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Alkali Elements The alkali elements are the elements in column one of the periodic table. H (Hydrogen), Li (Lithium), Na (Sodium), K (potassium), Rb (rubidium) … These elements are characterized by a single electron in the outermost s-subshell. This ns1 electron is the first electron in a new shell. Configuration Hydrogen: Lithium: Sodium: Potassium: 1s1 2 1s 2s1 1s22s22p63s1 2 2 1s 2s 2p63s23p64s1 Abbreviated Configuration 1s1 [He]2s1 [Ne]3s1 [Ar]4s1 “Nobel gas” core electrons “Optical” electron. Spherically symmetric Weakly bonded. Strongly bonded. Easy to excite and Shields nuclear charge. ionize. Do not participate in low energy excitation processes. Alkali Elements: Excited States We consider now excited states of the alkali elements in which the optically active electron is excited into higher subshells. e.g. for Sodium (Na): 1s22s22p63s1 1s22s22p63p1 1s22s22p64s1 1s22s22p63d1 … 2 1s 2s22p66f1 … For large n, (i.e. the optical electron is in “far-out” shells), the nuclear shielding is effectively complete. Z(r)=1 and the energies correspond to that of hydrogen. Figure constructed from optical line spectra. Alkali Elements: Spin-Orbit Coupling Close inspection of the optical transitions between these states reveals “multiplet” lines in the spectrum. Each state is split into two due to spin-orbit coupling of the optically active electron. Reminder: Spin-orbit coupling is the interaction of the magnetic moment due the electrons orbital motion with the magnetic moment due to electron spin. The spin orbit coupling energy is determined by the spin quantum number s, the orbital angular momentum number l, and the total angular momentum quantum number j. A constant for a given subshell (n,l). s is always 1/2 for one electron. l is given by the subshell that the optical electron is in. j determines how spin and orbital angular momentum are oriented relative to each other. e.g. Sodium Doublet Emission: 1s22s22p63p1 ! 1s22s22p63s1 Work out the spin coupling for the optical electron in both configurations: 3p1: " l=1, s=1/2 " Term symbol: j = 3/2, 1/2 2P , 2P 3/2 1/2 2P : 3/2 <ESO> = K [ 15/4 - 2 - 3/4 ] = +K 2P : 1/2 <ESO> = K [ 3/4 - 2 - 3/4 ] = -2K 3s1: " l=0, s=1/2 " Term symbol: 2S : 1/2 3p1 2P 3/2 2P 1/2 j = 1/2 2S 1/2 <ESO> = K [ 3/4 - 0 - 3/4 ] = 0 3s1 2S 1/2 Spin Orbit : Alkali Element Dependence Energy splitting into 2P1/2 and 2P3/2 as a function of the element. Spin orbit splitting increases massively with Z [which in turn affects V(r) and the integral over dV(r)/dr in the spin-orbit formula]. Element Li Na K Rb Cs Subshell 2p1 3p1 4p1 5p1 6p1 Spin-orbit Splitting (10-4 eV) 0.42 21 72 295 687 Atoms with several optically active electrons i.e. atoms with several electrons in partially filled subshells. e.g. carbon (ground state): e.g. nitrogen (ground state): e.g. beryllium (excited state): e.g. uranium (ground state): 1s22s22p2 1s22s22p3 1s22s12p1 [Rn]7s26d15f3 Most of the energetics of these atoms is well described by the Hartree model; however, in detail (e.g. in high-resolution spectroscopy), spin-orbit coupling and the residual coulomb interaction are important. Residual Coulomb Interaction: The Coulomb interaction that is not captured by the effective Hartree potential (remember: the Hartree method is an approximation in that it replaces the instantaneous, pairwise interaction of moving electrons with an averaged interaction potential !). The residual Coulomb interaction is the difference between the instantaneous Coulomb interaction between electrons in reality and the averaged Coulomb interaction in Hartree theory. Spin-Orbit Coupling: As before, the magnetic interaction between orbital- and spinmagnetic moments. However, now we have multiple electrons to consider (each with an orbital and an spin moment). Coupling of Angular Momenta Residual Coulomb Interaction: In a partially filled shell, the potential is no longer spherical symmetric, this has an effect on the relative motion of two optically active electrons in a partially filled shell. Tendency (i.e. energetic preference) to align orbital and spin angular momenta, so as to maximize total orbital angular momentum L’ and total spin momentum S’. LS-Coupling (elements with small Z; spin-orbit interaction << residual coulomb): First couple all single electron L together into L’, and all the single electron S into S’. Then couple L’ and S’ to get total angular momentum J’. JJ-Coupling (elements with large Z; spin-orbit interaction >> residual coulomb): For every electron, couple L and S to get J; then couple the single-electron J together into total angular momentum J’ e.g. Helium (two optical electrons) s’=0 1s12p1 1s12s1 Helium Ground State: 1s2 l’=1 s’=1 j’=1 1P 1 3P 2 3 P1 3 j’=2,1,0 P0 s’=0 j’=0 s’=1 j’=1 s’=0 j’=0 l’=0 l’=0 1S 0 3S 1 1S 0