Transcript 4. The Electromagnetic Spectrum and Blackbody Radiation

The Electromagnetic Spectrum
and Blackbody Radiation
Sources of light: gases, liquids, and solids
The electromagnetic spectrum
Long-wavelength sources
and applications
Visible light and the eye
Short-wavelength sources and
applications
Boltzmann's Law
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”:
P(t )  Nqxq (t )
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 xq (t ) is the position of the charge. Here,
we’ve assumed that each charge is identical has identical motion.
Maxwell's Equations in a Medium
The induced polarization, P, contains the effect of the medium and is
included in Maxwell’s Equations:
B
t
E  0
 E  
B  0
E
P
  B  0 0
 0
t
t
This extra term also adds to the wave equation, which is known as
the "Inhomogeneous Wave Equation:”
2

 E 1  E
 P xq
 2 2  0 Nq2 2
2
z
c t
t t
2
2
2
The polarization is the
“source term” and tells us
what light will be emitted.
Notice that the induced polarization, and hence xq (t ), gets
differentiated twice. But  2 xq / t 2 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
B
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
Atomic and molecular vibrations
correspond to excited energy levels in
quantum mechanics.
Energy levels are everything in quantum mechanics.
Excited level
Energy

DE = hn
Ground level
The atom is vibrating
at frequency, n.
The atom is in a “superposition”
of the ground and excited state.
This is true for all types of vibrations.
Molecules (and everything else) have many
energy levels and can emit light only by
making a transition from one level to another.
A typical molecule’s energy levels:
E = Eelectonic + Evibrational + Erotational
2nd excited
electronic state
Energy
1st excited
electronic state
Lowest vibrational and
rotational level of this
electronic “manifold”
Excited vibrational and
rotational level
Transition
Ground
electronic state
There are many other
complications, such as
spin-orbit coupling,
nuclear spin, etc.,
which split levels.
Different atoms emit light at different
widely separated frequencies.
Each colored
“emission” line
corresponds to
a difference
between two
energy levels.
Frequency (energy)
Atoms have simpler energy level systems (and hence simpler spectra)
because they have no nuclear vibrations (i.e., only electronic levels).
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.
Gases at atmospheric pressure have emission widths of ~ 1 GHz.
Solids and liquids emit much broader ranges of frequencies (~ 1013 Hz!).
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 molecules
that that, together,
have transitions
at every
wavelength.
Before
Absorption
Spontaneous
Emission
Stimulated
Emission
After
Boltzmann Population Factors
Energy
E3
E2
N3
Ni  exp Ei / kBT 
N2
N1
E1
Population density
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.
The Maxwell-Boltzman Distribution
In the absence of collisions,
molecules tend to remain
in the lowest energy state
available.
Collisions can knock a molecule into a higher-energy state.
The higher the temperature,
the more this happens.
3
2
High T
N2 exp   E2 / kBT 

N1 exp   E1 / kBT 
Energy
Energy
Low T
3
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.
Einstein A and B coefficients
In 1916, Einstein considered the various transition rates between
molecular states (say, 1 and 2) involving light of irradiance, I:
Absorption rate = B12 N1 I
Spontaneous emission rate = A N2
Stimulated emission rate = B21 N2 I
In equilibrium, the rate of upward transitions equals the rate of
downward transitions:
B12 N1 I = A N2 + B21 N2 I
Rearranging:
Recalling the MaxwellBoltzmann Distribution
(B12 I ) / (A + B21 I ) = N2 / N1 = exp[–DE/kBT ]
Einstein A and B coefficients and
Blackbody Radiation
Now solve for the irradiance in: (B12 I ) / (A + B21 I ) = exp[-DE/kBT ]
Rearrange to:
or:
or:
B12 I exp[DE/kBT] = A + B21 I
I = A / {B12 exp[DE/kBT] – 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  B
And:
I = [A/B] / {exp[DE/kBT ] – 1}
Eliminating A/B:
 Coeff up = coeff down!
 2 hv3 
I
exp  hv / kBT   1
using DE = hn
Blackbody Emission Spectrum
The higher the temperature, the more the emission and
the shorter the average wavelength.
"Blue hot" is hotter
than "red hot."
Wien's Law: Blackbody peak wavelength
scales as 1/Temperature.
Writing the Blackbody spectrum vs.
wavelength:
 2 hc 2 /  5 
I 
exp  hc /  kBT   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.
The Electromagnetic Spectrum
gamma-ray
microwave
2
10
1
106
10
visible
radio
infrared
0
105
10
-1
4
10
10
3
10
UV
2
10
wavelength (nm)
The transition wavelengths are a bit arbitrary…
X-ray
1
10
0
10
-1
10
The Electromagnetic Spectrum
Now, we’ll run through the entire electromagnetic spectrum, starting at
very low frequencies and ending with the highest-frequency gamma rays.
60-Hz radiation from
power lines
Yes, this very-low-frequency current
emits 60-Hz electromagnetic waves.
No, it is not harmful. A flawed epidemiological study in 1979 claimed
otherwise, but no other study has
ever found such results.
Also, electrical power generation has increased exponentially
since 1900; cancer incidence has remained essentially constant.
Also, the 60-Hz electrical fields reaching the body are small;
they’re greatly reduced inside the body because it’s conducting;
and the body’s own electrical fields (nerve impulses) are much
greater.
60-Hz magnetic fields inside the body are < 0.002 Gauss; the
earth’s magnetic field is ~ 0.4 G.
The LongWavelength
Electromagnetic
Spectrum
Arecibo radio
telescope
Radio & microwave regions (3 kHz – 300 GHz)
Global Positioning System (GPS)
It consists of 24 orbiting satellites in “half-synchronous orbits” (two
revolutions per day).
Four satellites per orbit,
equally spaced, inclined
at 55 degrees to equator.
Operates at 1.575 GHz
(1.228 GHz is a reference
to compensate for atmospheric water effects)
4 signals are required;
one for time, three for
position.
2-m accuracy
(100 m for us).
Microwave ovens
Microwave ovens operate at 2.45 GHz,
where water absorbs very well.
Percy LeBaron
Spencer, Inventor
of the microwave
oven
Geosynchronous communications
satellites
22,300 miles above the earth’s surface
6 GHz uplink, 4 GHz downlink
Each satellite is actually two (one is a spare)
Cosmic
Microwave
Background
Peak frequency is ~ 150 GHz
The 3° cosmic microwave
background is blackbody
radiation left over from
the Big Bang!
Wavenumber (cm-1)
Microwave background
vs. angle. Note the
variations.
Interestingly,
blackbody radiation
retains a blackbody
spectrum despite
the expansion the
universe. It does
get colder, however.
TeraHertz light (a region of microwaves)
TeraHertz light is light with a frequency of ~1 THz, that is, with a
wavelength of ~300 m.
THz light is heavily absorbed by water, but clothes are transparent
in this wavelength range.
Fortunately, I couldn’t get permission to show you the movies I
have of people with THz-invisible invisible clothes.
IR is useful for
measuring the
temperature of
objects.
Hotter and
hence brighter
in the IR
Old Faithful
Such studies help to confirm that Old
Faithful is in fact faithful and whether
human existence is interfering with it.
IR Liedetection
I don’t really buy
this, but I thought
you’d enjoy it…
He’s really sweating now…
The military uses IR to see objects it
considers relevant.
IR light penetrates fog and smoke better than visible light.
Jet engines emit infrared light from 3 to 5.5 µm
This light is easily distinguished from the ambient infrared, which peaks
near 10m and is relatively weak in this range
The Infrared Space Observatory
Stars that are just
forming emit light
mainly in the IR.
Using mid-IR laser light
to shoot down missiles
Wavelength =
3.6 to 4.2 m
The Tactical High Energy Laser uses a high-energy,
deuterium fluoride chemical laser to shoot down
short range unguided (ballistic flying) rockets.
Laser
welding
Near-IR
wavelengths
are commonly
used.
Atmospheric Penetration depth (from
space) vs. Wavelength
Visible Light
Wavelengths and
frequencies of visible
light
Dye lasers cover the entire visible spectrum.
Fluorescent lights
“Incandescent” lights (normal light bulbs) lack the emission lines.
The Human Retina
Rods
Cones
The retina is a mosaic of two basic types of photoreceptors, rods,
and cones.
Cones are highly concentrated in a region near the center of the
retina called the fovea. The maximum concentration of cones is
roughly 180,000 per mm2 there and the density decreases rapidly
outside of the fovea to less than 5,000 per mm2. Note the blind spot
caused by the optic nerve, which is void of any photoreceptors.
The eye’s response to light and color
The eye’s cones have three receptors, one for red, another for
green, and a third for blue.
Intermediate colors, such as yellow and orange, are perceived by
comparing relative responses of two or more different receptors.
The eye is poor at distinguishing spectra.
Because the eye perceives intermediate colors, such as orange and
yellow, by comparing relative responses of two or more different
receptors, the eye cannot distinguish between many spectra.
The various yellow spectra below appear the same (yellow), and the
combination of red and green also looks yellow!
The Ultraviolet
The UV is usually broken up into three regions, UVA (320-400
nm), UVB (290-320 nm), and UVC (220-290 nm).
UVC is almost completely absorbed by the atmosphere.
You can get sun burned by all three.
UV from the sun
The ozone layer absorbs wavelengths less than 320 nm (UVB and
UVC), and clouds scatter what isn’t absorbed.
But much UV (mostly UVA, but some UVB) penetrates the
atmosphere anyway.
IR, Visible, and UV Light and Humans
(Sunburn)
Tanning salons use UVA, but it can still cause a sunburn.
The very short-wavelength regions
Vacuum-ultraviolet (VUV)
180 nm > > 50 nm
Soft x-rays
5 nm > > 0.5 nm
Absorbed by <<1 mm of air
Ionizing to many materials
Strongly interacts with core
electrons in materials
Extreme-ultraviolet (XUV or EUV)
50 nm > > 5 nm
Ionizing radiation to all materials
Synchrotron Radiation
Formerly considered a nuisance to accelerators, it’s now often the
desired product!
Synchrotron radiation in all
directions around the circle
Synchrotron radiation only
in eight preferred directions
EUV Astronomy
The solar corona is very hot (30,000,000 degrees K) and so emits
light in the EUV region.
EUV astronomy requires satellites because the earth’s atmosphere is
highly absorbing at these wavelengths.
The sun also emits x-rays.
The sun seen in the x-ray region.
Matter falling into a black hole emits x-rays.
Nearby star
Black hole
A black hole accelerates particles to very high speeds.
Supernovas emit x-rays, even afterward.
A supernova
remnant in a
nearby galaxy (the
Small Magellanic
Cloud).
The false colors
show what this
supernova
remnant looks like
in x-rays (blue),
visible light (green)
and radio (red).
Some x-rays are created in auroras.
Atomic structure and x-rays
Ionization energy
~ 100 – 1000 e.v.
Ionization energy
~ .01 – 1 e.v.
Fast electrons
impacting a
metal generate
x-rays.
High voltage
accelerates electrons
to high velocity, which
then impact a metal.
Electrons displace electrons in
the metal, which then emit xrays.
The faster the electrons, the
higher the x-ray frequency.
X-rays penetrate tissue and do not
scatter much.
Roentgen’s x-ray image
of his wife’s hand (and
wedding ring)
X-rays for photo-lithography
You can only focus light to
a spot size of the light
wavelength. So x-rays are
necessary for integratedcircuit applications with
structure a small fraction
of a micron.
1 keV photons from a
synchrotron:
2 micron lines over a base
of 0.5 micron lines.
High-Harmonic Generation and x-rays
Amplified
femtosecond laser
pulse
x-rays
gas jet
An ultrashort-pulse x-ray beam can be generated by focusing a
femtosecond laser in a gas jet
Harmonic orders > 300, photon energy > 500 eV, observed to date
HHG is a highly nonlinear process
resulting from highly nonharmonic
motion of an electron in an intense field.
The strong field smashes the electron into the nucleus—a highly
non-harmonic motion!
Ion
x-ray
electron
How do we know this? Circularly polarized light (or even slightly
elliptically polarized light) yields no harmonics!
Gamma rays result from matterantimatter annihilation.
An electron and positron self-annihilate, creating two gamma
rays whose energy is equal to the electron mass energy, mec2.
ee+
hn = 511 kev
More massive particles create even more energetic gamma
rays. Gamma rays are also created in nuclear decay, nuclear
reactions and explosions, pulsars, black holes, and
supernova explosions.
Gamma-ray bursts emit massive
amounts of gamma rays.
The gamma-ray sky
A new one
appears almost
every day, and
it persists for
~1 second to
~1 minute.
No one knows
what they are.
In 10 seconds, they can emit more energy than our sun will in its
entire lifetime. Fortunately, there don’t seem to be any in our galaxy.
The universe in
different spectral
regions…
Gamma Ray
X-Ray
Visible
The universe in more spectral
regions…
IR
Microwave