 # 원자 구조 및 스펙트럼 수소 원자 다 전자 원자. 슈레딩거 방정식 파울리 배타 원리 주기율표

## Presentation on theme: "원자 구조 및 스펙트럼 수소 원자 다 전자 원자. 슈레딩거 방정식 파울리 배타 원리 주기율표"— Presentation transcript:

원자 구조 및 스펙트럼 수소 원자 다 전자 원자. 슈레딩거 방정식 파울리 배타 원리 주기율표
LS Coupling & jj Coupling

실험실 : 스펙트럼 시리즈 (수소의 Lyman, Balmer, Pachen, Brackett, Pfund, Humphrey, …

Lyman Series Balmer Series n λ(nm) 2 122 3 656 103 4 486 97.2 5 434 94.9 6 410 93.7 7 397 91.1 365

Paschen Series Brackett Series n λ(nm) 4 1870 5 4050 1280 6 2630 1090 7 2170 1000 8 1940 954 9 1820 820 1460

Pfund Series Humphreys Series n λ(nm) 6 7460 7 12372 4650 8 7503 3740 10 5129 9 3300 11 4673 3040 13 4171 2280 3282

보아 모형 전자의 각운동량(mvr)은 정수배의 h/2p mvr = n h/(2p) Coulomb force :
Z e2 / (4peo r2 ) = mv2 / r r = eo n2 h2 / (p Ze2m) 궤도의 크기는 n2 에 비례 (n=주양자수) 궤도상 전자의 총에너지 = 포텐셜 에너지 + 운동에너지 = - Ze2 /(4peor) + Ze2 /(8peor) = - Ze2 /(8peor) E = - Z2 e4m / (8eo2n2h2) 에너지는 n-2 에 비례, 두 에너지 준위는 (n1-2 – n2-2 ) 에 비례

Sommerfeld의 개선 전자의 질량 m  환원질량 m = mM/(m+M) 원의 궤도  타원 궤도(운동에너지 r, f)
 주 양자수 n + azimuthal 양자수 k (양자역학의 궤도 양자수 l = k-1) 상대론적 효과  에너지 준위의 약한 k에 대한 의존성

Hydrogen Energy Levels

Electron Volts 의 개념

쉬뢰딩거 방정식

파동방정식

파동 함수의 의미

수소 원자의 3 주양자 껍질에 전자 밀도 분포

파울리의 배타원리와 주기율표 주 양자수 n 궤도 양자수 l (k-1) 궤도 자기 양자수 mㅣ 스핀 자기 양자수 ms
 no two electrons in an atom may have the same four quantum mumbers, n, l, ml, ms  원소의 주기율표

하나의 원자가 전자를 갖는 원자

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 - 1 The 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 direction ml = -l, -l + 1 ... l and ms = ± 1/2.

Pauli exclusion principle
prohibits atomic states having two electrons with all four quantum numbers the same Electrons having both the same n value and l value : equivalent the maximum number of equivalent electrons is 2(2l + 1) parity of a configuration is even or odd : S lj is even or odd

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, respectively The Coulomb interaction between the nucleus and the single electron is dominant, so that the largest energy separations are associated with levels having different n hyperfine 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

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.

원자가(valance) 전자 하나인 알카리 금속

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.

Helium wavelengths (nm)
s=strong, m=med, w=weak

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 transitions Configuration (nili)Ni Transition array Polyad (nili)Ni g S1 L1 nl S L, S L... Supermultiplet Term (nili)Ni g S L multiplet Level (nili)Ni g S L J line State (nili)Ni g S L J M Line component a 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.

Ca I 3d4p 3D°2 level belongs to the 3D° term which, in turn, belongs to the 3d4p 3(P° D° F°) triplet triad 3d4p configuration also has a 1(P° D° F°) singlet triad. 3d4s configuration has only monads, one 1D and one 3D 3d4s 3D2 - 3d4p 3D°3 line belongs to the corresponding 3D - 3D° triplet multiplet, this multiplet belongs to the great Ca I 3d4s 3D - 3d4p 3(P° D° F°) supermultiplet of three triplet multiplets 3d4s - 3d4p transition array includes both the singlet and triplet supermultiplets, as well as any (LS-forbidden) intercombination or intersystem lines arising from transitions between levels of the singlet system and those of the triplet system

Allowed Terms of Levels for Equivalent Electrons
LS Coupling two 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

jj Coupling The 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).

Allowed J values for ljN equivalent electrons (jj coupling).
ljN Allowed J values l1/ ½ l 21/ l3/2 and l 33/ /2 l23/ , 2 l 43/ l5/2 and l 55/ /2 l 25/2 and l 45/ , 2, 4 l 35/ /2, 5/2, 9/2 l 65/ l7/2 and l 77/ /2 l 27/2 and l 67/ , 2, 4, 6 l 37/2 and l 57/ /2, 5/2, 7/2, 9/2, 11/2, 15/2 l 47/ , 22, 42, 24, 44, 5, 6, 8 l 87/

Notations for Different Coupling Schemes
LS Coupling (Russell-Saunders Coupling) jj Coupling J1  j or J1 J2 Coupling J1l or J1L2 Coupling (J1K Coupling) LS1 Coupling (LK Coupling)

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.

Coupling Schemes and Term Symbols
Coupling Quantum number Term Scheme for vectors that Symbol coupl to give J LS L, S S+1L J1J J1,, J (J1,, J2 ) J1L2(-> K) K, S S2+1 [K] LS1(->K) K, S S2+1 [K]

2개의 원자가 전자

탄소 ; 에너지 준위 표기

Zeeman Effect "weak" magnetic fields (the anomalous Zeeman effect) : split into magnetic sublevels : -J, -J + 1, ..., J: =M DE = 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)

자기장에 의한 영향

g value of a level bJ belonging to a pure LS-coupling term
g value for a pure electron spin as ge

Term Series, Quantum Defects, and Spectral-line Series
hydrogenic (one-electron) ion Zc is the charge of the core and n* = n -d is the effective principal quantum number

Sequences Isoelectronic 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 element The 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

Selection Rules

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 k For a homogeneous light source of length l and for the optically thin case, where all radiation escapes, the total emitted line intensity

숙제 3 수소의 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.

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 functions and Rik is the transition matrix element of the appropriate multipole operator P (Rik involves an integration over spatial and spin coordinates of all N electrons of the atom or ion).

천이 확율(Transition Probabilites)
선과 multiplet의 상대적 세기 <= 천이 확율                               <= 흡수, 방출계수 천이 확율 - 들뜸 에너지 준위의 수명 Life time of the transition = inverse of the sum of the transition probabilities life time 10-8s : allowed transitions           10-5s : intercombination transitions         > 10-3s : forbidden transitions                  21cm - 11 *106 years

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 A21 N2 = number density of atoms in upper level

Absorption Transition Probability
= need radiation  propotional to the radiation intensity number of absorptions/time/volume  = N1 B12 I21 B12 = absorption transition probability from 1 to 2 N1 = number Density of atoms in level 1 I21 = intensity of radiation (emitted 2->1)

Negative Absorption = Stimulated Emission
=>LASER (Light Amplification by Stimulated Emission of Radiation) =>MASER (Microwave Amplificaton by Stimulated Emission of Radiation) number of stimulated emisions/time/volume =N2 B21 I21 B21 = 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 photons cf: photons emitted spontaneouly ==> have random directions, polarization and phases

Principle of Detailed Balancing in TE
=every process balanced by its inverse =every upward transition have a downward transtion occurring nearby and nearly simultaneously ==> total number of absorptions      = total number of emissions N1 B12 I21 = N2 B21 I21 + N2 A21 TE : radiation field = BB radiation                      = Plank Function

여기서 m   는 매질의 굴절율로  보통 1 근처의 값 A21 = (N1/N2 B12 - B21) I21 =

아인슈타인 계수 transition probability 는 원자의 성질이므로 환경의 특성인 온도에 무관
주파수가 높아지면 ( 보통 적외선 보다)  stimulated emission 은 무시될 수 있다. 이로써  maser가 성간 가스 운에서는 발생하지만 높은 주파수의 laser는 발생하지않는 것을 보게된다.

A21        B21           B12 HI 1215(Ly )    6*108      2.8*1012      1.4*1012  1025(Ly )    1.7*108     4.5*1011      2.2*1011     972(Ly )    6.8*107     1.6*1011      7.9*1010 OI 3947          3.7*105     5.7*1010      8.0*1010 MgI 4571]       2.1*102      5.1*107      1.7*107 [NI 5198]        1.6*10-5     5.7          5.7

흡수와 방출 계수 아인슈타인 천이 확율은 각 천이가 일어날 확율을 결정하므로 스펙트럼의 흡수와 방출선의 결과가 된다
고전 회전자 - 약한 선의 경우 흡수 계수 :선윤곽 질량 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

양자 역학:진동자 세기(Oscillator Strength)
Damping 상수는 준위 수명과 관계가 고전과 다소 다르다 허가 천이의 경우 값이 108 로 가시 영역 천이의 g정도다. n lower than 준위 1,  m higher than 준위 1  G1 도 유사한 관계를 갖음

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

발머선의 진동자 세기 ( Oscillator Strength)
Bamer series n=2 Ha                  0.637 Hb                   0.119 Hg                   0.044 Hd                   0.021 He                   0.012 ... 흡수                 0.866 이온화               0.238 방출(Lyman )     총합(수소에 전자수) 1.000

양자역학적 자연 흡수 계수 선의 반폭 = G/2p 이며, G 가 선에 따라 달라지므로
고전적 결과와 달리 양자영학적 흡수 계수는 선에 따라 달라진다.

Oscillator Strength 와 아인슈타인 천이 확률계수들과의 관계

천이 확율과 흡수 및 방출 계수의 관계 stimulated emission, negative 흡수 포함 방출 계수
자연 선 윤곽의 형태 함수 stimulated emission, negative 흡수 포함 방출 계수  Kirchhoff's 법칙

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