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Optically polarized atoms

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Optically polarized atoms Marcis Auzinsh, University of Latvia Dmitry Budker, UC Berkeley and LBNL Simon M. Rochester, UC Berkeley* – PowerPoint PPT presentation

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Title: Optically polarized atoms


1
Optically polarized atoms
  • Marcis Auzinsh, University of Latvia
  • Dmitry Budker, UC Berkeley and LBNL
  • Simon M. Rochester, UC Berkeley

2
Chapter 2 Atomic states
  • A brief summary of atomic structure
  • Begin with hydrogen atom
  • The Schrödinger Eqn
  • In this approximation (ignoring spin and
    relativity)

Principal quant. Number n1,2,3,
3
  • Could have guessed me 4/?2 from dimensions
  • me 4/?2 1 Hartree
  • me 4/2?2 1 Rydberg
  • E does not depend on l or m ? degeneracy
  • i.e. different wavefunction have same E
  • We will see that the degeneracy is n2

4
Angular momentum of the electron in the hydrogen
atom
  • Orbital-angular-momentum quantum number l
    0,1,2,
  • This can be obtained, e.g., from the Schrödinger
    Eqn., or straight from QM commutation relations
  • The Bohr model classical orbits quantized by
    requiring angular momentum to be integer multiple
    of ?
  • There is kinetic energy associated with orbital
    motion ? an upper bound on l for a given value of
    En
  • Turns out l 0,1,2, , n-1

5
Angular momentum of the electron in the hydrogen
atom (contd)
  • In classical physics, to fully specify orbital
    angular momentum, one needs two more parameters
    (e.g., two angles) in addition to the magnitude
  • In QM, if we know projection on one axis
    (quantization axis), projections on other two
    axes are uncertain
  • Choosing z as quantization axis
  • Note this is reasonable as we expect projection
    magnitude not to exceed

6
Angular momentum of the electron in the hydrogen
atom (contd)
  • m magnetic quantum number because B-field can
    be used to define quantization axis
  • Can also define the axis with E (static or
    oscillating), other fields (e.g., gravitational),
    or nothing
  • Lets count states
  • m -l,,l i. e. 2l1 states
  • l 0,,n-1 ?

As advertised !
7
Angular momentum of the electron in the hydrogen
atom (contd)
  • Degeneracy w.r.t. m expected from isotropy of
    space
  • Degeneracy w.r.t. l, in contrast, is a special
    feature of 1/r (Coulomb) potential

8
Angular momentum of the electron in the hydrogen
atom (contd)
  • How can one understand restrictions that QM puts
    on measurements of angular-momentum components ?
  • The basic QM uncertainty relation ()
    leads to (and
    permutations)
  • We can also write a generalized uncertainty
    relation
  • between lz and f (azimuthal angle of the e)
  • This is a bit more complex than () because f is
    cyclic
  • With definite lz ,

9
Wavefunctions of the H atom
  • A specific wavefunction is labeled with n l m
  • In polar coordinates
  • i.e. separation of radial and angular parts
  • Further separation

Spherical functions (Harmonics)
10
Wavefunctions of the H atom (contd)
Legendre Polynomials
  • Separation into radial and angular part is
    possible for any central potential !
  • Things get nontrivial for multielectron atoms

11
Electron spin and fine structure
  • Experiment electron has intrinsic angular
    momentum --spin (quantum number s)
  • It is tempting to think of the spin classically
    as a spinning object. This might be useful, but
    to a point

Experiment electron is pointlike down to 10-18
cm
12
Electron spin and fine structure (contd)
  • Another issue for classical picture it takes a
    4p rotation to bring a half-integer spin to its
    original state. Amazingly, this does happen in
    classical world

from Feynman's 1986 Dirac Memorial Lecture
(Elementary Particles and the Laws of Physics,
CUP 1987)
13
Electron spin and fine structure (contd)
  • Another amusing classical picture spin angular
    momentum comes from the electromagnetic field of
    the electron
  • This leads to electron size

Experiment electron is pointlike down to 10-18
cm
14
Electron spin and fine structure (contd)
  • s1/2 ?
  • Spin up and down should be used with
    understanding that the length (modulus) of the
    spin vector is gt?/2 !

15
Electron spin and fine structure (contd)
  • Both orbital angular momentum and spin have
    associated magnetic moments µl and µs
  • Classical estimate of µl current loop
  • For orbit of radius r, speed p/m, revolution rate
    is

Gyromagnetic ratio
16
Electron spin and fine structure (contd)
Bohr magneton
  • In analogy, there is also spin magnetic moment

17
Electron spin and fine structure (contd)
  • The factor ?2 is important !
  • Dirac equation for spin-1/2 predicts exactly 2
  • QED predicts deviations from 2 due to vacuum
    fluctuations of the E/M field
  • One of the most precisely measured physical
    constants ?22?1.00115965218085(76)

(0.8 parts per trillion)
New Measurement of the Electron Magnetic Moment
Using a One-Electron Quantum Cyclotron, B.
Odom, D. Hanneke, B. D'Urso, and G. Gabrielse,
Phys. Rev. Lett. 97, 030801 (2006)
Prof. G. Gabrielse, Harvard
18
Electron spin and fine structure (contd)
19
Electron spin and fine structure (contd)
  • When both l and s are present, these are not
    conserved separately
  • This is like planetary spin and orbital motion
  • On a short time scale, conservation of individual
    angular momenta can be a good approximation
  • l and s are coupled via spin-orbit interaction
    interaction of the motional magnetic field in the
    electrons frame with µs
  • Energy shift depends on relative orientation of l
    and s, i.e., on

20
Electron spin and fine structure (contd)
  • QM parlance states with fixed ml and ms are no
    longer eigenstates
  • States with fixed j, mj are eigenstates
  • Total angular momentum is a constant of motion of
    an isolated system
  • mj ? j
  • If we add l and s, j l-s j ? ls
  • s1/2 ? j l ? ½ for l gt 0 or j ½ for l 0

21
Electron spin and fine structure (contd)
  • Spin-orbit interaction is a relativistic effect
  • Including rel. effects
  • Correction to the Bohr formula ??2
  • The energy now depends on n and j

22
Electron spin and fine structure (contd)
  • ??1/137 ? relativistic corrections are small
  • 10-5 Ry
  • ?E ? 0.366 cm-1 or 10.9 GHz for 2P3/2 , 2P1/2
  • ?E ? 0.108 cm-1 or 3.24 GHz for 3P3/2 , 3P1/2

23
Electron spin and fine structure (contd)
  • The Dirac formula
  • predicts that states of same n and j, but
    different l remain degenerate
  • In reality, this degeneracy is also lifted by QED
    effects (Lamb shift)
  • For 2S1/2 , 2P1/2 ?E ? 0.035 cm-1 or 1057 MHz

24
Electron spin and fine structure (contd)
  • Example n2 and n3 states in H (from C. J.
    Foot)

25
Vector model of the atom
  • Some people really need pictures
  • Recall for a state with given j, jz
  • We can draw all of this as (j3/2)

26
Vector model of the atom (contd)
  • These pictures are nice, but NOT problem-free
  • Consider maximum-projection state mj j
  • Q What is the maximal value of jx or jy that can
    be measured ?
  • A
  • that might be inferred from the picture is wrong

27
Vector model of the atom (contd)
  • So how do we draw angular momenta and coupling ?
  • Maybe as a vector of expectation values, e.g.,
    ?
  • Simple
  • Has well defined QM meaning
  • BUT
  • Boring
  • Non-illuminating
  • Or stick with the cones ?
  • Complicated
  • Still wrong

28
Vector model of the atom (contd)
  • A compromise
  • j is stationary
  • l , s precess around j
  • What is the precession frequency?
  • Stationary state
  • quantum numbers do not change
  • Say we prepare a state with
  • fixed quantum numbers l,ml,s,ms?
  • This is NOT an eigenstate
  • but a coherent superposition of eigenstates, each
    evolving as
  • Precession ? Quantum Beats
  • ? l , s precess around j with freq.
    fine-structure splitting

29
Multielectron atoms
  • Multiparticle Schrödinger Eqn. no analytical
    soltn.
  • Many approximate methods
  • We will be interested in classification of states
    and various angular momenta needed to describe
    them
  • SE
  • This is NOT the simple 1/r Coulomb potential ?
  • Energies depend on orbital ang. momenta

30
Gross structure, LS coupling
  • Individual electron sees nucleus and other es
  • Approximate total potential as central f(r)
  • Can consider a Schrödinger Eqn for each e
  • Central potential ? separation of angular and
    radial parts li (and si) are well defined !
  • Radial SE with a given li ? set of bound states
  • Label these with principal quantum number
    ni li 1, li 2, (in analogy with
    Hydrogen)
  • Oscillation Theorem of zeros of the radial
    wavefunction is ni - li -1

31
Gross structure, LS coupling (contd)
  • Set of ni , li for all electrons ? electron
    configuration
  • Different configuration generally have different
    energies
  • In this approximation, energy of a configuration
    is just sum of Ei
  • No reference to projections of li or to spins ?
    degeneracy
  • If we go beyond the central-field approximation
    some of the degeneracies will be lifted
  • Also spin-orbit (l?s) interaction lifts some
    degeneracies
  • In general, both effects need to be considered,
    but the former is more important in light atoms

32
Gross structure, LS coupling (contd)
  • Beyond central-field approximation (cfa)
  • Non-centrosymmetric part of electron repulsion
    (?1/rij ) residual Coulomb interaction
    (RCI)
  • The energy now depends on how li and si combine
  • Neglecting (l?s) interaction ? LS coupling or
    Russell-Saunders coupling
  • This terminology is potentially confusing..
  • .. but well motivated !
  • Within cfa, individual orbital angular momenta
    are conserved RCI mixes states with different
    projections of li
  • Classically, RCI causes precession of the orbital
    planes, so the direction of the orbital angular
    momentum changes

33
Gross structure, LS coupling (contd)
  • Beyond central-field approximation (cfa)
  • Projections of li are not conserved, but the
    total orbital momentum L is, along with its
    projection !
  • This is because li form sort of an isolated
    system
  • So far, we have been ignoring spins
  • One might think that since we have neglected
    (l?s) interaction, energies of states do not
    depend on spins

WRONG !
34
Gross structure, LS coupling (contd)
  • The role of the spins
  • Not all configurations are possible. For example,
    U has 92 electrons. The simplest configuration
    would be 1s92
  • Luckily, such boring configuration is impossible.
    Why ?
  • es are fermions ? Pauli exclusion principle
    no two es can have the same set
    of quantum numbers
  • This determines the richness of the periodic
    system
  • Note some people are looking for rare violations
    of Pauli principle and Bose-Einstein statistics
    ? new physics
  • So how does spin affect energies (of allowed
    configs) ?
  • ? Exchange Interaction

35
Gross structure, LS coupling (contd)
  • Exchange Interaction
  • The value of the total spin S affects the
    symmetry of the spin wavefunction
  • Since overall ? has to be antisymmetric ?
    symmetry of spatial wavefunction is affected ?
    this affects Coulomb repulsion between electrons
    ? effect on energies
  • Thus, energies depend on L and S. Term 2S1L
  • 2S1 is called multiplicity
  • Example He(g.s.) 1s2 1S

36
Gross structure, LS coupling (contd)
  • Within present approximation, energies do not
    depend on (individually conserved) projections of
    L and S
  • This degeneracy is lifted by spin-orbit
    interaction (also spin-spin and spin-other orbit)
  • This leads to energy splitting within a term
    according to the value of total angular momentum
    J (fine structure)
  • If this splitting is larger than the residual
    Coulomb interaction (heavy atoms) ? breakdown of
    LS coupling

37
Vector Model
  • Example a two-electron atom (He)
  • Quantum numbers
  • J, mJ good no restrictions
  • for isolated atoms
  • l1, l2 , L, S good in LS coupling
  • ml , ms , mL , mS not goodsuperpositions
  • Precession rate hierarchy
  • l1, l2 around L and s1, s2 around S
  • residual Coulomb interaction
  • (term splitting -- fast)
  • L and S around J
  • (fine-structure splitting -- slow)

38
jj and intermediate coupling schemes
  • Sometimes (for example, in heavy atoms),
    spin-orbit interaction gt residual
    Coulomb ? LS coupling
  • To find alternative, step back to central-field
    approximation
  • Once again, we have energies that only depend on
    electronic configuration lift approximations one
    at a time
  • Since spin-orbit is larger, include it first ?

39
jj and intermediate coupling schemes(contd)
  • In practice, atomic states often do not fully
    conform to LS or jj scheme sometimes there are
    different schemes for different states in the
    same atom ? intermediate coupling
  • Coupling scheme has important consequences for
    selection rules for atomic transitions, for
    example
  • L and S rules approximate only hold within
    LS coupling
  • J, mJ rules strict hold
    for any coupling scheme

40
Notation of states in multi-electron atoms
  • Spectroscopic notation
  • Configuration (list of ni and li )
  • ni integers
  • li code letters
  • Numbers of electrons with same n and l
    superscript, for example Na (g.s.)
    1s22s22p63s Ne3s
  • Term 2S1L ? State 2S1LJ
  • 2S1 multiplicity (another inaccurate
    historism)
  • Complete designation of a state e.g., Ba
    (g.s.) Xe6s2 1S0

41
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42
Fine structure in multi-electron atoms
  • LS states with different J are split by
    spin-orbit interaction
  • Example 2P1/2-2P3/2 splitting in the alkalis
  • Splitting ?Z2 (approx.)
  • Splitting ? with n

43
Hyperfine structure of atomic states
  • Nuclear spin I ? magnetic moment
  • Nuclear magneton
  • Total angular momentum

44
Hyperfine structure of atomic states (contd)
  • Hyperfine-structure splitting results from
    interaction of the nuclear moments with fields
    and
  • gradients produced by es ?
  • Lowest terms
  • M1 E2
  • E2 term B?0 only for I,Jgt1/2

45
Hyperfine structure of atomic states
  • A nucleus can only support multipoles of rank
    ??2I
  • E1, M2, . moments are forbidden by P and T
  • B?0 only for I,Jgt1/2
  • Example of hfs splitting (not to scale)

85Rb (I5/2)
87Rb (I3/2)
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