Download ATOMIC PHYSICS: Things You Should Remember Here are the

ATOMIC PHYSICS: Things You Should Remember
Here are the main points covered by the problem sheets. This is not a comprehensive list
covering all the material in the course (see the lecture sheets for this) but should contain
the core of what you should know.
1 Gross Structure of Atomic Levels
• Energy of 1s level = Z 2 e2 /8π0 aB where aB is the Bohr radius. 13.6eV for the hydrogen
atom.
• Excited state energies = Eground /n2 where n is the principal quantum number.
• Radius of orbit = aB /Z. Also proportional to 1/m (recall the muonic atom.)
• A single valence electrons is effectively screened from the nuclear charge by the electrons
in the completely filled “core” of inner shells — i.e. Zeff ' 1. Penetration of the
wavefunction into the core reduces the screening as well — remember the quantum
defect for alkali atoms and its dependence on l.
• The screening is not so effective if there is more than one electron in the outer shell: Z eff
is between 1 and m = number of electrons in outer shell. (Recall estimation of second
ionisation energies.)
2 Optical Spectra
• Emission spectra: experimental setup (e.g. sodium lamp). Highly excited atom emits
radiation, eventually returning to ground state.
• Absorption spectra: experimental setup. The initial state must be the ground state so
there are fewer lines than in emission spectra.
• Selection Rules for electric dipole transitions: ∆L = ±1, ∆S = 0, ∆J = 0, ±1 but not
J = 0 to 0.
• You should know how the lines are grouped into series and the information given by the
series limit and the lowest-energy line of each series. We have done this for hydrogen
and sodium.
• Linewidths: the uncertainty principle gives a natural linewidth to all lines because
of the finite lifetime of an excited state. The Doppler effect will give a linewidth to
spectral lines emitted from a gas because the emitting particles may be moving towards
or away from the observer. This Doppler broadening increases as the temperature is
raised. Other sources of broadening can be found in one of your lecture sheets, also
discussed in Haken and Wolf.
3 Fine Structure: Spin Orbit Interaction
• Physical origin of the interaction: electron spin experiences the magnetic field created
by the current loop of the nucleus around it.
• VSO = ξl · s. The semiclassical vector model indicates that l and s precesses around
j = l + s. Thus ml and ms are no longer good quantum numbers, i.e. the vectors l and
s are not constant of motion. Hence we should use j, mj , l and s as the appropriate
quantum numbers.
• In the LS coupling scheme for more than one electron in the outer shell, we treat the
spin-orbit interaction after dealing with the Coulomb repulsion. Since Coulomb repulsion
does not couple li and si for each electron, the total L and S are used in the formula for
spin-orbit interaction when we come to treat it in first-order perturbation theory. (You
have implicitly done this in the problem sheet on spin-orbit interaction.)
• Possible values for the total angular momentum J = L + S: |L − S| ≤ J ≤ L + S.
• Since ξ is positive, the state with highest J has the highest energy in each set of levels
of given L, S split by spin-orbit interaction.
• Interval Rule: ∆EJ,J −1 /∆EJ −1,J −2 = J/(J − 1) gives the ratio of the separation of
consecutive energy levels in the multiplet arising from spin-orbit splitting of a state with
given L, S. Also useful: ∆EJ,J −1 = ξJ ; ∆EJ,J −1 − ∆EJ −1,J −2 = ξ.
• Spin orbit interaction is only one of several relativistic corrections to the energy levels.
However the other ones do not split energy levels of given L, S. So the fine-structure
separation of spectral lines is due to spin-orbit effects only.
4 Atom in Magnetic Field
• Zeeman effect: Each energy level of a given J is split into 2J +1 levels. Energies shifted
by gJ MJ µB B where gJ = 1 + [J (J + 1) − L(L + 1) + S(S + 1)]/[2J (J + 1)] is the Landé
factor. The frequencies of spectral lines are obtained by the difference between these
energy levels. There is no fundamental difference between the normal and anomalous
effects. The former is the special case when the Landé factors for the multiplet of initial
states and the multiplet of final states are the same, in which case the line frequencies
depend only on ∆MJ . In general the Landé factors are different and the line frequencies
depend on the actual values of the initial and final MJ .
• Selection Rule for weak field: ∆MJ = 0, ±1 but not from 0 to 0 (in addition to the
selection rules for changes in the quantum numbers for the magnitudes L, S and J ).
Transition with ∆MJ = 0 comes from dipole oscillations of the electron charge cloud
along the field direction so that the radiation cannot be seen when observed in this
“longitudinal” direction. All these lines are seen in a “transverse” observation.
• Paschen Back Effect: magnetic field energy µB B much larger than spin-orbit energy
hVSO i. Treat the magnetic field first before spin-orbit, i.e. states are labelled by ML , MS
and not J, S, L, MJ as in the weak-field case.
• Selection Rule for strong field: ∆ML = 0, ±1 but not from 0 to 0; ∆MS = 0. Radiation
is polarised as in Zeeman effect but now it depends on ∆ML rather than ∆MJ .
5 Hyperfine Structure
• Physical origin: Nuclear spin experiences magnetic field due to current loop of electron.
Also dipolar interaction of the electronic and nuclear spins.
• VHFS = AI · J where I is the nuclear spin (analogous to S of the electron) and J is the
total electronic angular momentum. A can be positive or negative — it is proportional
to the magnitude of the magnetic moment of the nucleus µ = µN I. (Recall the problem
on the two isotopes of potassium.)
• By analogy with spin-orbit interaction, the energy level of the entire atom is now
described using by I, J, F and MF where F = I + J is the total angular momentum of
the entire atom. The energy levels of given I and J are split according to the different
values of F . (Vector addition gives |I − J | ≤ F ≤ I + J .)
• Note that the interval rule also applies for F . But since A can be of either sign, the
level with highest F may either be the highest or lowest in energy in each hyperfine
multiplet.
6 More Than One Valence Electron
• LS coupling scheme when Coulomb repulsion is much stronger than spin-orbit interaction. Use total L, S, J, MJ as quantum numbers.
• jj coupling when Coulomb repulsion is much weaker than spin-orbit interaction.
Applicable to heavy atoms (large Z) and nuclei. l and s coupled together for each
electron and the system should be described by the set l1 , j1 , mj1 ..... etc..
• Hund’s Rule for ground states only: Maximise S. Crude argument: Coulomb repulsion
is reduced electrons can be placed in different orbitals and hence occupy different regions
of space.
• Helium: 1s2 ground state must have S = 0; 1s2s excited state has S = 0 or 1. 3 S1 has
symmetric spin part and antisymmetric space part. The antisymmetry in the spatial
wavefunction means that electrons are avoiding each other and so 1s2s 3 S1 has lower
energy than 1s2s 1 S0 state.
7 X-Ray Spectra
• X-ray spectra probe the energies of electrons in the inner shell. You should understand
the features of the observed spectra, in particular the absorption edges indicate the actual
energies of these levels.
• Nomenclature of K, L, M series of edges
• Experimental setup for generating X rays and for the absorption experiments.
8 Lasers
• Einstein Coefficients for absorption (stimulated) and emission (stimulated and spontaneous). The rate equation for dynamic equilibrium and Planck’s law for ambient
radiation at thermodynamic equilibrium determines the relation between them. The
term for spontaneous emission is often dropped when considering coherent radiation
such as in a laser.
• Population inversion is necessary to satisfy for the operation of the laser. This
depends on the various factors as can be seen from the formula for the lasing condition:
frequency and linewidth for the relevant transition, mirror reflectivity, size of the cavity.
You should find out about how this is inversion is achieved e.g.optical pumping (see
Haken and Wolf)
• You should know how lasers work, e.g.gas laser, solid-state laser. Haken and Wolf has a
good chapter on these descriptive topics. The lecture sheets should indicate what topics
you are expected to know about.
9 Nuclear Magnetism
We did not cover this in the problem sheets. You should know a little bit about the
principles behind nuclear magnetic resonance (i.e. precession of nuclear spin in a weak
magnetic field) because of its important applications as an experimental technique.
10 General Problem Solving
• ALWAYS draw the scenario for the energy levels involved in transitions BEFORE
attempting to fit the spectral line frequencies to numerical formulae such as the interval
rule!
• Test for selection rules. This often narrows down possible quantum numbers for L, S
and J .
• Observed values of J (deduced using the interval rule for example), often gives an
indication of the values for L and S.
• Make sure that you DISTINGUISH between the vector J and the quantum number
J . J is always positive while MJ can be postive or negative. The magnitude of J is
[J (J + 1)]1/2 and not J .
• Convention in labelling atomic states:
2S+1
(L=S,P,D,F)J