5. Light-matter interactions: Blackbody radiation
The electromagnetic spectrum Sources of light Boltzmann's Law Blackbody radiation The cosmic microwave background The electromagnetic spectrum
All of these are electromagnetic waves. The amazing diversity is a result of the fact that the interaction of radiation with matter depends on the frequency of the wave.
The boundaries between regions are a bit arbitrary… Sources of light
Accelerating charges emit light
Linearly accelerating charge
http://www.cco.caltech.edu/~phys1/java/phys1/MovingCharge/MovingCharge.html
Vibrating electrons or nuclei of an atom or molecule
Synchrotron radiation—light emitted by charged particles whose motion is deflected by a DC magnetic field
Bremsstrahlung ("Braking radiation")—light emitted when charged particles collide with other charged particles The vast majority of light in the universe comes from molecular motions. Core (tightly bound) electrons vibrate in their motion around nuclei ~ 1018 -1020 cycles per second.
Valence (not so tightly bound) electrons vibrate in their motion around nuclei ~ 1014 -1017 cycles per second.
Nuclei in molecules vibrate with respect to each other ~ 1011 -1013 cycles per second.
Molecules rotate ~ 109 -1010 cycles per second. One interesting source of x-rays Sticky tape emits x-rays when you unpeel it.
Scotch tape generates enough x-rays to take an image of the bones in your finger.
This phenomenon, known as ‘mechanoluminescence,’ was discovered in 2008.
For more information and a video describing this: http://www.nature.com/nature/videoarchive/x-rays/ Three types of molecule-radiation interactions
key idea: conservation of energy Before After
Absorption
•Promotes molecule to a higher energy state •Decreases the number of photons Unexcited Excited molecule molecule
Spontaneous Emission
•Molecule drops from a high energy state to a lower state •Increases the number of photons
Stimulated Emission
•Molecule drops from a high energy state to a lower state •The presence of one photon stimulates the emission of a second one •This process has no analog in classical physics - it can only be understood with quantum mechanics! Atoms and molecules have discrete energy levels and can interact with light only by making a transition from one level to another.
A typical molecule’s energy levels:
2nd excited electronic state Lowest vibrational and rotational level of this electronic “manifold” 1st excited Excited vibrational and electronic state Energy rotational level
Transition Computing the energies of these levels is usually difficult, except Ground for very simple atoms or
electronic state molecules (e.g., H2). Different atoms emit light at different frequencies. Wavelength Frequency (energy)
the “Balmer series”
Johann Balmer (1825 - 1898) This is the emission spectrum from hydrogen. It is simple because hydrogen has only one electron.
high-pressure mercury lamp
Bigger atoms have much more complex spectra, because they are more complex objects. Molecules are even more complex. Ludwig Boltzmann computed the relative populations of energy levels In the absence of collisions, Collisions can knock a mole- molecules tend to remain cule into a higher-energy state. in the lowest energy state The higher the temperature, available. the more this happens. Low Temperature High Temperature
In equilibrium at a temperature T, the ratio of the populations of any two states (call them number 1 and number 2) is given by:
N expEkT / 2 2 B expEkT / B NEkT11exp / B Boltzmann population factor Boltzmann didn’t know quantum mechanics. But this result works equally well for quantum or classical systems.
A higher temperature A lower temperature
Ludwig Boltzmann (1844 - 1906)
E3 E3
E/kT E e high E 2 2 eE/kTlow energy
E1 E1 population population Blackbody radiation
Blackbody radiation is the radiation emitted from a hot body. It's anything but black!
It results from a balance between the absorption and emission of light by the atoms which comprise the object, at a given temperature.
Imagine a box, the inside of which is completely black. Consider the radiation emitted from the small hole. Inside: a gas of photons at some temperature T The classical physics approach to the blackbody question
Lord Rayleigh figured out how to count the electromagnetic modes inside a box.
Basically, the result follows from requiring that the electric field must be zero at the internal surfaces of the box. John William Strutt, 3rd Baron Rayleigh (1842 - 1919)
# modes ~ 2 And the energy
per mode is kBT, (this is called the “equipartition theorem”, also due to Boltzmann) The Rayleigh-Jeans Law Rayleigh-Jeans law (circa 1900):
energy density of a radiation field u() = 82kT/c3
Note: the units of this expression are correct. Strictly speaking, u() is an energy density per unit bandwidth, such that the integral ud gives an answer with units of energy density (energy per unit volume). Total energy radiated from a black body: ud uh-oh… the "ultraviolet catastrophe" ) Rayleigh (and others) knew this was wrong. This
keeps going failure of equipartition led Max Planck to propose up forever! (1900) that the energy emitted by the atoms in the solid must be quantized in multiples of h. This revolutionary idea troubled Planck deeply because energy density u( he had no explanation for why it should be true… frequency Einstein A and B coefficients
In 1916, Einstein considered the various transition rates between molecular states (say, 1 and 2) involving light of energy density u:
Absorption rate = B12N1 u
Spontaneous emission rate = A N2
Stimulated emission rate = B21 N2 u
In equilibrium, the rate of upward transitions equals the rate of downward transitions:
B12 N1 u = A N2 + B21 N2 u using Boltzmann’s result
Rearranging:
(B12 u ) / (A + B21 u ) = N2 / N1 = exp[–E/kBT ] Einstein A and B coefficients We can solve this expression for the energy density of the field:
(B12 u ) / (A + B21 u ) = exp[E/kBT ]
AB Rearrange to find: u 21 B12 expEkT /B 1 B21
For blackbody radiation, we would expect that u() should be infinite
in the limit that T = . But, as T , exp[E/kBT ] 1
So: B12 = B21 B Coeff. up = coeff. down!
33 We can eliminate* 8 hv c the ratio A/B to find: u (we have used E=h) exphv / kB T 1
* We require that this expression has to be the same as the Rayleigh-Jeans law in the limit of very low frequencies. This solves the ultraviolet catastrophe.
) Rayleigh-Jeans: u() ~ 2
3 Max Planck Planck-Einstein: u ~ 1858-1947 exphkT /B 1 energy density u(
frequency
At low frequencies, the two At high frequencies, Albert Einstein results are basically equivalent. 1879-1955 Einstein’s result goes (photo taken in 1916) back down to zero, so the integral of u() is finite. Blackbody emission
The higher the temperature, the more the emission and the shorter the average wavelength.
Wien’s Law: the peak wavelength is proportional to 1/T. "Blue hot" is hotter than "red hot."
Notice: for an object near room temperature, the peak of the blackbody spectrum is around = 10 microns – the so-called ‘thermal infrared’ region. Blackbody emission from hot objects
Molten glass and heated metal are hot enough to shift the blackbody spectrum into the visible range, so that they glow: “red hot”.
Cooler objects emit light at longer Even hotter wavelengths: infrared metal is “white hot”. The sun is a black body Irradiance of the sun vs. wavelength
When measured at sea level, one sees absorption lines due to molecules in the atmosphere. Cosmic microwave background As an object cools slowly, the radiation it emits retains the blackbody spectrum. But the peak shifts to lower and lower frequencies.
This is true of all objects. Even the universe.
The 2.7° cosmic microwave background is blackbody radiation left over from the Big Bang!
Arno Penzias and Bob Wilson Nobel prize, 1978 Cosmic microwave background
Measurements of the microwave background tell us about the conditions in the early formation of the universe.
Theory and observation agree so well that you cannot distinguish them on this plot!
The stunningly good fit (and the amazing level of isotropy) are considered to be convincing proof of the Big Bang description of the birth of the universe. The latest results: Planck satellite (European Space Agency)
The temperature of the blackbody is not the Tiny fluctuations in the same everywhere in the sky. The small temperature of the blackbody variations reflect the structure of the universe spectrum at different points in at the time that the radiation was emitted. the sky (hundreds of microKelvins).
Planck
data from 2013