Larmor Precession Magnetic field B exerts a torque on magnetic moment μ causing precession of μ and the associated angular momentum vector λ The additional angular velocity ω ’ changes the angular velocity and hence energy of the orbiting/spinning charge Δ E = - μ . B Oxford Physics: 3rd Year, Atomic Physics
Spin-Orbit interaction: Summary B parallel to l μ parallel to s Oxford Physics: 3rd Year, Atomic Physics
Perturbation energy Radial integral ? Angular momentum operator ^ How to find < s . l > using perturbation theory? ^ Oxford Physics: 3rd Year, Atomic Physics
Perturbation theory with degenerate states Perturbation Energy: Change in wavefunction: So won’t work if E i = E j i.e. degenerate states. We need a diagonal perturbation matrix, i.e. off-diagonal elements are zero New wavefunctions: New eignvalues: Oxford Physics: 3rd Year, Atomic Physics
The Vector Model Angular momenta represented by vectors: l 2 , s 2 and j 2, and l, s j and with magnitudes: l(l+1), s(s+1) and j(j+1). and l(l+1), s(s+1) and j(j+1). z Projections of vectors: m l h lh l, s and j on z -axis are m l , m s and m j Constants of the Motion Good quantum numbers Oxford Physics: 3rd Year, Atomic Physics
Summary of Lecture 3: Spin-Orbit coupling • Spin-Orbit energy • Radial integral sets size of the effect. • Angular integral < s . l > needs Degenerate Perturbation Theory • New basis eigenfunctions: s • j and j z are constants of the motion j l • Vector Model represents angular momenta as vectors • These vectors can help identify constants of the motion • These constants of the motion - represented by good quantum numbers Oxford Physics: 3rd Year, Atomic Physics
Z Z Fixed in (a) No spin-orbit j space j coupling s (b) Spin–orbit coupling gives precession l l around j s (c) Projection of l on z is not constant ( a ) i i ( b ) i i (d) Projection of s on z Z Z is not constant j j l z m l and m s are not good quantum numbers l s z Replace by j and m j s Oxford Physics: 3rd Year, Atomic Physics ( c ) i i ( d ) I i
Vector model defines: s j Vector triangle l Magnitudes Oxford Physics: 3rd Year, Atomic Physics
∼ β n,l x ‹ ½ { j 2 – l 2 – s 2 } › Using basis states: | n, l, s, j, m j › to find expectation value: The spin-orbit energy is: Δ E = β n,l x (1/2){ j(j+1) – l(l+1) – s(s+1) } Oxford Physics: 3rd Year, Atomic Physics
Δ E = β n,l x ( 1 / 2 ){ j(j+1) – l(l+1) – s(s+1) } Sodium 3s: n = 3, l = 0, no effect 3p: n = 3, l = 1, s = ½, -½, j = ½ or 3 / 2 Δ E( 1 / 2 ) = β 3p x ( - 1); Δ E( 3 / 2 ) = β 3p x ( 1 / 2 ) j = 3/2 1/2 2j + 1 = 4 3p (no spin-orbit) j = 1/2 -1 2j + 1 = 2 Oxford Physics: 3rd Year, Atomic Physics
Lecture 4 • Two-electron atoms: the residual electrostatic interaction • Adding angular momenta: LS-coupling • Symmetry and indistinguishability • Orbital effects on electrostatic interaction • Spin-orbit effects Oxford Physics: 3rd Year, Atomic Physics
Coupling of l i and s to form L and S: Electrostatic interaction dominates l 2 s 2 L S is 1 l 1 s s l l S = + L = + 1 1 2 2 Oxford Physics: 3rd Year, Atomic Physics
Coupling of L and S to form J S = 1 S = 1 L = 1 L = 1 S = 1 L = 1 J = 2 J = 1 J = 0 Oxford Physics: 3rd Year, Atomic Physics
Magnesium: “typical” 2-electron atom Mg Configuration: 1s 2 2s 2 2p 6 3s 2 Na Configuration: 1s 2 2s 2 2p 6 3s “Spectator” electron in Mg Mg energy level structure is like Na but levels are more strongly bound Oxford Physics: 3rd Year, Atomic Physics
Residual electrostatic interaction 3s4s state in Mg: Zero-order wave functions Perturbation energy: ? Degenerate states Oxford Physics: 3rd Year, Atomic Physics
Linear combination of zero-order wave-functions Off-diagonal matrix elements: Oxford Physics: 3rd Year, Atomic Physics
Off-diagonal matrix elements: Therefore as required! Oxford Physics: 3rd Year, Atomic Physics
Effect of Direct and Exchange integrals Singlet +K -K Triplet J Energy level with no electrostatic interaction Oxford Physics: 3rd Year, Atomic Physics
Orbital orientation effect on electrostatic interaction l 2 l 1 l 2 L l 1 l l L = + 1 2 Overlap of electron wavefunctions depends on orientation of orbital angular momentum: so electrostatic interaction depends on L Oxford Physics: 3rd Year, Atomic Physics
Residual Electrostatic and Spin-Orbit effects in LS-coupling Oxford Physics: 3rd Year, Atomic Physics
Term diagram of Magnesium Singlet terms Triplet terms 3p n 1 1 1 3 3 3 S P D S P D o 1 2 2 3p n s 5s 1 3s3d D 2 4s 1 3s3p P 1 3 3s3p P resonance line 2,1,0 (strong) intercombination line (weak) 2 1 3s S 0 Oxford Physics: 3rd Year, Atomic Physics
The story so far: Hierarchy of interactions H O H 1 H 2 H 3 : Nuclear Effects on atomic energy H 3 << H 2 << H 1 << H O Oxford Physics: 3rd Year, Atomic Physics
Lecture 5 • Nuclear effects on energy levels – Nuclear spin – addition of nuclear and electron angular momenta • How to find the nuclear spin •Isotope effects: – effects of finite nuclear mass – effects of nuclear charge distribution • Selection Rules
Nuclear effects in atoms Corrections Nucleus: Nuclear spin → magnetic dipole • stationary interacts with electrons orbits centre of mass with • infinite mass electrons charge spread over • point nuclear volume
Nuclear Spin interaction Magnetic dipole ~ angular momentum μ = - γλ ħ μ l = - g l μ B l μ s = - g s μ Β s μ Ι = - g I μ Ν I μ Ν = μ Β x m e / m P ~ μ Β / 2000 g I ~ 1 Perturbation energy: ^ Η 3 = − μ Ι . B el
Magnetic field of electrons: Orbital and Spin Closed shells: zero contribution s orbitals: largest contribution – short range ~1/r 3 l > 0, smaller contribution - neglect B el
B el = (scalar quantity) x J Usually dominated by spin contribution in s-states: Fermi “contact interaction”. Calculable only for Hydrogen in ground state, 1s
Coupling of I and J Depends on I Depends on J Nuclear spin interaction energy: empirical Expectation value
Vector model of nuclear interaction F = I + J I and J precess around F I F I I F J J J F
Hyperfine structure Hfs interaction energy: Vector model result: Hfs energy shift: Hfs interval rule:
Finding the nuclear spin, I • Interval rule – finds F, then for known J → I • Number of spectral lines (2 I + 1) for J > I, (2J + 1) for I > J • Intensity Depends on statistical weight (2F + 1) finds F, then for known J → I
Isotope effects reduced mass Orbiting about Fixed nucleus, infinite mass + Orbiting about centre of mass
Isotope effects reduced mass Orbiting about Fixed nucleus, infinite mass + Orbiting about centre of mass
Lecture 6 • Selection Rules • Atoms in magnetic fields – basic physics; atoms with no spin – atoms with spin: anomalous Zeeman Effect – polarization of the radiation
Parity selection rule r -r N.B. Error in notes eqn (161) Parity (-1) l must change Δ l = + 1
Configuration Only one electron “jumps”
Selection Rules: Conservation of angular momentum J 2 = J 1 h J 2 = J 1 h J 1 J 1 Δ L = 0, + 1 Δ S = 0 Δ M J = 0, + 1
Atoms in magnetic fields Oxford Physics: 3rd Year, Atomic Physics
Effect of B-field on an atom with no spin Interaction energy - Precession energy: Oxford Physics: 3rd Year, Atomic Physics
Normal Zeeman Effect Level is split into equally Spaced sub-levels (states) Selection rules on M L give a spectrum of the normal Lorentz Triplet Spectrum Oxford Physics: 3rd Year, Atomic Physics
Effect of B-field on an atom with spin-orbit coupling Precession of L and S around the resultant J leads to variation of projections of L and S on the field direction Oxford Physics: 3rd Year, Atomic Physics
Oxford Physics: 3rd Year, Atomic Physics
Total magnetic moment does not lie along axis of J. Effective magnetic moment does lie along axis of J, hence has constant projection on B ext axis Oxford Physics: 3rd Year, Atomic Physics
Perturbation Calculation of B ext effect on spin-orbit level Interaction energy Effective magnetic moment Perturbation Theory: expectation value of energy Energy shift of MJ level Oxford Physics: 3rd Year, Atomic Physics
Vector Model Calculation of B ext effect on spin-orbit level Projections of L and S on J are given by Oxford Physics: 3rd Year, Atomic Physics
Vector Model Calculation of B ext effect on spin-orbit level Perturbation Theory result Oxford Physics: 3rd Year, Atomic Physics
Anomalous Zeeman Effect: 3s 2 S 1/2 – 3p 2 P 1/2 in Na Oxford Physics: 3rd Year, Atomic Physics
Polarization of Anomalous Zeeman components associated with Δ m selection rules Oxford Physics: 3rd Year, Atomic Physics
Lecture 7 • Magnetic effects on fine structure - Weak field - Strong field • Magnetic field effects on hyperfine structure: - Weak field - Strong field Oxford Physics: 3rd Year, Atomic Physics
Summary of magnetic field effects on atom with spin-orbit interaction Oxford Physics: 3rd Year, Atomic Physics
Total magnetic moment does not lie along axis of J. Effective magnetic moment does lie along axis of J, hence has constant projection on B ext axis Oxford Physics: 3rd Year, Atomic Physics
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