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원자 구조 및 스펙트럼
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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)
n
λ(nm)
2
122
3
656
3
103
4
486
4
97.2
5
434
5
94.9
6
410
6
93.7
7
397
91.1
365
Paschen Series
Brackett
Series
n
λ(nm)
n
λ(nm)
4
1870
5
4050
5
1280
6
2630
6
1090
7
2170
7
1000
8
1940
8
954
9
1820
820
1460
Pfund Series
Humphreys
Series
n
λ(nm)
n
λ(nm)
6
7460
7
12372
7
4650
8
7503
8
3740
10 5129
9
3300
11 4673
10
3040
13 4171
2280
3282
보아 모형
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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)
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 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
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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
[1420.405 751 766 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
Hydrogen-like atoms
Other terms
Hyperfine structure
Closed shell
Terms
Equivalent electrons
Terms
Helium and Helium-like Ions; LS
Coupling
• condition for LS coupling
• (a) The orbital angular momenta of the electrons are coupled t
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 a
singlets (S = 0) or triplets (S =1)
• ionization energy 24.5874 eV, the 1s2s 3S - 1S separation is
0.7962 eV, the 1s2p 3P° - 1P° separation is 0.2539 eV, and the
1s2p 3P°2 - 3P°0 fine-structure spread is only 1.32 × 10-4 eV.
wavelengths
(nm)
438.793 w
443.755 w
447.148 s
471.314 m
492.193 m
501.567 s
504.774 w
587.562 s
667.815 m
s=strong,
m=med,
w=weak
LS
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
Equivalent electrons
• 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).
• Eg : (6p2 ½)0
Allowed J values for ljN equivalent
electrons (jj coupling).
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l jN
Allowed J values
l1/2
½
l 21/2
0
l3/2 and l 33/2
3/2
l23/2
0, 2
l 43/2
0
l5/2 and l 55/2
5/2
l 25/2 and l 45/2
0, 2, 4
l 35/2
3/2, 5/2, 9/2
l 65/2
0
l7/2 and l 77/2
7/2
l 27/2 and l 67/2
0, 2, 4, 6
l 37/2 and l 57/2
3/2, 5/2, 7/2, 9/2, 11/2, 15/2
l 47/2
0, 22, 42, 24, 44, 5, 6, 8
l 87/2
0
Allowed levels
• The allowed levels of the configuration
nl N may be obtained by dividing the
electrons into sets of two groups
nlQl+1/2 nlRl-1/2 , 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.
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)
• http://sed.nist.gov/Pubs/AtSpec/total.
html
Coupling Schemes and Term
Symbols
• Coupling
• Scheme
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Quantum number
for vectors that
coupl to give J
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L, S
J1,, J2
K, S2
K, S2
LS
J1J2
J1L2(-> K)
LS1(->K)
Term
Symbol
2S+1L
(J1,, J2 )
2S2+1 [K]
2S2+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 is
 Ds = 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
Selection Rules
Energy ordering of levels
Laporte Rule : all electric dipole transition connect states of
opposite parity.
Hund’ rules :
1st : For a given configuration, the state with the maximum
spin multiplicity is the lowest in energy.
For a given configuration and spin multiplicity, the term with
the largest value of L lies lowest in energy
2nd : The lowest energy is obtained for lowest value of J in
the normal case and for hight J value in the inverted case
(normal case = atoms with less than half-filled shells,
Inverted case = atoms with more than half-filled shells)
Emission Intensities (Transition
Probabilities)
• The total power e radiated in a spectral line of
frequency n per unit source volume and per unit
solid angle is
• 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 -1
• 수소의 Ha 가 n=3 -> n=2 로 천이되
는 모든 경우를 고려하여 선택 규율을
따르는 모든 천이를 에너지도를 그려 표
시하시오.
• 또 Zeeman 효과가 있을 경우 천이를
모두 도표로 그리고, 선의 분리를 보이
시오.
Absorption f values
fik is the atomic (absorption) oscillator strength
(dimensionless).
Line Strengths
Y i and Y 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).
Relationships between A, f, and
S
• The relationships between A, f, and S
for electric dipole (E1, or allowed)
transitions in SI units (A in s-1, l in m,
S in m2 C2)
units (A in s-1,
in Å, S in atomic units),
• S and E in atomic
units
천이 확율(Transition Probabilites)
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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
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10-15 s-1 for most extremely forbidden)
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• number of spontaneous transition /time/volume
• = N2 A21
• N2 = number density of atoms in upper level
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Absorption Transition Probability
• = need radiation
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propotional to the radiation intensity
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• number of absorptions/time/volume
• = N1 B12 I21
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• 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
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=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
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= Plank Function
TE : radiation field = BB radiation
= Plank Function
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여기서 m 는 매질의 굴절율로 보통 1 근처의 값
A21 = (N1/N2 B12 - B21) I21
=
아인슈타인 계수
• transition probability 는 원자의 성질이므
로 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
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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
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흡수와 방출 계수
• 아인슈타인 천이 확율은 각 천이가 일어날 확율을 결정하므로 스
펙트럼의 흡수와 방출선의 결과가 된다
• 고전 회전자 - 약한 선의 경우 흡수 계수 :선윤곽
• 질량 흡수 계수
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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
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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 발
생되는 모든 가능한 천이에 대한 진동자 세기
를 전부 합한 것(방출의 진동자 세기는 음으로
취급)은 원자나 이온의 전자 수와 같아진
다.==> Thomas-Reiche-Kuhn Sum Rule
발머선의 진동자 세기 ( Oscillator
Strength)
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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 ) -0.104
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 아인슈타
인 천이 확률계수들과의 관계
• 두 준위 사이의 흡수와
방출의 진동자 세기
(Oscillator Strength)는
두 준위의 통계 가중치
와 다음과 같이 관련된
다.
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• Oscillator Strength 는
천이 확율과의 관계
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천이 확율과 흡수 및 방출 계수의
관계
자연 선 윤곽의 형태 함수
stimulated emission, negative 흡수 포함
방출 계수
Kirchhoff's 법칙