBRIUM MAXWELL’S VELOCITY DISTRIBUTION THE SAHA-BOLTZMANN EQUATION THE BLACK-BODY RADIATION PROPERTIES OF THE BLACK-BODY RADIATION THE FERMI-DIRAC STATISTICS THE BOSE-EINSTEIN STATISTICS INTERACTION BETWEEN MATTER AND RADIATION THE INTERACTION HAMILTONIAN THE KINETIC EQUATIONS FERMI’S GOLDEN RULE THE MATRIX ELEMENT ELEMENTARY PROCESSES THE STATISTICAL EQUILIBRIUM EQUATIONS EINSTEIN COEF?CIENTS THE RADIATIVE TRANSFER EQUATION THE ABSORPTION AND EMISSION COEF?CIENTS 11 10 PRO?LE OF THE EMISSION COEF?CIENT SELECTION RULES AND LINE STRENGTHS SELECTION RULES FOR THE QUANTUM NUMBERS SELECTION RULES FOR THE CON?GURATIONS FORBIDDEN TRANSITIONS SEMI-FORBIDDEN TRANSITIONS FORBIDDEN LINES IN ASTRONOMICAL OBJECTS RELATIVE STRENGTHS WITHIN MULTIPLETS IN L-S COUPLING NON-EQUILIBRIUM PLASMAS THE KINETIC TEMPERATURE OF THE ELECTRONS ELECTRON-ATOM COLLISIONS THE EINSTEIN-MILNE RELATIONS THE TWO-LEVEL ATOM IN NON-EQUILIBRIUM CONDITIONS RADIATIVE TRANSFER FORMAL SOLUTION OF THE RADIATIVE TRANSFER EQUATION RADIATIVE TRANSFER IN STELLAR ATMOSPHERES THE GREY ATMOSPHERE THE HOPF EQUATION REALISTIC MODELS OF STELLAR ATMOSPHERES THE CONTINUUM SPECTRUM SPECTRAL LINES IN LOCAL THERMODYNAMIC EQUILIBRIUM SPECTRAL LINES IN NON-EQUILIBRIUM CONDITIONS SECOND ORDER PROCESSES INTRODUCTION THOMSON SCATTERING (QUANTUM APPROACH) THE RAYLEIGH AND RAMAN SCATTERING (QUANTUM APPROACH) COMPTON SCATTERING, KINEMATIC ASPECTS COMPTON SCATTERING, DYNAMIC ASPECTS THE KLEIN-NISHINA EQUATION THE TOTAL CROSS-SECTION FOR COMPTON SCATTERING POLARISATION PROPERTIES OF COMPTON SCATTERING ENERGY EXCHANGE BETWEEN PHOTONS AND ELECTRONS 15 10 THE INVERSE COMPTON SCATTERING CHAPTER C ?E ?T = 4? C J, (1 3) 1THE READER WHO IS NOT FAMILIAR WITH THE SYSTEM OF UNITS USED HERE CAN ?ND IN SECT 16 1 THE RELATIONS BETWEEN THIS SYSTEM AND THE SYSTÈME INTERNATIONAL (SI), MORE APPROPRIATE FOR PRACTICAL APPLICATIONS RATHER THAN THEORETICAL ONES GENERAL LAWS OF THE ELECTROMAGNETIC FIELD ROTE + 4? B ?B ?T RECALLING THE VECTOR IDENTITY (SEE SECT 16 2, EQ (16 7)) DIV(E × B) = B ROTE ?E ROTB, WE CAN WRITE ? ?T (W + U) + DIVS = 0, WHERE U = , S = C 4? E × B THIS EQUATION, KNOWN AS POYNTING’S THEOREM, REPRESENTS AN ENERGY BALANCE: THE ELECTROMAGNETIC ?ELD HAS AN ENERGY PER UNIT OF VOLUME GIVEN BY U AND IT TRANSPORTS ENERGY ALONG THE DIRECTION OF THE VECTOR S THE MAGNITUDE OF THIS VECTOR IS THE ENERGY ?UX DENSITY, I E THE ENERGY ?OWING PER UNIT OF TIME ACROSS THE UNIT AREA NORMAL TO THE DIRECTION OF THE VECTOR ITSELF S IS CALLED POYNTING VECTOR 1 3 MOMENTUM TRANSPORTED BY THE ELECTROMAGNETIC FIELD WE NOW DE?NE WITH Q THE MOMENTUM PER UNIT VOLUME ASSOCIATED WITH THE SAME SYSTEM OF CHARGES AND CURRENTS CONSIDERED IN THE PREVIOUS SECTION THIS MOMENTUM VARIES WITH TIME FOLLOWING THE EQUATION ?Q ?T = ?E + J C × B, WHERE THE ?RST TERM IN THE RIGHT-HAND SIDE IS DUE TO THE ELECTRIC FORCE, WHILE THE SECOND TERM IS DUE TO THE LORENTZ FORCE ELIMINATING ? AND J USING EQ (1 1) AND (1 3), WE OBTAIN ?Q ?T = 4? EDIVE + 4? (ROTB) × B ? 4?C ?E ?T × B, I E , RECALLING THE DE?NITION OF THE POYNTING VECTOR ?Q ?T = ?S ?T + 4? (EDIVE + BDIVB ?E × ROTE ?B × ROTB) WE NOW INTRODUCE THE TENSOR T, NAMED AFTER MAXWELL, DE?NED BY THE EXPRESSION T = U, WHERE U IS THE UNIT TENSOR (UIJ = ?IJ) IN COMPONENTS, WE HAVE TIJ = TJI = 1 4 ELECTROMAGNETIC POTENTIALS RECALLING EQ (1 2) AND THE FACT THAT THE DIVERGENCE OF THE CURL OF ANY VECTOR IS ZERO, THE MAGNETIC ?ELD2 CAN BE WRITTEN AS A FUNCTION OF A SUITABLE VECTOR POTENTIAL A B = ROTA SUBSTITUTING THIS EXPRESSION INTO EQ (1 4), WE OBTAIN ROT E + C ?A ?T WE NOW SUBSTITUTE THESE EXPRESSIONS FOR B AND E INTO THE NON-HOMOGENEOUS MAXWELL EQUATIONS FROM EQ (1 3) WE HAVE ROT(ROTA) + C ? ?T GRAD? + ?2A ?T2 ?GRAD DIVA + C ?? ?T = ?4? J C FROM EQ (1 1) WE ALSO HAVE ?2? + ?2? ?T2 + C ? ?T DIVA + C ?? ?T = ?4?? THE SYSTEM OF THESE LAST TWO EQUATIONS IN PRINCIPLE ALLOWS ONE TO OBTAIN THE PO- TENTIALS A AND ? (AND THEREFORE THE ?ELDS E AND B), ONCE ?, J, AND THE BOUNDARY CONDITIONS ARE KNOWN WE WILL SEE HOWEVER THAT THE EQUATIONS INVOLVING A AND ? CAN BE SIMPLI?ED USING A SUPPLEMENTARY CONDITION 2THE VECTOR B SHOULD BE CALLED MORE APPROPRIATELY “MAGNETIC INDUCTION” INSTEAD OF “MAGNETIC ?ELD”, THIS LAST NAME REFERRING TO THE VECTOR H HOWEVER IT HAS NOW BECOME CUSTOMARY TO CALL B THE MAGNETIC ?ELD VECTOR, WHEN NO AMBIGUITIES ARE PRESENT GENERAL LAWS OF THE ELECTROMAGNETIC FIELD 1 5 GAUGE INVARIANCE THE POTENTIALS A E ? UNIQUELY DETERMINE THE ELECTRIC AND MAGNETIC ?ELDS HOWEVER, THERE EXIST DIFFERENT A AND ? FUNCTIONS THAT CAN PROVIDE THE SAME E AND B INDEED, IF WE SUBSTITUTE A ?A? = A ?GRAD?, (1 5) ? ??? = ? + C ?? ?T , (1 6) WHERE ? IS AN ARBITRARY FUNCTION OF SPACE AND TIME, WE OBTAIN, FOR THE NEW ?ELD B?, B? = ROT(A ?GRAD?) = ROTA = B, AND, SIMILARLY FOR THE NEW ?ELD E?, E? = ?GRAD ? + C ? ?T (A ?GRAD?) = E THIS PROPERTY IS CALLED GAUGE INVARIANCE AND CAN BE CONVENIENTLY USED WITH AN AP- PROPRIATE CHOICE OF THE FUNCTION ? LET US SUPPOSE THAT A0 AND ?0 ARE THE VECTOR AND SCALAR POTENTIALS PRODUCING THE E AND B ?ELDS THAT WE WANT TO DETERMINE, GIVEN A SET OF BOUNDARY CONDITIONS AND A DISTRIBUTION OF CHARGES AND CURRENTS IF WE CONSIDER THE GAUGE TRANSFORMATION A = A0 ?GRAD?, ? = ?0 + ?2? ?T2 = DIVA0 + ?T , WE HAVE DIVA + ?2A ?T2 = ?4? J C + C GRAD ?? ?T , ?2? = ?4?? THIS GAUGE HAS THE ADVANTAGE THAT THE SCALAR POTENTIAL COINCIDES WITH THE (IN- STANTANEOUS) ONE OF ELECTROSTATICS, EVEN IN THE CASE OF NON-STATIONARY PHENOMENA INVOLVING THE PRESENCE OF CHARGES AND CURRENTS VARYING WITH TIME