ATOMIC PHYSICS: Things You Should Remember

Here are the main points covered by the problem sheets. This is not a comprehensive listcovering all the material in the course (see the lecture sheets for this) but should containthe core of what you should know.

1 Gross Structure of Atomic Levels

• Energy of 1s level = Z2e2/8πε0aB where aB is the Bohr radius. 13.6eV for the hydrogenatom.

• 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 electronsin the completely filled “core” of inner shells — i.e. Zeff ' 1. Penetration of thewavefunction into the core reduces the screening as well — remember the quantumdefect 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: Zeff

is between 1 and m = number of electrons in outer shell. (Recall estimation of secondionisation energies.)

2 Optical Spectra

• Emission spectra: experimental setup (e.g. sodium lamp). Highly excited atom emitsradiation, eventually returning to ground state.

• Absorption spectra: experimental setup. The initial state must be the ground state sothere are fewer lines than in emission spectra.

• Selection Rules for electric dipole transitions: ∆L = ±1, ∆S = 0, ∆J = 0,±1 but notJ = 0 to 0.

• You should know how the lines are grouped into series and the information given by theseries limit and the lowest-energy line of each series. We have done this for hydrogenand sodium.

• Linewidths: the uncertainty principle gives a natural linewidth to all lines becauseof the finite lifetime of an excited state. The Doppler effect will give a linewidth tospectral lines emitted from a gas because the emitting particles may be moving towardsor away from the observer. This Doppler broadening increases as the temperature israised. Other sources of broadening can be found in one of your lecture sheets, alsodiscussed in Haken and Wolf.

3 Fine Structure: Spin Orbit Interaction

• Physical origin of the interaction: electron spin experiences the magnetic field createdby the current loop of the nucleus around it.

• VSO = ξl · s. The semiclassical vector model indicates that l and s precesses aroundj = l + s. Thus ml and ms are no longer good quantum numbers, i.e. the vectors l ands are not constant of motion. Hence we should use j, mj , l and s as the appropriatequantum numbers.

• In the LS coupling scheme for more than one electron in the outer shell, we treat thespin-orbit interaction after dealing with the Coulomb repulsion. Since Coulomb repulsiondoes not couple li and si for each electron, the total L and S are used in the formula forspin-orbit interaction when we come to treat it in first-order perturbation theory. (Youhave 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 levelsof 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 ofconsecutive energy levels in the multiplet arising from spin-orbit splitting of a state withgiven 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-structureseparation 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 shiftedby gJMJµBB where gJ = 1 + [J(J + 1)−L(L + 1) + S(S + 1)]/[2J(J + 1)] is the Landefactor. The frequencies of spectral lines are obtained by the difference between theseenergy levels. There is no fundamental difference between the normal and anomalouseffects. The former is the special case when the Lande factors for the multiplet of initialstates and the multiplet of final states are the same, in which case the line frequenciesdepend only on ∆MJ . In general the Lande factors are different and the line frequenciesdepend 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 theselection 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 cloudalong 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 µBB much larger than spin-orbit energy〈VSO〉. 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. Radiationis 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 thetotal electronic angular momentum. A can be positive or negative — it is proportionalto the magnitude of the magnetic moment of the nucleus µ = µNI. (Recall the problemon the two isotopes of potassium.)

• By analogy with spin-orbit interaction, the energy level of the entire atom is nowdescribed using by I, J, F and MF where F = I + J is the total angular momentum ofthe entire atom. The energy levels of given I and J are split according to the differentvalues 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, thelevel with highest F may either be the highest or lowest in energy in each hyperfinemultiplet.

6 More Than One Valence Electron

• LS coupling scheme when Coulomb repulsion is much stronger than spin-orbit interac-tion. 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 repulsionis reduced electrons can be placed in different orbitals and hence occupy different regionsof space.

• Helium: 1s2 ground state must have S = 0; 1s2s excited state has S = 0 or 1. 3S1 hassymmetric spin part and antisymmetric space part. The antisymmetry in the spatialwavefunction means that electrons are avoiding each other and so 1s2s 3S1 has lowerenergy than 1s2s 1S0 state.

7 X-Ray Spectra

• X-ray spectra probe the energies of electrons in the inner shell. You should understandthe features of the observed spectra, in particular the absorption edges indicate the actualenergies 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 spon-taneous). The rate equation for dynamic equilibrium and Planck’s law for ambientradiation at thermodynamic equilibrium determines the relation between them. Theterm for spontaneous emission is often dropped when considering coherent radiationsuch as in a laser.

• Population inversion is necessary to satisfy for the operation of the laser. Thisdepends 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 (seeHaken and Wolf)

• You should know how lasers work, e.g.gas laser, solid-state laser. Haken and Wolf has agood chapter on these descriptive topics. The lecture sheets should indicate what topicsyou are expected to know about.

9 Nuclear Magnetism

We did not cover this in the problem sheets. You should know a little bit about theprinciples behind nuclear magnetic resonance (i.e. precession of nuclear spin in a weakmagnetic 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 BEFOREattempting to fit the spectral line frequencies to numerical formulae such as the intervalrule!

• Test for selection rules. This often narrows down possible quantum numbers for L, Sand J .

• Observed values of J (deduced using the interval rule for example), often gives anindication of the values for L and S.

• Make sure that you DISTINGUISH between the vector J and the quantum numberJ . 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