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Lectures 7-8: Fine and hyperfine structure of hydrogenLectures 7-8: Fine and hyperfine structure of hydrogen

o Fine structure

o Spin-orbit interaction.

o Relativistic kinetic energy correction

o Hyperfine structure

o The Lamb shift.

o Nuclear moments.

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Spin-orbit coupling in H-atomSpin-orbit coupling in H-atom

o Fine structure of H-atom is due to spin-orbit interaction:

o If L is parallel to S => J is a maximum => high energy configuration.

o Angular momenta are described in terms of quantum numbers, s, l and j:

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ΔE so = αZh

2m2cr3ˆ S ⋅ ˆ L

+Ze-e€

ˆ L

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ˆ S

is a max

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ˆ J

+Ze-e€

ˆ L

€

ˆ S

is a min

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ˆ J

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ˆ J = ˆ L + ˆ S

ˆ J 2 = ( ˆ L + ˆ S )( ˆ L + ˆ S ) = ˆ L ̂ L + ˆ S ̂ S + 2 ˆ S ⋅ ˆ L

ˆ S ⋅ ˆ L =1

2( ˆ J ⋅ ˆ J − ˆ L ⋅ ˆ L − ˆ S ⋅ ˆ S )

=> ˆ S ⋅ ˆ L =h2

2[ j( j +1) − l(l +1) − s(s +1)]

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∴ΔE so = αZh3

4m2c

1

r3[ j( j +1) − l(l +1) − s(s +1)]

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Spin-orbit effects in H fine structureSpin-orbit effects in H fine structure

o For practical purposes, convenient to express spin-orbit coupling as

where is the spin-orbit coupling constant. Therefore, for the 2p electron:

E

2p1

j = 3/2

j = 1/2

+1/2a Angular momenta aligned

-a Angular momenta opposite

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ΔE so =a

2

3

2

3

2+1

⎛

⎝ ⎜

⎞

⎠ ⎟−1(1+1) −

1

2

1

2+1

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥= +

1

2a

ΔE so =a

2

1

2

1

2+1

⎛

⎝ ⎜

⎞

⎠ ⎟−1(1+1) −

1

2

1

2+1

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥= −a

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ΔE so =a

2j j +1( ) − l(l +1) − s s +1( )[ ]

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a = Ze2μ0h2 /8πm2r3

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Spin-orbit coupling in H-atomSpin-orbit coupling in H-atom

o The spin-orbit coupling constant is directly measurable from the doublet structure of spectra.

o If we use the radius rn of the nth Bohr radius as a rough approximation for r (from Lectures 1-2):

o Spin-orbit coupling increases sharply with Z. Difficult for observed for H-atom, as Z = 1: 0.14 Å (H), 0.08 Å (H), 0.07 Å (H).

o Evaluating the quantum mechanical form,

o Therefore, using this and s = 1/2:

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r = 4πε0

n2h2

mZe2

=> a ~Z 4

n6

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a ~Z 4

n3 l(l +1)(2l +1)[ ]

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ΔE so =Z 2α 4

2n3mc 2 [ j( j +1) − l(l +1) − 3/4]

l(l +1)(2l +1)

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Term Symbols Term Symbols

o Convenient to introduce shorthand notation to label energy levels that occurs in the LS coupling regime.

o Each level is labeled by L, S and J: 2S+1LJ

o L = 0 => So L = 1 => Po L = 2 =>Do L = 3 =>F

o If S = 1/2, L =1 => J = 3/2 or 1/2. This gives rise to two energy levels or terms, 2P3/2 and 2P1/2

o 2S + 1 is the multiplicity. Indicates the degeneracy of the level due to spin.o If S = 0 => multiplicity is 1: singlet term.o If S = 1/2 => multiplicity is 2: doublet term.o If S = 1 => multiplicity is 3: triplet term.

o Most useful when dealing with multi-electron atoms.

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Term diagram for H fine structureTerm diagram for H fine structure

o The energy level diagram can also be drawn as a term diagram.

o Each term is evaluated using: 2S+1LJ

o For H, the levels of the 2P term arising from spin-orbit coupling are given below:

E

2p1 (2P)

2P3/2

2P1/2

+1/2a Angular momenta aligned

-a Angular momenta opposite

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Hydrogen fine structureHydrogen fine structure

o Spectral lines of H found to be composed of closely spaced doublets. Splitting is due to interactions between electron spin S and the orbital angular momentum L => spin-orbit coupling.

o H line is single line according to the Bohr or Schrödinger theory. Occurs at 656.47 nm for H and 656.29 nm for D (isotope shift, Δ~0.2 nm).

o Spin-orbit coupling produces fine-structure splitting of ~0.016 nm. Corresponds to an internal magnetic field on the electron of about 0.4 Tesla.

H

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Relativistic kinetic energy correctionRelativistic kinetic energy correction

o According to special relativity, the kinetic energy of an electron of mass m and velocity v is:

o The first term is the standard non-relativistic expression for kinetic energy. The second term is the lowest-order relativistic correction to this energy.

o Using perturbation theory, it can be show that

o Produces an energy shift comparable to spin-orbit effect.

o A complete relativistic treatment of the electron involves the solving the Dirac equation.

€

T ≈p2

2m−

p4

8m3c 2

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ΔE rel = −Z 2α 4

n3mc 2 1

2l +1−

3

8n

⎛

⎝ ⎜

⎞

⎠ ⎟

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Total fine structure correctionTotal fine structure correction

o For H-atom, the spin-orbit and relativistic corrections are comparable in magnitude, but much smaller than the gross structure.

Enlj = En + ΔEFS

o Gross structure determined by En from Schrödinger equation. The fine structure is determined by

o As En = -Z2E0/n2, where E0 = 1/22mc2, we can write

o Gives the energy of the gross and fine structure of the hydrogen atom.

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ΔEFS = ΔE so + ΔE rel = −Z 4α 4

2n3mc 2 1

2l +1−

3

8n

⎛

⎝ ⎜

⎞

⎠ ⎟

€

EH −atom = −Z 2E0

n21+

Z 2α 2

n

1

j +1/2−

3

4n

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟

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Fine structure of hydrogenFine structure of hydrogen

o Energy correction only depends on j, which is of the order of 2 ~ 10-4 times smaller that the principle energy splitting.

o All levels are shifted down from the Bohr energies.

o For every n>1 and l, there are two states corresponding to j = l ± 1/2.

o States with same n and j but different l, have the same energies (does not hold when Lamb shift is included). i.e., are degenerate.

o Using incorrect assumptions, this fine structure was derived by Sommerfeld by modifying Bohr theory => right results, but wrong physics!

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Hyperfine structure: Lamb shiftHyperfine structure: Lamb shift

o Spectral lines give information on nucleus. Main effects are isotope shift and hyperfine structure.

o According to Schrödinger and Dirac theory, states with same n and j but different l are degenerate. However, Lamb and Retherford showed in 1947 that 22S1/2 (n = 2, l = 0, j = 1/2) and 22P1/2 (n = 2, l = 1, j = 1/2) of H-atom are not degenerate.

o Experiment proved that even states with the same total angular momentum J are energetically different.

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Hyperfine structure: Lamb shiftHyperfine structure: Lamb shift

1. Excite H-atoms to 22S1/2 metastable state by e- bombardment. Forbidden to spontaneuosly decay to 12S1/2 optically.

2. Cause transitions to 22P1/2 state using tunable microwaves. Transitions only occur when microwaves tuned to transition frequency. These atoms then decay emitting Ly line.

3. Measure number of atoms in 22S1/2 state from H-atom collisions with tungsten (W) target. When excitation to 22P1/2, current drops.

4. Excited H atoms (22S1/2 metastable state) cause secondary electron emission and current from the target. Dexcited H atoms (12S1/2 ground state) do not.

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Hyperfine structure: Lamb shiftHyperfine structure: Lamb shift

o According to Dirac and Schrödinger theory, states with the same n and j quantum numbers but different l quantum numbers ought to be degenerate. Lamb and Retherford showed that 2 S1/2 (n=2, l=0, j=1/2) and 2P1/2 (n=2, l=1, j=1/2) states of hydrogen atom were not degenerate, but that the S state had slightly higher energy by E/h = 1057.864 MHz.

o Effect is explained by the theory of quantum electrodynamics, in which the electromagnetic interaction itself is quantized.

o For further info, see http://www.pha.jhu.edu/~rt19/hydro/node8.html

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Hyperfine structure: Nuclear momentsHyperfine structure: Nuclear moments

o Hyperfine structure results from magnetic interaction between the electron’s total angular momentum (J) and the nuclear spin (I).

o Angular momentum of electron creates a magnetic field at the nucleus which is proportional to J.

o Interaction energy is therefore

o Magnitude is very small as nuclear dipole is ~2000 smaller than electron (~1/m).

o Hyperfine splitting is about three orders of magnitude smaller than splitting due to fine structure.

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ΔEhyperfine = − ˆ μ nucleus ⋅ ˆ B electron ∝ ˆ I ⋅ ˆ J

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Hyperfine structure: Nuclear momentsHyperfine structure: Nuclear moments

o Like electron, the proton has a spin angular momentum and an associated intrinsic dipole moment

o The proton dipole moment is weaker than the electron dipole moment by M/m ~ 2000 and hence the effect is small.

o Resulting energy correction can be shown to be:

o Total angular momentum including nuclear spin, orbital angular momentum and electron spin is

where

o The quantum number f has possible values f = j + 1/2, j - 1/2 since the proton has spin 1/2,.

o Hence every energy level associated with a particular set of quantum numbers n, l, and j will be split into two levels of slightly different energy, depending on the relative orientation of the proton magnetic dipole with the electron state.

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ˆ μ p = gp

e

Mˆ I

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ˆ F = ˆ I + ˆ J

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F = f ( f +1)h

Fz = m f h

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ΔE p =gpe

2

mMc 2r3ˆ I ⋅ ˆ J

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Hyperfine structure: Nuclear momentsHyperfine structure: Nuclear moments

o The energy splitting of the hyperfine interaction is given by

where a is the hyperfine structure constant.

o E.g., consider the ground state of H-atom. Nucleus consists of a single proton, so I = 1/2. The hydrogen ground state is the 1s 2S1/2 term, which has J = 1/2. Spin of the electron can be parallel (F = 1) or antiparallel (F = 0). Transitions between these levels occur at 21 cm (1420 MHz).

o For ground state of the hydrogen atom (n=1), the energy separation between the states of F = 1 and F = 0 is 5.9 x 10-6 eV.

F = 1

F = 0

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

21 cm radio map of the Milky Way

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ΔE HFS =a

2f f +1( ) − i(i +1) − j j +1( )[ ]

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Selection rulesSelection rules

o Selection rules determine the allowed transitions between terms.

Δn = any integer Δl = ±1 Δj = 0, ±1 Δf = 0, ±1

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Summary of Atomic Energy ScalesSummary of Atomic Energy Scales

o Gross structure:o Covers largest interactions within the atom:

o Kinetic energy of electrons in their orbits.o Attractive electrostatic potential between positive nucleus and negative electronso Repulsive electrostatic interaction between electrons in a multi-electron atom.

o Size of these interactions gives energies in the 1-10 eV range and upwards.o Determine whether a photon is IR, visible, UV or X-ray.

o Fine structure:o Spectral lines often come as multiplets. E.g., H line.

=> smaller interactions within atom, called spin-orbit interaction.o Electrons in orbit about nucleus give rise to magnetic moment

of magnitude B, which electron spin interacts with. Produces small shift in energy.

o Hyperfine structure:o Fine-structure lines are split into more multiplets. o Caused by interactions between electron spin and nucleus spin.o Nucleus produces a magnetic moment of magnitude

~B/2000 due to nuclear spin. o E.g., 21-cm line in radio astronomy.