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Atomic Physics
P. Ewart
4.2.6 Evaluation of
ˆs · ˆl
- 1 Introduction Contents
- 2 Radiation and Atoms
- 2.1 Width and Shape of Spectral Lines
- 2.1.1 Lifetime Broadening
- 2.1.2 Collision or Pressure Broadening
- 2.1.3 Doppler Broadening
- 2.2 Atomic Orders of Magnitude
- 2.2.1 Other important Atomic quantities
- 2.3 The Central Field Approximation
- 2.4 The form of the Central Field
- 2.5 Finding the Central Field
- 3 The Central Field Approximation
- 3.1 The Physics of the Wave Functions
- 3.1.1 Energy
- 3.1.2 Angular Momentum
- 3.1.3 Radial wavefunctions
- 3.1.4 Parity
- 3.2 Multi-electron atoms
- 3.2.1 Electron Configurations
- 3.2.2 The Periodic Table
- 3.3 Gross Energy Level Structure of the Alkalis: Quantum Defect
- 4 Corrections to the Central Field: Spin-Orbit interaction
- 4.1 The Physics of Spin-Orbit Interaction
- 4.2 Finding the Spin-Orbit Correction to the Energy
- 4.2.1 The B-Field due to Orbital Motion
- 4.2.2 The Energy Operator
- 4.2.3 The Radial Integral
- 4.2.4 The Angular Integral: Degenerate Perturbation Theory
- 4.2.5 Degenerate Perturbation theory and the Vector Model
- using DPT and the Vector Model 〉
- 4.3 Spin Orbit Interaction: Summary
- 4.4 Spin-Orbit Splitting: Alkali Atoms
- 4.5 Spectroscopic Notation
- 5 Two-electron Atoms: Residual Electrostatic Effects and LS-Coupling
- 5.1 Magnesium: Gross Structure
- 5.2 The Electrostatic Perturbation
- 5.3 Symmetry
- 5.4 Orbital effects on electrostatic interaction in LS-coupling
- 5.5 Spin-Orbit Effects in 2-electron Atoms
- 6 Nuclear Effects on Atomic Structure
- 6.1 Hyperfine Structure
- 6.2 The Magnetic Field of Electrons
- 6.3 Coupling of I and J
- 6.4 Finding the Nuclear Spin, I
- 6.5 Isotope Effects
- 7 Selection Rules
- 7.1 Parity
- 7.2 Configuration
- 7.3 Angular Momentum Rules
- 8 Atoms in Magnetic Fields
- 8.1 Weak field, no spin
- 8.2 Weak Field with Spin and Orbit
- 8.2.1 Anomalous Zeeman Pattern
- 8.2.2 Polarization of the radiation
- 8.3 Strong fields, spin and orbit
- 8.4 Intermediate fields
- 8.5 Magnetic field effects on hyperfine structure
- 8.5.1 Weak field
- 8.5.2 Strong field
- 9 X-Rays: transitions involving inner shell electrons
- 9.1 X-ray Spectra
- 9.2 X-ray series
- 9.3 Fine structure of X-ray spectra
- 9.4 X-ray absorption
- 9.5 Auger Effect
- 10 High Resolution Laser Spectroscopy
- 10.1 Absorption Spectroscopy
- 10.2 Laser Spectroscopy
- 10.3 Spectral resolution
- 10.4 “Doppler Free” spectroscopy
- 10.4.1 Crossed beam spectroscopy
- 10.4.2 Saturation Spectroscopy
- 10.4.3 Two-photon-spectroscopy
- 10.5 Calibration of Doppler-free Spectra
- 10.6 Comparison of “Doppler-free” Methods
Atomic Physics, P. Ewart 2 Radiation and Atoms
y 1 y 2
y( ) =t y 1 +y 2
IY( )t I^2
Y( )t Y(^ t+t)
Oscillating charge cloud: Electric dipole
I Y( + t)t I
2
Figure 1: Evolution of the wavefunction of a system with time.
So the perturbation produces a charge cloud that oscillates in space – an oscillating dipole. This radiates dipole radiation. Whether or not we get a charge displacement or dipole will depend on the symmetry properties of the two states ψ 1 , and ψ 2. The rules that tell us if a dipole with be set up are called “selection rules”, a topic to which we will return later in the course.
2.1 Width and Shape of Spectral Lines
The radiation emitted (or absorbed) by our oscillating atomic dipole is not exactly monochromatic, i.e. there will be a range of frequency values for ω 12. The spectral line observed is broadened by one, or more, processes. A process that affects all the atoms in the same way is called “Homoge- neous Broadening”. A process that affects different individual atoms differently is “Inhomogeneous Broadening”. Examples of homogeneous broadening are lifetime (or natural) broadening or collision (or pres- sure) broadening. Examples of inhomogeneous broadening are Doppler broadening and crystal field broadening.
2.1.1 Lifetime Broadening
This effect may be viewed as a consequence of the uncertainty principle
∆E∆t ∼ ℏ (6)
Since E = ℏω, ∆E = ℏ∆ω and if the time uncertainty ∆t is the natural lifetime of the excited atomic state, τ , we get a spread in frequency of the emitted radiation ∆ω
∆ω τ ∼ 1 or ∆ω ∼
τ
Atomic Physics, P. Ewart 2 Radiation and Atoms
The lifetime, τ , is a statistical parameter related to the time taken for the population of the excited state to decay to 1/e of its initial value. This exponential decay is reflected in the experimental decay of the amplitude E(t) of the light wave emitted. The frequency (or power) spectrum of an exponentially decaying amplitude is a Lorentzian shape for the intensity as a function of frequency
I(ω) ∼
(ω − ω 0 )^2 + (1/τ )^2
The full width at half-maximum, FWHM, is then
∆ω ∼ 2
τ
A typical lifetime τ ∼ 10 −^8 sec.
N( ) t I( w)
Time, t frequency,^ w
Number of excited atoms^ Intensity spectrum
Exponential decay Lorentzian lineshape
Figure 2: Decay of excited state population N (t) leads to similar exponential decay of radiation amplitude, giving a Lorentzian spectrum.
2.1.2 Collision or Pressure Broadening
A collision with another atom while the atom is radiating (oscillating) disrupts the phase of the wave. The wave is therefore composed of various lengths of uninterrupted waves. The number of uninterrupted waves decays exponentially with a 1/e time τc, which is the mean time between collisions. At atmospheric pressure this is typically ∼ 10 −^10 sec. The exponential decay of the coherent oscillations again leads to a Lorentzian lineshape.
N( ) t I( w)
Time, t frequency,^ w
Number of uncollided atoms^ Intensity spectrum
Exponential decay Lorentzian lineshape
Figure 3: Decay in number of undisturbed atoms radiating leads to decay in amplitude of wave undisturbed by a phase changing collision. The associated frequency spectrum is again Lorentzian.
2.1.3 Doppler Broadening
Atoms in a gas have a spread of speeds given by the Maxwell-Boltzmann distribution. The Doppler shift of the light emitted is therefore different for the atoms moving at different speeds. There is then
Atomic Physics, P. Ewart 2 Radiation and Atoms
We can compare ∼ 2 eV with thermal energy or ∼ kT i.e. the mean Kinetic energy available from heat, 401 eV. This is not enough to excite atoms by collisions so atoms will mostly be in their ground state.
2.2.1 Other important Atomic quantities
Atomic size: Bohr radius
a 0 = 4 π 0 ℏ^2 me^2 = 0. 53 × 10 −^10 m (17)
Ionisation Energy of Hydrogen
EH =
me^4 (4π 0 )^2 2 ℏ^2
= 13.6 eV (18)
Rydberg constant
R =
EH
hc
= 1. 097 × 107 m−^1 (19)
R is useful is relating wavelengths λ to energies of transitions since wavenumber ¯ν = (^) λ^1 in units of m−^1.
Fine structure constant
α = e^2 4 π 0 ℏc
This is a dimensionless constant that gives a measure of the relative strength of the electromagnetic force. It is actually also the ratio of the speed ve of the electron in the ground state of H to c, the speed of light. α = ve/c and so it is a measure of when relativistic effects become important.
Bohr magneton
μB =
eℏ 2 m
∼ 9. 27 × 10 −^24 JT−^1 (21)
This is the basic unit of magnetic moment corresponding to an electron in a circular orbit with angular momentum ℏ, or one quantum of angular momentum. As well as having orbital angular momentum the electron also has intrinsic spin and spin magnetic moment μS = 2μB. A proton also has spin but because its mass is ∼ 2000 larger than an electron its magnetic moment is ∼ 2000 times smaller. Nuclear moments are in general ∼ 2000 times smaller than electron moments.
2.3 The Central Field Approximation
We can solve the problem of two bodies interacting with each other via some force e.g. a star and one planet with gravitational attraction, or a proton and one electron – the hydrogen atom. If we add an extra planet then things get difficult. If we add any more we have a many body problem which is impossible to solve exactly. Similarly, for a many electron atom we are in serious difficulty
- we will need to make some approximation to simplify the problem. We know how to do Hydrogen; we solve the Schr¨odinger equation:
ℏ^2
2 m
∇^2 ψ − Ze^2 4 π 0 r
ψ = Eψ (22)
We can find zero order solutions – wave functions ψ that we can use to calculate smaller perturbations e.g. spin-orbit interaction.
Atomic Physics, P. Ewart 2 Radiation and Atoms
The Hamiltonian for a many electron atom however, is much more complicated.
Hˆ =
∑^ N
i=
ℏ^2
2 m ∇^2 i − Ze^2 4 π 0 ri
i>j
e^2 4 π 0 rij
We ignore, for now, other interactions like spin-orbit. We have enough on our plate! The second term on the r.h.s. is the mutual electrostatic repulsion of the N electrons, and this prevents us from separating the equation into a set of N individual equations. It is also too large to treat as a small perturbation. We recall that the hydrogen problem was solved using the symmetry of the central Coulomb field – the 1/r potential. This allowed us to separate the radial and angular solutions. In the many electron case, for most of the time, a major part of the repulsion between one electron and the others acts towards the centre. So we replace the 1/r, hydrogen-like, potential with an effective potential due to the nucleus and the centrally acting part of the 1/rij repulsion term. We call this the Central Field U (r). Note it will not be a 1/r potential. We now write the Hamiltonian
Hˆ = Hˆ 0 + Hˆ 1 (24)
where Hˆ 0 =
i
ℏ^2
2 m ∇^2 i + U (ri)
and Hˆ 1 =
i>j
e^2 4 π 0 rij
i
Ze^2 4 π 0 ri
Hˆ 1 is the residual electrostatic interaction. Our approximation is now to assume Hˆ 1 << Hˆ 0 and then we can use perturbation theory. The procedure is to start with just Hˆ 0. Since this is a central potential the equations are separable. Solutions for the individual electrons will have the form:
ψ(n, l, ml, ms) = R′ n,l(r)Y (^) lm (θ, φ)χ(ms) (27)
So far, in this approximation the angular functions Y (^) lm (θ, φ) and spin functions χ(ms) will be the same as for hydrogen. The radial functions will be different but they will have some features of the hydrogenic radial functions. The wave function for the whole atom will consist of antisymmetric products of the individual electron wave functions. The point is that we can use these zero order wavefunctions as a basis set to evaluate the perturbation due to the residual electrostatic interactions Hˆ 1. We can then find new wavefunctions for the perturbed system to evaluate other, presumed smaller, perturbations such as spin-orbit interaction. When it comes to the test we will have to decide in any particular atom, which is the larger of the two perturbations – but more of that later.
2.4 The form of the Central Field
Calculating the form of U (ri) is a difficult problem. We can, however, get a feel for the answer in two limiting situations. Firstly, imagine one electron is taken far away from the nucleus, and the other electrons. What form of potential will it see? Clearly, there are then Z protons surrounded apparently by Z − 1 electrons in a roughly spherically symmetric cloud. Our electron then sees, at large r, a Hydrogen-like 1/r potential. Secondly, what happens when our electron goes “inside” the cloud of other electrons? Here, at small r, it sees Z protons and feels a Z/r potential. The potential then looks like it has the form
U (r) = Zeff (r)
e^2 4 π 0 r
Where Zeff (r) varies from Z at small r to 1 at large r. Note, for most of the important space U (r) is not a 1/r potential.
Atomic Physics, P. Ewart 2 Radiation and Atoms
states are not automatically orthogonal. The Hartree-Fock method uses an anti-symmetric basis set of wavefunctions. These are constructed using determinants where columns represent quantum states of individual electrons. This means that interchanging any column automatically changes the sign and makes the states correctly antisymmetric. The product states for each electron contain both space and spin functions. The potential is assumed to be the same for all the electrons. The potential is varied so as to produce the minimum energy for the system. This is the Variational Principle and has the same effect as finding a self-consistent field. The Hartree-Fock method is now the most commonly used way of finding wave functions and energy levels for many electron atoms. The wavefunctions produced are again numerical rather than analytic.
Atomic Physics, P. Ewart 3 The Central Field Approximation
3 The Central Field Approximation
To recap, we have lumped together the Coulomb attraction of the Z protons in the nucleus with the centrally acting part of the mutual electrostatic repulsion of the electrons into U (r). This Central Field goes like 1/r at large r and as Z/r at small r. At in-between values of r things are more complicated – but more interesting! The important point is that we can, in many cases, treat the residual electrostatic interaction as a perturbation, so the Hamiltonian for the Schr¨odinger equation will be
Hˆ = Hˆ 0 + Hˆ 1 + Hˆ 2 + ... (29)
Hˆ 1 will be the perturbation due to residual electrostatic interactions, Hˆ 2 that due to spin-orbit interactions. We will mostly deal with the cases where Hˆ 1 > Hˆ 2 but this won’t be true for all elements. We can also add smaller perturbations, Hˆ 3 etc due to, for example, interactions with external fields, or effects of the nucleus (other than its Coulomb attraction). The Central Field Approximation allows us to find solutions of the Schr¨odinger equation in terms of wave functions of the individual electrons: ψ(n, l, ml, ms) (30)
The zero-order Hamiltonian Hˆ 0 due to the Central Field will determine the gross structure of the energy levels specified by n, l. The perturbation Hˆ 1 , residual electrostatic, will split the energy levels into different terms. The spin orbit interaction, Hˆ 2 further splits the terms, leading to fine structure of the energy levels. Nuclear effects lead to hyperfine structures of the levels. Within the approximation we have made, so far, the quantum numbers ml, ms do not affect the energy.The energy levels are therefore degenerate with respect to ml, ms. The values of ml, ms or any similar magnetic quantum number, specify the state of the atom. There are 2l + 1 values of ml i.e. 2 l + 1 states and the energy level is said to be (2l + 1)-fold degenerate. The only difference between states of different ml (or ms) is that the axis of their angular momentum points in a different direction in space. We arbitrarily chose some z-axis so that the projection on this axis of the orbital angular momentum l would have integer number of units (quanta) of ℏ. (ms does the same for the projection of the spin angular momentum s on the z-axis). (Atomic physicists are often a bit casual in their use of language and sometimes use the words en- ergy level, and state (e.g. Excited state or ground state) interchangeably. This practice is regrettable but usually no harm is done and it is unlikely to change anytime soon.)
3.1 The Physics of the Wave Functions
You will know (or you should know!) how to find the form of the wave functions ψ(n, l, ml, ms) in the case of atomic hydrogen. In this course we want to understand the physics – the maths is done in the text books. (You may like to remind yourself of the maths after looking at the physics presented here!) Before we look at many electron atoms, we remind ourselves of the results for hydrogen.
3.1.1 Energy
The energy eigenvalues, giving the quantized energy levels are given by:
En =
ψn,l,ml
∣∣ Hˆ
∣∣ ψ n,l,ml
Z^2 me^4 (4π 0 )^2 2 ℏ^2 n^2
Note that the energy depends only on n, the Principal quantum number. The energy does not depend on l – this is true ONLY FOR HYDROGEN! The energy levels are degenerate in l. We represent the energy level structure by a (Grotrian) diagram.
Atomic Physics, P. Ewart 3 The Central Field Approximation
l ≥ 1. As l increases, the orbit becomes less and less elliptical until for the highest l = (n − 1) the orbit is circular. An important case (i.e. worth remembering) is l = 1 (ml = ± 1 , 0)
Y 11 = −
8 π
sin θeiφ^ (38)
Y 1 − 1 = +
8 π
sin θe−iφ^ (39)
Y 10 =
4 π
cos θ (40)
|Y 1
0
( q,f)|
2
(a)
|Y 1
( q,f)|
2
(b)
Figure 8: The angular functions, spherical harmonics, giving the angular distribution of the electron probability density.
If there was an electron in each of these three states the actual shapes change only slightly to give a spherically symmetric cloud. Actually we could fit two electrons in each l state provided they had opposite spins. The six electrons then fill the sub-shell (l = 1) In general, filled sub-shells are spherically symmetric and set up, to a good approximation, a central field. As noted above the radial functions determine the size i.e. where the electron probability is maximum.
Atomic Physics, P. Ewart 3 The Central Field Approximation
3.1.3 Radial wavefunctions
(^2 4 6 2 4 6 8 )
2 4 6 8 10 20
Zr a / (^) o (^) Zr a / (^) o
Zr a / (^) o
Ground state, n = 1, l = 0 1st excited state, n = 2, l = 0
2nd excited state, n = 3, l = 0
n = 3, l = 2
n = 2, l = 1
n = 3, l = 1
2 1.
1 0.
Figure 9: Radial functions giving the radial distribution of the probability amplitude.
NB: Main features to remember!
- l = 0, s-states do not vanish at r = 0.
- l 6 = 0, states vanish at r = 0 and have their maximum probability amplitude further out with increasing l.
- The size, position of peak probability, scales with ∼ n^2.
- The l = 0 function crosses the axis (n − 1) times ie. has (n − 1) nodes.
- l = 1 has (n − 2) nodes and so on.
- Maximum l = n − 1 has no nodes (except at r = 0)
3.1.4 Parity
The parity operator is related to the symmetry of the wave function. The parity operator takes r → −r. It is like mirror reflection through the origin.
θ → π − θ and φ → π + φ (41)
If this operation leaves the sign of ψ unchanged the parity is even. If ψ changes sign the parity is odd. Some states are not eigenstates of the parity operator i.e. they do not have a definite parity. The parity of two states is an important factor in determining whether or not a transition between them is allowed by emitting or absorbing a photon. For a dipole transition the parity must change.
Atomic Physics, P. Ewart 3 The Central Field Approximation
H: 1s He: 1s^2 Li: 1s^2 2s Be: 1s^2 2s^2 B: 1s^2 2s^2 2p C: 1s^2 2s^2 2p^2
... Ne: 1s^2 2s^2 2p^6 Na: 1s^2 2s^2 2p^6 3s
Everything proceeds according to this pattern up to Argon: 1s^2 2s^2 2p^6 3s^2 3p^6 At this point things get a little more complicated. We expect the next electron to go into the 3d sub-shell. As we have seen, however, a 3d electron is very like a 3d electron in hydrogen: it spends most of its time in a circular orbit outside the inner shell electrons. An electron in a 4s state however goes relatively close to the nucleus, inside the core, and so ends up more tightly bound - i.e. lower energy, than the 3d electron. So the next element, Potassium, has the configuration
K: 1s^2 2s^2 2p^6 3s^2 3p^6 4s Ca: 4s^2
The 3d shell now begins to fill
Sc: 1s^2 2s^2 2p^6 3s^2 3p^6 3d 4s^2
The 3d and 4s energies are now very similar and at Chromium a 3d electron takes precedence over a 4s electron
Va: 3s^2 3p^6 3d^3 4s^2 Cr: 3d^5 4s Mn: 3d^5 4s^2
As the 3d shell fills up, the successive elements, the transition elements, have interesting properties as a result of the partially filled outer shell – but this isn’t on the syllabus! There are, however, two features of the periodic table that are worth noting as consequences of the Central Field Approximation.
Rare gases These elements are chemically inert and have high ionization potentials (the energy needed to pull off a single electron). This is not because, as is sometimes (often) stated, that they have “closed shells”. They don’t all have closed shells i.e. all states for each n are full. They all do have filled s and p sub-shells
He: 1s^2 Kr: (... )4s^2 4p^6 Ne: 1s^2 2s^2 2p^6 Xe: (... )5s^2 5p^6 A: 1s^2 2s^2 2p^6 3s^2 3p^6 Rn: (... )6s^2 6p^6
As we noted earlier this leads to a spherically symmetric charge distribution. Since electrons are indistinguishable all the electrons take on a common wave function. The point is that this results in a higher binding energy for each one of the electrons. So it is harder for them to lose an electron in a chemical bond – they are chemically inert and have high ionization energies.
Atomic Physics, P. Ewart 3 The Central Field Approximation
Alkalis These are the next elements to the rare gases and have one electron outside the full (sp) sub-shells. This outer, or valence, electron therefore moves in a hydrogen like central potential. The electron is generally well-screened by (Z − 1) inner electrons from the nucleus and is easily lost to a chemical bond (ionic or co-valent). They are chemically reactive and have low ionization energies which don’t change much from one alkali to another.
3.3 Gross Energy Level Structure of the Alkalis: Quantum Defect
As noted already, the single outer electron in an alkali moves in a potential that is “central” to an excellent approximation. We are ignoring, at this stage, any other perturbations such as spin-orbit interaction. As an example we consider sodium. The ground stage (lowest energy level) has the configuration:
(closed shells) 3s.
We know the 3s penetrates the core (inner shells) and is therefore more tightly bound – lower energy – than a 3s electron would be in Hydrogen. When the electron is excited, say to 3p, it penetrates the core much less and in a 3d state its orbit is virtually circular and very close the n = 3 level of Hydrogen. The higher excited states n > 3 will follow a similar pattern. The thing to notice – and this turns out to be experimentally useful – is that the degree of core penetration depends on l and very little on n. As a result the deviation from the hydrogenic energy level is almost constant for a given l as n increases. The hydrogen energy levels can be expressed as
En = −
R
n^2
For alkalis, and to some degree for other atoms too, the excited state energies may be expressed as
En = −
R
(n∗)^2
Where n∗^ is an effective quantum number. n∗^ differs from the equivalent n-value in hydrogen by δ(l) i.e. n∗^ = n − δ(l). δ(l) is the quantum defect and depends largely on l only. It is found empirically, and it can be shown theoretically, to be independent of n. Thus all s-states will have the same quantum defect δ(s); all p-states will have the same δ(p), etc and δ(s) > δ(p) > δ(d).
For Na:
δ(s) ≈ 1. δ(p) ≈ 0. δ(d) ≈ 0
For heavier alkalis, the δ(l) generally increases as the core is less and less hydrogen-like. The ionization potential however is almost constant as noted previously. Although the increase in Z leads to stronger binding than in hydrogen the ground state electron starts in a higher n, and this almost exactly off-sets the increased attraction of the heavier nucleus. Everyone knows that the discrete wavelengths of light emitted, or absorbed, by an atom are discrete because they arise from transitions between the discrete energy levels. But how do we know which energy levels? This is where the quantum effect is very useful. We also need to remember that a transition involves a change in l of ± 1 (∆l = ±1). A transition from a lower state will therefore take the atom to an energy level with angular momentum differing
Atomic Physics, P. Ewart 4 Corrections to the Central Field: Spin-Orbit interaction
4 Corrections to the Central Field: Spin-Orbit interaction
The Central Field Approximation gives us a zero-order Hamiltonian Hˆ 0 that allows us to solve the Schr¨odinger equation and thus find a set of zero-order wavefunctions ψi. The hope is that we can treat the residual electrostatic interaction (i.e. the non-central bit of the electron-electron repulsion) as a small perturbation, Hˆ 1. The change to the energy would be found using the functions ψi. The residual electrostatic interaction however isn’t the only perturbation around. Magnetic inter- actions arise when there are moving charges. Specifically we need to consider the magnetic interaction between the magnetic moment due to the electron spin and the magnetic field arising from the elec- tron orbit. This field is due to the motion of the electron in the electric field of the nucleus and the other electrons. This spin-orbit interaction has an energy described by the perturbation Hˆ 2. The question is: which is the greater perturbation, Hˆ 1 or Hˆ 2? We may be tempted to assume Hˆ 1 > Hˆ 2 since electrostatic forces are usually much stronger than magnetic ones. However by setting up a Central Field we have already dealt with the major part of the electrostatic interaction. The remaining bit may not be larger than the magnetic spin-orbit interaction. In many atoms the residual electrostatic interaction, Hˆ 1 , does indeed dominate the spin- orbit. There is, however, a set of atoms where the residual electrostatic repulsion is effectively zero; the alkali atoms. In the alkalis we have only one electron orbiting outside a spherically symmetric core. The central field is, in this case, an excellent approximation. The spin-orbit interaction, Hˆ 2 will be the largest perturbation – provided there are no external fields present. So we will take the alkalis e.g. Sodium, as a suitable case for treatment of spin-orbit effects in atoms. You have already met the spin-orbit effect in atomic hydrogen, so you will be familiar with the quantum mechanics for calculating the splitting of the energy levels. There are, however, some important differences in the case of more complex atoms. In any case, we are interested in understanding the physics, not just doing the maths of simple systems. In what follows we shall first outline the physics of the electron’s spin magnetic moment μ interacting with the magnetic field, B, due to its motion in the central field (nucleus plus inner shell electrons). The interaction energy is found to be μ · B so the perturbation to the energy, ∆E, will be the expectation value of the corresponding operator
ˆμ · B
We then use perturbation theory to find ∆E. We will not, however, be able simply to use our zero order wavefunctions ψ 0 (n, l, ml, ms) derived from our Central Field Approximation, since they are degenerate in ml, ms. We then have to use degenerate perturbation theory, DPT, to solve the problem. We won’t have to actually do any complicated maths because it turns out that we can use a helpful model – the Vector Model, that guides us to the solution, and gives some insight into the physics of what DPT is doing.
4.1 The Physics of Spin-Orbit Interaction
What happens to a magnetic dipole in a magnetic field? A negatively charged object having a moment of inertia I, rotating with angular velocity ω, has angular momentum, Iω = λℏ. The energy is then E = 12 Iω^2 We suppose the angular momentum vector λℏ is at an angle θ to the z-axis. The rotating charge has a magnetic moment μ = −γλℏ (44)
The sign is negative as we have a negative charge. γ is known as the gyromagnetic ratio. (Classically γ = 1 for an orbiting charge, and γ = 2 for a spinning charge). If a constant magnetic field B is applied along the z-axis the moving charge experiences a force
- a torque acts on the body producing an extra rotational motion around the z-axis. The axis of rotation of λℏ precesses around the direction of B with angular velocity ω′. The angular motion of our rotating change is changed by this additional precession from ω to ω + ω′^ cos(θ). If the angular momentum λℏ was in the opposite direction then the new angular velocity would be ω − ω′^ cos(θ).
Atomic Physics, P. Ewart 4 Corrections to the Central Field: Spin-Orbit interaction
Figure 11: Illustration of the precession (Larmor precession) caused by the torque on the magnetic moment μ by a magnetic field B.
The new energy is, then
E′^ =
I(ω ± ω′^ cos θ)^2 (45)
=
Iω^2 +
I(ω′^ cos θ)^2 ± Iωω′^ cos θ (46)
We now assume the precessional motion ω′^ to be slow compared to the original angular velocity ω. ω′^ << ω so (ω′^ cos θ)^2 << ω^2 and we neglect the second term on the r.h.s. The energy change ∆E = E′^ − E is then
∆E = Iωω′^ cos θ (47) = λℏω′^ cos θ (48)
Now the precessional rate ω′^ is given by Larmor’s Theorem
ω′^ = −γB (49)
So ∆E = −γλℏB cos θ (50)
Hence ∆E = −μ · B (51)
So −μ · B is just the energy of the precessional motion of μ in the B-field. (Note that λ cos θℏ is the projection of the angular momentum on the z-axis (B-direction) Quantum mechanically this must be quantized in integer units of ℏ i.e. λ cos θ = m, the magnetic quantum number. So the angular momentum vector λℏ can take up only certain discrete direc- tions relative to the B-field. This is the space or orientation quantization behind the Stern-Gerlach experiment.)
