Many-electron-atoms.ppt
- 2. Identical Particles Identical Particles Example: all electrons have the same mass, electrical charge, magnetic properties… Therefore, we cannot distinguish one electron from another. Quantum theory then restricts the kinds of states for electrons
- 3. Two Families of Fundamental Particles Fermions Enrico Fermi 1901-1954 Bosons Satyendranath Bose 1894-1974
- 4. Fermions electrons, neutrinos, protons, quarks... At most one particle per quantum state Bosons photons, W, Z bosons, gluons,4He (nucleus and atom), 16O(nucleus), 85Rb (atom),… Possibly many particles in the same quantum state
- 5. Fermions versus Bosons Quantum theory tells us that angular momentum (amount of spin) = n (h/2) Experimental fact: Fermions have n = 1, 3, 5, … Bosons have n = 0, 2, 4, …
- 6. Many Particle Systems Schrodinger Equation for a many particle system with xi being the coordinate of particle i (r if 3D) the Hamiltonian has kinetic and potential energy if only two particles and V just depends on separation then can treat as “one” particle and use reduced mass in QM, H does not depend on the labeling. And so if any i j and j i, you get the same observables or state this as (for 2 particles) H(1,2)=H(2,1) t x x x i x x x H n n ) , ( ) , ( 2 1 2 1 ) , ( ) 2 1 2 1 ( 2 1 2 2 2 2 2 1 2 n x x x V x m x m H
- 7. Exchange Operator Using 2 particle as example. Use 1,2 for both space coordinates and quantum states (like spin) the exchange operator is We can then define symmetric and antisymmetric states for 2 and 3 particle systems 1 : ) 2 , 1 ( ) 1 , 2 ( )) 2 , 1 ( ( ) 1 , 2 ( ) 2 , 1 ( 12 12 12 12 s eigenvalue u u P u P P u u P )) 1 , 2 ( ) 2 , 1 ( ( 1 )) 1 , 2 ( ) 2 , 1 ( ( 1 N N A S )) 2 , 3 , 1 ( ) 2 , 1 , 3 ( ) 1 , 2 , 3 ( ) 1 , 3 , 2 ( ) 3 , 1 , 2 ( ) 3 , 2 , 1 ( ( 1 )) 2 , 3 , 1 ( ) 2 , 1 , 3 ( ) 1 , 2 , 3 ( ) 1 , 3 , 2 ( ) 3 , 1 , 2 ( ) 3 , 2 , 1 ( ( 1 N N A S
- 8. Particles in a Spin State If we have spatial wave function for 2 particles in a box which are either symmetric or antisymmetric there is also the spin. assume s= ½ as Fermions need totally antisymmetric: spatial Asymmetric + spin Symmetric (S=1) spatial symmetric + spin Antisymmetric (S=0) if Boson, need totally symmetric and so symmetric*symmetric or antisymmetric*antisymmetric ) ( 2 1 ) 0 ( ) 1 ( ) ( 2 1 ) 0 ( ) 1 ( z A z s z s z s S S S S S=1 S=0
- 9. Spectroscopic Notation n 2S + 1Lj This code is called spectroscopic or term symbols. For two electrons we have singlet states (S = 0) and triplet states (S = 1), which refer to the multiplicity 2S + 1.
- 10. Term Symbols Convenient to introduce shorthand notation to label energy levels that occurs in the LS coupling regime. Each level is labeled by L, S and J: 2S+1LJ ◦ L = 0 => S ◦ L = 1 => P ◦ L = 2 =>D ◦ L = 3 =>F If S = 1/2, L =1 => J = 3/2 or 1/2. This gives rise to two energy levels or terms, 2P3/2 and 2P1/2 2S + 1 is the multiplicity. Indicates the degeneracy of the level due to spin. ◦ If S = 0 => multiplicity is 1: singlet term. ◦ If S = 1/2 => multiplicity is 2: doublet term. ◦ If S = 1 => multiplicity is 3: triplet term.
- 12. Many-Electron Atoms Consider two electrons outside a closed shell. Label electrons 1 and 2. Then J = L1 + L2 + S1 + S2 There are two schemes, called LS coupling and jj coupling, for combining the four angular momenta to form J. The decision on which scheme to use depends on the relative strength of the interactions. JJ coupling predominates for heavier elements.
- 13. LS, or Russell-Saunders, Coupling The LS coupling scheme is used for most atoms when the magnetic field is weak. L = L1 + L2 S = S1 + S2 The L and S combine to form the total angular momentum: J = L + S Use upper case for many - electron atoms.
- 14. Allowed Transitions for LS Coupling DL = ± 1 DS = 0 DJ = 0, ± 1 (J = 0 J = 0 is forbidden)
- 15. jj Coupling J1 = L1 + S1 J2 = L2 + S2 J = S Ji Allowed Transitions for jj Coupling Dj1 = 0,± 1 Dj2 = 0 DJ = 0, ± 1 (J = 0 J = 0 is forbidden)
- 16. Helium Atom Spectra One electron is presumed to be in the ground state, the 1s state. An electron in an upper state can have spin anti parallel to the ground state electron (S=0, singlet state, parahelium) or parallel to the ground state electron (S=1, triplet state, orthohelium). The orthohelium states are lower in energy than the parahelium states. Energy difference between ground state and lowest excited is comparatively large due to tight binding of closed shell electrons. Lowest Ground state (13S) for triplet corresponding to lowest singlet state is absent.
- 17. DL=±1, DS = 0, DJ =0, ± 1
- 18. Sodium Atom Spectrum Selection Rule: Dl=±1 Various Transitions: Principal Series (np 3s, n=3,4,5,…) Sharp Series (ns 3p, n=3,4,5,…) Diffuse Series (nd 3p, n=3,4,5,…) Fundamental Series (nf 3d, n=3,4,5,…)
- 20. Sodium Atom Fine Structure Selection Rule: Dl=0, ±1