CHAPTER 8 Atomic Physics 8.1 Atomic Structure and the Periodic Table 8.2 Total Angular Momentum 8.3 Anomalous Zeeman Effect
8.1: Atomic Structure and Periodic Table What would happen if there are more than one electron? a nucleus with charge +2e attracting two electrons the two electrons repelling one another Can not solve problems exactly with the Schrödinger equation because of the complex potential interactions Can understand experimental results by applying the boundary conditions and selection rules, without computing the wave functions of many-electron atoms. (Two rules) Electrons occupy the lowest levels first. Pauli exclusion principle
Pauli Exclusion Principle To understand atomic spectroscopic data for optical frequencies, Wolfgang Pauli proposed an exclusion principle: No two electrons in an atom may have the same set of quantum numbers (n, l, m l, m s ). It applies to all particles of half-integer spin (fermions), including fermion particles in the nucleus; neutrons and protons. The periodic table can be understood by two rules: 1) The electrons in an atom tend to occupy the lowest energy levels available to them. 2) Only one electron can be in a state with a given (complete) set of quantum numbers (Pauli exclusion principle).
Atomic Structures: Electron shells and subshells The principle quantum number also has letter codes. n = 1 2 3 4... Letter = K L M N n = shells (K shell, L shell, ) nl = subshells (1s, 2p, 3d, ) 적은 l 값을가진전자는더큰 l 값을가진전자에비해훨씬덜가려지는위치인핵과좀더가까운위치에서발견될확률이높다. 작은 l 값을가진전자의총에너지는더낮아진다. ( 즉, 결합에너지가커진다.) 각각의껍질에있는전자는 l 이커짐에따라에너지도커진다. ( 즉, 결합에너지가줄어든다 ) 리드베리단위 (Rydberg unit) 로나타낸전자의속박에너지 (1 Ry = 13.6eV = 수소의바닥상태에너지 )
Atomic Structures: Electron shells and subshells The lower l values have more elliptical orbits than the higher l values. Electrons with higher l values are more shielded from the nuclear charge Electrons lie higher in energy than those with lower l values the shielding is so pronounced that the 4s fills before 3d even though it has a larger n
Atomic Structures: Electron shells and subshells (Ex) Hydrogen ( 1 H): (n, l, m l, m s ) = (1, 0, 0, ±½) in ground state In the absence of a magnetic field, the state m s = ½ is degenerate with the m s = ½ state. 1S 1 (Ex) Helium ( 2 He): (1, 0, 0, ½) for the first electron; (1, 0, 0, ½) for the second Electrons have anti-aligned (m s = +½ and m s = ½) spins as being paired. It supports Pauli exclusion principle. 1S 2 (Ex) Lithium ( 3 Li): (1, 0, 0, ½) for the first electron; (1, 0, 0, ½) for the second; (2, 0, 0, +½ or -½) for the third. 1S 2 2S 1 (Ex) Sodium ( 11 Na) 1s 2 2s 2 2p 6 3s 1
Hund s Rule - 일반적으로같은버금껍질안에들어있는전자들은가능한한짝을이루지않으려함 ( 즉, 평행한스핀을가지려함 ) 훈트의규칙 ( Hund s rule ) - 원자내의전자들사이의상호척력 (repulsion) 이훈트의규칙을나타나게하는원인 평행한스핀을갖는전자들은그들이짝을이루었을때보다공간적으로더멀리떨어져있게되며, 더낮은에너지를가지는이러한배열이좀더안정한상태가됨
How many electrons may be in each subshell? subshell Total For each m l : two values of m s 2 Order of Electron Filling in Atomic Subshells For each l: (2l + 1) values m l 2(2l + 1) Recall: l = 0 1 2 3 4 5 letter = s p d f g h l = 0, (s state) can have two electrons l = 1, (p state) can have six electrons.. N n 1 max (2 1) 2 0 2 n K L M N O 2 8 18 32 50 2
The Periodic Table Group ( 족 ) Period ( 주기 ) Transition elements ( 전이원소 ) Halogens ( 할로겐 ) Inert gas ( 불활성기체 ) Alkali metals ( 알카리금속 ) Lanthanides ( 란탄족,) or Rare earth ( 희토류 ) Actinides ( 악티늄족 )
The Periodic Table - 원소들의체계화 - Dmitri Mendeleev 는 1869년에 periodic law를체계화하였으며, 현대적인설명을다음과같이하였음. 원소들을원자번호순으로배열하면일정한간격으로비슷한화학적물리적성질을가진원소들이되풀이로나타난다. - 주기율표는원소들을원자번호에따라배열한일련의세로줄들로되어있고, 비슷한성질을가지는원소들이가로줄에함께묶이도록배열
Groups and Periods in Periodic Table Groups ( 족 ): Vertical columns Same number of electrons in an l orbit Can form similar chemical bonds Periods ( 주기 ): Horizontal rows Correspond to filling of the subshells
Groups ( 족 ) 족 (group) 주기율표의세로줄로비슷한성질을갖는원소들을묶어놓음 1족 알칼리금속으로부드럽고, 낮은녹는점을가지고, 활성이매우큼 Ex) Li, Na, K 수소는물리적으로금속이아니지만, 화학적으로는활성이큰금속과같이행동 7족 할로겐원소로휘발성비금속, 산화작용제로서큰활성을갖음 Ex) F, Cl, Br 8족 불활성기체로반응성이매우낮음 Ex) He, Ne, Ar 대부분의원소들은금속이다.
Period ( 주기 ) 주기 (period) 주기율표의가로줄로비슷한성질 ( ) 을갖는원소들을묶어놓음 처음의세주기는각원소들이아래의보다긴주기의원소들과가장밀접한관련을가지고배열할수있도록중간이띄어져있음각주기를가로질러보면처음에는활성이강한금속, 그다음은활성이약한금속, 활성이약한비금속그리고활성이아주큰비금속, 마지막으로불활성기체순으로원소의성질이변함 ex) 알칼리금속중에서원자번호가커지면화학적활성이증가하고, 할로겐에서는그반대가됨
The Periodic Table Inert Gases: Last group of the periodic table Closed p subshell except helium Zero net spin and large ionization energy Their atoms interact weakly with each other Alkalis: Single s electron outside an inner core Easily form positive ions with a charge +1e Lowest ionization energies Electrical conductivity is relatively good Alkaline Earths: Two s electrons in outer subshell Largest atomic radii High electrical conductivity
The Periodic Table Halogens: Need one more electron to fill outermost subshell Form strong ionic bonds with the alkalis More stable as the p subshell is filled Transition Metals: Three rows of elements in which the 3d, 4d, and 5d are being filled Properties primarily determined by s electrons, rather than by the d subshell being filled Have d-shell electrons with unpaired spins As the d subshell is filled, the magnetic moments, and the tendency for neighboring atoms to align spins are reduced Lanthanides (rare earths): Have the outside 6s 2 subshell completed As occurs in the 3d subshell, the electrons in the 4f subshell have unpaired electrons that align themselves The large orbital angular momentum contributes to the large ferromagnetic effects Actinides: Inner subshells are being filled while the 7s 2 subshell is complete Difficult to obtain chemical data because they are all radioactive Have longer half-lives
The Period Table
Ionization Energies 불활성기체 : 가장높은이온화에너지 알칼리금속 : 가장낮은이온화에너지 알칼리금속은최외각전자를잃어버리려는경향을보이는반면, 할로겐원자들은핵의전하가불완전하게차폐되어있기때문에, 전자를하나더얻어서바깥버금껍질 (subshell) 을완전하게채우려는경향이있음 원자가커질수록, 바깥쪽전자의위치는핵으로부터더멀어지고전자를붙잡아두려는힘은더약해짐 주기율표의주어진어느한족에서아래로갈수록이온화에너지는작아짐 어느한주어진주기에서내부에차폐하는전자들의수는일정한데비해오른쪽으로갈수록핵의전하가점점증가하기때문에주기의오른쪽으로갈수록이온화에너지가증가
Atomic Radii 관측된원자들간의거리를근거로보면, 원자마다크기가큰차이를보이지않음. ex) 90 개이상의전자를가진무거운원자도수소원자반지름의약 3 배밖에되지않고, 가장크다는 cesium 원자도수소의 4.4 배밖에되지않음. 크기변화에도주기성이있음. 주기성의원인은이온화에너지의경우와비슷하게, 외부전하가내부의전자들에의해차폐정도가커질수록외각전자의결합에너지는작아지고, 핵으로부터평균적으로더멀어짐. Atomic Radius Ionization Energies
8.2: Total Angular Momentum For an atom: Orbital angular momentum Spin angular momentum Total angular momentum L, L z, S, S z, J, and J z are quantized.
Total Angular Momentum (Case 1) No external magnetic field: Only J z can be known because the uncertainty principle forbids J x or J y from being known at the same time as J z (Case 2) With an internal magnetic field: will precess about
Single Electron Atoms (H, Alkali atoms) Total angular momentum is defined by All the quantum numbers are quantized. The total angular momentum quantum number, j for the single electron can only have the values:
Quantization of Total Angular Momentum
Spin-Orbit Coupling in a single-e atom An effect of interaction between the spin momentum and the orbital angular momentum of an electron is called spin-orbit coupling. (a) 핵이정지하고있는좌표계에서바라볼때 전자가핵주위를돌고있다. (a) 전자가정지하고있는좌표계에서바라볼때 핵이전자주위를돌고있다. 전자가느끼는핵에의한자기장은궤도면위로수직인방향이다. 이자기장과전자스핀과의상호작용에의해스핀 - 궤도결합현상이일어난다.
Term Symbols and Multiplicity for a single-electron atoms To represent a single-electron atoms with the total angular momentum J for a given orbital angular momentum L, 2S 1 n L J term symbols ( or, spectroscopic symbols) Where, 5 P, P, P 3 3 3 0,1,2 0,1,2 3 P, 3 P, 3 D 1 3 3 0 2 1 (2S + 1) Multiplicity : Possible number of J values for a given L. If L > S; J goes from (L S) to (L + S) (2S + 1) possible J values If L < S; possible J values are fewer than (2S + 1) For a single-electron atoms, S = ½ (2S + 1) = 2 Doublet state J = L + ½, L ½
Spin-Orbit Coupling Potential energy of the spin dipole moment under a magnetic field U m = - μbcosθ = z ±μ B B In an atom, the B field comes from the orbital motion of proton. U m = ±μ B B : Potential energy split due to spin-orbit coupling. 2P 2 2 P 1/2 2 2 P 3/2 1S 궤도 - 스핀결합의결과로수소 2p 상태가 E 만큼떨어져있는두상태로갈라진다. 2p 1s 전이선이단일선이아닌이중선 ( 가깝게붙어있는두개의선 ) 으로나타난다.
Spin-Orbit Coupling 2 2 P 3/2 2 2 P 1/2
Term Symbols and Multiplicity for a single-electron atoms
Single-electron atom allowed transition Now the selection rules for a single-electron atom become n = anything; l = ±1; j = 0, ±1; m j = 0, ±1 Hydrogen energy-level diagram for n = 2 and n = 3 with the spin-orbit splitting
Many-Electron Atoms 일반적으로같은버금껍질안에들어있는전자들은가능한한짝을이루지않으려함 ( 즉, 평행한스핀을가지려함 ) Hund s rules Hund s rules: the order in which a given subshell is filled 1) The total spin angular momentum S should be maximized to the extent possible without violating the Pauli exclusion principle. 1) Insofar as rule 1 is not violated, L should also be maximized. 2) For atoms having subshells less than half full, J should be minimized.
Many-Electron Atoms For two-electron atoms, 1 and 2, total angular momentum is There are two schemes: LS coupling and jj coupling to combine four angular momenta to form J. LS coupling : jj coupling : L L1 L2 J L S S S S 1 2 J1 L1 S1 J J1 J2 J L S 2 2 2
LS Coupling (Russel-Saunders coupling) This is used for most atoms when the magnetic field is weak. First consider the case of two electrons in a single subshell. Total spin angular momentum quantum number may be S = 0 or 1 depending on whether the spins are antiparallel or parallel. Term symbols for two electron atoms, 2S 1 n L J 1 3 3 3 P0,3 P2,3 D1 5 P, P, P 3 3 3 0,1,2 0,1,2 There are two multiplicity: S = 0 Singlet states (since J = L) S = 1 Triplet states (since J = L + 1, L, L - 1)
Term Symbols and Energy-level Splitting
Allowed Transition for the LS Coupling scheme Singlets Triplets There are separated energy levels according to S = 0 or 1 Allowed transitions must have S = 0 No allowed (forbidden) transitions are possible between singlet and triplet states In general, the allowed transitions for the LS coupling scheme are L = ±1 S = 0 J = 0, ±1 (J = 0 J = 0 is forbidden) A magnesium atom excited to the 3p triplet state has no lower triplet state to which it can decay. It is called metastable, because it lives for such a long time on the atomic scale.
jj Coupling It is for the heavier elements, where the nuclear charge causes the spin-orbit interactions to be as strong as the force between the individual and. J1 L1 S 1 J J1 J2 J L S 2 2 2 J i J i
8.3: Anomalous Zeeman Effect The spectral lines emitted by atoms in a magnetic field split into multiple energy levels. It is called the Zeeman effect. If line is split into three lines Normal Zeeman effect (Chapter 7) If line splits into more lines Anomalous Zeeman effect (Chapter 8) (Remind) Normal Zeeman effect (A spectral line is split into three lines): An electron as an orbiting circular current loop, has Magnetic moment In an external field, the dipole will experience a torque which tends to align it in the field. It has a potential energy of B Bohr magneton
(Note) Normal Zeeman effect and Selection rules - Concepts of Modern Pgysics, Chapter 6.10, Beiser- 자기장내에서는특정원자상태의에너지는 n 값뿐만아니라 m l 의값에도관계한다. 복사전이의경우, m l = 0, 1 로제한되기때문에서로다른 l 를가진갖는두상태에서생기는 spectrum 선은 3 개의진동수만갖게된다. 진동수가 0 인 spectrum 선이다음과같은진동수를갖는 3 개의성분으로분리됨. l = ±1 m l = 0, ±1 1 2 3 0 0 0 B B B h B h 0 0 e 4 m e 4 m B B Normal Zeeman effect
8.3: Anomalous Zeeman Effect Now, consider the LS coupling to form Total angular momentum, The magnetic moment depends on m ( J,,0, J) Orbital contribution and Spin magnetic moment The 2J + 1 degeneracy for a given total angular momentum state J is removed by the effect of the. If the is small compared to internal magnetic field, then and precess about while precesses slowly about. J
Anomalous Zeeman Effect The total magnetic moment is μ B is the Bohr magneton and it is called the Landé g factor The magnetic total angular momentum numbers, m J from J to J in integral steps. splits each state J into (2J + 1) equally spaced levels separated E = V. For photon transitions between energy levels m J = ±1, 0 but is forbidden when J = 0.
Anomalous Zeeman Effect Allowed transitions (Selection Rule) n = anything L = ±1 S = 0 J = 0, ±1 (J = 0 J = 0 is forbidden) m J = 0, ±1 (J = 0 J = 0 is forbidden)