1. CHAPTER 8 Many-electron atoms
Homework due this Friday Oct 23rd: Read Chapter 7: problems 1, 4, 5, 6, 7, 8, 10, 12, 14, 15
Dimitri Mendeleev
What distinguished Mendeleev was not only genius, but a passion for the elements. They became his personal friends; he knew every quirk and detail of their behavior.
- J. Bronowski

2. Multi-electron atoms What if there’s more than one electron?
Helium: a nucleus with charge +2e and two electrons,
the two electrons repelling one another.
Cannot solve problems exactly with the Schrödinger equation because of the complex potential interactions.
Can understand experimental results without computing the wave functions of many-electron atoms by applying the boundary conditions and selection rules.

3. Multi-electron atoms
When more than one electron is involved, the potential and the wave function are functions of more than one position:
Solving the Schrodinger Equation in this case can be very hard. But we can approximate the solution as the product of single-particle wave functions:
And it turns out that we’ll be able to approximate each Yi with a Hydrogen wave function.

4. Pauli Exclusion Principle
To understand atomic spectroscopic data, Pauli proposed his exclusion principle: No two electrons in an atom may have the same set of quantum numbers (n, ℓ, mℓ, ms). It applies to all particles of half-integer spin, which are called fermions, and particles in the nucleus are also fermions. The periodic table can be understood by two rules: The electrons in an atom tend to occupy the lowest energy levels available to them. The Pauli exclusion principle.

5. Atomic Structure
Hydrogen: (n, ℓ, mℓ, ms) = (1, 0, 0, ±½) in ground state. In the absence of a magnetic field, the state ms = ½ is degenerate with the ms = −½ state. Helium: (1, 0, 0, ½) for the first electron. (1, 0, 0, −½) for the second electron. Electrons have anti-aligned (ms = +½ and ms = −½) spins. The principle quantum number also has letter codes. n = Letter = K L M N… n = shells (eg: K shell, L shell, etc.) nℓ = subshells (e.g.: 1s, 2p, 3d)
Electrons for H and He atoms are in the K shell.
H: 1s
He: 1s2

6. Atomic Structure
How many electrons may be in each subshell? Recall: ℓ = … letter = s p d f g h … ℓ = 0, (s state) can have two electrons. ℓ = 1, (p state) can have six electrons, and so on.
Total
For each mℓ: two values of ms 2
For each ℓ: (2ℓ + 1) values of mℓ
2(2ℓ + 1)
Electrons with higher ℓ values are more shielded from the nuclear charge.
Electrons with higher ℓ values lie higher in energy than those with lower ℓ values.
4s fills before 3d.

7. The Periodic Table

8. The Periodic Table
Inert Gases: Last group of the periodic table Closed p subshell except helium Zero net spin and large ionization energy Their atoms interact weakly with each other Alkalis: Single s electron outside an inner core Easily form positive ions with a charge +1e Lowest ionization energies Electrical conductivity is relatively good Alkaline Earths: Two s electrons in outer subshell Largest atomic radii High electrical conductivity

9. The Periodic Table
Halogens: Need one more electron to fill outermost subshell Form strong ionic bonds with the alkalis More stable configurations occur as the p subshell is filled Transition Metals: Three rows of elements in which the 3d, 4d, and 5d are being filled Properties primarily determined by the s electrons, rather than by the d subshell being filled Have d-shell electrons with unpaired spins As the d subshell is filled, the magnetic moments, and the tendency for neighboring atoms to align spins are reduced

10. The Periodic Table
Lanthanides (rare earths): Have the outside 6s2 subshell completed As occurs in the 3d subshell, the electrons in the 4f subshell have unpaired electrons that align themselves The large orbital angular momentum contributes to the large ferromagnetic effects Actinides: Inner subshells are being filled while the 7s2 subshell is complete Difficult to obtain chemical data because they are all radioactive

11. Total Angular Momentum
Orbital angular momentum
Spin angular momentum
Total angular momentum
L, Lz, S, Sz, J, and Jz are quantized.

12. Total Angular Momentum
If j and mj are quantum numbers for the single-electron hydrogen atom: Quantization of the magnitudes: The total angular momentum quantum number for the single electron can only have the values

13. Spin-Orbit Coupling
An effect of the spins of the electron and the orbital angular momentum interaction is called spin-orbit coupling. is the magnetic field due to the electron’s orbital motion. where a is the angle between .
The dipole potential energy
The spin magnetic moment µ ●

14. Total Angular Momentum
Now the selection rules for a single-electron atom become Δn = anything Δℓ = ±1 Δmj = 0, ±1 Δj = 0, ±1 Hydrogen energy-level diagram for n = 2 and n = 3 with spin-orbit splitting.

15. Many-Electron Atoms
Hund’s rules: The total spin angular momentum S should be maximized to the extent possible without violating the Pauli exclusion principle. Insofar as rule 1 is not violated, L should also be maximized. For atoms having subshells less than half full, J should be minimized. For a two-electron atom There are LS coupling and jj coupling to combine four angular momenta J.

16. Lasers* Stimulated Emission Gain and Inversion Threshold The Laser
Laser Transition
Pump Transition
Fast decay
Stimulated Emission
Gain and Inversion
Threshold
The Laser
* Light Amplification by Stimulated Emission of Radiation

17. Spontaneous emission
When an atom in an excited state falls to a lower energy level, it emits a photon of light.
Excited level
Energy
Ground level
Molecules typically remain excited for no longer than a few nanoseconds. This is often also called fluorescence or, when it takes longer (because the transition is “forbidden”), phosphorescence.

18. Absorption
When an atom encounters a photon of light, it can absorb the photon’s energy and jump to an excited state.
Excited level
This is, of course, absorption.
Energy
Ground level
Image from
Absorption lines in an otherwise continuous light spectrum due to a cold atomic gas in front of a hot broadband source.

19. Einstein showed that another process, stimulated emission, can also occur.
When a photon encounters an atom in an excited state, the photon can induce the atom to emit its energy as another photon of light, resulting in two identical photons.
Excited level
Energy
Ground level
Einstein first proposed stimulated emission in 1916.

20. Population density (Number of molecules per unit volume)
In what energy levels do molecules reside? Boltzmann Population Factors
Ni is the number density (also known as the population density) of molecules in state i (i.e., the number of molecules per cm3).
T is the temperature, and kB is Boltzmann’s constant = ×  J/K E3 N3 E2 N2
Energy N1 E1
Population density (Number of molecules per unit volume)

21. The Maxwell-Boltzman distribution
In the absence of collisions,
molecules tend to remain
in the lowest energy state
available.
Collisions can knock a mole-
cule into a higher-energy state.
The higher the temperature,
the more this happens.
Low T
Energy
Molecules 3 2 1
High T
Energy
Molecules 3 2 1
The ratio of the population densities of two states is:
N2 / N1 = exp(–DE/kBT ), where DE = E2 – E1 = hn
As a result, higher-energy states are always less populated than the ground state, and absorption is stronger than stimulated emission.

22. Stimulated emission leads to a chain reaction and laser emission.
If a medium has many excited molecules, one photon can become many.
Excited medium
This is the essence of the laser. The factor by which an input beam is amplified by a medium is called the gain and is represented by G.

23. Laser medium with gain, G
The Laser
A laser is a medium that stores energy, surrounded by two mirrors.
A partially reflecting output mirror lets some light out.
R = 100%
R < 100% I0 I1 I2 I3
Output mirror
Back mirror
Laser medium with gain, G
I1 = G I0
I3 = G I2
A laser will lase if the beam increases in intensity during a round trip: that is if
Usually, additional losses in intensity occur, such as absorption, scattering, and reflections. In general, the laser will lase if, in a round trip:
Gain > Loss This called achieving Threshold.

24. Inversion B N2 I > B N1 I Inversion N2 > N1
In order to achieve G > 1, stimulated emission must exceed absorption:
Canceling the BI factors,
This condition is called inversion.
It does not occur naturally (it’s forbidden by the Boltzmann distribution). It’s inherently a non-equilibrium state.
In order to achieve inversion, we must hit the laser medium very hard in some way and choose our medium correctly.
B N2 I > B N1 I
Inversion
“Negative temperature”
Molecules 3 2 1 4
N2 > N1
Energy
Here, there is inversion from level 4 to levels 3 and 2.

25. Achieving Inversion: Pumping the Laser Medium
Now let I be the intensity of (flash lamp) light used to pump energy into the laser medium:
Back mirror I
Output mirror
Laser medium
Will this intensity be sufficient to achieve inversion, N2 > N1?
It’ll depend on the laser medium’s energy level system.

26. Types of Lasers
Solid-state lasers have lasing material distributed in a solid matrix (such as ruby or neodymium:yttrium-aluminum garnet "YAG"). Flash lamps are the most common power source. The Nd:YAG laser emits infrared light at 1,064 nm (1.064 mm).
Semiconductor lasers, sometimes called diode lasers, are pn junctions. Current is the pump source. Applications: laser printers or CD players.
Dye lasers use complex organic dyes, such as rhodamine 6G, in liquid solution or suspension as lasing media. They are tunable over a broad range of wavelengths.
Gas lasers are pumped by current. Helium-Neon lases in the visible and IR. Argon lases in the visible and UV. CO2 lasers emit light in the far-infrared (10.6 mm), and are used for cutting hard materials.
Excimer lasers (from the terms excited and dimers) use reactive gases, such as chlorine and fluorine, mixed with inert gases such as argon, krypton, or xenon. When electrically stimulated, a pseudo molecule (dimer) is produced. Excimers lase in the UV.
Slide provided by Optics I student Kham Ho, 2004

27. Diode Lasers