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W.N. Catford/P.H. Regan 1AMQ 83 Many Electron Atoms Spectroscopic Notation, Pauli Exculsion Principle Electron Screening, Shell and Sub-shell Structure Characteristic X-rays and Selection Rules. Optical Spectra of atoms and selection rules. Addition of Angular Momentum for Two electrons. K.Krane, Modern Physics, Chapter 8 Eisberg and Resnick, Quantum Physics, Chapters 9 and 10.

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W.N. Catford/P.H. Regan 1AMQ 84 Pauli Exclusion Principle and Spectroscopic Notation. A complete description of the state of an electron in an atom requires 4 quantum numbers, n, l, m l and m s. For each value of n, there are 2n 2 different combinations of the other quantum numbers which are allowed. The values of m l and m s have, at most, a very small effect on the energy of the states, so often only n and l are of interest for example for chemistry. Spectroscopic Notation uses letters to specify the l value, i.e. l = 0, 1, 2, 3, 4, 5…. have the designation s, p, d, f, g, h,…

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W.N. Catford/P.H. Regan 1AMQ 85 Spectroscopic Notation This notation has its origins in the early optical spectroscopy of atoms. The first few letters are named by the way the lines associated with them look in optical spectra. Thus: s---l = 0 related to lines that looked sharp p---l = 1 related to lines that are strong-Principal lines d—l = 2 related to lines that looked diffuse f---l = 3 related to lines that were narrow-Fine g h i k Then the others follow in alphabetical order

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W.N. Catford/P.H. Regan 1AMQ 86 Summary of the Quantum Numbers Specifying the Allowed States of Electrons in Atoms. Symbol Name n principal quantum number l orbital quantum number m l magnetic quantum number m s spin quantum number Symbol Allowed Values Physical Property n n=1,2,3,4,… size of orbit, r n =a 0 n 2 l l=0,1,2,3,…,(n-1) | L | & orbit shape m l -l, -l+1,…..,(l-1),+l projection of L on z m s +1/2 and -1/2 projection of S on z

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W.N. Catford/P.H. Regan 1AMQ 87 Atoms with Many Electrons Electrons in an atom fill the allowed states (a) beginning at the lowest energy (b) obeying the Exclusion Principle Electrons do not all collect in the lowest energy orbit (evident from chemistry). This experimental fact can be accounted for using the Pauli Exclusion Principle which states that “no two electrons in a single atom can have the same set of quantum numbers (n,l,m l,m s ).” (Wolfgang Pauli, 1929). For example the n=1 orbit (K-shell) can hold at most 2 electrons, n l m l m s / /2

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W.N. Catford/P.H. Regan 1AMQ 88 Pauli Principle Electrons do not all collect in the lowest energy orbit (evident from chemistry). This experimental fact can be accounted for using the Pauli Exclusion Principle which states that “no two electrons in a single atom can have the same set of quantum numbers (n,l,m l,m s ).” (Wolfgang Pauli, 1929). For example the n=2 orbit (L-shell) can hold at most 8 electrons, n l m l m s / / / / / /2

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W.N. Catford/P.H. Regan 1AMQ 89 Energies of Orbitals in Multielectron Atoms. The energies of outer “subshells” are affected by the presence of other electrons, particularly by screening of the nuclear charge. In high-Z atoms, the inner subshells are also affected by the electrons in the outer shells. Shell Structure of Atoms The n values dominates the determination of the radius of each subshell (as shown in the solutions to the Schrödinger equation). For the penetrating orbitals (s and p), the probability of being found at a small radius is balanced by some probability of also being found at a larger radius. We see that subshells with the same n but different l are grouped into “shells” with about the same average radius from the nucleus.

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W.N. Catford/P.H. Regan 1AMQ 90 n = 1n = 2 Mean values of radius, for various n values = 0 = 1 Hydrogen Atom

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W.N. Catford/P.H. Regan 1AMQ 91

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W.N. Catford/P.H. Regan 1AMQ 92 Note the notation-Principal Q.N followed by symbol for l

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W.N. Catford/P.H. Regan 1AMQ 93 Conventionally, the shells are designated by letter, eg, K shell, n=1 L shell, n=2 M shell, n=3 Subshells correspond to different l values within each shell. According to the Pauli Principle, each subshell has a maximum occupancy (number of electrons) which is given by, (2l+1) x 2 = number of possible m l values. × no. of m s values for each m l. Examples: s subshells (2 0 +1) × 2 = 2 electrons p subshells (2 ×1+1) × 2 = 6 electrons Periodic Table of the Elements. Inspection of the table of electronic structure shows that this determines the chemical properties of the elements (in particular, the number of valence electrons, i.e. number in outermost shell, is very important)

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W.N. Catford/P.H. Regan 1AMQ 94

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