Atomic Physics Revision

ATOMIC PHYSICS Revision Lectures Lecture 1 Schrödinger equation Atomic structure and notation Spin-orbit and fine structure Lecture 2 Atoms in magnetic fields Radiation and Lasers Lecture 3 Nuclear effects: hyperfine structure Two electron atoms X-rays

Stern Gerlach Experiment • Demonstrates quantization of direction for interacting vectors • A consequence of quantization of energy E • Interaction energy for magnetic dipole µ in field B is:

E = µ.B = µ.B cosθ • Since µ and B are constant in time, cosθ is quantized • Force on atoms is:

dB k F = −µ. dz

Directional quantization • Direction defined by magnetic field along the z-axis • The magnetic moment due to orbital angular momentum is m • The magnetic moment due to spin s is ms • Each orientation has different energy in field B • In zero field the states are degenerate

z

m +1 0 -1

ms +½

-½

z

The Schrödinger Equation −

2

∇ + V (r ) ψ (r ,θ , φ ) = Enψ (r ,θ , φ ) 2

2m

ψ ( r ,θ , φ ) = R (r )Φ (θ , φ ) 2

2

Ze e V (r ) = + ξ ( s.l ) + ζ ( µ .B ) + 4πε o r 4πε o r12 Nuclear Coulomb

Spin-Orbit

External Field

{electron-electron interaction}

Energy Level Diagrams SPECTROSCOPIC NOTATION [configuration)

2S+1

LJ

eg. Na: (1s22s22p63s) 32S1/2

2

2

1/2

and 3p P3/2

Spectroscopic Notation • One electron atoms e.g. Na (configuration) 2s+1 j

1s22s22p6 3s 2S1/2 1s22s22p6 3p 2P1/2 , 2P3/2

• Two electron atoms e.g. Mg (configuration) 2S+1LJ

1s22s22p6 3s2 1S0 1s22s22p6 3s3p 1P1 3P2,1,0

Vector Model: spin-orbit interaction VECTOR MODEL

• Orbital and spin s angular momenta give magnetic moments m and ms • Orientation of m and ms is quantized • Magnetic interaction gives precessional motion • Energy of precession shifts the energy level depending on relative orientation of and s

µ

µ

Spin-orbit precession (magnetic interaction)

Fine structure from spin-orbit splitting in n=2 level of Hydrogen VECTOR MODEL

Spin-orbit splitting of n=2 level in H Spin-orbit precession (magnetic interaction)

2

P3/2

2

n=2

S1/2

2

P1/2

Bohr + degenerate states

Spin-Orbit

2

Note: only the P term is split, 2

For n = 2,

2

2

= 1, P1/2 and P3/2

no splitting of S as = 0

=1

Fine structure in n=2 level in hydrogen 2

P3/2

n=2

2

S1/2

2

P3/2

2

P3/2

2

P1/2

2

2

Bohr + degenerate states

Spin-Orbit

+

2

2

S1/2 P1/2

Relativity

+

QED

S1/2 P1/2

•

•

•

•

Separation of 2S1/2 -2P1/2: the Lamb shift, a QED effect, was first measured by RF spectroscopy in Hydrogen; microwave transition at 1057 MHz Optical measurement uses emission of Balmer α line from Deuterium gas discharge

Fine Structure in Atomic Hydrogen Cooled Deuterium spectrum 2

n = 3 to n = 2 P3/2

n=3

Spectrum formed using Fabry-Perot interferometer for high resolution. Shift of 2S1/2 - 2P1/2 is resolved

2

2

S1/2 P1/2

2

S

2

P3/2

n=2

2

S1/2

2

P1/2

2

P

2

D

End lecture 1 • Example finals questions

(1996) A3 question 1

(1995) A3 Question 2

Normal Zeeman Effect Vector Model • Magnetic moment due to orbital motion µ = -g µB • Energy of precession in external field B is: ∆EZ = µ.B

Orbital magnetic moment in external magnetic field

Bext

µ

Magnetic dipole precession

Normal Zeeman Effect • Perturbation Energy (precession in B field) ‹∆EZ› = ‹µB.Bext›

Orbital magnetic moment in external magnetic field Bext 2

m +2

1

=2

• ‹∆EZ› = g µB.Bextm

+1 0 -1 -2

• 2 +1 sub-levels -2

• Separation of levels: µBBext

Normal Zeeman Effect m

• Selection rules:

2 1

∆m = 0, +1

0 -1 -2

=2

• Polarization of light: 1

∆m = 0, ∆m = +1

π

along z-axis

σ+ or

0 -1

=1

-

σ

σ−

σ, circular viewed along z-axis π, plane viewed along x,y-axis

ωο

π ωο

σ

Anomalous Zeeman Effect ext

Vector Model j

• Spin-orbit coupled motion in external magnetic field. s

• Projections of and s on Bext vary owing to precession around j. • m and ms are no longer good quantum numbers

ext

ext

j

j

s

Anomalous Zeeman Effect •

Total magnetic moment µTotal precesses around effective magnetic moment µ

•

Effective magnetic moment due to total angular momentum µ = -g µB

•

s

j

µ

Landé g-factor is:

g j = 1+ •

Bext

j ( j + 1) − l (l + 1) + s ( s + 1) 2 j ( j + 1)

Energy of precession in external field B is: ∆EAZ = − µ .Bext = g µBBextmj

µs µ

Total

µeff

Anomalous Zeeman Effect: Na D - lines mj 3/2 1/2 -1/2 -3/2

mj 1/2 -1/2 D2

D1 1/2

1/2

-1/2

-1/2

σπ πσ

σσ π π σσ

D1

D2

Selection rules: ∆mj = 0, +1

Magnetic effects in one - electron atoms

2

P

2

S

D1

D2

σπ πσ

σσπ πσσ

D1

D2

σ

π

σ

1

1/.2

0

1/2

-1 1

1/2 -1/2

0

-1/2

-1

-1/2

Atomic physics of lasers N2

ρ(ν)B12

ρ(ν)B21

A21 N1

Rate equation: dN 2 = B12 ρ (ν ) N1 − B21 ρ (ν ) N 2 − AN 2 dt Steady state: N2 B12 ρ (ν ) hν = = exp − N1 A + B21 ρ (ν ) kT ρ (ν ) = A

ρ(ν) Energy density of incident radiation ρ(ν)B12

Rate of stimulated absorption

ρ(ν)B21

Rate of stimulated emission

A21

Rate of spontaneous emission

1

[B12 exp(hν

kT ) − B21 ]

Planck law:

Hence:

8πhν 3 1 ρ (ν ) = c 3 [exp(hν kT ) − 1]

B12 = B 21 = B 8πhν 3 A= B 3 c

Laser operation Rate of change of photon number density n

Spectral mirror filter

dN dn =− 2 dt dt

∴

Amplifying medium

mirror

photon lifetime in cavity, τp

n dn = ( N 2 − N 1 ) B12 ρ (ν ) − ρ (ν ) ∆ν = n.hν and τp dt c3 A B12 ρ (ν ) = n = Qn 2 8πν ∆ν dn 1 n = ( N 2 − N1 )Q − dt τp

∴

n(t ) = n(0) exp ( N 2 − N1 )Q −

1

τp

t

Gain if ( N 2 − N1 ) >

1 Qτ p

End lecture 2

(1998) A3 question 1

Nuclear spin hyperfine structure • Nuclear spin µI interacts with magnetic field Bo from total angular momentum of electron, F

Vector model of Nuclear Spin interaction

I

F

µI = gIµNI • Interaction energy: H’ = AJ I.J • Shift in energy level: ∆Ehfs =

J

J = Total electronic angular momentum I = Nuclear spin F = Total atomic angular momentum F = I +J 2

2

2

I.J = 1/2 { F - I - U }

∆Ehfs = (1/2) AJ{F(F+1) – I(I+1) – J(J+1)

Hyperfine structure of ground state of hydrogen •

•

•

Hyperfine Energy shift: ∆Ehfs = AJ I.J

Hyperfine Structure of Hydrogen Ground State

Fermi contact interaction: spin of nucleus with spin of electron

F=1 A/4

J=1/ 2

Hyperfine interaction constant

1420 Mhz (21 cm.)

I=1/2

AJ → AS

3A/4

•

I = ½ , J = ½, F = 1 or 0

•

I.J = {F(F+1) – I(I+1) – J(J+1)} = ¼ or -¾

•

Hyperfine splitting ∆E = AS

F=0

A J = A S = µo g S µ B 2 3

µI

I

ψ (0) P

2

Determination of Nuclear Spin I from hfs spectra • Hyperfine interval rule ∆E(F) – ∆E(F-1) = AJF • Relative intensity in transition to level with no (or unresolved) hfs is proportional to 2F+1 • The number of hfs spectral components is (2I+1) for I < J (2J+1) for I > J

2-electron atoms Schrödinger equation:

−

2

∇1 +

2

2

2m

∇ 2 + V (r ) ψ (r ,θ , φ ) = Enψ (r ,θ , φ ) 2

2m

2

2

Ze e + + ξ i ( s.l ) V (r ) = 4πε o r12 i=1, 2 i =1, 2 4πε o ri When Electrostatic interaction between electrons is the dominant perturbation → LS labelled terms

2 electron atoms: LS coupling • Electrostatic interaction gives terms labelled by L and S

1

S

P

singlet

J

• L = 1 + 2, S = s1 + s2 • J=L+S

L

• Terms are split by electrostatic interaction into Singlet and Triplet terms

3

triplet

1

P1

P

singlet

S J 3

• Magnetic interaction (spin-orbit) splits only triplet term

L

2ξ triplet ξ

P2

3 3

P1 P0

2-electron atoms: symmetry considerations

Ψtotal = Φ(r ,θ , φ ) χ Spatial function

↑↓

Spin function

= Antisymmetric function

• Singlet terms (S = 0): χ is antisymmetric, Φ is symmetric spatial overlap allowed (Pauli Exclusion Principle) increases electrostatic repulsion e2/r12 • Triplet terms (S = 1): χ is symmetric, Φ is antisymmetric spatial overlap forbidden (Pauli Principle) reduces electrostatic repulsion e2/r12

Term diagram of Magnesium Singlet terms 1

So

1

P1

Triplet terms

1

3

D2

S

3

P

3

3pn

D 3p

ns 5s 1

3s3d D2

4s 1

3s3p P1 3

resonance line (strong)

3s3p P2,1,0

intercombination line (weak) 3s

• •

2 1

S0

Singlet and Triplet terms form separate systems Strong LS coupling: Selection Rule ∆S = 0 (weak intercombination lines) Singlet-Triplet splitting >> fine structure of triplet terms i.e Electrostatic interaction >> Magnetic (spin-orbit) Triplet splitting follows interval rule ∆E J

2

X-ray spectra • Wavelengths λ fit simple formula

Intensity

Bremstrahlung

• All lines of series appear together

• Above a certain energy no new series appear

0.1 nm

Characteristic X-rays Intensity

• Threshold energy for each series

Wavelength

Wavelength

0.1 nm

Generation of characteristic X-Rays

e

K

-

ejected electron

Moseley Plot LI LII

ν

LIII

R

e

-

X-ray Inner shell energy level

K – series: L – series:

Atomic Number, Z

νK = R

(Z - σ K )2 − (Z - σ i )2

νL = R

(Z - σ L )2 − (Z - σ i )2

12

22

ni2

ni2

Fine structure of X-Rays

X-Ray Series

∆

+1

∆j = 0, + 1

n=4

N

1

M-series M L-series L

αβγ

αβγ

n=3

L

K

αβγ

1 1/2 0 1/2

n=2

K-series

j 3/2

α2

α1

n=1

K

0 1/2

Series lines labelled by α,β, χ etc for decreasing wavelength λ Lines have fine structure due to spin-orbit effect of “hole” in filled shell

5.8Z 4 ∆Efs = 3 cm -1 n ( + 1)

Absorption of X-rays The Auger Effect

X-Ray absorption spectra

Absorption coefficient

L-edge K-edge

Kinetic Energy

M-edge

(EK - E L-) - EL

LI

e2-

e1-

LII LIII

L Wavelength

Potential Energy

• Absorption decreases below absorption edge due to effect of conservation of momentum • Fine structure seen at edges • Auger effect leads to emission of two electrons following X-ray absorption by inner shell electron

(EK - EL)

K X-ray absorption electron emitted

Second electron emitted