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Chap 5: From Stars in Galactic Clusters to Stars in Fireworks

Chap 5: From Stars in Galactic Clusters to Stars in Fireworks. New Office hours Tuesdays 1:00 to 2:30 pm 2033 YH Exam on Friday Chapters 3, 4, 5 and 6.0-6.6 One page of notes Review Session Thursday Lec (1) 5:00 - 5:50 and Lec (2) 6:00 - 6:50.

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Chap 5: From Stars in Galactic Clusters to Stars in Fireworks

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  1. Chap 5: From Stars in Galactic Clusters to Stars in Fireworks New Office hours Tuesdays 1:00 to 2:30 pm 2033 YH Exam on Friday Chapters 3, 4, 5 and 6.0-6.6 One page of notes Review Session Thursday Lec (1) 5:00 - 5:50 and Lec (2) 6:00 - 6:50

  2. Chap 5: Hartree Single Electron Orbitals nlm(r,,)/ r12 -e -e r1 r2 +2e The energy eigenvalues for a nl level enl as calculated using theHartree Self-Consistent-Field (SCF) orbital approximation.The SCF method uses a potential that is averaged over electron-electron repulsions experienced by the electron in a one electron, hydrogen like orbital nlm(r,,). Consider a two electron atom or ion; He, Li+, etc; this case He V( r1,r2)~(-2e2)/r1 +(-2e2)/r2 + (e2)/r12 Attraction to the Nucleus V( r1)~(-e)(2e)/r1 and V(r2) ~(-e)(2e)/r2 V( r1,r2 )~ (e)(e)/r12 and electron-electron repulsion

  3. Chap 5: Hartree Single Electron Orbitals nlm(r,,)/ r12 -e -e r1 r2 +2e The energy eigenvalues for a nl level Enl as calculated using theHartree Self-Consistent-Field (SCF) orbital approximation.The SCF method uses a potential that is averaged over electron-electron repulsions experienced by the electron in the one electron orbital nlm(r,,). Consider a two electron atom or ion; He, Li+, etc; this case He V( r1,r2)~(-2e2)/r1 +(-2e2)/r2 + (e2)/r12 Attraction to the Nucleus V( r1)~(-e)(2e)/r1 and V(r2) ~(-e)(2e)/r2 V( r1,r2 )~ (e)(e)/r12 and electron-electron repulsion -e +eZeff r1

  4. Chap 5: Hartree Single Electron Radial Orbitals Rnl for He, Li+,etc Average V( r1, r2) over positions r2 for electron (1) and avg r1 for electron (2) <V( r1,r2)>r2 ~ Vneff(r1)~ - Zeff(n)e2/r1Z(n=1)=1.69 for He: where Zeff represents the avg over e-e repulsion.The new two electron potential is now V( r1,r2) ~ - Zeff(n)e2/r1 - Zeff(n)e2/r2Separable solution: (r1, r2) =1s(1)1s(2) The ground state solution of the two electronic Schrödinger Eq. is:1s1s(1,2)= 1s(1)1s(2)  Singlet state spins are anti parallel S=0, Ms =c0 1s2s(1,2)= 1s(1)2s(2)Triplet state spins are parallel S=1, Ms =0, ±1 The Hartree one electron orbital energy n=-Zeff(n)2/n2(2.18x10-18J) = Zeff(n)2/n2( 13.6 eV)The ground state Energy n1n2 obtained form the Calculations using the Hartree 1electron orbital to construct a 2 electron orbital n1n2 =[-Zeff(n1)2/(n1)2] + [- Zeff(n2)2/(n2)2] n1n2(1,2)= n1(1)n2(2)  -e +eZeff r1

  5. Chap 5: Energy Eigen Values due to screen and e-e repulsion IE21~21 IE20~20 Koopman Approximation IEnl= - nl Singlet configuration 1s1s =[-Zeff2/(1)2] + [- Zeff2/(1)2] Fig. 5-14, p. 189

  6. Chap 5: Energy Eigen Values due to sheilding and e-e repulsion IE21 IE20 Koopman Approximation IEnl= - nl Excited atom Singlet configuration 1s2s =[-Zeff2/(1)2] + [- Zeff2/(2)2] Fig. 5-14, p. 189

  7. Chap 5: Energy Eigen Values due to screen and e-e repulsion IE21 IE20 Koopman Approximation IEnl= - nl Excited atom Triplet Configuration 1s2s =[-Zeff2/(1)2] + [- Zeff2/(2)2] Fig. 5-14, p. 189

  8. Chap 5: Energy Eigen Values due to screening and e-e repulsion The effective nuclear charge Zeff results in an effective coulomb potential Vneff( r )~ Zeff(n)/r electrons in different nl energy levels (ns and np) have different energy eigen-values Enl: Ens<Enp<End Electrons closer to the nucleus “screen” outer electrons from the full Z of the nucleus and electron-electron repulsion further lowers the Zeff

  9. Chap 5: Periodic Table reflects the Electron Configuration; Atomic Properties Alkali Metals Rare Gases Noble Metals Transition Metals Halogens Alkaline earth Lanthanides Actinides

  10. Chap 5: Hartree Single Electron Radial Orbitals Rnl for Ar r2|Rnl|2

  11. Chap 5: multi-electron Atoms (Ar)- Pauli Exclusion Principle One electron per state with the set of quantum numbers {n, l, m, ms} K L Shell model maximum number of electrons 2n2 (nl ) sub-shells 2(2l+1) electrons K shell n=1: (1s) 2 L shell n=2: (2s 2p) 8M shell n=3: (3s 3p 3d) 18 M

  12. Chap 6. Electronic Energy Eigen Values as a function inter-nuclear distance Electronic Energy as a Function of R(the distance between nuclei): Born-Oppenheimer Potential Curve Fig. 6-CO, p. 211

  13. Chap 6: Molecular Hydrogen Ion; H2+ the simplest Molecule (Diatomic) Due to the lack of Spherical Symmetry the angular momentum quantum number (l) is no longer Good so L2 cannot be measured. However, due to the Cylindrical Symmetry Lz can be measured and (Lz)2 LzmRr,=mlhmRr, Lz= mlh m- Magnetic Quantum number is still Good and can be used to label the electronic eigen states = m2 m=0,±1,±2,±3,±4, Use the Aufbau concept to build up the electron configuration of Homonucleardiatomics + L +

  14. Chap 6: H2+ Electronic Eigen States Classified by =m2 and Symmetry of V Since (l) is no longer useful, =m2 is now used to classify the molecular electronic Eigen States and Eigen Values E(R ) R is the internuclear distance.The magnetic quantum number m= 0, ±1, ±2, ±3labels the eigen values and eigen functions electronic states as well as the inversion symmetry eigen values: = +1(g, even, gerade) and  = -1(u, odd, ungerade) inversion symmetry In order of increasing energy (number of nodes) for the molecular orbital eigen states: 1g ,1u 2g2u1u3g1g3u

  15. Chap 5: multi-electron Atoms (Ar)- Pauli Exclusion Principle II III I V=∞ V=0 V=∞ -x 0 L +x Potential without inversion symmetry V(x)≠V(-x) Y1~ sin(πx/d) Y2~ sin(2πx/d) Potential with inversion symmetry V(x)=V(-x) II III I Y1~ cos(πx/d) even, g Y2~ sin(2πx/d) odd, u iYg/u=eYg/ue=±1 for g/u V=∞ V=0 V=∞ -x x=-d/2 x=0 +x x=d/2

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