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Ch5 Many-electron atoms • • • • • • Zeeman effect Exclusion principle Periodical table LS coupling jj coupling Two electron spectra (singlet and triplet) Some Words • • • • • • • Zeeman effect Normal & Anomalous Shell & subshell Pauli exclusion principle Electron configuration Fermions, Bosons Symmetric & antisymmetric • Subscript, superscript • Periodic table • Group, period • • • • • • • • • • Helium (氦), neon(氖) Lithium (锂), sodium(钠), Alkaline earth (碱土) lanthanide (rare earths) (稀土) Fluorine (氟), halogen(卤素) Nitrogen (氮) Hund’s rules Lande’s rule Inert (noble) gas Ion (离子) The Zeeman effect • The splitting of spectral lines due to the splitting of the energy terms of atoms in a magnetic field is called the Zeeman effect. • Selection rule: m 0, 1 • It is classified as the ordinary Zeeman effect and the anomalous Zeeman effect. – Ordinary: The splitting of states with pure orbital angular momentum (S=0), triplet, E ml B B – Anomalous: The splitting of states with the resultant angular momentum and non-zero spin. The Sodium Zeeman Effect Many electrons in an atom • The electrons interact not only with the nucleus but also among themselves. It is difficult to get the wave function. • Electron configuration: How do the electrons fill the shells and subshells? How to get its ground state? • Atoms with a close atomic number will have a smooth or regular variation in the atomic properties? Z=10, 11, 9? The Pauli exclusion principle • The Pauli exclusion principle requires that only one electron can be in a given state, which is labeled by quantum numbers that indicate the energy, orbital angular momentum, and spin angular momentum of an electron in that state (n,l,ml,ms). • This principle applies to fermions (next slide). • In the ground states of atoms, electrons occupy the lowest energy states available consistent with the exclusion principle. Fermions and Bosons • Particles of half-integral spin are often referred to as Fermi particles or fermions, such as protons, electrons and neutrons. They have wave functions that are antisymmetric to an exchange of any pair of them. A 12 [ a (1) b (2) a (2) b (1)] • Particles of 0 or an integral spin are referred to as Bose particles or Bosons, such as photons, α particles and helium atoms. They have wave functions that are symmetric to an exchange of 1 S 2 [ a (1) b (2) a (2) b (1)] any pair of them. Electron configuration (1) • Two basic rules determine the electron structure of many-electron atoms. – A system of particles is stable when its total energy is a minimum. – Only one electron can exist in any particular quantum state in an atom. • Atomic shells with different principle quantum number n are denoted by capital letters: – n=1(K), n=2(L), n=3(M), n=4(N), n=5(O), • Each subshell is identified by its principle quantum number n followed the letter corresponding to its orbital quantum number l, denoted by s,p,d,f, g,h and so on. Electron configuration (2) • Each subshell can contain a maximum of 2(2l+1) electrons, s(2), p(6), d(10)… • A superscript after the letter indicates the number of electrons in that subshell. For example, the electron configuration of sodium is expressed as 1s22s22p63s1. • The total number of electrons a shell can contain is equal to the number of electrons in all its closed subshells. The maximum number of electrons in the nth shell is equal to 2n2, n=1(2), n=2(8), n=3(18)… • With a given electron configuration, how to determine the atomic states? Total angular momentum • Each electron in an atom has a certain orbital angular momentum L and a certain spin angular momentum S, both of which contribute to the total angular momentum J of the atom. How? • In terms of vector addition, there are several ways to obtain the total angular momentum. The two main ways of coupling are LS coupling and jj coupling. – LS coupling holds for most atoms and for weak magnetic fields. – Jj coupling holds for the heavier atoms and for strong magnetic fields. LS coupling • When the orbit-spin coupling in an individual electron is smaller than the mutual interaction of l or s angular momenta, LS coupling takes places. • The orbital angular momenta Li of various electrons are coupled electrically into a single resultant L. So are the spin angular momenta Si into a single resultant S. Then the L & S interact magnetically to form the total angular momentum by orbit-spin coupling. L Li S Si J L S 2 S 1 LJ jj coupling • When the individual orbit-spin coupling is stronger than the mutual interaction of l or s angular momenta of different electron, j coupling takes places. • For heavy atoms, the nuclear charge becomes great enough to produce spin-orbit interaction within an electron. Each electron has a total angular momentum Ji resulting from the vector sum of Li and Si. Then Ji are Jcombined together to form the L S J J i i i i total angular momentum of the atom J. Atomic state : ( J i , J j ) J Example • Problem: Find the possible values of the total angular momentum number under LS coupling of two atomic electrons whose orbital quantum numbers are l1=1 and l2=2. • Solution: There are three ways to combine L1 and L2 to a single vector L, L=3,2,1. In the same way, there are two ways to combine S1 and S2, S=1,0. As shown in the table, the five possible values are J=0,1,2,3, and 4. S L 3 2 1 1 4,3,2 3,2,1 2,1,0 0 3 2 1 Atomic state of an atom • Atomic state: 2 S 1LJ • For a many-electron atom, capital letters are used to designate the entire electronic state of an atom according to its total orbital momentum quantum number L. L 0 1 2 3 4 5 6 … State S P D F G H I … • The spin angular momentum S is represented by the multiplicity of the state placed as a superscript. The total angular momentum quantum number is used as a subscript. For example, 2P3/2 S=1/2, L=1, J=3/2. Hund’s rules • The rules for determining the ground-state quantum numbers for LS coupling atoms are known as Hund’s rules. – For a given electron configuration, the state with maximum multiplicity (2S+1) lies lowest in energy. – For a given multiplicity (S), the state with the largest value of L lies lowest in in energy – For equivalent electrons, the states with the smallest J have the lowest energy in ‘normal’ multiplets, but otherwise the converse holds. “Normal” means that the subshells are less than half full. (np)(n’p) • l1 1, l2 1, L 2,1,0 s1 12 , s2 12 , S 1,0 1 S 0 ,1P1 ,1D2 ,3S1 ,3P0,1, 2 ,3D1, 2,3 S=0 1S 0 1P 1 1D 2 When n=n’, they are equivalent electrons. L+S=even number npn’p S=1 3S 1 3P 2,1,0 3D 3,2,1 (existing states) States in orange are forbidden. Example • Problem: Use Hund’s rules to find the groundstate quantum number of nitrogen. • Solutions: The electron configuration of the atom is 1s22s22p3. Three electrons in p subshell are permitted to have ms=1/2, and the maximum value of S is 3/2. Each electron has quantum number (2,1,ml,1/2). ml can only be =1, 0, -1, resulting in ML=0, L=0. Thus, L=0, S=3/2 & J=3/2 are the ground-state quantum numbers for 4 nitrogen, with an atomic S 3 state: 2 The periodic table (1) • When the elements are listed in order of atomic number, elements with similar chemical and physical properties recur at regular interval, known as the periodical law. • Elements with similar properties form the groups shown as vertical columns in the table. • The horizontal rows in the table are called periods. Across each period is a more or less steady transition from an active metal through less active metals and weakly active nonmetals to highly active nonmetals and finally to an inert gas. The periodic table (2) • For inert gases, atoms contain only closed shells. The atoms do not easily donate electrons to or accept electrons from other elements. • s-subshell elements form the first two column (groups) with the alkalis (ns1) and alkaline earths (ns2). Alkali metals have a single s electron in its outer shell, which can be easily lost. Elements in this group often form singly positive ions. • Transition metals are placed in the three rows in which the d subshell is filling. • The lanthanide (rare earths) series involves completing mainly the 5d and 4f subshells. Z=9, 10,11 • Z=10, neon, an inert gas, shell & subshells are full. It does not readily give up or to accept an electron, generally not combining with other elements to form compounds. Its boiling point are low and ionization energy is high. • Z=9, F, fluorine, halogens (卤素), p-subshell element np5, It is one electron shy of being full. It easily accepts an electron from other elements to form a compound, highly reactive. F-, Negative ions. • Z=11, Na, sodium, alkali, s-subshell element np1, The electron can be easily detached to other elements in a chemical reaction, highly reactive. Na+, Positive ions. Two-electron spectra • Selection rules under LS coupling are: S 0, L 0,1, J 0,1 • There is a division into singlet and triplet states. In singlet states, the two electrons are antiparallel (S=0). Triplet states have the two electrons in parallel (S=1). • According to the selection rules, no transition between singlet states and triplet states is allowed.