PowerPoint PresentationAtomic Structure and the Fine structure constant α
Niels Bohr Erwin Schrödinger Wolfgang Pauli Paul Dirac
Lecture Notes Fundamental Constants 2015; W. Ubachs
The Old Bohr Model
An electron is held in orbit by the Coulomb force: (equals centripetal force)
2 0
πε =
The size of the orbit is quantized, and we know the size of an atom !
CoulomblCentripeta FF =
nhnmvrL === π2
2
22
0
The Old Bohr Model: Energy Quantisation
∞−=−= R n Z
=∞ πε
The Rydberg constant is the scale unit of energies in the atom
2
2
2
2
2n ZR
n ZEn −⇒−= ∞
Energies in the atom in atomic units 1 Hartree = 2 Rydberg
22 2
The Old Bohr Model; velocity of the electron
c e 0
Velocity in Bohr orbit
Schrodinger Equation; Radial part: special case l=0
( ) ERR mr
ERR r
ZeR r
trial: ( ) arAerR /−=
a Re
0 4 0
0 1 a Z
∞−= RZE 2 Ground state in the Bohr model (n=1)
Quantum mechanics: same result
The effect of the proton-mass in the atom
Relative coordinates:
21 rrr −=
Lecture Notes Fundamental Constants 2015; W. Ubachs
Reduced mass in the old Bohr model isotope shifts
Quantisation of radius in orbit:
0
2
2
eZ nr e
n µµ πε
∞

µ
1. Isotope shift on an atomic transition 2. Effect of proton/electron mass ratio on the energy levels
µ µµ +
red
Conclusion: the atoms are not a good probe to detect a variation of µ
Lecture Notes Fundamental Constants 2015; W. Ubachs
General conclusions on atoms and atomic structure
Conclusion 2: the atoms are not a good probe to detect a variation of µ
22 2
dimensionless energy
Conclusion 1: All atoms have the Rydberg as a scale for energy; they cannot be used to detect a variation of α
µ µµ +
∞ ×=−= m hc ER I
Relativistic effects in atoms
No classical analogue for this phenomenon
Pauli: There is an additional “two-valuedness” in the spectra of atoms, behaving like an angular momentum
2 1
Goudsmit and Uhlenbeck This may be interpreted/represented as an angular momentum
Origin of the spin-concept -Stern-Gerlach experiment; space quantization -Theory: the periodic system requires an additional two-valuedness
Electron spin
2 1
( ) ss msssmsS ,1, 22 +=
In analogy with the orbital angular momentum of the electron
A spin (intrinsic) angular momentum can be defined:

b) in quantum electrodynamics
...00232.2=Sg
Note: the spin of the electron cannot be explained from a classically “spinning” electronic charge
Electron radius from EM-energy:
Spin-orbit interaction
Frame of electron:
The moving charged nucleus induces a magnetic field at the location of the electron, via Biot-Savart’s law
( ) 3
LZeB e
Sg B eS
LS rcm
ZeBV e
( ) LSrVLS
⋅= ζ
Fine structure in spectra due to Spin-orbit interaction
jnlj
jSLjSO
lsjmSLlsjm
lsjmVlsjmE
Evaluate the dot-product
( ) ( ) ( ) ( )( )



++
+−+−+ =
Kinetic Relativistic effects in atomic hydrogen
Relativistic kinetic energy
( )
−
+ −
change wave function
Relativistic energy levels:
( ) t
ihmcpc ∂
F=1
F=0

=µ
Interaction with electron spin, that may have density at the site of the nucleus (Fermi contact term)
( )222 2 1 IJFJISI −−=⋅=⋅

Splitting : F=1 ↔ F=0 1.42 GHz or λ = 21 cm
Magnetic dipole transition Scaling: µα /2
pg
Alkali Doublets
ns
The Alkali Doublet Method
The Many Multiplet Method
1. Strong transitions 2. Weak, narrow transitions 3. Hyperfine transitions
Lecture Notes Fundamental Constants 2015; W. Ubachs
The Many Multiplet Method
∞−= R n ZEn 2
( )

−
+ −=
(note: atomic units different)
α
with: En is the Rydberg energy scaling ν is effective quantum number
Further include Many body effects
( ) ( )

−
j ZEnn ,,
2/1 12
ν α
These effects separate light atoms (low Z) from heavy atoms (high Z)
( ) 6.0,, ≅ljZCIn many cases:
Many Multiplet Method
Dependence of the energy levels on α: (two values for different times)
in simplified form:
Advantages of MM-Method: 1) Many atoms can be “used” simultaneously
2) Transition frequencies can be used (not just splittings) 3) Combine heavy and light atoms
“q” given in frequency/energy units
2
Results
All allowed E1 transitions Negative signs for: d→p and p→s
Lecture Notes Fundamental Constants 2015; W. Ubachs
Quasar Lines
( )


“Quasar Absorptie Spectra”
Lyman limit Lyα
On weak and strong lines
E2
E1
BC =
3
πτ2 1
Lecture Notes Fundamental Constants 2015; W. Ubachs
Similar calculations for “laboratory lines”
Clock transitions
Lecture Notes Fundamental Constants 2015; W. Ubachs
“Accidental degeneracies”
Dy atom
Level A: q/(hc)= 6x103 cm-1
Level B: q/(hc)= -24x103 cm-1
ν(A-B) ~ 235 MHz
Hz
qq
1510~ −
Lecture Notes Fundamental Constants 2015; W. Ubachs
Modern Clock Comparisons
Lecture Notes Fundamental Constants 2015; W. Ubachs
Functional dependence on fundamental constants
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