24 one-electron atomic state defined by the quantum numbers nlmlms or nljmj, with n and l representing the principal quantum number and the orbital angular momentum quantum number, respectively.allowed values of n positive integers, and l = 0, 1, ..., n - 1The quantum number j represents the angular momentum obtained by coupling the orbital and spin angular momenta of an electron, i.e., j = l + s, so that j = l ± 1/2.magnetic quantum numbers ml, ms, and mj represent the projections of the corresponding angular momenta along a particular directionml = -l, -l + 1 ... l and ms = ± 1/2.
25 Pauli exclusion principle prohibits atomic states having two electrons with all four quantum numbers the sameElectrons having both the same n value and l value : equivalentthe maximum number of equivalent electrons is 2(2l + 1)parity of a configuration is even or odd : S lj is even or odd
26 Hydrogen and Hydrogen-like Ions A particular level is denoted either by nlj or by nl 2LJ with L = l and J = j.The multiplicity of the L term is equal to 2S + 1 = 2s + 1 = 2.: doublet : two levels, with J = L ± 1/2, respectivelyThe Coulomb interaction between the nucleus and the single electron is dominant, so that the largest energy separations are associated with levels having different nhyperfine splitting of the 1H 1s ground level [ 751 766 7(10) MHz] results from the interaction of the proton and electron magnetic moments. = 21cm line
27 Alkalis and Alkali-like Spectra In the central field approximation there exists no angular-momentum coupling between a closed subshell and an electron outside the subshell, since the net spin and orbital angular momenta of the subshell are both zero. nlj quantum numbers are appropriate for a single electron outside closed subshells. However, the electrostatic interactions of this electron with the core electrons and with the nucleus yield a strong l-dependence of the energy levels. The spin-orbit fine-structure separation between the nl (l > 0) levels having j = l - 1/2 and l + 1/2, respectively, is relatively small.
31 Helium and Helium-like Ions; LS Coupling condition for LS coupling(a) The orbital angular momenta of the electrons are coupled to give a total orbital angular momentum L = S ili.(b) The spins of the electrons are coupled to give a total spin S = Si si.combination of a particular S value with a particular L value , a spectroscopic term 2S+1L.(2S + 1 is the multiplicity of the term)total angular momentum, J = S + L : level is denoted as 2S+1LJ.For 1s2 nl configurations, L = l and S = 0 or 1, i.e., the terms are singlets (S = 0) or triplets (S =1)ionization energy eV, the 1s2s 3S - 1S separation is eV, the 1s2p 3P° - 1P° separation is eV, and the 1s2p 3P°2 - 3P°0 fine-structure spread is only 1.32 × 10-4 eV.
33 Hierarchy of Atomic Structure in LS Coupling Atomic structural hierarchy in LS coupling and names for the groups of all transitions between structural entities.Structural entity Quantum numbers a Group of all transitionsConfiguration (nili)Ni Transition arrayPolyad (nili)Ni g S1 L1 nl S L, S L... SupermultipletTerm (nili)Ni g S L multipletLevel (nili)Ni g S L J lineState (nili)Ni g S L J M Line componenta The configuration may include several open subshells, as indicated by the i subscripts. The letter g represents any additional quantum numbers, such as ancestral terms, necessary to specify a particular term.
34 Ca I 3d4p 3D°2 level belongs to the 3D° term which, in turn, belongs to the 3d4p 3(P° D° F°) triplet triad3d4p configuration also has a 1(P° D° F°) singlet triad.3d4s configuration has only monads, one 1D and one 3D3d4s 3D2 - 3d4p 3D°3 line belongs to thecorresponding 3D - 3D° triplet multiplet,this multiplet belongs tothe great Ca I 3d4s 3D - 3d4p 3(P° D° F°)supermultiplet of three triplet multiplets3d4s - 3d4p transition array includesboth the singlet and triplet supermultiplets,as well as any (LS-forbidden) intercombinationor intersystem lines arising from transitionsbetween levels of the singlet systemand those of the triplet system
36 Allowed Terms of Levels for Equivalent Electrons LS Couplingtwo nonequivalent groups of electrons coupling the S and L vectors of the groups in all possible ways, and the procedure may be extended to any number of such groups.The configuration l N has more than one allowed term of certain LS types if l > 1 and 2 < N < 4l (d 3 - d 7, f 3 - f 11, etc.).LS term type from d N and f N : tables of Nielson and Ko ster
37 jj CouplingThe allowed J values for a group of N equivalent electrons having the same j value, ljN, are given in the table below for j = 1/2, 3/2, 5/2, and 7/2 (sufficient for l < 3).
38 Allowed J values for ljN equivalent electrons (jj coupling). ljN Allowed J valuesl1/ ½l 21/l3/2 and l 33/ /2l23/ , 2l 43/l5/2 and l 55/ /2l 25/2 and l 45/ , 2, 4l 35/ /2, 5/2, 9/2l 65/l7/2 and l 77/ /2l 27/2 and l 67/ , 2, 4, 6l 37/2 and l 57/ /2, 5/2, 7/2, 9/2, 11/2, 15/2l 47/ , 22, 42, 24, 44, 5, 6, 8l 87/
39 Notations for Different Coupling Schemes LS Coupling (Russell-Saunders Coupling)jj CouplingJ1 j or J1 J2 CouplingJ1l or J1L2 Coupling (J1K Coupling)LS1 Coupling (LK Coupling)
40 The allowed levels of the configuration nl N may be obtained by dividing the electrons into sets of two groups , Q + R = N.The possible sets run from Q = N - 2l (or zero if N < 2l) up to Q = N or Q = 2l + 2, whichever is smaller.
41 Coupling Schemes and Term Symbols Coupling Quantum number TermScheme for vectors that Symbolcoupl to give JLS L, S S+1LJ1J J1,, J (J1,, J2 )J1L2(-> K) K, S S2+1 [K]LS1(->K) K, S S2+1 [K]
46 Zeeman Effect"weak" magnetic fields (the anomalous Zeeman effect) : split into magnetic sublevels : -J, -J + 1, ..., J: =MDE = gM µBB magnetic flux density is B, and µB is the Bohr magneton (µB = e /2me). wavenumber shift Ds corresponding to this energy shift = gM(0.466 86 B cm-1)
49 g value of a level bJ belonging to a pure LS-coupling term g value for a pure electron spin as ge
50 Term Series, Quantum Defects, and Spectral-line Series hydrogenic (one-electron) ionZc is the charge of the core and n* = n -d isthe effective principal quantum number
51 SequencesIsoelectronic Sequence :A neutral atom and those ions of other elements having the same number of electrons as the atom comprise an isoelectronic sequence : the Na I isolectronic sequence.Isoionic, Isonuclear, and Homologous Sequences :An isoionic sequence comprises atoms or ions of different elements having the same charge.The atom and successive ions of a particular element comprise the isonuclear sequence for that elementThe elements of a particular column and subgroup in the periodic table are homologous :C, Si, Ge, Sn, and Pb atoms belong to a homologous sequence having np2 ground configurations
53 Emission Intensities (Transition Probabilities) Emission Intensities (Transition Probabilities)Aki is the atomic transition probability and Nk the number per unit volume (number density) of excited atoms in the upper (initial) level kFor a homogeneous light source of length l and for the optically thin case, where all radiation escapes, the total emitted line intensity
54 숙제 3수소의 Ha 가 n=3 -> n=2 로 천이되는 모든 경우를 고려하여 선택 규율을 따르는 모든 천이를 에너지도를 그려 표시하시오.또 Zeeman 효과가 있을 경우 천이를 모두 도표로 그리고, 선의 분리를 보이시오.
55 fik is the atomic (absorption) oscillator strength (dimensionless). Absorption f values(17)fik is the atomic (absorption)oscillator strength (dimensionless).Line Strengths i and k are the initial- and final-state wave functionsand Rik is the transition matrix element of the appropriatemultipole operator P (Rik involves an integration overspatial and spin coordinates of all N electrons of the atomor ion).
56 천이 확율(Transition Probabilites) 선과 multiplet의 상대적 세기<= 천이 확율 <= 흡수, 방출계수천이 확율 - 들뜸 에너지 준위의 수명Life time of the transition= inverse of the sum of the transition probabilitieslife time 10-8s : allowed transitions 10-5s : intercombination transitions > 10-3s : forbidden transitions 21cm - 11 *106 years
57 아인슈타인 천이 확률 Spontaneous Emission Transition Probability = reciprocal of the lifetime of the transition= A21 ( 108 s-1 for allowed 10-15 s-1 for most extremely forbidden)number of spontaneous transition /time/volume= N2 A21N2 = number density of atoms in upper level
58 Absorption Transition Probability = need radiation propotional to the radiation intensitynumber of absorptions/time/volume = N1 B12 I21B12 = absorption transition probability from 1 to 2N1 = number Density of atoms in level 1I21 = intensity of radiation (emitted 2->1)
59 Negative Absorption = Stimulated Emission =>LASER (Light Amplification by Stimulated Emission of Radiation)=>MASER (Microwave Amplificaton by StimulatedEmission of Radiation)number of stimulated emisions/time/volume=N2 B21 I21B21 = Stimulated Emission Trasition Probability from 2 to 1==> photons added to the radiation field with the same direction, polarization and phase as those of the stimulating photonscf: photons emitted spontaneouly ==> have randomdirections, polarization and phases
60 Principle of Detailed Balancing in TE =every process balanced by its inverse=every upward transition have a downwardtranstion occurring nearby and nearlysimultaneously==> total number of absorptions = total number of emissionsN1 B12 I21 = N2 B21 I21 + N2 A21TE : radiation field = BB radiation = Plank Function
61 TE : radiation field = BB radiation = Plank Function 여기서 m 는 매질의 굴절율로 보통 1 근처의 값A21 = (N1/N2 B12 - B21) I21=
62 아인슈타인 계수 transition probability 는 원자의 성질이므로 환경의 특성인 온도에 무관 주파수가 높아지면 ( 보통 적외선 보다) stimulated emission 은무시될 수 있다. 이로써 maser가 성간 가스 운에서는 발생하지만높은 주파수의 laser는 발생하지않는 것을 보게된다.
65 흡수와 방출 계수 아인슈타인 천이 확율은 각 천이가 일어날 확율을 결정하므로 스펙트럼의 흡수와 방출선의 결과가 된다 고전 회전자 - 약한 선의 경우 흡수 계수 :선윤곽질량 흡수 계수 N : 흡수 회전자의 개수 밀도 m, e : 회전자(전자)의 질량과 전하 no : 공명 주파수 (즉 흡수선의 중심 주파수) g: 회전자 복사의 감쇄 상수
66 양자 역학:진동자 세기(Oscillator Strength) Damping 상수는 준위 수명과 관계가 고전과 다소 다르다허가 천이의 경우 값이 108 로 가시 영역 천이의 g정도다.n lower than 준위 1, m higher than 준위 1 G1 도 유사한 관계를 갖음
67 Thomas-Reiche-Kuhn Sum Rule N 대신 Nf 로 대치 :f는 고전을 양자 역학적 값으로 환원하는 보정 계수인 진동자 세기 ( Oscillator Strength) 관측되는 선세기를 만드는 고전 진동자의 수(보통 쪼각)따라서 한 원자 또는 이온에서 한 준위에서 발생되는 모든 가능한 천이에 대한 진동자 세기를 전부 합한 것(방출의 진동자 세기는 음으로 취급)은 원자나 이온의 전자 수와 같아진다.==> Thomas-Reiche-Kuhn Sum Rule