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o Fine structure o Spin-orbit interaction. o Relativistic kinetic energy correction o Hyperfine structure o The Lamb shift. o Nuclear moments. PY3P05 o Fine structure of H-atom is due to spin-orbit interaction: Jˆ is a max Z! ˆ ˆ "E so = # S$L 2m 2cr 3 Lˆ o If L is parallel to S => J is a maximum => high energy configuration. ! +Ze ! o Angular momenta are described in terms of quantum numbers, s, l and j: Jˆ = Lˆ + Sˆ Jˆ 2 = ( Lˆ + Sˆ )( Lˆ + Sˆ ) = Lˆ Lˆ + SˆSˆ + 2 Sˆ " Lˆ 1 Sˆ " Lˆ = ( Jˆ " Jˆ # Lˆ " Lˆ # Sˆ " Sˆ ) 2 !2 => Sˆ " Lˆ = [ j( j + 1) # l(l + 1) # s(s + 1)] 2 "#E so = $ ! -e ! ! ! Sˆ Jˆ is a min Lˆ ! +Ze -e ! Sˆ Z! 3 1 [ j( j + 1) % l(l + 1) % s(s + 1)] 4m 2c r 3 ! PY3P05 o For practical purposes, convenient to express spin-orbit coupling as a [ j ( j + 1) # l(l + 1) # s(s + 1)] 2 where a = Ze 2µ0 ! 2 /8"m 2 r 3 is the spin-orbit coupling constant. Therefore, for the 2p electron: ! "E = a * 3 # 3 + 1& )1(1+ 1) ) 1 # 1 + 1&- = + 1 a ( % (/ , % so 2 +2 $ 2 ' 2 $ 2 '. 2 ! "E so = a *1 # 1 & 1 # 1 &"E so = , % + 1( )1(1+ 1) ) % + 1(/ = )a 2 +2 $ 2 ' 2 $ 2 '. !E j = 3/2 +1/2a Angular momenta aligned j = 1/2 -a Angular momenta opposite 2p1 PY3P05 o The spin-orbit coupling constant is directly measurable from the doublet structure of spectra. o If we use the radius rn of the nth Bohr radius as a rough approximation for r (from Lectures 1-2): n 2!2 r = 4 "#0 2 mZe => a ~ Z4 n6 o Spin-orbit coupling increases sharply with Z. Difficult for observed for H-atom, as Z = 1: 0.14 Å (H!), 0.08 Å (H"), 0.07 Å (H#). ! o Evaluating the quantum mechanical form, o Therefore, using this and s = 1/2: ! "E so = a~ Z4 n 3 [ l(l + 1)(2l + 1)] Z 2# 4 [ j( j + 1) $ l(l + 1) $ 3/4] mc 2 3 2n l(l + 1)(2l + 1) PY3P05 ! o Convenient to introduce shorthand notation to label energy levels that occurs in the LS coupling regime. o Each level is labeled by L, S and J: o o o o 2S+1L J L = 0 => S L = 1 => P L = 2 =>D L = 3 =>F o 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 o 2S + 1 is the multiplicity. Indicates the degeneracy of the level due to spin. o If S = 0 => multiplicity is 1: singlet term. o If S = 1/2 => multiplicity is 2: doublet term. o If S = 1 => multiplicity is 3: triplet term. o Most useful when dealing with multi-electron atoms. PY3P05 o The energy level diagram can also be drawn as a term diagram. o Each term is evaluated using: 2S+1LJ o For H, the levels of the 2P term arising from spin-orbit coupling are given below: 2P E 3/2 +1/2a Angular momenta aligned 2p1 (2P) 2P 1/2 -a Angular momenta opposite PY3P05 o Spectral lines of H composed of closely spaced doublets. Splitting is due to interactions between electron spin S and the orbital angular momentum L => spin-orbit coupling. o H! line is single line according to the Bohr or Schrödinger theory. Occurs at 656.47 nm for H and 656.29 nm for D (isotope shift, !"~0.2 nm). o H! Spin-orbit coupling produces fine-structure splitting of ~0.016 nm. Corresponds to an internal magnetic field on the electron of about 0.4 Tesla. PY3P05 o According to special relativity, the kinetic energy of an electron of mass m and velocity v is: 2 4 T= p p " + ... 2m 8m 3c 2 o The first term is the standard non-relativistic expression for kinetic energy. The second term is the lowest-order relativistic correction to this energy. ! o Therefore, the correction to the Hamiltonian is "H rel = # o Using the fact that 1 p4 8m 3c 2 p2 = E " V we can write 2m ! 1 2 2 "H rel = # 2 ( E # 2EV + V ) 2mc ! PY3P05 ! o With V = -Z2e / r , applying first-order perturbation theory to previous Hamiltonian reduces the problem of finding the expectation values of r -1 and r -2. "E rel = # % 1 Z 2$ 4 3( mc 2 ' # * 3 & n 2l + 1 8n ) o Produces an energy shift comparable to spin-orbit effect. ! o Note that !Erel ~ "4 => (1/137)4 ~ 10-8 o A complete relativistic treatment of the electron involves the solving the Dirac equation. PY3P05 o For H-atom, the spin-orbit and relativistic corrections are comparable in magnitude, but much smaller than the gross structure. Enlj = En + $EFS o Gross structure determined by En from Schrödinger equation. The fine structure is determined by 4 4 "E FS = "E so + "E rel = # % 1 Z $ 3( mc 2 ' # * 3 & 2l + 1 8n ) 2n o As En = -Z2E0/n2, where E0 = 1/2!2mc2, we can write ! E H "atom = " Z 2 E 0 $ Z 2# 2 $ 1 3 '' 1+ " )) & 2 & n % n % j + 1/2 4n (( ! of the gross and fine structure of the hydrogen atom. o Gives the energy PY3P05 o Energy correction only depends on j, which is of the order of !2 ~ 10-4 times smaller that the principle energy splitting. o All levels are shifted down from the Bohr energies. o For every n>1 and l, there are two states corresponding to j = l ± 1/2. o States with same n and j but different l, have the same energies (does not hold when Lamb shift is included). i.e., are degenerate. o Using incorrect assumptions, this fine structure was derived by Sommerfeld by modifying Bohr theory => right results, but wrong physics! PY3P05 o Spectral lines give information on nucleus. Main effects are isotope shift and hyperfine structure. o According to Schrödinger and Dirac theory, states with same n and j but different l are degenerate. However, Lamb and Retherford showed in 1947 that 22S1/2 (n = 2, l = 0, j = 1/2) and 22P1/2 (n = 2, l = 1, j = 1/2) of are not degenerate. Energy difference is ~4.4 x 10-6 eV. o Experiment proved that even states with the same total angular momentum J are energetically different. PY3P05 1. Excite H-atoms to 22S1/2 metastable state by e- bombardment. Forbidden to spontaneuosly decay to 12S1/2 optically. 2. Cause transitions to 22P1/2 state using tunable microwaves. Transitions only occur when microwaves tuned to transition frequency. These atoms then decay emitting Ly! line. 3. Measure number of atoms in 22S1/2 state from H-atom collisions with tungsten (W) target. When excitation to 22P1/2, current drops. 4. Excited H atoms (22S1/2 metastable state) cause secondary electron emission and current from the target. Dexcited H atoms (12S1/2 ground state) do not. PY3P05 o According to Dirac and Schrödinger theory, states with the same n and j quantum numbers but different l quantum numbers ought to be degenerate. Lamb and Retherford showed that 2 S1/2 (n=2, l=0, j=1/2) and 2P1/2 (n=2, l=1, j=1/2) states of hydrogen atom were not degenerate, but that the S state had slightly higher energy by E/h = 1057.864 MHz. o Effect is explained by the theory of quantum electrodynamics, in which the electromagnetic interaction itself is quantized. o For further info, see http://www.pha.jhu.edu/~rt19/hydro/node8.html PY3P05 Energy scale Energy (eV) Effects Gross structure 1-10 electron-nuclear attraction Electron kinetic energy Electron-electron repulsion Fine structure 0.001 - 0.01 Spin-orbit interaction Relativistic corrections Hyperfine structure 10-6 - 10-5 Nuclear interactions PY3P05 o Hyperfine structure results from magnetic interaction between the electron’s total angular momentum (J) and the nuclear spin (I). o Angular momentum of electron creates a magnetic field at the nucleus which is proportional to J. o Interaction energy is therefore "E hyperfine = #µˆ nucleus $ Bˆ electron % Iˆ $ Jˆ o Magnitude is very small as nuclear dipole is ~2000 smaller than electron (µ~1/m). ! o Hyperfine splitting is about three orders of magnitude smaller than splitting due to fine structure. PY3P05 o o Like electron, the proton has a spin angular momentum and an associated intrinsic dipole moment e µˆ p = g p Iˆ M The proton dipole moment is weaker than the electron dipole moment by M/m ~ 2000 and hence the effect is small. gpe 2 ˆ ˆ I#J mMc 2 r 3 o ! be shown to be: Resulting energy correction can o Total angular momentum including nuclear spin, orbital angular momentum and electron spin is Fˆ = Iˆ + Jˆ ! F = f ( f + 1)! where "E p = Fz = m f ! o !The quantum number f has possible values f = j + 1/2, j - 1/2 since the proton has spin 1/2,. o !Hence every energy level associated with a particular set of quantum numbers n, l, and j will be split into two levels of slightly different energy, depending on the relative orientation of the proton magnetic dipole with the electron state. PY3P05 o The energy splitting of the hyperfine interaction is given by a "E HFS = [ f ( f + 1) # i(i + 1) # j ( j + 1)] 2 where a is the hyperfine structure constant. o ! E.g., consider the ground state of H-atom. Nucleus consists of a single proton, so I = 1/2. The hydrogen ground state is the 1s 2S1/2 term, which has J = 1/2. Spin of the electron can be parallel (F = 1) or antiparallel (F = 0). Transitions between these levels ! occur at 21 cm (1420 MHz). o For ground state of the hydrogen atom (n=1), the energy separation between the states of F = 1 and F = 0 is 5.9 x 10-6 eV. Prove this! F=1 F=0 21 cm radio map of the Milky Way PY3P05 o Selection rules determine the allowed transitions between terms. $n = any integer $l = ±1 $j = 0, ±1 $f = 0, ±1 PY3P05 o Gross structure: o Covers largest interactions within the atom: o Kinetic energy of electrons in their orbits. o Attractive electrostatic potential between positive nucleus and negative electrons o Repulsive electrostatic interaction between electrons in a multi-electron atom. o Size of these interactions gives energies in the 1-10 eV range and upwards. o Determine whether a photon is IR, visible, UV or X-ray. o Fine structure: o Spectral lines often come as multiplets. E.g., H! line. => smaller interactions within atom, called spin-orbit interaction. o Electrons in orbit about nucleus give rise to magnetic moment of magnitude µB, which electron spin interacts with. Produces small shift in energy. o Hyperfine structure: o Fine-structure lines are split into more multiplets. o Caused by interactions between electron spin and nucleus spin. o Nucleus produces a magnetic moment of magnitude ~µB/2000 due to nuclear spin. o E.g., 21-cm line in radio astronomy. PY3P05