5. Light-matter interactions: Blackbody radiation

The electromagnetic spectrum

Sources of light

Boltzmann's Law

Blackbody radiation

The cosmic microwave background

The electromagnetic spectrum

The boundaries between regions are a bit arbitrary…

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.

Sources of lightAccelerating charges emit light

Linearly accelerating chargehttp://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.

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.

Core (tightly bound) electrons vibrate in their motion around nuclei

~ 1018 - 1020 cycles per second.

One interesting source of x-rays

For more information and a video describing this:http://www.nature.com/nature/videoarchive/x-rays/

Sticky tape emits x-rays when you unpeel it.

This phenomenon, known as ‘mechanoluminescence,’ was discovered in 2008.

Scotch tape generates enough x-rays to take an image of the bones in your finger.

Three types of molecule-radiation interactions

•Promotes molecule to a higher energy state•Decreases the number of photons

•Molecule drops from a high energy state to a lower state•Increases the number of photons

•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!

Absorption

Spontaneous Emission

Stimulated Emission

key idea: conservation of energy Before After

Unexcited molecule

Excited molecule

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:

Ground electronic state

1st excited electronic state

2nd excited electronic state

Ene

rgy

Transition

Lowest vibrational and rotational level of this electronic “manifold”

Excited vibrational and rotational level

Computing the energies of these levels is usually difficult, except for very simple atoms or molecules (e.g., H2).

Different atoms emit light at different frequencies.

Frequency (energy)

This is the emission spectrum from hydrogen. It is simple because hydrogen has only one electron.

the “Balmer series”

Bigger atoms have much more complex spectra, because they are more complex objects. Molecules are even more complex.

high-pressure mercury lamp

Johann Balmer(1825 - 1898)

Wavelength

In the absence of collisions,molecules tend to remainin the lowest energy stateavailable.

Low Temperature

Collisions can knock a mole-cule into a higher-energy state.The higher the temperature, the more this happens.

High Temperature

22

1 1

exp /exp /

exp /

BB

B

E k TN E k TN E k T

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:

Ludwig Boltzmann computed the relative populations of energy levels

Boltzmann population factor

Ludwig Boltzmann(1844 - 1906)

Boltzmann didn’t know quantum mechanics. But this result works equally well for quantum or classical systems.

eE/kT

ener

gy

population

E1

E2

E3

eE/kT

population

E1

E2

E3

A lower temperatureA higher temperature

highlow

Blackbody radiation

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

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.

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 “equipartitiontheorem”, also due to Boltzmann)

Rayleigh-Jeans law (circa 1900):

energy density of a radiation field u() = 82kT/c3

Total energy radiated from a black body: u d uh-oh… the "ultraviolet catastrophe"

Note: the units of this expression are correct. Strictly speaking, u() is an energy density per unit bandwidth, such that the integral gives an answer with units of energy density (energy per unit volume).

u d

The Rayleigh-Jeans Law

Rayleigh (and others) knew this was wrong. This failure of equipartition led Max Planck to propose (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 he had no explanation for why it should be true…en

ergy

den

sity

u(

)

frequency

keeps going up forever!

Einstein A and B coefficients

In equilibrium, the rate of upward transitions equals the rate of downward transitions:

B12 N1 u = A N2 + B21 N2 u

Rearranging:(B12 u ) / (A + B21 u ) = N2 / N1

using Boltzmann’s result

= exp[–E/kBT ]

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

We can solve this expression for the energy density of the field:

(B12 u ) / (A + B21 u ) = exp[E/kBT ]

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!

Rearrange to find:

21

12

21

exp / 1

B

A Bu B E k TB

Einstein A and B coefficients

3 38

exp / 1

B

hv cu

hv k T

We can eliminate* the ratio A/B to find: (we have used E=h)

* 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.en

ergy

den

sity

u(

)

frequency

Rayleigh-Jeans: u() ~ 2

Planck-Einstein: 3

~exp / 1B

uh k T

At low frequencies, the two results are basically equivalent.

At high frequencies, Einstein’s result goes back down to zero, so the integral of u() is finite.

Max Planck1858-1947

Albert Einstein1879-1955

(photo taken in 1916)

Blackbody emissionThe higher the temperature, the more the emission and the shorter the average wavelength.

Wien’s Law: the peak wavelength is proportional to 1/T.

Notice: for an object near room temperature, the peak of the blackbody spectrum is around = 10 microns – the so-called ‘thermal infrared’region.

"Blue hot" is hotter than "red hot."

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”.

Even hotter metal is “white hot”.

Cooler objects emit light at longer wavelengths: infrared

The sun is a black bodyIrradiance of the sun vs. wavelength

When measured at sea level, one sees absorption lines due to molecules in the atmosphere.

Cosmic microwave backgroundAs 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 microwavebackground is blackbodyradiation left over fromthe Big Bang!

Arno Penzias and Bob WilsonNobel prize, 1978

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.

Cosmic microwave background

The latest results: Planck satellite (European Space Agency)

data from 2013

The temperature of the blackbody is not the same everywhere in the sky. The small variations reflect the structure of the universe at the time that the radiation was emitted.

Planck

Tiny fluctuations in the temperature of the blackbody spectrum at different points in the sky (hundreds of microKelvins).