Understanding Atomic Structure: Many-Electron Atoms and Quantum Numbers, Study notes of Thermodynamics

Download Understanding Atomic Structure: Many-Electron Atoms and Quantum Numbers and more Study notes Thermodynamics in PDF only on Docsity! Many-Electron Atoms Thornton and Rex, Ch. 8 In principle, can now solve Sch. Eqn for any atom. In practice, -> Complicated! Goal-- To explain properties of elements from principles of quantum theory (without exact solutions) n l ml ms Hydrogen 1 0 0 +1/2 Helium 1 0 0 -1/2 Lithium 2 0 0 +1/2 Beryllium 2 0 0 -1/2 Boron 2 1 -1 +1/2 Carbon 2 1 0 +1/2 Nitrogen 2 1 +1 +1/2 Oxygen 2 1 -1 -1/2 Flourine 2 1 0 -1/2 Neon 2 1 +1 -1/2 Sodium 3 0 0 +1/2 Magnesium 3 0 0 -1/2 Aluminum 3 1 -1 +1/2 Silicon 3 1 0 +1/2 Phosporus 3 1 +1 +1/2 Sulfur 3 1 -1 -1/2 Chlorine 3 1 0 -1/2 Argon 3 1 +1 -1/2 Potassium 4 0 0 +1/2 (L as t El ec tr on A dd ed ) Chemical properties of elements Electrons in outermost, largest n orbits are most weakly bound. They determine the chemical properties of the elements. Elements with similar electron structure have similar properties. • Inert or Noble Gases Closed p subshell (s for He). He (1s2), Ne (2s22p6), Ar (3s23p6) • Alkalis Have single electron electron outside closed shell. Li (2s1), Na (3s1), K (4s1) • Halogens Are one electron short of a closed shell. F (2s22p5), Cl (3s23p5) Spectroscopic notation nLj Examples: 2S1/2 3P3/2 etc. Principle Quantum Number Orbital Angular Momentum Letter Total Angular Momentum Number Spin-Orbit Coupling • Recall, coupling of spin to a magnetic field shifts the energy (VB = -ms• B). • Motion of electron produces an “internal” magnetic field. So there is an additional contribution to the energy: VSL = -ms• Bint VSL µ S • L Proportional to -S Proportional to L This is the Spin-Orbit Coupling: VSL µ S • L Now states with definite energy do not have unique L and S quantum numbers (ml, ms). We must use J quantum numbers (j, mj). States with j = l - 1/2 have slightly less energy than states with j = l + 1/2 . 2P3/2 2P 2P1/2 (States with different mj are still degenerate for each j.) Handwaving explanation: Electrons repel each other, so we want them as far from each other as possible. 1) If spins of two electrons are aligned (for maximum S), then Pauli Exclusion Principle says they must have different L orbits. They will tend to be farther apart. 2) If the L orbits are aligned (although with different magnitudes), then the electrons will travel around the nucleus in the same direction, so they don’t pass each other as often. Example: A d subshell (l =2) can contain 10 electrons. ms=+1/2 ml =+2 -1/2 +1/2 ml =+1 -1/2 +1/2 ml =0 -1/2 +1/2 ml =-1 -1/2 +1/2 ml =-2 -1/2 Example: A d subshell (l =2) can contain 10 electrons. ms=+1/2 ml =+2 -1/2 +1/2 ml =+1 -1/2 +1/2 ml =0 -1/2 +1/2 ml =-1 -1/2 +1/2 ml =-2 -1/2 5 4 3 2 1 Pu t fi rs t 5 el ec tr on s al l w it h sp in in sa m e di re ct io n. Fi rs t 2 el ec tr on s ha ve m l = -2 a nd - 1, e tc . Note that for S=0, there is 1 value of J, given by J=L. This state is called a Singlet. For S=1, there are 3 values of J, given by J=L-1, J=L, J=L+1. These states are called a Triplet. In general, the multiplicity of the states is given by (2S+1). The Spectroscopic notation is n(2S+1)LJ Example: 2 electrons, one in 4p, other in 4d. I.e., n=4, l1 =1, s1=1/2 l2 =2, s2=1/2 Possible values of S: S=0 or S=1 Possible values of L: L=1, 2, or 3 Possible values of J: for singlet (S=0): J=L for triplet (S=1): J=L-1 or J=L or J=L+1 Use Hund's rules to order the energies. —_—_—------ / é ¢ / ¢ ———ooOorrcccrc le i x 4 “ i So i i i 4p4d / \ ‘ \ \ 1 aa \ gi---- \ ‘ se \ ‘ \ / \ / 4 - \ -- fo o-——se2227- \__--" ~ a S S S S S 8 ae ~ ae - jj Coupling For high-Z elements the spin-orbit coupling is large for each electron. Now add the angular momentum: First, J1 = L1 + S1 J2 = L2 + S2 Then J = J1 + J2 Anomalous Zeeman Effect Recall, energy shift in external magnetic field: VB = -m• B The magnetic moment gets both orbital and spin contributions: -e m = mL+ mS = [ L + 2 S ] 2m If S=0, this is simple. It is just the Normal Zeeman effect. Energy levels split according to ml values: VB = ml mB B But. . . . . . . most atoms are not “Normal”. If both S and L are nonzero, the spin-orbit coupling requires us to use J-states. Projecting m onto J gives e VB = g J • B 2m = mB g mJ B where the projection factor (called the Lande g factor) is J(J+1) + S(S+1) - L(L+1) g = 1 + 2J(J+1) This is the Anomalous Zeeman Effect.