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Optically polarized atoms. Marcis Auzinsh, University of Latvia Dmitry Budker, UC Berkeley and LBNL Simon M. Rochester, UC Berkeley. Image from Wikipedia. Chapter 2: Atomic states. A brief summary of atomic structure Begin with hydrogen atom The Schr ö dinger Eqn :

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optically polarized atoms

Optically polarized atoms

Marcis Auzinsh, University of Latvia

Dmitry Budker, UC Berkeley and LBNL

Simon M. Rochester, UC Berkeley

chapter 2 atomic states

Image from Wikipedia

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

Principal quant. Number

n=1,2,3,…

slide3
Could have guessed me 4/2 from dimensions
  • me 4/2 =1Hartree
  • me 4/22 =1 Rydberg
  • E does not depend on lor m degeneracy

i.e.different wavefunction have same E

  • We will see that the degeneracy is n2
angular momentum of the electron in the hydrogen atom
Angular momentum of the electron in the hydrogen atom
  • Orbital-angular-momentum quantum numberl = 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 lfor a given value of En
  • Turns out: l = 0,1,2, …, n-1
angular momentum of the electron in the hydrogen atom cont d
Angular momentum of the electron in the hydrogen atom (cont’d)
  • In classical physics, to fully specify orbital angular momentum, one needs two more parameters (e.g., to 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
angular momentum of the electron in the hydrogen atom cont d1
Angular momentum of the electron in the hydrogen atom (cont’d)
  • 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
  • Let’s count states:
    • m = -l,…,l i. e. 2l+1 states
    • l = 0,…,n-1 

As advertised !

angular momentum of the electron in the hydrogen atom cont d2
Angular momentum of the electron in the hydrogen atom (cont’d)
  • 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
angular momentum of the electron in the hydrogen atom cont d3
Angular momentum of the electron in the hydrogen atom (cont’d)
  • 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 generalizeduncertainty relation

between lzand φ(azimuthal angle of the e):

  • This is a bit more complex than (*) because φis cyclic
  • With definite lz , φis completely uncertain…
wavefunctions of the h atom
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)

wavefunctions of the h atom cont d
Wavefunctions of the H atom (cont’d)
  • Separation into radial and angular part is possible for any central potential !
  • Things get nontrivial for multielectron atoms

Legendre Polynomials

electron spin and fine structure
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

electron spin and fine structure cont d
Electron spin and fine structure (cont’d)
  • Another issue for classical picture: it takes a 4πrotation 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)

electron spin and fine structure cont d1
Electron spin and fine structure (cont’d)
  • 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

electron spin and fine structure cont d2
Electron spin and fine structure (cont’d)
  • s=1/2 
  • “Spin up” and “down” should be used with understanding that the length (modulus) of the spin vector is >/2 !
electron spin and fine structure cont d3
Electron spin and fine structure (cont’d)
  • Both orbital angular momentum and spin have associated magnetic momentsμl and μs
  • Classical estimate of μl : current loop
  • For orbit of radius r, speed p/m, revolution rate is

Gyromagnetic ratio

electron spin and fine structure cont d4
Electron spin and fine structure (cont’d)
  • In analogy, there is also spin magnetic moment :

Bohr magneton

electron spin and fine structure cont d5
Electron spin and fine structure (cont’d)
  • 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: 2=2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

electron spin and fine structure cont d7
Electron spin and fine structure (cont’d)
  • 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 sare coupled via spin-orbit interaction: interaction of the motional magnetic field in the electron’s frame with μs
  • Energy shift depends on relative orientation of l and s, i.e., on
electron spin and fine structure cont d8
Electron spin and fine structure (cont’d)
  • QM parlance: states with fixed ml and ms are no longer eigenstates
  • States with fixed j, mjare 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  l+s
  • s=1/2 j = l ½ for l > 0 or j = ½ for l = 0
electron spin and fine structure cont d9
Electron spin and fine structure (cont’d)
  • Spin-orbitinteraction is a relativistic effect
  • Includingrel. effects :
  • Correction to the Bohr formula 2
  • The energy now depends on n and j
electron spin and fine structure cont d10
Electron spin and fine structure (cont’d)
  • 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
electron spin and fine structure cont d11
Electron spin and fine structure (cont’d)
  • 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
vector model of the atom

mj= 3/2

mj= 1/2

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 (j=3/2)
vector model of the atom cont d

mj= 3/2

Vector model of the atom (cont’d)
  • These pictures are nice, but NOT problem-free
  • Consider maximum-projection state mj= j
  • Q: What is the maximal value of jxor jy that can be measured ?
  • A:

that might be inferred from the picture is wrong…

vector model of the atom cont d1
Vector model of the atom (cont’d)
  • 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…
vector model of the atom cont d2
Vector model of the atom (cont’d)
  • 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
multielectron atoms
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 
  • Energiesdepend onorbital ang. momenta
gross structure ls coupling
Gross structure, LS coupling
  • Individual electron “sees” nucleus and other e’s
  • Approximate totalpotential as central: φ(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
gross structure ls coupling cont d
Gross structure, LS coupling (cont’d)
  • 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 orto 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
gross structure ls coupling cont d1
Gross structure, LS coupling (cont’d)

Beyond central-field approximation (cfa)

  • Non-centrosymmetric part of electron repulsion (1/rij) = residual Coulomb interaction (RCI)
  • The energy now depends on how li andsi 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
gross structure ls coupling cont d2
Gross structure, LS coupling (cont’d)

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 !

gross structure ls coupling cont d3
Gross structure, LS coupling (cont’d)

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 ?
  • e’s are fermions  Pauli exclusion principle: no two e’s 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
gross structure ls coupling cont d4
Gross structure, LS coupling (cont’d)

Exchange Interaction

  • The value of the total spinS affects the symmetry of the spin wavefunction
  • Since overall ψhas to be antisymmetric  symmetry of spatial wavefunction is affected  this affects Coulomb repulsion between electrons  effect on energies
  • Thus, energies depend on Land S. Term: 2S+1L
  • 2S+1 is called multiplicity
  • Example: He(g.s.): 1s2 1S
gross structure ls coupling cont d5
Gross structure, LS coupling (cont’d)
  • 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
vector model
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 good”=superpositions
  • “Precession” rate hierarchy:
    • l1, l2 around L and s1, s2 around S:

residual Coulomb interaction

(term splitting -- fast)

    • Land S around J

(fine-structure splitting -- slow)

jj and intermediate coupling schemes
jj and intermediate coupling schemes
  • Sometimes (for example, in heavy atoms), spin-orbit interaction > 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 
jj and intermediate coupling schemes cont d
jj and intermediate coupling schemes(cont’d)
  • 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
    • Land S rules: approximate; only hold within LS coupling
    • J, mJrules: strict; hold for any coupling scheme
notation of states in multi electron atoms
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 = [Ne]3s
  • Term 2S+1L State2S+1LJ
  • 2S+1 = multiplicity (another inaccurate historism)
  • Complete designation of a state [e.g., Ba (g.s.)]: [Xe]6s21S0
fine structure in multi electron atoms
Fine structure in multi-electron atoms
  • LS states with different J are split by spin-orbit interaction
  • Example: 2P1/2-2P3/2splitting in the alkalis
  • Splitting Z2(approx.)
  • Splitting  with n
hyperfine structure of atomic states
Hyperfine structure of atomic states
  • NuclearspinI  magnetic moment
  • Nuclear magneton
  • Total angular momentum:
hyperfine structure of atomic states cont d
Hyperfine structure of atomic states (cont’d)
  • Hyperfine-structure splitting results from interaction of the nuclear moments with fields and

gradients produced by e’s 

  • Lowest terms:

M1 E2

  • E2 term: B0 only for I,J>1/2
hyperfine structure of atomic states1
Hyperfine structure of atomic states
  • A nucleus can only support multipoles of rank κ2I
  • E1, M2, …. moments are forbidden by P and T

B0 only for I,J>1/2

  • Example of hfs splitting (not to scale)

85Rb (I=5/2)

87Rb (I=3/2)