Many-electron atoms CHAPTER 8 Many-electron atoms 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 Dimitri Mendeleev Homework next Friday Oct. 30th: Chapter 7: 18, 20, 24, 25 Chapter 8: 1, 2, 4, 8, 10, 19

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

Spontaneous emission When an atom in an excited state falls to a lower energy level, it emits a photon of light. 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. Energy Ground level Excited level

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

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. Einstein first proposed stimulated emission in Energy Ground level Excited level

In what energy levels do molecules reside? Boltzmann Population Factors N i is the number density (also known as the population density) of molecules in state i (i.e., the number of molecules per cm 3 ). T is the temperature, and k B is Boltzmann’s constant = × J/K Population density (Number of molecules per unit volume) Energy N1N1 N3N3 N2N2 E3E3 E1E1 E2E2

The Maxwell-Boltzman distribution The ratio of the population densities of two states is: N 2 / N 1 = exp (–  E/k B T ), where  E = E 2 – E 1 = h As a result, higher-energy states are always less populated than the ground state, and absorption is stronger than stimulated emission. 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 High T Energy Molecules 3 2 1

Calculating the Gain: Einstein A and B Coefficients In 1916, Einstein considered the various transition rates between molecular states (say, 1 and 2 ) involving light of intensity, I : Absorption rate = B N 1 I Spontaneous emission rate = A N 2 Stimulated emission rate = B N 2 I 2 1 where N i is the number density of molecules in the i th state, and I is the intensity.

A N 2 + B 21 N 2 I = Down = Up = B 12 N 1 I Einstein A and B Coefficients In 1916, Einstein considered the various transition rates between molecular states (say, 1 and 2) involving light of intensity, I : Spontaneous emission rate = A N 2 Absorption rate = B 12 N 1 I Stimulated emission rate = B 21 N 2 I In equilibrium, the rate of upward transitions equals the rate of downward transitions: Recalling the Maxwell- Boltzmann Distribution (B 12 I ) / (A + B 21 I ) = N 2 / N 1 = exp(–  E/k B T) at frequency Dividing by N 1 (A + B 21 I ) yields N 2 /N 1 :

Stimulated emission leads to a chain reaction and laser emission. Excited medium If a medium has many excited molecules, one photon can become many. 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.

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. The Laser A laser is a medium that stores energy, surrounded by two mirrors. A partially reflecting output mirror lets some light out. A laser will lase if the beam increases in intensity during a round trip: that is if R = 100% R < 100% I0I0 I1I1 I2I2 I3I3 Output mirror Back mirror Laser medium with gain, G I 1 = G I 0 I 3 = G I 2

Laser Gain Neglecting spontaneous emission: [Stimulated emission minus absorption] Proportionality constant is the absorption/gain cross-section,  If N 2 > N 1 : If N 2 < N 1 : There can be exponential gain or loss in intensity. Normally, N 2 < N 1, and there is loss (absorption). But if N 2 > N 1, there’s gain, and we define the gain, G : The solution is: Laser medium I(0) z L 0 I(L)I(L) g and  are the gain and absorption coefficients.

Inversion 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. N 2 > N 1 Energy Inversion “Negative temperature ” Molecules B N 2 I > B N 1 I Here, there is inversion from level 4 to levels 3 and 2. 4

Achieving Inversion: Pumping the Laser Medium Now let I be the intensity of (flash lamp) light used to pump energy into the laser medium: I Will this intensity be sufficient to achieve inversion, N 2 > N 1 ? It’ll depend on the laser medium’s energy level system. Output mirror Back mirror Laser medium

Two-, Three-, and Four-Level Systems Two-level system Laser Transition Pump Transition At best, you get equal populations. No lasing. It took laser physicists a while to realize that four-level systems are best. Four-level system Lasing is easy! Laser Transition Pump Transition Fast decay Three-level system If you hit it hard, it can lase. Laser Transition Pump Transition Fast decay

Inversion is Easy in a Four-Level System Laser Transition Pump Transition Fast decay Most laser materials are four-level systems.

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  m). 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. CO 2 lasers emit light in the far-infrared (10.6  m), 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.

Diode Lasers