Sources of Light and and Blackbody Radiation Prof. Rick Trebino Georgia Tech www.physics.gatech.edu/frog/lectures Sources of light: gases, liquids, and solids Boltzmann's Law and blackbody radiation

Where does light come from? We’ve seen that Maxwell’s Equations (i.e., the wave equation) describe the propagation of light. But where does light come from in the first place? Some matter must emit the light. It does so through the matter’s polarization: Note that matter’s polarization is analogous to the polarization of light. Indeed, it will cause the emission of light with the same polarization direction. where N is the number density of charged particles, q is the charge of each particle, and is the position of the charge. Here, we’ve assumed that each charge is identical and has identical motion.

Polarized and unpolarized media Unpolarized medium Polarized medium On the right, the displacements of the charges are correlated, so it is polarized at any given time (and its polarization is oscillating).

Maxwell's Equations in a Medium The induced polarization, , contains the effect of the medium and is included in Maxwell’s Equations: This extra term also adds to the wave equation, which is known as the Inhomogeneous Wave Equation: The polarization is the source term and tells us what light will be emitted. Notice that the induced polarization, and hence , gets differentiated twice. But is just the charge acceleration! So it’s accelerating charges that emit light!

Sources of light Accelerating charges emit light! Linearly accelerating charge Synchrotron radiation— light emitted by charged particles deflected by a magnetic field Bremsstrahlung (Braking radiation)— light emitted when charged particles collide with other charged particles

But the vast majority of light in the universe comes from molecular vibrations emitting light. Electrons vibrate in their motion around nuclei. High frequency: ~1014 - 1017 cycles per second. Nuclei in molecules vibrate with respect to each other. Intermediate frequency: ~1011 - 1013 cycles per second. Nuclei in molecules rotate. Low frequency: ~109 - 1010 cycles per second.

Water’s vibrations Movies from http://www.lsbu.ac.uk/water/vibrat.html

Atomic and molecular vibrations correspond to excited energy levels in quantum mechanics. Energy levels are everything in quantum mechanics. Excited level DE = hn Energy Ground level The atom is vibrating at frequency, n. The atom is at least partially in an excited state.

Excited atoms emit photons spontaneously. 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, phosphorescence.

Different atoms emit light at different widely separated frequencies. Each colored emission line corresponds to a difference between two energy levels. These are emission spectra from gases of hot atoms. Frequency (energy) Atoms have relatively simple energy level systems (and hence simple spectra).

Collisions broaden the frequency range of light emission. A collision abruptly changes the phase of the sine-wave light emission. So atomic emissions can have a broader spectrum. Quantum-mechanically speaking, the levels shift during the collision. Collision Electric field time New frequencies in the emission Gases at atmospheric pressure have emission widths of ~ 1 GHz. Solids and liquids emit much broader ranges of frequencies (~ 1013 Hz!).

Molecules have many energy levels. A typical molecule’s energy levels: E = Eelectonic + Evibrational + Erotational 2nd excited electronic state Lowest vibrational and rotational level of this electronic “manifold” Energy 1st excited electronic state Excited vibrational and rotational level Transition There are many other complications, such as spin-orbit coupling, nuclear spin, etc., which split levels. Ground electronic state As a result, molecules generally have very complex spectra.

Atoms and molecules can also absorb photons, making a transition from a lower level to a more excited one. Excited level This is, of course, absorption. Energy Ground level Image from http://phyun5.ucr.edu/~wudka/Physics7/Notes_www/node107.html Absorption lines in an otherwise continuous light spectrum due to a cold atomic gas in front of a hot source.

Decay from an excited state can occur in many steps. Infra-red Energy Ultraviolet Visible Microwave The light that’s eventually re-emitted after absorption may occur at other colors.

Einstein showed that stimulated emission can also occur. Before After Spontaneous emission Absorption Stimulated emission

In what energy levels do molecules reside? Boltzmann population factors Ni is the number density of molecules in state i (i.e., the number of molecules per cm3). T is the temperature, and kB is Boltzmann’s constant. E3 N3 E2 N2 Energy N1 E1 Population density

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 High T 3 3 Energy Energy 2 2 1 1 Molecules Molecules In equilibrium, the ratio of the populations 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.

Blackbody radiation Blackbody radiation is emitted from a hot body. It's anything but black! The name comes from the assumption that the body absorbs at every frequency and hence would look black at low temperature. It results from a combination of spontaneous emission, stimulated emission, and absorption occurring in a medium at a given temperature. It assumes that the box is filled with many different molecules that that, together, have transitions (absorptions) at every wavelength.

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 N2 Absorption rate = B12 N1 I Stimulated emission rate = B21 N2 I In equilibrium, the rate of upward transitions equals the rate of downward transitions: at frequency n Up = B12 N1 I = A N2 + B21 N2 I = Down Recalling the Maxwell- Boltzmann Distribution Solving for N2/N1: (B12 I ) / (A + B21 I ) = N2 / N1 = exp[–DE/kBT ]

Blackbody Radiation Now solve for the intensity in: (B12 I) / (A + B21 I) = exp[-DE/kBT ] Multiply by (A + B21 I) exp[DE/kBT] : B12 I exp[DE/kBT] = A + B21 I Solve for I: I = A / {B12 exp[DE/kBT] – B21} Dividing numerator and denominator by B21: I = [A/B21] / { [B12 /B21] exp[DE/kBT] – 1 } Now, when T ® ¥, I should also. As T ® ¥, exp[DE/kBT ] ® 1. So: [B12 /B21] – 1 = 0 And: B12 = B21 º B ¬ Coeff up = coeff down! And: I = [A/B] / {exp[DE/kBT ] – 1}

Blackbody Radiation I = [A/B] / {exp[DE/kBT ] – 1} We can eliminate A/B based on other information and, writing in terms of the intensity per unit frequency, In : also using DE = hn This is the total intensity per unit frequency (that is, in a range from n to n + dn ) emitted by an arbitrary blackbody. We considered only two levels, but our approach was general and so applies to any two levels and hence to the entire spectrum. Notice that it’s independent of the size of the object, but, to obtain the emitted power, multiply by the surface area of the blackbody.

Writing the blackbody spectrum vs. wavelength Units of In(n): intensity per unit frequency Total intensity Change variables from n to l: Units of Il(l): intensity per unit wavelength

Blackbody emission spectrum The higher the temperature, the more the emission and the shorter the average wavelength. Blue hot is hotter than red hot. The sun’s surface is 6000 degrees K, so its blackbody spectrum peaks at ~ 500 nm—in the green. However, blackbody spectra are broad, so it contains red, yellow, and blue, too, and so looks white.

Wien's Law: The blackbody peak wavelength scales as 1/Temperature. dlambda/dnu = -c/nu^2 = -lambda^2/c http://blogs.discovermagazine.com/badastronomy/files/2008/07/blackbody.jpg

We can tell how hot a star is by its blackbody emission spectrum. http://www.fas.org/irp/imint/docs/rst/Sect20/spic-sun-ant+.jpg

The earth is a blackbody, too. http://eesc.columbia.edu/courses/ees/slides/climate/blackbody.gif

Cosmic microwave background Microwave background vs. angle. Note the variations. Peak frequency is ~ 150 GHz The 3° cosmic microwave background is blackbody radiation left over from the Big Bang! Interestingly, blackbody radiation retains a blackbody spectrum despite the expansion the universe. It does get colder, however. http://imagers.gsfc.nasa.gov/ems/micro.html Wave-number (cm-1)

Color temperature Blackbodies are so pervasive that a light spectrum is often characterized in terms of its temperature even if it’s not exactly a blackbody. Keep in mind that blackbody spectra are broad, so they usually look white with a tint of the peak color (wavelength). Image from the magazine Digital PhotoPro, “White Balance,”, May/June 2004

Emissivity Our analysis assumed a perfectly black object. However, the object may not have transitions at all wavelengths. The emissivity takes this into account. It is the ratio of the actual emitted spectrum and the theoretical blackbody spectrum. It depends on the medium and its density.