Fine, Hyperfine and the Kinetic Moment of Photon

Zeeman Effect

Now we will consider the influence of electron movement around the positive charged nucleus. Classically the electron (Bohr atom model) is moving along the closed path. The current arising from this motion is as follows I = −e/τ (τ: orbital period). The magnetic moment is connected with this current using the following well-known expression µ = I · F. With F = π r² (the square that is made by moving radius-vector) one can obtain µ = I · F = −e/τ (π r²). And for the kinetic moment L = mevr = me2pr²/τ where v=2πr/τ

One can obtain after that:

L = − e/2me

Since the kinetic moment is represented by operator in quantum mechanics we obtain the eigenvalues µLz of z-component (Lz y = ml h y):

µLz = − e/2meh· ml where µB = eh/2me (Bohr Magneton)

µLz = −µB· ml where µB = eh/2me

µB = eh/2me = 9,273 · 10-24 J/K = 5,656 . 10-5 eV/T = 1,4 · 1010 Hz/T (T: Tesla)

The energy of such system in the magnetic field is as follows:

EB = −µLz· B = µB · ml· B

and the total energy is E = E0 + EB , i.e. the energy levels having kinetic moment with quantum number l are splitted into 2l + 1 new levels:

| This effect is also referred to as ordinary Zeeman Effect

The hydrogen ground state (ml = 0) stays uninfluenced. It's not accidentally because H is paramagnetic element! The reason is the electron spin. Now we consider the spin as classical rotating around its axis electron that give rise to spin itself. But we can find here

µSz = −gS µB · ms EB = gS· µB· ms· B

The so-called gyromagnetic factor gS is obtained from relativist Dirac-equation with gS = 2. Experiment gives

gS = 2,00231930438(6)

This value can be explained by quantum electrodynamics which is the full quantum theory of electromagnetic field. Moreover the the electromagnetic field is expanded into normal vibrations which are quantized as for harmonic oscillator. Moreover each normal vibration has its zero energy point h w/2. It means there are fluctuating electromagnetic fields even if there is no influence coming from external fields. Though the average field is zero the squared average field meaning isn't equal to 0 that leads to the squared average fluctuations of positions.

One can find more detailed information about the Zeeman effect here.

The value gS · µB≈ 28 GHz/T shows us which energy state is higher according to electron spin interaction with magnetic field. Since now it's possible to produce magnetic fields with strengths of few Tesla we expect to detect transitions in GHz region (microwaves) when applying such fields. That's why Electron-Spin-Resonance (ESR) Spectroscopy has a deal with microwaves. When applying NMR method we have a deal with MHz region. We will discuss it more details a bit later.

The Spin-Orbital Coupling

Two possible orientations of electron spin () relative to orbital moment () give rise to energy level doubling (certainly, it's not true for s-levels) which are spectroscopically resolved since the spectral lines are detected as a pair of lines (doublet) when having enough resolution. The most well-known example is the Na D-line on wavelengths 5890 Å and 5896 Å. To understand the physics of spin-orbital coupling (or interaction) let's "carry away" to atomic nucleus: electron revolves around nucleus having electronic orbital moment . Then let's carry away to an electron and we see that it produces the magnetic field that is parallel to (because of the positive nucleus charge). Since the electron (on which we're "sitting") is situated in a rest position then we have only the interaction between magnetic field and magnetic moment S of spin. This interaction is proportional to S· . Since || and S || the interaction is proportional to · , that explains the spin-orbital coupling. The electron energy is as follows according to this type of interaction

ESL = cSL· . /h²

where cSL is the proportionality constant which can be derived from the Dirac equation.

And what about . ? Now we have the total kinetic moment as observable value:

= +

Certainly we can't set random vector in the quantum mechanics. Squaring the above-mentioned equation one can get:

J² = L² + S² + 2 . ® . = ½ (J² − L² − S²)

Since the squared kinetic moment is a conservative quantity we can substitute the quantum mechanical operators with eigenvalues (now we gonna use the small letter for eigenvalues):

. º L. S = ½ [J(J + 1) − L(L + 1) − S(S + 1)] h²

Now we obtain for the energy of spin-orbital coupling:

ESL = cSL/2[J(J + 1) − L(L + 1) − S(S + 1)]

The value cSL can be directly observed in doublet structure of optical spectra.

For H-atom one can obtain (J = L ± S, S = ½)

ESL ( ≡ J = L + ½) = ½ cSL · L

ESL (¯ ≡ J = L − ½) = − ½ cSL (L + 1)

| J = L + S ≡ 1 + ½ = 3/2
J = L − S ≡ 1 − ½ = ½

The separation between two levels is as follows

ΔESL = ESL() − ESL (¯) = ½ cSL (2L + 1)

Together with proportionality constant cSL Dirac obtained the energy separation corresponding to spin-orbital interaction:

ΔESL = |En| Z²α²/n l(l +1) ≈ 5,3 · 10-5 Z²/n l(l + 1) |En|

where α = e²/4πh²ε0c = 1/137,0359895 is the Sommerfeld fine structure constant.

The energy eigenvalues (spin-orbital + relativistic effects) for H-atom are obtained from the full Dirac theory:

EnJ = − mc²/2α²Z²/n²{1 + α²Z²/n(1/J + ½− ¾ n)}

According to this theory energy levels having the same J and n are degenerated. For instance, states 2²S½ and 2²P½ would have the same energy. Lamb and Rutherford has proved in 1947 that this degeneracy doesn't correspond to experimental facts. They excited the transition ²S½ ← ²P½ using radiospectroscopy i.e. the energy levels are splitted and are not degenerated. Moreover this splitting is only about 0,03528 cm-1.

However it's not all concerning level splitting: there is one more spectral line splitting, the so-called hyperfine structure. It's caused by interaction between magnetic field (from electron movement) and nuclear spin. For instance, the hydrogen atom has one proton with spin I = ½ and corresponding magnetic moment P = −gI eh/2mp/h = µK/h

Since proton is heavier than electron its magneton µK (µK = gI eh/2mp) is lighter by factor me/mp !

Similar to fine structure coupling we must connect and now having "new" total kinetic moment:

= +

where F y = F(F +1) h² y and Fzy = mF h y ; mF = −F, −F+1, ... F−1, F

Squaring the equation for and solving it relatively . one can obtain the hyperfine energy structure:

EHFS = cHFS/2{F(F + 1) - J(J + 1) - I(I + 1)}

The lowest term of hydrogen 1²S½ is then splitted into two terms with F = 1 (spin electron and spin proton ) and F = 0 (spin electron and spin proton ¯ antiparallel). The transition between these two levels can only occur when the spin turns over. But it's utterly rare phenomenon (in can happen only once in 10 million years in hydrogen atom) which is not observable in our Universe. It lies near n» 1,4204 GHz (λ = 21 cm).

Since this transition changes the system kinetic moment from F = 1h into F = 0h but from the fact that the total kinetic moment should be the same follows photon must take the hydrogen atom kinetic moment. And so photon possesses also the kinetic moment.

However the photon spin is as follows

SPhoton = 1h

The photon spin isn't a good designation because the "photon spin" projection on the flight axis can take values m = +1 or m = -1 for right- and lefthand circular polarized light, correspondingly. And it can never be m=0, that would correspond to the longitudial polarization which is never realized!

The proton spin has interesting consequences for optical spectra since we obtain the selection rule for transitions. We will have for the final total kinetic moment f of the system that has been firstly at some initial state i according to the kinetic moment conservation law:

i = f + Photon

Certainly Photon shouldn't be necessarily equal to because photon can also possess (together with spin) the orbital moment relative to atom. The photon emission with unchanged orbital moment visually means that photon has left the boundary areas. Nevertheless this process is utterly improbable so that quantum numbers Ji and Jf has the following relation according to the kinetic moment conservation law:

ΔJ = Jf − Ji = 0 , ±1

ΔJ = −1 ΔM = 0 , ±1 P-Branch | ΔJ = 0 ΔM = ±1 Q-Branch | ΔJ = +1 ΔM = 0 , ±1 R-Branch

If Jf = 0 and Ji = 0 then ΔJ=0 is strictly forbidden according to the kinetic moment conservation law. If there is small interaction between and then there is special selection rule for spin that states:

ΔS = 0

The selection rule for electron orbital moment has been given here without any derivation:

Δl = ±1

The possible transitions are shown on the illustration to the left (Balmer a-Line, n = 3 ↔ n = 2). Since H atoms move there are Doppler distribution of these optical lines, that can be resolved by using Doppler free spectroscopy which can help us to prove the theory.

Certainly we must have the the kinetic moment conservation also for increasing rotation of molecule. In the simplest case of linear molecule for which electron contribution into kinetic moment is small so that ΔJ = ±1, i.e. the rotation can be increased by 1h or decreased by 1h when photon is absorbed. It's said about P-Branch (the rotation is decreased by 1h) and R-Branch (the rotation is increased by 1h).

If total electron kinetic moment isn't equal to 0 or if we have a deal with non-linear molecule then the kinetic impulse vector character according to the selection rule undergoes some changes, i.e. there is also transition DJ = 0 (Q-Branch) possible (when J = 0 → J = 0 !).

When we're talking about the Raman-Spectroscopy we're talking about two photons: the molecule undergoes transition from the initial state i into virtual state and then there is "emission" from this virtual state into the final state f. Since the kinetic moment of both photons should compensate or add up to each other (the kinetic moment conservation law) it follows the selection rules for the Raman spectroscopy: ΔJ = 0, ±2. When ΔJ = 0 the kinetic moments of both photons compensate each other (¯); and when ΔJ = ±2 they are added to each other ( or ¯¯). One can talk about S-Branch (ΔJ = +2) and O-Branch (ΔJ = -2) in this case.