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1AMQ, Part IV Many Electron Atoms 4 Lectures

1AMQ, Part IV Many Electron Atoms 4 Lectures. 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.

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1AMQ, Part IV Many Electron Atoms 4 Lectures

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  1. 1AMQ, Part IV Many Electron Atoms4 Lectures • 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. 1AMQ P.H. Regan & W.N.Catford

  2. Spectroscopic Notation, uses letters to specify the l value, ie. l= 0, 1, 2, 3, 4, 5…. has the designation s, p, d, f, g, h,…... Pauli Exclusion Principle and Spectroscopic Notation. A complete description of the state of an electron in an atom requires 4 quantum numbers, n, l, ml and ms. For each value of n, there are 2n2 different combinations of the other quantum numbers which are allowed. The values of ml and ms have, at most, a very small effect of the energy of the states, so often only n and l are of interest for chemistry. 1AMQ P.H. Regan & W.N.Catford

  3. Symbol Name n principal quantum number l orbital quantum number ml magnetic quantum number ms spin quantum number Symbol Allowed Values Physical Property nn=1,2,3,4,… size of orbit, rn=a0n2 l l=0,1,2,3,…,(n-1) AM & orbit shape ml -l, -l+1,…..,(l-1),+l projection of L ms +1/2 and -1/2 projection of s. Summary of the Allowed Quantum Numbers Specifying the Allowed Quantum Numbers of an Atoms. 1AMQ P.H. Regan & W.N.Catford

  4. Atoms with Many Electrons 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,ml,ms).’ (Wolfgang Pauli, 1929). For example the n=1 orbit (K-shell) can hold at most 2 electrons, n l ml ms 1 0 0 +1/2 1 0 0 -1/2 Electrons in an atom fill the allowed states (a) beginning at the lowest energy (b) obeying the Exclusion Principle 1AMQ P.H. Regan & W.N.Catford

  5. The energies of the subshells are affected by the presence of other electrons, particularly by the screening of the nuclear charge. In high-Z atoms, the inner subshells are also affected by the electrons in the higher subshells. Energies of Orbitals in Multielectron Atoms. Shell Structure of Atoms. The n values dominates the determination of the radius of each subshell (as shown in the solutions to the Schrodinger 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. Hence, subshells with the same n are grouped into `shells’ with about the same average radius from the nucleus. 1AMQ P.H. Regan & W.N.Catford

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  7. 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 mlvalues x x no. of ms values for each ml . Example: s subshells (2.0+1).2 =2 electrons p subshells (2.1+1).2=6 electrons Periodic Table of Elements. Inspection of the table of electronic structure show that this determines the chemical properties of that element (in particular, the number of valence electrons, ie. number in outermost shell is very important) 1AMQ P.H. Regan & W.N.Catford

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  9. n=4, N n=3, M n=2, L n=1, K L series K series M series La L series Ka K series Characteristic X-rays and Selection Rules. Characteristic X-rays: are emitted by atoms when electrons make transitions between inner shells (note a vacancy must be created before this can happen). An incident photon, electron or alpha-particle can knock out an e- from the atom. Electrons at higher excitation energies cascade down to fill the vacancy. A vacancy in the K shell can be filled with an e- from the L shell (a Katransition) or the M shell (Kb) etc. 1AMQ P.H. Regan & W.N.Catford

  10. We can calculate the Kaenergy for each element approximately. Consider an L-shell electron, about to fill a K-shell vacancy. K +Ze L Moseley’s Law Approximately, the L electron orbits an `effective’ charge of +(Z-1)e. To allow for penetrating orbits, say +(Z-b)e, where we expect b to be approximately equal to 1. We can use Bohr theory, for an electron orbiting a charge +(Z-b)e, to estimate the X-ray energies. This compares to experiment to within 0.5% with a value for the intercept, b of very close to 1. 1AMQ P.H. Regan & W.N.Catford

  11. Fine Structure of X-ray Spectra The subshells are split in energy by the spin-orbit interaction, into (l+1/2) and (l-1/2) levels. Note that not all the energetically allowed transitions are observed. The notation LI, LII, LIII etc. is used for these levels Only transitions which obey the selection rules are observed to occur. The selection rules are related to the underlying physics of (a) the spatial properties (symmetry) of the charge oscillations that produce transitions and (b) the angular momentum (spin) carried away by the photon itself (at least 1h). 1AMQ P.H. Regan & W.N.Catford

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  14. Optical Spectra of Atoms. Example: Sodium, Z=11, thus the electron configuration in the atomic ground state is 1s22s22p63s1 The first excited statehas 1s22s22p63p1 The lowest energy levels of the atom are due to excitations of electrons between outer levels (where the smallest energy gaps and the first vacancies exist). For electrons in the n=3 electron orbitals, the nuclear charge (Z=+11e) is screened by the inner 10 electrons from the n=1 and n=2 shell (Q=-10e). Thus the energy is similar to that of the n=3 Bohr orbit for hydrogen. Screening effects give an energy shift of the levels, which depends on the l of the orbital. 1AMQ P.H. Regan & W.N.Catford

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  16. Optical photonsare emitted when valence (outer) electrons make transitions between energy levels. Eg. 3p->3s , DE=2.10 eV (3.37 x10-19 J), thus l=(hc/DE) = 589 nm (ie. yellow street lamps). 1AMQ P.H. Regan & W.N.Catford

  17. Not all of the energetically possible transitions are allowed. The selection rules are (as in x-rays) The transition 3p->3s shows fine structure giving the sodium D lines, ie. D2, l=589.0 nm and D1, l=589.6 nm. If an external B field is applied, the Zeeman effectgives a further splitting of the levels. The Zeeman splitting is generally smaller (for typical laboratory size eternal fields) than the fine structure (which increases as approx. Z4). A further selection rule is observed, namely, • Summary: • Optical spectra arise from transitions of valence e-s. • Optical photon energies are approx. several eV. • Inner e-s are left undisturbed by optical transitions. • When more energy is injected, and inner electrons • are removed, x-rays (with energies ~keV) are • emitted when these vacancies are filled 1AMQ P.H. Regan & W.N.Catford

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  19. The Helium Atom: 2 Active Electrons While Sodium has, for low excitations, only one active electron for optical transitions, helium and other elements can have more. L2 L1 1AMQ P.H. Regan & W.N.Catford

  20. n l ml ms 1 0 0 +1/2 1 0 0 -1/2 Ground State of Helium: two electrons, both with l=0 (same (1s) orbital), gives configuration of 1s2 Using the coupling rules for l and s: i) Lmax = 0+0 = 0, Smax=1/2+1/2=1 ii) Lmin = |0-0| = 0, Smin=|1/2-1/2|=0 iii) L=0 (and thus ML=0), S=1 or 0 iv) ML=0, MS=-1,0,+1 are the only possibilities NB. Use capital `L’ and `S’ for > 1 electron. The Pauli Principle means that S=1 is not allowed, since would need ms=+1/2 for both e-s for S=1. ie. for 1s2, can only have (L=0, S=0), ie. ground state of helium has L=0, S=0, hence, J=0 1AMQ P.H. Regan & W.N.Catford

  21. n l ml ms 1 0 0 +1/2, -1/2 2 0 0 +1/2,-1/2 Excited State of the Helium Atom. The first excited state(above the ground state) has configuration 1s12s1. As for the ground state, L=0, S=0 or 1 from coupling of L1 and L2 vectors etc. Allowed states are (L=0, S=0) and (L=0, S=1). For S=0, only one state (singlet state) For S=1, three states (ie. MS=-1,0,+1) (triplet state) Hund’s rules lead to Etriplet<Esinglet. (This is related to the PEP, aligned electrons tend to `repel’ each other). 1AMQ P.H. Regan & W.N.Catford

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  24. 1s12s1, singlet and triplet 1s2 (singlet) Transitions in Helium Experimentally, the selection rules He are DL=0,+-1 and DS=0 (ie. no `spin-flips’). Singlet (S=0) and triplet (S=1) states CAN NOT BE CONNECTED by transitions. The triplet 1s12s1 state is rather long lived or metastable (=`isomeric’). This is due to the fact that no decay can occur to the ground state via single photon emission (DS=1 is forbidden). It subsequently decays by transferring kinetic energy in collisions. The singlet 1s12s1 state is also isomeric since a transition between 2s and 1s states would violate the Dl=+-1 selection rule for individual l values. 1AMQ P.H. Regan & W.N.Catford

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