Lecture Notes Fundamental Constants 2015; W. Ubachs

Atomic 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)

20

22

4 nn rZe

rmv

πε=

The size of the orbit is quantized, and we know the size of an atom !

CoulomblCentripeta FF =

nhnmvrL ===π2

Bohrs postulate: Quantization of angular momentum

2

22

0

222

4 mn

mrZerv

==πε

1

2

20

22

rZn

mZehnrn ==

πε m

mehr 10

20

2

1 10529.0 −×==π

ε

Lecture Notes Fundamental Constants 2015; W. Ubachs

The Old Bohr Model: Energy Quantisation

∞−=−= RnZ

rZemvE

nn 2

2

0

22

421

πεQuantisation of energy

2

2

0

2

24

emeR

=∞ πε

The Rydberg constant is the scale unit of energies in the atom

2

2

2

2

2nZR

nZEn −⇒−= ∞

Energies in the atom in atomic units 1 Hartree = 2 Rydberg

222

2

2

2

2mc

nZR

nZEn α−=−= ∞ c

e0

2

4πεα =with

dimensionless energy

Lecture Notes Fundamental Constants 2015; W. Ubachs

The Old Bohr Model; velocity of the electron

ce0

2

4πεα =

cZvn α==1

Limit on the number of elements ? Classical argument

Velocity in Bohr orbit

Lecture Notes Fundamental Constants 2015; W. Ubachs

Schrodinger Equation; Radial part: special case l=0

( ) ERRmr

rVdrdRr

drd

mr=

+++− 1

2)(

2 2

22

2

2

Find a solution for 0=

ERRr

ZeRr

R =−

+−

0

22

4'2"

2 πεµ

Physical intuition; no density for ∞→r

trial: ( ) arAerR /−=

aRe

aAR ar −=−= − /'

2/

2"aRe

aAR ar == −

Er

Zearam

=−

−−

0

2

2

2

421

2 πε

must hold for all values of r

04 0

22

=−πεZe

ma

Prefactor for 1/r:

mZea 2

204 πε

=Solution for the length scale paramater

01 aZ

a = with eme

a 2

20

04 πε

= Bohr radius

Solutions for the energy

2

2

0

22

2

242

emeZma

E

−=−=

πε

∞−= RZE 2 Ground state in the Bohr model (n=1)

Quantum mechanics: same result

Lecture Notes Fundamental Constants 2015; W. Ubachs

The effect of the proton-mass in the atom

Relative coordinates:

21 rrr −=

Centre of Mass

021 =+ rMrm

Position vectors:

rMm

Mr

+=1

rMm

mr

+−=2

Velocity vectors:

vMm

Mv

+=1

vMm

Mv

+−=2

Relative velocity

dtrdv

=

Kinetic energy

2222

211 2

121

21 vvmvmK µ=+=

With reduced mass

MmmM+

=µ

Angular momentum

vrrvmrvmL µ=+= 222111

Centripetal force

rv

rvm

rvmF

2

2

222

1

211 µ

===

Quantisation of angular momentum:

nhnvrL ===π

µ2

Problem is similar, but m µ r relative coordinate

Lecture Notes Fundamental Constants 2015; W. Ubachs

Reduced mass in the old Bohr model isotope shifts

Quantisation of radius in orbit:

0

2

2

20

2 4 amZn

eZnr e

n µµπε

==

Energy levels in the Bohr model:

∞

−= R

mnZE

en

µ2

2

Results

Rydberg constant:

∞

= R

mR

eH

µ

1. Isotope shift on an atomic transition 2. Effect of proton/electron mass ratio on the energy levels

µµµ+

=+

=+

=+

=1/1

//mM

mMMm

MmMm

mMme

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

222

2

2

2

2mc

nZR

nZEn α−=−= ∞

dimensionless energy

Conclusion 1: All atoms have the Rydberg as a scale for energy; they cannot be used to detect a variation of α

µµµ+

=+

=1/1

/mM

mMmred

Note units (different units in this equation): 1710)83(5490973731568.1 −

∞ ×=−= mhcER I

Lecture Notes Fundamental Constants 2015; W. Ubachs

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

21

=s

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

Lecture Notes Fundamental Constants 2015; W. Ubachs

Electron spin as an angular momentum operator

21

=s

Spin is an angular momentum, so it should satisfy

( ) ss msssmsS ,1, 22 +=

sssz msmmsS ,, =

21,

21

±== sms

Lg BL Lµµ −=

In analogy with the orbital angular momentum of the electron

A spin (intrinsic) angular momentum can be defined:

Sg BS S µµ −=

2=Sg

1=Lg

a) in relativistic Dirac theory

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:

ee r

ecm0

22

4πε= Angular momentum

from spin

21

52 2 ===

eeee r

vrmIL ω

Lecture Notes Fundamental Constants 2015; W. Ubachs

Spin-orbit interaction

Frame of nucleus:

+Ze

-e

v

+Ze

-e

v−

Frame of electron:

The moving charged nucleus induces a magnetic field at the location of the electron, via Biot-Savart’s law

( )3

04 r

rvZeB ×−

=π

µ

Use vrmL ×= 200

1c

=εµ;

Then 320int 4 rcm

LZeBe

πε=

Spin of electron is a magnet with dipole

Sg BeS

µµ −=

The dipole orients in the B-field with energy

LSrcm

ZeBVe

SLS

⋅=⋅−= 3220

2

4πεµ

A fully relativistic derivation (Thomas Precession) yields with

( ) LSrVLS

⋅= ζ

( )nle rcm

Zer 3220

2 18πε

ζ =

Use:

( )( )

( )( )12/12

12/121

3

3

333

++

=++

=

nnmcZ

nar

α

Lecture Notes Fundamental Constants 2015; W. Ubachs

Fine structure in spectra due to Spin-orbit interaction

jnlj

jSLjSO

lsjmSLlsjm

lsjmVlsjmE

⋅=

=

ζ

In first order correction to energy for state

Evaluate the dot-product

SLSLSLJ

⋅++=+= 22222

Then

( )( ) ( ) ( ){ } j

jj

sjmssjj

sjmSLJsjmSL

11121

21

2

222

+−+−+=

−−=⋅

Then the full interaction energy is:

( ) ( ) ( )( )( )

++

+−+−+=

12/12111

342

nssjjhcRZESO α

S-states 0=SOEsj == ,0

P-states

3

42

2nhcRZESO

α=

2/1,1 ±== j

jlsjm

Show that the “centre-of-gravity” does not shift

Lecture Notes Fundamental Constants 2015; W. Ubachs

Kinetic Relativistic effects in atomic hydrogen

Relativistic kinetic energy

+−+

=−+

=−+=

44

4

22

22

22222

24222

821

/1

cmp

cmpmc

mccmpmc

mccmcpErelkin

First relativistic correction term

23

4

8 cmpKe

rel −=

To be used in perturbation analysis:

( )

−

+−

=Ψ−Ψ=

nRhc

nZ

cmpK jmne

jmnrel

83

121

2

8

3

24

33

4

α

∇−=

ip operator does not

change wave function

Lecture Notes Fundamental Constants 2015; W. Ubachs

Relativistic effects in atomic hydrogen: SO + Kinetic

Relativistic energy levels:

( )

−

+−=

njRhc

nZEE nnj 4

312

22 3

24α

j=1/2 levels degenerate

P.A.M. Dirac

Also the outcome of the Dirac equation

( )t

ihmcpc∂

∂=+⋅

ψψβα 2

Fine structure splitting ~ Z4α2

Lecture Notes Fundamental Constants 2015; W. Ubachs

Hyperfine structure in atomic hydrogen: 21 cm

F=1

F=0

Nucleus has a spin as well, and therefore a magnetic moment

Ig NII µµ = ;

pN M

e2

=µ

Interaction with electron spin, that may have density at the site of the nucleus (Fermi contact term)

( )22221 IJFJISI −−=⋅=⋅

Splitting : F=1 ↔ F=0 1.42 GHz or λ = 21 cm

Magnetic dipole transition Scaling: µα /2

pg

Lecture Notes Fundamental Constants 2015; W. Ubachs

Alkali Doublets

3220

2

24 rcmLSZeVSL

⋅

=πε with

( )22221 SLJLS −−=⋅

Selection rules: 1±=∆ 1,0 ±=∆j0=∆s

ns

np 2P3/2

2P1/2

2S1/2

3

42

2nhcRZESO

α=

Na doublet

Lecture Notes Fundamental Constants 2015; W. Ubachs

The Alkali Doublet Method

Lecture Notes Fundamental Constants 2015; W. Ubachs

The Many Multiplet Method

1.

2. 3.

1. Strong transitions 2. Weak, narrow transitions 3. Hyperfine transitions

Lecture Notes Fundamental Constants 2015; W. Ubachs

The Many Multiplet Method

∞−= RnZEn 2

2

2

2

0

2

24

emeR

=∞ πε

Lecture Notes Fundamental Constants 2015; W. Ubachs

Relativistic corrections in the Many Multiplet Method

( )

−

+−=∆

njnZZme

n 43

122

2 3

2

2

24 α

Relativistic correction to energy level

(note: atomic units different)

( )( )2/1

2

+≅∆

jZEnn ν

α

with: En is the Rydberg energy scaling ν is effective quantum number

Further include Many body effects

( ) ( )

−

+≅∆ ljZC

jZEnn ,,

2/112

να

These effects separate light atoms (low Z) from heavy atoms (high Z)

( ) 6.0,, ≅ljZCIn many cases:

Lecture Notes Fundamental Constants 2015; W. Ubachs

Many Multiplet Method

Dependence of the energy levels on α: (two values for different times)

in simplified form:

with:

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

=

lab

xαα

Lecture Notes Fundamental Constants 2015; W. Ubachs

Results

All allowed E1 transitions Negative signs for: d→p and p→s

Lecture Notes Fundamental Constants 2015; W. Ubachs

Quasar Lines

Lecture Notes Fundamental Constants 2015; W. Ubachs

( )

+−= 2/30 1

11z

TT

Lecture Notes Fundamental Constants 2015; W. Ubachs

“Quasar Absorptie Spectra”

To Earth

Quasar

CIV SiIV CII SiII Lyαem

Lyman limit Lyα

NVem

SiIVem

Lyβem

Lyβ SiII

CIVem

Quasar absorption spectra

Lecture Notes Fundamental Constants 2015; W. Ubachs

On weak and strong lines

E2

E1

νhEE =− 12 νCu

E2

E1

νhEE =− 12 νCu A νBu

BC =

3

38ch

BA νπ

= A1

=τ

Einstein coefficients

22

0

2

3 ijeB µ

επ

=

Dipole strength Lifetime Heisenberg uncertainty

πτ21

=Γ

Strong lines broadened Weak lines narrow

Lecture Notes Fundamental Constants 2015; W. Ubachs

Similar calculations for “laboratory lines”

Clock transitions

Ion traps Optical lattice clock

Lecture Notes Fundamental Constants 2015; W. Ubachs

“Accidental degeneracies”

Dy atom

Cingoz et al, Phys. Rev. Lett. 98, 040801 (2007)

Level A: q/(hc)= 6x103 cm-1

Level B: q/(hc)= -24x103 cm-1

∆ν(A-B) ~ 235 MHz

∆q~ 30x 103 cm-1 ~ 9x105 GHz

Hz

qq

×=

∆=

∆=

αα

αα

ααδν

15

2

108.1

2

Look for “rate of change”

1510~ −

αα per year

Hz8.1=δν per year

Precision ~ 10-8

τΒ=200 µs τΑ=7.9 µs

ΓA~ 2x104 Hz ; Line split~ 10-4

Lecture Notes Fundamental Constants 2015; W. Ubachs

Modern Clock Comparisons

Further parametrization:

( )αFRyconstf ⋅⋅=

dtdA

dtRyd

dtfd αlnlnln

⋅+=

αlnln

dFdA =

Constraints from various experiments

Cf: Peik, Nucl. Phys B Supp. 203 (2010) 18

Lecture Notes Fundamental Constants 2015; W. Ubachs

Functional dependence on fundamental constants